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\begin{document}
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\title{Traffic-Redundancy Aware Network Design hanks{This work was
supported in part by NSF awards CCF-0643763 and CNS-0905134, and in part by a Sloan foundation fellowship.}
\begin{abstract}
We consider network design problems for information networks where
routers can replicate data but cannot alter it. This functionality allows the
network to eliminate data-redundancy in traffic, thereby saving on routing costs.
We consider two problems within this framework and design
approximation algorithms.
The first problem we study is the traffic-redundancy aware network
design (RAND) problem. We are given a weighted graph over a single
server and many clients. The server owns a number of different data
packets and each client desires a subset of the packets; the client
demand sets form a laminar set system. Our goal is to connect every
client to the source via a single path, such that the collective cost
of the resulting network is minimized. Here the transportation cost
over an edge is its weight times times the number of {\em distinct}
packets that it carries.
The second problem is a facility location problem that we call
RAFL. Here the goal is to find an assignment from clients to
facilities such that the total cost of routing packets from the
facilities to clients (along unshared paths), plus the total cost of
``producing'' one copy of each desired packet at each facility is
minimized.
We present a constant factor approximation for the RAFL and an $O(\log P)$ approximation for RAND, where $P$ is the total number of distinct packets. We remark that $P$ is always at most the number of different demand sets desired or the number of clients, and is generally much smaller.
\end{abstract}
\thispagestyle{empty}
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\section{Introduction}
We consider network design problems for information
networks where edges can replicate data but cannot otherwise alter it.
In this setting our goal is to exploit the redundancy in the given
traffic matrix to save on routing costs. Formally, we are given a graph over a single server and many clients. The server has a universe of data packets available, and each client desires a subset of the packets. The goal is to determine a collection of paths, one from the source to each client, such that the total cost of routing is minimized. Here the cost of routing on an edge is proportional to the total size of the {\em distinct} packets that the edge carries. For example, if the edge belongs to two paths that each carry the same packet, then the edge only needs to route the packet once and not twice. We call this the traffic-redundancy aware network design problem, or RAND for short.
RAND arises in networks that face a lot of data duplication. Consider for example a Netflix server serving movies to a large and varied clientele. Each client desires a certain subset of the movies. What routing paths should the server use to send the data to the clients so as to minimize its total bandwidth usage? If different movie streams involve disjoint sets of packets this boils down to setting up a single multicast tree, or solving the minimum Steiner tree problem, once per movie. However, the server may want to set up a single routing path for each client regardless of how many different movies the client desires. Moreover, clients desiring the same movie may desire it at different rates or qualities depending on their location or the device they are using. For example, a desktop user with a broadband connection may desire a high definition video, whereas a mobile phone user may be content with a much lower resolution. Then, the data sent to these clients is not identical but has some amount of overlap. The server can exploit this redundancy in traffic by using common paths for clients with similar demands, thereby saving on the actual amount of traffic routed. Redundancy in data can arise even across different movies when data streams are broken down into small enough packets. Anand et al.~\cite{AGA+08} show that this kind of traffic redundancy is highly prevalent in the Internet, and it can be eliminated across individual links by routers employing packet caches.
Given the cost structure that this redundancy generates, it makes sense to try to route the demands of clients desiring similar sets along overlapping paths. An extreme example of the benefit of merging paths is when all clients desire the same set of packets. In this case, the problem becomes equivalent to finding the minimum-cost Steiner tree over the clients and the server. At the other extreme, if all the clients desire disjoint sets of packets, then merging does not help at all, and it is optimal to pick the shortest path from every client to the source. There is thus a trade-off between routing demands along shortest paths and trying to merge the paths of clients with similar demands.
We also study a facility location version of the problem that we call traffic-redundancy aware facility location or RAFL. This problem is motivated by the prevalence of content distribution networks (CDNs) in the Internet. Netflix servers, instead of connecting to clients directly, cache their data at multiple servers spread around the network that are hosted by a CDN such as Akamai; each client then connects to a CDN server individually to obtain its data. The savings in this case comes from assigning clients with similar demand sets to the same CDN server and sending a single copy of the multiply desired packets to the server. Formally we are given a network over potential facilities and clients. Each client, as before, desires a subset of the available packets. Our goal is to assign each client to a facility and route to each facility the union of the demand sets desired by clients assigned to it. The cost of such a solution is the sum of the cost of routing packets from facilities to clients (that are proportional to the size of the respective client's demand set), and the cost of routing packets to facilities (that is proportional to the total size of the distinct packets being routed to the facility). The savings from redundancy in this case are realized in the facility opening costs where we assign multiple clients with similar demand sets to the same facility and pay for each of the common packets only once.
RAND and RAFL model information networks as opposed to commodity networks in traditional network design problems. They can therefore be considered as intermediate models between traditional network design and network coding. In the latter the information network is allowed to use coding to increase its capacity. In our context the network can eliminate redundant information but cannot otherwise alter the information. See \cite{nwcoding} and references therein for work on network coding.
\paragraph{Our results and techniques.}
We study the RAND and RAFL under a laminar demands assumption. In particular, we assume that the sets of packets demanded by clients form a laminar set family. In other words, every pair of demanded packet sets is either disjoint or one is a subset of the other. Such a structure arises, for example, in the Netflix problem described above when layered coding is used; the packets for a lower encoding rate are a subset of the packets for a higher encoding rate.
RAND generalizes the minimum Steiner tree problem and RAFL generalizes metric uncapacitated facility location (MUFL); therefore both problems are NP-hard. We study approximations.
We develop a constant factor approximation for the RAFL based on the natural LP-relaxation of the problem. Our algorithm follows the filtering approach developed by Lin and Vitter~\cite{LV92} and later exploited by Shmoys et al.~\cite{STA97} in the context of MUFL. In the MUFL setting, in the filtered solution, each client $t$ is associated with a set of facilities, say $\mathcal{F}(t)$, such that the routing cost of any of those facilities is comparable to the routing cost that the client pays in the LP solution. It is then sufficient to open a set of facilities in such a way that each can be charged to a client with a distinct set of associated facilities. Clients $t$ that are not charged (a.k.a. free riders) are rerouted to the facilities opened for other charged clients $t'$ such that $\mathcal{F}(t)$ and $\mathcal{F}(t')$ overlap, and furthermore the routing cost for $t'$ is smaller than that for $t$. Then, using the triangle inequality, the cost of rerouting can be bounded and this gives a constant factor approximation. In our setting, it is not sufficient to ensure that the routing cost of $t'$ is smaller than that of $t$. In addition, we must ensure that the facility opened by $t'$ can support the demand of $t$, otherwise every time we reroute a client we incur extra facility costs. Ensuring these two properties is tricky because clients with low routing costs may also have small demand sets; so essentially, these properties require us to consider clients according to two distinct and potentially conflicting orderings.
In order to deal with this issue, our algorithm is run in two phases. In the first phase we consider clients in order of increasing routing costs, in order to determine which clients will pay for their facilities and which ones are free riders. In the second phase, we consider free riders in decreasing order of the sizes of their demand sets. Each free rider is associated with a set of paying clients that pay for the facility that this client opens. Every time a free rider opens a facility, we reroute to it all of the other clients whose paying neighbors overlap with those of this facility. In this manner we can ensure that whenever a client is rerouted, it is routed to a facility that already produces the packets it needs. Unfortunately, our algorithm charges each paying client multiple times for different facilities opened. We ensure that the costs that a client pays each time it is charged form a geometrically decreasing sequence, and can therefore be bounded in terms of the LP solution. Overall we obtain a $27$-approximation.
For RAND an $O(\log n)$ randomized approximation can be obtained via tree embeddings~\cite{FRT03} because the problem is trivially solvable on trees; here $n$ is the number of nodes in the network. We give a simple deterministic combinatorial $O(\log P)$ approximation, where $P$ is the number of distinct packets to be routed. Note that if two or more packets are essentially identical in that they are desired by exactly the same set of clients, then we can combine them into a single packet (albeit with a larger size). Then, under the laminar demands assumption, $P$ is always at most the number of distinct demand sets or clients, which is at most $n$, the total number of nodes in the network. In fact in applications such as the Netflix multicast problem described above, we expect $n$ to be much larger than $P$.
Furthermore, our $O(\log P)$ approximation algorithm is a natural combinatorial algorithm that is simple and fast to implement. It is convenient to represent the laminar family of demand sets in the form of a tree where the demand set at any node is a proper subset of that at its parent and disjoint from those at its siblings. Our algorithm begins with some minor preprocessing of the demand tree to ensure that the tree has small (logarithmic) height. It then traverses the demand tree in a top-down fashion, and at each node of the tree constructs a (approximately optimal) Steiner tree over all the terminals with the corresponding demand set connecting them to the source. We show that the cost of the Steiner trees constructed at every level of the demand tree is bounded by a constant times the cost of the optimal solution, and therefore obtain an overall approximation factor proportional to the height of the tree. The height of the tree can however be much larger than $\log P$ because packets have different sizes. In order to prove an approximation factor of $O(\log P)$ we need to do a more careful bounding of the total cost spent on long chains of degree $2$ in the demand tree. Using the Steiner tree algorithm of Robins and Zelikovsky~\cite{RZ00} as a subroutine we obtain a $(6.2\log P)$-approximation.
\paragraph{Related work.}
RAND is closely related to single-source uniform buy-at-bulk network design (BaBND)~\cite{AA97, GKR03, MMP08} in that both problems involve a trade-off between picking short paths between the source and the clients and trying to merge different paths to avail of volume discounts. However the actual cost structure of the two problems is very different. In BaBND the cost on an edge is a concave function of the total load on the edge; In our setting the cost is a submodular function of the clients using that edge. Neither of the problems is a special case of the other. BaBND admits a constant approximation in the single-source version~\cite{GKR03, Talwar02} and an $O(\log n)$ approximation is known for the general multi-source multi-sink problem~\cite{AA97}.
One way of thinking about RAND is to break-up the problem and solution packet-wise: each packet $p$ defines a subset of the terminals, say $T_p$, that desire that packet; the solution restricted to these terminals is essentially solving a Steiner tree problem over the set $T_p\cup\{s\}$. Our goal is to pick a single collection of paths such that the sum over packets of the costs of these Steiner trees is minimized. In this respect, the problem is related to variants of the Steiner tree problem where the set of terminals is not precisely known before hand. This includes the maybecast problem of Karger and Minkoff~\cite{KM00} for which a constant factor approximation is known, as well as the universal Steiner tree problem~\cite{JLN+05} for which a randomized logarithmic approximation can again be obtained through tree embeddings and this is the best possible~\cite{BCK10}.
Facility location has been extensively studied under various models. The models most closely related to RAFL are the service installation costs model of Shmoys et al.~\cite{SSL04} and the heirarchical costs model of Svitkina and Tardos~\cite{ST06}. In the former, each client has a production cost associated with it; the cost of opening a facility is equal to a fixed cost associated with the facility plus the production costs of all the clients assigned to that facility. One way of representing these costs is in the form of a two-level tree for each facility with the fixed cost for the facility at the root of the tree and the client-specific production costs at the next level nodes. The cost of assigning a set of clients to the facility is the total cost of the subtree formed by the unique paths connecting the client nodes to the root of the tree. Svitkina and Tardos generalize this cost model to a tree of arbitrary depth, although in their setting the trees for different facilities are identical. Both the works present constant factor approximations for the respective versions, based on a primal-dual approach and local search respectively. Our model is similar to these models in that our facility costs are also submodular in the set of clients connecting to a facility. In our setting, the costs can again be modeled by a tree in which each node is associated with one or more clients; the cost of a collection of clients is given by the total cost of the union of subtrees rooted at those clients (as opposed to the portion of the tree ``above'' the clients). Therefore, neither of the two settings generalize the other. Moreover, facility costs in our setting are different for different facilities, although they are related through a multiplier per facility. Finally, while in Shmoys et al. and Svitkina et al. routing costs are given merely by the metric over facilities and clients, in our setting they are given by the distances times the total demand routed.
For modeling information flow, Hayrapetyan et al.~\cite{hayrapetyan2005network} have studied a single-source network design problem with monotone submodular costs on edges, and a group facility location problem. The network design setting generalizes ours in that edge costs can be arbitrary submodular functions of the clients using the edges; they note that an $O(\log n)$ approximation can be achieved via tree embeddings. In contrast, we obtain an $O(\log P)$ approximation, where $P$, the number of distinct packets in the system, is always at most $n$ and generally much smaller. In the group facility location problem, edge costs are identical to those in RAND, but neither of the problems subsumes the other: the former assumes there are multiple facilities (sources) with fixed opening costs, but also limits the number of distinct packets per client to at most one.
\section{Problem definition and preliminaries}
The traffic-redundancy aware network design (RAND) problem is defined as follows. We are given a graph $G=(V,E)$ with weights $c_e\in \Re^+$ on edges $e\in E$, and a special node $s$ called the source. In addition, we are given a set $T$ of clients or terminals located at different nodes in the graph. The source carries a set $\Pi$ of packets with $|\Pi|=P$. Each packet $p\in \Pi$ is associated with a weight $w_p$; we assume that the weights are integral. Each terminal $t\in T$ desires some subset of the packets; this is called the terminal's demand and is denoted by $\mathrm{D}(t)$. We use the convention $\mathrm{D}(s)=\Pi$. Also let $w(S)=\sum_{p\in S}w_p$ denote the total weight of a set $S$ of packets.
We assume that the collection of demand sets $\mathcal{D}=\{\mathrm{D}(t)\}_{t\in T}$ forms a laminar family of sets. In particular, for any two terminals $t_1, t_2\in T$, $\mathrm{D}(t_1)\cap\mathrm{D}(t_2)\ne \emptyset$ implies that either $\mathrm{D}(t_1)\subseteq \mathrm{D}(t_2)$ or $\mathrm{D}(t_2)\subseteq\mathrm{D}(t_1)$. We use a tree $\tau$ to represent the containment relationship between sets in the laminar family. The nodes of $\tau$ are sets in $\mathcal{D}$. A demand set $X$ is a parent of another set $Y$ if $Y\subset X$ and there is no set $Z\in\mathcal{D}$ with $Y\subset Z\subset X$. At the root of the tree is the universe $\Pi$ of packets. For a demand set $X$ in the tree $\tau$, we use $T_X$ to denote the terminals $t\in T$ with $\mathrm{D}(t)=X$.
We denote an instance of RAND by the tuple $(G,T,\mathcal{D},\tau)$.
The solution to RAND is a collection of paths $\mathcal{P}=\{P_t\}_{t\in T}$, with $P_t$ connecting the terminal $t$ to the source $s$. Given this solution, an edge $e\in E$ carries the set $S_{\mathcal{P}}(e) = \cup_{t\in T: P_t\ni e} \mathrm{D}(t)$ of packets. The load on the edge is $w_{\mathcal{P}}(e) = w(S_{\mathcal{P}}(e))$. We drop the subscript $\mathcal{P}$ when it is clear from the context. The cost of the solution $\mathcal{P}$ is $\mathrm{cost}(\mathcal{P})=\sum_{e\in E} c_e w_{\mathcal{P}}(e)$. Our goal is to find a solution of minimum cost.
Let $\mathrm{OPT}=\operatorname{argmin}_{\text{feasible }\mathcal{P}} \mathrm{cost}(\mathcal{P})$ be an optimal solution. For a subset $W$ of terminals, we use $\mathrm{OPT}(W)$ to denote the restriction of $\mathrm{OPT}$ to $W$, that is, the collection of paths $\{P_t\in\mathrm{OPT}\}_{t\in W}$.
In the traffic-redundancy aware facility location problem (RAFL), we are given a set $\mathcal{F}$ of facilities, and a graph over $T\cup \mathcal{F}$ with edge weights $c_e$. Let $c(u,v)$ denote the shortest path distance between nodes $u$ and $v$ in the graph under the metric $c$. Furthermore, each facility $f\in \mathcal{F}$ has a cost $\lambda_f$ associated with it.
A solution to this problem is an assignment $A$ from terminals to facilities. The assignment specifies the set of packets that a facility $f$ needs to produce in order to serve all the terminals connected to it: $S_{A}(f) = \cup_{t\in T: A(t)=f} \mathrm{D}(t)$. The load on the facility is $w_{A}(f) = w(S_{A}(f))$. We drop the subscript $A$ when it is clear from the context. The cost of the solution $A$ is $\mathrm{cost}(A)=\sum_{f\in\mathcal{F}} \lambda_f w_{A}(f) + \sum_{t\in T} w(\mathrm{D}(t)) c(t,A(t))$. The first component of the cost is called the facility opening cost $C_f(A)$, and the second the routing cost $C_r(A)$ of the solution. Once again our goal is to find a solution of minimum cost.
Both RAND and RAFL are NP-hard because they generalize Steiner tree and metric uncapacitated facility location respectively. Our goal is to find approximation algorithms.
\section{A constant factor approximation for RAFL}
In this section we present a constant factor approximation for the RAFL. For ease of exposition we assume that all packets have unit weight, and write $|S|$ for the total size or weight of a set $S$ of packets. This assumption is without loss of generality. We further assume without loss of generality that $\min_{f\in\mathcal{F}} \lambda_f=1$.
The following is a natural LP-relaxation of RAFL. Here $x_{t,f}$ is an indicator for whether terminal $t$ is assigned to facility $f$, and $y_{f,p}$ denotes the extent to which facility $f$ produces packet $p$.
\begin{align*}
\textrm{minimize} & \ \ \sum_{f \in \mathcal{F}} \sum_{p \in \Pi} \lambda_f y_{f,p} + \sum_{t \in T} \sum_{f\in \mathcal{F}} |\mathrm{D}(t)| x_{t,f} c(t,f) \\
\textrm{subject to } & \ \ \sum_{f \in \mathcal{F} } x_{t,f} \geq 1 \ \ \ \forall t \in T \\
& y_{f,p} \geq x_{t,f} \ \ \ \forall t, f, p \in \mathrm{D}(t)
\end{align*}
Our approach begins along the lines of the filtering approach developed by Lin and Vitter~\cite{LV92} and employs some of the rounding ideas of Shmoys et al.~\cite{STA97} developed for metric uncapacitated facility location. In particular, given an optimal solution to the LP, we preprocess the solution at a constant factor loss in performance such that each terminal is assigned to a non-zero extent only to facilities for which the terminal's routing cost is within a small constant factor of the corresponding average amount in the LP. At this point, each terminal can be assigned to any facility to which it is fractionally assigned by the filtered solution, at a low routing cost. The key part of the analysis is bounding the cost for producing packets at facilities.
Let $\mathcal{F}(t)$ denote the set of facilities to which $t$ is assigned fractionally by the filtered solution. In Shmoys et al.'s setting, in order to bound the cost of opening facilities, it is sufficient to find a ``paying'' terminal for each open facility such that the sets $\mathcal{F}(t)$ for paying terminals are mutually disjoint. For each terminal $t$ that is not paying, there exists at least one representative paying terminal $t'$ such that $\mathcal{F}(t)$ and $\mathcal{F}(t')$ overlap; any such terminal $t$ is assigned to the facility opened by its representative terminal $t'$ at a slight increase in routing cost as long as the average routing cost of $t'$ is no more than that of $t$. In order to accomplish this, we process terminals in order of increasing average routing cost.
In our setting, this approach has a basic flaw. The terminal $t'$ may have a much smaller demand set compared to the terminal $t$. Then, if we assign $t$ to the facility opened by $t'$, the facility needs to produce many more packets and its new larger opening cost can no longer be charged to $t'$. Unfortunately, it is not possible to ensure that $t'$ has both a small average routing cost than $t$ as well as a larger demand set. Instead, we divide the process of opening facilities into two parts. First we determine which terminals are paying and which ones are not by processing facilities in order of increasing average routing cost. Then we decide which facilities to open by processing the non-paying terminals in the order of decreasing demand set sizes.
The algorithm is described formally below. We first introduce some notation. Let $(x^*, y^*)$ denote the optimal solution to the RAFL LP given above. Let $C_r^*(t) = \sum_{f\in \mathcal{F}} x^*_{t,f}c(t,f)$ and $C_f^*(t)=\sum_{f\in \mathcal{F}} x^*_{t,f}\lambda_f$ denote the average routing and facility opening costs respectively under the solution $(x^*, y^*)$ associated with a terminal $t$. The total routing and facility opening costs of the solution are given by $C_r^*=\sum_{t\in T} |\mathrm{D}(t)| C_r^*(t)$ and $C_f^*=\sum_{f\in \mathcal{F}} \sum_{p\in \Pi} y^*_{f,p}\lambda_f$ respectively. Likewise, for a feasible solution $(x,y)$, we use $C_r^{(x,y)}(t)$ and $C_f^{(x,y)}(t)$ to denote the average routing and facility opening costs associated with a terminal $t$ respectively. We drop the superscript $(x,y)$ when it is clear from context. Also let $C_r(x,y)$ and $C_f(x,y)$ denote the total routing and facility opening costs of the solution $(x,y)$.
\begin{algorithm*}{Given: LP solution $(x^*,y^*)$; Return: Assignment $\mathcal{F}P$ from terminals to facilities}
\caption{Rounding algorithm for RAFL}
\label{Algorithm:Round}
\textbf{Phase 1: Filtering}
\begin{algorithmic}[1]
\mathcal{F}ORALL{$t\in T$}
\mathcal{S}ATE Let $C_r^*(t) = \sum_{f\in \mathcal{F}} x^*_{t,f}c(t,f)$ and $C_f^*(t)=\sum_{f\in \mathcal{F}} x^*_{t,f}\lambda_f$
\mathcal{S}ATE For all $f\in\mathcal{F}$, if $c(t,f)>\alpha C_r^*(t)$ set $x_{t,f}= 0$ else $x_{t,f}= x^*_{t,f}$.
\mathcal{S}ATE \label{step:renormalize} Renormalize $x_{t,f}$ so that $\sum_f x_{t,f}=1$.
\mathcal{S}ATE Let $\mathcal{F}(t) = \{f: x_{t,f}>0\}$, $C_f(t) =\sum_{f\in \mathcal{F}} x_{t,f}\lambda_f $, and $C_r(t) = \sum_{f\in \mathcal{F}} x_{t,f}c(t,f) $.
\ENDFOR
\mathcal{S}ATE For all $f\in\mathcal{F}$ and $p\in \Pi$, set $y_{f,p} = \max_{t\in T: \mathrm{D}(t)\ni p} x_{t,f}$.
\textbf{Phase 2: Classification of terminals into paying and free}
\mathcal{S}ATE Initialize paying and free terminal sets: $T^p= \emptyset$ and $T^f = \emptyset$.
\mathcal{S}ATE For all $t \in T$ initialize temporary assignment $\mathcal{F}T(t) = \emptyset$, permanent assignment $\mathcal{F}P(t) = \emptyset$, and cover $\mathrm{Cov}(t) = \emptyset$.
\mathcal{S}ATE For all $f \in \mathcal{F}$ initialize paying terminal set $\mathrm{Pay}(f) = \emptyset$ and final paying set $\mathcal{F}Pay(f)=\emptyset$.
\WHILE{ $T \setminus \left( T^p \cup T^f \right) \neq \emptyset $ }
\mathcal{S}ATE Let $t = \operatorname{argmin}_{j\in T\setminus(T^p\cupT^f)} C_r^*(j)$. \COMMENT{Select terminal with least connection cost}
\IF{ there exists $f \in \mathcal{F}(t)$ such that $ \mathrm{Pay}(f)$ covers $t$}
\mathcal{S}ATE $T^f =T^f \cup \{ t\}$ \COMMENT{$t$ is a free terminal}
\mathcal{S}ATE $\mathcal{F}T(t)= f$
\mathcal{S}ATE Assign covering set for $t$: $\mathrm{Cov}(t) = \{ j \in \mathrm{Pay} (f) \ | \ \mathrm{D}(j) \cap \mathrm{D}(t) \neq \emptyset \}$
\ELSE
\mathcal{S}ATE \label{step:add_to_TP} $T^p =T^p\cup\{ t\}$ \COMMENT{$t$ is a paying terminal}
\mathcal{S}ATE For all $f \in \mathcal{F}(t) $ update $\mathrm{Pay}(f) = \mathrm{Pay}(f)\cup\{ t\}$
\ENDIF
\ENDWHILE
\textbf{Phase 3: Opening facilities}
\mathcal{S}ATE For all $t \in T^p$ assign level $\ell(t) = \lceil \log_2 C_f(t) \rceil$;
for all $t \in T^f$ assign $\ell(t) = \min_{ j \in \mathrm{Cov}(t) } \ell(j) $.
\mathcal{S}ATE Initialize $T^f_d =\{ t \in T^f \mid \ell(t) = d \} $ and for all $t \in T^f_d$ set $\Gamma_d(t) = \{ t' \in T^f_d \mid \mathrm{Cov}(t) \cap \mathrm{Cov}(t') \neq \emptyset \} $.
\mathcal{F}OR{$d=0$ to $\max_{ j \in T^p} \ell(j)$}
\mathcal{S}ATE Initialize $W=T^f_d$ \COMMENT{Repeat till all terminals in $T^f_d$ have been assigned a permanent facility}.
\WHILE{$ W \neq \emptyset$ }
\mathcal{S}ATE Let $t \in \operatorname{argmax}_{ j \in W } |\mathrm{D}(j)| $ \COMMENT{Select any terminal with the largest demand set}.
\mathcal{S}ATE Let $\bar{t} \in \operatorname{argmin}_{j \in \Gamma_d(t)} C_r^*(j)$.
\mathcal{S}ATE Let facility $\phi(\bar{t}) \in \operatorname{argmin}_{ f \in \cup_{ j \in \mathrm{Cov}(\bar{t})} \mathcal{F}(j) } \lambda_f $.
\mathcal{S}ATE \label{step:open_fac} Let $\mathcal{F}P(t)= \phi(\bar{t})$\COMMENT{We say that $t$ opens the facility $\phi(\bar{t})$.}
\mathcal{S}ATE \label{step:open_fac_free} For all $t'\in\Gamma_d(t) $, assign $\mathcal{F}P(t')=\mathcal{F}P(t)$. Update $W=W\setminus \left( \Gamma_d(t) + \{ t \} \right)$.
\mathcal{S}ATE Assign final paying set for facility $\mathcal{F}P(t)$: $\mathcal{F}Pay( \mathcal{F}P(t) ) = \mathrm{Cov}(t)$.
\ENDWHILE
\ENDFOR
\mathcal{F}ORALL{ $t \in T^p$ }
\mathcal{S}ATE \label{step:open_fac_pay} Let $\mathcal{F}P(t) \in \operatorname{argmin}_{ f \in \mathcal{F}(t) } \lambda_f$.
\ENDFOR
\end{algorithmic}
\end{algorithm*}
Our algorithm proceeds in three stages. The first is a filtering stage in which we convert the solution $(x^*, y^*)$ into a fractional solution $(x,y)$ which satisfies the following property: for all $t, f$ with $x_{t,f}>0$, $c(t,f)\le\alpha C_r^*(t)$. Here $\alpha$ is a parameter that we fix later. For a terminal $t$, we use $\mathcal{F}(t)$ to denote all the facilities $f$ that are fractionally assigned to $t$ in $(x,y)$, that is, have $x_{t,f}>0$.
In the second stage of the algorithm, we classify terminals into paying terminals $T^p$ and free terminals $T^f$. Essentially, a terminal $t$ becomes a free terminal if any of the facilities in $\mathcal{F}(t)$ is already expected to produce a large fraction of $t$'s demand. We record this facility as $t$'s temporary assignment $\mathcal{F}T(t)$. To this end, we say that a set of terminals $W$ covers a terminal $t$ if $\left| \mathrm{D}(t) \setminus \left( \cup_{t' \in W } \mathrm{D}(t') \right) \right| < \frac{1}{2}|\mathrm{D}(t)|$. If a terminal $t$ is not covered at any of the facilities in $\mathcal{F}(t)$, then it becomes a paying terminal and can potentially pay for any of the facilities in $\mathcal{F}(t)$. $\mathrm{Pay}(f)$ tracks the set of terminals paying for a facility $f$.
Finally, in the third stage of the algorithm, we pick a permanent assignment from terminals to facilities by considering facilities in decreasing order of the sizes of their demand sets. As a first cut approach, suppose that we assign a free terminal $t$ to the facility at which it is covered ($\mathcal{F}T(t)$), and pay for that facility using the paying terminals associated with it. To ensure that no paying terminal $t'$ ends up paying for two or more opened facilities, we consider all the free terminals that this paying terminal covers and assign those also to the first facility that the paying terminal pays for. The order in which we assign free terminals to facilities ensures that in this last step we do not increase the facility opening cost of the solution. Here is the catch: which facility is actually opened is decided by the free terminal $t$ that starts this process and may be one of the more expensive facilities in the paying terminal $t'$'s set $\mathcal{F}(t')$. In this case, the paying terminal does not have enough charge in the LP solution to pay for this facility. In order to avoid this situation, we consider all of the facilities that are ``close'' to $t$ or to other free terminals covered by the paying terminals that cover $t$. Of these we open the facility with minimum cost and pay for it using the paying terminals associated with $f$.
While in our algorithm a paying terminal can end up paying for multiple opened facilities, in Lemma~\ref{lemma:fac_cost_one} below we argue that the costs of those facilities decrease geometrically and so the sum can be bounded.
We now formalize this argument. We begin by showing that the fractional solution $(x,y)$ is not too expensive.
\begin{lemma}
\label{lemma:filter_opt}
The solution $(x,y)$ is feasible for the RAFL LP. Moreover
$C_f(x,y) \leq \frac{\alpha}{\alpha -1 } C_f^*$.
\end{lemma}
\begin{proof}
$(x,y)$ is feasible by construction. Note that for all $t\in T$, $\sum_f x^*_{f,t}=1$, otherwise the cost of the solution can be improved. Therefore, by Markov's inequality, $\sum_{f: c(f,t)>\alpha C_r^*} x^*_{f,t} \le 1/\alpha$. Then, in the renormalization step (Step~\ref{step:renormalize}) we set $x_{t,f}$ to be no more than $\alpha/(\alpha-1) x^*_{t,f}$.
For all $f \in F$ and $p \in \Pi$ we set $y_{f,p}$ to be $\max_{t\in T: \mathrm{D}(t)\ni p} x_{t,f}$. Hence $y_{f,p}$ is no more than $ \max_{t\in T: \mathrm{D}(t)\ni p} \frac{\alpha} {\alpha - 1} x^*_{t,f}$, which is no more than $\alpha/(\alpha-1) y^*_{f,p}$. This in turn implies the second part of the claim.
\end{proof}
Next we bound the routing cost of the assignment $\mathcal{F}P$.
\begin{lemma}
\label{lemma:routing}
For all $ i \in T$, $c(i,\mathcal{F}P(i))\le 9 \alpha C_r^*(i)$.
\end{lemma}
\begin{proof}
Let $i$ be a paying terminal that is assigned a facility $f$ in Step~\ref{step:open_fac_pay} of the algorithm. Then, $f\in\mathcal{F}(i)$ and therefore, by the definition of $x$ and $\mathcal{F}(i)$, $c(i,f)\le \alpha C_r^*(i)$.
Now, let $i$ be a free terminal which is assigned a facility $f$ in Step~\ref{step:open_fac} or Step~\ref{step:open_fac_free} of the algorithm. Say level $\ell(i) = d$ and let $t$ be the terminal that opened $f$. Then there are two possibilities: either $i=t$ or $i\in\Gamma_d(t)$. In the former case, $\mathcal{F}T(t) \in \mathcal{F}(i)$ and so $c(i, \mathcal{F}T(t)) \leq \alpha C_r^*(i)$.
Next we show that if terminal $t$ opens a facility (in Step \ref{step:open_fac}) then for all $t' \in \Gamma_d(t)$ we have $c(t', \mathcal{F}T(t)) \leq 3 \alpha C_r^*(t')$. Note that $\mathrm{Cov}(t') \cap \mathrm{Cov}(t) \neq \emptyset$ and let $j$ be a paying terminal in $\mathrm{Cov}(t') \cap \mathrm{Cov}(t)$. We have $\mathcal{F}T(t) \in \mathcal{F}(j)$ and $\mathcal{F}T(t') \in \mathcal{F}(j)$. Since terminal $j$ was selected in the second phase before $t'$ we have $C^*_r(j) \leq C^*_r(t')$. By triangle inequality we get that $c(t', \mathcal{F}T(t)) \leq c(t', \mathcal{F}T(t') ) + c (\mathcal{F}T(t'), j) + c(j , \mathcal{F}T(t) ) \leq \alpha C_r^*(t') + 2 \alpha C_r^*(j) \leq 3 \alpha C_r^*(t')$.
In Step~\ref{step:open_fac} the opened facility, say $f$, is selected to be $\phi( \bar{t})$ for the terminal $\bar{t} \in \Gamma_d(t)$ with minimum $C_r^*$ value. Since $ f \in \mathcal{F}(j')$ for some $j' \in \mathrm{Cov}(\bar{t})$ we have $ c(f, \bar{t}) \leq 3 \alpha C_r^*(\bar{t}) $. Also $\bar{t}$ is contained in $\Gamma_d(t)$, which implies that $c(\bar{t}, \mathcal{F}T(t) ) \leq 3 \alpha C_r^*(\bar{t})$.
Again using triangle inequality we get the desired result: $c(i,f) \leq c(i , \mathcal{F}T(t) ) + c(\mathcal{F}T(t), \bar{t}) + c(\bar{t}, f) \leq 3 \alpha C_r^*(i) + 6 \alpha C_r^*(\bar{t}) \leq 9 \alpha C_r^*(i)$. Here the last inequality follows from the definition of $\bar{t}$.
\end{proof}
Finally we account for the facility opening cost of the solution. Recall that $S_{\mathcal{F}P}(f)$ denotes the set of packets produced at $f$ under the assignment $\mathcal{F}P$. We note that a facility $f$ may be ``opened'' multiple times by different free terminals in Step~\ref{step:open_fac} of the algorithm. In this case, we treat each subsequent opening as opening a new copy of $f$ and designate a distinct set of terminals, $\mathcal{F}Pay(f)$, to pay for all of the packets to be produced at the new copy freshly (even though some of them may already be assigned to the facility).
Henceforth, for ease of exposition we will assume that each facility is opened at most once.
The following lemma notes that $|S_{\mathcal{F}P}(f)|$ can be bounded in terms of the demand sets of the terminals finally paying for this facility.
\begin{lemma}
\label{lemma:saf}
Let $f$ be a facility opened by a free terminal $t$. Then $S_{\mathcal{F}P}(f) = \mathrm{D}(t)\cup \cup_{j\in\mathrm{Cov}(t)} \mathrm{D}(j)$. Furthermore, $|S_{\mathcal{F}P}(f)|\le 2|\cup_{j\in\mathrm{Cov}(t)} \mathrm{D}(j)|$.
\end{lemma}
\begin{proof}
Let $t\inT^f_d$ and consider a terminal $t' \in \Gamma_d(t)$. We claim that $\mathrm{D}(t')\subseteq \mathrm{D}(t)\cup \cup_{j\in\mathrm{Cov}(t)} \mathrm{D}(j)$, and this implies the first part of the lemma. Let $j\in\mathrm{Cov}(t)\cap\mathrm{Cov}(t')$. Then, by the definition of a covering set, $\mathrm{D}(t')\cap\mathrm{D}(j)\ne\emptyset$. By laminarity, either $\mathrm{D}(j)\supset\mathrm{D}(t')$ in which case our claim holds, or $\mathrm{D}(j)\subset\mathrm{D}(t')$. In the latter case, $\mathrm{D}(t)\cap\mathrm{D}(j)\ne\emptyset$ implies $\mathrm{D}(t)\cap\mathrm{D}(t')\ne\emptyset$. Once again by laminarity, either $\mathrm{D}(t')\subseteq\mathrm{D}(t)$, or $\mathrm{D}(t')\supset\mathrm{D}(t)$. The latter case cannot hold because $t$ is considered before $t'$ in phase $3$ and therefore $|\mathrm{D}(t')|\le|\mathrm{D}(t)|$. In the former case our claim holds.
The second part of the lemma follows from the definition of covering.
\end{proof}
Using this lemma we can bound the facility opening cost of the assignment $\mathcal{F}P$ in terms of the average facility opening costs $C_f(t)$ for the paying terminals $t\in T^p$ in the fractional solution $(x,y)$.
\begin{lemma}
\label{lemma:fac_cost_one}
$\sum_f |S_{\mathcal{F}P}(f)|\lambda_f \le 9 \sum_{t\inT^p} |\mathrm{D}(t)| C_f(t)$.
\end{lemma}
\begin{proof}
First we bound the opening cost of facilities that were opened by free terminals. Lemma~\ref{lemma:saf} implies that $|S_{\mathcal{F}P}(f)|\le 2|\cup_{j\in\mathcal{F}Pay(f)} \mathrm{D}(j)|\le 2\sum_{j\in\mathcal{F}Pay(f)}|\mathrm{D}(j)|$. For facility $f$ write $\ell(f) =d$ iff the terminal opening it has level $d$. Note that for facility $f$ with $\ell(f) = d$ we have $\lambda_f \leq 2^d$.
Fix a terminal $i \in T^p$, and write $\mathrm{Cov}^{-1}(i) = \{ j \in T^f \mid i \in \mathrm{Cov}(j) \}$. Say $\ell(i) = \ell$. Then $C_f(i) \in (2^{\ell-1}, 2^{\ell} ]$. By definition, for all $j \in \mathrm{Cov}^{-1}(i)$ we have $\ell(j) \leq \ell$.
We claim that $i$ pays for at most one facility at level $d$ for $d\le \ell$, and does not pay for any facilities at level $d>\ell$. We prove the first part by contradiction. Say there exits $f \neq f'$ such that $\ell(f) = \ell(f')=d$ and $i \in \mathcal{F}Pay(f) \cap \mathcal{F}Pay(f')$. Write $t$ as the terminal that opens $f$ and $t'$ as the terminal that opens $f'$. Both $t$ and $t'$ are in $\mathrm{Cov}^{-1}(i)$ and have level equal to $d$. Without loss of generality assume $t$ was processed before $t'$ by the algorithm. Then $t' \in \Gamma_d(t)$ and we would have $\mathcal{F}P(t') = \mathcal{F}P(t)$, contradicting the assumption that they are assigned to different facilities.
For the second part of the claim, suppose that $i \in \mathcal{F}Pay(f)$ for some facility $f$. Then $f$ is opened by a terminal $t \in \mathrm{Cov}^{-1}(i)$. As stated above, the level of such a terminal $t$ must be no more than the level of $i$, hence $\ell(f) \leq \ell(i)$.
For a level $d$ facility $f$ we have $\lambda_f \leq 2^d$ and hence $\sum_{f : \mathcal{F}Pay(f) \ni i } \lambda_f \leq \sum_{d=1}^\ell 2^d \leq 4 C_f(i) $. We therefore get the following chain of inequalities.
\begin{align*}
\sum_f \lambda_f |S_{\mathcal{F}P}(f)| & \le 2 \sum_f \lambda_f \sum_{j\in\mathcal{F}Pay(f)} |\mathrm{D}(j)| \\
& = 2 \sum_{ j \in T^p } | \mathrm{D}(j) | \sum_{f : \ \mathcal{F}Pay(f) \ni j } \lambda_f \\
& \le 8 \sum_{j\in T^p} C_f(j) \ |\mathrm{D}(j)|
\end{align*}
Here the first inequality follows from Lemma \ref{lemma:saf} and the second inequality follows from the bound $\sum_{f : \mathcal{F}Pay(f) \ni i } \lambda_f \leq 4 C_f(i)$.
To account for facilities that were opened by paying terminals in Step \ref{step:open_fac_pay} of the algorithm we note that for any such facility $h$, the set $S_{\mathcal{F}P} (h) = \mathrm{D}(t)$ where $t$ is the paying terminal that opened $h$. Moreover $\lambda_{h} \leq \sum_f x_{t,f} \lambda_f $ and hence the opening cost incurred by the algorithm is no more than the fractional value, $\lambda_h | \mathrm{D}(t) | \leq C_f(t) \ |\mathrm{D}(t)|$. Hence the total facility opening cost incurred by the algorithm is no more than $9 \sum_{t\inT^p} |\mathrm{D}(t)| C_f(t)$.
\end{proof}
To complete the argument, we relate the costs $C_f(t)$ to the total cost $C_f(x,y)$.
\begin{lemma}
\label{lemma:fac_cost_two}
$\sum_{t\inT^p} |\mathrm{D}(t)| C_f(t)\le 2C_f(x,y)$.
\end{lemma}
\begin{proof}
When a terminal $t$ is added to $T^p$ (Step~\ref{step:add_to_TP} of the algorithm) it is not covered at any of the facilities in $\mathcal{F}(t)$. For $f\in\mathcal{F}$ let $L(t,f) = \mathrm{D}(t)\setminus\cup_{t'\in\mathrm{Pay}(f)}\mathrm{D}(t')$ denote the set of packets in $\mathrm{D}(t)$ that is not covered at $f$ at the time that $t$ is considered. Here, $\mathrm{Pay}(f)$ denotes the set of terminals that is paying for $f$ at the time that $t$ is considered. Note that $|L(t,f)|\ge 1/2 |\mathrm{D}(t)|$ for $f\in\mathcal{F}(t)$. For any facility $f$ the sets $L(t,f)$ are disjoint and partition the support of $y_{f,p}$ and hence the following chain of inequalities hold:
\begin{eqnarray}
\sum_{p} y_{f,p} & \geq & \sum_{t \in T^p} \sum_{p \in L(t,f)} y_{f,p} \nonumber \\
& \geq & \sum_{t \in T^p} \sum_{p \in L(t,f)} x_{t,f} \nonumber \\
& = & \sum_{t \in T^p} |L(t,f)| x_{t,f} \label{eqn:Ltf}
\end{eqnarray}
We can now derive the desired bound:
\begin{align*}
C_f(x,y) & =\sum_f \sum_p \lambda_f y_{f,p} \\
& = \sum_f \lambda_f \sum_p y_{f,p} \\
& \geq \sum_f \lambda_f \sum_{t \in T^p} |L(t,f)| \ x_{t,f} \\
& \geq \sum_f \sum_{ t \in T^p} \frac{1}{2} \ |\mathrm{D}(t)| \ \lambda_f x_{t,f} \\
& = \sum_{t \in T^p} \frac{1}{2} \ |\mathrm{D}(t)| \sum_f \lambda_f x_{t,f} \\
& = \frac{1}{2} \sum_{t \in T^p} \ |\mathrm{D}(t)| C_f^{(x,y)}(t)
\end{align*}
Here the third step follows from inequality (\ref{eqn:Ltf}) and the fourth step holds because for any $t \in T^p$ and $f\in\mathcal{F}$, either $x_{t,f}=0$ or $f\in\mathcal{F}(t)$ and the size of the set $L(t,f)$ is at least half its demand: $|L(t,f)| \geq 1/2 |\mathrm{D}(t)|$.
\end{proof}
Putting together the above lemmas we obtain the following theorem:
\begin{theorem}
Algorithm~\ref{Algorithm:Round} gives a $27$-approximation to the RAFL.
\end{theorem}
\begin{proof}
Lemma \ref{lemma:routing} implies that for any terminal the routing cost is no more than $9 \alpha$ times the optimal. Also from Lemma \ref{lemma:fac_cost_one} and Lemma \ref{lemma:fac_cost_two} we get that the total facility opening cost is no more than $18$ times the total facility opening cost of the filtered solution $C_f(x,y)$. Finally from Lemma \ref{lemma:filter_opt} we have $C_f(x,y) \leq \alpha/( \alpha -1) \ C_f^*$, so the facility opening cost of the generated solution is no more than $18 \alpha / ( \alpha -1) $ times the optimal. Hence the algorithm achieves an approximation factor of $\max \{ 9 \alpha , \frac{18 \alpha}{\alpha -1 } \}$, which is minimized at $\alpha=3$ to give us a $27$-approximation.
\end{proof}
\section{An $O(\log P)$ approximation for RAND}
In this section we develop an $O( \log P)$-approximation algorithm for the RAND where $P=|\Pi|$. The basic observation that our algorithm hinges on is that if every pair of demand sets in $\mathcal{D}$ is either identical or disjoint, that is, the tree $\tau$ is a two-level tree, then the problem becomes easy and can be approximated to within a small constant factor. In particular, then the problem becomes one of finding optimal Steiner trees connecting the terminals in $T_X$ to $s$ for every set $X\in \mathcal{D}$. The cost of these Steiner trees is purely additive because the different sets in $\mathcal{D}$ are disjoint.
In particular, this implies that for any collection $\mathcal{X}$ containing disjoint sets of packets, we can construct a partial solution over terminals in $\cup_{X\in\mathcal{X}} T_X$ at a cost of constant times the cost of the optimal solution. This immediately suggests an algorithm with approximation ratio a constant times the depth of the tree $\tau$: define the level of a node in $\tau$ as its distance from the root $\Pi$; for every level $k$, consider the collection $\mathcal{X}_k$ of sets at level $k$ and construct a constant factor approximation over terminals in $\cup_{X\in\mathcal{X}_k} T_X$.
In order to obtain a small approximation ratio through this approach, our algorithm first performs a preprocessing of the demand tree $\tau$ to ensure, at small cost, that for any pair of demand sets in the tree where one is a parent of the other, the total weight of the parent is at least twice as large as the total weight of the child. Since the weight of every packet is integral, this implies that the depth of the preprocessed tree is at most $\log w(\Pi)$, and gives us an $O(\log w(\Pi))$ approximation.
To obtain an $O(\log P)$ approximation, we need to do a more clever analysis. As discussed in the introduction, $P$ denotes the number of effectively distinct packets in the instance. In particular, we can assume without loss of generality that $P$ is equal to the number of nodes in the tree $\tau$. Our next key observation is that we can collectively bound the total cost of Steiner trees for nodes in a long root to leaf chain of nodes by a constant times the cost of the optimal solution (rather than by the length of the chain times the optimal cost). Here we crucially use the fact that each subsequent node in the chain has a weight at most half that of the preceding node, a fact that is ensured through our preprocessing step. Given this, we break up the demand tree into $\log P$ collections of chains, each of which corresponds to disjoint packet sets. The total cost of Steiner trees over each collection of chains can then be bounded to within a constant factor of the optimal cost, and we get an overall $O(\log P)$ approximation.
We now present our algorithm and analysis formally.
\paragraph{Preprocessing the tree $\tau$.}
Our preprocessing phase is described in Algorithm~\ref{Algorithm:Preproc} below. The preprocessing phase makes two changes to the given instance. First, it changes the demand sets of some terminals to supersets of their original demands. Second, after these changes to demand sets, if there are nodes in $\tau$ that do not have any terminals associated with them, it merges these nodes with their parents.
\begin{algorithm}{Given: Instance $(G,T,\mathcal{D},\tau)$; Return: New instance $(G,T,\mathcal{D}',\tau')$}
\caption{Preprocessing algorithm}
\label{Algorithm:Preproc}
\begin{algorithmic}[1]
\mathcal{S}ATE Perform depth first search over the tree $\tau$.
\mathcal{F}ORALL{ nodes $X\in\mathcal{D}$ encountered during DFS}
\mathcal{S}ATE Let $Y$ be the parent of $X$ in $\tau$
\IF{$w(X)>\frac12 w(Y)$}
\mathcal{S}ATE For all $t\in T_X$, set $\mathrm{D}'(t)=Y$.
\mathcal{S}ATE Merge $X$ with $Y$. That is, set $T_Y=T_Y\cup T_X$, remove $X$ from $\tau$, and reattach the children of $X$ in $\tau$ as the children of $Y$ in the modified tree.
\ENDIF
\ENDFOR
\end{algorithmic}
\end{algorithm}
We obtain the following lemma.
\begin{lemma}
\label{lem:preproc}
Consider an instance $(G,T,\mathcal{D},\tau)$ of RAND. Then Algorithm~\ref{Algorithm:Preproc} returns an instance $(G,T,\mathcal{D}',\tau')$ with the following properties:
\begin{enumerate}
\item For every $t\in T$, $\mathrm{D}'(t)\supseteq \mathrm{D}(t)$ and $w(\mathrm{D}'(t))\le 2w(\mathrm{D}(t))$.
\item $\mathcal{D}'$ is a laminar family.
\item For every pair of sets $X,Y\in\mathcal{D}'$ such that $X$ is a child of $Y$ in $\tau'$, $w(Y)\ge 2w(X)$.
\item The cost of the optimal solution over $(G,T,\mathcal{D}',\tau')$ is at most twice that of the optimal solution over $(G,T,\mathcal{D},\tau)$.
\item Any feasible solution to $(G,T,\mathcal{D}',\tau')$ is also feasible for $(G,T,\mathcal{D},\tau)$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first remark that the demand of any terminal $t$ is modified at most once, when its original node $\mathrm{D}(t)$ is encountered in the DFS over $\tau$; after this, the new node $\mathrm{D}'(t)$ is never processed again by DFS. Then, it is easy to see that the first property holds. The second property holds because $\tau'$ is a tree over sets in $\mathcal{D}'$ and therefore any two sets in $\mathcal{D}'$ are either disjoint or one is an ancestor of another. The third property holds by construction: when the node $X$ is encountered by the DFS, if the node survives the preprocessing, then it holds that $w(X)\le \frac12w(Y)$.
The fifth property follows immediately from the first.
To prove the fourth property, we note that for any two terminals $t_1$ and $t_2$, $\mathrm{D}(t_1)\subset \mathrm{D}(t_2)$ implies $\mathrm{D}'(t_1)\subseteq\mathrm{D}'(t_2)$. Now consider any optimal solution $\mathcal{P}=\{P_t\}_{t\in T}$ to $(G,T,\mathcal{D},\tau)$. We will show that the cost of the solution $\mathcal{P}$ for the new instance $(G,T,\mathcal{D}',\tau')$ is no more than twice its cost for $(G,T,\mathcal{D},\tau)$. For every edge $e\in E$ consider the set of terminals, say $T(e)$, that use $e$: $T(e)=\{t: P_t\ni e\}$. Let $T'(e)$ be the subset of $T(e)$ of terminals whose demand is not contained inside the demand of any other terminals in $T(e)$; that is $T'(e)$ denotes the terminals with the maximal demand sets. Then, in the instance $(G,T,\mathcal{D},\tau)$, $w(S_{\mathcal{P}}(e)) = \sum_{t\in T'(e)} w(\mathrm{D}(t))$. Moreover by our observation above, in the new instance $(G,T,\mathcal{D}',\tau')$, terminals in $T'(e)$ are still the maximal demand terminals and $w(S_{\mathcal{P}}(e)) = \sum_{t\in T'(e)} w(\mathrm{D}'(t))$. The claim now follows from the first property.
\end{proof}
\paragraph{Main algorithm.}
We now proceed to describe the main algorithm. Henceforth we will assume that the given instance of RAND satisfies the properties listed in Lemma~\ref{lem:preproc}, in particular, property 3.
\begin{algorithm}{Given: Instance $(G,T,\mathcal{D},\tau)$ satisfying the properties in Lemma~\ref{lem:preproc}; Return: Collection of paths from each terminal to the source $s$.}
\caption{A logarithmic approximation for RAND}
\label{Algorithm:log-p}
\begin{algorithmic}[1]
\mathcal{S}ATE Perform depth first search over the tree $\tau$.
\mathcal{F}ORALL{ nodes $X\in\mathcal{D}$ encountered during DFS}
\mathcal{S}ATE Construct an approximately optimal Steiner tree $\mathcal{S}(X)$ in $G$ over $T_X\cup\{s\}$. \label{step:steiner-tree}
\mathcal{S}ATE For every $t\in T_X$, return the unique path in $\mathcal{S}(X)$ from $t$ to $s$.
\ENDFOR
\end{algorithmic}
\end{algorithm}
We first define some notation. For a demand set $Y$, let $\mathcal{S}^*(Y)$ denote the optimal Steiner tree over $T_Y\cup\{s\}$. The cost of this Steiner tree is $\mathrm{cost}(\mathcal{S}^*(Y)) = w(Y)\sum_{e\in\mathcal{S}^*(Y)} c_e$. Analogously we define the cost of an arbitrary Steiner tree $\mathcal{S}(Y)$ over $T_Y\cup\{s\}$ as $\mathrm{cost}(\mathcal{S}(Y))$.
In our analysis, we sometimes need to consider sets of nodes in $\tau$ and bound their cost collectively. To this end, we define a {\em chain} $\meta{Y}$ to be a set $\meta{Y} = \{Y_0, Y_1, \cdots, Y_k\}$ where for every $i<k$, $Y_i$ is a parent of $Y_{i+1}$. Recall that this implies $Y_0\supset Y_1\supset\cdots\supset Y_k$ and $w(Y_i)\ge 2w(Y_{i+1})$ for all $i<k$. We call the node $Y_0$ the start of the chain $\meta{Y}$. We say that two chains $\meta{Y_1}$ and $\meta{Y_2}$ are disjoint if $(\cup_{Y\in \meta{Y_1}} Y)\cap(\cup_{Y\in \meta{Y_2}} Y)=\emptyset$.
The following is the main lemma of this section and allows us to bound the cost of large collections of nodes in $\tau$.
\begin{lemma}
\label{lem:disjoint-sets}
Let $\mathcal{Y}=\{\meta{Y}_1, \meta{Y}_2, \cdots\}$ be a collection of mutually disjoint chains. That is, for any $\meta{Y}_i, \meta{Y}_j\in\mathcal{Y}$, $\meta{Y}_i$ and $\meta{Y}_j$ are disjoint. Then $\sum_{\meta{Y}\in\mathcal{Y}}\sum_{Y\in\meta{Y}} \mathrm{cost}(\mathcal{S}^*(Y))$ is at most $2\mathrm{cost}(\mathrm{OPT})$.
\end{lemma}
\begin{proof}
For $\meta{Y}\in\mathcal{Y}$, let $\mathrm{OPT}(\meta{Y})$ denote $\mathrm{OPT}(\cup_{Y\in\meta{Y}} T_Y)$, the restriction of the optimal solution to the terminals associated with sets in $\meta{Y}$. Note that because the chains in $\mathcal{Y}$ are disjoint, $\sum_{\meta{Y}\in\mathcal{Y}} \mathrm{cost}(\mathrm{OPT}(\meta{Y}))\le \mathrm{cost}(\mathrm{OPT})$. Therefore, for the rest of the proof we will argue that for any $\meta{Y}\in\mathcal{Y}$, $\sum_{Y\in\meta{Y}} \mathrm{cost}(\mathcal{S}^*(Y))\le 2\mathrm{cost}(\mathrm{OPT}(\meta{Y}))$, and this will imply the lemma.
To prove the claim, fix a chain $\meta{Y}\in\mathcal{Y}$, and let $\meta{Y} = \{Y_0, Y_1, \cdots, Y_k\}$ where $Y_0\supset Y_1\supset\cdots\supset Y_k$, and $w(Y_i)\ge 2w(Y_{i+1})$ for all $i<k$. Recall that $\mathrm{OPT}(\meta{Y})$ contains a path for every terminal in $\cup_{Y\in\meta{Y}} T_Y$. Let $\mathcal{P}_Y$ denote the collection of paths for terminals in $T_Y$. We say that $e\in\mathcal{P}_Y$ if $e\in \cup_{t\in T_Y} P_t$. Let $\mathcal{S}(Y)$ denote the Steiner tree over $T_Y\cup\{s\}$ defined by $\mathcal{P}_Y$ and note that $\mathrm{cost}(\mathcal{S}^*(Y)) \le \mathrm{cost}(\mathcal{S}(Y)) = w(Y)\sum_{e\in\mathcal{P}_Y} c_e$. Then,
\[\sum_{Y\in\meta{Y}} \mathrm{cost}(\mathcal{S}^*(Y))\le \sum_{e\in E} c_e \sum_{Y\in\meta{Y}: \mathcal{P}_Y\ni e} w(Y)\]
On the other hand, because of the containment structure of sets in $\meta{Y}$, \[\mathrm{cost}(\mathrm{OPT}(\meta{Y})) = \sum_{e\in E} c_e \max_{Y\in\meta{Y}: \mathcal{P}_Y\ni e} w(Y)\]
To conclude the proof we claim that for every edge $e$, $\sum_{Y\in\meta{Y}: \mathcal{P}_Y\ni e} w(Y)\le 2\max_{Y\in\meta{Y}: \mathcal{P}_Y\ni e} w(Y)$. But this is easy to see because weights of sets $Y\in\meta{Y}$ are geometrically decreasing by a factor of at least two.
\end{proof}
Next we show that the tree $\tau$ can be decomposed into at most $\log P$ different collections of mutually disjoint chains. This along with Lemma~\ref{lem:disjoint-sets} will allow us to prove our desired approximation.
\begin{lemma}
\label{lem:decomp}
Any demand tree $\tau$ can be decomposed into at most $\log P$ different collections of mutually disjoint chains $\mathcal{Y}_1, \mathcal{Y}_2, \cdots, \mathcal{Y}_k$, such that each node in $\tau$ belongs to exactly one collection. Here $P$ is the number of nodes in $\tau$.
\end{lemma}
\begin{proof}
We decompose the tree by finding a long chain, removing it from the tree, and then recursing on the remaining subtrees. Given a tree $\tau$ with root $Y_0$, we start at $Y_0$ and follow a path down to a leaf. Let $m$ denote the number of nodes in $\tau$. At any node, we consider the sizes of the subtrees rooted at the node in terms of the number of nodes in the tree; the path then moves to the child corresponding to the largest subtree. Note that all of the other remaining subtrees rooted at the node have at most $m/2$ nodes.
Let $\meta{Y}=\{Y_0, Y_1, \cdots, Y_k\}$ be the path obtained. This collection of nodes is a chain by definition. Consider removing the nodes in $\meta{Y}$ from $\tau$. We then note the following properties: (1) Every remaining connected component of the tree is of size at most $m/2$; (2) Let $\tau_1$ and $\tau_2$ denote any two connected components. Then $(\cup_{X\in\tau_1} X)\cap(\cup_{X\in\tau_2} X)=\emptyset$, that is, the connected components are mutually disjoint.
The second property can be proved by contradiction. If two of the components are not disjoint, then by laminarity, they contain nodes $X_1$ and $X_2$ respectively such that $X_1$ is an ancestor of $X_2$ in $\tau$. Since $X_1$ and $X_2$ belong to different connected components, there must be a node on the path between them in $\tau$ that is in the chain $\meta{Y}$. Then, all ancestors of this node are in $\meta{Y}$ including $X_1$ which contradicts the fact that $X_1$ belongs to a connected component left after removing the chain.
We output $\{\meta{Y}\}$ as the first collection of mutually disjoint chains. (Note that this first collection has only one chain in it.) Now let $\tau_1, \tau_2, \cdots$ be the connected components left over in $\tau$ after removing the nodes in $\meta{Y}$. We recursively find collections of mutually disjoint chains in each of the components. Call these $\mathcal{Y}_{i1}, \mathcal{Y}_{i2}, \cdots$ etc. for the $i$th connected component. Then, we output the collections $\cup_i \mathcal{Y}_{ij}$ for each $j$. Since the connected components are mutually disjoint, the collections we output are also mutually disjoint. Furthermore, since the sizes of the connected components decrease by a factor of $2$ each time we go down a level of recursion, it is easy to argue that the number of collections we output are bounded by $\log P$ where $P$ is the number of nodes in the tree $\tau$ that we started out with.
\end{proof}
We now present the main theorem for this section:
\begin{theorem}
\label{thm:main-log-p}
Let $(G,T,\mathcal{D},\tau)$ be an instance of RAND that satisfies the conditions in Lemma~\ref{lem:preproc}. Then Algorithm~\ref{Algorithm:log-p} obtains a $(2\alpha\log P)$-approximation for the RAND over this instance where $P$ is the number of effectively distinct packets in the instance, and $\alpha$ is the approximation factor of the Steiner tree algorithm used in Step~\ref{step:steiner-tree} of the algorithm.
\end{theorem}
\begin{proof}
For every set $X\in\mathcal{D}$, let $\mathcal{S}(X)$ denote the Steiner tree built by our algorithm over $T_X\cup\{s\}$. Then we have $\mathrm{cost}(\mathcal{S}(X))\le\alpha\mathrm{cost}(\mathcal{S}^*(X))$. We first use Lemma~\ref{lem:decomp} to decompose the tree $\tau$ into at most $\log P$ collections of mutually disjoint chains. Call these collections $\mathcal{Y}_1, \mathcal{Y}_2, \cdots, \mathcal{Y}_k$.
The total cost of our solution can now be written as
\begin{align*}
\sum_{Y\in\mathcal{D}} \mathrm{cost}(\mathcal{S}(Y)) & = \sum_{i\le k} \sum_{\meta{Y}\in\mathcal{Y}_i}\sum_{Y\in\meta{Y}} \mathrm{cost}(\mathcal{S}(Y))\\
& \le \alpha \sum_{i\le k} \sum_{\meta{Y}\in\mathcal{Y}_i}\sum_{Y\in\meta{Y}} \mathrm{cost}(\mathcal{S}^*(Y))\\
& \le \alpha \sum_{i\le k} 2\mathrm{cost}(\mathrm{OPT})\\
& = 2\alpha k\ \mathrm{cost}(\mathrm{OPT})
\end{align*}
Here the second inequality follows by applying Lemma~\ref{lem:disjoint-sets}. The theorem now follows by noting that $k\le\log P$.
\end{proof}
Combining Theorem~\ref{thm:main-log-p} with Lemma~\ref{lem:preproc} we get the following result.
\begin{corollary}
Algorithms~\ref{Algorithm:Preproc} and \ref{Algorithm:log-p} together obtain a $(4\alpha\log P)$-approximation for the RAND where $\alpha$ is the approximation factor of the Steiner tree algorithm used in the algorithm.
\end{corollary}
We conclude this section by noting that Algorithm~\ref{Algorithm:log-p} can be implemented in a simple combinatorial fashion in $O(n^3)$ time as a generalization of Prim's algorithm for the minimum spanning tree problem as follows. This version of the algorithm obtains an $(8\log P)$-approximation.
\begin{enumerate}
\item Let $\mathrm{anc}(t)$ denote the set of ancestors of $t$ (those with demands that are strict supersets of $\mathrm{D}(t)$), and $\mathrm{peer}(t)$ denote the set of its peers (those with demands identical to that of $t$).
\item At any step call a terminal $t$ eligible if all of its ancestors are already connected to the root.
\item Initialize $W=\{s\}$.
\item Let $\mathcal{D}elta(t)$ denote the distance in $G$ from $t$ to its closest node in $W\cap(\mathrm{anc}(t)\cup\mathrm{peer}(t))$.
\item While $W\ne T$, pick the eligible terminal with the smallest $\mathcal{D}elta(t)$ and connect it to its closest node in $W\cap(\mathrm{anc}(t)\cup\mathrm{peer}(t))$. Update $W=W\cup\{t\}$ and update $\mathcal{D}elta$.
\end{enumerate}
\end{document} |
\begin{document}
\thispagestyle{empty}
\begin{abstract}
In 1997 Bousquet-M\'elou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$. We provide a new view of their correspondence that allows results in one domain to be translated into the other. We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$.
\end{abstract}
\maketitle
\thispagestyle{empty}
\section{Introduction}
In 1997 Bousquet-M\'elou and Eriksson \cite{BME} introduced lecture hall partitions by showing that they are the inversion vectors of elements of the parabolic quotient ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}/C$. We provide a new way to understand their correspondence which allows results in one domain to be translated into the other. In Section~\ref{sec:proof} we distill the essence of the new proof into a correspondence between a $(2,4,\hdots,2n)$-inversion sequence (the excess of the lecture hall partition) and a signed permutation (the residues modulo $2n+2$ of the window of the element of the parabolic quotient). Indeed, a key insight is that the set of lecture hall partitions $L_n$ is the same as the set of generalized \mbox{$\s$-lecture} hall partitions for $\s=(2,4,\hdots,2n)$; this appears to be a more natural environment for understanding the correspondence with the affine Coxeter group ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$.
The structure of this paper is as follows. In this section we introduce just enough notation and background to state the result of Bousquet-M\'elou and Eriksson. In Section~\ref{sec:proof} we give our new proof of their result. In Section~\ref{sec:stats}, we show how this new correspondence allows us to find the equivalence of combinatorial statistics in the two regimes. This leads to Section~\ref{sec:results} in which we present new proofs and novel results in ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}/C$ corresponding to truncated lecture hall partitions, as well as a product-form generating function for a quadratic statistic $\mathsf{lhp}_C$ on $C_n$. Bj\"orner and Brenti \cite{BB} is our reference on Coxeter groups.
\subsection{The hyperoctahedral group}\
\noindent
The {\em hyperoctahedral group} $C_n$ (or $B_n$) is the (finite) group of symmetries of the $n$-cube, which can be defined as a reflection group with generators $\{s_0,\hdots,s_{n-1}\}$ and the relations $s_i^2=\id$ for $0\leq i\leq n-1$, $(s_is_{i+1})^3=\id$ for $1\leq i\leq n-2$, and $(s_0s_1)^4=\id$; all other pairs of generators commute. $C_n$ is also seen as the set of permutations $\sigma$ of \[\{-n,-(n-1),\hdots,n-1,n\}\] satisfying
$\sigma(-i)=-\sigma(i)$ for $1\leq i\leq n$. These permutations are completely defined by the values of the {\em window} $[\sigma(1), \hdots, \sigma(n)]$, which is a permutation of $\{\pm 1,\hdots,\pm n\}$ in which exactly one of $+i$ or $-i$ appears for $1\leq i\leq n$. In this way $C_n$ is realized as the set of {\em signed permutations}.
\subsection{The affine hyperoctahedral group}\lambdabdabel{sec:tldC}\
\noindent
The {\em affine hyperoctahedral group} ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ is an (infinite) reflection group that includes all the generators and relations of $C_n$ and along with one additional generator, $s_n$, satisfying $s_n^2=\id$ and one more non-commuting relation, $(s_{n-1}s_n)^4=\id$. Elements $w$ of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ can be written as permutations of $\bbZ$ satisfying
\begin{equation}\lambdabdabel{eq:transC1}
w_{i+(2n+2)}=w_{i}+(2n+2)
\end{equation}
and
\begin{equation}\lambdabdabel{eq:transC2}
w_{-i}=-w_i
\end{equation}
for all $i\in \bbZ$ \cite{Henrik, EE}. These conditions imply $w(i)=i$ for all $i\equiv 0\bmod (n+1)$. We will write $N=2n+2$.
As above, an element $w\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ is completely defined by the window $[w_1,\hdots,w_n]$. We know that a window corresponds to an element of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ if the values $\pm w_1, \hdots, \pm w_n$ are all distinct modulo $N$.
The group $C_n$ embeds as a parabolic subgroup of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$; as such each element $w\in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ has a parabolic decomposition $w=w^0w_0$ where $w^0\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ and $w_0\in C_n$. Bj\"orner and Brenti \cite[Proposition~8.4.4]{BB}, show that the entries of a window of an element ${w} \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ are all positive and sorted in increasing order.
\subsection{Class inversions}\
\noindent
Analogous to the concept of inversions for permutations is the idea of class inversions for $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$.
For $j \in \bbZ$, define the {\em class} $\ang{j}$ to be the set of all positions of a form $j+kN$ or $-j+kN$ for all $k\in \bbZ$. We will keep track of the number of {\em class inversions}, which are entries in positions of the class $\ang{j}$ that are inverted with a fixed entry in position $i$. More precisely, for $1 \leq j \leq i \leq n$ define
\begin{equation}
\lambdabdabel{eq:Iij}
\begin{aligned}
I_{i,j} &= \abs{\{k\in\bbZ_{> 0} : w_i>w_j+kN\}}+ \abs{\{k\in\bbZ_{> 0} : w_i>-w_{j}+kN\}}\\
&=\floor{\frac{w_i - w_j}{N}} + \floor{\frac{w_i + w_j}{N}}.\\
\end{aligned}
\end{equation}
For $1\leq i\leq n$, let $I_i = \sum_{1 \leq j \leq i} I_{i,j}$. Define the {\em class inversion vector} $I(w)$ of $w$ as
\[
I(w) = [I_1, I_2, \ldots, I_n].
\]
The {\em type-C affine inversion number} of $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ is given by
\[
\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w) = \sum_{i=1}^n I_i,
\]
which is equal to the Coxeter length $\ell(w)$ of $w$ \cite[Proposition~8.4.1]{BB}.
\subsection{Lecture hall partitions and class inversion vectors}\
\noindent
Lecture hall partitions, introduced by Bousquet-M\'elou and Eriksson in \cite{BME}, arose naturally in the study of inversions in ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$.
The set $L_n$ of {\em lecture hall partitions} of length $n$ is the set of integer sequences
\[
L_n = \left\{(\lambdabdam_1,\hdots,\lambdabdam_n) \in \mathbb{Z}^n: 0\leq \frac{\lambdabdam_1}{1} \leq \frac{\lambdabdam_2}{2}\leq \cdots \leq \frac{\lambdabdam_n}{n}\right\}.
\]
We will focus on the following characterization of lecture hall partitions and its implications.
\begin{theorem}[Bousquet-M\'elou and Eriksson \cite{BME}]
The mapping sending
$w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ to its class inversion vector $[I_1, \ldots, I_n]$ is a bijection
$${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n\rightarrow} \def\low{\leftarrow L_n.$$
\lambdabdabel{CtoL}
\end{theorem}
In the next section we provide a new view of this bijection that allows results in one domain to be translated into the other.
\section{A new view of the bijection}\lambdabdabel{sec:proof}
In this section, we will give a proof of Theorem \ref{CtoL} with the following overview.
\begin{itemize}
\item Show that the elements $\lambdabda \in L_n$ can be characterized as pairs in this set:
\[
T_n^{(\s)} = \left \{(b,e) \in \mathbb{Z}^n \times \mathbf{I}_n^{(\s)} : {\rm for} \ i \in \{0, \ldots, n-1\}, \ b_i \leq b_{i+1} \ {\rm and} \ i \in \mathsf{Asc}^{(\s)}(e) \implies b_i < b_{i+1} \right \},
\]
for the sequence $\s=(2,4,\ldots, 2n)$, in such a way that
$$\abs{\lambdabda} = \sum_{i=1}^n 2ib_i- \abs{e}.$$
Inversion sequences $\mathbf{I}_n^{(\s)}$ and the ascent statistic $\mathsf{Asc}^{(\s)}$ are defined in Section \ref{genlhp}.
\item Show that elements $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ can be characterized as pairs in this set:
\[
U_n = \left \{(c,\sigma) \in \mathbb{Z}^n \times C_n : {\rm for} \ i \in \{0, \ldots, n-1\}, \ c_i \leq c_{i+1} \ {\rm and} \ i \in \mathsf{Des}(\sigma) \implies c_i < c_{i+1} \right \}.
\]
in such a way that $$\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w)= \sum_{i=1}^n 2ic_i- \mathsf{inv}_C(\sigma).$$
\item Show a bijection $\Psi: C_n \rightarrow \mathbf{I}_n^{(\s)}$ with the property that for $\sigma \in C_n$,
\[
\mathsf{Des}(\sigma) = \mathsf{Asc}^{(\s)}(\Psi(\sigma))\ \ \ {\rm and} \ \ \ \mathsf{inv}_C(\sigma) = \abs{\Psi(\sigma))}=\abs{e}.
\]
\item Conclude that the following action sends $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ to its inversion vector $I(w)=\lambdabda$:
\[
w \longleftrightarrow (c, \sigma) \longleftrightarrow (c,\Psi(\sigma)) \longleftrightarrow \lambdabda
\]
\end{itemize}
\subsection{Generalized lecture hall partitions}\lambdabdabel{genlhp}\
\noindent
A natural generalization of $L_n$ is to consider the set $L_n^{(\mathbf{s})}$ of {\em $\s$-lecture hall partitions}, defined for a sequence $\s=(s_1,\hdots,s_n)$ of positive integers by
\[
L_n^{(\mathbf{s})} = \left\{(\lambdabdam_1,\hdots,\lambdabdam_n): 0\leq \frac{\lambdabdam_1}{s_1} \leq \frac{\lambdabdam_2}{s_2}\leq \cdots \leq \frac{\lambdabdam_n}{s_n}\right\}.
\]
In this paper we will be exploiting the fact that the sets $L_n$ and $L_n^{(2,4,\hdots,2n)}$ are equal.
For $\lambdabdam \in L_n^{(\mathbf{s})}$, define the ceiling of $\lambdabdam$ with respect to $\s$ by
\[
\ceil{\lambdabdam}^{(\s)}:=\left(
\Big\lceileil\! \frac{\lambdabdam_1}{s_1}\!\Big\rceileil,
\Big\lceileil\! \frac{\lambdabdam_2}{s_2}\!\Big\rceileil,
\hdots,
\Big\lceileil\! \frac{\lambdabdam_n}{s_n}\!\Big\rceileil\right)\]
and the {\em excess} of $\lambdabda$ with respect to $\s$ by
\[
e^{(\s)}(\lambdabdam) := \left(
s_1\Big\lceileil\! \frac{\lambdabdam_1}{s_1}\!\Big\rceileil-\lambdabdam_1,
s_2\Big\lceileil\! \frac{\lambdabdam_2}{s_2}\!\Big\rceileil-\lambdabdam_2,
\hdots,
s_n\Big\lceileil\! \frac{\lambdabdam_n}{s_n}\!\Big\rceileil-\lambdabdam_n\right).
\]
\begin{observation}
\lambdabdabel{lhpchar}
For any sequence $\s$ of positive integers, all of the following are consequences of $\lambdabdam \in L_n^{(\mathbf{s})}$.
\begin{enumerate}[label=(\alph*)]
\item
$\ceil{\lambdabdam}^{(\s)}$ is a nondecreasing sequence of positive integers.
\item
$0 \leq e_i^{(\s)}(\lambdabda) < s_i$ for $1 \leq i \leq n$.
\item
If $\ceil{\lambdabdam}_1^{(\s)}=0$ then $e_1^{(\s)}(\lambdabda)=0$.
\item
For $1 \leq i < n$,
\[
\text{If }\ceil{\lambdabdam}_i^{(\s)} = \ceil{\lambdabdam}_{i+1}^{(\s)} \text{ then } \frac{e_i^{(\s)}(\lambdabda)}{s_i} \geq \frac{e_{i+1}^{(\s)}(\lambdabda)}{s_{i+1}}.
\]
\end{enumerate}
\end{observation}
Observation \ref{lhpchar} motivated the following definitions from \cite{SS}.
For a sequence $\s=(s_1, \ldots, s_n)$ of positive integers, the {\em $\s$-inversion sequences} are defined by
\[
\mathbf{I}_n^{(\s)} = \{ (e_1, \ldots, e_n) \in \mathbb{Z}^n : 0 \leq e_i < s_i, \ 1 \leq i \leq n \}.
\]
For $e \in \mathbf{I}_n^{(\s)}$, the {\em ascent set} of $e$ with respect to $\s$ is
\begin{equation}\lambdabdabel{eq:ascs}
\mathsf{Asc}^{(\s)}(e) = \left \{ i \in \{0, \ldots, n-1\} \ :\ \frac{e_i}{s_i} < \frac{e_{i+1}}{s_{i+1}} \right \},
\end{equation}
where for the convenience of the definition we let $e_0=0$ and $s_0=1$.
If $\lambdabdam \in L_n^{(\mathbf{s})}$, Observation \ref{lhpchar}(b) says that $e^{(\s)}(\lambdabdam) \in \mathbf{I}_n^{(\s)}$ and
Observations \ref{lhpchar}(a,c,d) imply that if $i \in \mathsf{Asc}^{(\s)}(e)$ then $\ceil{\lambdabdam}_i^{(\s)} < \ceil{\lambdabdam}_{i+1}^{(\s)}$ and otherwise $\ceil{\lambdabdam}_i^{(\s)} \leq \ceil{\lambdabdam}_{i+1}^{(\s)}$.
\begin{example}
The sequence $(1,2,1,7,8)$ is a $(2,4,6,8,10)$-inversion sequence and its ascent set is $\mathsf{Asc}^{(2,4,6,8,10)}(1,2,1,7,8)= \{0,3\}$. We note that position 1 is not a $(2,4,6,8,10)$-ascent since $\frac{1}{2} \not < \frac{2}{4}$, while
4 is not a $(2,4,6,8,10)$-ascent because $\frac{7}{8} \not < \frac{8}{10}$
\end{example}
It was shown in \cite{SS} that these conditions characterize $ L_n^{(\mathbf{s})}$.
\begin{lemma}[\cite{SS}]
For any sequence $\s$ of positive integers, the mapping
\[
\lambdabda \longrightarrow \big(\ceil{\lambdabdam}^{(\s)}, e^{(\s)}(\lambdabdam)\big)
\]
is a bijection
\[
L_n^{(\mathbf{s})} \rightarrow T_n^{(\s)}
\]
where, under the convention that $b_0=0$,
\[
T_n^{(\s)} = \left \{(b,e) \in \mathbb{Z}^n \times \mathbf{I}_n^{(\s)} : {\rm for} \ i \in \{0, \ldots, n-1\}, \ b_i \leq b_{i+1} \ {\rm and} \ i \in \mathsf{Asc}^{(\s)}(e) \implies b_i < b_{i+1} \right \}.
\]
\end{lemma}
\subsection{A characterization of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$}\lambdabdabel{Cchar}\
\noindent
From Section \ref{sec:tldC}, each representative window $w = [w(1), \ldots, w(n)] \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ is an increasing sequence of positive integers with a unique representation
\[
[w(1), \ldots, w(n)] = [c_1 N + \sigma_1, c_2N + \sigma_2, \ldots, c_nN+\sigma_n]
\]
where $\sigma=(\sigma_1, \ldots, \sigma_n) \in C_n$. A {\em descent} of $\sigma \in C_n$ is a position $i \in [n-1]$ such that $\sigma_i > \sigma_{i+1}$. $\mathsf{Des}(\sigma)$ is the set of all descents of $\sigma$ and $\mathsf{des}(\sigma) = \abs{\mathsf{Des}(\sigma)}$.
\begin{observation}
\lambdabdabel{affCchar}
All of the following properties are consequences of the definitions:
\begin{enumerate}[label=(\alph*)]
\item
$0 \leq c_1 \leq c_2 \leq \ldots \leq c_n$.
\item
If $c_1=0$ then $\sigma_1 >0$.
\item
For $1 \leq i < n$ if $i \in \mathsf{Des}(\sigma)$ then $c_i < c_{i+1}$
\end{enumerate}
\end{observation}
We show these conditions characterize ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$.
\begin{lemma}
The mapping
\[
w=[w_1, \ldots, w_n] \longrightarrow (c,\sigma),
\]
where $w_i = c_iN+\sigma_i$ and $\sigma \in C_n$
is a bijection
\[
{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n \rightarrow U_n,
\]
where, under the convention that $c_0=0$,
\[
U_n = \left \{(c,\sigma) \in \mathbb{Z}^n \times C_n : {\rm for} \ i \in \{0, \ldots, n-1\}, \ c_i \leq c_{i+1} \ {\rm and} \ i \in \mathsf{Des}(\sigma) \implies c_i < c_{i+1} \right \}.
\]
\end{lemma}
\begin{proof}
Let $(c,\sigma)$ be an element of $U_n$. We need to verify that $[c_1N+\sigma_1, \ldots, c_nN+\sigma_n]$ is the window of a coset representative in ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$. Since the residues mod $N$ are distinct and not congruent to $0$ or $n+1$ we need only verify that the window elements are positive and in weakly increasing order.
To check that $c_1N+\sigma_1 > 0$, this is true if $\sigma_1>0$. But if $\sigma_1 < 0$, then position 0 is a descent of $\sigma$ and, by definition of $U_n$, $c_0 = 0 < c_1$. So, $c_1$ is positive and therefore $c_1N+\sigma_1 > 0$.
For $1 \leq i < n$, to check that $c_iN+\sigma_i \leq c_{i+1}N + \sigma_{i+1}$, this is clearly true if $c_i<c_{i+1}$. If not, by definition of $U_n$, it must be that $c_i=c_{i+1}$ and therefore that $i \not \in \mathsf{Des}(\sigma)$. Thus $\sigma_i < \sigma_{i+1}$ and therefore
$c_iN+\sigma_i \leq c_{i+1}N + \sigma_{i+1}$.
\end{proof}
\subsection{Inversion sequences, permutations, and signed permutations}\
\noindent
For a permutation $\pi=(\pi_1, \ldots, \pi_n) \in S_n$, an {\em inversion} of $\pi$ is a pair $i < j$ where $\pi_i > \pi_j$ and a {\em descent} of $\pi$ is a position $i \in [n-1]$ such that $\pi_i > \pi_{i+1}$. The number of inversions of $\pi$ is denoted $\mathsf{inv}(\pi)$ while $\mathsf{Des}(\pi)$ is the set of all descents of $\pi$ and $\mathsf{des}(\pi) = \abs{\mathsf{Des}(\pi)}$.
For a signed permutation $\sigma=(\sigma_1,\ldots, \sigma_n) \in C_n$, there are (at least) two notions of inversion number. One standard definition is
\[
\mathsf{inv}(\sigma) = \big\{(j,i) : 1 \leq j < i \leq n \ {\rm and} \ \sigma_j > \sigma_i\big\}.
\]
A more natural inversion number from \cite{BB} aligns with the Coxeter length of $\sigma$; for this we also need the number of {\em negative sum pairs},
\[
\mathsf{nsp}(\sigma) = \big\{(j,i) : 1 \leq j < i \leq n \ {\rm and} \ \sigma_j + \sigma_i<0\big\}
\]
and the number of negative entries
\[
\mathsf{neg}(\sigma) = \#\{i \in [n] : \sigma_i < 0\}.
\]
The inversion number for $\sigma \in C_n$ is then defined by
\[
\mathsf{inv}_C(\sigma) = \mathsf{inv}(\sigma) + \mathsf{neg}(\sigma) + \mathsf{nsp}(\sigma).
\]
Classical inversion sequences $e\in\mathbf{I}_n= \mathbf{I}_n^{(1,2, \ldots, n)}$ have been used in various ways to encode permutations. Define
$$\Theta: S_n \rightarrow \mathbf{I}_n$$
by $\Theta(\pi) = (e_1 \ldots, e_n)$ where
\[
e_i = \#\{j \in [i-1] : \pi_j > \pi_i\}.
\]
Clearly $\mathsf{inv}(\pi) = \abs{\Theta(\pi)}$. Moreover, it was shown in \cite{SS} that
\[
\mathsf{Des}(\pi) = \mathsf{Asc}(\Theta(\pi)).
\]
On the other hand, $(2,4,\hdots,2n)$-inversion sequences can be used to encode signed permutations.
Define
\[
\Psi: C_n \rightarrow \mathbf{I}_n^{(2,4, \ldots, 2n)}
\]
as follows.
For the signed permutation $\sigma \in C_n$, create the unsigned permutation $\abs\sigma=(\abs{\sigma_1}, \ldots, \abs{\sigma_n})$ and let
$e^*=\Theta(\abs\sigma)$. This means $e^*_i$ is the number of $j \in [i-1]$ such that
$\abs{\sigma_j} > \abs{\sigma_i}$.
Now define $\Psi(\sigma)=e=(e_1, \ldots, e_n)$ where
\[
e_i = \left \{
\begin{array}{ll}
e_i^* & {\rm if} \ \sigma_i > 0\\
2i-1-e_i^* & {\rm otherwise.}
\end{array}
\right .
\]
Lemma~\ref{lem:invsigma} will prove that $\mathsf{inv}_C(\sigma) = \abs{\Psi(\sigma)}$ and \cite[Theorem 3.12\,(4)]{SV} proves
\[\mathsf{Des}(\sigma)=\mathsf{Asc}^{(2,4,\hdots,2n)}\big(\Psi(\sigma)\big).\]
We remind the reader that this notion of ascent set from Equation~\eqref{eq:ascs} is more subtle than the classic notion of ascent set.
\begin{example}
For the signed permutation $\sigma=(-3,-1,2,-5,-4) \in C_5$, the unsigned permutation $\abs\sigma$ is $(3,1,2,5,4)$, whose inversion sequence $e^*=(0,1,1,0,1)$. As a consequence, \[\Psi(\sigma) = (1,2,1,7,8) \in \mathbf{I}_5^{(2,4,6,8,10)}.\] In addition, $\mathsf{inv}_C(\sigma)=19=1+2+1+7+8$ and
\[\mathsf{Des}(-3,-1,2,-5,-4) = \{0,3\} = \mathsf{Asc}^{(2,4,6,8,10)}(1,2,1,7,8).\]
\end{example}
For a Boolean function $f$, let $\chi(f)=1$ if $f$ is true and 0 otherwise. The following lemma is straightforward.
\begin{lemma}
\lambdabdabel{eij}
For $\sigma \in C_n$,
\[
\chi(\abs{\sigma_j} > \abs{\sigma_i}) =
\left \{
\begin{array}{ll}
\chi(\sigma_j > \sigma_i) + \chi(\sigma_j +\sigma_i < 0) & {\rm if} \ \sigma_i > 0\\
2-\chi(\sigma_j > \sigma_i) - \chi(\sigma_j +\sigma_i < 0) & {\rm if} \ \sigma_i < 0\\
\end{array}
\right .
\]
\end{lemma}
\begin{corollary}
\lambdabdabel{eidef}
Given $\sigma \in C_n$, define $e=\Psi(\sigma)=(e_1, \ldots, e_n)$. Then for $1 \leq i \leq n$,
\[
e_i = \sum_{j=1}^{i} \big(\chi(\sigma_j > \sigma_i) + \chi(\sigma_j +\sigma_i < 0)\big).
\]
\end{corollary}
\begin{proof}
By definition of $\Psi$, if $\sigma_i > 0$,
\[
e_i = \sum_{j=1}^{i-1} \chi(\abs{\sigma_j} > \abs{\sigma_i}).
\]
Since $\chi(\sigma_i>\sigma_i) = 0$ and $\chi(\sigma_i+\sigma_i<0) = 0$, the result follows from Lemma \ref{eij}.
If $\sigma_i < 0$, then by definition of $\Psi$,
\begin{eqnarray*}
e_i &= & 2i-1 - \sum_{j=1}^{i-1} \chi(\abs{\sigma_j} > \abs{\sigma_i})\\
& = & 2i-1 - \sum_{j=1}^{i-1} \big(2-\chi(\sigma_j > \sigma_i) - \chi(\sigma_j +\sigma_i < 0)\big)\\
& = & 1 + \sum_{j=1}^{i-1} \big(\chi(\sigma_j > \sigma_i) +\chi(\sigma_j +\sigma_i < 0)\big)
\end{eqnarray*}
and since $\chi(\sigma_i>\sigma_i) = 0$ and $\chi(\sigma_i+\sigma_i<0) = 1$, the result follows from Lemma \ref{eij}.
\end{proof}
\begin{lemma} \lambdabdabel{lem:invsigma}
$\Psi: C_n \rightarrow \mathbf{I}_n^{(2,4, \ldots, 2n)}$ satisfies, for $\sigma \in C_n$,
\begin{eqnarray*}
\mathsf{inv}_C(\sigma) & = & \abs{\Psi(\sigma)}
\end{eqnarray*}
\end{lemma}
\begin{proof}
For $\sigma \in C_n$ with $\Psi(\sigma)=e=(e_1, \ldots, e_n)$, applying Corollary \ref{eidef} gives:
\begin{eqnarray*}
\abs{\Psi(\sigma)}=\abs{e}& =& \sum_{i=1}^n e_i\\
& = & \sum_{i=1}^n \sum_{j=1}^{i} \big(\chi(\sigma_j > \sigma_i) + \chi(\sigma_j +\sigma_i < 0)\big)\\
& = & \sum_{i=1}^n \big(\chi(\sigma_i> \sigma_i) + \chi(\sigma_i+\sigma_i < 0)\big) + \sum_{1 \leq j < i \leq n}\chi(\sigma_j> \sigma_i) +
\sum_{1 \leq j < i \leq n}\chi(\sigma_j+\sigma_i <0) \\
& = & \mathsf{neg}(\sigma) + \mathsf{inv}(\sigma) + \mathsf{nsp}(\sigma) = \mathsf{inv}_C(\sigma).
\end{eqnarray*}
\end{proof}
We can now write a new expression to count class inversions of elements in the parabolic quotient ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ along with a new expression for entries of the class inversion vector.
\begin{lemma}
For $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ with window
$[w_1, \ldots, w_n] = [c_1 N + \sigma_1, c_2N + \sigma_2, \ldots, c_nN+\sigma_n]$
where $\sigma=(\sigma_1, \ldots, \sigma_n) \in C_n$ and for $1 \leq j \leq i$,
\[
I_{i,j}(w) = 2c_i - \chi(\sigma_j > \sigma_i) - \chi(\sigma_j + \sigma_i < 0).
\]
\end{lemma}
\begin{proof}
From \eqref{eq:Iij},
\begin{eqnarray*}
I_{i,j}(w) & = & \floor{\frac{{w}_i - {w}_j}{N}} + \floor{\frac{{w}_i + {w}_j}{N}}\\
& = & \floor{\frac{c_iN+\sigma_i - c_jN-\sigma_j}{N}} + \floor{\frac{c_iN+\sigma_i +c_jN+\sigma_j}{N}}\\
& = & 2c_i + \floor{\frac{\sigma_i - \sigma_j}{N}} + \floor{\frac{\sigma_i +\sigma_j}{N}}\\
& = & 2c_i - \chi(\sigma_j > \sigma_i) - \chi(\sigma_i + \sigma_j<0),
\end{eqnarray*}
since $\abs{\sigma_i}+\abs{\sigma_j} < N$.
\end{proof}
\begin{corollary}
For $w= [c_1 N + \sigma_1, c_2N + \sigma_2, \ldots, c_nN+\sigma_n] \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$,
\[
\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w) = \sum_{i=1}^n 2ic_i - \mathsf{inv}_C(\sigma).
\]
\end{corollary}
\subsection{Proof of Theorem \ref{CtoL}}
\noindent
Collecting the observations from the previous three subsections we have the following.
\begin{theorem}
\lambdabdabel{newCtoL}
The mapping
\[
w=[c_1N+\sigma_1, c_2N+\sigma_2, \ldots, c_nN+\sigma_n] \mapsto (2c_1-e_1, 4c_2-e_2, \ldots, 2nc_n-e_n) = \lambdabda,
\]
where $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n $, $(\sigma_1, \ldots, \sigma_n)=\sigma \in C_n$, and $(e_1, \ldots, e_n) = \Psi(\sigma)$, is a bijection
$$ {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n \rightarrow L_n$$
satisfying $\abs{\lambdabda} = \mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w)$ with $\lambdabda_i = I_i(w)$ for $1 \leq i \leq n$.
\end{theorem}
From this characterization, we know this is exactly the bijection of Bousquet-M\'elou and Eriksson.
\section{A thesaurus of statistics}\lambdabdabel{sec:stats}
In this section, we present the thesaurus between combinatorial statistics on an element
\[w=[w_1,w_2,\hdots,w_n]=[c_1N+\sigma_1, c_2N+\sigma_2, \ldots, c_nN+\sigma_n]\in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n\]
and the corresponding lecture hall partition
\[\lambdabda=(\lambdabda_1,\lambdabda_2,\hdots,\lambdabda_n)=(2c_1-e_1, 4c_2-e_2, \ldots, 2nc_n-e_n)\in L_n\]
given by Theorem~\ref{newCtoL}, in which $\sigma=(\sigma_1, \ldots, \sigma_n) \in C_n$, and $e=(e_1, \ldots, e_n) = \Psi(\sigma)$.
\subsection{The $\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}$ statistic}\
\noindent
The Coxeter length of an element $w\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ is its number of class inversions. This corresponds to the sum of the parts of $\lambdabda$, as shown in Theorem~\ref{newCtoL}.
\[\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w) = \abs{\lambdabda}.\]
\subsection{The $\mathsf{neg}$ statistic}\
\noindent
We define $\mathsf{neg}(w)$ for $w\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ to be $\mathsf{neg}(\sigma)$, the number of negative signs in its corresponding signed permutation $\sigma$. Equivalently, $\mathsf{neg}(w)$ is the number of values in its window whose value modulo $N$ is greater than $n+1$. This can be calculated directly from ceiling functions applied to~$\lambdabdam$.
\begin{lemma}
\lambdabdabel{neg}
$$\mathsf{neg}(w) = 2 \abs{\ceil{\lambdabda}^{(2,4,\ldots,2n)}} - \abs{\ceil{\lambdabda}^{(1,2,\ldots,n)}}.$$
\end{lemma}
\begin{proof}
From the definition of $\Psi$ we know that
$ \mathsf{neg}(w) = \mathsf{neg}(\sigma) = \sum_{i=1}^n \floor{\frac{e_i}{i}}$. So,
\begin{eqnarray*}
\mathsf{neg}(w) &=& \sum_{i=1}^n \floor{\frac{e_i}{i}}\\
& = & \sum_{i=1}^n\floor{\frac{2i \ceil{\frac{\lambdabda_i}{2i}}-\lambdabda_i}{i}}\\
& = & \sum_{i=1}^n 2\ceil{\frac{\lambdabda_i}{2i}} - \sum_{i=1}^n \ceil{\frac{\lambdabda_i}{i}}\\
& = & 2 \abs{\ceil{\lambdabda}^{(2,4,\ldots,2n)}} - \abs{\ceil{\lambdabda}^{(1,2,\ldots,n)}}.
\end{eqnarray*}
\end{proof}
In Lemma~\ref{lem:odd}, we show that this $\mathsf{neg}$ statistic equals the unexpected $o(\ceil{\lambdabda})$ statistic that occurs in the work of Bousquet-M\'elou and Eriksson in \cite{BME3}.
\subsection{The $\alpha$ and $\beta$ statistics}\
\noindent
The $\alpha$ and $\beta$ statistics on elements $w\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ originally appeared in a refinement of Bott's formula. The statistics $\alpha(w)$ and $\beta(w)$ are the number of occurrences of the generators $s_0$ and $s_n$ in any reduced word for $w$. Bousquet-M\'elou and Eriksson translated them into the language of lecture hall partitions in their combinatorial proof of a refinement of Bott's formula \cite{BME3}. We present a natural way in which the $\beta$ statistic appears.
\begin{lemma}
\lambdabdabel{beta}
$$\beta(w) = \sum_{i=1}^n c_i = \abs{\ceil{\lambdabda}^{(2,4,\ldots,2n)}}.$$
\end{lemma}
\begin{proof}
The application of an $s_n$ generator as an ascent to a window takes the entry $w_j=c_jN+n$ that is equal to $n$ modulo $N$ and replaces it by $w_j+2=(c_j+1)N-n$, equal to $-n$ modulo $N$, in effect increasing $c_j$ by $1$. From this we conclude that the number of $s_n$ generators is $\sum_{i=1}^n c_i$.
\end{proof}
The $\alpha$ statistic satisfies
\begin{lemma}
\lambdabdabel{alpha}
$$ \alpha(w)= \sum_{i=1}^n c_i \ - \mathsf{neg}(w)= \abs{\ceil{\lambdabda}^{(1,2,\ldots,n)}} - \abs{\ceil{\lambdabda}^{(2,4,\ldots,2n)}}.$$
\end{lemma}
\begin{proof}
The application of an $s_0$ generator as an ascent to a window takes the entry $w_j=c_jN-1$ which is equal to $-1$ modulo $N$ and replaces it by $w_j+2=c_jN+1$, equal to $+1$ modulo $N$. In particular, the number of $s_0$ generators is the number of times a negative residue modulo $N$ has become a positive residue modulo $N$. Furthermore, recognize that every application of an $s_n$ generator replaces one positive residue modulo $N$ by a negative residue modulo $N$. Since the identity permutation has all residues positive, and the number of $s_0$ applications would need to equal the number of $s_n$ applications in order to have all residues positive, then
$\alpha(w)+\mathsf{neg}(w)=\beta(w)$. Applying Lemmas~\ref{neg} and \ref{beta} completes the proof.
\end{proof}
Putting together Lemmas~\ref{beta} and \ref{alpha} gives the following.
\begin{observation} The statistic $\abs{\ceil{\lambdabda}^{(1,2,\ldots,n)}}$ on a lecture hall partition equals $\alpha(w) + \beta(w)$ for the corresponding element of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$.
\end{observation}
Other statistics have appeared in other related works. In his thesis, Mongelli \cite{Mongelli} defines $\mathsf{amaj}(w)$ and $\mathsf{ades}(w)$ statistics on elements $w\in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$. (Mongelli's affine major statistic $\mathsf{amaj}$ is not the same as the Savage-Schuster ascent major statistic $\mathsf{amaj}$ presented later.) Mongelli's statistics are equal to the statistics $\alpha(w)$ and $\beta(w)$, respectively. Bousquet-M\'elou and Eriksson in \cite{BME3} define a statistic for the number of odd parts in $\ceil{\lambdabda}^{(1,2,\ldots,n)}$, which has a simple interpretation in terms of $\alpha(\lambdabda)$ and $\beta(\lambdabda)$.
\begin{lemma}
\lambdabdabel{lem:odd}
$$ o(\lceileil\lambdabda\rceileil) = \mathsf{neg}(\sigma)= \beta(w)-\alpha(w).$$
\end{lemma}
\subsection{The $\max$ and $\lambdabdast$ statistics}\
\noindent
Two statistics pertaining to the largest element of $w\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ (and therefore $\lambdabdam\in L_n$) are $\lambdabdast$ and $\max$. Define $\lambdabdast(w)=c_n$, which corresponds to the statistic
\(
\ceil{\frac{\lambdabda_n}{2n}}
\) under the bijection of Theorem~\ref{newCtoL}. We also wish to define a statistic $\max$ on $w\in{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ that corresponds to the statistic $\lambdabda_n$ on lecture hall partitions.
\begin{lemma}
\lambdabdabel{lem:max}
Under the definition
\[
\max(w)= w_n-\floor{\frac{w_n}{n+1}}-n
\]
we have $\max(w)=\lambdabda_n$ when $w$ is the element of the parabolic quotient ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ that corresponds to the lecture hall partition $\lambdabda$.
\end{lemma}
\begin{proof}
Bradford et al \cite{Brant} approach lecture hall partitions through the use of the abacus model developed by Hanusa and Jones \cite{HJ12}, which in turn is a visualization of the work of \cite{EE} and \cite{BB} on representations of affine Coxeter groups permutations of $\mathbb{Z}$. Under the conventions of \cite{Brant}, they prove that the value of the highest-numbered bead is exactly the largest part of the corresponding lecture hall partition.
In order to convert our window notation to their set of lowest beads, we must take into account two differences in convention. First, they choose $N=2n$ while we choose $N=2n+2$ by omitting runners $0$ and $n+1$. Because of this, we must rescale every entry in the window by the ratio $\frac{n}{n+1}$ by subtracting $\big\lfloor{\frac{w_n}{n+1}}\big\rfloor$. Second, our identity element is $[1,2,\hdots n]$ while their identity element chooses lowest beads $\{ -n+1,-n+2,\hdots 0\}$, which requires an additional adjustment of $-n$.
\end{proof}
A related observation will be needed in Section~\ref{sec:smaller}.
\begin{observation}
\lambdabdabel{ob:max}
A lecture hall partition with largest part $\lambdabda_n=2tn$ corresponds to an element of the parabolic quotient ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ with largest entry $w_n=(n+1)(2t+1)$.
\end{observation}
\subsection{The $\mathsf{amaj}$ and $\mathsf{lhp}$ statistics}\
\noindent
The following statistics on inversion sequences allow us to relate them to lecture hall partitions via Ehrhart theory.
We have already defined $\mathsf{Asc}^{(\s)}(e)$ and $\mathsf{asc}^{(\s)}(e)$ for $e \in \mathbf{I}_n^{(\s)}$. We use two other statistics from \cite{SS} defined for a sequence $\s$: the ascent major index
$\mathsf{amaj}$, which is like a comajor index for the $\mathsf{asc}$ statistic, and the $\mathsf{lhp}$ statistic, which is inherited from the weight of a lecture hall partition.
They are defined as:
\[
\mathsf{amaj}^{(\s)}(e) = \sum_{i \in \mathsf{Asc}^{(\s)}(e)} (n-i);
\]
\[
\mathsf{lhp}^{(\s)}(e) = - \abs{e} + \sum_{i \in \mathsf{Asc}^{(\s)}(e)} (s_{i+1} + \ldots + s_n).
\]
Let $\mathsf{comaj}$ and $\mathsf{lhp}_C$ denote the type C version of the statistics, i.e., $\mathsf{amaj}^{(2,4, \ldots, 2n)}$ and
$\mathsf{lhp}^{(2,4, \ldots, 2n)}$, respectively. Then for $\sigma \in C_n$,
\[
\mathsf{comaj}(\sigma) = \sum_{i \in \mathsf{Des}(\sigma)} (n-i);
\]
and
\[
\mathsf{lhp}_C(\sigma) = -\mathsf{inv}_C(\sigma) + \sum_{i \in \mathsf{Des}(\sigma)} (2(i+1) + \ldots + 2n).
\]
The statistic $\mathsf{lhp}_C$ is {\em quadratic} in the sense that its summands are quadratic functions of descent positions $i$. In contrast, the summands of $\mathsf{comaj}$ are linear functions of descent positions $i$; for the statistic $\mathsf{des}$, the summands are constant functions of $i$.
We will use these corresponding statistics on signed permutations to translate results from lecture hall partitions and inversion sequences to $C_n$ and ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$. We show in Section 4 that $\mathsf{lhp}_C$ is the natural statistic to define on $C_n$ so that the distribution of $\mathsf{lhp}_C$ over $C_n$ inherits a nice product-form generating function from Bott's formula for ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$. The proof relies on an Ehrhart theory result for lecture hall partitions together with the bijection ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n \rightarrow L_n$.
\section{Translating results about lecture hall partitions into results about ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$}
\lambdabdabel{sec:results}
We can use this new insight on the bijection to translate results between lecture hall partitions and elements of the parabolic quotient ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$.
We use the standard $q$-notation, defining the $q$-bracket
\[[k]_q=(1+q+\cdots+q^{k-1}),\]
the $q$-factorial
\[[k]_q!=\prod_{i=1}^k[i]_q,\]
the $q$-binomial coefficients
\[\qbinom{n}{k}_q=
\frac{[n]_q!}{[k]_q![n-k]_q!},\] and the $q$-Pochhammer symbols \[(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k).\]
\subsection{Truncated coset representatives and Bott's Formula}\
Bousquet-M\'elou and Eriksson observed in \cite{BME} that Bott's formula \cite{Bott} for the Poincare series of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$ is equivalent to:
\[
\sum_{w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n } q^{\ell(w)} = \prod_{i=1}^n \frac{1}{1-q^{2i-1}}.
\]
As noted earlier, $\ell(w) = \mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w).$ They used Bott's formula and their correspondence between lecture hall partitions and class inversion vectors for elements of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ to give the first proof of the
{\em lecture hall theorem}:
\begin{theorem}[Lecture Hall Theorem, \cite{BME}]
\begin{equation*}\lambdabdabel{eq:lht}
\sum_{\lambdabda \in L_n} q^{|\lambdabda|} = \prod_{i=1}^n \frac{1}{1-q^{2i-1}}.
\end{equation*}
\end{theorem}
In a subsequent paper \cite{BME3}, Bousquet-M\'elou and Eriksson proved a refinement of the Lecture Hall Theorem corresponding to a refinement of Bott's formula. Following their notation, let
\[
{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n (q,a,b) = \sum_{w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n } q^{\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w)}a^{\alpha(w)}b^{\beta(w)}.
\]
They noted that
\begin{equation}
\lambdabdabel{eq:Bott}
{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n (q,a,b) = \prod_{i=1}^n \frac{
1+bq^i}
{1-abq^{n+i}}.
\end{equation}
follows from a refinement of Bott's formula for the Poincar{\'e} series of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n$, due to Macdonald, and is proved combinatorially by Eriksson and Eriksson in \cite{EE}. Bousquet-M\'elou and Eriksson proved the following as a consequence of Equation~\eqref{eq:Bott} and also gave an independent combinatorial proof.
\begin{theorem}[\cite{BME3}]
\lambdabdabel{refined}
\[
\sum_{\lambdabda \in L_n} q^{\abs{\lambdabda}}u^{\abs{\ceil{\lambdabda}}}v^{o(\ceil{\lambdabda})} = \prod_{i=1}^n \frac{1+uvq^i}{1-u^2q^{n+i}}.
\]
\end{theorem}
We can use our correspondence from Section~\ref{sec:proof} to give a formula for a refinement of Bott's formula. The so-called truncated lecture hall partitions $L_{n,k}$ are those in which a specified number of parts are required to be zero:
\[
L_{n,k} = \left\{(\lambdabdam_1,\hdots,\lambdabdam_n) \in \mathbb{Z}^n: 0 < \frac{\lambdabdam_{n-k+1}}{n-k+1} \leq \frac{\lambdabdam_{n-k+2}}{n-k+2}\leq \cdots \leq \frac{\lambdabdam_n}{n}\right\}.
\]
\begin{lemma}
Under the bijection of Section~\ref{sec:proof}, the truncated lecture hall partitions $L_{n,k}$ correspond to the set $T_{n,k}$ of truncated minimal length coset representatives of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ defined by
\[
T_{n,k} = \left \{w=[w_1,\hdots,w_n] \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n \ : \ w_{n-k} \leq n \textup{ and } w_{n-k+1}>n \right \}.
\]
\end{lemma}
\begin{proof}
Under the bijection of Section~\ref{sec:proof}, the elements of $T_{n,k}$ are of the form
\([c_1N+\sigma_1, \ldots, c_nN+\sigma_n]\) where \(c_1 = c_2 = \cdots = c_{n-k} = 0\) and $c_{n-k+1}>0$. Since the elements of the window are in increasing order, the lemma follows.
\end{proof}
The truncated lecture hall partitions were enumerated in \cite{CS} by the statistics $|\lambdabda|$, $\big\lvert\lceileil\lambdabda\rceileil\big\rvert$, and $o(\lceileil\lambdabda\rceileil)$. Using our thesaurus from Section~\ref{sec:stats}, we have the following refinement of Bott's formula to truncated minimal length coset representatives of ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$.
\begin{theorem}
\[
\sum_{w \in T_{n,k}} q^{\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w)} a^{\alpha(w)}b^{\beta(w)}= b^k q^{k+1 \choose 2} \qbinom{n}{k}_q
\frac{(-aq^{n-k+1};q)_k}{(abq^{2n-k+1};q)_k}
\]
\end{theorem}
\subsection{Odd and even window entries}\
In \cite{BME}, Bousquet-M\'elou and Eriksson gave a proof of the Lecture Hall Theorem independent of Bott's formula.
It involved counting separately the weights
\[
|\lambdabda|_o= \lambdabda_n + \lambdabda_{n-2} + \ldots
\]
and
\[
|\lambdabda|_e= \lambdabda_{n-1} + \lambdabda_{n-3} + \ldots
\]
of a lecture hall partition $\lambdabda \in L_n$. Indeed, they proved the following refinement of the lecture hall theorem:
\[
\sum_{\lambdabda \in L_n} x^{|\lambdabda|_o}y^{|\lambdabda|_e} = \prod_{i=1}^n \frac{1}{1-x^iy^{i-1}}.
\]
Using the correspondence between $L_n$ and ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ we can use this to state a different refinenent of Bott's formula in which, for $1 \leq i \leq n$, the class inversions with $w_i$ for $w=[w_1, \ldots, w_n] \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ are counted separately, depending on whether $i \in \{n, n-2, \ldots\}$ or $i \in \{n-1, n-3, \ldots \}$.
For $w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$, let $$|w|_o= I_n(w) + I_{n-2}(w) + \ldots$$ and let $$|w|_e= I_{n-1}(w) + I_{n-3}(w) + \ldots.$$
The correspondence gives us the following.
\begin{theorem}
\[
\sum_{w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n} x^{|w|_o}y^{|w|_e} = \prod_{i=1}^n \frac{1}{1-x^iy^{i-1}}.
\]
\end{theorem}
As in the preceding subsection, our tools allow us to exhibit a new further refinement of Bott's formula to truncated lecture hall partitions by reinterpreting Theorem~3 from \cite{CS}. The corresponding result for ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ becomes:
\begin{theorem} For $n\geq k\geq 0$,
\[\sum_{w \in T_{n,k}} x^{|w|_o}y^{|w|_e}=\frac{\big(x^{\lfloor k/2\rfloor+1}y^{\lceileil k/2\rceileil}\big)\displaystyle\qbinom{n-\lceileil k/2\rceileil}{\lfloor k/2\rfloor}_{xy}}{(x;xy)_{\lceileil k/2 \rceileil}\big(x^ny^{n-1};(xy)^{-1}\big)_{\lfloor k/2 \rfloor}}.\]
\end{theorem}
\subsection{Windows with smaller entries}\lambdabdabel{sec:smaller}\
\noindent
How many $\lambdabda \in L_n$ have largest part $\leq t$? This is the lecture hall analog of counting partitions in an $n$ by $t$ box. It has a nice answer if $t$ is expressed in the form $jn + k$.
\begin{theorem}[\cite{CLS}]
For $n \geq 0$, $j \geq 0$, and $0 \leq k \leq n$, the number of lecture hall partitions in $L_n$ satisfying $\lambdabda_n\leq jn + k$ is
$(j+1)^{n-k}(j+2)^k$.
\end{theorem}
As a consequence we have the following.
\begin{corollary}
The number of lecture hall partitions $\lambdabda$ with $\lambdabda_n \leq 2tn$ is $(2t+1)^n$.
\end{corollary}
By Observation~\ref{ob:max}, we define a corresponding set $S_{n,t}$ of these smaller windows by
\[
S_{n,t} = \big\{w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n \ : \ w_n \leq (2t+1)(n+1) \big\},
\]
and have the following corollary.
\begin{corollary}\lambdabdabel{cor:snt}
\[
|S_{n,t}|=(2t+1)^n.
\]
\end{corollary}
We can refine Corollary~\ref{cor:snt} to count via the statistics $\alpha$ and $\beta$. To do this, we will apply Equation~3.6 from \cite{CLS15} which says
\[
\sum_{\lambdabda \in L_n, \lambdabda_n \leq 2t}u^{|\ceil{\lambdabda}|}v^{o(\lambdabda)} =
\left (
\frac{(1+uv-u^{2t+1}-u^{2t+2}}{1-u^2}
\right )^n.
\]
Under our correspondence, it follows that
\begin{theorem}\lambdabdabel{thm:smallwindows}
\[
\sum_{w \in S_{n,t}} a^{\alpha(w)}b^{\beta(w) }=
\left (
\frac{(1+b-a^tb^{t+1}(1+a)}{1-ab}
\right )^n.
\]
\end{theorem}
Note that setting $a=b$ gives
\begin{eqnarray*}
\sum_{w \in S_{n,t}} a^{\alpha(w)+\beta(w) } & = &
\left (
\frac{(1+a)(1-a^{2t+1})}{1-a^2}
\right )^n\\
& = & [2t+1]_a^n.
\end{eqnarray*}
This proof of Theorem~\ref{thm:smallwindows} makes use of our correspondence. An alternate direct proof chooses independently for all $i$ from $1$ to $n$ which of the $2t+1$ values of $i$ or $-i$ modulo $2n+2$ appears in the window.
\subsection{A quadratic statistic for $C_n$}\
\noindent
Ehrhart theory was applied in \cite{SS} to give a further connection between lecture hall partitions and signed permutations.
The following is a specialization of the main result of that paper (Theorem~6).
\begin{theorem}[\cite{SS}]
\[
\sum_{\lambdabda \in L_n} u^{\abs{\ceil{\lambdabda}_{2n}}} q^{\abs{\lambdabda}}= \frac{
\sum_{\sigma \in C_n} u^{\mathsf{comaj}(\sigma)} q^{\mathsf{lhp}_C(\sigma)} }
{\prod_{i=0}^{n-1}(1-u^{n-i}q^{2(i+1)+ \cdots + 2n})}.
\]
\end{theorem}
Our correspondence between $L_n$ and and ${\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n$ gives
\begin{equation}
\lambdabdabel{eq:parabolicEhrhart}
\sum_{w \in {\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}_n/C_n} u^{\beta(w)} q^{\mathsf{inv}_{{\widetilde} \def\bar{\overline} \def\hat{\widehat{C}}}(w)}= \frac{
\sum_{\sigma \in C_n} u^{\mathsf{comaj}(\sigma)} q^{\mathsf{lhp}_C(\sigma)} }
{\prod_{i=0}^{n-1}(1-u^{n-i}q^{2(i+1)+ \cdots + 2n})}.
\end{equation}
Applying the refinement of Bott's formula from Equation~\eqref{eq:Bott} when $a=1$ to Equation~\eqref{eq:parabolicEhrhart} gives the joint distribution of $\mathsf{comaj}$ and $\mathsf{lhp}$ on $C_n$.
\begin{corollary}
\begin{equation}
\lambdabdabel{eq:comajlhp}
\sum_{\sigma \in C_n} u^{\mathsf{comaj}(\sigma)} q^{\mathsf{lhp}_C(\sigma)} = \prod_{i=1}^n \frac{1+uq^i}{1-uq^{n+i}}
\prod_{i=0}^{n-1}(1-u^{n-i}q^{2(i+1)+ \cdots + 2n}).
\end{equation}
\end{corollary}
Equation~\eqref{eq:comajlhp} is a refinement of the well-known distribution
\[
\sum_{\sigma \in C_n} u^{\mathsf{comaj}(\sigma)} = (1+u)^n\prod_{i=1}^n\, [i]_u.
\]
If instead we specialize Equation~\eqref{eq:comajlhp} to when $u=1$, we arrive at an interesting distribution for the quadratic statistic $\mathsf{lhp}_C$ on $C_n$, which appears to be new.
\begin{corollary}\lambdabdabel{cor:quadlhp}
\[
\sum_{\sigma \in C_n} q^{\mathsf{lhp}_C(\sigma)} = \prod_{k=1}^{n} [2k]_{q^{2(n-k)+1}}
.\]
\end{corollary}
\begin{proof}
Specializing Equation~\eqref{eq:comajlhp} to when $u=1$ and simplifying gives
\[\begin{aligned}
\sum_{\sigma \in C_n} q^{\mathsf{lhp}_C(\sigma)} &= \prod_{i=1}^n \frac{(1+q^i)(1-q^{2i+ \cdots + 2n})}{1-q^{n+i}} \\
&= \prod_{i=1}^n \frac{(1-q^{2i})(1-q^{2i+ \cdots + 2n})}{(1-q^i)(1-q^{n+i})} \\
&= \prod_{i=1}^n \frac{(1-q^{2i+ \cdots + 2n})}{(1-q^{2i-1})} \\
&= \prod_{k=1}^{\lceileil n/2 \rceileil} \frac{(1-q^{2(2k-1)(n-k+1)})}{(1-q^{2k-1})} \prod_{k=1}^{\lfloor n/2 \rfloor} \frac{(1-q^{2k(2(n-k)+1)})}{(1-q^{2(n-k)+1})} \\
&= \prod_{k=1}^{\lceileil n/2 \rceileil} [2(n-k+1)]_{q^{2k-1}} \prod_{k=1}^{\lfloor n/2 \rfloor} [2k]_{q^{2(n-k)+1}} \\
&= \prod_{k=n-\lfloor n/2 \rfloor}^{n} [2k]_{q^{2(n-k+1)-1}} \prod_{k=1}^{\lfloor n/2 \rfloor} [2k]_{q^{2(n-k)+1}} \\
&= \prod_{k=1}^{n} [2k]_{q^{2(n-k)+1}} \\
\end{aligned}
\]
The fourth equality follows because the sum $i+\cdots+n$ has two nice formulas depending on whether there are an even or an odd number of terms. If there are $2k-1$ terms, then the sum is $(2k-1)$ times the mean of the terms $(n-k+1)$. If there are $2k$ terms, then the sum is $k$ times the sum of the first and the last term, $(n+n-2k+1)$.
\end{proof}
Corollary~\ref{cor:quadlhp} can be viewed as a type~C analog of the following result from \cite{SS}.
\begin{theorem}[Corollary~5 of \cite{SS}]
\[
\sum_{\pi \in S_n} q^{\mathsf{lhp}(\pi)} = \prod_{k=1}^{n} [k]_{q^{2(n-k)+1}}
.\]
\end{theorem}
\section*{Acknowlegements}
Thanks to the Simons Foundation for a grant to the second author which supported the travel of the first author for collaboration. The first author also gratefully acknowledges support from PSC-CUNY Research Award TRADA-47-191.
\end{document} |
\begin{document}
\title{Simultaneous amplitude and phase damping of a kind of Gaussian states and
their separability}
\author{Xiao-yu Chen \\
{\small {Lab. of Quantum Information, China Institute of Metrology,
Hangzhou, 310018, China}}}
\date{}
\maketitle
\begin{abstract}
We give out the time evolution solution of simultaneous amplitude and phase
damping for any continuous variable state. For the simultaneous amplitude
and phase damping of a wide class of two- mode entangled Gaussian states,
two analytical conditions of the separability are given. One is the
sufficient condition of separability. The other is the condition of PPT
separability where the Peres-Horodecki criterion is applied. Between the two
conditions there may exist bound entanglement. The simplest example is the
simultaneous amplitude and phase damping of a two-mode squeezed vacuum
state. The damped state is non-Gaussian.
PACS 03.65.Ud 03.65.Yz 03.67.Mn 42.50.Dv
\end{abstract}
\section{Introduction}
Quantum entanglement or inseparability plays a major role in all branches of
quantum information and quantum computation. Peres\cite{Peres} proposed a
criterion for checking the inseparability of a state by introducing the
partial transpose operation. This condition is necessary and sufficient for
some lower dimensional discrete bipartite systems but is no longer
sufficient for higher dimensions\cite{Horodecki}. Despite many studies on
the discrete states, much attentions have been paid to the continuous
variable states \cite{Braunstein}. Recently, quantum teleportation of
coherent states has been experimentally realized by exploiting a two-mode
squeezed vacuum state as an entanglement resource\cite{Furusawa}. Due to the
decohence of the environment, a pure entanglement state will become mixed.
Thus it is important to know if a given bipartite continuous variable state
is entangled or not. The decoherence may be caused by coupling to the
thermal noise of the environment, amplitude damping, quantum dissipation and
phase damping. Besides the phase damping, the other three types of
decoherence preserve gaussian property of the state, and a two-mode squeezed
vacuum state will evolve to a two mode gaussian mixed state. For the
separability of two mode gaussian state, the positivity of the partially
transposed state is necessary and sufficient \cite{Duan}\cite{Simon}\cite
{Wang}. However, a gaussian state will evolve to a non-gaussian one by phase
damping, and the case of a two mode squeezed vacuum state under the only
decoherence of phase damping was perfectly solved\cite{Hiroshima}. In real
experiments, the general situation which should be taken into account is the
coexistence of noise, amplitude and phase damping. Theoretically, the
separability and entanglement of non-gaussian state are seldom investigated,
here we provide an example.
\section{Time Evolution of Characteristic Function}
Considering the simultaneous damping, the density matrix obeys the following
master equation in the interaction picture $\frac{d\rho }{dt}=(\mathcal{L}_1
\mathcal{+L}_2)\rho .$ Where $\mathcal{L}_1$ is the amplitude damping part
\begin{equation}
\mathcal{L}_1\rho =\sum_i\frac{\Gamma _i}2[(\overline{n}_i+1)(2a_i\rho
a_i^{+}-a_i^{+}a_i\rho -\rho a_i^{+}a_i)+\overline{n}_i(2a_i^{+}\rho
a_i-a_ia_i^{+}\rho -\rho a_ia_i^{+})],
\end{equation}
with $\overline{n}$ the average photon number of the thermal environment.
And $\mathcal{L}_2$ is the phase damping part (e.g. \cite{Hiroshima}),
\begin{equation}
\mathcal{L}_2=\sum_i\frac{\gamma _i}2[2a_i^{+}a_i\rho
a_i^{+}a_i-(a_i^{+}a_i)^2\rho -\rho (a_i^{+}a_i)^2].
\end{equation}
The state can be equivalently specified by its characteristic function.
Every operator $\mathcal{A}\in \mathcal{B(H)}$ is completely determined by
its characteristic function $\chi _{\mathcal{A}}:=tr[\mathcal{AD}(\mu )]$
\cite{Petz}, where $\mathcal{D}(\mu )=\exp (\mu a^{+}-\mu ^{*}a)$ is the
displacement operator, with $\mu =[\mu _1,\mu _2,\cdots ,\mu _s]^T$ $
,a=[a_1,a_2,\cdots ,a_s]$ and the total number of modes is $s.$ It follows
that $\mathcal{A}$ may be written in terms of $\chi _{\mathcal{A}}$ as \cite
{Perelomov}: $\mathcal{A}=\int [\prod_i\frac{d^2\mu _i}\pi ]\chi _{\mathcal{A
}}(\mu )\mathcal{D}(-\mu ).$ The density matrix $\rho $ can be expressed
with its characteristic function $\chi $. The amplitude damping equation of $
\chi $ and its solution are well known, they are $\frac{\partial \chi }{
\partial t}=-\frac 12\sum_i\Gamma _i(\left| \mu _i\right| \frac{\partial
\chi }{\partial \left| \mu _i\right| }+\left| \mu _i\right| ^2\chi ),$ $\chi
(\mu ,t)=\chi (\mu _ie^{-\frac{\Gamma _it}2},0)\exp [-\sum_i(\overline{n}
_i+\frac 12)(1-e^{-\Gamma _it})\left| \mu _i\right| ^2]$. We now give out
the phase damping equation of $\chi $, it will be
\begin{equation}
\frac{\partial \chi }{\partial t}=\frac 12\sum_i\gamma _i\frac{\partial
^2\chi }{\partial \theta _i^2}
\end{equation}
if we denote $\mu _i$ as $\left| \mu _i\right| e^{i\theta _i}$. We can see
that with the characteristic function the amplitude damping equation is
described by the amplitude of the parameter $\mu _i,$ the phase damping
equation is described by the phase of the parameter $\mu _i$. The solution
to the phase equation of $\chi $ then will be $\chi \left( \mu ,\mu
^{*},t\right) =\prod_i\left( 2\pi \gamma _it\right) ^{-1/2}\int \exp (-\sum_i
\frac{x_i^2}{2\gamma _it})$ $\chi \left( \mu e^{ix},\mu ^{*}e^{-ix},0\right)
dx,$ where $\mu e^{ix}$ is the abbreviation of $[\mu _1e^{ix_1},\mu
_2e^{ix_2},\cdots ,\mu _se^{ix_s}]$.The simultaneous amplitude and phase
damping to any initial characteristic function then will be
\begin{equation}
\chi \left( \mu ,\mu ^{*},t\right) =\prod_i\left( 2\pi \gamma _it\right)
^{-1/2}\int \exp \sum_i[-\frac{x_i^2}{2\gamma _it}+(\overline{n}_i+\frac
12)(1-e^{-\Gamma _it})\left| \mu _i\right| ^2]\chi \left( \mu _ie^{-\frac{
\Gamma _it}2+ix_i},\mu _i^{*}e^{-\frac{\Gamma _it}2-ix_i},0\right) dx.
\end{equation}
The density matrix then can be obtained as well by making use of operator
integral.
We will concentrate on the simultaneous amplitude and phase damping of
two-mode $x-p$ symmetric Gaussain state\cite{Jiang} whose characteristic
function is $\chi (\mu ,0)=\exp [-(A_{10}\left| \mu _1\right|
^2+A_{20}\left| \mu _2\right| ^2)+B_0\mu _1\mu _2+B_0^{*}\mu _1^{*}\mu
_2^{*}].$ . The time evolution of $\chi $ is
\begin{equation}
\chi \left( \mu ,\mu ^{*},t\right) =\int_x\exp [-(A_1^{\prime }\left| \mu
_1\right| ^2+A_2^{\prime }\left| \mu _2\right| ^2)+B\mu _1\mu
_2e^{ix}+B^{*}\mu _1^{*}\mu _2^{*}e^{-ix})].
\end{equation}
Where $A_i^{\prime }=e^{-\Gamma _it}A_{i0}+(\overline{n}_i+\frac
12)(1-e^{-\Gamma _it}),$ $B=e^{-\frac 12(\Gamma _1+\Gamma _2)t}B_0,\overline{
\gamma }=\frac 12(\gamma _1+\gamma _2)$. The integral on $x$ is one fold,
for simplicity, we denote $\frac 1{\sqrt{4\pi \overline{\gamma }t}}\int \exp
[-\frac{x^2}{4\overline{\gamma }t}]f(x)dx$ as $\int_xf(x).$ The $x-p$
symmetric Gaussain state set is a quite large set. It contains two-mode
squeezed vacuum state $\left| \Psi \right\rangle =\frac 1{\cosh
r}\sum_n(\tanh r)^n\left| n,n\right\rangle $ ( $r$ is the squeezing
parameter) and two-mode squeezed thermal state \cite{Chen} as its special
cases, with $A_{10}=A_{20}=\frac 12\cosh 2r$, $B_0=\frac 12\sinh 2r$ and $
A_{10}=A_{20}=(n_0+\frac 12)\cosh 2r$, $B_0=(n_0+\frac 12)\sinh 2r$
respectively.
\section{PPT separability}
For the sake of simplicity in description, let us firstly consider the
situation of $A_1^{\prime }=A_2^{\prime }=A^{\prime },B=B^{*}.$ The general
case will be obtained straightforward and be described at the end of this
section. The first question is that if the state after damping is entangled
or not. Then how much is the entanglement left? The necessary condition of a
bipartite state being entangled is that the partial transpose of the density
operator is not positive definite \cite{Peres}. The partial transpose
operation changes the characteristic function in the fashion of : $\chi
\left( \mu _1,\mu _2\right) \Longrightarrow $ $\chi \left( \mu _1,-\mu
_2^{*}\right) =\chi ^{PT}\left( \mu _1,\mu _2\right) .$ For the separability
of a non-gaussian bipartite state, a necessary condition was proposed by
Simon\cite{Simon} in terms of the second moment of the state. In the
original literature canonical operators were used, here we use creation and
annihilation operators instead. The necessary condition comes from the
non-negativity of $\rho ^{PT}$ and the commutation relations. For any $
Q=\eta \eta ^{+}$ with $\eta =c_1a_1+c_2a_2+c_3a_1^{+}+c_4a_2^{+}$ of every
set of complex coefficients $c_i$, one has $\left\langle Q\right\rangle
=tr(Q $ $\rho ^{PT})\geq 0,$ Hence the second moment matrix of $\rho ^{PT}$
should be semi-positive definite. The second moment such as $tr(\rho
^{PT}a_i^{+}a_j)$ can be obtained from the second derivative of $\chi $ with
respect to $\mu _i$ and $\mu _j^{*}$ , we have
\begin{equation}
A-1\geq Be^{-\gamma t} \label{wave1}
\end{equation}
Where $A=A^{\prime }+\frac 12$ . Other necessary conditions may come from
when $\eta $ is the linear combination of higher power of the creation and
annihilation operators, and they may be tighter than Ineq.(\ref{wave1}). And
this is really the case.
We will find a tighter condition by exploring the negative eigenvalues of
the partial transpose of the density operator. The eigenequation of $\rho
^{PT}$ can be simplified as the eigenequations of a serials of matrices (see
Appendix).
\begin{equation}
M_{ln}^{(m)}=(A^2-B^2)^{-1}C^m\sum_k\binom{m-n}{l-k}\binom nk\left( \frac
DC\right) ^{l+n-2k}e^{-\gamma t(n-l)^2},
\end{equation}
where $C=1-\frac A{A^2-B^2},$ $D=\frac B{A^2-B^2}.$ When $m=0$, one has the
first eigenvalue $\lambda ^{(0)}=(A^2-B^2)^{-1}$ which is always positive.
The matrix $M^{(m)}$ possesses the symmetry of $
M_{ln}^{(m)}=M_{m-l,m-n}^{(m)}$ , so that it can be reduced, and we get more
analytical solutions. The negative eigenvalues may appear at $\lambda
_0^{(1)}=(A^2-B^2)^{-1}(C-De^{-\gamma t}),$ $\lambda
_0^{(2)}=(A^2-B^2)^{-1}(C^2-D^2e^{-4\gamma t})$ and so on. Hence one of the
necessary conditions of the non-negativity of $\rho ^{PT}(t)$, so that the
necessary condition of a damped state $\rho (t)$ is PPT separable is that
\begin{equation}
C\geq De^{-\gamma t}. \label{wave2}
\end{equation}
We will prove that this condition is also sufficient for PPT separability.We
turn to the detail properties of matrix $M^{(m)}$. The necessary condition
of separability comes from $M^{(2)}$ is $C\geq De^{-2d}$, this is a trivial
result compared with Ineq.(\ref{wave2}) We have checked other solvable
eigenvalues for necessary condition of separability. They are also trivial
compared with Ineq.(\ref{wave2}). It may be anticipated that separable
conditions come from all other $M^{(m)}$ are weaker than Ineq.(\ref{wave2}),
that is Ineq.(\ref{wave2}) is also a sufficient condition of PPT
separability. To prove this, we just need to consider the PPT separability
at the case of $C=De^{-\gamma t}$. Because if a state corresponding to $
C=De^{-\gamma t}$ is PPT separable, then another state with stronger phase
damping (with increasing $\gamma $ while preserving all other parameters
unchanged) is definitely PPT separable, for this stronger phase damping
state we have $C>De^{-\gamma t}$ and it is PPT separable. Our proof of the
PPT separability of the state at $C=De^{-\gamma t}$ is not a most general
proof. We can only prove the non-negativity of $M^{(m)}$ by algebraic
programming up to $m=17$ at the case of $C=De^{-\gamma t}$. Denote $
M_{ln}^{(m)}=(A^2-B^2)^{-1}C^mN_{ln}^{(m)},$ and let $N^{(m,j)}$ (with its
elements $N_{ln}^{(m,j)}$ , $0\leq l,n\leq j$) be the $j$-th main submatrix
of $N^{(m)}$, then to prove the non-negativity of $M^{(m)}$ is to prove that
the determinants of all $N^{(m,j)}$ are not negative. The algebraic
programming gives $\det N^{(m,j)}=d^{-p}(d-1)^{j(j+1)/2}P^{(m,j)}(d)\geq 0,$
where $d=D/C=e^{\gamma t}\geq 1,$ and $P^{(m,j)}(d)$ is a polynomial of $d$
with all its coefficients being positive integer, $p$ is some integer rely
on $j$. The algebraic programming runs for all $m\leq 17$ and proves the
non-negativity of $M^{(m)}$. We suggest that $M^{(m)}$ is also non-negative
for $m>17$, but this is not verified because of the computing time of the
algebraic programming.
Another direct way of proving comes from perturbation theory. Firstly $
M^{(m)}$ can always be symmetrized. The zero order matrix is $
M_{(0)}^{(m)}=M^{(m)}(\gamma t=0)$ which is just the case of gaussian state,
and all its eigenvalues and eigenvectors are well known. So that the first
order and second order perturbation of the eigenvalues of $M^{(m)}$ can be
obtained. A more concise way to obtain the perturbation result is as
follows: In the eigenequation of characteristic function, if we use $
\left\langle \alpha \right| \left. \Phi \right\rangle =\exp (-\frac 12\left|
\alpha \right| ^2)\sum_{n=0}^mc_n^{\prime (m)}(\alpha _1^{*}+\alpha
_2^{*})^{m-n}(\alpha _1^{*}-\alpha _2^{*})^n$ as a test wave function, we
then get a matrix (see Appendix)
\begin{eqnarray}
M_{ln}^{\prime (m)} &=&\frac 1{(A^2-B^2)\sqrt{4\pi \gamma t}}\int dx\exp [-
\frac{x^2}{4\gamma t}]\sum_k\binom{m-n}{l-k}\binom nk \label{wave7} \\
&&(C+D\cos x)^{m-n-l+k}(C-D\cos x)^k(-1)^{l-k}(iD\sin x)^{l+n-2k} \nonumber
\end{eqnarray}
which is a linear transformation of $M^{(m)}$ and has the same eigenvalues.
The zero order of $M^{\prime (m)}$ is the matrix $M_{(0)}^{\prime
(m)}=M^{\prime (m)}(\gamma t=0)$ which is a diagonal matrix. Hence
eigenvalues up to the first order perturbation of $M^{\prime (m)}$ are
simply $M_{nn}^{\prime (m)}$. When $\gamma t$ is quite small, $
M_{nn}^{\prime (m)}$ can be approximated as $M_{nn}^{\prime (m)}\approx
(A^2-B^2)^{-1}(C+D)^{m-n}(C-D)^nf(m,n)$ with
\begin{equation}
f(m,n)=1-\gamma t[\frac D{D+C}(m-n)+\frac D{D-C}n+\frac{2D^2}{(D+C)(D-C)}
(m-n)n].
\end{equation}
For odd $n$, the $n-th$ eigenvalue of the zero order approximation is
negative. If $f(m,n)$ is also negative, then the $n-th$ eigenvalue of the
first order approximation becomes positive. For given $\gamma t$ and $D/C$
we can always find sufficient large $m$ and proper $n$ so that $f(m,n)$ is
negative, hence the eigenvalues with large $m$ become positive faster than
that with small $m$ under phase damping. We need to know at what condition
all original negative eigenvalues become positive or zero. It is easy to
obtained that when $m=n=1$, $f(m,n)$ reaches its maximum. Hence when $
f(1,1)=1-$ $\gamma t\frac D{D-C}\leq 0,$ that is
\begin{equation}
C\geq D(1-\gamma t), \label{wave5}
\end{equation}
all other odd $n$ negative eigenvalue become positive. Ineq.(\ref{wave5}) is
the first order approximation of Ineq.(\ref{wave2}). Hence at the sense of
first order approximation Ineq.(\ref{wave2}) is sufficient for a state to be
PPT\ separable.
As a by product, we can use matrices $M^{(m)}$ to calculate the logarithmic
negativity which is an entanglement measure itself \cite{Vidal} and provides
an upper bound to the distillable entanglement\cite{Audenaert}. It can be
seen in the figure when $t\geq t_2,$the logarithmic negativity is zero,
while it is positive when $t<t_2$. For sufficiently small $\gamma t$, the
negativity of the state can be estimated. It is the absolute of the
summation of all eigenvalues with odd $n$. The result will be $\mathcal{N}
(\rho )\approx \frac{D-C}{1+C-D}-\gamma t\frac{D(1-C+D)}{(1-C-D)(1+C-D)^2}=
\frac{1-A+B}{2(A-B)-1}-\gamma t\frac B{[2(A-B)-1]^2}.$
One of the most important quantities of a state $\rho $ is its entropy $
S(\rho )=-Tr\rho \log \rho $. The entropy of our damped state can be
obtained by solving the characteristic function eigenequation which is $\int
\frac{d^4\mu }{\pi ^2}\frac{d^4\alpha }{\pi ^2}\chi (\mu ,\mu
^{*},t)\left\langle \beta \right| D(-\mu )$ $\left| \alpha \right\rangle
\left\langle \alpha \right| \left. \Phi \right\rangle =\lambda \left\langle
\beta \right| \left. \Phi \right\rangle .$ After integrals on $\mu $, $
\alpha $ and $x$, then compare the coefficients of each $\beta ^{*}$ item of
the two side, one can get a series of matrices $L^{(m)}$ whose eigenvalues
are that of the damped state $\rho $ and
\begin{equation}
L_{ln}^{(m)}=\frac{C^m}{A^2-B^2}\sum_k\binom{m+n}{n-k}\binom
lkC^{2k}(-D)^{l+n-2k}e^{-\gamma t(n-l)^2}.
\end{equation}
The entropy of the state $\rho $ will be $S(\rho )=-TrL^{(0)}\log
(L^{(0)})-2\sum_{m=1}^\infty TrL^{(m)}\log (L^{(m)}).$ The reduced state of $
\rho $ is $\rho _1=Tr_2\rho $ with its characteristic function $\chi _1(\mu
_1,t)=\chi (\mu _1,0,t)=\exp [-A^{\prime }\left| \mu \right| ^2]$, hence its
entropy is $S(\rho _1)=A\log A-(A-1)\log (A-1).$ It has been proven that the
coherent information $I^i=\max (0,S(\rho _i)-S(\rho ))$ provides lower bound
of distillable entanglement of the state\cite{Devetak}. The coherent
information is calculated and plotted in the figure. At time $t_0$ the
coherent information turns to zero. In the figure we have $t_0<t_1$, but for
other parameters we may have $t_0>t_1$.
\begin{figure}
\caption{The upper solid line is for logarithmic negativity (LN), the down
solid line is for coherent information (CI). The parameters are $\Gamma
=\gamma =\overline{n}
\end{figure}
We have investigated the symmetric damping setting of the two mode squeezed
thermal state, that is, the two modes undergo the same damping and noise.
The generalization to asymmetric damping setting and $x-p$ symmetric
Gaussian state is straightforward with the method developed here. Denote $
C_i=1-\frac{A_i}{A_1A_2-\left| B\right| ^2},$ $D=\frac B{A_1A_2-\left|
B\right| ^2},$ the $M^{(m)}$ matrix will be
\begin{equation}
M_{ln}^{(m)}=(A_1A_2-B^2)^{-1}C_2^m\sum_k\binom{m-n}{l-k}\binom nk\left(
\sqrt{\frac{C_1}{C_2}}\right) ^{l+n}\left( \frac{\left| D\right| }{\sqrt{
C_1C_2}}\right) ^{l+n-2k}e^{-\gamma t(n-l)^2}, \label{wave6}
\end{equation}
The whole issue of the positivity of the asymmetric setting is equivalent
that of symmetric setting and omitted here. The necessary and sufficient
criterion of the PPT separability of the state will be
\begin{equation}
\sqrt{C_1C_2}\geq \left| D\right| e^{-\overline{\gamma }t}, \label{wave8}
\end{equation}
and the PPT sufficient criterion is again obtained at the sense of algebraic
programming and perturbation theory. Ineq.(\ref{wave1}) will be generalized
to
\begin{equation}
\sqrt{(A_1-1)(A_2-1)}\geq \left| B\right| e^{-\overline{\gamma }t},
\label{wave3}
\end{equation}
\section{Separability}
We will prove that
\begin{equation}
\text{ }\sqrt{(A_1-1)(A_2-1)}\geq \left| B\right| \Leftrightarrow \text{ }
\sqrt{C_1C_2}\geq \left| D\right| \label{wave4}
\end{equation}
is the sufficient condition of the separability of the damped state $\rho $.
Let us first consider a Gaussain density operator $\rho _G$ with its
characteristic function $\chi _G=\exp [-A_1^{\prime }\left| \mu _1\right|
^2-A_2^{\prime }\left| \mu _2\right| ^2+B\mu _1\mu _2e^{ix}+B^{*}\mu
_1^{*}\mu _2^{*}e^{-ix})]$. The Fourier transformation of $\chi _G\exp
(\frac 12\left| \mu \right| ^2)$ is a probability distribution function
(pdf) if $\sqrt{(A_1-1)(A_2-1)}\geq \left| B\right| $\cite{Duan}, where $x$
is absorbed into $\mu $.This pdf enables the P-representation of $\rho _G.$
Hence $\rho _G$ is separable when $\sqrt{C_1C_2}\geq \left| D\right| $. The
P-representation of $\rho $ is a positive integral of the P-representation
of $\rho _G$. Thus $\rho $ is separable. From physical consideration, we may
think $\rho $ is the phase damping of $\rho _G$, thus when $\rho _G$ is
separable, $\rho $ should be separable.
The problem left is that when $\sqrt{C_1C_2}<\left| D\right| $ and $\sqrt{
C_1C_2}\geq \left| D\right| e^{-\gamma t}$, the state is separable or not.
We have strong evidence to elucidate that the state is not separable,
although we do not give a full proof. The evidence is like this: the state
can not be expressed in P-representation for $\sqrt{C_1C_2}<\left| D\right| $
, $\rho =\int P(\alpha _1,\alpha _2)\left| \alpha _1\alpha _2\right\rangle
\left\langle \alpha _1\alpha _2\right| $ $d^2\alpha _1d^2\alpha _2/\pi ^2$
is only possible for $\sqrt{C_1C_2}\geq \left| D\right| $ $,$ where $
P(\alpha _1,\alpha _2)$ is a pdf and $\left| \alpha _1\alpha _2\right\rangle
$ denotes two-mode coherent state.
The Fourier transformation will be $P(\alpha )=\int \chi (\mu )\exp [\frac
12\left| \mu \right| ^2-\mu \alpha ^{*}+\mu ^{*}\alpha ]d^4\mu /\pi ^2.$
After the integral of $\mu $, we have
\begin{equation}
P(\alpha )=c\int_x\exp [-E_2\left| \alpha _1\right| ^2-E_1\left| \alpha
_2\right| ^2+Fe^{ix}\alpha _1\alpha _2+F^{*}e^{-ix}\alpha _1^{*}\alpha
_2^{*}],
\end{equation}
where $c=\frac 1{(A_1-1)(A_2-1)-\left| B\right| ^2}$ $E_i=\frac{A_i-1}{
(A_1-1)(A_2-1)-\left| B\right| ^2},F=\frac B{(A_1-1)(A_2-1)-\left| B\right|
^2}.$ Denote $\alpha _i=r_ie^{i\theta _i},F=\left| F\right| e^{i\phi }$ and $
\varphi =\theta _1+\theta _2+\phi ,$ then
\begin{equation}
P(\alpha )=c\exp [-E_2r_1^2-E_1r_2^2]\sum_{n=0}^\infty \frac{(\left|
F\right| r_1r_2)^n}{n!}\sum_{l=0}^n\binom nl\exp [-\overline{\gamma }t\left|
2l-n\right| ^2+i(2l-n)\varphi ].
\end{equation}
Clearly $P(\alpha )$ is real, and $P(\alpha )$ is positive. The positivity
of $P(\alpha )$ is warranted by the fact that when $\sqrt{C_1C_2}\geq \left|
D\right| $ , the state is separable, $P(\alpha )$ is a pdf. If we fix $F$
while decreasing $E_i$ to reach a state with $\sqrt{C_1C_2}<\left| D\right| $
, the sign of $P(\alpha )$ will not change by decreasing $E_i$. Thus $
P(\alpha )$ is positive even when $\sqrt{C_1C_2}<\left| D\right| .$
What left is the singularity of $P(\alpha ).$ We will prove that $P(\alpha )$
is singular if and only if $\sqrt{C_1C_2}<\left| D\right| .$ The singularity
may appear when $\left| \alpha _i\right| =r_i\rightarrow \infty ,$ $P(\alpha
)\rightarrow \infty .$ The maximum of $P(\alpha )$ reaches when $\varphi =0,$
thus in the following, we set $\varphi =0.$ Let $g(z)=\sum_{n=0}^\infty
\frac{\left( z/2\right) ^n}{n!}\sum_{l=0}^n\binom nl\exp [-\overline{\gamma }
t\left| 2l-n\right| ^2],$ then we have $g(z)\approx \sum_{n=0}^\infty \frac{
z^n}{n!\sqrt{1+\overline{\gamma }tn}}$ $\approx 1+\sum_{n=1}^\infty \frac{z^n
}{n!\sqrt{\overline{\gamma }tn}}>1+\frac 1{\sqrt{\overline{\gamma }t}
}\sum_{n=1}^\infty \frac{z^n}{(n+1)!}=1+\frac 1{\sqrt{\overline{\gamma }t}
}\frac 1z(e^z-1-z),$ where we have used DeMoirve-Laplace theorem $\binom
nlp^lq^{n-k}\approx $ $\frac 1{\sqrt{2\pi npq}}\exp [-\frac{(l-np)^2}{2npq}].
$ Thus $1+\frac 1{\sqrt{\overline{\gamma }t}}\frac 1z(e^z-1-z)\lesssim
g(z)\leq e^z,$ $\lim {}_{z\rightarrow \infty }\frac{\ln g(z)}z=1.$ We arrive
at $P(r_1,r_2)\rightarrow c\exp [-E_2r_1^2-E_1r_2^2+2\left| F\right|
r_1r_2]\leq c\exp [-2(\sqrt{E_2E_1}-\left| F\right| )r_1r_2]$ when $
r_1r_2\rightarrow \infty .$ The non-singularity condition of $P(\alpha )$ is
simply $\sqrt{E_2E_1}\geq \left| F\right| ,$ which is equivalent to Ineq.(
\ref{wave4}).
We now compare all three conditions of the separability of $\rho .$ If $
\sqrt{C_1C_2}\geq \left| D\right| ,$ the state $\rho $ is separable,
needlessly to say we have $\sqrt{C_1C_2}\geq \left| D\right| e^{-\overline{
\gamma }t},$ and $\sqrt{(A_1-1)(A_2-1)}\geq \left| B\right| e^{-\overline{
\gamma }t}$; if $\sqrt{C_1C_2}<\left| D\right| $ and $\sqrt{C_1C_2}\geq
\left| D\right| e^{-\overline{\gamma }t}$, we have $\sqrt{(A_1-1)(A_2-1)}
\geq \left| B\right| e^{-\overline{\gamma }t},$hence Ineq.(\ref{wave3}) can
be dropped as a necessary condition because it is weak than Ineq.(\ref{wave8}
), at this case we do not know the state $\rho $ is separable or not, we
suspect that the state is bound entangled; if $\sqrt{C_1C_2}<\left| D\right|
e^{-\overline{\gamma }t}$, the state $\rho $ is entangled. The conditions
are expressed with the curves $A-1-Be^{-\gamma t},C-De^{-\gamma t}$ and $
A-B-1$ in the figure for the special case of $A_1=A_2,B=B^{*}$. The zero
points of the curves are $t_1,t_2,t_3$, and $t_1\leq t_2\leq t_3$. For
channel without phase damping, the state is a gaussian state. All three
conditions will be the same, the zero points of the curves will coincide and
$t_1=t_2=t_3$.
\section{Conclusions and Discussions}
In conclusion, the phase damping equation of a state is obtained in the form
of characteristic function. It turns out to be a usual dissipation equation
with respect to the phase angle of the complex variable of the
characteristic function. The time evolution solution is given for any
continuous variable state undergo simultaneous amplitude and phase damping
and thermal noise. Two of the criteria are given for the amplitude and phase
damping of a two mode $x-p$ symmetric Gaussian state. One is the sufficient
condition of the damped state. The other is Peres-Horodecki criterion which
is not only necessary but also proved to be PPT sufficient. The proof is at
the sense of algebraic programming and also perturbation theory. The
logarithmic negativity and coherent information of the damped state are
investigated.
The evolution of the state is like this: the entanglement of the state (if
the state is prepared entangled initially) decreases with time, at some time
it reaches $0,$ this time is determined by Peres-Horodecki criterion. Then
the state may be bound entangled at the next time interval, we proved that
the state has not a P-representation at this time interval. The end of this
time interval is the time determined by the sufficient condition of the
separability. After this time the state is separable.
For a channel without phase damping, the state remains a Gaussian state. The
two criteria will coincide\cite{Duan}\cite{Simon}. For pure phase damping
channel, $\Gamma =0,$ $\overline{n}=0,$ we can see that the initially
two-mode vacuum state will never evolve to a separable state. Hiroshima
mentioned this result with numerical calculation\cite{Hiroshima}.
This work was supported by the National Natural Science Foundation of China
(under Grant No. 10347119), Zhejiang Province Natural Science Foundation
(under Grant No. R104265) and AQSIQ of China (under Grant No. 2004QK38)
\section{Appendix}
From the eigenequation of partial transposed density matrix $\rho
^{PT}(t)\left| \Phi \right\rangle =\lambda \left| \Phi \right\rangle ,$ the
eigenequation for characteristic function of it can be deduced as $\int
\frac{d^4\mu }{\pi ^2}\frac{d^4\alpha }{\pi ^2}\chi ^{PT}(\mu ,\mu
^{*},t)\left\langle \beta \right| D(-\mu )\left| \alpha \right\rangle
\left\langle \alpha \right| \left. \Phi \right\rangle =\lambda \left\langle
\beta \right| \left. \Phi \right\rangle .$ Let $\left\langle \alpha \right|
\left. \Phi \right\rangle =\exp (-\frac 12\left| \alpha \right|
^2)\sum_{n=0}^mc_n^{(m)}\alpha _1^{*m-n}\alpha _2^{*n}$, then
\begin{eqnarray*}
I_{mn} &=&\exp (\frac 12\left| \beta \right| ^2)\int \frac{d^4\mu }{\pi ^2}
\frac{d^4\alpha }{\pi ^2}\chi ^{PT}(\mu ,\mu ^{*},t)\left\langle \beta
\right| D(-\mu )\left| \alpha \right\rangle \exp (-\frac 12\left| \alpha
\right| ^2)\alpha _1^{*m-n}\alpha _2^{*n} \\
&=&\int_x\int \frac{d^4\mu }{\pi ^2}\exp [-A_1\left| \mu _1\right|
^2-A_2\left| \mu _2\right| ^2-B\mu _1\mu _2^{*}e^{ix}-B^{*}\mu _1^{*}\mu
_2e^{-ix}+\mu \beta ^{*}] \\
&&*(\beta _1^{*}-\mu _1^{*})^{m-n}(\beta _2^{*}-\mu _2^{*})^n.
\end{eqnarray*}
Where the integral formula
\[
\int \frac{d^2\tau }\pi \exp [-\left| \tau \right| ^2+\tau \sigma ]f\left(
\tau ^{*}\right) =f\left( \sigma \right)
\]
is used to integrate $\alpha .$ This formula can further be used to
integrate $\mu .$ After the integral of $\mu _1$ we have
\begin{eqnarray*}
I_{mn} &=&\frac 1{A_1}\int_x\int \frac{d^2\mu _2}\pi \exp [-A_2\left| \mu
_2\right| ^2-\frac 1{A_1}B^{*}\mu _2e^{-ix}(\beta _1^{*}-B\mu
_2^{*}e^{ix})+\mu _2\beta _2^{*}] \\
&&*[\beta _1^{*}-\frac 1{A_1}(\beta _1^{*}-B\mu _2^{*}e^{ix})]^{m-n}(\beta
_2^{*}-\mu _2^{*})^n.
\end{eqnarray*}
After the integral of $\mu _2$ we have
\[
I_{mn}=\frac 1{A_1A_2-\left| B\right| ^2}\int_x\left( C_2\beta
_1^{*}+De^{ix}\beta _2^{*}\right) ^{m-n}\left( C_1\beta
_2^{*}+D^{*}e^{-ix}\beta _1^{*}\right) ^n,
\]
with $C_i=1-\frac{A_i}{A_1A_2-\left| B\right| ^2}$ and $D=\frac
B{A_1A_2-\left| B\right| ^2}$. For each $m$ we expand the binomial and
complete the integral of $x,$ $I_{mn}$ will be a polynomial of $\beta _1^{*}
$ and $\beta _2^{*}$. The eigenequation will be
\[
\sum_{n=0}^mc_n^{(m)}I_{mn}(\beta _1^{*},\beta _2^{*})=\lambda
\sum_{k=0}^mc_k^{(m)}\beta _1^{*m-k}\beta _2^{*k}.
\]
By comparing the power of $\beta ^{*},$ and absorbing the phase of $D$ into
the coefficient $c_n^{(m)},$ we at last get a matrix $M^{(m)}$ (as in Eq.(
\ref{wave6}) ) whose eigenvalues are that of $\rho ^{PT}(t).$ Eq.(\ref{wave7}
) can be deduced in the same way.
\end{document} |
\begin{eqnarray}gin{document}
\title{Optimal verification of entanglement in a photonic cluster state experiment}
\author{H. Wunderlich$^1$, G. Vallone$^{2,3}$, P. Mataloni$^{3,4}$, M. B. Plenio$^1$}
\address{$^1$Institut f\"ur Theoretische Physik, Universit\"at Ulm, Albert Einstein-Allee 11, 89068 Ulm, Germany}
\address{$^2$Centro Studi e Ricerche ``Enrico Fermi'', Via Panisperna 89/A, Compendio del Viminale, Roma 00184, Italy}
\address{$^3$Dipartimento di Fisica della ``Sapienza'' Universit\`{a} di Roma,Roma 00185, Italy}
\address{$^4$Istituto Nazionale di Ottica (INO-CNR), L.go E. Fermi 6, I-50125 Florence, Italy}
\begin{eqnarray}gin{abstract}
We report on the quantification of entanglement by means of entanglement measures on a four- and a six- qubit cluster state
realized by using photons entangled both in polarization and linear momentum. This paper also addresses the question of the scaling of entanglement bounds from incomplete tomographic information on the density matrix under realistic experimental conditions.
\end{abstract}
\pacs{03.65.Ud, 03.67.Bg, 42.50.Ex}
\noindent{\it Keywords}: graph state experiment, entanglement measures
\maketitle
\section{Introduction}
Experiments in Quantum Information Science (QIS) rely heavily on multipartite entangled quantum states. Cluster states \cite{BR01}, or more generally graph states, are a particular class of multipartite states that offer a diversity of applications in QIS, ranging from measurement-based quantum computation and error-correction codes to nonlocality tests. Due to the importance of graph states,
a considerable experimental effort has been made to realize them using photons \cite{Walther05, Kiesel05, Lu07, Chen07, Vallone07, cecc09prl} and cold atoms \cite{Mandel03}. Proposals for trapped ions are also pursued \cite{Wunderlich09, Stock09, Ivanov08}.
In this paper we characterize the four- and six-qubit cluster states realized in \cite{Vallone07, cecc09prl,vall10pra} in terms of fidelity, purity and entanglement.
In particular, we quantify the amount of experimentally generated entanglement using entanglement measures \cite{PV07}.
Cluster states are uniquely defined by a set of correlation operators which generate a group called the stabilizer. An interesting question this paper addresses is how bounds based on measurement results of the generator of the stabilizer alone scale with increasing system size under realistic conditions using the results developed in Refs. \cite{WP09, WP10, WVP10}. We compare the optimal bounds based on such measurements for the fidelity, purity and the robustness of entanglement \cite{Robustness1, Robustness2, Robustness3} as well as the relative entropy of entanglement \cite{relentropy} with the density matrix obtained from all stabilizers in order to answer this question.
\section{Experimental Set-Up}
\begin{eqnarray}gin{figure}[b]
\centering
\includegraphics[width=16cm]{figure1.pdf}
\caption{left) Source of polarization-path hyperentangled state. right) Measurement setup for the path DOF.}
\label{fig:source}
\end{figure}
Cluster states are particular multiqubit entangled states associated to a graph. In the following we denote the Pauli spin matrices acting on the Hilbert space of qubit $q$ by $X_q$, $Y_q$, and $Z_q$.
Given a lattice with $n$ vertices and $L$ links, a $n$-qubit cluster state can be defined by associating
a qubit in the superposition state $\ket+=\frac{1}{\sqrt2}(\ket0+\ket1)$ to each vertex and
a control-Z gate CZ$_{ab}=\ket0_a\bra0\otimes\leavevmode\hbox{\small1\normalsize\kern-.33em1}_b+\ket1_a\bra1\otimes Z_b$
to each link between vertices $a$ and $b$.
In an equivalent way, the cluster state is defined as the
unique eigenvector with positive eigenvalues of the $n$ generators $g_a$ defined as
$g_a=X_a\prod_{b\in \mathcal N_a}Z_b$, where $\mathcal N_a$ is the set of neighbouring vertices linked with $a$. The set of operators $\{g_a\}$ generate an Abelian group called the stabilizer $\mathcal{S}$ of the underlying graph. Note that eigenstates of the generators with negative eigenvalues $g_a |\{i\}\rangle = (-1)^{i_a} |\{i\}\rangle$ with
$\ket{\{i\}}=\ket{i_1, ..., i_n}$ (the graph state basis states)
are also referred to as graph states in the literature. These states are equivalent to each other
up to single qubit unitary transformations and have therefore the same entanglement properties.
As an example, by considering a graph of four qubits linked in a row, the corresponding cluster is given by
\begin{eqnarray}
\ket{\Phi^{\mbox{lin}}_4}=\frac{1}{2}(\ket{+00+}+\ket{+01-}+\ket{-10+}-\ket{-11-})\,.
\end{eqnarray}
One way of generating cluster states is using photons. A useful tool to realize multiqubit
states is represented by the so-called hyperentanglement (HE), i.e. the entanglement of two (or more) particles
in several degrees of freedom (DOFs) \cite{kwia97jmo}.
Precisely, by using the source shown in figure \ref{fig:source}, we can generate two photons
hyperentangled in polarization and path:
\begin{eqnarray}\label{HE-pi-k}
\ket{\Xi_4}=\frac{1}{\sqrt2}({\ket{H}}_A{\ket{H}}_B+{\ket{V}}_A{\ket{V}}_B)\otimes
\frac{1}{\sqrt2}({\ket{\ell}}_A{\ket{r}}_B-{\ket{r}}_A{\ket{\ell}}_B)\,.
\end{eqnarray}
In the previous equation $\ket H$ ($\ket V$) represent the horizontal (vertical) polarization state and
$\ket r$ and $\ket\ell$ are the two modes (right and left) in which each photon ($A$ and $B$) can be emitted.
The two photons (at degenerate wavelength $\lambda = \unit[728]{nm}$)
are emitted by the spontaneous
parametric down conversion (SPDC) process in a
nonlinear type-I $\begin{eqnarray}ta$-barium-borate (BBO) crystal.
The BBO crystal is shined by a vertically polarized continuous wave (cw) Ar$^+$ laser ($\lambda_p = \unit[364]{nm})$.
Polarization entanglement is generated by the double passage (back and forth,
after the reflection on a spherical mirror) of the UV beam.
The backward emission generates the so called V-cone: the SPDC horizontally polarized photons
passing twice through the quarter waveplate (QWP) are transformed into vertical polarized photons.
The forward emission generates the H-cone.
Temporal and spatial superposition are respectively guaranteed by the long coherence time of the UV beam and
by aligning the crystal at a distance from the spherical mirror
which is equal to its radius of curvature.
In this way, the indistinguishability of the two perpendicularly polarized
SPDC cones creates polarization entanglement:
when two photons are detected it is impossible to know in which pump passage through
the crystal they have been generated.
It is worth noting that the probability of double pair emission (i.e. four photons) is
negligible due to the low power of the cw pump beam ($<\unit[100]{mW}$).
By translating the spherical mirror it is possible to change the relative phase between the
states $\ket{HH}_{AB}$ and $\ket{VV}_{AB}$. A lens $L$ located at a focal distance from the crystal
transforms the conical emission into a cylindrical one.
Path entanglement can be generated by exploiting the properties of Type-I phase matching.
The two polarization entangled photons are emitted over two opposite directions of the SPDC cone.
By selecting with a four-holed mask two pairs of correlated modes,
thanks to the spatial coherence property of the source, the two photons are also entangled in path.
We labeled the two pairs of correlated modes, as $r_A-\ell_B$ and $\ell_A-r_B$.
We set the relative phase between the two pair emissions to the value $\varphi = \pi$ by
tilting think glass on the photon paths.
The state expressed in (\ref{HE-pi-k}) encodes 4 qubits into 2 photons \cite{barb05pra,cine05lp}.
\subsection{4-qubit cluster}
The hyperentangled state may be transformed into a cluster state $\ket{C_4}$
by using a waveplate with
vertical optical axis and placed on the $\ket r$ mode of the $A$ photon.
The waveplate acts as a $\pi$ phase shift on the state $\ket{V}_B\ket{r}_B$. When applied to $\ket{\Xi_4}$
it generates the following cluster state:
\begin{eqnarray}
\ket{C_4}=
\frac{1}{2}({\ket{H\ell}}_A{\ket{Hr}}_B-{\ket{Hr}}_A{\ket{H\ell}}_B
+{\ket{V\ell}}_A{\ket{Vr}}_B+{\ket{Vr}}_A{\ket{V\ell}}_B)\,.
\end{eqnarray}
By using the correspondence $\ket H_{A,B}\leftrightarrow\ket0_{3,4}$, $\ket V_{A,B}\leftrightarrow\ket1_{3,4}$,
$\ket \ell_{A,B}\leftrightarrow\ket0_{2,1}$, $\ket r_{A,B}\leftrightarrow\ket1_{2,1}$,
the generated state $\ket{C_4}$ is equivalent
up to single qubit unitaries to $\ket{\Phi^{\mbox{lin}}_4}$: $\ket{C_4}=\mathcal U\ket{\Phi^{\mbox{lin}}_4}= X_1H_1\otimes Z_2\otimes\leavevmode\hbox{\small1\normalsize\kern-.33em1}_3\otimes H_4\ket{\Phi^{\mbox{lin}}_4}$, where
$H$ represents the Hadamard gate $H=\frac{1}{\sqrt2}(X+Z)$. The latter relation between $\ket{C_4}$ and $\ket{\Phi^{\mbox{lin}}_4}$,
implies that $\ket{C_4}$ is the only common eigenstate of the generators $\tilde g_a=\mathcal Ug_a\mathcal U^{-1}$ obtained
from $g_a$ by changing
$X_1\rightarrow Z_1$, $Z_1\rightarrow -X_1$, $X_2\rightarrow -X_2$ and $X_4\leftrightarrow Z_4$.
\subsection{6-qubit cluster}
\begin{eqnarray}gin{figure}[b]
\centering
\includegraphics[width=13cm]{figure2.pdf}
\caption{Generation and measurement of the 6-qubit linear cluster.
(a) By selecting 4 pairs of correlated SPDC modes, a 6-qubit polarization-path hyperentangled state can be generated.
Two half waveplates ($\lambda/2$) are used to transform it into a 6-qubit linear cluster state corresponding to the graph
shown in the inset. (b) Two cascade interferometers are used for path measurement. The first $BS_1$ performs the
measurement in the $\{\ket{r},\ket{\ell}\}$ qubit for both photons,
while the two beam splitters $BS_{2A}$ and $BS_{2B}$ perform the measurement in the $\{\ket{I},\ket{E}\}$ qubit for
Alice and Bob photon, respectively. Each detection stage $D_i$
is composed of a polarization analyzer
(waveplates and polarizing beam splitter) followed by a single photon detector.
Two translation stages change the optical delay $\Delta x_{1,2}$ to obtain the correct temporal superposition of the different modes.
}
\label{fig:6qubit}
\end{figure}
It is possible to add more qubits to the state by selecting more optical paths.
Precisely, by selecting four pairs of modes it is possible to generate a
two-photon six-qubit hyperentangled state \cite{vall09pra}. We labeled the four modes on which
each photon can be emitted as $\ket{Er}$, $\ket{E\ell}$, $\ket{Ir}$ and $\ket{I\ell}$,
where $E$ ($I$) stands for external (internal) mode (see figure \ref{fig:6qubit}a)).
The 6-qubit hyperentangled state can be written as
\begin{eqnarray}\label{HE-6}
\nonumber
\ket{\Xi_6}=\frac{1}{\sqrt2}({\ket{H}}_A{\ket{H}}_B-{\ket{V}}_A{\ket{V}}_B)&\otimes
\frac{1}{\sqrt2}({\ket{r}}_A{\ket{\ell}}_B+{\ket{\ell}}_A{\ket{r}}_B)\otimes
\\
& \otimes \frac{1}{\sqrt2}({\ket{E}}_A{\ket{E}}_B+{\ket{I}}_A{\ket{I}}_B)\,.
\end{eqnarray}
As shown in figure \ref{fig:6qubit}a), two half-waveplates are used to transform the previous state
into the 6-qubit linear cluster state \cite{cecc09prl, vall10pra}:
\begin{eqnarray}
\label{lc6}
\ket{\widetilde{\rm LC}_6} &= \frac{1}{2} \bigl[\ket{EE}\ket{\phi^{+}}_{\pi}\ket{r
\ell} + \ket{EE}\ket{\phi^{-}}_{\pi}\ket{\ell r} +
\ket{II}\ket{\psi^{+}}_{\pi}\ket{r \ell} -
\ket{II}\ket{\psi^{-}}_{\pi}\ket{\ell r}\bigr]
\end{eqnarray}
where $\ket{\psi^{\pm}}_{\pi}=1/\sqrt{2}(\ket{HV}\pm\ket{VH}$ and $\ket{\phi^{\pm}}_{\pi}=1/\sqrt{2}(\ket{HH}\pm\ket{VV}$.
$\ket{\widetilde{\rm LC}_6}$ corresponds to the graph shown in the inset of figure \ref{fig:6qubit}a) up to single qubit unitaries.
Precisely, $\ket{\widetilde{\rm LC}_6}$ is the only common eigenstate (with +1 eigenvalues) of
the generators $\widetilde g_i$ obtained from $g_i$ by changing $X_2 \leftrightarrow Z_2$, $X_3
\rightarrow -Z_3$, $Z_3 \rightarrow X_3$, $X_4 \leftrightarrow
Z_4$ and $X_5 \rightarrow -X_5$.
In order to measure Pauli path operators, two cascade interferometers are implemented (see figure \ref{fig:6qubit}b)).
\section{Results}
\subsection{Quantitative Entanglement Verification}
The detection and quantification of entanglement has become a standard part of quantum information experiments.
Methods for entanglement detection range from Bell inequalities over entanglement witnesses \cite{GT09} to semidefinite programs \cite{DPS02, NOP09}. In order to quantify entanglement, it is necessary to evaluate an entanglement measure for the state under scrutiny \cite{PV07}. Entanglement measures have the advantage that they do not only detect entanglement, but they may also provide an operational meaning to the amount of entanglement in a given quantum state. Until today, many entanglement measures have been invented, and the choice of the appropriate measure depends on the specific task \cite{PV07}.
Here, we choose the global robustness of entanglement \cite{Robustness1, Robustness2, Robustness3} and the relative entropy of entanglement \cite{relentropy}. Both measures are suitable to quantify graph state entanglement for the following reasons: cluster states were introduced as multipartite entangled states that exhibit a particular persistence against noise. While GHZ states become more vulnerable under noise with increasing system size, this is not the case for cluster states \cite{BR01}. Hence, the robustness of entanglement is a measure that quantifies this property. The relative entropy provides an operational meaning for cluster states in the sense that it `counts' the number of entangling gates. As shown in Ref. \cite{MMV07, Anders07} the relative entropy of entanglement for cluster states is proportional to the number of applied controlled-phase gates.
The relative entropy of entanglement is defined as \cite{relentropy}
\begin{eqnarray}gin{equation}
E_R (\rho) = \min_{\sigma \in Sep} S(\rho | \sigma),
\end{equation}
where $Sep$ denotes the set of fully separable states, and $S(\rho | \sigma) = tr[\rho (\log_2 \rho- \log_2 \sigma)]$.
The global robustness of entanglement measures how much noise must be mixed in to a given quantum state such that the mixture becomes separable \cite{Robustness1, Robustness2, Robustness3}:
\begin{eqnarray}gin{equation}
\label{robustness}
R_G(\rho) = \min_{\sigma \in \mathcal{D}, s\in {\mathbbm{R}}} \{ s:\frac{\rho + s \sigma}{1+s} \in Sep \},
\end{equation}
where $\mathcal{D}$ is the entire Hilbert space.
From a mathematical point of view it is more convenient to relax the global robustness by replacing the set of fully separable states by teh set of PPT states, thus obtaining the following semidefinite program:
\begin{eqnarray}gin{eqnarray}
\label{RPPT}
R_G^{PPT} (\rho) = & \min tr\{\sigma\} &
\\
\mathrm{subject~to}~ & \sigma & \ge 0,
\\
& (\rho + \sigma)^\Gamma & \ge 0 .
\end{eqnarray}
Here $\Gamma$ denotes partial transpostion with respect to a partition of choice. In principal, one could check all possible partitions. In this way we have relaxed the global robustness to a PPT version that can be formulated as a semidefinite program. Hence, numerical tools such as convex optimization solvers are instantly available to evaluate this measure \cite{sedumi}.
\subsection{Four-qubit cluster state}
\begin{eqnarray}gin{figure}
\centering
\includegraphics[width=12cm]{figure3.pdf}
\caption{Raw and optimized values of $p_{\{k\}}$ for the 4-qubit cluster state.
Each value corresponds to a given $p_{\{k\}}$ according to the correspondence
$\{k\}\leftrightarrow8k_4+4{k_3}+2{k_2}+{k_1}$.
}
\label{fig:data}
\end{figure}
To verify the creation of the four-qubit cluster state all elements of the stabilizer group were measured \cite{vall08pra}.
As the measurements were local measurements, also statistics of single Pauli operators are available. These local measurements do not contribute to the fidelity, but they allow us to improve bounds on entanglement measures as those are restricted minimizations that can only improve when more constraints are added \cite{IncompleteTomo1}. Using the measured data
we calculated the raw fidelity $F_{\ket{C_4}} = \frac{1}{16}\sum^1_{\{k_a\}=0}
\langle S_{k_1k_2k_3k_4}\rangle=0.880\pm0.013$ \cite{vall08pra},
where $S_{k_1k_2k_3k_4}=\prod^4_{a=1} (g_a)^{k_a}$ are the 16 stabilizers built as all possible products of generators.
The raw purity is found to be $P (\rho) = tr(\rho^2) = 0.779 \pm 0.005$.
From the raw data it is possible to obtain the fidelity with all possible graph state bases $\ket{\{i\}}\bra{\{i\}}$
since $\ket{\{i\}}\bra{\{i\}}=\frac{1}{16}\sum_{\{k\}}S_{\{k\}}(-1)^{\bf i\cdot k}$
with ${\bf i\cdot k}=\sum^4_{a=1} i_ak_a$. Some of the raw fidelities are negative
because of experimental inaccuracies and statistical fluctuations of coincidence counts
(the same problem arises in quantum state tomography \cite{jame01pra}). To solve the problem we applied
a maximum likelihood estimation. We determine the physical density matrix diagonal in the graph state basis and written as
\begin{eqnarray}gin{equation}
\rho_{phys}=\sum_{\{k\}}p_{\{k\}}\ket{\{k\}}\bra{\{k\}}\,,\qquad p_{\{k\}}\geq 0
\end{equation}
that is most compatible with the experimental data.
The value of the optimized $p_{0000}$ corresponds to the fidelity with the cluster state,
with value equal to 0.880 which is completely compatible with the raw fidelity.
The other values of the optimized $p_{\{k\}}$ are shown in figure
\ref{fig:data}. The optimized purity is given by $P_{opt}=\sum_{\{k\}}p^2_{\{k\}}=0.778$ again compatible with the raw value.
While for few-qubit systems the determination of the whole stabilizer is feasible, this is not the case for large graph states. Therefore, it is natural to ask which bounds on the fidelity, purity and entanglement can be obtained from incomplete information on the density matrix \cite{IncompleteTomo1, IncompleteTomo2, IncompleteTomo3}, e.g., from measuring generators of a graph state only; and how do such bounds scale with system size under realistic conditions?
Let us consider the estimation of the fidelity from information on the generators only. This is formulated as a worst-case estimation \cite{WP09}
\begin{eqnarray}
F_{min} = \min_{\rho} \{ F(\rho): tr(\rho g_i) = a_i, \rho \ge 0\},
\end{eqnarray}
where $g_i$ are the generators of the graph for $i=1,..,4$ with corresponding measurement outcomes $a_i$, and $g_0 = \mathds{1}$. Remarkably, this problem can be solved optimally, leading to a solution of the analytic form \cite{WP09}
\begin{eqnarray}
F_{min} = \max \{0, \frac{\sum_{i=1}^n |a_i|-n+2}{2} \}
\end{eqnarray}
for $n$ qubits and holds for all stabilizer operators with spectrum $\{+1,-1\}$. One may quickly check that the optimal lower bound on the fidelity consistent with the measurements of the generators (see Tab. (\ref{result4})) is given by $F_{min} = 0.846 \pm 0.009$. The relative loss of information on the fidelity is therefore only around 5\%,
even though only four out of sixteen elements of the stabilizer were determined.
It is also possible to optimally estimate the purity using only generator measurements. Following the techniques of Ref. \cite{WP10} we obtain a minimal purity consistent with such measurements of $P_{min} = 0.715 \pm 0.014$.
\begin{eqnarray}gin{table}
\caption{Four-qubit cluster state: measurement results of generators}
\label{result4}
\footnotesize\rm
\begin{eqnarray}gin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
Generator & Measurement Outcome \\
\mr
$g_1 =- Z_1 \otimes Z_2 \otimes \mathds{1}_3\otimes\mathds{1}_4$ & $0.994 \pm 0.001$ \\
$g_2 =- X_1 \otimes X_2 \otimes Z_3 \otimes\mathds{1}_4$ & $0.849 \pm 0.003$\\
$g_3 = \mathds{1}_1 \otimes Z_2 \otimes X_3 \otimes X_4$ & $0.937 \pm 0.003$ \\
$g_4 = \mathds{1}_1 \otimes \mathds{1}_2 \otimes Z_3 \otimes Z_4 $ & $0.911 \pm 0.002$ \\
\br
\end{tabular*}
\end{table}
Quantifying the experimentally created entanglement is achieved by evaluating the global robustness of entanglement and the relative entropy. The density matrix reconstructed from the stabilizer measurements and local observables obtained as a side product of the stabilizer measurements serves as the input for the semidefinite program (\ref{RPPT}), where we evaluate the global robustness with the constraint of positivity of the partial transpose with respect to all partitions.
We find that the PPT-Robustness is given by
$R_G^{PPT} = 2.519 \pm 0.012$. Note that this value represents a lower bound to the global robustness in its standard version (\ref{robustness}).
It is straightforward to compute the logartihmic global robustness:
$LR_G^{PPT} = 1.817 \pm 0.005$, which is not far from its desired value of $2$.
As in the case of the fidelity one might ask which bound on the entanglement can be obtained from generator measurements alone. Here the estimation is analogously formulated as a minimization of the measure over states consistent with the measurement data
\begin{eqnarray}
R_{G_{min}} = \min_{\rho} \{ R_G(\rho) : tr(\rho g_i) = a_i, \rho \ge 0\}.
\end{eqnarray}
A lower bound to this problem was derived in Ref. \cite{WVP10}, namely $R_{G_{min}} = \max\{0, 2^{|B|} (\frac{\sum_{i=1}^n |a_i|-n+2}{2})-1\}$. Here $B$ denotes the smaller set of qubits resulting from a coloring of the system into two colors, say Amber $A$ and Blue $B$ with $|A| \ge |B|$ (see Ref. \cite{MMV07} for more details). With this formula, we attain the following analytic bound on the global robustness based on the outcomes of the generators only \cite{WVP10}:
\begin{eqnarray}gin{equation}
R_{G_{min}} = 2.384 \pm 0.036 .
\end{equation}
In turn, one can then easily compute an analytic bound on the logarithmic global robustness:
$LR_{G_{min}} = 1.759$.
The relative entropy can also be bounded from below using techniques presented in Ref. \cite{WVP10}.
The problem reads:
\begin{eqnarray}
E_{R_{min}} = \min_{\rho} \{ E_R(\rho) : tr(\rho g_i) = a_i, \rho \ge 0\}.
\end{eqnarray}
A lower bound to this minimization is given by
\begin{eqnarray}
\label{bound_relent}
E_{R_{min}} = \max \{0,|B|-\sum_i H(p_i)\},
\end{eqnarray}
where $p_i = \frac{1+a_i}{2}$ and $H(x) = - x \log(x) - (1-x) \log (1-x)$ is the classical entropy function. Then, by merit of Eq. (\ref{bound_relent}) we achieve the following bound on the relative entropy of entanglement:
$ E_{R_{min}} = 1.120 \pm 0.021$.
Using all stabilizer measurements we find a lower bound to the relative entropy of $1.449 \pm 0.013$. Hence, the relative difference of the entanglement bounds of the relative entropy is considerably larger than the relative difference of the estimate of the robustness to its real value.
\subsection{Six-qubit cluster state}
The six-qubit cluster state is verified utilizing the same techniques as in the four-qubit case. All 64 stabilizer operators were measured and mapped to a density matrix via maximum likelihood, giving a fidelity of
$F= 0.645 \pm 0.006$.
Estimating the fidelity from the generators alone gives
$F_{min} = 0.545 \pm 0.027$.
Next, we obtain for the purity
$P (\rho) = tr(\rho^2) = 0.424 \pm 0.010$, and the worst-case purity estimate from the generators
$P_{min} = 0.297 \pm 0.015$.
We compute the PPT-Robustness for the reconstructed state and find
$R_G^{PPT} = 4.507 \pm 0.047$
resulting in a logarithmic PPT-Robustness of
$LR_G^{PPT} = 2.461$.
If one only measures the generators of the stabilizer, one obtains $R_{G_{min}} = 3.360 \pm 0.216$ and $LR_{G_{min}} = \log (1+ R_{G_{min}}) = 2.124$ respectively. This means that despite the lower fidelity and lower fidelity estimate one obtains a higher bound on the entanglement, even though only 6 out of the 64 elements of the stabilizer are used to obtain the bound.
To obtain a bound on a second measure, the relative entropy of entanglement, we use Eq. (\ref{bound_relent} ) to obtain $E_{R_{min}}=1.013 \pm 0.046$. Using all stabilizer measurements, one obtains a lower bound to the relative entropy of $1.492 \pm 0.027$.
\begin{eqnarray}gin{table}
\caption{Six-qubit cluster state: measurement results of generators}
\label{result6}
\footnotesize\rm
\begin{eqnarray}gin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}r}
\br
Generator & Measurement Outcome \\
\mr
$g_1 = X_1 \otimes X_2 \otimes \mathds{1}_3 \otimes X_4 \otimes\mathds{1}_5\otimes\mathds{1}_6 $ & $0.593 \pm 0.008$ \\
$g_2 = Z_1 \otimes Z_2 \otimes \mathds{1}_3\otimes\mathds{1}_4\otimes Z_5 \otimes\mathds{1}_6$ & $0.879 \pm 0.005$ \\
$g_3 =-\mathds{1}_1 \otimes \mathds{1}_2\otimes Z_3 \otimes \mathds{1}_4 \otimes\mathds{1}_5 \otimes Z_6 $ & $0.998 \pm 0.001$ \\
$g_4 = Z_1 \otimes \mathds{1}_2 \otimes \mathds{1}_3 \otimes Z_4 \otimes \mathds{1}_5 \otimes \mathds{1}_6$ & $0.997 \pm 0.001$ \\
$g_5 =-\mathds{1}_1 \otimes X_2 \otimes \mathds{1}_3 \otimes \mathds{1}_4 \otimes X_5 \otimes Z_6 $ & $0.791 \pm 0.006$ \\
$g_6 = \mathds{1}_1 \otimes \mathds{1}_2 \otimes X_3 \otimes\mathds{1}_4\otimes Z_5 \otimes X_6 $ & $0.831 \pm 0.006$ \\
\br
\end{tabular*}
\end{table}
\section{Conclusion}
We have presented the creation of four- and six-qubit cluster states using photons. The cluster state entanglement was encoded in path and polarization DOF, thus rendering the state hyperentangled. A summary of the relevant characteristics of the created states is given in Tab. \ref{resultsummary}. The created state could serve as the basis for one-way quantum computation and represents an important step in realizing optical quantum computing.
The experimentally created entanglement was quantified in terms of entanglement measures, namely the global robustness of entanglement and the relative entropy of entanglement. Our results also give insight into the question how analytic bounds from incomplete tomographic information scale with the system size under realistic noisy conditions. Our results demonstrate that despite the decreasing fidelity and purity of the state, one can still infer higher amounts of entanglement with a number of observables which is linear in the number of constituents.
\begin{eqnarray}gin{table}[b]
\caption{Summary of experimental results}
\label{resultsummary}
\footnotesize\rm
\begin{eqnarray}gin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
& Fidelity & Purity & Minimal Global Robustness & Minimal Relative Entropy \\
\mr
4 qubits & $0.880 \pm 0.006$ & $0.778 \pm 0.005$ & $2.519 \pm 0.012$ & $1.449 \pm 0.013$\\
6 qubits & $0.645 \pm 0.006$ & $0.424 \pm 0.010$ & $4.507 \pm 0.047$ & $1.492 \pm 0.027$\\
\br
\end{tabular*}
\end{table}
\section*{Acknowledgments}
We thank E. Pomarico, R. Ceccarelli and G. Donati for their contribution
in the measurements presented in \cite{Vallone07, vall08pra, cecc09prl, vall10pra}.
This work was supported by the EU Integrated Project Q-ESSENCE, the EU STREP project HIP and by an Alexander von Humboldt Professorship.
\section*{References}
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\end{document} |
\begin{document}
\title{Conditional implementation of asymmetrical universal quantum cloning machine}
\author{Radim Filip}
\affiliation{Department of Optics, Palack\' y University,\\
17. listopadu 50, 772~07 Olomouc, \\ Czech Republic}
\date{\today}
\begin{abstract}
We propose two feasible experimental implementations of an optimal
asymmetric $1\rightarrow 2$ quantum cloning of a polarization state of photon.
Both implementations are based on a partial and optimal reverse of recent conditional
symmetrical quantum cloning experiments. The reversion procedure is performed only by a local measurement
of one from the clones and ancilla followed by a local operation on the other clone.
The local measurement consists only of a single unbalanced beam splitter followed in one output by a single photon detector
and the asymmetry of fidelities in the cloning is controlled by a reflectivity of the beam splitter.
\end{abstract}
\pacs{03.67.-a}
\maketitle
Copying the quantum states is apparently dissimilar to
classical information processing since it is impossible to precisely
duplicate an unknown quantum state as a consequence of a
linearity of quantum mechanics \cite{nocloning}.
To clone an unknown quantum
state at least approximately, universal quantum cloning machines (UQCM) were developed \cite{UQCM}.
The UQCM is a device that universally and optimally produces a copy $\rho_{S'}$ from an
unknown quantum state $|\Psi\rangle_{S}$ of the original.
Specifically, an optimal symmetrical $1\rightarrow 2$ UQCM (SUQCM) for qubits
creates a copy $\rho_{S'}$ with a maximal state-independent fidelity $F_{S'}=_{S}\langle\Psi|\rho_{S'}|\Psi\rangle_{S}=5/6$.
Simultaneously, a pure state of original changes to mixed state $\rho_{S}$ exhibiting maximally the same fidelity $F_{S}=
_{S}\langle\Psi|\rho_{S}|\Psi\rangle_{S}=5/6$ as the clone.
To optimally control the fidelities $F_{S}$ and $F_{S'}$, a concept
of asymmetrical UQCM (AUQCM) has been theoretically developed \cite{Niu98,Buzek98,Cerf00}.
The optimal $1\rightarrow 2$ AUQCM produces the copies having state-independent fidelities controlled by a
parameter $R$ in such a way that for a given fidelity of the copy $F_{S'}(R)$, the fidelity $F_{S}(R)$ of the
original is maximal. More specifically, assuming a qubit in an unknown state $|\Psi\rangle$ then
the original $S$ and clone $S'$ leaving $1\rightarrow 2$ AUQCM can be represented by the following density matrices
\cite{Niu98,Buzek98,Cerf00}
\begin{eqnarray}
\rho_{S,S'}=F_{S,S'}|\Psi\rangle\langle\Psi|+(1-F_{S,S'})|\Psi_{\bot}\rangle\langle\Psi_{\bot}|
\end{eqnarray}
where the fidelities
$5/6\leq F_{S}\leq 1$ and $1/2\leq F_{S'}\leq 5/6$ satisfy the cloning relation
\begin{equation}\label{cond}
(1-F_{S})(1-F_{S'})\geq (1/2-(1-F_{S})-(1-F_{S'}))^{2},
\end{equation}
where the equality corresponds to an {\em optimal} AUQCM, in the sense that for a a larger $F_{S}$ cannot be
obtained for given $F_{S'}$. The Eq. (\ref{cond}) is the tightest no-cloning bound for the fidelities
of the $1\rightarrow 2$ cloner which copies an unknown qubit state to the another with an isotropic noise.
The recent experimental effort to build different quantum cloners is mainly stimulated
by their use as individual attacks in quantum communication and cryptography \cite{attack}.
More information about this practical application of the asymmetrical universal cloning as optimal attack
for a cryptographic protocol can be found in Ref.~\cite{asymmattack}.
To build quantum cloners, quantum networks using CNOT gates for both optimal SUQCM and AUQCM were proposed \cite{netclon}.
However, a strength of the state-of-the-art nonlinear interaction at a single photon
level is unfortunately too weak to produce a deterministic and
efficient CNOT operation only by a direct interaction
between photons. For this reason, the deterministic SUQCM and AUQCM still have not been experimentally
implemented in quantum optics. Netherless, stimulated or
spontaneous parametric down-conversions were used to realize a conditional implementation
of the SUQCM for a polarization state of photon \cite{Lamas-Linares02,DeMartini02,Ricci03}.
However, to the best of our knowledge, no feasible experimental setup for an
optimal $1\rightarrow 2$ AUQCM has been presented yet. On the other hand, an experimental realization of
optimal asymmetrical cloning machine
has already been proposed for coherent states \cite{CVasymm}.
In this paper, we propose optimal $1\rightarrow 2$ AUQCM which is a simple and experimentally feasible extension of
the recent experiments on the conditional symmetrical cloning of the polarization state of a photon.
Our method is based on a partial optimal reverse of SUQCM by a specific controllable
joint measurement on one of the copies and an auxiliary photon leaving the cloning process.
By this partial reverse the quantum information between the disturbed original and copy can be redistributed posteriori
only using the local operations and classical communication. It can be experimentally accomplished adding only
a single unbalanced beam splitter followed by a single-photon detector
in the recent cloning experiments \cite{Lamas-Linares02,DeMartini02,Ricci03}.
\begin{figure}
\caption{Setup for conditional AUQCM based on stimulated parametric down-conversion: L -- laser, BS1-BS3 beam splitters, PBS -- polarization beam splitter, 2$\omega$ -- frequency doubler, BBB -- nonlinear
BBO type II crystal, $\lambda/2,\lambda/4$ -- wave plates, D1 -- single photon detector.}
\end{figure}
An experimental realization of the conditional 1$\rightarrow$2 SUQCM \cite{DeMartini02,Lamas-Linares02}
was based on a nonlinear parametric down-conversion stimulated in a signal beam by
a single photon prepared in an unknown polarization state
$|\Psi\rangle_{S}=a|V\rangle_{S}+b|H\rangle_{S}$, where $V$ and $H$
denote the vertical and horizontal polarizations.
The experimental arrangement is depicted in Fig.~1. The input single
photon extracted from a laser pulse is prepared
in the state $|\Psi\rangle_{S}$ in a preparation device using $\lambda/2$ and $\lambda/4$ wave plates.
A more intensive part of the laser pulse is frequency doubled and used to
pump a BBO non-linear crystal. An action of a non-degenerate
type II parametric down-conversion process in the crystal can be described by the Hamiltonian
$H_{I}=i\hbar\chi(a^{\dag}_{H}b^{\dag}_{V}-a^{\dag}_{V}b^{\dag}_{H})+\mbox{h.c.}$,
where $\chi$ is proportional to
a nonlinear susceptibility of the crystal, and h.c. denotes the hermitian conjugation.
Here the annihilation operators $a,b$ are assumed to be acting on the selected signal mode $S$
and idler mode $I$, respectively. A short-time
approximation of the evolution operator
$U=\exp(-iH_{I}t/\hbar)\approx 1-\frac{it}{\hbar}H_{I}$ is used. This approximation
is correct for this kind of the experiments, since the gain $g=\chi t$ is usually very small $(|g|\ll 1)$. Within
the short-time approximation, polarization basis states $|H\rangle_{S}\equiv |1,0\rangle_{S}$ and
$|V\rangle_{S}\equiv |0,1\rangle_{S}$ of the input photon evolve according to the following rules
\begin{eqnarray}
U|1,0\rangle_{S}|0,0\rangle_{I}&\approx& |1,0\rangle_{S}|0,0\rangle_{I}+g(\sqrt{2}|2,0\rangle_{S}|0,1\rangle_{I}\nonumber\\
& &-|1,1\rangle_{S}|1,0\rangle_{I},\nonumber\\
U|0,1\rangle_{S}|0,0\rangle_{I}&\approx& |0,1\rangle_{S}|0,0\rangle_{I}-g(\sqrt{2}|0,2\rangle_{S}
|1,0\rangle_{I}\nonumber\\
& &-|1,1\rangle_{S}|0,1\rangle_{I}.
\end{eqnarray}
Here, the produced states $|2,0\rangle$ and $|0,2\rangle$
represent an effect of a stimulated emission which is used to prepare
the clone, and the state $|1,1\rangle$ corresponds to an unavoidable
effect of a spontaneous emission. After the amplification,
a balanced polarization-insensitive beam splitter $BS2$ in the signal mode
separates two photons in the states $|2,0\rangle$ or $|0,2\rangle$ to two distinguishable
spatial modes corresponding to the disturbed photon $S$ and clone $S'$.
An action of the beam splitter $BS2$ on a pair of photons is as follows
\begin{eqnarray}\label{BS}
|1,1\rangle_{S}|0,0\rangle_{S'}&\rightarrow &\frac{1}{2}(|1,1\rangle_{S}|0,0\rangle_{S'}+
|0,0\rangle_{S}|1,1\rangle_{S'}+\nonumber\\
& &|1,0\rangle_{S}|0,1\rangle_{S'}+|0,1\rangle_{S}|1,0\rangle_{S'}),\nonumber\\
|2,0\rangle_{S}|0,0\rangle_{S'}& \rightarrow &\frac{1}{2}(|2,0\rangle_{S}|0,0\rangle_{S'}+|0,0\rangle_{S}|2,0\rangle_{S'})+
\nonumber\\
& &\frac{1}{\sqrt{2}}|1,0\rangle_{S}|1,0\rangle_{S'},\nonumber\\
|0,2\rangle_{S}|0,0\rangle_{S'} &\rightarrow &\frac{1}{2}(|0,2\rangle_{S}|0,0\rangle_{S'}+|0,0\rangle_{S}|0,2\rangle_{S'})+
\nonumber\\
& &\frac{1}{\sqrt{2}}|0,1\rangle_{S}|0,1\rangle_{S'},
\end{eqnarray}
where $S$ and $S'$ are the signal modes. In the next
procedure only such cases when a single
photon is present in the mode $S$ are considered.
Returning to the previous notation $|1,0\rangle_{i}=|H\rangle_{i}$ and
$|0,1\rangle_{i}=|V\rangle_{i}$, the SUQCM transformation
\begin{eqnarray}\label{clontr}
|\Psi\rangle_{S}\rightarrow \sqrt{\frac{2}{3}}|\Psi\Psi\Psi^{\bot}\rangle_{SS'I}-\frac{1}{\sqrt{3}}
|\Psi_{+}\rangle_{SS'}|\Psi_{I}\rangle,
\end{eqnarray}
where $|\Psi_{+}\rangle=\frac{1}{\sqrt{2}}
(|\Psi\Psi^{\bot}\rangle_{SS'}+|\Psi^{\bot}\Psi\rangle_{SS'})$ and $|\Psi^{\bot}\rangle=a^{*}
|H\rangle-b^{*}|V\rangle$ is the orthogonal state to $|\Psi\rangle$, is actually performed. This SUQCM is optimal and transforms
an unknown state $|\Psi\rangle_{S}$ of the original to the disturbed one and a copy, with the fidelities $F_S=F_S'=5/6$.
Both the output states of photons $S$ and $S'$ have to be measured using the state analyzer composed from the
$\lambda/2$-wave plate and $\lambda/4$-wave plate, the polarization beam splitter $PBS$
and a pair of single photon detectors $D3,D4$, as depicted in Fig.~1.
Note, in the cloning experiment \cite{Lamas-Linares02},
the fidelities were approximately $F_{S},F_{S'}\approx 0.81$ which are really close to the optimal value of $5/6=0.833$.
\begin{figure}
\caption{Setup of conditional AUQCM based on spontaneous parametric down-conversion:
L -- laser, BS1-BS3 beam splitters, PBS -- polarization beam splitter, 2$\omega$ -- frequency doubler, BBB -- nonlinear
BBO type II crystal, $\lambda/2,\lambda/4$ -- wave plates, D1-D2 -- single-photon detectors.}
\end{figure}
Recently, a different implementation of the $1\rightarrow 2$ SUQCM was experimentally performed \cite{Ricci03}.
It is based on the following joint projection
\begin{equation}\label{sproj}
\Pi_{S}=(1_{SS'}-|\Psi_{-}\rangle_{SS'}\langle\Psi_{-}|)\otimes 1_{I}
\end{equation}
of an unknown polarization state $|\Psi\rangle_{S'}=a|V\rangle+B|H\rangle$ of the input photon and
the antisymmetric polarization Bell state
$|\Psi_{-}\rangle_{SI}=\frac{1}{\sqrt{2}}(|VH\rangle-|HV\rangle)$ of two photons produced by the spontaneous parametric
down-conversion from the same BBO nonlinear crystal as in the previous experiment.
The corresponding experimental setup is depicted in Fig.~2.
The initial state $|\Psi\rangle_{S'}=a|V\rangle_{S'}+b|H\rangle_{S'}$ is prepared by the same method.
The projection $\Pi_{S}$ to the symmetric subspace \cite{Werner98} on the state $|\Psi\rangle_{S'}|\Psi_{-}\rangle_{SI}$
can be accomplished by a sequence of two beam splitters $BS1$, $BS2$ and
a single-photon detector $D1$. If two input photons in the same state
$|0,1\rangle_{S}|0,1\rangle_{S'}$ or $|1,0\rangle_{S}|1,0\rangle_{S'})$
constructively interfere on the first balanced
beam splitter $BS1$ and no photon is detected by the detector $D1$, then a twin of photons in the state $|0,2\rangle_{S}$ or
$|2,0\rangle_{S}$ is produced in the mode $S$ with a success probability $1/2$.
The twins are further divided on the second balanced beam splitter
$BS2$ to separate the photons to the different spatial modes and if we selected only such cases when
exactly an single photon is in every mode $S,S',I$ the symmetric states
$|0,1\rangle_{S}|0,1\rangle_{S'}$ or $|1,0\rangle_{S}|1,0\rangle_{S'}$ with probability $1/4$ are obtained at a result.
On the other hand, if two orthogonal basis states $|0,1\rangle_{S}|1,0\rangle_{S'},|1,0\rangle_{S}
|0,1\rangle_{S'}$ are mixed at the beam splitter $BS1$, they do not mutually interfere and in addition,
if no photon is registered on the detector $D1$, then the state $|1,1\rangle_{S}$ with the
success probability $1/2$ is within the mode $S$.
Thus after splitting the photons by $BS2$ to separate spatial modes,
a symmetric state $\frac{1}{\sqrt{2}}(|1,0\rangle_{S}|0,1\rangle_{S'}+|0,1\rangle_{S}|1,0\rangle_{S'})$ is obtained with
the total success probability $1/2$. Thus, with the probability $1/4$ the following
transformation
\begin{eqnarray}\label{symm}
|\Psi\Psi\rangle_{SS'}&\rightarrow & \sqrt{2}
|\Psi\Psi\rangle_{SS'},\nonumber\\
|\Psi\Psi_{\bot}\rangle_{SS'}&\rightarrow &\frac{1}{\sqrt{2}}(
|\Psi\Psi_{\bot}\rangle_{SS'}+|\Psi_{\bot}\rangle_{SS'}),
\end{eqnarray}
of the states of the photons $S,S'$ is in fact conditionally implemented. Assuming that a state of the idler photon $I$
is selected only if this procedure is successful, the total projection
(\ref{sproj}) transforms the input state $|\Psi\rangle_{S'}|\Psi_{-}\rangle_{SI}$ to (\ref{clontr}). Thus
the optimal SUQCM is conditionally accomplished however now a spontaneous emission of maximally entangled pairs is used rather
than a stimulated emission in the previous experiment.
In the experiment based on this idea \cite{Ricci03}, the fidelities of the clone and disturbed original are
$F_{S},F_{S'}\approx 0.826$ which are even more closer to the theoretical value $5/6=0.833$ than it has been in the previous case.
An extension of both setups to achieve the optimal AUQCM can be presented.
It is known that the symmetrical quantum cloning is LOCC reversible \cite{Bruss01}.
If a projective measurement $\Pi_{-}=|\Psi_{-}\rangle\langle\Psi_{-}|$ on one clone and ancilla is performed,
the other clone returns back to the initial state $|\Psi\rangle$.
Thus we can guess that if an appropriate projection in a form
$\alpha 1+\beta \Pi_{-}$ is applied on the one clone and the ancilla, an intermediate case
corresponding to the optimal asymmetrical cloning machine could be obtained.
Now we show that this projection can be conditionally implemented if one mixes a pair of photons in the idler $I$
and signal $S'$ modes on unbalanced beam splitter $BS3$ having a variable reflectivity $0\leq R\leq 1/2$ and
select only the cases when both photons leaving the beam splitter are separated. Then
the beam splitter can be simply described by transformation
\begin{eqnarray}\label{unbaltr}
|VV\rangle_{S'I} &\rightarrow &(T-R)|VV\rangle_{S'I},\nonumber\\
|HH\rangle_{S'I} &\rightarrow & (T-R)|HH\rangle_{S'I},\nonumber\\
|HV\rangle_{S'I}&\rightarrow & T|HV\rangle_{S'I}-R|VH\rangle_{S'I},\nonumber\\
|VH\rangle_{S'I}&\rightarrow & T|VH\rangle_{S'I}-R|HV\rangle_{S'I},
\end{eqnarray}
where $T+R=1$. It can be simply proved that this transformation can be expressed in a covariant
way
\begin{eqnarray}
|\Psi\Psi\rangle_{S'I} &\rightarrow & (T-R)|\Psi\Psi\rangle_{S'I},\nonumber\\
|\Psi\Psi_{\bot}\rangle_{S'I}&\rightarrow & T|\Psi\Psi_{\bot}\rangle_{S'I}-R|\Psi_{\bot}\Psi
\rangle_{S'I}.
\end{eqnarray}
It is an asymmetrical projection controlled by the parameter $R$,
in a contrast to the symmetrizing projection (\ref{symm}). If an output state is selected only when there is
an single photon in each mode $S,S',I$, we obtain the following transformation
for the polarization states of the photon
\begin{eqnarray}\label{trans1}
|H\rangle &\rightarrow &\frac{1}{\sqrt{N(R)}}
((2-R)|HHV\rangle_{SS'I}-\nonumber\\
& &(1+R)|HVH\rangle_{SS'I}-
(1-2R)|VHH\rangle_{SS'I}),\nonumber\\
|V\rangle &\rightarrow &\frac{1}{\sqrt{N(R)}}((2-R)|VVH\rangle_{SS'I}-\nonumber\\
& &(1+R)|VHV\rangle_{SS'I}-
(1-2R)|HVV\rangle_{SS'I})\nonumber\\
\end{eqnarray}
where $N(R)=6(1-R(1-R))$. In a real experiment,
a detection of the photon in the mode $I$ can be performed destructively by a single-photon detector $D2$ whereas
the signal photons from total cloning operation are detected in the state analyzers.
It can be simply proved that the transformation (\ref{trans1})
is covariant and it can be written in a form
\begin{eqnarray}\label{trans1}
|\Psi\rangle &\rightarrow &\frac{1}{\sqrt{N(R)}}
((2-R)|\Psi\Psi\Psi_{\bot}\rangle_{SS'I}-\nonumber\\
& &(1+R)|\Psi\Psi_{\bot}\Psi\rangle_{SS'I}-
(1-2R)|\Psi_{\bot}\Psi\Psi\rangle_{SS'I})\nonumber\\
\end{eqnarray}
which corresponds to the following projection
\begin{equation}
\Pi_{A}(R)=\left((1-2R)1_{S'}\otimes 1_{I}+2R|\Psi_{-}\rangle_{S'I}\langle\Psi_{-}|\right)\otimes 1_{S},
\end{equation}
on the state (\ref{clontr}) produced by the SUQCM.
An interpretation of this projective measurement is straightforward: the asymmetrical cloning is obtained
as a partial optimal reverse of the symmetrical one. For $R=0$ the SUQCM is obtained and for $R=1/2$ the SQUCM is reversed and
an initial state of the original is precisely restored.
An optimal total reverse of the state after the symmetrical cloning was previously
theoretically already analyzed \cite{Bruss01}.
After the total reversion, any input state
is deterministically revealed by the complete Bell-state measurement on the clone and ancilla.
Thus this obtained result also represents a solution of the problem of a partial but still optimal reversion of
the symmetrical cloning. Further, this reversion is also obtained only using the
local operations on the clones and classical communication
between them. It enables the redistribution of the quantum information encoded in symmetric clones at a
distance without an additional quantum channel.
\begin{figure}
\caption{Fidelity of original and clone after AUCQM in dependence on beam splitter $BS3$ reflectivity.
The upper curve corresponds to $F_{S}
\end{figure}
Since the photon in the idler mode is detected in such a way
that no information about its polarization is acquired, we trace over the idler mode and obtain
the final output states of the modes $S$ and $S'$
\begin{eqnarray}
\rho_{SS'}&=&\frac{1}{N(R)}
\left[(2-R)^{2}|\Psi\Psi\rangle_{SS'}\langle\Psi\Psi|+\right.\nonumber\\
& &\left.\left((1+R)|\Psi\Psi_{\bot}\rangle_{SS'}+
(1-2R)|\Psi_{\bot}\Psi\rangle_{SS'}\right)\times\right.\nonumber\\
& &\left.\left((1+R)\langle\Psi\Psi_{\bot}|_{SS'}+
(1-2R)\langle\Psi_{\bot}\Psi|_{SS'}\right)\right].\nonumber\\
\end{eqnarray}
This state carries both the disturbed original and clone
\begin{eqnarray}\label{final}
\rho_{S}&=&\frac{1}{N(R)}(((2-R)^{2}+(1+R)^{2})|\Psi\rangle\langle\Psi|+\nonumber\\
& &(1-2R)^{2}|\Psi_{\bot}\rangle\langle\Psi_{\bot}|),\nonumber\\
\rho_{S'}&=&\frac{1}{N(R)}(((2-R)^{2}+(1-2R)^{2})|\Psi\rangle\langle\Psi|+\nonumber\\
& &(1+R)^{2}
|\Psi_{\bot}\rangle\langle\Psi_{\bot}|)
\end{eqnarray}
and is conditioned by a simultaneous detection of a single photon in each of the output modes $S,S',I$.
The marginal states of the disturbed original and clone in the selected sub-ensemble have the
following fidelities
\begin{eqnarray}\label{fid}
F_{S}&=&\frac{(2-R)^{2}+(1+R)^{2}}{6(1-R(1-R))},\nonumber\\
F_{S'}&=&\frac{(2-R)^{2}+(1-2R)^{2}}{6(1-R(1-R))},
\end{eqnarray}
with an initial state $|\Psi\rangle$ which vary with increasing $R\in \langle 0,1/2\rangle$
from a perfect SUQCM $(R=0)$ to the trivial non-cloning case $(R=1/2)$, as depicted in Fig.~3.
The output states of the original and clone can be measured by the state analyzers analogically as it was discussed for
the experiments with the SUQCMs. Inserting the fidelities (\ref{fid}) to the cloning inequality (\ref{cond}) which restricts
all the possible AUQCM, the equality is obtained in Eq.~(\ref{cond}) as can be straightforwardly proved.
In this paper an extension of the recent conditional
cloning experiment for a polarization state of photon toward the optimal asymmetrical $1\rightarrow 2$ quantum cloning machine
is proposed.
Our method is based on a conditional partial optimal reverse of the SUQCM controlled by an experimental parameter $R$.
We have applied this method in the recent symmetrical cloning
experiments \cite{Lamas-Linares02,DeMartini02,Ricci03} to obtain the optimal asymmetrical cloning.
In summary, the AUQCM can be described as the projection $\Pi(R)=\Pi_{A}(R)\Pi_{S}$, given explicitly by
\begin{equation}
\Pi(R)=\left((2-R)1_{S'}\otimes 1_{S}-2(1-2R)|\Psi_{-}\rangle_{S'S}\langle\Psi_{-}|\right)\otimes 1_{I},
\end{equation}
on the state $|\Psi\rangle_{S'}\otimes |\Psi_{-}\rangle_{SI}$ composed from an initial unknown
state and the antisymmetric Bell state produced from the spontaneous parametric down-conversion.
Since the fidelities obtained in these experiments
are very close to the optimal value $5/6$ and the proposed modification
is rather simple, this asymmetrical cloning machine is feasible and it could be
straightforwardly realized.
\noindent {\bf Acknowledgments}
The work was supported by the Post-Doc project 202/03/D239 of Grant agency of Czech Republic and
projects LN00A015 and CEZ: J14/98 of the Ministry of Education of the Czech Republic. I would like to
thank Jarom\' ir Fiur\' a\v sek and Petr Marek for the stimulating and fruitful discussions.
\end{document} |
\begin{document}
\title{On the Basis Property of the Root Functions of Some Class of Non-self-adjoint
Sturm--Liouville Operators.}
\author{Cemile Nur\\{\small Depart. of Math., Dogus University, Ac\i badem, Kadik\"{o}y, \ }\\{\small Istanbul, Turkey.}\ {\small e-mail: [email protected]}
\and O. A. Veliev\\{\small Depart. of Math., Dogus University, Ac\i badem, Kadik\"{o}y, \ }\\{\small Istanbul, Turkey.}\ {\small e-mail: [email protected]}}
\date{}
\maketitle
\begin{abstract}
We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of
the Sturm-Liouville operators with some regular boundary conditions. Using
these formulas, we find sufficient conditions on the potential $q$ such that
the root functions of these operators do not form a Riesz basis.
Key Words: Asymptotic formulas, Regular boundary conditions. Riesz basis.
AMS Mathematics Subject Classification: 34L05, 34L20.
\end{abstract}
\section{Introduction and Preliminary Facts}
Let $T_{1},T_{2},T_{3}$ and $T_{4}$ be the operators generated in $L_{2}[0,1]$
by the differential expression
\begin{equation}
l\left( y\right) =-y^{\prime\prime}+q(x)y
\end{equation}
and the following boundary conditions:
\begin{equation}
y_{0}^{\prime}+\beta y_{1}^{\prime}=0,\text{ }y_{0}-y_{1}=0,
\end{equation}
\begin{equation}
y_{0}^{\prime}+\beta y_{1}^{\prime}=0,\text{ }y_{0}+y_{1}=0,
\end{equation}
\begin{equation}
y_{0}^{\prime}-y_{1}^{\prime}=0,\text{ }y_{0}+\alpha y_{1}=0,
\end{equation}
and
\begin{equation}
y_{0}^{\prime}+y_{1}^{\prime}=0,\text{ }y_{0}+\alpha y_{1}=0
\end{equation}
respectively, where $q(x)$ is a complex-valued summable function on $[0,1]$,
$\beta\neq\pm1$ and $\alpha\neq\pm1.$
In conditions (2), (3), (4) and (5) if $\beta=1,$ $\beta=-1,$ $\alpha=1$ and
$\alpha=-1$ respectively, then any $\lambda\in
\mathbb{C}
$ is an eigenvalue of infinite multiplicity. In (2) and (4) if $\beta=-1$ and
$\alpha=-1$ then they are periodic boundary conditions; In (3) and (5) if
$\beta=1$ and $\alpha=1$ then they are antiperiodic boundary conditions.
These boundary conditions are regular but not strongly regular. Note that, if
the boundary conditions are strongly regular, then the root functions form a
Riesz basis (this result was proved independently in [6], [10] and [17]). In
the case when an operator is regular but not strongly regular, the root
functions generally do not form even usual basis. However, Shkalikov [20],
[21] proved that they can be combined in pairs, so that the corresponding
2-dimensional subspaces form a Riesz basis of subspaces.
In the regular but not strongly regular boundary conditions, periodic and
antiperiodic boundary conditions are the ones more commonly studied.
Therefore, let us briefly describe some historical developments related to the
Riesz basis property of the root functions of the periodic and antiperiodic
boundary value problems. First results were obtained by Kerimov and Mamedov
[8]. They established that, if
\[
q\in C^{4}[0,1],\ q(1)\neq q(0),
\]
then the root functions of the operator $L_{0}(q)$ form a Riesz basis in
$L_{2}[0,1],$ where $L_{0}(q)$ denotes the operator generated by (1) and the
periodic boundary conditions.
The first result in terms of the Fourier coefficients of the potential $q$ was
obtained by Dernek and Veliev [1]. They proved that if the conditions
\begin{align}
\lim_{n\rightarrow\infty}\frac{\ln\left\vert n\right\vert }{nq_{2n}} &
=0,\text{ }\\
q_{2n} & \sim q_{-2n}
\end{align}
hold, then the root functions of $L_{0}(q)$ form a Riesz basis in $L_{2}
[0,1]$, where $q_{n}=:(q,e^{i2\pi nx})$ is the Fourier coefficient of $q$ and
everywhere, without loss of generality, it is assumed that $q_{0}=0.$ Here
$(.,.)$ denotes the inner product in $L_{2}[0,1]$ and $a_{n}\sim b_{n}$ means
that $a_{n}=O(b_{n})$ and $b_{n}=O(a_{n})$ as $\ n\rightarrow\infty.$ Makin
[11] improved this result. Using another method he proved that the assertion
on the Riesz basis property remains valid if condition (7) holds, but
condition (6) is replaced by a less restrictive one: $q\in W_{1}^{s}[0,1],$
\[
q^{(k)}(0)=q^{(k)}(1),\quad\forall\,k=0,1,...,s-1
\]
holds and $\mid q_{2n}\mid>cn^{-s-1}$ with some$\ \,c>0$ for sufficiently
large $n,$ where $s$ is a nonnegative integer. Besides, some conditions which
imply the absence of the Riesz basis property were presented in [11].
Shkalilov and Veliev obtained in [22] more general results which cover all
results discussed above.
The other interesting results about periodic and antiperiodic boundary
conditions were obtained in [2-5, 7, 14-16, 24, 25].
The basis properties of regular but not strongly regular other some problems
are studied in [9,12,13]. It was proved in [12] that the system of the root
functions of the operator generated by (1) and the boundary conditions
\begin{align*}
y^{\prime}\left( 1\right) -\left( -1\right) ^{\sigma}y^{\prime}\left(
0\right) +\gamma y\left( 0\right) & =0\\
y\left( 1\right) -\left( -1\right) ^{\sigma}y\left( 0\right) & =0,
\end{align*}
forms an unconditional basis of the space $L_{2}[0,1]$, where $q\left(
x\right) $ is an arbitrary complex-valued function from the class
$L_{1}[0,1]$, $\gamma$ is an arbitrary nonzero complex constant and
$\sigma=0,1$. Kerimov and Kaya proved [9] that the system of the root
functions of the spectral problem
\begin{align*}
y^{\left( 4\right) }+p_{2}\left( x\right) y^{\prime\prime}+p_{1}\left(
x\right) y^{\prime}+p_{0}\left( x\right) y & =\lambda y,\text{ }0<x<1,\\
y^{\left( s\right) }\left( 1\right) -\left( -1\right) ^{\sigma
}y^{\left( s\right) }\left( 0\right) +\sum_{l=0}^{s-1}\alpha
_{s,l}y^{\left( l\right) }\left( 0\right) & =0,\text{ }s=1,2,3,\\
y\left( 1\right) -\left( -1\right) ^{\sigma}y\left( 0\right) & =0,
\end{align*}
forms a basis in the space $L_{p}\left( 0,1\right) $, $1<p<\infty$, when
$\alpha_{3,2}+\alpha_{1,0}\neq\alpha_{2,1}$, $p_{j}\left( x\right) \in
W_{1}^{j}\left( 0,1\right) $, $j=1,2$, and $p_{0}\left( x\right) \in
L_{1}\left( 0,1\right) $; moreover, this basis is unconditional for $p=2$,
where $\lambda$ is a spectral parameter; $p_{j}\left( x\right) \in
L_{1}\left( 0,1\right) $, $j=1,2,3$, are complex-valued functions;
$\alpha_{s,l}$, $s=1,2,3$, $l=\overline{0,s-1}$ are arbitrary complex
constants; and $\sigma=0,1$.
It was shown in [19] that if
\[
q\left( x\right) =q\left( 1-x\right) ,\text{ }\forall x\in\left[
0,1\right] ,
\]
then the spectrum of each of the problems $T_{1}$, and $T_{3}$, coincides with
the spectrum of the periodic problem and the spectrum of each of the problems
$T_{2},$ and $T_{4}$, coincides with the spectrum of the antiperiodic problem.
In this paper we prove that if
\begin{equation}
\lim_{n\rightarrow\infty}\dfrac{\ln\left\vert n\right\vert }{ns_{2n}}=0,
\end{equation}
where $s_{k}=\left( q,\sin2\pi kx\right) ,$ then the large eigenvalues of
the operators $T_{1}$ and $T_{3}$ are simple. Moreover, if there exists a
sequence $\left\{ n_{k}\right\} $ such that (8) holds when $n$ is replaced
by $n_{k},$ then the root functions of these operators do not form a Riesz basis.
Similarly, if
\begin{equation}
\lim_{n\rightarrow\infty}\dfrac{\ln\left\vert n\right\vert }{ns_{2n+1}}=0,
\end{equation}
then the large eigenvalues of the operators $T_{2}$ and $T_{4}$ are simple and
if there exists a sequence $\left\{ n_{k}\right\} $ such that (9) holds when
$n$ is replaced by $n_{k},$ then the root functions of these operators do not
form a Riesz basis.
Moreover we obtain asymptotic formulas of arbitrary order for the eigenvalues
and eigenfunctions of the operators $T_{1}$,$T_{2},T_{3}$ and $T_{4}$.
\section{Main Results}
We will focus only on the operator $T_{1}$. The investigations of the
operators $T_{2},T_{3}$ and $T_{4}$ are similar. It is well-known that ( see
formulas (47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators
$T_{1}(q)$ consist of the sequences $\{\lambda_{n,1}\},\{\lambda_{n,2}\}$
satisfying
\begin{equation}
\lambda_{n,j}=(2n\pi)^{2}+O(n^{1/2})
\end{equation}
for $j=1,2$. From this formula one can easily obtain the following inequality
\begin{equation}
\left\vert \lambda_{n,j}-(2\pi k)^{2}\right\vert =\left\vert 2(n-k)\pi
\right\vert \left\vert 2(n+k)\pi\right\vert +O(n^{\frac{1}{2}})>n
\end{equation}
for $j=1,2;$ $k\neq n;$ $k=0,1,...;$ and $n\geq N,$ where we denote by $N$ a
sufficiently large positive integer, that is, $N\gg1.$
It is easy to verify that if $q(x)=0$ then the eigenvalues of the operator
$T_{1},$ denoted by $T_{1}(0),$ are $\lambda_{n}=\left( 2\pi n\right) ^{2}$
for $n=0,1,\ldots$ The eigenvalue $0$ is simple and the corresponding
eigenfunction is $1.$ The eigenvalues $\lambda_{n}=\left( 2\pi n\right)
^{2}$ for $n=1,2,\ldots$ are double and the corresponding eigenfunctions and
associated functions are
\begin{equation}
y_{n}\left( x\right) =\cos2\pi nx\text{ }\And\text{ }\phi_{n}\left(
x\right) =\left( \frac{\beta}{1+\beta}-x\right) \frac{\sin2\pi nx}{4\pi n}
\end{equation}
respectively. Note that for any constant $c$, $\phi_{n}\left( x\right)
+cy_{n}\left( x\right) $ is also an associated function. It can be shown
that the adjoint operator $T_{1}^{\ast}(0)$ is associated with the boundary
conditions:
\begin{equation}
y_{1}+\overline{\beta}y_{0}=0,\text{ }y_{1}^{\prime}-y_{0}^{\prime}=0.
\end{equation}
It is easy to see that, $0$ is a simple eigenvalue of $T_{1}^{\ast}(0)$ and
the corresponding eigenfunction is $y_{0}^{\ast}\left( x\right) =x-\dfrac
{1}{1+\overline{\beta}}$ . The other eigenvalues $\lambda_{n}^{\ast}=\left(
2\pi n\right) ^{2}$ for $n=1,2,\ldots$, are double and the corresponding
eigenfunctions and associated functions are
\begin{equation}
y_{n}^{\ast}\left( x\right) =\sin2\pi nx\text{ }\And\text{ }\phi_{n}^{\ast
}\left( x\right) =\left( x-\dfrac{1}{1+\overline{\beta}}\right) \frac
{\cos2\pi nx}{4\pi n}\nonumber
\end{equation}
respectively.
Let
\begin{equation}
\varphi_{n}\left( x\right) :=\frac{16\pi n\left( \beta+1\right) }{\beta
-1}\phi_{n}\left( x\right) =\frac{4\left( \beta+1\right) }{\beta-1}\left(
\dfrac{\beta}{1+\beta}-x\right) \sin2\pi nx
\end{equation}
and
\begin{equation}
\varphi_{n}^{\ast}\left( x\right) :=\frac{16\pi n\left( \overline{\beta
}+1\right) }{\overline{\beta}-1}\phi_{n}^{\ast}\left( x\right)
=\frac{4\left( \overline{\beta}+1\right) }{\overline{\beta}-1}\left(
x-\dfrac{1}{1+\overline{\beta}}\right) \cos2\pi nx.
\end{equation}
The system of the root functions of $T_{1}^{\ast}(0)$ can be written as
$\{f_{n}:n\in\mathbb{Z}\},$ where
\begin{equation}
f_{-n}=\sin2\pi nx,\text{ }\forall n>0\And\text{ }f_{n}=\varphi_{n}^{\ast
}\left( x\right) ,\text{ }\forall n\geq0.
\end{equation}
One can easily verify that it forms a basis in $L_{2}[0,1]$ and the
biorthogonal system $\{g_{n}:n\in\mathbb{Z}\}$ is the system of the root
functions of $T_{1}(0),$ where
\begin{equation}
g_{-n}=\varphi_{n}\left( x\right) ,\forall n>0\text{ }\And\text{ }g_{n}
=\cos2\pi nx,\forall n\geq0,
\end{equation}
since $\left( f_{n},g_{m}\right) =\delta_{n,m}.$
To obtain the asymptotic formulas for the eigenvalues $\lambda_{n,j}$ and the
corresponding normalized eigenfunctions $\Psi_{n,j}(x)$ of $T_{1}(q)$ we use
(11) and the well-known relations
\begin{equation}
(\lambda_{N,j}-(2\pi n)^{2})(\Psi_{N,j},\sin2\pi nx)=(q\Psi_{N,j},\sin2\pi nx)
\end{equation}
and
\begin{equation}
\left( \lambda_{N,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi
_{N,j},\varphi_{n}^{\ast}\right) -\gamma_{1}n\left( \Psi_{N,j},\sin2\pi
nx\right) =\left( q\Psi_{N,j},\varphi_{n}^{\ast}\right) ,
\end{equation}
where
\[
\gamma_{1}=\frac{16\pi\left( \beta+1\right) }{\beta-1},
\]
which can be obtained by multiplying both sides of the equality
\[
-\left( \Psi_{N,j}\right) ^{\prime\prime}+q\left( x\right) \Psi
_{N,j}=\lambda_{N,j}\Psi_{N,j}
\]
by $\sin2\pi nx$ and $\varphi_{n}^{\ast}$ respectively. It follows from (18)
and (19) that
\begin{equation}
\left( \Psi_{N,j},\sin2\pi nx\right) =\frac{\left( q\left( x\right)
\Psi_{N,j},\sin2\pi nx\right) }{\lambda_{N,j}-\left( 2\pi n\right) ^{2}
};\text{ }N\neq n,
\end{equation}
\begin{equation}
\left( \Psi_{N,j},\varphi_{n}^{\ast}\right) =\frac{\gamma_{1}n\left(
q\left( x\right) \Psi_{N,j},\sin2\pi nx\right) }{\left( \lambda
_{N,j}-\left( 2\pi n\right) ^{2}\right) ^{2}}+\frac{\left( q\left(
x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right) }{\lambda_{N,j}-\left( 2\pi
n\right) ^{2}};\text{ }N\neq n.
\end{equation}
Moreover, we use the following relations
\begin{align}
\left( \Psi_{N,j},\overline{q}\sin2\pi nx\right) & =\sum_{n_{1}=0}
^{\infty}[\left( q\varphi_{n_{1}},\sin2\pi nx\right) \left( \Psi_{N,j}
,\sin2\pi n_{1}x\right) +\\
& +\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) \left( \Psi_{N,j}
,\varphi_{n_{1}}^{\ast}\right) ],\nonumber
\end{align}
\begin{equation}
\left( \Psi_{N,j},\overline{q}\varphi_{n}^{\ast}\right) =\sum_{n_{1}
=0}^{\infty}\left[ \left( q\varphi_{n_{1}},\varphi_{n}^{\ast}\right)
\left( \Psi_{N,j},\sin2\pi n_{1}x\right) +\left( q\cos2\pi n_{1}
x,\varphi_{n}^{\ast}\right) \left( \Psi_{N,j},\varphi_{n_{1}}^{\ast}\right)
\right] ,
\end{equation}
\begin{align}
\left\vert (q\Psi_{N,j},\sin2\pi nx)\right\vert & <4M,\\
\left\vert (q\Psi_{N,j},\varphi_{n}^{\ast})\right\vert & <4M,
\end{align}
for $N\gg1,$where $M=\sup\left\vert q_{n}\right\vert .$ These relations are
obvious for $q\in L_{2}(0,1),$ since to obtain (22) and (23) we can use the
decomposition of $\overline{q}\sin2\pi nx$ and $\overline{q}\varphi_{n}^{\ast
}$ by basis (16). For $q\in L_{1}(0,1)$\ see Lemma 1 of [23].
To obtain the asymptotic formulas for the eigenvalues and eigenfunctions we
iterate (18) and (19) by using (22), (23). First let us prove the following
obvious asymptotic formulas for the eigenfunctions $\Psi_{n,j}$. The expansion
of $\Psi_{n,j}$\ by basis (17) can be written in the form
\begin{equation}
\Psi_{n,j}=u_{n,j}\varphi_{n}\left( x\right) +v_{n,j}\cos2\pi nx+h_{n,j}
\left( x\right) ,
\end{equation}
where
\begin{equation}
u_{n,j}=\left( \Psi_{n,j},\sin2\pi nx\right) ,\text{ }v_{n,j}=\left(
\Psi_{n,j},\varphi_{n}^{\ast}\right) ,
\end{equation}
\[
h_{n,j}\left( x\right) =\sum_{\substack{k=0\\k\neq n}}^{\infty}\left[
\left( \Psi_{n,j},\sin2\pi kx\right) \varphi_{k}\left( x\right) +\left(
\Psi_{n,j},\varphi_{k}^{\ast}\right) \cos2\pi kx\right] .
\]
Using (20), (21), (24) and (25) one can readily see that, there exists a
constant $C$ such that
\begin{equation}
\sup\left\vert h_{n,j}\left( x\right) \right\vert \leq C\left( \sum_{k\neq
n}\left( \frac{1}{\mid\lambda_{n,j}-\left( 2\pi k\right) ^{2}\mid}+\frac
{n}{\left\vert \left( \lambda_{n,j}-\left( 2\pi k\right) ^{2}\right)
^{2}\right\vert }\right) \right) =O\left( \frac{\ln n}{n}\right)
\end{equation}
and by (26) we get
\begin{equation}
\Psi_{n,j}=u_{n,j}\varphi_{n}\left( x\right) +v_{n,j}\cos2\pi nx+O\left(
\frac{\ln n}{n}\right) .
\end{equation}
Since $\Psi_{n,j}$\ is normalized, we have
\[
1=\left\Vert \Psi_{n,j}\right\Vert ^{2}=\left( \Psi_{n,j},\Psi_{n,j}\right)
=\left\vert u_{n,j}\right\vert ^{2}\left\Vert \varphi_{n}\left( x\right)
\right\Vert ^{2}+\left\vert v_{n,j}\right\vert ^{2}\left\Vert \cos2\pi
nx\right\Vert ^{2}+
\]
\[
+u_{n,j}\overline{v_{n,j}}\left( \varphi_{n}\left( x\right) ,\cos2\pi
nx\right) +v_{n,j}\overline{u_{n,j}}\left( \cos2\pi nx,\varphi_{n}\left(
x\right) \right) +O\left( \frac{\ln n}{n}\right)
\]
\[
=\left( \frac{8}{3}\dfrac{\left\vert \beta\right\vert ^{2}-\operatorname{Re}
\beta+1}{\left\vert \beta-1\right\vert ^{2}}\right) \left\vert u_{n,j}
\right\vert ^{2}+\frac{1}{2}\left\vert v_{n,j}\right\vert ^{2}+O\left(
\frac{\ln n}{n}\right) ,
\]
that is,
\begin{equation}
a\left\vert u_{n,j}\right\vert ^{2}+\frac{1}{2}\left\vert v_{n,j}\right\vert
^{2}=1+O\left( \frac{\ln n}{n}\right) ,
\end{equation}
where
\[
a=\frac{8}{3}\dfrac{\left\vert \beta\right\vert ^{2}-\operatorname{Re}\beta
+1}{\left\vert \beta-1\right\vert ^{2}}.
\]
Note that $a\neq0$, since $\left\vert \beta\right\vert ^{2}+1>\left\vert
\beta\right\vert .$
Now let us iterate (18). Using (22) in (18) we get
\begin{gather*}
\left( \lambda_{n,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi
_{n,j},\sin2\pi nx\right) =\\
=\sum_{n_{1}=0}^{\infty}\left[ \left( q\varphi_{n_{1}},\sin2\pi nx\right)
\left( \Psi_{n,j},\sin2\pi n_{1}x\right) +\left( q\cos2\pi n_{1}x,\sin2\pi
nx\right) \left( \Psi_{n,j},\varphi_{n_{1}}^{\ast}\left( x\right) \right)
\right] .
\end{gather*}
Isolating the terms in the right-hand side of this equality containing the
multiplicands $\left( \Psi_{n,j},\sin2\pi nx\right) $ and $\left(
\Psi_{n,j},\varphi_{n}^{\ast}\left( x\right) \right) $ (i.e., case
$n_{1}=n$ ), using\ (20) and (21) for the terms $\left( \Psi_{n,j},\sin2\pi
n_{1}x\right) $ and \ $\left( \Psi_{n,j},\varphi_{n_{1}}^{\ast}\left(
x\right) \right) $ respectively (in the case $n_{1}\neq n$) we obtain
\begin{gather*}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n}
,\sin2\pi nx\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) -\left(
q\cos2\pi nx,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n}^{\ast}\right)
=\\
=\sum_{\substack{n_{1}=0\\n_{1}\neq n}}^{\infty}\left[ \left( q\varphi
_{n_{1}},\sin2\pi nx\right) \left( \Psi_{n,j},\sin2\pi n_{1}x\right)
+\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) \left( \Psi_{n,j}
,\varphi_{n_{1}}^{\ast}\left( x\right) \right) \right] \\
=\sum_{n_{1}}\left[ a_{1}\left( \lambda_{n,j}\right) \left( q\left(
x\right) \Psi_{n,j},\sin2\pi n_{1}x\right) +b_{1}\left( \lambda
_{n,j}\right) \left( q\left( x\right) \Psi_{n,j},\varphi_{n_{1}}^{\ast
}\right) \right] .
\end{gather*}
where
\begin{align*}
a_{1}\left( \lambda_{n,j}\right) & =\frac{\left( q\varphi_{n_{1}}
,\sin2\pi nx\right) }{\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}}
+\frac{\gamma_{1}n_{1}\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) }{\left(
\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}\right) ^{2}},\\
b_{1}\left( \lambda_{n,j}\right) & =\frac{\left( q\cos2\pi n_{1}
x,\sin2\pi nx\right) }{\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}}.
\end{align*}
Using (22) and (23) for the terms $\left( q\Psi_{n,j},\sin2\pi n_{1}x\right)
$ and $\left( q\Psi_{n,j},\varphi_{n_{1}}^{\ast}\right) $ of the last
summation we obtain
\begin{gather*}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n}
,\sin2\pi nx\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) -\left(
q\cos2\pi nx,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n}^{\ast}\right)
=\\
=\sum_{n_{1}}\left[ a_{1}\left( \lambda_{n,j}\right) \left( q\Psi
_{n,j},\sin2\pi n_{1}x\right) +b_{1}\left( \lambda_{n,j}\right) \left(
q\Psi_{n,j},\varphi_{n_{1}}^{\ast}\right) \right] =\\
=\sum_{n_{1}}a_{1}\left( \sum_{n_{2}=0}^{\infty}\left[ \left(
q\varphi_{n_{2}},\sin2\pi n_{1}x\right) \left( \Psi_{n,j},\sin2\pi
n_{2}x\right) +\left( q\cos2\pi n_{2}x,\sin2\pi n_{1}x\right) \left(
\Psi_{n,j},\varphi_{n_{2}}^{\ast}\left( x\right) \right) \right] \right)
+\\
+\sum_{n_{1}}b_{1}\left( \sum_{n_{2}=0}^{\infty}\left[ \left(
q\varphi_{n_{2}},\varphi_{n_{1}}^{\ast}\right) \left( \Psi_{n,j},\sin2\pi
n_{2}x\right) +\left( q\cos2\pi n_{2}x,\varphi_{n_{1}}^{\ast}\right)
\left( \Psi_{n,j},\varphi_{n_{2}}^{\ast}\left( x\right) \right) \right]
\right) .
\end{gather*}
Now isolating the terms for $n_{2}=n$ we get
\begin{gather*}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n}
,\sin2\pi nx\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) -\left(
q\cos2\pi nx,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n}^{\ast}\right)
=\\
=\sum_{n_{1}}\left[ a_{1}\left( q\varphi_{n},\sin2\pi n_{1}x\right)
+b_{1}\left( q\varphi_{n},\varphi_{n_{1}}^{\ast}\right) \right] \left(
\Psi_{n,j},\sin2\pi nx\right) +\\
+\sum_{n_{1}}\left[ a_{1}\left( q\cos2\pi nx,\sin2\pi n_{1}x\right)
+b_{1}\left( q\cos2\pi nx,\varphi_{n_{1}}^{\ast}\right) \right] \left(
\Psi_{n,j},\varphi_{n}^{\ast}\left( x\right) \right) +\\
=\sum_{n_{1},n_{2}}\left( \left[ a_{1}\left( q\varphi_{n_{2}},\sin2\pi
n_{1}x\right) +b_{1}\left( q\varphi_{n_{2}},\varphi_{n_{1}}^{\ast}\right)
\right] \left( \Psi_{n,j},\sin2\pi n_{2}x\right) +\right) +\\
+\sum_{n_{1},n_{2}}\left[ a_{1}\left( q\cos2\pi n_{2}x,\sin2\pi
n_{1}x\right) +b_{1}\left( q\cos2\pi n_{2}x,\varphi_{n_{1}}^{\ast}\right)
\right] \left( \Psi_{n,j},\varphi_{n_{2}}^{\ast}\right) .
\end{gather*}
Here and further the summations are taken under the conditions $n_{i}\neq n$
and $n_{i}=0,1,...$ for $i=1,2,...$ Introduce the notations
\begin{align*}
C_{1} & =:a_{1},\text{ }M_{1}=:b_{1},\\
C_{2} & =:a_{1}a_{2}+b_{1}A_{2}=C_{1}a_{2}+M_{1}A_{2},\text{ }M_{2}
=:a_{1}b_{2}+b_{1}B_{2}=C_{1}b_{2}+M_{1}B_{2},\\
C_{k+1} & =:C_{k}a_{k+1}+M_{k}A_{k+1},\text{ }M_{k+1}=:C_{k}b_{k+1}
+M_{k}B_{k+1};\text{ }k=2,3,\ldots,
\end{align*}
where
\begin{gather*}
a_{k+1}=a_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\varphi
_{n_{k+1}},\sin2\pi n_{k}x\right) }{\lambda_{n,j}-\left( 2\pi n_{k+1}
\right) ^{2}}+\dfrac{\gamma_{1}n_{k+1}\left( q\cos2\pi n_{k+1}x,\sin2\pi
n_{k}x\right) }{\left( \lambda_{n,j}-\left( 2\pi n_{k+1}\right)
^{2}\right) ^{2}},\\
b_{k+1}=b_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\cos2\pi
n_{k+1}x,\sin2\pi n_{k}x\right) }{\lambda_{n,j}-\left( 2\pi n_{k+1}\right)
^{2}},\\
A_{k+1}=A_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\varphi
_{n_{k+1}},\varphi_{n_{k}}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi
n_{k+1}\right) ^{2}}+\dfrac{\gamma_{1}n_{k+1}\left( q\cos2\pi n_{k+1}
x,\varphi_{n_{k}}^{\ast}\right) }{\left( \lambda_{n,j}-\left( 2\pi
n_{k+1}\right) ^{2}\right) ^{2}},\\
B_{k+1}=B_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\cos2\pi
n_{k+1}x,\varphi_{n_{k}}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi
n_{k+1}\right) ^{2}}.
\end{gather*}
Using these notations and repeating this iteration $k$ times we get
\begin{gather}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n}
,\sin2\pi nx\right) -\widetilde{A}_{k}\left( \lambda_{n,j}\right) \right]
\left( \Psi_{n,j},\sin2\pi nx\right) =\nonumber\\
=\left[ \left( q\cos2\pi nx,\sin2\pi nx\right) +\widetilde{B}_{k}\left(
\lambda_{n,j}\right) \right] \left( \Psi_{n,j},\varphi_{n}^{\ast}\left(
x\right) \right) +R_{k},
\end{gather}
where
\begin{align*}
\widetilde{A}_{k}\left( \lambda_{n,j}\right) & =\sum_{m=1}^{k}\alpha
_{m}\left( \lambda_{n,j}\right) \text{, }\widetilde{B}_{k}\left(
\lambda_{n,j}\right) =\sum_{m=1}^{k}\beta_{m}\left( \lambda_{n,j}\right)
,\\
\alpha_{k}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots,n_{k}}\left[
C_{k}\left( q\varphi_{n},\sin2\pi n_{k}x\right) +M_{k}\left( q\varphi
_{n},\varphi_{n_{k}}^{\ast}\right) \right] ,\\
\beta_{k}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots,n_{k}}\left[
C_{k}\left( q\cos2\pi nx,\sin2\pi n_{k}x\right) +M_{k}\left( q\cos2\pi
nx,\varphi_{n_{k}}^{\ast}\right) \right] ,\\
R_{k} & =\sum_{n_{1},\ldots,n_{k+1}}\left\{ C_{k+1}\left( q\Psi_{n,j}
,\sin2\pi n_{k+1}x\right) +M_{k+1}\left( q\Psi_{n,j},\varphi_{n_{k+1}}
^{\ast}\right) \right\} .
\end{align*}
It follows from (11), (24) and (25) that
\begin{equation}
\alpha_{k}\left( \lambda_{n,j}\right) =O\left( \left( \frac{\ln\left\vert
n\right\vert }{n}\right) ^{k}\right) ,\beta_{k}\left( \lambda_{n,j}\right)
=O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right) ^{k}\right)
,R_{k}=O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right)
^{k+1}\right) .
\end{equation}
Therefore if we take limit in (31) for $k\rightarrow\infty$, we obtain
\[
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-Q_{n}-A\left( \lambda
_{n,j}\right) \right] u_{n,j}=\left[ P_{n}+B\left( \lambda_{n,j}\right)
\right] v_{n,j},
\]
where
\begin{equation}
P_{n}=\left( q\cos2\pi nx,\sin2\pi nx\right) ,\text{ }Q_{n}=\left(
q\varphi_{n},\sin2\pi nx\right) ,
\end{equation}
\begin{equation}
A\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\alpha_{m}\left(
\lambda_{n,j}\right) =O\left( \frac{\ln\left\vert n\right\vert }{n}\right)
\text{, }B\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\beta_{m}\left(
\lambda_{n,j}\right) =O\left( \frac{\ln\left\vert n\right\vert }{n}\right)
.
\end{equation}
Thus iterating (18) we obtained (31). Now starting \ to iteration from (19)
instead of (18) and using (23), (22) and arguing as in the previous iteration,
we get
\begin{equation}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A_{k}^{\prime
}\left( \lambda_{n,j}\right) \right] v_{n,j}=\left[ \gamma_{1}
n+Q_{n}^{\ast}+B_{k}^{\prime}\left( \lambda_{n,j}\right) \right]
u_{n,j}+R_{k}^{\prime},
\end{equation}
where
\begin{equation}
P_{n}^{\ast}=\left( q\cos2\pi nx,\varphi_{n}^{\ast}\right) ,\text{ }
Q_{n}^{\ast}=\left( q\varphi_{n},\varphi_{n}^{\ast}\right) ,
\end{equation}
\begin{align*}
A_{k}^{\prime}\left( \lambda_{n,j}\right) & =\sum_{m=1}^{k}\alpha
_{m}^{\prime}\left( \lambda_{n,j}\right) \text{, }B_{k}^{\prime}\left(
\lambda_{n,j}\right) =\sum_{m=1}^{k}\beta_{m}^{\prime}\left( \lambda
_{n,j}\right) ,\\
\alpha_{k}^{\prime}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots
,n_{k}}\left[ \widetilde{C}_{k}\left( q\cos2\pi nx,\sin2\pi n_{k}x\right)
+\widetilde{M}_{k}\left( q\cos2\pi nx,\varphi_{n_{k}}^{\ast}\right) \right]
,\\
\beta_{k}^{\prime}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots,n_{k}
}\left[ \widetilde{C}_{k}\left( q\varphi_{n},\sin2\pi n_{k}x\right)
+\widetilde{M}_{k}\left( q\varphi_{n},\varphi_{n_{k}}^{\ast}\right) \right]
,\\
R_{k}^{\prime} & =\sum_{n_{1},\ldots,n_{k+1}}\left\{ \widetilde{C}
_{k+1}\left( q\Psi_{n,j},\sin2\pi n_{k+1}x\right) +\widetilde{M}
_{k+1}\left( q\Psi_{n,j},\varphi_{n_{k+1}}^{\ast}\right) \right\} ,
\end{align*}
\[
\widetilde{C}_{k+1}=\widetilde{C}_{k}a_{k+1}+\widetilde{M}_{k}A_{k+1},\text{
}\widetilde{M}_{k+1}=\widetilde{C}_{k}b_{k+1}+\widetilde{M}_{k}B_{k+1};\text{
}k=0,1,2,\ldots,
\]
\begin{align*}
\widetilde{C}_{1} & =A_{1}\left( \lambda_{n,j}\right) =\frac{\left(
q\varphi_{n_{1}},\varphi_{n}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi
n_{1}\right) ^{2}}+\frac{\gamma_{1}n_{1}\left( q\cos2\pi n_{1}x,\varphi
_{n}^{\ast}\right) }{\left( \lambda_{n,j}-\left( 2\pi n_{1}\right)
^{2}\right) ^{2}},\\
\widetilde{M}_{1} & =B_{1}\left( \lambda_{n,j}\right) =\frac{\left(
q\cos2\pi n_{1}x,\varphi_{n}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi
n_{1}\right) ^{2}}.
\end{align*}
Similar to (32) one can verify that
\begin{equation}
\alpha_{k}^{\prime}\left( \lambda_{n,j}\right) =O\left( \left( \frac
{\ln\left\vert n\right\vert }{n}\right) ^{k}\right) ,\beta_{k}^{\prime
}\left( \lambda_{n,j}\right) =O\left( \left( \frac{\ln\left\vert
n\right\vert }{n}\right) ^{k}\right) ,R_{k}^{\prime}=O\left( \left(
\frac{\ln\left\vert n\right\vert }{n}\right) ^{k+1}\right) .
\end{equation}
If we take limit in (35) for $k\rightarrow\infty$, we obtain
\[
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime
}\left( \lambda_{n,j}\right) \right] v_{n,j}=\left[ \gamma_{1}
n+Q_{n}^{\ast}+B^{\prime}\left( \lambda_{n,j}\right) \right] u_{n,j},
\]
where
\begin{equation}
A^{\prime}\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\alpha_{m}
^{\prime}\left( \lambda_{n,j}\right) =O\left( \frac{\ln\left\vert
n\right\vert }{n}\right) \text{, }B^{\prime}\left( \lambda_{n,j}\right)
=\sum_{m=1}^{\infty}\beta_{m}^{\prime}\left( \lambda_{n,j}\right) =O\left(
\frac{\ln\left\vert n\right\vert }{n}\right) .
\end{equation}
To get the main results of this paper we use the following system of
equations, obtained above, with respect to $u_{n,j}$ and $v_{n,j}$
\begin{gather}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-Q_{n}-A\left( \lambda
_{n,j}\right) \right] u_{n,j}=\left[ P_{n}+B\left( \lambda_{n,j}\right)
\right] v_{n,j},\\
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime
}\left( \lambda_{n,j}\right) \right] v_{n,j}=\left[ \gamma_{1}
n+Q_{n}^{\ast}+B^{\prime}\left( \lambda_{n,j}\right) \right] u_{n,j},
\end{gather}
where
\begin{gather}
Q_{n}=\left( q\varphi_{n},\sin2\pi nx\right) =\nonumber\\
=-\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right)
dx+\frac{2\left( \beta+1\right) }{\beta-1}\left( xq\left( x\right)
,\cos4\pi nx\right) -\frac{2\beta}{\beta-1}\left( q\left( x\right)
,\cos4\pi nx\right) \\
=-\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right)
dx+o\left( 1\right) ,
\end{gather}
\begin{gather}
P_{n}^{\ast}=\left( q\cos2\pi nx,\varphi_{n}^{\ast}\right) =\nonumber\\
=\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right)
dx+\frac{2\left( \beta+1\right) }{\beta-1}\left( xq\left( x\right)
,\cos4\pi nx\right) -\frac{2}{\beta-1}\left( q\left( x\right) ,\cos4\pi
nx\right) \\
=\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right)
dx+o\left( 1\right) ,
\end{gather}
\begin{equation}
P_{n}=\left( q\cos2\pi nx,\sin2\pi nx\right) =\frac{1}{2}\left( q,\sin4\pi
nx\right) =o\left( 1\right) ,
\end{equation}
\begin{equation}
Q_{n}^{\ast}=\left( q\varphi_{n},\varphi_{n}^{\ast}\right) =8\left(
\frac{\beta_{1}+1}{\beta_{1}-1}\right) ^{2}\int_{0}^{1}q\left( x\right)
\left( \dfrac{\beta_{1}}{1+\beta_{1}}-x\right) \left( x-\dfrac{1}
{1+\beta_{1}}\right) \sin4\pi nxdx=o\left( 1\right) .
\end{equation}
Note that (39), (40) with (34), (38) give us
\begin{gather}
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-Q_{n}+O\left( \dfrac
{\ln\left\vert n\right\vert }{n}\right) \right] u_{n,j}=\left[
P_{n}+O\left( \dfrac{\ln\left\vert n\right\vert }{n}\right) \right]
v_{n,j},\\
\left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}+O\left(
\dfrac{\ln\left\vert n\right\vert }{n}\right) \right] v_{n,j}=\left[
\gamma_{1}n+Q_{n}^{\ast}+O\left( \dfrac{\ln\left\vert n\right\vert }
{n}\right) \right] u_{n,j}.
\end{gather}
Introduce the notations
\begin{align}
c_{n} & =\left( q,\cos2\pi nx\right) \text{, }s_{n}=\left( q,\sin2\pi
nx\right) \nonumber\\
c_{n,1} & =\left( xq,\cos2\pi nx\right) \text{, }s_{n,1}=\left(
xq,\sin2\pi nx\right) \\
c_{n,2} & =\left( x^{2}q,\cos2\pi nx\right) \text{, }s_{n,2}=\left(
x^{2}q,\sin2\pi nx\right) .\nonumber
\end{align}
In these notations we have
\begin{equation}
Q_{n}=-\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left(
x\right) dx+\frac{2\left( \beta+1\right) }{\beta-1}c_{2n,1}-\frac{2\beta
}{\beta-1}c_{2n}
\end{equation}
\begin{equation}
P_{n}^{\ast}=\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left(
x\right) dx+\frac{2\left( \beta+1\right) }{\beta-1}c_{2n,1}-\frac{2}
{\beta-1}c_{2n}
\end{equation}
\begin{equation}
P_{n}=\frac{1}{2}s_{2n}
\end{equation}
\begin{equation}
Q_{n}^{\ast}=-8\left( \frac{\beta+1}{\beta-1}\right) ^{2}s_{2n,2}+8\left(
\frac{\beta+1}{\beta-1}\right) ^{2}s_{2n,1}-\frac{8\beta}{\left(
\beta-1\right) ^{2}}s_{2n}.
\end{equation}
\begin{theorem}
For $j=1,2$ the following statements hold:
$(a)$ Any eigenfunction $\Psi_{n,j}$ of $T_{1}$ corresponding to the
eigenvalue $\lambda_{n,j}$ defined in (10) satisfies
\begin{equation}
\Psi_{n,j}=\sqrt{2}\cos2\pi nx+O\left( n^{-1/2}\right) .
\end{equation}
Moreover there exists $N$ such that for all $n>N$ the geometric multiplicity
of the eigenvalue $\lambda_{n,j}$ is $1$.
$\left( b\right) $ A complex number $\lambda\in U(n)=:\{\lambda:\left\vert
\lambda-\left( 2\pi n\right) ^{2}\right\vert \leq n\}$ is an eigenvalue of
$T_{1}$ if and only if it is a root of the equation
\begin{gather}
\left[ \lambda-\left( 2\pi n\right) ^{2}-Q_{n}-A\left( \lambda\right)
\right] \left[ \lambda-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime
}\left( \lambda\right) \right] -\nonumber\\
-\left[ P_{n}+B\left( \lambda\right) \right] \left[ \gamma_{1}
n+Q_{n}^{\ast}+B^{\prime}\left( \lambda\right) \right] =0.
\end{gather}
Moreover $\lambda\in U(n)$ is a double eigenvalue of $T_{1}$ if and only if
\textit{it is a double root of} (55) .
\end{theorem}
\begin{proof}
$\left( a\right) $ By (10) the left hand side of (48) is $O(n^{1/2}),$ which
implies that $u_{n,j}=O(n^{-1/2}).$ Therefore from (29) we obtain (54). Now
suppose that there are two linearly independent eigenfunctions corresponding
to $\lambda_{n,j}$. Then there exists an eigenfunction satisfying
\[
\Psi_{n,j}=\sqrt{2}\sin2\pi nx+o\left( 1\right)
\]
which contradicts (54).
$(b)$ First we prove that the large eigenvalues $\lambda_{n,j}$ are the roots
of the equation (55). It follows from (54), (27) and (15) that $v_{n,j}\neq0.$
If $u_{n,j}\neq0$ then multiplying the equations (39) and (40) side by side
and then canceling $v_{n,j}u_{n,j}$ we obtain (55) . If $u_{n,j}=0$ then by
(39) and (40) we have $P_{n}+B\left( \lambda_{n,j}\right) =0$ \ and
$\lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime}\left(
\lambda_{n,j}\right) =0$ which mean that (55) holds. Thus in any case
$\lambda_{n,j}$ is a root of (55).
Now we prove that the roots of (55) lying in $U(n)$\ are the eigenvalues of
$T_{1}.$ Let $F(\lambda)$ be the left-hand side of (55) which can be written
as
\begin{gather}
F(\lambda)=(\lambda-\left( 2\pi n\right) ^{2})^{2}-\left( Q_{n}+A\left(
\lambda\right) +P_{n}^{\ast}+A^{\prime}\left( \lambda\right) \right)
\left( \lambda-\left( 2\pi n\right) ^{2}\right) +\\
+\left( Q_{n}+A\left( \lambda\right) \right) \left( P_{n}^{\ast
}+A^{\prime}\left( \lambda\right) \right) -\left( P_{n}+B\left(
\lambda\right) \right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime}\left(
\lambda\right) \right) \nonumber
\end{gather}
and
\[
G(\lambda)=(\lambda-\left( 2\pi n\right) ^{2})^{2}.
\]
One can easily verify that the inequality
\begin{equation}
\mid F(\lambda)-G(\lambda)\mid<\mid G(\lambda)\mid
\end{equation}
holds\ for all $\lambda$ from the boundary of $U(n).$ Since the function
$G(\lambda)$ has two roots in the set $U(n),$ by the Rouche's theorem we
obtain that $F(\lambda)$ has two roots in the same\ set.\ Thus\ $T_{1}$ has
two eigenvalues (counting with multiplicities) lying in $U(n)$ that are the
roots of (55). On the other hand, (55) has preciously two roots (counting with
multiplicities) in $U(n).$ Therefore $\lambda\in U(n)$ is an eigenvalue of
$T_{1}$ if and only if (55) holds.
If \textit{ }$\lambda\in U(n)$ is a double eigenvalue of $T_{1}$ then it has
no other eigenvalues\textit{ }in\textit{ }$U(n)$ and hence (55) has no other
roots. This implies that $\lambda$ is a double root of (55). By the same way
one can prove that if $\lambda$ is a double root of (55) then it is a double
eigenvalue of $T_{1}.$
\end{proof}
Let us consider (55) in detail. If we substitute $t=:\lambda-\left( 2\pi
n\right) ^{2}$ then it becomes
\begin{gather}
t^{2}-\left( Q_{n}+A\left( \lambda\right) +P_{n}^{\ast}+A^{\prime}\left(
\lambda\right) \right) t+\\
+\left( Q_{n}+A\left( \lambda\right) \right) \left( P_{n}^{\ast
}+A^{\prime}\left( \lambda\right) \right) -\left( P_{n}+B\left(
\lambda\right) \right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime}\left(
\lambda\right) \right) =0.\nonumber
\end{gather}
The solutions of this equation are
\[
t_{1,2}=\frac{\left( Q_{n}+P_{n}^{\ast}+A+A^{\prime}\right) \pm\sqrt
{\Delta\left( \lambda\right) }}{2},
\]
where
\begin{equation}
\Delta\left( \lambda\right) =\left( Q_{n}+P_{n}^{\ast}+A+A^{\prime}\right)
^{2}-4\left( Q_{n}+A\right) \left( P_{n}^{\ast}+A^{\prime}\right)
+4\left( P_{n}+B\right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime}\right)
\nonumber
\end{equation}
which can be written in the form
\begin{equation}
\Delta\left( \lambda\right) =\left( Q_{n}-P_{n}^{\ast}+A-A^{\prime}\right)
^{2}+4\left( P_{n}+B\right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime
}\right) .
\end{equation}
Clearly the eigenvalue $\lambda_{n,j}$\ is a root either of the equation
\begin{equation}
\lambda=\left( 2\pi n\right) ^{2}+\frac{1}{2}\left[ \left( Q_{n}
+P_{n}^{\ast}+A+A^{\prime}\right) -\sqrt{\Delta\left( \lambda\right)
}\right]
\end{equation}
or of the equation
\begin{equation}
\lambda=\left( 2\pi n\right) ^{2}+\frac{1}{2}\left[ \left( Q_{n}
+P_{n}^{\ast}+A+A^{\prime}\right) +\sqrt{\Delta\left( \lambda\right)
}\right] .
\end{equation}
Now let us examine $\Delta\left( \lambda_{n,j}\right) $ in detail. If (8)
holds then one can readily see from (34), (38), (50), (51) and (59) that
\begin{equation}
\Delta\left( \lambda_{n,j}\right) =2\gamma_{1}ns_{2n}(1+o(1)).
\end{equation}
Taking into account the last three equality and (34), (38), (50), (51), we see
that (60) and (61) have the form
\begin{equation}
\lambda=\left( 2\pi n\right) ^{2}-\frac{\sqrt{2\gamma_{1}}}{2}\sqrt{ns_{2n}
}(1+o(1)),
\end{equation}
\begin{equation}
\lambda=\left( 2\pi n\right) ^{2}+\frac{\sqrt{2\gamma_{1}}}{2}\sqrt{ns_{2n}
}(1+o(1)).
\end{equation}
\begin{theorem}
If (8) holds, then the large eigenvalues $\lambda_{n,j}$ are simple and
satisfy the following asymptotic formulas
\begin{equation}
\lambda_{n,j}=\left( 2\pi n\right) ^{2}+\left( -1\right) ^{j}\frac
{\sqrt{2\gamma_{1}}}{2}\sqrt{ns_{2n}}(1+o(1)).
\end{equation}
for $j=1,2.$ Moreover, if there exists a sequence $\left\{ n_{k}\right\} $
such that (8) holds when $n$ is replaced by $n_{k},$ then the root functions
of $T_{1}$ do not form a Riesz basis.
\end{theorem}
\begin{proof}
To prove that the large eigenvalues $\lambda_{n,j}$ are simple let us show
that one of the eigenvalues, say $\lambda_{n,1}$ satisfies (65) for $j=1$ and
the other $\lambda_{n,2}$ satisfies (65) for $j=2.$ Let us prove that each of
the equations (60) and (61) has a unique root in $U(n)$ by proving that
\[
\left( 2\pi n\right) ^{2}+\frac{1}{2}\left[ \left( Q_{n}+P_{n}^{\ast
}+A+A^{\prime}\right) \pm\sqrt{\Delta\left( \lambda\right) }\right]
\]
is a contraction mapping. For this we show that there exist positive real
numbers $K_{1},K_{2},K_{3}$ such that
\begin{equation}
\mid A\left( \lambda\right) -A(\mu)\mid<K_{1}\mid\lambda-\mu\mid,\text{
}\mid A^{\prime}(\lambda)-A^{\prime}(\mu)\mid<K_{2}\mid\lambda-\mu\mid,
\end{equation}
\begin{equation}
\left\vert \sqrt{\Delta\left( \lambda\right) }-\sqrt{\Delta\left(
\mu\right) }\right\vert <K_{3}\mid\lambda-\mu\mid,
\end{equation}
where $K_{1}+K_{2}+K_{3}<1$. The proof of (66) is similar to the proof of (56)
of the paper [26].
Now let us prove (67). By (62) and (8) we have
\[
\left( \sqrt{\Delta\left( \lambda\right) }\right) ^{-1}=o(1).
\]
On the other hand arguing as in the proof of (56) of the paper [26] we get
\[
\dfrac{d}{d\lambda}\Delta\left( \lambda\right) =O(1).
\]
Hence in any case we have
\[
\frac{d}{d\lambda}\sqrt{\Delta\left( \lambda\right) }=\frac{\dfrac
{d}{d\lambda}\Delta\left( \lambda\right) }{2\sqrt{\Delta\left(
\lambda\right) }}=o(1).
\]
Thus by the fixed point theorem, each of the equations (60) and (61) has a
unique root $\lambda_{1}$ and $\lambda_{2}$ respectively. Clearly by (63) and
(64), we have $\lambda_{1}\neq\lambda_{2}$ which implies that the equation
(55) has two simple root in $U\left( n\right) .$ Therefore by Theorem 1(b),
$\lambda_{1}$ and $\lambda_{2}$ are the eigenvalues of $T_{1}$ lying in
$U\left( n\right) ,$ that is, they are $\lambda_{n,1}$ and $\lambda_{n,2}$,
which proves the simplicity of the large eigenvalues and the validity of (65).
If there exists a sequence $\left\{ n_{k}\right\} $ such that (8) holds when
$n$ is replaced by $n_{k}$, then by Theorem 1(a)
\[
\left( \Psi_{n_{k},1},\Psi_{n_{k},2}\right) =1+O\left( n_{k}^{-1/2}\right)
.
\]
Now it follows from the theorems of [20,21] (see also Lemma 3 of [24]) that
the root functions of $T_{1}$ do not form a Riesz basis.
\end{proof}
Now let us consider the operators $T_{2}$, $T_{3}$ and $T_{4}.$ First we
consider the operator $T_{3}$.
It is well-known that ( see formulas (47a), (47b)) in page 65 of [18] ) the
eigenvalues of the operators $T_{3}(q)$ consist of the sequences
$\{\lambda_{n,1,3}\},\{\lambda_{n,2,3}\}$ satisfying (10) when $\lambda_{n,j}$
is replaced by $\lambda_{n,j,3}.$ The eigenvalues, eigenfunctions and
associated functions of $T_{3}$ are
\begin{align*}
\lambda_{n} & =\left( 2\pi n\right) ^{2};\text{ }n=0,1,2,\ldots\\
y_{0}\left( x\right) & =x-\dfrac{\alpha}{1+\alpha},\text{ }y_{n}\left(
x\right) =\sin2\pi nx;\text{ }n=1,2,\ldots\\
\phi_{n}\left( x\right) & =\left( x-\dfrac{\alpha}{1+\alpha}\right)
\frac{\cos2\pi nx}{4\pi n};\text{ }n=1,2,\ldots.
\end{align*}
respectively. The biorthogonal systems analogous to (16), (17) are
\begin{equation}
\left\{ \cos2\pi nx,\frac{4\left( 1+\overline{\alpha}\right) }
{1-\overline{\alpha}}\left( \dfrac{1}{1+\overline{\alpha}}-x\right) \sin2\pi
nx\right\} _{n=0}^{\infty}
\end{equation}
\begin{equation}
\left\{ \sin2\pi nx,\frac{4\left( 1+\alpha\right) }{1-\alpha}\left(
x-\dfrac{\alpha}{1+\alpha}\right) \cos2\pi nx\right\} _{n=0}^{\infty}
\end{equation}
respectively.
Analogous formulas to (18) and (19) are
\begin{equation}
\left( \lambda_{N,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi
_{N,j},\cos2\pi nx\right) =\left( q\left( x\right) \Psi_{N,j},\cos2\pi
nx\right)
\end{equation}
\begin{equation}
\left( \lambda_{N,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi
_{N,j},\varphi_{n}^{\ast}\right) -\gamma_{3}n\left( \Psi_{N,j},\cos2\pi
nx\right) =\left( q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right)
\end{equation}
respectively, where
\[
\gamma_{3}=\frac{16\pi\left( 1+\alpha\right) }{1-\alpha}.
\]
Instead of (16)-(19) using (68)-(71) and arguing as in the proofs of Theorem 1
and Theorem 2 we obtain the following results for $T_{3}.$
\begin{theorem}
If (8) holds, then the large eigenvalues $\lambda_{n,j,3}$ are simple and
satisfy the following asymptotic formulas
\begin{equation}
\lambda_{n,j,3}=\left( 2\pi n\right) ^{2}+\left( -1\right) ^{j}\frac
{\sqrt{2\gamma_{3}}}{2}\sqrt{ns_{2n}}(1+o(1)).
\end{equation}
for $j=1,2.$ The eigenfunctions $\Psi_{n,j,3}$ corresponding to $\lambda
_{n,j,3}$ obey
\begin{equation}
\Psi_{n,j,3}=\sqrt{2}\sin2\pi nx+O\left( n^{-1/2}\right) .
\end{equation}
Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (8)
holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{3}$ do
not form a Riesz basis.
\end{theorem}
Now let us consider the operator $T_{2}$. It is well-known that ( see formulas
(47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators $T_{2}(q)$
consist of the sequences $\{\lambda_{n,1,2}\},\{\lambda_{n,2,2}\}$ satisfying
\begin{equation}
\lambda_{n,j,2}=(2n\pi+\pi)^{2}+O(n^{1/2}),
\end{equation}
for $j=1,2$. The eigenvalues, eigenfunctions and associated functions of
$T_{2}$ are
\begin{align*}
\lambda_{n} & =\left( \pi+2\pi n\right) ^{2},\text{ }y_{n}\left( x\right)
=\cos\left( 2n+1\right) \pi x,\\
\phi_{n}\left( x\right) & =\left( \frac{\beta}{\beta-1}-x\right)
\frac{\sin\left( 2n+1\right) \pi x}{2\left( 2n+1\right) \pi}
\end{align*}
for $n=0,1,2,\ldots$respectively. The biorthogonal systems analogous to (16),
(17) are
\begin{equation}
\left\{ \sin\left( 2n+1\right) \pi x,\frac{4\left( \overline{\beta
}-1\right) }{\overline{\beta}+1}\left( x+\dfrac{1}{\overline{\beta}
-1}\right) \cos\left( 2n+1\right) \pi x\right\} _{n=0}^{\infty}
\end{equation}
\begin{equation}
\left\{ \cos\left( 2n+1\right) \pi x,\frac{4\left( \beta-1\right) }
{\beta+1}\left( \frac{\beta}{\beta-1}-x\right) \sin\left( 2n+1\right) \pi
x\right\} _{n=0}^{\infty}
\end{equation}
respectively.
Analogous formulas to (18) and (19) are
\begin{equation}
\left( \lambda_{N,j}-\left( \left( 2n+1\right) \pi\right) ^{2}\right)
\left( \Psi_{N,j},\sin\left( 2n+1\right) \pi x\right) =\left( q\left(
x\right) \Psi_{N,j},\sin\left( 2n+1\right) \pi x\right)
\end{equation}
\begin{equation}
\left( \lambda_{N,j}-\left( \left( 2n+1\right) \pi\right) ^{2}\right)
\left( \Psi_{N,j},\varphi_{n}^{\ast}\right) -\left( 2n+1\right) \gamma
_{2}\left( \Psi_{N,j},\sin\left( 2n+1\right) \pi x\right) =\left(
q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right)
\end{equation}
respectively, where
\[
\gamma_{2}=\frac{8\pi\left( \beta-1\right) }{\beta+1}.
\]
Instead of (16)-(19) using (75)-(78) and arguing as in the proofs of Theorem 1
and Theorem 2 we obtain the following results for $T_{2}.$
\begin{theorem}
If (9) holds, then the large eigenvalues $\lambda_{n,j,2}$ are simple and
satisfy the following asymptotic formulas
\begin{equation}
\lambda_{n,j,2}=\left( \left( 2n+1\right) \pi\right) ^{2}+\left(
-1\right) ^{j}\frac{\sqrt{2\gamma_{2}}}{2}\sqrt{\left( 2n+1\right)
s_{2n+1}}(1+o(1)).
\end{equation}
for $j=1,2.$ The eigenfunctions $\Psi_{n,j,2}$ corresponding to $\lambda
_{n,j,2}$ obey
\begin{equation}
\Psi_{n,j,2}=\sqrt{2}\cos\left( 2n+1\right) \pi x+O\left( n^{-1/2}\right)
.
\end{equation}
Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (9)
holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{2}$ do
not form a Riesz basis.
\end{theorem}
Lastly we consider the operator $T_{4}$. It is well-known that ( see formulas
(47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators $T_{4}(q)$
consist of the sequences $\{\lambda_{n,1,4}\},\{\lambda_{n,2,4}\}$ satisfying
(74) when $\lambda_{n,j,2}$ is replaced by $\lambda_{n,j,4}.$ The eigenvalues,
eigenfunctions and associated functions of $T_{4}$ are
\begin{align*}
\lambda_{n} & =\left( \pi+2\pi n\right) ^{2},\text{ }y_{n}\left( x\right)
=\sin\left( 2n+1\right) \pi x,\\
\phi_{n}\left( x\right) & =\left( \frac{\alpha}{1-\alpha}+x\right)
\frac{\cos\left( 2n+1\right) \pi x}{2\left( 2n+1\right) \pi}
\end{align*}
for $n=0,1,2,\ldots$respectively. The biorthogonal systems analogous to (16),
(17) are
\begin{equation}
\left\{ \cos\left( 2n+1\right) \pi x,\frac{4\left( 1-\overline{\alpha
}\right) }{1+\overline{\alpha}}\left( \dfrac{1}{1-\overline{\alpha}
}-x\right) \sin\left( 2n+1\right) \pi x\right\} _{n=0}^{\infty}
\end{equation}
\begin{equation}
\left\{ \sin\left( 2n+1\right) \pi x,\frac{4\left( 1-\alpha\right)
}{1+\alpha}\left( \dfrac{\alpha}{1-\alpha}+x\right) \cos\left( 2n+1\right)
\pi x\right\} _{n=0}^{\infty}
\end{equation}
respectively.
Analogous formulas to (18) and (19) are
\begin{equation}
\left( \lambda_{N,j}-\left( \pi+2\pi n\right) ^{2}\right) \left(
\Psi_{N,j},\cos\left( 2n+1\right) \pi x\right) =\left( q\left( x\right)
\Psi_{N,j},\cos\left( 2n+1\right) \pi x\right) ,
\end{equation}
\begin{equation}
\left( \lambda_{N,j}-\left( \left( 2n+1\right) \pi\right) ^{2}\right)
\left( \Psi_{N,j},\varphi_{n}^{\ast}\right) -\left( 2n+1\right) \gamma
_{4}\left( \Psi_{N,j},\cos\left( 2n+1\right) \pi x\right) =\left(
q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right)
\end{equation}
respectively, where
\[
\gamma_{4}=\frac{8\pi\left( 1-\alpha\right) }{1+\alpha}.
\]
Instead of (16)-(19) using (81)-(84) and arguing as in the proofs of Theorem 1
and Theorem 2 we obtain the following results for $T_{4}.$
\begin{theorem}
If (9) holds, then the large eigenvalues $\lambda_{n,j,4}$ are simple and
satisfy the following asymptotic formulas
\begin{equation}
\lambda_{n,j,4}=\left( \left( 2n+1\right) \pi\right) ^{2}+\left(
-1\right) ^{j}\frac{\sqrt{2\gamma_{4}}}{2}\sqrt{\left( 2n+1\right)
s_{2n+1}}(1+o(1)).
\end{equation}
for $j=1,2.$ The eigenfunctions $\Psi_{n,j,4}$ corresponding to $\lambda
_{n,j,4}$ obey
\begin{equation}
\Psi_{n,j,4}=\sqrt{2}\sin\left( 2n+1\right) \pi x+O\left( n^{-1/2}\right)
.
\end{equation}
Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (9)
holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{4}$ do
not form a Riesz basis.
\end{theorem}
\begin{remark}
Suppose that
\begin{equation}
\int_{0}^{1}xq\left( x\right) dx\neq0.
\end{equation}
If
\begin{equation}
\frac{1}{2}s_{2n}+B=o\left( \frac{1}{n}\right) ,
\end{equation}
where $B$ is defined by (34), then arguing as in the proof of Theorem 2, we
obtain that the large eigenvalues of the operator $T_{1}$ are simple. Moreover
if there exists a sequence $\left\{ n_{k}\right\} $ such that (88) holds
when $n$ is replaced by $n_{k},$ then the root functions of $T_{1}$ do not
form a Riesz basis. The similar results can be obtained for the operators
$T_{2},T_{3}$ and $T_{4}.$
\end{remark}
\begin{remark}
Using (31) and (35) and arguing as in the proof of Theorem 3 of [1] it can be
obtained asymptotic formulas of arbitrary order for the eigenvalues and
eigenfunctions of the operator $T_{1}.$ The similar formulas can be obtained
for the operators $T_{2},T_{3}$ and $T_{4}.$
\end{remark}
\end{document} |
\begin{document}
\begin{abstract}
We develop the notion of Lagrangian distribution on scattering manifolds, meaning on the compactified cotangent bundle, which is a manifold with corners equipped with a scattering symplectic structure. In particular, we study the notion of principal symbol of the arising class of distributions.
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
\label{sec:intro}
In this article we develop a theory of Lagrangian distributions on asymptotically Euclidean manifolds.
Lagrangian distributions were defined by H\"ormander \cite{HormanderFIO} as a tool to obtain a global calculus of Fourier integral operators.
The latter are widely applied, e.g. in the study of partial differential equations \cite{DH}, spectral theory \cite{DG}, index theory \cite{BaeStro} and mathematical physics \cite{GuSt}.
Motivating examples for the necessity of studying Lagrangian distributions on asymptotically Euclidean spaces include fundamental solutions to the Klein-Gordon equation, which exhibit Lagrangian behavior ``at infinity'', see \cite{CoSc2}, as well as simple or multi-layers which arise when solving partial differential equations along infinite boundaries or Cauchy hypersurfaces, see \cite{Cordes}.
In local coordinates, a classical Lagrangian distribution $u$ on a manifold $X$ is given by an oscillatory integral of the form
\begin{equation}
\label{eq:osciintproto}
I_\varphi(a)=\int_{{\mathbb{R}}s} e^{i\varphi}a(x,\theta)\,\dd\theta,
\end{equation}
for some symbol $a\in S^m({\mathbb{R}}d\times{\mathbb{R}}s)$ and a phase function $\varphi$ on a subset of ${\mathbb{R}}^d\times{\mathbb{R}}^s$ bounded in $x$. A class of oscillatory integrals on Euclidean spaces, the local model for our theory, was studied in \cite{CoSc}.
The key feature of the theory of Lagrangian distributions is that each such distribution is globally associated to a Lagrangian submanifold $\Lambda\subset T^*X$ and that its leading order behavior can be invariantly described by its principal symbol which is a section in a line bundle on $\Lambda$.
In this article, we prove that the situation on asymptotically Euclidean manifolds is similar, but with a more delicate structure ``at infinity''. To make this precise, we work within the framework of scattering geometry, developed in \cite{Melrose1,MZ}, see also \cite{HV,WZ}. In the article, we provide an extensive introduction to this theory and add to it a class of naturally arising morphisms, the \emph{scattering maps}. We note that the scattering manifolds may also be seen as Lie manifolds, and in this way our theory complements recent advances in the theory of Lagrangian distributions and Fourier integral operators on such singular spaces (via groupoid techniques), see \cite{Lescure}.
The prototype of a scattering geometry is the Euclidean space ${\mathbb{R}}d$, identified with a ball under radial compactification. For this setting, a fitting theory of Lagrangian submanifolds on ${\mathbb{R}}d$ was developed in \cite{CoSc2}. As a first step, we adapt this to general scattering manifolds with boundary $X=X^o\cup \partial X$, the boundary being viewed as infinity. On such manifolds, the environment for microlocalization is then the compactified scattering cotangent bundle ${}^\scat \,\overline{T}^*X$, a manifold with corners of codimension $2$ and its boundary $\mathcal{W}=\partial{}^\scat \,\overline{T}^*X$.
This boundary may be seen as a stratified space, and the two boundary faces of ${}^\scat \,\overline{T}^*X$, which intersect in the corner, inherit a type of contact structure.
The geometric objects of study in our theory are then Legendrian submanifolds of the faces $\mathcal{W}$ which intersect in the corner and are the boundary of some Lagrangian submanifold in the interior and smooth (distribution) densities thereon.
The link with Lagrangian distributions is now as follows. We prove that, despite the singular geometry, any Lagrangian submanifold $\Lambda \subset \mathcal{W}$ locally admits a parametrization through some phase function $\varphi$, via a generalization of the map
\[{\lambda_\varphi}:\mathcal{C}_\varphi\rightarrow\Lambda_\varphi\quad (x,\theta)\mapsto \big(x,\dd_x\varphi(x,\theta)\big),\]
where $\mathcal{C}_\varphi=(\dd_\theta\varphi)^{-1}\{0\}$.
For each such a phase function, a Lagrangian distribution can be expressed locally as an oscillatory integral as in \eqref{eq:osciintproto}.
Up to Maslov factors and some density identifications, the restriction of $a(x,\theta)$ to $C_\varphi$ yields the (principal) symbol $\sigma(u)$ of $u$ and is interpreted as a (density valued) function on $\Lambda$ by identification via ${\lambda_\varphi}$.
Indeed, the main theorem characterizing the principal symbol will be:
\begin{thm*}
Let $\Lambda$ be a ${\mathrm{sc}}$-Lagrangian on $X$. Then there exists a surjective principal symbol map
\begin{equation*}
j^\Lambda_{m_e,m_\psi}\colon I^{m_e,m_\psi}(X,\Lambda) \to {\mathscr{C}^\infty}(\Lambda, M_\Lambda\otimes\Omega^{1/2}),
\end{equation*}
where $M_\Lambda$ is the Maslov bundle and $\Omega^{1/2}$ denotes the half-density bundle over $\Lambda$.
Moreover, its null space is $I^{m_e-1,m_\psi-1}(X,\Lambda)$ and
we have the short exact sequence
\[
0 \longrightarrow
I^{m_e-1,m_\psi-1}(X,\Lambda) \longrightarrow
I^{m_e,m_\psi}(X,\Lambda) \xrightarrow{j^\Lambda_{m_e,m_\psi}}
{\mathscr{C}^\infty}(\Lambda,M_\Lambda\otimes\Omega^{1/2})
\longrightarrow 0.
\]
Equivalently,
\[I^{m_e,m_\psi}(X,\Lambda) / I^{m_e-1,m_\psi-1}(X,\Lambda) \simeq {\mathscr{C}^\infty}(\Lambda, M_\Lambda\otimes\Omega^{1/2}).\]
\end{thm*}
Summarizing, our results show that the theory of Lagrangian distributions, classically studied either locally or on compact manifolds, may be generalized to a theory of Lagrangian distributions on Euclidean spaces or manifolds with boundaries, hence a much wider class of geometries. It is formulated in a way that makes it easily transferable to other singular geometries as well as manifolds with corners, see \cite{Melrosemwc}.
The paper is organized as follows.
In Section \ref{sec:prelim} we give an introduction to scattering geometry.
In particular, we discuss the natural class of maps between scattering manifolds, compactification and scattering amplitudes.
In Section \ref{sec:phaseandlag} we define the Lagrangian submanifolds and phase functions that arise in our theory.
In Section \ref{sec:exchphase} we discuss the techniques of classifying phase functions which parametrize the same Lagrangian submanifold.
In Section \ref{sec:Lagdist} we define the Lagrangian distributions in this setting, starting from oscillatory integrals, and study their transformation properties.
Finally, in Section \ref{sec:symb}, we define the principal symbol of Lagrangian distributions and prove its invariance.
\section{Preliminary definitions}\label{sec:prelim}
In the following, we will recall some elements of the geometric theory known as ``scattering geometry'', cf. \cite{Melrose1,Melrose2,MZ,WZ}. To start with, we need to recall some groundwork on the analysis on manifolds with corners, for which we adopt the definition of \cite{MelroseAPS,Melrosemwc}, cf. also \cite{MO} and \cite{Joyce} for a discussion on the different notions of manifolds with corners in the literature.
\subsection{Manifolds with corners and scattering geometry}
\label{sec:scat}
We recall the following extrinsic definition of a (smooth) manifold with (embedded) corners.
\subsubsection*{Manifolds with corners and ${\mathscr{C}^\infty}$-functions}
Let $X$ be a paracompact Hausdorff space. As in the case of manifolds without boundary, a manifold with corners is defined in terms of local charts. A $d$-dimensional chart with corners (of codimension $k$) on $X$ is a pair $(U,\phi)$, where $U$ is an open subset of $[0,\infty)^k\times {\mathbb{R}}^{d-k}$ for some $0\leq k \leq d$, and $\phi : U \to \phi(U)\subset X$ is a homeomorphism. If $k = 1$ we call $(U,\phi)$ a chart with boundary.
As usual, we define compatibility between charts and an atlas of charts and therefore obtain a definition of manifolds with boundary and manifolds with corner (abbreviated mwb and mwc, respectively, in the following).
For every manifold with corners $X$ of dimension $d$ there exists a $d$-dimen\-sional ${\mathscr{C}^\infty}$-manifold $\wt{X}$ without boundary
with $X\subset\tilde{X}$, and the interior $X^o$ of $X$ is open in $\wt{X}$ and non-empty when $d>0$.
We denote by ${\mathscr{C}^\infty}(X)$ the space of the restrictions of the elements of ${\mathscr{C}^\infty}(\wt{X})$ to $X$. The tangent space $TX$ and differentials of maps $f:X\rightarrow Y$, $Tf:TX\rightarrow TY$, between manifolds with corners $X, Y$, are obtained as restrictions of the corresponding objects on $\wt{X}$ and $\wt{Y}$.
We always assume $X$ to be compact and assume that there is a finite collection of ${\mathscr{C}^\infty}$-functions on $\tilde{X}$, $\{\rho_i\}_{i\in I}$, called boundary defining functions (abbreviated bdf), such that $X=\bigcap_{i\in I}\{p\in \tilde{X}, \rho_i(p)\geq 0\}$, and at every point where $\rho_j=0$ for every $j\in J\subset I$, the differentials of these $\rho_j$ are supposed to be linearly independent. In particular, $\dd\rho_j\neq 0$ when $\rho_j=0$. We also always assume to be working in local coordinates of the form
$\mathbf{x}\colon p\mapsto (\rho_1,\dots,\rho_k,x_1,\dots,x_{d-k})(p)$, where $k$ is the number of boundary defining functions\footnote{Note that the $\rho_j$ cannot always be chosen as coordinates at interior points, since their differential may vanish in the interior. As it is customary, we disregard this minor technical inconvenience in order to allow for an easier consistent notation and think of the $\rho$ to be replaced by any other admissible coordinate function there.}.
\begin{rem}\label{rem:joycedef}
Joyce calls this notion a (compact) \emph{manifold with embedded corners} (cf. Remark 2.11 in \cite{Joyce}).
By Proposition 2.15 in \cite{Joyce}, we see that, locally, a boundary defining function always exists, and the property that all corners are embedded ensures that a global boundary defining function exists.
Most of the times the actual choice of boundary defining function is not relevant (cf. Proposition 2.15).
\end{rem}
Let $p\in X$. Then the depth of $p$, $\mathrm{depth}(p)$, is the number of independent boundary defining functions vanishing at $p$, which coincides with the co-dimension of the boundary stratum in which $p$ is contained. We recall that for $j\in\{0,\dots,d\}$ one sets
$\partial_jX=\{p\in X\,|\,\mathrm{depth}(p)= j\}.$
In particular, $X^o=\partial_0 X$ and $\partial X=\bigcup_{j>0} \partial_j X$. We note that as such, the boundary of a mwc is not a mwc itself, but rather a topological manifold. Nevertheless, it is possible to define smooth functions on $\partial X$ as the set of restrictions smooth functions on $X$ to $\partial X$.
Given a relatively open subset $U$ of a manifold with corner $X$, we say that $U$ is \emph{interior} if $\overline{U}\cap\partial X=\emptyset$. Otherwise, we always assume that $U$ contains all interior points of the boundary $\overline{U}\cap \partial X$ and call $U$ a \emph{boundary neighbourhood}.
We will write $f\in{\mathscr{C}^\infty}(U)$ if and only if there is an extension $\wt{f}\in{\mathscr{C}^\infty}(X)$ that coincides with $f$ on $U$. The space $\rho_1^{-m_1}\cdots\rho_k^{-m_k}{\mathscr{C}^\infty}(U)$ is the space of functions $h\in{\mathscr{C}^\infty}(U^o)$ such that $\rho_1^{m_1}\cdots\rho_k^{m_k}h$ extends to an element of ${\mathscr{C}^\infty}(U)$.
The class of mwc that interest us is that of (products of) fiber bundles where both the base as well as the fiber are allowed to be a compact manifold with boundary (abbreviated ``mwb''). The archetype of such a mwc is the product of two mwbs. Indeed, if $X$ and $Y$ are mwbs, $B=X\times Y$ is a mwc. We write $\mathcal{B}=\partial B$ and we have (adopting the notation of \cite{CoSc2,ES})
\begin{align*}
\mathcal{B}&=\underbrace{(\partial X\times Y^o) \cup (X^o\times \partial Y)}_{=\partial_1 B}\cup \underbrace{(\partial X\times \partial Y)}_{=\partial_2 B}=: \mathcal{B}e \cup \mathcal{B}p\cup \mathcal{B}pe.
\end{align*}
We now describe the basics of scattering geometry, cf. \cite{Melrose1,Melrose2,MZ,WZ}. We first recall the guiding example.
\begin{defn}[Radial compactification of ${\mathbb{R}}d$]
Pick any diffeomorphism $\iota:{\mathbb{R}}d\rightarrow ({\mathbb{B}}d)^o$ that, for $|x|>3$, is given by
\[\iota:x\mapsto \frac{x}{|x|}\left(1-\frac{1}{|x|}\right).\]
Then its inverse is given, for $|y|\geq \frac{2}{3}$, by
\[\iota^{-1}:y\mapsto \frac{y}{|y|}(1-|y|)^{-1}.\]
The map $\iota$ is called the \emph{radial compactification map}. We may hence view ${\mathbb{R}}d$ as the interior of the mwb ${\mathbb{B}}d$ and call $\partial {\mathbb{B}}d$ ``infinity''.
Denote by $\bdf{x}$ a smooth function ${\mathbb{R}}d\rightarrow (0,\infty)$ that, for $|x|>3$, is given by $x\mapsto |x|$. Then $(\iota^{-1})^*\bdf{x}^{-1}$ is a boundary defining function on ${\mathbb{B}}d$ (and we view $\bdf{x}^{-1}$ as a boundary defining function on ${\mathbb{R}}d$). Indeed, for $|y|>2/3$ it is given by
$y\mapsto 1-|y|=\rho_Y$.
\end{defn}
\begin{rem}
\label{rem:compequiv}
In scattering geometry, the explicit choice of compactification of ${\mathbb{R}}d$ often differs from ours, see \cite{MZ}. Write $\jap{x}=\sqrt{1+|x|^2}$ for $x\in{\mathbb{R}}d$ and define
\[x\mapsto \left(\frac{1}{\jap{x}},\frac{x}{\jap{x}}\right)=:\left(\wt{\rho_Y},\wt{y}\right).\]
This maps ${\mathbb{R}}d$ into the interior of the half-sphere with positive first component, and $\wt{\rho_Y}$ and $d-1$ of the $\wt{y} = \wt{\rho_Y}\cdot x$ functions may be chosen as local coordinates. Because of the following computation, both compactifications are equivalent, meaning they yield diffeomorphic manifolds. In fact, for $|x|>3$, we may write
\[\jap{x}^{-1}=\bdf{x}^{-1}\frac{1}{1+\bdf{x}^{-2}}, \qquad \bdf{x}^{-1}=\jap{x}^{-1}\frac{1}{\sqrt{1-\jap{x}^{-2}}}.\]
Hence, $\jap{x}^{-1}$ and $\bdf{x}^{-1}$ yield equivalent boundary defining functions on ${\mathbb{R}}d$.
\end{rem}
\begin{defn}[Scattering vector fields on mwbs]
Let $X$ be a mwb with bdf $\rho$. Consider the space $ {}^{b}\mathcal{V}(X)$ of vector fields tangential to $\partial X$.
Then ${}^\scat \,\mathcal{V}(X)$ is the space $\rho\, {}^{b}\mathcal{V}(X)$.
Near any point with $\rho=0$, the vector fields
$\{\rho^2\partial_\rho,\ \rho\partial_{x_j}\}$
generate ${}^\scat \,\mathcal{V}(X)$. In particular, ${}^\scat \,\mathcal{V}(X)$ contains vector fields supported in $X^o$.
By the Serre-Swan theorem, there exists a ${\mathscr{C}^\infty}$-vector bundle ${}^\scat \,T X$ such that ${}^\scat \,\mathcal{V}(X)$ are its ${\mathscr{C}^\infty}$-sections.
We have a natural inclusion map ${}^\scat \,T X\hookrightarrow TX$. Note that $\{\rho^2\partial_\rho,\ \rho\partial_{x_j}\}$ are, as elements of ${}^\scat \,T_pX$, non-vanishing at boundary points $p\in \partial X$ despite $\rho=0$.\\
The inclusion reverses for the dual bundles $ T^*X\hookrightarrow {}^\scat \,T^*X$. In coordinates, we denote the dual elements to $\{\rho^2\partial_\rho,\ \rho\partial_{x_j}\}$ by $\left\{\frac{\dd\rho}{\rho^2},\frac{\dd x_j}{\rho}\right\}$, and these span the sections of ${}^\scat \,T^*X$ near the boundary.
We now consider the the \textit{compactified scattering cotangent bundle} ${}^\scat \,\overline{T}^*X$, which is the fiber-wise radial compactification of ${}^\scat \,T^*X$, a compact manifold with corners.
The new-formed fiber boundary may be identified with a rescaling of the cosphere bundle, called ${}^\scat \,S^*X$.
The boundary of the new-formed mwc $W={}^\scat \,\overline{T}^*X$, which we denote\footnote{This is a slight change of notation compared to \cite{Melrose1} where it is denoted $C_{\mathrm{sc}}$.} by $\mathcal{W}$, splits into three components: the boundary faces
$$\Wt^e:={}^\scat \,T^*_{\partial X}X,\qquad \Wt^\psi:={}^\scat \,S^*_{X^o}X,\qquad \Wt^\psie:={}^\scat \,S^*_{\partial X}X.$$
This geometric situation (with $X$ identified as the zero section) near the boundary is summarised in Figure \ref{fig:Wt} (cf. \cite{CoSc2,MZ}).
\begin{figure}
\caption{The boundary faces and corner of ${}
\label{fig:Wt}
\end{figure}
The exterior derivative $\dd$ lifts to a well-defined scattering differential ${}^\scat \dd$ on the scattering geometric structure.
In coordinates, with $\rho$ a local boundary defining function, we write
\begin{align}
\label{def:scddef}
{}^\scat \dd f=\rho^2\partial_\rho f\,\frac{\dd\rho}{\rho^2}+\sum_{j=1}^{d-1}\rho\partial_{x_j} f\,\frac{\dd x_j}{\rho}.
\end{align}
Note that for $f\in {\mathscr{C}^\infty}(X)$, this means that as a section of ${}^\scat \,T^*X$, ${}^\scat \dd f$ necessarily vanishes on the boundary. In fact, we may extend ${}^\scat \dd$ to the space $\rho^{-1}{\mathscr{C}^\infty}(X)$ and obtain a map
$${}^\scat \dd:\rho^{-1}{\mathscr{C}^\infty}(X)\longrightarrow {}^\scat\, \varTheta(X)=\Gamma({}^\scat \,T^*X).$$
That is, in local coordinates near the boundary,
$${}^\scat \dd(\rho^{-1}f) = \rho^{-1}\, {}^\scat \dd f - f \frac{\dd\rho}{\rho^2}=(-f+\rho\partial_\rho f)\,\frac{\dd\rho}{\rho^2}+\sum_{j=1}^{d-1}\partial_{x_j} f\,\frac{\dd x_j}{\rho}.$$
\end{defn}
\begin{rem}
We note that $\rho^{-1}{\mathscr{C}^\infty}(X)$ and similarly defined spaces are independent of the actual choice of boundary defining function $\rho$ (cf. Remark \ref{rem:joycedef}).
\end{rem}
\begin{ex}
Outside a compact neighbourhood of the origin, polar coordinates provide an isomorphism ${\mathbb{R}}d\cong{\mathbb{R}}_+\times {\mathbb{S}}d$. The vector fields $\partial_r$ and $\frac{1}{r}\partial_{x_j}$, $x_j$ being coordinates on ${\mathbb{S}}d$, correspond (up to a sign) under radial inversion $\rho=\frac{1}{r}$ to $\rho^2\partial_\rho$ and $\rho\partial_{x_j}$. Hence, scattering vector fields on ${\mathbb{B}}d$ arise as the image of the vector fields of bounded length on ${\mathbb{R}}d$ under radial compactification.
\end{ex}
\begin{defn}
A \emph{scattering manifold} (also called asymptotically Euclidean manifold) is a compact manifold with boundary $(X, \rho)$ whose interior is equipped with a Riemannian metric $g$ that is supposed to take the form, in a tubular neighbourhood of the boundary,
$$g=\frac{{(\dd \rho)}^{\otimes 2}}{\rho^4}+\frac{g_\partial}{\rho^2}$$
where $g_\partial\in{\mathscr{C}^\infty}(X,\mathrm{Sym}^2 T^*X)$ restricts to a metric on $\partial X$.
\end{defn}
Any mwb may be equipped with a scattering metric.
\begin{ex}
In polar coordinates, the metric on ${\mathbb{R}}d\setminus\{0\}$ can be written as
$$g=(\dd r)^{\otimes 2}+r^2 g_{{\mathbb{S}}d}.$$
Pulled back to ${\mathbb{B}}d$ using $\iota$, that is $r=(1-|y|)^{-1}=\rho^{-1}$ near the boundary, this becomes
$$g_{{\mathbb{B}}d}=\frac{(\dd \rho)^{\otimes 2}}{\rho^4}+\frac{g_{{\mathbb{S}}d}}{\rho^2}.$$
\end{ex}
\begin{defn}[Scattering vector fields on product type manifolds]
For a product $B=X\times Y$, with $(X,\rho_X)$ and $(Y,\rho_Y)$ mwbs, we may introduce ${}^\scat \,\mathcal{V}(B)$ as $\rho_X\rho_Y ( {}^{b}\mathcal{V}(B))$.
Near a corner point the resulting bundle ${}^\scat \,T^*B$ is hence generated, if
$\mathbf{x}=(\rho_X,x)$ and $\mathbf{y}=(\rho_Y,y)$ are local coordinates on $X$ and $Y$ respectively, by
$$\rho_X^2\rho_Y\partial_{\rho_X},\ \rho_X\rho_Y\partial_{x_j},\ \rho_X\rho_Y^2\partial_{\rho_Y},\ \rho_X\rho_Y\partial_{y_k}.$$
The space ${}^\scat \,\mathcal{V}(B)$ splits into horizontal and vertical vector fields\footnote{Consider the projection ${\mathrm{pr}}_X:B\rightarrow X$. Then $v\in{}^\scat \,\mathcal{V}(B)$ satisfies $v\in{}^\scat \,\mathcal{V}X(B)$ if $v({\mathrm{pr}}_X^*f)=0$ for all $f\in{\mathscr{C}^\infty}(X)$. The set ${}^\scat \,\mathcal{V}Y(B)$ is defined in analogy.},
${}^\scat \,\mathcal{V}X(B)$ and ${}^\scat \,\mathcal{V}Y(B)$, respectively, and we define
${}^\scat\, \varTheta^X(B)$ as the set of (scattering) 1-forms $w \in {}^\scat\, \varTheta^1(B)$ such that
$w(v) = 0$ for all $v \in {}^\scat \,\mathcal{V}Y(B)$.
Given complete set of coordinates $\mathbf{x}=(\rho_X, x)$,
$\mathbf{y}=(\rho_Y, y)$ on $X$ and $Y$, respectively,
we see that ${}^\scat\, \varTheta^X(B)$ is the set of sections generated by
\[\frac{\dd\rho_X}{\rho_X^2\rho_Y}, \frac{\dd x_j}{\rho_X\rho_Y}.\]
The underlying vector bundle will be denoted by ${}^\scat H^X B$.
Similarly, we define ${}^\scat\, \varTheta^Y(B)$ and ${}^\scat H^Y B$.
It is important to note that we have the following ``rescaling identifications'':
\begin{equation}
\label{eq:rescal}
\begin{aligned}
{}^\scat\, \varTheta^X(B)\ni \frac{\dd \rho_X}{\rho_X^2 \rho_Y}&\,\longleftrightarrow\, \rho_Y^{-1}\frac{\dd\rho_X}{\rho_X^2}\in \rho_Y^{-1}{\mathscr{C}^\infty}(Y,{}^\scat\, \varTheta(X)),
\\
{}^\scat\, \varTheta^X(B)\ni \frac{\dd x_j}{\rho_X\rho_Y}&\,\longleftrightarrow\, \rho_Y^{-1}\frac{\dd x_j}{\rho_X}\in \rho_Y^{-1}{\mathscr{C}^\infty}(Y,{}^\scat\, \varTheta(X)).
\end{aligned}
\end{equation}
Again, we may define the scattering exterior differential
${}^\scat \dd$, induced by the usual exterior differential $\dd$, and extend it to a map
\[{}^\scat \dd : \rho_X^{-1}\rho_Y^{-1}{\mathscr{C}^\infty}(B)\longrightarrow {}^\scat\, \varTheta(B).\]
In terms of the scattering differentials on $X$ and $Y$ we may decompose ${}^\scat \dd$ as ${}^\scat \dd={}^\scat \ddX+{}^\scat \ddY$, where
\begin{align*}
{}^\scat \ddX : \rho_X^{-1}\rho_Y^{-1} {\mathscr{C}^\infty}(B) \to {}^\scat\, \varTheta^X(B),\\
{}^\scat \ddY : \rho_X^{-1}\rho_Y^{-1} {\mathscr{C}^\infty}(B) \to {}^\scat\, \varTheta^Y(B).
\end{align*}
\end{defn}
\subsection{Amplitudes}
\begin{defn}[Amplitudes of product-type]
Let $B$ be a mwc, $\{\rho_j\}_{j=1\dots k}$ a complete set of bdfs. Then $a$ is called an amplitude of order $m\in{\mathbb{R}}^{k}$ if
$$a\in \rho_1^{-m_1} \cdots \rho_k^{-m_k}{\mathscr{C}^\infty}(B).$$
For an open subset $U$ of $X$, a \emph{locally defined} amplitude of product type is an element of $\rho_1^{-m_1}\cdots\rho_k^{-m_k}{\mathscr{C}^\infty}(\overline{U})$.
For $p\in\partial X$ we call $a$ \emph{elliptic at $p$} if $\rho^{m_1}_1\cdots\rho_k^{m_k}a(p)\neq 0$.
We write
$${\dot{\mathscr{C}}^\infty}z(X):=\bigcap_{m\in{\mathbb{R}}^k}\rho_1^{-m_1}\cdots\rho_k^{-m_k}{\mathscr{C}^\infty}(B)$$
for the smooth functions vanishing at the boundary of infinite order.
For $p\in\partial B$ we call $a$ \emph{rapidly decaying at $p$} if there exists a neighbourhood $U$ of $p$ such that $a$ vanishes of infinite order on $U\cap \partial B$, that is $a \in {\dot{\mathscr{C}}^\infty}z(\overline U)$.
\end{defn}
We now study the leading boundary behavior of these amplitudes. For simplicity, we only consider $B=X\times Y$ for mwbs $X$ and $Y$.
\begin{defn}
\label{def:princsymbol}
Let $a \in \rho_X^{-m_e}\rho_Y^{-m_\psi} {\mathscr{C}^\infty}(B)$ and write $a = \rho_X^{-m_e}\rho_Y^{-m_\psi} f$ for some $f \in {\mathscr{C}^\infty}(B)$.
Given a coordinate neighbourhood $U$ of a point $p\in\mathcal{B}^\bullet$, we define symbols $\sigma^\bullet(a)$ of $a$ on $U$ by
\begin{align*}
\begin{cases}
\sigma^e(a)(\mathbf{x},\mathbf{y})=\rho_X^{-m_e}\rho_Y^{-m_\psi}f(0,x,\mathbf{y}), &p\in\mathcal{B}^e\cup\mathcal{B}^{\psi e} \\
\sigma^\psi(a)(\mathbf{x},\mathbf{y})=\rho_X^{-m_e}\rho_Y^{-m_\psi}f(\mathbf{x},0,y), &p\in\mathcal{B}^\psi\cup\mathcal{B}^{\psi e}\\
\sigma^{\psi e}(a)(\mathbf{x},\mathbf{y})=\rho_X^{-m_e}\rho_Y^{-m_\psi}f(0,x,0,y) &p\in\mathcal{B}^{\psi e}.
\end{cases}
\end{align*}
The tuple $(\sigma^\psi(a),\sigma^e(a), \sigma^{\psi e}(a))$ is denoted by $\sigma(a)$ and called the \emph{principal symbol}.
\end{defn}
Fix $\varepsilonilon>0$ so small that $\rho_X$ and $\rho_Y$ can be chosen as coordinates on $B$ respectively whenever $\rho_X<\varepsilonilon$ and $\rho_Y<\varepsilonilon$.
We choose a cut-off function $\chi \in {\mathscr{C}^\infty}({\mathbb{R}})$ such that $\chi(t) = 0$ for $t > \varepsilonilon/2$ and $\chi(t) = 1$ for $t < \varepsilonilon/4$.
\begin{defn}\label{def:princpart}
For any $a \in \rho_X^{-m_e}\rho_Y^{-m_\psi} {\mathscr{C}^\infty}(B)$ the amplitude
\begin{align*}
a_p(\mathbf{x},\mathbf{y}) = \chi(\rho_X) \sigma^e(a)(\mathbf{x},\mathbf{y}) + \chi(\rho_Y) \sigma^\psi(a)(\mathbf{x},\mathbf{y}) - \chi(\rho_X)\chi(\rho_Y) \sigma^{\psi e}(a)(\mathbf{x},\mathbf{y})
\end{align*}
is called the \emph{principal part} of $a$.
\end{defn}
While $a_p$ does depend on the choice of $\chi$, its leading boundary asymptotic do not. By Taylor expansion of $f$, we obtain:
\begin{lem}
\label{lem:princpart}
The principal part $a_p$ of $a$ satisfies $ a - a_p \in \rho_X^{-m_e+1}\rho_Y^{-m_\psi+1} {\mathscr{C}^\infty}(B).$
\end{lem}
\begin{ex}[Classical ${\mathrm{SG}}$-symbols]
Let $B={\mathbb{B}}d\times{\mathbb{B}}s$, where ${\mathbb{B}}d$ and ${\mathbb{B}}s$ are the radial compactifications of ${\mathbb{R}}d$ and ${\mathbb{R}}s$.
The space of so-called classical ${\mathrm{SG}}$-symbols, ${\mathrm{SG}}cl^{m_e,m_\psi}({\mathbb{R}}d\times{\mathbb{R}}s)$, is that of $a\in{\mathscr{C}^\infty}({\mathbb{R}}d\times{\mathbb{R}}s)$ such that
$(\iota^{-1}\times\iota^{-1})^*a\in \rho_X^{-m_e} \rho_Y^{-m_\psi}{\mathscr{C}^\infty}(B)$. These symbols are then precisely those that satisfy the estimates
\begin{equation}
\label{eq:SGest}
\left|\partial_x^{\alpha}\partial_\theta^{\beta} a(x,\theta)\right|\lesssim \jap{x}^{m_e-|\alpha|}\jap{\theta}^{m_\psi-|\beta|}
\end{equation}
and admit a polyhomogeneous expansion, see \cite{ES,Melrose1,WZ} and the principal symbol of $a$ corresponds to its homogeneous coefficients, see \cite[Chap. 8.2]{ES}.
\end{ex}
We will need to consider density-valued amplitudes and integrate amplitudes on mwbs. For this, we introduce the space of scattering $\sigma$-density bundles, cf. \cite{Melrose1}, where ${}^\scat\, \Omega^{\sigma}(X)=\rho^{-\sigma(d+1)}\Omega^\sigma(X)$ in terms of the usual $\sigma$-density bundle. Note that ${}^\scat\, \Omega^{\sigma}$ does not depend on the choice of boundary defining function.
\begin{ex}\label{ex:mugdensity}
Under the radial compactification, the canonical Lebesgue integration density on ${\mathbb{R}}^d$, $\dd x \in \Omega^1({\mathbb{R}}^d)$, is mapped to $\iota_*\dd x \in {}^\scat\, \Omega^1({\mathbb{B}}^d)$.
In particular, we obtain $\iota_*\dd x = \rho^{-(d+1)}\dd\rho\,\dd{\mathbb{S}}d$. More generally, if $(X,g)$ is a scattering manifold, then the metric induces a canonical volume scattering 1-density $\mu_g$.
\end{ex}
Since the density bundle is a line bundle, any choice of scattering density provides a section of it and allows for an identification of scattering densities on $X$ and ${\mathscr{C}^\infty}$-functions.
We denote the set of all smooth sections of the bundle ${}^\scat\, \Omega^\sigma(X)$ by ${\mathscr{C}^\infty}(X,{}^\scat\, \Omega^\sigma(X))$, and
the tempered distribution densities $({\dot{\mathscr{C}}^\infty}z)'(X, {}^\scat\, \Omega^\sigma(X))$ are the continuous linear functionals on ${\dot{\mathscr{C}}^\infty}z(X, {}^\scat\, \Omega^{1-\sigma}(X))$.
\begin{lem}\label{lem:intdensity}
Let $X$ be a mwb and $Y$ a manifold without boundary. Then, integration over $Y$ induces a map
\begin{align*}
\int_Y : {\mathscr{C}^\infty}c(X\times Y, {}^\scat\, \Omega^1(X\times Y)) \longrightarrow \rho_X^{-\dim Y} {\mathscr{C}^\infty}c(X,{}^\scat\, \Omega^1(X)).
\end{align*}
\end{lem}
\begin{rem}\label{rem:pushforward}
More generally, let $X,Y$ be mwbs and $Z$ a manifold without boundary.
Consider a differentiable fibration $f : X \to Y$ with typical fiber $Z$.
For every scattering density $\mu \in {\mathscr{C}^\infty}(X,{}^\scat\, \Omega^1(X))$ the pushforward
\[f_* \mu \in \rho_Y^{-\dim Z} {\mathscr{C}^\infty}c(Y, {}^\scat\, \Omega^1(Y))\]
is defined locally by integration along the fiber.
Let $(U, \psi)$ be a trivializing neighborhood of the fiber bundle,
that is $U \subset Y$ open, $\psi : X \to U \times Z$ smooth and $f|_{f^{-1}(U)} = {\mathrm{pr}}_M \circ \psi$.
Assume without loss of generality that $\mu$ is supported on $f^{-1}(U)$.
Then set
\[f_* \mu = \int_Z \mu\circ\psi_j.\]
\end{rem}
\subsection{Scattering maps}
We now introduce and characterize the class of maps whose pull-backs preserve amplitudes of product type. They are a special case of interior $b$-maps in the sense of \cite{MelroseAPS}, and humbly mimicking Melrose's naming conventions we call them ${\mathrm{sc}}$-maps. We first introduce them on manifolds with boundary and then generalize to manifolds with higher corner degeneracy, such as products of mwcs.
\begin{defn}[${\mathrm{sc}}$-maps on mwb]
Let $Y$ and $Z$ be mwbs. Suppose $\Psi:Y\rightarrow Z$. Then $\Psi$ is called an ${\mathrm{sc}}$-map if for any $m\in{\mathbb{R}}$ and $a\in \rho_Z^{-m}{\mathscr{C}^\infty}(Z)$ it holds that:
\begin{enumerate}
\item $\Psi^*a\in \rho_Y^{-m}{\mathscr{C}^\infty}(Y)$;
\item if $p\in \Psi(Y)$ with $p=\Psi(q)$ and $(\rho_Z^{m} a)(p)> 0$, then $(\rho_Y^{m} \Psi^*a)(q)> 0$.
\end{enumerate}
\end{defn}
\begin{rem}\label{rem:inward}
In particular, $\Psi$ maps the boundary of $Y$ into that of $Z$.
It also follows that $T\Psi$ maps inward pointing vectors at the boundary (meaning vectors with strictly positive $\partial_\rho$-component) to inward pointing vectors at the corresponding points. Indeed, we see that, at the boundary, $\Psi_*\partial_{\rho_Z}=h^{-1}\partial_{\rho_Y}$.
\end{rem}
\begin{rem}\label{rem:SGmapcomp}
It is obvious that the composition of two ${\mathrm{sc}}$-maps is again a ${\mathrm{sc}}$-map.
\end{rem}
It is straightforward to adapt this definition to that of a local ${\mathrm{sc}}$-map by replacing $Y$ and $Z$ with open subsets.
\begin{lem}[${\mathrm{sc}}$-maps in coordinates]
\label{lem:SGmapcoord}
Let $Y$ and $Z$ be mwbs, $U\subset Y$ and $V\subset Z$ open subsets.
A smooth map $\Psi:U\rightarrow V$ is a local ${\mathrm{sc}}$-map if and only if for the boundary defining functions on $Y$ and $Z$, $\rho_Y$ and $\rho_Z$, respectively, we have
\begin{equation}
\label{eq:locscmap}
\Psi^*\rho_Z=\rho_Yh\text{ for some }h\in{\mathscr{C}^\infty}(Y)\text{ with }h> 0.
\end{equation}
\end{lem}
Hence, any local diffeomorphism of mwbs is a local scattering map. Moreover:
\begin{lem}\label{lem:SGmapproj}
Let $X, Z$ be mwbs.
Given any open, bounded set $U \subset {\mathbb{R}}d$, define the projection ${\mathrm{pr}}_Z : Z \times U \to Z, (z,y) \mapsto z$.
Then $\mathrm{id}_X \times {\mathrm{pr}}_Z$ is a ${\mathrm{sc}}$-map.
\end{lem}
We now investigate the action of pull-backs by ${\mathrm{sc}}$-maps on the
objects introduced above. The following assertions can be verified in local coordinates.
\begin{lem}\label{lem:scdpullback}
Let $Y$ and $Z$ be mwbs, $U\subset Y$ and $V\subset Z$ open subsets.
Let $\Psi:U\rightarrow V$ be a local ${\mathrm{sc}}$-map. Then,
the following properties hold true.
\begin{itemize}
\item $\Psi^*$ yields a map $\rho_Z^m\,{}^\scat\, \varTheta^k(V)\rightarrow \rho_Y^m\,{}^\scat\, \varTheta^k(U)$ for any $m\in {\mathbb{R}}$ and $k\in {\mathbb{N}}$. Moreover, for $\theta\in\rho_{Z}^{m} \,{}^\scat\, \varTheta^k(V)$, we have
${}^\scat \dd (\Psi^*\theta) = \Psi^*({}^\scat \dd \theta)$.
\item $\Psi^*$ yields a map ${}^\scat\, \Omega^\sigma(V)\rightarrow {}^\scat\, \Omega^\sigma(U)$ for any $\sigma\in[0,1]$.
\item The map $T^*\Psi:T^*V\rightarrow T^*U$ lifts to a map ${}^\scat \,\overline{T}^*\Psi:{}^\scat \,\overline{T}^*V\rightarrow {}^\scat \,\overline{T}^*U$. In local coordinates, away from fiber-infinity, ${}^\scat \,\overline{T}^*\Psi$ is given by
$$(\Psi(\mathbf{y}),\boldsymbol{\zeta})\mapsto \big(\mathbf{y},\iota(^t(J \Psi)(\iota^{-1}\boldsymbol{\zeta}))\big),$$
wherein $J\Psi$ is the Jacobian of $\Psi$ at $\mathbf{y}.$
The extension to fiber-infinity is obtained by taking interior limits $|\zeta|^{-1}\rightarrow 0$.
\end{itemize}
\end{lem}
We observe that ${\mathrm{sc}}$-maps provide a natural class of maps between scattering manifolds.
\begin{cor}
Suppose $Y$ is a mwb, $(Z,\rho_Z,g)$ a scattering manifold, $\Psi$
a ${\mathrm{sc}}$-map $Y\rightarrow Z$ which is an immersion. Then $(Y,\Psi^*\rho_Z,\Psi^*g)$ is a scattering manifold.
\end{cor}
\begin{proof}
We first observe that $\Psi^*\rho_Z$ is a boundary defining function on $Y$. Indeed,
\begin{equation}
\label{eq:scTident}
\dd\Psi^*\rho_Z=h\,\dd\rho_Y+\rho_Y \dd h.
\end{equation}
This implies, at the boundary, $h\,\dd\rho_Y\neq 0$. The scattering metric on $Z$ pulls back to
$$\Psi^*g=\Psi^*\frac{(\dd \rho_Z)^{\otimes 2}}{\rho_Z^4}+\Psi^*\frac{g_\partial}{\rho_Z^2}=\frac{(\dd \Psi^*\rho_Z)^{\otimes 2}}{(\Psi^*\rho_Z)^4}+\frac{\Psi^*g_\partial}{(\Psi^*\rho_Z)^2},$$
which is again a scattering metric.
\end{proof}
\begin{cor}
\label{cor:realizeonball}
Any scattering manifold $Y$ of dimension $s$ is locally diffeomorphic to ${\mathbb{B}}s$. Moreover, any scattering density on $Y$ can locally be written as the pull-back by one on ${\mathbb{B}}s$.
\end{cor}
We now extend the notion of ${\mathrm{sc}}$-map to manifolds with corners.
\begin{defn}[${\mathrm{sc}}$-maps on mwc]
Let $Y$ and $Z$ be mwcs. Then, a smooth map $\Psi:Y\rightarrow Z$ is a local ${\mathrm{sc}}$-map for some complete sets of local bdfs $\{\rho_{Y_i}\}_{i\in I}$ and $\{\rho_{Z_i}\}_{i\in I}$ if:
$$\text{For all }i\in I\text{ we have }\Psi^*\rho_{Z_i}=\rho_{Y_i} h_i\text{ for some }h_{i}\in{\mathscr{C}^\infty}(Y)\text{ with }h_{i}> 0.$$
\end{defn}
\begin{rem}
In particular, $\Psi$ maps the boundary of $Y$ into that of $Z$.
As mentioned before, ${\mathrm{sc}}$-maps are special cases of $b$-maps. In fact, they are those \emph{interior} $b$-maps that are smooth maps in the sense of \cite{Joyce}. The only difference with the smooth maps in \cite{Joyce} is that, therein, $\Psi^*\rho_{Z_i}\equiv 0$ is allowed.
\end{rem}
\begin{ex}
In particular, if $\Psi_1:Y_1\rightarrow Z_1$ and $\Psi_2:Y_2\rightarrow Z_2$ are ${\mathrm{sc}}$-maps on mwb, then $\Psi_1\times \Psi_2:Y_1\times Y_2\rightarrow Z_1\times Z_2$ is a ${\mathrm{sc}}$-map on the resulting product mwc.
\end{ex}
\begin{rem}
Note that we fix the ordering of the boundary defining functions. This is important, in particular, when considering ${\mathrm{sc}}$-maps between products $X\times Y\rightarrow X\times Z$ or of the form $X\times Y\rightarrow{}^\scat \,\overline{T}^*X$. Most of the times, the choice of bdfs will be clear from the context.
\end{rem}
Note that, on a mwb, it is possible to extend any map $\partial X\mapsto \partial X$ with $x\mapsto x'$ to a scattering map, by setting $(\rho_X,x)\mapsto (\rho_X,x')$ in a collar neighbourhood of $\partial X$ given by $X\cong [0,\varepsilonilon)\times \partial X$. The following proposition grants us the ability to continue scattering maps from a corner into the interior.
\begin{prop}
\label{prop:cornerdiffeo}
Let $B_1=X_1\times Y_1$ and $B_2=X_2\times Y_2$ be products of mwbs. Let $\Psi^e$, $\Psi^\psi$ be two (local) scattering maps near a point $p\in\mathcal{B}pe_1$,
\begin{align*}
\Psi^e:\mathcal{B}e_1\longrightarrow \mathcal{B}e_2 \quad\text{ and }\quad
\Psi^\psi: \mathcal{B}p_1\longrightarrow \mathcal{B}p_2
\end{align*}
such that $\Psi^e=\Psi^\psi$ when restricted to $\mathcal{B}pe_1$. Then there exists a (local) scattering map $\Psi$ on a neighbourhood $U\subset B_1$ of $p$ with $\Psi^\bullet=\Psi|_{\mathcal{B}^\bullet}$ such that
\begin{equation}
\label{eq:strictscproperty}
\partial_{\rho_{X_1}}\Psi^*\rho_{Y_2}=\partial_{\rho_{Y_1}}\Psi^*\rho_{X_2}=0\quad \text{on }\mathcal{B}_1.
\end{equation}
If $\Psi^e$ and $\Psi^\psi$ are local diffeomorphisms near $p$ (in their respective boundary faces), then $\Psi$ is a local diffeomorphism near $p$.
\end{prop}
\begin{proof}
This is Whitney's extension theorem for smooth functions, applied to the system of functions (and their derivatives)
\begin{align*}
(\Psi^e)^*x,(\Psi^e)^*y,(\Psi^e)^*\rho_Y &\qquad\textrm{on }\mathcal{B}e_1,\\
(\Psi^\psi)^*\rho_X,(\Psi^\psi)^*x,(\Psi^\psi)^*y &\qquad\textrm{on }\mathcal{B}p_1,
\end{align*}
together with the conditions \eqref{eq:strictscproperty} and
\begin{align*}
\label{eq:strictscproperty}
D_{x,y}\Psi^*\rho_{Y_2}=0\quad \text{on }\mathcal{B}_1^\psi,\\
D_{x,y}\Psi^*\rho_{X_2}=0\quad \text{on }\mathcal{B}_1^e.
\end{align*}
Note that, if $\Psi^e$ and $\Psi^\psi$ are local diffeomorphisms at $p$, the differential of $\Psi$ is an invertible block matrix, and hence $\Psi$ is a local diffeomorphism.
\end{proof}
\begin{lem}\label{lem:trafoprinc}
Consider a ${\mathrm{sc}}$-map $\Psi : X \times Y \to X \times Y$ of product form $\Psi = \Psi_X \times \Psi_Y$, with ${\mathrm{sc}}$-maps on $X,Y$, $\Psi_X$ and $\Psi_Y$, respectively.
Assume $a \in \rho_Y^{-m_\psi} \rho_X^{-m_e} {\mathscr{C}^\infty}(X\times Y)$. With the notation of Definition \ref{def:princsymbol} and \ref{def:princpart}, we have:
\begin{align*}
\sigma^\psi(\Psi^*a) - \Psi^*(\sigma^\psi a) &\in \rho_Y^{-m_\psi + 1} \rho_X^{-m_e}{\mathscr{C}^\infty},\\
\sigma^e(\Psi^*a) - \Psi^*(\sigma^e a) &\in \rho_Y^{-m_\psi}\rho_X^{-m_e + 1} {\mathscr{C}^\infty},\\
(\Psi^*a)_p - \Psi^*(a_p) &\in \rho_Y^{-m_\psi + 1} \rho_X^{-m_e + 1}{\mathscr{C}^\infty}.
\end{align*}
\end{lem}
\begin{proof}
We will only prove the first identity, the others follows by similar arguments.
Write $(\Psi^*\rho_X)(\mathbf{x})=\rho_X h_X(\mathbf{x})$ and $(\Psi^*\rho_Y)(\mathbf{y})=\rho_Y h_Y(\mathbf{y})$. If $a = \rho_X^{-m_e} \rho_Y^{-m_\psi} f$ then
\begin{align*}
(\Psi^*a)(\mathbf{x},\mathbf{y}) = \rho_X^{-m_e} \rho_Y^{-m_\psi} h_X^{-m_e}(\mathbf{x}) h_Y^{-m_\psi}(\mathbf{y}) (\Psi^*f)(\mathbf{x}, \mathbf{y}).
\end{align*}
This implies
\begin{align*}
\sigma^{\psi}(\Psi^*a)(\mathbf{x},\mathbf{y}) &= \rho_X^{-m_e} \rho_Y^{-m_\psi} h_X^{-m_e}(\mathbf{x}) h_Y^{-m_\psi}(0,y) (\Psi^*f)(\mathbf{x}, 0, y),\\
\Psi^*(\sigma^{\psi}a)(\mathbf{x},\mathbf{y}) &= \rho_X^{-m_e} \rho_Y^{-m_\psi} h_X^{-m_e}(\mathbf{x}) h_Y^{-m_\psi}(\mathbf{y}) (\Psi^*f)(\mathbf{x}, 0, y).
\end{align*}
Using Taylor's theorem, we obtain that $h_Y^{-m_\psi}(\mathbf{y}) - h_Y^{-m_\psi}(0,y) \in \rho_Y {\mathscr{C}^\infty}(X\times Y)$, and therefore
$\sigma^\psi(\Psi^*a) - \Psi^*(\sigma^\psi a) \in \rho_Y^{-m_\psi + 1} \rho_X^{-m_e}{\mathscr{C}^\infty}(X\times Y)$, as claimed.
\end{proof}
\begin{cor}
The principal part of $a\in \rho_Y^{-m_\psi} \rho_X^{-m_e} {\mathscr{C}^\infty}(X\times Y)$ is well-defined as an element of
\[
\rho_X^{-m_e} \rho_Y^{-m_\psi} {\mathscr{C}^\infty}(X\times Y) / \rho_X^{-m_e+1} \rho_Y^{-m_\psi+1} {\mathscr{C}^\infty}(X\times Y),
\]
and does not depend on the choice of boundary-defining functions $\rho_X,\rho_Y$ on $X,Y$.
\end{cor}
\begin{rem}
Note that the space
\[\rho_X^{-m_e} \rho_Y^{-m_\psi} {\mathscr{C}^\infty}(X\times Y) / \rho_X^{-m_e+1} \rho_Y^{-m_\psi+1} {\mathscr{C}^\infty}(X\times Y)\]
can be identified with ${\mathscr{C}^\infty}(\partial(X\times Y))$, which identifies our notion of principal symbol with that of \cite[Section 6.4]{Melrose2}.
\end{rem}
The following lemma is one of the main technical tools in this article.
We have already observed that the local model of a scattering manifold near the boundary is the radial compactification of ${\mathbb{R}}d$. We now show that scattering maps arise naturally as the composition of vector-valued amplitudes and radial compactification. Furthermore, we clarify the relation between total derivative and the scattering differential under compactification.
\begin{lem}
\label{lem:horror}
Let $Y$ be a mwb.
Let $f\in \rho_Y^{-1}{\mathscr{C}^\infty}(Y,{\mathbb{R}}d)$ with $\rho_Y|f|\neq 0$ on $\partial Y$.\footnote{This means $\rho_Y f$ is the restriction to $Y^o$ of an element of $g\in{\mathscr{C}^\infty}(Y,{\mathbb{R}}d)$ with $g\neq 0$ on $\partial Y$.} Then,
$\Psi=\iota\circ f$ extends to a local ${\mathrm{sc}}$-map $Y\rightarrow {\mathbb{B}}d$. Moreover, the matrix of coefficients of
\[{}^\scat \dd f = \begin{pmatrix}
{}^\scat \dd f_1\\
\vdots\\
{}^\scat \dd f_d
\end{pmatrix}\]
has the same rank as the differential $T\Psi$ of $\Psi$.
\end{lem}
\begin{proof}
Since $\iota$ is a diffeomorphism, $\iota \circ f$ is a smooth map while $\rho_Y>\varepsilon$ and we may thus restrict our attention to a neighbourhood of $\partial Y$ where $\rho_Y |f|$ is everywhere non-vanishing.
As usual, we pick a suitable collar neighbourhood of product type such that locally $Y=[0,\varepsilon)\times \partial Y$, and we write $\dim(Y)=s$ and
$\mathbf{y}=(\rho_Y,y)$ for the coordinates. There we need to compute $\Psi^*\rho_{Z}$.
Write
$ f(\rho_Y, y) = \rho^{-1}_Y h(\rho_Y,y)$
for $h\in {\mathscr{C}^\infty}(Y,{\mathbb{R}}d)$ with $h(0,y)\neq 0$ for all $(0,y)\in\partial Y$. Since $\rho_Y$ is assumed sufficiently small,
$|f(\mathbf{y})|=\rho_Y^{-1}|h(\mathbf{y})|$ may be assumed sufficiently large and hence
$$\Psi(\mathbf{y})=(\iota\circ f)(\mathbf{y})=\frac{f(\mathbf{y})}{|f(\mathbf{y})|}\left(1-\frac{1}{|f(\mathbf{y})|}\right)=\frac{h(\mathbf{y})}{|h(\mathbf{y})|}\left(1-\frac{\rho_Y}{|h(\mathbf{y})|}\right).$$
In this form, $\Psi$ clearly extends up to the boundary.
The boundary defining function on ${\mathbb{B}}d$ is, in this coordinate patch,
$\rho_Z=1-|x|$. Thus,
$$\Psi^*\rho_Z=\frac{1}{|f(\mathbf{y})|}=\rho_Y \frac{1}{\rho_Y|f(\mathbf{y})|}.$$
By assumption, $\rho_Y|f(\mathbf{y})|=|h(\mathbf{y})|$ is smooth and non-vanishing, which proves that $\Psi$ is an ${\mathrm{sc}}$-map.
For the second half of the statement we first observe that, since $\iota$ is a diffeomorphism ${\mathbb{R}}d\rightarrow ({\mathbb{B}}d)^o$ and ${}^\scat \dd$ coincides, up to a rescaling by a non-vanishing factor, with the usual differential in the interior, we may restrict our attention to the boundary $\partial Y$. Then we compute
\begin{align*}
{}^\scat \dd f(\mathbf{y}) &=
\rho_Y^2\partial_{\rho_Y} f(\mathbf{y})\,\frac{\dd\rho_Y}{\rho_Y^2}+\sum_{j=1}^{s-1}\rho_Y\partial_{y_j} f(\mathbf{y})\,\frac{\dd y_j}{\rho_Y}\\
&=(-h(\mathbf{y})+ \rho_Y \partial_{\rho_Y}h(\mathbf{y}))\,\frac{\dd\rho_Y}{\rho_Y^2} + \sum_{j=1}^{s-1}\partial_{y_j} h(\mathbf{y}) \,\frac{\dd y_j}{\rho_Y}.
\end{align*}
We identify ${}^\scat \dd f$ with its coefficients ($s\times d$)-dimensional block matrix
$$\begin{pmatrix}
-h(\mathbf{y})+ \rho_Y \partial_{\rho_Y}h(\mathbf{y}) & (\partial_{y_j} h(\mathbf{y}))_{j=1,\dots,s-1}
\end{pmatrix}.$$
At the boundary $\rho_Y=0$ we obtain
\begin{align}
\label{eq:scdhorror}
\begin{pmatrix}
-h & (\partial_{y_j} h)_{j=1,\dots,s-1}
\end{pmatrix}\!(0,y).
\end{align}
We want to compare the rank of \eqref{eq:scdhorror} with that of the differential of $\Psi$ at the point $(0,y)\in\partial Y$.
As shown above, the function $\Psi$ is given, at an arbitrary point $\mathbf{y}=(\rho_Y,y)$ close enough to $\partial Y$, by
\begin{align*}
\frac{h(\mathbf{y})}{|h(\mathbf{y})|}\left(1-\frac{\rho_Y}{|h(\mathbf{y})|}\right),
\end{align*}
whose differential at $(0,y)$ is the block matrix
\begin{align}\label{eq:JPsi}
T \Psi(0, y) =
\begin{pmatrix}
-\frac{h}{|h|^2}+\partial_{\rho_Y}\frac{h}{|h|} & \left(\partial_{y_j} \frac{h}{|h|}\right)_{j=1,\dots,s-1}
\end{pmatrix}\!(0,y).
\end{align}
Now observe that, since they are derivatives of unit vectors,
$\partial_{y_j} \frac{h}{|h|}$ and $\partial_{\rho_Y} \frac{h}{|h|}$ are orthogonal to $h$, which is itself non-zero.\footnote{Recall that,
in fact, $|v(t)|=1 \Leftrightarrow v(t)\cdot v(t) = 1\Rightarrow 2v(t)\cdot v^{\mathrm{pr}}ime(t)=0\Leftrightarrow v(t) \perp v^{\mathrm{pr}}ime(t)$.}
Therefore, the rank of $T\Psi(0,y)$ equals that of the block matrix
\begin{align}\label{eq:JPsimod}
\begin{pmatrix}
-h &
\left(\partial_{y_j} \frac{h}{|h|}\right)_{j=1,\dots,s-1}
\end{pmatrix}\!(0,y).
\end{align}
Finally, we have that
$$\partial_{y_j} h= \partial_{y_j} \left(|h|\frac{h}{|h|}\right)=\underbrace{|h| \partial_{y_j} \frac{h}{|h|}}_{\text{collinear to }{\partial_{y_j} \frac{h}{|h|}}} + \underbrace{\frac{(h\cdot\partial_{y_j} h)}{|h|^2} h}_{\text{collinear to }h} .$$
This means that the null space (and hence the ranks) of \eqref{eq:scdhorror} and \eqref{eq:JPsimod} coincide.
\end{proof}
\begin{ex}
The simplest example for a map where Lemma \ref{lem:horror} applies is given by the map $f=\iota^{-1}:{\mathbb{B}}d\rightarrow {\mathbb{R}}d$.
\end{ex}
\begin{rem}
Recall (cf. \cite[App. C.3]{Hormander3}) that the intersection of two ${\mathscr{C}^\infty}$-sub\-mani\-folds $Y$ and $Z$ of a ${\mathscr{C}^\infty}$-manifold $X$ is \emph{clean} with excess $e\in{\NNz_0}$ if $Y\cap Z$ is a ${\mathscr{C}^\infty}$-submanifold of $X$ satisfying
\begin{align*}
T_x(Y\cap Z)&=T_xY\cap T_x Z,\qquad \forall x\in Y\cap Z,\\
\codim(Y)+\codim(Z)&=\codim(Y\cap Z)+e.
\end{align*}
\end{rem}
\begin{ex}\label{ex:embdball}
Let $X$ be a mwb and $a \in \rho_X^{-m_e} \rho_{{\mathbb{B}}^s}^{-m_\psi} {\mathscr{C}^\infty}(X \times {\mathbb{B}}^s)$. In this example, we extend $a$ to a local symbol on a suitable subset of $X \times {\mathbb{B}}^{s+1}$.
We view ${\mathbb{B}}^{s+1}$ as embedded in ${\mathbb{R}}^{s+1}$ with coordinates $(y_1,\dots,y_s,\tilde{y})$.
Define
\[\jmath : {\mathbb{B}}^{s+1} \to {\mathbb{B}}^s \times (-1,1), \qquad (y,\tilde{y}) \mapsto \left(\frac{y}{\sqrt{1 - \tilde{y}^2}}, \tilde{y}\right),\]
where $y = (y_1, \dotsc, y_s)$.
For every $\varepsilon \in (0,1)$, we obtain coordinates on
$$U = \jmath^{-1}\left\{{\mathbb{B}}^s \times (-\varepsilon, \varepsilon)\right\} = {\mathbb{B}}^{s+1} \cap \{|\tilde{y}| < \varepsilon\},$$
cf. Figure \ref{fig:fiberball}.
We note that $U$ is a fibration of base ${\mathbb{B}}^s$ and fiber $(-\varepsilon, \varepsilon)$.
\begin{figure}
\caption{The action of $\jmath$ visualized}
\label{fig:fiberball}
\end{figure}
We verify that $\jmath$ is a ${\mathrm{sc}}$-map. For this we now view ${\mathbb{B}}^s\times(-\varepsilon,\varepsilon)$ as a (non-compact) manifold with
boundary\footnote{This means we view ${\mathbb{B}}^s\times(-\varepsilon,\varepsilon)$ as embedded in the manifold with boundary ${\mathbb{B}}^s\times{\mathbb{S}}^1$, which can be embedded in ${\mathbb{S}}^s\times{\mathbb{S}}^1$.
For higher dimension, we embed $(-\varepsilon, \varepsilon)^r \hookrightarrow \mathbb{T}^r$.}
with boundary defining function $\rho_Z=1-[y]$. Observe that near the boundary we have
\begin{align*}
\jmath^*\rho_Z &= 1-\frac{[y]}{\sqrt{1-\tilde{y}^2}}\\
&=(1-\sqrt{[y]^2+\tilde{y}^2})\cdot \frac{1}{ \sqrt{1-\tilde{y}^2} } \cdot \frac{\sqrt{1 - \tilde{y}^2} - [y]}{1 - \sqrt{\tilde{y}^2 + [y]^2}}\\
&=\rho_{{\mathbb{B}}^{s+1}}h.
\end{align*}
Since $|\tilde{y}| < \varepsilonilon$, $h$ is positive and in ${\mathscr{C}^\infty}(U)$. Hence $\jmath$ is an ${\mathrm{sc}}$-map.
As usual, we may perform the same construction fiber-wise on a fiber bundle by considering local product decompositions to obtain a local ${\mathrm{sc}}$-map. Namely, let $X$ be an arbitrary mwb. Then $\Psi = \mathrm{id}_X \times \jmath$ is again a ${\mathrm{sc}}$-map on the product $X\times \big({\mathbb{B}}s\times(-\varepsilon,\varepsilon)\big)$. Using Lemma \ref{lem:SGmapproj} and Remark \ref{rem:SGmapcomp}, wee see that $\tilde{\Psi} = \Psi \circ (\mathrm{id}_X \times {\mathrm{pr}}_{{\mathbb{B}}^s}) : X \times U \to X \times {\mathbb{B}}^s$
is a ${\mathrm{sc}}$-map. Hence, $\tilde{\Psi}^* a \in \rho_X^{-m_e} \rho_{{\mathbb{B}}^{s+1}}^{-m_\psi} C^\infty(X \times U)$.
\end{ex}
\section{Phase functions and Lagrangian submanifolds}
\label{sec:phaseandlag}
\subsection{Clean phase functions}
\begin{defn}[Phase functions]
Let $X$ and $Y$ be mwbs, $B=X\times Y$. Let $U$ be an open subset in $B$. Then, a real valued $\varphi\in \rho_X^{-1}\rho_Y^{-1}{\mathscr{C}^\infty}(U)$ is a \emph{local} (${\mathrm{sc}}$-\-)\allowbreak phase function if it is the restriction of some $\widetilde{\varphi}\in \rho_X^{-1}\rho_Y^{-1}{\mathscr{C}^\infty}(B)$ to $U$ such that ${}^\scat \dd\tilde{\varphi}(p)\neq 0$ for all $p\in\overline{\mathcal{B}p}\cap\overline{\partial U}$.
If $U=B$, that is $\varphi\in \rho_X^{-1}\rho_Y^{-1}{\mathscr{C}^\infty}(B)$ with ${}^\scat \dd\varphi(p)|_{\overline{\mathcal{B}^\psi}}\neq 0$, we call $\varphi$ a \emph{global} ${\mathrm{sc}}$-phase function.
\end{defn}
\begin{rem}
Phrased differently, if $U$ is an interior open set, $\varphi$ is just a smooth function.
In the non-trivial case of $U$ being a boundary neighbourhood, the above definition means that, for every $p\in\partial B$ in the $\psi$- or $\psi e$-component of the boundary of $U$, there exists an element $\zeta \in{}^\scat \,\mathcal{V}(B)$ such that $\zeta (\varphi)$ is elliptic at $p$, meaning $\zeta (\varphi)\in{\mathscr{C}^\infty}(X\times Y)$ satisfies $\big(\zeta\varphi\big)(p)\neq 0$. It is, by compactness, bounded away from zero at the possible limit points in $\overline{\partial U}$. In the following, we usually do not write $\widetilde{\varphi}$ but simply identify $\widetilde{\varphi}$ and $\varphi$ at these limit points.
\end{rem}
\begin{ex}[${\mathrm{SG}}$-phase functions]
If $B={\mathbb{B}}d\times{\mathbb{B}}s$, such $\varphi$ correspond to so-called (classical) ${\mathrm{SG}}$-phase functions on ${\mathbb{R}}d\times{\mathbb{R}}s$, cf. \cite{CoSc,CoSc2}, but with a relaxed condition as $\|x\|\rightarrow \infty$. Indeed, in light of the ${\mathrm{SG}}$-estimates \eqref{eq:SGest}, the previous definition translates to
\begin{equation}
\label{eq:SGphaseineq}
|\jap{x}^{-1} \nabla_\theta\varphi|^2+|\jap{\theta}^{-1}\nabla_x\varphi|^2\geq C\quad \text{for} \quad |\theta|\gg 0.
\end{equation}
The relationship between these and ``standard'' phase functions which are homogeneous in $\theta$ is discussed in \cite{CoSc2}. Examples of ${\mathrm{SG}}$-phase functions are the standard Fourier phase $x\cdot \theta$ on ${\mathbb{R}}^d_x\times{\mathbb{R}}^d_\theta$ and $x_0\jap{\theta}-x\cdot \theta$ on
${\mathbb{R}}_{x_0,x}^{d+1}\times{\mathbb{R}}^d_\theta$.
\end{ex}
\begin{defn}[The set of critical points]
Let $B=X\times Y$, $\varphi\in \rho_X^{-1}\rho_Y^{-1}{\mathscr{C}^\infty}(B)$ a (local) phase function. A point $p\in B$ (in the domain of $\varphi$) is called a \emph{critical point} of $\varphi$ if ${}^\scat \ddY\varphi(p)=0$, that is, if
$\zeta(\varphi)(p)=0$ for every $\zeta \in{}^\scat \,\mathcal{V}Y(B).$
We define
\begin{equation}
C_\varphi=\{p\in B\,|\, {}^\scat \ddY\varphi(p) = 0 \}.
\end{equation}
We set $\mathcal{C}_\varphi=C_\varphi\cap \mathcal{B}$ and specify
$$\mathcal{C}_\varphi^\bullet = \mathcal{C}_\varphi\cap \mathcal{B}^\bullet \quad \text{for} \quad \bullet \in\{e,\psi,\psi e\}. $$
\end{defn}
We now adapt the usual definition of a \emph{clean} phase function from the classical setting to the case with boundary.
\begin{defn}[Clean phase functions]
\label{def:cleanphase}
A phase function $\varphi$
is called \emph{clean} if the following conditions hold:
\begin{itemize}
\item[1.)] there exists a neighbourhood $U\subset B$ of $\partial B$ such that $C_\varphi\cap U$ is a manifold with corners with $\partial C_\varphi\subset \partial B$;
\item[2.)] the tangent space of $T_pC_\varphi$ is at every point $p$ given by those vectors in $v\in T_p B$ such that $v(\zeta(\varphi))=0$ for all $\zeta\in {}^\scat \,\mathcal{V}Y$, that is, $T({}^\scat \ddY\varphi)v=0$;
\item[3.)] the intersections $\mathcal{C}_\varphi^\bullet=C_\varphi\cap\mathcal{B}^\bullet$ are clean.
\end{itemize}
\end{defn}
The last condition is equivalent to the existence of $w \in T_{\mathcal{C}_\varphi^\bullet}\mathcal{C}_\varphi^\bullet$ such that
\begin{align}\label{eq:cleanbdry}
(T{}^\scat \dd_Y\varphi)(w + \partial_{\rho_\bullet}) = 0.
\end{align}
This means that, for some $w$ tangent to $\mathcal{B}^\bullet$, we have $w + \partial_{\rho_\bullet} \in T_{\mathcal{C}_\varphi^\bullet} \mathcal{C}_\varphi$. Here, $\rho_\bullet$ is a bdf of $\mathcal{B}^\bullet$. We now discuss the implications of these conditions.
\begin{lem}
\label{lem:Cpprops}
Let $\varphi$ be a clean phase function. Then either we are in the ``non-corner crossing case'' $1a.)$ or in the ``corner crossing case'' $1b.)$, namely,
\begin{enumerate}[label=1\alph*.)]
\item both $\mathcal{C}_\varphie$ and $\mathcal{C}_\varphip$ are closed manifolds (without boundary) and $\mathcal{C}_\varphipe=\emptyset$;
\item $\mathcal{C}_\varphi$ consists of two components, $\overline{\mathcal{C}_\varphie}$ and $\overline{\mathcal{C}_\varphip}$, which are both submanifolds (with boundary), of the same dimension $\dim(C_\varphi)-1$, with joint boundary $\mathcal{C}_\varphipe=\partial \overline{\mathcal{C}_\varphie}=\partial \overline{\mathcal{C}_\varphip}$ of $\mathcal{B}$. The intersection of $\overline{\mathcal{C}_\varphie}$ and $\overline{\mathcal{C}_\varphip}$ in $\mathcal{C}_\varphipe$ is again clean.
\end{enumerate}
In both cases, the differential of ${}^\scat \ddY\varphi:B\rightarrow {}^\scat \,T^*B$, viewed as a map $T({}^\scat \ddY\varphi):TB\rightarrow T({}^\scat \,T^*B)$, characterizes $T\mathcal{C}_\varphi^\bullet$:
\begin{enumerate}[label=\arabic*.)]
\setcounter{enumi}{1}
\item \label{it:Cpprops2} The tangent space of $\overline{\mathcal{C}_\varphie}$ and $\overline{\mathcal{C}_\varphip}$ at each point $p$ is given by those vectors $v\in T\mathcal{B}^\bullet$ such that $v(\zeta(\varphi))=0$ for all $\zeta\in {}^\scat \,\mathcal{V}Y$, that is $T({}^\scat \ddY\varphi)v=0$.
\end{enumerate}
\end{lem}
By condition $3.)$ of Definition \ref{def:cleanphase}, we have
$\dim(\ker(T({}^\scat \ddY\varphi)))=\dim C_\varphi$. Hence,
the restrictions of $T({}^\scat \ddY\varphi)$ to the individual boundary components of $B$ on $\mathcal{C}_\varphi$ are of constant rank. Namely,
\[
\rk(T({}^\scat \ddY\varphi))=
\begin{cases}
s-e & \text{on $C_\varphi^o$},
\\
s-e-1 & \text{on $\mathcal{C}_\varphip$ and $\mathcal{C}_\varphie$},
\\
s-e-2 & \text{on $\mathcal{C}_\varphipe$},
\end{cases}
\]
for some fixed number $e$, called the excess of $\varphi$, which is given by
$$e=\dim C_\varphi - d.$$
\begin{rem}
Conversely, if the rank of $T({}^\scat \ddY\varphi)$ is constant \emph{in a neighborhood} of each critical point of ${}^\scat \ddY\varphi$, then $\varphi$ is clean by the constant rank theorem. In case $e=0$, $\varphi$ is called \emph{non-degenerate}, and the two characterizations coincide. The corresponding case of ${\mathrm{SG}}$-phase functions (on ${\mathbb{R}}d$) was studied in \cite{CoSc2}.
\end{rem}
\subsection{The associated Lagrangian}
In the classical local theory without boundary on subsets of ${\mathbb{R}}d\times({\mathbb{R}}^s\setminus\{0\})$, see \cite[Chapter XXI.2]{Hormander3},
the set of critical points $\mathcal{C}_\varphi$ is realized as an immersed Lagrangian in $T^*{\mathbb{R}}d$ by the map
$(x,\theta)\rightarrow (x,\varphi_x^{\mathrm{pr}}ime(x,\theta))$.
In the present setting, the situation is more complicated.
Following \cite{CoSc2}, we define an analogous map ${\lambda_\varphi}$ on the mwc $B=X\times Y$ into ${}^\scat \,\overline{T}^*X$.
For that, we consider the following sequence of maps: Using the ``rescaling identifications'' \eqref{eq:rescal}, we may view $(\mathbf{x},\mathbf{y})\rightarrow {}^\scat \ddX\varphi(\mathbf{x},\mathbf{y})$ as a map in
$\rho_Y^{-1}{\mathscr{C}^\infty}(Y,{}^\scat\, \varTheta(X))$. Since ${}^\scat\, \varTheta(X)$ are the sections of ${}^\scat \,\overline{T}^*X$, composing with the radial compactification yields, in view
of Lemma \ref{lem:horror}, a map into the compactified fibers of ${}^\scat \,\overline{T}^*X$.
\begin{defn}
The map $\lambda_\varphi: B\rightarrow {}^\scat \,\overline{T}^*X$ is defined by
$$(\mathbf{x},\mathbf{y})\mapsto \big(\mathbf{x},\iota({}^\scat \ddX\varphi(\mathbf{x},\mathbf{y}))\big).$$
\end{defn}
\begin{lem}
\label{lem:lpsct}
There is a neighbourhood $U\subset B$ of $\mathcal{C}_\varphi$ such that ${\lambda_\varphi}: U\rightarrow {}^\scat \,\overline{T}^*X$ is a local ${\mathrm{sc}}$-map.
\end{lem}
\begin{proof}
We write, $\mathbf{x}=(\rho_X,x)$, $\mathbf{y}=(\rho_Y,y)$ for coordinates in $B$, $\mathbf{x}$ and $\mathbf{x}i=(\rho_\Xi,\xi)$ for coordinates in ${}^\scat \,\overline{T}^*X$. Since $\lambda_\varphi$ is the identity in the first set of variables, we have
$\lambda_\varphi^*\mathbf{x}=\mathbf{x}.$
In the second set of variables, $\lambda_\varphi$ acts as $\iota\,\circ\,{}^\scat \ddX\varphi$, with ${}^\scat \ddX\varphi\in\rho_Y^{-1}{\mathscr{C}^\infty}(Y,{}^\scat\, \varTheta(X))$. Notice that on $\mathcal{C}_\varphip\cup\mathcal{C}_\varphipe$, we have ${}^\scat \ddX\varphi(\mathbf{x},\mathbf{y})\neq 0$, since ${}^\scat \dd \varphi\neq 0$ on $\mathcal{B}p\cup\mathcal{B}pe$ and ${}^\scat \ddY\varphi=0$ on $\mathcal{C}_\varphi$. Hence, due to compactness, we may find a neighbourhood of $\mathcal{C}_\varphip\cup\mathcal{C}_\varphipe$ on which ${}^\scat \ddX\varphi(\mathbf{x},\mathbf{y})\neq 0$.
Writing $\varphi=\rho_X^{-1}\rho_Y^{-1}f$ for $f\in{\mathscr{C}^\infty}(X\times Y)$, this means
$$(-f+\rho_X\partial_{\rho_X}f) \frac{\dd \rho_X}{\rho_X^2\rho_Y} + \sum_{j=1}^{d-1}\partial_{x_j}f\frac{\dd x_j}{\rho_X\rho_Y} \neq 0.$$
Rescaling and viewing ${}^\scat \ddX \varphi$ as a map in $\rho_Y^{-1}{\mathscr{C}^\infty}(Y,{}^\scat\, \varTheta(X))$, we express ${}^\scat \ddX\varphi$ as
\begin{equation}
\label{eq:scdxexpl}
{}^\scat \ddX\varphi=\rho_Y^{-1} \left((-f+\rho_X\partial_{\rho_X}f) \frac{\dd \rho_X}{\rho_X^2} + \sum_{j=1}^{d-1}\partial_{x_j}f\frac{\dd x_j}{\rho_X} \right).
\end{equation}
Composing with $\iota$, we are therefore in the situation of Lemma \ref{lem:horror}, up to additional smooth dependence on the $X$-variables, and conclude that $\lambda_\varphi$ is a local ${\mathrm{sc}}$-map.
On $\mathcal{C}_\varphie$, away from $\mathcal{C}_\varphipe$, we have that $\rho_Y\neq 0$ and correspondingly ${}^\scat \ddX\varphi(\mathbf{x},\mathbf{y})$ stays bounded. Since $\iota$ maps bounded arguments into the interior,
we find ${\lambda_\varphi}^*\rho_\Xi\neq 0$. Since ${\lambda_\varphi}$ is smooth, ${\lambda_\varphi}$ is an ${\mathrm{sc}}$-map.
\end{proof}
In particular, $\iota({}^\scat \ddX\varphi(\mathbf{x},\mathbf{y}))$ maps boundary points with $\rho_Y=0$ to boundary points of the fiber, that is to $\Wt^\psi\cup \Wt^\psie$.
\begin{defn}
We define $L_\varphi={\lambda_\varphi}(C_\varphi)$ and $\Lambda_\varphi:={\lambda_\varphi}(\mathcal{C}_\varphi)$. We further write $\Lambda_\varphi^\bullet$ for ${\lambda_\varphi}(\mathcal{C}_\varphi^\bullet)\subset \mathcal{W}^\bullet$ for $\bullet\in\{e,\psi,\psi e\}$. We say that $\varphi$ parametrizes $L_\varphi$ and $\Lambda_\varphi$.
\end{defn}
\begin{thm}
\label{thm:lpsubm}
The map ${\lambda_\varphi}: \mathcal{C}_\varphi \rightarrow {}^\scat \,\overline{T}^*X$ is of constant rank $d$. Its image $L_\varphi$ as well as the boundary and corner faces $\Lambda_\varphi^\bullet={\lambda_\varphi}(\mathcal{C}_\varphi^\bullet)$ are immersed manifolds of dimension $\dim\Lambda_\varphi^\bullet=\dim\mathcal{C}_\varphi^\bullet-e$. Furthermore, ${\lambda_\varphi}:\mathcal{C}_\varphi\rightarrow \Lambda_\varphi$ is a submersion.
\end{thm}
The proof is inspired by that of Lemma 2.3.2 in \cite{Duistermaat} (adapted to clean phase functions), but much more involved, due to the presence of the compactification. We treat this new phenomenon by carefully applying Lemma~\ref{lem:horror}.
\begin{proof}
We obtain the rank of $T{\lambda_\varphi}$ for ${\lambda_\varphi}: \mathcal{C}_\varphi \rightarrow {}^\scat \,\overline{T}^*X$ by computing the dimension of its null space.
Let $v = \delta\rho_X \cdot \partial_{\rho_X} + \delta x\cdot \partial_x + \delta\rho_Y\cdot \partial_{\rho_Y} + \delta y\cdot \partial_y$ be a vector at a point
$p = (\rho_X,x,\rho_Y,y) \in \mathcal{C}_\varphi$. For the moment, we assume $\rho_Y>0$. We write ${\lambda_\varphi} = (\mathrm{id} \times \iota) \circ {\ell_\varphi}$ with
\begin{align*}
{\ell_\varphi} : X\times Y^o \rightarrow {}^\scat \,T^*X\qquad
(x,y) \mapsto (x,{}^\scat \dd_X \varphi(x,y)).
\end{align*}
Assume that $T{\ell_\varphi}(p)v = 0$ and $v \in T_p \mathcal{C}_\varphi$.
The condition $T{\ell_\varphi}(p)v = 0$ implies that $\delta\rho_X = 0$ and $\delta x = 0$.
Let $\tilde{v} = \delta\rho_Y\cdot \partial_{\rho_Y} + \delta y\cdot \partial_y$. Hence the assumptions are reduced to
\begin{equation}\label{eq:VscdYX}
\begin{aligned}
\tilde{v} {}^\scat \dd_X \varphi(p) &= 0,\\
\tilde{v} {}^\scat \dd_Y \varphi(p) &= 0,
\end{aligned}
\end{equation}
where $\tilde{v}$ is interpreted as acting on the coefficient functions of the differentials.
In coordinates, these coefficient functions are given by
\begin{align*}
{}^\scat \dd_X \varphi(p) = \rho_Y^{-1}(-f + \rho_X\partial_{\rho_X} f, \partial_x f)(p), \qquad
{}^\scat \dd_Y \varphi(p) = (-f + \rho_Y \partial_{\rho_Y} f, \partial_y f)(p).
\end{align*}
On $\mathcal{C}_\varphi$, where $-f + \rho_Y \partial_{\rho_Y} f = 0$ and $\partial_y f = 0$ hold true, it is easily seen that \eqref{eq:VscdYX} is
equivalent to
\begin{align}\label{mat:scdX}
\begin{pmatrix}
\rho_X\rho_Y^{-2} (\rho_Y \partial_{\rho_Y} - 1) \partial_{\rho_X} f & \rho_X\rho_Y^{-1} \partial_{\rho_X}\partial_y f\\
\rho_Y^{-2}(\rho_Y \partial_{\rho_Y} - 1) \partial_x f & \rho_Y^{-1}\partial_x\partial_y f\\
\rho_Y \partial_{\rho_Y}\partial_{\rho_Y} f & \rho_Y\partial_{\rho_Y}\partial_y f\\
\partial_{\rho_Y}\partial_y f & \partial_y \partial_y f
\end{pmatrix}
\begin{pmatrix}
\delta \rho_Y\\
\delta y
\end{pmatrix}
= 0.
\end{align}
The cleanness condition translates to the dimension of the nullspace of $T{}^\scat \dd_X \varphi$ being constantly $e$. We identify $T{}^\scat \dd_Y\varphi$ with the matrix
\begin{align}\label{mat:clean}
J =
\begin{pmatrix}
(\rho_Y \partial_{\rho_Y}-1) \partial_{\rho_X}f & \partial_y \partial_{\rho_X} f\\
(\rho_Y \partial_{\rho_Y}-1) \partial_x f & \partial_y\partial_x f\\
\rho_Y\partial_{\rho_Y}\partial_{\rho_Y} f & \partial_y\partial_{\rho_Y} f\\
\rho_Y \partial_{\rho_Y}\partial_y f & \partial_y\partial_y f
\end{pmatrix}.
\end{align}
The matrices appearing in \eqref{mat:scdX} and \eqref{mat:clean} are related by
\begin{align*}
J
=
\begin{pmatrix}
\rho_Y\rho_X^{-1} & 0 & 0 & 0\\
0 & \rho_Y & 0 & 0\\
0 & 0 & \rho_Y^{-1} & 0\\
0 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
\rho_X\rho_Y^{-2} (\rho_Y \partial_{\rho_Y} - 1) \partial_{\rho_X} f & \rho_X\rho_Y^{-1} \partial_{\rho_X}\partial_y f\\
\rho_Y^{-2}(\rho_Y \partial_{\rho_Y} - 1) \partial_x f & \rho_Y^{-1}\partial_x\partial_y f\\
\rho_Y \partial_{\rho_Y}\partial_{\rho_Y} f & \rho_Y\partial_{\rho_Y}\partial_y f\\
\partial_{\rho_Y}\partial_y f & \partial_y \partial_y f
\end{pmatrix}
\begin{pmatrix}
\rho_Y & 0\\
0 & 1
\end{pmatrix}.
\end{align*}
This proves that \eqref{eq:VscdYX} is equivalent to $v \in \ker T{}^\scat \dd_Y \varphi$ under our assumptions $\rho_Y > 0$ and $\rho_X > 0$, and the rank of
${\ell_\varphi}$ is given by
\begin{align*}
\rk {\ell_\varphi} &= \dim T_p \mathcal{C}_\varphi - \dim \ker T{}^\scat \dd_Y \varphi
= (d + e) - e=d.
\end{align*}
Now assume that $\rho_X = 0$. We see that the first row of \eqref{mat:scdX} vanishes identically,
but we have the additional condition \eqref{eq:cleanbdry}, implying that, at $\rho_X=0$, the first row of \eqref{mat:clean} depends linearly on the other rows.
Therefore, the rank of ${\ell_\varphi}$ is still $d$ at points with $\rho_X=0$. The composition with $\mathrm{id} \times \iota$ changes nothing for $\rho_Y > 0$, since $\iota$ is a diffeomorphism there.
To perform the limit $\rho_Y \rightarrow 0$, we have to examine carefully the effect of the presence of the
compactification $\iota$, in the spirit of the proof of Lemma \ref{lem:horror}.
For $v \in T_p \mathcal{C}_\varphi$ such that $T{\lambda_\varphi}(p)v = 0$, that is, as above, of the form
\[v = \delta\rho_Y\cdot \partial_{\rho_Y} + \delta y\cdot \partial_y,\]
we now obtain the set of equations
\begin{equation}\label{eq:iotaVscdYX}
\begin{aligned}
v \big(\iota\,{}^\scat \dd_X \varphi\big)(p) &= 0,\\
v {}^\scat \dd_Y \varphi(p) &= 0,
\end{aligned}
\end{equation}
which are equivalent to the set of equations
\begin{align}\label{mat:iotaVscd}
\begin{pmatrix}
\partial_{\rho_Y} \iota {}^\scat \dd_X \varphi & \partial_y \iota {}^\scat \dd_X \varphi\\
\partial_{\rho_Y} \partial_y f & \partial_y \partial_y f
\end{pmatrix}
\begin{pmatrix}
\delta \rho_Y\\
\delta y
\end{pmatrix}
= 0.
\end{align}
We need to compare the rank of the coefficient matrix in \eqref{mat:iotaVscd} with that of $T{}^\scat \dd_Y \varphi$ at points of the form $(\rho_X,x,0,y)$.
For this purpose, we go through a series of ``reductions'', along the lines of the proof of Lemma \ref{lem:horror}, to simplify the comparison.
First, we can identify ${}^\scat \dd_X\varphi$ with
\[
\rho_Y^{-1}\begin{pmatrix}- f+\rho_X\partial_{\rho_X}f \\ \partial_x f \end{pmatrix}=:\rho_Y^{-1} h.
\]
Note that $h\neq 0$ near $\overline{\mathcal{C}_\varphip}$, since $\varphi$ is a phase function.
As in the proof of Lemma \ref{lem:horror}, the evaluation at $(\rho_X,x,0,y)$ then gives
\begin{align}
\label{eq:matma}
\begin{pmatrix}
\partial_{\rho_Y} \iota {}^\scat \dd_X \varphi & \partial_y \iota {}^\scat \dd_X \varphi\\
\partial_{\rho_Y} \partial_y f & \partial_y \partial_y f
\end{pmatrix}= \begin{pmatrix}
-\frac{h}{|h|^2}+\partial_{\rho_Y}\frac{h}{|h|} & \partial_{y} \frac{h}{|h|} \\
\partial_{\rho_Y} \partial_y f & \partial_y \partial_y f
\end{pmatrix}.
\end{align}
Since all derivatives of $\frac{h}{|h|}$ are orthogonal to $\frac{h}{|h|}$ and $h\neq 0$, the rank of the matrix \eqref{eq:matma} equals the one of
\begin{align}
\label{eq:matmat}
\begin{pmatrix}
-\frac{h}{|h|^2} & \partial_y \frac{h}{|h|}\\
0 & \partial_y \partial_y f
\end{pmatrix}.
\end{align}
In fact, in \eqref{eq:matma}, as well as in \eqref{eq:matmat}, the first column is linearly independent of the others.
Now we write
$$\partial_{y_j} \frac{h}{|h|}=\frac{1}{|h|}\partial_{y_j} h- \underbrace{\frac{(h\cdot\partial_{y_j} h)}{|h|^3} h}_{\text{collinear to }h},$$
and remove the collinear summands, which again does not change the rank of the matrix \eqref{eq:matmat}. Therefore, the rank of \eqref{eq:matma} is the same
as the one of
\begin{align}
\label{eq:matmat2}
\begin{pmatrix}
-\frac{h}{|h|^2} & \frac{1}{|h|}\partial_y h\\
0 & \partial_y \partial_y f
\end{pmatrix}.
\end{align}
Multiplying the first $d$ rows and the first column of \eqref{eq:matmat2} by the non-vanishing factor $|h|$, again the rank does not change, and we can look at
\begin{align}
\label{eq:matmat3}
\begin{pmatrix}
-h & \partial_y h\\
0 & \partial_y \partial_y f
\end{pmatrix}=\begin{pmatrix}
f-\rho_X\partial_{\rho_X}f & -\partial_yf + \rho_X\partial_y\partial_{\rho_X} f\\
-\partial_x f & \partial_y\partial_x f\\
0 & \partial_y \partial_y f
\end{pmatrix}.
\end{align}
On $\mathcal{C}_\varphi$ at $\rho_Y=0$ this equals
\begin{align}
\label{eq:matmat4}
\begin{pmatrix}
-\rho_X\partial_{\rho_X}f & \rho_X\partial_y\partial_{\rho_X} f\\
-\partial_x f & \partial_y\partial_x f\\
0 & \partial_y \partial_y f
\end{pmatrix}.
\end{align}
Finally, we observe that the dimension of the null space of \eqref{eq:matmat4} is,
by cleanness of $\varphi$ (in particular by \eqref{eq:cleanbdry} applied to $\mathcal{C}_\varphip$ or $\mathcal{C}_\varphipe$), the same as the one of
\begin{align}
\label{eq:cleandiff}
\begin{pmatrix}
- \partial_{\rho_X}f & \partial_y \partial_{\rho_X} f\\
- \partial_x f & \partial_y\partial_x f\\
0 & \partial_y\partial_{\rho_Y} f\\
0 & \partial_y\partial_y f
\end{pmatrix} = T{}^\scat \dd_Y \varphi|_{\mathcal{C}_\varphip},
\end{align}
namely $e$.
Therefore, the rank of ${\lambda_\varphi}$ equals $d=(d+e)-e$ near $\mathcal{C}_\varphi$, which concludes the proof.
\end{proof}
\begin{lem}
\label{lem:lpfibration}
The map ${\lambda_\varphi}: C_\varphi\rightarrow L_\varphi$ is a local fibration and the fiber is everywhere a smooth manifold without boundary.
\end{lem}
\begin{proof}
Since ${\lambda_\varphi}$ is locally an ${\mathrm{sc}}$-map, $T{\lambda_\varphi}$ maps the set of vectors at the boundary that are inwards pointing into itself, see Remark \ref{rem:inward}. Therefore ${\lambda_\varphi}$ is a so-called ``tame'' submersion in the sense of \cite[Lemma 1.3]{Nistor}. As such, it is a local fibration and the fiber is a manifold without boundary.
\end{proof}
\subsection{Symplectic properties of the associated Lagrangian}
\label{sec:symp}
As in the classical theory, $L_\varphi$ is an immersed Lagrangian submanifold, and its boundary faces $\Lambda^\bullet$ are immersed Legendrian submanifolds. Let us briefly recall these concepts. For more information, the reader is referred to \cite{CoSc2,MZ,HV}.
As a cotangent space, $T^*X^o$ carries a natural symplectic $2$-form $\omega$ induced by the canonical $1$-form $\alpha\in{\mathscr{C}^\infty}(T^*X^o,T^*(T^*X^o))$ as $\omega=\dd\alpha$. This $1$-form can be recovered from $\omega$ by setting $\alpha=\varrho^\psi\lrcorner\,\,\omega$ for the radial vector field $\varrho^\psi$ on ${\mathscr{C}^\infty}(T^*X^o)$, which is given by $\varrho^\psi=\xi\cdot\partial_{\xi}$ in canonical coordinates. \\
We now write $(\mathbf{x},\mathbf{x}i)=(\rho_X,x,\rho_\Xi,\xi)$ for the coordinates in the mwc ${}^\scat \,\overline{T}^*X$ which are obtained from the rescaled canonical coordinates under radial compactification in the fiber, cf. \cite{MZ}. Then $\varrho^\psi$ corresponds to $\rho_\Xi\partial_{\rho_\Xi}$ on ${\mathscr{C}^\infty}(\overline{T}^*X^o)$. For the purpose of scattering geometry, it is natural to rescale further and define, on $T^*({}^\scat \,\overline{T}^*X)^o$,
$$\alpha^\psi:=\rho_\Xi^2\partial_{\rho_\Xi}\lrcorner\,\omega.$$
There exists another form of interest, namely
\begin{align*}
\alpha^e:=\rho_X^2\partial_{\rho_X}\lrcorner\,\omega.
\end{align*}
We now extend these forms to $T^*({}^\scat \,\overline{T}^*X)$ and define the boundary restrictions of $\alpha^\bullet$.
Observe that, while their explicit form depends on the choice of bdfs, the induced contact structure at the boundary does not, see next Lemma \ref{lem:alphaext}
\begin{lem}\label{lem:alphaext}
The forms $\alpha^\bullet$ extend to $1$-forms on $\mathcal{W}^\bullet$, denoted by the same letter. The induced contact structures do not depend on the choice of bdfs.
\end{lem}
\begin{ex}
On $T^*{\mathbb{R}}d\cong {\mathbb{R}}d\times{\mathbb{R}}d$, with canonical coordinates $(x,\xi)$, the vector fields $\varrho^\psi$ and $\varrho^e$ correspond to $\varrho^\psi=\xi\cdot\partial_\xi$ and $\varrho^e=x\cdot\partial_x$. The symplectic $2$-form is $\sum_j\dd\xi_j\wedge\dd x_j$ and hence
$$\varrho^\psi\lrcorner\,\omega=\xi\cdot \dd x\quad \text{and}\quad\varrho^e\lrcorner\,\omega=-x\cdot \dd \xi.$$
Obviously, the coefficients of these forms diverge as $[\xi]\rightarrow \infty$ and $[x]\rightarrow \infty$. The rescaled forms ``at the boundary at infinity'' then correspond to
$$\alpha^\psi=\frac{\xi}{[\xi]}\cdot \dd x\quad \text{and}\quad \alpha^e=-\frac{x}{[x]}\cdot \dd \xi.$$
After a choice of coordinates near the respective boundaries, this is the general local geometric situation.
\end{ex}
We are now in the position to formulate the symplectic properties of $\Lambda_\varphi$, cf. \cite{CoSc}. Recall that a submanifold $N$ of a symplectic manifold $(M,\omega)$ is Lagrangian if $\omega|_{TN}=0$ and a submanifold $N$ of a contact manifold $(M,\alpha)$ is Legendrian if $\alpha|_{TN}=0$.
\begin{prop}
The immersed manifolds defined in Theorem \ref{thm:lpsubm} satisfy:
\begin{itemize}
\item[1.)] $L_\varphi^o$ is an immersed Lagrangian submanifold with respect to the $2$-form $\omega$ on $({}^\scat \,\overline{T}^* X)^o\cong T^*X$;
\item[2.)] $\Lambda_\varphip$ is Legendrian with respect to the canonical $1$-form $\alpha^\psi$ on $\Wt^\psi\cong S^*(X^o)$;
\item[3.)] $\Lambda_\varphie$ is Legendrian with respect to the $1$-form $\alpha^e$ on $\Wt^e\cong T^*_{\partial X}X$.
\end{itemize}
\end{prop}
We take this as the definition of an ${\mathrm{sc}}$-Lagrangian, cf. \cite{CoSc2}.
\begin{defn}[${\mathrm{sc}}$-Lagrangians]\label{def:scLagr}
Let $\Lambda:=\overline{\Lambda^\psi}\cup\overline{\Lambda^e}\subset \mathcal{W}$. $\Lambda$ is called an ${\mathrm{sc}}$-Lagrangian if:
\begin{itemize}
\item[1.)] $\Lambda^\psi=\Lambda\cap\Wt^\psi$ is Legendrian with respect to the canonical $1$-form $\alpha^\psi$ on $\Wt^\psi = {}^\scat \,S^*_{X^o}X$;
\item[2.)] $\Lambda^e=\Lambda\cap\Wt^e$ is Legendrian with respect to the $1$-form $\alpha^e$ on $\Wt^e = {}^\scat \,T^*_{\partial X}X$;
\item[3.)] $\overline{\Lambda^\psi}$ has a boundary if and only if $\overline{\Lambda^e}$ has a boundary, and, in this case,
$$\Lambda^{\psi e}:=\partial \overline{\Lambda^\psi}=\partial \overline{\Lambda^e}=\overline{\Lambda^\psi}\cap\partial \overline{\Lambda^e},$$
with clean intersection.
\end{itemize}
\end{defn}
Figure \ref{fig:Lpintersect}, which is taken from \cite{CoSc2}, summarizes, schematically, the relative positions of $\Lambda_\varphie$ and $\Lambda_\varphip$ near the corner in $W$.
\begin{figure}
\caption{Intersection of $\Lambda^\psi\subset\Wt^\psi$ and $\Lambda^e\subset\Wt^e$ at the corner $\Wt^\psie$}
\label{fig:Lpintersect}
\end{figure}
We may take the analysis one step further in order to stress the Legendrian character of the boundary components near the corner and to reveal the symplectic properties of $\Lambda^{\psi e}$ by blow-up. For the sake of brevity here, we move this analysis to the appendix, Section \ref{sec:blowup}.
We may sum up our previous analysis by stating the next Theorem \ref{thm:imphlagr}.
\begin{thm}\label{thm:imphlagr}
For a clean phase function $\varphi$, the image $\Lambda_\varphi$ under ${\lambda_\varphi}$ of $\mathcal{C}_\varphi$ is an immersed ${\mathrm{sc}}$-Lagrangian.
\end{thm}
\begin{defn}
We say that an ${\mathrm{sc}}$-Lagrangian $\Lambda$ is locally parametrized by a phase function $\varphi$ if, over the domain of definition of $\varphi$, we have $\Lambda=\Lambda_\varphi$.
\end{defn}
In particular, if $\Lambda$ is locally parametrized by a phase function, then it is admissible. Conversely, we have the following result, cf. \cite{CoSc2}.
\begin{prop}
\label{prop:locpar}
If $\Lambda$ is an ${\mathrm{sc}}$-Lagrangian, then it is locally parametrizable by a clean phase function $\varphi$, that is $\Lambda^\bullet\cap U^\bullet=\Lambda_\varphi^\bullet\cap U^\bullet$ for some open $U\subset \mathcal{W}^\bullet$. In particular, $\Lambda$ arises as the boundary of some Lagrangian submanifold $L_\varphi$ of ${}^\scat \,\overline{T}^* X$.
\end{prop}
\begin{rem}
The proof of Proposition \ref{prop:locpar} in \cite{CoSc2} is based on concrete parametrizations in ${\mathbb{R}}d\times{\mathbb{R}}d$. It applies here nonetheless, since any $d$-dimensional manifold with boundary $X$ can be locally modelled by ${\mathbb{B}}d$. Hence, ${}^\scat \,\overline{T}^*X$ can be locally modelled by ${\mathbb{B}}d\times{\mathbb{B}}d$ and thus, under inverse radial compactification (applied to both factors), by ${\mathbb{R}}d\times{\mathbb{R}}d$.
Note that in \cite{CoSc2} we imposed additional conditions, namely
\begin{equation}
\label{eq:nonzerosec}
\Lambda^e\cap (\partial X\times\iota(\{0\}))=\emptyset,
\end{equation}
and that $x\cdot \xi=0$ in local canonical coordinates on $\Lambda^{\psi e}$, since this is always true for a parametrized Lagrangian (see \eqref{eq:conormbi} below). However, condition \eqref{eq:nonzerosec} is equivalent to the stronger assumption that ${}^\scat \dd\varphi\neq 0$ also on $\mathcal{B}e$, which we do not impose here. The assumption $x\cdot \xi=0$, in turn, is superfluous, since it already follows from the symplectic assumptions on $\Lambda^{\psi e}$, as we now show.
Assume that both $\xi\cdot \dd x\equiv 0$ and $-x\cdot \dd \xi\equiv 0$ on a bi-conic submanifold $L$ of ${\mathbb{R}}^d\times{\mathbb{R}}^d$. Then we must have $\dd(x\cdot\xi)=0$. However, when $|x|$ and $|\xi|$ tend to $\infty$, this blows up unless $x\cdot \xi=0$. This shows that $x\cdot \xi=0$ is indeed automatically fulfilled.
This corresponds to the fact that, for the bi-homogenous principal symbol of a phase function $\varphi^{\psi e}$, we have, when
$\nabla_\theta\varphi(x,\theta)=0$, that (cf. \cite{CoSc2})
\begin{equation}
\label{eq:conormbi}
\langle x,\nabla_x\varphi(x,\theta)\rangle=\varphi(x,\theta)=\langle \theta,\nabla_\theta\varphi(x,\theta)\rangle=0,
\end{equation}
where we have used Euler's identity for homogeneous functions twice.
\end{rem}
\subsection{Scattering conormal bundles}
\label{sec:conorm}
In this section, we consider the simple example of a scattering conormal bundle. Consider a $k$-dimensional submanifold $X'\subset X$ which intersects the boundary of $X$ cleanly or not at all (called $p$-submanifold in \cite{Melrosemwc}). In the following, we assume an intersection with the boundary.
Then there exist local coordinates $(\rho_X,x^{\mathrm{pr}}ime,x'')$ such that $X'$ is locally given by
$$X'=\{(\rho_X,x^{\mathrm{pr}}ime,x'')\mid \rho_X\geq 0, x^{\mathrm{pr}}ime=0\in{\mathbb{R}}^{d-1-k}, x''\in{\mathbb{R}}^{k-1}\}.$$
We can now consider the compactified scattering conormal ${}^\scat \,\overline{T}^*X'\subset{}^\scat \,\overline{T}^*_{X'}X$. The boundary faces of ${}^\scat \,\overline{T}^*X'$ constitute a Lagrangian.
In fact, write $X=\iota({\mathbb{R}}d)$, so that $X'$ corresponds to a subspace of ${\mathbb{R}}d$ of the form
$$X'=\{(x^{\mathrm{pr}}ime,x'')\mid x'=0\in{\mathbb{R}}^{d-k}, x''\in{\mathbb{R}}^{k}\}.$$
We can then introduce $Y=\iota({\mathbb{R}}^{d-k})$ and $\phi(x,y)=x'\cdot y$ on ${\mathbb{R}}d\times{\mathbb{R}}^{d-k}$, which is an ${\mathrm{SG}}$-phase function, taking into account \eqref{eq:SGphaseineq}. The true phase function on $X\times Y$ is then $(\iota^{-1}\times\iota^{-1})^*\phi$. We can then compute $C_\varphi=X'\times Y$ and $\Lambda_\varphi={}^\scat \,\overline{T}^*X'$.
Indeed, in the Euclidean setting, $\Lambda_\varphi$ corresponds to the the three conic manifolds
\begin{align*}
\Lambda_\varphie&=\{(0,x'',\xi^{\mathrm{pr}}ime,0)\}\subset ({\mathbb{R}}dz)\times{\mathbb{R}}d\\
\Lambda_\varphipe&=\{(0,x'',\xi^{\mathrm{pr}}ime,0)\}\subset ({\mathbb{R}}dz)\times({\mathbb{R}}dz)\\
\Lambda_\varphip&=\{(0,x'',\xi^{\mathrm{pr}}ime,0)\}\subset {\mathbb{R}}d\times({\mathbb{R}}dz)
\end{align*}
which have the claimed symplectic properties. Compactification of the ${\mathbb{R}}d$-components and projection of the conic $({\mathbb{R}}dz)$-component to the corresponding sphere then yields the compactified notions in ${}^\scat \,\overline{T}^*X$.
\section{Phase functions which parametrize the same Lagrangian}
\label{sec:exchphase}
In this section, we adapt the classical techniques for exchanging the phase function locally parametrizing a given Lagrangian, see \cite[Chapter 8.1]{Treves}, to the setting with boundary. Since $\Lambda_\varphi$, not $L_\varphi$, is our true object of interest, we say that two phase functions $\varphi_i$, $i=1,2$, locally parametrize the same Lagrangian at $p_0\in\mathcal{W}$ if $\Lambda_{\varphi_1}=\Lambda_{\varphi_2}$ in a small (relatively) open neighbourhood of $p_0$ in the respective boundary faces.
Our first observation is the following:
\begin{lem}
\label{lem:phaseplussmooth}
If $\varphi\in\rho_X^{-1}\rho_{{\mathbb{B}}^s}^{-1}{\mathscr{C}^\infty}(X \times {\mathbb{B}}^s)$ is a local phase function and $r\in {\mathscr{C}^\infty}(X \times {\mathbb{B}}^s)$, then $\varphi+r$ is still a local phase function and it parametrizes the same Lagrangian as $\varphi$.
\end{lem}
\begin{proof}
Since $r\in{\mathscr{C}^\infty}(X \times {\mathbb{B}}^s)$, ${}^\scat \dd r=0$ when restricted to the boundary. Therefore, $\varphi+r$ is still a local phase function. By the same reason, $\mathcal{C}_{\varphi}=\mathcal{C}_{\varphi+r}$. Finally, we have
$$\lambda_{\varphi+r}(\mathbf{x},by)=(\mathbf{x},\iota({}^\scat \dd_X(\varphi + r))).$$
Computing ${}^\scat \dd_X(\varphi + r)$ in coordinates, see \eqref{eq:scdxexpl},
$$
{}^\scat \ddX\varphi=\rho_Y^{-1} \left((-f+\rho_X\partial_{\rho_X}f+\rho_Y\rho_X^2\partial_{\rho_X}r) \frac{\dd \rho_X}{\rho_X^2} +
\sum_{j=1}^{d-1}(\partial_{x_j}f+\rho_Y\rho_X\partial_{x_j}r)\frac{\dd x_j}{\rho_X} \right),
$$
we observe that at $\rho_X=0,$ the contribution from $r$ vanishes. The same is true in the limit of $\rho_Y\rightarrow 0$ under application of $\iota$, see also Lemma \ref{lem:horror}.
\end{proof}
\subsection{Increasing fiber variables}
Given a clean phase function $\varphi \in \rho_X^{-1}\rho_{{\mathbb{B}}^s}^{-1}C^\infty(X \times {\mathbb{B}}^s)$ with excess $e$, define $\widetilde{\psi} \in \rho_X^{-1} \rho_{{\mathbb{B}}^s}^{-1} C^\infty(X \times {\mathbb{B}}^s \times (-\varepsilon, \varepsilon))$ as follows:
\[\widetilde{\psi}(\mathbf{x},\mathbf{y}, \tilde y) = \varphi(\mathbf{x}, \mathbf{y}) + \frac{\tilde y^2}{\rho_X \rho_{{\mathbb{B}}^s}}.\]
We see that ${}^\scat \dd\widetilde{\psi} \neq 0$ when ${}^\scat \dd\varphi\neq 0$ and ${}^\scat \dd_{{\mathbb{B}}^s\times(-\varepsilonilon,\varepsilonilon)} \tilde{\psi} = 0$ if and only if $\tilde y = 0$ and ${}^\scat \dd_{{\mathbb{B}}^s} \varphi = 0$.
Thus,
$$C_{\widetilde{\psi}} = \left\{(\mathbf{x},\mathbf{y}, 0)\mid (\mathbf{x},\mathbf{y}) \in C_\varphi\right\},$$
which implies that the excess is not changed, and $\Lambda_{\widetilde{\psi}} = \Lambda_{\varphi}$. Summing up, $\psi$ is a local clean phase function in $s+1$ fiber variables with the same excess $e$ as $\varphi$ and (locally) parametrizing the same Lagrangian as $\varphi$.
This construction may once again be moved to balls, by using Example \ref{ex:embdball} and setting $\psi = \Psi^*\widetilde{\psi}$. Then $\psi \in \rho_X^{-1}\rho_{{\mathbb{B}}^{s+1}}^{-1}C^\infty(X \times U)$.
Using the fact that ${}^\scat \dd \psi = \Psi^*\widetilde{\psi}$, we see that $\psi$ is a clean phase function parametrizing $\Lambda_{\varphi}$ with excess $e$.
Again, $X\times{\mathbb{B}}^s$ can be exchanged by any relatively open subset, hence starting with local phase functions.
\subsection{Reduction of the fiber variables}\label{subs:fbred}
Starting again from a clean phase function $\varphi \in \rho_X^{-1}\rho_{{\mathbb{B}}^s}^{-1}C^\infty(X \times {\mathbb{B}}^s)$ with excess $e$, we now construct a (local) phase function $\psi$ in the smallest possible number of phase variables (without changing the excess) which (locally) parametrizes the same Lagrangian.
The argument is similar to the classical one, but extra attention needs to be paid at to what happens near points with $\rho_Y=0$, namely, we never seek to get rid of $\rho_Y$ as a parameter.
\begin{rem}
In the classical theory, meaning for homogeneous phase functions, it is possible to reduce the number of fiber variables under the assumption that the matrix $\partial^2_{\theta\theta}\varphi(x,\theta)$ has rank $r>0$ on $C_\varphi$.
However, since a classical phase function $\varphi$ is homogeneous in $\theta$, it holds that $\theta\cdot \nabla_\theta\varphi=\varphi$ and hence the second radial derivative is automatically zero on $C_\varphi$. Furthermore, the radial variable can always be chosen to parametrize $\Lambda_\varphi$.
\end{rem}
We proceed as in the proof of Theorem \ref{thm:lpsubm}. We first recall that, for $p_0\in C_\varphi$, writing
$\varphi=\rho_Y^{-1}\rho_X^{-1}f$ with $f\in C^\infty(X \times {\mathbb{B}}^s)$,
we have there
\begin{equation}\label{eq:scDyphi}
0={}^\scat \ddY\varphi=\left(-f+\rho_Y\partial_{\rho_Y}f,\partial_{y_k}f \right).
\end{equation}
We then identify $T_Y{}^\scat \ddY\varphi$ in coordinates with the matrix
\begin{equation}
\label{eq:DyDyphi}
J_Y\varphi = \begin{pmatrix}
\rho_Y\partial_{\rho_Y}^2f & -\partial_{y_j}f+\rho_Y \partial_{y_j}\partial_{\rho_Y} f \\
\partial_{\rho_Y}\partial_{y_k} f & \partial_{y_j}\partial_{y_k} f
\end{pmatrix}.
\end{equation}
We see, using \eqref{eq:scDyphi}, that on $\mathcal{C}_\varphip \subset \{\rho_Y = 0\}$ this becomes
\begin{equation}
\label{eq:Tyscdphipsi}
J_Y\varphi\big|_{\mathcal{C}_\varphi^\psi}= \begin{pmatrix}
0 & 0 \\
\partial_{\rho_Y}\partial_{y_k} f & \partial_{y_j}\partial_{y_k} f
\end{pmatrix}.
\end{equation}
Therefore, the rank of this matrix is at most $s-1$. Indeed, we observe that, by \eqref{eq:cleanbdry}, at $\rho_Y=0$ we have $\dd\rho_Y\neq 0$ on $TC_\varphi^\psi$ and hence we can always choose $\rho_Y$ as a parameter to locally describe $C^\psi_\varphi$.
\begin{rem}
By the same argument, $\rho_X$ can be chosen
as a parameter close to $\mathcal{B}e$, while, close to $\mathcal{B}pe$,
both $\rho_X$ and $\rho_Y$ can be chosen as parameters to represent
$C_\varphi$.
\end{rem}
We now seek to reduce the remaining set of variables under the assumption that
\begin{equation}
\label{eq:DyDyass0}
\text{The matrix }\big(\partial_{y_j}\partial_{y_k} \rho_X\rho_Y\varphi\big)_{jk} \text{ has rank }r>0\text{ at }p_0\in\mathcal{C}_\varphip\cup\mathcal{C}_\varphipe.
\end{equation}
Since at points where $\rho_Y\neq 0$ the variable $\rho_Y$ behaves like all other variables, the same restriction does not hold near a point $p\in \mathcal{C}_\varphie$. Here, we simply assume that
\begin{equation}
\label{eq:DyDyass1}
\text{The matrix } T_Y{}^\scat \ddY\varphi \text{ has rank }r>0\text{ at }p_0\in\mathcal{C}_\varphie.
\end{equation}
Since up to multiplication by $\rho_Y>0$ in one row, \eqref{eq:DyDyphi} is the Hessian of $h$ (with respect to $\mathbf{y}$), this is equivalent to
$\mathrm{rk}(H_Y f)=r>0$.
The two conditions may be summarized into one. Namely, consider the scattering Hessian (with respect to the $\mathbf{y}$-variables) of $\varphi$
\begin{equation}
\begin{aligned}
{}^\scat H_Y\varphi&=\begin{pmatrix}
\rho_Y^2\rho_X\partial_{\rho_Y}\rho_Y^2\rho_X\partial_{\rho_Y}\varphi & \rho_Y\rho_X\partial_{y_j}\rho_Y^2\rho_X\partial_{\rho_Y}\varphi \\
\rho_Y^2\rho_X\partial_{\rho_Y}\rho_Y\rho_X\partial_{y_k}\varphi & \rho_Y\rho_X\partial_{y_j}\rho_Y\rho_X\partial_{y_k}\varphi
\end{pmatrix}
\\
&= \rho_Y\rho_X\begin{pmatrix}
\rho_Y^2\partial_{\rho_Y}^2f & -\partial_{y_j}f+\rho_Y \partial_{y_j}\partial_{\rho_Y} f \\
\rho_Y\partial_{\rho_Y}\partial_{y_k} f & \partial_{y_j}\partial_{y_k} f
\end{pmatrix}.
\end{aligned}
\end{equation}
Then $\rho_Y^{-1}\rho_X^{-1}\,{}^\scat H_Y\varphi$ becomes, at a point in $\mathcal{C}_\varphi$:
\begin{align*}
\rho_Y^{-1}\rho_X^{-1}{}^\scat H_Y\varphi&=\begin{pmatrix}
0 & 0 \\
0 & \partial_{y_j}\partial_{y_k} f
\end{pmatrix},\quad\text{ if }p_0\in\mathcal{C}_\varphip\cup\mathcal{C}_\varphipe; \\
\rho_Y^{-1}\rho_X^{-1}\,^{\mathrm{sc}\hspace{-2pt}} H_Y\varphi&=\begin{pmatrix}
\rho_Y^2\partial_{\rho_Y}^2f & \rho_Y \partial_{y_j}\partial_{\rho_Y} f \\
\rho_Y\partial_{\rho_Y}\partial_{y_k} f & \partial_{y_j}\partial_{y_k} f
\end{pmatrix},\quad\text{ if }p_0\in\mathcal{C}_\varphie.
\end{align*}
Notice that we can factorize these matrices as
\begin{equation}
\label{eq:scatHessfactor}
\begin{pmatrix} \rho_Y & 0 \\
0 & \mathbbm{1} \end{pmatrix}
\begin{pmatrix}
\partial_{\rho_Y}^2f & \partial_{y_j}\partial_{\rho_Y} f \\
\partial_{\rho_Y}\partial_{y_k} f & \partial_{y_j}\partial_{y_k} f
\end{pmatrix}
\begin{pmatrix} \rho_Y & 0 \\
0 & \mathbbm{1} \end{pmatrix},
\end{equation}
the rank of which therefore is, for $\rho_Y\neq 0$, that of the standard Hessian of $f$, $H_Y f$. Therefore, our assumption may be expressed as:
\begin{equation}
\label{eq:DyDyass2}
\text{The matrix } \rho_Y^{-1}\rho_X^{-1}\,^{\mathrm{sc}\hspace{-2pt}} H_Y\varphi \text{ has rank }r>0\text{ at }p_0\in\mathcal{C}_\varphi.
\end{equation}
We may now proceed as in the standard theory and introduce a splitting of variables $\mathbf{y}=(\mathbf{y}^{\mathrm{pr}}ime,\mathbf{y}^{{\mathrm{pr}}ime {\mathrm{pr}}ime})$
such that $(\partial_{\mathbf{y}^{{\mathrm{pr}}ime{\mathrm{pr}}ime}}\partial_{\mathbf{y}^{{\mathrm{pr}}ime{\mathrm{pr}}ime}}f)_{jk}$ is an invertible $r\times r$ matrix. We can then apply the implicit function theorem to
$$0={}^\scat \ddY\varphi=\left(-f+\rho_Y\partial_{\rho_Y}f,\partial_{y_k}f \right)$$
at $p_0$. We obtain a map from an open neighbourhood of $p_0$,
$$k:(\mathbf{x},\mathbf{y}^{{\mathrm{pr}}ime})\mapsto \big(\mathbf{x},\mathbf{y}^{\mathrm{pr}}ime,\mathbf{y}^{{\mathrm{pr}}ime{\mathrm{pr}}ime}(\mathbf{x},\mathbf{y}^{\mathrm{pr}}ime)\big),$$
such that $C_\varphi$ and the range of $k$ locally coincide. Note that $k$ is a scattering map, since $\rho_Y$ is always one of the $\mathbf{y}^{\mathrm{pr}}ime$ near the $\psi$-face.
Then $\varphi_{\red}=\varphi\circ k$ is a clean local phase function in $d\times (s-r)$ variables with excess $e$, and $k$ provides a local isomorphism $C_{\varphi_{\red}}\rightarrow C_\varphi$. Furthermore, at stationary points $p_0$ and $k(p_0)$, we have that $\iota({}^\scat \ddX \varphi_{\red})=\iota({}^\scat \dd_X \varphi)$, since ${}^\scat \ddY\varphi=0$ there.
Hence, $\varphi_{\red}$ locally parametrizes the same Lagrangian as $\varphi$.
\begin{rem}
Note that, after applying a change of coordinates in the $\mathbf{y}$ variables,
$\varphi_{\red}$ may be assumed to be defined on ${\mathbb{B}}d\times{\mathbb{B}}^{s-r}$, see also Lemma \ref{lem:CLFstarff} below.
\end{rem}
Summing up, we can formulate the next Proposition \ref{prop:fiberred}.
\begin{prop}
\label{prop:fiberred}
Let $\varphi\in\rho_Y^{-1}\rho_X^{-1}{\mathscr{C}^\infty}(X\times{\mathbb{B}}^s)$ be a local clean phase function of excess $e$. Assume
$$\rho_Y^{-1}\rho_X^{-1}\,^{\mathrm{sc}\hspace{-2pt}} H_Y\varphi\text{ has rank }r>0\text{ at a stationary boundary point }p_0\in\mathcal{C}_\varphi.$$
We may then define a local phase function $\varphi\in\rho_Y^{-1}\rho_X^{-1}{\mathscr{C}^\infty}(X\times{\mathbb{B}}^{s-r})$ of excess $e$ parametrizing the same Lagrangian.
\end{prop}
We mention that, locally, the minimal number of fiber variables $y$ that a clean phase function of excess $e$ locally parametrizing $L_\varphi$ has to possess is $$s_{\mathrm{min}}=d+e-n,$$ where $n$ is the (local) number of independent $x$ variables on $L_\varphi$. This follows from a simple dimension argument: the dimension of $L_\varphi$ is $d$, that of $C_\varphi$ is $d+e$, and the one of the projection to $x$ of $C_\varphi$ coincides with that of $L_\varphi$. Note that, by cleanness of the intersection $\mathcal{C}_\varphi\cap\mathcal{B}p$, near $\Lambda^\psi$ we have $s_{\mathrm{min}}>0.$
\subsection{Increasing the excess}
Given a (local) clean phase function $\varphi \in \rho_X^{-1}\rho_{{\mathbb{B}}^s}^{-1}C^\infty(X \times {\mathbb{B}}^s)$ with excess $e$, define $\psi:={\mathrm{pr}}_{X\times{\mathbb{B}}^s}^*\varphi$ on $X\times({\mathbb{B}}^s\times(-\varepsilon,\varepsilon))$, viewing ${\mathbb{B}}^s\times(-\varepsilon,\varepsilon)$ as an open subset of ${\mathbb{B}}^s\times{\mathbb{S}}^1$, which is a manifold with boundary whose boundary defining function may be chosen as ${\mathrm{pr}}_{{\mathbb{B}}^s}^*\rho_{{\mathbb{B}}^s}$. In particular we have, with the obvious identifications,
$${}^\scat \dd_{{\mathbb{B}}^s\times(-\varepsilon,\varepsilon)}\psi={\mathrm{pr}}_{X\times{\mathbb{B}}^s}^*\left({}^\scat \dd_{{\mathbb{B}}^s}\varphi\right).$$
Then $C_\psi=C_\varphi\times(-\varepsilon,\varepsilon)$ and hence $\dim(C_\psi^\bullet)=\dim(C_\varphi^\bullet)+1.$ Furthermore, $\lambda_\psi={\mathrm{pr}}_{X\times{\mathbb{B}}^s}^*{\lambda_\varphi}$ and $\Lambda_\varphi=\Lambda_\psi$. Summing up, $\psi$ is a local clean phase function in $s+1$ fiber variables with excess $e+1$, defined and (locally) parametrizing the same Lagrangian as $\varphi$.
As before, we may choose to keep working on balls by invoking the construction from Example \ref{ex:embdball} and replacing $\psi$ with
$$\Psi^*\psi=\widetilde{\Psi}^*\varphi \in \rho_X^{-1}\rho_{{\mathbb{B}}^{s+1}}^{-1}{\mathscr{C}^\infty}(X \times U).$$
In this way, since $\Psi$ is a diffeomorphism, $\psi$ becomes a clean phase function with excess $e+1$ defined on a relatively open subset of $X\times{\mathbb{B}}^{s+1}$ and similarly we may raise the excess by any natural number.
\begin{ex}
The standard Fourier phase on ${\mathbb{R}}\times{\mathbb{R}}$, $\varphi(x,\xi)=x\cdot \xi$, cannot be seen as an ${\mathrm{SG}}$-phase on all of ${\mathbb{R}}\times{\mathbb{R}}^2$ by setting $\psi(x,\xi,\eta)=x\cdot\xi$. Indeed,
\begin{align}
\label{eq:SGphasextend}
\jap{x}^2|\nabla_x \varphi(x)|^2+\jap{(\xi,\eta)}|\nabla_{\xi,\eta}\varphi|^2&=(1+x^2)\xi^2+(1+\xi^2+\eta^2)x^2\\
\notag &=\jap{x}\jap{\xi}+x^2\eta^2-1
\end{align}
For $\xi=0$ and $x=0$ and $\eta\rightarrow \infty$, this vanishes but should be bounded from below by $c(1+|\eta|)^2$ if $\psi$ were an ${\mathrm{SG}}$-phase function, given \eqref{eq:SGphaseineq}.
Reviewing Example \ref{ex:embdball}, the ray $\xi=0$, $x=0$ and $\eta\neq 0$ corresponds precisely to the poles in Figure \ref{fig:fiberball} which were cut off. Indeed, \eqref{eq:SGphasextend} is bounded from below by $\jap{x}^2\jap{(\xi,\eta)}^2$ in any neighbourhood where $\frac{|\xi|}{|\eta|}>c $ and hence a local phase function in such sets.
\end{ex}
\subsection{Elimination of excess}
\label{sec:phaseexelim}
Assume now that $\varphi$ is a phase function on $X \times {\mathbb{B}}^s$ with excess $e$ and that at some point $p_0=(\rho_{X,0},x_0,\rho_{Y,0},y_0)\in \mathcal{C}_\varphi$ we have ${\lambda_\varphi}(p_0)=(\rho_{X,0},x_0,\rho_{\Xi,0},\xi_0)$. Then, by Lemma \ref{lem:lpfibration}, the preimage of $(\rho_{X,0},x_0,\rho_{\Xi,0},\xi_0)$ under ${\lambda_\varphi}$, meaning the fiber in $\mathcal{C}_\varphi$ through $p_0$, is an $e$-dimensional smooth submanifold. Locally, since ${\lambda_\varphi}$ is a submersion we may, by \cite[Prop. 5.1]{Joyce}, reduce to the case of a projection, that is, we may find a splitting $y=(y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$ near $p_0$ such that ${\lambda_\varphi}$ does not depend on $y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}$. Then,
$$\tilde{\varphi}(\rho_{X},x,\rho_{Y},y^{\mathrm{pr}}ime):=\varphi(\rho_{X},x,\rho_{Y},y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}_0)$$
defines a phase function without excess (i.e., a non-degenerate phase function) that parametrizes the same Lagrangian as $\varphi$. As usual, we may again reduce to the case of a ball and hence replace $\varphi$ by a phase function on an open subset of $X\times{\mathbb{B}}^{s-e}$.
\subsection{Equivalence of phase functions}
We will now discuss the changes of phase function under a change of coordinates and which phase functions can be considered equivalent. We first check how the stationary points of a phase function transform under changes by local diffeomorphisms.
\begin{lem}\label{lem:CLFstarff}
Let $X_1$, $Y_1$, $X_2$, $Y_2$ be mwbs, set $B_i=X_i\times Y_i$, $i\in\{1,2\}$, and let $\varphi \in \rho_{X_2}^{-1}\rho_{Y_2}^{-1}C^\infty(B_2)$ be a (local) phase function. Assume $g:X_1\rightarrow X_2$, $h:Y_1\rightarrow Y_2$ to be diffeomorphisms, and set $F=g\times h$. Then, $F^*\varphi\in \rho_{X_1}^{-1}\rho_{Y_1}^{-1}C^\infty(B_1)$ is a (local) phase function with the same excess of $\varphi$, and we have
\begin{align*}
C_{F^*\varphi}&=\big\{(\mathbf{x}_1,\mathbf{y}_1)\in B_1\,|\,F(\mathbf{x}_1,\mathbf{y}_1)
\in C_\varphi \big\},\quad
L_{F^*\varphi}=({}^\scat \,\overline{T}^*g)(L_\varphi).
\end{align*}
\end{lem}
\begin{rem}
This means that, while the boundary defining function $\rho_{\Xi_1}$ of ${}^\scat \,\overline{T}^* X_1$ does not vanish,
$L_{F^*\varphi}$ can then be computed as
\[
L_{F^*\varphi}=
\big\{(\mathbf{x}_1, \,\iota(^t (Jg) \iota^{-1}(\boldsymbol{\xi}_1))\in
{}^\scat \,\overline{T}^*X_1\mid (g(\mathbf{x}_1), \boldsymbol{\xi}_1)\in L_\varphi\big\}.
\]
As $\rho_\Xi\rightarrow 0$, $\Lambda^\psi_{F^*\varphi}$ is obtained by taking interior limits, see also Lemma \ref{lem:horror}.
\end{rem}
\begin{proof}[Proof of Lemma \ref{lem:CLFstarff}]
The result for $C_\varphi$ follows immediately from
the first assertion in Lemma \ref{lem:scdpullback}.
The statement for $L_\varphi$ then follows by writing
\begin{equation}\label{eq:Lfpullback}
\lambda_{F^*\varphi}(\mathbf{x}_1,\mathbf{y}_1)=
({}^\scat \,\overline{T}^*g)(\lambda_\varphi(\mathbf{x}_2,\mathbf{y}_2))
\end{equation}
near a point
$(\mathbf{x}_1,\mathbf{y}_1) \in (C_{F^*\varphi})^o$ such that
$(\mathbf{x}_2,\mathbf{y}_2)=(g(\mathbf{x}_1),h(\mathbf{x}_1,\mathbf{y}_1))$. Indeed, at
these stationary points,
${}^\scat \dd_X F^*\varphi=F^*({}^\scat \dd_X\varphi$),
since there ${}^\scat \dd_Y\varphi=0$.
Since equality \eqref{eq:Lfpullback} holds in the interior,
the result at the boundary faces can be
obtained as interior limits (see also Lemma \ref{lem:lpsct}).
\end{proof}
\begin{rem}
The diffeomorphism $g\times h$ may be replaced by a single diffeomorphism $F:X_1\times Y_1\rightarrow X_2\times Y_2$ locally of product type near the boundary faces of $X_2\times Y_2$, i.e., a (local) diffeomorphism that is
a fibered-map at the boundary.
\end{rem}
We now define in which sense two phase functions may be considered equivalent.
\begin{defn}\label{def:phequiv}
Let $X$, $Y_1$, $Y_2$ be mwbs, $B_i=X\times Y_i$. Let
$\varphi_i\in\rho_{X}^{-1}\rho_{Y_i}^{-1}C^\infty(B_i)$.
We say that $\varphi_1$ and $\varphi_2$ are equivalent at a pair of boundary points
$(\mathbf{x}^0,\mathbf{y}_1^0)\in\mathcal{B}_1$ and $(\mathbf{x}^0,\mathbf{y}_2^0)\in\mathcal{B}_2$ if there exists a local diffeomorphism
$F:X\times Y_2\rightarrow X\times Y_1$ of the form $F=\mathrm{id}\times g$ with $g(\mathbf{x}^0,\mathbf{y}_2^0)=\mathbf{y}_1^0$ such that the following two conditions are met:
\begin{equation}
\label{eq:eqv1}
F^*\varphi_1-\varphi_2\text{ is smooth in a neighbourhood $U$ of }(\mathbf{x}^0,\mathbf{y}_2^0),
\end{equation}
\begin{equation}
\label{eq:eqv2}
\rho_X\rho_{Y_2} \left(F^*\varphi_1-\varphi_2\right) \text{ restricted to } \mathcal{C}_{\varphi_2}\cap \partial U \text{ vanishes to second order.}
\end{equation}
\end{defn}
\begin{lem}
\label{lem:equivlag}
Equivalent phase functions parametrize the same Lagrangian, meaning $\Lambda_{F^*\varphi}=\Lambda_{\varphi}$ and we have $\mathcal{C}_{F^*\varphi_1}=\mathcal{C}_{\varphi_2}$.
\end{lem}
\begin{proof}
This follows from Lemmas \ref{lem:phaseplussmooth} and \ref{lem:CLFstarff}.
\end{proof}
We now associate to any local phase function its \emph{principal phase part}, which corresponds in the ${\mathrm{SG}}$-case to the leading homogeneous components of $\varphi$. From the fact that the principal part of Definition \ref{def:princpart} is obtained from the boundary restrictions of $\varphi$, we observe, using $F=\mathrm{id}\times\mathrm{id}$ and Lemma \ref{lem:princpart}:
\begin{lem}\label{lem:phpequiv}
A local phase function $\varphi$ and its principal part $\varphi_p$ are equivalent.
\end{lem}
\begin{rem}
In particular, each phase function is locally equivalent at the $e$- and $\psi$-face, respectively, to a homogeneous (w.r.t. $\rho_X$ or $\rho_Y$) phase function, after a choice of collar decomposition. In general, this is not true near the corner $\mathcal{B}pe$.
\end{rem}
Since the difference in condition \eqref{eq:eqv2} is restricted to the boundary, it does not restrict the behavior of $F^*\varphi_1-\varphi_2$ into the direction transversal to the boundary, e.g. $\partial_{\rho_X}\rho_X\rho_{Y_2}(F^*\varphi_1-\varphi_2)$ at $\mathcal{C}_{\varphi_2}^e$. The following lemma states the transformation behavior of this directional derivative.
\begin{lem}
Let $X,Y_1,Y_2$ be mwbs and let $F : X\times Y_2 \to X\times Y_1$ be a ${\mathrm{sc}}$-map of the form $F = \mathrm{id} \times \Psi$. Set $h = \rho_{Y_2}^{-1} F^*\rho_{Y_1}$.
Consider a clean phase function $\varphi$ on $X \times Y_1$. Write $f = \rho_X \rho_{Y_2} \varphi$.
Then we have the following transformation laws:
\begin{align*}
hF^*\partial_{\rho_{Y_1}} \rho_X^{-1}f &= \partial_{\rho_{Y_2}} F^*\rho_X^{-1}f, \quad\text{ on } F^*\mathcal{C}_{\varphi}^\psi,\\
F^*\rho_{Y_1}^{-1}\partial_{\rho_X} f &= \partial_{\rho_X} F^*\rho_{Y_1}^{-1}f, \quad\text{ on } F^*\mathcal{C}_{\varphi}^e.
\end{align*}
\end{lem}
\begin{proof}
On $F^*\mathcal{C}_\varphip$, we have that
\begin{align*}
\partial_{\rho_{Y_2}} F^*f &= hF^*\partial_{\rho_{Y_1}}f + F^*(\partial_{y_1} f)\partial_{\rho_{Y_2}}y_1= hF^*\partial_{\rho_{Y_1}}f,
\end{align*}
where we have used $\partial_{y_1} f = 0$ on $F^*\mathcal{C}_\varphip$.
This proves the first equality.
On $F^*\mathcal{C}_\varphie$, we compute
\begin{align*}
\partial_{\rho_X} F^*\rho_{Y_1}^{-1}f_1 &= F^*\rho_{Y_1}^{-1}\partial_{\rho_X} f_1 + F^*(\partial_{\rho_{Y_1}} \rho_{Y_1}^{-1}f_1)\, \partial_{\rho_X} F^*\rho_{Y_1} + F^*(\rho_{Y_1}^{-1}\partial_{y_1} f_1)\, \partial_{\rho_X} F^*y_1\\
&= \rho_{Y_2}^{-1}h^{-1}F^*\partial_{\rho_X} f_1.
\end{align*}
Therein, we used $\partial_{y_1}f_1 = 0$ and $\partial_{Y_1} \rho_{Y_1}^{-1} f_1 = 0$ on $\mathcal{C}_{\varphi_1}$.
\end{proof}
\begin{rem}\label{rem:strictness}
The previous lemma, combined with Lemma \ref{lem:phpequiv}, will imply that, away from the corner, any phase function can be replaced by an equivalent phase function without radial derivative (at $\mathcal{C}_\varphi$) and the vanishing of this derivative at $\mathcal{C}_\varphi$ is preserved under application of scattering maps.
This corresponds to the fact that, in the classical theory, one can always choose a homogeneous phase functions. The (non-homogeneous) terms of lower order which arise in transformations can be absorbed into the amplitude.
\end{rem}
The rest of this section will be dedicated to establishing a necessary and sufficient criterion for the local equivalence of phase functions.
\begin{lem}\label{lem:arrange}
Let $X$, $Y_1$, $Y_2$ be mwbs such that $\dim(Y_1)=\dim(Y_2)$, and set $B_i=X\times Y_i$, $i\in\{1,2\}$. Let $\varphi_i \in \rho_{X}^{-1}\rho_{Y_i}^{-1}C^\infty(B_i)$ be phase functions which have the same excess, and assume that there exist $p^0_i=(\mathbf{x}^0,\mathbf{y}^0_i)\in\mathcal{C}_{\varphi_i}$, $i\in\{1,2\}$, such that
\begin{align*}
\lambda_{\varphi_1}(\mathbf{x}^0,\mathbf{y}^0_1)&=\lambda_{\varphi_2}(\mathbf{x}^0,\mathbf{y}^0_2),
\end{align*}
and, close to $(\mathbf{x}^0,\mathbf{y}^0_i)$, $i\in\{1,2\}$, both phases parametrize the same Lagrangian $\Lambda$, i.e., locally $\Lambda=\Lambda_{\varphi_i}$,
$i\in\{1,2\}$.
Then, there exists a local diffeomorphism $F\colon B_2\to B_1$ of the
form $F=\mathrm{id}\times g$ with $F(\mathbf{x}^0,\mathbf{y}^0_2)=(\mathbf{x}^0,\mathbf{y}^0_1)$, such that
$F^*\varphi_1=\rho_X\rho_{Y_2}\widetilde{f}_1$
with $\mathcal{C}_{F^*\varphi_1}=\mathcal{C}_{\varphi_2}$,
locally. Moreover, locally near $(\mathbf{x}^0,\mathbf{y}^0_2)$,
\begin{equation}\label{eq:prsymbeq}
(f_2-\widetilde{f}_1)|_{\mathcal{B}_2}
\text{ vanishes of second order at any point of
$\mathcal{C}_{\varphi_2}$.
}
\end{equation}
\end{lem}
\begin{rem}
Notice that \eqref{eq:prsymbeq} means that the principal
part of $F^*\varphi_1$ and $\varphi_2$ in Lemma \ref{lem:arrange}
coincide on $\mathcal{C}_{\varphi_2}$.
\end{rem}
\begin{proof}[Proof of Lemma \ref{lem:arrange}]
Since $\lambda_{\varphi_i}$ are local fibrations from
$\mathcal{C}_{\varphi_i}$ to $\Lambda_{\varphi_i}$, $i\in\{1,2\}$,
and $\Lambda_{\varphi_1}=\Lambda_{\varphi_2}=\Lambda$,
there is a local fibered diffeomorphism $F\colon B_2\to B_1$
of the form $F=\mathrm{id}\times g$, locally
locally near $(\mathbf{x}^0,\mathbf{y}^0_1)=F(\mathbf{x}^0,\mathbf{y}^0_2)$,
such that the following diagram is
commutative.
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=2.5em,column sep=1.5em,minimum width=2em,minimum height=7mm,
text depth=0.5ex,
text height=2ex,
inner xsep=1pt,
outer sep=1pt]
{
\ & \Lambda & \ \\
\mathcal{C}_{\varphi_2} & & \mathcal{C}_{\varphi_1} \\
};
\path[->]
(m-2-1) edge node [left] {$\lambda_{\varphi_2}$}(m-1-2);
\path[->]
(m-2-3) edge node [right] {$\lambda_{\varphi_1}$}(m-1-2);
\path[->]
(m-2-1) edge node [above] {$\exists F$}(m-2-3);
\end{tikzpicture}
\end{center}
Note that $F$ is not uniquely determined, not even on $\mathcal{C}_{\varphi_2}$ when the phases are merely clean and not non-degenerate.
After application of $F$, we may assume that $Y_1=Y_2=:Y$, $\mathbf{y}_1^0=\mathbf{y}_2^0=:\mathbf{y}^0$ and,
locally, $\mathcal{C}_{\varphi_1}=\mathcal{C}_{\varphi_2}=:\mathcal{C}_\varphi$.
We now show that the restriction of $f_1$ and $f_2$ to a relative
neighbourhood of $(\mathbf{x}^0,\mathbf{y}^0)$ in $\mathcal{C}_\varphi$ vanishes of
second order. Recall that, since ${}^\scat \dd_Y\varphi_1={}^\scat \dd_Y\varphi_2=0$,
for any $p=(\mathbf{x},\mathbf{y})\in\mathcal{C}_\varphi$ we have
\begin{equation}\label{eq:scdmapphfncs}
\begin{pmatrix}\rho_Y\partial_{\rho_Y}f_1-f_1
&
\partial_{y_k}f_1
\end{pmatrix}
=
\begin{pmatrix}\rho_Y\partial_{\rho_Y}f_2-f_2
&
\partial_{y_k}f_2
\end{pmatrix}=0
\end{equation}
Furthermore, since $\varphi_1$ and $\varphi_2$ parametrize the same
Lagrangian, we also have
$\lambda_{\varphi_1}(p)=\lambda_{\varphi_2}(p)$, that is,
$\iota({}^\scat \dd_X\varphi_1(p))=\iota({}^\scat \dd_X\varphi_2(p))$.
We treat separately the cases $p\in\mathcal{C}_\varphie$ and $p\in\mathcal{C}_\varphip\cup\mathcal{C}_\varphipe$.
If $p\in\mathcal{C}_\varphie$, we then find
\begin{equation}\label{eq:scdxequal}
\iota((\rho_Y^{-1}\rho_X\partial_{\rho_X}f_1(p)-f_1(p),
\rho_Y^{-1}\partial_{x_k}f_1(p)))
=
\iota((\rho_Y^{-1}\rho_X\partial_{\rho_X}f_2(p)-f_2(p),
\rho_Y^{-1}\partial_{x_k}f_2(p))).
\end{equation}
Since $\rho_Y\not=0$ on $\mathcal{C}_\varphie$,
and $\iota$ is a diffeomorphism on the interior, this implies
\[
f_1(p)=f_2(p), \quad \partial_{x_k}f_1(p)=\partial_{x_k}f_2(p),
\; k = 1, \dots, d-1.
\]
Combining this with \eqref{eq:scdmapphfncs}, this further implies
\[
\partial_{\rho_Y}f_1(p)=\partial_{\rho_Y}f_2(p),
\quad
\partial_{y_k}f_1(p)=\partial_{y_k}f_2(p), \;
k=1, \dots, s-1.
\]
Since $(x,\mathbf{y})$ are a complete set of variables on $\mathcal{B}e$, we
can indeed
conclude that $f_1-f_2$ vanishes of second order along $\mathcal{C}_\varphie$.
If $p\in\mathcal{C}_\varphip$ or $p\in\mathcal{C}_\varphipe$, \eqref{eq:scdmapphfncs} implies that
\[
f_1(p)=f_2(p)=0, \quad
\partial_{y_k}f_1(p)=\partial_{y_k}f_2(p), \; k=1,\dots, s-1.
\]
We have to evaluate \eqref{eq:scdxequal} as a limit $\rho_Y\to0^+$,
using, as in Lemma \ref{lem:horror},
$\iota(z)=\frac{z}{|z|}(1-\frac{1}{|z|})$. We obtain that, with
\[
v_1=(\rho_X\partial_{\rho_X}f_1, \partial_{x_k}f_1),
\quad
v_2=(\rho_X\partial_{\rho_X}f_2, \partial_{x_k}f_2),
\]
$\frac{v_1}{\|v_1\|}=\frac{v_2}{\|v_2\|}$, but not necessarily
$v_1=v_2$, in which case the proof would be complete. We now modify
$F$ in order to achieve $v_1=v_2$. Notice that, since $\varphi_1$ and $\varphi_2$ are phase functions,
we have $v_1\not=0$ at $\mathcal{C}_\varphi$. We can therefore scale $\varphi_1$
by means of the local diffeomorphism (near $\mathcal{C}_\varphi$)
\[
\widetilde{F}\colon(\rho_Y, y)\to
(\rho_Y \, r(\rho_X,x,\rho_Y,y), y),
\]
where $r(\rho_X,x,\rho_Y,y)=\frac{\|v_2\|}{\|v_1\|}$. Notice that,
by our previous computations, $r|_{\mathcal{C}_\varphie\cup\mathcal{C}_\varphipe}=1$,
and $\widetilde{F}$
is the identity for $\rho_Y=0$. Therefore, by Lemma \ref{lem:CLFstarff},
\[
\mathcal{C}_{\widetilde{F}^*\varphi_1}=\mathcal{C}_{\varphi_1},
\;\text{ and }\;
\Lambda_{\widetilde{F}^*\varphi_1}=\Lambda_{\varphi_1}.
\]
By definition, for $\widetilde{F}^*\varphi_1$ we have
\[
\widetilde{f}_1:=\rho_X\rho_Y\widetilde{F}^*\varphi_1=
\frac{\|v_2\|}{\|v_1\|}(F^*f_1).
\]
Therefore,
\[
(\rho_X\partial_{\rho_X}\widetilde{f}_1, \partial_{x_k}\widetilde{f}_1)
=
\frac{\|v_2\|}{\|v_1\|}\cdot
(\rho_XF^*(\partial_{\rho_X}{f}_1), F^*(\partial_{x_k}\widetilde{f}_1))
=:\widetilde{v}_1,
\]
since the derivatives acting on $r$ produce a $\rho_Y$ factor, and
then vanish along $\mathcal{C}_\varphip$. Hence, $\widetilde{v}_1=v_2$, which completes
the proof.
\end{proof}
\begin{rem}
The additional computations in the proof of the previous lemma near the face $\mathcal{C}_\varphip$ correspond to the fact that, classically, $x\cdot\theta$ and $x\cdot(2\theta)$ both parametrize
\[
\Lambda=\left\{(0,\xi)\mid \xi\in{\mathbb{R}}d\setminus\{0\}\right\}.
\]
In fact, we observe from the same proof that we may choose
the norm of $(\rho_X\partial_{\rho_X}f_1,\partial_{x_k}f_1)$
at any point of $\Lambda_\varphip$ without changing $\Lambda_\varphi$.
\end{rem}
\begin{thm}[Equivalence of phase functions]
\label{thm:equivphase}
Let $X$, $Y_1$, $Y_2$ be mwbs such that $\dim(Y_1)=\dim(Y_2)$, and set $B_i=X\times Y_i$, $i\in\{1,2\}$. Let $\varphi_i \in \rho_{X}^{-1}\rho_{Y_i}^{-1}C^\infty(B_i)$, $i\in\{1,2\}$,
be phase functions which have the same excess, assume that there exist $(\mathbf{x}^0,\mathbf{y}^0_i)\in\mathcal{C}_{\varphi_i}$, $i\in\{1,2\}$, such that
\begin{align*}
\lambda_{\varphi_1}(\mathbf{x}^0,\mathbf{y}^0_1)&=\lambda_{\varphi_2}(\mathbf{x}^0,\mathbf{y}^0_2),
\end{align*}
and, close to $(\mathbf{x}^0,\mathbf{y}^0_i)$, $i\in\{1,2\}$, both phase functions parametrize the same Lagrangian $\Lambda$, i.e., locally $\Lambda=\Lambda_{\varphi_i}$,
$i\in\{1,2\}$. Then, it is necessary and sufficient
for $\varphi_1$ and $\varphi_2$ to be equivalent at $(\mathbf{x}^0,\mathbf{y}^0_1)$
and $(\mathbf{x}^0,\mathbf{y}^0_2)$ that there it holds that
\begin{equation}\label{eq:sgncond}
\mathrm{sgn}\left(\,\rho_{Y_1}^{-1}\rho_X^{-1}\,{}^\scat H_{Y_1}\varphi_1\right)
=\mathrm{sgn}\left(\rho_{Y_2}^{-1}\rho_X^{-1}\, {}^\scat H_{Y_2}\varphi_2\right).
\end{equation}
\end{thm}
\begin{rem}
\label{rem:scHess}
Before we go into the details of the proof, we recall the expression for
the differential in condition \eqref{eq:sgncond} in coordinates. By \eqref{eq:scatHessfactor} we have, writing
$\varphi=\rho_X^{-1}\rho_Y^{-1}f$,
$$
\rho_{Y}^{-1}\rho_X^{-1} \, {}^\scat H_Y\varphi=
\begin{pmatrix} \rho_Y & 0 \\
0 & \mathbbm{1} \end{pmatrix}
\begin{pmatrix}
\partial_{\rho_Y}^2f & \partial_{y_j}\partial_{\rho_Y} f \\
\partial_{\rho_Y}\partial_{y_k} f & \partial_{y_j}\partial_{y_k} f
\end{pmatrix}
\begin{pmatrix} \rho_Y & 0 \\
0 & \mathbbm{1} \end{pmatrix}.
$$
Hence, for $\rho_Y\neq 0$, the signature of this matrix is that of $H_Y f$, whereas for $\rho_Y=0$ it is that of the Hessian of $f$ \emph{restricted to $\rho_Y=0$}, that is,
only with respect to the boundary variables,
$\left(\partial_{y_j}\partial_{y_k} f(0,y)\right)_{jk}$.
\end{rem}
\begin{proof}[Proof of Theorem \ref{thm:equivphase}]
We first prove that condition \eqref{eq:sgncond} is necessary.
In view of Lemma \ref{lem:equivlag}, we only need to compare ${}^\scat H_{Y_1}\varphi_1$ and ${}^\scat H_{Y_2}\varphi_2$ by writing
\begin{equation}
{}^\scat H_{Y_2}\varphi_2={}^\scat H_{Y_2}F^*\varphi_1+{}^\scat H_{Y_2}(\varphi_2-F^*\varphi_1).
\end{equation}
We write $r=(\varphi_2-F^*\varphi_1)$, which, by assumption,
satisfies $r\in{\mathscr{C}^\infty}(X\times Y_2)$.
Therefore, $\rho_{Y_2}^{-1}\rho_X^{-1}\,{}^\scat H_{Y_2}r$
vanishes at the boundary.
Indeed, in local coordinates we have
$$
\rho_Y^{-1}\rho_{X}^{-1}\,{}^\scat H_{Y_2} r=\begin{pmatrix}
\rho_Y\rho_X \partial_{\rho_Y}\rho_Y^2\partial_{\rho_Y}r & \rho_Y^2\rho_X\partial_{y_j}\partial_{\rho_Y}r \\
\rho_Y\rho_X\partial_{\rho_Y}\rho_Y\partial_{y_k}r & \rho_Y\rho_X\partial_{y_j}\partial_{y_k}r
\end{pmatrix}.
$$
Thus, we have, at the boundary,
\begin{equation}\label{eq:sgneq}
\mathrm{sgn}\left(\,\rho_{Y_2}^{-1}\rho_X^{-1} \, {}^\scat H_{Y_2}F^*\varphi_1\right)
=\mathrm{sgn}\left(\rho_{Y_2}^{-1}\rho_X^{-1} \,{}^\scat H_{Y_2}\varphi_2\right).
\end{equation}
By computing these differentials in coordinates at corresponding stationary points, using \eqref{eq:scatHessfactor}, this implies \eqref{eq:sgncond}.
For the sufficiency of \eqref{eq:sgncond}, we assume familiarity of the reader with the equivalence of phase function theorem in the usual homogeneous setting, see \cite[Prop. 4.1.3]{Treves}, \cite[Prop. 4.1.3]{Treves} and sketch briefly that the argument goes through with little modification.
By Lemma \ref{lem:arrange} we may assume $Y_1=Y_2$. Note that equivalence is achieved for $\varphi_i=\rho_X\rho_Y f_i$ if the $f_i$ agree on the boundary. The condition on ${}^\scat H_Y\varphi_i$
means precisely that the signatures of the Hessians of the $f_i$ in the tangential derivatives agree in the interior and the signatures of the Hessians of the restriction of the $f_i$ to $\rho_Y=0$ as well, see Remark \ref{rem:scHess}. As such, we may use the same techniques as in the classical situation to construct a diffeomorphism on the boundary which transforms the restriction of $f_1$ into that of $f_2$, cf. also \cite{CoSc2}. This diffeomorphism is then extended by means of Proposition \ref{prop:cornerdiffeo} into the interior. For sake of brevity, we omit the details here.
\end{proof}
\begin{rem}
Note that near $(\mathbf{x}^0,\mathbf{y}^0)\in\mathcal{C}_\varphip$, we can also invoke the classical equivalence theorem directly. We need to find a transformation
$$F:(\mathbf{x},0,y)\mapsto (\mathbf{x},0,\tilde{y}(\mathbf{x},y))$$
such that $F^*\varphi_1=\varphi_2$. For $\lambda>0$ we set $\phi_j(\mathbf{x},\lambda,y)=\lambda f_j(\mathbf{x},0,y)$, $j\in\{1,2\}$. Then $\phi_j$ are equivalent \emph{phase functions in the usual homogeneous sense} on $X\times ({\mathbb{R}}_+\times Y)$. Indeed, evaluating $\dd\phi_j$ and ${}^\scat \dd\varphi_j$ in coordinates, we see that $\dd\phi_j\neq 0$ and $\phi_j$ is manifestly homogeneous. Furthermore, the signatures of $H_Y\phi_j$ are the same as those of ${}^\scat H_Y\varphi_j$. Since the $f_j$ are equal up to second order, the $\phi_j$ are equivalent in the usual sense and there exists a $\lambda$-homogeneous $G: (\mathbf{x},\lambda,y)\mapsto (\mathbf{x},\lambda,\tilde{y}(\lambda,\mathbf{x},y))$ which is homogeneous such that $G^*\phi_1=\phi_2$. Setting $F=G|_{\lambda=1}$ and possibly applying a scaling, as in the proof of Lemma \ref{lem:arrange}, concludes the proof for $(\mathbf{x}^0,\mathbf{y}^0)\in\mathcal{C}_\varphip$.
\end{rem}
\section{Lagrangian distributions}
\label{sec:Lagdist}
In this section, we will address the class of Lagrangian distributions on scattering manifolds.
First, we introduce oscillatory integrals associated with a phase function and show that they are well-defined in the usual sense.
Then, we define Lagrangian distributions as a locally finite sum of oscillatory integrals, where the phase function parametrizes a
Lagrangian submanifold.
Using the results from the previous section, we are able to reduce the number of fiber-variables to a minimum
and see that the order of the Lagrangian distribution is well-defined independently of the dimension of the fiber.
\subsection{Oscillatory integrals associated with a phase function}
\begin{defn}
Let $Y$ be a mwb. For the remainder of this section, $m_\varepsilon$ with $\varepsilon\in(0,1]$,
denotes a family of functions $m_\varepsilon \in{\dot{\mathscr{C}}^\infty}z(Y)$ such that for all $k\in{\NNz_0}$, $\alpha \in {\mathbb{N}}_0^{d-1}$ and $\varepsilonilon>0$,
\begin{equation}
\label{eq:approxone}
\left|(\rho_Y^2\partial_{\rho_Y})^k (\rho_Y\partial_y)^\alpha m_\varepsilon(\mathbf{y})\right| \leq C_{k,\alpha}\,\rho_Y^{k + |\alpha|},
\end{equation}
such that, for all $\mathbf{y} \in Y^o$, we have $m_\varepsilon(\mathbf{y}) \to 1$ as $\varepsilon \to 0.$
\end{defn}
\begin{rem}
We make the observation that \eqref{eq:approxone} does not depend on the choice of bdf and is preserved under pullbacks by ${\mathrm{sc}}$-maps.
It is possible to find such a family on any manifold with boundary. In fact, any choice of tubular neighbourhood $U$ of $\partial Y$ such that $U\cong [0,\delta)\times \partial Y$ with coordinates $(\rho_Y,y)$ introduces a dilation in the first variable.
Take a function $\chi \in {\mathscr{C}^\infty}c[0,\infty)$ such that $\chi(x) = 1$ on $[0,\delta]$.
Then set $m_\varepsilon=1$ on $Y\setminus U$ and
\[m_\varepsilon(\rho_Y,y)=\begin{cases}
\chi(\varepsilon\rho_Y^{-1}) & \text{if }\varepsilon\rho_Y^{-1}> \delta/2,\\
1 & \text{otherwise}.
\end{cases}\]
\end{rem}
\begin{defn}
Consider $X$, $Y$ mwbs, $U\subset X\times Y$ an open subset, $\varphi\in\rho_X^{-1}\rho_Y^{-1}{\mathscr{C}^\infty}(U)$ a phase function
and $a\in\rho_X^{-m_e}\rho_Y^{-m_\psi}{\mathscr{C}^\infty}(X\times Y, {}^\scat\, \Omega^{1/2}(X) \times {}^\scat\, \Omega^1(Y))$ an amplitude supported in $U$.
Then $I_\varphi(a)\in{\dot{\mathscr{C}}^\infty}dz(X,{}^\scat\, \Omega^{1/2}(X))$ is defined as the distributional $1/2$-density acting on $f\in{\dot{\mathscr{C}}^\infty}z(X,{}^\scat\, \Omega^{1/2}(X))$ by
\begin{equation}
\label{eq:oscidef}
\langle I_\varphi(a),f\rangle:= \lim_{\varepsilon\searrow 0} \iint_{X\times Y} \left(e^{i\varphi} a \cdot (f \otimes m_\varepsilon)\right).
\end{equation}
\end{defn}
\begin{rem}
If $X$ and $Y$ are equipped with a scattering metric, we have a canonical identification of functions and $1$-densities provided by the volume form.
Therefore, we can freely choose whether to view functions and distributions as matching (distributional) $1$-, $0$- or $\frac{1}{2}$-densities.
\end{rem}
\begin{rem}
When $X={\mathbb{B}}d$ and $Y={\mathbb{B}}s$, these oscillatory integrals correspond, under (inverse) radial compactification, to the tempered oscillatory integrals analyzed in \cite{CoSc2,Schulz}.
\end{rem}
\begin{lem}\label{lem:osciwelldef}
The expression \eqref{eq:oscidef} yields a well-defined tempered distribution (density) on $X$.
In particular, it is independent of the choice of $m_\varepsilon$.
\end{lem}
\begin{proof}
Assume, without loss of generality, that we have a fixed scattering metric and we can identify scattering densities and functions.
Let $U \subset X \times Y=: B$ be an open neighborhood of the boundary $\mathcal{B}^\psi$ such that ${}^\scat \dd \varphi \not = 0$ on $U$.
On $X\times Y \setminus U$, the dominated convergence theorem implies that \eqref{eq:oscidef} is well-defined. The integrand $u_\varepsilon = e^{i\varphi} a (f \otimes m_\varepsilon)$ converges pointwise and is dominated by $|a\cdot f|$, which is bounded for $\rho_Y>c$.
On $U$, as in the classical theory, we can define a first order scattering differential $L \in \mathrm{Diff}^1_{{\mathrm{sc}}}(U)$ which has the property that $Le^{i\varphi} = e^{i\varphi}$. By Proposition 1 from \cite{Melrose1}, we see that $L^t \in \mathrm{Diff}^1_{{\mathrm{sc}}}(U)$.
Using repeated integration by parts and \eqref{eq:approxone}, we are able to increase the order in $\rho_X$ and $\rho_Y$ to arbitrary powers, and an application of the dominated convergence theorem then finishes the proof.
\end{proof}
After an arbitrary choice of scattering metrics, we may locally identify $(X,g_X)$ and $(Y,g_Y)$ with subsets of ${\mathbb{B}}^d$ and ${\mathbb{B}}^s$, respectively. Then, using some explicit local isomorphism $\Psi=\Psi_X\times\Psi_Y$, we can identify densities with functions using the induced measures $\mu_X$ and $\mu_Y$. After use of a partition of unity, we may locally express \eqref{eq:oscidef} as
\begin{align}\label{eq:oscilocdef}
\langle I_\varphi(a),f\rangle:= \lim_{\varepsilon\searrow 0} \iint_{{\mathbb{B}}d\times {\mathbb{B}}s} \Psi^*\left(e^{i\varphi(\rho_X,x,\rho_Y,y)} a(\rho_X,x,\rho_Y,y) m_\varepsilon(\rho_Y,y) f(\rho_X,x)\right)\\
\label{eq:oscilocdefbis}
= \lim_{\varepsilon\searrow 0} \iint_{{\mathbb{B}}d\times {\mathbb{B}}s} e^{i\Psi^*\varphi(\rho_X,x,\rho_Y,y)} \wt m_\varepsilon(\rho_Y,y) \wt a(\rho_X,x,\rho_Y,y) \wt f(\rho_X,x) \dd \mu_{{\mathbb{B}}d} \dd \mu_{{\mathbb{B}}s}
\end{align}
where $\wt f=\Psi^*f |\dd \mu_{\mathbb{B}}d|^{-1/2}$ and $\wt a\in \rho_{\mathbb{B}}d^{-m_e}\rho_{\mathbb{B}}s^{-m_\psi}{\mathscr{C}^\infty}({\mathbb{B}}d\times {\mathbb{B}}s)$ satisfies $\wt a \wt f \dd \mu_{{\mathbb{B}}d} \dd \mu_{{\mathbb{B}}s}=a f$.
Summing up, we may always transform to locally work on ${\mathbb{B}}d\times{\mathbb{B}}s$ and in local coordinates we work with usual oscillatory integrals.
Since \eqref{eq:oscidef} does not depend on the choice of $m_\varepsilon$, as it is usual we drop it from the notation and write, \emph{in the sense of oscillatory integrals},
\begin{equation}\label{eq:osciformdef}
I_\varphi(a):= \int_{Y} e^{i\varphi}a.
\end{equation}
\subsubsection{Singularities of oscillatory integrals}
Recall that there is a notion of wavefront-set adapted to the pseudo-differential scattering calculus, called the scattering wavefront-set, cf. \cite{Cordes,Melrose1,CoMa}.
\begin{defn}
Let $u\in{\dot{\mathscr{C}}^\infty}dz(X,{}^\scat\, \Omega^{1/2})$.
A point $z_0 \in \mathcal{W}=\partial\big({}^\scat \,\overline{T}^*X\big)$ is not in the scattering wavefront-set, and we write $z_0 \notin \mathrm{WF}_\sct(u)$,
if there exists a scattering pseudo-differential operator $A$ whose symbol is elliptic at $z_0$ such that $Au\in{\dot{\mathscr{C}}^\infty}z(X,{}^\scat\, \Omega^{1/2}).$
\end{defn}
\begin{prop}
\label{prop:WFosci}
For the oscillatory integral in \eqref{eq:oscidef}, we have
$$\mathrm{WF}_\sct(I_\varphi(a))\subseteq \Lambda_\varphi.$$
Furthermore, if $z\in \Lambda_\varphi$ and $a$ is rapidly decaying near $\lambda_\varphi^{-1}(z)$, then $z\notin \mathrm{WF}_\sct(I_\varphi(a))$.
\end{prop}
\begin{rem}
\label{rem:css}
The (${\mathrm{sc}}$-)singular support of $u$ is defined as follows:
a point $p_0\in X$ is contained in $\mathrm{singsupp}_{\mathrm{sc}\hspace{-2pt}}(u)$ if and only if for every $f\in {\mathscr{C}^\infty}(X)$ with $f(p_0)=1$ we have $fu\notin {\dot{\mathscr{C}}^\infty}z(X)$.
Similar to the classical wavefront-set and singular support, we have that ${\mathrm{pr}}_1(\mathrm{WF}_\sct(u))=\mathrm{singsupp}_{\mathrm{sc}\hspace{-2pt}}(u)$.
Thus, in particular, if $a$ is rapidly decaying near $\mathcal{C}_\varphi$, then $I_\varphi(a)\in {\dot{\mathscr{C}}^\infty}z(X)$.
\end{rem}
We refer the reader to \cite{CoSc,Schulz} for the details of this analysis of the wavefront-sets. The proof is carried out as in the classical setting: first, a characterization of $\mathrm{WF}_\sct$ in terms of cut-offs and the Fourier transform is achieved, and then one estimates $\Fu I_\varphi(a)$ in coordinates.
Proposition \ref{prop:WFosci} gives another insight why we consider
$\Lambda_\varphi$ as the true object of interest associated with a phase function, not $L_\varphi$. In fact, considering \eqref{eq:oscidef} once more, we see that we may modify phase function and amplitude in the integral by any real valued function $\psi\in{\mathscr{C}^\infty}(X\times Y)$, writing
$$e^{i\varphi} a=e^{i(\varphi+\psi)} \left(e^{-i\psi}a\right).$$
Then $e^{-i\psi}a\in \rho_X^{-m_e}\rho_Y^{-m_\psi}{\mathscr{C}^\infty}(X\times Y)$, and hence it is still an amplitude, and $\varphi+\psi$ is a new local
phase function. Now, while in general $L_\varphi\neq L_{\varphi+\psi}$, we have $\Lambda_\varphi= \Lambda_{\varphi+\psi}$, by Lemma \ref{lem:phaseplussmooth}.
This underlines that only $\Lambda_\varphi$ and not $L_\varphi$ can be associated with $I_\varphi(a)$ in an intrinsic way.
Nevertheless, it is often convenient to have $L_\varphi$ available during the proofs.
\subsection{Definition of Lagrangian distributions}
The class of oscillatory integrals associated with a Lagrangian is -- as in the classical theory -- not a good distribution space, since in general it is not possible to find a single global phase function to parametrize $\Lambda$. Instead, we introduce the following class of Lagrangian distributions. Note that, by our previous findings, we may always reduce an oscillatory integral on $X\times Y$ into a finite sum of oscillatory integrals over $X\times{\mathbb{B}}^s$ for $s=\dim(Y)$.
\begin{defn}[${\mathrm{sc}}$-Lagrangian distributions]\label{def:Lagdist}
Let $X$ be a mwb, $\Lambda\subset \partial{}^\scat \,\overline{T}^*X$ a ${\mathrm{sc}}$-Lagrangian. Then, $I^{m_e,m_\psi}(X,\Lambda)$, $(m_e,m_\psi)\in {\mathbb{R}}^2$,
denotes the space of distributions that can be written as a finite sum of (local) oscillatory integrals as in \eqref{eq:osciformdef}, whose phase functions are clean and locally parametrize $\Lambda$, plus an element of ${\dot{\mathscr{C}}^\infty}z(X)$. More precisely, $u\in I^{m_e,m_\psi}(X,\Lambda)$ if, modulo a remainder in ${\dot{\mathscr{C}}^\infty}z(X)$,
\begin{equation}
\label{eq:Lagdistdef}
u=\sum_{j=1}^N \int_{Y_j} e^{i\varphi_j}a_j,
\end{equation}
where for $j=1,\dots,N$:
\begin{enumerate}
\item[1.)] $Y_j$ is a mwb of dimension $s_j$;
\item[2.)] $\varphi_j\in \rho_{Y_j}^{-1}\rho_X^{-1}{\mathscr{C}^\infty}(X\times Y_j)$ is a local clean phase function with excess $e_j$, defined on an open neighbourhood of the support of $a_j$,
which locally parametrizes $\Lambda$;
\item[3.)] $a_j\in \rho_{Y_j}^{-m_{\psi,j}}\rho_X^{-m_{e,j}}{\mathscr{C}^\infty}\big(X\times Y_j, {}^\scat\, \Omega^{1/2}(X) \times {}^\scat\, \Omega^1(Y)\big)$ with
\[
(m_{\psi,j},m_{e,j})=\left(m_\psi+\frac{d}{4}-\frac{s_j}{2}-\frac{e_j}{2},m_e-\frac{d}{4}+\frac{s_j}{2}-\frac{e_j}{2}\right).
\]
\end{enumerate}
We also set
\begin{align*}
I^{-\infty,-\infty}(X,\Lambda) &= \bigcap_{(m_\psi,m_e)\in{\mathbb{R}}^2} I^{m_\psi,m_e}(X,\Lambda),
\\
I(X,\Lambda) =I^{+\infty,+\infty}(X,\Lambda)&= \bigcup_{(m_\psi,m_e)\in{\mathbb{R}}^2} I^{m_\psi,m_e}(X,\Lambda).
\end{align*}
\end{defn}
\begin{rem}
The reason for the choice of the $a_j$ in the scattering amplitude densities spaces of order $(m_{e,j}, m_{\psi,j})$
will be explained in Section \ref{ssec:order}.
\end{rem}
The next result follows from Proposition \ref{prop:WFosci}.
\begin{prop}
Let $\Lambda\subset \partial\, {}^\scat \,\overline{T}^*X$ be a ${\mathrm{sc}}$-Lagrangian, and $u\in I(X,\Lambda)$. Then $\mathrm{WF}_\sct(u)\subseteq \Lambda.$
\end{prop}
As in the classical case, the class of Lagrangian distributions contains the globally regular functions (cf. Treves~\cite[Chapter VIII.3.2]{Treves}):
\begin{lem}\label{lem:smoothpart}
Let $\Lambda\subset \partial\, {}^\scat \,\overline{T}^*X$ be a ${\mathrm{sc}}$-Lagrangian. Then
\begin{equation}
{\dot{\mathscr{C}}^\infty}z(X,{}^\scat\, \Omega^{1/2}(X)) = I^{-\infty,-\infty}(X,\Lambda).
\end{equation}
\end{lem}
\begin{proof}
We first prove the inclusion ``$\supseteq$''. Choose a finite covering of ${}^\scat \,\overline{T}^*X$ with open sets $\{X_j\}_{j=1}^N$ such that there exists a clean phase function $\varphi_j$ on each $X_j$
parametrizing $\Lambda \cap {}^\scat \,\overline{T}^*X_j$, $j=1,\dots,N$. Let $\{g_j\}_{j=1}^N$ be a smooth partition of unity subordinate to such covering. We view $X_j$ as a subset
of $X \times {\mathbb{B}}^d$, $j=1,\dots,N$.
Let $\chi \in {\dot{\mathscr{C}}^\infty}z({\mathbb{B}}^d, {}^\scat\, \Omega^1({\mathbb{B}}^d))$ such that $\int \chi = 1$. For any $f \in {\dot{\mathscr{C}}^\infty}z(X,{}^\scat\, \Omega^{1/2}(X))$ we set
\begin{align*}
a_j = e^{-i\varphi_j}g_j \cdot (f \otimes \chi),\quad
f_j = \int_{{\mathbb{B}}^d} e^{i\varphi_j} a_j, \quad j=1,\dots,N.
\end{align*}
We see that
\[a_j \in {\dot{\mathscr{C}}^\infty}z(X\times {\mathbb{B}}^d, {}^\scat\, \Omega^{1/2}(X) \times {}^\scat\, \Omega^1({\mathbb{B}}^d)), \quad j=1,\dots,N,\]
and, summing up,
\begin{align*}
\sum_{j=1}^N f_j(x) &= \int_{{\mathbb{B}}^d} \left(\sum_{j=1}^N g_j(x,y)\right) \cdot (f(x) \otimes \chi(y))= f(x).
\end{align*}
The inclusion ``$\subseteq$'' is achieved by differentiation under the integral sign.
\end{proof}
\subsection{Examples}
We have the following examples of (scattering) Lagrangian distributions.
\begin{enumerate}
\item Standard Lagrangian distributions of compact support, \cite{HormanderFIO,Hormander4}, in particular Lagrangian distributions on compact manifolds $X$ without boundary, are scattering Lagrangian distributions, using the identification
\begin{align*}
\textrm{Fiber-conic sets in }T^*X\setminus\{0\}\longleftrightarrow \textrm{Sets in }S^*X
\stackrel{\textrm{rescaling}}{\longleftrightarrow} \textrm{Sets in }\Wt^\psi.
\end{align*}
\item Legendrian distributions of \cite{MZ}. Here, the distributions are smooth functions whose singularities at the boundary are of Legendrian type, meaning in $\Wt^e$.
\item Conormal distributions, meaning the distributions where the Lagrangian, see Section \ref{sec:conorm}, is $\partial\big({}^\scat \,\overline{T}^*X'\big)$ for a ($k$-dimensional) $p$-submanifold $X'\subset Y$. These distributions correspond, under compactification of base and fiber, to the oscillatory integrals given in local (pre-compactified) Euclidean coordinates by
$$u(x',x'')=\int e^{ix'\xi} a(x,\xi)\,\dd \xi, \qquad a(x,\xi)\in{\mathrm{SG}}_{\mathrm{cl}}^{m_e,m_\psi}({\mathbb{R}}d\times{\mathbb{R}}^{d-k}).$$
A prototypical example is given by (derivatives of) $\delta_{0}(x')\otimes 1$. These arise as (simple or multiple) layers when solving partial differential equations along infinite boundaries or Cauchy surfaces.
\item Examples of scattering Lagrangian distributions which are of none of the previous types arise in the parametrix construction to hyperbolic equations on unbounded spaces, for example the two-point function for the Klein-Gordon equation. For a discussion of this example consider \cite{CoSc2}.
\end{enumerate}
\begin{rem}
Note that, at this stage, the kernels of pseudo-differential operators on $X\times X$ are \emph{not} scattering conormal distributions associated with the diagonal $\Delta\subset X\times X$ when $X$ is a manifold with boundary. In fact, in this case $X\times X$ is a manifold with corners. Furthermore $\Delta\subset X\times X$ does not hit the corner $\partial X\times\partial X$ in a clean way, that is,
$\Delta\subset X\times X$ is not a $p$-submanifold. Similarly, the phase function associated to the ${\mathrm{SG}}$-phase $(x-y)\xi\in{\mathrm{SG}}^{1,1}_{\mathrm{cl}}({\mathbb{R}}^{2d}\times{\mathbb{R}}d)$ is not clean.
However, the formulation of the theory developed in this paper admits a natural extension to manifolds with corners. The geometric obstruction of $\Delta\subset X\times X$ -- or more generally the graphs of (scattering) canonical transformations -- not being a $p$-submanifold can be overcome by lifting the analysis to a blow-up space, see \cite{MZ,Melroseb}. We postpone this theory of compositions of canonical relations and calculus of scattering Fourier integral operators to a subsequent paper.
\end{rem}
\subsection{Transformations of oscillatory integrals}
In Section \ref{sec:exchphase} we have seen several procedures that allow
to switch from one phase function to others that parametrize the same Lagrangian. We will now exploit these to transform oscillatory integrals into ``standard form''. In the sequel, we will always assume, by
a partition of unity, that the support of the amplitude is suitably small.
\subsubsection{Transformation behavior and equivalent phase functions}
\label{sec:moves}
Now we reconsider \eqref{eq:oscilocdef}, to express the transformation
behavior of the oscillatory integrals under fiber-preserving
diffeomorphisms. With the chosen notation and a local
phase function $\varphi_1$, we have
\begin{equation}\label{eq:oscintsimpl}
I_{\varphi_1}(a)= \int_{Y_1} e^{i\varphi_1}a=\int_{Y_2} e^{iF^*\varphi_1}F^*a=I_{F^*\varphi_1}(F^*a)
\end{equation}
for any diffeomorphism $F:X\times Y_2\rightarrow X\times Y_1$ of the form $F=\mathrm{id}\times g$.
Assume that $\varphi_2$ is equivalent to $\varphi_1$ by $F$,
see Definition \ref{def:phequiv}. After the transformation,
we rewrite \eqref{eq:oscintsimpl} as
\begin{equation}
\int_{Y_2} e^{i\varphi_2}e^{i(F^*\varphi_1-\varphi_2)}F^*a.
\end{equation}
Now, since $F^*\varphi_1-\varphi_2$ is smooth up to the boundary,
the same holds for $e^{i(F^*\varphi_1-\varphi_2)}$ and this factor can be seen as part of the amplitude.
Therefore, we may write
\begin{equation}\label{eq:oscintequiv}
I_{\varphi_1}(a)=I_{\varphi_2}
\big((F^*a)\,\exp(i(F^*\varphi_1-\varphi_2))\big).
\end{equation}
In particular, we can express $I_\varphi(a)$,
near any boundary point of the domain of definition,
using the principal part of $\varphi$ introduced in Definition \ref{def:princpart}, namely
\begin{equation}\label{eq:oscintstd}
I_{\varphi_p}(\widetilde{a}), \text{ with }
\widetilde{a}=a\,\exp\big(i(\varphi-\varphi_p)\big).
\end{equation}
By Lemma \ref{lem:phpequiv}, $\varphi - \varphi_p \in {\mathscr{C}^\infty}$ and thus $\widetilde{a} \in \rho_X^{-m_e}\rho_Y^{-m_\psi} {\mathscr{C}^\infty}(B)$.
In the following constructions, we always assume that $\varphi$ is replaced by its principal part, cf. Remark \ref{rem:strictness}.
\subsubsection{Reduction of the fiber}\label{ssbs:redfbr}
We will now analyze the change of boundary behavior under a reduction of fiber variables near $p_0\in\operatorname{supp}(a)\cap\mathcal{C}_\varphi$. Hence, we assume that
$$\rho_Y^{-1}\rho_X^{-1}\,^{\mathrm{sc}\hspace{-2pt}} H_Y\varphi\text{ has rank }r>0\text{ at }p_0\in\mathcal{C}_\varphi.$$
We assume, as explained above, that the oscillatory integral
is in the form \eqref{eq:oscintstd}, namely, $\varphi$ is replaced
by its principal phase part. We observe that, at the boundary point $p_0$,
$$\rk(\rho_Y^{-1}\rho_X^{-1}\,^{\mathrm{sc}\hspace{-2pt}} H_Y\varphi)=\rk(\rho_Y^{-1}\rho_X^{-1}\,^{\mathrm{sc}\hspace{-2pt}} H_Y\sigma(\varphi_p)).$$
By Proposition \ref{prop:fiberred}, we can define a local phase function
$\varphi_{\red}$ parametrizing the same Lagrangian as
$\varphi$. In particular, after a change of coordinates by a scattering map,
we can assume $(\mathbf{x},\mathbf{y})\in X\times{\mathbb{B}}^{s-r}\times(-\varepsilon,\varepsilon)^{r}$,
and $\varphi_{\red}$ is given by
\[
\varphi_{\red}(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)=\varphi(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,0),
\]
where $\rho_Y=\rho_{{\mathbb{B}}^{s-r}}$ is the boundary defining function on ${\mathbb{B}}^{s-r}$ and on ${\mathbb{B}}^{s-r}\times(-\varepsilon,\varepsilon)^r$.
We introduce
\begin{equation}\label{eq:wtp}
\widetilde{\varphi}(\mathbf{x},\mathbf{y})=\varphi_{\red}(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)+
\frac{1}{2}\rho_X^{-1}\rho_Y^{-1}Q(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}),
\end{equation}
where $Q$ is a non-degenerate quadratic form with the same signature as
$\partial_{y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}}\partial_{y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}}f$ at $p_0$.
Then, by Theorem \ref{thm:equivphase},
$\varphi$ is equivalent to $\widetilde{\varphi}$ by a local
diffeomorphism $F=\mathrm{id}\times g$.
Note that $\varphi_\red$ is equal to its principal part, because we assumed that $\varphi$ is replaced by $\varphi_p$.
We may assume that $a$ is supported in an arbitrarily small neighbourhood of the stationary points of $\varphi$. Indeed, we may achieve this for a general amplitude $a$ by applying a cut-off in $y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}$ and writing $a=\phi a + (1-\phi) a$. The oscillatory integral with amplitude $(1-\phi)a$ produces a term in ${\dot{\mathscr{C}}^\infty}z(X, \Omega^{1/2}(X))$, by Remark \ref{rem:css}.
Therefore, choosing the support of $a$ small enough, we may perform the change of variables by the local diffeomorphism $F$ as in \eqref{eq:oscintequiv}. We write, motivated by Lemma \ref{lem:intdensity} and Example \ref{ex:embdball},
$$a_\red(\mathbf{x}, {\widetilde{\by}} )\,\frac{|\dd {\widetilde{\by}} ''|}{\rho_{ {\widetilde{Y}} }^{r} \cdot [h(\mathbf{x}, {\widetilde{\by}} )]^r}=(F^*a)(\mathbf{x}, {\widetilde{\by}} ),$$
which is assumed supported in some compact subset of $(-\varepsilonilon,\varepsilonilon)^r$. Then $I_\varphi(a)$ is transformed into
$I_{\varphi_{\red}}(b)$
where
\begin{equation}
b(\mathbf{x},\rho_{Y},y^{\mathrm{pr}}ime)=\rho_{Y}^{-r}\int_{(-\varepsilon,\varepsilon)^r} e^{\frac{i}{2}\rho_X^{-1}\rho_{Y}^{-1}Q(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}
\mathcal{B}ig(e^{i(F^*\varphi(\mathbf{x},\mathbf{y})-\widetilde{\varphi}(\mathbf{x},\mathbf{y}))}\,a_\red(\mathbf{x},\mathbf{y})\mathcal{B}ig)\dd y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}. \label{eq:fiberredsymb}
\end{equation}
We claim that
$b(\mathbf{x},\rho_{Y},y^{\mathrm{pr}}ime)$
is again a (density valued) amplitude. First, it is clear that $b$ decays rapidly at $(\mathbf{x},\rho_{Y},y^{\mathrm{pr}}ime)$ if $a$ decays rapidly at $(\mathbf{x},\rho_{Y},y^{\mathrm{pr}}ime,0)$. In particular, $b$ is smooth away from $\mathcal{B}$.
We now we apply the stationary phase lemma \cite[Lem. 7.7.3]{Hormander1} to \eqref{eq:fiberredsymb}, which yields the asymptotic equivalence, as $\rho_Y\rho_X\rightarrow 0$,
\begin{multline}
\label{eq:princred1}
b(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)= \rho_X^{r/2}\rho_Y^{-r/2}|\det Q|^{-1/2} e^{\frac{i}{4}\pi \mathrm{sgn}(Q)} e^{i(F^*\varphi(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,0)-\widetilde{\varphi}(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,0))} a_\red(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,0) \\
+{\mathcal{O}}\big(\rho_Y^{-m_\psi-\frac{r}{2}+1}\rho_X^{-m_e+\frac{r}{2}+1}\big).
\end{multline}
Similar asymptotics hold for all derivatives of $b$.
We may hence view $b$ as a (density valued) amplitude of the order
\begin{equation}\label{eq:order-fiber}
(m_e',m_\psi')=\left(m_e-\frac{r}{2},m_\psi+\frac{r}{2}\right).
\end{equation}
By Remark \ref{rem:strictness} we see that, away from the corner, $F^*\varphi-\widetilde{\varphi}$ vanishes at $\mathcal{C}_\varphi$. Therefore, the principal part of $b$ does not depend on $\varphi$.
Hence, by comparision of principal parts, cf. Lemma \ref{lem:princpart}, \eqref{eq:princred1} reduces to
\begin{equation}
\label{eq:princred}
b(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)\sim \rho_X^{r/2}\rho_Y^{-r/2}|\det Q|^{-1/2} e^{\frac{i}{4}\pi \mathrm{sgn}(Q)} a_\red(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,0)
\end{equation}
modulo terms of lower order.
\subsubsection{Elimination of excess}\label{subss:elexcess}
Assume now that $\varphi$ is a clean phase function of excess $e>0$. Near some point in $\mathcal{C}_\varphi$, as described in Section \ref{sec:phaseexelim}, we may make the following geometric assumptions after application of some diffeomorphism $F$: We assume that $Y={\mathbb{B}}^{s-e}\times(-\varepsilonilon,\varepsilonilon)^e$ and that the fibers of $\mathcal{C}_\varphi\rightarrow \Lambda_\varphi$ are given by constant $(\mathbf{x},\rho_Y,y')$ and arbitrary $y''$.
We proceed as in \cite{Treves} and define
\begin{equation}\label{eq:wtredex}
\wt{\varphi}(\rho_{X},x,\rho_{Y},y^{\mathrm{pr}}ime):=\varphi(\rho_{X},x,\rho_{Y},y^{\mathrm{pr}}ime,0).
\end{equation}
We observe that for any fixed $y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}$ the phase function $\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$, defined as
\begin{equation}\label{eq:phiypp}
[\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})](\mathbf{x},\rho_{Y},y^{\mathrm{pr}}ime)=\varphi(\mathbf{x},\rho_{Y},y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}),
\end{equation}
is equivalent to $ {\widetilde{\varphi}} $. Indeed, since $\partial_{y''}{}^\scat \ddY\varphi=0$, the differential ${}^\scat H_Y\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$
has the same signature as ${}^\scat H_{{\mathbb{B}}^{s-e}}\wt\varphi$ and both parametrize the same Lagrangian with the same number of phase variables $(s-e)$.
Therefore, Theorem \ref{thm:equivphase} guarantees the existence of a family of diffeomorphisms $G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}):(\mathbf{x},\rho_Y,y')\mapsto (\mathbf{x},g(\mathbf{x},\rho_Y,y',y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}))$
such that, defining $ \wt{G}\colon (\mathbf{x},\mathbf{y})=(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\mapsto(\mathbf{x}, g(\mathbf{x},\rho_Y,y',y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}), y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$,
\begin{equation}\label{eq:diffeoG}
\wt{G}^*\varphi-\wt{\varphi}
\end{equation}
is smooth everywhere, and vanishes on $\mathcal{C}_ {\widetilde{\varphi}} $ away from the corner by Remark~\ref{rem:strictness}.
Then we may express $I_\varphi(a)$ as $I_{\wt\varphi}(b)$, where
\begin{equation}\label{eq:excesssymb}
b(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)=\rho_Y^{-e} \int_{(-\varepsilon,\varepsilon)^e} e^{i(\wt{G}^*\varphi-\wt{\varphi})(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})} (\wt{G}^*a)_{\red}(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\,\dd y''
\end{equation}
and
$$ (\wt{G}^*a)_{\red}(\mathbf{x},\mathbf{y})\,\frac{|\dd y''|}{\rho_{ {\widetilde{Y}} }^{e} \cdot [h(\mathbf{x},\mathbf{y})]^e}=(\wt{G}^*a)(\mathbf{x},\mathbf{y}).$$
Since $\wt{G}^*\varphi-\wt\varphi$ is smooth, $b$ is again an amplitude of order
\begin{equation}\label{eq:order-excess}
(\tilde{m}_e,\tilde{m}_\psi)=\left(m_e,m_\psi+e\right).
\end{equation}
Notice that at points in $\mathcal{C}_\varphi$ away from the corner, $\wt{G}^*\varphi-\wt{\varphi}$ vanishes and hence \eqref{eq:excesssymb} reduces to
\begin{equation}\label{eq:excesssymbbis}
b(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)=\rho_Y^{-e} \int_{(-\varepsilon,\varepsilon)^e} (\wt{G}^*a)_{\red}(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\,\dd y''.
\end{equation}
\subsection{The order of a Lagrangian distribution}\label{ssec:order}
We will now obtain the definition of the order of $I_\varphi(a)$ which is invariant with respect to all the three steps
described above.
\begin{lem}
The numbers $\mu_\psi = m_\psi + s/2 + e/2$ and $\mu_e = m_e - s/2 +e/2$ remain constant under reduction of fiber-variables and elimination of excess.
\end{lem}
\begin{proof}
Consider a Lagrangian distribution $A = I_\varphi(a)$ where $a$ has order $m_\psi, m_e$ and $\dim Y = s$ with excess $e$ and $r$ reduceable fiber variables.
After the reduction of fiber, we obtain an amplitude $a'$ with order $m_e' = m_e -r/2, m_\psi' = m_\psi + r/2$ (cf. \eqref{eq:order-fiber}), with excess $e' = e$ and number of fiber variables $s' = s - r$.
The elimination of excess yields an amplitude $a^\#$ with order $m_e^\# = m_e, m_\psi^\# = m_\psi + e$ (cf. \eqref{eq:order-excess}), excess $e^\# = 0$ and $s^\# = s - e$.
It is now straightforward to check that
\begin{alignat*}{2}
m_\psi + s/2 + e/2 &= m_\psi' + s'/2 + e/2 & &= m_\psi^\# + s^\#/2+e^\#/2,\\
m_e - s/2 + e/2 &= m_e' - s'/2 + e/2 & &= m_e^\# - s^\# / 2+e^\#/2.
\end{alignat*}
\end{proof}
This shows that the tuple $(\mu_\psi, \mu_e)$ can be used to define the order of a Lagrangian distribution.
We still have the freedom to add arbitrary constants to both orders.
In order to choose these constants, we compare our class of Lagrangian distributions with H\"ormander's Lagrangian distributions and the Legendrian distributions of Melrose--Zworski~\cite{MZ}.
First, consider the Delta-distribution $\delta_0$, which is in the H\"ormander class $I^{d/4}$ and $\mu_\psi = d/2$. Therefore, we choose $m_\psi = \mu_\psi - d/4$ to obtain the same $\psi$-order for $\delta_0$.
Similarly, the constant function is a Legendrian distribution of order $-d/4$ and $\mu_e = 0$, and therefore we choose $m_e = \mu_e + d/4$.
Note that we use the opposite sign convention for the $m_e$-order then in \cite{MZ}.
\section{The principal symbol of a Lagrangian distribution}\label{sec:symb}
We will now define the principal symbol map $j^\Lambda_{m_e,m_\psi}$ on $I^{m_e,m_\psi}(X,\Lambda)$. Similarly to the classical theory,
it takes values in a suitable (density) bundle on $\Lambda$. This is coherent with the notion of principal symbol map $j_{m_e,m_\psi}$ for scattering
operators, see \cite{Melrose1,Melrose2}, as well as of principal part for classical ${\mathrm{SG}}$ symbols, see \cite{ES, Schulz},
which both provide smooth objects defined on $\mathcal{W}=\partial{}^\scat \,\overline{T}^*X\supset\Lambda$. We adapt the construction in \cite{Treves} (see also \cite{Hormander4,HormanderFIO}), starting from the simplest case of local non-degenerate phase functions parametrizing $\Lambda$, up to the general case of local clean functions.
Let $\Lambda\subset\mathcal{W}$ be an ${\mathrm{sc}}$-Lagrangian, which on $B=X\times Y$
is locally parametrized by a local non-degenerate phase function
$\varphi\in \rho_{Y}^{-1}\rho_X^{-1}{\mathscr{C}^\infty}(U)$, $U\subset B$.
Let $a\in \rho_{Y}^{-m_{\psi}}\rho_X^{-m_{e}}{\mathscr{C}^\infty}\big(X\times Y, {}^\scat\, \Omega^{1/2}(X)\times {}^\scat\, \Omega^{1}(Y)\big)$ be
supported in $U$, and let $I_\varphi(a)$ be a (micro-)local representation of $u\in I^{m_e,m_\psi}(X,\Lambda)$ as a single oscillatory integral.
We now fix a $1$-density $\mu_X$ on $X$. Any choice of $1$ density $\mu_Y$ on $Y$ then trivializes the one-dimensional bundle ${\mathscr{C}^\infty}(X\times Y, {}^\scat\, \Omega^{1/2}(X)\otimes{}^\scat\, \Omega^{1}(Y))$, and any element is given by a multiple of $\rho_X^{-(d+1)/2}\rho_Y^{-s-1}\sqrt{\mu_X}\otimes\mu_Y$.
Any choice of coordinates $(\rho_Y,y)$ in $Y$ allows for us to express $\mu_Y$ locally as $\frac{\partial \mu_Y}{\partial (\rho_Y,y)}\,\dd\rho_Y\dd y$, meaning as having a smooth density factor with respect to the (local) Lebesgue measure. As such, we rewrite the amplitude $a\in\rho_Y^{-m_\psi}\rho_X^{-m_e}{\mathscr{C}^\infty}(X\times Y, {}^\scat\, \Omega^{1/2}(X)\otimes{}^\scat\, \Omega^{1}(Y))$ in any choice of local coordinates as
\begin{align}
\label{eq:canampl}
\rho_Y^{m_\psi}\rho_X^{m_e}a(\mathbf{x},\mathbf{y})&={\mathfrak{a}}(\mathbf{x},\mathbf{y})\,\rho_X^{-(d+1)/2}\rho_Y^{-s-1}\sqrt{\mu_X}\dd\rho_Y\dd y.
\end{align}
for ${\mathfrak{a}}\in {\mathscr{C}^\infty}(X\times Y)$.
\subsection{Non-degenerate equivalent phase functions}\label{subs:ndg}
As above (cf. \eqref{eq:scdxexpl}), when $U$ is a neighbourhood of a point close to the boundary $\mathcal{B}$, we can there identify ${}^\scat \dd_Y\varphi$ with the map,
\[
(\mathbf{x},\mathbf{y})\mapsto\Phi(\mathbf{x},\mathbf{y})=\big(-f(\mathbf{x},\mathbf{y})+\rho_Y\partial_{\rho_Y}f(\mathbf{x},\mathbf{y}) \quad \partial_yf(\mathbf{x},\mathbf{y})\big) \in{\mathbb{R}}^s,
\]
locally well-defined on a neighbourhood of $C_\varphi$ within $U$.
In view of the non-degeneracy of $\varphi$, $\Phi$ has a surjective differential, so that we can consider the pullback of distributions $d_\varphi=\Phi^*\delta$, with $\delta=\delta_0\in\mathcal{D}^{\mathrm{pr}}ime({\mathbb{R}}^s)$
the Dirac distribution, concentrated at the origin, on ${\mathbb{R}}^s$ (cf. \cite[Ch. VI]{Hormander1}). More explicitly, choosing functions $(t_1, \dots, t_d)=:t$, which
restrict to a local coordinate system (up to the boundary) on $C_\varphi$, the pull-back $d_\varphi$ can be expressed locally as the density
\[
d_\varphi=\left| \det\frac{\partial(t,\Phi)}{\partial(\mathbf{x}, \mathbf{y} )}\right|^{-1}\dd t = \Delta_\varphi(t)\, \dd t.
\]
Consider another local non-degenerate
phase function $ {\widetilde{\varphi}} $ parametrizing $\Lambda$,
defined on an open subset $ {\widetilde{U}} \subset X\times {\widetilde{Y}} $, such that $ {\widetilde{\varphi}} =F^*\varphi$, with a (local, fibered)
diffeomorphism $F=\mathrm{id}\times g\colon X\times {\widetilde{Y}} \to X\times Y$.
Since $F$ is a ${\mathrm{sc}}$-map, there exists a function $h \in {\mathscr{C}^\infty}(X\times Y)$ such that $(F^*\rho_Y)(\mathbf{x}, {\widetilde{\by}} )=\rho_{ {\widetilde{Y}} }\cdot h(\mathbf{x}, {\widetilde{\by}} )$.
As above, we identify ${}^\scat \dd_Y\wt\varphi$ with the map $ {\widetilde{\Phi}} $ and define $d_ {\widetilde{\varphi}} $ and $\Delta_ {\widetilde{\varphi}} (\widetilde{t})$ in terms of the functions ${\wt t}_j=F^*t_j$, which are
local coordinates on $C_ {\widetilde{\varphi}} $, provided $ {\widetilde{U}} $ is small enough.
In the sequel, we show how objects defined in these two choices $(t,\varphi)$ and $(\wt t,\wt \varphi)$ are related. For that, we implicitly assume all objects evaluated at corresponding points $(\mathbf{x},\mathbf{y})\in C_\varphi$ (parametrized by $t$) and $(\mathbf{x},\wt \mathbf{y})=F(\mathbf{x},\mathbf{y})\in C_{ {\widetilde{\varphi}} }$ (parametrized by $\wt t$).
\begin{lem}\label{lem:trDelta}
The functions $\Delta_ {\widetilde{\varphi}} (\widetilde{t})$ and $\Delta_\varphi(t)$ are related by
\[
\Delta_ {\widetilde{\varphi}} (\widetilde{t}) =
h(\mathbf{x},\mathbf{y})^{s+1} \left| \det\frac{\partial g(\mathbf{x}, {\widetilde{\by}} )}{\partial {\widetilde{\by}} }\right|^{-2} \, \Delta_\varphi(t(\widetilde{t})).
\]
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem:trDelta}]
By direct computation, $ {\widetilde{\Phi}} $ and $\Phi$ are related by a matrix $M_{\Phi {\widetilde{\Phi}} }$ via
\begin{equation}\label{eq:MPtP}
{\widetilde{\Phi}} (\mathbf{x}, {\widetilde{\by}} )= \Phi(F(\mathbf{x}, {\widetilde{\by}} )) \cdot M_{\Phi {\widetilde{\Phi}} }(\mathbf{x}, {\widetilde{\by}} ),
\end{equation}
where
\begin{align*}
M_{\Phi {\widetilde{\Phi}} }(\mathbf{x}, {\widetilde{\by}} )&=
\begin{pmatrix}
[h(\mathbf{x}, {\widetilde{\by}} )]^{-2} \dfrac{\partial\rho_Y}{\partial\rho_{ {\widetilde{Y}} }}(\mathbf{x}, {\widetilde{\by}} )
&
[h(\mathbf{x}, {\widetilde{\by}} )]^{-2} \rho_{ {\widetilde{Y}} }^{-1}\dfrac{\partial\rho_Y}{\partial {\widetilde{y}} }(\mathbf{x}, {\widetilde{\by}} )
\\
[h(\mathbf{x}, {\widetilde{\by}} )]^{-1} \rho_{ {\widetilde{Y}} } \dfrac{\partial y}{\partial\rho_{ {\widetilde{Y}} }}(\mathbf{x}, {\widetilde{\by}} ) \rule{0mm}{9mm}
&
[h(\mathbf{x}, {\widetilde{\by}} )]^{-1} \dfrac{\partial y}{\partial {\widetilde{y}} }(\mathbf{x}, {\widetilde{\by}} )
\end{pmatrix}
\end{align*}
and
\[
|\det M_{\Phi {\widetilde{\Phi}} }(\mathbf{x}, {\widetilde{\by}} )|=h(\mathbf{x}, {\widetilde{\by}} )^{-s-1}\cdot\left| \det \frac{\partial g(\mathbf{x}, {\widetilde{\by}} )}{\partial {\widetilde{\by}} } \right|.
\]
Differentiating \eqref{eq:MPtP}, we obtain, using that $ {\widetilde{\Phi}} (\mathbf{x},\mathbf{y})=\Phi(F(\mathbf{x}, {\widetilde{\by}} ))=0$ on $C_{\widetilde{\varphi}}$,
\begin{equation}\label{eq:wpJ}
\begin{aligned}
\frac{\partial {\widetilde{\Phi}} }{\partial (\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} )
&={{}^t}\!M_{\Phi {\widetilde{\Phi}} }(\mathbf{x}, {\widetilde{\by}} ) \cdot \frac{\partial(\Phi(F(\mathbf{x}, {\widetilde{\by}} )))}{\partial(\mathbf{x}, {\widetilde{\by}} )}
\\
&={{}^t}\!M_{\Phi {\widetilde{\Phi}} }(\mathbf{x}, {\widetilde{\by}} ) \cdot \left[\frac{\partial\Phi}{\partial(\mathbf{x},\mathbf{y})}(F(\mathbf{x}, {\widetilde{\by}} ))\right]\cdot\frac{\partial F}{\partial(\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} ).
\end{aligned}
\end{equation}
Furthermore, we have
\[
\frac{\partial {\widetilde{t}} }{\partial(\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} )=\left[\frac{\partial t}{\partial(\mathbf{x},\mathbf{y})}(F(\mathbf{x}, {\widetilde{\by}} ))\right]\cdot\frac{\partial F}{\partial(\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} ).
\]
Summing up, we find
\begin{equation}\label{eq:wtwpJ}
\begin{aligned}
\frac{\partial( {\widetilde{t}} , {\widetilde{\Phi}} )}{\partial(\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} ) &=
\mathrm{diag}(\mathbbm{1}_{d}, {{}^t}\!M_{\Phi {\widetilde{\Phi}} }(\mathbf{x}, {\widetilde{\by}} )) \cdot \left[ \frac{\partial(t,{\Phi})}{\partial(\mathbf{x}, {\mathbf{y}})}(F(\mathbf{x}, {\widetilde{\by}} ))\right] \cdot
\frac{\partial F}{\partial (\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} ),
\end{aligned}
\end{equation}
which in turn implies, using $F=\mathrm{id}\times g$,
\begin{equation*}
\Delta_{ {\widetilde{\varphi}} }(\widetilde{t}) =
\left| \frac{\partial( {\widetilde{t}} , {\widetilde{\Phi}} )}{\partial(\mathbf{x}, {\widetilde{\by}} )}(\mathbf{x}, {\widetilde{\by}} ) \right|^{-1}
= [h(\mathbf{x}, {\widetilde{\by}} )]^{s+1} \left| \det \frac{\partial g(\mathbf{x}, {\widetilde{\by}} )}{\partial {\widetilde{\by}} } \right|^{-2} \Delta_\varphi(t(\widetilde{t})),
\end{equation*}
as claimed.
\end{proof}
We define
\begin{equation}\label{eq:gamma}
w_\varphi=(\rho_X^{-m_e}\rho_Y^{-m_\psi-(s+1)/2} {\mathfrak{a}})|_{C_\varphi} \cdot \sqrt{|d_\varphi|},
\end{equation}
with ${\mathfrak{a}}$ given in \eqref{eq:canampl}, which is a half-density on (the interior of) $C_\varphi$.
To define $w_ {\widetilde{\varphi}} $ accordingly, we check that $I_\varphi(a)$ transforms under the action of $F$ as
\begin{align*}
\int_Y e^{i\varphi}a&=\int_{ {\widetilde{Y}} } e^{i(F^*\varphi)(\mathbf{x}, {\widetilde{\by}} )}
F^*\!\!\left[\rho_X^{-m_e}\rho_{Y}^{-m_\psi}{\mathfrak{a}}\,\rho_X^{-(d+1)/2}\rho_Y^{-s-1}\sqrt{\mu_X}\otimes \dd\rho_Y \dd y\right](\mathbf{x}, {\widetilde{\by}} )
\\
&= \int_{ {\widetilde{Y}} } e^{i {\widetilde{\varphi}} (\mathbf{x}, {\widetilde{\by}} )}\rho_X^{-m_e}\rho_{ {\widetilde{Y}} }^{-m_\psi}{\widetilde{\mathfrak{a}}}(\mathbf{x}, {\widetilde{\by}} )\,
(\rho_X^{-(d+1)/2}\rho_{ {\widetilde{Y}} }^{-s-1}\sqrt{\mu_X}\otimes \dd\rho_ {\widetilde{Y}} \dd {\widetilde{y}} ),
\end{align*}
where
\begin{equation}\label{eq:trap}
{\widetilde{\mathfrak{a}}}(\mathbf{x}, {\widetilde{\by}} )={\mathfrak{a}}(F(\mathbf{x}, {\widetilde{\by}} )) h(\mathbf{x}, {\widetilde{\by}} )^{-m_\psi-s-1} \left| \det \frac{\partial g(\mathbf{x}, {\widetilde{\by}} )}{\partial {\widetilde{\by}} } \right|.
\end{equation}
We define, coherently with \eqref{eq:gamma}, $w_ {\widetilde{\varphi}} = \rho_X^{-m_e}\rho_{ {\widetilde{Y}} }^{-m_\psi-(s+1)/2}{\widetilde{\mathfrak{a}}}\sqrt{|d_{ {\widetilde{\varphi}} }|}$.
\begin{lem}\label{lem:wphi}
The half-densities $w_ {\widetilde{\varphi}} $ and $w_\varphi$ are related by
\begin{align*}
w_ {\widetilde{\varphi}} =
F^*w_{\varphi}
\end{align*}
in (the interior of) $C_{ {\widetilde{\varphi}} }$.
\end{lem}
\begin{proof}
We obtain from \eqref{eq:trap} and Lemma \ref{lem:trDelta} that
\begin{align*}
{\widetilde{\mathfrak{a}}}(\mathbf{x}, {\widetilde{\by}} )\left|\Delta_ {\widetilde{\varphi}} (\widetilde{t})\right|^{1/2} ={\mathfrak{a}}(F(\mathbf{x}, {\widetilde{\by}} )) h(\mathbf{x}, {\widetilde{\by}} )^{-m_\psi-(s+1)/2} \left| \Delta_\varphi(t(\widetilde{t}))\right|^{1/2}.
\end{align*}
Then, using the local coordinates $t$ and $\wt t=F^*t$ introduced above, on $C_ {\widetilde{\varphi}} $ we find
\begin{align*}
w_ {\widetilde{\varphi}} &= F^*\hspace*{-3pt}\left(\rho_X^{-m_e}\rho_Y^{-m_\psi-(s+1)/2} {\mathfrak{a}}\right) \left| \Delta_\varphi(t(\widetilde{t}))\right|^{1/2} \sqrt{\left|\dd \widetilde{t}\right|}\\
&=
F^*\hspace*{-3pt}\left(\rho_X^{-m_e}\rho_Y^{-m_\psi-(s+1)/2} {\mathfrak{a}}
\left|\Delta_\varphi(t)\right|^{1/2}
\sqrt{|\dd t|}\right)=
F^*w_{\varphi}.
\end{align*}
\end{proof}
As a half-density valued amplitude, $w_\varphi$ is
of order $(m_e,m_\psi-(s+1)/2)$, as shown by the computations above.
In accordance with the definition of the principal part (cf. Definition \ref{def:princpart}), we set
\[
{\mathfrak{w}_{\varphi}}=\left.\left({\mathfrak{a}} \cdot \sqrt{|d_\varphi|}\right)\right|_{\mathcal{C}_\varphi}.
\]
As seen above, ${\mathfrak{w}_{\varphi}}$ transforms to
${\mathfrak{w}_{ {\widetilde{\varphi}} }}$ under the pull-back via
$F$. Since $\lambda_\varphi$ is a local diffeomorphism $C_\varphi\to L_\varphi$, we can also consider
\[
\alpha_\varphi=(\lambda_\varphi)_*({\mathfrak{w}_{\varphi}}),
\]
which yields a local half-density on $\Lambda_\varphi$. The fact that, for the two
equivalent phase functions $\varphi$ and $ {\widetilde{\varphi}} $, we have
$\lambda_ {\widetilde{\varphi}} =\lambda_{\varphi}\circ F$, together with
the transformation properties of ${\mathfrak{w}_{\varphi}}$, shows
that
\[
\alpha_ {\widetilde{\varphi}} =\alpha_{\varphi}=\alpha,
\]
that is, $\alpha_ {\widetilde{\varphi}} $ and $\alpha_\varphi$ are equivalent local
representations of a half-density $\alpha$ defined on $\Lambda$,
in the local parametrizations $\Lambda_ {\widetilde{\varphi}} $ and $\Lambda_\varphi$,
respectively.
We now prove that the same holds true if $ {\widetilde{\varphi}} $ is merely a non-degenerate phase function equivalent to $\varphi$ in the sense of Definition \ref{def:phequiv}.
First, if we repeat the construction of $\sqrt{|d_ {\widetilde{\varphi}} |}$ described above, all the computations remain valid modulo terms, generated by $ {\widetilde{\Phi}} $,
which contain an extra factor $\rho_X\rho_{ {\widetilde{Y}} }$. This is due to
\begin{align*}
F^*\varphi- {\widetilde{\varphi}} &\in{\mathscr{C}^\infty}( {\widetilde{U}} )
\\
&\Leftrightarrow
\rho_X^{-1}\rho_{ {\widetilde{Y}} }^{-1}
\widetilde{f}(\mathbf{x}, {\widetilde{\by}} )
=\rho_X^{-1}\rho_{ {\widetilde{Y}} }^{-1}h(\mathbf{x}, {\widetilde{\by}} )^{-1}
(F^*f)(\mathbf{x}, {\widetilde{\by}} )+g(\mathbf{x}, {\widetilde{\by}} ),
g\in{\mathscr{C}^\infty}( {\widetilde{U}} )
\\
&\Leftrightarrow
\widetilde{f}(\mathbf{x}, {\widetilde{\by}} )
=h(\mathbf{x}, {\widetilde{\by}} )^{-1}(F^*f)(\mathbf{x}, {\widetilde{\by}} )
+\rho_X\rho_{ {\widetilde{Y}} } g(\mathbf{x}, {\widetilde{\by}} ),
g\in{\mathscr{C}^\infty}( {\widetilde{U}} ).
\end{align*}
Then, by rescaling $w_{ {\widetilde{\varphi}} }$ through multiplication by $\rho_X^{m_e}\rho_{ {\widetilde{Y}} }^{m_\psi+(s+1)/2}$ and then restricting ${\mathfrak{w}_{\varphi}}$ on $\mathcal{C}_ {\widetilde{\varphi}} $,
such additional terms identically vanish.
Moreover, by Lemma \ref{lem:phpequiv} and Remark \ref{rem:strictness},
we know that, in a neighbourhood $ {\widetilde{U}} $ of any point in the interior of
$\mathcal{C}_ {\widetilde{\varphi}} ^e$ or $\mathcal{C}_ {\widetilde{\varphi}} ^\psi$, which does not intersect
$\mathcal{C}_ {\widetilde{\varphi}} ^{\psi e}$, it can be assumed, after passage to the principal parts, that $ {\widetilde{\varphi}} =F^*\varphi$ on $\mathcal{C}_ {\widetilde{\varphi}} \cap\partial {\widetilde{U}} $, see Section \ref{sec:moves}. It follows that the factor
$\exp(i(F^*\varphi- {\widetilde{\varphi}} ))$, appearing in ${\widetilde{\mathfrak{a}}}$ (cf. \eqref{eq:oscintequiv}) also disappears, away from the corner, when
restricting to the faces $\mathcal{C}_ {\widetilde{\varphi}} ^e$ or $\mathcal{C}_ {\widetilde{\varphi}} ^\psi$.
Finally, we observe that ${\mathfrak{w}_{\varphi}}$ and ${\mathfrak{w}_{ {\widetilde{\varphi}} }}$ are obtained
as restrictions of smooth objects on $X\times Y$ and $X\times \wt Y$ to their respective boundaries. As such, their transformation behavior extends, by continuity, to the corner as well,
producing smooth objects on $\mathcal{C}_\varphi$ and $\mathcal{C}_ {\widetilde{\varphi}} $.
By push-forward
through $\lambda_ {\widetilde{\varphi}} $ and ${\lambda_\varphi}$, we find again that
$\alpha_ {\widetilde{\varphi}} =\alpha_\varphi=\alpha$ locally on
$\Lambda_ {\widetilde{\varphi}} =\Lambda_\varphi=\Lambda$.
\subsection{Non-degenerate phase functions, reduction of the fiber}\label{subs:ndgfbred}
We now consider a $\varphi$ such that reduction of fiber variables, see Section \ref{subs:fbred}, is possible. By the argument in Section \ref{subs:ndg}, we may then write $I_\varphi(a)=I_{{\mathrm{pr}}ed}(b)$
with $b$ from \eqref{eq:fiberredsymb}.
We now compare $\alpha_\varphi$ to the analogously defined half-density $\beta_{\mathrm{pr}}ed$. We can
replace the phase function $\varphi$ by the equivalent phase function given in \eqref{eq:wtp}, and this does not affect $\alpha_\varphi$. Hence we may assume that $\varphi$ is of the form $\varphi(\mathbf{x},\mathbf{y})={\mathrm{pr}}ed(\mathbf{x},\mathbf{y}')+\frac{1}{2}\rho_X^{-1}\rho_Y^{-1} \langle Qy'', y''\rangle$.
As such, we assume, in this splitting of coordinates, $C_\varphi\subset\{(\mathbf{x},\mathbf{y}',0)\}$.
We find:
\begin{lem}
\label{lem:dtransfnondeg}
Under the identification $C_{{\mathrm{pr}}ed}\times\{0\}=C_\varphi$, we have
\begin{equation*}
\sqrt{|d_{\varphi}|}=|\det Q|^{-\frac{1}{2}} \sqrt{|d_{{\mathrm{pr}}ed}|}.
\end{equation*}
\end{lem}
\begin{proof}
We compute
\begin{align*}
\Phi(\mathbf{x},\mathbf{y})
&=\big(- {f_{\red}} (\mathbf{x},\mathbf{y}p)+\rho_Y\partial_{\rho_Y} {f_{\red}} (\mathbf{x},\mathbf{y}p)\quad\partial_{y^{\mathrm{pr}}ime} {f_{\red}} (\mathbf{x},\mathbf{y}p)\quad 0\big)\\
&\quad+\big(-\dfrac{1}{2}\langle Qy'', y''\rangle\quad 0 \quad \partial_{ {y^{\prime\prime}} }Q( {y^{\prime\prime}} )\big)\\
&=:( {\Phi_{\red}} (\mathbf{x},\mathbf{y}p)\quad0)
+\left(\Psi( {y^{\prime\prime}} )\quad Q {y^{\prime\prime}} \right)\in{\mathbb{R}}^{s-r}\times{\mathbb{R}}^{r}.
\end{align*}
Therefore,
\begin{align}
\nonumber
\frac{\partial(t,\Phi)}{\partial(\mathbf{x},\mathbf{y})}(\mathbf{x},\mathbf{y})&=
\begin{pmatrix}
\dfrac{\partial t}{\partial\mathbf{x}}(\mathbf{x},\mathbf{y}) & \dfrac{\partial t}{\partial\mathbf{y}p}(\mathbf{x},\mathbf{y}) & \dfrac{\partial t}{\partial {y^{\prime\prime}} }(\mathbf{x},\mathbf{y})
\\
\rule{0mm}{9mm}\dfrac{\partial {\Phi_{\red}} }{\partial\mathbf{x}}(\mathbf{x},\mathbf{y}p) & \dfrac{\partial {\Phi_{\red}} }{\partial\mathbf{y}p}(\mathbf{x},\mathbf{y}p) & -\dfrac{1}{2}\dfrac{\partial \Psi}{\partial {y^{\prime\prime}} }( {y^{\prime\prime}} )
\\
\rule{0mm}{6mm}0 & 0 & Q
\end{pmatrix}.
\end{align}
Consequently,
\begin{align*}
\sqrt{|d_{\varphi}|}&= \left|\det\frac{\partial(t,\Phi)}{\partial(\mathbf{x},\mathbf{y})}\right|^{-1/2}_{C_ {\widetilde{\varphi}} }
\sqrt{|dt|}
\\
&=\left|\det\frac{\partial (t, {\Phi_{\red}} )}{\partial(\mathbf{x},\mathbf{y}p)}\right|^{-\frac{1}{2}}_{C_{{\mathrm{pr}}ed}}\cdot|\det Q|^{-\frac{1}{2}}
\sqrt{|dt|}
\\
&=|\det Q|^{-\frac{1}{2}} \sqrt{|d_{{\mathrm{pr}}ed}|}.
\end{align*}
\end{proof}
Notice that\footnote{Observe that $\mathfrak{a}_{red}$ is obtained by splitting of the density and weight factors in two steps.} $\mathfrak{a}=\mathfrak{a}_{\red}$. We compute, by \eqref{eq:princred1}, modulo amplitudes of lower order,
\begin{multline}
\label{eq:orderb}
b(\mathbf{x},\mathbf{y}p)= \rho_X^{-m_e+r/2}\rho_Y^{-m_\psi-r/2} |\det Q|^{-1/2} e^{i\frac{\pi}{4} \mathrm{sgn}(Q)} {\mathfrak{a}}(\mathbf{x},\mathbf{y}p,0)
\sqrt{\mu_X} (\rho_Y^{-(s-r+1)/2}|d\mathbf{y}p|).
\end{multline}
We observe that $b$ is an amplitude of order $(m_e-r/2, m_\psi+r/2)$ and find
\begin{align*}
{\mathfrak{b}}(\mathbf{x},\mathbf{y}p)&=|\det Q|^{-1/2} e^{i\frac{\pi}{4} \mathrm{sgn}(Q)} {\mathfrak{a}}(\mathbf{x},\mathbf{y}p,0) + {\mathcal{O}}\big(\rho_X \rho_Y\big),
\end{align*}
which implies, using Lemma \ref{lem:dtransfnondeg},
\begin{align*}
{\mathfrak{w}_{ {\varphi_{\red}} }}&=\left.\left( {\mathfrak{b}}(\mathbf{x},\mathbf{y}p) \sqrt{|d_{{\mathrm{pr}}ed}|} \right)\right|_{\mathcal{C}_{\mathrm{pr}}ed}
\\
&=e^{i\frac{\pi}{4} \mathrm{sgn}(Q)}\left.\left( \mathfrak{a}(\mathbf{x},\mathbf{y}) \sqrt{|d_\varphi|}\right)
\right|_{\mathcal{C}_ {\widetilde{\varphi}} }
\\
&=e^{i\frac{\pi}{4} \mathrm{sgn}(Q)}{\mathfrak{w}_{\varphi}}.
\end{align*}
This, in turn, finally gives
\[
\beta_{{\mathrm{pr}}ed}=(\lambda_{{\mathrm{pr}}ed})_*({\mathfrak{w}_{ {\varphi_{\red}} }})=e^{i\frac{\pi}{4} \mathrm{sgn}(Q)}\cdot(\lambda_\varphi)_*({\mathfrak{w}_{\varphi}})=e^{i\frac{\pi}{4} \mathrm{sgn}(Q)}\cdot\alpha_\varphi.
\]
\subsection{Clean phase functions, elimination of the excess}\label{subs:clnphf}
We now proceed with the last reduction step, namely, we consider a clean phase function and eliminate its excess. As in Section \ref{subss:elexcess}, we assume $Y={\mathbb{B}}^{s-e}\times(-\varepsilonilon,\varepsilonilon)^e$ with the fibers of $\mathcal{C}_\varphi\rightarrow \Lambda_\varphi$ given by constant $(\mathbf{x},\rho_Y,y')$ and arbitrary
$y''\in(-\varepsilonilon,\varepsilonilon)^e$.
Switching to the phase function $\wt{\varphi}$ in \eqref{eq:wtredex}, we may write $I_\varphi(a)=I_{\wt\varphi}(b)$ with $b$ defined in \eqref{eq:excesssymb}. We apply the
construction of the previous section, and obtain
the density $\beta_{\wt{\varphi}}=(\lambda_{ {\widetilde{\varphi}} })_*\left({\mathfrak{b}}\cdot \sqrt{|d_ {\widetilde{\varphi}} |}\right)_{\mathcal{C}_ {\widetilde{\varphi}} }$ from the data $( {\widetilde{\varphi}} ,b)$.
Alternatively, we may study the parameter dependent family of oscillatory integrals
$I_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}(a(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}))$ with phase functions $\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$ defined in \eqref{eq:phiypp}
and amplitudes
\[
a(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\colon(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)\mapsto \rho_Y^{-e}\,a(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime,y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})=\rho_Y^{-e}\,a(\mathbf{x},\mathbf{y}),
\]
with corresponding principal parts ${\mathfrak{a}}(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$. Since $\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$ is non-degenerate, we can define the
parameter dependent family of half-densities on $\Lambda$
$$
\alpha_\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})=(\lambda_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})})_*\left({\mathfrak{a}}(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}) \cdot \sqrt{|d_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}|}\right)_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}},
$$
and finally set
\begin{equation}\label{eq:prsymbexc}
\gamma_ {\widetilde{\varphi}} =\int_{(-\varepsilon,\varepsilon)^e}\alpha_\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\,dy^{{\mathrm{pr}}ime{\mathrm{pr}}ime}.
\end{equation}
\begin{prop}
The half-densities on $\Lambda_{ {\widetilde{\varphi}} }=\Lambda_{\varphi}=\Lambda$ given by $\gamma_ {\widetilde{\varphi}} $ and $\beta_ {\widetilde{\varphi}} $ coincide.
\end{prop}
\begin{proof}
We consider the smooth family of diffeomorphisms $G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})=\mathrm{id}\times g(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$, depending on the parameter $y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}$,
involved in $\wt{G}$ from \eqref{eq:diffeoG}. Assuming the amplitudes $a(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$ supported away from the corner points, we can suppose, as above,
$G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})^*\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}) - {\widetilde{\varphi}} =0.$
We now compute, using Lemma \ref{lem:CLFstarff} and the expression \eqref{eq:excesssymb}, together with the transformation properties of ${\mathfrak{w}_{\varphi}}$,
\begin{align*}
\left({\mathfrak{b}}_ {\widetilde{\varphi}} \cdot \sqrt{|d_ {\widetilde{\varphi}} |}\right)(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)|_{\mathcal{C}_ {\widetilde{\varphi}} } &=
{\mathfrak{b}}_ {\widetilde{\varphi}} (\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)|_{\mathcal{C}_ {\widetilde{\varphi}} }\left| \det\frac{\partial(\widetilde{t}, {\widetilde{\Phi}} )}{\partial(\mathbf{x}, \mathbf{y}^{\mathrm{pr}}ime )}\right|^{-\frac{1}{2}}_{\mathcal{C}_ {\widetilde{\varphi}} }\sqrt{|d\widetilde{t}|}
\\
(\eqref{eq:trap} \Rightarrow) \qquad &=\int_{(-\varepsilon,\varepsilon)^e}\hspace*{-12pt}
{\mathfrak{a}}(G(\mathbf{x},\mathbf{y}))|_{\mathcal{C}_ {\widetilde{\varphi}} }\,
\left|\det\frac{\partial g}{\partial\mathbf{y}^{\mathrm{pr}}ime}(\mathbf{x},\mathbf{y})\right|_{\mathcal{C}_ {\widetilde{\varphi}} } \hspace*{-2pt} [h(\mathbf{x},\mathbf{y})]_{\mathcal{C}_ {\widetilde{\varphi}} }^{-m_\psi-s-1}\times
\\
&\phantom{=\int_{(-\varepsilon,\varepsilon)^e}}
\times\left| \det\frac{\partial(\widetilde{t}, {\widetilde{\Phi}} )}{\partial(\mathbf{x}, \mathbf{y}^{\mathrm{pr}}ime )}\right|^{-\frac{1}{2}}_{\mathcal{C}_ {\widetilde{\varphi}} }\sqrt{|d\widetilde{t}|}\,d y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}
\\
(\text{Lemma }\ref{lem:trDelta} \Rightarrow) \qquad &=\int_{(-\varepsilon,\varepsilon)^e}\hspace*{-12pt}
G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})^*\hspace*{-3pt}\left[{\mathfrak{a}}(\mathbf{x},\mathbf{y})|_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}}\hspace*{-3pt}
\left| \det\frac{\partial(t,\Phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}))}{\partial(\mathbf{x}, \mathbf{y}^{\mathrm{pr}}ime )}\right|^{-\frac{1}{2}}_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}}\hspace*{-18pt}\sqrt{|dt|}\right]\hspace*{-2pt}d y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}
\\
(\text{Def. of }d_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}\Rightarrow) \qquad &=\int_{(-\varepsilon,\varepsilon)^e}\hspace*{-12pt}
G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})^*\hspace*{-3pt}\left[\left({\mathfrak{a}}(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\cdot \sqrt{|d_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}|}\right)(\mathbf{x},\rho_Y,y^{\mathrm{pr}}ime)\right]_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}}\hspace*{-3pt}d y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}.
\end{align*}
Applying $(\lambda_ {\widetilde{\varphi}} )_*$ to the left-hand side, we obtain $\beta_{ {\widetilde{\varphi}} }$.
To apply $(\lambda_ {\widetilde{\varphi}} )_*$ to the right-hand side, we first recall that $ {\widetilde{\varphi}} $ and $\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$ are equivalent by $G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})$. Using again Lemma \ref{lem:CLFstarff}
(see also Lemma \ref{lem:arrange}), this implies
\begin{equation}\label{eq:wtpphequiv}
\lambda_ {\widetilde{\varphi}} =\lambda_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}\circ G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\Rightarrow(\lambda_ {\widetilde{\varphi}} )_*=(\lambda_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})})_*\circ G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})_* .
\end{equation}
Since $\lambda_ {\widetilde{\varphi}} $ does not depend on $y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}$, we can take it inside the integral and use \eqref{eq:wtpphequiv}, finally obtaining
\begin{align*}
\beta_ {\widetilde{\varphi}} &=
(\lambda_ {\widetilde{\varphi}} )_*\left[\int_{(-\varepsilon,\varepsilon)^e}
G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})^*\hspace*{-3pt}\left[\left({\mathfrak{a}}(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\cdot \sqrt{|d_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}|}\right)\right]_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}}\hspace*{-3pt}d y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}\right]
\\
&=\int_{(-\varepsilon,\varepsilon)^e}(\lambda_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})})_*\circ G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})_*\circ
G(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})^*\hspace*{-3pt}\left[\left({\mathfrak{a}}(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\cdot \sqrt{|d_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}|}\right)\right]_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}}\hspace*{-3pt}
d y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}
\\
&=\int_{(-\varepsilon,\varepsilon)^e}(\lambda_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})})_*
\hspace*{-3pt}\left[\left({\mathfrak{a}}(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\cdot \sqrt{|d_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}|}\right)\right]_{\mathcal{C}_{\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})}}\hspace*{-3pt}
d y^{{\mathrm{pr}}ime{\mathrm{pr}}ime}=\int_{(-\varepsilon,\varepsilon)^e}\alpha_\phi(y^{{\mathrm{pr}}ime{\mathrm{pr}}ime})\,dy^{{\mathrm{pr}}ime{\mathrm{pr}}ime}=\gamma_ {\widetilde{\varphi}} .
\end{align*}
Extension to the corner points as in the previous subsections proves the claim.
\end{proof}
We already showed that the half-density $\alpha$ associated with $I_\varphi(a)$ is invariant under a change of equivalent non-degenerate phase functions.
Together with the argument above, this also shows that the half-density $\gamma$ associated with
$I_\varphi(a)$ remains the same under the change of equivalent phase functions which are clean with the same excess.
\subsection{Principal symbol and principal symbol map} Let $u\in I^{m_e,m_\psi}(X,\Lambda)$. Consider any local representation of $u$, as introduced in Definition \ref{def:Lagdist},
with clean phase function $\varphi$ with excess $e$ associated with $\Lambda$ and $a$ some local symbol density.
The arguments in the previous subsections show how to associate with these data a half-density $\gamma$,
defined on $\Lambda$. We also showed that switching to an equivalent phase function, as well as the elimination of the excess, do not change $\gamma$. The reduction of the fiber variables replaces $\gamma$
with $\gamma^{\mathrm{pr}}ime$ such that
\[
\gamma^{\mathrm{pr}}ime=e^{i\frac{\pi}{4}\mathrm{sgn}(Q)}\,\gamma,
\]
with $Q$ from \eqref{eq:wtp}. Let $\widetilde{\gamma}$ be the half-density defined by an integral representation $I_ {\widetilde{\varphi}} (\widetilde{a})$, with another
phase function $\widetilde{\varphi}$ associated with $\Lambda$. Then, similarly to \cite{Treves}, in general we have
\begin{equation}\label{eq:diffsigma}
\widetilde{\gamma}=e^{i(\sigma-\widetilde{\sigma})\frac{\pi}{4}}\,\gamma,
\end{equation}
where $\sigma=\mathrm{sgn}\left(\,\rho_{Y}^{-1}\rho_X^{-1}\,{}^\scat H_{Y}\varphi\right)$, and $\widetilde{\sigma}=\mathrm{sgn}\left(\,\rho_{ {\widetilde{Y}} }^{-1}\rho_X^{-1}\,{}^\scat H_{ {\widetilde{Y}} } {\widetilde{\varphi}} \right)$.
Denote by $\widetilde{r}$ the number of fiber variable for $ {\widetilde{\varphi}} $, $\widetilde{s}$ the dimension of $ {\widetilde{Y}} $ and $\widetilde{e}$ the excess of $ {\widetilde{\varphi}} $, and define the integer number
\[
\kappa=\frac{1}{2}(\sigma-\widetilde{\sigma}-s+\widetilde{s}+e-\widetilde{e}).
\]
Then, \eqref{eq:diffsigma} is equivalent to
\begin{equation}\label{eq:coherence}
i^\kappa e^{i(s-e)\frac{\pi}{4}}\,\gamma = e^{i(\widetilde{s}-\widetilde{e})\frac{\pi}{4}}\,\widetilde{\gamma}.
\end{equation}
We are then led to the following definition of principal symbol map.
\begin{defn}\label{def:prsymb}
Let $u\in I^{m_e,m_\psi}(X,\Lambda)$. We define $\mathscr{I}(u)=\{(Y_j,\varphi_j)\}$ as the collection of manifolds and associated clean phase functions $(Y_j,\varphi_j)$
locally parametrizing $\Lambda$, giving rise to local representations of $u$ in the form $I_{\varphi_j}(a_j)$.
With each pair $(Y,\varphi)\in\mathscr{I}(u)$ we associate the half-density $\gamma$, as described in Subsection \ref{subs:clnphf},
in such a manner that, for any other element $( {\widetilde{Y}} , {\widetilde{\varphi}} )\in\mathscr{I}(u)$, we have
the coherence relation \eqref{eq:coherence} in $\lambda_\varphi(Y)\cap\lambda_ {\widetilde{\varphi}} ( {\widetilde{Y}} )$. We call the collection of half-densities $\{\gamma_j\}$, each one associated
with $(Y_j,\varphi_j)\in\mathscr{I}(u)$, the \emph{principal symbol of $u$}, and write $j^\Lambda_{m_e,m_\psi}(u)=\{\gamma_j\}$.
\end{defn}
By an argument completely similar to the one in \cite{Treves}, we can prove the following result.
\begin{thm}
Let $\Lambda$ be a ${\mathrm{sc}}$-Lagrangian on $X$. Then, the map
\begin{equation}\label{eq:prsymbmap}
j^\Lambda_{m_e,m_\psi}\colon I^{m_e,m_\psi}(X,\Lambda)\ni u\mapsto \{\gamma_j\}
\end{equation}
given in Definition \ref{def:prsymb} is surjective.
Moreover, the null space of the map \eqref{eq:prsymbmap} is $I^{m_e-1,m_\psi-1}(X,\Lambda)$, and thus
\eqref{eq:prsymbmap} defines a bijection
\[
\text{classes in } I^{m_e,m_\psi}(X,\Lambda) / I^{m_e-1,m_\psi-1}(X,\Lambda) \mapsto \{\gamma_j\}.
\]
The image space of $j^\Lambda_{m_e,m_\psi}$ can be seen as ${\mathscr{C}^\infty}(\Lambda, M_\Lambda\otimes\Omega^{1/2})$, where $M_\Lambda$ is the Maslov bundle over $\Lambda$.
\end{thm}
{\mathfrak{a}}pendix
\section{Resolution of Lagrangian singularities near the corner}
\label{sec:blowup}
In this appendix, we show that $\Lambda^{\psi e}$ may be viewed as a Legendre manifold with respect to a (degenerate) contact form, well defined on the blow-up of the corner component $\Wt^\psie$ of ${}^\scat \,\overline{T}^*X$.
We have already stated that the forms
\begin{align*}
\alpha^\psi:=\rho_\Xi^2\partial_{\rho_\Xi}\lrcorner\,\omega \quad \text{and} \quad
\alpha^e:=\rho_X^2\partial_{\rho_X}\lrcorner\,\omega.
\end{align*}
are well-defined in the interior near the respective boundary face $\mathcal{W}^e$ or $\mathcal{W}^\psi$ and extend to it. The freedom in choosing the boundary defining function has as a consequence that these forms are merely well-defined up to a multiple by a positive function, however their contact structure at the boundary (which is all we need to characterize $\Lambda^\bullet$ as Legendrian) is independent of the choice of bdfs. Neither form extends to the corner component $\Wt^\psie$.
Instead of the rescaled $1$-forms, we now consider the non-rescaled forms
\begin{align*}
^{\mathrm{sc}}\alpha^\psi:=\rho_\Xi\partial_{\rho_\Xi}\lrcorner\,\omega\\
^{\mathrm{sc}}\alpha^e:=\rho_X\partial_{\rho_X}\lrcorner\,\omega
\end{align*}
as sections of ${}^\scat \,T^*({}^\scat \,T^*X^o)$. Then, these extend as \emph{scattering one forms} on ${}^\scat \,\overline{T}^*X$, cf. \cite[(2.11)]{MZ}.
\begin{lem}
\label{lem:scatforms}
The forms $^{\mathrm{sc}}\alpha^\psi$ and $^{\mathrm{sc}}\alpha^e$ extend from ${}^\scat \,T^*X^o$ to scattering one-forms on ${}^\scat \,\overline{T}^*X$.
In a particular choice of coordinates (see \cite{MZ} and Remark \ref{rem:compequiv}) they are given by
\begin{align*}
\,^{\mathrm{sc}}\alpha^e&=\frac{\dd \eta_1}{\rho_X\rho_\Xi}-\frac{\eta_1\dd \rho_\Xi}{\rho_X\rho_\Xi^2}+\eta''\frac{\dd x}{\rho_X\rho_\Xi},\\
\,^{\mathrm{sc}}\alpha^\psi&=\eta_1\frac{\dd \rho_X}{\rho_\Xi \rho_X^2}+\eta'' \frac{\dd x}{\rho_X\rho_\Xi}.
\end{align*}
Here, $\eta=(\eta_1,\eta'')$ are smooth functions of $(\rho_\Xi,\xi)$, $d-1$ of which may be chosen as coordinates.
\end{lem}
Again, the (scattering) contact structures of these forms, when restricted to the respective boundary faces, do not depend on the choice of bdf, since two choices of bdf only differ by positive factors. These forms $^{\mathrm{sc}}\alpha^\bullet$ will then vanish on $\Lambda^\bullet$, $\bullet\in\{e,\psi\}$, since one can identify the kernels of $^{\mathrm{sc}}\alpha^\bullet$ with that of $\alpha^\bullet$ by rescaling there. Furthermore, both $^{\mathrm{sc}}\alpha^\psi$ as well as $^{\mathrm{sc}}\alpha^e$ vanish when restricted to $\Lambda^{\psi e}$.
\begin{ex}
On $T^*{\mathbb{R}}d$ with canonical coordinates $(x,\xi)$, this corresponds to both the forms
$$\xi \cdot \dd x\quad \text{and}\quad -x\cdot \dd \xi$$
vanishing on the bi-conic (in $x$ and $\xi$) manifold with base $\Lambda^{\psi e}$, cf. \cite{CoSc2}.
\end{ex}
Hence, $\Lambda^{\psi e}$ is, in some sense, (scattering) isotropic.\footnote{Not with respect to the standard symplectic form, since it does not extend to the boundary, but to a rescaling of it.} We note, however, that the $\Lambda^{\psi e}$ is not Lagrangian with respect to any symplectic form on $\mathcal{W}^{\psi e}$, since
$$\dim(\Lambda^{\psi e})=d-2\neq d-1=\frac{\dim(\mathcal{W}^{\psi e})}{2}.$$
However, we may now blow-up the corner $\Wt^\psie$ in ${}^\scat \,\overline{T}(X)$ and consider the front face $\beta^{-1}(\mathcal{W}^{\psi e})$ in $[{}^\scat \,\overline{T}(X);\mathcal{W}^{\psi e}]$, which is a $2d-1$ dimensional manifold, see Figure \ref{fig:Lpblowup}. Here,
$$\beta: [{}^\scat \,\overline{T}(X);\mathcal{W}^{\psi e}]\rightarrow {}^\scat \,\overline{T}(X),$$
is the blow-down map.
\begin{figure}
\caption{Resolution of $\Lambda_\varphie$ near the corner}
\label{fig:Lpblowup}
\end{figure}
\begin{prop}
The lift of the form
\begin{align*}
\alpha^{\psi e} = \frac{\rho_X \rho_\Xi}{2} ({ }^{\mathrm{sc}}\alpha^\psi + { }^{\mathrm{sc}}\alpha^e)
\end{align*}
to the blowup space $$[{}^\scat \,\overline{T}^*X; \mathcal{W}^{\psi e}]\xrightarrow{\ \beta\ }{{}^\scat \,\overline{T}^*X}$$
restricts to a contact 1-form on the front face $\beta^{-1}\mathcal{W}^{\psi e}$.
Moreover, $\beta^{-1}(\Lambda^{\psi e})$ is Legendrian with respect to $\alpha^{\psi e}$.
\end{prop}
\begin{proof}
We note that
$$\alpha^{\psi e}=\rho_X\rho_\Xi\frac{1}{2}\left(\rho_X\partial_{\rho_X}+\rho_\Xi\partial_{\rho_\Xi}\right)
\lrcorner\,\omega.$$
In the special choice of coordinates of Lemma \ref{lem:scatforms}, we compute
$$\alpha^{\psi e}=\frac{1}{2}\eta_1\left(\frac{\dd \rho_X}{\rho_X}-\frac{\dd \rho_\Xi}{\rho_\Xi}\right)+\frac{1}{2} \dd\eta_1+\eta''\dd x$$
Now, smooth coordinates on the blow up of ${}^\scat \,\overline{T}^*X$ along $\Wt^\psie=\{\rho_X=\rho_\Xi=0\}$ are given by
\begin{equation}
\begin{cases}
\rho=\rho_X\quad \tau=\frac{\rho_\Xi}{\rho_X} \quad (x,\xi) & \rho_X>\rho_X\\
\rho=\rho_\Xi\quad \tau=\frac{\rho_X}{\rho_\Xi} \quad (x,\xi) & \rho_\Xi>\rho_X
\end{cases}
\end{equation}
In any case, $\beta^*\alpha^{\psi e}$ is of the form
$$\alpha^{\psi e}=\pm\frac{1}{2}\eta_1 \frac{\dd \tau}{\tau}+\frac{1}{2} \dd\eta_1+\eta''\dd x$$
Since $\tau=0$ marks the boundary of the front face $\beta^{-1}\mathcal{W}^{\psi e}$, $\alpha^{\psi e}$ is a $1$-form on the interior of $\beta^{-1}\mathcal{W}^{\psi e}$.
Finally, $\alpha^{\psi e}$ vanishes on $\beta^{-1}\Lambda^{\psi e}$ since $^{\mathrm{sc}}\alpha^\psi$ and $^{\mathrm{sc}}\alpha^e$ vanish on $\Lambda^{\psi e}$.
\end{proof}
\end{document} |
\begin{document}
\begin{abstract}
We give a survey on the concept of Poissonian pair correlation (PPC) of sequences in the unit interval, on existing and recent results and we state a list of open problems. Moreover, we present and discuss a quite recent multi-dimensional version of PPC.
\end{abstract}
\date{}
\maketitle
\section{The concept of Poissonian Pair Correlation (PPC) for sequences in $[0,1)$} \label{sect_1}
Let $x_1, x_2,\ldots,$ be a sequence of real numbers in the unit interval $[0,1)$. In the following, for some $x \in [0,1)$, we denote by $\| x \|$ the distance to the nearest integer, i.e., to be precise $\| x \|:= \min(x,1-x)$. Further, in the sequel, $\lbrace \cdot \rbrace$ will denote the fractional part of a real number.
\begin{definition}
We say that $\left(x_n\right)_{n \geq 1} \in [0,1)$ has Poissonian pair correlation (PPC) if for all real $s > 0$ we have
$$
\underset{N \rightarrow \infty}{\lim} \frac{1}{N} \# \left\{1 \leq k \neq l \leq N \left| \left\|x_k-x_l\right\|< \frac{s}{N}\right\}\right. = 2s.
$$
\end{definition}
To put this in intuitive words, PPC means to study small distances between sequence elements, i.e., the concept of PPC deals with a ``local'' distribution property of a sequence in the unit interval.\\ \\
It is natural to expect that the pair correlation function, $R_N$, defined as
$$
R_N(s):=\frac{1}{N} \# \left\{1 \leq k \neq l \leq N \left|\left\|x_k-x_l\right\|< \frac{s}{N} \right\}\right.
$$
tends to $2s$. We give the following heuristic explanation for this limit behaviour:
\begin{center}
\includegraphics[angle=0,width=100mm]{Fig1}
~\\
Figure 1
\end{center}
~\\
Consider a fixed $N$, and fix a sequence element $x_n$ for some $1\leq n\leq N$. Then, the region around $x_n$ with length $\frac{2s}{N}$ (see Figure 1) i expected to contain $2s\frac{N-1}{N}$ of the remaining $(N-1)$ points $x_i$, for $i=1,\ldots, N$ and $i \neq n$.\\
Consequently, this means, on average there are $2s \frac{N-1}{N}$ different indices $i = 1,\ldots, N$ with $i \neq n$, such that
\begin{equation*}
\left\|x_i - x_n\right\| < \frac{s}{N}.
\end{equation*}
Since $n$ can attain values between $1$ and $N$, we expect that there are $2s(N-1)$ pairs with
\begin{equation*}
\left\|x_k-x_l\right\|< \frac{s}{N}, \qquad \text{for } 1 \leq k \neq l \leq N.
\end{equation*}
Hence, we expect the quantity $R_N(s)$ to be approximately $2s \frac{N-1}{N}$, and therefore
\begin{equation*}
\underset{N \rightarrow \infty}{\lim} R_N (s) = 2s.
\end{equation*}
Indeed, it can be shown that, in a certain sense, almost every sequence $x_1, x_2, \ldots$ in $[0,1)$ has PPC. To be precise, if we consider a sequence $(X_n)_{n \geq 1}$ of i.i.d.~ random variables drawn from the uniform distribution on $[0,1)$, then
\begin{equation*}
\underset{N \rightarrow \infty}{\lim} \frac{1}{N} \# \left\{1 \leq k \neq l \leq N \left| \left\|X_k-X_l\right\|< \frac{s}{N}\right\}\right. = 2s,
\end{equation*}
almost surely. \\ \\
We want to briefly mention that the original motivation for the investigation of the PPC property comes from quantum physics. Roughly speaking, the concept is related to the distribution properties of the discrete energy spectrum $\lambda_1, \lambda_2, \ldots$ of a Hamiltonian operator of a quantum system. The famous Berry-Tabor-Conjecture in quantum physics now states, that this discrete energy spectrum (ingoring degenerate cases) has PPC. For more details on the connection to quantum physics, we refer the reader to, for example, \cite{AAL} and the references cited therein.\\ \\
Note that for several quantum systems the discrete energy spectrum $\lambda_1,\lambda_2,\ldots$ has the following special form
$$
\left(\lambda_n\right)_{n \geq 1} = \left(\left\{a_n \alpha\right\}\right)_{n \geq 1},
$$
where $\alpha$ is a real constant, and $\left(a_n\right)_{n \geq 1}$ is a given sequence of positive integers. Therefore, in the 1990's Rudnick, Sarnak and Zaharescu started to investigate the PPC property of sequences of the form $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ in $[0,1)$ from a purely mathematical point of view. \\
Whenever, in the following, we consider such sequences we restrict the setting to strictly increasing sequences of positive integers. The most basic example of such a sequence is the classical Kronecker sequence $\left(\left\{n \alpha\right\}\right)_{n \geq 1}$. This sequence does not have the PPC property for any choice of $\alpha$. In most of the seminal papers on PPC, this fact was argued by taking the famous Three-Gap-Theorem into account (see e.g., \cite{not1, not2, not3}). \\
The Three-Gap-Theorem states the following: For every choice of $\alpha$ and for every $N$, the gaps between neighbouring points of the set
\begin{equation*}
\left\{1 \alpha\right\}, \left\{2 \alpha\right\}, \ldots, \left\{N \alpha\right\}
\end{equation*}
can have at most three different lengths. A sequence with such a gap structure does not exhibit a random behaviour and therefore it is reasonable to expect that it cannot have PPC. Nonetheless, it is not immediately clear that this argument is indeed valid. \\
\begin{center}
\includegraphics[angle=0,width=100mm]{Fig2}
~\\
Figure 2
\end{center}
~\\
To argue that, we want to emphasize that the elements of a sequence satisfying such a weak gap structure could be ordered in a way, such that ``many different'' distances between (not necessarily neighbouring) elements can occur (see Figure 2). However, for the Kronecker sequence a very simple argument can be given to deduce the fact that it does not have PPC for any choice of $\alpha$: \\ \\
Let $\alpha \sim \frac{p_n}{q_n}$ where $\frac{p_n}{q_n}$ is a best approximation fraction to $\alpha$ with $\left(p_n,q_n\right)=1$. It is well-known from basic Diophantine approximation theory that $\alpha = \frac{p_n}{q_n} + \theta_n$, with either $0\leq \theta_n < \frac{1}{2q_n^2}$, or $-\frac{1}{2q_n^2} < \theta < 0$.\\
Let us assume the first case. Then, the set of points
\begin{equation*}
\left\{1 \alpha\right\}, \left\{2 \alpha\right\}, \ldots, \left\{N \alpha\right\}
\end{equation*}
equals the set of points
\begin{equation*}
\frac{0}{N} + \varphi_0, \frac{1}{N}+\varphi_1, \ldots, \frac{N-1}{N} + \varphi_{N-1},
\end{equation*}
with $0 \leq \varphi_i < \frac{1}{2N}$, for $i=0,\ldots,N-1$ (see red points in Figure 3).
\begin{center}
\includegraphics[angle=0,width=100mm]{Fig3}
~\\
Figure 3
\end{center}
~\\
Hence, two arbitrary elements of this point set have a distance of at least $\frac{1}{2N}$. Thus, for the choice $s=\frac{1}{4}$, we get that $R_N (s)=0$ and consequently the pair correlation function cannot tend to $2s =\frac{1}{2}$ for $N$ to infinity.\\ \\
In fact, a deeper investigation reveals that the Three-Gap-Theorem is indeed a valid argument to deduce the result on the PPC strucuture of the Kronecker sequence. It is an immediate consequence of the following theorem, which was proven in \cite{not4}, in combination with the Three-Gap-Theorem.
\begin{theorem}\label{th:gap}
Let $\left(x_n\right)_{n \geq 1}$ be a ``weak finite-gap-sequence'', i.e., there exists an integer $L$ and indices $N_1<N_2<N_3< \ldots$ such that for all $i$ the set $x_1, x_2, \ldots, x_{N_{i}}$ has at most $L$ different gap lengths between neighbouring elements. Then, $\left(x_n\right)_{n \geq 1}$ does not have PPC.
\end{theorem}
Let us now come to ``posititve results'' and to the study of the metrical pair correlation theory of sequences of the form $(\lbrace a_n \alpha \rbrace)_{n \geq 1}$. In \cite{not3} Rudnick and Sarnak showed the following:
\begin{theorem}
The sequence $\left(\left\{n^d\alpha\right\}\right)_{n \geq 1}$ with an integer $d \geq 2$ has PPC for almost all $\alpha$.
\end{theorem}
The case $d=2$ is of particular interest, as in this setting the spacings of sequence elements are related to the distances between the energy levels of the so-called ``boxed oscillator'', i.e., the study of the PPC property of $\left(\left\{n^2\alpha\right\}\right)_{n \geq 1}$ is of special importance in quantum physics. The PPC property and also the gap distribution of the sequence $\left(\left\{n^2\alpha\right\}\right)_{n \geq 1}$ was further investigated by several authors (see \cite{not5,not6,not2}) and they could also derive the metrical result for $d=2$. Heath-Brown could even show slightly more:
\begin{theorem}
The sequence $\left(\left\{n^2\alpha\right\}\right)_{n \geq 1}$ has PPC for almost all real numbers $\alpha$. Moreover, there is a dense set of constructible values of $\alpha$ for which the PPC property holds, i.e., there is an informal algorithm, which, for any closed interval $I$ of positive length, provides a convergent sequence of rational numbers belonging to $I$, whose limit $\alpha$ satisfies that $\left(\left\{n^2\alpha\right\}\right)_{n \geq 1}$ has PPC.
\end{theorem}
These three results are only metrical statements. However, until now, however, no single explicit $\alpha$ is known such that $\left(\left\{n^2 \alpha\right\}\right)_{n \geq 1}$ (or $\left(\left\{n^d \alpha\right\}\right)_{n \geq 1}$ for any integer $d \geq 2$) has PPC. Nonetheless, we know that it is \textbf{not true}, that $\left(\left\{n^2 \alpha\right\}\right)_{n \geq 1}$ has PPC for \textbf{all} irrational $\alpha$. Consider the following example of an $\alpha$ that is in a certain sense well-approximable: If $\alpha$ is an irrational number such that $\left|\alpha - \frac{a}{q}\right| < \frac{1}{4 q^3}$ for infinitely many integers $a$ and $q$, then $\left(\left\{n^2\alpha\right\}\right)_{n \geq 1}$ does not have PPC (see \cite{not5}). \\ \\
On the other hand, it is conjectured that for an $\alpha$ which is not too well approximable, we have PPC for $\left(\left\{n^2\alpha\right\}\right)_{n \geq 1}$. To be precise: Let $\alpha$ be such that for every $\varepsilon > 0$ there is a $c(\varepsilon)>0$ with $\left|\alpha - \frac{a}{q}\right|> c(\varepsilon) \frac{1}{q^{2+\varepsilon}}$ for all $a,q \in \mathbb{Z}$, then $\left(\left\{n^2 \alpha\right\}\right)_{n \geq 1}$ has PPC (see e.g., \cite{not5}). This property for an irrational $\alpha$ is often referred to as Diophantine. It is well-known that almost all irrationals are Diophantine, e.g., every real irrational algebraic number has this property. Above discussion illustrates that the pair correlation theory of sequences $\left(\left\{n^d\alpha\right\}\right)_{n \geq 1}$ is strongly related to the Diophantine properties of $\alpha$. \\ \\
The case of lacunary sequences $\left(a_n\right)_{n \geq 1}$ was considered for example by Rudnick and Zaharescu (see \cite{not1}) or by Berkes, Philipp and Tichy (see \cite{not7}). We recall that a sequence $\left(a_n\right)_{n \geq 1}$ is a lacunary sequence if there exists a $c > 1$ such that $\frac{a_{n+1}}{a_n} > c$ for all $n \geq N(c)$. Again, they obtained the following metrical result:
\begin{theorem}
Let $\left(a_n\right)_{n \geq 1}$ be a lacunary sequence of positive integers. Then $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$.
\end{theorem}
We may again ask for explicit examples of lacunary sequences $\left(a_n\right)_{n \geq 1}$ and $\alpha \in \mathbb{R}$ such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC.\\
One of the most basic examples of a lacunary sequence of integers certainly is the sequence $\left(2^n\right)_{n \geq 1}$. In a first step, we may restrict the possible candidates $\alpha$ for which $\left(\left\{2^n \alpha\right\}\right)_{n \geq 1}$ could have PPC. To do so, we consider the following result which has been shown independently by Grepstad and Larcher \cite{not8}, Aistleitner, Lachmann and Pausinger \cite{not9}, and Steinerberger \cite{not10}:
\begin{theorem}
If the sequence $\left(x_n\right)_{n \geq 1}$ in $[0,1)$ has PPC, then $\left(x_n\right)_{n \geq 1}$ is uniformly distributed in $[0,1)$.
\end{theorem}
\begin{remark}
The paper of Grepstad and Larcher also contains a quantitative version of this result. Roughly speaking: If $R_N(s)$ tends to $2s$ ``fast in some uniform sense'', then the discrepancy $D_N$ of the sequence $\left(x_n\right)_{n \geq 1}$ cannot tend to zero ``too slowly''.
\end{remark}
Having this result in mind, we can restrict the set of possible choices for $\alpha$, such that $\left(\left\{2^n \alpha\right\}\right)_{n \geq 1}$ has PPC. The above theorem implies that for such an $\alpha$ the sequence $\left(\left\{2^n \alpha\right\}\right)_{n \geq 1}$ has to be uniformly distributed, and hence $\alpha$ has to be normal in base 2.\\
The most well-known example of a real $\alpha$ which is normal in base 2 is the Champernowne number in base 2, i.e., the number $\alpha$ which has in base 2 the digit representation
$$
\alpha = 0.~01~10~11~100~101~110~111~1000\ldots
$$
However, it was shown by Pirsic and Stockinger in \cite{not11}, that for $\alpha$ the Champernowne number, the sequence $\left(\left\{2^n \alpha\right\}\right)_{n \geq 1}$ does not have PPC. Also for further concrete examples like Stoneham-numbers or infinite de Bruijn-words, the sequence $\left(\left\{2^n\alpha\right\}\right)_{n \geq 1}$ does not have PPC (see \cite{not4}).\\ \\
Indeed, until now we do not know any concrete example of $\left(a_n\right)_{n \geq1}$, a lacunary sequence, and a real $\alpha$ such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does have PPC.\\ \\
Recently, a much more general metric result on PPC of sequences of the form $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ was given in \cite{not12} which shows that there is an intimate connection between the concept of PPC of sequences $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ and the notion of additive energy of the sequence $\left(a_n\right)_{n \geq 1}$. The concept of additive energy plays a central role in additive combinatorics and also appears in the study of the metrical discrepancy theory of sequences $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ (see \cite{not26,not25}).\\
For a strictly increasing sequence $a_1<a_2 < a_3 < \ldots$ of positive integers, we consider the first $N$ elements $a_1, \ldots, a_N$. The additive energy of $a_1, \ldots, a_N$ is given by
$$
E\left(a_1, \ldots,a_N\right) := \sum_{\underset{a_i-a_j=a_k-a_l}{1 \leq i, j,k,l \leq N}} 1.
$$
It is obvious that $N^2 \leq E\left(a_1,\ldots,a_N\right) \leq N^3$ always holds. In \cite{not12} the following was shown:
\begin{theorem}
Let $\left(a_n\right)_{n \geq1}$ be a strictly increasing sequence of integers such that there exists $\varepsilon > 0$ with
$$
E\left(a_1, \ldots, a_N\right) = \mathcal{O} \left(N^{3-\varepsilon}\right),
$$
then $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$.
\end{theorem}
This result recovers all above mentioned metrical results and implies several new results and examples.
\begin{example}
If $\left(a_n\right)_{n \geq 1}$ is lacunary, then $E\left(a_1, \ldots,a_N\right) = \mathcal{O} \left(N^2\right)$, hence $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$.
\end{example}
\begin{example}
If $\left(a_n\right)_{n \geq 1}$ are the values of a polynomial $f(n) \in \mathbb{Z} \left[x\right]$ of degree $d \geq 2$, then $E\left(a_1, \ldots, a_N\right) = \mathcal{O} \left(N^{2 + \varepsilon}\right)$ for all $\varepsilon > 0$, hence $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$.
\end{example}
\begin{example}
Let $\left(a_n\right)_{n \geq 1}$ be a convex sequence, i.e., $a_n -a_{n-1} < a_{n+1}-a_n$ for all $n$, then it was shown by Konjagin \cite{not13}, that $E\left(a_1, \ldots,a_N\right) = \mathcal{O} \left(N^{\frac{5}{2}}\right)$, hence $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$.
\end{example}
\begin{example}
If $a_n = \left[\beta n^c\right]$ for some $\beta > 0$ and $c > 1$, then
$$
E\left(a_1,\ldots,a_N\right) = \mathcal{O} \left(\max\left(N^{\frac{5}{2}}, N^{4-c}\right)\right),
$$
hence $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$ (see \cite{not14}).
\end{example}
The above theorem immediately raises two natural questions:
\begin{question} \label{qu_a}
Is it possible for an increasing sequence of distinct integers $(a_n)_{n \geq 1}$ which satisfies $E(a_1, \ldots, a_N) = \Omega(N^3)$ that the sequence $(\lbrace a_n \alpha \rbrace)_{n \geq 1}$ has PPC for almost all $\alpha$?
\end{question}
\begin{question} \label{qu_b}
If, for almost all $\alpha$, $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does \textbf{not} have PPC, does this imply $E\left(a_1,\ldots,a_N\right) = \Omega \left(N^3\right)$?
\end{question}
Both questions were answered by J. Bourgain in an Appendix to \cite{not12}.\\ \\
Concerning Question~\ref{qu_a}, Bourgain showed:
\begin{itemize}
\item If $E\left(a_1, \ldots,a_N\right) = \Omega \left(N^3\right)$, then there exists a set of positive measure such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does not have PPC for every $\alpha$ in this set.
\end{itemize}
This result was improved by Lachmann and Technau \cite{not15}:
\begin{itemize}
\item If $E\left(a_1, \ldots,a_N\right) = \Omega \left(N^3\right)$, then there exists a set of full Lebesgue measure such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does not have PPC for every $\alpha$ contained in this set.
\end{itemize}
Finally, in \cite{not16} this result was improved to its final form:
\begin{theorem}
If $E\left(a_1, \ldots, a_N\right) = \Omega \left(N^3\right)$, then there is \textbf{no} $\alpha$ such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC.
\end{theorem}
Concerning Question~\ref{qu_b} Bourgain showed that the answer to this question is \textbf{``no''}: He gave a construction for a sequence $\left(a_n\right)_{n\geq 1}$ with $E\left(a_1, \ldots,a_N\right) = o\left(N^3\right)$, such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does not have PPC for almost all $\alpha$.\\ \\
Up to now, we have the following situation:\\
$E\left(a_1, \ldots,a_N\right) = \Omega \left(N^3\right)$ implies that there is \textbf{no} $\alpha$ such that the sequence $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC. \\
$E\left(a_1, \ldots,a_N\right) = \mathcal{O}\left(N^{3-\varepsilon}\right)$ for some $\varepsilon > 0$ implies PPC for almost all $\alpha$. \\ \\
The result by Aistleitner, Larcher and Lewko, was first extended by Bloom, Chow, Gafni and Walker, albeit under an additional density condition on the integer sequence $(a_n)_{n \geq 1}$ (see \cite{not27}).
\begin{theorem}
Let $a_1, \ldots, a_N$ be the first $N$ elements of an increasing sequence of positive integers $( a_n )_{n \geq 1}$ satisfying the following density condition:
\begin{equation*}
\delta(N) = \Omega_{\varepsilon} \left( \frac{1}{(\log N)^{2 +2\varepsilon}} \right),
\end{equation*}
where $\delta(N):=N^{-1} \#( \lbrace a_1, \ldots, a_N \rbrace \cap \lbrace 1, \ldots, N \rbrace)$ and suppose that
\begin{equation*}
E(a_1, \ldots, a_n) = \mathcal{O}_{\varepsilon}\left( \frac{N^3}{(\log N)^{2+ \varepsilon} } \right),
\end{equation*}
for some $\varepsilon>0$, then, for almost all $\alpha$, the sequence $(\lbrace a_n \alpha \rbrace)_{n \geq 1}$ has PPC.
\end{theorem}
Recently, Bloom and Walker (see \cite{not17}) improved over this result by showing the following theorem.
\begin{theorem}\label{TH:THWalk}
There exists an absolute positive constant $C$ such that the following is true. Suppose that
\begin{equation*}
E(a_1, \ldots, a_N) = \mathcal{O}\left(\frac{N^3}{(\log N)^C} \right),
\end{equation*}
then for almost all $\alpha$, $(\lbrace a_n \alpha \rbrace)_{n \geq 1}$ has PPC.
\end{theorem}
A consequence of this result is the following theorem:
\begin{theorem}
Let $(a_n)_{n \geq 1}$ be an arbitrary infinite subset of the squares. Then $(a_n)_{n \geq 1}$ is metric Poissonian, i.e., for almost all $\alpha$, $(\lbrace a_n \alpha \rbrace)_{n \geq 1}$ has PPC.
\end{theorem}
To see that this result is valid, we note that, if $a_1, \ldots, a_N$ denotes a finite set of squares, then $E(a_1, \ldots, a_N)= \mathcal{O}( N^3 \exp({-c_1 \log^{c_2} N)})$ for some absolute positive constants $c_1$ and $c_2$, see e.g., \cite{not24}. \\ \\
The proof of Theorem \ref{TH:THWalk} relies on a new bound for GCD sums with $\alpha=1/2$,
which improves over the bound by Bondarenko and Seip (see \cite{not18}),
if the additive energy of $a_1, \ldots, a_N$ is sufficiently large.
Note that the constant $C$ was not specified in the above mentioned paper,
but the authors thereof conjecture that Theorem~\ref{TH:THWalk} holds for $C>1$ already. This result would be best possible. To see this, consider the following result by Walker \cite{not19}:
\begin{theorem}
Let $\left(a_n\right)_{n \geq 1} = \left(p_n\right)_{n \geq 1}$ be the sequence of primes (note that for the primes we have $E\left(p_1, \ldots,p_N\right) \asymp \frac{N^3}{\log N}$). Then, $\left(\left\{p_n \alpha \right\}\right)_{n \geq 1}$ does not have PPC for almost all $\alpha$.
\end{theorem}
The region between $\mathcal{O}\left(\frac{N^3}{(\log N)^C} \right)$, $C>1$, and $\Omega\left(N^3\right)$ is therefore the interesting region and one might speculate about a sharp threshold which allows to fully describe the metrical pair correlation theory in terms of the additive energy. Further constructions and examples of sequences in this ``interesting region'', with an even smaller additive energy compared to the primes, were given by Lachmann and Technau \cite{not15}:
\begin{theorem}
There exists a strictly increasing sequence of positive integers $\left(a_n\right)_{n \geq1}$ with
$$
E\left(a_1, \ldots,a_N\right)=\mathcal{O} \left(\frac{N^3}{\log N (\log \log N)}\right)
$$
such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does not have PPC for almost all $\alpha$.
\end{theorem}
On the other hand they gave a positive result of the following form:
\begin{theorem}
There exists a strictly increasing sequence of positive integers $\left(a_n\right)_{n \geq 1}$ with
$$
E\left(a_1, \ldots,a_N\right) = \Omega \left(\frac{N^3}{\log N \left(\log \log\right)^{1+\varepsilon}}\right)
$$
for all $\varepsilon > 0$, such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$.
\end{theorem}
We have the following situation as illustrated in Figure 4.
\begin{center}
\includegraphics[angle=0,width=1\textwidth]{Fig7}
~\\
Figure 4
\end{center}
~\\
The following question therefore is near at hand:\\ \\
Is there a strict threshold $T$ such that an additive energy of magnitude smaller than $T$ implies PPC of $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ for almost all $\alpha$ and an additive energy of magnitude larger than $T$ implies PPC for $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ for almost no $\alpha$? The fundamental question concerning such a putative threshold was raised in \cite{not17}. The authors of this paper conjectured that there is a sharp Khintchine-type threshold, i.e., if $E (a_1, \ldots, a_N) =\Theta( N^3\psi(N))$, for some weakly decreasing function $\psi : \mathbb{Z}_{\geq 1} \to [0,1]$, then, for almost all $\alpha$, $(\lbrace a_n \alpha \rbrace)_{n \geq 1}$ has PPC if and only if
\begin{equation*}
\sum_{N \geq 1} \frac{\psi(N)}{N}
\end{equation*}
converges. \\ \\
The negative answer to this question was given by Aistleitner, Lachmann and Technau \cite{not20}:
\begin{theorem}
There exists a sequence $\left(a_n\right)_{n \geq 1}$ of integers with
$$
E\left(a_1, \ldots, a_N\right) = \Omega \left(\frac{N^3}{\left(\log N\right)^{\frac{3}{4}+\varepsilon}}\right)
$$
such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for almost all $\alpha$. Hence a threshold $T$ cannot exist.
\end{theorem}
To conclude, the additive energy is not enough to fully describe the metrical pair correlation theory. Some further number theoretic properties need to be considered to cope with that problem.
\section{The concept of Poissonian Pair Correlation (PPC) for sequences in $[0,1)^d$}
Of course it makes sense to generalize the concept of PPC to the multi-dimensional setting. One way to generalize the one-dimensional concept to a multi-dimensioanl setting was defined and discussed in \cite{not21} (for a more general analysis of a multi-dimensional PPC concept, we refer to the recent work \cite{not29}). Here, we present the definition of \cite{not21}.
\begin{definition}
Let $\left(x_n\right)_{n \geq 1}$ be a sequence in the $d$-dimensional unit-cube $\left[\left.0,1\right.\right)^d$. We say that $\left(x_n\right)_{n \geq 1}$ has PPC if for all $s > 0$ we have
$$
\underset{N \rightarrow \infty}{\lim} \frac{1}{N} \# \left\{1 \leq k \neq l \leq N \left| \left\|x_k-x_l\right\|_{\infty} < \frac{s}{N^{\frac{1}{d}}}\right\}\right. = \left(2s\right)^d.
$$
\end{definition}
For this definition of $d$-dimensional PPC it again follows that $\left(x_n\right)_{n \geq 1}$ with PPC is uniformly distributed in $\left[\left.0,1\right.\right)^d$. Moreover, for many of the above mentioned results in dimension $d=1$ we have analogous statements in dimension $d \geq 2$. For example the $d$-dimensional Kronecker sequence
\begin{equation*}
\left(\left\{n \alpha_1\right\}, \left\{n \alpha_2\right\}, \ldots,\left\{n \alpha_d\right\}\right)_{n \geq 1}
\end{equation*}
never has PPC. The proof of this fact however needs a bit more subtle arguments than in dimension $1$. \\ \\
Naturally, we would also expect that under the same condition on the additive energy as in Theorem \ref{TH:THWalk}, the sequence
\begin{equation*}
( \lbrace a_n \boldsymbol{\alpha} \rbrace)_{n \geq 1}
\end{equation*}
has Poissonian pair correlations for almost all instances and, in fact, we have the following even better result, which is a consequence of better bounds on GCD sums for larger exponents than $1/2$:
\begin{theorem}\label{TH:THMet}
Let $a_1, \ldots, a_N$ denote the first $N$ elements of $(a_n)_{n \geq 1}$ and suppose that
\begin{equation*}
E(a_1, \ldots, a_N) = \mathcal{O}\left(\frac{N^3}{(\log N)^{1+ \varepsilon}} \right), \text{ for any } \varepsilon > 0,
\end{equation*}
then for almost all choices of $\boldsymbol{\alpha}=(\alpha_1, \ldots, \alpha_d) \in \mathbb{R}^d$,
\begin{equation*}
( \lbrace a_n \boldsymbol{\alpha} \rbrace)_{n \geq 1}
\end{equation*}
has PPC.
\end{theorem}
However, if the additive energy is of maximal order, i.e., if we have $E(a_1, \ldots, a_N)=\Omega(N^3)$, then there is no $\boldsymbol{\alpha}$ such that $(\lbrace a_n \boldsymbol{\alpha} \rbrace)_{n \geq 1}$ has PPC:
\begin{theorem}\label{TH:THMAX}
If $E(a_1, \ldots, a_N)=\Omega(N^3)$, then for any choice of $\boldsymbol{\alpha} =(\alpha_1, \ldots, \alpha_d) \in \mathbb{R}^d$ the sequence
\begin{equation*}
( \lbrace a_n \boldsymbol{\alpha} \rbrace)_{n \geq 1},
\end{equation*}
does not have Poissonian pair correlations.
\end{theorem}
\section{Open Problems}
Many questions related to the concept of PPC are still open, and we will state some of them in this section as open problems.
\begin{problem}
Is it possible to extend the ``green region'' and/or the ``red region'' in Figure~4? That means: Are there functions $\varphi(n)$ (which increases slower than $(\log N)^C$, for $C$ the constant in Theorem \ref{TH:THWalk}) and $\psi (n)$ both tending to $+\infty$ for $n$ to infinity, such that:\\
If $E\left(a_1, \ldots,a_N\right) = \mathcal{O} \left(\frac{N^3}{\varphi(N)}\right)$, then, for almost \textbf{all} $\alpha$, $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC.\\
If $E\left(a_1, \ldots,a_N\right) = \Omega \left(\frac{N^3}{\psi(N)}\right)$, then there is \textbf{no} $\alpha$ such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC.
\end{problem}
\begin{problem}
We know that if $E\left(a_1, \ldots,a_N\right) = \Omega \left(N^3\right)$, then $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for \textbf{no} $\alpha$. We consider the following question to be of high interest: Is there a sequence $\left(a_n\right)_{n \geq 1}$ with the property that for almost all $\alpha$, $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ does not have PPC, but, there exists a set of zero measure such that $\left(\left\{a_n \alpha\right\}\right)_{n \geq 1}$ has PPC for every $\alpha$ contained in this set?
\end{problem}
Indeed, we believe, that this is not possible, i.e.,:
\begin{conj}
If, for almost all $\alpha$, $\left(\left\{a_n \alpha\right\}\right)_{n \geq1}$ does not have PPC, then it has PPC for \textbf{no} $\alpha$.
\end{conj}
For example (by the result of A. Walker) this would imply: $\left(\left\{p_n \alpha\right\}\right)_{n \geq 1}$ has PPC for \textbf{no} $\alpha$.
\begin{problem}
Although the metrical theory of sequences of the form $( \lbrace a_n \boldsymbol{\alpha} \rbrace)_{n \geq 1}$ seems to be well-established, we do not know any explicit construction of $\boldsymbol{\alpha}$ (not even in the one-dimensional case) such that $( \lbrace a_n \boldsymbol{\alpha} \rbrace)_{n \geq 1}$ has Poissonian pair correlations. It is in general very hard to construct sequences on the torus having the PPC property. The only known explicit examples -- to the best of our knowledge -- of sequences with this property are $\lbrace \sqrt{n} \rbrace_{n \geq 1}$ (see \cite{not22}) and certain directions of vectors in an affine Euclidean lattice (see \cite{not23}). Hence, of course, it would be of high interest to find more concrete examples of sequences with PPC.
\end{problem}
\begin{problem}
This problem concerns a possible extension of Theorem \ref{th:gap} mentioned above. We recall, that we have shown that a sequence $\left(x_n\right)_{n \geq 1}$ with a weak finite gap property never has PPC. We wonder whether this result can be improved by showing that it still holds if we have to deal with a sequence with not necessarily ``finite gap-'' but with a ``slowly-growing-gap-'' property, i.e., with a sequence with the following property:\\
There is a (slowly growing) function $L$ and a sequence of indices $N_1<N_2<N_3<\ldots$ such that $x_1,x_2,\ldots,x_{N_{i}}$ always has gaps of at most $L\left(N_i\right)$ different lengths.
\end{problem}
\begin{problem}
Let $\left(\boldsymbol{x}_n\right)_{n \geq 1}$ be the $d$-dimensional Halton-sequence in any bases $q_1, \ldots,q_d$, where $d \geq 2$. Does $\left(\boldsymbol{x}_n\right)_{n \geq 1}$ have PPC or not? Of course, we strongly conjecture that it does not have PPC. In dimension $d=1$ the Halton-sequence is the well-known van der Corput sequence. In this case the PPC property trivially does not hold. In fact, it should not be too hard to prove this in the multidimensional case, too.
\end{problem}
The last problem we want to state concerns a multi-dimensional version of the metrical PPC result for the primes.
\begin{problem}
Is it true that for almost all instances of $\boldsymbol{\alpha}$ the sequence $( \lbrace p_n \boldsymbol{\alpha} \rbrace)_{n \geq 1}$, where $(p_n)_{n \geq 1}$ denotes the primes, does not have PPC?
\end{problem}
\end{document} |
\begin{document}
\title{Skeleta in non-Archimedean and tropical geometry}
\begin{abstract}
I describe an algebro-geometric theory of \mathbf{e}mph{skeleta}, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the \mathbf{e}mph{spectra} of the same. The primary result of this paper is that the topological space $X$ underlying a non-Archimedean analytic space may locally be recovered from the sheaf $|{\sh S}h O_X|$ of \mathbf{e}mph{pointwise valuations} of its analytic functions; in other words, $(X,|{\sh S}h O_X|)$ is a skeleton.
\mathbf{e}nd{abstract}
{\sh S}etcounter{tocdepth}{2}
\tableofcontents
{\sh S}ection{Introduction}
There are several areas in modern geometry in which one is led to consider spaces with \mathbf{e}mph{affine} or \mathbf{e}mph{piecewise affine} structure. The three with which I am in particular concerned are, in order of increasing subtlety: \begin{itemize}\item skeleta of non-Archimedean analytic spaces (\cite{Berkpadic}); \item tropical geometry (\cite{Grisha},\cite{BPR}); \item affine manifolds with singularities (\cite{Grossbook},\cite{KoSo2}).\mathbf{e}nd{itemize}
These cases share the following features:
\begin{itemize}\item they are piecewise manifolds; \item it makes sense to ask which continuous, real-valued functions are \mathbf{e}mph{piecewise affine}; \item they admit a stratification on which it makes sense to ask which of these are \mathbf{e}mph{convex}.\mathbf{e}nd{itemize}
Moreover, in each case the structure is determined entirely by an underlying space $B$, together with a sheaf \[|{\sh S}h O_B|{\sh S}ubseteq C^0\left(B;{\mathbb R}{\sh S}qcup\{-\infty\}\right)\] of piecewise-affine, convex (where this is defined) functions.
The sheaf $|{\sh S}h O_B|$ is naturally a sheaf of \mathbf{e}mph{idempotent semirings} under the operations of pointwise maximum and addition. It has long been understood, at least in the tropical geometry community (cf. e.g. \cite{Grisha}) that such semirings are the correct algebraic structures to associate to piecewise-affine geometries like $B$.
A natural question to ask is whether this sheaf-theoretic language can be pushed further in this setting and, as in algebraic geometry, the underlying space $B$ recovered from the semirings of local sections of $|{\sh S}h O_B|$. In this paper, I provide an affirmative answer to this question, though, as for the passage from classical algebraic geometry to scheme theory, it will require us to alter our expectations of what type of space underlies a piecewise-affine geometry. The resulting theory is what I call the theory of \mathbf{e}mph{skeleta}.
The relationship between the theories of schemes and of skeleta goes beyond mere analogy: they can in fact be couched within the \mathbf{e}mph{same} theoretical framework (appendix \ref{TOPOS}), \`a la Grothendieck (cf. also \cite{Toen}, \cite{Durov}). Within this framework, one need only specify which semiring homomorphisms
\[ {\mathbb G}amma(U;|{\sh S}h O_U|)\rightarrow {\mathbb G}amma(V,|{\sh S}h O_V|) \]
are dual to open immersions $V{\mathfrak h }ookrightarrow U$ of skeleta. This is enough to associate to every semiring $\alpha$ a quasi-compact topological space, its \mathbf{e}mph{spectrum} $\Spec\alpha$. Skeleta can then be defined to be those semiringed spaces locally modelled by the spectra of semirings.
My main contention in this paper is that the \mathbf{e}mph{primary} source of skeleta is the non-Archimedean geometry, and this is why I have adopted the terminology of this field. The initial concept that links non-Archimedean and piecewise-affine geometry is that of a valuation. Indeed, semirings are the natural recipients of valuations, while topological rings are the sources.
The topology of skeleta is selected so as to ensure that there is a unique functor
\[ \mathrm{sk}:\mathbf{Ad}\rightarrow\mathbf{Sk} \]
from the category $\mathbf{Ad}$ of adic spaces to the category $\mathbf{Sk}$ of skeleta, a natural homeomorphism $X\widetilde\rightarrow\mathrm{sk}X$ for $X\in\mathbf{Ad}$, and a \mathbf{e}mph{universal valuation} ${\sh S}h O_X\rightarrow|{\sh S}h O_X|$. This \mathbf{e}mph{universal skeleton} $\mathrm{sk}X$ of an analytic space $X$ can be thought of as the skeleton whose functions are the pointwise logarithmic norms of analytic functions on $X$. In particular, $X$ is locally the spectrum of the semiring of these functions.
The existence of this functor is the primary result of this paper. I also recover within the category of skeleta certain further examples that already existed in the literature: the \mathbf{e}mph{dual intersection} or \mathbf{e}mph{Clemens complex} of a degeneration (\S\ref{EGS-Clemens}), and the \mathbf{e}mph{tropicalisation} of a subvariety of a toric variety (\S\ref{EGS-trop}).
{\sh S}ubsection*{Gist}
The categories of skeleta (section \ref{SKEL}) and of non-Archimedean analytic spaces may be constructed in the same way: as a category of \mathbf{e}mph{locally representable sheaves} on some site whose underlying category is opposite to a category of algebras (\`a la \cite{Toen}). As such, to build a bridge between the two categories, it is enough to build a bridge between the categories $\frac{1}{2}\mathbf{Ring}_t$ of \mathbf{e}mph{topological semirings} (defs. \ref{semiring}, \ref{tsemiring}) and $\mathbf{nA}$ of \mathbf{e}mph{non-Archimedean rings} (def. \ref{ADIC-def-na}), and to check that it satisfies certain compatibility conditions.
One can associate to any non-Archimedean ring $(A,A^+)$ a \mathbf{e}mph{free semiring} ${\mathbb B}^c(A;A^+)$, which, as a partially ordered set, is the set of finitely generated $A^+$-submodules of $A$. The addition on ${\mathbb B}^c(A;A^+)$ comes from the multiplication on $A$. It comes with a \mathbf{e}mph{valuation} \[ A\rightarrow {\mathbb B}^c(A;A^+),\quad f\mapsto (f) \] universal among continuous semivaluations of $A$ into a semiring whose values on $A^+$ are negative (or zero). In other words, ${\mathbb B}^c(A;A^+)$ corepresents the functor
\[ \mathrm{Val}(A,A^+;-):\frac{1}{2}\mathbf{Ring}_t\rightarrow\mathbf{Set} \]
which takes a topological semiring $\alpha$ to the set of continuous semivaluations $\val:A\rightarrow\alpha$ satisfying $\val|_{A^+}\leq0$.
In particular, if $A=A^+$, then ${\mathbb B}^cA^+:={\mathbb B}^c(A^+;A^+)$ is the set of finitely generated ideals of $A^+$, or of finitely presented subschemes of $\Spf A^+$.
Everything in the above paragraph may also be phrased in the internal logic of topoi so that, for example, it makes sense to replace $A$ and $\alpha$ with \mathbf{e}mph{sheaves} of non-Archimedean rings and semirings on a space. Thus if $X$ is a non-Archimedean analytic space, then \[|{\sh S}h O_X|:U\mapsto{\mathbb B}^c({\sh S}h O_U;{\sh S}h O_U^+)\] is a sheaf of topological semirings on $X$, universal among those receiving a continuous semivaluation from ${\sh S}h O_X$.
\begin{thrm}[\ref{SKEL-main}]Let $X$ be quasi-compact and quasi-separated. There is a natural homeomorphism \[X\widetilde\rightarrow\Spec{\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)\] which matches the structure sheaf on the right with $|{\sh S}h O_X|$ on the left.\mathbf{e}nd{thrm}
In particular, if $X$ is a qcqs formal scheme, then the spectrum of the semiring ${\mathbb B}^c{\sh S}h O_X$ of ideal sheaves on $X$ is naturally homeomorphic to $X$ itself.
This skeleton $\Spec{\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)$ is called the \mathbf{e}mph{universal skeleton} $\mathrm{sk}X$ of $X$. It follows from the universal property of its structure sheaf that the \mathbf{e}mph{real} points $\mathrm{sk}X({\mathbb R}_\vee)$ can be identified canonically with the Berkovich analytic space associated to $X$ \cite[\S1.6]{Berketale}, provided such a thing exists; see theorem \ref{thm-berk}.
\
A natural geometric counterpart to the universality of ${\mathbb B}^c$ might be to say that the universal skeleton of an analytic space is universal among skeleta $B$ equipped with a continuous map $\iota:B\rightarrow X$ and valuation ${\sh S}h O_X\rightarrow \iota_*|{\sh S}h O_B|$. However, my point of view is that the very construction of the universal semiring diminishes the importance of valuation theory in getting a handle on the geometry of $X$. It tends to be easier, and perhaps more natural, to construct skeleta $B$ with a morphism $X\rightarrow B$ in the \mathbf{e}mph{opposite} direction.
For example, let $X^+\rightarrow\Spf{\sh S}h O_K$ be a simple normal crossings degeneration over a DVR ${\sh S}h O_K$, with general fibre $j:X\rightarrow X^+$ (so $X$ is an analytic space, smooth over $K$). The irreducible components $E_i$ of the central fibre $X^+_0$ of the degeneration generate a subring $|{\sh S}h O_{\mathrm{sk}(X,X^+)}|^\circ{\sh S}ubseteq {\mathbb B}^c{\sh S}h O_{X^+}$ whose elements are the ideals \mathbf{e}mph{monomial} with respect to normal co-ordinates $(t=\prod_{i=1}^kx_i^{n_i})$. Their supports are the strata of $X^+_0$. Base-changing over $K$ yields the \mathbf{e}mph{dual intersection semiring}
\[ \mathrm{Cl}(X,X^+){\mathfrak h }ookrightarrow{\mathbb B}^c({\sh S}h O_{X^+}\mathbin{\otimes} K;{\sh S}h O_{X^+}) \]
and, dually, \mathbf{e}mph{dual intersection skeleton} $\Spec\mathrm{Cl}(X,X^+)=:{\mathbb D}elta(X,X^+){\sh S}tackrel{\mu}{\leftarrow} X$ (definition \ref{EGS-def-Clemens}).
That $X$ is defined over $K$ means that the universal skeleton, dual intersection skeleton, and morphism $\mu$ are defined over its value group: the \mathbf{e}mph{semifield of integers} ${\mathbb Z}_\vee:={\mathbb Z}{\sh S}qcup\{-\infty\}$. The real points $\mathrm{sk}(X,X^+)({\mathbb R}_\vee)$ of the dual intersection skeleton are ${\mathbb Z}_\vee$-semialgebra homomorphisms $|{\sh S}h O_{\mathrm{sk}(X,X^+)}|\rightarrow{\mathbb R}_\vee$ to the \mathbf{e}mph{real semifield} ${\mathbb R}_\vee=:={\mathbb R}{\sh S}qcup\{-\infty\}$. They can be identified with the points of the na\"ive dual intersection complex as defined in, for example, \cite[\S A.3]{KoSo2}. Indeed, the simplices of this complex are defined by the logarithms of local equations for the intersections of $X^+_0$:
\[ \frac{K\{x_1,\ldots,x_n\}}{(t=\prod_{i=1}^kx_i^{n_i})} \rightsquigarrow \frac{{\mathbb Z}_\vee\{X_1,\ldots,X_n\}}{(-1={\sh S}um_{i=1}^kn_iX_i)}, \]
where the curly braces on the right-hand side signify that $X_i\leq 0$. The latter equation $1+{\sh S}um_{i=1}^kn_iX_i=0$ cuts the dual intersection simplex
\[ \mathrm{conv}\{(0,\ldots,0,-1/n_i,0,\ldots,0)\}_{i=1}^k{\sh S}ubset {\mathbb R}_{\leq0}^n \] out of the negative orthant in ${\mathbb R}^n$.
Under this identification, the elements of the dual intersection semiring correspond to integral, piecewise-affine functions whose restriction to each cell is convex.
The construction of such skeleta, perhaps \mathbf{e}mph{partial} skeleta of $X$, is the crux of the theory. At this point I know of only a few examples (\S\ref{EGS}).
{\sh S}ubsection*{An elliptic curve}
Let us consider now the case that $X^+=E^+/{\sh S}h O_K$ is an elliptic curve degenerating semistably to a cycle of $n>3$ ${\mathbb P}^1_k$s, which I denote $\{D_i\}_{i=1}^n\in{\mathbb Z}_\vee\{X;X^+\}$. Its general fibre $E/K$ is a Tate elliptic curve. The dual intersection skeleton ${\mathbb D}elta(E,E^+)$ is, at the level of real points, a cycle of $n$ unit intervals joined at their endpoints. The vertices $\{v_i\}_{i=1}^n$ correspond to the lines $D_i$. Functions are allowed to be concave at these vertices. In particular, the function $D_i$ takes the value -1 at $v_i$ and zero at the other vertices.
Now let us collapse one of the $D_i$s \[p_i:E^+\rightarrow E^+_i\] to an $A_1$ singularity. The special fibre of $E^+_i$ is now a cycle of $(n-1)$ ${\mathbb P}^1_k$s meeting transversally except at the discriminant locus of the blow-up, which now has the local equation $(xy-t^2)$. With these co-ordinates, $p_i$ is the blow-up of the ideal $(x,y,t)$.
In the semialgebraic notation, the ideal is \[(x,t,y) = D^\prime_{i-1}\vee -1\vee D^\prime_{i+1}\in \mathrm{Cl}(E,E^+) ,\] where $D^\prime_j$ denotes the divisor whose strict transform under $p_i$ is $D_j$, so $p_i^*D_{i\pm1}^\prime=D_{i\pm 1}+D_i$. The blow-up is monomial, and hence induces a pullback homomorphism $p_i^*:\mathrm{Cl}(E,E^+)\rightarrow\mathrm{Cl}(E,E^+)$, and dually, a morphism
\[ p_i:{\mathbb D}elta(E,E^+)\rightarrow {\mathbb D}elta(E,E^+_i) \]
of the dual intersection skeleta.
\
The segment of the dual intersection complex corresponding to the singular intersection $D_{i-1}\cap D_{i+1}$ is an interval $I$ of affine length two. Considered as a function on $I$, the blow-up ideal $D^\prime_{i-1}\vee -1\vee D^\prime_{i+1}$ has real values as the absolute value
\[ |-|:I{\sh S}imeq [-1,1]\rightarrow{\mathbb R}. \]
It has a kink in the middle. Because, in ${\mathbb D}elta(E,E^+_i)$, there is no vertex here, the inverse of this function is not allowed; while of course the pullback $D_i$ \mathbf{e}mph{is} invertible on ${\mathbb D}elta(E,E^+)$.
In fact, $\mathrm{Cl}(E,E^+)$ is a \mathbf{e}mph{localisation} of $\mathrm{Cl}(E,E^+_i)$ at $D^\prime_{i-1}\vee -1\vee D^\prime_{i+1}$, and ${\mathbb D}elta(E,E^+)$ is an \mathbf{e}mph{open subset} of ${\mathbb D}elta(E,E^+_i)$.
Varying $i$, we obtain therefore an atlas
\[ \coprod_{i\neq j=1}^n{\mathbb D}elta(E,E^+) \rightrightarrows \coprod_{i=1}^n{\mathbb D}elta(E,E^+_i) \]
for a skeleton $B$ which \mathbf{e}mph{compactifies} ${\mathbb D}elta(E,E^+)$. Functions on $B$ are required to be convex everywhere, and $B({\mathbb R}_\vee)$ is, as an affine manifold, the flat circle ${\mathbb R}/n{\mathbb Z}$.
This skeleton is a kind of \mathbf{e}mph{Calabi-Yau skeleton} of $E$, and it depends only on the intrinsic geometry of $E$ and not on any choices of model. See also section \ref{EGS-ellipt}.
{\sh S}ubsection*{Mirror symmetry context}
Conjectures arising from homological mirror symmetry \cite{KoSo1} suggest that a Calabi-Yau $n$-fold $X$ approaching a so-called \mathbf{e}mph{large complex structure limit} acquires the structure of a completely integrable system $\mu:X\rightarrow B$ with singularities in (real) codimension one, shrinking to two in the limit. The base $B$ therefore acquires the structure of a Riemannian $n$-manifold with integral affine co-ordinates $y_i$, away from the singular fibres, given by the Hamiltonians of the system. The metric is locally the Hessian, with respect to these co-ordinates, of a convex function $K$, and satisfies the \mathbf{e}mph{Monge-Amp\`ere equation}
\[ d\det\left(\frac{\partial^2K}{\partial y_i\partial y_j}\right)=0 \]
which can be thought of as the `tropicalisation' of the complex Monge-Amp\`ere equation satisfied by the Yau metric.
The central idea of \cite{KoSo2} is that the skeleton $B$ can be constructed, with the \mathbf{e}mph{Legendre dual} affine structure $\check y_i$, from the non-Archimedean geometry of $X^\mathrm{an}$ or, what is the same thing, the birational geometry of its formal models. Indeed, Kontsevich noted that the Gromov-Hausdorff limit of $X$ should resemble the dual intersection complex of a certain `crepant' model thereof. To be precise, the real points of $B$ should be embeddable into $X^\mathrm{an}({\mathbb R}_\vee)$ as the dual intersection complex of any dlt minimal model of $X$ \cite{Nicaise}. Its structure as a dual intersection complex also endows it with the correct affine structure, away from a subset of codimension one which contains the singularities.
More subtle is to construct the correct \mathbf{e}mph{non-Archimedean torus fibration} $\mu:X\rightarrow B$. This would also determine the affine structure of $B$ in the sense that
\[ |{\sh S}h O_B|\cong\mathrm{Im}(\mu_*{\sh S}h O_X\rightarrow\mu_*|{\sh S}h O_X|). \]
Such a $\mu$ is determined by a choice of minimal model. Unfortunately, in dimensions greater than one, the morphisms $\mu$ coming from various models differ. The affine structures they induce are related by so-called \mathbf{e}mph{worm deformations}, which move the singularities of the affine structure along their monodromy-invariant lines. These deformations correspond to flops in birational geometry.
This forms the basis of a dictionary, motivated by mirror symmetry, between concepts in birational geometry and the tropical geometry of affine manifolds. This dictionary has been partially developed along combinatorial lines in the Gross-Siebert programme.\footnote{In general the Gross-Siebert programme \cite{GS} bypasses the non-Archimedean geometry to give a direct construction of the affine structure of $B$, up to worm deformations, in terms of toric geometry. Using this approach, they were able to obtain many results with a combinatorial flavour, and even a reconstruction of $X$ (as an algebraic variety) from $B$ together with some cocycle data. To mimic at least their basic construction in the context of skeleta is not difficult, but beyond the scope of this paper.}
However, geometrically interesting examples present enormous combinatorial complexity, already for the case of K3 surfaces. I propose that a more geometric approach, such as outlined in this paper, will be more robust in such applications.
There is some hope that, armed with a suitably flexible language, the birational geometry of $X$ together with a polarisation can be used to construct solutions to a real Monge-Amp\'ere equation on $B$.
{\sh S}ubsection{How to read this paper}
The structure of the paper is as follows. In the first three sections, we establish the theory of semirings and their (semi)modules as a theory of commutative algebras in a certain closed monoidal category, the category of \mathbf{e}mph{${\mathbb B}$-modules} $(\mathrm{Mod}_{\mathbb B},\oplus)$. The objects of $\mathrm{Mod}_{\mathbb B}$ are also known in the literature as `join-semilattices'. Since we wish to compare with non-Archimedean geometry, we actually need to work with \mathbf{e}mph{topological} ${\mathbb B}$-modules (\S\ref{TOP}). At this paper's level of sophistication, this causes few complications.
Apart from establishing the formal properties of the categories of ${\mathbb B}$-modules and semirings, the secondary thrust of this part is to introduce various versions of the \mathbf{e}mph{subobject} and \mathbf{e}mph{free} functors \begin{align}\nonumber {\mathbb B},{\mathbb B}^c:&\mathrm{Mod}_A\longrightarrow\mathrm{Mod}_{\mathbb B} \\
\nonumber &\mathbf{Ring}\longrightarrow\frac{1}{2}\mathbf{Ring} \\
\nonumber & etc. \mathbf{e}nd{align}
which will pave the major highway linking algebraic and `semialgebraic' geometry. I have spelled out in some detail the functoriality of these constructions, though they are mostly self-evident.
\
Section \ref{LOC} sets about defining the localisation theory of semirings, which is designed to parallel the one used for topological rings in non-Archimedean geometry. These \mathbf{e}mph{bounded localisations} factorise into two types: cellular, and free. The latter resemble ordinary localisations of algebras, and the algebraically-minded reader will be unsurprised by their presence. The cellular localisations, on the other hand, may be less familiar: they involve the non-flat operation of setting a variable equal to zero. Thinking of a skeleton as a polyhedral or cell complex, these localisations will be dual to the inclusions of cells (of possibly lower dimension). Perhaps confusingly, these are the semiring homomorphisms that correspond, under ${\mathbb B}^c$, to open immersions of formal schemes. The precise statements are the Zariski-open (\ref{LOC-Zar-open}) and cellular cover (\ref{LOC-Zar-cover}) formulas.
With some understanding of the `cellular topology' we are able, as an aside, to describe the spectrum of \mathbf{e}mph{contracting} semirings in terms of a na\"ive construction: the \mathbf{e}mph{prime spectrum} \S\ref{LOC-prime}. This makes clear the relationship between the topological space underlying a formal scheme $X^+$ and the spectrum of the ideal (sheaf) semiring ${\mathbb B}^c{\sh S}h O_X^+$.
It is also easy to describe the free localisations in terms of the polyhedral complex picture. Inverting a strictly convex function has the effect of destroying the affine structure along its non-differentiability locus (or `tropical set'); we therefore think of it as further subdividing our complex into the cells on which the function is affine. We can also give an algebro-geometric interpretation of these subdivisions: it is given by the \mathbf{e}mph{blow-up formula} (\ref{LOC-blow-up}). In the setting of a formal scheme $X^+$ over a DVR ${\sh S}h O_K$, it says that blowing up an ideal sheaf $J$ supported on the reduction has the effect of inverting $J$ in ${\mathbb B}^c({\sh S}h O_{X^+}\mathbin{\otimes} K;{\sh S}h O_{X^+})$. Intuitively, the blow-up of $J$ is the universal way to make it an invertible sheaf.
\
In \S\ref{SKEL} we meet the category $\mathbf{Sk}$ of skeleta, and introduce some universal constructions of certain skeleta from adic spaces and their models. The construction of this category follows the general programme of glueing objects inside a topos, as outlined in \cite{Toen}. The main result \ref{SKEL-main} - which concerns the main skeletal invariant of an analytic space $X$, the \mathbf{e}mph{universal skeleton} $\mathrm{sk}X$ - boils down to proving that for reasonable values of $X$, the topological space underlying $X$ can be identified with that of $\Spec{\mathbb G}amma(X;|{\sh S}h O_X|)$. The technical part of the proof is based on the Zariski-open, blow-up, and cellular cover formulas, which together allow us to explicitly match the open subsets of $X$ with those of its skeleton.
As an artefact of the proof, we may notice that a surprisingly many skeleta - those associated to any quasi-compact, quasi-separated analytic space - are affine. As an aside in $\S\ref{SKEL}$ I was able to obtain a kind of quantification (thm. \ref{SKEL-qc=aff}) of this observation. We also glance at the relationship (thm. \ref{thm-berk}) between skeleta and the theory of Berkovich.
In the examples section \ref{EGS}, we reconstruct some well-known `tropical spaces' as skeleta: the dual intersection complexes of locally toric degenerations (\S\ref{EGS-Clemens}), and the tropicalisations of subvarieties of a toric variety \cite{Payne} (\S\ref{EGS-trop}). I have also attempted to couch the construction of an affine manifold from a Tate elliptic curve, summarised above, in more general terms (\S\ref{EGS-ellipt}). This forms the first test case of an ongoing project.
{\sh S}ubsection*{Acknowledgements}
I would like to thank my PhD supervisors, Alessio Corti and Richard Thomas, for their support. I thank also Mark Gross, Sam Payne, and Jeff Giansiracusa, for interesting conversations.
I also thank the Cecil King foundation for funding my visit to Mark Gross in UCSD, where some of the aforementioned conversations, as well as part of the work writing this paper, occurred.
{\sh S}ection{Preliminaries and conventions}
{\sh S}ubsection{On sites and topoi}\label{TOPOS}
Our general notational conventions on sites and topoi follow the canonical \cite{SGA4}. The central glueing construction of the paper revolves around the notion of a \mathbf{e}mph{locally representable sheaf}, defined in \cite[def. 2.15]{Toen}. I only wish to replace the input, the authors' notion of \mathbf{e}mph{Zariski-open immersion}, with something a bit more flexible.
\begin{defn}\label{TOPOS-def}Let ${\sh S}h U$ be a class of monomorphisms in a category $\mathbf C$ stable for composition and base change. One defines the structure of a Grothendieck site on $\mathbf C$ whose generating coverings are finite families of morphisms in ${\sh S}h U$ that form a covering for the canonical site.
An \mathbf{e}mph{open immersion} in the associated topos $\mathbf C\hspace{4pt}\widetilde{}\hspace{4pt}$ is a morphism locally representable by morphisms in ${\sh S}h U$.
An object of $\mathbf C\hspace{4pt}\widetilde{}\hspace{4pt}$ is \mathbf{e}mph{locally representable} if it is a union of representable open subobjects.\mathbf{e}nd{defn}
Much of the theory of \cite{Toen} is valid with this more flexible input, notably proposition 2.18. I warn the reader only that without a \mathbf{e}mph{locality} requirement for our definition of affine open immersion, part 2 of \cite[prop. 2.17]{Toen} is false. This is the case, for example, for the category of adic spaces (\S\ref{ADIC}).
The resulting class of objects can also be characterised in terms of point-set topology via a modern analogue of Stone's construction:
\begin{enumerate}
\item By construction, $\mathbf C\hspace{4pt}\widetilde{}\hspace{4pt}$ is a coherent topos and so by Deligne's theorem \cite[\S IX.11.3]{Topos} it has enough points.
\item Since the morphisms in ${\sh S}h U$ were assumed to be monic, the small topos of an object $X\in\mathbf C\hspace{4pt}\widetilde{}\hspace{4pt}$ is localic.
\item Being localic and having enough points, the small topos of an object is therefore equivalent to a uniquely determined sober topological space \cite[\S IX.3.1-4]{Topos}. This determines a functor
\[ \mathbf C\hspace{4pt}\widetilde{}\hspace{4pt}\rightarrow\mathbf{Top} \]
that takes morphisms in ${\sh S}h U$ to open immersions.
\item The topological space associated to a representable (or more generally, compact) object is quasi-compact and quasi-separated.
\item Being locally representable corresponds to having a basis of open sets coming from open immersions with representable source.
\mathbf{e}nd{enumerate}
A covering - or, more precisely, two-term hypercovering - of a space $X$ will be denoted $U_\bullet\twoheadrightarrow X$, with the bullet ranging over a partially ordered set of indices.
{\sh S}ubsection{Non-Archimedean geometry}\label{ADIC}
The perspective on non-Archimedean geometry taken in this paper was influenced by the foundational works \cite{Hubook} and \cite{FujiKato}. Broadly speaking, I have adopted the categorical localisation constructions of the latter (after the approach of Raynaud), but the language and notation of the former - in particular, the nomenclature \mathbf{e}mph{adic spaces}.
I introduce the following innovations in terminology:
\begin{defns}\label{ADIC-def-na}A \mathbf{e}mph{marked formal scheme} is a pair $(X^+,Z)$ consisting of a formal scheme $X^+$ and a collection $Z$ of Cartier divisors. A morphism of marked formal schemes is a morphism $f:X^+_1\rightarrow X^+_2$ such that $(f^{-1}Z_2)^\mathrm{red}{\sh S}ubseteq Z_1$. An admissible blow-up is a finite type blow-up whose centre has underlying reduced scheme contained in $Z$.
A \mathbf{e}mph{non-Archimedean ring} is a pair $(A,A^+)$ consisting of an adic ring $A^+$ and a localisation $A$ of $A^+$. We only consider \mathbf{e}mph{locally convex} $(A,A^+)$-modules, that is, complete topological $A$-modules whose topology is generated by $A^+$-submodules; the category of such is denoted $\mathrm{LC}_{(A,A^+)}$, or just $\mathrm{LC}_A$ for short.\mathbf{e}nd{defns}
\begin{itemize}
\item The category $\mathbf{Ad}$ of adic spaces is defined by the same means as the category $\mathbf{Rf}$ of \cite[\S\textbf{II}.2]{FujiKato}, with the modification that the input to the localisation construction is instead the category of coherent \mathbf{e}mph{marked formal schemes} at admissible blow-ups, as in def. \ref{ADIC-def-na}. This ensures that the notion of adic space is a generalisation of that of formal scheme.
\item The glueing construction of \cite[\S\textbf{II}.2.2(c)]{FujiKato}, although expressed in less standard language, is identical to the locally representable sheaves story of \S\ref{TOPOS}.
\item Following Huber, the sheaf of functions extending over a model is denoted ${\sh S}h O^+$ (rather than ${\sh S}h O^\mathrm{int}$ as in \cite[\S\textbf{II}.3.2(a)]{FujiKato}). The structure sheaf of the adic topos $\mathbf{Ad}\hspace{4pt}\widetilde{}\hspace{4pt}$ is a \mathbf{e}mph{pair} $({\sh S}h O, {\sh S}h O^+)$. It is a sheaf of non-Archimedean rings in the sense of def. \ref{ADIC-def-na}.
\item Accordingly, we also adopt the notation $X^+$ for models of a adic space $X$. The category of models is denoted $\mathbf{Mdl}_{X^+}$; if $X$ is qcqs, it is cofiltered. The map $j:X\rightarrow X^+$ exhibiting the model is a morphism of adic spaces.
\item An \mathbf{e}mph{affine adic space} $X$ is one admitting an affine formal model whose marking divisors are principal. By definition, this space is the spectrum of the non-Archimedean ring $A:={\mathbb G}amma{\sh S}h O_X$; following Huber, this spectrum is denoted $\Spa A$.\mathbf{e}nd{itemize}
The key aspect of this construction that we will use is that for quasi-compact, quasi-separated $X$, as topologically ringed sites,
\begin{align}\label{important} (X,{\sh S}h O_X^+){\sh S}imeq\lim_{X^+\in\mathbf{Mdl}(X)}X^+ \mathbf{e}nd{align}
where the limit is over all formal models of $X$.
{\sh S}ection{Subobjects and ${\mathbb B}$-modules}\label{SPAN}
The theory of \mathbf{e}mph{${\mathbb B}$-modules} plays the same r\^ole in tropical geometry that the theory of Abelian groups plays in algebraic geometry: while rings are commutative monoids in the category of Abelian groups, semirings are commutative monoids instead in the category of ${\mathbb B}$-modules. This is the fundamental point of departure of the two theories. There is therefore a temptation to try to treat ${\mathbb B}$-modules as "broken" Abelian groups, and to literally translate as many concepts and constructions from the category $\mathbf{Ab}$ as will survive the transition.
In this paper, I adopt a different perspective. A ${\mathbb B}$-module is a particular type of partially ordered set which axiomatises some properties of \mathbf{e}mph{subobject posets} in Abelian and similar categories. In particular, there is a functor ${\mathbb B}:\mathbf{Ab}\rightarrow\mathrm{Mod}_{\mathbb B}$ which associates to an Abelian group its ${\mathbb B}$-module of subgroups. As such, I propose to treat ${\mathbb B}$-modules as though they are \mathbf{e}mph{lower categorical shadows} of structures in the \mathbf{e}mph{category} of Abelian groups, rather than simply as elements of a single Abelian group. The theory of ${\mathbb B}$-modules is a na\"ive form of category theory, rather than a weak form of group theory.
There is also a dual, or more precisely, \mathbf{e}mph{adjoint}, perspective, which is that a ${\mathbb B}$-module is the natural recipient of a \mathbf{e}mph{non-Archimedean seminorm} from an Abelian group. This fits well with traditional perspectives on non-Archimedean geometry. In keeping with the ahistorical nature of this paper, I barely touch upon this idea here (but see example \ref{eg-discs'}).
{\sh S}ubsection{${\mathbb B}$-modules}\label{SPAN-span}
\begin{defn}A \mathbf{e}mph{${\mathbb B}$-module} is an idempotent commutative monoid. In other words, it is a commutative monoid $(\alpha,\vee,-\infty)$ in which the identity
\[ X\vee X=X \]
holds for all $X\in\alpha$, and where $-\infty$ is the identity for $\vee$. The category of ${\mathbb B}$-modules and their homomorphisms will be denoted $\mathrm{Mod}_{\mathbb B}$.\mathbf{e}nd{defn}
A ${\mathbb B}$-module is automatically a partially ordered set with the relation
\[ X\leq Y \quad \Leftrightarrow \quad X\vee Y=Y. \]
It has all finite joins (suprema). Conversely, any poset with finite joins is a ${\mathbb B}$-module under the binary join operation. They are more commonly called \mathbf{e}mph{join semilattices} or simply \mathbf{e}mph{semilattices}.\footnote{I abandon this terminology for a number of reasons, but one could be the inconsistency of the r\^oles of the modifier \mathbf{e}mph{semi} in the words `semiring' and `semilattice'.}
We may therefore introduce immediately a path to category theory in the form of an essentially equivalent definition.
\begin{defn}\label{SPAN-def-preorder}A \mathbf{e}mph{${\mathbb B}$-module} is a preorder with finite colimits. A \mathbf{e}mph{${\mathbb B}$-module homomorphism} is a right exact functor.\mathbf{e}nd{defn}
\begin{egs}\label{eg-first}The \mathbf{e}mph{null} or \mathbf{e}mph{trivial} ${\mathbb B}$-module is the ${\mathbb B}$-module with one element $\{-\infty\}$. The \mathbf{e}mph{Boolean semifield}
is the partial order ${\mathbb B}=\{-\infty,0\}{\sh S}imeq\{\texttt{false,true}\}$.
The \mathbf{e}mph{integer, rational}, and \mathbf{e}mph{real semifields} ${\mathbb Z}_\vee,{\mathbb Q}_\vee,{\mathbb R}_\vee$ are obtained by disjointly affixing $-\infty$ to ${\mathbb Z},{\mathbb Q},{\mathbb R}$, respectively. More generally, we can obtain a semifield $H_\vee$ by adjoining $-\infty$ to any totally ordered Abelian group $H$. These semifields are totally ordered ${\mathbb B}$-modules (in fact, semirings; cf. e.g. \ref{semifields}).
If $X$ is a topological space, the set $C^0(X,{\mathbb R}_\vee)$ of continuous functions $X\rightarrow{\mathbb R}_\vee$, where ${\mathbb R}_\vee$ is equipped with the order topology, is a ${\mathbb B}$-module. So too are the subsets of bounded above functions, or of functions bounded above by some fixed constant $C\in{\mathbb R}$.
Suppose that $X$ is a manifold (with boundary). The subset $C^1(X,{\mathbb R}_\vee)$ of \mathbf{e}mph{differentiable} functions is not a ${\mathbb B}$-module, since the pointwise maximum $f\vee g$ of two differentiable functions $f,g$ needn't be differentiable. One must allow \mathbf{e}mph{piecewise} differentiable $\mathrm{P}C^1$ (or piecewise smooth $\mathrm{P}C^\infty$) functions to obtain submodules of $C^0(X,{\mathbb R}_\vee)$. Since convexity is preserved under $\vee$, the subsets of \mathbf{e}mph{convex} functions $\mathrm{CP}C^r(X;{\mathbb R}_\vee)$ are also submodules.
We can also endow $X$ with some kind of affine structure \cite[\S2.1]{KoSo2}, which gives rise to ${\mathbb B}$-modules $\mathrm{CPA}_*(X,{\mathbb R}_\vee),*={\mathbb R},{\mathbb Q},{\mathbb Z}$ of piecewise-affine, convex functions (with real, rational, or integer slopes, respectively).
If $X={\mathbb D}elta{\sh S}ubset{\mathbb R}^n$ is a polytope, then it has a notion of integer points, and so one can define a ${\mathbb B}$-module $\mathrm{CPA}_{\mathbb Z}(X,{\mathbb Z}_\vee)$ of piecewise-affine, convex functions with integer slopes and which take \mathbf{e}mph{integer values} on lattice points ${\mathbb Z}^n\cap{\mathbb D}elta$. Note that any function in this ${\mathbb B}$-module that attains the value $-\infty$ must in fact be constant.\mathbf{e}nd{egs}
\begin{eg}\label{PFR-set}Let $S$ be a set. The \mathbf{e}mph{subset ${\mathbb B}$-module} ${\mathbb B} S$ is the power set of $S$; its join operation is union. The \mathbf{e}mph{free ${\mathbb B}$-module} ${\mathbb B}^c S{\sh S}ubseteq {\mathbb B} S$ is the set of finite subsets of $S$. Its elements may be written uniquely (up to permutation of terms) as idempotent linear expressions ``with coefficients in ${\mathbb B}$,'' i.e.\ as $X_1\vee\cdots\vee X_k$ for some $X_1,\ldots,X_k\in S$.
Both constructions are functorial in $S$, so we have functors ${\mathbb B},{\mathbb B}^c:\mathbf{Set}\rightarrow\mathrm{Mod}_{\mathbb B}$; the latter is left adjoint to the forgetful functor.\mathbf{e}nd{eg}
The theory of ${\mathbb B}$-modules is a finitary algebraic theory, and so limits, filtered colimits, and quotients by groupoid relations are computed in $\mathbf{Set}$; this remains true with $\mathbf{Set}$ replaced by any topos. The following (proposition \ref{SPAN-prop-bicomp}) also remains true in that generality.
For any ${\mathbb B}$-modules $\alpha,\beta$, we can construct the \textit{direct join} $\alpha\vee\beta$ as the ${\mathbb B}$-module whose underlying set is the Cartesian product $\alpha\times\beta$ and whose join is defined by the law \[ (X_1,Y_1)\vee(X_2,Y_2):=(X_1\vee X_2,Y_1\vee Y_2). \] I simply write $X_1\vee X_2$ for $(X_1,X_2)$ where this is not likely to cause confusion.
There are natural ${\mathbb B}$-module homomorphisms $\alpha\rightarrow\alpha\vee\beta\rightarrow\alpha$ defined by \[X\mapsto X\vee(-\infty), \quad X\vee Y\mapsto X,\] and similarly for $\beta$, which make the direct join into a coproduct and product in $\mathrm{Mod}_{\mathbb B}$. In particular, there are natural homomorphisms \[\alpha{\sh S}tackrel{{\mathbb D}elta}{\longrightarrow}\alpha\vee\alpha {\sh S}tackrel{\vee}{\longrightarrow} \alpha,\] the diagonal and the map defining the ${\mathbb B}$-module structure, respectively. I use also the direct join notation for a pushout $\alpha\vee_\beta{\mathfrak g }amma:=\alpha{\sh S}qcup_\beta{\mathfrak g }amma$.
The null ${\mathbb B}$-module is the empty direct join, or zero object, of $\mathrm{Mod}_{\mathbb B}$.
The kernel and cokernel of a morphism $f:\alpha\rightarrow\beta$ of ${\mathbb B}$-modules are defined: $\ker f:=f^{-1}(-\infty),\coker f = \beta\vee_\alpha\{-\infty\}$.
If $f,g\in\mathrm{Hom}(\alpha,\beta)$, then their `sum' is given by the composition
\[\alpha{\sh S}tackrel{\id\times \id}\longrightarrow\alpha\vee\alpha\longrightarrow\beta\vee\beta{\sh S}tackrel{\id{\sh S}qcup\id}\longrightarrow\beta\]
which takes $X\in\alpha$ to $f(X)\vee g(X)$. This description establishes that the monoid $\mathrm{Hom}(\alpha,\beta)$ is in fact a ${\mathbb B}$-module in which $f\leq g$ if and only if $f(X)\leq g(X)\in\beta$ $\forall X\in\alpha$; moreover $\mathrm{Hom}(-,-)$ is a bifunctor from $\mathrm{Mod}_{\mathbb B}$ to itself.
\begin{prop}\label{SPAN-prop-bicomp}The category $\mathrm{Mod}_{\mathbb B}$ is semiadditive.\footnote{A category is \mathbf{e}mph{semiadditive} if it admits finite products and coproducts and the natural map $\times\rightarrow{\sh S}qcup$ is an isomorphism of bifunctors.} It is complete and cocomplete.\mathbf{e}nd{prop}
It is harder to obtain an explicit description of general coequalisers; see \S\ref{SPAN-quotient}.
{\sh S}ubsubsection{Subobjects}
Beyond the geometric examples \ref{eg-first}, the primary source of ${\mathbb B}$-modules are the \mathbf{e}mph{subobject posets} in certain finitely cocomplete categories. One could formulate a general theory of subobjects in certain kinds of categories; however, for the purposes of this paper we only need to know the version for modules over a commutative ring (possibly in a Grothendieck topos).
\begin{defn}\label{SPAN-def-B}Let $A$ be a ring, $M$ an $A$-module. I write ${\mathbb B}(M;A)$ for the \mathbf{e}mph{submodule lattice} of $M$, the partially ordered set of all $A$-submodules of $M$; its join operation is submodule sum. I abbreviate ${\mathbb B}(A;A)$ to ${\mathbb B} A$, the \mathbf{e}mph{ideal semiring} of $A$.\mathbf{e}nd{defn}
The submodule lattice is functorial in $A$-module homomorphisms $f:M_1\rightarrow M_2$
\[{\mathbb B} f:{\mathbb B}(M_1;A)\rightarrow{\mathbb B}(M_2;A),\quad N\mapsto\mathrm{Im}(f|_N) \]
and ring maps $g:A\rightarrow B$
\[ {\mathbb B} g:{\mathbb B}(M;A)\rightarrow{\mathbb B}(M\mathbin{\otimes}_AB;B),\quad N \mapsto\mathrm{Im}(N\mathbin{\otimes}_AB\rightarrow M\mathbin{\otimes}_AB).\]
In particular, ${\mathbb B} A\rightarrow {\mathbb B} B$.
Typically, $A=:{\sh S}h O_X$ will be a sheaf of rings on some space $X$ and $M$ an ${\sh S}h O_X$-module, in which case ${\mathbb B}(M;{\sh S}h O_X)$ is the lattice of ${\sh S}h O_X$-subsheaves of $M$. The submodule lattice is then functorial for maps defined in the sheaf category $X\hspace{4pt}\widetilde{}\hspace{4pt}$, but also for morphisms $g:(Y,{\sh S}h O_Y)\rightarrow (X,{\sh S}h O_X)$ of ringed spaces. In the latter case, I will write \[ g^*={\mathbb B} g:{\mathbb B}(M;{\sh S}h O_X)\rightarrow{\mathbb B}(g^*M;{\sh S}h O_Y),\quad N \mapsto\mathrm{Im}(g^*N\rightarrow g^*M) \]
for the induced map of lattices, though this should not be confused with the functor of pullback of ${\sh S}h O_Y$-modules, which it equals only when $g$ is flat.
\begin{eg}[Discs]\label{eg-discs}Let $K$ be a complete, valued field, $V$ a $K$-vector space. I would like to be able to say that the subobjects of $V$ are the \mathbf{e}mph{discs} \cite[\S 2.2]{Banach}. If $K$ is non-Archimedean with ring of integers ${\sh S}h O_K$, then a disc is the same thing as an ${\sh S}h O_K$-submodule, and so the set of discs is ${\mathbb B}(V;{\sh S}h O_K)$ (which in \mathbf{e}mph{loc. cit.} is called ${\sh S}h D(V)$).
If $K$ is Archimedean, then we need an alternative theory of `abstract discs' or `convex sets'. Following \cite{Durov}, one can describe it as a theory of modules for a certain algebraic monad. For instance, if $K={\mathbb R}={\mathbb Q}_\infty$, the corresponding monad is that ${\mathbb Z}_\infty$ (also written ${\sh S}h O_{\mathbb R}$) of convex, balanced sets \cite[\S2.14]{Durov}. An object of $\mathrm{Mod}_{{\mathbb Z}_\infty}$ is a set $M$ equipped with a way of evaluating convex linear combinations \[{\sh S}um_{i=1}^k\lambda_ix_i,\quad x_i\in M,\lambda_i\in{\mathbb R},{\sh S}um_{i=1}^k|\lambda_i|\leq 1\] of its elements. A subset of $V$ is a disc if and only if it is stable for the action of ${\mathbb Z}_\infty$. In other words, ${\sh S}h D(V)={\mathbb B}(V;{\mathbb Z}_\infty)$, in a mild generalisation of definition \ref{SPAN-def-B}.\mathbf{e}nd{eg}
{\sh S}ubsection{Orders and lattices}
The alternative definition \ref{SPAN-def-preorder} puts ${\mathbb B}$-module theory in the broader context of order theory. In particular, there are \mathbf{e}mph{sometimes} defined infinitary operations \[(X_i)_{i\in I}\mapsto {\sh S}up_{i\in I}X_i.\] I reserve the notation $\bigvee_{i=1}^kX_i$ for the (always defined) operation of finite supremum or join.
The following definitions are standard in order theory:
\begin{defns}A map of posets is \mathbf{e}mph{monotone} if it preserves the order. A monotone map of posets is the same as a functor of preorders. The category of posets and monotone maps is denoted $\mathbf{POSet}$.
A ${\mathbb B}$-module is a \mathbf{e}mph{complete lattice} if it has all suprema. A complete lattice is the same thing as a cocomplete poset. In particular, meets exist. A \mathbf{e}mph{lattice homomorphism} is a map of complete lattices preserving all suprema, that is, a colimit-preserving functor. The category of complete lattices and homomorphisms is denoted $\mathbf{Lat}{\sh S}ubset\mathbf{Span}$.
Let $\alpha$ be a ${\mathbb B}$-module, $S,T{\sh S}ubseteq\alpha$. The \textit{lower slice set}
\begin{align}\nonumber S_{\leq T} &:= \{Y\in S|\mathbf{e}xists X\in T \text{ s.t. }X\vee Y=X\}\\
& = \{Y\in S|\mathbf{e}xists X\in T\text{ s.t. } Y\leq X\}\mathbf{e}nd{align} is the ${\mathbb B}$-module of all elements contained in $S$ that are bounded above by an element of $T$. The \mathbf{e}mph{upper slice set} \[S_{{\mathfrak g }eq T} := \{Y\in S|\mathbf{e}xists X\in T \text{ s.t. }X\vee Y=Y\}\] is defined dually.
A subset $S$ is said to be \mathbf{e}mph{lower} (resp. \mathbf{e}mph{upper}) if $S=\alpha_{\leq S}$ (resp. $\alpha_{{\mathfrak g }eq S}$). A lower submodule of $\alpha$ is called an \mathbf{e}mph{ideal} of $\alpha$.
The subset $S$ is called \textit{coinitial} (resp. \mathbf{e}mph{cofinal}) if all lower (resp. upper) slice sets are non-empty, that is, $\forall X\in\alpha$, $\mathbf{e}xists Y\in S$ such that $Y\leq X$ (resp.\ $X\leq Y$).\mathbf{e}nd{defns}
\begin{eg}A quotient of a ${\mathbb B}$-module $\alpha$ by an ideal $\iota$, that is, the cokernel of the inclusion $\iota{\mathfrak h }ookrightarrow\alpha$, is easy to make explicit: it is simply the set-theoretic quotient $\alpha/\iota$ of $\alpha$ by the equivalence relation $\iota{\sh S}im-\infty$. If $\iota=\alpha_{\leq T}$ is a slice set, we may also write $\alpha/T$. The cokernel of a ${\mathbb B}$-module homomorphism $f:\alpha\rightarrow\beta$ is the quotient of $\beta$ by $\beta_{\leq f(\alpha)}$, the smallest ideal containing $f(\alpha)$. In particular, $\alpha$ is an ideal if and only if it is the kernel of its cokernel.\mathbf{e}nd{eg}
The set of all ideals of $\alpha$ can be thought of as a \mathbf{e}mph{subobject poset} in the category of ${\mathbb B}$-modules. It is a complete lattice.
\begin{defn}\label{SPAN-def-lat}The lattice of ideals ${\mathbb B}\alpha$ of a ${\mathbb B}$-module $\alpha$ is called the \mathbf{e}mph{lattice completion} of $\alpha$.\mathbf{e}nd{defn}
The lattice completion defines a left adjoint ${\mathbb B}:\mathrm{Mod}_{\mathbb B}\rightarrow\mathbf{Lat}$ to the inclusion of $\mathbf{Lat}$ into $\mathrm{Mod}_{\mathbb B}$.
The unit $\mathrm{id}\rightarrow{\mathbb B}$ of the adjunction is an injective homomorphism
\[ \alpha\rightarrow{\mathbb B}\alpha, \quad X\mapsto \alpha_{\leq X}. \]
As a preorder, the lattice completion of $\alpha$ is its category of ind-objects \cite[\S I.8.2]{SGA4}.
{\sh S}ubsection{Finiteness}\label{SPAN-fin}
In ordinary category theory, the notion of \mathbf{e}mph{finite presentation} of objects is captured by \mathbf{e}mph{compact objects}, that is, objects whose associated co-representable functor preserves filtered colimits. One then seeks to try to understand all objects of the category in terms of its compact objects. In particular, we like to work with \mathbf{e}mph{compactly generated} categories: those for which every object is a colimit of compact objects.
A compactly generated category $\mathbf C$ is equivalent
\[ \mathbf C \cong \mathrm{Ind}(\mathbf C^c) \]
to its category of ind-compact objects. In particular, filtered colimits are exact.
\
The order-theoretic version of compactness is \mathbf{e}mph{finiteness}. Its basic behaviour can be derived by applying the above results directly to the special case of objects in pre-orders.
\begin{defns}An element $X$ of a complete lattice $\alpha$ is \mathbf{e}mph{finite} if, for any formula $X\leq{\sh S}up_{i\in I}X_i$ in $\alpha$, with the $X_i$ a filtered family, there exists an index $i$ such that $X\leq X_i$.
A lattice is \mathbf{e}mph{algebraic} if every element is a supremum of finite elements.
A homomorphism $f:\alpha\rightarrow\beta$ \mathbf{e}mph{preserves finiteness} if $f(X)$ is finite whenever $X$ is.\mathbf{e}nd{defns}
Be warned that it is not, in general, equivalent to replace the inequalities in the above definition with equalities. An element $X\in\alpha$ can be finite as an element of $\alpha_{\leq X}$ without being finite in $\alpha$.
\begin{lemma}A finite join of finite elements is finite.\mathbf{e}nd{lemma}
Let $\alpha$ be a complete lattice. I denote by $\alpha^c$ its subset of finite elements; by the lemma, $\alpha^c$ is a ${\mathbb B}$-module. It is functorial for ${\mathbb B}$-module homomorphisms that preserve finiteness.
\begin{prop}Let $\alpha\in\mathbf{Lat}$. The following are equivalent:
\begin{enumerate}
\item $\alpha$ is algebraic;
\item ${\sh S}up:{\mathbb B}(\alpha^c)\rightarrow\alpha$ is an isomorphism;
\item Every element of $\alpha$ is a supremum of elements $X$ that are finite in their slice set $\alpha_{\leq X}$, and finite meets distribute over filtered suprema in $\alpha$.\mathbf{e}nd{enumerate}\mathbf{e}nd{prop}
Let $\alpha$ be any ${\mathbb B}$-module. A ${\mathbb B}$-module ideal $\iota{\mathfrak h }ookrightarrow\alpha$ is finite as an element of ${\mathbb B}\alpha$ if and only if it is \mathbf{e}mph{principal}, that is, equal to some slice set $\alpha_{\leq X}$. Therefore, $\alpha\rightarrow{\mathbb B}\alpha$ identifies $\alpha$ with the ${\mathbb B}$-module of finite elements of ${\mathbb B}\alpha$. This sets up an equivalence of categories
\[ {\mathbb B}:\mathrm{Mod}_{\mathbb B}\leftrightarrows \mathbf{Lat}_{al}:(-)^c \]
between $\mathrm{Mod}_{\mathbb B}$ and the category $\mathbf{Lat}_{al}$ of algebraic lattices.
\begin{egs}Let $S$ be a set. A subset of $S$ is finite as an element of ${\mathbb B} S$ if and only if it has finitely many elements; $({\mathbb B} S)^c\cong{\mathbb B}^cS$ in the notation of example \ref{eg-first}. The power set ${\mathbb B} S\cong{\mathbb B}{\mathbb B}^cS$ is an algebraic lattice.
A submodule of a module $M$ over a ring $A$ is finite if and only if it is finitely generated; ${\mathbb B}(M;A)$ is an algebraic lattice.\mathbf{e}nd{egs}
\begin{defn}\label{SPAN-def-fin}The \mathbf{e}mph{finite submodule} or \mathbf{e}mph{free} ${\mathbb B}$-module on $M$ is the ${\mathbb B}$-module ${\mathbb B}^c(M;A)\cong ({\mathbb B}(M;A))^c$ of finitely generated $A$-submodules of $M$; since a sum of finite submodules is finite, this is closed in ${\mathbb B}(M;A)$ under joins. By algebraicity, ${\mathbb B}{\mathbb B}^c(M;A)\cong{\mathbb B}(M;A)$. We abbreviate ${\mathbb B}^c(A;A)$ to ${\mathbb B}^cA$.\mathbf{e}nd{defn}
\begin{eg}[Seminorms]\label{eg-seminorms}Let $A$ be an Abelian group. A \mathbf{e}mph{(logarithmic) non-Archimedean seminorm} on $A$ with values in a ${\mathbb B}$-module $\alpha$ is a map of sets $\val:A\rightarrow\alpha$ satisfying the \mathbf{e}mph{ultrametric inequality} $\val(f+g)\leq\val f\vee\val g$. One can take the supremum of any (non-Archimedean) seminorm on $A$ over any finitely generated subgroup $X{\sh S}ubseteq A$; indeed, if $X=(x_1,\ldots,x_n)$, then \[ {\sh S}up_{f\in X}\val f = \bigvee_{i=1}^n\val x_n. \] This supremum defines a \mathbf{e}mph{${\mathbb B}$-module homomorphism} ${\mathbb B}^c(A;{\mathbb Z})\rightarrow\alpha$.
This correspondence exhibits the natural seminorm \[A\rightarrow{\mathbb B}^c(A;{\mathbb Z}),\quad a\mapsto (a) \] as \mathbf{e}mph{universal} among seminorms of $A$ into any ${\mathbb B}$-module. In other words, ${\mathbb B}^c(A;{\mathbb Z})$ corepresents the functor \[\frac{1}{2}\mathrm{Nm}(A,-):\mathrm{Mod}_{\mathbb B}\rightarrow\mathbf{Set}\] of seminorms on $A$.\mathbf{e}nd{eg}
\begin{eg}\label{eg-discs-field}Let $K$ be a non-Archimedean field with ring of integers ${\sh S}h O_K$ and value group $|K|{\sh S}ubseteq{\mathbb R}$. The given valuation induces a ${\mathbb B}$-module isomorphism ${\mathbb B}^c(K;{\sh S}h O_K)\widetilde\rightarrow |K|_\vee$. In fact, the same holds if $K$ is Archimedean, cf. e.g. \ref{eg-discs}.\mathbf{e}nd{eg}
\begin{eg}[Not enough finites]Let $K$ be a complete, discrete valuation field with uniformiser $t$. Let $\overline K$ be an algebraic closure with ring of integers ${\sh S}h O_{\overline K}$. Then ${\mathbb B}^c{\sh S}h O_{\overline K}\cong{\mathbb Q}_\vee^\circ={\mathbb Q}_{\leq0}{\sh S}qcup\{-\infty\}$ (cf. def. \ref{.5RING-integers}) is the set of principal ideals generated by positive rational powers $t^q$ of the uniformiser. The `traditional' way to complete ${\mathbb Q}_\vee\circ$ would be to embed it in its set ${\mathbb R}_\vee^\circ$ of Dedekind cuts. The latter is a complete lattice with no finite elements.
Of course, it is more sensible in this case to consider ${\mathbb Q}_\vee^\circ$ as the set of finite elements in the well-behaved lattice ${\mathbb B}{\mathbb Q}_\vee^\circ\in\mathbf{Lat}_{al}$.\mathbf{e}nd{eg}
One can show that if the above statements are interpreted in the usual semantics within the topos of sheaves on a space $X$, one obtains the following set-theoretic characterisation of the finite submodule ${\mathbb B}$-module (sheaf). Let ${\sh S}h O_X$ be a sheaf of rings on $X$, $M$ an ${\sh S}h O_X$-module.
\begin{defn}\label{SPAN-def-fin'}A submodule $N{\mathfrak h }ookrightarrow M$ is \mathbf{e}mph{locally finitely generated} if there exists a covering $\{f_i:U_i\rightarrow X\}_{i\in I}$ and epimorphisms ${\sh S}h O_{U_i}^{n_i}\twoheadrightarrow f_i^*N$ for some numbers $n_i\in{\mathbb N}$.
The \mathbf{e}mph{finite submodule} or \mathbf{e}mph{free} ${\mathbb B}$-module on $M$ is the sheaf \[{\mathbb B}^c(M;{\sh S}h O_X):U\mapsto{\mathbb B}^c(M(U);{\sh S}h O_X(U))\]
of locally finitely generated ${\sh S}h O_X$-submodules of $M$.\mathbf{e}nd{defn}
One may simply take this as a set-theoretic definition of ${\mathbb B}^c$, verifying directly that ${\mathbb B}^c(M;{\sh S}h O_X)$ is a sheaf.
\begin{eg}[Local seminorms]\label{eg-sheafseminorms}Let $X$ be a space, $A$ a sheaf of Abelian groups on $X$. A \mathbf{e}mph{seminorm} on $A$ with values in a sheaf $\alpha$ of ${\mathbb B}$-modules is a map $A\rightarrow\alpha$ of sheaves which induces over each $U{\sh S}ubseteq X$ a non-Archimedean seminorm on $A(U)$ (e.g. \ref{eg-seminorms}).
One can define a \mathbf{e}mph{universal seminorm} $A\rightarrow{\mathbb B}^c(A;\mathrm{Mod}_{{\sh S}h O_X})$, which, for a given $U{\sh S}ubseteq X$, takes $f\in A(U)$ to the subsheaf of $A|_U$ that it locally generates. Any seminorm $\val:A\rightarrow\alpha$ factors uniquely through this universal one, with the factoring arrow taking any finite subsheaf \(F{\sh S}ubseteq A|_V\) to
\[ {\sh S}up_{f^\bullet\in F(U_\bullet)}\left|\val f^\bullet\right| = \left|\bigvee_{i=1}^{n^\bullet}\val f_i^\bullet\right| \in |\alpha(U_\bullet)|\cong \alpha(V) \]
In this formula, $U_\bullet\twoheadrightarrow V$ is a covering on which $F$ is defined, and $(f_1^\bullet,\ldots,f_{n^\bullet}^\bullet)$ denotes a locally finite system of generators for $F(U_\bullet)$. (Note that the numbers $n^\bullet$ need not be bounded.)\mathbf{e}nd{eg}
{\sh S}ubsection{Noetherian}
\begin{defn}A ${\mathbb B}$-module is called \textit{Noetherian} if the slice sets satisfy the ascending chain condition, that is, if every bounded, totally ordered subset has a maximum.\mathbf{e}nd{defn}
\begin{prop}Let $\{X_i\}_{i\in I}{\sh S}ubseteq\alpha$ be a bounded family of elements of a ${\mathbb B}$-module $\alpha$. If $\alpha$ is Noetherian, then ${\sh S}up_{i\in I}X_i=\bigvee_{i\in J}X_i$ for some finite $J{\sh S}ubseteq I$.\mathbf{e}nd{prop}
\begin{proof}We proceed by contraposition. Suppose that for all finite $J{\sh S}ubseteq I$, there is some $i(J)\in I{\sh S}etminus J$ such that $X_{i(J)}\not\leq\bigvee_{j\in J}X_j$, that is, such that $\bigvee_{j\in J}X_j < X_{i(J)}\vee\bigvee_{j\in J}X_j$. Then $I$ is infinite, and starting from any index $0\in I$ we can inductively construct an infinite, strictly increasing sequence
\[ X_0 < \left(X_1 \vee X_0\right) < \left(X_2 \vee X_1 \vee X_0\right) < \cdots \]where $n:=i(\{0,\ldots,n-1\})\in I$. Therefore $\alpha$ is not Noetherian.\mathbf{e}nd{proof}
\begin{cor}The following are equivalent for a bounded ${\mathbb B}$-module $\alpha$:
\begin{enumerate}\item $\alpha$ is Noetherian;
\item $\alpha$ is a complete lattice, and $\alpha^c=\alpha$;
\item $\alpha\widetilde\rightarrow{\mathbb B}\alpha$.\mathbf{e}nd{enumerate}
A ${\mathbb B}$-module is Noetherian if and only if its every bounded ideal is Noetherian.\mathbf{e}nd{cor}
\begin{eg}Let $A$ be a ring.
The following are equivalent:
\begin{enumerate}
\item $A$ is Noetherian;
\item ${\mathbb B} A$ is Noetherian;
\item ${\mathbb B}^cA$ is Noetherian;
\item ${\mathbb B}^cM$ is Noetherian for all $A$-modules $M$;
\mathbf{e}nd{enumerate}
In this case, ${\mathbb B}^cM={\mathbb B} M$ if and only if $M$ is finitely generated.\mathbf{e}nd{eg}
{\sh S}ubsection{Adjunction}
As in category theory, the notion of adjoint map is central to the theory of ${\mathbb B}$-modules.
\begin{defn}Let $f:\alpha\rightarrow\beta$ be a monotone map of ${\mathbb B}$-modules. We say that a monotone map $g:\beta\rightarrow\alpha$ is \mathbf{e}mph{right adjoint} to $f$, and write $f^\dagger:=g$, if $\mathrm{id}_\alpha\leq gf$ and $fg\leq\mathrm{id}_\beta$. In this situation, we also say ${}^\dagger g:=f$ is \mathbf{e}mph{left adjoint} to $g$.\mathbf{e}nd{defn}
If $\alpha$ is a complete lattice, then by the adjoint functor theorem a right adjoint exists for $f$ if and only if it preserves arbitrary suprema. We have the formula
\[ X\mapsto f^\dagger X={\sh S}up\alpha_{\leq f^{-1}(X)}. \]
Alternatively, ${\mathbb B} f$ always preserves suprema, and therefore we can always find an adjoint
\[({\mathbb B} f)^\dagger:{\mathbb B}\beta\rightarrow{\mathbb B}\alpha, \quad \iota\mapsto f^{-1}\iota \]
at the level of the lattice completions. The restriction of $({\mathbb B} f)^\dagger$ to $\beta$ is an ind-adjoint in the sense of \cite[\S I.8.11]{SGA4}. If an ordinary right adjoint to $f$ exists, then the ind-adjoint is the composite of this with the inclusion $\alpha\rightarrow{\mathbb B}\alpha$; I therefore denote the ind-adjoint also by $f^\dagger$ in general, since no confusion can arise.
In particular, any ${\mathbb B}$-module homomorphism gives rise to a diagram
\[\xymatrix{ {\mathbb B}\alpha \\ \alpha\ar[r]^f\ar[u] & \beta\ar[ul]_{f^\dagger} }\]
in $\mathbf{POSet}$, and $\mathrm{id}_\alpha\leq f^\dagger f$.
{\sh S}ubsubsection{Pullback and pushforward}
Suppose that $A$ is a ring, $f:M_1\rightarrow M_2$ an $A$-module homomorphism. If $N{\mathfrak h }ookrightarrow M_2$ is a submodule, then so is $N\times_{M_2}M_1\rightarrow M_1$. The fibre product is a monotone map
\[ f^\dagger=f^{-1}:{\mathbb B}(M_2;A)\rightarrow{\mathbb B}(M_1;A),\quad N\mapsto N\times_{M_2}M_1, \]
\mathbf{e}mph{right adjoint} to the image functor ${\mathbb B} f$. It happens to be a lattice homomorphism.
Secondly, let $g:X\rightarrow Y$ be a morphism of ringed spaces, $A={\sh S}h O_X$. Then the pushforward functor $f_*$ is right adjoint to $f^*$ on the category $\mathrm{Mod}_{{\sh S}h O}$ of modules. Correspondingly,
\[ g_*:{\mathbb B}(M;{\sh S}h O_X)\rightarrow{\mathbb B}(g_*M;{\sh S}h O_Y),\quad N\mapsto g_*N \]
is right adjoint to the lattice homomorphism $g^*={\mathbb B} g$. Since pushforward is left exact, this lattice homomorphism \mathbf{e}mph{does} agree with the functor on modules.
\begin{eg}\label{eg-closure}If $X{\mathfrak h }ookrightarrow Y$ is an open immersion of schemes, then the right adjoint to $f^*:{\mathbb B}{\sh S}h O_Y\rightarrow{\mathbb B}{\sh S}h O_X$ sends a closed subscheme of $X$ to its scheme-theoretic closure in $Y$.\mathbf{e}nd{eg}
{\sh S}ubsubsection{${\mathbb B}$-module quotients}\label{SPAN-quotient}
In the theory of categorical localisation, certain types of adjunction can provide a substitute for a linear calculus of quotients of categories. One can apply a similar technique to semilinear algebra in order to provide explicit descriptions of ${\mathbb B}$-module coequalisers and quotients.
Let $s,t:\alpha\rightrightarrows\beta$ be a pair of ${\mathbb B}$-module homomorphisms.
\begin{defn}An ideal $\iota{\mathfrak h }ookrightarrow\beta$ is \mathbf{e}mph{invariant} for the pair $s,t$ if, for all $X\in\alpha$, $sX\in\iota\Leftrightarrow tX\in\iota$.\mathbf{e}nd{defn}
Since $s$ and $t$ are ${\mathbb B}$-module homomorphisms, the subset ${\mathbb B}\beta/(s{\sh S}im t){\sh S}ubseteq{\mathbb B}\beta$ of invariant ideals is closed under arbitrary suprema. The right adjoint to the inclusion is a self-homomorphism
\[ p:={\sh S}up_{n\in{\mathbb N}}\left( (ts^\dagger)^n \vee (st^\dagger)^n \right):{\mathbb B}\beta\rightarrow{\mathbb B}\beta \]
taking an ideal to the smallest invariant ideal containing it. It coequalises $s,t$. In fact, for any ${\mathbb B}$-module homomorphism $f:\beta\rightarrow{\mathfrak g }amma$ coequalising $s,t$, ${\mathbb B} f$ is independent of the action of $s,t$, that is, factors uniquely through $p$. Setting
\[ p:\beta\rightarrow\beta/(s=t):=p(\beta){\sh S}ubseteq{\mathbb B}\beta/(s{\sh S}im t){\sh S}ubseteq{\mathbb B}\beta\]
where $p(\beta)$ is the set-theoretic image, we therefore obtain:
\begin{lemma}\label{SPAN-adj-coeq}$\beta/(s= t)$ is a coequaliser for $s,t$.\mathbf{e}nd{lemma}
In the special case $\alpha={\mathbb B}$, where $s,t$ are determined by some elements $S,T\in\beta$, we write also as usual $\beta/(S=T)$ for the ${\mathbb B}$-module quotient (by the congruence relation generated by the relation $S=T$).
Specialisations of the above construction will come into play in later sections; see, for example, \S\ref{.5RING-contract}.
{\sh S}ection{Topological lattices}\label{TOP}
A topological space with linear structure is \mathbf{e}mph{linearly topologised} if its topology is generated by linear subspaces. In other words, a linear topology on a space is one that can be defined in terms of a certain decoration - a \mathbf{e}mph{principal topology} - on its subobject lattice.
Let $\alpha$ be a complete lattice, $\alpha^u{\sh S}ubseteq\alpha$ a non-empty, upper subset, closed under finite meets. Such an $\alpha^u$ is called a \mathbf{e}mph{fundamental system of opens}, or just \mathbf{e}mph{fundamental system}, for short.
\begin{lemma}The collection of slice sets $\alpha_{\leq X}$ for $X$ in a fundamental system, together with $\mathbf{e}mptyset$, are a topology on $\alpha$ for which $\vee$ is continuous.\mathbf{e}nd{lemma}
\begin{proof}It is clear that these sets define a topology; for continuity of $\vee:\alpha\times\alpha\rightarrow\alpha$, note simply that $\vee^{-1}(\alpha_{\leq X})=\alpha_{\leq X}\times\alpha_{\leq X}$.\mathbf{e}nd{proof}
The topology in the lemma is that \mathbf{e}mph{defined by the fundamental system}.
Any intersection of fundamental systems is a fundamental system. Therefore, for any family $f_i:\alpha\rightarrow\beta_i$ of maps of complete lattices and fundamental systems $\beta_i^u$ on $\beta_i$, there is a \mathbf{e}mph{smallest} fundamental system on $\alpha$ such that the $f_i$ are \mathbf{e}mph{continuous} for the induced topology. Explicitly, it is given by the closure of the upper set
\[\bigcup_{i,X\in\beta_i^u}\alpha_{{\mathfrak g }eq f_i^\dagger(X)}\] under finite meets.
Dually, any union of fundamental systems generates a new fundamental system under finite meets. This coincides with the usual notion of generation of new topologies. Hence, for any family $g_i:\alpha_i\rightarrow\beta$ of homomorphisms and fundamental systems $\alpha_i^u$, there is a \mathbf{e}mph{largest} fundamental system
\[\beta^u:=\bigcap_i\beta_{{\mathfrak g }eq f_i(\alpha_i^u)}\]on $\beta$ making the $g_i$ continuous, and the topology it defines is simply the strong topology on the underlying set.
\begin{defns}A complete lattice equipped with a \mathbf{e}mph{principal topology}, that is, a topology defined by a fundamental system, is called simply a \mathbf{e}mph{topological lattice}. A \mathbf{e}mph{topological ${\mathbb B}$-module} is a ${\mathbb B}$-module $\alpha$ equipped with an \mathbf{e}mph{ideal topology}, that is, is the subspace topology with respect to some principal topology on ${\mathbb B}\alpha{\sh S}upseteq\alpha$. A \mathbf{e}mph{fundamental system for $\alpha$} is a fundamental system for ${\mathbb B}\alpha$, and we write $\alpha^u:=({\mathbb B}\alpha)^u$.
A homomorphism of topological ${\mathbb B}$-modules (resp. lattices), is a continuous ${\mathbb B}$-module homomorphism (resp. lattice homorphism). The category of topological lattices is denoted $\mathbf{Lat}_t$, the category of topological ${\mathbb B}$-modules $\mathrm{Mod}_{{\mathbb B},t}$.\mathbf{e}nd{defns}
A topological ${\mathbb B}$-module is a ${\mathbb B}$-module whose inhabited open sets are ideals, and in which every neighbourhood (of $-\infty$) is open. A topological lattice is the same, except that inhabited open sets are principal ideals. There is also an obvious notion of principal topology on a possibly incomplete ${\mathbb B}$-module.
\begin{eg}\label{semifields-top}The semifields $H_\vee$ (e.g. \ref{eg-first}) will always come equipped with the (principal) topology $H_\vee^u=H$. (In particular, ${\mathbb B}$ is discrete.) A net $\{X_i\}_{i\in I}$ converges to $-\infty$ if and only if it does so with respect to the order; in other words, if $\forall\lambda\in H$ $\mathbf{e}xists i\in I$ such that $X_j\leq\lambda$ for all $j>i$.\mathbf{e}nd{eg}
The category of topological ${\mathbb B}$-modules (resp. lattices) comes with a forgetful functor \[?:\mathrm{Mod}_{{\mathbb B},t}\rightarrow\mathrm{Mod}_{\mathbb B}\] which I do not suppress from the notation. Its left adjoint is given by equipping a lattice $\alpha$ with the \mathbf{e}mph{discrete topology} $\alpha^u=\alpha$, its right adjoint by the \mathbf{e}mph{trivial topology} $\alpha_\mathrm{triv}^u=\{{\sh S}up\alpha\}$. Both adjoints are fully faithful. We will treat the category of ${\mathbb B}$-modules as the (full) subcategory of discrete objects inside $\mathrm{Mod}_{{\mathbb B},t}$.
In particular, limits (resp. colimits) in $\mathrm{Mod}_{{\mathbb B},t}$ are computed by equipping the limits (resp. colimits) of the underlying discrete ${\mathbb B}$-modules with weak (resp. strong) topologies.
\begin{eg}\label{TOP-Hausdorff}A non-trivial topological ${\mathbb B}$-module is never Hausdorff in the sense of point-set topology, since every open set contains $-\infty$. Let us instead say that a ${\mathbb B}$-module $\alpha$ is \mathbf{e}mph{Hausdorff} if $\inf\alpha^u=-\infty$. The category $\mathrm{Mod}_{{\mathbb B},\dot t}$ of Hausdorff ${\mathbb B}$-modules is a reflective subcategory of $\mathrm{Mod}_{{\mathbb B},t}$.\footnote{The notation $\dot{t}$ follows Bourbaki \cite{Banach}.}\mathbf{e}nd{eg}
\begin{defns}Let $f_i:\alpha_i\rightarrow\beta$ be a family of continuous ${\mathbb B}$-module homomorphisms. We say that $\beta$ carries the \mathbf{e}mph{strong topology} with respect to the $f_i$, or that the family $f_i$ is \mathbf{e}mph{strong}, if its topology is the strongest ideal topology such that the $f_i$ are continuous.
In particular, if $f$ is just a single map, $f:\alpha\rightarrow\beta$ is strong if and only if it sends $\alpha^u$ into $\beta^u{\sh S}ubseteq{\mathbb B}\beta$. In this case, we may also say that $f$ is \mathbf{e}mph{open} - although beware that it may fail to be open in the sense of general topology.
If $g_j:\alpha\rightarrow\beta_j$ are a family of continuous ${\mathbb B}$-module homomorphisms, then $\alpha$ carries the \mathbf{e}mph{weak topology} with respect to the $g_j$, or that the family $g_i$ is \mathbf{e}mph{weak}, if its topology is the weakest ideal topology such that the $g_i$ are continuous.\mathbf{e}nd{defns}
From the definition of ideal topology, it follows:
\begin{lemma}A family $f_i$ is weak (resp. strong) if and only if the induced family ${\mathbb B} f_i$ on the lattice completions is weak (resp. strong).\mathbf{e}nd{lemma}
In particular, weak and strong topologies, and hence limits and colimits, always exist.
\begin{prop}\label{TOP-span-complete}The category $\mathrm{Mod}_{{\mathbb B},t}$ is complete, cocomplete, and semi-additive. Filtered colimits are exact.\mathbf{e}nd{prop}
\begin{proof}
We need to check that the product and coproduct topology on the direct join agree. The explicit formulas show that
\[ (\alpha\times\beta)^u=\{ (X,{\sh S}up\beta)\wedge({\sh S}up\alpha,Y) | X\in\alpha^u,Y\in\beta^u \}
= \alpha^u\times\beta^u = (\alpha{\sh S}qcup\beta)^u \]
which proves that $\mathrm{Mod}_{{\mathbb B},t}$ is semi-additive.
Now let $\alpha_i,\beta_i\rightarrow{\mathfrak g }amma_i$ be a filtered system, with $\alpha,\beta\rightarrow{\mathfrak g }amma$ its colimit. We will confuse $\alpha_i,\beta_i$ with their image in $\alpha\times_{\mathfrak g }amma\beta$. To show that the natural map $\colim_i(\alpha_i\times_{{\mathfrak g }amma_i}\beta_i)\rightarrow\alpha\times_{\mathfrak g }amma\beta$ is a homeomorphism, it will suffice to show that it is open. Let
\[ U\in \left(\colim_i(\alpha_i\times_{{\mathfrak g }amma_i}\beta_i)\right)^u = \bigcap_i \alpha\times_{\mathfrak g }amma\beta_{{\mathfrak g }eq\alpha^u_i\wedge\beta^u_i} \]
so there exist $X_i\in\alpha^u_i,Y_i\in\beta^u_i$ such that ${\sh S}up_i(X_i\wedge Y_i)\leq U$. Since, in ${\mathbb B}(\alpha\times_{\mathfrak g }amma\beta)$, filtered suprema distribute over meets (cf. \ref{SPAN-fin}),
\[ U{\mathfrak g }eq ({\sh S}up_iX_i)\wedge({\sh S}up_iY_i)\in \left(\bigcap_i \alpha\times_{\mathfrak g }amma\beta_{{\mathfrak g }eq\alpha^u_i}\right) \wedge \left(\bigcap_i \alpha\times_{\mathfrak g }amma\beta_{{\mathfrak g }eq\beta^u_i}\right) = (\alpha\times_{\mathfrak g }amma\beta)^u \]
and is therefore open.\mathbf{e}nd{proof}
\begin{eg}\label{eg-second}There are two obvious ways to topologise the function ${\mathbb B}$-module $C^0(X,{\mathbb R}_\vee)$ on a topological space $X$ (and similarly $\mathrm{P}C^r(X;-)$, $\mathrm{CP}C^r(X;-)$, etc., cf. e.g. \ref{eg-first}): a topology of \mathbf{e}mph{pointwise convergence}, which is the weak topology with respect to evaluation maps
\[ \mathrm{ev}_x:C^0(X,{\mathbb R}_\vee)\rightarrow{\mathbb R}_\vee, \]
and one of \mathbf{e}mph{uniform convergence}, which is the strong topology with respect to the inclusion ${\mathbb R}_\vee{\mathfrak h }ookrightarrow C^0(X,{\mathbb R}_\vee)$ of constants. In the important case $\mathrm{CPA}_*(X,{\mathbb R}_\vee)$ of convex, piecewise-affine functions, when $X$ is compact with affine structure, these two topologies agree.\mathbf{e}nd{eg}
{\sh S}ubsection{Topological modules}
Let $A$ be a non-Archimedean ring (def. \ref{ADIC-def-na}), $M$ a (complete locally convex) $A$-module. We equip ${\mathbb B}(M;A^+)$ with a principal topology
\[ ({\mathbb B} M)^u:=\{ U{\mathfrak h }ookrightarrow M|U\text{ open} \} ,\]
which, by definition of local convexity, is enough to recover the topology on $M$. We also consider ${\mathbb B}^c(M;A^+)$ as a topological ${\mathbb B}$-module with respect to the subspace topology. This topology is natural for continuous module homomorphisms, and hence lifts ${\mathbb B}^{(c)}$ to a functor
\[ {\mathbb B}^{(c)}(-;M):\mathrm{LC}_A\rightarrow\mathrm{Mod}_{{\mathbb B},t}. \]
If $g:A\rightarrow B$ is a ring homomorphism, then the base extension must be replaced with a \mathbf{e}mph{completed} base extension $-\widehat\mathbin{\otimes}_AB:\mathrm{LC}_A\rightarrow\mathrm{LC}_B$ (that is, ordinary base extension followed by topological completion with respect to the projective tensor product topology). Correspondingly, there is a lattice (resp. ${\mathbb B}$-module) homomorphism
\[ {\mathbb B} g:{\mathbb B}^{(c)}(M;A^+)\rightarrow{\mathbb B}^{(c)}(M\widehat\mathbin{\otimes}_{A^+}B^+;B^+),\quad N \mapsto\mathrm{Im}(N\mathbin{\otimes}_{A^+}B^+\rightarrow M\widehat\mathbin{\otimes}_AB); \]
in the case $M=A$ this agrees with the homomorphism ${\mathbb B}^{(c)}(A;A^+)\rightarrow{\mathbb B}^{(c)}(B;B^+)$ defined previously without taking into account the topology. The same functoriality extends to morphisms of nA-ringed spaces.
Beware that the elements of ${\mathbb B}^c(M;A^+)$ correspond to not necessarily \mathbf{e}mph{closed} submodules of $M$, and hence might not be represented by a subobject in $\mathrm{LC}_A$.
\begin{eg}[Continuous seminorms]Let $A$ be a linearly topologised Abelian group. It follows immediately from the definition of the topology on ${\mathbb B}(A;{\mathbb Z})$ that the universal seminorm $A\rightarrow{\mathbb B}^c(A;{\mathbb Z})$ (e.g. \ref{eg-seminorms}) on $A$ is continuous. In fact, ${\mathbb B}^c(A;{\mathbb Z})$ carries the \mathbf{e}mph{strong} topology with respect to this map. In other words, if $A\rightarrow\alpha$ is any continuous seminorm into some $\alpha\in\mathrm{Mod}_{{\mathbb B},t}$, then the factorisation ${\mathbb B}^c(A;{\mathbb Z})\rightarrow\alpha$ is also continuous.
The topological free ${\mathbb B}$-module ${\mathbb B}^c(A;{\mathbb Z})$ corepresents the functor of continuous seminorms
\[ \frac{1}{2}\mathrm{Nm}(A,-):\mathrm{Mod}_{{\mathbb B},t}\rightarrow\mathbf{Set}. \]
As we know, we may also use a seminorm $\nu:A\rightarrow\alpha$ to induce a coarser topology on $A$, the weak topology with respect to ${\mathbb B}^c(A\;{\mathbb Z})rightarrow\alpha$. This is called the \mathbf{e}mph{induced} topology with respect to $\nu$. It is Hausdorff if and only if the image of ${\mathbb B}^c(A;{\mathbb Z})$ in $\alpha$ is.\mathbf{e}nd{eg}
\begin{eg}Let $K$ be a complete, rank one valuation field. The isomorphism ${\mathbb B}^c(K;{\sh S}h O_K)\cong|K|_\vee$ of example \ref{eg-discs-field} is a homeomorphism.\mathbf{e}nd{eg}
One can also formulate a theory of \mathbf{e}mph{pro-discrete completion} for ${\mathbb B}$-modules and lattices to correspond to the completion operation for non-Archimedean rings and their modules. Followed to the conclusion of this paper, this would yield a different category of skeleta.
However, in situations typically encountered in geometry, one only has to deal with rings $A$ that have an ideal of definition $I$, and are therefore in particular \mathbf{e}mph{first countable}. In this situation, one can use the axiom of dependent choice to show that ${\mathbb B}(-;A^+)$ is automatically pro-discrete.
Moreover, Nakayama's lemma ensures that in these situations, even the free ${\mathbb B}$-module ${\mathbb B}^c(-;A^+)$ is pro-discrete. Indeed, if $M\twoheadrightarrow M/I$ is a quotient of discrete $A^+$-modules, any finite system of generators for $M/I$ lifts to generators for $M$. A pro-finite, $I$-adic $A^+$-module is therefore finitely generated. Conversely, any finite topological $A$-module is $I$-adically complete. It follows that ${\mathbb B}^c(M;A^+)$ is pro-discrete for any complete $A$-module $M$.
The main results \ref{SKEL-main}, \ref{thm-berk} of this paper remain true, under such first-countability hypotheses, if we work instead with pro-discrete ${\mathbb B}$-modules.
{\sh S}ection{Semirings}\label{.5RING}
Any symmetric monoidal category $\mathbf C$ gives rise to a theory of \mathbf{e}mph{commutative algebras} $\mathrm{Alg(C)}$ and their \mathbf{e}mph{modules}. In this section, I describe a closed, symmetric monoidal structure on the category of ${\mathbb B}$-modules; the corresponding theories are those of \mathbf{e}mph{semirings} and their \mathbf{e}mph{semimodules}. This semialgebra will provide the algebraic underpinning of the theory of skeleta.
Let $\mathbf C$ a category equipped with a monoidal structure $\mathbin{\otimes}$ with unit $1=1_\mathbf{C}$. One has a category $\mathrm{Alg}(\mathbf C)$ of \mathbf{e}mph{monoids} or \mathbf{e}mph{algebras} in $\mathbf C$, which are objects $A$ of $\mathbf C$ equipped with structural morphisms
\[ A\mathbin{\otimes} A {\sh S}tackrel{\mu}{\rightarrow} A {\sh S}tackrel{e}{\leftarrow} 1 \]
satisfying various usual constraints, and morphisms respecting these. If $A\in\mathrm{Alg}(\mathbf C)$, there is also a category $\mathrm{Mod}_A(\mathbf C)$ of \mathbf{e}mph{$A$-modules in $\mathbf C$}, which comes equipped with a free-forgetful adjunction
\[ -\mathbin{\otimes} A:\mathbf C \rightleftarrows \mathrm{Mod}_A(\mathbf C):?_A. \]
The (right adjoint) forgetful functor $?_A$ is conservative.
If $\mathbin{\otimes}$ is \mathbf{e}mph{symmetric}, then there is also a category $\mathrm{CAlg}(\mathbf C)$ of \mathbf{e}mph{commutative} algebras. The module category $\mathrm{Mod}_A(\mathbf C)$ over $A\in\mathrm{CAlg}(\mathbf C)$ acquires its own symmetric monoidal structure, the \mathbf{e}mph{relative tensor product}
\[ -\mathbin{\otimes}_A-=\mathrm{coeq}(-\mathbin{\otimes}-\mathbin{\otimes} A\rightrightarrows -\mathbin{\otimes} -) \]
(as long as $\mathbf C$ has coequalisers).
If $\mathbin{\otimes}$ is \mathbf{e}mph{closed}, that is, $-\mathbin{\otimes} A$ has a right adjoint $\mathrm{Hom}_{\mathbf C}(-,A)$, then $?_A$ also commutes with colimits and therefore $-\mathbin{\otimes} A$ preserves compactness. Limits and colimits of modules are computed in the underlying category.
{\sh S}ubsection{The tensor sum} \label{.5RING-sum}
The category of ${\mathbb B}$-modules carries a closed symmetric monoidal structure given by the \textit{tensor sum} operation $\oplus$ which, by definition, is characterised by a natural isomorphism
\[ \mathrm{Hom}_{\mathbb B}(\alpha\oplus\beta,{\mathfrak g }amma) \cong \mathrm{Hom}_{\mathbb B}(\alpha,\mathrm{Hom}_{\mathbb B}(\beta,{\mathfrak g }amma)) \]
where we use the internal $\mathrm{Hom}$ functor defined in section $\ref{SPAN}$. Alternatively, it is characterised as universal with respect to order-preserving maps $\alpha\times\beta\rightarrow{\mathfrak g }amma$ that are right exact in each variable, that is, such that for each $X\in\beta$ the composite $\alpha\rightarrow\alpha\times\{X\}\rightarrow{\mathfrak g }amma$ is a ${\mathbb B}$-module homomorphism, and similarly the transpose of this property. There is a canonical monotone map $\alpha\times\beta\rightarrow\alpha\oplus\beta$ such that for any such map, there is a unique extension
\[\xymatrix{ \alpha\oplus\beta \ar[dr] \\ \alpha\times\beta\ar[u]\ar[r] & {\mathfrak g }amma}\]
to a commuting diagram of sets. It identifies $\alpha\times\{-\infty\}\cup\{-\infty\}\times\beta$ with $\{-\infty\}$.
Explicitly, $\alpha\oplus\beta$ is generated by symbols $X\oplus Y$ with $X\in\alpha,Y\in\beta$ subject to the relations
\begin{align}\nonumber X\oplus(Y_1\vee Y_2) &= (X\oplus Y_1)\vee(X\oplus Y_2); \\
\nonumber X\oplus(-\infty) &=-\infty\mathbf{e}nd{align}
which ensure that the map
\[ [f:\alpha\oplus\beta\rightarrow{\mathfrak g }amma] \quad \mapsto \quad [X \mapsto [Y\mapsto f(X\oplus Y)]] \]
is well-defined and determines the promised adjunction.
\begin{prop}\label{.5RING-closed}The tensor sum defines a closed, symmetric monoidal structure on $\mathrm{Mod}_{\mathbb B}$.\mathbf{e}nd{prop}
\begin{proof}The argument is routine; I reproduce here the unit and counit of the adjunction $-\oplus\alpha\dashv\mathrm{Hom}(\alpha,-)$. First, we have maps
\[ \beta\rightarrow\mathrm{Hom}(\alpha,\alpha\oplus\beta),\quad X\mapsto [Y\mapsto Y\oplus X] \]
which is a ${\mathbb B}$-module homomorphism by the relations above. Second, one checks that the map
\[ \mathrm{ev}:\mathrm{Hom}(\alpha,\beta)\times\alpha\rightarrow\beta \]
preserves joins in each variable, and so descends to a homomorphism $\mathrm{Hom}(\alpha,\beta)\oplus\alpha\rightarrow\beta$.\mathbf{e}nd{proof}
\
The definitions of semirings and semimodules are those of algebras and their modules in the category $(\mathrm{Mod}_{\mathbb B},\oplus)$. I spell out some of these definitions here, in order to fix notation.
\begin{defns}\label{semiring}An \textit{idempotent semiring} $\alpha$, or, more briefly, \textit{semiring}, is a commutative monoid object $(\alpha,+,0)$ in the monoidal category $(\mathrm{Mod}_{\mathbb B},\oplus)$. Explicitly, it is a ${\mathbb B}$-module equipped with an additional commutative monoidal operation $+$, called \mathbf{e}mph{addition}, with identity $0$, that satisfies
\[ X+(Y_1\vee Y_2)=(X+Y_1)\vee(X+Y_2) \qquad X+(-\infty)=-\infty \quad\forall X. \]
In notation, addition takes priority over joins: $X+Y\vee Z=(X+Y)\vee Z$. A semiring homomorphism is a monoid homomorphism. The category of semirings is denoted $\frac{1}{2}\mathbf{Ring}$.
We will also have occasion to use a category $\frac{1}{2}\mathbf{Alg}:=\mathrm{Alg}(\mathrm{Mod}_{\mathbb B},\oplus)$ of possibly non-commutative \mathbf{e}mph{semialgebras}.
A \mathbf{e}mph{right semimodule} over a semiring $\alpha$, or (right) $\alpha$-module, is a ${\mathbb B}$-module $\mu$ equipped with an action $\mu\oplus\alpha\rightarrow\mu$ of $\alpha$, written $X\oplus Z\mapsto X+Z$. A homomorphism of semimodules is a module homomorphism. The category of $\alpha$-modules is denoted $\mathrm{Mod}_\alpha$.
The \mathbf{e}mph{relative} tensor sum $\oplus_\alpha$ on $\mathrm{Mod}_\alpha$ is the quotient
\[ \mu\oplus_\alpha\nu \cong \mathrm{coeq}[\mu\oplus\nu\oplus\alpha \rightrightarrows \mu\oplus\nu]\]
in $\mathrm{Mod}_{\mathbb B}$. A commutative monoid in $\mathrm{Mod}_\alpha$ is an \mathbf{e}mph{$\alpha$-algebra}; it consists of the same data as a semiring $\beta$ equipped with a semiring homomorphism $\alpha\rightarrow\beta$. The category of $\alpha$-algebras is denoted $\frac{1}{2}\mathbf{Ring}_\alpha$. The tensor sum of two $\alpha$-algebras over $\alpha$ has a semiring structure.
\mathbf{e}nd{defns}
\begin{egs}\label{semifields}The Boolean semifield ${\mathbb B}=\{-\infty,0\}$ is a unit for the tensor sum operation. It therefore carries a unique semiring structure, of which the notation is indicative, rendering it initial in the category of semirings. That is, ${\mathbb B}$ plays the r\^ole in the category of semirings that ${\mathbb Z}$ plays in the category of rings.
Any ${\mathbb B}$-module is in a canonical and unique way a module over ${\mathbb B}$, with $0$ acting as the identity and $-\infty$ as the constant map $-\infty$; whence the terminology of ${\mathbb B}$-modules.
More generally, the semifield $H_\vee=H{\sh S}qcup\{-\infty\}$ associated to a totally ordered Abelian group $H$ (e.g. \ref{eg-first}) carries an addition induced by the group operation on $H$.
If $H$ can be embedded into the additive group ${\mathbb R}$, $H_\vee$ is a \mathbf{e}mph{rank one} semifield; these semifields play the r\^ole in tropical geometry that ordinary fields play in algebraic geometry. Of particular interest are ${\mathbb Z}_\vee,{\mathbb Q}_\vee,{\mathbb R}_\vee$, the value semifields of DVFs, their algebraic closures, and of Novikov fields, respectively. Other semifields that arise from geometry, for example in Huber's work \cite{Hubook}, include those with $H$ of the form ${\mathbb Z}^k_\mathrm{lex}$, that is, ${\mathbb Z}^k$ with the lexicographic ordering and $-\infty$ adjoined. These semifields are non-Noetherian. They fit into a tower
\[ ({\mathbb Z}^k_\mathrm{lex})_\vee\rightarrow({\mathbb Z}^{k-1}_\mathrm{lex})_\vee\rightarrow \cdots \rightarrow {\mathbb Z}_\vee \]
of semiring homomorphisms which successively kill each irreducible convex subgroup. See also \cite[\S\textbf{0}.6.1.(a)]{FujiKato}.
\mathbf{e}nd{egs}
From general principles about algebra in monoidal categories, it follows:
\begin{prop}The category of semirings is complete and cocomplete. Limits and filtered colimits are computed in $\mathrm{Mod}_{\mathbb B}$, and the latter are exact. Pushouts are computed by the relative tensor sum.\mathbf{e}nd{prop}
{\sh S}ubsubsection{Free semimodules}\label{fungus}
Let $A$ be a ring, $M_1,M_2\in\mathrm{Mod}_A$. There are natural homomorphisms
\[ m:{\mathbb B}^{(c)}(M_1;A)\oplus{\mathbb B}^{(c)}(M_2;A)\rightarrow {\mathbb B}^{(c)}(M_1\mathbin{\otimes}_AM_2;A),\quad [N_1]\oplus [N_2]\mapsto \mathrm{Im}(N_1\mathbin{\otimes} N_2\rightarrow M_1\mathbin{\otimes}_AM_2)\]
which in the case of the subobject ${\mathbb B}$-module ${\mathbb B}$ is a lattice homomorphism. These homomorphisms upgrade ${\mathbb B}^{(c)}$ to \mathbf{e}mph{lax monoidal} functors
\[ {\mathbb B}^{(c)}:(\mathrm{Mod}_A,\mathbin{\otimes}_A)\rightarrow(\mathrm{Mod}_{\mathbb B},\oplus). \]
It is therefore compatible with algebra on both sides, in the following ways:
\begin{enumerate}\item If $B$ is an $A$-algebra, then the multiplication $\mu$ on $B$ induces a semiring structure on ${\mathbb B}^{(c)}(B;A)$
\[ [N_1]+[N_2]=\mu(N_1\mathbin{\otimes} N_2){\sh S}ubseteq B \]
and therefore the subobject (resp. free) ${\mathbb B}$-module functors are upgraded to functors
\[ {\mathbb B}^{(c)}:\mathrm{CAlg}_A\rightarrow\frac{1}{2}\mathbf{Ring}. \]
Beware that the sum $[N_1]+[N_2]$ of elements of this submodule semiring corresponds to a \mathbf{e}mph{product} in $B$, and should not be confused with the set of sums of elements of $N_1$ and $N_2$, which corresponds instead to $\vee$.
\item If $M$ is a $B$-module, then the $B$-action on $M$ induces a ${\mathbb B}^{(c)}(B;A)$-module structure on ${\mathbb B}^{(c)}(M;A)$.
\[ {\mathbb B}^{(c)}:\mathrm{Mod}_B\rightarrow \mathrm{Mod}_{{\mathbb B}^{(c)}(B;A)} \]
With respect to the relative tensor sum $\oplus_{{\mathbb B}^{(c)}(B;A)}$, these functors are lax monoidal. In particular, ${\mathbb B}^{(c)}(B;A)$ is a ${\mathbb B}^{(c)}A$-algebra.
\mathbf{e}nd{enumerate}
Be warned that ${\mathbb B}^{(c)}$ is not \mathbf{e}mph{strongly} monoidal: usually
\[{\mathbb B}^{(c)} (M_1;A) \oplus_{{\mathbb B}^{(c)}A} {\mathbb B}^{(c)}(M_2;A) \not\cong {\mathbb B}^{(c)}(M_1\mathbin{\otimes}_AM_2;A).\]
Similarly, it does not commute with most base changes - but see prop. \ref{.5RING-prop-contract}.
\begin{eg}[Seminormed vector spaces]\label{eg-discs'}Let $V$ be a vector space over a complete, valued field $K$, considered as an ${\sh S}h O_K$-module as in example \ref{eg-discs}. Let us discuss seminorms on $V$ with values in $|K|_\vee=|K|{\sh S}qcup\{-\infty\}$, the value semifield of $K$. Note that $|K|_\vee$ acts on the set of discs ${\mathbb B}^{(c)}(V;{\sh S}h O_K)$ (cf. \S\ref{fungus}).
If $K$ is non-Archimedean, then in the same vein as the previous example \ref{eg-seminorms}, the ultrametric inequality for a seminorm can be rephrased as \[{\sh S}up_{z\in\langle x,y\rangle}\nu z = \nu x\vee\nu y, \] where $\langle x,y\rangle$ denotes the ${\sh S}h O_K$-module span of $x$ and $y$. In other words, a seminorm is the same thing as a ${\mathbb B}$-module homomorphism ${\mathbb B}^c(V;{\sh S}h O_K)\rightarrow|K|_\vee$, compatible with the actions of $|K|_\vee$ on both sides.
On the other hand, if $K$ is Archimedean, and therefore either ${\mathbb R}$ or ${\mathbb C}$, then the subobjects are the convex, balanced discs. The join of two discs is their convex hull, and a disc is finite if it is the convex hull of finitely many `vertices'. Note that this implies that, for example, the unit disc of a $K$-Banach space $V$ is infinite as soon as $\dim V>1$.
The same triangle inequality as for the non-Archimedean case works if we replace the ${\sh S}h O_K$-module span $\langle x,y\rangle$ by the \mathbf{e}mph{convex hull} $\mathrm{conv}(x,y)$. An Archimedean seminorm is therefore once again a $|K|_\vee$-module homomorphism ${\mathbb B}^c(V;{\sh S}h O_K)\rightarrow |K|_\vee$.
In either case, the valuation on $K$ induces a semiring isomorphism ${\mathbb B}^c(K;{\sh S}h O_K)\widetilde\rightarrow |K|_\vee$ (e.g. \ref{eg-discs-field}).
The space of seminorms is the hom-space $\mathrm{Hom}({\mathbb B}^cV,|K|_\vee)$. The \mathbf{e}mph{unit disc} associated to a seminorm $\nu$ is $\nu^\dagger0$. Conversely, if $D\in{\mathbb B}(V;{\sh S}h O_K)$ is a disc, then the $|K|_\vee$-action thereon determines a homomorphism \[|K|_\vee\rightarrow{\mathbb B}(V;{\sh S}h O_K),\quad r\mapsto rD,\] where we interpret $r$ as the disc of radius $r$ in $K$. Since $\bigcap_{r>r_0}rD=r_0D$, this homomorphism preserves infima. If $|K|={\mathbb Z}$ or ${\mathbb R}$, then $|K|_\vee$ has all infima, and hence this homomorphism has a left adjoint $\nu$. Its behaviour on elements of $V$ is
\[ \nu x=\inf\left\{r\in|K|_\vee|x\in r \right\}. \]
It therefore maps ${\mathbb B}^cV$ into $|K|_\vee$ if and only if the disc $D$ \mathbf{e}mph{absorbs} in the sense that $KD=V$; in this case, $\nu$ is a seminorm.
This correspondence recovers the well-known dictionary between seminorms and absorbing discs in the theory of vector spaces over valued fields \cite[\S2.1.2]{Banach}.\mathbf{e}nd{eg}
{\sh S}ubsubsection{Free semirings}
Let $\alpha$ be a semiring. The forgetful functor $\frac{1}{2} \mathbf{Ring}_\alpha\rightarrow\mathbf{Set}$ commutes with limits and therefore has a left adjoint $\alpha[-]$. It is the set of `tropical polynomials'
\[ \alpha[S]\cong\left\{\left. \bigvee_{n\in{\mathbb N}^S}{\sh S}um_{X\in S}n_XX+C_n \right|C_n\in\alpha, C_n=-\infty\text{ for }n{\mathfrak g }g0 \right\} \]
with the evident join and plus operations.
\begin{defn}Let $\alpha$ be a semiring, $S$ a set; $\alpha[S]$ is called the \mathbf{e}mph{free semiring} on $S$.\mathbf{e}nd{defn}
The free semiring construction commutes with colimits; in particular we have the base change
\[\alpha[S]\cong\alpha\oplus{\mathbb B}[S]\]
and composition
\[ \alpha[S{\sh S}qcup T]=\alpha[S]\oplus_\alpha\alpha[T]\] for any $\alpha\in\frac{1}{2}\mathbf{Ring}$.
There is similarly a free functor $T\mapsto{\mathbb B}[T]$ for a \mathbf{e}mph{${\mathbb B}$-module} $T$; intuitively, it is the free semiring generated by the set $T$, subject to the order relations that exist in $T$.
{\sh S}ubsection{Action by contraction}\label{.5RING-contract}
The concept of \mathbf{e}mph{contracting operator} is natural in analysis, and is intimately related to the operator norm. In the context of this paper, we use this concept to control the \mathbf{e}mph{bounds} of tropical functions, and hence the radii of convergence of analytic functions.
\begin{defn}An endomorphism $f$ of a ${\mathbb B}$-module $\alpha$ is \mathbf{e}mph{contracting} if, for each ideal $\iota{\mathfrak h }ookrightarrow\alpha$, $f(\iota){\sh S}ubseteq\iota$. That is, $f$ is contracting if and only if $f(X)\leq X$ for all $X\in\alpha$.\mathbf{e}nd{defn}
\begin{eg}Let $A$ be an algebra and $M$ an $A$-module. An $A$-linear endomorphism of $M$ induces a contracting endomorphism of ${\mathbb B}(M;A)$ if and only if it preserves all $A$-submodules; that is, if it is an element of $A$.\mathbf{e}nd{eg}
Let now $\alpha$ be a semiring, $\mu$ a semimodule. Let $\iota{\mathfrak h }ookrightarrow\alpha$ be an ideal.
\begin{defn}We say that $\iota$ \mathbf{e}mph{contracts $\mu$} if it acts by contracting endomorphisms, or equivalently, every ideal of $\mu$ is $\iota$-invariant.
If $\iota=\alpha$, we say that $\mu$ is a \mathbf{e}mph{contracting} $\alpha$-module. If also $\mu=\alpha$, we say simply that $\alpha$ is \mathbf{e}mph{contracting} (as a semiring).\mathbf{e}nd{defn}
In particular, $\alpha$ is contracted by an ideal $\iota$ if and only if $\iota\leq0$, and $\alpha$ itself is contracting if and only if $0$ is a maximal element.
Let $\mathrm{Mod}_{\alpha\{\iota\}}$ denote the full subcategory of $\mathrm{Mod}_\alpha$ on whose objects $\iota$ contracts. This subcategory is closed under limits and the tensor sum, and so its inclusion has a lax monoidal left adjoint
\[\mathrm{Mod}_\alpha\rightarrow \mathrm{Mod}_{\alpha\{\iota\}},\quad \mu\mapsto \mu\{\iota\}, \] the \mathbf{e}mph{contraction} functor. In particular, $\alpha\{\iota\}$ is an $\alpha$-algebra, and an $\alpha$-module $\mu$ is contracting if and only if its action factors through the structure homomorphism $\alpha\rightarrow\alpha\{\iota\}$. In other words, $\mathrm{Mod}_{\alpha\{\iota\}}$ really is the category of modules over the contraction $\alpha\{\iota\}$ of $\alpha$.
The inclusion into $\frac{1}{2}\mathbf{Ring}$ of the full subcategory $\frac{1}{2}\mathbf{Ring}_{\leq0}$ of contracting semirings commutes with limits and colimits, and hence has left and right adjoints
\begin{align} \nonumber \mathrm{Left}:\alpha &\mapsto {}^\circ{\mathfrak h }space{-1pt}\alpha:=\alpha\{\alpha\} \\
\nonumber \mathrm{Right}:\alpha &\mapsto \alpha^\circ:=\alpha_{\leq0} \mathbf{e}nd{align}
and unit and counit $\alpha^\circ{\mathfrak h }ookrightarrow\alpha\rightarrow{}^\circ{\mathfrak h }space{-1pt}\alpha$. We will also write \[{}^\circ(-):=(-)\{\alpha\}\cong -\oplus_\alpha{}^\circ\alpha \] for the corresponding functor $\mathrm{Mod}_\alpha\rightarrow \mathrm{Mod}_{{}^\circ\alpha}$; but beware that this notation hides the dependence on $\alpha$.
\begin{defn}\label{.5RING-integers}The subring $\alpha^\circ$ is the \mathbf{e}mph{semiring of integers} of $\alpha$. The \mathbf{e}mph{(universal) contracting quotient} is ${}^\circ\alpha$.\mathbf{e}nd{defn}
\begin{prop}The semiring of integers functor commutes with limits and filtered colimits.\mathbf{e}nd{prop}
The contraction functor $\mathrm{Mod}_\alpha\rightarrow\mathrm{Mod}_{\alpha\{\iota\}}$ defined above can be described explicitly in terms of the ind-adjoint to $\mu\rightarrow\mu\{\iota\}$ (compare \S\ref{SPAN-quotient}). To be precise, the semiring homomorphism $\alpha^\circ\rightarrow\alpha^\circ[\iota]$ induces a homomorphism
\[ (-)[\iota]:{\mathbb B}\mu={\mathbb B}(\mu;\alpha^\circ)\rightarrow{\mathbb B}(\mu;\alpha^\circ[\iota]), \]
where we write ${\mathbb B}(\mu;\alpha)$ for the set of ideals of $\mu$ that are also $\alpha$-submodules. Its right adjoint identifies the term on the right with the set of $\iota$-invariant ideals of $\mu$. Any $\alpha$-module homomorphism $\mu\rightarrow\nu$ to a semimodule $\nu$ contracted by $\iota$ factors uniquely through the image of $\mu$ in ${\mathbb B}(\mu,\alpha^\circ[\iota])$. Thus, $\mu\{\iota\}{\sh S}ubseteq{\mathbb B}\mu$ is the subset of $\iota$-invariant ideals that are generated as such by a single element.
\begin{lemma}\label{.5RING-contract-construction}The image of $\mu$ in ${\mathbb B}(\mu,\alpha^\circ[\iota])$ is uniquely isomorphic to $\mu\{\iota\}$.\mathbf{e}nd{lemma}
\begin{eg}The ideal semiring ${\mathbb B}^{(c)}A$ of a ring $A$ is a contracting semiring. If $B$ is any $A$-algebra, then ${\mathbb B}(B;A)^\circ$ is the image of ${\mathbb B} A\rightarrow{\mathbb B}(B;A)$. Indeed, the additive identity of ${\mathbb B} B$ is precisely the image of the unit $A\rightarrow B$ of the algebra.\mathbf{e}nd{eg}
\begin{eg}[Semivaluations]\label{eg-vals}Let $A$ be a ring. A \mathbf{e}mph{semivaluation} on $A$ is a map $\val:A\rightarrow\alpha$ into a semiring $\alpha$ which is a seminorm of the underlying Abelian group, and for which
\[ \val(fg)=\val f + \val g.\]
It is said to be \mathbf{e}mph{contracting} or \mathbf{e}mph{integral} if $\alpha$ is a contracting semiring.
Let $A$ now be a non-Archimedean ring. A (non-Archimedean) semivaluation of $A$ is a continuous valuation on $A$ whose restriction to $A^+$ is integral. Any such valuation factors uniquely through the adic semiring ${\mathbb B}^c(A;A^+)$ (def. \ref{def-adic}). That is, this semiring corepresents the functor
\[ \mathrm{Hom}({\mathbb B}^c(A;A^+),-)\cong\frac{1}{2}\mathrm{Val}(A,A^+,-):\frac{1}{2}\mathbf{Ring}_t\rightarrow\mathbf{Set} \]
of continuous semivaluations on $A$.\mathbf{e}nd{eg}
\begin{prop}\label{.5RING-prop-contract}Let $f:A\rightarrow B$ be a ring homomorphism. The extension of scalars transformation ${\mathbb B}(-;A)\rightarrow{\mathbb B}(-;B)$ induces an isomorphism
\[ {\mathbb B}(-;A)\oplus_{{\mathbb B}(B;A)}{\mathbb B} B \cong {}^\circ{\mathbb B}(-;A) \cong {\mathbb B}(-;B) \]
of functors $\mathrm{Mod}_B\rightarrow \mathrm{Mod}_{{\mathbb B}(B;A)}$, and similarly
\[ {}^\circ{\mathbb B}^c(-;A)\cong {\mathbb B}^c(-;B)\]
as functors $\mathrm{Mod}_B\rightarrow \mathrm{Mod}_{{\mathbb B}^c(B;A)}$.\mathbf{e}nd{prop}
\begin{proof}Let $M\in\mathrm{Mod}_B$. We will see that the morphism ${\mathbb B}(M;A)\rightarrow{\mathbb B}(M;B)$ satisifies the universal property of ${}^\circ{\mathbb B}(M;A)$.
Let $p:{\mathbb B}(M;A)\rightarrow\alpha$ be a ${\mathbb B}(B;A)$-module map. Precomposing with the forgetful map ${\mathbb B} f^\dagger:{\mathbb B}(-;B)\rightarrow{\mathbb B}(-;A)$ gives a map
\[p{\mathbb B} f^\dagger:{\mathbb B}(M;B)\rightarrow\alpha.\]
Now ${\mathbb B} f^\dagger{\mathbb B} f$ is not the identity on ${\mathbb B}(-;A)$, but the endomorphism $\mathrm{id}+A{\mathfrak g }eq \mathrm{id}$. However, since $\alpha$ is contracting, the diagram
\[\xymatrix{ {\mathbb B}(M;B)\ar[dr]^{p{\mathbb B} f^\dagger} \\ {\mathbb B}(M;A) \ar[u]^{{\mathbb B} f}\ar[r]_-p & \alpha }\]
nonetheless commutes. In other words, $p{\mathbb B} f^\dagger$ exhibits ${\mathbb B}(-;B)$ as ${}^\circ{\mathbb B}(-;A)$.
As for the finite version, since ${\mathbb B}{\mathbb B}^c(-;A)\cong{\mathbb B}(-;A)$, applying ${\mathbb B}$ across the board embeds the picture into the one above.\mathbf{e}nd{proof}
{\sh S}ubsubsection{Freely contracting semirings}
Let $\alpha$ be a contracting semiring. The forgetful functor $\left(\frac{1}{2} \mathbf{Ring}_{\leq0}\right)_{\alpha}\rightarrow \frac{1}{2} \mathbf{Ring}_\alpha\rightarrow\mathbf{Set}$ commutes with limits and therefore has a left adjoint $\alpha\{-\}$. It is the composite of left adjoints $\alpha\mapsto\alpha[-]\mapsto{}^\circ\alpha[S]$.
\begin{defn}\label{.5RING-def-freec}Let $\alpha$ be a contracting semiring, $S$ a set (or $\alpha$-module); $\alpha\{S\}$ is called the \mathbf{e}mph{freely contracting semiring} on $S$. If $\alpha$ is any semiring, we may also write $\alpha\{S\}:=\alpha\oplus_{\alpha^\circ}\alpha^\circ\{S\}$.\mathbf{e}nd{defn}
Note $\alpha\{S\}\cong{}^\circ(\alpha^\circ[S])\oplus_{\alpha^\circ}\alpha \cong \alpha[S]/(S\leq 0)=\alpha[S]/(S\vee0=0)$ (semiring quotient).
The freely contracting functor commutes with colimits; in particular we have the base change
\[\alpha\{S\}\cong\alpha\oplus{\mathbb B}\{S\}\]
and composition
\[ \alpha\{S{\sh S}qcup T\}=\alpha\{S\}\oplus_\alpha\alpha\{T\}\] for any $\alpha\in\frac{1}{2}\mathbf{Ring}$.
\begin{eg}If $A$ is a complete DVR with maximal ideal $\lie m$, then its ideal semiring ${\mathbb B}^cA$ is freely contracting on the element $\lie m$.
This can be understood as an explicit construction of a freely contracting semiring on one element. More generally, ${\mathbb B}\{S\}$ for arbitrary $S$ can be described as the semiring of \mathbf{e}mph{monomial} ideals in a polynomial ring $k[S]$ on the same set of variables.\mathbf{e}nd{eg}
\begin{eg}\label{non-eg}Let ${\mathbb D}elta=[-\infty,0]$ denote the infinite half-line, and consider the semiring $\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$ of its convex, piecewise-affine functions with integer slopes (e.g. \ref{eg-first}). It is generated over ${\mathbb R}_\vee$ by a single, contracting element $X$. However, this generation is \mathbf{e}mph{not} free: it satisfies additional relations, such as
\[ n(Y_1\vee Y_2)=nY_1\vee nY_2 \]
for all $n\in{\mathbb N}$ and $Y_i\in\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$. We can see that these relations are not satisfied in ${\mathbb R}_\vee\{X\}$ by thinking of it as the set of monomial ${\sh S}h O_K\{x\}$-submodules of $K\{x\}$, where $K$ is any non-Archimedean field with value group $|K|={\mathbb R}$.
The key difference between free semirings and function semirings is that the latter are \mathbf{e}mph{cancellative}, while the former are not. In the present example, cancellativity can be enforced by imposing the above list of relations in ${\mathbb R}_\vee\{X\}$. The resulting \mathbf{e}mph{universal cancellative quotient} ${\mathbb R}_\vee\{X\}\rightarrow\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$ is infinitely presented. In particular, $\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$ is not a finitely presented ${\mathbb R}_\vee$-algebra.\mathbf{e}nd{eg}
{\sh S}ubsection{Projective tensor sum}
The join of two continuous ${\mathbb B}$-module homomorphisms is continuous. The category of topological ${\mathbb B}$-modules is therefore enriched over $\mathrm{Mod}_{\mathbb B}$. We extend this to an \mathbf{e}mph{internal} Hom functor by equipping the continuous homomorphism ${\mathbb B}$-module $\mathrm{Hom}_{\mathrm{Mod}_{{\mathbb B},t}}(\alpha,\beta)$ with the weak topology with respect to the evaluation maps \[\mathrm{ev}_X:f\mapsto f(X)\] for $X\in\alpha$. In other words, it carries the topology of \mathbf{e}mph{pointwise convergence}. A fundamental system for this topology is given \[ \mathrm{Hom}_{\mathrm{Mod}_{{\mathbb B},t}}(\alpha,\beta)^u:=\{U_{X,Y}:=\{f|f(X){\sh S}ubseteq Y\}| X\in\alpha, Y\in\beta^u\}, \] a formula that should evoke the compact-open topology of mapping spaces in general topology.
\begin{eg}This is not the only reasonable way of topologising the continuous Hom ${\mathbb B}$-module, though it is of course the weakest. For instance, one could also define a topology of \mathbf{e}mph{uniform convergence} as the weak topology with respect to the natural embedding
\[ \mathrm{Hom}(\alpha,\beta)\rightarrow \mathrm{Hom}({\mathbb B}\alpha,{\mathbb B}\beta), \] where the right-hand term is equipped with the usual topology. These topologies are in general inequivalent; in fact, this embedding is not always continuous in the topology of pointwise convergence.
For example, a net $\{f_n\}_{n\in{\mathbb N}}$ in $\mathrm{Hom}({\mathbb Z}_\vee,{\mathbb Z}_\vee)$ tends to $-\infty$ as $n\rightarrow\infty$ if and only if $f_n(x)\rightarrow-\infty$ for all $x\in{\mathbb Z}$. For the same net to die away in $\mathrm{Hom}({\mathbb B}{\mathbb Z}_\vee,{\mathbb B}{\mathbb Z}_\vee)$, in addition $\{{\sh S}up_{x\in{\mathbb Z}}f_n(x)\}_{n\in{\mathbb N}}$ must tend to $-\infty$ (and in particular, be finite for cofinal $n\in{\mathbb N}$).\mathbf{e}nd{eg}
We can also extend the monoidal structure to $\mathrm{Mod}_{{\mathbb B},t}$. The \mathbf{e}mph{projective topological tensor sum} of topological ${\mathbb B}$-modules $\alpha,\beta$ is tensor sum $?\alpha\oplus ?\beta$ equipped with the strong topology with respect to the maps \[e_Y:\alpha\rightarrow\alpha\oplus\beta,\quad X\mapsto X\oplus Y\]
for $Y\in\beta$ and $e_X$ for $X\in\alpha$. If $\alpha,\beta$ are lattices, a fundamental system is generated by elements \[X\oplus \beta \vee \alpha\oplus Y,\quad X\in\alpha^u,Y\in\beta^u.\] It is more difficult to give a fundamental system for general $\alpha$ and $\beta$.
\begin{eg}\label{TOP-injective}The ideal ${\mathbb B}$-module functor ${\mathbb B}$ is not lax monoidal for the projective topology. For instance, the ${\mathbb B}$-module ${\mathbb Z}_\vee\oplus{\mathbb Z}_\vee$ is topologised so that a net $X_n\oplus Y_n$ dies away if and only if either $X_n$ dies and $Y_n$ is bounded, or vice versa. However, from the description of the fundamental system it follows that for the same net to die away in ${\mathbb B}{\mathbb Z}_\vee\oplus{\mathbb B}{\mathbb Z}_\vee$ it is enough that either $X_n$ or $Y_n$ does. The natural lattice homomorphism
\[ {\mathbb B}{\mathbb Z}_\vee\oplus{\mathbb B}{\mathbb Z}_\vee \rightarrow {\mathbb B}({\mathbb Z}_\vee\oplus{\mathbb Z}_\vee) \]
is discontinuous.
It is, however, lax monoidal on bounded ${\mathbb B}$-modules, and in particular, lattices.\mathbf{e}nd{eg}
\begin{prop}The topological tensor sum and continuous internal Hom define a closed, symmetric monoidal structure on $\mathrm{Mod}_{{\mathbb B},t}$ extending that of $\mathrm{Mod}_{\mathbb B}$.\mathbf{e}nd{prop}
\begin{proof}We only need to check that the unit and counit maps of proposition \ref{.5RING-closed} are continuous. For the unit $\alpha\rightarrow\mathrm{Hom}(\beta,\alpha\oplus\beta)$, which by the definition of the projective topology factors through the continuous Hom module, it is enough that the compositions $e_X:\alpha\rightarrow\alpha\oplus\beta$ with the evaluations at $X\in\beta$ are continuous. Continuity of the counit is similarly tautological.\mathbf{e}nd{proof}
\begin{prop}\label{RING-open}Let $\alpha\rightarrow\beta$ be strong. Then for any topological ${\mathbb B}$-module ${\mathfrak g }amma$, $\alpha\oplus{\mathfrak g }amma\rightarrow\beta\oplus{\mathfrak g }amma$ is strong.\mathbf{e}nd{prop}
\begin{proof}This follows from the fact that if $fg$ and $g$ are strong (families of) maps, then $f$ is strong.\mathbf{e}nd{proof}
\begin{defns}\label{tsemiring}A \mathbf{e}mph{topological semiring} is a commutative algebra in $(\mathrm{Mod}_{{\mathbb B},t},\oplus)$. A topological semiring $\alpha$ is \mathbf{e}mph{adic} if $\alpha^u$ is stable in ${\mathbb B}\alpha$ under addition, that is, if addition by an open element is an open map (def.\ \ref{TOP-open}). The category of adic semirings and continuous homomorphisms is denoted $\frac{1}{2}\mathbf{Ring}_t$. By proposition \ref{RING-open}, it is stable in the category of all topological semirings under tensor sum.
\label{def-adic}\mathbf{e}nd{defns}
In the sequel, all semirings will be assumed adically topologised, and so we will typically omit the adjectives `topological' and `adic'. A non-Archimedean ring $A$ (def. \ref{ADIC-def-na}), resp. homomorphism $f:A\rightarrow B$, is adic if and only if ${\mathbb B}^c(A;A^+)$ is adic, resp. ${\mathbb B} f$ is strong.
\begin{eg}\label{semifields'}The semifields $H_\vee$ associated to totally ordered Abelian groups (e.g. \ref{semifields}) are adic with respect to the topology of e.g. \ref{semifields-top}. All our examples of adic semirings will be adic over some $H_\vee$. The convergence condition for such semirings will therefore be that a net $X_n\in\alpha$ converges to $-\infty$ if and only if for each `constant' $r\in H_\vee$, cofinally many $X_n\leq r$ in $\alpha$.
For instance, the semirings ${\mathbb R}_\vee\rightarrow\mathrm{CPA}_*(X,{\mathbb R}_\vee)$ (e.g. \ref{eg-second}) are of this form.
Any continuous semiring homomorphism $H_\vee\rightarrow{\mathbb B}$ (where ${\mathbb B}$ is as always discrete) is an isomorphism. On the other hand, if $H{\sh S}ubseteq{\mathbb R}$ has rank one, then there is always a unique homomorphism $H_\vee^\circ\rightarrow{\mathbb B}$, the \mathbf{e}mph{reduction} map. One can still define this map for general totally ordered semifields, but it is no longer unique.\mathbf{e}nd{eg}
\begin{eg}\label{eg-adic-Noetherian}An \mathbf{e}mph{element of definition} of an adic semiring $\alpha$ is a principal open $I\in\alpha^u\cap\alpha$ such that $\alpha$ is ${\mathbb Z}_\vee^\circ$-adic with respect to the induced homomorphism
\[ {\mathbb Z}_\vee^\circ\rightarrow\alpha,\quad -1\mapsto I. \]
The join of two elements of definition is an element of definition. If $\alpha$ is Noetherian and has an element of definition, there is therefore also a \mathbf{e}mph{largest} element of definition, and hence a canonical \mathbf{e}mph{largest} ${\mathbb Z}_\vee^\circ$-algebra structure on $\alpha$. It thereby attains also a canonical reduction $\overline\alpha=\alpha^\circ\oplus_{{\mathbb Z}_\vee^\circ}{\mathbb B}$. Note that this ${\mathbb Z}_\vee^\circ$-algebra structure need not be unique or functorial, even for adic semiring homomorphisms.
If $X$ is any Noetherian formal scheme, ${\mathbb B}^c{\sh S}h O_X$ attains a canonical ${\mathbb Z}_\vee^\circ$-algebra structure, and the reduction $\overline{{\mathbb B}^c{\sh S}h O_X}\cong{\mathbb B}^c{\sh S}h O_{\overline X}$. Again, this is not to say that ${\mathbb B}^c$ defines a functor with values in $\mathrm{Alg}_{{\mathbb Z}_\vee^\circ}$.\mathbf{e}nd{eg}
\begin{eg}\label{eg-convergent-series}The free and freely contracting semirings $\alpha[X],\alpha\{X\}$ over an adic semiring $\alpha$ are topologised adically over $\alpha$.
Let $A$ be a non-Archimedean ring. The convergent power series ring $A\{x\}$ may be constructed as a certain completion of $A[x]$; in terms of semirings, it is the completion with respect to the topology induced by \[A[x]\rightarrow{\mathbb B}^cA[X]\rightarrow{\mathbb B}^cA\{X\},\] where the left-hand map is the unique valuation sending $x$ to $X$.\mathbf{e}nd{eg}
\begin{eg}[Discrete valuations]Let $X$ be an irreducible variety over a field $k$. A classic result of birational geometry states that `algebraic' discrete valuations $\val:K\rightarrow{\mathbb Z}_\vee$ on the function field $K$ of $X$, integral on ${\sh S}h O_X$, are in one-to-one correspondence with prime Cartier divisors on blow-ups of $X$.
More specifically, let $\widetilde X\rightarrow X$ be a blow-up, $D{\sh S}ubset\widetilde X$ a prime Cartier divisor, and consider the formal completion $i:\widehat D\rightarrow\widetilde X$. Then the order of vanishing against $D$ is a continuous discrete valuation on the sheaf $i^*K$ of ${\sh S}h O_{\widehat D}$-modules. Conversely, given any discrete valuation $v$ on $K$, then provided that the associated residue field is of the correct dimension over $k$ (the algebraicity condition), one can construct the generic point of a $\widehat D$ giving rise to $v$ in this way as the formal spectrum of the completed ring of integers.
We can couch this correspondence in terms of semiring theory as follows. Let $U:=\widetilde X{\sh S}etminus D$, and consider $(\widehat{{\sh S}h O}_U;\widehat{{\sh S}h O}_{\widetilde X})$ as a sheaf of non-Archimedean ${\sh S}h O$-algebras on the completion $\widehat D$. The reduction $D$ corresponds to an invertible element $I\in{\mathbb B}^c(\widehat{{\sh S}h O}_U;\widehat{{\sh S}h O}_{\widetilde X})$, and induces an adic homomorphism \[\nu^\dagger:{\mathbb Z}_\vee{\mathfrak h }ookrightarrow {\mathbb B}^c(\widehat{{\sh S}h O}_U;\widehat{{\sh S}h O}_{\widetilde X})\] of semirings over $\widehat D$; here ${\mathbb Z}_\vee$ denote the locally constant sheaf.
By Krull's intersection theorem, $\bigcap_{n\in{\mathbb N}}I^n=0$, that is, $\nu^\dagger$ preserves infima. It therefore has a left adjoint
\[\nu:{\mathbb B}^c(\widehat{{\sh S}h O}_U;\widehat{{\sh S}h O}_{\widetilde X})\rightarrow {\mathbb B}{\mathbb Z}_\vee={\mathbb Z}_\vee{\sh S}qcup\{\infty\}, \quad J \mapsto \inf\{n\in{\mathbb N}|J\leq nI\}. \]
In fact, this adjoint is finite (i.e. does not achieve the value $\infty$), since every section of ${\sh S}h O_U$ becomes a section of ${\sh S}h O_{\widetilde X}$ after multiplication by a power of $I$; moreover $\nu^{-1}(-\infty)=\{-\infty\}$. We have therefore defined a \mathbf{e}mph{complete, discrete norm}
\[ \nu:\widehat{{\sh S}h O}_U\rightarrow{\mathbb Z}_\vee\]
over $\widehat D$.
For this norm to define a \mathbf{e}mph{valuation}, the left adjoint $\nu$ must commute with addition. In general this property is much more delicate than the existence and finiteness of $\nu$. In our setting, a study of the local algebra shows directly that this happens exactly when $D$ is prime.
In this case, if $\Spec A=V{\sh S}ubseteq X$ is an affine subset meeting $D$, then localisation induces an extension $K\rightarrow{\mathbb Z}_\vee$ of the induced discrete valuation on $A$. This extension is \mathbf{e}mph{not} left adjoint to the obvious map ${\mathbb Z}_\vee\rightarrow{\mathbb B}^c(K;{\sh S}h O_X)$, which is typically infinite (not to mention discontinuous).
For the converse statement, note only that discrete valuations on $K$, integral on some model $X=\Spec A$, are the same thing as homomorphisms \[ v:{\mathbb B}^c(K;A)\rightarrow{\mathbb Z}_\vee. \] This homomorphism has a (discontinuous) right ind-adjoint $v^\dagger$; the algebraicity condition is equivalent to this ind-adjoint being the extension of an ordinary adjoint, in which case $v^\dagger(-1)$ is a finitely generated ideal on $\Spec A$ which may be blown up to obtain $D$.\mathbf{e}nd{eg}
{\sh S}ubsubsection{Projective tensor product}\label{.5RING-proj}
Let $A$ be a non-Archimedean ring. The projective tensor product $M\mathbin{\otimes}_AN$ of locally convex $A$-modules $M$ and $N$ is strongly topologised with respect to the map
\[ {\mathbb B}^c(M;A^+)\oplus{\mathbb B}^c(N;A^+) \rightarrow {\mathbb B}(M\mathbin{\otimes}_AN;A^+).\]
We can describe this topology in terms of linear algebra alone: it is the strong topology with respect to the maps
\[ e_y:M\rightarrow M\mathbin{\otimes}_AN,\quad x\mapsto x\mathbin{\otimes} y \]
for $y\in N$, and similarly $e_x$ for $x\in M$. A sequence converges to zero in $M\mathbin{\otimes}_AN$ if and only if it is a sum of sequences of the form $x_n\mathbin{\otimes} y$ and $x\mathbin{\otimes} y_n$, where $x_n$ and $y_n$ converge to zero in $M$ and $N$, respectively.
With this definition, the monoidal functoriality of the \mathbf{e}mph{free} ${\mathbb B}$-module ${\mathbb B}^c$ spelled out in \S\ref{fungus} lifts to the topological setting; for example, ${\mathbb B}^c(M;A^+)$ is a topological ${\mathbb B}^c(A;A^+)$-module. The corresponding statements for ${\mathbb B}$ are false unless $A=A^+$.
Similarly, we topologise $\mathrm{Hom}(M_1,M_2)$ weakly with respect to
\[ \mathrm{Hom}_A(M_1,M_2)\rightarrow\mathrm{Hom}_{\mathrm{Mod}_{{\mathbb B},t}}({\mathbb B}^cM_1;{\mathbb B}^cM_2),\quad f\mapsto {\mathbb B}^c f. \] A sequence of maps $\{f_n\}_{n\in{\mathbb N}}$ converges to zero if and only if for every finitely generated submodule $N{\sh S}ubseteq M_1$, every sequence $x_n\in f_n(N)$ converges to zero. This `finite-open' topology is the weak topology with respect to the evaluation maps
\[ \mathrm{ev}_x:\mathrm{Hom}_A(M_1,M_2)\rightarrow M_2,\quad f\mapsto f(x)\]
for $x\in M_1$.
{\sh S}ection{Localisation}\label{LOC}
Let $\mathbf C$ be a category with filtered colimits, $M$ an object. In this setting, we can define the (free) \mathbf{e}mph{localisation} of $M$ at an endomorphism $s\in\End_{\mathbf C}(M)$ as the sequential colimit
\[ M[s^{-1}]:=\colim\left[ M {\sh S}tackrel{s}{\rightarrow} M {\sh S}tackrel{s}{\rightarrow} \cdots \right] \]
It is universal among objects under $M$ for which $s$ extends to an automorphism. More generally, by composing colimits the localisation with respect to any set $S$ of commuting endomorphisms is defined.
If $\mathbf C$ is a category of modules over some algebra $A$, then in particular we can localise modules with respect to an element $s\in A$. If $A$ is commutative, and $M$ carries its own $A$-algebra structure, then the localisation $M[s^{-1}]$ is also an ($M$-)algebra.
The general theory specialises to the case of topological semirings; we write $\alpha[-S]$ for the localisation of $\alpha$ at an element $S$.
\begin{eg}Let $A$ be a domain, ${\mathbb B}^cA$ the finite ideal semiring. If $s\in A$, then ${\mathbb B}^cA[-(s)]\cong{\mathbb B}^c(A[s^{-1}];A)$. In order to obtain the ideal semiring of $A[s^{-1}]$, we need to enforce a contraction $(s)\leq0$.\mathbf{e}nd{eg}
\begin{eg}Suppose that $S\in\alpha$ is open. Then $\alpha[-S]$ is an adic $\alpha$-algebra (def. \ref{def-adic}).
This corresponds to the fact that if $A$ is an adic, linearly topologised ring, and $f\in A$ generates an open ideal, then $A[f^{-1}]$ is an adic $A$-algebra.\mathbf{e}nd{eg}
\begin{defn}\label{LOC-Tate}A topological semiring is \mathbf{e}mph{Tate} if $\alpha^\circ$ is adic, and $\alpha$ is a free localisation of $\alpha^\circ$ at an additive family of open elements. The full subcategory of $\frac{1}{2}\mathbf{Ring}_t$ whose objects are Tate is denoted $\frac{1}{2}\mathbf{Ring}_T$.\mathbf{e}nd{defn}
In particular, any contracting semiring is Tate. A non-Archimedean ring $A$ is Tate (def. \ref{ADIC-def-na}) if and only if ${\mathbb B}^c (A;A^+)$ is.
{\sh S}ubsection{Bounded localisation}
In non-Archimedean geometry, localisations must be supplemented by certain completions, which control the radii of convergence of the inverted functions. For the geometry of skeleta to reflect analytic geometry, there must therefore be a corresponding concept for semirings.
\begin{defn}\label{LOC-def-radius}Let $\alpha$ be an adic semiring. An element $T\in\alpha^\circ$ that is invertible in $\alpha$ is called an \mathbf{e}mph{admissible bound}, or simply a \mathbf{e}mph{bound}.\mathbf{e}nd{defn}
Invertible elements $S=(+(-S))^{-1}(0)$ in adic semirings - in particular, bounds - are always open.
A localisation $\mu\rightarrow\mu[-S]$ is adic if and only if $S$ is open.
If $T\in\alpha^u$ is an open ideal, then $S$ is open as an endomorphism of the semiring quotient
\[ \mu/(T\leq S) = \mu/(T\vee S=S), \]
since $T\leq S$ forces $S$ to be open. The \mathbf{e}mph{bounded} localisation $\mu\rightarrow\mu/(T\leq S)[-S]$ is therefore adic.
\begin{defn}Let $S\in\alpha^\circ$ and $T\in\alpha$ a bound. Let $\mu$ be an $\alpha$-module. A \mathbf{e}mph{bounded localisation of $\mu$ at $S$ with bound $T$} is an $\alpha$-module homomorphism
\[\mu\rightarrow\mu\{T-S\}=\mu[-S]\{T-S\}, \]
universal among those under which $S$ becomes invertible with inverse bounded (above) by $-T$.
It is called a \mathbf{e}mph{cellular} localisation if $T=0$.
If $T\leq S$ in $\alpha^\circ$, then the bounded localisation is isomorphic to an ordinary, or \mathbf{e}mph{free} localisation. In this case, we will often call it a \mathbf{e}mph{subdivision}. Note that only free localisations at elements that are bounded below by an admissible bound are allowed.
More generally, the above definition makes sense if we replace $S$ with an arbitrary additive subset of $\alpha^\circ$ and $T$ with an additive set of bounds in bijection with $S$.\mathbf{e}nd{defn}
\begin{lemma}Any bounded localisation can be factored as a cellular localisation followed by a subdivision.\mathbf{e}nd{lemma}
\begin{proof}Factor $\alpha\rightarrow\alpha\{T-S\}$ as \[\alpha\rightarrow \alpha\{T-(S\vee T)\}\{-(S-(S\vee T))\}. \] In fact, this factorisation is natural in $\alpha,S$, and $T$.\mathbf{e}nd{proof}
\begin{eg}[Intervals]\label{eg-int}Consider the semiring $\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$ (e.g. \ref{eg-first}), and for simplicity, specialise to the case that ${\mathbb D}elta=[a,b]$ is an interval with $a,b\in{\mathbb Z}$ (but see also \S\ref{EGS-poly}).
The admissible bounds of $\mathrm{CPA}_{\mathbb Z}([a,b],{\mathbb R}_\vee)$ are the affine functions $mX+c$, $m\in{\mathbb Z},c\in{\mathbb R}$. Since every convex function on $[a,b]$ is bounded below by an affine function, any element of $\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$ may be freely inverted by a bounded localisation.
Let us invert the function $X\vee r$ for some $r\in[a,b]$. The resulting semiring, which we denote $\mathrm{CPA}_{\mathbb Z}([a,r,b],{\mathbb R}_\vee)$, now consists of integer-sloped, piecewise-affine functions on $[a,b]$ which are convex \mathbf{e}mph{except possibly at $r$}. I would like to think of this as a ring of functions on the polyhedral complex obtained by joining the intervals $[a,r]$ and $[r,b]$ at their endpoints, or alternatively, by \mathbf{e}mph{subdividing} $[a,b]$ into two subintervals meeting at $r$. The affine structure does not extend over the join point. This is the motivation for the terminology `subdivision'.
More generally, the free bounded localisations of $\mathrm{CPA}_{\mathbb Z}([a,b],{\mathbb R}_\vee)$ are in one-to-one correspondence with finite sequences of rationals $r_1,\ldots,r_k\in(a,b)$:
\[ \mathrm{CPA}_{\mathbb Z}([a,b],{\mathbb R}_\vee)\rightarrow\mathrm{CPA}_{\mathbb Z}([a,r_1,\ldots,r_k,b],{\mathbb R}_\vee),\]
that is with subdivisions of $[a,b]$ in the sense of rational polyhedral complexes.
Now let's compose this with the \mathbf{e}mph{cellular} localisation at $S=-(0\vee(X-r))$. This has the effect of imposing the relation $X\leq r$. In other words, the localisation is naturally $\mathrm{CPA}_{\mathbb Z}([a,r],{\mathbb R}_\vee)$, the semiring of functions on the lower \mathbf{e}mph{cell} $[a,r]$. In particular, when $r=a$, the subdivision has no effect (since in that case $X\vee-a=X$ is already invertible), and the cellular localisation is just the evaluation $\mathrm{CPA}_{\mathbb Z}([a,b],{\mathbb R}_\vee)\rightarrow{\mathbb R}_\vee$ at $a$.
The composite of both localisations can be expressed more succinctly as $\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)\{X-r\}$, from which we can read that $r$ is the upper bound for the interval they cut out.
More generally, every cellular localisation of $\mathrm{CPA}_{\mathbb Z}([a,r_1,\ldots,r_k,b],{\mathbb R}_\vee)$ is determined by the union of cells on which a defining function vanishes.
\mathbf{e}nd{eg}
\begin{eg}\label{eg-int'}In the limiting case of the above example ${\mathbb D}elta={\mathbb R}$, the only functions bounded by zero are the constants ${\mathbb R}_\vee^\circ=\mathrm{CPA}_*({\mathbb R},{\mathbb R}_\vee)^\circ$. The semiring $\mathrm{CPA}_*({\mathbb R},{\mathbb R}_\vee)$ therefore has no completed localisations; it is a poor semialgebraic model for the real line.\mathbf{e}nd{eg}
\begin{eg}\label{non-eg'}We have seen (e.g. \ref{non-eg}) that the semirings $\mathrm{CPA}$ are not finitely presented over ${\mathbb R}_\vee$. It may therefore be easier to work instead with finitely presented models of them; for example, ${\mathbb R}_\vee\{X\}$ instead of $\mathrm{CPA}_{\mathbb Z}({\mathbb R}_\vee^\circ,{\mathbb R}_\vee)$.
However, the free localisation theory of these semirings is much more complicated than their cancellative counterparts - it depends on more than just the `kink set' of the function being inverted. For example, inverting $X\vee(-1)$ and $nX\vee(-n)$ define non-isomorphic localisations for $n>1$ (though the former factors through the latter).
This could be regarded as a problem with the theory as I have set it up. I will not make any serious attempt to address it in this paper, as it does not directly affect the main results - but see e.g. \ref{circnorm}.\mathbf{e}nd{eg}
\begin{eg}Let $K$ be a non-Archimedean field with uniformiser $t$, $K\{x\}$ the Tate algebra in one variable. It is complete with respect to the valuation $K\{x\}\rightarrow |K|_\vee\{X\}$ of example \ref{eg-convergent-series}.
A completed localisation of the Tate algebra at $x$ has the form $K\{x,t^{-k}x^{-1}\}$ for some $k\in|K|$. This $k$ is a bound in the sense of definition \ref{LOC-def-radius}. The completed localisation is a completion of $K\{x\}[x^{-1}]$ with respect to the topology induced by its natural valuation into $|K|_\vee\{X,k-X\}$.
The number $e^k$ (or $p^k$ when the residue characteristic is $p>0$) is conventionally called the \mathbf{e}mph{inner radius} of the annulus $\Spa K\{x,t^kx^{-1}\}$. In other words, bounds in semiring theory arise intuitively as the `logarithms' of radii of convergence in analytic geometry.\mathbf{e}nd{eg}
\begin{eg}[Admissible blow-ups]Let $X$ be a quasi-compact adic space, $T\in{\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)$ an admissible bound. Let $j:X\rightarrow X^+$ be a formal model on which $T$ is defined. Then $T\leq 0$ corresponds to a subscheme of $X^+$ whose pullback to $X$ is empty. In other words, the admissible bounds of ${\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)$ that are defined on $X^+$ are exactly the centres for admissible blow-ups of $X^+$ (cf. \ref{ADIC}).\mathbf{e}nd{eg}
The following elementary properties of bounded localisation are a consequence of the universal properties.
\begin{lemma}Let $\alpha$ be a topological semiring, $\mu$ an $\alpha$-module.\label{LOC-properties}
\begin{enumerate}
\item $\alpha\{T-S\}$ is a semiring, and $\mu\{T-S\}\cong\mu\oplus_\alpha\alpha\{T-S\}$ as an $\alpha\{T-S\}$-module.
\item Localisation commutes with contraction. That is, $\mu\{\iota\}[-S]\cong\mu[-S]\{\iota\}$.
\item Let $S_1,S_2\in\alpha^\circ$, $T_1,T_2$ two bounds. Then $\mu\{T_1-S_1,T_2-S_2\}\cong\mu\{T_1-S_1\}\{T_2-S_2\}$.
\mathbf{e}nd{enumerate}\mathbf{e}nd{lemma}
It follows also from the discussion above:
\begin{lemma}Let $\alpha$ be adic. Then $\mu\rightarrow\mu\{T-S\}$ is adic.\label{LOC-strong}\mathbf{e}nd{lemma}
{\sh S}ubsection{Cellular localisation}
Let $\alpha$ be a contracting semiring. Then the only invertible element, and hence only admissible bound, is $0$. All localisations of a contracting semiring are therefore cellular: $\alpha\rightarrow\alpha/(S=0)$.
\begin{eg}\label{eg-topol}Let $X$ be a coherent topological space \cite[def. \textbf{0}.2.2.1]{FujiKato}, so that the ${\mathbb B}$-module $|{\sh S}h O_X|$ of quasi-compact open subsets of $X$ has finite meets that distribute over joins. Its lattice completion ${\mathbb B}|{\sh S}h O_X|$ is the lattice of all open subsets of $X$ (or the opposite to the lattice of all closed subsets of $X$).
If $X$ is quasi-compact, then it is an identity for the meet operation on $|{\sh S}h O_X|$; in other words, intersection of open subsets is a \mathbf{e}mph{contracting semiring} operation on $|{\sh S}h O_X|$, and $X=0$. Note that this addition is idempotent. Let us describe the localisations of $|{\sh S}h O_X|$.
Let $S\in|{\sh S}h O_X|$. The inclusion $\iota:S{\mathfrak h }ookrightarrow X$ induces adjoint \mathbf{e}mph{semiring} homomorphisms
\[ \iota_!:|{\sh S}h O_S|\leftrightarrows|{\sh S}h O_X|:\iota^* \] by composition with and pullback along $\iota$, respectively. They satisfy the identities $\iota^*\iota_!=\mathrm{id}$ and $\iota_!\iota^*=(-)+S$. The right adjoint $\iota^*$ identifies $S$ with $0$. Moreover, any semiring homomorphism $f:|{\sh S}h O_X|\rightarrow\alpha$ with this effect admits a factorisation $f=f+S=f\iota_!\iota^*$ through $|{\sh S}h O_S|$, necessarily unique since $\iota^*$ is surjective. In other words, $|{\sh S}h O_S|$ is a cellular localisation of $|{\sh S}h O_X|$ at $S$.
Alternatively, and more in the spirit of what follows, one can argue this using the \mathbf{e}mph{right ind-adjoint} \[f^\dagger:|{\sh S}h O_X|\{-S\}\rightarrow{\mathbb B}|{\sh S}h O_X|\] to the localisation $f$. This map is easier to describe in terms of closed subsets: if $Z\in|{\sh S}h O_X|\{-S\}$, then $f^\dagger Z$ is the smallest closed subset of $X$ whose image in $|{\sh S}h O_X|\{-S\}$ is $Z$. It identifies the localised semiring with the image of the composite $f^\dagger f$, which is the set of subsets $K{\sh S}ubseteq X$ equal to the closure of their intersections with $S$, $K=\overline{K\cap S}$. Closure puts $|{\sh S}h O_S|$ in one-to-one correspondence with this set.\mathbf{e}nd{eg}
The latter method of this example can be abstracted, in line with the methods of \S\ref{SPAN-quotient} and \S\ref{.5RING-contract}. Let $\alpha\in\frac{1}{2}\mathbf{Ring}_t$, $\mu$ an $\alpha$-module, $S\in\alpha^\circ$.
\begin{defns}An ideal $\iota{\mathfrak h }ookrightarrow\mu$ is \mathbf{e}mph{$-S$-invariant} if $X+S\in\iota{\mathbb R}ightarrow X\in\iota$. The \mathbf{e}mph{$-S$-span} of an ideal $\iota$ is \[ \bigcup_{n\in{\mathbb N}}(+S)^{-n}(\iota), \] that is, the smallest $-S$-invariant ideal containing $\iota$.\mathbf{e}nd{defns}
If $S$ is invertible, then being $-S$-invariant is the same as being invariant under the action of $-S$. In particular, the set of $-S$-invariant ideals of $\mu[-S]$ is the lattice ${\mathbb B}(\mu[-S];\alpha^\circ[-S])$ of $\alpha^\circ[-S]$-submodule ideals of $\mu[-S]$. Moreover,
\begin{lemma}The right adjoint to the localisation map
\[ {\mathbb B}\mu{\sh S}tackrel{f}{\rightarrow}{\mathbb B}(\mu[-S];\alpha^\circ[-S]) \]
identifies the latter with the set of $-S$-invariant ideals of $\mu$.\mathbf{e}nd{lemma}
\begin{proof}Let $\iota{\mathfrak h }ookrightarrow\mu$ be $-S$-invariant. Every element of $\iota[-S]{\mathfrak h }ookrightarrow\mu[-S]$ is of the form $X-nS$ with $X\in\iota$. If $X-nS=f(Y)$ for some $Y\in\mu$, then $f(Y+nS)=X\in\iota$ and hence $Y\in\iota$. This proves that $f^\dagger f\iota=\iota$.
\mathbf{e}nd{proof}
Since in the cellular localisation, $-S\leq 0$, every ideal is automatically $-S$-invariant. By lemma \ref{.5RING-contract-construction}, the contraction $(-)\{-S\}$ induces isomorphisms
\[ {\mathbb B}(\mu[-S];\alpha^\circ[-S])\widetilde\rightarrow{\mathbb B}(\mu\{-S\};\alpha^\circ\{-S\})\cong{\mathbb B}\mu\{-S\}. \]
This identifies the cellular localisation $\mu\{-S\}$ with the image of $\mu$ in ${\mathbb B}(\mu[-S];\alpha^\circ[-S])$.
We have obtained a characterisation of cellular localisations in terms of ideals:
\begin{lemma}\label{LOC-Zar-construction}A homomorphism $f:\mu\rightarrow\nu$ of $\alpha$-modules is a cellular localisation of $\mu$ at $S\in\alpha^\circ$ if and only if $f^\dagger$ identifies ${\mathbb B}\nu$ with the $-S$-invariants of ${\mathbb B}\mu$.\mathbf{e}nd{lemma}
Note only that the `if' part of the statement follows from the fidelity of ${\mathbb B}$.
\begin{eg}[Zariski-open formula]\label{LOC-Zar-open}
Let $X$ be a quasi-compact formal scheme, $i:U{\mathfrak h }ookrightarrow X$ a quasi-compact open subset. Let $I$ be a finite ideal sheaf cosupported inside $X{\sh S}etminus U$. The restriction $\rho:{\mathbb B}^c{\sh S}h O_X\rightarrow i_*{\mathbb B}^c{\sh S}h O_U$ evidently factors through ${\mathbb B}^c{\sh S}h O_X\{-I\}$.
Now suppose that $U=X{\sh S}etminus Z(I)$ is exactly the complement of the zeroes of $I$. Then $\rho^\dagger$ identifies $i_*{\mathbb B}^c{\sh S}h O_U$ with the sheaf of subschemes $Z{\mathfrak h }ookrightarrow X$ equal to the scheme-theoretic closure of their intersection with $U$ (cf. e.g. \ref{eg-closure}). These subschemes are the $-I$-invariants of ${\mathbb B}^c{\sh S}h O_X$. Indeed, suppose that $f$ is some local function on $X$ such that $fI$ vanishes on $Z$. Then over $U$, $fI=(f)$, that is, $f$ vanishes on $Z\cap U$ and therefore on $Z$.
By lemma \ref{LOC-Zar-construction}, the natural semiring homomorphism
\[ {\mathbb B}^c{\sh S}h O_X\{-I\}\widetilde\rightarrow i_*{\mathbb B}^c{\sh S}h O_U\]
is an isomorphism.
\mathbf{e}nd{eg}
{\sh S}ubsection{Prime spectrum}\label{LOC-prime}
The purpose of this section is to discuss a special case of the general theory of the following section \ref{SKEL}, in which constructions can be made particularly explicit. It therefore perhaps would logically have its place after that section. For this reason, the discussion here is relatively informal.
In algebraic geometry, the underlying space of a formal scheme can be described in terms of open primes. A strong analogy holds in the setting of contracting semirings.
\begin{defn}Let $\alpha$ be a semiring. A \mathbf{e}mph{semiring ideal} $\iota{\mathfrak h }ookrightarrow\alpha$ is an ideal and an $\alpha$-submodule. It is further a \mathbf{e}mph{prime ideal} if $\alpha{\sh S}etminus\iota$ is closed under addition.\mathbf{e}nd{defn}
Let $\alpha$ be a contracting semiring, $p:\alpha\rightarrow{\mathbb B}$ a (continuous) semiring homomorphism. The kernel $p^{-1}(-\infty)$ is an open prime ideal. Conversely, given an open prime ideal $\lie p\trianglelefteq\alpha$, one can define a semiring homomorphism
\[ \alpha\rightarrow{\mathbb B},\quad X\mapsto\left\{
\begin{array}{cc}
-\infty, & X\in\lie p \\
0, & X\notin\lie p
\mathbf{e}nd{array}\right.
\]
This sets up an order-reversing, bijective correspondence between the poset $\Spec_{\lie p}\alpha:=\mathrm{Hom}(\alpha,{\mathbb B})$ and that of open prime ideals $\lie p\triangleleft\alpha$. In other words, every point in the \mathbf{e}mph{prime spectrum} of a contracting semiring is represented by a ${\mathbb B}$-point.
Let us write ${\mathbb D}_{\mathbb B}^1$ for the Sierpinski space, whose underlying set is the Boolean semifield, but equipped with the topology is generated instead by the open set $\{0\}$ instead of the semiring topology. The Sierpinski space underlies the \mathbf{e}mph{unit disc over ${\mathbb B}$}.
We now topologise the prime spectrum of a contracting semiring $\alpha$ weakly with respect to the evaluation maps $\Spec^{\lie p}\alpha\rightarrow{\mathbb D}_{\mathbb B}^1$, defined by identifying the underlying set of ${\mathbb B}$ with that of ${\mathbb D}_{\mathbb B}^1$. In other words, a sub-base for the topology is given by the open sets \[U_X:=\{f:\alpha\rightarrow{\mathbb B}|f(X)=0\}, \]
and $U_{X\vee Y}=U_X\cup U_Y$. This upgrades the prime spectrum to a contravariant functor
\[ \Spec^{\lie p}:\frac{1}{2}\mathbf{Ring}_{\leq0}\rightarrow\mathbf{Top}. \]
The continuous map of prime spectra induced by a homomorphism $f:\alpha\rightarrow\beta$ can be described in terms of prime ideals as
\[ \Spec^{\lie p}f:\beta\triangleright\lie p \mapsto f^{-1}(\lie p)\triangleleft \alpha, \]
just as in the case of formal schemes.
By construction the localisation morphism $\Spec^{\lie p}\alpha\{-S\}\rightarrow\Spec^{\lie p}\alpha$ induces an identification
\[ \Spec^{\lie p}\alpha\{-S\}\cong U_S{\sh S}ubseteq\Spec^{\lie p}\alpha \]
as topological spaces. This allows us to define a presheaf $|{\sh S}h O|$ of semirings on the site ${\sh S}h U_{/\Spec^{\lie p}\alpha}$ of affine subsets of the prime spectrum. By proposition \ref{LOC-Zar-cover}, below, it is actually a sheaf.
In summary, the \mathbf{e}mph{prime spectrum} construction allows us to contravariantly associate to each \mathbf{e}mph{contracting} topological semiring $\alpha$ a topological space $\Spec^\lie{p}\alpha$ equipped with a sheaf of semirings whose global sections are naturally $\alpha$.
\begin{egs}
First, it is of course easy to describe the spectrum of a freely contracting semiring: by the adjoint property, $\mathrm{Hom}({\mathbb B}\{X_1,\ldots,X_k\},{\mathbb B})={\mathbb D}^k_{\mathbb B}:=\prod_{i=1}^k{\mathbb D}^1_{\mathbb B}$ is the \mathbf{e}mph{polydisc of dimension $k$} over ${\mathbb B}$. The open subset defined by $\bigvee_{i=1}^kX_i=0$ is a kind of combinatorial simplex, in the sense that its poset of irreducible closed subsets is isomorphic to that of the faces of a $k$-simplex. See also \S\ref{EGS-poly}.
Similar statements hold for free contracting $H_\vee^\circ$-algebras, where $H_\vee$ is a rank one semifield. Indeed, the unique continuous homomorphism $H_\vee^\circ\rightarrow{\mathbb B}$ induces a homeomorphism
\[ \Spec^{\lie p}\alpha\oplus_{H_\vee^\circ}{\mathbb B} \rightarrow\Spec^{\lie p}\alpha \]
for any $\alpha$ over $H_\vee^\circ$. If $\alpha$ is of finite type, then in particular the set underlying the spectrum is finite.\mathbf{e}nd{egs}
\begin{eg}\label{LOC-prime-Noetherian}
The prime spectrum of a Noetherian semiring is a Noetherian topological space. As such, it has well-behaved notions of dimension and decomposition into irreducible components, cf. \cite[\S\textbf{0}.2]{EGA}. In particular, it is quasi-compact.\footnote{In fact, one can conclude from Zorn's lemma that \mathbf{e}mph{any} prime spectrum is quasi-compact. I omit an argument, since anyway the definitions of this section will ultimately be superseded.}\mathbf{e}nd{eg}
{\sh S}ubsection{Blow-up formula}\label{blowup}
Let $X$ be a formal scheme, $I$ a finite ideal sheaf. The blow-up $p:\widetilde X\rightarrow X$ of $X$ along $I$ is constructed as ${\mathbb P}roj_X R_I$, where $R_I$ is the Rees algebra
\[ R_I:=\bigoplus_{n\in{\mathbb N}}I^nt^n{\sh S}ubseteq {\sh S}h O_X[t]. \] One associates in the usual fashion \cite[\S\textbf{II}.2.5]{EGA} a quasi-coherent sheaf on $\widetilde X$ to any quasi-coherent, graded $R_I$-module on $X$; in particular, if $M$ is quasi-coherent over ${\sh S}h O_X$, then $p^*M$ is associated to $M\mathbin{\otimes}_{{\sh S}h O_X}R_I$. If we write ${\mathbb B}^{(c)}(M;R_I)$ for the set of (finitely generated) \mathbf{e}mph{homogeneous} $R_I$-submodules of $M$, the associated module functor is induces a natural transformation
\[ {\mathbb B}^{(c)}(-;R_I)\rightarrow{\mathbb B}^{(c)}(-;{\sh S}h O_{\widetilde X}) \]
of functors $\mathrm{Mod}_{R_I}\rightarrow\mathrm{Mod}_{{\mathbb B},t}$ over $X$.
By following the algebra through, we can obtain an explicit formula relating the subobjects of quasi-coherent sheaves on $X$ to those of their pullbacks to $\widetilde X$.
The dependence of the associated sheaf to a graded module is only `up to' the irrelevant ideal $R_I^+=\bigoplus_{n>0}I^nt^n$. For example, let $M$ be quasi-coherent and homogeneous over $R_I$, and let $N_1,N_2{\mathfrak h }ookrightarrow M$ be finite, homogeneous submodules. Then $N_1=N_2$ as sections of ${\mathbb B}^c(M;{\sh S}h O_{\widetilde X})$ if and only if
\[ N_i+kR_I^+\leq N_j\text{ for all }i,j \text{ and }k{\mathfrak g }g0 \]
in ${\mathbb B}^c(M;R_I)$.
It is equivalent that the high degree graded pieces $(N_i)_k,k{\mathfrak g }g0$ agree.
In other words, ${\mathbb B}^c(M;R_I)\rightarrow{\mathbb B}^c(M;{\sh S}h O_{\widetilde X})$ descends to an isomorphism
\[ {\mathbb B}^c(M;R_I)/(R_I^+=0)\widetilde\rightarrow {\mathbb B}^c(M;{\sh S}h O_{\widetilde X}). \]
Now suppose that $M$ is quasi-coherent on $X$. The ${\mathbb B}$-module ${\mathbb B}^c(M\mathbin{\otimes} R_I;{\sh S}h O_X)$ of finite, homogeneous ${\sh S}h O_X$-submodules of $M\mathbin{\otimes} R_I$ is itself graded
\[ {\mathbb B}^c(M\mathbin{\otimes} R_I;{\sh S}h O_X)\cong\bigvee_{n\in{\mathbb N}}{\mathbb B}^c(M\mathbin{\otimes} I^n;{\sh S}h O_X)+nT, \]
where $T=(t)$ is a formal variable to keep track of the grading. It is a module over the graded semiring
\[{\mathbb B}^c(R_I;{\sh S}h O_X)\cong\bigvee_{n\in{\mathbb N}}{\mathbb B}^c(I^n;{\sh S}h O_X)+nT \cong \bigvee_{n\in{\mathbb N}}({\mathbb B}^c{\sh S}h O_X)_{\leq nI}+nT\]
in which the irrelevant ideal is written $R_I^+=\bigvee_{n\in{\mathbb Z}_{>0}}n(I+T)$.
By proposition \ref{.5RING-prop-contract},
\[ {\mathbb B}^c(M\mathbin{\otimes} R_I;{\sh S}h O_X)\{R_I^+\}\cong {}^\circ{\mathbb B}^c(M\mathbin{\otimes} R_I;{\sh S}h O_X)\widetilde\rightarrow{\mathbb B}^c(M\mathbin{\otimes} R_I;R_I) \]
in the category of ${\mathbb B}^c(R_I;{\sh S}h O_X)$-modules (cf. def. \ref{.5RING-integers} for notation).
Composing these identifications, we therefore have for any $M$ a factorisation
\[ {\mathbb B}^c(M;{\sh S}h O_X)\rightarrow \left(\bigvee_{n\in{\mathbb N}}{\mathbb B}^c(M\mathbin{\otimes} I^n)+nT\right)\left\{\pm\bigvee_{n\in{\mathbb Z}_{>0}}n(I+T)\right\}
\widetilde\rightarrow{\mathbb B}^c\left(p^*M;{\sh S}h O_{\widetilde X}\right) \]
of ${\mathbb B}^c{\sh S}h O_X$-module homomorphisms. The isomorphism on the right is the general blow-up formula.
In the context of adic spaces and their models, a more elegant form is available.
\begin{prop}[Blow-up formula]\label{LOC-blow-up}Let $X$ be an adic space, $j:X\rightarrow X^+$ a quasi-compact formal model. Let $I\in{\mathbb B}^cj_*{\sh S}h O_X^+$ be an ideal sheaf cosupported away from $X$, i.e.\ such that $j^*I={\sh S}h O_X$. Let $\tilde j:X\rightarrow \widetilde X^+\rightarrow X^+$ be the blow-up of $X^+$ along $I$. Then the pullback homomorphism \[{\mathbb B}^c(j_*{\sh S}h O_X;{\sh S}h O_{X^+})\rightarrow{\mathbb B}^c(\tilde j_*{\sh S}h O_X;{\sh S}h O_{\widetilde X^+})\] is a free localisation at $I$.\mathbf{e}nd{prop}
\begin{proof}
First, the preimage of $I$ on $\widetilde X^+$ is an invertible sheaf, and therefore invertible in ${\mathbb B}^c(\tilde j_*{\sh S}h O_X;{\sh S}h O_{\widetilde X^+})$; hence this semiring homomorphism at least factors though the localisation \[\varphi:{\mathbb B}^c(j_*{\sh S}h O_X;{\sh S}h O_{X^+})[-I]\rightarrow{\mathbb B}^c(\tilde j_*{\sh S}h O_X;{\sh S}h O_{\widetilde X^+}).\] Moreover, $\varphi$ is injective; if two sections $J_i-n_iI,i=1,2$ become identical on $\widetilde X^+$, then by the general blow-up formula, $J_i+kI$ are already equal on $X$ for $k{\mathfrak g }g0$.
Surjectivity, on the other hand, follows from this
\begin{lemma}\label{lemma}If $I$ is finite, then for any $N\in{\mathbb B}^cp^*M$, $N+kI$ is in the image of $p^*$ for $k{\mathfrak g }g0$.\mathbf{e}nd{lemma}
Suppose that $N$ is generated in degrees less than $k$. Then \[N^\prime=\bigvee_{i=0}^kN_i+(k-i)I\] is finite, and satisfies the inequalities
\[ p^*N^\prime \leq N+kI \leq p^*N^\prime + kR^+. \] It is therefore a lift for $N+kI$.
\mathbf{e}nd{proof}
In fact, the proof of this lemma shows more: it gives a recipe for exactly which modules on $X$ pull back to which modules on $\mathrm{Bl}_IX$. Following this recipe yields a generalisation.
First, observe that $\widetilde j_*{\sh S}h O_X$ is the ${\sh S}h O_{\widetilde X^+}$-algebra associated to the graded $R_I$ algebra \[K_I:=j_*{\sh S}h O_X[t]{\sh S}imeq\bigoplus_{n{\mathfrak g }g0}j_*{\sh S}h O_Xt^n\] on
$X^+$. We therefore obtain surjective homomorphisms
\[ {\mathbb B}^c(j_*{\sh S}h O_X;{\sh S}h O_{X^+})[T] \twoheadrightarrow{\mathbb B}^c(K_I;R_I)\twoheadrightarrow {\mathbb B}^c(\widetilde j_*{\sh S}h O_X;{\sh S}h O_{\widetilde X^+}) \]
in which the left-hand arrow associates to a polynomial $\bigvee_{i=0}^kiT+J_i$ the $R_I$-submodule of $K_I$ that the $J_it^i$ generate.
\begin{defn}\label{LOC-def-sta}Let $\alpha{\sh S}ubseteq{\mathbb B}^c(j_*{\sh S}h O_X;{\sh S}h O_{X^+})$ be a subring containing $I$. The \mathbf{e}mph{strict transform semiring} $\widetilde\alpha$ of $\alpha$ is subring of ${\mathbb B}^c(\widetilde j_*{\sh S}h O_X;{\sh S}h O_{\widetilde X^+})$ whose objects can be written as graded $R_I$-submodules of $K_I$ in the form
\[ \bigoplus_{n\in{\mathbb N}}J_nt^n{\sh S}ubseteq K_I \]
with $J_n\in\alpha$. It is the image of $\alpha[T]\rightarrow{\mathbb B}^c(\widetilde j_*{\sh S}h O_X;{\sh S}h O_{\widetilde X^+})$.\mathbf{e}nd{defn}
The strict transform semiring contains the inverse of $I$: it is defined by the formula
\[-(p^*I) {\sh S}imeq \bigoplus_{n{\mathfrak g }g0}I^{n-1}t^n. \]
The argument of lemma \ref{lemma} therefore establishes:
\begin{cor}\label{LOC-strict-transform}The strict transform semiring $\widetilde\alpha$ is a free localisation of $\alpha$ at $I$.\mathbf{e}nd{cor}
{\sh S}ection{Skeleta}\label{SKEL}
{\sh S}ubsection{Spectrum of a semiring}\label{SKEL-spec}
Let $\frac{1}{2}\mathbf{Ring}$ denote the category of Tate semirings (def. \ref{LOC-Tate}; the subscript $T$ is held to be implicit from hereon in), $\mathbf{Sk}^\mathrm{aff}$ its opposite. We say that a morphism $f:X\rightarrow Y$ in $\mathbf{Sk}^\mathrm{aff}$ is an \mathbf{e}mph{open immersion} if it is dual to a bounded localisation
\[ f^{\sh S}harp:|{\sh S}h O_Y|\rightarrow |{\sh S}h O_Y|\{T_i-S_i\}_{i=1}^k \]
of the semiring $|{\sh S}h O_Y|$ dual to $Y$ at finitely many variables $S_i,T_i\in|{\sh S}h O_Y|$.
Paraphrasing lemma \ref{LOC-properties} above:
\begin{lemma}The class of open immersions is closed under composition and base change.\mathbf{e}nd{lemma}
Following the general principles outlined in the preliminaries \S\ref{TOPOS}, and in more detail in \cite{Toen}, we obtain the structure of a Grothendieck site on $\mathbf{Sk}^\mathrm{aff}$ generated by those finite canonical covers of the form
\[ \{U_i\rightarrow X\}_{i=1}^k \]
where $U_i\rightarrow X$ is an open immersion for each $i\in[k]$. The tautological presheaf $|{\sh S}h O|$ of Tate semirings on $\mathbf{Sk}^\mathrm{aff}$ is a sheaf, by the definition of canonical coverings.
\begin{defns}\label{SKEL-skel}The category $\mathbf{Sk}^\mathrm{aff}$, considered equipped with this topology, is called the \mathbf{e}mph{skeletal site}. Its sheaf category is denoted $\mathbf{Sk}\hspace{4pt}\widetilde{}\hspace{4pt}$.
An \mathbf{e}mph{affine skeleton}, resp. \mathbf{e}mph{skeleton}, is a representable, resp. locally representable sheaf on the skeletal site (cf. def. \ref{TOPOS-def}, \cite[def. 2.15]{Toen}). If $\alpha$ is a semiring, the dual affine skeleton is called its \mathbf{e}mph{spectrum} and denoted $\Spec\alpha$ The category of skeleta is denoted $\mathbf{Sk}$.\mathbf{e}nd{defns}
Of course, the Yoneda embedding identifies $\mathbf{Sk}^\mathrm{aff}$ with the category of affine skeleta.
More general arguments (cf. \S\ref{TOPOS}) equip each skeleton $X$ with a small topos $X\hspace{4pt}\widetilde{}\hspace{4pt}$, equivalent to the category of sheaves on a uniquely determined sober topological space with lattice of open sets ${\sh S}h U_{/X}$. I will abuse notation and denote this topological space also by $X$.
This fact allows us to alternatively interpret the Grothendieck site structure on $\mathbf{Sk}^\mathrm{aff}$ in terms of a contravariant functor
\[ \mathbf{Sk}^\mathrm{aff}\rightarrow\mathbf{Top} \]
into the category of sober, quasi-compact, and quasi-separated topological spaces equipped with a sheaf of Tate semirings.
A skeleton is then a topological space $X$ equipped with a sheaf $|{\sh S}h O_X|$ of Tate semirings, locally isomorphic to an affine skeleton. The sections of $|{\sh S}h O_X|$ may be called \mathbf{e}mph{convex functions} on $X$.
\begin{prop}[\cite{Toen}]An affine skeleton is qcqs and sober, and affine open subsets form a basis for the topology.
The category of skeleta has all fibre products.\mathbf{e}nd{prop}
\begin{eg}\label{eg-cpa}The spectrum ${\mathbb D}elta_{[a,b]}=\Spec\mathrm{CPA}_{\mathbb Z}([a,b],{\mathbb R}_\vee)$ of the semiring of convex, piecewise-affine functions on an interval $[a,b]{\sh S}ubseteq{\mathbb R}$ with rational endpoints (cf. e.g. \ref{eg-int}) is homeomorphic to a certain Grothendieck site structure on the poset of closed subintervals of ${\mathbb D}elta{\sh S}ubset{\mathbb R}$.
Indeed, we already observed in example \ref{eg-int} that every subdivision of ${\mathbb D}elta_{[a,b]}$ is determined by a subdivision of $[a,b]$ as a rational polyhedron; meanwhile, by the cellular cover formula of the next section (proposition \ref{LOC-Zar-cover}), the cellular topology of ${\mathbb D}elta_{[a,r_1,\ldots,r_k,b]}$ is generated by the inclusions $[r_i,r_{i+1}]\rightarrow[r_0,r_k]$.
It remains to say when a collection of affine subsets $U_j=\{[a_{ji},b_{ji}]\}_{i=1}^{k_j}$ covers $X$.
\begin{conj}The $U_j$ cover $X$ if and only if $[a,b]=\bigcup_{i,j}[a_{ji},b_{ji}]$ and $(a,b)=\bigcup_{i,j}(a_{ji},b_{ji})$.\mathbf{e}nd{conj}
With the cellular cover fomula, the only part in question is the condition for a family of subdivisions to cover ${\mathbb D}elta$. The proposed criterion says that a collection of subdivisions covers if and only if there are no common `kink' points, that is, if \[\bigcap_j\bigcup_{i=1}^{k_j}\{a_{ji},b_{ji}\}=\mathbf{e}mptyset.\] Indeed, in that case, the intersection over $j$ (in, say, the set of continuous functions) of the semirings of piecewise-affine functions convex on $U_j$ is exactly the set of such functions convex on ${\mathbb D}elta$. In other words,
\[ \mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee) \rightarrow\prod_i|{\sh S}h O_{U_i}| \rightrightarrows\prod_{i,j}|{\sh S}h O_{U_i\cap U_j}| \]
is an equaliser of semirings. I do not know how to show that this equaliser is universal.
As was pointed out in e.g. \ref{eg-int'}, the spectrum of $\mathrm{CPA}_*({\mathbb R},{\mathbb R}_\vee)$ consists of a single point. One can obtain a better model for the affine real line ${\mathbb R}$ as the increasing union
\[ \mathrm{sk}{\mathbb R}:=\bigcup_{a\rightarrow\infty}[-a,a] \]
in $\mathbf{Sk}$. Like the analytic torus over a non-Archimedean field, it is not quasi-compact.\mathbf{e}nd{eg}
\begin{eg}[Dichotomy]\label{circnorm}The skeleton constructed in the above example \ref{eg-cpa}, although relatively easy to describe, is not finitely presented over $\Spec{\mathbb R}_\vee$ (cf. \ref{non-eg}). In the vein of example \ref{non-eg'}, we can replace $\mathrm{CPA}_{\mathbb Z}([a,b],{\mathbb R}_\vee)$ with its finitely presented cousin
\[ {\mathbb R}_\vee\{[a,b]\}:={\mathbb R}_\vee\{X-b,a-X\}. \]
They are related by an (infinitely presented) morphism $\Spec\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)\rightarrow\Spec{\mathbb R}_\vee\{[a,b]\}$.
We also saw in \ref{non-eg'} that this morphism is not a homeomorphism, and that in fact the topology of the target is difficult to describe.
It seems to be possible to modify the definition of skeleton, by introducing another condition into our definition of semiring, so as to make this morphism a homeomorphism. This condition is the semiring version of the algebraic notion of relative normality (integral closure of $A^+$ in $A$), which is used in non-Archimedean geometry to define a good Spec functor. However, I wish to defer a serious pursuit of this approach to a later paper, since this issue does not directly affect any of the results here.
The examples in \S\ref{EGS} all more closely resemble finitely presented skeleta like $\Spec{\mathbb R}_\vee\{{\mathbb D}elta\}$, but I will often only describe open subsets in a way that depends only on their pullbacks to a `geometric' counterpart $\Spec\mathrm{CPA}_{\mathbb Z}({\mathbb D}elta,{\mathbb R}_\vee)$.
\mathbf{e}nd{eg}
{\sh S}ubsection{Integral skeleta and cells}
\begin{defn}A skeleton that admits a covering by spectra of contracting semirings is said to be \mathbf{e}mph{integral}. The full subcategory of $\mathbf{Sk}$ whose objects are integral is denoted $\mathbf{Sk}^\mathrm{int}$.\mathbf{e}nd{defn}
The tropical site $\mathbf{Sk}^\mathrm{aff}$ carries a tautological sheaf $|{\sh S}h O|^\circ$ of contracting semirings, whose sections over $\Spec\alpha$ are $\alpha^\circ$. Taking the spectrum defines a functor \[ \Spec|{\sh S}h O|^\circ: \mathbf{Sk}^\mathrm{aff}\rightarrow\mathbf{Sk}^\mathrm{int}. \] Moreover, any covering of an object $\Spec|{\sh S}h O|^\circ(X)$ lifts, by base extension $|{\sh S}h O|^\circ\rightarrow|{\sh S}h O|$, to a covering of $X$, that is, $\Spec|{\sh S}h O|^\circ$ is cocontinuous. It therefore extends to the pushforward functor of a morphism
\[ (-)^\circ:\mathbf{Sk}\hspace{4pt}\widetilde{}\hspace{4pt}\rightarrow(\mathbf{Sk}^\mathrm{int})\hspace{4pt}\widetilde{}\hspace{4pt} \]
of the corresponding topoi.
This functor takes a skeleton $X$ to an integral skeleton $X^\circ$ if and only if there exists an affine open cover $X=\bigcup_iU_i$ such that $(U_i\cap U_j)^\circ{\mathfrak h }ookrightarrow U_i^\circ$ is an open immersion, in which case $U_\bullet^\circ$ provides an atlas for $X^\circ$. In algebraic terms, we need that $X$ admit an affine atlas each of whose structure maps is dual to a localisation $\alpha\rightarrow\alpha\{T-S\}$ that restricts to a localisation $\alpha^\circ\rightarrow\alpha\{T-S\}^\circ\cong\alpha^\circ\{T-S\}$ of the semiring of integers. This occurs if and only if the localisation is cellular, that is, if (up to isomorphism) $T$ is invertible in $\alpha^\circ$ and therefore zero.
\begin{defns}\label{SKEL-def-cel}An open immersion of skeleta is \mathbf{e}mph{cellular} if it is locally dual to a cellular localisation of semirings.
A skeleton that admits a cover by affine, cellular-open subsets is said to be a \mathbf{e}mph{cell complex}. In particular, any affine skeleton is a cell complex.
If \mathbf{e}mph{every} open subset is cellular, it is a \mathbf{e}mph{spine}. By the discussion above, any integral skeleton is a spine.
The categories of spines, resp. cell complexes are denoted $\mathbf{Sk}^\mathrm{sp}{\mathfrak h }ookrightarrow\mathbf{Sk}^\mathrm{cel}$.
There is a functor
\[ (-)^\circ:\mathbf{Sk}^\mathrm{cel}\rightarrow\mathbf{Sk}^\mathrm{int}, \]
left adjoint to the inclusion, which associates to a cell complex $X$ its \mathbf{e}mph{integral model} $X^\circ$. The unit of the adjunction is a morphism $j:X\rightarrow X^\circ$. The cellular open subsets of $X$ are those pulled back along $j$.\mathbf{e}nd{defns}
We have access to a reasonably concrete description of the `cellular topology'.
\begin{lemma}Let $\alpha$ be a semiring, $\{S_i\}_{i=1}^k{\sh S}ubseteq\alpha^\circ$ a finite list of contracting elements. Write $S=\bigvee_{i=1}^kS_i$. Then
\[ \alpha\{-S\}\rightarrow\prod_i\alpha\{-S_i\}\rightrightarrows\prod_{i,j}\alpha\{-S_i,-S_j\}\]
is a universal equaliser of semirings.\mathbf{e}nd{lemma}
\begin{proof}The lemma \ref{LOC-Zar-construction} yields an embedding of forks
\[ \xymatrix{ \alpha \ar[r]\ar[d] & \prod_i\alpha\{-S_i\} \ar@<2pt>[r]\ar@<-2pt>[r]\ar[d] & \prod_{i,j}\alpha\{-S_i,-S_j\}\ar[d] \\
{\mathbb B}\alpha \ar[r] & \prod_i{\mathbb B}\alpha \ar@<2pt>[r]\ar@<-2pt>[r] & \prod_{i,j}{\mathbb B}\alpha } \]
in which the $i$th arrow in the lower row takes an ideal to its $-S_i$-span.
Let $\mathrm{eq}{\sh S}ubseteq\prod_i{\mathbb B}\alpha$ denote the equaliser of the second row, $f:{\mathbb B}\alpha\rightarrow\mathrm{eq}$ the natural ${\mathbb B}$-module homomorphism. An element of $\mathrm{eq}$ is a finite list $\{\iota_i\}_{i=1}^k$ of $-S_i$-invariant ideals, such that for each $i$ and $j$ the $-S_j$-span of $\iota_i$ is equal to the $-S_i$-span of $\iota_j$. The right adjoint $f^\dagger$ to $f$ sends such a list to their intersection in $\alpha$.
Since localisation commutes with base change, the fork in the statement is a universal equaliser as soon as it is an equaliser. By \ref{LOC-Zar-construction}, it is equivalent to show that $f^\dagger$ identifies $\mathrm{eq}$ with the set of $-S$-invariant ideals of $\alpha$.
On the one hand, the elements $f^\dagger(\mathrm{eq})$ are certainly $-S$-invariant. Indeed, suppose $X+nS\in\iota=\cap_i\iota_i$. Then $X+nS_i\leq X+nS\in\iota_i$, and so $X\in\iota_i$ for each $i$. Furthermore, since the $-S_i$-span of $\iota_j$ contains $\iota_i$, if $X\in\iota_i$, then $X+nS_i\in\iota_j$ for some $n$. Therefore, for $n{\mathfrak g }g0$, $X+nS_i\in\iota$, and $\iota_i$ is the $-S_i$-span of $\iota$.
Conversely, suppose that $\iota$ is $-S$-invariant. Let $X\in f^\dagger f\iota{\sh S}upseteq\iota$. Then for $n{\mathfrak g }g0$, $X+nS_i\in\iota$ for all $i$, and therefore $X+nS\in\iota$, so $X\in\iota$. Therefore, $f$ and $f^\dagger$ are inverse.\mathbf{e}nd{proof}
In geometric terms:
\begin{prop}[Cellular cover formula]\label{LOC-Zar-cover}Let $\alpha$ be a semiring, $\{S_i\}_{i=1}^k{\sh S}ubseteq\alpha^\circ$ a finite list of contracting elements. Write $S=\bigvee_{i=1}^kS_i$. Then
\[ \Spec\alpha\{-S\}=\bigcup_{i=1}^k\Spec\alpha\{-S_i\} \]
as subsets of $\Spec\alpha$.\mathbf{e}nd{prop}
\begin{cor}\label{SKEL-qco=aff}Let $U$ be a quasi-compact cell complex. If $U$ can be embedded as an open subset of an affine skeleton, then $U$ is affine.\mathbf{e}nd{cor}
In fact, this result can be greatly improved.
\begin{thm}\label{SKEL-qc=aff}Let $X$ be a quasi-separated cell complex, $j:X\rightarrow X^\circ$ its integral model. Let us confuse $X^\circ$ with its site ${\sh S}h U^\mathrm{qc}_{/X^\circ}$ of quasi-compact open subsets. Then:
\begin{enumerate}\item ${\mathbb B}\left(j_*|{\sh S}h O_X|\right)$ is flabby;
\item $X$ is affine if and only if it is quasi-compact and $j_*|{\sh S}h O_X|$ is flabby.\mathbf{e}nd{enumerate}\mathbf{e}nd{thm}
\begin{cor}Any quasi-compact, integral skeleton with Noetherian structure sheaf is affine.\mathbf{e}nd{cor}
\begin{proof}In this case $X=X^\circ$ and $|{\sh S}h O_X|={\mathbb B}{\sh S}h O_X|$.\mathbf{e}nd{proof}
\begin{cor}Let $H_\vee$ be a rank one semifield. Any integral skeleton finitely presented over $\Spec H_\vee^\circ$ is affine.\mathbf{e}nd{cor}
\begin{proof}Such admits a model over some finitely generated subring of $H_\vee$.\mathbf{e}nd{proof}
If we make the assumption that all $H_\vee$-algebras $\alpha$ satisfy $\alpha\cong\alpha^\circ\oplus_{H_\vee^\circ}H_\vee$, then this last corollary applies also to any cell complex finitely presented over $\Spec H_\vee$ (which is, in this case, simply the base change of its integral model).
\begin{proof}[Proof of \ref{SKEL-qc=aff}]
Let $f:U_\bullet\twoheadrightarrow V$ be a finite, affine, cellular hypercover of some quasi-compact $V{\sh S}ubseteq X$. By corollary \ref{SKEL-qco=aff}, we may in fact assume that $U_\bullet$ is the nerve of an ordinary cover ${\sh S}qcup_if_i:\coprod_{i=1}^kU_i\twoheadrightarrow X$. Let $\alpha={\mathbb G}amma(V,|{\sh S}h O_X|)$. We have an equaliser
\[ \alpha\rightarrow \prod_{i=1}^k\alpha_i\rightrightarrows \prod_{i,j=1}^k\alpha_i\{-S_{ij}\} \]
commuting with isomorphisms $\alpha_i\{-S_{ij}\}\cong\alpha_j\{-S_{ji}\}$. There are unique elements $S_j\in\alpha$ whose images in $\alpha_i$ are $S_{ij}$.
Since $\alpha_i\rightarrow \alpha_i\{-S_{ij}\}$ is surjective, its right ind-adjoint is injective. The compositions
\[ \alpha_i {\sh S}tackrel{\rho}{\rightarrow} \alpha_i\{-S_{ij}\} \widetilde\rightarrow \alpha_j\{-S_{ji}\} {\sh S}tackrel{\rho^\dagger}{\rightarrow} {\mathbb B}\alpha_j \]
therefore together yield a section ${\mathbb B}\alpha_i\rightarrow|{\mathbb B}\alpha_\bullet|$ of the projection. Since this holds for any quasi-compact $V$, ${\mathbb B}\left(j_*|{\sh S}h O_X|\right)$ is flabby.
Now set $V=X$. For the second part, it will be enough to show that each $f_i$ is a cellular localisation of $\alpha$ at $S_i$, since in this case the equaliser will be a covering, and hence induce an isomorphism $\Spec\alpha\widetilde\rightarrow X$. We will show this using the characterisation \ref{LOC-Zar-construction}.
Certainly, $f_i^\dagger:{\mathbb B}\alpha_i\rightarrow{\mathbb B}\alpha$ has image in the set of $-S_i$-invariant ideals. Since $|{\sh S}h O_X|$ is flabby, $f_i^\dagger$ is also injective. We need only show that it is surjective. The argument is based on two lemmata.
\begin{lemma}\label{SKEL-Zar-inv}Let $f:\alpha\rightarrow\beta$, $S\in\alpha^\circ$. If $f$ is surjective, ${\mathbb B} f$ preserves $-S$-invariance.\mathbf{e}nd{lemma}
\begin{lemma}\label{SKEL-push-pull}Let $f:\mu\rightarrow\mu\{-S\}$ be a cellular localisation of $\alpha$-modules, $g:\mu\rightarrow\nu$ a surjective homomorphism. The diagram
\[\xymatrix{ \mu \ar[d]^g & \mu\{-S\}\ar[d]^g\ar[l]_{f^\dagger} \\ \nu & \nu\{-S\}\ar[l]_{f^\dagger} }\] commutes.\mathbf{e}nd{lemma}
\begin{proof}The right adjoints embed $\mu\{-S\},\nu\{-S\}$ into ${\mathbb B}\mu,{\mathbb B}\nu$ as the set of $-S$-invariant ideals, which are preserved under $g$ by \ref{SKEL-Zar-inv}.
\mathbf{e}nd{proof}
Let $\iota{\mathfrak h }ookrightarrow\alpha$ be a $-S_i$-invariant ideal, $\iota_\bullet$ its image in $\alpha_\bullet$. By \ref{SKEL-Zar-inv}, $\iota_\bullet$ is $-S_{\bullet i}$-invariant, and hence
\[ \iota_\bullet=f_i^\dagger f_i\iota_\bullet=f_i^\dagger f_\bullet\iota_i=f_\bullet f_i^\dagger\iota_i\]
where the last equality follows from \ref{SKEL-push-pull}. Therefore $\iota=|\iota_\bullet|=f_i^\dagger\iota_i$.
\mathbf{e}nd{proof}
\
Let $\alpha$ be a contracting semiring. The arguments of \S\ref{LOC-prime} show that we have a natural continuous functor
\[ {\sh S}h U_{/\Spec\alpha} \rightarrow {\sh S}h U_{/\Spec^\lie{p}\alpha} \]
and hence a morphism of semiringed spaces $\Spec^\lie{p}\alpha\rightarrow\Spec\alpha$.
\mathbf{e}mph{If} $\Spec^\lie{p}\alpha$ is quasi-compact for all $\alpha\in\frac{1}{2}\mathbf{Ring}_{\leq0}$, then this is an isomorphism by the unicity of canonical topologies.
This is true for Noetherian semirings by \ref{LOC-prime-Noetherian}. The general case is implied by Zorn's lemma.
\begin{prop}\label{contentious}Let $\alpha$ be a (Noetherian) contracting semiring. Then $\Spec^\lie{p}\alpha\widetilde\rightarrow\Spec\alpha$, as semiringed spaces.\mathbf{e}nd{prop}
Topologising as before the set $X({\mathbb B})$ weakly with the respect to evaluations $X({\mathbb B})\rightarrow{\mathbb D}^1_{\mathbb B}$, we obtain:
\begin{cor}\label{cool}Let $X$ be an integral skeleton. Then $X({\mathbb B})\rightarrow X$ is a homeomorphism.\mathbf{e}nd{cor}
{\sh S}ubsection{Universal skeleton of a formal scheme}
\begin{eg}Let us return to the quasi-compact, coherent space $X$ of example \ref{eg-topol} and its semiring $|{\sh S}h O_X|$ of quasi-compact open subsets. We saw there that the inclusion
$S{\mathfrak h }ookrightarrow X$ of a quasi-compact subspace induces a localisation $|{\sh S}h O_X|\rightarrow|{\sh S}h O_S|$ at $S$. The cellular cover formula \ref{LOC-Zar-cover} implies that if $S=\bigcup_{i=1}^kS_i$ is a finite union of open subsets, then
\[ \Spec|{\sh S}h O_S|=\bigcup_{i=1}^k\Spec|{\sh S}h O_{S_i}|. \]
It follows that the functor
\[ \Spec|{\sh S}h O|:{\sh S}h U_{/X}^\mathrm{qc}\rightarrow{\sh S}h U_{/\mathrm{sk}X}^\mathrm{qc} {\mathfrak h }ookrightarrow \mathbf{Sk}_{/\mathrm{sk}X}^\mathrm{aff} \]
preserves coverings, and hence induces a homeomorphism of $X$ with $\mathrm{sk}X:=\Spec|{\sh S}h O_X|$.
As we have seen (cor. \ref{cool}), every point of $\mathrm{sk}X$ is represented uniquely by a ${\mathbb B}$-point. In fact, the stalk of the structure sheaf $|{\sh S}h O_X|$ at any point $p\in\mathrm{sk}X$ is canonically isomorphic to ${\mathbb B}$, with $0$ (resp. $-\infty$) represented by an open subset containing (resp. not containing) $p$.
Under the homeomorphism $\mathrm{sk}X\widetilde\rightarrow X$, $|{\sh S}h O_X|$ can be identified with the semiring $C^0(X,{\mathbb D}_{\mathbb B}^1)$ of continuous maps from $X$ to the Sierpinski space ${\mathbb D}^1_{\mathbb B}$, that is, with the set of \mathbf{e}mph{indicator functions} of open subsets.\mathbf{e}nd{eg}
It follows from the functoriality of the sheaves ${\mathbb B}^c{\sh S}h O_X$ associated on formal schemes $X$, as outlined in sections \ref{SPAN}, \ref{TOP}, \ref{.5RING}, that they assemble to a sheaf $|{\sh S}h O|$ of contracting semirings on the large formal site (in fact, with the fpqc topology). Its sections over a quasi-compact, quasi-separated formal scheme $X$ are the semiring of finite type ideal sheaves on $X$. This can be thought of as a \mathbf{e}mph{geometric} version of the sheaf $|{\sh S}h O|$ of the above example, which is simply an avatar of the correspondence between (certain) frames and locales.
Let $X$ be any formal scheme, ${\sh S}h U_{/X}^\mathrm{qc}$ its corresponding small site, $|{\sh S}h O_X|$ the restriction of $|{\sh S}h O|$. The Zariski-open formula \ref{LOC-Zar-open} implies that if $V{\mathfrak h }ookrightarrow X$ is a quasi-compact open subset, then $|{\sh S}h O_X|$ puts the (necessarily cellular) bounded localisations of $|{\sh S}h O_X|(V)$ into one-to-one correspondence with quasi-compact Zariski-open subsets of $V$. The cellular cover formula implies that if $I_i\in|{\sh S}h O_X|(V)$ is a finite family of finite-type ideal sheaves on $X$, $U_i=V{\sh S}etminus Z(I_i)$ the complementary quasi-compact opens, and
\[ U=V{\sh S}etminus Z\left(\bigvee_{i=1}^kI_i\right)=\bigcup_{i=1}^kU_i, \]
then
\[ \Spec|{\sh S}h O_X|(U)=\bigcup_{i=1}^k\Spec|{\sh S}h O_X|(U_i) \]
as subsets of $\Spec|{\sh S}h O_X|(V)$. In other words, $U\mapsto\Spec|{\sh S}h O_X|(U)$ defines a cover-preserving equivalence of categories between ${\sh S}h U_{/\Spec|{\sh S}h O_X|(U)}$ and ${\sh S}h U_{/X}$. This proves:
\begin{lemma}Let $X$ be a quasi-compact formal scheme. Then $\Spec|{\sh S}h O(X)|\rightarrow X$ is a homeomorphism.\mathbf{e}nd{lemma}
\begin{thm}\label{SKEL-min}Let $X$ be any formal scheme. Then $\mathrm{sk}X:=(X,|{\sh S}h O_X|)$ is a skeleton.\mathbf{e}nd{thm}
Of course, $\mathrm{sk}X$ is actually an \mathbf{e}mph{integral} skeleton.
{\sh S}ubsection{Universal skeleton of an adic space}
Let $\mathbf{Ad}^\mathrm{qcqs}$ denote the quasi-compact, quasi-separated adic site. The sheaves ${\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)$ on each adic space $X$ assemble to a sheaf $|{\sh S}h O|$ of Tate semirings on $\mathbf{Ad}^\mathrm{qcqs}$, extending the one with the same name introduced in the previous section.
Note that, unlike the case of formal schemes, this sheaf does not restrict to the presheaf
\[ |{\sh S}h O|^\mathrm{pre}={\mathbb B}^c:\mathbf{nA}\rightarrow\frac{1}{2}\mathbf{Ring} \]
defined in terms of the section spaces of ${\sh S}h O$, since the ${\sh S}h O^+$-submodules it parametrises are, on the whole, not quasi-coherent. Naturally, $|{\sh S}h O|$ is the sheafification of $|{\sh S}h O|^\mathrm{pre}$.
In this section, we will derive the following generalisation of theorem \ref{SKEL-min}:
\begin{thm}\label{SKEL-main}Let $X$ be any adic space. Then $\mathrm{sk}X:=(X,|{\sh S}h O_X|)$ is a skeleton.\mathbf{e}nd{thm}
\begin{defn}The skeleton $\mathrm{sk}X$ is called the \mathbf{e}mph{universal skeleton} of $X$.\mathbf{e}nd{defn}
The proof rests on a limit formula, following from the fundamental limit \ref{important} of \S\ref{ADIC}.
\begin{lemma}\label{SKEL-lim}Let $X$ be a qcqs adic space. Then
\[ {\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)\cong\colim_{j\in\mathbf{Mdl}(X)}{\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+) \]
in $\frac{1}{2}\mathbf{Ring}_t$.\mathbf{e}nd{lemma}
\begin{proof}Indeed, the limit formula states explicitly that $j:X\widetilde\rightarrow\lim X^+$ as locales, and that ${\sh S}h O_X^+=\colim j^*j_*{\sh S}h O_X^+$ as sheaves on $X$. Any finitely generated ideal of ${\sh S}h O_X^+$ is therefore pulled back from some level $j_*{\sh S}h O_X^+$.
Since, by ${}^+$normality, the morphisms $j^*j_*{\sh S}h O_X^+\rightarrow{\sh S}h O_X$ are injective, then any two such ideals have the same image in ${\sh S}h O_X$ if and only if they agree on any cover, that is, on any model on which they are both defined.\mathbf{e}nd{proof}
\begin{proof}[Proof of \ref{SKEL-main}]Let $X\in\mathbf{Ad}^\mathrm{qcqs}$. We need to show that the localisations of $|{\sh S}h O|(X)$ are in one-to-one correspondence with the quasi-compact subsets of $X$.
Let $S{\mathfrak h }ookrightarrow X$ be a quasi-compact subset. There exists a formal model $j:X\rightarrow X^+$ and open subset $S^+{\mathfrak h }ookrightarrow X^+$ such that $S\cong X\times_{X^+}S^+$, and
\[ {\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+)\rightarrow {\mathbb B}^c(j_*{\sh S}h O_S;j_*{\sh S}h O_S^+) \]
is a cellular localisation at some (any) finite ideal $I$ cosupported on $X^+{\sh S}etminus S^+$. This remains true when we modify $X^+$. Since $S$ is quasi-compact, every formal model $j_S:S\rightarrow S^+$ can be extended to a model $j$ of $X$, and so the colimit formula \ref{SKEL-lim} implies that
\[ |{\sh S}h O|(X)\cong\colim_{j\in\mathbf{Mdl}(X)_{/X^+}}{\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+)\rightarrow \colim_{j\in\mathbf{Mdl}(X)_{/X^+}}{\mathbb B}^c(j_{S*}{\sh S}h O_S;j_{S*}{\sh S}h O_S^+)\cong|{\sh S}h O|(S) \]
is a localisation at $I$.
Conversely, any $I\in|{\sh S}h O|(X)$ is representable by some finite ideal sheaf on a qcqs formal model $j:X\rightarrow X^+$ of $X$, whence $|{\sh S}h O|(X)\{-I\}\cong|{\sh S}h O|(U)$, where $U\cong X\times_{X^+}(X^+{\sh S}etminus Z(I))$.
The cellular cover formula \ref{LOC-Zar-cover} shows that this correspondence preserves coverings, and hence induces a homeomorphism of $X$ with $\Spec|{\sh S}h O|(X)$.\mathbf{e}nd{proof}
The argument also shows:
\begin{cor}The universal skeleton of an adic space is a spine (def. \ref{SKEL-def-cel}).\mathbf{e}nd{cor}
{\sh S}ubsubsection{Real points of the universal skeleton}
Let $X$ be any skeleton. We can topologise the set $X({\mathbb R}_\vee)$ of \mathbf{e}mph{real points} of $X$ with respect to the evaluation maps $f:X({\mathbb R}_\vee)\rightarrow{\mathbb R}_\vee$ associated to functions $f\in|{\sh S}h O_X|$, where on the right-hand side ${\mathbb R}_\vee$ is equipped with the usual \mathbf{e}mph{order} topology (rather than the semiring topology). If $X$ is defined over some rank one semifield $H_\vee{\sh S}ubseteq{\mathbb R}_\vee$, then we may rigidify by considering ${\mathbb R}_\vee$-points over $H_\vee$; the subset $X_{H_\vee}({\mathbb R}_\vee){\sh S}ubseteq X({\mathbb R}_\vee)$ similarly acquires a topology.
The natural map $X({\mathbb R}_\vee)\rightarrow X$ is often discontinuous with respect to this topology.
If $X$ is now an adic space, we can consider (following e.g. \ref{eg-vals}) the space $\mathrm{sk}X({\mathbb R}_\vee)$ of real points of the universal skeleton as a space of \mathbf{e}mph{real valuations} of ${\sh S}h O_X$. For this to be geometrically interesting, we usually want to consider this equipped with some $H_\vee$-structure. For instance, if $X$ is Noetherian, then $\mathrm{sk}X$ carries a canonical `maximal' morphism to $\Spec{\mathbb Z}_\vee$ (e.g. \ref{eg-adic-Noetherian}). The corresponding valuations send irreducible topological nilpotents to $-1$. Alternatively, if $X$ is defined over a rank one non-Archimedean field $K\rightarrow H_\vee$, then $\mathrm{sk}X$ is defined over $H_\vee$, and the real points are valuations extending the valuation of the ground field.
Where there is no possibility of confusion, I will abbreviate $\mathrm{sk}X_{H_\vee}({\mathbb R}_\vee)$ to $X({\mathbb R}_\vee)$.
To the reader familiar with analytic geometry in the sense of Berkovich \cite{Berketale} the following theorem will come as no surprise:
\begin{thm}\label{thm-berk}Let $X^\mathrm{Berk}$ be a Hausdorff Berkovich analytic space over a non-Archimedean field $K$, $X$ the corresponding quasi-separated adic space \mathbf{e}mph{\cite[thm. 1.6.1]{Berketale}}. The composition
\[ X({\mathbb R}_\vee)\rightarrow X\rightarrow X^\mathrm{Berk} \]
is a homeomorphism.\mathbf{e}nd{thm}
\begin{proof}It is enough to show that the restriction of this map to every affinoid subdomain is a homeomorphism. This follows from the definitions and the identity
\[ \mathrm{Hom}({\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+),{\mathbb R}_\vee)\widetilde\rightarrow\mathrm{Hom}({\mathbb B}^c(A;A^+),{\mathbb R}_\vee) \]
for affine $X=\Spa A$, which holds because ${\mathbb B}^c({\sh S}h O_X;{\sh S}h O_X^+)$ is a localisation of ${\mathbb B}^c(A;A^+)$.\mathbf{e}nd{proof}
\
\begin{prop}\label{SKEL-rat}Let $X$ be integral and adic over an adic space with a Noetherian formal model. Every function on $\mathrm{sk}X$ is determined by its rational values.\mathbf{e}nd{prop}
In classical terms this means the following: let $j:X\rightarrow X^+$ be a formal model of $X$, $Z_1,Z_2{\mathfrak h }ookrightarrow X^+$ two finitely presented subschemes; then if for all continuous rational valuations $\val:{\sh S}h O_X\rightarrow{\mathbb Q}_\vee$, $\val(I_1)=\val(I_2)$, then $Z_1=Z_2$ after some further blow-up of $X^+$.
\begin{proof}The statement is clear when $X^+$ is Noetherian and the $Z_i$ are both supported away from $X$; in this case, we may blow-up each $Z_i$ to obtain Cartier divisors, which by the Noetherian hypothesis factorise into prime divisors. Knowing that the $Z_i$ are Cartier divisors, they are therefore determined by the multiplicities of each prime divisor therein, that is, the values of local functions for the $Z_i$ under the corresponding discrete valuations.
Moreover, any formal subschemes $Z_i$ are, by definition, formal inductive limits of subschemes supported away from $X$, and so determined by a (possibly infinite) set of valuations.
Finally, for the general case we may assume that $Z_i$ are pulled back from some Noetherian formal scheme $X^+\rightarrow Y^+$ over which $X^+$ is integral. Since rational valuations admit unique extensions along integral ring maps, the discrete valuations on ${\sh S}h O_Y$ determining the $Z_i$ extend to rational valuations on ${\sh S}h O_X$.\mathbf{e}nd{proof}
\begin{cor}The universal skeleton of an adic space is cancellative.\mathbf{e}nd{cor}
\begin{cor}Let $X$ be as in \ref{SKEL-rat}. Then $X({\mathbb R}_\vee)$ satisfies the conclusion of Urysohn's lemma.\mathbf{e}nd{cor}
\begin{proof}The proposition implies that $|{\sh S}h O_X|$ injects into the the set $C^0(X({\mathbb R}_\vee),{\mathbb R}_\vee)$ of continuous, real-valued functions. By definition, two points of $X({\mathbb R}_\vee)$ agree only every element of $|{\sh S}h O_X|$ takes the same value at both points. In other words, distinct points are separated by continuous functions.\mathbf{e}nd{proof}
This last result can be understood as a cute proof of the corresponding property for Hausdorff Berkovich spaces, that is, that they are \mathbf{e}mph{completely} Hausdorff.
{\sh S}ubsection{Shells}
Let $X$ be an adic space. The universal skeleton of $X$ is a spine, so that any function with an admissible lower bound is invertible. If, for example, the skeleton is adic over ${\mathbb Z}_\vee$, then this is the same as every bounded function being invertible. Intuitively, this means that we have not defined a good notion of convexity for functions on $\mathrm{sk}X$.
We obtain a more restrictive notion of convexity by embedding $\mathrm{sk}X$ into a \mathbf{e}mph{shell}, that is, a skeleton $B$ inside which $\mathrm{sk}X$ is a subdivision - in fact, the intersection of all subdivisions. At the level of the Berkovich spectrum $X({\mathbb R}_\vee)$, this is akin to choosing a kind of `pro-affine structure' (a concept that I do not define here).
Suppose that $X$ is qcqs, and let $j:X\rightarrow X^+$ be a formal model of $X$. Write
\[ \mathrm{sk}(X;X^+):=\Spec{\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+), \] for the \mathbf{e}mph{$X^+$-shell} of $X$. It is an affine skeleton whose integral model is the universal skeleton $\mathrm{sk}X^+$ of $X^+$.
More generally, if $X$ is any adic space admitting a formal model $X^+$, then a qcqs cover $U_\bullet^+\twoheadrightarrow X^+$ with generisation $U_\bullet=X\times_{X^+}U^+_\bullet$ gives rise to an \mathbf{e}mph{$X^+$-shell}
\[ \mathrm{sk}(X;X^+):=|\mathrm{sk}(U_\bullet;U^+_\bullet)|, \]
which is a cell complex whose integral model, again, is $\mathrm{sk}X^+$.
The blow-up formula \ref{LOC-blow-up} shows that the colimit \ref{SKEL-lim}, for each qcqs $U{\mathfrak h }ookrightarrow X$, is in fact over all possible \mathbf{e}mph{free localisations} of ${\mathbb B}^c(j_*{\sh S}h O_U;j_*{\sh S}h O_U^+)$. In other words,
\begin{prop}Let $X$ be an adic space, $j:X\rightarrow X^+$ a formal model; $\mathrm{sk}X{\sh S}ubseteq\mathrm{sk}(X;X^+)$ is the intersection of all subdivisions of $\mathrm{sk}(X;X^+)$.\mathbf{e}nd{prop}
Any open subset of the $X^+$-shell is induced by a blow-up $X^+_i\rightarrow X^+$ followed by a Zariski-open immersion $U^+{\mathfrak h }ookrightarrow X^+_i$. I do not know of any easily-checked \mathbf{e}mph{necessary} criterion to determine when a family of blow-ups $\{X^+_i\rightarrow X^+\}_{i=1}^k$ gives rise to a cover $\mathrm{sk}(X;X^+_\bullet)\twoheadrightarrow\mathrm{sk}(X;X^+)$ of the corresponding shells; it is certainly \mathbf{e}mph{sufficient} that the blow-up centres have no common point.
Note that the formal model $X^+$ can be recovered from the data of $X$ and the shell $\mathrm{sk}X{\mathfrak h }ookrightarrow\mathrm{sk}(X;X^+)$. Indeed, one obtains from these data the continuous map $j:X\widetilde\rightarrow\mathrm{sk}X\rightarrow\mathrm{sk}X^+$ to the integral model of $\mathrm{sk}(X;X^+)$, $X^+$ is the formal scheme with the same underlying space as $\mathrm{sk}X^+$ and structure sheaf $j_*{\sh S}h O_X^+$.
\
Finally, the fact that any two models of $X$ are dominated by a third means that any two shells of $\mathrm{sk}X$ have a common open subshell; the shells can therefore be glued together to create a \mathbf{e}mph{universal shell} $\mathbf{sk}X$. In abstract terms, the functor $\mathbf{sk}$ is obtained by left Kan extension along the inclusion $\mathbf{Ad}^\mathrm{aff}{\mathfrak h }ookrightarrow\mathbf{Ad}$ of
\[ \Spec|{\sh S}h O|^\mathrm{pre}:\mathbf{Ad}^\mathrm{aff}\rightarrow\mathbf{Sk}, \]
where $|{\sh S}h O|^\mathrm{pre}$, as before, denotes the presheaf $\Spa A\mapsto{\mathbb B}^c(A;A^+)$. Again, the universal shell $\mathbf{sk}X$ contains the spine $\mathrm{sk}X$ as the intersection of all subdivisions.
The universal shell is a universal way of defining a `pro-affine structure' on $X({\mathbb R}_\vee)$ with respect to which the valuations of sections of ${\sh S}h O_X$ are convex. It also supports convex potentials for semipositive metrics on $X$.
{\sh S}ection{Examples \& applications}\label{EGS}
I conclude this paper with some abstract constructions of skeleta which are already well-known via combinatorial means in their respective fields.
{\sh S}ubsection{Polytopes and fans}\label{EGS-poly}
Let $N$ be a lattice with dual $M$, and let ${\mathbb D}elta{\sh S}ubset N\mathbin{\otimes}{\mathbb R}$ be a rational polytope with supporting half-spaces
$\{\langle-,f_i\rangle\leq\lambda_i\}_{i=1}^k,\lambda_i\in{\mathbb Q}$. We will allow ${\mathbb D}elta$ to be non-compact, as long as it has at least one vertex; this means that the submonoid $M_{\mathbb D}elta{\sh S}ubseteq M$ of functions bounded above on ${\mathbb D}elta$ separates its points. In this case, we can compactify ${\mathbb D}elta{\sh S}ubseteq\overline{\mathbb D}elta$ in, for example, the real projective space ${\mathbb R}{\mathbb P}(N\oplus{\mathbb Z})$.
The semiring of `tropical functions' on ${\mathbb D}elta$ is presented
\[ {\mathbb Z}_\vee\{{\mathbb D}elta\}:={\mathbb Z}_\vee[M_{\mathbb D}elta]/(f_i\leq\lambda_i)_{i=1}^k; \]
its elements have the form $\bigvee_{j=1}^dX_i+n_i$, with $X_i\in M_{\mathbb D}elta$ and $n_i\in{\mathbb Z}$.
\begin{defn}The semiring ${\mathbb Z}_\vee\{{\mathbb D}elta\}$ is the \mathbf{e}mph{polytope semiring} associated to ${\mathbb D}elta$. Its spectrum $\mathrm{sk}{\mathbb D}elta$ is the corresponding \mathbf{e}mph{polytope skeleton}, or just \mathbf{e}mph{polytope} if the skeletal structure is implied by the context.\mathbf{e}nd{defn}
The construction $\mathrm{sk}$ is functorial for morphisms $\phi:M_1\rightarrow M_2,\phi({\mathbb D}elta_1){\sh S}ubseteq{\mathbb D}elta_2$ of polytopes. In particular, every sub-polytope ${\mathbb D}elta^\prime{\sh S}ubseteq{\mathbb D}elta$ (with $N$ fixed) induces an open immersion of skeleta $\mathrm{sk}{\mathbb D}elta^\prime{\mathfrak h }ookrightarrow\mathrm{sk}{\mathbb D}elta$. The morphism induced by a refinement $N\rightarrow \frac{1}{d}N$ can be thought of as a degree $d$ `base extension' $\mathrm{sk}{\mathbb D}elta\rightarrow\mathrm{sk}d{\mathbb D}elta$.
The polytope skeleton $\mathrm{sk}{\mathbb D}elta$ is a skeletal enhancement of $\overline{\mathbb D}elta$, in the sense that there is a canonical homeomorphism
\[ \mathrm{sk}{\mathbb D}elta({\mathbb R}_\vee){\sh S}imeq\overline{\mathbb D}elta \]
of the real points, and surjective homomorphism ${\mathbb Z}_\vee\{{\mathbb D}elta\}\rightarrow\mathrm{CPA}_{\mathbb Z}(\overline{\mathbb D}elta,{\mathbb Z}_\vee)$ onto the semiring of integral, convex piecewise-affine functions on $\overline{\mathbb D}elta$ (that is, the semiring of integral, convex piecewise-affine, and bounded above functions on ${\mathbb D}elta$).
We can produce a continuous map $\mathrm{sk}{\mathbb D}elta\rightarrow\overline{\mathbb D}elta$, right inverse to the natural inclusion, whose inverse image functor sends an open $U{\sh S}ubseteq\overline{\mathbb D}elta$ to the union
\[ \bigcup_{{\sh S}igma{\sh S}ubseteq U}\mathrm{sk}{\sh S}igma {\mathfrak h }ookrightarrow \mathrm{sk}{\mathbb D}elta, \]
ranging over all polytopes ${\sh S}igma$ contained in $U$. This map presents $\overline{\mathbb D}elta$ as a \mathbf{e}mph{Hausdorff quotient} of $\mathrm{sk}{\mathbb D}elta$ (cf. thm. \ref{thm-berk}).
\
Polytope semirings admit an alternate presentation, related to the theory of toric degenerations. Let $N^\prime=N\oplus{\mathbb Z}$, with dual $M^\prime\cong M\oplus{\mathbb Z}$, and take the closed cone \[{\sh S}igma:=\overline{\bigcup_{\lambda>0}\lambda{\mathbb D}elta\times\{\lambda\}}{\sh S}ubset N^\prime\mathbin{\otimes}{\mathbb R}\] over the polytope placed at height one. The inclusion of the factor ${\mathbb Z}$ induces a homomorphism $i:{\mathbb N}\rightarrow{\sh S}igma^\vee\cap M^\prime$ of monoids; we topologise ${\mathbb N}$ linearly with ideal of definition $1$, and the cone monoid adically with respect to $i$. In other words, a fundamental system of open ideals of ${\sh S}igma^\vee\cap M^\prime$ is given by the subsets ${\sh S}igma^\vee\cap M^\prime + i(n)$ for $n\in{\mathbb N}$.
We find that \[{\mathbb Z}_\vee^\circ\{{\mathbb D}elta\}={\mathbb B}\{{\sh S}igma^\vee\cap M^\prime\}={\mathbb B}^c({\sh S}igma^\vee\cap M^\prime)\] (see definitions \ref{.5RING-def-freec} and \ref{SPAN-def-fin} for notation) is the semiring of integers (def. \ref{.5RING-integers}) in ${\mathbb Z}_\vee\{{\mathbb D}elta\}$. Its elements are idempotent expressions $\bigvee_{i=1}^kX_i$ with $X_i\in{\sh S}igma^\vee\cap M^\prime$, subject to $X_i\leq 0$. Note that under this notation $-1\in{\mathbb Z}_\vee^\circ$ corresponds, perhaps somewhat confusingly, to $(0,1)\in M^\prime$.
An element $S=\bigvee_{i=1}^kX_i\in{\mathbb Z}_\vee^\circ\{{\mathbb D}elta\}$ corresponds to a finite union of subcones ${\sh S}igma_S=\bigcup_{i=1}^k(X=0){\sh S}ubseteq{\sh S}igma$ and hence of faces ${\mathbb D}elta_S$ of ${\mathbb D}elta$, and the induced restriction
\[ {\mathbb Z}_\vee^\circ\{{\mathbb D}elta\}\rightarrow{\mathbb Z}_\vee^\circ\{{\mathbb D}elta_S\} \]
is a localisation at $S$. The topology of the integral model $\mathrm{sk}{\mathbb D}elta^\circ=\Spec{\mathbb Z}_\vee^\circ\{{\mathbb D}elta\} $ of $\mathrm{sk}{\mathbb D}elta$ is therefore equal, as a partially ordered set, to the set of unions of faces of ${\mathbb D}elta$. In particular, $\mathrm{sk}{\mathbb D}elta^\circ$ is a finite topological space.
A refinement of the lattice $N\mapsto\frac{1}{k}N$ commutes with base extension $\Spec\frac{1}{k}{\mathbb Z}_\vee\rightarrow\Spec{\mathbb Z}_\vee$:
\[ \frac{1}{k}{\mathbb Z}_\vee\{{\mathbb D}elta\}:=\frac{1}{k}{\mathbb Z}_\vee\oplus_{{\mathbb Z}_\vee}{\mathbb Z}_\vee\{{\mathbb D}elta\}\widetilde\rightarrow {\mathbb Z}_\vee\{k{\mathbb D}elta\}. \]
The dual morphism \[\mathrm{sk}k{\mathbb D}elta^\circ\cong\frac{1}{k}{\mathbb Z}_\vee^\circ\times_{{\mathbb Z}_\vee^\circ}\mathrm{sk}{\mathbb D}elta^\circ\rightarrow \mathrm{sk}{\mathbb D}elta^\circ\] of integral skeleta is a homeomorphism.
\
Let $k[\![{\sh S}igma^\vee\cap M^\prime]\!]$ denote the completed monoid algebra of ${\sh S}igma^\vee\cap M^\prime$, and write $z^m$ for the monomial corresponding to an element $m\in{\sh S}igma^\vee\cap M^\prime$. I introduce the special notation $t:=z^{(0,1)}$ for the uniformiser; the completion is with respect to the $t$-adic topology. The monoid inclusion ${\sh S}igma^\vee\cap M^\prime{\sh S}ubset k[\![{\sh S}igma^\vee\cap M^\prime]\!]$ induces a continuous embedding
\[ {\mathbb Z}_\vee^\circ\{{\mathbb D}elta\} {\mathfrak h }ookrightarrow {\mathbb B}^c\left(k[\![{\sh S}igma^\vee\cap M^\prime]\!]\right) \]
into the ideal semiring of $k[\![{\sh S}igma^\vee\cap M^\prime]\!]$, matching $-1\in{\mathbb Z}_\vee^\circ$ with the ideal of definition $(t)$.
The formal spectrum ${\mathbb D}^+_{k[\![t]\!]}{\mathbb D}elta$ of $k[\![{\sh S}igma^\vee\cap M^\prime]\!]$ is an affine \mathbf{e}mph{toric degeneration} in the sense of Mumford. That is, it is a flat degeneration
\[\xymatrix{ {\mathbb D}^+_{k[\![t]\!]}{\mathbb D}elta\ar[d] \\ \Spf k[\![t]\!] }\]
of varieties over the formal disc arising as the formal completion of a toric morphism of toric varieties.
\begin{defn}Let $k$ be a ring. The \mathbf{e}mph{polyhedral algebra} of functions \mathbf{e}mph{convergent over ${\mathbb D}elta$} is the finitely presented $k(\!(t)\!)$-algebra
\[k(\!(t)\!)\{{\mathbb D}elta\} := k[\![{\sh S}igma^\vee\cap M^\prime]\!][t^{-1}]. \]
Its (analytic) spectrum ${\mathbb D}_{k(\!(t)\!)}{\mathbb D}elta$ is called the \mathbf{e}mph{polyhedral domain over $k(\!(t)\!)$ associated to ${\mathbb D}elta$}.\mathbf{e}nd{defn}
For example, if ${\mathbb D}elta$ is the negative orthant in ${\mathbb R}^n$, then ${\mathbb D}_{k(\!(t)\!)}{\mathbb D}elta$ is just the ordinary unit polydisc ${\mathbb D}^n_{k(\!(t)\!)}$ over $k(\!(t)\!)$. Note that the polyhedral algebra has relative dimension equal to the rank of $N$, while the polyhedral semiring depends only on the lattice points of ${\mathbb D}elta$ and not on the ambient lattice.
A similar construction is possible in mixed characteristic.
In light of the main result \ref{SKEL-main}, there is a commuting diagram
\[\xymatrix{ {\mathbb D}_{k(\!(t)\!)}{\mathbb D}elta \ar[r]\ar[d]_{\mu_{\mathbb D}elta} & {\mathbb D}^+_{k[\![t]\!]}{\mathbb D}elta \ar[d] \\
\mathrm{sk}{\mathbb D}elta \ar[r] & \mathrm{sk}{\mathbb D}elta^\circ }\]
in which the top and bottom horizontal arrows are morphisms of adic spaces and of skeleta, respectively, and the vertical arrows are continuous maps. If ${\mathbb D}elta$ spans $M$, then I would like to call the leftmost arrow $\mu_{\mathbb D}elta$ a \mathbf{e}mph{standard non-Archimedean torus fibration over ${\mathbb D}elta$}.
\
This construction can be globalised to obtain torus fibrations on toric varieties and on certain possibly non-compact analytic subsets, in analogy with (and, more precisely, mirror to) the symplectic theory. Let $\Sigma$ be a fan in a lattice $N$, and let $X=X_\Sigma$ be the associated toric variety over a non-Archimedean field $K$, considered as an analytic space. Each cone ${\mathbb D}elta$ of $\Sigma$ corresponds to a Zariski-affine subset $U_{\mathbb D}elta{\sh S}ubseteq X$. Considering the cone as a polytope
\[{\mathbb D}elta=\bigcap_{i=1}^k\{\langle-,f_i\rangle\leq\lambda_i\}_{i=1}^k, \] embed it in a filtered family of \mathbf{e}mph{expansions}
\[{\mathbb D}elta_{\mathbf r}=\bigcap_{i=1}^k\{\langle-,f_i\rangle\leq\lambda_i+r_i\}_{i=1}^k \]
for $\mathbf r=(r_i)\in{\mathbb R}_{{\mathfrak g }eq0}^k$; the analytic version of the subset $U_{\mathbb D}elta$ fits into the increasing union
\[ \xymatrix{ {\mathbb D}_{k(\!(t)\!)}{\mathbb D}elta_{\mathbf r} \ar[r]\ar[d] & U_{\sh S}igma \ar[d] \\
\mathrm{sk}{\mathbb D}elta_{\mathbf r} \ar[r] & \bigcup_{\mathbf r\rightarrow\infty}\mathrm{sk}{\mathbb D}elta_{\mathbf r} } \]
of standard non-Archimedean torus fibrations. Note that it is not quasi-compact unless $N=0$. By glueing, we obtain a skeleton $\mathrm{sk}\Sigma$ and torus fibration
\[\xymatrix{ X_\Sigma\ar[d]_{\mu_\Sigma} \\ \mathrm{sk}\Sigma }\]
which is covered by the standard fibrations over affine polyhedral domains $\mathrm{sk}{\mathbb D}elta_{\mathbf r}{\sh S}ubseteq\mathrm{sk}\Sigma$, where ${\mathbb D}elta_{\mathbf r}$ ranges over all expansions of cones of $\Sigma$.
{\sh S}ubsection{Dual intersection skeleta}\label{EGS-Clemens}
Let $X^+$ be a reduced, Noetherian formal scheme, and let $X$ be the analytic space obtained by puncturing $X^+$ along its reduction $X^+_0$.
\begin{defns}\label{EGS-def-Clemens}The \mathbf{e}mph{dual intersection} or \mathbf{e}mph{Clemens semiring} of $X^+$ is the subring
\[\mathrm{Cl}(X;X^+){\mathfrak h }ookrightarrow{\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+) \]
generated by the additive units of ${\mathbb B}^c(j_*{\sh S}h O_X,j_*{\sh S}h O_X^+)$, that is, the invertible fractional ideals of $j_*{\sh S}h O_X^+$ in $j_*{\sh S}h O_X$. It is a sheaf of semirings on $X^+$, and it is functorial in both $X$ and $X^+$. The elements of the semiring of integers $\mathrm{Cl}(X;X^+)^\circ$ correspond to \mathbf{e}mph{monomial} subschemes of $X^+$.
The \mathbf{e}mph{dual intersection} or \mathbf{e}mph{Clemens skeleton} of $X^+$ is \[ \mathrm{sk}{\mathbb D}elta(X,X^+):= \Spec {\mathbb G}amma(X^+;\mathrm{Cl}(X;X^+)). \]
It comes equipped with a \mathbf{e}mph{collapse map} $X\rightarrow\mathrm{sk}X\rightarrow\mathrm{sk}{\mathbb D}elta(X;X^+)$.\mathbf{e}nd{defns}
It is possible, where confusion cannot occur, to drop $X$ and/or $\mathrm{sk}$ from the notation. I also write $\mathrm{Cl}^\circ(X;X^+)$ and $\mathrm{sk}{\mathbb D}elta^\circ(X;X^+)$ for the semiring of integers and integral model, respectively.
The flattening stratification decomposes $X^+=\coprod_{i\in I}E_i$ into locally closed, irreducible subsets such that the restriction of the normalisation $\nu:\widetilde X^+_0\rightarrow X^+_0$ to each $E_i$ is flat. In particular, the set underlying each monomial subscheme appears in this stratification.
\begin{lemma}\label{i can't be bothered to think of good labels any more}If $E{\mathfrak h }ookrightarrow X^+$ is a monomial subscheme with complement $V^+$, then $\mathrm{Cl}X^+\rightarrow\mathrm{Cl}V^+$ is a cellular localisation at $E$.\mathbf{e}nd{lemma}
\begin{proof}For this we may repeat the argument of \ref{LOC-Zar-open} with `ideal' replaced by `monomial ideal' throughout.\mathbf{e}nd{proof}
\begin{eg}
If $X^+$ is an affine toric degeneration associated to some polytope ${\mathbb D}elta$, with general fibre $X={\mathbb D}_{{\sh S}h O_K}{\mathbb D}elta$, then the Clemens skeleton is $\mathrm{sk}{\mathbb D}elta(X,X^+)=\mathrm{sk}{\mathbb D}elta$ and the collapse map $\mu$ is a standard torus fibration $\mu_{\mathbb D}elta$. Its real points $\mathrm{sk}{\mathbb D}elta({\mathbb R}_\vee)$ are the dual intersection complex of $X^+$ in the classical sense: its $n$-dimensional faces correspond to codimension $n$ toric strata of $X^+$.
The commuting diagram \[\xymatrix{ {\mathbb D}^+_{{\sh S}h O_K}{\mathbb D}elta^\prime \ar[r]\ar[d] & {\mathbb D}^+_{{\sh S}h O_K}{\mathbb D}elta \ar[d] \\
{\mathbb D}elta^\prime \ar[r] & {\mathbb D}elta }\] coming from the open inclusion of a face ${\mathbb D}elta^\prime$ of ${\mathbb D}elta$ can be seen as an instance of lemma \ref{i can't be bothered to think of good labels any more}.
\mathbf{e}nd{eg}
\begin{defn}\label{EGS-toric}A normal formal scheme $X^+$ over a field $k$ is said to have \mathbf{e}mph{toroidal crossings} if it admits a cover $\{f_i:U_i^+\rightarrow X^+\}_{i=1}^k$ by open strata such that each $U_i^+$ is isomorphic to an open subset of some affine toric degeneration ${\mathbb D}^+_{k[\![t]\!]}{\mathbb D}elta_i$.\mathbf{e}nd{defn}
One can choose whether to consider \'etale or Zariski-open subsets for the covering, with the former being the usual choice. Zariski-local toroidal crossings is a very restrictive notion - for instance, it forces the irreducible components of $X^+_0$ to be rational. For simplicity, I will nonetheless work with this latter notion in this section, though the arguments may be generalised with some additional work.
Let $X^+$ be a formal scheme with Zariski-local toroidal crossings, and select model data as in the definition. Write $U_i=U_i^+\times_{X^+}X$. Assume, without loss of generality, that the given inclusions $U_i^+{\mathfrak h }ookrightarrow{\mathbb D}^+_{k[\![t]\!]}{\mathbb D}elta_i$ induce a bijection on the sets of strata, and hence isomorphisms ${\mathbb Z}_\vee\{{\mathbb D}elta_i\}\widetilde\rightarrow{\mathbb G}amma(U_i^+;\mathrm{Cl}(U_i;U_i^+))$. They identify ${\mathbb D}elta_i$ with the dual intersection complex of $U^+_i$. By lemma \ref{i can't be bothered to think of good labels any more}, the inclusion of the open substratum $U_{ij}^+:=U_i^+\cap U_j^+$ identifies its dual intersection complex with a face ${\mathbb D}elta_{ij}$ common to ${\mathbb D}elta_i$ and ${\mathbb D}elta_j$.
It follows from this and the cellular cover formula \ref{LOC-Zar-cover} that $\{{\mathbb D}elta(U_i;U_i^+)\rightarrow{\mathbb D}elta(X;X^+)\}_{i=1}^k$ is a (cellular) open cover.
\[\xymatrix{
X\ar[d]_\mu & \coprod_{i=1}^k U_i\ar@{->>}[l]\ar@{^{(}->}[r]\ar[d] & \coprod_{i=1}^k{\mathbb D}_i{\mathbb D}elta_i\ar[d]^{\mu_{{\mathbb D}elta_i}} \\
{\mathbb D}elta (X;X^+) & \coprod_{i=1}^k {\mathbb D}elta(U_i;U_i^+)\ar@{->>}[l]\ar@{=}[r] & \coprod_{i=1}^k{\mathbb D}elta_i }\]
Intuitively, ${\mathbb D}elta(X;X^+)$ is constructed by glueing together the dual intersection polytopes ${\mathbb D}elta_i$ of the affine pieces $U_i^+$ along their faces ${\mathbb D}elta_{ij}$ corresponding to the intersections $U_{ij}^+$.
\begin{prop}Let $X^+$ be a locally toric formal scheme. Then
\[ \coprod_{i,j=1}^k{\mathbb D}elta_{ij}\rightrightarrows \coprod_{i=1}^k{\mathbb D}elta_i\rightarrow{\mathbb D}elta(X;X^+) \]
is a cellular-open cover. In particular, ${\mathbb D}elta(X;X^+)$ is a cell complex (def. \ref{SKEL-def-cel}).
\mathbf{e}nd{prop}
The collapse map $\mu$ is affine in the sense that ${\mathbb D}elta X^+$ admits an open cover that pulls back to an affine open cover of $X^+$. It follows that $X=\Spa\mu_*{\sh S}h O_X$ and $X^+=\Spf\mu^\circ_*{\sh S}h O_X^+$, where $\mu^\circ:X^+\rightarrow{\mathbb D}elta^\circ X^+$ is the integral model of $\mu$. It is locally isomorphic to standard torus fibrations $\mu_{{\mathbb D}elta_i}$.
\
Suppose that $X^+={\mathbb D}^+{\mathbb D}elta$ is an affine toric degeneration, and let $p:\widetilde X^+\rightarrow X^+$ be a toric blow-up with monomial centre $Z{\sh S}ubseteq X^+_0$. The toric affine open cover of $\widetilde X^+$ induces a decomposition
\[ \coprod_{i,j=1}^k{\mathbb D}elta_{ij} \rightrightarrows \coprod_{i=1}^k \rightarrow {\mathbb D}elta(X;\widetilde X^+)\]
of the dual intersection skeleton of $\widetilde X^+$ into polyhedral cells. The map
\[ {\mathbb D}elta(X;\widetilde X^+) \rightarrow {\mathbb D}elta(X;X^+) \]
induced by the blow-up is a subdivision at the function $Z\in\mathrm{Cl}(X;X^+)$.
More geometrically, the Clemens semiring of a monomial blow-up is the \mathbf{e}mph{strict transform semiring} of the Clemens semiring of $X^+$ (cf. \ref{LOC-strict-transform}). It follows that monomial blow-ups induce subdivisions of the dual intersection skeleta.
{\sh S}ubsection{Tropicalisation}\label{EGS-trop}
Let $X$ be a toric variety, so that following \S\ref{EGS-poly} it comes with a canonical `tropicalisation' $X\rightarrow\mathrm{sk}\Sigma$. Let $f:C{\mathfrak h }ookrightarrow X$ be a closed subspace of $X$. We would like to complete the composite $\mathrm{sk}C{\mathfrak h }ookrightarrow\mathrm{sk}X\rightarrow\mathrm{sk}\Sigma$ to a commuting square
\[\xymatrix{ C \ar[r]\ar[d]_{\mathrm{trop}} & X\ar[d] \\ \mathrm{Trop}(C/X/\Sigma) \ar@{^{(}->}[r] & \mathrm{sk}\Sigma }\]
and to call $C\rightarrow\mathrm{Trop}(C/X/\Sigma)$ the \mathbf{e}mph{amoeba} or \mathbf{e}mph{tropicalisation} of $C$ in $\mathrm{sk}\Sigma$, after (in chronological order) \cite{Kap} and \cite{Payne}.
Let us begin in the affine setting: let $X={\mathbb D}_K{\mathbb D}elta$ be a polyhedral domain, and let $I_C$ be the ideal defining $C$ in ${\sh S}h O_X$. There is an associated toric degeneration $j:X\rightarrow X^+$ over ${\sh S}h O_K$, and we may close the subspace $C$ to obtain an integral model $C^+$ with ideal $I_C\cap j_*{\sh S}h O_X^+$. Let us set $\alpha_{\mathbb D}elta$ to be the image in ${\mathbb B}^c(j_*{\sh S}h O_C;j_*{\sh S}h O_C^+)$ of ${\mathbb Z}_\vee\{{\mathbb D}elta\}$, so that
\[\xymatrix{ {\mathbb B}^c(j_*{\sh S}h O_C;j_*{\sh S}h O_C^+) & {\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+)\ar[l] \\
\alpha_{\mathbb D}elta \ar[u] & {\mathbb Z}_\vee\{{\mathbb D}elta\}\ar[l]\ar[u] }\]
commutes (here we confuse the sheaves ${\mathbb B}^c({\sh S}h O;{\sh S}h O^+)$ with their global sections). The elements of $\alpha_{\mathbb D}elta^\circ$ are subschemes of $C^+$ \mathbf{e}mph{monomial} in the sense that they are defined by monomials from ${\sh S}h O_X$. We set $\mathrm{Trop}(C/X/{\mathbb D}elta):=\Spec\alpha_{\mathbb D}elta$.
\begin{eg}[Plane tropical curves]Let ${\mathbb D}elta$ be the lower quadrant
\[ \{\mathbf r=(r_1,r_2)\in{\mathbb R}^2|r_1,r_2\leq\lambda \} \]
with $0\ll\lambda\in{\mathbb Z}$. The polyhedral domain ${\mathbb D}_K{\mathbb D}elta=\Spa K\{t^{r_1}x,t^{r_2}y\}$ is an arbitrarily large polydisc in the affine plane $X={\mathbb A}^2_K$. Let $X^+$ be the corresponding formal model (which is isomorphic to ${\mathbb A}^2_{{\sh S}h O_K}$).
Let $f={\sh S}um_{i,j,k}c_{ijk}t^kx^iy^j\in K\{t^{r_1}x,t^{r_2}y\}$ be some series, where $(c_{ijk})$ is a matrix of constants in $k$. The `tropicalisation' $F$ of the function $f$ in the polytope semiring ${\mathbb Z}_\vee\{{\mathbb D}elta\}={\mathbb Z}_\vee\{X-r_1,Y-r_2\}$ is \[\iota^\dagger(f)=\bigvee_{i,j,k}iX+jY-k\]
where $\iota:{\mathbb Z}_\vee\{{\mathbb D}elta\}\rightarrow{\mathbb B}^c(j_*{\sh S}h O_X;j_*{\sh S}h O_X^+)$ is the inclusion. Note that $\iota^\dagger$ is a norm, but not a valuation.
Suppose that $C{\mathfrak h }ookrightarrow{\mathbb D}_K{\mathbb D}elta$ is a plane curve. Let $J$ be a monomial ideal of $C^+$, $\{t^kx^iy^j\}$ a finite list of generators. A generator $t^kx^iy^j$ may be removed from the list if and only if it is expressible in terms of the other generators, which occurs exactly when the coefficient $c_{ijk}$ of that monomial in some $f\in I_C$ is non-zero. In other words, the relations of the quotient ${\mathbb Z}_\vee\{{\mathbb D}elta\}\rightarrow\alpha_{\mathbb D}elta$ are generated by those of the form
\[ F=\iota^\dagger(f)=\bigvee_{(i,j,k)\neq(i_0,j_0,k_0)}iX+jY-k \]
where $F$ is the tropicalisation of $f$ and $c_{i_0j_0k_0}\neq0$. There are in general infinitely many such relations. The image of $\mathrm{Trop}(C/X/{\mathbb D}elta)$ in the Hausdorff quotient $\mathrm{sk}{\mathbb D}elta\rightarrow\overline{\mathbb D}elta{\sh S}ubset\overline{{\mathbb R}^2}$ is the \mathbf{e}mph{non-differentiability locus} of the convex piecewise-affine function on $\overline{\mathbb D}elta$ defined by $F$.
These relations were also obtained by different means in \cite{Giansiracusa}.\mathbf{e}nd{eg}
In order to globalise this procedure, we need to check the functoriality of the amoeba under inclusion ${\mathbb D}elta^\prime{\sh S}ubseteq{\mathbb D}elta$ of lattice polytopes. The corresponding open immersion $\mathrm{sk}{\mathbb D}elta^\prime{\mathfrak h }ookrightarrow\mathrm{sk}{\mathbb D}elta$ may be factored into a subdivision at some element $Z\in{\mathbb Z}_\vee^\circ\{{\mathbb D}elta\}$ followed by a cell inclusion. This is the combinatorial shadow of the operation of taking the toric blow-up $\widetilde X^+\rightarrow X^+$ along $Z$, and then restricting to an affine subset.
\begin{lemma}Let $C{\mathfrak h }ookrightarrow X$ be a closed embedding of adic spaces, $j:X\rightarrow X^+$ a formal model of $X$, $C^+{\sh S}ubseteq X^+$ the closure of $C$ in $X^+$. Let $\widetilde X^+\rightarrow X^+$ be an admissible blow-up with ideal $J$. Then the closure $\widetilde C^+$ of $C$ in $\widetilde X^+$ is the blow-up of $C^+$ along ${\sh S}h O_{C^+}J$.\mathbf{e}nd{lemma}
\begin{proof}The definitions directly imply the following identity
\[ \frac{R_J}{I_C\cap R_J}\cong\frac{\bigoplus_{n\in{\mathbb N}}J^nt^n}{\bigoplus_{n\in{\mathbb N}}I_C\cap J^nt^n}\cong\bigoplus_{n\in{\mathbb N}}\frac{J^n}{I_C\cap J^n}t^n \cong R_{{\sh S}h O_{C^+}J} \]
of the Rees algebras on $C^+$.\mathbf{e}nd{proof}
As we observed in the previous section, the Clemens semiring $\mathrm{Cl}(X;\widetilde X^+)$ is the strict transform semiring of ${\mathbb Z}_\vee\{{\mathbb D}elta\}$ under the monomial blow-up $\widetilde X^+\rightarrow X^+$ (def. \ref{LOC-def-sta}). Furthermore, the formation of the strict transform semiring commutes with the tropicalisation of ideals on $C$:
\[ j_*{\sh S}h O_C^+\bigoplus_{n\in{\mathbb N}}J_nt^n\cong \bigoplus_{n\in{\mathbb N}}\frac{J_n}{I_C\cap J_n}t^n\cong \bigoplus_{n\in{\mathbb N}}j_*{\sh S}h O_C^+J_n. \]
By corollary \ref{LOC-strict-transform}, the image of ${\mathbb B}^c(j_*{\sh S}h O_C;j_*{\sh S}h O_C^+)$ in $\mathrm{Cl}(X;\widetilde X^+)$ is a free localisation of $\alpha_{\mathbb D}elta$.
Now writing $X^\prime={\mathbb D}_K{\mathbb D}elta^\prime$ and $C^\prime=C\times_XX^\prime$, we obtain a natural morphism of skeleta $\mathrm{Trop}(C^\prime/X^\prime/{\mathbb D}elta^\prime)\rightarrow\mathrm{Trop}(C/X/{\mathbb D}elta)$. The above arguments, together with lemma \ref{i can't be bothered to think of good labels any more} show:
\begin{prop}$\mathrm{Trop}(C^\prime/X^\prime/{\mathbb D}elta^\prime)\rightarrow\mathrm{Trop}(C/X/{\mathbb D}elta)$ is an open immersion.\mathbf{e}nd{prop}
We can therefore glue tropicalisations as we glue polytopes. In particular, we can construct the amoeba
\[ \mathrm{Trop}(C/X/\Sigma)=\bigcup_{{\sh S}igma_{\mathbf r}{\sh S}ubset\Sigma}\mathrm{Trop}\left(C\times_X{\mathbb D}_k{\sh S}igma_{\mathbf r}/{\mathbb D}_K{\sh S}igma_{\mathbf r}/{\sh S}igma_{\mathbf r}\right) \]
of any subscheme of a toric variety, as promised above.
{\sh S}ubsection{Circle}\label{EGS-ellipt}
Returning to the situation of \ref{EGS-Clemens}, let us specialise to the case of an elliptic curve. Let $K$ be a DVF with residue field $k$, $E/K$ an elliptic curve; write $\overline E/\overline K$ for the base change to the algebraic closure. Let $\Omega=\Omega^{1,0}\in{\mathbb G}amma(E;\omega_{E/K}){\sh S}etminus\{0\}$ be a holomorphic volume form.
\begin{defn}A formal model $E^+$ of $E$ is \mathbf{e}mph{crepant} if it is ${\mathbb Q}$-Gorenstein and one of the following equivalent conditions are true:
\begin{enumerate}\item there exists a log resolution $f:(E^+)^\prime\rightarrow E^+$ on which $\overline{(E^+)^\prime}+f^*\Omega=t^k$ as ${\mathbb Q}$-divisors on $(E^+)^\prime$, where $k\in{\mathbb Z}$ and $\overline{(E^+)^\prime}$ denotes the reduction of $(E^+)^\prime$;
\item The log canonical threshold is equal to a constant $k$ on $E^+$ (in equal characteristic zero);
\item The canonical bundle $\omega_{\overline E^+/{\sh S}h O_{\overline K}}$ over the algebraic closure is trivial.\mathbf{e}nd{enumerate}
A formal model of $\overline E$ is \mathbf{e}mph{crepant} if it is finitely presented with trivial canonical bundle over ${\sh S}h O_{\overline K}$, or equivalently, it is obtained by flat base extension from a crepant formal model over some finite extension of ${\sh S}h O_K$.\mathbf{e}nd{defn}
A simple normal crossings model $E^+$ of $E$ is crepant if and only if its reduction is a cycle of projective lines. The multiplicity of a line in the central fibre $E^+_0:=k\times_{{\sh S}h O_K}E^+$ is one more than its multiplicity in the canonical divisor. One can make the multiplicities all one, and hence trivialise the canonical bundle, by effecting a finite base change followed by a normalisation. In particular, $E^+$ is semistable if and only if $\omega_{E^+/{\sh S}h O_K}$ is trivial, that is, if and only if it is a minimal model.
On the other hand, a formal model of $E$ is locally toric if and only if it has at worst monomial cyclic quotient singularities and the components of its central fibre are smooth rational curves. It is automatically ${\mathbb Q}$-Gorenstein. Such a model exists only if $E$ has bad, but semistable reduction; let us assume this.
Let $\mathbf{Mdl}^\mathrm{clt}(E)$ denote the category of crepant, locally toric models of $E$. Let $E^+$ be an object of this category. Its singularities occur at the intersections of components, and they have the form
\[ {\mathbb D}^+_{k[\![t]\!]}{\mathbb D}elta\rightarrow{\mathbb D}elta^\circ \]
where ${\mathbb D}elta=[a,b]$ is an interval with rational endpoints. They may be resolved explicitly, and crepantly, by subdividing the interval at all its integer points.
\
Since, by assumption, a crepant resolution of $E^+$ exists, $(E^+)^\prime$ must be a crepant model. Its reduction is therefore a cycle of ${\mathbb P}^1_k$s. Since ${\mathbb D}elta(E^+)^\prime\rightarrow{\mathbb D}elta E^+$ is a subdivision, it follows that both are cycles of intervals; ${\mathbb D}elta E^+({\mathbb R}_\vee)$ has the topology of a circle.
The Clemens functor
\[ {\mathbb D}elta:\mathbf{Mdl}^\mathrm{clt}(E) \rightarrow \mathbf{Sk}^\mathrm{aff} \]
is defined, and its image is a diagram of subdivisions. It may therefore be \mathbf{e}mph{glued} to obtain the \mathbf{e}mph{Kontsevich-Soibelman} (or \mathbf{e}mph{KS}) skeleton $\mathrm{sk}(E;\Omega):=\colim{\mathbb D}elta$. It is a \mathbf{e}mph{shell} of any crepant Clemens skeleton, and hence comes with a collapse map
\[ \mu:E\rightarrow\mathrm{sk}(E;\Omega), \]
which is a \mathbf{e}mph{torus fibration}: every point of $\mathrm{sk}(E;\Omega)$ has an overconvergent neighbourhood over which $\mu$ is isomorphic over ${\mathbb Z}_\vee$ to a standard torus fibration on an interval. In the introduction we introduced an explicit `atlas' for $\mathrm{sk}(E;\Omega)$ under the assumption (which may be lifted) that the minimal model of $E$ consist of at least three reduced lines.\footnote{I insert the word `atlas' between inverted commas because I do not prove that these open sets really cover $\mathrm{sk}(E;\Omega)$. That statement would be equivalent to the conjecture raised in example \ref{eg-cpa}.}
If, more generally, $E$ has only bad reduction, we can still define $\mathrm{sk}(\overline E;\Omega)$ as the colimit of the Clemens functor on $\mathbf{Mdl}^\mathrm{clt}(\overline E)$. If $L{\sh S}upseteq K$ is a finite extension over which $E_L:=L\times_KE$ has semistable reduction, then
\[ \mathrm{sk}(\overline E;\Omega)\cong {\mathbb Q}_\vee\times_{{\mathbb Z}_\vee}\mathrm{sk}(E_L;\Omega) \]
by the base change property for polytopes. In particular, the collapse map $\overline E\rightarrow\mathrm{sk}(\overline E;\Omega)$ is again a torus fibration. The KS skeleton is of finite presentation over ${\mathbb Q}_\vee$.
There is a continuous projection $\pi:\mathrm{sk}(\overline E;\Omega)\rightarrow B:=\mathrm{sk}(\overline E;\Omega)({\mathbb R}_\vee){\sh S}imeq S^1$. The local models for $\mu$ induce a canonical smooth structure on $B$ with respect to which the \mathbf{e}mph{affine functions}, that is, invertible sections $\mathrm{Aff}_{\mathbb Z}(B,{\mathbb Q})$ of $\mathrm{CPA}_{\mathbb Z}(B,{\mathbb Q}_\vee):=\pi_*|{\sh S}h O|^\mathrm{canc}$, are smooth. It therefore attains an \mathbf{e}mph{affine structure} in the sense of \cite[\S2.1]{KoSo2} defined by the exact sequence
\[ 0\longrightarrow{\mathbb Q}\longrightarrow \mathrm{Aff}_{\mathbb Z}(B,{\mathbb Q}) \longrightarrow \Lambda^\vee \longrightarrow 0 \]
of Abelian sheaves on $B$ and the induced embedding $\Lambda^\vee{\mathfrak h }ookrightarrow T^\vee B$.
Let $E^+_L$ be a semistable minimal model of $E_L$. By writing $\Omega$ locally in the form $\lambda d\log x$ for some monomial $x\in{\sh S}h O_E^\times$ and $\lambda\in L^\times$, we can think of $\Omega$ as a non-zero section of $L\mathbin{\otimes}_{\mathbb Z}\Lambda^\vee$. It induces a $\overline K$-orientation of $B$ that does not depend on the choice of $L$ or $x$.
\mathbf{e}nd{document} |
\begin{document}
\title{Unifying type systems for mobile processes}
\begin{abstract}
We present a unifying framework for type systems for process calculi.
The core of the system provides an accurate correspondence between
essentially functional processes and linear logic proofs; fragments of this
system correspond to previously known connections between proofs and
processes.
We show how the addition of extra logical axioms can widen the class of
typeable processes in exchange for the loss of some computational properties
like lock-freeness or termination, allowing us to see various well studied
systems (like i/o types, linearity, control) as instances of a general
pattern.
This suggests unified methods for extending existing type systems with new
features while staying in a well structured environment and constitutes a
step towards the study of denotational semantics of processes using
proof-theoretical methods.
\end{abstract}
\section{Introduction}
Process calculi are a wide range of formalisms designed to model concurrent
systems and reason about them by means of term rewriting.
Their applications are diverse, from the semantics of proof systems to the
conception of concrete programming languages.
Type systems for such calculi are therefore a wide domain, with systems of
different kinds designed to capture different behaviours and ensure different
properties of processes: basic interfacing,
input-output discipline \cite{pierce-1993-typing},
linearity \cite{kobayashi-1999-linearity},
lock-freeness \cite{kobayashi-2002-type},
termination \cite{deng-2004-ensuring},
respect of communication protocols \cite{honda-1993-types,honda-2008-multiparty},
functional or sequential
behaviour \cite{berger-2001-sequentiality,honda-2004-control,yoshida-2001-strong}.
In order to better understand the diversity of calculi and uncover basic
structures and general patterns, many authors have searched for languages
with simpler or more general theory in which the most features could be
expressed by means of restrictions or codings:
asynchrony \cite{boudol-1992-asynchrony},
internal mobility \cite{sangiorgi-1996-calculus},
name fusions \cite{fu-1997-proof,parrow-1998-fusion},
solos \cite{laneve-1999-solos}, etc.
It is natural to search for similar unification in the realm of type systems,
and the aim of this paper is to make a step towards this long-term objective.
Our ideal system would be simple enough so that general properties could be
reasonably easy to obtain and expressive enough so that most interesting type
systems could naturally be expressed in it in a structured way.
For this purpose, we will take inspiration and tools in proof theory.
A useful analogue is the famous and fruitful Curry-Howard correspondence: at
the core is the simply typed λ-calculus, which matches minimal intuitionistic
logic and ensures strong normalisation.
The type language can be extended for expressiveness (with quantifiers,
dependent types, polymorphism, etc.), classical logic can be embedded in it by
CPS translation or by adding logical rules.
Furthermore, extending it with a simple type equation $D=D→D$ yields the full
untyped calculus where normalisation is lost, but the identification of this
equation leads to the definition of abstract structures that are useful for
denotational semantics.
We claim that the analogue of simple types for process calculi is to be found
in linear logic \cite{girard-1987-linear}, and we propose a new implementation
of this idea.
Of course, term assignment systems for linear logic proofs have been proposed
in the past by various
authors \cite{bellin-1994-calculus,beffara-2006-concurrent,caires-2010-session}
but no such system has yet appeared as a satisfactory type system for
processes (because of too much constraint on the syntactic structure of terms),
with the notable exception of Honda and Laurent's
result \cite{honda-2010-exact} which precisely matches a proper type system
for the π-calculus with a meaningful class of proof nets.
The novelty of our approach is to distinguish two aspects in typing: firstly
we have a typing rule for each syntactic construct independently, secondly we
have a subtyping relation that implements logical reasoning without affecting
the structure of terms; this subtyping is nothing else than entailment in
linear logic (actually a reasonable fragment of it), which allows to use all
existing theory for reasoning about it.
In this method, we insist on treating seriously the fundamental structures of
both the process calculus and the logic, the fundamental example being that
typing is preserved both by structural congruence on processes and by logical
isomorphism between types (and these are closely related).
This is necessary for developing logic-based semantics of processes in future
works, using existing tools and methods from the semantics of proofs and of
processes.
This paper is divided in two parts.
In section \ref{sec:basic} we define our basic type system for the polyadic
π-calculus and we discuss variations around the same principles for
alternative calculi.
In section \ref{sec:existing} we review several type systems and term
assignment systems and show how they fit in our framework, by means of extra
logical axioms and syntactic restrictions.
Section \ref{sec:discussion} concludes by discussing shortcomings, extensions
and ideas for future work.
\section{The basic typed calculus}
\label{sec:basic}
\subsection{Syntax}
Processes are terms of the standard polyadic π-calculus with input-guarded
replication (and no sum in the present paper), with type annotations on name
creation.
In our type system, we will derive judgements of the form $E⊢P$ where $E$ is
an environment type and $P$ is a process term.
Such a judgement is to be understood as “$P$ is well-formed under the contract
of the environment $E$”.
Environment types are made of capability assignments of the shape $x:T$ where
$x$ is a channel name and $T$ a capability type, combined using logical
connectives.
Capability types consist of an input or output capability (written $↓$ and $↑$
respectively) together with a behaviour type for the data that is
communicated, and behaviour types are capability types combined using logical
connectives.
\begin{definition}[typed terms]
The grammar of types and processes is defined in table \ref{table:syntax}.
\end{definition}
\begin{table}
\begin{syntax}
\define[Capabilities:] T, U
\case ↓A \comment{input}
\case ↑A \comment{output}
\define[Behaviours:] A, B
\case T \altcase \oc T \altcase \wn T
\comment{capability (linear, replicable, multiple)}
\case A ⊗ B \altcase A ⅋ B
\comment{concatenation (independent, correlated)}
\case 1 \altcase ⊥
\comment{empty tuple (neutral for $⊗$ or $⅋$)}
\define[Environments:] E, F
\case x:T \altcase \oc x:T \altcase \wn x:T
\comment{capability assignment}
\case E ⊗ F \altcase E ⅋ F
\comment{union (independent or correlated)}
\case 1 \altcase ⊥
\comment{empty environment (neutral for $⊗$ or $⅋$)}
\define[Processes:] P, Q
\case \inb{u}{¤x}.P \comment{input prefix}
\case \inr{u}{¤x}.P \comment{replicated input prefix}
\case \outf{u}{¤v}.P \comment{output prefix}
\case \nop \comment{inactive process}
\case P \para Q \comment{parallel composition}
\case \newk{x}{A}{k}P \comment{name creation, with $k∈\{1,ω\}$}
\end{syntax}
\caption{The syntax of types and process terms}
\label{table:syntax}
\end{table}
Remark that we do \emph{not} force each channel name to occur only
once in an environment type, and this is a fundamental feature of our system.
It notably allows name substitution $E[¤u/¤x]$ to make sense even when it
equalises some names.
The name creation operator $\new{x}$ is annotated with a type $A$ and a kind
$k$ that distinguishes between linear and non-linear channels.
Contrary to usual practice, the type $A$ is not the type that $x$ itself will
have, but the type of the data that $x$ will transport.
In statements where the kind and type of a channel are unimportant, we use the
standard notation $\new{x}$.
The logical connectives used in environments and behaviours are those of
multiplicative-exponential linear logic.
This logic, recalled in table \ref{table:mell}, is used to reason about
behaviours of processes.
The key ingredient is that logical consequence is interpreted as
subtyping: if $E$ and $F$ are environments such that $E$ entails $F$, then
a process that respects $F$ will respect $E$.
\begin{definition}[subtyping]
The subtyping preorder $≤$ over environments is such that $E≤F$ holds when
$⊢E^⊥,F$ is provable in MELL using capability assignments as atomic
formulas.
The associated equivalence is written $≃$.
\end{definition}
\begin{table}
Formulas, assuming a given set of atoms written $α,β…$:
\begin{syntax}
\define A, B
\case α
\altcase α^⊥
\altcase A ⊗ B
\altcase A ⅋ B
\altcase 1
\altcase ⊥
\altcase \oc A
\altcase \wn A
\end{syntax}
Linear negation $(⋅)^⊥$ is the involution over formulas such that:
\begin{align*}
(α^⊥)^⊥ &= α &
(A ⊗ B)^⊥ &= A^⊥ ⅋ B^⊥ &
1^⊥ &= ⊥ &
(\oc A)^⊥ &= \wn A^⊥
\end{align*}
Sequents are finite multisets of formulas, they are proved using the
following rules:
\begin{center}
\begin{prooftree}
\Infer0[ax]{ ⊢ A^⊥, A }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ ⊢ Γ, A }
\Hypo{ ⊢ A^⊥, Δ }
\Infer2[cut]{ ⊢ Γ, Δ }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ ⊢ Γ, A }
\Hypo{ ⊢ Δ, B }
\Infer2[$⊗$]{ ⊢ Γ, Δ, A⊗B }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ &⊢ Γ, A, B }
\Infer1[$⅋$]{ &⊢ Γ, A⅋B }
\end{prooftree}
\\[1ex]
\begin{prooftree}
\Infer0[$1$]{ ⊢ 1 }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ &⊢ Γ }
\Infer1[$⊥$]{ &⊢ Γ, ⊥ }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ &⊢ Γ, A }
\Infer1[$\wn$]{ &⊢ Γ, \wn A }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ &⊢ Γ }
\Infer1[w]{ &⊢ Γ, \wn A }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ &⊢ Γ, \wn A, \wn A }
\Infer1[c]{ &⊢ Γ, \wn A }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ &⊢ \wn A_1, …, \wn A_n, B }
\Infer1[$\oc$]{ &⊢ \wn A_1, …, \wn A_n, \oc B }
\end{prooftree}
\end{center}
\caption{Multiplicative-exponential linear logic (MELL)}
\label{table:mell}
\end{table}
Formulas of MELL with capability assignments as atoms and where modalities
$\wn$ and $\oc$ are only applied to literals (atoms and atom negations) will
be called \emph{environment formulas}, they will be useful for reasoning about
typed processes.
Environment types correspond to such formulas with only positive atoms, {i.e.}
without negation.
\begin{definition}[typing judgement]
Typing judgements have the shape $E⊢P$ where $E$ is an environment type and
$P$ is a process term.
They are derived using the rules of table \ref{table:typing}.
The notation $¤x:A$ where $A$ is a behaviour type stands for the
environment type obtained by annotating each capability in $A$ by a name
in the sequence $¤x$, respecting the left-to-right order, assuming that
the length of $¤x$ matches the number of capabilities in $A$.
For instance, $xyz:(T⅋1)⊗\wn U⊗⊥⊗V$ stands for $((x:T)⅋1)⊗\wn(y:U)⊗⊥⊗(z:V)$.
\end{definition}
\begin{table}
\begin{center}
\begin{prooftree}
\Infer0[nop]{ ⊥ ⊢ \nop }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ E ⊢ P }
\Hypo{ F ⊢ Q }
\Infer2[para]{ E ⅋ F ⊢ P\para Q }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ E ≤ F }
\Hypo{ F ⊢ P }
\Infer2[sub]{ E ⊢ P }
\end{prooftree}
\\[1ex]
\begin{prooftree}
\Hypo{ ¤x:A ⊗ E ⊢ P }
\Infer1[in]{ u:↓A ⊗ E ⊢ \inb{u}{¤x}.P }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ ¤x:A ⊗ E^! ⊢ P }
\Infer1[in!]{
\wn u:↓A ⊗ E^! ⊢ \inr{u}{¤x}.P }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ E ⊢ P }
\Infer1[out]{ u:↑A ⊗ (¤v:A ⅋ E) ⊢ \outf{u}{¤v}.P }
\end{prooftree}
\\[1ex]
\begin{prooftree}
\Hypo{ (x:↑A ⅋ x:↓A) ⊗ E ⊢ P }
\Infer1[new1]{ E ⊢ \newl{x}{A}P }
\end{prooftree}
\hfil
\begin{prooftree}
\Hypo{ (\oc x:↑A ⅋ \wn x:↓A) ⊗ E ⊢ P }
\Infer1[newω]{ E ⊢ \newr{x}{A}P }
\end{prooftree}
\end{center}
\begin{itemize}
\item In the \rulename{new} rules, the name $x$ must not
occur in the environment $E$.
\item In the \rulename{in} rules, the names in $¤x$ must not occur in the
environments $E$ and $E^!$.
\item In \rulename{in!}, $E^!$ stands for an environment of the shape
$\oc y_1:T_1 ⊗ … ⊗ \oc y_n:T_n$.
\end{itemize}
\caption{Typing rules for processes}
\label{table:typing}
\end{table}
Note that process terms have no type, in other words there is a unique type
for processes which means “well-formed”; it is also the case for instance in
i/o types with linearity, as studied in section \ref{sec:kpt}.
Of course, it would be strictly equivalent to consider that, in $E⊢P$, the
formula $E^⊥$ is the type of $P$: this is what usually happens in systems more
oriented towards logic, like those studied in sections \ref{sec:hyb}
and \ref{sec:session}.
Remark that input and output capabilities are logically not dual, in the sense
of being a negation of each other: $(u:↓A)^⊥$ and $u:↑A$ are just distinct
literals.
Actual duality between input and output is established by the typing rules for
name creation, for instance \rulename{new1} corresponds to setting the formula
$x:↑A⅋x:↓A$ as an axiom, which does represent the creation of a name $x$ with
one occurrence of each capability, where capabilities are dual.
In the statements and proofs, the types for channels in premisses of
\rulename{new} rules is used in many places.
For readability and conciseness, we introduce the following notations:
\begin{align*}
[x]^1_A &:= x:↑A ⅋ x:↓A, &
[x]^ω_A &:= \oc x:↑A ⅋ \wn x:↓A.
\end{align*}
In $[x]^k_A$, we may keep $k$ or $A$ implicit when the details are
unimportant.
This way, the rules \rulename{new1} and \rulename{newω} are simplified into a
single form:
\[
\begin{prooftree}
\Hypo{ [x]^k_A ⊗ E &⊢ P }
\Infer1[new$k$]{ E &⊢ \newk{x}{A}{k}P }
\end{prooftree}
\qquad\text{where $x$ does not occur in $E$.}
\]
Moreover, in applications of the \rulename{sub} rule, we will keep the premiss $E≤F$
implicit since $E$ and $F$ are the environments of the conclusion and premiss
respectively.
\begin{lemma}\label{lemma:subtyping-commutation}
A judgement $E⊢P$ holds if and only if it has a derivation where
\rulename{sub} rules appear only right above \rulename{in} and
\rulename{new} rules and at the root of the proof.
\end{lemma}
\begin{proof}
Firstly, it is clear that successive uses of the \rulename{sub} rule can
always be gathered into one thanks to the cut rule of MELL.
We may thus assume without loss of generality that no \rulename{sub} rule
occurs above another \rulename{sub} rule.
Then one easily checks by case analysis on the proofs that each
\rulename{sub} rule can be commuted down with any rule except \rulename{new}
and \rulename{in} rules because these impose constraints on the context of
their premiss.
\end{proof}
This lemma allows us to consider a restricted form of derivation when
reasoning on typed processes.
In order to establish subject reduction in the next section, we will also need
the following general properties of MELL proofs:
\begin{lemma}[substitutivity]\label{lemma:mell-subst}
If $⊢Γ$ is a provable sequent in MELL, then for all propositional variable
$α$ and formula $A$ the sequent $⊢Γ[A/α]$ is also provable.
\end{lemma}
\begin{lemma}[interpolation]\label{lemma:interpolation}
Let $Γ$ and $Δ$ be two multisets of MELL formulas.
If $⊢Γ,Δ$ is provable, then there exists a formula $F$ that contains only
literals present in both $Γ^⊥$ and $Δ$ such that the sequents $⊢Γ,F$ and
$⊢F^⊥,Δ$ are provable.
\end{lemma}
Both lemmas are easily proved by structural induction over proofs (a detailed
proof for lemma \ref{lemma:interpolation} can be found in
appendix \ref{app:interpolation}).
They actually hold for full linear logic but we state them in MELL because it
is the fragment we use in this paper.
\subsection{Execution}
Our presentation of execution uses structural congruence and reduction,
because it provides simpler statements than a presentation using a labelled
transition system.
\begin{definition}[structural congruence]
The congruence $≡$ over process terms is defined by abelian monoid laws for
parallel composition and the standard scoping rules:
\begin{gather*}
(P \para Q) \para R ≡ P \para (Q \para R) \qquad
P \para Q ≡ Q \para P \qquad
P \para 0 ≡ P \\
\newk{x}{A}{k}\newk{y}{B}{ℓ}P ≡ \newk{y}{B}{ℓ}\newk{x}{A}{k}P \qquad
P \para \newk{x}{A}{k}Q ≡ \newk{x}{A}{k}(P \para Q)
\end{gather*}
where $x≠y$ and $x$ does not occur free in $P$ in the last rule.
\end{definition}
\begin{lemma}\label{lemma:congruence}
Typing is preserved by structural congruence.
\end{lemma}
\begin{proof}
This is proved by checking each axiom of structural congruence.
Most cases are direct, the only technical point is the proof that if
$\newk{x}{A}{k}(P\para Q)$ is typeable and $x$ does not occur in $P$, then
$P\para\newk{x}{A}{k}Q$ has the same type.
We transform a generic typing of $\newk{x}{A}{k}(P\para Q)$ into a typing of
$P\para\newk{x}{A}{k}Q$ as follows:
\[
\begin{prooftree}
\Hypo{ E ⊢ P }
\Hypo{ F ⊢ Q }
\Infer2[para]{ E ⅋ F ⊢ P \para Q }
\Infer1[sub]{ [x]^k_A ⊗ G &⊢ P \para Q }
\Infer1[new$k$]{ G &⊢ \newk{x}{A}{k}(P \para Q) }
\end{prooftree}
\quad\to\quad
\begin{prooftree}
\Hypo{ E ⊢ P }
\Hypo{ F &⊢ Q }
\Infer1[sub]{ [x]^k_A ⊗ H &⊢ Q }
\Infer1[new$k$]{ H &⊢ \newk{x}{A}{k}Q }
\Infer2[para]{ E ⅋ H ⊢ P \para \newk{x}{A}{k}Q }
\Infer1[sub]{ G ⊢ P \para \newk{x}{A}{k}Q }
\end{prooftree}
\]
In order to find the environment $H$, remark that the subtyping on the left
is justified by an MELL proof of $⊢([x]^k_A)^⊥,G^⊥,E,F$.
By lemma \ref{lemma:interpolation}, there is a formula $H$ such that
$⊢G^⊥,E,H$ and $⊢H^⊥,([x]^k_A)^⊥,F$ are provable and the literals in $H$ occur
both in $G,E^⊥$ and in $([x]^k_A)^⊥,F$.
By hypothesis $x$ does not occur in $P$ hence not in $E$, and not in $G$
either by the side-condition on \rulename{new$k$}, so there is no $x$ in $H$.
Hence the literals in $H$ occur in $F$ so they are positive and $H$ is an
environment type.
The proofs of $⊢G^⊥,E,H$ and $⊢H^⊥,([x]^k_A)^⊥,F$ justify the \rulename{sub}
rules on the right.
The other cases are detailed in appendix \ref{app:congruence}.
\end{proof}
\begin{definition}[reduction]
Reduction is the relation $\redpi{ℓ}$ where $ℓ$ is either a name or the
symbol $τ$.
It is generated by the rules
\begin{align*}
\outf{u}{¤v}.P \para \inb{u}{¤x}.Q
&\redpi{u} P \para Q[¤v/¤x] &
\outf{u}{¤v}.P \para \inr{u}{¤x}.Q
&\redpi{u} P \para Q[¤v/¤x] \para \inr{u}{¤x}.Q
\end{align*}
extended to arbitrary contexts as
\[
\begin{prooftree}
\Hypo{ P \redpi{ℓ} P' }
\Infer1{ P\para Q \redpi{ℓ} P'\para Q }
\end{prooftree}
\;
\begin{prooftree}
\Hypo{ P \redpi{ℓ} P' }
\Hypo{ ℓ ≠ u }
\Infer2{ \new{u}P \redpi{ℓ} \new{u}P' }
\end{prooftree}
\;
\begin{prooftree}
\Hypo{ P \redpi{u} P' }
\Infer1{ \newl{u}{A}P \redpi{τ} P' }
\end{prooftree}
\;
\begin{prooftree}
\Hypo{ P \redpi{u} P' }
\Infer1{ \newr{u}{A}P \redpi{τ} \newr{u}{A}P' }
\end{prooftree}
\]
and saturated under structural congruence.
\end{definition}
The only difference with standard reduction is that we delete linear name
creations as soon as their name is used.
This is consistent with the linearity requirement, moreover in typed processes
this requirement is fulfilled.
In plain π-calculus this operation would be handled by the congruence rule
$\new{x}P≡P$ if $x$ is not free in $P$, but we choose not to use this approach
here in order to avoid an extra kind of “new” operator just for this case.
\begin{theorem}[subject reduction]\label{thm:subject-reduction}
For all typed term $Γ⊢P$ and execution step $P\redpi{τ}P'$,
the judgement $Γ⊢P'$ is derivable.
\end{theorem}
\begin{proof}
Thanks to lemma \ref{lemma:congruence}, we can reason up to structural
congruence.
For an interaction step on a linear channel, we have
$\newl{u}{A}(\outf{u}{¤v}.P\para\inb{u}{¤x}.Q)\redpi{τ}P\para Q[¤v/¤x]$.
The left-hand side is typed as follows (using the simplification from
lemma \ref{lemma:subtyping-commutation}):
\begin{prooftree*}
\Hypo{ E &⊢ P }
\Infer1[out]{ u:↑A ⊗ (¤v:A ⅋ E) &⊢ \outf{u}{¤v}.P }
\Hypo{ ¤x:A ⊗ F &⊢ Q }
\Infer1[in]{ u:↓A ⊗ F &⊢ \inb{u}{¤x}.Q }
\Infer2[para]{ (u:↑A⊗(¤v:A⅋E))⅋(u:↓A⊗F) ⊢ \outf{u}{¤v}.P\para\inb{u}{¤x}.Q }
\Infer1[sub]{ (u:↑A⅋u:↓A) ⊗ H ⊢ \outf{u}{¤v}.P\para\inb{u}{¤x}.Q }
\Infer1[new1]{ H ⊢ \newl{u}{A}(\outf{u}{¤v}.P\para\inb{u}{¤x}.Q) }
\end{prooftree*}
with the hypothesis that no name in $¤x$ occurs in $F$.
The \rulename{sub} rule is justified by an MELL proof of
$⊢(((u:↑A)^⊥⊗(u:↓A)^⊥)⅋H^⊥),(u:↑A⊗(¤v:A⅋E))⅋(u:↓A⊗F)$.
By lemma \ref{lemma:mell-subst}, we can replace the atomic formula $u:↓A$ by
$¤v:A$ and the atomic formula $u:↑A$ by $(¤v:A)^⊥$, then we get a proof of
\[
⊢ (¤v:A ⊗ (¤v:A)^⊥) ⅋ H^⊥, ((¤v:A)^⊥ ⊗ (¤v:A ⅋ E)) ⅋ (¤v:A ⊗ F)
\]
The following sequents are easily proved in MELL:
\begin{gather*}
⊢ H^⊥, (¤v:A⅋(¤v:A)^⊥)⊗H \\
⊢ (¤v:A ⅋ ((¤v:A)^⊥ ⊗ E^⊥)) ⊗ ((¤v:A)^⊥ ⅋ F^⊥), E ⅋ (¤v:A ⊗ F)
\end{gather*}
so we get a proof of $⊢H^⊥,(¤v:A⊗F)⅋E$ by composition, which justifies the
typing:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ ¤v:A ⊗ F ⊢ Q[¤v/¤x] }
\Infer2[para]{ E ⅋ (¤v:A ⊗ F) &⊢ P\para Q[¤v/¤x] }
\Infer1[sub]{ H &⊢ P\para Q[¤v/¤x] }
\end{prooftree*}
The case of a reduction on a non-linear channel is similar, with some extra
work to handle duplication; details can be found in
appendix \ref{app:subject-reduction}.
\end{proof}
Remark that the introduction of negated atoms in the proof above makes us go
through environment \emph{formulas} that are not proper types, although
composition by cut provides a subtyping between environment types.
These intermediate steps correspond to the introduction in our system of axiom
rules that transport arbitrary behaviours (here the $¤v:A$) with no
counterpart in the terms, as a decomposition of the name passing mechanism.
This is similar to the central role of axioms in the
proofs-as-schedules \cite{beffara-2014-proof} paradigm.
\subsection{The role of “new”}
The subject reduction property is formulated for reductions on
private channels, {i.e.} names that are explicitly created in the term.
Indeed, the property fails without this assumption: not only is the type not
preserved (which is expected in the case of linear capabilities), but
communicated data may not have proper types.
For instance, in a typed term like
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Infer1[out]{ u:↑A ⊗ (v:A ⅋ E) ⊢ \outf{u}{v}.P }
\Hypo{ x:B ⊗ F ⊢ Q }
\Infer1[in]{ u:↓B ⊗ F ⊢ \inb{u}{x}.Q }
\Infer2[para]{ (u:↑A ⊗ (v:A ⅋ E)) ⅋ (u:↓B ⊗ F)
⊢ \outf{u}{v}.P \para \inb{u}{x}.Q }
\end{prooftree*}
the name $v$ has type $A$ but the name $x$ has type $B$, and there is no
reason that $A$ and $B$ are compatible, thus in general we cannot type the
reduct $P\para Q[v/x]$.
We do not consider this a serious defect of the system, it is mostly a matter
of presentation.
Indeed, the purpose of typing is to ensure proper composition of processes,
and the creation of channels is part of the composition operation.
Therefore, composition only makes sense in the presence of name creation
operators, and in the example above neither \rulename{new1} nor
\rulename{newω} applies if $A$ and $B$ do not match.
We could reformulate our system so that situations like the one above are
forbidden by typing.
A natural approach would be to enforce syntactic constraints on environment
types, for instance that linear capability assignments occur at most once,
that dual capability assignments have matching types, etc.
We chose not to include such restrictions for simplicity, relying on the
above justification.
\subsection{Properties of typed processes}
It can be proved that processes typed in our system are well-behaved:
\begin{theorem}[termination]
Typed processes have no infinite sequence of transitions on
private names.
\end{theorem}
\begin{theorem}[lock-freeness]
In every execution of a typed closed term, every active output eventually
interacts with an input.
\end{theorem}
Proofs of these facts can be obtained by realisability techniques, as in
previous work by the
author \cite{beffara-2005-logique,beffara-2006-concurrent}, or by syntactic
means by relating execution with the cut-elimination procedure of linear
logic.
We do not include proofs because they are out of the scope of this paper.
Nevertheless, a fundamental point in the arguments is that they rely on the
consistency of linear logic (through the cut-elimination property).
In relaxations of the system introduced in section \ref{sec:existing}, we will
express systems which do not enjoy those properties, by means of inconsistent
extensions of this logic.
\subsection{Variations}
The choice of the polyadic π-calculus in the presentation of our system is
justified by the fact that it is very expressive and also very standard.
However, we can adapt our approach to most variants of the calculus.
\subparagraph*{Asynchrony}
This is the restriction on outputs to have no
continuations \cite{boudol-1992-asynchrony}.
The typing of a free output atom $\outf{u}{¤v}$,
considered as a simple process $\outf{u}{¤v}.\nop$, is as follows:
\[
\begin{prooftree}
\Infer0[nop]{ ⊥ ⊢ \nop }
\Infer1[out]{ u:↑A ⊗ (¤v:A ⅋ ⊥) ⊢ \outf{u}{¤v}.\nop }
\end{prooftree}
\quad\leadsto\quad
\begin{prooftree}
\Infer0[out-async]{ u:↑A ⊗ ¤v:A ⊢ \outf{u}{¤v} }
\end{prooftree}
\]
where the simplified type is appropriate since it is linearly equivalent to
the one on the left, because of neutrality of $⊥$ for $⅋$.
Apart from this rule, nothing is changed in the system for the asynchronous
π-calculus.
\subparagraph*{Internal mobility}
This is the restriction that output prefixes only communicate distinct bound
names \cite{sangiorgi-1996-calculus}.
This simplifies the theory of the calculus and makes it symmetric like CCS.
In our type system, we also get the symmetry in typing rules.
For this purpose we can introduce duality over behaviour types:
\begin{definition}[duality]
For a behaviour type $A$, the dual $\dual{A}$ is defined inductively as:
\begin{align*}
\dual{↑A} &:= ↓A &
\dual{\oc A} &:= \wn\dual{A} &
\dual{A ⊗ B} &:= \dual{A} ⅋ \dual{B} &
\dual{1} &:= ⊥ \\
\dual{↓A} &:= ↑A &
\dual{\wn A} &:= \oc\dual{A} &
\dual{A ⅋ B} &:= \dual{A} ⊗ \dual{B} &
\dual{⊥} &:= 1
\end{align*}
\end{definition}
The dual $\dual{A}$ of a formula $A$ is a form of linear negation, except that
the dual of a capability $↑A$ is the capability $↓A$, whereas negations keep
capabilities unaffected in our environment formulas.
Note that we do not apply duality inside the capability, since we follow the
approach of i/o types, where this convention is the appropriate one.
Nevertheless, logically, the output capability contains a negation, as
illustrated by the bound output rule below.
\begin{lemma}[generalised new]\label{lemma:new*}
The following rule is derivable, assuming the tuple $¤x$ is made of
pairwise distinct names:
\begin{prooftree*}
\Hypo{ (¤x:A ⅋ ¤x:\dual{A}) ⊗ E ⊢ P }
\Infer1[new*]{ E ⊢ \new{¤x}P }
\end{prooftree*}
\end{lemma}
\begin{proof}
This is proved by induction $A$.
The base case is when $A$ is a linear or exponential capability, then one of
the \rulename{new} rules applies directly.
If $A=⊥$, then $¤x$ is empty and we have
$(¤x:A⅋¤x:\dual{A})⊗E=(⊥⅋1)⊗E≃1⊗E≃E$ so the rule holds by linear
equivalence.
The case $A=1$ is similar.
If $A=B⅋C$ then $¤x$ splits as $¤y,¤z$ so that we have
\begin{prooftree*}
\Hypo{ ((¤y:B ⅋ ¤z:C) ⅋ (¤y:\dual{B} ⊗ ¤z:\dual{C})) ⊗ E ⊢ P }
\Infer1[sub]{ (¤z:C ⅋ ¤z:\dual{C}) ⊗ (¤y:B ⅋ ¤y:\dual{B}) ⊗ E ⊢ P }
\Infer1[new*]{ (¤y:B ⅋ ¤y:\dual{B}) ⊗ E ⊢ \new{¤z}P }
\Infer1[new*]{ E ⊢ \new{¤y}\new{¤z}P }
\end{prooftree*}
where the \rulename{sub} rule is justified by a simple MLL proof.
The case $A=B⊗C$ is similar.
\end{proof}
Using this lemma, we can derive a typing rule for bound output:
\[
\begin{prooftree}
\Hypo{ ¤x:\dual{A} ⊗ E ⊢ P }
\Infer1[out]{ u:↑A ⊗ (¤x:A ⅋ (¤x:\dual{A} ⊗ E)) ⊢ \outf{u}{¤x}.P }
\Infer1[sub]{ u:↑A ⊗ (¤x:A ⅋ ¤x:\dual{A}) ⊗ E ⊢ \outf{u}{¤x}.P }
\Infer1[new*]{ u:↑A ⊗ E ⊢ \new{¤x}\outf{u}{¤x}.P }
\end{prooftree}
\quad\leadsto\quad
\begin{prooftree}
\Hypo{ ¤x:\dual{A} ⊗ E ⊢ P }
\Infer1[out-bound]{ u:↑A ⊗ E ⊢ \outb{u}{¤x}.P }
\end{prooftree}
\]
\subparagraph*{Fusions}
Our system can be extended to calculi with free input, such as
the fusion calculus \cite{parrow-1998-fusion}.
The appropriate formulation is with a preorder over
names \cite{hirschkoff-2013-name} generated by “arcs” $a/b$ which are explicit
substitution atoms.
The logical meaning of an arc is an implication $\oc(a:T⊸b:T)$ for any
capability type $T$: it allows a capability on $a$ to be used as a
capability on $b$; the modality is because the substitution is permanently
available.
The typing rule would be an axiom like $\wn(a:T⊗(b:T)^⊥)⊢a/b$; this
implies the handling of negative atoms, which may have an impact on the
structure of the system.
We defer the formal development of this extension to future work, since it
exceeds the scope of the present paper.
\section{Existing systems as fragments and extensions}
\label{sec:existing}
In this section, we describe formally how our system can express known type
systems for processes, using relaxations and identifying fragments.
By \emph{relaxation}, we mean that we add new logical rules to MELL in order
to prove more subtypings.
The resulting system need not be logically consistent, the minimal
requirements are that the new rules preserve
\begin{itemize}
\item the interpolation lemma, so that typing is still preserved under
structural congruence,
\item the substitution lemma, so that subject reduction still holds.
\end{itemize}
\subsection{Linearity and i/o types}
\label{sec:kpt}
We show here how our system can express plain i/o types à la Pierce and
Sangiorgi \cite{pierce-1993-typing} and their extension with linearity by
Kobayashi, Pierce and Turner \cite{kobayashi-1999-linearity} (hereafter
referred to as KPT).
We develop the relationship only with KPT, since plain i/o types are its
fragment without linear types.
We refer the reader to the original paper for the notations.
\begin{definition}
Let $ℒ$ be the fragment of KPT where
\begin{itemize}
\item in channel types, only pure input or output capabilities are used,
\item linear channel creations must create both capabilities,
\item the boolean data type is not used.
\end{itemize}
The translation $⟦⋅⟧$ maps channel types of $ℒ$ to channel types, tuples of
channel types of $ℒ$ to behaviour types and contexts of $ℒ$ to environment
types as follows:
\begin{gather*}
⟦ \oc^1¤T ⟧ := ↑⟦¤T⟧ \quad
⟦ \wn^1¤T ⟧ := ↓⟦¤T⟧ \quad
⟦ \oc^ω¤T ⟧ := \oc↑⟦¤T⟧ \quad
⟦ \wn^ω¤T ⟧ := \wn↓⟦¤T⟧ \\
⟦ T_1…T_n ⟧ := ⟦T_1⟧ ⊗ … ⊗ ⟦T_n⟧ \\
⟦ x:\oc^m¤T ⟧ := x:⟦\oc^m¤T⟧ \quad
⟦ x:\wn^m¤T ⟧ := x:⟦\wn^m¤T⟧ \quad
⟦ x:↕^m¤T ⟧ := x:⟦\oc^m¤T⟧ ⅋ x:⟦\wn^m¤T⟧ \\
⟦ x_1:T_1, … x_n:T_n ⟧ := ⟦x_1:T_1⟧ ⊗ … ⊗ ⟦x_n:T_n⟧
\end{gather*}
\end{definition}
The restriction on channel types is of minor importance as it can be lifted by
a simple coding: communicating channels with no capabilities is useless so it
can be removed, and instead of communicating a channel with both capabilities,
one can communicate each capability as distinct arguments.
As for the restriction on channel creation, it is harmless since a channel
created without both capabilities will never have any communication.
The exclusion of booleans is simply because our system, for simplicity of
presentation, does not include base data types; extension of the system with
such types is not problematic.
\begin{theorem}
A typing judgement $Γ⊢P$ holds in $ℒ$ if and only if the judgement $⟦Γ⟧⊢P$
holds in our system extended with the logical equivalences
\begin{align*}
A ⊗ B &≃ A ⅋ B &
1 &≃ ⊥ &
\oc A &≃ \wn A
\end{align*}
\end{theorem}
\begin{proof}[Sketch of proof]
Using these equivalences, environment types, up to associativity,
commutativity and neutrality, are just multisets of capability assignments of
the shape $x:T$ or $\oc x:T$.
Moreover, the multiplicity of each $\oc x:T$ does not matter.
Similarly, behaviour types are now just tuples of capabilities.
This provides a reverse mapping from our types to those of $ℒ$.
Then it is easy to check that each typing rule in $ℒ$ can be derived in our
system, which proves the direct implication.
For the reverse implication, we just have to check that our rules are also
valid in $ℒ$, only taking care of multiple occurrences of a name in an
environment type by appropriate constraints on the use of contraction.
\end{proof}
The addition of the logical equivalences can be achieved by adding to the
proof rules of MELL any rules that implement these equations as linear
equivalences (as new axiom rules or as new introduction rules for the
connectives involved; these methods are equivalent).
It is not hard to check that this relaxation does preserve the interpolation
and substitution lemmas.
Of course, lock-freeness and termination are lost, and this is directly
related to the fact that the equivalences make the logic inconsistent: cut
elimination is lost.
\subsection{Control, sequentiality, etc.}
\label{sec:hyb}
In a series of
works \cite{berger-2001-sequentiality,berger-2005-genericity,honda-2004-control,yoshida-2001-strong},
Berger, Honda and Yoshida studied refinements of i/o types with linearity
where various properties are enforced including sequentiality, strong
normalisation, or the behaviour of functional computation with control.
The latter system (hereafter called HYB, we refer the reader to the
paper \cite{honda-2004-control} for the notations) was put in precise
correspondence with proof nets for polarised linear logic by Honda and
Laurent \cite{honda-2010-exact} and this correspondence fits in our system.
\begin{definition}
The translation $⟦⋅⟧$ from HYB types to behaviour types, HYB contexts to
environment types and processes to processes is defined as follows:
\begin{gather*}
⟦ (¤{τ})^? ⟧ := ↑\dual{⟦¤{τ}⟧}, \quad
⟦ (¤{τ})^! ⟧ := ↓⟦¤{τ}⟧, \quad
⟦ τ_1 … τ_n ⟧ :=
(\oc⟦τ_1⟧ ⅋ \wn\dual{⟦τ_1⟧}) ⊗ … ⊗ (\oc⟦τ_n⟧ ⅋ \wn\dual{⟦τ_n⟧}) \\
⟦ x:τ_O ⟧ := \oc x:⟦τ_O⟧, \quad
⟦ x:τ_I ⟧ := \oc x:\dual{⟦τ_I⟧} ⅋ \wn x:⟦τ_I⟧, \\
⟦ \inr{x}{y_1…y_n}.P ⟧ := \inr{x}{y_1y'_1…y_ny'_n}.⟦ P ⟧
\qquad\text{with $y'_1,…,y'_n$ fresh,} \\
⟦ \outb{x}{y_1…y_n}P ⟧ :=
\new{y_1…y_n}(\outf{x}{y_1y_1…y_ny_n} \para ⟦ P ⟧) .
\end{gather*}
\end{definition}
This translation is essentially the isomorphism between π-calculus types à la
HYB and formulas of LLP, plus the capability indications.
A crucial difference is that we have to code every communication of a single
name as the communication of a pair for the input and the output capabilities,
since in HYB an input type $(¤{τ})^!$ actually allows the presence of outputs,
while our type system does not allow sending both capabilities as a single
argument.
Through this translation, we do capture HYB's typing, and the following
theorem is proved by writing translations between the two systems:
\begin{theorem}
A judgement $⊢P\triangleright x_1:τ_1,…,x_n:τ_n$ is derivable in HYB if and
only if the judgement $⟦x_1:τ_1⟧⊗…⊗⟦x_n:τ_n⟧⊢P$ is derivable in our system
extended with the equations $A⊗B≃A⅋B$ and $1≃⊥$.
\end{theorem}
Again, the identification of dual connectives makes the underlying logic
degenerate, and indeed the logic above does not ensure normalisation.
Honda and Laurent enumerate several restrictions of this system: acyclicity of
name dependence, input or output determinism, etc; in our system, these
restrictions mean than we do not identify dual connectives, then the theorem
above extends as an embedding of LLP/$π^c$ into our system.
The same approach can be used to handle other type systems of the same family,
we leave the formalisation of the correspondence for those systems to future
work.
\subsection{Session types}
\label{sec:session}
Caires and Pfenning \cite{caires-2010-session} formulated an equivalence
between dyadic session types \cite{honda-1993-types} and intuitionistic linear
logic, using a suitable interpretation of the connectives: $u:A⊸B$ means “on
$u$, receive a channel of type $A$ then proceed according to $B$”, dually
$u:A⊗B$ means “send a channel of type $A$, then proceed according to $B$”.
This implies that the type of a channel must change during an interaction,
following the progress of the session.
This seems to be incompatible with type systems in which a type is assigned to
each channel in a static way, including the present work, however the same
authors with DeYoung and Toninho \cite{deyoung-2012-cut} found a
reformulation of their correspondence (hereafter called DCPT) in the
asynchronous π-calculus where this contradiction vanishes.
The trick is that these channels must never have more than one active
occurrence per polarity and this can be turned into linearity by applying to
synchronous processes a translation $⟦⋅⟧$ defined as follows:
\begin{align*}
⟦ \inb{u}{x}.P ⟧ &:= \inb{u}{xu'}.⟦P⟧[u'/u] &
⟦ \outf{u}{v}.P ⟧ &:= \new{u'}(\outf{u}{vu'} \para ⟦P⟧[u'/v])
\end{align*}
where $u'$ is a fresh name that represents the state of $u$ at the next step
of interaction.
Of course, this translation does not make sense for general processes, but in
the case of the interaction discipline enforced by session types, this
transformation is perfectly adequate.
\begin{theorem}
Let $⟦⋅⟧$ be the following translation from LL formulas to channel types:
\begin{gather*}
⟦ 1 ⟧ := ↑⊥ \qquad
⟦ A ⊗ B ⟧ := ↑(⟦A^⊥⟧ ⅋ ⟦B^⊥⟧) \qquad
⟦ \oc A ⟧ := ↑\wn⟦A⟧ \\
⟦ ⊥ ⟧ := ↓⊥ \qquad
⟦ A ⅋ B ⟧ := ↓(⟦A⟧ ⅋ ⟦B⟧) \qquad
⟦ \wn A ⟧ := ↓\wn⟦A⟧ \\
⟦ x_1:A_1, …, x_n:A_n ⟧ := x_1:⟦A_1⟧ ⊗ … ⊗ x_n:⟦A_n⟧
\end{gather*}
If $Γ⊢P::x:A$ is derivable in DCPT then $⟦Γ⟧⊗x:⟦A^⊥⟧⊢P$ holds in our
system. \\
If $⟦Γ⟧⊗x:⟦A^⊥⟧⊢P$ holds, then $Γ⊢P'::x:A$ is derivable in DCPT for some
$P'≡P$.
\end{theorem}
\begin{proof}
The direct implication simply consists in checking that each rule of DCPT
translates in our system, which is straightforward.
For the reverse implication, we establish a standardisation result for our
type system (applied to the π-calculus with internal mobility) which
essentially eliminates the \rulename{sub} rule by cut elimination; we just
have to check that all permutations involved are structural congruences.
\end{proof}
\section{Discussion}
\label{sec:discussion}
\subparagraph*{More systems}
Our results are formulated in a π-calculus without choice using MELL as a
subtyping logic.
We chose to present this system since it illustrates the fundamental ideas of
our approach, but it can be naturally extended to a type system for the
π-calculus with choice, more liberal replication,
genericity \cite{berger-2005-genericity} etc using full linear logic, with
additives and second-order quantification.
We also conjecture that it should be possible to embed
systems of a different kind using modalities different from the $\oc$ and
$\wn$ of linear logic.
In particular, type systems that ensure termination by stratification of
names \cite{deng-2004-ensuring} should correspond to using our basic system
but replacing MELL with a form of light logic \cite{girard-1998-light} where
the operations on exponentials are constrained using stratification techniques
that are (at least superficially) similar.
\subparagraph*{Synchrony, or lack thereof}
The lock-freeness property that the system ensures is important but it implies
a serious defect of our system: it is very weak at dealing with prefixing.
A witness of this fact can be seen in the following derivation:
\[
\begin{prooftree}
\Hypo{ y:B ⊗ E ⊢ P }
\Infer1[in]{ v:↓B ⊗ E ⊢ \inb{v}{y}.P }
\Infer1[sub]{ x:A ⊗ F ⊢ \inb{v}{y}.P }
\Infer1[in]{ u(x):↓A ⊗ F ⊢ \inb{u}{x}.\inb{v}{y}.P }
\end{prooftree}
\]
Assuming that the names $u,v,x,y$ are all distinct, it is easy to prove (by
reasoning on the MELL proof of $⊢(x:A)^⊥⅋F^⊥,v:↓B⊗E$) that $F$ can actually be
written $v:↓B⊗F'$ up to associativity and commutativity, and that subsequently
the subtypings $x:A⊗F'≤E$ and hence $x:A⊗y:B⊗F'≤y:B⊗E$ hold.
Therefore the term $\inb{v}{y}.\inb{u}{x}.P$ will also by typeable by the same
type as above.
Hence our types are preserved by the equivalence
\[
\inb{u}{x}.\inb{v}{y}.P ≃ \inb{v}{y}.\inb{u}{x}.P
\]
The same argument applies to output prefixes and commutation between inputs
and outputs.
A consequence of this observation is that any typed equivalence over processes
must include the rule above, in other words our type system actually tells
about a very asynchronous calculus (this is nearly the calculus of
solos \cite{laneve-1999-solos} with restrictions on scopes, except that
prefixes can freely commute but not interact).
A deep reason for this state of things is that the discipline on names in
process composition stems from proof composition in linear logic, which
fundamentally works by enforcing acyclicity and connectedness in connections
between proofs \cite{danos-1989-structure}, in a \emph{commutative} context.
Indeed, the multiplicative connectives can be interpreted as follows:
\begin{itemize}
\item $E⊗F⊢P$ means that $P$ is expected to behave well in an environment that
provides some behaviour for $E$ and some behaviour for $F$, and those are
\emph{independent}.
\item $E⅋F⊢P$ means that $P$ is expected to behave well when these two
behaviours are \emph{correlated}, {i.e.} some events in $E$ can be prefixed
by events in $F$ and vice-versa.
\end{itemize}
With only this kind of information, there is no hope to have a type system
that would accept $\inb{a}{}.\inb{b}{}\para\outf{a}{}.\outf{b}{}$ but would
reject $\inb{a}{}.\inb{b}{}\para\outf{b}{}.\outf{a}{}$.
The only way out of this problem is either to extend the logic with
non-commutative connectives, or to introduce other forms of dependencies, for
instance through quantification.
\subparagraph*{Semantics}
This paper does not discuss semantic aspects of logic and processes, however
these are fundamental motivations of our approach.
We claim that the method of starting with a very constrained system and the
relaxing it in a controlled way using logical axioms should be fruitful
in this respect.
Realisability can be used to extract interpretations of formulas and terms
from syntax itself, using orthogonality as a generic form of testing.
It is efficient, in particular, for specifying operational properties of
processes, among which termination and lock-freeness.
Capabilities get interpreted by basic operational definitions while logic is
interpreted as in phase semantics, which justifies our use of entailment as
subtyping since, in such semantics, $E⊢F$ does imply the inclusion of $E$ into
$F$.
Besides, consistency of phase interpretation accepts some axioms (like the mix
rule or arbitrary weakening) but not others, which justifies the effects of
adding those axioms in our subtyping logic.
Another promising direction is the use of denotational semantics of proofs as
a way to build semantics of processes.
Evidence for this can be found, for instance, in the relational model of
linear logic: it is a non-trivial model of proofs, yet it supports the
identification of opposite types, as used in section \ref{sec:kpt} to rebuild
i/o types.
Using an appropriate interpretation for capability types, this should provide
meaningful denotational models for i/o-typed processes.
Besides, the flexibility of the relational model makes it suitable to
interpret differential linear logic, in which it is possible to formulate
encodings of processes of the calculus of solos \cite{ehrhard-2010-acyclic}.
Our approach thus provides new tools for the study of denotational models of
processes.
This could for instance extend a line of work of Varacca and
Yoshida \cite{varacca-2006-typed} interpreting the π-calculus in event
structures using logical constructs.
\appendix
\section{Technical appendix}
\subsection{Interpolation lemma (lemma \ref{lemma:interpolation})}
\label{app:interpolation}
\begin{proof}
We reason by induction on a cut-free proof $π$ of $⊢Γ,Δ$.
\begin{itemize}
\item If $π$ is an axiom rule, then three cases may occur:
\begin{itemize}
\item either $Γ$ and $Δ$ are equal and are a single formula, then $F:=Δ$
works,
\item or $Γ=A^⊥,A$ for some $A$ and $Δ$ is empty, then $F:=⊥$ works,
\item or $Γ$ is empty and $Δ=A^⊥,A$ for some $A$, then $F:=1$ works.
\end{itemize}
\item If $π$ is a $1$ rule then either $Γ=∅$ and $Δ=1$ or $Γ=1$ and $Δ=∅$.
In either case, $Γ^⊥,Δ$ is a singleton $\{F\}$ where $F$ provides the
expected result.
\item It $π$ ends with a $⊥$ rule, it has the shape
\[
\begin{prooftree}
\Hypo{ &⊢ Γ', Δ' }
\Infer1[$⊥$]{ &⊢ Γ', Δ', ⊥ }
\end{prooftree}
\qquad\text{with}\qquad
\begin{aligned}
& Γ = Γ', \quad Δ = ⊥, Δ' &&\text{or} \\
& Γ = Γ', ⊥, \quad Δ = Δ'
\end{aligned}
\]
We can apply the induction hypothesis on $⊢Γ',Δ'$, yielding proofs of
$⊢Γ',F$ and $⊢F^⊥,Δ'$, and conclude by adding a $⊥$ rule on the
appropriate side.
\item If $π$ ends with a $⊗$ rule, it has the shape
\[
\begin{prooftree}
\Hypo{ ⊢ Γ_1, A_1, Δ_1 }
\Hypo{ ⊢ Γ_2, A_2, Δ_2 }
\Infer2[$⊗$]{ ⊢ Γ_1, Γ_2, A_1⊗A_2, Δ_1, Δ_2 }
\end{prooftree}
\qquad\text{with}\qquad
\begin{aligned}
& Γ = Γ_1, Γ_2, \quad Δ = A_1⊗A_2, Δ_1, Δ_2 &&\text{or} \\
& Γ = Γ_1, Γ_2, A_1⊗A_2, \quad Δ = Δ_1, Δ_2
\end{aligned}
\]
In the first case, we proceed as follows using the induction hypothesis on
$Γ_i$ and $Δ_i,A_i$ for each $i$:
\[
\begin{prooftree}
\Infer0[IH]{ ⊢ Γ_1, F_1 }
\Infer0[IH]{ ⊢ Γ_2, F_2 }
\Infer2[$⊗$]{ ⊢ Γ_1, Γ_2, F_1⊗F_2 }
\end{prooftree}
\qquad
\begin{prooftree}
\Infer0[IH]{ ⊢ F_1^⊥, A_1, Δ_1 }
\Infer0[IH]{ ⊢ F_2^⊥, A_2, Δ_2 }
\Infer2[$⊗$]{ ⊢ F_1^⊥, F_2^⊥, A_1⊗A_2, Δ_1, Δ_2 }
\Infer1[$⅋$]{ ⊢ F_1^⊥⅋F_2^⊥, A_1⊗A_2, Δ_1, Δ_2 }
\end{prooftree}
\]
These proofs provide the expected conclusions, with $F:=F_1⊗F_2$.
As for the constraints on literals, the induction hypothesis gives us that
the atoms in each $F_i$ are present both in $Γ_i^⊥$ and in $Δ_i,A_i$, hence
the atoms in $F$ are present both in $Γ^⊥$ and in $Δ$.
The second case is similar except that we get $F:=F_1⅋F_2$.
\item If $π$ ends with a $⅋$ rule, it has the shape
\[
\begin{prooftree}
\Hypo{ &⊢ Γ', A, B, Δ' }
\Infer1[$⅋$]{ &⊢ Γ', A⅋B, Δ' }
\end{prooftree}
\qquad\text{with}\qquad
\begin{aligned}
& Γ = Γ', \quad Δ = A⅋B, Δ' &&\text{or} \\
& Γ = Γ', A⅋B, \quad Δ = Δ'
\end{aligned}
\]
In the first case we get a formula $F$ and proofs of $⊢Γ',F$ and
$⊢F^⊥,A,B,Δ'$ by induction hypothesis, and from the second one we
immediately deduce a proof of $⊢F^⊥,A⅋B,Δ'$, so the same $F$ is appropriate.
The second case is similar.
The constraint on atoms is immediately satisfied.
\item If $π$ ends with a dereliction, weakening or contraction rule, we get
the expected formula immediately by induction hypothesis on the premiss.
\item If $π$ ends with a promotion, it has the shape
\[
\begin{prooftree}
\Hypo{ &⊢ \wn Γ', A, \wn Δ' }
\Infer1[$\oc$]{ &⊢ \wn Γ', \oc A, \wn Δ' }
\end{prooftree}
\qquad\text{with}\qquad
\begin{aligned}
& Γ = \wn Γ', \quad Δ = \oc A, \wn Δ' &&\text{or} \\
& Γ = \wn Γ', \oc A, \quad Δ = \wn Δ'
\end{aligned}
\]
In the first case we get a formula $F$ and proofs of $⊢\wn Γ,F$ and
$⊢F^⊥,A,\wn Δ'$, then by dereliction and promotion we get $⊢\wn Γ,\oc F$ and
$⊢\wn F^⊥,\oc A,\wn Δ'$ (promotion on $A$) so $\oc F$ is appropriate.
In the second case, similarly, we get $\wn F$ as the intermediate formula.
\qedhere
\end{itemize}
\end{proof}
\subsection{Typing and structural congruence (lemma \ref{lemma:congruence})}
\label{app:congruence}
\begin{proof}
Thanks to lemma \ref{lemma:subtyping-commutation}, it is enough to
consider typing derivations where \rulename{sub} rules only occur right above \rulename{new}
rules (since no structural congruence rule involves inputs).
For associativity, commutativity and neutrality in parallel composition, the
associated properties for $⅋$ and $⊥$ are easily provable in multiplicative
linear logic.
For scope extrusion, consider a typed term $P\para\newk{x}{A}{k}Q$ where $x$
does not occur in $P$.
The typing derivation has the following shape:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ [x]_A^k ⊗ F &⊢ Q }
\Infer1[new$k$]{ F &⊢ \newk{x}{A}{k}Q }
\Infer2[para]{ E ⅋ F ⊢ P \para \newk{x}{A}{k}Q }
\end{prooftree*}
where $x$ does not occur in $E$ (since it does not occur in $P$) nor in $F$
(by the side condition in \rulename{new$k$}).
Then we can write the following derivation:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ [x]_A^k ⊗ F ⊢ Q }
\Infer2[para]{ E ⅋ ([x]_A^k ⊗ F) ⊢ P \para Q }
\Infer1[sub]{ [x]_A^k ⊗ (E ⅋ F) ⊢ P \para Q }
\Infer1[new]{ E ⅋ F ⊢ \newk{x}{A}{k}(P \para Q) }
\end{prooftree*}
where the subtyping is easily proved in MLL.
For the reverse rule, the typing of a term $\newk{x}{A}{k}(P\para Q)$ has the
following shape:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ F ⊢ Q }
\Infer2[para]{ E ⅋ F ⊢ P \para Q }
\Infer1[sub]{ [x]_A^k ⊗ G &⊢ P \para Q }
\Infer1[new]{ G &⊢ \newk{x}{A}{k}(P \para Q) }
\end{prooftree*}
The subtyping judgement is a proof in MELL of $⊢([x]_A^k)^⊥⅋G^⊥,E⅋F$, which
is equivalent to $⊢([x]_A^k)^⊥,G^⊥,E,F$.
By lemma \ref{lemma:interpolation}, we can deduce that there exists a MELL
formula $H$ such that $⊢G^⊥,E,H$ and $⊢H^⊥,([x]_A^k)^⊥,F$ are provable and
the literals in $H$ occur both in $G,E^⊥$ and in $([x]_A^k)^⊥,F$.
By hypothesis $x$ does not occur in $P$ so it does not occur in $E$, by the
side-condition on \rulename{new$k$} it does not occur in $G$ either,
therefore $x$ does not occur in $H$.
Therefore the literals in $H$ occur in $F$, which proves that $H$ only has
positive literals, so it is an environment type.
The proofs of $⊢G^⊥,E,H$ and $⊢H^⊥,([x]_A^k)^⊥,F$ induce subtypings $G≤E⅋H$
and $[x]_A^k⊗H≤F$ so we can conclude this case by the following typing:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ F &⊢ Q }
\Infer1[sub]{ [x]_A^k ⊗ H &⊢ Q }
\Infer1[new]{ H &⊢ \newk{x}{A}{k}Q }
\Infer2[para]{ E ⅋ H ⊢ P \para \newk{x}{A}{k}Q }
\Infer1[sub]{ G ⊢ P \para \newk{x}{A}{k}Q }
\end{prooftree*}
For commutation of restrictions, a typed term $\newk{x}{A}{k}\newk{y}{B}{ℓ}P$ must have a
derivation of the following shape:
\begin{prooftree*}
\Hypo{ [y]_B^ℓ ⊗ E &⊢ P }
\Infer1[new]{ E &⊢ \newk{y}{B}{ℓ}P }
\Infer1[sub]{ [x]_A^k ⊗ F &⊢ \newk{y}{B}{ℓ}P }
\Infer1[new]{ F &⊢ \newk{x}{A}{k}\newk{y}{B}{ℓ}P }
\end{prooftree*}
From $[x]_A^k⊗F≤E$ we deduce $[x]_A^k⊗[y]_B^ℓ⊗F≤[y]_B^ℓ⊗E$, so we
have the following typing:
\begin{prooftree*}
\Hypo{ [y]_B^ℓ ⊗ E &⊢ P }
\Infer1[sub]{ [x]_A^k ⊗ [y]_B^ℓ ⊗ F &⊢ P }
\Infer1[new]{ [y]_B^ℓ ⊗ F &⊢ \newk{x}{A}{k}P }
\Infer1[new]{ F &⊢ \newk{y}{B}{ℓ}\newk{x}{A}{k}P }
\end{prooftree*}
which validates the case of commutation.
\end{proof}
\subsection{Subject reduction (theorem \ref{thm:subject-reduction})}
\label{app:subject-reduction}
\begin{proof}
Thanks to lemma \ref{lemma:congruence}, we can reason up to structural
congruence.
For an interaction step between linear actions, we have
$\newl{u}{A}(\outf{u}{¤v}.P\para\inb{u}{¤x}.Q)→P\para Q[¤v/¤x]$.
The left-hand side is typed as follows:
\begin{prooftree*}
\Hypo{ E &⊢ P }
\Infer1[out]{ u:↑A ⊗ (¤v:A ⅋ E) &⊢ \outf{u}{¤v}.P }
\Hypo{ ¤x:A ⊗ F &⊢ Q }
\Infer1[in]{ u:↓A ⊗ F &⊢ \inb{u}{¤x}.Q }
\Infer2[para]{ (u:↑A⊗(¤v:A⅋E))⅋(u:↓A⊗F)
⊢ \outf{u}{¤v}.P\para\inb{u}{¤x}.Q }
\Infer1[sub]{ [u]_A^1 ⊗ H ⊢ \outf{u}{¤v}.P\para\inb{u}{¤x}.Q }
\Infer1[new1]{ H ⊢ \newl{u}{A}(\outf{u}{¤v}.P\para\inb{u}{¤x}.Q) }
\end{prooftree*}
with the hypothesis that no name in $¤x$ occurs in $F$.
The natural typing for the reduct is obtained as follows:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ ¤v:A ⊗ F ⊢ Q[¤v/¤x] }
\Infer2[para]{ E ⅋ (¤v:A ⊗ F) ⊢ P\para Q[¤v/¤x] }
\end{prooftree*}
By lemma \ref{lemma:mell-subst}, in the subtyping
$[u]_A^1⊗H≤(u:↑A⊗(¤v:A⅋E))⅋(u:↓A⊗F)$ we can replace the atomic formula
$u:↓A$ by $¤v:A$ and the atomic formula $u:↑A$ by $(¤v:A)^⊥$, then we get a
proof of
\[
⊢ ((¤v:A)^⊥ ⊗ ¤v:A) ⅋ H^⊥, ((¤v:A)^⊥⊗ (¤v:A ⅋ E)) ⅋ (¤v:A ⊗ F)
\]
The sequents
\begin{gather*}
⊢ H^⊥, (¤v:A⅋(¤v:A)^⊥)⊗H \\
⊢ (¤v:A ⅋ ((¤v:A)^⊥ ⊗ E^⊥)) ⊗ ((¤v:A)^⊥ ⅋ F^⊥), E ⅋ (¤v:A ⊗ F)
\end{gather*}
are easily provable in MLL so by the cut rule we get a proof of
$⊢H^⊥,(¤v:A⊗F)⅋E$ by which we can conclude with the typing of the reduct:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ ¤v:A ⊗ F ⊢ Q[¤v/¤x] }
\Infer2[para]{ E ⅋ (¤v:A ⊗ F) ⊢ P\para Q[¤v/¤x] }
\Infer1[sub]{ H ⊢ P\para Q[¤v/¤x] }
\end{prooftree*}
For an interaction step involving a replicated input, we have
\[
\newr{u}{A}(\outf{u}{¤v}.P\para\inr{u}{¤x}.Q\para R)
→ \newr{u}{A}(P\para Q[¤v/¤x]\para\inr{u}{¤x}.Q\para R) .
\]
The left-hand side is typed as follows:
\begin{prooftree*}
\Hypo{ E &⊢ P }
\Infer1[out]{ u:↑A ⊗ (¤v:A ⅋ E) &⊢ \outf{u}{¤v}.P }
\Hypo{ ¤x:A ⊗ F^! &⊢ Q }
\Infer1[in!]{ \wn u:↓A ⊗ F^! &⊢ \inr{u}{¤x}.Q }
\Hypo{ G ⊢ R }
\Infer3[para]{ (u:↑A ⊗ (¤v:A ⅋ E)) ⅋ (\wn u:↓A ⊗ F^!) ⅋ G
&⊢ \outf{u}{¤v}.P \para \inr{u}{¤x}.Q \para R }
\Infer1[sub]{ [u]_A^ω ⊗ H &⊢ \outf{u}{¤v}.P \para \inr{u}{¤x}.Q \para R }
\Infer1[newω]{ H &⊢ \newr{u}{A}(\outf{u}{¤v}.P \para \inr{u}{¤x}.Q \para R) }
\end{prooftree*}
Then we can deduce the following typing for the reduct without $\newr{u}{A}$:
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ ¤v:A ⊗ F^! ⊢ Q[¤v/¤x] }
\Hypo{ ¤x:A ⊗ F^! &⊢ Q }
\Infer1[rep]{ \wn u:↓A ⊗ F^! &⊢ \inr{u}{¤x}.Q }
\Hypo{ G ⊢ R }
\Infer4[para]{ E ⅋ (¤v:A ⊗ F^!) ⅋ (\wn u:↓A ⊗ F^!) ⅋ G
⊢ P \para Q[¤v/¤x] \para \inr{u}{¤x}.Q \para R }
\end{prooftree*}
The instance of \rulename{sub} in the first typing uses a proof of
\[
⊢ (\wn(u:↑A)^⊥ ⊗ \oc(u:↓A)^⊥) ⅋ H^⊥,
(u:↑A ⊗ (¤v:A ⅋ E)) ⅋ (\wn u:↓A ⊗ F^!) ⅋ G
\]
Let $O:=u:↑A$ (output), $I:=u:↓A$ (input) and $V:=¤v:A$ (value).
The sequent above is
\[
⊢ (\wn O^⊥ ⊗ \oc I^⊥) ⅋ H^⊥,
(O ⊗ (V ⅋ E)) ⅋ (\wn I ⊗ F^!) ⅋ G
\]
Because of the side condition in the rule \rulename{newω}
the name $u$ does not occur in $H$ so there is no other occurrence of the
literal $O^⊥$ in the above sequent, hence the linear atom $O$ can
only be introduced as follows, up to permutations of rules:
\begin{prooftree*}
\Infer0[ax]{ ⊢ O^⊥, O }
\Infer1[$\wn$]{ ⊢ \wn O^⊥, O }
\end{prooftree*}
If we replace this with
\begin{prooftree*}
\Infer0[ax]{ ⊢ V^⊥, V }
\Infer1[w]{ ⊢ \wn O^⊥, V^⊥, V }
\end{prooftree*}
and we introduce a $⅋$ between $\wn O^⊥$ and $V^⊥$ just before
$\wn O^⊥$ is involved in a tensor rule, we replace $O$ with $V$ in the
proof above and we get a proof of
\[
⊢ ((\wn O^⊥ ⅋ V^⊥) ⊗ \oc I^⊥) ⅋ H^⊥,
(V ⊗ (V ⅋ E)) ⅋ (\wn I ⊗ F^!) ⅋ G
\]
The subformula $\wn I$ is necessarily introduced by a (possibly
$η$-expanded) axiom rule that introduces $\oc I^⊥$, besides the latter
only occurs once so $\wn I$ is only introduced once and thus is not
involved in any contraction (except possibly with formulas introduced by
weakening, but this case can be eliminated), so if we replace this axiom by
an axiom on any formula and get another valid proof.
Using the formula $V^⊥⅋\wn I$ we get
\[
⊢ ((\wn O^⊥ ⅋ V^⊥) ⊗ (V ⊗ \oc I^⊥)) ⅋ H^⊥,
(V ⊗ (V ⅋ E)) ⅋ ((V^⊥ ⅋ \wn I) ⊗ F^!) ⅋ G
\]
Composing this with the following proofs:
\begin{prooftree*}
\Infer0[ax]{ ⊢ \wn O^⊥, \oc O }
\Infer0[ax]{ ⊢ V, V^⊥ }
\Infer2[$⊗$]{ ⊢ \wn O^⊥, \oc O ⊗ V, V^⊥ }
\Infer0[ax]{ ⊢ \oc I^⊥, \wn I }
\Infer2[$⊗$]{ ⊢ \wn O^⊥ ⊗ \oc I^⊥,
\oc O ⊗ V, V^⊥, \wn I }
\Infer[double]1[$⅋$]{ ⊢ \wn O^⊥ ⊗ \oc I^⊥,
(\oc O ⊗ V) ⅋ (V^⊥ ⅋ \wn I) }
\Infer0[ax]{ ⊢ H^⊥, H }
\Infer2[$⊗$]{ ⊢ \wn O^⊥ ⊗ \oc I^⊥, H^⊥,
((\oc O ⊗ V) ⅋ (V^⊥ ⅋ \wn I)) ⊗ H }
\Infer1[$⅋$]{ ⊢ (\wn O^⊥ ⊗ \oc I^⊥) ⅋ H^⊥,
((\oc O ⊗ V) ⅋ (V^⊥ ⅋ \wn I)) ⊗ H }
\end{prooftree*}
and
\begin{prooftree*}
\Infer0[ax,ax,$⊗$]{ ⊢ V^⊥, V ⊗ E^⊥, E }
\Infer1[$⅋$]{ ⊢ V^⊥ ⅋ (V ⊗ E^⊥), E }
\Infer0[ax]{ ⊢ V^⊥, V }
\Infer0[ax]{ ⊢ (F^!)^⊥, F^! }
\Infer2[$⊗$]{ ⊢ V^⊥, (F^!)^⊥, V ⊗ F^! }
\Infer0[ax]{ ⊢ \oc I^⊥, \wn I }
\Infer0[ax]{ ⊢ (F^!)^⊥, F^! }
\Infer2[$⊗$]{ ⊢ \oc I^⊥, (F^!)^⊥, \wn I ⊗ F^! }
\Infer2[$⊗$]{ ⊢ V^⊥ ⊗ \oc I^⊥, (F^!)^⊥, (F^!)^⊥, V ⊗ F^!, \wn I ⊗ F^! }
\Infer1[c]{ ⊢ V^⊥ ⊗ \oc I^⊥, (F^!)^⊥, V ⊗ F^!, \wn I ⊗ F^! }
\Infer1[$⅋$]{ ⊢ (V^⊥ ⊗ \oc I^⊥) ⅋ (F^!)^⊥, V ⊗ F^!, \wn I ⊗ F^! }
\Infer[separation=-4em]2[$⊗$]{ ⊢
(V^⊥ ⅋ (V ⊗ E^⊥)) ⊗ ((V^⊥ ⊗ \oc I^⊥) ⅋ (F^!)^⊥),
E, V ⊗ F^!, \wn I ⊗ F^! }
\Infer0[ax]{ ⊢ G^⊥, G }
\Infer[separation=-4em]2[$⊗$]{ ⊢
(V^⊥ ⅋ (V ⊗ E^⊥)) ⊗ ((V^⊥ ⊗ \oc I^⊥) ⅋ (F^!)^⊥) ⊗ G^⊥,
E, V ⊗ F^!, \wn I ⊗ F^!, G }
\Infer[double]1[$⅋$]{ ⊢
(V^⊥ ⅋ (V ⊗ E^⊥)) ⊗ ((V^⊥ ⊗ \oc I^⊥) ⅋ (F^!)^⊥) ⊗ G^⊥,
E ⅋ (V ⊗ F^!) ⅋ (\wn I ⊗ F^!) ⅋ G }
\end{prooftree*}
we get
\[
⊢ (\wn O^⊥ ⊗ \oc I^⊥) ⅋ H^⊥, E ⅋ (V ⊗ F^!) ⅋ (\wn I ⊗ F^!) ⅋ G
\]
that is
\[
⊢ (\wn (u:↑A)^⊥ ⊗ \oc(u:↓A)^⊥) ⅋ H^⊥,
E ⅋ (¤v:A ⊗ F^!) ⅋ (\wn(u:↓A) ⊗ F^!) ⅋ G
\]
hence we have
\begin{prooftree*}
\Hypo{ E ⊢ P }
\Hypo{ ¤v:A ⊗ F^! ⊢ Q[¤v/¤x] }
\Hypo{ ¤x:A ⊗ F^! &⊢ Q }
\Infer1[rep]{ \wn u:↓A ⊗ F^! &⊢ \inr{u}{¤x}.Q }
\Hypo{ G ⊢ R }
\Infer4[para]{ E ⅋ (¤v:A ⊗ F^!) ⅋ (\wn u:↓A ⊗ F^!) ⅋ G
⊢ P \para Q[¤v/¤x] \para \inr{u}{¤x}.Q \para R }
\Infer1[sub]{ (\wn (u:↑A) ⅋ \oc(u:↓A)) ⊗ H
⊢ P \para Q[¤v/¤x] \para \inr{u}{¤x}.Q \para R }
\Infer1[newω]{ H
⊢ \newr{u}{A}(P \para Q[¤v/¤x] \para \inr{u}{¤x}.Q \para R) }
\end{prooftree*}
which concludes the proof.
\end{proof}
\end{document} |
\begin{enumerate}gin{document}
\articletitle{Clustering stability: an overview}
\authorname1{Ulrike von Luxburg}
\author1email{[email protected]}
\author1address2ndline{Max Planck Institute for Biological
Cybernetics, T{\"u}bingen, Germany}
\abstract{A popular method for selecting the number of clusters is
based on stability arguments: one chooses the
number of clusters such that the corresponding clustering results
are ``most
stable''. In recent years, a series of papers has analyzed the
behavior of this method from a theoretical point of view. However,
the results are very technical and difficult to interpret for
non-experts. In this paper we give a high-level overview about the
existing literature on clustering stability. In addition to
presenting the results in a slightly informal but accessible way, we
relate them to each other and discuss their different
implications.
}
\maketitle
\cleardoublepage
\pagenumbering{roman}
\tableofcontents
\setcounter{page}{235}
\pagenumbering{arabic}
\setcounter{tocdepth}{4}
\chapter{Introduction} \label{sec-intro}
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\includegraphics[width=0.6\textwidth]{./figures/stability_idea.pdf}
\end{center}
\caption{Idea of clustering stability. Instable clustering solutions if the number of clusters is too small
(first row) or too large (second row). See text for
details. }
\label{fig-stability-idea}
\end{figure}
Model selection is a difficult problem in non-parametric
clustering. The obvious reason is that, as opposed to supervised
classification, there is no ground truth against which we could
``test'' our clustering results. One of the most pressing questions in
practice is how to determine the number of clusters. Various ad-hoc
methods have been suggested in the literature, but none of them is
entirely convincing. These methods usually suffer from the fact that
they implicitly have to define ``what a clustering is'' before they
can assign different scores to different numbers of clusters. In
recent years a new method has become increasingly popular:
selecting the number of clusters based on clustering
stability. Instead of defining ``what is a clustering'', the basic
philosophy is simply that a clustering should be a structure on the
data set that is ``stable''. That is, if applied to several data
sets from the same underlying model or of the same data generating
process, a clustering algorithm should obtain similar results. In this
philosophy it is not so important
how the clusters look (this is taken care of by the
clustering algorithm), but that they can be constructed
in a stable manner. \\
The basic intuition of why people believe that this
is a good principle can be described by \fig{fig-stability-idea}.
Shown is a data distribution with four underlying clusters
(depicted by the black circles), and different samples from this
distribution (depicted by red diamonds). If we cluster this data set into $K=2$
clusters, there are two reasonable solutions: a horizontal and a
vertical split. If a clustering algorithm is applied repeatedly to
different samples from this distribution, it might sometimes construct the
horizontal and sometimes the vertical solution. Obviously, these two
solutions are very different from each other, hence the clustering
results are instable. Similar effects take place if we start with
$K=5$. In this case, we necessarily have to split an existing cluster
into two clusters, and depending on the sample this could happen to
any of the four clusters. Again the clustering solution is
instable. Finally, if we apply the algorithm with the correct number
$K=4$, we observe stable results (not shown in the figure):
the clustering algorithm always discovers the correct clusters
(maybe up to a few outlier points). In this example, the
stability principle detects the correct number of clusters. \\
At first glance, using stability-based principles for model selection
appears to be very attractive. It is elegant as it avoids to define what
a good clustering is. It is a meta-principle that can be applied to
any basic clustering algorithm and does not require a particular
clustering model. Finally, it sounds ``very fundamental'' from a
philosophy of inference point of view. \\
However, the longer one thinks about this principle, the less obvious
it becomes that model selection based on clustering stability ``always
works''. What is clear is that solutions that are completely instable
should not be considered at all. However, if there are several stable
solutions, is it always the best choice to select the one
corresponding to the most stable results? One could
conjecture that the most
stable parameter always corresponds to the simplest solution, but clearly there exist
situations where the most simple solution is not what we are looking
for. To find
out how model selection based on clustering stability works we need
theoretical results. \\
In this paper we discuss a series of theoretical results on clustering
stability that have been obtained in recent years. In Section
\ref{sec-implementation} we review different protocols for how clustering
stability is computed and used for model selection. In Section
\ref{sec-kmeans} we concentrate on theoretical results for the
$K$-means algorithm and discuss their various relations. This is the
main section of the paper. Results for more general clustering
algorithms are presented in Section
\ref{sec-beyond}. \\
\chapter{Clustering stability: definition and implementation} \label{sec-implementation}
A {\em clustering of a data set $S = \{X_1, \hdots, X_n\}$} is a function that assigns labels
to all points of $S$, that is
$
\mathcal{C}_K: S \to \{1, \hdots, K\}.
$
Here $K$ denotes the number of clusters.
A {\em clustering algorithm} is a procedure that takes a set $S$ of
points as input and outputs a clustering of $S$.
The clustering algorithms considered in this paper take an additional
parameter as input, namely the number $K$ of clusters they are
supposed to construct.
We analyze clustering stability in a {\em statistical setup}. The
data set $S$ is assumed to consist of $n$ data points $X_1, \hdotsots, X_n$ that have been drawn
independently from some unknown underlying distribution $P$ on some
space $\mathcal{X}$. The final goal is to use these sample points to
construct a good partition of the underlying space $\mathcal{X}$. For some
theoretical results it will be easier to ignore sampling effects and
directly work on the underlying space $\mathcal{X}$ endowed with the
probability distribution $P$. This can be considered as the case of
having ``infinitely many'' data points. We sometimes call this the
limit case for $n \to
\infty$. \\
Assume we agree on a way to compute distances $d(\mathcal{C}, \mathcal{C}')$ between different
clusterings $\mathcal{C}$ and $\mathcal{C}'$ (see
below for details). Then, for a fixed probability distribution $P$, a fixed number $K$ of clusters
and a fixed sample size $n$, the {\em instability of a clustering
algorithm} is defined as the expected distance between two
clusterings $\mathcal{C}_K(S_n)$, $\mathcal{C}_K(S_n')$ on different data sets $S_n$, $S_n'$ of size $n$, that is
\banum \label{def-instab}
\instab(K, n) :=
E\begin{enumerate}gin{itemize}g( \;
d( \mathcal{C}_K(S_n),\mathcal{C}_K(S_n') ) \; \begin{enumerate}gin{itemize}g)
\eanum
The expectation is taken with respect to the drawing of the two
samples. \\
In practice, a large variety of methods has been devised to compute
stability scores and use them for model selection. On a very general
level they works as follows: \\
\ulesquote{
\small\tt
Given: a set $S$
of data points, a clustering
algorithm $\mathcal{A}$ that takes the number $k$ of clusters as input
\sloppy
\begin{enumerate}gin{enumerate}
\item For $k=2, \hdots, k_{\max}$
\setlength{\rightmargin}{0pt}
\begin{enumerate}gin{enumerate}
\item Generate perturbed versions $S_b$ $(b = 1,\hdots,b_{\max})$ of the original data set (for
example by subsampling or adding noise, see below)
\item For $b = 1, \hdots, b_{\max}$: \\
Cluster the data set $S_b$ with algorithm $\mathcal{A}$
into $k$ clusters to obtain clustering $\mathcal{C}_b$
\item For $b , b' = 1, \hdots, b_{\max}$: \\
Compute pairwise distances $d(\mathcal{C}_{b}, \mathcal{C}_{b'})$ between
these clusterings (using one of the distance functions described
below)
\item Compute instability as the mean distance between clusterings~$\mathcal{C}_b$:
\ba
\widehat{\instab}(k,n) = \frac{1}{b_{\max}^2}
\sum_{b, b' =1}^{b_{max}} \; d(\mathcal{C}_{b}, \mathcal{C}_{b'})
\ea
\end{enumerate}
\item Choose the parameter $k$ that gives the best stability, in
the simplest case as follows:
\ba
K := \argmin_k \widehat{\instab}(k,n)
\ea
(see below for more options).
\end{enumerate}
}
This scheme gives a very rough overview of how clustering stability can
be used for model selection. In practice, many details have to be
taken into account, and they will be discussed in the next
section. Finally, we want to mention an approach that is vaguely
related to clustering stability, namely the ensemble method
\citep{StrGho02}. Here, an
ensemble of {\em algorithms} is applied to one fixed data set. Then a
final clustering is built from the results of the individual
algorithms. We are not going to discuss this approach in our paper. \\
{\bf Generating perturbed versions of the data set. }
To be able to evaluate the stability of a fixed clustering algorithm
we need to run the clustering algorithm several times on slightly
different data sets. To this end we need to generate perturbed versions
of the original data set. In practice, the following schemes have
been used:
\begin{enumerate}gin{itemize}
\item Draw a random subsample of the original data set without
replacement
\citep{LevDom01,BenEliGuy02,FriDud01,LanRotBraBuh04}. \\
\item Add random noise to the original data
points \citep{Bittner00_long,MolRad06}. \\
\item If the original data set is high-dimensional, use different random
projections in low-dimensional spaces, and then cluster the
low-dimensional data sets \citep{SmoGho03}. \\
\item If we work in a model-based framework, sample data from the
model \citep{KerChu01}. \\
\item Draw a random sample of the original data with
replacement. This approach has not been reported in the literature
yet, but it avoids the problem of setting the size of the
subsample. For good reasons, this kind of sampling is the standard in the bootstrap
literature \citep{EfrTib93} and might also have advantages in the stability
setting. This scheme requires that the algorithm
can deal with weighted data points (because some data points will
occur several times in the sample). \\
\end{itemize}
In all cases, there is a trade-off that has to be treated
carefully. If we change the data set too much (for example, the
subsample is too small, or the noise too large), then we might destroy
the structure we want to discover by clustering. If we change the data
set too little, then the clustering algorithm will always obtain the
same results, and we will observe trivial stability. It is hard to
quantify this trade-off in practice. \\
{\bf Which clusterings to compare? }
Different protocols are used to compare the clusterings on the
different data sets $S_b$.
\begin{enumerate}gin{itemize}
\item Compare the clustering of the original data set with the
clusterings obtained on subsamples \citep{LevDom01}. \\
\item Compare clusterings of overlapping subsamples on the data
points where both clusterings are defined.
\citep{BenEliGuy02}. \\
\item Compare clusterings of disjoint subsamples
\citep{FriDud01, LanRotBraBuh04}. Here we first need to apply an
extension operator to extend each clustering to the domain of the other
clustering. \\
\end{itemize}
{\bf Distances between clusterings. } If two clusterings are defined
on the same data points, then it is straightforward to compute a
distance score between these clusterings based on any of the
well-known clustering distances such as the Rand index, Jaccard index,
Hamming distance, minimal matching distance, Variation of Information
distance \citep{Meila03_colt}. All these distances count, in some way or
the other, points or pairs of points on which the two clusterings
agree or disagree. The most convenient choice from a theoretical point
of view is the minimal matching distance. For two clusterings $\mathcal{C},
\mathcal{C}'$ of the same data set of $n$ points it is defined as
\banum \label{def-dmm}
d_{\text{MM}}(\mathcal{C}, \mathcal{C}') :=
\min_\pi
\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{ \{\mathcal{C}(X_i) ^{(n)}eq \pi( \mathcal{C}'(X_i)) \}}
\eanum
where the minimum is taken over all permutations $\pi$ of the $K$
labels. Intuitively, the minimal matching distance measures the same
quantity as the 0-1-loss used in supervised classification. For a
stability study involving the adjusted Rand index or an adjusted
mutual information index see \citet{VinEpp09}. \\
If two clusterings are defined on different data sets one has two
choices. If the two data sets have a big overlap one can use a {\em
restriction operator} to restrict the clusterings to the points
that are contained in both data sets. On this restricted set one can
then compute a standard distance between the two clusterings. The
other possibility is to use an
{\em extension operator} to extend both clusterings from their domain
to the domain of the other clustering. Then one can compute a standard
distance between the two clusterings as they are now both defined on the
joint domain. For center-based clusterings, as constructed by the
$K$-means algorithm, a natural extension operator exists. Namely, to a
new data point we simply assign the label of the closest cluster
center. A more general scheme to extend an existing clustering to new
data points is to train a classifier on the old data points and use
its predictions as labels on the new data points. However, in the
context of clustering stability it is not obvious what kind of bias we
introduce with this approach. \\
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\includegraphics[width=0.49\textwidth]{./figures/normalization_uniform.pdf}
\includegraphics[width=0.49\textwidth]{./figures/normalization_four_gaussians.pdf}
\end{center}
\caption{Normalized stability scores. Left plots: data points from a
uniform density on $[0,1]^2$. Right plots: data points from a
mixture of four well-separated Gaussians in $\RR^2$. The first row
always shows the unnormalized instability $\widehat{\instab}$ for $K=2, ...,
15$. The second row shows the instability $\widehat{\instab}_{\text{norm}}$
obtained on a reference distribution (uniform distribution). The
third row shows the normalized stability $\widehat{\instab}_{\text{norm}}$.}
\label{fig-stability-normalized}
\end{figure}
{\bf Stability scores and their normalization. }
The stability protocol outlined above results in a set of distance
values $(d(\mathcal{C}_{b}, \mathcal{C}_{b'}))_{b,b' = 1, \hdots, b_{max}}$. In most approaches, one summarizes
these values by taking their mean:
\ba
\widehat{\instab}(k,n) = \frac{1}{b_{\max}^2}
\sum_{b, b' =1}^{b_{max}} \; d(\mathcal{C}_{b}, \mathcal{C}_{b'})
\ea
Note that the mean is the simplest summary statistic one can compute based on the distance
values $d(\mathcal{C}_{b}, \mathcal{C}_{b'})$. A different approach is to use the area under the cumulative
distribution function of the distance values as the stability score, see
\citet{BenEliGuy02} or \citet{BerVal07} for details. In principle one could also
come up with more elaborate statistics based on distance values.
To the best of our knowledge, such concepts
have not been used so far. \\
The simplest way to select the number $K$
of clusters is to minimize the instability:
\ba
K = \argmin_{k=2,\hdots,k_{\max}} \widehat{\instab}(k,n).
\ea
This approach has been suggested in \citet{LevDom01}. However, an
important fact to note is that $\widehat{\instab}(k,n)$ trivially scales
with $k$, regardless of what the underlying data structure is. For
example, in the top left plot in
Figure~\ref{fig-stability-normalized} we can see that even
for a completely unclustered data set, $\widehat{\instab}(n,k)$
increases with $k$.
When using stability for model selection, one should correct for
the trivial scaling of $\widehat{\instab}$, otherwise it might be
meaningless to take the minimum afterwards. There exist several different {\em normalization}
protocols:
\begin{enumerate}gin{itemize}
\item Normalization using a reference null distribution
\citep{FriDud01,BerVal07}. One repeatedly samples data sets from
some reference null distribution. Such a distribution is defined on
the same domain as the data points, but does not possess any cluster
structure. In simple cases one can use the uniform distribution on
the data domain as null distribution. A more practical approach is
to scramble the individual dimensions of the existing data points
and use the ``scrambled points'' as null distribution (see
\citealp{FriDud01,BerVal07} for details). Once we have drawn several data sets from
the null distribution, we cluster them using our clustering
algorithm and compute the corresponding stability score
$\widehat{\instab}_{\text{null}}$ as above. The {\em normalized stability} is then
defined as $\widehat{\instab}_{\text{norm}} := \widehat{\instab} /
\widehat{\instab}_{\text{null}}$.\\
\item Normalization by random labels \citep{LanRotBraBuh04}. First, we
cluster each of the data sets $S_b$ as in the protocol above to
obtain the clusterings $\mathcal{C}_b$. Then, we randomly permute these
labels. That is, we assign the label to data point $X_i$ that
belonged to $X_{\pi(i)}$, where $\pi$ is a permutation of
$\{1,\hdots,n\}$. This leads to a permuted clustering $\mathcal{C}_{b, \text{
perm}}$. We then compute the stability score $\widehat{\instab}$ as above,
and similarly we compute $\widehat{\instab}_{\text{perm}}$ for the permuted
clusterings. The {\em normalized stability} is then defined as
$\widehat{\instab}_{\text{norm}} := \widehat{\instab} / \widehat{\instab}_{\text{perm}}$.\\
\end{itemize}
Once we computed the normalized stability scores $\widehat{\instab}_{\text{norm}}$ we
can choose the number of clusters that has smallest normalized instability, that
is
\ba
K = \argmin_{k=2,\hdots,k_{\max}} \widehat{\instab}_{\text{norm}}(k,n)
\ea
This approach has been taken for example in \citet{BenEliGuy02,LanRotBraBuh04}. \\
{\bf Selecting $K$ based on statistical tests. }
A second approach to select the final number of clusters is to use
a statistical test. Similarly to the normalization
considered above, the idea is to compute stability scores not only on the
actual data set, but also on ``null data sets'' drawn from some
reference null distribution. Then one tests whether,
for a given parameter $k$,
the stability on the actual data is significantly larger than the one
computed on the null data. If there are several values $k$ for which this
is the case, then one selects the one that is most
significant. The most well-known implementation of such a procedure
uses bootstrap methods \citep{FriDud01}. Other authors use a
$\chi^2$-test
\citep{BerVal07}
or a test based on the Bernstein inequality
\citep{BerVal08}. \\
To summarize, there are many different implementations for selecting the
number $K$ of clusters based on stability scores. Until now, there
does not exist any convincing empirical study that thoroughly
compares all these approaches on a variety of data sets. In my
opinion, even fundamental issues such as the normalization have not
been investigated in enough detail. For example, in my experience
normalization often has no effect whatsoever
(but I did not conduct a thorough study either). To put stability-based model selection
on a firm ground it would be crucial to compare the different
approaches with each other in an extensive case study. \\
\chapter{Stability analysis of the $K$-means algorithm} \label{sec-kmeans}
The vast majority of papers about clustering stability use the
$K$-means algorithm as basic clustering algorithm. In this section we
discuss the stability results for the $K$-means algorithm in depth. Later, in Section \ref{sec-beyond} we will see how these results can be
extended to other clustering algorithms. \\
For simpler reference we briefly recapitulate the $K$-means algorithm
(details can be found in many text books, for example
\citealp{HasTibFri01}). Given a set of $n$ data points $X_1, \hdotsots,
X_n \in \RR^d$ and a fixed number $K$ of clusters to construct, the
$K$-means algorithm attempts to minimize the clustering objective
function
\banum \label{eq-kmeans-objective-finite}
Q^{(n)}_K(c_1, \hdotsots, c_K) = \frac{1}{n}
\sum_{i=1}^n
\min_{k=1,..,K}
\|X_i - \text{c}_k\|^2
\eanum
where $c_1, \hdotsots, c_K$ denote the centers of the $K$ clusters.
In the limit $n \to \infty$, the $K$-means clustering
is the one that minimizes the limit objective
function
\banum \label{eq-kmeans-objective-limit}
Q^{(\infty)}_K(c_1, \hdotsots, c_K) =
\int
\min_{k=1,..,K}
\|X - \text{c}_k\|^2 \; dP(X)
\eanum
where $P$ is the underlying probability distribution. \\
Given an
initial set $c^{<0>} = \{c_1^{<0>}, \hdotsots, c_K^{<0>}\}$ of centers, the $K$-means
algorithm iterates the following two steps until convergence: \\
\ulesquote{\small\tt
\begin{enumerate}gin{enumerate}
\item Assign data points to closest cluster centers:
\ba
\forall i=1, \hdotsots, n: \;\;\; \mathcal{C}^{<t>}(X_i) := \argmin_{k = 1, \hdotsots K} \| X_i - c_k^{<t>} \|
\ea
\item Re-adjust cluster means:
\ba
\forall k=1, \hdotsots, K: \;\;\;
c_k^{<t+1>} : =
\frac{1}{N_k}
\sum_{ \{i \;|\; \mathcal{C}^{<t>}(X_i) = k\} }
X_i
\ea
where $N_k$ denotes the number of points in cluster $k$. \\
\end{enumerate}
}
It is well known that, in general, the $K$-means algorithm terminates in a
local optimum of $Q^{(n)}_K$ and does not necessarily find the global
optimum. We study the $K$-means algorithm in two
different scenarios: \\
{\bf The idealized scenario: } Here we assume an idealized algorithm
that always finds the {\em global} optimum of the $K$-means objective
function $Q^{(n)}_K$. For simplicity, we
call this algorithm the idealized $K$-means algorithm. \\
{\bf The realistic scenario: } Here we analyze the actual $K$-means
algorithm as described above. In particular, we take into account its
property of getting stuck in local optima. We also take into
account the initialization of the algorithm.\\
Our theoretical investigations are based on the following simple protocol to compute the
stability of the $K$-means algorithm:
\begin{enumerate}
\item We assume to have access to as many samples of size $n$ of the
underlying distribution as we want. That is, we ignore artifacts
introduced by computing stability on artificial perturbations of a
fixed, given sample.
\item As distance between two $K$-means clusterings of two samples
$S$, $S'$ we use the minimal matching distance between the extended
clusterings on the domain $S \cup S'$.
\item We work with the expected
minimal matching distance as in Equation \ref{def-instab}, that
is we analyze $\instab$ rather than the practically
used $\widehat{\instab}$. This does not do
much harm as instability scores are highly concentrated around their
means anyway.
\end{enumerate}
\section{The idealized $K$-means algorithm} \label{sec-idealized}
In this section we focus on the idealized $K$-means algorithm, that is
the algorithm that always finds the global optimum $c^{(n)}$ of the $K$-means
objective function:
\ba
c^{(n)} := (c^{(n)}_1, \hdots, c^{(n)}_K) \; := \;\argmin_c \; Q^{(n)}_K(c).
\ea
\subsection{First convergence result and the role of symmetry} \label{sec-convergence-simple}
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
a. \includegraphics[width=0.2\textwidth]{./figures/too_few_clusters_stable.pdf}
b. \includegraphics[width=0.2\textwidth]{./figures/too_few_clusters_unstable.pdf}
c. \includegraphics[width=0.2\textwidth]{./figures/too_many_clusters_stable.pdf}
d. \includegraphics[width=0.2\textwidth]{./figures/too_many_clusters_unstab.pdf}
\end{center}
\caption{If data sets are not symmetric, idealized
$K$-means is stable even if the number $K$ of clusters is too small
(Figure a) or too large (Figure c). Instability of the wrong number
of clusters only occurs in symmetric data sets (Figures b and d). }
\label{fig-stability-wrong}
\end{figure}
The starting point for the results in this section is the following
observation \citep{BenLuxPal06}.
Consider the situation in \fig{fig-stability-wrong}a. Here the data
contains three clusters, but two of them are closer to each other than
to the third cluster. Assume we run the idealized $K$-means algorithm
with $K=2$ on such a data set. Separating the left two clusters from
the right cluster (solid line) leads to a much better value of $Q^{(n)}_K$
than, say, separating the top two clusters from the bottom one (dashed
line). Hence, as soon as we have a reasonable amount of data,
idealized (!) $K$-means with $K=2$ always constructs the first
solution (solid line). Consequently, it is stable in spite of the fact
that $K=2$ is the wrong number of clusters. Note that this would not
happen if the data set was symmetric, as depicted in
\fig{fig-stability-wrong}b. Here neither the solution depicted by the
dashed line nor the one with the solid line is clearly superior, which
leads to instability if the idealized $K$-means algorithm is applied to
different samples. Similar examples can be constructed to detect that
$K$ is too large, see \fig{fig-stability-wrong}c and d. With
$K=3$ it is clearly the best solution to split the big cluster in
\fig{fig-stability-wrong}c, thus clustering this data set is
stable. In \fig{fig-stability-wrong}d, however, due to symmetry
reasons neither splitting the top nor the bottom cluster
leads to a clear advantage. Again this leads to instability.\\
These informal observations suggest that unless the data set contains
perfect symmetries, the idealized $K$-means algorithm is stable even
if $K$ is wrong. This can be formalized with the following theorem.
\begin{enumerate}gin{theorem}[Stability and global optima of the objective function]
\label{th-convergence-simple}
Let $P$ be a
probability distribution on $\RR^d$ and $Q^{(\infty)}_K$ the limit $K$-means objective
function as defined in \eq{eq-kmeans-objective-limit}, for some
fixed value $K > 1$.
\begin{enumerate}gin{enumerate}
\item If $Q^{(\infty)}_K$ has a unique global minimum, then the idealized $K$-means
algorithm is perfectly stable when $n \to \infty$, that is
\ba
\lim_{n \to \infty} \instab(K,n) = 0.
\ea
\item If $Q^{(\infty)}_K$ has several global minima (for example, because the probability
distribution is symmetric), then the idealized $K$-means
algorithm is instable, that is
\ba
\lim_{n \to \infty} \instab(K,n) > 0.
\ea
\end{enumerate}
\end{theorem}
This theorem has been proved (in a slightly more general setting) in
\citet{BenLuxPal06} and \citet{BenPalSim07}. \\
{\em Proof sketch, Part 1. } It is well known that if the
objective function $Q^{(\infty)}_K$ has a unique global minimum, then the centers
$c^{(n)}$
constructed by the idealized $K$-means algorithm on a sample of $n$ points almost surely
converge to the true population centers $c^{(*)}$ as $n \to
\infty$ \citep{Pollard81}. This means that given some $\ensuremath{\varepsilon} > 0$ we can find some large
$n$ such that $c^{(n)}$ is $\ensuremath{\varepsilon}$-close to $c^{(*)}$ with high probability. As a
consequence, if we compare two clusterings on different samples of
size $n$, the
centers of the two clusterings are at most $2\ensuremath{\varepsilon}$-close to each
other. Finally, one can show that if the cluster centers of two
clusterings are $\ensuremath{\varepsilon}$-close, then their minimal matching distance is small
as well. Thus, the expected distance between the clusterings
constructed on two samples of size $n$ becomes arbitrarily small and
eventually converges to 0 as $n \to \infty$. \\
{\em Part 2.} For simplicity, consider the symmetric situation in
\fig{fig-stability-wrong}a. Here the probability distribution has
three axes of symmetry. For $K=2$ the objective function $Q^{(\infty)}_2$ has three
global minima $c^{(*1)},c^{(*2)},c^{(*3)}$ corresponding to the three
symmetric solutions. In such a situation, the idealized $K$-means
algorithm on a sample of $n$ points gets arbitrarily close to one of
the global optima, that is $\min_{i=1, \hdots, 3} d(c^{(n)}, c^{(*i)}) \to 0$
\citep{Lember03}. In particular, the sequence $(c^{(n)})_n$ of empirical
centers has three convergent subsequences, each of which converge to
one of the global solutions. One can easily conclude that if we
compare two clusterings on random samples, with probability 1/3 they
belong to ``the same subsequence'' and thus their distance will become
arbitrarily small. With probability 2/3 they ``belong to different
subsequences'', and thus their distance remains larger than a constant
$a >0$.
From the latter we can conclude that
$\instab(K,n)$ is always larger than $2a/3$.
\smiley \\
The interpretation of this theorem is distressing. The
stability or instability of parameter $K$ does not depend on whether
$K$ is ``correct'' or ``wrong'', but only on whether the $K$-means
objective function for this particular value $K$ has one or several
global minima. However, the number of global minima is usually not
related to the number of clusters, but rather to the fact that
the underlying probability distribution has symmetries. In
particular, if we consider ``natural'' data distributions, such
distributions are rarely perfectly symmetric. Consequently, the
corresponding functions $Q^{(\infty)}_K$
usually only have one global minimum, for any value
of $K$. In practice this means that for a large sample size $n$, the
idealized $K$-means algorithm {\em is stable for any value of
$K$}. This seems to suggest that model selection based on clustering
stability does not work. However, we will see later in
\sec{sec-relationships} that this result is essentially an artifact of
the idealized clustering setting and does not carry over to the
realistic setting. \\
\subsection{Refined convergence results for the case of a unique
global minimum} \label{sec-convergence-refined}
Above we have seen that if, for a particular distribution $P$ and a
particular value $K$, the objective function $Q^{(\infty)}_K$ has a unique global
minimum, then the idealized $K$-means algorithm is stable in the sense
that $\lim_{n \to \infty} \instab(K,n) = 0$. At first glance, this
seems to suggest that stability cannot distinguish between
different values $k_1$ and $k_2$ (at least for large $n$). However,
this point of view is too simplistic. It can happen that
even though both $\instab(k_1, n)$ and $\instab(k_2, n)$ converge to 0
as $n \to \infty$, this happens ``faster'' for $k_1$ than for $k_2$.
If measured relative to the absolute
values of $\instab(k_1,n)$ and $\instab(k_2,n)$, the difference
between $\instab(k_1, n)$ and $\instab(k_2,n)$ can still be large
enough to be
``significant''. \\
The key in verifying this intuition is to study the limit process more
closely. This line of work has been established by Shamir and Tishby
in a series of papers
\citep{ShaTis08_colt,ShaTis08_nips,ShaTis09_nips}. The main idea is
that instead of studying the convergence of $\instab(k,n)$ one needs
to consider the rescaled instability $\sqrt{n} \cdot \instab(k,
n)$. One can prove that the rescaled instability converges in
distribution, and the limit distribution depends on $k$. In
particular, the means of the limit distributions are different for
different values of $k$. This can be formalized as follows.
\begin{enumerate}gin{theorem}[Convergence of rescaled stability]
\label{th-convergence-refined}
Assume that the probability distribution $P$ has a density
$p$. Consider a fixed parameter $K$, and assume that the
corresponding limit objective function $Q^{(\infty)}_K$ has a unique global
minimum $c^{(*)} = (c^{(*)}_1,\hdots, c^{(*)}_K)$. The boundary between clusters $i$ and $j$ is
denoted by $B_{ij}$. Let $m \in \mathbb{N}$, and $S_{n,1}, \hdots, S_{n,
2m}$ be samples of size $n$ drawn independently from $P$. Let
$\mathcal{C}_K(S_{n,i})$ be the result of the idealized $K$-means
clustering on sample $S_{n,i}$. Compute the instability as mean
distance between clusterings of disjoint pairs of samples, that is
\banum \label{def-instab-ohad}
\overline{\instab}(K,n) :=
\frac{1}{m} \sum_{i=1}^m d_{\text{MM}} \begin{enumerate}gin{itemize}g(\mathcal{C}_K(S_{n,2i-1}),
\mathcal{C}_K(S_{n,2i}) \begin{enumerate}gin{itemize}g).
\eanum
Then, as $n \to \infty$ and $m \to \infty$,
the rescaled instability $\sqrt{n} \cdot \overline{\instab}(K,n)$
converges in probability to
\banum \label{eq-rinstab}
\text{RInstab}(K) :=
\sum_{1 \leq i < j \leq K} \;
\int_{B_{ij}} \;
\frac{ V_{ij}}{ \| c^{(*)}_i - c^{(*)}_j \| } \; p(x) dx,
\eanum
where $V_{ij}$ stands for a term describing the
asymptotics of the random fluctuations of the cluster boundary between
cluster $i$ and cluster $j$ (exact formula given in
\citealp{ShaTis08_colt,ShaTis09_nips}).
\end{theorem}
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\fbox{\parbox[t]{\textwidth}{
\scriptsize
\begin{enumerate}gin{minipage}{0.15\textwidth}
\mbox{}
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\scriptsize
distribution of $d(\mathcal{C}, \mathcal{C}')$\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.2\textwidth}
\mbox{}
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\scriptsize
distribution of $\sqrt{n} \cdot d(\mathcal{C}, \mathcal{C}')$\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.15\textwidth}
\scriptsize
$k$ fixed \\
$n=10^2$:
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{figures/wide_distribution.pdf}\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.24\textwidth}
\small$\xrightarrow{\text{scale with } \sqrt{n}=10}$
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{figures/scaled_distribution3.pdf}\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.15\textwidth}
\scriptsize
$n=10^4$:
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{figures/narrow_distribution.pdf}\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.24\textwidth}
\small$\xrightarrow{\text{scale with } \sqrt{n}=100}$
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{figures/scaled_distribution3.pdf}\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.15\textwidth}
\centerline{$\downarrow$}
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\centerline{$\downarrow$}
\end{minipage}
\begin{enumerate}gin{minipage}{0.24\textwidth}
\mbox{}
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\centerline{$\downarrow$}
\end{minipage}
\begin{enumerate}gin{minipage}{0.15\textwidth}
\scriptsize
$n=\infty$:
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{figures/converged_distribution.pdf}\\
\end{minipage}
\begin{enumerate}gin{minipage}{0.24\textwidth}
\mbox{}
\end{minipage}
\begin{enumerate}gin{minipage}{0.29\textwidth}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{figures/scaled_distribution2.pdf}\\
\end{minipage}
}}
\end{center}
\caption{Different convergence processes. The left column shows the
convergence studied in \th{th-convergence-simple}.
As the sample size $n \to \infty$, the distribution of distances $d_{\text{MM}}(\mathcal{C}, \mathcal{C}')$
is degenerate, all mass is concentrated on 0. The right column shows
the convergence studied in \th{th-convergence-refined}. The
rescaled distances converge to a non-trivial distribution, and its
mean (depicted by the cross) is positive. To go from the left to the
right side one has to rescale by $\sqrt{n}$.
}
\label{fig-convergence}
\end{figure}
Note that even though the definition of instability in
\eq{def-instab-ohad}
differs slightly from the definition in \eq{def-instab}, intuitively
it measures the same quantity. The definition in \eq{def-instab-ohad}
just has the technical advantage that all pairs of samples are
independent from one another. \\
{\em Proof sketch. } It is well known that if $Q^{(\infty)}_K$ has a unique global
minimum, then the centers constructed by the idealized $K$-means
algorithm on a finite sample satisfy a central limit theorem \citep{Pollard82}. That is,
if we rescale the distances between the sample-based centers and the
true centers with the factor $\sqrt{n}$, these rescaled distances
converges to a normal distribution as $n \to \infty$. When the cluster centers converge, the same can be
said about the cluster boundaries. In this case, instability
essentially counts how many points change side when the cluster
boundaries move by some small amount. The points that potentially
change side are the points close to the boundary of the true limit
clustering. Counting these points is what the integrals $\int_{B_{ij}}
... p(x) dx$ in the definition of $\text{RInstab}$
take care of. The exact characterization of how the cluster
boundaries ``jitter'' can be derived from the central limit
theorem. This leads to the term $V_{ij} / \|c_i^{(*)} - c_j^{(*)}\|$ in the
integral. $V_{ij}$ characterizes how the cluster centers themselves
``jitter''. The normalization $\|c_i^{(*)} - c_j^{(*)}\|$ is needed to transform
jittering of cluster centers to jittering of cluster boundaries: if
two cluster centers are very far apart from each other, the cluster
boundary only jitters by a small amount if the centers move by
$\ensuremath{\varepsilon}$, say. However, if the centers are very close to each other (say, they
have distance $3\ensuremath{\varepsilon}$), then moving the centers by $\ensuremath{\varepsilon}$ has a large
impact on the cluster boundary. The details of this proof are very
technical, we refer the interested reader to
\citealt{ShaTis08_colt,ShaTis09_nips}.
\smiley\\
Let us briefly
explain how the result in \th{th-convergence-refined} is compatible
with the result in \th{th-convergence-simple}. On a high level, the
difference between both results resembles the difference between the
law of large numbers and the central limit theorem in probability
theory. The LLN studies the convergence of the mean of a sum of random
variables to its expectation (note that $\instab$ has the form of a
sum of random variables). The CLT is concerned with the same
expression, but rescaled with a factor $\sqrt{n}$. For the rescaled
sum, the CLT then gives results on the convergence in distribution.
Note that in the particular case of instability, the distribution of
distances lives on the non-negative numbers only. This is why the
rescaled instability in \th{th-convergence-refined} is positive and
not 0 as in the limit of $\instab$ in
Theorem~\ref{th-convergence-simple}. A toy figure explaining the
different convergence processes can be seen in \fig{fig-convergence}.\\
Theorem \ref{th-convergence-refined} tells us that different parameters $k$
usually lead to different rescaled stabilities in the limit for $n \to
\infty$. Thus we can hope that if the sample size $n$ is large enough
we can distinguish between different values of $k$ based on the
stability of the corresponding clusterings. An important question is
now which values of $k$ lead to stable and which ones lead to instable results,
for a given distribution $P$. \\
\subsection{Characterizing stable clusterings} \label{sec-characterization}
It is a straightforward consequence of \th{th-convergence-refined}
that if we consider different values $k_1$ and $k_2$ and the clustering objective
functions $Q^{(\infty)}_{k_1}$ and $Q^{(\infty)}_{k_2}$ have unique global minima, then the
rescaled stability values $\text{RInstab}(k_1)$ and $\text{RInstab}(k_2)$ can
differ from each other. Now we want to investigate which values of $k$ lead to
high stability and which ones lead to low stability. \\
\begin{enumerate}gin{conclusion}[Instable clusterings] \label{conclusion-instable}
Assume that $Q^{(\infty)}_K$ has a unique global optimum. If $\instab(K,n)$
is large, the idealized $K$-means clustering tends to have
cluster boundaries in high density regions of the space.
\end{conclusion}
There exist two different derivations of this conclusion, which have been
obtained independently from each other by completely different
methods \citep{BenLux08,ShaTis08_nips}.
On a high level, the reason why the conclusion tends to hold is that
if cluster boundaries jitter in a region of high density, then more
points ``change side'' than if the boundaries jitter in a region of low
density. \\
{\em First derivation, informal, based on
\citet{ShaTis08_nips,ShaTis09_nips}. } Assume that $n$ is large
enough such that we are already in the asymptotic regime (that is,
the solution $c^{(n)}$ constructed on the finite sample is close to the
true population solution $c^{(*)}$). Then the rescaled instability
computed on the sample is close to the expression given in
\eq{eq-rinstab}. If the cluster boundaries $B_{ij}$ lie in a high
density region of the space, then the integral in \eq{eq-rinstab} is
large --- compared to a situation where the cluster boundaries lie in
low density regions of the space. From a high level point of view,
this justifies the conclusion above. However, note that it is
difficult to identify how exactly the quantities $p$, $B_{ij}$ and
$V_{ij}$ influence $\text{RInstab}$, as they are not independent of each other.\\
{\em Second derivation, more formal, based on \citet{BenLux08}}. A
formal way to prove the conclusion is as follows. We introduce a new
distance $d_{\text{boundary}}$ between two clusterings. This distance measures
how far the cluster boundaries of two clusterings are apart from each
other. One can prove that the $K$-means quality function $Q^{(\infty)}_K$ is
continuous with respect to this distance function. This means that if
two clusterings $\mathcal{C}, \mathcal{C}'$ are close with respect to
$d_{\text{boundary}}$, then they have similar quality values. Moreover, if
$Q^{(\infty)}_K$ has a unique global optimum, we can invert this argument and
show that if a clustering $\mathcal{C}$ is close to the optimal limit
clustering $\mathcal{C}^*$, then the distance $d_{\text{boundary}}(\mathcal{C}, \mathcal{C}^*)$
is small. Now consider the clustering $\mathcal{C}^{(n)}$ based on a sample
of size $n$. One can prove the following key statement. If
$\mathcal{C}^{(n)}$ converges uniformly (over the space of all probability
distributions) in the sense that with probability at least $1 -
\delta$ we have $d_{\text{boundary}}(\mathcal{C}_n, \mathcal{C}) \leq \gamma$, then
\banum \label{eq-tube}
\instab(K,n) \leq 2 \delta + P(T_\gamma(B)).
\eanum
Here $P(T_\gamma(B))$ denotes the probability mass of a tube of width
$\gamma$ around the cluster boundaries $B$ of $\mathcal{C}$. Results in
\citet{Bendavid07} establish the uniform convergence of the idealized
$K$-means algorithm. This proves the conjecture: \eq{eq-tube}
shows that if $\instab$ is high, then there is a lot of mass
around the cluster boundaries, namely the cluster boundaries are in
a region of high density.\\
For stable clusterings, the situation is not as simple. It is tempting
to make the following conjecture.
\begin{enumerate}gin{conjecture}[Stable clusterings] \label{conjecture-stable}
Assume that $Q^{(\infty)}_K$ has a unique global optimum. If $\instab(K,n)$
is ``small'', the idealized $K$-means clustering tends to have
cluster boundaries in low density regions of the space.\\
\end{conjecture}
{\em Argument in favor of the conjecture: } As in the first approach above, considering the limit
expression of $\text{RInstab}$ reveals that if the cluster boundary lies in
a low density area of the space, then the integral in $\text{RInstab}$ tends to
have a low value. In the extreme case where the cluster boundaries go
through a region of zero density, the rescaled instability is even 0.\\
{\em Argument against the conjecture: counter-examples! }
One can construct artificial examples
where clusterings are stable although their
decision boundary lies in a high density region of the space (\citealp{BenLux08}). The way
to construct such examples is to ensure that the variations of the
cluster centers happen in parallel to cluster boundaries and not
orthogonal to cluster boundaries. In this case, the sampling variation
does not lead to jittering of the cluster boundary, hence the result
is rather stable. \\
These counter-examples show that Conjecture~\ref{conjecture-stable} cannot be true in
general. However, my personal opinion is that the counter-examples are
rather artificial, and that similar situations will rarely be
encountered in practice. I believe that the conjecture ``tends to
hold'' in practice. It might be possible to formalize this intuition
by proving that the statement of the conjecture holds on a subset of ``nice'' and ``natural''
probability distributions. \\
The important consequence of Conclusion \ref{conclusion-instable} and
Conjecture~\ref{conjecture-stable} (if true) is the following.
\begin{enumerate}gin{conclusion} \label{conclusion-idealized}
{\bf (Stability of idealized $K$-means detects whether
$K$ is too large)}
Assume that the underlying distribution $P$ has $K$ well-separated clusters,
and assume that these clusters can be represented by a center-based
clustering model. Then the following statements tend to hold for the idealized
$K$-means algorithm.
\begin{enumerate}gin{enumerate}
\item If $K$ is too large, then the clusterings obtained by the
idealized $K$-means algorithm tend to be instable.
\item If $K$ is correct or too small, then the clusterings obtained by the
idealized $K$-means algorithm tend to be stable (unless the
objective function has several global minima, for example due to
symmetries).
\end{enumerate}
\end{conclusion}
Given Conclusion \ref{conclusion-instable} and
Conjecture~\ref{conjecture-stable} it is easy to see why
Conclusion~\ref{conclusion-idealized} is true. If $K$ is larger than
the correct number of clusters, one necessarily has to split a true
cluster into several smaller clusters. The corresponding boundary goes
through a region of high density (the cluster which is being split).
According to Conclusion~\ref{conclusion-instable} this leads to
instability. If $K$ is correct, then the idealized (!) $K$-means
algorithm discovers the correct clustering and thus has decision
boundaries between the true clusters, that is in low density regions
of the space. If $K$ is too small, then the $K$-means algorithm has to
group clusters together. In this situation, the cluster boundaries are
still between true clusters, hence in a low density
region of the space.\\
\section{The actual $K$-means algorithm} \label{sec-actual}
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\includegraphics[width=0.8\textwidth]{./figures/configurations2.pdf}
\end{center}
\caption{Different initial configurations and the corresponding outcomes of the
$K$-means algorithm. Figure a: the two boxes in the top row depict a data
set with three clusters and four initial centers. Both boxes show
different realizations of the same initial configuration. As can be
seen in the bottom, both initializations lead to the same $K$-means
clustering. Figure b: here the initial configuration is different
from the one in Figure a, which leads to a different $K$-means
clustering. }
\label{fig-init-configuration}
\end{figure}
In this section we want to study the actual $K$-means algorithm. In
particular, we want to investigate when and how it gets stuck in
different local optima. The general insight is that even though, from
an algorithmic point of view, it is an annoying property of the
$K$-means algorithm that it can get stuck in different local optima,
this property might
actually help us for the purpose of model selection.
We now want to focus on the effect of the random
initialization of the $K$-means algorithm. For simplicity, we ignore
sampling artifacts and assume that we always work with
``infinitely many'' data points; that is, we work on the underlying
distribution directly. \\
The following observation is the key to our analysis. Assume we are
given a data set with $K_{\text{true}}$ well-separated clusters, and assume
that we initialize the $K$-means algorithm with $K_{\text{init}} \geq K_{\text{true}}$
initial centers. The key observation is that if there is at least
one initial center in each of the underlying clusters, then {\em the
initial centers tend to stay in the clusters they had been placed
in. } This means that during the course of the $K$-means algorithm,
cluster centers are only re-adjusted within the underlying
clusters and do not move between them. If this property is true,
then {the final clustering result is essentially determined by the
{\em number} of initial centers in each of the true clusters. In
particular, if we call the number of initial centers per cluster
the {\em initial configuration}, one can say that each initial
configuration leads to a unique clustering, and different
configurations lead to different clusterings; see
Figure~\ref{fig-init-configuration} for an illustration. Thus, if
the initialization method used in
$K$-means regularly leads to different initial configurations, then we observe instability. \\
In \citet{BubMeiLux09}, the first results in this direction were
proved. They are still preliminary in the sense that so far, proofs
only exist for a simple setting. However, we believe that the
results also hold in a more general context.
\begin{enumerate}gin{theorem}[Stability of the actual $K$-means algorithm]\label{th-actual-stability}
Assume that the underlying distribution $P$ is a mixture of two
well-separated Gaussians on $\RR$. Denote the means of the Gaussians by
$\mu_1$ and $\mu_2$.
\begin{enumerate}gin{enumerate}
\item
Assume that we run the $K$-means algorithm with
$K=2$ and that we use an initialization scheme that places one initial center
in each of the true clusters (with high probability). Then the
$K$-means algorithm is stable in the sense that with high probability,
it terminates in a solution with one center close to $\mu_1$ and one
center close to $\mu_2$.
\item
Assume that we run the $K$-means algorithm with
$K=3$ and that we use an initialization scheme that places at least
one of the initial centers in each of the true clusters (with high probability). Then the
$K$-means algorithm is instable in the sense that with probability
close to 0.5 it terminates in a solution that considers the first Gaussian
as cluster, but splits the second Gaussian into two clusters;
and with probability close to 0.5 it does it the other way round.
\end{enumerate}
\end{theorem}
{\em Proof idea. }
The idea of this proof is best described with \fig{fig-proof-actual}.
In the case of $K_{\text{init}}=2$ one has to prove that if the one center lies in a
large region around $\mu_1$ and the second center in a similar region
around $\mu_2$, then the next step of $K$-means does not move the
centers out of their regions (in \fig{fig-proof-actual}, these
regions are indicated by the black bars). If this is true, and if we know that there
is one initial center in each of the regions, the same is true when
the algorithm stops. Similarly, in the case of $K_{\text{init}}=3$, one proves
that if there are two initial centers in the first region and one
initial center in the second region, then all centers stay in their
regions in one step of $K$-means.
\smiley \\
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\includegraphics[width=0.5\textwidth]{./figures/stable_region_proof.pdf}
\end{center}
\caption{Stable regions used in the proof of
Theorem~\ref{th-actual-stability}. See text for details. }
\label{fig-proof-actual}
\end{figure}
All that is left to do now is to find an initialization scheme that
satisfies the conditions in
Theorem~\ref{th-actual-stability}. Luckily, we can adapt a scheme
that has already been used in \citet{DasSch07}.
For simplicity, assume that all clusters have similar weights (for the
general case see \citealp{BubMeiLux09}), and that we want to
select $K$ initial centers for the
$K$-means algorithm. Then the following initialization should be used: \\
{\small\tt
Initialization (I):
\ulesquote{
\begin{enumerate}gin{enumerate}
\item Select $L$ preliminary centers uniformly at random
from the given data set, where $L \approx K \log (K )$.
\item Run one step of $K$-means, that is assign the data points to the
preliminary centers and re-adjust the centers once.
\item Remove all centers for which the mass of the assigned data
points is smaller than $p_0 \approx 1 / L$.
\item Among the remaining centers, select $K$ centers by the
following procedure:
\begin{enumerate}gin{enumerate}
\item Choose the first center uniformly at random.
\item Repeat until $K$ centers are selected: Select the next
center as the one that maximizes the minimum distance to the
centers already selected.
\end{enumerate}
\end{enumerate}
}
}
One can prove that this initialization scheme satisfies the conditions
needed in Theorem~\ref{th-actual-stability} (for exact details see
\citealp{BubMeiLux09}).
\begin{enumerate}gin{theorem}[Initialization]\label{th-init}
Assume we are given a mixture of $K_{\text{true}}$ well-separated Gaussians
in $\RR$, and denote the centers of the Gaussians by $\mu_i$. If we
use the Initialization (I) to select $K_{\text{init}}$
centers, then there
exist $K_{\text{true}}$ disjoint regions ${A}_k$ with
$\mu_k\in {A}_k$, so that
all $K_{\text{init}}$ centers are contained in one of the $A_k$ and
\begin{enumerate}gin{itemize}
\item if $K_{\text{init}}=K_{\text{true}}$, each $A_k$ contains exactly one center,
\item if $K_{\text{init}}<K_{\text{true}}$, each $A_k$ contains at most one center,
\item if $K_{\text{init}}>K_{\text{true}}$, each $A_k$ contains at least one center.
\end{itemize}
\end{theorem}
{\em Proof sketch. } The following statements can be proved to hold
with high probability. By selecting $K_{\text{true}} \log(K_{\text{true}})$ preliminary
centers, each of the Gaussians receives at least one of these
centers. By running one step
of $K$-means and removing the
centers with too small mass, one removes all preliminary centers that
sit on outliers. Moreover, one can prove that ``ambiguous centers''
(that is, centers that sit between two clusters) attract only few
data points and will be removed as well. Next one
shows that centers that are ``unambiguous'' are reasonably close to a
true cluster center $\mu_k$. Consequently, the method for selecting
the final center from the remaining preliminary ones ``cycles though
different Gaussians'' before visiting a particular Gaussian for the second time.
\smiley \\
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\includegraphics[width=0.3\textwidth]{./figures/too_small_and_stable.pdf}\hspace{1cm}
\includegraphics[width=0.3\textwidth]{./figures/too_small_and_instable.pdf}
\end{center}
\caption{Illustration for the case where $K$ is too small.
We consider two data sets that have been drawn from a mixture of three
Gaussians with means
$\mu_1 = (-5,-7)$,
$\mu_2 = (-5,7)$,
$\mu_3 = (5,7)$ and unit variances. In the left figure, all clusters
have the same weight $1/3$, whereas in the right figure the top right
cluster has larger weight 0.6 than the other two clusters with weights
0.2 each. If we run $K$-means with $K=2$, we can verify experimentally
that the algorithm is pretty stable if applied to points from the distribution in the
left figure. It nearly always merges the top two
clusters. On the distribution shown in the right figure, however, the algorithm is
instable. Sometimes the top two clusters are merged, and sometimes the
left two clusters. }
\label{fig-too-small}
\end{figure}
When combined, the results of Theorems~\ref{th-actual-stability} and
\ref{th-init} show that if the data set contains $K_{\text{true}}$
well-separated clusters, then the $K$-means algorithm is stable if it
is started with the true number of clusters, and instable if the
number of clusters is too large. Unfortunately, in the case where $K$
is too small one cannot make any useful statement about stability
because the aforementioned configuration argument does not hold any
more. In particular, initial cluster centers do not
stay inside their initial clusters, but move out of
the clusters. Often, the final centers constructed by the $K$-means
algorithm lie in between several true clusters, and it is very hard to
predict the final positions of the centers from the initial ones. This can be seen
with the example shown in Figure \ref{fig-too-small}. We
consider two data sets from a mixture of three Gaussians. The only
difference between the two data sets is that in the left plot all mixture
components have the same weight, while in the right plot the top right
component has a larger weight than the other two components. One can
verify experimentally that if initialized with $K_{\text{init}} = 2$, the
$K$-means algorithm is rather stable in the left figure (it always
merges the top two clusters). But it is instable in the right figure
(sometimes it merges the top clusters, sometimes the left two
clusters). This example illustrates that if the number of clusters is
too small, subtle differences in the distribution can decide on
stability or instability of the actual $K$-means algorithm. \\
In general, we expect that the following statements hold
(but they have not yet been proved in a context more general than in
Theorems~\ref{th-actual-stability} and \ref{th-init}). \\
\begin{enumerate}gin{conjecture}[Stability of the actual $K$-means algorithm] \label{conjecture-stable-actual}
Assume that the underlying distribution has $K_{\text{true}}$ well-separated
clusters, and that these clusters can be represented by a center-based
clustering model. Then, if one uses Initialization (I) to
construct $K_{\text{init}}$ initial centers, the
following statements hold:
\mbox{}\\[1mm]\mbox{}\hspace{1mm}$\bullet$ b{If $K_{\text{init}} = K_{\text{true}}$, we have one center per cluster, with
high probability. The clustering results are stable.}
\mbox{}\\[1mm]\mbox{}\hspace{1mm}$\bullet$ b{If $K_{\text{init}} > K_{\text{true}}$, different initial configurations
occur. By the above argument, different configurations lead to different
clusterings, so we observe instability. }
\mbox{}\\[1mm]\mbox{}\hspace{1mm}$\bullet$ b{If $K_{\text{init}} < K_{\text{true}}$, then depending on subtle differences in
the underlying distribution we can have
either stability or instability. \\}
\end{conjecture}
\section{Relationships between the results} \label{sec-relationships}
In this section we discuss conceptual aspects of the results and relate them to each other.
\subsection{Jittering versus jumping}
\begin{enumerate}gin{figure}[t]
\begin{enumerate}gin{center}
\includegraphics[width=0.7\textwidth]{./figures/wiggling_global_all_soltutions.pdf}
\end{center}
\caption{The $x$-axis depicts the space of all clusterings for a fixed
distribution $P$ and for a fixed parameter $K$ (this is an
abstract sketch only). The $y$-axis shows the value
of the objective function of the different solutions. The solid line
corresponds to the true limit objective function $Q^{(\infty)}_K$, the dotted lines
show the sample-based function $Q^{(\infty)}_K$ on different samples. The
idealized $K$-means algorithm only studies the jittering of the global
optimum, that is how far the global optimum varies due to the sampling
process. The jumping between different local optima is induced by
different random initializations, as investigated for the actual
$K$-means algorithm. }
\label{fig-jittering}
\end{figure}
There are two main effects that
lead to instability of the $K$-means algorithm. Both effects
are visualized in \fig{fig-jittering}. \\
{\em Jittering of the cluster boundaries. } Consider a fixed local (or
global) optimum of $Q^{(\infty)}_K$ and the corresponding clustering on different random
samples. Due to the fact that different samples lead to slightly
different positions of the cluster centers, the cluster boundaries
``jitter''. That is, the cluster boundaries corresponding to different
samples are slightly shifted with respect to one another. We call this behavior the ``jittering'' of a
particular clustering solution. For the special case of the global
optimum, this jittering has been investigated in Sections
\ref{sec-convergence-refined} and \ref{sec-characterization}. It has
been established that different parameters $K$ lead to different
amounts of jittering (measured in terms of rescaled instability). The
jittering is larger if the cluster boundaries are in a high density
region and smaller if the cluster boundaries are in low density
regions of the space. The main ``source'' of jittering is the
sampling variation. \\
{\em Jumping between different local optima. } By ``jumping'' we refer
to the fact that an algorithm terminates in different local
optima. Investigating jumping has been the major goal in Section
\ref{sec-actual}. The main source of jumping is the
random initialization. If we initialize the $K$-means algorithm in
different configurations, we end in different local optima. The key
point in favor of clustering stability is that one can relate the
number of local optima of $Q^{(\infty)}_K$ to whether the number $K$ of
clusters is correct or too large (this has happened implicitly in
Section~\ref{sec-actual}).
\subsection{Discussion of the main theorems}
{\em Theorem~\ref{th-convergence-simple}} works in the idealized
setting. In Part 1 it shows that if the underlying distribution is not
symmetric, the idealized clustering results are stable in the sense
that
different samples always lead
to the same clustering. That is, no jumping between different
solutions takes place. In hindsight, this result can be considered as an
artifact of the idealized clustering scenario. The idealized $K$-means
algorithm artificially excludes the possibility of ending in different
local optima. Unless there exist several global optima, jumping
between different solutions cannot happen. In particular, the
conclusion that clustering results are stable for all values of $K$ does not carry over
to the realistic $K$-means
algorithm (as can be seen from the results in \sec{sec-actual}). Put
plainly, even though the idealized $K$-means algorithm with $K=2$ is
stable in the example of \fig{fig-stability-wrong}a, the actual $K$-means algorithm
is instable. \\
Part 2 of Theorem~\ref{th-convergence-simple} states that if the
objective function has several global optima, for example due to
symmetry, then jumping takes place even for the idealized $K$-means
algorithm and results in instability. In the setting of the theorem,
the jumping is merely induced by having different random
samples. However, a similar result can be shown to hold for the actual
$K$-means algorithm, where it is induced due to random
initialization. Namely, if the underlying distribution is perfectly
symmetric, then ``symmetric initializations'' lead to the different
local optima corresponding to the different symmetric solutions.\\
To summarize, Theorem~\ref{th-convergence-simple} investigates whether
jumping between different solutions takes place due to the random
sampling process.
The
negative connotation of Part 1 is an artifact of the idealized setting
that does not carry over to the actual $K$-means algorithm, whereas the
positive connotation of Part 2 does carry
over. \\
{\em Theorem~\ref{th-convergence-refined}} studies how
different samples affect the jittering of a unique solution of the
idealized $K$-means algorithm. In general, one can expect that
similar jittering takes place for the actual $K$-means algorithm as
well. In this sense, we believe that the results of this theorem can
be carried over to the actual $K$-means algorithm. \\
However, if we reconsider the intuition stated in the introduction and
depicted in \fig{fig-stability-idea}, we realize that
jittering was not really what we had been looking for. The main intuition
in the beginning was that the algorithm might jump between different
solutions, and that such jumping shows that the underlying parameter
$K$
is wrong. In practice, stability is usually computed for the actual
$K$-means algorithm with random initialization and on different
samples. Here both effects (jittering and jumping) and both random
processes (random samples and random initialization) play a role. We
suspect that the effect of jumping to different local optima due to
different initialization has higher impact on stability than the
jittering of a particular solution due to sampling variation. Our
reason to believe so is that the distance between two clusterings is
usually higher if the two clusterings correspond to different local
optima than if they correspond to the same solution with a slightly
shifted boundary. \\% Preliminary experimental results corroborate this
To summarize, Theorem~\ref{th-convergence-refined} describes the jittering
behavior of an individual solution of the idealized $K$-means
algorithm. We believe that similar effects take place for the actual
$K$-means algorithm. However, we also believe that the influence of
jittering on stability plays a minor role compared to the one of jumping. \\
{\em Theorem~\ref{th-actual-stability}} investigates
the jumping behavior of the actual $K$-means algorithm. As the source of
jumping, it considers the random initialization only. It does not take
into account variations due to random samples (this is hidden in the
proof, which works on the underlying distribution rather than with
finitely many sample points). However, we believe that the
results of this theorem also hold for finite
samples. Theorem~\ref{th-actual-stability} is not yet as general as we
would like it to be. But we believe that studying the jumping
behavior of the actual $K$-means algorithm is the key to understanding
the stability of the $K$-means algorithm used
in practice, and Theorem~\ref{th-actual-stability} points in the right
direction. \\
{\em Altogether, } the results obtained in the idealized and realistic setting
perfectly complement each other and describe two sides of the same coin. The
idealized setting mainly studies what influence the different samples
can have on the stability of one particular solution. The realistic
setting focuses on how the random initialization makes the algorithm
jump between different local optima. In both settings, stability
``pushes'' in the same direction: If the number of clusters is too
large, results tend to be instable. If the number of clusters is
correct, results tend to be stable. If the number of clusters is too
small, both stability and instability can occur, depending on subtle
properties of the underlying distribution. \\
\chapter{Beyond $K$-means} \label{sec-beyond}
Most of the theoretical results in the literature on clustering stability have been
proved with the $K$-means algorithm in mind. However, some of them hold for
more general clustering algorithms. This is mainly the case for
the idealized clustering setting. \\
Assume a general clustering objective function $Q$ and an ideal
clustering algorithm that globally minimizes this objective
function. If this clustering algorithm is consistent in the sense that
the optimal clustering on the finite sample converges to the optimal
clustering of the underlying space, then the results of Theorem
\ref{th-convergence-simple} can be carried over to this general
objective function \citep{BenLuxPal06}. Namely, if the objective
function has a unique global optimum, the clustering algorithm is
stable, and it is instable if the algorithm has several global minima
(for example due to symmetry). It is not too surprising that one can
extend the stability results of the $K$-means algorithm to more
general vector-quantization-type algorithms. However, the setup of
this theorem is so general that it also holds for completely different
algorithms such as spectral clustering. The consistency requirement
sounds like a rather strong assumption. But note that clustering algorithms that are
not consistent are completely unreliable and should not be used
anyway. \\
Similarly as above, one can also generalize the characterization of
instable clusterings stated in Conclusion \ref{conclusion-instable},
cf. \citet{BenLux08}. Again we are dealing with algorithms that minimize an
objective function. The consistency requirements are slightly
stronger in that we need uniform consistency over the space (or a
subspace) of probability distributions. Once such uniform consistency
holds, the characterization that instable clusterings tend to
have their boundary in high density regions of the space can be
established. \\
While the two results mentioned above can be carried over to a huge
bulk of clustering algorithms, it is not as simple for the refined
convergence analysis of Theorem \ref{th-convergence-refined}. Here we
need to make one crucial additional assumption, namely the existence
of a central limit type result. This is a rather strong assumption
which is not satisfied for many clustering objective
functions. However, a few results can be established
\citep{ShaTis09_nips}: in addition to the traditional $K$-means
objective function, a central limit theorem can be proved for other
variants of $K$-means such as kernel $K$-means (a kernelized version
of the traditional $K$-means algorithm) or Bregman divergence
clustering (where one selects a set of centroids such that the average
divergence between points and centroids is minimized). Moreover,
central limit theorems are known for maximum likelihood
estimators, which leads to stability results for certain types of
model-based clusterings using maximum likelihood
estimators. Still the results of Theorem
\ref{th-convergence-refined} are limited to a small number of clustering
objective functions, and one cannot expect to be able to
extend them to a wide range of clustering algorithms. \\
Even stronger limitations hold for the results about the actual
$K$-means algorithm. The methods used in Section \ref{sec-actual} were
particularly designed for the $K$-means algorithm. It might be
possible to extend them to more general centroid-based algorithms, but
it is not obvious how to advance further. In spite of this shortcoming, we
believe that these results hold in a much more general context of
randomized clustering algorithms.
From a high level point of view, the actual $K$-means algorithm is a
randomized algorithm due to its random initialization. The
randomization is used to explore different local optima of the
objective function. There were two key insights in our stability
analysis of the actual $K$-means algorithm: First, we could describe
the ``regions of attraction'' of different local minima, that is we
could prove which initial centers lead to which solution in the end
(this was the configurations idea).
Second, we could relate the ``size'' of the
regions of attraction to the number of clusters.
Namely, if the number of
clusters is correct, the global minimum will have a huge region of
attraction in the sense that it is very likely that we will end in the
global minimum. If the number of clusters is too large, we could show
that there exist several local optima with large regions of
attraction. This leads to a significant likelihood of
ending in different local optima and observing instability.\\
We believe that similar arguments can be used to investigate stability
of other kinds of randomized clustering algorithms. However, such an
analysis always has to be adapted to the particular algorithm under
consideration. In particular, it is not obvious whether the number of
clusters can always be related to the number of large regions of
attraction. Hence it is an open question whether
results similar to the ones for the actual $K$-means algorithm also hold for completely
different randomized clustering algorithms. \\
\chapter{Outlook} \label{sec-outlook}
Based on the results presented above one can draw a cautiously
optimistic picture about model selection based on clustering
stability for the $K$-means algorithm. Stability can
discriminate between different values of $K$, and the values of $K$
that lead to stable results have desirable properties. If the
data set contains a few well-separated clusters that can be
represented by a center-based clustering, then stability has the
potential to discover the correct number of clusters. \\
An important point to stress is that stability-based model selection
for the $K$-means algorithm can only lead to convincing results if the
underlying distribution can be represented by center-based
clusters. If the clusters are very elongated or have complicated
shapes, the $K$-means algorithm cannot find a good representation of
this data set, regardless what number $K$ one uses. In this case,
stability-based model selection breaks down, too. It is a
legitimate question what implications this has in practice. We
usually do not know whether a given data set can be represented
by center-based clusterings, and often the $K$-means algorithm is
used anyway. In my opinion, however, the question of selecting the
``correct'' number of clusters is not so important in this case. The
only way in which complicated structure can be represented using
$K$-means is to break each true cluster in several small, spherical
clusters and either live with the fact that the true clusters are split in
pieces, or use some mechanism to join these pieces afterwards to form
a bigger cluster of general shape. In
such a scenario it is not so important what number of
clusters we use in the $K$-means step: it does not really matter whether we split an
underlying cluster into, say, 5 or 7 pieces.\\
There are a few technical questions that deserve further
consideration. Obviously, the results in \sec{sec-actual} are still
somewhat preliminary and should be worked out in more generality. The
results in Section~\ref{sec-idealized} are large sample results. It is
not clear what ``large sample size'' means in practice, and one can
construct examples where the sample size has to be arbitrarily large
to make valid statements \citep{BenLux08}. However, such examples can
either be countered by introducing assumptions on the underlying
probability distribution, or one can state that the sample size has to
be large enough to ensure that the cluster structure is
well-represented in the data and that we don't miss any clusters. \\
There is yet another limitation that is more severe, namely
the number of clusters to which the results apply. The conclusions in
\sec{sec-idealized} as well as the results in \sec{sec-actual} only
hold if the true number of clusters is relatively small (say, on the
order of 10 rather than on the order of 100), and if the parameter $K$
used by $K$-means is in the same order of magnitude. Let us briefly
explain why this is the case.
In the idealized setting, the limit results in Theorems \ref{th-convergence-simple} and
\ref{th-convergence-refined} of course hold regardless of what the true
number of clusters is. But the subsequent interpretation regarding
cluster boundaries in high and low density areas breaks down if the
number of clusters is too large. The reason is that the influence of
one tiny bit of cluster boundary between two clusters is negligible
compared to the rest of the cluster boundary if there are many
clusters, such that other factors might dominate the behavior of
clustering stability.
In the realistic setting of
Section~\ref{sec-actual}, we use an initialization scheme
which, with high probability, places centers in different clusters
before placing them into the same cluster. The procedure works well if
the number of clusters is small. However, the larger the number of
clusters, the higher the likelihood to fail with this
scheme. Similarly problematic is the situation where the true number of
clusters is small, but the $K$-means algorithm is run with a very
large $K$.
Finally, note that similar limitations hold for all model selection
criteria. It is simply a very difficult (and pretty useless) question
whether
a data set contains 100 or 105 clusters, say. \\
While stability is relatively well-studied for the $K$-means
algorithm, there does not exist much work on the stability of
completely different clustering mechanisms. We have seen in Section
\ref{sec-beyond} that some of the results for the idealized $K$-means
algorithm also hold in a more general context. However, this is not
the case for the results about the actual $K$-means algorithm. We
consider the results about the actual $K$-means algorithm as the
strongest evidence in favor of stability-based model selection for
$K$-means. Whether this principle can be proved to work well for
algorithms very different from $K$-means is an open question.\\
An important point we have not discussed in depth is how clustering
stability should be implemented in practice. As we have outlined in
Section \ref{sec-implementation} there exist many different protocols
for computing stability scores. It would be very important to compare
and evaluate all these approaches in practice, in particular as there
are several unresolved issues (such as the
normalization). Unfortunately, a thorough study that
compares all different protocols in practice does not
exist. \\
\begin{enumerate}gin{itemize}bliography{general_bib,ules_publications,ules_publications_submitted}
\end{document} |
\begin{document}
\newcommand{\hbox{Re}\,feq}[1]{(\hbox{Re}\,f{#1})}
\deltaef\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1}{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1}}
\deltaef\widehat}\deltaef\cal{\mathcal{\widehat}\deltaef\cal{\mathcal}
\deltaef \beq {\begin {eqnarray}}
\deltaef \varepsiloneq {\varepsilonnd {eqnarray}}
\deltaef \ba {\begin {eqnarray*}}
\deltaef \varepsilona {\varepsilonnd {eqnarray*}}
\deltaef \bfo {\ba}
\deltaef \varepsilonfo {\varepsilona}
\deltaef{\cal O}mega{{\cal O}megaega}
\deltaef{\cal O}{{\mathcal O}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{Mtheorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem {adefinition}[theorem]{Assumption}
\newtheorem {definition}[theorem]{Definition}
\newtheorem {example}[theorem]{Example}
\newtheorem {corollary}[theorem]{Corollary}
\newtheorem {problem}[theorem]{Problem}
\newtheorem {ex}[theorem]{Exercise}
\deltaef\noindent{\bf Proof.}\quad {\noindent{\bf Proof.}\quad}
\deltaef \cA {{\cal A}}\deltaef\varrho{\varrho}
\deltaef \fg{{\bf g}}
\deltaef \cB {{\cal B}}
\deltaef \cD {{\cal D}}
\deltaef \A {{\cal A}}
\deltaef \B {{\cal B}}
\deltaef \D {{\cal D}}
\deltaef \Disc {{\Bbb D}}
\deltaef \tX {{\widetilde X}}
\deltaef \t {{\tau}}
\deltaef \Box {\quad $\square$
}
\deltaef \ssskip {\vskip -23pt \hskip 40pt {\bf.} }
\deltaef \sskip {\vskip -23pt \hskip 33pt {\bf.} }
\deltaef\alpha{\alphalpha}
\deltaef \varepsilon{\varepsilon}
\deltaef \b{\beta}
\deltaef \bn{\underline n}
\deltaef\hbox{Grad}\,}\deltaef\Div{\hbox{Div}\,{\hbox{Grad}\,}\deltaef\Div{\hbox{Div}\,}
\deltaef \cW {{\cal W}}
\deltaef{\rm Hess}\,{{\rm Hess}\,}
\deltaef{\rm sign}\,{{\rm sign}\,}
\deltaef\hbox{Re}\,{\hbox{Re}\,}
\deltaef\hbox{Im}\,{\hbox{Im}\,}
\deltaef\hbox{Re}\,{\hbox{Re}\,}
\deltaef\hbox{Im}\,{\hbox{Im}\,}
\deltaef\noindent{\bf Proof.}\quad {\noindent{\bf Proof.}\quad}
\title{CALDER\'ON'S INVERSE PROBLEM FOR ANISOTROPIC CONDUCTIVITY IN THE PLANE}
\alphauthor {Kari Astala}
\alphaddress{Rolf Nevanlinna Institute, University of Helsinki,
P.O.~Box~4 (Yliopistonkatu~5), FIN-00014 University of Helsinki,
Finland}
\varepsilonmail{[email protected], [email protected],
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ \sigmainebreak
\hbox{ } $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [email protected]}
\alphauthor{Matti Lassas}
\alphauthor{Lassi P\"aiv\"arinta}
\maketitle
\deltaefH^{s+1}_0(\partialartial M\times [0,T]){H^{s+1}_0(\partialartial M\times [0,T])}
\deltaefH^{s+1}_0(\partialartial M\times [0,T/2]){H^{s+1}_0(\partialartial M\times [0,T/2])}
\deltaefC^\inftynfty_0(\partialartial M\times [0,2r]){C^\inftynfty_0(\partialartial M\times [0,2r])}
\deltaef\hbox{supp }{\hbox{supp }}
\deltaef\hbox{diam }{\hbox{diam }}
\deltaef\hbox{dist}{\hbox{dist}}
\deltaef{\mathbb R}{{\mathbb R}}
\deltaef \fR{{\mathbb R}}
\deltaef{ \mathbb Z}{{ \mathbb Z}}
\deltaef{\mathbb C}{{\mathbb C}}
\deltaef\varepsilon{\varepsilon}
\deltaef\tau{\tau}
\deltaef{\cal F}{{\cal F}}
\deltaef{\cal N}{{\cal N}}
\deltaef{\cal U}{{\cal U}}
\deltaef{\cal W}{{\cal W}}
\deltaef{\cal O}{{\cal O}}
\deltaef\varepsilonxp{\text{exp}}
\deltaef\sigma{\sigma}
\deltaef\over{\overver}
\deltaef\infty{\inftynfty}
\deltaef\delta{\deltaelta}
\deltaef\Gamma{\Gammaamma}
\deltaef\varepsilonxp{\hbox{exp}}
\deltaef\langle{\sigmaangle}
\deltaef\rangle{\rangle}
\deltaef\partial{\partialartial}
\deltaef\tilde M{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} M}
\deltaef\tilde R{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} R}
\deltaef\tilde r{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} r}
\deltaef\tilde z{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} z}
\deltaef\varepsilonxpec{\hskip1pt{\mathbb E}\hskip2pt}
\deltaef\partialrob{{\cal P}}
\deltaef\tauood{{\cal A}}
\deltaef\noindent{\bf Proof.}\quad {\noindent{\bf Proof.}\quad }
\deltaef\deltas{\deltaisplaystyle}
\deltaef\partialz{\partial_z}
\deltaef\partialbz{\partial_{\oververline z}}
\deltaef\overline{\oververline}
\deltaef{\cal H}{{\cal H}}
{\bf Abstract:} We study inverse conductivity problem for an
anisotropic conductivity $\sigma\inftyn L^\inftynfty$ in bounded and unbounded domains.
Also, we give applications of the results in the case when
Dirichlet-to-Neumann and
Neumann-to-Dirichlet maps are given only on a part of the boundary.
\section{INTRODUCTION}
Let us consider the
anisotropic conductivity equation in two dimensions
\beq\sigmaabel{conduct}
\nabla\cdotp \sigma\nabla u= \sum_{j,k=1}^2
\frac \partial{\partial x^j}\sigma^{jk}(x) \frac \partial{\partial x^k} u &=& 0\hbox{ in } {\cal O}megaega,\\
u|_{\partial {\cal O}megaega}&=&\partialhi.\nonumber
\varepsiloneq
Here
${\cal O}megaega\subset{\mathbb R}^2$ is a simply connected domain.
The
conductivity
$\sigma=[\sigma^{jk}]_{j,k=1}^2$ is a
symmetric, positive definite matrix function, and
$\partialhi \inftyn H^{1/2}(\partial {\cal O}megaega)$ is the prescribed voltage on the boundary.
Then it is well known that equation (\hbox{Re}\,f{conduct}) has
a unique solution $u\inftyn H^1({\cal O}mega)$.
In the case when $\sigma$ and $\partial {\cal O}megaega$
are smooth, we can define the voltage-to-current
(or Dirichlet-to-Neumann) map by
\beq
\Lambda_\sigma(\partialhi)= Bu|_{\partial {\cal O}megaega}
\varepsiloneq
where
\beq
Bu=\nu \cdotp \sigma \nabla u,
\varepsiloneq
$u\inftyn H^1({\cal O}mega) $ is the solution of (\hbox{Re}\,f{conduct}), and $\nu$
is the unit
normal vector of $\partial{\cal O}megaega$.
Applying
the divergence theorem, we have
\beq\sigmaabel{Q_l}
Q_{\sigma,{\cal O}mega} (\partialhi):=\inftynt_{\cal O}megaega \sum_{j,k=1}^2\sigma^{jk}(x)\frac {\partial u}{\partial x^j}
\frac {\partial u}{\partial x^k}
dx=\inftynt_{\partial {\cal O}megaega} \Lambda_\sigma(\partialhi) \partialhi\, dS,
\varepsiloneq
where $dS$ denotes the arc lenght on $\partial {\cal O}megaega$.
The quantity $Q_{\sigma,{\cal O}mega} (\partialhi)$ represents the
power needed to maintain the potential $\partialhi$ on $\partial {\cal O}megaega$. By symmetry
of $\Lambda_\sigma$,
knowing $Q_{\sigma,{\cal O}mega}$ is
equivalent with knowing
$\Lambda_\sigma$. For general ${\cal O}megaega$ and $\sigma\inftyn L^\inftynfty({\cal O}mega)$,
the trace $u|_{\partial {\cal O}mega}$ is defined as the equivalence class
of $u$ in $H^1({\cal O}mega)/H^1_0({\cal O}mega)$ (see \cite{AP}) and
formula (\hbox{Re}\,f{Q_l}) is used to define the map $\Lambda_\sigma$.
If $F:{\cal O}megaega\to {\cal O}megaega,\quad F(x)=(F^1(x),F^2(x))$, is a
diffeomorphism with $F|_{\partial
{\cal O}megaega}=\hbox{Identity}$, then by making the change of variables $y=F(x)$
and setting $v=u\circ F^{-1}$ in the first
integral in (\hbox{Re}\,f{Q_l}), we obtain
\ba
\nabla\cdotp (F_*\sigma)\nabla v=0\quad\hbox{in }{\cal O}mega,
\varepsilona
where
\beq\sigmaabel{cond and metr}
(F_*\sigma)^{jk}(y)=\sigmaeft.
\frac 1{\deltaet [\frac {\partial F^j}{\partial x^k}(x)]}
\sum_{p,q=1}^2 \frac {\partial F^j}{\partial x^p}(x)
\,\frac {\partial F^k}{\partial x^q}(x) \sigma^{pq}(x)\right|_{x=F^{-1}(y)},
\varepsiloneq
or
\beq\sigmaabel{5 1/2}
F_*\sigma(y)=\sigmaeft.\frac 1{J_F(x)} DF(x)\,\sigma(x)\,DF(x)^t\right|_{x=F^{-1}(y)},
\varepsiloneq
is the push-forward of the conductivity $\sigma$ by $F$.
Moreover, since $F$ is identity at $\partial {\cal O}mega$, we obtain
from (\hbox{Re}\,f{Q_l}) that
\ba
\Lambda_{F_*\sigma}=\Lambda_\sigma.
\varepsilona
Thus, the change of coordinates shows that there is a large
class of conductivities which give rise to the
same electrical
measurements at the boundary.
We consider here the converse question, that if we have two
conductivities which have the same Dirichlet-to-Neumann map, is it the case
that each of them can be obtained by
pushing forward the other.
In applied terms,
this inverse problem to determine $\sigma$ (or its properties)
from $\Lambda_{\sigma}$ is
also known as {\varepsilonm Electrical Impedance Tomography}.
It has been proposed as a valuable diagnostic, see \cite{CIN99}.
In the case where $\sigma^{jk}(x)=\sigma(x)\deltaelta^{jk}$, $\sigma(x)\inftyn {\mathbb R}_+$,
the metric is said to be isotropic.
In 1980 it was proposed by A.~Calder\'on
\cite {Cl} that in the isotropic case any
bounded conductivity $\sigma(x)$ might be determined
solely from the boundary measurements, i.e., from $\Lambda_\sigma$.
Recently this has been confirmed in the two dimensional case
(c.f. \cite{AP}). In the case when isotropic $\sigma$ is smoother
than just a $L^\inftynfty$-function, the same conclusion
is known to hold also in higher dimensions.
The first global
uniqueness result was obtained for a $C^\inftynfty$--smooth
conductivity in dimension $n\taueq 3$ by
J.~Sylvester and G.~Uhlmann in 1987 \cite{SylUhl}.
In dimension two A.~Nachman \cite{Nach2D} produced in 1995 a uniqueness
result for conductivities with two derivatives.
The corresponding algorithm has been successfully implemented and
proven to work efficiently even with real data \cite{Siltanen1,Siltanen2}.
The reduction of
regularity assumptions has since been under active study.
In dimension two the optimal $L^\inftynfty$-regularity
was obtained in \cite{AP}.
In dimension $n\taueq 3$ the uniqueness has presently been shown for
isotropic conductivities $\sigma \inftyn W^{3/2,\inftynfty}({\cal O}mega)$
in \cite{PPU}
and for globally
$C^{1+\varepsilon}$--smooth isotropic conductivities having only
co-normal singularities in \cite{GLU1}.
Also, the stability of reconstructions of the inverse conductivity problem
have been extensively studied. For these results, see \cite{Al,AlS,BBR}
where stability results are based on reconstruction
techniques of \cite{BU} in dimension two and
those of \cite{Na1} in dimensions $n\taueq 3$.
In anisotropic case, where $\sigma$ is a matrix function and the
problem is to recover the conductivity $\sigma$ up to the action
of a class of diffeomorphisms, much
less is known. In dimensions $n\taueq 3$ it is generally known
only that piecewise analytic conductivities can be constructed
(see \cite{KV1,KV2}). For Riemannian manifolds this kind of
technique has been generalized in \cite{LeU,LaU,LTU}.
In dimension $n=2$ the inverse problem has been considered
by J.~Sylvester \cite{Sy1} for $C^3$ and Z.~Sun and G.~Uhlmann \cite{SU}
for $W^{1,p}$-conductivities. The idea of
\cite{Sy1} and \cite{SU} is that under quasiconformal change
of coordinates (cf. \cite{Alf,IM}) any anisotropic
conductivity can be changed to isotropic one, see also
section \hbox{Re}\,f{Sec: proofs} below. The
purpose of this paper is to carry this technique over
to the $L^\inftynfty$--smooth case and then use the result of
\cite{AP} to obtain uniqueness up to the group of diffeomorphisms.
The advantage of the reduction of the smoothness assumptions
up to $L^\inftynfty$
does not lie solely on the fact that many conductivities
have jump-type singularities but it also allows us to consider
much more complicated singular structures such as porous rocks
\cite{Che}. Moreover it is important that this approach
enables us to consider general diffeomorphisms.
Thus anisotropic inverse problems in half-space or exterior
domains can be solved simultaneously. This will be considered
in Section \hbox{Re}\,f{sec: con}.
If ${\cal O}mega\subset {\mathbb R}^2$ is a bounded domain, it is convenient
to consider the class of matrix functions $\sigma=[\sigma^{jk}]$
such that
\beq\sigmaabel{basic ass}
[\sigma^{ij}]\inftyn L^\inftynfty({\cal O}mega;{\mathbb R}^{2\times 2}),
\quad [\sigma^{ij}]^t=[\sigma^{ij}],\quad
C_0^{-1}I \sigmaeq [\sigma^{ij}]\sigmaeq C_0I
\varepsiloneq
where $C_0>0$. In sequel, the minimal possible value of $C_0$
is denoted by $C_0(\sigma)$.
We use the notation
\ba
\Sigma({\cal O}mega)=\{\sigma\inftyn L^{\inftynfty}({\cal O}mega;{\mathbb R}^{2\times 2})& |&
\ C_0(\sigma)<\inftynfty\}.
\varepsilona
Note that it is necessary
to require $C_0(\sigma)<\inftynfty$
as otherwise there would be counterexamples showing
that even the equivalence class of the conductivity can not be recovered
\cite{GLU2a,GLU2}.
Our main goal in this paper is to show that
an anisotropic $L^\inftynfty$--conductivity can be determined
up to a $W^{1,2}$-diffeomorphism:
\begin{Mtheorem}\sigmaabel{theorem1}
Let ${\cal O}mega\subset{\mathbb R}^{2}$ be a simply connected bounded domain and
$\sigma\inftyn L^{\inftynfty}({\cal O}mega;{\mathbb R}^{2\times 2})$. Suppose that
the assumptions (\hbox{Re}\,f{basic ass}) are valid. Then
the Dirichlet-to-Neumann map
$\Lambda_{\sigma}$
determines the equivalence class
\ba
E_\sigma=\{\sigma_1\inftyn \Sigma({\cal O}mega)& |&
\hbox{$\sigma_1 = F_*\sigma$,
$F:{\cal O}mega\to {\cal O}mega$ is $W^{1,2}$-diffeomorphism and}\\
& &F|_{\partial {\cal O}mega}=I\}.
\varepsilona
\varepsilonnd{Mtheorem}
We prove this result in Section \hbox{Re}\,f{Sec: proofs}.
Finally, note that the $W^{1,2}$--diffeomorphisms $F$
preserving the class $\Sigma({\cal O}mega)$ are precisely the quasiconformal
mappings. Namely, if $\sigma_0\inftyn \Sigma({\cal O}mega)$ and $\sigma_1=F_*(
\sigma_0)\inftyn \Sigma(F({\cal O}mega))$ then
\beq\sigmaabel{Kari 8}
\frac 1{C_0}||DF(x)||^2I\sigmaeq DF(x)\, \sigma_0(x)\, DF(x)^t\sigmaeq
C_1J_F(x)I
\varepsiloneq
where $I=[\deltaelta^{ij}]$ and we obtain
\beq\sigmaabel{Kari 9}
||DF(x)||^2\sigmaeq K J_F(x),\quad \hbox{for a.e. }\ x\inftyn {\cal O}mega
\varepsiloneq
where $K=C_1C_0<\inftynfty$. Conversely, if (\hbox{Re}\,f{Kari 9}) holds
and $F$ is $W^{1,2}_{loc}$-homeomorphism then
$F_*\sigma\inftyn \Sigma(F({\cal O}mega))$ whenever $\sigma\inftyn \Sigma({\cal O}mega)$.
Furhtermore, recall that a map $F:{\cal O}mega \to \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$ is quasiregular
if $F\inftyn W^{1,2}_{loc}({\cal O}mega)$ and the condition (\hbox{Re}\,f{Kari 9}) holds.
Moreover, a map $F$ is quasiconformal if it is
quasiregular and a $W^{1,2}$--homeomorphism.
\section{CONSEQUENCES AND APPLICATIONS OF THEOREM \hbox{Re}\,f{theorem1}}\sigmaabel{sec: con}
Here we consider applications of the diffeomorphism-technique
to various inverse problem. The formulated results,
Theorems \hbox{Re}\,f{Lem: A1}--\hbox{Re}\,f{Lem: A3} are proven in
Section \hbox{Re}\,f{sec: proof of con}.
\subsection{Inverse Problem in the Half Space}
Inverse problem in half space is of crucial importance
in geophysical prospecting,
seismological imaging, non-destructive testing etc.
For instance, the imaging of soil was the original
motivation of Calder\'on's seminal paper \cite {Cl}.
As we can use a diffeomorphism to map the open half space
to the unit disc, we can apply the previous result for the half space case.
One should observe that in this deformation even
infinitely smooth conductivities can become
non-smooth at the boundary
(e.g. conductivity oscillating near infinity produces
a non-Lipschitz conductivity in push-forward) and thus
the low-regularity result \cite{AP} is essential for the problem.
Thus, for $\sigma\inftyn \Sigma({\mathbb R}^2_-)$ let us consider the problem
\beq\sigmaabel{conduct D1a}
\nabla\cdotp \sigma\nabla u&=& 0\hbox{ in } {\mathbb R}^2_-=\{(x^1,x^2)\,|\, x^2<0\}
,\\
u|_{\partial {\mathbb R}^2_-}&=&\partialhi,\sigmaabel{conduct D1b}\\
u&\inftyn& L^\inftynfty({\mathbb R}^2_-)\sigmaabel{conduct D1c}.
\varepsiloneq
Notice that here the radiation condition at infinity (\hbox{Re}\,f{conduct D1c})
is quite simple. We assume just that the potential $u$
does not blow up at infinity.
The equation (\hbox{Re}\,f{conduct D1a}--\hbox{Re}\,f{conduct D1c}) is uniquely solvable
and as before we can define
\ba
\Lambda_\sigma:
H_{comp}^{1/2}(\partial {\mathbb R}^2_-)\to H^{-1/2}(\partial {\mathbb R}^2_-),\quad
\partialhi\mapsto \nu\cdotp\sigma\nabla u|_{\partial {\mathbb R}^2_-}.
\varepsilona
\begin{theorem}\sigmaabel{Lem: A1} The map $\Lambda_\sigma$
determines the
equivalence class
\ba
E_{\sigma}=\{\sigma_1\inftyn \Sigma({\mathbb R}^{2}_-)& |&
\hbox{$\sigma_1 = F_*\sigma$,
$F:{\mathbb R}^2_-\to {\mathbb R}^2_-$ is $W^{1,2}$-diffeomorphism,}\\
& &F|_{\partial {\mathbb R}^2_-}=I\}.
\varepsilona
Moreover, each orbit $E_{\sigma}$ contains at most
one isotropic conductivity, and consequently
if $\sigma$ is known to be isotropic, it is determined uniquely by $\Lambda_
\sigma$.
\varepsilonnd{theorem}
Note that the natural growth requirement
$ \sigmaim_{|z|\to \inftynfty} |F(z)|=\inftynfty$
follows automatically from the above assumptions on $F$.
\subsection{Inverse Problem in the Exterior Domain}
An inverse problem similar to that of the half space can be considered
in an exterior domain where one wants to find the conductivity in
a complement of
a bounded simply connected domain. This type of problem is encountered in
cases where measurement devices are embedded to an unknown domain.
In the case of $S={\mathbb R}^2\setminus \oververline D$,
where $D$ is a bounded Jordan domain,
we consider the problem
\beq\sigmaabel{conduct S}
\nabla\cdotp \sigma\nabla u&=& 0\quad \hbox{ in } S,\\
u|_{\partial S}&=&\partialhi\inftyn H^{1/2}(\partial S),\sigmaabel{conduct S1} \\
u&\inftyn& L^\inftynfty(S).\sigmaabel{conduct S2a}
\varepsiloneq
Again, the radiation condition (\hbox{Re}\,f{conduct S2a}) of infinity
is only that the solution is uniformly bounded.
For this equation we define
\ba
\Lambda_\sigma:
H^{1/2}(\partial S)\to H^{-1/2}(\partial S),\quad
\partialhi\mapsto \nu\cdotp\sigma \nabla u|_{\partial S}.
\varepsilona
Surprisingly, the result is different from the half-space case.
The reason for this is the phenomenon that
the group of diffeomorphisms preserving the data does not fix the point
of the infinity. More precisely,
there are two points $x_0,x_1\inftyn S\cup\{\inftynfty\}$
such that $F(x_0)=\inftynfty$, $F^{-1}(x_1)=\inftynfty$, and
$F:S\setminus \{x_0\}\to S\setminus \{x_1\}$.
In particular, this means that the uniqueness does not hold up to
diffeomorphisms mapping the exterior domain to itself.
For convenience, we compactify $S$ by adding one infinity point,
denote $\oververline S=S\cup \{\inftynfty\}$, and define $\sigma(\inftynfty)=1$.
We say that $F:\oververline S\to \oververline S$ is a $W^{1,2}$-diffeomorphism
if $F$ is homeomorphism and a $W^{1,2}$-diffeomorphism
in spherical metric \cite{Alf}.
\begin{theorem}\sigmaabel{Lem: A2} Let $\sigma\inftyn \Sigma(S)$.
Then the map $\Lambda_\sigma$
determines the
equivalence class
\ba
E_{\sigma,S}=\{\sigma_1\inftyn \Sigma(S)& |&
\hbox{$\sigma_1 = F_*\sigma$,
$F:\oververline S\to \oververline S$ is a $W^{1,2}$-diffeomorphism,}\\
& &F|_{\partial S}=I\,\}.
\varepsilona
Moreover, if $\sigma$ is known to be isotropic, it is determined uniquely
by $\Lambda_\sigma$.
\varepsilonnd{theorem}
\subsection{Data on Part of the Boundary}
In many inverse problems data is measured
only on a part of the boundary. For the conductivity
equation in dimensions $n\taueq 3$ it
has been shown that if the measurements are done
on a part of the boundary, then the integrals of the unknown conductivity over
certain 2-planes can be determined \cite{GU}.
In one-dimensional inverse problems partial data is
often considered with two different boundary conditions,
see e.g. \cite{Le,Ma}.
For instance, in the inverse spectral problem for
a one-dimensional Schr\"odinger operator,
it is known that measuring spectra corresponding to two different boundary
conditions determine the potential uniquely.
Here we consider similar results for the 2-dimensional conductivity
equation assuming that we know measurements on part of the boundary
for two different boundary conditions.
Let us consider the conductivity equation with the Dirichlet
boundary condition
\beq\sigmaabel{conduct D}
\nabla\cdotp \sigma\nabla u&=& 0\hbox{ in } {\cal O}megaega,\\
u|_{\partial {\cal O}megaega}&=&\partialhi\nonumber
\varepsiloneq
and with the Neumann
boundary condition
\beq\sigmaabel{conduct N}
\nabla\cdotp \sigma\nabla v&=& 0\hbox{ in } {\cal O}megaega,\\
\nu\cdotp\sigma \nabla v|_{\partial {\cal O}megaega}&=&\partialsi,\nonumber
\varepsiloneq
normalized by $\inftynt_{\partial {\cal O}mega}v\,dS=0$.
Let $\Gammaamma\subset \partial{\cal O}mega$ be open.
We denote by
$H_0^{s}(\Gammaamma)$ the space of functions $f\inftyn H^{s}(\partial {\cal O}mega)$
that are supported on $\Gammaamma$ and by
$H^{s}(\Gammaamma)$ the space of restrictions $f|_\Gammaamma$ of
$f\inftyn H^{s}(\partial {\cal O}mega)$.
We define the Dirichlet-to-Neumann map
$\Lambda_\Gammaamma$ and Neumann-to-Dirichlet map
$\Sigma_\Gammaamma$ by
\ba
& &\Lambda_\Gammaamma:H_0^{1/2}(\Gammaamma)\to H^{-1/2}(\Gammaamma),
\quad \partialhi\mapsto (\nu\cdotp\sigma \nabla u)|_{\Gammaamma},\\
& &\Sigma_\Gammaamma:H_0^{-1/2}(\Gammaamma)\to H^{1/2}(\Gammaamma),
\quad \partialsi\mapsto v|_{\Gammaamma}.
\varepsilona
\begin{theorem}\sigmaabel{Lem: A3} Let $\Gammaamma\subset \partial{\cal O}mega$ be open.
Then knowing $\partial {\cal O}mega$ and both of the maps
$\Lambda_\Gammaamma$ and
$\Sigma_\Gammaamma$ determine the
equivalence class
\ba
E_{\sigma,\Gammaamma}=\{\sigma_1\inftyn \Sigma({\cal O}mega)& |&
\hbox{$\sigma_1 = F_*\sigma$,
$F:{\cal O}mega\to {\cal O}mega$ is a $W^{1,2}$-diffeomorphism,}\\
& &F|_{\Gammaamma}=I\}.
\varepsilona
Moreover, if $\sigma$ is known to be isotropic, it is determined uniquely
by $\Lambda_\Gammaamma$ and
$\Sigma_\Gammaamma$.
\varepsilonnd{theorem}
\section{PROOF OF THEOREM \hbox{Re}\,f{theorem1}} \sigmaabel{Sec: proofs}
\subsection{Preliminary Considerations}
In the following we identify ${\mathbb R}^2$ and ${\mathbb C}$ by the
map $(x^1,x^2)\mapsto x^1+ix^2$
and denote $z=x^1+ix^2$. We use the standard notations
\ba
\partial_z=\frac 12(\partial_1-i\partial_2),\quad \partial_{\oververline z}=\frac 12(\partial_1+i\partial_2),
\varepsilona
where $\partial_j=\partial/\partial x^j$.
Below we consider
$\sigma:{\cal O}mega\to {\mathbb R}^{2\times 2}$ to be extended as a function
$\sigma:{\mathbb C}\to {\mathbb R}^{2\times 2}$
by defining $\sigma(z)=I$ for $z\inftyn {\mathbb C}\setminus {\cal O}mega$.
In following, we denote $C_0=C_0(\sigma)$.
For the conductivity $\sigma=\sigma^{jk}$
we define the corresponding Beltrami coefficient (see \cite{Sy1,AP,IM})
\beq\sigmaabel{mu}
\mu_{1}(z)=
\frac {-\sigma^{11}(z)+\sigma^{22}(z)-2i\,\sigma^{12}(z)}
{\sigma^{11}(z)+\sigma^{22}(z)+2\sqrt{\deltaet(\sigma(z))}}.
\varepsiloneq
The coefficient $\mu_{1}(z)$ satisfies $|\mu_{1}(z)|\sigmaeq \kappa<1$
and is compactly supported.
Next we introduce a $W^{1,2}$-diffeomorphism (not necessarily
preserving the boundary)
that transforms the conductivity
to an isotropic one.
\begin{lemma}\sigmaabel{lem: 1}
There is a quasiconformal homeomorphism $F:{\mathbb C}\to {\mathbb C}$
such that
\beq\sigmaabel{asympt 1}
F(z)=z+{\cal O}(\frac 1{z})\quad\hbox{as }|z|\to \inftynfty
\varepsiloneq
and such that $F\inftyn W^{1,p}_{loc}({\mathbb C};{\mathbb C})$, $2<p<p(C_0)=\frac {2C_0}{C_0-1}$
for which
\beq\sigmaabel{isotropic}
(F_*\sigma)(z)=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma(z):=
\deltaet(\sigma(F^{-1}(z)))^{\frac 12}.
\varepsiloneq
\varepsilonnd{lemma}
\noindent{\bf Proof.}\quad
The proof can be found from \cite{Sy1} for
$C^3$-smooth conductivities, see also \cite{IM}.
Because of varying sign conventions, we
sketch here the proof for readers convenience. We
need to find a quasiconformal map $F$ such that
\beq\sigmaabel{lassi 1}
DF\,\sigma\,DF^t=\sqrt{\deltaet(\sigma)}\, J_F I
\varepsiloneq
where $J_F=\deltaet(DF)$ is the Jacobian of $F$. Denoting by
$G=[g_{ij}]_{i,j=1}^2$ the inverse of the matrix $\sigma/\sqrt{\deltaet(\sigma)}$
we see that the claim is equivalent to proving the following:
For any symmetric matrix $G$ with $\deltaet(G)=1$ and $\frac 1 K I
\sigmaeq G\sigmaeq K I$ there exists a quasiconformal map
$F$ such that
\beq\sigmaabel{lassi 2}
J_F G= DF^t DF.
\varepsiloneq
Next, the non-linear equation
(\hbox{Re}\,f{lassi 2}) can be replaced in complex notation
by a linear one. Indeed, if $F=u+iv$ then
(\hbox{Re}\,f{lassi 2}) is equivalent to
\beq\sigmaabel{lassi 3}
\nabla v=J G^{-1}\nabla u,\quad \hbox{where }J=
\sigmaeft(\begin{array}{cc}
0 & -1 \\
1 & 0
\varepsilonnd{array}\right).
\varepsiloneq
This follows readily from the identity
\ba
DF^tJ=\deltaet(DF)\,J\,(DF)^{-1}=JG^{-1}DF^t
\varepsilona
where the latter equality uses (\hbox{Re}\,f{lassi 2}).
The matrix $J$ corresponds to the multiplication with the imaginary
unit $i$ in complex notation. Denoting by
$R=
\sigmaeft(\begin{array}{cc}
1 & 0 \\
0 & -1
\varepsilonnd{array}\right)
$
(the matrix corresponding to complex conjugation) we see that
(\hbox{Re}\,f{lassi 3}) is equivalent to
\beq\sigmaabel{lassi 4}
\nabla u+J\nabla v=(G-1)(G+1)^{-1}(\nabla u-J\nabla v).
\varepsiloneq
But, $\nabla u+J\nabla v=2\partialbz F$ and
$R (\nabla u-J\nabla v) =2\partialz F$
in complex notation and hence (\hbox{Re}\,f{lassi 4}) becomes
\beq\sigmaabel{lassi 5}
& &\partial_{\oververline z} F=\mu_{1}(z)\partial_{z} F
\varepsiloneq
where
\ba \mu_{1}=(G-1)(G+1)^{-1}R=(\sqrt {\deltaet \sigma} I-\sigma)
(\sqrt {\deltaet \sigma} I+\sigma )^{-1}R.
\varepsilona
A direct calculation gives
\ba
\mu_{1} =\frac 1{2+g_{11}+g_{22}}\sigmaeft(\begin{array}{cc}
g_{11}-g_{22} & -2g_{12} \\
2g_{12} & g_{11}-g_{22}
\varepsilonnd{array}\right)
\varepsilona
which shows that the matrix $\mu_{1}=(G-1)(G+1)^{-1}R$
corresponds to a multiplication
operator (in complex notation) by the function
\ba
\mu_{1}(z)=\frac {g_{11}(z)-g_{22}(z)+2ig_{12}(z)}
{2+g_{11}(z)+g_{22}(z)}.
\varepsilona
This gives (\hbox{Re}\,f{mu}) since $G^{-1}=\sigma/\sqrt{\deltaet(\sigma)}$.
Since $|\mu_{1}(z)|\sigmaeq \kappa <1$ for every $z\inftyn {\mathbb C}$
it is well known by \cite[Thm. V.1, V.2]{Alf} that the equation
(\hbox{Re}\,f{lassi 5}) with asymptotics
\ba
F(z)=z+{\cal O}(\frac 1z ),\quad\hbox{as }z\to \inftynfty
\varepsilona
has a unique (quasiconformal) solution $F$. The fact that
$F\inftyn W^{1,p}_{loc}({\mathbb C};{\mathbb C})$, $2<p<\frac {2C_0}{C_0-1}$ follows from
\cite{Astala}.
\Box
In this section we denote by $F=F_\sigma$ the diffeomorphism
determined by Lemma \hbox{Re}\,f{lem: 1}.
We also denote $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega=F({\cal O}mega)$ where
$F$ is as in Lemma \hbox{Re}\,f{lem: 1}.
Note that (\hbox{Re}\,f{asympt 1}) implies also that
\beq\sigmaabel{asympt 2}
F^{-1}(z)=z+{\cal O}(\frac 1{|z|})\quad\hbox{as }|z|\to \inftynfty.
\varepsiloneq
Later we will use the obvious fact that
the knowledge of map $\Lambda_{\sigma}$ is equivalent
to the knowledge of the Cauchy data pairs
\ba
C_{\sigma}=\{(u|_{\partial {\cal O}megaega},\nu\cdotp \sigma \nabla u|_{\partial {\cal O}megaega})\ |\ u\inftyn H^1({\cal O}mega),
\ \nabla\cdotp \sigma \nabla u=0\}.
\varepsilona
In addition to the anisotropic conductivity equation (\hbox{Re}\,f{conduct})
we consider the corresponding conductivity equation with isotropic
conductivity. For these considerations,
we observe that if $u$ satisfies equation (\hbox{Re}\,f{conduct})
and $\widetilde \sigma$ is as in (\hbox{Re}\,f{isotropic}) then the function
\ba
w(x)=u(F^{-1}(x))\inftyn H^1(\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega)
\varepsilona
satisfies the isotropic conductivity equation
\beq\sigmaabel{EQ 1}
& &\nabla\cdotp \widetilde\sigma \nabla w=0\quad \hbox{ in }\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega,\\
& &w|_{\partial \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}megaega}=\partialhi\circ F^{-1}.\nonumber
\varepsiloneq
Thus, $\widetilde\sigma$ can be considered as a scalar, isotropic $L^\inftynfty$--smooth
conductivity $\widetilde\sigma I$. We
continue also the function
$\widetilde\sigma:\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega\to {\mathbb R}_+$ to a function
$\widetilde\sigma:{\mathbb C}\to {\mathbb R}_+$
by defining $\widetilde\sigma(x)=1$ for $x\inftyn {\mathbb C}\setminus \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$.
\subsection{Conjugate Functions}
While solving the isotropic inverse problem in \cite{AP}, the interplay of
the scalar conductivities
$\sigma(x)$ and $\frac 1 {\sigma(x)}$ played a crucial role.
Motivated by this, we define
\ba
\widehat}\deltaef\cal{\mathcal \sigma^{jk}(x)=\frac 1{\deltaet(\sigma(x))}\sigma^{jk}(x).
\varepsilona
Note that for a isotropic conductivity $\widehat}\deltaef\cal{\mathcal \sigma=1/\sigma$.
Let now $F$ be the quasiconformal map defined in Lemma \hbox{Re}\,f{lem: 1}
and $\widetilde\sigma=F_*\sigma$ as in (\hbox{Re}\,f{isotropic}). We say that
$\widehat}\deltaef\cal{\mathcal w\inftyn H^1(\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega)$ is a
$\widetilde\sigma$-harmonic conjugate of $w$ if
\beq\sigmaabel{EQ 2}
& &\partial_1\widehat}\deltaef\cal{\mathcal w(z)=-\widetilde\sigma(z)\partial_2 w(z),\\
& &\partial_2\widehat}\deltaef\cal{\mathcal w(z)=\widetilde\sigma(z)\partial_1 w(z)\nonumber
\varepsiloneq
for $z=x^1+ix^2\inftyn {\mathbb C}$.
Using $\widehat}\deltaef\cal{\mathcal w$ we define the function $\widehat}\deltaef\cal{\mathcal u$
that we call the $\sigma$-harmonic conjugate of $u$,
\ba
\widehat}\deltaef\cal{\mathcal u(x)=\widehat}\deltaef\cal{\mathcal w(F(x)).
\varepsilona
To find the equation governing $\widehat}\deltaef\cal{\mathcal u$, it easily follows that
(c.f. \cite{AP})
\beq\sigmaabel{EQ 3}
\nabla\cdotp \frac 1 {\widetilde\sigma} \nabla\widehat}\deltaef\cal{\mathcal w=0\quad \hbox{ in }\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega,
\varepsiloneq
and by changing coordinates to $y=F(x)$ we see that
$1/\widetilde\sigma=F_*\widehat}\deltaef\cal{\mathcal \sigma$.
These facts imply
\beq\sigmaabel{EQ 4}
\nabla\cdotp \widehat}\deltaef\cal{\mathcal \sigma \nabla\widehat}\deltaef\cal{\mathcal u=0\quad \hbox{ in }{\cal O}mega.
\varepsiloneq
Thus $\widehat}\deltaef\cal{\mathcal u$ is the $\widehat\sigma$-harmonic conjugate
function of $u$ and we have
\beq\sigmaabel{EQ 5}
\nabla \widehat}\deltaef\cal{\mathcal u=J \sigma \nabla u,\quad
\nabla u=J\widehat \sigma \nabla \widehat}\deltaef\cal{\mathcal u.
\varepsiloneq
Since $u$ is a solution of the
conductivity equation if and only if $u+c$, $c\inftyn {\mathbb C}$, is
solution, we see from (\hbox{Re}\,f{EQ 5}) that the Cauchy data
pairs $C_{\sigma}$ determine the pairs $C_{\widehat}\deltaef\cal{\mathcal \sigma}$
and vice versa. Thus we get, almost free, that $\Lambda_\sigma$
determines $\Lambda_{\widehat}\deltaef\cal{\mathcal \sigma}$, too.
Let us next consider the function
\beq\sigmaabel{EQ 6}
f(z)=w(z)+i\widehat}\deltaef\cal{\mathcal w(z).
\varepsiloneq
By \cite{AP}, it satisfies the pseudo-analytic equation
of second type,
\beq\sigmaabel{EQ 7}
\partialbz f=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_{2}\, \oververline {\partialz f}
\varepsiloneq
where
\beq\sigmaabel{EQ 7b}
\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_{2}(z)=\frac {1-\widetilde\sigma(z)}{1+\widetilde\sigma(z)},\quad
|\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_{2}(z)|\sigmaeq
\frac {C_0-1}{C_0+1}<1.
\varepsiloneq
Using this Beltrami coefficient,
we define $\mu_2=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_2\circ F$.
We will need the following:
\begin{lemma}\sigmaabel{lemma21}
Let $g=f\circ F$ where
$F:{\cal O}mega\to \widetilde {\cal O}mega$ is a quasiconformal homeomorphism and $f$
is a quasiregular map satisfying
\beq\sigmaabel{L99}
\partialbz f=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_2 \oververline{\partialz f}\quad\
\hbox{and}\quad
\partialbz F=\mu_1 \partialz F,
\varepsiloneq
where $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_2=\mu_2\circ F^{-1}$ and $\mu_1$ satisfies
$|\mu_j|\sigmaeq \kappa<1$ and $\mu_2$ is real.
Then $g$ is quasiregular and satisfies
\beq\sigmaabel{L100}
\partialbz g= \nu_1{\partialz g}+ \nu_2 \oververline{\partialz g},
\varepsiloneq
where
\beq\sigmaabel{L101}
\nu_1=\mu_1 \frac {1-\mu_2^2}{1-|\mu_1|^2\mu_2^2}
,\quad\
\hbox{and}\quad
\nu_2=\mu_2 \frac {1-|\mu_1|^2}{1-|\mu_1|^2\mu_2^2}.
\varepsiloneq
Conversely, if $g$ satisfies (\hbox{Re}\,f{L100}) with
$\nu_2$ real and $|\nu_1|+|\nu_2|\sigmaeq \kappa' <1$
then there exists unique $\mu_1$ and $\mu_2$
such that (\hbox{Re}\,f{L101}) holds and
$f=g\circ F^{-1}$ satisfies (\hbox{Re}\,f{L99}).
\varepsilonnd{lemma}
\noindent{\bf Proof.}\quad
We apply the chain rule
\ba
\partial(f\circ F)&=&
(\partial f)\circ F\cdotp\partial F+(\overline \partial f)\circ F\cdotp\overline{\overline \partial F},\\
\overline \partial(f\circ F)
&=&
(\partial f)\circ F\cdotp\overline \partial F+(\overline \partial f)\circ F\cdotp\overline{ \partial F},
\varepsilona
and obtain
\ba
\nu_1\partialz g+ \nu_2 \oververline{\partialz g}=
\partial f\circ F\cdotp \partial F \cdotp (\nu_1+\nu_2\mu_1\mu_2)+
\oververline {\partial f\circ F}\cdotp {\oververline\partial F}\cdotp (\nu_2+\nu_1\oververline \mu_1\mu_2)
\varepsilona
and
\ba
\partialbz g= \mu_1 \cdotp \partial f\circ F\cdotp \partial F
+
\mu_2 \cdotp \oververline {\partial f\circ F}\cdotp {\oververline\partial F}.
\varepsilona
Hence, if $\mu_1,\mu_2,\nu_1$, and $\nu_2$ are related
so that
\ba
\mu_1=\nu_1+ \nu_1\mu_2,\quad
\mu_2=\nu_2+\oververline \nu_1\mu_2,
\varepsilona
we see that (\hbox{Re}\,f{L100}) and (\hbox{Re}\,f{L101}) are satisfied.
It is not difficult to see that for each $\nu_1$ and
$\nu_2$ (\hbox{Re}\,f{L101}) has a unique solution
$\mu_1,\mu_2$ with $|\mu_j|\sigmaeq \kappa'<1$, $j=1,2$.
Again, the general theory of quasiregular maps \cite{Alf} implies
that (\hbox{Re}\,f{L99}) has a solution and the factorization
$g=f\circ F$ holds.
\Box
Note that (\hbox{Re}\,f{L101}) implies that
\beq\sigmaabel{L103}
|\nu_1|+|\nu_2|= \frac {|\mu_1|+|\mu_2|}{1+|\mu_1|\,|\mu_2|}
\sigmaeq
\frac {2\kappa}{1+\kappa^2}<1.
\varepsiloneq
Lemma \hbox{Re}\,f{lemma21} has the following important corollary,
that is the main goal of this subsection.
\begin{corollary}\sigmaabel{cor21}
If $u\inftyn H^1({\cal O}mega)$ is a real solution
of the conductivity equation (\hbox{Re}\,f{conduct}),
there exists $\widehat}\deltaef\cal{\mathcal u\inftyn H^1({\cal O}mega)$, unique up to
a constant, such that $g=u+i\widehat}\deltaef\cal{\mathcal u$ satisfies
(\hbox{Re}\,f{L100}) where
\beq\sigmaabel{L104}
\nu_1= \frac {\sigma^{22}-\sigma^{11}-2i\sigma^{12}}
{1+\hbox{\rm tr}\, \sigma+\deltaet(\sigma)}
,\quad\
\hbox{and}\quad
\nu_2= \frac {1-\deltaet(\sigma)}
{1+\hbox{\rm tr}\, \sigma+\deltaet(\sigma)}.
\varepsiloneq
Conversely, if
$\nu_1$ and $\nu_2$, $|\nu_1|+|\nu_2|\sigmaeq \kappa' <1$
are as in Lemma \hbox{Re}\,f{lemma21}
then there are unique $\sigma$ and $\widehat}\deltaef\cal{\mathcal\sigma$ such
that for any solution
$g$ of (\hbox{Re}\,f{L100})
$u=\hbox{Re}\, g$ and $\widehat}\deltaef\cal{\mathcal u=\hbox{Im}\, g$ satisfy the
conductivity equations
\beq\sigmaabel{cond eq 2}
\nabla\cdotp \sigma\nabla u=0,\quad\hbox{and}\quad
\nabla\cdotp \widehat}\deltaef\cal{\mathcal \sigma\nabla \widehat}\deltaef\cal{\mathcal u=0.
\varepsiloneq
\varepsilonnd{corollary}
\noindent{\bf Proof.}\quad Since $g=f\circ F$ where $
f=w+i\widehat}\deltaef\cal{\mathcal w$ according to (\hbox{Re}\,f{EQ 6}),
we obtain immediately the existence of $\widehat}\deltaef\cal{\mathcal u=\widehat}\deltaef\cal{\mathcal w\circ F$.
Thus we need only to calculate $\nu_1$ and $\nu_2$
in terms of $\sigma$. Note that by
(\hbox{Re}\,f{mu}),
\beq\sigmaabel{L106}
|\mu_1|^2= \frac {\text{tr}\,(\sigma)-2\deltaet(\sigma)^{1/2}}
{\text{tr}\,(\sigma)+2\deltaet(\sigma)^{1/2}}.
\varepsiloneq
We recall that
\beq\sigmaabel{L107}
\mu_2=
\frac {1-\deltaet(\sigma)^{1/2}}
{1+\deltaet(\sigma)^{1/2}}
\varepsiloneq
and thus
\ba
1-|\mu_1|^2\mu_2^2=
\frac {4(\deltaet(\sigma)^{1/2}\text{tr}\,(\sigma)+
(1+\deltaet(\sigma))\deltaet(\sigma)^{1/2})}
{(1+\deltaet(\sigma)^{1/2})^2(\text{tr}\,(\sigma)+
2\deltaet(\sigma)^{1/2})}
\varepsilona
which readily yields (\hbox{Re}\,f{L104}) from (\hbox{Re}\,f{L101}).
Note that since $\nu_1$ and $\nu_2$ uniquely
determine $\mu_1$ and $\mu_2$, they by (\hbox{Re}\,f{L106})
and (\hbox{Re}\,f{L107}) also determine $\deltaet (\sigma)$
and tr$(\sigma)$. After observing this, it is clear from
(\hbox{Re}\,f{L104}) that $\sigma$ is uniquely determined by $\nu_1
$ and $\nu_2$.
\Box
Now one can write equations (\hbox{Re}\,f{EQ 5}) in
more explicit
form
\beq\sigmaabel{EQ 9}
\tau\cdotp \nabla \widehat}\deltaef\cal{\mathcal u|_{\partial {\cal O}mega}=\Lambda_\sigma (u|_{\partial {\cal O}mega})
\varepsiloneq
where $\tau=(-\nu_2,\nu_1)$ is a unit tangent vector
of $\partial {\cal O}mega$. As $\hbox{Re}\, g|_{\partial {\cal O}mega}=u|_{\partial {\cal O}mega}$
and $\hbox{Im}\, g|_{\partial {\cal O}mega}=\widehat}\deltaef\cal{\mathcal u|_{\partial {\cal O}mega}$, we see that
$\Lambda_\sigma$ determines
the $\sigma$-Hilbert transform ${\cal H}_\sigma$ defined by
\beq\sigmaabel{EQ 9b}
{\cal H}_\sigma&:& H^{1/2}(\partial {\cal O}mega)\to H^{1/2}(\partial {\cal O}mega)/{\mathbb C},\\
\nonumber
& &\hbox{Re}\, g|_{\partial {\cal O}mega}\mapsto \hbox{Im}\, g|_{\partial {\cal O}mega}+{\mathbb C}.
\varepsiloneq
Put yet in another terms, for $u,\widehat}\deltaef\cal{\mathcal u\inftyn H^{1/2}(\partial {\cal O}mega)$,
$\widehat}\deltaef\cal{\mathcal u={\cal H}_\sigma u$ if and only if the map
$g(\xi)=(u+i\widehat}\deltaef\cal{\mathcal u)(\xi),$ $\xi \inftyn \partial {\cal O}mega$, extends
to ${\cal O}mega$ so that (\hbox{Re}\,f{L100}) is satisfied.
Summarizing the previous results, we have
\begin{lemma}\sigmaabel{lem: 2}
The Dirichlet-to-Neumann map $\Lambda_\sigma$
determines the maps $\Lambda_{\widehat}\deltaef\cal{\mathcal \sigma}$ and ${\cal H}_\sigma$.
\varepsilonnd{lemma}
\subsection{Solutions of Complex Geometrical Optics}
Next we consider exponentially growing solutions,
i.e., solutions of complex geometrical optics originated
by Calder\'on for linearized inverse problems and by Sylvester and
Uhlmann for non-linear inverse problems.
In our case, we seek solutions
$G(z,k)$, $z\inftyn {\mathbb C}\setminus {\cal O}mega$, $k\inftyn {\mathbb C}$ satisfying
\beq\sigmaabel{EQ 10}
& &\partialbz G(z,k)=0 \quad \hbox{for }z\inftyn {\mathbb C}\setminus \oververline {\cal O}mega,\\
\sigmaabel{EQ 10 asym}
& &G(z,k)=e^{ikz}(1+{\cal O}_k(\frac 1{z})),\\
\sigmaabel{EQ 10 bnd}
& &\hbox{Im}\, G(z,k)|_{z\inftyn \partial {\cal O}mega}={\cal H}_\sigma
(\hbox{Re}\, G(z,k)|_{z\inftyn \partial {\cal O}mega}).
\varepsiloneq
Here ${\cal O}_k(h(z))$ means a function of $(z,k)$ that satisfies
$|{\cal O}_k(h(z))|\sigmaeq C(k)|h(z)|$ for all $z$ with some constant $C(k)$
depending only on
$k\inftyn {\mathbb C}$.
For the conductivity $\widetilde\sigma$ we consider the
corresponding exponentially growing solutions
$W(z,k)$, $z\inftyn {\mathbb C}$, $k\inftyn {\mathbb C}$ where
\beq\sigmaabel{EQ 11}
& &\partialbz W(z,k)=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_{2}(z)\oververline {\partialz W(z,k)}, \quad \hbox{for }z\inftyn {\mathbb C},\\
\sigmaabel{EQ 11 b}
& &W(z,k)=e^{ikz}(1+{\cal O}_k(\frac 1 z)).
\varepsiloneq
Note that in this stage, $z\mapsto G(z,k)$ is defined only
in the exterior domain ${\mathbb C}\setminus {\cal O}mega$ but $z\mapsto W(z,k)$ in the whole complex plane.
These two solutions are closely related:
\begin{lemma}\sigmaabel{lem: 3} For all $k\inftyn {\mathbb C}$ we have:
i. The system (\hbox{Re}\,f{EQ 11}) has a unique solution
$W(z,k)$ in $ {\mathbb C}$.\\
ii. The system (\hbox{Re}\,f{EQ 10}--\hbox{Re}\,f{EQ 10 bnd}) has a unique solution
$G(z,k)$ in ${\mathbb C}\setminus {\cal O}mega$.\\
iii. For $z\inftyn {\mathbb C}\setminus {\cal O}mega$ we have
\beq\sigmaabel{EQ 11b}
G(z,k)=W(F(z),k).
\varepsiloneq
\varepsilonnd{lemma}
\noindent{\bf Proof.}\quad
For the claim i. we refer to \cite[Theorem 4.2]{AP}.
Next we consider ii. and iii. simultaneously.
Assume that $G(z,k)$ is a solution
of (\hbox{Re}\,f{EQ 10}--\hbox{Re}\,f{EQ 10 bnd}). By Lemma \hbox{Re}\,f{lemma21} and
boundary condition (\hbox{Re}\,f{EQ 10 bnd}) we
see that equation
\ba
& &\overline \partial h(z,k)
= \nu_1\partialz h+\nu_2 \oververline{\partialz h}, \quad \hbox{in }{\cal O}mega, \\
& &h(\cdotp,k)|_{\partial {\cal O}megaega}=G(\cdotp,k)|_{\partial {\cal O}megaega}
\varepsilona
has a unique solution where $\nu_1$ and $\nu_2$ are given
in (\hbox{Re}\,f{L104}).
Let
\beq\sigmaabel{extensions}
H(z,k)=\begin{cases}
G(z,k)&\quad \hbox{for } z\inftyn {\mathbb C}\setminus {\cal O}mega\\
h(z,k)&\quad \hbox{for } z\inftyn {\cal O}mega\varepsilonnd{cases}
\varepsiloneq
and $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)=H(F^{-1}(z),k)$. Then
$\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)$ satisfies equations
\ba
& &\partialbz \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)=0, \quad \hbox{for }z\inftyn {\mathbb C}\setminus \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega,\\
& &\partialbz \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \mu_{2}(z)\oververline {\partialz \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)}, \quad \hbox{for }z\inftyn
\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega,
\varepsilona
and traces from both sides of $\partial\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$ coincide.
Thus $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)$ satisfies equation
in (\hbox{Re}\,f{EQ 11}).
Now (\hbox{Re}\,f{asympt 2}) and (\hbox{Re}\,f{EQ 10 asym}) yield
that
\beq\sigmaabel{EQ 11c}
\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)&=&H(F^{-1}(z),k)\\
&=& \nonumber
\varepsilonxp(ikF^{-1}(z))(1+{\cal O}_k(\frac 1{1+|F^{-1}(z)|}))\\
&=& \nonumber
\varepsilonxp(ikz)(1+{\cal O}_k(\frac 1{1+|z|}))
\varepsiloneq
showing that $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H$ satisfies
(\hbox{Re}\,f{EQ 11}--\hbox{Re}\,f{EQ 11 b}). Thus
by i., $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} H(z,k)=W(z,k)$.
This proves both ii. and iii.
\Box
\subsection{Proof of Theorem \hbox{Re}\,f{theorem1}}
As $G(z,k)$ is the unique solution of (\hbox{Re}\,f{EQ 10}--\hbox{Re}\,f{EQ 10 bnd})
and the operator appearing in boundary condition
(\hbox{Re}\,f{EQ 10 bnd}) is known,
Lemmata \hbox{Re}\,f{lem: 2} and \hbox{Re}\,f{lem: 3}
imply the following:
\begin{lemma}\sigmaabel{lem: 3b} The Dirichlet-to-Neumann map
$\Lambda_\sigma$ determines
$G(z,k)$, $z\inftyn {\mathbb C}\setminus {\cal O}mega$, $k\inftyn {\mathbb C}$.
\varepsilonnd{lemma}
Next we use this to find the diffeomorphism $F_\sigma$ outside ${\cal O}mega$.
\begin{lemma}\sigmaabel{lem: 4} The Dirichlet-to-Neumann map
$\Lambda_\sigma$ determines the values the restriction
$F_\sigma|_{{\mathbb C}\setminus {\cal O}mega}$.
\varepsilonnd{lemma}
\noindent{\bf Proof.}\quad
By (\hbox{Re}\,f{EQ 11b}), $G(z,k)=W(F(z),k)$,
where $W(z,k)$ is the exponentially growing solution corresponding
to the isotropic conductivity $\widetilde\sigma$. Thus by applying
the sub-exponential growth results for such solutions,
\cite[Lemma 7.1 and Thm. 7.2]{AP}, we have representation
\beq\sigmaabel{EQ 17}
W(z,k)=\varepsilonxp(ik\varphi(z,k))
\varepsiloneq
where
\beq\sigmaabel{EQ 18}
\sigmaim_{k\to \inftynfty} \sup_{z\inftyn {\mathbb C}}|\varphi(z,k)-z|=0.
\varepsiloneq
As $F(z)=z+{\cal O}(1/z)$, and $G(z,k)=W(F(z),k)$ we have
\beq\sigmaabel{EQ 19}
\sigmaim_{k\to \inftynfty} \frac{\sigmaog G(z,k)}{ik}=
\sigmaim_{k\to \inftynfty} \varphi(F(z),k)=F(z).
\varepsiloneq
By Lemma \hbox{Re}\,f{lem: 3b} we know the values
of limit (\hbox{Re}\,f{EQ 19}) for any $z\inftyn {\mathbb C}\setminus{\cal O}megaega$. Thus the claim is
proven. \Box
We are ready to prove Theorem \hbox{Re}\,f{theorem1}.
\noindent{\bf Proof.}\quad
As we know $F|_{{\mathbb C}\setminus {\cal O}mega}\inftyn W^{1,p}$, $2<p<p(C_0)$,
we in particular know
$\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega={\mathbb C}\setminus (F({\mathbb C}\setminus {\cal O}mega)).$
When $u$ is the solution of conductivity equation (\hbox{Re}\,f{conduct})
with Dirichlet boundary value $\partialhi$ and $w$ is the solution
of (\hbox{Re}\,f{EQ 1}) with Dirichlet boundary value $\widetilde \partialhi
=\partialhi\circ h$, where $h=F^{-1}|_{\partial \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega}$ we see that
\beq\sigmaabel{kaava 1 apu}
\inftynt_{\partial \widetilde {\cal O}mega}\widetilde \partialhi
\Lambda_{\widetilde\sigma}\widetilde \partialhi\, dS =
Q_{\widetilde\sigma,\widetilde{\cal O}mega}(w)=
Q_{\sigma,{\cal O}mega}(u)=\inftynt_{ \partial {\cal O}mega}\partialhi
\Lambda_{\sigma}\partialhi\, dS.
\varepsiloneq
Here, the second identity is justified by the fact that $F$
is quasiconformal and hence satisfies (\hbox{Re}\,f{Kari 9}).
Since $
\Lambda_{\sigma}$ and
$\Lambda_{\widetilde\sigma}$ are symmetric, this implies
\beq\sigmaabel{kaava 1}
\inftynt_{\partial \widetilde {\cal O}mega}\widetilde \partialsi
\Lambda_{\widetilde\sigma}\widetilde \partialhi\, dS =
\inftynt_{ \partial {\cal O}mega}\partialsi
\Lambda_{\sigma}\partialhi\, dS
\varepsiloneq
for any $\widetilde \partialsi ,\widetilde \partialhi\inftyn H^{1/2}(\partial
\widetilde {\cal O}mega)$ and $ \partialsi , \partialhi\inftyn H^{1/2}(\partial
{\cal O}mega)$ are related by $\widetilde \partialsi= \partialsi\circ h$
and $\widetilde \partialsi= \partialsi\circ h$. Note that
$\partialhi\inftyn H^{1/2}(\partial {\cal O}mega)$ if and only if
$\widetilde \partialhi=\partialhi \circ h\inftyn H^{1/2}(\partial \widetilde{\cal O}mega)$.
To see this, extend $\partialhi$ to a $H^1({\cal O}mega)$ function
and after that define $\widetilde \partialhi$ in the interior
of $\widetilde{\cal O}mega$ by $\widetilde \partialhi=\partialhi\circ F^{-1}$.
Now
\ba
||\nabla \partialhi||_{L^2({\cal O}mega)}^2
\sim
\inftynt_{{\cal O}mega } \nabla \partialhi\cdotp \sigma \oververline {\nabla \partialhi}\,dx
\sim
\inftynt_{\widetilde{\cal O}mega } \nabla \widetilde\partialhi\cdotp
\widetilde\sigma \oververline {\nabla \widetilde\partialhi}\,dx
\sim ||\nabla \widetilde\partialhi||_{L^2(\widetilde{\cal O}mega)}^2
\varepsilona
and hence
\ba
||\partialhi||_{H^{1/2}(\partial {\cal O}mega)}^2
\sim
|| \partialhi||_{H^1({\cal O}mega)}^2
\sim
||\widetilde \partialhi||_{H^1(\widetilde {\cal O}mega)}^2
\sim
||\widetilde \partialhi||_{H^{1/2}(\partial \widetilde {\cal O}mega)}^2.
\varepsilona
As we know $F|_{{\mathbb C}\setminus {\cal O}mega}$ and $\Lambda_\sigma$,
we can find $\Lambda_{\widetilde \sigma}$ using formula
(\hbox{Re}\,f{kaava 1}).
By \cite{AP}, the map $\Lambda_{\widetilde \sigma}$
determines uniquely the conductivity $\widetilde\sigma$
on $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$ in a constructive manner.
Knowing ${\cal O}mega$, $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$, and the boundary value
$f=F|_{\partial {\cal O}mega}$ of the map $F:{\cal O}mega\to \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$, we next
construct a sufficiently smooth diffeomorphism
$H:\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega\to {\cal O}mega$. First, by the Riemann mapping
theorem we can map ${\cal O}mega$ and $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega$ to the unit
disc $\Disc$ by the conformal maps $R$ and $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} R$, respectively.
Now
\ba
G=\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} R\circ F\circ R^{-1}:\Disc\to\Disc
\varepsilona
is a quasiconformal map and since we know $R$ and $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} R$,
we know the function $g=G|_{\partial \Disc}$ mapping
$\partial \Disc$ onto itself. The map $g$ is quasisymmetric
(cf. \cite{Alf}) and by
Ahlfors-Beurling extension theorem \cite[Thm. IV.2]{Alf}
it has a quasiconformal
extension ${\mathcal AB}(g)$ mapping $\oververline \Disc$
onto itself. Note that one can obtain ${\mathcal AB}(g)$
from $g$ constructively by an explicit formula
\cite[p. 69]{Alf}. Thus we may find a quasiconformal diffeomorphism
$H=R^{-1} \circ[{\mathcal AB}(g)]^{-1}\circ \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} R$, $H:\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega\to {\cal O}mega$
satisfying $H|_{\partial \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega}=
F^{-1}|_{\partial \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} {\cal O}mega}$.
Combining the above results, we can find
$H_*\widetilde\sigma$ that is a representative of the equivalence class
$E_\sigma$.
\Box
In the above proof the Riemann mappings can not be found as explicitly
as the Ahlfors-Beurling extension. However, there are
numerical packages for approximative construction of
Riemann mappings, see e.g. \cite{webbi}.
\section{PROOFS OF CONSEQUENCES OF MAIN RESULT}
\sigmaabel{sec: proof of con}
Here we give proofs of Theorems \hbox{Re}\,f{Lem: A1}--\hbox{Re}\,f{Lem: A3}.
{\bf Proof of Theorem \hbox{Re}\,f{Lem: A1}.}
Let $F:{\mathbb R}_-^2={\mathbb R}+i{\mathbb R}_-\to \Disc $ be the M\"obius transform
\ba
F(z)=\frac {z+i}{z-i}.
\varepsilona
Since this map is conformal, we see that
$C_0(F_*\sigma)=C_0(\sigma).$
Let $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma=F_*\sigma$ be the conductivity in $\Disc $.
Then $\Lambda_{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma}\partialhi$ is determined as in
(\hbox{Re}\,f{kaava 1}) for all $\partialhi\inftyn C^\inftynfty_0(\partial \Disc \setminus \{1\})$.
Since $\Lambda_{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma}1=0$ and functions
${\mathbb C}\overplus C^\inftynfty_0(\partial \Disc \setminus \{1\})$ are dense
in the space $H^{1/2}(\partial \Disc )$, we see that
$\Lambda_{\sigma}$ determines the Dirichlet-to-Neumann map
$\Lambda_{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma}$ on $\partial \Disc $. Thus we
can find the equivalence class of the conductivity on $\Disc $.
Pushing these conductivities forward with $F^{-1}$ to ${\mathbb R}^2_-$, we obtain the claim.
\Box
{\bf Proof of Theorem \hbox{Re}\,f{Lem: A2}.}
Let $F:S\to \Disc \setminus \{0\}$ be the conformal map
such that
\ba
\sigmaim_{z\to \inftynfty}F(z)=0.
\varepsilona
Again, since this map is conformal we have for $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma=F_*\sigma$
the equality
$C_0(\sigma)=C_0(F_*\sigma).$
Moreover, if $u$ is a solution of (\hbox{Re}\,f{conduct S}),
we have that $w=u\circ F^{-1}$ is solution of
\beq\sigmaabel{conduct S2}
\nabla\cdotp \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma\nabla w&=& 0\hbox{ in } \Disc \setminus \{0\},
\\
w|_{\partial \Disc}&=&\partialhi\circ F^{-1},\nonumber\\
w&\inftyn& L^\inftynfty( \Disc)\nonumber.
\varepsiloneq
Since set $\{0\}$ has capacitance zero in $\Disc $, we
see that $w=W|_{\Disc \setminus \{0\}}$ where
\beq\sigmaabel{conduct S3}
\nabla\cdotp \widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma\nabla W&=& 0\hbox{ in } \Disc ,
\\
W|_{\partial \Disc}&=&\partialhi\circ F^{-1}.\nonumber
\varepsiloneq
Since $F$ can be constructed via the Riemann mapping theorem,
we see that $\Lambda_\sigma$ determines
$\Lambda_{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma}$ on $\partial \Disc $ and thus
the equivalence class $E_{\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma}$. When
$\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} F:\Disc \to \Disc $ is a boundary preserving
diffeomorphism, we see that $F^{-1}\circ\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} F\circ F$
defines a diffeomorphism $\oververline S\to \oververline S$.
Since we have determined
the conductivity $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma$ up to a
boundary preserving
diffeomorphism, the claim follows easily.
\Box
{\bf Proof of Theorem \hbox{Re}\,f{Lem: A3}.}
Let $\Disc\subset {\mathbb C}$ be the unit disc and
$\Disc_+=\{z\inftyn \Disc\ | \ \hbox{Re}\, z>0\}$.
Let $F:{\cal O}mega\to \Disc_+$ be a Riemann mapping
such that
\ba
\Disc_+\subset {\mathbb R}\times {\mathbb R}_+,\quad F(\Gammaamma)=\partial \Disc_+\setminus
({\mathbb R}\times \{0\}),\quad F(\partial {\cal O}mega \setminus \Gammaamma)=\partial \Disc_+\cap
({\mathbb R}\times \{0\}).
\varepsilona
Let $\varepsilonta:(x^1,x^2)\mapsto (x^1,-x^2)$
and define $\Disc_-=\varepsilonta (\Disc_+)$,
and $\widetilde}\newcommand{{\cal H}OX}[1]{\marginpar{\footnotesize #1} \sigma=F_*\sigma$. Let
\ba
\widehat}\deltaef\cal{\mathcal \sigma(x)=
\begin{cases}
\sigma(x) &\quad \hbox{for } x\inftyn \Disc_+,
\\
(\varepsilonta_*\sigma)(x) &\quad \hbox{for } x\inftyn \Disc_-.\varepsilonnd{cases}
\varepsilona
Consider equation
\beq\sigmaabel{conduct D+-}
\nabla\cdotp \widehat}\deltaef\cal{\mathcal \sigma\nabla w&=& 0\hbox{ in } \Disc.
\varepsiloneq
Using formula (\hbox{Re}\,f{kaava 1}) we see that
$F$ and $\Lambda_\Gammaamma$ determine the corresponding
map $\Lambda_{F(\Gammaamma)}$ for $\widehat}\deltaef\cal{\mathcal \sigma$.
Similarly, we can find $\Sigma_{F(\Gammaamma)}$ for $\widehat}\deltaef\cal{\mathcal \sigma$.
Then $\Lambda_{F(\Gammaamma)}$ determines the Cauchy data
on the boundary for the solutions
of (\hbox{Re}\,f{conduct D+-}) for which
$w\inftyn H^1(\Disc)$, $w=-w\circ \varepsilonta$. On the other hand,
$\Sigma_{F(\Gammaamma)}$ determines the Cauchy data
on the boundary of the solutions
of (\hbox{Re}\,f{conduct D+-}) for which
$w\inftyn H^1(\Disc)$ and $w=w\circ \varepsilonta$.
Now each solution $w$ of (\hbox{Re}\,f{conduct D+-}) can be
written as a linear combination
\ba
w(x)=\frac 12(w(x)+w(\varepsilonta(x)))+\frac 12(w(x)-w(\varepsilonta(x))).
\varepsilona
Thus the maps $\Lambda_{F(\Gammaamma)}$ and $\Sigma_{F(\Gammaamma)}$
together determine $C_{\widehat}\deltaef\cal{\mathcal \sigma}$, and hence we
can find $\widehat}\deltaef\cal{\mathcal \sigma$ up to a diffeomorphism.
We can choose
a representative $\widehat}\deltaef\cal{\mathcal \sigma_0$ of the equivalence class
$E_{\widehat}\deltaef\cal{\mathcal \sigma}$ such that
$\widehat}\deltaef\cal{\mathcal \sigma_0= \widehat}\deltaef\cal{\mathcal \sigma_0\circ \varepsilonta$.
In fact, choosing a symmetric
Ahlfors-Beurling extension in the construction
given in the proof of Theorem \hbox{Re}\,f{theorem1},
we obtain such a conductivity.
Pushing the conductivity $\widehat}\deltaef\cal{\mathcal \sigma_0$
from $\Disc_+$ to ${\cal O}mega$ with $F^{-1}$, we
obtain the claim.
\Box
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\begin{equation}gin{document}
\title{A generalized forward-backward splitting operator: Degenerate analysis and applications}
\begin{equation}gin{abstract}
In this paper, we study the nonexpansive properties of a generalized forward-backward splitting (G-FBS) operator, particularly under the setting of degenerate metric, from which follow the convergence results in terms of degenerate metric of the associated fixed-point iterations. The descent lemma and sufficient decrease property are also extended to degenerate case. It is further shown that the G-FBS operator provides a simplifying and unifying framework to model and analyze a great variety of operator splitting algorithms, many existing results are exactly recovered or relaxed from our general results.
\mathbf{e}nd{abstract}
\begin{equation}gin{keywords}
Generalized forward-backward splitting (G-FBS), nonexpansive properties, degenerate metric, operator splitting algorithms
\mathbf{e}nd{keywords}
\begin{equation}gin{AMS}
68Q25, 47H05, 90C25, 47H09
\mathbf{e}nd{AMS}
\section{Introduction} \label{sec_intro}
\subsection{Forward-backward splitting algorithms}
A classical optimization problem is to find a zero point of the sum of a (set-valued) maximally monotone operator $\mathcal{A}: \mathcal{H} \mapsto 2^\mathcal{H}$ and a $(1/\begin{equation}ta)$-cocoercive operator $\mathcal{B}: \mathcal{H} \mapsto \mathcal{H}$ with $\begin{equation}ta \in [0,+\infty[$\mathbf{f}ootnote{In our context, the problem \mathbf{e}qref{inclusion} encompasses the case of $\begin{equation}ta=0$, which corresponds to $\mathcal{B}=0$, that can be understood as $+\infty$-cocoercive. See further Remark \mathrm{e}f{r_assume_1}-(iv).}:
\begin{equation} \label{inclusion}
0 \in (\mathcal{A} +\mathcal{B}) x^\star.
\mathbf{e}e
{\mathrm{e}d A standard solver of the problem \mathbf{e}qref{inclusion} is the classical forward-backward splitting algorithm \cite{attouch_2018}:
\begin{equation} \label{fbs}
x^{k+1} := \big(\mathcal{A} + \mathbf{f}rac{1}{\tau} \mathcal{I} \big)^{-1}
\big(\mathbf{f}rac{1}{\tau} \mathcal{I} - \mathcal{B} \big) x^k .
\mathbf{e}e
We in this paper study a generalized version of \mathbf{e}qref{fbs}, by extending $\mathbf{f}rac{1}{\tau} \mathcal{I}$ to more general metric $\mathcal{Q}$
\cite{plc_vu_2014}:
\begin{equation} \label{gfbs}
x^{k+1} := \mathcal{T} x^k, \quad \text{where\ }
\mathcal{T} = (\mathcal{A} + \mathcal{Q})^{-1} (\mathcal{Q} - \mathcal{B}),
\mathbf{e}e
where $\mathcal{T}$ is called a generalized forward-backward splitting (G-FBS) operator.}
The scheme \mathbf{e}qref{gfbs} unifies and extends several typical iterative methods shown in Table 1.1. The proximal FBS has been extensively studied in the literature, e.g., \cite{plc,plc_chapter}, in a convex setting, and further discussed in \cite{bolte_2009,bolte_2013,bolte_2014} for the nonconvex case. The metric proximal FBS was recently studied in \cite{pesquet_2016,pesquet_2014} in a nonconvex setting.
\begin{equation}gin{table} [h!] \label{table}
\centering
\caption{The related schemes to G-FBS \mathbf{e}qref{gfbs} }
\hspace*{-.2cm}
\mathrm{e}sizebox{.97\columnwidth}{!} {
\begin{equation}gin{tabular}{|l|l|l|}
\hline
algorithms & iterative schemes & correspondence with \mathbf{e}qref{gfbs} \\
\hline
\tabincell{l}{ classical \\ FBS \cite{attouch_2018,plc_vu_2014} } &
\tabincell{l} { $x^{k+1} := (\mathcal{I} +\tau \mathcal{A})^{-1} (\mathcal{I} -\tau \mathcal{B}) x^k $ \\ \hspace*{.7cm} $:= J_{\tau \mathcal{A}} \circ (\mathcal{I} - \tau \mathcal{B}) x^k $ } & $\mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}$ \\
\hline
\tabincell{l}{ proximal \\ FBS \cite{plc,plc_chapter} } &
\tabincell{l} { $
x^{k+1} := (\mathcal{I} +\tau \partial g)^{-1} (\mathcal{I} - \tau \nabla f) x^k $ \\ \hspace*{.7cm} $ := \prox_{\tau g} \big( x^k - \tau \nabla f(x^k) \big) $ } &
$\mathcal{A} = \partial g$, $\mathcal{B} = \nabla f$, $\mathcal{Q}=\mathbf{f}rac{1}{\tau}\mathcal{I}$ \\
\hline
\tabincell{l}{ metric proximal \\ FBS \cite{pesquet_2016,pesquet_2014,repetti} } &
\tabincell{l} { $ x^{k+1} : = (\partial g + \mathcal{Q})^{-1} (\mathcal{Q} - \nabla f ) x^k $ \\ \hspace*{.7cm} $ := \prox_g^\mathcal{Q} \big( x^k - \mathcal{Q}^{-1} \nabla f(x^k) \big) $ } &
$\mathcal{A} = \partial g$, $\mathcal{B} = \nabla f$ \\
\hline
\tabincell{l}{ PPA \cite{rtr_1976,ppa_guler} } &
\tabincell{l} { $ x^{k+1} : = (\mathcal{I} + \tau \partial g )^{-1} x^k $ \\ \hspace*{.7cm} $ :=\prox_{\tau g} (x^k) $ } &
$\mathcal{A} = \partial g$, $\mathcal{B} = 0$, $\mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}$ \\
\hline
\tabincell{l}{ metric PPA \cite{warp} } &
\tabincell{l} { $ x^{k+1} : = (\mathcal{A} + \mathcal{Q})^{-1} \mathcal{Q} x^k $ } & $\mathcal{B} = 0$ \\
\hline
\mathbf{e}nd{tabular} }
\vskip 0.5em
\mathbf{e}nd{table}
The first-order operator splitting algorithms have been revitalized in the past decade, due to the high demand in solving large and huge scale problems arising in a wide range of fundamental applications (e.g., signal processing and machine learning).
Several frameworks and tools have recently been proposed for the unification of these algorithms, e.g., nonexpansive operator \cite{ljw_mapr}, Fej\'{e}r monotonicity \cite{plc_vu} and fixed-point theory \cite{plc_fixed}. However, these concepts are rather abstract, and not directly connected to the specific algorithms at hand. It is not easy to reinterpret many complicated splitting algorithms using these tools. The works of \cite{teboulle_2018,plc_bregman} discussed the Bregman proximal mapping and the associated Bregman proximal gradient algorithm. It remains unclear how to use the framework to analyze the splitting algorithms.
Asymmetric forward-backward adjoint (AFBA) splitting scheme was shown in \cite{latafat_2017,latafat_chapter} to be a unified structure for many splitting algorithms. However, it fails to provide a standard pattern of convergence analysis for these algorithms. In a series of works of He and Yuan \cite{hbs_siam_2012,hbs_siam_2012_2,hbs_jmiv_2017}, the metric PPA have been shown as a unified framework for Douglas-Rachford splitting (DRS), alternating direction methods of multipliers (ADMM) and primal-dual hybrid gradient (PDHG) algorithms, by the characterization of variational inequality. This approach was further extended in \cite{fxue_gopt}, which unifies and simplifies the analysis of these algorithms based on an equivalent inclusion form. The problem setting, which was characterized by the variational inequality in \cite{hbs_siam_2012,hbs_siam_2012_2,hbs_jmiv_2017} on a case-by-case basis, is completely and uniformly encoded in the maximally monotone operator $\mathcal{A}$ in \cite{fxue_gopt}. The similar idea of using monotone operator theory was also developed in a most recent book draft \cite{ywt_book}. However, this method fails to cover the proximal FBS \cite{plc} and many primal-dual splitting (PDS) algorithms \cite{plc_vu_2014,condat_2013,plc_2012}, which involve a Lipschitz continuous gradient. This limitation was also mentioned in \cite[Sect. 8]{fxue_gopt}.
The purpose of this paper is to show that the G-FBS operator \mathbf{e}qref{gfbs} provides a unified framework and analysis for various splitting algorithms. However, we found that the metric $\mathcal{Q}$ associated with several splitting algorithms is often positive {\it semi-}definite, rather than {\it strictly} positive definite (see Examples \mathrm{e}f{eg_radmm}, \mathrm{e}f{eg_alm} and \mathrm{e}f{eg_lb} in Sect. \mathrm{e}f{sec_eg}). This phenomenon is referred to as ‘{\it degeneracy}’, which implies that the information of $x$ lying in the non-trivial kernel space of $\mathcal{Q}$ is redundant and inactive in the scheme \mathbf{e}qref{gfbs}. Consequently, the convergence of $\{x^k\}_{k\in\mathbb{N}}$ generated by \mathbf{e}qref{gfbs} in the whole space may not hold, since the distance in a sense of the degenerate metric\mathbf{f}ootnote{Correspondingly, the term {\it non-degenerate} used in this paper exactly means {\it strictly positive definite}.} cannot measure the closeness between two points in whole space. Thus, it is not a trivial extension from the classical scalar case of $\mathcal{Q}=\mathbf{f}rac{1}{\tau}\mathcal{I}$ and deserves particular attention. Many recent works related to variable metric version of algorithms assumed the metric $\mathcal{Q}$ to be non-degenerate, e.g., \cite{pesquet_2016,pesquet_2014,repetti,
fxue_gopt,condat_tour}. The degenerate case, to the best of our knowledge, has never been discussed before. This is the focus of this paper.
\subsection{Contributions}
The contributions are in order.
\begin{equation}gin{itemize}
\item We study the nonexpansive properties of the G-FBS operator \mathbf{e}qref{gfbs} and its relaxed version \mathbf{e}qref{rgfbs} under degenerate setting (cf. Sect. \mathrm{e}f{sec_operator}). The weak convergence and metric-based asymptotic regularity of the associated fixed-point iterations are established in Sect. \mathrm{e}f{sec_gfbs}. All of the existing results presented in \cite{plc_vu_2014,pesquet_2016,pesquet_2014,repetti} are extended to the non-degenerate case.
\item We generalize the descent lemma and sufficient decrease property to the degenerate setting (cf. Lemmas \mathrm{e}f{l_descent} and \mathrm{e}f{l_decrease}), and then, the convergence in terms of objective value is established in Proposition \mathrm{e}f{p_gpfbs_obj}. Many related results under non-degenerate setting can be exactly recovered from our results.
\item In Sect. \mathrm{e}f{sec_extension}, the G-FBS operator is further generalized to arbitrary relaxation operator $\mathcal{M}$, the convergence behaviours of the relaxed G-FBS algorithm \mathbf{e}qref{gppa} are discussed.
\item It is shown in Sect. \mathrm{e}f{sec_eg} that a great variety of popular algorithms can be uniformly represented by the G-FBS operator \mathbf{e}qref{gfbs} or its relaxed version \mathbf{e}qref{t_relaxed}, by specifying the operators $\mathcal{A}$ and $\mathcal{B}$, (degenerate) metric $\mathcal{Q}$ (and relaxation operator $\mathcal{M}$, if necessary). There are three big advantages over other existing frameworks: (1) it is easy to fit specific algorithm into the G-FBS operator \mathbf{e}qref{gfbs}, without basically changing the algorithmic structures and exploring the contractive properties as in \cite{ljw_mapr,plc_vu}; (2) the convergence analysis is unified in Sect. \mathrm{e}f{sec_gfbs} and \mathrm{e}f{sec_extension}: one does not need to perform the {\it ad-hoc} analysis for specific algorithms as in \cite{plc_fixed,condat_tour}; (3) it covers proximal FBS and its related algorithms, which cannot be analyzed by metric PPA framework of \cite{fxue_gopt,ywt_book}. This unification and simplification provides a much easier way to understand these algorithms, compared to the original proofs in literature.
\mathbf{e}nd{itemize}
\subsection{Notations and definitions} \label{sec_notation}
We use standard notations and concepts from convex analysis and variational analysis, which, unless otherwise specified, can all be found in the classical and recent monographs \cite{rtr_book,rtr_book_2,plc_book,beck_book}.
A few more words about our notations are in order. Let $\mathcal{H}$ be a real Hilbert space, equipped with inner product $\langle \cdot |\cdot \mathrm{a}ngle$ and induced norm $\|\cdot \|$. The classes of linear, self-adjoint, self-adjoint and positive (semi-)definite operators are denoted by $xS$, $xS_+$ and $xS_{++}$, respectively. The $\mathcal{Q}$-norm is defined as: $\|\cdot\|_\mathcal{Q}^2 := \langle \mathcal{Q}\cdot | \cdot \mathrm{a}ngle$. Note that $\mathcal{Q}$ here is allowed to be degenerate, and thus, $\|x\|_\mathcal{Q}=0$ does not necessarily imply $x=0$. $\mathcal{Q}^\top$ denotes the adjoint of $\mathcal{Q}$. $\mathcal{Q}^\dagger$ stands for the pseudo-inverse of $\mathcal{Q}$, if $\mathcal{Q}$ is singular. The strong and weak convergences are denoted by $\rightarrow$ and $\mathrm{w}eak$, respectively. {\mathrm{e}d Following \cite[Definition 11.3]{plc_book}, the set of minimizers of a function $f$ is denoted by $\Arg\min f$. If $\Arg \min f$ is a singleton, its unique element is denoted by $\arg\min_x f(x)$. }
\vskip.1cm
Note that our expositions will be largely based on the nonexpansive properties in the context of arbitrary degenerate metric $\mathcal{Q}$. Thus, it is necessary to extend the classical notions of Lipschitz continuity \cite[Definition 1.47]{plc_book}, nonexpansiveness \cite[Definition 4.1]{plc_book}, cocoerciveness \cite[Definition 4.10]{plc_book} and averagedness \cite[Definition 4.33]{plc_book}
to this setting.
\begin{equation}gin{definition} \label{def_nonexpansive}
Let the metric $\mathcal{Q}$ be self-adjoint and at least degenerate. Then, the operator $\mathcal{T}$ is:
\begin{equation}gin{itemize}
\item[\rm (i)] {\it $\mathcal{Q}$-based $\xi$-Lipschitz continuous}, if
$ \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q} \le \xi
\big\| x_1 - x_2 \big\|_\mathcal{Q}$.
\item[\rm (ii)] {\it $\mathcal{Q}$-nonexpansive}, if
$ \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q} \le
\big\| x_1 - x_2 \big\|_\mathcal{Q}$.
\item[\rm (iii)] {\it $\mathcal{Q}$-firmly nonexpansive}, if
\[
\langle \mathcal{Q} (x_1 - x_2) | \mathcal{T}x_1-\mathcal{T}x_2 \mathrm{a}ngle
\ge \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2,
\]
or equivalently,
\[
\big\| \mathcal{T}x_1-\mathcal{T}x_2 \big\|_\mathcal{Q}^2 +
\big\|(\mathcal{I} - \mathcal{T}) x_1 - (\mathcal{I} - \mathcal{T}) x_2 \big\|_\mathcal{Q}^2 \le
\big\| x_1 - x_2 \big\|_\mathcal{Q}^2.
\]
\item[\rm (iv)] {\it $\mathcal{Q}$-based $\begin{equation}ta$-cocoercive}, if $\begin{equation}ta \mathcal{T}$ is $\mathcal{Q}$-firmly nonexpansive:
\[
\langle \mathcal{Q} (x_1 - x_2) | \mathcal{T}x_1-\mathcal{T}x_2 \mathrm{a}ngle
\ge \begin{equation}ta \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2.
\]
\item[\rm (v)] {\it $\mathcal{Q}$-based $\alpha$-averaged} with $\alpha \in \ ]0,1 [$, if there exists a $\mathcal{Q}$-nonexpansive operator $\mathcal{K}$, such that $\mathcal{T} = (1-\alpha) \mathcal{I} + \alpha \mathcal{K}$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{definition}
\vskip.1cm
It is easy to check that the standard results of \cite[Remark 4.34, Propositions 4.35, 4.39, 4.40]{plc_book} also hold for the (degenerate) metric $\mathcal{Q}$. This is merely a trivial extension, and the degeneracy of $\mathcal{Q}$ is not troublesome here. For example, similar to \cite[Proposition 4.35]{plc_book}, the $\mathcal{Q}$-based $\alpha$-averaged $\mathcal{T}$ should satisfy
\begin{equation} \label{ave}
\big\|\mathcal{T} x_1 -\mathcal{T} x_2 \big\|_\mathcal{Q}^2 \le
\big\| x_1 - x_2 \big\|_\mathcal{Q}^2
- \mathbf{f}rac{1-\alpha}{\alpha} \big\| (\mathcal{I}-\mathcal{T}) x_1
-(\mathcal{I}-\mathcal{T}) x_2 \big\|_\mathcal{Q}^2.
\mathbf{e}e
\section{The nonexpansive properties under degenerate setting}
\label{sec_operator}
\subsection{Assumptions and basic results}
In this part, we make the following assumption regarding the operator $\mathcal{T}$ in \mathbf{e}qref{gfbs}, with particular focus on the degenerate metric.
\begin{equation}gin{assumption} [{\mathrm{e}d Degenerate} setting] \label{assume_1}
\begin{equation}gin{itemize}
\item [\rm (i)] $\mathcal{A}$ is (set-valued) maximally monotone;
\item [\rm (ii)] $\mathcal{Q}$ is linear, self-adjoint and bounded;
\item [\rm (iii)] $\mathcal{Q}$ is degenerate, such that $\ker \mathcal{Q} \mathbf{a}ckslash \{0\} \ne \mathbf{e}mptyset$;
\item [\rm (iv)] $\mathcal{Q}$ has a closed range, i.e., $\overline{\mathrm{a}n \mathcal{Q}} = \mathrm{a}n \mathcal{Q}$;
\item [\rm (v)] $\mathcal{B}$ is $\begin{equation}ta^{-1}$-cocoercive with $\begin{equation}ta \in\ [0,+\infty[$, and $\mathrm{dom} \mathcal{B} =\mathcal{H}$;
\item [\rm (vi)] $\mathrm{a}n \mathcal{B} \subseteq \mathrm{a}n \mathcal{Q}$;
\item [\rm (vii)] $\mathrm{a}n(\mathcal{A}+\mathcal{Q}) \supseteq \mathrm{a}n (\mathcal{Q}-\mathcal{B})$;
\item [\rm (viii)] $\mathrm{zer}(\mathcal{A}+\mathcal{B}) \ne \mathbf{e}mptyset$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{assumption}
One will see in Sect. \mathrm{e}f{sec_eg} that Assumption \mathrm{e}f{assume_1} is rather mild, which can be satisfied by most typical splitting algorithms. We further have the following remarks regarding Assumption \mathrm{e}f{assume_1}.
\begin{equation}gin{remark} \label{r_assume_1}
{\rm (i)} Assumption \mathrm{e}f{assume_1}-(iv) implies, by \cite[Fact 2.26]{plc_book}, that $\mathbf{e}xists \nu \in\ ]0,+\infty[$, $\| \mathcal{Q} x\| \ge \nu \|x\|$, $\mathbf{f}orall x \in (\ker \mathcal{Q})^\perp = \mathrm{a}n \mathcal{Q}$. This shows: (1) $\mathcal{Q}$ is strictly positive definite in $\mathrm{a}n \mathcal{Q}$; (2) This constant $\nu$ can be understood as the smallest eigenvalue restricted to $\mathrm{a}n\mathcal{Q}$, which will be frequently used in the remainder of this paper. If $\mathcal{Q}$ is non-degenerate and closed, this assumption simply becomes: $\mathbf{e}xists \nu \in\ ]0,+\infty[$, such that $\mathcal{Q} \succeq \nu \mathcal{I}$.
{\rm (ii)} Regarding Assumption \mathrm{e}f{assume_1}-(v), by the classical Baillon-Haddad Theorem \cite[Corollary 18.17]{plc_book}, $\mathcal{B}$ is $\begin{equation}ta$-Lipschitz continuous. By \cite[Example 20.31]{plc_book}, $\mathcal{B}$ is also maximally monotone, and so is $\mathcal{A} +\mathcal{B}$ due to $\mathrm{dom} \mathcal{B}=\mathcal{H}$, by \cite[Corollary 25.5]{plc_book}. This observation will be essential for proving weak convergence in Theorem \mathrm{e}f{t_dist}.
{\rm (iii)} Assumption \mathrm{e}f{assume_1}-(vi) is essential for our degenerate analysis. To understand this, one can think of a simple, but non-trivial example of $ \mathcal{B} = \begin{equation}gin{bmatrix}
1 & 0 \\ 0 & 0 \mathbf{e}nd{bmatrix}$, which is 1-cocoercive, and satisfies $\mathrm{a}n \mathcal{B} \subseteq \mathrm{a}n \mathcal{Q}$ for a degenerate metric $ \mathcal{Q} = \begin{equation}gin{bmatrix}
1 & 0 \\ 0 & 0 \mathbf{e}nd{bmatrix}$.
{\rm (iv)} Assumption \mathrm{e}f{assume_1}-(v) and (vi) also encompass a trivial, but important case of $\mathcal{B}=0$, since $\mathcal{B}=0$ is obviously $0$-Lipschitz continuous (i.e., $\begin{equation}ta=0$), and thus $+\infty$-cocoercive. Also note $\mathrm{a}n \mathcal{B} =\{0\} \subseteq \mathrm{a}n \mathcal{Q}$ for any linear operator $\mathcal{Q}$. This special case reduces the G-FBS operator \mathbf{e}qref{gfbs} to $(\mathcal{A}+\mathcal{Q})^{-1}\mathcal{Q}$---the warped resolvent \cite{warp} or $D$-resolvent \cite{hhb_resolvent,hhb_2003}, which has many important applications in DRS, ADMM and PDHG (see \cite{fxue_gopt} and also Examples \mathrm{e}f{eg_ppa}, \mathrm{e}f{eg_cp}, \mathrm{e}f{eg_plc_dual}, \mathrm{e}f{eg_alm}, \mathrm{e}f{eg_linear_alm} and \mathrm{e}f{eg_lb} in Sect. \mathrm{e}f{sec_eg}).
{\rm (v)} Assumption \mathrm{e}f{assume_1}-(vii) guarantees that $\mathcal{T}$ given by \mathbf{e}qref{gfbs} is well-defined everywhere, i.e., $\mathrm{dom} \mathcal{T} = \mathcal{H}$. This assumption was carefully discussed in \cite{warp,arias_infimal}. If $\mathcal{Q}$ is non-degenerate, by the classical Minty's theorem \cite[Theorem 21.1]{plc_book}, one can conclude $\mathrm{a}n(\mathcal{A}+\mathcal{Q}) = \mathcal{H}$ for any maximally monotone operator $\mathcal{A}$. This automatically fulfills this assumption. However, this assumption is needed here, since $\mathcal{A}+\mathcal{Q}$ may not be surjective in $\mathcal{H}$ for degenerate $\mathcal{Q}$.
{\rm (vi)} Assumption \mathrm{e}f{assume_1}-(viii) ensures the existence of the solution to \mathbf{e}qref{inclusion}.
\mathbf{e}nd{remark}
The following basic results are essential for later developments.
\begin{equation}gin{fact} \label{f_1}
Under Assumption \mathrm{e}f{assume_1}, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $ \big\| \mathcal{B} x \big\|^2 \ge \nu \big\| \mathcal{Q}^\dagger \mathcal{B} x \big\|_{\mathcal{Q}}^2$, $\mathbf{f}orall x\in\mathcal{H}$;
\item[\rm (ii)]
$\big\langle \mathcal{B} x | y \big\mathrm{a}ngle
\le \big\| \mathcal{Q}^\dagger \mathcal{B} x \big\|_{\mathcal{Q}} \cdot
\big\| y\big\|_\mathcal{Q}$, $\mathbf{f}orall (x,y) \in \mathcal{H} \times \mathcal{H}$;
\item[\rm (iii)]
$\big\langle \mathcal{B} x | y \big\mathrm{a}ngle
\le \mathbf{e}ta \big\| \mathcal{Q}^\dagger \mathcal{B} x \big\|_{\mathcal{Q}}^2
+\mathbf{f}rac{1}{4\mathbf{e}ta} \big\| y\big\|_\mathcal{Q}^2$, $\mathbf{f}orall \mathbf{e}ta\in\ ]0,+\infty[$, $\mathbf{f}orall (x,y) \in \mathcal{H} \times \mathcal{H}$,
\mathbf{e}nd{itemize}
where $\nu$ is defined in Remark \mathrm{e}f{r_assume_1}-(i).
\mathbf{e}nd{fact}
\begin{equation}gin{proof}
(i) Since $\mathcal{B} x\in \mathrm{a}n \mathcal{Q}$, we have
\begin{equation}gin{eqnarray}
\big\| \mathcal{B} x \big\|^2 &=&
\big\|\mathcal{P}_{\mathrm{a}n\mathcal{Q}} ( \mathcal{B} x )\big\|^2
\quad \text{[$\mathcal{P}_{\mathrm{a}n\mathcal{Q}}$---projection onto $\mathrm{a}n\mathcal{Q}$]}
\nonumber \\
&= & \big\| \mathcal{Q} \mathcal{Q}^\dagger \mathcal{B} x \big\|^2
\quad \text{[by $\mathcal{P}_{\mathrm{a}n \mathcal{Q}} = \mathcal{Q} \mathcal{Q}^\dagger$]}
\nonumber\\
& \ge & \nu \big\| \mathcal{Q}^\dagger \mathcal{B} x \big\|_{\mathcal{Q}}^2.
\quad \text{[by Remark \mathrm{e}f{r_assume_1}-(i)]}
\nonumber
\mathbf{e}nd{eqnarray}
(ii) $\big\langle \mathcal{B} x | y \big\mathrm{a}ngle =
\big\langle \mathcal{Q} \mathcal{Q}^\dagger \mathcal{B} x | y \big\mathrm{a}ngle
= \big\langle \sqrt{ \mathcal{Q}} \mathcal{Q}^\dagger \mathcal{B} x |
\sqrt{ \mathcal{Q}} y \big\mathrm{a}ngle
\le \big\| \mathcal{Q}^\dagger \mathcal{B} x \big\|_{\mathcal{Q}} \cdot
\big\| y\big\|_\mathcal{Q}$.
(iii) By (ii) and {\mathrm{e}d Fenchel--Young inequality}\mathbf{f}ootnote{\mathrm{e}d The Fenchel--Young inequality reads $f(a)+f^*(b) \ge \langle a|b\mathrm{a}ngle$ for a proper function $f$ \cite[Proposition 13.15]{plc_book}. Particularly, if $f = \mathbf{e}ta \|\cdot\|^2$ with $\mathbf{e}ta>0$, it becomes $\langle a|b \mathrm{a}ngle \le \mathbf{e}ta\|a\|^2 + \mathbf{f}rac{1}{4\mathbf{e}ta} \|b\|^2$, $\mathbf{f}orall \mathbf{e}ta >0$, from which follows Fact \mathrm{e}f{f_1}-(iii).}, we have, $\mathbf{f}orall \mathbf{e}ta\in\ ]0,+\infty[$:
\[
\big\langle \mathcal{B} x | y \big\mathrm{a}ngle =
\big\langle \sqrt{ \mathcal{Q}} \mathcal{Q}^\dagger \mathcal{B} x |
\sqrt{ \mathcal{Q}} y \big\mathrm{a}ngle
\le \mathbf{e}ta \big\| \mathcal{Q}^\dagger \mathcal{B} x \big\|_{\mathcal{Q}}^2
+\mathbf{f}rac{1}{4\mathbf{e}ta} \big\| y\big\|_\mathcal{Q}^2.
\]
\mathbf{e}nd{proof}
\begin{equation}gin{lemma} \label{l_t}
Under Assumption \mathrm{e}f{assume_1}, the G-FBS operator given in \mathbf{e}qref{gfbs} can also be written as
\[
\mathcal{T} = (\mathcal{A}+\mathcal{Q})^{-1} \mathcal{Q} \circ (\mathcal{I} - \mathcal{Q}^\dagger \mathcal{B}).
\]
\mathbf{e}nd{lemma}
\begin{equation}gin{proof}
We have, $\mathbf{f}orall x \in \mathcal{H}$:
\begin{equation}gin{eqnarray}
\mathcal{T} x &=& (\mathcal{A}+\mathcal{Q})^{-1} (\mathcal{Q}-\mathcal{B}) x
= (\mathcal{A}+\mathcal{Q})^{-1} (\mathcal{Q} x - \mathcal{B} x)
\nonumber \\
&= & (\mathcal{A}+\mathcal{Q})^{-1} (\mathcal{Q} x - \mathcal{P}_{\mathrm{a}n \mathcal{Q}} (\mathcal{B} x) )
\quad \text{[by $\mathrm{a}n \mathcal{B} \subseteq \mathrm{a}n \mathcal{Q}$]}
\nonumber \\
&= & (\mathcal{A}+\mathcal{Q})^{-1} (\mathcal{Q} x - \mathcal{Q} \mathcal{Q}^\dagger \mathcal{B} x )
\quad \text{[by $\mathcal{P}_{\mathrm{a}n \mathcal{Q}} = \mathcal{Q} \mathcal{Q}^\dagger$]}
\nonumber \\
&= & (\mathcal{A}+\mathcal{Q})^{-1} \mathcal{Q} ( x - \mathcal{Q}^\dagger \mathcal{B} x ).
\nonumber
\mathbf{e}nd{eqnarray}
Thus, the desired result follows, since $x$ is arbitrary.
\mathbf{e}nd{proof}
\vskip.1cm
In particular, if $\mathcal{Q}$ is non-degenerate, it is easy to derive
$ \mathcal{T} = (\mathcal{I}+\mathcal{Q}^{-1}\mathcal{A})^{-1} (\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B})
= J_{\mathcal{Q}^{-1}\mathcal{A}} \circ (\mathcal{I} - \mathcal{Q}^{-1} \mathcal{B})$, where $J_{\mathcal{Q}^{-1}\mathcal{A}} $ denotes the resolvent of $\mathcal{Q}^{-1}\mathcal{A}$. However, for degenerate case, $(\mathcal{A}+\mathcal{Q})^{-1}\mathcal{Q}$ cannot be viewed as a well-defined resolvent since $\mathrm{a}n(\mathcal{A}+\mathcal{Q}) \supseteq \mathrm{a}n\mathcal{Q}$ and $\mathrm{a}n(\mathcal{A}+\mathcal{Q}) =\mathcal{H}$ may not hold.
\vskip.1cm
\begin{equation}gin{lemma} \label{l_cocoercive}
Under Assumption \mathrm{e}f{assume_1}, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $\mathcal{Q}^{\dagger} \mathcal{B}$ is $\mathcal{Q}$-based $ \mathbf{f}rac{\nu} {\begin{equation}ta}$-cocoercive;
\item[\rm (ii)] $\mathcal{Q}^{\dagger} \mathcal{B}$ is $\mathcal{Q}$-based $ \mathbf{f}rac{\begin{equation}ta} {\nu} $-Lipschitz continuous,
\mathbf{e}nd{itemize}
where $\nu$ is specified in Remark \mathrm{e}f{r_assume_1}-(i).
\mathbf{e}nd{lemma}
\begin{equation}gin{proof}
We deduce that:
\begin{equation}gin{eqnarray}
\big\| x_1 - x_2 \big\|_\mathcal{Q} \cdot
\big\| \mathcal{Q}^\dagger \mathcal{B} x_1 - \mathcal{Q}^\dagger \mathcal{B} x_2 \big\|_\mathcal{Q}
& \ge & \big\langle x_1 - x_2 \big|
\mathcal{B} x_1 - \mathcal{B} x_2 \big\mathrm{a}ngle
\quad \text{[by Fact \mathrm{e}f{f_1}-(ii)]}
\nonumber \\
& \ge & \mathbf{f}rac{1}{\begin{equation}ta} \big\| \mathcal{B} x_1 - \mathcal{B} x_2 \big\|^2
\quad \text{[by cocoerciveness of $\mathcal{B}$]}
\nonumber\\
& \ge & \mathbf{f}rac{\nu}{\begin{equation}ta} \big\| \mathcal{Q}^\dagger \mathcal{B} x_1 -
\mathcal{Q}^\dagger \mathcal{B} x_2 \big\|_{\mathcal{Q}}^2.
\quad \text{[by Fact \mathrm{e}f{f_1}-(i)]}
\nonumber
\mathbf{e}nd{eqnarray}
The proof is completed by Definition \mathrm{e}f{def_nonexpansive}.
\mathbf{e}nd{proof}
\vskip.1cm
Lemma \mathrm{e}f{l_cocoercive} extends the classical Baillon-Haddad theorem \cite[Corollary 18.17]{plc_book} to the degenerate metric case. If $\mathcal{Q}$ is non-degenerate, this result still holds, by replacing the pseudo-inverse $\mathcal{Q}^\dagger$ with a simple inverse $\mathcal{Q}^{-1}$. In addition, Lemma \mathrm{e}f{l_cocoercive} gives an answer to the question raised in \cite[Theorem 3.2]{condat_tour}, where it is required to check the cocoerciveness of $\mathcal{Q}^{-1}\mathcal{B}$ in the context of non-degenerate $\mathcal{Q}$.
\subsection{Nonexpansive properties}
The nonexpansive properties are given below.
\begin{equation}gin{lemma} \label{l_T}
Given the operator $\mathcal{T}: \mathcal{H}\mapsto \mathcal{H}$ defined in \mathbf{e}qref{gfbs}, then, under Assumption \mathrm{e}f{assume_1}, the following hold.
\begin{equation}gin{itemize}
\item [\rm (i)] $\big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2 \le
\langle \mathcal{Q}( x_1 - x_2) | \mathcal{T}x_1 - \mathcal{T} x_2 \mathrm{a}ngle
- \langle \mathcal{B} x_1 - \mathcal{B} x_2 | \mathcal{T}x_1-\mathcal{T}x_2 \mathrm{a}ngle$;
\item[\rm (ii)] $\langle \mathcal{Q}( x_1 - x_2) | \mathcal{T}x_1 - \mathcal{T} x_2 \mathrm{a}ngle \ge \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2 - \mathbf{f}rac{\begin{equation}ta}{4\nu} \big\| (\mathcal{I} - \mathcal{T}) x_1 -
(\mathcal{I} - \mathcal{T}) x_2 \big\|_\mathcal{Q}^2$;
\item[\rm (iii)] $ \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2
+ \Big( 1-\mathbf{f}rac{\begin{equation}ta}{2 \nu } \Big)
\big\|(\mathcal{I}- \mathcal{T}) x_1 - (\mathcal{I} -\mathcal{T}) x_2 \big\|_\mathcal{Q}^2
\le \big\| x_1 - x_2 \big\|_\mathcal{Q}^2 $,
\mathbf{e}nd{itemize}
where $\nu$ is defined in Remark \mathrm{e}f{r_assume_1}-(i).
\mathbf{e}nd{lemma}
\begin{equation}gin{proof}
(i) We develop
\begin{equation}gin{eqnarray}
&& \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2
\nonumber \\ &=& \big \langle \mathcal{Q} \mathcal{T} x_1 - \mathcal{Q} \mathcal{T} x_2 | \mathcal{T} x_1 - \mathcal{T} x_2 \big \mathrm{a}ngle
\nonumber \\
& \le &\big \langle \mathcal{Q} \mathcal{T} x_1 - \mathcal{Q} \mathcal{T} x_2 | \mathcal{T} x_1- \mathcal{T} x_2 \big \mathrm{a}ngle +
\big \langle \mathcal{A} \mathcal{T} x_1 - \mathcal{A} \mathcal{T} x_2 | \mathcal{T}x_1 - \mathcal{T}x_2 \big \mathrm{a}ngle
\quad \text{[by monotonicity of $\mathcal{A}$]}
\nonumber \\ &= &
\big \langle (\mathcal{Q} - \mathcal{B}) x_1 - (\mathcal{Q} - \mathcal{B}) x_2 | \mathcal{T}x_1 - \mathcal{T}x_2\big \mathrm{a}ngle
\quad \text{[since $\mathcal{Q} - \mathcal{B} \in \mathcal{Q} \mathcal{T} +\mathcal{A} \mathcal{T} $ by \mathbf{e}qref{gfbs}]}
\nonumber \\
& = &
\big \langle \mathcal{Q}(x_1 - x_2) | \mathcal{T}x_1-\mathcal{T}x_2 \big \mathrm{a}ngle
- \big \langle \mathcal{B} x_1 - \mathcal{B} x_2 | \mathcal{T}x_1-\mathcal{T}x_2 \big \mathrm{a}ngle.
\nonumber
\mathbf{e}nd{eqnarray}
\vskip.1cm
(ii) Denoting $\mathcal{R} :=\mathcal{I} - \mathcal{T}$, we deduce that
\begin{equation}gin{eqnarray}
&& \big \langle \mathcal{B} x_1 - \mathcal{B} x_2 | \mathcal{T}x_1 - \mathcal{T} x_2 \big \mathrm{a}ngle
\nonumber \\ &=& \big \langle \mathcal{B} x_1 - \mathcal{B} x_2 | x_1 - x_2 \big \mathrm{a}ngle -
\big \langle \mathcal{B} x_1 - \mathcal{B} x_2 | \mathcal{R}x_1 - \mathcal{R} x_2 \big \mathrm{a}ngle
\quad \text{[by $\mathcal{T} = \mathcal{I} - \mathcal{R}$]}
\nonumber \\
& \ge & \mathbf{f}rac{\nu} {\begin{equation}ta} \big\|\mathcal{Q}^\dagger \mathcal{B} x_1 - \mathcal{Q}^\dagger \mathcal{B} x_2 \big\|_\mathcal{Q}^2
- \big \langle \mathcal{B} x_1 - \mathcal{B} x_2 | \mathcal{R}x_1 - \mathcal{R} x_2 \big \mathrm{a}ngle \quad \text{[by Lemma \mathrm{e}f{l_cocoercive}]}
\nonumber \\
& \ge & \mathbf{f}rac{\nu} {\begin{equation}ta} \big\|\mathcal{Q}^\dagger \mathcal{B} x_1 - \mathcal{Q}^\dagger \mathcal{B} x_2 \big\|_\mathcal{Q}^2
- \mathbf{f}rac{\nu}{\begin{equation}ta} \big\|\mathcal{Q}^\dagger \mathcal{B} x_1 -\mathcal{Q}^\dagger \mathcal{B} x_2 \big\|_\mathcal{Q}^2
- \mathbf{f}rac{\begin{equation}ta}{4\nu} \big\| \mathcal{R} x_1 - \mathcal{R} x_2 \big\|_\mathcal{Q}^2
\quad \text{[by Fact \mathrm{e}f{f_1}-(iii)]}
\nonumber \\
& = & - \mathbf{f}rac{\begin{equation}ta}{4\nu} \big\| \mathcal{R} x_1 - \mathcal{R} x_2 \big\|_\mathcal{Q}^2 .
\nonumber
\mathbf{e}nd{eqnarray}
Then, (ii) follows by substituting the above into (i).
\vskip.1cm
(iii) We develop
\begin{equation}gin{eqnarray}
&& \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2
+ \big\|(\mathcal{I}- \mathcal{T}) x_1 - (\mathcal{I} -\mathcal{T}) x_2 \big\|_\mathcal{Q}^2
\nonumber \\
& = & 2 \big\| \mathcal{T} x_1 - \mathcal{T} x_2 \big\|_\mathcal{Q}^2
+ \big\| x_1 - x_2 \big\|_\mathcal{Q}^2
-2 \big\langle \mathcal{Q}(x_1 - x_2) | \mathcal{T} x_1 - \mathcal{T} x_2 \big\mathrm{a}ngle
\nonumber \\
& \le & \big\| x_1 - x_2 \big\|_\mathcal{Q}^2
+\mathbf{f}rac{\begin{equation}ta}{2 \nu} \big\| (\mathcal{I} - \mathcal{T}) x_1 -
(\mathcal{I} - \mathcal{T}) x_2 \big\|_\mathcal{Q}^2,
\quad \text{[by (ii)]}
\nonumber
\mathbf{e}nd{eqnarray}
which yields (iii).
\mathbf{e}nd{proof}
\begin{equation}gin{theorem} \label{t_T}
Let $\mathcal{T}$ be defined as \mathbf{e}qref{gfbs}. Under Assumption \mathrm{e}f{assume_1}, if $\nu$ specified in Remark \mathrm{e}f{r_assume_1}-(ii) satisfies $\nu> \begin{equation}ta/ 2 $, then, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $\mathcal{T}$ is $\mathcal{Q}$-based $\mathbf{f}rac{2\nu } {4 \nu - \begin{equation}ta } $-averaged.
\item[\rm (ii)] $\mathcal{T}$ is always $\mathcal{Q}$-nonexpansive, and in particular, $\mathcal{Q}$-firmly nonexpansive, if and only if $\begin{equation}ta=0$.
\item[\rm (iii)] $\mathcal{I} - \mathcal{T}$ is $\mathcal{Q}$-based $(1-\mathbf{f}rac{\begin{equation}ta}{4\nu})$-cocoercive.
\item[\rm (iv)] $\mathcal{I} - \gamma(\mathcal{I} - \mathcal{T})$ is $\mathcal{Q}$-based $\mathbf{f}rac{2 \gamma \nu } {4 \nu - \begin{equation}ta} $-averaged, if $\gamma \in \ ]0, 2-\mathbf{f}rac{\begin{equation}ta}{2 \nu}[$.
\item[\rm (v)] $\mathcal{I} - \gamma(\mathcal{I} - \mathcal{T})$ is $\mathcal{Q}$-nonexpansive, and in particular, $\mathcal{Q}$-firmly nonexpansive, if $\gamma \in\ ]0, 1-\mathbf{f}rac{\begin{equation}ta}{4 \nu } [$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{theorem}
{\mathrm{e}d
\begin{equation}gin{proof}
(i) Comparing Lemma \mathrm{e}f{l_T}--(iii) with \mathbf{e}qref{ave}, the averagedness $\alpha$ satisfies $\mathbf{f}rac{1-\alpha}{\alpha} = 1-\mathbf{f}rac{\begin{equation}ta}{2\nu}$, which yields $\alpha = \mathbf{f}rac{2\nu } {4 \nu - \begin{equation}ta } $.
\vskip.1cm
(ii) If $\nu>\begin{equation}ta/2$, the averagedness $\alpha = \mathbf{f}rac{2\nu } {4 \nu - \begin{equation}ta} \in [\mathbf{f}rac{1}{2}, 1[$, which shows that $\mathcal{T}$ is always $\mathcal{Q}$-nonexpansive, and in particular, $\mathcal{Q}$-firmly nonexpansive, only when $\alpha=\mathbf{f}rac{1}{2}$, i.e., $\begin{equation}ta = 0$.
\vskip.1cm
(iii) By \cite[Proposition 4.39]{plc_book}, $\mathcal{T}$ is $\mathcal{Q}$-based $\alpha$-averaged, if and only if $\mathcal{I} - \mathcal{T}$ is $\mathcal{Q}$-based $\mathbf{f}rac{1}{2\alpha}$-cocoercive. Substituting $\alpha = \mathbf{f}rac{2\nu } {4 \nu - \begin{equation}ta } $ according to (i) completes the proof.
\vskip.1cm
(iv)--(v): direct results of applying \cite[Proposition 4.40]{plc_book} to (i).
\mathbf{e}nd{proof} }
\begin{equation}gin{remark} \label{r_nonexpansive}
{\rm (i)} Lemma \mathrm{e}f{l_T} and Theorem \mathrm{e}f{t_T} also hold for non-degenerate case, where the condition $\nu >\begin{equation}ta/ 2$ is simply equivalent to $\mathcal{Q} \succ \mathbf{f}rac{\begin{equation}ta}{2} \mathcal{I}$.
{\rm (ii)} If $\begin{equation}ta=0$ (i.e., $\mathcal{B}=0$), the basic nonexpansive properties of the warped resolvent $\mathcal{T} = (\mathcal{A}+\mathcal{Q})^{-1} \mathcal{Q}$ are recovered: (1) $\mathcal{T}$ is $\mathcal{Q}$-based $\mathbf{f}rac{1}{2}$-averaged ($\mathcal{Q}$-firmly nonexpansive); (2) $\mathcal{I} - \gamma(\mathcal{I} - \mathcal{T})$ is $\mathbf{f}rac{\gamma}{2}$-averaged, if $\gamma \in\ ]0, 2[$.
\mathbf{e}nd{remark}
\subsection{Another view on the averagedness}
Lemma \mathrm{e}f{l_T} can also be verified by the composition of $\mathcal{T}$.
\begin{equation}gin{proposition} \label{p_com}
Given $\mathcal{T}$ given as Lemma \mathrm{e}f{l_t}, under Assumption \mathrm{e}f{assume_1}, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $(\mathcal{A}+\mathcal{Q})^{-1} \mathcal{Q}$ is $\mathbf{f}rac{1}{2}$-averaged (or simply $\mathcal{Q}$-firmly nonexpansive).
\item[\rm (ii)] $\mathcal{I} - \mathcal{Q}^\dagger \mathcal{B}$ is $\mathcal{Q}$-based $\mathbf{f}rac {\begin{equation}ta} {2\nu }$-averaged, if $\nu > \begin{equation}ta/ 2$. In particular, if $\nu \ge \begin{equation}ta$, $\mathcal{I} - \mathcal{Q}^\dagger \mathcal{B}$ is $\mathcal{Q}$-firmly nonexpansive.
\item[\rm (iii)] $\mathcal{T}$ is $\mathcal{Q}$-based $\mathbf{f}rac{2\nu} {4 \nu -\begin{equation}ta }$-averaged, if $\nu > \begin{equation}ta/ 2$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
{\rm (i)} Remark \mathrm{e}f{r_nonexpansive}-(ii).
\vskip.1cm
(ii) First, $\mathcal{Q}^\dagger \mathcal{B}$ is $\mathcal{Q}$-based $ \mathbf{f}rac{\nu} {\begin{equation}ta}$-cocoercive by Lemma \mathrm{e}f{l_cocoercive}. Then, $\mathcal{I} - \mathcal{Q}^\dagger \mathcal{B}$ is $\mathcal{Q}$-based $ \mathbf{f}rac {\begin{equation}ta} {2\nu} $-averaged by \cite[Proposition 4.39]{plc_book}.
\vskip.1cm
(iii) If $\begin{equation}ta>0$, since $\mathcal{T} = (\mathcal{A}+\mathcal{Q})^{-1} \mathcal{Q} \circ (\mathcal{I} - \mathcal{Q}^\dagger \mathcal{B}) $, with $\alpha_1 = \mathbf{f}rac{1}{2}$ and $\alpha_2 = \mathbf{f}rac {\begin{equation}ta} {2\nu}$, the averagedness of $\mathcal{T}$, by \cite[Theorem 3]{yamada} or \cite[Proposition 2.4]{plc_yamada}, is given as $\alpha = \mathbf{f}rac{\alpha_1+\alpha_2 -2\alpha_1\alpha_2}
{1-\alpha_1\alpha_2} = \mathbf{f}rac{2\nu} {4 \nu -\begin{equation}ta }$.
If $\begin{equation}ta=0$, $\mathcal{T}$ becomes $(\mathcal{A}+\mathcal{Q})^{-1}\mathcal{Q}$, which is $\mathcal{Q}$-based $\mathbf{f}rac{1}{2}$-averaged by the assertion (i).
Both cases are merged as (iii).
\mathbf{e}nd{proof}
\begin{equation}gin{remark} \label{r_ave}
Proposition \mathrm{e}f{p_com}--(iii) is an extended version of \cite[Proposition 4.14]{pfbs_siam}, from a scalar parameter $\gamma$ to (degenerate) metric $\mathcal{Q}$. As observed in \cite[Remark 1]{ljw_mapr}, Proposition \mathrm{e}f{p_com}--(iii) is sharper than \cite[Proposition 4.32]{plc_book}, which gives the averagedness of $\mathcal{T}$ as
\[
\alpha = \mathbf{f}rac{2} {1+ \mathbf{f}rac{1}{\max \{\mathbf{f}rac{1}{2},
\mathbf{f}rac{2\nu} {\begin{equation}ta} \} } } =
\mathbf{f}rac{2\begin{equation}ta} {\begin{equation}ta + 2\times \min\{\begin{equation}ta,
\nu \} } = \max \bigg\{ \mathbf{f}rac{2}{3},
\mathbf{f}rac{2\begin{equation}ta} {\begin{equation}ta+ 2\nu } \bigg\}.
\]
\mathbf{e}nd{remark}
\subsection{Short summary} \label{sec_summary}
This section presents the nonexpansive properties of the G-FBS operator \mathbf{e}qref{gfbs} under degenerate metric setting, which extends the discussions of classical FBS operator in \cite{ljw_mapr}. Most existing works related to metric-based FBS algorithms, e.g., \cite{plc_vu_2014,pesquet_2016,pesquet_2014}, are concerned with the convergence issue under the problem setting of minimization of $f+g$.
Our exposition here deals with more general operators $\mathcal{A}$ and $\mathcal{B}$ (not limited to the subdifferential or gradient of some functions, see further Remark \mathrm{e}f{r_assume_2}-(iv)) under more general metric (in particular, degenerate case). We will see in Sect. \mathrm{e}f{sec_eg} that these generalizations are essential for the broad applications.
\section{The G-FBS algorithm under degenerate setting}
\label{sec_gfbs}
We now consider the G-FBS scheme \mathbf{e}qref{gfbs} or its equivalent implicit form \cite[Eq.(54)]{condat_tour}:
\begin{equation} \label{gfbs_eq}
0 \in \mathcal{A} x^{k+1} +\mathcal{B} x^k +\mathcal{Q} (x^{k+1}-x^k).
\mathbf{e}e
The convergence has been investigated in \cite{plc_vu_2014} for the non-degenerate metric. We here focus on the degenerate case.
\subsection{Convergence in terms of metric distance}
\label{sec_dist}
First, it is easy to recognize the following
\begin{equation}gin{fact} \label{f_2}
$\mathbf{F}ix \mathcal{T} = \mathrm{zer} (\mathcal{A} + \mathcal{B})$.
\mathbf{e}nd{fact}
\begin{equation}gin{proof}
Indeed,
$x^\star \in \mathbf{F}ix \mathcal{T} \Longleftrightarrow
x^\star = (\mathcal{A} + \mathcal{Q})^{-1} (\mathcal{Q} - \mathcal{B}) x^\star
\Longleftrightarrow (\mathcal{Q} - \mathcal{B}) x^\star \in
(\mathcal{A} + \mathcal{Q}) x^\star \Longleftrightarrow 0 \in (\mathcal{A} +\mathcal{B}) x^\star \Longleftrightarrow x^\star \in \mathrm{zer} (\mathcal{A} +\mathcal{B})$.
\mathbf{e}nd{proof}
The following theorem is a main result of this paper, which shows the weak convergence of $\{x^k\}_{k\in\mathbb{N}}$ in $\mathrm{a}n\mathcal{Q}$. The proof adopts some techniques in \cite[Theorem 2.1]{fxue_gopt}.
\begin{equation}gin{theorem}[Weak convergence in $\mathrm{a}n \mathcal{Q}$]
\label{t_dist}
Let $x^0\in \mathcal{H}$, $\{x^k\}_{k \in \mathbb{N}}$ be a sequence generated by \mathbf{e}qref{gfbs}. Under Assumption \mathrm{e}f{assume_1}, if $\nu > \begin{equation}ta/2$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] {\rm [Finite length in $\mathrm{a}n\mathcal{Q}$]} $\{\mathcal{Q} x^k\}_{k\in\mathbb{N}}$ has a finite length in a sense that $\sum_{k=0}^\infty \|x^{k+1}-x^k\|_\mathcal{Q}^2 < \infty$.
\item[\rm (ii)] {\rm [$\mathcal{Q}$-based asymptotic regularity]} $\mathcal{Q} (x^k-x^{k+1}) \rightarrow 0$, as $k\rightarrow \infty$.
\item[\rm (iii)] {\rm [Rate of $\mathcal{Q}$-based regularity]} $\|x^{k+1 } -x^{k} \|_\mathcal{Q}$ has the pointwise rate of $\mathcal{O}(1/\sqrt{k})$:
\[
\big\|x^{k +1} -x^{k} \big\|_\mathcal{Q}
\le \mathbf{f}rac{1}{\sqrt{k+1}} \sqrt{ \mathbf{f}rac{2\nu}{2\nu-\begin{equation}ta} }
\big\|x^{0} -x^\star \big\|_\mathcal{Q}, \quad
\mathbf{f}orall k \in \mathbb{N}.
\]
\item[\rm (iv)] {\rm [Weak convergence in $\mathrm{a}n \mathcal{Q}$]} There exists $x^\star \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$, such that $ \mathcal{Q} x^k \mathrm{w}eak \mathcal{Q} x^\star$, as $k\rightarrow \infty$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{theorem}
\vskip.2cm
\begin{equation}gin{proof}
(i)-(ii) Taking $x_1 = x^k$ and $x_2 = x^\star \in \mathbf{F}ix \mathcal{T} $ in Lemma \mathrm{e}f{l_T}-(iii), we obtain
\begin{equation} \label{x12}
\big\| x^{k+1} -x^\star \big\|_\mathcal{Q}^2
\le \big\| x^k - x^\star \big\|_\mathcal{Q}^2
- \Big( 1-\mathbf{f}rac{\begin{equation}ta}{2 \nu } \Big)
\big\| x^k - x^{k+1} \big\|_\mathcal{Q}^2.
\mathbf{e}e
Summing up \mathbf{e}qref{x12} from $k=0$ to $K$ yields
\begin{equation} \label{x34}
\sum_{k=0}^{K} \big\| x^k - x^{k+1} \big\|
_\mathcal{Q}^2 \le \mathbf{f}rac{2\nu}{2\nu-\begin{equation}ta}
\big\| x^0 - x^\star \big\|_\mathcal{Q}^2.
\mathbf{e}e
Taking $K \rightarrow \infty$, we have: $\sum_{k=0}^{\infty} \big\| x^k - x^{k+1} \big\|_\mathcal{Q}^2 \le \mathbf{f}rac{2\nu}{2\nu-\begin{equation}ta} \big\| x^{0} - x^\star \big\|_\mathcal{Q}^2 < +\infty$, which implies that $\lim_{k\rightarrow \infty} \|x^k-x^{k+1}\|_\mathcal{Q} = 0$.
\vskip.1cm
(iii) Taking $x_1 = x^k$ and $x_2 = x^{k+1}$ in Lemma \mathrm{e}f{l_T}-(iii), we have
\begin{equation} \label{x33}
\big\| x^{k+1} -x^{k+2} \big\|_\mathcal{Q}
\le \big\| x^k - x^{k+1} \big\|_\mathcal{Q},
\mathbf{e}e
which implies that $\| x^k - x^{k+1} \|_\mathcal{Q}$ is non-increasing. Then, (iii) follows from \mathbf{e}qref{x34}.
\vskip.1cm
(iv) Following the reasoning of the well-known Opial's lemma \cite{opial}\mathbf{f}ootnote{Refer to \cite[Lemma 2.47]{plc_book} or \cite[Lemma 2.1]{attouch_2001} for the Opial's argument.}, the weak convergence proof is divided into 3 steps\mathbf{f}ootnote{This line of reasoning is very similar to Fej\'{e}r monotonicity, see \cite[Proposition 5.4, Theorem 5.5]{plc_book} for example.}:
{\mathrm{e}d
Step-1: show that $\lim_{k\rightarrow \infty} \| x^{k} - x^\star \|_\mathcal{Q} $ exists for any given $x^\star \in \mathrm{zer} (\mathcal{A}+\mathcal{B}) $; }
Step-2: show that $\{\sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ has at least one weak sequential cluster point lying in $\sqrt{\mathcal{Q}} \mathrm{zer} (\mathcal{A}+\mathcal{B}) $;
Step-3: show that the cluster point of $\{\sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ is unique.
\vskip.2cm
Step-1: \mathbf{e}qref{x12} shows that the sequence
$\{ \| x^{k} - x^\star \|_\mathcal{Q} \}_{k\in\mathbb{N}} $ is non-increasing, and bounded from below (always being non-negative), and thus, convergent, i.e. $\lim_{k\rightarrow \infty} \| x^{k} - x^\star \|_\mathcal{Q}$ exists.
\vskip.2cm
Step-2: \mathbf{e}qref{x12} also implies that the sequence $\{\sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ is bounded, since $\big\|\sqrt{\mathcal{Q}} x^k- \sqrt{\mathcal{Q}}x^\star \big\| = \|x^k-x^\star\|_\mathcal{Q} \le \|x^0 - x^\star\|_\mathcal{Q}
= \big\|\sqrt{\mathcal{Q}} x^0- \sqrt{\mathcal{Q}}x^\star \big\| $, $\mathbf{f}orall k \in \mathbb{N}$. Then, by \cite[Lemma 2.37]{plc_book}, $\{\sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ has at least one weak sequential cluster point, i.e. $\{\sqrt{\mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ has a subsequence $\{\sqrt{ \mathcal{Q}} x^{k_i}\}_{i\in\mathbb{N}}$ that weakly converges to a point $v^*$, denoted by $\sqrt{\mathcal{Q}} x^{k_i} \rightharpoonup v^*$, as $k_i \rightarrow \infty$. Our aim in Step-2 is to show that $v^* = \sqrt{\mathcal{Q}} x^*$ for some $x^* \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$, and more generally, every weak sequential cluster point of $\{ \sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ belongs to the set $ \sqrt{\mathcal{Q}} \mathrm{zer} (\mathcal{A}+\mathcal{B}) := \{v\in \mathcal{H}: v=\sqrt{\mathcal{Q}} x\text{\ and\ } x \in \mathrm{zer} (\mathcal{A}+\mathcal{B}) \}$. To this end, we first note that (i) asserts that $\mathcal{Q} (x^k- x^{k+1}) \rightarrow 0$ strongly in $\mathcal{H}$ as $k \rightarrow \infty$, and further, $\mathrm{dist} (\mathcal{A} x^{k+1}+\mathcal{B} x^k, 0) \rightarrow 0$, as $k\rightarrow \infty$ by the scheme \mathbf{e}qref{gfbs}. Then, since $\mathcal{B}$ is $\begin{equation}ta$-Lipschitz continuous (by Remark \mathrm{e}f{r_assume_1}-(ii)), and due to the sequential closedness of the graph of $\mathcal{A}+\mathcal{B}$ in $\mathcal{H}_\text{weak} \times \mathcal{H}_\text{strong}$ (by the maximality of $\mathcal{A}+\mathcal{B}$ in Remark \mathrm{e}f{r_assume_1}-(ii) and \cite[Proposition 20.38]{plc_book}), the weak sequential cluster point of $\{x^{k} \}_{k\in \mathbb{N}}$ lies in $\mathrm{zer} (\mathcal{A}+\mathcal{B})$. Owing to the closedness of $\mathrm{a}n \mathcal{Q}$ (i.e., Assumption \mathrm{e}f{assume_1}-(iv)), $\{\sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ has at least one weak sequential cluster point lying in $\sqrt{\mathcal{Q}} \mathrm{zer} (\mathcal{A}+\mathcal{B}) $.
\vskip.2cm
Step-3: We need to show that $\{\sqrt{\mathcal{Q}} x^k\}_{k\in\mathbb{N}}$ cannot have two distinct weak sequential cluster point in $\sqrt{ \mathcal{Q}} \mathrm{zer} (\mathcal{A}+\mathcal{B})$. To this end, let $\sqrt{\mathcal{Q}} x_1^*, \sqrt{\mathcal{Q}} x_2^{*} \in \sqrt{\mathcal{Q}} \mathrm{zer} (\mathcal{A}+\mathcal{B})$ be two cluster points of $\{ \sqrt{ \mathcal{Q}} x^k\}_{k\in\mathbb{N}}$. Since $\lim_{k\rightarrow \infty} \| x^{k} - x^\star \|_\mathcal{Q} $ exists as proved in Step-1, set $l_1 = \lim_{k\rightarrow \infty} \| x^k - x_1^*\|_\mathcal{Q} = \|\sqrt{\mathcal{Q}} x^k - \sqrt{\mathcal{Q}} x_1^*\| $, and $l_2 = \lim_{k\rightarrow \infty} \| x^k - x_2^*\|_\mathcal{Q} = \|\sqrt{\mathcal{Q}} x^k - \sqrt{\mathcal{Q}} x_2^*\| $. Take a subsequence $\{\sqrt{\mathcal{Q}} x^{k_i} \}$ weakly converging to $ \sqrt{\mathcal{Q}} x_1^*$, as $k_i \rightarrow \infty$. From the identity of
\[
\big\| x^k-x_1^*\big\|_\mathcal{Q}^2 - \big\|x^k-x_2^* \big\|_\mathcal{Q}^2
=\big\|x_1^*- x_2^*\big\|_\mathcal{Q}^2 +2 \big\langle
x_1^*- x_2^* \big| x_2^* - x^k \big\mathrm{a}ngle_\mathcal{Q},
\]
we deduce that $l_1 -l_2 =- \big\| x_1^*- x_2^*\big\|_\mathcal{Q}^2$ by taking $k\rightarrow \infty$ on both sides. Similarly, take a subsequence $\{ \sqrt{\mathcal{Q}} x^{k_j} \}$ weakly converging to $\sqrt{\mathcal{Q}} x_2^*$, as $k_j \rightarrow \infty$, which yields that $l_1 -l_2 = \big\|x_1^* - x_2^*\big\|_\mathcal{Q}^2$. Consequently, $\big\|x_1^*-x_2^*\big\|_\mathcal{Q} =
\big\|\sqrt{\mathcal{Q}} x_1^* - \sqrt{\mathcal{Q}} x_2^*\big\| =0$, which establishes the uniqueness of the weak sequential cluster point, denoted by $\sqrt{\mathcal{Q}} x^\star$.
Finally, with a trivial replacement of $\sqrt{\mathcal{Q}}$ by $\mathcal{Q}$ (see, for instance, \cite[Fact 2.25]{plc_book}), we summarize that $\{\mathcal{Q} x^k\}_{k\in\mathbb{N}}$, is bounded and possesses a unique weak sequential cluster point $\mathcal{Q} x^\star \in \mathcal{Q} \mathrm{zer} (\mathcal{A}+\mathcal{B})$. By \cite[Lemma 2.38]{plc_book}, $\mathcal{Q} x^k \rightharpoonup \mathcal{Q} x^\star \in \mathcal{Q} \mathrm{zer} (\mathcal{A}+\mathcal{B}) $, as $k\rightarrow \infty$.
\mathbf{e}nd{proof}
\vskip.1cm
If $\mathcal{Q}$ is non-degenerate, one can safely conclude $x^{k+1}-x^k\rightarrow 0$ and $x^k \mathrm{w}eak x^\star$, as $k\rightarrow \infty$. This coincides with \cite[Theorem 4.1]{plc_vu_2014} and \cite[Theorem 4.8]{repetti}. The convergence results of the basic FBS algorithm with $\mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}$ presented in \cite{rtr_1976,ppa_guler,corman,taomin_2018} are also exactly recovered. However, if $\mathcal{Q}$ is degenerate, one cannot establish the weak convergence of $x^k \rightharpoonup x^\star \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$. It is indeed true that the weak sequential cluster point of $\{x^{k} \}_{k\in \mathbb{N}}$ lies in $\mathrm{zer} (\mathcal{A}+\mathcal{B})$ as proved in Step-2. However, the cluster point may not be unique. As observed in Step-3, one can only obtain $\big\|x_1^*-x_2^*\big\|_\mathcal{Q} =0$, which yields $x_1^*- x_2^* \in \ker \mathcal{Q}$ rather than $x_1^*=x_2^*$. This is essentially due to the existence of $\lim_{k\rightarrow \infty}\big\|x^k- x^\star\big\|_\mathcal{Q}$ only (as shown in Step-1), while the existence of $\lim_{k\rightarrow \infty} \big\|x^k- x^\star\big\|$ is not guaranteed.
In addition, in view of \cite[Lemma 2.7]{corman}, the rate of asymptotic regularity in Theorem \mathrm{e}f{t_dist}-(iii) can be refined to $o(1/\sqrt{k})$.
\vskip.1cm
If $\mathcal{B}=0$, \mathbf{e}qref{gfbs} becomes the (degenerate) metric PPA:
\begin{equation} \label{ppa}
x^{k+1} := (\mathcal{A}+\mathcal{Q})^{-1} \mathcal{Q} x^k,
\mathbf{e}e
whose convergence result is stated below.
\begin{equation}gin{corollary}[Weak convergence in $\mathrm{a}n \mathcal{Q}$]
\label{c_ppa}
Let $x^0\in \mathcal{H}$, $\{x^k\}_{k \in \mathbb{N}}$ be a sequence generated by \mathbf{e}qref{ppa}. Under Assumption \mathrm{e}f{assume_1} with $\mathcal{B}=0$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] {\rm [Rate of $\mathcal{Q}$-based regularity]} $\|x^{k+1 } -x^{k} \|_\mathcal{Q}$ has the pointwise rate of $\mathcal{O}(1/\sqrt{k})$:
\[
\big\|x^{k +1} -x^{k} \big\|_\mathcal{Q}
\le \mathbf{f}rac{1}{\sqrt{k+1}}
\big\|x^{0} -x^\star \big\|_\mathcal{Q}, \quad
\mathbf{f}orall k \in \mathbb{N}.
\]
\item[\rm (ii)] {\rm [Weak convergence in $\mathrm{a}n \mathcal{Q}$]} There exists $x^\star \in \mathrm{zer} \mathcal{A}$, such that $ \mathcal{Q} x^k \mathrm{w}eak \mathcal{Q} x^\star$, as $k\rightarrow \infty$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{corollary}
\begin{equation}gin{proof}
Substituting $\begin{equation}ta=0$ into Theorem \mathrm{e}f{t_dist}.
\mathbf{e}nd{proof}
\subsection{Convergence of objective value}
\subsubsection{Assumptions and basic descent results}
If $\mathcal{A} = \partial g$, $\mathcal{B} =\nabla f$, where the functions $f$ and $g$ satisfy the following
\begin{equation}gin{assumption} \label{assume_2}
\begin{equation}gin{itemize}
\item[\rm (i)] $f: \mathcal{H} \mapsto \mathbb{R}$ and $g: \mathcal{H} \mapsto \mathbb{R} \cup \{ +\infty\}$ are proper and lower semi-continuous (l.s.c.);
\item[\rm (ii)] $f$ is Fr\'{e}chet differentiable with $\begin{equation}ta$-Lipschitz continuous gradient $\nabla f$;
\item[\rm (iii)] $\mathrm{dom} f\cap \mathrm{dom} g \ne \mathbf{e}mptyset$, $\Arg\min (f+g) \ne \mathbf{e}mptyset$;
\item[\rm (iv)] $\mathcal{Q}$ is linear, bounded, closed, self-adjoint and degenerate, such that $\ker\mathcal{Q} \mathbf{a}ckslash \{0\} \ne \mathbf{e}mptyset$;
\item[\rm (v)] $\mathrm{a}n \nabla f \subseteq \mathrm{a}n \mathcal{Q}$;
\item[\rm (vi)] $\mathrm{a}n (\partial g+\mathcal{Q}) \supseteq \mathrm{a}n (\mathcal{Q}-\nabla f)$;
\item[\rm (vii)] $g$ is convex;
\item[\rm (viii)] $f$ is convex.
\mathbf{e}nd{itemize}
\mathbf{e}nd{assumption}
\begin{equation}gin{remark} \label{r_assume_2}
{\rm (i)} Assumption \mathrm{e}f{assume_2} is parallel to Assumption \mathrm{e}f{assume_1}, by the correspondence of $\mathcal{A}=\partial g$ and $\mathcal{B} = \nabla f$. Under Assumption \mathrm{e}f{assume_2}-(iii), the problem \mathbf{e}qref{inclusion} is equivalent to minimizing $f+g$, if $\mathrm{dom} f\cap \mathrm{dom} g \ne \mathbf{e}mptyset$, by \cite[Proposition 16.42]{plc_book}.
{\rm (ii)} To understand Assumption \mathrm{e}f{assume_2}-(v), consider a function
$f:\mathbb{R}^2\mapsto \mathbb{R}: x=(a,b)\mapsto \mathbf{f}rac{1}{2}a^2$. Then, $\nabla f(x) = \begin{equation}gin{bmatrix}
\partial_a f(x) \\ \partial_b f(x) \mathbf{e}nd{bmatrix} =
\begin{equation}gin{bmatrix}
a \\ 0 \mathbf{e}nd{bmatrix}$, which lies in the range of a degenerate metric $ \mathcal{Q} = \begin{equation}gin{bmatrix}
1 & 0 \\ 0 & 0 \mathbf{e}nd{bmatrix}$, i.e. $\mathbb{R} \times \{0\}$---a proper subspace of $\mathcal{H}=\mathbb{R}^2$. Note that here $\mathcal{B}=\nabla f = \begin{equation}gin{bmatrix}
1 & 0 \\ 0 & 0 \mathbf{e}nd{bmatrix}$, i.e., the previous example in Remark \mathrm{e}f{r_assume_1}-(iii).
{\rm (iii)} The convexity of $f$ and $g$ is assumed in separate items, since some of the following results do not require the convexity.
{\rm (iv)} It should be stressed that Assumption \mathrm{e}f{assume_1} is more general than Assumption \mathrm{e}f{assume_2}. In many examples in Sect. \mathrm{e}f{sec_eg}, $\mathcal{A}$ is not {\it cyclically} maximally monotone. Consequently, by \cite[Theorem 22.18]{plc_book}, there does not exist a proper, l.s.c. and convex function $f$, such that $\mathcal{A} = \partial f$. In this sense, the original problem \mathbf{e}qref{inclusion} is essentially beyond the scope of minimizing a certain cost function. This part actually deals with a special case of Sect. \mathrm{e}f{sec_dist}, if there exists an objective function $f+g$ to minimize.
{\rm (v)} Assumption \mathrm{e}f{assume_2} also covers a special case of $g=0$ or $f=0$, where the G-FBS iteration \mathbf{e}qref{gfbs}, under the case of $\mathcal{Q}=\mathbf{f}rac{1}{\tau}\mathcal{I}$, becomes classical gradient descent (see Example \mathrm{e}f{eg_grad}) or classical PPA (see Example \mathrm{e}f{eg_ppa} in Sect. \mathrm{e}f{sec_eg}).
\mathbf{e}nd{remark}
Let us first show an important descent lemma restricted to the range space of $\mathcal{Q}$.
\begin{equation}gin{lemma} [Degenerate descent lemma]
\label{l_descent}
Under Assumption \mathrm{e}f{assume_2}-(ii), (iii) and (iv), it holds that
\[
f(x_2) \le f(x_1) +
\big\langle \mathcal{Q} (x_2-x_1) | \mathcal{Q}^\dagger
\nabla f(x_1) \big\mathrm{a}ngle
+ \mathbf{f}rac{\begin{equation}ta} {2\nu} \big\| x_2-x_1 \big\|_\mathcal{Q}^2 ,\
\mathbf{f}orall (x_1,x_2) \in \mathcal{H} \times \mathcal{H}.
\]
where $\nu$ is specified in Remark \mathrm{e}f{r_assume_1}-(i).
\mathbf{e}nd{lemma}
\begin{equation}gin{proof}
We adopt similar technique with the proof of \cite[Lemma 2.64]{plc_book}:
\begin{equation}gin{eqnarray}
&& \big| f(x_2) -f(x_1)-
\big\langle x_2-x_1 |\nabla f(x_1) \big\mathrm{a}ngle \big|
\nonumber \\
& =& \int_0^1 \mathbf{f}rac{1}{t} \big\langle t( x_2-x_1) \big|
\nabla f(x_1 +t( x_2-x_1)) - \nabla f(x_1) \big\mathrm{a}ngle
\mathrm{d} t
\nonumber \\
& \le & \int_0^1 \mathbf{f}rac{1}{t} \big\| t ( x_2-x_1) \big\|_\mathcal{Q} \cdot \big\| \mathcal{Q}^\dagger \nabla f(x_1 +t( x_2-x_1)) - \mathcal{Q}^\dagger \nabla f(x_1) \big\|_\mathcal{Q} \mathrm{d} t
\ \text{[by Fact \mathrm{e}f{f_1}-(ii)]}
\nonumber \\
& \le & \int_0^1 \mathbf{f}rac{1}{t} \big\| t ( x_2-x_1) \big\|_\mathcal{Q} \cdot \mathbf{f}rac{\begin{equation}ta} {\nu} \big\| t ( x_2-x_1) \big\|_\mathcal{Q} \mathrm{d} t
\ \text{[by Lemma \mathrm{e}f{l_cocoercive}]}
\nonumber \\
& = & \mathbf{f}rac{\begin{equation}ta} {2 \nu}
\big\| x_2-x_1 \big\|_\mathcal{Q}^2.
\nonumber
\mathbf{e}nd{eqnarray}
The desired inequality follows by noting that
$\big\langle x_2-x_1 |\nabla f(x_1) \big\mathrm{a}ngle
= \big\langle \mathcal{Q} (x_2-x_1) | \mathcal{Q}^\dagger
\nabla f(x_1) \big\mathrm{a}ngle $, due to Fact \mathrm{e}f{f_1}-(ii).
\mathbf{e}nd{proof}
\vskip.1cm
The well-known {\it Descent Lemma} \cite[Lemma 2.64, Theorem 18.15-(iii)]{plc_book} gives
\begin{equation} \label{descent_simple}
f(x_2) \le f(x_1) +
\big\langle x_2-x_1 | \nabla f(x_1) \big\mathrm{a}ngle
+ \mathbf{f}rac{\begin{equation}ta} {2} \big\| x_2-x_1 \big\|^2 ,\
\mathbf{f}orall (x_1,x_2) \in \mathcal{H} \times \mathcal{H}.
\mathbf{e}e
This is instrumental for proving the convergence of $f+g$, e.g., \cite[Lemma 3.1]{pesquet_2014}. Lemma \mathrm{e}f{l_descent} extends this standard result to the case of degenerate metric.
By the proof of Lemma \mathrm{e}f{l_descent}, one can see that, if $\mathrm{a}n \nabla f \subseteq \mathrm{a}n \mathcal{Q}$ and $x_2-x_1\in\ker\mathcal{Q}$,
then,
\[
\big| f(x_2) -f(x_1)-
\big\langle x_2-x_1 |\nabla f(x_1) \big\mathrm{a}ngle \big|
= \big| f(x_2) -f(x_1) \big|\le \mathbf{f}rac{\begin{equation}ta} {2 \nu}
\big\| x_2-x_1 \big\|_\mathcal{Q}^2 =0,
\]
i.e., $f(x_1)=f(x_2)$. Here, the first equality comes from $\big\langle x_2-x_1 |\nabla f(x_1) \big\mathrm{a}ngle = 0$, due to $x_2-x_1 \in \ker\mathcal{Q} = (\mathrm{a}n \mathcal{Q})^\perp$ and $\nabla f(x_1) \in \mathrm{a}n \mathcal{Q}$. This implies that the directional derivative of the function $f$ at any point $x$ along the direction of $s\in\ker \mathcal{Q}$ is 0. In other words, by \cite[Definition 17.1]{plc_book}, the directional derivative is given as
\[
f'(x; s) = \inf_{\xi \in \ ]0,+\infty[} \mathbf{f}rac{f(x+\xi s) -f(x)}{\xi}
=0,\quad \mathbf{f}orall x\in\mathcal{H},\ s\in \ker\mathcal{Q}.
\]
To understand this, it is helpful to recall the previous example in Remark \mathrm{e}f{r_assume_2}-(ii): $f(x)=f(a,b)=\mathbf{f}rac{1}{2}a^2$. It is clear that $f(x_1)=f(x_2)$ for $x_1=(a,b_1)$ and
$x_2=(a,b_2)$.
The following lemma extends the {\it sufficient decrease property} \cite[Lemma 2]{bolte_2014} to the case of arbitrary (degenerate) metric $\mathcal{Q}$.
\begin{equation}gin{lemma} [Degenerate sufficient decrease property] \label{l_decrease}
Let $x^0\in \mathcal{H}$, $\{x^k\}_{k \in \mathbb{N}}$ be a sequence generated by \mathbf{e}qref{gfbs}. {\mathrm{e}d Define $h:=f+g$.} Then, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] Under Assumption \mathrm{e}f{assume_2}-(i)--(v), we have
\[
h (x^k) - h (x^{k+1}) \ge
\mathbf{f}rac{1}{2} \Big(1- \mathbf{f}rac{\begin{equation}ta} { \nu} \Big)
\big\| x^{k+1} - x^k\big\|_\mathcal{Q}^2;
\]
\item[\rm (ii)] Under Assumption \mathrm{e}f{assume_2}-(i)--(vi), we have
\[
h (x^k) - h (x^{k+1}) \ge
\Big(1- \mathbf{f}rac{\begin{equation}ta}{2\nu} \Big)
\big\| x^{k+1} - x^k\big\|_\mathcal{Q}^2,
\]
\mathbf{e}nd{itemize}
where $\nu$ is specified in Remark \mathrm{e}f{r_assume_1}-(i).
\mathbf{e}nd{lemma}
\begin{equation}gin{proof}
Rewrite \mathbf{e}qref{gfbs_eq} as
\begin{equation} \label{gpfbs_eq}
0 \in \nabla f(x^k) +\partial g(x^{k+1})
+\mathcal{Q} (x^{k+1} - x^k).
\mathbf{e}e
By Lemma \mathrm{e}f{l_descent}, we have
\begin{equation}gin{eqnarray} \label{x1}
h(x^{k+1}) & = & f(x^{k+1}) + g(x^{k+1} )
\nonumber \\
& \le & f(x^k) + \big \langle \nabla f(x^k) | x^{k+1} - x^k \big \mathrm{a}ngle
+\mathbf{f}rac{\begin{equation}ta}{2\nu} \big \|x^{k+1} - x^k\big \|_\mathcal{Q}^2
+ g(x^{k+1}).
\mathbf{e}nd{eqnarray}
(i) By the definition of generalized proximity operator, we have
\[
x^{k+1} =\arg\min_x g(x) + \mathbf{f}rac{1}{2} \big\|x - x^k \big\|_\mathcal{Q}^2 + \big \langle x - x^k | \nabla f(x^k)
\big \mathrm{a}ngle,
\]
which implies
\begin{equation} \label{x3}
g(x^{k+1}) + \mathbf{f}rac{1}{2} \big\|x^{k+1} - x^k \big\|_\mathcal{Q}^2 + \big \langle x^{k+1} - x^k | \nabla f(x^k)
\big \mathrm{a}ngle \le g(x^k).
\mathbf{e}e
Combining \mathbf{e}qref{x1} with \mathbf{e}qref{x3} yields
\[
h( x^{k+1}) \le
h(x^{k }) - \mathbf{f}rac{1}{2} \Big(1- \mathbf{f}rac{\begin{equation}ta}{\nu} \Big)
\big\| x^k - x^{k+1} \big\|^2_{\mathcal{Q}}.
\]
\vskip.2cm
(ii) By convexity of $g$, we have
\begin{equation} \label{x2}
h(x^k) = f(x^k) + g(x^{k} ) \ge f(x^k)
+ g(x^{k+1}) + \big \langle \partial g (x^{k+1}) | x^k - x^{k+1} \big \mathrm{a}ngle.
\mathbf{e}e
Combining \mathbf{e}qref{x1} with \mathbf{e}qref{x2} yields
\begin{equation}gin{eqnarray}
h( x^k) - h(x^{k+1}) & \ge & - \big \langle
\nabla f(x^{k}) + \partial g (x^{k+1}) | x^{k+1} - x^k \big \mathrm{a}ngle - \mathbf{f}rac{\begin{equation}ta}{2\nu} \big\| x^k - x^{k+1} \big\|^2_\mathcal{Q}
\nonumber \\
& = & \big \langle x^{k+1} - x^k | x^{k+1} - x^k \big \mathrm{a}ngle_\mathcal{Q} - \mathbf{f}rac{\begin{equation}ta}{2\nu} \big\| x^k - x^{k+1} \big\|_\mathcal{Q}^2 \quad \text{[by \mathbf{e}qref{gpfbs_eq}]}
\nonumber \\
& = & \Big(1- \mathbf{f}rac{\begin{equation}ta}{2\nu} \Big) \big\| x^{k+1} - x^k\big\|_\mathcal{Q}^2.
\nonumber
\mathbf{e}nd{eqnarray}
\mathbf{e}nd{proof}
\begin{equation}gin{remark} \label{r_obj}
{\rm (i)} If $\mathcal{Q}$ is non-degenerate and closed, combining \mathbf{e}qref{descent_simple} with $\|x^{k+1}-x^k\|^2 \le \mathbf{f}rac{1}{\nu} \|x^{k+1}-x^k\|_\mathcal{Q}^2$, one can reach exactly the same result as Lemma \mathrm{e}f{l_decrease}. This similar results can also be found in \cite[Lemma 4.1, Proposition 4.3]{repetti}, \cite[Lemma 4.1]{pesquet_2014} and \cite[Lemma 3.1]{pesquet_2016}.
{\rm (ii)} Lemma \mathrm{e}f{l_decrease} extends the existing results of sufficient decrease properties to the degenerate setting. More importantly, Lemma \mathrm{e}f{l_decrease} is valid without the convexity of $f$.
{\rm (iii)} Without convexity of $g$ (i.e. Lemma \mathrm{e}f{l_decrease}--(i)), the sufficient decreasing requires $\nu \ge \begin{equation}ta $. If $g$ is convex (i.e. Lemma \mathrm{e}f{l_decrease}--(ii)), this condition is relaxed to $\nu \ge \mathbf{f}rac{\begin{equation}ta}{2}$. This coincides with the observation in \cite[Remark 4--(iii)]{bolte_2014}. In addition, combining \mathbf{e}qref{gpfbs_eq} with \mathbf{e}qref{x2}, we obtain
\[
g(x^{k+1}) + \big\langle x^{k+1} - x^k | \nabla f(x^k) \big \mathrm{a}ngle + \big\|x^{k+1} - x^k \big\|_\mathcal{Q}^2 \le
g(x^k),
\]
which is in agreement with the {\it sufficient decrease condition} \cite[Eq.(3.6)]{repetti}, \cite[Eq.(7a)]{pesquet_2014}, \cite[Remark 2.7]{pesquet_2016}.
{\rm (iv)} If $f=0$, Lemma \mathrm{e}f{l_decrease} reduces to
$ g (x^k) - g (x^{k+1}) \ge
\mathbf{f}rac{1}{2} \big\| x^{k+1} - x^k\big\|_\mathcal{Q}^2$, which can be further improved as
$ g (x^k) - g (x^{k+1}) \ge
\big\| x^{k+1} - x^k\big\|_\mathcal{Q}^2$, if $g$ is convex.
If $g=0$, we obtain
$ f (x^{k+1}) \le f (x^k) -
\big\| x^{k+1} - x^k\big\|_{\mathcal{Q}-\mathbf{f}rac{\begin{equation}ta}{2}\mathcal{I}}^2$, for which the decrease of $f$ requires $\mathcal{Q} \succ \mathbf{f}rac{\begin{equation}ta}{2}\mathcal{I}$.
\mathbf{e}nd{remark}
\subsubsection{Convergence result}
The result is given below.
\begin{equation}gin{proposition}[Non-ergodic rate in terms of objective value] \label{p_gpfbs_obj}
Let $x^\star \in \Arg\min (f+g)$, $x^0\in \mathcal{H}$, $\{x^k\}_{k \in \mathbb{N}}$ be a sequence generated by \mathbf{e}qref{gfbs}. Under Assumption \mathrm{e}f{assume_2}, if $\nu \ge \begin{equation}ta $, the objective value $(f+g)(x^k)$ converges to $(f+g) (x^\star)$ with the {\it non-ergodic} rate of $\mathcal{O}(1/k)$, i.e.,
\[
(f+g) (x^{k}) - (f+g) (x^\star) \le \mathbf{f}rac{1}{2k}
\big\| x^0 - x^\star \big\|_\mathcal{Q}^2.
\]
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
Denote $h := f + g $. By convexity of $f$ and $g$, we have
\begin{equation}gin{eqnarray} \label{x4}
h( x^\star) &=& f( x^\star) + g( x^\star)
\nonumber \\
&\ge & f(x^{k}) + \big \langle
\nabla f (x^{k}) | x^\star - x^{k} \big \mathrm{a}ngle
+ g(x^{k+1}) + \big \langle \partial g (x^{k+1}) | x^\star - x^{k+1}\big \mathrm{a}ngle.
\mathbf{e}nd{eqnarray}
Combining \mathbf{e}qref{x4} with \mathbf{e}qref{x1} yields
\begin{equation}gin{eqnarray} \label{x15}
h( x^\star) - h(x^{k+1}) & \ge & - \big \langle
\nabla f(x^{k}) + \partial g (x^{k+1}) | x^{k+1} - x^\star \big \mathrm{a}ngle - \mathbf{f}rac{\begin{equation}ta}{2\nu} \big\| x^k - x^{k+1} \big\|_\mathcal{Q}^2
\nonumber \\
& = & \big \langle x^{k+1} - x^k | x^{k+1} - x^\star \big \mathrm{a}ngle_\mathcal{Q} - \mathbf{f}rac{\begin{equation}ta}{2\nu} \big\| x^k - x^{k+1} \big\|_\mathcal{Q}^2 \quad \text{[by \mathbf{e}qref{gpfbs_eq}]}
\nonumber \\
& = & \mathbf{f}rac{1}{2} \big( 1-\mathbf{f}rac{\begin{equation}ta}{\nu} \big) \big\| x^{k+1} - x^k\big\|_{\mathcal{Q}}^2 +
\mathbf{f}rac{1}{2} \big\| x^{k+1} - x^\star \big\|_\mathcal{Q}^2
-\mathbf{f}rac{1}{2} \big\|x^k -x^\star \big\|_\mathcal{Q}^2
\nonumber \\
& \ge & \mathbf{f}rac{1}{2} \big\| x^{k+1} - x^\star \big\|_\mathcal{Q}^2
-\mathbf{f}rac{1}{2} \big\|x^k -x^\star \big\|_\mathcal{Q}^2.
\quad \text{[by $\nu \ge \begin{equation}ta$]}
\mathbf{e}nd{eqnarray}
{\mathrm{e}d Adding \mathbf{e}qref{x15} from $k=0$ to $k=K-1$ and dividing by $K$ yields
\[
\mathbf{f}rac{1}{K}\sum_{k=0}^{K-1} h(x^k) - h(x^\star)
\le \mathbf{f}rac{1}{2K} \big\| x^{0} - x^\star \big\|^2_\mathcal{Q}.
\]
Then, the pointwise rate of $\mathcal{O}(1/k)$ is obtained, by combining with the fact that $\mathbf{f}rac{1}{K}\sum_{k=0}^{K-1} h(x^k) \ge h(x^K)$ due to Lemma \mathrm{e}f{l_decrease}--(ii).} The convergence of $h(x^k)$ to $h(x^\star)$ is proved by \cite[Theorem 2.1]{ppa_guler}.
\mathbf{e}nd{proof}
\begin{equation}gin{remark} \label{r_obj_2}
{\rm (i)} Proposition \mathrm{e}f{p_gpfbs_obj} extends the standard result of the PFBS algorithm---\cite[Theorem 3.1]{fista}---to arbitrary (degenerate) metric. This non-ergodic rate in terms of cost value was never discussed in variable metric FBS algorithms, e.g., \cite{plc_vu_2014,pesquet_2016,pesquet_2014,repetti}.
{\rm (ii)} In view of \cite[Lemma 2.7]{corman}, the rate can be refined to $(f+g)(x^k) - (f+g)(x^\star) \sim o(1/\sqrt{k})$.
{\rm (iii)} If $f=0$, Proposition \mathrm{e}f{p_gpfbs_obj} boils down to
$ g (x^{k}) - g (x^\star) \le \mathbf{f}rac{1}{2k}
\big\| x^0 - x^\star \big\|_\mathcal{Q}^2$. This is an extended result of \cite[Theorem 2.1]{ppa_guler} under arbitrary degenerate metric. If $g=0$, $f (x^{k}) - f(x^\star) \le \mathbf{f}rac{1}{2k}
\big\| x^0 - x^\star \big\|_\mathcal{Q}^2$.
\mathbf{e}nd{remark}
\section{Relaxations of the G-FBS operator}
\label{sec_extension}
\subsection{The Krasnosel'ski\u{\i}-Mann iteration}
The Krasnosel'ski\u{\i}-Mann iteration of $\mathcal{T}$ in \mathbf{e}qref{gfbs} is given as
\begin{equation} \label{rgfbs}
x^{k+1} := x^k + \gamma \big( \mathcal{T} x^k - x^k \big),
\mathbf{e}e
where $\gamma$ is a relaxation parameter. The scheme \mathbf{e}qref{rgfbs} is also a fixed-pooint iteration of $\mathcal{T}_\gamma := \mathcal{I} - \gamma (\mathcal{I} - \mathcal{T})$.
\begin{equation}gin{fact} \label{f_3}
$\mathbf{F}ix \mathcal{T}_\gamma = \mathbf{F}ix \mathcal{T} = \mathrm{zer} (\mathcal{A} + \mathcal{B})$.
\mathbf{e}nd{fact}
The convergence properties of \mathbf{e}qref{rgfbs} are given below.
\begin{equation}gin{corollary}[Convergence in terms of metric distance] \label{c_rgfbs}
Let $x^0\in \mathcal{H}$, $\{x^k\}_{k \in \mathbb{N}}$ be a sequence generated by \mathbf{e}qref{rgfbs}. Under Assumption \mathrm{e}f{assume_1}, if $\nu > \begin{equation}ta/ 2 $, $\gamma \in\ ] 0, 2-\mathbf{f}rac{\begin{equation}ta}{2\nu } [$, then, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] {\rm [Weak convergence in $\mathrm{a}n\mathcal{Q}$]} There exists $x^\star \in \mathrm{zer} (\mathcal{A} +\mathcal{B})$, such that $\mathcal{Q} x^k \mathrm{w}eak \mathcal{Q}x^\star$, as $k\rightarrow \infty$.
\item[\rm (ii)] {\rm [$\mathcal{Q}$-based asymptotic regularity]} $\|x^{k+1 } -x^{k} \|_\mathcal{Q}$ has the pointwise convergence rate of $\mathcal{O}(1/\sqrt{k})$:
\[
\big\|x^{k +1} -x^{k} \big\|_\mathcal{Q}
\le \mathbf{f}rac{1}{\sqrt{k+1}} \sqrt{ \mathbf{f}rac{2\gamma \nu } {(4-2\gamma) \nu - \begin{equation}ta} }
\big\|x^{0} -x^\star \big\|_\mathcal{Q}, \quad
\mathbf{f}orall k \in \mathbb{N}.
\]
\mathbf{e}nd{itemize}
\mathbf{e}nd{corollary}
\begin{equation}gin{proof}
First, $\mathcal{T}_\gamma $ is $\mathcal{Q}$-based $\mathbf{f}rac{2 \gamma \nu }
{4 \nu - \begin{equation}ta }$-averaged, if $\gamma \in\ ] 0, 2-\mathbf{f}rac{\begin{equation}ta}{2\nu} [$, by Theorem \mathrm{e}f{t_T}--(iii). Then, the proof is completed by the similar reasoning with Theorem \mathrm{e}f{t_dist} and Fact \mathrm{e}f{f_3}.
\mathbf{e}nd{proof}
\begin{equation}gin{remark} \label{r_rgfbs}
{\rm (i)} Corollary \mathrm{e}f{c_rgfbs} extends the existing result of the relaxed PFBS \cite[Theorem 25.8]{plc_book} to arbitrary (degenerate) metric $\mathcal{Q}$, under milder condition. In \cite[Theorem 25.8]{plc_book}, the condition is $\gamma < \min\{1, \mathbf{f}rac{\nu}{\begin{equation}ta} \} +\mathbf{f}rac{1}{2}$, which is obtained by the rough estimate of averagedness of $\mathcal{T}_\gamma$, given as $\alpha = \gamma \cdot \max\{ \mathbf{f}rac{2}{3}, \mathbf{f}rac{2\begin{equation}ta}{\begin{equation}ta +2\nu} \}$ (see Remark \mathrm{e}f{r_ave}). By contrast, our result corresponds to the sharper estimate of $\alpha = \mathbf{f}rac{2\gamma \nu}
{4\nu -\begin{equation}ta}$ (i.e. Theorem \mathrm{e}f{t_T}--(iv)).
{\rm (ii)} If $\mathcal{Q}$ is non-degenerate, the weak convergence of \mathbf{e}qref{rgfbs} holds for the whole space $\mathcal{H}$, i.e., $x^{k+1}-x^k\rightarrow 0$ and $x^k \mathrm{w}eak x^\star$, as $k\rightarrow \infty$.
{\rm (iii)} If $\begin{equation}ta=0$ (i.e., $\mathcal{B}=0$), \mathbf{e}qref{rgfbs} is a relaxed PPA. The rate of asymptotic regularity becomes
\[
\big\|x^{k +1} -x^{k} \big\|_\mathcal{Q}
\le \sqrt{ \mathbf{f}rac{\gamma}{ (k+1)( 2-\gamma) } }
\big\|x^{0} -x^\star \big\|_\mathcal{Q},\quad
\mathbf{f}orall k \in \mathbb{N}.
\]
\mathbf{e}nd{remark}
\subsection{Arbitrary relaxation operator}
In this sequel, we further consider a more general relaxation operator $\mathcal{M}$:
\begin{equation} \label{t_relaxed}
\mathcal{T}_\mathcal{M} := \mathcal{I} - \mathcal{M} (\mathcal{I} - \mathcal{T}),
\mathbf{e}e
where $\mathcal{T}$ is given by \mathbf{e}qref{gfbs}.
Then, the fixed-point iteration $x^{k+1} : = \mathcal{T}_\mathcal{M} x^k$ can be rewritten as the following {\it relaxed G-FBS} algorithm:
\begin{equation} \label{gppa}
\left\lfloor \begin{equation}gin{array}{llll}
0 & : \in & \mathcal{A} xtilde^k + \mathcal{B} x^k +
\mathcal{Q} (xtilde^k - x^k), & \text{(FBS step)} \\
x^{k+1} & := & x^k +\mathcal{M} (xtilde^k - x^k) .
& \text{(relaxation step)}
\mathbf{e}nd{array} \right.
\mathbf{e}e
To the best of our knowledge, \mathbf{e}qref{gppa} has never been discussed before in the literature. The applications of \mathbf{e}qref{gppa} will be illustrated in Section \mathrm{e}f{sec_eg}.
\begin{equation}gin{assumption} \label{assume_3}
\begin{equation}gin{itemize}
\item[\rm (i)] $\mathcal{A}$ is (set-valued) maximally monotone;
\item [\rm (ii)] $\mathcal{B}$ is $\begin{equation}ta^{-1}$-cocoercive with $\begin{equation}ta \in [0,+\infty[$, and $\mathrm{a}n \mathcal{B} \subseteq \mathrm{a}n \mathcal{Q}$;
\item[\rm (iii)] $\mathcal{M}$ is invertible, i.e., $\mathcal{M}^{-1}$ exists;
\item[\rm (iv)] $\mathcal{S}: = \mathcal{Q} \mathcal{M}^{-1} \in xS_+$, such that $\ker \mathcal{S} \mathbf{a}ckslash \{0\} \ne \mathbf{e}mptyset$;
\item[\rm (v)] $\mathcal{Q}tilde: =\mathbf{f}rac{1}{2} ( \mathcal{Q} + \mathcal{Q}^\top)$ is at least degenerate;
\item[\rm (vi)] $\mathcal{Q}tilde$ is closed, i.e., $\mathbf{e}xists \nu \in \ ]0,+\infty[$, such that $\| \mathcal{Q}tilde x\| \ge \nu \|x\|$, $\mathbf{f}orall x \in \mathrm{a}n \mathcal{Q}tilde$;
\item[\rm (vii)] $ \mathcal{G} := ( 1 - \mathbf{f}rac{\begin{equation}ta}{4\nu} )
(\mathcal{Q} +\mathcal{Q}^\top) - \mathcal{M}^\top \mathcal{Q} \in xS_+$;
\item [\rm (viii)] $\mathrm{a}n(\mathcal{A}+\mathcal{Q}) \supseteq \mathrm{a}n (\mathcal{Q}-\mathcal{B})$ and $\mathrm{zer}(\mathcal{A}+\mathcal{B}) \ne \mathbf{e}mptyset$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{assumption}
\vskip.1cm
For Assumption \mathrm{e}f{assume_3}-(ii), the case of $\begin{equation}ta=0$ has been discussed in Remark \mathrm{e}f{r_assume_1}-(iv). The definitions of $\mathcal{S}$, $\mathcal{G}$ and $\tilde{\mathcal{Q}}$ stem from the following Lemma \mathrm{e}f{l_gppa}, for sake of convenience. The non-singularity of $\mathcal{M}$ keeps the basic rationale of the iteration of \mathbf{e}qref{gppa}: $x^{k+1}$ should contain the information of $\tilde{x}^k$ for the logical update. A simplest case is $\mathcal{M}=\mathcal{I}$, which yields $x^{k+1}=\tilde{x}^k$ and reduces \mathbf{e}qref{gppa} to \mathbf{e}qref{gfbs}.
Other items in Assumption \mathrm{e}f{assume_3} are the same as Assumptions \mathrm{e}f{assume_1} and \mathrm{e}f{assume_2}.
Lemma \mathrm{e}f{l_gppa} presents several key ingredients, which are the `recipe' for proving the convergence of \mathbf{e}qref{gppa}.
\begin{equation}gin{lemma} \label{l_gppa}
Let $\mathcal{T}$ defined as \mathbf{e}qref{gfbs}. Denote $\mathcal{R}:=\mathcal{I}-\mathcal{T}$. Let $x^\star \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$, $x^0\in \mathcal{H}$ and $\{x^k\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{gppa}. Under Assumption \mathrm{e}f{assume_3}, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $ \big\| x^{k+1} - x^\star \big\|_\mathcal{S}^2
\le \big\| x^k - x^\star \big\|_\mathcal{S}^2
- \big\| x^k - x^{k+1}\big\|_{\mathcal{M}^{-\top}
\mathcal{G} \mathcal{M}^{-1} }^2 $;
\item[\rm (ii)] $ \big \langle \mathcal{R} x^k \big|\mathcal{M}^\top \mathcal{Q} (\mathcal{R} x^k - \mathcal{R} x^{k+1} ) \big \mathrm{a}ngle \ge \mathbf{f}rac{1}{2}
\big( 1 - \mathbf{f}rac{\begin{equation}ta}{4\nu} \big)
\big\| \mathcal{R} x^k - \mathcal{R} x^{k+1} \big\|
_{\mathcal{Q}+\mathcal{Q}^\top}^2$ ;
\item[\rm (iii)] $ \big \| x^k - x^{k+1} \big\|_\mathcal{S}^2 -
\big\| x^{k+1} - x^{k+2} \big\|_\mathcal{S}^2
\ge \big\| \mathcal{R} x^k - \mathcal{R} x^{k+1} \big\|_\mathcal{G}^2 $,
\mathbf{e}nd{itemize}
where $\nu$ is defined in Assumption \mathrm{e}f{assume_3}-(vi).
\mathbf{e}nd{lemma}
\begin{equation}gin{proof}
(i) By monotonicity of $\mathcal{A}$, we develop
\begin{equation}gin{eqnarray} \label{x5}
0 & \le & \langle \mathcal{A} xtilde^k - \mathcal{A} x^\star \big|
xtilde^k - x^\star \mathrm{a}ngle
\nonumber \\
& =& \langle -\mathcal{B} x^k +\mathcal{Q}(x^k - xtilde^k) +\mathcal{B} x^\star
\big| xtilde^k - x^\star \mathrm{a}ngle
\quad \text{[by \mathbf{e}qref{gppa} and $x^\star \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$]}
\nonumber \\
& =& \langle \mathcal{Q}(x^k - xtilde^k) \big| xtilde^k - x^\star \mathrm{a}ngle - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle
\nonumber \\
& =& \langle \mathcal{Q} \mathcal{M}^{-1} (x^k - x^{k+1}) \big| xtilde^k - x^\star \mathrm{a}ngle - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle
\quad \text{[by \mathbf{e}qref{gppa}]}
\nonumber \\
& =& \langle \mathcal{S} (x^k - x^{k+1}) \big| x^k - x^\star +\mathcal{M}^{-1} (x^{k+1} - x^k) \mathrm{a}ngle - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle
\quad \text{[by \mathbf{e}qref{gppa}]}
\nonumber \\
& =& \langle \mathcal{S} (x^k - x^{k+1}) \big| x^k - x^\star \mathrm{a}ngle
+ \langle \mathcal{S} (x^k - x^{k+1}), \mathcal{M}^{-1} (x^{k+1} - x^k) \mathrm{a}ngle - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle
\nonumber \\
& =& \mathbf{f}rac{1}{2} \big\| x^k - x^{k+1}\big\|^2_\mathcal{S}
+\mathbf{f}rac{1}{2} \big\| x^k - x^\star \big\|_\mathcal{S}^2
-\mathbf{f}rac{1}{2} \big\| x^{k+1} - x^\star \big\|_\mathcal{S}^2
- \mathbf{f}rac{1}{2} \big\| x^k - x^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{S}
+\mathcal{S} \mathcal{M}^{-1}}^2
\nonumber \\
& - & \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle
\nonumber \\
& =& \mathbf{f}rac{1}{2} \big\| x^k - x^\star \big\|_\mathcal{S}^2
-\mathbf{f}rac{1}{2} \big\| x^{k+1} - x^\star \big\|_\mathcal{S}^2
- \mathbf{f}rac{1}{2} \big\| x^k - x^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{S}
+\mathcal{S} \mathcal{M}^{-1} -\mathcal{S} }^2
\nonumber \\
& - & \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle.
\mathbf{e}nd{eqnarray}
By adopting similar techniques with \cite[Theorem 1]{lorenz}, the last term of \mathbf{e}qref{x5} becomes
\begin{equation}gin{eqnarray}
&& - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^\star \mathrm{a}ngle
\nonumber \\
& = & - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k -
x^k +x^k - x^\star \mathrm{a}ngle
\nonumber \\
& = & - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k -
x^k \mathrm{a}ngle
- \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| x^k - x^\star \mathrm{a}ngle
\nonumber \\
& \le & - \langle \mathcal{B} x^k - \mathcal{B} x^\star \big| xtilde^k - x^k \mathrm{a}ngle
- \mathbf{f}rac{1}{ \begin{equation}ta} \big\| \mathcal{B} x^k - \mathcal{B} x^\star \big\|^2
\nonumber \\
& \le & \mathbf{f}rac{\nu}{ \begin{equation}ta} \big\| \mathcal{Q}tilde^\dagger \mathcal{B} x^k - \mathcal{Q}tilde^\dagger\mathcal{B} x^\star \big\|_{\mathcal{Q}tilde}^2 + \mathbf{f}rac{\begin{equation}ta}{4\nu} \big\| xtilde^k - x^k \big\|_{\mathcal{Q}tilde}^2
- \mathbf{f}rac{\nu}{ \begin{equation}ta} \big\| \mathcal{Q}tilde^\dagger \mathcal{B} x^k -\mathcal{Q}tilde^\dagger \mathcal{B} x^\star \big\|_{\mathcal{Q}tilde}^2
\ \text{[by Fact \mathrm{e}f{f_2}]}
\nonumber \\
& = & \mathbf{f}rac{\begin{equation}ta}{4\nu} \big\| xtilde^k -
x^k \big\|_{\mathcal{Q}tilde}^2 = \mathbf{f}rac{\begin{equation}ta}{8\nu } \big\| x^k - x^{k+1} \big\|_{\mathcal{M}^{-\top} (\mathcal{Q}+\mathcal{Q}^\top)\mathcal{M}^{-1}}^2.
\quad \text{[by \mathbf{e}qref{gppa}]}
\nonumber
\mathbf{e}nd{eqnarray}
By substituting into \mathbf{e}qref{x5}, it yields
\[
\mathbf{f}rac{1}{2} \big\| x^k - x^\star \big\|_\mathcal{S}^2
-\mathbf{f}rac{1}{2} \big\| x^{k+1} - x^\star \big\|_\mathcal{S}^2
- \mathbf{f}rac{1}{2} \big\| x^k - x^{k+1}\big\|_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge 0,
\]
where $\mathcal{G}$ is defined in Assumption \mathrm{e}f{assume_3}.
\vskip.2cm
(ii) By carefully checking the proof of Lemma \mathrm{e}f{l_T}-(i), one can see that Lemma \mathrm{e}f{l_T}--(i) is valid for arbitrary (not necessarily self-adjoint) metric $\mathcal{Q}$. More specifically, we have ($\mathcal{Q}$ here is not necessarily self-adjoint)
\[
0 \le \langle \mathcal{Q}(\mathcal{R} x_1 - \mathcal{R} x_2) | \mathcal{T}x_1 - \mathcal{T} x_2 \mathrm{a}ngle
- \langle \mathcal{B} x_1 - \mathcal{B} x_2 | \mathcal{T}x_1-\mathcal{T}x_2 \mathrm{a}ngle.
\]
Adding $\|\mathcal{R} x_1 - \mathcal{R} x_2 \|_\mathcal{Q}^2$ on both sides, we develop
\begin{equation}gin{eqnarray}
&& \big\|\mathcal{R} x_1 - \mathcal{R} x_2 \big\|_\mathcal{Q}^2
=\big\|\mathcal{R} x_1 - \mathcal{R} x_2 \big\|_{\mathcal{Q}tilde}^2 \quad \text{[by definition of $\mathcal{Q}tilde$]}
\nonumber \\
& \le & \big \langle x_1 - x_2 \big|\mathcal{Q}( \mathcal{R} x_1 - \mathcal{R} x_2 ) \big \mathrm{a}ngle
- \big\langle \mathcal{B} x_1 - \mathcal{B} x_2 \big| \mathcal{T}x_1-\mathcal{T}x_2
\big \mathrm{a}ngle
\quad \text{[by $\mathcal{R} + \mathcal{T} = \mathcal{I}$]}
\nonumber \\
& = & \big \langle x_1 - x_2 \big| \mathcal{Q}(\mathcal{R} x_1 - \mathcal{R} x_2 ) \big \mathrm{a}ngle
+ \big\langle \mathcal{B} x_1 - \mathcal{B} x_2 \big| \mathcal{R}x_1-\mathcal{R}x_2
\big \mathrm{a}ngle
- \big\langle \mathcal{B} x_1 - \mathcal{B} x_2 \big| x_1 - x_2
\big \mathrm{a}ngle
\nonumber \\
& \le & \big \langle x_1 - x_2 \big| \mathcal{Q}(\mathcal{R} x_1 - \mathcal{R} x_2 )\big \mathrm{a}ngle
+ \mathbf{f}rac{\nu}{ \begin{equation}ta} \big\| \mathcal{Q}tilde^\dagger \mathcal{B} x_1 -\mathcal{Q}tilde^\dagger \mathcal{B} x_2 \big\|_{\mathcal{Q}tilde}^2 +
\mathbf{f}rac{\begin{equation}ta}{4\nu} \big\| \mathcal{R}x_1-\mathcal{R}x_2 \big \|_{\mathcal{Q}tilde}^2
\nonumber \\
&- & \mathbf{f}rac{\nu}{ \begin{equation}ta} \big\| \mathcal{Q}tilde^\dagger \mathcal{B} x_1 - \mathcal{Q}tilde^\dagger \mathcal{B} x_2 \big\|_{\mathcal{Q}tilde}^2
\nonumber \\
& = & \big \langle x_1 - x_2 \big|\mathcal{Q}( \mathcal{R} x_1 - \mathcal{R} x_2 )\big \mathrm{a}ngle
+ \mathbf{f}rac{\begin{equation}ta} {4\nu} \big\| \mathcal{R}x_1-\mathcal{R}x_2 \big \|_{\mathcal{Q}tilde}^2,
\nonumber
\mathbf{e}nd{eqnarray}
which leads to
\begin{equation}gin{eqnarray}
\big \langle x^k - x^{k+1} \big| \mathcal{Q}(
\mathcal{R} x^k - \mathcal{R} x^{k+1} ) \big \mathrm{a}ngle
& \ge & \big( 1 - \mathbf{f}rac{\begin{equation}ta}{4\nu} \big)
\big\| \mathcal{R} x^k - \mathcal{R} x^{k+1} \big\|_{\mathcal{Q}tilde}^2.
\nonumber
\mathbf{e}nd{eqnarray}
Then, (ii) follows from $x^k - x^{k+1} = \mathcal{M}\mathcal{R} x^k$ by \mathbf{e}qref{gppa} and the definition of $\mathcal{Q}tilde$.
\vskip.2cm
(iii) From Lemma \mathrm{e}f{l_gppa}-(ii), we have
\begin{equation}gin{eqnarray} \label{dd}
& & \big \| x^k - x^{k+1} \big\|_\mathcal{S}^2 -
\big\| x^{k+1} - x^{k+2} \big\|_\mathcal{S}^2
\nonumber \\
& = & \big \|\mathcal{M} \mathcal{R} x^k \big\|_\mathcal{S}^2 -
\big\| \mathcal{M} \mathcal{R}x^{k+1} \big\|_\mathcal{S}^2
\quad \text{[by \mathbf{e}qref{gppa}] }
\nonumber \\
&=& 2 \big \langle \mathcal{R} x^k \big| \mathcal{M}^\top \mathcal{S} \mathcal{M} (
\mathcal{R} x^k - \mathcal{R} x^{k+1} ) \big \mathrm{a}ngle
- \big\| \mathcal{R}x^k - \mathcal{R}x^{k+1} \big\|^2
_{\mathcal{M}^\top \mathcal{S} \mathcal{M} }
\nonumber \\
& \ge & \big\| \mathcal{R} x^k - \mathcal{R} x^{k+1} \big\|_{
( 1 - \mathbf{f}rac{\begin{equation}ta}{4\nu} )
(\mathcal{Q} +\mathcal{Q}^\top) - \mathcal{M}^\top \mathcal{S}\mathcal{M} }^2. \qquad
\text{[by Lemma \mathrm{e}f{l_gppa}-(ii)] }
\nonumber
\mathbf{e}nd{eqnarray}
This completes the proof.
\mathbf{e}nd{proof}
\vskip.1cm
In particular, if $\mathcal{M}=\gamma \mathcal{I}$, Lemma \mathrm{e}f{l_gppa}-(i) is simplified to
\[
\big\| x^{k+1} - x^\star \big\|_\mathcal{Q}^2
\le \big\| x^k - x^\star \big\|_\mathcal{Q}^2
- \mathbf{f}rac{1}{\gamma} \big( 2 - \gamma - \mathbf{f}rac{\begin{equation}ta}{2\nu} \big) \big\| x^k - x^{k+1}\big\|_\mathcal{Q}^2,
\]
which exactly leads to Corollary \mathrm{e}f{c_rgfbs}-(ii).
\vskip.1cm
The following theorem gives the convergence result.
\begin{equation}gin{theorem}[Convergence in terms of metric distance] \label{t_gppa}
Let $x^0 \in \mathcal{H}$, $\{x^k\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{gppa}. Under Assumption \mathrm{e}f{assume_3}, if $\mathbf{e}xists \mathbf{e}ta \in\ ]0, +\infty[$, s.t. $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] {\rm [Weak convergence in $\mathrm{a}n\mathcal{S}$]} There exists $x^\star\in \mathrm{zer} (\mathcal{A}+\mathcal{B})$, such that $\mathcal{S} x^k \mathrm{w}eak \mathcal{S} x^\star$, as $k \rightarrow \infty$.
\item[\rm (ii)] {\rm [Rate of $\mathcal{S}$-asymptotic regularity]} $\| x^{k } - x^{k+1 } \|_\mathcal{S}$ has the non-ergodic convergence rate of $\mathcal{O}(1/\sqrt{k})$, i.e.,
\[
\big\| x^{k+1 } - x^{k } \big\|_\mathcal{S}
\le \mathbf{f}rac{1}{ \sqrt{k+1 } } \mathbf{f}rac{1} {\sqrt{\mathbf{e}ta}}
\big\|x^{0} -x^\star \big\|_\mathcal{S},
\quad \mathbf{f}orall k \in \mathbb{N}.
\]
\mathbf{e}nd{itemize}
\mathbf{e}nd{theorem}
\begin{equation}gin{proof}
(i) Lemma \mathrm{e}f{l_gppa}--(i) becomes
\[
\big\| x^{k+1} - x^\star \big\|_\mathcal{S}^2
\le \big\| x^k - x^\star \big\|_\mathcal{S}^2
- \mathbf{e}ta \big\| x^k - x^{k+1}\big\|_\mathcal{S}^2,
\]
{\mathrm{e}d
which is in spirit the same as \mathbf{e}qref{x12}. By the similar argument of Theorem \mathrm{e}f{t_dist}-(iv), it is easy to prove that:
\begin{equation}gin{itemize}
\item $\lim_{k\rightarrow \infty} \| x^{k} - x^\star \|_\mathcal{S}$ exists for any given $x^\star \in \mathrm{zer} (\mathcal{A}+\mathcal{B}) $;
\item $\{\sqrt{ \mathcal{S}} x^k\}_{k\in\mathbb{N}}$ has at least one weak sequential cluster point lying in $\sqrt{\mathcal{S}} \mathrm{zer} (\mathcal{A}+\mathcal{B}) $;
\item the cluster point of $\{\sqrt{ \mathcal{S}} x^k\}_{k\in\mathbb{N}}$ is unique.
\mathbf{e}nd{itemize}
Finally, we summarize that $\{\mathcal{S} x^k\}_{k\in\mathbb{N}}$, is bounded and possesses a unique weak sequential cluster point $\mathcal{S} x^\star \in \mathcal{S} \mathrm{zer} (\mathcal{A}+\mathcal{B})$. By \cite[Lemma 2.38]{plc_book}, $\mathcal{S} x^k \rightharpoonup \mathcal{S} x^\star \in \mathcal{S} \mathrm{zer} (\mathcal{A}+\mathcal{B}) $, as $k\rightarrow \infty$. }
\vskip.1cm
(ii) in view of Lemma \mathrm{e}f{l_gppa}--(i) and (iii), similar to the proof of Theorem \mathrm{e}f{t_dist}-(iii).
\mathbf{e}nd{proof}
\begin{equation}gin{remark} \label{r_gppa}
{\rm (i)} The condition of $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $ is in general much milder than
$ \mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} \succeq \mathbf{e}ta \mathcal{S}$. In Example \mathrm{e}f{eg_radmm} of Sect. \mathrm{e}f{sec_eg}, we will see that this condition is satisfied, but $ \mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} \succeq \mathbf{e}ta \mathcal{S}$ is not guaranteed for any $\mathbf{e}ta \in \ ]0, +\infty[$.
{\rm (ii)} In particular, if $\mathcal{M} = \gamma \mathcal{I}$, $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $ is equivalent to
$ \mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} \succeq \mathbf{e}ta \mathcal{S}$. This yields that $\mathbf{f}rac{1}{\gamma^2} (2-\mathbf{f}rac{\begin{equation}ta}{2\nu}) - \mathbf{f}rac{1}{\gamma} \ge \mathbf{f}rac{\mathbf{e}ta} {\gamma}$. One can safely choose the best possible estimate of $\mathbf{e}ta = \mathbf{f}rac{1}{\gamma} (2-\mathbf{f}rac{\begin{equation}ta}{2\nu} - \gamma)$. Thus, Theorem \mathrm{e}f{t_gppa}-(ii) boils down to Corollary \mathrm{e}f{c_rgfbs}-(ii).
{\rm (iii)} If $\mathcal{S}$ and $\mathcal{G}$ are non-degenerate, $\mathcal{G}$ can be redefined as $\mathcal{G}: = \mathcal{Q}+\mathcal{Q}^\top - \mathcal{M}^\top \mathcal{Q} -\mathbf{f}rac{\begin{equation}ta}{2}\mathcal{I}$; the weak convergence of $x^k \mathrm{w}eak x^\star$ is guaranteed without the additional assumption of $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $.
\mathbf{e}nd{remark}
\section{Applications to the first-order operator splitting algorithms}
\label{sec_eg}
This part shows that a great variety of operator splitting algorithms falls into the G-FBS category. More importantly, we show that all the properties of each algorithm can be readily obtained from our general results in Sect. \mathrm{e}f{sec_gfbs} and \mathrm{e}f{sec_extension}.
\subsection{The ADMM/DRS algorithms}
ADMM is one of the most commonly used algorithms for solving the structured constrained optimization \cite{boyd_admm}:
\begin{equation} \label{problem1}
\min_{u,v} f(u) +g(v),\quad
\text{s.t.}\ \ Au+Bv = c,
\mathbf{e}e
where $u \in \mathcal{U}$, $v \in \mathcal{V}$, the operators $A: \mathcal{U} \mapsto \mathcal{Z}$ and $B: \mathcal{V} \mapsto \mathcal{Z}$ are linear and bounded. The functions $f: \mathcal{U} \mapsto \mathbb{R}\cup\{+\infty\}$ and $g: \mathcal{V} \mapsto \mathbb{R}\cup\{+\infty\}$ are proper, l.s.c. and convex.
Two typical ADMM algorithms are listed below.
\begin{equation}gin{example} [Relaxed-ADMM] \label{eg_radmm}
The relaxed-ADMM, or equivalent relaxed-DRS applied to the dual problem \cite{self_eq}, is given as \cite[Eq.(3)]{fang_2015}
\begin{equation} \label{radmm}
\left\lfloor \begin{equation}gin{array}{lll}
u^{k+1} & :\in & \Arg \min_u f(u) +
\mathbf{f}rac{\tau}{2} \big\| Au +Bv^k - c - \mathbf{f}rac{1}{\tau} p^k \big\|^2, \\
v^{k+1} & :\in & \Arg \min_v g(v) + \mathbf{f}rac{\tau}{2} \big\| B (v - v^k) + \gamma (A u^{k+1}
+ B v^{k} - c ) - \mathbf{f}rac{1}{\tau} p^k \big\|^2, \\
p^{k+1} & := & p^k -\tau B (v^{k+1} - v^k)
- \tau\gamma (Au^{k+1} + Bv^{k} - c) ,
\mathbf{e}nd{array} \right.
\mathbf{e}e
which fits into the relaxed G-FBS operator \mathbf{e}qref{t_relaxed} as:
\[
x^k = \begin{equation}gin{bmatrix}
u^k \\ v^k \\ p^k \mathbf{e}nd{bmatrix},\
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial f & 0 & -A^\top \\
0 & \partial g & -B^\top \\
A & B & \partial l \mathbf{e}nd{bmatrix} , \
\mathcal{B}=0,
\]
\[
\mathcal{Q} = \begin{equation}gin{bmatrix}
0 & 0 & 0 \\
0 & \tau B^\top B & (1-\gamma) B^\top \\
0 & -B & \mathbf{f}rac{1}{\tau} I
\mathbf{e}nd{bmatrix},\
\mathcal{M} = \begin{equation}gin{bmatrix}
I & 0 & 0 \\ 0 & I & 0 \\
0 & -\tau B & \gamma I \mathbf{e}nd{bmatrix},
\]
where the function $l$ is $l = -\langle \cdot|c \mathrm{a}ngle$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_radmm}
Let $\{(u^k,v^k,p^k)\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{radmm}. If $\tau \in\ ]0,+\infty[$ and $\gamma \in \ ]0,2[$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $\big\| \xbar^{k+1 } - \xbar^{k } \big\|_\mathcal{S}bar
\le \sqrt{ \mathbf{f}rac{\gamma} {(2-\gamma)(k+1)} }
\big\|\xbar^{0} - \xbar^\star \big\|_\mathcal{S}bar$,
$\mathbf{f}orall k \in \mathbb{N}$, where $\xbar = (v,p)$, and $ \mathcal{S}bar =
\begin{equation}gin{bmatrix}
\tau B^\top B & (1- \gamma) B^\top \\
(1-\gamma) B & \mathbf{f}rac{1}{\tau } I
\mathbf{e}nd{bmatrix}$.
\item[\rm (ii)] There exists a solution $(u^\star, v^\star, p^\star)$ to the problem \mathbf{e}qref{problem1}, such that $(A u^k, B v^k, p^k) \mathrm{w}eak ( Au^\star, Bv^\star, p^\star)$, as $k \rightarrow \infty$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
To apply Theorem \mathrm{e}f{t_gppa}, let us check if Assumption \mathrm{e}f{assume_3} is satisfied. First, it is easy to see that
$\mathcal{A}$ is maximally monotone,
$\mathcal{B} = 0$ is $0$-Lipschitz continuous (i.e., $\begin{equation}ta=0$), and $\mathcal{M}^{-1}$ exists. We then compute $\mathcal{S}$ and $\mathcal{G}$ as
\[
\mathcal{S} = \mathbf{f}rac{1}{\gamma} \begin{equation}gin{bmatrix}
0 & 0 & 0 \\
0 & \tau B^\top B & (1- \gamma) B^\top \\
0 & (1-\gamma) B & \mathbf{f}rac{1}{\tau } I
\mathbf{e}nd{bmatrix},\ \mathcal{G} = \begin{equation}gin{bmatrix}
0 & 0 & 0 \\ 0 & 0 & 0 \\
0 & 0 & \mathbf{f}rac{2-\gamma}{\tau} I
\mathbf{e}nd{bmatrix},
\]
which are positive semi-definite, if $\tau \in\ ]0,+\infty[$ and $\gamma \in \ ]0,2[$.
Now, we check if $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $ holds for some $\mathbf{e}ta \in\ ]0,+\infty[$. The operator $ \mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} - \mathbf{e}ta \mathcal{S}$ is given as
\[
\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} - \mathbf{e}ta \mathcal{S} = \begin{equation}gin{bmatrix}
0 & 0 & 0 \\
0 & (\mathbf{f}rac{\tau (2-\gamma)}{\gamma^2} - \mathbf{f}rac{\tau \mathbf{e}ta}{\gamma} ) B^\top B
& (\mathbf{f}rac{ 2-\gamma }{\gamma^2} - \mathbf{e}ta \mathbf{f}rac{1-\gamma }{\gamma} ) B^\top \\
0 & (\mathbf{f}rac{ 2-\gamma }{\gamma^2} - \mathbf{e}ta \mathbf{f}rac{1-\gamma }{\gamma} ) B & (\mathbf{f}rac{ 2-\gamma } {\tau \gamma^2} - \mathbf{f}rac{ \mathbf{e}ta} {\tau \gamma} ) I
\mathbf{e}nd{bmatrix},
\]
which can be decomposed as $\mathcal{S}_1 +\mathcal{S}_2$, where
\[
\mathcal{S}_1 = \mathbf{f}rac{2 - (\mathbf{e}ta +1)\gamma } {\gamma^2} \begin{equation}gin{bmatrix}
0 & 0 & 0 \\
0 & \tau B^\top B & B^\top \\
0 & B & \mathbf{f}rac{ 1}{\tau} I
\mathbf{e}nd{bmatrix},\quad
\mathcal{S}_2 = \mathbf{e}ta \begin{equation}gin{bmatrix}
0 & 0 & 0 \\ 0 & 0 & B^\top \\
0 & B & 0 \mathbf{e}nd{bmatrix}.
\]
It is easy to verify that $\mathcal{S}_1 \in xS_+$ for $\gamma \in \ ]0,2[$ and $\mathbf{e}ta \in \ ]0, \mathbf{f}rac{2}{\gamma} - 1]$, since $\|x\|^2_{\mathcal{S}_1} = \mathbf{f}rac{2 - (\mathbf{e}ta+1) \gamma} {\gamma^2} \big\|\sqrt{\tau} Bv +\mathbf{f}rac{1}{\sqrt{\tau} } p \big\|^2 \ge 0$, $\mathbf{f}orall x=(u,v,p)$.
What remains to prove is that $ \big \| x^k - x^{k+1} \big \|^2_{ \mathcal{S}_2 } \ge 0$, i.e., $\langle B(v^k-v^{k+1}) | p^k-p^{k+1}\mathrm{a}ngle \ge 0$. To see this, we consider the convexity of $g$:
\[
g(v) \ge g(v^{k+1}) + \langle \partial g(v^{k+1}) |v-v^{k+1} \mathrm{a}ngle,\quad
g(v) \ge g(v^{k}) + \langle \partial g(v^{k}) | v-v^{k} \mathrm{a}ngle.
\]
By the $v$-update of \mathbf{e}qref{radmm}, we have $B^\top p^{k+1} \in \partial g(v^{k+1})$, and thus, $B^\top p^{k} \in \partial g(v^{k})$. Then, the above inequalities become, $\mathbf{f}orall p \in \mathcal{Z}$:
\[
g(v) \ge g(v^{k+1}) + \langle B^\top p^{k+1}| v-v^{k+1} \mathrm{a}ngle,\quad
g(v) \ge g(v^{k}) + \langle B^\top p^{k} |v-v^{k} \mathrm{a}ngle.
\]
Taking $v=v^k$ in the first inequality, $v=v^{k+1}$ in the second, and summing up both yields $\langle p^k-p^{k+1}| B(v^k-v^{k+1})\mathrm{a}ngle \ge 0$. Thus, we prove that $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $ holds for $\mathbf{e}ta\in \ ]0,\mathbf{f}rac{2}{\gamma}-1]$, if $\gamma \in\ ]0,2[$.
{\rm (i)} In view of Theorem Theorem \mathrm{e}f{t_gppa}-(ii) and choosing $\mathbf{e}ta =\mathbf{f}rac{2}{\gamma} -1$. This shows the strong convergences of $Bv^{k+1}-Bv^k \rightarrow 0$ and $p^{k+1}-p^k \rightarrow 0$.
{\rm (ii)} By Theorem \mathrm{e}f{t_gppa}-(i), we obtain $Bv^k\mathrm{w}eak Bv^\star$ and $p^k \mathrm{w}eak p^\star$. Then, $Au^k \mathrm{w}eak Au^\star$ follows from the $p$-update of \mathbf{e}qref{radmm} and the strong convergences of $Bv^{k+1}-Bv^k \rightarrow 0$ and $p^{k+1}-p^k \rightarrow 0$.
Finally, note that $(u^\star,v^\star,p^\star)\in \mathrm{zer} \mathcal{A}$ satisfies the Karush-Kuhn-Tucker conditions of \mathbf{e}qref{problem1}.
\mathbf{e}nd{proof}
\vskip.1cm
The scheme \mathbf{e}qref{radmm} is also known as the relaxed-DRS, and the standard DRS/ADMM is exactly recovered by letting $\gamma=1$ \cite{self_eq}. Since $\mathcal{S}$ is degenerate, one can only conclude the strong convergences of $B(v^{k+1}-v^k) \rightarrow 0$ and $p^{k+1}-p^k \rightarrow 0$, but except for $A(u^{k+1}-u^k) \rightarrow 0$. Furthermore, one cannot conclude $u^k \mathrm{w}eak u^\star$ and $v^k \mathrm{w}eak v^\star$. Actually, $u^k$ and $v^k$ may not be uniquely determined by \mathbf{e}qref{radmm}. Hence, we use `$:\in \Arg\min$' instead of `$:= \arg\min$' in the $(u,v)$-updates. This is related to the notion of {\it infimal postcomposition} \cite{arias_infimal,self_eq}, which is beyond the scope of this paper, and not discussed in details here.
Historically, the weak convergence of $p^k \mathrm{w}eak p^\star$ has long been proved in the seminal work of \cite{lions}, while the weak convergence of $Au^k\mathrm{w}eak Au^\star$ (i.e., in terms of the solution itself) was recently settled in \cite{svaiter}. Our proof addresses this intricate problem in a much easier way and under more general problem setting of \mathbf{e}qref{problem1}.
Finally, we stress that the relaxed-ADMM \mathbf{e}qref{radmm} is a good example to show that $\big \| x^k - x^{k+1} \big \|
_{\mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} }^2 \ge \mathbf{e}ta \big \| x^k - x^{k+1} \big \|_\mathcal{S}^2 $, but $ \mathcal{M}^{-\top} \mathcal{G} \mathcal{M}^{-1} \succeq \mathbf{e}ta \mathcal{S}$ does not hold for any $\mathbf{e}ta \in \ ]0, +\infty[$.
\begin{equation}gin{example} [Proximal-ADMM] \label{eg_padmm}
The proximal-ADMM is given as \cite[modified SPADMM]{lxd_2016}\mathbf{f}ootnote{The solutions to $u$ and $v$-steps in \mathbf{e}qref{padmm} are guaranteed to be unique, if $P_1, P_2 \in xS_{++}$. Hence, we use `$:= \arg\min$' here.}
\begin{equation} \label{padmm}
\left\lfloor \begin{equation}gin{array}{lll}
u^{k+1} & := & \arg \min_u f(u) +
\mathbf{f}rac{\tau}{2} \big\| Au +Bv^k - c - \mathbf{f}rac{1}{\tau} p^k \big\|^2 + \mathbf{f}rac{1}{2} \big\| u - u^k \big\|_{P_1}^2 , \\
v^{k+1} & := & \arg \min_v g(v) + \mathbf{f}rac{\tau}{2} \big\|A u^{k+1} + Bv - c - \mathbf{f}rac{1}{\tau} p^k \big\|^2
+ \mathbf{f}rac{1}{2} \big\| v - v^k \big\|_{P_2}^2 , \\
p^{k+1} & := & p^k -\tau (A u^{k+1} + Bv^{k+1} - c) ,
\mathbf{e}nd{array} \right.
\mathbf{e}e
It fits into the relaxed G-FBS operator \mathbf{e}qref{t_relaxed} as:
\[
\mathcal{Q} = \begin{equation}gin{bmatrix}
P_1 & 0 & 0 \\
0 & P_2 + \tau B^\top B & 0 \\
0 & -B & \mathbf{f}rac{1}{\tau} I
\mathbf{e}nd{bmatrix},\
\mathcal{M} = \begin{equation}gin{bmatrix}
I & 0 & 0 \\ 0 & I & 0 \\
0 & -\tau B & I \mathbf{e}nd{bmatrix},
\]
where $x^k$, $\mathcal{A}$ and $\mathcal{B}$ are same as Example \mathrm{e}f{eg_radmm}.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_padmm}
Let $\{(u^k,v^k,p^k)\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{padmm}. If $P_1,P_2 \inxS_{++}$ and $\tau >0$,
then, there exists a solution $(u^\star, v^\star, p^\star) $ to \mathbf{e}qref{problem1}, such that $(u^k, v^k, p^k) \mathrm{w}eak ( u^\star, v^\star, p^\star)$, as $k \rightarrow \infty$.
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
The proof is similar to Proposition \mathrm{e}f{p_radmm}. We compute $\mathcal{S}$ and $\mathcal{G}$ as
\[
\mathcal{S} = \begin{equation}gin{bmatrix}
P_1 & 0 & 0 \\
0 & P_2 + \tau B^\top B & 0 \\
0 & 0 & \mathbf{f}rac{1}{\tau } I
\mathbf{e}nd{bmatrix},\ \mathcal{G} = \begin{equation}gin{bmatrix}
P_1 & 0 & 0 \\ 0 & P_2 & 0 \\
0 & 0 & \mathbf{f}rac{1}{\tau} I
\mathbf{e}nd{bmatrix} .
\]
Then, the proof is completed by Theorem \mathrm{e}f{t_gppa} and Remark \mathrm{e}f{r_gppa}-(iii).
\mathbf{e}nd{proof}
\vskip.1cm
Comparing Examples \mathrm{e}f{eg_radmm} and \mathrm{e}f{eg_padmm}, one can see that the preconditioning technique of using $P_1$ and $P_2$ basically changes the algorithmic structure. By the proximization, $(u^k,v^k)$ are uniquely determined in \mathbf{e}qref{padmm}. The corresponding $\mathcal{S}$ and $\mathcal{G}$ become non-degenerate, and thus, the weak convergence of all variables $(u^k,v^k,p^k)$ is guaranteed.
\subsection{Gradient descent, P-FBS and PDS algorithms}
\label{sec_grad}
Consider the primal problem \cite[Problem 4.1]{vu_2013}:
\begin{equation} \label{p}
\min_x f(x) + \sum_{i=1}^m
(g_i \square l_i ) (A_i x - r_i)
+ h(x) + \langle x | z \mathrm{a}ngle,
\mathbf{e}e
where $x \in \mathcal{X}$, $A_i: \mathcal{X}\mapsto \mathcal{Y}_i $, $r_i \in \mathcal{Y}_i$, $z \in \mathcal{X}$. The functions are defined as $f: \mathcal{X} \mapsto \mathbb{R}$, $g_i: \mathcal{Y}_i \mapsto \mathbb{R}\cup\{+\infty\}$, $l_i: \mathcal{Y}_i \mapsto \mathbb{R}\cup\{+\infty\}$, $h: \mathcal{X}\mapsto \mathbb{R}\cup\{+\infty\}$. {\mathrm{e}d The symbol $\square$ denotes the infimal convolution of both functions $g_i$ and $l_i$, which is defined by
$(g_i \square l_i)(x) = \inf_{u \in \mathcal{Y}_i} \{ g_i (u) + l_i(x-u)\}$, $\mathbf{f}orall x\in \mathcal{Y}_i$.} We assume the functions in \mathbf{e}qref{p} satisfy
{\mathrm{e}d
\begin{equation}gin{assumption} \label{assume_4}
\begin{equation}gin{itemize}
\item[\rm (i)] $f$, $g_i$, $l_i$ and $h$ are proper, l.s.c. and convex for $i=1,...,m$;
\item [\rm (ii)] $f$ is differentiable with $\begin{equation}ta$-Lipschitz continuous gradient;
\item[\rm (iii)] $l_i$ is $\mu_i$-strongly convex for $i=1,...,m$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{assumption} }
There are various classes of algorithms for solving \mathbf{e}qref{p} or the special cases, listed below. Note that for all algorithms in Sect. \mathrm{e}f{sec_grad}, $\begin{equation}ta$ always stands for the Lipschitz constant of $\nabla f$.
\begin{equation}gin{example} [Gradient descent]
\label{eg_grad}
Consider $\min_x f(x)$, which is a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{ 0\} }$ (i.e., the indicator function of the set $C=\{ 0\}$), $g= 0$, $h=0$, $ z = 0$. If $\Arg \min f\ne \mathbf{e}mptyset$, the gradient descent method is given by \cite[Sect. 1.2.1]{bert_book_nonlinear}
\begin{equation} \label{grad}
x^{k+1} := x^k - \tau \nabla f( x^k),
\mathbf{e}e
which fits into the G-FBS operator \mathbf{e}qref{gfbs} with
$\mathcal{A} = 0$, $\mathcal{B} = \nabla f$ and $\mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_grad}
Let $x^\star\in\Arg\min f$, $x^0\in \mathcal{H}$, $\{x^k\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{grad}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau \in\ ]0, 2/\begin{equation}ta[$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $f(x^{k+1}) \le f(x^k) - \big( \mathbf{f}rac{1}{\tau} - \mathbf{f}rac{\begin{equation}ta}{2}\big) \big\|x^{k+1}-x^k \big\|^2$.
\item[\rm (ii)] $f(x^{k}) -f(x^\star) \le \mathbf{f}rac{1}{2k\tau}
\big\|x^{0}-x^\star \big\|^2$, and $f(x^k) \downarrow f(x^\star)$.
\item[\rm (iii)] $f(x^{k}) -f(x^\star) \sim o(1/k)$.
\item[\rm (iv)] $ \big\| x^{k+1 } - x^{k } \big\|
\le \sqrt{ \mathbf{f}rac{2}{ (k+1 ) (2-\tau\begin{equation}ta) } }
\big\|x^{0} -x^\star \big\|$.
\item[\rm (v)] $x^k\mathrm{w}eak x^\star$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
{\rm (i)} Lemma \mathrm{e}f{l_decrease}-(ii) and Remark \mathrm{e}f{r_obj}-(iv).
{\rm (ii)} Proposition \mathrm{e}f{p_gpfbs_obj} and Remark \mathrm{e}f{r_obj_2}-(iii).
{\rm (iii)} In view of (ii) and \cite[Lemma 2.7]{corman}.
{\rm (iv)} Substituting $\nu=1/\tau$ into Theorem \mathrm{e}f{t_dist}-(iii).
{\rm (v)} Theorem \mathrm{e}f{t_dist}-(iv).
\mathbf{e}nd{proof}
\vskip.1cm
Proposition \mathrm{e}f{p_grad} exactly recovers the classical convergence condition of $\tau \in\ ]0, 2/\begin{equation}ta[$, which coincides with \cite[Proposition 63]{plc_fixed} and \cite[Propositions 1.2.2 and 1.3.3]{bert_book_nonlinear}.
\begin{equation}gin{example} [Classical PPA]
\label{eg_ppa}
Consider $\min_x g(x)$, which is a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{ 0\} }$, $f=0$, $h=0$, $z = 0$. Assuming $\Arg\min g \ne \mathbf{e}mptyset$, the classical PPA is given by \cite[Proposition 64]{plc_fixed}
\begin{equation} \label{classic_ppa}
x^{k+1} := x^k +\gamma \big( \prox_{\tau g}(x^k) - x^k \big),
\mathbf{e}e
which fits the relaxed G-FBS operator \mathbf{e}qref{rgfbs} with
$\mathcal{A}=\partial h$, $\mathcal{B}=0$ and $ \mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_ppa}
Let $x^0\in\mathcal{H}$, $\{x^k\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{classic_ppa}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau >0$ and $\gamma \in\ ]0,2[$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $x^k\mathrm{w}eak x^\star \in \Arg\min g$.
\item[\rm (ii)] $\big\| x^{k+1 } - x^{k } \big\|
\le \sqrt{ \mathbf{f}rac{\gamma}{ (k+1)(2-\gamma) } }
\big\|x^{0} -x^\star \big\|$, and thus, $x^{k+1}-x^k \rightarrow 0$ as $k \rightarrow \infty$.
\item[\rm (iii)] If $\gamma=1$, $g(x^k) -g(x^\star) \le \mathbf{f}rac{1}{2\tau k}
\big\|x^{0} -x^\star \big\|^2$, $\mathbf{f}orall k \in \mathbb{N}$.
\item[\rm (iv)] If $\gamma=1$, $g(x^k) -g(x^\star) \sim o(1/k)$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
{\rm (i)--(ii)} In view of Remark \mathrm{e}f{r_rgfbs}-(iii) or
substituting $\nu=\mathbf{f}rac{1}{\tau}$ and $\begin{equation}ta=0$ into Corollary \mathrm{e}f{c_rgfbs}.
{\rm (iii)} Proposition \mathrm{e}f{p_gpfbs_obj} or Remark \mathrm{e}f{r_obj_2}-(iii).
{\rm (iv)} Remark \mathrm{e}f{r_obj_2}-(ii).
\mathbf{e}nd{proof}
\vskip.1cm
Many classical results presented in the seminal work \cite{ppa_guler} can be retrieved: Proposition \mathrm{e}f{p_ppa}-(i) and (iii) recovers \cite[Theorem 2.1]{ppa_guler}; (iv) recovers \cite[Theorem 3.1]{ppa_guler}; (ii) extends \cite[Corollary 2.3]{ppa_guler} to any relaxation parameter $\mathbf{e}ta \in\ ]0,2[$.
\begin{equation}gin{example} [Classical proxmal FBS \cite{plc,plc_chapter}]
Consider the problem $\min_x f(x) + g(x)$, which is a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{0\} }$, $h= 0$, $z = 0$. Assuming $0\in \mathrm{sri}(\mathrm{dom} g- \mathrm{dom} f)$, the error-free version of the classical proximal FBS is given by \cite[Eq.(3.6)]{plc}:
\begin{equation} \label{pfbs}
x^{k+1} := x^k + \gamma \big( \prox_{\tau g}(x^k - \tau \nabla f(x^k)) - x^k \big),
\mathbf{e}e
which fits the relaxed G-FBS operator \mathbf{e}qref{rgfbs} with
$\mathcal{A}=\partial g$, $\mathcal{B} = \nabla f$ and $\mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_pfbs}
Let $x^0 \in \mathcal{H}$, $\{x^k\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{pfbs}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau> \begin{equation}ta/2$ and $\gamma \in\ ]0,2[$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $x^k\mathrm{w}eak x^\star \in \Arg\min (f+g)$.
\item[\rm (ii)] $\big\| x^{k+1 } - x^{k } \big\|
\le \mathbf{f}rac{1}{ \sqrt{k+1 } }
\sqrt{\mathbf{f}rac{2\gamma} {(4-2\gamma)-\tau \begin{equation}ta } }
\big\|x^{0} -x^\star \big\|$, $\mathbf{f}orall k \in \mathbb{N}$.
\item[\rm (iii)] If $\gamma=1$, $(f+g)(x^k) - (f+g)(x^\star) \le \mathbf{f}rac{1}{2\tau k}
\big\|x^{0} -x^\star \big\|^2$, $\mathbf{f}orall k \in \mathbb{N}$.
\item[\rm (iv)] If $\gamma=1$, $(f+g)(x^k) - (f+g)(x^\star) \sim o(1/k)$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
{\rm (i)--(ii)} Substituting $\nu=\mathbf{f}rac{1}{\tau}$ into Corollary \mathrm{e}f{c_rgfbs}.
{\rm (iii)} Proposition \mathrm{e}f{p_gpfbs_obj}.
{\rm (iv)} Remark \mathrm{e}f{r_obj_2}-(ii).
\mathbf{e}nd{proof}
\vskip.1cm
This algorithm \mathbf{e}qref{classic_ppa} is also known as the {\it proximal gradient method} \cite{boyd_prox}. Proposition \mathrm{e}f{p_pfbs} loosens the convergence condition in \cite[Theorem 3.4]{plc} and \cite[Proposition 10.4]{plc_chapter} from $\gamma \in \ ]0,1]$ to $\gamma \in\ ]0, 2[$.
\begin{equation}gin{example} [Chambolle-Pock algorithm \cite{cp_2011}] \label{eg_cp}
Consider $\min_u h(u) + g(Au)$, which is a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{ 0\} } $, $f= 0$, $r = 0$, $z = 0$. Assuming $0\in\mathrm{sri}(\mathrm{dom} g -A(\mathrm{dom} h))$, the Chambolle-Pock algorithm is \cite[Algorithm 1]{cp_2011}:
\begin{equation} \label{cp}
\left\lfloor \begin{equation}gin{array}{llll}
s^{k+1} & := & \prox_{\sigma g^*} \big(s^k +\sigma
A (2u^k - u^{k-1} ) \big), & \text{\rm (dual step)} \\
u^{k+1} & := & \prox_{\tau h} \big( u^k - \tau
A^\top s^{k+1} \big). & \text{\rm (primal step)}
\mathbf{e}nd{array} \right.
\mathbf{e}e
The corresponding G-FBS operator \mathbf{e}qref{gfbs} is:
\[
x^k = \begin{equation}gin{bmatrix} s^{k} \\ u^{k-1}
\mathbf{e}nd{bmatrix}, \
\mathcal{A}= \begin{equation}gin{bmatrix}
\partial g^* & -A \\
A^\top & \partial h \mathbf{e}nd{bmatrix},\
\mathcal{B}=0, \ \mathcal{Q} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{ \sigma} I & - A \\
- A^\top & \mathbf{f}rac{1}{ \tau} I
\mathbf{e}nd{bmatrix} .
\]
\mathbf{e}nd{example}
In the above G-FBS fitting, we use a mismatch of iteration indices between $u$ and $s$: $x^k := (s^k, u^{k-1})$. This technique can also be found in \cite{bot_2015}.
\begin{equation}gin{proposition}
Let $\{ (s^k, u^{k})\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{cp}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau \in\ ]0, \mathbf{f}rac{1}{\sigma \|A^\top A\|}[$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $ (s^k, u^{k}) \mathrm{w}eak (s^\star,u^\star) \in \mathrm{zer} \mathcal{A}$;
\item[\rm (ii)] $\big\| x^{k+1 } - x^{k } \big\|_\mathcal{Q}
\le \mathbf{f}rac{1}{ \sqrt{k+1 } }
\big\|x^{0} -x^\star \big\|_\mathcal{Q}$,
$ \mathbf{f}orall k \in \mathbb{N}$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
In view of Corollary \mathrm{e}f{c_ppa} and \cite[Theorem 16.47]{plc_book}.
\mathbf{e}nd{proof}
\vskip.1cm
Comparing with \cite[Theorem 1]{cp_2011}, which only depends on the diagonal part of $\mathcal{Q}$, our result takes into account the off-diagonal components of $\mathcal{Q}$. This is consistent with the result of \cite{cp_2016}---an improved version of \cite{cp_2011}.
\begin{equation}gin{example} [Arias-Combettes algorithm \cite{arias_2011}]
Consider the problem
$\min_u h(u) + g(Au -r) + \langle u| z\mathrm{a}ngle$, which is a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{ 0\} } $, $f= 0$. Assuming $\Arg\min (h+g\circ (A\cdot -r) - \langle \cdot|z\mathrm{a}ngle)\ne \mathbf{e}mptyset$ and $r \in \mathrm{sri}(A(\mathrm{dom} h)-\mathrm{dom} g)$, the error-free version of \cite[Proposition 4.2]{arias_2011} is given as
\begin{equation} \label{arias}
\left\lfloor \begin{equation}gin{array}{lll}
\tilde{u}^{k} & := & \prox_{\tau h} \big( u^k - \tau
A^\top s^k - \tau z \big), \\
\tilde{s}^{k} & := & \prox_{\tau g^*} \big( s^k +\tau A u^k -\tau r \big), \\
u^{k+1} & := & \tilde{u}^k - \tau A^\top (\tilde{s}^k - s^k), \\
s^{k+1} & := & \tilde{s}^k +\tau A (\tilde{u}^k - u^k) ,
\mathbf{e}nd{array} \right.
\mathbf{e}e
which corresponds to the following relaxed G-FBS operator:
\[
x^k =\begin{equation}gin{bmatrix} u^{k } \\ s^{k }\mathbf{e}nd{bmatrix},
\ \mathcal{A} = \begin{equation}gin{bmatrix}
\partial \mathbf{f}tilde & A^\top \\
- A & \partial \tilde{g}^* \mathbf{e}nd{bmatrix},\
\mathcal{B}= 0, \ \mathcal{Q} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{\tau} I & -A^\top \\
A & \mathbf{f}rac{1}{\tau} I \mathbf{e}nd{bmatrix}, \
\mathcal{M} = \begin{equation}gin{bmatrix}
I & -\tau A^\top \\ \tau A & I
\mathbf{e}nd{bmatrix},
\]
where $\mathbf{f}tilde = f+\langle \cdot | z\mathrm{a}ngle$,
$\tilde{g}^* = g^* +\langle \cdot | r\mathrm{a}ngle$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition}
Let $\{ (s^k, u^{k})\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{arias}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau \in\ ]0, \mathbf{f}rac{1}{ \|A\| }[$, then, $ (s^k, u^{k}) \mathrm{w}eak (s^\star,u^\star) \in \mathrm{zer} \mathcal{A}$.
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
We compute $\mathcal{S}$ and $\mathcal{G}$ as:
\[
\mathcal{S} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{ \tau} I & 0 \\
0 & \mathbf{f}rac{1}{\tau } I \mathbf{e}nd{bmatrix},\ \mathcal{G} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{\tau} I - \tau A^\top A & 0 \\
0 & \mathbf{f}rac{1}{\tau} I - \tau AA^\top
\mathbf{e}nd{bmatrix} .
\]
Then, $\mathcal{S},\mathcal{G} \in xS_{++}$ requires $\tau \in\ ]0, \mathbf{f}rac{1}{ \|A\| }[$. The weak convergence follows from Remark \mathrm{e}f{r_gppa}-(iii).
\mathbf{e}nd{proof}
\vskip.1cm
The condition of original \cite[Proposition 4.2]{arias_2011} is $\tau\in [\mathbf{e}psilon, \mathbf{f}rac{1-\mathbf{e}psilon}{\|A\|} ]$ with $\mathbf{e}psilon\in\ ]0, \mathbf{f}rac{1}{\|A\|+1}[$, which is equivalent to our result, when $\mathbf{e}psilon \rightarrow 0^+$.
\begin{equation}gin{example} [Generalized Dykstra-like algorithm \cite{plc_dual_2010}]
\label{eg_plc_dual}
Consider \cite[Problem 1.2]{plc_dual_2010}
\[
\min_u h(u) + g(A u -r)+
\mathbf{f}rac{1}{2} \|u - w\|^2,
\]
which is a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{ 0\} } $, $f = \mathbf{f}rac{1}{2} \| \cdot - w\|^2$, $ z = 0$. Assuming $r\in\mathrm{sri} (A(\mathrm{dom} h) - \mathrm{dom} g)$, \cite[Algorithm 3.5]{plc_dual_2010} is given as
\begin{equation} \label{x45}
\left\lfloor \begin{equation}gin{array}{lll}
u^{k} &: = & \prox_{h} \big( w - A^\top s^k \big), \\
s^{k+1} & := & s^k +\gamma \big(
\prox_{\tau g^*} ( s^k + \tau ( Au^k - r) ) - s^k \big).
\mathbf{e}nd{array} \right.
\mathbf{e}e
By \cite[Proposition 3.1]{plc_dual_2010}, the dual problem is $\min_s q(s) +g^*(s) +\langle s|r \mathrm{a}ngle$, where $q(s)= \mathbf{f}rac{1}{2} \big\| w - A^\top s \big\|^2 - \big( \inf_u h(u) +\mathbf{f}rac{1}{2} \big\|u - w + A^\top s \big\|^2 \big)$.
By \cite[Theorem 3.7]{plc_dual_2010}, \mathbf{e}qref{x45} is equivalent to a simple proximal FBS for solving the dual problem:
\begin{equation} \label{w4}
s^{k+1} := s^k+ \gamma \big( \prox_{\tau \tilde{g}^*}
\big( s^k - \tau \nabla q( s^k) \big) - s^k \big),
\mathbf{e}e
where $\tilde{g}^* = g^* +\langle \cdot |r\mathrm{a}ngle$. This algorithm fits the relaxed G-FBS operator \mathbf{e}qref{rgfbs} with
\[
x = s, \ \mathcal{A}= \partial \tilde{g}^* , \ \mathcal{B}= \nabla q , \
\mathcal{Q} = \mathbf{f}rac{1}{\tau} \mathcal{I}.
\]
\mathbf{e}nd{example}
\begin{equation}gin{proposition}
Let $\{ (s^k, u^{k})\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{x45}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau \in\ ]0, 2/ \|A\|^2[$, $\gamma \in\ ]0,2-\mathbf{f}rac{\tau}{2} \|A\|^2[ $, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $ s^k \mathrm{w}eak s^\star$, as $k \rightarrow \infty$;
\item[\rm (ii)] $\big\| s^{k+1 } - s^{k } \big\|
\le \mathbf{f}rac{1}{ \sqrt{k+1 } } \sqrt{\mathbf{f}rac{2\gamma}{4-2\gamma -\tau \|A\|^2} }
\big\|s^{0} - s^\star \big\|$,
$\mathbf{f}orall k \in \mathbb{N}$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
$\mathcal{B} = \nabla q$ is $\|A\|^2$-Lipschitz continuous, and thus, $\mathbf{f}rac{1}{\|A\|^2}$-cocoercive (i.e., $\begin{equation}ta = \|A\|^2$). Then, the results follow by Corollary \mathrm{e}f{c_rgfbs}.
\mathbf{e}nd{proof}
\vskip.1cm
Our convergence condition is milder than the original version of
$\tau \in \ ]0, 2/\|A\|^2[$ and $\gamma \in \ ]0, 1]$ presented in \cite[Theorems 3.6 and 3.7]{plc_dual_2010} and \cite[Theorem 3.4]{plc}. Moreover, since the variable $u$ is merely intermediate update of \mathbf{e}qref{x45}, and is afterwards removed in \mathbf{e}qref{w4}. The convergence of $\{u^k\}_{k\in\mathbb{N}}$ cannot be concluded by Corollary \mathrm{e}f{c_rgfbs}, which requires additional work, see \cite[Theorem 3.7-(ii)]{plc_dual_2010}.
\begin{equation}gin{example} [PAPC \cite{zxq_ip,teboulle_2015,gist}]
Consider $\min_u f(u) + g(Au)$, which is
a special case of \mathbf{e}qref{p} with $m=1$, $h= 0$, $l=\iota_{ \{ 0 \} }$, $r = 0 $, $z = 0$. Assuming $f+g$ is coercive, the PAPC scheme is given as
\begin{equation} \label{papc}
\left\lfloor \begin{equation}gin{array}{lll}
s^{k+1} & := & \prox_{\sigma g^*} \big(
(I - \sigma \tau A A^\top ) s^k +\sigma A
(u^k - \tau \nabla f(u^k)) \big), \\
u^{k+1} & := & u^k - \tau \nabla f(u^k) - \tau A^\top
s^{k+1},
\mathbf{e}nd{array} \right.
\mathbf{e}e
which can be interpreted by the G-FBS operator \mathbf{e}qref{gfbs}:
\[
x^k = \begin{equation}gin{bmatrix}
s^{k } \\ u^k \mathbf{e}nd{bmatrix}, \
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial g^* & -A \\
A^\top & 0 \mathbf{e}nd{bmatrix},\
\mathcal{B} = \begin{equation}gin{bmatrix}
0 & 0 \\ 0 & \nabla f \mathbf{e}nd{bmatrix},\
\mathcal{Q} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{\sigma} I - \tau A A^\top & 0 \\
0 & \mathbf{f}rac{1}{\tau} I \mathbf{e}nd{bmatrix}.
\]
\mathbf{e}nd{example}
\begin{equation}gin{proposition}
Let $\{ (s^k, u^{k})\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{papc}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $0< \tau < \min \big\{\mathbf{f}rac{2}{\begin{equation}ta}, \mathbf{f}rac{2-\begin{equation}ta\sigma} {2\sigma \|A^\top A\|} \big\}$,
the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $ (s^k, u^{k}) \mathrm{w}eak (s^\star,u^\star) \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$.
\item[\rm (ii)]
$\big\| x^{k+1 } - x^{k } \big\|_\mathcal{Q}
\le \sqrt{ \mathbf{f}rac{2\nu}{ (k+1) (2\nu -\begin{equation}ta)} }
\big\|x^{0} -x^\star \big\|_\mathcal{Q}$,
where $\nu = \min\big\{ \mathbf{f}rac{1}{\sigma} - \tau \|A^\top A\|,
\mathbf{f}rac{1}{\tau} \big\}$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
Since $f+g$ is coercive, then $\Arg\min (f+g) \ne \mathbf{e}mptyset$ by \cite[Proposition 3.1-(i)]{plc}. By Theorem \mathrm{e}f{t_dist},
the convergence condition follows from $\mathcal{Q} \succ \mathbf{f}rac{\begin{equation}ta}{2}\mathcal{I} $. The value of $\nu$ chosen in (ii) satisfies $\mathcal{Q} \succeq \nu\mathcal{I}$.
\mathbf{e}nd{proof}
\begin{equation}gin{example} [AFBA \cite{latafat_2017,latafat_chapter}] \label{afba}
Consider $\min_u f(u) + h(u) + g(Au)$, which is
a special case of \mathbf{e}qref{p} with $m=1$, $l=\iota_{ \{ 0 \} }$, $r = 0$, $z = 0$. The AFBA scheme is given by
\begin{equation} \label{afba}
\left\lfloor \begin{equation}gin{array}{lll}
s^{k+1} & := & \prox_{\sigma g^*} \big( s^k
+ \sigma Aw^k \big), \\
u^{k+1} & := & w^k - \tau A^\top ( s^{k+1} - s^k), \\
w^{k+1} & := & \prox_{\tau h} \big( u^{k+1} - \tau \nabla f (u^{k+1}) -\tau A^\top s^{k+1} \big) .
\mathbf{e}nd{array} \right.
\mathbf{e}e
To interpret it by the G-FBS operator, we remove $w$ and obtain the equivalent form:
\[
\left\lfloor \begin{equation}gin{array}{lll}
s^{k+1} & := & \prox_{\sigma g^*} \big( s^k
+ \sigma A \big(u^{k+1} +\tau A^\top (s^{k+1} - s^k) \big) \big), \\
u^{k+1} & := & \prox_{\tau h} \big( u^{k } - \tau \nabla f (u^{k }) -\tau A^\top s^{k } \big) -\tau A^\top (s^{k+1} - s^k) ,
\mathbf{e}nd{array} \right.
\]
which corresponds exactly to the relaxed G-FBS operator \mathbf{e}qref{t_relaxed} with
\[
x^k = \begin{equation}gin{bmatrix}
\mathbf{s} ^{k } \\ u^{k } \mathbf{e}nd{bmatrix},\
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial g^* & -A \\
A^\top & \partial h \mathbf{e}nd{bmatrix}, \
\mathcal{B} = \begin{equation}gin{bmatrix}
0 & 0 \\ 0 & \nabla f \mathbf{e}nd{bmatrix},
\quad \mathcal{Q} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{\sigma} I & 0 \\
-A^\top & \mathbf{f}rac{1}{ \tau} I
\mathbf{e}nd{bmatrix},\
\mathcal{M} = \begin{equation}gin{bmatrix}
I & 0 \\ -\tau A^\top & I
\mathbf{e}nd{bmatrix}.
\]
\mathbf{e}nd{example}
\begin{equation}gin{proposition}
Let $\{ (s^k, u^{k})\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{afba}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $0< \tau < \min \big\{\mathbf{f}rac{2}{\begin{equation}ta}, \mathbf{f}rac{2-\begin{equation}ta\sigma} {2\sigma \|A^\top A\|} \big\}$,
$ (s^k, u^{k}) \mathrm{w}eak (s^\star,u^\star) \in \mathrm{zer} (\mathcal{A}+\mathcal{B})$.
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
By Remark \mathrm{e}f{r_gppa}-(iii), $\mathcal{S}$ and $\mathcal{G}$ are computed as
\[
\mathcal{S} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{ \sigma} I & 0 \\
0 & \mathbf{f}rac{1}{\tau } I \mathbf{e}nd{bmatrix},\
\mathcal{G} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{\sigma} I - \tau A A^\top -\mathbf{f}rac{\begin{equation}ta}{2} I & 0 \\
0 & \mathbf{f}rac{1}{\tau} I - \mathbf{f}rac{\begin{equation}ta}{2} I
\mathbf{e}nd{bmatrix} .
\]
Then, by Theorem \mathrm{e}f{t_gppa}, the convergence condition follows from $\mathcal{S},\mathcal{G} \in xS_{++}$.
\mathbf{e}nd{proof}
\vskip.1cm
The convergence condition of AFBA \mathbf{e}qref{afba} is same as that of PAPC \mathbf{e}qref{papc}. This result is consistent with \cite[Eq.(5.10)]{latafat_chapter} and \cite[Proposition 5.2]{latafat_2017}.
\begin{equation}gin{example} [Condat algorithm \cite{condat_2013}]
Consider the problem $\min_u f(u) + h(u) +\sum_{i=1}^m g_i(A_i u)$, which is a special case of \mathbf{e}qref{p} with $l_i = \iota_{\{ 0 \} }$, $r = 0$, $z = 0$.
\cite[Algorithms 3.1 and 5.1]{condat_2013} is given by
\begin{equation} \label{condat}
\left\lfloor \begin{equation}gin{array}{lll}
\tilde{u}^{k} & = & \prox_{\tau h} \big(u^k - \tau
\nabla f(u^k) - \tau \sum_{i=1}^m A_i^\top s_i^k \big), \\
\tilde{s}_i^{k} & = & \prox_{\sigma g_i^*} \big( s_i^k +\sigma A_i ( 2\tilde{u}^k - u^k ) \big), \quad (i=1,2,...,m) \\
u^{k+1} & = & u^k +\gamma (\tilde{u}^k - u^k), \\
s_i^{k+1} & = & s_i^k +\gamma (\tilde{s}_i^k - s_i^k), \quad (i=1,2,...,m)
\mathbf{e}nd{array} \right.
\mathbf{e}e
which can be compactly expressed in a relaxed G-FBS form \mathbf{e}qref{rgfbs}:
\[
x^{k} = \begin{equation}gin{bmatrix}
u^k \\ s_1^k \\ \vdots \\ s_m^k
\mathbf{e}nd{bmatrix},\
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial h & A_1^\top & \cdots & A_m^\top \\
- A_1 & \partial g_1^* & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
- A_m & 0 & \cdots & \partial g_m^* \\
\mathbf{e}nd{bmatrix},\
\mathcal{B} = \begin{equation}gin{bmatrix}
\nabla f & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 0
\mathbf{e}nd{bmatrix},
\]
\[
\mathcal{Q} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{\tau} I & -A_1^\top & \cdots & - A_m^\top \\
- A_1 & \mathbf{f}rac{1}{ \sigma} I & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
- A_m & 0 & \cdots & \mathbf{f}rac{1}{ \sigma} I \\
\mathbf{e}nd{bmatrix}.
\]
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_condat}
Let $\{ (u^k, s_1^k, ..., s_m^k)\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{condat}. {\mathrm{e}d Under Assumption \mathrm{e}f{assume_4}}, if $\tau \in\ ]0, \mathbf{f}rac{1}{\sigma \|\sum_{i=1}^m A_i^\top A_i\|} [$, $\gamma \in\ ]0, 2-\mathbf{f}rac{\begin{equation}ta}{2} (\mathbf{f}rac{1}{\tau} -\sigma \|\sum_i^m A_i^\top A_i\|)^{-1} [$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $ (u^{k},s_1^k,...,s_m^k) \mathrm{w}eak (u^\star,s_1^\star,...,s_m^\star) \in \mathrm{zer}( \mathcal{A}+\mathcal{B})$ as $k \rightarrow \infty$.
\item[\rm (ii)]
$\big\| x^{k+1 } - x^{k } \big\|_\mathcal{Q}
\le \mathbf{f}rac{1}{\sqrt{k+1}} \sqrt{ \mathbf{f}rac{2\gamma\nu}{ (4-2\gamma)\nu -\begin{equation}ta}}
\big\|x^{0} -x^\star \big\|_\mathcal{Q}$,
where $\nu = \mathbf{f}rac{1}{\tau} - \sigma \big\| \sum_{i=1}^m A_i^\top A_i \big\|$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
The condition $\mathcal{Q} \in xS_{++}$ requires $\tau\sigma \in\ ]0, \mathbf{f}rac{1}{\|\sum_i A_i^\top A_i \|} [ $, and $\nu$ could be chosen as $ \mathbf{f}rac{1}{\tau} - \sigma \big\| \sum_{i=1}^m A_i^\top A_i \big\|$, such that $\mathcal{Q} \succeq \nu \mathcal{I}$. Then the results follow by Corollary \mathrm{e}f{c_rgfbs}.
\mathbf{e}nd{proof}
\vskip.1cm
This result is consistent with \cite[Theorem 5.1]{condat_2013}. Another counterpart algorithm to \mathbf{e}qref{condat} is given by \cite[Algorithms 3.2 and 5.2]{condat_2013}:
\begin{equation} \label{t6}
\left\lfloor \begin{equation}gin{array}{lll}
\tilde{s}_i^{k} & = & \prox_{\sigma g_i^*} \big(s_i^k + \sigma A_i u^k \big), \quad (i=1,2,...,m) \\
\tilde{u}^{k} & = & \prox_{\tau h} \big(u^k - \tau
\nabla f(u^k) - \tau \sum_{i=1}^m
A_i^\top (2\tilde{s}_i^k - s_i^k) \big) ,\\
u^{k+1} & = & u^k +\gamma (\tilde{u}^k - u^k) ,\\
s_i^{k+1} & = & s_i^k +\gamma (\tilde{s}_i^k - s_i^k). \quad (i=1,2,...,m)
\mathbf{e}nd{array} \right.
\mathbf{e}e
The corresponding relaxed G-FBS form is with the same $x$, $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{M}$ as the above example, and $\mathcal{Q}$ is given as
\[
\mathcal{Q} = \begin{equation}gin{bmatrix}
\mathbf{f}rac{1}{ \tau } I & A_1^\top & \cdots & A_m^\top \\
A_1 & \mathbf{f}rac{1}{ \sigma } I & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
A_m & 0 & \cdots & \mathbf{f}rac{1}{ \sigma } I\\
\mathbf{e}nd{bmatrix}.
\]
Proposition \mathrm{e}f{p_condat} also applies for \mathbf{e}qref{t6}.
\subsection{Other examples}
Other classes of algorithms can also be expressed by the G-FBS operator \mathbf{e}qref{gfbs}. Let us now consider a typical optimization problem with a linear equality constraint:
\begin{equation} \label{problem_alm}
\min_u h(u),\qquad \text{s.t.\ } Au = c ,
\mathbf{e}e
where $A: \mathcal{X} \mapsto \mathcal{Y}$, the function $h: \mathcal{X}\mapsto \mathbb{R}\cup \{+\infty\}$ is proper, l.s.c. and convex.
\begin{equation}gin{example} [Basic ALM] \label{eg_alm}
The augmented Lagrangian method (ALM) is (see \cite[Eq.(1.2)]{mafeng_2018}, \cite[Eq.(7.2)]{taomin_2018} and \cite[Algorithm 6.1]{hbs_2014} for example)
\begin{equation} \label{alm}
\left\lfloor \begin{equation}gin{array}{lll}
u^{k+1} & :\in & \arg \min_u h(u) + \mathbf{f}rac{\tau}{2}
\big\| Au - c - \mathbf{f}rac{1}{\tau} s^k \big\|^2 , \\
s^{k+1} & := & s^k - \tau (Au^{k+1} - c ).
\mathbf{e}nd{array} \right.
\mathbf{e}e
It can be customized by the G-FBS operator with
\[
x^k = \begin{equation}gin{bmatrix}
u^{k } \\ s^{k } \mathbf{e}nd{bmatrix},\
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial h & - A^\top \\ A & \partial l
\mathbf{e}nd{bmatrix},\
\mathcal{B}= 0, \
\mathcal{Q} = \begin{equation}gin{bmatrix}
0 & 0 \\
0 & \mathbf{f}rac{1}{\tau} I_M \mathbf{e}nd{bmatrix},
\]
where $l = -\langle \cdot |c \mathrm{a}ngle$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_alm}
Let $\{ (u^k, s^k)\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{alm}. If $\tau >0$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $\big\| s^{k+1 } - s^{k } \big\|
\le \mathbf{f}rac{1}{ \sqrt{k+1 } }
\big\| s^{0} - s^\star \big\|$, $\mathbf{f}orall k \in \mathbb{N}$;
\item[\rm (ii)] $ s^{k} \mathrm{w}eak s^\star$, as $k \rightarrow \infty$;
\item[\rm (iii)] $ Au^{k} \rightarrow c$, as $k \rightarrow \infty$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
{\rm (i)--(ii)} follow from Corollary \mathrm{e}f{c_ppa}.
{\rm (iii)} follows from the $s$-update of \mathbf{e}qref{alm} and strong convergence of $s^{k+1}-s^k\rightarrow 0$.
\mathbf{e}nd{proof}
\vskip.1cm
The variable $u$ in \mathbf{e}qref{alm} is merely intermediate result that is not proximally regularized. That is why the metric $\mathcal{Q}$ is degenerate here. Moreover, the $u$-step of \mathbf{e}qref{alm} may not have the unique solution. However, $Au^k$ is always unique \cite{self_eq}. This is also observed in \cite[Remark 6.1]{hbs_2014}. The tractability of the $u$-step can be solved by linearization, as shown in the next example.
\begin{equation}gin{example} [Linearized ALM] \label{eg_linear_alm}
The linearlized ALM is given as \cite{yang_yuan_2013}
\begin{equation} \label{linear_alm}
\left\lfloor \begin{equation}gin{array}{lll}
u^{k+1} & := & \arg \min_u h(u) + \mathbf{f}rac{\rho } {2}
\big\| u - u^k + \mathbf{f}rac{1}{ \rho} A^\top \big( \tau
(A u^k -c) - s^k \big) \big\|^2, \\
s^{k+1} & := & s^k - \tau ( Au^{k+1} - c) .
\mathbf{e}nd{array} \right.
\mathbf{e}e
Its G-FBS interpretation \mathbf{e}qref{gfbs} is given as
\[
x^k = \begin{equation}gin{bmatrix}
u^{k } \\ s^{k } \mathbf{e}nd{bmatrix},\
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial h & - A^\top \\ A & \partial l
\mathbf{e}nd{bmatrix},\
\mathcal{B} = 0 , \
\mathcal{Q} = \begin{equation}gin{bmatrix}
\rho I - \tau A^\top A & 0 \\
0 & \mathbf{f}rac{1}{\tau} I \mathbf{e}nd{bmatrix},
\]
where $l=-\langle \cdot| c\mathrm{a}ngle$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition} \label{p_lalm}
Let $\{ (u^k, s^k)\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{linear_alm}. If $\tau \in \ ]0, \rho/ \|A^\top A\|[$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $\big\| x^{k+1 } - x^{k } \big\|_\mathcal{Q}
\le \mathbf{f}rac{1}{ \sqrt{k+1 } }
\big\| x^{0} - x^\star \big\|_\mathcal{Q} $,
$\mathbf{f}orall k \in \mathbb{N}$.
\item[\rm (ii)] $(u^k, s^{k}) \mathrm{w}eak (u^\star, s^\star) \in \mathrm{zer} \mathcal{A}$, as $k \rightarrow \infty$, where $u^\star$ is a solution to the problem \mathbf{e}qref{problem_alm}.
\item[\rm (iii)] $ Au^{k} \rightarrow c$, as $k \rightarrow \infty$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
{\rm (i)--(ii)} follow from Corollary \mathrm{e}f{c_ppa}.
{\rm (iii)} follows from the $s$-update of \mathbf{e}qref{linear_alm} and strong convergence of $s^{k+1}-s^k\rightarrow 0$.
\mathbf{e}nd{proof}
\vskip.1cm
The linearization strategy can be viewed as a preconditioning, which guarantees the uniqueness of the $u$-step and the weak convergence of $u^k \mathrm{w}eak u^\star$.
\begin{equation}gin{example} [Linearized Bregman algorithm \cite{cjf_3}]
\label{eg_lb}
The scheme reads as \cite[Eq.(2.2)]{cjf_3}
\begin{equation} \label{lb}
\left\lfloor \begin{equation}gin{array}{lll}
u^{k+1} & := & \arg \min_u \rho \tau h(u) +
\mathbf{f}rac{1}{2} \big\| u - \rho A^\top s^k \big\|^2, \\
s^{k+1} & := & s^k - (Au^{k+1} - c).
\mathbf{e}nd{array} \right.
\mathbf{e}e
Its G-FBS interpretation \mathbf{e}qref{gfbs} is given as
\[
x^k = \begin{equation}gin{bmatrix}
u^{k} \\ s^{k} \mathbf{e}nd{bmatrix},\
\mathcal{A} = \begin{equation}gin{bmatrix}
\tau \partial \tilde{h} + \mathbf{f}rac{1}{\rho} I - A^\top A
& -A^\top \\ A & \partial l
\mathbf{e}nd{bmatrix},\
\mathcal{B} = 0,\ \mathcal{Q} = \begin{equation}gin{bmatrix}
0 & 0 \\ 0 & I \mathbf{e}nd{bmatrix} ,
\]
where $\tilde{h} = h+\langle \cdot| A^\top c\mathrm{a}ngle$, $l=-\langle \cdot |c \mathrm{a}ngle$.
\mathbf{e}nd{example}
\begin{equation}gin{proposition}
Let $\{ (u^k, s^k)\}_{k\in\mathbb{N}}$ be a sequence generated by \mathbf{e}qref{lb}. If $\tau>0$ and $\rho \in\ ]0, \mathbf{f}rac{1}{\|A^\top A\|} [$, the following hold.
\begin{equation}gin{itemize}
\item[\rm (i)] $\big\| s^{k+1 } - s^{k } \big\|
\le \mathbf{f}rac{1}{ \sqrt{k+1 } }
\big\| s^{0} - s^\star \big\|$, $ \mathbf{f}orall k \in \mathbb{N}$;
\item[\rm (ii)] $ s^{k} \mathrm{w}eak s^\star$, as $k\rightarrow\infty$;
\item[\rm (iii)] $Au^k \rightarrow c$, as $k\rightarrow\infty$.
\mathbf{e}nd{itemize}
\mathbf{e}nd{proposition}
\begin{equation}gin{proof}
Similar to Proposition \mathrm{e}f{p_alm} and \mathrm{e}f{p_lalm}.
\mathbf{e}nd{proof}
\subsection{Short summary}
Many first-order operator splitting algorithms have been shown as the customized applications of a simple G-FBS operator \mathbf{e}qref{gfbs} or its relaxed version \mathbf{e}qref{t_relaxed}. One can verify that more existing algorithms, e.g., \cite{fang_2015,mafeng_2018,hbs_2014,hbs_yxm_2018,
cch_2016,bai_2018}, belong to the G-FBS class, which are not detailed here.
We stress the simplicity of the G-FBS fitting, compared to the complicated characterization by variational inequality \cite{hbs_siam_2012,hbs_siam_2012_2} or procedure of constructing Fej\'{e}r monotone sequence \cite{plc_vu}. More importantly, there is no need to perform the convergence analysis case-by-case: All the results are the immediate consequences of the general results in Sect. \mathrm{e}f{sec_gfbs} and \mathrm{e}f{sec_extension}.
In many examples listed above, the cocoercive operator $\mathcal{B}=0$, and the problem \mathbf{e}qref{inclusion} becomes $0\in\mathcal{A} x$. Notice that it cannot be simply understood as a minimization of only one convex function. In fact, $0\in \mathcal{A} x$ in our setting also encompasses the minimization of the sum of multiple convex (not necessarily smooth) functions.
Moreover, the monotone operators $\mathcal{A}$ in many examples bear the typical (diagonal) monotone + (off-diagonal) skew structure:
\[
\mathcal{A} = \begin{equation}gin{bmatrix}
\partial f & -A^\top \\
A & \partial g \mathbf{e}nd{bmatrix} = \underbrace{
\begin{equation}gin{bmatrix}
\partial f & 0 \\
0 & \partial g \mathbf{e}nd{bmatrix} }_\text{monotone} +
\underbrace{ \begin{equation}gin{bmatrix}
0 & - A^\top \\ A & 0
\mathbf{e}nd{bmatrix} }_\text{skew},
\]
which coincides with the observations in \cite{plc_fixed,arias_2011,bredies_2017}. It can be further verified that $\mathcal{A}$ with such structure fails to be cyclically monotone, and cannot be viewed as a subdifferential of a convex function, by \cite[Theorem 22.18]{plc_book}. This was also mentioned in Sect. \mathrm{e}f{sec_summary} and Remark \mathrm{e}f{r_assume_2}-(iv). Therefore, there are no conclusions regarding the convergence of objective value in Examples \mathrm{e}f{eg_radmm}--\mathrm{e}f{eg_padmm} and \mathrm{e}f{eg_cp}--\mathrm{e}f{eg_lb}.
\section{Conclusions}
In this paper, we performed a systematic study of the G-FBS operator and its associated fixed-point iterations, particularly under degenerate setting. A great variety of operator splitting algorithms were illustrated as the concrete examples of the G-FBS operator.
Last, it seems interesting to further extend the proposed framework to the case when the metric $\mathcal{Q}$ and relaxation operator $\mathcal{M}$ are allowed to vary over the iterations. Another limitation of this G-FBS operator is that $\mathcal{Q}$ and $\mathcal{M}$ are assumed as linear here. Thus, it fails to cover Bregman proximal algorithms \cite{teboulle_2018,plc_bregman} and a few PDS algorithms \cite{plc_2012,bot_jmiv_2014,ywt_2017}, which may correspond to nonlinear metric $\mathcal{Q}$ and $\mathcal{M}$. It is worthwhile to extend the G-FBS operator \mathbf{e}qref{gfbs} to nonlinear case.
\section{Data availability}
There is no associated data with this manuscript.
\section{Disclosure statement}
The author declares there are no conflicts of interest regarding the publication of this paper.
\small{
}
\mathbf{e}nd{document} |
\begin{document}
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\newcommand\eg {{\em e.g., }} \newcommand\re {{\em r.e. }}
\title {\vspace*{-6pc}\ttl} \date{} \author{\aut\\
Boston University\thanks {Computer Science dept., 111 Cummington Mall,
Boston, MA 02215; Home page: \hreff{www.cs.bu.edu/fac/Lnd}}}\maketitle
\vspace*{-3pc}\begin{flushright}\parbox{1pc}{\begin{tabbing}
С этой безмерностью в мире мер.\\*\em -- Марина Цветаева
\footnotemark\end{tabbing}}\end{flushright}\footnotetext
{Measureless in this world of measures. -- Marina Tsvetaeva}
\vspace*{-2pc}\begin{abstract}\noindent Mutual information $\I$ in infinite
sequences (and in their finite prefixes) is essential in theoretical analysis
of many situations. Yet its right definition has been elusive for a long time.
I address it by generalizing Kolmogorov Complexity theory from measures
to {\bf semimeasures} \ie infimums of sets of measures. Being concave
rather than linear functionals, semimeasures are quite delicate to handle.
Yet, they adequately grasp various theoretical and practical scenaria.
A simple lower bound $\i(\a:\b)\edf\sup_{x\in\N}(\K(x)- \K(x|\a)-\K(x|\b))$
of information turns out tight for Martin-L\"of random $\a,\b\in\{0,1\}^\N$.
For all sequences $\I(\a:\b)$ is characterized by the minimum of $\i(\a':\b')$
over random $\a',\b'$ with $U(\a'){=}\a$, $U(\b'){=}\b$.\end{abstract}
\section {Introduction}
Kolmogorov Information theory applies to individual objects, in contrast to
Shannon theories that apply to the models of processes that generated such
objects. It thus has a much wider domain since many objects (\eg Shakespeare
plays) have no realistic generation models. For completed objects, such as
integers, the concept is simple and robust: $\I(x:y)=\K(x)+\K(y)-\K(x,y)$.
Yet, the concept is also needed for emerging objects, such as, \eg prefixes of
infinite sequences. Encoding prefixes as integers distorts the information by
specifying their (arbitrary) cut-off point. This cut-off information is not a
part of the original sequence and can be smaller in a longer prefix. In fact,
this distortion can overwhelm the actual mutual information between the
sequences.
This issue complicates many studies forcing one to use (as, \eg in \cite{fi})
concepts of information that are merely lower bounds, differ between
applications, and known not to be tight.
For the related concept of rarity (randomness deficiency) Per Martin-L\"of
proposed an extention that works well for infinite sequences under computable
distributions. Yet, computability of distributions requires a running time
limit for the processes generating them. Such limits then must be accounted for
in all formulas, obscuring the simplicity of purely informational values, at a
great cost to elegance and transparency. Without such limit many important
distributions are only lower-enumerable (r.e.). For instance, universal
probability {\M} is the largest within a constant factor \re distribution. It
is extraordinarily flat: all sequences are random with respect to it.
Yet {\M} is instrumental in defining other interesting distributions. In
particular, Mutual Information in two sequences is their \trm {dependence}, \ie
rarity with respect to the distribution $\M\otimes\M$ generating them
independently with universal probability each. R.e. distributions are of
necessity semimeasures: concave rather than linear functionals. Semimeasures
also are relevant in more mundane and widespread situations where the specific
probability distribution is not fully known (\eg due to interaction with a
party that cannot be modeled). They require much more delicate handling than
measures. This article considers many subtleties that arise in such
generalization of complexity theory. The concept of rarity for such
distribution considered here respects randomness conservation inequalities and
is the strongest (\ie largest) possible such definition. The definition of
mutual information arising from this concept is shown to allow rather simple
descriptions.
\section {Conventions and Background}
Let \R, \Q, \N, $\St{=}\{0,1\}^*$, $\W{=}\{0,1\}^\N$ be, respectively,
the sets of reals, rationals, integers, finite, and infinite binary sequences;
$x_{[n]}$ is the $n$-bit prefix and $\|x\|$ is the bit-length of $x{\in}\St$;
for $a{\in}\Re^+$, $\|a\|{\edf}|\,\ceil{\log a}{-}1|$. A function $f$
and its values are \trm {enumerable} or \trm\re ($-f$ is \trm{co-r.e.}) if its
subgraph $\{(x,t):t<f(x)\}$ is r.e., i.e. a union of an \re set of open balls.
$X^+$ means $X\cap\{x{\ge}0\}$. \trm {Elementary} ($f{\in}\Es$) are functions
$f:\W\to\Q$ depending on a finite number of digits; $\one\in\Es$ is their
unity: $\one(\a)=1$. $\tld E$~is the set of all supremums of subsets of $E$.
$f{\uparrow}$ for $f:\W\to\R$, denotes $\sup\{g:f>g\in\Es\}$.
\trm {Majorant} is an \re function largest, up to constant factors, among
\re functions in its class.\\ ${\lea}f$, ${\gea}f$, ${\eqa}f$, and ${\lel}f$,
${\gel} f$, ${\sim}f$ denote ${\le}f{+}O(1)$, ${\ge}f{-}O(1)$, ${=}f{\pm}O(1)$,
and ${\le}f{+}O(\|f{+}1\|)$,\\ ${\ge}f{-}O(\|f{+}1\|)$, ${=}f{\pm}O(\|f{+}1\|)$,
respectively. $[A]\edf1$ if statement $A$ holds, else $[A]\edf0$.
When unambiguous, I identify objects in clear correspondence: \eg prefixes with
their codes or their sets of extensions, sets with their characteristic
functions, etc.
\subsection {Integers: Complexity, Randomness, Rarity}
Let us define Kolmogorov \trm {complexity} $\K(x)$ as $\|\m(x)\|$ where
$\m:\N\to\R$ is the \trm {universal distribution}, \ie a majorant \re function
with $\sum_x\m(x){\le}1$. It was introduced in \cite{ZL}, and noted in \cite
{L73,L74,g74} to be a modification of the least length of binary programs for
$x$ defined in \cite {K65}. The modification restricts the domain $D$ of the
universal algorithm $u$ to be prefixless. While technically different, {\m}
relies on intuition similar to that of \cite {Sol}. The proof of the existence
of a majorant function was a direct modification of \cite {Sol, K65} proofs
which have been a keystone of the informational complexity theory.
For $x{\in}\N,y{\in}\N$ or $y{\in}\W$, similarly, $\m(\cdot|\cdot)$ is a
majorant \re real function with $\sum_x\m(x|y){\le}1$; $\K(x|y)\edf\|\m(x|y)\|$
($=$ the least length of prefixless programs transforming $y$ into $x$).
\cite {K65} considers \trm {rarity} $\d(x)\edf\|x\|{-}\K(x)$ of uniformly
distributed $x{\in}\{0,1\}^n$.\\ Our modified {\K} allows extending this to
other measures $\mu$ on~$\N$. A $\mu$-test is $f:\N\to\R$ with mean
$\mu(f){\le}1$ (and, thus, small values $f(x)$ on randomly chosen~$x$). For
computable $\mu$, a majorant \re test is $\tb(x)\edf\m(x)/\mu(x)$. This
suggests defining $\d_\mu(x)$ as $\|\ceil{\tb(x)}\|\eqa \|\mu(x)\|-\K(x)$.
\subsection {Integers: Information}
In particular, $x{=}(a,b)$ distributed with $\mu{=}\m\otimes\m$, is a pair
of two independent, but otherwise completely generic, finite objects.
Then, $\I(a:b)\edf\d_{\m\otimes\m}((a,b)){\eqa}\K(a){+}\K(b){-}\K (a,b)$ measures
their \trm {dependence} or \trm {mutual information}. It was shown
(see \cite{ZL}) by Kolmogorov and Levin to be close (within
${\pm}O(\log\K(a,b))$) to the expression $\K(a){-}\K(a|b)$ of \cite{K65}.
Unlike\\ this earlier expression (see \cite {g74}), our {\I} is symmetric and
monotone: $\I(a:b)\lea\I((a,x):b)$ (which will allow extending {\I} to $\W$);
it equals $\eqa\K(a)-\K(a|\ov b)$, where by $\ov b$ we will denote $(b,\K(b))$.
\\ (The $\I_z$ variation of $\I$ with all algorithms accessing oracle $z$,
works similarly.)\\ $\I$ satisfies the following Independence Conservation
Inequalities \cite{L74,L84}:\\ For any computable transformation $A$ and measure
$\mu$, and some family $t_{a,b}$ of $\mu$-tests
\[(1)\ \I(A(a):b)\lea \I(a:b); \hspace{4pc} (2)\ \I((a,w):b)\lea
\I(a:b)+\log t_{a,b}(w).
\]
(The $O(1)$ error terms reflect the constant complexities of $A,\mu$.)
So, independence of $a$ from $b$ is preserved in random processes,
in deterministic computations, their combinations, etc. These
inequalities are not obvious (and false for the original 1965 expression
$\I(a:b){=}\K(a){-}\K(a/b)$~) even with $A$, say, simply cutting off
half of $a$. An unexpected aspect of $\I$ is that $x$ contains all
information about $k{=}\K(x)$, $\I(x:k)\eqa\K(k)$, despite
$\K(k|x)$ being ${\sim}\|k\|$, or ${\sim}\log\|x\|$ in the worst case
\cite{g74}. One can view this as an "Occam Razor'' effect: with no
initial information about it, $x$ is as hard to obtain as its simplest
($k$-bit) description.
\subsection {Reals: Measures and Rarity}\label{ML}
\p {A measure} on $\W$ is a function $\mu(x){=}\mu(x0){+}\mu
(x1)$, for $x{\in}\St$. Its mean $\mu(f)$ is a functional on \Es, linear:
$\mu(cf{+}g){=}c\mu(f){+}\mu(g)$ and \trm {normal:} $\mu(\pm\one){=}\pm1$,
$\mu(|f|)\ge0$. It extends to other functions, as usual.
An example is $\l(x\W)\edf 2^{-\|x\|}$ (or $\l(x)$ for short).
I use $\mu_{(\a)}(A)$ to treat the expression $A$ as a
function of $\a$, taking other variables as parameters.
$\mu$-\trm{tests} are functions $f\in\Ess$, $\mu(f){\le}1$; computable
$\mu$ have \trm {universal} (\ie majorant {\em r.e.}) tests $\T_\mu(\a)
{=}\sum_i\m(\a_{[i]})/\mu(\a_{[i]})$, called \trm {Martin-L\"of tests.}\footnote
{The condition $\mu(\T_\mu){\le}1$, slightly stronger (in log scale)
than the original one of \cite {ML}, was\\ required in \cite{L76}
in order to satisfy conservation of randomness. Both types of tests
diverge simultaneously.\\ \cite {Schn73} (for divergence of $\T_\l$),
\cite {L73}, \cite {g80} characterized the tests in complexity terms.}
Indeed, let $t$ be an \re $\mu$-test, and $S_k$ be an \re family of
prefixless subsets of $\St$ such that $\cup_{x\in S_k}x\W=\{\a:t
(\a){>}2^{k+1}\}$. Then $t(\a)=\Theta(\sum_{k,x{\in}S_k}(2^k[\a{\in}x\W]))
=\Theta(\sup_{k,x{\in}S_k}(2^k[\a{\in}x\W]))$. Now, $\sum_{k,x{\in}S_k}
(2^k\mu(x)) <\mu(t)\le1$, so $2^k\mu(x){=}O(\m(x))$ for $x{\in}S_k$
and $t(\a){=}O(\sup_{k,x{\in}S_k}([\a{\in}x\W]\m(x)/\mu(x))){=}
O(\sup_i(\m(\a_{[i]})/\mu(\a_{[i]})))$.
\trm{Martin-L\"of random} are $\a$ with finite \trm{rarity}
$\d_\mu(\a)\edf\|\ceil {\T_\mu(\a)}\|\eqa\sup_i(\|\mu(\a_{[i]})\|-\K(\a_{[i]}))$
and we also use $\d_\mu(\a|x)\edf\sup_i(\|\mu(\a_{[i]})\|-\K(\a_{[i]}|x))$.
\p {Continuous transformations} $A:\W{\to}\W$ induce normal linear
operators $A^*:f{\mapsto}g$ over $\Es$, where $g(\w){=}f(A(\w))$. So obtained,
$A^*$ are \trm {deterministic}: $A^*(\min\{f,f'\})=\min\{A^*(f),A^*(f')\}$.
Operators that are not, correspond to probabilistic transformations (their
inclusion is the benefit of the dual representation), and $g(\w)$ is then the
expected value of $f(A(\w))$. Such $A$ also induce $A^{**}$ transforming input
distributions $\mu$ to output distributions $\ph=A^{**}(\mu):\ph(f)=\mu(A^*(f))$.
I treat $A,A^*,A^{**}$ as one function $A$ acting as $A^*$, or $A^{**}$ on the
respective (disjoint) domains. Same for partial transformations below and their
concave duals. I also identify $\w{\in}\W$ with measures $f\mapsto f(\w)$.
\section {Partial Operators, Semimeasures, Complexity of Prefixes}
Not all algorithms are total: narrowing down the output to a single
sequence may go slowly and fail (due to divergence or missing
information in the input), leaving a compact set of eligible results:
\BD\label{op}\BE \item
{\em Partial} continuous transformations \trm {(PCT)} are compact subsets
$A\subset\W{\times}\W$ with $A(\a)\edf\{\b:(\a,\b){\in} A\}\ne\emptyset$.
When not confusing I identify singletons $\{\b\}$ with $\b{\in}\W$.\\
\trm{Computable} PCT are r.e., \ie enumerate the open complement of $A$;
\item a PCT $A$ is \trm{clopen} if co-images $A^{-1}(s)=\{\a: A(\a)\subset s\}$
of all clopen $s\subset\W$ are clopen.\\ $A$ is $t$-clopen
if $A^{-1}(x\W)$ depend only on $\a_{[t(x_{[i]})]}$ for some $i$.
\item\trm {Dual} of PCT $A$ is the operator $A^*:\Es\to\Ess$,
where $A^*(f)=g:\a\mapsto\min_{\b{\in}A(\a)}f(\b)$. \EE\ED
An important example is a \trm {universal} algorithm $U$.
It enumerates all algorithms $A_i$ with a prefixless set $P$ of
indexes $i$ and sets $(i\a,\b)\in U$ iff $(\a,\b){\in}A_i,i{\in}P$.
\BR\label{cmp} Composing PCT with linear operators produces normal concave
operators, all of them by Hahn–Banach theorem. Indeed, each such $C(f)$ is a
composition $A(R(B(f)))$: Here a PCT $A(\a)$ relates each $\a$ to the binary
encodings $\{\mu\}$ of measures $\mu\ge C(\a)$; $R$ transforms $\{\mu\}$ into a
distribution $\{\mu\}\otimes\l$; and $B(\{\mu\}, \b)$ relates $\l$-distributed
$\b$ to $\mu$-distributed $\g$ with $\mu[0,\g)\le\b\le\mu[0, \g]$.\ER
Normal concave operators transform measures into \trm {semimeasures}:
\BD\label{sm}\BE\item A \trm {semimeasure} $\mu:\Es{\to}\R$ is
a normal ($\mu(\pm\one){=}{\pm}1,\,\mu(|f|){\ge}0$) functional\\
that is concave: $\mu(cf{+}g)\ge c\mu(f){+}\mu(g),\,c\in\R^+$,
\eg $\mu(x)\ge\mu(x0){+}\mu(x1)$, for $x\in\St$.\\ $\mu$ extends to
$f{\in}{-}\Ess$ as $\inf\{\mu(g):f\le g{\in}\Es\}$, and to other functions
as $\sup\{\mu(g): f\ge g{\in}{-}\Ess\}$, as is usual for inner measures.
$\mu$ is \trm {deterministic} if $\mu(\min \{f,g\})=\min\{\mu(f),\mu(g)\}$.
\item Normal ($A(\pm\one)=\pm\one$, $A(|f|)\ge0$) concave operators
$A:\Es\to\tld\Es$ transform input points $\a$ and distributions $\ph$ (measures
or semimeasures) into their output distributions $A(\ph):f{\mapsto}\ph(A(f))$.
Operators $A$ are deterministic if semimeasures $A(\a)$ are.\\
\trm {Regular} are semimeasures $A(\l)$ for deterministic \re $A$;
$t$-regular for a $t$-clopen $A$. \EE\ED
\BP\label{id}\BE\itemsep0pt\item\label{id1} Each deterministic $\mu$
is $\mu(f)=\min_{\w\in S}f(\w)$ for some compact $S\subset\W$. \item\label{id2}
Dual of PCT are those and only those operators that are normal, concave,
and deterministic.
\item\label{id3} Each $f{\in}\Es$ has a unique form $f{=}\sum r_if_i$
with distinct boolean $f_i{\ge}f_{i+1},f_0{=}\one$, $r_i{>}0$ for $i{>}0$.\\
Then $\un\mu(f)\edf\sum_i r_i\mu(f_i)$. $\mu{=}\un\mu$
if $\mu$ is regular. All \re measures are regular.
\item\label{id4} Each \re semimeasure $\mu$ has a regular
\re $\mu'{\le}\mu$ with $\mu'(x){=}\mu(x)$ for all $x\in\St$.\\
$\mu'$ is $t$-regular for a computable $t$ if $\mu(x)$ have $<t(x)$ bits.\EE\EP
\BPR \ref{id1}: Note, $p(\b)\edf\inf_{g:\mu(g){\ge}1}|g(\b)|\in\{0,1\}$.
Indeed, if $\mu(f){-}f(\b)=t{>}0$ and $g=(f{-}f(\b)\one)/t$ then $g(\b){=}0$,
$\mu(g){\ge}1$. Then $S$ is $\{\b:p(\b){=}1\}$. \ref{id2}: $\mu{=}A(\a)$ are
deterministic, so $\mu(f){=}\min_{\b\in S}f(\b)$. \ref{id3} is since regular
$\mu$ are averages of deterministic ones. \ref{id4} is by Theorem 3.2 of
\cite{ZL}.\EPR
\subsection {Complexity: General Case}
\BP\label{um} There exists a \trm {universal}, \ie majorant (on $\Es^+$) \re
semimeasure {\M}. The values $\M(x)$ can have $\K(x)$ bits. (Thus $t$-clopen
PCT can generate $\M$ for any computable $t(x){>}\K(x)$).\footnote
{For $t(x)\sim\|x\|$ shown in \cite{L71}, Th.13;
also mentioned in \cite {ZL}, Prp.3.2.}\EP
\BPR For an \re family $\mu_i$ of all \re semimeasures, take $\M(x)=
\sum_i\mu_i(x)/2i^2$. $\M(x)$ can be rounded-up to $\K(x)$ bits after adding
$\sum_{y\ne\{\}}\m(xy)$ (to keep $\M(x)\ge\M(x0)+\M(x1)$).\EPR
As in \cite {ZL}, $\KM(x)\edf\|\M(x\W)\|$.
Same for $\M_\a$, \re w.r.t. $\a$ and $\KM(x|\a)\edf\|\M_\a(x\W)\|$.
$\K(x|y)$, $\KM(x)$ are examples of the many types of complexity measures on
$\St$.\\ \cite{L76b} gives the general construction of Kolmogorov-like
complexities $\K_v$. I summarize it here.
$\K_v$ are associated with classes $v$ of functions $m{:}\;\St{\to}[0,1]$, in
linear scale, and their logarithmic scale projections $\ov v\edf\{K=\|m\|:
m{\in}v\}$. Thus, $\K(x|y)$ is $\K_v$ for $v=\{m:\sup_y\sum_xm(x|y)\le1\}$.
These $v$ are closed-down, weakly compact, and decidable on tables with finite
support. $\ov v$ will have a minimal, up to $\eqa$, co-\re function $\K_v$.
This justifies the logarithmic scale where the values of $\K_v$ are well
defined up to $O(1)$ adjacent integers. (Though linear scale is often clearer
analytically.)
$\K_v$ minimality requires $\min\{K',K''\}{+} O(1)\in\ov v$ for any $K',K''$
in $\ov v$. In the linear scale of $m$ this comes
to $(m'{+}m'')/c\in v$ for some $c{=}O(1)$. I tightened this to convexity
with $c{=}2$; this changes $K$ in $\ov v$ by just $\Theta(1)$ factors:
a matter of choosing bits as units of complexity.
Similarly to Proposition~\ref{um}, this condition suffices for $\ov v$ to have
a minimal, up to $\eqa$, co-\re $\K_v$. Each such $\K_v{<}\infty$ has a
computable lower bound $B_v(x)=\min_{K{\in}\ov v}K(x)$, largest up to $\eqa$,
among \re bounds. And $\K_v{-}B_v$, too, is such a $\K_{v'}$; I call $v'$ \trm
{normal}, as $B_{v'}{=}0$. Let $\Es_1{=}\Es^+\cap\{f{:}\:\max_\a f(\a){=}1\}$.
$\KM(f){=}\|\M(f)\|,\,f{\in}\Es_1$ is a normal complexity measure and all
others are its special cases:
\BP For each normal $v$ a computable representation $t_x\in\Es_1$
for $x\in\St$ exists such that $\K_v(x)\eqa\KM(t_x)\lea\K(x)$.\EP
\BPR $\K_v(x){\lea}\K(x)$ follows from normality ($B_v{\eqa}0$) and convexity
of $v$. Thus $\m_v(x)$ needs $\lea\K(x)\lea2\|x\|$ bits. Let $m'$ be $m{\in}v$
so rounded-down. For $m{\in}v$, let $m_x$ be a prefixless code of $(x,m'(x))$,
and $m_{[x]}$ be $m_1m_2\ldots m_x$. Then $t_x(\a)\edf m'(x)$ if
$\a{=}m_{[x]}\b,m{\in}v$; otherwise $t_x(\a)\edf0$.
The measure concentrated in a single $\a$ has some $m{\in}v$ for which it maps
each $t_x$ to $m'(x)$.\\ Other measures $\mu$ also have $\tau_\mu:x\mapsto\mu
(t_x)$ in $v$ by convexity of $v$.\\ As $v$ is closed down, $\tau_\M\in v$, too,
and so, $\tau_\M=O(\m_v)$. Conversely, some measure $\a$ has $\tau_\a{=}\m_v$.
As $\m_v$ is r.e., the minimal semimeasure $\mu$ with $\tau_\mu\ge\m_v$ is
r.e., too, and so, $\m_v\le\tau_\mu=O(\tau_\M)$.\EPR
\section {Complete Sequences} \label{cmpl}
\cite{L76a} calls \trm {complete} sequences $\a$ that are $\mu$-random for a
computable $\mu$. This class is closed under all total recursive operators.
Here I use this term \trm {complete} also for $\a'$ Turing-equivalent to such
\a. This is identical to $\a'$ being either recursive or Turing-equivalent to a
$\l$-random sequence.
By \cite {ku,g86,BL}, each $\a{\in}\W$ is w.t.t.-reducible to a \l-random~\w.
Indeed, for $P(x,\a)=x\a$, let measure $\r$ be $\l$-integral of $\T_\l$:
$\r=P(\m\otimes\l)$. Let $R=\{\a:\T_\l(\a)\le c\}$ for a convenient constant
$c$. When $A(\l)$ generates $\M(x)$, the co-images of all prefixes intersect
$R$. (Otherwise $A(\r)$ would exceed $\M=A(\l)$.) But for clopen $A$ (see
Prp.~\ref{um}), co-image of any $\a{\in}\W$ is the intersection of (non-empty
in $R$) clopen co-images of its prefixes $\a_n$, so intersects $R$, too.
Yet partial algorithms can generate incomplete sequences with positive
probability: \cite {Vyugin}.
I extend $\K(\b|\a)$ to $\a,\b\in\W$ using a universal PCT $U(p,\a)$ that runs
on $\a$ a program $p$ given on a separate tape; $\a_p$ combines bits of $p,\a$
in order read by $U$. $p$ must be prefixless: $U$ diverges and $\a_p$ is
undefined unless $U$ detects the end of $p$ and does not try to move beyond
its end of tape.
\BD Here $\a,\b\in\W$. $\K(\b|\a)\,\edf\,\min_p\{\|p\|:U(p,\a){=}\b\}$.\\
The \trm {codeset} $R_\a$ for $\a$ is $\{\b:U(\b){=}\a,\,\d_\l(\b){<}c\}$
where $c$ is a constant such that\\ the \trm {incompleteness}\footnote
{For some applications of $\Ks$ its lower bound $\|\M_\a(R_\a)\|$ may suffice.}
$\Ks(\a)\edf\min_{\b\in R_\a}\K(\b|\a)$ of
any $\a$ is $\lel\|\d_\l(\a)\|$\footnote
{By finding $p$ to replace a prefix $q{=}U(p)$ where $\|q\|{-}\|p\|$
is the rarity. \newcounter{cmpr}\setcounter{cmpr}\thefootnote}.\\
\trm {Tight complexity} $\Ki(x|\a)$ is $\|\mf(x|\a)\|$ where $x{\in}\N$,
$\widehat\m_x(\a)\edf\min_{\b\in R_\a}\m(x|\b)$,
$\mf(x|\a)\edf\widehat\m_x{\uparrow}(\a)$.\ED
These concepts satisfy many properties similar to
those given (for integers) in \cite {g74,L74}:
\BP\label{kfs}\BE\itemsep0pt \item\label{kf1} $\K(\b|\a)\sim\KM(\b|\a)$.
\item\label{kf2} $\d_\l(\b_q)\eqa\d_\l(\b)+\|q\|{-}\K(q|\b,\d_\l(\b))$.
\item\label{kf3} $\Ks(\a)\eqa\min_{\b}\{\K(\a|\b){+}\K(\b|\a){+}\d_\l(\b)\}$.
\item\label{kf4} $\Ki(x|\a)\eqa\Ki(\ov x|\a)$. (Recall: $\ov x$ is $(x,\K(x))$.)
\item\label{kf5} $\If(\a:x)\edf\K(x)-\Ki(x|\a)\lea\If(\a:(x,y))$.
\item\label{kf6} $\If(\a:x)\eqa(\min_{\b\in R_\a}\d_\l(\b|\ov x)){\uparrow}
\eqa(\min_{\b\in U^{-1}(\a)}\d_\l(\b|\ov x)){\uparrow}$.\EE\EP
\BPR\ref{kf1}.
Let $k{=}\KM(\b|\a)$, $s_{k,\a}\edf\{x0,x1:\KM(x0\W|\a){<}k,\KM(x1\W|\a){<}k\}$,
so, $|s_{k,\a}|<2^k$.\\ Let $x$ be the longest prefix of $\b$ in $s_{k,\a}$.
Then $\K(x|\a,k)\lea k$, and $\b$ can be computed from $x,k,\a$.
\ref{kf2}. "$\d_\l(\b_q)\gea$'' is by $t_{\b_q}\edf\T_\l(\b)2^{\|q\|}
\m(q|\b,\d_\l(\b))$ being \re with $\l_{(\b_q)}(t_{\b_q})\le1$. For "$\lea$''
take a distribution $\mu_{\b,d}(q)\edf\T_\l(\b_q)/2^{\|q\|+d}$ enumerated
for each $\b,d$ only while $\dl_\b\edf\|\sum_q 2^d\mu_{\b,d}(q)\|{\le}d$;\\
so enumeration of $\mu_{\b,\dl_\b}$ is not stopped.
Now, $\dl_\b\eqa\d_\l(\b)$ since $\l_{(\b)}(2^{\dl_\b}){\le}1$.
Also, $\sum_q\mu_{\b,d}(q){=}O(1)$, so $\mu_{\b,d}(q){=}O(\m(q|\b,d))$. Thus,
$\d_\l(\b)+\|q\|-\d_\l(\b_q)\gea\|\mu_{\b,\dl_\b}(q)\|\gea\K(q|\b,\d_\l(\b))$.
\ref{kf3}. Take $p,q,\b{=}U(p,\a)$ with $U(q,\b){=}\a$, $\Ks(\a){\eqa}
\|p\|{+}\|q\|{+}\d_\l(\b)$.\\ Then $\d_\l(\b)\eqa0$, $\K(q|\b)\eqa\|q\|$,
else $\b$ or $q$ could be shrunk decreasing $\Ks(\a)$.\\
Then $\d_\l(\b_q)\eqa0$ by \ref{kf2}, and the claim follows
by appending $q$ to $p$ to map $\a\mapsto(q,\b){\mapsto\b_q}$.
\ref{kf4}. Let $\b{=}v\w,\,\d_\l(\b){\eqa}0,\,\|p\|{=}\K(x|\b)$
(and so, ${\eqa}\K(p|\b)$), and $U(p,\b){=}x$ reads only $p,v$, so,\\
$\K(p,v){\lea}\|pv\|$. Then $\|pv\|{-}\K(p,v)\lea\d_\l(v_p){\eqa}0$
by \ref{kf2}. So, $\K(x){+} \K((p,v)|\ov x)\eqa\K(p,v)\eqa\|pv\|$.\\
Thus, finding $i,j$ with $\K(x){<}i,\K((p,v)|x,i){<}j$,
$i{+}j{\lea}\|pv\|$ computes $\K(x)\eqa i$ from $p,v$.
\ref{kf5}. By \ref{kf4} and $\K(\ov x|\ov{(x,y)}){\eqa}0$, we can replace $x$
with $\ov x$. Let $\d_\l(\b){\eqa}0$.\\ Then $\K(\ov x)-\K(\ov x|\b)-\K(\ov x,y)
+\K((\ov x,y)|\b)\eqa\K(y|\b,\ov x,\K(\ov x|\b)){-}\K(y|\ov x)\lea0$.
\ref{kf6}. For $\b{\in}R_\a$, $\K(x)-\K(x|\b)\gea\d_\l(\b|x)$, \ie
$\m(x)\T_\l(\b|x)=O(\m(x|\b))$.\\ Indeed, the \re $\sum_x\m(x)\T_\l(\b|x)$
is $O(\T_\l(\b)){=}O(1)$ since $\l_{(\b)}(\sum_x\m(x)\T_\l(\b|x))=\\
\sum_x\m(x)\l_{(\b)}(\T_\l(\b|x))\le\sum_x\m(x){\le}1$.
Also for all $\b$, $x{=}U(p)$ with $\K(x){=}\|p\|{\eqa}\K(p)$, the \re\\
$\m(U(p)|\b)p$ is $O(\T_\l(\b|p))$ since $\l_{(\b)}\m(x|\b)/\m(x){=}O(1)$.
So and $\K(x)-\K(x|\b)\lea\d_\l(\b|\ov x)$.\\
And any $\b\in U^{-1}(\a)$ can be compressed\footnotemark[\thecmpr]
to $\b'{\in}R_\a$ with $\d_\l(\b'|\ov x)\lea\d_\l(\b|\ov x)$.\EPR
\section {Rarity}
\subsection {Non-algorithmic Distributions}
\cite {L73} considered a definition of rarity $\T_\mu(\a)$ for arbitrary
measures $\mu$ where $\T_\mu$ is \re only relative to $\mu$ used as an oracle.
This concept gives interesting results on testing for co-\re classes of
measures such as, \eg Bernoulli measures. Yet, for individual $\mu$ it is
peculiar in its strong dependence on insignificant digits of $\mu$ that have
little effect on probabilities. \cite {L76,g80} confronted this aspect by
restrictions making $1/\T_\mu(\a)$ monotone, homogeneous, and concave in
$\mu$.\footnote
{The Definition in \cite {L76} has a typo: "$Q(f)$'' meant to be "$Q(g)$''.
Also, in English version "concave relative to $P$''
would be clearer as "for any measure $Q$ concave over $P$''.
So, its $\T_\mu(\a)$ is $\sup_{f,g\in\Es}(t(f|g)f(\a)/\mu(g))$,
for\\ a $t$ majorant among \re functions that keep $\T_\mu(\mu)\le1$
for all measures $\mu$, where $\T_\mu(\ph)\edf\ph_{(\a)}(\T_\mu(\a))$.\par
Restrictions on $t$ (\eg $t\subset\St{\times}\Es$, $\T_\mu(\a)\edf\sup_
{(f,g){\in}t}f(\a)/\mu(g)$) can reduce redundancy with no loss of generality.}
\cite {L84} used another construction for $\T_\mu(\a)$. It generates
$\mu$-tests by randomized algorithms and averages their values on $\a$. For
computable $\mu$ the tests' ${\le}1$-mean can be forced by the generating
algorithm, so the definition agrees with the standard one. But for other $\mu$
the ${\le}1$-mean needs to be imposed externally. \cite {L84} does this by just
replacing the tests of higher mean with $\one$ (thus tarnishing the purity of
the algorithmic generation aspect). That definition respects the conservation
inequalities, so for \re semimeasures it gives a lower bound for our
$\d_\mu(\a)$ below (by Prop.\ref{mx}).
\subsection{R.E. Semimeasures}
\p {Coarse Graining.} I use $\l$ as a typical continuous computable
measure on \W, though any of them can be equivalently used instead. Also, any
recursive tree of clopen subsets can serve in place of $\St$.
Restricting inputs $\w$ of a PCT $A$ to those with converging outputs (\ie a
singletons $A(\w)\in\W$) truncate the output semimeasure to a smaller {\em
linear} functional: a maximal measure $\mu^\Es\le\mu{=}A(\l)$. Yet, much
information is lost this way: \eg $\|\M^\Es(x)\|,x{\in}\St$ has no recursive in
$\|\M(x)\|$ upper bound. To keep information about generated prefixes, I will
require linearity of $\mu^E$ only on a subspace $E{\subset}\Es$. $E$ will play
a role of space of $\mu^E$-measurable functions. E.g., relaxing $A(\w)$
restriction from singletons to sets of radius ${\le}2^{-n}$, produces a
semimeasure linear on the subspace of $f$ with $f(\a)$ dependent only on
$\a_{[n]}$. Subspaces $E\subset\Es$ used below are generated by
subtrees\footnote
{If a non-binary tree is used instead of $\St$ then any
$x{\in}S$ must have either all its children in $S$ or none.}
$S{\subset}\St$, \ie are spaces of linear combinations of functions in $S$.
By \trm {$E$-measures} I call semimeasures linear on such $E$.
\BP\label{cg} Each semimeasure $\mu$, for each $E$,
has the largest (on $\Es^+$) $E$-measure $\mu^E\le\mu$.\EP
\BPR Let $X$ be the set of all measures $\ph$ which, for some
$F{\subset}E^+$ with $\sum_{f\in F}f>0$\\ and all $g\in\Es^+$, $g\le f\in F$,
have $\ph(g)\ge\mu(g)$. Then $\mu^E(f)=\inf_{\ph\in X}\ph(f)$.\EPR
Now, I will extend the concept of rarity $\T_\mu$, $\d{\edf}\|\ceil{\T}\|$
from computable measures $\mu$ to r.e. semimeasures. The idea is for
$\d_\mu(\a)$ to be bounded by $\d_\l(\w)$ if $\a{=}A(\w)$, $\mu{\ge}A(\l)$.
Coarse graining on a space rougher than the whole $\Es$, allows to define
rarity not only for $\a{\in}\W$ but also for its prefixes. For semimeasures,
rarity of extensions does not determine the rarity of a prefix.
$\T_\mu$ for a computable measure $\mu$ is a single \re function $\W\to\R^+$
with $\le1$ mean. It is obtained by averaging the \re family of all such
functions. This fails if $\mu$ is a semimeasure: its mean of sum can exceed the
sum of means. So, our extended $\T_\mu$ will be refined with a subspace
$E{\subset}\Es$ parameter.
\BD\label{d1} For an $E{\subset}\Es$ and a PCT $A$,
$t^E_A$ is $\sup\{f{\in}E:A(f)\le\T_\l\}$.\ED
\BP\label{uo} Each \re $\mu$, among all \re PCT $A$ with $A(\l)\le\mu$,
has a universal one $U_\mu$, \ie such that $t^E_{U_\mu}=O(t^E_A)$ for each $A$
and all $E$. $\mu(f)\le\l(2U_\mu(f))$ if $f\in\St$ or $\mu$ is regular.\EP
\BPR $U(i\w)\edf A_i(\w)$ for a prefixless enumeration $A_i$ of all such $A$.
\EPR
\BD\label{rm} $\T_\mu^E(\ph)$ for semimeasures $\ph$, \re $\mu$ is the
mean: $\ph^E(t^E_{2U_\mu})$ for $U_\mu$ defined in Prop.\ref{uo}.\ED
\BL\label{dM} (1) $\d_\mu^\Es\eqa\d_\mu$ for computable measures $\mu$.
(So, if $E=\Es$, we omit $E$ in $\d_\mu^E\edf\|\ceil{\T_\mu^E}\|$.)\\
(2) $\d_\mu^E(\mu){=}0$.\hspace{3pc}
(3) $\d_\M\eqa0$ for the universal semimeasure $\M$.\EL
\BPR (1) follows from \cite {ZL} Th.~3.1 and enumerability of $\T_\mu$.
(2) Let $A{=}U_\mu$. By Prop.\ref{uo}, $\mu^E(f)/2\le\l(A(f))$
for $f{\in}\St$, and thus for $f{\in}E^+$. Also any $f{<}t_A^E$ is
$<\sum_if_i$ where $f_i{\in}E^+$, $f_if_{j{\ne}i}=0$, and $A(f_i)\le\T_\l$.
Now, $\T_\mu^E(\mu)=\sup_{f{\in}\Es^+,f{<}t_A^E}\mu^E(f)/2$,\\ and $\mu^E(f)/2
\le\sum_i\mu^E(f_i)/2\le\l(\sum_i A(f_i))=\l(\sup_i A(f_i))\le\l(\T_\l)\le1$.
(3) By \cite{g86,ku}, an \re PCT $A$ exists such that any {\a} is $A(\w)$ with
$d_\l(\w){=}0$. Then $g{=}A(f)\le\T_\l$ means $g(\w){=}f(A(\w))=f(\a)\le\T_\l(\w)
\le2$. For a universal $\M$, $\d_\M\lea\d_{A(\l)}\eqa0$.\EPR
Let the semimeasure $\nu{=}\mu{\otimes}\ph$ on $\W^2$ be the minimum of
$\mu'{\otimes}\ph'$ over all measures $\mu'{\ge}\mu,\,\ph'{\ge}\ph$.
Then $\nu(h){=}\mu(f)\ph(g)$ for $h(\a,\b){=}f(\a)g(\b)$, and for all $h$, if
$\ph$ is a measure, $\nu(h){=}$ $\mu(\ph_{(\b)}(h(\a,\b)))$. Let $E\otimes\Es$
be the space generated by $\{f(\a)g(\b),\,g{\in}E,f{\in}\Es\}$.
Adding coin-flips preserves randomness:
\BL\label{cir} $\d_{\mu{\otimes}\l}^{E\otimes\Es}(\ph{\otimes}\l)\lea
\d_\mu^E(\ph)$ for all $\ph$, \re $\mu$, space $E{\subset}\Es$.\EL
\BPR Let $\phi\edf\ph{\otimes}\l$, $\nu\edf\mu{\otimes}\l$, $E'\edf
E{\otimes}\Es$, $A(\a,\b)\edf(U_\mu(\a),\b)$, $t\edf\T_\nu^{E'}(\phi)=
\phi^{E'}(t^{E'}_{U_\nu})$. Then\\ for some $c{\in}\Q^+$,
$t/c<\phi^{E'}(t^{E'}_{A})=\phi^{E'}(\sup H)$ where $H=\{h{\in}E':A(h)
{\le}\T_{\l^2}\}$. So $t/c<\phi^{E'} (\sup G)$ for a finite set
$G=\{f_i(\a)g_i(\b)\}\subset H$ with $\l(g_i){=}1$ and $f_if_{j\ne i}{=}0$,
thus $\sup G=\sum G$.\\ Now, $U_\mu(f_i)g_i<\T_{\l^2}$,
thus $U_\mu(f_i)<\l_{(\b)}(\T_{\l^2}(\a,\b))=O(\T_\l(\a))$.
Then, $t/c<\phi^{E'}(\sum_if_i g_i)=\sum_i\phi^{E'}(f_i g_i)=\sum_i\ph^E(f_i)=
\ph^E(\sum_if_i)=\ph^E(\sup_if_i)=O(\ph^E(t^E_{U_\mu}))=O(\T_\mu^E(\ph))$.\EPR
Let $A(E)$ be $\{f{\in}\Es:A(f){\in}\tld E{\subset}\Ess\}$.
Deterministic processing preserves randomness, too:
\BL\label{cid} $\d_{A(\mu)}^{A(E)}(A(\ph))\lea\d_\mu^E(\ph)$ for
each \re PCT $A$, all $\ph$, \re $\mu$, space $E{\subset}\Es$.\EL
\BPR Let $E'\edf A(E)$, $\phi\edf A(\ph)^{E'}\le A(\ph^E)$,
$A_\mu(f)\edf U_\mu(A(f))$. So, $t\edf\T_{A(\mu)}^{E'}(A(\ph))=\\
\phi(t^{E'}_{U_{A(\mu)}})<c\,\phi(t^{E'}_{A_\mu})<c\,\phi(\sup F)$
for $F\edf\{f{\in}E'^+:U_\mu(A(f))\le\T_\l\}$ and some $c\in\Q^+$.\\
Then $t<c\phi (\sup G)$ for a finite set $G\subset F$ that can
be made disjoint, \ie $gg'=0$\\ for $g{\ne}g'$ in $G$ (and thus
$A(g)A(g')=0$ as $A$ is deterministic), so $\sup G=\sum G$.\\
Now, $U_\mu(h){\le}\T_\l$ for $h\edf\sup\{A(f):f{\in}F\}{\in}\tld E^+$,
so $h\le t^E_{U_\mu}$. Then $t/c<\phi(\sup G)= \phi(\sum G)=\\
\sum_{g{\in G}}\phi(g)\le \sum_{g{\in G}}\ph^E(A(g))= \ph^E(\sum_{g{\in G}}A(g))=
\ph^E(\sup_{g{\in G}}A(g)) \le \ph^E(h) \le 2\T_\mu^E(\ph)$. \EPR
By the remark~\ref{cmp}, Lemmas~\ref{cir}, \ref{cid} imply the following
theorem:
\BT[Randomness Conservation]\label{t1} The test $\d$ satisfies
$\d_{A(\mu)}^{A(E)}(A(\ph))\lea \d_\mu^E(\ph)$\\ for each normal concave
\re operator $A$, all $\ph$, \re $\mu$, space $E{\subset}\Es$.\ET
These tests $\d_\mu^E$ are the strongest (largest)
extensions of Martin-L\"of tests for computable $\mu$:
\BP $\T_\mu^E(\w)$ is majorant among extensions $\tau_\mu {\in}\tld E^+$
of Martin-L\"of test $\T_\l=\tau_\l$\\ that are non-increasing
on $\mu$ and obey Lemma~\ref{cid} for $\|\ceil\tau\|$
with $\tau^E_\mu(\ph)\edf\ph^E(\tau_\mu)$.\label{mx}\EP
\BPR With $A{\edf}U^*_\mu$, $A(\tau_\mu){\le}A(\tau_{A(\l)})$ and Lemma~\ref
{cid} for $\|\ceil\tau\|$ gives $A(\tau_{A(\l)})(\w)=\tau_{A(\l)}(A^*(\w))\le
c\,\tau_\l(\w){=}c\T_\l(\w)$ for some $c\,{\in}\Q^+$. If $\tau_\mu{>}2c\,f{\in}
E^+$ then $2c\,A(f){<}A(\tau_\mu){\le}c\T_\l$, so $\T_\mu^E{>}f$ as defined.\EPR
\section {Information and its Bounds}
Now, like for the integer case, mutual information $\I(\a:\b)$ can be
defined as the deficiency of independence, \ie rarity for the distribution
where $\a,\b$ are assumed each universally distributed (a vacuous
assumption, see \eg Lemma~\ref{dM}(3)) but independent of each other:
\[\I(\a:\b)\edf\d_{\M\otimes\M}((\a,\b)).\]
Its conservation inequalities are just special cases of Theorem~\ref{t1}
and supply $\I(\a:\b)$ with lower bounds $\I(A(\a):B(\b))$
for various operators $A,B$. In particular transforming $\a,\b$ into
distributions $\m(\cdot|\a),\m(\cdot|\b)$, gives $\I(\a:\b)\gea\i(\a:\b)\edf
\|\ceil{\sum_{x,y\in\N}\m(x|\a)\m(y|\b)2^{\I(x:y)}}\|$.\footnote
{This $\i$ was used as the definition of information in \cite {L74}.}
Same for $\If(\a:\b)\edf\|\ceil{\sum_{z\in\N}
\mf(z|\a)\mf(z|\b)/\m(z)}\|\gea\i(\a:\b)$.\footnote
{$\If\gea\i$ since for $z{=}(x,y)$, by Prop.\ref{kfs}.\ref{kf4},
$\Ki(z|\a)\lea\Ki(\ov y|\a){+}\K(x|\ov y)\eqa
\Ki(y|\a){+}\K(x|\ov y)\lea\K(y|\a){+}\K(x,y){-}\K(y)$.} These bounds also
satisfy the conservation inequalities, and agree with $\I(\a:\b)$ for $\a,
\b\in\N$. While $\I$ is the largest such extension from $\N$, $\i$ is the
smallest one. Interestingly, not only for integers, but also for all complete
sequences this simple bound $\i$ is tight, as is an even simpler one
$\i'(\a:\b)\edf\sup_{x\in\N}(\K(x){-}\K(x|\a){-}\K(x|\b))\lea\i(\a{:}\b)$:
\BP\label{cmpl-i} For $\a,\b\,{\in}\W,\,b\,{\in}\N$:
(1) $\I(\a:b)\eqa\K(b){-}\Ki(b|\a)$ (follows from Prop.\ref{kfs}.\ref{kf6});\\
(2) $\I(\a:\b)\lea(\min_{\a'\in R_\a,\b'\in R_\b}\i'(\a':\b')){\uparrow}
\lea\i'(\a:\b)+\Ks(\a){+}\Ks(\b)$.\EP
In particular, this can be used for $\a$ being the Halting Problem sequence
(which is complete, being Turing-equivalent to any random \re real,
such as, \eg one constructed in sec.~4.4 of \cite {ZL}).
\BPR We can replace $\a,\b$ with $\a'{\in}R_\a,\b'{\in}R_\b$.
Let~$h_n\edf(\a_{[n]},\b_{[n]})$.\\ $\l^2\edf\l{\otimes}\l\,{=}\,O(\M^2)$,
so $\I(\a{:}\b){\lea}\d_{\l^2}((\a,\b)){\eqa}\|\ceil{\sup_n4^n\m(h_n)}\|
\eqa\sup_n(\K(h_n)-2(\K(h_n){-}n))$.\\ Also $t\edf\sum_{n,v}2^n\m((\a_n,v))=
\Theta(\T_\l(\a))$, so $2^n\m((\a_n,v))/t=O(\m((n,v)|\a,\|t\|))$, and\\
$\K(h_n|\a)-(\K(h_n)-n)\lel\|t\|\eqa0$. Thus $\K(h_n|\a)\lea\K(h_n)-n$ and
$\K(h_n|\b)\lea\K(h_n)-n$.\\ Then $\I(\a:\b)\lea\sup_n(\K(h_n)-2(\K(h_n){-}n))
\lea\sup_n(\K(h_n)-\K(h_n|\a){-}\K(h_n|\b))\lea\i'(\a:\b)$.\EPR
\BP\label{M} Let $A\subset\W$. Then $\M^\Es(A)=0$ iff
$\exists\a\forall\b_{\in A}\I(\b:\a)=\infty$.\EP
\BPR "If'' is by Theorem\ref{t1}. Now, any $A$ with $\M^\Es(A){=}0$ has a
sequence $\a$ of clopen sets $\a_i\subset\W$ with shrinking $\M(\a_i)$, \ie
$\l(\{\g:\exists x\, U(\g)\subset x\W{\subset}\a_i\})<2^{-i}$, and s.t. each
$\b{\in}A$ is in infinitely many $\a_i$. Then, by Prop.\ref{kfs}.\ref{kf6},
$\If(\b:(i,\a_i))\gea(\min_{\g\in U^{-1}(\b)}\d_\l(\g|i,\a_i)){\uparrow}\gea i$
and so $\I(\b:\a)\gea\If(\b:\a)=\infty$.\EPR
\end{document} |
\betaegin{equation}gin{document}
\title{No-iteration of unknown quantum gates}
\alphauthor{Mehdi Soleimanifar}
\epsilonmail{[email protected]}
\alphauthor{Vahid Karimipour}
\epsilonmail{[email protected]}
\alphaffiliation{Department of Physics, Sharif University of Technology, Tehran, Iran}
\betaegin{equation}gin{abstract}
We propose a new no-go theorem by proving the impossibility of constructing a deterministic quantum circuit that iterates a unitary oracle by calling it only once. Different schemes are provided to bypass this result and to approximately realize the iteration. The optimal scheme is also studied. An interesting observation is that for large number of iterations, a trivial strategy like using the identity channel has the optimal performance, and preprocessing, postprocessing, or using resources like entanglement does not help at all. Intriguingly, the number of iterations, when being large enough, does not affect the performance of the proposed schemes.
\epsilonnd{abstract}
\pacs{03.67.-a, 03.67.Lx, 03.67.Ac}
\muaketitle
\section{Introduction}
\lambdaanglebel{sec:intro}
\indent
No-go theorems play a major role in quantum information science. The impossibility of perfect cloning of an unknown pure state, the \epsilonmph{no-cloning theorem}, is one of the striking features of quantum mechanics \cite{Wootters}. This no-go result is fundamental to key distribution \cite{Scarani}, quantum secret sharing \cite{Hillery}, and quantum error correction \cite{Bennett}. A similar no-go theorem is valid for cloning of an \epsilonmph{unknown quantum gate} from one to two copies \cite{unitary-cloning}; that is to say, given a set of distinct states $\betaigotimes_{i=1}^m \ket{\psi_i}$ and an unknown unitary channel $\muathcal{U}$, it is impossible to prepare $\betaigotimes_{i=1}^mU\ket{\psi_i}$ by a quantum circuit that uses $\muathcal{U}$ only once. This result has implications in cryptographic protocols where the secret is encoded in unitary transformations instead of quantum states, e.g., an alternative version of BB84 protocol where Alice uses two orthogonal bases of unitary transformations instead of states \cite{unitary-cloning}. \\
The no-cloning of states is about the impossibility of realizing a specific transformation of \epsilonmph{states}, while the no-cloning of gates is about a transformation of \epsilonmph{unitary channels}. Other examples of no-go theorems on transformations of quantum channels are: The impossibility of realizing the \epsilonmph{switch} circuit defined by $\muathcal{Z}(\muathcal{V},\muathcal{W})=\ket{0}\betara{0}\pr{\muathcal{VW}}+\ket{1}\betara{1}\pr{\muathcal{WV}}$, in which a pair of input unitary blackboxes $\muathcal{V}$ and $\muathcal{W}$ are connected in two different orders conditioned on the value of an input bit \cite{causal-order}. By generalizing the conventional quantum circuit model to bypass this no-go result, a computational advantage can be obtained \cite{computational-advantage}. Another example is the no-go theorem on controlling a unitary gate given as a blackbox discussed in \cite{Brukner,Friis,Bisio}, or failure of programming a quantum gate array $\muathcal{G}$ that deterministically implements the unitary operation $\muathcal{U}$ determined by the quantum program $\ket{P_U}$ or strictly speaking $\muathcal{G}(\ket{\psi}\pr{\ket{P_U}})=U\ket{\psi}\pr{\ket{P'_U}}$ \cite{no-programming}. \\
In this paper, we introduce and investigate a new no-go theorem on \epsilonmph{iterations of an unknown quantum gate}. The iteration of a unitary gate is widely used in quantum algorithms. Quantum search algorithms like Grover algorithm \cite{Grover} or quantum random walk search algorithm \cite{random-walk-search} are based on the repetition of a unitary oracle. They use iterations of the oracle to amplify the amplitude of a desired state in a superposition of states \cite{Amplification}. Quantum phase estimation \cite{phase-estimation} is another algorithm in which successive iterations of a unitary gate are used to generate states appropriate for an inverse quantum Fourier transform. These algorithms are bases of other quantum computations like order finding \cite{phase-estimation}, integer factorization and discrete logarithms \cite{Shor} or the collision problem \cite{collision}.\\
The question we try to answer is whether it is possible to avoid iterations of a unitary oracle using a deterministic quantum circuit. One possible scenario for doing this is that an apparatus called \epsilonmph{gate iterator} of the $n$'th order ($n\in \muathbb{N}/\{1\}$), denoted by $\muathsf{Iter_n}$, takes a unitary oracle $\muathcal{U}$, an arbitrary state $\ket{\psi}$ and the state of the rest of the world $\ket{0}$ as inputs, and by calling $\muathcal{U}$ once, gives $U^n\ket{\psi}$ as the output. The state $\ket{0}$ may also change to another state $\ket{0^{\prime}}$ at the end, see Fig. \ref{fig:ckt-1}. In a more general scenario, the input system could be mixed and the output state be entangled with the ancillary system.\\
\betaegin{equation}gin{figure}
\centering
\includegraphics[width=.3\textwidth]{0.eps}
\caption{In a possible scenario, the gate iterator apparatus, $\muathsf{Iter_n}$, takes $\muathcal{U}$, $\ket{\psi}$ and $\ket{0}$ as inputs and gives $U^n\ket{\psi}$ as the output.}
\lambdaanglebel{fig:ckt-1}
\epsilonnd{figure}
We prove that it is impossible to realize $\muathsf{Iter}_n$ which consists of a deterministic quantum circuit. We first consider the most general circuit for doing the iteration consists of a \epsilonmph{preprocessing} and a \epsilonmph{postprocessing} channel. We then show that such a procedure contradicts the linearity of quantum mechanics. \\
Although it is not possible to construct $\muathsf{Iter_n}$ perfectly, it is natural to ask for strategies that realize it in an approximate way. We propose such schemes and then, by using the notion of \epsilonmph{fidelity} as a figure of merit, we investigate their performance and how it scales with the number of iterations $n$, and the dimensionality of the state $d$. We also address the problem of finding the optimal iterator. We show that the optimal fidelity is the answer of a semidefinite programming. We solve this problem numerically for $d=2, 3$, see Fig. \ref{fig:data_plot_3}.\\
As we will see, the approximate realization of a gate iterator has interesting characteristics. We show that in all strategies, including the optimal, by increasing the dimension of the system $d$, the fidelity decreases. Intriguingly, the fidelity reaches a constant value by increasing $n$, so when we are allowed to query a unitary oracle only once, there is no difference in quality of high-order imperfect iterations of that. Another interesting observation is that when $n>d$, a trivial strategy like approximating $\muathcal{U}^n$ by $\muathcal{U}$ or even by the identity channel has the optimal performance, and preprocessing, postprocessing, or using resources like entanglement does not help at all. These results are depicted in Fig. \ref{fig:plot1}, \ref{fig:plot_data_2} and \ref{fig:data_plot_3}. \\
An anticipated result of our no-go theorem is that when the oracle is completely unknown, the only way to perform iterations of that, seems to be calling the oracle each time and give the state to that and do this repeatedly. For a large number of iterations, this makes the algorithm inefficient. In fact, one way of comparing the complexity of different algorithms is to count the necessary number of querying within the program \cite{querying}. \\
The rest of the paper is organized as follows: In the next section, notations and some basic definitions are presented, and a convenient figure of merit is introduced to measure how good a quantum circuit approximates a given unitary gate. In Sec. \ref{sec:no-repetition}, we prove the no-iteration theorem for an arbitrary order $n$. The fidelity and details of the \epsilonmph{random guess} and \epsilonmph{measure-and-prepare} strategies are discussed in Sec. \ref{sec:quantum circuit of a repeater} and \ref{sec:random-guess}. Then, two trivial but important methods are introduced in Sec. \ref{sec:idn strategies}, and the optimum fidelity is obtained numerically in Sec. \ref{sec:Optimal strategies}. Finally, we conclude the paper in Sec. \ref{sec:Conclusion}.
\section{Notations and conventions}\lambdaanglebel{Notations and conventions}
In this section, we gather some well-known facts which we will frequently use in the sequel. We denote the complex $d-$dimensional Hilbert space by ${\cal H}_d$, and the linear space of operators acting on it by $L({\cal H}_d)$ and the set of density matrices by $D({\cal H}_d)$. A basis for ${\cal H}_d$ is denoted by $\{|j\rangle: j=1,2,\deltaots,n\}$ and any linear operator $V$ in $L({\cal H}_d)$ is expanded as $V=\sum_{j,k}V_{jk}|j\rangle\lambdaangle k|$.
A correspondence between this operator and a vector $\kett{V} \in \muathcal{H}_d\pr{\muathcal{H}_d}$ can be established by defining
\betaegin{equation}gin{align}
\kett{V}:=\frac{1}{\sqrt{d}} \sum_{j,k} V_{jk}\ket{j}\ket{k},
\epsilonnd{align}
where $\kett{V}$ is called the \epsilonmph{vectorized form} of the operator $V$.
Therefore, the maximally entangled state $|\phi^+\rangle:=\frac{1}{\sqrt{d}}\sum_{j}|j\rangle |j\rangle$ is the vectorized form of the identity operator, i.e., $|\phi^+\rangle=\kett{\muathds{1}}$. The inner product between two operators $A$ and $B$ defined as $\Tr(A^\deltaagger B)$ can equally be written as the ordinary vector product of their vectorized form, that is $\Tr(A^{\deltaagger}B)=\betarakett{A}{B}$. Finally, we note that a vector $\kett{V}$ can be prepared by performing $V$ on the maximally entangled state $\kett{\muathds{1}}$:
\betaegin{equation}gin{align}\lambdaanglebel{eq:-8}
\kett{V}=V\pr{\muathds{1}}\ \kett{\muathds{1}}.
\epsilonnd{align}
\betaegin{equation}gin{comment}
Using Lemma \ref{Lemma:1} and replacing $M$ with $\kett{\muathds{1}}\betaraa{\muathds{1}}$, we get
\betaegin{equation}gin{align}
\int dV \kett{V}\betaraa{V}=\frac{\muathds{1}}{d^2},
\epsilonnd{align}
\epsilonnd{comment}
The Choi operator $R_{\muathcal{T}}$ associated with a quantum channel $\muathcal{T}:D(\muathcal{H}_d)\rightarrow D(\muathcal{K}_d)$ is defined on $\muathcal{K}_d\pr{\muathcal{H}_d}$ by
\betaegin{equation}gin{align}\lambdaanglebel{eq:3}
R_{\muathcal{T}}:=(\muathcal{T}\pr{\muathcal{I}})\ (\kett{\muathds{1}}\betaraa{\muathds{1}}),
\epsilonnd{align}
where $\muathcal{I}$ is the identity channel. Obviously, we have $R_{\muathcal{I}}:=\kett{\muathds{1}}\betaraa{\muathds{1}}$, that is to say, the Choi operator of the identity channel is the Bell state. \\
A unitary quantum channel (quantum gate) $\muathcal{U}$ is defined as
\betaegin{equation}gin{align}\lambdaanglebel{unitary channel}
\muathcal{U}(\rho):=U\rho U^{\deltaagger},
\epsilonnd{align}
that according to Eq. (\ref{eq:-8}), its Choi operator is the pure state $R_{\muathcal{U}}=\kett{U}\betaraa{U}$. \\
When we want to evaluate the performance of a process $\muathcal{E}$ that approximates a gate $\muathcal{U}$, we need to introduce a figure of merit. The fidelity between two channels $\muathcal{G}$ and $\muathcal{E}$ is defined to be the state fidelity between the Choi operators of these channels \cite{Raginsky}:
\betaegin{equation}gin{align}
\muathcal{F}(\muathcal{G},\muathcal{E}):= \lambdaeft(\Tr \lambdaeft(\sqrt{\sqrt{R_{\muathcal{G}}}R_\muathcal{E}\sqrt{R_{\muathcal{G}}}} \right) \right)^2,
\epsilonnd{align}
which reduces to the following when one of them is a unitary channel of the form (\ref{unitary channel})
\betaegin{equation}gin{align}\lambdaanglebel{eq:-10}
\muathcal{F}(\muathcal{U}, \muathcal{E})=\betaraa{U} R_{\muathcal{E}}\kett{U}.
\epsilonnd{align}
Now, we assume that instead of a single gate, a specific set of gates $S$, consists of a finite or infinite collection of unitary gates, are to be approximated with a process $\muathcal{E}$, and each gate $U\in S$ occurs with probability $P(U)$. The input of $\muathcal{E}$ is a given $U\in S$ and the output is ${\muathcal E}_{U}$. Then, a figure of merit that determines the performance of process $\muathcal{E}$ is given by:
\betaegin{equation}gin{align} \lambdaanglebel{eq:-17}
F(\muathcal{E}):=\int dU P(U) \muathcal{F}(\muathcal{U},\muathcal{E}_{U}).
\epsilonnd{align}
Here, $dU$ is an invariant Haar measure, that is $d(UV)=d(VU)=dU,\ \forall V\in \muathbb{U}(d)$. When $S$ is the unitary group $\muathbb{U}(d)$, and gates are chosen uniformly, $P(U)=1,\ \forall U\in \muathbb{U}(d)$.
\section{No-iteration of unknown quantum gates}
\lambdaanglebel{sec:no-repetition}
We now prove the impossibility of implementing $\muathsf{Iter}_n$. We call this no-go result, the \epsilonmph{no-iteration of unknown quantum gates} and provide two proofs for that. One, for the case that the output states are product states, is based on the linearity of the quantum circuit that implements $\muathsf{Iter_n}$, and a more general proof, available in Appendix \ref{app:-1}, is a corollary of a lower bound on the performance of quantum search algorithms. As another confirmation for the validity of this theorem, the optimum fidelity for approximating $\muathsf{Iter_n}$ is obtained numerically for $d=2, 3$ in Sec. \ref{sec:Optimal strategies}, and as expected, it is less than $1$.
\betaegin{equation}gin{theorem}\lambdaanglebel{thm: no-rep}
The universal deterministic gate iterator of order $n$, $\muathsf{Iter_n}$, cannot be implemented perfectly.
\epsilonnd{theorem}
\betaegin{equation}gin{proof}[\textbf{Proof}]
The most general quantum circuit that uses a single copy of $\muathcal{U}$ to implement $\muathcal{U}^n$ is depicted in Fig. \ref{fig:ckt} \cite{Arch, Supermaps}. $\muathcal{A}_n$ and $\muathcal{B}_n$ are preprocessing and postprocessing gates respectively, and $\ket{0}$ shows the ancillary system. This circuit transforms inputs to $B_n(U\otimes\muathds{1})A_n\ \ket{\psi}\otimes \ket{0}$. In this proof, it is assumed that input states are pure and output states are product states (this is relaxed in the alternate proof, see Appendix \ref{app:-1}), so the output is of the form $U^n \ket{\psi}\otimes \ket{a_U}$ where $\ket{a_U}$ is the output ancillary system that possibly depends on U.\\
To prove the theorem, it must be shown that no quantum gates $\muathcal{A}_n$ and $\muathcal{B}_n$ can be found such that for all unitary gates $\muathcal{U}$
\betaegin{equation}gin{align}
B_n(U\otimes\muathds{1})A_n\ \ket{\psi}\otimes \ket{0}=U^n\ket{\psi}\otimes\ket{a_U}. \lambdaanglebel{eq:5}
\epsilonnd{align}
This can be seen by noticing the linearity of the LHS of Eq. (\ref{eq:5}) with respect to $U$, while the RHS seems not to be so.
\betaegin{equation}gin{figure}
\centering
\includegraphics[width=0.45\textwidth]{1}
\caption{The Stinespring realization of the quantum circuit of $\muathsf{Iter}_n$}
\lambdaanglebel{fig:ckt}
\epsilonnd{figure}
In order to show that this is not possible, we use the linearity of quantum mechanics. To proceed, we need two unitary operators so that their linear combinations is also unitary. We take these operators to be $\muathds{1}$ and $\Omega=\sum_{k=0}^{d-1} \ket{k}\betara{d-k-1}= \lambdaeft(\betaegin{equation}gin{smallmatrix}
• & • & 1 \\
• & \iddots & • \\
1 & • & •
\epsilonnd{smallmatrix}\right) $. Note that $\Omega$ is both unitary and Hermitian, so $\Omega^2=\muathds{1}$, which makes
$U:= \cos\theta \ \muathds{1}+i \sin\theta \ \Omega$, also unitary for every $\theta$. Therefore, we should have
\betaegin{equation}gin{align}
B_n(\muathds{1}\otimes\muathds{1})A_n\ \ket{\psi}\otimes \ket{0} &=\muathds{1}\ket{\psi}\otimes \ket{a_{\muathds{1}}} \cr
B_n(\Omega\otimes\muathds{1})A_n\ \ket{\psi}\otimes\ket{0} &=\Omega^n \ket{\psi}\otimes \ket{a_{\Omega}} \\
B_n\lambdaeft((\cos\theta \ \muathds{1} + i\sin\theta \ \Omega)\otimes\muathds{1}\right) A_n\ \ket{\psi}\otimes \ket{0} & =(\cos\theta \ \muathds{1} + i\sin\theta \ \Omega)^n\ket{\psi}\otimes \ket{a_{\theta}},\nuonumber
\epsilonnd{align}
where $\ket{a_{\theta}}$ stands for $\ket{a_{\cos\theta \muathds{1} + i\sin\theta \Omega}}$. Using the first two equations in the LHS of the third, we find
\betaegin{equation}gin{align}
\cos \theta\ \muathds{1}\ket{\psi}\otimes \ket{a_{\muathds{1}}} + i \sin\theta\ \Omega^n \ket{\psi}\otimes \ket{a_{\Omega}} =\lambdaeft(\cos n\theta\ \muathds{1} + i\sin n\theta\ \Omega\right) \ket{\psi}\otimes \ket{a_{\theta}}.
\epsilonnd{align}
By looking at a specific entry of the first factor, we get
\betaegin{equation}gin{align}
\cos \theta\ \betara{0}\muathds{1} \ket{d-1}\ \ket{a_{\muathds{1}}}+ i \sin\theta\ \betara{0}\Omega^n
\ket{d-1}\ \ket{a_{\Omega}} =\betara{0}\lambdaeft(\cos n\theta\ \muathds{1} + i\sin n \theta\ \Omega\right) \ket{d-1}\ \ket{a_{\theta}},
\epsilonnd{align}
but since $\Omega^n=\Omega$ for odd $n$, and $\Omega^n=\muathds{1}$ for even $n$, and $\betara{0}\muathds{1} \ket{d-1}=0$, $\betara{0} \Omega \ket{d-1}=1$, we find
\betaegin{equation}gin{align}
\sin n\theta\ =\lambdaeft\{
\betaegin{equation}gin{array}{c l}
0& n~ even\\
\pm \sin \theta\ & n~ odd
\epsilonnd{array}\right.
\epsilonnd{align}
that cannot be satisfied for arbitrary $\theta$.\\
\epsilonnd{proof}
Although it is impossible to perfectly iterate an unknown gate in $\muathbb{U}(d)$, if a unitary is randomly picked from a set of \epsilonmph{jointly perfectly discriminable unitaries}, then clearly it is possible to iterate it. Whether or not the set of perfectly discriminable unitaries is the only set with this property, is an open question and remains for further investigation \cite{Referee_note}. \\
In the forthcoming sections, we explore different schemes that not perfectly but approximately bypass the introduced no-go result.
\section{The random guess strategy}
\lambdaanglebel{sec:random-guess}
In this section, we investigate the random guess strategy in which the input gate is discarded and iterations of a randomly chosen unitary channel are applied to the input state. The random gate is chosen according to a probability distribution induced by normalized Haar measure on $\muathbb{U}(d)$. Therefore, it can be expressed with the following process
\betaegin{equation}gin{align}\lambdaanglebel{eq:-16}
\muathcal{J}_n(\rho)&=\int dV\ V^n\rho V^{n \deltaagger}.
\epsilonnd{align}
The motivation for studying this rather simple or blind strategy is that it plays an important role for understanding and comparing the performance of other strategies discussed in the following sections.\\
Let us begin by a theorem on the fidelity of this process:
\betaegin{equation}gin{theorem}\lambdaanglebel{thm:random-guess}
The fidelity of the random guess strategy is
\betaegin{equation}gin{align}
F_{rand,n}=p_n^2+\frac{1-p_n^2}{d^2},
\epsilonnd{align}
where
\betaegin{equation}gin{align}
p_n=\frac{min(n,d)-1}{d^2-1}.\lambdaanglebel{eq:6}
\epsilonnd{align}
\epsilonnd{theorem}
Before we give the proof of this theorem, we need two lemmas.
\betaegin{equation}gin{lemma}\lambdaanglebel{Lemma:1}
For all self-adjoint matrices $M\in L(\muathcal{H}_d)$
\betaegin{equation}gin{align}
\muathcal{J}_n(M):=\int dU\ U^n M U^{n \deltaagger}= p_n M +(1-p_n)\Tr(M)\frac{\muathds{1}}{d},
\epsilonnd{align}
where the integration is with respect to the normalized Haar measure on $\muathbb{U}(d)$, and $p_n$ is the same as Eq. (\ref{eq:6}).
\epsilonnd{lemma}
\betaegin{equation}gin{proof}[\textbf{Proof of Lemma}]
Let $\muathcal{E}$ be any quantum channel. The \epsilonmph{twirled transformation} associated with $\muathcal{E}$ is defined as
\betaegin{equation}gin{align}
\tilde{\muathcal E}(M):=\int dV\ V \muathcal{E} (V^{\deltaagger}M V) V^{\deltaagger}. \lambdaanglebel{eq:7}
\epsilonnd{align}
It is shown in in \cite{Emerson}, that the twirled transformation acts like a depolarizing channel with parameter $p_{n,\muathcal{E}}$ that depends on the original channel $\muathcal{E}$:
\betaegin{equation}gin{align}
\tilde{\muathcal{E}}(M)=p_{n,\muathcal{E}} M +(1-p_{n,\muathcal{E}})\Tr(M)\frac{\muathds{1}}{d}.
\epsilonnd{align}
For the specific channel $\muathcal{J}_n$, the twirled channel $\tilde{\muathcal{J}_n}$ equals $\muathcal{J}_n$ itself. To see this, we note that
\betaegin{equation}gin{align}
\tilde{\muathcal{J}}_n(\rho)=\int dV \int dU\ (VU^nV^{\deltaagger}) \rho (VU^{n \deltaagger}V^{\deltaagger})\lambdaanglebel{eq:8}.
\epsilonnd{align}
By defining $W:=VUV^{\deltaagger}$ and using right and left invariance of Haar measure, we find
\betaegin{equation}gin{align}
\tilde{\muathcal{J}_n}(\rho)&=\int dV \int dW\ W^n \rho W^{n \deltaagger}=\int dV \muathcal{J}_n(\rho)=\muathcal{J}_n(\rho),
\epsilonnd{align}
where we have used the normalization $\int dV=\muathds{1}$. It remains to determine the value of the parameter $p_n:=p_{n,\muathcal{J}_n}$. To do this, we enact the channel $\muathcal{J}_n$ on the matrix $|i\rangle\lambdaangle j|$ to obtain
\betaegin{equation}
\int dU\ U^n |i\rangle \lambdaangle j| U^{n \deltaagger}= p_n |i\rangle \lambdaangle j| +(1-p_n)\deltaelta_{ij}\frac{\muathds{1}}{d}.
\epsilonnd{equation}
Multiplying both sides by $\lambdaangle i|$ and $|j\rangle$ and summing over $i$ and $j$, we find
\betaegin{equation}
\int dU |\Tr(U^n)|^2=p_n d^2+(1-p_n).
\epsilonnd{equation}
Using theorem 2.1 of \cite{Diaconis}, according to which $\int dU |\Tr(U^n)|^2=min(n,d)$, we finally find the value of $p_n$:
\betaegin{equation}gin{align}
p_n=\frac{min(n,d)-1}{d^2-1}.
\epsilonnd{align}
This completes the proof.
\epsilonnd{proof}
\betaegin{equation}gin{corollary}\lambdaanglebel{Lemma:2}
For all self-adjoint matrices $M \in L(\muathcal{H}^{(1)}_{d}\pr{\muathcal{H}^{(2)}_{d}})$
\betaegin{equation}gin{align}
\muathcal{K}_n(M):&=\int dU (U^n \pr{\muathds{1})}\ M\ (U^{n \deltaagger}\pr{\muathds{1}}) \lambdaanglebel{eq:12} \nuonumber \\
&=p_n M+(1-p_n)\frac{\muathds{1}\pr \Tr_1(M)}{d}.
\epsilonnd{align}
\epsilonnd{corollary}
\betaegin{equation}gin{corollary}\lambdaanglebel{cor:1}
By substituting $M=\kett{\muathds{1}}\betaraa{\muathds{1}}$ in Eq. (\ref{eq:12}), we get
\betaegin{equation}gin{align}
\int dU\ \kett{U^n}\betaraa{U^n}=p_n \kett{\muathds{1}}\betaraa{\muathds{1}}+(1-p_n)\frac{\muathds{1}}{d^2}.\lambdaanglebel{eq:-14}
\epsilonnd{align}
\epsilonnd{corollary}
\betaegin{equation}gin{proof}[\textbf{Proof of Theorem \ref{thm:random-guess}}]
According to Eq. (\ref{eq:-10}), approximating the iteration of a given gate $\muathcal{U}$ with the iteration of a Haar-distributed random gate $\muathcal{V}$ has the fidelity $\muathcal{F}(\muathcal{U}^n, \muathcal{V}^n)=|\betarakett{U^n}{V^n}|^2$, so its expected value is
\betaegin{equation}gin{align}
\muathbb{E}\lambdaeft[ \muathcal{F}(\muathcal{U}^n,\muathcal{V}^n) \right]&=\betaraa{U^n}\lambdaeft(\int dU \kett{V^n}\betaraa{V^n}\right)\kett{U^{n}} \\
&=p_n\ |\betarakett{U^n}{\muathds{1}}|^2+(1-p_n)\frac{1}{d^2},
\epsilonnd{align}
where the second equality follows from Eq. (\ref{eq:-14}). The fidelity of the random guess strategy can be obtained by taking the average over all $\muathcal{U}$'s:
\betaegin{equation}gin{align}
F_{rand,n}&=\int dU\ \muathbb{E}\lambdaeft[ \muathcal{F}(\muathcal{U}^n,\muathcal{V}^n) \right] \lambdaanglebel{eq:-15}\\
&=p_n \int dU |\betarakett{U^n}{\muathds{1}}|^2+(1-p_n)\frac{1}{d^2}.
\epsilonnd{align}
Again, the integral is simplified using Eq. (\ref{eq:-14}):
\betaegin{equation}gin{align}
F_{rand,n} =p_n^2+\frac{1-p_n^2}{d^2}.
\epsilonnd{align}
\epsilonnd{proof}
As stated in Theorem \ref{thm:random-guess} and depicted in Fig. \ref{fig:plot1}, the fidelity of the random guess decreases quadratically with growth of the dimension $d$, but less intuitively is a kind of \epsilonmph{phase transition} occurs at $n=d$: the fidelity increases with growth of $n$ for $n<d$, and reaches a constant value for $n\gammaeq d$. The random guess is not the only scheme with such phase transition, and as we shall see in next sections, this is a characteristic of all of our approximating strategies. The same phenomena is observed in a similar context when dealing with the joint distribution $f(\theta_1,\deltaots,\theta_d)$ of eigenvalues $\{e^{i \theta_j}\}_{j=1}^d$ of a Haar-distributed unitary matrix in $\muathbb{U}(d)$ \cite{Diaconis_2}, that is
\betaegin{equation}gin{align}\lambdaanglebel{eq:-9}
f(\theta_1,\deltaots,\theta_d)=\frac{1}{(2\pi)^d d!}\prod_{j<k} |e^{i\theta_j}-e^{i\theta_k}|^2,
\epsilonnd{align}
so when $\theta_j\rightarrow \theta_k$, $f\rightarrow 0$, and eigenvalues somehow repel each other. To see this more intuitively, consider $d$ identically charged particles confined to move on the unit circle with Coulomb interaction between them. Their associated Gibbs distribution is
\betaegin{equation}gin{align}
f(\theta_1,\deltaots,\theta_d)=\frac{1}{(2\pi)^d d!}e^{-\betaegin{equation}ta H(\theta_1,\deltaots,\theta_d)},
\epsilonnd{align}
with the Hamiltonian $H=-\sum_{j<k} \lambdaongrightarrowg|e^{i\theta_j}-e^{i\theta_k}|$ and $\betaegin{equation}ta=2$.
This is the same distribution as in Eq. (\ref{eq:-9}) and the repulsion of eigenvalues comes to have a clear physical meaning, and is similar to the repulsion between particles in the ordinary Coulomb gas. \\
When the joint distribution of eigenvalues of higher powers is considered, a phase transition occurs, and for $n\gammaeq d$, the eigenvalues of $U^n$ are exactly distributed as $d$ points chosen independently and uniformly on the unit circle \cite{Rains}. Thus, the eigenvalues that seem to have an ordered structure and are very neatly spaced for $n=1$ have no structure for $n\gammaeq d$.\\
To see the connection of this result to the fidelity of the random guess, notice that from the proof of Lemma \ref{Lemma:1}, the parameter $p_n$ in Eq. (\ref{eq:6}) is
\betaegin{equation}gin{align}
p_n=\frac{\int dU\ |\Tr(U^n)|^2-1}{d^2-1},
\epsilonnd{align}
but $\int dU\ |\Tr(U^n)|^2$ depends on the joint distribution of eigenvalues of $U^n$. For $n\gammaeq d$, this distribution remains the same and $\int dU\ |\Tr(U^n)|^2=d$, so $p_n$ and fidelity also remain constant. \\
Finally, we prove that there exists a depolarizing channel whose fidelity equals the fidelity of the random guess and may be considered as an implementation of that.
\betaegin{equation}gin{theorem}\lambdaanglebel{thm:ckt for random}
The fidelity of the random guess strategy for approximating the $n$'th iteration of a given unknown unitary $\muathcal{U}$ is equal to the fidelity of the following depolarizing channel
\betaegin{equation}gin{align}
\muathcal{J}_n(\rho)&=\int dV\ V^n\rho V^{n \deltaagger}\nuonumber\\
&=p_n \rho +(1-p_n)\frac{\muathds{1}}{d},
\lambdaanglebel{eq:20}
\epsilonnd{align}
with $p_n=\frac{min(n,d)-1}{d^2-1}$, the same as in Eq. (\ref{eq:6}).
\epsilonnd{theorem}
\betaegin{equation}gin{proof}[\textbf{Proof}]
Consider the quantum channel
\betaegin{equation}gin{align}
\muathcal{J}_n(\rho)=\int dV\ V^n\rho V^{n \deltaagger},
\epsilonnd{align}
with $V^n$ as its Kraus operators. The Choi operator of this channel is
\betaegin{equation}gin{align}
R_{\muathcal{J}_n}=\int dV\ \kett{V^n}\betaraa{V^n},
\epsilonnd{align}
so the fidelity of $\muathcal{J}_n$ is
\betaegin{equation}gin{align}
F(\muathcal{J}_n)&=\int dU \betaraa{U^n}R_{\muathcal{J}_n}\kett{U^n}\nuonumber\\
&=\int dU \int dV \ |\betarakett{U^n}{V^n}|^2,
\epsilonnd{align}
which is exactly the same as Eq. (\ref{eq:-15}), so
\betaegin{equation}gin{align}
F(\muathcal{J}_n)=F_{rand,n}.
\epsilonnd{align}
It is also clear from Lemma \ref{Lemma:1} that $\muathcal{J}_n$ is a polarizing channel
\betaegin{equation}gin{align}
\muathcal{J}_n(\rho)=p_n\rho+(1-p_n)\frac{\muathds{1}}{d},
\epsilonnd{align}
with $p_n$ as in Eq. (\ref{eq:6}).
\epsilonnd{proof}
\section{The estimation strategy}\lambdaanglebel{sec:quantum circuit of a repeater}
In the previous scheme, we blindly iterated a random gate and found its fidelity. We now discuss a more prepared and discriminating strategy in which we use more resources. Namely, we estimate the given unitary blackbox and based on the result of the estimation, we choose a gate and perform its iteration on the input state. \\
To make things clear, we can compare it with the random guess circuit that is described by the channel $\muathcal{J}_n(\rho)=\int dV\ V^n\rho V^{n \deltaagger}$, Eq. (\ref{eq:-16}). $\muathcal{J}_n(\rho)$ is the average of all states $V^n\rho V^{n \deltaagger}$, and each unitary gate $V$ has the same weight in it. We can use estimation results to give higher weights to more preferable gates. Let us denote the weight of unitary gate $V$ by $\omega_V$, so the action of our approximate gate iterator is to take $\rho$ and gives $\int dV \omega_V V^n\rho V^{n \deltaagger}$ as the output. As much as these weights are decided correctly, our circuit has higher fidelity than that of the random guess and approximates $\muathsf{Iter_n}$ better. \\
One idea for obtaining reasonable weights $\omega_V$ from a unitary channel with a single try is to encode the effect of the channel into a maximally entangled state and then perform a measurement on the state, the \epsilonmph{measure-and-prepare} method \cite{Estimation}. To see how this works, we first notice that according to Corollary \ref{cor:1}
\betaegin{equation}gin{align}
\int dV \kett{V}\betaraa{V}=\frac{\muathds{1}}{d}, \lambdaanglebel{eq:-18}
\epsilonnd{align}
so the set of operators $d\ \kett{V}\betaraa{V}$ provides bases for a non-orthogonal measurement. On the other hand, the state $\kett{U}$ can be prepared by a single use of $\muathcal{U}$, Eq. (\ref{eq:-8}). Obviously, this measurement cannot perfectly discriminate $\kett{U}$ from other states, and after the measurement, a vector $\kett{V}$ is obtained with probability density $d^2|\betarakett{V}{U}|^2$. Thus, the output state is a weighted mean of states $V^n\rho V^{n \deltaagger}$ with $\omega_V=d^2|\betarakett{V}{U}|^2$. \\
As depicted in Fig. \ref{fig:ckt}, the circuit may include a preprocessing unit ($\muathcal{A}_n$) for preparing necessary states for estimation of $\muathcal{U}$, and a postprocessing unit ($\muathcal{B}_n$) for performing the measurement and preparing the output state based on the estimation. Let the input state on which the iteration is performed be $\rho$ and the ancillary system be a bipartite state $\ket{00}\betara{00}$. The preprocessing channel $\muathcal{A}_n$, prepares a maximally entangled state $\kett{\muathds{1}}\betaraa{\muathds{1}}$ from the ancillary system and swaps that with $\rho$, so that the input state remains unchanged until the estimation results are ready. In other words,
\betaegin{equation}gin{align}
\muathcal{A}_n(\rho\pr{\ket{00}\betara{00}})
&=\kett{\muathds{1}}\betaraa{\muathds{1}}\pr{\rho}.
\epsilonnd{align}
Then, $\muathcal{U}\otimes \muathcal{I}$ acts on the entangled ancillary system and state $\kett{U}\betaraa{U}$ is prepared. In channel $\muathcal{B}_n$, according to the results of the measurement of $\kett{U}\betaraa{U}$, unitary gates are performed on $\rho$, so the output is
\betaegin{equation}gin{align}
\muathcal{B}_n(\kett{U}\betaraa{U}\pr{\rho})&=\int dV (d^2|\betarakett{V}{U}|^2)\ \kett{V}\betaraa{V}\pr{V^n \rho V^{n \deltaagger}}.
\epsilonnd{align}
The action of the whole circuit on $\rho$ is obtained by tracing the output of $\muathcal{B}_n$ over the ancillary system:
\betaegin{equation}gin{align}\lambdaanglebel{eq:19}
\muathcal{D}_{n,\muathcal{U}}(\rho)&:=\Tr_a\lambdaeft(\muathcal{B}_n(\kett{U}\betaraa{U}\pr{\rho})\right)\nuonumber\\
&=d^2 \int dV |\betarakett{V}{U}|^2 \ V^n \rho V^{n \deltaagger}.
\epsilonnd{align}
\betaegin{equation}gin{theorem}
The fidelity of this strategy is
\betaegin{equation}gin{align}
F_{est,n}=d^2 \Tr \lambdaeft(M_n^2\right), \lambdaanglebel{eq:18}
\epsilonnd{align}
with
\betaegin{equation}gin{align}
M_n:=\int dU\ \kett{U}\betaraa{U}\ \pr{\kett{U^n}\betaraa{U^n}}.
\epsilonnd{align}
\epsilonnd{theorem}
\betaegin{equation}gin{proof}[\textbf{Proof}]
The Choi operator associated with the map $\muathcal{D}_{n,\muathcal{U}}$ is
\betaegin{equation}gin{align}
R_{\muathcal{D}_{n,\muathcal{U}}}=d^2\ \betaraa{U} \lambdaeft(\int dV\ \kett{V}\betaraa{V}\ \pr{\kett{V^n}\betaraa{V^n}}\right) \kett{U}.
\epsilonnd{align}
The fidelity of this strategy is $F_{est,n}=\int dU\ \Tr(R_{\muathcal{D}_{n,\muathcal{U}}}R_{\muathcal{U}^n})$. By replacing $R_{\muathcal{U}^n}=\kett{U^n}\betaraa{U^n}$, it is immediate to get Eq. (\ref{eq:18}).
\epsilonnd{proof}
The matrix $M_n$ can be calculated numerically using Monte Carlo method. One approach is to generate Haar-distributed random unitary matrices in $\muathbb{U}(d)$. Then, the integral in Eq. (\ref{eq:18}) can be approximated by averaging the integrand over these random matrices. A simple algorithm exists for uniform generation of random matrices \cite{Random}. The idea is to generate a matrix $Z$ with \epsilonmph{QR-decomposition}
\betaegin{equation}gin{align}
Z=QR,
\epsilonnd{align}
where $Q$ is unitary and $R$ is upper-triangular and invertible. Let $D$ be the diagonalization of $R$ whose entries are divided by their absolute value, then it turns out that if entries of $Z$ are \epsilonmph{i.i.d} standard complex normal random variables, the matrix $U=QD$ is distributed according to Haar measure \cite{Random}. \\
The fidelity of the estimation strategy for different iteration orders is depicted in Fig. \ref{fig:plot1}. It can be seen that by increasing the dimension of the input system, fidelity of the proposed circuit decreases and performance of this circuit tends to that of the random guess method. Similar to the case of the random guess, the fidelity of this circuit reaches a constant value and remains the same for higher order iterations.\\
\betaegin{equation}gin{figure}
\includegraphics[width=8.6 cm]{plot_1_}
\caption{
Fidelity of the random guess and the estimation strategies for various orders of iteration in dimensions $d=2, 3$ and $4$. The advantage of estimation strategy over random guess is clear. It is also seen that this advantage tends to decrease with increasing the dimension $d$.
}
\lambdaanglebel{fig:plot1}
\epsilonnd{figure}
Note that in the estimation strategy we could have replaced the measurement in the over-complete basis in Eq. (\ref{eq:-18}), by a measurement over an orthonormal basis
\betaegin{equation}gin{align}\lambdaanglebel{eq:-19}
&\betarakett{U_j}{U_k}=\Tr (U^{\deltaagger}_j U_k)=0 \quad \forall j, k\in\{1,\deltaots,d^2\},\ j\nueq k, \nuonumber \\
&\sum_{j=1}^{d^2} \kett{U_j}\betaraa{U_j}=\muathds{1}.
\epsilonnd{align}
which is a basis of jointly perfectly discriminable gates. In this case, the quantum channel (\ref{eq:19}) would have been replaced by
\betaegin{equation}gin{align}
\tilde{\muathcal{D}}_{n,\muathcal{U}}(\rho)=\sum_{j=1}^{d^2} |\betarakett{U_j}{U}|^2 \ U_j^n \rho U_j^{n \deltaagger}.
\epsilonnd{align}
However, as we will show in the next section, none of these two kinds of estimation are optimal. The same is true in the case of cloning of unitary gates \cite{unitary-cloning}, where $F_{est}$ is even worse than $F_{rand}$ for $d>2$. However, for $n=-1$, i.e, when the unknown gate is to be inverted, the estimation strategy is the optimal scheme \cite{Inverse}. \\
\section{The identity and direct channels strategies}\lambdaanglebel{sec:idn strategies}
An apparently trivial strategy, called the \epsilonmph{identity channel strategy}, is to take the identity channel as an approximation of $\muathsf{Iter}_n$, i.e., to neglect the given gate $\muathcal{U}$, and to put the input state $\rho$ directly in the output. To see why this approximation is \epsilonmph{reasonable}, we note that by using Eq. (\ref{eq:-10}) and Corollary \ref{cor:1}, we get
\betaegin{equation}gin{align}\lambdaanglebel{eq:-11}
\int dU \muathcal{F}(\muathcal{V},\muathcal{U}^n)=p_n|\betarakett{V}{\muathds{1}}|^2+\frac{1-p_n}{d^2},
\epsilonnd{align}
which immediately gives
\betaegin{equation}gin{align}
\muax_{V\in \muathbb{U}(d)} \int dU \muathcal{F}(\muathcal{V},\muathcal{U}^n)=\muathcal{F}(\muathds{1},\muathcal{U}^n)
\epsilonnd{align}
so on average, the identity channel has the maximum similarity to all $\muathcal{U}^n$'s, and in this sense, it is a reasonable approximation of $\muathcal{U}^n$.\\
The fidelity of this channel is obtained by replacing $\muathcal{V}$ with $\muathds{1}$ in Eq. (\ref{eq:-11}), that gives:
\betaegin{equation}gin{theorem}
The fidelity of the identity channel $F_{iden,n}$ is
\betaegin{equation}gin{align}
F_{iden,n}&=\int dU \muathcal{F}(\muathcal{I},\muathcal{U}^n)\nuonumber\\
&=p_n+\frac{1-p_n}{d^2}
.
\epsilonnd{align}
where $p_n$ is given by Eq. (\ref{eq:6}).
\epsilonnd{theorem}
As it can be seen in Fig. \ref{fig:plot_data_2}, the performance of this process is better than the estimation method, and for $n\gammaeq d$, $F_{idn,n}=\frac{1}{d}$. In fact, we will see in the next section that this channel achieves the optimum fidelity in certain cases. \\
The \epsilonmph{direct channel strategy} is another trivial method with the similar performance for high enough orders n. In this case, $\muathcal{U}^n$ is approximated by $\muathcal{U}$ and the given gate is performed \epsilonmph{directly} on the input state by replacing $\muathcal{A}_n$ and $\muathcal{B}_n$ with identity channels. Numerical results for this scheme is depicted in Fig. \ref{fig:plot_data_2}. Note that the same phase transition as in case of the estimation and random guess strategies occurs. \\
\betaegin{equation}gin{figure}
\centering
\includegraphics[width=8.6 cm]{plot_2_}
\caption{Fidelity of the identity and direct channel strategies. Both schemes have equal performances for $n>d$, and, in fact, they achieve the optimum fidelity in this case.}
\lambdaanglebel{fig:plot_data_2}
\epsilonnd{figure}
\section{Optimum fidelity}\lambdaanglebel{sec:Optimal strategies}
The most general form of the quantum circuit of $\muathsf{Iter}_n$ may be described by concatenation of different unitary channels and some ancillary systems, namely, the Stinespring realization shown in Fig. \ref{fig:ckt}. Therefore, one way to find the optimal process that faithfully realizes $\muathsf{Iter}_n$ is to maximize the fidelity over all quantum channels $\muathcal{A}_n$ and $\muathcal{B}_n$. This is not the only way, and in fact, a more suitable way to describe $\muathsf{Iter}_n$ exists: the quantum comb notion \cite{Arch}.\\
In this method, instead of considering separate channels $\muathcal{A}_n$ and $\muathcal{B}_n$, they are merged and replaced with a channel $\muathcal{C}_n$ from $D(\muathcal{H}^{(0)})\pr{D(\muathcal{H}^{(2)})}$ to $D(\muathcal{H}^{(1)})\pr{D(\muathcal{H}^{(3)})}$, see Fig. \ref{fig:Picture4}. This channel has an open slot in which the given unitary gate $\muathcal{U}$ is inserted and the $n$'th iteration of $\muathcal{U}$ is realized, Fig \ref{fig:Picture5}. \\
\betaegin{equation}gin{figure}
\hskip 1cm space*{.1 cm}
\subfloat[]{\lambdaanglebel{fig:Picture4}}
\includegraphics[width=0.3\textwidth]{2.eps}
\hskip 1cm space{.5 cm}
\subfloat[]{ \lambdaanglebel{fig:Picture5}}
\includegraphics[width=0.35 \textwidth]{3.eps}
\caption{(a) The channel $\muathcal{C}_n$ with one open slot is a replacement for the Stinespring realization shown in Fig. \ref{fig:ckt}, (b) the unitary gate is inserted into the slot to realize $\muathsf{Iter}_n$.}
\epsilonnd{figure}
In the circuit shown in Fig. \ref{fig:Picture5}, the processing of information is from left to right as time passes, and the outputs may depend on the previous but not the later times inputs. Thus, not every quantum channel from $D(\muathcal{H}^{(0)})\pr{D(\muathcal{H}^{(2)})}$ to $D(\muathcal{H}^{(1)})\pr{D(\muathcal{H}^{(3)})}$ can be realized with such a circuit, and they need to meet additional \epsilonmph{causality constraints}. \\
The \textit{quantum comb} $R_{\muathcal{C}_n}$ is defined as the Choi operator associated with $\muathcal{C}_n$ acting on $D(H^{(1)})\pr{D(H^{(3)})}\pr{D(H^{(0)})}\pr{D(H^{(2)})}$. As it is proven in \cite{Arch}, the causality constraint is equivalent to the following set of linear constraints on the quantum comb $R_{\muathcal{C}_n}$
\betaegin{equation}gin{align}
\Tr_3(R_{\muathcal{C}_n})&=\frac{\muathds{1}_2}{d}\pr{R_{\muathcal{C}_n}^{(1)}} \nuonumber\\
\Tr_1(R_{\muathcal{C}_n}^{(1)})&=\frac{\muathds{1}_0}{d}, \lambdaanglebel{eq:-12}
\epsilonnd{align}
where $\Tr_i$ means the partial trace over $H^{(i)}$ and $R_{\muathcal{C}}^{(1)}$ is a Choi operator on $\muathcal{H}^{(1)}\pr{\muathcal{H}^{(0)}}$, and subscripts of operators represent the related Hilbert space of them.\\
The benefit of using the quantum comb notion is clearly seen when the composition of $\muathcal{C}_n$ with the unitary gate $\muathcal{U}$, denoted by $\muathcal{C}_n \star \muathcal{U}$, is to be described, Fig. \ref{fig:Picture5}. It can be proven that (see Ref. \cite{Arch}) the Choi operator of the channel $\muathcal{C}_n \star \muathcal{U}$ is :
\betaegin{equation}gin{align}
R_{\muathcal{C}_n \star \muathcal{U}}=d^2\betaraa{U^*_{21}}R_{\muathcal{C}_n}\kett{U^*_{21}},
\epsilonnd{align}
where $R_{\muathcal{C}_n \star \muathcal{U}}\in L(\muathcal{H}^{(3)}\otimes \muathcal{H}^{(0)})$ and $U^*$ is the conjugate complex of $U$. The subscripts $21$ in $\kett{U^*_{21}}$ denotes the domain and image Hilbert spaces of the operator $U$. Thus, the fidelity (Eq. (\ref{eq:-17})) is
\betaegin{equation}gin{align}
F(\muathcal{C}_n) &=\int dU\ \Tr( d^2 \betaraa{U^*_{21}} R_{\muathcal{C}_n} \kett{U^*_{21}}\ \kett{U^n_{30}}\betaraa{U^n_{30}}\ )\nuonumber\\
&=\Tr(d^2 R_{\muathcal{C}_n} \int dU\ \kett{U^n_{30}}\betaraa{U^n_{30}}\ \pr{\kett{U^*_{21}}\betaraa{U^*_{21}}}\ ).
\epsilonnd{align}
Let $\tilde{M}_n:=d^2 \int dU\ \kett{U^n_{30}}\betaraa{U^n_{30}}\ \pr{\kett{U^*_{21}}\betaraa{U^*_{21}}}$, then
\betaegin{equation}gin{align}
F(\muathcal{C}_n)=\Tr(R_{\muathcal{C}_n}\tilde{M}_n). \lambdaanglebel{eq:-13}
\epsilonnd{align}
Therefore, to find optimal strategies for realizing $\muathsf{Iter}_n$, the following optimization problem should be solved:
\betaegin{equation}gin{align}
\muax_{R_{\muathcal{C}_n}}\ & \Tr(R_{\muathcal{C}_n}\tilde{M}_n)\nuonumber \\
\sth \ &\Tr_3(R_{\muathcal{C}_n})=\frac{\muathds{1}_2}{d}\pr{R_{\muathcal{C}_n}^{(1)}}, \\
&\Tr_1(R_{{\muathcal{C}_n}}^{(1)})=\frac{\muathds{1}_0}{d},\nuonumber \\
&R_{\muathcal{C}_n}\gammaeq0,\ R_{\muathcal{C}_n}^{(1)}\gammaeq 0\nuonumber.
\epsilonnd{align}
This is an example of Semidefinite Programming (SDP) \cite{Boyd}, which is numerically solvable using packages like CVX \cite{CVX}. The optimum fidelity obtained by this method and fidelity of the identity channel are shown for different cases in Fig. \ref{fig:data_plot_3}. For $d=2$, $n>2$, the identity and direct channels discussed in Sec. \ref{sec:idn strategies} achieve the optimum fidelity, and this is quite unanticipated, since both are trivial methods where resources like entanglement or general preprocessing or postprocessing units are not used. \\
As in the case of other approximating processes investigated earlier, the optimum fidelity reaches a constant value, and the optimal iterator has the same performance for high enough orders $n$. This phenomena is not observed in the case of $1$-to-$n$ cloning of unitary gates where the fidelity seems to decreases monotonically with growth of $n$ \cite{unitary-cloning}. In addition, in that problem, the performance of the optimal cloner depends crucially on the entanglement of input states with the ancillary system, and the identity channel has a by-far-worse fidelity than the optimal cloner.\\
\betaegin{equation}gin{figure}
\centering
\includegraphics[width=8.6 cm]{plot_3_}
\caption{The optimum fidelity in approximating $\muathsf{Iter}_n$.}
\lambdaanglebel{fig:data_plot_3}
\epsilonnd{figure}
\section{Conclusion}\lambdaanglebel{sec:Conclusion}
We have shown that it is impossible to iterate an unknown quantum gate by using it once, what we called it the no-iteration theorem. We have also investigated different schemes to approximately bypass this no-go result: (1) The random guess strategy in which iterations of a randomly chosen gate is performed. (2) The measure-and-prepare method where the given gate $\muathcal{U}$ is first estimated using the state $\kett{U}$, and then unitary processes are performed on the input state accordingly. (3) Approximating with the identity channel or by performing the given unitary process directly on the input system. In addition, by using the notion of quantum comb, we have been able to state the problem of finding the optimal iterator as a semidefinite programming, which we have solved numerically for $d=2, 3$.\\
The iteration problem has some unique features that make it different from similar problems like cloning of unitary channels. One is that the performance of all discussed methods including the optimal one, remains the same for highly enough orders $n$. In the case of random guess, we saw the connection of this phase transition to the joint distribution of eigenvalues of a random unitary matrix, which changes from being highly ordered to having no structure for $n\gammaeq d$. The other feature is that the performance of trivial processes like identity or direct channels is comparable to the optimal strategies, and at least for $d=2,3$, numerical solutions show they achieve the optimum performance for $n >d$.\\
This no-go theorem is another example of transformations of quantum channels that cannot be realized perfectly. Providing these examples helps us to understand the characteristics of quantum operations as \epsilonmph{carriers of information}, and shows us how laws of quantum mechanics act when evolution of operations is considered instead of states.\\
Interesting behaviors of gate iterators discussed in this paper, motivates a more general study of powers of unitary operators in $n \gammag 1$ limit. The iteration problem and the performance of the optimal iterator might also be explored when multiple copies of the oracle is provided.
\alphappendix
\section{Alternate Proof of Theorem \ref{thm: no-rep}}\lambdaanglebel{app:-1}
\betaegin{equation}gin{proof}[\textbf{Alternate proof}]
According to the following lemma, proved as a theorem in \cite{bounds_search}, there exists a lower bound on the performance of quantum search algorithms. This lower bound is only a few percent smaller than the number of iterations required by Grover’s algorithm \cite{Grover}.
\betaegin{equation}gin{lemma}
Let $T$ be any set of $N$ strings, and $M$ be any oracle quantum machine with bounded error probability. Let $y \in_R T$ be a randomly and uniformly chosen element from $T$. Put $\muathcal{O}$ to be the oracle where $\muathcal{O}(x) = 1$ if and only if $x = y$. Then the expected number of times $M$ must query $\muathcal{O}$ in order to determine $y$ with probability at least $\frac{1}{2}$ is at least $\lambdafloor\sin(\frac{\pi}{8})\sqrt{N}\rfloor$.
\epsilonnd{lemma}
Now imagine that $\muathsf{Iter_n}$ can be constructed perfectly, then for an appropriate number of strings $N$, the required number of queries can be reduced. This can be done easily by replacing each $n$ successive queries in Grover search algorithm with a single use of $\muathsf{Iter_n}$. Thus, the lower bound of the last Lemma is violated and this is a contradiction.
\epsilonnd{proof}
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\epsilonnd{thebibliography}
\epsilonnd{document} |
\begin{document}
\title{Decidable fragments of the Simple Theory of Types with Infinity and $\mathrm{NF}$ \thanks{The research of the first and the third author was supported in part by EPSRC grant EP/H026835.}}
\author[1]{Anuj Dawar}
\author[2]{Thomas Forster}
\author[1]{Zachiri McKenzie}
\affil[1]{University of Cambridge Computer Laboratory\\ \texttt{[email protected]}}
\affil[2]{DPMMS, University of Cambridge\\
\texttt{[email protected]}}
\maketitle
\begin{abstract}
We identify complete fragments of the Simple Theory of Types with Infinity ($\mathrm{TSTI}$) and Quine's $\mathrm{NF}$ set theory. We show that $\mathrm{TSTI}$ decides every sentence $\phi$ in the language of type theory that is in one of the following forms:
\begin{itemize}
\item[(A)] $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ where the superscripts denote the types of the variables, $s_1 > \ldots > s_l$ and $\theta$ is quantifier-free,
\item[(B)] $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ where the superscripts denote the types of the variables and $\theta$ is quantifier-free.
\end{itemize}
This shows that $\mathrm{NF}$ decides every stratified sentence $\phi$ in the language of set theory that is in one of the following forms:
\begin{itemize}
\item[(A')] $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\phi$ admits a stratification that assigns distinct values to all of the variable $y_1, \ldots, y_l$,
\item[(B')] $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\phi$ admits a stratification that assigns the same value to all of the variables $y_1, \ldots, y_l$.
\end{itemize}
\end{abstract}
\section[Introduction]{Introduction}
\indent Roland Hinnion showed in his thesis \cite{hin75} that {\sl Every consistent $\exists^*$ sentence in the language of set theory is a theorem of $\mathrm{NF}$} or, equivalently: {\sl Every finite binary structure can be embedded in every model of $\mathrm{NF}$}. Both these formulations invite generalisations. On the one hand we find results like {\sl every countable binary structure can be embedded in every model of $\mathrm{NF}$} (this is theorem 4 of \cite{for87}) and on the other we can ask about the status of sentences with more quantifiers: $\forall^*\exists^*$ sentences in the first instance; it is the second that will be our concern here.\\
\\
\indent It is elementary to check that $\mathrm{NF}$ does not decide all $\forall^*\exists^*$ sentences, since the existence of Quine atoms ($x = \{x\}$) is consistent with, and independent of, $\mathrm{NF}$. However `$(\forall x)(x \not= \{x\})$' is not stratified, and this invites the conjecture that (i) $\mathrm{NF}$ decides all stratified $\forall^*\exists^*$ sentences and that (ii) all unstratified $\forall^*\exists^*$ sentences can be proved both relatively consistent and independent by means of Rieger-Bernays permutation methods. It's with limb (i) of this conjecture that we are concerned here.\\
\\
\indent The foregoing is all about $\mathrm{NF}$; the connection with the Simple Theory of Types with Infinity ($\mathrm{TSTI}$) arises because of work of Ernst Specker \cite{spe62} and \cite{spe53}: $\mathrm{NF}$ decides all stratified $\forall^*\exists^*$ sentences of the language of set theory if and only if $\mathrm{TSTI} + \mathrm{Ambiguity}$ decides all $\forall^*\exists^*$ sentences of the language of type theory.
\begin{quote}{\bf Conjecture}: All models of $\mathrm{TSTI}$ agree on all $\forall^*\exists^*$ sentences.\end{quote}
\noindent It is towards a proof of this conjecture that our efforts in this paper are directed.\\
\\
\indent Observe that {\sl there is a total order of $V$} is consistent with and independent of $\mathrm{TST}$ and it can be said with three blocks of quantifiers: $$(\exists O)[(\forall x y \in O)(x \subseteq y \lor y \subseteq x) \land (\forall u v)( u \not= v \to (\exists x \in O)( u \in x \iff v \not\in x))]$$ making it $\exists^1\forall^6\exists^1$.
\section[Background and definitions]{Background and definitions} \label{Sec:Background}
The Simple Theory of Types is the simplification of the Ramified Theory of Types, the underlying system of \cite{rw08}, that was independently discovered by Frank Ramsey and Leon Chwistek. Following \cite{mat01} we use $\mathrm{TSTI}$ and $\mathrm{TST}$ to abbreviate the Simple Theory of Types with and without an axiom of infinity respectively. These theories are naturally axiomatised in a many-sorted language with sorts for each $n \in \mathbb{N}$.
\begin{Definitions1}
We use $\mathcal{L}_{\mathrm{TST}}$ to denote the $\mathbb{N}$-sorted language endowed with binary relation symbols $\in_n$ for each sort $n \in \mathbb{N}$. There are variables $x^n, y^n, z^n, \ldots$ for each sort $n \in \mathbb{N}$ and well-formed $\mathcal{L}_{\mathrm{TST}}$-formulae are built-up inductively from atomic formulae in the form $x^n \in_n y^{n+1}$ and $x^n = y^n$ using the connectives quantifiers of first-order logic.
\end{Definitions1}
\noindent We refer to sorts of $\mathcal{L}_{\mathrm{TST}}$ as types. We will attempt to stick to the convention of denoting $\mathcal{L}_{\mathrm{TST}}$-structures using calligraphy letters ($\mathcal{M}, \mathcal{N}, \ldots$). A $\mathcal{L}_{\mathrm{TST}}$-structure $\mathcal{M}$ consists of domains $M_n$ for each type $n \in \mathbb{N}$ and interpretations of the relations $\in_n^{\mathcal{M}} \subseteq M_n \times M_{n+1}$ for each type $n \in \mathbb{N}$; we write $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$. If \mbox{$\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$} is an $\mathcal{L}_{\mathrm{TST}}$-structure then we call the elements of $M_0$ atoms.
\begin{Definitions1}
We use $\mathrm{TST}$ to denote the $\mathcal{L}_{\mathrm{TST}}$-theory with axioms
\begin{itemize}
\item[](Extensionality) for all $n \in \mathbb{N}$,
$$\forall x^{n+1} \forall y^{n+1} (x^{n+1}= y^{n+1} \iff \forall z^n(z^{n} \in_n x^{n+1} \iff z^n \in y^{n+1})),$$
\item[](Comprehension) for all $n \in \mathcal{N}$ and for all well-formed $\mathcal{L}_{\mathrm{TST}}$-formulae $\phi(x^n, \vec{z})$,
$$\forall \vec{z} \exists y^{n+1} \forall x^n (x^n \in_n y^{n+1} \iff \phi(x^n, \vec{z})).$$
\end{itemize}
\end{Definitions1}
Comprehension ensures that every successor type is closed under the set-theoretic operations: union ($\cup$), intersection ($\cap$), difference ($\backslash$) and symmetric difference ($\triangle$). For all $n \in \mathbb{N}$, we use $\emptyset^{n+1}$ to denote the point at type $n+1$ which contains no points from type $n$ and we use $V^{n+1}$ to denote the point at type $n+1$ that contains every point from type $n$. The Wiener-Kuratowski ordered pair allows us to code ordered pairs in the form $\langle x, y \rangle$ as objects in $\mathrm{TST}$ which have type two higher than the type of $x$ and $y$. Functions, as usual, are thought of as collections of ordered pairs. This means that a function $f: X \longrightarrow Y$ will be coded by an object in $\mathrm{TST}$ that has type two higher than the type of $X$ and $Y$. The theory $\mathrm{TSTI}$ is obtained from $\mathrm{TST}$ by asserting the existence of a Dedekind infinite collection at type $1$.
\begin{Definitions1}
We use $\mathrm{TSTI}$ to denote the $\mathcal{L}_{\mathrm{TST}}$-theory obtained from $\mathrm{TST}$ by adding the axiom
$$\exists x^1 \exists f^3(f^3: x^1 \longrightarrow x^1 \textrm{ is injective but not surjective}).$$
\end{Definitions1}
Let $X$ be a set. If the $\mathcal{L}_{\mathrm{TST}}$-structure $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0, \in_1, \ldots \rangle$ is defined by $M_n= \mathcal{P}^n(X)$ and $\in_n^\mathcal{M}= \in \upharpoonright \mathcal{P}^n(X) \times \mathcal{P}^{n+1}(X)$ for all $n \in \mathbb{N}$, then $\mathcal{M} \models \mathrm{TST}$. If $m \in \mathbb{N}$ and $|X|= m$ then $\mathcal{M}$ is the unique, up to isomorphism, model of $\mathrm{TST}$ with exactly $m$ atoms and we say that $\mathcal{M}$ is finitely generated by $m$ atoms. Alternatively, if $X$ is Dedekind infinite then $\mathcal{M} \models \mathrm{TSTI}$. This shows that $\mathrm{ZFC}$ proves the consistency of $\mathrm{TSTI}$. In fact, in \cite{mat01} it is shown that $\mathrm{TSTI}$ is equiconsistent with Mac Lane Set Theory.\\
\\
\indent We say that an $\mathcal{L}^\prime$-theory $T$ decides an $\mathcal{L}^\prime$-sentence $\phi$ if and only if $T \vdash \phi$ or $T \vdash \neg \phi$. The Completeness Theorem implies that $T$ decides $\phi$ if and only if $\phi$ holds in all $\mathcal{L}^\prime$-structures $\mathcal{M} \models T$, or $\neg \phi$ holds in all $\mathcal{L}^\prime$-structures $\mathcal{M} \models T$.
\begin{Definitions1}
We say that a $\mathcal{L}_{\mathrm{TST}}$-sentence $\phi$ is $\exists^* \forall^*$ if and only if\\
\mbox{$\phi= \exists x_1^{r_1} \cdots \exists x_k^{r_k} \forall y_1^{s_1} \cdots \forall y_l^{s_l} \theta$} where $\theta$ is quantifier-free.
\end{Definitions1}
\begin{Definitions1}
We say that an $\mathcal{L}_{\mathrm{TST}}$-sentence $\phi$ is $\forall^* \exists^*$ if and only if\\
\mbox{$\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$} where $\theta$ is quantifier-free.
\end{Definitions1}
We will show that $\mathrm{TSTI}$ decides a significant fragment of the $\forall^* \exists^*$ sentences (and thus it also decides the $\exists^* \forall^*$ sentences that are logically equivalent to the negation of these $\forall^* \exists^*$ sentences). We achieve this result by showing that every sentence or negation of a sentence in this fragment that is true in some model of $\mathrm{TSTI}$ is true in all models of $\mathrm{TST}$ that are finitely generated by sufficiently many atoms.
\begin{Definitions1}
We say that an $\mathcal{L}_{\mathrm{TST}}$-sentence $\phi$ has the finitely generated model property if and only if, if there exists an $\mathcal{N} \models \mathrm{TSTI}+\phi$ then there exists a $k \in \mathbb{N}$ such that for all $m \geq k$, if $\mathcal{M} \models \mathrm{TST}$ is finitely generated by $m$ atoms then $\mathcal{M} \models \phi$.
\end{Definitions1}
\noindent Note that if $\Gamma$ is class of $\mathcal{L}_{\mathrm{TST}}$-sentences that have the finitely generated model property and $\Gamma$ is closed under negations then $\mathrm{TST}$ decides every sentence in $\Gamma$.\\
\\
\indent In \cite{qui37} Willard van Orman Quine introduces a set theory by identifying a syntactic condition on formulae in the single sorted language of set theory that captures the restricted comprehension available in $\mathrm{TST}$. This set theory has been dubbed `New Foundations' ($\mathrm{NF}$) after the title of \cite{qui37}. We will use $\mathcal{L}$ to denote the language of set theory --- the language of first-order logic endowed with a binary relation symbol $\in$ whose intended interpretation is membership. Before giving the axioms of $\mathrm{NF}$ we first recall Quine's definition of a stratified formulae. If $\phi$ is an $\mathcal{L}$-formula then we use $\mathbf{Var}(\phi)$ to denote the set of variables (both free and bound) which appear in $\phi$.
\begin{Definitions1}
Let $\phi(x_1, \ldots, x_n)$ be an $\mathcal{L}$-formula. We say that $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ if and only if
\begin{itemize}
\item[(i)] if `$x \in y$' is a subformula of $\phi$ then $\sigma(\textrm{`}y\textrm{'})= \sigma(\textrm{`}x\textrm{'})+1$,
\item[(ii)] if `$x = y$' is a subformula of $\phi$ then $\sigma(\textrm{`}y\textrm{'})= \sigma(\textrm{`}x\textrm{'})$.
\end{itemize}
If there exists a stratification of $\phi$ then we say that $\phi$ is stratified.
\end{Definitions1}
Let $\phi$ be an $\mathcal{L}$-formula. Note that $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ if and only if the formula obtained by decorating every variable appearing in $\phi$ with the type given by $\sigma$ yields a well-formed $\mathcal{L}_{\mathrm{TST}}$-formula. Conversely, let $\theta$ be a well-formed $\mathcal{L}_{\mathrm{TST}}$-formula and let $\phi$ an $\mathcal{L}$-formula obtained for $\theta$ by deleting the types from the variables appearing in $\theta$ while ensuring (by relabeling variables) that no two distinct variables in $\theta$ become the same variable in $\phi$. Then the $\mathcal{L}$-formula $\phi$ is stratified and the function which sends a variable in $\phi$ to the type index of the corresponding variable in $\theta$ is a stratification.
\begin{Definitions1}
Let $\phi$ be an $\mathcal{L}$-formula with stratification $\sigma: \mathbf{Var}(\phi)$. We use $\phi^{(\sigma)}$ to denote the $\mathcal{L}_{\mathrm{TST}}$-formula obtained by assigning each variable `$x$' appearing $\phi$ the type $\sigma(\textrm{`}x\textrm{'})$.
\end{Definitions1}
$\mathrm{NF}$ is the $\mathcal{L}$-theory with the axiom of extensionality and comprehension for all stratified $\mathcal{L}$-formulae.
\begin{Definitions1}
We use $\mathrm{NF}$ to denote the $\mathcal{L}$-theory with axioms
\begin{itemize}
\item[](Extensionality) $\forall x \forall y (x=y \iff \forall z(z \in x \iff z \in y))$,
\item[](Stratified Comprehension) for all stratified $\phi(x, \vec{z})$,
$$\forall \vec{z} \exists y \forall x (x \in y \iff \phi(x, \vec{z})).$$
\end{itemize}
\end{Definitions1}
We direct the interested reader to \cite{for95} for detailed treatment of $\mathrm{NF}$. One interesting feature of $\mathrm{NF}$ is that it refutes the Axiom of Choice and so proves the Axiom of Infinity (see \cite{spe53}). There is a strong connection between the theories $\mathrm{NF}$ and $\mathrm{TSTI}$. \cite{spe62} shows that models of $\mathrm{NF}$ can be obtained from models of $\mathrm{TSTI}$ plus the scheme $\phi \iff \phi^+$, for all $\mathcal{L}_{\mathrm{TST}}$-sentences $\phi$, where $\phi^+$ is obtained from $\phi$ by incrementing the types of all the variables appearing in $\phi$. Conversely, let $\mathcal{M}= \langle M, \in^{\mathcal{M}} \rangle$ be an $\mathcal{L}$-structure with $\mathcal{M} \models \mathrm{NF}$. The $\mathcal{L}_{\mathrm{TST}}$-structure $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ defined by $N_n= M$ and $\in_n^{\mathcal{N}}= \in^{\mathcal{M}}$ is such that $\mathcal{N} \models \mathrm{TSTI}$. Moreover, if $\phi$ is an $\mathcal{L}$-sentence with stratification $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ and $\mathcal{M} \models \phi$ then $\mathcal{N} \models \phi^{(\sigma)}$. This immediately shows that a decidable fragment of $\mathrm{TSTI}$ yields a decidable fragment of $\mathrm{NF}$.
\begin{Theorems1} \label{Th:DecidableFragmentsOfNF}
Let $\phi$ be an $\mathcal{L}$-sentence with stratification $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$. If $\mathrm{TSTI}$ decides $\phi^{(\sigma)}$ then $\mathrm{NF}$ decides $\phi$.
\Square
\end{Theorems1}
\section[$\exists^* \forall^*$ sentences have the finitely generated model property]{$\exists^* \forall^*$ sentences have the finitely generated model property}
In this section we prove that all $\exists^* \forall^*$ sentences have the finitely generated model property. This result follows from the fact that if $\mathcal{N}$ is a model of $\mathrm{TSTI}$, $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$ with $r_1 \leq \ldots \leq r_k$ and $\mathcal{M}$ is a model of $\mathrm{TST}$ that is finitely generated by sufficiently many atoms then there is an embedding of $\mathcal{M}$ into $\mathcal{N}$ with $a_1^{r_1}, \ldots, a_k^{r_k}$ in the range. Given $k \in \mathbb{N}$ we define the function $\mathbf{G}_k: \mathbb{N} \longrightarrow \mathbb{N}$ by recursion
\begin{equation}
\mathbf{G}_k(0)= k \textrm{ and } \mathbf{G}_k(n+1)= \binom{\mathbf{G}_k(n)}{2}+ k.
\end{equation}
\begin{Lemma1} \label{Th:EmeddingProperty}
Let $\mathcal{N} \models \mathrm{TSTI}$ and let \mbox{$a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$} with $r_1 \leq \ldots \leq r_k$. If $\mathcal{M} \models \mathrm{TST}$ is finitely generated by at least $\mathbf{G}_k(r_k)$ atoms then there exists a sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ such that for all $n \in \mathbb{N}$,
\begin{itemize}
\item[(i)] $f_n: M_n \longrightarrow N_n$ is injective,
\item[(ii)] for all $x \in M_n$ and for all $y \in M_{n+1}$,
$$\mathcal{M} \models x \in_n y \textrm{ if and only if } \mathcal{N} \models f_n(x) \in_n f_{n+1}(y),$$
\item[(iii)] $$a_1^{r_1}, \ldots, a_k^{r_k} \in \bigcup_{m \in \mathbb{N}} \mathrm{rng}(f_m).$$
\end{itemize}
\end{Lemma1}
\begin{proof}
Let $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ be such that $\mathcal{N} \models \mathrm{TSTI}$ and let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$ with $r_1 \leq \ldots \leq r_k$. Let $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$ be such that $\mathcal{M} \models \mathrm{TST}$ is finitely generated and $|M_0| \geq \mathbf{G}_k(r_k)$. We begin by defining $C \subseteq \mathcal{N}$ such that $|C \cap N_0| \leq \mathbf{G}_k(r_k)$ and for any two points $x \neq y$ in $C$ that are not atoms, there exists a point $z$ in $C$ which $\mathcal{N}$ believes is in the symmetric difference of $x$ and $y$. Define $C_0= \{a_1^{r_1}, \ldots, a_k^{r_k} \} \subseteq \mathcal{N}$. Note that $|C_0 \cap N_{r_k}| \leq \mathbf{G}_k(0)= k$ and for all $0\leq m < r_k$, $|C_0 \cap N_m| \leq k$. For $0 < n \leq r_k$ we recursively define $C_n \subseteq \mathcal{N}$ which satisfies
\begin{itemize}
\item[(I)] $|C_n \cap N_{r_k-n}| \leq \mathbf{G}_k(n)$,
\item[(II)] for all $0 \leq m < r_k-n$, $|C_n \cap N_m| \leq k$.
\end{itemize}
Suppose that $n < r_k$ and $C_n \subseteq \mathcal{N}$ has been defined and satisfies (I) and (II). For all $y, z \in N_{r_k-n}$ with $y \neq z$, let $\gamma_{\{y, z\}} \in N_{r_k-(n+1)}$ be such that
$$\mathcal{N} \models \gamma_{\{y, z\}} \in_{r_k-(n+1)} y \triangle z.$$
Define
$$C_{n+1}= C_n \cup \{ \gamma_{\{y, z\}} \mid \{y, z\} \in [N_{r_k-n} \cap C_n]^2 \}.$$
It follows from (I) and (II) that
$$|C_{n+1} \cap N_{r_k-(n+1)}| \leq |C_n \cap N_{r_k-(n+1)}|+\binom{|C_n \cap N_{r_k-n}|}{2} \leq k + \binom{\mathbf{G}_k(n)}{2}= \mathbf{G}_k(n+1)$$
and for all $0 \leq m < r_k-(n+1)$, $|C_{n+1} \cap N_m| \leq k$. Now, let $C= C_{r_k}$. This recursion ensures that $|C \cap N_0| \leq \mathbf{G}_k(r_k)$.\\
We now turn to defining the family of maps $\langle f_n \mid n \in \mathbb{N} \rangle$ which embed $\mathcal{M}$ into $\mathcal{N}$. We define the sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ by induction. Let $C^\prime= C \cap N_0$. Let $f_0: M_0 \longrightarrow N_0$ be an injection such that $C^\prime \subseteq \mathrm{rng}(f_0)$. Suppose that $\langle f_0, \ldots, f_n \rangle$ has been defined such that
\begin{itemize}
\item[(I')] for all $0 \leq j \leq n$, $f_j: M_j \longrightarrow N_j$ is injective,
\item[(II')] for all $0 \leq j < n$, for all $x \in M_j$ and for all $y \in M_{j+1}$,
$$\mathcal{M} \models x \in_j y \textrm{ if and only if } \mathcal{N} \models f_j(x) \in_j f_{j+1}(y),$$
\item[(III')] for all $0 \leq j \leq n$, $C \cap N_j \subseteq \mathrm{rng}(f_j)$.
\end{itemize}
If $0 \leq j \leq n$ and $x \in M_{j+1}$ then we use $f_j``x$ to denote the point in $N_{j+1}$ such that $\mathcal{N} \models f_j``x= \{ f_j(y) \mid \mathcal{M} \models y \in_j x \}$. Note that, since $\mathcal{M}$ is finitely generated, for all $x \in M_{j+1}$, $f_j``x$ exists in $\mathcal{N}$. We define $f_{n+1}: M_{n+1} \longrightarrow N_{n+1}$ by
$$f_{n+1}(x)= \left\{ \begin{array}{ll}
\gamma & \textrm{if } \gamma \in C \cap N_{n+1} \textrm{ and } \mathcal{N} \models f_n``x= \gamma \cap f_n``(V^{n+1})^\mathcal{M}\\
f_n``x & \textrm{otherwise}
\end{array} \right.$$
We first need to show that the map $f_{n+1}$ is well-defined. Suppose that $\xi_1, \xi_2 \in C \cap N_{n+1}$ with $\xi_1 \neq \xi_2$ and $x \in M_{n+1}$ are such that
$$\mathcal{N} \models f_n``x= \xi_1 \cap f_n``(V^{n+1})^\mathcal{M} \textrm{ and } \mathcal{N} \models f_n``x= \xi_2 \cap f_n``(V^{n+1})^\mathcal{M}.$$
Now, there is a $\gamma \in C \cap N_n$ such that $\mathcal{N} \models \gamma \in_n \xi_1 \triangle \xi_2$. By (III'), $\gamma \in \mathrm{rng}(f_n)$, which is a contradiction. Therefore $f_{n+1}$ is well-defined.\\
The fact that $f_n$ is injective ensures that $f_{n+1}$ is injective.\\
We now turn to showing that the sequence $\langle f_0, \ldots, f_{n+1} \rangle$ satisfies (II'). Let $x \in M_n$ and let $y \in M_{n+1}$. There are two cases. Firstly, suppose that $f_{n+1}(y)= \gamma \in C$. Therefore $\mathcal{N} \models f_n``y= \gamma \cap f``(V^{n+1})^\mathcal{M}$. If $\mathcal{M} \models x \in_n y$ then $\mathcal{N}\models f_n(x) \in_n f_n``y$ and so $\mathcal{N} \models f_n(x) \in_n f_{n+1}(y)$. Conversely, if $\mathcal{N} \models f_n(x) \in_n \gamma$ then $\mathcal{N} \models f_n(x) \in_n f_n``y$ and so $\mathcal{M} \models x \in_n y$. The second case is when $f_{n+1}(y)= f_n``y$. In this case it is clear that
$$\mathcal{M} \models x \in_n y \textrm{ if and only if } \mathcal{N} \models f_n(x) \in_n f_{n+1}(y).$$
This shows that the sequence $\langle f_0, \ldots, f_{n+1} \rangle$ satisfies (II').\\
This concludes the induction step of the construction and shows that we can construct a sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ that satisfies (i)-(iii).
\Square
\end{proof}
This embedding property allows us to show that every $\exists^* \forall^*$ sentence has the finitely generated model property.
\begin{Theorems1} \label{Th:ExistentialUniversalSentencesHaveFinitelyGeneratedModelProperty}
Let $\phi= \exists x_1^{r_1} \cdots \exists x_k^{r_k} \forall y_1^{s_1} \cdots \forall y_l^{s_l} \theta$ where $r_1 \leq \ldots \leq r_k$ and $\theta$ is quantifier-free. If $\mathcal{N} \models \mathrm{TSTI}+\phi$ and $\mathcal{M} \models \mathrm{TST}$ is finitely generated by at least $\mathbf{G}_k(r_k)$ atoms then $\mathcal{M} \models \phi$.
\end{Theorems1}
\begin{proof}
Let $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ be such that $\mathcal{N} \models \mathrm{TSTI}+\phi$. Let $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$ be such that $\mathcal{M} \models \mathrm{TST}$ and $\mathcal{M}$ is finitely generated by at least $\mathbf{G}_k(r_k)$ atoms. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{N}$ be such that
$$\mathcal{N} \models \forall y_1^{s_1} \cdots \forall y_l^{s_l} \theta[a_1^{r_1}, \ldots, a_k^{r_k}].$$
Using Lemma \ref{Th:EmeddingProperty} we can find a sequence $\langle f_n \mid n \in \mathbb{N} \rangle$ such that
\begin{itemize}
\item[(i)] $f_n: M_n \longrightarrow N_n$ is injective,
\item[(ii)] for all $x \in M_n$ and for all $y \in M_{n+1}$,
$$\mathcal{M} \models x \in_n y \textrm{ if and only if } \mathcal{N} \models f_n(x) \in f_{n+1}(y),$$
\item[(iii)] $$a_1^{r_1}, \ldots, a_k^{r_k} \in \bigcup_{m \in \mathbb{N}} \mathrm{rng}(f_m).$$
\end{itemize}
Let $b_1^{r_1}, \ldots, b_k^{r_k} \in \mathcal{M}$ be such that for all $1 \leq j \leq k$, $f_{r_j}(b_j^{r_j})= a_j^{r_j}$. Let $c_1^{s_1}, \ldots, c_l^{s_l} \in \mathcal{M}$. Since $\mathcal{N} \models \theta[a_1^{r_1}, \ldots, a_k^{r_k}, f_{s_1}(c_1^{s_1}), \ldots, f_{s_l}(c_l^{s_l})]$, it follows that
$$\mathcal{M} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, c_1^{s_1}, \ldots, c_l^{s_l}].$$
Therefore
$$\mathcal{M} \models \forall y_1^{s_1} \cdots \forall y_l^{s_l}\theta[b_1^{r_1}, \ldots, b_k^{r_k}],$$
which proves the theorem.
\Square
\end{proof}
\section[Decidable fragments of the $\forall^* \exists^*$ sentences]{Decidable fragments of the $\forall^* \exists^*$ sentences}
In this section we will show that $\mathrm{TSTI}$ decides every $\forall^* \exists^*$ sentence $\phi$ that is in one of the following forms:
\begin{itemize}
\item[(A)] $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ where $s_1 > \ldots > s_l$ and $\theta$ is quantifier-free,
\item[(B)] $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ where $\theta$ is quantifier-free.
\end{itemize}
By applying Theorem \ref{Th:DecidableFragmentsOfNF} it then follows that $\mathrm{NF}$ decides every stratified $\mathcal{L}$-sentence $\phi$ that is in one of the following forms:
\begin{itemize}
\item[(A')] $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns distinct values to all of the variables $y_1, \ldots, y_l$,
\item[(B')] $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ where $\theta$ is quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns the same value to all of the variables $y_1, \ldots, y_l$.
\end{itemize}
Throughout this section we will fix $k, l \in \mathbb{N}$ and a sequence $r_1 \leq \ldots \leq r_k$ that will represent the types of the universally quantified variables in a $\forall^* \exists^*$ sentence. Let $k^\prime$ be the number of distinct elements in the list $r_1, \ldots, r_k$. Let $K_1, \ldots, K_{k^\prime}$ be the multiplicities of the elements in the list $r_1, \ldots, r_k$, so $k= \sum_{1 \leq i \leq k^\prime} K_i$, and let $K= \max\{ K_1, \ldots, K_{k^\prime}, l\}$. We also fix structures $\mathcal{N}= \langle N_0, N_1, \ldots, \in_0^{\mathcal{N}}, \in_1^{\mathcal{N}}, \ldots \rangle$ with $\mathcal{N} \models \mathrm{TSTI}$ and $\mathcal{M}= \langle M_0, M_1, \ldots, \in_0^{\mathcal{M}}, \in_1^{\mathcal{M}}, \ldots \rangle$ with $\mathcal{M} \models \mathrm{TST}$ finitely generated by at least $(2^K)^{k^\prime+2}$ atoms. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$.\\
\\
\indent Our approach will be to define colour classes $\mathcal{C}_{i, j}$, the elements of which we will call colours, and functions $c_{i, j}^{\mathcal{M}}: M_i \longrightarrow \mathcal{C}_{i, j}$ and $c_{i, j}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{i, j}$, which we will call colourings, for all $i \in \mathbb{N}$ and for all $0 \leq j \leq k^\prime$. For all $0 < j \leq k^\prime$, the colourings $c_{i, j}^{\mathcal{M}}$ will be defined using the elements $a_1^{r_1}, \ldots, a_{j^\prime}^{r_{j^\prime}}$ where $j^\prime= \sum_{1 \leq m \leq j} K_m$, and in the process of defining the colourings $c_{i, j}^{\mathcal{N}}$ we will construct corresponding elements $b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}} \in \mathcal{N}$. The colourings will be designed with the following properties:
\begin{itemize}
\item[(i)] For a fixed colour $\alpha$ in some $\mathcal{C}_{i, j}$, the property of being an element of $\mathcal{N}$ that is given colour $\alpha$ by $c_{i, j}^{\mathcal{N}}$ will be definable by an $\mathcal{L}_{\mathrm{TST}}$-formula, $\Phi_{i, j, \alpha}$, with parameters over $\mathcal{N}$.
\item[(ii)] The colour given to an element $x$ in $\mathcal{M}$ (or $\mathcal{N}$) by the colouring $c_{i, j}^{\mathcal{M}}$ (respectively $c_{i, j}^{\mathcal{N}}$) will tell us which quantifier-free $\mathcal{L}_{\mathrm{TST}}$-formulae with parameters $a_1^{r_1}, \ldots, a_{j^\prime}^{r_{j^\prime}}$ (respectively $b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}$), where $j^\prime= \sum_{1 \leq m \leq j} K_m$, are satisfied by $x$ in $\mathcal{M}$ (respectively $\mathcal{N}$).
\item[(iii)] For every colour $\beta$ in $\mathcal{C}_{i, j}$, the colour given to an element $x$ in $\mathcal{M}$ (or $\mathcal{N}$) by the colouring $c_{i+1, j}^{\mathcal{M}}$ (respectively $c_{i+1, j}^{\mathcal{N}}$) will tell us whether or not there is an element $y$ in $\mathcal{M}$ (respectively $\mathcal{N}$) such that $\mathcal{M} \models y \in_i x$ (respectively $\mathcal{N} \models y \in_i x$) and $y$ is given colour $\beta$ by $c_{i, j}^{\mathcal{M}}$ (respectively $c_{i, j}^{\mathcal{N}}$).
\item[(iv)] For every colour $\beta$ in $\mathcal{C}_{i, j}$, the colour given to an element $x$ in $\mathcal{M}$ (or $\mathcal{N}$) by the colouring $c_{i+1, j}^{\mathcal{M}}$ (respectively $c_{i+1, j}^{\mathcal{N}}$) will tell us whether or not there is an element $y$ in $\mathcal{M}$ (respectively $\mathcal{N}$) such that $\mathcal{M} \models y \notin_i x$ (respectively $\mathcal{N} \models y \notin_i x$) and $y$ is given colour $\beta$ by $c_{i, j}^{\mathcal{M}}$ (respectively $c_{i, j}^{\mathcal{N}}$).
\end{itemize}
Note that since $\mathcal{M}$ is finitely generated, the analogue of condition (i) automatically holds for $\mathcal{M}$.\\
\\
\indent Before defining the colour classes $\mathcal{C}_{i, j}$ and the colourings $c_{i,j}^{\mathcal{M}}$ and $c_{i, j}^{\mathcal{N}}$ we first introduce the following definitions:
\begin{Definitions1}
Let $m \in \mathbb{N}$. We say that a colour $\alpha \in \mathcal{C}_{i, j}$ is $m$-special with respect to a colouring $f: X \longrightarrow \mathcal{C}_{i, j}$ if and only if
$$|\{ x \in X \mid f(x)= \alpha\}|= m.$$
If $\alpha \in \mathcal{C}_{i, j}$ is $0$-special then we say that $\alpha$ is forbidden.
\end{Definitions1}
\begin{Definitions1}
Let $m \in \mathbb{N}$. We say that a colour $\alpha \in \mathcal{C}_{i, j}$ is $m$-abundant with respect to a colouring $f: X \longrightarrow \mathcal{C}_{i, j}$ if and only if
$$|\{ x \in X \mid f(x)= \alpha\}|\geq m.$$
\end{Definitions1}
\begin{Definitions1}
Let $J \in \mathbb{N}$. We say that colourings $f: X \longrightarrow \mathcal{C}_{i, j}$ and $g: Y \longrightarrow \mathcal{C}_{i, j}$ are $J$-similar if and only if for all $0 \leq m < J$ and for all $\alpha \in \mathcal{C}_{i, j}$,
$$\alpha \textrm{ is } m \textrm{-special w.r.t. } f \textrm{ if and only if } \alpha \textrm{ is } m \textrm{-special w.r.t. } g.$$
\end{Definitions1}
The colour classes $\mathcal{C}_{i, j}$ and colourings $c_{i, j}^\mathcal{M}$ and $c_{i, j}^\mathcal{N}$ for all $i \in \mathbb{N}$ and for all $0 \leq j \leq k^\prime$ will be defined by a two-dimensional recursion. At each stage of the construction we will ensure that $c_{i, j}^\mathcal{M}$ and $c_{i, j}^\mathcal{N}$ are $(2^K)^{k^\prime-j+2}$-similar.\\
\\
Let $\mathcal{C}_{0, 0}= \{0\}$. Define $c_{0, 0}^{\mathcal{M}}: M_0 \longrightarrow \mathcal{C}_{0, 0}$ by
$$c_{0, 0}^{\mathcal{M}}(x)= 0 \textrm{ for all } x \in M_0.$$
Define $c_{0, 0}^{\mathcal{N}}: N_0 \longrightarrow \mathcal{C}_{0, 0}$ by
$$c_{0, 0}^{\mathcal{N}}(x)= 0 \textrm{ for all } x \in N_0.$$
Let $\Phi_{0, 0, 0}(x^0)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula $x^0=x^0$. Note that for all $x \in N_0$,
$$\mathcal{N} \models \Phi_{0, 0, 0}[x] \textrm{ if and only if } c_{0, 0}^\mathcal{N}(x)= 0.$$
\begin{Lemma1}
The colourings $c_{0, 0}^{\mathcal{M}}$ and $c_{0, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar.
\end{Lemma1}
\begin{proof}
This follows immediately from the fact that $|M_0| \geq (2^K)^{k^\prime+2}$.
\Square
\end{proof}
We now turn to defining the colour classes $\mathcal{C}_{i, 0}$ and colourings $c_{i, 0}^{\mathcal{M}}: M_i \longrightarrow \mathcal{C}_{i, 0}$ and $c_{i, 0}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{i, 0}$ for all $i \in \mathbb{N}$. Suppose that we have defined the colour class $\mathcal{C}_{n, 0}$ with a canonical ordering, colourings $c_{n, 0}^{\mathcal{M}}: M_n \longrightarrow \mathcal{C}_{n, 0}$ and $c_{n, 0}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{n, 0}$ and $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{n, 0, \alpha}(x^n)$ for all $\alpha \in \mathcal{C}_{n, 0}$ with the following properties:
\begin{itemize}
\item[(I)] $c_{n, 0}^{\mathcal{M}}$ and $c_{n, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar,
\item[(II)] for all $\alpha \in \mathcal{C}_{n, 0}$ and for all $x \in N_n$,
$$\mathcal{N} \models \Phi_{n, 0, \alpha}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha.$$
\end{itemize}
Let $\mathcal{C}_{n, 0}= \{ \alpha_1, \ldots, \alpha_q \}$ be the enumeration obtained from the canonical ordering. Define $\mathcal{C}_{n+1, 0}= 2^{2 \cdot q}$ --- the set of all 0-1 sequences of length $2 \cdot q$. Define $c_{n+1, 0}^{\mathcal{M}}: M_{n+1} \longrightarrow \mathcal{C}_{n+1, 0}$ such that for all $x \in M_{n+1}$,
$$c_{n+1, 0}^{\mathcal{M}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
$$\textrm{where } f_i= \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in M_n, \textrm{ if } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ then } \mathcal{M}\models y \notin_n x\\
1 & \textrm{if there exists } y \in M_n, \textrm{ s.t. } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ and } \mathcal{M} \models y \in_n x
\end{array} \right.$$
$$\textrm{and } g_i= \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in M_n, \textrm{ if } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ then } \mathcal{M} \models y \in_n x\\
1 & \textrm{if there exists } y \in M_n \textrm{ s.t. } c_{n, 0}^{\mathcal{M}}(y)= \alpha_i \textrm{ and } \mathcal{M} \models y \notin_n x
\end{array} \right.$$
\begin{Examp1}
Using this definition we get $\mathcal{C}_{1, 0}= \{ \langle 0, 0 \rangle, \langle 1, 0 \rangle, \langle 0, 1 \rangle, \langle 1, 1 \rangle \}$. There are no $x \in M_1$ which are given the colour $\langle 0, 0 \rangle$ by $c_{1, 0}^{\mathcal{M}}$. The only point in $M_1$ which is given the colour $\langle 1, 0 \rangle$ by $c_{1, 0}^{\mathcal{M}}$ is $(V^1)^{\mathcal{M}}$. Similarly, the only point in $M_1$ which is given the colour $\langle 0, 1 \rangle$ by $c_{1, 0}^{\mathcal{M}}$ is $(\emptyset^1)^{\mathcal{M}}$. Every other point in $M_1$ is given the colour $\langle 1, 1 \rangle$ by $c_{1, 0}^{\mathcal{M}}$.
\end{Examp1}
We define the colouring $c_{n+1, 0}^{\mathcal{N}}: N_{n+1} \longrightarrow \mathcal{C}_{n+1, 0}$ identically. Define $c_{n+1, 0}^{\mathcal{N}}: N_{n+1} \longrightarrow \mathcal{C}_{n+1, 0}$ such that for all $x \in N_{n+1}$,
$$c_{n+1, 0}^{\mathcal{N}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
$$\textrm{where } f_i= \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in N_n, \textrm{ if } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ then } \mathcal{N} \models y \notin_n x\\
1 & \textrm{if there exists } y \in N_n, \textrm{ s.t. } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ and } \mathcal{N} \models y \in_n x
\end{array} \right.$$
$$\textrm{and } g_i= \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in N_n, \textrm{ if } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ then } \mathcal{N} \models y \in_n x\\
1 & \textrm{if there exists } y \in N_n \textrm{ s.t. } c_{n, 0}^{\mathcal{N}}(y)= \alpha_i \textrm{ and } \mathcal{N} \models y \notin_n x
\end{array} \right.$$
We first show that there are $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{n+1, 0, \beta}$, for all $\beta \in \mathcal{C}_{n+1, 0}$, that satisfy condition (II) above for the colouring $c_{n+1, 0}^{\mathcal{N}}$.
\begin{Lemma1} \label{Th:LiftedColouringDefinable}
For all $\beta \in \mathcal{C}_{n+1, 0}$, there is an $\mathcal{L}_{\mathrm{TST}}$-formula $\Phi_{n+1, 0, \beta}(x^{n+1})$ such that for all $x \in N_{n+1}$,
$$\mathcal{N} \models \Phi_{n+1, 0, \beta}[x] \textrm{ if and only if } c_{n+1, 0}^{\mathcal{N}}(x)= \beta.$$
\end{Lemma1}
\begin{proof}
For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$,
$$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$
Let $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle \in \mathcal{C}_{n+1, 0}$. For all $1 \leq i \leq q$ and $j \in \{ 0, 1 \}$ define the $\mathcal{L}_{\mathrm{TST}}$-formula $\Theta_{i, j}^\beta(x^{n+1})$ by:
$$\Theta_{i, 0}^\beta(x^{n+1}) \textrm{ is } \left\{ \begin{array}{ll}
\forall y^n(\Phi_{n, 0, \alpha_i}(y^n) \Rightarrow y^n \notin x^{n+1}) & \textrm{if } f_i= 0\\
\exists y^n( y^n \in x^{n+1} \land \Phi_{n, 0, \alpha_i}(y^n)) & \textrm{if } f_i= 1
\end{array}\right.$$
$$\Theta_{i, 1}^\beta(x^{n+1}) \textrm{ is } \left\{ \begin{array}{ll}
\forall y^n(\Phi_{n, 0, \alpha_i}(y^n) \Rightarrow y^n \in x^{n+1}) & \textrm{if } g_i= 0\\
\exists y^n( y^n \notin x^{n+1} \land \Phi_{n, 0, \alpha_i}(y^n)) & \textrm{if } g_i= 1
\end{array}\right.$$
Define $\Phi_{n+1, 0, \beta}(x^{n+1})$ to be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigwedge_{1 \leq i \leq q} \bigwedge_{j \in \{0, 1\}} \Theta_{i, j}^\beta(x^{n+1}).$$
It follows from the definition of $c_{n+1, 0}^{\mathcal{N}}$ that for all $x \in N_{n+1}$,
$$\mathcal{N} \models \Phi_{n+1, 0, \beta}[x] \textrm{ if and only if } c_{n+1, 0}^{\mathcal{N}}(x)= \beta.$$
\Square
\end{proof}
We now turn to showing that $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar. In order to prove this we introduce the following sets:
$$\mathrm{FOR}_n= \{ i \in [q] \mid \alpha_i \textrm{ is forbidden w.r.t. } c_{n, 0}^{\mathcal{M}} \textrm{ and } c_{n, 0}^{\mathcal{N}} \},$$
$$m\textrm{-}\mathrm{SPC}_n= \{ i \in [q] \mid \alpha_i \textrm{ is } m\textrm{-special w.r.t. } c_{n, 0}^{\mathcal{M}} \textrm{ and } c_{n, 0}^{\mathcal{N}} \} \textrm{ for } 1 \leq m < (2^K)^{k^\prime+2},$$
$$\mathrm{ABN}_n= \{ i \in [q] \mid \alpha_i \textrm{ is } (2^K)^{k^\prime+2}\textrm{-abundant w.r.t. } c_{n, 0}^{\mathcal{M}} \textrm{ and } c_{n, 0}^{\mathcal{N}} \}.$$
We classify the colours in $\mathcal{C}_{n+1, 0}$ which are forbidden, $1$-special and abundant with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$.
\begin{Lemma1} \label{Th:ClassifyForbiddenBase}
Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ if and only if either
\begin{itemize}
\item[(i)] there exists an $i \in [q]$ with $i \notin \mathrm{FOR}_n$ such that $f_i= g_i= 0$ OR,
\item[(ii)] there exists an $i \in 1\textrm{-}\mathrm{SPC}_n$ such that $f_i= g_i= 1$ OR,
\item[(iii)] there exists an $i \in \mathrm{FOR}_n$ such that $f_i= 1$ or $g_i=1$.
\end{itemize}
\end{Lemma1}
\begin{proof}
It is clear that if any of the conditions (i)-(iii) hold then the colour $\beta$ is forbidden. Conversely, suppose that none of the conditions (i)-(iii) hold. We need to show that $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$. We first construct a point in $\mathcal{N}$ that is given colour $\beta$ by $c_{n+1, 0}^{\mathcal{N}}$. For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$,
$$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$
Let $\Theta_1(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigvee_{g_i=0} \Phi_{n, 0, \alpha_i}(x^n).$$
We work inside $\mathcal{N}$. Let $X_1= \{ x^n \mid \Theta_1(x^n) \}$. Note that comprehension ensures that $X_1$ exists. Let
$$B= \mathrm{ABN}_n \cup \bigcup_{2 \leq m < (2^K)^{k^\prime+2}} m \textrm{-}\mathrm{SPC}_n$$
and let $A= \{ i \in B \mid f_i=g_i=1 \}$. Let $\Theta_2(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigvee_{i \in A} \Phi_{n, 0, \alpha_i}(x^n).$$
Let $X_2= \{ x^n \mid \Theta_2(x^n) \}$. Again, comprehension ensures that $X_2$ exists. For all $i \in A$, let $x_i \in N_n$ be such that $c_{n, 0}^{\mathcal{N}}(x_i)= \alpha_i$. Now, let $X= X_1 \cup (X_2 \backslash \{ x_i \mid i \in A \})$. Comprehension guarantees that $X$ exists in $\mathcal{N}$ and our construction ensures that $c_{n+1, 0}^{\mathcal{N}}(X)= \beta$. An identical construction shows that if none of the conditions (i)-(iii) hold then there is a point $X$ in $\mathcal{M}$ such that $c_{n+1, 0}^{\mathcal{M}}(X)= \beta$.
\Square
\end{proof}
\begin{Lemma1} \label{Th:ClassifyOneSpecialBase}
Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is $1$-special with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ if and only if $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and for all $i \in [q]$ with $i \notin \mathrm{FOR}_n$, $f_i= 0$ or $g_i= 0$.
\end{Lemma1}
\begin{proof}
Suppose $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and for all $i \in [q]$ with $i \notin \mathrm{FOR}_n$, $f_i= 0$ or $g_i= 0$. If $x$ is a point that is given colour $\beta$ by $c_{n+1, 0}^{\mathcal{M}}$ or $c_{n+1, 0}^{\mathcal{N}}$ then $x$ is completely determined in $\mathcal{M}$ or $\mathcal{N}$ respectively. Therefore $\beta$ is $1$-special.\\
Conversely, suppose that $\beta$ is not forbidden and there exists an $i \in [q]$ with $i \notin \mathrm{FOR}_n$ such that $f_i= g_i= 1$. We will show that $\beta$ is not $1$-special with respect to $c_{n+1, 0}^{\mathcal{M}}$ or $c_{n+1, 0}^{\mathcal{N}}$. We first construct two distinct points of $\mathcal{N}$ that are given colour $\beta$ by $c_{n+1, 0}^{\mathcal{N}}$. For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$,
$$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$
We work inside $\mathcal{N}$. Let $A= \{i \in [q] \mid f_i= g_i= 1 \}$. Since $\beta$ is not forbidden, for all $i \in A$, we can find $x_i, y_i \in N_n$ such that $c_{n, 0}^{\mathcal{N}}(x_i)= c_{n, 0}^{\mathcal{N}}(y_i)= \alpha_i$ and $x_i \neq y_i$. Let $\Theta_1(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigvee_{g_i=0} \Phi_{n, 0, \alpha_i}(x^n).$$
Let $\Theta_2(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigvee_{i \in A} \Phi_{n, 0, \alpha_i}(x^n).$$
Let $X_1= \{ x^n \mid \Theta_1(x^n) \}$ and let $X_2= \{ x^n \mid \Theta_2(x^n) \}$. Comprehension guarantees that both $X_1$ and $X_2$ exist. Let $X= X_1 \cup (X_2 \backslash \{ x_i \mid i \in A \})$ and let $Y= X_1 \cup (X_2 \backslash \{ y_i \mid i \in A \})$. Now, this construction ensures that $c_{n+1, 0}^{\mathcal{N}}(X)= c_{n+1, 0}^{\mathcal{N}}(Y)= \beta$ and $X \neq Y$. Therefore $\beta$ is not $1$-special with respect to $c_{n+1, 0}^{\mathcal{N}}$. An identical construction shows that $\beta$ is not $1$-special with respect to $c_{n+1, 0}^{\mathcal{M}}$.
\Square
\end{proof}
\begin{Lemma1} \label{Th:ClassifyAbundantBase}
Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. If $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and there exists an $i \in \mathrm{ABN}_n$ such that $f_i= g_i= 1$ then $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$.
\end{Lemma1}
\begin{proof}
Suppose that $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and there exists an $i \in \mathrm{ABN}_n$ such that $f_i= g_i= 1$. We first construct $(2^K)^{k^\prime+2}$ distinct points in $\mathcal{N}$ that are given colour $\beta$ by $c_{n+1, 0}^{\mathcal{N}}$. For all $1 \leq i \leq q$, let $\Phi_{n, 0, \alpha_i}(x^{n})$ be such that for all $x \in N_n$,
$$\mathcal{N} \models \Phi_{n, 0, \alpha_i}[x] \textrm{ if and only if } c_{n, 0}^{\mathcal{N}}(x)= \alpha_i.$$
We work inside $\mathcal{N}$. Let $u \in \mathrm{ABN}_n$ be such that $f_u= g_u= 1$. Let $A= \{ i \in [q] \mid f_i= g_i= 1 \}$. For all $i \in A$ with $i \neq u$, let $x_i \in N_n$ be such that $c_{n, 0}^{\mathcal{N}}(x_i)= \alpha_i$. Let $y_1, \ldots, y_{(2^K)^{k^\prime+2}} \in N_n$ be such that for all $1 \leq v \leq (2^K)^{k^\prime+2}$, $c_{n, 0}^{\mathcal{N}}(y_v)= \alpha_u$ and for all $1 \leq v_1 < v_2 \leq (2^K)^{k^\prime+2}$, $y_{v_1} \neq y_{v_2}$. Let $\Theta_1(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigvee_{g_i=0} \Phi_{n, 0, \alpha_i}(x^n).$$
Let $\Theta_2(x^n)$ be the $\mathcal{L}_{\mathrm{TST}}$-formula
$$\bigvee_{i \in A} \Phi_{n, 0, \alpha_i}(x^n).$$
Let $X_1= \{ x^n \mid \Theta_1(x^n) \}$ and let $X_2= \{ x^n \mid \Theta_2(x^n) \}$. Comprehension guarantees that $X_1$ and $X_2$ exist. For all $1 \leq v \leq (2^K)^{k^\prime+2}$, let
$$Y_v= X_1 \cup (X_2 \backslash (\{ x_i \mid i \in A \land i \neq u \} \cup \{y_v\})).$$
This construction ensures that for all $1 \leq v_1 < v_2 \leq (2^K)^{k^\prime+2}$, $Y_{v_1} \neq Y_{v_2}$ and for all $1 \leq v \leq (2^K)^{k^\prime+2}$, $c_{n+1, 0}^{\mathcal{N}}(Y_v)= \beta$. Therefore $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to $c_{n+1, 0}^{\mathcal{N}}$. An identical construction shows that $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to $c_{n+1, 0}^{\mathcal{M}}$.
\Square
\end{proof}
\noindent This allows us to show that the colourings $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar.
\begin{Lemma1} \label{Th:BaseColouringsSimilar}
The colourings $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar.
\end{Lemma1}
\begin{proof}
Lemma \ref{Th:ClassifyForbiddenBase} shows that for all $\beta \in \mathcal{C}_{n+1,0}$,
$$\beta \textrm{ is forbidden w.r.t. } c_{n+1, 0}^{\mathcal{M}} \textrm{ if and only if } \beta \textrm{ is forbidden w.r.t. } c_{n+1, 0}^{\mathcal{N}}.$$
Lemma \ref{Th:ClassifyOneSpecialBase} shows that for all $\beta \in \mathcal{C}_{n+1,0}$,
$$\beta \textrm{ is } 1\textrm{-special w.r.t. } c_{n+1, 0}^{\mathcal{M}} \textrm{ if and only if } \beta \textrm{ is } 1\textrm{-special w.r.t. } c_{n+1, 0}^{\mathcal{N}}.$$
Let $\beta \in \mathcal{C}_{n+1, 0}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. Lemma \ref{Th:ClassifyAbundantBase} shows that if $\beta$ is not forbidden with respect to $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ and there is an $i \in \mathrm{ABN}_n$ such that $f_i= g_i= 1$ then $\beta$ is $(2^K)^{k^\prime+2}$-abundant with respect to both $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$. The remaining case is if $\beta$ is not forbidden or $1$-special and for all $i \in \mathrm{ABN}_n$, $f_i= 0$ or $g_i= 0$. Let
$$B= \bigcup_{2 \leq m < (2^K)^{k^\prime+2}} m \textrm{-}\mathrm{SPC}.$$
In this case the number of $x \in M_{n+1}$ ($\in N_{n+1}$) with colour $\beta$ is completely determined by the number of $y \in M_{n}$ ($\in N_n$ respectively) with colour $\alpha_i$ such that $i \in B$ and $f_i= g_i= 1$. Therefore, the colourings $c_{n+1, 0}^{\mathcal{M}}$ and $c_{n+1, 0}^{\mathcal{N}}$ are $(2^K)^{k^\prime+2}$-similar.
\Square
\end{proof}
\noindent Therefore, by induction, for all $i \in \mathbb{N}$, the colourings $c_{i, 0}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, 0}$ and $c_{i, 0}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, 0}$ are $(2^K)^{k^\prime+2}$-similar.\\
\\
\indent We now turn to defining the colour classes $\mathcal{C}_{i, j}$, and the colourings $c_{i, j}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, j}$ and $c_{i, j}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, j}$ for $1 \leq j \leq k^\prime$ and $i \in \mathbb{N}$. Let $0 \leq n < k^\prime$. Suppose that the colour classes $\mathcal{C}_{i, n}$ have been defined for all $i \in \mathbb{N}$ and that each of these colour classes has a canonical ordering. Let $j^\prime= \sum_{1 \leq m \leq n} K_m$ and suppose that $b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}} \in \mathcal{N}$ have been chosen. Moreover, suppose that for all $i \in \mathbb{N}$ and for all $\alpha \in \mathcal{C}_{i, n}$, the colourings $c_{i, n}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, n}$ and $c_{i, n}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, n}$, and the $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{i, n, \alpha}(x^i, \vec{z})$ have been defined with the following properties
\begin{itemize}
\item[(I')] $c_{i, n}^\mathcal{M}$ and $c_{i, n}^\mathcal{N}$ are $(2^K)^{k^\prime-n+2}$-similar,
\item[(II')] for all $x \in N_i$,
$$\mathcal{N} \models \Phi_{i, n, \alpha}[x, b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}] \textrm{ if and only if } c_{i, n}^\mathcal{N}(x)= \alpha.$$
\end{itemize}
Observe that $r_{j^\prime+1}= \ldots = r_{j^\prime+K_{n+1}}$ and let $r= r_{j^\prime+1}$. We will define the colour classes $\mathcal{C}_{i, n+1}$ and colourings $c_{i, n+1}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, n+1}$ and $c_{i, n+1}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, n+1}$ such that for all $i \in \mathbb{N}$, $c_{i, n+1}^\mathcal{M}$ and $c_{i, n+1}^\mathcal{N}$ are $(2^K)^{k^\prime-n+1}$-similar and the colouring $c_{i, n+1}^\mathcal{N}$ is definable in $\mathcal{N}$. In the process of achieving this goal we will identify points $b_{j^\prime+1}^{r}, \ldots, b_{j^\prime+K_{n+1}}^{r} \in N_{r}$.\\
\\
For all $0 \leq i < r-1$, define
$$\mathcal{C}_{i, n+1}= \mathcal{C}_{i, n},$$
$$c_{i, n+1}^\mathcal{M}= c_{i, n}^\mathcal{M},$$
$$c_{i, n+1}^\mathcal{N}= c_{i, n}^\mathcal{N}.$$
We now define the colour class $\mathcal{C}_{r-1, n+1}$, and the colourings $c_{r-1, n+1}^\mathcal{M}: M_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$ and $c_{r-1, n+1}^\mathcal{N}: N_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$. Let $\mathcal{C}_{r-2, n+1}= \mathcal{C}_{r-2, n}= \{ \alpha_1, \ldots, \alpha_q \}$ be obtained from the canonical ordering. Consider $a_{j^\prime+1}^{r}, \ldots, a_{j^\prime+K_{n+1}}^{r} \in M_{r}$ and use $\bar{a}_1, \ldots, \bar{a}_{K_{n+1}}$ to denote this sequence of elements. Define $\mathcal{C}_{r-1, n+1}= 2^{K_{n+1}} \times \mathcal{C}_{r-1, n}$ --- the set of all 0-1 sequences of length $K_{n+1}+2\cdot q$. Define $c_{r-1, n+1}^{\mathcal{M}}: M_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$ such that for all $x \in M_{r-1}$,
$$c_{r-1, n+1}^{\mathcal{M}}(x)= \langle F_1, \ldots, F_{K_{n+1}}, f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
$$\textrm{where } c_{r-1, n}^{\mathcal{M}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
$$\textrm{and } F_p= \left\{ \begin{array}{ll}
0 & \textrm{if } \mathcal{M} \models x \notin_{r-1} \bar{a}_p\\
1 & \textrm{if } \mathcal{M} \models x \in_{r-1} \bar{a}_p
\end{array}\right. \textrm{ for all } 1 \leq p \leq K_{n+1}.$$
\begin{Lemma1} \label{Th:ColourRefinement}
There exists $\bar{b}_1, \ldots, \bar{b}_{K_{n+1}} \in N_{r}$ such that $c_{r-1, n+1}^{\mathcal{M}}$ and the colouring $c_{r-1, n+1}^{\mathcal{N}}: N_{r-1} \longrightarrow \mathcal{C}_{r-1, n+1}$, defined such that for all $x \in N_{r-1}$,
$$c_{r-1, n+1}^{\mathcal{N}}(x)= \langle F_1, \ldots, F_{K_{n+1}}, f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
\begin{equation} \label{Eq:RefinedNColouring}
\textrm{where } c_{r_{j^\prime+1}-1, n}^{\mathcal{N}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle
\end{equation}
$$\textrm{and } F_p= \left\{ \begin{array}{ll}
0 & \textrm{if } \mathcal{N} \models x \notin_{r-1} \bar{b}_p\\
1 & \textrm{if } \mathcal{N} \models x \in_{r-1} \bar{b}_p
\end{array}\right. \textrm{ for all } 1 \leq p \leq K_{n+1},$$
are $(2^K)^{k^\prime-n+1}$-similar.
\end{Lemma1}
\begin{proof}
Let $\mathcal{C}_{r-1, n}= \{ \alpha_1, \ldots, \alpha_{q^\prime} \}$ be obtained from the canonical ordering. For all $1 \leq i \leq q^\prime$ and for all $\sigma \in 2^{K_{n+1}}$ define $X_\sigma^i \subseteq M_{r-1}$ by
$$X_\sigma^i= \{ x \in M_{r-1} \mid (c_{r-1, n}^{\mathcal{M}}(x)= \alpha_i) \land (\forall v \in K_{n+1})(\sigma(v)=1 \iff x \in \bar{a}_v)\}.$$
Note that for all $1 \leq i \leq q^\prime$, the sets $\langle X_\sigma^i \mid \sigma \in 2^{K_{n+1}} \rangle$ partition the elements of $M_{r-1}$ that are given colour $\alpha_i$ by $c_{r-1, n}^{\mathcal{M}}$ into $2^{K_{n+1}}$ pieces. For each $1 \leq i \leq q^\prime$ choose a sequence $\langle Z_\sigma^i \mid \sigma \in 2^{K_{n+1}} \rangle$ such that for all $\sigma \in 2^{K_{n+1}}$,
\begin{itemize}
\item[(i)] $Z_\sigma^i \in N_{r}$,
\item[(ii)] for all $z \in N_{r-1}$ with $\mathcal{N} \models z \in_{r-1} Z_\sigma^i$, $c_{r-1, n}^{\mathcal{N}}(z)= \alpha_i$,
\item[(iii)] if $|X_\sigma^i| < (2^K)^{k^\prime-n+1}$ then $|\{z \in \mathcal{N} \mid \mathcal{N} \models z \in_{r-1} Z_\sigma^i \}|= |X_\sigma^i|$,
\item[(iv)] if $|X_\sigma^i| \geq (2^K)^{k^\prime-n+1}$ then $|\{z \in \mathcal{N} \mid \mathcal{N} \models z \in_{r-1} Z_\sigma^i \}| \geq (2^K)^{k^\prime-n+1}$.
\end{itemize}
To see that we can make this choice we work inside $\mathcal{N}$. For all $1 \leq i \leq q^\prime$, let $\Phi_{r-1, n, \alpha_i}(x^{r-1}, \vec{z})$ be such that for all $x \in N_{r-1}$,
$$\mathcal{N} \models \Phi_{r-1, n, \alpha_i}[x, b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}] \textrm{ if and only if } c_{r-1, n}^\mathcal{N}(x)= \alpha_i.$$
For all $1 \leq i \leq q^\prime$, let $W_i= \{ x^{r-1} \mid \Phi_{r-1, n, \alpha_i}(x^{r-1}, b_1^{r_1}, \ldots, b_{j^\prime}^{r_{j^\prime}}) \}$. Comprehension ensures that the $W_i$s exist. For all $1 \leq i \leq q^\prime$ and for all $\sigma \in 2^{K_{n+1}}$, $Z_\sigma^i$ can be chosen to be a finite or cofinite subset of $W_i$. Moreover, the fact that $c_{r-1, n}^\mathcal{M}$ and $c_{r-1, n}^\mathcal{N}$ are $(2^K)^{k^\prime-n+2}$-similar ensures that for all $1 \leq i \leq q^\prime$ we can choose the sequence $\langle Z_\sigma^i \mid \sigma \in 2^{K_{n+1}} \rangle$ to satisfy condition (iii) above.\\
Now, for all $1 \leq p \leq K_{n+1}$, let $\bar{b}_p \in N_{r}$ be such that
$$\mathcal{N} \models \bar{b}_p= \bigcup_{1 \leq i \leq q^\prime} \bigcup_{^{\sigma \in 2^{K_{n+1}}}_{\textrm{s.t. } \sigma(p)=1}} Z_\sigma^i.$$
This construction ensures that the colourings $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$ define by (\ref{Eq:RefinedNColouring}) are $(2^K)^{k^\prime-n+1}$-similar.
\Square
\end{proof}
Let $b_{j^\prime+1}^{r}, \ldots, b_{j^\prime+K_{n+1}}^{r} \in \mathcal{N}$ be the points $\bar{b}_1, \ldots, \bar{b}_{K_{n+1}}$ produced in the proof of Lemma \ref{Th:ColourRefinement} and let $c_{r-1, n+1}^{\mathcal{N}}$ be defined by (\ref{Eq:RefinedNColouring}). Therefore $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar. We can immediately observe that the colouring $c_{r-1, n+1}^{\mathcal{N}}$ is definable in $\mathcal{N}$ by an $\mathcal{L}_{\mathrm{TST}}$-formula using parameters $b_{1}^{r_{1}}, \ldots, b_{j^\prime+K_{n+1}}^{r_{j^\prime+K_{n+1}}}$.
\begin{Lemma1} \label{Th:ColourRefinmentDefinable}
For all $\alpha \in \mathcal{C}_{r-1, n+1}$, there exists an $\mathcal{L}_{\mathrm{TST}}$-formula $\Phi_{r-1, n+1, \alpha}(x^{r-1}, \vec{z})$ such that for all $x \in N_{r-1}$,
$$\mathcal{N} \models \Phi_{r-1, n+1, \alpha}[x, b_{1}^{r_{1}}, \ldots, b_{j^\prime+K_{n+1}}^{r_{j^\prime+K_{n+1}}}] \textrm{ if and only if } c_{r-1, n+1}^{\mathcal{N}}(x)= \alpha.$$
\Square
\end{Lemma1}
Let $t= \sum_{1 \leq m \leq n+1} K_m$. Lemma \ref{Th:ColourRefinement} and Lemma \ref{Th:ColourRefinmentDefinable} show that we can define colourings $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$, and $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{r-1, n+1, \alpha}(x^{r-1}, \vec{z})$ for all $\alpha \in \mathcal{C}_{r-1, n+1}$ which satisfy the following properties:
\begin{itemize}
\item[(I'')] $c_{r-1, n+1}^{\mathcal{M}}$ and $c_{r-1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar,
\item[(II'')] for all $x \in N_{r-1}$,
$$\mathcal{N} \models \Phi_{r-1, n+1, \alpha}[x, b_1^{r_1}, \ldots, b_t^{r_t}] \textrm{ if and only if } c_{r-1, n+1}^{\mathcal{N}}(x)= \alpha.$$
\end{itemize}
\indent We now turn to defining the colour classes $\mathcal{C}_{i, n+1}$, and the colourings $c_{i, n+1}^\mathcal{M}: M_i \longrightarrow \mathcal{C}_{i, n+1}$ and $c_{i, n+1}^\mathcal{N}: N_i \longrightarrow \mathcal{C}_{i, n+1}$ for all $i \geq r$. Let $i \geq r-1$. Suppose that the colour class $\mathcal{C}_{i, n+1}$ has been defined with a canonical ordering. Suppose, also, that the colourings $c_{i, n+1}^{\mathcal{M}}: M_i \longrightarrow \mathcal{C}_{i, n+1}$ and $c_{i, n+1}^{\mathcal{N}}: N_i \longrightarrow \mathcal{C}_{i, n+1}$, and the $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{i, n+1, \alpha}(x^{i}, \vec{z})$ have been defined and satisfy:
\begin{itemize}
\item[(I''')] $c_{i, n+1}^{\mathcal{M}}$ and $c_{i, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar,
\item[(II''')] for all $x \in N_i$,
$$\mathcal{N} \models \Phi_{i, n+1, \alpha}[x, b_1^{r_1}, \ldots, b_t^{r_t}] \textrm{ if and only if } c_{i, n+1}^{\mathcal{N}}(x)= \alpha.$$
\end{itemize}
We `lift' the colour class $\mathcal{C}_{i, n+1}$ and the colourings $c_{i, n+1}^{\mathcal{M}}$ and $c_{i, n+1}^{\mathcal{N}}$ in the same way that we `lifted' the colour classes $\mathcal{C}_{i, 0}$ and the colourings $c_{i, 0}^{\mathcal{M}}$ and $c_{i, 0}^{\mathcal{N}}$ above. Let $\mathcal{C}_{i, n+1}= \{ \alpha_1, \ldots, \alpha_q \}$ be obtained from the canonical ordering. Define $\mathcal{C}_{i+1, n+1}= 2^{2 \cdot q}$--- the set of all 0-1 sequence of length $2 \cdot q$. Define $c_{i+1, n+1}^{\mathcal{M}}: M_{i+1} \longrightarrow \mathcal{C}_{i+1, n+1}$ such that for all $x \in M_{i+1}$,
$$c_{i+1, n+1}^{\mathcal{M}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
$$\textrm{where } f_p = \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in M_i, \textrm{ if } c_{i,n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ then } \mathcal{M} \models y \notin_i x\\
1 & \textrm{if there exists } y \in M_i \textrm{ such that } c_{i, n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ and } \mathcal{M} \models y \in_i x
\end{array}\right.$$
$$\textrm{and } g_p = \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in M_i, \textrm{ if } c_{i, n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ then } \mathcal{M} \models y \in_i x\\
1 & \textrm{if there exists } y \in M_i \textrm{ such that } c_{i, n+1}^{\mathcal{M}}(y)= \alpha_p \textrm{ and } \mathcal{M} \models y \notin_i x
\end{array}\right.$$
Again, we define $c_{i+1, n+1}^{\mathcal{N}}$ identically. Define $c_{i+1, n+1}^{\mathcal{N}}: N_{i+1} \longrightarrow \mathcal{C}_{i+1, n+1}$ such that for all $x \in N_{i+1}$,
$$c_{i+1, n+1}^{\mathcal{N}}(x)= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$$
$$\textrm{where } f_p= \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in N_i, \textrm{ if } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ then } \mathcal{N} \models y \notin_i x\\
1 & \textrm{if there exists } y \in N_i \textrm{ such that } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ and } \mathcal{N} \models y \in_i x
\end{array}\right.$$
$$\textrm{and } g_p= \left\{ \begin{array}{ll}
0 & \textrm{if for all } y \in N_i, \textrm{ if } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ then } \mathcal{N} \models y \in_i x\\
1 & \textrm{if there exists } y \in N_i \textrm{ such that } c_{i, n+1}^{\mathcal{N}}(y)= \alpha_p \textrm{ and } \mathcal{N} \models y \notin_i x
\end{array}\right.$$
We first observe that there exists $\mathcal{L}_{\mathrm{TST}}$-formulae $\Phi_{i+1, n+1, \beta}(x^{i+1}, \vec{z})$ for each $\beta \in \mathcal{C}_{i+1, n+1}$ which witness the fact that the colouring $c_{i+1, n+1}^{\mathcal{N}}$ satisfies condition (II''').
\begin{Lemma1}
For all $\beta \in \mathcal{C}_{i+1, n+1}$, there is an $\mathcal{L}_{\mathrm{TST}}$-formula $\Phi_{i+1, n+1, \beta}(x^{i+1}, \vec{z})$ such that for all $x \in N_{i+1}$,
$$\mathcal{N} \models \Phi_{i+1, n+1, \beta}[x, b_1^{r_1}, \ldots, b_t^{r_t}] \textrm{ if and only if } c_{i+1, n+1}^{\mathcal{N}}(x)= \beta.$$
\end{Lemma1}
\begin{proof}
Identical to the proof Lemma \ref{Th:LiftedColouringDefinable} using the fact that $c_{i, n+1}^{\mathcal{N}}$ satisfies condition (II''').
\Square
\end{proof}
We now turn to showing that $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar. To do this we prove analogues of Lemmata \ref{Th:ClassifyForbiddenBase}, \ref{Th:ClassifyOneSpecialBase} and \ref{Th:ClassifyAbundantBase}.
$$\mathrm{FOR}_i^{n+1}= \{ v \in [q] \mid \alpha_v \textrm{ is forbidden w.r.t. } c_{i, n+1}^{\mathcal{M}} \textrm{ and } c_{i, n+1}^{\mathcal{N}} \},$$
$$m\textrm{-}\mathrm{SPC}_i^{n+1}= \{ v \in [q] \mid \alpha_v \textrm{ is } m \textrm{-special w.r.t. } c_{i, n+1}^{\mathcal{M}} \textrm{ and } c_{i, n+1}^{\mathcal{N}} \} \textrm{ for } 1 \leq m < (2^K)^{k^\prime-n+1},$$
$$\mathrm{ABN}_i^{n+1}= \{ v \in [q] \mid \alpha_v \textrm{ is } (2^K)^{k^\prime-n+1} \textrm{-abundant w.r.t. } c_{i, n+1}^{\mathcal{M}} \textrm{ and } c_{i, n+1}^{\mathcal{N}} \}.$$
\begin{Lemma1} \label{Th:ClassifyForbiddenRefined}
Let $\beta \in \mathcal{C}_{i+1, n+1}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is forbidden with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ if and only if either
\begin{itemize}
\item[(i)] there exists a $v \in [q]$ with $v \notin \mathrm{FOR}_i^{n+1}$ such that $f_v= g_v= 0$ OR,
\item[(ii)] there exists a $v \in 1\textrm{-}\mathrm{SPC}_i^{n+1}$ such that $f_v= g_v= 1$ OR,
\item[(iii)] there exists a $v \in \mathrm{FOR}_i^{n+1}$ such $f_v= 1$ or $g_v=1$.
\end{itemize}
\end{Lemma1}
\begin{proof}
Identical to the proof of Lemma \ref{Th:ClassifyForbiddenBase}.
\Square
\end{proof}
\begin{Lemma1} \label{Th:ClassifyOneSpecialRefined}
Let $\beta \in \mathcal{C}_{i+1, n+1}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. The colour $\beta$ is $1$-special with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ if and only if $\beta$ is not forbidden with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ and for all $v \in [q]$ with $v \notin \mathrm{FOR}_i^{n+1}$, $f_v=0$ or $g_v=0$.
\end{Lemma1}
\begin{proof}
Identical to the proof of Lemma \ref{Th:ClassifyOneSpecialBase}.
\Square
\end{proof}
\begin{Lemma1} \label{Th:ClassifyAbundantRefined}
Let $\beta \in \mathcal{C}_{i+1, n+1}$ with $\beta= \langle f_1, \ldots, f_q, g_1, \ldots, g_q \rangle$. If $\beta$ is not forbidden with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ and there exists a $v \in \mathrm{ABN}_i^{n+1}$ with $f_v= g_v= 1$ then $\beta$ is $(2^K)^{k^\prime-n+1}$-abundant with respect to $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$.
\end{Lemma1}
\begin{proof}
Identical to the proof of Lemma \ref{Th:ClassifyAbundantBase}.
\end{proof}
\noindent These results allow us to show that $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar.
\begin{Lemma1} \label{Th:RefinedColouringsSimilar}
The colourings $c_{i+1, n+1}^{\mathcal{M}}$ and $c_{i+1, n+1}^{\mathcal{N}}$ are $(2^K)^{k^\prime-n+1}$-similar.
\end{Lemma1}
\begin{proof}
Identical to the proof of Lemma \ref{Th:BaseColouringsSimilar} using Lemmata \ref{Th:ClassifyForbiddenRefined}, \ref{Th:ClassifyOneSpecialRefined} and \ref{Th:ClassifyAbundantRefined}.
\Square
\end{proof}
This recursion allows us to define the colour classes $\mathcal{C}_{n, k^\prime}$ and colourings $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$ for all $n \in \mathbb{N}$, and elements $b_1^{r_1}, \ldots, b_1^{r_k} \in \mathcal{N}$. The above arguments show that for all $n \in \mathbb{N}$, $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$ are $2^K$-similar. We have constructed the colourings $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$ so as the colour assigned to a point $x \in \mathcal{M}$ (or $\mathcal{N}$) completely captures the set of quantifier-free formulae with parameters $a_1^{r_1}, \ldots, a_k^{r_k}$ (respectively $b_1^{r_1}, \ldots, b_k^{r_k}$) that are satisfied by $x$.
\begin{Lemma1} \label{Th:ColouringsCaptureOneTypes}
Let $n \in \mathbb{N}$ and let $\theta(x_1^{r_1}, \ldots, x_k^{r_k}, x^n)$ be a quantifier-free $\mathcal{L}_{\mathrm{TST}}$-formula. If $x \in M_n$ and $y \in N_n$ are such that $c_{n, k^\prime}^{\mathcal{M}}(x)= c_{n, k^\prime}^{\mathcal{M}}(y)$ then
$$\mathcal{M} \models \theta[a_1^{r_1}, \ldots, a_k^{r_k}, x] \textrm{ if and only if } \mathcal{N} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, y]$$
\end{Lemma1}
\begin{proof}
This follows immediately from the definition of the colourings $c_{n, k^\prime}^{\mathcal{M}}$ and $c_{n, k^\prime}^{\mathcal{N}}$.
\Square
\end{proof}
\noindent Our construction also ensures that if $x \in M_{n+1}$ (or $N_{n+1}$) then the colour assigned to $x$ by $c_{n+1, k^{\prime}}^{\mathcal{M}}$ (respectively $c_{n+1, k^\prime}^{\mathcal{N}}$) tells us, for all $\alpha \in \mathcal{C}_{n, k^\prime}$, whether there exists a point $y \in M_n$ (respectively $N_n$) such that $c_{n, k^\prime}^{\mathcal{M}}(y)= \alpha$ (respectively $c_{n, k^\prime}^{\mathcal{N}}(y)= \alpha$) and $y$ is in the relationship $\in_n$ or $\notin_n$ to $x$ in $\mathcal{M}$ (respectively $\mathcal{N}$).
\begin{Lemma1} \label{Th:KeepingTrackOfColousLemma}
Let $x \in M_{n+1}$ and $y \in N_{n+1}$, and let $\alpha \in \mathcal{C}_{n, k^\prime}$. If $c_{n+1, k^\prime}^{\mathcal{M}}(x)= c_{n+1, k^\prime}^{\mathcal{N}}(y)$ then
$$(\exists z \in M_n)(c_{n, k^\prime}^{\mathcal{M}}(z)= \alpha \land \mathcal{M} \models z \in_n x) \textrm{ if and only if } (\exists z \in N_n)(c_{n, k^\prime}^{\mathcal{N}}(z)= \alpha \land \mathcal{N} \models z \in_n y),$$
$$\textrm{and }(\exists z \in M_n)(c_{n, k^\prime}^{\mathcal{M}}(z)= \alpha \land \mathcal{M} \models z \notin_n x) \textrm{ if and only if } (\exists z \in N_n)(c_{n, k^\prime}^{\mathcal{N}}(z)= \alpha \land \mathcal{N} \models z \notin_n y).$$
\end{Lemma1}
\begin{proof}
This follows immediately from the definition of the colourings $c_{n+1, k^\prime}^{\mathcal{M}}$ and $c_{n+1, k^\prime}^{\mathcal{N}}$.
\Square
\end{proof}
\noindent This allows us to show that an $\mathcal{L}_{\mathrm{TST}}$-sentence in the form (A) or (B) which is true $\mathcal{N}$ is also true in $\mathcal{M}$.
\begin{Theorems1} \label{Th:SentencesOfTypeBHoldInM}
Let $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ be an $\mathcal{L}_{\mathrm{TST}}$-formula with $\theta$ is quantifier-free. If $\mathcal{N} \models \phi$ then $\mathcal{M} \models \phi$.
\end{Theorems1}
\begin{proof}
Suppose that $\mathcal{N} \models \phi$. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$. Using $a_1^{r_1}, \ldots, a_k^{r_k}$ and the construction we presented above we can define the colour classes $\mathcal{C}_{n, k^\prime}$ and colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ for all $n \in \mathbb{N}$, and elements $b_1^{r_1}, \ldots, b_k^{r_k} \in \mathcal{N}$. The colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ are $2^K$-similar and satisfy Lemma \ref{Th:ColouringsCaptureOneTypes}. Let $e_1, \ldots, e_l \in N_s$ be such that
$$\mathcal{N} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, e_1, \ldots, e_l].$$
For all $1 \leq i \leq l$, let $d_i \in M_s$ such that $c_{s, k^\prime}^\mathcal{M}(d_i)= c_{s, k^\prime}^{\mathcal{N}}(e_i)$ and for all $1 \leq j < i$, $d_j \neq d_i$ if and only if $e_i \neq e_j$. The fact that $l < 2^K$ and $c_{s, k^\prime}^\mathcal{M}$ and $c_{s, k^\prime}^\mathcal{N}$ are $2^K$-similar ensures we can find $d_1, \ldots, d_l \in M_s$ satisfying these conditions. Now, since the variables $y_1^s, \ldots y_l^s$ all have the same type in $\theta$, the only atomic or negatomic subformulae of $\theta$ are in the form $y_i^s = y_j^s$, $y_i^s \in_s x_j^{r_j}$ if $r_j= s+1$, $x_i^{r_i} \in_{r_i} y_j^s$ if $s= r_i +1$ or $x_i^{r_i} \in_{r_i} x_j^{r_j}$ if $r_j= r_i +1$ or one of negations of these. Therefore, by Lemma \ref{Th:ColouringsCaptureOneTypes},
$$\mathcal{M} \models \theta[a_1^{r_1}, \ldots, a_k^{r_k}, d_1, \ldots, d_l].$$
Since the $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$ were arbitrary this shows that $\mathcal{M} \models \phi$.
\Square
\end{proof}
\begin{Theorems1} \label{Th:SentencesOfTypeAHoldInM}
Let $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ be an $\mathcal{L}_{\mathrm{TST}}$-sentence with $s_1 > \ldots > s_l$ and $\theta$ quantifier-free. If $\mathcal{N} \models \phi$ then $\mathcal{M} \models \phi$.
\end{Theorems1}
\begin{proof}
Suppose that $\mathcal{N} \models \phi$. Let $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$. Using $a_1^{r_1}, \ldots, a_k^{r_k}$ and the construction we presented above we can define the colour classes $\mathcal{C}_{n, k^\prime}$ and colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ for all $n \in \mathbb{N}$, and elements $b_1^{r_1}, \ldots, b_k^{r_k} \in \mathcal{N}$. The colourings $c_{n, k^\prime}^\mathcal{M}$ and $c_{n, k^\prime}^\mathcal{N}$ are $2^K$-similar and satisfy Lemma \ref{Th:ColouringsCaptureOneTypes}. Let $e_1^{s_1}, \ldots, e_l^{s_l} \in \mathcal{N}$ be such that
$$\mathcal{N} \models \theta[b_1^{r_1}, \ldots, b_k^{r_k}, e_1^{s_1}, \ldots, e_l^{s_l}].$$
We inductively choose $d_1^{s_1}, \ldots, d_l^{s_l} \in \mathcal{M}$. Let $d_1^{s_1} \in \mathcal{M}$ be such that $c_{s_1, k^\prime}^\mathcal{M}(d_1^{s_1})= c_{s_1, k^\prime}^{\mathcal{N}}(e_1^{s_1})$. Suppose that $1 \leq i < l$ and we have chosen $d_i^{s_i} \in \mathcal{M}$ with $c_{s_i, k^\prime}^\mathcal{M}(d_i^{s_i})= c_{s_i, k^\prime}^{\mathcal{N}}(e_i^{s_i})$. If $s_i \neq s_{i+1} + 1$ then let $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ be such that $c_{s_{i+1}, k^\prime}^\mathcal{M}(d_{i+1}^{s_{i+1}})= c_{s_{i+1}, k^\prime}^{\mathcal{N}}(e_{i+1}^{s_{i+1}})$. If $s_i= s_{i+1} + 1$ and $\mathcal{N} \models e_{i+1}^{s_{i+1}} \in_{s_{i+1}} e_i^{s_i}$ then let $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ be such that $c_{s_{i+1}, k^\prime}^\mathcal{M}(d_{i+1}^{s_{i+1}})= c_{s_{i+1}, k^\prime}^{\mathcal{N}}(e_{i+1}^{s_{i+1}})$ and $\mathcal{M} \models d_{i+1}^{s_{i+1}} \in_{s_{i+1}} d_i^{s_i}$. If $s_i= s_{i+1} + 1$ and $\mathcal{N} \models e_{i+1}^{s_{i+1}} \notin_{s_{i+1}} e_i^{s_i}$ then let $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ be such that $c_{s_{i+1}, k^\prime}^\mathcal{M}(d_{i+1}^{s_{i+1}})= c_{s_{i+1}, k^\prime}^{\mathcal{N}}(e_{i+1}^{s_{i+1}})$ and $\mathcal{M} \models d_{i+1}^{s_{i+1}} \notin_{s_{i+1}} d_i^{s_i}$. Lemma \ref{Th:KeepingTrackOfColousLemma} and the fact that $1 < 2^K$, and $c_{s_{i+1}, k^\prime}^\mathcal{M}$ and $c_{s_{i+1}, k^\prime}^\mathcal{N}$ are $2^K$-similar ensure that we can find $d_{i+1}^{s_{i+1}} \in \mathcal{M}$ satisfying these conditions. Now, since the variables $y_1^{s_1}, \ldots y_l^{s_l}$ all have distinct types in $\theta$, the only atomic or negatomic subformulae of $\theta$ are in the form $y_{i+1}^{s_{i+1}} \in_{s_{i+1}} y_i^{s_i}$ if $s_i= s_{i+1}+1$, $y_i^{s_i} \in_{s_i} x_j^{r_j}$ if $r_j= s_i+1$, $x_i^{r_i} \in_{r_i} y_j^{s_j}$ if $s_j= r_i+1$, or $x_i^{r_i} \in_{r_i} x_j^{r_j}$ if $r_j= r_i+1$, or one of the negations of these. Therefore, by Lemma \ref{Th:ColouringsCaptureOneTypes},
$$\mathcal{M} \models \theta[a_1^{r_k}, \ldots, a_k^{r_k}, d_1^{s_1}, \ldots, d_l^{s_l}].$$
Since the $a_1^{r_1}, \ldots, a_k^{r_k} \in \mathcal{M}$ were arbitrary this shows that $\mathcal{M} \models \phi$.
\Square
\end{proof}
\noindent Since $\mathcal{N}$ is an arbitrary model of $\mathrm{TSTI}$ and $\mathcal{M}$ is an arbitrary sufficiently large finitely generated model of $\mathrm{TST}$, Theorems \ref{Th:SentencesOfTypeBHoldInM} and \ref{Th:SentencesOfTypeAHoldInM} show that any $\mathcal{L}_{\mathrm{TST}}$-sentence in the form (A) or (B) has the finitely generated model property. Combining this with Theorem \ref{Th:ExistentialUniversalSentencesHaveFinitelyGeneratedModelProperty} shows that $\mathrm{TSTI}$ decides any sentence in the form (A) or (B).
\begin{Coroll1}
If $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} \theta$ is an $\mathcal{L}_{\mathrm{TST}}$-sentence with $s_1 > \ldots > s_l$ and $\theta$ quantifier free then $\mathrm{TST}$ decides $\phi$.
\Square
\end{Coroll1}
\begin{Coroll1}
If $\phi= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} \theta$ is an $\mathcal{L}_{\mathrm{TST}}$-sentence with $\theta$ quantifier-free then $\mathrm{TST}$ decides $\phi$.
\Square
\end{Coroll1}
\noindent Combining these results with Theorem \ref{Th:DecidableFragmentsOfNF} shows that sentences in the form (A') or (B') are decided by $\mathrm{NF}$.
\begin{Coroll1}
If $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ is an $\mathcal{L}$-formula with $\theta$ quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns the same value to all of the variables $y_1, \ldots, y_l$ then $\mathrm{NF}$ decides $\phi$.
\Square
\end{Coroll1}
\begin{Coroll1}
If $\phi= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l \theta$ is an $\mathcal{L}$-formula with $\theta$ quantifier-free and $\sigma: \mathbf{Var}(\phi) \longrightarrow \mathbb{N}$ is a stratification of $\phi$ that assigns distinct values to all of the variable $y_1, \ldots, y_l$ then $\mathrm{NF}$ decides $\phi$.
\Square
\end{Coroll1}
\noindent It is interesting to note that the only use of the Axiom of Infinity in the above arguments was to ensure that the bottom type is externally infinite. Thus our arguments show that all models of $\mathrm{TST}$ with infinite bottom type agree on all sentences in the form (A) and all sentences in the form (B).
\end{document} |
\betaginin{document}
\text{\mathbf rm{d}}eltaf{\bf m}athbb R{{\bf m}athbb R}
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\text{\mathbf rm{d}}eltaf{\bf m}athbb E{{\bf m}athbb E}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{H}{{\bf m}athbb H}
\text{\mathbf rm{d}}eltaf{\bf m}athbb Q{{\bf m}athbb Q}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{H}{{\bf m}athbb{H}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{P}{{\bf m}athbb{P}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{S}{{\bf m}athbb{S}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{Y}{{\bf m}athbb{Y}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{W}{{\bf m}athbb{W}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{D}{{\bf m}athbb{D}}
\text{\mathbf rm{d}}eltaf{\bf m}athcal{H}{{\bf m}athcal{H}}
\text{\mathbf rm{d}}eltaf{\bf m}athcal{S}{{\bf m}athcal{S}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{A}{{\bf m}athscr{A}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr {B}{{\bf m}athscr {B}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr {C}{{\bf m}athscr {C}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr {D}{{\bf m}athscr {D}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{F}{{\bf m}athscr{F}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{G}{{\bf m}athscr{G}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{L}{{\bf m}athscr{L}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{P}{{\bf m}athscr{P}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{S}{{\bf m}athscr{S}}
\text{\mathbf rm{d}}eltaf{\bf m}athscr{M}{{\bf m}athscr{M}}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{e}}quation{\text{\mathbf rm{e}}quationuation}
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\text{\mathbf rm{d}}eltaf\text{\mathbf rm{e}}psilon{\text{\mathbf rm{e}}psilonsilon}
\text{\mathbf rm{d}}eltaf\varepsilon{\varepsilon}
\text{\mathbf rm{d}}eltaf\varphi{\varphi}
\text{\mathbf rm{d}}eltaf\varrho{\varrho}
\text{\mathbf rm{d}}eltaf\omega{\omegaega}
\text{\mathbf rm{d}}eltaf\Omega{\Omegaega}
\text{\mathbf rm{d}}eltaf\sigma{\sigmagma}
\text{\mathbf rm{d}}eltaf\frac{\frac}
\text{\mathbf rm{d}}eltaf\sqrt{\sqrtrt}
\text{\mathbf rm{d}}eltaf\kappa{\kappa}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{d}}elta{\text{\mathbf rm{d}}eltalta}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{e}}llangle{\text{\mathbf rm{e}}llangle}
\text{\mathbf rm{d}}eltaf\mathbf rightarrowngle{\mathbf rightarrowngle}
\text{\mathbf rm{d}}eltaf\Gamma{\Gammamma}
\text{\mathbf rm{d}}eltaf\gamma{\gammamma}
\text{\mathbf rm{d}}eltaf\nabla{\nablabla}
\text{\mathbf rm{d}}eltaf\beta{\betata}
\text{\mathbf rm{d}}eltaf\alpha{\alphapha}
\text{\mathbf rm{d}}eltaf\partial{\partial}
\text{\mathbf rm{d}}eltaf\tilde{\widetilde}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{e}}llesssim{\text{\mathbf rm{e}}llesssim}
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\text{\mathbf rm{d}}eltaf{\bf m}{{\bf m}}
\text{\mathbf rm{d}}eltaf\mathbf B{{\bf m}athbf B}
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\text{\mathbf rm{d}}eltaf\text{\mathbf rm{e}}lla{\text{\mathbf rm{e}}llambda}
\text{\mathbf rm{d}}eltaf\theta{\thetaeta}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{d}}{\text{\mathbf rm{d}}}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{e}}ss{\text{\mathbf rm{ess}}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb Ric{\text{\mathbf rm{Ric}}}
\text{\mathbf rm{d}}eltaf {\bf m}athbb{H}ess{\text{\mathbf rm{Hess}}}
\text{\mathbf rm{d}}eltaf\underline a{\underline a}
\text{\mathbf rm{d}}eltaf{\bf m}athbb Ric{\text{\mathbf rm{Ric}}}
\text{\mathbf rm{d}}eltaf\text{\mathbf rm{cut}}{\text{\mathbf rm{cut}}}
\text{\mathbf rm{d}}eltaf\alphaphaa{{\bf m}athbf{r}}
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\text{\mathbf rm{d}}eltaf\pi_{{\bf m},\varrho}{\pi_{{\bf m},\varrho}}
\text{\mathbf rm{d}}eltaf\mathbf r{{\bf m}athbf r}
\text{\mathbf rm{d}}eltaf\tilde{\widetilde}
\text{\mathbf rm{d}}eltaf\tilde{\widetilde}
\text{\mathbf rm{d}}eltaf\mathbb I{{\bf m}athbb I}
\text{\mathbf rm{d}}eltaf{\rm in}{{\mathbf rm in}}
\text{\mathbf rm{d}}eltaf{\bf m}athbb{S}ect{{\mathbf rm Sect}}
\mathbf renewcommand{\overline}{\overline}
\mathbf renewcommand{\widehat}{\widehat}
\mathbf renewcommand{\widetilde}{\widetilde}
\alphalowdisplaybreaks
{\bf m}aketitle
\betaginin{abstract}
In this paper we are concerned with distribution dependent backward stochastic differential equations (DDBSDEs) driven
by Gaussian processes. We first show the existence and uniqueness of solutions to this type of equations. This is done by
formulating a transfer principle to transfer the well-posedness problem to an auxiliary DDBSDE driven by Brownian motion.
Then, we establish a comparison theorem under Lipschitz condition and boundedness of Lions derivative imposed on the
generator. Furthermore, we get a new representation for DDBSDEs driven by Gaussian processes, this representation is
even new for the case of the equations driven by Brownian motion. The new obtained representation enables us to prove
a converse comparison theorem. Finally, we derive transportation inequalities and Logarithmic-Sobolev inequalities via the
stability of the Wasserstein distance and the relative entropy of measures under the homeomorphism condition.
\text{\mathbf rm{e}}nd{abstract}
AMS Subject Classification: 60H10, 60G15, 60G22
{\bf m}edskip
\par\noindent
Keywords: Distribution dependent BSDEs; Gaussian processes; comparison theorem; converse comparison theorem; transportation inequality;
Logarithmic-Sobolev inequality.
\section{Introduction}
Backward stochastic differential equations (BSDEs) were first introduced in their linear form by Bismut in \cite{Bismut73} to investigate stochastic
control problems and their connections with a stochastic version of the Pontryagin maximum principle.
Afterwards, BSDEs was generally formalised and developed in the seminal work \cite{PP90}.
In the last decades, BSDEs have been the subject of growing interest in stochastic analysis,
as these equations naturally arise in stochastic control problems in mathematical finance and they provide Feynman-Kac type formulas for
semi-linear PDEs (see, e.g., \cite{EPQ9705,MY99,PP92,ZJ17}).
On the other hand, distribution dependent stochastic differential equations (DDSDEs), also known as mean-field equations or McKean-Vlasov
equations, are the It\^{o} equations whose coefficients depend upon the law of the solution. As DDSDEs can provide a probabilistic representation
for the solutions of a class of nonlinear PDEs, in which a typical example is the propagation of chaos, they are widely used as models in statistical
physics and in the study of large scale social interactions within the memory of mean-field games, for which we refer to, e.g., \cite{BT97,CD15,HMC13}
and references therein. Furthermore, nonlinear DDBSDEs were first introduced by Buckdahn, Djehiche, Li and Peng in \cite{BDLP09}.
Since then, the DDBSDEs have received increasing attentions and have been investigated in a variety of settings. Let us just mention a few here.
Chassagneux, Crisan and Delarue \cite{CCD15} showed the existence and uniqueness of solutions to fully coupled DDBSDEs; Carmona and Delarue \cite{CD15} studied DDBSDEs via the stochastic maximum principle; Li \cite{Li18} considered the well-posedness problem of DDBSDEs driven by a
Brownian motion and an independent Poisson random measure, and provided a probabilistic representation for a class of nonlocal PDEs of mean-field
type; Li, Liang and Zhang \cite{LLZ18} obtained a comparison theorem for DDBSDEs.
In this paper, we want to study the following DDBSDEs driven by Gaussian processes
\betaginin{equation}\text{\mathbf rm{e}}llabel{Bsde-In}
\text{\mathbf rm{e}}lleft\{
\betaginin{array}{ll}
\text{\mathbf rm{d}} Y_t=-f(t,X_t,Y_t,Z_t,{\bf m}athscr{L}_{(X_t,Y_t,Z_t)})\text{\mathbf rm{d}} V_t+Z_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_t,\\
Y_T=g(X_T,{\bf m}athscr{L}_{X_T}),
\text{\mathbf rm{e}}nd{array} \mathbf right.
\text{\mathbf rm{e}}nd{equation}
where $X$ is a centered one-dimensional Gaussian process such that $V_t:={\bf m}athrm{Var}X_t, t{\rm in}n[0,T]$, is a strictly increasing, continuous function with $V(0)=0$ introduced in \cite{Bender14}, ${\bf m}athscr{L}_{(X_t,Y_t,Z_t)}$ and ${\bf m}athscr{L}_{X_T}$ denote respectively the laws of $(X_t,Y_t,Z_t)$ and $X_T$,
and the stochastic integral is the Wick-It\^{o} integral defined by the $S$-transformation and the Wick product (see Section 2.1).
Precise assumptions on the generator $f:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_\theta({\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$ and the terminal value function $g:{\bf m}athbb R\tildemes{\bf m}athscr{P}_\theta({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$ will be specified in later sections, where ${\bf m}athscr{P}_\theta({\bf m}athbb R^m)$ stands for the totality of probability measures on ${\bf m}athbb R^m$ with finite $\theta$-th moment. We would like to mention that the driving noise $X$ in \text{\mathbf rm{e}}quationref{Bsde-In} includes fractional Brownian motion $B^H$ with Hurst parameter $H{\rm in}n(0,1)$ (while $B^{1/2}$ is the standard Brownian motion), and fractional Wiener integral (see Remark \mathbf ref{Re(Inte)}).
The main objectives of the present paper are to show the well-posedness and (converse) comparison theorems, and then to establish functional inequalities including transportation inequalities and Logarithmic-Sobolev inequalities for \text{\mathbf rm{e}}quationref{Bsde-In}. Our strategy is as follows.
Based on a new transfer principle that extends \cite[Theroem 3.1]{Bender14} to the distribution dependent setting,
we first prove general existence and uniqueness results for \text{\mathbf rm{e}}quationref{Bsde-In} (see Theorem \mathbf ref{Th1}), which are then applied to the case of Lipschitz
generator $f$.
Second, with the help of a formula for the $L$-derivative, we are able to derive a comparison theorem
which generalises and improves the corresponding one in the existing literature (see Theorem \mathbf ref{Th(com)} and Remark \mathbf ref{Re-comp1}).
It is worth stressing that comparing with the works in the distribution-free cases, here we need to impose an additional condition on $f$ that involves
the Lions derivative of $f$.
Moreover, we obtain a converse comparison theorem which is roughly speaking a converse to the comparison theorem obtained above (see Theorem \mathbf ref{Th(Conve)} and Remark \mathbf ref{Re(Conve)}).
To this end, we provide a representation theorem for the generator $f$ which is even new for DDBSDEs driven by Brownian motions which is also
interesting in itself. Finally, by utilising the stability of the Wasserstein distance and relative entropy of measures under the homeomorphism,
we establish several functional inequalities including transportation inequalities and Logarithmic-Sobolev inequalities (see Theorems \mathbf ref{Th(TrIn)} and \mathbf ref{Th(LS)}). Here, let us point out that the transportation inequality for the law of the control solution $Z$ of \text{\mathbf rm{e}}quationref{Bsde-In} stated in
Theorem \mathbf ref{Th(TrIn)} is of the form
\betaginin{align*}
{\bf m}athbb{W}_p({\bf m}athscr{L}_{Z},{\bf m}u)\text{\mathbf rm{e}}lleq C\text{\mathbf rm{e}}lleft(H({\bf m}u|{\bf m}athscr{L}_{Z})\mathbf right)^{\frac 1 {2p}}
\text{\mathbf rm{e}}nd{align*}
with any $p\geq1$.
In particular, when $p=2$, this inequality reduces to
\betaginin{align*}
{\bf m}athbb{W}_2({\bf m}athscr{L}_{Z},{\bf m}u)\text{\mathbf rm{e}}lleq C\text{\mathbf rm{e}}lleft(H({\bf m}u|{\bf m}athscr{L}_{Z})\mathbf right)^{\frac 1 {4}},
\text{\mathbf rm{e}}nd{align*}
which is clearly different from the usual quadratic transportation inequality (also called the Talagrand inequality). On the other hand,
as shown in \cite{BT20}, this type of inequalities allows one to derive deviation inequality. However, it does not allow to get other important inequalities
such as Poincar\'{e} inequality etc. Hence, an interesting problem is whether our results can be further improved in the sense of ${\bf m}athbb{W}_2({\bf m}athscr{L}_{Z},{\bf m}u)\text{\mathbf rm{e}}lleq C\sqrtrt{H({\bf m}u|{\bf m}athscr{L}_{Z})}$. Our techniques are currently not enough to give a full answer, since Lemma \mathbf ref{FI-Le2} below cannot applied to the case of $Z$.
We will leave this topic for the future work.
The remaining of the paper is organised as follows. Section 2 presents some basic facts on Gaussian processes, the Lions derivative,
and introduce a transfer result which allows us to build a relation between DDBSDEs concerned and DDBSDEs driven by Brownian motion.
In Section 3, we show the existence and uniqueness of a solution to DDBSDE driven by Gaussian process.
In Section 4, we establish a comparison theorem, and also provide a converse comparison theorem via a new representation theorem.
Section 5 is devoted to deriving functional inequalities, including transportation inequalities and Logarithmic-Sobolev inequalities.
Section 6 is designed as an appendix that we prove an auxiliary result needed in Section 5 (cf. Proposition 5.1).
\section{Preliminaries}
\subsection{Wick-It\^{o} integral for Gaussian processes}
In this part, we shall recall some important definitions and facts concerning the Wick-It\^{o} integral for Gaussian processes.
Further detailed and deep discussions can be found, e.g., \cite[Section 2]{Bender14} and references therein.
Let $(\Omegaega,{\bf m}athscr{F},({\bf m}athscr{F}_t^X)_{t{\rm in}n[0,T]},{\bf m}athbb{P})$ be a filtered probability space with $({\bf m}athscr{F}_t^X)_{t{\rm in}n[0,T]}$ the natural completed and right continuous filtration generated by a centered Gaussian process $(X_t)_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}$,
whose covariance function $V_t:={\bf m}athrm{Var}X_t, t{\rm in}n[0,T]$, is a strictly increasing and continuous function with $V(0)=0$.
The first chaos associated to $X$ is
\betaginin{align*}
{\bf m}athfrak{C}_X:=\overline{{\bf m}athrm{span}\{X_t:t{\rm in}n[0,T]\}},
\text{\mathbf rm{e}}nd{align*}
where the closure is taken in $L^2_X:=L^2(\Omegaega,{\bf m}athscr{F}_T^X,{\bf m}athbb{P})$.
It is obvious that the elements in ${\bf m}athfrak{C}_X$ are centered Gaussian variables.
We define the map ${\bf m}athcal{R}:{\bf m}athfrak{C}_X\mathbf rightarrow{\bf m}athbb R^{[0,T]}$ by
\betaginin{align*}
({\bf m}athcal{R}f)(t)={\bf m}athbb E(X_tf).
\text{\mathbf rm{e}}nd{align*}
It is readily checked that ${\bf m}athcal{R}$ is injective, whose image ${\bf m}athcal{R}({\bf m}athfrak{C}_X)$ is called the Cameron-Martin space of $X$ and is equipped with the inner product
\betaginin{align*}
\text{\mathbf rm{e}}llangle f,g\mathbf rightarrowngle_X:={\bf m}athbb E\text{\mathbf rm{e}}lleft[{\bf m}athcal{R}^{-1}(f){\bf m}athcal{R}^{-1}(g)\mathbf right].
\text{\mathbf rm{e}}nd{align*}
Now, we let ${\bf m}athfrak{H}_X$ be the set of all those ${\bf m}athfrak{h}{\rm in}n{\bf m}athcal{R}({\bf m}athfrak{C}_X)$ that are absolutely continuous with respect to $\text{\mathbf rm{d}} V$ with square integrable density, i.e.
\betaginin{align*}
{\bf m}athfrak{h}(t)={\rm in}nt_0^t\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}(s)\text{\mathbf rm{d}} V_s, \ \ \text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}{\rm in}n L^2([0,T],\text{\mathbf rm{d}} V).
\text{\mathbf rm{e}}nd{align*}
Throughout the paper, we suppose that ${\bf m}athfrak{H}_X$ and $\{\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}:{\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X\}$ are respectively the dense subsets of
${\bf m}athcal{R}({\bf m}athfrak{C}_X)$ and $L^2([0,T],\text{\mathbf rm{d}} V)$.
\betagin{rem}\text{\mathbf rm{e}}llabel{Re(Inte)}
As pointed out in \cite[Theorem 2.2]{Bender14}, the Gaussian processes concerned consists of a large class of examples which includes, e.g.,
fractional Brownian motion $B^H$ with $H{\rm in}n(0,1)$ and fractional Wiener integral of the form ${\rm in}nt_0^t\sigma(s)\text{\mathbf rm{d}} B^H_s$ with $H{\rm in}n(1/2,1)$
and a deterministic function $\sigma$ satisfying $c^{-1}\text{\mathbf rm{e}}lleq\sigma\text{\mathbf rm{e}}lleq c$ for some $c>0$.
\text{\mathbf rm{e}}nd{rem}
Next, we shall introduce the construction of the Wick-It\^{o} integral with respect to Gaussian process $X$.
Due to \cite[Corollary 3.40]{Janson97}, the random variables of the form
\betaginin{align*}
\text{\mathbf rm{e}}^{\text{\mathbf rm{d}}iamond{\bf m}athfrak{h}}:=\text{\mathbf rm{e}}xp\text{\mathbf rm{e}}lleft\{{\bf m}athcal{R}^{-1}({\bf m}athfrak{h})-\frac 1 2 {\bf m}athrm{Var}{\bf m}athcal{R}^{-1}({\bf m}athfrak{h})\mathbf right\},\ \ {\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X,
\text{\mathbf rm{e}}nd{align*}
form a total subset of $L^2_X$, in which $\text{\mathbf rm{e}}^{\text{\mathbf rm{d}}iamond{\bf m}athfrak{h}}$ is called Wick exponential.
Then for each random variable $\text{\mathbf rm{e}}ta{\rm in}n L^2_X$, it can be uniquely determined by its ${\bf m}athcal{S}$-transform
\betaginin{align*}
({\bf m}athcal{S}\text{\mathbf rm{e}}ta)({\bf m}athfrak{h}):={\bf m}athbb E(\text{\mathbf rm{e}}ta\text{\mathbf rm{e}}^{\text{\mathbf rm{d}}iamond{\bf m}athfrak{h}}), \ \ {\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X.
\text{\mathbf rm{e}}nd{align*}
That is, if $\text{\mathbf rm{e}}ta$ and $\zeta$ belong to $L^2_X$ satisfying $({\bf m}athcal{S}\text{\mathbf rm{e}}ta)({\bf m}athfrak{h})=({\bf m}athcal{S}\zeta)({\bf m}athfrak{h})$ for every ${\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X$, then there holds $\text{\mathbf rm{e}}ta=\zeta, {\bf m}athbb{P}$-a.s..
In addition, observe that for each ${\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X$,
\betaginin{align*}
({\bf m}athcal{S}X_t)({\bf m}athfrak{h})={\bf m}athbb E(X_t{\bf m}athcal{R}^{-1}({\bf m}athfrak{h}))={\bf m}athfrak{h}(t)={\rm in}nt_0^t\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}(s)\text{\mathbf rm{d}} V_s, \ \ t{\rm in}n[0,T]
\text{\mathbf rm{e}}nd{align*}
is a bounded variation function and then can be regarded as an integrator in a Lebesgue-Stieltjes integral, which allows us to introduce the following Wick-It\^{o} integral.
\betagin{defn}\text{\mathbf rm{e}}llabel{De-WI}
A measurable map $Z:[0,T]\mathbf rightarrow L^2_X$ is said to have a Wick-It\^{o} integral with respect to $X$, if for any ${\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X$,
\betaginin{align*}
{\rm in}nt_0^T({\bf m}athcal{S}Z_t)({\bf m}athfrak{h})\text{\mathbf rm{d}} {\bf m}athfrak{h}(t)
\text{\mathbf rm{e}}nd{align*}
exists and there is a random variable $\xi{\rm in}n L^2_X$ such that
\betaginin{align*}
({\bf m}athcal{S}\xi)({\bf m}athfrak{h})={\rm in}nt_0^T({\bf m}athcal{S}Z_t)({\bf m}athfrak{h})\text{\mathbf rm{d}} {\bf m}athfrak{h}(t).
\text{\mathbf rm{e}}nd{align*}
In this case, we denote $\xi$ by ${\rm in}nt_0^TZ_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_t$ and call it the Wick-It\^{o} integral of $Z$ with respect to $X$.
Besides, we often use ${\rm in}nt_a^bZ_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_t$ to denote ${\rm in}nt_0^T{\bf m}athrm{I}_{[a,b]}(t)Z_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_t$.
\text{\mathbf rm{e}}nd{defn}
\betagin{rem}\text{\mathbf rm{e}}llabel{Re(WI)}
(i) Suppose that $Z:[0,T]\mathbf rightarrow L^2_X$ is continuous and $\{\pi^n\}_{n\geq1}$ is a sequence of partition of $[0,T]$. Then, we have
\betaginin{align}\text{\mathbf rm{e}}llabel{1Re(WI)}
{\rm in}nt_0^TZ_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_t=\text{\mathbf rm{e}}llim_{|\pi^n|\mathbf rightarrow0}\sum_{t_i{\rm in}n\pi^n}Z_{t_i}\text{\mathbf rm{d}}iamond(X_{t_{i+1}}-X_{t_i}),
\text{\mathbf rm{e}}nd{align}
provided that the above limit exists in $L^2_X$.
Here, $Z_{t_i}\text{\mathbf rm{d}}iamond(X_{t_{i+1}}-X_{t_i})$ is a Wick product defined as follows:
\betaginin{align*}
({\bf m}athcal{S}(Z_{t_i}\text{\mathbf rm{d}}iamond(X_{t_{i+1}}-X_{t_i})))({\bf m}athfrak{h})=({\bf m}athcal{S}Z_{t_i})({\bf m}athfrak{h})({\bf m}athcal{S}(X_{t_{i+1}}-X_{t_i}))({\bf m}athfrak{h}),\ \ {\bf m}athfrak{h}{\rm in}n{\bf m}athfrak{H}_X.
\text{\mathbf rm{e}}nd{align*}
In view of \text{\mathbf rm{e}}quationref{1Re(WI)}, one can see that the Wick-It\^{o} integral can be interpreted as a limit of Riemann sums in terms of the Wick product.
(ii) If $X=B^{1/2}$, i.e. $X$ is a Brownian motion, then we obtain that $V_t=t$ and
\betaginin{align*}
{\bf m}athcal{R}({\bf m}athfrak{C}_{B^{1/2}})={\bf m}athfrak{H}_{B^{1/2}}=\text{\mathbf rm{e}}lleft\{{\bf m}athfrak{h}:{\bf m}athfrak{h}(t)={\rm in}nt_0^t\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}(s)\text{\mathbf rm{d}} s, \ \ \text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}{\rm in}n L^2([0,T],\text{\mathbf rm{d}} t)\mathbf right\}.
\text{\mathbf rm{e}}nd{align*}
Suppose that $Z$ is progressively measurable satisfying ${\bf m}athbb E{\rm in}nt_0^TZ_t^2\text{\mathbf rm{d}} t<{\rm in}nfty$. Then, it is easy to verify that the Wick-It\^{o} integral ${\rm in}nt_0^TZ_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond B^{1/2}_t$ coincides with the usual It\^{o} integral ${\rm in}nt_0^TZ_t\text{\mathbf rm{d}} B^{1/2}_t$, and then
\betaginin{align}\text{\mathbf rm{e}}llabel{2Re(WI)}
\text{\mathbf rm{e}}lleft({\bf m}athcal{S}{\rm in}nt_0^TZ_t\text{\mathbf rm{d}} B^{1/2}_t\mathbf right)({\bf m}athfrak{h})={\rm in}nt_0^T({\bf m}athcal{S}Z_t)({\bf m}athfrak{h})\text{\mathbf rm{d}} {\bf m}athfrak{h}(t),\ \ {\bf m}athfrak{h}{\rm in}n{\bf m}athcal{R}({\bf m}athfrak{C}_{B^{1/2}})={\bf m}athfrak{H}_{B^{1/2}}.
\text{\mathbf rm{e}}nd{align}
More details can be found in \cite[Remark 2.4]{Bender14}.
\text{\mathbf rm{e}}nd{rem}
\subsection{The transfer principle}
This part is devoted to establishing a transfer principle, which connects DDSDEs driven by Gaussian processes and DDSDEs driven by Brownian motion,
and will play a crucial role in the proofs of our main results.
In order to state the principle, we let $U$ be the inverse of $V$ defined as
\betagin{align*}
U_s:={\rm in}nf\{r\geq0: V_r\geq s\}, \ \ s{\rm in}n[0,V_T],
\text{\mathbf rm{e}}nd{align*}
and introduce an auxiliary Brownian motion $(\tilde{W})_{t{\rm in}n[0,V_T]}$ on a filtered probability space $(\tilde\Omegaega,\tilde{\bf m}athscr{F},(\tilde{\bf m}athscr{F}_t^X)_{t{\rm in}n[0,V_T]},\tilde{\bf m}athbb{P})$, is the filtration generated by $\tilde{W}$.
Similar to Section 2.1, we can define the ${\bf m}athcal{S}$-transform on this auxiliary probability space as follows:
for each random variable $\tilde\text{\mathbf rm{e}}ta{\rm in}n L^2(\tilde\Omegaega,\tilde{\bf m}athscr{F},\tilde{\bf m}athbb{P})$,
\betaginin{align*}
(\widetilde{{\bf m}athcal{S}}\tilde\text{\mathbf rm{e}}ta)({\bf m}athfrak{h}):=\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\tilde\text{\mathbf rm{e}}ta\text{\mathbf rm{e}}xp\text{\mathbf rm{e}}lleft\{{\rm in}nt_0^{V_T}\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}(s)\text{\mathbf rm{d}}\tilde{W}_s-\frac 1 2{\rm in}nt_0^{V_T}\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}^2(s)\text{\mathbf rm{d}} s\mathbf right\}\mathbf right), \ \ {\bf m}athfrak{h}{\rm in}n{\bf m}athcal{R}({\bf m}athfrak{C}_{\tilde{W}}).
\text{\mathbf rm{e}}nd{align*}
Here, we recall that owing to Remark \mathbf ref{Re(WI)} (ii), ${\bf m}athcal{R}({\bf m}athfrak{C}_{\tilde{W}})$ is of the form
\betaginin{align*}
{\bf m}athcal{R}({\bf m}athfrak{C}_{\tilde{W}})={\bf m}athfrak{H}_{\tilde{W}}=\text{\mathbf rm{e}}lleft\{{\bf m}athfrak{h}:{\bf m}athfrak{h}(t)={\rm in}nt_0^t\text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}(s)\text{\mathbf rm{d}} s, \ \ \text{\mathbf rm{d}}ot{{\bf m}athfrak{h}}{\rm in}n L^2([0,V_T],\text{\mathbf rm{d}} t)\mathbf right\}.
\text{\mathbf rm{e}}nd{align*}
Now, we have the following transfer principle, which is a distribution dependent version of \cite[Theorem 3.1]{Bender14}.
\betagin{prp}\text{\mathbf rm{e}}llabel{Pr1}
Assume that $\phi:{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R, b:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R^m)\mathbf rightarrow{\bf m}athbb R, \vartheta:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R^m$ and $\sigma:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$
are measurable functions satisfying
\betagin{align}\text{\mathbf rm{e}}llabel{Pr1-0}
{\bf m}athbb E\text{\mathbf rm{e}}lleft[\phi^2(X_t,{\bf m}athscr{L}_{X_t})+{\rm in}nt_0^t(b^2(s,X_s,{\bf m}athscr{L}_{\vartheta(s,X_s,{\bf m}athscr{L}_{X_s})})+\sigma^2(s,X_s,{\bf m}athscr{L}_{X_s}))\text{\mathbf rm{d}} V_s\mathbf right]<{\rm in}nfty \text{\mathbf rm{e}}nd{align}
for some $t{\rm in}n[0,T]$, and
\betagin{align}\text{\mathbf rm{e}}llabel{Pr1-1}
\phi(\tilde{W}_{V_t},{\bf m}athscr{L}_{\tilde{W}_{V_t}})={\rm in}nt_0^{V_t}b(U_s,\tilde{W}_s,{\bf m}athscr{L}_{\vartheta(U_s,\tilde{W}_s,{\bf m}athscr{L}_{\tilde{W}_s})})\text{\mathbf rm{d}} s+{\rm in}nt_0^{V_t}\sigma(U_s,\tilde{W}_s,{\bf m}athscr{L}_{\tilde{W}_s})\text{\mathbf rm{d}}\tilde{W}_s, \ \ \widetilde{{\bf m}athbb{P}}\textit{-}a.s.
\text{\mathbf rm{e}}nd{align}
Then, ${\rm in}nt_0^t\sigma(s,X_s,{\bf m}athscr{L}_{X_s})\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s$ is well-defined and there holds in $L^2(\Omegaega,{\bf m}athscr{F}_T^X,{\bf m}athbb{P})$
\betagin{align}\text{\mathbf rm{e}}llabel{Pr1-2}
\phi(X_t,{\bf m}athscr{L}_{X_t})={\rm in}nt_0^tb(s,X_s,{\bf m}athscr{L}_{\vartheta(s,X_s,{\bf m}athscr{L}_{X_s})})\text{\mathbf rm{d}} V_s+{\rm in}nt_0^t\sigma(s,X_s,{\bf m}athscr{L}_{X_s})\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s.
\text{\mathbf rm{e}}nd{align}
\text{\mathbf rm{e}}nd{prp}
Before proving Proposition \mathbf ref{Pr1}, we first give a useful lemma whose proof is identical to \cite[Lemma 3.2]{Bender14} and therefore omitted here.
\betagin{lem}\text{\mathbf rm{e}}llabel{Le1}
Assume that $\vartheta:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R^m$ and $\psi:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R^m)\mathbf rightarrow{\bf m}athbb R$ are two measurable functions
such that ${\bf m}athbb E\psi^2(t,X_t,{\bf m}athscr{L}_{\vartheta(t,X_t,{\bf m}athscr{L}_{X_t})})<{\rm in}nfty$ with some $t{\rm in}n[0,T]$.
Then for any $\hbar{\rm in}n{\bf m}athfrak{H}_X$,
\betagin{align}\text{\mathbf rm{e}}llabel{1Le1}
({\bf m}athcal{S}\psi(t,X_t,{\bf m}athscr{L}_{\vartheta(t,X_t,{\bf m}athscr{L}_{X_t})}))(\hbar)=(\widetilde{{\bf m}athcal{S}}\psi(t,\widetilde{W}_{V_t},{\bf m}athscr{L}_{\vartheta(t,\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}})}))(\hbar\circ U).
\text{\mathbf rm{e}}nd{align}
\text{\mathbf rm{e}}nd{lem}
Let us stress that if $\hbar{\rm in}n{\bf m}athfrak{H}_X$, then $\hbar\circ U$ belongs to ${\bf m}athcal{R}({\bf m}athfrak{C}_{\tilde{W}})$, and thus the right-hand side of \text{\mathbf rm{e}}quationref{1Le1} is well-defined.
Indeed, observe that
\betagin{align*}
(\hbar\circ U)(t)={\rm in}nt_0^{U_t}\text{\mathbf rm{d}}ot{\hbar}(s)\text{\mathbf rm{d}} V_s={\rm in}nt_0^t(\text{\mathbf rm{d}}ot{\hbar}\circ U)(s)\text{\mathbf rm{d}} s, \ \ t{\rm in}n[0,V_T]
\text{\mathbf rm{e}}nd{align*}
and
\betagin{align*}
{\rm in}nt_0^{V_T}(\text{\mathbf rm{d}}ot{\hbar}\circ U)^2(s)\text{\mathbf rm{d}} s={\rm in}nt_0^T\text{\mathbf rm{d}}ot{\hbar}^2(s)\text{\mathbf rm{d}} V_s<{\rm in}nfty.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}mph{Proof of Proposition \mathbf ref{Pr1}.}
According to the linearity of ${\bf m}athcal{S}$, Lemma \mathbf ref{Le1} and the change of variables, we have for each $\hbar{\rm in}n{\bf m}athfrak{H}_X$,
\betagin{align}\text{\mathbf rm{e}}llabel{Pf(Pr1)-1}
&\text{\mathbf rm{e}}lleft({\bf m}athcal{S}\text{\mathbf rm{e}}lleft(\phi(X_t,{\bf m}athscr{L}_{X_t})-{\rm in}nt_0^tb(s,X_s,{\bf m}athscr{L}_{\vartheta(s,X_s,{\bf m}athscr{L}_{X_s})})\text{\mathbf rm{d}} V_s\mathbf right)\mathbf right)(\hbar)\cr
&=({\bf m}athcal{S}\phi(X_t,{\bf m}athscr{L}_{X_t}))(\hbar)-{\rm in}nt_0^t({\bf m}athcal{S}b(s,X_s,{\bf m}athscr{L}_{\vartheta(s,X_s,{\bf m}athscr{L}_{X_s})}))(\hbar)\text{\mathbf rm{d}} V_s\cr
&=(\widetilde{{\bf m}athcal{S}}\phi(\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}}))(\hbar\circ U)-
{\rm in}nt_0^t(\widetilde{{\bf m}athcal{S}}b(s,\widetilde{W}_{V_s},{\bf m}athscr{L}_{\vartheta(s,\widetilde{W}_{V_s},{\bf m}athscr{L}_{\widetilde{W}_{V_s}})}))(\hbar\circ U)\text{\mathbf rm{d}} V_s\cr
&=(\widetilde{{\bf m}athcal{S}}\phi(\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}}))(\hbar\circ U)-
{\rm in}nt_0^{V_t}(\widetilde{{\bf m}athcal{S}}b(U_r,\widetilde{W}_{r},{\bf m}athscr{L}_{\vartheta(U_r,\widetilde{W}_{r},{\bf m}athscr{L}_{\widetilde{W}_{r}})}))(\hbar\circ U)\text{\mathbf rm{d}} r\cr
&=\text{\mathbf rm{e}}lleft(\widetilde{{\bf m}athcal{S}}\text{\mathbf rm{e}}lleft(\phi(\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}})-
{\rm in}nt_0^{V_t}b(U_r,\widetilde{W}_{r},{\bf m}athscr{L}_{\vartheta(U_r,\widetilde{W}_{r},{\bf m}athscr{L}_{\widetilde{W}_{r}})})\text{\mathbf rm{d}} r\mathbf right)\mathbf right)(\hbar\circ U)\cr
&=\text{\mathbf rm{e}}lleft(\widetilde{{\bf m}athcal{S}}{\rm in}nt_0^{V_t}\sigma(U_r,\tilde{W}_r,{\bf m}athscr{L}_{\tilde{W}_r})\text{\mathbf rm{d}}\tilde{W}_r\mathbf right)(\hbar\circ U),
\text{\mathbf rm{e}}nd{align}
where the last equality is due to \text{\mathbf rm{e}}quationref{Pr1-1}.\\
Since the classical It\^{o} integral coincides with the Wick-It\^{o} integral (see Remark \mathbf ref{Re(WI)} (ii))
and $\hbar\circ U$ belongs to ${\bf m}athfrak{H}_{\tilde W}={\bf m}athcal{R}({\bf m}athfrak{C}_{\tilde W})$, we get from \text{\mathbf rm{e}}quationref{2Re(WI)}
\betagin{align*}
\text{\mathbf rm{e}}lleft(\widetilde{{\bf m}athcal{S}}{\rm in}nt_0^{V_t}\sigma(U_r,\tilde{W}_r,{\bf m}athscr{L}_{\tilde{W}_r})\text{\mathbf rm{d}}\tilde{W}_r\mathbf right)(\hbar\circ U) ={\rm in}nt_0^{V_t}\text{\mathbf rm{e}}lleft(\widetilde{{\bf m}athcal{S}}\sigma(U_r,\tilde{W}_r,{\bf m}athscr{L}_{\tilde{W}_r})\mathbf right)(\hbar\circ U)\text{\mathbf rm{d}}(\hbar\circ U)(r).
\text{\mathbf rm{e}}nd{align*}
Plugging this into \text{\mathbf rm{e}}quationref{Pf(Pr1)-1} and using the change of variables and Lemma \mathbf ref{Le1} again, we obtain
\betagin{align*}
&\text{\mathbf rm{e}}lleft({\bf m}athcal{S}\text{\mathbf rm{e}}lleft(\phi(X_t,{\bf m}athscr{L}_{X_t})-{\rm in}nt_0^tb(s,X_s,{\bf m}athbb{P}_{\vartheta(s,X_s,{\bf m}athscr{L}_{X_s})})\text{\mathbf rm{d}} V_s\mathbf right)\mathbf right)(\hbar)\cr
&={\rm in}nt_0^{V_t}\text{\mathbf rm{e}}lleft(\widetilde{{\bf m}athcal{S}}\sigma(U_r,\tilde{W}_r,{\bf m}athscr{L}_{\tilde{W}_r})\mathbf right)(\hbar\circ U)\text{\mathbf rm{d}}(\hbar\circ U)(r)\cr
&={\rm in}nt_0^t\text{\mathbf rm{e}}lleft(\widetilde{{\bf m}athcal{S}}\sigma(r,\tilde{W}_{V_r},{\bf m}athscr{L}_{\tilde{W}_{V_r}})\mathbf right)(\hbar\circ U)\text{\mathbf rm{d}}\hbar(r)\cr
&={\rm in}nt_0^t\text{\mathbf rm{e}}lleft({\bf m}athcal{S}\sigma(r,X_r,{\bf m}athscr{L}_{X_r})\mathbf right)(\hbar)\text{\mathbf rm{d}}\hbar(r),
\text{\mathbf rm{e}}nd{align*}
which yields that ${\rm in}nt_0^t\sigma(s,X_s,{\bf m}athbb{P}_{X_s})\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s$ is well-defined and moreover the relation \text{\mathbf rm{e}}quationref{Pr1-2} holds due to Definition \mathbf ref{De-WI}.
The proof is now complete.
\qed
\subsection{The Lions derivative}
For later use, we state some basic facts about the Lions derivative.
For any $\theta{\rm in}n[1,{\rm in}nfty)$, ${\bf m}athscr{P}_\theta({\bf m}athbb R^d)$ denotes the set of $\theta$-integrable probability measures on ${\bf m}athbb R^d$,
and the $L^\theta$-Wasserstein distance on ${\bf m}athscr{P}_\theta({\bf m}athbb R^d)$ is defined as follows:
\betaginin{align*}
{\bf m}athbb{W}_\theta({\bf m}u,\nu):={\rm in}nf_{\pi{\rm in}n{\bf m}athscr {C}({\bf m}u,\nu)}\text{\mathbf rm{e}}lleft({\rm in}nt_{{\bf m}athbb R^d\tildemes{\bf m}athbb R^d}|x-y|^\theta\pi(\text{\mathbf rm{d}} x, \text{\mathbf rm{d}} y)\mathbf right)^\frac 1 \theta,\ \ {\bf m}u,\nu{\rm in}n{\bf m}athscr{P}_\theta({\bf m}athbb R^d),
\text{\mathbf rm{e}}nd{align*}
where ${\bf m}athscr {C}({\bf m}u,\nu)$ stands for the set of all probability measures on ${\bf m}athbb R^d\tildemes{\bf m}athbb R^d$ with marginals ${\bf m}u$ and $\nu$.
It is well known that $({\bf m}athscr{P}_\theta({\bf m}athbb R^d),{\bf m}athbb{W}_\theta)$ is a Polish space, usually referred to as the $\theta$-Wasserstein space on ${\bf m}athbb R^d$.
We use $\text{\mathbf rm{e}}llangle\cdot,\cdot\mathbf rightarrowngle$ for the Euclidean inner product, and $\|\cdot\|_{L^2_{\bf m}u}$ for the $L^2({\bf m}athbb R^d\mathbf rightarrow{\bf m}athbb R^d,{\bf m}u)$ norm.
Let ${\bf m}athscr{L}_X$ be the distribution of random variable $X$.
\betagin{defn}
Let $f:{\bf m}athscr{P}_2({\bf m}athbb R^d)\mathbf rightarrow{\bf m}athbb R$.
\betaginin{enumerate}
{\rm in}tem[(1)] $f$ is called $L$-differentiable at ${\bf m}u{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R^d)$, if the functional
\betaginin{align*}
L^2({\bf m}athbb R^d\mathbf rightarrow{\bf m}athbb R^d,{\bf m}u)\ni\phi{\bf m}apsto f({\bf m}u\circ({\bf m}athrm{Id}+\phi)^{-1}))
\text{\mathbf rm{e}}nd{align*}
is Fr\'{e}chet differentiable at $0{\rm in}n L^2({\bf m}athbb R^d\mathbf rightarrow{\bf m}athbb R^d,{\bf m}u)$. That is, there exists a unique $\gammamma{\rm in}n L^2({\bf m}athbb R^d\mathbf rightarrow{\bf m}athbb R^d,{\bf m}u)$ such that
\betaginin{align*}
\text{\mathbf rm{e}}llim_{\|\phi\|_{L^2_{\bf m}u}\mathbf rightarrow0}\frac{f({\bf m}u\circ({\bf m}athrm{Id}+\phi)^{-1})-f({\bf m}u)-{\bf m}u(\text{\mathbf rm{e}}llangle\gammamma,\phi\mathbf rightarrowngle)}{\|\phi\|_{L^2_{\bf m}u}}=0.
\text{\mathbf rm{e}}nd{align*}
In this case, $\gammamma$ is called the $L$-derivative of $f$ at ${\bf m}u$ and denoted by $D^Lf({\bf m}u)$.
{\rm in}tem[(2)] $f$ is called $L$-differentiable on ${\bf m}athscr{P}_2({\bf m}athbb R^d)$, if the $L$-derivative $D^Lf({\bf m}u)$ exists for all ${\bf m}u{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R^d)$.
Furthermore, if for every ${\bf m}u{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R^d)$ there exists a ${\bf m}u$-version $D^Lf({\bf m}u)(\cdot)$ such that $D^Lf({\bf m}u)(x)$ is jointly continuous in $({\bf m}u,x){\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R^d)\tildemes{\bf m}athbb R^d$, we denote $f{\rm in}n C^{(1,0)}({\bf m}athscr{P}_2({\bf m}athbb R^d))$.
\text{\mathbf rm{e}}nd{enumerate}
\text{\mathbf rm{e}}nd{defn}
In addition, according to \cite[Theorem 6.5]{Cardaliaguet13} and \cite[Proposition 3.1]{RW}, we get the following useful formula for the $L$-derivative.
\betagin{lem}\text{\mathbf rm{e}}llabel{FoLD}
Let $(\Omegaega,{\bf m}athscr{F},{\bf m}athbb{P})$ be an atomless probability space and $\xi,\text{\mathbf rm{e}}ta{\rm in}n L^2(\Omegaega\mathbf rightarrow{\bf m}athbb R^d,{\bf m}athbb{P})$.
If $f{\rm in}n C^{1,0}({\bf m}athscr{P}_2({\bf m}athbb R^d))$, then
\betaginin{align*}
\text{\mathbf rm{e}}llim_{\varepsilon\text{\mathbf rm{d}}ownarrow0}\frac {f({\bf m}athscr{L}_{\xi+\varepsilon\text{\mathbf rm{e}}ta})-f({\bf m}athscr{L}_\xi)} \varepsilon={\bf m}athbb E\text{\mathbf rm{e}}llangleD^Lf({\bf m}athscr{L}_\xi)(\xi),\text{\mathbf rm{e}}ta\mathbf rightarrowngle.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{lem}
\section{Well-posedness of DDBSDE by Gaussian processes}
In this section, we consider the following DDBSDE driven by Gaussian process:
\betaginin{equation}\text{\mathbf rm{e}}llabel{Bsde}
\text{\mathbf rm{e}}lleft\{
\betaginin{array}{ll}
\text{\mathbf rm{d}} Y_t=-f(t,X_t,Y_t,Z_t,{\bf m}athscr{L}_{(X_t,Y_t,Z_t)})\text{\mathbf rm{d}} V_t+Z_t\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_t,\\
Y_T=g(X_T,{\bf m}athscr{L}_{X_T}),
\text{\mathbf rm{e}}nd{array} \mathbf right.
\text{\mathbf rm{e}}nd{equation}
where $f:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$ and $g:{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$ are measurable functions.
Now, let $\Upsilon$ be the space of pair of $({\bf m}athscr{F}_t^X)_{t{\rm in}n[0,T]}$-adapted processes $(Y,Z)$ on $[0,T]$ satisfying
\betaginin{align*}
{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|Y_t|^2+{\rm in}nt_0^T|Z_t|^2\text{\mathbf rm{d}} V_t\mathbf right)<{\rm in}nfty
\text{\mathbf rm{e}}nd{align*}
and
\betagin{align*}
Y_t=u(t,X_t,{\bf m}athscr{L}_{X_t}), \ \ {\bf m}athbb{P}{\textit-}a.s., \ t{\rm in}n[0,T],\ \ Z_t=v(t,X_t,{\bf m}athscr{L}_{X_t}), \ \text{\mathbf rm{d}}{\bf m}athbb{P}\otimes\text{\mathbf rm{d}} V{\textit-}a.e.
\text{\mathbf rm{e}}nd{align*}
with two deterministic functions $u,v:[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$.
\betagin{defn}\text{\mathbf rm{e}}llabel{De-sol}
(1) A pair of stochastic processes $(Y,Z)=(Y_t,Z_t)_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}$ is called a solution to \text{\mathbf rm{e}}quationref{Bsde}, if $(Y,Z){\rm in}n\Upsilon$ satisfies
\betagin{align*}
{\bf m}athbb E\text{\mathbf rm{e}}lleft[g^2(X_T,{\bf m}athscr{L}_{X_T})+{\rm in}nt_0^Tf^2(s,X_s,Y_s,Z_s,{\bf m}athscr{L}_{(X_s,Y_s,Z_s)})\text{\mathbf rm{d}} V_s\mathbf right]<{\rm in}nfty,
\text{\mathbf rm{e}}nd{align*}
the Wick-It\^{o} integral ${\rm in}nt_0^tZ_s\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s$ exists for any $t{\rm in}n[0,T]$ and ${\bf m}athbb{P}$-a.s.
\betagin{align*}
Y_t=g(X_T,{\bf m}athscr{L}_{X_T})+{\rm in}nt_t^Tf(s,X_s,Y_s,Z_s,{\bf m}athscr{L}_{(X_s,Y_s,Z_s)})\text{\mathbf rm{d}} V_s-{\rm in}nt_t^TZ_s\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s,\ \ t{\rm in}n[0,T].
\text{\mathbf rm{e}}nd{align*}
(2) \text{\mathbf rm{e}}quationref{Bsde} is said to have uniqueness, if for any two solutions $(Y^i,Z^i){\rm in}n\Upsilon, i=1,2$,
\betaginin{align*}
{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|Y^1_t-Y^2_t|^2+{\rm in}nt_0^T|Z^1_t-Z^2_t|^2\text{\mathbf rm{d}} V_t\mathbf right)=0.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{defn}
In order to study \text{\mathbf rm{e}}quationref{Bsde}, we shall introduce the following auxiliary BSDE
\betaginin{equation}\text{\mathbf rm{e}}llabel{Aueq}
\text{\mathbf rm{e}}lleft\{
\betaginin{array}{ll}
\text{\mathbf rm{d}}\widetilde{Y}_t=-f(U_t,\widetilde{W}_t,\widetilde{Y}_t,\widetilde{Z}_t,{\bf m}athscr{L}_{(\widetilde{W}_t,\widetilde{Y}_t,\widetilde{Z}_t)})\text{\mathbf rm{d}} t+\widetilde{Z}_t\text{\mathbf rm{d}}\widetilde{W}_t,\\
\widetilde{Y}_{V_T}=g(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\widetilde{W}_{V_T}}),
\text{\mathbf rm{e}}nd{array} \mathbf right.
\text{\mathbf rm{e}}nd{equation}
where we recall that $\{\widetilde{W}_t\}_{t{\rm in}n[0,V_T]}$ is a standard Brownian motion stated in Section 2.2.
Let $\widetilde{\Upsilon}$ be the space of pair of $({\bf m}athscr{F}_t^{\tilde W})_{t{\rm in}n[0,V_T]}$-adapted processes $(\tilde Y,\tilde Z)$ on $[0,V_T]$ satisfying
\betaginin{align*}
\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq V_T}|\tilde Y_t|^2+{\rm in}nt_0^{V_T}|\tilde Z_t|^2\text{\mathbf rm{d}} t\mathbf right)<{\rm in}nfty
\text{\mathbf rm{e}}nd{align*}
and
\betagin{align*}
\tilde Y_t=\tilde u(t,\tilde W_t,{\bf m}athscr{L}_{\tilde W_t}), \ \tilde{\bf m}athbb{P}{\textit-}a.s., \ t{\rm in}n[0,V_T], \ \ \tilde Z_t=\tilde v(t,\tilde W_t,{\bf m}athscr{L}_{\tilde W_t}), \ \text{\mathbf rm{d}}\tilde{\bf m}athbb{P}\otimes\text{\mathbf rm{d}} t{\textit-}a.e.
\text{\mathbf rm{e}}nd{align*}
with two deterministic functions $\tilde u,\tilde v:[0,V_T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)\mathbf rightarrow{\bf m}athbb R$.
Similar to Definition \mathbf ref{De-sol}, we can give the notion of a solution $(\tilde Y,\tilde Z)=(\tilde Y_t,\tilde Z_t)_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq V_T}$ for \text{\mathbf rm{e}}quationref{Aueq} in the space $\widetilde{\Upsilon}$.
\betagin{thm}\text{\mathbf rm{e}}llabel{Th1}
Suppose that ${\bf m}athbb E [g^2(X_T,{\bf m}athscr{L}_{X_T})]<{\rm in}nfty$ and \text{\mathbf rm{e}}quationref{Aueq} has a unique solution $(\widetilde{Y},\widetilde{Z}){\rm in}n\widetilde{\Upsilon}$.
Then \text{\mathbf rm{e}}quationref{Bsde} has a unique solution $(Y,Z){\rm in}n\Upsilon$.
More precisely, if
\betagin{align*}
\widetilde{Y}_t=\widetilde{u}(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t}), \ \ \widetilde{Z}_t=\widetilde{v}(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t}), \ \ t{\rm in}n[0,V_T]
\text{\mathbf rm{e}}nd{align*}
is a unique solution of \text{\mathbf rm{e}}quationref{Aueq}, then
\betagin{align*}
Y_t=u(t,X_t,{\bf m}athscr{L}_{X_t}), \ \ Z_t=v(t,X_t,{\bf m}athscr{L}_{X_t}), \ \ t{\rm in}n[0,T],
\text{\mathbf rm{e}}nd{align*}
is a unique solution of \text{\mathbf rm{e}}quationref{Bsde}, where
\betagin{align*}
u(t,x,{\bf m}u)=\widetilde{u}(V_t,x,{\bf m}u),\ \ v(t,x,{\bf m}u)=\widetilde{v}(V_t,x,{\bf m}u),\ \ (t,x,{\bf m}u){\rm in}n[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R).
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{thm}
\betagin{proof}
The proof is divided into two steps.
\textit{Step 1. Existence.}
Since $(\widetilde{Y},\widetilde{Z}){\rm in}n\widetilde{\Upsilon}$ is a solution to \text{\mathbf rm{e}}quationref{Aueq}, we let
$$\widetilde{Y}_t=\widetilde{u}(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t}),\ \tilde{\bf m}athbb{P}{\textit-}a.s., \ t{\rm in}n[0,V_T], \ \
\widetilde{Z}_t=\widetilde{v}(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t}),\ \text{\mathbf rm{d}}\tilde{\bf m}athbb{P}\otimes\text{\mathbf rm{d}} t{\textit-}a.e., $$
and then we have for any $t{\rm in}n[0,T]$,
\betagin{align*}
&\widetilde{u}(V_t,\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}})-\widetilde{u}(0,0,\text{\mathbf rm{d}}eltalta_0)\cr
&=-{\rm in}nt_0^{V_t}f(U_s,\widetilde{W}_s,\widetilde{u}(s,\widetilde{W}_{s},{\bf m}athscr{L}_{\widetilde{W}_{s}}),\widetilde{v}(s,\widetilde{W}_{s},{\bf m}athscr{L}_{\widetilde{W}_{s}}),
{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{u}(s,\widetilde{W}_{s},{\bf m}athscr{L}_{\widetilde{W}_{s}}),\widetilde{v}(s,\widetilde{W}_{s},{\bf m}athscr{L}_{\widetilde{W}_{s}}))})\text{\mathbf rm{d}} s\cr
&\quad+{\rm in}nt_0^{V_t}\widetilde{v}(s,\widetilde{W}_{s},{\bf m}athscr{L}_{\widetilde{W}_{s}})\text{\mathbf rm{d}}\widetilde{W}_s,
\text{\mathbf rm{e}}nd{align*}
where $\text{\mathbf rm{d}}eltalta_0$ is the Dirac measure.\\
Because of the definition of $\widetilde{\Upsilon}$,
it is readily checked that \text{\mathbf rm{e}}quationref{Pr1-0} with $(\phi,b,\sigma)$ replaced by $(\widetilde{u}(V_t,\cdot,\cdot),f,\widetilde{v})$ holds.
So, by Proposition \mathbf ref{Pr1} we derive for any $t{\rm in}n[0,T]$,
\betagin{align*}
&\widetilde{u}(V_t,X_t,{\bf m}athscr{L}_{X_t})-\widetilde{u}(0,0,\text{\mathbf rm{d}}eltalta_0)\cr
&=-{\rm in}nt_0^tf(s,X_s,\widetilde{u}(V_s,X_{s},{\bf m}athscr{L}_{X_s}),\widetilde{v}(V_s,X_s,{\bf m}athscr{L}_{X_s}),
{\bf m}athscr{L}_{(X_s,\widetilde{u}(V_s,X_{s},{\bf m}athscr{L}_{X_{s}}),\widetilde{v}(V_s,X_{s},{\bf m}athscr{L}_{X_{s}}))})\text{\mathbf rm{d}} V_s\cr
&\quad+{\rm in}nt_0^t\widetilde{v}(V_s,X_s,{\bf m}athscr{L}_{X_s})\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s.
\text{\mathbf rm{e}}nd{align*}
Let $u(t,x,{\bf m}u)=\widetilde{u}(V_t,x,{\bf m}u), v(t,x,{\bf m}u)=\widetilde{v}(V_t,x,{\bf m}u),(t,x,{\bf m}u){\rm in}n[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}_2({\bf m}athbb R)$.
Then $(Y_t,Z_t):=(u(t,X_t,{\bf m}athscr{L}_{X_t}),v(t,X_t,{\bf m}athscr{L}_{X_t}))$ solves the following BSDE:
\betagin{align}\text{\mathbf rm{e}}llabel{Pf(Th1)-1}
Y_t=Y_T+{\rm in}nt_t^Tf(s,X_s,Y_s,Z_s,{\bf m}athscr{L}_{(X_s,Y_s,Z_s)})\text{\mathbf rm{d}} V_s+{\rm in}nt_t^TZ_sd^\text{\mathbf rm{d}}iamond X_s.
\text{\mathbf rm{e}}nd{align}
On the other hand, noting that
$g(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\widetilde{W}_{V_T}})=\widetilde{u}(V_T,\widetilde{W}_{V_T},{\bf m}athscr{L}_{\widetilde{W}_{V_T}}),\ {\bf m}athbb{P}{\textit-}a.s.$ and the Gaussian law of $\widetilde{W}_{V_T}$, we get
$$g(x,{\bf m}athscr{L}_{\widetilde{W}_{V_T}})=\widetilde{u}(V_T,x,{\bf m}athscr{L}_{\widetilde{W}_{V_T}}),\ \ \text{\mathbf rm{d}} x{\textit-}a.e.,$$
which implies
\betagin{align}\text{\mathbf rm{e}}llabel{Pf(Th1)-2}
&g(X_T,{\bf m}athscr{L}_{X_T})=g(X_T,{\bf m}athscr{L}_{\widetilde{W}_{V_T}})=\widetilde{u}(V_T,X_T,{\bf m}athscr{L}_{\widetilde{W}_{V_T}})\cr
&=\widetilde{u}(V_T,X_T,{\bf m}athscr{L}_{X_T})=u(T,X_T,{\bf m}athscr{L}_{X_T})=Y_T,\ \ {\bf m}athbb{P}{\textit-}a.s.
\text{\mathbf rm{e}}nd{align}
In addition, we easily obtain
\betagin{align}\text{\mathbf rm{e}}llabel{Pf(Th1)-3}
&{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|Y_t|^2+{\rm in}nt_0^T|Z_t|^2\text{\mathbf rm{d}} V_t\mathbf right)\cr
&={\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|u(t,X_t,{\bf m}athscr{L}_{X_t})|^2+{\rm in}nt_0^T|v(t,X_t,{\bf m}athscr{L}_{X_t})|^2\text{\mathbf rm{d}} V_t\mathbf right)\cr
&={\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|\widetilde{u}(V_t,X_t,{\bf m}athscr{L}_{X_t})|^2+{\rm in}nt_0^T|\widetilde{v}(V_t,X_t,{\bf m}athscr{L}_{X_t})|^2\text{\mathbf rm{d}} V_t\mathbf right)\cr
&=\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|\widetilde{u}(V_t,\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}})|^2+{\rm in}nt_0^T|\widetilde{v}(V_t,\widetilde{W}_{V_t},{\bf m}athscr{L}_{\widetilde{W}_{V_t}})|^2\text{\mathbf rm{d}} V_t\mathbf right)\cr
&=\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq V_T}|\widetilde{u}(t,\widetilde{W}_{t},{\bf m}athscr{L}_{\widetilde{W}_{t}})|^2+{\rm in}nt_0^{V_T}|\widetilde{v}(t,\widetilde{W}_{t},{\bf m}athscr{L}_{\widetilde{W}_{t}})|^2\text{\mathbf rm{d}} t\mathbf right)\cr
&=\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq V_T}|\widetilde{Y}_t|^2+{\rm in}nt_0^{V_T}|\widetilde{Z}_t|^2\text{\mathbf rm{d}} V_t\mathbf right)<{\rm in}nfty,
\text{\mathbf rm{e}}nd{align}
which, together with \text{\mathbf rm{e}}quationref{Pf(Th1)-1}-\text{\mathbf rm{e}}quationref{Pf(Th1)-2}, yields that $(Y,Z){\rm in}n\Upsilon$ is a solution to \text{\mathbf rm{e}}quationref{Bsde}.
\textit{Step 2. Uniqueness.}
Let $(Y^i,Z^i){\rm in}n\Upsilon,i=1,2$ be two solutions of \text{\mathbf rm{e}}quationref{Bsde}.
Then there exist $u^i,v^i,i=1,2$ such that $(Y_t^i,Z_t^i)=(u^i(t,X_t,{\bf m}athscr{L}_{X_t}),v^i(t,X_t,{\bf m}athscr{L}_{X_t})),i=1,2$.
Along the same arguments as in step 1, we can conclude that
$(\widetilde{Y}_t^i,\widetilde{Z}_t^i)=(u^i(U_t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t}),v^i(U_t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t})){\rm in}n\widetilde{\Upsilon},i=1,2$ are two solutions of \text{\mathbf rm{e}}quationref{Aueq}.
Similar to \text{\mathbf rm{e}}quationref{Pf(Th1)-3} and by the uniqueness of \text{\mathbf rm{e}}quationref{Aueq}, we arrive at
\betagin{align*}
&{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|Y_t^1-Y_t^2|^2+{\rm in}nt_0^T|Z_t^1-Z_t^2|^2\text{\mathbf rm{d}} V_t\mathbf right)\cr
&={\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq T}|u^1(t,X_t,{\bf m}athscr{L}_{X_t})-u^2(t,X_t,{\bf m}athscr{L}_{X_t})|^2+{\rm in}nt_0^T|v^1(t,X_t,{\bf m}athscr{L}_{X_t})-v^2(t,X_t,{\bf m}athscr{L}_{X_t})|^2\text{\mathbf rm{d}} V_t\mathbf right)\cr
&=\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq V_T}|u^1(U_t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t})-u^2(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t})|^2+{\rm in}nt_0^{V_T}|v^1(U_t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t})-v^2(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t})|^2\text{\mathbf rm{d}} t\mathbf right)\cr
&=\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq t\text{\mathbf rm{e}}lleq V_T}|\widetilde{Y}_t^1-\widetilde{Y}_t^2|^2+{\rm in}nt_0^{V_T}|\widetilde{Z}_t^1-\widetilde{Z}_t^2|^2\text{\mathbf rm{d}} t\mathbf right)=0,
\text{\mathbf rm{e}}nd{align*}
which finishes the proof.
\text{\mathbf rm{e}}nd{proof}
Next, we are devoted to applying our general Theorem \mathbf ref{Th1} to the case of Lipschitz continuous functions $(g,f)$.
More precisely, we assume the following conditions:
\betaginin{enumerate}
{\rm in}tem[\textsc{\textbf{(H1)}}] (i) $g,f$ are Lipschitz continuous,
i.e. there exist two constants $L_g, L_f>0$ such that
for all $t{\rm in}n[0,T],x_i,y_i,z_i{\rm in}n{\bf m}athbb R,{\bf m}u,\widetilde{{\bf m}u}{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R),{\bf m}u_i{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R),i=1,2$,
\betaginin{align*}
|g(x_1,{\bf m}u)-g(x_2,\widetilde{{\bf m}u})|\text{\mathbf rm{e}}lleq L_g(|x_1-x_2|+{\bf m}athbb{W}_2({\bf m}u,\widetilde{{\bf m}u}))
\text{\mathbf rm{e}}nd{align*}
and
\betaginin{align*}
|f(t,x_1,y_1,z_1,{\bf m}u_1)-f(t,x_2,y_2,z_2,{\bf m}u_2)|\text{\mathbf rm{e}}lleq L_f(|x_1-x_2|+|y_1-y_2|+|z_1-z_2|+{\bf m}athbb{W}_2({\bf m}u_1,{\bf m}u_2)).
\text{\mathbf rm{e}}nd{align*}
{\rm in}tem[(ii)] $|g(0,\text{\mathbf rm{d}}eltalta_0)|+\sup_{t{\rm in}n [0,T]}|f(t,0,0,0,\text{\mathbf rm{d}}eltalta_0)|<{\rm in}nfty$.
\text{\mathbf rm{e}}nd{enumerate}
Owing to \cite[Theorem A.1]{Li18} where the driven noises are a Brownian motion and an independent Poisson random measure,
the auxiliary equation \text{\mathbf rm{e}}quationref{Aueq} has a unique solution $(\widetilde{Y},\widetilde{Z}){\rm in}n\widetilde{\Upsilon}$.
Hence, with the help of Theorem \mathbf ref{Th1}, \text{\mathbf rm{e}}quationref{Bsde} admits a unique solution $(Y,Z){\rm in}n\Upsilon$.
\betagin{rem}\text{\mathbf rm{e}}llabel{Re1}
(i) (Stability estimate) Let $(Y^i,Z^i){\rm in}n\Upsilon$ be the unique solution of \text{\mathbf rm{e}}quationref{Bsde} with the coefficients $(g^i,f^i),i=1,2$,
which satisfy \textsc{\textbf{(H1)}}. Then, there exists a constant $C>0$ such that
\betaginin{align*}
&{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup_{s{\rm in}n[0,T]}|Y^1_s-Y^2_s|^2+{\rm in}nt_0^{T}|Z^1_s-Z^2_s|^2\text{\mathbf rm{d}} V_s\mathbf right)\cr
\text{\mathbf rm{e}}lleq& C{\bf m}athbb E\text{\mathbf rm{e}}lleft[\text{\mathbf rm{e}}lleft|(g^1-g^2)(X_T,{\bf m}athscr{L}_{X_{T}})\mathbf right|^2+
\text{\mathbf rm{e}}lleft({\rm in}nt_0^{T}|(f^1-f^2)(s,X_s,{Y}^1_s,{Z}^1_s,{\bf m}athscr{L}_{(X_s,{Y}^1_s,{Z}^1_s)})|\text{\mathbf rm{d}} V_s\mathbf right)^2\mathbf right].
\text{\mathbf rm{e}}nd{align*}
Indeed, we denote by $(\tilde Y^i,\tilde Z^i){\rm in}n\tilde\Upsilon$ the unique solution of \text{\mathbf rm{e}}quationref{Aueq} with the coefficient $(g^i,f^i),i=1,2$.
By \cite[Theorem A.2]{Li18}, we derive that there is a constant $C>0$ such that
\betaginin{align*}
&\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup_{s{\rm in}n[0,V_T]}|\tilde Y^1_s-\tilde Y^2_s|^2+{\rm in}nt_0^{V_T}|\tilde Z^1_s-\tilde Z^2_s|^2\text{\mathbf rm{d}} s\mathbf right)\cr
\text{\mathbf rm{e}}lleq& C\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft[\text{\mathbf rm{e}}lleft|(g^1-g^2)(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\widetilde{W}_{V_T}})\mathbf right|^2+
\text{\mathbf rm{e}}lleft({\rm in}nt_0^{V_T}|(f^1-f^2)(U_s,\widetilde{W}_s,\widetilde{Y}^1_s,\widetilde{Z}^1_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s,\widetilde{Z}^1_s)})|\text{\mathbf rm{d}} s\mathbf right)^2\mathbf right].
\text{\mathbf rm{e}}nd{align*}
Then, the desired stability estimate follows from straightforward calculations via transfer principle Proposition \mathbf ref{Pr1}.
(ii) In the distribution-free case, namely the coefficients $g$ and $f$ in \text{\mathbf rm{e}}quationref{Bsde} are independent of distributions,
Bender \cite[Theorems 4.2 and 4.4]{Bender14} proved the existence and uniqueness result, and our recent work \cite[Theorem 5.1]{FW21}
investigated the existence of densities and moreover derived their Gaussian estimates for the marginal laws of the solution.
So, our work in this note can be regarded as a continuation and generalization of \cite{Bender14,FW21}.
\text{\mathbf rm{e}}nd{rem}
\section{Comparison theorem and converse comparison theorem}
In this section, we first obtain a comparison theorem by imposing on the condition involved the Lions derivative of the generator.
In the second part, we concern with the converse problem of comparison theorem.
More precisely, we shall establish a representation theorem for the generator, and show how to combine this result to derive a converse comparison theorem.
\subsection{Comparison theorem}
In the Brownian motion case, i.e. $X=B^{1/2}$, the counter examples given in \cite[Examples 3.1 and 3.2]{BLP09} or \cite[Example 2.1]{LLZ18} show that
if the generator $f$ depends on the law of $Z$ or is not increasing with respect to the law of $Y$, comparison theorems usually don't hold for equation \text{\mathbf rm{e}}quationref{Bsde}.
We consider now the following special version of (3.1):
\betaginin{align}\text{\mathbf rm{e}}llabel{Bsde-com}
Y_t=g(X_T,{\bf m}athscr{L}_{X_T})+{\rm in}nt_t^Tf(s,X_s,Y_s,Z_s,{\bf m}athscr{L}_{(X_s,Y_s)})\text{\mathbf rm{d}} V_s-{\rm in}nt_t^TZ_s\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s,\ \ t{\rm in}n[0,T].
\text{\mathbf rm{e}}nd{align}
Correspondingly, the auxiliary equation of \text{\mathbf rm{e}}quationref{Bsde-com} is of the form
\betaginin{align}\text{\mathbf rm{e}}llabel{AuBsde-com}
\tilde Y_t=&g(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})+{\rm in}nt_t^{V_T}f(U_s,\tilde{W}_s,\tilde{Y}_s,\tilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s)})\text{\mathbf rm{d}} s
-{\rm in}nt_t^{V_T}\widetilde{Z}_s\text{\mathbf rm{d}}\widetilde{W}_s,\ \ t{\rm in}n[0,V_T].
\text{\mathbf rm{e}}nd{align}
\betagin{thm}\text{\mathbf rm{e}}llabel{Th(com)}
Assume that two generators $f^i,i=1,2$ satisfy \textsc{\textbf{(H1)}}, and that for any $(t,x,y,z){\rm in}n[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$, $f^1(t,x,y,z,\cdot)$ belongs to $C^{(1,0)}({\bf m}athscr{P}_2({\bf m}athbb R\tildemes{\bf m}athbb R))$ with $0\text{\mathbf rm{e}}lleq (D^Lf^1)^{(2)}\text{\mathbf rm{e}}lleq K$ for some constant $K>0$, where $(D^Lf^1)^{(2)}$ denotes the second component of $D^Lf^1$.
Let $(Y^i,Z^i){\rm in}n\Upsilon, i=1,2$ be the solutions of \text{\mathbf rm{e}}quationref{Bsde-com} with data $(g^i,f^i),i=1,2$, respectively.
Then, if
\betaginin{align}\text{\mathbf rm{e}}llabel{1Th(com)}
g^1(x,{\bf m}u)\text{\mathbf rm{e}}lleq g^2(x,{\bf m}u),\ \ x{\rm in}n{\bf m}athbb R,{\bf m}u{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R)
\text{\mathbf rm{e}}nd{align}
and
\betaginin{align}\text{\mathbf rm{e}}llabel{2Th(com)}
f^1(t,x,y,z,\nu)\text{\mathbf rm{e}}lleq f^2(t,x,y,z,\nu),\ \ t{\rm in}n[0,T], x,y,z{\rm in}n{\bf m}athbb R,\nu{\rm in}n{\bf m}athscr{P}_2({\bf m}athbb R\tildemes{\bf m}athbb R),
\text{\mathbf rm{e}}nd{align}
there holds that for every $t{\rm in}n[0,T], Y_t^1\text{\mathbf rm{e}}lleq Y_t^2, {\bf m}athbb{P}$-a.s..
\text{\mathbf rm{e}}nd{thm}
\betagin{proof}
Let $(\widetilde{Y}^i,\widetilde{Z}^i){\rm in}n\widetilde{\Upsilon}$ be the unique solutions of \text{\mathbf rm{e}}quationref{AuBsde-com} associated with $(g^i,f^i), i=1,2$,
and denote by $(\tilde u^i,\tilde v^i),i=1,2$ their representation functions, respectively.
We first give a comparison result for $(\widetilde{Y}^i,\widetilde{Z}^i), i=1,2$.
Our strategy hinges on the It\^{o}-Tanaka formula applied to $[(\tilde Y_t^1-\tilde Y_t^2)^+]^2$ (see, e,g., \cite[Proposition 2.35 and Remark 2.36]{PR14}), which gives for any $t{\rm in}n[0,V_T]$,
\betaginin{align*}
&\tilde{\bf m}athbb E[(\tilde Y_t^1-\tilde Y_t^2)^+]^2+\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}|\tilde Z_s^1-\tilde Z_s^2|^2{\bf m}athrm{I}_{\{\tilde Y_s^1-\tilde Y_s^2\geq0\}}\text{\mathbf rm{d}} s\nonumber\\
=&\tilde{\bf m}athbb E[(g^1-g^2)^+(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})]^2\nonumber\\
&+2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+\text{\mathbf rm{e}}lleft(f^1(U_s,\Theta_s^1,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})-f^2(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})\mathbf right)\text{\mathbf rm{d}} s\nonumber\\
=&2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+\text{\mathbf rm{e}}lleft(f^1(U_s,\Theta_s^1,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})-f^2(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})\mathbf right)\text{\mathbf rm{d}} s,
\text{\mathbf rm{e}}nd{align*}
Here, we have set $\Theta_s^i=(\widetilde{W}_s,\tilde{Y}^i_s,\tilde{Z}^i_s), i=1,2$ and used \text{\mathbf rm{e}}quationref{1Th(com)}.\\
Using the Lipschitz continuity of $f^1$ and the fact that $f^1(U_t,\cdot,\cdot,\cdot,\cdot)\text{\mathbf rm{e}}lleq f^2(U_t,\cdot,\cdot,\cdot,\cdot)$ for each $t{\rm in}n[0,V_T]$ implied by \text{\mathbf rm{e}}quationref{2Th(com)}, we get
\betaginin{align}\text{\mathbf rm{e}}llabel{1PfTh(com)}
&\tilde{\bf m}athbb E[(\tilde Y_t^1-\tilde Y_t^2)^+]^2+\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}|\tilde Z_s^1-\tilde Z_s^2|^2{\bf m}athrm{I}_{\{\tilde Y_s^1-\tilde Y_s^2\geq0\}}\text{\mathbf rm{d}} s\nonumber\\
=&2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+
\text{\mathbf rm{e}}lleft(f^1(U_s,\Theta_s^1,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})-f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})\mathbf right)\text{\mathbf rm{d}} s\cr
&+2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+
\text{\mathbf rm{e}}lleft(f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})-f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})\mathbf right)\text{\mathbf rm{d}} s\cr
&+2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+
\text{\mathbf rm{e}}lleft(f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})-f^2(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})\mathbf right)\text{\mathbf rm{d}} s\cr
\text{\mathbf rm{e}}lleq&2L_{f^1}\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+\text{\mathbf rm{e}}lleft(|\tilde Y_s^1-\tilde Y_s^2|+|\tilde Z_s^1-\tilde Z_s^2|\mathbf right)\text{\mathbf rm{d}} s
\cr
&+2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+
\text{\mathbf rm{e}}lleft(f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})-f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})\mathbf right)\text{\mathbf rm{d}} s.
\text{\mathbf rm{e}}nd{align}
Observe that by Lemma \mathbf ref{FoLD}, we obtain
\betaginin{align*}
&f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^1_s)})-f^1(U_s,\Theta_s^2,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}^2_s)})\cr
=&{\rm in}nt_0^1\frac \text{\mathbf rm{d}} {\text{\mathbf rm{d}} r}f^1(U_s,\Theta_r^2,{\bf m}athscr{L}_{\chi_s(r)})\text{\mathbf rm{d}} r\cr
=&{\rm in}nt_0^1\tilde{\bf m}athbb E\text{\mathbf rm{e}}llangle D^Lf^1(U_s,x,\cdot)({\bf m}athscr{L}_{\chi_s(r)})(\chi_s(r)),(0,\widetilde{Y}^1_s-\widetilde{Y}^2_s)\mathbf rightarrowngle|_{x=\Theta_r^2}\text{\mathbf rm{d}} r\cr
=&{\rm in}nt_0^1\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft[(D^Lf^1)^{(2)}(U_s,x,\cdot)({\bf m}athscr{L}_{\chi_s(r)})(\chi_s(r))\cdot(\widetilde{Y}^1_s-\widetilde{Y}^2_s)\mathbf right]\big|_{x=\Theta_r^2}\text{\mathbf rm{d}} r\cr
\text{\mathbf rm{e}}lleq& K\tilde{\bf m}athbb E(\widetilde{Y}^1_s-\widetilde{Y}^2_s)^+,
\text{\mathbf rm{e}}nd{align*}
where for any $r{\rm in}n[0,1],\chi_s(r)=(\widetilde{W}_s,\widetilde{Y}^2_s)+r(0,\widetilde{Y}^1_s-\widetilde{Y}^2_s)$.\\
Therefore, plugging this into \text{\mathbf rm{e}}quationref{1PfTh(com)} and applying the H\"{o}lder and the Young inequalities, we have
\betaginin{align*}
&\tilde{\bf m}athbb E[(\tilde Y_t^1-\tilde Y_t^2)^+]^2+\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}|\tilde Z_s^1-\tilde Z_s^2|^2{\bf m}athrm{I}_{\{\tilde Y_s^1-\tilde Y_s^2\geq0\}}\text{\mathbf rm{d}} s\nonumber\\
\text{\mathbf rm{e}}lleq&2(L_{f^1}\varepsilone K)\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}(\tilde Y_s^1-\tilde Y_s^2)^+\text{\mathbf rm{e}}lleft(|\tilde Y_s^1-\tilde Y_s^2|+|\tilde Z_s^1-\tilde Z_s^2|+\tilde{\bf m}athbb E(\widetilde{Y}^1_s-\widetilde{Y}^2_s)^+\mathbf right)\text{\mathbf rm{d}} s\cr
\text{\mathbf rm{e}}lleq&C{\rm in}nt_t^{V_T}\tilde{\bf m}athbb E[(\tilde Y_s^1-\tilde Y_s^2)^+]^2\text{\mathbf rm{d}} s+\frac 1 2\tilde{\bf m}athbb E{\rm in}nt_t^{V_T}|\tilde Z_s^1-\tilde Z_s^2|^2{\bf m}athrm{I}_{\{\tilde Y_s^1-\tilde Y_s^2\geq0\}}\text{\mathbf rm{d}} s,
\text{\mathbf rm{e}}nd{align*}
where and in what follows C denotes a generic constant.\\
Then, the Gronwall inequality implies $\tilde{\bf m}athbb E[(\tilde Y_t^1-\tilde Y_t^2)^+]^2=0$ for every $t{\rm in}n[0,V_T]$.
Consequently, it holds that for any $t{\rm in}n[0,V_T]$,
\betaginin{align*}
\widetilde{u}^1(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t})=\tilde Y_t^1\text{\mathbf rm{e}}lleq \tilde Y_t^2=\widetilde{u}^2(t,\widetilde{W}_t,{\bf m}athscr{L}_{\widetilde{W}_t}),\ \ \tilde{\bf m}athbb{P}\textit{-}a.s..
\text{\mathbf rm{e}}nd{align*}
Since $\widetilde{W}_t$ has the Gaussian law, we derive that for any $t{\rm in}n[0,V_T]$,
\betaginin{align*}
\widetilde{u}^1(t,x,{\bf m}athscr{L}_{\widetilde{W}_t})\text{\mathbf rm{e}}lleq\widetilde{u}^2(t,x,{\bf m}athscr{L}_{\widetilde{W}_t}),\ \ \text{\mathbf rm{d}} x\textit{-}a.e.,
\text{\mathbf rm{e}}nd{align*}
which, together with the fact that ${\bf m}athscr{L}_{\widetilde{W}_{V_t}}={\bf m}athscr{L}_{X_t}$, yields that for each $t{\rm in}n[0,T]$,
\betaginin{align*}
\widetilde{u}^1(V_t,x,{\bf m}athscr{L}_{X_t})\text{\mathbf rm{e}}lleq\widetilde{u}^2(V_t,x,{\bf m}athscr{L}_{X_t}),\ \ \text{\mathbf rm{d}} x\textit{-}a.e.,
\text{\mathbf rm{e}}nd{align*}
According to Theorem \mathbf ref{Th1}, we conclude that $Y_t^1\text{\mathbf rm{e}}lleq Y_t^2, {\bf m}athbb{P}$-a.s. for every $t{\rm in}n[0,T]$.
This completes the proof.
\text{\mathbf rm{e}}nd{proof}
\betagin{rem}\text{\mathbf rm{e}}llabel{Re-comp1}
(i) In light of the proof above, one can see that if the conditions for $f^1$ are replaced by that for $f^2$,
Theorem \mathbf ref{Th(com)} also holds.
(ii) Compared with the relevant result on DDBSDEs driven by the standard Brownian motion $B^{1/2}$ shown in \cite[Theorem 2.2 and Remark 2.2]{LLZ18},
it is easy to see that our above result Theorem \mathbf ref{Th(com)} applies to more general BSDEs since we replace $B^{1/2}$ with a general
Gaussian process $X$ as driving process.
In addition, in contrast to the distribution-free case (see, e.g., \cite[Theorem 4.6 (iii)]{Bender14}, \cite[Theorem 2.2]{EPQ9705} or \cite[Theorem 12.3]{HCS12}), we need to overcome difficulty induced by the appearance of ${\bf m}athscr{L}_{(X_s,Y_s)}$ in the generator via a formula for the $L$-derivative (Lemma \mathbf ref{FoLD}).
\text{\mathbf rm{e}}nd{rem}
\subsection{Converse comparison theorem}
In this part, we investigate a kind of converse comparison problem:
if for each $t{\rm in}n[0,T],\text{\mathbf rm{e}}psilon{\rm in}n(0,T-t)$ and $\xi{\rm in}n L^2_X$, $Y_t^1(t+\text{\mathbf rm{e}}psilon,\xi)\text{\mathbf rm{e}}lleq Y_t^2(t+\text{\mathbf rm{e}}psilon,\xi)$ (see the definition at the beginning of Theorem \mathbf ref{Th(Conve)}), do we have $f^1(t,y,z,\nu)\text{\mathbf rm{e}}lleq f^2(t,y,z,\nu)$ for every $(t,y,z,\nu){\rm in}n[0,T]\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athscr{P}({\bf m}athbb R\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R)$?
That is, if we can compare the solutions of two DDBSDEs with the
same terminal condition, for all terminal conditions, can we compare the generators?
In order to study this question, we first give a representation theorem for the generator $f$ initiated in \cite{BCHMP00}.
Now, given $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$ and let $0<\text{\mathbf rm{e}}psilonsilon<T-t$, and denote by $(Y^\text{\mathbf rm{e}}psilonsilon,Z^\text{\mathbf rm{e}}psilonsilon)$ the unique solution of the following DDBSDE on $[t,t+\text{\mathbf rm{e}}psilonsilon]$:
\betaginin{align}\text{\mathbf rm{e}}llabel{1-Repr}
Y^\text{\mathbf rm{e}}psilonsilon_r=y+z(X_{t+\text{\mathbf rm{e}}psilon}-X_t)+{\rm in}nt_r^{t+\text{\mathbf rm{e}}psilon}f(s,Y^\text{\mathbf rm{e}}psilon_s,Z^\text{\mathbf rm{e}}psilon_s,{\bf m}athscr{L}_{(X_s,Y_s^\text{\mathbf rm{e}}psilon,Z_s^\text{\mathbf rm{e}}psilon)})\text{\mathbf rm{d}} V_s-{\rm in}nt_r^{t+\text{\mathbf rm{e}}psilonsilon}Z^\text{\mathbf rm{e}}psilonsilon_s\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s.
\text{\mathbf rm{e}}nd{align}
Our representation theorem is formulated as follows
\betagin{thm}\text{\mathbf rm{e}}llabel{Th(Repre)}
Assume that \textsc{\textbf{(H1)}} holds and $V_{t+\text{\mathbf rm{e}}psilon}-V_t=O(\text{\mathbf rm{e}}psilon)$ as $\text{\mathbf rm{e}}psilon\mathbf rightarrow0$ for any $t{\rm in}n[0,T)$.
Then for any $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$, the following two statements are equivalent:
\betagin{align}\text{\mathbf rm{e}}llabel{1Th(Re)}
(i) \text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}&\frac{Y^\text{\mathbf rm{e}}psilon_t-y}{\text{\mathbf rm{e}}psilon}=f(t,y,z,{\bf m}athscr{L}_{(X_t,y,z)});\\
(ii) \text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}&\frac 1 \text{\mathbf rm{e}}psilon{\rm in}nt_t^{t+\text{\mathbf rm{e}}psilon}f(r,y,z,{\bf m}athscr{L}_{(X_t,y,z)})\text{\mathbf rm{d}} r=f(t,y,z,{\bf m}athscr{L}_{(X_t,y,z)}).\text{\mathbf rm{e}}llabel{2Th(Re)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{\mathbf rm{e}}nd{align}
\text{\mathbf rm{e}}nd{thm}
\betagin{proof}
We split the proof into two steps.
First, we prove this theorem for the case of the auxiliary equation of \text{\mathbf rm{e}}quationref{1-Repr};
then we extend this result to \text{\mathbf rm{e}}quationref{1-Repr} by applying the transfer principle.
\textit{Step 1. The auxiliary equation case.}
Observe first that the corresponding auxiliary equation of \text{\mathbf rm{e}}quationref{1-Repr} is given by
\betaginin{align}\text{\mathbf rm{e}}llabel{1PfTh(Re)}
\tilde{Y}^\text{\mathbf rm{e}}psilon_{V_r}=&y+z(\tilde{W}_{V_{t+\text{\mathbf rm{e}}psilon}}-\tilde{W}_{V_t})+{\rm in}nt_{V_r}^{V_{t+\text{\mathbf rm{e}}psilon}}f(U_s,\tilde{Y}^\text{\mathbf rm{e}}psilon_s,\tilde{Z}^\text{\mathbf rm{e}}psilon_s,{\bf m}athscr{L}_{(\tilde{W}_s,\tilde{Y}^\text{\mathbf rm{e}}psilon_s,\tilde{Z}_s^\text{\mathbf rm{e}}psilon)})\text{\mathbf rm{d}} s\cr
&-{\rm in}nt_{V_r}^{V_{t+\text{\mathbf rm{e}}psilon}}\tilde{Z}^\text{\mathbf rm{e}}psilon_s\text{\mathbf rm{d}}\tilde{W}_s,\ \ r{\rm in}n[t,t+\text{\mathbf rm{e}}psilon].
\text{\mathbf rm{e}}nd{align}
We claim that for any $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$, the following two statements are equivalent:
\betaginin{align*}
(I) \text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}&\frac{\tilde Y^\text{\mathbf rm{e}}psilonsilon_{V_t}-y}{\text{\mathbf rm{e}}psilonsilon}=f(t,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)});\\
(II) \text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}&\frac 1 \text{\mathbf rm{e}}psilon{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}f(U_r,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\text{\mathbf rm{d}} r=f(t,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)}).
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{\mathbf rm{e}}nd{align*}
Indeed, for $s{\rm in}n[V_t,V_{t+\text{\mathbf rm{e}}psilon}]$, we put $\Gammamma^\text{\mathbf rm{e}}psilon_s:=\tilde{Y}^\text{\mathbf rm{e}}psilon_s-(y+z(\tilde{W}_s-\tilde{W}_{V_t}))$.
Then, applying the It\^{o} formula to $\Gammamma^\text{\mathbf rm{e}}psilon_s$ on the interval $[V_t,V_{t+\text{\mathbf rm{e}}psilon}]$ yields
\betaginin{align}\text{\mathbf rm{e}}llabel{2PfTh(Re)}
\Gammamma^\text{\mathbf rm{e}}psilon_s=&{\rm in}nt_s^{V_{t+\text{\mathbf rm{e}}psilon}}f(U_r,\Gammamma^\text{\mathbf rm{e}}psilon_r+y+z(\tilde{W}_r-\tilde{W}_{V_t}),\tilde{Z}^\text{\mathbf rm{e}}psilon_r,{\bf m}athscr{L}_{(\tilde{W}_r,\Gammamma^\text{\mathbf rm{e}}psilon_r+y+z(\tilde{W}_r-\tilde{W}_{V_t}),\tilde{Z}_r^\text{\mathbf rm{e}}psilon)})\text{\mathbf rm{d}} r\cr
&-{\rm in}nt_s^{V_{t+\text{\mathbf rm{e}}psilon}}(\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z)\text{\mathbf rm{d}}\tilde{W}_r\cr
=:&{\rm in}nt_s^{V_{t+\text{\mathbf rm{e}}psilon}}f^\text{\mathbf rm{e}}psilon(r)\text{\mathbf rm{d}} r-{\rm in}nt_s^{V_{t+\text{\mathbf rm{e}}psilon}}(\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z)\text{\mathbf rm{d}}\tilde{W}_r.
\text{\mathbf rm{e}}nd{align}
Using the facts that $\tilde{Y}^\text{\mathbf rm{e}}psilon_{V_t}-y=\Gammamma^\text{\mathbf rm{e}}psilon_{V_t}$ and $\Gammamma^\text{\mathbf rm{e}}psilon_{V_t}$ is deterministic thanks to \cite[Proposition 4.2]{EPQ9705} and taking the conditional expectation with respect to $\tilde{\bf m}athscr{F}_{V_t}$ in \text{\mathbf rm{e}}quationref{2PfTh(Re)}, we arrive at
\betaginin{align}\text{\mathbf rm{e}}llabel{6PfTh(Re)}
&\frac{\tilde Y^\text{\mathbf rm{e}}psilonsilon_{V_t}-y}{\text{\mathbf rm{e}}psilonsilon}-f(t,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\cr
=&\frac{\Gammamma^\text{\mathbf rm{e}}psilon_{V_t}}\text{\mathbf rm{e}}psilon-f(t,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\cr
=&\frac 1 \text{\mathbf rm{e}}psilon\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft({\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}f^\text{\mathbf rm{e}}psilon(r)\text{\mathbf rm{d}} r|\tilde{\bf m}athscr{F}_{V_t}\mathbf right)-f(t,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\cr
=&\frac 1 \text{\mathbf rm{e}}psilon\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft[{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}\text{\mathbf rm{e}}lleft(f^\text{\mathbf rm{e}}psilon(r)-f(U_r,y+z(\tilde{W}_r-\tilde{W}_{V_t}),z,{\bf m}athscr{L}_{(\tilde{W}_r,y+z(\tilde{W}_r-\tilde{W}_{V_t}),z)})\mathbf right)\text{\mathbf rm{d}} r\big|\tilde{\bf m}athscr{F}_{V_t}\mathbf right]\cr
&+\frac 1 \text{\mathbf rm{e}}psilon\tilde{\bf m}athbb E\bigg[{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}\mathbf Big(f(U_r,y+z(\tilde{W}_r-\tilde{W}_{V_t}),z,{\bf m}athscr{L}_{(\tilde{W}_r,y+z(\tilde{W}_r-\tilde{W}_{V_t}),z)})\cr
&\qquad\qquad\qquad\quad -f(U_r,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\mathbf Big)\text{\mathbf rm{d}} r\big|\tilde{\bf m}athscr{F}_{V_t}\bigg]\cr
&+\frac 1 \text{\mathbf rm{e}}psilon{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}f(U_r,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\text{\mathbf rm{d}} r-f(t,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\cr
=:&I_1+I_2+I_3.
\text{\mathbf rm{e}}nd{align}
By the H\"{o}lder inequality and \textsc{\textbf{(H1)}}, we have
\betaginin{align}\text{\mathbf rm{e}}llabel{3PfTh(Re)}
\tilde{\bf m}athbb E|I_1|^2\text{\mathbf rm{e}}lleq&3L_f^2\frac {V_{t+\text{\mathbf rm{e}}psilon}-V_t}{\text{\mathbf rm{e}}psilon^2}{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft[|\Gammamma^\text{\mathbf rm{e}}psilon_r|^2+|\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z|^2+\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(|\Gammamma^\text{\mathbf rm{e}}psilon_r|^2+|\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z|^2\mathbf right)\mathbf right]\text{\mathbf rm{d}} r\cr
=&6L_f^2\frac {V_{t+\text{\mathbf rm{e}}psilon}-V_t}{\text{\mathbf rm{e}}psilon^2}{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(|\Gammamma^\text{\mathbf rm{e}}psilon_r|^2+|\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z|^2\mathbf right)\text{\mathbf rm{d}} r
\text{\mathbf rm{e}}nd{align}
and
\betaginin{align}\text{\mathbf rm{e}}llabel{4PfTh(Re)}
\tilde{\bf m}athbb E|I_2|^2\text{\mathbf rm{e}}lleq&2L_f^2\frac {V_{t+\text{\mathbf rm{e}}psilon}-V_t}{\text{\mathbf rm{e}}psilon^2}{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}{\bf m}athbb E\text{\mathbf rm{e}}lleft[z^2|\tilde{W}_r-\tilde{W}_{V_t}|^2+(1+z^2)\tilde{\bf m}athbb E|\tilde{W}_r-\tilde{W}_{V_t}|^2\mathbf right]\text{\mathbf rm{d}} r\cr
=&2L_f^2(1+2z^2)\frac {V_{t+\text{\mathbf rm{e}}psilon}-V_t}{\text{\mathbf rm{e}}psilon^2}{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}(r-V_t)\text{\mathbf rm{d}} r\cr
=&L_f^2(1+2z^2)\frac {(V_{t+\text{\mathbf rm{e}}psilon}-V_t)^3}{\text{\mathbf rm{e}}psilon^2}.
\text{\mathbf rm{e}}nd{align}
Similar to \cite[Proposition 2.2]{BCHMP00}, using the It\^{o} formula applied to $\text{\mathbf rm{e}}^{\beta s}|\Gammamma^\text{\mathbf rm{e}}psilon_s|^2$ with some constant $\betata>0$ depending only on $L_f$, we obtain the following a priori estimate for \text{\mathbf rm{e}}quationref{2PfTh(Re)} (its solution is regarded as $(\Gammamma^\text{\mathbf rm{e}}psilon_\cdot,\tilde{Z}^\text{\mathbf rm{e}}psilon_\cdot-z)$)
\betaginin{align*}
&\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup_{V_t\text{\mathbf rm{e}}lleq s\text{\mathbf rm{e}}lleq V_{t+\text{\mathbf rm{e}}psilon}}|\Gammamma^\text{\mathbf rm{e}}psilon_s|^2+{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}|\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z|^2\text{\mathbf rm{d}} r\big|\tilde{\bf m}athscr{F}_{V_t}\mathbf right)\cr
\text{\mathbf rm{e}}lleq&C\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft[\text{\mathbf rm{e}}lleft({\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}|
f(U_r,y+z(\tilde{W}_r-\tilde{W}_{V_t}),z,{\bf m}athscr{L}_{(\tilde{W}_r,y+z(\tilde{W}_r-\tilde{W}_{V_t}),z)})|\text{\mathbf rm{d}} r\mathbf right)^2\big|\tilde{\bf m}athscr{F}_{V_t}\mathbf right].
\text{\mathbf rm{e}}nd{align*}
Then, it follows from the H\"{o}lder inequality and \textsc{\textbf{(H1)}} that
\betaginin{align*}
&\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft(\sup_{V_t\text{\mathbf rm{e}}lleq s\text{\mathbf rm{e}}lleq V_{t+\text{\mathbf rm{e}}psilon}}|\Gammamma^\text{\mathbf rm{e}}psilon_s|^2+{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}|\tilde{Z}^\text{\mathbf rm{e}}psilon_r-z|^2\text{\mathbf rm{d}} r\mathbf right)\cr
\text{\mathbf rm{e}}lleq&C(V_{t+\text{\mathbf rm{e}}psilon}-V_t)
{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}\tilde{\bf m}athbb E\text{\mathbf rm{e}}lleft[|f(U_r,y,z,{\bf m}athscr{L}_{(\tilde{W}_{V_t},y,z)})|^2+z^2|\tilde{W}_r-\tilde{W}_{V_t}|^2+(1+z^2)\tilde{\bf m}athbb E|\tilde{W}_r-\tilde{W}_{V_t}|^2\mathbf right]\text{\mathbf rm{d}} r\cr
=&C(V_{t+\text{\mathbf rm{e}}psilon}-V_t)
\text{\mathbf rm{e}}lleft[{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}|f(U_r,y,z,{\bf m}athscr{L}_{(\tilde{W}_{V_t},y,z)})|^2\text{\mathbf rm{d}} r+(1+2z^2)(V_{t+\text{\mathbf rm{e}}psilon}-V_t)^2\mathbf right].
\text{\mathbf rm{e}}nd{align*}
Substituting this into \text{\mathbf rm{e}}quationref{3PfTh(Re)} yields
\betaginin{align}\text{\mathbf rm{e}}llabel{5PfTh(Re)}
\tilde{\bf m}athbb E|I_1|^2\text{\mathbf rm{e}}lleq& C\text{\mathbf rm{e}}lleft(\frac{V_{t+\text{\mathbf rm{e}}psilon}-V_t}{\text{\mathbf rm{e}}psilon}\mathbf right)^2(1+V_{t+\text{\mathbf rm{e}}psilon}-V_t)\cr
&\tildemes\text{\mathbf rm{e}}lleft[{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}|f(U_r,y,z,{\bf m}athscr{L}_{(\tilde{W}_{V_t},y,z)})|^2\text{\mathbf rm{d}} r+(1+2z^2)(V_{t+\text{\mathbf rm{e}}psilon}-V_t)^2\mathbf right].
\text{\mathbf rm{e}}nd{align}
Note that by the absolute continuity of integral, we have
\betaginin{align*}
\text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}{\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}|f(U_r,y,z,{\bf m}athscr{L}_{(\tilde{W}_{V_t},y,z)})|^2\text{\mathbf rm{d}} r=0.
\text{\mathbf rm{e}}nd{align*}
Therefore, owing to the condition $V_{t+\text{\mathbf rm{e}}psilon}-V_t=O(\text{\mathbf rm{e}}psilon)$ as $\text{\mathbf rm{e}}psilon\mathbf rightarrow0$ and \text{\mathbf rm{e}}quationref{4PfTh(Re)}-\text{\mathbf rm{e}}quationref{5PfTh(Re)}, we conclude that
\betaginin{align*}
\text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}\text{\mathbf rm{e}}lleft(\tilde{\bf m}athbb E|I_1|^2+\tilde{\bf m}athbb E|I_2|^2\mathbf right)=0,
\text{\mathbf rm{e}}nd{align*}
which, along with \text{\mathbf rm{e}}quationref{6PfTh(Re)} and the fact that $\tilde Y^\text{\mathbf rm{e}}psilonsilon_{V_t}$ is deterministic due to \cite[Proposition 4.2]{EPQ9705} again, implies the desired assertion.
\textit{Step 2. The equation with Gaussian noise case.}
With a slight modification of the proof of Theorem \mathbf ref{Th1}, we know that for every $\text{\mathbf rm{e}}psilon>0$, there exists a representation function $\tilde u^\text{\mathbf rm{e}}psilon$ such that the solutions of \text{\mathbf rm{e}}quationref{1-Repr} and \text{\mathbf rm{e}}quationref{1PfTh(Re)} can be written as the following form: for any $r{\rm in}n[t,t+\text{\mathbf rm{e}}psilon]$,
\betaginin{align*}
Y^\text{\mathbf rm{e}}psilon_r=\tilde u^\text{\mathbf rm{e}}psilon(V_r,X_r-X_t,{\bf m}athscr{L}_{X_r-X_t}) \ \ \ {\bf m}athrm{and} \ \ \ \tilde Y^\text{\mathbf rm{e}}psilon_{V_r}=\tilde u^\text{\mathbf rm{e}}psilon(V_r,\tilde W_{V_r}-\tilde W_{V_t},{\bf m}athscr{L}_{\tilde W_{V_r}-\tilde W_{V_t}}).
\text{\mathbf rm{e}}nd{align*}
Then, we get
\betaginin{align}\text{\mathbf rm{e}}llabel{7PfTh(Re)}
\frac {Y^\text{\mathbf rm{e}}psilon_t-y}\text{\mathbf rm{e}}psilon=\frac{\tilde u^\text{\mathbf rm{e}}psilon(V_t,0,\text{\mathbf rm{d}}elta_0)-y}\text{\mathbf rm{e}}psilon=\frac{\tilde Y^\text{\mathbf rm{e}}psilon_{V_t}-y}{\text{\mathbf rm{e}}psilon}.
\text{\mathbf rm{e}}nd{align}
On the other hand, by a change of variables it is easy to see that
\betaginin{align}\text{\mathbf rm{e}}llabel{8PfTh(Re)}
{\rm in}nt_{t}^{t+\text{\mathbf rm{e}}psilon}f(r,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t},y,z)})\text{\mathbf rm{d}} r={\rm in}nt_{V_t}^{V_{t+\text{\mathbf rm{e}}psilon}}f(U_r,y,z,{\bf m}athscr{L}_{(\tilde W_{V_t}.y,z)})\text{\mathbf rm{d}} r.
\text{\mathbf rm{e}}nd{align}
Finally, taking into account of the claim in step 1 and using \text{\mathbf rm{e}}quationref{7PfTh(Re)}-\text{\mathbf rm{e}}quationref{8PfTh(Re)} and the fact that ${\bf m}athscr{L}_{(\tilde W_{V_t},y,z)}={\bf m}athscr{L}_{(X_t,y,z)}$,
we deduce that \text{\mathbf rm{e}}quationref{1Th(Re)} is equivalent to \text{\mathbf rm{e}}quationref{2Th(Re)}.
Our proof is now finished.
\text{\mathbf rm{e}}nd{proof}
\betagin{rem}\text{\mathbf rm{e}}llabel{Re(Rep)}
Although the computations become much more involved, it is not hard to extend the result in step 1 to \text{\mathbf rm{e}}quationref{1PfTh(Re)} with $\tilde W$ being replaced by a diffusion process, which is a generalisation of \cite[Proposition 2.3]{BCHMP00} and \cite[Theorem 3.3]{Jiang05} that handled the distribution-free BSDEs
driven by Brownian motion.
\text{\mathbf rm{e}}nd{rem}
With the help of Theorem \mathbf ref{Th(Repre)}, we can establish a converse comparison theorem.
To this end, we denote by $(Y_t^i(T,g(X_T,{\bf m}athscr{L}_{X_T})),Z_t^i(T,g(X_T,{\bf m}athscr{L}_{X_T})))_{t{\rm in}n[0,T]}$ the solution of
\betagin{align}\text{\mathbf rm{e}}llabel{3Th(Conve)}
Y_t^i=&g(X_T,{\bf m}athscr{L}_{X_T})+{\rm in}nt_t^Tf^i(s,Y_s^i,Z_s^i,{\bf m}athscr{L}_{(X_s,Y^i_s,Z^i_s)})\text{\mathbf rm{d}} V_s\cr
&-{\rm in}nt_t^TZ_s^i\text{\mathbf rm{d}}^\text{\mathbf rm{d}}iamond X_s,\ \ t{\rm in}n[0,T],\ \ i=1,2.
\text{\mathbf rm{e}}nd{align}
Our converse comparison theorem reads as follows
\betagin{thm}\text{\mathbf rm{e}}llabel{Th(Conve)}
Let \textsc{\textbf{(H1)}} hold for $f^i,i=1,2$ and $V_{t+\text{\mathbf rm{e}}psilon}-V_t=O(\text{\mathbf rm{e}}psilon)$ as $\text{\mathbf rm{e}}psilon\mathbf rightarrow0$ for any $t{\rm in}n[0,T)$.
Assume moreover that for each $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$ and $\text{\mathbf rm{e}}psilon{\rm in}n(0,T-t]$,
\betagin{align}\text{\mathbf rm{e}}llabel{1Th(Conve)}
Y_t^1(t+\text{\mathbf rm{e}}psilon,y+z(X_{t+\text{\mathbf rm{e}}psilon}-X_t))\text{\mathbf rm{e}}lleq Y_t^2(t+\text{\mathbf rm{e}}psilon,y+z(X_{t+\text{\mathbf rm{e}}psilon}-X_t)).
\text{\mathbf rm{e}}nd{align}
Then for each $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$, we have
\betagin{align}\text{\mathbf rm{e}}llabel{2Th(Conve)}
f^1(t,y,z,{\bf m}athscr{L}_{(X_t,y,z)})\text{\mathbf rm{e}}lleq f^2(t,y,z,{\bf m}athscr{L}_{(X_t,y,z)}).
\text{\mathbf rm{e}}nd{align}
\text{\mathbf rm{e}}nd{thm}
\betagin{proof}
Since \textsc{\textbf{(H1)}} holds for $f^i,i=1,2$, one can see that for any $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$,
\betagin{align*}
\text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}&\frac 1 \text{\mathbf rm{e}}psilon{\rm in}nt_t^{t+\text{\mathbf rm{e}}psilon}f^i(r,y,z,{\bf m}athscr{L}_{(X_t,y,z)})\text{\mathbf rm{d}} r=f^i(t,y,z,{\bf m}athscr{L}_{(X_t,y,z)}),\ \ i=1,2.
\text{\mathbf rm{e}}nd{align*}
Then, in light of Theorem \mathbf ref{Th(Repre)}, we have
\betagin{align}\text{\mathbf rm{e}}llabel{1PfTh(Conve)}
\text{\mathbf rm{e}}llim\text{\mathbf rm{e}}llimits_{\text{\mathbf rm{e}}psilon\mathbf rightarrow0^+}&\frac{Y_t^i(t+\text{\mathbf rm{e}}psilon,y+z(X_{t+\text{\mathbf rm{e}}psilon}-X_t))-y}{\text{\mathbf rm{e}}psilon}=f^i(t,y,z,{\bf m}athscr{L}_{(X_t,y,z)}),\ \ i=1,2.
\text{\mathbf rm{e}}nd{align}
By the hypothesis \text{\mathbf rm{e}}quationref{1Th(Conve)} we derive that for any $(t,y,z){\rm in}n[0,T)\tildemes{\bf m}athbb R\tildemes{\bf m}athbb R$ and $\text{\mathbf rm{e}}psilon{\rm in}n(0,T-t]$,
\betagin{align*}
Y_t^1(t+\text{\mathbf rm{e}}psilon,y+z(X_{t+\text{\mathbf rm{e}}psilon}-X_t))-y\text{\mathbf rm{e}}lleq Y_t^2(t+\text{\mathbf rm{e}}psilon,y+z(X_{t+\text{\mathbf rm{e}}psilon}-X_t))-y,
\text{\mathbf rm{e}}nd{align*}
which, along with \text{\mathbf rm{e}}quationref{1PfTh(Conve)}, yields the desired relation.
This then completes the proof.
\text{\mathbf rm{e}}nd{proof}
We conclude this section with a remark.
\betagin{rem}\text{\mathbf rm{e}}llabel{Re(Conve)}
By Theorems \mathbf ref{Th(Repre)} and \mathbf ref{Th(Conve)}, it is a surprise to us for finding that the relations \text{\mathbf rm{e}}quationref{1Th(Re)}, \text{\mathbf rm{e}}quationref{2Th(Re)} and \text{\mathbf rm{e}}quationref{2Th(Conve)} all depend on ${\bf m}athscr{L}_{(X_t,y,z)}$, which has a big difference from distribution-free case (see, e.g., \cite[Theorem 4.1]{BCHMP00}, \cite[Theorem]{Chen98a} or
\cite[Theroem 5.1]{Jiang05}).
This means that there is no arbitrariness for the measure of the generator when dealing with representation theorem or converse comparison theorem,
which is due to the appearance of the distribution dependent terms ${\bf m}athscr{L}_{(X_s,Y_s^\text{\mathbf rm{e}}psilon,Z_s^\text{\mathbf rm{e}}psilon)}$ in \text{\mathbf rm{e}}quationref{1-Repr} and ${\bf m}athscr{L}_{(X_s,Y^i_s,Z^i_s)}$ in \text{\mathbf rm{e}}quationref{3Th(Conve)}, respectively.
\text{\mathbf rm{e}}nd{rem}
\section{Functional inequalities}
In this section, we aim to establish functional inequalities for \text{\mathbf rm{e}}quationref{Bsde}, including mainly transportation inequalities and Logarithmic-Sobolev inequalities.
Our arguments consist of utilising stability of the Wasserstein distance and the relative entropy of measures under the homeomorphism, together with the transfer principle.
\subsection{Transportation inequalities}
Let $(E,d)$ be a metric space and let ${\bf m}athscr{P}(E)$ denote the set of all probability measures on $E$.
For $p{\rm in}n[1,{\rm in}nfty)$, we say that a probability measure ${\bf m}u{\rm in}n{\bf m}athscr{P}(E)$ satisfies $p$-transportation inequality on $(E,d)$ (noted ${\bf m}u{\rm in}n T_p(C)$) if there is a constant $C\geq0$ such that
\betaginin{align*}
{\bf m}athbb{W}_p({\bf m}u,\nu)\text{\mathbf rm{e}}lleq C\sqrtrt{H(\nu|{\bf m}u)}, \ \ \nu{\rm in}n{\bf m}athscr{P}(E),
\text{\mathbf rm{e}}nd{align*}
where $H(\nu|{\bf m}u)$ is the relative entropy (or Kullback-Leibler divergence) of $\nu$ with respect to ${\bf m}u$ defined as
\betaginin{equation*}
H(\nu|{\bf m}u)=\text{\mathbf rm{e}}lleft\{
\betaginin{array}{ll}
{\rm in}nt\text{\mathbf rm{e}}llog\frac {\text{\mathbf rm{d}}\nu}{\text{\mathbf rm{d}}{\bf m}u}\text{\mathbf rm{d}}\nu,\ \ {\bf m}athrm{if}\ \nu\text{\mathbf rm{e}}lll{\bf m}u,\\
+{\rm in}nfty,\ \ \ \ \ \ \ \ \ \ {\bf m}athrm{otherwise}.
\text{\mathbf rm{e}}nd{array} \mathbf right.
\text{\mathbf rm{e}}nd{equation*}
The transportation inequality has found numerous applications, for instance, to quantitative finance, the concentration of measure phenomenon and various problems of probability in higher dimensions.
We refer the reader e.g. to \cite{BT20,DGW04,Lacker18,Rid17,Saussereau12,SYZ22} and references therein.
Before moving to \text{\mathbf rm{e}}quationref{Bsde}, we first show the transportation inequalities for the auxiliary equation \text{\mathbf rm{e}}quationref{Aueq}, which is a distribution dependent version of \cite[Theorem 1.3 and Lemma 4.1]{BT20}.
\betagin{prp}\text{\mathbf rm{e}}llabel{Prp(Tr)}
Assume that \textsc{\textbf{(H1)}} holds. Then we have
\betaginin{itemize}
{\rm in}tem[(i)] The law of $(\tilde Y_t)_{t{\rm in}n[0,V_T]}$ satisfies $T_2(C_{Tr,\tilde Y})$ on $\tilde\Omega$ with $C_{Tr,\tilde Y}=2(L_g+L_fV_T)^2\text{\mathbf rm{e}}^{2L_fV_T}$.
{\rm in}tem[(ii)] For any $p\geq1$, the law of $(\tilde Z_t)_{t{\rm in}n[0,V_T]}$, denoted by ${\bf m}athscr{L}_{\tilde Z}$, satisfies
\betaginin{align*}
{\bf m}athbb{W}_p({\bf m}athscr{L}_{\tilde Z},{\bf m}u)\text{\mathbf rm{e}}lleq C_{Tr,\tilde Z}\text{\mathbf rm{e}}lleft(H({\bf m}u|{\bf m}athscr{L}_{\tilde Z})\mathbf right)^{\frac 1 {2p}},\ \ {\bf m}u{\rm in}n{\bf m}athscr{P}(\tilde\Omega),
\text{\mathbf rm{e}}nd{align*}
where
\betaginin{align*}
C_{Tr,\tilde Z}=2{\rm in}nf_{\alpha>0}\text{\mathbf rm{e}}lleft\{\frac 1{2\alpha}\text{\mathbf rm{e}}lleft[1+\alpha\text{\mathbf rm{e}}^{2pL_fV_T}(L_g+L_fV_T)^{2p}\mathbf right]\mathbf right\}^{\frac 1{2p}}.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{itemize}
\text{\mathbf rm{e}}nd{prp}
For the sake of conciseness, we defer the proof to the Appendix.
With Proposition \mathbf ref{Prp(Tr)} in hand, along with the transfer principle, we now state and prove the transportation inequalities for \text{\mathbf rm{e}}quationref{Bsde} as follows.
\betagin{thm}\text{\mathbf rm{e}}llabel{Th(TrIn)}
Assume that \textsc{\textbf{(H1)}} holds. Then for every $t{\rm in}n[0,T]$, we have
\betaginin{itemize}
{\rm in}tem[(i)] The law of $Y_t$ satisfies $T_2(C_{Tr,Y_t})$ on ${\bf m}athbb R$ with $C_{Tr,Y_t}=2(L_g+L_f(V_T-V_t))^2\text{\mathbf rm{e}}^{2L_f(V_T-V_t)}$.
{\rm in}tem[(ii)] For any $p\geq1$, the law of $Z_t$ satisfies
\betaginin{align*}
{\bf m}athbb{W}_p({\bf m}athscr{L}_{Z_t},{\bf m}u)\text{\mathbf rm{e}}lleq C_{Tr,Z_t}\text{\mathbf rm{e}}lleft(H({\bf m}u|{\bf m}athscr{L}_{Z_t})\mathbf right)^{\frac 1 {2p}},\ \ {\bf m}u{\rm in}n{\bf m}athscr{P}({\bf m}athbb R),
\text{\mathbf rm{e}}nd{align*}
where
\betaginin{align*}
C_{Tr,Z_t}=2{\rm in}nf_{\alpha>0}\text{\mathbf rm{e}}lleft\{\frac 1{2\alpha}\text{\mathbf rm{e}}lleft[1+\alpha\text{\mathbf rm{e}}^{2pL_f(V_T-V_t)}(L_g+L_f(V_T-V_t))^{2p}\mathbf right]\mathbf right\}^{\frac 1{2p}}.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{itemize}
\text{\mathbf rm{e}}nd{thm}
\betagin{proof}
By Proposition \mathbf ref{Prp(Tr)} and its proof (see \text{\mathbf rm{e}}quationref{add4PfPrp(Tr)} and \text{\mathbf rm{e}}quationref{6PfPrp(Tr)} in the Appendix), it is easy to see that for each $t{\rm in}n[0,V_T]$,
${\bf m}athscr{L}_{\tilde Y_t}$ satisfies $T_2(C_{Tr,\tilde Y_t})$ on ${\bf m}athbb R$ with $C_{Tr,\tilde Y_t}=2(L_g+L_f(V_T-t))^2\text{\mathbf rm{e}}^{2L_f(V_T-t)}$, and ${\bf m}athscr{L}_{\tilde Z_t}$ satisfies
\betaginin{align*}
{\bf m}athbb{W}_p({\bf m}athscr{L}_{\tilde Z_t},{\bf m}u)\text{\mathbf rm{e}}lleq C_{Tr,\tilde Z_t}\text{\mathbf rm{e}}lleft(H({\bf m}u|{\bf m}athscr{L}_{\tilde Z_t})\mathbf right)^{\frac 1 {2p}},\ \ {\bf m}u{\rm in}n{\bf m}athscr{P}({\bf m}athbb R)
\text{\mathbf rm{e}}nd{align*}
with any $p\geq1$ and
\betaginin{align*}
C_{Tr,\tilde Z_t}=2{\rm in}nf_{\alpha>0}\text{\mathbf rm{e}}lleft\{\frac 1{2\alpha}\text{\mathbf rm{e}}lleft[1+\alpha\text{\mathbf rm{e}}^{2pL_f(V_T-t)}(L_g+L_f(V_T-t))^{2p}\mathbf right]\mathbf right\}^{\frac 1{2p}}.
\text{\mathbf rm{e}}nd{align*}
Noting that for any $t{\rm in}n[0,T]$, the laws of $Y_t$ and $Z_t$ are respectively the same as those of $\tilde Y_{V_t}$ and $\tilde Z_{V_t}$ due to Theorem \mathbf ref{Th1},
we obtain the desired assertions (i) and (ii).
\text{\mathbf rm{e}}nd{proof}
\betagin{rem}\text{\mathbf rm{e}}llabel{Re(TrIn)}
If $f=0$ and $g(x,{\bf m}u)=x$, then one can check that \text{\mathbf rm{e}}quationref{Bsde} and \text{\mathbf rm{e}}quationref{Aueq} have unique solutions $(Y,Z)=(X,1)$ and $(\tilde Y,\tilde Z)=(\tilde W,1)$, respectively.
By Theorem \mathbf ref{Th(TrIn)} and Proposition \mathbf ref{Prp(Tr)}, we have $C_{Tr,Y_t}=2$ and $C_{Tr,\tilde Y}=2$ which are known to be optimal for Gaussian processes and Brownian motion.
This means that the constants $C_{Tr,Y_t}$ and $C_{Tr,\tilde Y}$ above are sharp.
Let us also mention that it is not clear here, if the laws of the paths of $Y$ and $Z$ satisfy the transportation inequality or not.
Indeed, the transfer principle may fail in this situation because of the difference between spaces ${\bf m}athscr{P}(C([0,T]))$ and ${\bf m}athscr{P}(C([0,V_T]))$.
\text{\mathbf rm{e}}nd{rem}
\subsection{Logarithmic-Sobolev inequality}
For introducing our result for \text{\mathbf rm{e}}quationref{Bsde}, let us first give the Logarithmic-Sobolev inequality for the auxiliary equation \text{\mathbf rm{e}}quationref{Aueq}.
\betagin{prp}\text{\mathbf rm{e}}llabel{Prp(LS)}
Assume that \textsc{\textbf{(H1)}} holds. Then for every $t{\rm in}n[0,V_T]$,
\betaginin{align}\text{\mathbf rm{e}}llabel{1Prp(LS)}
{\bf m}athrm{Ent}_{{\bf m}athscr{L}_{\tilde Y_t}}(f^2)\text{\mathbf rm{e}}lleq C_{LS,\tilde Y_t}{\rm in}nt_{\bf m}athbb R|f'|^2\text{\mathbf rm{d}}{\bf m}athscr{L}_{\tilde Y_t}
\text{\mathbf rm{e}}nd{align}
holds for all ${\bf m}athscr{L}_{\tilde Y_t}$-integrable and differentiable function $f:{\bf m}athbb R\mathbf rightarrow{\bf m}athbb R$,
where
\betaginin{align*}
C_{LS,\tilde Y_t}=2V_T(L_g+L_f(V_T-t))^2\text{\mathbf rm{e}}^{2L_f(V_T-t)}.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{prp}
Here the entropy of $0\text{\mathbf rm{e}}lleq F{\rm in}n L^1({\bf m}u)$ with respect to the probability measure ${\bf m}u$ is defined as
\betaginin{align*}
{\bf m}athrm{Ent}_{{\bf m}u}(F)={\rm in}nt_{\bf m}athbb R F\text{\mathbf rm{e}}llog F\text{\mathbf rm{d}}{\bf m}u-{\rm in}nt_{\bf m}athbb R F\text{\mathbf rm{d}}{\bf m}u\cdot\text{\mathbf rm{e}}llog{\rm in}nt_{\bf m}athbb R F\text{\mathbf rm{d}}{\bf m}u.
\text{\mathbf rm{e}}nd{align*}
When ${\bf m}u$ satisfies \text{\mathbf rm{e}}quationref{1Prp(LS)} for ${\bf m}u$ replacing ${\bf m}athscr{L}_{\tilde Y_t}$, we shall say that ${\bf m}u$ satisfies the $LSI(C_{LS})$.
This inequality, initiated by Gross \cite{Gross75}, has become a crucial tool in infinite dimensional stochastic analysis.
It had been well investigated in the context of forward diffusions, and was related with the 2-transportation inequality (see for instance \cite{CGW10,GW06,Ledoux99,OV00,Wang01,Wang09}).
To prove Proposition \mathbf ref{Prp(LS)}, we recall a result which asserts that the Logarithmic-Sobolev inequality satisfies stability under push-forward by Lipschitz maps (see \cite[Section 1]{CFJ09} or \cite[Lemma 6.1]{BT20}).
\betagin{lem}\text{\mathbf rm{e}}llabel{LS-Le}
If $\psi:(\tilde\Omega,\|\cdot\|_{\rm in}nfty)\mathbf rightarrow{\bf m}athbb R^d$ is Lipschitzian, i.e.
\betaginin{align*}
|\psi(\omega_1)-\psi(\omega_2)|\text{\mathbf rm{e}}lleq L_\psi\|\omega_1-\omega_2\|_{\rm in}nfty:=L_\psi\sup\text{\mathbf rm{e}}llimits_{r{\rm in}n[0,V_T]}|\omega_1(r)-\omega_2(r)|,\ \ \omega_1, \omega_2{\rm in}n\tilde\Omega,
\text{\mathbf rm{e}}nd{align*}
where $L_\psi>0$ is a constant.
Then $\tilde{\bf m}athbb{P}\circ\psi^{-1}$ satisfies LSI($2V_TL_\psi^2$).
\text{\mathbf rm{e}}nd{lem}
In light of the proof of Proposition \mathbf ref{Prp(Tr)} (see the Appendix), we have shown that
$\tilde Y:\tilde\Omega\mathbf rightarrow C([0,V_T])$ is Lipschitz continuous with Lipschitzian constant $(L_g+L_fV_T)\text{\mathbf rm{e}}^{L_fV_T}$,
which actually implies that for any $t{\rm in}n[0,V_T], \tilde Y_t:\tilde\Omega\mathbf rightarrow{\bf m}athbb R$ is also Lipschitz continuous with Lipschitzian constant $(L_g+L_f(V_T-t))\text{\mathbf rm{e}}^{L_f(V_T-t)}$ thanks to \text{\mathbf rm{e}}quationref{add4PfPrp(Tr)} .
So, owing to Lemma \mathbf ref{LS-Le}, we get the desired assertion stated in Proposition \mathbf ref{Prp(LS)}.
Observe that by Theorem \mathbf ref{Th1}, the law of $Y_t$ is the same as that of $\tilde Y_{V_t}$ for every $t{\rm in}n[0,T]$.
Therefore, by Proposition \mathbf ref{Prp(LS)} we have the following Logarithmic-Sobolev inequality for \text{\mathbf rm{e}}quationref{Bsde}.
\betagin{thm}\text{\mathbf rm{e}}llabel{Th(LS)}
Assume that \textsc{\textbf{(H1)}} holds. Then for any $t{\rm in}n[0,T]$, the law of $Y_t$ satisfies the LSI($C_{LS,Y_t}$),
where
\betaginin{align*}
C_{LS,Y_t}=2V_T(L_g+L_f(V_T-V_t))^2\text{\mathbf rm{e}}^{2L_f(V_T-V_t)}.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{thm}
\section{Appendix: proof of Proposition \mathbf ref{Prp(Tr)}}
In order to prove the proposition, we first present three lemmas that are needed later on.
The first one concerns the stability of transportation inequalities under push-forward by Lipschitz maps, which is due to \cite[Lemma 2.1]{DGW04} (see, e.g., \cite[Lemma 4.1]{Rid17} and \cite[Corollary 2.2]{SYZ22} for a generalisation).
\betagin{lem}\text{\mathbf rm{e}}llabel{FI-Le2}
Let $(E,d_E)$ and $(\overline{E},d_{\overline{E}})$ be two metric spaces.
Assume that ${\bf m}u{\rm in}n T_p(C)$ on $(E,d_E)$ and $\chi:(E,d_E)\mathbf rightarrow(\overline{E},d_{\overline{E}})$ is Lipschitz continuous with Lipschitzian constant $L_\chi$.
Then ${\bf m}u\circ\chi^{-1}{\rm in}n T_p(CL^2_\chi)$ on $(\overline{E},d_{\overline{E}})$.
\text{\mathbf rm{e}}nd{lem}
Our second lemma below provides a sufficient condition expressed in terms of exponential moment for a probability measure satisfying transportation
inequality of the form \text{\mathbf rm{e}}quationref{1FI-Le3}.
\betagin{lem}\text{\mathbf rm{e}}llabel{FI-Le3}
(\cite[Corollary 2.4]{BV05}) Let $(E,d_E)$ be a metric space, and let $\nu$ be a probability measure on $E$ and $p\geq 1$.
Assume that there exist $x_0{\rm in}n E$ and $\alpha>0$ such that
\betaginin{align*}
{\rm in}nt_E\text{\mathbf rm{e}}xp\text{\mathbf rm{e}}lleft\{\alpha d^{2p}_E(x_0,x)\mathbf right\}\text{\mathbf rm{d}}\nu(x)<{\rm in}nfty.
\text{\mathbf rm{e}}nd{align*}
Then
\betaginin{align}\text{\mathbf rm{e}}llabel{1FI-Le3}
{\bf m}athbb{W}_p({\bf m}u,\nu)\text{\mathbf rm{e}}lleq C(H({\bf m}u|\nu))^{\frac 1 {2p}},\ \ {\bf m}u{\rm in}n{\bf m}athscr{P}(E)
\text{\mathbf rm{e}}nd{align}
holds with
\betaginin{align*}
C=2{\rm in}nf_{x_0{\rm in}n E,\alpha>0}\text{\mathbf rm{e}}lleft[\frac 1{2\alpha}\text{\mathbf rm{e}}lleft(1+\text{\mathbf rm{e}}llog{\rm in}nt_E\text{\mathbf rm{e}}xp\text{\mathbf rm{e}}lleft\{\alpha d^{2p}_E(x_0,x)\mathbf right\}\text{\mathbf rm{d}}\nu(x)\mathbf right)\mathbf right]^{\frac 1{2p}}<{\rm in}nfty.
\text{\mathbf rm{e}}nd{align*}
\text{\mathbf rm{e}}nd{lem}
Before stating the third lemma, we need some notations from \cite{BT20,EKTZ14}.
For $t{\rm in}n[0,V_T]$, let $\tilde\Omega^t$ be the shifted space of $\tilde\Omega$ given by
\betaginin{align*}
\tilde\Omega^t:=\{\gamma{\rm in}n C([t,V_T]):\gamma(t)=0\}.
\text{\mathbf rm{e}}nd{align*}
Denote by $\tilde W^t$ and $\tilde{\bf m}athbb{P}^t$ the canonical process and the Wiener measure on $\tilde\Omega^t$, respectively,
and by $(\tilde{\bf m}athscr{F}^t_s)_{s{\rm in}n[t,V_T]}$ the filtration generated by $\tilde W^t$.
For $\omega{\rm in}n\tilde\Omega,t{\rm in}n[0,V_T]$ and $\gamma{\rm in}n\tilde\Omega^t$, define the concatenation $\omega\otimes_t\gamma{\rm in}n\tilde\Omega$ by
\betaginin{equation*}
(\omega\otimes_t\gamma)(s):=\text{\mathbf rm{e}}lleft\{
\betaginin{array}{ll}
\omega(s),\ \ \ \ \ \ \ \ \ \ \ s{\rm in}n[0,t),\\
\omega(t)+\gamma(s),\ \ s{\rm in}n[t,V_T],
\text{\mathbf rm{e}}nd{array} \mathbf right.
\text{\mathbf rm{e}}nd{equation*}
and for $\zeta:\tilde\Omega\tildemes[0,V_T]\mathbf rightarrow{\bf m}athbb R$, define its shift $X^{t,\omega}$ as follows
\betaginin{align*}
\zeta^{t,\omega}:&\ \tilde\Omega^t\tildemes[t,V_T]\mathbf rightarrow{\bf m}athbb R,\cr
&(\gamma,s){\bf m}apsto \zeta_s(\omega\otimes_t\gamma)=:\zeta^{t,\omega}_s(\gamma).
\text{\mathbf rm{e}}nd{align*}
As pointed out in \cite{BT20}, we have
\betaginin{align*}
\tilde{\bf m}athbb E(\zeta|\tilde{\bf m}athscr{F}_t)(\omega)={\rm in}nt_{\Omega^t}\zeta^{t,\omega}(\gamma)\tilde{\bf m}athbb{P}^t(\text{\mathbf rm{d}}\gamma)=:\tilde{\bf m}athbb E_{\tilde{\bf m}athbb{P}^t}\zeta^{t,\omega}.
\text{\mathbf rm{e}}nd{align*}
The following lemma gives a result which is a distribution dependent version of \cite[Lemma 2.2]{BT20}. The proof is pretty similar to that of \cite[Lemma 2.2]{BT20} and we omit it here.
\betagin{lem}\text{\mathbf rm{e}}llabel{FI-Le1}
Let $(\tilde{Y},\tilde{Z}){\rm in}n\widetilde{\Upsilon}$ be the solution to \text{\mathbf rm{e}}quationref{Aueq}.
Then for any $t{\rm in}n[0,V_T]$, there exists a $\tilde{\bf m}athbb{P}$-zero set $N\subset\tilde\Omegaega$ such that for $\omega{\rm in}n N^c$,
\betaginin{align*}
\tilde{Y}_s^{t,\omega}=&g(\tilde{W}^{t,\omega}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})
+{\rm in}nt_s^{V_T}f(U_r,\tilde{W}^{t,\omega}_r,\tilde{Y}^{t,\omega}_r,\tilde{Z}^{t,\omega}_r,{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})\text{\mathbf rm{d}} r\cr
&-{\rm in}nt_s^{V_T}\tilde{Z}^{t,\omega}_r\text{\mathbf rm{d}}\tilde{W}^t_r,
\ \ \tilde{\bf m}athbb{P}^t \textit{-}a.s., \ s{\rm in}n[t,V_T]
\text{\mathbf rm{e}}nd{align*}
and $\tilde{Y}_t^{t,\omega}=\tilde{Y}_t(\omega), \tilde{\bf m}athbb{P}^t$-a.s..
\text{\mathbf rm{e}}nd{lem}
We are now ready to prove Proposition \mathbf ref{Prp(Tr)}.
\text{\mathbf rm{e}}mph{Proof of Proposition \mathbf ref{Prp(Tr)}.}
Owing to \textsc{\textbf{(H1)}}, we know that there exists a unique solution $(\tilde{Y},\tilde{Z}){\rm in}n\tilde{\Upsilon}$ to \text{\mathbf rm{e}}quationref{Aueq}.
Moreover, it easily follows that $\tilde{Y}$ has $\tilde{\bf m}athbb{P}$-almost surely continuous paths and $\tilde{Z}$ is square integrable.
The rest of the proof is divided into two steps.
\textsl{Step 1. Transportation inequality for $\tilde Y$}.
Using arguments from the proofs of \cite[Proposition 5.4]{EKTZ14} and \cite[Theorem 1.3]{BT20}, we intend to show that $\tilde Y:\tilde\Omega\mathbf rightarrow{\bf m}athbb R$ is Lipschitz continuous.
Let $t{\rm in}n[0,V_T]$.
According to Lemma \mathbf ref{FI-Le1}, there exists a $\tilde{\bf m}athbb{P}$-zero set $N\subset\tilde\Omega$ such that for $\omegaega{\rm in}n N^c$,
\betaginin{align}\text{\mathbf rm{e}}llabel{1PfPrp(Tr)}
\tilde{Y}_t^{t,\omega}=\tilde{Y}_t(\omega),\ \ \tilde{\bf m}athbb{P}^t\textit{-}a.s.
\text{\mathbf rm{e}}nd{align}
and
\betaginin{align}\text{\mathbf rm{e}}llabel{2PfPrp(Tr)}
\tilde{Y}_s^{t,\omega}=&g(\tilde{W}^{t,\omega}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})
+{\rm in}nt_s^{V_T}f(U_r,\tilde{W}^{t,\omega}_r,\tilde{Y}^{t,\omega}_r,\tilde{Z}^{t,\omega}_r,{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})\text{\mathbf rm{d}} r\cr
&-{\rm in}nt_s^{V_T}\tilde{Z}^{t,\omega}_r\text{\mathbf rm{d}}\tilde{W}^t_r,
\ \ \tilde{\bf m}athbb{P}^t\textit{-}a.s., \ s{\rm in}n[t,V_T].
\text{\mathbf rm{e}}nd{align}
Then for any $w_1,w_2{\rm in}n N^c$, by \text{\mathbf rm{e}}quationref{2PfPrp(Tr)} and \textsc{\textbf{(H1)}} we derive
\betaginin{align*}
&\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2}\cr
=&g(\tilde{W}^{t,\omega_1}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})-g(\tilde{W}^{t,\omega_2}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})\cr
&+{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}lleft(f(U_r,\tilde{W}^{t,\omega_1}_r,\tilde{Y}^{t,\omega_1}_r,\tilde{Z}^{t,\omega_1}_r,{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})
-f(U_r,\tilde{W}^{t,\omega_2}_r,\tilde{Y}^{t,\omega_2}_r,\tilde{Z}^{t,\omega_2}_r,{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})\mathbf right)
\text{\mathbf rm{d}} r\cr
&-{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r\mathbf right)\text{\mathbf rm{d}}\tilde{W}^t_r\cr
=&g(\tilde{W}^{t,\omega_1}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})-g(\tilde{W}^{t,\omega_2}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})\cr
&+{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}lleft[\alpha_r(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r)+\beta_r(\tilde{Y}^{t,\omega_1}_r-\tilde{Y}^{t,\omega_2}_r)+\mathbf rho_r(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r) \mathbf right]\text{\mathbf rm{d}} r\cr
&-{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r\mathbf right)\text{\mathbf rm{d}}\tilde{W}^t_r,\ \ {\bf m}athbb{P}^t\textit{-}a.s., \ s{\rm in}n[t,V_T],
\text{\mathbf rm{e}}nd{align*}
where
\betaginin{align*}
\alpha_r&:={\rm in}nt_0^1\partial_xf(U_r,\tilde{W}^{t,\omega_2}_r+\theta(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r),\tilde{Y}^{t,\omega_1}_r,\tilde{Z}^{t,\omega_1}_r,{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})\text{\mathbf rm{d}}\theta,\cr
\beta_r&:={\rm in}nt_0^1\partial_yf(U_r,\tilde{W}^{t,\omega_2}_r,\tilde{Y}^{t,\omega_2}_r+\theta(\tilde{Y}^{t,\omega_1}_r-\tilde{Y}^{t,\omega_2}_r),\tilde{Z}^{t,\omega_1}_r,{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})\text{\mathbf rm{d}}\theta,\cr
\mathbf rho_r&:={\rm in}nt_0^1\partial_zf(U_r,\tilde{W}^{t,\omega_2}_r,\tilde{Y}^{t,\omega_2}_r,\tilde{Z}^{t,\omega_2}_r+\theta(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r),{\bf m}athscr{L}_{(\tilde{W}_r,\widetilde{Y}_r,\tilde{Y}_r)})\text{\mathbf rm{d}}\theta.
\text{\mathbf rm{e}}nd{align*}
We claim that the product $\text{\mathbf rm{e}}^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}(\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2})$ yields a more suitable representation for $\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2}$.
Indeed, for $s{\rm in}n(t,V_T]$,
\betaginin{align*}
&\text{\mathbf rm{d}}\text{\mathbf rm{e}}lleft[\text{\mathbf rm{e}}^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}(\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2})\mathbf right]\cr
=&\bigg[(\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2})\text{\mathbf rm{e}}^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}\beta_s\cr
&-\text{\mathbf rm{e}}^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}
\text{\mathbf rm{e}}lleft(\alpha_s(\tilde{W}^{t,\omega_1}_s-\tilde{W}^{t,\omega_2}_s)+\beta_s(\tilde{Y}^{t,\omega_1}_s-\tilde{Y}^{t,\omega_2}_s)+\mathbf rho_s(\tilde{Z}^{t,\omega_1}_s-\tilde{Z}^{t,\omega_2}_s)\mathbf right)\bigg]\text{\mathbf rm{d}} s\cr
&+e^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_s-\tilde{Z}^{t,\omega_2}_s\mathbf right)\text{\mathbf rm{d}}\tilde{W}^t_s\cr
=&\text{\mathbf rm{e}}lleft[
-\text{\mathbf rm{e}}^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}
\text{\mathbf rm{e}}lleft(\alpha_s(\tilde{W}^{t,\omega_1}_s-\tilde{W}^{t,\omega_2}_s)+\mathbf rho_s(\tilde{Z}^{t,\omega_1}_s-\tilde{Z}^{t,\omega_2}_s)\mathbf right)\mathbf right]\text{\mathbf rm{d}} s
+e^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_s-\tilde{Z}^{t,\omega_2}_s\mathbf right)\text{\mathbf rm{d}}\tilde{W}^t_s.
\text{\mathbf rm{e}}nd{align*}
Then, integrating from $s$ to $V_T$ we get
\betaginin{align*}
&\text{\mathbf rm{e}}^{{\rm in}nt_t^{V_T}\beta_r\text{\mathbf rm{d}} r}(\tilde{Y}_{V_T}^{t,\omega_1}-\tilde{Y}_{V_T}^{t,\omega_2})-\text{\mathbf rm{e}}^{{\rm in}nt_t^s\beta_r\text{\mathbf rm{d}} r}(\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2})\cr
=&-{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}^{{\rm in}nt_t^r\beta_\theta\text{\mathbf rm{d}} \theta}\text{\mathbf rm{e}}lleft(\alpha_r(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r)+\mathbf rho_r(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r)\mathbf right)\text{\mathbf rm{d}} r\cr
&+{\rm in}nt_s^{V_T}e^{{\rm in}nt_t^r\beta_\theta\text{\mathbf rm{d}}\theta}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r\mathbf right)\text{\mathbf rm{d}}\tilde{W}^t_r,\ \ \tilde{\bf m}athbb{P}^t\textit{-}a.s..
\text{\mathbf rm{e}}nd{align*}
Observing that $\tilde{Y}_{V_T}^{t,\omega_i}=g(\tilde{W}^{t,\omega_i}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}}),i=1,2$, we thus obtain
\betaginin{align}\text{\mathbf rm{e}}llabel{3PfPrp(Tr)}
\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2}
=&\text{\mathbf rm{e}}^{{\rm in}nt_s^{V_T}\beta_r\text{\mathbf rm{d}} r}\text{\mathbf rm{e}}lleft(g(\tilde{W}^{t,\omega_1}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})-g(\tilde{W}^{t,\omega_2}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})\mathbf right)\cr
&+{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}^{{\rm in}nt_s^r\beta_\theta\text{\mathbf rm{d}} \theta}\text{\mathbf rm{e}}lleft(\alpha_r(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r)+\mathbf rho_r(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r)\mathbf right)\text{\mathbf rm{d}} r\cr
&-{\rm in}nt_s^{V_T}e^{{\rm in}nt_s^r\beta_\theta\text{\mathbf rm{d}}\theta}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r\mathbf right)\text{\mathbf rm{d}}\tilde{W}^t_r\cr
=&\text{\mathbf rm{e}}^{{\rm in}nt_s^{V_T}\beta_r\text{\mathbf rm{d}} r}\text{\mathbf rm{e}}lleft(g(\tilde{W}^{t,\omega_1}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})-g(\tilde{W}^{t,\omega_2}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})\mathbf right)\nonumber\\
&+{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}^{{\rm in}nt_s^r\beta_\theta\text{\mathbf rm{d}} \theta}\alpha_r(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r)\text{\mathbf rm{d}} r
-{\rm in}nt_s^{V_T}e^{{\rm in}nt_s^r\beta_\theta\text{\mathbf rm{d}}\theta}\text{\mathbf rm{e}}lleft(\tilde{Z}^{t,\omega_1}_r-\tilde{Z}^{t,\omega_2}_r\mathbf right)\text{\mathbf rm{d}}\overline{W}^t_r,
\text{\mathbf rm{e}}nd{align}
where $\overline{W}^t_r:=\tilde W_r-{\rm in}nt_t^r\mathbf rho_r\text{\mathbf rm{d}} r$.\\
Now, for $s{\rm in}n[t,V_T]$, we set
\betaginin{align*}
R_s=\text{\mathbf rm{e}}xp\text{\mathbf rm{e}}lleft\{{\rm in}nt_t^s\mathbf rho_r\text{\mathbf rm{d}}\tilde W_r-\frac 1 2 {\rm in}nt_t^s|\mathbf rho_r|^2\text{\mathbf rm{d}} r\mathbf right\}.
\text{\mathbf rm{e}}nd{align*}
By \textsc{\textbf{(H1)}}, it is easy to verify that the Novikov condition holds, which implies that $\overline{W}^t_\cdot$ is a Brownian motion under the probability $R_{V_T}\tilde{\bf m}athbb{P}^t$ due to the Girsanov theorem.
Then, conditioning by $\tilde{\bf m}athscr{F}^t_s$ under $R_{V_T}\tilde{\bf m}athbb{P}^t$ on both sides of \text{\mathbf rm{e}}quationref{3PfPrp(Tr)} yields
\betaginin{align}\text{\mathbf rm{e}}llabel{4PfPrp(Tr)}
&\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}\text{\mathbf rm{e}}lleft(\tilde{Y}_s^{t,\omega_1}-\tilde{Y}_s^{t,\omega_2}|\tilde{\bf m}athscr{F}^t_s\mathbf right)\cr
=&\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}\text{\mathbf rm{e}}lleft[\text{\mathbf rm{e}}^{{\rm in}nt_s^{V_T}\beta_r\text{\mathbf rm{d}} r}\text{\mathbf rm{e}}lleft(g(\tilde{W}^{t,\omega_1}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})-g(\tilde{W}^{t,\omega_2}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})\mathbf right)|\tilde{\bf m}athscr{F}^t_s\mathbf right]\nonumber\\
&+\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}\text{\mathbf rm{e}}lleft[{\rm in}nt_s^{V_T}\text{\mathbf rm{e}}^{{\rm in}nt_s^r\beta_\theta\text{\mathbf rm{d}} \theta}\alpha_r(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r)\text{\mathbf rm{d}} r|\tilde{\bf m}athscr{F}^t_s\mathbf right],\ \ \tilde{\bf m}athbb{P}^t\textit{-}a.s., \ s{\rm in}n[t,V_T].
\text{\mathbf rm{e}}nd{align}
Consequently, we have
\betaginin{align*}
\tilde{Y}_t^{t,\omega_1}-\tilde{Y}_t^{t,\omega_2}
=&\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}\text{\mathbf rm{e}}lleft[\text{\mathbf rm{e}}^{{\rm in}nt_t^{V_T}\beta_r\text{\mathbf rm{d}} r}\text{\mathbf rm{e}}lleft(g(\tilde{W}^{t,\omega_1}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})-g(\tilde{W}^{t,\omega_2}_{V_T},{\bf m}athscr{L}_{\tilde{W}_{V_T}})\mathbf right)\mathbf right]\cr
&+\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}\text{\mathbf rm{e}}lleft[{\rm in}nt_t^{V_T}\text{\mathbf rm{e}}^{{\rm in}nt_t^r\beta_\theta\text{\mathbf rm{d}} \theta}\alpha_r(\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r)\text{\mathbf rm{d}} r\mathbf right],\ \ \tilde{\bf m}athbb{P}^t\textit{-}a.s..
\text{\mathbf rm{e}}nd{align*}
This allows us to deduce from \text{\mathbf rm{e}}quationref{1PfPrp(Tr)} and \textsc{\textbf{(H1)}} that
\betaginin{align}\text{\mathbf rm{e}}llabel{add4PfPrp(Tr)}
&|\tilde{Y}_t(\omega_1)-\tilde{Y}_t(\omega_2)|=|\tilde{Y}_t^{t,\omega_1}-\tilde{Y}_t^{t,\omega_2}|\cr
\text{\mathbf rm{e}}lleq&\text{\mathbf rm{e}}^{L_f(V_T-t)}\text{\mathbf rm{e}}lleft(L_g\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}|\tilde{W}^{t,\omega_1}_{V_T}-\tilde{W}^{t,\omega_2}_{V_T}|
+L_f{\rm in}nt_t^{V_T}\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}|\tilde{W}^{t,\omega_1}_r-\tilde{W}^{t,\omega_2}_r|\text{\mathbf rm{d}} r\mathbf right)\cr
=&\text{\mathbf rm{e}}^{L_f(V_T-t)}\bigg(L_g\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}|(\omega_1\otimes_t\cdot)(V_T)-(\omega_2\otimes_t\cdot)(V_T)|\cr
&\ \ \ \ \ \ \ \ \ \ \ \ \ +L_f{\rm in}nt_t^{V_T}\tilde{\bf m}athbb E_{R_{V_T}\tilde{\bf m}athbb{P}^t}|(\omega_1\otimes_t\cdot)(r)-(\omega_2\otimes_t\cdot)(r)|\text{\mathbf rm{d}} r\bigg)\cr
\text{\mathbf rm{e}}lleq&\text{\mathbf rm{e}}lleft(L_g+L_f(V_T-t)\mathbf right)\text{\mathbf rm{e}}^{L_f(V_T-t)}\sup\text{\mathbf rm{e}}llimits_{0\text{\mathbf rm{e}}lleq r\text{\mathbf rm{e}}lleq t}|\omega_1(r)-\omega_2(r)|,
\text{\mathbf rm{e}}nd{align}
where the last inequality is due to the definitions of concatenation variables $\omega_i\otimes_t\cdot, i=1,2$.
Noting that $t{\rm in}n[0,V_t]$ and $w_1,w_2{\rm in}n N^c$ are arbitrary and $\tilde{\bf m}athbb{P}(N)=0$, we conclude that $\tilde Y:\tilde\Omega\mathbf rightarrow C([0,V_T])$ is Lipschitz continuous with Lipschitzian constant $(L_g+L_fV_T)\text{\mathbf rm{e}}^{L_fV_T}$.
Therefore, taking into account of the fact that the law of Wiener process satisfies $T_2(2)$ (see \cite[Theorem 3.1]{FU04}),
we obtain the first assertion due to Lemma \mathbf ref{FI-Le2}.
\textsl{Step 2. Transportation inequality for $\tilde Z$}.
We first suppose that $g(x,{\bf m}u)$ and $f(t,x,y,z,\nu)$ are differentiable with respect to $x,y$ and $z$.
Then $(\tilde Y,\tilde Z)$ is differentiable, and moreover $(\nablabla\tilde{Y},\nablabla\tilde{Z})$ solves the following linear DDBSDE
\betaginin{align*}
\nablabla\tilde{Y}_t=&\nablabla_xg(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\widetilde{W}_{V_T}})-{\rm in}nt_t^{V_T}\nablabla\widetilde{Z}_s\text{\mathbf rm{d}}\widetilde{W}_s\cr
&+{\rm in}nt_t^{V_T}\mathbf Big[\nablabla_xf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})
+\nablabla_yf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})\nablabla\tilde{Y}_s\cr
&\qquad\qquad+\nablabla_zf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})\nablabla\tilde{Z}_s\mathbf Big]\text{\mathbf rm{d}} s.
\text{\mathbf rm{e}}nd{align*}
Along the same lines as in \text{\mathbf rm{e}}quationref{4PfPrp(Tr)}, applying the product $\text{\mathbf rm{e}}^{{\rm in}nt_0^t\nablabla_yf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})\text{\mathbf rm{d}} s}\nablabla\tilde{Y}_t$
and the Girsanov theorem we deduce that there exists some probability $\tilde{\bf m}athbb Q$ under which
\betaginin{align*}
\widetilde{W}_\cdot-{\rm in}nt_0^\cdot\nablabla_zf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})\text{\mathbf rm{d}} s
\text{\mathbf rm{e}}nd{align*}
is a Brownian motion, and $\nablabla\tilde{Y}_t$ has the representation
\betaginin{align*}
\nablabla\tilde{Y}_t=&\tilde{\bf m}athbb E_{\tilde{\bf m}athbb Q}\bigg[e^{{\rm in}nt_t^{V_T}\nablabla_yf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})\text{\mathbf rm{d}} s}\nablabla_xg(\widetilde{W}_{V_T},{\bf m}athscr{L}_{\widetilde{W}_{V_T}})\cr
&+{\rm in}nt_t^{V_T}e^{{\rm in}nt_t^s\nablabla_yf(U_r,\widetilde{W}_r,\widetilde{Y}_r,\widetilde{Z}_r,{\bf m}athscr{L}_{(\widetilde{W}_r,\widetilde{Y}_r,\widetilde{Z}_r)})\text{\mathbf rm{d}} r}\nablabla_xf(U_s,\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s,{\bf m}athscr{L}_{(\widetilde{W}_s,\widetilde{Y}_s,\widetilde{Z}_s)})\text{\mathbf rm{d}} s\big|\tilde{\bf m}athscr{F}_t\bigg].
\text{\mathbf rm{e}}nd{align*}
Consequently, it is easy to see that
\betaginin{align}\text{\mathbf rm{e}}llabel{5PfPrp(Tr)}
|\nablabla\tilde{Y}_t|\text{\mathbf rm{e}}lleq\text{\mathbf rm{e}}^{L_f(V_T-t)}(L_g+L_f(V_T-t)).
\text{\mathbf rm{e}}nd{align}
On the other hand, it is classical to show that there exists a version of $(\tilde Z_t)_{t{\rm in}n[0,T]}$ given by $(\nablabla\tilde{Y}_t)_{t{\rm in}n[0,T]}$
(see, e.g., \cite[Remark 9.1]{Li18}).
Hence, combining this with \text{\mathbf rm{e}}quationref{5PfPrp(Tr)} leads to
\betaginin{align}\text{\mathbf rm{e}}llabel{6PfPrp(Tr)}
|\tilde Z_t|\text{\mathbf rm{e}}lleq\text{\mathbf rm{e}}^{L_f(V_T-t)}(L_g+L_f(V_T-t)).
\text{\mathbf rm{e}}nd{align}
When $g$ and $f$ are not differentiable, we can also obtain \text{\mathbf rm{e}}quationref{6PfPrp(Tr)} via a standard approximation and stability results (see Remark \mathbf ref{Re1} (i)).
Therefore, with the help of Lemma \mathbf ref{FI-Le3}, we derive the second assertion.
\qed
\textbf{Acknowledgement}
X. Fan is partially supported by the Natural Science Foundation of Anhui Province (No. 2008085MA10) and the National Natural Science Foundation of China (No. 11871076, 12071003).
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\text{\mathbf rm{e}}nd{document} |
\begin{document}
\let\labeloriginal\label
\let\reforiginal\ref
\begin{abstract}
We prove that the existence of a Dowker filter at $\kappa^+$, where $\kappa$ is regular and uncountable, is consistent with $2^\kappa=\kappa^+$.
We also prove the consistency of a Dowker filter at $\mu^+$ where $\mu>{\rm cf}(\mu)>\omega$.
This can be forced with $2^\mu=\mu^+$ as well.
\end{abstract}
\title{Dowker filters and Magidor forcing}
\section{Introduction}
Let $f:\kappa\rightarrow\mathcal{P}(\kappa)$ be a set-mapping.
We shall say that $f$ is \emph{anti-free} iff there are $\alpha<\beta<\kappa$ such that $\alpha\in f(\beta)\wedge\beta\in f(\alpha)$.
Let $\mathscr{F}$ be a filter over $\kappa, e\in{}^\kappa 2$.
A set-mapping $f:\kappa\rightarrow\mathscr{F}$ codes $e$ iff for every $\alpha<\beta<\kappa$, if $\alpha\in f(\beta)\wedge \beta\in f(\alpha)$ then $e(\alpha)=e(\beta)$.
Notice that this is only a partial coding, since if $\alpha\notin f(\beta)$ or $\beta\notin f(\alpha)$ then $f$ gives no information about the relationship between $e(\alpha)$ and $e(\beta)$.
Assume that $\mathscr{F}$ is a filter over $\kappa$.
Is it possible that every set-mapping $f:\kappa\rightarrow\mathscr{F}$ would be anti-free?
A trivial positive answer is given by $\mathscr{F}=\{\kappa\}$, in which case the only function $f:\kappa\rightarrow\mathscr{F}$ is the constant function $f(\alpha)=\kappa$ for every $\alpha\in\kappa$.
Hence for producing an anti-free filter, the simpler the better.
Consider now the coding property.
If $\mathscr{F}=\{\kappa\}$ and $e\in{}^\kappa 2$ is not constant then $e$ cannot be coded by a function $f:\kappa\rightarrow\mathscr{F}$.
Indeed, if $e(\alpha)\neq e(\beta)$ then the only possible $f$ (which is the constant function) fails to code $e$ at $\{\alpha,\beta\}$.
We conclude that $\mathscr{F}$ must be somewhat complicated in order to code $\kappa$-reals, whence we infer that anti-freeness and coding functions are enemies to some extent.
Dowker, \cite{MR0047741}, asked if they can be friends, notwithstanding.
Let $\mathscr{F}$ be a filter over $\kappa$.
Call $\mathscr{F}$ a Dowker filter iff every $f:\kappa\rightarrow\mathscr{F}$ is anti-free and every $e\in{}^\kappa 2$ is coded by some function $f_e:\kappa\rightarrow\mathscr{F}$.
Dowker proved that no Dowker filters live over $\omega_1$ and asked about larger cardinals.
The first positive answer in the environment of the failure of the axiom of choice has been published in \cite{MR973098}.
A few years later, Balogh and Gruenhage forced Dowker filters over $\omega_2$ without violating AC, see \cite{MR1136457}.
An interesting feature of $\omega_2$ is that if $\mathscr{F}$ is a Dowker filter over $\omega_2$ then $\mathscr{F}$ must be uniform.
Namely, if $x\in\mathscr{F}$ then $|x|=\aleph_2$.
Indeed, if $\mathscr{F}$ is a Dowker filter over $\kappa$ then every countable $x\subseteq\kappa$ is $\mathscr{F}$-small.
Likewise, the size of the smallest element of $\mathscr{F}^+$ is strictly less than the minimal cardinality of an element in $\mathscr{F}$.
Hence if $\kappa=\omega_2$ then for some $x\in\mathscr{F}^+$ we have $|x|=\aleph_1$ and every $x\in\mathscr{F}^+$ is uncountable.
It follows that the minimal size of an element of $\mathscr{F}$ is greater than $\aleph_1$ so if $x\in\mathscr{F}$ then $|x|=\aleph_2$.
Non-uniform Dowker filters over every $\kappa>\omega_2$ are obtained immediately from \cite{MR1136457}.
Let $\mathscr{F}$ be a Dowker filter over $\omega_2$, let $\kappa$ be greater than $\aleph_2$ and define $\mathscr{D}=\{A\subseteq\kappa:A\cap\omega_2\in\mathscr{F}\}$.
Such a filter is not uniform, of course.
So the interesting question with respect to cardinals larger than $\omega_2$ takes uniformity as a necessary assumption.
Let us incorporate this property in the formal definition.
\begin{definition}
\label{defdowker} Dowker filters. \newline
Assume that $\kappa\geq\omega_2$ and let $\mathscr{F}$ be a filter over $\kappa$.
We shall say that $\mathscr{F}$ is a Dowker filter iff:
\begin{enumerate}
\item [$(\aleph)$] $\mathscr{F}$ is uniform.
\item [$(\beth)$] $\mathscr{F}$ is anti-free, that is for every $f:\kappa\rightarrow\mathscr{F}$ there are $\alpha<\beta<\kappa$ such that $\alpha\in f(\beta)\wedge\beta\in f(\alpha)$.
\item [$(\gimel)$] $\mathscr{F}$ codes $\kappa$-reals, that is for every $e\in{}^\kappa 2$ there exists $f_e:\kappa\rightarrow\mathscr{F}$ such that $\alpha\in f_e(\beta)\wedge\beta\in f_e(\alpha)\Rightarrow e(\alpha)=e(\beta)$.
\end{enumerate}
\end{definition}
The proof of \cite{MR1136457} has been considerably simplified by Cummings and Morgan in \cite{MR3717964}.
They proved that if $\kappa={\rm cf}(\kappa)>\aleph_0$ and $\mathbb{P}=Add(\kappa,\kappa^{++})$ then there is a Dowker filter over $\kappa^+$ in $V[G]$ whenever $G\subseteq\mathbb{P}$ is $V$-generic.
Notice that $V[G]\models 2^\kappa=\kappa^{++}$ in these models, and thence they raised the following:
\begin{question}
\label{qcm} Is it consistent that $2^\kappa=\kappa^+$ and $\kappa^+$ carries a Dowker filter?
\end{question}
The first possible value of $\kappa=\aleph_0$ which gives a Dowker filter over $\omega_2$ is suggestive, since Dowker filters do not exist over $\omega_1$.
Nevertheless, we shall give a positive answer in the next section by proving that $2^\kappa=\kappa^+$ is consistent with a Dowker filter over $\kappa^+$.
This brings to light another fundamental question, namely whether the existence of a Dowker filter over some successor cardinal is a theorem of ZFC.
An easy negative answer, using the GCH, is excluded by our results.
The previous results concerning Dowker filters were obtained at successors of regular cardinals, and in some sense the forcing constructions are more adapted to this setting.
In particular, Cummings and Morgan raised in \cite{MR3717964} the following:
\begin{question}
\label{qcmsing} Is it consistent that $\mu>{\rm cf}(\mu)$ and there is a Dowker filter over $\mu^+$?
\end{question}
If the question is phrased in this way, then the answer is yes.
One can start with a singular cardinal $\mu$ and force with $Add(\theta,\mu^+)$ where $\theta$ is some regular cardinal below $\mu$.
However, it is clear that this is not what the poet meant, and the interesting question is whether one can force a Dowker filter over $\mu^+$ where $\mu$ is a \emph{strong limit} singular cardinal.
Assuming that there is a measurable cardinal $\mu$ with $o(\mu)\geq\omega_1$ in the ground model we will be able to give a positive answer.
Namely, one can force a Dowker filter over $\mu^+$ where $\mu$ is a singular cardinal and $\mu$ is strong limit.
Our notation is standard.
We shall say that $\kappa$ is \emph{strongly regular} iff $\kappa=\kappa^{<\kappa}$.
If $e\in{}^\alpha 2$ then we shall say that $\alpha=\len(e)$.
We use the Jerusalem forcing notation, so $p\leq q$ means that $p$ is weaker than $q$.
The forcing notion $Add(\kappa,\lambda)$ is the usual Cohen forcing for adding $\lambda$-many subsets of $\kappa$ using partial functions from $\lambda\times\kappa$ into $\{0,1\}$ of size less than $\kappa$.
If $\mathscr{F}$ is a filter over $\kappa$ then $\mathcal{I}(\mathscr{F})=\{A\subseteq\kappa:(\kappa-A)\in\mathscr{F}\}$ and $\mathscr{F}^+=\mathcal{P}(\kappa)-\mathcal{I}(\mathscr{F})$.
The elements of $\mathscr{F}^+$ will be called $\mathscr{F}$-positive sets.
If $\mathbb{P}$ is a forcing notion and $\mathunderaccent\tilde-3 {\tau}$ is a $\mathbb{P}$-name then a \emph{nice name} for a subset of $\mathunderaccent\tilde-3 {\tau}$ is a name of the form $\bigcup\{\{\sigma\}\times A_\sigma:\sigma\in\dom \tau\}$, every $A_\sigma$ being an antichain of $\mathbb{P}$.
Nice names are helpful when one has to reduce the number of names of some object.
The following lemma will be used in the next two sections:
\begin{lemma}
\label{lemcm} Assume that:
\begin{enumerate}
\item [$(a)$] $\mathscr{G}$ is an $\aleph_1$-complete filter over $\mu$.
\item [$(b)$] $C\in\mathscr{G}^+$.
\item [$(c)$] $h:C\rightarrow[{}^{<\mu}{\rm Ord}]^{<\omega}, h(\alpha)\in[{}^{\alpha+1}{\rm Ord}]^{<\omega}$ for every $\alpha\in\mu$.
\end{enumerate}
Then there exist $a\in[{}^\mu{\rm Ord}]^{<\omega}$ and $D\in\mathscr{G}^+, D\subseteq C$ such that for every $f\in{}^\mu{\rm Ord}$ if $f\notin a$ then
\[\{\beta\in D:f\upharpoonright(\beta+1)\in h(\beta) \text { and }\forall g \in a, f \upharpoonright (\beta + 1) \neq g \upharpoonright (\beta + 1)\}\notin\mathscr{G}^+.\]
\end{lemma}
\par\noindent\emph{Proof}. \newline
Assume toward contradiction that the conclusion fails.
For every $n\in\omega$ let $C_n=\{\alpha\in C:|h(\alpha)|=n\}$, so $C=\bigcup_{n\in\omega}C_n$.
By $(a)$ we can choose $n\in\omega$ so that $C_n\in\mathscr{G}^+$.
By finite induction on $i\leq n+1$ we shall define a triple $(a_i,D_i,\eta_i)$ such that:
\begin{enumerate}
\item [$(\aleph)$] $a_i=\{\eta_j:j<i\}\subseteq[{}^\mu{\rm Ord}]^{<\omega}$.
\item [$(\beth)$] $\eta_i\in{}^\mu{\rm Ord}-a_i$.
\item [$(\gimel)$] The set, $D_{i+1}$ which consists of all $\beta \in D_i$ such that $\eta_i\upharpoonright(\beta+1)\in h(\beta)$ and $\forall j < i, \eta_j \upharpoonright (\beta + 1) \neq \eta_i \upharpoonright (\beta + 1)$ is in $\mathscr{G}^{+}$.
\end{enumerate}
Notice that $D_{i+1}$ is determined at the $i$th stage, and we stipulate $D_0=C_n$.
For choosing the elements of the triple at the $(i+1)$th stage we apply the assumption toward contradiction with respect to $(a_i,D_i)$.
This application gives an element $f\in{}^\mu{\rm Ord}, f\notin a_i$ such that $D_{i+1}=\{\beta:\in D_i:f\upharpoonright(\beta+1)\in h(\beta)\}\in\mathscr{G}^+$.
We let $\eta_i=f$ and $a_i=\{\eta_j:j<i\}$.
Choose any ordinal $\beta\in D_{n+1}$ and notice that $\beta\in\bigcap_{i\leq n+1}D_i$.
Hence $\eta_i \upharpoonright (\beta + 1) \in h(\beta)$ for every $i\leq n+1$, so $|h(\beta)|\geq n+1$ by part $(\beth)$ which implies that $i\neq j\Rightarrow \eta_i\upharpoonright (\beta + 1) \neq\eta_j \upharpoonright (\beta + 1)$.
On the other hand $|h(\beta)|=n$ since $\beta\in D_n\subseteq D_0$, a contradiction.
\qedref{lemcm}
Remark that $h:\mu\rightarrow[{}^{<\mu}{\rm Ord}]^{<\omega}$ but the same statement holds if $h:\mu\rightarrow[Z]^{<\omega}$ for some set $Z$, as in \cite{MR3717964}.
We shall apply this lemma also where $Z=\mu$, in which case the set $a$ will be simply a finite set of ordinals in $\mu$.
Finally, we will use Magidor forcing from \cite{MR0465868} in order to give a positive answer to Question \ref{qcmsing}.
It is not clear whether Prikry forcing can serve in this context, and actually we do not know if a Dowker filter over a successor of a singular cardinal with countable cofinality can be forced.
The essential property of Magidor forcing which we use is a good covering property of new $\omega$-subsets in the Magidor extension.
An explicit proof of the following lemma can be found in \cite[Claim 3.5]{MR3957389}.
\begin{lemma}
\label{lemcovering} Assume that $\kappa$ is measurable, $o(\kappa)\geq\omega_1$ and $\lambda\geq\kappa$. \newline
Let $\mathbb{M}$ be Magidor forcing to make $\kappa>{\rm cf}(\kappa)>\omega$, and let $\mathunderaccent\tilde-3 {\tau}$ be an $\mathbb{M}$-name of an element in $[\lambda]^{\aleph_0\text{-bd}}$. \newline
Then there are $p\in\mathbb{M}, \theta<\kappa$ and $x\in V$ such that $|x|=\theta$ and $p\Vdash\mathunderaccent\tilde-3 {\tau}\subseteq\check{x}$.
\end{lemma}
\qedref{lemcovering}
The rest of the paper contains two additional sections.
In the first one we shall give a positive answer to Question \ref{qcm} and in the second one we shall try to give a positive answer to Question \ref{qcmsing}.
\section{Dowker filters and the continuum hypothesis}
In this section we prove that the existence of Dowker filters over $\kappa^+$ is consistent with $2^\kappa=\kappa^+$ where $\kappa={\rm cf}(\kappa)>\aleph_0$.
We shall use the ideas of \cite{MR3717964}, which in turn employs the ideas of \cite{MR1136457}.
\begin{theorem}
\label{thmdowkerconthyp} Let $\kappa>\aleph_0$ be strongly regular, let $\mathbb{P}=Add(\kappa,\kappa^{+})$ and let $G\subseteq\mathbb{P}$ be generic over $V$ where $V$ is assumed to satisfy GCH. \newline
Then there is a Dowker filter over $\kappa^+$ in $V[G]$.
\end{theorem}
\par\noindent\emph{Proof}. \newline
Let $\mathunderaccent\tilde-3 {e}=(\mathunderaccent\tilde-3 {e}_\zeta:\zeta\in\kappa^+)$ be an enumeration of all the nice names of the elements of ${}^{<\kappa^+}2\cap V[G]$ whose length is a successor ordinal.
We will assume that $\ell g(\mathunderaccent\tilde-3 {e}_\zeta)$ is decided by the empty condition for every $\zeta\in\kappa^+$.
We require also that for every $\zeta\in\kappa^+$ and every $\delta<\ell g(\mathunderaccent\tilde-3 {e}_\zeta)$ there is a unique $\xi\in\kappa^+$ such that $\mathunderaccent\tilde-3 {e}_\xi=\mathunderaccent\tilde-3 {e}_\zeta\upharpoonright(\delta+1)$.
This is a requirement on the names $\mathunderaccent\tilde-3 {e}_\xi,\mathunderaccent\tilde-3 {e}_\zeta$, not on the interpretations of them.
It is possible that for some $\xi\neq\xi'$ there will be a condition $p$ such that $p\Vdash\mathunderaccent\tilde-3 {e}_\xi=\mathunderaccent\tilde-3 {e}_{\xi'}$.
Rather than $Add(\kappa,\kappa^+)$ we shall use the forcing notion $\mathbb{P}$ defined as follows.
A condition $p\in\mathbb{P}$ is a partial function from $\kappa^+\times\kappa^+$ into $\{0,1\}$ so that $|p|<\kappa$.
If $p,q\in\mathbb{P}$ then $p\leq_{\mathbb{P}}q$ iff $p\subseteq q$.
One verifies easily that $\mathbb{P}$ and $Add(\kappa,\kappa^+)$ are isomorphic.
If $G\subseteq\mathbb{P}$ is $V$-generic then $g=\bigcup G$ is a function from $\kappa^+\times\kappa^+$ into $\{0,1\}$.
For each $\xi\in\kappa^+$ let $g_\xi = g\upharpoonright\{\xi\}\times\kappa^+$, so $g_\xi:\kappa^+\rightarrow 2$.
If $y\subseteq\kappa^+\times\kappa^+$ then $\pi_0(y)=\{\alpha\in\kappa^+:\exists\beta,(\alpha,\beta)\in y\}$ and $\pi_1(y)=\{\beta\in\kappa^+:\exists\alpha,(\alpha,\beta)\in y\}$.
We shall use these conventions mainly with respect to the domain of conditions in $\mathbb{P}$.
For every $\zeta\in\kappa^+$ we define a set $A_\zeta\subseteq\kappa^+$ by describing its characteristic function $\chi_{A_\zeta}$ as follows.
If $\beta<\len(e_\zeta)=\eta_\zeta+1$ and $e_\zeta\upharpoonright(\beta+1)=e_\xi$ and $g_\zeta(\beta)=g_\xi(\eta_\zeta)=1$ and $e_\zeta(\beta)\neq e_\zeta(\eta_\zeta)$ then let $\chi_{A_\zeta}(\beta)=0$.
In all other cases let $\chi_{A_\zeta}(\beta)=g_\zeta(\beta)$.
The idea is much simpler than it looks like.
Our default for $\chi_{A_\zeta}$ is the function $g_\zeta(\beta)$, but we rewrite the values of $g_\zeta(\beta)$ in case where we anticipate troubles with the coding property.
This happens, basically, if $g_\zeta(\beta)=g_\zeta(\eta_\zeta)=1$ and concomitantly $e_\zeta(\beta)\neq e_\zeta(\eta_\zeta)$.
Now by removing $\beta$ from $A_\zeta$, to wit by setting $\chi_{A_\zeta}(\beta)=0$, we need not worry about coding at $e_\zeta(\beta)$.
Notice that the length of each $e_\zeta$ and the identity of the ordinal $\xi$ so that $e_\xi=e_\zeta\upharpoonright(\beta+1)$ are computed on names and this can be rendered in $V$, but the values of $g_\zeta(\beta),g_\xi(\eta_\zeta)$ and the inequality $e_\zeta(\beta)\neq e_\zeta(\eta_\zeta)$ are done in $V[G]$.
The sets of the form $A_\zeta$ will generate our Dowker filter.
We shall also define another filter $\mathscr{G}$ from these sets, needed for the proof.
Let $\mathscr{D}$ be the filter obtained by taking all the finite intersections of $A_\zeta$s and closing upwards.
Let $\mathscr{G}$ be the filter obtained by $(<\kappa)$-intersections of $A_\zeta$s, clubs of $\kappa^+$ and the set $S^{\kappa^+}_\kappa$.
It follows that $\mathscr{G}$ is a $\kappa$-complete filter over $\kappa^+$ and it extends the club filter of $\kappa^+$ restricted to $S^{\kappa^+}_\kappa$.
The fact that $\mathscr{D}$ and $\mathscr{G}$ are proper filters (that is, $\varnothing\notin\mathscr{D}$ and $\varnothing\notin\mathscr{G}$) follows essentially from the fact that $|p|<\kappa$ for every $p\in\mathbb{P}$.
A detailed argument appears in \cite[Lemma 4.7]{MR3717964} and \cite[Corollary 4.9]{MR3717964}.
Observe that $\mathscr{D}$ will satisfy Definition \ref{defdowker}($\gimel$) by the definition of the sets $A_\zeta$.
Likewise, each $A_\zeta$ is of size $\kappa^+$, hence $\mathscr{D}$ is uniform.
To see that $\mathscr{D}$ satisfies \ref{defdowker}($\gimel$), assume that $\mathunderaccent\tilde-3 {d}$ is a name of a function from $\kappa^+$ into $\{0,1\}$, so it is interpreted as an element of ${}^{\kappa^+}2\cap V[G]$.
Without loss of generality, $\mathunderaccent\tilde-3 {d}$ is a nice name.
By our assumption on the enumeration $\mathunderaccent\tilde-3 {e}$, for every $\alpha\in\kappa^+$ there is a unique $\zeta(\alpha)\in\kappa^+$ such that $\mathunderaccent\tilde-3 {d}\upharpoonright(\alpha+1)=\mathunderaccent\tilde-3 {e}_{\zeta(\alpha)}$.
Notice that $\zeta(\alpha)$ can be computed from the name $\mathunderaccent\tilde-3 {d}$ in the ground model and define $F(\alpha)=A_{\zeta(\alpha)}$ for every $\alpha\in\kappa^+$.
Assume now that $\alpha<\beta<\kappa^+$ and $\alpha\in F(\beta)\wedge\beta\in F(\alpha)$.
It follows that $g_{\zeta(\alpha)}(\beta)=g_{\zeta(\beta)}(\alpha)=1$.
Recall that $\alpha<\beta+1=\ell g(e_{\zeta(\beta)})$ and by our choice $e_{\zeta(\beta)}\upharpoonright(\alpha+1)=e_{\zeta(\alpha)}$.
Since $g_{\zeta(\alpha)}(\beta)=g_{\zeta(\beta)}(\alpha)=1$ we conclude that $e_{\zeta(\beta)}(\beta)=e_{\zeta(\beta)}(\alpha)$, so $\mathunderaccent\tilde-3 {d}$ is coded correctly in $V[G]$ as required.
The burden of the proof is to show that \ref{defdowker}($\beth$) holds in the generic extension.
Since the sets $A_\zeta$ form a base for $\mathscr{D}$, suffice it to deal with functions from $\kappa^+$ into intersections of $A_\zeta$s.
These can be represented by functions from $\kappa^+$ into $[\kappa^+]^{<\omega}$, where for every $\alpha\in\kappa^+$ the function assigns a finite set $\{\zeta_1,\ldots,\zeta_n\}$ and gives rise to $\bigcap_{1\leq i\leq n}A_{\zeta_i}$.
Proving anti-freeness for the intersections $\bigcap_{1\leq i\leq n}A_{\zeta_i}$ will accomplish the proof of Dowkerness.
Suppose that $\mathunderaccent\tilde-3 {t}:\kappa^+\rightarrow[\kappa^+]^{<\omega}$.
Define $h:\kappa^+\rightarrow[{}^{<\kappa^+}2]^{<\omega}$ by letting $h(\alpha)=\{e_{\xi_1},\ldots,e_{\xi_n}\}$ whenever $\mathunderaccent\tilde-3 {t}(\alpha)=\{\xi_1,\ldots,\xi_n\}$.
We apply Lemma \ref{lemcm} to the filter $\mathscr{G}$ and the function $h$ (where $\mu=\kappa^+$ and $C=\kappa^+$).
By the conclusion of the lemma we infer that there are a set $D\in\mathscr{G}^+$ and a finite set $a\subseteq{}^{\kappa^+}2$ such that:
\begin{enumerate}
\item [$(i)$] For some fixed $n\in\omega$ and for every $\beta\in D$ we have $|h(\beta)|=n$.
\item [$(ii)$] For every $f:\kappa^+\rightarrow 2$ if $f\notin a$ then $\{\beta\in D:f\upharpoonright(\beta+1)\in h(\beta)\}\notin\mathscr{G}^+$.
\end{enumerate}
Note that we can remove the assumption that $f \upharpoonright (\beta + 1) \neq g \upharpoonright (\beta + 1)$ for all $g \in a$ that appears in the formulation of the lemma, since the filter $\mathscr{G}$ contains all the tails of $\kappa^+$, and thus, by removing a bounded initial segment from $D$, we may assume that the initial segments of elements of $a$ are different from the initial segments of $f$.
Let $a=\{f_0,\ldots,f_{k-1}\}$.
There are only finitely many possibilities for the sequence $(f_0(\alpha),\ldots,f_{k-1}(\alpha))$.
Hence by shrinking $D$ further we may assume that the sequence $(f_0(\alpha),\ldots,f_{k-1}(\alpha))$ is the same fixed sequence for every $\alpha\in D$.
We apply the lemma once again to obtain a finite set of ordinals $a'\subseteq\kappa^+$ such that if $\gamma\notin a'$ then $\{\beta\in D:\gamma\in\mathunderaccent\tilde-3 {t}(\beta)\}\notin\mathscr{G}^+$.
For this, see the paragraph immediately after Lemma \ref{lemcm}.
Let us work in $V$. Let $\chi$ be a sufficiently large regular cardinal.
Choose $M\prec\mathcal{H}(\chi)$ which enjoys the following properties:
\begin{enumerate}
\item [$(\aleph)$] $|M|=\kappa$.
\item [$(\beth)$] ${}^{<\kappa}M\subseteq M$.
\item [$(\gimel)$] $a,a'\in M$.
\item [$(\daleth)$] $\beta=M\cap\kappa^+\in\kappa^+$.
\end{enumerate}
Let $p_0\in\mathbb{P}$ be a condition which forces a value to $n$ and determines the elements of $a'$.
Choose a condition $q\in\mathbb{P}$ such that $p_0\leq q$ and $q\Vdash\check{\beta}\in\mathunderaccent\tilde-3 {D}$.
To obtain this model $M$ and the ordinal $\beta$, let $(M_i:i\in\kappa^+)$ be an increasing continuous sequence of elementary submodels of $\mathcal{H}(\chi)$ satisfying the above requirements and notice that $E=\{M_i\cap\kappa^+:i\in\kappa^+\}$ is a club of $\kappa^+$.
Recall that $\mathscr{G}$ extends the club filter of $\kappa^+$ and $D\in\mathscr{G}^+$, hence for some $i\in\kappa^+$ we will have a condition $q\geq p_0$ such that $q\Vdash M_i\cap\kappa^+\in E\cap D$.
Set $M=M_i$ and $\beta=M_i\cap\kappa^+$.
Observe that by the same reasoning, we can pick $M_i$ in a way that $q\Vdash\beta\in\bigcap_{\zeta\in a'}A_\zeta$ since $\bigcap_{\zeta\in a'}A_\zeta\in\mathscr{G}$.
By extending $q$ if needed we can assume that $q$ forces a value to $\mathunderaccent\tilde-3 {t}(\beta)$, that is $q\Vdash\mathunderaccent\tilde-3 {t}(\beta)=\{\xi_1,\ldots,\xi_n\}$.
Likewise, $q\Vdash\len(e_{\xi_i})=\gamma_i$ for every $1\leq i\leq n$.
Observe that ${\rm cf}(\beta)=\kappa$ since $\beta=M\cap\kappa^+$ and ${}^{<\kappa}M\subseteq M$.
Hence requirement $(\beth)$ implies that $q\cap M\in M$ bearing in mind that $|q|<\kappa$.
Working within $M$ we are asking whether there is a condition $r\in M$ such that $q\cap M\leq r$ and for some ordinal $\alpha\in\kappa^+$ we can extend $r$ to a condition which forces $\alpha\in A_{t(\beta)}\wedge \beta\in A_{t(\alpha)}$.
More specifically, we wish to find an ordinal $\alpha>\sup(q\cap M), \alpha\in D\cap(\bigcap_{\xi_i\in M}A_{\xi_i}-\bigcup_{\gamma_i\in M}\gamma_i)$ and a condition $r\geq q\cap M$ such that $r\Vdash\alpha\in A_{t(\beta)}\wedge\beta\in A_{t(\alpha)}$.
Let us assume that there are no such $r$ and $\alpha$. Let $D'$ be the set of all $\alpha \in D\cap(\bigcap_{\xi_i\in M}A_{\xi_i}-\bigcup_{\gamma_i\in M}\gamma_i)$ such that $\mathunderaccent\tilde-3 {t}(\alpha)-a'$ is disjoint from $\pi_0(\dom q) \cap M$.
Since $|q| < \kappa$, we remove $<\kappa$ many null sets from a positive set and thus $D'$ is forced to be positive and in particular, unbounded in $\kappa^{+}$. Since $D' \in M$, we can find many ordinals $\alpha$ and conditions $r \geq q \cap M$ such that $\alpha, r\in M$ and $r \Vdash \alpha \in D'$. By taking a stronger condition if needed we may assume that $r$ decides the value of $t(\alpha) = \{\zeta_1, \dots, \zeta_n\}$ and the lengths of $e_{\zeta_i}$ for $1 \leq i \leq n$. Note that all those ordinals are going to be in $M$ and in particular, below $\beta$.
By the definition of $A_{\zeta_i}$, $\beta \in A_{\zeta_i}$ if and only if $g_{\zeta_i}(\beta) = 1$. Let us look at the condition $q \cup r$. In order for the pair $(\zeta_i, \beta)$ to be in the domain of this condition, it has to be in $q$.
Indeed, $(\zeta_i,\beta)\notin\dom r$ since $r\in M$ and $\beta\notin M$ so necessarily $(\zeta_i,\beta)\in\dom q$.
Note that $\zeta_i \in a'$. Otherwise, it contradicts our choice of $D'$. In this case, $g_{\zeta_i}(\beta) = 1$, by our choice of $\beta$. In case that $(\zeta_i,\beta)\notin \dom q$, we can extend $q \cup r$ to a condition $q_1$ by adding $(\zeta_i, \beta)$ to the domain with value $1$. So in any case we found a condition, $q_1$, which forces that $\beta$ is in $A_{t(\alpha)} = \bigcap_i A_{\zeta_i}$.
Let us try to obtain also the other requirement, $\alpha \in A_{t(\beta)}$. Let $\xi_i \in t(\beta)$, and $\gamma_i = \len(e_{\xi_i})$. If $\gamma_i < \beta$, it is in $M$ and thus $\alpha > \gamma_i$. In this case, in order to get $\alpha$ to be in $A_{\xi_i}$, we need to have $g_{\xi_i}(\alpha) = 1$. If further $\xi_i \in M$ then this is part of the definition of $D'$.
Otherwise, $\xi_i \notin M$ and in particular $(\xi_i, \alpha) \notin \dom q \cup \dom r$ since $\alpha \notin \pi_1(\dom q)$ and $\xi_i \notin \pi_0(\dom r)$.
Moving to $q_1$ did not change this fact, and thus we can extend $q_1$ to $q_2$ by adding a single pair into the domain of the condition, sending $(\xi_i, \alpha)$ to $1$.
Thus, we are left with the case $\gamma_i \geq \beta$. This implies that $\xi_i$ is not in $M$ as otherwise, by elementarity, the length of $e_{\xi_i}$ would be inside $M$. Thus, the pair $(\xi_i, \alpha)$ is not in the domain of $q_2$ so let $q_3$ be a condition that sends this pair to $1$. Recall that in the definition of $A_{\xi_i}$, in order for $q_3$ not to force $\alpha$ to be in $A_{\xi_i}$ the following must hold: let $e_{\rho} = e_{\xi_i} \upharpoonright(\alpha + 1)$, then $g_\rho(\gamma_i) = g_{\xi_i}(\alpha) = 1$ and $e_\rho(\alpha) = e_{\xi_1}(\alpha) \neq e_{\xi_i}(\gamma_i)$. Since $\gamma_i \notin M$, in order for $q_3$ to force $g_{\rho}(\gamma_i) = 1$, either $\rho \in \pi_0(\dom q)$ or $\gamma_i = \beta$ and $\rho = \zeta_j$ for some $j$. The first case can be avoided easily, since $\len(e_\rho) = \alpha + 1$ and thus there are at most $<\kappa$ many possible values of $\alpha$ for which this can occur.
Note that if $e_{\xi_i}$ is a restriction of $f_k$ for some $f_k\in a$, then since we stabilized the values of $f_k$ on $D'$ and they equal to the values which are obtained in $\beta$, $e_{\xi_1}(\alpha) \neq e_{\xi_i}(\gamma_i)$ is impossible.
Thus, $e_{\xi_i}$ is not a restriction of an element in $a$.
We conclude that this is the only possible obstacle. Let us summarize the discussion so far:
\begin{claim}
\label{clmobstacle}
In any generic extension, for all sufficiently large $\alpha \in D' \cap M$, there are $i, j \leq n$ such that $e_{\xi_i} \upharpoonright (\alpha + 1) = e_{\zeta_j}$ and $e_{\xi_i}$ is not a restriction of an element in $a$.
\end{claim}
Let us work now in the generic extension $V[G]$. By standard arguments, $M[G \cap M] \prec \mathcal{H}(\chi)[G]$. In $V[G]$, one can partition an end segment of $D' \cap M$ into $\leq n^2$ many pieces according to the values of $i$ and $j$ in Claim \ref{clmobstacle}. In particular, in $M[G \cap M]$, for every finite collection of ordinals in $D' \cap M$, $F$, there is a partition into $n^2$ many parts $F = \bigcup F_{i,j}$, such that for every $\alpha, \beta \in F_{i,j}$, if $\alpha < \beta$ then $e_{\zeta_j^\alpha}$ is an initial segment of length $\alpha + 1$ of $e_{\zeta_j^\beta}$.
By elementarity, the same holds in $V[G]$. Thus, in $V[G]$, every finite subset of $D'$ can be colored by $n^2$ colors such that if $\alpha < \beta$ are both colored in the color $(i,j)$ then $e_{\zeta_j}^\alpha$ is an initial segment of $e_{\zeta_j}^\beta$ (with the right lengths), and $e_{\zeta_j^\alpha}$ is not an initial segment of any $f\in a$.
By compactness, we conclude that there is a coloring of all $D'$ into $n^2$ many colors with this property.
Pick a positive homogeneous set, say $H_{i,j}$ and take $f = \bigcup_{\alpha \in H_{i,j}} e_{\zeta_j^\alpha}$. Then $f\notin a$, which is a contradiction to Lemma \ref{lemcm}.
We conclude, therefore, that for some $r\in M, \alpha\in {\rm Ord}$ it is true that $r\Vdash\alpha\in A_{\mathunderaccent\tilde-3 {t}(\beta)} \wedge \beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$.
Hence $\mathscr{D}$ is a Dowker filter and we are done.
\qedref{thmdowkerconthyp}
We conclude this section by mentioning the original question of Dowker, which remains open:
\begin{question}
\label{qdowkerzfc} Does there exist a successor cardinal $\kappa^+$ which carries a Dowker filter in ZFC?
\end{question}
\section{Successor of a singular cardinal}
The theorem in the previous section gives a positive answer to Question \ref{qcm} when $\kappa={\rm cf}(\kappa)>\aleph_0, 2^\kappa=\kappa^+$ and a Dowker filter is forced over $\kappa^+$.
The reason for $\kappa>\aleph_0$ is that we need the filter $\mathscr{G}$ within the proof to be $\omega_1$-complete.
This plays a key-role in the proof, and if $\kappa=\aleph_0$ or more generally if ${\rm cf}(\kappa)=\omega$ then this essential point becomes problematic.
The concrete case of $\aleph_1$ shows that this issue is not merely a technical problem.
For this reason, a simple attempt to force a Dowker filter over $\kappa^+$ when $\kappa$ is measurable and then to add a Prikry sequence to $\kappa$ will probably fail.
However, this obstacle is also suggestive, since we can singularize a large cardinal with Magidor forcing.
\begin{theorem}
\label{thmmagforcing} Suppose that $\mu$ is a measurable cardinal and $o(\mu)\geq\omega_1$. \newline
Then one can force $\mu>{\rm cf}(\mu), \mu$ is strong limit along with the existence of a Dowker filter over $\mu^+$.
\end{theorem}
\par\noindent\emph{Proof}. \newline
Let $\mathbb{P}$ be an iteration with Easton support of $Add(\delta,\delta^+)$ at every inaccessible cardinal $\delta\leq\mu$, and let $G\subseteq\mathbb{P}$ be generic over $V$.
Remark that $\mu$ is still measurable and $o(\mu)\geq\omega_1$ in $V[G]$.
Working in $V[G]$, let $\mathbb{M}$ be Magidor forcing to make $\mu>{\rm cf}(\mu)=\omega_1$ and let $H\subseteq\mathbb{M}$ be generic over $V[G]$.
We claim that in $V[G\ast H]$ there is a Dowker filter over $\mu^+$.
Let $\mathunderaccent\tilde-3 {e}=(\mathunderaccent\tilde-3 {e}_\zeta:\zeta\in\mu^+)$ be an enumeration of all the nice names of the elements of ${}^{<\mu^+}2\cap V[G\ast H]$ whose length is a successor ordinal.
We may assume that $\ell g(\mathunderaccent\tilde-3 {e}_\zeta)$ is decided by the empty condition for every $\zeta\in\mu^+$.
As in the previous section we let $g=\bigcup G$ and $g_\zeta=g\upharpoonright\{\zeta\}\times\mu^+$ for every $\zeta\in\mu^+$, so each $g_\zeta$ is a function from $\mu^+$ to $2$.
We define $A_\zeta\subseteq\mu^+$ as done in Theorem \ref{thmdowkerconthyp}, for every $\zeta\in\mu^+$.
Explicitly, if $\beta<\ell g(e_\zeta)=\eta_\zeta+1, e_\xi=e_\zeta\upharpoonright(\beta+1), g_\zeta(\beta)=g_\xi(\eta_\zeta)=1$ and $e_\zeta(\beta)\neq e_\zeta(\eta_\zeta)$ then $\chi_{A_\zeta}(\beta)=0$.
Otherwise, $\chi_{A_\zeta}(\beta)=g_\zeta(\beta)$ and of course $A_\zeta$ is the set whose $\chi_{A_\zeta}$ is the characteristic function.
Working in $V[G]$, let $\mathscr{G}'$ be the filter generated by the sets $A_\zeta$ for $\zeta\in\mu^+$, the club filter of $\mu^+$ and $S^{\mu^+}_\mu$.
Let $\mathscr{G}$ be the filter generated by $\mathscr{G}'$ in $V[G\ast H]$.
We shall prove in Lemma \ref{lemcompleteness} below that $\mathscr{G}$ is $\aleph_1$-complete in $V[G\ast H]$ and notice that it contains the $\aleph_1$-complete filter generated by the $A_\zeta$s.
Therefore, we will be able to apply Lemma \ref{lemcm} to $\mathscr{G}$.
Let $\mathscr{D}$ be the filter generated by finite intersections of $A_\zeta$s.
Notice that $\mathscr{D}$ is uniform and satisfies \ref{defdowker}($\gimel$) by the properties of each $A_\zeta$.
The argument is exactly as in the previous section.
It remains to prove that $\mathscr{D}$ satisfies \ref{defdowker}($\beth$) as well.
Assume that $\mathunderaccent\tilde-3 {t}:\mu^+\rightarrow[\mu^+]^{<\omega}$ so $\mathunderaccent\tilde-3 {t}(\alpha)\in[\mu^+]^{<\omega}$ for each $\alpha\in\mu^+$, and let $A_{\mathunderaccent\tilde-3 {t}(\alpha)}=\bigcap\{A_\zeta:\zeta\in\mathunderaccent\tilde-3 {t}(\alpha)\}$.
We wish to prove that for every $\mathunderaccent\tilde-3 {t}$ there are $\alpha<\beta<\mu^+$ and a condition $r$ so that $r\Vdash\alpha\in A_{\mathunderaccent\tilde-3 {t}(\beta)} \wedge \beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$.
Given $\mathunderaccent\tilde-3 {t}$ we define $h=h(\mathunderaccent\tilde-3 {t}):\mu^+\rightarrow[{}^{<\mu^+}2]^{<\omega}$ by $h(\alpha)=\{e_{\xi_1},\ldots,e_{\xi_n}\}$ where $\mathunderaccent\tilde-3 {t}(\alpha)=\{\xi_1,\ldots,\xi_n\}$ and let $C=\mu^+$.
Applying Lemma \ref{lemcm} to the triple $(\mathscr{G},h,C)$ we obtain $D\subseteq\mu^+,D\in\mathscr{G}^+$ and a finite set $a$ of functions as guaranteed by the lemma.
For every $\beta\in D$ and every $q\in\mathbb{P}$ we define:
$$
R_{q\beta} = \{\gamma\in\beta:(\gamma,\beta)\in\dom q, q(\gamma,\beta)=0\}.
$$
Working in $V[G\upharpoonright\mu]$ we fix a sufficiently large regular cardinal $\chi$ and an increasing continuous sequence $(M_i:i\in\mu^+)$ of elementary submodels of $\mathcal{H}(\chi)$ such that $|M_i|=\mu, \mu+1\subseteq M_i$ for every $i\in\mu^+$ and each $M_{i+1}$ is $<\mu$-closed.
Notice that if ${\rm cf}(i)=\mu$ then $M_i$ will be $<\mu$-closed as well.
We shall prove in Lemma \ref{lemrqbeta} that there exist an ordinal $\beta\in\mu^+$ and a condition $(q,p)\in\mathbb{P}\ast\mathbb{M}$ with the following properties:
\begin{enumerate}
\item [$(a)$] $\beta=M_i\cap\mu^+$ for some $i\in\mu^+$ and ${\rm cf}(\beta)=\mu$.
\item [$(b)$] $(q,p)\Vdash\check{\beta}\in\mathunderaccent\tilde-3 {D}$.
\item [$(c)$] $(q,p)\Vdash\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q\beta}=\varnothing$ for a $\mathscr{G}$-positive set of $\alpha\in D$.
\end{enumerate}
By extending $(q,p)$ if needed we may assume that $(q,p)$ forces a value to $\mathunderaccent\tilde-3 {t}(\beta)$, determines the size of $a$ and forces the value of $\ell g(e_\zeta)$ for every $\zeta\in\mathunderaccent\tilde-3 {t}(\beta)$.
Namely, $(q,p)\Vdash\mathunderaccent\tilde-3 {t}(\beta)=\{\xi_1,\ldots,\xi_n\}$ and $\ell g(e_{\xi_i})=\gamma_i$ for every $1\leq i\leq n$.
Let $M=M_i$ where $i\in\mu^+$ is the ordinal provided by part $(a)$.
Notice that ${\rm cf}(i)=\mu$ and hence ${}^{<\mu}M\subseteq M$.
Every $q\in\mathbb{P}$ satisfies $|\dom q|<\mu$ hence $|R_{q\beta}|<\mu$ as well, and therefore $R_{q\beta}\in M$.
Likewise, the name of the set of all $\alpha\in D$ such that $\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q\beta}=\varnothing$ belongs to $M$ since it is definable in $M$.
For every $1\leq i\leq n$ let $E_i$ be $\{\alpha<\beta:\exists\rho, \mathunderaccent\tilde-3 {e}_\rho=\mathunderaccent\tilde-3 {e}_{\xi_i}\upharpoonright(\alpha+1), (\rho,\gamma_i)\in\dom q\}$.
Again, $|E_i|<\mu$ and hence the set $E=\bigcup\{E_i:1\leq i\leq n\}$ is of size less than $\mu$, and consequently $E\in M$.
Define $D'_0=\bigcap\{A_{\xi_j}:\xi_j\in M\}-(\bigcup_{\gamma_j\in M}\gamma_j\cup E)$, so $D'_0\in M$.
Finally, let $D'$ be $D\cap D'_0$ intersected with the set of $\alpha\in D$ for which $\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q\beta}=\varnothing$.
Notice that $D'\in M$ and it is forced (by the rest of $\mathbb{P}\ast\mathbb{M}$) to be in $\mathscr{G}^+$.
We work now in $V[G\ast H]$ and we assume that $(q,p)\in G\ast H$.
Observe that $M[(G\ast H)\cap M]\prec\mathcal{H}(\chi)[G\ast H]$, since $\mu\subseteq M$ and $\mathbb{P}\ast\mathbb{M}$ is $\mu^+$-cc.
As noted above, in $M[G\ast H]$ we know that the set of $\alpha\in D$ such that $\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q\beta}=\varnothing$ is in $\mathscr{G}^+$, so in particular it is unbounded in $\mu^+$.
For every $\alpha$ in this positive set we choose a condition $(r_\alpha,p_\alpha)\in G\ast H$ so that $(r_\alpha,p_\alpha)$ forces $\alpha$ to be in this set and it forces the values of $\mathunderaccent\tilde-3 {t}(\alpha)$ and the lengths of its elements.
Applying the pigeonhole principle, there are a condition $r\in\mathbb{P}$ and a fixed stem $s_\star$ such that for some $B_{s_\star}\subseteq\mu^+$, $r$ forces that $|B_{s_\star}|=\mu^+$ and if $\alpha\in B_{s_\star}$ then $r_\alpha \geq r$ and the stem of $p_\alpha$ is $s_\star$.
By the directness of $H$ we may assume that the stem of $p$ is $s_\star$ as well.
We claim that there are a condition $(q',p')\in\mathbb{P}\ast\mathbb{M}$ and an ordinal $\alpha\in B_{s_\star}\cap\beta$ such that $(q,p)\leq_{\mathbb{P}\ast\mathbb{M}}(q',p')$ and $(q',p')\Vdash\alpha\in A_{\mathunderaccent\tilde-3 {t}(\beta)} \wedge \beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$.
The easy part is to force $\beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$.
To this end, we choose a condition $(r,p^+)\in M$ and some $\alpha\in M$ such that $\alpha$ is forced to be in $B_{s_\star}, s_\star$ is the stem of $p^+$ and $q\cap M\leq r$.
We indicate that the Cohen part of the condition forces $p\parallel p^+$ since $s_\star$ is the Magidor stem of both conditions.
Hence $r\cup q$ also forces this fact.
Now $(r,p^+)\in M$ and hence $\mathunderaccent\tilde-3 {t}(\alpha)\in M$ and for every $\xi\in\mathunderaccent\tilde-3 {t}(\alpha)$ we have $\ell g(e_\xi)\in M$ so $\ell g(e_\xi)<\beta$.
By a finite sequence of extensions of $r\cup q$ we can make sure that $\beta$ is forced into $A_\xi$ for every $\xi\in\mathunderaccent\tilde-3 {t}(\alpha)$.
To see this, observe first that $\beta$ is absent from the ordinals mentioned in $r$ since $r\in M$ and $\beta\notin M$.
For the $q$ part either $(\xi,\beta)\in\dom q$, in which case $q\Vdash\beta\in A_\xi$ since $\alpha\in D'$, or $(\xi,\beta)\notin\dom q$ and we can add the triple $(\xi,\beta,1)$ to $q$.
As $\mathunderaccent\tilde-3 {t}(\alpha)$ is finite we can extend $r\cup q$ and obtain $q'$ and $p'\geq p,p^+$ so that $(q',p')\Vdash\beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$.
The harder part is to ensure that $(q',p')\Vdash\alpha\in A_{\mathunderaccent\tilde-3 {t}(\beta)}$.
Now if $\xi\in\mathunderaccent\tilde-3 {t}(\beta)\cap M$ then $\alpha\in A_\xi$ since $\alpha\in A\subseteq D'\subseteq\bigcap\{A_\zeta:\zeta\in M\}$.
Assume, therefore, that $\xi\in\mathunderaccent\tilde-3 {t}(\beta)$ and $\xi\notin M$.
Let $\gamma$ be such that $\ell g(e_\xi)=\gamma+1$.
We distinguish three cases:
\par\noindent\emph{Case 1}: $\gamma<\beta$.
In this case necessarily $\alpha>\gamma$ since $\alpha\in D'$ and by the definition of $D'$.
We claim that one can add the triple $(\xi,\alpha,1)$ to the finite sequence of extenstions of $r\cup q$ rendered to obtain $\beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$, which is exactly what we need.
For this, observe that $\xi\notin\pi_0(\dom r)$ since $r\in M$ and $\xi\notin M$.
Likewise, $\alpha\notin\pi_1(\dom q)$ since $\alpha\in D'$ and hence $\alpha\notin E$.
Finally, the finite sequence of extensions of $r\cup q$ needed to force $\beta\in A_{\mathunderaccent\tilde-3 {t}(\alpha)}$ involve $\beta$ at each step, and here $\alpha\neq\beta$ so there is no problem with our triple.
\par\noindent\emph{Case 2}: $\gamma>\beta$.
Let $\rho$ be such that $e_\rho=e_\xi\upharpoonright(\alpha+1)$.
Since $\alpha\notin E$ we see that $(\rho,\gamma)\notin\dom q$.
Since $\gamma>\beta$ and $r\in M$ we see that $(\rho,\gamma)\notin\dom r$.
Hence we can add the triple $(\rho,\gamma,0)$ to the extension of $r\cup q$ obtained so far.
By doing so we conclude that our extension of $r\cup q$ does not force $\alpha\notin A_\xi$, hence we can add $(\xi,\alpha,1)$ to this extension.
\par\noindent\emph{Case 3}: $\gamma=\beta$.
In this case, one of the extensions of $r\cup q$ (made for ensuring that $\beta$ is forced into $A_{\mathunderaccent\tilde-3 {t}(\alpha)}$) might be problematic when trying to force $\alpha$ into $A_{\mathunderaccent\tilde-3 {t}(\beta)}$.
Let us analyze the situation and see how the problem can be avoided.
We shall need the concept of a \emph{thread}.
A set $F=\{f_0, \dots, f_{k-1}\}$ of functions from $\mu^+$ into $2$ is a thread covering for $A$, where $A$ is an unbounded subset of $\mu^+$, iff for every $\alpha\in A$ there is $\zeta\in\mathunderaccent\tilde-3 {t}(\alpha)$ and $i < k$ such that $e_\zeta=f_i\upharpoonright(\alpha+1)$. We will always assume that our threads are \emph{minimal}, namely that for every $i < k$, $\{f_j \mid j \neq i\}$ is not a thread.
It may happen that for some $\alpha\in B_{s_\star}$ we will have $\zeta\in\mathunderaccent\tilde-3 {t}(\alpha),\xi\in\mathunderaccent\tilde-3 {t}(\beta), e_\zeta=e_\xi\upharpoonright(\alpha+1)$ and $\ell g(e_\xi)=\beta+1$.
In such cases $e_\zeta\lhd e_\xi$ (recall that $\alpha<\beta$) and maybe we cannot add $(\xi,\alpha,1)$ to our condition.
We can try a different $\alpha\in B_{s_\star}$, but perhaps the same problem occurs at every $\alpha\in B_{s_\star}$.
In such cases there is a thread covering $F$ for $B_{s_\star}$, using the compactness argument from the proof of Theorem \ref{thmdowkerconthyp}. More precisely, $|F| \leq n$.
Lest $F\cap a \neq \emptyset$ there will be no problem, by the definition of $D'$.
Therefore, our purpose is to show that there exists a stem $s$ such that if there is a thread covering $F = \{f_0, \dots, f_{k-1}\}$, $k \leq n$ for $B_s$ then necessarily $f_i\in a$ for some $i$.
Assume towards contradiction that for every Magidor-stem $s$ there is a thread covering $F_s$ for the set $B_s$ and $F_s \cap a = \emptyset$, $|F_s| \leq n$.
Notice that if $s$ is fixed then there are at most finitely many minimal thread coverings for $B_s$ of size $\leq n$. Recall that $n$ is the size of $\mathunderaccent\tilde-3 {t}(\alpha)$ for every $\alpha\in B_s$. Indeed, let $\{F_i \mid i < \omega\}$ be an infinite set of thread coverings. Using the finite $\Delta$-system lemma (the sunflower lemma), we may assume that $F_i \cap F_{i'} = F_\star$ for all $i \neq i'$. For every sufficiently large $\alpha$ which is not covered by $F_\star$, for all $i \neq i'$ and $f \in F_i \setminus F_\star, g \in F_{i'} \setminus F_\star$ $f\restriction \alpha \neq g \restriction \alpha$. For such $\alpha$, for every $i$ there is some $f \in F_i \setminus F_\star$ such that $f \restriction \alpha + 1 = e_\zeta$, $\zeta \in \mathunderaccent\tilde-3 {t}(\alpha)$. Since all of them are distinct, the size of $\mathunderaccent\tilde-3 {t}(\alpha)$ must be infinite, which is a contradiction.
Now if $s_0$ and $s_1$ are two Magidor-stems and $s_1$ is stronger than $s_0$ then every thread covering for $B_{s_1}$ is also a thread covering for $B_{s_0}$ since $B_{s_0}\subseteq B_{s_1}$.
In particular, there is a Magidor-stem $s_\star$ such that if $s$ is a stronger stem than $s_\star$ then the set of thread coverings for $B_s$ is identical with the set of thread coverings for $B_{s_\star}$.
Since $D'$ is expressible as a union of $B_s$s where every two Magidor-stems $s,s'$ are compatible, we conclude that in the generic extension there is a thread covering $F$ for $D'$.
It follows from Lemma \ref{lemcm} that $F\subseteq a$, contradicting our assumption.
Hence the last case is covered, and we are done.
\qedref{thmmagforcing}
In order to accomplish the proof we need two lemmata.
\begin{lemma}
\label{lemcompleteness} The completeness lemma. \newline
The filter $\mathscr{G}$ is $\aleph_1$-complete in $V[G\ast H]$.
\end{lemma}
\par\noindent\emph{Proof}. \newline
Suppose that $(\zeta_n:n\in\omega)$ is a sequence of ordinals of $\mu^+$ in $V[G\ast H]$.
Let $A=\bigcap\{A_{\zeta_n}:n\in\omega\}$.
We need showing that $|A|=\mu^+$.
Fix an ordinal $\alpha_0\in\mu^+$.
We shall find an ordinal $\gamma\in\mu^+$ so that $\alpha_0<\gamma$ and $\gamma\in A$ in $V[G\ast H]$.
To begin with, choose a set $x\in V$ so that $|x|<\mu$ and it is forced by $\mathbb{P}\ast\mathbb{M}$ that $\{\zeta_n:n\in\omega\}\subseteq x$.
Such a set $x$ exists by virtue of Lemma \ref{lemcovering}.
Now choose an ordinal $\alpha_1\in\mu^+$ such that $\alpha_0<\alpha_1$ and moreover $\alpha_1>\ell g(e_{\zeta_n})$ for every $n\in\omega$.
By the genericity of the Cohen functions we can find an ordinal $\gamma\in\mu^+$ such that $\alpha_1\leq\gamma$ and $g_\alpha(\gamma)=1$ for every $\alpha\in x$.
It follows that $\gamma\in A_{\zeta_n}$ for every $n\in\omega$, namely $\gamma\in A$ in $V[G\ast H]$, so we are done.
\qedref{lemcompleteness}
Our second lemma provides us with the following property:
\begin{lemma}
\label{lemrqbeta} The evasion lemma. \newline
There are $(q,p)\in\mathbb{P}\ast\mathbb{M}$ and $\beta\in\mu^+$ such that:
\begin{enumerate}
\item [$(a)$] $\beta=M_i\cap\mu^+$ for some $i\in\mu^+$, and ${\rm cf}(\beta)=\mu$.
\item [$(b)$] $(q,p)\Vdash\check{\beta}\in\mathunderaccent\tilde-3 {D}$.
\item [$(c)$] $(q,p)\Vdash\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q\beta}=\varnothing$ for a $\mathscr{G}$-positive set of $\alpha\in D$.
\end{enumerate}
\end{lemma}
\par\noindent\emph{Proof}. \newline
We commence with a simple observation concerning the sets $R_{q\beta}$.
Suppose that $q_0\in\mathbb{P},\beta_0\in\mu^+$ and $A_0=\bigcap\{A_\xi:\xi\in R_{q_0\beta_0}\}$.
Since $|R_{q_0\beta_0}|<\mu$ we see that $A_0\in\mathscr{G}$.
Suppose that $q_0\leq q_1$ and $\beta_1\in A_0$.
We claim that $R_{q_0\beta_0}\cap R_{q_1\beta_1}=\varnothing$.
Indeed, if $\gamma\in R_{q_0\beta_0}\cap R_{q_1\beta_1}$ then $\beta_1\in A_\gamma$ since $\gamma\in R_{q_0\beta_0}$.
Hence $q_0\Vdash g_\gamma(\beta_1)=1$, so $q_1\Vdash g_\gamma(\beta_1)=1$ as well.
On the other hand, $\gamma\in R_{q_1\beta_1}$ so $q_1\Vdash g_\gamma(\beta_1)=0$, a contradiction.
Working in $V[G]$ we assume toward contradiction that the lemma fails.
Let $S=S^{\mu^+}_\mu$ and let $C=\{M_i\cap\mu^+:i\in\mu^+\}$, so $C\cap S$ is a stationary subset of $\mu^+$.
Every element of $C\cap S$ satisfies $(a)$, so for each $\beta\in C\cap S$ either $(b)$ fails or $(c)$.
Choose $(q_0,p_0)$ and $\beta_0\in C\cap S$ such that $(q_0,p_0)\Vdash\check{\beta}\in\mathunderaccent\tilde-3 {D}$.
By our assumption toward contradiction we see that $(q,p)\Vdash\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q_0\beta_0}\neq\varnothing$ for $\mathscr{G}$-almost every $\alpha\in D$.
Let $A_0=\bigcap\{A_\xi:\xi\in R_{q_0\beta_0}\}$.
Now by induction on $0<m\in\omega$ we choose $(q_m,p_m)$ and $\beta_m$ so that $(q_{m-1},p_{m-1})\leq(q_m,p_m), \beta_m\in C\cap S$ and $(q_m,p_m)\Vdash\beta_m\in\bigcap_{j<m}A_j\cap\mathunderaccent\tilde-3 {D}$.
Pick up some $\alpha\in\bigcap_{m\in\omega}A_m\cap D$ such that $(q_m,p_m)\Vdash \mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q_m\beta_m}\neq\varnothing$ for every $m\in\omega$, and let $n=|\mathunderaccent\tilde-3 {t}(\alpha)|$.
By the construction, $(q_{n+1},p_{n+1})\Vdash\mathunderaccent\tilde-3 {t}(\alpha)\cap R_{q_j\beta_j}\neq\varnothing$ for every $j\leq n$.
By the observation at the beginning of the proof, if $j<\ell\leq n$ then $R_{q_j\beta_j}\cap R_{q_\ell\beta_\ell}=\varnothing$, so $(q_{n+1},p_{n+1})\Vdash|\mathunderaccent\tilde-3 {t}(\alpha)|\geq n+1$, a contradiction.
\qedref{lemrqbeta}
The main result of this section points to the possibility that singular cardinals with countable cofinality behave in a different way than their friends with uncountable cofinality.
As mentioned in the introduction, it is easy to force a Dowker filter at $\mu^+$ even if $\mu>{\rm cf}(\mu)=\omega$ if we drop the requirement of strong limitude.
\begin{question}
\label{qcofomega} Is it consistent that $\mu$ is a strong limit singular cardinal with countable cofinality and there exists a Dowker filter over $\mu^+$?
\end{question}
\end{document} |
\begin{document}
\title{The Tutte polynomial via lattice point counting}
\begin{abstract}
We recover the Tutte polynomial of a matroid, up to change of coordinates,
from an Ehrhart-style polynomial
counting lattice points in the Minkowski sum of its base polytope and scalings of simplices.
Our polynomial has coefficients of alternating sign
with a combinatorial interpretation closely tied to the Dawson partition.
Our definition extends in a straightforward way to polymatroids,
and in this setting our polynomial has K\'alm\'an's internal and external activity polynomials as its univariate specialisations.
\end{abstract}
\section{Introduction}
\label{sec:in}
The Tutte polynomial $T_M(x,y)$ of a matroid $M$,
formulated by Tutte for graphs and generalised to matroids by Crapo,
is perhaps the invariant most studied by researchers in either field,
on account of its diverse applications.
Most straightforwardly, any numerical function of matroids or graphs (or function valued in another ring) which can be computed by a deletion-contraction recurrence
can also be obtained as an evaluation of the Tutte polynomial.
Examples include the number of bases and the number of independent sets in a given matroid, and the number of acyclic orientations of a graph.
A matroid is expediently encoded by its \emph{matroid (base) polytope},
the convex hull of the indicator vectors of the bases.
One useful feature of this polytope is that no other \emph{lattice points}, i.e.\ points with integer coordinates, are caught in the convex hull.
Therefore, counting the lattice points in the base polytope of $M$ tells us the number of bases of $M$,
which equals $T_M(1,1)$.
The Tutte polynomial can be written as a generating function for bases of a matroid according to their \emph{internal} and \emph{external activity},
two statistics that count elements which permit no basis exchanges of certain shapes.
K\'alm\'an \cite{kalman} observed that the active elements of a basis can be readily discerned
from the matroid polytope:
given a lattice point of this polytope, one inspects in which ways one can increment one coordinate, decrement another, and remain in the polytope.
The lattice points one reaches by incrementing a coordinate are those contained in the Minkowski sum with the standard simplex $\operatorname{Conv}\{e_1,\ldots,e_n\}$,
and decrementing a coordinate is similarly encoded by the upside-down simplex $\operatorname{Conv}\{-e_1,\ldots,-e_n\}$.
Our main result, Theorem~\ref{lpg}, proves that the bivariate matroid polynomial
that counts the lattice points in the Minkowski sum of the matroid polytope and scalings of the above simplices
contains the same information as the Tutte polynomial.
To be precise, either polynomial is an evaluation of the other.
K\'alm\'an's interest in this view of activity arose from the question of
enumerating spanning trees of bipartite graphs according to their vector of degrees at the vertices of one colour.
The set of such vectors is rarely a matroid polytope, but it is always a \emph{polymatroid} polytope.
Polymatroids were introduced in Edmonds' work in optimisation circa 1970, as an extension of matroids formed by
relaxing one matroid rank axiom so as to allow singletons to have arbitrarily large rank.
Postnikov's \emph{generalised permutohedra} \cite{postnikov} are virtually the same object.
(Integer generalised permutohedra are exactly translates of polymatroid polytopes,
and they are polymatroid polytopes just when they lie in the closed positive orthant.)
One would like an analogue of the Tutte polynomial for polymatroids.
Unlike the case for matroids, the simultaneous generating function for internal and external activity does not answer to this desire,
as it is not even an invariant of a polymatroid:
Example \ref{bivaractivity} shows that it varies with different choices of ordering of the ground set.
Only the generating functions for either activity statistic singly are in fact invariants,
as K\'alm\'an proved, naming them $I_r(\xi)$ and $X_r(\eta)$.
Our lattice point counting polynomial can be applied straightforwardly to a polymatroid,
and we prove in Theorem~\ref{thm:activity} that it specialises to $I_r(\xi)$ and~$X_r(\eta)$.
That is, the invariant we construct is a bivariate analogue of K\'{a}lm\'{a}n's activity polynomials, which answers a question of~\cite{kalman}.
In seeking to generalise the Tutte polynomial to polymatroids one might be interested in other properties than activity, especially its universal property with respect to deletion-contraction invariants.
Like matroids, polymatroids have a well-behaved theory of \emph{minors}, analogous to graph minors:
for each ground set element $e$ one can define a deletion and contraction,
and knowing these two determines the polymatroid.
The deletion-contraction recurrence for the Tutte polynomial reflects this structure.
The recurrence has three cases,
depending on whether $e$ is a loop, a coloop, or neither of these.
The number of these cases grows quadratically with the maximum possible rank of a singleton,
making it correspondingly hairier to write down a universal invariant.
In 1993 Oxley and Whittle \cite{whittle} addressed the case of \emph{$\it{2}$-polymatroids},
where singletons have rank at most~2, and prove that
the corank-nullity polynomial is still universal for a form of deletion-contraction recurrence.
The general case, with a natural assumption on the coefficients of the recurrence but one slightly stronger than in~\cite{whittle},
is addressed using coalgebraic tools in~\cite{DFM},
fitting into the tradition of applications of coalgebras and richer structures in combinatorics:
see \cite{AA,KMT} for other work of particular relevance here.
For our polymatroid invariant we give a recurrence that involves not just the deletion and contraction,
but a whole array of ``slices'' of which the deletion and the contraction are the extremal members (Theorem~\ref{delcont}).
We do not know a recurrence relation where only the deletion and the contraction appear.
We would be remiss not to mention the work of K\'alm\'an and Postnikov \cite{kalman2} proving the central conjecture of \cite{kalman},
that swapping the two colours in a bipartite graph leaves $I_r(\xi)$ unchanged.
Their proof also exploits Ehrhart-theoretic techniques, but the key polytope is the \emph{root polytope} of the bipartite graph.
We expect that it should be possible to relate this to our machinery via the Cayley trick.
Oh \cite{oh} has also investigated a similar polyhedral
construction, as a way of proving Stanley's pure O-sequence
conjecture for cotransversal matroids.
This paper is organised as follows.
Section~\ref{sec:prelim} introduces the definitions of our main objects.
In Section \ref{sec:hints} we begin by explaining the construction of our polynomial for matroids, followed by how this is related to the Tutte polynomial (our main theorem, Theorem \ref{lpg}).
In Section \ref{sec:coeffs}, we give a geometric interpretation of the coefficients of our polynomial, by way of a particular subdivision of the relevant polytope, which has a simple interpretation in terms of Dawson partitions.
In Section~\ref{sec:polymatroids}, we discuss the extension to polymatroids,
including properties our invariant satisfies in this generality.
Section~\ref{sec:kalman} is dedicated to the relationship to K\'alm\'an's univariate activity invariants.
\subsection*{Acknowledgments}
We thank Tam\'as K\'alm\'an, Madhusudan Manjunath, and Ben P.\ Smith for fruitful discussions,
Iain Moffatt for very useful expository advice on a draft,
and the anonymous reviewers for FPSAC 2016 for their suggested improvements.
The authors were supported by EPSRC grant EP/M01245X/1.
\section{Preliminaries}
\label{sec:prelim}
We assume the reader has familiarity with basic matroid terminology, and
recommend \cite{Oxley} as a reference for this material.
Given a set $E$, let $\mathcal P(E)$ be its power set.
\begin{dfn}
A \emph{polymatroid} $M=(E,r)$ on a finite \emph{ground set} $E$
consists of the data of a rank function $r:\mathcal{P}(E)\rightarrow \mathbb{Z^+}\cup\{0\}$ such that, for $X,Y\in\mathcal{P}(E)$, the following conditions hold:
\begin{itemize}
\item[{\rm P1}.] $r(\emptyset)=0$
\item[{\rm P2}.] If $Y\subseteq X$, then $r(Y)\leq r(X)$
\item[{\rm P3}.] $r(X\cup Y)+r(X\cap Y)\leq r(X)+r(Y)$
\end{itemize}
\end{dfn}
A \emph{matroid} is a polymatroid such that $r(i)\leq1$ for all $i\in E$.
Like matroids, polymatroids can be defined \emph{cryptomorphically} in other equivalent ways,
the way of most interest to us being as polytopes (Definition~\ref{dfn:polytope}).
We will not be pedantic about which axiom system we mean when we say ``polymatroid''.
The following three
definitions of activity for polymatroids are from \cite{kalman}.
\begin{dfn}
A vector ${\bf x}\in\mathbb Z^E$ is called a \emph{base} of a polymatroid $M=(E,r)$ if ${\bf x}\cdot{\bf e}_E=r (E)$
and ${\bf x}\cdot{\bf e}_S\leq r (S)$ for all subsets $S\subseteq E$.
\end{dfn}
Let $\mathcal B_M$ be the set of all bases of a polymatroid $M=(E,r)$.
\begin{dfn}
A \emph{transfer} is possible from $u_1\in E$ to $u_2\in E$
in the base ${\bf x}\in \mathcal B_M\cap\mathbb{Z}^E$
if by decreasing the $u_1$-component of ${\bf x}$ by $1$ and increasing its $u_2$-component by $1$ we get another base.
\end{dfn}
Like matroids, polymatroids have a base exchange property \cite[Theorem 4.1]{hibi}.
If $\bf x$ and $\bf y$ are in $\mathcal B_M$ and ${\bf x}_i>{\bf y}_i$ for some $i\in E$,
then there exists $l$ such that ${\bf x}_l< {\bf y}_l$ and ${\bf x}-{\bf e}_i+{\bf e}_l$ is again in $\mathcal B_M$,
or in other words, such that a transfer is possible from $i$ to $l$ in~$\bf x$.
Fix a total ordering of the elements of~$E$.
\begin{dfn}\label{dfn:activity}
\begin{enumerate}[i.]
\item We say that $u\in E$ is \emph{internally active} with respect to the base $x$ if no transfer is possible in~$x$ \emph{from} $u$ to a smaller element of $E$.
\item We say that $u\in E$ is \emph{externally active} with respect to $x$ if no transfer is possible in~$x$ \emph{to} $u$ from a smaller element of $E$.
\end{enumerate}
\end{dfn}
For $x\in \mathcal B_M\cap\mathbb{Z}^E$\!, let the set of internally active elements with respect to $x$ be denoted with $\mathrm{Int}(x)$, and let $\iota(x)=|\mathrm{Int}(x)|$;
likewise, let the set of externally active elements be denoted with $\mathrm{Ext}(x)$
and $\varepsilon(x)=|\mathrm{Ext}(x)|$.
Let $\ol{\iota}(x),\ol{\varepsilon}(x)$ denote
the respective numbers of inactive elements.
When $M$ is a matroid, the following definitions of activity are more commonly used,
analogous to Tutte's original formulation using spanning trees of graphs.
\begin{dfn}\label{dfn:activity 2}
Take a matroid $M=(E,r)$. Let $B$ be a basis of $M$.
\begin{enumerate}[i.]
\item We say that $e\in E-B$ is \emph{externally active} with respect to $B$ if $e$ is the smallest element in the unique circuit contained in $B\cup e$, with respect to the ordering on $E$.
\item We say that $e\in B$ is \emph{internally active} with respect to $B$ if $e$ is the smallest element in the unique cocircuit in $(E\setminus B)\cup e$.
\end{enumerate}
\end{dfn}
\begin{remark}\label{rem:activity}
In the cases where it is set forth, namely $e\in E-B$ for external activity and $e\in B$ for internal activity,
Definition~\ref{dfn:activity 2} agrees with Definition~\ref{dfn:activity}.
But it will be crucial that we follow Definition~\ref{dfn:activity} where Definition~\ref{dfn:activity 2} doesn't apply:
when $M$ is a matroid and $B$ a basis thereof,
we consider all elements $e\in B$ externally active, and all elements $e\in E-B$ internally active, with respect to~$B$.
\end{remark}
\begin{dfn}\label{dfn:Tutte}
Let $M=(E,r)$ be a matroid with ground set $E$ and rank function $r:\mathcal{P}(E)\rightarrow \mathbb{Z^+}\cup\{0\}$. The \emph{Tutte polynomial} of $M$ is
\begin{equation}\label{eq:Tutte}
T_M(x,y) = \sum_{S\subseteq E} (x-1)^{r(M)-r(S)}(y-1)^{|S|-r(S)}.
\end{equation}
\end{dfn}
The presentation of the Tutte polynomial in Definition~\ref{dfn:Tutte}
is given in terms of the \emph{corank-nullity polynomial}:
up to a change of variables, it is the generating function for subsets $S$ of the
ground set by their corank $r(M)-r(S)$ and nullity $|S|-r(S)$.
When $M$ is a matroid, the Tutte polynomial is equal to a generating function for activities:
\begin{equation}\label{eq:act}
T_M(x,y)=\frac1{x^{|E|-r(E)}y^{r(E)}}\sum_{B\in\mathcal{B}_M} x^{\iota(B)}y^{\varepsilon(B)}.
\end{equation}
The unfamiliar denominator in this formula appears on account of Remark~\ref{rem:activity}.
Although on its face the right hand side of the formula depends on the ordering imposed on~$E$,
its equality with the right hand side of equation~\ref{eq:Tutte} shows that there is no such dependence.
For polymatroids, activity invariants can be defined as well: see Definition~\ref{activity} and following discussion.
In this paper we will be principally viewing polymatroids as polytopes.
These polytopes live in the vector space $\mathbb R^E$,
where the finite set $E$ is the ground set of our polymatroids.
For a set $U\subseteq E$, let ${\bf e}_U\in\mathbb R^E$ be the indicator vector of $U$,
and abbreviate ${\bf e}_{\{i\}}$ by ${\bf e}_{i}$.
Let $r:\mathcal{P}(E)\rightarrow\mathbb{Z}^+\cup\{0\}$ be a rank function,
and $M=(E,r)$ the associated polymatroid.
The \emph{extended polymatroid} of $M$ is defined to be the polytope
\begin{displaymath}EP(M)=\{{\bf x}\in\mathbb{R}^E \ | \ {\bf x}\geq 0\mbox{ and }
{\bf x}\cdot{\bf e}_U\leq r(U) \ \mathrm{for \ all} \ U\subseteq E\}.\end{displaymath}
\begin{dfn}\label{dfn:polytope}
The \emph{polymatroid (base) polytope} of $M$
is a face of the extended polymatroid:
\[P(M) = EP(M)\cap\{{\bf x}\in\mathbb{R}^E \ | \ {\bf x}\cdot{\bf e}_E= r(E)\}
= \operatorname{conv} \mathcal B_M.\]
\end{dfn}
Either one of these polytopes contains all the information in the rank function.
In fact, they can be used as cryptomorphic axiomatisations of polymatroids:
a polytope whose vertices have nonnegative integer coordinates is a polymatroid polytope
if and only if all its edges are parallel to a vector of the form ${\bf e}_{i}-{\bf e}_{j}$ for some $i,j\in E$.
Extended polymatroids permit a similar characterisation;
moreover, they can be characterised as those polytopes over which a greedy algorithm
correctly optimises every linear functional with nonnegative coefficients,
which was the perspective of their inventor Edmonds \cite{edmonds}.
\section{Our invariant}
\label{sec:hints}
This section describes the construction of our matroid polynomial, to be denoted $Q'_M$,
which counts the lattice points of a particular Minkowski sum of polyhedra,
and explains its relation to the Tutte polynomial.
In anticipation of Section~\ref{sec:polymatroids} we set out the definition in the generality of polymatroids.
\subsection{Construction}
\label{sec:const}
Let $\Delta$ be the standard simplex in $\mathbb R^E$ of dimension $|E|-1$,
that is
\[\Delta = \operatorname{conv}\{\mathbf e_{i}:i\in E\},\]
and $\nablala$ be its reflection through the origin,
$\nablala = \{-x : x\in\Delta\}$.
The faces of $\Delta$ are the polyhedra
\[\Delta_S = \operatorname{conv}\{\mathbf e_{i}:i\in S\}\]
for all nonempty subsets $S$ of~$E$; similarly,
the faces of $\nablala$ are the polyhedra $\nablala_S$ given as the reflections of the $\Delta_S$.
(We exclude the empty set as a face of a polyhedron.)
We consider $P(M)+u\Delta+t\nablala $ where $M=(E,r)$ is any polymatroid and $u,t\in\mathbb{Z}^+\cup\{0\}$.
We are interested in the lattice points in this sum.
These can be interpreted as the vectors
that can be turned into bases of~$M$ by incrementing a coordinate $t$ times
and decrementing one $u$ times.
By Theorem 7 of \cite{mcmullan}, the number
\begin{equation}\label{qdef}
Q_{M}(t,u):=\#(P(M)+u\Delta+t\nablala )\cap\mathbb{Z}^E
\end{equation}
of lattice points in the sum
is a polynomial in $t$ and~$u$, of degree $\dim(P(M)+u\Delta+t\nablala) = |E|-1$.
We will mostly work with this polynomial after a change of variables:
letting the coefficients $c_{ij}$ be defined by \begin{displaymath}Q_{M}(t,u)=\sum_{i,j} c_{ij}\binom{u}{j}\binom{t}{i},\end{displaymath}
we use these to define the polynomial \begin{displaymath}Q'_M(x,y)=\sum_{ij}c_{ij}(x-1)^i(y-1)^j.\end{displaymath}
The change of variables is chosen so that
applying it to $\#(u\Delta_X+t\nablala_Y)$ yields $x^iy^j$,
where $\Delta_X$ and $\nablala_Y$ are faces of $\Delta$ and $\nablala$
of respective dimensions $i$ and~$j$. This will allow for a
combinatorial interpretation of the coefficients of~$Q'$ in Theorem~\ref{coeffs}.
\subsection{Relation to the Tutte polynomial}
\label{tutte}
For the remainder of Section~\ref{sec:hints}, we assume that $M$ is a matroid.
The main theorem of this section is that $Q'_M(x,y)$ is an evaluation of the Tutte polynomial, and in fact one that contains
precisely the same information.
As such, the Tutte polynomial can be evaluated by lattice point
counting methods.
\begin{restatable}{thm}{lpg}
\label{lpg}
Let $M=(E,r)$ be a matroid. Then
\begin{multline*}
T_M(x,y)=(xy-x-y)^{|E|}(-x)^{r(M)}(-y)^{|E|-r(M)}\cdot\\ \sum_{u,t\geq 0} Q_M(t,u)\cdot\left(\dfrac{y-xy}{xy-x-y}\right)^t\left(\dfrac{x-xy}{xy-x-y}\right)^u
\end{multline*}
\end{restatable}
Our proof of Theorem~\ref{lpg} arrives first at the relationship between $Q'_M(x,y)$ and the Tutte polynomial.
\begin{thm}\label{thm:T to Q}
Let $M=(E,r)$ be a matroid. Then we have that
\begin{equation}
Q'_M(x,y)=\dfrac{x^{|E|-r(M)}y^{r(M)}}{x+y-1}\cdot T_M\left(\dfrac{x+y-1}{y},\dfrac{x+y-1}{x}\right)
\end{equation}
\end{thm}
An observation is in order before we embark on the proof.
Since lattice points and their enumeration are our foremost concerns in this work,
we prefer not to have to think of the points of our polyhedra with non-integral coordinates.
It is the following lemma that lets us get away with this.
\begin{lemma}[{\cite[Corollary 46.2c]{Schrijver}}]\label{712}
Let $P$ and $Q$ be generalised permutohedra whose vertices are lattice points.
Then if $x\in P+Q$ is a lattice point, there exist lattice points $p\in P$ and $q\in Q$ such that $x=p+q$.
\end{lemma}
By repeated use of the lemma, if $q\in u\Delta_{X}+{\bf e}_B+t\nablala_{Y}$ is a lattice point,
then $q$ has an expression of the form
\[
q = {\bf e}_B+{\bf e}_{i_1}+\cdots+{\bf e}_{i_t}-{\bf e}_{j_1}-\cdots-{\bf e}_{j_u}.
\]
Since all of the summands in $u\Delta_{X}+{\bf e}_B+t\nablala_{Y}$ are translates of matroid polytopes,
where the scalings are treated as repeated Minkowski sums,
this can also be proved using the matroid partition theorem, as laid out by Edmonds \cite{edmondspartition}.
\begin{proof}[Proof of Theorem~\ref{thm:T to Q}]
Let $q=e_B+{\bf e}_{x_1}+\cdots+{\bf e}_{x_i}-{\bf e}_{y_1}-\cdots -{\bf e}_{y_j}$ be a point in $P(M)+i\Delta+j\nablala$, where ${\bf e}_B\in P(M)$,
and Lemma~\ref{712} guarantees the existence of some such expression for any~$q$.
We say that the expression for $q$ has a \emph{cancellation} if ${x_k}={y_l}$ for some $k,l$. Let $k$ be the number of cancellations in the expression for $q$, allowing a summand to appear in only one cancellation. For instance, if ${x_k}={y_l}={y_m}$ is the complete set of equalities, there is one cancellation, while ${x_k}={x_n}={y_l}={y_m}$ would give two. We will partition the set of lattice points of $P(M)+i\Delta+j\nabla$ according to how many coordinates are non-negative, and then construct $Q$ by counting the lattice points in each part of the partition. Given a lattice point $q$, let $S=\{1\leq i\leq n \ | \ q_i> 0\}$. In order to construct ${\bf e}_B$ from ${\bf e}_S$ we need to use $|S|-r(S)$ of the $\Delta$ summands: as $B$ is spanning, we must have used a $\Delta$-summand every time $|S|$ rises above $r(S)$. Similarly, we must use $r(M)-r(S)$ $\nabla$-summands to account for any fall in rank. Now, we will set the remaining $\Delta$-summands equal to those coordinates already positive, that is, set them equal to indicator vectors of elements of $S$. The remaining $\nabla$-summands we will set equal to indicator vectors of elements not in $S$.
We will ensure, through choice of $B$ and $k$, that this is the largest $q$ (in terms of sum of coordinates) we can find given $i$ and $j$. There are two ways the expression can fail to be maximal in this sense:
\begin{itemize}
\itemsep0em
\item when we decrease $k$, we could construct $q$ using fewer $\Delta$ and $\nabla$ summands, and
\item if we write $q$ using $B'$ where we have summands ${\bf e}_a-{\bf e}_b$ such that $(B'\cup a)-b$ is a valid basis exchange, we would again be able to construct $q$ using fewer summands.
\end{itemize}
We will choose $B$ in the expression for $q$ and the maximal $k$ so that describing all lattice points $q$ can be done uniquely in the way described.
Now we have that $|S|$ is the number of non-negative coordinates in at least one point of $P(M)+i\Delta+j\nablala$, and all our positive summands of such a point are assigned to such coordinates. The sum of these summands must be $r(M)+i-k-|S|=i-k-\operatorname{null}(S)$. If we ensure that $|E-S|$ is the number of negative integers in the respective points of $P(M)+i\Delta+j\nablala$, summing over these sets $S$ will give a count of all lattice points. By the reasoning above, we must have that the $|E-S|$ non-negative integers sum to $j-k-r(M)+r(S)=j-k-\operatorname{cork}(S)$. Thus,
$$ \#(P(M)+i\Delta+j\nablala)=\sum_S \sum_k[ \#(|S| \ \text{non-negative integers summing to} \ i-k-\operatorname{null}(S))$$
$$\qquad\qquad\qquad\qquad\times\#(|E-S| \ \text{non-negative integers summing to} \ j-k-\operatorname{cork}(S))]$$
which is
\begin{multline}
\label{points} \#(P(M)+i\Delta+j\nablala)=\sum_S \sum_k \binom{i-k+|S|-\operatorname{null}(S)-1}{|S|-1}\\\qquad\qquad\times\binom{j-k+|E-S|-\operatorname{cork}(S)-1}{|E-S|-1}.
\end{multline}
Now form the generating function $$\sum_{i,j} \#(P(M)+i\Delta+j\nablala)v^iw^j=\sum_i \sum_j \#(P(M)+i\Delta+j\nablala)v^iw^j.$$
Substituting Equation \ref{points} into the generating function gives
\begin{align*}
\sum_i \sum_j\sum_S \sum_{k\geq 0} &\binom{i-k+|S|-\operatorname{null}(S)-1}{|S|-1}v^{i-k}\\
&\times\binom{j-k+|E-S|-\operatorname{cork}(S)-1}{|E-S|-1}w^{j-k}(vw)^k.
\end{align*}
Using the identity $\sum\limits_i\binom{i+a}{b}x^i=\dfrac{x^{b+a}}{(1-x)^{b+1}}$ simplifies this to
$$\sum_S \sum_k \dfrac{v^{\operatorname{null}(S)}}{(1-v)^{|S|}}\cdot\dfrac{w^{\operatorname{cork}(S)}}{(1-w)^{|E-S|}}\cdot (vw)^k$$
which we can write as
$$\sum_S \dfrac{v^{\operatorname{null}(S)}}{(1-v)^{\operatorname{null}(S)-\operatorname{cork}(S)+r(M)}}\cdot\dfrac{w^{\operatorname{cork}(S)}}{(1-w)^{\operatorname{cork}(S)-\operatorname{null}(S)+|E|-r(M)}}\cdot\sum_k(vw)^k.$$
Collecting like exponents, we end up with
\begin{align}
\sum_{i,j} \#(P(M)+i\Delta+j\nablala)v^iw^j &=\dfrac{1}{1-vw}\cdot\dfrac{1}{(1-v)^{r(M)}(1-w)^{|E|-r(M)}}\nonumber\\&\qquad\qquad\times\sum_S \left(\dfrac{v(1-w)}{1-v}\right)^{\operatorname{null}(S)}\left(\dfrac{w(1-v)}{1-w}\right)^{\operatorname{cork}(S)} \nonumber\\
&= \dfrac{1}{1-vw}\cdot\dfrac{1}{(1-v)^{r(M)}(1-w)^{|E|-r(M)}}\nonumber\\&\qquad\qquad\times T\left(\dfrac{w(1-v)}{1-w}+1,\dfrac{v(1-w)}{1-v}+1\right) \nonumber\\
& = \dfrac{1}{1-vw}\cdot\dfrac{1}{(1-v)^{r(M)}(1-w)^{|E|-r(M)}}\nonumber \\ &\qquad\quad\qquad\times T_M\left(\dfrac{1-vw}{1-w},\dfrac{1-vw)}{1-v}\right)
\end{align}
where $T_M$ is the Tutte polynomial of $M$. Now it remains to be shown that the left-hand side contains an evaluation of our polynomial $Q'_M$.
Using our original definition of $Q_M$, Equation \eqref{qdef}, we have that
\begin{align*}
\sum_{i,j} \#(P(M)+i\Delta+j\nablala)v^iw^j &= \sum_{i,j,k,l}c_{kl}\binom{i}{l}\binom{j}{k}v^iw^j \\
&= \sum_{k,l} c_{kl}\cdot\dfrac{v^l}{(1-v)^{l+1}}\cdot\dfrac{w^k}{(1-w)^{k+1}}.
\end{align*}
If we let $\dfrac{w}{1-w}=x-1$ and $\dfrac{v}{1-v}=y-1$, then
\begin{align*}
\sum_{i,j} \#(P(M)+i\Delta+j\nablala)v^iw^j &= \sum_{k,l} c_{kl}\cdot\dfrac{v^l}{(1-v)^{l+1}}\cdot\dfrac{w^k}{(1-w)^{k+1}} \\
&= (1-v)(1-w)\sum_{k,l}c_{kl}(x-1)^k(y-1)^l\\
&= (1-v)(1-w)Q_M'(x,y).
\end{align*}
So, from Equation (5.8.2), we have that
\begin{align*}
(1-v)(1-w)Q'_M(x,y)=\dfrac{1}{1-vw}\cdot\dfrac{1}{(1-v)^{r(M)}(1-w)^{|E|-r(M)}}\cdot T_M\left(\dfrac{1-vw}{1-w},\dfrac{1-vw}{1-v}\right).
\end{align*}
Solving for $w$ and $v$ in terms of $x$ and $y$ gives that $w=\dfrac{x-1}{x},v=\dfrac{y-1}{y}$. Substitute these into the above equation to get
\begin{equation}
Q_M'(x,y)=\dfrac{x^{|E|-r(M)}y^{r(M)}}{x+y-1}\cdot T_M\left(\dfrac{x+y-1}{y},\dfrac{x+y-1}{x}\right).
\end{equation}
\end{proof}
We can invert this formula by setting $x'=\dfrac{x+y-1}{y},y'=\dfrac{x+y-1}{x}$, rearranging, and then relabelling.
\begin{thm}\label{thm:Q to T}
Let $M=(E,r)$ be a matroid. Then
\begin{equation}
T_M(x,y)=-\,\dfrac{(xy-x-y)^{|E|-1}}{(-y)^{r(M)-1}(-x)^{|E|-r(M)-1}}\cdot Q_M'(\dfrac{-x}{xy-x-y},\dfrac{-y}{xy-x-y})
\end{equation}
\end{thm}
We conjecture that there is a relationship between
our formula for the Tutte polynomial and
the algebro-geometric formula for the Tutte polynomial in \cite{speyer}.
The computations on the Grassmannian in that work are done
in terms of~$P(M)$, the moment polytope of a certain torus orbit closure,
and $\Delta$ and $\nablala$ are the moment polytopes of the
two dual copies of~$\mathbb P^{n-1}$, the $K$-theory ring of whose product
$\mathbb Z[x,y]/(x^n,y^n)$ is identified with the ambient ring of the Tutte polynomial.
\begin{ex}\label{ex:1}
Let $M$ be the matroid on ground set $[3]=\{1,2,3\}$ with
$\mathcal B_M = \{\{1\},\{2\}\}$.
When $u=2$ and $t=1$, the sum $P(M)+u\Delta+t\nablala $ is the polytope of Figure~\ref{fig:1}, with 16 lattice points.
\begin{figure}
\caption{The polytope $P(M)+u\Delta+t\nablala $ of Example~\ref{ex:1}
\label{fig:1}
\end{figure}
To compute $Q_M(x,y)$, it is enough to count the lattice points in
$P(M)+u\Delta+t\nablala $ for a range of $u$ and $t$, and interpolate.
Since $Q_M$ is a polynomial of degree~2, it is sufficient to take
$t$ and $u$ nonnegative integers with sum at most~2.
These are the black entries in the table below:
\begin{center}
\begin{tabular}{l|ccc}
$t$ $\setminus$ $u$ & 0 & 1 & 2 \\\hline
0 & 2 & 5 & 9 \\
1 & 5 & 10 & \textcolor{Gray}{16} \\
2 & 9 & \textcolor{Gray}{16} & \textcolor{Gray}{24}
\end{tabular}
\end{center}
The unique degree~$\leq2$ polynomial with these evaluations is
\[Q_M(t,u) = \binom t2 + 2tu + \binom u2 + 3t + 3u + 2,\]
so
\begin{align*}
Q'_M(x,y) &= (x-1)^2 + 2(x-1)(y-1) + (y-1)^2 + 3(x-1) + 3(y-1) + 2
\\&= x^2 + 2xy + y^2 - x - y.
\end{align*}
Finally, by Theorem~\ref{thm:Q to T},
\begin{align*}
T_M(x,y) &= -\,\frac{(xy-x-y)^2}{(-y)^0(-x)^1}\cdot
\left(\frac{y^2 + 2xy + x^2}{(xy-x-y)^2}+\frac{y + x}{xy-x-y}\right)
\\&= xy + y^2
\end{align*}
which is indeed the Tutte polynomial of~$M$.
\end{ex}
Given Theorem \ref{thm:Q to T}, we can now prove Theorem \ref{lpg}.
\lpg*
\begin{proof}
Consider the power series $\Sigma:=\sum\limits_{u,t\geq 0}Q_M(t,u)\,a^tb^u$. Note that
$$\sum_{u,t\geq 0}\binom{t}{i}\binom{u}{j}\,a^tb^u=\frac{1}{ab}\cdot\left(\dfrac{a}{1-a}\right)^{i+1}\left(\dfrac{b}{1-b}\right)^{j+1}.$$
We can thus write $\Sigma$ as
$$\frac{1}{ab}\sum_{i,j}c_{ij}\left(\dfrac{a}{1-a}\right)^{i+1}\left(\dfrac{b}{1-b}\right)^{j+1}.$$
Substituting $a=(v-1)/v$ and $b=(w-1)/w$ turns this into
\begin{align*}
\Sigma &= \dfrac{vw}{(v-1)(w-1)}\sum_{i,j}c_{ij}(v-1)^{i+1}(w-1)^{j+1}
\\ &= vw \sum_{i,j}c_{ij}(v-1)^i(w-1)^j
\\ &= vw\, Q_M'(v,w).
\end{align*}
We can now apply Theorem \ref{thm:T to Q}:
\begin{align*}
&\mathrel{\phantom=}\sum_{u,t\geq 0} Q_M(t,u)\left(\dfrac{v-1}{v}\right)^t\left(\dfrac{w-1}{w}\right)^u \\
&= vw\, Q'_M(v,w)\\
&= \dfrac{v^{|E|-r(M)+1}w^{r(M)+1}}{v+w-1}\cdot T_M\left(\dfrac{v+w-1}{w},\dfrac{v+w-1}{v}\right).\\
\end{align*}
Substitute $v=-x/(xy-x-y)$ and $w=-y/(xy-x-y)$ to get the stated result.
\end{proof}
A further substitution and simple rearrangement gives the following corollary, included for the sake of completeness.
\begin{cor}
Let $M=(E,r)$ be a matroid. Then
\[\sum_{u,t\geq 0}Q_M(t,u)v^tw^u=\dfrac{1}{(1-v)^{|E|-r(M)}(1-w)^{r(M)}(1-vw)}\cdot T_M\left(\dfrac{1-vw}{1-v},\dfrac{1-vw}{1-w}\right).\]
\end{cor}
Being a Tutte evaluation, $Q'$ must have a deletion-contraction recurrence.
We record the form it takes.
\begin{prop}
\label{recurrence}
Let $M=(E,r)$ be a matroid with $|E|=n$. Then, for $e\in E$,
\begin{enumerate}[i.]
\item $Q'_M(x,y)=xQ_{M\setminusckslash e}(x,y)+yQ'_{M/e}(x,y)$ when $e$ is not a loop or coloop, and
\item $Q'_M(x,y)=(x+y-1)Q'_{M/e}(x,y)=(x+y-1)Q'_{M\setminusckslash e}(x,y)$ otherwise.
\end{enumerate}
\end{prop}
\begin{proof}Part \emph{ii} is a consequence of Proposition~\ref{prop:direct sum} below (which does not depend on the present section).
When $e$ is a (co)loop, $M = M_e\oplus M\setminusckslash e = M_e\oplus M/e$,
where $M_e$ is the restriction of $M$ to~$\{e\}$ (or the equivalent contraction).
For part \emph{i}, recall that if $e$ is neither a loop nor a coloop, then $E(M\setminusckslash e)=E-e=E(M/e)$, $r(M\setminusckslash e)=r(M)$, and $r(M/e)=r(M)-1$. Take the equation $T_M(x,y)=T_{M\setminusckslash e}(x,y)+T_{M/e}(x,y)$ and rewrite it in terms of $Q'$, as per Theorem~\ref{thm:Q to T}:
\begin{multline*}
-\dfrac{(xy-x-y)^{n-1}}{(-y)^{r(M)-1}(-x)^{n-r(M)-1}}\cdot Q'_M(x,y)= \\
-\dfrac{(xy-x-y)^{n-2}}{(-y)^{r(M)-1}(-x)^{n-r(M)-2}}\cdot Q'_{M\setminusckslash e}(x,y) -\dfrac{(xy-x-y)^{n-2}}{(-y)^{r(M)-2}(-x)^{n-r(M)-1}}\cdot Q'_{M/e}(x,y)
\end{multline*}
Multiplying through by $-\dfrac{(-y)^{r(M)-1}(-x)^{n-r(M)-1}}{(xy-x-y)^{n-1}}$ gives the result.
\end{proof}
\section{Coefficients}
\label{sec:coeffs}
Some coefficients of the Tutte polynomial provide structural information about the matroid in question. Let $b_{i,j}$ be the coefficient of $x^iy^j$ in $T_M(x,y)$. The best-known case is that $M$ is connected only if $b_{1,0}$, known as the \emph{beta invariant}, is non-zero; moreover, $b_{1,0} = b_{0,1}$ when $|E|\geq 2$.
Not every coefficient yields such an appealing result, though of course
they do count the bases with internal and external activity of fixed sizes.
In like manner,
we are able to provide a enumerative interpretation of the coefficients of $Q'_M(x,y)$, which is the focus of this section.
In order to do this, we will make use of a regular mixed subdivision $\mathcal{F}$ of $u\Delta +P(M)+t\nablala$.
Let $\alpha_1<\cdots <\alpha_n$ and $\beta_1<\cdots <\beta_n$ be positive reals.
Our regular subdivision will be that determined by
projecting the ``lifted'' polytope
\[\mathit{Lift}=\operatorname{Conv}\{(u{\bf e}_i,\alpha_i)\}+(P(M)\times \{0\})+\operatorname{Conv}\{(-t{\bf e}_i,\beta_i)\}\subseteq\mathbb R^E\times\mathbb R\]
to $\mathbb R^E$.
Let $\mathfrak{F}$ be the set of ``lower'' facets of~$\mathit{Lift}$ which maximise some linear function $\langle a,x\rangle$, where $a\in(\mathbb R^E\times\mathbb R)^*$ is a linear functional with last coordinate $a_{n+1}=-1$.
For each face $F\in\mathfrak{F}$, let $\pi(F)$ be its projection back to $\mathbb{R}^n$.
Now $\mathcal{F}:=\{\pi(F) \ | \ F\in\mathfrak{F}\}$ is a regular subdivision of $u\Delta+P(M)+t\nablala$.
We will write $\mathcal F$ as $\mathcal F(t,u)$ when we need to make the dependence on the parameters explicit.
Note however that the structure of the face poset of~$\mathcal{F}$
does not depend on $t$ and $u$ as long as these are positive.
Since $\mathcal{F}$ is a mixed subdivision of $u\Delta +P(M)+t\nablala$,
each of its cells bears a canonical decomposition as a Minkowski sum of
a face of $u\Delta$, a face of $P(M)$, and a face of $t\nablala$.
When we name a face of $\mathcal F$ as a sum of three polytopes $F+G+H$,
we mean to invoke this canonical decomposition.
These decompositions are compatible between faces:
if $m_a(P)$ denotes the face of a polytope~$P$ on which a linear functional $a$ is maximised,
then the canonical decomposition for $m_a(F+G+H)$ is $m_a(F)+m_a(G)+m_a(H)$.
We now state the main result of this section:
\begin{thm}
\label{coeffs}
Take the regular mixed subdivision $\mathcal F$ of $u\Delta+P(M)+t\nablala$ as described above.
The unsigned coefficient $|[x^iy^j]Q'_M|$ counts the cells $F+G+H$ of~$\mathcal F$ where $i=dim(F)$, $j=dim(H)$,
and $G$ is a vertex of $P(M)$ and there exists no cell $F+G'+H$ where $G'\supsetneq G$.
\end{thm}
The key fact in the proof is the following.
\begin{dfn}
A maximal cell $F+G+H$ of the mixed subdivision $\mathcal F$ is a \emph{top degree face}
when $G$ is a vertex of $P(M)$.
\end{dfn}
\begin{prop}\label{prop:top degree}
In the subdivision $\mathcal{F}$,
each of the lattice points of $u\Delta+P(M)+t\nablala$ lies in a top degree face.
\end{prop}
To expose the combinatorial content of this proposition, we need to describe
the top degree faces more carefully. All top degree faces are of dimension $|E|-1$
and have the form $u\Delta_{X}+{\bf e}_B+t\nablala_{Y}$.
By Lemma~\ref{712}, if $p\in u\Delta_{X}+{\bf e}_B+t\nablala_{Y}$ is a lattice point,
then $p$ has an expression of the form
\begin{equation}\label{eq:tdf sum}
p = {\bf e}_B+{\bf e}_{i_1}+\cdots+{\bf e}_{i_t}-{\bf e}_{j_1}-\cdots-{\bf e}_{j_u}.
\end{equation}
The subdivision $\mathcal F$ determines a height function $h(x)$
on the lattice points $x$ of~$u\Delta +P(M)+t\nablala$,
where $h(x)$ is the minimum real number such that $(x,h(x))\in\mathit{Lift}$.
This height function is
\[h(x):=\text{min}\{\alpha_{i_1}+\cdots+\alpha_{i_t}+\beta_{j_1}+\cdots+\beta_{j_u}
\ | \ x-{\bf e}_{i_1}-\cdots-{\bf e}_{i_t}+{\bf e}_{j_1}+\cdots+{\bf e}_{j_u}\in \mathcal B_M\}.\]
If $x$ is a lattice point of a top-degree face then choosing the $i_k$ and $j_l$ in accord with \eqref{eq:tdf sum} achieves the minimum.
Let $\Pi=u\Delta_{X}+{\bf e}_B+t\nablala_{Y}$ be a top-degree face.
If $i$ and $j$ were distinct elements of $X\cap Y$,
then $\Pi$ would have edges of the form $\operatorname{Conv}\{x,x+k({\bf e}_i-{\bf e}_j)\}$ whose preimages in the corresponding lower face of $\mathit{Lift}$ were not edges,
since they would contain the sum of the nonparallel segments
$\operatorname{Conv}\{(u{\bf e}_i,\alpha_i),(u{\bf e}_j,\alpha_j)\}$ and $\operatorname{Conv}\{(-t{\bf e}_i,\beta_i),(-t{\bf e}_j,\beta_j)\}$.
Therefore we must have $|X\cap Y|\leq1$.
Together with the fact that the dimensions of $\Delta_{X}$ and~$\nablala_{Y}$ sum to $|E|-1$,
this implies that $X\cup Y=E$ and $|X\cap Y|=1$. In fact
the conditions on the $\alpha$ and $\beta$ imply that $X\cap Y=\{1\}$,
because replacing equal subscripts $i_k=j_l>1$ by~$1$ in the definition of $h(x)$ decreases the right hand side.
We thus potentially have $2^{|E|-1}$ top degree faces, one for each remaining valid choice of $X$ and $Y$ -- each element except $1$ is either in $X$ but not $Y$, or it is in $Y$ but not $X$.
In fact, all $2^{|E|-1}$ of these do appear in~$\mathcal{F}$.
\begin{lemma}\label{lem:B}
Take subsets $X$ and $Y$ of~$E$ with $X\cup Y=E$ and $X\cap Y=\{1\}$.
There is a unique basis $B$ such that
$u\Delta_{X}+{\bf e}_B+t\nablala_{Y}$ is a top-degree face.
It is the unique basis $B$ such that
no elements of $X$ are externally inactive and no elements of $Y$ are internally inactive with respect to $B$, with reversed order on $E$.
\end{lemma}
The basis $B$ can be found using the simplex algorithm for linear programming
on $P(M)$, applied to a linear functional constructed from the $\alpha$ and $\beta$
encoding the activity conditions.
This procedure can be completely combinatorialised, giving a way to start from a randomly chosen initial basis and make a sequence of exchanges which yields a unique output $B$ regardless of the input choice.
The proof is as follows.
\begin{proof}
Choose any basis, $B_0$, and order the elements $b_1,\ldots, b_r$ lexicographically. Perform the following algorithm to find the basis $B$.
The algorithm makes a sequence of replacements of the elements of~$B$, of two kinds, until it is unable to make any more.
\begin{algorithm}\label{alg:1}\mbox{}
\begin{enumerate}[(1)]
\item[\textbf{Input:}] a basis $B_0$ of~$M$.
\item[\textbf{Output:}] the basis $B$ of~$M$ called for in the lemma.
\item Let $i=0$.
\item Attempt to produce new bases $B_1,B_2,\ldots$ as follows.
Let the elements of $B_i$ be $b_1,\ldots,b_r$, where $b_1<\cdots<b_r$. For each $j=1,\ldots,r$:
\begin{enumerate}[(a)]
\item If there exists $x\in X$ greater than $b_j$ such that $B_i\setminus\{b_j\}\cup\{x\}$ is a basis of~$M$,
then choose the maximal such $x$, let $B_{i+1} = B_i\setminus\{b_j\}\cup\{x\}$, increment $i$, and repeat step~(2).
Call this a check of type (a).
\item If not, and $b_j\in Y$, and there exists $z$ less than $b_j$ such that $B_i\setminus\{b_j\}\cup\{z\}$ is a basis of~$M$,
then choose the minimal such $z$, let $B_{i+1} = B_i\setminus\{b_j\}\cup\{z\}$, increment $i$, and repeat step~(2).
Call this a check of type (b).
\end{enumerate}
\item Terminate and return $B=B_i$.
\end{enumerate}
\end{algorithm}
The remainder of the proof analyses this algorithm.
Let $\gamma_1,\ldots,\gamma_n\in\mathbb{R}$ be such that $0=|\gamma_1|\ll \cdots \ll |\gamma_n|$, and $\gamma_a>0$ if $a\in X$ while $\gamma_a<0$ if $a\in Y$.
\begin{sublemma}
\label{claim3}
Let $B_i$ and $B_{i+1}$ be two bases of $M$ found consecutively by the algorithm. Then $\sum\limits_{a\in B_i}\gamma_a < \sum\limits_{a\in B_{i+1}}\gamma_a $ for all $i$. That is, the sum $\sum\limits_{a\in B}\gamma_a $ is increasing with the algorithm.
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{claim3}]
Moves of type (a) replace an element $b$ of a basis with a larger element $c$ in $X$, so regardless of whether $b$ was in $X$ or $Y$, this must increase the sum as $\gamma_b>0$. Moves of type (b) replace an element $y\in Y$ in the basis with a smaller element $d$. If $d\in Y$, we are replacing $\gamma_y$ with a smaller negative, as $|\gamma_d|\ll|\gamma_y|$. If $d\in X$, we are replacing a negative $\gamma_y$ with a positive $\gamma_d$. So $\sum\limits_{a\in B}\gamma_a$ is increasing in every case.
\end{subproof}
We will write the symmetric difference of two sets $A$ and $B$ as $A\operatorname{\triangle} B$. The next result follows as a corollary of the previous claim.
\begin{sublemma}
\label{delta}
$B\operatorname{\triangle} Y$ written with the largest elements first is lexicographically increasing with the algorithm.
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{delta}]
A move of type (a) replaces an element of $B$ with a larger element in $X$. This puts a larger element into $B\operatorname{\triangle} Y$ than was in $B$ originally, and so must cause a lexicographic increase. A move of type (b) removes a $Y$ element in $B$ (and so, an element not in $B\operatorname{\triangle} Y$), and puts a smaller element $z$ into $B$. Removing the $Y$ element from $B$ adds it to $B\operatorname{\triangle} Y$, and adding an element to a set cannot decrease the lexicographic order. If the smaller element $z$ is in $Y$, then this move removes $z$ from $B\operatorname{\triangle} Y$. As we have replaced it with a larger element, the lexicographic order of $B\operatorname{\triangle} Y$ is increased. If $z$ is in $X$, the move adds $z$ to $B\operatorname{\triangle} Y$, increasing the lexicographic order of $B\operatorname{\triangle} Y$.
\end{subproof}
\begin{sublemma}
\label{claim4}
The algorithm described above terminates and gives an output independent of $B_0$.
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{claim4}]
Order all bases of the matroid based on the increasing lexicographic order of $B\operatorname{\triangle} Y$. As we chose the elements $\Deltata_a$ to be much greater than the previous element, only the largest element of $B\operatorname{\triangle} Y$ determines the total ordering. We have shown in the previous corollary that this sequence is increasing with the algorithm. As there is a finite number of bases, there must be a greatest element, and thus the algorithm terminates.
We now need to show that there is a unique basis for which the algorithm can terminate.
The structure $(E,\{B_i\operatorname{\triangle} Y \ | \ B_i\in\mathcal{B}_M\})$ is what is known as a \emph{delta-matroid}, a generalisation of a matroid allowing bases to have different sizes. This delta-matroid is a \emph{twist} of $M$ by the set $Y$ \cite{bouchet}. The subsets $B_i\operatorname{\triangle} Y$ are called the feasible sets. It is a result of Bouchet (\cite{bouchet}) that feasible sets of largest size form the bases of a matroid.
In order to show uniqueness of termination bases, we will first show that if $B$ is a termination basis and $B\operatorname{\triangle} Y$ is not of largest size, then any basis $B'$ with $|B'\operatorname{\triangle} Y|>|B\operatorname{\triangle} Y|$ is terminal. Suppose this is not the case. As $|B\operatorname{\triangle} Y|=|B|+|Y|-2|B\cap Y|$, this requires that $|B'\cap Y|<|B\cap Y|$.
As $B'$ is not a termination basis, there is either an element $b\in B'$ such that $(B'-b)\cup c\in\mathcal{B}$ for some element $c\in X$, where $c>b$, or there is an element $b\in Y\cap B$ such that $(B'-b)\cup a\in\mathcal{B}$ for some $a<b$. If we have $b,c\in X$, $|((B'-b)\cup c)\cap Y|=|B'\cap Y|$. If $b\in Y$, $|((B'-b)\cup c)\cap Y|<|B'\cap Y|$. If $a,b\in Y$, then $|((B'-b)\cup a)\cap Y|=|B'\cap Y|$. Finally, if $b\in Y$ and $a\in X$, then $|((B'-b)\cup a)\cap Y|<|B'\cap Y|$. In every case we have a contradiction.
As the algorithm terminates, we know that after a finite number of such exchanges, we produce $B$ from $B'$. Let the bases constructed in each step form a chain $$B',B_1,B_2,\ldots,B_n, B.$$ From above, we have that $|B'\cap Y|\geq |B_1\cap Y|\geq\cdots\geq |B_n\cap Y|\geq |B\cap Y|$. This contradicts the initial assumption that $|B'\cap Y|<|B\cap Y|$.
Now assume the algorithm can terminate with two bases $B_1,B_2$. Take $B_1\operatorname{\triangle} Y$ and $B_2\Delta Y$, and choose the earliest element $b\in B_1\operatorname{\triangle} Y-B_2\operatorname{\triangle} Y$ (assuming this comes lexicographically first in $B_1\operatorname{\triangle} Y$). If $b\in X$, then $b\in B_1-B_2$. If $b\in Y$, then $b\in B_2-B_1$. Similarly, if $c\in B_2\operatorname{\triangle} Y-B_1\operatorname{\triangle} Y$, if $c\in X$ then $c\in B_2-B_1$, or if $c\in Y$ then $c\in B_1-B_2$.
Apply the delta-matroid exchange algorithm to $B_1\operatorname{\triangle} Y$ and $B_2\operatorname{\triangle} Y$ to get that $(B_1\operatorname{\triangle} Y)\operatorname{\triangle} \{b,c\}$ is a feasible set, for some element $c\in (B_1\operatorname{\triangle} Y)\operatorname{\triangle} (B_2\operatorname{\triangle} Y)$. Given we have a twist of a matroid, we must have that $(B_1\operatorname{\triangle} Y)\operatorname{\triangle} \{b,c\}=B_3\operatorname{\triangle} Y$ for some basis $B_3$, and so $|(B_1\operatorname{\triangle} \{b,c\}|=|B_3|=|B_1|$ as $\operatorname{\triangle} $ is associative. This means we must have that exactly one of $\{b,c\}$ is in $B_1$. If $b\in X$, $(B_1\operatorname{\triangle} Y-b)\cup c=((B_1-b)\cup c)\operatorname{\triangle} Y$, so $(B_1-b)\cup c\in\mathcal{B}$ and we must have $c\in X$ by the above paragraph. As $b$ was the earliest element different in either basis, we must have $c>b$, and so $B_1$ was not a termination basis of the original algorithm. If $b\in Y$, $(B_1\operatorname{\triangle} Y-b)\cup c=B_1\operatorname{\triangle} ((Y-b)\cup c)$. But we cannot change $Y$, so must have $[(B_1-c)\cup b]\operatorname{\triangle} Y$ and $c\in Y$. This means that again $B_1$ was not a termination basis, as we are replacing an element of $Y$ with a smaller one. This completes the proof of Claim~\ref{claim4}.
\end{subproof}
This, in turn, completes the proof of Lemma \ref{lem:B}.
\end{proof}
Before we can get to the proof of Theorem \ref{coeffs}, we first need two results on how these top degree cells interact.
Note that in the service of readability we write $1$ instead of $\{1\}$ in subscripts.
When we say that a polytope contains a basis $B$, we mean that it contains the indicator vector ${\bf e}_B$.
\begin{lemma}
\label{meet}
Take two distinct partitions $(X_1,Y_1)$, $(X_2,Y_2)$ of $[n]\setminus\{1\}$.
Let $B_1$, $B_2$ be the bases found by Algorithm~\ref{alg:1} such that we have top degree cells $T_i={\bf e}_{B_i}+\Delta_{1\cup X_i}+\nablala_{1\cup Y_i}$, $i\in\{1,2\}$. Suppose that $T_1\cap T_2\neq\emptyset$. Then $B_1=B_2$.
\end{lemma}
\begin{proof}
As we have noted, the combinatorial type of the subdivision $\mathcal F(t,u)$ is independent of the values of $t$ and~$u$, as long as these are positive.
Also, if $t\leq t'$ and $u\leq u'$,
then each cell of $\mathcal F(t,u)$ is a subset of the corresponding cell of $\mathcal F(t',u')$,
up to translation of the latter by $(t'-t-u'+u){\bf e}_1$.
Thus if the top-degree cells indexed by $(X_1,Y_1)$ and $(X_2,Y_2)$ intersect, they will continue to intersect if $t$ or $u$ are increased.
So we may assume that none of $t$, $u$, $t-u$ lie in $\{-1,0,1\}$, by increasing $t$ and $u$ as necessary.
Because $\mathcal F$ is a cell complex, $T_1\cap T_2$ is a face of~$\mathcal F$, and it therefore contains a vertex $p$ of~$\mathcal F$.
For each $i=1,2$, the point $p$ is the sum of ${\bf e}_{B_i}$, a vertex of $u\Delta$, and a vertex of $t\nablala$.
Because every subset of the list $1,u,-t$ has a different sum,
$p$ can be decomposed as a zero-one vector plus a vertex of $u\Delta$ plus a vertex of $t\nablala$ in only one way,
and it follows that ${\bf e}_{B_1} = {\bf e}_{B_2}$.
\end{proof}
\begin{lemma}
\label{lattice}
Take two distinct partitions $P_1=(X_1,Y_1)$, $P_2=(X_2,Y_2)$ of $[n]\setminus\{1\}$ such that their corresponding top degree cells $T_1$ and $T_2$ contain a common point $p$. Now let $P_3=(X_3,Y_3)$ be a partition of $[n]\setminus\{1\}$ such that
$X_1\cap X_2\subseteq X_3$ and $Y_1\cap Y_2\subseteq Y_3$. Then $p$ is in the top degree cell $T_3$ indexed by~$P_3$,
and Algorithm~\ref{alg:1} finds the same basis $B^*$ for each of $P_1$, $P_2$ and $P_3$.
\end{lemma}
\begin{proof}
By Lemma \ref{meet}, we have $T_1={\bf e}_{B^*}+\Delta_{1\cup X_1}+\nablala_{1\cup Y_1}$ and $T_2={\bf e}_{B^*}+\Delta_{1\cup X_2}+\nablala_{1\cup Y_2}$ where the basis $B^*$ found by Algorithm~\ref{alg:1} is common to both expressions.
The lexicographically greatest set of form $B\operatorname{\triangle} Y_1$ is that with $B=B^*$, likewise for $B\operatorname{\triangle} Y_2$.
Let $B'\operatorname{\triangle} Y_3$ be the lexicographically greatest set of form $B\operatorname{\triangle} Y_3$. Our objective is to show that $B' = B^*$.
Assume otherwise for a contradiction, and let $e$ be the largest element in $B'\operatorname{\triangle} B^* = (B'\operatorname{\triangle} Y_3)\operatorname{\triangle}(B^*\operatorname{\triangle} Y_3)$.
By choice of~$B'$, we have $e\in B'\operatorname{\triangle} Y_3$ and $e\not\in B^*\operatorname{\triangle} Y_3$.
The latter implies that $e\not\in B^*\operatorname{\triangle} Y_i$ for at least one of $i=1,2$; without loss of generality, say $e\not\in B^*\operatorname{\triangle} Y_1$.
Then $B'\operatorname{\triangle} Y_1$ is lexicographically earlier than $B^*\operatorname{\triangle} Y_1$, because the former but not the latter contains $e$ and they agree in which elements greater than $e$ they contain.
This is the desired contradiction.
We conclude that $T_k = {\bf e}_{B^*}+\Delta_{1\cup X_k}+\nabla_{1\cup Y_k}$ for each $k=1,2,3$.
Expanding $p-{\bf e}_{B^*}$ in the basis ${\bf e}_2-{\bf e}_1, \ldots, {\bf e}_n-{\bf e}_1$ of the affine span of the $T_k$,
we see that the coefficient of ${\bf e}_i-{\bf e}_1$ is nonnegative if $i\in 1\cup X_k$ and nonpositive if $i\in 1\cup Y_k$, for $k=1,2$.
Therefore this coefficient is zero unless $i\in X_1\cap X_2$ or $i\in Y_1\cap Y_2$, and this implies $p\in T_3$.
\end{proof}
The following result is an immediate corollary of Lemma \ref{lattice}:
\begin{cor}
Define $T_Y=u\Delta_X+{\bf e}_B+t\nabla_Y$. For every face $F$ of the mixed subdivision, if $F$ is contained in any top degree face, then the set of $Y$ such that $F$ is contained in $T_Y$ is an interval in the boolean lattice.
\end{cor}
We now have all the ingredients we need to prove the main result of this section, restated here:
\begin{thmcoeffs}
Take the regular mixed subdivision $\mathcal F$ of $u\Delta+P(M)+t\nablala$ as described above.
The unsigned coefficient $|[x^iy^j]Q'_M|$ counts the cells $F+G+H$ of~$\mathcal F$ where $i=dim(F)$, $j=dim(H)$,
and $G$ is a vertex of $P(M)$ and there exists no cell $F+G'+H$ where $G'\supsetneq G$.
\end{thmcoeffs}
\begin{proof}
First, we must show that all the lattice points of $P(M)+u\Delta+t\nablala$ lie in a top degree face.
Recall that $\pi$ is the projection map from $\mathbb{R}^{n+1}\rightarrow\mathbb{R}^n$.
\begin{sublemma}
\label{claim5}
Any $\operatorname{\pi}(x)\in (t\nablala+P(M)+u\Delta)\cap\mathbb{Z}^n$ on $\mathfrak{F}$ is of the form $(-{\bf e}_{i_1},\beta_1)+\cdots +(-{\bf e}_{i_t},\beta_t)+({\bf e}_B,0)+({\bf e}_{j_1},\alpha_1)+\cdots +({\bf e}_{j_u},\alpha_u)$.
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{claim5}]
Lemma~\ref{712} provides an expression for $x$ of the form
\[x=-{\bf e}_{i_1}-\cdots -{\bf e}_{i_t}+{\bf e}_B+{\bf e}_{j_1}+\cdots +{\bf e}_{j_u}.\]
Taking arbitrary preimages under $\pi$ gives the requisite expression except that the sum may be incorrect in the last coordinate.
To obtain an expression where the last coordinate is correct, we will now rewrite this to show that $x$ in fact lies on $\mathfrak {F}$: this is equivalent to showing that there exists a partition $X\sqcup Y=[n]\setminus 1$ such that every $i$ is in $1\cup Y$, every $j$ is in $1\cup X$, and the algorithm of Theorem \ref{lem:B}, given $X$ and $Y$, yields $B$. This is because, as we know the height function used to lift the top-degree faces, finding this $X$ and $Y$ will give a top-degree face containing $\operatorname{\pi}(x)$, and we would then have the correct last coordinate.
The postconditions of Algorithm~\ref{alg:1} require that
\begin{enumerate}
\item if there exists an element $d\notin B$ such that $d<e\in B$ and $(B-e)\cup d\in\mathcal{B}$ then $e\in X$, and
\item if there exists an element $e\in B$ such that $e<f\notin B$ and $(B-e)\cup f\in\mathcal{B}$ then $f\in Y$.
\end{enumerate}
Choose any $X$ and any $Y$. Given these, we will construct $X'$ and $Y'$ such that $X'\sqcup Y'=[n]\setminus 1$.
\begin{itemize}
\item Suppose $e\in B$ and $e=i_k$, and there exists $d<e$ with $d\in B$ such that $(B-e)\cup d\in\mathcal{B}$. In the expression for $x$, replace $-{\bf e}_e+{\bf e}_B$ with $-{\bf e}_d+{\bf e}_{(B-e)\cup d}$. Add $d$ to $Y'$.
\item Suppose $f\in B$ and $f=j_l$, and there exists $e<f$ with $e\in B$ such that $(B-e)\cup f\in\mathcal{B}$. In the expression for $x$, replace ${\bf e}_B+{\bf e}_f$ with ${\bf e}_{(B-e)\cup f}+{\bf e}_e$. Add $e$ to $X'$.
\item If we have an element $i\in X\cap Y$, in the expression for $x$ replace $-{\bf e}_i+{\bf e}_i$ with $-{\bf e}_1+{\bf e}_1$. Remove $i$ from both $X'$ and $Y'$.
\end{itemize}
The above three operations always replace a term $\pm {\bf e}_a$ with a smaller term. As we have a finite ground set, there is a finite amount of such operations, and so this construction must terminate with a $X',Y'$ which fits the criteria. At this point, the expression we have for $x$ will be that required by the claim.
\end{subproof}
Continuing the proof of the theorem, we now form a poset $P$ where the elements are the top degree faces and all nonempty intersections of sets of these, ordered by containment.
This poset is a subposet of the face lattice of the $(|E|-1)$-dimensional cube
whose vertices correspond to the top degree faces.
Proposition~\ref{prop:top degree} shows that every lattice point of
$u\Delta+P(M)+t\nablala$ lies in at least one face in~$P$.
The total number of lattice points is given by inclusion-exclusion on
the function on~$P$ assigning to each element of~$P$ the number of lattice points
in that face. Let $[\cdot]$ denote the number of lattice points of the corresponding face. So we have that
\begin{equation}
\label{eqn}
Q_M(t,u)=\sum_{i,j}c_{ij}\binom{u}{j}\binom{t}{i}=\sum_{k\geq 1}(-1)^k\sum_{\substack{S\subseteq \textrm{atoms}(P)\\ |S|=k}}\left[\bigwedge S\right]
\end{equation}
$$\qquad\qquad=\sum_{x\in P}\mu (0,x)[x]$$
where $\mu$ is the M\"{o}bius function. Now, as the face poset of the cubical complex $\mathcal{C}$ is Eulerian, we have that $\mu (x,y)=(-1)^{r(y)-r(x)}$. This means that
\begin{equation}
\label{eqn2}
\sum_{k\geq 1}(-1)^k\sum_{\substack{S\subseteq \textrm{atoms}(P)\\ |S|=k}}\left[\bigwedge S\right]=\sum_{E \ \textrm{face \ of} \ \mathcal{C}} (-1)^{\textrm{codim} E}\binom{t+i}{i}\binom{u+j}{j}
\end{equation}
where $E$ is the product of an $i$-dimensional face of $\Delta$ with a $j$-dimensional face of $\nabla$, that is, $E$ corresponds to a face of type $tF+G+uH$, where $G$ is a basis of $P(M)$.
$Q'_M$ expands as
$$Q'_M(x,y)=\sum_{i,j}c_{ij}(x-1)^i(y-1)^j=\sum_{i,j,k,l}c_{ij}\binom{i}{k}x^k(-1)^{i-k}\binom{j}{l}y^l(-1)^{j-l}$$
in which the coefficient of $x^ky^l$ is $\sum_{i,j}c_{ij}\binom{i}{k}(-1)^{i-k}\binom{j}{l}(-1)^{j-l}$. To compare this to the count in the lattice, we need to expand $\binom{t}{i}$ (and $\binom{u}{j}$) in the basis of $\binom{t+i}{i}$ (and $\binom{u+j}{j}$). This gives that $$\binom{t}{i}=\sum_{k=0}^i (-1)^{i-k}\binom{i}{k}\binom{t+k}{k},$$ as proven below.
\begin{sublemma}
\label{claim6}
For any positive integers $i,t$,
$$\binom{t}{i}=\sum_{k=0}^i (-1)^{i-k}\binom{i}{k}\binom{t+k}{k}.$$
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{claim6}]
The Vandermonde identity gives that $$\binom{t+i}{i}=\sum_{k=0}^i \binom{t}{k}\binom{i}{i-k}=\sum_{k=0}^i\binom{t}{k}\binom{i}{k}.$$ We will use the binomial inversion theorem to get $\binom{t}{i}$. Rewriting the above identity in this language, we have that $$f_i=\sum_{k=0}^ig_k\binom{i}{k},$$ where $f_i=\binom{t+i}{i}$ and $g_k=\binom{t}{k}$. Then, $$g_i=\sum_{k=0}^i(-1)^{i+k}f_k\binom{i}{k}.$$ Substituting in values again gives that $$\binom{t}{i}=\sum_{k=0}^i(-1)^{i+k}\binom{t+k}{k}\binom{i}{k},$$ as in the statement of the claim.
\end{subproof}
Substitute this into Equation \ref{eqn} to get
$$\sum_{k\geq 1}(-1)^k\sum_{\substack{S\subseteq \textrm{atoms}(P)\\ |S|=k}}\left[\bigwedge S\right]=\sum_{i,j,k,l}c_{ij}(-1)^{i-k}\binom{i}{k}\binom{t+k}{k}(-1)^{j-l}\binom{j}{l}\binom{u+l}{l}$$
$$\qquad\qquad = \sum_{k,l}[x^ky^l]Q'_M(x,y)\binom{t+k}{k}\binom{u+l}{l}.$$
Comparing this to Equation \ref{eqn2} proves Theorem \ref{coeffs}.
\end{proof}
The above proof immediately yields as a corollary that the signs of the coefficients of $Q'_M(x,y)$ are alternating.
\begin{cor}\label{cor:alternating}
$(-1)^{|E|-1}Q'_M(-x,-y)$ has nonnegative coefficients in $x$ and~$y$.
\end{cor}
This is not dissimilar to the Tutte polynomial, whose coefficients are all nonnegative.
The coefficients of $Q'_M$, up to sign, have the combinatorial interpretation
of counting elements of~$P$ of form $u\Delta_X + {\bf e}_B + t\nablala_Y$
by the cardinalities of $X\setminus\{1\}$ and~$Y\setminus\{1\}$.
In particular the top degree faces are counted by the collection
of coefficients of~$Q'_M$ of top degree (hence the name),
and the degree $|E|-1$ terms of $Q'_M$ are always $(x+y)^{|E|-1}$.
The appearance of basis activities in Lemma~\ref{lem:B} reveals that
$P$ is intimately related to a familiar object in matroid theory,
the \emph{Dawson partition} \cite{dawson}.
Give the lexicographic order to the power set $\mathcal P(E)$.
A partition of $\mathcal P(E)$ into intervals $[S_1,T_1],\ldots,[S_p,T_p]$ with indices such that $S_1<\ldots<S_p$ is a Dawson partition if and only if $T_1<\ldots < T_p$.
Every matroid gives rise to a Dawson partition in which these intervals are $[B\setminus\mathrm{Int}(B),B\cup\mathrm{Ext}(B)]$ for all $B\in\mathcal B_M$.
\begin{prop}\label{prop:Dawson}
Let $[S_1,T_1],\ldots,[S_p,T_p]$ be the Dawson partition of~$M$.
The poset $P$ is a disjoint union of face posets of cubes $C_1,\ldots,C_p$
where the vertices of $C_i$ are the top-degree faces $u\Delta_X+{\bf e}_B+t\nablala_Y$
such that $X\in[S_i,T_i]$.
\end{prop}
The description of the cubes comes from Lemma \ref{lattice}. Note that the element $1$ is both internally and externally active with respect to every basis, due to it being the smallest element in the ordering. So, even though $1$ is in both $X$ and $Y$, it is in $T_i-S_i$ for all $i$.
\section{Polymatroids}
\label{sec:polymatroids}
In this section we investigate the invariants $Q_M$ and $Q'_M$ when $M=(E,r)$ is a polymatroid.
Many familiar matroid operations have polymatroid counterparts,
and we describe the behaviour of $Q'_M$ under these operations.
We see that it retains versions of several formulae true of the Tutte polynomial.
For instance, there is a polymatroid analogue of the direct sum of matroids:
given two polymatroids $M_1=(E_1,r_1),M_2=(E_2,r_2)$ with disjoint ground sets,
their \emph{direct sum} $M=(E,r)$ has ground set $E = E_1\sqcup E_2$
and rank function $r(S) = r_1(S\cap E_1)+r_2(S\cap E_2)$.
This definition extends the usual direct sum of matroids.
\begin{prop}\label{prop:direct sum}
Let $M_1\oplus M_2$ be the direct sum of two polymatroids $M_1$ and $M_2$.
Then
\[Q'_{M_1\oplus M_2}(x,y)=(x+y-1)\cdot Q'_{M_1}(x,y)\cdot Q'_{M_2}(x,y).\]
\end{prop}
\begin{proof}
We will need to use distinguished notation for our standard simplices according to the matroid under consideration.
So we write $\Delta_i = \operatorname{Conv}\{\mathbf e_j : j\in E_i\}$ for $i=1,2$
and reserve the unadorned name $\Delta$ for $\operatorname{Conv}\{\mathbf e_j : j\in E\}$.
Define $\nabla_i$ and $\nabla$ similarly.
The basic relationship between $M=M_1\oplus M_2$ and its summands in terms of our lattice point counts is the following equality:
\begin{equation}\label{eq:ziggurat}
\sum_{k=0}^{\min\{t,u\}} Q_M(t-k,u-k)=
\sum_{t_1=0}^{t}\sum_{u_1=0}^{u}Q_{M_1}(t_1,u_1)\cdot Q_{M_2}(t-t_1,u-u_1).
\end{equation}
The right hand side counts tuples $(t_1,u_1,q_1,q_2)$
where $q_1\in (P(M_1)+u_1\Delta_1+t_1\nabla_1)\cap\mathbb Z^{E_1}$
and $q_2\in (P(M_2)+(u-u_1)\Delta_2+(t-t_1)\nabla_2)\cap\mathbb Z^{E_2}$.
Because the coordinate inclusions of $\Delta_1$ and $\Delta_2$ are subsets of $\Delta$,
and similarly for $\nabla$, the concatenation $(q_1,q_2)\in\mathbb Z^E$
is a lattice point of $P(M)+u\Delta+t\nabla$.
The left hand side counts pairs $(k,q)$ where
$q$ is a lattice point of $P(M)+(u-k)\Delta+(t-k)\nabla$;
this polyhedron is a subset of $P(M)+u\Delta+t\nabla$.
To prove equation~\eqref{eq:ziggurat} we will show that each $q=(q_1,q_2)\in\mathbb Z^E$
occurs with the same number of values of $k$ on the left as values of $(t_1,u_1)$ on the right.
If $q\in(P(M)+u\Delta+t\nabla)\cap\mathbb Z^E$ then there is some maximal integer $K$ such that
$q\in(P(M)+(u-K)\Delta+(t-K)\nabla)\cap\mathbb Z^E$, and then $q$ is counted just $K+1$ times on the left hand side,
namely for $k=0,1,\ldots,K$.
This $K$ is what we called the number of \emph{cancellations} in $q$ when proving Theorem~\ref{thm:T to Q}.
Choose an expression
\[q=e_B+{\bf e}_{x_1}+\cdots+{\bf e}_{x_{u-K}}-{\bf e}_{y_1}-\cdots -{\bf e}_{y_{t-K}},\]
where $B$ is a basis of $M$ and the $x_i$ and $y_i$ are elements of $E$.
Suppose the $x_i$ and $y_i$ are ordered such that
$x_i\in E_1$ if and only if $i\leq t'$ and $y_i\in E_1$ if and only if $i\leq u'$.
Then we have
\begin{align*}
q_1 &= e_{B\cap E_1}+{\bf e}_{x_1}+\cdots+{\bf e}_{x_{u'}}-{\bf e}_{y_1}-\cdots -{\bf e}_{y_{t'}},\\
q_2 &= e_{B\cap E_2}+{\bf e}_{x_{u'+1}}+\cdots+{\bf e}_{x_{u-K}}-{\bf e}_{y_{t'+1}}-\cdots -{\bf e}_{y_{t-K}},
\end{align*}
and both of these expressions also have the minimal number of ${\bf e}_{x_i}$ and ${\bf e}_{y_i}$ summands,
or else our expression for $q$ would not have done.
Thus $q_1$ is in $P(M_1)+u'\Delta+t'\nabla$ but not $P(M_1)+(u'-1)\Delta+(t'-1)\nabla$,
and $q_2$ is in $P(M_1)+(u-u'-K)\Delta+(t-t'-K)\nabla$ but not $P(M_1)+(u-u'-K-1)\Delta+(t-t'-K-1)\nabla$.
So the possibilities for $t_1$ and $u_1$ on the right hand side of \eqref{eq:ziggurat}
are those that arrange $t_1\geq t'$ and $t-t_1\geq t-t'-K$, and the corresponding equations for the $u$ variables,
together with $t_1-u_1=t'-u'$.
There are exactly $K+1$ solutions here as well, namely $t_1 = t', t'+1, \ldots, t'+K$.
Using equation~\eqref{eq:ziggurat} within a generating function for $t$ and~$u$, we have that
\begin{align*}
&\mathrel{\phantom=}
\sum_{t,u}\sum_{k=0}^{\min\{t,u\}}Q_{M_1\oplus M_2}(t-k,u-k)v^uw^t\\ &=\sum_{t,u}\sum_{t_1=0}^{t}\sum_{u_1=0}^{u}Q_{M_1}(t_1,u_1)v^{u_1}w^{t_1} \cdot Q_{M_2}(t-t_1,u-u_1)v^{u-u_1}w^{t-t_1} \\
& = \sum_{t_1,u_1}Q_{M_1}(t_1,u_1)v^{u_1}w^{t_1}\\
&\hspace{15mm}\cdot\sum_{t-t_1,u-u_1}Q_{M_2}(t-t_1,u-u_1)v^{u-u_1}w^{t-t_1} \\
& = \left(\sum_{t,u}Q_{M_1}(t,u)v^{u}w^{t}\right)\left(\sum_{t,u}Q_{M_2}(t,u)v^{u}w^{t}\right).
\end{align*}
The left-hand side can also be simplified as
\begin{align*}
&\mathrel{\phantom=}
\sum_{t,u}\sum_{k=0}^{\min\{t,u\}}Q_{M_1\oplus M_2}(t-k,u-k)v^uw^t\\
&=\sum_{k\geq 0}\sum_{t,u\geq k}Q_{M_1\oplus M_2}(t-k,u-k)v^{u-k}w^{t-k}(vw)^k\\
&= \sum_{k\geq 0}\sum_{t,u}Q_{M_1\oplus M_2}(t,u)v^uw^t(vw)^k\\
&= \dfrac{1}{1-vw}\cdot\sum_{t,u}Q_{M_1\oplus M_2}(t,u)v^uw^t
\end{align*}
and thus we have that
$$\dfrac{1}{1-vw}\cdot\sum_{t,u}Q_{M_1\oplus M_2}(t,u)v^uw^t= \left(\sum_{t,u}Q_{M_1}(t,u)v^{u_1}w^{t_1}\right)\left(\sum_{t,u}Q_{M_2}(t,u)v^{u_1}w^{t_1}\right).$$
The generating functions above allow us easily to change basis from $Q_M$ to $Q'_M$:
\begin{align*}
\sum_{t,u}Q_M(t,u)v^uw^t &= \sum_{t,u}\sum_{i,j}c_{ij}\binom{u}{j}\binom{t}{i}v^uw^t\\
&= \sum_{i,j}c_{ij}\dfrac{v^j}{(1-v)^{j+1}}\cdot\dfrac{w^i}{(1-w)^{i+1}}\\
&= \dfrac{1}{(1-v)(1-w)}\cdot Q'_M\left(\dfrac{w}{1-w}+1,\dfrac{v}{1-v}+1\right)
\end{align*}
where the last line follows from the definition of $Q'_M$. Hence we have
$$Q'_{M_1\oplus M_2}\left(\dfrac{1}{1-w},\dfrac{1}{1-v}\right)=\dfrac{1-vw}{(1-v)(1-w)}Q'_{M_1}\left(\dfrac{1}{1-w},\dfrac{1}{1-v}\right)Q'_{M_2}\left(\dfrac{1}{1-w},\dfrac{1}{1-v}\right)$$
and substituting $w=\dfrac{x-1}{x}$ and $v=\dfrac{y-1}{y}$ gives the result.
\end{proof}
\begin{remark}
It follows from Proposition~\ref{prop:direct sum}
that the rescaled matroid invariant $(x+y-1)\cdot Q'_M(x,y)$ is exactly multiplicative under direct sum.
Recasting Theorem~\ref{thm:T to Q} in terms of this rescaled invariant also eliminates a denominator.
And, by Proposition~\ref{prop:Dawson}, its coefficients can be interpreted as
counting intervals in the Boolean lattice $\mathcal P(E)$ contained in a single part of a Dawson partition
according to the ranks of their minimum and maximum,
with no need to accord a special role to one element.
\end{remark}
Moving on to the next polymatroid operation,
it is apparent from the symmetry of Theorem~\ref{lpg} under switching $x$ and~$y$
that $Q'_M$, like the Tutte polynomial, exchanges its two variables under matroid duality.
The best analogue of duality for polymatroids requires a parameter $s$
greater than or equal to the rank of any singleton; then if $M=(E,r)$ is a polymatroid,
its \emph{$s$-dual} is the polymatroid $M^*=(E,r^*)$ with
\[r^*(S) = r(E) + s|E\setminus S| - r(E\setminus S).\]
The 1-dual of a matroid is its usual dual.
\begin{prop}\label{prop:duality}
For any polymatroid $M=(E,r)$ and any $s$-dual $M^*$ of~$M$,
$Q'_{M^*}(x,y)=Q'_M(y,x)$.
\end{prop}
\begin{proof}
Let $\phi:\mathbb R^E\to\mathbb R^E$ be the involution that subtracts every coordinate from~$s$.
Definition~\ref{dfn:polytope} implies that
$\phi$ is a bijection which takes elements of $P(M)$ to elements of $P(M^*)$;
it also clearly preserves the property of being a lattice point.
This gives that
\begin{align*}
\#(P(M)+u\Delta+t(-\Delta))&=\#(\phi(P(M))+u\phi(\Delta)+t\phi(-\Delta))\\
& =\#(P(M^*)+u(\nablala+1_E)+t(\Delta+1_E)\\
& =\#(P(M^*)+(t-u)1_E+u\nabla+t\Delta)\\
&= \#(P(M^*)+u\nabla+t\Delta)
\end{align*}
where the last line is true due to the polytope being a translation of the one in the line above. The statement follows.
\end{proof}
Given a subdivision $P_1,\ldots,P_n$ of a polytope $P$, a \emph{valuation} is a function $f$ such that
\[f(P)=\sum\limits_{P_i}f(P_i)-\sum\limits_{P_i,P_j}f(P_i\cap P_j)+\ldots+(-1)^{n-1} f(P_1\cap\cdots \cap P_n).\]
The number of lattice points in a polytope is a valuation: if $f(P)$ is this counting function,
then it is easily checked that each lattice point counted in $f(P)$ also contributes exactly one to the sum on the right hand side.
Thus the invariant $Q'_M$ is a valuation as well.
\begin{prop}
Let $\mathcal F$ be a polyhedral complex whose total space is a polymatroid base polytope
$P(M)$, and each of whose faces $F$ is a polymatroid base polytope $P(M(F))$. Then
\[Q'_M(x,y) = \sum_{\mbox{\scriptsize $F$ a face of $\mathcal F$}}
(-1)^{\dim(P(M))-\dim F} Q'_{M(F)}(x,y).\]
\end{prop}
For example, if $M$ is a matroid and we relax a circuit-hyperplane, we get the following result:
\begin{cor}
Take a matroid $M=(E,r)$ and let $C\subset E$ be a circuit-hyperplane of $M$. Let $M'$ be the matroid formed by relaxing $C$. Then $Q'_M(x,y)=Q'_{M'}(x,y)-x^{n-r(M)-1}y^{r(M)-1}$.
\end{cor}
Now consider deletion and contraction in polymatroids.
We have that $P(M\setminusckslash a)=\{p\in P(M) \ | \ p_a=\underline{k}\}$, where $\underline{k}$ is the minimum value $p_a$ takes (this will be $0$ unless $a$ is a coloop), and that $P(M/a)=\{p\in P(M) \ | \ p_a=\overline{k}\}$, where $\overline{k}$ is the maximum value $p_a$ takes. When $M$ is a matroid, these two sets partition $P(M)$. However, when $M$ is a polymatroid, we can have points in $P(M)$ where $\underline{k}<p_a<\overline{k}$. Let $N_k:=\{p\in P(M) \ | \ p_a=k\}$, and let $P(N_k)$ be the polytope consisting of the convex hull of such points. Now we have that $P(M\setminusckslash a)$, $P(M/a)$, and the collection of $P(N_k)$ for $k\in\{\underline{k}+1,\ldots,\overline{k}-1\}$ partition $P(M)$. We will refer to each of these parts, when they exist, as an \emph{$a$-slice} of $P(M)$. When we do not include the deletion and contraction slices, we can talk about \emph{(strictly) interior} slices.
\begin{thm}
\label{delcont}
Let $M=(E,r)$ be a polymatroid and take $a\in E(M)$. Then
$$Q'_M(x,y)=(x-1)Q'_{M\setminusckslash a}(x,y)+(y-1)Q'_{M/a}(x,y)+\sum_{N} Q'_N(x,y).$$
where $N$ ranges over $a$-slices of $P(M)$.
\end{thm}
Note that when $M$ is a matroid, the statement simplifies to the formulae given in Lemma \ref{recurrence}: if $a$ is neither a loop nor coloop, then the $a$-slices are $P(M\setminusckslash a)$ and $P(M/a)$, so
\begin{align*}
Q'_M(x,y)&=(x-1)Q'_{M\setminusckslash a}(x,y)+(y-1)Q'_{M/a}(x,y)+\sum_{N_k} Q'_{N_k}(x,y)\\
&=(x-1)Q'_{M\setminusckslash a}(x,y)+(y-1)Q'_{M/a}(x,y)+Q'_{M\setminusckslash a}(x,y)+Q'_{M/a}(x,y)\\
&=xQ'_{M\setminusckslash a}(x,y)+yQ'_{M/a}(x,y).
\end{align*}
When $a$ is a loop or coloop, $M\setminusckslash a=M/a$, and we have only one $a$-slice: $P(M\setminusckslash a)=P(M/a)$. So we get that
\begin{align*}
Q'_M(x,y)&=(x-1)Q'_{M\setminusckslash a}(x,y)+(y-1)Q'_{M/a}(x,y)+\sum_{N_k} Q'_{N_k}(x,y)\\
&= (x+y-1)Q'_{M/a}(x,y)
\end{align*}
as in Lemma \ref{recurrence}.
Also note that this result gives another proof of Theorem \ref{thm:T to Q} as a corollary.
\begin{proof}
In this proof, we make constant use of Lemma~\ref{712} in order to express lattice points as sums of lattice points.
Let $M$ be a polymatroid. If the rank function of $M$ is a matroid rank function summed with a function of the form $S\mapsto\sum\limits_{i\in S} c_i$, then $P(M)$ will be a translate of a matroid polytope, and the same argument as above will hold. Assume now that this is not the case. This means that for any $a\in E(M)$, there will be at least one $a$-slice of $P(M)$, $P(N_k)$, which is not equal to $P(M/a)$ or $P(M\setminusckslash a)$.
\begin{sublemma}
\label{claim1}
Define $R$ to be the polytope $\{q\in P(M)+u\Delta_E+t\nablala_E \ | \ q_a=k\}$, and define $S$ to be $P(N_k)+u\Delta_{E-a}+t\nablala_{E-a}$. If $R$ intersects the set of lattice points of $P(M)$, then $R=S$.
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{claim1}]
It is clear that the lattice points of $S$ are contained in $R$. Take a point in $R$, $q_1=p_1+{\bf e}_{i_1}+\cdots+{\bf e}_{i_u}-{\bf e}_{j_1}-\cdots -{\bf e}_{j_t}$. We will show that we can write this as a point contained in $S$, $q_2=p_2+{\bf e}_{m_1}+\cdots +{\bf e}_{m_{u}}-{\bf e}_{n_1}-\cdots -{\bf e}_{n_t}$, where no ${m_i}$ or ${n_j}$ can be equal to $a$.
If $(p_1)_a=k$, then we simply choose $p_2$ to be $p_1$ and choose $m_k=i_k$, $n_k=j_k$ for all $k\in\{1,\ldots,t\}$, with one possible change: if we have ${i_k}={j_l}=a$ in $q_1$, in $q_2$ replace ${m_k}$ and ${n_l}$ with $b$, where $b$ is any other element in $E(M)$. Note ${\bf e}_a$ must always appear paired in this way, such that $(q_1)_a=k$, and so this change does not affect the coordinate values of $q_2$.
If $(p_1)_a\neq k$, we first must rewrite the expression for $q_1$.
By the base exchange property for polymatroids (\cite[Theorem 4.1]{hibi}), given $p_1$ and any point $p_3\in P(M)$, if $(p_1)_i>(p_3)_i$ there exists $l$ such that $(p_1)_l< (p_3)_l$ and $p_1-{\bf e}_i+{\bf e}_l\in P(M)$.
Let $(p_1)_a=k+\lambda$, where $\lambda>0$. Then, by repeatedly applying the exchange property, we get that $p_1-\lambda {\bf e}_a+{\bf e}_{l_1}+\cdots+{\bf e}_{l_\lambda}\in P(M)$. Then we can find $q_2$ by setting $p_2=p_1-\lambda {\bf e}_a+{\bf e}_{l_1}+\cdots+{\bf e}_{l_\lambda}$, so $$q_2=p_2+\lambda {\bf e}_a-{\bf e}_{l_1}-\cdots-{\bf e}_{l_\lambda}+{\bf e}_{i_1}+\ldots+{\bf e}_{i_u}-{\bf e}_{j_1}-\cdots -{\bf e}_{j_t}=q_1.$$ Note that as $(q_1)_a=k$ and $(p_2)_a=k$, there must be $\lambda$ $-{\bf e}_{j_k}$ terms equal to $-{\bf e}_a$, so
$$q_2=p_2-{\bf e}_{l_1}-\cdots-{\bf e}_{l_\lambda}+{\bf e}_{i_1}+\ldots+{\bf e}_{i_u}-{\bf e}_{j_1}-\cdots -{\bf e}_{j_{t-\lambda}}$$
which is of the correct form, completing the proof of Claim~\ref{claim1}.
\end{subproof}
\begin{sublemma}
\label{6.12}
Let $N_i$ be a strictly interior slice of $P(M)$. Then $P(M)+t\Delta_E+t\nablala_{E}=(P(M/a)+u\Delta_E+t\nabla_{E-a}) \ \sqcup \ \bigsqcup\limits_i (P(N_i)+t\Delta_{E-a}+t\nabla_{E-a}) \ \sqcup \ (P(M\setminusckslash a)+t\Delta_{E-a}+t\nabla_E)$.
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{6.12}]
Take $P(M)+u\Delta_E+t\nabla_E$ and split it into a collection of polytopes according to the value of $q_a$ for all points $q\in P(M)+u\Delta_E+t\nabla_E$. The disjoint union of the lattice points of these parts clearly will give back those of the original polytope. By the previous result, if one of these parts intersects $P(M)$ we can write it as $P(N_k)+u\Delta_{E-a}+t\nabla_{E-a}$. Otherwise, we must be able to write the part as $P(M/a)+(u-\lambda)\Delta_{E-a}+\lambda {\bf e}_a+t\nabla_{E-a}$, where $\lambda>\ol{k}$, or as $P(M\setminusckslash a)+t\Delta_{E-a}-\mu {\bf e}_a+(t-\mu)\nabla_{E-a}$, where $\mu>\underline{k}$.
We will show that
\begin{equation}
\label{raise}
\bigsqcup_\lambda P(M/a)+(u-\lambda)\Delta_{E-a}+\lambda {\bf e}_a+t\nabla_{E-a}=P(M/a)+u\Delta_E+t\nabla_{E-a}.
\end{equation}
It is clear that the sets of lattice points of the summands are pairwise disjoint as the $a$-coordinates in each set must be different. It is also clear that the lattice points contained in the polytope on the left hand side are contained in that of the right hand side. Take a point $q_1=p_1+{\bf e}_{i_1}+\cdots+{\bf e}_{i_u}-{\bf e}_{j_1}-\cdots -{\bf e}_{j_t}$ contained in $P(M/a)+u\Delta_E+t\nabla_{E-a}$. Let $(q_1)_a=\ol{k}+\mu$, where $\mu>0$. We need to write $q_1$ as $p_2+{\bf e}_{m_1}+\cdots +{\bf e}_{m_{u-\lambda}}+\lambda {\bf e}_a-{\bf e}_{n_1}-\cdots -{\bf e}_{n_t}$, a lattice point contained in one of the summands on the left hand side. Choose $\mu=\lambda$, $p_2=p_1$, $\{j_\alpha\}=\{n_\alpha\}$, and $\{{i_\beta} \ | \ i_\beta\neq a\}=\{{m_\beta}\}$ and the equality follows.
The same arguments show that
\begin{equation}
\label{lower}
\bigsqcup_\mu P(M\setminusckslash a)+t\Delta_{E-a}-\mu {\bf e}_a+(t-\mu)\nabla_{E-a}=P(M\setminusckslash a)+u\Delta_{E-a}+t\nabla_E
\end{equation}
and the claim follows.
\end{subproof}
\begin{sublemma}
\label{claim2}
We have that
$$\#(P(M/a)+u\Delta_E+t\nabla_{E-a})=\sum_{j=0}^u\#(P(M/a)+j\Delta_{E-a}+t\nabla_{E-a})$$
and
$$\#(P(M\setminusckslash a)+u\Delta_{E-a}+t\nabla_{E})=\sum_{j=0}^u\#(P(M\setminusckslash a)+u\Delta_{E-a}+i\nabla_{E-a}).$$
\end{sublemma}
\begin{subproof}[Proof of Claim~\ref{claim2}]
Take the cardinalities of both sides of Equations \ref{raise} and \ref{lower}.
\end{subproof}
Continuing the proof of the theorem, we now that have
\begin{equation}
\label{almost}
Q_M(t,u)=\sum_{N_k} Q_{N_k}(t,u)+\sum_{j=0}^u Q_{M/a}(t,j)+\sum_{i=0}^t Q_{M\setminusckslash a}(i,u)
\end{equation}
where $k\in\{\underline{k}+1,\ldots,\ol{k}-1\}$, that is, $N_k$ is always a strictly interior slice of $P(M)$.
We now work out how the change of basis from $Q$ to $Q'$ transforms the sums in Equation \ref{almost}. Take a term in $Q_{M/a}$, $c_{ik}\binom{j}{k}\binom{t}{i}$. We have that
\begin{align*}
\sum_{j=0}^uc_{ik}\binom{j}{k}\binom{t}{i} &=c_{ik}\binom{u+1}{k+1}\binom{t}{i}\\
&= c_{ik}\binom{t}{i}\left(\binom{u}{k}+\binom{u}{k+1}\right).
\end{align*}
Now apply the change of basis to get
\begin{align*}
c_{ik}(x-1)^i((y-1)^k+(y-1)^{k+1}) & = c_{ik}(x-1)^i\left((y-1)^k(1+y-1)\right)\\
&=c_{ik}(x_1)^i(y-1)^ky.
\end{align*}
Thus
$$\sum_{j=0}^u Q_{M/a}(t,j)=yQ'_{M/a}(t,u)$$
and similarly, $$\sum_{i=0}^t Q_{M\setminusckslash a}(i,u)=xQ'_{M\setminusckslash a}(t,u).$$
Finally, putting this together with Claim~\ref{6.12} and Equations \eqref{raise}, \eqref{lower} gives:
\begin{align*}
Q'_M(t,u)&=xQ'_{M/a}(t,u)+yQ'_{M/a}(t,u)+\sum_{\textrm{interior} \ N_k} Q'_{N_k}(t,u) \\
&= (x-1)Q'_{M/a}(t,u)+(y-1)Q'_{M/a}(t,u)+\sum_{N_k} Q'_{N_k}(t,u).
\end{align*}
This completes the proof of Theorem \ref{delcont}.
\end{proof}
Unfortunately, when $M$ is a polymatroid, there is no analogue to Corollary~\ref{cor:alternating}:
the coefficients of~$Q'_M$ do not have sign independent of~$M$, and thus
there can be no straightforward enumerative interpretation of the coefficients.
This is a consequence of the failure of Theorem~\ref{coeffs} for polymatroids.
Here is an example to illustrate this.
\begin{ex}
The left of Figure~\ref{fig:2} displays the subdivision $\mathcal F$ for the sum of Example~\ref{ex:1}.
\begin{figure}
\caption{At left, the regular subdivision $\mathcal F$ associated
to the Minkowski sum of Example~\ref{ex:1}
\label{fig:2}
\end{figure}
We see that the four grey top degree faces contain all the lattice points between them,
and the poset $P$ contains two other faces which are pairwise intersections thereof,
the horizontal segment on the left with $(X,Y) = (1,12)$ and the one on the right with $(X,Y) = (12,1)$.
These are indeed enumerated, up to the alternation of sign,
by the polynomial $Q'_M(x,y) = x^2 + 2xy + y^2 - x - y$ found earlier.
By contrast, the right of the figure displays $\mathcal F$
for the polymatroid $M_2$ obtained by doubling the rank function of~$M$.
The corresponding polynomial is $Q'_{M_2}(x,y) = x^2 + 2xy + y^2 - 1$,
in which the signs are not alternating, dashing hopes of a similar enumerative interpretation.
In the figure we see that there are lattice points not on any grey face.
\end{ex}
\section{K\'alm\'an's activities}
\label{sec:kalman}
One motivation for the particular Minkowski sum we have employed in our definition
is that it provides a polyhedral translation of K\'alm\'an's construction of activities in a polymatroid.
This section explains the connection.
We first recall the definitions of K\'alm\'an's univariate activity invariants of polymatroids.
These polynomials do not depend on the order on~$E$ that was used to define them \cite[Theorem~5.4]{kalman}.
\begin{citedfn}[{\cite{kalman}}]
\label{activity}
Define the \emph{internal polynomial} and \emph{external polynomial} of a polymatroid $M=(E,r)$ by
\begin{displaymath}I_M(\xi)=\sum_{x\in \mathcal B_M\cap\mathbb{Z}^E} \xi^{\ol{{\iota}}(x)} \quad \mathrm{and} \quad X_M(\eta)=\sum_{x\in \mathcal B_M\cap\mathbb{Z}^E} \eta^{\ol{\varepsilon}(x)}.\end{displaymath}
\end{citedfn}
\begin{lemma}
\label{old}Let $P$ be a polymatroid polytope. At every lattice point $f\in P$, attach the scaled simplex \[f+t\operatorname{conv}(\{-e_i \ | \ i \ \textrm{is internally active in} \ f \ \textrm{or} \ i\notin f \}).\] This operation partitions $P+t\nablala $
into a collection of translates of faces of $t\nablala $,
with the simplex attached at $f$ having codimension $\ol{\iota}(f)$
within $P$.
\end{lemma}
\begin{figure}
\caption{An example of the partition of Lemma~\ref{old}
\label{activitypartition}
\end{figure}
Figure \ref{activitypartition} shows a case of this operation, to illustrate why we speak of ``attaching'' a simplex.
Our polymatroid $P(M)$ is coloured grey, and the polytope drawn is $P(M)+2\Delta+\nabla$.
In the picture we translate $P(M)$ by a multiple of ${\bf e}_1$, namely $1{\bf e}_1$, in order to allow both polygons to reside in the same plane.
Coordinate labels are written without parentheses and commas. The blue areas are faces of the scaled simplices $2\Delta_f$, and can be seen to be a partition of the lattice points.
(Don't overlook the blue dot at 410!)
\begin{proof}
We will first show that our simplices covers all the lattice points of $P+t\nabla$.
Let $g\in P+t\nabla$ be a lattice point. We will find $f$ such that $g\in t\Delta_f$.
Let $g_t=g$. For $i\in\{0,\ldots,t-1\}$, define
\begin{displaymath}
g_i = \left\{
\begin{array}{ll}
g_{i+1}+{\bf e}_1 & \textrm{if} \ g_{i+1}+{\bf e}_1 \in P+(t-i)\nabla\\
g_{i+1}+{\bf e}_2 & \textrm{if} \ g_{i+1}+{\bf e}_1 \notin P+(t-i)\nabla, g_{i+1}+{\bf e}_2 \in P+(t-i)\nabla\\
\qquad \vdots & \\
g_{i+1}+{\bf e}_n & \textrm{if} \ g_{i+1}+{\bf e}_{h} \notin P+(t-i)\nabla, \ \forall j\in [n], \ g_{i+1}+{\bf e}_n \in P+(t-i)\nabla
\end{array}
\right.
\end{displaymath}
In other words, at each iteration $i$, we are adding an element ${\bf e}_j$ which is internally active with respect to $g_{i+1}$. We cannot replace ${\bf e}_j$ with ${\bf e}_i$ where $i<j$ and remain inside $P+t\nabla$. Let ${\bf e}_{j_i}$ be the element added in iteration $t$. We get that $$g=g_t=g_0-{\bf e}_{j_1}-\ldots -{\bf e}_{j_t}\in P+t\nabla.$$ Note that if we added ${\bf e}_i$ at some stage $g_s$ of the iteration, and ${\bf e}_j$ at stage $g_{s-1}$, then $j\geq i$. Thus if we take a tuple ${\bf e}_{k}$ such that $(k_1,\ldots,k_t)<(j_1,\ldots,j_t)$ with respect to the lexicographic ordering, then $g_0-\sum \limits_{t}{\bf e}_{j_t}+\sum\limits_{t} {\bf e}_{k_t}\notin P$, so each ${\bf e}_j$ is internally active. Thus $g_0=f$ and the ${\bf e}_j$ found define a simplex $\Delta_f$ such that $g\in t\Delta_f$.
Now we will show that this operation gives disjoint sets. We have that $\{t\Delta_f\}$ covers $P+t\nabla$, and that $\{(t-1)\Delta_f\}$ partitions $P+(t-1)\nabla$. Thus in order to show that $\{t\Delta_f\}$ is in fact a partition of the lattice points of $P$, it suffices to prove that if $g_t\in t\Delta_f$, then $g_{t-1}\in(t-1)\Delta_{f}$. Say that $f=g_t+{\bf e}_{i_1}+\cdots+{\bf e}_{i_t}$. This means that each element ${\bf e}_{i_k}$ is internally active at $f$ for all $k\in\{0,\ldots,t\}$. Now, for a contradiction, let $g_0=f'\neq f$, so that $g_{t-1}=g_t+{\bf e}_i\in (t-1)\Delta_{f'}$. Apply the same iterative process as before to get
$$
\begin{array}{ll}
f'=g_0& = g_{t-1}+{\bf e}_{i_1}+\cdots+{\bf e}_{i_{t-1}}\\
& = g_t+{\bf e}_{i}+{\bf e}_{i_1}+\cdots+{\bf e}_{i_{t-1}}\\
& = f-{\bf e}_{i_t}+{\bf e}_i.
\end{array}$$
Thus ${\bf e}_{i_t}$ was internally inactive at $f$, contradicting our construction of $t\Delta_f$.
\end{proof}
The following is a direct consequence of this lemma,
and its exterior analogue which arises from replacing $\iota$ and $\nablala$
with $\varepsilon$ and $\Delta$.
\begin{thm}\label{thm:activity}
Let $M$ be a polymatroid.
Then $I_M(\xi) = \xi\cdot Q'_M(\xi,1)$
and $X_M(\eta) = \eta\cdot Q'_M(1,\eta)$.
\end{thm}
It is this result which first motivated
the particular change of basis we have made from $Q_M$ to~$Q'_M$,
since an $i$-dimensional face of $t\Delta$
has $\displaystyle\binom{t+i}i = \sum_{k=0}^i\binom ik\binom ti$ lattice points.
The bivariate enumerator of internal and external activities for
polymatroids is not order-independent, and so we do not have that $Q'_M=\sum_{x\in\mathcal{B}_M\cap\mathbb{Z}^{|E|}}\xi^{\iota (x)}\eta^{\varepsilon (x)}$.
\begin{ex}\label{bivaractivity}
Take the polymatroid with bases $\{(0,2,1),(1,1,1),(1,2,0),(2,1,0),(2,0,1)\}$.
Using the natural ordering on $[3]$, we have that $\sum\limits_{x\in\mathcal{B}_M\cap\mathbb{Z}^{|E|}}\xi^{\iota (x)}\eta^{\varepsilon (x)}=\xi^3\eta+2\xi^2\eta^2+\xi\eta^2+\xi\eta^3.$ If we instead use the ordering $2<3<1$, the enumerator is $\xi^3\eta^2+\xi^2\eta^2+\xi^2\eta+\xi\eta^2+\xi\eta^3.$
\begin{figure}
\caption{The polymatroid of Example \ref{bivaractivity}
\end{figure}
\end{ex}
\begin{question}
Section~10 of~\cite{kalman} is dedicated to the behaviour of K\'alm\'an's activity invariants
in trinities in the sense of Tutte.
One can obtain six hypergraphs from a properly three-coloured triangulation of the sphere
by deleting one colour class and regarding the second and third as vertices and hyperedges of a hypergraph.
K\'alm\'an considered the relationships between values of his invariants on these six hypergraphs.
With the proof of his main conjecture in~\cite{kalman2},
we know that besides the internal and external invariants $I_G$ and $X_G$ of a hypergraph $G$ with plane embedding,
there exists a third invariant $Y_G$
such that the values of the three invariants are permuted by the action of the symmetric group $S_3$ on the colour classes in the natural way.
Our work has cast $I_M$ and $X_M$ as univariate evaluations of a bivariate polynomial $Q'_M$,
and the content of Proposition~\ref{prop:duality} is that polymatroid duality,
i.e.\ exchanging the deleted and vertex colour classes, exchanges the two variables of $Q'_M$.
Is there a good trivariate polynomial of a three-coloured triangulation of the sphere which similarly encapsulates the above observations on trinities?
Permuting the three colours should permute its three variables,
and three of its univariate evaluations should be $I_G$, $X_G$, and $Y_G$.
We would of course be even happier if $Q'_G$ were among its bivariate evaluations.
\end{question}
\label{sec:biblio}
\end{document} |
\begin{document}
\begin{center}
\Large \bf{Parameter estimation by implicit sampling}
\end{center}
\begin{center}
Matthias Morzfeld$^{1,2,*}$, Xuemin Tu$^{3}$, Jon Wilkening$^{1,2}$ and Alexandre J. Chorin$^{1,2}$
$^1$Department of Mathematics, University of California, Berkeley, CA 94720,~USA.\\
$^2$Lawrence Berkeley National Laboratory, Berkeley, CA 94720,~USA.\\
$^3$Department of Mathematics, University of Kansas, Lawrence, KS 66045,~USA.\\
\let\thefootnote\relax\footnote{$^*$ Corresponding author. Tel: +1~510~486~6335. Email address: [email protected].
Lawrence Berkeley National Laboratiry, 1 Cyclotron Road, Berkeley, California 94720,~USA.}
\emph{Abstract}
\end{center}
Implicit sampling is a weighted sampling method that is used in data assimilation,
where one sequentially updates estimates of the state of a stochastic model
based on a stream of noisy or incomplete data.
Here we describe how to use implicit sampling in parameter estimation problems,
where the goal is to find parameters of a numerical model, e.g.~a
partial differential equation (PDE),
such that the output of the numerical model is compatible with (noisy) data.
We use the Bayesian approach to parameter estimation, in which a posterior probability density
describes the probability of the parameter conditioned on data
and compute an empirical estimate of this posterior with implicit sampling.
Our approach generates independent samples,
so that some of the practical difficulties one encounters with Markov Chain Monte Carlo methods,
e.g.~burn-in time or correlations among dependent samples, are avoided.
We describe a new implementation of implicit sampling for parameter estimation problems
that makes use of multiple grids (coarse to fine) and BFGS optimization coupled to adjoint equations
for the required gradient calculations.
The implementation is ``dimension independent'',
in the sense that a well-defined finite dimensional subspace is sampled
as the mesh used for discretization of the PDE is refined.
We illustrate the algorithm with an example
where we estimate a diffusion coefficient in an elliptic equation
from sparse and noisy pressure measurements.
In the example, dimension\slash mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions.
\section{Introduction}
We take the Bayesian approach to
parameter estimation and compute the probability density function (pdf)
$p(\theta|z)$, where $\theta$ is a set of parameters (an $m$-dimensional vector)
and $z$ are data (a $k$-dimensional vector, see, e.g.~\cite{StuartInverse}).
We assume a prior pdf $p(\theta)$ for the parameters,
which describes what one knows about the parameters before
collecting the data.
For example, one may know a priori that a parameter is positive.
We further assume a likelihood $p(z|\theta)$, which describes
how the parameters are connected with the data.
Bayes' rule combines the prior and likelihood to find
$p(\theta|z)\propto p(\theta)p(z|\theta)$ as a posterior density.
If the prior and likelihood are Gaussian,
then the posterior is also Gaussian,
and it is sufficient to compute the
mean and covariance of $\theta|z$
(because the mean and covariance define the Gaussian).
Moreover, the mean and covariance
are the minimizer and the inverse of the Hessian of
the negative logarithm of the posterior.
If the posterior is not Gaussian,
e.g.~because the numerical model is nonlinear,
then one can compute the posterior mode,
often called the maximum a posteriori (MAP) point,
by minimizing the negative logarithm of the posterior,
and use the MAP point
as an approximation of the parameter $\theta$.
The inverse of the Hessian of the negative
logarithm of the posterior can be used to measure
the uncertainty of the approximation.
This method is sometimes called linearization about the MAP point (LMAP)
or the Laplace approximation~\cite{Iglesias2,Oliver2011,Dean2007,Bui}.
One can also use Markov Chain Monte Carlo (MCMC)
and represent the posterior by a collection of samples,
see, e.g.~\cite{Efendiev,Martin,Petra,Stuart2}.
The samples form an empirical estimate of the posterior,
and statistics, e.g.~the mean or mode,
can be computed from this empirical estimate
by averaging over the samples.
Under mild assumptions, the moments one computes from the samples
converge to the moments of the posterior
(as the number of samples goes to infinity).
In practice, one often encounters difficulties with MCMC.
For example, the samples may have a distribution
which converges slowly to the posterior,
or there could be strong correlations among the samples.
In these cases, it is difficult to determine
how many samples are ``enough samples'' for an accurate empirical estimate.
We propose to use importance sampling to avoid some of these issues,
in particular the estimation of burn-in or correlation times.
The idea in importance sampling is to generate independent
samples from a density that one knows how to sample, the importance function,
rather than from the one one wants to sample.
Weights are attached to each sample to
account for the imperfection of the importance function.
Under mild assumptions, the weighted samples also form an
empirical estimate of the posterior pdf~\cite{ChorinHald}.
The efficiency and applicability of an importance sampling scheme
depends on the choice of the importance function.
Specifically, a poorly chosen importance function can be (nearly) singular
with respect to the posterior,
in which case most of the samples one generates
are unlikely with respect to the posterior,
so that
the number of samples required
becomes large and importance sampling, therefore, becomes impractical
\cite{Blb,Blb2,Sny}.
We show how to use implicit sampling
for constructing importance functions that are
large where the posterior is large,
so that a manageable number of samples forms an accurate empirical estimate.
Implicit sampling has been studied before in the context of
online-filtering\slash data assimilation, i.e.~state estimation of a stochastic
model in \cite{amc,Morzfeld2012,cmt,CT1,Morzfeld2011,cmt2},
and for parameter estimation in stochastic models in \cite{Brad}.
Here we describe how to use implicit sampling
for Bayesian parameter estimation.
In principle, using implicit sampling for parameter estimation is straightforward
(since it is a technique for sampling arbitrary, finite-dimensional probability densities~\cite{amc}),
however its implementation in the context of parameter estimation requires attention.
We discuss how to sample the posterior of parameter estimation
with implicit sampling (in general),
as well as a specific implementation that
is suitable for parameter estimation,
where the underlying model
that connects the parameters with the data is
a partial differential equation (PDE).
We show that the sampling algorithm is
independent of the mesh one uses for discretization of the PDE.
The idea of mesh-independence is also discussed in,e.g.,~\cite{Bui,Martin,Petra}.
Mesh-independence means that the sampling algorithm
``converges'' as the mesh is refined in the sense that
the same finite dimensional subspace is sampled.
We further show how to use multiple grids
and adjoints during the required optimization
and during sampling, and discuss approximations
of a Hessian to reduce the computational costs.
The optimization in implicit sampling represents the link
between implicit sampling and LMAP, as well as recently
developed stochastic Newton MCMC methods~\cite{Bui,Martin,Petra}.
The optimization and Hessian codes used in these codes can be used
for implicit sampling, and the weighing of the samples
can describe non-Gaussian characteristics of the posterior
(which are missed by LMAP).
We illustrate the efficiency of our implicit sampling
algorithm with numerical experiments
in which we estimate the diffusion coefficient in
an elliptic equation using sparse and noisy data.
This problem is a common test problem for MCMC algorithms
and has important applications in reservoir simulation\slash management and in pollution modeling \cite{Bear,Dean2007}.
Moreover, the conditions for the existence of a posterior measure and its continuity are well understood \cite{Stuart2}.
Earlier work on this problem includes \cite{Efendiev}, where Metropolis-Hastings MC sampling is used,
\cite{Iglesias} where an ensemble Kalman filter is used,
and~\cite{MarzoukOptimalMaps}, which uses optimal maps and is further discussed below.
The remainder of this paper is organized as follows.
In section 2 we explain how to use implicit sampling
for parameter estimation and discuss an efficient implementation.
A numerical example is provided in section 3.
The numerical example involves an elliptic PDE
so that the dimension of the parameter we estimate is infinite.
We discuss its finite dimensional approximation
and achieve mesh-independence via KL expansions.
Conclusions are offered in section 4.
\section{Implicit sampling for parameter estimation}
We wish to estimate an $m$-dimensional parameter vector $\theta$ from data
which are obtained as follows.
One measures a function of the parameters $h(\theta)$,
where $h$ is a given $k$-dimensional function;
the measurements are noisy, so that the data $z$ satisfy the relation:
\begin{equation}
\label{equation:obs}
z=h(\theta)+r,
\end{equation}
where $r$ is a random variable with a known distribution
and the function $h$ maps the parameters onto the data.
Often, the function $h$ involves solving a PDE.
In a Bayesian approach,
one obtains the pdf $p(\theta |z)$ of the conditional random
variable $\theta | z$ by Bayes' rule:
\begin{equation}
p(\theta|z)\propto p(\theta)p(z|\theta),
\label{bbe}
\end{equation}
where the likelihood $p(z|\theta)$ can be read off (\ref{equation:obs})
and the prior $p(\theta)$ is assumed to be known.
The goal is to compute this posterior.
This can be done with importance sampling as follows~\cite{ChorinHald,Kalos}.
One can represent the posterior by $M$ weighted samples.
The samples~$\theta_j$, $j=1,\dots,M$ are obtained
from an importance function $\pi(\theta)$
(which is chosen such that it is easy to sample from),
and the $j$th sample is assigned the weight
\begin{equation*}
w_j\propto \frac{p(\theta_j)p(z|\theta_j)}{\pi(\theta_j)}.
\end{equation*}
The location of a sample corresponds to a set of possible parameter values
and the weight describes how likely this set is in view of the posterior.
The weighted samples $\left\{ \theta_j,w_j \right\}$ form an empirical estimate of $p(\theta|z)$,
so that for a smooth function $u$, the sum
\begin{equation*}
E_M(u) = \sum_{j=0}^{M}u(\theta_j)\hat{w}_j,
\end{equation*}
where $\hat{w}_j=w_j/\sum_{j=0}^{M}w_j$,
converges almost surely to the expected value of $u$
with respect to $p(\theta|z)$ as $M \rightarrow \infty$,
provided that the support of $\pi$ includes the support of~$p(\theta|z)$ \cite{Kalos,ChorinHald}.
The importance function must be chosen carefully
or else sampling is inefficient \cite{Blb,Blb2,Sny,CM2013}.
We construct the importance function via implicit sampling,
so that the importance function is large where the posterior pdf is large.
This is done by computing the maximizer of $p(\theta|z)$,
i.e.~the MAP point.
If the prior and likelihood are exponential functions (as they often are in applications),
it is convenient to find the MAP point by minimizing the function
\begin{equation}
\label{equation:FDef}
F(\theta) = -\log\left( p(\theta)p(z|\theta)\right).
\end{equation}
After the minimization,
one finds samples in the neighborhood of the MAP point, $\mu = \argmin F$, as follows.
Choose a reference variable $\xi$ with pdf $g(\xi)$ and let $G(\xi)=-\log(g(\xi))$ and $\gamma=\min_\xi G$.
For each member of a sequence of samples of $\xi$ solve the equation
\begin{equation}
\label{gen}
F(\theta)-\phi=G(\xi)-\gamma,
\end{equation}
to obtain a sequence of samples $\theta$, where $\phi$ is the minimum of $F$.
The sampling weight is
\begin{equation}
w_j\propto J(\theta_j),
\end{equation}
where $J$ is the Jacobian of the one-to-one and onto map $\theta \rightarrow \xi$.
There are many ways to choose this map since (\ref{gen}) is underdetermined \cite{cmt,CT1,Morzfeld2011};
we describe two choices below.
The sequence of samples we obtain by solving (\ref{gen}) is in the neighborhood of the minimizer $\mu$ since,
by construction, equation~(\ref{gen}) maps a likely $\xi$ to a likely $\theta$:
the right hand side of (\ref{gen}) is small with a high probability since
$\xi$ is likely to be close to the mode (the minimizer of $G$);
thus the right hand side is also likely to be small and, therefore,
the sample is in the neighborhood of the MAP point $\mu$.
An interesting construction, related to implicit sampling,
has been proposed in \cite{MarzoukOptimalMaps,Tabak}.
Suppose one wants to generate samples with the pdf $p(\theta|z)$, and have
$\theta$ be a function of a reference variable $\xi$ with pdf $g$, as above.
If the samples are all to have equal weights, one must have, in the notations above,
\begin{equation*}
p(\theta|z)=g(\xi)/J(\xi),
\end{equation*}
where, as above, $J$ is the Jacobian of a map $\theta \rightarrow \xi$.
Taking logs, one finds
\begin{equation}
\label{Marzouk}
F(\theta)+\log\beta=G(\xi)-\log\left(J(\xi)\right),
\end{equation}
where $\beta=\int p(z|\theta)p(\theta)d\theta$ is the
proportionality constant that has been elided in~(\ref{bbe}).
If one can find a one-to-one mapping from $\xi$ to $\theta$ that satisfies this equation,
one obtains an optimal sampling strategy,
where the pdf of the samples matches exactly the posterior pdf.
In \cite{MarzoukOptimalMaps}, this map is found globally by choosing $g=p(\theta)$ (the prior),
rather than sample-by-sample as in implicit sampling.
The main differences between the implicit sampling equation (\ref{gen}) and equation~(\ref{Marzouk})
are the presence of the Jacobian $J$ and of the normalizing constant $\beta$ in the latter;
$J$ has shifted from being a weight to being a term in the equation that picks the samples,
and the optimization that finds the probability mass has shifted to the computation of the map.
If the reference variable is Gaussian and the problem is linear,
equation~(\ref{Marzouk}) can be solved by a linear map with a constant Jacobian
and this map also solves (\ref{gen}),
so that one recovers implicit sampling.
In particular, in a linear Gaussian problem,
the local (sample-by-sample) map (\ref{gen}) of implicit sampling
also solves the global equation (\ref{Marzouk}), which, for the linear problem,
is a change of variables from one Gaussian to another.
If the problem is not linear,
the task of finding a global map that satisfies (\ref{Marzouk})
is difficult (see also \cite{Doucet2000,liuchen1995,Zaritski,Tabak}).
The determination of optimal maps in~\cite{MarzoukOptimalMaps},
based on nonlinear transport theory,
is elegant but can be computationally intensive,
and requires approximations that reintroduce non-uniform weights.
Using (simplified) optimal maps and re-weighing the samples
from approximate maps is discussed in~\cite{Tabak}.
In \cite{Oliver14}, further optimal transport maps from
prior to posterior are discussed.
These maps are exact in linear Gaussian problems,
however in general they are approximate, due to the neglect of a Jacobian,
when the problem is nonlinear.
\subsection{Implementation of implicit sampling for parameter estimation}
Above we assume that the parameter $\theta$ is finite-dimensional.
However, if $h$ involves a PDE, then the parameter may be infinite-dimensional.
In this case, one can discretize the parameter,
e.g.~using Karhunen-Lo\`{e}ve expansions (see below).
The theory of implicit sampling then immediately applies.
A related idea is dimension-\slash mesh-independent MCMC, see,e.g.,
\cite{StuartInverse,Stuart2,Bui,Petra,Martin},
where MCMC sampling algorithms operate efficiently,
independently of the mesh that is used to discretize the PDE.
In particular, the algorithms sample the same finite dimensional sub-space
as the mesh-size is refined.
Below we present an implementation of implicit sampling
that is mesh independent.
\subsubsection{Optimization and multiple-grids}
The first step in implicit sampling is to find the MAP point by
minimizing $F$ in (\ref{equation:FDef}).
Upon discretization of the PDE,
this can be done numerically by Newton, quasi-Newton, or Gauss-Newton methods
(see, e.g.~\cite{Nocedal}).
The minimization requires derivatives of the function $F$,
and these derivatives may not be easy to compute.
When the function $h$ in~(\ref{equation:obs}) involves solving a PDE,
then adjoints are efficient for computing the gradient of $F$.
The reason is that the complexity of solving the adjoint equation is similar to that of
solving the original ``forward'' model.
Thus, the gradient can be computed at the cost of (roughly) two forward solutions of (\ref{equation:obs}).
The adjoint based gradient calculations can be used in connection with a quasi-Newton method, e.g.~BFGS,
or with Gauss-Newton methods.
We illustrate how to use the adjoint method for BFGS optimization in the example below.
One can make use of multiple grids during the optimization.
This idea first appeared in the context of online state estimation in \cite{amc},
and is similar to a multi-grid finite difference method \cite{Fedorenko}
and multi-grid Monte Carlo~\cite{Goodman}.
The idea is as follows.
First, initialize the parameters and pick a coarse grid.
Then perform the minimization on the coarse grid and use the minimizer
to initialize a minimization on a finer grid.
The minimization on the finer grid should require only a few steps,
since the initial guess is informed by the computations on the coarser grid,
so that the number of fine-grid forward and adjoint solves is small.
This procedure can be generalized to use more than two grids (see the example below).
For the minimization, we distinguish between two scenarios.
First, suppose that the physical parameter can have a mesh-independent representation
even if it is defined on a grid.
This happens, for example, if the parameter is represented using
Karhunen-Lo\`{e}ve expansions, where the expansions are later evaluated on the grid.
In this case, no interpolation of the parameter is required in the multiple-grid approach.
On the other hand, if the parameter is defined on the grid,
then the solution on the coarse mesh must be interpolated onto the fine grid,
as is typical in classical multi-grid algorithms (e.g.~for linear systems).
We illustrate the multiple-grid approach in the context
of the numerical example in section~3.
\subsubsection{Solving the implicit equations with linear maps}
Once the optimization problem is done,
one needs to solve the random algebraic equations (\ref{gen}) to generate samples.
There are many ways to solve~(\ref{gen})
because it is an underdetermined equation (one equation in $m$ variables).
We describe and implement two strategies for solving~(\ref{gen}).
For a Gaussian reference variable $\xi$ with mean $0$ and covariance matrix $H^{-1}$,
where $H$ is the Hessian of the function $F$ at the minimum,
the equation becomes
\begin{equation}
\label{equation:SamplingEquation}
F(\theta)-\phi =\frac{1}{2}\xi^TH\xi.
\end{equation}
In implicit sampling with linear maps
one approximates $F$
by its Taylor expansion to second order
\begin{equation}
F_0(\theta) =\phi + \frac{1}{2}(\theta-\mu)^TH(\theta-\mu),
\nonumber
\end{equation}
where $\mu=\mbox{arg min}\, F$ is the minimizer of $F$ (the MAP point) and
$H$ is the Hessian at the minimum.
One can then solve the quadratic equation
\begin{equation}
\label{eq:QuadApprox}
F_0(\theta) -\phi = \frac{1}{2}\xi^TH\xi,
\end{equation}
instead of (\ref{equation:SamplingEquation}),
using
\begin{equation}
\label{eq:LM}
\theta = \mu + \xi.
\end{equation}
The bias created by solving the quadratic equation (\ref{eq:QuadApprox})
instead of (\ref{equation:SamplingEquation})
can be removed by the weights \cite{cmt,amc}
\begin{equation}
\label{eq:weightsLinear}
w\propto \exp\left(F_0(\theta)-F(\theta)\right).
\end{equation}
Note that the algorithm is mesh-independent
in the sense of~\cite{Martin,Bui,Petra}
due to the use of Hessian of~$F$.
Specifically,
the eigenvectors associated with non-zero eigenvalues
of the discrete Hessian span the same
stochastic sub-space as the mesh is refined.
\subsubsection{Solving the implicit equations with random maps}
A second strategy for solving (\ref{gen}) is to use random maps~\cite{Morzfeld2011}.
The idea is to solve (\ref{equation:SamplingEquation})
in a random direction, $\xi$, where $\xi\sim\mathcal{N}(0,H^{-1})$ as before:
\begin{equation}
\label{equation:Ansatz}
\theta = \mu +\lambda(\xi)\,\xi.
\end{equation}
We look for a solution of (\ref{equation:SamplingEquation})
in a $\xi$-direction by substituting~(\ref{equation:Ansatz}) into~(\ref{equation:SamplingEquation}),
and solving the resulting equation for the scalar~$\lambda(\xi)$ with Newton's method.
A formula for the Jacobian of the random map defined by~(\ref{equation:SamplingEquation})
and~(\ref{equation:Ansatz}) was derived in \cite{Morzfeld2011,Goodman},
\begin{equation}
\label{eq:Jacobian}
w \propto \left\vert J(\xi)\right\vert = \left\vert\lambda^{m-1} \;\frac{\xi^TH\xi}{\nabla_\theta F \cdot\xi}\right\vert
\end{equation}
where $m$ is the number of non-zero eigenvalues of $H$,
making it easy to evaluate the weights of the samples if the gradient of $F$ is easy to compute,
e.g.~using the adjoint method (see below).
Note that the random map algorithm is affine invariant and, therefore,
capable of sampling within flat and narrow valleys of $F$.
It is also mesh-independent in the sense of~~\cite{Martin,Bui,Petra},
for the same reasons as the linear map method above.
\section{Application to subsurface flow}
We illustrate the applicability of our implicit sampling method
by a numerical example from subsurface flow,
where we estimate subsurface structures from pressure measurements of flow through a porous medium.
This is a common test problem for MCMC and has applications in
reservoir simulation\slash management (see e.g.~\cite{Dean2007})
and pollution modeling (see e.g.~\cite{Bear}).
We consider Darcy's law
\begin{equation*}
\kappa \nabla p = - \nu u,
\end{equation*}
where $\nabla p$ is the pressure gradient across the porous medium,
$\nu$ is the viscosity and $u$ is the average flow velocity;
$\kappa$ is the permeability and describes the subsurface structures we are interested in.
Assuming, for simplicity, that the viscosity and density are constant,
we obtain, from conservation of mass, the elliptic problem
\begin{equation}\label{equation:upeqn}
-\nabla\cdot \left(\kappa\nabla p\right) = g,
\end{equation}
on a domain $\Omega$,
where the source term $g$ represents externally prescribed inward or outward flow rates.
For example, if a hole were drilled and a constant inflow were applied through this hole,
$g$ would be a delta function with support at the hole.
Here we choose $g=200\pi^2 \sin(\pi x)\sin(\pi y)$.
Equation (\ref{equation:upeqn}) is supplemented with Dirichlet boundary conditions.
The uncertain quantity in this problem is the permeability,
i.e.~$\kappa$ is a random variable,
whose realizations we assume to be smooth enough
so that for each one a solution of \EQ{upeqn} uniquely exists.
We would like to update our knowledge about $\kappa$ on the basis of
noisy measurements of the pressure at $k$ locations
within the domain~$\Omega$ so that \EQ{obs} becomes
\begin{equation}\label{equation:obseqn}
z = h(p(\kappa),x,y) + r.
\end{equation}
Computation requires a discretization of the forward problem \EQ{upeqn}
as well as a characterization of the uncertainty in the permeability before data are collected,
i.e.~a prior for $\kappa$.
We describe our choices for the discretization and prior below.
\subsection{Discretization of the forward problem}
In the numerical experiments below we consider a 2D-problem
and choose the domain $\Omega$ to be the square $[0,1]\times [0,1]$,
and discretize \EQ{upeqn} with a piecewise linear
finite element method on a uniform $(N+1)\times (N+1)$ mesh of triangular elements
with $2(N+1)^2$ triangles~\cite{Braess}.
Solving the (discretized) PDE thus
amounts to solving the linear system
\begin{equation}\label{equation:DEQ}
AP=G,
\end{equation}
where $A$ is a $N^2\times N^2$ matrix,
and where $P$ and $G$
are $N^2$ vectors;
$P$ is the pressure and $G$ contains the discretized right
hand side of the equation \EQ{upeqn}.
For a given permeability~$\kappa$, the matrix $A$ is
symmetric positive definite (SPD) and
we use the balancing domain decomposition by constraints method
\cite{BDDC}
to solve (\ref{equation:DEQ}),
i.e.~we first decompose the computational domain into smaller subdomains
and then solve a subdomain interface problem.
For details of the linear solvers, see~\cite{BDDC}.
In the numerical experiments below,
a $64\times 64$ grid is our finest grid,
and the data, i.e.~the pressure measurements,
are arranged such that they align with grid points
of our finest grids, as well as with the coarse grids
we use in the multiple-grid approach.
Specifically, the data equation~\EQ{obseqn} becomes
\begin{equation*}
z = MP + r,
\end{equation*}
where $M$ is a $k\times N^2$ matrix
that defines at which locations on the (fine) grid we
collect the pressure.
Here we collect the pressure every four grid points,
however exclude a 19 grid points deep layer around the boundary
(since the boundary conditions are known),
so that the number of measurement points is 49.
Collecting data this way allows us to use all
data directly in our multiple
grids approach with $16\times 16$ and $32\times 32$ grids
(see below).
The data are perturbed with a Gaussian random variable
$r\sim\mathcal{N}(0,R)$, with a diagonal covariance matrix $R$
(i.e.~we assume that measurement errors are uncorrelated).
The variance at each measurement location is set
to 30\% of the reference solution.
This relatively large variance brings about significant
non-Gaussian features in the posterior pdf.
\subsection{The log-normal prior, its discretization and dimensional reduction}
The prior for permeability fields is often assumed to be log-normal
and we follow suit.
The logarithm of the permeability $\kappa$ is thus a Gaussian field
with a squared exponential covariance function~\cite{Rasmussen2006},
\begin{equation}\label{equation:correlation}
R(x_1,x_2,y_1,y_2)=\sigma_x^2\sigma_y^2\exp \left(-\frac{\left(x_1-x_2\right)^2}{l_x^2}-\frac{\left(y_1-y_2\right)^2}{l_y^2}\right),
\end{equation}
where $(x_1,y_1)$, $(x_2,y_2)$ are two points in $\Omega$,
and where the correlation length $l_x$ and $l_y$
and the parameters $\sigma_x,\sigma_y$ are given scalars.
In the numerical experiments below, we choose $\sigma_x=\sigma_y=1$
and $l_x=l_y=\sqrt{0.5}$.
With this prior,
we assume that the (log-) permeability is a smooth function of $x$ and $y$,
so that solutions of~(\ref{equation:upeqn}) uniquely exist.
Moreover, the theory presented in \cite{StuartInverse,Stuart2} applies
and a well defined posterior also exists.
We approximate the lognormal prior on the regular $N\times N$ grid
by an $N^2$ dimensional log-normal random variable with
covariance matrix $\Sigma$ with elements $\Sigma(i,j)=R(x_i,x_j,y_i,y_j)$,
$i,j=1,\dots,N$ where $N$ is the number of
grid points in each direction.
To keep the computations manageable (for fine grids and large $N$),
we perform the matrix computations with a low-rank approximation of $\Sigma$
obtained via Karhunan-Lo\`{e}ve~(KL) expansions~\cite{Ghanem,LeMaitre}.
Specifically, the factorization of the covariance function $R(x_1,x_2,y_1,y_2)$
allows us to compute the covariance matrices in $x$ and $y$ directions separately,
i.e.~we compute the matrices
\begin{equation*}
\Sigma_x(i,j) = \sigma_x^2\exp \left(-\frac{(x_i-x_j)^2}{l_x^2}\right),\quad \Sigma_y(i,j)=\sigma_y^2\exp \left(-\frac{(y_i-y_j)^2}{l_y^2}\right).
\end{equation*}
We then compute singular value decompositions (SVD) in each
direction to form low-rank approximations $\hat{\Sigma}_x\approx \Sigma_x$
and $\hat{\Sigma}_y\approx \Sigma_y$ by neglecting small eigenvalues.
These low rank approximations define a low rank approximation of the covariance matrix
\begin{equation*}
\Sigma \approx \hat{\Sigma}_x \otimes \hat{\Sigma}_y,
\end{equation*}
where $\otimes$ is the Kronecker product.
Thus, the eigenvalues and eigenvectors of $\hat{\Sigma}$ are
the products of the eigenvalues and eigenvectors of $\hat{\Sigma}_x$ and $\hat{\Sigma}_y$.
The left panel of Figure 1
shows the spectrum of $\hat{\Sigma}$,
and it is clear that the decay of the eigenvalues of $\Sigma$
is rapid and suggests a natural model reduction.
\begin{figure}
\caption{Spectrum of the covariance matrix of lognormal prior.}
\label{fig:prior}
\end{figure}
If we neglect small eigenvalues (and set them to zero),
then
\begin{equation*}
\hat{\Sigma} = V^T\Lambda V,
\end{equation*}
approximates $\Sigma$ (in a least squares sense in terms
of the Frobenius norms of $\Sigma$ and $\hat{\Sigma}$);
here $\Lambda$ is a diagonal matrix whose diagonal elements
are the $m$ largest eigenvalues of $\Sigma$ and $V$ is an $m\times N$
matrix whose columns are the corresponding eigenvectors.
With $m=30$, we capture $99.9\%$ of the variance
(in the sense that the sum of the first 30 eigenvalues
is 99\% of the sum of all eigenvalues).
In reduced coordinates, the prior is
\begin{equation*}
\hat{K} \sim \ln \mathcal{N}\left(\hat{\mu},\hat{\Sigma}\right).
\end{equation*}
The linear change of variables
\begin{equation*}
\theta = V^T\Lambda^{-0.5}\hat{K},
\end{equation*}
highlights that it is sufficient to estimate $m\ll N^2$ parameters
(the remaining parameters are constrained by the prior).
We will carry out the computations in the reduced
coordinates $\theta$, for which the prior is
\begin{equation}
\label{eq:PriorTheta}
p(\theta)=\mathcal{N}\left(\mu,I_m\right),
\end{equation}
where $\mu= V^T\Lambda^{-0.5}\hat{\mu}$.
Note that this model reduction follows naturally
from assuming that the
permeability is smooth,
so that errors correlate,
and the probability mass localizes
in parameter space.
A similar observation, in connection with
data assimilation, was made in \cite{CM2013}.
Note that we achieve mesh-independence in the
implicit sampling algorithm we propose
by sampling in the $\theta$-coordinates
rather than in the physical coordinate system.
We consider this scenario because it
allows us to compare our approach with MCMC
that also samples in $\theta$-coordinates
and, therefore, also is mesh-independent.
\subsection{BFGS optimization with adjoints and multiple grids}
Implicit sampling requires minimization of $F$ in \EQ{FDef} which,
for this problem and in reduced coordinates, takes the form
\begin{equation*}
F(\theta)= \frac{1}{2} \theta^T\theta+\frac{1}{2}\left(z-MP(\theta)\right)^TR^{-1}\left(z-MP(\theta)\right).
\end{equation*}
We solve the optimization problem using BFGS
coupled to an adjoint code to compute the gradient of $F$ with respect to $\theta$
(see also, e.g.~\cite{Oliver2011,Hinze}).
The adjoint calculations are as follows.
The gradient of $F$ with respect to $\theta$ is
\begin{equation*}
\nabla_{\theta} F(\theta)= \theta +\left(\nabla_{\theta} P(\theta)\right)^T W,
\end{equation*}
where $W=-M^TR^{-1}(z-MP(\theta))$. We use the chain rule to derive
$\left(\nabla_{\theta} P(\theta)\right)^T W $ as follows:
\begin{equation*}
(\nabla_{\theta} P(\theta))^TW=\left(\nabla_{K}
P(\theta)\frac{\partial K}{\partial \hat{K}}\frac{\partial
\hat{K}}{\partial \theta} \right)^T W
=\left(\nabla_{K}
P(\theta) e^{\hat{K}} V\Lambda^{0.5}\right)^TW
=\left(V\Lambda^{0.5}\right)^T\left(\nabla_{K}
P(\theta)e^{\hat{K}} \right)^T W,
\end{equation*}
where $e^{\hat{K}}$ is a $N^2\times N^2$ diagonal matrix whose elements are the exponentials of the components of $\hat{K}$.
The gradient $\nabla_{K} P(\theta)$ can be obtained directly from our
finite element discretization. Let $P=P(\theta)$ and let $K_{l}$ be the $l$th component of $K$, and take the derivative with respect to $K_{l}$ of \EQ{DEQ} to obtain
\begin{equation*}
\frac{\partial P}{\partial K_{l} } =- A^{-1}
\frac{\partial A}{\partial K_{l}} P
\end{equation*}
where $\partial A/\partial K_{l}$ are component-wise derivatives. We
use this result to obtain the following expression
\begin{equation}\label{gradF}
\left(\nabla_{K}
P(\theta) e^{\hat{K}}\right)^T W=-\left(e^{\hat{K}}\right)^T\left[ \begin{array}{c}
P^T\frac{\partial A}{\partial K_{1}}\left(A^{-T} W \right)\\
\vdots \\
P^T\frac{\partial A}{\partial K_{N^2}} \left(A^{-T} W \right)
\end{array}
\right].
\end{equation}
When $P$ is available, the most expensive part in (\ref{gradF}) is to evaluate $A^{-T}W$,
which is equivalent to solving the adjoint
problem (which is equal to itself for this self-adjoint problem). The
rest can be computed element-wise by the definition of $A$. Note that
there are only a fixed number of nonzeros in each $\frac{\partial
A}{\partial K_l}$, so that the additional work for solving the adjoint
problem in (\ref{gradF}) is about $O(N^2)$,
which is small compared to the work required for the adjoint solve.
Collecting terms we finally obtain the gradient
\begin{eqnarray*}
\nabla_\theta F(\theta)= \theta+\left(V\Lambda^{0.5}\right)^T\left(\nabla_{K} P(\theta) e^{\hat{K}}\right)^TW
&=&\theta-\left(V\Lambda^{0.5}\right)^T\left(e^{\hat{K}}\right)^T\left[ \begin{array}{c}
P^T\frac{\partial A}{\partial K_{1}}\left(A^{-T} W \right)\\
\vdots \\
P^T\frac{\partial A}{\partial K_{N^2}} \left(A^{-T} W \right)
\end{array}
\right]
.
\end{eqnarray*}
Multiplying by
$\left(V\Lambda^{0.5}\right)^T$ to
go back to physical coordinates will require an additional work of $O(mN^2)$.
Note that the adjoint calculations for the gradient
require only one adjoint solve because the forward solve
(required for $P$) has already been done before the gradient calculation in the BFGS algorithm.
This concludes our derivation of an adjoint method for gradient computations.
The gradient is used in a BFGS method
with a cubic interpolation line search (see \cite[Chapter~3]{Nocedal}).
We chose this method here because it defaults to taking the full step (of length 1)
without requiring additional computations,
if the full step length satisfies the Wolfe conditions.
To reduce the number of fine-grid solves we use
the multiple grid approach described above with $16\times 16$,
$32\times 32$ and $64\times 64$ grids.
We initialize the minimization on the course grid with the mean of the prior,
and observe a convergence after about 9 iterations,
requiring 16 function and 16 gradient evaluations,
which corresponds to a cost of 32 coarse grid solves
(estimating the cost of adjoint solves with the cost of forward solves).
The result is used to initialize an optimization on a finer $32\times 32$ grid.
The optimization on $32\times 32$ grid converges in 6 iterations,
requiring 7 function and 7 gradient evaluations
(at a cost of 14 medium grid solves).
The solution on the medium grid is then used to initialize
the finest $64\times 64$ grid optimization.
This optimization converges in 5 iterations, requiring
12 fine-grid solves.
We find the same minimum without the multiple grid approach,
i.e.~if we solve the minimization on the fine grid,
however these computations require 36 fine grid solves.
The multiple-grids approach we propose
requires about 17 fine grid solves
(converting the cost of coarse-grid solves to fine-grid solves)
and, thus, could significantly reduce
the number of required fine-grid solves.
\subsection{Implementation of the random and linear maps}
Once the minimization is completed,
we generate samples using either the linear map
or random map methods described above.
Both require the Hessian of $F$ at
the minimum.
We have not coded second-order adjoints,
and computing the Hessian with finite differences
requires $m(m+1) = 930$ forward solutions,
which is expensive (and if $m$ becomes large,
this becomes infeasible).
Instead, we approximate the Hessian.
We found that the approximate Hessian of our BFGS
is not accurate enough to lead to a good implicit sampling method.
However, the Hessian approximation proposed in
\cite{Iglesias2} and often used in LMAP,
\begin{equation}
\label{eq:approxHess}
H \approx \hat{H} = I - Q^T(QQ^T+R)^{-1}Q,
\end{equation}
where $Q = M\,\nabla_\theta P$,
leads to good results (see below).
Here the gradient of the pressure (or the Jacobian)
is computed with finite differences,
which requires $m+1$ forward solves.
Note that the approximation is exact for linear Gaussian problems.
With the approximate Hessian, we define $L$
in (\ref{eq:LM}) and (\ref{equation:Ansatz})
as a Cholesky factor of $\hat{H}=LL^T$.
Generating samples with the random map method requires solving
(\ref{equation:SamplingEquation}) with the ansatz (\ref{equation:Ansatz}).
We use a Newton method for solving these equations
and observe that it usually converges quickly
(within 1-4 iterations).
Each iteration requires a derivative of $F$
with respect to $\lambda$,
which we implement using the adjoint method,
so that each iteration requires two forward solutions.
In summary, the random map method requires
between 2-8 forward solutions per sample.
The linear map method requires generating a sample
using (\ref{eq:LM}) and weighing it by (\ref{eq:weightsLinear}).
Evaluation of the weights thus requires one forward solve.
Neglecting the cost for the remaining linear algebra,
the linear map has a cost of 1 PDE solve per sample.
We assess the quality of the weighted samples
by the variance of the weights:
the sampling method is good if the
variance of the weights is small.
In particular, if the weights are constant,
then this variance is zero and the sampling method is perfect.
The variance of the weights is equal to $R-1$, where
\begin{equation*}
R = \frac{E(w^2)}{E(w)^2}.
\end{equation*}
In fact, $R$ itself can be used to measure the quality
of the samples \cite{VEW12,AMGC02}.
If the variance of the weights is small,
then $R\approx 1$.
Moreover, the effective sample size,
i.e.~the number of unweighted samples that would be
equivalent in terms of statistical accuracy to the
set of weighted samples,
is about $M/R$ \cite{VEW12},
where $M$ is the number of samples we draw.
In summary, an $R$ close to one indicates
a well distributed weighted ensemble.
We evaluate $R$ for 10 runs with $M=10^4$ samples
for each method and find, for the linear map method,
a mean of $1.79$ and standard deviation $0.014$,
and for the random map method a mean of $1.77$
and standard deviation $0.013$.
The random map method thus performs slightly better,
however the cost per sample is also slightly larger
(because generating a sample requires solving
(\ref{equation:SamplingEquation}), which in turn
requires solving the forward problem).
Because the linear map method produces weighted ensembles of
about the same quality as the random map,
and since the linear map is less expensive and easier to program,
we conclude that the linear map is a more natural choice for this example.
We have also experimented with symmetrization of implicit sampling~\cite{GLM}.
The idea is similar to the classic trick of antithetic variates~\cite{Kalos}.
The symmetrization of the linear map method is as follows.
Sample the reference variable to obtain a $\xi$ and compute a sample $x^+$ using~(\ref{eq:LM}).
Use the same $\xi$ to compute $x^- = \mu-L^{-T}\xi$.
Then pick $x$ with probability $p^+ = w(x)/(w(x)+w(x^-))$
and pick $x^-$ with probability $p^- =w(x^-)/(w(x)+w(x^-))$,
and assign the weight $w^s = (w(x^+)+w(x^-))/2$.
This symmetrization can lead to a smaller $R$,
i.e.~a better distributed ensemble, in the small noise limit.
In our example, we compute the quality measure $R$ of 1.67.
While this $R$ is smaller than for the non-symmetrized methods,
the symmetrization does not pay off in this example,
since each sample of the symmetrized method requires two
forward solves (to evaluate the weights).
Note that we neglect computations other than the forward model
evaluations when we estimate the computational cost of the sampling algorithms
(as we did with the BFGS optimization as well).
This is justified because computations with $\theta$
(e.g.~generating a sample using the linear map method)
is inexpensive due to the model reduction via Karhunen-Lo\`{e}ve.
We illustrate how our algorithms can be used
by presenting results of a typical run and for a typical problem set-up
in terms of e.g.~strength of the observation noise and the number of observations.
We tested our algorithms in a variety of other settings as well,
and observed that our methods operate reliably in different problem set-ups,
however found that many of the problems one can set up are
almost Gaussian problems and therefore easy to solve.
We present here a case where the large observation noise (see above)
brings about significant non-Gaussian features in the posterior.
Shown in Figure 2 are the true permeability
(the one we use to generate the data) on the left,
the mean of the prior in the center,
and the conditional mean we computed with the linear map
method and $10^4$ samples on the right.
\begin{figure}
\caption{Left: true permeability that generated the data.
Center: mean of prior.
Right: conditional mean computed with implicit sampling with random maps.}
\label{fig:result}
\end{figure}
We observe that the prior is not very informative,
in the sense that it underestimates the permeability considerably.
The conditional mean captures most of the large scale features,
such as the increased permeability around $x=0.7$, $y=0.8$,
however, there is considerable uncertainty in the posterior.
Figure 3 illustrates this uncertainty
and shows four samples of the posterior.
\begin{figure}
\caption{Four samples of the posterior generated by implicit sampling with random maps.}
\label{fig:PosteriorSamples}
\end{figure}
These samples are obtained by resampling the weighted ensemble,
so that one is left with an equivalent unweighted set of samples,
four of which are shown in Figure 4.
The four samples correspond to rather different subsurface structures.
If more accurate and more reliable estimates of the permeability
are required, one must increase the resolution of the data
or reduce the noise in the data.
\subsection{Connections with other methods}
We discuss connections of our implicit sampling schemes
with other methods that are in use in subsurface flow
parameter estimation problems.
\subsubsection{Connections with linearization about the MAP}
One can estimate parameters by computing the MAP point,
i.e.~the most likely parameters in view of the data
\cite{Iglesias2,Oliver2011}.
This method, sometimes called the MAP method,
can make use of the multiple grids approach presented here,
however represents an incomplete solution to the Bayesian parameter estimation problem,
because the uncertainty in the parameters may be large,
however the MAP point itself contains no information
about this uncertainty.
To estimate the uncertainty of the MAP point,
one can use linearization about the MAP point (LMAP)
\cite{Iglesias2,Oliver2011,Dean2007,Bui},
in which one computes the MAP point and the Hessian
of $F$ at the MAP point and uses the inverse of this Hessian
as a covariance.
The cost of this method is the cost
of the optimization plus the cost of computing the Hessian.
For the example above,
LMAP overestimates the uncertainty
and gives a standard deviation of $0.61$
for the first parameter $\theta_1$.
The standard deviation we compute with the
linear map and random map methods however is $0.31$.
The reason for the over-estimation of the uncertainty with LMAP
is that the posterior is not Gaussian.
This effect is illustrated in Figure~4
where we show histograms of the marginals
of the posterior for the first four parameters
$\theta_1,\theta_2,\theta_3,\theta_4$,
along with their Gaussian approximation as in LMAP.
\begin{figure}
\caption{Marginals of the posterior computed with
implicit sampling with random maps and their Gaussian
approximation obtained via LMAP.
Top left: $p(\theta_1|z)$.
Top right: $p(\theta_2|z)$.
Bottom left: $p(\theta_3|z)$.
Bottom right: $p(\theta_4|z)$.
}
\label{fig:Gaussian}
\end{figure}
We also compute the skewness and excess kurtosis
for these marginal densities.
While the marginals for the parameters may become
``more Gaussian'' for the higher order coefficients
of the KL expansion, the joint posterior exhibits significant non-Gaussian behavior.
Since implicit sampling (with random or linear maps)
does not require linearizations
or Gaussian assumptions, it can correctly capture these
non-Gaussian features.
In the present example, accounting for the non-Gaussian\slash
nonlinear effects brings about a reduction of the uncertainty
(as measured by the standard deviation) by a factor of two in the parameter $\theta_1$.
Note that code for LMAP,
can be easily converted into an implicit sampling code.
In particular, implicit sampling with linear maps
requires the MAP point and an approximation of the Hessian
at the minimum. Both can be computed with LMAP codes.
Non-Gaussian features of the posterior can then be captured
by weighted sampling, where each sample comes at a cost
of a single forward simulation.
\subsubsection{Connections with Markov Chain Monte Carlo}
Another important class of methods
for solving Bayesian parameter estimation problems is MCMC
(see e.g.~\cite{Iglesias2} for a discussion of MCMC in subsurface flow problems).
First we compare implicit sampling with Metropolis MCMC \cite{Liu2008},
where we use an isotropic Gaussian proposal density,
for which we tuned the variance to achieve an acceptance rate of about 30\%.
This method requires one forward solution per step
(to compute the acceptance probability).
We start the chain at the MAP (to reduce burn-in time).
In figure~5 we show the approximation of the conditional mean
of the variables $\theta_1,\theta_2$, and $\theta_5$,
as a function of the number of steps in the chain (left)
to illustrate the behavior of the MCMC chain.
\begin{figure}
\caption{Expected value as a function of the number of samples (left),
and as a function of required forward solves (right).
Red: MCMC. Turquoise: implicit sampling with random maps
and approximate Hessian (dashed) and finite difference Hessian (solid).
Blue: implicit sampling with linear maps
and approximate Hessian (dashed) and finite difference Hessian (solid).}
\end{figure}
We observe that, even after $10^4$ steps,
the chain has not settled,
in particular for the parameter $\theta_2$ (see bottom left).
With implicit sampling we observe a faster convergence,
in the sense that the approximated conditional mean does not
change significantly with the number of samples.
In fact, about $10^2$ samples are sufficient for accurate estimates
of the conditional mean.
As a reference solution, we also show results we obtained
with implicit sampling (with both random and linear maps)
for which we used a Hessian computed with
finite differences (rather than with the approximation in
equation~(\ref{eq:approxHess})).
The cost per sample of implicit sampling
and the cost per step of Metropolis MCMC are different,
and a fair comparison of these methods should take
these costs into account.
In particular, the off-set cost of the minimization and
computation of the Hessian, required for implicit sampling
must be accounted for.
We measure the cost of the algorithms by the number of
forward solves required (because all other computations
are negligible due to the model reduction).
The results are shown for the parameters $\theta_1,\theta_2$
and $\theta_5$ in the right panels of Figure~5.
We find that the fast convergence
of implicit sampling makes up for the
relatively large a priori cost
(for minimization and Hessian computations).
In fact, the figure suggests that the random and linear map methods require
about $10^3$ forward solves while Metropolis MCMC converges slower
and shows significant errors even after running the chain for $10^4$ steps,
requiring $10^4$ forward solves.
The convergence of Metropolis MCMC can perhaps be increased
by further tuning, or by choosing a more advanced transition density.
Implicit sampling on the other hand requires little tuning
other than deciding on standard tolerances for the optimization.
Moreover, implicit sampling generates independent samples
with a known distribution, so that issues such as
determining burn-in times, auto-correlation times and
acceptance ratios, do not arise.
It should also be mentioned that implicit sampling
is easy to parallelize (it is embarrassingly parallel
once the optimization is done).
Parallelizing Metropolis MCMC on the other hand is not trivial,
because it is a sequential technique.
Finally, we discuss connections of our proposed implicit sampling methods to a new MCMC
method, stochastic Newton MCMC \cite{Martin}.
In stochastic Newton one first finds the MAP point (as in implicit sampling)
and then starts a number of MCMC chains from the MAP point.
The transition probabilities are based on local information about
$F$ and make use of the Hessian of $F$, evaluated at the location of the chain.
Thus, at each step, a Hessian computation is required which,
with our finite difference scheme,
requires 31 forward solves (see above) and, therefore,
is expensive (compared to generating samples with implicit sampling,
which requires computing the Hessian only once).
Second-order adjoints (if they were available)
do not reduce that cost significantly.
We have experimented with stochastic Newton in our example
and have used 10--50 chains
and taking about 200 steps per chain.
Without significant tuning,
we find acceptance rates of only a few percent,
leading to a slow convergence of the method.
We also observe that the Hessian may not be positive definite
at all locations of the chain and, therefore,
can not be used for a local Gaussian transition probability.
In these cases, we use a modified Cholesky algorithm
(for affine invariance) to obtain a definite matrix that can be used
as a covariance of a Gaussian.
In summary, we find that stochastic Newton MCMC
is impractical for this example because
the cost of computing the Hessian is too large with our
finite differences approach.
Variations of stochastic Newton were explained in~\cite{Petra}.
The stochastic Newton MCMC with MAP-based Hessian
is the stochastic Newton method as above,
however the Hessian is computed only at the MAP point
and then kept constant throughout the chain.
The ``independence sampling with a MAP point-based Gaussian proposal'' (ISMAP)
is essentially an MCMC version of the linear map method described above.
The ISMAP MCMC method is to use the Gaussian approximation
of the posterior probability at the MAP point as the proposal density for MCMC.
The samples are accepted or rejected based on the weights of the linear map method
described above.
ISMAP is also easier to parallelize than stochastic Newton or stochastic Newton
with MAP-based Hessian.
\section{Conclusions}
We explained how to use implicit sampling
to estimate the parameters in PDE
from sparse and noisy data.
The idea in implicit sampling is to find
the most likely state, often called
the maximum a posteriori (MAP) point,
and generate samples that explore the
neighborhood of the MAP point.
This strategy can work well if
the posterior probability mass localizes around the MAP point,
which is often the case when the data constrain the parameters.
We discussed how to implement these ideas efficiently
in the context of parameter estimation problems.
Specifically, we demonstrated that our approach is
mesh-independent in the sense that we sample
finite dimensional subspaces even when the grid is refined.
We further showed how to use multiple grids
to speed up the required optimization,
and how to use adjoints for the optimization and during sampling.
Our implicit sampling approach has the advantage that
it generates independent samples, so that issues connected with MCMC,
e.g.~estimation of burn-in times, auto-correlations of the samples, or tuning of
acceptance ratios, are avoided.
Our approach is also fully nonlinear and captures non-Gaussian
features of the posterior (unlike linear methods such as
the linearization about the MAP point) and is easy to parallelize.
We illustrated the efficiency of our approach
in numerical experiments with an elliptic inverse problem
that is of importance in applications to
reservoir simulation\slash management and pollution modeling.
The elliptic forward model is discretized using finite elements,
and the linear equations are solved by balancing domain decomposition by constraints.
The optimization required by implicit sampling is done with
with a BFGS method coupled to an adjoint code for gradient calculations.
We use the fact that the solutions are expected to be smooth
for model order reduction based on Karhunan-Lo\`{e}ve expansions,
and found that our implicit sampling approach can exploit this low-dimensional structure.
Moreover, implicit sampling is about an order of magnitude
faster than Metropolis MCMC sampling (in the example we consider).
We also discussed connections and differences of our approach
with linear\slash Gaussian methods, such as linearization about the MAP,
and with stochastic Newton MCMC methods.
\end{document} |
\begin{document}
\title{A Complete Equational Theory for Quantum~Circuits}
\thispagestyle{plain}
\pagestyle{plain}
\author{\IEEEauthorblockN{Alexandre Clément\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\href{mailto:[email protected]}{[email protected]}\\
\url{https://orcid.org/0000-0002-7958-5712}\\
\url{https://members.loria.fr/AClement}}
\and
\IEEEauthorblockN{Nicolas Heurtel\IEEEauthorrefmark{2}\IEEEauthorrefmark{3}}
\IEEEauthorblockA{\href{mailto:[email protected]}{[email protected]}\\
\url{https://orcid.org/0000-0002-9380-8396}}
\and
\IEEEauthorblockN{Shane Mansfield\IEEEauthorrefmark{2}}
\IEEEauthorblockA{\href{mailto:[email protected]}{[email protected]}}
\linebreakand
\IEEEauthorblockN{Simon Perdrix\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\href{mailto:[email protected]}{[email protected]}\\
\url{https://orcid.org/0000-0002-1808-2409}\\
\url{https://members.loria.fr/SPerdrix}}
\and
\IEEEauthorblockN{Beno\^\i t Valiron\IEEEauthorrefmark{3}}
\IEEEauthorblockA{\href{mailto:[email protected]}{[email protected]}\\
\url{https://orcid.org/0000-0002-1008-5605}\\
\url{https://www.monoidal.net}}
\linebreakand
\IEEEauthorblockA{\IEEEauthorrefmark{1}Universit\'e de Lorraine, CNRS,\\
Inria, LORIA\\
F-54000 Nancy, France}
\and
\IEEEauthorblockA{\IEEEauthorrefmark{2}Quandela\\
7 Rue Léonard de Vinci\\
91300 Massy, France}
\and
\IEEEauthorblockA{\IEEEauthorrefmark{3}Université Paris-Saclay, CentraleSupélec,\\
Inria, CNRS, ENS Paris-Saclay,\\
Laboratoire Méthodes Formelles\\
91190, Gif-sur-Yvette, France}
}
\maketitle
\begin{abstract}
We introduce the first complete equational theory for
quantum circuits. More precisely, we introduce a set of circuit
equations that we prove to be sound and complete: two circuits
represent the same unitary map if and only if they can be
transformed one into the other using the equations. The proof is
based on the properties of multi-controlled gates -- that are
defined using elementary gates -- together with an encoding of
quantum circuits into linear optical circuits, which have been
proved to have a complete axiomatisation.
\end{abstract}
\section{Introduction}
Quantum computation is the art of manipulating the states of objects
governed by the laws of quantum physics in order to perform
computation. The standard model for quantum computation is the
\emph{quantum co-processor model}: an auxiliary device, hosting a
quantum memory. This coprocessor is then interfaced with a classical
computer: the classical computer sends the co-processor a series of
instructions to update the state of the memory. The standard formalism
for these instructions is the \emph{circuit model}
\cite{deutsch-circuit}. Akin to boolean
circuits, in quantum circuits wires represent \emph{quantum bits} and
boxes elementary operations -- \emph{quantum gates}. The mathematical
model is however very different: quantum bits (qubits) correspond to vectors in
a $2$-dimensional
Hilbert space, gates to unitary maps and parallel composition to the
tensor product -- the Kronecker product.
Quantum circuits currently form the \emph{de facto} standard for
representing low-level, logical operations on a quantum memory. They
are used for everything: resource estimation
\cite{green2013quipper}, optimization
\cite{amy2014polynomial,duncan2020graph,PhysRevA.102.022406,maslov2005quantum,maslov2008quantum,nam2018automated}, satisfaction of hardware
constraints \cite{kissinger2019cnot,nash2020quantum}, \textit{etc}.
However, as ubiquitous to quantum computing as they are, the graphical
language of quantum circuits has never been fully
formalized.
In particular, a \emph{complete
equational theory} has been a longstanding open problem for 30 years
\cite{aaronson-slide}. It would make it possible to directly prove
properties such as circuit equivalence without having to rely on
ad-hoc set of equations. So far, complete equational theories were
only known for non-universal fragments, such as circuits acting on at
most two qubits \cite{bian2022generators,coecke2018zx}, the stabilizer
fragment \cite{makary2021generators,ranchin2014complete}, the
CNot-dihedral fragment \cite{Amy_2018}, or fragments of reversible
circuits~\cite{iwama2002transformation,cockett2018categorycnot,cockett2018categorytof}.
Interestingly enough, other diagrammatic languages for quantum
computation have been developed on sound foundations: it is
reasonable to think
that this could help in developing a complete equational theory for circuits.
Arguably the strongest candidate has been the ZX-calculus
\cite{coecke2008interacting,coecke2011interacting},\footnote{or its
variants like ZH \cite{Backens_2019} and ZW \cite{DBLP:conf/lics/HadzihasanovicN18}, sharing several similar
properties.} equipped with complete equational theories
\cite{jeandel2018complete,hadzihasanovic2018two,jeandel2018diagrammatic,lmcs:6532,vilmart2019near}. The ZX-calculus shares the same
underlying mathematical representation for states: wires corresponds
to Hilbert spaces and parallel composition to the tensor
operation. Nonetheless, the completeness of the
ZX-calculus does not lead \textit{a priori} to a complete equational theory for
quantum circuits. The reason lies in the expressiveness of the
ZX-calculus and the \emph{non-unitarity} of some of its generators. Any
quantum circuit can be straightforwardly seen as a ZX-diagram. On the
other hand, a ZX-diagram does not necessarily represent a unitary map, and
even when it does, extracting a corresponding quantum circuit
is known to be a hard task in general \cite{duncan2020graph,de2022circuit}.
Another example of a quantum language with a complete equational theory
is the LOv-calculus, a language for linear optical quantum circuits
for which a simple complete equational theory has recently been
introduced \cite{clement2022LOv}. While both linear optical and
regular quantum circuits are universal for unitary transformations,
they do not share the same structure. In particular, if the parallel
composition of quantum circuits corresponds to the tensor product, for
linear optical circuits it stands for the \emph{direct sum}.
In this paper, we introduce the first complete equational theory for quantum
circuits, by first closing the gap between regular and linear optical quantum
circuits.
Despite the seemingly incompatible approaches to parallel
composition, our completeness result derives from the completeness result
for linear optical circuits. Indeed, unlike ZX-generators, linear
optical components are unitary, making it possible write a translation
in both directions.
The complete equational theory for quantum circuits is derived from
the completeness of the LOv-calculus as follows: equipped with maps
for encoding (from quantum circuits to linear optical circuits) and decoding
(from linear optical circuits to quantum circuits), one can roughly speaking
prove completeness for quantum circuits as long as its equational
theory is powerful enough to derive a finite number of equations,
those corresponding to the decoding of the equations of the complete
equational theory for linear optical circuits.
Due to the difference in its interpretation in both kinds of circuits, the parallel composition is not preserved by the encoding nor the decoding maps. The translations are actually based on a sequentialisation of circuits, since the translation of a local gate (acting on at most two wires) is translated as a piece of circuit acting potentially on all wires. Technically, it forces to work with a raw version of circuits, as a circuit may lead to a priori distinct translations depending on the choice of the sequentialisation. Moreover, a single linear optical gate like a phase shifter (which consists in applying a phase on a particular basis state) is decoded as a piece of circuits that can be interpreted as a multi-controlled gate acting on all qubits. As we choose to stick with the usual generators of quantum circuits acting on at most two qubits, multi-controlled gates are inductively defined and we introduce an equational theory powerful enough to prove the basic algebra of multi-controlled gates, necessary to finalise the proof of completeness.
The paper is structured as follows. We first introduce a set of
structural relations for quantum circuits generated by the standard
elementary gates: Hadamard, Phase-rotations, and CNot. We define
multi-controlled gates using these elementary gates, and show that the
basic algebra of multi-controlled gates can be derived from the
structural relations. In addition to the structural equations, we
introduce Euler-angle-based equations. We then proceed to the proof of
completeness, based on a back-and-forth translation from quantum
circuits to linear optical circuits.
\section{Quantum Circuits}
\label{sec:QC}
In quantum computation, circuits ---such as quantum circuits or
optical quantum circuits--- are graphical descriptions of quantum
processes. Akin to (conventional) boolean circuits, circuits in
quantum computations are built from wires (oriented from left to
right), representing the flow of information, and gates, representing
operations to update the state of the system.
Every circuit comes with a set of input wires (incoming the circuit
from the left) and a set of output wires (exiting the circuit on the
right).
\subsection{Graphical languages}\label{sec:raw}
To provide a formal definition of circuits, we first use the notion of
\emph{raw circuits}.\footnote{\emph{Raw terms} are for instance
similarly used \cite{Pawel} as an intermediate step in the defintion
of prop.} Given a set of generators, one can generate a \emph{raw
circuit} by means of iterative sequential ($\circ$) and parallel
($\otimes$) compositions. For instance, given the elementary gates
$\gh$ and $\gppifour$ (with one input and one output) and $\gcnot$ (with
two inputs and two outputs), one can construct the raw circuit
$\gcnot \circ ((\gh\otimes \gppifour)\circ \gcnot)$. Notice that a
sequential composition $C'\circ C$ requires that the number of outputs
of $C$ matches the number of inputs of $C'$. This raw circuit can be
depicted by gluing the generators together and using boxes to witness
how the generators have been composed:
\[{\scalebox{.8}{\tikzfig{bare-circuit-ex3}}}\]
To avoid the use of boxes and recover the intuitive notion of
circuits, we formally define circuits as a prop \cite{prop}, which
consists in considering the raw circuits up to the rules given in
Figure \ref{fig:axiom}. More precisely, a prop generated by a set $\mathcal G$ of
elementary gates is the collection of raw circuits generated by
$\mathcal G\cup \{\gid, \gswap,\tikzfig{diagrammevide-ss}\}$\footnote{
$\gid$ denotes the identity, $\gswap$ the swap and
$\tikzfig{diagrammevide-ss}$ the empty circuit.} quotiented by the
equations of Figure \ref{fig:axiom}.
\begin{figure*}
\caption{Definition of $\equiv$ for raw circuits (either raw quantum
circuits or raw optical circuits).\label{fig:axiom}
\label{idneutre}
\label{assoccomp}
\label{videneutre}
\label{naturaliteswap}
\label{assoctens}
\label{mixedprod}
\label{doubleswap}
\label{fig:axiom}
\end{figure*}
The use of the prop formalism guarantees that circuits can be depicted
graphically without ambiguity. Circuits are thus defined up to
deformations, as for instance:
\[
\scalebox{.7}{\tikzfig{ex1Lb}\ =\ \tikzfig{ex1Rb}}.
\]
\subsection{Quantum circuits: Syntax and semantics}
We consider quantum circuits defined on the following standard set of
generators: Hadamard, Control-Not, and Phase-gates together with
global phases.
\begin{definition}
\label{def:QC}
Let $\textup{\textbf {QC}}$ be the prop
generated
by $\gh$, $\gcnot$, and for any $\varphi \in \mathbb R$, $\gp$ and
$\gs$.
\end{definition}
\begin{figure*}
\caption{Usual abbreviations of quantum circuits.\label{fig:shortcut}
\label{zgate}
\label{xgate}
\label{RXgate}
\label{CNot-def}
\label{NotC-def}
\label{fig:shortcut}
\end{figure*}
The gates $\gh$ and $\gp$ have one input and one output,
while $\gcnot$ has two and $\gs$ zero. A quantum circuit $C$ with $n$ inputs
and $n$ outputs is called a $n$-qubit circuit. Given an $n$-qubit
circuit $C$, the corresponding unitary map $\interp C$ is acting on the
Hilbert space
$\mathbb C^{\{0,1\}^n}=\textup{span}(\ket{x}, x\in
\{0,1\}^n)$:\footnote{We use the standard Dirac
notations.}
\begin{definition}[Semantics]
For any $n$-qubit quantum circuit $C$, let
$\interp C: \mathbb C^{\{0,1\}^n} \to \mathbb C^{\{0,1\}^n}$ be the
linear map inductively defined as follows:
$\interp{C_2\circ C_1} = \interp{C_2}\circ\interp{ C_1}$,
$\interp{C_1\otimes C_3} = \interp{C_1}\otimes\interp{ C_3}$, and
$\forall x,y\in \{0,1\}$, $\forall \varphi \in \mathbb R$,
\begin{align*}
\interp{\gh} &=\ket x \mapsto {\textstyle\frac{1}{\sqrt2}}(\ket 0 +(-1)^x\ket 1),
\\
\interp{\gp} &= \ket x\mapsto e^{ix\varphi}\ket x,
\\
\interp\gid &= \ket x\mapsto \ket x,
\\
\interp{\gcnot} &=\ket{x,y}\mapsto \ket{x,x\oplus y},
\\
\interp{\gswap} &= \ket{x,y}\mapsto \ket{y,x},
\\
\interp{\gs} &= 1\mapsto e^{i\varphi},
\\
\interp{\emptyc} &= 1\mapsto 1.
\end{align*}
\end{definition}
\begin{remark}
Although the definition of $\interp .$ relies on the inductive
structure of raw quantum circuits, it is well-defined on quantum
circuits as for any raw quantum circuits $C,C'$, whenever $C\equiv C'$
we have $\interp C = \interp{C'}$.
\end{remark}
\begin{proposition}[Universality \cite{Barenco1995gates}]
For any $n$-qubit
unitary map $U$ acting on $\mathbb C^{\{0,1\}^n}$,
there exists an $n$-qubit circuit $C$ such that $\interp C=U$.\qed
\end{proposition}
We use standard shortcuts in the description of quantum circuits,
given in \cref{fig:shortcut}.
In textual description, we sometimes use CNot, $s(\varphi)$, $X$, $P(\varphi)$, \emph{etc} to denote respectively $\gcnot$, $\gs$, $\gx$, $\gp$, \emph{etc}.
Moreover, when the parameters (e.g. $\varphi$) are not specific values they can take arbitrary ones.
We write $R_X(\theta)$ for the so-called
$X$-rotation \cite{NielsenChuang}, whereas the standard phase gate $P(\varphi)$ is a $Z$-rotation
only up to a global phase. As a consequence, they have a slightly
different behaviour: $P$ is $2\pi$-periodic: $\interp{P(2\pi)}=I$,
whereas $R_X$ is $4\pi$-periodic, and we instead have
$\interp{R_X(2\pi)} = -I$.
\subsection{Structural equations}
We introduce a set $\qczero$ of \emph{structural equations} on quantum
circuits in \cref{eq:qc0}. These equations are structural in the sense
that the transformations on the parameters are only based on the fact
that $\mathbb R$ is an additive group. In particular, these equations
are valid for any reasonable\footnote{I.e. which forms an additive
group and contains $\pi/2$.} restriction on the angles.
We write $\qczero\vdash C_1 = C_2$ when $C_1$ can be transformed into
$C_2$ using the equations of \cref{eq:qc0}.\footnote{More formally,
$\qczero\vdash \cdot = \cdot$ is defined as the smallest congruence
which satisfies equations of Figures \ref{fig:axiom} and
\ref{eq:qc0}.}
\begin{proposition}
The structural equations of \cref{eq:qc0} are sound, i.e. if
$\qczero\vdash C_1 = C_2$ then $\interp{C_1} = \interp{C_2}$.
\end{proposition}
\begin{figure*}
\caption{\label{eq:qc0}
\label{HH}
\label{S0}
\label{SS}
\label{P0}
\label{CNotCNot}
\label{CNotX}
\label{CNotlift}
\label{tripleCNotswap}
\label{commutationPctrl}
\label{commutationCNothaut}
\label{PP}
\label{XPX}
\label{CZ}
\label{commutationdecompctrlPRY}
\label{commutationdecompctrlRXpasenface}
\label{eq:qc0}
\end{figure*}
\begin{proof}
By inspection of the equations of
\cref{eq:qc0}.
\end{proof}
Equations (\ref{HH}) to (\ref{XPX}) are fairly standard in quantum
computing. \cref{CZ}, which is used for instance in
\cite{abdessaied2014quantum}, describes two equivalent ways to define
a controlled-Z gate. Notice that this equation cannot be derived from
the other axioms as it is the only equation on 2 qubits which does not
preserve the parity of the number of CNots plus the number of
swaps. Equations (\ref{commutationdecompctrlPRY}) and
(\ref{commutationdecompctrlRXpasenface}) are more involved and account
for some specific commutation properties of controlled gates (see
\cref{commctrl} and \cref{commctrlpasenface}).
The axioms of $\textup{QC}_0$, i.e.~the equations given in
\cref{eq:qc0}, are sufficient to derive standard elementary circuit
identities like those given in \cref{usefull_eq}.
One can also prove that some particular circuits, called phase-gadgets
\cite{Cowtan_2020}, can be flipped vertically:
\begin{equation}\label{CNotPCNotreversible}\qczero\vdash ~\scalebox{0.8}{\tikzfig{CNotPbCNot}}~=~\scalebox{0.8}{\tikzfig{CNotPbCNotalenvers}}\end{equation}
\begin{equation}\label{NotCRXNotCreversible}\qczero\vdash~\scalebox{0.8}{\tikzfig{RXhconjCNot}}~=~\scalebox{0.8}{\tikzfig{RXbconjCNot}}\end{equation}
The derivations are given in \cref{proof:usefuleq}. Combining
\cref{CNotPCNotreversible} and \cref{commutationPctrl}, one can easily
prove the following equation, used for instance in
\cite{nam2018automated} in the context of circuit optimisation:
\[
\qczero\vdash \scalebox{.8}{\tikzfig{CPL}}
{=}\!\scalebox{.8}{\tikzfig{CPR}}
\]
\begin{figure*}
\caption{Standard circuit identities that can be derived from the
axioms of $\textup{QC}
\label{commutationCNotsbas}
\label{CNotHH}
\label{XX}
\label{CNotliftvar}
\label{commutationXCNot}
\label{ZZ}
\label{NotClift}
\label{ZCNot}
\label{commutationRXCNot}
\label{RX0}
\label{RXRX}
\label{CNotHCNot}
\label{usefull_eq}
\end{figure*}
Notice that when $\varphi = -\varphi' = \alpha/2$ the above circuits are two
equivalent standard implementations of a controlled-phase gate of angle
$\alpha$. We show in the next section how the basic algebra of (multi-) controlled gates can be derived.
\subsection{Controlled gates}
Multi-controlled gates are useful to describe more elaborate quantum
circuits. We use the notations ``$\lambda$'' and ``$\Lambda$'' for
controls.
Given a 1-qubit gate $G$, $\lambda^1 G$ is a 2-qubit positively
controlled gate: if the control qubit (the top one) is in state
$\ket 1$ (resp.\ $\ket 0$) then $G$ (resp.\ the identity) is applied
on the target qubit (the bottom one). $\lambda^2 G$ is a 3-qubit
positively controlled gate, where the two upper qubits are controls:
they both need to be in state $\ket{1}$ for the gate $G$ to fire on
the bottom qubit.
We also consider more general multi-controlled gates
$\Lambda^{x_1\ldots x_k}G$ with positive (when $x_i=1$) and negative
(when $x_i=0$) controls: if the first qubit is in the state
$\ket{x_1}$ (resp. $\ket {\bar x_1}$) then $\Lambda^{x_2\ldots x_k}G$
(resp. the identity) is applied on the remaining qubits. Finally,
$\Lambda ^x_y G$ denotes a multi-controlled gate with control qubits
on both sides -- above and below -- of the target qubit.
We will follow a standard construction for multi-controls
using a decomposition into elementary $1$- and $2$-qubit gates (see
for instance \cite{Barenco1995gates}). Note that we do not aim here at defining
\emph{all} controlled operators: as this construction is the main
apparatus for the completeness result, we only focus on the operations
$s(\varphi)$, $X$, $R_X(\theta)$ and $P(\varphi)$. Other controlled
operations can then be derived if needed.
We first define in \cref{def:multicontrolled-pos} circuits
implementing regular, all-positive multi-controlled gates
$\lambda^nG$. We then present in \cref{def:multicontrolled-oriented}
how to handle positive and negative controls. In
\cref{def:multicontrolled} we finally introduce controlled gates with
controls both above and below the gate $G$.
\begin{definition}[Positively multi-controlled gates]
\label{def:multicontrolled-pos}
For all $n\in\mathbb{N}$ and
$G\in \{s(\varphi),X,R_X(\theta),P(\varphi)\}$, we define a quantum
circuit $\lambda^n G$.\footnote{Note that $G$ spans non-elementary
gates, the constructor $\lambda$ is not considered as a gate
operator, and the fact that the circuit $\lambda^nG$ happens to be
related to $G$ is a corollary of its definition, as
discussed further in the article.} This circuit acts on
$n$ wires when $G=s(\varphi)$ and $n+1$ otherwise. We define each
circuit $\lambda^nG$ as follows.
\begin{itemize}
\item $\lambda^{n} R_X(\theta)$ is defined by induction:
\[
\lambda^{0} R_X(\theta)\coloneqq R_X(\theta),
\]
\[
\lambda^{n+1} R_X(\theta)\coloneqq\scalebox{0.8}{\tikzfig{mctrlXthetaOnlyBlack}}.
\]
\item $\lambda^{n} P(\varphi)$ is defined by induction using
$\lambda^nR_X(\varphi)$:
\[
\lambda^{0} P(\varphi)\coloneqq P(\varphi),
\]
\[
\lambda^{n+1}
P(\varphi)\coloneqq\scalebox{0.8}{\tikzfig{mctrlPphi2OnlyBlack}}.
\]
\item $\lambda^{n}X$ is a simple macro:
\[ \lambda^{n}X\coloneqq
\scalebox{0.8}{\tikzfig{mctrlPHOnlyBlack}} \]
\item Finally, $\lambda^{0}s(\psi)\coloneqq s(\psi)$ and
$\lambda^{n+1}s(\psi)\coloneqq \lambda^nP(\psi)$.
\end{itemize}
\end{definition}
\begin{definition}[Multi-controlled gates]
\label{def:multicontrolled-oriented}
For any $k$-length list of booleans $x = x_1,\dots, x_k$ ($x_i\in\{0,1\}$), for any
$G\in \{s(\varphi),X,R_X(\theta),P(\varphi)\}$ we define the quantum circuit $\Lambda^x
G$ as
\[
\Lambda^{x}
G\coloneqq\scalebox{0.8}{\tikzfig{generalControledGate}}
\]
when $G\in \{X,R_X(\theta),P(\varphi)\}$, and
\[\Lambda^{x}
s(\varphi)\coloneqq\scalebox{0.8}{\tikzfig{generalControledGateS}}.
\]
where $\overline x = 1-x$, $\!\scalebox{0.8}{\tikzfig{X1}}\!=\!\scalebox{0.8}{\tikzfig{X}}$, and $\!\scalebox{0.8}{\tikzfig{X0}}\!=\!\scalebox{0.8}{\tikzfig{filcourt-s}}$.
\end{definition}
\begin{definition}[General multi-controlled gates]
\label{def:multicontrolled}
Given two lists of booleans $x\in \{0,1\}^k$ and $y\in
\{0,1\}^\ell$, if $xy$ is the concatenation of $x$ and $y$ we define the
two quantum circuits
\begin{itemize}
\item for any $G\in \{X,R_X(\theta),P(\varphi)\}$
\[\Lambda^x_y G\coloneqq \scalebox{0.8}{\tikzfig{multixy}}\]
\item $\Lambda^x_y s(\varphi)\coloneqq \Lambda^{xy} s(\varphi)$.
\end{itemize}
\end{definition}
One can double check using the semantics that $\Lambda^x_y G$ is
actually a multi-controlled gate:
\begin{proposition}
For any $x,u\in \{0,1\}^k$, $y,v\in \{0,1\}^\ell$, $a\in\{0,1\}$ and
$G\in
\{X,R_X(\theta),P(\varphi)\}$,
\begin{align*}
\interp{\Lambda^x_yG}\ket{u,a,v} &= \begin{cases}\ket u\otimes
(\interp G \ket a)\otimes \ket{v}&\text{if $uv=xy$,
}\\\ket{u,a,v} &\text{otherwise,}\end{cases}
\intertext{and}
\interp{\Lambda^x_ys(\varphi)}\ket{u,v}
&= \begin{cases}e^{i\varphi}\ket {u,v}&\text{if $uv=xy$,
}\\\ket{u,v} &\text{otherwise.}\end{cases}
\end{align*}
\end{proposition}
We use the standard bullet-based graphical notation for multi-controlled gates:
the $i^\text{th}$ control is black (resp.\ white) when $x_i=1$ (resp.\ $x_i=0$), and
the $j^\text{th}$ from the end control is black (resp.\ white) when
$y_{\ell-j+1}=1$ (resp.\ $=0$), e.g.:
\begin{align*}
\Lambda^{11}_1 X&: \hspace{1.4ex}\scalebox{.9}{\tikzfig{C11X1}},
&
\Lambda^{0}_{10} R_X(\theta)&: \scalebox{.9}{\tikzfig{C0X10}},
\\[0.3cm]
\Lambda^{10} P(\varphi)&: \scalebox{.9}{\tikzfig{C10P}},
&
\Lambda^{1\ldots1} R_X(\theta)&: \scalebox{.9}{\tikzfig{CCRX}}.
\end{align*}
To avoid ambiguity with CNot we will not use this notation in the particular case of $\Lambda^1 X$ and $\Lambda_1X$. Notice however that
$\Lambda^1X$ is provably equivalent to $\cnot$:
\begin{proposition}\label{ctrlXCNot}$\qczero \vdash \Lambda^1 X = \gcnot$.
\end{proposition}
\begin{proof}
The proof is given in \cref{preuvectrlXCNot}.
\end{proof}
\subsection{Properties of multi-controlled gates}\label{propertiesmultictrl}
In a multi-qubit controlled gate, all control qubits play a similar
role. This can be expressed as the following commuting property:
\[
\scalebox{0.8}{\tikzfig{swapCCRX3}} =
\scalebox{0.8}{\tikzfig{CCRXswap3}}
\]
This property is provable in $\qczero$, considering three cases
depending whether the exchanged control qubits are either above or
below the target qubit:
\begin{proposition}\label{cor:swap}
For any $x\in \{0,1\}^k, y\in \{0,1\}^\ell, z\in \{0,1\}^m$,
$a,b\in \{0,1\}$ and any $G\in \{s(\psi),X,R_X(\theta),P(\varphi)\}$,
\begin{alignat}{100}
\qczero&\vdash& \scalebox{0.9}{\tikzfig{swapMctrl}}
&{=} \scalebox{0.9}{\tikzfig{Mctrlswap}}
\label{swapCCZ}
\\
\qczero&\vdash& \scalebox{0.9}{\tikzfig{swapMctrl-down}}
&{=} \scalebox{0.9}{\tikzfig{Mctrlswap-down}}
\label{swapCCZ2}
\\
\qczero&\vdash& \scalebox{0.9}{\tikzfig{swapMctrl-middle}}
&{=} \scalebox{0.9}{\tikzfig{Mctrlswap-middle}}
\label{swapCCZ3}
\end{alignat}
\end{proposition}
A peculiar property of controlled phase gates (and hence controlled
scalars) is that the target qubit is actually equivalent to the
control qubits, e.g.:
\[ \scalebox{0.8}{\tikzfig{CCP3}} = \scalebox{0.8}{\tikzfig{CCP3-}}\]
This property is also provable in $\qczero$:
\begin{proposition}\label{prop:CP}
For any $x\in \{0,1\}^k, y\in \{0,1\}^\ell$,
\begin{equation}
\qczero\vdash\Lambda^x_{y1} P(\varphi)=\Lambda^{x1y}P(\varphi)
\label{phasemobile}
\end{equation}
\end{proposition}
\begin{proof}[Proof of \cref{cor:swap} and \cref{prop:CP}]
The two properties are proved at once. The proof relies on the
following commutation property which can be proved by induction (see
Appendix \ref{sec:proofcomRX}).
\begin{multline}
\label{eq:comRX}
\qczero\vdash \scalebox{0.8}{\tikzfig{com-L}} \\= \scalebox{0.8}{\tikzfig{com-R}}
\end{multline}
The proof of Equations~\eqref{swapCCZ}-\eqref{swapCCZ3} for $G=R_X(\theta)$ follows by
induction. We then prove \cref{phasemobile} which requires a few
technical developments. The proof of Eq.~\eqref{swapCCZ}-\eqref{swapCCZ3} for the other
gates then follows from the $R_X(\theta)$ case and
\cref{phasemobile} (see \cref{preuvesswapsmultictrl}).
\end{proof}
The gates $P(\varphi)$ form a monoid, i.e.
$P(\varphi+\varphi') = P(\varphi)\circ P(\varphi')$ (\cref{PP}) and
$P(0)=\gidspace$ (\cref{P0}). Notice that $R_X(\theta)$ and $s(\varphi)$ also form
monoids. It is provable in $\qczero$ that their multi-controlled
versions enjoy the same property:
\begin{proposition}\label{prop:sum}
For any $x\in \{0,1\}^k$, $y\in \{0,1\}^\ell$,
\begin{align*}
\qczero &\vdash \Lambda^{x}_y R_X(\theta')\circ \Lambda^{x}_y
R_X(\theta) = \Lambda^{x}_y R_X(\theta+\theta'),
\\
\qczero &\vdash \Lambda^{x}_y P(\varphi')\circ \Lambda^{x}_y
P(\varphi)= \Lambda^{x}_y P(\varphi+\varphi'),
\\
\qczero &\vdash \Lambda^{x}_y s(\varphi')\circ \Lambda^{x}_y
s(\varphi)= \Lambda^{x}_y s(\varphi+\varphi'),
\\
\qczero &\vdash \Lambda^{x}_y R_X(0)=id_{k+\ell+1},
\\
\qczero &\vdash \Lambda^{x}_y P(0)= id_{k+\ell+1},
\\
\qczero &\vdash \Lambda^{x}_y s(0)= id_{k+\ell},
\end{align*}
where $id_{k}$ is defined as in \cref{fig:axiom}.
\end{proposition}
\begin{proof}
First, proving that multi-controlled gates with angle $0$ are equivalent to the identity is straightforward by induction.
To prove the rest of the proposition, we first prove that
$\qczero \vdash \Lambda^{1..1}R_X(\theta')\circ
\Lambda^{1..1}R_X(\theta)= \Lambda^{1..1} R_X(\theta+\theta')$. The proof is by induction:
we unfold the two multi-controlled gates, use \cref{eq:comRX} to put
the multi-controlled gates with angles $\theta/2$ and $\theta'/2$
side by side, and merge them using the induction hypothesis. We use
again \cref{eq:comRX} to allow the combination of the
multi-controlled gates with angle $-\theta/2$ and $-\theta'/2$,
closing the case.
The cases with more general controls are derived from this one using \cref{def:multicontrolled-oriented,def:multicontrolled}. The cases of $P$ and $s$ are derived from the $R_X$ case using \cref{def:multicontrolled-pos} and an ancillary lemma stating that a multi-controlled phase commutes with the controls of another multi-controlled gate. The details of the proof are given in \cref{proofpropsum}.
\end{proof}
\begin{remark}
Note that \cref{prop:sum} does not imply the periodicity of
controlled gates. The latter is proven in \cref{prop:period} with
the help of the rules of \cref{fig:euler}.
\end{remark}
Combining a control and anti-control on the same qubit makes the
evolution independent of this qubit, as in the following example in
which the evolution is independent of the second
qubit:\footnote{Notice that in the above example we implicitly use
\cref{cor:swap} to swap the first two qubits and apply
\cref{prop:comb}. As a consequence, the resulting multi-controlled gate
acts on non-adjacent qubits. Similarly to the \cnot{} case (see
\cref{CNot-def,NotC-def}), we use some syntactic sugar to represent such
multi-controlled gates acting on non-adjacent qubits.}
\[
\scalebox{0.8}{\tikzfig{com-fu-L}} =
\scalebox{0.8}{\tikzfig{com-fu-R}}
\]
Such simplifications can be derived in $\qczero$.
\begin{proposition}\label{prop:comb}
For all bitstrings $x\in \{0,1\}^k$, $y\in \{0,1\}^\ell$, and for all
$G\in\{s(\varphi),X,R_X(\theta),P(\varphi)\}$,
\[
\qczero \vdash \Lambda^{0x}_y G\circ \Lambda^{1x}_{y} {G} =
\gidspace\otimes\Lambda^{x}_{y} {G}.
\]
\end{proposition}
\begin{proof}
Without loss of generality, we assume $y$ as the empty string $\epsilon$ and
$G = R_X(\theta)$, as it can derive the other cases. The proof is by induction: we unfold the multi-controlled and
multi-anti-controlled gates. We can then move the $X$ gate through $H$
and $\cnot$ gates due to the anti-control, changing the sign of an
$R_X$ rotation from $-\theta/2$ to $\theta/2$. The rest of the
proof is similar to the one of \cref{prop:sum}, except that two
$R_X$ gates cancel out, leading to the identity on the first qubit
and the desired multi-controlled gate on the second
one. The details of the proof are given in \cref{preuvepropcomb}.
\end{proof}
\cref{prop:comb} shows how control and anti-control can be combined on the
first qubit of a multi-controlled gate. Note, however, that it can be
generalised to any control qubit thanks to \cref{cor:swap}.
Another useful property of multi-controlled gates is that they commute when
there is a control and anti-control on the same qubit, as in the following
example in which their controls differ on the third (and last) qubit:
\[
\scalebox{0.8}{\tikzfig{com-ex-L}} =\scalebox{0.8}{
\tikzfig{com-ex-R}}
\]
When the target qubit is the same, such a commutation property can be
derived in $\qczero$, using in particular
\cref{commutationdecompctrlPRY}.
\begin{proposition}\label{commctrl}
For any $x,x'\in \{0,1\}^k$, $y,y'\in \{0,1\}^\ell$, and
$G,G'\in \{X,R_X(\theta),P(\varphi)\}$, if
$xy\neq x'y'$\footnote{$xy\neq x'y'$ iff
$\exists i, x_i\neq x'_i \vee y_i\neq y'_i$.} then
$$\qczero \vdash \Lambda^x_y G\circ \Lambda^{x'}_{y'} {G'} =
\Lambda^{x'}_{y'} {G'}\circ \Lambda^x_y G \, .$$
\end{proposition}
\begin{proof}
The proof relies on a generalisation of \cref{eq:comRX}, and follows
by an induction argument whose base case can be derived thanks to
\cref{commutationdecompctrlPRY}. The details of the proof are given in \cref{preuvecommctrl}.
\end{proof}
Controlled and anti-controlled gates also commute when the target
qubits are not the same in both gates, as in:
\[
\scalebox{0.8}{\tikzfig{com-ex-L-diff}} =\scalebox{0.8}{
\tikzfig{com-ex-R-diff}}.
\]
This property can also be derived in $\qczero$, using in particular
\cref{commutationdecompctrlRXpasenface}:
\begin{proposition}\label{commctrlpasenface}
For any $a,b\in \{0,1\}$, $x,x'\in \{0,1\}^k$,
$y,y'\in \{0,1\}^\ell$, $z,z'\in \{0,1\}^m$ and
$G,G'\in \{X,R_X(\theta),P(\varphi)\}$, if $xyz\neq x'y'z'$ then
$$\qczero \vdash \Lambda^x_{yaz} G\circ \Lambda^{x'by'}_{z'} {G'} =
\Lambda^{x'by'}_{z'} {G'}\circ \Lambda^x_{yaz} G$$
\end{proposition}
\begin{proof}
The proof is also based on the generalisation of \cref{eq:comRX},
using an inductive argument whose base case can be derived thanks to
\cref{commutationdecompctrlRXpasenface}. The details of the proof are given in \cref{preuvecommctrlpasenface}.
\end{proof}
\subsection{Euler angles and Periodicity}\label{Eulerandperiod}
$\qczero$ is not complete. In particular equations based on Euler
angles, which require non-trivial calculations on the angles, cannot
be derived. As a consequence we add to the equational theory the three rules shown in \cref{fig:euler},
leading to the equational theory $\textup{QC}$. We write $\textup{QC}\vdash C_1=C_2$ when $C_1$ can be
rewritten into $C_2$ using equations of \cref{eq:qc0} and
\cref{fig:euler} (together with the deformation rules).
\begin{figure*}
\caption{\textbf{Non-structural equations.}
\label{EulerH}
\label{Euler2d}
\label{Euler3dmulticontrolled}
\label{fig:euler}
\end{figure*}
The Euler decomposition of $H$ (\cref{EulerH}) is not unique:
\begin{proposition}\label{EulerHmoins}
$\textup{QC}\vdash \!\!\!\scalebox{0.8}{\tikzfig{H}} {=}
\scalebox{0.75}{\tikzfig{EulerHmoins}}$
\end{proposition}
\begin{proof}The proof is given in \cref{preuveEulerHmoins}.
\end{proof}
More generally the Euler angles are not unique, but can be made unique by adding some constraints on the angles, like choosing them in the appropriate intervals (see \cref{fig:euler}).
\begin{proposition}\label{soundnessEuleraxioms}
Equations (\ref{Euler2d}) and (\ref{Euler3dmulticontrolled}) are
sound. Moreover, the choice of parameters in the RHS-circuits
to make the equations sound is unique (under the constraints given
in \cref{fig:euler}).
\end{proposition}
\begin{proof}
The soundness and uniqueness of \cref{Euler2d} are well-known
properties. Regarding \cref{Euler3dmulticontrolled}, we first notice
that the semantics of both circuits is of the form
$\left(\begin{array}{c|c}I&0\\\hline 0&U\end{array}\right)$ where
$U$ is a $3\times 3$ matrix. We then use the fact that this matrix
can be decomposed into basic rotations that can be proved to be
unique \cite{clement2022LOv}. The details of the proof are given in
\cref{appendix:euler3d}.
\end{proof}
Notice that \cref{Euler2d} subsumes Equations (\ref{PP}) and
(\ref{XPX}), which can now be derived using the other axioms of
$\textup{QC}$.
\begin{proposition}\label{klfollowfromEuler}The following two equations of $\textup{QC}$,
\[\begin{array}{rclr}
\scalebox{0.9}{\tikzfig{PP}}&{=}&\scalebox{0.9}{\tikzfig{Pphiplusphiprime}} &\text{~~~~(\ref{PP})} \\[0.4cm]
\scalebox{0.9}{\tikzfig{XPX}}&{=}&\scalebox{0.9}{\tikzfig{Pmoinsphieiphi}}&\text{~~~~(\ref{XPX})}
\end{array}\]
can be
derived from the other axioms of $\textup{QC}$.
\end{proposition}
\begin{proof}
The proofs are given in \cref{preuveklfollowfromEuler}.
\end{proof}
The introduction of the additional equations of \cref{fig:euler} allows us to
prove some extra properties about multi-controlled gates, like periodicity (for
those with a parameter) in \cref{prop:period} and the fact that a
multi-controlled X gate is self-inverse.
\begin{proposition}\label{prop:CCX}
For any $x\in \{0,1\}^k$, $y\in \{0,1\}^\ell$,
\[
\textup{QC} \vdash \Lambda^{x}_y X\circ \Lambda^{x}_y X=
id_{k+\ell+1}
\]
\end{proposition}
\begin{proof}
The case $x=y=\epsilon$ is a direct consequence of \cref{XX}.
For the other cases, by \cref{def:multicontrolled-pos,def:multicontrolled-oriented,def:multicontrolled,HH,XX,prop:sum}, it is equivalent to
show that, for any $x\in \{0,1\}^k$,
\[
\qc \vdash \Lambda^x P(2\pi) = id_{k+1}.
\]
Without loss of generality, we can consider $x\in \{1\}^k$.
Then the result is a consequence of \cref{prop:sum} and
\cref{Euler3dmulticontrolled}.
Indeed, by taking
$\gamma_1=\gamma_3=\gamma_4=0$ and $\gamma_2=2\pi$ in the
LHS of \cref{Euler3dmulticontrolled}, the unique angles on the
right are all zeros:
$\delta_1=\delta_2=\delta_3=\delta_4=\delta_5=\delta_6=\delta_7=\delta_8=\delta_9=0$.
By \cref{prop:sum}, any multi-controlled gate with zero angle is the identity, which gives us the desired equality.
Further
details can be found in \cref{proof:CCX}.
\end{proof}
\begin{proposition}\label{prop:period}
For any $x\in \{0,1\}^k$, $y\in \{0,1\}^\ell$,
$\theta\in \mathbb R$,
\begin{align*}
\qc &{}\vdash \Lambda^x_y R_X(\theta+4\pi) = \Lambda^x_y R_X(\theta)
\\
\qc &{}\vdash \Lambda^x_y P(\theta+2\pi) = \Lambda^x_y
P(\theta)
\\
\qc &{}\vdash \Lambda^x_y s(\theta+2\pi) =
\Lambda^x_y s(\theta)
\end{align*}
\end{proposition}
\begin{proof}
Following the additivity of \cref{prop:sum}, it is sufficient to
show that for any $x\in \{0,1\}^k$, $y\in \{0,1\}^\ell$,
\begin{align*}
\qc &{}\vdash \Lambda^x_y R_X(4\pi) = id_{k+\ell+1},
\\
\qc &{}\vdash \Lambda^x_y P(2\pi) = id_{k+\ell+1},
\\
\qc &{}\vdash \Lambda^x_y s(2\pi) = id_{k+\ell}.
\end{align*}
Also, with Equations \eqref{XX} and
\cref{def:multicontrolled-oriented,def:multicontrolled}, it is sufficient to
show that for any $x\in \{1\}^k$,
\begin{align*}
\qc &{}\vdash \Lambda^x R_X(4\pi) = id_{k+1},
\\
\qc &{}\vdash \Lambda^x P(2\pi) = id_{k+1},
\\
\qc &{}\vdash \Lambda^x s(2\pi) = id_{k}.
\end{align*}
First, we prove the three cases with $x=\epsilon$. Then,
we use $\qc \vdash \Lambda^x P(2\pi) = id_{k+1}$, proven in
\cref{prop:CCX}
using \cref{Euler3dmulticontrolled}. We obtain the
other
statements as direct consequences of the $2\pi$-periodicity of
$P$. Further details are provided in \cref{proof:propperiod}.
\end{proof}
\section{Completeness}\label{sec:completeness}
In this section we prove the main result of the paper, namely the completeness
of $\textup{QC}$. To this end, a back and forth encoding of quantum circuits
into linear optical quantum circuits is introduced. We use the graphical
language for linear optical circuits introduced in \cite{clement2022LOv}.
\subsection{Optical circuits}\label{sec:loppcircuits}
A \emph{linear optical polarisation-preserving} (LOPP for short)
circuit is an optical circuit made of beam splitters
($\tikzfig{bs-xs}$) and phase shifters
($\tikzfig{convtp-phase-shift-xs}$):
\begin{definition}
\label{def:LOPP}
Let $\textup{\textbf {LOPP}}$ be the prop generated by
$\tikzfig{convtp-phase-shift-xs}$, $\tikzfig{bs-xs}$ with $\varphi, \theta \in \mathbb R$.
\end{definition}
Like quantum circuits, LOPP-circuits are defined as a prop: one can see them as raw circuits quotiented by the $\equiv$-equivalence given in \cref{fig:axiom}.
In the following, we consider the single photon case, hence each input mode (or
wire) represents a possible input position for the photon. The photon moves
from left to right in the circuit. The state of the photon is entirely defined
by its position, and as a consequence the state space is of the form $\mathbb
C^n$ when there are $n$ possible modes. We consider the standard orthonormal
basis $\{\ket{p}\}_{p\in [0,n)}$ of $\mathbb C^n$. The semantics is defined as
follows.
\begin{definition}[Semantics]
\label{semLOPP}
For any $n$-mode $\textup{LOPP}$-circuit $C$, let
$\interp C: \mathbb C^{n} \to \mathbb C^{n}$ be a linear map
inductively defined as follows:
$\interp{C_2\circ C_1} \coloneqq \interp{C_2}\circ\interp{ C_1}$,
$\interp{C_1\otimes C_3} \coloneqq \interp{C_1}\oplus\interp{
C_3}=\left(\begin{array}{c|c}\interp{C_1}&0\\\hline0&\interp{C_3}\end{array}\right)$,
\begin{align*}
\interp{\tikzfig{bs-xs}}
&\coloneqq\ket p
\mapsto\cos(\theta)\ket{p}+i\sin(\theta)\ket{{1-p}}
\\
&=
\begin{pmatrix}
\cos(\theta)&i\sin(\theta)\\
i\sin(\theta)&\cos(\theta)
\end{pmatrix}
\\
\interp{\gswap}
&\coloneqq \ket{p}\mapsto
\ket{1-p}
=
\begin{pmatrix}
0&1\\1&0
\end{pmatrix}
\end{align*}
\[
\interp{\tikzfig{convtp-phase-shift-xs}}
\coloneqq e^{i\varphi}
\qquad \interp\gid \coloneqq1
\qquad
\interp{\emptyc} \coloneqq0
\]
\end{definition}
\begin{figure*}
\caption{Axioms of the LOPP-calculus. In \cref{Eulerbsphasebs}
\label{phase0}
\label{bs0}
\label{swapbspisur2}
\label{phaseaddition}
\label{globalphasepropagationbs}
\label{Eulerbsphasebs}
\label{Eulerscalaires}
\label{axiomsLOPP}
\end{figure*}
\begin{remark}
The definition of $\interp .$ relies on the inductive structure of raw \LOPP-circuits, it is however well-defined on \LOPP-circuits as for any raw \LOPP-circuits $C,C'$, $C\equiv C'$ implies $\interp C = \interp{C'}$.
\end{remark}
We consider a simple equational theory for \LOPP-circuits
(\cref{axiomsLOPP}), which is derived from the rewriting system
introduced in \cite{clement2022LOv}. Contrary to the rewriting system
of \cite{clement2022LOv}, the swap is part of \LOPP-circuits.
Moreover, the most elaborate equation -- \cref{Eulerscalaires} -- is
slightly simplified in the present paper to have one parameter less.
We use the notation $\textup{LOPP}\vdash C_1=C_2$ whenever $C_1$ can
be transformed into $C_2$ using the equations of \cref{axiomsLOPP} (and
circuit deformations of \cref{fig:axiom}).
\begin{theorem} \label{thm:LOPPcompleteness} The equational theory
given by \cref{axiomsLOPP} is sound and complete: for any
$\textup{LOPP}$-circuits $C_1, C_2$, $\textup{LOPP}\vdash C_1=C_2$
iff $\interp {C_1} = \interp{C_2}$.
\end{theorem}
\begin{proof}
The soundness can be shown with the semantics given in
\cref{semLOPP}. Regarding completeness, we show that we can derive
from \cref{axiomsLOPP} the rules of the strongly normalising
rewriting system of \cite{clement2022LOv}. The full proof is given
in \cref{proof:thmLOPPcompleteness}.
\end{proof}
\subsection{Forgetting the monoidal structure}\label{sec:bare}
The proof of completeness for quantum circuits is based on a back and
forth translation from linear optical circuits. While both kinds of
circuits form a prop, so both have a monoidal structure, these
monoidal structures do not coincide. The monoidal structure of quantum
circuits corresponds to the tensor product, whereas that of linear
optical circuits is a direct sum. Hence the translations do not
preserve the monoidal structure.
As a consequence there is a technical issue around defining the
translation directly on circuits. We instead define the transformations
on \emph{raw} circuits (cf. \cref{sec:raw}). The collection of raw quantum (resp. LOPP) circuits is denoted $\QCbarebf$ (resp. $\LOPPbarebf$).
Notice that we
recover the standard circuits by considering the raw circuits up to
the equivalence relation $\equiv$ given in \cref{fig:axiom}:
$ \QCbf ={\QCbarebf}{/{\equiv}}$ and
$\LOPPbf = {\LOPPbarebf}{/{\equiv}} $.
To avoid ambiguity in the graphical representation of raw circuits
one can use boxes like $\scalebox{0.7}{\tikzfig{XXXbox}}$ for
$(\gx \otimes \gx)\otimes \gx$. We also use box-free graphical
representation that we interpret as a layer-by-layer description of a
raw circuit, more precisely we associate with any box-free graphical
representation, a raw-circuit of the form
$C=(\ldots((L_1\circ L_2)\circ L_3)\circ \ldots )\circ L_k$ where
$L_i=(\ldots((g_{i,1}\otimes g_{i,2})\otimes g_{i,3})\otimes \ldots
)\otimes g_{i,\ell_i}$.
For instance, $((id_1\otimes id_1)\otimes X)\circ (CNot
\otimes H)$ is
\[
\scalebox{.8}{\tikzfig{bare-circuit-ex}} = \scalebox{.8}{\tikzfig{bare-circuit-ex-2}} \circ
\scalebox{.8}{\tikzfig{bare-circuit-ex-1}}
\]
We extend the notation $\textup{QC}\vdash \cdot = \cdot$ and
$\textup{LOPP}\vdash \cdot = \cdot$ to raw circuits. For any raw
quantum circuits (resp.\ raw optical circuits) $C_1,C_2$, we write
$\textup{QC}\vdash C_1 = C_2$ (resp. $\textup{LOPP}\vdash C_1=C_2$) if
$C_1$ and $C_2$ are equivalent by the congruence defined in
\cref{eq:qc0}, \cref{fig:euler} and \cref{fig:axiom} (resp.\
\cref{axiomsLOPP} and \cref{fig:axiom}).\footnote{In this context, the circuits depicted in Figures \ref{eq:qc0}, \ref{fig:euler} and \ref{axiomsLOPP} are interpreted as box-free graphical representations of raw circuits.\label{rawinfigures}}
Notice that there exists a derivation between two circuits if and only
if there exists a derivation between two of their representative raw
circuits. Indeed, intuitively the only difference is that the
derivation on raw circuits is more fine-grained as the equivalence
relation $\equiv$ is made explicit.
\subsection{Encoding quantum circuits into optical ones}
We are now ready to define the encoding of (raw) quantum circuits
into (raw) linear optical circuits. For dimension reasons, an $n$-qubit system is encoded into $2^n$ modes.
One can
naturally choose to encode $\ket x$, with $x\in \{0,1\}^n$, into the
mode $\ket{\underline x}$ where $\underline x =\sum_{i=1}^{n}x_i2^{n-i}$ is the
usual binary encoding. Alternatively, we use Gray codes to produce
circuits with a simpler connectivity, in particular two adjacent modes
encode basis qubit states which differ on exactly one qubit.
\begin{definition}[Gray code]\label{defgraycode}
Let
$\mathfrak G_n: \mathbb C^{2^n} \to \mathbb C^{\{0,1\}^n}$ be the map $\ket k
\mapsto \ket{G_n(k)}$ where $G_n(k)$ is the Gray code of $k$,
inductively defined by $G_0(0)=\epsilon$ and
\[
G_n(k)=\begin{cases}
0G_{n-1}(k)&\text{if $k<2^{n-1}$,}
\\
1G_{n-1}(2^n-1-k)
&
\text{if $k\geq 2^{n-1}$.}
\end{cases}
\]
\end{definition}
For instance $G_3$ is defined as follows:
\[\begin{array}{rclrcl}
0&\mapsto& 000&\qquad 4&\mapsto 110\\
1&\mapsto& 001&\qquad 5&\mapsto 111\\
2&\mapsto& 011&\qquad 6&\mapsto 101\\
3&\mapsto& 010&\qquad 7&\mapsto 100\\
\end{array}\]
In order to get around the fact
that the encoding an $n$-qubit circuit into a $2^n$-mode optical circuit cannot preserve the parallel composition, we
proceed by `sequentialising' the circuit:
roughly speaking, an $n$-qubit circuit is seen as a sequential composition of layers, each layer being an $n$-qubit circuit made of an elementary gate $g$ acting on at most two qubits in parallel with the identity on all other qubits, \emph{e.g.} $id_k\otimes g
\otimes id_{l}$. The encoding of such a layer, denoted $E_{k,l}(g)$, is a $2^n$-mode optical circuit acting non-trivially on potentially all the modes.
For instance, consider a $3$-qubit layer which consists in applying $P(\varphi)$ on the second qubit. Its semantics is $\ket{x,y,z}\mapsto e^{i\varphi y}\ket{x,y,z}$. Such a circuit is encoded into an $8$-mode optical circuit $E_{1,1}(P(\varphi))$
made of $4$ phase shifters acting on the modes $p\in[2,5]$ (those s.t. $G_3(p)=x1z$).
Indeed, the semantics of $E_{1,1}(P(\varphi))$ is $\ket p\mapsto \begin{cases}e^{i\varphi }\ket{p} & \text{if~} p\in[2,5]\\ \ket{p} &\text{otherwise}\end{cases}$.
The encoding map is formally defined as follows:
\begin{definition}[Encoding]\label{def:encoding}
Let $E:\QCbarebf \to \LOPPbarebf$ be defined as follows: for any
$n$-qubit circuit $C$, $E(C)=E_{0,0}(C)$ where $E_{k,\ell}$ is
inductively defined as:
\begin{itemize}
\item $E_{k,\ell}(C_1\otimes C_2) = E_{k+n_1,\ell}(C_2)\circ
E_{k,\ell+n_2}(C_1)$, where $C_1$ (resp.\ $C_2$) is acting on $n_1$ (resp.\ $n_2$) qubits;
\item $E_{k,\ell}(C_2\circ C_1) = E_{k,\ell}(C_2)\circ E_{k,\ell}(C_1)$;
\end{itemize}
Let us define $\sigma_{k,n,\ell}$ as a $2^{k+n+\ell}$-mode linear optical
circuit made only of swaps (that is, without any
$\tikzfig{phase-shift-xs}$ or $\tikzfig{bs-xs}$) such that
$\mathfrak G_n\circ\interp{\sigma_{k,n,\ell}}\circ\mathfrak
G_n^{-1}(\ket{x,y,z})=\ket{x,z,y}$ for any $x\in \{0,1\}^k$,
$y\in \{0,1\}^n$ and $z\in \{0,1\}^\ell$. We then define
\begin{align*}
E_{k,\ell}(\gswap)
&= \sigma_{k,\ell,2}\circ\sigma_{k+\ell,1,1}\circ \sigma_{k,2,\ell},
\\
E_{k,\ell}(\tikzfig{diagrammevide-s})
&=
(\tikzfig{filcourt-s})^{\otimes
{2^{k+\ell}}},
\\
E_{k,\ell}(\gid)
&= (\tikzfig{filcourt-s})^{\otimes {2^{k+\ell+1}}},
\\
E_{k,\ell}(s(\varphi))
&=
\left(\tikzfig{convtp-phase-shift-xs}\right)^{\otimes
{2^{k+\ell}}}.
\end{align*}
where $C^{\otimes n}$ means $C$ $n$ times in parallel: $C^{\otimes 0} = \tikzfig{diagrammevide-s} $ and $C^{\otimes n+1} = C\otimes C^{\otimes n}$.
For the remaining generators, we have:
\begin{align*}
E_{0,0}(\gh) &= \tikzfig{H-LOPP-xs},
\\
E_{0,0}(\gp) &= \tikzfig{Z-PHOL},
\\
E_{0,0}(\gcnot) &= \tikzfig{CNot-PHOL},
\intertext{and whenever $(k,\ell)\neq(0,0)$:}
E_{k,\ell}(\gh) &=
\sigma_{k,\ell,1}\circ \left(\tikzfig{H-LOPP-doublesym-xs}\right)^{\otimes {2^{k+\ell-1}}}\circ \sigma_{k,1,\ell},
\\
E_{k,\ell}(\gp) &=
\sigma_{k,\ell,1}\circ
\left(\tikzfig{Z-PHOL-doublesym}\right)^{\otimes
{2^{k+\ell-1}}}\circ \sigma_{k,1,\ell},
\\
E_{k,\ell}(\gcnot) &=
\sigma_{k,\ell,2}\circ \left(\tikzfig{CNot-PHOL-doublesym}\right)^{\otimes {2^{k+\ell-1}}}\circ \sigma_{k,2,\ell}.
\end{align*}
\end{definition}
\begin{remark}
Note that for any $n$-qubit circuit $C$, $E_{k,\ell}(C)$ is a $2^{k+n+\ell}$-mode optical circuit. Also note that $\sigma_{k,n,\ell}$ is nothing but a permutation of wires. By
\cref{decodingtoporules} -- which is independent from the definition
of $E$ -- any actual circuit satisfying the above property ($\mathfrak G_n\circ\interp{\sigma_{k,n,\ell}}\circ\mathfrak
G_n^{-1}(\ket{x,y,z})=\ket{x,z,y}$) is convenient for our purposes.
A formal definition of $\sigma_{k,n,\ell}$ is however given in \cref{defsigma}.
\end{remark}
\begin{figure*}
\caption{Encoding of the circuit discussed in \cref{ex:encod}
\label{fig:ex:encod}
\end{figure*}
\begin{example}\label{ex:encod}
Consider the simple circuit $C_0=\gcnotexqc$. The
encoding is as shown in \cref{fig:ex:encod}.
Using the topological rules (\cref{fig:axiom}), one can simplify $E (C_0)$ into the circuit $C_1$:
\[
\scalebox{1.1}{\gcnotexloppgray} \\[0.2cm]
\]
\end{example}
The encoding of quantum circuits into linear optical circuits preserves the
semantics, up to Gray codes.
\begin{proposition}
\label{prop:sem-preserving} For any $n$-qubit
quantum circuit $C$,
\[
\mathfrak G_n \circ \interp{E(C)} = \interp{C}\circ \mathfrak
G_n
\]
\end{proposition}
\begin{proof}
By induction.
\end{proof}
\subsection{Decoding}
Regarding the decoding, i.e.\ the translation back from linear optical
circuits to quantum circuits, we use the same sequentialisation
approach. Note that such a decoding is defined only for optical
circuits with a power of two number of modes.
The decoding of a $2^n$-mode layer $id_{k}\otimes g \otimes id_{l}$ is a $n$-qubit circuit denoted $D_{k,n}(g)$. For instance consider a $16$-mode layer which consists in applying $\tikzfig{convtp-phase-shift-xs}$ on the fourth mode.
Its semantics is $\ket{p}\mapsto\begin{cases}e^{i\varphi }\ket{p} & \text{if~} p=3\\ \ket{p} &\text{otherwise}\end{cases}$. Such a circuit is decoded into a $4$-qubit circuit $D_{3,4}(\tikzfig{convtp-phase-shift-xs})$
implementing the multi-controlled phase $\Lambda^{G_4(3)}s(\varphi)$, whose semantics is $\ket{x,y,z,t}\mapsto \begin{cases}e^{i\varphi }\ket{x,y,z,t} & \text{if~} xyzt = G_4(3)\\ \ket{x,y,z,t} &\text{otherwise}\end{cases}$.
The decoding map is formally defined as follows:
\begin{definition}[Decoding]\label{defdecoding}
Let $D: \LOPPbarebf\to \QCbarebf$ be defined as follows: for any
$2^n$-mode circuit $C$, $D(C)=D_{0,n}(C)$ where for any $n,k,\ell$
with $k+\ell\leq2^n$ and $C:\ell\to\ell$, $D_{k,n}(C)$ is
inductively defined as follows.
\begin{itemize}
\item $D_{k,n}(C_1\otimes C_2) = D_{k+\ell_1,n}(C_2)\circ D_{k,n}(C_1)$, where $C_1$ is acting on $\ell_1$ modes;
\item $D_{k,n}(C_2\circ C_1) = D_{k,n}(C_2)\circ D_{k,n}(C_1)$;
\item $D_{k,n}(\tikzfig{filcourt-s}) = id_{n}$.
\end{itemize}
The remaining generators are treated as follows.
\begin{align*}
D_{k,n}(\tikzfig{diagrammevide-s}) &= id_{n},
&
D_{k,n}(\tikzfig{convtp-phase-shift-xs}) &= \Lambda^{G_n(k)}
s(\varphi),
\\
D_{k,n}(\tikzfig{swap-s}) &= \Lambda^{x_{k,n}}_{y_{k,n}} X,
&
D_{k,n}(\tikzfig{bs-xs}) &= \Lambda^{x_{k,n}}_{y_{k,n}} R_X(-2\theta),
\end{align*}
where $x_{2k,n}\coloneqq G_{n-1}(k)$, $y_{2k,n}\coloneqq\epsilon$, $x_{2k+1,n}\coloneqq w$ and $y_{2k+1,n}\coloneqq 1.0^q$, where $q\in\{0,...,n-2\}$ and $w\in\{0,1\}^{n-q-2}$ are such that $G_{n}(2k+1)=wa1.0^q$ for some $a\in\{0,1\}$.
\end{definition}
\begin{example}\label{ex:decod}
We consider the optical circuit $C_1$ obtained in \cref{ex:encod}.
With all of the gates $P$ and $R_X$ parametrized with
$\frac{-\pi}{2}$, we can show that $D(C_1)\equiv$
\[
\scalebox{1}{\gcnotdec}
\]
\end{example}
Similarly to the encoding function, the decoding function preserves the
semantics up to Gray codes.
\begin{proposition}
For any $2^n$-mode optical circuit $C$,
$$\interp{D(C)}\circ \mathfrak G_n = \mathfrak G_n \circ
\interp{C}.$$
\end{proposition}
\begin{proof}
The proof is by induction.
\end{proof}
\subsection{Quantum circuit completeness}\label{sec:QCcompleteness}
The proof of completeness is based on the encoding/decoding of quantum
circuits into optical circuits. Intuitively, given two quantum
circuits representing the same unitary map, one can encode them as
linear optical circuits. Since the encoding preserves the semantics
and LOPP is complete, there exists a derivation proving the
equivalence of the encoded circuits. In order to lift this proof to
quantum circuits, it remains to prove that the decoding of an
encoded quantum circuit is provably equivalent to the original quantum
circuit, and that each axiom of LOPP can be mimicked in
\textup{QC}. Notice that since the encoding/decoding is defined on
raw circuits, an extra step in the proof consists in showing that the
axioms of $\equiv$ can also be mimicked in \textup{QC}.
Examples (\ref{ex:encod}) and (\ref{ex:decod}) point out that
composing encoding and decoding does not lead, in general, to the
original circuit, the decoded circuit being made of multi-controlled
gates. However, we show that the equivalence with the initial circuit can always be
derived in $\textup{QC}$:
\begin{lemma}\label{DE}
For any $n$-qubit raw quantum circuit $C$,
\[\textup{QC}\vdash D(E(C))=C.\]
\end{lemma}
\begin{proof}
We prove by structural induction on $C$ that
\[\forall k,\ell,\ \textup{QC}\vdash D(E_{k,\ell}(C))=id_k\otimes C\otimes id_\ell.\]
For any two $n$-qubit raw circuits $C_1,C_2$, one has
\[D(E_{k,\ell}(C_2\circ C_1))= D(E_{k,\ell}(C_2))\circ D(E_{k,\ell}(C_1))\]
and for any $m$-qubit raw circuit $C_3$,
\[D(E_{k,\ell}(C_1\otimes C_3))=D(E_{k+n,\ell}(C_3))\circ D(E_{k,\ell+m}(C_1)).\]
Hence, it remains the basis cases which are proved as \cref{basecaseDE} in \cref{lemmasforbasecaseDE}.
\end{proof}
Note that in general, the decoding function does not preserve the
topological equivalence. For instance, with the raw circuits
$C_1=\scalebox{0.56}{$\tikzfig{exnaturalitebsL}$}$ and
$C_2=\scalebox{0.56}{$\tikzfig{exnaturalitebsR}$}$, we have
$C_1\equiv C_2$ but
$D(C_1)=\scalebox{0.8}{$\tikzfig{exdecodagenaturalitebsL}$}$ and
$D(C_2)=\scalebox{0.8}{$\tikzfig{exdecodagenaturalitebsR}$}$
. Thus,
the topological rules also have to be mimicked in \textup{QC}:
\begin{lemma}\label{decodingtoporules}
For any $2^n$-mode raw optical circuits $C_1, C_2$, if
${C_1} \equiv C_2$ then $\textup{QC}\vdash D(C_1) = D(C_2)$.
\end{lemma}
\begin{proof}
The proof consists intuitively in verifying that the decoding of
every equation of \cref{fig:axiom} is provable in $\textup{QC}$. The
proof is given in
\cref{preuvedecodingtoporules}.
\end{proof}
\begin{lemma}\label{decodingLOPPrules}
For any $2^n$-mode raw optical circuits $C_1, C_2$, if
$\textup{LOPP}\vdash C_1 = C_2$ then
$\textup{QC} \vdash D(C_1) = D(C_2)$.
\end{lemma}
\begin{proof}
The proof consists intuitively in verifying that the decoding of
every equation of \cref{axiomsLOPP} is provable in
$\textup{QC}$. The proof is given in \cref{preuvedecodingLOPPrules}.
\end{proof}
We are now ready to prove the main result of the paper.
\begin{theorem}[Quantum circuit
completeness] \label{thm:QCcompleteness} \textup{QC} is a complete
equational theory for quantum circuits: for any quantum circuits
$C_1$, $C_2$, if $\interp{C_1}=\interp {C_2}$ then
$\textup{QC}\vdash C_1=C_2$.
\end{theorem}
\begin{proof}
Given two quantum circuits $C_1$, $C_2$ s.t.
$\interp{C_1}=\interp {C_2}$, let $C_1'$ (resp.\ $C_2'$) be a raw
quantum circuit, representative of $C_1$ (resp.\ $C_2$). Thanks to
\cref{prop:sem-preserving} we have
$\interp{E(C'_1)}=\interp {E(C'_2)}$. The completeness of
$\textup{LOPP}$ implies $\textup{LOPP}\vdash E(C'_1)=E(C'_2)$. By
\cref{decodingLOPPrules}, we have
$\textup{QC}\vdash D(E(C'_1)) = D(E(C'_2))$. Moreover \cref{DE}
implies $\textup{QC}\vdash C'_1 = C'_2$. From this derivation we
obtain a derivation of $\textup{QC}\vdash C_1 = C_2$, where the
steps corresponding to the equivalence relation $\equiv$ are
trivialised.
\end{proof}
\section{Discussions}
We have introduced the first complete equational theory for quantum
circuits. Although this equational theory is fairly simple,
\cref{Euler3dmulticontrolled} is an unbounded family of equations
---one for each possible number of control qubits. Such a family of
equations is a natural byproduct of our proof technique: The decoding
of each axiom of LOPP produces an equation made of multi-controlled
gates that has to be derived using QC.
It is actually quite surprising that \cref{Euler3dmulticontrolled} is
the only remaining equation with multi-controlled
gates.
Notice that one can get rid of these multi-controlled gates by
extending the context rule as described below. Indeed,
\cref{Euler3dmulticontrolled} can be derived from its 2-qubit case
\begin{multline}\label{Euler3d}
\tag{r'}
\scalebox{.6}{\tikzfig{Euler3D-2qubit-L}}=\\
\scalebox{.6}{\tikzfig{Euler3D-2qubit-R}}
\end{multline}
\noindent if one allows the following control context rule $\vdash \Lambda C_1 =
\Lambda C_2$ when $\vdash C_1=C_2$.
Notice that it requires extending the $\Lambda$-construction to any circuit -- which can be done in an inductive way like $\Lambda(C_2\circ C_1) = \Lambda C_2 \circ \Lambda C_1$ and $\Lambda (C_1\otimes C_2) = (\Lambda C_1\otimes id_m)\circ (id_1\otimes \sigma_{m,n})\circ (\Lambda C_2\otimes id_n)\circ (id_1\otimes \sigma_{n,m})$.
A natural application of
the completeness result is to design procedures for quantum circuit
optimisation based on this equational theory. One can take advantage of the terminating and confluent rewriting system for optical circuits \cite{clement2022LOv} by mimicking the applications of the rewrite rules on quantum circuits. However, the exponential blowup of the encoding map makes this approach probably inefficient as it is and requires some improvements.
Another future work is to prove (upper or lower) bounds on the size of
a derivation between two given equivalent circuits, as well as a bound
on the size of the intermediate quantum circuits. This might be
useful for providing a verifiable quantum advantage, in particular if
there exist polysize quantum circuits requiring exponentially
many rewrites \cite{aaronson-slide}.
\onecolumn
\appendices
\crefalias{section}{appendix}
\crefalias{subsection}{appendix}
\crefalias{subsubsection}{appendix}
\crefalias{paragraph}{appendix}
\crefalias{subparagraph}{appendix}
\section{Contents}
All the equations from Equation \eqref{CNotPCNotreversible} to Equation \eqref{CNotHCNot} are proven either directly from the axioms of $\textup{QC}_0$, given in
\cref{eq:qc0}, or from the equations already proven. Those proofs are in \cref{proof:usefuleq}.
Proposition \ref{ctrlXCNot} is proven in \cref{preuvectrlXCNot}.
In Appendix \ref{proofinductiveprop}, we highlight the inductive properties of multicontrolled gates which will be used in the inductive proofs of the following appendices, in the form of Lemmas \ref{lem:Lambda0} to \ref{lem:Lambda-Rx}.
Lemmas \ref{commctrlciblesdistinctes} to \ref{decompctrlblancRX} are introduced and proven by induction in \cref{proofmultictrl}. Alongside with Equation \eqref{eq:comRX} proven in \cref{sec:proofcomRX}, those properties are used to prove Propositions \ref{cor:swap} and \ref{prop:CP} in \cref{preuvesswapsmultictrl}.
To prove \cref{prop:sum}, we introduce Lemma \ref{commctrlphaseenhaut}.
We do a proof by induction with both hypotheses, to prove at the same time \cref{prop:sum} and Lemma \ref{commctrlphaseenhaut}, as detailed in \cref{proofpropsum}.
Appendices \ref{usefulmulctrl}, \ref{proof:commctrlPNotCvar} and \ref{commutationsXtarget} introduce and prove
Equations \eqref{mctrlX} and \eqref{Euler2dmulticontrolled} and Lemmas \ref{commctrlPNotCvar} to \ref{commctrlphaseenhautP}. Those properties on multi-controlled gates are to be used in other later proofs.
Propositions \ref{prop:comb}, \ref{commctrl}, \ref{commctrlpasenface}, \ref{EulerHmoins}, \ref{soundnessEuleraxioms}, \ref{klfollowfromEuler}, \ref{prop:CCX} and \ref{prop:period} are respectively proven in Appendix \ref{preuvepropcomb}, \ref{preuvecommctrl}, \ref{preuvecommctrlpasenface}, \ref{preuveEulerHmoins}, \ref{appendix:euler3d}, \ref{preuveklfollowfromEuler}, \ref{proof:CCX} and \ref{proof:propperiod}.
Theorem \ref{thm:LOPPcompleteness} is proven in \cref{proof:thmLOPPcompleteness}.
Appendices \ref{usefuldef} and \ref{proof:usefullemmas} introduce convenient notations and Lemmas \ref{antisymmetriecontrolee} to \ref{decodagefilsdisjoints}, useful for proving the main result.
Finally, Lemmas \ref{DE}, \ref{decodingtoporules} and \ref{decodingLOPPrules} are proven in Appendices \ref{lemmasforbasecaseDE}, \ref{preuvedecodingtoporules} and \ref{preuvedecodingLOPPrules}.
The $\sigma_{k,n,\ell}$ are defined in \cref{defsigma}.
\section{Useful Quantum Circuits Equations}
\subsection{Proofs of Equations \eqref{CNotPCNotreversible} to \eqref{CNotHCNot}}\label{proof:usefuleq}
\noindent Proof of \cref{CNotPCNotreversible}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{CNotPbCNot}&=&\tikzfig{CNotPbCNot1}\\[0.5cm]
&\eqeqref{tripleCNotswap}&\tikzfig{CNotPbCNot2}\\[0.5cm]
&\eqeqref{CNotCNot}&\tikzfig{CNotPbCNot3}\\[0.5cm]
&\eqeqref{commutationPctrl}&\tikzfig{CNotPbCNot4}\\[0.5cm]
&\eqeqref{CNotCNot}&\tikzfig{CNotPbCNotalenvers}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{commutationCNotsbas}:
\begin{eqnarray*} \label{proofcommutationCNotsbas}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{CNotgrandCNotbas}&\eqeqref{CNotCNot}&\tikzfig{CNotgrandCNotbas1}\\[0.5cm]
&\eqeqref{CNotlift}&\tikzfig{CNotgrandCNotbas2}\\[0.5cm]
&\eqeqref{CNotCNot}&\tikzfig{CNotbasCNotgrand}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{CNotHH}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{CNotHH}&\eqeqref{HH}&\tikzfig{CNotHH1}\\[0.5cm]
&\eqeqref{CZ}&\tikzfig{CNotHH2}\\[0.5cm]
&\eqeqref{CNotPCNotreversible}&\tikzfig{CNotHH3}\\[0.5cm]
&\eqeqref{CZ}&\tikzfig{CNotHH4}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{HHNotC}
\end{eqnarray*}
Note that the second use of \cref{CZ} relies on the fact that \gcnot{} is defined as \minitikzfig{NotCdef-s}, and
uses a few topological rules
.
\noindent Proof of \cref{NotCRXNotCreversible}:
\begin{longtable}{RCL}
\tikzfig{RXhconjCNot}&\eqeqref{HH}&\tikzfig{RXhconjCNot1}\\\\
&\eqeqref{CNotHH}&\tikzfig{RXhconjCNot2}\\\\
&\eqeqref{CNotPCNotreversible}&\tikzfig{RXhconjCNot3}\\\\
&\eqeqref{CNotHH}&\tikzfig{RXhconjCNot4}\\\\
&\eqeqref{HH}&\tikzfig{RXbconjCNot}.
\end{longtable}
\noindent Proof of \cref{XX}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{XX}&\eqeqref{P0}&\tikzfig{XX1}\\[0.5cm]
&\eqeqref{XPX}&\tikzfig{XX2}\\[0.5cm]
&\eqdeuxeqref{S0}{P0}&\tikzfig{filcourt-s}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{CNotliftvar}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{CNotgrandCNotbas}&\eqeqref{commutationCNotsbas}&\tikzfig{CNotgrandconjNotChauts1}\\[0.5cm]
&\eqeqref{CNotlift}&\tikzfig{CNotgrandconjNotChauts2}\\[0.5cm]
&=&\tikzfig{CNotgrandconjNotChauts3}\\[0.5cm]
&=&\tikzfig{CNotgrandconjNotChauts}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{commutationXCNot}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{XbCNot}&\eqdeuxeqref{HH}{xgate}&\tikzfig{XbCNot1}\\[0.5cm]
&\eqdeuxeqref{CNotHH}{zgate}&\tikzfig{XbCNot2}\\[0.5cm]
&\eqdeuxeqref{commutationPctrl}{CNotHH}&\tikzfig{XbCNot3}\\[0.5cm]
&\eqdeuxeqref{HH}{xgate}&\tikzfig{CNotXb}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{ZZ}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{ZZ}&\eqdeuxeqref{xgate}{HH}&\tikzfig{ZZ1}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{ZZ2}\\[0.5cm]
&\eqeqref{XX}&\tikzfig{HH}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{filcourt-s}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{NotClift}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{NotCgrandNotCbas}&=&\tikzfig{NotCNotCNotC1}\\[0.5cm]
&\eqdeuxeqref{commutationCNotsbas}{CNotlift}&\tikzfig{NotCNotCNotC2}\\[0.5cm]
&\eqeqref{CNotCNot}&\tikzfig{NotCNotCNotC3}\\[0.5cm]
&=&\tikzfig{NotCNotCNotC}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{ZCNot}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{ZbCNot}&\eqdeuxeqref{xgate}{HH}&\tikzfig{ZbCNot1}\\[0.5cm]
&\eqeqref{CNotHH}&\tikzfig{ZbCNot2}\\[0.5cm]
&\eqeqref{CNotX}&\tikzfig{ZbCNot3}\\[0.5cm]
&\eqeqref{CNotHH}&\tikzfig{ZbCNot4}\\[0.5cm]
&\eqdeuxeqref{xgate}{HH}&\tikzfig{CNotZZ}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{commutationRXCNot}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{XCNot}&\eqdeuxeqref{HH}{RXgate}&\tikzfig{XCNot1}\\[0.5cm]
&\eqeqref{CNotHH}&\tikzfig{XCNot2}\\[0.5cm]
&\eqeqref{commutationPctrl}&\tikzfig{XCNot3}\\[0.5cm]
&\eqeqref{CNotHH}&\tikzfig{XCNot4}\\[0.5cm]
&\eqdeuxeqref{HH}{RXgate}&\tikzfig{CNotX}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{RX0}:
\begin{longtable}{RCL}
\tikzfig{RX0}&\eqeqref{RXgate}&\tikzfig{HP0H}\\\\
&\eqdeuxeqref{S0}{P0}&\tikzfig{HH}\\\\
&\eqeqref{HH}&\tikzfig{filcourt-s}\\\\
\end{longtable}
\noindent Proof of \cref{RXRX}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{RXRX}&\eqeqref{RXgate}&\tikzfig{RXRX1}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{RXRX2}\\[0.5cm]
&\eqdeuxeqref{SS}{PP}&\tikzfig{RXRX3}\\[0.5cm]
&\eqeqref{RXgate}&\tikzfig{RXthetaplusthetaprime}\\[0.5cm]
\end{eqnarray*}
\noindent Proof of \cref{CNotHCNot}:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT}&\eqeqref{HH}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT1}\\[0.5cm]
&\eqdeuxeqref{CNotHH}{HH}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT2}\\[0.5cm]
&\eqeqref{CZ}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT3}\\[0.5cm]
&\eqdeuxeqref{CNotPCNotreversible}{CNotCNot}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT4}\\[0.5cm]
&\eqeqref{commutationPctrl}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT4bis}\\[0.5cm]
&\eqdeuxeqref{ZZ}{ZCNot}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT5}\\[0.5cm]
&\eqtroiseqref{zgate}{PP}{ZZ}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT6}\\[0.5cm]
&\eqeqref{commutationPctrl}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT6bis}\\[0.5cm]
&\eqdeuxeqref{CNotPCNotreversible}{commutationPctrl}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT6ter}\\[0.5cm]
&\eqeqref{CZ}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT7}\\[0.5cm]
&\eqdeuxeqref{zgate}{commutationPctrl}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT8}\\[0.5cm]
&\eqdeuxeqref{xgate}{HH}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT9}\\[0.5cm]
&\eqdeuxeqref{XX}{CNotHH}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT10}\\[0.5cm]
&\eqdeuxeqref{HH}{CNotCNot}&\tikzfig{XbHhCNotHhCNotHhCNOTHhCNOT11}\\[0.5cm]
\end{eqnarray*}
It follows that
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \tikzfig{CNotHhCNot}&\eqdeuxeqref{CNotCNot}{HH}&\tikzfig{XbHhCNotHhCNotHh}\\[0.5cm]
\end{eqnarray*}
\subsection{Proof of Proposition \ref{ctrlXCNot}}\label{preuvectrlXCNot}
First, we can notice that
\begin{longtable}{RCL}
\Lambda^{1} P(\pi)&\eqdef&\tikzfig{Ppidec1} \\\\
&\overset{\text{def}}{=}& \tikzfig{Ppidec2} \\\\
&\eqeqref{RXgate}& \tikzfig{Ppidec2bis1} \\\\
&\eqdeuxeqref{HH}{SS}& \tikzfig{Ppidec2bis2} \\\\
&\eqeqref{S0}& \tikzfig{Ppidec3} \\\\
&\eqeqref{CNotHH}& \tikzfig{Ppidec4} \\\\
&\eqdeuxeqref{HH}{CNotHH}& \tikzfig{Ppidec5} \\\\
&\eqeqref{HH}& \tikzfig{Ppidec6} \\\\
&\eqeqref{CZ}& \tikzfig{CZ}
\end{longtable}
It follows that
\begin{longtable}{RRCL}
\qczero\vdash&\Lambda^1 X &\eqdef& \tikzfig{Lambda1Xdec1} \\\\
&&=& \tikzfig{Lambda1Xdec2} \\\\
&&\eqeqref{HH}& \tikzfig{CNot-m}.
\end{longtable}
\subsection{Inductive properties for multi-controls: Lemmas \ref{lem:Lambda0} to \ref{lem:Lambda-Rx}}
\label{proofinductiveprop}
The following technical lemmas highlight the inductive properties of
the circuits $\Lambda^xG$. They are at the heart of the proof of the
completeness result.
\begin{lemma}[Base case for the inductive properties]
\label{lem:Lambda0}
For all $G\in\{s(\psi),X,R_X(\theta),P(\varphi)\}$, if
$\epsilon$ is the empty list, $\Lambda^{\epsilon} G = G$.
\end{lemma}
\begin{proof}
In the case of an empty list, in
\cref{def:multicontrolled-oriented} there are no gates $X^{\overline{x_i}}$, and
$\Lambda^\epsilon G = \lambda^0G$. We can then check in
\cref{def:multicontrolled-pos} that each $\lambda^0G$ is
$G$: by definition this is true for $R_X(\theta)$, $s(\psi)$ and
$P(\theta)$. For $X$ we fall back on the definition of $X$ as
$HP(\pi)H=HZH$.
\end{proof}
\begin{lemma}[Inductive properties for $\Lambda^xG$]
\label{lem:Lambda0x}
For all $x\in \{0,1\}^k$,
and $G\in \{s(\varphi),X,R_X(\theta),P(\varphi)\}$,
\[\Lambda^{0x} G=\scalebox{0.8}{\tikzfig{neg-control}}\]
\end{lemma}
\begin{proof}
This is directly derived from the definition of $\Lambda^xG$: the
$X^{\overline{x_1}}$'s on the top wire are $X$ for $\Lambda^{0x} G$ and the identity
for $\Lambda^{1x} G$, while the $X^{\overline{x_i}}$'s on the lower wires are the
same.
\end{proof}
\begin{lemma}[Inductive properties for $\Lambda^xs(\varphi)$]
\label{lem:Lambda-s}
Suppose that $x$ is a $k$-length list of booleans. We then have
$\Lambda^1 s(\varphi) = P(\varphi)$, $\Lambda^{1x1}s(\varphi) =
\Lambda^{1x} P(\varphi)$, and
\[
\Lambda^{1x0}s(\varphi)= \scalebox{0.8}{\tikzfig{mctrlsphi0}}
\]
\end{lemma}
\begin{proof}
By definition, $\Lambda^1 s(\varphi)$ is $\lambda^1 s(\varphi)$:
there are no $X^{\overline{x_i}}$ since the list only contains a single $1$. By
definition, $\lambda^1 s(\varphi)$ is $\lambda^0 P(\varphi)$, which
is $P(\varphi)$.
Suppose now that $x$ is a $k$-length list of booleans, and $b$ is a
single boolean. Consider $\Lambda^{1xb}s(\varphi)$: by definition
it is
\[
\scalebox{0.8}{\tikzfig{generalControledGateS1xb}}.
\]
By definition, $\lambda^{k+2}s(\varphi) =
\lambda^{k+1}P(\varphi)$. Now, $\Lambda^{1x}P(\varphi)$ is
\[
\scalebox{0.8}{\tikzfig{generalControledGateP1x}}.
\]
We directly recover $\Lambda^{1x1}s(\varphi)$, i.e. when $b=1$, and
the case $b=0$ since this just amounts to add the two gates $X^{\overline{0}}=X^1 =
X$ on the bottom wire.
\end{proof}
\begin{lemma}[Inductive properties of $\Lambda^xX$]
\label{lem:Lambda-X}
Suppose that $x$ is a $k$-length list of boolean. Then
\[\Lambda^{1x}X = \scalebox{0.8}{\tikzfig{mctrlPH}}.\]
\end{lemma}
\begin{proof}
By definition,
\[
\Lambda^{1x}X = \scalebox{0.8}{\tikzfig{mctrlPHdef}}
= \scalebox{0.8}{\tikzfig{mctrlPHdef2}},
\]
which is exactly the right-hand-side of the desired equation.
\end{proof}
\begin{lemma}[Inductive properties of $\Lambda^xP(\varphi)$]
\label{lem:Lambda-P}
Suppose that $x$ is a $k$-length list of boolean. Then
\[\qczero\vdash \Lambda^{1x} P(\varphi) = \scalebox{0.8}{\tikzfig{mctrlPphi2}} \]
\end{lemma}
\begin{proof}
By definition,
\[ \Lambda^{1x} P(\varphi)
= \scalebox{0.8}{\tikzfig{mctrlP1xdef}}
= \scalebox{0.8}{\tikzfig{mctrlP1xdef2}}
\]
Since $XX$ is the identity according to \cref{XX}, this is equal to
\[
\scalebox{0.8}{\tikzfig{mctrlP1xdef3}}.
\]
We can conclude by noting that
\[
\Lambda^{1x} s(\frac\varphi2) =
\scalebox{0.8}{\tikzfig{mctrlP1xdef3-2}}
\text{ and }
\Lambda^{1x} R_X(\varphi) =
\scalebox{0.8}{\tikzfig{mctrlP1xdef3-1}}.
\]
\end{proof}
\begin{lemma}[Inductive properties of $\Lambda^xR_X(\varphi)$]
\label{lem:Lambda-Rx}
Suppose that $x$ is a $k$-length list of boolean. Then
\[\qczero\vdash \Lambda^{1x} R_X(\theta)=\scalebox{0.8}{\tikzfig{mctrlXtheta}}.
\]
\end{lemma}
\begin{proof}
By definition of $\Lambda^{1x} R_X(\theta)$ and $\lambda^{k+1}
R_X(\theta)$, we have:
\[ \Lambda^{1x} R_X(\theta) =
\scalebox{0.8}{\tikzfig{mctrlXtheta2}}.
\]
Using \cref{XX}, we infer that
\[ \Lambda^{1x} R_X(\theta) =
\scalebox{0.8}{\tikzfig{mctrlXtheta3}}.
\]
We can then conclude by using the definition of $\Lambda^{x}
R_X(\frac\theta2)$ and $\Lambda^{x} R_X(\text{-}\frac\theta2)$ (and
the deformation of circuits coming from the prop structure).
\end{proof}
Since these lemmas are essentially consequences of the definitions (except for the use of \cref{XX} in \cref{lem:Lambda-P,lem:Lambda-Rx}), in the following we will mostly keep their uses implicit.
\subsection{Ancillary lemmas: Lemmas \ref{commctrlciblesdistinctes} to \ref{decompctrlblancRX}}
\label{proofmultictrl}
\newcommand{\tripleindice}[3]{\begingroup\arraycolsep=0pt\def\arraystretch{0.5}\begin{scriptarray}{c}#1\\#2\\#3\end{scriptarray}\kern.08333em\endgroup}
For the following lemmas, it is convenient to introduce a graphical notation of multi-controlled gate which allows for more flexibility in the position of the target qubit, relatively to the control qubits:
\[ \scalebox{1}{\tikzfig{LambdaGxynew}} \coloneqq \scalebox{1}{\tikzfig{LambdaGxynewdef}}\]
\begin{lemma}\label{commctrlciblesdistinctes}
For any $x\in\{0,1\}^k$,
\[\tikzfig{LambdaRXhLambdaRXb}\ =\ \tikzfig{LambdaRXbLambdaRXh}.\]
\end{lemma}
\begin{proof}
We proceed by induction on $k$. If $k=0$, then the equality is a consequence of the topological rules. If $k\geq1$, by \cref{XX} we can assume without loss of generality that $x=1z$ with $z\in\{0,1\}^{k-1}$. One has
\[\tikzfig{LambdaRXhLambdaRXbgrand}\ \overset{\text{\cref{lem:Lambda-Rx}}}{=}\ \tikzfig{LambdaRXhLambdaRXbdecomp}\]
then it is easy to see that the two parts commute by induction hypothesis and \cref{HH,commutationCNotsbas}, together with topological rules.
\end{proof}
\begin{lemma}\label{CZRXCZreversiblecontrole}
For any $x\in\{0,1\}^k$,
\[\tikzfig{LambdaRXhconjCNot}\ =\ \tikzfig{LambdaRXbconjCNot}.\]
\end{lemma}
\begin{proof}
We proceed by induction on $k$. If $k=0$, then
the result is just \cref{NotCRXNotCreversible}.
If $k\geq1$, then we can assume without loss of generality that $x=1z$ with $z\in\{0,1\}^{k-1}$. One has
\begin{longtable}{RCL}
\tikzfig{LambdaRXhconjCNotgrand}&=&\tikzfig{LambdaRXhconjCNot1}\\\\
&\eqeqref{CNotCNot}&\tikzfig{LambdaRXhconjCNot2}\\\\
&\eqeqref{commutationCNothaut}&\tikzfig{LambdaRXhconjCNot3}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{LambdaRXhconjCNot4}\\\\
&\eqeqref{CNotCNot}&\tikzfig{LambdaRXhconjCNot5}\\\\
&\eqdeuxeqref{NotClift}{commutationCNothaut}&\tikzfig{LambdaRXhconjCNot6}\\\\
&\eqeqref{CNotCNot}&\tikzfig{LambdaRXhconjCNot7}\\\\
&=&\tikzfig{LambdaRXbconjCNotgrand}
\end{longtable}
\end{proof}
\begin{lemma}\label{decompctrlblancRX}
For any $x\in\{0,1\}^k$,
\[\qczero\vdash\Lambda^{0x}R_X(\theta)=\tikzfig{mctrlXthetablanc}.\]
\end{lemma}
\begin{proof}
The proof relies on the following property:
\begin{equation}\label{ZctrlRX}
\qczero\vdash\tikzfig{ZmctrlRX}\ =\ \tikzfig{mctrlRXZmoins}
\end{equation}
that we prove by induction on the length of $x$ as follows:
If $x=\epsilon$, then
\begin{longtable}{RCL}
\tikzfig{ZRX}&\overset{\eqref{RXgate}}{=}&\tikzfig{ZHPH}\\\\
&\eqdeuxeqref{HH}{xgate}&\tikzfig{HXPH}\\\\
&\eqtroiseqref{XX}{XPX}{SS}&\tikzfig{HPXH}\\\\
&\eqtroiseqref{xgate}{RXgate}{HH}&\tikzfig{RXZmoins}
\end{longtable}
If $x\neq\epsilon$, then the commutation is a direct consequence of the induction hypothesis and \cref{commutationPctrl}.
Given this property, the result can be deduced as follows:
\begin{longtable}{RCL}
\Lambda^{0x}R_X(\theta)&=&\tikzfig{mctrlRX1}\\\\
&\eqdeuxeqref{xgate}{HH}&\tikzfig{mctrlRX2}\\\\
&\eqeqref{ZCNot}&\tikzfig{mctrlRX3}\\\\
&\eqeqref{ZctrlRX}&\tikzfig{mctrlRX4}\\\\
&\eqquatreeqref{ZCNot}{zgate}{commutationPctrl}{ZZ}&\tikzfig{mctrlRX5}\\\\
&\eqtroiseqref{xgate}{HH}{ZZ}&\tikzfig{mctrlXthetablanc}
\end{longtable}
\end{proof}
\subsection{Proof of Equation \eqref{eq:comRX}}
\label{sec:proofcomRX}
We actually prove a slightly more general result: for any $x,x'\in\{0,1\}^k$,
\begin{equation}\label{commctrlRXconjCNots}\qczero\vdash\tikzfig{com-L-prime}\ =\ \tikzfig{com-R-prime}\end{equation}
\Cref{eq:comRX} corresponds to the case where $x=x'$.
\begin{proof}[Proof of \cref{commctrlRXconjCNots}]
The proof is by induction on $x$.
\\\noindent If $x=\epsilon$ (i.e. $k=0$),
\begin{eqnarray*}\tikzfig{RXNotCRXNotC}&\eqeqref{RXgate}&\tikzfig{RXNotCRXNotC-1}\\[0.5cm]
&\eqdeuxeqref{HH}{CNotHH}&\tikzfig{RXNotCRXNotC-2}\\[0.5cm]
&\eqquatreeqref{CNotPCNotreversible}{RXgate}{HH}{SS}&\tikzfig{RXNotCRXNotC-3}\\[0.5cm]
&\eqeqref{commutationPctrl}&\tikzfig{RXNotCRXNotC-4}\\[0.5cm]
&\eqquatreeqref{CNotPCNotreversible}{HH}{SS}{RXgate}&\tikzfig{RXNotCRXNotC-5}\\[0.5cm]
&\eqtroiseqref{CNotHH}{HH}{RXgate}&\tikzfig{NotCRXNotCRX}\end{eqnarray*}
If $k\geq 1$, then we can write $x=az$ and $x'=a'z'$ with $a,a'\in\{0,1\}$. One has (where the $\pm$ signs correspond respectively to $(-1)^a$ and $(-1)^{a'}$):
\begin{longtable}{CL}
\tikzfig{com-L-prime}\hspace*{-15em}\\\\
\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{ctrlzRXctrlzprimeRXCNots1}\\\\
\eqtroiseqref{HH}{commutationCNothaut}{NotClift}&\tikzfig{ctrlzRXctrlzprimeRXCNots2}\\\\
\eqdeuxeqref{CNotCNot}{NotClift}&\tikzfig{ctrlzRXctrlzprimeRXCNots3}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{ctrlzRXctrlzprimeRXCNots4}\\\\
\eqeqref{CNotlift}&\tikzfig{ctrlzRXctrlzprimeRXCNots5}\\\\
\eqdeuxeqref{commutationCNotsbas}{commutationCNothaut}&\tikzfig{ctrlzRXctrlzprimeRXCNots6}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{ctrlzRXctrlzprimeRXCNots7}\\\\
\eqdeuxeqref{commutationCNotsbas}{commutationCNothaut}&\tikzfig{ctrlzRXctrlzprimeRXCNots8}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{ctrlzRXctrlzprimeRXCNots9}\\\\
\eqeqref{commutationCNotsbas}&\tikzfig{ctrlzRXctrlzprimeRXCNots10}\\\\
\eqeqref{CNotlift}&\tikzfig{ctrlzRXctrlzprimeRXCNots11}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{ctrlzRXctrlzprimeRXCNots12}\\\\
\eqeqref{NotClift}&\tikzfig{ctrlzRXctrlzprimeRXCNots13}\\\\
\eqdeuxeqref{commutationCNotsbas}{CNotCNot}&\tikzfig{ctrlzRXctrlzprimeRXCNots14}\\\\
=&\tikzfig{ctrlzRXctrlzprimeRXCNots15}\\\\
\eqdeuxeqref{NotClift}{commutationCNothaut}&\tikzfig{ctrlzRXctrlzprimeRXCNots16}\\\\
\eqeqref{HH}&\tikzfig{ctrlzRXctrlzprimeRXCNots17}\\\\
\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{com-R-prime}.
\end{longtable}
\end{proof}
\subsection{Proof of Propositions \ref{cor:swap} and \ref{prop:CP}}\label{preuvesswapsmultictrl}
First, we consider the case $G=R_X(\theta)$ of Equations~\eqref{swapCCZ}-\eqref{swapCCZ3}, for which the proof is a direct induction based on \cref{eq:comRX} that is proven in \cref{sec:proofcomRX}.
Next, we prove \cref{phasemobile} in the case $y=\epsilon$.
We can assume without loss of generality that $x=1^k$. We proceed by induction on $k$.
If $k=0$, then
\begin{longtable}{RCL}
\Lambda^1P(\varphi)&\eqdef&\tikzfig{ctrlPdecbrut}\\\\
&\eqquatreeqref{SS}{S0}{HH}{CNotHH}&\tikzfig{ctrlPdecsimple}\\\\
&\eqeqref{CNotPCNotreversible}&\tikzfig{ctrlPdecsimplealenvers}\\\\
&\eqquatreeqref{SS}{S0}{HH}{CNotHH}&\Lambda^\epsilon_1 P(\varphi).
\end{longtable}
If $k\geq1$,
then one has
\begin{longtable}{RCL}
\Lambda^{x1}P(\varphi)&\eqdef&\tikzfig{LambdaPmove1}\\\\\\
&\overset{\text{Equations~\eqref{swapCCZ}-\eqref{swapCCZ3}}}{\overset{\text{(case $G=R_X(\theta)$)}}{=}}&\tikzfig{LambdaPmove2}\\\\\\
&\overset{\text{def}}{=}&\tikzfig{LambdaPmove3}\\\\\\
&=&\tikzfig{LambdaPmove4}\\\\\\
&\overset{\text{def}}{=}&\scalebox{0.8}{$\tikzfig{LambdaPmove5}$}\\\\\\
&\overset{\text{\cref{commctrlciblesdistinctes,CZRXCZreversiblecontrole}}}{=}&\scalebox{0.8}{$\tikzfig{LambdaPmove6}$}\\\\\\
&=&\scalebox{0.8}{$\tikzfig{LambdaPmove7}$}\\\\\\
&=&\tikzfig{LambdaPmove8}\\\\\\
&=&\Lambda^x_1 P(\varphi).
\end{longtable}
Now, we can prove Equations~\eqref{swapCCZ}-\eqref{swapCCZ3} in the case $G=s(\psi)$ (the cases $G=P(\varphi)$ and $G=X$ are direct consequences of this case). Without loss of generality we can assume $y=\epsilon$ and consider only \cref{swapCCZ}.
The proof is by induction on the number $r$ of input qubits of $\Lambda^{xabz}G$. If $z=\epsilon$, which is necessarily the case in the base case $r=2$, then the result is a direct consequence of the case $y=\epsilon$ of \cref{phasemobile}. If $z\neq\epsilon$, then using \cref{def:multicontrolled-pos,def:multicontrolled-oriented} (in particular in the case of $\Lambda^{1x}P(\varphi)$), the result is a direct consequence of the induction hypothesis and the case $G=R_X(\theta)$ of Equations~\eqref{swapCCZ}-\eqref{swapCCZ3}.
Finally, using the definition of $\Lambda^x_{y1} P(\varphi)$ in terms of $\Lambda^{xy1} P(\varphi)$, the general case of \cref{phasemobile} follows directly from the case $y=\epsilon$ and Equations~\eqref{swapCCZ}-\eqref{swapCCZ3}.
\subsection{Proof of Proposition \ref{prop:sum}}
\label{proofpropsum}
It remains to treat the $\Lambda^xP$ and $\Lambda^xs$ cases of \cref{prop:sum}.
Those cases are a direct consequence of the following lemma:
\begin{lemma}\label{commctrlphaseenhaut}
For any $x\in\{0,1\}^k$ and $y\in\{0,1\}^\ell$ with $\ell\geq k$,
\[\qczero\vdash\tikzfig{mctrlxeiphihmctrlyRX}\ =\ \tikzfig{mctrlyRXmctrlxeiphih}.\]
\end{lemma}
To prove the previous lemma, we do a proof by induction on $k$.
However, to prove the induction step for $k\geq 2$, we use $\qczero \vdash \Lambda^{1^{k-2}} s(\varphi) \circ \Lambda^{1^{k-2}} s(\varphi')= \Lambda^{1^{k-2}} s(\varphi+\varphi')$ and
$\qczero\vdash \Lambda^{1^{k-2}} s(0)= id_{k-1}$, which are the statements of \cref{prop:sum}.
Therefore, we will do a common induction proof for both the other cases of \cref{prop:sum} and for \cref{commctrlphaseenhaut}.
The plan of the proof is the following. First we prove an ancillary equation (\cref{passageCNOtcontroles}) which is derived from previous lemmas. Then we proceed with the induction proof: for $k\geq2$, \cref{commctrlphaseenhaut} is proved with \cref{prop:sum} for $k-2$, while the induction step of \cref{prop:sum} is directly a consequence of \cref{commctrlphaseenhaut} and \cref{prop:sum} for $k-1$, and the $\Lambda^{x}R_X$ case which is already proven.
\begin{proof}
First we prove the following property, which is true for any $a,b\in\{0,1\}$, $z\in\{0,1\}^m$ and $G\in\{s(\varphi),P(\varphi),R_X(\theta),X\}$:
\begin{equation}\label{passageCNOtcontroles}
\qczero\vdash\tikzfig{CNothmctrlG}\ =\ \tikzfig{mctrlGCNoth}\qquad\text{where $c=\begin{cases}b&\text{if $a=0$}\\\bar b&\text{if $a=1$}\end{cases}$}
\end{equation}
To prove \cref{passageCNOtcontroles}, by \cref{CNotX,commutationXCNot,XX} we can assume without loss of generality that $a=b=1$. If $G=R_X(\theta)$, then
\begin{longtable}{CL}
\tikzfig{CNothmctrl11zRX}\hspace*{-7em}\\\\
=&\tikzfig{CNotmctrlRX1}\\\\
\eqdeuxeqref{CNotHH}{HH}&\tikzfig{CNotmctrlRX2}\\\\
\eqdeuxeqref{CNotCNot}{NotClift}&\tikzfig{CNotmctrlRX3}\\\\
\eqdeuxeqref{CNotCNot}{NotClift}&\tikzfig{CNotmctrlRX4}\\\\
\eqtroiseqref{commutationCNotsbas}{commutationCNothaut}{CNotCNot}&\tikzfig{CNotmctrlRX5}\\\\
\eqdeuxeqref{CNotCNot}{NotClift}&\tikzfig{CNotmctrlRX6}\\\\
\eqdeuxeqref{CNotCNot}{NotClift}&\tikzfig{CNotmctrlRX7}\\\\
\eqtroiseqref{commutationCNotsbas}{commutationCNothaut}{CNotCNot}&\tikzfig{CNotmctrlRX8}\\\\
\eqeqref{CNotHH}&\tikzfig{CNotmctrlRX9}\\\\
\eqeqref{CNotCNot}&\tikzfig{CNotmctrlRX10}\\\\
\eqeqref{NotClift}&\tikzfig{CNotmctrlRX11}\\\\
\eqeqref{CNotCNot}&\tikzfig{CNotmctrlRX11-1}\\\\
\eqtroiseqref{NotClift}{commutationCNothaut}{commutationCNotsbas}&\tikzfig{CNotmctrlRX12}\\\\
\eqeqref{eq:comRX}&\tikzfig{CNotmctrlRX13}\\\\
\eqdeuxeqref{commutationCNothaut}{CNotCNot}&\tikzfig{CNotmctrlRX14}\\\\
\eqeqref{eq:comRX}&\tikzfig{CNotmctrlRX15}\\\\
\eqeqref{CNotlift}&\tikzfig{CNotmctrlRX16}\\\\
\eqeqref{NotClift}&\tikzfig{CNotmctrlRX17}\\\\
\eqeqref{eq:comRX}&\tikzfig{CNotmctrlRX18}\\\\
\eqdeuxeqref{commutationCNothaut}{HH}&\tikzfig{CNotmctrlRX19}\\\\
\overset{\text{\cref{decompctrlblancRX},}}{\overset{\text{def}}{=}}&\tikzfig{mctrl10zRXCNoth}
\end{longtable}
Now, to prove \cref{prop:sum} and \cref{commctrlphaseenhaut}, by \cref{XX} we can assume without loss of generality that $x=1^k$. We proceed by induction on $k$. If $k=0$, then \cref{prop:sum} is a consequence of \cref{S0,,SS,,P0,,PP}, and \cref{commctrlphaseenhaut} is a consequence of the topological rules. If $k=1$, then $\Lambda^x s(\varphi)=P(\varphi)$. Let $y=az$ with $a\in\{0,1\}$. By \cref{decompctrlblancRX}, one has
\begin{longtable}{RCL}
\qczero\vdash\tikzfig{PphihmctrlyRX}&=&\tikzfig{PphihmctrlyRXdecomp}\\\\
&\eqquatreeqref{HH}{S0}{SS}{RXgate}&\tikzfig{PphihmctrlyRXdecomp1}\\\\
&\eqeqref{commutationRXCNot}&\tikzfig{PphihmctrlyRXdecomp2}\\\\
&\eqquatreeqref{RXgate}{SS}{S0}{HH}&\tikzfig{PphihmctrlyRXdecomp3}\\\\
&\overset{\text{\cref{decompctrlblancRX}}}=&\tikzfig{mctrlyRXPphih}
\end{longtable}
where the $\pm$ sign is $(-1)^a$.
The case of $k=1$ for \cref{prop:sum} is then a direct consequence of the previous result, the case with $R_X$, \cref{def:multicontrolled-pos} (case $\lambda^nP(\varphi)$) and \cref{HH,P0,,PP}.
If $k\geq2$, let $z=1^{k-1}$ and $t=1^{k-2}$. To prove \cref{commctrlphaseenhaut}, one has
\begin{longtable}{RCL}
\Lambda^xs(\varphi)&=&\tikzfig{mctrlxeiphi1}\\\\
&\overset{\text{induction hypothesis}}{\overset{\text{of \cref{prop:sum}}}{=}}&\tikzfig{mctrlxeiphi2}\\\\
&\overset{\text{induction hypothesis}}{\overset{\text{of \cref{commctrlphaseenhaut}}}{=}}&\tikzfig{mctrlxeiphi3}\\\\
&\overset{\text{\eqref{HH}, def}}{=}&\tikzfig{mctrlxeiphi4}\\\\
&\eqdeuxeqref{CNotHH}{HH}&\tikzfig{mctrlxeiphi5}\\\\
&\overset{\text{def}}{=}&\tikzfig{mctrlxeiphi6}.
\end{longtable}
\noindent Hence, the commutation with $\Lambda^yR_X(\theta)$ follows by induction hypothesis and \cref{passageCNOtcontroles}, together with \cref{cor:swap}.
Then to prove the $\Lambda^xP$ case of \cref{prop:sum}, one has
\begin{longtable}{RCL}
\Lambda^xP(\varphi')\circ\Lambda^xP(\varphi)&=&\tikzfig{propsumcasP1}\\\\
&\eqeqref{HH}&\tikzfig{propsumcasP2}\\\\
&\overset{\text{induction hypothesis}}{\overset{\text{of \cref{commctrlphaseenhaut}}}{=}}&\tikzfig{propsumcasP3}\\\\
&\overset{\text{$\Lambda^{x}R_X$ case and}}{\overset{\text{induction hypothesis}}{\overset{\text{of \cref{prop:sum}}}{=}}}&\tikzfig{propsumcasP4}\\\\
&=&\lambda^xP(\varphi+\varphi').
\end{longtable}
Finally, the $\Lambda^xs$ case is a direct consequence of the $\Lambda^zP$ case.
\end{proof}
\subsection{Ancillary equations: Equations \eqref{mctrlX} and \eqref{Euler2dmulticontrolled}}
\label{usefulmulctrl}
\begin{lemma}
The following equations can be derived in $\textup{QC}$:
\begin{equation}\label{mctrlX}\begin{array}{rcl}\Lambda^xX&=&\tikzfig{mctrlXsimpl}\end{array}\end{equation}
\begin{equation}\label{Euler2dmulticontrolled}\begin{array}{rcl}\tikzfig{Euler2dleft-multicontrolled}&=&\tikzfig{Euler2dright-multicontrolled}\end{array}\end{equation}
where in \cref{Euler2dmulticontrolled}, the angles are the same as in \cref{Euler2d}.
\end{lemma}
\begin{proof}
If $x=\epsilon$, then \cref{mctrlX} is a direct consequence of \crefnosort{lem:Lambda0,xgate,S0,SS,RXgate}. If $x\neq\epsilon$, then \cref{mctrlX} is a direct consequence of \cref{lem:Lambda0x,lem:Lambda-X,lem:Lambda-P,XX,HH}.
\noindent Proof of \cref{Euler2dmulticontrolled}:
\begin{longtable}{RCL}
\scalebox{0.8}{$\tikzfig{Euler2dleft-multicontrolled}$}&\equiv&\scalebox{0.8}{$\tikzfig{Euler2dleft-multicontrolled1}$}\\\\
&\overset{\text{\cref{prop:CP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{Euler2dleft-multicontrolled2}$}\\\\
&\eqeqref{Euler3dmulticontrolled}&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas-swaps}$}
\end{longtable}
By uniqueness of the right-hand side in \cref{Euler2d,Euler3dmulticontrolled}, the $\delta_i$ are such that the last circuit is equal to \vspacebeforeline{0.5em}$\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-anglesEuler2d}$}$
, where the $\beta_j$ are computed in the same way as in \cref{Euler2d}. It follows from \cref{prop:CP,prop:sum} that this is equal modulo $\qczero$ to the right-hand side of \cref{Euler2dmulticontrolled}.
\end{proof}
\section{Proofs of Sections \ref{propertiesmultictrl} and \ref{Eulerandperiod}}
\label{}
\subsection{Proof of Proposition \ref{prop:comb}}\label{preuvepropcomb}
Without loss of generality, we can assume that $y=\epsilon$.
The case where $G=s(\varphi)$ and $x=\epsilon$ follows directly from \crefnosort{XPX,,PP,,P0}. The cases where $G=s(\varphi)$ and $x\neq\epsilon$ follow directly from the case $G=P(\varphi)$, together with \cref{XX}.
By \cref{HH,XX}, the case $G=X$ follows directly from the case $G=P(\pi)$.
The case $G=P(\varphi)$ follows from the case $G=R_X(\theta)$ by a straightforward induction, using \cref{lem:Lambda-P,HH,commctrlphaseenhaut}.
Thus, it suffices to treat the case where $G=R_X(\theta)$. One has
\begin{longtable}{RCL}
\Lambda^{0x}R_X(\theta)\circ\Lambda^{1x}R_X(\theta)&\overset{\text{\cref{lem:Lambda-Rx,decompctrlblancRX}}}{=}&\tikzfig{propcomb1}\\\\
&\eqeqref{HH}&\tikzfig{propcomb2}\\\\
&\eqeqref{eq:comRX}&\tikzfig{propcomb3}\\\\\\
&\overset{\text{\eqref{CNotCNot}, \cref{prop:sum},}}{\overset{\eqref{CNotCNot}\eqref{HH}}{=}}&\tikzfig{propcombfinal}.
\end{longtable}
\subsection{Ancillary lemmas: Lemmas \ref{commctrlPNotCvar} to \ref{commutationctrldotsCNotRX}}
\label{proof:commctrlPNotCvar}
\begin{lemma}\label{commctrlPNotCvar}
For any $x\in\{0,1\}^k$,
\[\tikzfig{HmctrlRXHNotC}\ =\ \tikzfig{NotCHmctrlRXH}\]
\end{lemma}
\begin{proof}
We proceed by induction on $k$. If $k=0$ then the result is a direct consequence of \cref{RXgate,HH,commutationPctrl}. If $k\geq1$, then without loss of generality we can assume that $x=1z$ with $z\in\{0,1\}^{k-1}$. One has
\begin{longtable}{RCL}
\tikzfig{HmctrlRXHNotC-grand}&=&\tikzfig{HmctrlRXHNotCdecomp}\\\\
&\eqeqref{HH}&\tikzfig{HmctrlRXHNotCdecomp1}\\\\
&\eqeqref{CNotHH}&\tikzfig{HmctrlRXHNotCdecomp2}\\\\
&\eqeqref{HH}&\tikzfig{HmctrlRXHNotCdecomp3}\\\\
&\eqdeuxeqref{CNotCNot}{CNotliftvar}&\tikzfig{HmctrlRXHNotCdecomp4}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{HmctrlRXHNotCdecomp5}\\\\
&\eqdeuxeqref{CNotliftvar}{CNotCNot}&\tikzfig{HmctrlRXHNotCdecomp7}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{HmctrlRXHNotCdecomp8}\\\\
&\eqtroiseqref{HH}{CNotHH}{HH}&\tikzfig{HmctrlRXHNotCdecomp9}\\\\
&=&\tikzfig{NotCHmctrlRXH-grand}.
\end{longtable}
\end{proof}
\begin{lemma}\label{commutationctrldotsCNotRX}
For any $x\in\{0,1\}^k$,
\[\tikzfig{cLambdaRXCNoth}\ =\ \tikzfig{CNothcLambdaRX}.\]
\end{lemma}
\begin{proof}~
\begin{longtable}{RCL}
\tikzfig{cLambdaRXCNoth}&=&\tikzfig{cLambdaRXCNoth1}\\\\
&\eqeqref{HH}&\tikzfig{cLambdaRXCNoth2}\\\\
&\eqeqref{CNotHH}&\tikzfig{cLambdaRXCNoth3}\\\\
&\eqeqref{commutationCNotsbas}&\tikzfig{cLambdaRXCNoth4}\\\\
&\eqeqref{CNotHH}&\tikzfig{cLambdaRXCNoth5}\\\\
&\eqeqref{HH}&\tikzfig{cLambdaRXCNoth6}\\\\
&=&\tikzfig{CNothcLambdaRX}
\end{longtable}
\end{proof}
\subsection{Proof of Proposition \ref{commctrl}}\label{preuvecommctrl}
We assume without loss of generality that $y=y'=\epsilon$.
First, for the case where $G=R_X(\theta)$ and $G'=R_X(\theta')$, we prove by induction on $k$
that for any $x,x'\in\{0,1\}^k$,
\begin{equation}\label{commctrlRX}\qczero\vdash\Lambda^xR_X(\theta)\circ\Lambda^{x'}R_X(\theta')=\Lambda^{x'}R_X(\theta')\circ\Lambda^xR_X(\theta).\end{equation}
The desired result corresponds to \cref{commctrlRX} with $x\neq x'$. Note that when $x=x'$, \cref{commctrlRX} is a consequence of \cref{prop:sum}.
If $k=0$, then
\cref{commctrlRX} is a direct consequence of \cref{RXRX}. If $k\geq 1$, then we can write $x=az$ and $x'=a'z'$ with $a,a'\in\{0,1\}$. One has (where the $\pm$ signs correspond respectively to $(-1)^a$ and $(-1)^{a'}$):
\begin{longtable}{RCL}
\Lambda^{x'}R_X(\theta')\circ\Lambda^xR_X(\theta)&\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{ctrlzRXctrlzprimeRX1}\\\\
&\eqeqref{HH}&\tikzfig{ctrlzRXctrlzprimeRX2}\\\\
&\eqeqref{commctrlRXconjCNots}&\tikzfig{ctrlzRXctrlzprimeRX3}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{ctrlzRXctrlzprimeRX4}\\\\
&\eqeqref{CNotCNot}&\tikzfig{ctrlzRXctrlzprimeRX5}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis,}}{\eqeqref{CNotCNot}}}&\tikzfig{ctrlzRXctrlzprimeRX6}\\\\
&\eqeqref{commctrlRXconjCNots}&\tikzfig{ctrlzRXctrlzprimeRX7}\\\\
&\eqeqref{HH}&\tikzfig{ctrlzRXctrlzprimeRX8}\\\\
&\overset{\text{\cref{decompctrlblancRX}}}{=}&\Lambda^xR_X(\theta)\circ\Lambda^{x'}R_X(\theta')
\end{longtable}
If $G=P(\theta)$ and $G'=P(\theta')$, we prove by induction on $k$ that for any $z,z'\in\{0,1\}^k$,
\begin{equation}\label{commutationLambdaeiphi}
\Lambda^zs(\varphi)\circ\Lambda^{z'}s(\varphi')=\Lambda^{z'}s(\varphi')\circ\Lambda^zs(\varphi).
\end{equation}
The result corresponds to the case where
$z=x1$ and $z'=x'1$ with $x\neq x'$. Note that the case where $x=x'$ is a consequence of \cref{prop:sum}.
If $k=0$, then
\cref{commutationLambdaeiphi} is a consequence of the topological rules.
If $k=1$, then
it is a consequence of \cref{PP,XPX}.
If $k\geq2$, note first that by Equations \eqref{xgate}, \eqref{HH}, \eqref{ZctrlRX}, and \eqref{ZZ} (or \eqref{XPX}, \eqref{HH} and \eqref{RXgate} if $m=0$), for any $x\in\{0,1\}^m$,
\begin{equation}\label{ctrlblancbaseiphi}\qczero\vdash\Lambda^{x0}s(\varphi)\ =\ \tikzfig{mctrleiphiblancbas}.\end{equation}
Let $z=xa$ and $z'=x'a'$ with $a,a'\in\{0,1\}$ and $x,x'\in\{0,1\}^{k-1}$.
One has (with the $\pm$ signs being $(-1)^{1-a}$ and $(-1)^{1-a'}$ respectively):
\begin{longtable}{RCL}
\Lambda^{z'}s(\varphi')\circ\Lambda^zs(\varphi)&
\eqtroiseqref{XX}{HH}{ctrlblancbaseiphi}&\tikzfig{mctrlPphicomp-pm}\\\\
&\overset{\text{\cref{commctrlphaseenhaut}}}{=}&\tikzfig{mctrlPphicomp1-pm}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{mctrlPphicomp2-pm}\\\\
&\eqeqref{commctrlRX}&\tikzfig{mctrlPphicomp3-pm}\\\\
&\overset{\text{\cref{commctrlphaseenhaut}}}{=}&\tikzfig{mctrlPphicomp4-pm}\\\\
&
\eqtroiseqref{XX}{HH}{ctrlblancbaseiphi}&
\Lambda^zs(\varphi)\circ\Lambda^{z'}s(\varphi').
\end{longtable}
For the case where $G=R_X(\theta)$ and $G'=P(\theta')$, we prove by induction on $k\geq1$ that for any $x,x'\in\{0,1\}^k$ with $x\neq x'$,
\begin{equation}\label{commctrlRXHRXH}\qczero\vdash\tikzfig{mctrlRXHRXH}=\tikzfig{mctrlHRXHRX}\end{equation}
Note that by \cref{commctrlphaseenhaut} and the preceding case, \cref{commctrlRXHRXH} is equivalent to the desired result.
If $k=1$, then without loss of generality we can assume that $x=1$ and $x'=0$. One has
\begin{longtable}{RCL}
\tikzfig{cRXHcbRXH}&\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{cRXHcbRXHdecomp}\\\\
&\eqdeuxeqref{HH}{NotCRXNotCreversible}&\tikzfig{cRXHcbRXHdecomp1}\\\\
&\eqeqref{commutationRXCNot}&\tikzfig{cRXHcbRXHdecomp2}\\\\
&\eqeqref{commutationdecompctrlPRY}&\tikzfig{cRXHcbRXHdecomp3}\\\\
&\eqeqref{commutationRXCNot}&\tikzfig{cRXHcbRXHdecomp4}\\\\
&\eqdeuxeqref{NotCRXNotCreversible}{HH}&\tikzfig{cRXHcbRXHdecomp5}\\\\
&\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{HcbRXHcRX}
\end{longtable}
If $k\geq2$, then by \cref{cor:swap}, we can assume without loss of generality that we can write $x=az$ and $x'=az'$ with $a,a'\in\{0,1\}$ and $z\neq z'$. One has (where the $\pm$ signs correspond respectively to $(-1)^a$ and $(-1)^{a'}$):
\begin{longtable}{CL}
\tikzfig{mctrlRXHRXH-grand}\hspace*{-15em}\\\\
\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{mctrlRXHRXHdecomp}\\\\
\eqeqref{HH}&\tikzfig{mctrlRXHRXHdecomp1}\\\\
\overset{\text{\cref{commctrlPNotCvar}}}{=}&\tikzfig{mctrlRXHRXHdecomp2}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{mctrlRXHRXHdecomp3}\\\\
\overset{\text{\cref{commctrlPNotCvar}}}{=}&\tikzfig{mctrlRXHRXHdecomp4}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{mctrlRXHRXHdecomp5}\\\\
\eqeqref{CNotHCNot}&\tikzfig{mctrlRXHRXHdecomp6}\\\\
\eqeqref{HH}&\scalebox{0.9}{$\tikzfig{mctrlRXHRXHdecomp7}$}\\\\
\overset{\text{\cref{commctrlPNotCvar},}}{\eqeqref{HH}}&\tikzfig{mctrlRXHRXHdecomp8}\\\\
\overset{\text{\cref{commctrlPNotCvar}}}{=}&\tikzfig{mctrlRXHRXHdecomp9}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis,}}{\eqeqref{HH}}}&\tikzfig{mctrlRXHRXHdecomp10}\\\\
\overset{\eqref{commutationXCNot}\eqref{CNotHCNot}}{\eqeqref{XX}}&\tikzfig{mctrlRXHRXHdecomp11}\\\\
\overset{\text{\cref{commctrlPNotCvar},}}{\eqeqref{HH}}&\tikzfig{mctrlRXHRXHdecomp12}\\\\
\overset{\text{\cref{commctrlPNotCvar}}}{=}&\tikzfig{mctrlRXHRXHdecomp13}\\\\
\overset{\text{induction}}{\overset{\text{hypothesis,}}{\eqeqref{HH}}}&\tikzfig{mctrlRXHRXHdecomp14}\\\\
\eqdeuxeqref{CNotHCNot}{HH}&\tikzfig{mctrlRXHRXHdecomp15}\\\\
\eqeqref{HH}&\tikzfig{mctrlRXHRXHdecomp16}\\\\
\overset{\text{\cref{commctrlPNotCvar},}}{\eqeqref{HH}}&\tikzfig{mctrlRXHRXHdecomp17}\\\\
\overset{\eqref{commutationXCNot}\eqref{CNotHCNot}}{\eqdeuxeqref{XX}{HH}}&\tikzfig{mctrlRXHRXHdecomp18}\\\\
\overset{\text{\cref{commctrlPNotCvar},}}{\eqeqref{HH}}&\tikzfig{mctrlRXHRXHdecomp19}\\\\
\overset{\text{\cref{decompctrlblancRX}}}{=}&\tikzfig{mctrlHRXHRX-grand}
\end{longtable}
If $G=X$ or $G'=X$, then by \cref{mctrlX}, the result follows from the preceding cases together with \cref{commctrlphaseenhaut} and \cref{commutationLambdaeiphi}.
\subsection{Ancillary lemmas: Lemmas \ref{symmetriesemicontrolee} to \ref{commctrlphaseenhautP}}\label{commutationsXtarget}
\begin{lemma}\label{symmetriesemicontrolee}
For any $x\in\{0,1\}^k$ and $y\in\{0,1\}^\ell$,
\[\qczero\vdash(id_k\otimes X\otimes id_\ell)\circ\Lambda^x_yX=\Lambda^x_yX\circ(id_k\otimes X\otimes id_\ell)\]
and
\[\qczero\vdash(id_k\otimes X\otimes id_\ell)\circ\Lambda^x_yR_X(\theta)=\Lambda^x_yR_X(\theta)\circ(id_k\otimes X\otimes id_\ell)\]
\end{lemma}
\begin{proof}
The case of $\Lambda^x_yX$ is a direct consequence of \cref{prop:comb,commctrl}. Indeed, using \cref{prop:comb}, $(id_k\otimes X\otimes id_\ell)$ can be decomposed into a product of multi-controlled gates of the form $\Lambda^{x'}_{y'}X$ with $x'\in\{0,1\}^k$ and $y'\in\{0,1\}^\ell$. Then these multi-controlled gates commute with $\Lambda^x_yX$, trivially in the case where $x'y'=xy$, and by \cref{commctrl} in the other cases.
For the case of $\Lambda^x_yR_X(\theta)$, note that $\eqref{S0},\eqref{SS}\vdash\tikzfig{QgateX}=\tikzfig{eipisur2RXpi}$. Then $s(\frac\pi2)$ commutes by the topological rules, while the commutation of $(id_k\otimes R_X(\pi)\otimes id_\ell)$ is a direct consequence of \crefnosort{prop:comb,commctrl,prop:sum}: using \cref{prop:comb}, it can be decomposed into a product of multi-controlled gates of the form $\Lambda^{x'}_{y'}R_X(\pi)$ with $x'\in\{0,1\}^k$ and $y'\in\{0,1\}^\ell$. Then these multi-controlled gates commute with $\Lambda^x_yR_X(\theta)$, by \cref{commctrl} in the cases where $x'y'\neq xy$, and by \cref{prop:sum} in the case where $x'y'=xy$.
\end{proof}
\begin{lemma}\label{antisymmetriesemicontrolee}
\[\qczero\vdash\tikzfig{XmctrlPX}\ =\ \tikzfig{mctrlPmoinsphictrlPphihaut}\]
\end{lemma}
\begin{proof}~
\begin{longtable}{RCL}
\tikzfig{XmctrlPX}&=&\tikzfig{XmctrlPX1}\\\\
&\eqdeuxeqref{xgate}{HH}&\tikzfig{XmctrlPX2}\\\\
&\eqtroiseqref{ZctrlRX}{HH}{ZZ}&\tikzfig{XmctrlPX3}\\\\
&\overset{\text{\crefnosort{prop:sum,commctrlphaseenhaut}}}{=}&\tikzfig{XmctrlPX4}\\\\
&=&\tikzfig{mctrlPmoinsphictrlPphihaut}
\end{longtable}
\end{proof}
\begin{lemma}\label{commctrlphaseenhautP}
For any $x\in\{0,1\}^k$ and $y\in\{0,1\}^\ell$ with $\ell\geq k$,
\[\qczero\vdash\tikzfig{mctrlxeiphihmctrlyeiphiprime}\ =\ \tikzfig{mctrlyeiphiprimemctrlxeiphih}.\]
\end{lemma}
\begin{proof}
We proceed by induction on $\ell-k$. If $\ell=k$ then the result is a consequence of \cref{prop:sum} or \ref{commctrl} (or just of the topological rules if $k=\ell=0$). If $\ell\geq k+1$, then
without loss of generality, we can assume that $y=t1$ for some $t\in\{0,1\}^{\ell-1}$. Then by \cref{lem:Lambda-P} (together with \cref{lem:Lambda0x} and \cref{XX}),
\begin{multline*}
\qczero\vdash\tikzfig{mctrlxeiphihmctrlyeiphiprime}\\=\tikzfig{mctrlxeiphihmctrlyeiphiprime1}
\end{multline*}
so that the commutation follows by induction hypothesis and \cref{commctrlphaseenhaut}.
\end{proof}
\subsection{Proof of Proposition \ref{commctrlpasenface}}\label{preuvecommctrlpasenface}
First, the cases where $G$ or $G'=X$ follow from the other cases. Indeed, using \cref{mctrlX} and \cref{prop:comb} (together with \cref{cor:swap}), and then \cref{prop:CP}, one gets that for any $t\in\{0,1\}^p$,
\[\qczero\vdash\Lambda^tX=\tikzfig{mctrlXsimplmemetarget}.\]
Then, if $G$ or $G'=X$, one can use this decomposition and make the multi-controlled parts commute using the other cases. The non-controlled $X$ gates commute with the control dots by changing their colour, with the help of \cref{XX}. This does not alter the fact that the multi-controlled gates commute, since the $X$ gates are not on the same wire than the control dots of different colours. And since the decomposition produces each time two $X$ gates on the same wire, any control dot gets changed twice, so that it is the same at the end as at the beginning.
Thus, it suffices to treat the cases where $G,G'\in \{R_X(\theta),P(\varphi)\}$.
If $G=R_X(\theta)$ and $G'=P(\varphi)$ (or conversely), then by \cref{prop:CP}, the result is a consequence of \cref{symmetriesemicontrolee,commctrl}.
If $G=P(\varphi)$ and $G'=P(\varphi')$, then by \cref{prop:CP}, the result is a consequence of \cref{antisymmetriesemicontrolee,commctrlphaseenhautP} (together with \cref{XX}) and \cref{commctrl}.
It remains to treat the case where $G=R_X(\theta)$ and $G'=R_X(\theta')$. By \cref{symmetriesemicontrolee}, we can assume without loss of generality that $a=b=1$. By definition of $\Lambda^t_u$, we can also assume without loss of generality that $k=m=0$. Then the hypothesis $xyz\neq x'y'z'$ becomes $y\neq y'$. We proceed by induction on $\ell$. If $\ell=1$, then without loss of generality we can assume that $x=1$ and $x'=0$. One has
\begin{longtable}{CL}
&\tikzfig{ccRXRXcbc}\\\\
\overset{\text{\cref{cor:swap},}}{\overset{\text{def}}{=}}&\tikzfig{ccRXRXcbcsemidec1}\\\\
\eqeqref{eq:comRX}&\tikzfig{ccRXRXcbcsemidec2}\\\\
\overset{\text{\cref{decompctrlblancRX},}}{\overset{\text{def}}{=}}&\scalebox{0.8}{$\tikzfig{ccRXRXcbcdec}$}\\\\
\eqeqref{HH}&\scalebox{0.8}{$\tikzfig{ccRXRXcbcdec1}$}\\\\
\eqeqref{NotCRXNotCreversible}&\scalebox{0.8}{$\tikzfig{ccRXRXcbcdec2}$}\\\\
\eqeqref{commutationRXCNot}&\tikzfig{ccRXRXcbcdec4}\\\\
\eqdeuxeqref{CNotHH}{CNotCNot}&\tikzfig{ccRXRXcbcdec5}\\\\
\eqeqref{commutationdecompctrlRXpasenface}&\tikzfig{ccRXRXcbcdec6}\\\\
\eqdeuxeqref{CNotCNot}{CNotHH}&\tikzfig{ccRXRXcbcdec8}\\\\
\eqeqref{commutationRXCNot}&\scalebox{0.8}{$\tikzfig{ccRXRXcbcdec9}$}\\\\
\eqeqref{NotCRXNotCreversible}&\scalebox{0.8}{$\tikzfig{ccRXRXcbcdec10}$}\\\\
\eqeqref{HH}&\scalebox{0.8}{$\tikzfig{ccRXRXcbcdec11}$}\\\\
\overset{\text{\cref{decompctrlblancRX},}}{\overset{\text{def}}{=}}&\tikzfig{ccRXRXcbcsemidec3}\\\\
\eqeqref{eq:comRX}&\tikzfig{ccRXRXcbcsemidec4}\\\\
\overset{\text{\cref{cor:swap},}}{\overset{\text{def}}{=}}&\tikzfig{RXcbcccRX}.
\end{longtable}
If $k\geq2$, by \cref{cor:swap} we can assume without loss of generality that $y=at$ and $y'=a't'$ with $a,a'\in\{0,1\}$ and $t\neq t'$. One has (with the $\pm$ signs being $(-1)^a$ and $(-1)^{a'}$ respectively):
\begin{longtable}{RCL}
\tikzfig{RXhLambdaccLambdaRxbgrand}&=&\tikzfig{RXhLambdaccLambdaRxbdecomp}\\\\
&\eqeqref{HH}&\tikzfig{RXhLambdaccLambdaRxbdecomp1}\\\\
&\overset{\text{\cref{commutationctrldotsCNotRX}}}{=}&\tikzfig{RXhLambdaccLambdaRxbdecomp2}\\\\
&\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}&\tikzfig{RXhLambdaccLambdaRxbdecomp3}\\\\
&\overset{\text{\cref{commutationctrldotsCNotRX},}}{\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}}&\tikzfig{RXhLambdaccLambdaRxbdecomp4}\\\\
&\eqeqref{commutationCNotsbas}&\tikzfig{RXhLambdaccLambdaRxbdecomp5}\\\\
&\overset{\text{\cref{cor:swap},}}{\overset{\text{\cref{commutationctrldotsCNotRX}}}{=}}&\tikzfig{RXhLambdaccLambdaRxbdecomp6}\\\\
&\overset{\eqref{commutationCNotsbas},}{\overset{\text{\cref{cor:swap},}}{\overset{\text{\cref{commutationctrldotsCNotRX}}}{=}}}&\tikzfig{RXhLambdaccLambdaRxbdecomp7}\\\\
&\overset{\text{\cref{commutationctrldotsCNotRX},}}{\overset{\text{induction}}{\overset{\text{hypothesis}}{=}}}&\tikzfig{RXhLambdaccLambdaRxbdecomp8}\\\\
&\overset{\eqref{commutationCNotsbas},}{\overset{\text{\cref{cor:swap},}}{\overset{\text{\cref{commutationctrldotsCNotRX}}}{=}}}&\tikzfig{RXhLambdaccLambdaRxbdecomp9}\\\\
&\eqeqref{HH}&\tikzfig{RXhLambdaccLambdaRxbdecomp10}\\\\
&=&\tikzfig{cLambdaRxbRXhLambdacgrand}.
\end{longtable}
\subsection{Proof of Proposition \ref{EulerHmoins}}\label{preuveEulerHmoins}
\begin{longtable}{RCL}
\tikzfig{H}&\eqquatreeqref{P0}{PP}{RX0}{RXRX}&\tikzfig{EulerHmoins1}\\\\
&\eqeqref{EulerH}&\tikzfig{EulerHmoins2}\\\\
&\eqeqref{HH}&\tikzfig{EulerHmoins}
\end{longtable}
\subsection{Proof of Proposition \ref{soundnessEuleraxioms}}
\label{appendix:euler3d}
The proof is inspired by the proofs of Lemmas 10 and 11 of \cite{clement2022LOv}.
Given any $n$-qubit quantum circuit $C$, let $\interpg{C}\coloneqq\mathfrak G_n^{-1}\circ\interp{C}\circ \mathfrak G_n$.
\subsubsection{Soundness of Equation \eqref{Euler2d}}
Given any $\alpha_1,\alpha_2,\alpha_3\in\mathbb R$, let $U\coloneqq\interpg{\minitikzfig{RXPRXalphas}}$. We have to prove that there exist unique $\beta_0,\beta_1,\beta_2,\beta_3$ satisfying the conditions of \cref{fig:euler} such that $\interpg{\minitikzfig{RXPRXbetas}}=U$. We are going to first prove that assuming that such $\beta_j$ exist, their values are uniquely determined by $U$. Since we are going do so by giving explicit expressions of the unique possible value of each $\beta_j$ in terms of the entries of $U$, it will then be easy to check that these expressions indeed define angles with the desired properties.
One has
\[U=\interpg{\minitikzfig[0.83]{RXPRXbetas}}=e^{i\beta_0}\begin{pmatrix}\cos\bigl(\frac{\beta_2}2\bigr)&-ie^{i\beta_1}\sin\bigl(\frac{\beta_2}2\bigr)\\-ie^{i\beta_3}\sin\bigl(\frac{\beta_2}2\bigr)&e^{i(\beta_1+\beta_3)}\cos\bigl(\frac{\beta_2}2\bigr)\end{pmatrix}\]
If $U$ has a null entry, then since it is unitary, it is either diagonal or anti-diagonal. If it is diagonal, then $\sin\bigl(\frac{\beta_2}2\bigr)=0$, which, since $\beta_2\in[0,2\pi)$, implies that $\beta_2=0$, which by the constraint on $\beta_1$ and $\beta_2$, implies that $\beta_1=0$. Consequently, $\beta_0=\arg(U_{0,0})$ and $\beta_3=\arg\left(\frac{U_{1,1}}{U_{0,0}}\right)$. If $U$ is anti-diagonal, then $\cos\bigl(\frac{\beta_2}2\bigr)=0$, which, since $\beta_2\in[0,2\pi)$, implies that $\beta_2=\pi$, which by the constraint on $\beta_1$ and $\beta_2$, implies that $\beta_1=0$. Consequently, $\beta_0=\arg\left(\frac{U_{0,1}}{-i}\right)$ and $\beta_3=\arg\left(\frac{U_{1,0}}{U_{0,1}}\right)$.
If $U$ has no null entry, then one has $\beta_2\neq\pi$ and $\dfrac{ie^{-i\beta_1}U_{0,1}}{U_{0,0}}=\tan\bigl(\frac{\beta_2}2\bigr)$. Hence, $\beta_1$ is the unique angle in $[0,\pi)$ such that $\dfrac{ie^{-i\beta_1}U_{0,1}}{U_{0,0}}\in\mathbb R$, namely $\arg\left(\frac{iU_{0,1}}{U_{0,0}}\right)\bmod\pi$. In turn, $\beta_2$ is the unique angle in $[0,2\pi)\setminus\{\pi\}$ such that $\tan\bigl(\frac{\beta_2}2\bigr)=\dfrac{ie^{-i\beta_1}U_{0,1}}{U_{0,0}}$. Finally, one has $e^{i\beta_3}=\frac{\cos(\frac{\beta_2}2)U_{1,0}}{-i\sin(\frac{\beta_2}2)U_{0,0}}$, so that $\beta_3=\arg\left(\frac{\cos(\frac{\beta_2}2)U_{1,0}}{-i\sin(\frac{\beta_2}2)U_{0,0}}\right)$, and $e^{i\beta_0}=\frac{U_{0,0}}{\cos(\frac{\beta_2}2)}$, so that $\beta_0=\arg\left(\frac{U_{0,0}}{\cos(\frac{\beta_2}2)}\right)$.
\subsubsection{Soundness of Equation \eqref{Euler3dmulticontrolled}}
Given any $n$-qubit quantum circuit $C$ such that $\interpg{C}$ is of the form $\left(\begin{array}{c|c}I&0\\\hline 0&U\end{array}\right)$ with $U\in\mathbb C^{3\times3}$, let $\interpt{C}\coloneqq U$.
Given any $\gamma_1,\gamma_2,\gamma_3,\gamma_4\in\mathbb R$, let $U\coloneqq\interpt{\minitikzfig{Euler3dleft-multicontrolled-simp-gammas}}$. We have to prove that there exist unique $\delta_1,\delta_2,\delta_3,\delta_4,\delta_5,\delta_6,\delta_7,\delta_8,\delta_9$ satisfying the conditions of \cref{fig:euler} such that
\[\interpt{\minitikzfig{Euler3dright-multicontrolled-simp-deltas}}=U,\]
or equivalently,
\[\interpt{\minitikzfig{Euler3D-2qubit-R}}=U.\]
We are going to first prove that assuming that such $\delta_j$ exist, their values are uniquely determined by $U$. Since we are going do so by giving explicit expressions of the unique possible value of each $\delta_j$ in terms of the entries of $U$, it will then be easy to check that these expressions indeed define angles with the desired properties.
Let $U_{123}\coloneqq\interpt{\minitikzfig{Euler3dright-123}}=\begin{pmatrix}e^{i\delta_2}&0&0\\0&e^{i(\delta_1+\delta_2)}\cos\bigl(\frac{\delta_3}2\bigr)&-i\sin\bigl(\frac{\delta_3}2\bigr)\\0&-ie^{i(\delta_1+\delta_2)}\sin\bigl(\frac{\delta_3}2\bigr)&\cos\bigl(\frac{\delta_3}2\bigr)\end{pmatrix}$, $U_{4}\coloneqq\interpt{\minitikzfig{Euler3dright-4}}=\begin{pmatrix}\cos\bigl(\frac{\delta_4}2\bigr)&-i\sin\bigl(\frac{\delta_4}2\bigr)&0\\-i\sin\bigl(\frac{\delta_4}2\bigr)&\cos\bigl(\frac{\delta_4}2\bigr)&0\\0&0&1\end{pmatrix}$ and $U_{56}\coloneqq\interpt{\minitikzfig{Euler3dright-56}}=\begin{pmatrix}1&0&0\\0&e^{i\delta_5}\cos\bigl(\frac{\delta_6}2\bigr)&-i\sin\bigl(\frac{\delta_6}2\bigr)\\0&-ie^{i\delta_5}\sin\bigl(\frac{\delta_6}2\bigr)&\cos\bigl(\frac{\delta_6}2\bigr)\end{pmatrix}$. Let also $U_{\mathrm{I}}\coloneqq U_{123}\circ U^\dag$, $U_{\mathrm{II}}\coloneqq U_4\circ U_{\mathrm{I}}$ and $U_{\mathrm{III}}\coloneqq U_{56}\circ U_{\mathrm{II}}$.
By construction,
\begin{eqnexpr}\label{U3}U_{\mathrm{III}}=\interpt{\minitikzfig{Euler3dright-789}}^\dag=\begin{pmatrix}e^{-i\delta_9}&0&0\\0&e^{-i(\delta_7+\delta_8+\delta_9)}&0\\0&0&e^{-i\delta_8}\end{pmatrix}\end{eqnexpr}
so that
\begin{eqnexpr}\label{U2}U_{\mathrm{II}}=U_{56}^\dag\circ U_{\mathrm{III}}=\begin{pmatrix}e^{-i\delta_9}&0&0\\0&e^{-i(\delta_5+\delta_7+\delta_8+\delta_9)}\cos\bigl(\frac{\delta_6}2\bigr)&ie^{-i(\delta_5+\delta_8)}\sin\bigl(\frac{\delta_6}2\bigr)\\0&ie^{-i(\delta_7+\delta_8+\delta_9)}\sin\bigl(\frac{\delta_6}2\bigr)&e^{-i\delta_8}\cos\bigl(\frac{\delta_6}2\bigr)\end{pmatrix}\end{eqnexpr}
and $U_{\mathrm{I}}=U_4^\dag\circ U_{\mathrm{II}}$. Since $U_4$ acts as the identity on the last entry, this implies that $(U_{\mathrm{I}})_{2,0}=0$.\footnote{Where we denote by $M_{i,j}$ the entry of indices $(i,j)$ of any matrix $M$, the index of the first row and column being $0$.} That is, by definition of $U_{\mathrm{I}}$,
\begin{eqnexpr}\label{premiercoef}\textstyle-ie^{i(\delta_1+\delta_2)}\sin\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag+\cos\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag=0.\end{eqnexpr}
By direct calculation using the definitions of $U_{\mathrm{I}}$ and $U_{\mathrm{II}}$, one gets $(U_{\mathrm{I}})_{0,0}=e^{i\delta_2}U_{0,0}^\dag$ and $(U_{\mathrm{I}})_{1,0}=e^{i(\delta_1+\delta_2)}\cos\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag-i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag$, so that $(U_{\mathrm{II}})_{1,0}=-i\sin\bigl(\frac{\delta_4}2\bigr)(U_{\mathrm{I}})_{0,0}+\cos\bigl(\frac{\delta_4}2\bigr)(U_{\mathrm{I}})_{1,0}=-i\sin\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag+\cos\bigl(\frac{\delta_4}2\bigr)(e^{i(\delta_1+\delta_2)}\cos\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag-i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag)$. That is, since by \eqref{U2}, $(U_{\mathrm{II}})_{1,0}=0$:
\begin{eqnexpr}\label{deuxiemecoef}\textstyle-i\sin\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag+\cos\bigl(\frac{\delta_4}2\bigr)\left(e^{i(\delta_1+\delta_2)}\cos\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag-i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag\right)=0\end{eqnexpr}
\begin{itemize}
\item If $U_{0,1}=U_{0,2}=0$, then since $U$ is unitary, $U_{0,0}\neq0$ and \eqref{deuxiemecoef} becomes $-i\sin\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag=0$, that is $\sin\bigl(\frac{\delta_4}2\bigr)=0$. Since $\delta_4\in[0,2\pi)$, this implies that $\delta_4=0$, which by the conditions of \cref{fig:euler}, implies that $\delta_1=\delta_2=\delta_3=0$.
\item If $(U_{0,1},U_{0,2})\neq(0,0)$, then $e^{i(\delta_1+\delta_2)}\cos\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag-i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag\neq0$. Indeed, if this expression was equal to $0$, by \eqref{premiercoef} this would mean that the non-zero vector $\begin{pmatrix}e^{i(\delta_1+\delta_2)}U_{0,1}^\dag\\U_{0,2}^\dag\end{pmatrix}$ is in the kernel of the matrix $\begin{pmatrix}\cos\bigl(\frac{\delta_3}2\bigr)&-i\sin\bigl(\frac{\delta_3}2\bigr)\\-i\sin\bigl(\frac{\delta_3}2\bigr)&\cos\bigl(\frac{\delta_3}2\bigr)\end{pmatrix}$, whereas this matrix is invertible. Then:
\begin{itemize}
\item If $U_{0,0}=0$, then \eqref{deuxiemecoef} implies that $\cos\bigl(\frac{\delta_4}2\bigr)=0$, which, since $\delta_4\in[0,2\pi)$, implies that $\delta_4=\pi$. By the conditions of \cref{fig:euler}, this implies that $\delta_2=0$. Then:
\begin{itemize}
\item If $U_{0,2}=0$, then $U_{0,1}\neq0$, and \eqref{premiercoef} implies that $\sin\bigl(\frac{\delta_3}2\bigr)=0$, that is, since $\delta_3\in[0,2\pi)$, that $\delta_3=0$. By the conditions of \cref{fig:euler}, together with the fact that $\delta_4=\pi$, this implies that $\delta_1=0$.
\item If $U_{0,1}=0$, then $U_{0,2}\neq0$, and \eqref{premiercoef} implies that $\cos\bigl(\frac{\delta_3}2\bigr)=0$, that is, since $\delta_3\in[0,2\pi)$, that $\delta_3=\pi$. By the conditions of \cref{fig:euler}, this implies that $\delta_1=0$.
\item If $U_{0,1},U_{0,2}\neq0$, then \eqref{premiercoef}, on the one hand, implies that $\delta_3\neq\pi$, and on the other hand, is equivalent to
\[\tan\bigl(\tfrac{\delta_3}2\bigr)=\frac{e^{-i\delta_1}U_{0,2}^\dag}{iU_{0,1}^\dag}.\]
Hence, $\delta_1$ is the unique angle in $[0,\pi)$ such that $\frac{e^{-i\delta_1}U_{0,2}^\dag}{iU_{0,1}^\dag}\in\mathbb R$. In turn, $\delta_3$ is the unique angle in $[0,2\pi)$ such that $\tan\bigl(\frac{\delta_3}2\bigr)=\frac{e^{-i\delta_1}U_{0,2}^\dag}{iU_{0,1}^\dag}$.
\end{itemize}
\item If $U_{0,0}\neq0$, then \eqref{deuxiemecoef} can be simplified into
\begin{eqnexpr}\label{deuxiemecoefbis}\textstyle-i\tan\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag+e^{i(\delta_1+\delta_2)}\cos\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag-i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag=0.\end{eqnexpr}
\begin{itemize}
\item If $U_{0,2}=0$, then $U_{0,1}\neq0$, and \eqref{premiercoef}
implies that $\sin\bigl(\frac{\delta_3}2\bigr)=0$, that is, since $\delta_3\in[0,2\pi)$, that $\delta_3=0$. By the conditions of \cref{fig:euler}, this implies that $\delta_2=0$. Then \eqref{deuxiemecoefbis} becomes
\[\textstyle-i\tan\bigl(\frac{\delta_4}2\bigr)U_{0,0}^\dag+e^{i\delta_1}U_{0,1}^\dag=0\]
that is,
\[\tan\bigl(\tfrac{\delta_4}2\bigr)=\frac{e^{i\delta_1}U_{0,1}^\dag}{iU_{0,0}^\dag}.\]
Hence, $\delta_1$ is the unique angle in $[0,\pi)$ such that $\frac{e^{i\delta_1}U_{0,1}^\dag}{iU_{0,0}^\dag}\in\mathbb R$. In turn, $\delta_4$ is the unique angle in $[0,2\pi)$ such that $\tan\bigl(\frac{\delta_4}2\bigr)=\frac{e^{i\delta_1}U_{0,1}^\dag}{iU_{0,0}^\dag}$.
\item If $U_{0,1}=0$, then $U_{0,2}\neq0$, and \eqref{premiercoef}
implies that $\cos\bigl(\frac{\delta_3}2\bigr)=0$, that is, since $\delta_3\in[0,2\pi)$, that $\delta_3=\pi$. By the conditions of \cref{fig:euler}, this implies that $\delta_1=0$. Then \eqref{deuxiemecoefbis} becomes
\[\textstyle-i\tan\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag-iU_{0,2}^\dag=0\]
that is,
\[\tan\bigl(\tfrac{\delta_4}2\bigr)=-\frac{e^{-i\delta_2}U_{0,2}^\dag}{U_{0,0}^\dag}.\]
Hence, $\delta_2$ is the unique angle in $[0,\pi)$ such that $\frac{e^{-i\delta_2}U_{0,2}^\dag}{U_{0,0}^\dag}\in\mathbb R$. In turn, $\delta_4$ is the unique angle in $[0,2\pi)$ such that $\tan\bigl(\frac{\delta_4}2\bigr)=-\frac{e^{-i\delta_2}U_{0,2}^\dag}{U_{0,0}^\dag}$.
\item If $U_{0,1},U_{0,2}\neq0$, then \eqref{premiercoef}, on the one hand, implies that $\delta_3
\notin\{0,\pi\}$, and on the other hand, is equivalent to
\begin{eqnexpr}\label{eidelta12}e^{i(\delta_1+\delta_2)}=\frac{\cos\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag}{i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,1}^\dag}.\end{eqnexpr}
Then by substituting in \eqref{deuxiemecoefbis}, we get
\[\textstyle-i\tan\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag+\dfrac{\cos^2\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag}{i\sin\bigl(\frac{\delta_3}2\bigr)}-i\sin\bigl(\frac{\delta_3}2\bigr)U_{0,2}^\dag=0\]
which can be simplified into
\[\textstyle-i\tan\bigl(\frac{\delta_4}2\bigr)e^{i\delta_2}U_{0,0}^\dag+\dfrac{U_{0,2}^\dag}{i\sin\bigl(\frac{\delta_3}2\bigr)}=0\]
which is equivalent to
\begin{eqnexpr}\label{tandelta4}\textstyle\tan\bigl(\frac{\delta_4}2\bigr)=-\dfrac{e^{-i\delta_2}U_{0,2}^\dag}{\sin\bigl(\frac{\delta_3}2\bigr)U_{0,0}^\dag}.\end{eqnexpr}
Hence, $\delta_2$ is the unique angle in $[0,\pi)$ such that $\dfrac{e^{-i\delta_2}U_{0,2}^\dag}{U_{0,0}^\dag}\in\mathbb R$. Then \eqref{eidelta12} can be rephrased into
\[\tan\bigl(\tfrac{\delta_3}2\bigr)=\frac{e^{-i(\delta_1+\delta_2)}U_{0,2}^\dag}{iU_{0,1}^\dag}.\]
Hence, $\delta_1$ is the unique angle in $[0,\pi)$ such that $\frac{e^{-i(\delta_1+\delta_2)}U_{0,2}^\dag}{iU_{0,1}^\dag}\in\mathbb R$. In turn, $\delta_3$ is the unique angle in $[0,2\pi)$ such that $\tan\bigl(\tfrac{\delta_3}2\bigr)=\frac{e^{-i(\delta_1+\delta_2)}U_{0,2}^\dag}{iU_{0,1}^\dag}$. Finally,
$\delta_4$ is the unique angle in $[0,2\pi)$ satisfying \eqref{tandelta4}.
\end{itemize}
\end{itemize}
\end{itemize}
Thus, assuming that the $\delta_j$ exist, since $U_{\mathrm{I}}$ and $U_{\mathrm{II}}$ only depend on $\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$ and $U$, they are uniquely determined by $U$. Then \eqref{U2} implies that
\begin{itemize}
\item If $(U_{\mathrm{II}})_{1,2}=0$, then $\sin\bigl(\frac{\delta_6}2\bigr)=0$, which means, since $\delta_6\in[0,2\pi)$, that $\delta_6=0$. By the conditions of \cref{fig:euler}, this implies that $\delta_5=0$.
\item If $(U_{\mathrm{II}})_{2,2}=0$, then $\cos\bigl(\frac{\delta_6}2\bigr)=0$, which means, since $\delta_6\in[0,2\pi)$, that $\delta_6=\pi$. By the conditions of \cref{fig:euler}, this implies that $\delta_5=0$.
\item If $(U_{\mathrm{II}})_{1,2}=0,(U_{\mathrm{II}})_{2,2}\neq0$, then
\[\tan\bigl(\tfrac{\delta_6}2\bigr)=\frac{e^{i\delta_5}(U_{\mathrm{II}})_{1,2}}{i(U_{\mathrm{II}})_{2,2}}.\]
Hence, $\delta_5$ is the unique angle in $[0,\pi)$ such that $\frac{e^{i\delta_5}(U_{\mathrm{II}})_{1,2}}{i(U_{\mathrm{II}})_{2,2}}\in\mathbb R$. In turn, $\delta_6$ is the unique angle in $[0,2\pi)$ such that $\tan\bigl(\frac{\delta_6}2\bigr)=\frac{e^{i\delta_5}(U_{\mathrm{II}})_{1,2}}{i(U_{\mathrm{II}})_{2,2}}$.
\end{itemize}
Thus, assuming that the $\delta_j$ exist, since $U_{\mathrm{III}}$ only depends on $\delta_5$, $\delta_6$ and $U_{\mathrm{II}}$, it is uniquely determined by $U$. Then by \eqref{U3}, $\delta_8=\arg((U_{\mathrm{III}})_{2,2}^\dag)$, $\delta_9=\arg((U_{\mathrm{III}})_{0,0}^\dag)$ and $\delta_7=\arg\left(\frac{(U_{\mathrm{III}})_{0,0}(U_{\mathrm{III}})_{2,2}}{(U_{\mathrm{III}})_{1,1}}\right)$.
\subsection{Proof of Proposition \ref{klfollowfromEuler}}\label{preuveklfollowfromEuler}
\noindent Proof of Equation \eqref{PP}:
\begin{longtable}{RCL}
\tikzfig{PP}&\eqeqref{HH}&\tikzfig{PP1}\\\\
&\eqtroiseqref{S0}{SS}{RXgate}&\tikzfig{PP2}\\\\
&\eqeqref{P0}&\tikzfig{PP3}\\\\
&\eqeqref{Euler2d}&\tikzfig{PP4}\\\\
&\eqeqref{Euler2d}&\tikzfig{PP5}\\\\
&\eqdeuxeqref{P0}{RX0}&\tikzfig{PP6}\\\\
&\eqquatreeqref{RXgate}{HH}{SS}{S0}&\tikzfig{Pphiplusphiprime}
\end{longtable}
The first use of \cref{Euler2d} is valid since \cref{Euler2d} is applied from the left to the right. The second use of \cref{Euler2d} is valid since it
preserves the semantics. Note that one can show that $\beta_1=\beta_3=0$, $\beta_2=\varphi_1+\varphi_2\bmod 2\pi$ and $\beta_0=\begin{cases}0&\text{if $(\varphi_1+\varphi_2\bmod 4\pi)\in[0,2\pi)$}\\\pi&\text{if $(\varphi_1+\varphi_2\bmod 4\pi)\in[2\pi,4\pi)$}\end{cases}$
.
\noindent Proof of Equation \eqref{XPX}:
\begin{longtable}{RCL}
\tikzfig{XPX}&\eqdeuxeqref{xgate}{zgate}&\tikzfig{XPX0}\\\\
&\eqtroiseqref{S0}{SS}{RXgate}&\tikzfig{XPX1}\\\\
&\eqdeuxeqref{Euler2d}{SS}&\tikzfig{RXPRXbetas1}
\end{longtable}
One has $\beta_1=\beta_2=0$,
$\beta_3=-\varphi \bmod 2\pi$ and $\beta_0=\varphi-\pi \bmod 2\pi$
. Indeed, this choice of angles
satisfies the conditions of \cref{Euler2d} and is sound with respect to the semantics, and \cref{soundnessEuleraxioms} guarantees that this is the only possible choice.
Thus, by \crefnosort{P0,RX0}, this implies that one can
transform $\scalebox{0.8}{\tikzfig{XPX}}$ into
$\scalebox{0.8}{\tikzfig{RXPRXbetas4bis}}\eqdeuxeqref{S0}{SS}
\scalebox{0.8}{\tikzfig{RXPRXbetas5}}$
. Finally,
$\scalebox{0.8}{\tikzfig{RXPRXbetas6}}\eqeqref{RX0}
\scalebox{0.8}{\tikzfig{RXPRXbetas7}}\eqdeuxeqref{Euler2d}{S0}
\scalebox{0.8}{\tikzfig{RXPRXbetas3}} \eqdeuxeqref{P0}{RX0} $
$\scalebox{0.8}{\tikzfig{RXPRXbetas8}}$, which terminates the proof.
\subsection{Proof of Proposition \ref{prop:CCX}}
\label{proof:CCX}
First, we can show that $\qc \vdash \Lambda^x P(2\pi) = id_{k+1}$ as follows:
\begin{eqnarray*}\tikzfig{CCP2Pi1}&\overset{\text{Proposition }\ref{prop:sum}}{=}&\tikzfig{CCP2Pi2}\\[0.5cm]
&\eqeqref{Euler3dmulticontrolled}&\tikzfig{CCP2Pi3}\\[0.5cm]
&\overset{\text{Proposition }\ref{prop:sum}}{=}&\tikzfig{idk1}\\[0.5cm]
\end{eqnarray*}
It follows that, for $x\in \{1\}^k$:
\begin{eqnarray*}\tikzfig{lambdaXlambdaX1}&\overset{\text{def}}{=}&\tikzfig{lambdaXlambdaX2}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{lambdaXlambdaX3}\\[0.5cm]
&\overset{\text{Proposition }\ref{prop:sum}}{=}&\tikzfig{lambdaXlambdaX4}\\[0.5cm]
&\overset{\textup{QC} \vdash \Lambda^{x}P(2\pi)=id_{k+1}}{\overset{}{=}}&\tikzfig{lambdaXlambdaX5}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{idk1}\\[0.5cm]
\end{eqnarray*}
\subsection{Proof of Proposition \ref{prop:period}}
\label{proof:propperiod}
First, we prove the case for $x=\epsilon$ :
\[\qczero \vdash R_X(4\pi) = id_1 \qquad\qczero \vdash P(2\pi) = id_1 \qquad\qczero \vdash s(2\pi) = id_0\].
\begin{eqnarray*}\tikzfig{P2pi}&\eqdeuxeqref{zgate}{PP}&\tikzfig{ZZ}\\[0.5cm]
&\eqeqref{ZZ}&\tikzfig{filcourt-s}\\[0.5cm]
\end{eqnarray*}
\begin{eqnarray*}\tikzfig{s2pi}&\eqeqref{S0}&\tikzfig{diagrammevide-s}\\[0.5cm]
\end{eqnarray*}
It follows that:
\begin{eqnarray*}\tikzfig{RX4pi}&\eqeqref{RXgate}&\tikzfig{HP4piH}\\[0.5cm]
&=&\tikzfig{HH}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{filcourt-s}\\[0.5cm]
\end{eqnarray*}
We can now prove the general case, first by noticing that $\qc \vdash \Lambda^x P(2\pi)= id_{k+1}$, as proven in \cref{proof:CCX}.
As $\Lambda^{x1}s(2\pi) = \Lambda^x P(2\pi)$, we have for any $x \in \{1\}^k$, $\qc \vdash \Lambda^{x}s(2\pi)=id_{k}$.
Finally:
\begin{eqnarray*}\tikzfig{lambdaxRX4pi}&=&\tikzfig{lambdaxRX4pi1}\\[0.5cm]
&\overset{\qc \vdash id_{k+1}=\Lambda^{1x}s(2\pi)}{\overset{}{=}}&\tikzfig{lambdaxRX4pi2}\\[0.5cm]
&\overset{\text{Proposition \ref{prop:sum}}}{=}&\tikzfig{lambdaxRX4pi3}\\[0.5cm]
&\overset{\qc \vdash \Lambda^{1x}s(2\pi)=id_{k+1}}{\overset{\qc \vdash \Lambda^{1x}P(2\pi)=id_{k+2}}{\overset{}{=}}}&\tikzfig{lambdaxRX4pi4}\\[0.5cm]
&\eqeqref{HH}&\tikzfig{id1k1}\\[0.5cm]
\end{eqnarray*}
\section{Proofs of Section \ref{sec:completeness}}
\subsection{Proof of Theorem \ref{thm:LOPPcompleteness}}
\label{proof:thmLOPPcompleteness}
One can easily show that every equation of \cref{axiomsLOPP} is sound with respect to the semantics. Regarding the completeness proof, we use the rewriting system of \cref{PPRS} that has been introduced in
\cite{clement2022LOv}.
This rewriting system has been proved to be strongly normalising, moreover it has been proved that any two swap-free circuits having the same semantics are reduced to the same normal form \cite{clement2022LOv}.
Using \cref{swapbspisur2} one can transform any circuit into a swap-free circuit. As a consequence, to prove the completeness it only remains to show that every rule of \cref{PPRS} can be derived using the equations of \cref{axiomsLOPP}.
\begin{figure}
\caption{Rewriting rules of PPRS. $\protect\minitikzfig[0.75]{convtp-phase-shift-etoile-s}
\label{phasemod2pi}
\label{bsmod2pi}
\label{fusionphaseshifts}
\label{zerophaseshifts}
\label{zerobs}
\label{removebottomphase}
\label{passagepisur2}
\label{passagephasepi}
\label{soustractionpi}
\label{glissadeEulerscalaires}
\label{fusionEulerbsphasebs}
\label{PPRS}
\end{figure}
First we can notice that Rule \eqref{fusionEulerbsphasebs} is exactly the same as \cref{Eulerbsphasebs} (up to \cref{phase0}).
Rule \eqref{phasemod2pi} is derived from \cref{phase0} and \cref{phaseaddition}.
Rule \eqref{bsmod2pi} is derived from \cref{Eulerbsphasebs} with $\alpha_1=\alpha_2=0$ and $\alpha_3=\psi+2k\pi$.
Rule \eqref{fusionphaseshifts} is derived from \cref{phaseaddition}.
Rule \eqref{zerophaseshifts} is derived from \cref{phase0}.
Rule \eqref{zerobs} is derived from \cref{bs0}.
Rule \eqref{removebottomphase} is derived from \cref{phaseaddition}, \cref{phase0} and \cref{globalphasepropagationbs}.
Rule \eqref{passagepisur2} is derived from \cref{swapbspisur2} and \cref{phaseaddition}.
Rule \eqref{passagephasepi} is derived from \cref{Eulerbsphasebs} with $\alpha_1=0$, $\alpha_2=\varphi_0$ and $\alpha_3=\theta_0$.
Rule \eqref{soustractionpi} is derived from \cref{Eulerbsphasebs} with $\alpha_1=\alpha_2=0$ and $\alpha_3=\theta_4$.
Regarding Rule \eqref{glissadeEulerscalaires}, its LHS can be transformed as follows:
\begin{eqnarray*}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\tikzfig{convtp-bsyangbaxterpointeenbas-etoiles}&\eqtroiseqref{phase0}{phaseaddition}{globalphasepropagationbs}&\tikzfig{convtp-bsyangbaxterpointeenbas-etoiles1}\\[0.5cm]
&\eqeqref{Eulerscalaires}&\tikzfig{bsyangbaxterpointeenhaut1}\\[0.5cm]
&\eqeqref{phaseaddition}&\tikzfig{bsyangbaxterpointeenhaut2}\\[0.5cm]
\end{eqnarray*}
Note that the angles in the resulting circuit are not necessarily those of the RHS of Rule \eqref{glissadeEulerscalaires}.
However, one can show that
it can be put in normal
form using the rules of \cref{PPRS} except
Rule \eqref{glissadeEulerscalaires}.
As we have seen above that each of these rules can be derived using equations of \cref{axiomsLOPP}, this shows that Rule \eqref{glissadeEulerscalaires} can also be derived using the equations of \cref{axiomsLOPP}.
\subsection{Useful Definitions}\label{usefuldef}
\begin{definition}
Given $x\in\{0,1\}^k$, $y\in\{0,1\}^\ell$ and $G\in \{s(\psi),X,R_X(\theta),P(\varphi)\}$, we define
\[\bar\Lambda^x_y G\coloneqq\prod_{\begin{scriptarray}{c}\\[-1.5em]x'\in\{0,1\}^k\\[-0.2em]y'\in\{0,1\}^\ell\\[-0.2em]x'y'\neq xy\end{scriptarray}}\Lambda^{x'}_{y'}G\]
where the product denotes a sequential composition taken in an arbitrary order.
\end{definition}
\begin{definition}
Given $x\in\{0,1\}^k$, $y\in\{0,1\}^\ell$ and $z\in\{0,1\}^m$, we define
\[\Lambda\tripleindice xyz\gcnot\coloneqq\Lambda^{x1y}_z X,\qquad \Lambda\tripleindice xyz\gnotc\coloneqq\Lambda^x_{y1z}X,\qquad \bar\Lambda\tripleindice xyz\gcnot\coloneqq\prod_{\begin{scriptarray}{c}\\[-1.5em]x'\in\{0,1\}^k\\[-0.2em]y'\in\{0,1\}^\ell\\[-0.2em]z'\in\{0,1\}^m\\[-0.2em]x'y'z'\neq xyz\end{scriptarray}}\Lambda^{x'1y'}_{z'}X\quad\text{ and }\quad\bar\Lambda\tripleindice xyz\gnotc\coloneqq\prod_{\begin{scriptarray}{c}\\[-1.5em]x'\in\{0,1\}^k\\[-0.2em]y'\in\{0,1\}^\ell\\[-0.2em]z'\in\{0,1\}^m\\[-0.2em]x'y'z'\neq xyz\end{scriptarray}}\Lambda^{x'}_{y'1z'}X.\]
\end{definition}
\subsection{Ancillary lemmas: Lemmas \ref{antisymmetriecontrolee} to \ref{decodagefilsdisjoints}}
\label{proof:usefullemmas}
\begin{lemma}\label{antisymmetriecontrolee}
\[\qc\vdash\tikzfig{mctrlXmctrlPmctrlX}\ =\ \tikzfig{mctrlPmoinsphictrlPphihaut}\]
\end{lemma}
\begin{proof}~
\begin{longtable}{RCL}
\tikzfig{mctrlXmctrlPmctrlX}&\overset{\text{\crefnosort{XX,prop:CCX,commctrl,prop:comb}}}{=}&\tikzfig{mctrlXmctrlPmctrlX1}\\\\
&\overset{\text{\crefnosort{commctrl,prop:CCX}}}{=}&\tikzfig{XmctrlPX}\\\\
&\overset{\text{\cref{antisymmetriesemicontrolee}}}{=}&\tikzfig{mctrlPmoinsphictrlPphihaut}
\end{longtable}
where $\vec 1$ denotes a list of appropriate length whose elements are all equal to $1$.
\end{proof}
\begin{lemma}\label{passagephasepicircuits}
\[\qczero\vdash\tikzfig{mctrlZmctrlRX}\ =\ \tikzfig{mctrlRXmoinsthetamctrlZ}\]
\end{lemma}
\begin{proof}~
\begin{longtable}{RCL}
\tikzfig{mctrlZmctrlRX}&\overset{\text{\crefnosort{zgate,prop:sum,commctrl,prop:comb}}}{=}&\tikzfig{mctrlZmctrlRX1}\\\\
&\overset{\text{\crefnosort{commctrl}}}{=}&\tikzfig{mctrlZmctrlRX2}\\\\
&
\eqeqref{ZctrlRX}&\tikzfig{mctrlZmctrlRX3}\\\\
&\overset{\text{\crefnosort{prop:comb,commctrl,prop:sum,zgate}}}{=}&\tikzfig{mctrlRXmoinsthetamctrlZ}
\end{longtable}
\end{proof}
\begin{lemma}\label{CRX2pi}
For any $x\in\{0,1\}^k$,
\[\qc\vdash\Lambda^{x}R_X(2\pi)=\Lambda^xs(\pi)\otimes\gid\]
\end{lemma}
\begin{proof}~
\begin{longtable}{RCL}
\Lambda^{x}R_X(2\pi)&\overset{\text{\eqref{XX}, \eqref{HH}, \crefnosort{prop:sum,lem:Lambda-P}}}{=}&\tikzfig{mctrlRX2pi1}\\\\
&\overset{\text{\crefnosort{prop:period,prop:sum,HH}}}{=}&\Lambda^xs(\pi)\otimes\gid
\end{longtable}
\end{proof}
\begin{lemma}\label{passagephasepihbcircuits}
\[\qc\vdash\tikzfig{lemmepassagepihb1}\ =\ \tikzfig{lemmepassagepihb6}\]
\end{lemma}
\begin{proof}~
\begin{longtable}{RCL}
\tikzfig{lemmepassagepihb1}&\overset{\text{\crefnosort{passagephasepicircuits,prop:period}}}{=}&\tikzfig{lemmepassagepihb2}\\\\
&\overset{\text{\crefnosort{prop:sum,prop:period}}}{=}&\tikzfig{lemmepassagepihb3}\\\\
&\overset{\text{\cref{CRX2pi}}}{=}&\tikzfig{lemmepassagepihb4}\\\\
&\overset{\text{\cref{prop:CP}}}{=}&\tikzfig{lemmepassagepihb5}\\\\
&\overset{\text{\crefnosort{prop:comb,commctrl,prop:sum}}}{=}&\tikzfig{lemmepassagepihb6}
\end{longtable}
\end{proof}
\begin{lemma}\label{decodagefilsdisjoints}
For any raw optical circuits $C_1:\ell_1\to\ell_1$ and $C_2:\ell_2\to\ell_2$, and any $k,\ell,n$ with $\ell\geq\ell_1$ and $k+\ell\leq 2^n$,
\[\qczero\vdash D_{k+\ell,n}(C_2)\circ D_{k,n}(C_1)=D_{k,n}(C_1)\circ D_{k+\ell,n}(C_2).\]
\end{lemma}
\begin{proof}
We proceed by structural induction on $C_1$ and $C_2$.
\begin{itemize}
\item If $C_1=C_1''\circ C_1'$, then
\[D_{k+\ell,n}(C_2)\circ D_{k,n}(C_1)= D_{k+\ell,n}(C_2)\circ (D_{k,n}(C_1'')\circ D_{k,n}(C_1'))\]
while
\[D_{k,n}(C_1)\circ D_{k+\ell,n}(C_2)= (D_{k,n}(C_1'')\circ D_{k,n}(C_1'))\circ D_{k+\ell,n}(C_2)\]
so the result follows by \cref{assoccomp} of quantum circuits and the induction hypothesis.
\item The case $C_2=C_2''\circ C_2'$ is similar to the previous one.
\item If $C_1=C_1'\otimes C_1''$ with $C_1':\ell_1'\to\ell_1'$, then
\[D_{k+\ell,n}(C_2)\circ D_{k,n}(C_1)= D_{k+\ell,n}(C_2)\circ (D_{k+\ell_1',n}(C_1'')\circ D_{k,n}(C_1'))\]
while
\[D_{k,n}(C_1)\circ D_{k+\ell,n}(C_2)= (D_{k+\ell_1',n}(C_1'')\circ D_{k,n}(C_1'))\circ D_{k+\ell,n}(C_2)\]
so the result follows by \cref{assoccomp} of quantum circuits and the induction hypothesis.
\item The case $C_2=C_2'\otimes C_2''$ is similar to the previous one.
\item If $C_1$ or $C_2$ is $\tikzfig{diagrammevide-s}$ or $\gid$, then the results follows from \cref{idneutre} of quantum circuits.
\item If $C_1,C_2\in\{\tikzfig{phase-shift-xs},\tikzfig{bs-xs},\tikzfig{swap-xs}\}$, then $D_{k,n}(C_1)=\Lambda^{G_n(k)}s(\varphi)$, $\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\theta)$ or $\Lambda^{x_{k,n}}_{y_{k,n}}X$ and $D_{k+\ell,n}(C_2)=\Lambda^{G_n(k+\ell)}s(\varphi)$, $\Lambda^{x_{k+\ell,n}}_{y_{k+\ell,n}}R_X(-2\theta)$ or $\Lambda^{x_{k+\ell,n}}_{y_{k+\ell,n}}X$.
Using the definitions of $G_n(k)$, $x_{k,n}$ and $y_{k,n}$, it is easy to check that in any case, $D_{k,n}(C_1)$ and $D_{k+\ell,n}(C_2)$ satisfy the premises of either \cref{commctrl} or \ref{commctrlpasenface} and therefore commute.
\end{itemize}
\end{proof}
\subsection{Proof of Lemma \ref{DE}}\label{lemmasforbasecaseDE}
For the sake of clarity, the proofs are given separately in \cref{preuveDupsilonLambda,preuvesigmatoswaps,preuvebasecaseDE}.
\begin{lemma}\label{DupsilonLambda}
For any $N\geq1$, $i\in\{0,...,N-1\}$, $b\in\{0,...,2^i-1\}$ and $a\in\{0,...,2^{N-i-1}-1\}$,
\[\textup{QC}\vdash D(\upsilon_{N,i,b,a})=\Lambda^{G_i(b)}_{G_{N-i-1}(2^{N-i-1}-a-1)}X\]
where $\upsilon_{N,i,b,a}$ is defined in \cref{defsigma}, and given $n\in\mathbb N$ and $k\in\{0,...,2^n-1\}$, $G_n(k)\in\{0,1\}^n$ is the $n$-bit Gray code of $k$, defined in \cref{defgraycode}. Note that $G_{N-i-1}(2^{N-i-1}-a-1)$ differs from $G_{N-i-1}(a)$ by only the first bit.
\end{lemma}
\begin{lemma}\label{sigmatoswaps}
For any $k,\ell,n\in\mathbb N$,
\[\textup{QC}\vdash D(\sigma_{k,n,\ell})=id_k\otimes\sigma_{n,\ell}.\]
where $\sigma_{0,0}\coloneqq\emptyc$ and $\sigma_{n,\ell}\coloneqq\sigma_{n+\ell-1}^\ell$, where $\sigma_{n+\ell-1}$ is defined in \cref{fig:axiom}.
\end{lemma}
\begin{lemma}\label{basecaseDE}
For any $g\in
\{\emptyc,\gid,s(\varphi),\gh,\gp,\gcnot,\gswap\}$,
\[\textup{QC}\vdash D(E_{k,\ell}(g))=id_k\otimes g\otimes id_\ell.\]
\end{lemma}
\subsubsection{Proof of Lemma \ref{DupsilonLambda}}\label{preuveDupsilonLambda}
We proceed by induction on $a$.
It follows from the definition of $D$ that
\[D(\upsilon_{N,i,b,0})\eqdef D\left(\tikzfig{swapib2pNi1-PHOL}\right)=\Lambda^{G_i(b)}_{G_{N-i-1}(2^{N-i-1}-1)}X.\]
Assuming for some $a\in\{1,...,2^{N-i-1}-1\}$ that $\qczero\vdash D(\upsilon_{N,i,b,a-1})=\Lambda^{G_i(b)}_{G_{N-i-1}(2^{N-i-1}-a)}X$, by definition of $\upsilon_{N,i,b,a}$, one has\footnote{Note that this product of five raw circuits should be written with more parentheses since the composition is not associative. We have omitted these parentheses by abuse of language in order to lighten the notations. In the following, we will similarly omit the associativity parentheses whenever this does not create ambiguity.}
\[\qczero\vdash D(\upsilon_{N,i,b,a})=D(s_{-a})\circ D(s_{+a})\circ\left(\Lambda^{G_i(b)}_{G_{N-i-1}(2^{N-i-1}-a)}X\right)\circ D(s_{+a})\circ D(s_{-a})
\]
where $s_{+a}=\tikzfig{swapib2pNi1plusa-PHOL}$\quad and\quad $s_{-a}=\tikzfig{swapib2pNi1moinsa-PHOL}$
.
Due to the properties of Gray codes, $G_{N-i-1}(2^{N-i-1}-a-1)$ differs from $G_{N-i-1}(2^{N-i-1}-a)$ by only one bit. That is, there exist $k,\ell\geq0$ with $k+\ell=N-i-2$, $x\in\{0,1\}^k$, $y\in\{0,1\}^\ell$ and $\alpha\in\{0,1\}$, such that
\[G_{N-i-1}(2^{N-i-1}-a-1)=x\alpha y\quad\text{and}\quad G_{N-i-1}(2^{N-i-1}-a)=x\bar\alpha y\]
where $\bar\alpha\coloneqq1-\alpha$.
Additionally, $G_{N-i}(2^{N-i-1}-a-1)$ differs from $G_{N-i}(2^{N-i-1}+a)$ by only the first bit, and $G_{N-i}(2^{N-i-1}-a)$ also differs from $G_{N-i}(2^{N-i-1}+a-1)$ by only the first bit. Therefore, there exists $\beta\in\{0,1\}$ such that
\[\begin{array}{ll}G_{N-i}(2^{N-i-1}-a-1)=\beta x\alpha y,&G_{N-i}(2^{N-i-1}+a)=\bar\beta x\alpha y,\\[0.8em]
G_{N-i}(2^{N-i-1}-a)=\beta x\bar\alpha y\quad\text{and}&G_{N-i}(2^{N-i-1}+a-1)=\bar\beta x \bar\alpha y.\end{array}\]
It follows from the definition of $D$ that $D(s_{-a})=\Lambda^{G_i(b).\beta x}_yX$ and $D(s_{+a})=\Lambda^{G_i(b).\bar\beta x}_yX$. Hence, by \cref{commctrl,cor:swap,prop:comb}, $\qczero\vdash D(s_{-a})\circ D(s_{+a})=D(s_{+a})\circ D(s_{-a})=
\scalebox{0.69}{$\tikzfig{LambdaGibxyX}$}=(\sigma_{1,i}\otimes id_{N-i-1})\circ\left(\gid\otimes\Lambda^{G_i(b) x}_yX\right)\circ(\sigma_{i,1}\otimes id_{N-i-1})$, so that
\[\qc\vdash D(\upsilon_{N,i,b,a})=(\sigma_{1,i}\otimes id_{N-i-1})\circ\left(\gid\otimes\Lambda^{G_i(b) x}_yX\right)\circ\left(\Lambda^{\epsilon}_{G_i(b)x\bar\alpha y}X\right)\circ \left(\gid\otimes\Lambda^{G_i(b) x}_yX\right)\circ(\sigma_{i,1}\otimes id_{N-i-1})\]
with
\begin{longtable}{CL}
\multicolumn{1}{R}{\qc\vdash}&\left(\gid\otimes\Lambda^{G_i(b) x}_yX\right)\circ\left(\Lambda^{\epsilon}_{G_i(b)x\bar\alpha y}X\right)\circ \left(\gid\otimes\Lambda^{G_i(b) x}_yX\right)\\\\
\overset{\text{\crefnosort{prop:CCX,commctrl,prop:comb}}}{=}&(id_{N-1}\otimes X)\circ\left(\gid\otimes\bar\Lambda^{G_i(b) x}_yX\right)\circ\left(\Lambda^{\epsilon}_{G_i(b)x\bar\alpha y}X\right)\circ \left(\gid\otimes\bar\Lambda^{G_i(b) x}_yX\right)\circ(id_{N-1}\otimes X)\\\\
\overset{\text{\cref{commctrl,prop:comb}}}{=}&(id_{N-1}\otimes X)\circ\left(\gid\otimes\bar\Lambda^{G_i(b) x}_yX\right)\circ \left(\gid\otimes\bar\Lambda^{G_i(b) x}_yX\right)\circ\left(\Lambda^{\epsilon}_{G_i(b)x\bar\alpha y}X\right)\circ(id_{N-1}\otimes X)\\\\
\overset{\text{\cref{prop:CCX,commctrl}}}{=}&(id_{N-1}\otimes X)\circ\left(\Lambda^{\epsilon}_{G_i(b)x\bar\alpha y}X\right)\circ(id_{N-1}\otimes X)
\end{longtable}
In other words,
\[\qc\vdash D(\upsilon_{N,i,b,a})=\left(id_{i+k+1}\otimes X\otimes id_{\ell}\right)\circ\left(\Lambda^{G_i(b)}_{x\bar\alpha y}X\right)\circ \left(id_{i+k+1}\otimes X\otimes id_{\ell}\right).\]
By definition of $\Lambda^{G_i(b)}_{x\bar\alpha y}X$ and \cref{XX}, this implies that \[\qc\vdash D(\upsilon_{N,i,b,a})=\Lambda^{G_i(b)}_{x\alpha y}X\]
which, since $x\alpha y=G_{N-i-1}(2^{N-i-1}-a-1)$, is the desired property.
\begin{remark}
By defining $\upsilon_{N,i,b,a}$ in a less natural way using not only $\gid$ and $\gswap$ but also $\tikzfig{phase-shift-xs}$ and $\tikzfig{bs-xs}$, one could avoid using \cref{prop:CCX} and get the stronger result that $\qczero\vdash D(\upsilon_{N,i,b,a})=\Lambda^{G_i(b)}_{G_{N-i-1}(2^{N-i-1}-a-1)}X$, which would in turn imply that the equalities of \cref{sigmatoswaps,basecaseDE} can also be taken modulo $\qczero$ instead of $\textup{QC}$.
\end{remark}
\subsubsection{Proof of Lemma \ref{sigmatoswaps}}\label{preuvesigmatoswaps}
First, if $n=1$, by definition (see \cref{defdecoding,defsigma}), one has
\[D(\sigma_{k,1,\ell})=\prod_{j=k+1}^{k+\ell}P_jQ_jP_j\]
where $M\coloneqq k+\ell+1$, $\displaystyle P_j\coloneqq\prod_{\begin{scriptarray}{c}\\[-1.5em]b=0\\[-0.4em]b\bmod 4\in\{1,2\}\end{scriptarray}}^{2^{j}-1}\hspace{-1.5em}\prod_{a=0}^{2^{M-j-1}-1}\hspace{-1em}D(\upsilon_{M,j,b,a})$ and $\displaystyle Q_j\coloneqq\prod_{b=0}^{2^{j-1}-1}\ \ \prod_{a=0}^{2^{M-j-3}-1}\hspace{-1em}D(\upsilon_{M,j-1,b,a})$.
By \cref{DupsilonLambda}, for all $j$,
\[\textup{QC}\vdash P_j=\prod_{\begin{scriptarray}{c}\\[-1.5em]b=0\\[-0.4em]b\bmod 4\in\{1,2\}\end{scriptarray}}^{2^{j}-1}\hspace{-1.5em}\prod_{a=0}^{2^{M-j-1}-1}\hspace{-1em}\Lambda^{G_j(b)}_{G_{M-j-1}(2^{M-j-1}-a-1)}X\]
It is easy to check that when $a$ goes
from $0$ to $2^{M-j-1}-1$,
$G_{M-j-1}(2^{M-j-1}-a-1)$ takes all possible values in $\{0,1\}^{M-j-1}$, once each, and that when $b$ takes all possible values between $0$ and $2^j-1$ that are congruent to $1$ or $2$ modulo $4$, $G_j(b)$ takes, once each, all values in $\{0,1\}^j$ in which the last bit has value $1$. Hence, it follows from \crefnosort{prop:comb,commctrl,ctrlXCNot} that
\[\textup{QC}\vdash P_j=id_{j-1}\otimes \gcnot\otimes id_{M-j-1}.\]
Again by \cref{DupsilonLambda}, for all $j$,
\[\textup{QC}\vdash Q_j=\prod_{b=0}^{2^{j-1}-1}\ \ \prod_{a=0}^{2^{M-j-3}-1}\hspace{-1em}\Lambda^{G_{j-1}(b)}_{G_{M-j}(2^{M-j}-a-1)}X\]
Similarly, it is easy to check that when $b$ goes from $0$ to $2^{j-1}-1$, $G_{j-1}(b)$ takes all values in $\{0,1\}^{j-1}$, once each, and that when $a$ goes from $0$ to $2^{M-j-3}$, $G_{M-j}(2^{M-j}-a-1)$ takes, once each, all values in $\{0,1\}^{M-j}$ in which the first bit has value $1$. Hence, it follows from \crefnosort{prop:comb,commctrl,ctrlXCNot} that
\[\textup{QC}\vdash Q_j=id_{j-1}\otimes \gnotc\otimes id_{M-j-1}.\]
Thus,
\[\textup{QC}\vdash D(\sigma_{k,1,\ell})=\prod_{j=k+1}^{k+\ell}id_{j-1}\otimes\scalebox{0.69}{$\tikzfig{tripleCNot-s}$}\otimes id_{M-j-1}.\]
By \cref{tripleCNotswap}, this implies that
\begin{equation}\label{sigmatoswapscase1}\qc\vdash D(\sigma_{k,1,\ell})=\prod_{j=k+1}^{k+\ell}id_{j-1}\otimes \gswap\otimes id_{M-j-1}\equiv id_k\otimes\sigma_{1,\ell}.\end{equation}
Finally, if $n>1$, then
\begin{longtable}{RCL}
D(\sigma_{k,n,\ell})&\eqdef&D(\sigma_{k,1,\ell+n-1}^n)\\[0.5em]
&\eqdef&D(\sigma_{k,1,\ell+n-1})^n\\[0.5em]
&\eqeqref{sigmatoswapscase1}&(id_k\otimes\sigma_{1,\ell+n-1})^n\\[0.5em]
&\equiv&id_k\otimes\sigma_{n,\ell}.
\end{longtable}
\subsubsection{Proof of Lemma \ref{basecaseDE}}\label{preuvebasecaseDE}
If $g=\tikzfig{diagrammevide-s}$ or $\tikzfig{filcourt-s}$ then the result follows directly from the definitions.
If $g=s(\varphi)$, then it follows from the definitions of $E_{k,
\ell}$ and $D$ that
\[D(E_{k,
\ell}(s(\varphi)))=\prod_{x\in\{0,1\}^{k+\ell}}\Lambda^xs(\varphi)\]
where we use the notation $\prod_{x\in\{0,1\}^{k+\ell}}$ to denote the product without specifying the order of the factors. By \cref{prop:comb,commctrl}, this implies that
\[\qc\vdash D(E_{k,
\ell}(s(\varphi)))=
id_{k+\ell}\otimes s(\varphi)\]
which is equal to $id_k\otimes s(\varphi)\otimes id_{\ell}$ by the topological rules of
quantum circuits
.
If $g=
\gp$, then it follows from the definitions that if $k=\ell=0$,
\[D(E_{0,0}(\gp))=D(\tikzfig{Z-PHOL})\equiv\Lambda^1s(\varphi)=P(\varphi).\]
and if $(k,\ell)\neq(0,0)$,
\[D(E_{k,\ell}(P(\varphi))=D(\sigma_{k,\ell,1})
\circ D\left(\left(\tikzfig{Z-PHOL-doublesym}\right)^{\otimes {2^{k+\ell-1}}}\right)
\circ D(\sigma_{k,1,\ell})\]
with
\[D\left(\left(\tikzfig{Z-PHOL-doublesym}\right)^{\otimes {2^{k+\ell-1}}}\right)=\prod_{x\in\{0,1\}^{k+\ell}}\Lambda^{x1}s(\varphi)=\prod_{x\in\{0,1\}^{k+\ell}}\Lambda^xP(\varphi).\]
By \cref{prop:comb,commctrl}, this product is equal modulo $\qczero$ to $id_{k+\ell}\otimes P(\varphi)$. Then, \cref{sigmatoswaps} together with topological rules
of quantum circuits gives us the result
.
If $g=\gh$, then it follows from the definitions that if $k=\ell=0$,
\begin{longtable}{R@{\ }L}D(E_{0,0}(\gh))=D(\tikzfig{H-LOPP-xs})\equiv&\Lambda^1s(-\frac\pi2)\circ\Lambda^\epsilon_\epsilon R_X(-\frac\pi2)\circ\Lambda^1s(-\frac\pi2)\\\\
=&\tikzfig{EulerHmoins}\\\\
\eqeqref{EulerHmoins}&\tikzfig{H}\end{longtable}
and if $(k,\ell)\neq(0,0)$,
\[D(E_{k,\ell}(\gh)=D(\sigma_{k,\ell,1})\circ D\left(\left(\tikzfig{H-LOPP-doublesym-xs}\right)^{\otimes {2^{k+\ell-1}}}\right)\circ D(\sigma_{k,1,\ell})\]
with
\[D\left(\left(\tikzfig{H-LOPP-doublesym-xs}\right)^{\otimes {2^{k+\ell-1}}}\right)\equiv\prod_{x\in\{0,1\}^{k+\ell}}\left(\left(\prod_{a\in\{0,1\}}\Lambda^{xa1}s(-\frac\pi2)\right)\circ\left(\prod_{a\in\{0,1\}}\Lambda^{xa}R_X\bigl(-\frac\pi2\bigr)\right)\circ\left(\prod_{a\in\{0,1\}}\Lambda^{xa1}s(-\frac\pi2)\right)\right).\]
By \cref{prop:comb,commctrl}, this product is equal modulo $\qczero$ to $id_{k+\ell}\otimes \scalebox{0.69}{$\tikzfig{EulerHmoins}$}$, which by \cref{EulerHmoins} is equal modulo $\qczero$ to \gh. Then, \cref{sigmatoswaps} together with topological rules of quantum circuits gives us the result
.
If $g=\gcnot$, then it follows from the definitions that if $k=\ell=0$,
\[D(E_{0,0}(\gcnot))=D\left(\tikzfig{CNot-PHOL}\right)\equiv\Lambda^1_\epsilon X\]
which is equal to $\gcnot$ modulo $\qczero$ by \cref{ctrlXCNot};
and if $(k,\ell)\neq(0,0)$,
\[D(E_{k,\ell}(\gcnot)=D(\sigma_{k,\ell,2})
\circ D\left(\left(\tikzfig{CNot-PHOL-doublesym}\right)^{\otimes {2^{k+\ell-1}}}\right)
\circ D(\sigma_{k,2,\ell})\]
with
\[D\left(\left(\tikzfig{CNot-PHOL-doublesym}\right)^{\otimes {2^{k+\ell-1}}}\right)\equiv\prod_{x\in\{0,1\}^{k+\ell}}\Lambda^{x1}X.\]
By \cref{prop:comb,commctrl}, this product is equal modulo $\qczero$ to $id_{k+\ell}\otimes \Lambda^1X$, which by \cref{ctrlXCNot} is equal modulo $\qczero$ to $id_{k+\ell}\otimes \gcnot$.
Then, \cref{sigmatoswaps} together with topological rules
of quantum circuits gives us the result
.
If $g=\gswap$, then it follows from the definitions that
\[D(E_{k,2,\ell}(\gswap)=D(\sigma_{k,\ell,2})
\circ D(\sigma_{k+\ell,1,1})
\circ D(\sigma_{k,2,\ell})\]
By \cref{sigmatoswaps}, this is equal modulo $\textup{QC}$ to $(id_k\otimes\sigma_{\ell,2})\circ(id_{k+\ell}\otimes \gswap)\otimes(id_k\otimes\sigma_{\ell,2})$, which by the topological rules of
quantum circuits, is equal to $id_k\otimes \gswap\otimes id_\ell$.
\subsection{Proof of Lemma \ref{decodingtoporules}}\label{preuvedecodingtoporules}
\begin{definition}[Context]
A $N$-mode raw context $\C[\cdot]_i$ with $i\in\mathbb N$ is inductively defined as follows:
\begin{itemize}
\item $[\cdot]_i$ is a $i$-mode raw context
\item if $\C[\cdot]_i$ is a $N$-mode raw context and $C$ is a $M$-mode raw optical circuit then $\C[\cdot]_i\otimes C$ and $C\otimes \C[\cdot]_i$ are $N{+}M$-mode raw contexts
\item if $\C[\cdot]_i$ is a $N$-mode raw context and $C$ is a $N$-mode raw optical circuit then $\C[\cdot]_i\circ C$ and $C\circ \C[\cdot]_i$ are $N$-mode raw contexts.
\end{itemize}
\end{definition}
\begin{definition}[Substitution]
Given a $N$-mode raw context $\C[\cdot]_i$ and a $i$-mode raw circuit $C$, we define the substituted circuit $\C[C]$ as the $N$-mode raw circuit obtained by replacing the hole $[\cdot]_i$ by $C$ in $\C[\cdot]_i$.
\end{definition}
To prove \cref{decodingtoporules}, it suffices to prove that for each rule of \cref{fig:axiom}, of the form $C_1=C_2$ with $C_1,C_2\in\LOPPbarebf(i,i)$, and any $2^n$-mode raw context $\C[\cdot]_i$, one has $\qc\vdash D(\C[C_1])=D(\C[C_2])$. For this purpose, we prove a slightly more general result, namely that for any $k,n$ and any $\ell$-mode raw context $\C[\cdot]_i$ with $k+\ell\leq 2^n$, one has $\qc\vdash D_{k,n}(\C[C_1])=D(\C[C_2])$. We proceed by induction on $\C[\cdot]_i$:
\begin{itemize}
\item If $\C[\cdot]_i=C\circ \C'[\cdot]_i$, then $D_{k,n}(\C[C_1])=D_{k,n}(C)\circ D_{k,n}(\C'[C_1])$ and $D_{k,n}(\C[C_2])=D_{k,n}(C)\circ D_{k,n}(\C'[C_2])$, so the result follows by induction hypothesis. The case $\C[\cdot]_i=\C'[\cdot]_i\circ C$ is similar.
\item If $\C[\cdot]_i=C\otimes \C'[\cdot]_i$ with $C:\ell_1\to\ell_1$, then $D_{k,n}(\C[C_1])=D_{k+\ell_1,n}(\C'[C_1])\circ D_{k,n}(C)$ and $D_{k,n}(\C[C_2])=D_{k+\ell_1,n}(\C'[C_2])\circ D_{k,n}(C)$, so the result follows by induction hypothesis. The case $\C[\cdot]_i=\C'[\cdot]_i\otimes C$ is similar.
\end{itemize}
It remains to prove for each rule of \cref{fig:axiom}, of the form $C_1=C_2$ with $C_1,C_2\in\LOPPbarebf(i,i)$, that for any $k,n$ with $k+i\leq 2^n$, one has $\qc\vdash D_{k,n}(C_1)=D_{k,n}(C_2)$.
For \cref{assoccomp}, for any $C_1,C_2,C_3:\ell\to \ell$,
\[D_{k,n}((C_3\circ C_2)\circ C_1)=(D_{k,n}(C_3)\circ D_{k,n}(C_2))\circ D_{k,n}(C_1)\]
and
\[D_{k,n}(C_3\circ (C_2\circ C_1))=D_{k,n}(C_3)\circ (D_{k,n}(C_2)\circ D_{k,n}(C_1)).\]
Both are equal according to \cref{assoccomp} of quantum circuits.
For \cref{assoctens}, for any optical circuits $C_1:\ell_1\to\ell_1$, $C_2:\ell_2\to\ell_2$ and $C_3:\ell_3\to\ell_3$,
\[D_{k,n}((C_1\otimes C_2)\otimes C_3)=D_{k+\ell_1+\ell_2,n}(C_3)\circ (D_{k+\ell_1,n}(C_2)\circ D_{k,n}(C_1))\]
and
\[D_{k,n}(C_1\otimes (C_2\otimes C_3))=(D_{k+\ell_1+\ell_2,n}(C_3)\circ D_{k+\ell_1,n}(C_2))\circ D_{k,n}(C_1).\]
Again, both are equal according to \cref{assoccomp} of quantum circuits.
For \cref{idneutre}, for any $\ell$-mode optical circuit $C$, by definition of $id_\ell$ and $D_{k,n}$,
\[D_{k,n}(id_\ell\circ C)=(id_n\circ(id_n\circ(\cdots\circ(id_n\circ id_n))\cdots))\circ D_{k,n}(C)\]
with $\ell+1$ occurences of $id_n$ in the right-hand side. This is equal to $D_{k,n}(C)$ according to \cref{idneutre} of quantum circuits. Similarly, $D_{k,n}(C\circ id_\ell)\equiv D_{k,n}(C)$.
For \cref{videneutre}, for any $\ell$-mode optical circuit $C$,
\[D_{k,n}(\tikzfig{diagrammevide-s}\otimes C)=D_{k,n}(C)\circ id_\ell\]
which is equal to $D_{k,n}(C)$ according to \cref{idneutre} of quantum circuits. Similarly, $D_{k,n}(C\otimes \tikzfig{diagrammevide-s})\equiv D_{k,n}(C)$.
For \cref{mixedprod}, for any optical circuits $C_1,C_2:\ell\to\ell$ and $C_3,C_4:m\to m$,
\[D_{k,n}((C_2\circ C_1)\otimes (C_4\circ C_3))=(D_{k+\ell,n}(C_4)\circ D_{k+\ell,n}(C_3))\circ(D_{k,n}(C_2)\circ D_{k,n}(C_1))\]
and
\[D_{k,n}((C_2\otimes C_4)\circ (C_1\otimes C_3))=(D_{k+\ell,n}(C_4)\circ D_{k,n}(C_2))\circ(D_{k+\ell,n}(C_3)\circ D_{k,n}(C_1)).\]
The result follows from \cref{assoccomp} of quantum circuits and \cref{decodagefilsdisjoints}.
For \cref{doubleswap}, one has
\[D_{k,n}(\gswap\circ\gswap)=\Lambda^{x_{k,n}}_{y_{k,n}}X\circ\Lambda^{x_{k,n}}_{y_{k,n}}X\]
which by \cref{prop:CCX}, implies that
\[\qc\vdash D_{k,n}(\gswap\circ\gswap)=id_n.\]
On the other hand,
\[D_{k,n}(\tikzfig{filcourt-s}\otimes\tikzfig{filcourt-s})=id_n\circ id_n\equiv id_n.\]
For \cref{naturaliteswap}, we proceed by induction on $C$.
\begin{itemize}
\item If $C=C_1\circ C_2$, then $\sigma_k\circ((C_1\circ C_2)\otimes\gid)\equiv(\sigma_k\circ(C_1\otimes\gid))\circ(C_2\otimes\gid)$, and the derivation of the equivalence does not use \cref{naturaliteswap}. Hence it follows from the paragraphs above that
\[\qc\vdash D_{k,n}(\sigma_k\circ((C_1\circ C_2)\otimes\gid))=D_{k,n}((\sigma_k\circ(C_1\otimes\gid))\circ(C_2\otimes\gid)).\]
It follows similarly from those paragraphs that
\[\qc\vdash D_{k,n}((\gid\otimes(C_1\circ C_2))\circ\sigma_k)=D_{k,n}((C_1\otimes\gid)\circ((C_2\otimes\gid)\circ\sigma_k)).\]
The equality modulo $\qc$ of the two right-hand sides follows from the induction hypothesis, together with the compatibility of $D_{k,n}$ with \cref{assoccomp} modulo $\qc$, which is proved above.
\item If $C=C_1\otimes C_2$ with $C_1:\ell_1\to\ell_1$ and $C_2:\ell_2\to\ell_2$, then
\[\sigma_k\circ((C_1\otimes C_2)\otimes\gid)\equiv((\sigma_{\ell_1}\circ(C_1\otimes\gid))\otimes id_{\ell_2})\circ(id_{\ell_1}\otimes(\sigma_{\ell_2}\circ(C_2\otimes\gid)))\]
and the derivation of the equivalence does not use \cref{naturaliteswap}, so that by the paragraphs above (together with \cref{idneutre} of quantum circuits),
\[\qc\vdash D_{k,n}(\sigma_k\circ((C_1\otimes C_2)\otimes\gid))=D_{k,n}(\sigma_{\ell_1}\circ(C_1\otimes\gid))\circ D_{k+\ell_1}(\sigma_{\ell_2}\circ(C_2\otimes\gid)).\]
The result follows by applying a similar transformation to the right-hand side of \cref{naturaliteswap} and applying the induction hypothesis.
\item If $C=\emptyc\text{ or }\gid$, then the result follows from \cref{idneutre,videneutre} of quantum circuits.
\item If $C=\tikzfig{phase-shift-xs}$, let us write $G_n(k)$ as $xay$ with $a\in\{0,1\}$ and $y=\epsilon$ if $k$ is even or $y=1.0^q$ for some $q$ if $k$ is odd. Then by definition of $D_{k,n}$ and \cref{phasemobile}, if $a=1$ then
\[\qc\vdash D_{k,n}(\sigma_1\circ(\tikzfig{phase-shift-xs}\otimes\gid))=\Lambda^x_y X\circ\Lambda^x_y P(\varphi)\]
and
\[\qc\vdash D_{k,n}((\gid\otimes\tikzfig{phase-shift-xs})\circ\sigma_1)=(id_{|x|}\otimes X\otimes id_{|y|})\circ\Lambda^x_y P(\varphi)(id_{|x|}\otimes X\otimes id_{|y|})\circ\Lambda^x_y X.\]
By \cref{prop:CCX,commctrl,prop:comb}, the following equalities are true modulo $\qc$:
\begin{longtable}{RCL}
\Lambda^x_y X\circ\Lambda^x_y P(\varphi)&=&(id_{|x|}\otimes X\otimes id_{|y|})\circ\bar\Lambda^x_y X\circ\Lambda^x_y P(\varphi)\\
&=&(id_{|x|}\otimes X\otimes id_{|y|})\circ\Lambda^x_y P(\varphi)\circ\bar\Lambda^x_y X\\
&=&(id_{|x|}\otimes X\otimes id_{|y|})\circ\Lambda^x_y P(\varphi)\circ(id_{|x|}\otimes X\otimes id_{|y|})\circ\Lambda^x_y X
\end{longtable}
which gives us the result. The case $a=0$ is similar.
\item If $C=\tikzfig{bs-xs}$, by the properties of the Gray code, exactly one bit differs between $G_n(k)$ and $G_n(k+1)$, as well as between $G_n(k+1)$ and $G_n(k+2)$, and in exactly one of the two cases this is the last bit that differs (namely between $G_n(k)$ and $G_n(k+1)$ if $k$ is even, and between $G_n(k+1)$ and $G_n(k+2)$ if $k$ is odd). Hence we can write $G_n(k)$ as $xayb$ with $a,b\in\{0,1\}$, in such a way that $G_n(k+2)=x\bar ay\bar b$ and $G_n(k+1)=xay\bar b\text{ or }x\bar ayb$ depending on the parity of $k$. We treat the case where $k$ is even, the case with $k$ odd is similar. Then
\[D_{k,n}(\sigma_2\circ(\tikzfig{bs-xs}\otimes\gid))\equiv\Lambda^{xay}X\circ\Lambda^x_{y\bar b}X\circ\Lambda^{xay}R_X(-2\theta)\]
and
\[D_{k,n}((\gid\otimes\tikzfig{bs-xs})\circ\sigma_2)\equiv\Lambda^x_{y\bar b}R_X(-2\theta)\circ\Lambda^{xay}X\circ\Lambda^x_{y\bar b}X\]
so by \cref{symmetriesemicontrolee,XX}, it suffices to prove that for any $\theta$,
\[\qc\vdash\Lambda^{x1y}X\circ\Lambda^x_{y1}X\circ\Lambda^{x1y}R_X(\theta)=\Lambda^x_{y1}R_X(\theta)\circ\Lambda^{x1y}X\circ\Lambda^x_{y1}X.\]
To prove this, one has, modulo $\qc$ (together with the topological rules of quantum circuits):
\begin{longtable}{RCL}
\Lambda^{x1y}X\circ\Lambda^x_{y1}X\circ\Lambda^{x1y}R_X(\theta)&\overset{\text{\crefnosort{prop:CCX,commctrl,cor:swap,prop:comb,ctrlXCNot}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite1}$}\\\\
&\overset{\text{\cref{commctrl,commctrlpasenface}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite2}$}\\\\
&\overset{\text{\crefnosort{prop:CCX,commctrl,cor:swap,prop:comb,ctrlXCNot}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite3}$}\\\\
&\overset{\text{\cref{commctrlpasenface}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite4}$}\\\\
&\eqeqref{CNotCNot}&\scalebox{0.85}{$\tikzfig{decodagenaturalite5}$}\\\\
&\eqeqref{tripleCNotswap}&\scalebox{0.85}{$\tikzfig{decodagenaturalite6}$}\\\\
&\overset{\text{\cref{cor:swap}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite7}$}\\\\
&\eqeqref{tripleCNotswap}&\scalebox{0.85}{$\tikzfig{decodagenaturalite8}$}\\\\
&\eqtroiseqref{XX}{CNotX}{commutationXCNot}&\scalebox{0.85}{$\tikzfig{decodagenaturalite9}$}\\\\
&\overset{\text{\cref{ctrlXCNot,prop:comb,commctrl}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite10}$}\\\\
&\overset{\text{\cref{symmetriesemicontrolee}, \eqref{XX}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite11}$}\\\\
&\overset{\text{\cref{prop:CCX,commctrl,cor:swap,prop:comb,ctrlXCNot}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite12}$}\\\\
&\overset{\text{\cref{commctrlpasenface}}}{=}&\scalebox{0.85}{$\tikzfig{decodagenaturalite13}$}\\\\
&\overset{\text{\crefnosort{prop:CCX,commctrl,cor:swap,prop:comb,ctrlXCNot}}}{=}&
\Lambda^x_{y1}R_X(\theta)\circ\Lambda^{x1y}X\circ\Lambda^x_{y1}X.
\end{longtable}
\item The case $C=\tikzfig{swap-xs}$ is similar to the preceding one, with $R_X(\theta)$ replaced by $X$.
\end{itemize}
\subsection{Proof of Lemma \ref{decodingLOPPrules}}\label{preuvedecodingLOPPrules}
By \cref{decodingtoporules}, to prove \cref{decodingLOPPrules}, it suffices to prove that for each rule of \cref{axiomsLOPP}, of the form $C_1=C_2$ with $C_1,C_2\in\LOPPbarebf(i,i)$ (see \cref{rawinfigures}), and any $2^n$-mode raw context $\C[\cdot]_i$, one has $\qc\vdash D(\C[C_1])=D(\C[C_2])$. For this purpose, we prove a slightly more general result, namely that for any $k,n$ and any $\ell$-mode raw context $\C[\cdot]_i$ with $k+\ell\leq 2^n$, one has $\qc\vdash D_{k,n}(\C[C_1])=D(\C[C_2])$. We proceed by induction on $\C[\cdot]_i$:
\begin{itemize}
\item If $\C[\cdot]_i=C\circ \C'[\cdot]_i$, then $D_{k,n}(\C[C_1])=D_{k,n}(C)\circ D_{k,n}(\C'[C_1])$ and $D_{k,n}(\C[C_2])=D_{k,n}(C)\circ D_{k,n}(\C'[C_2])$, so the result follows by induction hypothesis. The case $\C[\cdot]_i=\C'[\cdot]_i\circ C$ is similar.
\item If $\C[\cdot]_i=C\otimes \C'[\cdot]_i$ with $C:\ell_1\to\ell_1$, then $D_{k,n}(\C[C_1])=D_{k+\ell_1,n}(\C'[C_1])\circ D_{k,n}(C)$ and $D_{k,n}(\C[C_2])=D_{k+\ell_1,n}(\C'[C_2])\circ D_{k,n}(C)$, so the result follows by induction hypothesis. The case $\C[\cdot]_i=\C'[\cdot]_i\otimes C$ is similar.
\end{itemize}
It remains to prove for each rule of \cref{axiomsLOPP}, of the form $C_1=C_2$ with $C_1,C_2\in\LOPPbarebf(i,i)$, that for any $k,n$ with $k+i\leq 2^n$, one has $\qc\vdash D_{k,n}(C_1)=D_{k,n}(C_2)$. Again by \cref{decodingtoporules}, it suffices to prove that $\qc\vdash D_{k,n}(C'_1)=D_{k,n}(C'_2)$ for arbitrary $C'_1$ and $C'_2$ such that $C'_1\equiv C_1$ and $C'_2\equiv C_2$.
For \cref{phase0}, one has $D_{k,n}(\scalebox{0.69}{$\tikzfig{phase-shift0}$})=\Lambda^{G_n(k)}s(0)$, $D_{k,n}(\scalebox{0.69}{$\tikzfig{phase-shift2pi}$})=\Lambda^{G_n(k)}s(2\pi)$ and $D_{k,n}(\gid)=id_n$. The three are equal modulo $\textup{QC}$ by \cref{prop:sum,prop:period}.
For \cref{bs0}, one has $D_{k,n}(\scalebox{0.69}{$\tikzfig{bs0-s}$})=\Lambda^{x_{k,n}}_{y_{k,n}}R_X(0)$ (where $x_{k,n}$ and $y_{k,n}$ are defined in \cref{defdecoding}) and $D_{k,n}(\scalebox{0.4}{$\tikzfig{filsparalleleslongbs-m}$})=id_n\circ id_n\equiv id_n$. The two are equal modulo $\textup{QC}$
by \cref{prop:sum}.
For \cref{swapbspisur2}, one has $D_{k,n}(\scalebox{0.69}{$\tikzfig{swap-s}$})=\Lambda^{x_{k,n}}_{y_{k,n}}X$, and $D_{k,n}(\scalebox{0.575}{$\tikzfig{bspissur2-ms}$})=
\left(\displaystyle\prod_{j\in\{k,k+1\}}\Lambda^{G_n(j)}s(-\frac\pi2)\right)\circ\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-\pi)$.
Note that the definitions imply that
\begin{equation}\label{Grayxy}\{G_n(k),G_n(k+1)\}=\{x_{k,n}0y_{k,n},x_{k,n}1y_{k,n}\}.\end{equation}
Therefore,
\begin{longtable}{RCL}D_{k,n}(\scalebox{0.575}{$\tikzfig{bspissur2-ms}$})&=&\sigma_{1,|x_{k,n}|}\circ\left(\displaystyle\prod_{a\in\{0,1\}}\Lambda^{ax_{k,n}y_{k,n}}s(-\frac\pi2)\right)\circ\Lambda^{\epsilon}_{x_{k,n}y_{k,n}}R_X(-\pi)\circ \sigma_{|x_{k,n}|,1}\\\\
&\overset{\text{\cref{prop:comb,commctrl}}}{=}&\sigma_{1,|x_{k,n}|}\circ\left(\gid\otimes\Lambda^{x_{k,n}y_{k,n}}s(-\frac\pi2)\right)\circ\Lambda^{\epsilon}_{x_{k,n}y_{k,n}}R_X(-\pi)\circ \sigma_{|x_{k,n}|,1}
\end{longtable}
which by \cref{commctrlphaseenhaut,def:multicontrolled,prop:period,HH}, is equal modulo $\qc$ to $\Lambda^{x_{k,n}}_{y_{k,n}}X$.
For \cref{phaseaddition}, one has $D_{k,n}(\scalebox{0.69}{$\tikzfig{convtp-phase-shifts-12}$})=\Lambda^{G_n(k)}s(\varphi_2)\circ\Lambda^{G_n(k)}s(\varphi_1)$ and $D_{k,n}(\scalebox{0.69}{$\tikzfig{convtp-phase-shift-1plus2}$})=\Lambda^{G_n(k)}s(\varphi_1+\varphi_2)$. Both are equal modulo $\qc$ by \cref{prop:sum}.
For \cref{globalphasepropagationbs}, one has
\begin{longtable}{RCL}D_{k,n}(\scalebox{0.69}{$\tikzfig{thetathetabs-ms}$})&=&\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\theta)\circ\left(\displaystyle\prod_{j\in\{k,k+1\}}\Lambda^{G_n(j)}s(\varphi)\right)\\\\
&\eqeqref{Grayxy}&\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\theta)\circ\left(\displaystyle\prod_{a\in\{0,1\}}\Lambda^{x_{k,n}ay_{k,n}}s(\varphi)\right)\\\\
&=&\sigma_{1,|x_{k,n}|}\circ\Lambda^{\epsilon}_{x_{k,n}y_{k,n}}R_X(-2\theta)\circ\left(\displaystyle\prod_{a\in\{0,1\}}\Lambda^{ax_{k,n}y_{k,n}}s(\varphi)\right)\circ \sigma_{|x_{k,n}|,1}\\\\
&\overset{\text{\cref{prop:comb,commctrl}}}{=}&\sigma_{1,|x_{k,n}|}\circ\Lambda^{\epsilon}_{x_{k,n}y_{k,n}}R_X(-2\theta)\circ\left(\gid\otimes\Lambda^{x_{k,n}y_{k,n}}s(\varphi)\right)\circ \sigma_{|x_{k,n}|,1}\\\\
&\overset{\text{\cref{commctrlphaseenhaut}}}{=}&\sigma_{1,|x_{k,n}|}\circ\left(\gid\otimes\Lambda^{x_{k,n}y_{k,n}}s(\varphi)\right)\circ\Lambda^{\epsilon}_{x_{k,n}y_{k,n}}R_X(-2\theta)\circ \sigma_{|x_{k,n}|,1}\\\\
&\overset{\text{\cref{prop:comb,commctrl}}}{=}&\sigma_{1,|x_{k,n}|}\circ\left(\displaystyle\prod_{a\in\{0,1\}}\Lambda^{ax_{k,n}y_{k,n}}s(\varphi)\right)\circ\Lambda^{\epsilon}_{x_{k,n}y_{k,n}}R_X(-2\theta)\circ \sigma_{|x_{k,n}|,1}\\\\
&=&\left(\displaystyle\prod_{a\in\{0,1\}}\Lambda^{x_{k,n}ay_{k,n}}s(\varphi)\right)\circ\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\theta)\\\\
&\eqeqref{Grayxy}&\left(\displaystyle\prod_{j\in\{k,k+1\}}\Lambda^{G_n(j)}s(\varphi)\right)\circ\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\theta)\\\\
&=&D_{k,n}(\scalebox{0.69}{$\tikzfig{bsthetatheta-ms}$}).
\end{longtable}
For \cref{Eulerbsphasebs}, one has
\[D_{k,n}(\scalebox{0.69}{$\tikzfig{bsphasebsalpha-ms}$})\equiv\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\alpha_3)\circ\Lambda^{G_n(k)}s(\alpha_2)\circ\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\alpha_1)\]
and
\[D_{k,n}(\scalebox{0.69}{$\tikzfig{phasebsphasebeta-ms}$})\equiv\Lambda^{G_n(k+1)}s(\beta_4)\circ\Lambda^{G_n(k)}s(\beta_3)\circ\Lambda^{x_{k,n}}_{y_{k,n}}R_X(-2\beta_2)\circ\Lambda^{G_n(k)}s(\beta_1).\]
Note that for some $a_k\in\{0,1\}$, one has $G_n(k)=x_{k,n}a_ky_{k,n}$ and $G_n(k+1)=x_{k,n}\bar a_ky_{k,n}$. Therefore, by \cref{prop:CP}, for any $\varphi\in\mathbb R$, one has $\qc\vdash\Lambda^{G_n(k)}s(\varphi)=\Lambda^{x_{k,n}}_{y_{k,n}}P(\varphi)$ and $\qc\vdash\Lambda^{G_n(k+1)}s(\varphi)=(id_{|x_{k,n}|}\otimes X\otimes id_{|y_{k,n}|})\circ\Lambda^{x_{k,n}}_{y_{k,n}}P(\varphi)\circ(id_{|x_{k,n}|}\otimes X\otimes id_{|y_{k,n}|})$, or conversely. Thus, up to using \cref{XX} and possibly \cref{symmetriesemicontrolee}, it suffices to prove that $\lambda^{n-1}R_X(-2\alpha_3)\circ\lambda^{n-1}P(\alpha_2)\circ\lambda^{n-1}R_X(-2\alpha_1)=(id_{n-1}\otimes X)\circ\lambda^{n-1}P(\beta_4)\circ(id_{n-1}\otimes X)\circ\lambda^{n-1}P(\beta_3)\circ\lambda^{n-1}R_X(-2\beta_2)\circ\lambda^{n-1}P(\beta_1)$. One has
\begin{longtable}{RCL}
\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit}$}&\overset{\text{\cref{antisymmetriesemicontrolee}}}{=}&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit1}$}\\\\
&\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit2}$}\\\\
&\overset{\text{\cref{prop:period}}}{=}&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit3}$}
\end{longtable}
Because of the conditions on the angles in the right-hand side of \cref{Eulerbsphasebs}, if $\beta_2=0$ then the angles of the last circuit satisfy the conditions so that it matches the right-hand side of \cref{Euler2dmulticontrolled}. Hence, since it has the same semantics as $\lambda^{n-1}R_X(-2\alpha_3)\circ\lambda^{n-1}P(\alpha_2)\circ\lambda^{n-1}R_X(-2\alpha_1)$, both circuits are equal according to \cref{Euler2dmulticontrolled}.
If $\beta_2\neq0$, then
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit3}$}\\\\
\overset{\text{\cref{prop:period,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit4}$}\\\\
\overset{\text{\cref{CRX2pi}}}{=}&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit5}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP,prop:sum,prop:period}}}{=}&\scalebox{0.8}{$\tikzfig{phasebsphasebeta-circuit6}$}
\end{longtable}
Because of the conditions on the angles in the right-hand side of \cref{Eulerbsphasebs}, one has $\beta_2\in(0,\pi)$, so that $2\pi-2\beta_2\in(0,2\pi)$, and if $2\pi-2\beta_2=\pi$ then $\beta_2=\frac\pi2$, so that $\beta_1=0$. Hence, the angles of the last circuit satisfy the conditions so that it matches the right-hand side of \cref{Euler2dmulticontrolled}. Again, since it has the same semantics as $\lambda^{n-1}R_X(-2\alpha_3)\circ\lambda^{n-1}P(\alpha_2)\circ\lambda^{n-1}R_X(-2\alpha_1)$, both circuits are equal according to \cref{Euler2dmulticontrolled}.
For \cref{Eulerscalaires}, by the properties of the Gray code, exactly one bit differs between $G_n(k)$ and $G_n(k+1)$, as well as between $G_n(k+1)$ and $G_n(k+2)$, and in exactly one of the two cases this is the last bit that differs (namely between $G_n(k)$ and $G_n(k+1)$ if $k$ is even, and between $G_n(k+1)$ and $G_n(k+2)$ if $k$ is odd). Hence we can write $G_n(k)$ as $xayb$ with $a,b\in\{0,1\}$, in such a way that $G_n(k+2)=x\bar ay\bar b$ and $G_n(k+1)=xay\bar b\text{ or }x\bar ayb$ depending on the parity of $k$. We treat the case where $k$ is even, the case with $k$ odd is similar. One has
\[D_{k,n}\left(\scalebox{0.69}{$\tikzfig{bsyangbaxterpointeenbas-simp-gammas-ms}$}\right)\equiv\Lambda^{xay}R_X(-2\gamma_4)\circ\Lambda^x_{y\bar b}R_X(-2\gamma_3)\circ\Lambda^{xayb}s(\gamma_2)\circ\Lambda^{xay}R_X(-2\gamma_1)\]
and
\[D_{k,n}\left(\scalebox{0.69}{$\tikzfig{bsyangbaxterpointeenhaut-deltas-ms}$}\right)\equiv\begin{array}[t]{l}\Lambda^{x\bar ay\bar b}s(\delta_9)\circ\Lambda^{xay\bar b}s(\delta_8)\circ\Lambda^{xayb}s(\delta_7)\circ\Lambda^x_{y\bar b}R_X(-2\delta_6)\circ\Lambda^{xay\bar b}s(\delta_5)\\
\circ\Lambda^{xay}R_X(-2\delta_4)\circ\Lambda^x_{y\bar b}R_X(-2\delta_3)\circ\Lambda^{xayb}s(\delta_2)\circ\Lambda^{xay\bar b}s(\delta_1).\end{array}\]
Up to using \cref{XX}, we can assume that the components of $x$ and $y$ are all equal to $1$. Up to using additionally \cref{symmetriesemicontrolee}, we can assume that $a=1$ and $b=0$. Finally, up to deforming the circuits, we can assume that $y=\epsilon$. Thus, it suffices to prove that
\[\qc\vdash\Lambda^{x1}R_X(-2\gamma_4)\circ\Lambda^x_{1}R_X(-2\gamma_3)\circ\Lambda^{x10}s(\gamma_2)\circ\Lambda^{x1}R_X(-2\gamma_1)=\begin{array}[t]{l}\Lambda^{x01}s(\delta_9)\circ\Lambda^{x11}s(\delta_8)\circ\Lambda^{x10}s(\delta_7)\circ\Lambda^x_{1}R_X(-2\delta_6)\circ\Lambda^{x11}s(\delta_5)\circ\\
\Lambda^{x1}R_X(-2\delta_4)\circ\Lambda^x_{1}R_X(-2\delta_3)\circ\Lambda^{x10}s(\delta_2)\circ\Lambda^{x11}s(\delta_1)\end{array}\]
where $x=1^{n-2}$.
The left-hand side is equal to
\begin{longtable}{RCL}
\scalebox{0.8}{$\tikzfig{Eulerscalairesleft-circuit}$}&\overset{\text{\cref{prop:sum,commctrl,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{Eulerscalairesleft-circuit1}$}\\\\
&\overset{\text{\cref{commctrlphaseenhaut}}}{=}&\scalebox{0.8}{$\tikzfig{Eulerscalairesleft-circuit2}$}\\\\
&\equiv&\scalebox{0.8}{$\tikzfig{Eulerscalairesleft-circuit-arrange}$}
\end{longtable}
while the right-hand side is equal to
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Eulerscalairesright-circuit}$}\\\\
\overset{\text{\cref{prop:sum,commctrl,
prop:comb}}}=&\scalebox{0.8}{$\tikzfig{Eulerscalairesright-circuit2}$}\\\\
\equiv&\scalebox{0.8}{$\tikzfig{Eulerscalairesright-circuit3}$}
\end{longtable}
Hence, it suffices to prove that
\[\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledleft-preuve}$}=\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve}$}.\]
The left-hand side matches the left-hand side of \cref{Euler3dmulticontrolled}, hence it suffices to prove that the right-hand side can be put in the form of the right-hand side of \cref{Euler3dmulticontrolled} with the angles satisfying the conditions. One has
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve}$}\\\\
\overset{\text{\cref{prop:sum,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve1}$}\\\\
\overset{\text{\cref{prop:CP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve2}$}\\\\
\overset{\text{\cref{prop:comb,commctrl,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve3}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve4}$}\\\\
\overset{\text{\cref{prop:comb,commctrl,prop:CP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{Euler3dmulticontrolledright-preuve5}$}
\end{longtable}
It remains to
prove that any circuit of the form $\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}$ can be transformed using the axioms of $\qc$ in such a way that the angles satisfy the conditions given in \cref{fig:euler}. We treat the conditions in the following order (note that some of the conditions of \cref{fig:euler} have been split into two parts):
\begin{itemize}
\item \hyperref[delta3in02pi]{$\delta_3\in[0,2\pi)$}
\item \hyperref[delta4in02pi]{$\delta_4\in[0,2\pi)$}
\item \hyperref[delta6in02pi]{$\delta_6\in[0,2\pi)$}
\item \hyperref[delta3zeroimpldelta2zero]{if $\delta_3=0$ then $\delta_2=0$}
\item \hyperref[delta4piimpldelta2zeroifdelta3notzero]{if $\delta_3\neq0$ but $\delta_4=\pi$ then $\delta_2=0$}
\item \hyperref[delta3zerodelta4piimpldelta1zero]{if $\delta_3=0$ and $\delta_4=\pi$ then $\delta_1=0$}
\item \hyperref[delta3piimpldelta1zero]{if $\delta_3=\pi$ then $\delta_1=0$}
\item \hyperref[delta4zeroimpldelta123zero]{if $\delta_4=0$ then $\delta_1=\delta_2=\delta_3=0$}
\item \hyperref[delta1in0piifdelta3notzero]{if $\delta_3\neq0$ then $\delta_1\in[0,\pi)$}
\item \hyperref[delta1in0piifdelta3zero]{if $\delta_3=0$ then $\delta_1\in[0,\pi)$}
\item \hyperref[delta6zeroimpldelta5zero]{if $\delta_6=0$ then $\delta_5=0$}
\item \hyperref[delta6piimpldelta5zero]{if $\delta_6=\pi$ then $\delta_5=0$}
\item \hyperref[delta2in0pi]{$\delta_2\in[0,\pi)$}
\item \hyperref[delta5in0pi]{$\delta_5\in[0,\pi)$}
\item \hyperref[delta789in02pi]{$\delta_7,\delta_8,\delta_9\in[0,2\pi)$}.
\end{itemize}
For each of them, we prove that given a circuit satisfying the previous conditions, we can transform it into a circuit satisfying also the considered condition.
\phantomsection\label{delta3in02pi}If $\delta_3\notin[0,2\pi)$, then by \cref{prop:period}, we can assume that it is in $[0,4\pi)$, and then if it is in $[2\pi,4\pi)$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles1bis}$}\\\\
\overset{\text{\cref{CRX2pi}}}{=}&
\scalebox{0.8}{$\tikzfig{anglesbonsintervalles3}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles4}$}\\\\
\overset{\text{\cref{commctrl,passagephasepicircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles5}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles6}$}\\\\
\overset{\text{\cref{commctrlphaseenhaut,commctrlphaseenhautP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles7}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles8}$}
\end{longtable}
with $\delta_3-2\pi\in[0,2\pi)$. Hence, we can assume that $\delta_3\in[0,2\pi)$.
\phantomsection\label{delta4in02pi}If $\delta_4\notin[0,2\pi)$, then by \cref{prop:period}, we can ensure that it is in $[0,4\pi)$, and then if it is in $[2\pi,4\pi)$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles9}$}\\\\
\overset{\text{\cref{CRX2pi}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles10}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles11}$}\\\\
\overset{\text{\crefnosort{commctrl,prop:sum,passagephasepicircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles12}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP,prop:comb,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles13}$}
\end{longtable}
with $\delta_4-2\pi\in[0,2\pi)$. Hence, we can assume additionally that $\delta_4\in[0,2\pi)$.
\phantomsection\label{delta6in02pi}If $\delta_6\notin[0,2\pi)$, then by \cref{prop:period}, we can ensure that it is in $[0,4\pi)$, and then if it is in $[2\pi,4\pi)$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles14}$}\\\\
\overset{\text{\cref{CRX2pi}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles15}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles16}$}
\end{longtable}
with $\delta_6-2\pi\in[0,2\pi)$. Hence, we can assume additionally that $\delta_6\in[0,2\pi)$.
\phantomsection\label{delta3zeroimpldelta2zero}If $\delta_3=0$ but $\delta_2\neq0$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles16etdemi}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles17}$}\\\\
\overset{\text{\cref{commctrlphaseenhaut}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles18}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles19}$}\\\\
\overset{\text{\crefnosort{commctrl,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles20}$}\\\\
\overset{\text{\cref{prop:sum,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles21}$}\\\\
\overset{\text{\cref{prop:comb,commctrl,prop:CP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles22}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles23}$}.
\end{longtable}
Hence, we can assume additionally that if $\delta_3=0$ then $\delta_2=0$.
\phantomsection\label{delta4piimpldelta2zeroifdelta3notzero}If $\delta_3\neq0$, and $\delta_4=\pi$ but $\delta_2\neq0$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles44}$}\\\\
\overset{\text{\cref{prop:sum,mctrlX}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles45}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles46}$}\\\\
\overset{\text{\cref{prop:sum,commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles47}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles48}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles49}$}\\\\
\overset{\text{\cref{antisymmetriesemicontrolee,antisymmetriecontrolee}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles50}$}\\\\
\overset{\text{\cref{prop:CCX}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles51}$}\\\\
\overset{\text{\crefnosort{prop:CP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles52}$}\\\\
\overset{\text{\cref{mctrlX,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles53}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles54}$}
\end{longtable}
Hence, we can assume additionally that if $\delta_4=\pi$ then $\delta_2=0$ (note that by the previous assumption we already had $\delta_2=0$ when $\delta_3=0$).
\phantomsection\label{delta3zerodelta4piimpldelta1zero}If $\delta_3=0$ and $\delta_4=\pi$, then by assumption, $\delta_2=0$. If we do not have additionally that $\delta_1=0$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles55}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles56}$}\\\\
\overset{\text{\cref{prop:sum,mctrlX}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles57}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles58}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles59}$}\\\\
\overset{\text{\crefnosort{prop:CCX,antisymmetriecontrolee,antisymmetriesemicontrolee}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles60}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles61}$}\\\\
\overset{\text{\cref{commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles62}$}\\\\
\overset{\text{\cref{prop:sum,commctrlphaseenhautP,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles63}$}\\\\
\overset{\text{\cref{mctrlX,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles64}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles65}$}
\end{longtable}
Hence, we can assume additionally that if $\delta_3=0$ and $\delta_4=\pi$ then $\delta_1=0$.
\phantomsection\label{delta3piimpldelta1zero}If $\delta_3=\pi$ but $\delta_1\neq0$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles24}$}\\\\
\overset{\text{\cref{prop:sum,mctrlX}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles25}$}\\\\
\overset{\text{\cref{prop:comb,prop:sum,commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles26}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles27}$}\\\\
\overset{\text{\crefnosort{prop:CCX,antisymmetriecontrolee,antisymmetriesemicontrolee}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles28}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles29}$}\\\\
\overset{\text{\cref{commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles30}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles31}$}\\\\
\overset{\text{\cref{commctrl,prop:sum,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles32}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles33}$}\\\\
\overset{\text{\cref{commctrlphaseenhaut,commctrlphaseenhautP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles34}$}\\\\
\overset{\text{\cref{mctrlX,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles35}$}
\end{longtable}
Hence, we can assume additionally that if $\delta_3=\pi$ then $\delta_1=0$.
\phantomsection\label{delta4zeroimpldelta123zero}If $\delta_4=0$ but $(\delta_1,\delta_2,\delta_3)\neq(0,0,0)$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles36}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles37}$}\\\\
\eqeqref{Euler2dmulticontrolled}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles38}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles39}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles40}$}\\\\
\overset{\text{\crefnosort{commctrl,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles41}$}\\\\
\overset{\text{\cref{prop:sum,commctrlphaseenhautP,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles42}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles43}$}
\end{longtable}
where $\beta_0,\beta_1,\beta_2\text{ and }\beta_3$ satisfy the conditions given in \cref{fig:euler}. In particular, $\beta_2\in[0,2\pi)$, so that the previous assumptions are preserved. This implies that we can assume additionally that if $\delta_4=0$ then $\delta_1=\delta_2=\delta_3=0$.
\phantomsection\label{delta1in0piifdelta3notzero}If $\delta_1\notin[0,\pi)$, then by \cref{prop:period}, we can ensure that it is in $[0,2\pi)$, and then if it is in $[\pi,2\pi)$, then, if $\delta_3\neq0$:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles80}$}\\\\
\overset{\text{\cref{prop:comb,prop:sum,commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles81}$}\\\\
\overset{\text{\cref{passagephasepihbcircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles86}$}\\\\
\overset{\text{\cref{commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles87}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles88}$}\\\\
\overset{\text{\cref{commctrl,prop:sum,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles89}$}\\\\
\overset{\text{\cref{commctrlphaseenhaut,commctrlphaseenhautP,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles90}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles91}$}
\end{longtable}
with $\delta_1-\pi\in[0,\pi)$. Moreover, since $\delta_3\neq0$, one has $2\pi-\delta_3\in[0,2\pi)$, so that the previous assumptions are preserved.
\phantomsection\label{delta1in0piifdelta3zero}And, still in the case where $\delta_1\in[\pi,2\pi)$, if $\delta_3=0$, then by assumption, $\delta_2=0$, and one has:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles92}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles93}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles93etdemi}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles94}$}\\\\
\overset{\text{\cref{passagephasepihbcircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles95}$}\\\\
\overset{\text{\cref{commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles96}$}\\\\
\overset{\text{\cref{prop:sum,commctrlphaseenhautP,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles97}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles98}$}
\end{longtable}
with $\delta_1-\pi\in[0,\pi)$.
\phantomsection\label{delta6zeroimpldelta5zero}If $\delta_6=0$ but $\delta_5\neq0$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles66}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles67}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles68}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles69}$}
\end{longtable}
Hence, we can assume additionally that if $\delta_6=0$ then $\delta_5=0$.
\phantomsection\label{delta6piimpldelta5zero}If $\delta_6=\pi$ but $\delta_5\neq0$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles70}$}\\\\
\overset{\text{\cref{prop:sum,mctrlX}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles71}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles72}$}\\\\
\overset{\text{\crefnosort{prop:CCX,antisymmetriecontrolee,antisymmetriesemicontrolee}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles73}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles74}$}\\\\
\overset{\text{\crefnosort{commctrl,prop:sum,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles75}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles76}$}\\\\
\overset{\text{\cref{prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles77}$}\\\\
\overset{\text{\cref{mctrlX,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles78}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles79}$}
\end{longtable}
Hence, we can assume additionally that if $\delta_6=\pi$ then $\delta_5=0$.
\phantomsection\label{delta2in0pi}If $\delta_2\notin[0,\pi)$, then by \cref{prop:period}, we can ensure that it is in $[0,2\pi)$, and then if it is in $[\pi,2\pi)$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles99}$}\\\\
\overset{\text{\cref{prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles100}$}\\\\
\overset{\text{\crefnosort{commctrl,prop:CP}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles101}$}\\\\
\overset{\text{\cref{symmetriesemicontrolee}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles102}$}\\\\
\overset{\text{\cref{passagephasepihbcircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles103}$}\\\\
\overset{\text{\cref{symmetriesemicontrolee,XX}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles104}$}\\\\
\overset{\text{\cref{passagephasepihbcircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles105}$}\\\\
\overset{\text{\cref{commctrl}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles106}$}\\\\
\overset{\text{\crefnosort{prop:CP,commctrl,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles107}$}\\\\
\overset{\text{\cref{commctrlphaseenhautP,commctrlphaseenhaut,prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles108}$}
\end{longtable}
with $\delta_2-\pi\in[0,\pi)$. Moreover, since $\delta_2\neq0$, by assumption $\delta_3\neq0$ and $\delta_4\neq0$, so that $2\pi-\delta_3$ and $2\pi-\delta_4$ are still in $[0,2\pi)$ and the previous assumptions are preserved.
\phantomsection\label{delta5in0pi}If $\delta_5\notin[0,\pi)$, then by \cref{prop:period}, we can ensure that it is in $[0,2\pi)$, and then if it is in $[\pi,2\pi)$, then:
\begin{longtable}{CL}
&\scalebox{0.8}{$\tikzfig{Euler3dright-multicontrolled-simp-deltas}$}\\\\
\overset{\text{\cref{prop:sum}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles109}$}\\\\
\overset{\text{\cref{passagephasepihbcircuits}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles110}$}\\\\
\overset{\text{\cref{prop:CP,commctrl,prop:sum,prop:comb}}}{=}&\scalebox{0.8}{$\tikzfig{anglesbonsintervalles111}$}
\end{longtable}
with $\delta_5-\pi\in[0,\pi)$. Moreover, since $\delta_5\neq0$, by assumption $\delta_6\neq0$, so that $2\pi-\delta_6\in[0,2\pi)$ and the previous assumptions are preserved.
\phantomsection\label{delta789in02pi}Finally, by \cref{prop:period} we can put $\delta_7$, $\delta_8$ and $\delta_9$ in $[0,2\pi)$ without modifying the other angles.
\subsection[Definition of sigma\_k,n,l]{Definition of $\sigma_{k,n,\ell}$}\label{defsigma}
$\sigma_{k,0,\ell}\coloneqq(\tikzfig{filcourt-s})^{\otimes 2^{k+\ell}}$ and $\forall n\geq2,\ \displaystyle\sigma_{k,n,\ell}\coloneqq
\sigma_{k,1,\ell+n-1}^n$, with
\[\sigma_{k,1,\ell}=\prod_{j=k+1}^{k+\ell}\mathcal P_j\mathcal Q_j\mathcal P_j\]
where
\begin{itemize}
\item given a family of $N$-mode circuits $C_A,...,C_B$,\quad $\displaystyle\prod_{i=A}^BC_i\coloneqq
(\ldots((C_B\circ C_{B-1})\circ C_{B-2})\circ \ldots )\circ C_A$,
\item $M\coloneqq k+\ell+1$
\item $\mathcal P_j$ is a raw optical circuit such that $\mathfrak G_n\circ\interp{\mathcal P_j}\circ\mathfrak G_n^{-1}=id_{j-1}\otimes\interp{\gcnot}\otimes id_{M-j-1}$, defined as\newline $\displaystyle \mathcal P_j\coloneqq\hspace{-1.5em}\prod_{\begin{scriptarray}{c}\\[-1.5em]b=0\\[-0.4em]b\bmod 4\in\{1,2\}\end{scriptarray}}^{2^{j}-1}\hspace{-1.5em}\prod_{a=0}^{2^{M-j-1}-1}\hspace{-1em}\upsilon_{M,j,b,a}$
\item $\mathcal Q_j$ is a raw optical circuit such that $\mathfrak G_n\circ\interp{\mathcal Q_j}\circ\mathfrak G_n^{-1}=id_{j-1}\otimes\interp{\gnotc}\otimes id_{M-j-1}$, defined as\newline $\displaystyle \mathcal Q_j\coloneqq\prod_{b=0}^{2^{j-1}-1}\ \ \prod_{a=0}^{2^{M-j-3}-1}\hspace{-1em}\upsilon_{M,j-1,b,a}$
\item $\upsilon_{N,i,b,a}$ is a raw optical circuit such that $\upsilon_{N,i,b,a}\equiv\scalebox{0.69}{\tikzfig{swap-ibaN-etroit}}$
. It is defined for any $N\geq1$, $i\in\{0,...,N-1\}$, $b\in\{0,...,2^i-1\}$ and $a\in\{0,...,2^{N-i-1}-1\}$,
by finite induction on $a$ by \[\upsilon_{N,i,b,0}\coloneqq\tikzfig{swapib2pNi1-PHOL},\] and for $a\in\{1,...,2^{N-i-1}-1\}$,
\[\upsilon_{N,i,b,a}\coloneqq s_{-a}\circ s_{+a}\circ\upsilon_{N,i,b,a-1}\circ s_{+a}\circ s_{-a}
,\] where $s_{+a}\coloneqq\tikzfig{swapib2pNi1plusa-PHOL}$\quad and\quad $s_{-a}\coloneqq\tikzfig{swapib2pNi1moinsa-PHOL}$
.
\end{itemize}
\end{document} |
\begin{document}
\begin{abstract}
We present a construction of two infinite graphs $G_1$ and $G_2$, and of an infinite set $\mathscr{F}$ of graphs such that $\mathscr{F}$ is an antichain with respect to the immersion relation and, for each graph $G$ in $\mathscr{F}$, both $G_1$ and $G_2$ are subgraphs of $G$, but no graph properly immersed in $G$ admits an immersion of $G_1$ and of $G_2$. This shows that the class of infinite graphs ordered by the immersion relation does not have the finite intertwine property.
\end{abstract}
\date{\today}
\title{A Note On Immersion Intertwines of Infinite Graphs}
\begin{section}{Introduction}
A \emph{graph} $G$ is a pair $(V(G),E(G))$ where $V(G)$, the set of vertices, is an arbitrary and possibly infinite set, and $E(G)$, the set of edges, is a subset of the set of two-element subsets of $V(G)$.
In particular, this definition implies that all graphs in this paper are simple, that is, with no loops or multiple edges.
The class of finite graphs will be denoted $\mathscr{G}_{<\infty}$ and the class of graphs whose vertex set is infinite will be denoted by $\mathscr{G}_\infty$.
Let $G$ and $H$ be graphs, and let $\mathscr{P}(G)$ denote the set of all nontrivial, finite paths of $G$.
We say $H$ is \emph{immersed} in $G$ if there is a map $\varphi : V(H) \cup E(H) \rightarrow V(G) \cup \mathscr{P}(G)$, sometimes abbreviated as $\varphi : H \rightarrow G$, such that:
\begin{enumerate}
\item if $v\in V(H)$, then $\varphi (v)\in V(G)$;
\item if $v$ and $v'$ are distinct vertices of $H$, then $\varphi(v) \neq \varphi (v') $;
\item if $e = \{v,v'\} \in E(H)$, then $\varphi(e) \in \mathscr{P}(G)$ and the path $\varphi (e)$ connects $\varphi(v)$ with $\varphi(v')$;
\item if $e$ and $e'$ are distinct edges of $H$, then the paths $\varphi(e)$ and $\varphi(e')$ are edge-disjoint; and
\item if $e=\{v,v'\}\in E(H)$ and $v''$ is a vertex of $H$ other than $v$ and $v'$, then $\varphi (v'') \notin V(\varphi(e))$.
\end{enumerate}
We call $\varphi$ an \emph{immersion} and write $H \im G$. It is easy to prove (see~\cite{siamd}) that the relation $\im$ is transitive.
If $C$ is a subgraph of $H$, then the restriction of $\varphi$ to $V(C)\cup E(C)$ will be abbreviated by $\varphi|_C$.
If $\varphi |_{V(H)}$ is a bijection such that two vertices, $v$ and $v'$, of $H$ are adjacent if and only if their images, $\varphi(v)$ and $\varphi(v')$, are adjacent in $G$, then we say that $\varphi$ induces an isomorphism between $H$ and $G$; otherwise $\varphi$ is \emph{proper}.
If $H = G$, then $\varphi$ is a \emph{self-immersion}, and, if additionally, it induces the identity map, then it is \emph{trivial}.
It is worth noting that immersion, as defined above, is sometimes called \emph{strong immersion}.
Let $S$ be a possibly infinite set of pairwise edge-disjoint paths in a graph $G$.
We say that $S$ is \emph{liftable} if no end-vertex of path in $S$ is an internal vertex of another path in $S$.
The operation of \emph{lifting} $S$ consists of deleting all internal vertices of all paths in $S$, and adding edges joining every pair of non-adjacent vertices of $G$ that are end-vertices of the same path in $S$.
It is easy to see that a graph $H$ is immersed in $G$ if and only if $H$ is isomorphic to a graph obtained from $G$ by deleting a set $V$ of vertices, deleting a set $E$ of edges, and then lifting a liftable set $S$ of paths.
Furthermore, a self-immersion of $G$ is proper if and only if at least one of the sets $V$, $E$, and $S$ is nonempty.
Given a graph $G$, a {\em blob} is a maximal 2-edge-connected subgraph of $G$. Note that if a graph is 2-edge-connected, the graph itself is also a blob.
An easy lemma about the immersion relation can be stated as follows.
\begin{lemma}
\label{immersionblob}
Let $H\im G$ via the immersion $\varphi$ and let $C$ be a blob of $H$. Then there is a blob $D$ of $G$ such that $C\im D$ via the immersion $\varphi|_C$.
\end{lemma}
A pair $(\mathscr{G}, \leq)$, where $\mathscr{G}$ is a class of graphs and $\leq$ is a binary relation on $\mathscr{G}$, is called a \emph{quasi-order} if the relation $\leq$ is both reflexive and transitive.
A quasi-order $(\mathscr{G}, \leq)$ is a \emph{well-quasi-order} if it admits no infinite antichains and no infinite descending chains.
Suppose $(\mathscr{G}, \leq)$ is a quasi-order and $G_1$ and $G_2$ are two elements of $\mathscr{G}$.
An \emph{intertwine} of $G_1$ and $G_2$ is an element $G$ of $\mathscr{G}$ satisfying the following conditions:
\begin{itemize}
\item $G_1 \leq G$ and $G_2 \leq G$, and
\item if $G'\leq G$ and $G \nleq G'$, then $G_1 \nleq G$ or $G_2 \nleq G$.
\end{itemize}
The class of all intertwines of $G_1$ and $G_2$ is denoted by $\mathscr{I}_{\leq}(G_1,G_2)$.
A quasi-order $(\mathscr{G},\leq)$ satisfies the \emph{finite intertwine property} if for every pair $G_1$ and $G_2$ of elements of $\mathscr{G}$, the class of intertwines $\mathscr{I}_{\leq}(G_1, G_2)$ has no infinite antichains.
It is clear that if $(\mathscr{G},\leq)$ is a well-quasi-order, then it also satisfies the finite intertwine property.
However, it is well known that the converse is not true; for example, see~\cite{anoioig}.
Nash-Williams conjectured, and Robertson and Seymour later proved~\cite{gm23nwic} that
$(\mathscr{G}_{<\infty}, \im)$ is a well-quasi-order, and so it follows that
$(\mathscr{G}_{<\infty}, \im)$ satisfies the finite intertwine property.
In~\cite{anoioig}, the second author showed that $(\mathscr{G}_\infty , \m)$, where $\m$ denotes the minor relation, does not satisfy the finite intertwine property.
Andreae showed \cite{oioug} that $(\mathscr{G}_\infty , \im)$ is not a well-quasi-order.
In a result analogous to \cite{anoioig}, we strengthen Andreae's result by showing that $(\mathscr{G}_\infty , \im)$ does not satisfy the finite intertwine property.
In particular, we construct two graphs $G_1$ and $G_2$, and an infinite class $\mathscr{F}$ in $\mathscr{G}_\infty $ such that:
\begin{enumerate}
\item[(IT1)] $\mathscr{F}$ is an immersion antichain;
\item[(IT2)] every graph in $\mathscr{F}$ is connected;
\item[(IT3)] both $G_1$ and $G_2$ are subgraphs of each graph in $\mathscr{F}$;
\item[(IT4)] if $G'$ is properly immersed in a graph $G$ in $\mathscr{F}$, then $G_1 \nim G'$ or $G_2 \nim G'$.
\end{enumerate}
Note that~(IT3) implies that $G_1$ and $G_2$ are immersed in $G$.
Hence, the existence of graphs $G_1$, $G_2$ and a class of graphs $\mathscr{F}$ satisfying~(IT1)--(IT4) implies the following statement, which is the main result of the paper.
\begin{theorem}
\label{mainthm}
The quasi-order $(\mathscr{G}_{\infty}, \im)$ does not satisfy the finite intertwine property.
\end{theorem}
\end{section}
\begin{section}{The Construction}
We will exhibit two graphs $G_1$ and $G_2$ in $\mathscr{G}_{\infty}$ such that $\mathscr{I}_{\im}(G_1, G_2)$ is infinite. The construction of $G_1$ and $G_2$ begins with the following results, which are immediate consequences of, respectively, Lemmas 3 and 4, and Lemmas 1 and 2 of \cite{osioig}.
\begin{theorem}
\label{thm1}
There is an infinite set $\mathscr{H}$ of pairwise-disjoint infinite blobs such that $|H| \leq |\mathscr{H}|$ for all $H\in \mathscr{H}$, and $\mathscr{H}$ forms an immersion antichain.
\end{theorem}
\begin{theorem}
\label{thm2}
Given an immersion antichain $\mathscr{H}$ of pairwise-disjoint infinite blobs such that $|H| \leq |\mathscr{H}|$ for all $H\in \mathscr{H}$, there is a connected graph $G$ such that the set of blobs of $G$ is $\mathscr{H}$ and $G$ admits no self-immersion except for the trivial one.
\end{theorem}
Let $\mathscr{H}$ be an antichain as described in Theorem~\ref{thm1}.
Partition $\mathscr{H}$ into countably many sets $\{\mathscr{H}_i\}_{i\in \Z}$ with the cardinality of each $\mathscr{H}_i$ equal to $|\mathscr{H}|$.
Then, by Theorem~\ref{thm2}, for each $i\in \Z$, there is a connected graph $B_i$ whose set of blobs is $\mathscr{H}_i$, and that admits no proper self-immersion.
Furthermore, Lemma \ref{immersionblob} implies that if $i$ and $j$ are distinct integers, then $B_i \nim B_j$, as no blob of $B_i$ is immersed in a blob of $B_j$. Therefore, the set of graphs $\{B_i\}_{i\in \mathbb{Z}}$ is an immersion antichain.
For each graph $B_i$, label one vertex $u_i$.
Let $P$ be a two-way infinite path with vertices labeled $\{v_i\}_{i\in \Z}$ such that, for each integer $i$, the vertex $v_i$ is adjacent to $v_{i+1}$ and $v_{i-1}$.
We construct the graph $G_1$ by taking the disjoint union of $P$ and the graphs $B_i$ for which $i$ is odd, and then identifying the vertices $u_i$ and $v_j$ for $i=j$.
Similarly, we construct the graph $G_2$ by taking the disjoint union of $P$ and the graphs $B_i$ for which $i$ is even, and then identifying the vertices $u_i$ and $v_j$ for $i=j$.
Now let $j$ be an integer.
Take the disjoint union of $G_1$ and all the graphs $B_i$ for which $i$ is even.
Then, for each even integer $i$, identify the vertex $v_i$ of $G_1$ with the vertex $u_{i+2j}$ of the graph $B_{i+2j}$.
Let $F_j$ be the resulting graph (see Figure~\ref{f1}) and define $\mathscr{F}$ as the set $\{F_j\}_{j\in \mathbb{Z}}$.
\end{section}
\begin{figure}
\caption{The graph $F_j$}
\label{f1}
\end{figure}
The following lemma immediately implies our main result, Theorem~\ref{mainthm}.
\begin{lemma}
The set of graphs $\mathscr{F}=\{F_j\}_{j\in \mathbb{Z}}$ is an immersion antichain. Furthermore, each $F_j \in \mathscr{F}$ is an immersion intertwine of the graphs $G_1$ and $G_2$.
\end{lemma}
\begin{proof}
Let $j$ be an integer.
It is easy to see that $F_j$ satisfies (IT2) and (IT3).
Therefore, in order to show that $F_j$ is an immersion intertwine of $G_1$ and $G_2$, it suffices to prove that it also satisfies (IT4).
Suppose, for contradiction, that $F'_j$ is a graph that is properly immersed in $F_j$ via a map $\varphi$, and both $G_1$ and $G_2$ are immersed in $F'_j$.
Then we can obtain $F'_j$ from $F_j$ by deleting a set of vertices $V$, deleting a set of edges $E$, and then lifting a liftable set of paths $S$, with at least one of these sets being nonempty. We consider two cases depending on whether there is an integer $i$ for which $B_i$ meets $V\cup E \cup S$.
First, assume that no $B_i$ meets $V\cup E \cup S$.
Then the sets $V$ and $S$ are empty, as all the vertices of $F_j$ are contained in the subgraphs $\{B_n\}_{n\in \Z}$, and $E$ consists of some edges of $P$.
Suppose the edge~$e=\{v_k, v_{k+1}\}$ is in $E$ where $k$ is odd; the argument is symmetric when $k$ is even.
The graph $F_j \setminus e$ has exactly two components, with the subgraphs $B_k$ and $B_{k+2}$ in distinct components.
Label the component containing $B_k$ as $C_1$ and the component containing $B_{k+2}$ as $C_2$.
Let $A$ be a blob of $B_k$.
As $A$ and each blob of $C_2$ are members of the antichain $\mathscr{A}$, by Lemma~\ref{immersionblob}, we have $A\nim C_2$.
Hence, by transitivity, $B_k\nim C_2$. It follows similarly that $B_{k+2}\nim C_1$.
But as $G_1$ is connected and the only components of $F_j\setminus e$ are $C_1$ and $C_2$, we have that $G_1 \nim F_j \setminus e$.
Furthermore, as $F'_j \im F_j \setminus e$, by transitivity, $G_1 \nim F'_j$; a contradiction.
Now suppose that, for some odd integer $i$, the graph $B_i$ meets $V\cup E \cup S$; again, the argument is symmetric if $i$ is even.
As $G_1$ is immersed in $F'_j$, so is $B_i$. Let $T$ be the subgraph of $F'_j$ induced by $\varphi^{-1} (V(B_i)\cup \mathscr{P}(B_i))$, and let $\psi$ be the immersion of $B_i$ into $F'_j$.
As $B_i$ admits no proper self-immersion, there must be some vertex $v$ of $B_i$ such that $\psi(v)$ is a vertex of $F'_j - T$.
Let $A_v$ be the blob of $B_i$ containing $v$.
By Lemma~\ref{immersionblob}, the blob $A_v$ is immersed in some blob of $F'_j - T$. But, again by Lemma~\ref{immersionblob}, each blob of $F'_j - T$ is immersed in a graph of the antichain $\mathscr{A} \setminus \{A_v\}$. So $A_v$ cannot be immersed in $F'_j - T$. Therefore, $B_i$ is not immersed in $F'_j$ and neither is $G_1$.
Hence, $\mathscr{F}$ satisfies the condition (IT4).
To show that $\mathscr{F}$ is an antichain in $(\mathscr{G}_\infty, \im)$, suppose that $F_i$ is immersed in $F_j$ for some distinct integers $i$ and $j$.
By construction, $F_i$ and $F_j$ are not isomorphic.
Therefore, $F_i$ is properly immersed in the intertwine $F_j$ and so either $G_1 \nim F_i$ or $G_2 \nim F_i$.
But both $G_1$ and $G_2$ are immersed in $F_i$ by construction; a contradiction. The conclusion follows.
\end{proof}
The graphs $\{B_i\}_{i\in \Z}$ used in our construction, whose existence was proved in~\cite{oioug}, have vertex sets of very large cardinality. In fact, the cardinal in question is the first limit cardinal greater than the cardinality of the continuum. It is not known whether the class of graphs of smaller cardinality ordered by the strong immersion relation is a well-quasi-ordering, whether it has the finite intertwine property, and whether there exists a infinite graph of smaller cardinality that admits only the trivial self-immersion.
\end{document} |
\begin{document}
\title[SUBALGEBRAS IN POSITIVE CHARACTERISTIC]
{SUBALGEBRAS OF THE POLYNOMIAL ALGEBRA IN POSITIVE CHARACTERISTIC
AND THE JACOBIAN}
\author{Alexey~V.~Gavrilov}
\email{[email protected]}
\maketitle
{\small
Let $\ks$ be a field of characteristic $p>0$ and
$R$ be a subalgebra of $\ks[X]=\ks[x_1,\dots,x_n]$. Let
$J(R)$ be the ideal in $\ks[X]$ defined by
$J(R)\Omega_{\ks[X]/\ks}^n=\ks[X]\Omega_{R/\ks}^n$. It is shown that if
it is a principal ideal then $J(R)^q\subset R[x_1^p,\dots,x_n^p]$,
where $q=\frac{p^n(p-1)}{2}$.}
\newline
{\bf Key words:} Polynomial ring; Jacobian; generalized Wronskian.
\newline
{\bf 2000 Mathematical Subject Classification:}13F20;13N15.
\section{INTRODUCTION}
Let $\ks$ be a field and $\ks[X]=\ks[x_1,\dots,x_n]$ be the polynomial ring.
For the most of the paper the number $n\ge 1$ and the basis
$X=\{x_1,\dots,x_n\}$ are fixed.
Let $R$ be a subalgebra of $\ks[X]$.
Denote by $J(R)$ the ideal in $\ks[X]$ generated by the Jacobians
of sets of elements of $R$ (a formal definition will be given below).
The main result is the following theorem.
{\bf Theorem}
{\it Let $\ks$ be a field of characteristic $p>0$ and
$R$ be a subalgebra of $\ks[X]$. If $J(R)$ is a principal ideal then
$$J(R)^{q}\subset R[X^p],$$
where $R[X^p]=R[x_1^p,\dots,x_n^p]$ and $q=\frac{p^n(p-1)}{2}$.
}
Presumably the statement holds for a nonprincipal ideal as well.
However, a proof of this conjecture probably requires another technique.
{\bf Corollary }
{\it
Let $\ks$ be a field of characteristic $p>0$ and
$R$ be a subalgebra of $\ks[X]$. If $J(R)=\ks[X]$ then
$$R[X^p]=\ks[X].$$
}
Nousiainen [2] proved this in the case $R=\ks[F]=\ks[f_1,\dots,f_n]$
(see also [1]). In this case the condition $J(R)=\ks[X]$ is
equivalent to $j(F)\in\ks^{\times}$, where $j(F)$ is the Jacobian of
the polynomials $f_1,\dots,f_n$. Thus, Nousiainen's result is a
positive characteristic analogue of the famous Jacobian conjecture:
if $\cha(\ks)=0$ and $j(F)\in\ks^{\times}$ then $\ks[F]=\ks[X]$. The
zero characteristic analogue of Corollary is obviously false.
(Consider, for example, the subalgebra $R=\ks[t-t^2,t-t^3]$ of
$\ks[t]$. Then $R\neq\ks[t]$ but $J(R)=\ks[t]$).
Nousiainen's method is based on properties of the derivations
$\frac{\partial}{\partial f_i}\in\Der(\ks[X])$, which are the
natural derivations of $\ks[F]$ extended to $\ks[X]$. Probably his
method can be applied to a more general case as well. However, our
approach is based on the calculation of generalized Wronskians of a
special kind. This calculation may be an interesting result in
itself.
\section{PRELIMINARIES AND NOTATION}
An element $\alpha=(\alpha_1,\dots,\alpha_n)$ of $\Ns^n$, where
$\Ns$ denotes the set of non-negative integers, is called a multiindex.
For $F\in\ks[X]^n$ and two multiindices
$\alpha,\beta\in\Ns^n$ we will use the following notation
$$|\alpha|=\sum_{i=1}^n\alpha_i,\,\alpha!=\prod_{i=1}^n\alpha_i!,\,
{\alpha\choose\beta}=\prod_{i=1}^n{\alpha_i\choose\beta_i},$$
$$X^{\alpha}=\prod_{i=1}^n x_i^{\alpha_i},\,
F^{\alpha}=\prod_{i=1}^n f_i^{\alpha_i},\,
\partial^{\alpha}=\prod_{i=1}^n \partial_i^{\alpha_i},$$
where $\partial_i=\frac{\partial}{\partial x_i}\in\Der(\ks[X]).$
In several places multinomial coefficients will appear, denoted by
${\alpha\choose\beta^1\dots\beta^k}$, where
$\alpha,\beta^1,\dots,\beta^k\in\Ns^n$ and
$\alpha=\beta^1+\dots+\beta^k$. They are defined exactly by the same
way as the binomial ones. The set of multiindices possess the
partial order; by definition, $\alpha\le\beta$ iff
$\alpha_i\le\beta_i$ for all $1\le i\le n$. For the sake of
convenience we introduce the "diagonal" intervals
$[k,m]=\{\alpha\in\Ns^n:k\le\alpha_i\le m, 1\le i\le n\}$, where
$k,m\in\Ns$.
Let $R$ be a subalgebra of $\ks[X]$. Since $\Omega_{\ks[X]/\ks}^n$
is a free cyclic module, we can make the following definition.
{\bf Definition 1}
{\it
Let $R$ be a subalgebra of $\ks[X]$, where $\ks$ is a field.
The Jacobian ideal of $R$ is the ideal
$J(R)$ in $\ks[X]$ defined by the equality
$$J(R)\Omega_{\ks[X]/\ks}^n=\ks[X]\Omega_{R/\ks}^n,$$
where $\Omega_{R/\ks}^n$ is considered a submodule of $\Omega_{\ks[X]/\ks}^n$
over $R$.
}
The exact meaning of the words "is considered a submodule" is that we write
$\ks[X]\Omega_{R/\ks}^n$ instead of
$\ks[X]{\rm Im}(\Omega_{R/\ks}^n\to\Omega_{\ks[X]/\ks}^n)$. This is a slight
abuse of notation, because the natural $R$ - module homomorphism
$\Omega_{R/\ks}^n\to\Omega_{\ks[X]/\ks}^n$ is not injective in general.
There is also a more explicit definition. For $F\in\ks[X]^n$, the
Jacobian matrix and the Jacobian are defined by
$$JF=\left\|\frac{\partial f_i}{\partial x_j}\right\|_{1\le i,j\le n}\in
M(n,\ks[X]),\,
j(F)=\det JF\in\ks[X].$$
The module $\ks[X]\Omega_{R/\ks}^n$ is generated by
$df_1\wedge\dots\wedge df_n=j(F)dx_1\wedge\dots\wedge dx_n$, where
$F=(f_1,\dots,f_n)\in R^n$. Thus
$$J(R)=\langle\{j(F):F\in R^n\}\rangle,$$
where $\langle S\rangle$ denotes the ideal in $\ks[X]$ generated by a set $S$.
It is an easy consequence of the chain rule that the Jacobian ideal
of a subalgebra generated by $n$ polynomials is a principal ideal:
$$J(\ks[F])=\ks[X]j(F),\,F\in\ks[X]^n.$$
Clearly, the ring $\ks[X]$ is a free module over $\ks[X^p]$ of rank
$p^n$: the set of monomials $\{X^{\alpha}:\alpha\in[0,p-1]\}$ is a
natural basis of this module. This construction became important
when $\cha(\ks)=p$ (note that in this case $\ks[X^p]$ does not
depend on the choice of generators of $\ks[X]$ i.e. it is an
invariant).
{\bf Definition 2}
{\it
Let $\ks$ be a field of characteristic $p>0$ and $F\in\ks[X]^n$.
The matrix $U(F)\in M(p^n,\ks[X^p])$ is defined by
$$F^{\alpha}=\sum_{\beta\in[0,p-1]}U(F)_{\alpha\beta}X^{\beta},
\,\alpha\in[0,p-1].$$
}
\section{GENERALIZED WRONSKIANS}
In this section we compute generalized Wronskians of a special form.
The key tool for this computation is the following simple lemma.
{\bf Lemma 1}
{\it Let $R$ be a ring and $f\in R$. If $D_1,\dots,D_l\in\Der(R)$
and if $m\ge l\ge 0$ then
$$\sum_{k=0}^m{m\choose k}(-f)^{m-k}D_1\dots D_l f^k
=\begin{cases}
0 & {\rm if}\, m>l \\
l!\prod_{k=1}^lD_kf & {\rm if}\, m=l
\end{cases}
\eqno{(1)}$$
}
In the case $l=0$ there are no derivations and the formula becomes
$$\sum_{k=0}^m{m\choose k}(-f)^{m-k} f^k
=\begin{cases}
0 & {\rm if}\, m>0 \\
1 & {\rm if}\, m=0
\end{cases}
$$
which is obviously true.
\newline
{\it Proof}
Denote
$$S_{m,l}=\sum_{k=0}^m{m\choose k}(-f)^{m-k}D_l\dots D_1 f^k.$$
This sum coincides with the left hand side of (1), except for the
reverse order of the derivations.
We have $S_{0,0}=1$ and $S_{m,0}=0,\,m>0$.
The following equality can easily be verified
$$S_{m+1,l+1}=D_{l+1}S_{m+1,l}+(m+1)(D_{l+1}f)S_{m,l}.$$
By induction on $l$, for $m>l$ we have $S_{m,l}=0$, and
$$S_{l,l}=l(D_{l}f)S_{l-1,l-1}=
l!\prod_{k=1}^lD_kf.$$
\qed
Let $R$ be a ring. Denote by $R[Z_{ij}]$ the polynomial ring in the
$n^2$ indeterminates $Z_{ij},\,1\le i,j\le n$. If $h\in R[Z_{ij}]$ and
$A\in M(n, R)$ then $h(A)$ denotes the result of the substitution
$Z_{ij}\mapsto A_{ij}$.
{\bf Proposition 1} {\it For any $r\ge 1$ there exists the
homogeneous polynomial $H_r\in \Zs[Z_{ij}]$ of degree
$\frac{nr^n(r-1)}{2}$ with the following property. Let $R$ be a ring
with derivations $D_1,\dots,D_n\in\Der(R)$. If $[D_i,D_j]=0$, for
all $i,j$, then for any $F=(f_1,\dots,f_n)\in R^n$ the following
equality holds
$$\det W=H_r(JF),\eqno{(2)}$$
where $W\in M(r^n,R)$ and $JF\in M(n,R)$ are defined by
$$W_{\alpha\beta}=D^{\alpha}F^{\beta},\,\alpha,\beta\in[0,r-1];\,
(JF)_{ij}=D_if_j,\,1\le i,j\le n.$$
}
Here $JF$ is an obvious generalization of the Jacobian matrix. The
determinant $\det W$ is a generalized Wronskian of the polynomials
$F^{\beta}$.
\newline
{\it Proof}
By the Leibniz formula,
$$W_{\alpha\beta}=\sum
{\alpha\choose\theta^1\dots\theta^n}\prod_{i=1}^n(D^{\theta^i}
f_i^{\beta_i}),$$
where $\theta^1,\dots,\theta^n\in\Ns^n$ and the sum is taken over the
multiindices satisfying the equality
$\sum_{i=1}^n\theta^i=\alpha$.
Let $T\in M(r^n,R)$ be determined by
$$T_{\alpha\beta}={\beta\choose\alpha}(-F)^{\beta-\alpha},\,
\alpha,\beta\in[0,r-1].$$ Let $W^{\prime}=WT\in M(r^n,R)$. Then
$$W_{\alpha\beta}^{\prime}=\sum_{\gamma}W_{\alpha\gamma}T_{\gamma\beta}=
\sum{\alpha\choose\theta^1\dots\theta^n}\prod_{i=1}^n
S_i(\beta_i,\theta^i),\eqno{(3)}$$
where
$$S_i(m,\theta)=\sum_{k=0}^m{m\choose k}(-f_i)^{m-k}D^{\theta}f_i^k.$$
By Lemma 1, if $m\ge|\theta|$ then
$$S_i(m,\theta)
=\begin{cases}
0 & {\rm if}\, m>|\theta| \\
m!\prod_{j=1}^n(D_j f_i)^{\theta_j} & {\rm if}\, m=|\theta|
\end{cases}
$$
Thus if the product in the right hand side of (3) is not zero
then $\beta_i\le|\theta^i|$ for all $1\le i\le n$. The latter implies
$|\beta|\le|\alpha|$. It follows that if $|\alpha|<|\beta|$ then
$W_{\alpha\beta}^{\prime}=0$. In the case $|\alpha|=|\beta|$ the product
is zero unless $\beta_i=|\theta^i|,\,1\le i\le n$. Thus, if
$|\alpha|=|\beta|$ then
$$W_{\alpha\beta}^{\prime}=\beta!
\sum_{|\theta^1|=\beta_1}\dots\sum_{|\theta^n|=\beta_n}
{\alpha\choose\theta^1\dots\theta^n}\prod_{i=1}^n\prod_{j=1}^n
(D_jf_i)^{\theta^i_j}.\eqno{(4)}$$
Put the multiindices in a total order compatible with the partial
order (e.g. in the lexicographic order). Then the matrix $T$ becomes
upper triangular with the unit diagonal, hence $\det T=1$. The
matrix $W^{\prime}$ becomes block lower triangular, hence $\det
W^{\prime}$ is equal to the product of the $nr-n+1$ determinants of
the blocks. Each block determinant
$\det\left\|W_{\alpha\beta}^{\prime}\right\|_{|\alpha|=|\beta|=l}$
(where $0\le l\le nr-n$) is by (4) a homogeneous polynomial in the
variables $D_if_j$ of degree $ls_l$, where $s_l$ is the size of the
block. Clearly $s_l=\#\{\alpha\in\Ns^n:\alpha\in[0,r-1],
|\alpha|=l\}$. The determinant $\det W=\det W^{\prime}$ is then a
homogeneous polynomial of degree
$$\sum_{l=0}^{nr-n}ls_l=\sum_{\alpha\in[0,r-1]}|\alpha|=\frac{nr^n(r-1)}{2}.$$
One can see from (4) that all the coefficients of this polynomial are integers.
\qed
When $n=1$, the determinant in the left hand side of (2) is a common
Wronskian $W(1,f,\dots,f^{r-1})$. In this case $H_r$ is a polynomial
in one variable, which can be easily computed. The matrix
$W^{\prime}\in M(r,R)$ is a triangular matrix with the diagonal
elements $W_{kk}^{\prime}=k!(DF)^k,\,0\le k\le r-1$, hence $\det
W=\det W^{\prime}$ is equal to the product of these elements. So, we
have the following equality
$$\det\left\|D^kf^l\right\|_{0\le k,l\le r-1}=(Df)^{\frac{r(r-1)}{2}}
\prod_{k=1}^{r-1} k!\eqno{(5)}$$ where $f\in R$ and $D\in\Der(R)$.
The formula (5) may be considered a consequence of the known
Wronskian chain rule [3,Part Seven, Ex. 56].
\section{ THE DETERMINANT}
For the rest of the paper $\ks$ is a field of fixed characteristic $p>0$, and
$q=\frac{p^n(p-1)}{2}$.
For $F\in\ks[X]^n$, denote
$$\Delta(F)=\det U(F)\in\ks[X^p].$$
Our aim is to compute this determinant.
Denote by $\phi_F$ the algebra endomorphism
$$\phi_F\in\End(\ks[X]),\,\phi_F:x_i\mapsto f_i,\,1\le i\le n.$$
{\bf Lemma 2}
{\it Let $F,G\in\ks[X]^n$. Let
$\phi_F G=(\phi_FG_1,\dots,\phi_FG_n)\in\ks[X]^n$.
Then
$$\Delta(\phi_FG)=(\phi_F\Delta(G))\cdot\Delta(F).\eqno{(6)}$$
}
\newline
{\it Proof}
By definition,
$$G^{\alpha}=\sum_{\gamma}U(G)_{\alpha\gamma}X^{\gamma},\,\alpha\in[0,p-1].$$
Applying the endomorphism $\phi_F$ to the both sides of this equality, we get
$$(\phi_FG)^{\alpha}=\sum_{\gamma}(\phi_FU(G)_{\alpha\gamma})F^{\gamma}=
\sum_{\beta\gamma}(\phi_FU(G)_{\alpha\gamma})U(F)_{\gamma\beta}X^{\beta}.$$
On the other hand,
$$(\phi_FG)^{\alpha}=\sum_{\beta}U(\phi_FG)_{\alpha\beta}X^{\beta}.$$
As the monomials $X^{\beta}$ form a basis, the coefficients are the same:
$$U(\phi_FG)=(\phi_FU(G))U(F).$$
Taking the determinant, we have (6). \qed
If $F\in\ks[X]^n$ consists of linear forms, then
$$F_i=\sum_{j=1}^n A_{ij}X_j,\,1\le i\le n$$
for some matrix $A\in M(n,\ks)$. We write this as
$$F=AX;$$
in this notation $X$ and $F$ are considered column vectors.
{\bf Lemma 3}
{\it Let $A\in M(n,\ks)$. If $F=AX$, then
$$\Delta(F)=(\det A)^{q}.\eqno{(7)}$$
}
\newline
{\it Proof}
Let $A,B\in M(n,\ks)$. Let $G=BX$ and $F=AX$. Then $\phi_FG=BAX$.
One can see that the elements of the matrix $U(BX)$ belong to the
field $\ks$. It follows that $\phi_F\Delta(BX)=\Delta(BX)$, and by
(6) we have
$$\Delta(BAX)=\Delta(BX)\Delta(AX).$$
It is well known that any matrix over a field is a product of diagonal matrices and
elementary ones. Thus, because of the latter equality,
it is sufficient to prove (7) for diagonal and elementary matrices.
If $A$ is elementary, then for some $j\neq k$ we have
$f_j=x_j+\lambda x_k$, where $\lambda\in\ks$, and
$f_i=x_i,\,i\neq j$. Then $U(F)$ is a (upper or lower) triangular matrix
with the unit diagonal, hence $\Delta(F)=\det A=1$, and (7) holds.
If $A$ is diagonal then $f_i=A_{ii}x_i,\,1\le i\le n$. In this case
$U(F)$ is diagonal as well and
$U(F)_{\alpha\alpha}=\prod_{i=1}^nA_{ii}^{\alpha_i}$. The equality (7)
can be easily checked.
\qed
{\bf Lemma 4}
{\it Let $F\in\ks[X]^n$ and
$W=\left\|\partial^{\alpha}F^{\beta}\right\|_{\alpha,\beta\in[0,p-1]}$.
Then
$$\det W=
c_p^n\Delta(F),$$
where $c_p=\prod_{k=1}^{p-1}k!\in\ks^{\times}$.
}
\newline
{\it Proof}
Denote $Q=\left\|\partial^{\alpha}X^{\beta}\right\|
_{\alpha,\beta\in[0,p-1]}\in M(p^n,\ks[X])$. This is a Kronecker
product
$$Q=Q_1\otimes\dots\otimes Q_n;\,
Q_i=\left\|\partial_i^{a}x^{b}_i\right\| _{0\le a,b\le p-1}\in
M(p,\ks[x_i]),1\le i\le n.$$ Applying the equality (5) to the rings
$\ks[x_i]$, we have $\det Q_i=c_p, 1\le i\le n$. Then
$$\det Q=\prod_{i=1}^n(\det Q_i)^{p^{n-1}}=c_p^{np^{n-1}}=c_p^n.$$
The elements of $U(F)$ belong to the kernels of derivations $\partial_i$, hence
$$\partial^{\alpha}F^{\beta}=\sum_{\gamma}U(F)_{\beta\gamma}
\partial^{\alpha}X^{\gamma}.$$
This can be written in the matrix notation as
$$W=QU(F)^{\rm T}.$$ The formula is a consequence.
\qed
\section{PROOF OF THE THEOREM}
{\bf Proposition 2}
{\it
Let $F\in\ks[X]^n$. Then
$$\Delta(F)=j(F)^{q}.$$
}
\newline
{\it Proof}
Let $W=\left\|\partial^{\alpha}F^{\beta}\right\|
_{\alpha,\beta\in[0,p-1]}\in M(p^n,\ks[X])$.
By Lemma 4 and Proposition 1,
$$\det W=c_p^{n}\Delta(F)=H_{p}(JF).$$
If $A\in M(n,\ks)$ and $F=AX$ then $JF=A$, hence
$$H_{p}(A)=c_p^{n}(\det A)^{q}$$
by Lemma 3. Without loss of generality, $\ks$ is infinite.
The latter equality holds for any matrix, hence it is valid as a formal
equality in the ring $\ks[Z_{ij}]$. Thus
$$H_{p}(JF)=c_p^{n}(\det JF)^{q}=c_p^{n}j(F)^{q}.$$
\qed
We have the following corollary, proved first by Nousiainen [2].
{\bf Corollary}
{\it
Let $\ks$ be a field of characteristic $p>0$ and
$F\in\ks[X]^n$. Then the set
$\{F^{\alpha}:\alpha\in[0,p-1]\}$ is a basis of $\ks[X]$
over $\ks[X^p]$ if and only if $j(F)\in\ks^{\times}$.
}
{\bf Proposition 3}
{\it
Let $F\in\ks[X]^n$. Then
$$\ks[X]j(F)^{q}
\subset\ks[X^p][F].$$
}
\newline
{\it Proof}
From linear algebra we have
$$\Delta(F)U(F)^{-1}={\rm adj}\,U(F)\in M(p^n,\ks[X^p]).$$
Thus
$$j(F)^qX^{\alpha}=
\Delta(F)X^{\alpha}=\Delta(F)\sum_{\beta}U(F)^{-1}_{\alpha\beta}F^{\beta}
\in\ks[X^p][F]$$
for any multiindex $\alpha\in[0,p-1]$.
The set $\{X^{\alpha}\}$ is a basis of $\ks[X]$ over $\ks[X^p]$, hence
the inclusion follows.
\qed
\newline
{\it Proof of Theorem}
We have
$$J(R)=\langle\{j(F):F\in R^n\}\rangle=\langle P\rangle,\,P\in\ks[X].$$
If $P=0$, the statement is trivial. Suppose $P\neq 0$.
Since $P\in J(R)$, there exists a number $m\ge 1$, such that
$$P=\sum_{i=1}^mg_ij(F_i),$$
where $g_i\in\ks[X],\,F_i\in R^n,1\le i\le m$.
On the other hand, $J(R)\subset\langle P\rangle$, hence
$$\mu_i=j(F_i)/P\in\ks[X],\,1\le i\le m.$$
Consider the following two ideals:
$$J=\{f\in \ks[X]: \ks[X]f\subset R[X^p]\};\,
I=\langle \mu_1^q,\dots,\mu_m^q\rangle.$$
By Proposition 3, $\mu_i^qP^q=j(F_i)^q\in J$ for all $1\le i\le m$, hence
$$IP^q\subset J.$$
Raising the equality $\sum_{i=1}^mg_i\mu_i=1$ to the power $qm$, it
follows that $1\in I$, whence $P^q\in J$. \qed
\section*{ACKNOWLEDGMENT}
The author thanks Prof. Arno van den Essen for giving him the
information about the Nousiainen's preprint.
\section*{REFERENCES}
[1] H. Bass, E.H. Connell, D. Wright. The Jacobian Conjecture:
Reduction of degree and formal expansion of the inverse, Bull.
A.M.S. 7(2)(1982) 287-330.
[2] P. Nousiainen. On the Jacobian problem in positive
characteristic. Pennsylvania State Univ. preprint. 1981
[3] G. Polya, G. Szeg\"o. Problems and Theorems in Analysis, Vol II.
Springer-Verlag. 1976
\end{document} |
\begin{document}
\title{Sasakian Manifold and $*$-Ricci Tensor}
\author{Venkatesha $\cdot$ Aruna Kumara H}
\address{Department of Mathematics, Kuvempu University,\\
Shankaraghatta - 577 451, Shimoga, Karnataka, INDIA.\\}
\email{[email protected], [email protected]}
\begin{abstract} The purpose of this paper is to study $*$-Ricci tensor on Sasakian manifold. Here, $\varphi$-confomally flat and confomally flat $*$-$\eta$-Einstein Sasakian manifold are studied. Next, we consider $*$-Ricci symmetric conditon on Sasakian manifold. Finally, we study a special type of metric called $*$-Ricci soliton on Sasakian manifold.
\end{abstract}
\subjclass[2010]{53D10 $\cdot$ 53C25 $\cdot$ 53C15 $\cdot$ 53B21.}
\keywords{Sasakian metric $\cdot$ $*$-Ricci tensor $\cdot$ Conformal curvature tensor $\cdot$ $\eta$-Einstein manifold.}
\maketitle
\section{Introduction} The notion of contact geometry has evolved from the mathematical
formalism of classical mechanics\cite{Gei}. Two important classes of contact manifolds are $K$-contact manifolds and Sasakian manifolds\cite{DEB}. An odd dimensional analogue of Kaehler geometry is the Sasakian geometry. Sasakian manifolds were firstly studied by the famous geometer Sasaki\cite{Sasa} in 1960, and for long time focused on this, Sasakian manifold have been extensively studied under several points of view in \cite{Chaki,De,Ikw,Olz,Tanno}, and references therein.
\par On the other hand, it is mentioned that the notion
of $*$-Ricci tensor was first introduced by Tachibana \cite{TS} on almost
Hermitian manifolds and further studied by Hamada and Inoguchi \cite{HT} on real hypersurfaces of non-flat complex space forms.
\par Motivated by these studies the present paper is organized as follows: In section 2, we recall some basic formula and result concerning Sasakian manifold and $*$-Ricci tensor which we will use in further sections. A $\varphi$-conformally flat Sasakian manifold is studied in section 3, in which we obtain some intersenting result. Section 4 is devoted to the study of conformally flat $*$-$\eta$-Einstein Sasakian manifold. In section 5, we consider $*$-Ricci symmetric Sasakian manifold and found that $*$-Ricci symmetric Sasakian manifold is $*$-Ricci flat, moreover, it is $\eta$-Einstein manifold. In the last section, we studied a special type of metric called $*$-Ricci soliton. Here we have proved a important result on Sasakian manifold admitting $*$-Ricci soliton.
\section{Preliminaries}
In this section, we collect some general definition and basic formulas on contact metric manifolds and Sasakian manifolds which we will use in further sections. We may refer to \cite{Ar,Bo,Kus} and references therein for more details and information about Sasakian geometry.
\par A (2n+1)-dimensional smooth connected manifold $M$ is called almost contact manifold if it admits a triple $(\varphi, \xi, \eta)$, where $\varphi$ is a tensor field of type $(1,1)$, $\xi$ is a global vector field and $\eta$ is a 1-form, such that
\begin{align}
\label{2.1} \varphi^2X=-X+\eta(X)\xi, \qquad \eta(\xi)=1, \qquad \varphi\xi=0, \qquad \eta\circ\varphi=0,
\end{align}
for all $X, Y\in TM$. If an almost contact manifold $M$ admits a structure $(\varphi, \xi, \eta, g)$, $g$ being a Riemannian metric such that
\begin{align}
\label{2.2} g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y),
\end{align}
then $M$ is called an almost contact metric manifold. An almost contact metric manifold $M(\varphi, \xi, \eta, g)$ with $d\eta(X,Y)=\Phi(X,Y)$, $\Phi$ being the fundamental 2-form of $M(\varphi, \xi,\eta,g)$ as defined by $\Phi(X,Y)=g(X,\varphi Y)$, is a contact metric manifold and $g$ is the associated metric. If, in addition, $\xi$ is a killing vector field (equivalentely, $h=\frac{1}{2}L_\xi \varphi=0$, where $L$ denotes Lie differentiation), then the manifold is called $K$-contact manifold. It is well known that\cite{DEB}, if the contact metric structure $(\varphi,\xi,\eta,g)$ is normal, that is, $[\varphi,\varphi]+2d\eta\otimes\xi=0$ holds, then $(\varphi,\xi,\eta,g)$ is Sasakian. An almost contact metric manifold is Sasakian if and only if
\begin{align}
\label{2.3} (\nabla_X \varphi)Y=g(X,Y)\xi-\eta(Y)X,
\end{align}
for any vector fields $X, Y$ on $M$, where $\nabla$ is Levi-Civita connection of $g$. A Sasakian manifold is always a $K$-contact manifold. The converse also holds when the dimension is three, but which may not be true in higher dimensions\cite{JB}. On Sasakian manifold, the following relations are well known;
\begin{align}
\label{a2.4}\nabla_X \xi&=-\varphi X\\
\label{2.4} R(X,Y)\xi&=\eta(Y)X-\eta(X)Y,\\
\label{2.5} R(\xi, X)Y&=g(X,Y)\xi-\eta(Y)X,\\
\label{2.6} Ric(X,\xi)&=2n\eta(X)\qquad (or\,\, Q\xi=2n\xi),
\end{align}
for all $X,Y\in TM$, where $R, Ric$ and $Q$ denotes the curvature tensor, Ricci tensor and Ricci operator, respectively.
\par On the other hand, let $M(\varphi,\xi,\eta,g)$ be an almost contact metric manifold with Ricci tensor $Ric$. The $*$-Ricci tensor and $*$-scalar curvature of $M$ repectively are defined by
\begin{align}
\label{2.7}Ric^*(X,Y)=\sum_{i=1}^{2n+1}R(X,e_i,\varphi e_i, \varphi Y),\qquad r^*=\sum_{i=1}^{2n+1}Ric^*(e_i,e_i),
\end{align}
for all $X,Y\in TM$, where ${e_1,...,e_{2n+1}}$ is an orthonormal basis of the tangent space $TM$. By using the first Bianchi identity and \eqref{2.7} we get
\begin{align}
Ric^*(X,Y)=\frac{1}{2}\sum_{i=1}^{2n+1}g(\varphi R(X,\varphi Y)e_i,e_i).
\end{align}
An almost contact metric manifold is said to be $*$-Einstein if $Ric^*$ is a constant
multiple of the metric $g$. One can see $Ric^*(X,\xi)=0$, for all $X \in TM$. It should be remarked that $Ric^*$ is not symmetric, in general. Thus the
condition $*$-Einstein automatically requires a symmetric property of the $*$-Ricci tensor\cite{HT}.
Now we make an effort to find $*$-Ricci tensor on Sasakian manifold.
\begin{Lemma}
In a (2n+1)-dimensional Sasakian manifold $M$, the $*$-Ricci tensor is given by
\begin{align}
\label{2.9} Ric^*(X,Y)=Ric(X,Y)-(2n-1)g(X,Y)-\eta(X)\eta(Y).
\end{align}
\end{Lemma}
\begin{proof}
In a (2n+1)-dimensional Sasakian manifold $M$, the Ricci tensor $Ric$ satisfies the relation (see page 284, Lemma 5.3 in \cite{YKK}):
\begin{align}
\label{2.10} Ric(X,Y)=\frac{1}{2}\sum_{i=1}^{2n+1}g(\varphi R(X,\varphi Y)e_i,e_i)+(2n-1)g(X,Y)+\eta(X)\eta(Y).
\end{align}
Using the definition of $Ric^*$ in \eqref{2.10}, we obtain \eqref{2.9}
\end{proof}
\begin{defi}\label{d2.1} \cite{JTC} An almost contact metric manifold $M$ is said to be weakly $\varphi$-Einstein if
\begin{align*}
Ric^\varphi(X,Y)=\beta g^\varphi(X,Y),\quad X,Y\in TM,
\end{align*}
for some function $\beta$. Here $Ric^\varphi$ denotes the symmetric part of $Ric^*$, that is,
\begin{align*}
Ric^\varphi(X,Y)=\frac{1}{2}\{Ric^*(X,Y)+Ric^*(Y,X)\}, \quad X,Y\in TM,
\end{align*}
we call $Ric^\varphi$, the $\varphi$-Ricci tensor on $M$ and the symmetric tensor $g^\varphi$ is defined by $g^\varphi(X,Y)=g(\varphi X, \varphi Y)$. When $\beta$ is constant, then $M$ is said to be $\varphi$-Einstein.
\end{defi}
\begin{defi}
If the Ricci tensor of a Sasakian manifold $M$ is of the form
\begin{align*}
Ric(X,Y)=\alpha g(X,Y)+\gamma \eta(X)\eta(Y),
\end{align*}
for any vector fields $X, Y$ on $M$, where $\alpha$ and $\gamma$ being constants, then $M$ is called an $\eta$-Einstein manifold.
\end{defi}
Let $M(\varphi,\xi,\eta,g)$ be a Sasakian $\eta$-Einstein manifold with constants $(\alpha, \gamma)$. Consider a $D$-homothetic Sasakian structure $S = (\varphi',\xi',\eta',g')=(\varphi,a^{-1}\xi,a\eta,ag + a(a-1)\eta\otimes\eta)$. Then $(M,S)$ is also $\eta$-Einstein with constants
$\alpha'=\frac{\alpha+2-2a}{a}$ and $\gamma'=2n-\alpha'$ (see proposition 18 in \cite{Cha}).
Here we make a remark that the particular value: $\alpha=-2$ remains fixed under a
$D$-homothetic deformation\cite{AG}. Thus, we state the following definition.
\begin{defi}
A Sasakian $\eta$-Einstein manifold with $\alpha=-2$ is said to be
D-homothetically fixed.
\end{defi}
\section{$\varphi$-conformally Flat Sasakian Manifold}
The Weyl conformal curvature tensor\cite{YKK} is defined as a map $C:TM\times TM\times TM \longrightarrow TM$ such that
\begin{align}
\nonumber C(X,Y)Z=&R(X,Y)Z-\frac{1}{2n-1}\{Ric(Y,Z)X-Ric(X,Z)Y+g(Y,Z)QX\\
\label{3.1} &-g(X,Z)QY\}+\frac{r}{2n(2n-1)}\{g(Y,Z)X-g(X,Z)Y\}, \quad X,Y\in TM.
\end{align}
In\cite{CAR}, Cabrerizo et al proved some necessary condition for $K$-contact manifold to be $\varphi$-conformally flat. In the following theorem we find a condition for $\varphi$-conformally flat Sasakian manifold.
\begin{Th}
If a (2n+1)-dimensional Sasakian manifold $M$ is $\varphi$-conformally flat, then $M$ is $*$-$\eta$-Einstein manifold. Moreover, $M$ is weakly $\varphi$-Einstein.
\end{Th}
\begin{proof}
It is well known that (see in \cite{CAR}), if a $K$-contact manifold is $\varphi$-conformally flat then we get the following relation:
\begin{align}
\label{3.2} R(\varphi X, \varphi Y, \varphi Z, \varphi W)=\frac{r-4n}{2n(2n-1)}\{g(\varphi Y,\varphi Z)g(\varphi X, \varphi W)-g(\varphi X,\varphi Z)g(\varphi Y, \varphi W)\}.
\end{align}
In a Sasakian manifold, in view of \eqref{2.4} and \eqref{2.5} we can verify that
\begin{align}
\nonumber R(\varphi^2 X, \varphi^2Y,\varphi^2Z,\varphi^2W)=&R(X,Y,Z,W)-g(Y,Z)\eta(X)\eta(Y)+g(X,Z)\eta(Y)\eta(W)\\
\label{3.3} &+g(Y,W)\eta(X)\eta(Z)-g(X,W)\eta(Y)\eta(Z),
\end{align}
for all $X,Y,Z,W \in TM$. Replacing $X, Y,Z,W$ by $\varphi X, \varphi Y, \varphi Z, \varphi W$ respectively in \eqref{3.2} and making use of \eqref{2.2} and \eqref{3.3} we get
\begin{align}
\nonumber R(X,Y,Z,W)&=\frac{r-4n}{2n(2n-1)}\{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\}\\
\nonumber&-\frac{r-2n(2n+1)}{2n(2n-1)}\{g(Y,Z)\eta(X)\eta(W)-g(X,Z)\eta(Y)\eta(W)\\
&+g(X,W)\eta(Y)\eta(Z)-g(Y,W)\eta(X)\eta(Z)\}.
\end{align}
By the definition of $Ric^*$, direct computation yields
\begin{align}
\label{3.5} Ric^*(X,Y)=\sum_{i=1}^{2n+1}R(X,e_i,\varphi e_i, \varphi Y)=\beta g(X,Y)-\beta\eta(X)\eta(Y),
\end{align}
where $\beta=\frac{r-4n}{2n(2n-1)}$, showing that $M$ is $*$-$\eta$-Einstein. Next, in view of \eqref{2.2} we have
\begin{align}
\label{3.6} Ric^*(X,Y)=\frac{r-4n}{2n(2n-1)} g^\varphi(X,Y),
\end{align}
for all $X,Y \in TM$. Hence $Ric^*=Ric^\varphi$ and hence it is weakly $\varphi$-Einstein. This completes the proof.
\end{proof}
Suppose the scalar curvature of the manifold is constant. Then in view of \eqref{3.6}, we have
\begin{Cor}
If a $\varphi$-conformally flat Sasakian manifold has constant scalar curvature, then it is $\varphi$-Einstein.
\end{Cor}
In a Sasakian manifold, the $*$-Ricci tensor is given by \eqref{2.9} and so in view of \eqref{3.5}, we state the following;
\begin{Cor}
A $\varphi$-conformally flat Sasakian manifold is $\eta$-Einstein.
\end{Cor}\
The notion of $\eta$-parallel Ricci tensor was introduced in the context of Sasakian manifold by Kon\cite{KON} and is defined by $(\nabla_Z Ric)(\varphi X,\varphi Y)=0$, for all $X,Y \in TM$. From this definition, we define a $\eta$-parallel $*$-Ricci tensor by $(\nabla_Z Ric^*)(\varphi X,\varphi Y)=0$.
Replacing $X$ by $\varphi X$ and $Y$ by $\varphi Y$ in \eqref{3.5}, we obtain
$Ric^*(\varphi X,\varphi Y)=\beta g(\varphi X,\varphi Y)$.
Now taking covariant differention with respect to $W$, we get $(\nabla_W Ric^*)(\varphi X,\varphi Y)=dr(W) g(\varphi X, \varphi Y)$. Therefore we have the following;
\begin{Cor}
A (2n+1)-dimensional $\varphi$-conformally flat Sasakian manifold has $\eta$-parallel $*$-Ricci tensor if and only if the scalar curvature of the
manifold is constant.
\end{Cor}
\section{Conformally Flat $*$-$\eta$-Einstein Sasakian Manifold}
Suppose $M$ is conformally flat Sasakian manifold, then from \eqref{3.1} we have
\begin{align}
\nonumber R(X.Y)Z=&\frac{1}{2n-1}\{Ric(Y,Z)X-Ric(X,Z)Y+g(Y,Z)QX\\
\label{4.1} &-g(X,Z)QY\}-\frac{r}{2n(2n-1)}\{g(Y,Z)X-g(X,Z)Y\}.
\end{align}
If we set $Y=Z=\xi\perp X$, we find $QX=\frac{r-2n}{2n}X$. From this equation, \eqref{4.1} becomes
\begin{align}
\label{4.2} R(X,Y)Z=\frac{r-4n}{2n(2n-1)}\{g(Y,Z)X-g(X,Z)Y\}.
\end{align}
By definition of $*$-Ricci tensor, direct computation yields
\begin{align}
\label{4.3} Ric^*(X,Y)=\frac{r-4n}{2n(2n-1)}g(\varphi X, \varphi Y).
\end{align}
Since M is a conformally flat Sasakian manifold, we have
the following equations from the definition of $Ric^*$ and equation \eqref{4.2}:
\begin{align*}
Ric^*(\varphi Y, \varphi X)&=\sum_{i=1}^{2n+1}R(\varphi Y, e_i, \varphi e_i, \varphi^2X)\\
&=\sum_{i=1}^{2n+1}\{-R(X,\varphi e_i,e_i,\varphi Y)+\eta(X)R(\varphi Y,e_i,\varphi e_i,\xi)\}\\
&=Ric^*(X,Y).
\end{align*}
If we set $Y=\varphi X$ such that $X$ is unit, we obatin $Ric^*(\varphi^2X,\varphi X)=Ric^*(X,\varphi X)$ which implies that $Ric^*(X,\varphi X)=0$. Thus, from the definition of $*$-$\eta$-Einstein and \eqref{4.3}, we obtain $a g(X,Y)+b \eta(X)\eta(Y)=\frac{r-4n}{2n(2n-1)}g(\varphi X,\varphi Y)$. If we choose $X=Y=\xi$, we find $a+b=0$. If we set $Y=X\perp \xi$ such that $X$ and $Y$ are units, we get
\begin{align}
\label{4.4} a=\frac{r-4n}{2n(2n-1)}=K(X,\varphi X).
\end{align}
In\cite{OKU}, the author proved that every conformally flat Sasakian manifold has a constant curvature +1, that is, $R(X,Y)Z=g(Y,Z)X-g(X,Z)Y$. From this result and \eqref{4.2}, we find $r=2n(2n-1)+4n$. In view of \eqref{4.4}, we obtain $a=1$. Therefore we have the following:
\begin{Th}\label{t4.2}
Let $M$ be a (2n+1)-dimensional conformally flat Sasakian manifold. If $M$ is $*$-$\eta$-Einstein, then it is of constant curvature +1.
\end{Th}
We know that every Riemannian manifold of constant sectional curvature is locally symmetric. From theorem \eqref{t4.2}, we have
\begin{Cor}
A conformally flat $*$-$\eta$-Einstein Sasakian manifold is locally symmetric.
\end{Cor}
\begin{Th}
A (2n+1)-dimensional conformally flat Sasakian manifold is $\varphi$-Einstein.
\end{Th}
\begin{proof}
The theorem follows from \eqref{4.3} and definition \eqref{d2.1}.
\end{proof}
\section{$*$-Ricci Semi-symmetric Sasakian Manifold}
A contact metric manifold is called Ricci semi-symmetric if $R(X,Y)\cdot Ric=0$, for all $X,Y \in TM$. Analogous to this definition, we define $*$-Ricci semi-symmetric by $R(X,Y)\cdot Ric^*=0$.
\begin{Th}
If a (2n+1)-dimensional Sasakian manifold $M$ is $*$-Ricci semi-symmetric, then $M$ is $*$-Ricci flat. Moreover, it is $\eta$-Einstein manifold and
the Ricci tensor can be expressed as
\begin{align*}
Ric(X,Y)=(2n-1)g(X,Y)+\eta(X)\eta(Y).
\end{align*}
\end{Th}
\begin{proof}
Let us consider (2n+1)-dimensional Sasakian manifold which satisfies the condition $R(X,Y).Ric^*=0$. Then we have
\begin{align}
\label{5.1} Ric^*(R(X,Y)Z,W)+Ric^*(Z,R(X,Y)W)=0.
\end{align}
Putting $X=Z=\xi$ in \eqref{5.1}, we have
\begin{align}
\label{5.2} Ric^*(R(\xi, Y)\xi,W)+Ric^*(\xi,R(\xi,Y)W)=0.
\end{align}
It is well known that $Ric^*(X,\xi)=0$. Making use of \eqref{2.4} in \eqref{5.2} and by virtue of last equation, we find
\begin{align}
\label{5.3} Ric^*(Y,W)=0, \qquad Y,W\in TM,
\end{align}
showing that $M$ is $*$-Ricci flat. Moreover, in view of \eqref{5.3} and \eqref{2.9}, we have the required result.
\end{proof}
\section{Sasakian Manifold Admitting $*$-Ricci Soliton}
\par Ricci flows are intrinsic geometric flows on a Riemannian manifold,
whose fixed points are solitons and it was introduced by Hamilton\cite{HRS}. Ricci solitons also correspond to self-similar solutions of Hamilton's Ricci
flow. They are natural generalization of Einstein metrics and is defined by
\begin{align}
\label{R1}(L_Vg)(X,Y)+2Ric(X,Y)+2\lambda g(X,Y)=0,
\end{align}
for some constant $\lambda$, a potential vector field $V$. The Ricci soliton is said to be shrinking, steady, and expanding according as $\lambda$ is negative, zero, and positive respectively.
\par The notion of $*$-Ricci soliton was introduced by George and Konstantina\cite{Geo}, in 2014, where they essentially modified the definition of Ricci soliton by replacing the Ricci tensor in \eqref{R1} with the $*$-Ricci tensor. Recently, the authors studied $*$-Ricci soliton on para-Sasakian manifold in the paper \cite{Prak} and obtain several interesting result. A Riemannian metric $g$ on $M$ is called $*$-Ricci soliton, if there is vector field $V$, such that
\begin{align}
\label{R2} (L_Vg)(X,Y)+2Ric^*(X,Y)+2\lambda g(X.Y)=0,
\end{align}
for all vector fields $X,Y$ on $M$. In this section we study a special type of metric called $*$-Ricci soliton on Sasakian manifold. Now we prove the following result:
\begin{Th}
If the metric $g$ of a (2n+1)-dimensional Sasakian manifold $M(\varphi,\xi,\eta,g)$
is a $*$-Ricci soliton with potential vector field $V$, then (i) $V$ is Jacobi along geodesic of $\xi$, (ii) M is an $\eta$-Einstein manifold and the Ricci tensor can be expressed as
\begin{align}
\label{6.1} Ric(X,Y)=\left[2n-1-\frac{\lambda}{2}\right]g(X,Y)+\left[1+\frac{\lambda}{2}\right]\eta(X)\eta(Y).
\end{align}
\end{Th}
\begin{proof}
Call the following commutation formula (see \cite{YK}, page 23):
\begin{align}
(L_V\nabla_X g-\nabla_X L_V g-\nabla_{[V,X]}g)(Y,Z)=-g((L_V\nabla)(X,Y),Z)-g((L_V \nabla)(X,Z),Y),
\end{align}
and is well known for all vector fields $X,Y,Z$ on $M$. Since $g$ is parallel with respect to Levi-Civita connection $\nabla$, then the above relation becomes
\begin{align}
\label{6.2}(\nabla_XL_V g)(Y,Z)=g((L_V \nabla)(X,Y),Z)+g((L_V \nabla)(X,Y),Z).
\end{align}
Since $L_V\nabla$ is a symmetric tensor of type $(1,2)$, i.e., $(L_V \nabla)(X,Y)=(L_V\nabla)(Y,Z)$, it follows from \eqref{6.2} that
\begin{align}
\label{6.3}g((L_V\nabla)(X,Y),Z)=\frac{1}{2}\{(\nabla_XL_V g)(Y,Z)+(\nabla_YL_V g)(Z,X)-(\nabla_ZL_Vg)(X,Y)\}.
\end{align}
Next, taking covariant differentiation of $*$-Ricci soliton equation \eqref{R2} along a vector field $X$, we obatin $(\nabla_X L_V g)(X,Y)=-2(\nabla_X Ric^*)(X,Y)$. Substitutimg this relation into \eqref{6.3} we have
\begin{align}
\label{6.4} g((L_V\nabla)(X,Y),Z)=(\nabla_Z Ric^*)(X,Y)-(\nabla_X Ric^*)(Y,Z)-(\nabla_Y Ric^*)(X,Z).
\end{align}
Again, taking covariant differentiation of \eqref{2.9} with respect to $Z$, we get
\begin{align}
\label{6.5} (\nabla_Z Ric^*)(X,Y)=(\nabla_Z Ric)(X,Y)-\{g(Z,\varphi X)\eta(Y)+g(Z,\varphi Y)\eta(X)\}.
\end{align}
Combining \eqref{6.5} with \eqref{6.4}, we find
\begin{align}
\nonumber g((L_V\nabla)(X,Y),Z)=&(\nabla_Z Ric)(X,Y)-(\nabla_X Ric)(Y,Z)-(\nabla_Y Ric)(X,Z)\\
\label{6.6} &+2g(X,\varphi Z)\eta(Y)+2g(Y,\varphi Z)\eta(X).
\end{align}
In Sasakian manifold we know the following relation\cite{AG}:
\begin{align}
\label{6.7} \nabla_\xi Q=Q\varphi-\varphi Q=0, \qquad (\nabla_X Q)\xi=Q\varphi X-2n\varphi X.
\end{align}
Replacing $Y$ by $\xi$ in \eqref{6.6} and then using \eqref{6.7} we obtain
\begin{align}
\label{6.8} (L_V \nabla)(X,\xi)=-2Q\varphi X+2(2n-1)\varphi X.
\end{align}
From the above equation, we have
\begin{align}
\label{6.9} (L_V\nabla)(\xi,\xi)=0.
\end{align}
Now, substituting $X=Y=\xi$ in the well known formula \cite{YK}:
\begin{align*}
(L_V \nabla)(X,Y)=\nabla_X\nabla_Y V-\nabla_{\nabla_X Y}V+R(V,X)Y,
\end{align*}
and then making use of equation \eqref{6.9} we obtain
\begin{align*}
\nabla_\xi\nabla_\xi V+R(V,\xi)\xi=0,
\end{align*}
which proves part (i).
Further, differentiating \eqref{6.8} covariantly along an arbitrary vector field $Y$ on $M$ and then using \eqref{2.3} and last equation of \eqref{2.6}, we obtain
\begin{align}
\nonumber &(\nabla_Y L_V \nabla)(X,\xi)-(L_V\nabla)(X,\varphi Y)\\
\label{6.10} &=2\{-(\nabla_Y Q)\varphi X-g(X,Y)\xi+\eta(X)QY-(2n-1)\eta(X)Y\}.
\end{align}
According to Yano\cite{YK}, we have the following commutation formula:
\begin{align}
\label{c1} (L_V R)(X,Y)Z=(\nabla_X L_V \nabla)(Y,Z)-(\nabla_YL_V\nabla)(X,Z).
\end{align}
Substituting $\xi$ for $Z$ in the foregoing equation and in view of \eqref{6.10}, we obtain
\begin{align}
\nonumber &(L_V R)(X,Y)\xi-(L_V\nabla)(Y,\varphi X)+(L_V\nabla)(X,\varphi Y)\\
\label{6.11} &=2\{(\nabla_Y Q)\varphi X-(\nabla_X Q)\varphi Y+\eta(Y)QX-\eta(X)QY+(2n-1)(\eta(X)Y-\eta(Y)X)\}.
\end{align}
Replacing $Y$ by $\xi$ in \eqref{6.11} and then using last equation of \eqref{2.6}, \eqref{6.7} and \eqref{6.8}, we have
\begin{align}
\label{6.12} (L_V R)(X,\xi)\xi=4\{QX-\eta(X)\xi-(2n-1)X\}.
\end{align}
Since $Ric^*(X,\xi)=0$, then $*$-Ricci soliton equation \eqref{R2} gives $(L_V g)(X,\xi)+2\lambda\eta(X)=0$, which gives
\begin{align}
\label{6.13} (L_V\eta)(X)-g(X,L_V\xi)+2\lambda\eta(X)=0.
\end{align}
Lie-derivative of $g(\xi,\xi)=1$ along $V$ gives $\eta(L_V \xi)=\lambda$. Next, Lie-differentiating the formula $R(X,\xi)\xi=X-\eta(X)\xi$ along $V$ and then by virtue of last equation, we obtain
\begin{align}
\label{6.14} (L_V R)(X,\xi)\xi-g(X,L_V\xi)\xi+2\lambda X=-((L_V\eta)X)\xi.
\end{align}
Combining \eqref{6.14} with \eqref{6.12}, and making use of \eqref{6.13}, we obtain part (ii). This completes the proof.
\end{proof}
By virtue of \eqref{2.9} and \eqref{6.1}, the $*$-Ricci soliton equation \eqref{R2} takes the form
\begin{align}
\label{617}(L_V g)(X,Y)=-\lambda\{g(X,Y)+\eta(X)\eta(Y)\}.
\end{align}
Now, differentiating this equation covariantly along an arbitrary vector field $Z$ on $M$, we have
\begin{align}
\label{6.17} (\nabla_Z L_V g)(X,Y)=-\lambda\{g(Z,\varphi X)\eta(Y)+g(Z,\varphi Y)\eta(X)\}.
\end{align}
Substitute this equation in commutation formula \eqref{6.3}, we find
\begin{align}
\label{6.18} (L_V\nabla)(X,Y)=\lambda\{\eta(Y)\varphi X+\eta(X)\varphi Y\}.
\end{align}
Taking covariant differentiation of \eqref{6.18} along a vector field $Z$ and then using \eqref{2.3}, we obtain
\begin{align}
\nonumber (\nabla_Z L_V\nabla)(X,Y)=&\lambda\{g(Z,\varphi Y)\varphi X+g(Z,\varphi X)\varphi Y+g(X,Z)\eta(Y)\xi\\
\label{6.19} &+g(Y,Z)\eta(X)\xi-2\eta(X)\eta(Y)Z\}.
\end{align}
Making use of \eqref{6.19} in commutation formula \eqref{c1}, we have
\begin{align}
\nonumber (L_V R)(X,Y)Z=&\lambda\{g(X,\varphi Z)\varphi Y+2g(X,\varphi Y)\varphi Z-g(Y,\varphi Z)\varphi X+g(X,Z)\eta(Y)\xi\\
\label{6.21} &-g(Y,Z)\eta(X)\xi+2\eta(X)\eta(Z)Y-2\eta(Y)\eta(Z)X\}.
\end{align}
Contracting \eqref{6.21} over $Z$, we have
\begin{align}
\label{6.22} (L_V Ric)(Y,Z)=2\lambda\{g(Y,Z)-(2n+1)\eta(Y)\eta(Z)\}.
\end{align}
On other hand, taking Lie-differentiation of \eqref{6.1} along the vector field $V$ and using \eqref{617}, we obtain
\begin{align}
\nonumber (L_V Ric)(Y,Z)=&\left[1+\frac{\lambda}{2}\right]\{(L_V\eta)(Y)\eta(Z)+
\eta(Y)(L_V\eta)(Z)\}\\
\label{6.23} &-\lambda\left[2n-1-\frac{\lambda}{2}\right]\{g(Y,Z)+\eta(Y)\eta(Z)\}.
\end{align}
Comparison of \eqref{6.22} and \eqref{6.23} gives
\begin{align}
\nonumber &\left[1+\frac{\lambda}{2}\right]\{(L_V\eta)(Y)\eta(Z)+
\eta(Y)(L_V\eta)(Z)\} -\lambda\left[2n-1-\frac{\lambda}{2}\right]\{g(Y,Z)+\eta(Y)\eta(Z)\}\\
\label{6.24} &=2\lambda\{g(Y,Z)-(2n+1)\eta(Y)\eta(Z)\}.
\end{align}
Taking $Y$ by $\xi$ in the foregoing equation, we get
\begin{align}
\label{6.25} \left[1+\frac{\lambda}{2}\right](L_V\eta)(Y)=-\left[\lambda+\frac{\lambda^2}{2}\right]\eta(Y).
\end{align}
Substitute \eqref{6.25} in \eqref{6.24} and then replacing $Z$ by $\varphi Z$, we obtain
\begin{align}
\lambda\left[2n+1-\frac{\lambda}{2}\right]g(Y,\varphi Z)=0.
\end{align}
Since $g(Y,\varphi Z)$ is non-vanishing everywhere on $M$, thus we have either $\lambda=0$, or $\lambda=2(2n+1)$.\\
\textbf{Case I:} If $\lambda=0$, then from \eqref{617} we can see that $L_V g=0$, i.e., $V$ is Killing. From \eqref{6.1}, we have
\begin{align*}
Ric(X,Y)=(2n-1)g(X,Y)+\eta(X)\eta(Y).
\end{align*}
This shows that $M$ is $\eta$-Einstein manifold with scalar curvature $r=2n(\alpha+1)=4n^2$.\\
\textbf{Case II:} If $\lambda=2(2n+1)$, then plugging $Y$ by $\varphi Y$ in \eqref{6.25} we have the relation $\left[1+\frac{\lambda}{2}\right](L_V\eta)(\varphi Y)=0$. Since $\lambda=2(2n+1)$, by virtue of last equation we have $\lambda\neq-2$, thus we must have $(L_V\eta)(\varphi Y)=0$. Replacing $Y$ by $\varphi Y$ in the foregoing equation and then using \eqref{2.1}, we have
\begin{align}
\label{6.27} (L_V\eta)(Y)=-2(2n+1)\eta(Y).
\end{align}
This shows that $V$ is a non-strict infinitesimal contact transformation. Now, substituting $Z$ by $\xi$ in \eqref{617} and using \eqref{6.27} we immediately get $L_V\xi=2(2n+1)\xi$. Using this in the commutation formula (see \cite{YK}, page 23)
\begin{align*}
L_V\nabla_X\xi-\nabla_XL_V\xi-\nabla_{[V,X]}\xi=(L_V\nabla)(X,\xi),
\end{align*}
for an arbitrary vector field $X$ on $M$ and in view of \eqref{a2.4} and \eqref{6.18} gives $L_V\varphi=0$. Thus, the vector field $V$ leaves the structure tensor $\varphi$ invariant.
\par Other hand, using $\lambda=2(2n+1)$ in \eqref{6.1} if follows that
\begin{align*}
Ric(X,Y)=-2g(X,Y)+2(n+1)\eta(X)\eta(Y),
\end{align*}
showing that $M$ is $\eta$-Einstein with $\alpha=-2$. Thus $M$ is a $D$-homothetically fixed. In\cite{Cha}, the authors give a wonderful information on $\eta$-Einstein Sasakian geometry; in this paper authors says, when $M$ is null (transverse Calabi-Yau) then always $(\alpha,\gamma)=(-2, 2n+2)$ (see page 189). By this we conclude that $M$ is a $D$-homothetically fixed null $\eta$-Einstein manifold. Therefore, we have the following;
\begin{Th}
Let $M$ be a (2n+1)-dimensional Sasakian manifold. If $M$ admits $*$-Ricci soliton, then either $V$ is Killing, or $M$ is $D$-homothetically fixed null $\eta$-Einstein manifold. In the first case, $M$ is $\eta$-Einstein manifold of constant scalar curvature $r=2n(\alpha+1)=4n^2$ and in second case, $V$ is a non-strict infinitesimal contact transformation and leaves the structure tensor $\varphi$ invariant.
\end{Th}
Now, we prove the following result, which gives some remark on $*$-Ricci soliton.
\begin{Th}
Let $M$ be a (2n+1)-dimensional Sasakian manifold admitting $*$-Ricci soliton with $Q^*\varphi=\varphi Q^*$. Then the soliton vector field $V$ leaves the structure tensor $\varphi$ invariant if and only if $g(\varphi(\nabla_V \varphi)X,Y)=(dv)(X,Y)-(dv)(\varphi X,\varphi Y)-(dv)(X,\xi)\eta(Y)$.
\end{Th}
\begin{proof}
The $*$-Ricci soliton equation can be written as
\begin{align}
\label{6.28} g(\nabla_XV,Y)+g(\nabla_YV,X)+2Ric^*(X,Y)+2\lambda g(X,Y)=0.
\end{align}
Suppose $v$ is 1-form, metrically equivalent to $V$ and is given by $v(X) = g(X, V)$, for any arbitrary vector field $X$, then the exterior derivative $dv$ of $v$ is given by
\begin{align}
\label{6.29} 2(dv)(X,Y)=g(\nabla_X V,Y)-g(\nabla_YV,X)
\end{align}
As $dv$ is a skew-symmetric, if we define a tensor fieeld $F$ of type (1, 1) by
\begin{align}
(dv)(X,Y)=g(X,FY),
\end{align}
then $F$ is skew self-adjoint i.e., $g(X, FY)=-g(FX, Y)$. The equation \eqref{6.29} takes the form $2g(X,FY)=g(\nabla_X V,Y)-g(\nabla_YV,X)$. Adding it to equation \eqref{6.28} side by side and factoring out $Y$ gives
\begin{align}
\label{6.31}\nabla_XV=-Q^*X-\lambda X-FX,
\end{align}
where $Q^*$ is $*$-Ricci operator. Applying $\varphi$ on \eqref{6.31}, we have
\begin{align}
\label{6.32} \varphi\nabla_XV=-\varphi Q^*X-\varphi\lambda X-\varphi FX.
\end{align}
Next, Replacing $X$ by $\varphi X$ in \eqref{6.31}, we obtain
\begin{align}
\label{6.33} \nabla_{\varphi X}V=-Q^*\varphi X-\lambda\varphi X-F\varphi X.
\end{align}
Substracting \eqref{6.32} and \eqref{6.33}, we have
\begin{align}
\varphi\nabla_X V-\nabla_{\varphi X}V=(Q^*\varphi-\varphi Q^*)X+(F\varphi-\varphi F)X.
\end{align}
By our hypothesis, noting that $\varphi$ commutes with the $*$-Ricci operator $Q^*$ for Sasakian manifold, we have
\begin{align}
\label{6.35}\varphi\nabla_X V-\nabla_{\varphi X}V=(F\varphi-\varphi F)X.
\end{align}
Now, we note that
\begin{align*}
(L_V\varphi)X&=L_V\varphi X-\varphi L_VX\\
&=\nabla_V\varphi X-\nabla_{\varphi X}V-\varphi\nabla_VX+\varphi\nabla_XV\\
&=(\nabla_V\varphi)X-\nabla_{\varphi X}V+\varphi\nabla_XV.
\end{align*}
The use of foregoing equation in \eqref{6.35} gives
\begin{align}
\label{6.36}(L_V\varphi)X-(\nabla_V\varphi)X=(F\varphi-\varphi F)X.
\end{align}
Operating $\varphi$ on both sides of the equation \eqref{6.35} and then making use of \eqref{2.1}, \eqref{6.31} and \eqref{6.33}, we find
\begin{align*}
(dv)(\varphi X,\varphi Y)-(dv)(X,Y)+(dv)(X,\xi)\eta(Y)=g(\varphi(F\varphi-\varphi F)X,Y).
\end{align*}
Using \eqref{6.36} in the above equation provides
\begin{align*}
(dv)(\varphi X,\varphi Y)-(dv)(X,Y)+(dv)(X,\xi)\eta(Y)=g(\varphi(L_V\varphi)X-\varphi(\nabla_V\varphi)X, Y).
\end{align*}
This shows that $L_V\varphi=0$ if and only if $g(\varphi(\nabla_V \varphi)X,Y)=(dv)(X,Y)-(dv)(\varphi X,\varphi Y)-(dv)(X,\xi)\eta(Y)$, completing the proof.
\end{proof}
\end{document} |
\begin{document}
\baselineskip=0.20in
\baselineskip=0.30in
\begin{center}
{\large \bf Relativistic Treatment of the Spin-Zero Particles Subject to the
q-Deformed Hyperbolic Modified P\"{o}schl-Teller Potential.}
{ \large K.\,J. Oyewumi\footnote{E-Mail:[email protected]} and T.\,T. Ibrahim\footnote{Department of Physics, University of Stellenbosch, Matieland, PO 1529, Stellenbosch, 7599, South
Africa.
} }
{ Theoretical Physics Section, Department of Physics\\ University of Ilorin, P. M. B. 1515, Ilorin, Nigeria. \\}
{\large S.\, O. Ajibola}
{
Department of Mathematics, Faculty of Science and Technology, \\National Open University of Nigeria, Lagos, Nigeria.
}
{\large D.\, A. Ajadi }
{
Department of Physics, Ladoke Akintola University of Technology, \\Ogbomoso, Nigeria.
}
\baselineskip=0.20in
\end{center}
\begin{abstract}
\noindent
In this study, we solve the Klein-Gordon equation with equal scalar and vector q-deformed hyperbolic modified P\"{o}schl-Teller potential. The explicit expressions of bound state spectra and the normalized eigenfunctions for s-wave bound states are obtained analytically. The energy equations and the corresponding wave functions for the special cases of the equally mixed q-deformed hyperbolic modified P\"{o}schl-Teller potential for spinless particle are briefly discussed.
\end{abstract}
\baselineskip=0.28in
\noindent
{ {\bf KEY WORDS}}: Klein- Gordon equation, q-deformed hyperbolic modified P\"{o}schl-Teller potential, reflectionless-type potential, q-deformed symmetric hyperbolic modified P\"{o}schl-Teller potential, symmetric modified P\"{o}schl-Teller potential, PT-symmetric hyperbolic modified P\"{o}schl-Teller potential.
\baselineskip=0.28in
\noindent
{\bf PACS}: 03.65.Ge, 03.65.Pm, 02.30.Gp
\baselineskip=0.28in
\noindent
\section{\bf Introduction}
In recent years, there has been increasing interest in finding the analytical solutions of relativistic and non-relativistic quantum mechanical problems. The presence of strong fields or high speeds introduces relativistic phenomena that cannot be described using the Schr\"{o}dinger equation. In relativistic quantum mechanics, the solutions of the Klein-Gordon and Dirac equations with various physical potential play an important role in nuclear physics and other related areas. The Klein-Gordon equation has also been used to understand the motion of the spin-zero particles in various potentials.
Many studies have been carried out to explore the relativistic energy eigenvalues and the corresponding wave functions by solving the Klein-Gordon and Dirac equations (Greiner 2000, Simsek and E$\check{g}$rifes 2004, Qiang 2004, Chen 2005, Zhao et al. 2005, Alhaidari et al. 2006, Berkdermir 2007, Durmus and Yasuk 2007, Qiang et al. 2007, Soylu et al. 2008, Liu et al. 2009, Motavalli 2009 and Xu et al. 2010, Oyewumi 2010). For analytic solutions of these equations of motion with various potentials, some authors have assumed equal scalar and vector potentials (see Alhaidari et al. 2006 for details).
Several methods ranging from exact analytical technique to approximate analytical methods have also been used in solving the relativistic quantum mechanical problems with different potentials.
The generalized symmetrical double-well potential (Zhao et al. 2005); Kratzer-type and generalized ring-shaped Kratzer potentials (Qiang 2004, Berkdermir 2007, Oyewumi 2010); exponential-type potentials (Simsek and E$\check{g}$rifes 2004); double ring-shaped Kratzer potential (Durmus and Yasuk 2007); Hulth$\acute{e}$n (Qiang et al. 2004, Guo et al. 2003); ring-shaped harmonic oscillator (Qiang 2003); Rosen-Morse-type potential (Yi et al. 2004, Soylu et al. 2008); Eckart potential (Zou et al. 2005, Dong et al. 2007, Liu et al. 2009, Yahya et al. 2010); Manning-Rosen potential (Qiang and Dong 2007) and Scarf-type potential (Zhang et al. 2005, Motavalli 2009) are examples of potentials which finds application in various aspects of modern physics. These have been studied in both the relativistic and non-relativistic limits.
In this paper, we study analytical bound state solutions of the Klein-Gordon equation with the equal scalar and vector q-deformed hyperbolic modified P\"{o}schl-Teller potential. P\"{o}schl-Teller potential (P\"{o}schl and Teller 1933, Landau and Liftshitz 1977) is found very useful in the study of the $\Lambda$-hypernuclear in nuclear physics and other related areas (Oyewumi and Bangudu 1999, 2000, Grypeos et al. 2004, Oyewumi et al. 2004, Efthymiou et al. 2008, Oyewumi 2009).
We have also considered the special cases of this potential in the Klein-Gordon equation, the bound state energy spectra and the corresponding wave functions of the resulting potentials which include the reflectionless-type potential, q-deformed symmetric hyperbolic modified P\"{o}schl-Teller potential, symmetric modified P\"{o}schl-Teller potential and the PT-symmetric version of the hyperbolic modified P\"{o}schl-Teller potential are obtained.
This paper is organized as follows. Section $2$ contains the bound state solutions of the q-deformed hyperbolic modified P\"{o}schl-Teller potential. In Section $3$, we investigate the special cases of this potential and finally, conclusion is given in Section $4$.
\section{Bound State Solutions of the q-Deformed Hyperbolic Modified P\"{o}schl-Teller Potential.}
The time-independent three-dimensional Klein-Gordon equation with the scalar potential $V_{s}(r)$ and vector potential $V_{v}(r)$ is given by (Greiner 2000, Simsek and E$\check{g}$rifes 2004, Qiang 2004, Chen 2005, Zhao et al. 2005, Alhaidari et al. 2006, Berkdermir 2007, Durmus and Yasuk 2007, Qiang et al. 2007, Soylu et al. 2008, Liu et al. 2009, Motavalli 2009 and Xu et al. 2010)
\begin{equation}
\left[\hbar^{2} c^{2} \nabla^{2} + \left(\mu c^{2} + \frac{V_{s}(r)}{2}\right)^{2} - \left(E_{R} - \frac{V_{r}(r)}{2}\right)^{2} \right]\Psi(r, \theta, \phi) = 0
\label{r1},
\end{equation}
where $E_{R}$ is the relativistic energy of the system and $\mu$ denotes the rest mass of particle.
For spherically symmetric scalar and vector potentials and by putting
\begin{equation}
\Psi_{n,~\ell,~m}(r, \theta, \phi) = \frac{U_{n,~\ell}(r) }{2} Y_{\ell}^{m}(\theta, \phi),
\label{r2}.
\end{equation}
where $Y_{\ell}^{m}(\theta, \phi)$ is the spherical harmonic function, the radial Klein-Gordon equation for $\ell = 0$ is obtained as
\begin{equation}
\hbar^{2} c^{2} \frac{d^{2} U_{n, \ell}(r)}{dr^{2}} + \left \{ \left[ E_{R}^{2} - \mu^{2} c^{4}\right] - \left[ \mu c^{2}V_{s}(r) + E_{R} V_{s}(r)\right] + \left[ \frac{V_{v}^{2}(r)}{4} -\frac{V_{s}^{2}(r)}{4} \right] \right\}U_{n, \ell}(r) = 0
\label{r3}.
\end{equation}
We consider the case when the scalar and vector potentials are equal to a q-deformed hyperbolic modified P\"{o}schl-Teller potential model, which is expressed as (P\"{o}schl and Teller 1933, Landau and Liftshitz 1977, Fl\"{u}gge 1994, Grosche and Steiner 1998, E$\check{g}$rifes et al. 1999, Dong and Dong 2002, Grosche 2005, Zhao et al. 2005)
\begin{equation}
V(r) = - \frac{D}{\cosh_{q}^{2}(\alpha r)}
\label{r4}.
\end{equation}
In Figure $1$, we show the radial as well as the deformation dependence of the deformed modified P\"{o}schl-Teller potential. On the graph, blue line is the graph of $q = 1$; green line is the graph of $q = 2$; red line is the graph of $q = 3$; dark line is graph of $q = 4$; short dash dark line is the graph of $q = 5$ and long dash dark line is the graph of $q = 6$.
Using the definitions of the deformed hyperbolic functions (Arai 1991), we have:
\begin{eqnarray}
&\sinh_{q}x = \frac{1}{2}(e^{x} - qe^{-x}),~\cosh_{q}x = \frac{1}{2}(e^{x} + qe^{-x}),~ \tanh_{q}x = \frac{\sinh_{q}x}{\cosh_{q}x},\\ \nonumber
& \coth_{q}x = \frac{1}{\tanh_{q}x}
\label{r5},
\end{eqnarray}
We have,
\begin{eqnarray}
&\cosh_{q}^{2}x - \sinh_{q}^{2}x = q,~ \frac{d}{dx}\cosh_{q}x = \sinh_{q}x,~ \frac{d}{dx}\sinh_{q}x = \cosh_{q}x \\ \nonumber
&\frac{d}{dx}\tanh_{q}x = \frac{q}{\cosh_{q}^{2}x}, ~\frac{d}{dx}\coth_{q}x = -\frac{q}{\sinh_{q}^{2}x}
\label{r6},
\end{eqnarray}
where $q>0$ is a real parameter. When $q$ is complex, the above deformed hyperbolic functions are called the generalized deformed (q-deformed) hyperbolic functions (E$\check{g}$rifes et al. 1999, Yi et al. 2004, Grosche 2005, Zhao et al. 2005).
With equation (\ref{r4}), equation (\ref{r3}) becomes
\begin{equation}
U_{n,~\ell}^{~''}(r) + \left[\frac{(E_{R}^{2} - \mu^{2}c^{4})}{\hbar^{2}c^{2}} + \frac{D(\mu c^{2} + E_{R})}{\hbar^{2}c^{2}\cosh_{q}^{2}(\alpha r)}\right]U_{n,~\ell}= 0
\label{r7}.
\end{equation}
Introducing the parameters $\tilde{E_{R}^{2}} = \frac{(E_{R}^{2} - \mu^{2}c^{4})}{\hbar^{2}c^{2}}$, $ \gamma(\gamma + 1) = \frac{D(\mu c^{2} + E_{R})}{ \alpha^{2} \hbar^{2}c^{2}} $ and by using a new variable $y = \tanh_{q}(\alpha r)$, equation (\ref{r7}) becomes
\begin{equation}
\frac{d}{dy}\left[ (1 - y^{2})\frac{d}{dy}U_{n, ~\ell}(y)\right] + \left[\frac{\gamma(\gamma + 1)}{q} - \frac{\xi^{2}}{1 - y^{2}} \right]U_{n, ~\ell}(y) = 0
\label{r8},
\end{equation}
where
\begin{equation}
\xi = \sqrt{\frac{- \tilde{E_{R}^{2}} }{\alpha^{2}}}
\label{r9}.
\end{equation}
We seek for the exact solution of equation (\ref{r8}) via the following ansatz:
\begin{equation}
U_{n, ~\ell}(y) = (1 - y^{2})^{\frac{\xi}{2}}f(y)
\label{r10},
\end{equation}
with equation (\ref{r10}), equation (\ref{r8}) becomes
\begin{equation}
(1 - y^{2}) f''(y) - 2y(\xi + 1)f'(y) + \left[\frac{\gamma(\gamma + 1)}{q} + \frac{\xi(\xi - 2)y^{2}}{(1 - y^{2})} - \xi + \frac{2 \xi y^{2}}{(1 - y^{2})} \right]f(y) = 0
\label{r11}.
\end{equation}
By defining the variable y as
\begin{equation}
\eta = \frac{1}{2}(1- y)
\label{r12},
\end{equation}
then, equation (\ref{r11}) can be rewritten as
\begin{equation}
\eta(1 - \eta) f''(\eta) + \left[(\xi + 1) - 2 \eta(\xi + 1) \right]f'(\eta) + \left[\frac{\gamma(\gamma + 1)}{q} - \xi(\xi + 1) \right]f(\eta) = 0
\label{r13}.
\end{equation}
Equation (\ref{r13}) is the hypergeometric differential equation and has the solutions (Abramowitz and Stegun 1970, Arfken and Weber 1994, Gradsyteyn and Ryzhik 1994, Andrew et al. 2000):
\begin{equation}
\Psi_{n}^{q}(y) = N^{q}(1 - y^{2})^{\frac{\xi}{2}}~ _{2}F_{1} \left(-n,~-n + 2k, ~-n + k + \frac{1}{2};~ \frac{1 - y}{2}\right)
\label{r14},
\end{equation}
where $ _{2}F_{1} \left(-n,~-n + 2k, ~-n + k + \frac{1}{2};~ \frac{1 - y}{2}\right)$ are the hypergeometric polynomials of degree $n$ and $k = \sqrt{\frac{1}{4} + \frac{\gamma (\gamma + 1)}{q}}$.
We next consider the relationship between the hypergeometric functions and the Gegenbauer polynomials (Abramowitz and Stegun 1970, Arfken and Weber 1994, Gradsyteyn and Ryzhik 1994, Andrew et al. 2000), that is,
\begin{equation}
C_{n}^{\lambda}(x) = \frac{\Gamma{(2\lambda + n)}}{n! \Gamma{2 \lambda}} ~ _{2}F_{1} \left(-n,~-n + 2\lambda, ~\lambda + \frac{1}{2};~ \frac{1 - x}{2}\right)
\label{r15},
\end{equation}
On writing equation (\ref{r14}) in terms of the Gegenbauer functions, we have
\begin{equation}
\Psi_{n}^{q}(y) = N^{q}(1 - y^{2})^{\frac{k - n - \frac{1}{2}}{2}} C_{n}^{~ -n + k}(x)
\label{r16},
\end{equation}
where $N^{q}$ is the normalization constant to be determined from the normalization condition
\begin{equation}
\int_{-\infty}^{\infty} [\Psi_{n}^{q}(x)]^{2}dx = \frac{[N^{q}]^{2}}{\alpha} \int_{-1}^{1} (1 - y^{2})^{k - n - \frac{1}{2}} [ C_{n}^{~ -n + k}(y)]^{2}dy =1
\label{r17}.
\end{equation}
The integrand in equation (\ref{r17}) can be evaluated by using the integrals (Abramowitz and Stegun 1970, Arfken and Weber 1994, Gradsyteyn and Ryzhik 1994, Andrew et al. 2000, Dong and Dong 2002, Dong 2007):
\begin{equation}
\int_{-1}^{1} (1 - x)^{\nu - \frac{3}{2}} (1 + x)^{\nu - \frac{1}{2}} [ C_{n}^{ \nu}(x)]^{2}dx = \frac{\pi^{1/2} \Gamma{(\nu - \frac{1}{2})} \Gamma{(2 \nu + n)}} { n! \Gamma{(\nu )} \Gamma{(2 \nu)}}; ~~Re~ \nu~>1
\label{r18},
\end{equation}
since
\begin{eqnarray}
\int_{-1}^{1} (1 - x)^{\nu - \frac{3}{2}} (1 + x)^{\nu - \frac{1}{2}} [ C_{n}^{ \nu}(x)]^{2}dx
= \int_{-1}^{1} (1 - x^{2})^{\nu - \frac{3}{2}} (1 + x) [ C_{n}^{ \nu}(x)]^{2}dx \\ \nonumber
= \int_{-1}^{1} (1 - x^{2})^{\nu - \frac{3}{2}} [ C_{n}^{ \nu}(x)]^{2}dx + \int_{-1}^{1} (1 - x^{2})^{\nu - \frac{3}{2}} x [ C_{n}^{ \nu}(x)]^{2}dx \\ \nonumber
=\frac{\pi^{1/2} \Gamma{(\nu - \frac{1}{2})} \Gamma{(2 \nu + n)}} { n!\Gamma{(\nu )} \Gamma{(2 \nu)}}
\label{r19}.
\end{eqnarray}
Note that $\int_{-1}^{1} (1 - x^{2})^{\nu - \frac{3}{2}} x [ C_{n}^{ \nu}(x)]^{2}dx = 0$, due to the odd parity of the integrand. Then, $N^{q}$, which is the normalization constant in equation (\ref{r17}), gives
\begin{equation}
N^{q} =\sqrt{\frac{\alpha n!(k - n - 1)!(2k - 2n - 1)! }{\pi^{1/2} (k - n - \frac{3}{2})!(2k - n - 1)! } }
\label{r20}.
\end{equation}
Therefore, the wave function of the Klein-Gordon equation for the q-deformed hyperbolic modified P\"{o}schl-
Teller potential in terms of the Gegenbauer functions is
\begin{equation}
\Psi_{n}^{q}(r) = N^{q} ~~ q^{\frac{k - n - \frac{1}{2}}{2}} [\sec h_{q}^{2}(\alpha r)]^{\frac{k - n - \frac{1}{2}}{2}} C_{n}^{~ -n + k}(\tanh_{q}(\alpha r))
\label{r21},
\end{equation}
where $N^{q}$ is as given in equation (\ref{r20}). The corresponding relativistic bound state energy spectra are obtained from equation (\ref{r9}) as
\begin{equation}
E_{R}^{2} - \mu^{2} c^{4} = - \alpha^{2} \hbar^{2} c^{2}\left[ \left(n + \frac{1}{2}\right) -k \right]^{2}; n = 0, 1, 2, \ldots
\label{r22},
\end{equation}
where $k = \sqrt{\frac{1}{4} + \frac{\gamma (\gamma + 1)}{q} }$.
\section{Discussions}
In this section, in the framework of the Klein-Gordon theory with equal scalar and vector potentials, the relativistic bound state energy spectra and the corresponding wave functions for the reflectionless-type potential, q-deformed symmetric hyperbolic modified P\"{o}schl-Teller potential, symmetric modified P\"{o}schl-Teller potential and the PT-symmetric version of the hyperbolic modified P\"{o}schl-Teller potential are obtained by choosing appropriate parameters in the q-deformed hyperbolic modified P\"{o}schl-Teller potential.
\subsection{Reflectionless-type potential}
On putting $q = 1$, $\alpha = 1$ and $D = \frac{1}{2} \lambda(\lambda + 1)$ in the q-deformed hyperbolic modified P\"{o}schl-Teller potential, then, equation (\ref{r4}) reduces to the reflectionless-type potential (Zhao et al. 2005, Setare and Haidari 2010).
\begin{equation}
V(r) = -\frac{1}{2}\lambda (\lambda + 1)\sec h^{2}r
\label{r23},
\end{equation}
where $\lambda$ is an integer, the wave function and the relativistic bound state energy spectra for the Klein-
Gordon equation with equal scalar and vector reflectionless-type potential are respectively obtained as:
\begin{equation}
\Psi_{n}^{RT}(r) = N^{RT} ~~ [\sec h^{2}(r)]^{\frac{k - n - \frac{1}{2}}{2}} C_{n}^{~ -n + k}(\tanh( r))
\label{r24},
\end{equation}
and
\begin{equation}
E_{RT}^{2} - \mu^{2} c^{4} = - \hbar^{2} c^{2}\left[ \left(n + \frac{1}{2}\right) -k \right]^{2}; n = 0, 1, 2, \ldots
\label{r25},
\end{equation}
where $k = \sqrt{\frac{1}{4} + \frac{\lambda (\lambda + 1)(E_{RT} + \mu c^{2})}{2\hbar^{2} c^{2}} }$ and $N^{RT} =\sqrt{\frac{ n!(k - n - 1)!(2k - 2n - 1)! }{\pi^{1/2} (k - n - \frac{3}{2})!(2k - n - 1)! } } $~~.
\subsection{q-deformed symmetric hyperbolic Modified P\"{o}schl-Teller potential}
Choosing $\alpha = 1$ and $D = \lambda^{2} - \frac{1}{4}$ in equation (\ref{r4}) (Grosche 2005), we have
\begin{equation}
V(r) = -\frac{\lambda^{2} - \frac{1}{4}}{\cosh_{q}^{2}(r)}
\label{r26},
\end{equation}
again, the relativistic bound state energy spectra and the wave function for the Klein-Gordon equation with equal scalar and vector symmetric potential are :
\begin{equation}
E_{qs}^{2} - \mu^{2} c^{4} = - \hbar^{2} c^{2}\left[ \left(n + \frac{1}{2}\right) -k \right]^{2}; n = 0, 1, 2, \ldots
\label{r27},
\end{equation}
and
\begin{equation}
\Psi_{n}^{qs}(r) = N^{qs} ~~ q^{\frac{k - n - \frac{1}{2}}{2}} [\sec h_{q}^{2}(r)]^{\frac{k - n - \frac{1}{2}}{2}} C_{n}^{~ -n + k}(\tanh_{q}(r))
\label{r28},
\end{equation}
where $k = \sqrt{\frac{1}{4} + \frac{(\lambda^{2} - \frac{1}{4})(E_{qs} + \mu c^{2})}{ q \hbar^{2} c^{2}} }$ and $N^{qs} =\sqrt{\frac{ n!(k - n - 1)!(2k - 2n - 1)! }{\pi^{1/2} (k - n - \frac{3}{2})!(2k - n - 1)! } } $~~.
\subsection{Symmetric hyperbolic modified P\"{o}schl-Teller potential}
Choosing $q = 1$, $\alpha = 1$ and $D = \lambda^{2} - \frac{1}{4}$ in the q-deformed hyperbolic modified P\"{o}schl-Teller potential (Grosche and Steiner 1998, Oyewumi and Bangudu 1999, 2000, Grypeos et al. 2004, Oyewumi et al. 2004, Efthymiou et al. 2008, Oyewumi 2009), equation (\ref{r4}) becomes
\begin{equation}
V(r) = -\frac{\lambda^{2} - \frac{1}{4}}{\cosh^{2}(r)}
\label{r29},
\end{equation}
with the following solutions:
\begin{equation}
E_{s}^{2} - \mu^{2} c^{4} = - \hbar^{2} c^{2}\left[ \left(n + \frac{1}{2}\right) -k \right]^{2}; n = 0, 1, 2, \ldots
\label{r30},
\end{equation}
and
\begin{equation}
\Psi_{n}^{s}(r) = N^{s} [\sec h^{2}(r)]^{\frac{k - n - \frac{1}{2}}{2}} C_{n}^{~ -n + k}(\tanh(r))
\label{r31},
\end{equation}
where $k = \sqrt{\frac{1}{4} + \frac{(\lambda^{2} - \frac{1}{4})(E_{s} + \mu c^{2})}{ \hbar^{2} c^{2}} }$ and $N^{s} =\sqrt{\frac{ n!(k - n - 1)!(2k - 2n - 1)! }{\pi^{1/2} (k - n - \frac{3}{2})!(2k - n - 1)! } } $~~.
\subsection{PT-Symmetric version of the hyperbolic modified P\"{o}schl-Teller potential}
If we substitute $D = - Dq_{c}$, $q = - q_{c}$ and $q_{c} = e^{2 i \alpha \epsilon}$, then, potential in equation (\ref{r4}) becomes the PT-symmetric version of the hyperbolic modified P\"{o}schl-Teller potential, here $D>0$, $|\epsilon| > \pi/4$, that is,
\begin{equation}
V(r) = \frac{Dq_{c}}{\frac{1}{4} [e^{2 \alpha r} - 2q_{c} + q_{c}^{2}e^{-2 \alpha r}] } = Dq_{c} cosec{h_{q_{c}}^{2}}
(\alpha r)
\label{r32}.
\end{equation}
With these substitutions, it follows that the relativistic bound state energy spectra and the corresponding wave functions are respectively obtained as:
\begin{equation}
E_{q_{c}}^{2} - \mu^{2} c^{4} = - \alpha \hbar^{2} c^{2}\left[ \left(n + \frac{1}{2}\right) -k \right]^{2}; n = 0, 1, 2, \ldots
\label{r33},
\end{equation}
and
\begin{equation}
\Psi_{n}^{q_{c}}(r) = N^{q_{c}}~~ (-q_{c})^{\frac{k - n - \frac{1}{2}}{2}}[cosec h_{q_{c}}^{2}(\alpha r)]^{\frac{k - n - \frac{1}{2}}{2}} C_{n}^{~ -n + k}(\coth_{q_{c}}(\alpha r))
\label{r34},
\end{equation}
where $k = \sqrt{\frac{1}{4} + \frac{D(E_{q_{c}} + \mu c^{2})}{ \alpha \hbar^{2} c^{2}} }$ and $N^{q_{c}} =\sqrt{\frac{ \alpha n!(k - n - 1)!(2k - 2n - 1)! }{\pi^{1/2} (k - n - \frac{3}{2})!(2k - n - 1)! } } $~~.
\section{Conclusion}
In this research work, we have investigated the s-wave bound states of the Klein-Gordon equation with equal scalar and vector q-deformed hyperbolic P\"{o}schl-Teller potential. The energy eigenvalue equation and the normalized radial wave function are obtained analytically. The radial wave functions of spin $0$ particles are expressed in terms of the Gegenbauer functions by exploiting the relationship between the Gegenbauer functions and the hypergeometric functions.
The energy equations and the corresponding wave functions of the reflectionless-type potential, the q-deformed symmetric hyperbolic P\"{o}schl-Teller potential, the symmetric P\"{o}schl-Teller potential, and the PT-symmetric version of the hyperbolic P\"{o}schl-Teller potential are obtained as special cases of the q-deformed hyperbolic P\"{o}schl-Teller potential in the Klein-Gordon theory with equally mixed scalar and vector potentials.
{\bf Aknowledgements}
\noindent
The authors are indebted to the following people for the motivations received from them: Emeritus Prof. K. T. Hecht (USA), Prof. S. H. Dong (Mexico), Prof. C. Berkdermir (Turkey), Prof. C. S. Jia (China), Prof. J. L.
L$\grave{o}$pez-Bonila (Mexico), Prof. Dr. C. Grosche (Germany), Prof. M. N. Hounkonnou (Benin Republic), Prof. J. Govaerts (Belgium) and Prof. W. C. Qiang (China). Also, the authors thank the anonymous kind referees and editors for the constructive comments and suggestions.
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{\bf ACKNOWLEDGEMENTS.}\\
The authors thank Prof. S. H. Dong for many useful and fruitful discussions and for communicating most of his papers to us during the preparation of this work. Also, we express our thanks to Profs. D. Popov, S. Ikhdair and R. Sever for communicating to us most of their works. We also thank the kind referees for the positive and invaluable suggestions which have improved the manuscript greatly.
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\end{document} |
\begin{document}
\title{Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system}
\author{Anupam Pal Choudhury\footnote{APC: Departement Mathematik und Informatik, Universit\"at Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland. Email: [email protected]}, Gianluca Crippa\footnote{GC: Departement Mathematik und Informatik, Universit\"at Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland. Email: [email protected]}, Laura V. Spinolo\footnote{LVS: IMATI-CNR, via Ferrata 1, I-27100 Pavia, Italy. Email: [email protected]}
}
\date{}
\maketitle
\begin{abstract}
We establish existence and uniqueness results for initial boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness
of the initial-boundary value problem for the Keyfitz and Kranzer system of conservation laws in several space dimensions.
\end{abstract}
\section{Introduction}
The Keyfitz and Kranzer system is a system of conservation laws in several space dimensions that was introduced in~\cite{KK} and takes the form
\begin{equation}
\label{e:KK}
\partial_{t} U+\sum_{i=1}^{d} \partial_{x_{i}} (f^{i}(\vert U \vert) U) =0.
\notag
\end{equation}
The unknown is $U: \mathbb R^d \to \mathbb R^N$ and $|U|$ denotes its modulus. Also, for every $i=1, \dots, d$ the function $f^i: \mathbb R \to \mathbb R^N$ is smooth. In this work we establish existence and uniqueness results for the initial-boundary value problem associated to~\eqref{e:KK}.
The well-posedness of the Cauchy problem associated to~\eqref{e:KK} was established by Ambrosio, Bouchut and De Lellis in~\cite{ABD,AD} by relying on a strategy suggested by Bressan in~\cite{Br}. Note that the results in~\cite{ABD,AD} are one of the very few well-posedness results that apply to systems of conservation laws in several spaces dimensions. Indeed, establishing either existence or uniqueness for a general system of conservation laws in several space dimensions is presently a completely open problem, see~\cite{Daf,Serre1,Serre2} for an extended discussion on this topic.
The basic idea underpinning the argument in~\cite{ABD,AD} is that~\eqref{e:KK} can be (formally) written as the coupling between a \emph{scalar} conservation law and a transport equation with very irregular coefficients. The scalar conservation law is solved by using the foundamental work by Kru{\v{z}}kov~\cite{Kr}, while the transport equation is handled by relying on Ambrosio's celebrated
extension of the DiPerna-Lions' well-posendess theory, see~\cite{A} and~\cite{Diperna-Lions}, respectively, and~\cite{AC,Delellis2} for an overview. Note, however, that Ambrosio's theory~\cite{A} does not directly apply
to~\eqref{e:KK} owing to a lack of control on the divergence of the vector fields. In order to tackle this issue, a theory of \textit{nearly incompressible vector fields} was developed, see~\cite{Delellis1} for an extended discussion. Since we will need it in the following, we recall the definition here.
\begin{definition}\label{near-incom}
Let $\Omega \subseteq \mathbb{R}^{d} $ be an open set and $T>0 $. We say that a vector field
$b \in L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d})$ is \textbf{nearly incompressible} if there are a density function $\rho \in L^{\infty}((0,T) \times \Omega) $ and a constant $C > 0 $ such that
\begin{itemize}
\item[i.] $0 \leq \rho \leq C, \ \mathcal{L}^{d+1}-a.e. \ \text{in} \ (0,T) \times \Omega, $ and
\item[ii.] the equation
\begin{equation}
\label{e:continuityrho}
\partial_{t}\rho +\mathrm{div}(\rho b)=0
\end{equation}
holds in the sense of distributions in $(0, T) \times \Omega$.
\end{itemize}
\end{definition}
The analysis in~\cite{ABD, AD, Delellis1} ensures that, if $b \in L^\infty ((0, T) \times \mathbb R^d; \mathbb R^d) \cap BV ((0, T) \times \mathbb R^d; \mathbb R^d)$ is a nearly incompressible vector field with density $\rho \in BV ((0, T) \times \mathbb R^d)$, then the Cauchy problem
$$
\left\{
\begin{array}{ll}
\partial_{t}[ \rho u] +\mathrm{div}[ \rho bu ]=0 &
\text{in $(0, T) \times \mathbb R^d$}\\
u = \overline{u} &
\text{at $t=0$}\\
\end{array}
\right.
$$
is well-posed for every initial datum $\overline{u} \in L^\infty (\mathbb R^d)$. This result is pivotal to the proof of the well-posedness of the Cauchy problem for the Keyfitz and Kranzer system~\eqref{e:KK}. See also~\cite{ACFS} for applications of nearly incompressible vector fields to the so-called chromatography system of conservation laws. Note, furthermore, that here and in the following we denote by $BV$ the space of functions with \emph{bounded variation}, see~\cite{AFP} for an extended introduction.
The present paper aims at extending the analysis in~\cite{ABD, AD, Delellis1} to the case of initial-boundary value problems. First, we establish the well-posedness of initial-boundary value problems with $BV$, nearly incompressible
vector fields, see Theorem~\ref{IBVP-NC} below for the precise statement. In doing so, we rely on well-posedness results for continuity and transport equations with weakly differentiable vector fields established in~\cite{CDS1}, see also~\cite{CDS2} for related results. Next, we discuss the applications to the Keyfitz and Kranzer system~\eqref{e:KK}.
We now provide a more precise description of our results concerning nearly incompressible vector fields. We fix an open, bounded set $\Omega$ and a nearly incompressible vector field $b$ with density $\rho$ and we consider the initial-boundary value problem
\begin{equation}
\left\{
\begin{array}{ll}
\partial_{t} [\rho u]+ \text{div}[\rho u b]=0 &
\text{in $(0,T)\times \Omega$}\\
u=\overline{u} & \text{at $t=0$}\\
u= \overline{g} & \text{on $ \Gamma^{-}$},
\end{array}
\right.
\label{prob-2}
\end{equation}
where $\Gamma^{-}$ is the part of the boundary $(0,T) \times \partial \Omega $ where the characteristic lines of the vector field $\rho b $ are \emph{inward pointing}. Note that, in general, if $b$ and $\rho$ are only weakly differentiable, one cannot expect that the solution $u$ is a regular function. Since $\Gamma^-$ will in general be negligible, then assigning the value of $u$ on $\Gamma^-$ is in general not possible. In~\S~\ref{s:formu} we provide the rigorous (distributional) formulation of the initial-boundary value problem~\eqref{prob-2} by relying on the theory of normal traces for low regularity vector fields, see~\cite{ACM,Anz,CF,CTZ}.
We can now state our well-posedness result concerning~\eqref{prob-2}.
\begin{theorem}\label{IBVP-NC}
Let $T > 0 $ and $\Omega \subseteq \mathbb{R}^{d}$ be an open, bounded set with $C^2 $ boundary. Also, let $b \in BV((0,T) \times \Omega; \mathbb{R}^{d}) \cap L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d})$ be a nearly incompressible vector field with density $\rho \in BV((0,T) \times \Omega) \cap L^{\infty}((0,T) \times \Omega)$, see Definition~\ref{near-incom}. Further, assume that
$\overline{u} \in L^{\infty}(\Omega) $ and $\overline{g} \in L^{\infty}(\Gamma^{-}) $.
Then there is a distributional solution $u \in L^{\infty}((0,T) \times \Omega) $
to \eqref{prob-2} satisfying the maximum principle
\begin{equation}
\label{e:maxprin}
\| u \|_{L^\infty} \leq \max \{ \| \overline u \|_{L^\infty}, \| \overline g \|_{L^\infty} \}.
\end{equation}
Also, if $u_1, \; u_2 \in L^\infty ((0, T) \times \Omega)$ are two different distributional solutions of the
same initial-boundary value problem, then $\rho u_1 = \rho u_2$
$a.e.$ in $(0,T) \times \Omega$.
\end{theorem}
Note that the reason why we do not exactly obtain uniqueness of the function $u$ is because $\rho$ can attain the value $0$. If $\rho$ is bounded away from $0$, then the distributional solution $u$ of~\eqref{prob-2} is unique.
Also, we refer to~\cite{Bar,Boyer,CDS1,CDS2,GS,Mis} for related results on the well-posedness of initial-boundary value problems for continuity and transport equation with weakly differentiable vector fields.
In~\S~\ref{s:KK} we discuss the applications of Theorem~\ref{IBVP-NC} to the Keyfitz and Kranzer system and our main well-posedness result is Theorem~\ref{t:KK}. Note that the proof of Theorem~\ref{t:KK} combines Theorem~\ref{IBVP-NC}, the analysis in~\cite{Delellis1}, and well-posedness results for the initial-boundary value problems for scalar conservation laws established in~\cite{BLN, CR, Serre2}.
\subsection*{Paper outline}
In~\S~\ref{s:prel} we go over some preliminary results concerning normal traces of weakly differentiable vector fields. By relying on these results, in~\S~\ref{s:formu} we provide the rigorous formulation of the initial-boundary value problem~\eqref{prob-2}. In~\S~\ref{s:exi} we establish the existence part of Theorem~\ref{IBVP-NC}, and in~\S~\ref{s:uni} the uniqueness.
In~\S~\ref{s:ssc} we establish some stability and space continuity property results. Finally, in~\S~\ref{s:KK} we discuss the applications to the Keyfitz and Kranzer system.
\subsection*{Notation}
For the reader's convenience, we collect here the main notation used in the present paper.
\begin{itemize}
\item $\mathrm{div}$: the divergence, computed with respect to the $x$ variable only.
\item $\mathrm{Div}$: the complete divergence, i.e. the divergence computed with respect to the $(t, x)$ variables.
\item $\mathrm{Tr} ( B, \partial \Lambda)$: the normal trace of the bounded, measure-divergence vector field $B$ on the boundary of the set $\Lambda$, see \S~\ref{s:prel}.
\item $(\rho u)_0$,
$\rho_0$: the initial datum of the functions $\rho u$ and $\rho$, see Lemma~\ref{trace-existence} and Remark~\ref{r} .
\item $ T (f)$: the trace of the $BV$ function $f$, see Theorem~\ref{bv-trace}.
\item $\mathcal H^s$: the $s$-dimensional Hausdorff measure.
\item $f_{|_E}$: the restriction of the function $f$ to the set $E$.
\item $\mu \llcorner E$: the restriction of the measure $\mu$ to the measurable set $E$.
\item $a.e.$: almost everywhere.
\item $|\mu|$: the total variation of the measure $\mu$.
\item $a \cdot b$: the (Euclidean) scalar product between $a$ and $b$.
\item $\mathbf{1}_E:$ the characteristic function of the measurable set $E$.
\item $\Gamma, \Gamma^-, \Gamma^+, \Gamma^0$: see~\eqref{e:gamma}.
\item $\vec n$: the outward pointing, unit normal vector to $\Gamma$.
\end{itemize}
\section{Preliminary results}
\label{s:prel}
In this section, we briefly recall some notions and results that shall be used in the sequel.
First, we discuss the notion of normal trace for weakly differentiable vector fields, see~\cite{ACM,Anz,CF,CTZ}. Our presentation here closely follows that of \cite{ACM}. Let $\Lambda \subseteq \mathbb{R}^{N} $ be an open set and let us denote by $\mathcal{M}_{\infty}(\Lambda) $, the family of bounded, measure-divergence vector fields. The space $\mathcal{M}_{\infty}(\Lambda) $, therefore, consists of bounded functions $B \in L^{\infty}(\Lambda;\mathbb{R}^{N})$ such that the distributional divergence of $B$ (denoted by $\text{Div} B $) is a locally bounded Radon measure on $\Lambda$.
The normal trace of $B \in \mathcal{M}_{\infty}(\Lambda)$ on the boundary $\partial \Lambda $ can be defined as follows.
\begin{definition}
Let $\Lambda \subseteq \mathbb{R}^{N}$ be an open and bounded set with Lipschitz continuous boundary and let $B \in \mathcal{M}_{\infty}(\Lambda)$. The normal trace of $B$ on $\partial \Lambda $ is a distribution
defined by the identity
\begin{equation}
\Big\langle \emph{Tr}(B,\partial \Lambda), \psi \Big\rangle = \int_{\Lambda} \nabla \psi \cdot B \ dy + \int_{\Lambda} \psi\ d(\emph{Div} B) , \qquad \forall \ \psi \in C^{\infty}_{c}(\mathbb{R}^{N}).
\label{prel-1}
\end{equation}
Here $\emph{Div} B $ denotes the distributional divergence of $B$ and is a bounded Radon measure on $\Lambda $.
\end{definition}
Note that, owing to the Gauss-Green formula, if $B$ is a smooth vector field, then $\mathrm{Tr}(B,\partial \Lambda) = B \cdot \vec n$,
where $\vec n$ denotes the outward pointing, unit normal vector to $\partial \Lambda$.
Note, furthermore, that the analysis in~\cite{ACM} shows that the normal trace distribution satisfies the following properties.
\begin{itemize}
\item[(a)] The normal trace distribution is induced by an $L^{\infty}$ function on $\partial \Lambda $, which we shall continue to refer to as $\mathrm{Tr}(B,\partial \Lambda) $. The bounded function $\mathrm{Tr}(B,\partial \Lambda) $ satisfies \[\Vert \mathrm{Tr}(B,\partial \Lambda) \Vert_{L^{\infty}(\partial \Lambda)} \leq \Vert B \Vert_{L^{\infty}(\Lambda)}. \]
\item[(b)] Let $\Sigma $ be a Borel set contained in $\partial \Lambda_{1} \cap \partial \Lambda_{2} $, and let $\vec{n}_{1}=\vec{n}_{2} \ \text{on}\ \Sigma$ (here $\vec{n}_{1},\vec{n}_{2} $ denote the outward pointing, unit normal vectors to $\partial \Lambda_{1},\partial \Lambda_{2} $ respectively). Then
\begin{equation}
\mathrm{Tr}(B, \partial \Lambda_{1}) = \mathrm{Tr}(B, \partial \Lambda_{2}) \qquad \text{$\mathcal{H}^{N-1}$-a.e.~on $\Sigma$.}
\label{prel-2}
\end{equation}
\end{itemize}
In the following we will use several times the following renormalization result, which was established in~\cite{ACM}.
\begin{theorem}\label{trace-renorm}
Let $B \in BV (\Lambda;\mathbb{R}^{N})\cap L^{\infty}(\Lambda;\mathbb{R}^{N}) $ and $w \in L^{\infty}(\Lambda) $ be such that $\emph{Div} (wB )$ is a Radon measure. If $\Lambda' \subset \subset \Lambda $ is an open set with bounded and Lipschitz continuous boundary and $h \in C^{1}(\mathbb{R})$, then
\begin{equation}
\emph{Tr}(h(w)B,\partial \Lambda')=h\left(\frac{\emph{Tr}(wB,\partial \Lambda')}{\emph{Tr}(B,\partial \Lambda')}\right) \emph{Tr}(B,\partial \Lambda') \qquad \text{$\mathcal{H}^{N-1}$-a.e.~on~$\partial \Lambda'$,}
\notag
\end{equation}
where the ratio $\displaystyle{\frac{\emph{Tr}(wB,\partial \Lambda')}{\emph{Tr}(B,\partial \Lambda')} }$ is arbitrarily defined at points where the trace $\emph{Tr}(B,\partial \Lambda') $ vanishes.
\end{theorem}
We can now introduce the notion of normal trace on a general bounded, Lipschitz continuous, oriented hypersurface $\Sigma \subseteq \mathbb{R}^{N}$ in the following manner. Since $\Sigma $ is oriented, an orientation of the normal vector $\vec{n}_{\Sigma} $ is given. We can then find a domain $\Lambda_{1} \subseteq \mathbb{R}^{N} $ such that $\Sigma \subseteq \partial \Lambda_{1} $ and the normal vectors $\vec{n}_{\Sigma}, \vec{n}_{1} $ coincide. Using \eqref{prel-2}, we can then define
\[\text{Tr}^{-}(B,\Sigma):= \text{Tr}(B,\partial \Lambda_{1}). \]
Similarly, if $\Lambda_{2} \subseteq \mathbb{R}^{N} $ is an open set satisfying $\Sigma \subseteq \partial \Lambda_{2} $, and $\vec{n}_{2}=-\vec{n}_{\Sigma} $, we can define
\[\text{Tr}^{+}(B,\Sigma):=- \text{Tr}(B,\partial \Lambda_{2}). \]
Furthermore we have the formula
\[(\text{Div} B)\llcorner \Sigma= \Big( \text{Tr}^{+}(B,\Sigma)-\text{Tr}^{-}(B,\Sigma) \Big) \mathcal{H}^{N-1} \llcorner \Sigma. \]
Thus $\text{Tr}^{+} $ and $\text{Tr}^{-} $ coincide $\mathcal{H}^{N-1}- $a.e. on $\Sigma$ if and only if $\Sigma $ is a $(\text{Div} B)$-negligible set.\\
We next recall some results from \cite{ACM} concerning space continuity.
\begin{definition}\label{graph}
A family of oriented surfaces $\{\Sigma_{r} \}_{r \in I} \subseteq \mathbb{R}^{N} $ (where $I \subseteq \mathbb{R}$ is an open interval) is called a family of graphs if there
exist
\begin{itemize}
\item a bounded open set $D \subseteq \mathbb{R}^{N-1}$,
\item a Lipschitz function $f:D \rightarrow \mathbb{R}$,
\item a system of coordinates $(x_{1},\cdots,x_{N})$
\end{itemize}
such that the following holds true:
For each $r \in I$, we can write
\begin{equation}
\Sigma_{r}=\big\{(x_{1},\cdots,x_{N}): f(x_{1},\cdots,x_{N-1})-x_{N}=r \big\},
\label{ACM-99}
\end{equation}
and the orientation of $\Sigma_{r}$ is determined by the normal $\displaystyle{\frac{(-\nabla f,1)}{\sqrt{1+\vert \nabla f \vert^{2}}} }$.
\end{definition}
We now quote a space continuity result.
\begin{theorem}[see \cite{ACM}]\label{Weak-continuity}
Let $B \in \mathcal{M}_{\infty}(\mathbb{R}^{N})$ and let $\{\Sigma_{r} \}_{r \in I} $ be a family of graphs as above. For a fixed $r_{0} \in I$, let us define the functions $\alpha_{0}, \alpha_{r}: D \rightarrow \mathbb{R} $ as
\begin{equation}
\begin{aligned}
\alpha_{0}(x_{1},\cdots,x_{N-1})&:=\emph{Tr}^{-}(B,\Sigma_{r_{0}})(x_{1},\cdots,x_{N-1},f(x_{1},\cdots,x_{N-1})-r_{0}), \ \text{and} \\
\alpha_{r}(x_{1},\cdots,x_{N-1})&:=\emph{Tr}^{+}(B,\Sigma_{r})(x_{1},\cdots,x_{N-1},f(x_{1},\cdots,x_{N-1})-r) .
\end{aligned}
\label{ACM-100}
\end{equation}
Then $\alpha_{r} \stackrel{*}{\rightharpoonup} \alpha_{0} $ weakly$^{*}$ in $L^{\infty}(D,\mathcal{L}^{N-1} \llcorner D) $ as $r \rightarrow r^{+}_{0}$.
\end{theorem}
We will also need the following result, which was originally established in~\cite{CDS1}.
\begin{lemma}\label{extension}
Let $\Lambda \subseteq \mathbb{R}^{N}$ be an open and bounded set with bounded and Lipschitz continuous boundary and let $B$ belong to
$\mathcal{M}_{\infty}(\Lambda)$. Then the vector field
\begin{equation}
\tilde{B}(z):=
\left\{\begin{array}{ll}
B(z) & z \in \Lambda \\
0 & \text{otherwise}
\end{array}\right.
\notag
\end{equation}
belongs to $\mathcal{M}_{\infty}(\mathbb{R}^{N})$.
\end{lemma}
We conclude by recalling some results concerning traces of $BV$ functions and we refer to~\cite[\S 3]{AFP} for a more extended discussion.
\begin{theorem}
\label{bv-trace}
Let $\Lambda \subseteq \mathbb{R}^{N}$ be an open and bounded set with bounded and Lipschitz continuous boundary. There exists a bounded linear mapping
\begin{equation}
T: BV(\Lambda) \rightarrow L^{1}(\partial \Lambda;\mathcal{H}^{N-1})
\label{ACM-101}
\end{equation}
such that $T (f) = f_{|_{\partial \Lambda}}$ if $f$ is continuous up to the boundary. Also,
\begin{equation}
\int_{\Lambda} \nabla \psi \cdot f \ dy = - \int_{\Lambda} \psi \ d(\emph{\text{Div}} f) + \int_{\partial \Lambda} \psi \ Tf \cdot \vec n \ d\mathcal{H}^{N-1},
\label{ACM-102}
\end{equation}
for all $f \in BV(\Lambda)$ and $\psi \in C^{\infty}_{c}(\mathbb{R}^{N})$. In the above expression, $\vec n$ denotes the outward pointing, unit normal vector to $\partial \Lambda$.
\end{theorem}
By comparing~\eqref{prel-1} and~\eqref{ACM-102} we conclude that
\begin{equation}
\label{e:equal}
\mathrm{Tr} (f, \partial \Lambda) = T (f) \cdot \vec n, \quad \text{for every $f \in BV (\Lambda)$}.
\end{equation}
By combining Theorems~3.9 and~3.88 in~\cite{AFP} we get the following result.
\begin{theorem}[\cite{AFP}]
\label{t:traceafp}
Assume $\Lambda \subseteq \mathbb R^N$ is an open set with bounded and Lipschitz continuous boundary. If $f \in BV (\Lambda; \mathbb R^m)$, then there is a sequence $\{\tilde f_m \} \subseteq C^\infty (\Lambda)$ such that
\begin{equation}
\label{e:tracefp}
\tilde f_m \to f \; \text{ strongly in $L^1 (\Lambda; \mathbb R^m)$},
\qquad
T (\tilde f_m) \to T(f) \text{ strongly in $L^1 (\partial \Lambda; \mathbb R^m)$}.
\end{equation}
Also, we can choose $\tilde f_m$ in such a way that
\begin{itemize}
\item
$\tilde f_m \ge 0$ if $f \ge 0$,
\item if $f \in L^\infty (\Lambda; \mathbb R^m)$, then
\begin{equation}
\label{e:four}
\| \tilde f_m \|_{L^\infty} \leq 4 \| f \|_{L^\infty}.
\end{equation}
\end{itemize}
\end{theorem}
A sketch of the proof of Theorem~\ref{t:traceafp} is provided in~\S~\ref{s:proof1}.
\section{Distributional formulation of the problem}
\label{s:formu}
In this section, we follow~\cite{Boyer,CDS1} and we provide the distributional formulation of the problem \eqref{prob-2}. We first establish a preliminary result.
\begin{lemma}\label{trace-existence}
Let $\Omega \subseteq \mathbb{R}^{d}$ be an open bounded set with $C^2$ boundary
and let $T > 0 $. We assume that
$b \in L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d}) $ is a nearly incompressible vector field with density $\rho \in L^{\infty}((0,T) \times \Omega) $, see Definition \ref{near-incom}. If $u \in L^{\infty}((0,T) \times \Omega)$ satisfies
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho u (\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt= 0, \quad \forall \ \phi \in \mathcal{C}^{\infty}_{c} ((0,T) \times \Omega),
\label{iden-2}
\end{equation}
then there are two unique functions, which we henceforth denote by $\emph{Tr}(\rho u b) \in L^{\infty}((0,T) \times \partial \Omega) $ and $(\rho u)_{0} \in L^{\infty}(\Omega)$, that satisfy
\begin{equation}
\int_{0}^{T} \! \! \int_{\Omega} \rho u (\partial_{t} \psi+ b \cdot \nabla \psi) \ dx dt= \int_{0}^{T} \! \! \int_{\partial \Omega} \emph{Tr}(\rho u b) \psi \
d\mathcal{H}^{d-1}\ dt - \int_{\Omega} \psi(0,\cdot) (\rho u)_{0}\ dx, \quad \forall \psi \in \mathcal{C}^{\infty}_{c} ([0,T) \times \mathbb{R}^{d}).
\label{iden-3}
\end{equation}
Also, we have the bounds
\begin{equation}
\label{e:maxprintraces2}
\| \emph{Tr}(\rho u b) \|_{L^\infty((0,T) \times \partial \Omega) } , \;
\| (\rho u)_{0} \|_{ L^{\infty}(\Omega)}
\leq \max\{ \| \rho u \|_{L^\infty((0,T) \times \Omega) } ; \| \rho u b \|_{L^\infty((0,T) \times \Omega) } \}.
\end{equation}
\end{lemma}
\begin{proof}
First of all, let us note that the uniqueness of such functions follow from the liberty in choosing the test functions $\psi$. Therefore
it is enough to discuss the existence of the functions with the above properties.
Let us define
\begin{equation}
B(t,x):= \left\{
\begin{array}{ll}
(u \rho, u \rho b) & (t,x) \in (0,T)\times \Omega \\
0 &\text{elsewhere in}\ \mathbb{R}^{d+1}.\\
\end{array} \right.
\label{e:extend}
\end{equation}
Then $B \in L^{\infty}(\mathbb{R}^{d+1})$ and from \eqref{iden-2}, it also follows that $\big[\text{Div} B \llcorner {(0,T) \times \Omega} \big]=0 $.
We can now apply Lemma \ref{extension} with $\Lambda= (0,T) \times \Omega $ to conclude that $B \in \mathcal{M}_{\infty}(\mathbb{R}^{d+1}).$
Hence $B$ induces the existence of normal trace on $\partial \Lambda$.
Let
\begin{equation}
\text{Tr}(\rho u b):= \text{Tr} (B,\partial \Lambda) \Big\vert_{(0,T) \times \partial \Omega}, \ \
(\rho u)_{0}:= -\text{Tr}(B,\partial \Lambda) \Big\vert_{\{0 \} \times \Omega}.
\notag
\end{equation}
The identity \eqref{iden-3} then follows from \eqref{prel-1} by virtue of the fact that $\text{Div}B=0 \ \text{in}\ (0,T)\times \Omega $.
\end{proof}
\begin{remark}
\label{r}
We define the vector field $P:=(\rho,\rho b) $ and we point out that $P \in {L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d+1})}$ since
$\rho$ and $b$ are both bounded functions. By introducing the same extension as in~\eqref{e:extend} and using the fact that
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho (\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt= 0, \quad \forall \ \phi \in \mathcal{C}^{\infty}_{c} ((0,T) \times \Omega),
\notag
\end{equation}
we can argue as in the proof of the above lemma to establish the existence of unique functions $\emph{Tr}(\rho b) \in L^{\infty}((0,T) \times \partial \Omega) $ and
$\rho_0 \in L^\infty(\Omega)$ defined as
$$
\emph{Tr}(\rho b):= \emph{Tr}(P, \partial \Lambda) \Big\vert_{(0,T)
\times \partial \Omega}, \quad
\rho_0 : = - \emph{Tr}(P, \partial \Lambda) \Big\vert_{\{ 0 \} \times
\Omega}.
$$
In this way, we can give a meaning to the normal trace $\mathrm{Tr} (\rho b)$ and to the initial datum $\rho_0$. Also, we have the bounds
\begin{equation}
\label{e:maxprintraces1}
\| \emph{Tr}(\rho b) \|_{L^\infty((0,T) \times \partial \Omega) } , \;
\| \rho_{0} \|_{ L^{\infty}(\Omega)}
\leq \max\{ \| \rho \|_{L^\infty((0,T) \times \Omega) } ; \| \rho b \|_{L^\infty((0,T) \times \Omega) }\}.
\end{equation}
\end{remark}
We can now introduce the distributional formulation to the problem \eqref{prob-2} by using Lemma \ref{trace-existence}.
We introduce the following notation:
\begin{equation}
\left.
\begin{array}{ll}
\Gamma: = (0, T) \times \partial \Omega,
& \Gamma^{-}:= \{(t,x) \in \Gamma: \ \text{Tr} (\rho b)(t,x)<0 \},\\
\Gamma^{+}:=\{(t,x) \in \Gamma: \ \text{Tr} (\rho b)(t,x) > 0 \}, &
\Gamma^0:=\{(t,x) \in \Gamma: \ \text{Tr} (\rho b)(t,x) = 0 \}. \\
\end{array}
\right.
\label{e:gamma}
\end{equation}
\begin{definition}
\label{d:distrsol}
Let $\Omega \subseteq \mathbb{R}^{d}$ be an open bounded set with $C^2$ boundary and let $T > 0 $. Let $b \in
L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d}) $ be a nearly incompressible vector field
with density $\rho $, see Definition~\ref{near-incom}. Fix $\overline{u} \in L^\infty (\Omega)$ and $\overline{g} \in L^\infty (\Gamma^-)$.
We say that a function $u \in L^{\infty}((0,T)\times \Omega)$ is a distributional solution of \eqref{prob-2} if the following conditions are satisfied:
\begin{itemize}
\item[i.] $u$ satisfies \eqref{iden-2};
\item[ii.] $(\rho u)_{0}= \overline{u} \rho_0 $;
\item[iii.] $\emph{Tr}(\rho u b)= \overline{g} \emph{Tr}(\rho b) $ on the set $\Gamma^{-}$.
\end{itemize}
\end{definition}
\section{Proof of Theorem~\ref{IBVP-NC}: existence of solution}
\label{s:exi}
In this section we establish the existence part of Theorem~\ref{IBVP-NC}, namely we prove the existence of functions $u \in L^{\infty}((0,T)\times \Omega) $ and $w \in L^{\infty}(\Gamma^{0}\cup \Gamma^+ ) $ such that for every $\psi \in C^{\infty}_{c}([0,T)\times \mathbb{R}^{d})$,
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho u (\partial_{t} \psi+b \cdot \nabla \psi) \ dx dt= \int_{\Gamma^{-}} \overline{g} \text{Tr}(\rho b) \psi \ d\mathcal{H}^{d-1} dt +\int_{\Gamma^{+}\cup \Gamma^0} \! \!\text{Tr}(\rho b) \psi w \ d\mathcal{H}^{d-1} dt-\int_{\Omega} \rho_0
\ \overline{u}\ \psi(0,\cdot)\ dx .
\label{weak-exist1}
\end{equation}
We proceed as follows: first, in~\S~\ref{ss:as} we introduce an approximation scheme. Next, in~\S~\ref{ss:limit} we pass to the limit and establish existence.
\subsection{Approximation scheme}
\label{ss:as}
In this section we rely on the analysis in~\cite[\S~3.3]{Delellis1}, but we employ a more refined approximation scheme which guarantees strong convergence of the traces.
We set $\Lambda: =(0, T) \times \Omega$ and we recall that by assumption $\rho \in BV(\Lambda) \cap L^\infty (\Lambda).$ We apply Theorem~\ref{t:traceafp} and we select a sequence $\{ \tilde \rho_m \}
\subseteq C^\infty (\Lambda)$ satisfying~\eqref{e:tracefp} and~\eqref{e:four}. Next, we set
\begin{equation}
\label{e:rhoenne}
\rho_m: = \frac{1}{m} + \tilde \rho_m \ge \frac{1}{m}.
\end{equation}
We then apply Theorem~\ref{t:traceafp} to the function $b \rho$ and we set
\begin{equation}
\label{e:benne}
b_m : = \frac{\widetilde{(b \rho)}_m}{\rho_m}.
\end{equation}
Owing to Theorem~\ref{t:traceafp} we have
\begin{equation}
\label{e:elle1conv}
\rho_m \to \rho \;
\text{strongly in $L^1 ((0, T) \times \Omega)$}, \quad
b_m \rho_m \to b \rho
\;
\text{strongly in $L^1 ((0, T) \times \Omega;\mathbb R^d)$}.
\end{equation}
and, by using the identity~\eqref{e:equal},
\begin{equation}
\label{e:traceconv}
\begin{split}
\mathrm{Tr} (\rho_m) \to \mathrm{Tr} (\rho)& \; \text{strongly in $L^1 (\Gamma)$}, \quad
\mathrm{Tr} (\rho_mb_m) \to \mathrm{Tr} (\rho b) \;
\text{strongly in $L^1 (\Gamma)$}, \\
& \quad
\rho_{m0} \to \rho_0 \;
\text{strongly in $L^1 (\Omega)$}.
\end{split}
\end{equation}
Note, furthermore, that
\begin{equation}
\label{e:linftytraces}
\| \mathrm{Tr} (b_m \rho_m ) \|_{L^\infty} \stackrel{\eqref{e:maxprintraces1}}{\leq} \| b_m \rho_m \|_{L^\infty}
\stackrel{\eqref{e:four}}{\leq} 4 \| b \rho \|_{L^\infty}.
\end{equation}
In the following, we will use the notation
\begin{equation}
\label{e:gammadef}
\Gamma_m^- : = \big\{(t, x) \in \Gamma: \; \mathrm{Tr} (\rho_m b_m)
< 0 \big\},
\qquad
\Gamma_m^+ : = \big\{(t, x) \in \Gamma: \; \mathrm{Tr} (\rho_m b_m) > 0 \big\}
\end{equation}
Finally, we extend the function $\overline{g}$ to the whole $\Gamma$ by setting it equal to $0$ outside $\Gamma^-$ and we construct two sequences $\{ \overline{g}_m \} \subseteq C^1 (\Gamma)$ and
$\{\overline{u}_m \} \subseteq C^\infty (\Omega)$ such that
\begin{equation}
\label{e:convbdata}
\overline{g}_m \to \overline{g} \; \text{strongly in $L^1 (\Gamma)$}, \quad
\overline{u}_m \to \overline{u} \; \text{strongly in $L^1 (\Omega)$}
\end{equation}
and
\begin{equation}
\label{e:tomaxprin}
\| \overline{g}_m \|_{L^\infty} \leq \| \overline{g} \|_{L^\infty}, \quad
\| \overline{u}_m \|_{L^\infty} \leq \| \overline{u} \|_{L^\infty}.
\end{equation}
We can now define the function $u_m$ as the solution of the initial-boundary value problem
\begin{equation}
\left\{
\begin{array}{ll}
\partial_{t} u_m+b_m \cdot \nabla u_m=0 & \text{on $(0, T) \times \Omega$} \\
u_m=\overline{u}_m & \text{at $t=0$}\\
u_m= \overline{g}_m & \text{on} \; \tilde \Gamma^{-}_m,
\end{array}
\right.
\label{exist3}
\end{equation}
where $\tilde \Gamma^-_m$ is the subset of $\Gamma$ such that the characteristic lines of $b_m$ starting at a point in $\tilde \Gamma^-_m$ are entering $(0, T) \times \Omega$. We recall~\eqref{e:gammadef} and we point out that
$$
\Gamma^-_m
\subseteq
\tilde \Gamma^-_m \subseteq
\big\{ (t, x) \in \Gamma:
\; b_m \cdot \vec n \leq 0 \big\}.
$$
In the previous expression, $\vec n$ denotes as the outward pointing, unit normal vector to $\partial \Omega$. By using the classical method of characteristics (see also~\cite{Bar}) we establish the existence of a solution $u_m$ satisfying
\begin{equation}
\Vert u_m\Vert_{\infty} \leq \max\{\Vert \overline{u}_m \Vert_{\infty}, \Vert \overline{g}_m \Vert_{\infty} \}
\stackrel{\eqref{e:tomaxprin}}{\leq} \max \{\Vert \overline{u} \Vert_{\infty}, \Vert \overline{g} \Vert_{\infty} \}.
\label{mp}
\end{equation}
We now introduce the function $h_m$ by setting
\begin{equation}
\label{e:accaenne}
h_m : = \partial_t \rho_m + \mathrm{div} (b_m \rho_m)
\end{equation}
and by using the equation at the first line of~\eqref{exist3} we get that
$$
\partial_t (\rho_m u_m ) + \mathrm{div} (b_m \rho_m u_m ) = h_m u_m.
$$
Owing to the Gauss-Green formula, this implies that,
for every $\psi \in C^\infty_c ([0, T) \times \mathbb R^d)$,
\begin{equation}
\begin{aligned}
&\int_{0}^{T} \int_{\Omega} \rho_m u_m [\partial_{t} \psi+ b_m \cdot \nabla \psi ] \ dx dt
+ \int_0^T \int_\Omega h_m u_m \psi \, dx dt
\\
&\quad = -\int_{\Omega} \psi(0,x) \overline{\rho}_{m0} \overline{u}_{m} \ dx- \int_{0}^{T} \! \! \int_{\partial \Omega}
\psi u_m \rho_m b_m \cdot \vec n \, d\mathcal{H}^{d-1} dt \\
& \quad =
-\int_{\Omega} \psi(0,x) \overline{\rho}_{m0} \overline{u}_{m} \ dx-
\int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^-}
\overline{g}_{m} \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt
- \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^+}
u_m \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt.
\end{aligned}
\label{weak-exist2}
\end{equation}
In the above expression, we have used the notation introduced in~\eqref{e:gammadef} and the fact that
$\mathrm{Tr} (\rho_m b_m )=0$ on~${\Gamma \setminus (\Gamma^-_m \cup \Gamma^+_m)}$.
\subsection{Passage to the limit}
\label{ss:limit}
Owing to the uniform bound~\eqref{mp}, there are a subsequence of $\{ u_m \}$ (which, to simplify notation, we do not relabel) and a function $u \in L^\infty ((0, T) \times \Omega$
such that
\begin{equation}
\label{e:uweaks}
u_m \weaks u \; \text{weakly$^\ast$ in $L^\infty ((0, T) \times \Omega)$. }
\end{equation}
The goal of this paragraph is to show that the function $u$ in~\eqref{e:uweaks} is a distributional solution of~\eqref{prob-2} by passing to the limit in~\eqref{weak-exist2}.
We first introduce a technical lemma.
\begin{lemma}
\label{l:meyerserrin}
We can construct the approximating sequences $\{ \rho_m \}$ and $\{ b_m \}$
in such a way that the sequence $\{ h_m \}$ defined as in~\eqref{e:accaenne} satisfies
\begin{equation}
\label{e:convaccaemme}
h_m \to 0 \; \text{strongly in $L^1 ((0, T) \times \Omega)$}.
\end{equation}
\end{lemma}
The proof of Lemma~\ref{l:meyerserrin} is deferred to~\S~\ref{s:proof1} . For future reference, we state the next simple convergence result as a lemma.
\begin{lemma}
\label{l:traces}
Assume that
\begin{equation}
\label{e:hyp}
\mathrm{Tr} (\rho_m b_m )\to \mathrm{Tr}(\rho b)
\; \text{strongly in $L^1 (\Gamma)$}.
\end{equation}
Let $\Gamma^-_m$ and $\Gamma^+_m$ as in~\eqref{e:gammadef} and $\Gamma^-$ and $\Gamma^+$ as in~\eqref{e:gamma}, respectively. Then, up to subsequences,
\begin{equation}
\label{e:convchar1}
\mathbf{1}_{\Gamma^-_m} \to \mathbf{1}_{\Gamma^-} +
\mathbf{1}_{\Gamma'} \; \text{strongly in $L^1 (\Gamma)$}
\end{equation}
and
\begin{equation}
\label{e:convchar2}
\mathbf{1}_{\Gamma^+_m} \to \mathbf{1}_{\Gamma^+} +
\mathbf{1}_{\Gamma''} \; \text{strongly in $L^1 (\Gamma)$},
\end{equation}
where $\Gamma'$ and $\Gamma''$ are (possibly empty) measurable sets satisfying
\begin{equation}
\label{e:subsetgamma0}
\Gamma', \Gamma'' \subseteq \Gamma^0.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{l:traces}]
Owing to~\eqref{e:hyp} we have that, up to subsequences, the sequence
$\{ \mathrm{Tr} (\rho_m b_m) \}$ satisfies
$$
\mathrm{Tr} (\rho_m b_m) (t, x) \to \mathrm{Tr} (\rho b)(t, x),
\quad \text{for $\mathcal{L}^1 \otimes \mathcal{H}^{d-1}$-almost
every $(t, x) \in \Gamma.$}
$$
Owing to the Lebesgue Dominated Convergence Theorem, this implies~\eqref{e:convchar1} and~\eqref{e:convchar2}.
\end{proof}
We can now pass to the limit in all the terms in~\eqref{weak-exist2}. First, by combining~\eqref{e:elle1conv},~\eqref{mp},~\eqref{e:uweaks} and~\eqref{e:convaccaemme} we get that
\begin{equation}
\label{e:conv11}
\int_{0}^{T} \! \! \int_{\Omega} \rho_m u_m [\partial_{t} \psi+ b_m \cdot \nabla \psi ] \ dx dt
+ \int_0^T \! \! \int_\Omega h_m u_m \psi \, dx dt
\to \int_{0}^{T} \! \! \int_{\Omega}
\rho u [\partial_{t} \psi+ b \cdot \nabla \psi ] \ dx dt,
\end{equation}
for every $ \psi \in C^\infty_c ([0, T) \times \mathbb R^d)$. Also, by combining the second line of~\eqref{e:traceconv} with~\eqref{e:convbdata} and~\eqref{e:tomaxprin} we arrive at
\begin{equation}
\label{e:conv21}
\int_{\Omega} \psi(0,x) {\rho}_{m0} \overline{u}_{m} \ dx
\to
\int_{\Omega} \psi(0,x) {\rho}_{0} \overline{u} \ dx ,
\end{equation}
for every $ \psi \in C^\infty_c ([0, T) \times \mathbb R^d). $ Next, we combine~\eqref{e:traceconv},~\eqref{e:convbdata},~\eqref{e:tomaxprin},~\eqref{e:convchar1},~\eqref{e:subsetgamma0} and the fact that
$\mathrm{Tr}(\rho b) =0 $ on $\Gamma^0$ to get that
\begin{equation}
\label{e:conv4}
\begin{split}
\int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^-}
\overline{g}_{m} \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt \to
&
\int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma^-}
\overline{g} \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt \\
& =
\int_{0}^{T} \! \! \int_{\Gamma^-}
\overline{g} \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt,
\end{split}
\end{equation}
for every $\psi \in C^\infty_c ([0, T) \times \Omega; \mathbb R^d)$. We are left with the last term in~\eqref{weak-exist2}: first, we denote by $u_{m{|_\Gamma}}$ the restriction of $u_m$ to $\Gamma$. Since $u_m$ is a smooth function, then
$$
\| u_{m{|_\Gamma}} \|_{L^\infty (\Gamma)} \leq
\| u_m \|_{L^\infty ((0, T) \times \Omega)}
\stackrel{\eqref{mp}}{\leq}
\max \big\{ \| \bar u \|_{L^\infty}, \| \bar g \|_{L^\infty} \big\}
$$
and hence there is a function $w \in L^\infty (\Gamma)$ such that, up to subsequences,
\begin{equation}
\label{e:convw}
u_{m{|_\Gamma}} \weaks w \; \text{weakly$^\ast$ in $L^\infty (\Gamma)$}.
\end{equation}
By combining~\eqref{e:traceconv},~\eqref{e:convchar2},~\eqref{e:convw} and the fact that $\mathrm{Tr} (\rho b) =0$ on $\Gamma^0$ we get that
\begin{equation}
\begin{split}
\label{e:conv5}
\int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^+}
u_m \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt \to &
\int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma^+}
w \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt \\
& = \! \! \int_{\Gamma^+ \cup \Gamma^0}
w \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt .
\end{split}
\end{equation}
By combining~\eqref{e:conv11},~\eqref{e:conv21},~\eqref{e:conv4} and~\eqref{e:conv5} we get that $u$ satisfies~\eqref{weak-exist1} and this establishes existence of a distributional solution of~\eqref{prob-2}.
\subsection{Proof of Lemma~\ref{l:meyerserrin}}
\label{s:proof1}
To ensure that~\eqref{e:convaccaemme} holds we use the same approximation \emph{\`a la} Meyers-Serrin as in~\cite[pp.122-123]{AFP}. We now recall some details of the construction. First, we fix a countable family of open sets
$\big\{ \Lambda_h \big\}$ such that
\begin{itemize}
\item[i.] $\Lambda_h$ is compactly contained in $\Lambda$, for every $h$;
\item[ii.] $\big\{ \Lambda_h \big\}$ is a covering of $\Lambda$, namely
$$
\bigcup_{h=1}^\infty \Lambda_h = \Lambda;
$$
\item[iii.] every point in $\Lambda$ is contained in at most $4$ sets $\Lambda_h$.
\end{itemize}
Next, we consider a partition of unity associated to $\big\{ \Lambda_h \big\}$, namely a countably family of smooth, nonnegative functions $\{ \zeta_h \}$ such that
\begin{itemize}
\item[iv.] we have
\begin{equation}
\label{e:isone}
\sum_{h=1}^\infty \zeta_h \equiv1
\quad \text{in $\Omega$}
;
\end{equation}
\item[v.] for every $h>0$, the support of $\zeta_h$ is contained in $\Lambda_h$.
\end{itemize}
Finally, we fix a convolution kernel $\eta: \mathbb R^{d+1} \to \mathbb R^+$ and we define $\eta_\ee$ by setting
$$
\eta_\ee (z) : = \frac{1}{\ee^{d+1}} \eta
\left(
\frac{z}{\ee}
\right)
$$
For every $m>0$ and $h>0$ we can choose $\ee_{mh}$ in such a way that
$(\rho \zeta_h) \ast \eta_{\ee_{mh}} $ is supported in $\Lambda_h$ and furthermore
\begin{equation}
\label{e:ms2}
\int_0^T \! \! \int_\Omega
| \rho \zeta_h - ( \rho \zeta_h) \ast \eta_{\ee_{mh}} |+
| \rho \, \partial_t \zeta_h - ( \rho \, \partial_t \zeta_h) \ast \eta_{\ee_{mh}}|
+
| \rho b \cdot \nabla \zeta_h - ( \rho b \cdot \nabla \zeta_h) \ast \eta_{\ee_{mh}}| dx dt
\leq \frac{2^{-h}}{m}.
\end{equation}
We then define $\tilde \rho_m$ by setting
\begin{equation}
\label{e:ms3}
\tilde \rho_m : = \sum_{h=1}^\infty
(\rho \zeta_h) \ast \eta_{\ee_{mh}} .
\end{equation}
The function $(\widetilde{\rho b})_m$ is defined analogously. Next, we proceed as in~\cite[p.123]{AFP} and we point out that
\begin{equation*}
\begin{split}
h_m \stackrel{\eqref{e:accaenne}}{=} &
\partial_t \rho_m + \mathrm{div} ({\rho_m b_m})
\stackrel{\eqref{e:accaenne}}{=}
\underbrace{\sum_{h=1}^\infty
(\partial_t \rho \zeta_h) \ast \eta_{\ee_{mh}} +
\sum_{h=1}^\infty
(\mathrm{div} (\rho b) \zeta_h) \ast \eta_{\ee_{mh}}}_{= 0
\; \text{by~\eqref{e:continuityrho} } }
\\ &\quad +
\sum_{h=1}^\infty
(\rho \, \partial_t \zeta_h) \ast \eta_{\ee_{mh}} +
\sum_{h=1}^\infty
(\rho b \cdot \nabla \zeta_h) \ast \eta_{\ee_{mh}}
\\ & \stackrel{\eqref{e:isone}}{=}
\sum_{h=1}^\infty
(\rho \, \partial_t \zeta_h) \ast \eta_{\ee_{mh}} -
\rho \sum_{h=1}^\infty \partial_t \zeta_h
\quad +
\sum_{h=1}^\infty
(\rho b \cdot \nabla \zeta_h) \ast \eta_{\ee_{mh}}-
\rho b \cdot \sum_{h=1}^\infty
\nabla \zeta_h
\end{split}
\end{equation*}
By using~\eqref{e:ms2} we then get that
$$
\int_0^T \! \! \int_\Omega |h_m| dx dt \leq \sum_{h=1}^\infty
\frac{2^{-h}}{m} = \frac{1}{m}
$$
and this establishes~\eqref{e:convaccaemme}.
\label{s:proof2}
\section{Proof Theorem~\ref{IBVP-NC}: comparison principle and uniqueness}
\label{s:uni}
In this section we complete the proof of Theorem~\ref{IBVP-NC}. More precisely, we establish the following comparison principle.
\begin{lemma}
\label{l:uni}
Let $\Omega$, $b$ and $\rho$ as in the statement of
Theorem~\ref{IBVP-NC}. Assume $u_1$ and $u_2 \in
L^{\infty}((0,T) \times \Omega)$ are distributional
solutions (in the sense of Definition~\ref{d:distrsol}) of the initial-boundary value problem~\eqref{prob-2}
corresponding to the initial and boundary data
$\overline{u}_{1} \in L^{\infty}(\Omega)$, $\overline{g}_1 \in L^\infty(\Gamma^-)$ and
$\overline{u}_2 \in L^{\infty}(\Omega)$, $\overline{g}_2 \in L^\infty(\Gamma^-)$, respectively. If $\overline{u}_1 \ge \overline{u}_2$ and $\overline{g}_1 \ge \overline{g}_2$, then
\begin{equation}
\label{e:compa}
\rho u_1 \ge \rho u_2 \quad a.e. \; \text{in} \; (0, T) \times \Omega.
\end{equation}
\end{lemma}
Note that the uniqueness of $\rho u$, where $u$ is a distributional solution of the initial-boundary value problem~\eqref{prob-2}, immediately follows from the above result.
\begin{proof} [Proof of Lemma~\ref{l:uni}]
Let us define the function
$$
\tilde{\beta}(u)=
\left\{
\begin{array}{ll}
u^2 & u \geq 0 \\
0 & u<0.
\end{array}\right.
$$
In what follows, we shall prove that the identity $\rho\ \tilde{\beta}(u_{2}-u_{1})=0 $ holds almost everywhere, whence the comparison principle follows. To see this, we proceed as described below.
First, we point out that, since the equation at the
first line of~\eqref{prob-2} is linear, then $u_2-u_1$ is a distributional solution of the initial boundary value problem with data $\overline{u}_2 - \overline{u}_1$, $\overline{g}_2 - \overline{g}_1$. In particular, for every $ \psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d} )$ we have
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho (u_{2}-u_{1}) (\partial_{t} \psi +b \cdot \nabla \psi) \ dx dt= \int_{0}^{T} \int_{\partial \Omega} [\text{Tr}(\rho u_2 b) - \text{Tr}(\rho u_{1} b)] \ \psi \ d\mathcal{H}^{d-1} dt -\int_{\Omega} \psi(0,\cdot) {\rho}_0 (\overline{u}_{2}-\overline{u}_{1}) \ dx
\label{e7}
\end{equation}
and
\begin{equation}
\label{e:ntraces}
\text{Tr}(\rho u_2 b) = \overline{g}_2 \text{Tr}(\rho b), \quad
\text{Tr}(\rho u_1 b) = \overline{g}_1 \text{Tr}(\rho b)
\quad \text{on $\Gamma^-$}.
\end{equation}
Note that~\eqref{e7} implies that
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho (u_{2}-u_{1}) (\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt=0, \quad \forall \phi \in C^{\infty}_{c}((0,T) \times \Omega).
\label{e5}
\end{equation}
By using~\cite[Lemma 5.10]{Delellis1} (renormalization property inside the domain), we get
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho \ \tilde{\beta}(u_{2}-u_{1})(\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt=0, \qquad \forall \phi \in C^{\infty}_{c}((0,T) \times \Omega).
\label{e10}
\end{equation}
We next apply Lemma \ref{trace-existence} to the function $\tilde{\beta}(u_{2}-u_{1})$ to infer that there are bounded functions $\text{Tr}(\rho \tilde{\beta}(u_{2}-u_{1}) b)$ and $(\rho \tilde{\beta}(u_{2}-u_{1}))_{0} $ such that, for every
$ \psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d} ),$ we have
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho \ \tilde{\beta}(u_{2}-u_{1}) (\partial_{t} \psi +b \cdot \nabla \psi) \ dx dt= \int_{0}^{T} \int_{\partial \Omega} \text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) \ \psi \ d\mathcal{H}^{d-1} dt -\int_{\Omega} \psi(0,\cdot) (\rho \ \tilde{\beta}(u_{2}-u_{1}))_{0} \ dx.
\label{e11}
\end{equation}
We recall~\eqref{e7} and we apply Lemma \ref{trace-renorm} (trace renormalization property) with $w= u_2 -u_1$, $h= \tilde \beta$, $B=(\rho,\rho b) $, $\Lambda = \mathbb R^{d+1}$ and $\Lambda'=(0,T)\times \Omega$. We recall that the vector field $P$ is defined by setting $P:= (\rho, \rho b)$ and we get
\begin{equation}
\begin{aligned}
(\rho\ \tilde{\beta} (u_{2}-u_{1}))_{0}=-
\text{Tr}(\tilde{\beta}(u_{2}-u_{1}) P,\partial \Lambda')
\Big\vert_{\{0\} \times \Omega}&= - \tilde{\beta}\left(\frac{(\rho (u_{2}-u_{1}))_{0}}{\text{Tr}(P,\partial \Lambda')\Big\vert_{\{0\}\times \Omega}} \right)
\text{Tr}(P,\partial \Lambda')\Big\vert_{\{ 0\} \times \Omega}\\
&=-\tilde{\beta}\left( \frac{\rho_0 (\overline{u}_{2}-\overline{u}_{1})}{\overline{\rho}} \right) \rho_0 \\
& =0, \; \text{since} \ \overline{u}_{1} \geq \overline{u}_{2} \phantom{\int}
\end{aligned}
\label{e12}
\end{equation}
and
\begin{equation}
\begin{aligned}
\text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) &=
\text{Tr}(\tilde{\beta}(u_{2}-u_{1}) \rho, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega} =
\tilde{\beta} \left(
\frac{\text{Tr}((u_{2}-u_{1})\rho, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega}}{\text{Tr}(P, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega}}
\right) \text{Tr}(P, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega}\\
&=\tilde{\beta}\left(\frac{\text{Tr}(\rho (u_{2}-u_{1}) b)}{\text{Tr}(\rho b)} \right) \text{Tr}(\rho b).
\end{aligned}
\notag
\end{equation}
By recalling~\eqref{e:ntraces} and the inequality $\bar g_1 \ge \bar g_2$, we conclude that
$$
\text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) = 0 \quad \text{on $\Gamma^-$}
$$
and, since $\tilde \beta \ge 0$, that
\begin{equation}
\label{e:ntracein}
\text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) \ge 0
\quad \text{on $\Gamma$.}
\end{equation}
We now choose a test function $\nu \in C^\infty_c (\mathbb R^d)$ in such a way that $\nu \equiv 1$ on the bounded set $\Omega$. Note that
\begin{equation}
\label{e:zerozero}
\partial_t \nu + b \cdot \nabla \nu =0 \quad \text{on $(0, T) \times \Omega$.}
\end{equation}
Next we choose a sequence of functions $\chi_{n} \in {C}^{\infty}_{c}([0,+\infty))$ that satisfy
\[\chi_{n} \equiv 1 \ \text{on}\ [0,\bar{t}],\ \chi_{n}\equiv 0\ \text{on}\ [\bar{t}+\frac{1}{n},+\infty),\ \chi'_{n} \leq 0, \]
and we define
\[\psi_{n}(t,x):= \chi_{n}(t) \nu(x), \ (t,x)\in [0,T)\times \mathbb{R}^{d}.\]
Note that $\psi$ is smooth, non-negative and compactly supported in
$[0,T)\times \mathbb{R}^{d}$. By combining the identities \eqref{e11},~\eqref{e12} and the inequality~\eqref{e:ntracein}, we get
\begin{equation}
\begin{aligned}
0 &\leq \int_{0}^{T} \int_{\Omega} \rho\ \tilde{\beta}(u_{2}-u_{1}) [\partial_{t}(\chi_{n} \nu)+b \cdot \nabla (\chi_{n} \nu)] \ dx dt \\
& = \int_{0}^{T} \int_{\Omega} \nu \rho\ \tilde{\beta}(u_{2}-u_{1}) \chi'_{n} \ dx dt+ \int_{0}^{T} \int_{\Omega} \chi_{n} \rho \ \tilde{\beta}(u_{2}-u_{1}) (\partial_{t} \nu
+b \cdot \nabla \nu) \ dx dt \\
& \stackrel{\eqref{e:zerozero}}{=} \int_{0}^{T} \int_{\Omega} \nu \rho \ \chi'_{n} \ \tilde{\beta}(u_{2}-u_{1}) \ dx dt. \\
\end{aligned}
\notag
\end{equation}
Passing to the limit as $n \rightarrow +\infty $ and noting that $\chi'_{n} \rightarrow -\delta_{\bar{t}} $ as $n \rightarrow \infty $ in the sense of distributions
and recalling that $\nu \equiv 1$ on $\Omega$ we obtain
\begin{equation}
\int_{\Omega} \rho(\bar{t},\cdot) \tilde{\beta}(u_{2}-u_{1})(\bar{t},\cdot) \leq 0.
\notag
\end{equation}
Since the above inequality is true for arbitrary $\bar t \in [0, T]$, we can conclude that
\begin{equation}
\begin{aligned}
\rho \ \tilde{\beta}(u_2-u_1)=0,\ \text{for almost every}\ (t,x)
\mathbb Rightarrow \rho u_{1} \geq \rho u_{2}, \ \text{for almost every}\ (t,x).
\end{aligned}
\label{e14}
\end{equation}
This concludes the proof of Lemma~\ref{l:uni}.
\end{proof}
\section{Stability and space continuity properties}
\label{s:ssc}
In this section, we discuss some qualitative properties of solutions of the initial-boundary value problem~\eqref{prob-2}. First, we establish Theorem~\ref{stability-weak}, which establishes (weak) stability of solutions with respect to perturbations in the vector fields and the data. Theorem~\ref{stability-strong} implies that, under stronger hypotheses, we can establish strong stability. Finally, Theorem~\ref{space-continuity} establishes space continuity properties.
\begin{theorem}\label{stability-weak}
Let $T>0$ and let $\Omega \subseteq \mathbb R^d$ be an open and bounded set with $C^2$ boundary.
Assume that
$$
b_{n}, b \in BV((0,T) \times \Omega; \mathbb{R}^{d}) \cap L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d}), \qquad \rho_{n},\rho \in BV((0,T) \times \Omega) \cap L^{\infty}([0,T) \times \Omega)
$$ satisfy
\begin{equation}
\begin{aligned}
\partial_{t} \rho_{n}+\mathrm{div} (b_{n}\rho_{n})=0,\\
\partial_{t} \rho+\mathrm{div} (b \rho)=0,
\end{aligned}
\label{stability-1}
\end{equation}
in the sense of distributions on $(0, T) \times \Omega$. Assume furthermore that
\begin{equation}
0 \leq \rho_{n}, \rho \leq C \; \text{and} \; \Vert b_{n} \Vert_{\infty}\ \text{is uniformly bounded},
\label{stability-2}
\end{equation}
\begin{equation}
(b_{n},\rho_{n}) \xrightarrow[n \rightarrow \infty]{} (b,\rho) \ \text{strongly in} \ L^{1}((0,T) \times \Omega; \mathbb R^{d+1}),
\label{stability-3}
\end{equation}
\begin{equation}
\rho_{n0} \xrightarrow[n \rightarrow \infty]{} \rho_0
\; \text{strongly in}\ L^{1}(\Omega),
\label{stability-4}
\end{equation}
\begin{equation}
\emph{Tr}(\rho_{n} b_n) \xrightarrow[n \rightarrow \infty]{} \emph{Tr}(\rho b)\ \text{strongly in}\ L^{1}(\Gamma),
\label{stability-5}
\end{equation}
Let $u_{n} \in L^{\infty}((0,T) \times \Omega) $ be a distributional solution
(in the sense of Definition~\ref{d:distrsol}) of the initial-boundary value problem
\begin{equation}
\label{e:ibvpapp}
\left\{
\begin{array}{lll}
\partial_{t}(\rho_{n} u_{n})+\mathrm{div}(\rho_{n} u_{n} b_{n})=0 &
\text{in} \ (0,T)\times \Omega \\
u_{n}=\overline{u}_{n} & \text{at $t=0$}\\
u_{n} =\overline{g}_{n} & \text{on}
\ \Gamma_{n}^{-} \\
\end{array}
\right.
\end{equation}
and $u \in L^{\infty}((0,T) \times \Omega) $ be a distributional solution of the equation
\begin{equation}
\label{e:ibvplimit}
\left\{
\begin{array}{lll}
\partial_{t}(\rho u)+\mathrm{div}(\rho u b)=0 & \text{in} \ (0,T)\times \Omega \\
u=\overline{u} & \text{at $t=0$}\\
u=\overline{g} & \text{on}\ \Gamma^{-}.
\end{array}
\right.
\end{equation}
If $u_m, \bar u \in L^\infty (\Omega)$ and $ \overline{g}_{n}, \bar g \in L^\infty (\Gamma)$ satisfy
\begin{equation}
\overline{u}_{n} \stackrel{\ast}{\rightharpoonup} \overline{u}\ \text{weak-$^\ast$ in}\ L^{\infty}(\Omega),
\label{stability-7}
\end{equation}
\begin{equation}
\overline{g}_{n} \stackrel{\ast}{\rightharpoonup} \overline{g} \;
\text{weak-$^\ast$ in}\ L^{\infty}(\Gamma) ,
\label{stability-8}
\end{equation}
then
\begin{equation}
\rho_{n} u_{n} \stackrel{*}{\rightharpoonup}
\rho u \ \text{weak-* in}\ L^{\infty}((0,T) \times \Omega)
\label{stability-10}
\end{equation}
and
\begin{equation}
\emph{Tr}(\rho_{n} u_{n} b_{n}) \stackrel{*}{\rightharpoonup}
\emph{Tr}(\rho u b) \ \text{weak-* in}\ L^{\infty}(\Gamma).
\label{stability-9}
\end{equation}
\end{theorem}
Note that in the statement of the above theorem $\overline{g}_m$ and $\overline{g}$ are functions defined on the whole $\Gamma$, although the values of $\rho_m u_m$ and $\rho u$ are only determined by their values on $\Gamma^-_m$ and $\Gamma^-$, respectively.
\begin{proof}
We proceed according to the following steps. \\
{\sc Step 1:} we apply Theorem~\ref{IBVP-NC} and we infer that the function $\rho_n u_n$ satisfying~\eqref{e:ibvpapp} is unique. Also, without loss of generality, we can redefine the function $u_n$ on the set $\{\rho_n=0\}$ in such a way that $u_n$ satisfies the maximum principle~\eqref{e:maxprin}. Owing to~\eqref{stability-9}, the sequences $\| \overline{u}_m \|_{L^\infty}$ and $\| \overline{g}_m \|_{L^\infty}$ are both uniformly bounded and by the maximum principle so is $\| u_m \|_{L^\infty}$. Also, by combining~\eqref{e:maxprintraces2} and~\eqref{stability-2} we infer that the sequence $\Vert \text{Tr}(\rho_{n} b_{n} u_{n}) \Vert_{\infty} $ is also uniformly bounded. We conclude that, up to subsequences (which we do not label to simplify the notation), we have
\begin{comment}
We begin with the preliminary observation that if we are able to prove that
\begin{equation}
u_{n} \stackrel{*}{\rightharpoonup} u \ \text{weak-* in}\ L^{\infty}((0,T) \times \Omega),
\label{stability-10}
\end{equation}
we can combine the fact that $\Vert u_{n} \Vert_{\infty} $ are uniformly bounded (this follows from the maximum principle and \eqref{stability-7}-\eqref{stability-8}) with \eqref{stability-3} to infer \eqref{stability-9}. Therefore it is sufficient to establish \eqref{stability-10} which we pursue next.
We note that since $\Vert u_{n} \Vert_{\infty} $ and $\Vert \text{Tr}(\rho_{n} b_{n} u_{n}) \Vert_{\infty} $ are uniformly bounded (see Remark $2.2 (a)$), there exist $R_{1} \in L^{\infty}((0,T) \times \Omega)$ and $R_{2} \in L^{\infty}((0,T)\times \partial \Omega)$ such that as $n \rightarrow \infty $,
\end{comment}
\begin{equation}
\begin{aligned}
& u_{n} \stackrel{*}{\rightharpoonup} r_{1} \ \text{weak-* in} \ L^{\infty}((0,T) \times \Omega),\\
& \text{Tr}(\rho_{n} u_{n} b_{n}) \stackrel{*}{\rightharpoonup} r_{2} \ \text{weak-* in} \ L^{\infty}(\Gamma)
\end{aligned}
\label{stability-11}
\end{equation}
for some $r_{1} \in L^{\infty}((0,T) \times \Omega)$ and $r_{2} \in L^{\infty}(\Gamma)$.
By using \eqref{iden-2} and \eqref{iden-3}, we get that
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho r_{1} (\partial_{t} \phi+b \cdot \nabla \phi)\ dx dt=0, \quad \forall \phi \in C^{\infty}_{c}((0,T) \times \Omega),
\label{stability-12}
\end{equation}
and
\begin{equation}
\int_{0}^{T} \int_{\Omega} \rho r_{1} (\partial_{t} \psi+b \nabla \psi)\ dx dt= \int_{0}^{T} \int_{\partial \Omega} r_{2} \psi\ d\mathcal{H}^{d-1} dt -\int_{\Omega} \psi(0,\cdot) {\rho}_0\ \overline{u}\ dx ,\ \forall \psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d}).
\label{stability-13}
\end{equation}
From Lemma \ref{trace-existence}, it also follows that
\begin{equation}
r_{2}= \text{Tr}(\rho r_{1} b).
\label{stability-14}
\end{equation}
Assume for the time being that we have established the equality
\begin{equation}
\label{e:whatww}
r_{2}=\overline{g} \text{Tr}(\rho b), \quad \text{on}\ \Gamma^{-} ,
\end{equation}
then by recalling~\eqref{stability-14} and the uniqueness part in Theorem~\ref{IBVP-NC} we conclude that $r_1= \rho u$ and $r_2 = \text{Tr}(\rho bu) $. Owing to~\eqref{stability-11}, this concludes the proof of the theorem. \\
{\sc Step 2:} we establish~\eqref{e:whatww}. First, we decompose $\text{Tr}(\rho_m u_m b_m)
$ as
\begin{equation}
\label{e:decompo}
\begin{split}
\text{Tr}(\rho_n u_n b_n) &= \text{Tr}(\rho_n u_n b_n)
\mathbf{1}_{\Gamma^{-}_{n}}+\text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{+}_{n}} + \text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{0}_{n}}
\\
& = \overline{g}_n \text{Tr}(\rho_n b_n)
\mathbf{1}_{\Gamma^{-}_{n}}+\text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{+}_{n}} + \text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{0}_{n}} ,
\end{split}
\end{equation}
where $\Gamma^-_n$, $\Gamma^+_n$ and $\Gamma^0_n$ are defined as in~\eqref{e:gamma}.
By using Lemma~\ref{trace-renorm} (trace renormalization), one could actually prove that the last term in the above expression is actually $0$. This is actually not needed here. Indeed, it suffices to recall~\eqref{stability-5} and Lemma~\ref{l:traces} and point out that by combining~\eqref{e:convchar1} and~\eqref{e:convchar2} we get
\begin{equation}
\label{e:conchar3}
\mathbf{1}_{\Gamma^{0}_{n}} \to \mathbf{1}_{\Gamma^0}
- \mathbf{1}_{\Gamma'} - \mathbf{1}_{\Gamma''}.
\end{equation}
Next, we recall that
the sequence $\| \text{Tr}(\rho_n u_n b_n)\|_{L^\infty}$ is uniformly bounded owing to the uniform bounds on $\| \rho_n \|_{L^\infty}$ and $\| u_n \|_{L^\infty}$. By recalling~\eqref{stability-8}, we conclude that
\begin{equation}
\label{e:conv1}
\overline{g}_n \text{Tr}(\rho_n b_n)
\mathbf{1}_{\Gamma^{-}_{n}} \stackrel{*}{\rightharpoonup}
\overline{g} \, \text{Tr}(\rho b)
\Big( \mathbf{1}_{\Gamma^{-}} + \mathbf{1}_{\Gamma'} \Big)
\qquad \text{weak-* in $L^\infty (\Gamma)$.}
\end{equation}
By recalling that $\Gamma' \subseteq \Gamma^0$ we get that $\text{Tr}(\rho b)
\mathbf{1}_{\Gamma'} =0$. We now pass to the weak star limit in~\eqref{e:decompo} and using~\eqref{e:convchar1},~\eqref{e:convchar2},~\eqref{stability-11},~\eqref{stability-8} and~\eqref{e:conv1} we get
\begin{equation}
\label{e:conv2}
r_2 = \overline{g} \text{Tr}(\rho b)
\mathbf{1}_{\Gamma^{-}} + r_2
\Big( \mathbf{1}_{\Gamma^{+}} + \mathbf{1}_{\Gamma'} \Big)+ r_2
\Big( \mathbf{1}_{\Gamma^{0}} - \mathbf{1}_{\Gamma'} -\mathbf{1}_{\Gamma''} \Big),
\end{equation}
which owing to the properties
$$
\Gamma^- \cap \Gamma^{0}= \emptyset,
\quad \Gamma^- \cap \Gamma'=\emptyset, \quad
\Gamma^- \cap \Gamma''= \emptyset
$$
implies~\eqref{e:whatww}. This concludes the proof Theorem~\ref{stability-weak}.
\end{proof}
\begin{theorem}\label{stability-strong}
Under the same assumptions as in Theorem~\ref{stability-weak}, if we furthermore assume that
\begin{equation}
\overline{u}_{n} \xrightarrow[n \rightarrow \infty]{} \overline{u}\ \text{strongly in}\ L^{1}(\Omega),
\label{stability-35}
\end{equation}
\begin{equation}
\overline{g}_{n} \xrightarrow[n \rightarrow \infty]{} \overline{g} \ \text{strongly in}\ L^{1}(\Gamma) ,
\label{stability-36}
\end{equation}
then we get
\begin{equation}
\begin{aligned}
&\rho_{n} u_{n} \xrightarrow[n \rightarrow \infty]{} \rho u \ \text{strongly in}\ L^{1}((0,T) \times \Omega),\\
&\emph{Tr}(\rho_{n} u_{n} b_{n}) \xrightarrow[n \rightarrow \infty]{} \emph{Tr}(\rho u b) \ \text{strongly in}\ L^{1}(\Gamma).
\end{aligned}
\label{stability-37}
\end{equation}
\end{theorem}
\begin{proof}
First, we point out that the first equation
in~\eqref{stability-9} implies that
\begin{equation}
\label{e:ell2}
\rho_n u_m
{\rightharpoonup} \rho {u}\ \text{weakly in}\ L^{2}((0, T) \times \Omega ).
\end{equation}
Next, by using Lemma \ref{trace-renorm} (trace-renormalization property), we get that $\rho_m u^{2}_{n}$ and $\rho u^{2}$ satisfy (in the sense of distributions)
\begin{equation}
\left\{
\begin{array}{lll}
\partial_{t}(\rho_{n} u_{n}^{2})+\text{div}(\rho_{n} u_{n}^{2} b_{n})=0 &
\text{in} \ (0,T)\times \Omega \\
u_{n}^{2}=\overline{u}_{n}^{2} & \text{at $t=0$}\\
u^{2}_{n} =\overline{g}^{2}_{n} & \text{on}\ \Gamma_{n}^{-}, \\
\end{array}
\right.
\notag
\end{equation}
and
\begin{equation}
\left\{
\begin{array}{lll}
\partial_{t}(\rho u^{2})+\text{div}(\rho u^{2} b)=0 &
\text{in} \ (0,T)\times \Omega \\
u^{2}=\overline{u}^{2} & \text{at $t=0$} \\
u^{2} =\overline{g}^{2} & \text{on}\ \Gamma^{-}, \\
\end{array}
\right.
\notag
\end{equation}
respectively. Also, by combinig~\eqref{stability-7},\eqref{stability-8}, \eqref{stability-35} and \eqref{stability-36}, we get that
\begin{equation}
\overline{u}^2_{n} \stackrel{*}{\rightharpoonup} \overline{u}^2\ \text{weak-$^\ast$ in}\ L^{\infty}(\Omega), \qquad
\overline{g}^2_{n} \stackrel{*}{\rightharpoonup} \overline{g}^2 \;
\text{weak-$^\ast$ in}\ L^{\infty}(\Gamma)
\notag
\end{equation}
and by applying Theorem~\ref{stability-weak} to $\rho_m u_m^2$ we conclude that
$$
\rho_m u_m^2
\stackrel{*}{\rightharpoonup} \rho {u}^2\ \text{weak-$^\ast$ in}\ L^{\infty}((0, T) \times \Omega )
$$
and that
\begin{equation}
\label{e:convbur}
\text{Tr}(\rho_{n} u^2_{n} b_{n})
\stackrel{*}{\rightharpoonup}
\text{Tr}(\rho u^2 b)
\ \text{weak-$^\ast$ in}\ L^{\infty}(\Gamma ).
\end{equation}
Since the sequence $\| \rho_m \|_{L^\infty}$ is uniformly bounded, then
by recalling~\eqref{stability-3} we get
$$
\rho^2_m u_m^2
\stackrel{*}{\rightharpoonup} \rho^2 {u}^2\ \text{weak-$^\ast$ in}\ L^{\infty}((0, T) \times \Omega )
$$
and hence
\begin{equation}
\label{e:square}
\rho^2_m u_m^2
{\rightharpoonup} \rho^2 {u}^2\ \text{weakly in}\ L^2((0, T) \times \Omega ).
\end{equation}
By combining~\eqref{e:ell2} and~\eqref{e:square} we get that
$\rho^2_m u_m^2 \longrightarrow \rho u$ strongly in
$L^2((0, T) \times \Omega )$ and this implies the first convergence in~\eqref{stability-37}.
Next, we establish the second convergence in $L^2((0, T) \times \Omega )$. Since $\Gamma $ is a set of finite measure, from \eqref{stability-9} and \eqref{e:convbur} we can infer that
\begin{equation}
\begin{aligned}
& \text{Tr}(\rho_{n} u_{n} b_{n}) \rightharpoonup \text{Tr}(\rho u b) \ \text{weakly in} \ L^{2}(\Gamma),\\
& \text{Tr}(\rho_{n} u^{2}_{n} b_{n}) \rightharpoonup \text{Tr}(\rho u^{2} b) \ \text{weakly in} \ L^{2}(\Gamma).
\end{aligned}
\label{stability-39}
\end{equation}
By using the uniform bounds for $\Vert \text{Tr}(\rho_{n} b_{n})\Vert_{\infty} $, we infer from the $L^{1}$ convergence of $\text{Tr}(\rho_{n} b_{n}) $ to $\text{Tr}(\rho b)$ that
\begin{equation}
\text{Tr}(\rho_{n} b_{n}) \xrightarrow[n \rightarrow \infty]{} \text{Tr}(\rho b) \ \text{strongly in}\ L^{2}(\Gamma).
\label{stability-40}
\end{equation}
Next, we apply Lemma \ref{trace-renorm} (trace renormalization property) and we get that
\begin{equation}
[\text{Tr}(\rho_{n} u_{n} b_{n})]^{2}= \left[\frac{\text{Tr}(\rho_{n} u_{n} b_{n})}{\text{Tr}(\rho_{n} b_{n})} \right]^{2} [\text{Tr}(\rho_{n} b_{n})]^{2}= \text{Tr}(\rho_{n} u_{n}^{2} b_{n}) \text{Tr}(\rho_{n}b_{n})
\notag
\end{equation}
and
\begin{equation}
[\text{Tr}(\rho u b)]^{2}= \left[\frac{\text{Tr}(\rho u b)}{\text{Tr}(\rho b)} \right]^{2} [\text{Tr}(\rho b)]^{2}= \text{Tr}(\rho u^{2} b) \text{Tr}(\rho b).
\notag
\end{equation}
From \eqref{stability-39} and \eqref{stability-40}, we can then conclude that
\begin{equation}
[\text{Tr}(\rho_{n} u_{n} b_{n})]^{2} \rightharpoonup [\text{Tr}(\rho u b)]^{2} \ \text{weakly in}\ L^{2}(\Gamma),
\label{stability-41}
\end{equation}
and by recalling~\eqref{stability-39} the second convergence in \eqref{stability-37} follows.
\end{proof}
Finally, we establish space-continuity properties of the vector field $(\rho u, \rho u b)$
similar to those established in~\cite{Boyer,CDS1}.
\begin{theorem}\label{space-continuity}
Under the same assumptions as in Theorem~\ref{IBVP-NC}, let $P$ be the vector field $P : = ( \rho, \rho b)$, $u$ be a distributional solution of~\eqref{prob-2} and
$\{\Sigma_{r} \}_{r \in I} \subseteq \mathbb R^d$ be a family of graphs as in Definition \ref{graph}.
Also, fix $r_{0} \in I $ and let $\gamma_{0}, \gamma_{r}: (0,T) \times D \rightarrow \mathbb{R} $ be defined by
\begin{equation}
\begin{aligned}
\gamma_{0}(t,x_{1},\cdots,x_{d-1})&:= \emph{Tr}^{-}(uP,(0,T)\times \Sigma_{r_{0}})(t,x_{1},\cdots,x_{d-1},f(x_{1},\cdots,x_{d-1})-r_{0}),\\
\gamma_{r}(t,x_{1},\cdots,x_{d-1})&:=\emph{Tr}^{+}(uP,(0,T) \times \Sigma_{r})(t,x_{1},\cdots,x_{d-1},f(x_{1},\cdots,x_{d-1})-r) .
\end{aligned}
\label{space1}
\end{equation}
Then $\gamma_{r} \rightarrow \gamma_{0}$ strongly in $L^{1}((0,T)\times D) $ as $r \rightarrow r^{+}_{0} $.
\end{theorem}
The proof of the above result follows the same strategy as the proof of~\cite[Proposition 3.5]{CDS1} and is therefore omitted.
\section{Applications to the Keyfitz and Kranzer system}
\label{s:KK}
In this section, we consider the initial-boundary value problem for the Keyfitz and Kranzer system~\cite{KK} of conservation laws in several space dimensions, namely
\begin{equation}
\left\{
\begin{array}{lll}
\partial_{t} U+\displaystyle{
\sum_{i=1}^{d} \partial_{x_{i}} (f^{i}(\vert U \vert) U)=0}
& \text{in}\ (0,T) \times \Omega \\
U = U_{0} & \text{at $t=0$} \\
U = U_{b} & \text{on} \ \Gamma.
\displaystyle{\phantom{\int}} \\
\end{array}
\right.
\label{KK1}
\end{equation}
Note that, in general, we cannot expect that the boundary datum is pointwise attained on the whole boundary $\Gamma$. We come back to this point in the following.
We follow the same approach as in~\cite{ABD,AD,Br,Delellis2} and we formally split the equation at the first line of~\eqref{KK1} as the coupling between a scalar conservation law and a linear transport equation. More precisely, we set $F:=(f^{1},\cdots,f^{d})$ and we point out that the modulus
$\rho: = |U|$ formally solves the initial-boundary value problem
\begin{equation}
\left\{
\begin{array}{ll}
\partial_{t} \rho+\text{div} (F(\rho) \rho)=0 & \text{in}\ (0,T) \times \Omega\\
\rho =\vert U_{0} \vert & \text{at $t=0$}\\
\rho= \vert U_{b} \vert & \text{on}\, \Gamma.
\end{array}
\right.
\label{KK2}
\end{equation}
We follow~\cite{BLN,CR,Serre2} and we extend notion of \emph{entropy admissible} solution (see~~\cite{Kr}) to initial boundary value problems.
\begin{definition}
A function $\rho \in L^{\infty}((0,T) \times \Omega) \cap BV((0,T) \times \Omega) $ is an entropy solution of \eqref{KK2} if for all $k \in \mathbb{R}$,
\begin{equation}
\begin{aligned}
&\int_{0}^{T} \int_{\Omega} \Big\{\vert \rho(t,x)-k \vert\ \partial_{t} \psi + \emph{sgn}(\rho-k)[F(\rho)-F(k)] \cdot \nabla \psi \Big\} \ dx dt \\
&+ \int_{\Omega} \vert \rho_{0}-k \vert\ \psi(0, \cdot) \ dx -\int_{0}^{T} \int_{\partial \Omega} \emph{sgn}(\vert U_{b} \vert(t,x)-k)\ [F(T(\rho))-F(k)]\cdot \vec n \ \psi \ dx dt \geq 0,
\end{aligned}
\notag
\end{equation}
for any positive test function $\psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d}; \mathbb{R}^{+}).$ In the above expression $T(\rho)$ denotes the trace of the function $\rho$ on the boundary $\Gamma$ and $\vec n$ is the outward pointing, unit normal vector to $\Gamma$.
\end{definition}
Existence and uniqueness results for entropy admissible solutions of the above systems were obtained by Bardos, le Roux and N{\'e}d{\'e}lec~\cite{BLN} by extending the analysis by Kru{\v{z}}kov to initial-boundary value problems (see also~\cite{CR,Serre2} for a more recent
discussion). Note, however, that one cannot expect that the boundary value $|U_b|$ is pointwise attained on the whole boundary $\Gamma$, see again~\cite{BLN,CR,Serre2} for a more extended discussion.
Next, we introduce the equation for the \emph{angular part} of the solution of~\eqref{KK1}. We recall that, if $|U_b|$ and $|U_0|$ are of bounded variation, then so is $\rho$ and hence the trace of $F(\rho) \rho$ on $\Gamma$ is well defined. As usual, we denote it by $T(F(\rho) \rho)$. In particular, we can introduce the set
$$
\Gamma^- : = \big\{ (t, x) \in \Gamma: \; T(F(\rho) \rho) \cdot \vec n <0 \big\},
$$
where as usual $\vec n$ denotes the outward pointing, unit normal vector to $\Gamma$. We consider the vector $\theta=(\theta_{1},\cdots,\theta_{N}) $ and we impose
\begin{equation}
\left\{
\begin{array}{llll}
\partial_{t}(\rho \theta)+\text{div}(F(\rho) \rho \theta)=0 & \text{in}\ (0,T) \times \Omega \phantom{\displaystyle{\int}}\\
\theta=\displaystyle{\frac{U_{0}}{\vert U_{0} \vert}}& \text{at $t=0$}\\
\theta=\displaystyle{\frac{U_{b}}{\vert U_{b} \vert}} & \text{on} \ \Gamma^{-},
\end{array}
\label{KK5}
\right.
\end{equation}
where the ratios $U_0 / |U_0|$ and $U_b / |U_b|$ are defined to be an arbitrary unit vector when $|U_0|=0$ and $|U_b|=0$, respectively. Note that the product $U=\theta \rho$ formally satisfies the equation at the first line of~\eqref{KK1}. We now extend the notion of \emph{renormalized entropy solution} given in~\cite{ABD,AD,Delellis2} to initial-boundary value problems.
\begin{definition}
\label{d:res}
A renormalized entropy solution of~\eqref{KK1} is a function $U \in L^\infty ( (0, T) \times \Omega; \mathbb R^N)$ such that $U = \rho \theta$, where
\begin{itemize}
\item $\rho = |U|$ and $\rho$ is an entropy admissible solution of~\eqref{KK2}.
\item $\theta = (\theta_1, \dots, \theta_N)$ is a distributional solution, in the sense of Definition~\ref{d:distrsol}, of~\eqref{KK5}.
\end{itemize}
\end{definition}
Some remarks are here in order. First, we can repeat the proof of \cite[Proposition 5.7]{Delellis1} and conclude that, under fairly general assumptions, any renormalized entropy solution is an entropy solution. More precisely, let us fix a renormalized entropy solution $U$ and an \emph{entropy-entropy flux pair} $(\eta, Q)$, namely a couple of functions $\eta: \mathbb R^N \to \mathbb R$, $Q: \mathbb R^N \to \mathbb R^d$ such that
$$
\nabla \eta D f^i = \nabla Q^i, \quad \text{for every $i=1, \dots, d$.}
$$
Assume that
$$
\mathcal L^1 \big\{ r \in \mathbb R: \; (f^1)'(r) = \dots = (f^d)' (r)=0 \big\}=0. $$
By arguing as in~\cite{Delellis1} we conclude that, if $\eta$ is convex, then
$$
\int_0^T \int_\Omega \eta (U) \partial_t \phi + Q(U)\cdot \nabla \phi \, dx dt \ge 0
$$
for every \emph{entropy-entropy flux pair} $(\eta, Q)$ and for every nonnegative test function $\phi \in C^\infty_c ((0, T) \times \Omega)$.
Second, we point out that, as the Bardos, le Roux and N{\'e}d{\'e}lec~\cite{BLN} solutions of scalar initial-boundary value problems, renormalized entropy solutions of the Keyfitz and Kranzer system do not, in general pointwise attain the boundary datum $U_0$ on the whole boundary $\Gamma$.
We now state our well-posedness result.
\begin{theorem}
\label{t:KK} Assume $\Omega$ is a bounded open set with $C^2$ boundary. Also, assume that $U_0 \in L^\infty (\Omega; \mathbb R^N)$ and $U_b \in L^\infty (\Gamma; \mathbb R^N)$ satisfy $|U_0| \in
BV ( \Omega)$, $|U_b| \in BV (\Gamma).$ Then there is a unique renormalized entropy solution of~\eqref{KK1} that satisfies $U \in L^\infty ((0, T)\times \Omega; \mathbb R^N)$.
\end{theorem}
\begin{proof} We first establish existence, next uniqueness. \\
{\sc Existence:} first, we point out that the results in~\cite{BLN,CR,Serre2} imply that there is an entropy admissible solution of~\eqref{KK2} satisfying
$\rho \in L^\infty ((0, T) \times \Omega) \cap BV ((0, T) \times \Omega).$
Also, $\rho$ satisfies the maximum principle, namely
\begin{equation}
\label{e:rhomaxprin}
0 \leq \rho \leq \max \big\{ \| U_0 \|_{L^\infty}, \|U_b \|_{L^\infty} \big\}.
\end{equation}
For every $j=1, \dots, N$ we consider the initial-boundary value problem
\begin{equation}
\left\{
\begin{array}{llll}
\partial_{t}(\rho \theta_j)+\text{div}(F(\rho) \rho \theta_j)=0 & \text{in}\ (0,T) \times \Omega \phantom{\displaystyle{\int}}\\
\theta_j=\displaystyle{\frac{U_{0j}}{\vert U_{0} \vert}}& \text{at $t=0$}\\
\theta_j=\displaystyle{\frac{U_{bj}}{\vert U_{b} \vert}} & \text{on} \ \Gamma^{-},
\end{array}
\label{KK6}
\right.
\end{equation}
where $U_{0j}$ and $U_{bj}$ is the $j$-th component of $U_0$ and $U_b$, respectively. The existence of a distributional solution $\theta_j$ follows from the existence part in Theorem~\ref{IBVP-NC}.
We now set $U: = \rho \theta$, where $\theta = (\theta_1, \dots, \theta_N)$. To conclude the existence part we are left to show that $|U|=\rho$. To this end, we point out that, by combining~\cite[Lemma 5.10]{Delellis1} (renormalization property inside the domain) with Theorem~\ref{trace-renorm} (trace renormalization property) and by arguing as in \S~\ref{s:uni}, we conclude that, for every $j=1, \dots, N$, $\theta^2_j$ is a distributional solution, in the sense of Definition~\ref{d:distrsol}, of the initial-boundary value problem
\begin{equation*}
\left\{
\begin{array}{llll}
\partial_{t}(\rho \theta^2_j)+\text{div}(F(\rho) \rho \theta^2_j)=0 & \text{in}\ (0,T) \times \Omega \phantom{\displaystyle{\int}}\\
\theta_j=\displaystyle{\frac{U^2_{0j}}{\vert U_{0} \vert^2}}& \text{at $t=0$}\\
\theta=\displaystyle{\frac{U^2_{bj}}{\vert U_{b} \vert^2}} & \text{on} \ \Gamma^{-}.
\end{array}
\right.
\end{equation*}
By adding from $1$ to $N$, we conclude that $|\theta|^2$ is a distributional solution of
\begin{equation*}
\left\{
\begin{array}{llll}
\partial_{t}(\rho |\theta|^2)+\text{div}(F(\rho) \rho |\theta|^2)=0 & \text{in}\ (0,T) \times \Omega \\
\theta_j=1& \text{at $t=0$}\\
\theta=1 & \text{on} \ \Gamma^{-}.
\end{array}
\right.
\end{equation*}
By recalling the equation at the first line of~\eqref{KK2} we infer that $|\theta|^2 =1$ is a solution of the above initial-boundary value problem. By the uniqueness part of Theorem~\ref{IBVP-NC}, we then deduce that $\rho |\theta|^2= \rho$ and this concludes the proof of the existence part. \\
{\sc Uniqueness:} assume $U_1$ and $U_2$ are two renormalized entropy solutions, in the sense of Definition~\ref{d:res}, of the initial-boundary value
problem~\eqref{KK1}. Then $\rho_1: = |U_1|$ and $\rho_2 : =|U_2|$ are two
entropy admissible solutions of the initial-boundary value problem~\eqref{KK2} and hence $\rho_1=\rho_2$. By applying the uniqueness part of Theorem~\ref{IBVP-NC} to the initial-boundary value problem~\eqref{KK6}, for every $j=1, \dots, N$, we can then conclude that $U_1 =U_2$.
\end{proof}
\section*{Acknowledgments}
This paper has been written while APC was a postdoctoral fellow at the University of Basel supported by a ``Swiss Government Excellence Scholarship'' funded by the State Secretariat for Education, Research and Innovation (SERI). APC would like to thank the SERI for the support and the Department of Mathematics and Computer Science of the University of Basel for the kind hospitality. GC was partially supported by the Swiss National Science Foundation (Grant 156112). LVS is a member of the GNAMPA group of INdAM (``Istituto Nazionale di Alta Matematica"). Also, she would like to thank the Department of Mathematics and Computer Science of the University of Basel for the kind hospitality during her visit, during which part of this work was done.
\small
\end{document} |
\begin{document}
\RUNAUTHOR{Byeon, Van Hentenryck, Bent, and Nagarajan}
\RUNTITLE{Communication-Constrained Expansion Planning}
\TITLE{Communication-Constrained Expansion Planning \\ for Resilient Distribution Systems}
\ARTICLEAUTHORS{
\AUTHOR{Geunyeong Byeon and Pascal Van Hentenryck}
\AFF{Industrial and Operations Engineering, University of Michigan \mathcal{E}MAIL{} \mathcal{U}RL{}}
\AUTHOR{Russell Bent and Harsha Nagarajan}
\AFF{Los Alamos National Laboratory \mathcal{E}MAIL{} \mathcal{U}RL{}}
}
\ABSTRACT{
Distributed generation and remotely controlled switches have
emerged as important technologies to improve the resiliency
of distribution grids against extreme weather-related
disturbances. Therefore it becomes important to study how
best to place them on the grid in order to meet a
resiliency criteria, while minimizing costs and capturing
their dependencies on the associated communication systems
that sustains their distributed operations. This paper
introduces the Optimal Resilient Design Problem for
Distribution and Communication Systems (ORDPDC) to address
this need. The ORDPDC is formulated as a two-stage
stochastic mixed-integer program that captures the physical
laws of distribution systems, the communication connectivity
of the smart grid components, and a set of scenarios which
specifies which components are affected by potential
disasters. The paper proposes an exact branch-and-price
algorithm for the ORDPDC which features a strong lower bound
and a variety of acceleration schemes to address degeneracy.
The ORDPDC model and branch-and-price algorithm were
evaluated on a variety of test cases with varying disaster
intensities and network topologies. The results demonstrate
the significant impact of the network topologies on the
expansion plans and costs, as well as the computational
benefits of the proposed approach.
}
\KEYWORDS{Planning for Resiliency, Power Systems, Branch and Price}
\maketitle
\section{Introduction}
The last decades have highlighted the vulnerability of the current
electric power system to weather-related extreme events. Between 2007
and 2016, outages caused by natural hazards, such as thunderstorms,
tornadoes, and hurricanes, amounted to 90 percent of major electric
disturbances, each affecting at least 50,000 customers (derived from
Form OE-417 of U.S. DOE). It is also estimated that 90 percent of all
outages occur along distribution systems \citep{Executive2013}.
Moreover, the number of weather-related outages is expected to rise as
climate change increases the frequency and intensity of extreme
weather events \citep{Executive2013}. Accordingly, it is critical to
understand how to harden and modernize distribution grids to
prepare for potential natural disasters.
Distributed Generation (DG) is one of the advanced technologies that
can be utilized to enhance grid resilience. DG refers to electric
power generation and storage performed by a collection of distributed
energy resources (DER). DG decentralizes the electric power
distribution by supplying power to the loads closer to where it is
located. The potential of DGs is realized via a system approach that
views DGs and associated loads as a microgrid
\citep{lasseter2002certs}. A microgrid is often defined as a
small-scale power system on medium- or low- voltage distribution
feeder that includes loads and DG units, together with an appropriate
management and control scheme supported by a communication
infrastructure \citep{resende2011service}. When faults occur in the
main grid, microgrids can be detached from the main grid and act in
island mode to serve critical loads by utilizing local DGs or work in
the grid-connected mode to provide ancillary services for the bulk
system restoration \citep{wang2016research}. Remotely controlled
switches (RCS), another advanced technology, can be used to increase
the grid flexibility by controlling the grid topology through a
communication network and facilitate microgrid formations in
emergencies. Other than the aforementioned operational enhancement
measures, a grid can also be hardened physically by installing
underground cables and/or upgrading the overhead lines with stronger
materials, which reduces the physical impact of catastrophic events
\citep{panteli2017}.
A critical issue in building resilient distribution grids is to
determine where to place such advanced devices (i.e., DGs, RCSs, and
underground cables) and which existing lines to harden. It is also
important to understand the dependency between the distribution grid
and its associated communication network, which is critical to the
effective operation of a modernized grid during emergency situations
and is also vulnerable to extreme events
\citep{falahati2012reliability, gholami2016front,
martins2017interdependence, li2017networked}.
To address this pivotal and pressing issue, this paper introduces the
Optimal Resilient Design Problem for Distribution and Communication Systems
(ORPDDC). The ORDPDC determines how to harden and
modernize an interdependent network to ensure its resilience against
extreme weather events. Like recent papers (e.g.,
\citet{yamangil2015resilient, Barnes2017Tools}, the ORDPDC
takes into account a set of disaster scenarios, each defining a set of
power system components that are damaged during an extreme
event. These scenarios are generated from historical data or
probabilistic models of how power system components respond to
hazard-specific stress (e.g., wind speed and flood depth). The ORDPDC
considers the following upgrade options: a set of hardening
options on existing power lines and communication links and a set of
new components that can be added to the system---new lines, new
communication pathways, remotely controlled switches, and distributed
generation. The objective of the ORDPDC is to find the cheapest set of
upgrade options that can be placed on the grid in order to guarantee that a
minimal amount of critical and non-critical load be served in each
scenario. These guarantees are called the reslience criteria.
The ORDPDC is modeled with a two-stage stochastic mixed integer
program. The first stage decides an upgrade profile and the second
stage decides how to utilize the DGs, RCSs, and power
lines/communication links, whose availability is decided in the
first-stage, to restore critical loads up to resiliency criteria
(e.g., 98 \%) in each disaster scenario. For each scenario, the second
stage is viewed as a restoration model that identifies how to
reconfigure the grid. Within this second stage problem, the physics of
power flows is modeled with the steady-state, unbalanced three phase
AC power equations and constraints that ensure that the radial structure of
distribution grids is maintained. When the grid is reconfigured due to
some disturbances, each island or microgrid must be connected to at
least one control center that coordinates its DGs and loads and
operates its RCSs. This communication requirement is modeled with a
single-commodity flow model.
Several solution methods can be used to solve the ORDPDC,
taking advantage of its block diagonal structure.
\citet{yamangil2015resilient} proposed a Scenario-Based
Decomposition (SBD) that restricts attention to a smaller set of
scenarios and adds new ones on an as needed basis (see Section
\ref{sec:sbd}). However, in the worst case, SBD must solve the
large-scale ORDPDC as a whole. Branch and Price (B\&P),
which combines column generation and branch-and-bound, is another
solution method for approaching large-scale mixed-integer programming
\citep{lubbecke2005selected}. Although widely successful on many
applications, it may suffer from degeneracy and long-tail effects as
problems become larger. To address these difficulties, several
stabilization techniques have been proposed and proven to be effective
in many applications (e.g., \citep{du1999stabilized,
oukil2007stabilized, amor2009choice}). Nevertheless, the high degree
of degeneracy and the significant scale of the ORDPDC create
significant challenges for dual stabilization techniques.
To address these computational challenges, this paper proposes a B\&P
algorithm that systematically exploits the structure of the
ORDPDC. The algorithm starts with a compact reformulation that results
in strong lower bounds on the test cases and pricing subproblems that
are naturally solved in parallel. Moreover, the B\&P algorithm
tackles the degeneracy inherent in the ORDPDC through a variety of
acceleration schemes for the pricing subproblems: A pessimistic
reduced cost, an optimality cut, and a lexicographic objective. The
resulting B\&P algorithm produces significant computational
improvements compared to existing approaches.
The key contributions of this paper can be summarized as follows:
\begin{enumitemize}
\item The paper proposes the first planning model for resilient distribution
networks that combines the use of advanced technologies (e.g., DGs,
RCSs, and undergrounding) with traditional hardening options and
captures the dependencies between the distribution grid and its
associated communication system.
\item The paper proposes an exact B\&P algorithm for solving the ORDPDC
problem, which systematically exploits the ORDPDC structure to
obtain strong lower bounds and address its significant degeneracy
issues.
\item The paper evaluates the impact of grid and communication system
topologies on potential expansion plans. It also reports extensive
computational results demonstrating the benefits of the proposed
B\&P algorithm on the test cases.
\end{enumitemize}
The remainder of this paper is organized as follows. Section
\ref{sec:review} reviews related work on the ORDPDC. Section
\ref{sec:formulation} formalizes the ORDPDC and Section
\ref{sec:approxi} presents a tight linear approximation. Section
\ref{sec:sbd} briefly reviews the SBD algorithm. Section \ref{sec:cg}
presents the new B\&P algorithm. Section \ref{sec:data} describes the
test cases. Lastly, Section \ref{sec:casestudy} analyzes the behavior
of the model on the case studies and Section \ref{sec:performance}
reports on the computational performance of the proposed algorithm.
Section \ref{sec:conclusions} concludes the paper.
\section{Literature Review}
\label{sec:review}
There has been a considerable progress in advancing methods that
address weather-related issues at distribution level
\citep{wang2016research}. Many studies develop post-fault distribution
system restoration (DSR) models to bring power back as soon as
possible and restore critical loads after a severe outage. Recently,
DGs, RCSs, and redundant lines were utilized to leverage microgrids in
load restoration. Most of the studies assume the existence of those
devices beforehand \citep{chen2016resilient, ding2017new,
gao_resilience-oriented_2016,yuan2017modified}. \citet{wang2016networked}
proposed a DSR model that utilizes the placement of dispatchable DGs.
The above-mentioned studies however propose post-contingency models. To
facilitate these novel restoration methods, the devices should be
placed in suitable places in advance. This paper focuses on the
optimal placement of those devices so that the grid survives potential
weather-related events.
Only a limited number of studies have discussed how to optimally add
resilience to distribution networks. Most relevant is the work by
\citet{Barnes2017Tools} and \citet{yamangil2015resilient} who propose
multi-scenario models for making a distribution grid resilient with
respect to a set of potential disaster scenarios. They propose
decomposition-based exact and heuristic solution approaches. However,
theses studies do not consider some of the upgrade options discussed
in this paper, and communication networks are not taken into
account. \citet{yuan2016robust} proposed a two-stage robust
optimization model by utilizing a bi-level network interdiction model
that identifies the critical components to upgrade for the resilience
against the $N-K$ contingency criterion. However, as pointed out in
\citet{Barnes2017Tools}, in practice, the computational complexity of
this approach grows quickly with the number of allowable faults. The
study also did not explicitly consider the dependency on the
communication network: A DG can supply power to the node it is placed
on and its children if they are not damaged by the attack.
\citet{carvalho2005decomposition} and \citet{xu2016placement} discuss
how to place RCSs in distribution systems, but only single fault
scenarios are assumed, which is not suitable for capturing
weather-related extreme events.
As the instrumentation of the grid increases, frameworks for modeling its
dependence on communication networks from a resilience viewpoint have
been studied \citep{martins2017interdependence,
parhizi2015state}. \citet{resende2011service} proposed a
hierarchical control system, which assumes the existence of a controller
in each microgrid to allow for the coordination among distributed
generation units in the microgrid, while multiple microgrids are
organized by a central management controller. On the other hand,
distributed control systems are applied to microgrids where there are
many devices with their own controllers. Accordingly,
\citet{chen2016resilient} assumed that RCSs have local communication
capabilities to exchange information with neighboring switches over
short-range low-cost wireless networks and proposed a global
information discovery scheme to get the input parameters for a DSR
model. However, the assumption that RCSs are installed in all lines is
premature for current distribution systems. \citet{wang2016networked}
proposed a two-layered communication framework where the lower-layer
cyber network supports microgrids where local control systems are
installed, while the upper-layer network is composed of multiple local
control systems that only communicate with their neighboring
counterparts. The study can be viewed as a hybrid of centralized and
decentralized framework: At a microgrid level, it is operated in a
centralized fashion, while the upper-level network is operated in a
decentralized manner. However, it did not consider fault scenarios in
communication networks. This paper only assumes the lower-layer cyber
network proposed in \citet{wang2016networked} by dynamically
allocating a local control system to each microgrid in islanding mode.
Moreover, this paper also considers potential faults in the
communication system.
To the best of our knowledge, this paper proposes, for the first time,
an exact optimization algorithm for expanding an integrated
distribution grid and communication network through the placement of
new DGs and RCSs and the hardening of existing lines in order to
ensure resilience against a collection of disaster scenarios.
\section{The ORDPDC}
\label{sec:formulation}
The ORDPDC considers an unbalanced three-phase distribution
grid coupled with a communication network, as illustrated in Figure
\ref{fig:physical_cyber}. In the figure, blue- and red-colored arrows
represent regular and critical loads. Nodes in the communication
networks may control a generator or a switch in the distribution
network, as indicated by dotted lines. The figure also
highlights how the line phases are interconnected at the buses and the
communication centers that will send instructions to generators and
switches remotely.
\begin{figure}
\caption{The Cyber-Physical Network for Electricity Distribution. Solid lines represent power lines and dotted lines represent communication links.}
\label{fig:physical_cyber}
\end{figure}
Let $G = (V,E)$ be an undirected graph that represents a distribution
grid and its available upgrade options: $V$ and $E$ denote the set of
buses and the set of distribution lines. The communication network,
along with its potential upgrade options, is represented by a
undirected graph $\mathcal{G}c = (\mathcal{N}c , \mathcal{E}c )$, where $\mathcal{N}c$ and $\mathcal{E}c$ are the
set of communication nodes and a set of communication links. A
communication node is either a control point or an intermediate
point. Each control point is associated with some device in $G$ and
some nodes in $\mathcal{N}c$ are designated as control centers.
The power grid $G$ depends on its communication network $\mathcal{G}c$ in the
following way: A device in $G$ (e.g., a generator or a RCSs) is
operable only when its associated control point can receive a signal
from some control center in $\mathcal{G}c$. This modeling enables islands to
form and to be operated independently only when at least one control
center can communicate to the island and, in particular, its
generator(s).
Let $\mathcal{G} = (\mathcal{N}, \mathcal{E})$ be the integrated system of $G$ and $\mathcal{G}c$ with $\mathcal{N}
= N \cup \mathcal{N}c$ and $\mathcal{E} = E \cup \mathcal{E}c$. Let $\mathcal{D}$ be a set of damage
scenarios for $\mathcal{G}$ indexed with $\mathcal{S} := \{1, \cdots, |\mathcal{D}|\}.$ Each
scenario $s \in \mathcal{S}$ is a set of edges of $\mathcal{E}$ that are damaged under
$s$. The goal of the ORDPDC is to find an optimal upgrade
profile for the cyber-physical system $\mathcal{G}$ that is resilient with
respect to the damage scenarios in $\mathcal{D}$. The upgrade options include
a) the building of new edges in $\mathcal{E}$ (i.e., distribution lines or
communication links); b) the building of RCSs on some lines in $E$ to
provide operational flexibility; c) the hardening of existing edges in
$\mathcal{E}$ to lower the probability of damage, and d) the building of DGs at
some buses of the grid.
The ORDPDC is a two-stage mixed integer stochastic program. The
first-stage variables represent potential infrastructure enhancements
for the coupled network $\mathcal{G}$ and the second-stage variables capture
how upgrades serve the loads in each disaster scenario.
\subsection{Mathematical Formulation}
Table \ref{table:param} specifies the input data for the ORDPDC
problem, while Table \ref{table:var} describes the model
variables. The formulation assumes that all new lines come with
switches (i.e., $\mathcal{E}_x^n \subseteq \mathcal{E}_t^0$) which reflects current
industry practice. Throughout this paper, an edge $e \in \mathcal{E}$ is
represented as an ordered pair $(e_h,e_t)$ for some $e_h, e_t \in \mathcal{N}$
and $\delta(e) = \{e_h, e_t\}$. The set of all edges incident to a
node $i \in \mathcal{N}$ is denoted by $\delta(i)$. The notation
$x_{\mathcal{A}}$ represents the projection of a vector $x$ to the
space of some index set $\mathcal A$, i.e., $(x_a)_{a \in
\mathcal{A}}$: For instance, $x^s_{\mathcal{E}_x} = (x^s_e)_{e \in \mathcal{E}_x}.$
\begin{table}[!t]
\TABLE{The Parameters of the ORDPDC. \label{table:param}}
{\begin{tabular}{ll}
\hline
\up $G=(N, E)$ & an undirected extended distribution grid with available upgrade options \\%$N$ represents a set of buses and $E$ represents a set of lines.\\
$\mathcal{U} := \mathcal{U}^0 \cup \mathcal{U}^n$ & a set of generators, indexed with $l$\\
$ \qquad \mathcal{U}^0$ & a set of existing generators\\
$ \qquad \mathcal{U}^n$ & a set of generators that can be installed\\
$i(l) \in N$ & the bus in which the generator $l \in \mathcal{U}$ is located\\
$\mathcal{U}_i \subseteq \mathcal{U}$ & the set of generators connected to bus $i \in N$\\
$E_V \subseteq E$ & a set of transformers\\
$\beta_e$ & maximum flow variation allowed between different phases on line $e \in E_V$\\
$\mathcal{C} \subseteq 2^{|E|}$ & a collection of a set of edges which forms a cycle with a distinct node set \\
$\mathcal P_e,\mathcal P_i,\mathcal P_l$ & a set of phases on line $e \in E$, bus $i \in N$, and generator $l \in \mathcal{U}$, respectively \\
$T_{e}^k$ & a thermal limit on line $e \in E$ for phase $k \in \mathcal P_e$ \\
$\underline V_{i}^k, \overline V_{i}^k$ & lower and upper bound on voltage magnitude at bus $i \in N$ on phase $k \in \mathcal P_i$ \\
$Z_{e} = R_{e}+\mathbf{i} \ X_{e}$ & phase impedance matrix of line $e \in E$ \\
$\mathcal{L} \subseteq N$ & a set of buses with critical loads \\
$D_{i,p}^k + \mathbf{i}\ D_{i,q}^k$ & complex power demand at bus $i \in N$ on phase $k \in \mathcal P_i$ \\
$\eta_{c}, \eta_{t}$ & resiliency criteria in percentage for critical and total loads respectively \\
$\overline g^{k}_{l,p} + \mathbf{i} \ \overline g^{k}_{l,q}$ & complex power generation capacity of generator $l \in \mathcal{U}$ on phase $k \in \mathcal P_l$ \\
$ \mathcal{G}c=(\mathcal{N}c, \mathcal{E}c)$ & an extended associated communication network with potential upgrade options \\%$\mathcal{N}c$ represents a set of communication nodes and $\mathcal{E}c$ represents a set of communication links. \\
$\mathcal{N}c_c := \mathcal{N}c_t \cup \mathcal{N}c_u$ &\\
$\qquad \mathcal{N}c_t \subseteq \mathcal{N}c$ & a set of control points for switches \\
$\qquad\mathcal{N}c_u \subseteq \mathcal{N}c$ & a set of control points for generators\\
$\tilde{i}(e) \in \mathcal{N}c_t, \tilde{i}(l) \in \mathcal{N}c_u $ & the control point in $\mathcal{G}c$ of a switch $e \in \mathcal{E}_t$ and a generator $l \in \mathcal{U}$, respectively\\
$\tilde i_d \in \mathcal{N}c$ & an artificial dummy node in $\mathcal{G}c$\\
$ \mathcal{G}=(\mathcal{N}, \mathcal{E})$ & the integrated system of $G$ and $\mathcal{G}c$\\%, where $\mathcal{N} = N \cup \mathcal{N}c$, indexed with $i$, and $\mathcal{E} = E \cup \mathcal{E}c$, indexed with $e$. \\
$\mathcal{E}_x := \mathcal{E}_x^0 \cup \mathcal{E}_x^n$ & \\%a set of lines and links that are existing or can be installed.\\
$\qquad \mathcal{E}_x^0\subseteq \mathcal{E}$ & a set of existing lines and links\\
$\qquad \mathcal{E}_x^n\subseteq \mathcal{E}$ & a set of lines and links that can be installed \\
$\mathcal{E}_t := \mathcal{E}^0_t \cup \mathcal{E}^{n}_t$ & \\%a set of lines in which a switch is installed,
$\qquad \mathcal{E}^0_t \subseteq E$ & a set of lines in which a switch is installed\\
$\qquad\mathcal{E}^n_t\subseteq E$ & a set of lines in which a switch can be installed \\
$\mathcal{E}_h \subseteq E$ & a set of lines or links that can be hardened \\
$c^x_e$ & installation cost of $e \in \mathcal{E}_x^n$ \\
$c^t_e$ & installation cost of switch on $e \in \mathcal{E}^n_t$ \\
$c^h_e$ & line hardening cost of $e \in \mathcal{E}_h$ \\
$c^u_l$ & installation cost of $l \in \mathcal{U}^n$ on the corresponding bus \\
\down $\mathcal{D}$ & a collection of sets of damaged lines for each scenario, indexed with $\mathcal{S}:=\{1, \cdots, |\mathcal{D}|\}$ \\%i.e., $\mathcal{D}_s$ corresponds to a subset of $\mathcal{E}$ consisting of damaged lines under scenario $s \in \mathcal{S}$ \\
\hline
\end{tabular}}{}
\end{table}
\begin{table}
\TABLE{The Variables of the ORDPDC. \label{table:var}}
{\begin{tabular}{ll}
\hline
\multicolumn{2}{l}{\up\down \textbf{Binary variables}} \\
$x_e$ & 1 if $e \in \mathcal{E}_x^n$ is built\\
$t_e$ & 1 if a switch is built on $e \in \mathcal{E}^n_t$ \\
$h_e$ & 1 if $e \in \mathcal{E}_h$ is hardened \\
$u_l$ & 1 if a generator $l \in \mathcal{U}^n$ is built. \\
\multicolumn{2}{l}{\up For each disaster scenario $s \in \mathcal{S}$,} \\
$z^s_e$ & 1 if $e \in \mathcal{E}$ is active during $s$ \\
$x^s_e$ & 1 if $e \in \mathcal{E}_x$ exists during $s$ \\
$t^s_e$ & 1 if a switch on $e$ is used or not during $s$ \\
$h^s_e$ & 1 if $e \in \mathcal{E}_h$ is hardened during $s$ \\
$u^s_l$ & 1 if a generator $l \in \mathcal{U}^n$ is available during $s$\\
$y^s_e$ & 1 if $i, \ j \in N$ can be disconnected, for $e = (i,j) \in C, \ C \in \mathcal{C}$, during $s$\\
$b_e$ & 1 if the real power on line $e = (i,j) \in E$ flows from $j$ to $i$ during $s$\\
$b'_e$ & 1 if the reactive power on line $e = (i,j) \in E$ flows from $j$ to $i$ during $s$\\
\multicolumn{2}{l}{\up \down \textbf{Continuous variables}} \\
\multicolumn{2}{l}{For each disaster scenario $s \in \mathcal{S}$,} \\
$d^{s,k}_{i} = d^{s,k}_{i,p} + \mathbf{i} \ d^{s,k}_{i,q}$ & amount of power delivered at bus $i \in N$ on phase $k \in \mathcal P_i$ during $s$ \\
$g^{s,k}_{l} = g^{s,k}_{l,p} + \mathbf{i} \ g^{s,k}_{l,q}$ & amount of power generation of $l \in \mathcal{U}$ on phase $k \in \mathcal P_l$ during $s$ \\
$s^{s,k}_{e,i} = p^{s,k}_{e,i} + \mathbf{i} \ q^{s,k}_{e,i}$ & power flow on $i$-end of line $ e \in E$, where $i \in \delta(e)$, on phase $k \in \mathcal P_e$ during $s$ \\
$V^{s,k}_{i}$ & complex voltage at bus $i \in N$ on phase $k \in \mathcal P_i$ during $s$ \\
$I^{s,k}_e$ & complex current on line $e \in E$ on phase $k \in \mathcal P_e$ during $s$ \\
$v_{i}^{s,k}$ & squared voltage magnitude at bus $i \in N$ on phase $k \in \mathcal P_i$ during $s$ \\
$f^{s}_{e}$ & the amount of artificial flow on $e \in \mathcal{E}c$ during $s$\\
\down$\gamma_{\tilde{i}}^s$ & indicator of connectivity of control point $\tilde i \in \mathcal{N}c$ to some control center during $s$\\
\hline
\end{tabular}}{}
\end{table}
The presentation uses $w= ( x_{\mathcal{E}^n_x}, t_{\mathcal{E}^n_t}, h_{\mathcal{E}_h},
u_{\mathcal{U}^n})$ to denote upgrade profiles, $m$ the dimension of $w$, $c =
(c^x_{\mathcal{E}^n_x}, c^t_{\mathcal{E}^n_t}, c^h_{\mathcal{E}_h}, c^u_{\mathcal{U}^n}) \in \mathbb{R}^m$ the
cost vector, and $w^s = ( x^s_{\mathcal{E}^n_x}, t^s_{\mathcal{E}^n_t}, h^s_{\mathcal{E}_h},
u^s_{\mathcal{U}^n})$ feasible upgrade profiles for each scenario $s \in
\mathcal{S}$. For each $s \in \mathcal{S},$ $\mathcal Q(s)$ denotes the set of upgrade
profiles that enable the grid to maintain the predetermined load
satisfaction (resiliency) level $\eta_c, \eta_t$ (e.g., $\eta_c =
0.98$ and $\eta_t = 0.5$) under disaster scenario $s$.
With these notations, the ORDPDC is formulated as follows:
\begin{subequations}
\begin{alignat}{3}
(P) \quad &&\min \quad& c^T w \label{eq:first_obj}\\
&&\mbox{s.t.}\quad & w \geq w^s, & \forall s \in \mathcal{S}, \label{eq:link_constr}\\
&& &w^s \in \mathcal Q(s), \ &\forall s \in \mathcal{S}, \label{eq:second_stage}\\
&& &w \in \{0,1\}^{m}.\nonumber
\end{alignat}
\label{eq:first}
\end{subequations}
\noindent
Problem ($P$) tries to find the optimal upgrade profile $w^* =
(x^*_{\mathcal{E}^n_x}, t^*_{\mathcal{E}^n_t}, h^*_{\mathcal{E}_h}, u^*_{\mathcal{U}^n})$ that
ensures resilient operations for each disaster scenario. Equation
\eqref{eq:link_constr} ensures that an upgrade profile is feasible if
it dominates a feasible solution $w^s \in \mathcal Q(s)$ for each scenario
$s$, i.e., if the grid survives each of the extreme events in $\mathcal{S}$.
The set $\mathcal Q(s)$ is specified by resiliency constraints that are
expressed in terms of the AC power flow equations, load satisfaction
requirements, the communication network, and the grid topology:
\[
\mathcal Q(s) = \{ w^s \in \{0,1\}^m: (2),(3), \eqref{eq:generator/demand}, (5), \mbox{ and } \eqref{eq:topology} \}
\]
where Constraints (2), (3), \eqref{eq:generator/demand}, (5), and
\eqref{eq:topology} are stated in detail in the following. The
variables in each $\mathcal Q(s)$ are indexed by $s$. For simplicity, this
section omits index $s$.
\subsubsection{Power Flow Constraints}
\label{sec:formulation:powerflow}
\begin{figure}
\caption{Notations for the Power Flow Equations.}
\label{fig:powerflow}
\end{figure}
Figure \ref{fig:powerflow} specifies the power flow equations and
summarizes some of the notations. Let $\mathcal P = \{a, b, c\}$ denote the
three phases of the network. For each bus $i \in N$, define $V_i =
(V_i^k)_{k \in \mathcal P_i}$ and, for each line $e \in E$, define $I_e =
(I_e^k)_{k \in \mathcal P_e}$ and $s_{e,i} = (s_{e,i}^k)_{k \in \mathcal P_e}$. The
notations also use a superscript $\mathcal P' \subseteq \mathcal P$ to represent the
{\em projection} or the {\em extension} of a vector to the space of
$\mathcal P'$. For example, if $\mathcal P_i = \{a,b,c\}$ and $\mathcal P' = \{a,b\}$, then
$V_i^{\mathcal P'} = \left( V_i^a, V_i^b \right)^T$. If $\mathcal P_i = \{a,c\}$ and
$\mathcal P' = \{a,b, c\}$, then $V_i^{\mathcal P'} = \left( V_i^a, 0, V_i^c
\right)^T.$
For each line $e = (i,j) \in E$, Ohm's law for 3-phase lines states
the relationship $V_j^{\mathcal P_e} = V_i^{\mathcal P_e} - Z_eI_e$ between $I_e$,
$V_i$, and $V_j$. For each line $e \in E$ and bus $i \in \delta(e)$,
the electric power flow equation $s_{e,i} =
\mbox{diag}(V_i^{\mathcal P_e}I_e^H)$ describes the relationship between
$s_{e,i}$, $V_i^{\mathcal P_e}$, and $I_e$, where superscript $H$ indicates
the conjugate transpose. In Figure \ref{fig:powerflow}, the big-M
method is used in Equations (2a) to apply Ohm's law only for available
lines; the big-M can be set as $\max_{j' \in \{i,j\}, k \in \mathcal P_e}
\overline{V}_{j'}^k - \min_{j' \in \{i,j\}, k \in
\mathcal P_e}\underline{V}^k_{j'}$. Equations (2c) is the balance equation
for power flow at each bus $i \in N$, i.e., the sum of incoming flows
equals the sum of the outgoing flows.
Let $p_{e,i} + \textbf{i} q_{e,i}$ be the rectangular representation of
$s_{e,i}$, where $p_{e,i} = (p_{e,i}^k)_{k \in \mathcal P_i}$ and $q_{e,i} =
(q_{e,i}^k)_{k \in \mathcal P_i}$ denote the real and reactive power at the
$i$-end of line $e$. Equations (2d) and (2e) specify the thermal
limits on lines and the voltage bounds on buses.
In some disaster scenarios when some of the lines are broken, power
flows of different phases on the same line can have opposite
directions, which is a very undesirable operationally. Equations (2f)
and (2g) prevent this behavior from happening.
The real and reactive power on different phase must stay within a
certain limit. Let $\widehat p_{e,i} = \sum_{\tilde{k} \in \mathcal P_e}
p_{e,i}^{\tilde{k}}$ and $\widehat q_{e,i} = \sum_{\tilde{k} \in
\mathcal P_e} q_{e,i}^{\tilde{k}}$. Then, these limits are formulated as
follows:
\addtocounter{equation}{1}
\begin{subequations}
\begin{alignat}{2}
&\left(\underline{\beta}_{e}(1-b_{e}) + \overline{\beta}_{e} b_{e}\right) \frac{\widehat p_{e,i}}{|\mathcal P_{e}|} \leq {p}_{e,i}^k \leq \left(\underline{\beta}_{e}b_{e} + \overline{\beta}_{e}(1-b_{e})\right) \frac{\widehat p_{e,i}}{|\mathcal P_{e}|}, &\ \forall e \in E_V, k \in \mathcal P_e, \label{eq:ph_bal_p}\\
&\left(\underline{\beta}_{e} (1-b'_{e}) + \overline{\beta}_{e}b'_{e}\right) \frac{\widehat q_{e,i}}{|\mathcal P_{e}|} \leq {q}_{e,i}^k \leq \left(\underline{\beta}_{e}b'_{e} + \overline{\beta}_{e}(1-b'_{e})\right) \frac{\widehat q_{e,i}}{|\mathcal P_{e}|}, &\ \forall e \in E_V, k \in \mathcal P_e, \label{eq:ph_bal_q}
\end{alignat}
\label{eq:ph_bal}
\end{subequations}
where $\underline \beta_e = 1-\beta_e$ and $\overline \beta_e = 1+ \beta_2$.
\subsubsection{Generator/resiliency Constraints}
\label{sec:formulation:gen}
Moreover, each generator $l \in \mathcal{U}$ has its own capacity and at least
some percentage of critical and total loads must be satisfied
as specified by the resiliency criteria $\eta_c$ and $\eta_t$.
\begin{subequations}
\label{eq:generator/demand}
\begin{alignat}{2}
& 0 \le g_{l,p}^k \leq \overline g^k_{l,p} u_l, \ g_{l,q}^k \leq \overline g^k_{l,q} u_l, & \forall l \in \mathcal{U}, k \in \mathcal P_l, \label{eq:capa_generator}\\
&0 \le d_{i, p}^k \le D_{i,p}^{k}, \ 0 \le d_{i, q}^k \le D_{i,q}^{k}, &~\forall i \in N, k \in \mathcal P_i\label{eq:demand}\\
& \sum_{i\in \mathcal{L}} d^{k}_{i,p} \geq \eta_{c} \sum_{i\in \mathcal{L}} D^k_{i,p},\quad \sum_{i\in \mathcal{L}} d^{k}_{i,q} \geq \eta_{c} \sum_{i\in \mathcal{L}} D^k_{i,q}, &\quad \forall k \in \mathcal P, \label{eq:critical_load_met}\\
& \sum_{i\in N} d^{k}_{i,p} \geq \eta_{t} \sum_{i\in N} D^k_{i,p},\quad \sum_{i\in N} d^{k}_{i,q} \geq \eta_{t} \sum_{i\in N} D^k_{i,q}, & \forall k \in \mathcal P. \label{eq:ncritical_load_met}
\end{alignat}
\end{subequations}
Equation \eqref{eq:capa_generator} captures the power generation
capacity constraints. Equation \eqref{eq:demand} states that the
delivered power at each bus $i$ should not exceed the load. Equations
\eqref{eq:critical_load_met}-\eqref{eq:ncritical_load_met} enforce
the resiliency constraints.
\subsubsection{Communication Constraints}
\label{sec:formulation:comm}
\begin{figure}
\caption{The Single-Commodity Flow Model for $\mathcal{G}
\label{fig:single_flow}
\end{figure}
The operation of generators and RCSs depend on the communication
network: A generator $l \in \mathcal{U}$ and a RCS on line $e \in \mathcal{E}_t$ is
operable only if their associated control points $\tilde i(l) \in \mathcal{N}c$
and $\tilde i (e) \in \mathcal{N}c$ can receive a control signal from some
control centers through $\mathcal{G}c$. To capture the connectivity of a
vertex to some control centers, the formulation uses a
single-commodity flow model summarized in Equations (5) in Figure
\ref{fig:single_flow}. The formulation uses a dummy node $\tilde i_d$
to $\mathcal{N}c$ and connect $\tilde i_d$ to all control centers with
additional links. The flow $f \in \mathbb{R}^{|\mathcal{E}c|}$ originating from
the dummy node $\tilde i_d$ then is used to check the connectivity of
every node. By Equation (5c), the flow passes only through available
links during disaster $s$ (the big-M value is set to $|\mathcal{N}c_c|$ in the
implementation). If a control point $i \in \mathcal{N}c_c$ is connected with
some control center through some path, it can borrow a unit of flow
from $f$ to make $\gamma_{ i}$ 1, as specified in Equations (5a) and
(5b). In other words, $\gamma_{ i}$ indicates whether control point
$i \in \mathcal{N}c_c$ can receive a control signal. If $\gamma_{ i}$ is 1,
the associated device in $G$ is operable by Equations (5d) and (5e).
Some communication network may be affected by a failure in
distribution grid, e.g., when the grid fails to supply power to
communication centers. This kind of dependencies is not considered in
this paper but it can be easily captured if needed. Indeed, first
assign a small critical load to each communication center and add
constraints that restrict the auxiliary arcs between the dummy node
and each communication center to have positive flow only when the
associated communication center has a positive power supply. The
constraints can be expressed in terms of an extra binary variable for
each bus at which a communication center is located. The extra binary
variable determines if there is a positive power supply to the
communication center.
\subsubsection{Topological constraints.}
\label{sec:formulation:topo}
The final set of constraints captures the topology restrictions in distribution systems:
\addtocounter{equation}{1}
\begin{subequations}
\begin{alignat}{2}
& x_e \ge t_e, &\forall e \in \mathcal{E}, \label{eq:topology_switch} \\
& z_e = x_e - t_e, \hspace{4cm}& \forall e \in \mathcal{E}, \label{eq:line_availability}\\
& x_e = h_e, &\forall e \in \mathcal{D}_s, \label{eq:topology_damage}\\
& \sum_{e \in C} y_e \leq |C|-1,&\forall C \in \mathcal{C}, \label{eq:topology_cycle1}\\
& z_{\hat e} \le y_e, & \forall \hat e \in E: \delta(\hat{e}) = \delta(e), \ e \in C, \ C \in \mathcal{C}. \label{eq:topology_cycle2}
\end{alignat}
\label{eq:topology}
\end{subequations}
Constraint \eqref{eq:topology_switch} restrict switches to be operable
only on existing lines. In Equation \eqref{eq:line_availability},
$z_e$ represents whether line $e \in \mathcal{E}$ is active under scenario
$s$. A line is active when it exists and its switch is off. Equation
\eqref{eq:topology_damage} states that a damaged line during scenario
$s \in \mathcal{S}$ is inoperable unless it is hardened. Constraints
\eqref{eq:topology_cycle1} and \eqref{eq:topology_cycle2} ensures that
the distribution grid should operate in a radial manner. Accordingly,
Constraint \eqref{eq:topology_cycle1} eliminates the sub-tours within
$\mathcal{C}$. Since $G$ is usually sparse, the implementation enumerates all
the sub-tours $\mathcal{C}$ and variable $y_e$ indicates whether $i,j \in
\delta(e)$ are disconnected. If they are disconnected, then all the
lines between $i$ and $j$ are inactive by Constraint
\eqref{eq:topology_cycle2}.
Note also that, for existing lines not damaged under scenario $s$,
$x_e$ is fixed as one. For each line $e \in E \setminus \mathcal{E}_t$, $t_e$
is set to zero. Finally, for each line $e \in \mathcal{E} \setminus \mathcal{E}_h,$
$h_e$ is fixed as 0 and all the existing generators have $u_l = 1$.
This paper assumes perfect hardening, i.e., a hardened line survives
all disaster scenarios. This assumption can be naturally generalized
to imperfect hardening \citep{yamangil2015resilient}.
\section{ Linearization of the ORDPDC}
\label{sec:approxi}
The formulation of the ORDPDC is nonlinear. This section discusses how to obtain an accurate linearization.
\subsection{Linear Approximation of the AC Power Flow Equations for Radial Networks}
\label{sec:approxi:powerflow}
The main difficulty lies in linearizing constraints (2a--2b) for which
the formulation uses the tight linearization from
\citet{gan2014convex}. The linearization is based on two assumptions:
(A1) line losses are small, i.e., $Z_{e}I_e I_e^H \approx 0$ for $e =
(i,j) \in E$ and (A2) voltages are nearly balanced, i.e., if $\mathcal P_i =
\{a, b, c\}$, then $V^a_i/V^b_i \approx V^b_i/V^c_i \approx
V^c_i/V^a_i \approx e^{i 2 \pi / 3}.$ Informally speaking, the
approximation generalizes the distflow equations to 3 phases, drops
the quadratic terms, and eliminates the current variables using the
balance assumption. The derivation assumes that all phases are
well-defined for simplicity. Moreover, if $A$ is an $n\times n$
matrix, then diag($A$) denotes the $n$-dimensional vector that
represents its diagonal entries. If $a$ is an $n$-dimensional vector,
then diag($a$) denotes the $n\times n$ matrix with $a$ in its diagonal
entries and zero for the off-diagonal entries.
Let $s_i = \sum_{l \in \mathcal{U}_i} g_l - d_i$ denote the
power injection at bus $i$. By (A1), $s_{e,i} = s_{e,j}$ for
all $e \in (i,j) \in E$ and therefore, given $s_i$, $s_{e,i}$ ($i \in
\delta(e))$ is uniquely determined by Equation (2c).
Now define $S_{e,i} := V_i I_e^H$, whose diagonal entries are $s_{e,i}$.
Multipling both sides of $V_j = V_i - Z_e I_e$ with their conjugate transposes
gives
\begin{equation}
V_j V_j^H = V_iV_i^H - S_{e,i} Z_e^H - Z_e S_{e,i}^H + Z_e I_e I_e^H Z_e^H.
\end{equation}
By assumption (A1), this becomes
\begin{equation}
V_j V_j^H = V_iV_i^H - S_{e,i} Z_e^H - Z_e S_{e,i}^H
\end{equation}
and, by restricting attention to diagonal elements only,
\begin{equation}
v_j = v_i - \mbox{diag}(S_{e,i} Z_e^H - Z_e S_{e,i}^H).
\label{eq:Ohm2}
\end{equation}
where $(v^k_i)_{k \in \mathcal P_i} = \mbox{diag}(V_i V_i^H)$ represents the
squared voltage magnitude at bus $i \in N$.
By (A2), we have $S_{e,i} \approx \gamma^{\mathcal P_e} \mbox{diag}(s_{e,i}),$
where
$$\gamma = \left[\begin{array}{ccc} 1 & \alpha^2 & \alpha \\
\alpha & 1 & \alpha^2 \\
\alpha^2 & \alpha &1 \end{array} \right] \mbox{ and } \alpha = e^{-i 2 \pi /3}.$$
As a result, Equation \eqref{eq:Ohm2} can now be simplified as
follows: for each line $e = (i,j) \in E$ and $k \in \mathcal P_e$,
\begin{equation}
v^k_{i} = v^k_{j} - \sum_{k' \in \mathcal P_e} 2\left[(\alpha^{n_k - n_{k'}}R_{e})^{kk'} p_{e,i}^{k'} + (\alpha^{n_k - n_{k'}}X_{e})^{kk'} q_{e,i}^{k'}\right],
\label{eq:lindist}
\end{equation}
where $n_a = 2, n_b = 1, n_c = 0$, $R_e + \textbf{i} X_e = Z_e$, and superscript $kk'$ of a matrix denotes its $(k,k')$-entry.
In summary, Ohm's law and the power flow equation in Constraints (2a) and (2b) are approximated by Eq. \eqref{eq:lindist} for all $e = (i,j) \in E$ and $k \in \mathcal P_e$
and the big-$M$ is set to $\max_{j' =i,j} (\overline V_{j', k})^2 - \min_{j' =i,j} (\underline V_{j', k})^2$, along with Equation (2c). Accordingly, Constraint (2e) is replaced by the following constraint:
\begin{displaymath}
(\underline V_i^k)^2\le v_i^k \le (\overline V_i^k)^2, \quad \forall i \in N, k \in \mathcal P_i.
\end{displaymath}
\subsection{Linearization of \eqref{eq:ph_bal_p}-\eqref{eq:ph_bal_q}}
\label{sec:appendix:ph_bal}
Constraints \eqref{eq:ph_bal_p} and \eqref{eq:ph_bal_q} contain
products of a binary variable and a bounded real variable. These
constraints are linearized without loss of accuracy using McCormick
inequalities \cite{mccormick1976computability}.
\subsection{Piecewise-Linear Inner Approximation of Thermal Limits}
\label{sec:approxi:thermal}
\begin{figure}
\caption{The Piecewise-Linear Inner Approximation of a Circle.}
\label{fig:thermal}
\end{figure}
The quadratic thermal limit constraints (Constraint (2d)) can be approximated with $K$ linear inequalities as shown in Figure \ref{fig:thermal}.
The resulting inequalities are as follows: for all $e \in E$, $i \in \delta(e)$, $k \in \mathcal P_e$:
\begin{subequations}
\label{eq:linearlized_thermal}
\begin{alignat}{2}
&\left(\sin\left(\frac{2n\pi}{K}\right) - \sin\left(\frac{2(n-1)\pi}{K}\right)\right) p_{e,i}^k \nonumber\\
&- \left(\cos\left(\frac{2n\pi}{K}\right) - \cos\left(\frac{2(n-1)\pi}{K}\right)\right) q_{e,i}^k \le \sin\left(\frac{2\pi}{K}\right)T_{e,k}, &\ \forall n = 1, \cdots, K,\\
&-M z^s_e \le p_{e,i}^{k} \le M z^s_e,\quad -M z^s_e \le q_{e,i}^{k} \le M z^s_e, & \quad \forall e \in E, k \in \mathcal P_e. \label{eq:lin_ther_i}
\end{alignat}
\end{subequations}
where the big-M is set to $\sum_{i \in N} D^p_{i,k}$. Our implementation
uses $K = 28$.
\section{Scenario-Based Decomposition}
\label{sec:sbd}
In Section \ref{sec:performance}, the branch and price algorithm
presented in the next section is compared to the Scenario-Based
Decomposition (SBD) algorithm proposed by
\citet{nagarajan2016optimal}. SBD iteratively solves a master problem
$P(\mathcal{S}')$ which only includes the constraints of a subset of scenarios
$\mathcal{S}' \subseteq \mathcal{S}$. The algorithm terminates when the optimal
solution to $P(\mathcal{S}')$ is feasible (and hence optimal) for the
remaining scenarios $\mathcal{S} \setminus \mathcal{S}'$. Otherwise, at least one
scenario $s \in \mathcal{S} \setminus \mathcal{S}'$ is infeasible. Scenario $s$ is
added to $\mathcal{S}'$ and the process is repeated.
\section{The Branch-and-Price Algorithm}
\label{sec:cg}
This paper proposes a branch-and-price (B\&P) algorithm for the
ORDPDC. The B\&P exploits the special structure of the ORDPDC
in several ways. First, it uses a compact reformulation that yields a
better lower bound than the LP relaxation. The reformulation also
makes it possible to use column generation and solve independent
pricing problems associated with each scenario in parallel. Finally,
several additional techniques are used to accelarate the column
generation significantly. Section \ref{sec:reformulation} presents the
problem reformulation and Section \ref{sec:cg:basic} briefly reviews
the basic column generation of the B\&P algorithm. Section
\ref{sec:accelerating} introduces several acceleration schemes. The
implementation of the B\&P algorithm is presented in Section
\ref{sec:implementation}.
\subsection{The Problem Reformulation }
\label{sec:reformulation}
Letting $\widetilde{\mathcal Q}(s)$ be the linearization of $\mathcal Q(s)$, the problem
($P$) is rewritten as
\begin{subequations}
\begin{alignat}{3}
(P) \quad & \mbox{min} & c^T w \nonumber\\
& \mbox{ s.t. } & w - w^s \ge 0, &\ \forall s \in \mathcal{S}, \label{eq:link}\\
&& w^s \in \widetilde{\mathcal Q}(s), &\ \forall s \in \mathcal{S}, \label{eq:sub}\\
&& w^s \in \{0,1\}^m, &\ \forall s \in \mathcal{S}. \label{eq:binary}
\end{alignat}
\label{eq:CG:original}
\end{subequations}
\noindent
Without the linking constraint \eqref{eq:link}, ($P$) can be
decomposed into $|\mathcal{S}|$ independent problems, each of which has a
feasible region defined by
$$
\mathcal{P}^s = \left\{w^s \in \mathbb{R}^m: \eqref{eq:sub} \mbox{ and } \eqref{eq:binary}\right\}, \ \forall s \in \mathcal{S}.
$$
Observe that $\mathcal{P}^s$ is bounded and let $\mathcal{J}^s =
\{\hat w^s_j \in \mathbb{R}^m: \hat w^s_j \mbox{ is a vertex of }
\mbox{conv} (\mathcal{P}^s) \}$ be the set of all vertices of
conv($\mathcal{P}^s$). Letting $\mathcal{J} = \cup_s \mathcal{J}_s$,
consider the following problem:
\begin{subequations}
\begin{alignat}{6}
(\widetilde P) \quad &\mbox{min} \quad & c^T w \nonumber\\
&\mbox{s.t.} & w & \quad - \sum_{j \in \mathcal{J}^s} \lambda_j^s \hat w_j^s \geq 0, \ & &&\forall s \in \mathcal{S},\label{eq:link2}\\
& & &\quad \sum_{j \in \mathcal{J}^s} \lambda_j^s = 1, & && \forall s \in \mathcal{S},\label{eq:convex}\\
& & &\quad w \in \{0,1\}^{m},\label{eq:binary2}\\
& & & \quad \lambda^s_j \ge 0, & && \forall j \in \mathcal{J}^s, s \in \mathcal{S}.
\end{alignat}
\label{eq:CG:reformulation}
\end{subequations}
\begin{theorem}
($P$) and ($\widetilde{P}$) are equivalent.
\end{theorem}
\textbf{Proof.} Since ($P$) and ($\widetilde{P}$) have the same objective
function, it suffices to show that ($P$) has an optimal solution that
is feasible to ($\widetilde{P}$) and vice versa. Let $(\bar{w},
\{\bar{w}^{s}\}_{s \in \mathcal{S}})$ be the optimal solution of ($P$). By the
Farkas-Minkowski-Weyl theorem \citep{schrijver1998theory}, $\bar w^s$
can be expressed as a convex combination of some extreme points in
$\mathcal J^s$, for each $s \in \mathcal{S}$. Hence, we can construct a
feasible solution of ($\widetilde{P}$) from $(\bar{w}, \{\bar{w}^{s}\}_{s \in
\mathcal{S}})$.
Consider now an optimal solution of ($\widetilde{P}$), $(\bar{w}^\prime,
\{\bar{\lambda}^{s \prime}_j\}_{j \in \mathcal{J}^s} \mbox{ for } s
\in \mathcal{S})$. By \eqref{eq:link2}, if $\bar{\lambda}^{s \prime}_j > 0$
for $j \in \mathcal{J}^s$, $\hat{w}^s_j$ is dominated by
$\bar{w}^\prime.$ Therefore, it is possible to construct another
optimal solution to ($\widetilde{P}$) by choosing a single $j^*$ for which
$\bar{\lambda}^{s \prime}_{j^*} > 0$ for each $s \in \mathcal{S}$, setting
$\bar{\lambda}^{s \prime}_{j^*}$ to one and setting the other
$\bar{\lambda}^{s \prime}_{j}$ to zero. By definition of
$\mathcal{J}_s$, the constructed optimal solution is feasible for
($P$). $\square$
\noindent
This paper uses a branch-and-price algorithm to solve ($\widetilde{P}$). Let
$LP_{\widetilde{P}}$ denote the LP relaxation of ($\widetilde{P}$). Since the feasible
region of ($\widetilde{P}$) is the intersection of the convex hulls of each
subproblem, $LP_{\widetilde{P}}$ yields a stronger lower bound than the LP
relaxation of ($P$).
\subsection{The Basic Branch and Price }
\label{sec:cg:basic}
The B\&P algorithm uses a restricted master problem ($M$) with a
subset of columns of ($\widetilde{P}$) and $|\mathcal{S}|$ independent subproblems
($P_s$) for $s \in \mathcal{S}$, instead of handling $LP_{\widetilde{P}}$ globally. The
column generation starts with an initial basis that consists of the
first-stage variables $w$, a column associated with a feasible
solution for each subproblem, and some slack variables. Let
$\widetilde{\mathcal{J}}^s$ be the corresponding subset of
$\mathcal{J}^s$. The restricted mater problem ($M$) is
as follows:
\begin{subequations}
\begin{alignat}{5}
(M) \quad &\mbox{min} \ & c^T w \nonumber\\
&\mbox{s.t.} \ & w - & \sum_{j \in \widetilde{\mathcal{J}}^s} \lambda_j^s \hat w_j^s \geq 0, & \ \forall s \in \mathcal{S},\label{eq:m_link}\\
& & &\sum_{j \in \widetilde{\mathcal{J}}^s} \lambda_j^s = 1, & \forall s \in \mathcal{S},\label{eq:m_convex}\\
& && w \ge 0,\ \\
& && \lambda^s_j \ge 0, \quad & \forall j \in \widetilde{\mathcal{J}}^s, \ \forall s \in \mathcal{S}.
\end{alignat}
\label{eq:master}
\end{subequations}
\noindent
and the pricing problem for scenario $s$ is specified as follows:
\begin{equation*}
\begin{array}{lll}
(P_s) \quad & \ \min & \ - \bar{\sigma}^s \ + {\bar{y}^{sT}} \ w^s \\
& \mbox{s.t.} & \ w^s \in \widetilde{\mathcal Q}(s), \\
& & \ w^s \in \{0,1\}^m,
\end{array}
\end{equation*}
\noindent
where, for a scenario $s$, $\bar{y}^{s}$ is the dual solution for
constraints \eqref{eq:m_link} and $\bar{\sigma}^s$ is the dual
solution of the convexity constraint (\ref{eq:m_convex}).
\subsection{Acceleration Schemes }
\label{sec:accelerating}
The performance of column generation deteriorates when the master
problem exhibits degeneracy, leading to multiple dual solutions which
may significantly influence the quality of columns generated by the
pricing problem. The master problem $(M)$ suffers from degeneracy,
especially early in the column-generation process. Initially, ($M$)
has $(m+1) |\mathcal{S}|$ constraints, $m$ columns corresponding to the
first-stage variables $w$, and $|\mathcal{S}|$ columns for the second-stage
variables $\{\lambda^s\}_{s \in \mathcal{S}}$. Therefore, in early iterations,
linear solvers have a natural tendency to select $m(|\mathcal{S}|-1)$ columns
from the slack variables in Constraints \eqref{eq:m_link}. For example, assume that
the slack variable is in basis for the constraint involving a non-basic first-stage variable $w_k$ and
a scenario $s$ in Constraints \eqref{eq:m_link}. By complementary
slackness, this implies that the dual variable is zero. Consider a
vertex $\hat w^s$ whose $k$-th entry is non-zero. The value $\bar
y^s_k w^s_k$ is zero in the pricing problem. However, for this vertex
to enter the basis, it must incur the cost $c_k$ of $w_k$, which is
ignored in the pricing subproblem. As a result, subproblem ($P_s$)
prices many columns too optimistically and generates columns that do
not improve the current objective value, resulting in a large number
of iterations.
\subsubsection{Pessimistic Reduced Cost }
\label{sec:accelerating:revised_rc}
In order to overcome the poor pricing of columns, this section first
proposes a pessimistic pricing scheme that selects more meaningful
columns in early iterations. Consider a solution $w^s$ to the pricing
problem. If $w^s_k = 1$ but the first-stage variable $w_k$ is not in
basis, then by the relevant constraint from $\eqref{eq:m_link}$, the variable
$\lambda_j^s$ corresponding to $w^s$ can only enter in the basis at 1
if $w_k$ is also in the basis at 1. As a result, the pessimistic
pricing scheme adds the reduced cost
$
c_k - \sum_{s \in \mathcal{S}} \bar{y}^s_k
$
to the pricing objective, which becomes
\[
- \bar{\sigma}^s + (\bar{y}^s)^T w^s + \sum_{k \in \eta} (c_k - \sum_{s \in \mathcal{S}}\bar{y}^s_k) w^s_k \label{eq:revised_rc}
\]
where $\eta$ is the set of non-basic first-stage variables, i.e., $
\eta = \{ k \ | \ w_k \mbox{ is non-basic} \}. $ Note that column
generation with this pessimistic pricing subproblem is not guaranteed
to converge to the optimal linear relaxation. Hence, the
implementation switches to the standard pricing problem in later
iterations.
\subsubsection{Optimality Cut}
\label{sec:accelerating:opt_cut}
A solution to the master problem ($M$) where the first-stage variables
take integer value gives an upper bound to the optimal solution. The
B\&P algorithm periodically solves the integer version of ($M$) to
obtain its objective value $\bar v(M)$. The constraint
\[
c^T w^s \le \bar v(M)
\]
can then be added to the pricing subproblem for scenario $s$ since any
solution violating this constraint is necessarily suboptimal. As shown
later on, this optimal cut is critical to link the two phases of the
column generation, preventing many potential columns to be generated
in the second phase.
\subsubsection{A Lexicographic Objective for Pricing Subproblems}
\label{sec:accelerating:multi_obj}
In general, sparse columns are more likely to enter the basis in the
master problem ($M$). As a result, the B\&P algorithm uses a
lexicographic objective in the pricing subproblem. First, it minimizes
the (pessimistic or standard) reduced cost. Then it maximizes sparsity
by minimizing $1^T w^s$ subject to the constraint that the reduced
cost must be equal to the optimal objective value of the first stage.
\subsection{The Final Branch and Price Implementation}
\label{sec:implementation}
\subsubsection{Column Generation}
\label{sec:implementation:cg}
The column generation starts with an initial basis built from the
optimal solutions of each subproblems under the objective function of
$c^T w^s$. It then proceeds with two phases of column generation,
first using the pessimistic reduced cost and then switching to the
standard one.
The second phase terminates when the optimality gap becomes lower than
the predetermined tolerance, e.g., $0.1\%$. The lower bound is based
on Lagrangian relaxation. Given a pair $\bar{w}$ and $(\bar{y},
\bar{\sigma})$ of optimal primal and dual solutions for ($M$), the
Lagrangian relaxation is given by
\[
L(\bar{w},\bar{y},\bar{\sigma}) = c^T \bar{w} + \sum_{s \in \mathcal{S}} {\cal O}_s(\bar{y},\bar{\sigma})
\]
where ${\cal O}_s(\bar{y},\bar{\sigma})$ is the optimal solution of
the pricing problem for scenario $s$ under dual variables
$(\bar{y},\bar{\sigma})$. The first phase uses the same technique for
termination, although the resulting formula is no longer guaranteed to
be a lower bound. Once the gap between the upper bound and the
``approximate'' lower bound is smaller than the tolerance, the column
generation process moves to the second phase.
The column generation also avoids generating dominated columns. Assume
that $[w^s_1 = 1, w^s_2 = 1]$ is a feasible solution of $(P_s)$ and
the corresponding column has been added to the master problem ($M$).
Then, there is no need to consider a solution $[w^s_1 = 1, w^s_2 = 1,
w^s_3 =1]$. The column generation adds the constraint of $w^s_1 +
w^s_2 \le 1$ to $(P_s)$ when such a dominated solution is produced and
does not include it in the master problem.
\subsubsection{The Branch and Bound}
\label{sec:implementation:bnb}
After convergence of the column generation to $LP_{\widetilde{P}}$, the branch
and bound algorithm solves the restricted master problem ($M$) with the
integral condition $w \in \{0,1\}^m$ to obtain a strong primal bound.
In general, this incumbent solution is of very high quality and the
average optimality gap is 0.19\%. Therefore, the branch and price
algorithm uses a depth-first branch and bound. Moreover, at each branching node,
it selects the variable that minimizes the optimality gap.
\section{Description of the Data Sets}
\label{sec:data}
This section describes the distribution test systems. The data set is
available from \url{https://github.com/lanl-ansi/micot/} in the
\url{application_data/lpnorm} directory. Details of the data format
are available from
\url{https://github.com/lanl-ansi/micot/wiki/Resilient-Design-Executable}.
The first two sets, the {\em Rural} and {\em Urban} systems, is from
\citet{yamangil2015resilient}. They are based on the IEEE 34 bus
system \citep{Kersting1991} (see Figure \ref{fig:ieee34}) and
replicate the 34-bus distribution feeder three times. All three
feeders are connected to a single transmission bus and candidate new
lines were added to the network to allow back-feeds. In the rural
model, the distribution feeder was geolocated to model feeders with
long distances between nodes. Similarly, the urban network was
geolocated to model compact feeders typical of urban
environments. Geolocation of these networks has the net effect of
adjusting the lengths of the power lines and their associated
impedance values. Spreading the network out also increases the
hardening and new line costs. As a result, the rural system is
expected to favor solutions with distributed generation and the urban
system solutions with new lines and switches (in addition to hardening
lines). The fixed cost of installing a new distributed generator is
set at \$500k. The cost of a distributed generator is set at \$1,500k
per MW based on the 2025 projections from
\citet{U.S.EnergyInformationAdministration2014}. The cost of
installing new switches for 3-phase lines is set between 10k and 50k
\citep{Ba}. The cost of new underground 3-phase lines is set at about
\$500k per mile and the cost of new underground single phase lines is
set at about \$100k per mile. The hardening cost was set at roughly
\$50k and \$10k per mile for multi-phase and single-phase lines
\citep{Of2005}. The third network, \textsc{network123}, is based on
the 123-node network of \citet{Kersting1991}. This network was
unaltered except for adding new line candidates and labeling large
loads as critical.
\begin{figure}
\caption{The urban and rural distribution systems which contain three copies of the IEEE 34 system to mimic situations where there are three normally independent distribution circuits that support each other during extreme events. These test cases include 109 nodes, 118 generators, 204 loads, and 148 edges.}
\label{fig:1:a}
\label{fig:1:b}
\label{fig:ieee34}
\end{figure}
The communication network $\mathcal{G}c$ is built to conform to $G$. Let $G' =
(N',E')$ be the duplicate of $G$. For each generator $l \in \mathcal{U}$, its
duplicate $i(l)$ represents its control point. Consider $\mathcal{E}'_t
\subseteq E'$, the duplicate of $\mathcal{E}_t$. To represent the control point
for a switch, $e \in \mathcal{E}'_t$ is divided in the middle and a new vertex
$v_e$ is added to represent the control point for the switch. In
other words, the edge $e = (e_h, e_t) \in \mathcal{E}'_t$ is replaced by a new
vertex $ v_e$ and two new edges $ e_1 = (e_h, v_e), e_2 = ( v_e,
e_t)$. The test cases assume that the damage, installation, and
hardening of a line in $G$ are also incurred for the corresponding
line in $\mathcal{G}c$. These assumptions can be easily generalized without changing
the nature of the model.
The experimental evaluation considers 100 scenarios per damage
intensity for all three networks and the damage intensities are taken
in the set $\{1\%,$ $2\%,$ $3\%,$ $4\%,$ $5\%,$ $10\%,$ $15\%,$
$20\%,$ $25\%,$ $30\%,$ $35\%,$ $40\%,$ $45\%,$ $50\%,$ $55\%,$
$60\%,$ $65\%,$ $70\%,$ $75\%,$ $80\%,$ $85\%,$ $90\%,$ $95\%,$
$100\%\}$. The scenario generation procedure is based on damage caused
by ice storms. The intensity tends to be homogeneous on the scale of
distribution systems \citep{Sa2002}. Ice storm intensity is modeled as
a per-mile damage probability, i.e. the probability at least one pole
fails in a one mile segment of power line. Each line is segmented into
1-mile segments and a scenario is generated by randomly failing each
segment with the specified probability. This probability is normalized
for any line segment shorter than 1 mile. A line is ``damaged'' if any
segment fails.
\section{ Case Study}
\label{sec:casestudy}
This section analyzes the behavior of the optimization model on a
variety of test cases. In particular, it studies how the topology of
the distribution grid and the dispersion level of its communication
network affect the optimal design. For each network described in
Section \ref{sec:data}, this section analyzes the optimal design under
different settings of damage probability, the resiliency level, and
the number of communication centers. The default value of $\eta_c$ and
$\eta_t$ are 98\% and 50\% respectively, the default number of
communication centers is 4, and the phase variation parameter $\beta$
is set to 15\% for $E_V$ and $\infty$ otherwise. Unless specified otherwise, the comparisons are based
on these default values.
\subsection{Impact of grid topology} \label{sec:impact_topology:grid}
Let $n_h, n_x, n_t,$ and $n_u$ be the number of hardened lines, new
lines, new switches, and new generators in the optimal design. Figure
\ref{fig:opt} reports these values for various damage levels and
the three networks. The red line indicates the optimal upgrade costs,
and the counts of the upgrade options are represented as a bar. The
results show that hardening lines is the major component of each
optimal design and that its share increases with the disaster
intensity. The results also show that DGs are used in significant
numbers in the rural network, while new lines and switches
complement hardening in the urban model. This was expected given the
length of the lines in these two networks. The third network only
needs line hardenings.
\begin{figure}
\caption{Statistics on the Optimal Grid Designs.}
\label{fig:opt_sol:rural}
\label{fig:opt_sol:urban}
\label{fig:opt_sol:network123}
\label{fig:opt}
\end{figure}
\begin{figure}
\caption{Optimal Designs of the Rural and Urban Networks (3\% damage level).}
\label{fig:opt_design:rural}
\label{fig:opt_design:urban}
\label{fig:opt_design}
\end{figure}
\begin{figure}
\caption{Optimal Design of Network {\sc network123}
\label{fig:opt_design:network123}
\end{figure}
\subsection{Impact of the Communication Network}
\label{sec:impact_commu}
First note that ignoring the communication network is equivalent to
assuming that every bus has its own communication center. In the
following, $\mathcal{G}c(k)$ denotes a communication network with $k$ centers
and $\mathcal{G}c(\infty)$ the case where each bus has a center.
Figure \ref{fig:cost_cc} and Table \ref{table:sol_cc} report the
impact of the communication system: They report optimal objective
values and solution statistics under various numbers of communication
centers. Fewer communication centers lead to significant
cost increases in the rural network, but have limited effect on
the urban network and {\sc network123}. In the rural network,
resiliency comes from forming microgrids with DGs, which require
their own communication centers. When these are not available,
optimal designs harden existing lines and build new lines and switches,
which are more costly as substantiated in Table \ref{table:sol_cc}.
Figure \ref{fig:optimal_design} illustrates the resulting designs on
the rural network for scenarios with a damage level of 3\%. The top
row depicts some of the scenarios and shows the affected lines. The
bottom row depicts the optimal designs for various configurations of
the communication network. For $\mathcal{G}c(\infty)$, the optimal design
features three new DGs in the west-, north-, and east-end of the
network to meet the critical loads of each region. These regions are
then islanded under various scenarios. For $\mathcal{G}c(4)$, the optimal
design installs a new line linking critical loads in the north side to
the west side of the network, instead of using DG in the north
side. This stems from Scenario 100 where a DG in the bus with critical
loads cannot be operated since it has no communication center. For
$\mathcal{G}c(1)$, scenario 1 prevents the operation of an east-end DG and
scenario 100 the operation of a west-end DG. Hence, the optimal design
only considers hardening and new lines and switches. On the other
hand, the urban network and {\sc Network123} achieve resiliency by
increasing grid connectivity for all communication networks.
\begin{figure}
\caption{Cost Analysis For the Number of Communication Centers}
\label{fig:cost_cc:rural}
\label{fig:cost_cc:urban}
\label{fig:cost_cc:network123}
\label{fig:cost_cc}
\end{figure}
\begin{table}[!t]
\TABLE{Impact of the Communication Network on Optimal Grid Designs. \label{table:sol_cc}}
{
\begin{tabular}{lcccccc}
\hline
& \up \down \makecell{Comm. Network} & Obj. & $n_h$ & $n_x$ & $n_t$ & $n_u$\\
\hline
\up \multirow{4}{1cm}{Rural, 3\% damage}& $\mathcal{G}c(1)$& 2095.74 & 12 & 3 & 1 &0 \\
&$\mathcal{G}c(4)$& 1948.09 &6 &2& 1 &2\\
&$\mathcal{G}c(8)$& 1948.09 &6 &2 &1 &2\\
\down &$\mathcal{G}c(\infty)$& 1914.99& 5& 1& 0& 3\\
\hline
\end{tabular}
}
{}
\end{table}
\begin{figure}
\caption{Optimal Designs of the Rural Network under 3\% Damage and Various Communication Network Configurations.}
\label{fig:opt_design:scenario}
\label{fig:opt_design:cc_1}
\label{fig:optimal_design}
\end{figure}
\section{Performance Analysis of the Branch and Price Algorithm}
\label{sec:performance}
This section studies the performance of the B\&P algorithm. All
computations were implemented with the C++/Gurobi 6.5.2 interface. They
use a Haswell architecture compute node configured with 24 cores (two
twelve-core 2.5 GHz Intel Xeon E5-2680v3 processors) and 128 GB RAM.
\subsection{ Computational Performance}
Figure \ref{fig:time_comparison:scatter} reports the computation time
of the B\&P and SBD algorithms for all the instances described in
Section \ref{sec:data}, where the reference line (in red) serves to
delineate when an algorithm is faster than the other. Their statistics
are displayed in Figure \ref{fig:time_comparison:bar}. In average, the
B\&P algorithm is faster than the SBD algorithm by a factor of
3.25. These figures also indicate that the SBD algorithm has a high
degree of performance variance. This comes from the nature of the
scenario set $\mathcal S$. If $S$ contains a dominating scenario and
the scenario has low index in $\mathcal{S}$, then the SBD algorithm
solves the problem quickly. Otherwise, the SBD may need a large number
of iterations and the MIP model keeps growing in size with each
iteration. For 2 out of 1120 instances, the SBD algorithm times out
(wallclock time limit of 4 hours). On the other hand, the B\&P
algorithm is stable across all instances. The B\&P algorithm also has
the additional benefit that it produces improving feasible solutions
continuously. In contrast, the SBD algorithm only produces a feasible
solution at optimality. Finally, the B\&P algorithm appears more
stable numerically than the SBD algorithm. For 5 out of 1120
instances, the B\&P algorithm yields a better optimal solution than
the SBD algorithm as shown in Table \ref{table:opt_sol_gap}. Each such
solution was validated for feasibility.
\begin{figure}
\caption{Comparison of Computation Times: SBD versus B\&P.}
\label{fig:time_comparison:scatter}
\label{fig:time_comparison:bar}
\label{fig:time_comparison}
\end{figure}
\begin{table}[!t]
\TABLE{Numerical Stability of the B\&P Algorithm.\label{table:opt_sol_gap}}
{\begin{tabular}[h]{cccc}
\hline
\up \down \multirow{2}{2cm}{Instance}& \multicolumn{2}{c}{Opt. obj.val} & \\
\cline{2-3}
\up & SBD & B\&P & Gap \\
\hline
\up \makecell{Rural, 30\% damage,
$\eta_t = 0.5,$ $\mathcal{G}c(4)$} & 2458.49 & 2453.79 & -0.19 \%\\
\makecell{Rural, 30\% damage,
$\eta_t = 0.6,$ $\mathcal{G}c(4)$} & 2458.49 & 2453.79 & -0.19 \%\\
\makecell{Rural, 30\% damage,
$\eta_t = 0.7,$ $\mathcal{G}c(4)$} &2524.68 & 2519.98 & -0.19 \%\\
\makecell{Rural, 30\% damage,
$\eta_t = 0.8,$ $\mathcal{G}c(4)$} & 2572.31 & 2567.60 & -0.19 \%\\
\down \makecell{Network123, 55\% damage,
$\eta_t = 0.8,$ $\mathcal{G}c(8)$} & 232.48 & 227.27 & -2.24 \%\\
\hline
\end{tabular}}
{}
\end{table}
\subsection{Solution Quality at the Root Node.}
\label{sec:result:tree}
The problem reformulation produces a strong lower bound and the
majority of the instances are proven optimal at the root node. Table
\ref{table:tree} summarizes the average number of branching nodes and
the average optimality gap at the root node.
\begin{table}[!t]
\TABLE
{Branching Tree Statistics.\label{table:tree}}
{\begin{tabular}{cc}
\hline
\up \down Avg. \# of branching nodes & Avg. opt. gap at the root node \\
\hline
\up \down 1.8 & 0.19 \%\\
\hline
\end{tabular}}
{}
\end{table}
\subsection{Benefits of the Accelerating Schemes}
\label{sec:result:accelerating}
To highlight its design choices, the B\&P algorithm is compared to a
column generation with dual stabilization \citep{du1999stabilized}.
In addition, the benefit of each of the accelerating schemes is
investigated independently by running the B\&P algorithm without the
considered extension. We sample 90 instances by setting $\eta_{t} =
0.5$ and $0.8$, and the damage level to $5\%,$ $30 \%,$ $65\%,$
$85\%,$ $100\%$ for the three networks $\mathcal{G}c(0)$, $\mathcal{G}c(1),$ and
$\mathcal{G}c(4)$. Dual stabilization prevents dual variables from fluctuating
too much, which is often the case in column generation. It tries to
confine dual variables in a box that contains the current best
estimate of the optimal dual solution and penalizes solutions that
deviate from the box. See, for instance, \citet{du1999stabilized,
lubbecke2005selected} for details about stabilized column
generation. Our implementation updates the box whenever the Lagrangian
lower bound is updated.
Table \ref{table:dual} summarizes the computational performance of the
stabilized column generation in comparison with the B\&P
algorithm. B\&P$_B$ denotes the branch-and-price algorithm with the
basic scheme only (Section \ref{sec:cg:basic}) and B\&P$_S$ stands for
the branch and price algorithm with dual stabilization. The symbol
$\dag$ is used to denote that the algorithm reaches the wallclock time
limit for some instances. For more than one third of the sampled
instances, B\&P$_B$ and B\&P$_S$ exceed the wallclock time limit. For
instances where both algorithms terminate within the time limit,
B\&P$_S$ is faster than B\&P$_B$ by a factor of around 4. Although the
dual stabilization does improve the computation time of the basic
algorithm, it is still not adequate to solve the ORDPDC
practically. The B\&P algorithm, on the other hand, shortens
computation times by a factor of 26.35.
\begin{table}[!t]
\TABLE
{Comparison to a Column Generation with Dual Stabilization.
\label{table:dual}}
{\begin{tabular}{lrr}
\hline
& \up \down Avg. computation time (sec) & Avg. number of iterations \\
\hline
\up B\&P$_{B}$& 12857.97$^\dag$ &3122.57$^\dag$ \\
B\&P$_{S}$ & 11563.44$^\dag$ &1514.58$^\dag$\\
\down B\&P&488.03 & 96.12\\
\hline
\end{tabular}}{}
\end{table}
The next results investigate the performance gain of each accelerating
scheme by removing them one at a time from the B\&P algorithm. Table
\ref{table:CG:comparison} describes the computational performance and
Figure \ref{fig:convergence} illustrates the impact of each
accelerating schemes on the convergence rate of the rural network
under 6\% damage level. In the table and figure, $R$ denotes the
revised reduced cost, $C$ the optimality cut, $O$ the lexicographic objective
pricing problem, B\&P$_{\setminus k}$ the B\&P algorithm without
scheme $k$, with $k \in \{R, C, O\}$, and CG$_{\setminus k}$ the
column generation of B\&P without scheme $k$.
The results in Table \ref{table:CG:comparison} indicate that all the
accelerating schemes contribute to the computational performance of
the B\&P algorithm. Figure \ref{fig:convergence:O} illustrates the
key role of the optimality cut. Without this cut, the second stage of
the column generation which uses the traditional pricing objective
does not take advantage of the columns generated in the first stage
and its lower bound drastically drops. Figure \ref{fig:convergence:R}
compares the convergence behavior of CG and CG$_{\setminus R}$,
showing that CG reaches the optimal objective value faster than
CG$_{\setminus R}$. Figure \ref{fig:convergence:S} highlights the
impact of the lexicographic objective function and shows that it
significantly contributes to the fast convergence of the algorithm.
\begin{table}[!t]
\TABLE{Benefits of the Accelerating Schemes. \label{table:CG:comparison}}
{\begin{tabular}[h]{lrr}
\hline
& \up \down Avg. computation time (sec) & \up \down Avg. number of iterations\\
\hline
\up B\&P&488.03 & 96.12\\
B\&P$_{\setminus R}$ & 844.24 & 96.39 \\
B\&P$_{\setminus C}$ & 2589.55$^\dag$& 215.94$^\dag$\\
\down B\&P$_{\setminus O}$ & 2979.84$^\dag$& 544.65$^\dag$\\
\hline
\end{tabular}}
{}
\end{table}
\begin{figure}
\caption{Comparison of Convergence Rates (rural network, 6\% level of damage).}
\label{fig:convergence:O}
\label{fig:convergence:R}
\label{fig:convergence:S}
\label{fig:convergence}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
This paper proposed an expansion planning model to improve the
resiliency of distribution systems facing natural disasters. The
planning model considers the hardening of existing lines and the
addition of new lines, switches, and distributed generators that would
allow a subpart of the system to operate as a microgrid. The expansion
model uses a 3-phase model of the distribution system. In addition, it
also considers damages to the communication system which may prevent
generators and switches to be controlled remotely. The input of the
expansion model contains a set of damage scenarios, each of which
specifying how the disaster affects the distribution system.
The paper proposed a branch and price algorithm for this model where
the pricing subproblem generates new expansions for each damage
scenario. The branch and price uses a number of acceleration schemes
to address significant degeneracy in the model. They include
a new pricing objective, an optimality cut, and a multi-objective
function to encourage sparsity in the generated expansions. The
resulting branch and price algorithm significantly improves the
performance of a scenario-based decomposition algorithm and a branch
and price with a stabilized column generation. The case studies show
that optimal solutions strongly depend on the grid topology and the
sophistication of the communication network. In particular, the
results highlight the importance of distributed generation for rural
networks, which necessitates a resilient communication system.
The acceleration techniques presented in this paper are not limited to
the electricity distribution grid planning problem; They can be used
on problems with similar structure, i.e, two-stage stochastic problems
with feasibility recourse.
Future work will be devoted to applying and scaling these techniques
to instances with thousands of components.
\end{document} |
\begin{document}
\title{Numerical study of Galerkin--collocation approximation in time for the wave
equation}
\mathsf{a}uthor{Mathias Anselmann$^{\star}$, Markus Bause$^{\star}$\\
{\small ${}^\star$ Helmut Schmidt University, Faculty of
Mechanical Engineering, Holstenhofweg 85,}\\
{\small 22043 Hamburg, Germany}\\
}
\mathrm date{}
\maketitle
\begin{abstract}
\textbf{Abstract.} The elucidation of many physical problems in science and engineering is subject
to the accurate numerical modelling of complex wave propagation phenomena. Over the last
decades, high-order numerical approximation for partial differential equations has become
a well-established tool. Here we propose and study numerically the implicit
approximation in time of wave equations by a Galerkin--collocation approach that relies
on a higher order space-time finite element approach. The conceptual basis is the
establishment of a direct connection between the Galerkin method for the time
discretization and the classical collocation methods, with the perspective of achieving
the accuracy of the former with reduced computational costs provided by the latter in
terms of less complex linear algebraic systems. For the fully discrete solution, higher
order regularity in time is further ensured which can be advantageous in the
discretization of multi-physics systems. The accuracy and efficiency of the variational
collocation approach is carefully studied by numerical experiments.
\end{abstract}
\section{Introduction}
The accurate and efficient numerical simulation of wave phenomena continues to remain a
challenging task and attract researchers' interest. Wave phenomena are studied in various
branches of natural sciences and technology. For instance, fluid-structure interaction,
acoustics, poroelasticity, seismics, electro-magnetics and non-destructive material
inspection represent prominent fields in that wave propagation is studied. One of our key
application for wave propagation is structural health monitoring of lightweight
material (for instance, carbon-fibre reinforced polymers) by ultrasonic waves in
aerospace engineering. The conceptional idea of this new and intelligent approach is
sketched in \cref{fig:elastic_wave}.
\begin{figure}
\caption{Concept of structural health monitoring with finite
element simulation (scaled displacement field) illustrating the expansion of elastic
waves.}
\label{fig:elastic_wave}
\end{figure}
The structure is equipped with an integrated actuator-sensor network. The ultrasonic
waves that are emitted by the actuators interact with material defects of the solid
structure. By means of an inverse modelling, the signals that are recorded by the sensors
monitor material failure (cf. \cite{Koecher15}) and, as perspective for the future, may
allow prognoses about the structure's residual lifetime. The design of such monitoring
systems and the signal interpretation require the elucidation of wave propagation in
composite material which demands for highly advanced and efficient numerical simulation
techniques.
High-order numerical approximation of partial differential equations has been strongly
focused and investigated in the last decades. High order methods are known to be
efficient if they approximate functions with large elements of high polynomial degree in
regions of high regularity. Prominent examples are $hp$- and spectral element methods in
application areas such as computational fluid dynamics or computational mechanics.
Their theoretical convergence analysis and adaptive $hp$- and spectral
element versions still experience strong development. Whereas high-order approaches have
been considered for the approximation of the spatial variables, first- or second-order
implicit schemes are often still used for the discretization of the time variable. We
note that in this work only implicit time discretization schemes are in the scope of
interest. Thus, all remarks refer to this class of methods. Our motivation for
using implicit time discretization schemes comes from the overall goal to apply the
proposed Galerkin--collocation techniques to mixed systems like, for
instance, fluid-structure interaction for free flow modelled by the Navier--Stokes
equations \cite{Richter2017} or fully dynamic poroelasticity \cite{Wheeler2012}.
Driven by the tremendous increase in computing power of modern high performance computing
systems and recent progress in the technology of algebraic solver, including efficient
techniques of preconditioning, space-time finite element approaches have recently
attracted high attention and have been brought to application maturity; cf., e.g.,
\cite{Hussain2014,Doerfler2016,Ernesti2018,Steinbach2015}. Space-time finite element methods offer
appreciable advantages over discretizations of mixed type based on finite difference
techniques for the discretization of the time variable (e.g., by Runge-Kutta methods)
and, for instance, finite element methods for the discretization of the space variables.
In particular, advantages are the natural embedding of higher order members in the
various families of schemes, the applicability of functional analysis techniques in their
analyses due to the uniform space-time framework and the applicability of well-known
adaptive mesh refinement techniques, including goal-oriented error control
\cite{Bangerth2010}. In the meantime a broad variety of implementations of space-time
finite element methods does exist. The families of schemes
differ by the choices of the trial and test spaces. This leads to continuous or
discontinuous approximations of the time variable (cf. e.g.,
\cite{Matthies2011,Zhao2014}). Further, the fully coupled treatment of all time
steps versus time-marching approaches is discussed. In particular, the simultaneous
computation of all time steps imposes high demands on the linear solver technology (cf.,
e.g., \cite{Doerfler2016,Ernesti2017,Ernesti2018}).
In this work, we propose the Galerkin--collocation method for the numerical solution of
wave equations. This approach combines variational approximation in time by finite
element techniques with the concepts of collocation methods and follows the ideas of
\cite{Becher2018}. By imposing collocation conditions, the test space of the variational
condition is downsized. The key ingredients and innovations of the approach are:
\begin{itemize}
\kappatem[A.] Higher order regularity in time of the fully discrete
approximation;
\kappatem[B.] Linear systems of reduced complexity;
\end{itemize}
Ingredient [A] is a direct consequence of the construction of the schemes. Higher order
regularity is ensured by imposing collocation conditions at the discrete time nodes and
endpoints of the subintervals $[t_{n-1},t_n]$, for $n=1,\ldots , N$, of the global time
interval $[0,T]$. Higher order regularity in time might offer appreciable advantages
for future approximations of coupled multi-physics systems if higher order time
derivatives of the discrete solution of one subproblem arise as coefficient functions in
other subproblems. Ingredient [B] is ensured by the proper choice of a special for the
discrete in time function spaces. Thereby, simple vector identities for the degrees of
freedom in time are obtained at the left endpoints of the subintervals without generating
computational costs. These vector identities can then be exploited to eliminate
conditions from the algebraic systems and reduce its size compared to the standard
continuous Galerkin--Petrov approximation in time; cf.~\cite{Bause2018}. In a further
work of the authors \cite{Bause2019}, it is shown that the optimal order of convergence
in time (and space) of the underlying finite element discretization is preserved by the
Galerkin--collocation approach. The numerical example
of \cref{sec:Challenging_Example} that mimics typical studies of
structural health monitoring (cf.\ \cref{fig:elastic_wave}), demonstrates the
superiority of the Galerkin--collocation approach over a standard continuous
Galerkin--Petrov method admitting continuity and no
differentiability in time of the discrete solution.
For the sake of brevity, standard conforming finite element methods are used for the
discretization of the spatial variables in this work. This is done since we focus here on
time discretization. In the literature it has been mentioned that discontinuous
finite element methods in space offer appreciable advantages over continuous ones for the
discretization of wave equations; cf., e.g., \cite{Basabe2008,Grote2006}. The
application of, for instance, the symmetric interior penalty disccontinuous Galerkin
method (cf.\ \cite{Bause2014,Koecher15} along with a Galerkin--collocation
discretization in time, is straightforward.
This work is organized as follows. In \cref{sec:ProblemDescription} we
introduce our prototype model. In \cref{sec:Gal-col_scheme} we present its
discretization by two families of Galerkin--collocation methods. In \cref{sec:c1_solution}, the discrete form of a member of theses families with
$C^1$-regularity in time is derived. The resulting algebraic system is built and our
algebraic solver is described. In \cref{sec:C2_solution} the discrete form of a member
of the Galerkin--collocation family with $C^2$-regularity in time is derived.
For both methods, the results of our numerical experiments are presented and evaluated.
\section{Mathematical problem and notation}
\label{sec:ProblemDescription}
As a prototype model, we study the wave problem
\begin{equation}
\label{eq:ScalarWaveSO}
\begin{array}{r@{\;}c@{\;}ll}
\partial_t^2 u - c^2 \Delta{u} & = f\,, & \kappan \Omega \times I\,,\\[1ex]
u(0) = u_0\,, \quad \partial_t u(0) & = v_0 \,, & \text{in}\; \Omega\,,\\[1ex]
u = g^u \,, \;\; \text{on}\; \partial \Omega_D \times I\,,\quad \partial_n u & = 0
\,, & \text{on}\; \partial \Omega_N \times I\,.\\
\end{array}
\end{equation}
In our application of structural health monitoring (cf.\ \cref{fig:elastic_wave}), $u$ denotes the scalar valued displacement field, $c\kappan
R$ with $c>0$, is a material parameter and $f$ an external force acting
on the domain
$\Omega\subset \mathbb{R}^d$, with $d=2,3$. Further, $g^u$ is a prescribed trace on the Dirichlet
part $\partial \Omega_D$ of the boundary $\partial \Omega=\partial \Omega_D \cup \partial
\Omega_N$, with $\partial \Omega_D \cap \partial \Omega_N = \emptyset$. By $\partial_n$
we denote the normal derivative with outer unit normal vector $\boldsymbol n$. Homogeneous
Neumann boundary conditions on $\partial \Omega_N$ are prescribed for brevity. Finally,
$I=(0,T]$ denotes the time domain. Problem \eqref{eq:ScalarWaveSO} is
well-posed and admits a unique solution $(u,\partial_t u) \kappan L^2(0,T;H^1(\Omega))\times
L^2(0,T;L^2(\Omega))$ under appropriate assumptions about the data; cf.\ \cite{Lions1968}.
By imbedding, $u \kappan C([0,T];H^1(\Omega))$ and $v \kappan C([0,T];L^2(\Omega))$ is ensured;
cf.\ \cite{Lions1971}. Throughout, we tacitly assume that the solution admits all the
(improved) regularity being necessary in the arguments.
Our notation is standard. By $H^m(\Omega)$ we denote the Sobolev space of $L^2(\Omega)$
functions with derivatives up to order $m$ in $L^2(\Omega)$. For brevity, we let $H :=
L^2(\Omega)$ and $V=H^1_{0,D}(\Omega)$ be the space of all $H^1$-functions with
vanishing trace on the Dirichlet part $\partial \Omega_D$ of $\partial
\Omega$. By $\langle \cdot,\cdot \rangle_{\Omega}$ we denote the inner
product in $L^2(\Omega)$. For the norms we use $\| \cdot \| := \| \cdot\|_{L^2(\Omega)}$
and $\| \cdot \|_m := \| \cdot\|_{H^m(\Omega)}$ for $m\kappan \mathbb{N}$ and $m\geq 1$. Finally, the
expression $a\lesssim b$ stands for the inequality $a \leq C\, b$ with a generic constant
$C$ that is indepedent of the size of the space and time meshes.
By $L^2(0,T;B)$, $C([0,T];B)$ and $C^q([0,T];B)$, for $q\kappan \mathbb{N}$, we denote the
standard Bochner spaces of $B$-valued functions for a Banach space $B$, equipped with
their natural norms. Further, for a subinterval $J\subseteq [0,T]$, we will use the
notations $L^2(J;B)$, $C^m(J;B)$ and $C^0(J;B):= C(J;B)$ for the corresponding Bochner
spaces.
To derive our Galerkin--collocation approach, we first rewrite problem
\eqref{eq:ScalarWaveSO} as a first order system in time for the unknowns $(u,v)$, with
$v=\partial_t u$,
\begin{equation}
\label{eq:InitialProblem}
\partial_t u - v = 0\,, \quad \partial_t v - c^2 \Delta{u} = f\,.
\end{equation}
Further, we represent the unknowns $u$ and $v$ in terms of
\begin{equation}
\label{eq:SplitSol}
u = u^0 + u^D \quad \text{and} \quad v = v^0 + v^D\,.
\end{equation}
Here, $u^D$, $v^D\kappan C(\overline I;H^1(\Omega))$ are supposed to be (extended) functions
with traces $u^D=g^u$ and $v^D=g^v:=\partial_t g^u$ on the Dirichlet part $\partial
\Omega_D$ of
$\partial \Omega$.
Using \eqref{eq:InitialProblem} and \eqref{eq:SplitSol}, we then consider solving the
following variational problem: \textit{Find $(u^0,v^0)\kappan
L^2(0,T;H^1_{0,D}(\Omega))\times L^2(0;T;H^1_{0,D}(\Omega))$ such that}
\begin{equation}
u^0(0)=u_0-u^D(0)\,, \qquad v^0(0)=v_0-v^D(0)
\end{equation}
\textit{and, for all $(\phi,\psi)\kappan (L^2(0;T;H^1_{0,D}(\Omega)))^2$,}
\begin{align}
\label{eq:cont_weak_formulation_1}
\kappant_{I} \bigl< \partial_t \mat uC^0, \phi \bigr>_{\Omega}
-
\bigl< \mat vC^0, \phi \bigr>_{\Omega} \, \mathrm d t
={}&
0\,, \\
\begin{split}
\label{eq:cont_weak_formulation_2}
\kappant_{I} \bigl< \partial_t \mat vC^0, \psi \bigr>_{\Omega} +
\langle c^2 \nabla \mat uC^0, \nabla \psi \rangle_{\Omega} \, \mathrm d t
={}&
\kappant_{I} \Big( \bigl< {\mat f}C, \psi \bigr>_{\Omega}
+
\bigl< \partial_n u, \psi \bigr>_{\partial \Omega_N}
\\[1ex]
&\quad -
\bigl< \partial_t \mat vC^D, \psi \bigr>_{\Omega}
-
\langle c^2 \nabla \mat uC^D, \nabla \psi \rangle_{\Omega}\Big)
\, \mathrm d t\,.
\end{split}
\end{align}
\begin{remark} i) We note that the correct treatment of inhomogeneous time-dependent
boundary conditions is an import issue in the application of variational space-time
methods. The space-time discretization that is derived below (cf.\ \cref{sec:Gal-col_scheme}) and based on the variational problem
\eqref{eq:cont_weak_formulation_1}, \eqref{eq:cont_weak_formulation_2} ensures
convergence rates of optimal order in space and time, also for time-dependent boundary
conditions. This is confirmed by the second of the numerical experiments given in \cref{sec:convergence_test_C1}. \\
ii) Our Galerkin--collocation approach is based on solving, along with some collocation
conditions, the variational equations \eqref{eq:cont_weak_formulation_1},
\eqref{eq:cont_weak_formulation_2} in finite dimensional subspaces. In
particular, the same approximation space will be used for $u^0$ and $v^0$. For this
reason, the solution space for $v^0$ and the test space in \cref{eq:cont_weak_formulation_2} are chosen slightly stronger than usually; cf.\
\cite{Bangerth2010}.
Choosing $L^2(0;T;L^2(\Omega)))$ instead, would have been sufficient.
\end{remark}
\section{Galerkin-collocation schemes}
\label{sec:Gal-col_scheme}
In this section we introduce two families of Galerkin--collocation schemes. These
families combine the concept of collation condition methods applied to the spatially
discrete counterpart of the equations \eqref{eq:InitialProblem} with the
finite element discretization of the variational equations
\eqref{eq:cont_weak_formulation_1}, \eqref{eq:cont_weak_formulation_2}. The collocation
constrains then allow us to reduce the size of the discrete test spaces for the
variational conditions compared to a standard Galerkin--Petrov approach; cf.\
\cite{Bause2014}.
First, we need some notation. For the time discretization we decompose the time interval
$I=(0,T]$ into $N$ subintervals
$I_n=(t_{n-1},t_n]$, where $n\kappan \{1,\ldots ,N\}$ and $0=t_0<t_1< \cdots < t_{n-1} < t_n
= T$ such that $I=\bigcup_{n=1}^N I_n$. We put $\tau = \max_{n=1,\ldots N} \tau_n$ with
$\tau_n = t_n-t_{n-1}$. Further, the set of time intervals $\boldsymbolhcal M_\tau :=
\{I_1,\ldots, I_n\}$ is called the time mesh. For a Banach space $B$ and any $k\kappan \mathbb{N}$,
we let
\begin{equation}
\label{Def:Pk}
\boldsymbolhbb P_k(I_n;B) = \bigg\{w_\tau : I_n \mapsto B \; \Big|\; w_\tau(t) = \sum_{j=0}^k
W^j t^j
\,, \; \forall t\kappan I_n\,, \; W^j \kappan B\; \forall j \bigg\}\,.
\end{equation}
For an integer $k\kappan \mathbb{N}$ we introduce the space of globally continuous functions
in time
\begin{equation}
\label{Eq:DefXk}
X_\tau^k (B) := \left\{w_\tau \kappan C(\overline I;B) \mid w_\tau{}_{|I_n} \kappan \boldsymbolhbb
P_k(I_n;B)\; \forall I_n\kappan \boldsymbolhcal M_\tau \right\}\,,
\end{equation}
and for an integer $l\kappan \mathbb{N}_0$ the space of globally $L^2$-functions in time
\[
\label{Eq:DefYk}
Y_\tau^{l} (B) := \left\{w_\tau \kappan L^2(I;B) \mid w_\tau{}_{|I_n} \kappan \boldsymbolhbb
P_{l}(I_n;B)\; \forall I_n\kappan \boldsymbolhcal M_\tau \right\}\,.
\]
For the space discretization, let $\boldsymbolhcal T_h=\{K\}$ be a shape-regular mesh of
$\Omega$
consisting of quadrilateral or hexahedral elements with mesh size $h>0$. Further, for
some integer $p\kappan \mathbb{N}$ let $V_h=V_h^{(p)}$ be the finite element space that is given by
\begin{equation}
\label{Eq:DefVh}
V_h = V_h^{(p)}=\left\{v_h \kappan C(\overline \Omega)
\mid v_h{}_{|T} \circ T_K \kappan \boldsymbolhbb Q_p \,
\forall K
\kappan \boldsymbolhcal \Tau_h \right\}\cap H^1_{0,D}(\Omega)\,,
\end{equation}
where $T_K$ is the invertible mapping from the reference cell $\hat K$ to the cell $K$ of
$\Tau_h$ and $\boldsymbolhbb Q_p$ is the space of all polynomials of maximum degree $p$ in
each variable. We let $\boldsymbolhcal A_h:
H^1_{0,D}(\Omega) \mapsto V_h$ be the discrete operator that is defined by
\begin{equation}
\label{Eq:DefAh}
\langle \boldsymbolhcal A_h w , v_h \rangle = \langle \nabla w, \nabla v_h\rangle \quad
\text{for all}\; v_h\kappan V_h\,.
\end{equation}
Moreover, $(u_{0,h},v_{0,h})\kappan V_h^2$ and $(u^D_{\tau,h},v^D_{\tau,h})\kappan
(C([0,T];V_h))^2$ define suitable finite element approximations of the initial values
$(u_0,v_0)$ and the extended boundary values $(u^D,v^D)$ in \cref{eq:SplitSol}. Here, we use interpolation in $V_h$ of the given data.
Now we define our classes of Galerkin--collocation schemes. We follow the
lines of \cite{Bause2019,Becher2018}. We restrict ourselves to the schemes studied in
the numerical experiments presented in Secs.\
\ref{sec:convergence_test_C1}, \ref{sec:Challenging_Example} and \ref{sec:NumTestC2}. The
definition of classes of Galerkin--collocation
schemes
with even higher regularity in time is straightforward, but not done here.
\begin{definition}[$\boldsymbol{C^l}$--regular in time Galerkin-collocation schemes GCC$\boldsymbol{
{}^l(k)}$]
\label{Def:GCC}
Let $l\kappan \{1,2\}$ be fixed and $k\geq 2l+1$. For $n=1,\ldots, N$ and
given $(u_{\tau,h}{}_|{}_{I_{n-1}}(t_{n-1})$, $v_{\tau,h}{}_|{}_{I_{n-1}}(t_{n-1}))\kappan
V_h^2$ for $n>1$ and $u_{\tau,h}{}_|{}_{I_0}(t_0)=u_{0,h}$,
$v_{\tau,h}{}_|{}_{I_0}(t_0)=v_{0,h}$ for $n=1$, find $(u^0_{\tau,h}{}_|{}_{I_n},
v^0_{\tau,h}{}_|{}_{I_n}) \kappan
(\boldsymbolhbb P_k (I_n;V_h))^2$ such that, for $s_0\kappan \mathbb{N}_0$, $s_1\kappan \mathbb{N}$ with $s_0,s_1\leq l$,
\begin{align}
& \partial_t^{s_0} w^0_{\tau,h}{}_|{}_{I_n}(t_{n-1})
= \partial_t^{s_0} w^0_{\tau,h}{}_|{}_{I_{n-1}}(t_{n-1})
\,, \quad \text{for} \; w^0_{\tau,h} \kappan \left\{u^0_{\tau,h},v^0_{\tau,h}\right\}\,,
\label{Eq:SemiDisLocalcGPC_6}
\\[1ex]
&\partial_t^{s_1} u^0_{\tau,h}{}_|{}_{I_n}(t_{n})
-\partial_t^{s_1-1}v^0_{\tau,h}{}_|{}_{I_n}(t_{n})
= 0 \,,
\label{Eq:SemiDisLocalcGPC_7}
\\[1ex]
&
\begin{aligned}
\partial_t^{s_1} v^0_{\tau,h}{}_|{}_{I_n}(t_{n})
+ \boldsymbolhcal A_h \partial_t^{s_1-1} & u^0_{\tau,h}{}_|{}_{I_n}(t_{n})
= \partial_t^{s-1} f(t_{n})\\
& - \partial_t^{s_1}
v^D_{\tau,h}{}_|{}_{I_n}(t_{n}) - \boldsymbolhcal A_h \partial_t^{s_1-1} u^D_{\tau,h}{}_|{}_{I_n}(t_{n})
\,,
\label{Eq:SemiDisLocalcGPC_8}
\end{aligned}
\end{align}
and, for all $(\varphi_{\tau,h}, \psi_{\tau,h}) \kappan
(\boldsymbolhbb P_{0} (I_n;V_h))^2$,
\begin{align}
\label{Eq:SemiDisLocalcGPC_9}
&\kappant_{I_n} \Big(\langle \partial_t u^0_{\tau,h} , \varphi_{\tau,h} \rangle_{\Omega}
- \langle v^0_{\tau,h} , \varphi_{\tau,h} \rangle_{\Omega} \Big) \, \mathrm d t
=
0\,,
\\
& \begin{alignedat}{1}
\kappant_{I_n} \Big(\langle \partial_t v^0_{\tau,h} , \psi_{\tau,h} \rangle_{\Omega}
& + \langle \boldsymbolhcal A u^0_{\tau,h} , \psi_{\tau,h} \rangle_{\Omega} \Big) \, \mathrm d t
=
\kappant_{I_n}
\langle f,\psi_{\tau,h}\rangle_{\Omega} \, \mathrm d t
\\
& -
\kappant_{I_n} \Big(\langle \partial_t v^D_{\tau,h} , \psi_{\tau,h} \rangle_{\Omega}
+ \langle \boldsymbolhcal A_h u^D_{\tau,h} , \psi_{\tau,h} \rangle_{\Omega} \Big) \, \mathrm d t\,.
\label{Eq:SemiDisLocalcGPC_10}
\end{alignedat}
\end{align}
\end{definition}
\begin{remark}
\begin{itemize}
\kappatem In \cref{Eq:SemiDisLocalcGPC_6}, the discrete
initial values $(\partial_t u_{\tau,h}(0),\partial_t v_{\tau,h}(0))$ arise for $s_0=1$. For
$\partial_t u_{\tau,h}(0)$ we use a suitable finite element approximation $v_{0,h}\kappan
V_h$ (here, an interpolation) of $v_0\kappan V$. For $\partial_t v_{\tau,h}(0)$
we evaluate the wave equation in the initial time point and use a suitable finite element
approximation (here, an interpolation) of $\partial_t^2 u(0)= c^2 \Delta u(0) + f(0)$. For $s_0=2$ in \cref{Eq:SemiDisLocalcGPC_6}, the initial value $\partial_t^2
v_{\tau,h}(0)$ is computed as a suitable finite element approximation (here, an
interpolation) of $\partial_t^3 u(0)= c^2 \Delta \partial_t u(0) + \partial_t f(0)$.
Mathematically, this approach requires that the partial equation and its time derivative
are satisfied up to the initial time point and, thereby, sufficient regularity of the
continuous solution. Without such regularity assumptions, the application of higher order
discretization schemes cannot be justified rigorously. Nevertheless, in practice such
methods often show a superiority over lower-order ones, even for solutions without the
expected high regularity (cf.\ \cref{sec:Challenging_Example}).
\kappatem From \cref{Eq:SemiDisLocalcGPC_6}, $(u_{\tau,h},v_{\tau,h})\kappan
(C^l(\overline I;V_h))^2$, for fixed $l \kappan \{1,2\}$, is easily concluded.
\end{itemize}
\end{remark}
An optimal order error analysis for the GCC$^1(k)$ family of schemes of Def.\
\ref{Def:GCC} is provided in \cite{Bause2019}. The following theorem is proved.
\begin{theorem}[Error estimates for ${\boldsymbol{(u_{\tau,h}$, $v_{\tau,h})}}$ of
GCC$^1(\boldsymbol k)$]
\label{th:error}
Let $l=1$ and $k\geq 3$. For the error $(e^{\; u}, e^{\; v})=(u-u_{\tau,h}, v-v_{\tau,h})$ of the fully discrete scheme GCC$^l(k)$ of Def.~\ref{Def:GCC} there holds that
\begin{equation}
\begin{aligned}
\label{eq:error_1a}
\| e^{\; u}(t)\| + \| e^{\; v}(t)\|
&\lesssim
\tau^{k+1}+h^{p+1} \, , \;\; t\kappan \overline I\,,
\\
\|\nabla e^{\; u}(t)\|
&\lesssim
\tau^{k+1}+h^{p} \, , \;\; t\kappan \overline I\,,
\end{aligned}
\end{equation}
as well as
\begin{equation}
\begin{aligned}
\label{eq:error_2a}
\| e^{\; u}(t)\|_{L^2(I;H)} + \| e^{\; v}(t)\|_{L^2(I;H)}
&\lesssim \tau^{k+1}+h^{p+1} \, ,
\\
\|\nabla e^{\; u}(t)\|_{L^2(I;H)}
&\lesssim
\tau^{k+1}+h^{p} \, .
\end{aligned}
\end{equation}
\end{theorem}
Error estimates for the GCC$^2(k)$ family remain as a work for the future. In \cref{sec:NumTestC2}, the convergence of GCC$^2(5)$ is demonstrated numerically.
Further, we note that a computationally cheap post-processing of improved regularity and
accuracy for continuous Galerkin--Petrov methods is presented and studied in
\cite{Bause2018}.
In the next sections we study the schemes GCC$^1(3)$ and GCC$^2(5)$ of
Def.~\ref{Def:GCC} in detail. Their algebraic forms are derived and the algebraic linear
solver are presented. Finally, the results of our numerical experiments with the proposed
methods are presented. Here, we restrict ourselves to the lowest-order cases with $k=3$
for $l=1$ and $k=5$ for $l=2$ of Def.~\ref{Def:GCC}. This is sufficient to demonstrate the
potential of the Galerkin--collocation approach and its superiority over the standard
continuous Galerkin approach in space and time \cite{KM04,Bause2018}. An implementation of
GCC$^l(k)$ for higher values of $k$ along with efficient algebraic solvers is currenty
still missing.
\section{\texorpdfstring{Galerkin--collocation GCC$\boldsymbol{{}^1(3)}$}
{Galerkin--collocation GCC1(3)}}
\label{sec:c1_solution}
Here, we derive the algebraic system of the GCC$^1(3)$ approach and discuss our algebraic solver for the arising block system. For brevity, the derivation is done for $k=3$ only. The generalization to larger values of $k$ is straightforward; cf.\ \cite{Bause2019}.
\subsection{Fully discrete system}
\label{sec:Deriving_C1_System}
To derive the discrete counterparts of the variational conditions \eqref{Eq:SemiDisLocalcGPC_9}, \eqref{Eq:SemiDisLocalcGPC_10} and the collocation constrains \eqref{Eq:SemiDisLocalcGPC_6} to \eqref{Eq:SemiDisLocalcGPC_8}, we let $\{\phi_j\}_{j=1}^J \subset V_h$, denote a (global) nodal Lagrangian basis of $V_h$. The mass matrix $\boldsymbol M$ and the stiffness matrix $\boldsymbol A$ are defind by
\begin{align}
\label{eq:DefM_A}
\boldsymbol{M} &\coloneqq
\left(\bigl<
\phi_i, \phi_j
\bigr>_{\Omega}\right)_{i,j=1}^J\,,
&
\boldsymbol{A} &\coloneqq
\left(\bigl<
\nabla \phi_i, \nabla \phi_j
\bigr>_{\Omega}\right)_{i,j=1}^J\,,
\end{align}
On the reference time interval $\hat I = [0,1]$ we define a Hermite-type basis $\{\hat
\xi_l\}_{l=0}^3 \subset \boldsymbolhbb P_3 (\hat I;\mathbb{R})$ of $\boldsymbolhbb P_3 (\hat I;\mathbb{R})$ by the
conditions
\begin{equation}
\label{eq:CondXi}
\begin{aligned}
\hat{\xi_{0}}(0) &= 1\,,
& \hspace{1em}
\hat{\xi_{0}}(1) &= 0\,,
& \hspace{1em}
\partial_t\hat{\xi_{0}}(0) &= 0\,,
& \hspace{1em}
\partial_t\hat{\xi_{0}}(1) &= 0\,,
\\
\hat{\xi_{1}}(0) &= 0\,,
&
\hat{\xi_{1}}(1) &= 0\,,
&
\partial_t\hat{\xi_{1}}(0) &= 1\,,
&
\partial_t\hat{\xi_{1}}(1) &= 0\,,
\\
\hat{\xi_{2}}(0) &= 0\,,
&
\hat{\xi_{2}}(1) &= 1\,,
&
\partial_t\hat{\xi_{2}}(0) &= 0\,,
&
\partial_t\hat{\xi_{2}}(1) &= 0\,,
\\
\hat{\xi_{3}}(0) &= 0\,,
&
\hat{\xi_{3}}(1) &= 0\,,
&
\partial_t\hat{\xi_{3}}(0) &= 0\,,
&
\partial_t\hat{\xi_{3}}(1) &= 1\,.
\end{aligned}
\end{equation}
\
These conditions then define the basis of $\boldsymbolhbb P_3(\hat I;\mathbb{R})$ by
\begin{align}
\label{eq:basisxi}
\hat{\xi_{0}} &= 1 - 3 t^2 + 2 t^3, &
\hat{\xi_{1}} &= t - 2 t^2 + t^3, &
\hat{\xi_{2}} &= 3 t^2 - 2 t^3, &
\hat{\xi_{3}} &= -t^2 + t^3.
\end{align}
By means of the affine transformation $T_n(\hat t):= t_{n-1} + \tau_n\cdot \hat t$,
with $\hat t \kappan \hat I$, from the reference interval $\hat I$ to $I_n$ such that
$t_{n-1}=T_n(0)$ and $t_{n}=T_n(1)$, the basis $\{\xi_l\}_{l=0}^3 \subset \boldsymbolhbb P_3
(I_n;\mathbb{R})$ is given by $\xi_l = \hat \xi_l \circ T_n^{-1}$ for $l=0,\ldots, 3$. In terms of
basis functions, $w_{\tau,h}\kappan \boldsymbolhbb P_3 (I_n;V_h)$ is thus represented by
\begin{align}
\label{eq:full_discrete_ansatz}
w_{\tau,h}(\boldsymbol x, t) = \sum_{l=0}^{3} \sum_{j=1}^{J}w_{n,l,j} \phi_j
(\boldsymbol{x})\xi_{l}(t)\,, \quad (\boldsymbol x,t)\kappan \Omega \times \overline{I_n}\,.
\end{align}
For $\zeta_0 \equiv 1$ on $\overline{I_n}$, a test basis of $\boldsymbolhbb P_0 (I_n;V_h)$ is then given by
\begin{equation}
\label{eq:testbasis}
\boldsymbolhcal B = \{\phi_1 \zeta_0, \ldots, \phi_J \zeta_0\}\,.
\end{equation}
To evaluate the time integrals on the right-hand side of \cref{Eq:SemiDisLocalcGPC_10}
we still use the Hermite-type interpolation operator $I_{\tau|I_n}$, on $I_n$, defined
by
\begin{equation}
\label{eq:hermite_interpolation}
\begin{alignedat}{2}
I_{\tau|I_n}g(t)
&\coloneqq
\sum_{s=0}^{l} \tau_n^s \hat \xi_s (0)
\underbrace{\partial_t^s g_|{}_{I_n}(t_{n-1})}_{=:g_s}
&&+ \sum_{s=0}^{l} \tau_n^s \hat \xi_{s+l+1} (1)
\underbrace{\partial_t^s g|_{I_n}(t_{n})}_{=:g_{s+l+1}}\,.
\end{alignedat}
\end{equation}
Here, the values $\partial_t^s g_|{}_{I_n}(t_{n-1})$ and
$\partial_t^s g_|{}_{I_n}(t_{n})$ in \eqref{eq:hermite_interpolation} denote the corresponding
one-sided limits of values $\partial_t^s g(t)$ from the interior of $I_n$.
Now, we can put the equations of the proposed GCC$^1(3)$ approach in their algebraic
forms. In the variational equations \eqref{Eq:SemiDisLocalcGPC_9} and
\eqref{Eq:SemiDisLocalcGPC_10}, we use the representation
\eqref{eq:full_discrete_ansatz} for each component of $(u_{\tau,h}, v_{\tau,h})
\kappan (\boldsymbolhbb P_3(I_n;V_h))^2$, choose the test functions \eqref{eq:testbasis}
and interpolate the right-hand side of \eqref{Eq:SemiDisLocalcGPC_10} by applying
\eqref{eq:hermite_interpolation}. All of the arising time integrals are evaluated
analytically. Then, we can recover the variational conditions
\eqref{Eq:SemiDisLocalcGPC_9} and \eqref{Eq:SemiDisLocalcGPC_10} on the subinterval $I_n$
in their algebraic forms
\begin{align}
\label{eq:time_scheme_a1}
&\boldsymbol{M} \left(
-\boldsymbol{u}_{n,0}^{0} + \boldsymbol{u}_{n,2}^{0}
\right)
-
\tau_n \boldsymbol{M} \left(
\frac{1}{2} \boldsymbol{v}_{n,0}^{0} + \frac{1}{12} \boldsymbol{v}_{n,1}^{0} + \frac{1}{2}
\boldsymbol{v}_{n,2}^{0}
- \frac{1}{12} \boldsymbol{v}_{n,3}^{0}
\right)
=
\boldsymbol 0\,,
\\
\begin{split}
\label{eq:time_scheme_a2}
&\boldsymbol{M} \left(
-\boldsymbol{v}_{n,0}^{0} + \boldsymbol{v}_{n,2}^{0}
\right)
+
\tau_n \boldsymbol{A}
\left(
\frac{1}{2} \boldsymbol{u}_{n,0}^{0} + \frac{1}{12} \boldsymbol{u}_{n,1}^{0} + \frac{1}{2}
\boldsymbol{u}_{n,2}^{0}
- \frac{1}{12} \boldsymbol{u}_{n,3}^{0}
\right)
=
\\
&
\hphantom{
\boldsymbol{M} \left(
-\boldsymbol{v}_{n,0}^{D}\right)
}
\tau_n \boldsymbol{M}
\left( \frac{1}{2} \boldsymbol{f}_{n,0}^{} + \frac{1}{12} \boldsymbol{f}_{n,1}^{} + \frac{1}{2}
\boldsymbol{f}_{n,2}^{} - \frac{1}{12} \boldsymbol{f}_{n,3} \right)
-
\boldsymbol{M} \left(
-\boldsymbol{v}_{n,0}^{D} + \boldsymbol{v}_{n,2}^{D}
\right)
\\
&
\hphantom{
\boldsymbol{M} \left(
-\boldsymbol{v}_{0}^{D} + \boldsymbol{v}_{2}^{K,I} + \boldsymbol{v}_{2} +\boldsymbol{v}_{2}
\right)
}
-
\tau_n \boldsymbol{A}
\left(
\frac{1}{2} \boldsymbol{u}_{n,0}^{D} + \frac{1}{12} \boldsymbol{u}_{n,1}^{D} + \frac{1}{2}
\boldsymbol{u}_{2}^{D}
- \frac{1}{12} \boldsymbol{u}_{n,3}^{D}
\right).
\end{split}
\end{align}
This gives us the first two equations for the set of eight unknown solution vectors
$\boldsymbolhcal L = \{\boldsymbol u_{n,0}^0,\ldots, \boldsymbol u_{n,3}^0,\boldsymbol v_{n,0}^0,\ldots, \boldsymbol
v_{n,3}^0\}$ on each subinterval $I_n$, where each of these vectors is
defined by means of \eqref{eq:full_discrete_ansatz} through $\boldsymbol w =
(w_1,\ldots,w_J)^\top$ for $\boldsymbol w \kappan \boldsymbolhcal L$.
Next, we study the algebraic forms of the collocations conditions
\eqref{Eq:SemiDisLocalcGPC_6} to \eqref{Eq:SemiDisLocalcGPC_8}. By means
of the definition \eqref{eq:CondXi} of the basis of $\boldsymbolhbb P(I_n;\mathbb{R})$, the
constraints \eqref{Eq:SemiDisLocalcGPC_6} read as
\begin{equation}
\label{eq:constraint_left_a}
\boldsymbol{u}_{n,0}^{0} = \boldsymbol{u}_{n-1,2}^{0}\,, \;\; \boldsymbol{u}_{n,1}^{0} =
\boldsymbol{u}_{n-1,3}^{0}\,, \quad \boldsymbol{v}_{n,0}^{0} = \boldsymbol{v}_{n-1,2}^{0}\,, \;\;
\boldsymbol{v}_{n,1}^{0} = \boldsymbol{v}_{n-1,3}^{0}\,.
\end{equation}
By means of \eqref{eq:CondXi} along with \eqref{eq:DefM_A}, the conditions
\eqref{Eq:SemiDisLocalcGPC_7} and \eqref{Eq:SemiDisLocalcGPC_8} can be recovered as
\begin{align}
\label{eq:C1_Col2}
\boldsymbol{M}
\frac{1}{\tau_n} \boldsymbol{u}_{n,3}^{0}
-
\boldsymbol{M}
\boldsymbol{v}_{n,2}^{0}
& = \boldsymbol 0\,,
\\
\label{eq:C1_Col2_2}
\boldsymbol{M} \frac{1}{\tau_n} \boldsymbol{v}_{n,3}^{0}
+
\boldsymbol{A} \boldsymbol{u}_{n,2}^{0}
& =
\boldsymbol{M}
\boldsymbol{f}_{n,2}
-
\boldsymbol{M} \frac{1}{\tau_n} \boldsymbol{v}_{n,3}^{D}
-
\boldsymbol{A} \boldsymbol{u}_{n,2}^{D}\,.
\end{align}
Putting relations \eqref{eq:constraint_left_a} into the identities
\eqref{eq:time_scheme_a1} and \eqref{eq:time_scheme_a2} and combining the resulting
equations with \eqref{eq:C1_Col2} and \eqref{eq:C1_Col2_2} yields for the subinterval
$I_n$ the linear block system
\begin{equation}
\label{eq:Sx=b}
\boldsymbol{S}\boldsymbol{x} = \boldsymbol{b}
\end{equation}
for the vector of unknowns
\begin{equation}
\label{def:vec_x}
\boldsymbol{x} =
\left(\left(\boldsymbol{v}_{n,2}^{0}\right)^{\top}\,, \left(\boldsymbol{v}_{n,3}^{0}\right)^\top\,,
\left(\boldsymbol{u}_{n,2}^{0}\right)^\top\,, \left(\boldsymbol{u}_{n,3}^{0}\right)^\top\right)^\top
\end{equation}
and the system $\boldsymbol S$ and right-hand side $\boldsymbol b$ given by
\begin{align}
\label{eq:system_matrix_c1}
\boldsymbol{S} &=
\begin{pmatrix}
\boldsymbol{M} & \boldsymbol{0} & \boldsymbol{0} & \frac{1}{\tau_n}\boldsymbol{M} \\[1ex]
\boldsymbol{0} & \frac{1}{\tau_n}\boldsymbol{M} & \boldsymbol{A} & \boldsymbol{0} \\[1ex]
-\frac{\tau_n}{2}\boldsymbol{M} & \frac{\tau_n}{12}\boldsymbol{M} & \boldsymbol{M} & \boldsymbol{0} \\[1ex]
\boldsymbol{M} & \boldsymbol{0} & \frac{\tau_n}{2}\boldsymbol{A} & -\frac{\tau_n}{12}\boldsymbol{A}
\end{pmatrix},
&
\boldsymbol{b} &=
\begin{pmatrix}
\boldsymbol{0}
\\[1ex]
\boldsymbol{M}
\left(\boldsymbol{f}_{n,2}
- \frac{1}{\tau_n} \boldsymbol{v}^D_{n,3}
\right)
-
\boldsymbol{A} \boldsymbol u^D_{n,2}
\\[1ex]
\boldsymbol{M}
\left(
\boldsymbol{u}^0_{n,0}
+ \frac{\tau_n}{2}\boldsymbol{v}^0_{n,0}
+ \frac{\tau_n}{12}\boldsymbol{v}^0_{n,1}
\right)
\\[1ex]
\boldsymbol{b}_{n,4}
\end{pmatrix},
\end{align}
with $\boldsymbol{b}_{n,4} = \boldsymbol{M} \bigl(
\boldsymbol{v}^0_{n,0}
+ \boldsymbol{v}^D_{n,0}
- \boldsymbol{v}^D_{n,2}
+ \frac{\tau_n}{2} (\boldsymbol{f}_{n,0} + \boldsymbol{f}_{n,2})
+ \frac{\tau_n}{12} (\boldsymbol{f}_{n,1} - \boldsymbol{f}_{n,3})
\bigr)
- \boldsymbol{A}
\bigl(
\frac{\tau_n}{2} (\boldsymbol{u}^0_{n,0} + \boldsymbol{u}^D_{n,0} + \boldsymbol{u}^D_{n,2})
+
\frac{\tau_n}{12} (\boldsymbol{u}^0_{n,1} + \boldsymbol{u}^D_{n,1} - \boldsymbol{u}^D_{n,3})
\bigr)$. By means of the collocation constraints \eqref{eq:constraint_left_a}, the number of unknown coefficient vectors for the discrete
solution $(u_{\tau,h}{}_{|I_n},v_{\tau,h}{}_{|I_n})\kappan (\boldsymbolhbb P_3(I_n;V_h))^2$ is
thus effectively reduced from eight to four vectors, assembled now in $\boldsymbol x$ by \eqref{def:vec_x}.
We note that the first two rows of \cref{eq:system_matrix_c1} represent the
collocation conditions \eqref{eq:C1_Col2} and \eqref{eq:C1_Col2_2}. They have a sparser
structure then the last two rows representing the variational conditions which can be
advantageous or exploited for the construction of efficient iterative solvers for
\eqref{eq:Sx=b}. Compared with a pure variational approach (cf.\
\cite{Hussain2011,Hussain2011_2,KM04}), more degrees or freedom are obtained directly
by computaionally cheap vector identities (cf.\ \eqref{eq:time_scheme_a1}) in GCC$^l(k)$ such that they
can be eliminated from the overall linear system and, thereby, used to reduce the systems size.
\subsection{Solver technology}
\label{sec:solver_technology}
In the sequel, we present two different iterative approaches for solving the linear
system \eqref{eq:Sx=b} with the non-symmetric matrix $\boldsymbol S$. In \cref{sec:Challenging_Example}, a runtime comparison between the two concepts is
provided. As basic toolbox we use the deal.II finite element
library \cite{DealIIReference} along with the Trilinos library \cite{TrilinosReference} for parallel
computations.
\subsubsection{1.\ Approach: Condensing the linear system}
\label{sec:method_1}
The first method for solving \eqref{eq:Sx=b} is based on the concepts developed in
\cite{Koecher15}. The key idea is to use Gaussian block elimination within the system
matrix $\boldsymbol{S}$ and end up with a linear system with matrix $\boldsymbol{S}_r$ of reduced size
for one of the subvectors in $\boldsymbol x$ in \eqref{def:vec_x} only, and to compute the
remaining subvectors of \eqref{def:vec_x} by computationally cheap post-processing
steps afterwards. The reduced system matrix $\boldsymbol{S}_r$ should have sufficient potenial that
an efficient preconditioner for the iterative solution of the reduced
system can be constructed. Of course, the Gauss elimination on the block level can be done in different
ways. The goal of our approach is to avoid the inversion of the stiffness matrix
$\boldsymbol A$ in \eqref{eq:system_matrix_c1} in the computation of the condensed system
matrix $\boldsymbol{S}_r$ such that a matrix-vector multiplication with $\boldsymbol{S}_r$ just involves
calculating $\boldsymbol{M}^{-1}$. At least for discontinuous Galerkin methods in space, where $\boldsymbol{M}$ is block diagonal, this is computationally
cheap; cf.\ \cite{Bause2014,Koecher15}. We note that a continuous Galerkin approach in
space is used here only in order to simplify the notation and since the discretization in time
by the combined Galerkin--collocation approach is in the scope of interest.
Here, we choose the subvector $\boldsymbol u_{n,2}^0$ of $\boldsymbol x$ in \eqref{def:vec_x} as the
essential unknown, i.e.\ as the unknown solution vector of the condensed system with
matrix $\boldsymbol{S}_r$. By block Gaussian elemination we then end up with solving the linear
system,
\begin{equation}
\label{eq:recuced_c1_system}
\left(
\boldsymbol{M} + \frac{\tau_n^2}{12} \boldsymbol{A} + \frac{\tau_n^4}{144} \boldsymbol{A}\boldsymbol{M}^{-^1}\boldsymbol{A}
\right)
\boldsymbol{u}_{n,2}^{0}
= \boldsymbol{b}_{n,r}
\end{equation}
with right-hand side vector
\begin{multline}
\label{def:br}
\boldsymbol{b}_{n,r} =
\boldsymbol{M} \Biggl(
\frac{1}{2} \boldsymbol{f}_{n,0} + \frac{1}{12} \boldsymbol{f}_{n,1} + \frac{1}{3} \boldsymbol{f}_{n,2} - \frac{1}{12}
\boldsymbol{f}_{n,3}
\Biggr)
+ \boldsymbol{M} \Biggl(
2 \boldsymbol{v}_{n,0}^0 + \frac{1}{6} \boldsymbol{v}_{n,1}^0 + \frac{2}{\tau_n} \boldsymbol{u}_{n,0}^0
\Biggr)
\\
- \boldsymbol{A} \Biggl(
\frac{2}{3} \tau_n \boldsymbol{u}_{n,0}^0 \! + \! \frac{1}{12} \tau_n \boldsymbol{u}_{n,1}^0 \!+ \!\frac{1}{12} \tau_n ^2
\boldsymbol{v}_{n,2}^0 \!+ \!\frac{1}{72} \tau_n^2 \boldsymbol{v}_{n,1}^0
\Biggr)
\!+ \! \boldsymbol{M} \Biggl(
2 \boldsymbol{v}_{n,0}^D \! + \!\frac{1}{6} \boldsymbol{v}_{n,1}^D \!+ \!\frac{2}{\tau_n} \boldsymbol{u}_{n,0}^D \!- \! \frac{2}{\tau_n}
\boldsymbol{u}_{n,2}^D
\Biggr)
\\
+\boldsymbol{A} \Biggl(
\frac{1}{72} \tau_n^3 \boldsymbol{f}_{n,2} - \boldsymbol{M}^{-1} \frac{1}{72} \tau_n^3 \boldsymbol{u}_{n,2}^D
\Biggr)
\! + \!
\boldsymbol{A} \Biggl(
- \frac{2}{3} \tau_n \boldsymbol{u}_{n,0}^D \!- \! \frac{1}{12} \tau_n \boldsymbol{u}_{n,1}^D \! -\! \frac{\tau_n}{6}
\boldsymbol{u}_{n,2}^D \! - \! \frac{\tau_n^2}{12} \boldsymbol{v}_{n,0}^D \! -\! \frac{\tau_n^2}{72} \boldsymbol{v}_{n,1}^D
\Biggr)\,.
\end{multline}
The product of $\boldsymbol A \boldsymbol{M}^{-1}\boldsymbol{A}$ in \eqref{eq:recuced_c1_system} mimics the
discretization of a fourth order operator due to the appearance of the product of $\boldsymbol A$
with its ''weighted'' form $\boldsymbol{M}^{-1}\boldsymbol{A}$. Thereby, the conditioning number of the condensed system is strongly
increased (cf.\ \cite{Koecher15}) which is the main drawback in this concept of condensing the
overall system \eqref{eq:Sx=b} to \eqref{eq:recuced_c1_system} for the essential unknown
$\boldsymbol u_{n,2}^0$. On the other hand, since $\boldsymbol{M}$ and $\boldsymbol{A}$ are symmetric and,
thus, $\boldsymbol{A}\boldsymbol{M}^{-1}\boldsymbol{A} = (\boldsymbol{A}\boldsymbol{M}^{-1}\boldsymbol{A})^\top$, the condensed
matrix $\boldsymbol{S}_r$ is symmetric such that the preconditioned conjugate gradient method
can be applied. Solving systems of type \eqref{eq:recuced_c1_system} is carefully studied
in \cite{Bause2014,Koecher15} and the references given therein.
We solve \eqref{eq:recuced_c1_system} by the conjugate gradient method. The left
preconditioning operator
\begin{align}
\boldsymbol{P} &= \boldsymbol{K}_{\mu} \boldsymbol{M}^{-1} \boldsymbol{K}_{\mu}
= \left(\mu \boldsymbol{M} + \frac{\tau_n^2}{4} \boldsymbol{A}\right) \boldsymbol{M}^{-1}
\left(\mu \boldsymbol{M} + \frac{\tau_n^2}{4} \boldsymbol{A}\right)\,,
\end{align}
with positive $\mu \kappan \mathbb{R}$ chosen such that the spectral norm of
$\boldsymbol{P}^{-1}\boldsymbol{S}_r$ is
minimised, is applied. For details
of the choice of the parameter $\mu$, we refer to \cite{Bause2014,Koecher15}. Here, we
use $\mu = \sqrt{11/2}$. In order to apply the preconditioning operator $\boldsymbol P$ in the
conjugate gradient iterations, without assembling $\boldsymbol P$ explicitly, i.e.\ to solve the auxiliary system with matrix $\boldsymbol P$,
we have to solve linear systems for the mass matrix $\boldsymbol{M}$ and the stiffness
matrix $\boldsymbol{A}$. For this, we use embedded conjugate gradient iterations combined with
an algebraic multigrid preconditioner of the Trilinos library \cite{TrilinosReference}.
The overvall algorithm for solving \eqref{eq:recuced_c1_system} is sketched in \cref{fig:Solver_condensed}. The advantage of this approach is that we just have to store
$\boldsymbol{M}$ and $\boldsymbol{A}$ as sparse matrices in the computer memory. We never have to
assemble the full matrix $ \boldsymbol S$ from \eqref{eq:system_matrix_c1}, nor do we have to store
the reduced matrix $\boldsymbol{S}_r$ from \eqref{eq:recuced_c1_system}. Finally, the
remaining unknown subvectors $\boldsymbol{v}_{n,2}^{0}, \boldsymbol{v}_{n,3}^{0}$ and
$\boldsymbol{u}_{n,3}^{0}$ in \eqref{eq:Sx=b} are successively computed in post-processing steps.
\begin{figure}
\caption{Preconditioning and solver for the condensed system \eqref{eq:recuced_c1_system}
\label{fig:Solver_condensed}
\end{figure}
\subsubsection{2.\ Approach: Solving the non-symmetric system}
\label{sec:method_2}
The second approach used to solve \eqref{eq:Sx=b} relies on assembling the system
matrix $\boldsymbol{S}$ of \eqref{eq:system_matrix_c1} as a sparse matrix and solving the
resulting non-symmetric system. For smaller dimensions of $\boldsymbol S$ a parallel direct
solver \cite{Demmel2003} is used. For constant time step sizes $\tau_n$ the matrix $\boldsymbol S$ needs to be
factorized once only, which results in excellent performance properties for large
sequences of time steps. For high-dimensional problems with interest in practice,
we use the Generalized Minimal Residual (GMRES) method, an iterative Krylov subspace
method, to solve \eqref{eq:Sx=b}. The drawback of this approach then comes through the
necessity to provide an efficient preconditioner, i.e.\ an approximation to the inverse
of $\boldsymbol{S}$, for the complex block matrix $\boldsymbol S$ of \eqref{eq:system_matrix_c1}.
Here, we use the algebraic multigrid method as preconditioning technique. We use the
MueLue preconditioner \cite{MueLueReference}, which is part of the Trilinos project, with
non-symmetric smoothed aggregation. We use an usual V-cycle algorithm along with a
symmetric successive over-relaxation (SSOR) smoother with
a damping factor of $1.33$. The design of efficient
algebraic solvers for block systems like \eqref{eq:Sx=b}, and for higher order
variational time discretizations in general, is still an active field of research. We
expect further improvement in the future.
\subsection{Numerical convergence tests}
\label{sec:convergence_test_C1}
In this section we present a numerical convergence test for the proposed GCC$^1(3)$ approach of
Def.\ \ref{Def:GCC} and \cref{sec:Deriving_C1_System}, respectively. For the
solution $\{u, v\}$ of \cref{eq:InitialProblem} and the fully discrete
approximation GCC$^1(3)$ of Def.\ \ref{Def:GCC} we let
\begin{align}
e^{u} &\coloneqq u(\boldsymbol{x},t) - u_{\tau,h}(\boldsymbol{x},t),
&
e^{v} &\coloneqq v(\boldsymbol{x},t) - v_{\tau,h}(\boldsymbol{x},t).
\end{align}
We study the error $(e^u,e^v)$ with respect to the norms
\begin{equation}
\begin{aligned}
\| e^w \|_{L^\kappanfty(L^2)} &\coloneqq \max_{t \kappan I} \left( \kappant_{\Omega} \| e^w \|^2 \mathrm d
x \right)^{\frac{1}{2}} \,,
\quad
\| e^w \|_{L^2(L^2)} \coloneqq \biggl( \kappant_{I} \kappant_{\Omega} \| e^w \|^2 \mathrm d x \, \mathrm d t
\biggr)^{\frac{1}{2}}\,,
\end{aligned}
\end{equation}
where $w \kappan (u, v)$, and in the energy quantities
\begin{equation}
\begin{aligned}
||| E \, |||_{L^\kappanfty} &\coloneqq \max_{t \kappan I} \biggl(\| \nabla e^{u} \|^2 + \| e^{v}
\|^2
\biggr)^{\frac{1}{2}} \,,
\quad
||| E\, |||_{L^2} \coloneqq \Biggl( \kappant_I
\kappant_{\Omega} \| \nabla e^{u} \|^2 + \| e^{v} \|^2 \mathrm d \boldsymbol{x} \, \mathrm d t
\Biggr)^{\frac{1}{2}} \, .
\end{aligned}
\end{equation}
All $L^{\kappanfty}$-norms in time are computed on the discrete time grid
\begin{equation}
\label{eq:discrete_time_grid}
I = \big\{
t_n^d | t_n^d = t_{n-1} + d \cdot k_n \cdot \tau_n,
\quad
k_n = 0.001, d = 0, \ldots, 999, n = 1, \ldots, N
\big\}.
\end{equation}
For our first convergence test we prescribe the solution
\begin{equation}
\label{eq:conv_test_1}
u_1(\boldsymbol{x},t) = \sin(4 \pi t) \cdot x_1 \cdot (x_1 - 1) \cdot x_2 \cdot (x_2 - 1).
\end{equation}
on $\Omega \times I = (0,1)^2 \times [0,1]$. We let $c = 1$, use a constant mesh size
$h_0 = 0.25$ and start with the time step size $\tau_0 = 0.1$. We compute the errors on a
sequence of successively refined time meshes by halving the step sizes in each
refinement step. We choose a bicubic discretization of the space
variables in $V_h^3$ (cf.\ \eqref{Eq:DefVh}) such that the spatial part of the solution
is resolved exactly by its numerical approximation. Table
\ref{tab:conv_1} summarizes the computed errors and experimental orders of convergence.
The expected convergence rates of Thm.~\ref{th:error} are nicely confirmed.
\begin{table}[!htb]
\centering
\small
\begin{tabular}{c@{\,\,\,\,}c c@{\,}c c@{\,}c c@{\,}c}
\toprule
{$\tau$} & {$h$} &
{ $\| e^{u} \|_{L^\kappanfty(L^2)} $ } & EOC &
{ $\| e^{v} \|_{L^\kappanfty(L^2)} $ } & EOC &
{ $||| E\, |||_{L^\kappanfty} $ } & EOC \\
\midrule
$\tau_0/2^0$ & $h_0$ & 2.318e-04 & {--} & 1.543e-03 & {--} & 1.574e-03 &
{--} \\
$\tau_0/2^1$ & $h_0$ & 1.541e-05 & 3.91 & 9.694e-05 & 3.99 & 1.004e-04 &
3.97 \\
$\tau_0/2^2$ & $h_0$ & 9.825e-07 & 3.97 & 6.260e-06 & 3.95 & 6.478e-06 &
3.95 \\
$\tau_0/2^3$ & $h_0$ & 6.185e-08 & 3.99 & 3.946e-07 & 3.99 & 4.082e-07 &
3.99 \\
$\tau_0/2^4$ & $h_0$ & 3.873e-09 & 4.00 & 2.472e-08 & 4.00 & 2.557e-08 &
4.00 \\
$\tau_0/2^5$ & $h_0$ & 2.422e-10 & 4.00 & 1.548e-09 & 4.00 & 1.609e-09 &
3.99 \\
\midrule
\multicolumn{8}{c}{}\\[-3ex]
\midrule
{$\tau$} & {$h$} &
{ $\| e^{u} \|_{L^2(L^2)} $ } & EOC &
{ $\| e^{v} \|_{L^2(L^2)} $ } & EOC &
{ $||| E\, |||_{L^2} $ } & EOC \\
\midrule
$\tau_0/2^0$ & $h_0$ & 1.634e-04 & {--} & 1.232e-03 & {--} & 1.441e-03 &
{--} \\
$\tau_0/2^1$ & $h_0$ & 1.070e-05 & 3.93 & 7.864e-05 & 3.97 & 9.269e-05 &
3.96 \\
$\tau_0/2^2$ & $h_0$ & 6.765e-07 & 3.98 & 4.943e-06 & 3.99 & 5.836e-06 &
3.99 \\
$\tau_0/2^3$ & $h_0$ & 4.240e-08 & 4.00 & 3.094e-07 & 4.00 & 3.654e-07 &
4.00 \\
$\tau_0/2^4$ & $h_0$ & 2.652e-09 & 4.00 & 1.934e-08 & 4.00 & 2.285e-08 &
4.00 \\
$\tau_0/2^5$ & $h_0$ & 1.659e-10 & 4.00 & 1.212e-09 & 4.00 & 1.433e-09 &
3.99 \\
\bottomrule
\end{tabular}
\caption{Calculated errors for GCC$^1(3)$ with solution \eqref{eq:conv_test_1}.}
\label{tab:conv_1}
\end{table}
In our second numerical experiment we study the space-time convergence behavior of a
solution satisfying non-homogeneous Dirichlet boundary conditions,
\begin{equation}
\label{eq:u2}
u_2(\boldsymbol{x},t) = \sin(2 \cdot \pi \cdot t + x_1) \cdot \sin(2 \cdot \pi \cdot t \cdot
x_2)
\end{equation}
on $\Omega \times I = (0,1)^2 \times [0,1]$. We choose a bicubic discretization in
$V_h^3$ (cf.\ \eqref{Eq:DefVh}) of the space variable. We refine the space-time mesh by
halving both step sizes in each refinement step. Table~\ref{tab:conv_2} shows the computed
errors and experimental orders of convergence for this example. In all measured norms,
optimal rates in space and time (cf.\ Thm.~\ref{th:error}) are confirmed. This underlines
the correct treatment of the prescribed non-homogeneous Dirichlet boundary conditions.
\begin{table}[!htb]
\centering
\small
\begin{tabular}{c@{\,\,\,\,}c c@{\,}c c@{\,}c c@{\,}c}
\toprule
{$\tau$} & {$h$} &
{ $\| e^{u} \|_{L^\kappanfty(L^2)} $ } & EOC &
{ $\| e^{v} \|_{L^\kappanfty(L^2)} $ } & EOC &
{ $||| E\, |||_{L^\kappanfty} $ } & EOC \\
\midrule
$\tau_0/2^0$ & $h_0/2^0$ & 3.486e-03 & {--} & 3.602e-02 & {--} &
5.013e-02 & {--} \\
$\tau_0/2^1$ & $h_0/2^1$ & 2.329e-04 & 3.90 & 2.392e-03 & 3.90 &
3.338e-03 & 3.92 \\
$\tau_0/2^2$ & $h_0/2^2$ & 1.483e-05 & 3.97 & 1.527e-04 & 3.97 &
2.128e-04 & 3.98 \\
$\tau_0/2^3$ & $h_0/2^3$ & 9.320e-07 & 3.99 & 9.609e-06 & 3.99 &
1.338e-05 & 3.99 \\
$\tau_0/2^4$ & $h_0/2^4$ & 5.837e-08 & 4.00 & 6.022e-07 & 4.00 &
8.383e-08 & 4.00 \\
$\tau_0/2^5$ & $h_0/2^5$ & 3.649e-09 & 4.00 & 3.767e-08 & 4.00 &
5.243e-08 & 4.00 \\
\midrule
\multicolumn{8}{c}{}\\[-3ex]
\midrule
{$\tau$} & {$h$} &
{ $\| e^{u} \|_{L^2(L^2)} $ } & EOC &
{ $\| e^{v} \|_{L^2(L^2)} $ } & EOC &
{ $||| E\, |||_{L^2} $ } & EOC \\
\midrule
$\tau_0/2^0$ & $h_0/2^0$ & 2.700e-03 & {--} & 2.568e-02 & {--} &
3.458e-02 & {--} \\
$\tau_0/2^1$ & $h_0/2^1$ & 1.771e-04 & 3.93 & 1.689e-03 & 3.93 &
2.278e-03 & 3.92 \\
$\tau_0/2^2$ & $h_0/2^2$ & 1.120e-05 & 3.98 & 1.070e-04 & 3.98 &
1.444e-04 & 3.98 \\
$\tau_0/2^3$ & $h_0/2^3$ & 7.020e-07 & 4.00 & 6.713e-06 & 3.99 &
9.061e-06 & 3.99 \\
$\tau_0/2^4$ & $h_0/2^4$ & 4.391e-08 & 4.00 & 4.199e-07 & 4.00 &
5.669e-07 & 4.00 \\
$\tau_0/2^5$ & $h_0/2^5$ & 2.744e-09 & 4.00 & 2.624e-08 & 4.00 &
3.543e-08 & 4.00 \\
\bottomrule
\end{tabular}
\caption{Calculated errors for GCC$^1(3)$ with solution \eqref{eq:u2}.}
\label{tab:conv_2}
\end{table}
\subsection{Test case of structural health monitoring}
\label{sec:Challenging_Example}
Next, we consider a test problem that is based on \cite{Bangerth2010} and
related to typical problems of structural health monitoring by ultrasonic waves (cf.\ \cref{fig:elastic_wave}). We aim to compare the GCC$^1(3)$ approach with a standard
continuous in time Galerkin--Petrov approach cGP(2) of piecewise quadratic polynomials in
time; cf.\ \cite{Bause2018,Hussain2011} for details. The cGP(2) scheme has superconvergence
properties in the discrete time nodes $t_n$ for $n=1,\ldots, N$ as shown in
\cite{Bause2018}. Thus,the errors $\max_{n=1,\ldots, N} \|e^u(t_n)\|$ and
$\max_{n=1,\ldots, N} \|e^v(t_n)\|$ for the GCC$^1(3)$ and the cGP(2) scheme admit the
same fourth order rate of convergence in time and, thus, are comparable with respect to
accuracy.
The test setting is sketched in \cref{fig:Challenging_Plate}. We consider $\Omega
\times I = (-1,1)^2 \times (0,1)$, let $f = 0$ and, for simplicity, prescribe homogeneous
Dirichlet boundary conditions such that $u^D = 0$. For the initial value we prescribe
a regularized Dirac impulse by
\begin{equation}
\begin{aligned}
u_0(\boldsymbol{x}) &= e^{-|\boldsymbol{x}_s|^2}
\big(1 - | \boldsymbol{x}_s |^2 \big) \Theta \big( 1 - | \boldsymbol{x}_s | \big),
&
\boldsymbol{x}_s &= 100 \boldsymbol{x}\,,
\end{aligned}
\end{equation}
where $\Theta$ is the Heaviside function. The coefficient function $c(\boldsymbol{x})$, mimicing
a material parameter, has a jump discontinuity and is given by $c(\boldsymbol x) = 1$ for $x_2
< 0.2$ and $c(\boldsymbol x) = 9$ for $x_2\geq 0.2$. Further we put $v_0=0$ for the second
initial value. Finally, we define the control region $\Omega_c = (0.75 - h_c, 0.75 + h_c)
\times (-h_c, h_c)$ where we calculate the signal arrival, at a sensor position for
instance, in terms of
\begin{equation}
\label{eq:control_quantity}
u_c(t) = \kappant_{\Omega_c} u_{\tau,h} (\boldsymbol{x}, t) \mathrm d \boldsymbol{x}.
\end{equation}
\begin{figure}
\caption{Test case of structural health monitoring}
\label{fig:Challenging_Plate}
\label{fig:Sol_t_0_5}
\label{fig:setup_preview}
\end{figure}
We choose a spatial mesh of \num{65536} cells and $\boldsymbolhbb Q_7$ elements; cf.\ \cref{fig:Sol_t_0_5}. This leads to more than \num{3.2E6} degrees of freedom in space in
each time step for each of the solution vectors. For each computation of the control
quantity \eqref{eq:control_quantity}, with $t\kappan (0,1]$, we use a constant time step size
$\tau_n$ for all time steps and compare the computation with the initially chosen
reference time step size of $\tau_0 = \num{2e-5}$.
\mathcal{C}ref{fig:res_all} shows the signal arrival and control quantity
\eqref{eq:control_quantity} over $t \kappan (0.6, 1)$ with different choices of the time
step sizes for the Galerkin--collocation scheme GCC$^1(3)$ and the standard
Galerkin--Petrov approach cGP(2) (cf.\ \cite{Bause2018,Hussain2011}) of a continuous in
time approximation. For the cGP(2) approach, very small time step sizes are required to avoid over- and undershoots in the control quantity $u_c(t)$. For the GCC$^1(3)$ approach with $C^1$ regularity in time, much larger time steps, approximately 100 times larger, can be applied without loss of accuracy compared to the fully converged reference solution given by GCC$^1()3$ with step size $\tau_0$. This clearly shows the superiority of the Galerkin--collocation scheme GCC$^1(3)$.
\begin{figure}
\caption{Control quantity \eqref{eq:control_quantity}
\label{fig:res_all}
\end{figure}
In Table~\ref{tab:runtime_comparison} the computational costs are summarized, where $r_1$
is the runtime for solving the condensed system by the approach of \cref{sec:method_1} and $r_2$ is the runtime for solving the block system by the
approach of \cref{sec:method_2}. For the cGP(2) approach, only the first of
the either iterative solver techniques was implemented. Recalling
from \cref{fig:res_all} that GCC$^1(3)$ with $\tau_n = 100 \times \tau_0$ leads to
the fully converged solution whereas cGP(2) with $\tau_n = \tau_0$ already shows over-
and undershoots, a strong superiority of GCC$^1(3)$ over cGP(2) is observed in in
Table~\ref{tab:runtime_comparison}. For both solver, a reduction in the wall clock
time by a factor of about 25 is shown.
\begin{table}[h!tb]
\centering
\small
\begin{tabular}{cccrSS[table-number-alignment=center]}
\toprule
DoF (space) & cores & method & \multicolumn{1}{c}{$\tau_n$} &
{$r_1$[h]} & {$r_2$[h]}\\
\midrule
\num{3.2E6} & 224 & $C^0$ & $\num{0.25} \times \tau_0$ & 219.3 &
{-} \\
& & $C^0$ & $\tau_0$ & 40.0 & {-} \\
\mathsf{a}ddlinespace
& & $C^1$ & $\tau_0$ & 46.6 & 25.3 \\
& & $C^1$ & $2 \times \tau_0$ & 33.1 & 19.4 \\
& & $C^1$ & $25 \times \tau_0$ & 4.5 & 2.3 \\
& & $C^1$ & $35 \times \tau_0$ & 3.7 & 2.2 \\
& & $C^1$ & $50 \times \tau_0$ & 2.9 & 1.6\\
& & $C^1$ & $100 \times \tau_0$ & 1.7 & 0.9\\
& & $C^1$ & $200 \times \tau_0$ & 1.1 & 0.7 \\
\mathsf{a}ddlinespace
\num{4.2E6} & 336 & $C^1$ & $50 \times \tau_0$ & 3.3 & 1.7\\
\bottomrule
\end{tabular}
\caption{Runtime (wall clock time) for GCC$^1(3)$ ({method
$C^1$}) and cGP(2) ({method $C^0$}) for different time step sizes and solvers
of \cref{sec:method_1} ($r_1$) and \ref{sec:method_2} ($r_2$) .}
\label{tab:runtime_comparison}
\end{table}
\section{\texorpdfstring{Galerkin--collocation GCC$\boldsymbol{{}^2(5)}$}
{Galerkin--collocation GCC2(5)}}
\label{sec:C2_solution}
Here, we briefly derive the algebraic form of the Galerkin--collocation scheme GCC$^2(k)$ of Def.~\ref{Def:GCC} with fully discrete solutions $(u_{\tau,h}{}_{|I_n},v_{\tau,h}{}_{|I_n})\kappan (X_\tau^k (V_h))^2$ such that $(u_{\tau,h},v_{\tau,h})\kappan (C^2(\overline I;V_h))^2$. For brevity, we restrict ourselves to the lowest polynomial degree in time $k=5$ that is possible to get $C^2$-regularity. The convergence properties are then demonstrated numerically.
\subsection{Fully discrete system}
We follow the lines of \cref{sec:Deriving_C1_System} and use the notation introduced there. The six basis function of $\boldsymbolhbb P_5 (\hat I;\mathbb{R})$ on the reference interval $\hat I$ are defined by the conditions
\begin{align}
\hat{\xi_{i}}^{(l)}(j)
&=
\mathrm delta_{i-2*j-l, j}
&&
\forall \,
i \kappan \{0, \cdots, 5\}
\quad \land \quad
j \kappan \{0, 1\}\,,\quad
l \kappan \{0, 1, 2\}\,,
\end{align}
where $\mathrm delta_{i,j}$ denotes the usual Kronecker symbol. This gives us
\begin{equation}
\label{eq:basis_C2}
\mbox{}\hspace*{-3cm}
\begin{array}{r@{\;}c@{\;}lr@{\;}c@{\;}lr@{\;}c@{\;}l}
\hat{\xi_{0}} &= & -6 t^5+15 t^4-10 t^3+1\,,
& \hat{\xi_{1}} &= & -3 t^5+8 t^4-6 t^3+t\,,
& \hat{\xi_{2}} &= & -\frac{1}{2} t^5 +\frac{3}{2} t^4 - \frac{3}{2} t^3 + \frac{1}{2} t^2\,,\\[2ex]
\hat{\xi_{3}} &= & 6 t^5-15 t^4+10 t^3\,,
& \hat{\xi_{4}} &= & -3 t^5+7 t^4-4 t^3\,,
& \hat{\xi_{5}} &= & \frac{1}{2}t^5 - t^4 + \frac{1}{2} t^3\,.
\end{array}\hspace*{-3cm}\mbox{}
\end{equation}
For this basis of $\boldsymbolhbb P_5(\hat I;\mathbb{R})$, the discrete variational conditions \eqref{Eq:SemiDisLocalcGPC_9}, \eqref{Eq:SemiDisLocalcGPC_10} then read as
\begin{align}
\label{eq:time_scheme_C2_a1}
& \boldsymbol{M} \biggl(
-\boldsymbol{u}_{n,0}^{0} + \boldsymbol{u}_{n,3}^{0}
\biggr)
- \tau_n \boldsymbol{M} \biggl(
\frac{1}{2} \boldsymbol{v}_{n,0}^{0} + \frac{1}{10} \boldsymbol{v}_{n,1}^{0}
+ \frac{1}{120} \boldsymbol{v}_{n,2}^{0} + \frac{1}{2} \boldsymbol{v}_{n,3}^{0}
- \frac{1}{10} \boldsymbol{v}_{n,4}^{0} + \frac{1}{120} \boldsymbol{v}_{n,5}^{0}
\biggr)
= \boldsymbol 0\,,
\\[1ex]
&\begin{multlined}
\label{eq:time_scheme_C2_a2}
\boldsymbol{M} \biggl(
-\boldsymbol{v}_{n,0}^{0} + \boldsymbol{v}_{n,3}^{0}
\biggr)
+
\tau_n \boldsymbol{A}
\biggl(
\frac{1}{2} \boldsymbol{u}_{n,0}^{n,0} + \frac{1}{10} \boldsymbol{u}_{n,1}^{0}
+ \frac{1}{120} \boldsymbol{u}_{n,2}^{0} + \frac{1}{2} \boldsymbol{u}_{n,3}^{0}
- \frac{1}{10} \boldsymbol{u}_{n,4}^{0} + \frac{1}{120} \boldsymbol{u}_{n,5}^{0}
\biggr)
=
\\
\tau_n \boldsymbol{M}
\biggl( \frac{1}{2} \boldsymbol{f}_{n,0}^{} + \frac{1}{10} \boldsymbol{f}_{n,1}^{}
+ \frac{1}{120} \boldsymbol{f}_{n,2}^{} + \frac{1}{2} \boldsymbol{f}_{n,3}
- \frac{1}{10} \boldsymbol{f}_{n,4}^{} + \frac{1}{120} \boldsymbol{f}_{n,5}
\biggr)
-
\boldsymbol{M} \biggl(
-\boldsymbol{v}_{n,0}^{D} + \boldsymbol{v}_{n,3}^{D}
\biggr)
\\
-
\tau_n \boldsymbol{A}
\biggl(
\frac{1}{2} \boldsymbol{u}_{n,0}^{D} + \frac{1}{10} \boldsymbol{u}_{n,1}^{D}
+ \frac{1}{120} \boldsymbol{u}_{n,2}^{D} + \frac{1}{2} \boldsymbol{u}_{n,3}^{D}
- \frac{1}{10} \boldsymbol{u}_{n,4}^{D} + \frac{1}{120} \boldsymbol{u}_{n,5}^{D}
\biggr)\,.
\end{multlined}
\end{align}
In the basis, the first collocation conditions \eqref{Eq:SemiDisLocalcGPC_6} yield for $\boldsymbol{w}_{n,i}^0 \kappan \{\boldsymbol{u}_{n,i}^0, \boldsymbol{v}_{n,i}^0\}$ that
\begin{align}
\label{eq:constraint_left_C2_a}
\boldsymbol{w}_{n,0}^{0} &= \boldsymbol{w}_{n-1,3}^{0}\,,
&
\boldsymbol{w}_{n,1}^{0} &= \boldsymbol{w}_{n-1,4}^{0}\,,
&
\boldsymbol{w}_{n,2}^{0} &= \boldsymbol{w}_{n-1,5}^{0}\,,
\end{align}
which reduces the number of unknown solution vectors by $6$ on each subinterval $I_n$. For the collocation conditions \eqref{Eq:SemiDisLocalcGPC_7}, \eqref{Eq:SemiDisLocalcGPC_8} at $t_n$ and $s=1$ we deduce that
\begin{align}
\label{eq:C2_Col2a1}
&\boldsymbol{M}
\frac{1}{\tau_n} \boldsymbol{u}_{n,4}^{0}
-
\boldsymbol{M}
\boldsymbol{v}_{n,3}^{0}
= \boldsymbol 0\,,\quad
\begin{split}
\label{eq:C2_Col2a2}
& \boldsymbol{M} \frac{1}{\tau_n} \boldsymbol{v}_{n,4}^{0}
+
\boldsymbol{A} \boldsymbol{u}_{n,3}^{0}
=
\boldsymbol{M}
\boldsymbol{f}_{n,3}^{}
-
\boldsymbol{M} \frac{1}{\tau_n} \boldsymbol{v}_{n,4}^{D}
-
\boldsymbol{A} \boldsymbol{u}_{n,3}^{D}\,.
\end{split}
\end{align}
Similarly, for $s=2$ the collocation conditions \eqref{Eq:SemiDisLocalcGPC_7}, \eqref{Eq:SemiDisLocalcGPC_8} at $t_n$ read as
\begin{align}
\label{eq:C2_Col2b1}
& \boldsymbol{M}
\frac{1}{\tau_n} \boldsymbol{u}_{n,5}^{0}
-
\boldsymbol{M}
\boldsymbol{v}_{n,4}^{0}
=\boldsymbol 0\,, \quad
\begin{split}
\label{eq:C2_Col2b2}
&\boldsymbol{M} \frac{1}{\tau_n} \boldsymbol{v}_{n,5}^{0}
+
\boldsymbol{A} \boldsymbol{u}_{n,4}^{0}
=
\boldsymbol{M}
\boldsymbol{f}_{n,4}^{}
-
\boldsymbol{M} \frac{1}{\tau_n} \boldsymbol{v}_{n,5}^{D}
-
\boldsymbol{A} \boldsymbol{u}_{n,4}^{D}.
\end{split}
\end{align}
Finally, we recover the previous conditions as the linear system $\boldsymbol S \boldsymbol x = \boldsymbol
b$ for the vector of unknowns $\boldsymbol x =\left((\boldsymbol{u}_{n,3}^{0})^\top,
(\boldsymbol{u}_{n,4}^{0})^\top, (\boldsymbol{v}_{n,5}^{0})^\top,
\boldsymbol{u}_{n,5}^{0})^\top\right)^\top$ and with the system matrix $\boldsymbol S$ and right-hand side
vector $\boldsymbol b$ given by
\begingroup
\renewcommand*{\mathsf{a}rraystretch}{1.5}
\begin{align}
\label{eq:system_matrix_c2}
\boldsymbol{S} &=
\begin{pmatrix}
\boldsymbol{A} & \boldsymbol{0} & \boldsymbol{0} & \frac{1}{\tau_n^2}\boldsymbol{M} \\
\boldsymbol{0} & \boldsymbol{A} & \frac{1}{\tau_n}\boldsymbol{M} & \boldsymbol{0} \\
\boldsymbol{M} & -\frac{1}{2}\boldsymbol{M} & -\frac{\tau_n}{120} \boldsymbol{M} & \frac{1}{10}\boldsymbol{M} \\
\frac{\tau_n}{2}\boldsymbol{A} & \frac{1}{\tau_n}\boldsymbol{M} -\frac{\tau_n}{10} \boldsymbol{A} & \boldsymbol{0} &
\frac{\tau_n}{120}\boldsymbol{A}
\end{pmatrix}.
&
\boldsymbol{b} &=
\begin{pmatrix}
\boldsymbol{f}_{n,3} - \boldsymbol{A} \boldsymbol{u}_{n,3}^D - \frac{1}{\tau_n} \boldsymbol{M} \boldsymbol{v}_{n,4}^D
\\
\boldsymbol{f}_{n,4} - \boldsymbol{A} \boldsymbol{u}_{n,4}^D - \frac{1}{\tau_n} \boldsymbol{M} \boldsymbol{v}_{n,5}^D
\\
\boldsymbol{b}_{n,3}
\\
\boldsymbol{b}_{n,4}
\end{pmatrix},
\end{align}
\endgroup
with $\boldsymbol{b}_{n,3} =
\boldsymbol{M} \bigl(
\boldsymbol{u}_{n,0}^0 + \boldsymbol{u}_{n,0}^D - \boldsymbol{u}_{n,3}^D
+ \frac{\tau_n}{2} (\boldsymbol{v}_{n,0}^0 + \boldsymbol{v}_{n,0}^D)
+ \frac{\tau_n}{10} (\boldsymbol{v}_{n,1}^0 + \boldsymbol{v}_{n,1}^D)
+ \frac{\tau_n}{120} (\boldsymbol{v}_{n,2}^0 + \boldsymbol{v}_{n,2}^D)
+ \tau_n (\frac{1}{2} \boldsymbol{v}_{n,3}^D - \frac{1}{10} \boldsymbol{v}_{n,4}^D + \frac{1}{120} \boldsymbol{v}_{n,5}^D)
$
and
$
\boldsymbol{b}_{n,4} =
\boldsymbol{M} \bigl( \boldsymbol{v}_{n,0}^0 + \boldsymbol{v}_{n,0}^D - \boldsymbol{v}_{n,3}^D \bigr)
+ \tau_n \bigl( \frac{1}{2} \boldsymbol{f}_{n,3} + \frac{1}{10} \boldsymbol{f}_{n,1} + \frac{1}{120} \boldsymbol{f}_{n,2} + \frac{1}{2} \boldsymbol{f}_{n,3} - \frac{1}{10} \boldsymbol{f}_{n,4} + \frac{1}{120} \boldsymbol{f}_{n,5}\bigr)
-
\tau_n \boldsymbol{A} \bigl(
\frac{1}{2} (\boldsymbol{u}_{n,0}^0 + \boldsymbol{u}_{n,0}^D)2
+ \frac{1}{10} (\boldsymbol{u}_{n,1}^0 + \boldsymbol{u}_{n,1}^D)
+ \frac{1}{120} (\boldsymbol{u}_{n,2}^0 + \boldsymbol{u}_{n,2}^D)
+ \frac{1}{2} \boldsymbol{u}_{n,3}^D
- \frac{1}{10} \boldsymbol{u}_{n,4}^D
+ \frac{1}{120} \boldsymbol{u}_{n,5}^D
\bigr)
$.
\subsection{Iterative solver and convergence study}
\label{sec:NumTestC2}
To solve the linear system $\boldsymbol S \boldsymbol x = \boldsymbol b$ with $\boldsymbol{S}$ from \eqref{eq:system_matrix_c2}, we use block Gaussian elimination, as sketched in \cref{sec:method_1}, to find a reduced system $\boldsymbol S_r \boldsymbol{u}_{n,4}^{0} = \boldsymbol b_r$ for the essential unknown $\boldsymbol{u}_{n,4}^{0}$. All remaining unknown subvectors of $\boldsymbol x$ can be computed in post-processing steps. In explicit form, the condensed system reads as
\begin{equation}
\label{eq:recuced_c2_system}
\left(
14400 \boldsymbol{M} + 720 \tau_n^2 \boldsymbol{A} + 24 \tau_n^4 \boldsymbol{A}\boldsymbol{M}^{-^1}\boldsymbol{A} + \tau_n^6
\boldsymbol{A}\boldsymbol{M}^{-^1}\boldsymbol{A}\boldsymbol{M}^{-^1}\boldsymbol{A}
\right)
\boldsymbol{u}_{n,4}^{0}
=
\boldsymbol{b}_r\,.
\end{equation}
For brevity, we omit the exact definition of $\boldsymbol b_r$ that can be deduced easily from
\eqref{eq:system_matrix_c2}.
The matrix $\boldsymbol S_r$ is symmetric such that preconditioned conjugate gradient iterations are used for its solution. The preconditioner is constructed along the lines of \cref{sec:method_1}. The remainder part $\tau_n^6 \boldsymbol{A}\boldsymbol{M}^{-^1}\boldsymbol{A}\boldsymbol{M}^{-^1}\boldsymbol{A}$ is still ignored in the construction of the preconditioner. Even though the remainder is weighted by the small factor $\tau_n^6$, numerical experiments indicate that this scaling is not sufficient to balance its impact on the interation process. For the construction of an efficient preconditioning technique for $\boldsymbol S_r$ of GCC$^2(5)$ further improvements are still necessary.
To illustrate the convergence behavior and performance of the GCC$^2(5)$ Galerkin--collocation approach, we present in Table~\ref{tab:conv_C2_1} our numerical results for the test problem \eqref{eq:conv_test_1}. The expected convergence of sixth order in time is nicely observed in all norms.
\begin{table}[!htb]
\centering
\small
\begin{tabular}{c@{\,\,\,\,}c c@{\,}c c@{\,}c c@{\,}c}
\toprule
{$\tau$} & {$h$} &
{ $\| e^{u} \|_{L^\kappanfty(L^2)} $ } & EOC &
{ $\| e^{v} \|_{L^\kappanfty(L^2)} $ } & EOC &
{ $||| E\, |||_{L^\kappanfty} $ } & EOC \\
\midrule
$\tau_0/2^0$ & $h_0$ & 8.748e-06 & {--} & 4.355e-05 & {--} & 4.985e-05 &
{--} \\
$\tau_0/2^1$ & $h_0$ & 1.370e-07 & 6.00 & 7.404e-07 & 5.88 & 8.043e-07 &
5.95 \\
$\tau_0/2^2$ & $h_0$ & 2.165e-09 & 5.98 & 1.202e-08 & 5.95 & 1.266e-08 &
5.99 \\
$\tau_0/2^3$ & $h_0$ & 3.388e-11 & 6.00 & 1.883e-10 & 6.00 & 1.980e-10 &
6.00 \\
$\tau_0/2^4$ & $h_0$ & 5.301e-13 & 6.00 & 2.940e-12 & 6.00 & 3.093e-12 &
6.00 \\
\midrule
\multicolumn{8}{c}{}\\[-3ex]
\midrule
{$\tau$} & {$h$} &
{ $\| e^{u} \|_{L^2(L^2)} $ } & EOC &
{ $\| e^{v} \|_{L^2(L^2)} $ } & EOC &
{ $||| E\, |||_{L^2} $ } & EOC \\
\midrule
$\tau_0/2^0$ & $h_0$ & 4.022e-06 & {--} & 2.996e-05 & {--} & 3.502e-05 &
{--} \\
$\tau_0/2^1$ & $h_0$ & 6.353e-08 & 5.98 & 4.808e-07 & 5.96 & 5.599e-07 &
5.97 \\
$\tau_0/2^2$ & $h_0$ & 9.957e-10 & 6.00 & 7.565e-09 & 5.99 & 8.800e-09 &
5.99 \\
$\tau_0/2^3$ & $h_0$ & 1.557e-11 & 6.00 & 1.184e-10 & 6.00 & 1.377e-10 &
6.00 \\
$\tau_0/2^4$ & $h_0$ & 2.431e-13 & 6.00 & 1.849e-12 & 6.00 & 2.151e-12 &
6.00 \\
\bottomrule
\end{tabular}
\caption{Calculated errors for GCC$^2(5)$ with solution \eqref{eq:conv_test_1}.}
\label{tab:conv_C2_1}
\end{table}
\vspace*{-6ex}
\end{document}
\mathsf{a}ppendix
\section{Algebraic forms of the discrete variational and collocation conditions}
Here algebraic counter the Galerkin-collocation schemes CGC$^1(k)$ and
CGC$^1(k)$ is described.
schemes is uilding blocks for our Galerkin--collocation approach, the
discrete variational
and collocation conditions, are now derived separately. By the combination of these
conditions, members of $C^1$- and $C^2$-regular in time classes of approximation
schemes
are then obtained in Secs.\ \ref{sec:c1_solution} and \ref{sec:C2_solution} in a unified
framework.
\subsection{Variational discretization}
\label{sec:variational_discretization}
Here we derive the space-time discretization of the variational equations
\eqref{eq:cont_weak_formulation_1}, \eqref{eq:cont_weak_formulation_2}.
\subsubsection{Space discretization}
\label{sec:Space}
We start with the discretization in space. For this, let $\Omega_h$ be a shape-regular
mesh of $\Omega$ with mesh size $h>0$. We replace $H^1_0$ now with the finite dimensional
subspace $V_h^p$, which is the finite element space that is built on the mesh $\Omega_h$
of quadrilateral or hexahedral elements $K$ and is given by
\begin{equation}
\begin{aligned}
\label{eq:DefVh}
V_h^p = \left\{v_h \kappan C(\overline \Omega) \mid v_h{}_{|T}\kappan \boldsymbolhbb Q_p(K) \, \forall K
\kappan \Omega_h \right\} \subset H^1_0(\Omega)\,.
\end{aligned}
\end{equation}
where $\boldsymbolhbb Q_p(K)$ is the space defined by the reference mapping of polynomials on
the reference element with maximum degree of $p$ in each variable.\marginpar{Ändern!!}
Let $\phi_j$ build a local basis of $V_h^p$ on each element $K$. The space discrete
ansatz for the displacement $u_h$ and the velocity $v_h$ on every element $K$ can then be
written as linear combination of the basis functions:
\begin{align}
\label{eq:ansatzuv_space}
w^0_{h_{|_{K}}} &= \sum_{j=1}^{N_{loc}}w_{j}^{0,K}(t) \phi_j^K(\boldsymbol{x}).
\end{align}
Where $w_h \kappan \{u_h, v_h\}$, as mentioned in the introduction of
\cref{sec:ProblemDescription}. Here $N_{loc}$ is a placeholder for the number of local
degrees of freedom on each $K$, which depends on the spatial dimension of the problem and
the polynomial degree $p$.
We use consistent spatial test functions $\omega_h$, which are therefore span on each
element $k$ by:
\begin{equation}
\omega_{h_{|K}} \kappan \spanset{\phi_1^K, \ldots, \phi_{N_{loc}}^K}, \quad \forall K \kappan
\Omega_h.
\end{equation}
Let the interpolant of ${\mat f}C(\boldsymbol{x},t)$ and the inhomogeneous Dirichlet boundary
conditions $u^D$ and $v^D$ in $V_h$ on every element $K$ be:
\begin{align}
\label{eq:l2_projection_force}
\boldsymbolhrm{I}_{h_{|K}} f(\boldsymbol{x},t) &=
\sum_{j=1}^{N_{loc}}
f_{j}^{K}(t) \phi_j(\boldsymbol{x}),
&
\boldsymbolhrm{I}_{h_{|K}} w^D(\boldsymbol{x},t) &=
\sum_{j=1}^{N_{loc}}
w_{j}^{D,K}(t) \phi_j(\boldsymbol{x}).
\end{align}
Inserting this space discrete ansatz into
\cref{eq:cont_weak_formulation_1,eq:cont_weak_formulation_2}, testing with the same basis
functions $\phi_i$ and integrating over the space domain, using the divergence theorem,
leads to:
\begin{equation}
\begin{aligned}
\sum_{j=1}^{N_{loc}}\partial_t \mat uC_{j}^{0,K}
\bigl< \phi_j , \phi_i \bigr>_{\Omega_h}
-
\sum_{j=1}^{N_{loc}} \mat vC_{j}^{0,K}
\bigl< \phi_j, \phi_i \bigr>_{\Omega_h}
={}&
0
\\
\sum_{j=1}^{N_{loc}} \partial_t \mat vC_{j}^{0,K}
\bigl< \phi_j, \phi_i \bigr>_{\Omega_h}
+
c^2 \sum_{j=1}^{N_{loc}} \mat uC_{j}^{0,K}
\bigl< \nabla \phi_j, \nabla \phi_i \bigr>_{\Omega_h}
={}&
{\mat f}C_j^K \bigl< \phi_j, \phi_i \bigr>_{\Omega_h}
\\
-
\sum_{j=1}^{N_{loc}} \partial_t \mat vC_{j}^{D,K}
\bigl< \phi_j, \phi_i \bigr>_{\Omega_h}
-
c^2 \sum_{j=1}^{N_{loc}} &\mat uC_{j}^{D,K}
\bigl< \nabla \phi_j, \nabla \phi_i \bigr>_{\Omega_h}
\end{aligned}
\end{equation}
for all $i \kappan \{1,\ldots, N_{loc} \}$. We can identify the space integrals over the
inner product of the basis functions with the mass matrix $\boldsymbol{M}$ and the integrals over
the gradients of the basis functions as the stiffness matrix $\boldsymbol{A}$ and rewrite the
system in matrix vector notation as:
\begin{align}
\boldsymbol{M} \partial_t \mat u^{0,K}
-
\boldsymbol{M} \mat v^{0,K} &
=
0,
\\
\boldsymbol{M} \partial_t \mat v^{0,K}
+
c^2 \boldsymbol{A} \mat u^{0,K} &=
{\mat f}^K
-
\boldsymbol{M} \partial_t \mat v^{D,K}
-
c^2 \boldsymbol{A} \mat u^{D,K}.
\end{align}
\subsubsection{Time discretization}
\label{sec:Time}
For the discretization in time we also use the finite element method. The interval $I$ is
therefore divided into $N$ subintervals $I_n$ with $n \kappan \{1, \ldots, N\}$ and $I_n =
\left]t_{n-1}, t_n \right]$. The length of each time
subinterval is defined as $\tau_n := t_n - t_{n-1}$.
The practical calculations on the time domain $I$ are done on the unit time interval
$\hat{I} \kappan \left]0, 1\right]$ for each $I_n$ with the linear affine transformation
$\Tau_n$ from the unit time domain $\hat{I}$ to $I$ and it's inverse function:
\begin{align}
\label{eq:Transform_Unit_Time}
\Tau_n &= \tau_n \cdot \hat{t} + t_{n-1},
&
\Tau_n^{-1} &= \frac{t - t_{n-1}}{\tau_n}.
\end{align}
We denote the Bochner space:
\[
\boldsymbolhbb P_r(I_n;B) = \bigg\{u_\tau : I_n \mapsto B \; \Big|\; u_\tau(t) = \sum_{\iota=0}^r
U_\iota t^\iota,
\; \forall t\kappan I_n, \; U_\iota \kappan B\; \forall \iota \bigg\},
\]
with $r\kappan \mathbb{N}$ to be the space of all $B$-valued polynomials in time of order $r$ over
$I_n$.
Assuming that we are deriving a $C^q$ regular solution in time we need the possibility to
express the function values an the derivatives up to the order of $q$ at the beginning
and the end of each time interval $I_n$. Therefore the polynomial degree $r$ is coupled
with:
\begin{equation}
\label{eq:min_r}
r \geqq 2 q + 1
\end{equation}
to the regularity $q$ of the solution and we have $r + 1 = 2q + 2$ unknowns on each $I_n$.
For the time semi-discrete approximation we introduce the following solution spaces:
\begin{equation}
\begin{aligned}
\label{Eq:DefXk}
X_{\tau}^{r} (B) := \left\{ u_{\tau} \kappan C(\overbar{I};B) \mid u_{\tau_{|I_n}} \kappan
\mathbb{P}_r(I_n;B) \right\}
\cap C^k(\overbar{I};B),
\\
\quad \forall
n \kappan \left[1, \ldots, N\right],
k \kappan \left[1, \ldots, q\right].
\end{aligned}
\end{equation}
These are time continuous and differentiable up to the order $q$ on $\overbar{I}$ and
locally consist of polynomials with degree $r$ on every $I_n$.
Next we introduce a time basis $\xi_{\iota}$ for $X_{\tau}^{r}$, so that:
\begin{equation}
\label{eq:ansatz_time}
\begin{aligned}
w_{\tau_{|I_n}} \kappan \spanset{\xi_0, \ldots, \xi_r}.
\end{aligned}
\end{equation}
We can then express our time discrete solution as a linear combination of our time basis:
\begin{align}
w_{\tau_{|I_n}} &= \sum_{\iota=0}^{r}w_{\iota}^{\tau}(\boldsymbol{x}) \xi_\iota^\tau(t).
\label{eq:ansatzuv_space}
\end{align}
Next we define the test space as:
\begin{equation}
\label{Eq:DefYk}
Y_\tau^{l} (B) := \left\{w_\tau \kappan L^2(I;B) \mid w_\tau{}_{|I_n} \kappan \mathbb{P}_{l}(I_n;B)\;
\quad \forall n \kappan \left[1, N\right] \right\},
\end{equation}
which is time discontinuous and allows us the formulation of time marching schemes.
The polynomial degree $l$ of the test space is linked to the degree of the ansatz space
$r$ and the demanded regularity $q$ with:
\begin{equation}
\label{eq:Degree_Test_Space}
l = 2 \left( r - 2 q - 1 \right).
\end{equation}
We can express every test function as linear combination of the test basis functions:
\begin{equation}
\label{eq:test_space_time}
\omega_{\tau_{|I_n}} \kappan \spanset{\zeta_0, \ldots, \zeta_l}.
\end{equation}
Next we define the interpolation of ${\mat f}C(\boldsymbol{x},t)$, $u^D$ and $v^D$ in time on
every subinterval: $I_n$:
\begin{align}
\label{eq:l2_interp_force_time}
\boldsymbolhrm{I}_{\tau_{|I_n}} f(\boldsymbol{x},t)
&
=
\sum_{\iota=0}^{r}
f_{\iota}^{I_n} \xi_{n,\iota}(t),
&
\boldsymbolhrm{I}_{\tau_{|I_n}} w^D(\boldsymbol{x},t) &=
\sum_{\iota=0}^{r}
w_{\iota}^{D,I_n} \xi_{n,\iota}(t).
\end{align}
If we use our time discrete ansatz from \cref{eq:ansatz_time} in
\cref{eq:cont_weak_formulation_1,eq:cont_weak_formulation_2}, test with
\cref{eq:test_space_time} and integrate over the time domain we get, after transforming
the integrals to the unit time interval $\hat{I}$, the following time discrete system on
each time interval $I_n$ for $\mat uC_{\iota}^{0,I_n}$ and $\mat vC_{\iota}^{0,I_n}$
\begin{align}
\sum_{\iota=0}^{r} \mat uC_{\iota}^{0,I_n}
&\kappant_{\hat{I}} \left<
\partial_{\hat{t}} \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa \right> \mathrm d \hat{t}
-
\sum_{\iota=0}^{r} \mat vC_{\iota}^{0,I_n} \tau_n
\kappant_{\hat{I}} \left< \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa \right> \mathrm d \hat{t}
=
0,
\\[1ex]
\begin{split}
\sum_{\iota=0}^{r} \mat vC_{\iota}^{0, I_n}
&\kappant_{\hat{I}} \left<
\partial_{\hat{t}} \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa \right> \mathrm d \hat{t}
+
c^2 \sum_{\iota=0}^{r} \mat uC_{\iota}^{0, I_n} \tau_n
\kappant_{\hat{I}} \left< \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa \right> \mathrm d \hat{t}
=
\sum_{\iota=0}^{r} f_{\iota}^{I_n} \kappant_{\hat{I}} \left< \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa
\right> \mathrm d \hat{t}
\\[1ex]
&-
\sum_{\iota=0}^{r} \mat vC_{\iota}^{D,I_n}
\kappant_{\hat{I}} \left<
\partial_{\hat{t}} \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa \right> \mathrm d \hat{t}
+
c^2 \sum_{\iota=0}^{r} \mat uC_{\iota}^{D,I_n} \tau_n
\kappant_{\hat{I}} \left< \hat{\xi}_{n,\iota} , \hat{\zeta}_\kappa \right> \mathrm d \hat{t}
\qquad
\end{split}
\end{align}
for all $\kappa \kappan (0,\ldots,l)$.
\subsubsection{Space and time discretization}
\label{sec:space_time_discretization}
Now we put together the results from \cref{sec:Space} and \cref{sec:Time}.
The full discrete solution ansatz for the velocity and the displacement on every element
$K$ and in every time step $I_n$ is:
\begin{equation}
\label{eq:full_discrete_ansatz}
w_{\tau,h}^{C1}\mid_{K \times I_n} = \sum_{\iota=0}^{r} \sum_{j=1}^{N_{loc}}w_{j,\iota}^{K,I_n}
\phi_j^K(\boldsymbol{x})\xi_{\iota}^{I_n}(t).
\end{equation}
Similar the full discrete test space is:
\begin{equation}
\label{eq:test_space_full}
\omega_{h,\tau_{|K \times I_n}} \kappan \spanset{\phi_1 \zeta_0, \ldots, \phi_{N_{loc}}
\zeta_l}.
\end{equation}
We let interpolation of ${\mat f}C(\boldsymbol{x},t)$ and the inhomogeneous boundary parts in
space and time be:
\begin{flalign}
\label{eq:l2_projection_force}
\boldsymbolhrm{I}f(\boldsymbol{x},t) &=
\sum_{\iota=0}^{r}
\sum_{j=1}^{N_{DoF}}
f_{j,\iota}^{K,I_n} \xi_{n,\iota}(t) \phi_j(\boldsymbol{x}),
&
\boldsymbolhrm{I}w(\boldsymbol{x},t) &=
\sum_{\iota=0}^{r}
\sum_{j=1}^{N_{DoF}}
w_{j,\iota}^{K,I_n} \xi_{n,\iota}(t) \phi_j(\boldsymbol{x}).
\end{flalign}
Next we put \cref{eq:full_discrete_ansatz,eq:test_space_full} into
\cref{eq:cont_weak_formulation_1,eq:cont_weak_formulation_2}, so that we can combine the
results from \cref{sec:Space,sec:Time} and derive the fully discrete variational system:
Find $\mat u_{\iota}^{0,K,I_n}$ and $\mat v_{\iota}^{0,K,I_n}$, so that
\begin{align}
\label{eq:Fully_discrete_variational_1}
\boldsymbol{M} \sum_{\iota} \mat u_{\iota}^{0,K,I_n}
&\kappant_{\hat{I}} \left<
\partial_t \hat{\xi}_{j}^{I_n} , \hat{\zeta}_\kappa \right> \mathrm d t
-
\tau_n \boldsymbol{M} \sum_{\iota} \mat v_{\iota}^{0,K,I_n}
\kappant_{\hat{I}} \left< \hat{\xi}_{\iota}^{I_n} , \hat{\zeta}_\kappa \right> \mathrm d t
=
\boldsymbol{0},
\\
\begin{split}
\label{eq:Fully_discrete_variational_2}
\boldsymbol{M} \sum_{\iota} \mat v_{\iota}^{0,K,I_n}
&\kappant_{\hat{I}} \left<
\partial_t \hat{\xi}_{j}^{I_n} , \hat{\zeta}_\kappa \right> \mathrm d t
-
\tau_n \boldsymbol{A} \sum_{\iota} \mat u_{\iota}^{0,K,I_n}
\kappant_{\hat{I}} \left< \hat{\xi}_{\iota}^{I_n} , \hat{\zeta}_\kappa \right> \mathrm d t
=
\sum_{\iota} \boldsymbol{f}_{\iota}^{K,I_n}
\kappant_{\hat{I}} \left< \hat{\xi}_{\iota}^{I_n} , \hat{\zeta}_\kappa \right>
\\
-
&\boldsymbol{M} \sum_{\iota} \mat v_{\iota}^{D,K,I_n}
\kappant_{\hat{I}} \left<
\partial_t \hat{\xi}_{j}^{I_n} , \hat{\zeta}_\kappa \right> \mathrm d t
-
\tau_n \boldsymbol{A} \sum_{\iota} \mat u_{\iota}^{D,K,I_n}
\kappant_{\hat{I}} \left< \hat{\xi}_{\iota}^{I_n} , \hat{\zeta}_\kappa \right> \mathrm d t
\end{split}
\end{align}
is fulfilled $\forall \hat{\zeta}_\kappa \kappan \mathbb{P}_{l}(I_n, V_h^p)$, with the solution space
$X_{\tau}^{r} (V_h^p)$ and test space $Y_\tau^{l} (V_h^p)$.
\mathcal{C}ref{eq:Fully_discrete_variational_1,eq:Fully_discrete_variational_2} set $l + 1$ out of
the $r + 1$ degrees of freedom on each $I_n$.
\subsection{Collocation conditions}
\label{sec:collocation_conditions}
With the results of \cref{sec:space_time_discretization} we still have $2q + 1$ "unfixed"
degrees of freedom on each $I_n$.
Assuming that we are looking for a $C^q$ regular solution in time, we can use these
degrees of freedom now to enforce the demand for continuity in the function values and
derivatives up to the order $q$ with the following collocation conditions:
\begin{equation}
\label{eq:col1}
\begin{aligned}
\partial_t^{(k)} w_{\tau,h}|_{I_n} (t_{n-1}) &=
\partial_t^{(k)} w_{\tau,h}|_{I_{n-1} } (t_{n-1})
, \qquad \forall k \kappan \left[0, \ldots, q \right],
\end{aligned}
\end{equation}
where $\partial_t^{(k)}$ means the $k$-th derivative in time.
Due to the choosing of our ansatz functions, the global solution, as a linear combination
of the ansatz functions, is of the order $2 q + 1$ continuous differentiable in every
time interval $I_n$. With these set of conditions the solution will globally be
continuous differentiable up to the order $q$ in time.
With \cref{eq:col1} we can effectively reduce the degrees of freedom in each time
interval $I_n$ by $q + 1$ and transfer information from the solution of the previous time
interval $I_{n-1}$, which reduces the effective costs for the method presented here.
The second set of collocation conditions is the demand for fulfilling the system of
partial differential equations at all discrete time points $t_n$ up to the demanded order
of regularity $q$ in time.
Here we suppress the indices "$^{K,I_n}$" on the $\boldsymbol{u}$ and $\boldsymbol{v}$ in order to
increase the readability of the formulas:
\begin{align}
\boldsymbol{M}
&\sum_{\iota} \mat u_{\iota}^{0}
\partial_t^{(k)} \xi_{\iota}^{I_n}(t_n)
-
\boldsymbol{M}
\sum_{\iota} \mat v_{\iota}^{0}
\partial_t^{(k-1)} \xi_{\iota}^{I_n}(t_n)
=
\boldsymbol{0},
\\
\begin{split}
\boldsymbol{M}
&\sum_{\iota} \mat v_{\iota}^{0}
\partial_t^{(k)} \xi_{\iota}^{I_n}(t_n)
+
\boldsymbol{A}
\sum_{\iota} \mat u_{\iota}^{0}
\partial_t^{(k-1)} \xi_{\iota}^{I_n}(t_n)
=
\boldsymbol{f}_{\iota}^{} \xi_{\iota}^{I_n}(t_n)
\\
&-
\boldsymbol{M}
\sum_{\iota} \mat v_{\iota}^{D}
\partial_t^{(k)} \xi_{\iota}^{I_n}(t_n)
-
\boldsymbol{A}
\sum_{\iota} \mat u_{\iota}^{D}
\partial_t^{(k-1)} \xi_{\iota}^{I_n}(t_n)
, \qquad \forall k \kappan \left[1, \ldots, q \right].
\end{split}
\end{align}
With \cref{eq:Transform_Unit_Time} we can transform these conditions to the unit time
interval and, due to the chain rule, derive:
\begin{align}
\label{eq:col2_a}
\frac{1}{\tau_n^k} \boldsymbol{M}
&\sum_{\iota} \mat u_{\iota}^{0}
\partial_t^{(k)} \hat{\xi}_{\iota}(1)
-
\frac{1}{\tau_n^{k-1}} \boldsymbol{M}
\sum_{\iota} \mat v_{\iota}^{0}
\partial_t^{(k-1)} \hat{\xi}_{\iota}(1)
=
\boldsymbol{0},
\\
\begin{split}
\label{eq:col2_b}
\frac{1}{\tau_n^k} \boldsymbol{M}
&\sum_{\iota} \mat v_{\iota}^{0}
\partial_t^{(k)} \hat{\xi}_{\iota}(1)
+
\frac{1}{\tau_n^{k-1}} \boldsymbol{A}
\sum_{\iota} \mat u_{\iota}^{0}
\partial_t^{(k-1)} \hat{\xi}_{\iota}(1)
=
\frac{1}{\tau_n^{k-1}}
\boldsymbol{f}_{\iota}^{} \hat{\xi}_{\iota}(1)
\\
&-
\frac{1}{\tau_n^k} \boldsymbol{M}
\sum_{\iota} \mat v_{\iota}^{D}
\partial_t^{(k)} \hat{\xi}_{\iota}(1)
+
\frac{1}{\tau_n^{k-1}} \boldsymbol{A}
\sum_{\iota} \mat u_{\iota}^{D}
\partial_t^{(k-1)} \hat{\xi}_{\iota}(1),
\qquad
\forall k \kappan \left[1, \ldots, q \right].
\end{split}
\end{align}
This gives another set of $q$ conditions, so that all of the $r + 1$ degrees of freedom
are fixed now.
\section{Conclusions and outlook}
\label{sec:Conclusions}
In this paper we derived a variational formulation in combination with a collocation
method to get a finite element solution with global regularity of any desired order in
time. With this method, the disadvantage of more degrees of freedom, that are needed to
compute higher order solutions, is at least partly outweighed by the fact, that parts of
the solution can simply be transferred from the previous time interval, as shown in
\cref{sec:collocation_conditions}.
In three convergence tests we confirmed the results of \cite{Bause2019} and demonstrated
in \cref{sec:Challenging_Example} the powerfulness of the method, when it comes to
problems with high dynamics. In this challenging example the main advantage of this mixed
formulations could be shown: due to the better convergence of higher order
approximations, we could increase the time step size clearly further than with a regular,
pure variational approximation and decrease the computational costs dramatically.
The main drawback of the described method is, that the arising problems have a structure
that has not been thoroughly researched yet. The problems even get more complicated, the
higher the order of the required method is. In \cref{sec:solver_technology} we showed
two possible ways of dealing with this problem. The results of
\cref{sec:Challenging_Example} let us suppose, that the benefits of a higher order in time
approximation outweigh the effort to solve these systems, at least when it comes to
problems with high dynamics. By using a $C^1$ method we could immensely lower the
computational costs.
One task of future research is clearly to find appropriate solvers and preconditioners in
order to deal with the arising problems with good performance. Another field of research
is to apply the described methods to other problems and investigate these arising
structures as well.
In addition, the method gives the possibility to solve higher order in time (partial)
differential equations directly, without transforming them into a first order system
beforehand. By doing this we might be able to get finite element solutions with even less
degrees of freedom, which might mean even less computational costs.
$C^2$ reduced right-hand side:
$\boldsymbol b_r = \biggl(
\tau_n^2 \bigl(3000 \boldsymbol{f}_{n,0} + 600 \boldsymbol{f}_{n,1} + 50 \boldsymbol{f}_{n,2} + 3000 \boldsymbol{f}_{n,3} - 600 \boldsymbol{f}_{n,4} + 50 \boldsymbol{f}_{n,5}\bigr)
- \boldsymbol{A} \Bigl(
\tau_n^2\bigl(
5040 \boldsymbol{u}_{n,0}^0 - 4440 \boldsymbol{u}_{n,0}^D
+ 720 \boldsymbol{u}_{n,1}^0 - 600 \boldsymbol{u}_{n,1}^D
+ 60 \boldsymbol{u}_{n,2}^0 - 50 \boldsymbol{u}_{n,2}^D
- 3000 \boldsymbol{u}_{n,3}^D + 720 \boldsymbol{u}_{n,4}^D - 50 \boldsymbol{u}_{n,5}^D
\bigr)
+ \tau_n^3\bigr(
840 \boldsymbol{v}_{n,0}^0 - 720 \boldsymbol{v}_{n,0}^D
+ 144 \boldsymbol{v}_{n,1}^0 - 144 \boldsymbol{v}_{n,1}^D
+ 12 \boldsymbol{v}_{n,2}^0 - 12 \boldsymbol{v}_{n,2}^D
- 720 \boldsymbol{v}_{n,3}^D + 228 \boldsymbol{v}_{n,4}^D - 24 \boldsymbol{v}_{n,5}^D
\bigr)
+ \tau_n^4 \bigl( 60 \boldsymbol{f}_{n,0} + 12 \boldsymbol{f}_{n,1} + \boldsymbol{f}_{n,2} - 24 \boldsymbol{f}_{n,3} + \boldsymbol{f}_{n,5} \bigr)
\Bigr)
+ \boldsymbol{M} \Bigl(
14400 \boldsymbol{u}_{n,0}^0 - 14400 \boldsymbol{u}_{n,0}^D + 14400 \boldsymbol{u}_{n,3}^D
+ \tau_n \bigl(
14400 \boldsymbol{v}_{n,0}^0 - 13200 \boldsymbol{v}_{n,0}^D + 1440 \boldsymbol{v}_{n,1}^0 - 1440 \boldsymbol{v}_{n,1}^D + 120 \boldsymbol{v}_{n,2}^0 - 120 \boldsymbol{v}_{n,2}^D - 1200 \boldsymbol{v}_{n,3}^D
+ 2880 \boldsymbol{v}_{n,4}^D - 240 \boldsymbol{v}_{n,5}^D
\bigr)
+ \tau_n^2 (
600 \boldsymbol{f}_{n,0} + 120 \boldsymbol{f}_{n,1} + 10 \boldsymbol{f}_{n,2} - 840 \boldsymbol{f}_{n,3} + 10 \boldsymbol{f}_{n,5} + 50 \boldsymbol{f}_{n,5}
)
\Bigr)
+ \boldsymbol{A} \boldsymbol{M}^{-1} \boldsymbol{A} \Bigl(
\tau_n^4 \bigl(60 \boldsymbol{u}_{n,0}^0 + 12 \boldsymbol{u}_{n,1}^0 + \boldsymbol{u}_{n,2}^0 - 84 \boldsymbol{u}_{n,3}^D + 12 \boldsymbol{u}_{n,4}^D\bigr)
- \tau_n^5 \boldsymbol{v}_{n,4}^D
+ \tau_n^6 \boldsymbol{f}_{n,3}
\Bigr)
- \tau_n^6 \boldsymbol{A} \boldsymbol{M}^{-1} \boldsymbol{A}\boldsymbol{M}^{-1} \boldsymbol{A} \boldsymbol{u}_{n,3}^D
\biggr)$.
\end{document} |
\begin{document}
\title{Quantum limits to polarization measurement of classical light}
\author{Marcin Jarzyna}
\affiliation{Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, ul. Banacha 2c, 02-097 Warszawa}
\date{\today}
\begin{abstract}
Polarization of light is one of the fundamental concepts in optics. There are many ways to measure and characterise this feature of light but at the fundamental level it is quantum mechanics that imposes ultimate limits to such measurements. Here, I calculate the quantum limit to a precision of a polarization measurement of classical coherent light. This is a multiparameter estimation problem with a crucial feature of noncommuting optimal observables corresponding to each parameter which prohibits them to be measured at the same time. I explicitly minimize the quantum Holevo-Cramer-Rao bound which tackles this issue and show that it can be locally saturated by two types of conventional receivers.
\end{abstract}
\maketitle
\section{Introduction}
Polarization measurement lies at the core of many scientific and technical endeavors. It is a standard venture in many applications in communication \cite{Han2005, Martinelli2006, Zhou2009, Rosskopf2020, Kikuchi2020}, metrology \cite{Angelsky2009, Salvail2013, Toeppel2014, Goldberg2021} and imaging \cite{Demos1997, Colomb2002, Sattar2020}. Increasing precision of light polarization measurements is of crucial importance for enhancing performance in various aspects of these fields. In order to accomplish this goal one may employ different detection schemes and eliminate various kinds of systematic errors and noise. However, at the fundamental level both light and the detection process are described by the principles of quantum mechanics which is known to impose ultimate limits on the estimation precision for a variety of physical parameters like optical phase, magnetic field strength, light source separation etc. \cite{Giovannetti2006, DemkowiczDobrzanski2012, Baumgratz2016, Tsang2016}. It should be therefore expected that similar bounds can be constructed for the polarization estimation.
The issue of finding ultimate limits to polarization measurement precision has been extensively studied in the literature. For single photon states, which polarization states can be described by a qubit, itself characterized by the position of a point in the Bloch ball, the bounds were found in parameter estimation and tomographic pictures \cite{Rehacek2004, Bagan2006, Ling2006, Salvail2013}. More general states of light and approaches have been considered recently \cite{Klimov2010, Rudnicki2020, Norrman2020, Martin2020, Goldberg2021, Goldberg2021a} indicating plethora of interesting results, such as precision enhancement in the presence of entanglement \cite{Toeppel2014}. Still, however, the exact ultimate bounds on the precision of polarization measurement for many important instances have not been properly studied or are unknown at all.
Polarization estimation is in general an example of a multiparameter estimation problem. As such, a proper care needs to be taken in order to find meaningful bounds on estimation precision. In particular, the multiparameter quantum Cramer-Rao bound (QCRB) \cite{Helstrom1976, Braunstein1994} often used in the literature to ascertain ultimate precision limits allowed by quantum mechanics is known to be in fact not attainable in general. This is because in quantum picture, unlike its classical counterpart, measurements of different physical quantities may not commute with each other, preventing them to be performed simultaneously on the same system. Another issue is the assumption of resources that may not be present in practical realizations, such as global phase reference. Lastly, the bounds itself may just serve as benchmarks of optimal performance but it is an equally vital problem to identify realistic measurement schemes that allow to saturate them. It is therefore of crucial importance to use proper tools, like Holevo-Cramer-Rao bound (HCRB) \cite{Holevo1982}, and keep in mind physical limitations of realistic systems in order to find actual saturable bounds on precision and means to attain them.
In this paper I consider polarization measurements of a classical light. The latter is conventionally characterized by the amplitudes of electromagnetic field in two orthogonal polarization modes. In the quantum picture a classical light state with arbitrary polarization is described by a two mode coherent state $\ket{\alpha_H}_H\otimes\ket{\alpha_V}_V$, where $\alpha_H,\,\alpha_V$ denote complex amplitudes in horizontal and vertical polarization modes $H$, $V$ respectively. The exact state of polarization depends on the absolute values of the amplitudes and the relative phase between them. This description covers all fully polarized classical states. One may also consider partially polarized light which includes an incoherent admixture, however, this is outside the scope of this work.
Polarization of light is usually described in terms of the Stokes vector $\vec{S}=(S_0,S_1,S_2,S_3)$ \cite{Stokes1852, Born1985} which is illustrated in Fig.~\ref{fig:poincare}. For the fully polarized classical light the coordinates $S_j$ can be expressed by the amplitudes $\alpha_{H,V}$ as
\begin{gather}
S_0=|\alpha_H|^2+|\alpha_V|^2,\nonumber\\
S_1=|\alpha_H|^2-|\alpha_V|^2,\nonumber\\
S_2=\alpha_H^*\alpha_V+\alpha_V^*\alpha_H,\nonumber\\
S_3=i(\alpha_H^*\alpha_V-\alpha_V^*\alpha_H).\label{eq:stokes_parameter3}
\end{gather}
Note that $S_0=\sqrt{S_1^2+S_2^2+S_3^2}$ and it is proportional to the average energy carried by the light field. Horizontal and vertical polarization states are obtained for $S_1>0$ and $S_1<0$ respectively while $S_2,\,S_3=0$. Similarly diagonal and antidiagonal polarization occurs whenever $S_2>0$ or $S_2<0$ with $S_1,\,S_3=0$ and the light is left or right circularly polarized for $S_3>0$ and $S_3<0$ with $S_1,\,S_2=0$ respectively. Various other kinds of elliptical polarization states may be obtained in intermediate scenarios.
\begin{figure}
\caption{Poincare ball representing states of polarization. For fully polarized light $S_0$ is the radius of the ball. Poles on axes $S_1,\,S_2,\,S_3$ represent respectively states of vertical-horizontal, diagonal-antidiagonal and right -left circular polarized light, as indicated by the arrows. $\varphi_-=\varphi_H-\varphi_V$ denotes relative phase between the polarization modes.}
\label{fig:poincare}
\end{figure}
In the quantum picture Stokes vector coordinates are promoted to Stokes operators \cite{Jauch1980, Korolkova2002} given by
\begin{gather}
\hat{S}_0=\hat{a}^\dagger\hat{a}+\hat{b}^\dagger\hat{b},\nonumber\\
\hat{S}_1=\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b},\nonumber\\
\hat{S}_2=\hat{a}^\dagger\hat{b}+\hat{b}^\dagger\hat{a},\nonumber\\
\hat{S}_3=i(\hat{a}^\dagger\hat{b}-\hat{b}^\dagger\hat{a}),\label{eq:stokes_operator3}
\end{gather}
where $\hat{a},\hat{a}^\dagger$ and $\hat{b},\hat{b}^\dagger$ denote annihilation and creation operators for the horizontal and vertical light modes respectively. The Stokes parameters in eqs.~(\ref{eq:stokes_parameter3}) are obtained as expectation values of respective observables in eqs.~(\ref{eq:stokes_operator3}) on the quantum state of light $\ket{\alpha_H}\ket{\alpha_V}$. Each such observable has a non-vanishing variance $\Delta S_j^2=S_0$, meaning that polarization components cannot be measured exactly. Crucially, this is not the only limitation set by quantum mechanics on the polarization measurement. This is because observables $\hat{S}_j$ for $j=1,2,3$ do not commute with each other, $[\hat{S}_j,\hat{S}_k]=2i\epsilon_{jkl} \hat{S}_l$, which means that they cannot be measured simultaneously on a given quantum state. Therefore, in general a naive bound on the precision $\Delta S_1^2+\Delta S_2^2+\Delta S_3^2\geq 3S_0$ which one could write using just uncertainties of each component cannot be saturated. Finding the actual attainable lower bound on the precision is the main goal of this manuscript.
\section{Quantum estimation theory}
As mentioned in the previous section, because of noncommutativity of the Stokes operators one cannot simply measure polarization directly and hope for optimal precision. Instead, a detection of some particular quantity, described by a positive operator valued measure (POVM) $\{\Pi_y\}$, needs to be performed first. Next, from the results of the measurement an estimate of the Stokes parameters can be found through estimators $\tilde{S}_j(y),\,j=1,2,3$. In order to find the best estimation precision allowed by the laws of quantum mechanics it is then necessary to optimize the precision over the POVMs and estimators. This task seems difficult at the first glance, but it can be vastly simplified by using tools from quantum estimation theory.
In a general quantum estimation problem \cite{Helstrom1976} one is concerned about estimating a set of parameters $\theta_j$ combined in a vector $\pmb{\theta}=(\theta_1,\theta_2,\dots)$ from a parameter-dependent quantum state $\rho_{\pmb{\theta}}$. For a particular measurement described by a POVM $\Pi_y$ performed on the state the distribution of outcomes is given by $p(y|\pmb{\theta})=\textrm{Tr}(\rho_{\pmb{\theta}}\Pi_y)$. In order to obtain estimates of the parameter values one utilizes an estimator function $\tilde{\pmb{\theta}}(y)=(\tilde{\theta}_1(y),\tilde{\theta}_2(y),\dots)$. The precision of such procedure is quantified by the cost $\Delta\tilde{\pmb{\theta}}^2=\sum_j\Delta\tilde{\theta}_j^2$ i.e. sum of mean squared errors for each parameter $\Delta\tilde{\theta}_j^2=\sum_y p(y|\pmb{\theta})(\tilde{\theta}_j(y)-\theta_j)^2$. In such setting the lower bound on precision for any unbiased estimator is given by the Cramer-Rao bound \cite{Kay1993}
\begin{equation}\label{eq:class_CR}
\Delta\tilde{\pmb{\theta}}^2\geq \textrm{Tr} \left(\mathcal{F}^{-1}\right),
\end{equation}
where $\mathcal{F}$ is called the Fisher information matrix and equal to
\begin{equation}\label{eq:class_FI}
\mathcal{F}_{jk}=\sum_y\frac{1}{p(y|\pmb{\theta})}\frac{\partial p(y|\pmb{\theta})}{\partial\theta_j}\frac{\partial p(y|\pmb{\theta})}{\partial\theta_k}.
\end{equation}
This bound can be in principle saturated by a maximum likelihood estimator.
Importantly, the bound in \eqnref{eq:class_CR}, is optimized only over estimators and depends on the measurement. Therefore, in order to find the best possible precision one needs to additionally maximize the Fisher information matrix over the POVMs. This is a hard task, so in order to avoid this cumbersome optimization one often uses a quantum Cramer-Rao bound (QCRB)\cite{Helstrom1976, Braunstein1994}
\begin{equation}\label{eq:quantum_CR}
\Delta\tilde{\pmb{\theta}}^2\geq \textrm{Tr}\left(\mathcal{F}^{-1}\right)\geq \textrm{Tr}\left(\mathcal{Q}^{-1}\right),
\end{equation}
where $Q$ is the quantum Fisher information matrix equal to
\begin{equation}\label{eq:quantum_FI}
\mathcal{Q}_{jk}=\frac{1}{2}\textrm{Tr}[\rho_{\pmb{\theta}}(L_j L_k+L_k L_j)].
\end{equation}
Operators $L_j$ are called symmetric logarithmic derivatives (SLD) and are implicitly defined as solutions to the equation
\begin{equation}
\frac{\partial \rho_{\pmb{\theta}}}{\partial\theta_j}=\frac{1}{2}(\rho_{\pmb{\theta}}L_j+L_j\rho_{\pmb{\theta}}).
\end{equation}
The quantum Cramer-Rao bound for single parameter problems, i.e. when $\pmb{\theta}=\theta$, may always be saturated by performing a projective measurement in the eigenbasis of SLD operators. However, if there is more than one parameter, finding the optimal POVM posses a challenge since if any two SLDs do not commute with each other then it is not possible to find their common eigenbasis. Therefore the bound in \eqnref{eq:quantum_CR} is in general not attainable.
The actual saturable lower bound for multiparameter quantum estimation problem is given by the Holevo-Cramer-Rao bound (HCRB) \cite{Holevo1982, Nagaoka1989, Ragy2016, Demkowicz2020}, which unfortunately does not enjoy any kind of elegant and simple formula like the QCRB in \eqnref{eq:quantum_FI}. The expression for HCRB reads
\begin{equation}\label{eq:HCRB}
\Delta\tilde{\pmb{\theta}}^2\geq \min_{\{X_j\}}\{\textrm{Tr} (\textrm{Re} Z[X])+||\textrm{Im}Z[X]||_1\},
\end{equation}
where the matrix $Z[X]_{jk}=\textrm{Tr}(\rho_{\pmb{\theta}}X_j X_k)$, $||A||_1=\textrm{Tr}\sqrt{A^\dagger A}$ is the trace norm and the minimization is performed over all Hermitian matrices satisfying $\textrm{Tr}(X_j\partial_{\theta_k}\rho_{\pmb{\theta}})=\delta_{jk}$ and $\textrm{Tr}(X_j\rho_{\pmb{\theta}})=0$. The QCRB and HCRB are equivalent if and only if the commutators of all SLDs evaluated on the state vanish $\textrm{Tr}(\rho_{\pmb{\theta}}[L_j,L_k])=0$. It is known that QCRB results in a bound that is at most twice as large as HCRB \cite{Carollo2019, Tsang2020}. Importantly, if one is interested in estimating quantities $\pmb{\theta}'$ which are given by a function of the original parameters $\pmb{\theta}'=\pmb{f}(\pmb{\theta})$ the HCRB transforms according to the rules $X'=JX$ and $Z[X]'=J Z[X] J^T$ where $J_{jk}=\partial \theta_j'/\partial \theta_k$ is the Jacobian matrix.
A crucial feature of the HCRB bound is that it takes care of the issue of noncommutativity of the optimal observables corresponding to different parameters. A profound consequence of this fact is the existence of the so called nuisance parameters \cite{Yang2019, Suzuki2020, Suzuki2020a, Demkowicz2020}. These are any unobserved, i.e. not estimated, parameters in the model for which corresponding observables do not commute with the ones related to the estimated quantities. Because of the noncommutativity, the existence of nuisance parameters can affect the form of the optimal observable for the observed parameters and the precision bound for them. In multiparameter estimation problems it is therefore important to carefully analyze all the parameters describing the state, even if they are not being estimated in a particular scenario, in order to find valid bounds on precision.
\section{HCRB for polarization estimation}
\label{sec:hcrb}
In the case of polarization measurement the Stokes parameters in \eqnref{eq:stokes_parameter3} depend on the amplitudes $\alpha_H,\,\alpha_V$ of the polarization modes. One could therefore naively conclude that an equivalent estimation problem is to estimate both complex amplitudes. The optimal measurement for the latter problem is known to be a double homodyne measurement of each polarization mode \cite{Holevo1982}. The bound on the polarization estimation precision can be then shown to be equal to $6S_0$ \cite{Kikuchi2020, Mecozzi2021}, twice as much as would follow from separate optimal measurement of each Stokes parameter and summing up the precisions. However, by writing coherent states amplitudes explicitly as $\alpha_H=|\alpha_H| e^{i\varphi_H}$ and $\alpha_V=|\alpha_V|e^{i\varphi_V}$ the Stokes parameters can be expressed as
\begin{gather}
S_0=|\alpha_H|^2+|\alpha_V|^2,\nonumber\\
S_1=|\alpha_H|^2-|\alpha_V|^2,\nonumber\\
S_2=2|\alpha_H||\alpha_V|\cos\varphi_-,\nonumber\\
S_3=2|\alpha_H||\alpha_V|\sin\varphi_-,\label{eq:stokes_parameter_ph3}
\end{gather}
where $\varphi_-=\varphi_H-\varphi_V$ is the relative phase between the polarization modes. It can be seen in eqs.~(\ref{eq:stokes_parameter_ph3}), that only the absolute values of the amplitudes and the relative phase between the modes enter the expression for the Stokes parameters. One can therefore rephrase estimation of $\vec{S}$ as estimation of just three real parameters $(|\alpha_H|,|\alpha_V|,\varphi_-)$ rather than the full complex amplitudes $\alpha_H,\alpha_V$. On the other hand, in order to fully describe the quantum state $\ket{\alpha_H}_H\otimes\ket{\alpha_V}_V$ one needs to add to those three quantities also the global phase $\varphi_+=\varphi_H+\varphi_V$. It is well known that amplitude, or more likely the number of photons, and phase exhibit a Heisenberg-like uncertainty relation \cite{Smithey1993}, meaning that their corresponding observables are not commuting. This strongly suggests that $\varphi_+$ is a nontrivial nuisance parameter in the polarization estimation problem and that double homodyne detection is in principle suboptimal for polarization measurement. Another important problem is that in order to meaningfully consider global phase one needs to posses an external reference beam with respect to which this phase is defined \cite{Molmer1997, Jarzyna2012}.
The detailed calculations leading to HCRB on polarization estimation precision are quite complicated and given in the appendix \ref{app:hcrb}. Importantly, the inclusion of a global phase as an unobserved nuisance parameter manifests in an additional constraint on the form of matrices $X_j$ in \eqnref{eq:HCRB}
\begin{equation}\label{eq:XH0const1}
\textrm{Re}\left[\bra{\alpha_H}\bra{\alpha_V}X_j\,\partial_{\varphi_+}\left(\ket{\alpha_H}\ket{\alpha_V}\right)\right]=0,
\end{equation}
for $j=|\alpha_H|,|\alpha_V|,\varphi_-$. The presence of the above constraint completely fixes the form of matrices $X_j$, removing the need of optimization in\eqnref{eq:HCRB}. The final bound, after rephrasing it in terms of Stokes coefficients reads
\begin{equation}\label{eq:HCRB_pol}
\Delta S_1^2+\Delta S_2^2 +\Delta S_3^2\geq 5S_0.
\end{equation}
Crucially, the above expression lies below the value of $6S_0$ attainable by double homodyne detection and is the lower limit imposed by the laws of quantum mechanics. The limit in \eqnref{eq:HCRB_pol} has been conjectured recently in \cite{Mecozzi2021}, where a concrete estimation scheme locally saturating this bound was found. Note also, that the value of HCRB in \eqnref{eq:HCRB_pol} firmly lies in the region allowed by the QCRB $\textrm{Tr}(\mathcal{Q}^{-1})\leq \textrm{Tr}(C_{HCRB})\leq 2\textrm{Tr}(\mathcal{Q}^{-1})$ with $\textrm{Tr}(\mathcal{Q}^{-1})=3S_0$, as discussed in appendix~\ref{app:qfi}.
An important instance of polarization estimation is the case in which the total power, $S_0$, is known. In such a case one can use the fact that $|\alpha_V|=\sqrt{S_0-|\alpha_H|^2}$ which means that there are only two parameters to estimate $(|\alpha_H|,\,\varphi_-)$, still, however, one must include global phase as a nuisance parameter. The precision bound in this scenario is given by
\begin{equation}\label{eq:HCRB_const}
\Delta S_1^2+\Delta S_2^2+\Delta S_3^2\geq4S_0.
\end{equation}
The detailed derivation is presented in appendix \ref{app:hcrb_s0}. This bound is lower than the one in \eqnref{eq:HCRB_pol} which is not surprising since one estimates only two instead of three parameters. It is also two times worse than the corresponding QCRB, which in turn is equal to $\textrm{Tr}(\mathcal{Q}^{-1})=2S_0$, see Appendix~\ref{app:qfi}.
\section{Attaining the HCRB}
\label{sec:stokes}
A nontrivial question in many estimation tasks is what is the measurement that saturates the HCRB bound. This problem has a solution for a general estimation procedure for pure states \cite{Matsumoto2002}, however, the resulting POVM is often of limited practical use as it is complicated and its physical realization is rather difficult to obtain. In this work I will take a different approach, that is, I will consider a specific POVM and show that it attains the HCRB bound in \eqnref{eq:HCRB_pol}. The measurement I will mainly consider is a standard Stokes measurement \cite{Kikuchi2020}, presented schematically in Fig.~\ref{fig:stokes}(a). The setup first uniformly splits the signal into three separate light beams. The first beam then travels undisturbed while the second one passes through a $45^\circ$ polarization rotator, turning the polarization to a diagonal-antidiagonal basis. In the case of the third beam the rotator is additionally preceded by a quater-waveplate which sets the polarization basis to a left and right circular. After these operations each beam is separately split by a polarizing beam splitter and photon number resolving detectors measure numbers of photons in each output arm of the respective system.
\begin{figure}
\caption{Schemes of Stokes (a) and tetrahedron (b) detection schemes. Here $\alpha_\pm$ and $\alpha_{R,L}
\label{fig:stokes}
\end{figure}
In order to find what is the best precision attained by a given measurement one can calculate the Fisher information matrix in \eqnref{eq:class_FI} and evaluate the cost. The bound on estimation precision utilizing the Stokes receiver is given by
\begin{equation}\label{eq:stokes_bound}
\Delta S_1^2+\Delta S_2^2 +\Delta S_3^2\geq \frac{11}{2}S_0-\frac{9}{2S_0}\left(\frac{1}{S_1^2}+\frac{1}{S_2^2}+\frac{1}{S_3^2}\right)^{-1},
\end{equation}
and the detailed calculations can be found in appendix~\ref{app:crb}. It can be seen in Fig.~\ref{fig:kulki}(a) that the expression in \eqnref{eq:stokes_bound} lies between the values of HCRB $\Delta\vec{S}^2=5S_0$ from \eqnref{eq:HCRB_pol} attained for $S_1=S_2=S_3=S_0/\sqrt{3}$ and $\Delta\vec{S}^2=\frac{11}{2}S_0$ which can be obtained when one of the coordinates $S_1,S_2,S_3$ vanishes. Clearly, the bound for Stokes measurement depends on the Stokes parameters and one can achieve the HCRB only locally. This can be in principle overcome if one allows for an additional rotation operation in front of the receiver which would adjust the Stokes vector to have all three coordinates equal. However, from a practical point of view this would require prior knowledge of the signal polarization which means that only small deviations from an otherwise known value can be measured. Note that such parameter dependence of the measurement is usually a feature of general POVMs optimizing the HCRB bound for most estimation problems \cite{Demkowicz2020}. On the other hand even in the worst case scenario pure Stokes receiver attains precision worse than HCRB by only 10\% so the prior knowledge is rather not critical.
In the scenario when the total signal power is fixed the conventional Stokes receiver attains precision lower bounded by
\begin{equation}\label{eq:Stokes_const}
\Delta\vec{S}^2\geq \frac{9}{2S_0}\frac{\left(1+\frac{S_3^2}{S_2^2}\right)\left(1+\frac{S_2^2}{S_1^2}\right)\left(1+\frac{S_1^2}{S_3^2}\right)}{\frac{1}{S_1^2}+\frac{1}{S_2^2}+\frac{1}{S_3^2}}.
\end{equation}
It can be seen in Fig.~\ref{fig:kulki} that for $S_1=S_2=S_3=S_0/\sqrt{3}$ the bound in \eqnref{eq:Stokes_const} saturates HCRB in \eqnref{eq:HCRB_const}. On the other hand whenever one of the Stokes coordinates is vanishing \eqnref{eq:Stokes_const} attains its worst value $9S_0/2$, worse by $12.5\%$ than the corresponding HCRB. Interestingly, note that even the worst case scenario still predicts better precision than the optimal HCRB in the unconstrained case in \eqnref{eq:HCRB}. Knowledge of the total signal power can therefore considerably increase the polarization measurement performance.
Importantly, it was recently shown \cite{Mecozzi2021} that the Stokes receiver, together with a particular estimation procedure can indeed reach the value of $5S_0$ and $4S_0$ precision locally in the general and constrained power cases respectively. Based on this it was conjectured that these values represent quantum limits on polarization measurements. However, the employed estimator in the general scenario attains only $7S_0$ in the worst case instance, which does not saturate the classical Cramer-Rao bound for the Stokes measurement. Although it is known that CRB can be always saturated by the maximum likelihood estimator \cite{Kay1993} the question remains if there exist any simple estimation procedure that would allow to attain the bounds in \eqnsref{eq:stokes_bound}{eq:Stokes_const} everywhere.
\begin{figure}
\caption{Precision attained as a function of the Stokes vector direction for Stokes receiver in the general case (a), Stokes receiver in the case of known total power $S_0$ (b), tetrahedron receiver in the general case (c), tetrahedron receiver for the known $S_0$ (d). }
\label{fig:kulki}
\end{figure}
Another type of measurement scheme frequently considered in the context of polarization measurement that may seem promising is the so called tetrahedron measurement. This is inspired by a single photon quantum tomography case in which quantum states can be characterized in the Bloch ball picture, isomorphic to the Poincare sphere for polarization \cite{Rehacek2004, Ling2006}. In tetrahedron measurement, presented schematically in Fig.~\ref{fig:stokes}(b), the incoming light is first divided on a partially polarizing beam splitter (PPBS) such that the state evolves according to $\ket{\alpha_H}\ket{\alpha_V}\to(\ket{x\alpha_H}\ket{y\alpha_V})\otimes(\ket{y\alpha_H}\ket{x\alpha_V})$, where $x,y$ are the transmissivity and reflectivity of the beam splitter and brackets indicate separate beams leaving the device. The first beam is then rotated by $45^\circ$ to the diagonal-antidiagonal basis and the second one travels both through a quarter wave plate and a $45^\circ$ rotator in order to end up in right-left circular polarized basis. Each beam is then split by a polarizing beam splitter and one performs photon number measurements on the resulting output ports. Importantly, one can optimize this scheme over the coefficients $x,y$ of the first beam splitter in order to get the optimal performance. The resulting precision bound is quite complicated, but it can be found that
\begin{equation} \label{eq:stokes_tetra}
5S_0\leq\Delta S_1^2+\Delta S_2^2+\Delta S_3^2\leq (4+2\sqrt{2}) S_0
\end{equation}
in the general case and
\begin{equation}\label{eq:HCRB_tet}
4S_0 \leq \Delta S_1^2+\Delta S_2^2+\Delta S_3^2\leq (3+2\sqrt{2})S_0
\end{equation}
for the constrained total power scenario. In both instances the minimum is attained when one of the Stokes coordinates is equal to $S_0$ and others vanish whereas the maximal value is obtained for $S_1=0, S_2=\pm S_0/\sqrt{2}, S_3=\pm S_0/\sqrt{2}$. Note that \eqnsref{eq:stokes_tetra}{eq:HCRB_tet} assume optimal transmission coefficients $x,y$ of the first beam splitter for each value of the Stokes parameters. For both considered polarization estimation scenarios such an optimized tetrahedron detector can locally saturate respective HCRBs in \eqnref{eq:HCRB_pol} and \eqnref{eq:HCRB_const}. Moreover, as can be seen in Fig.~\ref{fig:kulki}(c)(d), it offers precision very close to the ultimate bounds for a considerably larger region of Stokes coefficients than the Stokes receiver, although with a caveat of weaker performance in the worst case scenario. Importantly, one can also saturate respective HCRBs using tetrahedron detection scheme with a fixed non-optimized value of $x$, although this can be accomplished just for a limited range of Stokes parameters.
\section{HCRB in the presence of reference frame}
In most conventional scenarios one usually does not have knowledge about the global phase of the signal. This is because the phase is a relative concept and requires presence of some external phase reference, such as local oscillator, in order to be physically meaningful \cite{Molmer1997}. However, in the context of polarization estimation an interesting question is what would be the bound on precision of Stokes vector measurement had prior knowledge on the global phase been included. An example of such situation would be if one had access to additional reference beam and performed some kind of coherent measurement on a small fraction of the light before the actual measurement of Stokes coefficients. This would enable global phase estimation which could then be used to boost Stokes vector estimation.
Let me assume that the experimentalist knows the global phase. In such case the state entering the detection apparatus can be described by only three parameters $\{|\alpha_H|,|\alpha_V|,\varphi_-\}$ as $\ket{\psi}=\ket{|\alpha_H|e^{i\varphi_-/2}}\ket{|\alpha_V|e^{-i\varphi_-/2}}$, where I choose global phase to be equal to $\varphi_+=0$. Since global phase is known to the receiver and one estimates all other quantities describing the state there is no nuisance parameter. Therefore in order to find HCRB it is necessary to perform calculations without additional constraint in \eqnref{eq:XH0const1}. The resulting matrices in \eqnref{eq:HCRB} have free variables over which they have to be optimized, resulting in a complicated expression for HCRB. Numerical calculations show that HCRB takes values in the range given by the inequality
\begin{equation}\label{eq:HCRB_phase}
2S_0\leq \Delta S_1^2+\Delta S_2^2+\Delta S_3^2\leq 5S_0.
\end{equation}
The left hand side value is saturated for $S_2=S_3=0$, i.e. horizontally or vertically polarized light, whereas the maximum value on the right hand side is attained for $S_1=0$, i.e. Stokes vector lying in a plane spanned by $S_2,S_3$. If the total power is known in advance the above bound can be improved by an additional factor of $S_0$ subtracted on both sides.
\begin{figure}
\caption{HCRB for Stokes vector estimation as a function of Stokes vector direction in the presence of prior knowledge about the global phase in the general case of unknown $S_0$ (a) and known $S_0$ (b).}
\label{fig:hcrb_phase}
\end{figure}
It is seen in Fig.~\ref{fig:hcrb_phase} that precision in the presence of prior knowledge about global phase posses a cylindrical symmetry and depends just on the Stokes vector inclination. Interestingly, for some directions of the Stokes vector, the bound is substantially improved with respect to the case of unknown global phase. A similar effect is present in quantum interferometry, where it is known that presence of an external reference beam, serving as a global phase reference, can change achievable bounds on precision of the relative phase delay inside the interferometer \cite{Jarzyna2012, Ataman2020}. Note also, that HCRB in \eqnref{eq:HCRB_phase} in general is not equal to corresponding QCRB in appendix \ref{app:qfi} except at the poles $S_1=\pm S_0$. This means that at these two points one is able to saturate HCRB with conventional measurement in the eigenbasis of SLD. Finally, in contrast to previous cases described in Sec.~\ref{sec:stokes}, classical Stokes receiver or tetrahedron measurement cannot saturate HCRB bound in \eqnref{eq:HCRB_phase}. This is because measurement of the number of photons, which is eventually performed in each arm of these receivers, is insensitive to the global phase. One is therefore forced to use a more sophisticated measurement utilizing a reference beam.
\section{Conclusion}
In conclusion I have showed that the ultimate quantum limit on Stokes vector estimation is equal to $5S_0$, where $S_0$ is the average number of photons in the signal. This bound can be attained locally using conventional Stokes or tetrahedron receivers utilizing photon number resolving detectors. In the case when the total power of the signal is known in advance the bound can be improved to $4S_0$ which is also saturable with Stokes and tetrahedron receivers. Interestingly, these bounds apply in the case in which global phase is unknown to the experimentalist, even though Stokes coefficients do not depend on this parameter. In the opposite regime in which the global phase is an element of prior knowledge the quantum precision bound depends on the signal polarization and in the best case scenario can be improved up to $2S_0$ in the general case and $S_0$ when the total power is known. These latter limits, however, make any physical sense only in the presence of an external reference frame. An interesting question is if there exist any realistic procedure that allows to attain these bounds. Another important problem that remains to be solved is to find the quantum limit on polarization estimation precision in the case of non fully polarized light and for more general quantum states.
\acknowledgements
I acknowledge insightful discussions with R. Demkowicz-Dobrza\'{n}ski and M. Karpi\'{n}ski. This work was supported by the Foundation for Polish Science under the â€Quantum Optical Technologies†project carried out within the International Research Agendas programme co-financed by the European Union under the European Regional Development Fund.
\appendix
\section{HCRB for polarization estimation}
\label{app:hcrb}
In order to find the HCRB for polarization measurement it is convenient to parametrize the quantum states by global and relative phases and absolute values of the amplitudes $\ket{|\alpha_H| e^{\frac{i}{2}(\varphi_++\varphi_-)}}\otimes\ket{|\alpha_V| e^{\frac{i}{2}(\varphi_+-\varphi_-)}}$. As noted in sec.~\ref{sec:hcrb}, the parameters one wants to estimate are $\pmb{\theta}=(|\alpha_H|,|\alpha_V|,\varphi_-)$ with $\varphi_+$ being the nuisance parameter that affects the HCRB bound due to noncommutativity of optimal observables. A first step to obtain the matrix $Z[X]$ in \eqnref{eq:HCRB} is to find the orthonormal basis in which matrices $X_j$ can be written. In general, if the state $\rho=\ket{\psi}\bra{\psi}$ is pure, as is the case for coherent states considered in this paper, the constraints on $X_j$ are given by
\begin{equation}\label{eq:X_constr}
\bra{\psi}X_j\ket{\psi}=0,\quad \bra{\partial_{\theta_j}\psi}X_k\ket{\psi}+\bra{\psi}X_k\ket{\partial_{\theta_j}\psi}=\delta_{jk},
\end{equation}
where $\ket{\partial_{\theta_j}\psi}=\partial_{\theta_j}\ket{\psi}$ denotes the derivative of state with respect to parameter $\theta_j$. Eq.~(\ref{eq:X_constr}) suggests that a useful orthonormal basis is the one characterizing the subspace spanned by vectors $\ket{\psi}$ and $\{\ket{\partial_{\theta_j}\psi}\}$. In order to find this basis one needs to perform Gram-Schmidt orthogonalization of the set of vectors spanning this subspace. Taking $\ket{\psi}=\ket{\alpha_H}\otimes\ket{\alpha_V}$ and parameters $|\alpha_H|,|\alpha_V|,\varphi_-,\varphi_+$ one obtains then the basis
\begin{gather}\label{eq:e0}
\ket{e_0}=\ket{\alpha_H}\otimes\ket{\alpha_V},\\
\ket{e_1}=\partial_{|\alpha_H|}(\ket{\alpha_H}\otimes\ket{\alpha_V})=\ket{\partial_{|\alpha_H|}\alpha_H}\otimes\ket{\alpha_V},\\
\ket{e_2}=\partial_{|\alpha_V|}(\ket{\alpha_H}\otimes\ket{\alpha_V})=\ket{\alpha_H}\otimes\ket{\partial_{|\alpha_V|}\alpha_V}.\label{eq:e2}
\end{gather}
Note that this basis has only three elements, meaning that the subspace spanned by $\{\ket{\alpha_H}\otimes\ket{\alpha_V},\partial_{|\alpha_H|}(\ket{\alpha_H}\otimes\ket{\alpha_V}),\partial_{|\alpha_V|}(\ket{\alpha_H}\otimes\ket{\alpha_V}),\partial_{\varphi_-}(\ket{\alpha_H}\otimes\ket{\alpha_V}),\partial_{\varphi_+}(\ket{\alpha_H}\otimes\ket{\alpha_V})\}$ is actually three dimensional. Therefore, vectors $\partial_{\varphi_-}(\ket{\alpha_H}\otimes\ket{\alpha_V})$ and $\partial_{\varphi_+}(\ket{\alpha_H}\otimes\ket{\alpha_V})$ are linear combinations of vectors found in \eqnsref{eq:e0}{eq:e2}
\begin{multline}
\partial_{\varphi_\pm}(\ket{\alpha_H}\otimes\ket{\alpha_V})=\\=\frac{i}{2}(|\alpha_H|^2\pm|\alpha_V|^2)\ket{e_0}+\frac{i|\alpha_H|}{2}\ket{e_1}\pm\frac{i|\alpha_V|}{2}\ket{e_2}.
\end{multline}
The constraints on $X_{|\alpha_H|}$ due to derivatives with respect to the estimated parameters $|\alpha_H|,\,|\alpha_V|,\varphi_-$ from \eqnref{eq:X_constr} can be therefore written as
\begin{align}
&(X_{|\alpha_H|})_{00}=0,\quad 2\textrm{Re}(X_{|\alpha_H|})_{01}=1, \quad 2\textrm{Re}(X_{|\alpha_H|})_{02}=0, \nonumber\\ &|\alpha_H|\textrm{Im}(X_{|\alpha_H|})_{01}-|\alpha_V|\textrm{Im}(X_{|\alpha_H|})_{02}=0.
\label{eq:cons}
\end{align}
Based on the above equations operator $X_{|\alpha_H|}$ has a following form
\begin{equation}\label{eq:X_H0}
X_{|\alpha_H|}=\left(
\begin{array}{c c c}
0 & \frac{1}{2}+ib & i\frac{|\alpha_H|}{|\alpha_V|}b\\
\frac{1}{2}-ib & 0 & 0\\
-i\frac{|\alpha_H|}{|\alpha_V|}b & 0 &0
\end{array}\right),
\end{equation}
where $b\in\mathbb{R}$ is a free parameter over which one has to optimize the HCRB. The matrix elements not constrained by \eqnref{eq:cons} were set to $0$ since they do not enter the expression for $Z[X]$. As mentioned above, the global phase, as a nuisance parameter, also has an impact on the form of $X_j$ due to noncommutativity. This manifests as an additional constraint corresponding to a derivative with respect to $\varphi_+$ which reads
\begin{equation}\label{eq:XH0const_ap}
|\alpha_H|\textrm{Im}(X_{|\alpha_H|})_{01}+|\alpha_V|\textrm{Im}(X_{|\alpha_H|})_{02}=0.
\end{equation}
The above equation further constraints the form of $X_{|\alpha_H|}$ in \eqnref{eq:X_H0} and a unique solution can be found, equal to
\begin{equation}\label{eq:X_H}
X_{|\alpha_H|}=\left(
\begin{array}{c c c}
0 & \frac{1}{2} & 0\\
\frac{1}{2} & 0 & 0\\
0 & 0 &0
\end{array}\right),
\end{equation}
with $b=0$ in \eqnref{eq:X_H0}. Using similar method, one can obtain that operators $X_{|\alpha_V|}$ and $X_{\varphi_-}$ are given by
\begin{equation}\label{eq:X_V}
X_{|\alpha_V|}=\left(
\begin{array}{c c c}
0 & 0 & \frac{1}{2}\\
0 & 0 & 0\\
\frac{1}{2} & 0 &0
\end{array}\right),
\end{equation}
and
\begin{equation}\label{eq:X_ph}
X_{\varphi_-}=\left(
\begin{array}{c c c}
0 & -\frac{i}{2|\alpha_H|} & \frac{i}{2|\alpha_V|}\\
\frac{i}{2|\alpha_H|} & 0 & 0\\
-\frac{i}{2|\alpha_V|} & 0 &0
\end{array}\right).
\end{equation}
Note that none of the operators $X_j$ posses any kind of free parameter over which one could perform optimization in HCRB. This is because the constraint originating from the lack of knowledge about global phase $\textrm{Tr}[\partial_{\varphi_+}\rho X_j]=0$ fixes any kind of freedom in the choice of $X_j$'s.
Using \eqnsref{eq:X_H}{eq:X_ph} the matrix $Z[X]$ is equal to
\begin{equation}\label{eq:zx}
Z[X]=\frac{1}{4}\left(
\begin{array}{c c c}
1 & 0 & \frac{i}{|\alpha_H|}\\
0 & 1 & -\frac{i}{|\alpha_V|}\\
-\frac{i}{|\alpha_H|} & \frac{i}{|\alpha_V|} &\frac{1}{|\alpha_H|^2}+\frac{1}{|\alpha_V|^2}
\end{array}\right).
\end{equation}
In order to find the HCRB bound for polarization one can now transform \eqnref{eq:zx} through Jacobian of the transformation in eqs.~(\ref{eq:stokes_parameter_ph3}) of the main text
\begin{equation}\label{eq:jacobian}
J=2\left(
\begin{array}{c c c}
|\alpha_V|\cos\varphi_- & |\alpha_H|\cos\varphi_- & -|\alpha_H||\alpha_V|\sin\varphi_- \\
|\alpha_V|\sin\varphi_- & |\alpha_H|\sin\varphi_- & |\alpha_H||\alpha_V|\cos\varphi_-\\
|\alpha_H| & -|\alpha_V| & 0
\end{array}\right),
\end{equation}
resulting in
\begin{equation}\label{eq:zxj}
JZ[X]J^T=\left(
\begin{array}{c c c}
S_0 & -i S_1 & iS_3\\
i S_1 & S_0 & -iS_2\\
- iS_3& iS_2 &S_0
\end{array}\right).
\end{equation}
Plugging \eqnref{eq:zxj} into HCRB one obtains that precision is lower bounded by
\begin{equation}\label{eq:HCRB_pol_ap}
\Delta S_1^2+\Delta S_2^2 +\Delta S_3^2\geq 5S_0.
\end{equation}
In order to find the HCRB in the presence of prior knowledge about the global phase the only change in the above reasoning is to drop the constraint in \eqnref{eq:XH0const_ap}. In such a case the resulting matrices $X_j$ will have free variables, such as in \eqnref{eq:X_H0}, over which one has to optimize in \eqnref{eq:HCRB}.
\section{HCRB for constrained total power}
\label{app:hcrb_s0}
If the total power of light, $S_0$, is known a priori, one can use the relation $|\alpha_V|=\sqrt{S_0-|\alpha_H|^2}$ which means that there are only two parameters to estimate $(|\alpha_H|,\,\varphi_-)$. Similarly as in the general case, in order to find HCRB in the first step one needs to find an orthonormal basis of the space spanned by $\ket{\psi},\,\ket{\partial_{|\alpha_H|}\psi},\,\ket{\partial_{\varphi_-}\psi},\,\ket{\partial_{\varphi_+}\psi}$ which is equal to
\begin{gather}\label{eq:e0_const}
\ket{e_0}=\ket{\psi},\\
\ket{e_1}=\sqrt{\beta}\ket{\partial_{|\alpha_H|}\psi},\\
\ket{e_2}=\frac{\sqrt{S_0}}{\gamma}\left[\ket{\partial_{\varphi_-}\psi}-i\gamma\ket{\psi}-i\beta|\alpha_H|\ket{\partial_{|\alpha_H|}\psi}\right],\label{eq:e2_const}
\end{gather}
where $\beta=1-\frac{|\alpha_H|^2}{S_0}$ and $\gamma=|\alpha_H|^2-\frac{S_0}{2}$. As can be seen this subspace is three dimensional and the derivative with respect to the global phase can be expressed by linear combination of vectors in \eqnsref{eq:e0_const}{eq:e2_const}
\begin{equation}
\ket{\partial_{\varphi_+}\psi}=\frac{1}{2}\left(\sqrt{S_0}\ket{e_2}+iS_0\ket{e_0}\right).
\end{equation}
Therefore, the constraints on $X_{|\alpha_H|}$ originating from the estimated parameters are given by
\begin{align}
(X_{|\alpha_H|})_{00}=0,\quad 2\textrm{Re}(X_{|\alpha_H|})_{01}=\sqrt{\beta}, \nonumber\\
\frac{\gamma}{\sqrt{S_0}}\textrm{Re}(X_{|\alpha_H|})_{02}-\sqrt{\beta}|\alpha_H|\textrm{Im}(X_{|\alpha_H|})_{01}=0,
\end{align}
whereas the constraint coming from the global phase reads
\begin{equation}\label{eq:const_s0}
2\textrm{Re}(X_{|\alpha_H|})_{02}=0.
\end{equation}
A general solution to these equations is given by
\begin{equation}\label{eq:X_H_S0}
X_{|\alpha_H|}=\left(
\begin{array}{c c c}
0 & \frac{\sqrt{\beta}}{2} & -i b\\
\frac{\sqrt{\beta}}{2} & 0 & 0\\
i b & 0 &0
\end{array}\right),
\end{equation}
where $b\in\mathbb{R}$ is a free parameter. Similarly one obtains
\begin{equation}\label{eq:X_p_S0}
X_{\varphi_-}=\left(
\begin{array}{c c c}
0 & -\frac{i}{2|\alpha_H|\sqrt{\beta}} & -i c\\
\frac{i}{2|\alpha_H|\sqrt{\beta}} & 0 & 0\\
i c & 0 &0
\end{array}\right),
\end{equation}
where once again $c\in\mathbb{R}$ is a free parameter. The resulting $Z[X]$ matrix in HCRB is equal to
\begin{equation}
Z[X]=\left(
\begin{array}{c c}
\frac{\beta}{4}+b^2 & \frac{i}{4|\alpha_H|} +bc\\
-\frac{i}{4|\alpha_H|}+bc & \frac{1}{4|\alpha_H|^2\beta}+c^2
\end{array}\right).
\end{equation}
The transformation between Stokes coordinates and coherent states parameters is given by
\begin{equation}
|\alpha_H|=\sqrt{\frac{S_0+S_1}{2}},\quad
\varphi_-=\arctan\frac{S_3}{S_2},
\end{equation}
which results in a Jacobian
\begin{equation}\label{eq:jacobian_s0}
J=2\left(
\begin{array}{c c}
\frac{S_0-2|\alpha_H|^2}{\sqrt{S_0-|\alpha_H|^2}}\cos\varphi_- & -|\alpha_H|\sqrt{S_0-|\alpha_H|^2}\sin\varphi_-\\
\frac{S_0-2|\alpha_H|^2}{\sqrt{S_0-|\alpha_H|^2}}\sin\varphi_- & |\alpha_H|\sqrt{S_0-|\alpha_H|^2}\cos\varphi_-\\
2|\alpha_H| & 0
\end{array}\right).
\end{equation}
After reparametrization the HCRB has to be optimized over $b$ and $c$ resulting in optimal values $b=c=0$. The precision bound is then given by
\begin{equation}\label{eq:HCRB_const_app}
\Delta S_1^2+\Delta S_2^2+\Delta S_3^2\geq4S_0.
\end{equation}
Similarly as in app.~\ref{app:hcrb}, in order to find the HCRB in the presence of prior knowledge about the global phase, one needs to repeat the calculations without the constraint in \eqnref{eq:const_s0} and optimize the resulting expression in \eqnref{eq:HCRB} over all free variables.
\section{Quantum Cramer-Rao bounds}
\label{app:qfi}
In order to find QCRB one needs to first calculate SLDs for all parameters. In a general quantum multiparameter estimation model with pure states $\ket{\psi_{\pmb{\theta}}}$ SLD for the $j$-th parameter is equal to
\begin{equation}
L_{\theta_j}=2(\ket{\partial_{\theta_j}\psi}\bra{\psi}+\ket{\psi}\bra{\partial_{\theta_j}\psi}).
\end{equation}
Assuming one posses knowledge or access to the global phase, the state in the polarization measurement problem is given by $\ket{\psi_{\pmb{\theta}}}=\ket{|\alpha_H|e^{i\varphi_-/2}}\ket{|\alpha_V|e^{-i\varphi_-/2}}$, where I have chosen global phase to be equal to $\varphi_+=0$. One can easily calculate resulting SLDs which leads to a following quantum Fisher information matrix
\begin{equation}
\mathcal{Q}=\left(
\begin{array}{c c c}
4 &0 & 0\\
0 & 4 & 0\\
0 & 0 & S_0
\end{array}\right)
\end{equation}
in the $(|\alpha_H|,|\alpha_V|,\varphi_-)$ parametrization. The resulting cost, after transforming to Stokes coefficients parametrization reads
\begin{equation}\label{eq:QCRB_phase}
\textrm{Tr}(\mathcal{Q}^{-1})=3S_0-\frac{S_1^2}{S_0},
\end{equation}
which takes the values in the region $2S_0\leq \textrm{Tr}(\mathcal{Q}^{-1})\leq 3S_0 $. The minimal value is obtained for $S_1=S_0$ whereas the worst one for $S_1=0$. The QCRB bound, however, cannot be saturated in this case as $\textrm{Tr}(\rho_{\pmb{\theta}}[L_{j},L_k])\neq 0$ for $j=\varphi_-$ and $k=|\alpha_H|,\,|\alpha_V|$. In particular
\begin{gather}
\textrm{Tr}(\ket{\psi_{\pmb{\theta}}}\bra{\psi_{\pmb{\theta}}}[L_{\varphi_-},L_{|\alpha_H|}])=-2i|\alpha_H|,\\
\textrm{Tr}(\ket{\psi_{\pmb{\theta}}}\bra{\psi_{\pmb{\theta}}}[L_{\varphi_-},L_{|\alpha_V|}])=2i|\alpha_V|.
\end{gather}
Note that this in principle means that HCRB is different than QCRB for all possible values of amplitudes, except $|\alpha_H|=|\alpha_V|=0$, which is a vacuum state. On the other hand, it is seen in Fig.~\ref{fig:kulki} that on the poles of the Poincare sphere $S_1=\pm S_0$ the respecitve bounds coincide. The reason for this apparent contradiction is that exactly at these two points one has either $|\alpha_V|=0$ (northern pole) or $|\alpha_H|=0$ (sourthern pole) and the Stokes coefficients in \eqnref{eq:stokes_parameter_ph3} do not depend on the realitve phase in any way. Therefore, the respective contribution for the relative phase estimation does not enter the expressions for both bounds and one can perform estimation neglecting this parameter, meaning that only SLDs for absolute values of both amplitudes matter. These two operator commute, therefore, QCRB is equal to HCRB on poles of the Poincare sphere.
In a realistic scenario, when one does not have access to an external reference frame, the state should be averaged over the global phase \cite{Molmer1997, Jarzyna2012}. In such a case one obtains
\begin{equation}
\rho_{\pmb{\theta}}=\int\frac{d\varphi_+}{2\pi}\ket{\psi_{\pmb{\theta}}}\bra{\psi_{\pmb{\theta}}}
=\sum_{N=0}^\infty p_N(\pmb{\theta}) \ket{v^N_{\pmb{\theta}}}\bra{v^N_{\pmb{\theta}}},
\end{equation}
where
\begin{gather}
p_N(\pmb{\theta})=e^{-S_0}\frac{S_0^N}{N!},\\
\ket{v^N_{\pmb{\theta}}}=\frac{1}{S_0^{N/2}}\sum_{n=0}^N\sqrt{{N}\choose{n}}|\alpha_H|^n|\alpha_V|^{N-n}e^{i\varphi_-n}\ket{n,N-n}.
\end{gather}
Note, that different $\ket{v^N_{\pmb{\theta}}}$ are orthogonal $\braket{v^N_{\pmb{\theta}}}{v^M_{\pmb{\theta}}}=\delta_{N,M}$ as they live in subspaces with different total numbers of photons. One can evaluate SLD's directly and obtain
\begin{multline}
L_{|\alpha_H|}=2|\alpha_H|\sum_{N=0}^\infty\left(\frac{N}{S_0}-1\right)\ket{v^N_{\pmb{\theta}}}\bra{v^N_{\pmb{\theta}}}+\\+2\sum_{N=0}^\infty\left(\ket{\partial_{|\alpha_H|}v^N_{\pmb{\theta}}}\bra{v^N_{\pmb{\theta}}}+\ket{v^N_{\pmb{\theta}}}\bra{\partial_{|\alpha_H|}v^N_{\pmb{\theta}}}\right),
\end{multline}
\begin{multline}
L_{|\alpha_V|}=2|\alpha_V|\sum_{N=0}^\infty\left(\frac{N}{S_0}-1\right)\ket{v^N_{\pmb{\theta}}}\bra{v^N_{\pmb{\theta}}}+\\+2\sum_{N=0}^\infty\left(\ket{\partial_{|\alpha_V|}v^N_{\pmb{\theta}}}\bra{v^N_{\pmb{\theta}}}+\ket{v^N_{\pmb{\theta}}}\bra{\partial_{|\alpha_V|}v^N_{\pmb{\theta}}}\right)
\end{multline}
and
\begin{equation}
L_{\varphi_-}=2\sum_{N=0}^\infty\left(\ket{\partial_{\varphi_-}v^N_{\pmb{\theta}}}\bra{v^N_{\pmb{\theta}}}+\ket{v^N_{\pmb{\theta}}}\bra{\partial_{\varphi_-}v^N_{\pmb{\theta}}}\right).
\end{equation}
The resulting quantum Fisher information matrix is equal to
\begin{equation}
\mathcal{Q}=\left(
\begin{array}{c c c}
4 &0 & 0\\
0 & 4 & 0\\
0 & 0 & \frac{4|\alpha_H|^2|\alpha_V|^2}{S_0}
\end{array}\right),
\end{equation}
which, after transformation into Stokes coefficients parametrization gives the cost equal to
\begin{equation}
\textrm{Tr}(\mathcal{Q}^{-1})=3S_0.
\end{equation}
Similarly to the previous case, in the phase averaged scenario SLDs do not commute on the state. One obtains
\begin{gather}
\textrm{Tr}(\ket{\psi_{\pmb{\theta}}}\bra{\psi_{\pmb{\theta}}}[L_{\varphi_-},L_{|\alpha_H|}])=-\frac{8i|\alpha_H||\alpha_V|^2}{S_0},\\
\textrm{Tr}(\ket{\psi_{\pmb{\theta}}}\bra{\psi_{\pmb{\theta}}}[L_{\varphi_-},L_{|\alpha_V|}])=-\frac{8i|\alpha_V||\alpha_H|^2}{S_0}.
\end{gather}
Therefore, QCRB for the phase averaged state is also unattainable.
Finally, one can also consider scenario with known total power $S_0$, where the estimated parameters are $(|\alpha_H|,\varphi_-)$. The resulting expressions for quantum Fisher information matrix when one has access to the global phase is given by
\begin{equation}
\mathcal{Q}=\left(
\begin{array}{c c}
\frac{8S_0}{S_0-S_1} &0 \\
0 & S_0
\end{array}\right),
\end{equation}
which, after transformation to Stokes coefficients parametrization gives precision equal to
\begin{equation}
\Delta S_1^2+\Delta S_2^2+\Delta S_3^2=2S_0-\frac{S_1^2}{S_0}.
\end{equation}
which takes values in the interval $[S_0,2S_0]$, the minimal obtained for $S_1=S_0$ and maximal for $S_1=0$. For the phase averaged scenario the respective quantities are given by
\begin{equation}
\mathcal{Q}=\left(
\begin{array}{c c}
\frac{8S_0}{S_0-S_1} &0 \\
0 & S_0-\frac{S_1^2}{S_0}
\end{array}\right),
\end{equation}
which, after transformation to Stokes coefficients parametrization gives precision equal to
\begin{equation}
\Delta S_1^2+\Delta S_2^2+\Delta S_3^2=2S_0.
\end{equation}
In both cases SLDs do not commute on the respective states.
\section{Classical Fisher information matrix for Stokes receiver}
\label{app:crb}
The quantum states of light impinging on photon number detectors in each arm of the Stokes receiver in Fig.~\ref{fig:stokes}(a) are coherent states with estimated parameters encoded in their amplitude. The photon number distribution of a coherent state with amplitude $|\alpha_{\pmb{\theta}}|=\sqrt{\bar{n}_{\pmb{\theta}}}$ is a Poisson distribution
\begin{equation}
p_k(\pmb{\theta})=e^{-\bar{n}_{\pmb{\theta}}}\frac{\bar{n}_{\pmb{\theta}}^k}{k!}.
\end{equation}
Plugging this expression into the formula for Fisher information matrix in \eqnref{eq:class_FI} gives a following expression for its elements
\begin{equation}
F_{jl}=\sum_{k=0}^\infty \frac{1}{p_k(\pmb{\theta})}\left(\frac{\partial p_k(\pmb{\theta})}{\partial \bar{n}_{\pmb{\theta}}}\right)^2\frac{\partial\bar{n}_{\pmb{\theta}}}{\partial \theta_j}\frac{\partial\bar{n}_{\pmb{\theta}}}{\partial \theta_l}=\frac{1}{\bar{n}_{\pmb{\theta}}}\frac{\partial\bar{n}_{\pmb{\theta}}}{\partial \theta_j}\frac{\partial\bar{n}_{\pmb{\theta}}}{\partial \theta_l}.
\end{equation}
The amplitudes of light impinging on each detector are given by
\begin{equation}\label{eq:amplitudes}
\tilde{\alpha}_{H,V}=\frac{\alpha_{H,V}}{\sqrt{3}},\quad\tilde{\alpha}_\pm=\frac{\alpha_H\pm\alpha_V}{\sqrt{6}},\quad \tilde{\alpha}_{R,L}=\frac{\alpha_H\pm i\alpha_V}{\sqrt{6}},\\
\end{equation}
in the first, second and third arm respectively, where the factor $1/\sqrt{3}$ comes from the division of the signal on a three-way splitter.
The Fisher information matrices for each of the three arms of the Stokes receiver are given as sums of Fisher matrices on both detectors in these respective arms. For estimation of $|\alpha_H|,|\alpha_V|,\varphi_-$ they read
\begin{equation}\label{eq:FI_HV}
\mathcal{F}^{HV}=\left(
\begin{array}{c c c}
\frac{4}{3} &0 & 0\\
0 & \frac{4}{3} & 0\\
0 & 0 & 0
\end{array}\right),
\end{equation}
\begin{widetext}
\begin{equation}
\mathcal{F}^{+-}=\frac{1}{3}\left(
\begin{array}{c c c}
\frac{|\alpha_H|^2S_0+|\alpha_V|^2(|\alpha_V|^2-3|\alpha_H|^2)\cos^2\varphi_-}{|\alpha_+|^2|\alpha_-|^2} & |\alpha_H||\alpha_V|\frac{S_0\sin^2\varphi_-}{|`\alpha_+|^2|\alpha_-|^2} & \frac{|\alpha_H||\alpha_V|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_+|^2|\alpha_-|^2}\sin2\varphi_-\\
|\alpha_H||\alpha_V|\frac{S_0\sin^2\varphi_-}{|\alpha_+|^2|\alpha_-|^2} & \frac{|\alpha_V|^2S_0+|\alpha_H|^2(|\alpha_H|^2-3|\alpha_V|^2)\cos^2\varphi_-}{|\alpha_+|^2|\alpha_-|^2} & -\frac{|\alpha_V||\alpha_H|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_+|^2|\alpha_-|^2}\sin2\varphi_-\\
\frac{|\alpha_H||\alpha_V|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_+|^2|\alpha_-|^2}\sin2\varphi_- & -\frac{|\alpha_V||\alpha_H|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_+|^2|\alpha_-|^2}\sin2\varphi_- & \frac{|\alpha_H|^2|\alpha_V|^2S_0}{|\alpha_+|^2|\alpha_-|^2}\sin^2\varphi_-
\end{array}\right),
\end{equation}
\end{widetext}
and
\begin{widetext}
\begin{equation}
\mathcal{F}^{RL}=\frac{4}{3}\left(
\begin{array}{c c c}
\frac{|\alpha_H|^2S_0+|\alpha_V|^2(|\alpha_V|^2-3|\alpha_H|^2)\sin^2\varphi_-}{|\alpha_R|^2|\alpha_L|^2} & |\alpha_H||\alpha_V|\frac{S_0\cos^2\varphi_-}{|\alpha_R|^2|\alpha_L|^2} & -\frac{|\alpha_H||\alpha_V|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_R|^2|\alpha_L|^2}\sin2\varphi_-\\
|\alpha_H||\alpha_V|\frac{S_0\cos^2\varphi_-}{|\alpha_R|^2|\alpha_L|^2} & \frac{|\alpha_V|^2S_0+|\alpha_H|^2(|\alpha_H|^2-3|\alpha_V|^2)\sin^2\varphi_-}{|\alpha_R|^2|\alpha_L|^2} & \frac{|\alpha_V||\alpha_H|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_R|^2|\alpha_L|^2}\sin2\varphi_-\\
-\frac{|\alpha_H||\alpha_V|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_R|^2|\alpha_L|^2}\sin2\varphi_- & \frac{|\alpha_V||\alpha_H|^2(|\alpha_H|^2-|\alpha_V|^2)}{2|\alpha_R|^2|\alpha_L|^2}\sin2\varphi_- & \frac{|\alpha_H|^2|\alpha_V|^2S_0}{|\alpha_R|^2|\alpha_L|^2}\cos^2\varphi_-
\end{array}\right).
\label{eq:FI_RL}
\end{equation}
\end{widetext}
Upon adding \eqnsref{eq:FI_HV}{eq:FI_RL} one obtains classical Fisher information for Stokes receiver, which then has to be reperamterized through Jacobian given in \eqnref{eq:jacobian} in order to obtain precision bound for Stokes vector estimation, giving
\begin{equation}
\Delta S_1^2+\Delta S_2^2 +\Delta S_3^2\geq \frac{11}{2}S_0-\frac{9}{2S_0}\left(\frac{1}{S_1^2}+\frac{1}{S_2^2}+\frac{1}{S_3^2}\right)^{-1}.
\end{equation}
Similarly, one can obtain FI matrix for the Stokes receiver in the case of known total signal power which has been analyzed in Sec.~\ref{sec:stokes}(a). Once again, using \eqnref{eq:amplitudes} and \eqnref{eq:class_FI} together with the constraint $|\alpha_V|=\sqrt{S_0-|\alpha_H|^2}$ one gets
\begin{equation}\label{eq:FI_HV_pow}
\mathcal{F}_{HV}=\left(
\begin{array}{c c}
\frac{4}{3(S_0-|\alpha_H|^2)} &0 \\
0 & 0
\end{array}\right),
\end{equation}
\begin{widetext}
\begin{equation}
\mathcal{F}_{+-}=\frac{4}{3}\left(
\begin{array}{c c}
\frac{S_0(S_0-2|\alpha_H|^2)^2\cos^2\varphi_-}{(S_0-|\alpha_H|^2)(S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\cos^2\varphi_-)} & -\frac{|\alpha_H|S_0(S_0-2|\alpha_H|^2)\sin \varphi_-\cos\varphi_-}{S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\cos^2\varphi_-}\\
-\frac{|\alpha_H|S_0(S_0-2|\alpha_H|^2)\sin \varphi_-\cos\varphi_-}{S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\cos^2\varphi_-} & \frac{|\alpha_H|^2S_0(S_0-2|\alpha_H|^2)\sin^2\varphi_-}{S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\cos^2\varphi_-}
\end{array}\right),
\label{eq:FI_RL_pow}
\end{equation}
\end{widetext}
and
\begin{widetext}
\begin{equation}
\mathcal{F}_{RL}=\frac{4}{3}\left(
\begin{array}{c c}
\frac{S_0(S_0-2|\alpha_H|^2)^2\sin^2\varphi_-}{(S_0-|\alpha_H|^2)(S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\sin^2\varphi_-)} & \frac{|\alpha_H|S_0(S_0-2|\alpha_H|^2)\sin \varphi_-\cos\varphi_-}{S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\sin^2\varphi_-}\\
\frac{|\alpha_H|S_0(S_0-2|\alpha_H|^2)\sin \varphi_-\cos\varphi_-}{S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\sin^2\varphi_-} & \frac{|\alpha_H|^2S_0(S_0-2|\alpha_H|^2)\cos^2\varphi_-}{S_0^2-4|\alpha_H|^2(S_0-|\alpha_H|^2)\sin^2\varphi_-}
\end{array}\right)
\end{equation}
\end{widetext}
for parameters $|\alpha_H|,\varphi_-$ respectively. This has to be then reparametrized through Jacobian in \eqnref{eq:jacobian_s0}, resulting in the bound
\begin{equation}
\Delta\vec{S}^2\geq \frac{9}{2S_0}\frac{\left(1+\frac{S_3^2}{S_2^2}\right)\left(1+\frac{S_2^2}{S_1^2}\right)\left(1+\frac{S_1^2}{S_3^2}\right)}{\frac{1}{S_1^2}+\frac{1}{S_2^2}+\frac{1}{S_3^2}}.
\end{equation}
A similar analysis can be performed for tetrahedron measurement with additional optimization of the Cramer-Rao bound over the transmission and reflection coefficients of the first beam splitter in Fig.~\ref{fig:stokes}(b).
\end{document} |
\begin{document}
\centerline{}
\begin{frontmatter}
\selectlanguage{english}
\title{On a price formation free boundary model by Lasry \& Lions: The Neumann problem}
\selectlanguage{english}
\author[authorlabel1]{Luis A. Caffarelli}
\ead{[email protected]}
\author[authorlabel2,authorlabel3]{Peter A. Markowich}
\ead{[email protected]}
\author[authorlabel3]{Marie-Therese Wolfram}
\ead{[email protected]}
\address[authorlabel1]{Department of Mathematics, Institute for Computational Engineering and Sciences, University of Texas at Austin, USA}
\address[authorlabel2]{DAMTP, University of Cambridge, Cambridge CB3 0WA, UK}
\address[authorlabel3]{Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria}
\begin{center}
{\small Received *****; accepted after revision +++++\\
Presented by }
\end{center}
\begin{abstract}
\selectlanguage{english}
We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry \& Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.
\vskip 0.5\baselineskip
\selectlanguage{francais}
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
Nous discutons l'existence locale et globale, ainsi que l'unicit\'e des solutions pour le mod\`ele de formation des prix \`a fronti\`ere libre avec des conditions aux bords de Neumann homog\`enes introduit par Lasry \& Lions en 2007. Nos r\'esultats sont bas\'es sur une transformation de ce probl\`eme en une \'equation de la chaleur avec des conditions aux bords non standard. La fronti\`ere libre devient la ligne de niveau z\'ero de la solution de l'\'equation de la chaleur. Cette transformation nous permet de construire une solution explicite et de discuter le comportement de la fronti\`ere libre. L'existence globale peut \^etre v\'erifi\'ee sous certaines conditions sur la fronti\`ere libre, et nous donnons des exemples de non-existence.
\end{abstract}
\end{frontmatter}
\selectlanguage{english}
\section{Introduction}
\label{sec:intro}
In this paper we discuss the mathematical analysis of a price formation mean field model, as stated in \cite{Lasry2007}. The equation models the trading of an economic good between a large group of buyers and a large group of vendors. It consists of a non-linear parabolic free boundary evolution equation which describes the evolution of the densities of buyers and vendors and also determines the price of the good. Here we consider the problem on the real interval $(-L,L)$, where $L$ denotes the maximum and $-L$ the minimum price. The model is given by the free boundary value problem
\begin{subequations}\label{eq:pfm}
\begin{align}
f_t - f_{xx} &= \lambda(t)(\delta(x-p(t)+a) - \delta(x-p(t)-a)),\, x\in(-L,L),t>0,\label{eq:pfm1}\\
\lambda(t) &= -f_x(p(t),t),\; f(p(t),t)=0,\label{eq:pfm2}\\
f(x,0) &= f_I,\, p(0) = p_0,\;\text{for some $p_0$ in $(-L+a,L-a)$},\label{eq:initial}\\
f_x(\pm L,t) &= 0,~t>0,
\end{align}
\end{subequations}
with $ 0 < a < L$ and compatibility conditions at time $t=0$: $ f_I(p_0) = 0\text{ and }f_I(x) > 0\text{ for }x < p_0 \text{ and }f_I(x) < 0\text{ for }x > p_0$. Throughout this paper we assume that $f_I$ is in $L^2(-L,L)$ and denote the $L^2$ norm on $(-L,L)$ by $\lVert \cdot \rVert$. System \eqref{eq:pfm} has been studied in a number of papers, see \cite{Chayes2009,Markowich2009,Caffarelli2011}. Here we present global (non-)existence results of a solution of \eqref{eq:pfm} on the bounded interval $(-L,L)$.
\section{Analysis of the Neumann problem - Transformation to the Heat Equation}
In the sequel we denote the positive and negative part of a function $f$ defined for $x \in (-L,L)$ by $f^+:= \max(f,0)$, $f^-:= \max(-f,0)$. We extend $f^+$,$f^-$ by the value $0$ outside the interval $(-L,L)$.
\begin{proposition}
Let $f$ be a solution of the modified price formation problem consisting of equation \eqref{eq:pfm1} with conditions \eqref{eq:pfm2}, initial datum \eqref{eq:initial} and modified boundary conditions (bc):
\begin{align}\label{eq:newbc}
~~f_x(-L,t) =
\begin{cases}
0 \quad & p(t) > -L+a\\
f_x(-L+a,t) \quad & p(t) \leq -L+a.
\end{cases}
f_x(L,t) =
\begin{cases}
0 \quad & p(t) < L-a\\
f_x(L-a,t) & p(t) \geq L-a.
\end{cases}
\end{align}
Then the function
\begin{align}\label{eq:trans}
F(x,t) =
\begin{cases}
\phantom{-}\sum\nolimits^{\infty}_{n=0} f^+(x+na,t), x < p(t)\\
-\sum\nolimits ^{\infty}_{n=0} f^-(x-na,t), x > p(t),
\end{cases}
\end{align}
is a solution of the following BVP for the heat equation
\begin{subequations}\label{eq:heat}
\begin{align}
F_t &= F_{xx}, ~\text{ for all } x \in (-L,L), t>0,\\
F_x(\pm L,t) &= F_x(\pm L\mp a,t), ~t>0\\
F(x,t=0) &= F_I(x), ~\text{ for all } x \in (-L,L),
\end{align}
\end{subequations}
where $F_I$ is constructed from $f_I$ according to \eqref{eq:trans}. Conversely, if $F$ is a solution of \eqref{eq:heat} then $f(x,t) = F(x,t)-F^+(x+a,t)+F^-(x-a,t)$ satisfies \eqref{eq:pfm1}, \eqref{eq:pfm2}, \eqref{eq:initial}, \eqref{eq:newbc}.
\end{proposition}
The construction of the function $F$ is motivated by the fact that $f^+$ ($f^-$) has jump-discontinuities at $x = p(t)-a$ ($x = p(t)+a$) and $x = p(t)$ of equal magnitude but opposite signs. The summation procedure then moves the jumps in the derivatives of $f$ outwards. Note that the sum in \eqref{eq:trans} actually consists only of finitely many terms and that the free boundary $p=p(t)$ of the price formation problem becomes the zero level set of the heat solution and vice versa. In principle the free boundary can leave and reenter the interval $(-L+a, L-a)$ without impeding the existence of a global solution of \eqref{eq:pfm1}, \eqref{eq:pfm2}, \eqref{eq:initial}, \eqref{eq:newbc}.\\
We shall now construct an 'explicit' solution of \eqref{eq:heat} by separation of variables. We set $F(x,t) = \varphi(x) \psi(t)$ and find $\varphi''/\varphi = \dot{\psi}/\psi = -z^2$. The boundary conditions give the equation $G(z) := \cos(zL) - \cos(z(L-a)) = 0$ with the eigenfunctions $\varphi(x) = A \sin(zx)$ and $H(z) := \sin(zL)-\sin(z(L-a)) = 0$ with eigenfunctions $\varphi = B \cos(zx)$. We easily compute $G(z) = 0$ iff $z = (2\pi l) / a, (2\pi l)/(2L-a)$ and $H(z) = 0$ iff $z = (2\pi l)/a, (\pi(2l-1)/(2L-a)$ for $l \in \mathbb{Z}$. Next we conclude that $H=H(z)$ is a sine-type function of type $L$ with simple and separated zeros. Therefore $\lbrace \exp(i \frac{2\pi l}{a} x), \exp(i \frac{\pi(2l-1)}{2L-a}x)\rbrace$ is a Riesz basis in $L^2(-L,L)$, see \cite{Avdonin1988}. Also $G = G(z)$ is a sine-type function of type L, its zero are separated except $z=0$, which is a zero of order $2$. Hence $\lbrace \exp(i \frac{2\pi l}{a} x), \exp(i \frac{2 \pi l}{2L-a} x), x\rbrace_{l \in \mathbb{Z}}$ is a Riesz basis in $L^2(-L,L)$, see Theorem D in \cite{Horvath1987}. By separately considering even and odd parts we conclude that $\lbrace\cos(\frac{2\pi l}{a} x), \sin(\frac{2 \pi l}{a} x), \sin(\frac{2\pi l}{2L-a} x), \cos(\frac{\pi (2l-1)}{2L-a} x), 1, x \rbrace$, $l=1,2, \ldots$ is a Riesz basis of eigenfunctions of the heat equation \eqref{eq:heat}. The solution of \eqref{eq:heat} is
\begin{align}\label{eq:solF}
\begin{split}
F(x,t) = \sum_{l=1}^{\infty} &[(A_l \sin(\omega_{1,l} x) + B_l \cos(\omega_{1,l} x)) e^{-\omega_{1,l}^2t} + C_l \sin(\omega_{2,l} x) e^{-\omega_{2,l}^2 t} + D_l \cos(\omega_{3,l}x) e^{-\omega_{3,l}^2t}]{}\\
+ &A_0 x + B_0,
\end{split}
\end{align}
with $\omega_{1,l} = (2\pi l)/a, ~\omega_{2,l} = (2\pi l)/(2L-a),~\omega_{3,l} = ((2l-1)\pi)/(2L-a)$. Note that $A_l, B_l, C_l, D_l, A_0, B_0$ can be determined uniquely for every initial datum $F_I \in L^2(-L,L)$.
\begin{theorem}[Global Existence] The BVP \eqref{eq:pfm1}, \eqref{eq:pfm2}, \eqref{eq:initial}, \eqref{eq:newbc} has a unique global solution $f = f(x,t)$ for $t>0$. Furthermore the free boundary $p=p(t)$ is a smooth graph $p(t) \in (-L,L)$ for all $t > 0$.
\end{theorem}
The proof follows from the construction described above. The min-max principle implies that $p=p(t)$ is a graph for $t>0$ and the Hopf principle implies that $p$ is smooth. Finite-time oscillations of $p(t)$ are excluded by the $x$-analyticity of solutions of the heat equations (see \cite{Caffarelli2011}). Since $0 < \int_{-L}^{-L+a} F(x,t) dx$ and $0 > \int_{L-a}^L F(x,t) dx$ are conserved in time (by the equation and the boundary conditions) we conclude that $p(t) \in (-L,L)$ for all $t>0$. \\
Clearly, a solution of \eqref{eq:pfm1}, \eqref{eq:pfm2}, \eqref{eq:initial}, \eqref{eq:newbc} is a solution of \eqref{eq:pfm} on a time interval $[0,T]$ iff $p(t) \in (-L+a, L-a)$ for $t \in [0,T]$. Then the total mass of buyers $M_B = \int_{-\infty}^{p(t)} f(x,t) dx$ and vendors $M_V = -\int_{p(t)}^\infty f(x,t) dx$ are time conserved quantities.
\begin{theorem}\label{t:mass}
The BVP \eqref{eq:pfm} has a global solution conserving the total mass of buyers and vendors iff the zero level set $p$ of the solution of \eqref{eq:heat} satisfies $p(t) \in (-L+a, L-a)$ for all $t>0$. Then the free boundary $p(t)$ converges to $p_{\infty} \in (-L+a, L-a)$.
\end{theorem}
The proof of Theorem \ref{t:mass} is based on the following lemmas.
\begin{lemma}\label{l:exp}
The solution $F$ converges exponentially fast to $\Fi = \ai{A}x + \ai{B}$ in $L^2(-L,L)$.
\end{lemma}
{\em Proof:} $~$ From the Riesz base property we deduce that there exist $c_1, c_2 \in \mathbb{R}^+$ such that $c_1 \lVert F_I\rVert^2 \leq \sum_{l=1}^\infty (A_l^2 + B_l^2 + C_l^2 + D_l^2) + A_0^2 + B_0^2 \leq c_2 \lVert F_I \rVert^2$. For $\tilde{F} = F-(A_0x + B_0)$ we deduce
\begin{align*}
c_1 \lVert \tilde{F} \rVert^2 \leq \sum\nolimits_{l=1}^{\infty} [(A_l^2 + B_l^2 + C_l^2+D_l^2)] e^{-2 \gamma_{l} t} \leq e^{-2 \gamma_1 t} \sum\nolimits_{l=1}^{\infty} (A_l^2 + B_l^2 + C_l^2+D_l^2) \leq c_2 e^{-2 \gamma_1 t} \lVert F_I \rVert^2,
\end{align*}
with the sequence $\gamma_l = \min((4\pi^2 l^2)/a^2, (4\pi^2 l^2)/(2L-a)^2, ((2l-1)^2 \pi^2)/(2L-a)^2)$, $l=1,2,\ldots$.\qed
\begin{lemma}\label{l:F}
The solution \eqref{eq:solF} satisfies $\displaystyle\lvert F_x(x,t) \rvert\leq \sup_{x\in(-L,L)} \lvert (F_I)_x \rvert$ and $\lvert F(x,t) \rvert \leq c$, $ \forall x \in (-L,L), ~ t > 0$.
\end{lemma}
{\em Proof:} The function $V=F_x$ satisfies the heat equation with $V(-L,t) = V(-L+a,t),~V(L,t)=V(L-a,t)$ and initial condition $V(x,t=0) = (F_I)_x(x)$. If $V$ assumes its maximum on the cylinder $[-L,L] \times [0,T)$ at either boundary $x = \pm L$, then it must also assume a maximum (with the same value) in the interior (due to bc on $V$). This contradicts the maximum principle, thus $V$ must assume its maximum at $t=0$. The same arguments hold for the minimum. \\
Since $F(x,t) = F(-L,t) + \int_{-L}^x F_x(y,t) dy$ we deduce $a F(-L,t) = -\int_{-L}^{-L+a} F(x,t) dx + \int_{-L}^{-L+a}\int_{-L}^x F_x(y,t) dy dx = I_1 + I_2$.
We know that $F_x$ is bounded, therefore $\lvert I_2 \rvert \leq K$. In addition $\frac{d}{dt} \int_{-L}^{-L+a} F(x,t) dx = 0$ and therefore $\int_{-L}^{-L+a} F(x,t) dx = \int_{-L}^{-L+a} F_I(x) dx$. Thus $F(-L,t)$ as well as $F = F(x,t)$ are bounded uniformly on $(-L,L) \times \mathbb{R}^+$.\qed\\
{\em Proof of Theorem \ref{t:mass}:} We know that $F$ converges exponentially fast to $\Fi = \ai{A} x + \ai{B}$, and that there exists a smooth graph $p = p(t)$ such that $F(p(t),t) = 0$. Now we assume that $p(t) \in (-L+a, L-a)$ for $t>0$, choose any sequence $t_n \rightarrow \infty$ and conclude that there is a subsequence $t_{n_k}$ such that $p(t_{n_k}) \rightarrow p_{\infty} \in [-L+a, L-a]$. Let $\varphi$ be a test function in $\mathcal{D}(p_{\infty},L)$. If $k$ is sufficiently large we conclude $f(x,t_{n_k}) < 0$ for $x \in \supp \varphi$. Therefore $\int_{-L}^L f(x,t_{n_k}) \varphi(x) dx < 0$. Since $f(\cdot,t) \rightarrow \ai{f}$ there is a subsequence $t_{n_{k_l}}$ such that $f(x,t_{n_{k_l}})$ converges to $\ai{f}(x)$ pointwise a.e. in $(-L,L)$. The function $\lvert f(x,t_{n_k})\rvert \leq K$ on $[-L,L]$ for all $k$. Then we deduce from Lebesgue's' dominated convergence theorem that $\int_{-L}^L f(x,t_{n_k}) \varphi(x) dx \rightarrow \int_{-L}^L f_{\infty} \varphi(x) \leq 0.$ Since $f_{\infty} = \ai{A} x + \ai{B} + (\ai{A}(x-a) + \ai{B})^- - (\ai{A} (x+a) + \ai{B})^+$ we conclude $\ai{f} \leq 0$ for $x > \ai{p}$ and $\ai{f} \geq 0$ for $x < \ai{p}$. From Lebesgue's' dominated convergence theorem, we conclude $-\int_{p(t_{n_{k_l}})}^L f(x,t) dx \rightarrow -\int_{\ai{p}}^L \ai{f}(x) dx \geq 0.$ Analogously$ \int_{-L}^{\ai{p}} \ai{f} dx \geq 0$. \\
Next we show that $\ai{F}$ has a unique zero in $(-L,L)$. If $\ai{F} = \ai{A} x + \ai{B} \geq 0$ on $(-L,L)$ then $\ai{F}^-=0,~\ai{F}^+=\ai{F}$ and therefore $\ai{f}$ is constant in $(-L,L)$, which is a contradiction. The same argument holds for $\ai{F} \leq 0$. Therefore the function $\ai{f}$ is given by $\ai{f}(x) = \pm \alpha$ for $x \in (-L, \ai{p}-a)$ and $ x\in (\ai{p}+a,L)$ respectively and $\ai{f}= -\alpha/a(x-\ai{p})$ for $x \in [\ai{p}-a, \ai{p}+a]$, with $\alpha \in \mathbb{R}^+$. We conclude that $\ai{p}$ is unique and that $p(t) \rightarrow \ai{p}$ as $t \rightarrow \infty$, since every sequence has a subsequence which converges to the same limit. \qed
\begin{theorem}
Let $f_I$ be such that $M_B/M_V \notin \left[a/(4L-3a), (4L-3a)/a\right]$, where $M_B, M_V$ denotes the initial mass of buyers and vendors.
Then \eqref{eq:pfm} does not have a global-in-time solution, which conserves both buyers and vendors masses.
\end{theorem}
{\em Proof:} The result follows since $\alpha$, $\ai{p}$ can not be adjusted such that $\ai{p} \in [-L+a,L-a]$, where $M_B = \int_{-L}^{\ai{p}} \ai{f} dx,~M_V = -\int_{\ai{p}}^L \ai{f} dx$. \qed\\
Note that the choice of $L$, which corresponds to the maximally attainable price, is more or less arbitrary but a bad (too small) choice of $L$ might impede global existence. In this case the model clearly looses its 'practical' significance. We remark that global existence results for \eqref{eq:pfm} (with a free boundary which remains in $(-L+a,L-a)$) for initial data which are small perturbations of stationary solutions are straightforward, without using the analytical machinery of \cite{Gonzalez2011}.
\section*{Acknowledgments}
PM acknowledges support by the King Abdullah University of Science and Technology, the Leverhulme Trust and the Royal Society. LC acknowledges support from the Division of Mathematical Sciences of the NSF, MTW from the Austrian Science Foundation FWF.
\end{document} |
\begin{document}
\author[Palak Arora]{Palak Arora${}^1$}
\thanks{${}^1$Research supported by NSF grant DMS-2154494}
\address{University of Florida}
\email{[email protected]}
\author[Meric Augat]{Meric Augat${}^2$}
\thanks{${}^2$Research supported by NSF grant DMS-2155033}
\address{University of South Florida}
\email{[email protected]}
\author[Michael T. Jury]{Michael T. Jury${}^3$}
\thanks{${}^3$Research supported by NSF grant DMS-2154494}
\address{University of Florida}
\email{[email protected]}
\author[Meredith Sargent]{Meredith Sargent${}^4$}
\address{University of Manitoba}
\thanks{${}^4$Supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute.}
\email{[email protected]}
\title{An optimal approximation problem for free polynomials}
\begin{abstract}
Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n := \|p_nf-1\|^2$. (Here the norm is the $\ell^2$ norm on coefficients.) We show that $c_n\to 0$ if and only if $f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift.
\end{abstract}
\maketitle
\section{Introduction}
\iffalse
\fi
This paper concerns an approximation problem for polynnomials in noncommuting indeterminates. To state the problem, let $\mathbbm{x} = \set{x_1,\dots, x_d}$ be a set of freely noncommuting indeterminates and write $\bm{x} = (x_1,\dots, x_d)$.
We write ${\mathbb C}\fralg{\mathbbm{x}}$ for the \textbf{free algebra} in the indeterminates $x_1, \dots, x_d$ and its elements are \textbf{free polynomials}. We write elements $p\in {\mathbb C}\fralg{\mathbbm{x}}$ as $p(x) = \sum_{w\in \langle \bbx \rangle} p_w x^w$ (with $p_w\in {\mathbb C}$ nonzero for all but finitely many $w$). Here, $w$ is a ``word" $w=i_1 i_2\cdots i_m$ in the ``letters" ${1, \dots, d}$, and $x^w:= x_{i_1}x_{i_2}\cdots x_{i_m}$. In this case the length of the word is defined to be $m$, and the {\bf degree} of $p$ is defined to be the maximum length of words $w$ for which $p_w\neq 0$.
We equip ${\mathbb C}\fralg{\mathbbm{x}}$ with the natural inner product
obtained by declaring the ``words'' in the letters $x_i$ to be an orthonormal system:
\[
\ip{x_{i_1}\dots x_{i_n}}{x_{j_1}\dots x_{j_m}} := \begin{cases}
1 & m=n \text{ and } i_\ell = j_\ell \text{ for } 1\leq \ell \leq n;\\
0 & \text{otherwise}
\end{cases}
\]
Thus for $p=\sum_{w\in\langle \bbx \rangle} p_w x^w$, $q=\sum_{w\in\langle \bbx \rangle} q_w x^w$, we have
\[
\langle p,q\rangle = \sum_{w\in\langle \bbx \rangle} p_w \overline{q_w}
\]
and the associated $\ell^2$ norm on coefficients
\[
\norm{p}^2 = \sum_{w\in\langle \bbx \rangle} |p_w|^2.
\]
Given a polynomial $f\in{\mathbb C}\fralg{\mathbbm{x}}$, for each $n$ there is a unique polynomial $p_n\in {\mathbb C}\fralg{\mathbbm{x}}$, of degree at most $n$, which minimizes the norm
\[
c_n:=\norm{p_nf-1}^2
\]
We call this $p_n$ the {\em ${n}^{th}$ optimal polynomial approximant} ({\em OPA}) of $f^{-1}$.
The sequence $(c_n)$ is evidently non-increasing. We can ask two questions about the numbers $(c_n)$:
{\bf Question 1:} {\em Under what conditions on $f$ do we have $c_n\to 0$? }
{\bf Question 2:} {\em If $c_n\to 0$, how fast? (That is, give bounds on the rate of decay.)}
Our main theorem gives answers to Questions 1 and 2, in the more general setting of polynomials with matrix coefficients. (For precise definitions see Section~\ref{sec:preliminaries}).
\begin{theorem}\label{mainRODthm-repeat}
Let $F\in M_k({\mathbb C})\otimesimes {\mathbb C}\fralg{\mathbbm{x}}$ be a $k\times k$-matrix valued free polynomial, let $(P_n)$ denote the sequence of optimal polynomial approximants, and put
\[
c_n:=\|P_n(x)F(x)-I\|_2^2.
\]
Then: \begin{enumerate}
\item $c_n\to 0$ if and only if $F$ is nonsingular in the row ball.
\item If $F$ is nonsingular in the row ball, then
\[
c_n\lesssim \frac{1}{n^p}
\]
for some $p>0$ depending on $F$. In particular, if $F$ is a product of $\ell$ atomic factors, we can take $p=\frac{1}{3^{\ell-1}}$.
\end{enumerate}
\end{theorem}
Let us describe the motivation for these questions. The first motivation comes from the problem of characterizing the so-called {\em cyclic} free polynomials. We say $f$ is {\em cyclic} if every free polynomial $q$ can be approximated arbitrarily well in the $\ell^2$ norm by polynomials of the form $pf$. It was proved in \cite{jury2020noncommutative} that a free polynomial $f$ is cyclic if and only if it is nonsingular in the row ball, but the proof is quite abstract and indirect. It turns out that cyclicity of $f$ is equivalent to the statement that $c_n\to 0$ (see Section~\ref{sec:preliminaries}). In this light, Question 1 asks about ``qualitative cyclicity,'' and Question 2 asks about ``quantitative cyclicity.'' We refer to \cite{jury2020noncommutative} for further discussion and motivation for the cyclicity question.
Thus, one of our motivations was to try to see if there is a more elementary proof of the characterization of cyclic $f$ obtained in \cite{jury2020noncommutative}. It turns out that such a proof is possible, and in the proof we present in fact allows us to give some answer to Question 2 (namely, item (2) of the theorem), though we do not know if this answer is sharp (see the remarks at the end of Section~\ref{sec:proof-of-main}).
Besides this, the problem can be motivated by considering what is already known for polynomials in one complex variable, and asking to what extent this result can be generalized to the free setting. In particular, Beneteau et al. \cite{JAM15} considered the analogous approximation problem for polynomials in one complex variable: given a polynomial $f\in{\mathbb C}[z]$, let $p_n$ be the polynomial of degree at most $n$ minimizing the $\ell^2$ norm $c_n:=\|p_nf-1\|^2$, and estimate the $c_n$. Among other things, they proved the following (we have stated their result in a slightly different form):
\begin{theorem}\label{thm:original} Let $f\in {\mathbb C}[z]$ be a polynomial with no zeroes in the disk $|z|<1$. Then:
\[
c_n:=\|p_n f-1\|^2 \lesssim \frac{1}{n}.
\]
\end{theorem}
The study of optimal polynomial approximants has roots tracing back to at least the 1970's where engineering and applied mathematics researchers investigated polynomial least-squares inverses in the context of digital filters in signal processing. A survey for mathematical audiences can be found in \cite{BCSurvey}. Roughly speaking, one considers the polynomials $p_n$ as approximations to $1/f$, even though $1/f$ will not in general have $\ell^2$ coefficients and therefore cannot be approximated directly by Hilbert space methods.
In the mathematical context, OPAs were introduced in \cite{JAM15} as a tool to study cyclicity in a family of Dirichlet-type spaces on the disk ${\mathbb D}$ which includes the classical Hardy space of the disk $H^2({\mathbb D})$. (This was then extended to a more general reproducing kernel Hilbert space context in \cite{FMS14}. Surveys of OPA results can be found in \cite{SecoSurvey, BCSurvey}.) A function $f$ is said to be cyclic (for the shift operator $Sf:=zf$) in $H^2({\mathbb D})$ if
\begin{equation}
[f]=\overline{\text{span}\set{z^kf\::\:k=1,2,3,\dots}}
\end{equation}
is the entire space $H^2({\mathbb D})$. It is not hard to see that to be cyclic, the function $f$ must be zero-free in the disk, and if the polynomials are dense, the function $1$ is cyclic. One can then observe that for a cyclic function $f$, the optimal norms $\norm{p_nf-1}$ converge to zero. Because optimal approximants are unique, the rate of decay of these optimal norms can then be used to quantify and compare ``how cyclic'' different functions are, e.g. a fast rate of decay would mean a function is strongly cyclic.
The questions about rates of decay and locations of zeros for optimal approximants have been considered in several contexts beyond the spaces mentioned above. The sequence spaces $\ell^p_A$ are considered in \cite{MR4159768,MR4406262}, and \cite{MR4448804} considers $L^p$ and $H^p$, and the pairs of papers \cite{PJM15,TAMS16} and \cite{MSAS1,MSAS2} consider approximations problems adapted to the bidisk and unit ball, respectively.
The role of the hypothesis on the zeroes of $f$ in Theorem~\ref{thm:original} is explained by the following observation: note that polynomials $p$ and complex numbers $z$, the evaluation functional $p\to p(z)$ is continuous for the $\ell^2$ norm if and only if $|z|<1$. Thus, it quickly follows that a necessary condition for $c_n\to 0$ is that $f$ be nonvanishing in the disk $|z|<1$. It is a folklore theorem that this condition is also sufficient for $c_n\to 0$, but as far as we are aware this result from \cite{JAM15} is the first to obtain a quantitative bound. The result can be proved by explicit calculation in the affine linear case $f(z)=1-\mu z$ ($|\mu|\leq 1$), and then extended to general $f$ by factoring and induction (though some care is required in handling repeated roots). Thus, in the free case, one may suspect that some sort of nonvanishing condition on $f$ will be necessary for cyclicity, and indeed this is the case, for essentially the same reason that certain evaluaion functionals $f\to f(X)$ (now at matrix points $X$) are bounded for the $\ell^2$ norm. The details may be found in Section~\ref{sec:preliminaries}.
Of course, in the free setting we will have no recourse to the fundamental theorem of algebra, so rather different techniques will be required. In particular, while any free polynomial $f$ can be factored into irreducibles, these irreducibles will not in general be linear (and indeed the factorization need not be unique, e.g. $x-xyx=x(1-yx)=(1-xy)x$). Nonetheless, it turns out we will be able to reduce the general question to the case of (affine) linear $f$, though at the cost of introducing matrix coefficients. The technique rests on the (by now) well-understood technique of {\em linearizations} or {\em realizations} of free polynomials and rational functions.
\subsection{Reader's Guide}
The next section will give a short tour of our setting in the nc (noncommutative) universe, including the row ball and the Fock space, along with the definition of an nc function.
We also discuss the mechanics of the approximation problem in this universe, including the definitions of the d-shift and cyclicity, and some simple conditions.
Our results hold in the general case of polynomials with matrix coefficients, so subsection \ref{sec:matrix-coefficients} gives background on these, while subsection \ref{sec:stabeq} sets out the lemmas used to decompose a polynomial. We round out the preliminaries with a discussion of the outer spectral radius and some related lemmas.
Section \ref{sec:proof-of-main} contains the proof of Theorem \ref{mainRODthm-repeat}, as well as further questions and examples.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{The Fock space $ {\mathcal{F}} _d$, nc functions and nc domains}
\begin{definition}
Let $\mathbbm{x} = \set{x_1,\dots, x_d}$ be a set of freely noncommuting indeterminates and write $\bm{x} = (x_1,\dots, x_d)$.
We write ${\mathbb C}\fralg{\mathbbm{x}}$ for the \textbf{free algebra} in the indeterminates $x_1, \dots, x_d$ and its elements are \textbf{free polynomials}.
We define an inner product on ${\mathbb C}\fralg{\mathbbm{x}}$ by declaring the ``words'' in the letters $x_i$ to be an orthonormal system:
\[
\ip{x_{i_1}\dots x_{i_n}}{x_{j_1}\dots x_{j_m}} := \begin{cases}
1 & m=n \text{ and } i_\ell = j_\ell \text{ for } 1\leq \ell \leq n;\\
0 & \text{otherwise}
\end{cases}
\]
This inner product induces a norm $\norm{\cdot}$ and the completion of ${\mathbb C}\fralg{\mathbbm{x}}$ in terms of this norm is $ {\mathcal{F}} _d$, the \textbf{full Fock space}
in $d$ letters. We may identify $ {\mathcal{F}} _d$ with the space of formal power series in the words $w$, with square-summable coefficients:
\[
{\mathcal{F}} _d = \set{\sum_{w\in \langle \bbx \rangle} a_w w \, : \, \sum_{w\in \langle \bbx \rangle} \abs{a_w}^2 < \infty }.
\]
\end{definition}
When $d=1$, every power series $\sum a_n z^n$ with square-summable coefficients has radius of convergence at least $1$, and hence defines a function $f(z)=\sum a_n z^n$ in the unit disk $|z|<1$. In our free setting, it turns out that the formal power series we are considering can also be viewed as {\em noncommutative (nc) function} on an appropriate {\em nc domain}. To make this precise we begin with some definitions.
The noncommutative analog of the disk $|z|<1$ will be the {\em row ball}:
\begin{definition} Let $d\geq 1$ be an integer. For a $d$-tuple of $k\times k$ matrices $X=(X_1, \dots, X_d)$ we define the {\em row norm} of $X$ to be
\[
\|X\|_{row}:= \|X_1X_1^*+\cdots +X_dX_d^*\|^{1/2}
\]
(that is, the usual operator norm of the $k\times dk$ matrix $(X_1 \ X_2\ \cdots X_d)$).
Similarly we define the {\em column norm}
\[
\|X\|_{col}:= \|X_1^*X_1+\cdots+ X_d^*X_d\|^{1/2}.
\]
We say $X$ is a {\em row contraction} if $\|X\|_{row}\leq 1$ and a {\em strict row contraction} if $\|X\|_{row}<1$. Column contraction and strict column contraction are defined similarly.
For fixed $d$ and each $k\geq 1$, we denote
\[
{\mathfrak{B}} ^d_k = \{X\in M_k({\mathbb C})^d : \|X\|_{row}<1\}
\]
and put
\[
{\mathfrak{B}} ^d := \bigsqcup_{k=1}^\infty {\mathfrak{B}} _k^d.
\]
The set $ {\mathfrak{B}} ^d$ is called the {\em row ball}, and we refer to $ {\mathfrak{B}} ^d_k$ as the $k^{th}$ ``level" of the row ball.
\end{definition}
We note that each $ {\mathfrak{B}} ^d_k$ is an open set in the usual topology on $M_k({\mathbb C})^d$, and $ {\mathfrak{B}} ^d$ is closed under direct sums: if $X\in {\mathfrak{B}} ^d_{k_1}$ and $Y\in {\mathfrak{B}} ^d_{k_2}$ then $X\mathrm{op}lus Y := (X_1\mathrm{op}lus Y_1, \dots, X_d\mathrm{op}lus Y_d)\in {\mathfrak{B}} ^d_{k_1+k_2}$. The row ball is thus an {\em nc domain}.
\iffalse
\fi
With these definitions, we have:
\begin{theorem} \cite[Theorem 1.1]{MR2264252}
If $\sum_{w}|a_w|^2<\infty$ and $X$ is a $d$-tuple of $k\times k$ matrices with $\|X\|_{row}<1$, then the series
\[
f(X):=\sum_{w\in \langle \bbx \rangle} a_w X^w
\]
converges in the usual topology of $M_k({\mathbb C})$.
\end{theorem}
Thus, every element of the Fock space $ {\mathcal{F}} _d$ determines a graded function on the row ball $f: {\mathfrak{B}} ^d_k\to M_k({\mathbb C})$. This function respects direct sums: $f(X\mathrm{op}lus Y)=f(X)\mathrm{op}lus f(Y)$ and similarities: if $S\in GL_k({\mathbb C})$ and both $\|X\|_{row}<1$, $\|S^{-1}XS\|_{row}<1$, then $f(S^{-1}XS)=S^{-1}f(X)S$. This $f$ is then an {\em nc function} on the row ball. (We remark that the theorem and subsequent remarks also hold with the column ball in place of the row ball).
In particular, for any $X$ in the row ball at level $k$, the map from $ {\mathcal{F}} _d$ to $M_k({\mathbb C})$ given by
\[
f\to f(X)
\]
is continuous for the norm topologies on each space. (One thinks of this as ``bounded point evaluation'' analogous to the point evaluation $f\to f(z)$ for $f$ in the Hardy space and $|z|<1$.) As a consquence we obtain the following proposition, relevant to our approximation problem:
\begin{theorem}\label{nonsingular-is-necessary} If $f$ is an nc polynomial and $p_n$ is a sequence of nc polynomials such that
\[
\lim_{n\to \infty}\norm{p_nf-1}_2 =0,
\]
then $f$ is nonsingular in the row ball.
\end{theorem}
\begin{proof}
Suppose $X$ is in the row ball and $\det f(X)=0$. Then also $\det(p_n(X)f(X))=0$ for all $n$, so each of the matrices $p_n(X)f(X)$ is singular, and hence these cannot converge to $I$.
\end{proof}
\subsection{The $d$-shift and cyclicity} From now on we think of the Fock space $ {\mathcal{F}} _d$ in this way, as a space of nc functions represented as convergent powers series in the row ball, in analogy with spaces of holomorphic functions in the disk. Let us introduce an important class of operators acting in the space $ {\mathcal{F}} _d$:
\begin{definition}
The \textbf{left $d$-shift} is the tuple of operators $L = (L_1,\dots, L_d)$ where each $L_i: {\mathcal{F}} _d\to {\mathcal{F}} _d$ is given by $L_i:f\mapsto x_if$.
We similarly define the \textbf{right $d$-shift}.
\end{definition}
In the nc function picture, for each $f\in {\mathcal{F}} _d$ and each $X\in {\mathfrak{B}} ^d$
\[
(L_if)(X) = X_i f(X).
\]
It is evident that each $L_i$ is a bounded operator for the $\ell^2$ norm on coefficients. In fact, it follows quickly from definitions that each of the operators $L_i$ is an isometry for the $\ell^2$ norm, and their ranges are mutually orthogonal. These facts are summarized algebraically in the relations
\[
L_i^*L_j = \delta{ij}I.
\]
For any polynomial $p\in {\mathbb C}\fralg{\mathbbm{x}}$, the ``left multiplication'' operator
\[
f(X)\to p(X)f(X)
\]
is bounded, since this operator is just $p(L_1, \dots, L_d)$. Similarly the ``right multiplication operators'' $f(X)\to f(X)p(X)$ are bounded.
Analogous to the notion of cyclicity in Hardy space, we make the following definition:
\begin{definition}\label{def:free-cyclic}
An element $f\in {\mathcal{F}} _d$ is said to be {\bf cyclic} for the $d$-shift if the set
\(
\{ pf: p\in {\mathbb C}\fralg{\mathbbm{x}}\}
\)
is dense in $ {\mathcal{F}} _d$.
\end{definition}
For a given free polynomial $f$, it is evident that $f$ is cyclic if and only if $c_n:=\|p_nf-1\|^2\to 0$. Indeed, if $f$ is cyclic then there is some sequence of polynomials $q_n$ so that $\|q_nf-1\|^2\to 0$, so by the optimality of the $p_n$ we must have $\|p_nf -1\|^2\to 0$ as well. On the other hand, if $\|p_nf-1\|^2\to 0$ and $q$ is any polynomial, then since $q$ is a bounded left multiplier, we have $\|qp_nf-q\|^2\to 0$ as well, so that $f$ is cyclic.
\subsection{Polynomials with matrix coefficients}\label{sec:matrix-coefficients} We will also have cause to deal with matrix-valued versions of $ {\mathcal{F}} _d$. For fixed $m,n$ we view $M_{m\times n}({\mathbb C})$ as a Hilbert space with the tracial inner product $\langle A, B\rangle = tr(B^*A)$, so that $M_{m\times n}({\mathbb C})$ is a Hilbert space with the Hilbert-Schmidt or {\em Frobenius} norm. Elements of the Hilbert space tensor product $M_{m\times n}({\mathbb C})\otimesimes {\mathcal{F}} _d$ may be identified with formal power series with $m\times n$ matrix coefficients
\[
\sum_{w\in \langle \bbx \rangle} A_w x^w
\]
with $\sum_{w\in \langle \bbx \rangle} tr(A_w^*A_w)<\infty$. These may in turn be identified with ``matrix-valued" functions on the row ball, where at each level $k$ for $\|X\|_{row}<1$ we have a convergent power series in $M_{m\times n}\otimesimes M_k$
\[
F(X): = \sum_{w\in\langle \bbx \rangle} A_w\otimesimes X^w.
\]
We again obtain bounded point evaluations $f\to f(X)$, and the analog of Theorem~\ref{nonsingular-is-necessary} holds in the (square) matrix-valued case as well.
Also as before, each matrix-valued polynomial $P\in M_{m\times n}( {\mathbb C}\fralg{\mathbbm{x}})$ will define bounded left and right multipliers $F\mapsto PF$, $F\mapsto FP$ between matrix-valued $ {\mathcal{F}} _d$ spaces of appropriate sizes. We will need the following lemma about multiplication by linear polynomials:
\begin{lemma}\label{lem:contractive-pencil}
Let $A=(A_1, \dots, A_d)\in M_m({\mathbb C})^d$ let and $P(x):= A_1x_1+\cdots +A_dx_d$. Then the left multiplication opeartor $F\mapsto PF$ has norm equal to $\|A\|_{col}$.
\end{lemma}
\begin{proof}
Writing $L_i$ for the left shifts as above, the operator $F\to PF$ acting in the Hilbert space tensor product $M_m\otimesimes {\mathcal{F}} _d$ is given by $\sum_{j=1}^d A_j\otimesimes L_j$. (Here we identify $A_j$ with the operator $B\mapsto A_jB$ acting in the Hilbert space $M_m({\mathbb C})$.) From definitions, the operator $B\mapsto CB$ in the tracial Hilbert space $M_m({\mathbb C})$ has norm equal to $\|C\|_{op}$. Since the $L_j$ are isometries with orthogonal ranges, the relations $L_i^*L_j=\delta_{ij}$ entail
\begin{align*}
\norm{\bm{r}kt{\sum_{j=1}^d A_j\otimesimes L_j}}^2
&= \norm{\bm{r}kt{\sum_{j=1}^d A_j\otimesimes L_j}^*\bm{r}kt{\sum_{j=1}^d A_j\otimesimes L_j}} \\
&=\norm{\sum_{j=1}^d (A_j^*A_j)\otimesimes I}
= \norm{\sum_{j=1}^d (A_j^*A_j)}_{op} \\
&=\norm{A}_{col}^2.
\end{align*}
\end{proof}
\iffalse
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{\varsigma}kip.2in
\iffalse
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\iffalse
\fi
\subsection{Stable equivalence and linearization}\label{sec:stabeq}
\begin{definition}
An\, $m\times\ell$\, nc\, linear pencil (in $d$ indeterminates) is an expression of the form
\[L_A(\mathbbm{x})=A_0+A\mathbbm{x}\] where $A\mathbbm{x}=A_1x_1\,+\,\cdots\,+\,A_dx_d$ and $A_i\in M_{m\times\ell}({\mathbb C})$ for $i = 1, \dots, d$. If $A_0=I$ \,then\, we call \,$L_A(\mathbbm{x})$\, a monic linear pencil. A matrix tuple $A = (A_1,\dots, A_d)\in M_m({\mathbb C})^d$ is \textbf{irreducible} if $A$ generates $M_m({\mathbb C})$ as a (unital) algebra, i.e.
$M_m({\mathbb C}) = \set{p(A) \, : \, p\in {\mathbb C}\fralg{\mathbbm{x}}}$.
The monic linear pencil $L_A$ is \textbf{irreducible} if $A$ is irreducible.
\end{definition}
The protagonist of our proof is stable associativity. Let us define that as well:
\begin{definition}
Given\, $A\in M_{k\times k}({\mathbb C}\fralg{\mathbbm{x}})$ and $B\in M_{\ell\times \ell}({\mathbb C}\fralg{\mathbbm{x}})$.\newline
We say $A$ and $B$ are \textbf{stably associated} if there exists $\, N\in {\mathbb Z}^+$ and $P,Q\in \mathrm{GL}_N({\mathbb C}\fralg{\mathbbm{x}})$
\, such\, that
\[
P\begin{pmatrix} A & \\ & I {\varepsilon}m Q = \begin{pmatrix} B & \\ & I {\varepsilon}m.
\]{\varsigma}pace{2mm}
We use the notation $A\sim_{\mathrm{sta}} B$ for $A$ being stable associated to $B$; it is evident that this is an equivalence relation.
\end{definition}
\begin{definition}
If $f\in M_{k\times k}({\mathbb C}\fralg{\mathbbm{x}})$, then \[ {\mathscr{Z}} _n(f) = \set{X\in M_n({\mathbb C})^d \, : \, \det(f(X)) = 0},\] and \[ {\mathscr{Z}} (f) :=\bigsqcup_{n\geq1} {\mathscr{Z}} _n(f).\]
The set $ {\mathscr{Z}} (f)$ is the \textbf{free zero locus} of $f$.
\end{definition}
It is also evident from the definitions that if $f\sim_{sta} g$ then $ {\mathscr{Z}} (f)= {\mathscr{Z}} (g)$. The concept of stable associativity is useful to us because of the following classical fact (we refer to \cite{HKVfactor} for a discussion of this in the context of factorization of free polynomials):
\begin{lemma}[Linearization trick]\label{linearization trick}
Suppose $F\in M_k({\mathbb C}\fralg{\mathbbm{x}})$ and $F(0) = I$. Then $F$ is stably associated to a monic linear pencil. Moreover, by Burnside's theorem we get that every monic pencil is similar to a pencil of the form
\[ L_A(x)=
\begin{bmatrix}
L_1(x) & * & * & \bm{h}dots & * \\
0 & L_2(x) & * & \bm{h}dots & * \\
0 & 0 & L_3(x) & \bm{h}dots & * \\
\vdots & \vdots &\vdots & \ddots & \vdots\\
0 & 0 & 0 & \bm{h}dots & L_l(x)
\end{bmatrix}
\]
where for every $k$, $L_k(x) = I-\sum_{j=1}^d {\mathcal{B}} ar{A}^{(k)}_jx_j = I- {\mathcal{B}} ar{A}^{(k)}x$ for $ {\mathcal{B}} ar{A}^{(k)}=( {\mathcal{B}} ar{A}^{(k)}_1\: {\mathcal{B}} ar{A}^{(k)}_2\: \cdots \: {\mathcal{B}} ar{A}^{(k)}_d)$ and either $ {\mathcal{B}} ar{A}^{(k)}=0$ or irreducible.
\end{lemma}
For example, the $1\times 1$ polynomial $F(x,y)=1-xy$ is stably associated to the $2\times 2$ monic linear pencil $\begin{pmatrix} 1 & x \\ y & 1{\varepsilon}m$ via
\[
\begin{pmatrix} 1 & x \\ 0 & 1 {\varepsilon}m \begin{pmatrix} 1 - xy & 0 \\ 0 & 1 {\varepsilon}m \begin{pmatrix} 1 & 0 \\ y & 1 {\varepsilon}m = \begin{pmatrix} 1 & x \\bm{y} & 1 {\varepsilon}m.
\]
(This calculation is sometimes known as ``Higman's trick.'') In fact the full lemma can be proved by iteratively applying this trick (increasing the size of the matrices at each step) to gradually reduce the degree of each monomial appearing in the entries of $F$.
We say $F\in M_k({\mathbb C}\fralg{\mathbbm{x}})$ is \textbf{regular} if it is not a zero divisor. In particular, if $F(0) = I$ then $F$ is regular.
A regular non-invertible matrix polynomial $F$ is an \textbf{atom} if it is not the product of two non-invertible matrices in $M_k({\mathbb C}\fralg{\mathbbm{x}})$.
When $k=1$, every nonzero polynomial is regular and a nonconstant polynomial $p$ is an atom if it is not the product of two nonconstant
polynomials.
If $F_1,F_2$ are regular matrices over ${\mathbb C}\fralg{\mathbbm{x}}$ (not necessarily of the same size) and $F_1\sim_{\mathrm{sta}} F_2$, then $F_1$
is an atom if and only if $F_2$ is an atom. The following lemma from \cite{HKVfactor} relates irreducibility of the polynomial $F$ to irreducibility of the pencil $L_A$.
\begin{lemma}\cite[Lemma 4.2]{HKVfactor}
If $F\in M_k({\mathbb C}\fralg{\mathbbm{x}})$ is an atom and $F(0)=I$, then $F$ is stably associated to an irreducible monic linear pencil $L$.
That is, if $F$ is an atom, then there exists an irreducible tuple $A\in M_m({\mathbb C})^d$ such that $F$ is stably associated to $L_A = I_m
- \sum_{i=1}^d A_ix_i$.
\end{lemma}
It is worth noting that in general, $L_A$ is not guaranteed to have a nonempty free zero locus.
In fact, $ {\mathscr{Z}} (L_A) = \varnothing$ if and only if $A$ is a jointly nilpotent tuple (i.e. there exists $N\in {\mathbb N}$ such that $w(A) = 0$ for
all $\abs{w}\geq N$).
However, if $A$ is irreducible, then $A$ cannot be jointly nilpotent, thus $ {\mathscr{Z}} (L_A) \neq \varnothing$.
If $F$ is stably associated to $L_A$, then for any invertible $S$ of the same size as $A$, $F$ is stably associated to $L_{S^{-1}AS}$.
\iffalse
\fi
The following lemma shows that our rate-of-decay problem is invariant under stable associativity. Precisely:
\begin{lemma}\label{same ROD}
Suppose $F$ and $G$ are matrix polynomials.
If $F$ and $G$ are stably associated, then there exists $C_1,\,C_2\in {\mathbb R}R$ and $D_1, D_2\in {\mathbb N}$ such that for $P_n$,\,$Q_n$, the $n$th degree OPAs of $F$ and $G$ respectively, the following holds:
\[
C_1\norm{P_{n+D_1}F - I}_F^2 \leq \norm{Q_nG- I}_F^2 \leq C_2\norm{P_{n-D_2}F - I}_F^2.
\]
\end{lemma}
\begin{proof}
Since $F$ and $G$ are stably associated, there exists $N\in {\mathbb Z}^+$ and $A,B\in \mathrm{GL}_N({\mathbb C}\fralg{\mathbbm{x}})$ such that
\[
(F\mathrm{op}lus I) = A(G\mathrm{op}lus I)B.
\]
Since $B$ and $B^{-1}$ are both polynomials, it follows that the map $H\mapsto BHB^{-1}$ is bounded in $ {\mathcal{F}} _d$, so there there exists $C_1\in {\mathbb R}R$ such that
$\norm{BHB^{-1}}_F\leq C_1\norm{H}_F$ for all $H\in M_N({\mathbb C}\fralg{\mathbbm{x}})$.
Next, set $D_1 = \deg(A) + \deg(B)$ (here, $\deg(A)$ is the maximum of the degrees taken over the entries of $A$).
Observe
\begin{align*}
B\begin{pmatrix} P_n & \\ & I {\varepsilon}m A \begin{pmatrix} G & \\ & I {\varepsilon}m - \begin{pmatrix} I & \\ & I {\varepsilon}m
&= B\left(\begin{pmatrix} P_n & \\ & I {\varepsilon}m A \begin{pmatrix} G & \\ & I {\varepsilon}m B - \begin{pmatrix} I & \\ & I{\varepsilon}m\right)B^{-1} \\
&= B\left(\begin{pmatrix} P_n & \\ & I {\varepsilon}m \begin{pmatrix} F & \\ & I {\varepsilon}m - \begin{pmatrix} I & \\ & I{\varepsilon}m\right)B^{-1}.
\end{align*}
Hence,
\begin{align*}
\Norm{B\begin{pmatrix} P_n & \\ & I {\varepsilon}m A \begin{pmatrix} G & \\ & I {\varepsilon}m - \begin{pmatrix} I & \\ & I{\varepsilon}m}_F^2
&\leq C\Norm{\begin{pmatrix} P_n & \\ & I {\varepsilon}m \begin{pmatrix} F & \\ & I {\varepsilon}m - \begin{pmatrix} I & \\ & I{\varepsilon}m}_F^2 \\
&= C\Norm{P_nF - I}_F^2
\end{align*}
The matrix $B(P_n\mathrm{op}lus I)A$ has a block structure of $\bm{s}bm Q & * \\ * & * \varnothingbm$, and we note that the nature of the Frobenius norm
implies that
\[
\norm{QG-I}_F^2\leq \Norm{\begin{pmatrix} Q & * \\ * & * {\varepsilon}m \begin{pmatrix} G & \\ & I {\varepsilon}m - \begin{pmatrix} I & \\ & I {\varepsilon}m}_F^2.
\]
Thus,
\[
\norm{QG-I}_F^2\leq C\norm{P_nF-I}_F^2.
\]
Finally, note that
\[
\deg(Q)\leq \deg(B(P_n\mathrm{op}lus I)A) \leq \deg(B) + \deg(P_n) + \deg(A) = \deg(P_n) + D_1.
\]
For $n$th degree OPA, $Q_n$ of $G$, $\|Q_nG-I\|$ decreases with $n$, thus, we can write
\[\norm{Q_nG- I}_F^2 \leq C_1\norm{P_{n-D_1}F - I}_F^2.\]
The other inequality follows by interchanging the roles of $F$ and $G$, and chasing through the proof, we find that in this case we can take $D_1=\deg(A^{-1})+\deg(B^{-1})$.
\end{proof}
\subsection{The Outer Spectral Radius}\label{sec:OuterSpecRad}
The last technical tool we will need is the {\em outer spectral radius}, which is one of several possible notions of a ``joint spectral radius" for a system of matrices $A=(A_1, \dots, A_d)$.
\begin{definition}
Let $n\in\mathbb{N}$ and $X\in M_{n\times n}({\mathbb C})^d$. We associate to $X$ the completely positive map on $M_{n\times n}$ defined by
\[
\Psi_X(T) \,=\, \sum^d_{j=1}X_j T X_j^*.
\]
The \textbf{outer spectral radius} is defined to be $\rho(X) \,:=\,\lim_{k\to \infty} \|\Psi_X^k(I)\|^{1/2k}$ (that is, the square root of the spectral radius of $\Psi_X$ viewed as a linear transformation on $M_n$).
\end{definition}
\begin{remark}
Observe that if we equip $M_n(\mathbb{C})$ with tracial inner product then $\Psi_X^*=\Psi_{X^*}$, so $\rho(X)=\rho(X^*)$.
\end{remark}
There are two other equivalent definitions of outer spectral radius \cite{MR4344050}:
\iffalse
\fi
Several properties of the outer spectral radius are pointed out in \cite{SSS20} Section 4;
\begin{enumerate}
\item $\rho(X) = \rho(S^{-1}XS)$ for any $S\in \mathrm{GL}_n({\mathbb C})$, follows from the definition $2.16$;
\item $\rho(X)=\rho(X^*)$, follows from remark $2.17$;
\item $\rho(X)<1$ if and only if $X$ is similar to an element of the row ball, this follows from Theorem $3.8$, \cite{PopescuSimilaritypoly} and Lemma $2.5$, \cite{SSS20};
\item if $X$ is irreducible, then $\rho(X)=\min\set{\Norm{S^{-1}XS} \, : \, S\in \mathrm{GL}_n({\mathbb C})}$, follows from Lemma $2.4$, \cite{SSS20};
\item if $X$ is irreducible and $\rho(X) = 1$, then $X$ is similar to a row coisometry, from Lemma $2.9$. \cite{SSS20}.
\end{enumerate}
We remark in passing that the last item is proved using the so-called ``quantum Perron-Frobenius theorem'' of Evans and H{\o}egh-Krohn \cite{EvansAndHK}.
\begin{lemma}\label{M is contraction}
If $A = (A_1,\dots,A_d)$ is irreducible with outer spectral radius $\rho(A)\leq 1$ then $A$ is similar to a column contraction.
\end{lemma}
\begin{proof}
As $A$ is irreducible then $A^*$ is irreducible as well. Also by definition of spectral radius, $\rho(A^*)=\rho(A)\leq 1$. Then by properties $(3)$ and $(5)$ above, we have that $A^*$ is similar to a row contraction, say $X$. That is, there exists an invertible, $S$ such that $A^* = SXS^{-1}$, taking adjoints gives $A=S^{-*}X^*S^*$. If $X$ is a row contraction then $X^*$ is a column contraction, and so we see that $A$ is similar to a column contraction.
\end{proof}
We also require the following lemma from \cite{jury2020noncommutative}:
\begin{lemma} \cite[Proposition 4.1]{jury2020noncommutative}\label{lem:radius-1-no-zeroes}
A monic linear pencil $I-Ax$ is nonsingular in the row ball if and only if $\rho(A)\leq 1$.
\end{lemma}
\section{Proof of Theorem~\ref{mainRODthm-repeat} \label{sec:proof-of-main}}
The proof of the theorem consists of first proving it in the special case where $F$ is a contractive, monic linear pencil (accomplished by the following lemma), then reducing the general case to that one by means of the algebraic and analytic machinery of the previous section, and finishing by induction.
Throughout the proof, for a matrix polynomial $P(x)=\sum_{w\in \langle \bbx \rangle} P_w x^w$ we will write $\|P\|_2$ for the Hilbert space norm $(\sum_{w\in\langle \bbx \rangle} tr(P_w^*P_w))^{1/2}$ and $\|P\|_{mult}$ for the norm of $P$ as a left multiplier $F\to PF$.
\begin{lemma}\label{The Lemma}
Assume that $M$ is column contraction: $\|M\|_{{col}}\leq 1$ and let $\pi_n$ be the one variable optimal polynomial approximants for $f=1-z$. Then
\begin{enumerate}[(a)]
\item \(\norm{\pi_n(Mx)}_{mult} \lesssim n\), and \label{infnorm}
\item \(\norm{\pi_n(Mx)(I-Mx)-I}^2_2 \lesssim \frac{1}{n}.\) \label{twonorm}
\end{enumerate}
\end{lemma}
\begin{proof}
We begin by recalling the one variable commutative case \(f(z)=1-z\) where the \(H^2({\mathbb D})\) optimal polynomial approximants are given in~\cite{FMS14} by
\begin{equation}
\pi_n(z) = \sum_{k=0}^n \bm{r}kt{1-\frac{k+1}{n+2}}z^k
\end{equation}
To show (\ref{infnorm}), observe that
\begin{equation}
\norm{\pi_n}_\infty := \sup\{|\pi_n(z)|:|z|<1\}= \sum_{k=0}^n \bm{r}kt{1-\frac{k+1}{n+2}} = \frac{n+1}{2}. \label{FMSsupnorm}
\end{equation}
Because \(M\) is a column contraction, by Lemma~\ref{lem:contractive-pencil} we have that multiplication by \(Mx\) is a contraction, so we can apply von Neumann's inequality to see that
\begin{equation*}
\norm{\pi_n(Mx)}_{mult} \leq \norm{\pi_n}_\infty \lesssim n
\end{equation*}
as needed.
For part (\ref{twonorm}), again consider the one variable case and use algebra to see that
\begin{align}
\pi_n(z)(1-z)-1
&= (1-z)\sum_{k=0}^n \bm{r}kt{1-\frac{k+1}{n+2}}z^k -1 \nonumber \\
&= -\frac{1}{n+2} \sum_{k=1}^{n+1} z^k . \label{1voptimalnorm}
\end{align}
Again, since multiplication by $Mx$ is contractive, we have $\|(Mx)^k\|_2\leq \|Mx\|_2:=c$ for all $k\geq 1$. Moreover, since the \((Mx)^k\) are orthogonal (they are homogeneous polynomials of different degrees), by the Pythagorean theorem we have
\begin{align}
\norm{\pi_n(Mx)(I-Mx)-I}_2^2
&= \norm{-\frac{1}{n+2} \sum_{k=1}^{n+1} (Mx)^k}_2^2 \nonumber \\
&= \frac{1}{(n+2)^2} \sum_{k=1}^{n+1} \norm{(Mx)^k}_2^2 \leq \frac{c^2(n+1)}{(n+2)^2}, \label{MXoptimalnorm}
\end{align}
which gives \(\norm{\pi_n(Mx)(I-Mx)-I}^2_2 \lesssim \frac {1}{n}\).
\end{proof}
Let us now prove Theorem~\ref{mainRODthm-repeat}. We recall the statement:
\begin{theorem*}(Theorem \ref{mainRODthm-repeat})
Let $F\in M_k({\mathbb C})\otimesimes {\mathbb C}\fralg{\mathbbm{x}}$ be a $k\times k$-matrix valued free polynomial, let $(P_n)$ denote the sequence of optimal polynomial approximants, and put
\[
c_n:=\|P_n(x)F(x)-I\|_2^2.
\]
Then: \begin{enumerate}
\item $c_n\to 0$ if and only if $F$ is nonsingular in the row ball.
\item If $F$ is nonsingular in the row ball, then
\[
c_n\lesssim \frac{1}{n^p}
\]
for some $p>0$ depending on $F$. In particular, if $F$ is a product of $\ell$ atomic factors, we can take $p=\frac{1}{3^{\ell-1}}$.
\end{enumerate}
\end{theorem*}
\begin{proof}[Proof of Theorem~\ref{mainRODthm-repeat}]
By Theorem~\ref{nonsingular-is-necessary} (and the remarks in Section~\ref{sec:matrix-coefficients}), if $c_n\to 0$ then necessarily $F$ is nonsingular in the row ball. The other direction of item (1) will evidently follow from item (2).
Let
$F\in M_k({\mathbb C}\fralg{\mathbbm{x}})$ and assume it is nonsingular on $ {\mathfrak{B}} ^d$.
We can assume, without loss of generality, that $F(0)=I$.
Then by lemma \ref{linearization trick} we have that $F$ is stably associated to a monic linear pencil, say, $L_A(x)$ and
\[
L_A(x)\sim\begin{bmatrix}
L_1(x) & * & * & \bm{h}dots & * \\
0 & L_2(x) & * & \bm{h}dots & * \\
0 & 0 & L_3(x) & \bm{h}dots & * \\
\vdots & \vdots &\vdots & \ddots & \vdots\\
0 & 0 & 0 & \bm{h}dots & L_l(x)
\end{bmatrix}
\]
where for each $k$ the linear pencil $L_k(x)=I- {\mathcal{B}} ar{A}^{(k)}x$ has $ {\mathcal{B}} ar{A}^{(k)}$ either $0$ or irreducible.
By hypothesis, our original $F$ was nonsingular in the row ball, and since stable associativity preserves the zero locus, the pencil $L_A$, and hence also each of the $L_{ {\mathcal{B}} ar{A}^{(k)}}$, is also nonsingular in the row ball. Thus by Lemma~\ref{lem:radius-1-no-zeroes}, each of the pencils $ {\mathcal{B}} ar{A}^{(k)}$ has outer spectral radius at most $1$. It follows, in turn, by Lemma~\ref{M is contraction}, that each $ {\mathcal{B}} ar{A}^{(k)}$ is similar to a column contractive pencil $M^{(k)}$.
In summary, we conclude that that our $F$ is stably associated to a monic linear pencil of the following block upper-triangular form:
\begin{equation}\label{eqn:main-reduction-step}
F\sim_{\mathrm{sta}}\begin{bmatrix}
I-M^{(1)}(x) & * & * & \bm{h}dots & * \\
0 & I-M^{(2)}(x) & * & \bm{h}dots & * \\
0 & 0 & I-M^{(3)}(x) & \bm{h}dots & * \\
\vdots & \vdots &\vdots & \ddots & \vdots\\
0 & 0 & 0 & \bm{h}dots & I-M^{(l)}(x)
\end{bmatrix}
\end{equation}
where each $M^{(k)}$ is a column contraction. Note that the off-diagonal entries are arbitrary linear pencils $Y_{ij}(x)$ (genuinely linear, with no constant term). \\
\begin{comment}
We prove our result via induction on $l$ that is, we prove that there exists a matrix polynomial $g^{(l)}$ such that $|| g^{(l)}F-1 ||\longrightarrow 0$.\\
For $l=2$, let us assume
\[
F\sim\begin{bmatrix}
I-M^{(1)}(x) & Y^{(2)}(x) \\
0 & I-M^{(2)}(x) \\
\end{bmatrix}
\]
where for $k=1,\:2$, $M^{(k)}$ is a column contraction.\\
\newline
Consider the polynomial matrix, $g^{(2)}=\begin{pmatrix}
p_2 & r_2\\
0 & q_2
\end{pmatrix}$ where $p_2,\,r_2,\,q_2$ are polynomials of any degree.
Then
\[
g^{(2)}F=\begin{pmatrix}
p_2(x)(I-M^{(1)}(x)) & p_2(x)Y^{(2)}(x)+r_2(x)(I-M^{(2)}(x))\\
0 & q_2(x)(I-M^{(2)}(x))
\end{pmatrix}.
\]
Now since $I-M^{(1)}(x)$ is cyclic in Fock space, there exists a matrix polynomial, $p_2(x)$ such that $p_2(x)(I-M^{(1)}(x))$ can be made close to $I$.
Similarly as $(I-M^{(2)})$ is cyclic in Fock space we can find a matrix polynomials, $q_2(x)$ and $r_2(x)$ such that $q_2(x)(I-M^{(2)}(x))$ and $r_2(x)(I-M^{(2)}(x))$ can be made as close to $I$ and $p_2(x)Y^{(2)}(x)$ respectively, as we want.
(Induction hypothesis) Let us assume that the result is true for all matrices of size $l-1$.
For \[
f\sim\begin{bmatrix}
I-M^{(1)}(x) & * & * & \bm{h}dots & Y_1(x) \\
0 & I-M^{(2)}(x) & * & \bm{h}dots & Y_2(x) \\
0 & 0 & I-M^{(3)}(x) & \bm{h}dots & Y_3(x)\\
\vdots & \vdots &\vdots & \ddots & \vdots\\
0 & 0 & 0 & \bm{h}dots & I-M^{(l)}(x)
\end{bmatrix}
\]
where $M^{(k)}$ is a column contraction.\\
Assume \[g^{(l)}=\begin{bmatrix}
g_1^{(1)} & g_1^{(2)} & \bm{h}dots & g_1^{(l-2)} & g_1^{(l-1)} & g_1^{(l)}\\
0 & g_2^{(2)} & \bm{h}dots & g_2^{(l-2)} & g_2^{(l-1)} & g_2^{(l)}\\
\vdots & \vdots & \ddots & & \vdots & \vdots \\
\vdots & \vdots & & \ddots & \vdots & \vdots \\
0 & 0 & \bm{h}dots & \bm{h}dots & g_{l-1}^{(l-1)} & g_{l-1}^{(l)}\\
0 & 0 & \bm{h}dots & \bm{h}dots & 0 & g_{l}^{(l)}\\
\end{bmatrix} \]
Let us rewrite the matrix polynomials, $f$ and $g$ by decomposing them in the following manner: \\
$F\sim\begin{bmatrix}
\begin{array}{cccc|c}
I-M^{(1)}(x) & * & * & \bm{h}dots & Y_1(x) \\
0 & I-M^{(2)}(x) & * & \bm{h}dots & Y_2(x) \\
0 & 0 & I-M^{(3)}(x) & \bm{h}dots & Y_3(x) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\bm{h}line
0 & 0 & 0 & \bm{h}dots & I-M^{(l)}(x)
\end{array}
\end{bmatrix}=\begin{bmatrix}
\begin{array}{c|c}
f^{(l-1)}(x) & Y^{(l)}(x) \\
\bm{h}line
0 & I-M^{(l)}(x)
\end{array}
\end{bmatrix}$
\newline
and
\newline
$g^{(l)}=\begin{bmatrix}
\begin{array}{ccccc|c}
g_1^{(1)} & g_1^{(2)} & \bm{h}dots & g_1^{(l-2)} & g_1^{(l-1)} & g_1^{(l)}\\
0 & g_2^{(2)} & \bm{h}dots & g_2^{(l-2)} & g_2^{(l-1)} & g_2^{(l)}\\
\vdots & \vdots & \ddots & & \vdots & \vdots \\
\vdots & \vdots & & \ddots & \vdots & \vdots \\
0 & 0 & \bm{h}dots & \bm{h}dots & g_{l-1}^{(l-1)} & g_{l-1}^{(l)}\\
\bm{h}line
0 & 0 & \bm{h}dots & \bm{h}dots & 0 & g_{l}^{(l)}
\end{array}
\end{bmatrix}=\begin{bmatrix}
\begin{array}{c|c}
p_{l}(x) & r_{l}(x) \\
\bm{h}line
0 & g_{l}^{(l)}
\end{array}
\end{bmatrix}$.\\
Then
\[
g^{(l)}F = \begin{bmatrix}
p_l f^{(l-1)} & p_l Y^{(l)} + r_l (I-M^{(l)}(x))\\
0 & g_l^{(l)}(I-M^{(l)}(x))
\end{bmatrix}.
\]
\newline
Again since $(I-M_l(x))$ is cyclic in Fock space we can find a matrix polynomial $q_l^{(l)}$ such that $q_l^{(l)}(I-M_l(x))$ is close to $I$.
Now by induction hypothesis we have that as $F^{(l)}$ is a $l-1$ order matrix, we can find a matrix polynomial, $p_l(x)$, such that $F^{(l-1)}p_l$ can be made very close to $I$. Similarly there exists a matrix polynomial, $r_l(x)$ such that $F^{(l-1)}r_l+Yq_l^{(l)}$ can be made to close to $0$.
Improved proof:\,\emph{Proof by carefully choosing polynomials, $g_k^{(m)}$ in block matrix polynomial, $g^{(l)}$.} \newline
From above proof we get that our $F$ is similar to the following block upper-triangular form:
\begin{equation}
F\sim\begin{bmatrix}
I-M^{(1)}(x) & * & * & \bm{h}dots & * \\
0 & I-M^{(2)}(x) & * & \bm{h}dots & * \\
0 & 0 & I-M^{(3)}(x) & \bm{h}dots & * \\
\vdots & \vdots &\vdots & \ddots & \vdots\\
0 & 0 & 0 & \bm{h}dots & I-M^{(l)}(x)
\end{bmatrix}
\end{equation}
where $M^{(k)}$ is a column contraction.\\
\end{comment}
By Remark $5.2$ and Section $6.2$ of \cite{HKVfactor}, the number of blocks $\ell$ will be equal to the number of atomic factors of $F$.
We now consider a monic linear pencil $L_A(x)$ of the form appearing in the right-hand side of (\ref{eqn:main-reduction-step}), in $\ell\times \ell$ block upper triangular form, with contractive pencils down the diagonal. We prove the following
{\bf Claim:} {\em for each such pencil $L_A(x)$, there exists a sequence of matrix polynomials $\sigma_{n,\ell}(x)$, $n=1, 2, \dots, $ satisfying the following conditions:
\begin{enumerate}
\item $\deg \sigma_{n,\ell} \leq (\ell-1) +n +n^3+ \cdots +n^{3^{\ell-1}} \lesssim n^{3^{\ell-1}}$,
\item $\|\sigma_{n,\ell}\|_{mult}\lesssim n^{(1+3+3^2+\dots+3^{l-1})}$, and
\item $\|\sigma_{n,\ell}(x) (I-Ax)-I\|_2^2 \lesssim \frac1n$
\end{enumerate}
for all $n$, where the implied constants are allowed to depend on $\ell$ but not on $n$. }
Assuming the claim for the moment, we conclude that for this fixed $A$ (which fixes $\ell$) there exist, for each $n$, matrix polynomials $\sigma_N:=\sigma_{n, \ell}$ of degree $N\lesssim n^{3^{\ell-1}}$ such that
\[
\|\sigma_N(x) (I-Ax)-I\|_2^2 \lesssim \frac1n
\]
It follows that for the optimal approximants $P_N$ at this same degree $N$, we will also have $\| Q_N(x)(I-Ax)-I\|_2^2\lesssim \frac1n\lesssim \frac{1}{N^p}$ (where $p=\frac{1}{3^{\ell-1}}$), and thus (since these quantities decrease as the degree $N$ increases), we conclude that for all degrees $n$
\[
\|Q_n(x) (I-Ax)-I\|_2^2 \lesssim \frac{1}{n^p}
\]
where $p=\frac{1}{3^{\ell-1}}$. Finally, since $I-Ax$ was stably equivalent to our original matrix polynomial $F$, we conclude by Lemma \ref{same ROD} that the optimal approximants $P_n$ of the original $F$ achieve the same rate of decay.
To complete the proof, it remains to prove the claim, which we will do via induction on $\ell$.
In the case $\ell=1$, our pencil $L_A(x)$ has the form $1-M^{(1)}x$ for some column contractive irreducible pencil $M^{(1)}$, so we can take $\sigma_{n,1}$ to be the polynomials provided by Lemma \ref{The Lemma}, namely $\sigma_{n,1}(x) =\pi_n(M^{(1)}x)$ where $\pi_n$ are the $1$-variable OPAs for the polynomial $f(z)=1-z$.
Now suppose the claim is proved for $\ell$, let us prove it for $\ell+1$. We write our pencil in the form
\begin{equation*}
L_A(x) = \begin{bmatrix} I-\bm{w}idetilde{A}x & Yx \\ 0 & I-M^{(\ell+1)}x\end{bmatrix}
\end{equation*}
where $\bm{w}idetilde{A}$ has at most $\ell$ irreducible blocks.
We will choose $\sigma_{n,\ell+1}$ of the form
\[
\sigma_{n, \ell+1} (x) =\begin{bmatrix}\sigma_{n,\ell}(x) & r(x) \\ 0 & q(x)\end{bmatrix}
\]
where $\sigma_{n,\ell}$ is chosen by applying the induction hypothesis to the block $\ell\times\ell$ pencil $I-\bm{w}idetilde{A}x$.
We choose $q(x)$ to be $\pi_n(M^{(\ell+1)}(x))$, as in Lemma \ref{The Lemma}, so $\deg(q)\leq n$ and $\|q\|_{mult}\lesssim n$ by the lemma.
Finally we define
\[
r(x) = -\sigma_{n,\ell}(x) \cdot (Yx) \cdot \pi_{N}(M^{(\ell+1)}x),
\]
where again $\pi_N$ is the one-variable OPA for $f(z)=1-z$, at degree $N=n^{3^\ell}$. Let us now verifty the claims (1)-(3). First, the degree of $\sigma_{n, \ell+1}$ will be the maximum of the degrees of $\sigma_{n,\ell}, r(x)$, and $q(x)$, and we see by inspection that $r(x)$ has the largest degree, which is
\begin{align}
\deg r(x) &= \deg(\sigma_{n, \ell}) + 1 + n^{3^\ell}\\
&\leq \ell + n+ n^3 + \cdots + n^{3^{\ell}}.
\end{align}
which proves (1).
For (2), observe that the $\|\cdot\|_{mult}$ norm of $\sigma_{n, \ell+1}$ is comparable to the maximum of the $\|\cdot\|_{mult}$ norms of its entries; by the induction hypothesis, Lemma \ref{The Lemma}, and the definition of $r(x)$, we have
\begin{align*}
\|r(x)\|_{mult} &\leq \|\sigma_{n, \ell}\|_{mult} \|Yx\|_{mult} \|\pi_N(M^{(\ell+1)}x)\|_{mult} \\
&\lesssim n^{1+3+\cdots +3^{\ell}}.
\end{align*}
Since this is larger than $\|\sigma_{n, \ell}\|_{mult}$ and $\|q(x)\|_{mult}$, (2) is proved.
Finally for (3), we have
\begin{equation*}
\sigma_{n, \ell+1}(x) (I-A^{(\ell+1)}x) = \begin{bmatrix} \sigma_{n, \ell}(x) (I-A^{(\ell)}x) & \sigma_{n, \ell}(x)\cdot Yx + r(x) (I-M^{(\ell+1)}x) \\ 0 & q(x) (I-M^{(\ell+1)}x)\end{bmatrix}
\end{equation*}
Subtracting $\begin{bmatrix} I & 0 \\ 0 & I\end{bmatrix}$ and taking the (squared) Hilbert space norm, in the diagonal entries we have by the induction hypothesis and the Lemma
\[
\| \sigma_{n, \ell}(x) (I-A^{(\ell)}x)-I \|_2^2 \lesssim \frac1n
\]
and
\[
\|q(x) (I-M^{(\ell+1)}x) -I\|_2^2 = \|\pi_{n}(M^{(\ell+1)}x) (I-M^{(\ell+1)}x) -I\|_2^2 \lesssim \frac1n.
\]
In the off-diagonal entry, we have by the definition of $r(x)$, the induction hypothesis, and the Lemma (applied at degree $N=n^{3^\ell}$),
\begin{align*}
\|\sigma_{n, \ell}(x)\cdot Yx + &r(x) (I-M^{(\ell+1)}x) \|_2^2 \leq \\ &\leq\|\sigma_{n,\ell}\|_{mult}^2 \|\|Yx\|_{mult}^2 \|I - \pi_N(M^{(\ell+1)}x) (I-M^{(\ell+1)}x\|_2^2 \\
&\lesssim n^{2(1+3+\cdots 3^{(\ell-1)})} \cdot \frac{1}{n^{3^\ell}}\\
&= n^{3^\ell -1} \cdot \frac{1}{n^{3^\ell}}\\
&=\frac1n.
\end{align*}
This completes the proof of the claim.
\end{proof}
\iffalse
\fi
As an immediate consequence of the theorem (and the remarks following Definition~\ref{def:free-cyclic}), we obtain a new proof of the following fact, first proved in \cite{jury2020noncommutative}:
\begin{corollary} Let $f$ be an nc polynomial with scalar coefficients. If $f$ is nonvanishing in the row ball, then $f$ is cyclic for the $d$-shift.
\end{corollary}
\iffalse
\fi
\subsection{Remarks and Questions} In general, the exponent $p=\frac{1}{3^{\ell-1}}$ appearing in Theorem~\ref{mainRODthm-repeat} (where $\ell$ is the number of irreducible factors of $F$) is not sharp. Already in one variable, the bound $c_n\lesssim \frac{1}{n}$ holds regardless of the number of factors, as was already mentioned in the introduction.
It is possible to run the proof presented here in the one-variable case, making more careful choices of the polynomials constructed in the induction step, to recover the uniform $\frac{1}{n}$ bound in that case, but this relies crucially on the commutativity of the polynomial ring ${\mathbb C} [z]$.
At the moment, we do not know if the same is true in the free setting, or even if there is a uniform power $p$ so that $\|P_nF-I\|_2^2\lesssim \frac{1}{n^p}$ independently of $F$. Also, observe that our proof does not actually construct the optimal approximants for any given $f$, we only construct a sequence $p_n$ obeying the claimed bounds, so that the optimal approximants must be at least as good. We have computed the optimal approximants (with computer assistance) in some simple examples, but the results do not appear enlightening and we have not included them.
The sharp value of $p$ seems difficult to calculate even for relatively simple cases. For example, for the free polynomial in two variables $f(x,y)=(1-x)(1-y)$ one can construct polynomials $p_n$ (by essentially the method used in the induction step of the proof, but making more careful choices of $\sigma_n$, $r_n$) for which
\[
\norm{ p_n(x,y)(1-x)(1-y)-1}_2^2\lesssim \frac{1}{n^{1/2}},
\]
which improves on the bound $\frac{1}{n^{1/3}}$ provided by the theorem, but we do not know if this value $p=1/2$ is sharp.
Finally, it would be of interest to know 1) whether the methods used here can be modified to handle the case of nc rational functions $F$, and 2) what can be said in the case of non-square polynomial matrix functions $F$.
\end{document} |
\begin{document}
\draft
\title{Simulating nonlinear spin models in an ion trap}
\author{G.J. Milburn}
\address{Isaac Newton Institute , University of Cambridge,\\
Clarkson Road, Cambridge, CB3 0HE, UK.\\
and Department of Physics, The University of Queensland,QLD 4072 Australia.}
\date{\today}
\maketitle
\begin{abstract}
We show how a conditional displacement of the vibrational mode of trapped ions can be used to simulate
nonlinear collective and interacting spin systems including nonlinear tops and Ising models ( a universal two qubit gate),
independent of the vibrational state of the ion. Thus cooling to the vibrational ground state is unnecessary provided the
heating rate is not too large.
\end{abstract}
\pacs{ 42.50.Vk,03.67.Lx, 05.50.+q}
One of the paths leading to the current interest in quantum computation begins with attempts to answer Feynman's
question\cite{Feynman}: can quantum physics be efficiently simulated on a classical computer? It is generally believed that the
answer is no, although there is no explicit proof of this conjecture. It then follows that a computer operating entirely by
quantum means could be more efficient than a classical computer. Our belief in this conjecture stems from a number of
algorithms, such as Shor's factorisation algorithm\cite{Shor1994}, which appear to be substantially (even exponentially)
more efficient than the classical algorithms. A number of schemes have now been proposed for a quantum
computer, and some have been implemented in a very limited way. What kinds of simulations might these schemes enable? A
number of investigators have attempted to answer this question\cite{Lloyd96,Abrams97,Lidar97,Boghosian98,Zalka98}. In this
paper we consider this question in the context of the ion trap quantum computer model and show that there is a class of
nonlinear collective and interacting spin models that can be simulated with current technology.
Nonlinear collective and interacting spin
models have long endured as tractable, nonlinear quantum models with wide ranging relevance. Such models have appeared in
nuclear physics\cite{Ring1980}, laser physics\cite{Drummond}, condensed matter physics\cite{Mahan} and of course as a
theoretical laboratory to investigate aspects of nonlinear field theories\cite{Kaku}. In many cases however the match between
model and experiment is only qualitative. In this paper I will show how some of these models may be directly simulated on a
linear ion trap with individual ion addressing as in the quantum computation architecture.
The interaction Hamiltonian for N ions interacting with the centre of mass vibrational mode can be controlled by using
different kinds of Raman laser pulses. A considerable variety of interactions has already been achieved
or proposed \cite{NIST,LANL,James}. Consider first the simplest interaction that does not change the vibrational mode
of the ions. Each ion is assumeds to be driven by a resonant laser field which couples two states, the ground state $|g\rangle$
and an excited state $|e\rangle$. The interaction Hamiltonian is
\begin{equation}
H_I=-\frac{i}{2}\hbar\sum_{i=1}^N(\Omega_i\sigma^{(i)}_+-\Omega_i^*\sigma^{(i)}_-)
\end{equation}
where $\Omega_i$ is the effective Rabi frequency at the i'th ion and we have assumed the dipole and rotating wave
approximation as usual. The raising and lowering operators for each ion are defined by $\sigma_-=|g\rangle \langle e|$ and
$\sigma_+=|e\rangle\langle g|$. If we now assume that each ion is driven by an
identical field and chose the phase appropriately, the interaction may be written as
\begin{equation}
H_I=\hbar\Omega \hat{J}_y
\label{rotation}
\end{equation}
where we have used the definition of the collective spin operators,
\begin{equation}
\hat{J}_\alpha=\sum_{i=1}^N\sigma^{(i)}_\alpha
\end{equation}
where $\alpha=x,y,z$ and
\begin{eqnarray}
\sigma^{(i)}_x & = & \frac{1}{2}(\sigma^{(i)}_++\sigma^{(i)}_-)\\
\sigma^{(i)}_y & = & -\frac{i}{2}(\sigma^{(i)}_+-\sigma^{(i)}_-)\\
\sigma^{(i)}_x & = & \frac{1}{2}(|e\rangle\langle e|-|g\rangle\langle g|)
\end{eqnarray}
The interaction Hamiltonian in Eq \ref{rotation} corresponds to a single collective spin of value $j=N/2$ precessing around
the $\hat{J}_y$ direction due to an applied field. By choosing the driving field on each ion to be the same we have imposed a
permutation symmetry in the ions reducing the dimension of the Hilbert space from $2^N$ to $2N+1$. The eigenstates of
$\hat{J}_z$ may be taken as a basis in this reduced Hilbert space. In ion trap quantum computers it is more usual to designate
the electronic states with a binary number as $|g\rangle=|0\rangle,\ \ |e\rangle=|1\rangle$. The product basis for all N ions
is then specified by a single binary string, or the corresponding integer code if the ions can be ordered. Each eigenstate,
$|j,m\rangle_z$, of $\hat{J}_z$ is a degenerate eigenstate of the Hamming weight operator ( the sum of the number of ones in
a string) on the binary strings labelling the product basis states in the $2^N$ dimensional Hilbert space of all possible
binary strings of length N. Collective spin models of this kind were considered many decades ago in quantum
optics\cite{Drummond} and are sometimes called Dicke models after the early work on superradiance of Dicke\cite{Dicke}. In
much of that work however the collective spin underwent an irreversible decay. In the case of an ion trap model however we can
neglect such decays due to the long lifetimes of the excited states. However when the electronic and vibrational motion is
coupled heating of the vibrational centre-of-mass mode can induce irreversible dynamics in the collective spin variables.
The natural variable to measure is $\hat{J}_z$ as a direct determination of the state of each ion via shelving techniques will
give such a measurement. These measurements are highly efficient, approaching ideal projective measurements. The
result of the measurement is a binary string which is an eigenstate of
$\hat{J}_z$. Repeating such measurements it is possible to construct the distribution for $\hat{J}_z$ and corresponding
averages. Other components may also be measured by first using a collective rotation of the state of the ions.
We now show how to realise nonlinear Hamiltonians using N trapped ions. By appropriate choice of Raman lasers it is possible
to realise the conditional displacement operator for the i'th ion\cite{Monroe1996,NIST}
\begin{equation}
H=-i\hbar(\alpha_ia^\dagger-\alpha_i^* a)\sigma_z^{(i)}
\end{equation}
If the ion is in the excited (ground) state this Hamiltonian displaces the vibrational mode by a complex amplitude $\alpha$
($-\alpha$). In the case of N ions with each driven by identical Raman lasers, the total Hamiltonian is
\begin{equation}
H=-i\hbar(\alpha a^\dagger-\alpha^* a)\hat{J}_z
\end{equation}
By an appropriate choice of Raman laser pulse phases we can then implement the following sequence of unitary transformations
\begin{equation}
U_{NL}=e^{i\kappa_x\hat{X}\hat{J}_z}e^{i\kappa_p\hat{P}\hat{J}_z}e^{-i\kappa_x\hat{X}\hat{J}_z}e^{i\kappa_p\hat{P}\hat{J}_z}
\end{equation}
where $\hat{X}=(a+a^\dagger)/\sqrt{2},\ \hat{P}=-i(a-a^\dagger)/\sqrt{2}$. Noting that
\begin{equation}
e^{i\kappa_p\hat{P}\hat{J}_z} \hat{X}e^{-i\kappa_p\hat{P}\hat{J}_z} =\hat{X}+\kappa_p\hat{J}_z
\end{equation}
it is easy to see that
\begin{equation}
U_{NL}=e^{-i\theta\hat{J}_z^2}
\label{UNL}
\end{equation}
where $\theta=\kappa_x\kappa_p$ which is the unitary transformation generated by a nonlinear top Hamiltonian describing
precession around the $\hat{J}_z$ axis at a rate dependant on the $z$ component of angular momentum. Such nonlinear tops have
appeared in collective nuclear models\cite{Ring1980} and form the basis of a well known quantum chaotic system\cite{Haake91}.
It should be noted that the transformation in Eq(\ref{UNL}) contains no operators that act on the vibrational state. It is thus
completely independent of the vibrational state and it does not matter if the vibrational state is cooled to the ground state or
not. However Eq(\ref{UNL}) only holds if the heating of the vibrational mode can be neglected over the time it takes to
apply the conditional displacement operators. We discuss below what this implies for current experiments.
In itself the unitary transformation in Eq (\ref{UNL}) can generate interesting states. For example if we begin with all the
ions in the ground state so that the collective spin state is initially $|j,-j\rangle_z$ and apply laser pulses to each
electronic transition according to the Hamiltonian in Eq (\ref{rotation}) for a time $T$ such that $\Omega T=\pi/2$ the
collective spin state is just the $\hat{J}_x$ eigenstate $|j,-j\rangle_x$. If we now apply the nonlinear unitary transformation
in Eq (\ref{UNL}) so that $\theta=\pi/2$ we find that the system evolves to the highly entangled state
\begin{equation}
|+\rangle=\frac{1}{\sqrt{2}}(e^{-i\pi/4}|j,-j\rangle_x+(-1)^je^{i\pi/4}|j,j\rangle_x)
\end{equation}
Such states have been considered by Bollinger et al.\cite{Bollinger} in the context of high precision frequency
measurements,and also by Sanders\cite{Sanders}. They exhibit interference fringes for measurements of
$\hat{J}_z$. As noted above a measurement of $\hat{J}_z$ is easily made simply by reading out the state of each ion using
highly efficient fluorescence shelving techniques. This particular nonlinear model is a well known system for studying quantum
chaos, as we now discuss.
The nonlinear top model was introduced by Haake\cite{Haake91,Sanders89} as a system that could exhibit chaos in the classical
limit on a compact phase space, and which could be treated quantum mechanically with a finite Hilbert space. This removed the
necessity of truncating the Hilbert space and the possibility of thereby introducing spurious quantum features. The nonlinear
top is defined by the collective spin Hamiltonian,
\begin{equation}
H = \frac{\kappa}{2 j \tau} \hat{J}_z^2 + p \hat{J}_y \sum_{n = -\infty}^\infty
\delta(t-n\tau),
\label{eq.top.hamiltonian}
\end{equation}
where $\tau$ is the duration between kicks, ${\bf \hat{J}} = (\hat{J}_x, \hat{J}_y, \hat{J}_z)$
is the angular momentum vector, and $\hat{J}^2=j(j+1)$ is a constant of the motion.
As the Hamiltonian is time periodic the appropriate quantum description is via the Floquet operator
\begin{equation}
U = \exp\left(-i\frac{3}{2 j} \hat{J}_z^2\right)
\exp\left(-i\frac{\pi}{2} \hat{J}_y\right), \label{eq.U.top}
\end{equation}
which takes a state from just before one kick to just before the next,
i.e.,
$\left|\psi\right> \longrightarrow U\left|\psi\right>$,
where $J_z$ and $J_y$ are the usual angular momentum operators, and
$j$ is the angular momentum quantum number. The first exponential,
$U_P = \exp\left(-i\frac{3}{2 j} \hat{J}_z^2\right )$, describes the
precession about the $z$-axis, and the second,
$U_K = \exp\left(-i\frac{\pi}{2} \hat{J}_y\right)$, describes the kick.
The classical dynamics can be
reduced to a two dimensional map of points on a sphere of radius $j$
\cite{Sanders89}, and the angular momentum vector can be parameterised
in
polar coordinates as
\begin{equation}
{\bf J} = j(\sin\Theta \cos\Phi, \sin\Theta \sin\Phi, \cos \Theta).
\end{equation}
The first term in the Hamiltonian (\ref{eq.top.hamiltonian})
describes a non-linear precession of
the top about the z-axis, and the second term describes
periodic kicks around the y-axis.
The classical
map for $p = \pi /2$ and $\kappa = 3$ has a mixed phase space with periodic elliptical fixed points and chaotic regions.
It is now clear that this model can be simulated by the
sequence of pulses in Eq ({\ref{UNL}) with appropriate values for the pulse area, together with a single linear rotation. This
presents the possibility of directly testing a number of ideas in the area of quantum chaos, particularly the idea of
hypersensitivity to perturbation introduced by Schack and Caves \cite{Caves94}. Of particular interest here is the
ability to very precisely simulate the measurement induced hypersensitivity discussed in \cite{Breslin99}. In that paper the
kicked top was subjected to a readout using a single spin that could be prepared in a variety of states. The interaction
between the top and the readout spin is described by
\begin{equation}
U_I = \exp\left(-i \mu J_y \sigma_z^{(R)}\right),
\end{equation}
where we regard one ion as set aside to do the readout and label it with a superscript.
It is relatively straight forward to generate this interaction via the pulse sequence of conditional phase shifts
\begin{equation}
U_{NL}=e^{i\kappa_x\hat{X}\sigma_z^{(R)}}e^{i\kappa_p\hat{P}\sigma_z^{(R)}}e^{-i\kappa_x\hat{X}\hat{J}_z}e^{i\kappa_p\hat{P}\sigma_z^{(R)}}
\end{equation}
with $\mu=\kappa_x\kappa_p$. It is now possible to consider a long sequence of measurements made at the end of each nonlinear
kick and record the resulting binary strings of measurement results.
Initial states of the kicked top can be easily be
prepared as coherent angular momentum states by appropriate linear rotations. In the basis of orthonormal $\hat{J}_z\mbox{
eigenstates}$, and $\hat{{\bf J}}^2\left|j,m\right> = j(j+1)\left|j,m\right>$.
the spin coherent states can be written as a rotation of the collective ground state\cite{Haake91,Arrechi72}
through the spherical polar angles $(\theta,\phi)$,
\begin{equation}
|\gamma\rangle = \exp\left [i\theta(\hat{J}_x\sin\phi-\hat{J}_y\cos\phi))\right ]|j,-j\rangle
\label{eq.coherent}
\end{equation}
where $\gamma = e^{i\Phi} \tan\left(\frac{\Theta}{2}\right)$. This can be achieved by identical, appropriately phased pulses
on each ion separately. Initial states localised in either the regular or chaotic regions of the classical phase space may thus
be easily prepared.
Using a sequence of conditional displacement operators that does distinguish different ions we can simulate
various interacting spin models. As interacting spins are required for general quantum logic gates, these models may be seen as
a way to perform quantum logical operations without first cooling the ions to the ground state of some collective vibrational
mode.
Suppose for example we wish to simulate the interaction of two spins with the Hamiltonian
\begin{equation}
H_{int}=\hbar\chi\sigma_z^{(1)}\sigma_z^{(2)}
\end{equation}
The required pulse sequence is
\begin{eqnarray}
U_{int} & = &
e^{i\kappa_x\hat{X}\sigma_z^{(1)}}e^{i\kappa_p\hat{P}\sigma_z^{(2)}}e^{-i\kappa_x\hat{X}\sigma_z^{(1)}}e^{i\kappa_p\hat{X}\sigma_z^{(2)}}\\
\nonumber & = & e^{-i\chi\sigma_z^{(1)}\sigma_z^{(2)}}
\end{eqnarray}
This transformation may be used together with single spin rotations to simulate a two spin transformation that is one of the
universal two qubit gates for quantum computation. For example the controlled phase shift operation
\begin{equation}
U_{cp}=e^{-i\pi|e\rangle_1\langle e|\otimes|e\rangle_2\langle e|}
\end{equation}
may be realised with $\chi=\pi$ as
\begin{equation}
U_{cp}=e^{-i\frac{\pi}{2}\sigma_z^{(1)}}e^{-i\frac{\pi}{2}\sigma_z^{(2)}}U_{int}
\end{equation}
Once again this transformation does not depend on the vibrational state and so long as it is applied faster than the heating
rate of the collective vibrational mode it can describe the effective interaction between two qubits independent of the
vibrational mode.
We have proposed a scheme, based on conditional displacements of a collective vibrational mode, to simulate a variety of
nonlinear spin models using a linear ion trap in the quantum computing architecture and which does not require that the
collective vibrational mode be cooled to the ground state. However the scheme does require that the heating of the collective
vibrational mode is negligible over the time of the application of the Raman conditional displacement pulses. It does not
matter that the ion heats up between pulses. If the pulses were applied for times comparable to the heating times the pulse
sequences described above would not be defined by a product of unitary transformations but rather by the completely positive
maps which include the unitary part as well as the nonunitary heating part. Such maps provide a means to test various
thermodynamic limits of nonlinear spin models and will be discussed in a future paper. In current experiments the heating time
is estimated to be of the order of 1 ms, which is much shorter than the theoretically expected values that are as long as
seconds\cite{NIST}. The source of this heating is unclear but efforts are under way to eliminate it, so we can
expect heating times to eventually be sufficiently long to ignore. In current experiments however the sequence of conditional
displacements would need to be applied on time scales of less than 1 ms. This is achievable using Raman pulses. We thus
conclude that simple collective and interacting spin models with a few spins are within reach of current ion trap quantum
computer experiments.
\acknowledgements
I would like to thank Daniel James and David Wineland for useful discussions.
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\end{document} |
\begin{document}
\title{Continuation Methods for Riemannian Optimization}
\begin{abstract}
Numerical continuation in the context of optimization can be used to mitigate convergence issues due to a poor initial guess. In this work, we extend this idea to Riemannian optimization problems, that is, the minimization of a target function on a Riemannian manifold. For this purpose, a suitable homotopy is constructed between the original problem and a problem that admits an easy solution. We develop and analyze a path-following numerical continuation algorithm on manifolds for solving the resulting parameter-dependent problem. To illustrate our developments, we consider two classical applications of Riemannian optimization: the computation of the Karcher mean and low-rank matrix completion. We demonstrate that numerical continuation can yield improvements for challenging instances of both problems.
\text{end}d{abstract}
\section{Introduction}\label{s.fromHomotopyToOptimization}
\normalem
This work aims at developing and analyzing numerical continuation for Riemannian optimization. Let us first recall the setting of numerical continuation for nonlinear equations. Given a nonlinear equation
\begin{equation}\label{eq.EuclideanEquation}
F(x) = 0,
\text{end}d{equation}
for a smooth function $F:\mathbb{R}^d\to\mathbb{R}^d$, numerical continuation~\cite{allg, deuflhard} is used to track solutions of~\eqref{eq.EuclideanEquation} when the problem is smoothly perturbed. This can be useful for, e.g., ensuring global convergence of the Newton method by progressively transforming a simple problem with a known solution into~\eqref{eq.EuclideanEquation}. More specifically, one considers a parametrized family of equations,
\begin{equation}\label{eq.eculdieanParametricProblem}
G(x,\lambda) = 0, \quad\forall\lambda\in\pac{0,1},
\text{end}d{equation}
with $G:\mathbb{R}^d\times\pac{0,1} \to \mathbb{R}^d$ such that $G(x,1) = F(x)$ holds and a solution $x_0\in\mathbb{R}^d$ of $G(x_0,0) = 0$ can be easily determined.
The function $G$ is also known as a \emph{homotopy}. Under suitable assumptions, the solution set
\begin{equation}
G^{-1}(0) = \paa{(x,\lambda)\in \mathbb{R}^d\times\pac{0,1} : G(x,\lambda) = 0}
\text{end}d{equation}
to the parametric problem~\eqref{eq.eculdieanParametricProblem} contains a smooth $x(\lambda)$, $\lambda\in\pac{0,1}$, connecting $x_1 = x(1)$, the solution to the original problem, to $x_0 = x(0)$.
Homotopy methods are also relevant in optimization.
Optimization methods for a given target function $f:\mathbb R^n \to \mathbb R$ often aim at retrieving critical points, that is, solutions to
\begin{equation}
F(x) = \nabla f (x) = 0.
\text{end}d{equation}
Homotopy methods can be useful for, e.g., ensuring global convergence (to a critical point) by tracking critical points of a parametrized optimization problem, which amounts to considering
\begin{equation}\label{eq.EuclideanParametricCriticalPoint}
G(x,\lambda) = \nabla f(x,\lambda) = 0, \quad\forall\lambda\in\pac{0,1}.
\text{end}d{equation}
This approach to optimization problems has been widely studied in the literature, both for unconstrained and constrained optimization problems~\cite{deformationOfLinearPrograms1,deformationOfLinearPrograms2}. Among others, this has led to almost always globally convergent methods for non convex optimization~\cite{prob1homotopies} and nonlinear programming~\cite{deformationOfLinearPrograms2,prob1homotopies2}. Another use of homotopy methods is to improve the convergence behavior of a method by, e.g. defining a homotopy in which a regularization term is reduced progressively~\cite{sparsereghomotopy}.
Riemannian optimization~\cite{absilbook, boumalBook} is concerned with optimizing a target function $f:\mathcal{M} \to \mathbb{R}$ on a smooth manifold $\mathcal{M}$ equipped with a Riemannian metric. The geometry of $\mathcal{M}$ gives the tools to design optimization methods that produce the iterates guaranteed to stay on the manifold.
The Riemannian counterpart of the homotopy~\eqref{eq.EuclideanParametricCriticalPoint} is
\begin{equation}\label{eq.riemannianParametricCriticalPoint}
\grad f(x,\lambda) = 0, \quad\forall\lambda\pac{0,1},
\text{end}d{equation}
where $f: \mathcal{M}\times\pac{0,1}\to\mathbb{R}$ and $\grad f(x,\lambda)$ denotes the Riemannian gradient of $f(\cdot,\lambda)$ at $x$. Continuation methods for~\eqref{eq.riemannianParametricCriticalPoint} need to ensure that $x$ stays on $\mathcal{M}$. In this work, we use tools from Riemannian optimization to design path-following algorithms achieving this demand. A related question has been explored in the more restricted setting of time-varying convex optimization on Hadamard manifolds~\cite{maass}, making use of the exponential map.
In~\cite{manton}, a theoretical study of parameter-dependent Riemannian optimization is performed; the resulting homotopy-based algorithm involves local charts in order to utilize standard continuation algorithms on Euclidean spaces. In this work, we develop continuation methods within the framework of Riemannian optimization as presented in~\cite{absilbook}, which allows for the convenient design of efficient numerical methods in a general setting.
\paragraph{Outline} After recalling in Section~\ref{s.Euclideancontinuation} the general structure of a path-following predictor-corrector continuation algorithm for nonlinear equations on Euclidean spaces, we introduce in Section~\ref{s.riemannianContinuation} the setting of parametric Riemannian optimization and provide sufficient conditions for the numerical continuation problem to be well-posed. We then translate to the Riemannian setting the predictor-corrector algorithm to address them. We analyse the prediction phase, a key step of the algorithm and also propose a step size adaptivity strategy. Finally, Sections~\ref{s.karcherMean} and~\ref{s.matrixCompletion} are dedicated to the application of the algorithm to two classical Riemannian optimization problems, respectively the computation of the Karcher mean and the low-rank matrix completion problem.
\section{Euclidean predictor-corrector continuation}\label{s.Euclideancontinuation}
To motivate our Riemannian continuation algorithm, let us first recall the standard predictor-corrector continuation approach; see, e.g.~\cite[chapter 2]{allg}.
Considering the parametric nonlinear equation~\eqref{eq.eculdieanParametricProblem}, let us assume that $0$ is a regular value of $G$, that is, the differential
\begin{equation*}
\operatorname{D}\hspace{-0.08cm} G(x,\lambda) = \pac{G_x(x,\lambda)\lvert G_\lambda(x,\lambda)}\in\matr{d}{d+1},
\text{end}d{equation*}
has full rank for each $\pa{x,\lambda}\in G^{-1}(0)$. Then the constant-rank level set theorem~\cite[Theorem 5.12]{lee} asserts the set $G^{-1}(0)$ is an embedded submanifold of $\mathbb{R}^{d+1}$ of dimension $1$ or, in other words, the union of disjoint curves.
Under the stronger assumption that $G_x(x,\lambda)\in\matr{d}{d}$ has full rank, the implicit function theorem~\cite[Theorem 1.3.1]{implicitFunctionTheorem} implies that it is possible to parametrize each solution curve as a function $x(\lambda)$. Moreover, its derivative is given by
\begin{equation}\label{eq.tangentVector}
x'(\lambda) = - G_x(x(\lambda),\lambda)^{-1}\pac{G_\lambda(x(\lambda),\lambda)}.
\text{end}d{equation}
In turn, the solution curve in~\eqref{eq.eculdieanParametricProblem} can be obtained from solving the following implicit ODE:
\begin{equation} \label{eq:davidenko}
\begin{cases*}
G_x(x,\lambda)\pac{x'} + G_\lambda(x,\lambda) = 0,\quad \forall\lambda\in\pac{0,1}\\
x(0) = x_0.
\text{end}d{cases*}
\text{end}d{equation}
This equation is sometimes called Davidenko equation~\cite{davidenko}. The path-following approach consists of numerically integrating~\eqref{eq:davidenko} from time $\lambda = 0$ to $\lambda = 1$. The existence of the solution to~\eqref{eq:davidenko} is discussed in~\cite[Theorem 4.2.1]{implicitFunctionTheorem}; see also Theorem~\ref{teo.curveExistence} below.
Given an approximation $x_k\simeq x(\lambda_k)$ of the solution curve at point $\lambda_k$, a predictor-corrector continuation algorithm first performs a prediction step, which obtains a possibly very rough estimate $y_{k+1}$ of the solution curve at the next point $\lambda_{k+1}$. This is followed by a correction phase which aims at projecting this estimate back to the solution curve.
The most common choices for the \emph{prediction step} are:
\begin{align}
\text{\textit{classical prediction}}\: &: \: y_{k+1} = x_k\label{eq.EuclideanClassicalPred}\\
\text{\textit{tangential prediction}}\: &: \: y_{k+1} = x_k + (\lambda_{k+1}-\lambda_k) t(x_k,\lambda_k)\label{eq.EuclideanTangentialPred},
\text{end}d{align}
where the tangent vector $t(x_k,\lambda_k):=x^\prime(\lambda_k)$ is obtained from~\eqref{eq.tangentVector}. This requires the solution of a linear system, a cost that is offset by increased prediction accuracy, see~\cite[p.238-239]{deuflhard} and Section~\ref{ss.predictionOrder}. Note that~\eqref{eq.EuclideanTangentialPred} coincides with one step of the Euler method applied to~\eqref{eq:davidenko}.
In the \textit{correction phase}, the refinement of the estimate $y_{k+1}$ is performed by applying a nonlinear equation solver, typically a Newton-type method, on the equation $G(x,\lambda_{k+1}) = 0$ with initial guess $y_{k+1}$. A sufficiently small step size $\lambda_{k+1} - \lambda_k$ leads to a prediction that is accurate enough to yield (very) fast convergence. Various step size selection strategies have been developed in the literature, see~\cite{allg, deuflhard} and Section~\ref{ss.stepSizeAdaptivity}.
\section{Continuation for Riemannian optimization}\label{s.riemannianContinuation}
In this section, we consider a Riemannian optimization problem depending on a scalar parameter. The parameter can be intrinsic to the problem (e.g., time) or has been artificially added to form a homotopy. Examples of homotopies for Riemannian optimization problems will be given in Sections~\ref{s.karcherMean} and~\ref{s.matrixCompletion}.
\subsection{Riemannian Davidenko equation}
We consider a $d$-dimensional Riemannian manifold $\mathcal{M}$ endowed with the Riemannian metric $\scalp{\cdot}{\cdot}$ and let $\nabla$ denote the Riemannian connection.
The parameter-dependent objective function
\begin{align*}
\map{f}{\mathcal{M}\times\pac{0,1}}{\mathbb{R}}{(x,\lambda)}{f(x,\lambda)}
\text{end}d{align*}
is assumed to be smooth in both arguments (at least of class $C^2$).
For fixed $\lambda \in [0,1]$,
the Riemannian gradient $\grad{f}(x,\lambda)$ of $f(\cdot,\lambda)$ at $x\in\mathcal{M}$
is defined to be the vector in the tangent space $T_x\mathcal{M}$ satisfying
\begin{equation*}
\diff{f}{x,\lambda}{\xi} :=
\text{d}t{f(\gamma_{x,\xi}(t),\lambda)}{t}{\big\vert}_{t=0} = \scalp{\grad{f}(x,\lambda)}{\xi}_x, \quad \forall \xi\in T_x\mathcal{M},
\text{end}d{equation*}
where $\gamma_{x,\xi}$ is a manifold curve of $\mathcal{M}$ such that $\gamma_{x,\xi}(0) = x$ and $\dot\gamma_{x,\xi}(0) = \xi$. Likewise, the Riemannian Hessian $\hessx{f}(x,\lambda)$ of $f(\cdot,\lambda)$ at $x\in\mathcal{M}$ is the linear map on the tangent space $T_x\mathcal{M}$ satisfying
\begin{equation*}
\hessx{f}(x,\lambda)\pac{\xi} = \nabla_\xi\grad{f}(x,\lambda), \quad \forall \xi\in T_x\mathcal{M}.
\text{end}d{equation*}
Consider the numerical continuation problem~\eqref{eq.riemannianParametricCriticalPoint} of tracking critical points of the objective function as the parameter $\lambda$ varies. Theorem~\ref{teo.curveExistence} below is inspired by~\cite[Theorem 4.2.1]{implicitFunctionTheorem} and gives sufficient conditions for the existence and parametrizability with respect to $\lambda$ of a differentiable manifold curve joining a critical point $x_0\in\mathcal{M}$ at $\lambda = 0$ and a critical point at $\lambda = 1$. Note that we let $B(x_0,L):=\{x\in M\colon d(x_0,x) < L\}$ denote a ball on the manifold, where $d(\cdot,\cdot)$ is the manifold distance function induced by the metric. We recall the manifold distance function is defined as
\begin{equation}\label{eq.distanceFunction}
d(x,y) = \underset{\gamma\in\Gamma_{xy}}{\inf}\paa{L(\gamma)}
\text{end}d{equation}
where $\Gamma_{xy} = \paa{\gamma:\pac{0,1}\to\mathcal{M}\,:\, \gamma(0) = x, \,\gamma(1) = y}$ is the set of piecewise smooth curves joining $x$ and $y$ and $L(\gamma) = \int_{0}^{1}\|\gamma'(\tau)\|_{\gamma(\tau)}d\tau$ is the length of the curve.
For the purpose of the analysis, we will assume that $\mathcal{M}$ is \emph{complete}.
\begin{theorem}\label{teo.curveExistence}
Let $\mathcal{M}$ be a complete Riemannian manifold, $\mathcal{U}$ be an open subset of $\mathcal{M}$ and $V$ an open subset of $\mathcal{M}\times \mathbb{R}$ such that ${\mathcal{U} \times\pac{0,1}\subset V}$. Consider a scalar field $f\in C^2(V,\mathbb{R})$.
Assume that there exist $x_0\in \mathcal{U}$ such that $\grad{f}(x_0,0) = 0$ and a constant $L>0$ such that $B(x_0,L)\subseteq \mathcal{U}$. Moreover, suppose that for every $(z,\lambda)\in \mathcal{U} \times\pac{0,1}$ it holds that
\begin{enumerate}[label=(\roman*)]
\item $\rank{\hessx{f}(z,\lambda)} = d$, \label{hyp.hessFullRank}
\item $\|\hessx{f}(z,\lambda)^{-1}\pac{\text{d}p{}{\lambda}\grad{f}(z,\lambda)}\|_z<L$. \label{hyp.boundedDerivative}
\text{end}d{enumerate}
Then there exist an open interval $J \supset [0,1]$ and a curve $x \in C^1(J,\mathcal{M})$ verifying
\begin{equation}\label{eq.criticalPointsCurve}
x(0) = x_0, \quad \grad{f}(x(\lambda),\lambda)= 0,\quad\forall \lambda\in\pac{0,1}.
\text{end}d{equation} This curve satisfies the initial value problem
\begin{equation}\label{eq.riemannianDavidenko}
\begin{cases*}
\hessx{f}(x(\lambda),\lambda)\pac{ \dot x(\lambda)} + \text{d}p{\grad f(x(\lambda),\lambda)}{\lambda} = 0,\quad \forall\lambda\in\pac{0,1},\\
x(0) = x_0.
\text{end}d{cases*}
\text{end}d{equation}
\text{end}d{theorem}
Hypothesis~\ref{hyp.hessFullRank} guarantees the parametrizability with respect to $\lambda$ by ensuring the implicit ODE~\eqref{eq.riemannianDavidenko} is well-defined. For a fixed $\lambda$, it is an analogous assumption guaranteeing local quadratic convergence of the Riemannian Newton method~\cite[Theorem 6.3.2]{absilbook}. Hypothesis~\ref{hyp.boundedDerivative} ensures that the manifold curve can be parametrized up to $\lambda = 1$ as the limit point of the curve for $\lambda\to \lambda^*$, for any $0<\lambda^*<1$, is guaranteed to stay in the region $\mathcal{U}$ where the Riemannian Hessian is still of full rank. These hypotheses are global a priori assumption that are difficult to verify in practice. Yet, for a large class of problems it is reasonable to assume the Riemannian Hessian is of full rank at the starting point $(x_0,0)$, and therefore the solution curve is at least parametrizable on a possibly smaller interval $\pac{0,\tau}\subseteq \pac{0,1}$. In the following, we call the initial value problem~\eqref{eq.riemannianDavidenko} the \emph{Riemannian Davidenko equation}. Note that by Hypothesis~\ref{hyp.hessFullRank}, if $x_0$ is a local minimum, then the solution curve to the Riemannian Davidenko equation is a manifold curve of local minima. If we further assume the objective function to be geodesically convex~\cite[Chapter 11]{boumalBook} for each $\lambda\in\pac{0,1}$, this implies that the solution curve consists of global minima.
The following proof of Theorem~\ref{teo.curveExistence} is an adaptation of the proof for the Euclidean case~\cite[Theorem 4.2.1]{implicitFunctionTheorem}.
\begin{proof}(of Theorem~\ref{teo.curveExistence})
Consider a local chart $\varphi: \mathcal{N} \to \mathbb{R}^d$ such that $x_0\in\mathcal{N}$ and ${\mathcal{N}\times\pac{0,1}\subseteq V}$. We give a local coordinate representation of the gradient vector field through this local chart by defining
\begin{equation*}
F({\hat x},\lambda) := \operatorname{D}\hspace{-0.08cm}\varphi(\varphi^{-1}\pa{{\hat x}})\pac{\grad{f}(\varphi^{-1}({\hat x}),\lambda)},\quad\forall ({\hat x},\lambda)\in\varphi\pa{\mathcal{N}}\times\pac{0,1}.
\text{end}d{equation*}
The Jacobian of this vector field along the vector $\hat v\in\mathbb{R}^d$ is
\begin{align*}
\operatorname{D}\hspace{-0.08cm}_{{\hat x}} F({\hat x},\lambda)\pac{\hat v} &= \operatorname{D}\hspace{-0.08cm}^2\varphi(\varphi^{-1}({\hat x}))\pac{\grad{f}(\varphi^{-1}({\hat x}),\lambda),\operatorname{D}\hspace{-0.08cm}\varphi^{-1}({\hat x})\pac{\hat v}}\\
&+\operatorname{D}\hspace{-0.08cm}\varphi(\varphi^{-1}({\hat x}))\pac{\nabla_{\operatorname{D}\hspace{-0.08cm}\varphi^{-1}({\hat x})\pac{\hat v}}\grad{f}(\varphi^{-1}({\hat x}),\lambda)}.
\text{end}d{align*}
Letting ${\hat x}_0 = \varphi(x_0)$, we find
\begin{align*}
F({\hat x}_0,0) = 0,\qquad
\operatorname{D}\hspace{-0.08cm}_{{\hat x}} F({\hat x}_0,0) = \operatorname{D}\hspace{-0.08cm}\varphi(\varphi^{-1}({\hat x}_0))\circ{\hessx{f}(\varphi^{-1}({\hat x}_0),\lambda)}\circ\operatorname{D}\hspace{-0.08cm}\varphi^{-1}({\hat x}_0).
\text{end}d{align*}
Since local charts are diffeormorphisms, hypothesis (i) implies that $\operatorname{D}\hspace{-0.08cm}_{\hat x} F({\hat x}_0,0)$ has full rank $d$. Then by applying the implicit function theorem to $F$ at $({\hat x}_0,0)$ there exist an open interval $I$ containing $0$
and ${\hat x} \in C^1(I,\varphi(\mathcal{N}))$ such that
\begin{align*}
{\hat x}(0) = {\hat x}_0,\qquad
F({\hat x}(\lambda),\lambda) = 0, \quad\forall\lambda\in I.
\text{end}d{align*}
Defining $x(\lambda) := \varphi^{-1}({\hat x}(\lambda))$ for $\lambda \in I$, it holds that $x(0) = x_0$. Moreover, there exists
$\lambda_0>0$ such that
\begin{enumerate}[label=(\arabic*)]
\item $x$ is defined on $\left[0, \lambda_0\right)$,
\item $\grad{f}(x(\lambda),\lambda) = 0\quad\forall\lambda\in\left[0, \lambda_0\right)$,
\item $x$ is continuously differentiable on $\left[0, \lambda_0\right)$,
\item $x(\lambda)\in \mathcal{U}, \quad\forall\lambda\in\left[0, \lambda_0\right)$.
\text{end}d{enumerate}
Define the following
\begin{equation*}
\lambda^* = \sup\paa{\lambda_0: \text{there exists $x$ such that (1), (2), (3) and (4) are verified}}.
\text{end}d{equation*}
By the discussion above, $\lambda^*>0$.
If $\lambda^*>1$, the result is proved. Therefore assume that $0<\lambda^*\le 1$. Differentiation with respect to $\lambda$ of (2) yields
\begin{equation*}
x'(\lambda) =- \hessx{f}(x(\lambda),\lambda)^{-1}\pac{\text{d}p{}{\lambda}\grad{f}(x(\lambda),\lambda)},\quad\forall\lambda\in\left[0, \lambda^*\right).
\text{end}d{equation*}
Due to condition (4) and hypothesis $(ii)$ we have $\|x'(\lambda)\|_{x(\lambda)}<L$ for every $\lambda\in\left[0, \lambda^*\right)$.
This implies
\begin{align} \label{eq:inequalityL}
\tilde L:= \lim_{\lambda\uparrow\lambda^*} d(x_0,x(\lambda))\leq \lim_{\lambda\uparrow\lambda^*} \int_{0}^{\lambda}\|x'(\tau)\|_{x(\tau)}\text{d}\tau< \lim_{\lambda\uparrow\lambda^*} \int_{0}^{\lambda}L\text{d}\tau\leq L.
\text{end}d{align}
Given a sequence $\{\lambda_k\}$ with $\lambda_k \to \lambda^*$, it follows in an analogous fashion that $\{x(\lambda_k)\}$ is a Cauchy sequence. Because of~\eqref{eq:inequalityL},
$\{x(\lambda_k)\}$ is contained in the closed ball
$\overline{B(x_0,\tilde L)} \subset \mathcal{U}$ and therefore converges to some $x^* \in \mathcal{U}$.
Now, using a local chart $\psi: \mathcal{N}' \to \mathbb{R}^d$
such that $x^*\in\mathcal{N}'$ we can apply the implicit function theorem to
\begin{equation*}
\tilde F(\hat z,\lambda) = \operatorname{D}\hspace{-0.08cm}\psi(\psi^{-1}\pa{\hat z})\pac{\grad{f}(\psi^{-1}(\hat z),\lambda)}
\text{end}d{equation*}
at $(\psi(x^*),\lambda^*)$ and thus extend $x(\lambda)$ to a larger interval. This contradicts the definition of $\lambda^*$.
\text{end}d{proof}
\subsection{Riemannian predictor-corrector continuation}
The Riemannian \linebreak predictor-corrector continuation algorithm mimics the Euclidean version from Section~\ref{s.Euclideancontinuation} by numerically integrating the Riemannian Davidenko equation~\eqref{eq.riemannianDavidenko}. For the moment, we consider $N$ steps with fixed step size $h_k = 1 / N$, for $k = 1,\dots,N$. A suitable adaptive step size strategy will be discussed in Section~\ref{ss.stepSizeAdaptivity}.
\paragraph{Prediction}
The classical continuation scheme~\eqref{eq.EuclideanClassicalPred} can be trivially extended to the Riemannian case without any adjustment. The initial guess for the subsequent correction phase is simply
\begin{equation}\label{eq.classicalPred}
y_{k+1} = x_k,
\text{end}d{equation}
the iterate at the previous step of the algorithm.
The Riemannian extension of the tangential prediction strategy~\eqref{eq.EuclideanTangentialPred} is more involved. It consists of performing a step in the direction of the tangent vector of the solution curve. This tangent vector can be computed from the Davidenko equation as
\begin{equation}\label{eq.predictionDirection}
t(x_k,\lambda_k) := -\hessx{f}(x_k,\lambda_k)^{-1}\pac{ \text{d}p{\grad f(x_k,\lambda_k)}{\lambda}} \in T_{x_k}\mathcal{M}.
\text{end}d{equation}
We note that this involves the solution of a linear system with the Riemannian Hessian. If its solution by a direct solver (e.g., via the Cholesky decomposition) is too expensive, especially for
manifolds of higher dimension, matrix-free Krylov type methods~\cite[chapter 5]{krylovMethodBook} can be used instead.
In the Euclidean case, a tangent vector was simply added to the current iterate. In the manifold setting, this needs to be combined with a retraction in order to make sure that the result is again on the manifold.
A retraction is a smooth mapping $R\,:\, T\mathcal{M}\to\mathcal{M}$ with the following two properties:
\begin{itemize}
\item[1)] $R_x(0_x) = x$, where $0_x$ is the zero element of $T_x\mathcal{M}$ and $R_x$ denotes the restriction of $R$ to $T_x\mathcal{M}$;
\item[2)] $\operatorname{D}\hspace{-0.08cm} R_x(0_x) = \operatorname{Id}_{T_x\mathcal{M}}$, where we have identified $T_{0_x}T_x\mathcal{M}\simeq T_x\mathcal{M}$ and $\operatorname{Id}_{T_x\mathcal{M}}$ is the identity mapping on $T_x\mathcal{M}$.
\text{end}d{itemize}
These properties ensure that the retraction is a first order approximation of the Riemannian exponential map~\cite[section 10.2]{boumalBook}; the second property is also known as \emph{local rigidity}.
More details can be found in~\cite[Chapter 4]{absilbook}; see also Sections~\ref{s.karcherMean} and~\ref{s.matrixCompletion} for examples.
The Riemannian tangential prediction step is defined as
\begin{equation}\label{eq.tangentialPred}
y_{k+1} = R_{x_k}(h_k t(x_k,\lambda_k)),
\text{end}d{equation}
where we recall that $h_k$ denotes the step size.
In the case of a manifold embedded into an Euclidean space, the metric projection yields the particular retraction $R_x^\pi(v) := \pi\pa{x + v}$; see~\cite[Chapter 4]{absilbook}, which is also used in the context of numerically integrating differential equations on embedded submanifolds~\cite{hairer}.
\paragraph{Correction} In analogy to the Euclidean case
from Section~\ref{s.Euclideancontinuation},
we rely on a second order method for refining the estimate $y_{k+1}$ such that it becomes a (nearly) critical point of $f(\cdot,\lambda_{k+1})$. The tolerance on the Riemannian gradient norm is chosen small enough to closely track the solution curve, typically $10^{-6}$.
The Riemannian Newton (RN) method~\cite[chapter 6]{absilbook} can be used for this purpose; its basic form is described in Algorithm~\ref{alg.riemannianNewton}. Note that the Riemannian Newton method can be replaced by any locally superlinearly convergent method, e.g., the Riemannian Trust Region (RTR) method \cite{absilbook}[AMS08, Chapter 7] or the Riemannian BFGS method \cite{bfgs}. These methods can take full advantage of sufficiently accurate initial guess provided by the prediction, yielding a fast correction phase. Although a first order method such as steepest descent could in principle be used, they would not benefit the warmstarting fully as they do not exhibit accelerated convergence near a critical point. \\
\begin{algorithm}
\caption{\aheader{x^*}{RiemannianNewton}{x^{(0)},f,\mathrm{tol}, N_{\text{inner}}}}
\label{alg.riemannianNewton}
\begin{algorithmic}[1]
\While{$\|\grad f({x^{(j)}})\|> \mathrm{tol}$ $\wedge$ $j \leq N_{\text{inner}}$}
\State Solve $\hessx{f}(x^{(j)})\pac{n^{(j+1)}} = -\grad{f}(x^{(j)})$;
\State $x^{(j+1)} = R_{x^{(j)}}(n^{(j+1)})$;
\EndWhile\\
\textbf{end}
\State \textbf{return} $x^* = x^{(j)}$;
\text{end}d{algorithmic}
\text{end}d{algorithm}
\paragraph{Riemannian-Newton Continuation (RNC)}
The whole predictor-corrector scheme for Riemannian manifolds is sketched in Algorithm~\ref{alg.riemannianContinuation}. The optional adaptive step size strategy in line~\ref{line:adaptive} will be explained in Section~\ref{ss.stepSizeAdaptivity} below.
\begin{algorithm}
\caption{$\paa{x_k,\lambda_k}$ = RiemannianNewtonContinuation$\pa{x_0, f, N_{\text{steps}}, \mathrm{tol}, N_{\text{inner}}}$}
\label{alg.riemannianContinuation}
\begin{algorithmic}[1]
\State $h_0 = \frac{1}{N_{\text{steps}}}$, $\lambda_0 = 0$, $k = 0$;
\While{$\lambda_k<1$}
\If{\text{tangentialPrediction}}
\State Solve $ \hessx f(x_k,\lambda_k)\pac{t_{k}} = - \text{d}p{\grad{f}(x_k,\lambda_k)}{\lambda} $;
\If{\text{adaptStepSize}}
\State \label{line:adaptive} Determine the new step size $h_{k}$ with Algorithm~\ref{alg.stepSizeAdaptivity}.
\EndIf
\State $y_{k+1} = R_{x_{k}}(h_k t_{k})$;
\Else
\State $y_{k+1} =x_{k}$;
\EndIf
\State $\lambda_{k+1} = \min\paa{\lambda_{k}+h_k,1}$
\State $x_{k+1} = \operatorname{RiemannianNewton}\pa{y_{k+1}, f(\cdot,\lambda_{k+1}),\mathrm{tol}, N_{\text{inner}}}$;
\If{$ \|\grad f(x_{k+1},\lambda_{k+1})\|>\operatorname{tol}$}
\State Error("Traversing failed at step k.");
\Else
\State $k = k+1$;
\EndIf
\EndWhile\\
\textbf{end}
\State \textbf{return} $\paa{x_j,\lambda_j}_{j = 1,\dots,k}$
\text{end}d{algorithmic}
\text{end}d{algorithm}
\subsection{Prediction order analysis} \label{ss.predictionOrder}
An accurate prediction step leads to fast convergence in the correction step (Algorithm~\ref{alg.riemannianNewton}). The concept of order is used in the Euclidean case~\cite[p.238-239]{deuflhard} to qualitatively capture this accuracy. The following definition extends this concept to the Riemannian case by considering the prediction path $y(h) \in \mathcal{M}$, $h>0$, obtained from the prediction step by varying the step size $h$.
\begin{definition}[Prediction order]
Let $x(\lambda)$ be the solution curve defined by~\eqref{eq.riemannianDavidenko} for ${\lambda\in\pac{0,1}}$. A prediction path $y(h)$ such that $y(0) = x(\lambda)$ is said to be of \emph{order p} if there exists a constant $\,\& \,a_p>0$, such that
\begin{equation*}
d(x(\lambda+h),y(h))\leq \,\& \,a_p h^p,\quad\forall\lambda\in[0,1),
\text{end}d{equation*}
holds for all sufficiently small $h > 0$.
\text{end}d{definition}
In the following we will prove that the prediction orders for the Riemannian classical and tangential prediction schemes match the ones in the Euclidean case. More specifically, the following lemmas show that classical prediction~\eqref{eq.classicalPred} has order 1 while tangential prediction~\eqref{eq.tangentialPred} has order 2.
\begin{lemma}
The classical prediction path $y_c(h) = x(\lambda)$ has order 1.
\text{end}d{lemma}
\begin{proof}Applying the definition of distance function, we obtain for sufficiently small $h>0$ that
\begin{align*}
d(x(\lambda + h),y_c(h)) &= d\pa{x(\lambda+h),x(\lambda)}\leq \int_{\lambda}^{\lambda+h}\|x'(\tau)\|\text{d}\tau\leq h \underset{\tau\in\pac{\lambda,\lambda+h}}{\max}\paa{\|x'(\tau)\|}\\
&\leq h\underset{\tau\in\pac{0,1}}{\max}\paa{\|x'(\tau)\|}
\text{end}d{align*}
\text{end}d{proof}
\begin{lemma}
If $x(\cdot)\in C^2([0,1))$, the tangential prediction path $$y_t(h) = R_{x(\lambda)}(ht(x(\lambda),\lambda))$$ has order 2.
\text{end}d{lemma}
\begin{proof}
We choose $h$ sufficiently small such that $\lambda + h < 1$, and $x(\lambda)$, $x(\lambda+h)$, $R_{x(\lambda)}(h x'(\lambda))$ lie in the same open neighborhood $\mathcal{U}\in\mathcal{M}$, corresponding to the local chart $\varphi$.
We denote the coordinate representations of $x(\lambda + h)$ and $R_{x(\lambda)}(h x'(\lambda))$ by
\begin{equation*}
\hat x(h) = \varphi(x(\lambda + h)), \quad \quad \hat r(h) = \varphi(R_{x(\lambda)}(hx'(\lambda))).
\text{end}d{equation*}
By the smoothness assumptions on $x$, $\varphi$, and $R$, it follows that $\hat r$ and $\hat x$ are both two times continuously differentiable. This allows us to write their second order Taylor expansion with Lagrange remainder as :
\begin{align*}
\hat x(h) &= \hat x(0) + h\hat x'(0) + \frac{h^2}{2}\hat x''(h_x),\\
\hat r(h) &= \hat r(0) + h\hat r'(0) + \frac{h^2}{2}\hat r''(h_r),
\text{end}d{align*}
for some $h_x,h_r\in\pa{0,h}$. By the retraction definition, note that $\hat x(0) = \hat r(0) = \varphi(x(\lambda))$ and using the local rigidity property
\begin{equation*}
\hat x'(0) = \hat r'(0) = \operatorname{D}\hspace{-0.08cm}\varphi(x(\lambda))\pac{x'(\lambda)}.
\text{end}d{equation*}
We now define the line
\begin{align*}
\hat e(\tau) &= (1-\tau)\cdot\hat x(h) + \tau\cdot\hat r(h)\\
&= \hat x(0) + h\hat x'(0) + \tau \cdot \frac{h^2}{2}\pa{\hat r''(h_r)-\hat x''(h_x)}.
\text{end}d{align*}
Because $\varphi\pa{\mathcal{U}}$ is open, this line is contained in $\varphi\pa{\mathcal{U}}$ for sufficiently small $h$.
This allows us to define
\begin{equation*}
e(\tau) = \varphi^{-1}(\hat e(\tau)),
\text{end}d{equation*}
which is a smooth curve on $\mathcal{M}$ joining $x(\lambda + h)$ and $R_{x(\lambda)}(hx'(\lambda))$:
\begin{equation*}
e(0) = x(\lambda + h)\quad\quad e(1) = R_{x(\lambda)}(hx'(\lambda)).
\text{end}d{equation*}
Taking the derivative with respect to $\tau$ we have that
\begin{equation*}
e'(\tau) = \operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat e(\tau))\pac{\hat e'(\tau)} = h^2\cdot \operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat e(\tau))\pac{\tfrac{1}{2}\pa{\hat r''(h_r)-\hat x''(h_x)}}.
\text{end}d{equation*}
This concludes the proof by noting that
\begin{align*}
d(R_{x(\lambda)}(hx'(\lambda)),x(\lambda + h))\! &\leq\! \int_{0}^{1}\!\!\!\|e'(\tau)\|\text{d}\tau \\
\! &\leq\! h^2\! \int_{0}^{1}\!\!\!\|\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat e(\tau))\pac{\tfrac{1}{2}\pa{\hat r''(h_r)-\hat x''(h_x)}}\!\|\,\text{d} \tau = O(h^2).
\text{end}d{align*}
\text{end}d{proof}
\subsection{Step size adaptivity via asymptotic expansion}\label{ss.stepSizeAdaptivity}
The selection of the step size $h_k$ in Algorithm~\ref{alg.riemannianContinuation} is of crucial importance for its efficiency. A good step size selection should find a balance between the two conflicting goals of attaining fast convergence in each correction step and maintaining a low number of correction steps.
An overview of existing strategies for the Euclidean case can be found in~\cite{allg, deuflhard}. In the following, we focus on the case of tangential prediction. We propose to generalize to the Riemannian setting a step size selection scheme which is summarized in~\cite[section 6.1]{allg}. It aims at guaranteeing the three following conditions: (i) the distance between the prediction point $y_{k+1}$ and the corresponding solution point $x_{k+1}$ is below a prescribed tolerance, (ii) the RN method on $f(\cdot, \lambda_k + h_k)$ started at the prediction point $y_{k+1}$ is sufficiently contractive and (iii) the curvature of the solution curve between $x_k$ and $x_{k+1}$ is below a prescribed tolerance. For the Euclidean case, an analogous approach intended to fulfill condition (ii) is used in the numerical continuation software package HOMPACK~\cite{hompackLibrary}, while the strategy we now describe targets the three above conditions simultaneously.
Given any $(w,\lambda)\in\mathcal{M}\times\pac{0,1}$ such that $\hessx{f}(w,\lambda)$ is full rank, we denote
\begin{itemize}
\item $t(w,\lambda) = -\hessx{f}(w,\lambda)^{-1}\pac{\text{d}p{\grad{f}(w,\lambda)}{\lambda}}$ : the prediction vector,
\item $n(w,\lambda) = -\hessx{f}(w,\lambda)^{-1}\pac{\grad{f}(w,\lambda)}$ : the RN update vector.
\text{end}d{itemize}
Given $(x(\lambda),\lambda)$ on the solution curve, recall the tangential prediction point as a function of step size $h > 0$ is
\begin{equation}\label{eq.predictionPath}
y( h) = R_{x(\lambda)}( h t(x(\lambda),\lambda)).
\text{end}d{equation}
An approximation of the distance between $y(h)$ and $x(\lambda + h)$ can be obtained from the norm of the first RN update vector. We shall denote it
\begin{equation}\label{eq.delta}
\delta(x(\lambda), \lambda, h) := \|n(y(h),\lambda + h)\|.
\text{end}d{equation}
If we let $z(h) = R_{y(h)}\pa{n(y(h), \lambda + h)}$ indicate the first iterate of the RN method, the first contraction rate of the RN is defined as
\begin{equation}\label{eq.kappa}
\kappa(x(\lambda), \lambda, h) := \frac{\|n(z(h),\lambda + h)\|}{\|n(y(h),\lambda + h)\|}.
\text{end}d{equation}
Upon convergence of the RN method for $f(\cdot, \lambda + h)$, this ratio is smaller than 1. Finally, the curvature of the solution curve between two points $x(\lambda)$ and $x(\lambda + h)$ can be approximated with
\begin{equation}\label{eq.alpha}
\alpha(x(\lambda), \lambda, h) := \operatorname{acos}\pa{\scalp{\frac{t(x(\lambda),\lambda)}{\| t(x(\lambda),\lambda)\|}}{\frac{\mathcal{T}_{y(h)\to x(\lambda)}( t(y(h),\lambda+ h))}{\|\mathcal{T}_{y(h)\to x(\lambda)}( t(y(h),\lambda+ h))\|}}_{x(\lambda)}},
\text{end}d{equation}
the angle between the prediction vector at the solution curve point $(x(\lambda),\lambda))$ and the prediction vector at the prediction point $(y(h),\lambda + h)$. In order to measure their relative angle we transport ${t(y(h),\lambda+ h)\in T_{y(h)}\mathcal{M}}$ to ${T_{x(\lambda)}\mathcal{M}}$ using a linear map ${\mathcal{T}_{y(h)\to x} : T_{y(h)}\mathcal{M}\to T_{x(h)}\mathcal{M}}$ which can be either parallel transport along the prediction curve $y(h)$ or, more generally, a transporter~\cite[Definition 10.61]{boumalBook}. Note that~\eqref{eq.alpha} is well defined only if $t(x,\lambda)\ne 0$, which also guarantees $t(y(h),\lambda+ h))$ is non zero for sufficiently small $h$.
The following lemma inspired by~\cite[Lemmas 6.1.2, 6.1.8]{allg} is the cornerstone of the step selection strategy. It provides a Taylor expansion with respect to $h$ around $h=0$ of the indicators~\eqref{eq.delta},~\eqref{eq.kappa},~\eqref{eq.alpha}.
\begin{lemma}\label{lem.asymptExpansion}
Assume $f\in C^4$. If for each $(x,\lambda)$ of the solution curve we have
\begin{equation}\label{eq.nonDegeneracy}
\operatorname{D}\hspace{-0.08cm}ndtz{n(y(h),\lambda+h)}{h}{2}\ne 0,
\text{end}d{equation}
where $\operatorname{D}\hspace{-0.08cm}ndt{}{h}{2}$ denote the second covariant derivative along the prediction path~\eqref{eq.predictionPath}. Then there exist functions $\delta_2(x, \lambda)$, $\kappa_2(x, \lambda)$, $\alpha_1(x, \lambda)$ only depending on $x$ and $\lambda$ such that the following holds:
\begin{enumerate}[label=(\roman*)]
\item The norm of the first Newton update vector $\delta(x, \lambda, h) = \|n(y(h),\lambda + h)\|$ verifies
\begin{equation*}
\delta(x, \lambda, h) = \delta_2(x, \lambda)h^2 + \Ocal{h^3}.
\text{end}d{equation*}
\item Newton's method contraction rate $\kappa(x,h) = \dfrac{\|n(z(h),\lambda + h)\|}{\|n(y(h),\lambda + h)\|}$ verifies
\begin{equation*}
\kappa(x, \lambda, h) = \kappa_2(x, \lambda)h^2 + \ocal{h^2}.
\text{end}d{equation*}
\item If $t(x,\lambda) \ne 0$, the prediction angle $$\alpha(x,h) = \operatorname{acos}\pa{\scalp{\frac{t(x,\lambda)}{\| t(x,\lambda)\|}}{\frac{\mathcal{T}_{y(h)\to x}( t(y(h),\lambda+ h))}{\|\mathcal{T}_{y(h)\to x}( t(y(h),\lambda+ h))\|}}_{x(\lambda)}}$$ is well defined, and provided that
\begin{equation}\label{eq:missingAssumption}
\operatorname{D}\hspace{-0.08cm}dtz{\mathcal{T}_{y(h)\to x}( t(y(h),\lambda+ h))}{h} \ne c t(x,\lambda),\quad\forall\,c\in\mathbb{R},
\text{end}d{equation}
it verifies
\begin{equation*}
\alpha(x, \lambda, h) = \alpha_1(x, \lambda)h + \Ocal{h^2}.
\text{end}d{equation*}
\text{end}d{enumerate}
\text{end}d{lemma}
The proof of Lemma~\ref{lem.asymptExpansion} can be found in appendix~\ref{a:proofLemma3}. We now describe the step size selection strategy inspired by this result. Given positive constants $\delta_{\max}$, $\kappa_{\max}$ and $\alpha_{\max}$, we aim at finding the largest $h_k>0$ such that
\[
\delta(x_k,\lambda_k, h_k) \leq \delta_{\max},\quad
\kappa(x_k,\lambda_k, h_k) \leq \kappa_{\max},\quad
\alpha(x_k,\lambda_k, h_k) \leq \alpha_{\max}.
\]
Given a trial step size $\tilde h_k$ (obtained, e.g., from the previous step), Lemma~\ref{lem.asymptExpansion} allows us to estimate
\begin{align}
\label{eq.deltaTilde}\delta_2(x_k,\lambda_k) \simeq\tilde \delta_2(x_k, \lambda_k) := \sqrt{\frac{\delta(x_k, \lambda_k,\tilde h_k)}{\tilde h_k^2}},\\
\label{eq.kappaTilde}\kappa_2(x_k,\lambda_k) \simeq\tilde \kappa_2(x_k, \lambda_k) = \sqrt{\frac{\kappa(x_k, \lambda_k,\tilde h_k)}{\tilde h_k^2}}.\\
\label{eq.alphaTilde}\alpha_1(x_k,\lambda_k) \simeq\tilde \alpha_1(x_k, \lambda_k) = \frac{\alpha(x_k, \lambda_k,\tilde h_k)}{\tilde h_k},
\text{end}d{align}
Then, imposing
\begin{align*}
\tilde \delta_2(x_k, \lambda_k) h_k^2 \leq \delta_{\max},\quad
\tilde \kappa_2(x_k, \lambda_k) h_k^2 \leq \kappa_{\max},\quad
\tilde \alpha_1(x_k, \lambda_k) h_k \leq \alpha_{\max},
\text{end}d{align*}
yields
\begin{equation*}
h_k \leq \tilde h_k \min\paa{ \sqrt{\frac{\delta_{\max}}{\tilde\delta(x_k,\lambda_k)}}, \sqrt{\frac{\kappa_{\max}}{\tilde\kappa(x_k,\lambda_k)}}, \frac{\alpha_{\max}}{\tilde\alpha(x_k,\lambda_k)}}.
\text{end}d{equation*}
This is the criterion to adjust step size, but not to make too drastic changes in the step size, the increase is limited to a factor of 2 and the decrease to a factor $\frac{1}{2}$. The resulting procedure is summarized in Algorithm~\ref{alg.stepSizeAdaptivity}. Note that this comes at the non-negligible cost of (approximately) solving 3 extra linear systems involving the Riemannian Hessian.
\begin{algorithm}
\caption{\aheader{ h_{k}}{AdaptiveStepSize}{\tilde h_k,x_k,\lambda_k,t(x_k,\lambda_k),f, \alpha_{\max},\delta_{\max},\kappa_{\max}}}
\label{alg.stepSizeAdaptivity}
\begin{algorithmic}[1]
\State $y_k = R_{x_{k}}(\tilde h_k t(x_k,\lambda_k))$;
\State Solve $ \hessx f(y_k,\lambda_k+\tilde h_k)\pac{t(y_k,\lambda + \tilde h_k)} = - \text{d}p{\grad f(y_k,\lambda_k + \tilde h_k)}{\lambda} $;
\State Solve $\hessx{f}(y_k,\lambda_{k} + \tilde h_k)\pac{n(y_k,\lambda + \tilde h_k)} = -\grad{f}(y_{k},\lambda_k + \tilde h_k)$;
\State $z_k = R_{y_k}(n(y_k,\lambda + \tilde h_k))$;
\State Solve $\hessx{f}(z_k,\lambda_{k} + \tilde h_k)\pac{n(z_k, \lambda + \tilde h_k)} = -\grad{f}(z_{k},\lambda_k + \tilde h_k)$;
\State Compute $\tilde \delta_2$, $\tilde \kappa_2$ and $\tilde \alpha_1$ using~\eqref{eq.deltaTilde},~\eqref{eq.kappaTilde} and~\eqref{eq.alphaTilde}.
\State $h_{k} = \tilde h_k\max\paa{\frac{1}{2},\min\paa{ \sqrt{\dfrac{\delta_{\max}}{\tilde\delta_2}}, \sqrt{\dfrac{\kappa_{\max}}{\tilde\kappa_2}}, \dfrac{\alpha_{\max}}{\tilde\alpha_1}, 2}}$;
\State \textbf{return} $h_{k}$
\text{end}d{algorithmic}
\text{end}d{algorithm}
\section{Application to the Karcher mean of symmetric positive definite matrices}\label{s.karcherMean}
In this section, we apply RNC, Algorithm~\ref{alg.riemannianContinuation}, to a classical problem of Riemannian optimization: the computation of the Karcher mean, also referred to as Riemannian center of mass~\cite{karchermeanfirstpaper}.
Given $K$ points $p_1,\dots,p_K\in\mathcal{M}$ the Karcher mean (with uniform weights) is defined as
\begin{equation} \label{eq:karcher}
\underset{q\in\mathcal{M}}{\arg\min}\paa{\sum_{i=1}^{K}d(q,p_i)^2},
\text{end}d{equation}
where $d(q,p_i)$ is the distance function on $\mathcal{M}$.
This optimization problem admits a unique solution for any manifold provided that all $p_i$ are sufficiently close to each other. This requirement can be dropped for instance in the case of complete Riemannian manifolds of non-positive sectional curvature, also called Cartan-Hadamard manifolds, for which the Karcher mean is always uniquely defined for any set of points~\cite{karcherMeanUniqueness}.
We will focus on the Karcher mean of $n\times n$ real symmetric positive definite matrices
\begin{equation*}
\SPD{n} = \paa{A\in \mathbb{R}^{n\times n}\,:\,A = A^T,\, v^TAv>0\,\forall v\in\mathbb{R}^n\text{ if } v\ne0}.
\text{end}d{equation*}
In the following, we recall facts from~\cite{bhatia} on a suitably chosen Riemannian manifold structure of $\SPD{n}$.
Clearly, $\SPD{n}$ is an open cone of the vector space of symmetric matrices $\SYM{n}$. The tangent space at $A\in\SPD{n}$ can be identified with this vector space:
\begin{equation*}
T_A\SPD{n} \simeq \SYM{n}.
\text{end}d{equation*}
The Thompson or statistical metric
makes $\SPD{n}$ a Cartan-Hadamard manifold; it has the following expression
\begin{equation}
\scalp{V}{W}_A = \operatorname{Tr}\pa{A^{-1}VA^{-1}W},\quad \forall \,V,W \in T_A\SPD{n}\simeq\SYM{n}.
\text{end}d{equation}
With this metric, a geodesic joining ${A,B\in \SPD{n}}$ is given by
\begin{equation}\label{eq.SPDgeodesic}
\gamma_{AB}(t) = A\exp(t\log(A^{-1}B)),
\text{end}d{equation}
where $\exp$ and $\log$ are the matrix exponential and logarithm. In turn, the distance function reads as
$
d(A,B) = \|\log(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\|_F
$
and the Karcher mean problem~\eqref{eq:karcher} becomes
\[
\underset{X\in \SPD{n}}{\arg\min} f(X), \qquad f(X):= \sum_{i = 1}^{K}\|\log(A_i^{-\frac{1}{2}}XA_i^{-\frac{1}{2}})\|_F^2,
\]
with $A_1,\dots,A_K \in \SPD{n}$.
An expression for the Riemannian gradient $f$ and for the Riemannian Hessian of $f$ associated to the Levi-Civita connection compatible with the Thompson metric can be found in~\cite[Equations 4.6 and 4.16]{surveyGeometricMean}. For numerical experiments, we consider the second order retraction~\cite[Equation 4.10]{surveyGeometricMean} and the transporter given by parallel transport along geodesics~\cite[Equation 3.4]{vecTranspSPDPaper}.
\subsection{Homotopy for the Karcher mean problem}
The Riemannian manifold structure for $\SPD{n}$ introduced in the previous section, makes the Karcher mean of positive definite matrices a strictly geodesically convex problem.
This implies that standard Riemannian optimization algorithms can successfully solve the problem without the need of numerical continuation. Nevertheless, we use this application as a model problem for the purpose of testing the RNC algorithm and illustrating its behavior.
We propose the following homotopy for the Karcher mean of $A_1,\dots,A_K \in \SPD{n}$.
We define $K$ smooth curves $B_i\::\:\pac{0,1}\to\SPD{n}$ such that
\begin{equation*}
B_i(1) = A_i,\quad\forall i = 1, \dots, K,
\text{end}d{equation*}
and such that the Karcher mean of $B_1(0),\dots,B_K(0)$ can be solved easily. In particular, this is the case when all starting points are equal, $B_1(0) = \cdots = B_K(0) = A_0$. In our experiments, we choose $A_0 = I_{n \times n}$. For $B_i$, we choose the geodesic curve~\eqref{eq.SPDgeodesic} joining $A_0$ to $A_i$, that is,
\begin{equation*}
B_i(\lambda) = A_0 \exp(\lambda \log(A_0^{-1}A_i)).
\text{end}d{equation*}
We can now write the parametric Karcher mean optimization problem as
\begin{equation}\label{eq.KarcherMeanHomotopy}
\underset{X\in\SPD{n}}{\arg\min}\paa{f(X,\lambda) = \sum_{i = 1}^{K}\|\log(B_i(\lambda)^{-\frac{1}{2}}XB_i(\lambda)^{-\frac{1}{2}})\|_F^2},\quad \forall\lambda\in\pac{0,1}.
\text{end}d{equation}
Using the parameter dependent expression of the Riemannian gradient of \eqref{eq.KarcherMeanHomotopy}, its derivative with respect to the parameter $\lambda$, needed for performing tangential continuation, is given by
\begin{equation*}
\text{d}p{\grad f(X,\lambda)}{\lambda} = - 2 \sum_{i = 1}^K X\operatorname{D}\hspace{-0.08cm}\log(X^{-1}B_i(\lambda))\pac{X^{-1}B_i'(\lambda)},
\text{end}d{equation*}
where $B_i^\prime(\lambda) = A_0 \exp(\lambda \log(A_0^{-1}A_i))\log(A_0^{-1}A_i)$ and $\operatorname{D}\hspace{-0.08cm}\hspace{0.02cm}\log(X)[\cdot]$ is the Fréchet derivative of the matrix logarithm; see~\cite{dlog} for its computation.
\subsection{Numerical results}
All numerical experiments presented in this paper have been performed in Matlab 2019b, using the Matlab Riemannian optimization library Manopt~\cite{manoptPaper}.
In all experiments, we consider computing the Karcher mean for a set of $K = 75$ symmetric positive definite matrices of size $n = 10$ that are built from their eigenvalue decomposition
\begin{equation*}
A_i = V_i D_i V_i^T,\quad \forall i = 1,\dots, K,
\text{end}d{equation*}
where $V_i$ is a random orthogonal matrix and $D_i$ a diagonal matrix. For the diagonal entries, 9 are chosen at random in the interval $\pac{1,2}$ and the last one is chosen such that the matrices have a large but still moderate condition number (approximately $10^3$). Figure~\ref{fig.KarcherMeanEasyInstance} compares the direct optimization with the standard RN method and the continuation approach (tangential RNC with fixed step size $N_{\text{steps}} = 3$) using the homotopy~\eqref{eq.KarcherMeanHomotopy}. For all experiments, we used the identity matrix as initial condition, $\operatorname{tol} = 10^{-6}$ and $\operatorname{N_{\text{inner}}} = 5000$. Note that other choices, like the planar approximations of the Karcher mean discussed in~\cite{surveyGeometricMean}, are possible. For this example, it turns out that the RN method enters
a superlinear convergence regime from the beginning (as seen from the concavity of the black convergence curve) and thus solves the problem in very few iterations. For such a simple instance, the continuation approach does not offer advantages.
\begin{figure}
\centering
\includegraphics[trim=4.1cm 0cm 4.1cm 0,clip,width=0.85\textwidth]{KarcherMeanEasyInstance.eps}
\caption{\footnotesize
Convergence of the Riemannian gradient norm versus RN iterations for a non-pathological instance of the Karcher mean problem. The iterations needed by the (plain) RN method is compared to the total number of RN correction steps needed by fixed step size classical and tangential prediction RNC ($N_{\text{steps}} = 3$). The Riemannian gradient norm for $\lambda = 1$ is plotted with solid lines, whereas we use dashed lines for intermediate values of $\lambda$. }
\label{fig.KarcherMeanEasyInstance}
\text{end}d{figure}
In order to better highlight the advantage of the RNC algorithm, we choose a somewhat pathological instance: the diagonal matrices $D_i$ are chosen such that their condition number is $10^8$. Half of the diagonal entries are exponentially distributed in $\pac{0.1, 1}$ and the other half exponentially distributed in $\pac{10^6, 10^7}$. In turn, the optimization problem is highly ill-conditioned, leading to stagnation in the initial phase of the RN method; see Figure~\ref{fig.KarcherMeanFixedStepSize}. In contrast, the RNC algorithm~\ref{alg.riemannianContinuation} with fixed number of steps $N_{\text{steps}} = 2$ does not suffer from such stagnation during the correction phase. In turn, the total number of RN iterations is reduced. Tangential prediction leads to slightly better convergence compared to classical prediction, but it also comes at the cost of solving an extra linear system, which leads to a less favorable computational time; see Table~\ref{tab.summaryKarcherMeanExperiments}. The number of fixed steps in Figure~\ref{fig.KarcherMeanFixedStepSize} is chosen to best highlight the slight improvement of RNC over direct RN optimization. However, for this particular application the advantage disappears when an automatic step sizing strategy is used. Nevertheless, the step size adaptivity results for different set of hyperparameters $(\kappa_{\max}, \alpha_{\max}, \delta_{\max})$ in Table~\ref{tab.summaryKarcherMeanExperiments} illustrate the need for a compromise to be found between the number of corrections and the length of each correction. This is further demonstrated by Figure~\ref{fig.KarcherMeanVaryingNSteps} where the computational effort for fixed step size RNC is reported for different values of $N_{\mathrm{steps}}$.
\begin{figure}
\centering
\includegraphics[width=0.85\textwidth]{KarcherMeanFixedStepSize.eps}
\caption{\footnotesize
Convergence of the Riemannian gradient norm (of the original problem in solid lines and of each intermediate problem in dashed lines) versus RN iterations for the pathological instance of the Karcher mean problem. RN method is compared with fixed step size classical and tangential prediction RNC algorithm ($N_{\text{steps}} = 2$).}
\label{fig.KarcherMeanFixedStepSize}
\text{end}d{figure}
\begin{figure}\label{fig.KarcherMeanVaryingNSteps}
\centering
\begin{minipage}{0.495\textwidth}
\includegraphics[trim=0.75cm 0cm 1.75cm 0.5cm,clip,width=\textwidth]{KarcherMeanCorrectionsVsNSteps.eps}
\text{end}d{minipage}
\begin{minipage}{0.495\textwidth}
\includegraphics[trim=0.75cm 0cm 1.75cm 0.5cm,clip,width=\textwidth]{KarcherMeanCPUTimeVsNSteps.eps}
\text{end}d{minipage}
\caption{\footnotesize
RN iterations (left) and computation time (right) versus the number of continuation steps for the fixed step size RNC on the pathological instance of the Karcher mean problem.}
\text{end}d{figure}
\begin{table}
\centering
\caption{\footnotesize Summary of the number of iterations and computation times for the numerical experiments on the Karcher mean pathological instance. The hyperparameters $(\kappa_{\max}, \alpha_{\max}, \delta_{\max})$ for the step size adaptive experiments (1), (2) and (3) are respectively $(0.6, 3^\circ, 10)$, $(0.3, 1.5^\circ, 5)$ and $(0.15, 0.75^\circ, 2.5)$.}
\begin{tabular}{c|c|c|c|}
\cline{2-4}
& \multicolumn{3}{c|}{\textbf{Karcher mean}} \\ \hline
\multicolumn{1}{|c|}{Direct Optimization (RN)} & 1 & 17 & 20.04 \\ \hline
\multicolumn{1}{|c|}{Fixed step size classical RNC} & 2 & 11 & 6.65 \\ \hline
\multicolumn{1}{|c|}{Fixed step size tangential RNC} & 2 & 9 & 6.32 \\ \hline
\multicolumn{1}{|c|}{Step size adaptive RNC (1)} & 3 & 22 & 45.66 \\ \hline
\multicolumn{1}{|c|}{Step size adaptive RNC (2)} & 3 & 17 & 32.36 \\ \hline
\multicolumn{1}{|c|}{Step size adaptive RNC (3)} & 6 & 25 & 56.96 \\ \hline
& Corrections & Correction iterations & Time (s) \\ \cline{2-4}
\text{end}d{tabular}
\label{tab.summaryKarcherMeanExperiments}
\text{end}d{table}
\section{Application to low-rank matrix completion}\label{s.matrixCompletion}
In matrix completion, only some entries of a matrix $A \in\mathbb{R}^{m\times n}$ are available and the goal is to determine the rest of the entries. This is clearly an ill-posed problem and one way to regularize it is to impose low-rank constraints; see \cite{matrixCompletionSurvey} for a recent review on existing methods.
In the following, we describe the Riemannian optimization setting introduced by~\cite{bart}.
We let ${\Omega \subset \paa{1,\dots,m}\times\paa{1,\dots,n}}$ contain the indices $(i,j)$ for which $A_{ij}$ is known and define the projection
\begin{equation}
P_\Omega(A)=
\begin{cases}
A_{ij} & \text{if }\pa{i,j}\in\Omega\\
0 & \text{if }\pa{i,j}\notin\Omega.
\text{end}d{cases}
\text{end}d{equation}
We aim at approximating $A$ by a matrix of a given fixed rank $k\ll \min\{m,n\}$ or, equivalently, by a matrix from the set
\begin{equation*}
\mathcal{M}_k = \paa{U\Sigma V^T\!\in\!\matr{m}{n}\! : U\!\in\! \operatorname{St}(m,k),V\!\in\!\operatorname{St}(n,k), \Sigma = \operatorname{diag}(\sigma_i), \sigma_1\geq\dots\geq\sigma_k>0},
\text{end}d{equation*}
where $\operatorname{St}(m,k) = \paa{U\in \matr{m}{k}:U^TU = I_k}$ is the Stiefel manifold. It can be shown that $\mathcal{M}_k$ is a smooth manifold of dimension $k(m+n-k)$. This leads to the following smooth Riemannian optimization formulation:
\begin{equation}\label{eq:RiemannianCompletionProblem}
\underset{X\in\mathcal{M}_k}{\min} f(X), \quad f(X) := \frac{1}{2}\|P_{\Omega}(X)-A_\Omega\|_F^2,
\text{end}d{equation}
with $A_\Omega = P_\Omega(A)$.
The fixed rank manifold $\mathcal{M}_k$ is endowed the standard structure of Riemannian submanifold of $\matr{m}{n}$ as presented in \cite[Section 2]{bart}. The expressions for the Riemannian gradient and the Riemannian Hessian are given in \cite[Equation 11 and Proposition 2.2]{bart}. For the numerical experiments, we opted for the metric projection retraction \cite[Equation 13]{bart} and the orthogonal projection to the destination tangent space \cite[Equation 14]{bart} for the transporter.
\subsection{Homotopy for the matrix completion}\label{ss.homotopyCurveMatrixCompletion}
The homotopy we propose for the matrix completion problem consists of replacing $A_\Omega$ in~\eqref{eq:RiemannianCompletionProblem} with a smooth curve $B_\Omega(\lambda)\in\matr{m}{n}$, $\lambda\in\pac{0,1}$, such that $B_\Omega(1) = A_\Omega$. If we take $B_\Omega(0) = P_\Omega(A_0)$, for some known matrix $A_0$ of rank $k$, then the first point of the continuation solution curve is $A_0$ itself. If we let $\pi:\matr{m}{n}\to\mathcal{M}_k$ denote the rank-$k$ truncated singular value decomposition, we use $A_0 = \pi\pa{\mathcal{F}(A_\Omega)}$, where $\mathcal{F}$ does not alter the known entries of $A_\Omega$ and imputes the unknown entries via a heuristic procedure. For example, it is common to use zeros for the unknown entries when initializing Riemannian optimization applied to~\eqref{eq:RiemannianCompletionProblem} \cite{matrixCompletionSurvey, boumalOnMatrixCompletion}. In our experiments, we found it more effective to replace missing entries by averaging neighboring known values.
The parametric matrix completion problem is given by
\begin{equation}\label{eq.matrixCompletionHomotopy}
\underset{X\in\mathcal{M}_{k}}{\min}\paa{f(X,\lambda) = \frac{1}{2}\|P_{\Omega}(X)-B_{\Omega}(\lambda)\|_F^2},\quad \forall\lambda\in\pac{0,1},
\text{end}d{equation}
with
\begin{equation}\label{eq.matrixCompletionInstanceCurve}
B_{\Omega}(\lambda) = (1-\lambda)P_\Omega(\pi(\mathcal{F}(A_{\Omega}))) + \lambda A_\Omega.
\text{end}d{equation}
From the parameter dependent expression of the Riemannian gradient of \eqref{eq.matrixCompletionHomotopy}, the linearity of $P_\Omega$ and of the tangent space projection $\Pi(X):\matr{m}{n}\to T_X\mathcal{M}_{k}$, we obtain
\begin{equation}
\text{d}p{\grad{f}(X,\lambda)}{\lambda} = \Pi(X) \pa{A_\Omega - P_\Omega(\pi(\mathcal{F}(A_{\Omega})))}.
\text{end}d{equation}
\subsection{Numerical results}
We apply the RNC Algorithm to an instance of the matrix completion problem where the matrix $A$ is obtained by sampling a bivariate smooth function $g$ on a regular grid of ${\pac{a,b}\times\pac{c,d}}$,
\begin{equation*}
A_{i,j} = g\pa{a + i\frac{(b-a)}{m-1}, c + j\frac{(d-c)}{n-1}}, \quad \forall\, i = 0,\dots, m-1, \forall \,j = 1,\dots, n-1.
\text{end}d{equation*}
We then set $A_\Omega = P_\Omega(A)$, with a randomly generated observation operator $P_\Omega$. We choose the number of known entries accordingly with the rank chosen for $\mathcal{M}_k$ using the oversampling rate defined as
\begin{equation*}
\operatorname{OS} = \frac{|\Omega|}{\operatorname{dim}(\mathcal{M}_k)} = \frac{|\Omega|}{k(m+n-k)},
\text{end}d{equation*}
where $|\Omega|$ is the cardinality of $\Omega$.
The matrix $A$ is known to exhibit exponentially decaying singular values, which -- as we will see -- deteriorates the convergence of direct Riemannian optimization methods for~\eqref{eq:RiemannianCompletionProblem}. In particular, we consider the function
\begin{equation*}
g(x,y) = e^{-\frac{(x-y)^2}{\sigma}}
\text{end}d{equation*}
with $\sigma = 0.1$.
This function is sampled on $\pac{-1,1}^2$ with a regular grid of $m = n = 300$ points in each direction. We choose the rank $k = 15$ and set $\operatorname{OS} = 3$, implying that $29.25\%$ of the entries are observed.
As the standard RN method tends to fail for this kind of problems, we substituted it with the Riemannian Trust Region algorithm (RTR), both as a corrector at line 11 of algorithm \ref{alg.riemannianContinuation} and as a direct optimization scheme.
\begin{figure}
\centering
\includegraphics[trim=4cm .25cm 3.5cm 0,clip,width=0.85\textwidth]{MatrixCompletionFixedStepSize.eps}
\caption{\footnotesize Convergence of the Riemannian gradient norm (of the original problem in solid lines and of each intermediate problem in dashed lines) versus RTR iterations on the matrix completion problem. We compare (plain) RTR optimization initialized at $A_0$ with fixed step size classical and tangential prediction RNC algorithm ($N_{\text{steps}} = 5$) on the matrix completion problem. }
\label{fig.MatrixCompletionFixedStepSize}
\text{end}d{figure}
The results of the direct optimization with RTR initialized at $A_0$ compared with fixed step size continuation $N_{\mathrm{steps}} = 5$ on the homotopy using the instance curve~\eqref{eq.matrixCompletionInstanceCurve} can be seen in Figure~\ref{fig.MatrixCompletionFixedStepSize}. For all experiments we set $\operatorname{tol} = 10^{-7}$ and $N_{\text{inner}} = 5000$. The direct method suffers a long stagnation before entering the superlinear convergence regime. The same stagnation occurs in the last corrections of the continuation procedures, yet less severely and thus the continuation scheme showed to be globally faster both in number of RTR iterations and computation time as summarized in Table~\ref{tab.summaryMatrixCompletionExperiments}. The table also report experiments conducted with two other widely used methods for low-rank matrix completion, namely the Riemannian Conjugate Gradient, referred to as LRGeomCG~\cite{bart}, and the alternating least-squares approach LMAFit~\cite{lmafit}. To make a fair comparison, both use the same initial condition $A_0$ and the stopping criterion is based on the final relative residual on the known entries that the direct RTR method achieves. In Figure~\ref{fig.MatrixCompletionVaryingNSteps}, the best compromise in terms of computation time of fixed step size RNC between the number of continuation steps and the number of steps of each correction is found to be for $N_{\mathrm{steps}} = 3$. If we increase the number of continuation steps, convergence on each correction requires less steps so the total number of RTR does not increase significantly, however the computation time increases due to the fixed costs of each correction.
\begin{figure}\label{fig.MatrixCompletionVaryingNSteps}
\centering
\begin{minipage}{0.495\textwidth}
\includegraphics[trim=0.75cm 0cm 1.75cm 0.5cm,clip,width=\textwidth]{MatrixCompletionCorrectionsVsNSteps.eps}
\text{end}d{minipage}
\begin{minipage}{0.495\textwidth}
\includegraphics[trim=0.75cm 0cm 1.75cm 0.5cm,clip,width=\textwidth]{MatrixCompletionCPUTimeVsNSteps.eps}
\text{end}d{minipage}
\caption{\footnotesize RTR iterations (left) and computation time (right) versus the number of continuation steps for the fixed step size RNC on the matrix completion problem.}
\text{end}d{figure}\begin{figure}
\centering
\includegraphics[trim=7.8cm 1.5cm 6.6cm 0,clip,width=\textwidth]{MatrixCompletionIndicatorsVsLambda.eps}
\caption{\footnotesize Step size selection on the matrix completion problem. Indicators~\eqref{eq.delta},~\eqref{eq.kappa},~\eqref{eq.alpha} measured after running algorithm~\ref{alg.stepSizeAdaptivity} for selecting the step size (bottom plot), are plotted against the corresponding continuation parameter $\lambda$. The dashed lines are the hyperparameters $\kappa_{\max}$, $\alpha_{\max}$, $\delta_{\max}$ used in the step size adaptivity procedure for each experiment. }
\label{fig.MatrixCompletionPerformanceIndicators}
\text{end}d{figure}\begin{figure}
\centering
\includegraphics[trim=1.75cm 1.2cm 1.75cm 0,clip,width=\textwidth]{MatrixCompletionStepSizeAdaptive.eps}
\caption{\footnotesize Convergence of the Riemannian gradient norm of each intermediate problem versus RTR iterations on the matrix completion problem. The step size adaptive RNC algorithm is compared for different step size adaptivity hyperparameters. }
\label{fig.MatrixCompletionStepSizeAdaptive}
\text{end}d{figure} As also confirmed by the step size adaptivity experiments (Figures~\ref{fig.MatrixCompletionPerformanceIndicators} and~\ref{fig.MatrixCompletionStepSizeAdaptive}), the solution curve to the homotopy generated by the instance curve~\eqref{eq.matrixCompletionInstanceCurve} is initially trivial to trace. Indeed, in the first part of the homotopy very few RTR iterations per correction are necessary for the classical prediction and even less for the tangential prediction. We clearly get a sense of the increasing difficulty by observing the results of Figure~\ref{fig.MatrixCompletionPerformanceIndicators}. Shorter and shorter step sizes are chosen in order to satisfy the step size selection criteria. Finally, as seen from the last plot in Figure~\ref{fig.MatrixCompletionStepSizeAdaptive}, completely removing the stagnation from the correction phase requires to enforce very strict step size selection criteria causing very small step sizes to be taken and numerous intermediate corrections to be performed. All in all, the most effective setting is the step size adaptive configuration with a permissive step size selection criteria (first plot in Figure~\ref{fig.MatrixCompletionStepSizeAdaptive}), which still exhibited transient stagnations. We therefore conclude that continuation is effective when the stagnation in the correction is mitigated, while removing completely this behavior requires an effort that is not worthwhile.
\begin{table}
\centering
\caption{\footnotesize Summary of the number of iterations and computation time for the numerical experiments on the matrix completion problem. The parameters for the step size adaptive experiments (1), (2) and (3) are the same as in Figure~\ref{fig.MatrixCompletionStepSizeAdaptive}, from top to bottom. }
\begin{tabular}{c|c|c|c|}
\cline{2-4}
& \multicolumn{3}{c|}{\textbf{Matrix completion}} \\ \hline
\multicolumn{1}{|c|}{Direct (RTR)} & 1 & 159 & 10.67 \\ \hline
\multicolumn{1}{|c|}{Direct (LRGeomCG)} & 1 & 1117 & 5.78 \\ \hline
\multicolumn{1}{|c|}{Direct (LMAFit)} & 1 & 17309 & 15.65 \\ \hline
\multicolumn{1}{|c|}{Fixed step size classical RNC} & 5 & 68 & 3.39 \\ \hline
\multicolumn{1}{|c|}{Fixed step size tangential RNC} & 3 & 32 & 1.96 \\ \hline
\multicolumn{1}{|c|}{Step size adaptive RNC (1)} & 4 & 46 & 4.96 \\ \hline
\multicolumn{1}{|c|}{Step size adaptive RNC (2)} & 32 & 154 & 70.01 \\ \hline
\multicolumn{1}{|c|}{Step size adaptive RNC (3)} & 143 & 175 & 259.30 \\ \hline
& Corrections & Correction iterations & Time (s) \\ \cline{2-4}
\text{end}d{tabular}
\label{tab.summaryMatrixCompletionExperiments}
\text{end}d{table}
\section{Conclusions}
In this work, we have proposed a generalization of numerical continuation to the setting of Riemannian optimization and stated sufficient conditions for the existence of a solution curve. The central contribution is the RNC Algorithm~\ref{alg.riemannianContinuation}, a path-following predictor-corrector algorithm relying on the concept of retraction for the prediction combined with superlinearly converging Riemannian optimization routines such as Riemannian Newton method or the Riemannian Trust Region algorithm for the correction. This method can track a curve of critical points of a parametric Riemannian optimization problem when an initial point on the curve is given. We have generalized to the Riemannian case an adaptive step size strategy relying on the asymptotic expansion of the some performance indicators of the correction. Furthermore, we have provided the analysis of the prediction phase motivating the choice of tangential prediction over classical prediction.
The behavior of our algorithm has been illustrated for the problem of computing the Karcher mean of positive definite matrices and for low-rank matrix completion. Particular homotopies have been proposed for both problems, thereby suggesting a more general approach for achieving this task: defining smooth curves of problem instances starting from an easily solvable one and ending at the instance of interest. This proved to be successful in particular for the matrix completion problem, where a fast decay of singular values leads to a challenging optimization task. The step size adaptivity proved to effectively control the Newton update vector norm, the Newton contraction rate and the prediction vectors angle allowing for the correction algorithms to directly exhibit superlinear convergence. However this came at a relatively high computational cost due to the small step sizes required making the fixed step size continuation or permissive step size selection more competitive.
\addcontentsline{toc}{section}{References}
\begin{appendices}
\section{Proof of lemma~\ref{lem.asymptExpansion}}\label{a:proofLemma3}
Our proof mimics the proof of the Euclidean case from~\cite[Section 6.1]{allg}, making use of a local chart to map the problem to $\mathbb{R}^d$.
We assume that $h$ is sufficiently small such that $x$, $y(h)$ and $z(h)$ are contained in the domain of the same local chart $\pa{\mathcal{U},\varphi}$. All the involved points and functions are expressed in local coordinates~\cite[Chapter 1]{lee} associated with $\varphi$ and the tangent vectors are decomposed into the the local coordinate vector fields~\cite[Exmaple 8.2]{lee} induced by $\varphi$. The same strategy is used in the local convergence proof of the Riemannian Newton in~\cite[Theorem 6.3.2]{absilbook}. In the following, for the convenience of the reader, we recall how the different entities are mapped to local coordinates; see~\cite{absilbook} for details.
\begin{itemize}
\item For a point $w\in \mathcal{U}$, we write $\hat w := \varphi(w)$. The solution point $x$, the predicted point $y(h)$ and the first RN iterate $z(h)$ become respectively,
\begin{align*}
{\hat x} &:= \varphi(x),\\
{\hat y}(h) &:= \varphi(y(h)),\\
\hat z(h) &:=\varphi(z(h)).
\text{end}d{align*}
Conversely, for a vector $\hat w \in \hat\mathcal{U}:= \varphi(\mathcal{U})$, we write $w:=\varphi^{-1}(\hat w)$.
\item For a tangent vector $\xi\in T_w\mathcal{M}$ for some $w\in \mathcal{U}$, we write $\hat \xi := \operatorname{D}\hspace{-0.08cm}\varphi(w)\pac{\xi}\in\mathbb{R}^d$. The Riemannian gradient $\grad f(w,\lambda)$, its differential with respect to lambda $\text{d}p{\grad f(w,\lambda)}{\lambda}$, the tangential prediction vector $t(w,\lambda)$ and the RN update vector $n(w,\lambda)$ translate respectively to
\begin{align*}
\hat F({\hat w},\lambda) &:= \operatorname{D}\hspace{-0.08cm}\varphi(w)\pac{\grad{f}(w,\lambda)},\\
\hat F_\lambda({\hat w},\lambda) &:= \operatorname{D}\hspace{-0.08cm}\varphi(w)\pac{\text{d}p{\grad{f}(w,\lambda)}{\lambda}},\\
\hat t({\hat w},\lambda) &:= \operatorname{D}\hspace{-0.08cm}\varphi(w)\pac{t(w,\lambda)},\\
\hat n({\hat w},\lambda)&:=\operatorname{D}\hspace{-0.08cm}\varphi(w)\pac{n(w,\lambda)},
\text{end}d{align*}
for any $({\hat w},\lambda)\in\hat \mathcal{U} \times \pac{0,1}$. Conversely, given $\hat\xi \in \mathbb{R}^d$ and some ${\hat w}\in\varphi\pa{\mathcal{U}}$, we write $\xi := \operatorname{D}\hspace{-0.08cm}\varphi^{-1}({\hat w})\pac{\hat \xi}\in T_w\mathcal{M}$.
\item The coordinate representation of the Riemannian Hessian is \begin{equation*}
\hat H : \hat\mathcal{U}\times\pac{0,1}\to \mathbb{R}^{d\times d}: (\hat w,\lambda)\mapsto \operatorname{D}\hspace{-0.08cm}\varphi(w)\pac{\hessx{f}(w,\lambda)\pac{\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{\cdot}}}.
\text{end}d{equation*}
\item As discussed in~\cite[Chapter 2]{leeRiemManifs}, the Riemannian metric can be represented with the Gramian matrix in the basis of coordinate vector field as
\begin{equation*}
{\hat G_{\hat w}:\hat\mathcal{U}\to \mathbb{R}^{d\times d}:\hat w\mapsto \pa{\scalp{\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{e_i}}{\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{e_j}}_w}_{i,j = 1,\dots,d}},
\text{end}d{equation*}where $e_i$ are the canonical vectors of $\mathbb{R}^d$. By smoothness of the Riemannian metric, this function is also smooth. Furthermore, given $\xi\in T_x\mathcal{M}$ for some $w\in\mathcal{U}$ it holds that $\left\|\xi\right\|_x = \sqrt{\hat\xi^\top \hat G_{{\hat x}} \hat\xi}.$
\item Given a sufficiently small $\xi\in T_w\mathcal{M}$ for some $w\in\mathcal{U}$, the retraction point $R_w(\xi)$ is well defined and $R_w(\xi)\in\mathcal{U}$. For the local representation of such vectors, the coordinate representation of the retraction is
\begin{equation*}
\hat R_{\hat w} (\hat\xi) = \varphi(R_w(\operatorname{D}\hspace{-0.08cm} \varphi^{-1}(\hat w)\pac{\hat\xi})).
\text{end}d{equation*}
\item Finally, the coordinate representation of the transporter is
\begin{equation*}
\hat \tcal_{{\hat y}\to {\hat x}}:\mathbb{R}^d\to \mathbb{R}^d: \hat{\xi}\mapsto \operatorname{D}\hspace{-0.08cm}\varphi(x)\pac{\mathcal{T}_{y\to x}\pa{\operatorname{D}\hspace{-0.08cm} \varphi^{-1}({\hat y})\pac{\hat{\xi}}}},\quad \forall{\hat y},{\hat x}\in\hat \ucal.
\text{end}d{equation*}
\text{end}d{itemize}
Note that the function $\hat F_\lambda$ defined above coincides with the derivative of $\hat F$ with respect to $\lambda$. However, the differential of $\hat F$ with respect to its first argument, denoted $\hat F_{{\hat x}}$, does \emph{not} coincide with the $\hat H$, the coordinate representation of the Hessian. Indeed, one obtains
\begin{equation}\label{eq:Fxhat}
\begin{aligned}
\hat F_{\hat x}({\hat w},\lambda)[\cdot] &= \hat H({\hat w},\lambda)[\cdot] + \operatorname{D}\hspace{-0.08cm}^2\varphi(w)\pac{\grad{f}(w,\lambda),\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{\cdot}}\\
&= \hat H({\hat w},\lambda)[\cdot] + \operatorname{D}\hspace{-0.08cm}^2\varphi(w)\pac{\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{\hat F({\hat w},\lambda)},\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{\cdot}} \\
&= \hat H({\hat w},\lambda)[\cdot] + \hat A({\hat w})\pac{\hat F({\hat w},\lambda),\cdot},
\text{end}d{aligned}
\text{end}d{equation}
where we defined the bilinear form
\begin{equation} \label{eq:Ahat}
\hat A({\hat w})\pac{\cdot,\cdot} = \operatorname{D}\hspace{-0.08cm}^2\varphi(w)\pac{\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{\cdot},\operatorname{D}\hspace{-0.08cm}\varphi^{-1}(\hat w)\pac{\cdot}}.
\text{end}d{equation}
On the solution curve, we have $\hat F({\hat x},\lambda) = 0$, so the second term in~\eqref{eq:Fxhat} vanishes for $\hat w = \hat x$ and we find ${\hat F_{\hat x}({\hat x},\lambda) = \hat H({\hat x},\lambda)}$. By the definitions above, the coordinate representations of $y(h)$, $z(h)$, $t(w,\lambda)$ and $n(w, \lambda)$ have the following convenient expressions
\begin{align}
\label{eq:yhat}{\hat y}(h) &= \hat R_{\hat x}(h\hat t(\hat x,\lambda)),\\
\label{eq:zhat}\hat z(h) &= \hat R_{\hat y(h)}(\hat n(\hat y(h),\lambda + h)),\\
\label{eq:that} \hat t(\hat w, \lambda) & = -\hat H(\hat w,\lambda)^{-1}\pac{\hat F_\lambda(\hat w,\lambda)},\\
\label{eq:nhat}\hat n(\hat w, \lambda) & = -\hat H(\hat w,\lambda)^{-1}\pac{\hat F(\hat w,\lambda)}.
\text{end}d{align}
As noted in~\cite[Theorem 6.3.2]{absilbook}, we point out that the local rigidity property of the retraction transfers to its local chart version, i.e.
\begin{equation*}
\operatorname{D}\hspace{-0.08cm} \hat R_{{\hat w}}(0)\pac{\hat \xi} = \hat \xi,\quad \forall {\hat w}\in\hat \mathcal{U}, \:\hat\xi\in\operatorname{dom}(\hat R_{\hat w}).
\text{end}d{equation*}
Using the previous definitions, we conclude the following expressions:
\begin{align*}
\delta(x, \lambda,h)\! &=\! \sqrt{\hat n(\hat y(h),\lambda + h)^T \hat G_{\hat y(h)}\hat n(\hat y(h),\lambda + h)},\\
\kappa(x, \lambda,h)\! &=\! \frac{\sqrt{\hat n(\hat z(h),\lambda + h)^T \hat G_{\hat z(h)}\hat n(\hat z(h),\lambda + h)}}{\delta(x,\lambda,h)},\\
\alpha(x, \lambda,h)\! &=\! \operatorname{acos}\pa{\!\frac{\hat t({\hat x},\lambda)^T}{\sqrt{\hat t({\hat x},\lambda)^T\hat G_{\hat x}\hat t({\hat x},\lambda)}}\hat G_{{\hat x}}\frac{\hat \tcal_{{\hat y}\to {\hat x}}(\hat t({\hat y},\lambda+h))}{\sqrt{\hat \tcal_{{\hat y}\to {\hat x}}(\hat t({\hat y},\lambda+h))^T\hat G_{\hat x}\hat \tcal_{{\hat y}\to {\hat x}}(\hat t({\hat y},\lambda+h))}}}
\text{end}d{align*}
We are now in the position to perform Taylor expansion with respect to $h$ of these functions.
\noindent \emph{Result (i)}\\
By combining~\eqref{eq:yhat} and~\eqref{eq:nhat} we have
\begin{equation} \label{eq:nhatOfY}
\hat n(\hat y(h),\lambda + h) = -\hat H(\hat R_{\hat x}(h\hat t(\hat x,\lambda)),\lambda +h)^{-1}\pac{\hat F(\hat R_{\hat x}(h\hat t(\hat x,\lambda)),\lambda+h)}.
\text{end}d{equation}
Let us expand both terms separately.
\begin{equation} \label{eq:FhatExpansion}
\begin{aligned}
\hat F(\hat R_{\hat x}(h\hat t(\hat x,\lambda)),\lambda+h) \!&=\! \hat F(\hat x,\lambda)\! + \! h\!\pa{\!\hat F_{\hat x}(\hat x,\lambda)\!\pac{\hat t(\hat x,\lambda)}\! + \!\hat F_\lambda(\hat x,\lambda)} \! + \! h^2 c_1(\hat x, \lambda) \! + \! \Ocal{h^3} \\
\!&=\! h^2 c_1(\hat x, \lambda) \! + \! \Ocal{h^3},
\text{end}d{aligned}
\text{end}d{equation}
where the second equality follows from $\hat F({\hat x},\lambda) = 0$, $\hat F_{{\hat x}}({\hat x},\lambda) = \hat H({\hat x},\lambda)$ and~\eqref{eq:that}.
For later purposes, let us note the explicit expression
\begin{equation}\label{eq:c1}
\begin{aligned}
c_1(\hat x, \lambda) = \frac{1}{2}\Big(\hat F_{{\hat x}{\hat x}}({\hat x},\lambda)\pac{\hat t({\hat x},\lambda),\hat t({\hat x},\lambda)} &+ \hat F_{\hat x}({\hat x},\lambda)\pac{\operatorname{D}\hspace{-0.08cm}^2\hat R_{\hat x}(0)\pac{\hat t({\hat x},\lambda),\hat t({\hat x},\lambda)}} \\ + 2 \hat F_{{\hat x}\lambda}(\hat x,\lambda)\pac{\hat t({\hat x},\lambda)} &+ \hat F_{\lambda\lambda}({\hat x},\lambda)\Big).
\text{end}d{aligned}
\text{end}d{equation}
Now note that
\begin{equation*}
\hat H(\hat R_{\hat x}(h\hat t(\hat x,\lambda)),\lambda +h) = \hat H(\hat x,\lambda) + \Ocal{h}.
\text{end}d{equation*}
Then by smoothness of matrix inversion
\begin{equation*}
\hat H(\hat R_{\hat x}(h\hat t(\hat x,\lambda)),\lambda +h)^{-1} = \hat H(\hat x,\lambda)^{-1}+ \Ocal{h}.
\text{end}d{equation*}
Combined with~\eqref{eq:FhatExpansion} one has
\begin{equation}\label{eq:nhatOfYExpansion}
\hat n(\hat y(h),\lambda + h) = h^2 c_2(\hat x, \lambda) + \Ocal{h^{3}}.
\text{end}d{equation}
with $c_2(\hat x, \lambda) = -\hat H(\hat x,\lambda)^{-1}\pac{c_1(\hat x, \lambda)}$. Noting that $ \hat G_{\hat y(h)} = \hat G_{\hat x} + \Ocal{h}$ we obtain
\begin{align*}
\delta(x, \lambda,h) =& \sqrt{\hat n(\hat y(h),\lambda + h)^T \hat G_{\hat y(h)}\hat n(\hat y(h),\lambda + h)} \\ =& (h^4c_3({\hat x}, \lambda)^2 + \Ocal{h^5})^{1/2} \\ =& h^2 c_3({\hat x}, \lambda) + \Ocal{h^3},
\text{end}d{align*}
where $c_3({\hat x}, \lambda) := \sqrt{c_2(\hat x, \lambda)^T \hat G_{\hat x} c_2(\hat x, \lambda)}$. The last equality follows from the Taylor expansion of the square root in $c_3({\hat x},\lambda)^2$. This is possible provided the $c_3({\hat x}, \lambda)$ does not vanish. By hypothesis~\eqref{eq.nonDegeneracy}, it can be shown that $c_1({\hat x}, \lambda)$ is not zero. Hence, $c_2({\hat x}, \lambda)$ and $c_3({\hat x}, \lambda)$ are also not zero.
Setting $\delta_2(x, \lambda) := c_3(\varphi(x), \lambda)$, this concludes the proof of (i). \\
\noindent \emph{Result (ii)}\\
To obtain the expansion for $\kappa$, we combine result (i) with the expansion of the Newton direction evaluated in $\hat z(h)$.
For this purpose, note that by combining~\eqref{eq:zhat} and~\eqref{eq:nhat}
\begin{equation} \label{eq:nhatOfZ}
\hat n(\hat z(h),\lambda + h) = -\hat H(\hat R_{{\hat y}(h)}(\hat n({\hat y}(h),\lambda + h),\lambda +h)^{-1}\pac{\hat F(\hat R_{{\hat y}(h)}(\hat n({\hat y}(h),\lambda + h),\lambda+h)}.
\text{end}d{equation}
The Taylor expansion with respect to $\hat n({\hat y}(h),\lambda + h)$ of the right-hand side term gives
\begin{align*}
\hat F\pa{\hat R_{{\hat y}(h)}(\hat n({\hat y}(h),\lambda + h)),\lambda + h} = \hat F({\hat y}(h),\lambda + h) + \hat H({\hat y}(h),\lambda + h)\pac{\hat n({\hat y}(h),\lambda + h)}\\ + \hat A({\hat y}(h))\pac{\hat F({\hat y}(h),\lambda + h),\hat n({\hat y}(h),\lambda + h)}\\+
\frac{1}{2}\hat F_{{\hat x}{\hat x}}({\hat y}(h),\lambda + h)\pac{\hat n({\hat y}(h),\lambda + h),\hat n({\hat y}(h),\lambda + h)} \\
+ \frac{1}{2}\hat F_{\hat x}({\hat y}(h),\lambda + h)\!\pac{\operatorname{D}\hspace{-0.08cm}^2\hat R_{\hat x}(0)\pac{\hat n({\hat y}(h),\lambda + h),\hat n({\hat y}(h),\lambda + h)}}\! + \Ocal{\|\hat n({\hat y}(h),\lambda + h)\|^3}.
\text{end}d{align*}
The first two summands cancel out owing to~\eqref{eq:nhat}. Furthermore, by smoothness of the retraction and of the local charts, we have \begin{align*}
&\hat A({\hat y}(h)) = \hat A({\hat x}) + \Ocal{h},\\
&\hat F_{{\hat x}{\hat x}}({\hat y}(h),\lambda + h) = \hat F_{{\hat x}{\hat x}}(\hat x,\lambda) + \Ocal{h},\\
&\hat F_{\hat x}({\hat y}(h),\lambda + h)\circ\operatorname{D}\hspace{-0.08cm}^2\hat R_{{\hat y}(h)}(0) = \hat F_{\hat x}({\hat x},\lambda)\circ\operatorname{D}\hspace{-0.08cm}^2\hat R_{{\hat x}}(0) + \Ocal{h}.
\text{end}d{align*}
By plugging in the Taylor expansions of $\hat n({\hat y}(h),\lambda + h)$ and $\hat F({\hat y}(h),\lambda + h)$ given by~\eqref{eq:FhatExpansion} and~\eqref{eq:nhatOfYExpansion} respectively we obtain
\begin{equation*}
\hat F(\hat z(h),\lambda + h) = h^4c_4({\hat x}, \lambda) + \Ocal{h^5},
\text{end}d{equation*}
for some $c_4({\hat x}, \lambda)$ not depending on $h$.
Now, for the left-hand side term in~\eqref{eq:nhatOfZ}, the Taylor expansion with respect to $\hat n({\hat y}(h),\lambda + h)$ gives
\begin{equation*}
\hat H(\hat z(h),\lambda + h) = \hat H({\hat y}(h),\lambda + h) + \Ocal{\|\hat n({\hat y}(h),\lambda + h)\|} = \hat H({\hat x},\lambda) + \Ocal{h},
\text{end}d{equation*}
and thus
\begin{equation*}
\hat H(\hat z(h),\lambda + h)^{-1} = \hat H({\hat x},\lambda)^{-1} + \Ocal{h}.
\text{end}d{equation*}
Therefore
\begin{equation*}
\hat n(\hat z(h),\lambda + h) = h^4c_5({\hat x}, \lambda) + \Ocal{h^5},
\text{end}d{equation*}
where $c_5({\hat x}, \lambda) = -\hat H({\hat x},\lambda)^{-1}\pac{c_4({\hat x}, \lambda)}$. Finally, noticing that $\hat G_{\hat z(h)} = \hat G_{{\hat x}} + \Ocal{h}$, we can approximate the numerator of $\kappa$ as
\begin{equation}\label{eq:nHatOfZNormExpansion}
\sqrt{\hat n(\hat z(h),\lambda\! +\! h)^T G_{\hat z(h)}\hat n(\hat z(h),\lambda \!+\! h)} = \pa{h^8c_6({\hat x}, \lambda)^2\! +\! \Ocal{h^9}}^{1/2}\! = h^4c_6({\hat x}, \lambda)\! +\! \ocal{h^4},
\text{end}d{equation}
with $c_6({\hat x}, \lambda) = \sqrt{c_5({\hat x}, \lambda) G_{\hat x} c_5({\hat x}, \lambda)}$. This allows to conclude that
\begin{equation*}
\kappa(x, \lambda,h) = \frac{h^4c_6({\hat x}, \lambda) + \ocal{h^4}}{h^2 c_3({\hat x}, \lambda) + \Ocal{h^3}} = h^2 c_7({\hat x}, \lambda) + \ocal{h^2},
\text{end}d{equation*}
with $c_7({\hat x}, \lambda) = \frac{c_6({\hat x}, \lambda)}{c_3({\hat x}, \lambda)}$, where we used the Taylor expansion of the inverse function in $c_3({\hat x},\lambda)$, which is non-zero as noted for result (i). This proves the expansion (ii) with ${\kappa_2(x, \lambda) = c_7(\varphi(x), \lambda)}$.
\noindent \emph{Result (iii)}\\
The proof for the prediction angle requires to expand the argument of the arcosine at second order and exploit the following Puiseux series expansion, for $q>0$:
\begin{equation}\label{eq.PuiseuxSeries}
\operatorname{acos}(1-q) = \sqrt{2q} + \frac{q^{3/2}}{6\sqrt{2}} + \Ocal{q^2}.
\text{end}d{equation}
By combining~\eqref{eq:yhat} and~\eqref{eq:that}, we find
\begin{equation*}
\hat t({\hat y}(h),\lambda+h) = -\hat H(\hat y(h),\lambda+h)^{-1}\pac{\hat F_\lambda(\hat y(h),\lambda+h)}.
\text{end}d{equation*}
Concerning the transport of this vector, we exploit the smoothness of ${\hat y}$, $\hat H$, $\hat F_\lambda$ and of the transporter operator to conclude
\begin{equation}\label{eq:transportExpansion}
\hat \tcal_{{\hat y}(h)\to {\hat x}}\pa{\hat t({\hat y}(h),\lambda + h)} = \hat t({\hat x},\lambda) + h \hat \tcal^{(1)}({\hat x},\lambda) + h^2\hat \tcal^{(2)}({\hat x},\lambda) + \Ocal{h^3},
\text{end}d{equation}
for some $\hat \tcal^{(1)}({\hat x},\lambda)$ and $\hat \tcal^{(2)}({\hat x},\lambda)$ depending smoothly only on $({\hat x},\lambda)$.
Let us from now on omit the dependence on $({\hat x},\lambda)$ of these vectors (i.e. $\hat t = \hat t({\hat x},\lambda)$, $\hat \tcal^{(1)} = \hat \tcal^{(1)}({\hat x},\lambda)$, $\hat \tcal^{(2)} = \hat \tcal^{(2)}({\hat x},\lambda)$). Computing the inner product of ~\eqref{eq:transportExpansion} with itself and using the Taylor expansion of the square root in $\|\hat t\|^2$ we have
\begin{equation}\label{eq:transportNormExpansion}
\!\!\!\|\hat \tcal_{{\hat y}(h)\to {\hat x}}\!\pa{\hat t({\hat y}(h),\lambda\!+\! h)}\!\|\!=\! \|\hat t\| + h\frac{\hat t^T\hat G_{{\hat x}}\hat \tcal^{(1)}}{\|\hat t\|} + h^2\!\!\pa{\!\frac{2\hat t^T\hat G_{\hat x}\hat \tcal^{(2)}\!\!+\!\! \|\hat \tcal^{(1)}\|^2}{2\|\hat t\|}\!-\!\frac{\pa{\!\hat t^T\hat G_{\hat x}\hat \tcal^{(1)}\!}\!\!^2}{2\|\hat t\|^3}\!}\!.
\text{end}d{equation}
Then, combing~\eqref{eq:transportExpansion} and~\eqref{eq:transportNormExpansion} with the expansion of the inverse function we get
\begin{align*}
\frac{\hat \tcal_{{\hat y}(h)\to {\hat x}}\pa{\hat t({\hat y}(h),\lambda + h)}}{\|\hat \tcal_{{\hat y}(h)\to {\hat x}}\pa{\hat t({\hat y}(h),\lambda + h)}\|} = \frac{\hat t}{\|\hat t\|} + h\pa{\frac{\hat \tcal^{(1)}}{\|t\|} - \frac{\hat t^T\hat G_{\hat x}\hat \tcal^{(1)}}{\|\hat t\|^3}\hat t}+\\ \frac{h^2}{2}\pa{\frac{3(\hat t^T\hat G_{\hat x}\hat \tcal^{(1)})^2}{\|\hat t\|^5}\hat t-\frac{2\hat t^T\hat G_{\hat x}\hat \tcal^{(2)} + \|\hat \tcal^{(1)}\|^2}{\|\hat t\|^3}\hat t - \frac{2\hat t^T\hat G_{\hat x}\hat \tcal^{(1)}}{\|\hat t\|^3}\hat \tcal^{(1)} + \frac{2\hat \tcal^{(2)}}{\|\hat t\|}} + \Ocal{h^3}.
\text{end}d{align*}
Computing the inner product of this expression with $\frac{\hat t}{\|\hat t\|}$ with respect to the metric $\hat G_{{\hat x}}$ we get $\cos\pa{\alpha(x,\lambda, h)}$ and it can be see that the term proportional to $h$ vanishes. Thus if we denote $\cos(\theta_{\hat x}(\hat\xi,\hat\,\& \,a)) = \frac{\hat{\xi}^T\hat G_{\hat x}\hat{\,\& \,a}}{\|\hat{\xi}\|\|\hat{\,\& \,a}\|}$ we find
\begin{align*}
\cos(\alpha(x,\lambda)) &= \frac{\hat t^T}{\|\hat t\|}\hat G_{\hat x}\frac{\hat \tcal_{{\hat y}(h)\to {\hat x}}\pa{\hat t({\hat y}(h),\lambda + h)}}{\|\hat \tcal_{{\hat y}(h)\to {\hat x}}\pa{\hat t({\hat y}(h),\lambda + h)}\|} \\
&= 1 + \frac{h^2}{2}\pa{\frac{\pa{\hat t^T\hat G_{\hat x}\hat \tcal^{(1)}}^2}{\|\hat t\|^4} - \frac{\|\hat \tcal^{(1)}\|^2}{\|\hat t\|^2}} + \Ocal{h^3}\\
&= 1-h^2\frac{\sin(\theta_{\hat x}(\hat t,\hat \tcal^{(1)}))^2\|\hat \tcal^{(1)}\|^2}{2\|\hat t\|^2}+\Ocal{h^3}.
\text{end}d{align*}
Assumptions~\eqref{eq.nonDegeneracy} and~\eqref{eq:missingAssumption} imply that coefficient multiplied by $h^2$ is not zero. Finally, using the Puiseux series~\eqref{eq.PuiseuxSeries} for the arcosine, we conclude
\begin{equation*}
\alpha(x,\lambda, h) = h\alpha_1(x, \lambda) + \Ocal{h^2},\text{ with } \alpha_1(x, \lambda) = \frac{|\sin(\theta_{\hat x}(\hat t,\hat \tcal^{(1)}))|\|\hat \tcal^{(1)}\|}{\|\hat t\|}.
\text{end}d{equation*}
\flushright$\square$
\text{end}d{appendices}
\text{end}d{document} |
\text{b}egin{equation}gin{document}
\title{Collective Modes in the Cooperative Jahn-Teller Model: Path Integral Approach}
\pacs{64.70.Tg, 03.67.Ac, 37.10.Ty, 71.70.Ej}
\author{Peter A. Ivanov }
\email{[email protected]}
\affiliation{Department of Physics, St. Kliment Ohridski University of Sofia, James
Bourchier 5 Boulevard, 1164 Sofia, Bulgaria}
\text{b}egin{equation}gin{abstract}
We discuss analytical approximations to the ground-state phase diagram and the elementary excitations of the cooperative Jahn-Teller model describing strongly correlated spin-boson system on a lattice in various quantum optical systems. Based on the mean-field theory approach we show that the system exhibits quantum magnetic structural phase transition which leads to magnetic ordering of the spins and formation of the bosonic condensates. We determine existing of one gapless Goldstone mode and two gapped amplitude modes inside the symmetry-broken phase.
\end{abstract}
-aketitle
+ection{Introduction}
Over the last few years there has been a great deal of interest in studying the many-body physics of strongly correlated spins-bosons lattice models using a table top experimental quantum-optical systems \cite{Georgescu2014}. A prominent example is the recent experimental demonstration of the quantum phase transition from Mott insulator state to superfluid state of spin-boson excitations in a system of trapped ions \cite{Toyoda2013}. It was shown that with the spontaneous breaking of continuous $U(1)$ symmetry at the quantum phase transition in such models, including for example Jaynes-Cummings-Hubbard system \cite{Schmidt2010} and Dicke-like system \cite{Baksic2014,Xiang2013}, two-types of collective excitations emerge. One is the the gapless Goldstone mode, and the other is the gapped amplitude mode corresponding to phase and density fluctuations. Such amplitude modulation of the order parameter is referred to as Higgs mode generated by a physical mechanism analogous as the Higgs boson in high energy physics. Recently, the amplitude mode was experimentally observed in a system of strongly interacting condensate of ultracold atoms near the superfluid-insulator phase transition \cite{Bissbort2011,Endres2012} opened fascinating prospect for exploring the condensed matter excitations under controlled conditions.
In this work we present study of the collective hybrid spin-boson excitations in the cooperative Jahn-Teller (cJT) model. Originally, the Jahn-Teller model was introduced to explain the distortions and the nondegenerate energy levels in molecules, via the strong interaction between the localized electronic states and the vibrations of the nuclei \cite{Englman,Bersuker}. In solids, the cJT effect leads to structural phase transition and magnetic ordering of the spins. Furthermore, the collective effects induced by the Jahn-Teller coupling may explain the transition of some solids, such as fullerene compounds, to high-temperature superconductors \cite{Dunn2004}. With the current quantum optical technologies the cJT model can be realized in laser or magnetically driven ion crystal \cite{Porras2012,Ivanov2013} and cavity/circuit QED systems based on superconducting qubits in transmission line resonators \cite{Larson2008,Dereli}. Here we focus on $E\otimes e$ Jahn-Teller model which possess continuous $U(1)$ symmetry. We use path integral approach to describe analytically the quantum magnetic structural phase transition with the formation of bosonic condensates and magnetic ordering of the spins in the cJT model. Within the framework of the saddle-point approximation we determine the mean-field solution and then consider the quantum fluctuations around the mean-field result. We show that the energy spectrum of the cJT system consists of three collective excitations branches. In the symmetry broken phase we find a linear gapless Goldstone mode and two gapped amplitude modes.
The paper is organized as follows. In Sec. \text{r}ef{CJTm} we introduce the cJT model and consider the associated continuous $U(1)$ symmetry. In Sec. \text{r}ef{implementation} we discuss a possible scheme for the experimental realization of the cJT model using quantum optical systems. In Sec. \text{r}ef{PIF} we turn to the path integral treatment of the model and determine its saddle-point. In Sec. \text{r}ef{QF} we consider the quantum fluctuations around the mean-field solution and find the elementary excitations of the cJT model. In Sec. \text{r}ef{MP} we discuss many-body spectroscopy protocol to detect the quantum phases and the collective excitations. Finally, the conclusions are presented in Sec. \text{r}ef{C}.
+ection{Cooperative Jahn-Teller Model}\label{CJTm}
We consider a chain of $N$ spins with states $\left|\uparrow_{j}\text{r}ight\text{r}angle$, $\left|\downarrow_{j}\text{r}ight\text{r}angle$ each one coupled symmetrically with two boson species ($\epsilon=x,y$ and $\hbar=1$ from now on),
\text{b}egin{equation}gin{eqnarray}
&&\hat{H}_{\text{r}m cJT}=\hat{H}_{\text{r}m s}+\hat{H}_{\text{r}m t-b}+\hat{H}_{\text{r}m I},\quad \hat{H}_{\text{r}m s}=\text{f}rac{\omega_{z}}{2}+um_{j=1}^{N}+igma_{j}^{z}, \notag\\
&&\hat{H}_{\text{r}m t-b}=+um_{\epsilon}+um_{j=1}^{N}\Delta_{j}\hat{a}_{\epsilon,j}^{\dag}\hat{a}_{\epsilon,j}++um_{\epsilon}+um_{j>l}^{N}t_{j,l}(\hat{a}_{\epsilon,j}^{\dag}\hat{a}_{\epsilon,l}+{\text{r}m H.c}),\notag\\
&&\hat{H}_{\text{r}m I}=\text{f}rac{g}{+qrt{2}}+um_{j=1}^{N}\{+igma_{j}^{x}(\hat{a}_{x,j}^{\dag}+\hat{a}_{x,j})++igma_{j}^{y}(\hat{a}_{y,j}^{\dag}+\hat{a}_{y,j})\}.\label{HcJT}
\end{eqnarray}
The term $\hat{H}_{\text{r}m s}$ describes the energy of the effective spins with frequency $\omega_{z}$, where $+igma_{j}^{c}$ ($c=x,y,z$) are the Pauli matrices for the spin at site $j$. Note that this term also can represent the coupling with applied external magnetic field. The tight-binding term $\hat{H}_{\text{r}m t-b}$ describes the delocalization of the two bosonic species $\epsilon=x,y$ between different lattice sites with hopping matrix elements $t_{j,l}<0$ and on-site boson energy $\Delta_{j}$, where $\hat{a}_{\epsilon,j}^{\dag}$, $\hat{a}_{\epsilon,j}$ are the respective creation and annihilation operators of boson at site $j$. The last term in (\text{r}ef{HcJT}) describes the $E\otimes e$ symmetrical Jahn-Teller interaction between the spins and two boson species with coupling strength $g$.
Alternatively, the spin-boson interaction can be expressed in terms of right $\hat{a}_{{\text{r}m r},j}^{\dag}=(\hat{a}_{x,j}^{\dag}+{\text{r}m i}\hat{a}_{y,j}^{\dag})/+qrt{2}$ and left $\hat{a}_{{\text{r}m l},j}^{\dag}=(\hat{a}_{x,j}^{\dag}-{\text{r}m i}\hat{a}_{y,j}^{\dag})/+qrt{2}$ chiral operators, which yield
\text{b}egin{equation}gin{equation}
\hat{H}_{\text{r}m I}=g+um_{j=1}^{N}\{+igma_{j}^{+}(\hat{a}_{{\text{r}m r},j}+\hat{a}_{{\text{r}m l},j}^{\dag})++igma_{j}^{-}(\hat{a}_{{\text{r}m r},j}^{\dag}+\hat{a}_{{\text{r}m l},j})\},\label{HI}
\end{equation}
where $+igma_{j}^{\pm}$ are the corresponding spin raising and lowering operators. We note that because of the contra-rotating terms $+igma_{j}^{+}\hat{a}_{{\text{r}m l},j}^{\dag}$ and $+igma_{j}^{-}\hat{a}_{{\text{r}m r},j}$ in (\text{r}ef{HI}) the total number of spin and boson excitations is not conserved. Instead of that, the Hamiltonian $H_{\text{r}m cJT}$ commute with the operator $\hat{C}=+um_{j=1}^{N}(\hat{a}_{{\text{r}m r},j}^{\dag}\hat{a}_{{\text{r}m r},j}-\hat{a}_{{\text{r}m l},j}^{\dag}\hat{a}_{{\text{r}m l},j}++igma_{j}^{z}/2)$. The latter implies that the cJT Hamiltonian (\text{r}ef{HcJT}) possesses continuous $U(1)$ symmetry implemented by the action of the operator $\hat{R}(\phi)=e^{{\text{r}m i}\phi \hat{C}}$, which gives
\text{b}egin{equation}gin{eqnarray}
&&\hat{R}(\phi)\hat{a}_{{\text{r}m r},j}^{\dag}\hat{R}(\phi)^{\dag}=e^{{\text{r}m i}\phi}\hat{a}_{{\text{r}m r},j}^{\dag},\quad \hat{R}(\phi)\hat{a}_{{\text{r}m l},j}^{\dag}\hat{R}(\phi)^{\dag}=e^{-{\text{r}m i}\phi}\hat{a}_{{\text{r}m l},j}^{\dag},\notag\\
&&\hat{R}(\phi)+igma_{j}^{\pm}\hat{R}(\phi)^{\dag}=e^{\pm {\text{r}m i}\phi}+igma_{j}^{\pm},\label{symmetry}
\end{eqnarray}
such that we have $\hat{R}(\phi)\hat{H}_{{\text{r}m cJT}}\hat{R}(\phi)^{\dag}=\hat{H}_{{\text{r}m cJT}}$. Finally, it is convenient to work in representation, where the tight-binding term $\hat{H}_{\text{r}m t-b}$ in (\text{r}ef{HcJT}) is diagonal. Indeed, performing the transformation in to the momentum space $\hat{a}_{\gamma,k}=+um_{j=1}^{N}b_{k,j}\hat{a}_{\gamma,j}$, ($\gamma={\text{r}m r,l}$ from now on), where $b_{k,j}$ are the normal mode wave functions yield $\hat{H}_{\text{r}m t-b}=+um_{\gamma}+um_{k=1}^{N}\Delta_{k}\hat{a}_{\gamma,k}^{\dag}\hat{a}_{\gamma,k}$ with $\Delta_{k}$ being the collective mode energies. Hereafter we assume that $\Delta_{k}>0$ with minimum at $k=0$ corresponding to the center-of-mass mode.
In the following we discuss the realization of the cJT model using quantum optical systems.
+ection{Implementation with quantum optical systems}\label{implementation}
The cJT model comprises of two bosonic species and spin degrees of freedom, the specific interpretation of which depends on the actual physical system. One possible experimental setup for the realization of the model is based on the laser cooled trapped ions \cite{Wineland,Schneider2012}. In that case the bosonic degrees of freedom represent the local phonons, which quantify the small radial ion oscillations around the equilibrium positions \cite{Ivanov2009,Porras2004}. The hopping term $\hat{H}_{\text{r}m t-b}$ in (\text{r}ef{HcJT}) describes the Coulomb-mediated long-range phonon hopping dynamics with hopping elements $t_{j,l}$ and on-site frequency $\Delta_{j}$. The spins are implemented by the internal two metastable levels of the trapped ions where $\omega_{z}$ is the effective spin frequency. The desired Jahn-Teller coupling can be realized by the interaction of the spins with an oscillating magnetic field gradient \cite{Porras2012,Ivanov2013}. Alternative realization of the Jahn-Teller coupling is based on the interaction of the ions with laser beams propagating in two orthogonal directions tuned near the respective red and blue sidebands \cite{Ivanov2014}. The ion trap based realization of the cJT model offers unique opportunity to easy tuning the parametric regime of the couplings by adjusting for example the trap frequencies and the laser intensity. Although with the current ion technologies the realization of the model is restricted to one-dimension where the ions are placed in a chain, considerable progress is achieved to scaling to two-dimensional ion trap network where the ions are trapped in individual potential wells \cite{Sterling2014,Wilson2014}.
On the other hand lattice spin-boson models can be realized naturally in cavity and circuit QED systems \cite{Greentree2006,Angelakis2007,Hartmann2007}. Here the bosons represent a single or several quantized modes inside of an electromagnetic resonators, while the spin degrees of freedom are implemented either by real two-level atom, or artificial atoms such as quantum dot or superconducting circuit. Recently it was shown that the interaction between the quantum dot \cite{Larson2008} or superconducting circuit \cite{Dereli} with two cavity modes leads to the Jahn-Teller coupling. Arranging atoms and resonators in the form of lattice can realize our model (\text{r}ef{HcJT}), with coupling between the resonators provided by the photon hopping.
In the next section, we turn to the field-theoretical treatment of the cJT model. This method allows us to determine the stationary saddle-point of the model and then consider the small fluctuations around the mean-field result.
+ection{Functional Integral Representation of the cooperative Jahn-Teller model}\label{PIF}
+ubsection{Path integral approach to the cJT model}
In the functional integral treatment, the second quantized Hamiltonian of the model is translated to the phase representation with the help of the path integral formalism. In this approach the boson operators are replaced with their associated fields, namely $\hat{a}_{\gamma,k}\text{r}ightarrow \alpha_{\gamma,k}(\tau)$, $\hat{a}_{\gamma,k}^{\dag}\text{r}ightarrow \alpha_{\gamma,k}^{*}(\tau)$ where $\tau$ is the imaginary time \cite{Auerbach}. For the spin-degrees of freedom we choose a spin-coherent representation with coherent-state parameterized by the independent polar $\theta_{j}$ and azimuthal $\varphi_{j}$ angles, respectively,
\text{b}egin{equation}gin{equation}
|n_{j}\text{r}angle=\cos\left(\text{f}rac{\theta_{j}}{2}\text{r}ight)\left|\uparrow_{j}\text{r}ight\text{r}angle+e^{{\text{r}m i}\varphi_{j}}+in\left(\text{f}rac{\theta_{j}}{2}\text{r}ight)\left|\downarrow_{j}\text{r}ight\text{r}angle.\label{nvector}
\end{equation}
The spin operators are replaced by the corresponding Bloch vector $\vec{n}_{j}=[n_{x,j},n_{y,j},n_{z,j}]$ whose components are the expectation values of the Pauli matrices with respect to the state (\text{r}ef{nvector}) which gives
\text{b}egin{equation}gin{equation}
\vec{n}_{j}=[+in(\theta_{j})\cos(\varphi_{j}),+in(\theta_{j})+in(\varphi_{j}),\cos(\theta_{j})].
\end{equation}
The Bloch vector has a unit length $\vec{n}_{j}^{2}=1$ and specifies the orientation of spin at site $j$. Having this in hand the partition function for the cJT model can be expressed as
\text{b}egin{equation}gin{equation}
Z(\text{b}egin{equation}ta)=\text{i}nt\prod_{\gamma,k}\prod_{j}D\alpha_{\gamma, k}^{*}(\tau)D\alpha_{\gamma, k}(\tau)D\vec{n}_{j}(\tau)
\delta(\vec{n}_{j}^{2}-1)e^{-S},
\end{equation}
with the Euclidian action given by
\text{b}egin{equation}gin{eqnarray}
S&=&\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau\{+um_{\gamma,k}(\alpha_{\gamma,k}^{*}\text{f}rac{\partial\alpha_{\gamma,k}}{\partial\tau}+\Delta_{k}\alpha_{\gamma,k}^{*}\alpha_{\gamma,k})
+\text{f}rac{\omega_{z}}{2}+um_{j}\cos(\theta_{j})\notag\\
&&+\text{f}rac{g}{2}+um_{j,k}+in(\theta_{j})\{e^{{\text{r}m i}\varphi_{j}}(b_{k,j}^{*}\alpha_{{\text{r}m r},k}+b_{k,j}\alpha_{{\text{r}m l},k}^{*})+{\text{r}m c.c}\}\}+S_{\text{r}m B},\label{S}
\end{eqnarray}
where $\text{b}egin{equation}ta=1/T$ is the inverse temperature.
The Berry phase contribution to the action (\text{r}ef{S}) from the spin-degrees of freedom is given by
\text{b}egin{equation}gin{equation}
S_{\text{r}m B}=+um_{j}\text{i}nt_{0}^{\text{b}egin{equation}ta}\langle n_{j}|\text{f}rac{\partial}{\partial\tau}|n_{j}\text{r}angle={\text{r}m i}+um_{j}\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau
+in^{2}\left(\text{f}rac{\theta_{j}}{2}\text{r}ight)\text{f}rac{\partial\varphi_{j}}{\partial\tau}.\label{berry}
\end{equation}
Finally, we require that the corresponding bosonic fields have periodic boundary conditions $\alpha_{\gamma,k}(\text{b}egin{equation}ta)=\alpha_{\gamma,k}(0)$ and $\alpha_{\gamma,k}^{*}(\text{b}egin{equation}ta)=\alpha_{\gamma,k}^{*}(0)$. The same condition is hold and for the spin variables, where we have $\theta_{j}(\text{b}egin{equation}ta)=\theta_{j}(0)$ and $\varphi_{j}(\text{b}egin{equation}ta)=\varphi_{j}(0)$.
+ubsection{Saddle-Point Approximation}
Next we consider the classical equation of motion, which are determined by the condition, that the variation of the action (\text{r}ef{S}) with respect to the field variables should vanish,
\text{b}egin{equation}gin{equation}
\text{f}rac{\delta S}{\delta \theta_{j} }=0, \quad \text{f}rac{\delta S}{\delta \varphi_{j} }=0, \quad \text{f}rac{\delta S}{\delta \alpha_{\gamma,k}^{*}}=0.\label{Var}
\end{equation}
Note that the same condition is also satisfied for the bosonic field $\alpha_{\gamma,k}$. The term classical refers to a mean-field solution, i.e., disregarding the quantum fluctuations. The variation of the action $S$ with respect of the spin-degrees of freedom gives the following equations of motion
\text{b}egin{equation}gin{eqnarray}
{\text{r}m i}+in(\theta_{j})\text{f}rac{\partial\varphi_{j}}{\partial\tau}&=&\omega_{z}+in(\theta_{j})-g\cos(\theta_{j})
+um_{k}\{e^{{\text{r}m i}\varphi_{j}}(b_{k,j}^{*}\alpha_{{\text{r}m r},k} \notag\\
&&+b_{k,j}\alpha_{{\text{r}m l},k}^{*})+{\text{r}m c.c}\},\notag\\
\text{f}rac{\partial\theta_{j}}{\partial\tau}&=&g+um_{k}\{e^{{\text{r}m i}\varphi_{j}}(b_{k,j}^{*}\alpha_{{\text{r}m r},k}
+b_{k,j}\alpha_{{\text{r}m l},k}^{*})-{\text{r}m c.c}\})\label{spin_eq},
\end{eqnarray}
where the dynamics follow from the Berry phase term (\text{r}ef{berry}). The third condition in (\text{r}ef{Var}) reads
\text{b}egin{equation}gin{eqnarray}
\text{f}rac{\partial\alpha_{{\text{r}m r},k}}{\partial\tau}&=&-\Delta_{k}\alpha_{{\text{r}m r},k}-\text{f}rac{g}{2}+um_{j}b_{k,j}(n_{x,j}-{\text{r}m i}n_{y,j}),\notag\\
\text{f}rac{\partial\alpha_{{\text{r}m l},k}}{\partial\tau}&=&-\Delta_{k}\alpha_{{\text{r}m l},k}-\text{f}rac{g}{2}+um_{j}b_{k,j}(n_{x,j}+{\text{r}m i}n_{y,j}).\label{alpha}
\end{eqnarray}
We note that because the action $S$ (\text{r}ef{S}) is invariant with respect to $U(1)$ transformation specified in Eq. (\text{r}ef{symmetry}), the corresponding equations of motion (\text{r}ef{spin_eq}) and (\text{r}ef{alpha}) obey the same symmetry. Although the Eq. (\text{r}ef{spin_eq}) describes the dynamics of the spin-degree of freedom it is not expressed in terms of $\vec{n}_{j}$. One way to remedy this is to introduce a new set of two orthogonal to $\vec{n}_{j}$ vectors, namely $\vec{\theta}_{j}=[\cos(\theta_{j})\cos(\varphi_{j}),\cos(\theta_{j})+in(\varphi_{j}),-+in(\theta_{j})]$ and $\vec{\varphi}_{j}=[-+in(\varphi_{j}),\cos(\varphi_{j}),0]$ which form an orthogonal triad $\vec{\theta}_{j}\times\vec{\varphi}_{j}=\vec{n}_{j}$. Then the Eq. (\text{r}ef{spin_eq}) is rewritten as follows
\text{b}egin{equation}gin{eqnarray}
\text{f}rac{{\text{r}m i}}{2}\vec{\varphi}_{j}\cdot\text{f}rac{\partial \vec{n}_{j}}{\partial\tau}&=&\text{f}rac{\omega_{z}}{2}(\vec{\varphi}_{j}+\vec{\theta}_{j})\cdot\text{f}rac{\partial\vec{n}_{j}}{\partial\varphi_{j}}-g\vec{\theta}_{j}\cdot\vec{\alpha}_{j},\notag\\
\text{f}rac{{\text{r}m i}}{2}\vec{\theta}_{j}\cdot\text{f}rac{\partial\vec{n}_{j}}{\partial\tau}&=&g\vec{\varphi}_{j}\cdot\vec{\alpha}_{j}.\label{spin1}
\end{eqnarray}
Here we have introduce the vector notation $\vec{\alpha}_{j}=+qrt{2}[\Re{\alpha_{x,j}},\Re{\alpha_{y,j}},0]$ for the two bosonic fields. Finally, one can combine the two equations in (\text{r}ef{spin1}) in a vector form which yield
\text{b}egin{equation}gin{equation}
\text{f}rac{{\text{r}m i}}{2}\text{f}rac{\partial\vec{n}_{j}}{\partial\tau}=\text{f}rac{\omega_{z}}{2}\vec{B}\times\vec{n}_{j}+g\vec{\alpha}_{j}\times\vec{n}_{j},\label{spin_vector}
\end{equation}
where we use $\text{f}rac{\partial\vec{n}_{j}}{\partial\varphi_{j}}=\vec{B}\times \vec{n}_{j}$. Now we are in position to interpret the dynamical equation for the spins. The first term in (\text{r}ef{spin_vector}) represent the effect of the externally applied magnetic field along the $z$ direction, $\vec{B}=[0,0,1]$, such that for $g=0$ the spins will perform precession with frequency determined by $\omega_{z}$. On the other hand the spin-boson interaction gives rise to an effective magnetic field $\vec{\alpha}_{j}$ experienced by the spin at lattice site $j$. In the case when the bosonic fields describe the motional degrees of freedom of the spins in two orthogonal directions, the effective magnetic field $\vec{\alpha}_{j}$ becomes position-dependent, which is in close analogy with the Rashba spin-orbit coupling in the quantum spin Hall effect \cite{Sinova2004}. Because the vectors $\vec{B}$ and $\vec{\alpha}_{j}$ are always orthogonal, the spins execute precession along the axis $45^{0}$ to both magnetic fields. We note that although Eq. (\text{r}ef{spin_vector}) is purely local in a sense that it only depends on the lattice index $j$, the components of $\vec{\alpha}_{j}$ depend on the boson fields at different sites due to the tunneling elements $t_{j,l}$. As we will see below such a tight-binding lattice dynamics of the two bosonic species strongly coupled to the spins are capable of forming magnetic ordering and bosonic condensates.
The stationary saddle-point is obtained by the solution of Eq. (\text{r}ef{Var}) with the requirement that $\alpha_{\gamma,k}(\tau)=\text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}$ and $\vec{n}_{j}(\tau)=\vec{\text{b}egin{equation}gin{align}r{n}}_{j}$. Then we derive the following set of algebraic equations for the spin-degrees of freedom
\text{b}egin{equation}gin{eqnarray}
&&\omega_{z}+in(\text{b}egin{equation}gin{align}r{\theta}_{j})=-\cos(\text{b}egin{equation}gin{align}r{\theta}_{j})+um_{l}J_{j,l}+in(\text{b}egin{equation}gin{align}r{\theta}_{l})\cos(\text{b}egin{equation}gin{align}r{\varphi}_{j}-\text{b}egin{equation}gin{align}r{\varphi}_{l}),\notag\\
&&+um_{l}J_{j,l}+in(\text{b}egin{equation}gin{align}r{\theta}_{l})+in(\text{b}egin{equation}gin{align}r{\varphi}_{j}-\text{b}egin{equation}gin{align}r{\varphi}_{l})=0,\label{mf_s}
\end{eqnarray}
where $J_{j,l}=2+um_{k}\text{f}rac{g^{2}}{\Delta_{k}}\Re\{b_{k,j}b_{k,l}^{*}\}$ and respectively for the bosonic fields
\text{b}egin{equation}gin{eqnarray}
&&\text{b}egin{equation}gin{align}r{\alpha}_{{\text{r}m r},k}=-\text{f}rac{g}{2\Delta_{k}}+um_{j}b_{k,j}+in(\text{b}egin{equation}gin{align}r{\theta}_{j})e^{-{\text{r}m i}\text{b}egin{equation}gin{align}r{\varphi}_{j}},\notag\\
&&\text{b}egin{equation}gin{align}r{\alpha}_{{\text{r}m l},k}=-\text{f}rac{g}{2\Delta_{k}}+um_{j}b_{k,j}+in(\text{b}egin{equation}gin{align}r{\theta}_{j})e^{{\text{r}m i}\text{b}egin{equation}gin{align}r{\varphi}_{j}}\label{mf_b}.
\end{eqnarray}
Apparently, the system (\text{r}ef{mf_s}) has a trivial solution $+in(\text{b}egin{equation}gin{align}r{\theta}_{j})=0$ for ($j=1,2,\ldots,N$) which implies $\text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}=0$. Assuming $+in(\text{b}egin{equation}gin{align}r{\theta}_{j})\neq 0$, the condition $\text{b}egin{equation}gin{align}r{\varphi}_{j}=\text{b}egin{equation}gin{align}r{\varphi}_{l}=\text{b}egin{equation}gin{align}r{\varphi}$ solved the second equation in (\text{r}ef{mf_s}). The latter is the arbitrary choice for a direction of spontaneous symmetry breaking where the system chooses a direction along which to order. Hear after we assume $\text{b}egin{equation}gin{align}r{\varphi}=0$, such that the Bloch vector becomes $\vec{\text{b}egin{equation}gin{align}r{n}}_{j}=[+in(\text{b}egin{equation}gin{align}r{\theta}_{j}),0,\cos(\text{b}egin{equation}gin{align}r{\theta}_{j})]$ indicating that the spins are aligned in the $xz$ plane. Let us now discuss the homogenous limit $\text{b}egin{equation}gin{align}r{\theta}_{j}=\text{b}egin{equation}gin{align}r{\theta}$ neglecting any boundary effect. In this limit Eqs. (\text{r}ef{mf_s}) and (\text{r}ef{mf_b}) can be solved exactly, which yield
\text{b}egin{equation}gin{eqnarray}
&&\cos(\text{b}egin{equation}gin{align}r{\theta})=-1,\quad \text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}=0,\quad g< g_{{\text{r}m c}},\notag\\
&&\cos(\text{b}egin{equation}gin{align}r{\theta})=-\text{f}rac{g_{{\text{r}m c}}^{2}}{g^{2}},\quad \text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}=-\text{f}rac{g+qrt{N}}{2\Delta_{k}}+in(\text{b}egin{equation}gin{align}r{\theta})\delta_{k,0},\quad
g>g_{\text{r}m {c}}\label{sp}
\end{eqnarray}
The solution (\text{r}ef{sp}) corresponds to the classical ground-state of the cJT model. For a coupling smaller than the critical value of $g_{{\text{r}m c}}=+qrt{\Delta_{0}\omega_{z}/2}$ ($g<g_{\text{r}m c}$) the system is in a normal state where the Bloch vector for each spin points along the $-z$ direction and $\text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}=0$. Increasing the coupling through $g_{{\text{r}m c}}$ ($g>g_{\text{r}m c}$) drives the system to undergo a quantum phase transition to a ferromagnetic ordering of spins in $xz$ plane and condensation of the two boson species in the lowest energy mode $k=0$.
Here we emphasize that for a linear ion crystal with positive hopping amplitude the saddle-point approximation is not applied straightforward. This problem can be overcome by applying a canonical transformation to the operators $\hat{a}_{\epsilon,j}\text{r}ightarrow (-1)^{j}\hat{a}_{\epsilon,j}$ and $+igma_{j}^{\epsilon}\text{r}ightarrow (-1)^{j}+igma_{j}^{\epsilon}$ in (\text{r}ef{HcJT}) \cite{Porras2012,Mering2009}. After this transformation to the staggered spin-boson basis the cJT model (\text{r}ef{HcJT}) is unchanged, but the tunneling is modified to $t_{j,l}^{\text{r}m stagg}=(-1)^{j-l}t_{j,l}$. The ferromagnetic spin order in the new basis, corresponds to an antiferromagnetic order in the physical basis, in which ions alternate spin direction and position.
In the following section we study the low-energy spectra of the cJT model in terms of collective excitations. We expand the action of the system around its saddle-point up to second order in the spin and bosonic fields. This leads to a Gaussian integral which can be evaluated.
+ection{Quantum Fluctuations around the saddle-point}\label{QF}
+ubsection{Linear parametrization}
Having described the saddle-point solution, we now consider the low-energy excitations of the cJT model in the symmetry-broken phase. For the bosonic fields we can use the standard linear parametrization
\text{b}egin{equation}gin{equation}
\alpha_{\gamma,k}=\text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}+\delta\alpha_{\gamma,k},
\end{equation}
where $\delta\alpha_{\gamma,k}$ describes the quantum fluctuations around the order parameter $\text{b}egin{equation}gin{align}r{\alpha}_{\gamma,k}$. In order to account the spin fluctuations around the state $\vec{\text{b}egin{equation}gin{align}r{n}}_{j}$ for each spin at site $j$ we first perform rotation of the Bloch vector $\vec{n}_{j}=\hat{R}(\text{b}egin{equation}gin{align}r{\theta})\vec{n}_{j}^{\prime}$ with rotation matrix given by
\text{b}egin{equation}gin{figure}[h]
\text{i}ncludegraphics[width=0.45\textwidth]{ecjt.eps}
\caption{(Color online) The dispersions of the three branch frequencies versus $k$. We set $\Delta/g=1$, $t/g=0.5$, and $\omega_{z}/g=1$. We have assumed position independent bosonic frequency $\Delta_{j}=\Delta+2t$.}
\label{fig1}
\end{figure}
\text{b}egin{equation}gin{equation}
\hat{R}(\text{b}egin{equation}gin{align}r{\theta}) =\left[
\text{b}egin{equation}gin{array}{ccc}
\cos(\text{b}egin{equation}gin{align}r{\theta}) & 0 & +in(\text{b}egin{equation}gin{align}r{\theta})\\
0 & 1 & 0 \\
-+in(\text{b}egin{equation}gin{align}r{\theta}) & 0 & \cos(\text{b}egin{equation}gin{align}r{\theta})
\end{array}
\text{r}ight].\label{R}
\end{equation}
Note that transformation of the Bloch vector implies rotation of the spin-coherent state specified by $|n_{j}^{\prime}\text{r}angle=e^{{\text{r}m i}\text{f}rac{\text{b}egin{equation}gin{align}r{\theta}}{2}+igma_{j}^{y}}|n_{j}\text{r}angle$. The rotation matrix $\hat{R}(\text{b}egin{equation}gin{align}r{\theta})$ is determined in a such a way that transform $\vec{\text{b}egin{equation}gin{align}r{n}}_{j}$ to a new reference Bloch vector $\vec{\text{b}egin{equation}gin{align}r{n}}_{j}^{\prime}=[0,0,1]$ which points along the $z$ direction.
Assuming that the spin and bosonic fluctuations are small in a sense that one can keep only the quadratic terms, such that $n_{z,j}^{\prime}=+qrt{1-n_{x,j}^{\prime 2}-n_{y,j}^{\prime2}}\approx 1-(n_{x,j}^{\prime 2}+n_{y,j}^{\prime 2})/2$ we obtain
\text{b}egin{equation}gin{eqnarray}
S&=&S_{\text{r}m B}^{\prime}+\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau\{+um_{\epsilon,k}(\delta\alpha_{\epsilon,k}^{*}\text{f}rac{\partial\delta\alpha_{\epsilon,k}}{\partial\tau}+\Delta_{k}\delta\alpha_{\epsilon,k}^{*}\delta\alpha_{\epsilon,k})\notag\\
&&+\text{f}rac{\Omega}{4}+um_{j}(n_{x,j}^{\prime 2}+n_{y,j}^{\prime 2})+\text{f}rac{g}{+qrt{2}}+um_{j,k}n_{y,j}^{\prime}
\{b_{k,j}^{*}\delta\alpha_{y,k}+b_{k,j}\delta\alpha_{y,k}^{*}\}
\notag\\
&&+\text{f}rac{g}{+qrt{2}}\cos(\text{b}egin{equation}gin{align}r{\theta})+um_{j,k}n_{x,j}^{\prime}
\{b_{k,j}^{*}\delta\alpha_{x,k}+b_{k,j}\delta\alpha_{x,k}^{*}
\}\},\label{S_f}
\end{eqnarray}
where the linear terms in the field fluctuations vanishes due to the conditions Eqs. (\text{r}ef{mf_s}) and (\text{r}ef{mf_b}) for $g>g_{\text{r}m c}$. Here $\Omega=\omega_{z}/|\cos(\text{b}egin{equation}gin{align}r{\theta})|$ is the renormalised spin frequency and $S_{\text{r}m B}^{\prime}$ is the Berry phase term in the rotating basis. Note that the $S_{\text{r}m B}^{\prime}$ is invariant with respect to the rotation transformation, which implies that $S_{{\text{r}m B}}^{\prime}=+um_{j}\text{i}nt_{0}^{\text{b}egin{equation}ta}\langle n_{j}^{\prime}|\text{f}rac{\partial}{\partial\tau}|n_{j}^{\prime}\text{r}angle$. Up to quadratic terms in the spin fluctuation fields the Berry phase can be written as
\text{b}egin{equation}gin{equation}
S_{\text{r}m B}^{\prime}=\text{f}rac{{\text{r}m i}}{4}+um_{j}\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau\left(n_{x,j}^{\prime}\text{f}rac{\partial n_{y,j}^{\prime}}{\partial\tau}-n_{y,j}^{\prime}\text{f}rac{\partial n_{x,j}^{\prime}}{\partial\tau}\text{r}ight).
\end{equation}
We emphasize that in order to describe the collective excitations around the mean-field solution one needs to identify the number of conjugate pairs. In our model presented here the bosonic fields $\delta\alpha_{\epsilon,k}$ and $\delta\alpha_{\epsilon,k}^{*}$ in (\text{r}ef{S_f}) are canonically conjugate variables, which leads to two independent degree of freedom. On the other hand the pairs $n_{+,j}^{\prime}=(n_{x,j}^{\prime}+{\text{r}m i}n_{y,j}^{\prime})/2$ and $n_{-,j}^{\prime}=(n_{x,j}^{\prime}-{\text{r}m i}n_{y,j}^{\prime})/2$ represent conjugate quantities corresponding to the spin fluctuations, which implies that one can expect in total three collective modes. In order to obtain the low-energy excitations, it is convenient to transform the spin fields in the momentum representation using $n_{+,j}^{\prime}=+um_{k}b_{k,j}n_{+,k}^{\prime}$ and $n_{-,j}^{\prime}=+um_{k}b_{k,j}^{*}n_{-,k}^{\prime}$, which yield
\text{b}egin{equation}gin{eqnarray}
S&=&\text{i}nt_{0}^{\text{b}egin{equation}ta}+um_{k}\{n_{-,k}^{\prime}\text{f}rac{\partial n_{+,k}^{\prime}}{\partial\tau} ++um_{\epsilon}(\delta\alpha_{\epsilon,k}^{*}\text{f}rac{\partial\delta\alpha_{\epsilon,k}}{\partial\tau}+\Delta_{k}\delta\alpha_{\epsilon,k}^{*}\delta\alpha_{\epsilon,k})\notag\\
&&+\Omega n_{+,k}^{\prime}n_{-,k}^{\prime}-\text{f}rac{g}{+qrt{2}}(n_{+,k}^{\prime}-n_{-,-k}^{\prime})(\delta\alpha_{y,-k}^{*}-\delta\alpha_{y,k})
\notag\\
&&+\text{f}rac{g}{+qrt{2}}\cos(\text{b}egin{equation}gin{align}r{\theta})(n_{+,k}^{\prime}+n_{-,-k}^{\prime})(\delta\alpha_{x,-k}^{*}+\delta\alpha_{x,k}).\label{S_k}
\end{eqnarray}
The action (\text{r}ef{S_k}) is quadratic in the field fluctuations, which lead to Gaussian functional integral. To diagonalize (\text{r}ef{S_k}) one can introduce harmonic oscillator degrees of freedom for each pair of conjugate variables, such that we obtain (see, Appendix \text{r}ef{A})
\text{b}egin{equation}gin{eqnarray}
S&=&\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau+um_{k}\{{\text{r}m i}+um_{a=1}^{3}\text{f}rac{\partial p_{a,k}}{\partial\tau}q_{a,k}+\text{f}rac{1}{2}+um_{a=1}^{3}p_{a,k}p_{a,-k}\notag\\
&&+\text{f}rac{1}{2}+um_{a,a^{\prime}=1}^{3}B_{a,a^{\prime}}^{(k)}q_{a,k}q_{a^{\prime},-k}\},\label{S_final}
\end{eqnarray}
where the coupling matrix $B_{a,a^{\prime}}^{(k)}$ is given by
\text{b}egin{equation}gin{equation}
B_{a,a^{\prime}}^{(k)} =\left[
\text{b}egin{equation}gin{array}{ccc}
\Delta_{k}^{2} & -g_{\text{r}m c}^{2}+qrt{\text{f}rac{2\Delta_{k}}{\Delta_{0}m_{+}(k)}} & -g_{\text{r}m c}^{2}+qrt{\text{f}rac{2\Delta_{k}}{\Delta_{0}m_{-}(k)}}\\
-g_{\text{r}m c}^{2}+qrt{\text{f}rac{2\Delta_{k}}{\Delta_{0}m_{+}(k)}} & \text{f}rac{\varepsilon_{k}^{2}}{m_{+}(k)} & \text{f}rac{\varepsilon_{k}^{2}-\Delta_{k}^{2}}{+qrt{m_{+}(k)m_{-}(k)}} \\
-g_{\text{r}m c}^{2}+qrt{\text{f}rac{2\Delta_{k}}{\Delta_{0}m_{-}(k)}} & \text{f}rac{\varepsilon_{k}^{2}-\Delta_{k}^{2}}{+qrt{m_{+}(k)m_{-}(k)}} & \text{f}rac{\varepsilon_{k}^{2}}{m_{-}(k)}
\end{array}
\text{r}ight],\label{B_matrix}
\end{equation}
with $\varepsilon_{k}^{2}=(\Delta_{k}^{2}+\Omega^{2})/2$ and $m_{\pm}(k)=(1\pm+qrt{\Delta_{0}/\Delta_{k}})^{-1}$. The dispersion relation of field fluctuations around the ground-state configuration, i.e., the collective spin-boson excitations, can be found by solving the eigenvalue problem $+um_{a}B_{a,a^{\prime}}^{(k)}u_{a}^{(p)}(k)=\omega_{p}^{2}(k)u_{a^{\prime}}^{(p)}(k)$ with $p=1,2,3$. The result is summarized in Fig. \text{r}ef{fig1} where are shown the three branches of collective excitations, assuming periodic boundary conditions with nearest-neighbours bosonic tunneling $t_{j,l}=-t(\delta_{j,l+1}+\delta_{j,l-1})$ and bosonic dispersion $\Delta_{k}=\Delta+2t\{1-\cos(2\pi k/N)\}$ \cite{Nevado2013}. The lowest-lying branch correspond to the gapless Goldstone mode $\omega_{\text{r}m G}$, which is linear for small $k$, i.e., $\omega_{\text{r}m G}= c_{\text{r}m s}2\pi k/N+O(k^{2})$ with characteristic slope $c_{\text{r}m s}=2g^{2}+in(\text{b}egin{equation}gin{align}r{\theta})+qrt{t\Delta/(\Delta^{4}+4g^{4}+in^{2}(\text{b}egin{equation}gin{align}r{\theta}))}$. The other two excitations, the so-called amplitude modes, remain gapped with $\omega_{{\text{r}m A},\pm}=\Delta_{\pm}+O(k^{2})$, where the gaps are given by
\text{b}egin{equation}gin{equation}
\Delta_{\pm}^{2}=\text{f}rac{\Omega^{2}}{2}+\Delta^{2}\pm+qrt{\text{f}rac{\Omega^{4}}{4}+4g_{\text{r}m c}^{4}}.
\end{equation}
So far, we have discussed the quantum fluctuations of the bosonic fields $\alpha_{\gamma,k}$ around their classical configuration. Because, in the symmetry broken phase the saddle-point solution predicts formation of bosonic condensates, it is naturally to express the cJT action in terms of density and phase of the respective condensate. Such a treatment allows us to connect the density fluctuations and the local phase of the condensates with the creation of the energy gaps in the spectra of cJT model. A convenient way to do this is to adopt the polar parametrization of the bosonic fields.
+ubsection{Polar decomposition}
Let us choose the nonlinear polar parametrization
\text{b}egin{equation}gin{equation}
\alpha_{\gamma,j}=+qrt{\text{b}egin{equation}gin{align}r{\text{r}ho}+\delta\text{r}ho_{\gamma,j}}e^{{\text{r}m i}\zeta{\gamma,j}},\label{polar}
\end{equation}
of the bosonic fields entering the path integral (\text{r}ef{S}). Here the conjugate variables $\zeta_{\gamma,j}$ and $\delta\text{r}ho_{\gamma,j}$ describe, respectively, the local phase and the density fluctuation of the bosonic condensates around the mean-field solution $\text{b}egin{equation}gin{align}r{\text{r}ho}=(g/2\Delta_{0})^{2}+in^{2}(\text{b}egin{equation}gin{align}r{\theta})$. In the limit of $\delta\text{r}ho_{\gamma,j}/\text{b}egin{equation}gin{align}r{\text{r}ho}\ll 1$ one can expand the square root in Eq. (\text{r}ef{polar}) and keep only the quadratic terms of the density fluctuations. The latter condition can be fulfilled for large coupling $g\gg g_{\text{r}m c}$ ($\text{b}egin{equation}gin{align}r{\text{r}ho}\gg 1$), where the quantum fluctuations are suppressed \cite{Porras2012}. Assuming that the spin and bosonic fields vary smoothly on the scale of lattice constant $a$ in a $d$ dimensional cubic lattice one can perform gradient expansion, such that in the symmetry broken phase $g>g_{{\text{r}m c}}$ the continuum action becomes
\text{b}egin{equation}gin{eqnarray}
S&=&a^{-d}\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau\text{i}nt d^{d}x\{S_{\text{r}m B}^{\prime}++um_{\gamma}\{{\text{r}m i}\delta\text{r}ho_{\gamma}\text{f}rac{\partial\zeta_{\gamma}}{\partial\tau}+t\text{b}egin{equation}gin{align}r{\text{r}ho}a^{2}(\nabla\zeta_{\gamma})^{2}\notag\\
&&+\text{f}rac{ta^{2}}{4\text{b}egin{equation}gin{align}r{\text{r}ho}}(\nabla\delta\text{r}ho_{\gamma})^{2}++qrt{\text{b}egin{equation}gin{align}r{\text{r}ho}}\text{f}rac{g+in(\text{b}egin{equation}gin{align}r{\theta})}{2}(\zeta_{\gamma}^{2}+\text{f}rac{\delta\text{r}ho_{\gamma}^{2}}
{4\text{b}egin{equation}gin{align}r{\text{r}ho}^{2}})\}+\Omega n_{+}^{\prime}n_{-}^{\prime}\notag\\
&&-\text{f}rac{g\cos(\text{b}egin{equation}gin{align}r{\theta})}{2+qrt{\text{b}egin{equation}gin{align}r{\text{r}ho}}}n_{x}^{\prime}(\delta\text{r}ho_{{\text{r}m r}}+\delta\text{r}ho_{{\text{r}m l}})+g+qrt{\text{b}egin{equation}gin{align}r{\text{r}ho}}n_{y}^{\prime}(\zeta_{{\text{r}m r}}-\zeta_{{\text{r}m l}})\}.\label{S_con}
\end{eqnarray}
Observe that the term $t\text{b}egin{equation}gin{align}r{\text{r}ho}a^{2}(\nabla\zeta_{\gamma})^{2}$ corresponds to the kinetic energy of a free particle with quadratic dispersion relation $+im k^{2}$. Additionally, the spontaneous symmetry breaking gives rise to terms proportional to $\zeta_{\gamma}^{2}$ such that the system gain an energy gaps which are not vanish in the limit $k\text{r}ightarrow 0$. As a result of that the conjugate pairs $(\zeta_{{\text{r}m r}},\delta\text{r}ho_{{\text{r}m r}})$ and $(\zeta_{{\text{r}m l}},\delta\text{r}ho_{{\text{r}m l}})$ lead to two gapped amplitude modes in the spectra of cJT model. Indeed, the subsequent diagonalization of the action (\text{r}ef{S_con}) in the position-momentum representation gives an identical to $B_{a,a^{\prime}}^{(k)}$ matrix Eq. (\text{r}ef{B_matrix}) where the bosonic dispersion is replaced by its long-wave length limit, $\Delta_{k}\approx \Delta-td+t(ka)^{2}$.
+ection{Measurement Protocol}\label{MP}
At the end we discuss the experimental verifiability of the presented results. For concreteness, we focus on the trapped ion-based realization of the cJT model. The protocol starts with the initialization of the linear ion crystal in the normal phase ground-state $g=0$, by laser cooling of the radial phonons to the motional ground-state and pumping spins to $\left|\downarrow_{j}\text{r}ight\text{r}angle$ state. After that the Jahn-Teller coupling is switched on and increases adiabatically to the desired regime $g>g_{\text{r}m c}$. The quantum magnetic structural phase transition predicted by the mean-field solution (\text{r}ef{sp}) can be detected by measuring either the spin population or the phonon number. The antiferromagnetic spin order can be verified experimentally using a spin-dependent laser fluorescence, where the spins in the upper state emit light and appear bright, while the spins on the down state remain dark. The structural phase transition is related with the position reordering of the ion crystal into the zigzag configuration and creation of phonons in the lowest energy vibrational mode. These can be measured by laser induced fluorescence, which is imaged on a CCD camera and, respectively, with sideband spectroscopy which allows to determine the mean phonon number \cite{Haffner2008}. Finally, following the recent proposals \cite{Kurcz2014prl,Kurcz}, the dispersion relations of the excitations can be measured by the dynamical response of the system due to the weak coupling to the quantum probe.
+ection{Conclusion}\label{C}
In conclusion, we have provided the path-integral formalism of the cJT model. Within the saddle-point approximation we have obtained the classical ground-state of the cJT model. The solution predicts a quantum magnetic structural phase transition with formation of ferromagnetic spin order and condensations of the two bosonic species in the lowest energy mode. We have calculated the elementary excitations of our model and found a linear gapless Goldstone mode and two gapped amplitude modes in the symmetry-broken phase.
\acknowledgments
This work has been supported by the EC Seventh Framework Programme under Grant Agreement No.270843 (iQIT).
\appendix
+ection{Derivation of the energy spectrum}\label{A}
In order to determine the collective modes of the cJT model we define the harmonic oscillator degrees of freedom for each pair of conjugate variables using the relations
\text{b}egin{equation}gin{eqnarray}
&&\tilde{q}_{\epsilon,k}=\text{f}rac{1}{+qrt{2\Delta_{k}}}(\delta\alpha_{\epsilon,-k}^{*}+\delta\alpha_{\epsilon,k}), \notag\\ &&\tilde{q}_{z,k}=\text{f}rac{1}{+qrt{2\Omega}}(n_{+,-k}^{\prime}+n_{-,k}^{\prime}),
\end{eqnarray}
and respectively
\text{b}egin{equation}gin{eqnarray}
&&\tilde{p}_{\epsilon,k}={\text{r}m i}+qrt{\text{f}rac{\Delta_{k}}{2}}(\delta\alpha_{\epsilon,k}^{*}-\delta\alpha_{\epsilon,-k}), \notag\\ &&\tilde{p}_{z,k}={\text{r}m i}+qrt{\text{f}rac{\Omega}{2}}(n_{-,-k}^{\prime}-n_{+,k}^{\prime}).
\end{eqnarray}
These variables are used to express the action (\text{r}ef{S_k}) in the position-momentum representation which yield
\text{b}egin{equation}gin{eqnarray}
S&=&\text{i}nt_{0}^{\text{b}egin{equation}ta}d\tau+um_{k}\{+um_{c}\{{\text{r}m i}\text{f}rac{\partial \tilde{p}_{c,k}}{\partial\tau}\tilde{q}_{c,k}+\text{f}rac{1}{2}\tilde{p}_{c,k}\tilde{p}_{c,-k}\}+\text{f}rac{\Omega^{2}}{2}\tilde{q}_{z,k}\tilde{q}_{z,-k}\notag\\
&&+\text{f}rac{\Delta_{k}^{2}}{2}(\tilde{q}_{x,k}\tilde{q}_{x,-k}+\tilde{q}_{y,k}\tilde{q}_{y,-k})-2g_{{\text{r}m c}}^{2}+qrt{\text{f}rac{\Delta_{k}}{\Delta_{0}}}\tilde{q}_{x,k}\tilde{q}_{z,-k}\notag\\
&&-+qrt{\text{f}rac{\Delta_{0}}{\Delta_{k}}}\tilde{p}_{y,-k}\tilde{p}_{z,k}.\label{S_pm}
\end{eqnarray}
The action (\text{r}ef{S_pm}) describes a collection of coupled oscillators. In order to decouple momentum dependent couplings between the different oscillators in (\text{r}ef{S_pm}) we perform transformation of the position variables
\text{b}egin{equation}gin{equation}
\left[\text{b}egin{equation}gin{array}{c}
\tilde{q}_{x,k}\\\tilde{q}_{y,k}\\\tilde{q}_{z,k}
\end{array}\text{r}ight]=\text{f}rac{1}{+qrt{2}}\left[
\text{b}egin{equation}gin{array}{ccc}
+qrt{2} & 0 & 0\\
0 & -m_{+}(k)^{-1/2} & m_{-}(k)^{-1/2} \\
0 & m_{+}(k)^{-1/2} & m_{-}(k)^{-1/2}
\end{array}
\text{r}ight]\left[\text{b}egin{equation}gin{array}{c}
q_{1,k}\\q_{2,k}\\q_{3,k}
\end{array}\text{r}ight]\label{q}
\end{equation}
and respectively of the momentum variables
\text{b}egin{equation}gin{equation}
\left[\text{b}egin{equation}gin{array}{c}
\tilde{p}_{x,k}\\\tilde{p}_{y,k}\\\tilde{p}_{z,k}
\end{array}\text{r}ight]=\text{f}rac{1}{+qrt{2}}\left[
\text{b}egin{equation}gin{array}{ccc}
+qrt{2} & 0 & 0\\
0 & -m_{+}(k)^{1/2} & m_{-}(k)^{1/2} \\
0 & m_{+}(k)^{1/2} & m_{-}(k)^{1/2}
\end{array}
\text{r}ight]\left[\text{b}egin{equation}gin{array}{c}
p_{1,k}\\p_{2,k}\\p_{3,k}
\end{array}\text{r}ight].\label{p}
\end{equation}
Using Eqs. (\text{r}ef{S_pm}), (\text{r}ef{q}) and (\text{r}ef{p}) we arrive to Eq. (\text{r}ef{S_final}).
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\begin{document}
\title {On the symbolic powers of binomial edge ideals}
\author {Viviana Ene, J\"urgen Herzog}
\address{Viviana Ene, Faculty of Mathematics and Computer Science, Ovidius University, Bd.\ Mamaia 124,
900527 Constanta, Romania} \email{[email protected]}
\address{J\"urgen Herzog, Fachbereich Mathematik, Universit\"at Duisburg-Essen, Campus Essen, 45117
Essen, Germany} \email{[email protected]}
\begin{abstract} We show that under some conditions, if the initial ideal $\ini_<(I)$ of an ideal $I$ in a polynomial ring has the property that its symbolic and ordinary powers coincide, then the ideal $I$ shares the same property. We apply this result to prove the equality between symbolic and ordinary powers for binomial edge ideals with quadratic Gr\"obner basis.
\end{abstract}
\subjclass[2010]{05E40,13C15}
\keywords{symbolic power, binomial edge ideal, chordal graphs}
\maketitle
\section{Introduction}
Binomial edge ideals were introduced in \cite{HHHKR} and, independently, in \cite{Oh}. Let $S=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$ be the polynomial ring in $2n$ variables over a field $K$ and $G$ a simple graph on the vertex set $[n]$ with edge set $E(G).$ The binomial edge ideal of $G$ is generated by the set of $2$-minors of the generic matrix
$X=\left(
\begin{array}{cccc}
x_1 & x_2 & \cdots & x_n\\
y_1 & y_2 & \cdots & y_n
\end{array}\right)
$ indexed by the edges of $G$. In other words,
\[
J_G=(x_iy_j-x_jy_i: i<j \text{ and }\{i,j\}\in E(G)).
\] We will often use the notation $[i,j]$ for the maximal minor $x_iy_j-x_jy_i$ of $X.$
In the last decade, several properties of binomial edge ideals have been studied. In \cite{HHHKR}, it was shown that, for every graph $G,$ the ideal $J_G$ is a radical ideal and the minimal prime ideals are characterized in terms of the combinatorics of the graph. Several articles considered the Cohen-Macaulay property of binomial edge ideals; see, for example, \cite{BMS, EHH, RR, Ri, Ri2}. A significant effort has been done for studying the resolution of binomial edge ideals. For relevant results on this topic we refer to the recent survey
\cite{Sara} and the references therein.
In this paper, we consider symbolic powers of binomial edge ideals. The study and use of symbolic powers have been a reach topic of research in commutative algebra for more than 40 years. Symbolic powers and ordinary powers do not coincide in general. However, there are classes of homogeneous ideals in polynomial rings for which the symbolic and ordinary powers coincide. For example, if $I$ is the edge ideal of a graph, then $I^k=I^{(k)}$ for all $k\geq 1$ if and only if the graph is bipartite. More general, the facet ideal $I(\Delta)$ of a simplicial complex $\Delta$ has the property that $I(\Delta)^k=I(\Delta)^{(k)}$ for all $k\geq 1$ (equivalently, $I(\Delta)$ is normally torsion free) if and only if $\Delta$ is a Mengerian complex; see \cite[Section 10.3.4]{HH10}. The ideal of the maximal minors of a generic matrix shares the same property, that is, the symbolic and ordinary powers coincide \cite{DEP}.
To the best of our knowledge, the comparison between symbolic and ordinary powers for binomial edge ideals was considered so far only in \cite{Oh2}. In Section 4 of this paper, Ohtani proved that if $G$ is a complete multipartite graph, then $J_G^k=J_G^{(k)}$ for all integers
$k\geq 1.$
In our paper we prove that, for any binomial edge ideal with quadratic Gr\"obner basis, the symbolic and ordinary powers of $J_G$ coincide. The proof is based on the transfer of the equality for symbolic and ordinary powers from the initial ideal to the ideal itself.
The structure of the paper is the following. In Section~\ref{one} we survey basic results needed in the next section on symbolic powers of ideals in Noetherian rings and on binomial edge ideals and their primary decomposition.
In Section~\ref{three} we discuss symbolic powers in connection to initial ideals. Under some specific conditions on the homogeneous ideal $I$ in a polynomial ring over a field, one may derive that if $\ini_<(I)^k=\ini_<(I)^{(k)}$ for some integer $k\geq 1,$ then $I^k=I^{(k)}$; see Lemma~\ref{inilemma}. By using this lemma and the properties of binomial edge ideals, we show in Theorem~\ref{iniconseq} that if
$\ini_<(J_G)$ is a normally torsion-free ideal, then the symbolic and ordinary powers of $J_G$ coincide. This is the case, for example, if
$G$ is a closed graph (Corollary~\ref{closed}) or the cycle $C_4.$ However, in general, $\ini_<(J_G)$ is not a normally torsion-free ideal. For example, for the binomial edge ideal of the $5$--cycle, we have $J_{C_5}^2=J_{C_5}^{(2)}$, but $(\ini_<(J_{C_5}))^2\subsetneq (\ini_<(J_{C_5}))^{(2)}.$
\section{Preliminaries}
\label{one}
In this section we summarize basic facts about symbolic powers of ideals and binomial edge ideals.
\subsection{Symbolic powers of ideals}
Let $I\subset R$ be an ideal in a Noetherian ring $R,$ and let $\Min(I)$ the set of the minimal prime ideals of $I.$ For an iteger $k\geq 1,$ one defines the \emph{$k^{th}$ symbolic power} of $I$ as follows:
\[
I^{(k)}=\bigcap_{{\frk p}\in\Min(I)}(I^kR_{\frk p}\cap R)=\bigcap_{{\frk p}\in\Min(I)}\ker(R\to (R/I^k)_{\frk p})=\]
\[=\{a\in R: \text{ for every }{\frk p}\in \Min(I), \text{ there exists }w_{\frk p}\not\in {\frk p} \text{ with }w_{\frk p} a\in I^k\}=
\]
\[=\{a\in R: \text{ there exists }w\not\in \bigcup_{{\frk p}\in \Min(I)}{\frk p}\text{ with } wa\in I^k\}.
\]
By the definition of the symbolic power, we have $I^k\subseteq I^{(k)}$ for $k\geq 1. $ Symbolic powers do not, in general, coincide with the ordinary powers. However, if $I$ is a complete intersection or it is the determinantal ideal generated by the maximal minors
of a generic matrix, then it is known that $I^k= I^{(k)}$ for $k\geq 1;$ see \cite{DEP} or \cite[Corollary 2.3]{BC03}.
Let $I=Q_1\cap \cdots \cap Q_m$ an irredundant primary decomposition of $I$ with $\sqrt{Q_i}={\frk p}_i$ for all $i.$ If the minimal prime ideals of $I$ are ${\frk p}_1,\ldots {\frk p}_s,$ then
\[I^{(k)}=Q_1^{(k)}\cap\cdots \cap Q_s^{(k)}. \]
In particular, if $I\subset R=K[x_1,\ldots,x_n]$ is a square-free monomial ideal in a polynomial ring over a field $K$, then
\[I^{(k)}=\bigcap_{{\frk p}\in \Min(I)} {\frk p}^k.
\]
Moreover, $I$ is normally torsion-free (i.e. $\Ass(I^m)\subseteq \Ass(I)$ for $m\geq 1$) if and only if $I^k= I^{(k)}$ for all $k\geq 1,$ if and only if $I$ is the Stanley-Reisner ideal of a Mengerian simplicial complex; see \cite[Theorem 1.4.6, Corollary 10.3.15]{HH10}. In particular, if $G$ is a bipartite graph, then its monomial edge ideal $I(G)$ is normally torsion-free \cite[Corollary 10.3.17]{HH10}.
In what follows, we will often use the binomial expansion of symbolic powers \cite{HNTT}. Let $I\subset R$ and $J\subset R^\prime$ be two homogeneous ideals in the polynomial algebras $R,R^\prime$ in disjoint sets of variables over the same field $K$. We write $I,J$ for the extensions of these two ideals in $R\otimes_K R^\prime.$ Then, the following binomial expansion holds.
\begin{Theorem}\cite[Theorem 3.4]{HNTT} In the above settings, \[ (I+J)^{(n)}=\sum_{i+j=n}I^{(i)}J^{(j)}.\]
\end{Theorem}
Moreover, we have the following criterion for the equality of the symbolic and ordinary powers.
\begin{Corollary}\cite[Corollary 3.5]{HNTT} \label{corh} In the above settings, assume that $I^t\neq I^{t+1}$ and $J^t\neq J^{t+1}$ for $t\leq n-1.$ Then
$(I+J)^{(n)}=(I+J)^n$ if and only if $I^{(t)}=I^t$ and $J^{(t)}=J^t$ for every $t\leq n.$
\end{Corollary}
\subsection{Binomial edge ideals}
Let $G$ be a simple graph on the vertex set $[n]$ with edge set $E(G)$ and let $S$ be the polynomial ring $K[x_1,\ldots,x_n,y_1,\ldots,y_n]$ in $2n$ variables over a
field $K.$ The binomial edge ideal $J_G\subset S$ associated with $G$ is
\[
J_G=(f_{ij}: i<j, \{i,j\}\in E(G)),
\] where $f_{ij}=x_iy_j-x_jy_i$ for $1\leq i<j\leq n.$ Note that $f_{ij}$ are exactly the maximal minors of the $2\times n$ generic matrix
$X=\left(
\begin{array}{cccc}
x_1 & x_2 & \cdots & x_n\\
y_1 & y_2 & \cdots & y_n
\end{array}\right).
$ We will use the notation $[i,j]$ for the $2$- minor of $X$ determined by the columns $i$ and $j.$
We consider the polynomial ring $S$ endowed with the lexicographic order induced by the natural order of the variables, and $\ini_<(J_G)$ denotes the initial ideal of $J_G$ with respect to this monomial order. By \cite[Corollary 2.2]{HHHKR}, $J_G$ is a radical ideal. Its minimal prime ideals may be characterized in terms of the combinatorics of the graph $G.$ We introduce the following notation.
Let ${\mathcal S}\subset [n]$ be a (possible empty) subset of $[n]$, and let $G_1,\ldots,G_{c({\mathcal S})}$ be the connected components of $G_{[n]\setminus {\mathcal S}}$ where
$G_{[n]\setminus {\mathcal S}}$ is the induced subgraph of $G$ on the vertex set $[n]\setminus {\mathcal S}.$ For $1\leq i\leq c({\mathcal S}),$ let $\tilde{G}_i$ be the complete
graph on the vertex set $V(G_i).$ Let \[P_{{\mathcal S}}(G)=(\{x_i,y_i\}_{i\in {\mathcal S}}) +J_{\tilde{G_1}}+\cdots +J_{\tilde{G}_{c({\mathcal S})}}.\]
Then $P_{{\mathcal S}}(G)$ is a prime ideal. Since the symbolic powers of an ideal of maximal minors of a generic matrix coincide with the ordinary powers, and by using
Corollary~\ref{corh}, we get
\begin{equation}\label{eqprime}
P_{{\mathcal S}}(G)^{(k)}=P_{{\mathcal S}}(G)^k \text{ for } k\geq 1.
\end{equation}
By \cite[Theorem 3.2]{HHHKR}, $J_G=\bigcap_{{\mathcal S}\subset [n]}P_{{\mathcal S}}(G).$ In particular, the minimal primes of $J_G$ are among the prime ideals $P_{{\mathcal S}}(G)$ with ${\mathcal S}\subset [n].$
The following proposition characterizes the sets ${\mathcal S}$ for which the prime ideal $P_{{\mathcal S}}(G)$ is minimal.
\begin{Proposition}\label{cpset}\cite[Corollary 3.9]{HHHKR}
$P_{{\mathcal S}}(G)$ is a minimal prime of $J_G$ if and only if either ${\mathcal S}=\emptyset$ or ${\mathcal S}$ is non-empty and for each $i\in {\mathcal S},$ $c({\mathcal S}\setminus\{i\})<c({\mathcal S})$.
\end{Proposition}
In combinatorial terminology, for a connected graph $G$, $P_{{\mathcal S}}(G)$ is a minimal prime ideal of $J_G$ if and only if ${\mathcal S}$ is empty or ${\mathcal S}$ is non-empty and is a \emph{cut-point set} of $G,$ that is, $i$ is a cut point of the restriction $G_{([n]\setminus{\mathcal S})\cup\{i\}}$ for every $i\in {\mathcal S}.$ Let ${\mathcal C}(G)$ be the set of all sets ${\mathcal S}\subset [n]$ such that $P_{{\mathcal S}}(G)\in \Min(J_G).$
Let us also mention that, by \cite[Theorem 3.1]{CDeG} and \cite[Corollary 2.12]{CDeG}, we have
\begin{equation}\label{intersectini}
\ini_<(J_G)=\bigcap_{{\mathcal S} \in {\mathcal C}(G)} \ini_< P_{{\mathcal S}}(G).
\end{equation}
\begin{Remark} {\em The cited results of \cite{CDeG} require that $K$ is algebraically closed. However, in our case, we may remove this condition on the field $K.$ Indeed, neither the Gr\"obner basis of $J_G$ nor the primary decomposition of $J_G$ depend on the field $K,$ thus we may extend the field $K$ to its algebraic closure $\bar{K}.$}
\end{Remark}
When we study symbolic powers of binomial edge ideals, we may reduce to connected graphs. Let $G=G_1\cup \cdots \cup G_c$ where $G_1,\ldots,G_c$ are the connected components of $G$ and $J_G\subset S$ the binomial edge ideal of $G.$ Then we may write
\[J_G=J_{G_1}+\cdots +J_{G_c}
\] where $J_{G_i}\subset S_i=K[{x_j,y_j: j\in V(G_i)}]$ for $1\leq i\leq c.$ In the above equality, we used the notation $J_{G_i}$ for
the extension of $J_{G_i}$ in $S$ as well.
\begin{Proposition}\label{connected}
In the above settings, we have $J_G^k=J_G^{(k)}$ for every $k\geq 1$ if and only if $J_{G_i}^k=J_{G_i}^{(k)}$ for every $k\geq 1.$
\end{Proposition}
\begin{proof}
The equivalence is a direct consequence of Corollary~\ref{corh}.
\end{proof}
\section{Symbolic powers and initial ideals}
\label{three}
In this section we discuss the transfer of the equality between symbolic and ordinary powers from the initial ideal to the ideal itself.
Let $R=K[x_1,\ldots,x_n]$ be the polynomial ring over the field $K$ and $I\subset R$ a homogeneous ideal. We assume that there exists a monomial order $<$ on $R$ such that $\ini_<(I)$ is a square-free monomial ideal. In particular, it follows that $I$ is a radical ideal. Let
$\Min(I)=\{{\frk p}_1,\ldots,{\frk p}_s\}.$ Then $I=\bigcap_{i=1}^s {\frk p}_i.$
\begin{Lemma}\label{inilemma}
In the above settings, we assume that the following conditions are fulfilled:
\begin{itemize}
\item [(i)] $\ini_<(I)=\bigcap_{i=1}^s \ini_<({\frk p}_i);$
\item [(ii)] For an integer $t\geq 1$ we have:
\begin{itemize}
\item [(a)] ${\frk p}_i^{(t)}={\frk p}_i^t$ for $1\leq i\leq s;$
\item [(b)] $\ini_<({\frk p}_i^t)=(\ini_<({\frk p}_i))^t$ for $1\leq i\leq s;$
\item [(c)] $(\ini_<(I))^{(t)}=(\ini_<(I))^t.$
\end{itemize}
\end{itemize}
Then $I^{(t)}=I^t.$
\end{Lemma}
\begin{proof}
In our hypothesis, we obtain:
\[\ini_<(I^t)\supseteq (\ini_<(I))^t=(\ini_<(I))^{(t)}=\bigcap_{i=1}^s(\ini_<({\frk p}_i))^{(t)}\supseteq
\bigcap_{i=1}^s(\ini_<({\frk p}_i))^{t}=\bigcap_{i=1}^s \ini_<({\frk p}_i^t)\supseteq\]
\[\supseteq \ini_<(\bigcap_{i=1}^s {\frk p}_i^t)=\ini_<(\bigcap_{i=1}^s {\frk p}_i^{(t)})=\ini_<(I^{(t)})\supseteq \ini_<(I^t).
\] Therefore, it follows that $\ini_<(I^{(t)})=\ini_<(I^t).$ Since $I^t\subseteq I^{(t)},$ we get $I^t= I^{(t)}.$
\end{proof}
We now investigate whether one may use the above lemma for studying symbolic powers of binomial edge ideals. Note that, by (\ref{intersectini}), the first condition in Lemma~\ref{inilemma} holds for any binomial edge ideal $J_G.$ In addition, as we have seen in (\ref{eqprime}), condition (a) in Lemma~\ref{inilemma} holds for any
prime ideal $P_{\mathcal S}(G)$ and any integer $t\geq 1.$
\begin{Lemma}\label{inipowers}
Let ${\mathcal S}\subset [n].$ Then $\ini_<(P_{{\mathcal S}}(G)^{t})=(\ini_<(P_{\mathcal S}(G)))^t,$ for every $t\geq 1.$
\end{Lemma}
\begin{proof}
To shorten the notation, we write $P$ instead of $P_{{\mathcal S}}(G)$, $c$ instead of $c({\mathcal S}),$ and $J_i$ instead of $J_{\tilde{G_i}}$ for $1\leq i\leq c.$
Let ${\mathcal R}(P),$ respectively ${\mathcal R}(\ini_<(P))$ be the Rees algebras of $P,$ respectively $\ini_<(P).$ Then, as the sets of variables
$\{x_j,y_j:j\in V(\tilde{G_i})\}$ are pairwise disjoint, we get
\begin{equation}\label{eqRees1}
{\mathcal R}(P)={\mathcal R}((\{x_i,y_i\}_{i\in {\mathcal S}}))\otimes_K (\otimes_{i=1}^c{\mathcal R}(J_i)).
\end{equation}
On the other hand, since $\ini_<(P)=(\{x_i,y_i\}_{i\in {\mathcal S}})+\ini_<(J_1)+\cdots+\ini_<(J_c),$ due to the fact that
$J_1,\ldots,J_c$ are ideals in disjoint sets of variables different from $\{x_i,y_i\}_{i\in {\mathcal S}}$ (see \cite{HHHKR}), we obtain
\begin{eqnarray}\label{eqRees2}
{\mathcal R}(\ini_<P)={\mathcal R}((\{x_i,y_i\}_{i\in {\mathcal S}}))\otimes_K (\otimes_{i=1}^c{\mathcal R}(\ini_<J_i))=\\ \nonumber
={\mathcal R}((\{x_i,y_i\}_{i\in {\mathcal S}}))\otimes_K (\otimes_{i=1}^c\ini_<{\mathcal R}(J_i)).
\end{eqnarray}
For the last equality we used the equality $\ini_<(J_i^t)=(\ini_<J_i)^t$ for all $t\geq 1$ which is a particular case of \cite[Theorem 2.1]{Con} and
the equality ${\mathcal R}(\ini_<J_i)=\ini_<{\mathcal R}(J_i)$ due to \cite[Theorem 2.7]{CHV}.
We know that ${\mathcal R}(P)$ and $\ini_<({\mathcal R}(P))$ have the same Hilbert function. On the other hand, equalities~(\ref{eqRees1}) and
(\ref{eqRees2}) show that ${\mathcal R}(P)$ and ${\mathcal R}(\ini_<P)$ have the same Hilbert function since ${\mathcal R}(J_i)$ and $\ini_<{\mathcal R}(J_i)$ have the same Hilbert function for every $1\leq i\leq s.$ Therefore, ${\mathcal R}(\ini_<P)$ and $\ini_<{\mathcal R}(P)$ have the same Hilbert function. As
${\mathcal R}(\ini_<P)\subseteq \ini_<({\mathcal R}(P))$, we have ${\mathcal R}(\ini_<P)= \ini_<({\mathcal R}(P))$, which implies by \cite[Theorem 2.7]{CHV} that
$\ini_<(P^t)=(\ini_<P)^t$ for all $t.$
\end{proof}
\begin{Theorem}\label{iniconseq}
Let $G$ be a connected graph on the vertex set $[n].$ If $\ini_<(J_G)$ is a normally torsion-free ideal, then $J_G^{(k)}=J_G^k$ for $k\geq 1.$
\end{Theorem}
\begin{proof}
The proof is a consequence of Lemma~\ref{inipowers} combined with relations (\ref{intersectini}) and (\ref{eqprime}).
\end{proof}
There are binomial edge ideals whose initial ideal with respect to the lexicographic order are normally torsion-free.
For example, the binomial edge ideals which have a quadratic Gr\"obner basis have normally torsion-free initial ideals. They were characterized in
\cite[Theorem 1.1]{HHHKR} and correspond to the so-called closed graphs. The graph $G$ is \textit{closed} if there exists a labeling of its vertices such that for any edge $\{i,k\}$ with $i<k$ and for every $i<j<k$, we have $\{i,j\}, \{j,k\}\in E(G).$ If $G$ is closed with respect to its labeling, then, with respect to the lexicographic order $<$ on $S$ induced by the natural ordering of the indeterminates, the initial ideal of $J_G$ is $\ini_<(J_G)=(x_iy_j: i<j \text{ and }\{i,j\}\in E(G)).$ This implies that $\ini_<(J_G)$ is the edge ideal of a bipartite graph, hence it is normally torsion-free. Therefore, we get the following.
\begin{Corollary}\label{closed}
Let $G$ be a closed graph on the vertex set $[n].$ Then $J_G^{(k)}=J_G^k$ for $k\geq 1.$
\end{Corollary}
Let $C_4$ be the $4$-cycle with edges
$\{1,2\},\{2,3\},\{3,4\},\{1,4\}.$ Let $<$ be the lexicographic order on $K[x_1,\ldots,x_4,y_1,\ldots,y_4]$ induced by
$x_1>x_2>x_3>x_4>y_1>y_2>y_3>y_4.$ With respect to this monomial order, we have
\[
\ini_<(J_{C_4})=(x_1x_4y_3,x_1y_2,x_1y_4,x_2y_1y_4,x_2y_3,x_3y_4).
\]
Let $\Delta$ be the simplicial complex whose facet ideal $I(\Delta)=\ini_<(J_{C_4}).$ It is easily seen that $\Delta$ has no special odd cycle, therefore, by \cite[Theorem 10.3.16]{HH10}, it follows that $I(\Delta)$ is normally torsion-free. Note that the $4$-cycle is a complete bipartite graph, thus the equality $J_{C_4}^k=J_{C_4}^{(k)}$ for all $k\geq 1$ follows also from \cite{Oh2}.
In view of this result, one would expect that initial ideals of binomial edge ideals of cycles are normally torsion-free. But this is not the case. Indeed, let $C_5$ be the $5$-cycle with edges $\{1,2\},\{2,3\},\{3,4\},\{4,5\},\{1,5\}$ and $I=\ini_<(J_{C_5})$ the initial ideal of $J_{C_5}$ with respect to the lexicographic order on $K[x_1\ldots,x_5,y_1,\ldots,y_5].$ By using \textsc{Singular} \cite{Soft}, we checked that $I^2\subsetneq I^{(2)}.$ Indeed, the monomial $x_1^2x_4x_5y_3y_5\in I^2$ is a minimal generator of $I^2.$ On the other hand,
the monomial $x_1x_4x_5y_3y_5\in I^{(2)}$, thus $I^2\neq I^{(2)}$, and $I$ is not normally torsion-free. On the other hand, again with \textsc{Singular}, we have checked that $J_{C_5}^2=J_{C_5}^{(2)}.$
\end{document} |
\begin{document}
\title{Efficient bounds on quantum communication
rates {\it via} their reduced variants}
\author{Marcin L. Nowakowski and Pawel Horodecki\footnote{Electronic address: [email protected]}}
\affiliation{Faculty of Applied Physics and Mathematics, ~Gdansk
University of Technology, 80-952 Gdansk, Poland}
\affiliation{National Quantum Information Centre of Gda´nsk,
Andersa 27, 81-824 Sopot, Poland}
\pacs{03.67.-a, 03.67.Hk}
\begin{abstract}
We investigate one-way communication scenarios where Bob manipulating on his parts can transfer some sub-system to the environment. We define reduced versions of quantum communication rates and further, prove new upper bounds on one-way quantum secret key, distillable entanglement and quantum channel capacity by means of their reduced versions. It is shown that in some cases they drastically improve their estimation.
\end{abstract}
\maketitle Recently years have seen enormous advances in quantum
information theory proving it has been well established as a basis
for a concept of quantum computation and communication. Much work
\cite{BennettDiVincenzo, BennettSmolin, Barnum3, Barnum4,
DevetakW1, DevetakW2, DevetakW3} has been performed to understand
how to operate on quantum states and distill entanglement enabling
quantum data processing or establish quantum secure communication
between two or more parties. One of the central problems of
quantum communication field is to estimate efficiency of
communication protocols establishing secure communication between
involved parties or distilling quantum entanglement \cite{Renner,
KHPH, KHPH2, DevetakW1, DevetakW2, DevetakW3, Smith}. Most simple
communication scenarios are those that do not use classical side
channel or use it only in one-way setup. The challenge for the
present theory is to determine good bounds on such quantities like
the secret key rate or quantum channel capacity and distillable
entanglement of a quantum state, that allow to estimate the
communication capabilities. In this paper we provide efficient
upper bounds avoiding massive overestimation of communication
rates. We are inspired by classical information and entanglement
measures theory where so-called reduced quantities have been used
\cite{Renner, KHPH, DiVincenzo}. Herewith we consider two pairs of
quantities: private capacity $\mathcal{P}$, quantum one-way secret
key $K_{\rightarrow}$ and one-way quantum channel capacity
$\mathcal{Q}_{\rightarrow}$, one-way distillable entanglement
$D_{\rightarrow}$ providing new efficient upper bounds. We prove
that in some cases the bounds explicitly show that the
corresponding quantity is relatively small if compared to sender
and receiver systems. The main method is again the fact that all
the above quantities vanish on some classes of systems. Moreover,
we introduce 'defect' parameters $\Delta$ for the reduced
quantities resulting from possible transfer of sub-systems on
receivers' side which are (sub)additive and hence, can be
exploited in case of composite systems and regularization.
\textit{\bf Reduced one-way secret key.} A secret key is a quantum
resource allowing two parties Alice and Bob private communication
over a public channel. In an ideal scenario they generate a pair
of maximally correlated classical secure bit-strings such that Eve
representing the adversary in the communication is not able to
receive any sensible information from further communication
between Alice and Bob. In this section we will elaborate on
generation of a one-way secret key from a tripartite quantum state
shared by the parties with Eve that means Alice and Bob can use
only protocols consisting of local operations and one-way public
communication. We propose a new reduced measure of the one-way
secret key that simplify in many cases analysis of one-way
security of quantum states.
To derive new observations about one-way quantum secret key we utilize in this section fundamental information notions engaging entropy \cite{Entropy} and quantum mutual information \cite{MInformation} which play a vital role in quantum information theory.
We state a new result about the upper bound on the Holevo function \cite{HolevoFunction} $\chi(\cdot)$:
\textbf{Observation 1.}\label{reducedholevo2} \textit{For any
ensemble of density matrices $\mathfrak{A}=\{\lambda_{i},
\rho^{i}_{BB'}\}$ with average density matrix
$\rho_{BB'}=\sum_{i}\lambda_{i}\rho^{i}_{BB'}$ there holds:
\begin{equation}\label{reducedholevo}
\chi(\rho_{BB'}) \leq \chi(\rho_{B}) + 2S(\rho_{B'})
\end{equation}}
\textit{Proof. }
Basing on subadditivity and concavity of quantum entropy we can easily show that:
\begin{eqnarray*}\label{LHS1}
&&|S(\rho_{BB'})-\sum_{i}p_{i}S(\rho^{i}_{BB'})-S(\rho_{B})+\sum_{i}p_{i}S(\rho^{i}_{B})| \leq \\
&\leq&|S(\rho_{BB'})-S(\rho_{B})|+ |\sum_{i}p_{i}S(\rho^{i}_{BB'})-\sum_{i}p_{i}S(\rho^{i}_{B})\\
&\leq&S(\rho_{B'})+\sum_{i}p_{i}S(\rho^{i}_{B'})\leq 2S(\rho_{B'})
\end{eqnarray*}
which completes the proof. $\Box$
One can use \cite{DevetakW1, DevetakW2} a general tripartite pure
state $\rho_{ABE}$ to generate a secret key between Alice and Bob.
Alice engages a particular strategy to perform a quantum
measurement (POVM) described by $Q=(Q_{x})_{x \in \cal X}$ which
leads to: $\widetilde{\rho}_{ABE}=\sum_{x}|x\rangle\langle x|_{A}
\otimes Tr_{A}(\rho_{ABE}(Q_{x})\otimes I_{BE})$. Therefore,
starting from many copies of $\rho_{ABE}$ we obtain many copies of
cqq-states $\widetilde{\rho}_{ABE}$ and we restate the theorem
defining one-way secret key $K_{\rightarrow}$:
\textbf{Theorem 1.}\cite{DevetakW1}
\textit{For every state $\rho_{ABE}$,
$K_{\rightarrow}(\rho) = \lim_{n\rightarrow\infty}\frac{K_{\rightarrow} ^{(1)}(\rho^{\otimes n})}{n}$,
with
$K_{\rightarrow}^{(1)}(\rho)=\max_{Q,T|X} I(X:B|T) - I (X:E|T)$
where the maximization is over all POVMs $Q=(Q_{x})_{x \in \cal X}$ and channels R
such that $T=R(X)$ and the information quantities refer to the state:
$\omega_{TABE}=\sum_{t,x} R(t|x)P(x)
|t\rangle\langle t|_{T}\otimes |x\rangle\langle x|_{A} \otimes
Tr_{A}(\rho_{ABE}(Q_{x})\otimes I_{BE}).$
The range of the measurement Q and the random variable T may be assumed to be bounded as follows:
$|T|\leq d^{2}_{A}$ and $|\cal X|\leq d^{2}_{A}$ where T can be taken a (deterministic) function
of $\cal X$.
}
Following we define a modified version of the one-way secret key rate $K_{\rightarrow}$ basing on the
results of \cite{Renner,KHPH} for reduced intrinsic information and reduced entanglement
measure.
\textbf{Definition 1.} \textit{For the one-way secret key rate
$K_{\rightarrow}^{(1)}(\rho_{AB})$ of a bipartite state
$\rho_{AB}\in B(\cal{H}_{A}\otimes \cal{H}_{B})$ shared between
Alice and Bob the reduced one-way secret key rate
$K_{\rightarrow}^{(1)}\downarrow(\rho_{AB})$ is defined as:
\begin{equation}
K_{\rightarrow}^{(1)}\downarrow(\rho_{AB})=\inf_{\cal{U}}[K_{\rightarrow}^{(1)}(\cal{U}(\rho_{AB}))+\Delta_{K_{\rightarrow}}
]
\end{equation}
where $\cal{U}$ denotes unitary operations on Bob's system with a
possible transfer of subsystems from Bob to Eve, i.e.
$\cal{U}(\rho_{AB})=Tr_{B'}(I\otimes \cal{U})\rho_{ABB'}$.
$\Delta_{K_{\rightarrow}} =4 S (\rho_{B'})$ denotes the defect
parameter related to increase of entropy produced by the transfer
of B'-subsystem from Bob's side to Eve.}
The reduced one-way secret key rate is an upper bound on $K_{\rightarrow}$ which we prove now
for every cqq-state $\rho$:
\textbf{Theorem 2.} \textit{For every cqq-state $\rho_{ABE}$ there
holds:
\begin{equation}
K_{\rightarrow}(\rho)=\lim_{n\rightarrow\infty} \frac
{K_{\rightarrow}^{(1)}(\rho^{\otimes n})}{n} \leq
K_{\rightarrow}\downarrow(\rho)
\end{equation}
where $K_{\rightarrow}\downarrow(\rho)=\lim_{n\rightarrow\infty}
\frac { K_{\rightarrow}^{(1)}\downarrow(\rho^{\otimes
n})}{N}$.Particularly, for identity operation $\cal{U}=id$ on
Bob's side one obtains: $K_{\rightarrow}(\rho_{ABB'}) \leq
K_{\rightarrow}(\rho_{AB})+ 4S(\rho_{B'})$.}
To prove this theorem one can start showing how the formula behaves for one-copy secret key:
\textbf{Lemma 2.}
\textit{
For every cqq-state $\rho_{ABE}$ there holds:
\begin{equation}\label{lemma2}
K_{\rightarrow}^{(1)}(\rho)
\leq K_{\rightarrow}^{(1)}\downarrow(\rho)
\end{equation}}
\begin{proof}
Since
\[
\left\lbrace
\begin{array}{l}
I(A:B|C)=S(AC)+S(BC)-S(ABC)-S(C)\\
I(A:E|C)=S(AC)+S(EC)-S(AEC)-S(C)\\
\end{array}
\right.
\]
then:
\[
K_{\rightarrow}^{(1)}(\rho)=\max_{Q,C|A}[S(BC)-S(ABC)-S(EC)+S(AEC)]
\]
To prove the thesis of this lemma it suffices to show that:
\begin{equation}\label{key1}
K_{\rightarrow}^{(1)}(\rho_{A(BB')E})\leq K_{\rightarrow}^{(1)}(\rho_{AB(B'E)})+4S(B')
\end{equation}
due to the fact that in case of application of $\cal{U}$ without discarding subsystem $B'$ one
obtains equality. We denote by $\rho_{AB(B'E)}$ transition of $B'$-subsystem to the environment.
For (\ref{key1}) we can omit maximization that is
performed on both side of the inequality representing an application of a chosen 1-LOCC protocol distilling
a secret key that invokes:
\begin{eqnarray*}
&&S(BB'C)-S(ABB'C)-S(EC)+S(AEC) \leq \\
&&S(BC)-S(ABC)-S(B'EC)+S(AB'EC)+ 4S(B')
\end{eqnarray*}
It is easy to note that application of unitary operations on Bob's side do not change the inequality mainly
due to property of unitary invariancy of the von Neumann entropy.
To simplify the proof one can decompose this inequality into following two inequalities:
\begin{equation}\label{key2}
\left\lbrace
\begin{array}{l}
S(BB'C)-S(ABB'C)\leq S(BC)-S(ABC) + 2S(B')\\
S(B'EC)-S(AB'EC)\leq S(EC)-S(AEC) + 2S(B')\\
\end{array}
\right.
\end{equation}
or equivalently considering the assumption that the initial state is of cqq-type and 'A' represents
classical distribution we can rewrite the first inequality into the form:
\begin{eqnarray*}
&&S(\sum_{i}p_{i}\rho_{i}^{BB'})-H(p_{i})-\sum_{i}p_{i}S(\rho_{i}^{BB'})-S(\sum_{i}p_{i}\rho_{i}^{B})\\
&&+H(p_{i})+\sum_{i}p_{i}S(\rho_{i}^{B})\leq 2S(B')
\end{eqnarray*}
and similarly for the second inequality which gives in result a more compact structure:
\[
\left\lbrace
\begin{array}{l}
\chi(\sum_{i}p_{i}\rho_{i}^{BB'C})-\chi(\sum_{i}p_{i}\rho_{i}^{BC})\leq 2S(B') \\
\chi(\sum_{i}p_{i}\rho_{i}^{B'EC})-\chi(\sum_{i}p_{i}\rho_{i}^{EC})\leq 2S(B') \\
\end{array}
\right.
\]
However, the above was proved in Lemma 1 that completes the proof.
\end{proof}
Finally, we will extend this result in the asymptotic regime
proving Theorem 2.
\begin{proof} To prove Theorem 2 it suffices to notice that (\ref{lemma2})
holds under 1-LOCC and an arbitrary chosen $\mathcal{U}$ for any
$\rho_{n}=\rho^{\otimes n}$. Moreover, existence of the defect
parameter $\Delta_{K_{\rightarrow}}$ enables regularization of the
reduced one-way secret rate since in the asymptotic regime after
application of unitary operations on Bob side one can apply
subadditivity of entropy to estimate entropy of the transferred B'
part which implies $K_{\rightarrow}(\rho_{ABB'}) \leq
K_{\rightarrow}(\rho_{AB})+ 4S(\rho_{B'})$. \end{proof}
It is interesting that our results reflect E-nonlockability of the
secret key rate \cite{Ekert} which means that the rate cannot be
locked with information on Eve's side.
Monogamy of entanglement has been used to prove that for some
region quantum depolarizing channel has zero capacity even if does
not destroy entanglement \cite{Bruss} which is a particular
application of symmetric extendibility of states to evaluation of
the quantum channel capacity. The following examples will show
application of the concept:
\textit{Example 1.} As an example of application of Theorem 2 we
present a state which after discarding a small B' part on Bob's
side becomes a symmetric extendible state \cite{MNPH}. This
example is especially important since the presented state does not
possess \cite{MNPH2} any symmetric extendible component in its
decomposition for symmetric and
non-symmetric parts, thus, one cannot use the method
\cite{Lutk3} to find an upper bound on $K_{\rightarrow}$ by means of linear optimization.
Let us consider a bipartite quantum state shared between Alice and Bob on the Hilbert
space $\cal{H}_{A}\otimes\cal{H}_{B}\cong \cal{C}^{d+2}\otimes\cal{C}^{d+2}$:
\begin{equation}
\rho_{AB}=\frac{1}{2} \left[
\begin{array}{cccc}
\Upsilon_{AB} & 0 & 0 & \cal A\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\cal A^{\dagger} & 0 & 0 & \Upsilon_{AB}\\
\end{array}
\right]
\end{equation}
where $\cal A$ is an arbitrary chosen operator so that $\rho_{AB}$
represents a correct quantum state. This matrix is represented in
the computational basis $|00\rangle, |01\rangle, |10\rangle,
|11\rangle$ held by Alice and Bob and possess a singlet-like
structure.
Whenever one party (Alice or Bob) measures the state, the state
decoheres and off-diagonal elements vanish which leads to a
symmetric extendible state \cite{MNPH}:
\begin{equation}
\Upsilon_{AB}=\frac{d}{2d-1}P_{+}+\frac{1}{2d-1}\sum^{d-1}_{i=1}|i\;0\rangle\langle
i\;0|
\end{equation}
from which no entanglement nor secret key can be distilled by
means of 1-LOCC \cite{D1,D2,Lutk3,MNPH}. Therefore, applying
Theorem 2 one derives $K_{\rightarrow}(\Upsilon_{AB})=0$ and
$K_{\rightarrow}(\rho_{AB})\leq
K_{\rightarrow}\downarrow(\rho_{AB})=4$.
\textit{Example 2.} Let us consider a graph state \cite{Hein}
$|\mathcal{G}\rangle$ of a $3n+1$-qubit system associated with a
mathematical graph $\mathcal{G}= \{\mathcal{V},\mathcal{E}\}$,
composed of a set $\mathcal{V}$ of $3n+1$ vertices and a set
$\mathcal{E}$ of edges $\{i,j\}$ connecting each vertex $i$ with
some other $j$:
\begin{equation}
|\mathcal{G}\rangle=\bigotimes_{{i,j}\in\mathcal{E}}CZ_{ij}|\mathcal{G}_{0}\rangle
\end{equation}
where $3n+1$ qubits are initialized in the product state
$|\mathcal{G}_{0}\rangle=\bigotimes_{i\in\mathcal{V}}|\psi_{i}\rangle$
with $|\psi_{i}\rangle=|0_{i}\rangle+|1_{i}\rangle$. Afterwards,
one applies a maximally-entangling control-Z (CZ) gate to all
pairs $\{i,j\}$ of qubits joined by an edge:
$CZ_{ij}=|0_{i}0_{j}\rangle\langle0_{i}0_{j}| +
|0_{i}1_{j}\rangle\langle0_{i}1_{j}|+|1_{i}0_{j}\rangle\langle
1_{i}0_{j}|-|1_{i}1_{j}\rangle\langle1_{i}1_{j}|$. If Alice takes
no more than $n$ qubits from the graph system that will use to
establish communication with Bob who uses other $n$ qubits in this
graph state, then they will be not able by any means to set secure
one-way communication. This results from the fact that the state
$\rho^{AB}_{2n}$ (with n qubits on Alice side and n qubits on
Bob's side) is symmetric extendible to a state $\rho^{AB}_{3n}$
which means that $K_{\rightarrow}(\rho^{AB}_{2n})=0$. A natural
symmetric extension of $\rho^{AB}_{2n}$ is a state
$\rho^{AB}_{3n}=Tr_{B'}|\mathcal{G}\rangle\langle\mathcal{G}|$
resulting from tracing out an arbitrary chosen qubit B' from graph
$\mathcal{G}$. However, if Alice takes $n$ qubits and Bob takes
$n+1$ qubits from the graph system, the resulting state
$\rho^{AB}_{2n+1}$ is not symmetric extendible anymore. Exemplary,
for $n=2$ this state has spectral representation:
\begin{equation}\label{state1}
\rho^{AB}_{2n+1}=\frac{1}{2}(|\phi_{0}\rangle\langle\phi_{0}|+|\phi_{1}\rangle\langle\phi_{1}|)
\end{equation}
where
$|\phi_{0}\rangle=|0_{A}\rangle|0_{B}\rangle+|1_{A}\rangle|1_{B}\rangle$,
$|\phi_{1}\rangle=|0_{A}\rangle|1_{B}\rangle-|1_{A}\rangle|0_{B}\rangle$
and
$\{|0\rangle_{A}=|00-01-10-11\rangle_{A},|1\rangle_{A}=|00+01+10-11\rangle_{A},
|0\rangle_{B}=|001+010+100-111\rangle_{B},|1\rangle_{B}=|000-011-101-110\rangle_{B}\}$.
This state is isomorphic to qubit bipartite state and meets the
condition \cite{Lutk1, Lutk2} for $\cal{C}^{2}\otimes\cal{C}^{2}$
Bell-diagonal states to be symmetric extendible:
$4\sqrt{det(\rho_{AB})} \geq Tr(\rho^{2}_{AB})-\frac{1}{2}$. One
can easily show the isomorphism of $\rho^{AB}_{2n+1}$ for any n
with a qubit bipartite state structure (\ref{state1}). Thus, for
one-way secret key of the state there holds:
$K_{\rightarrow}(\rho^{AB}_{2n+1}) \leq
K_{\rightarrow}\downarrow(\rho^{AB}_{2n+1})=4$, since after
discarding one qubit B' on Bob's side his system would become
symmetric extendible.
\textit{\bf An upper bound on quantum channel capacity.} The best
known definition of the one-way quantum channel capacity
$\mathcal{Q}_{\rightarrow}(\Lambda)$ \cite{Bennett2, Barnum3} is
expressed as an asymptotic regularization of coherent information:
$\mathcal{Q}_{\rightarrow}(\Lambda)=\lim_{n\rightarrow
\infty}\frac{1}{n}\sup_{\rho_{n}} I_{c}(\rho_{n}, \Lambda^{\otimes
n})$ with parallel use of N copies of $\Lambda$ channel. Coherent
information for a channel $\Lambda$ and a source
state $\sigma$ transferred through the channel is defined as:
$I_{c}(\sigma, \Lambda)=I^{B}(I\otimes \Lambda)(|\Psi\rangle\langle\Psi|)$ where $\Psi$ is a pure state with reduction
$\sigma$ and coherent information of a bipartite state $\rho_{AB}$ shared between Alice and Bob is defined as:
$I^{B}(\rho_{AB})=S(B)-S(AB)$. We will use further the following notation: $I_{c}(A \rangle B)=I^{B}(\rho_{AB})$.
\textbf{Observation 1. }\textit{For a bipartite state
$\rho_{ABB'}\in B(\cal{H}_{A}\otimes \cal{H}_{B}\otimes
\cal{H}_{B'})$ shared between Alice and Bob (B and B' system)
there holds:}
\begin{equation}
I_{c}(A \rangle BB')\leq I_{c}(A \rangle B) + 2S(B')
\end{equation}
\textit{Proof.} One can easily observe that for subadditivity of
entropy $S(BB')\leq S(B)+S(B')$ and for the Araki-Lieb inequality
$|S(AB)-S(B')|\leq S (ABB')$, the left hand side can be bounded as
follows: $S(BB')-S(ABB')\leq S(B)+S(B')-S(AB)+ S(B')=I_{c}(A
\rangle B) + 2S(B')$ which completes the proof. $\Box$
Motivated by the reduced quantity of secret key rate and above
observation we derive further the reduced version of quantum
channel capacity and show that it is a good bound on quantum
channel capacity:
\textbf{Definition 4.} \textit{For a one-way quantum channel
$\Lambda_{BB'}:B(\cal{H}_{BB'})\rightarrow
B(\cal{H}_{\widetilde{B}\widetilde{B'}})$ the reduced one-way
quantum channel capacity is defined as:
\begin{equation}
\mathcal{Q}_{\rightarrow}^{(1)}\downarrow(\Lambda_{BB'}) =
\inf_{\cal{U}}[\mathcal{Q}_{\rightarrow}^{(1)}(\cal{U}(\Lambda_{B}))+
\Delta_{\mathcal{Q}_{\rightarrow}}]
\end{equation}
where $\cal{U}$ denotes unitary operations on Bob's system with a
possible transfer of subsystems from Bob to Eve after action of
$\Lambda_{BB'}$ channel, i.e.
$\cal{U}(\Lambda_{B}(\rho_{B}))=Tr_{B'}\cal{U}\Lambda_{BB'}(\rho_{BB'})$.
$\Delta_{\mathcal{Q}_{\rightarrow}}=2
\sup_{\rho_{BB'}}S(Tr_{B}\Lambda_{BB'}(\rho_{BB'}))$ denotes the
defect parameter related to increase of entropy produced by the
transfer of B'-subsystem from Bob's side to Eve.}
\textbf{Theorem 3. }\textit{For any one-way quantum channel
$\Lambda_{BB'}:B(\cal{H}_{BB'})\rightarrow
B(\cal{H}_{\widetilde{B}\widetilde{B'}})$ there holds:
\begin{equation}
\mathcal{Q}_{\rightarrow}(\Lambda_{BB'}) \leq
\mathcal{Q}_{\rightarrow}\downarrow(\Lambda_{BB'})
\end{equation}
where
$\mathcal{Q}_{\rightarrow}\downarrow(\Lambda_{BB'})=\lim_{n}\mathcal{Q}_{\rightarrow}^{(1)}\downarrow(\Lambda_{BB'}^{\otimes
n })/n$ denotes the reduced quantum capacity. Particularly, for
identity operation $\cal{U}=id$ on Bob's side one obtains:
$\mathcal{Q}_{\rightarrow}(\Lambda_{BB'}) \leq
\mathcal{Q}_{\rightarrow}(\Lambda_{B})+
2\sup_{\rho_{BB'}}S(Tr_{B}\Lambda_{BB'}(\rho_{BB'}))$}.
To prove this inequality for regularized quantum capacity and its
reduced version it is sufficient to derive the below lemma for a
single copy case in analogy to the lemma for one-way secret key
rate above:
\textbf{Lemma 4. }\textit{For any one-way quantum channel
$\Lambda_{BB'}:B(\cal{H}_{BB'})\rightarrow
B(\cal{H}_{\widetilde{B}\widetilde{B'}})$ there holds:
\begin{equation}
\mathcal{Q}_{\rightarrow}^{(1)}(\Lambda_{BB'}) \leq
\mathcal{Q}_{\rightarrow}^{(1)}\downarrow(\Lambda_{BB'})
\end{equation}
}
\textit{Proof.} The proof of this lemma is straightforward with
application of Observation 1 that for a state $\rho_{BB'}$
maximizing coherent information on the left hand side of the
observation the above formula holds also for a possible transfer
of B' to the environment. It is worth recalling that action of
unitary operator on a state does not change its entropy and in a
result coherent information for any partition of the system.$\Box$
Further, one can complete the proof of the theorem in the
asymptotic regime:
\textit{Proof.} To prove the inequality of Theorem 3
asymptotically it suffices to notice that statements of Lemma 4
hold also for arbitrary chosen state $\rho_{n}=\rho^{\otimes n}$.
Now we can prove that: $\mathcal{Q}_{\rightarrow}(\Lambda_{BB'})
\leq \mathcal{Q}_{\rightarrow}(\Lambda_{B})+
\Delta_{\mathcal{Q}_{\rightarrow}}$. Let $\rho^{BB'}_{n}$ be a
state maximizing $\mathcal{Q}_{\rightarrow}(\Lambda_{BB'})$ as an
asymptotic regularization of coherent information, i.e.
$\mathcal{Q}_{\rightarrow}(\Lambda_{BB'})=\lim_{n\rightarrow
\infty}\frac{1}{n}I_{c}(\rho^{BB'}_{n}, \Lambda_{BB'}^{\otimes
n})$ which one can represent as $I_{c}(A \rangle BB')$ for the
aforementioned Choi-Jamiolkowski isomorphism between states and
channels. Basing on Observation 1, one can immediately derive for
the maximizing state $\rho^{BB'}_{n}$: $\frac{1}{n}I_{c}(A \rangle
BB') \leq \frac{1}{n}[I_{c}(A \rangle B) +2S(\rho^{B'}_{n})]$
where $I_{c}(A \rangle
B)=I_{c}(Tr_{B'}\rho^{BB'}_{n},\Lambda^{\otimes n}_{B})$ and
$\rho^{B'}_{n}=Tr_{B}\Lambda_{BB'}^{\otimes n}(\rho^{BB'}_{n})$.
However, if there exists a state $\sigma^{B}_{n}$ for which $I_{c}(\sigma^{B}_{n},\Lambda^{\otimes n}_{B}) > I_{c}(Tr_{B'}\rho^{BB'}_{n},\Lambda^{\otimes n}_{B})$, then it proves that right hand side of the inequality
in the lemma can be only larger than in case of the chosen state
$\rho^{BB'}_{n}$ which completes the proof. Finally, as in the
aforementioned proof for key subadditivity of entropy can be
applied to verify that in case of the regularized reduced secret
key its defect parameter cannot be larger than
$\Delta_{\mathcal{Q}_{\rightarrow}}=2
\sup_{\rho_{BB'}}S(Tr_{B}\Lambda_{BB'}(\rho_{BB'}))$, since
$\sup_{\rho_{BB'}^{n}}S(Tr_{B^{n}}\Lambda_{BB'}^{\otimes n
}(\rho_{BB'}^{n}))\leq
n\sup_{\rho_{BB'}}S(Tr_{B}\Lambda_{BB'}(\rho_{BB'})$. $\Box$
\textit{Example 3.} As an example we will use the aforementioned
graph state from Example. 2 and we will search for one-way channel
capacity of a channel $\Lambda_{BB'}$, isomorphic due to
Choi-Jamiolkowski isomorphism, with a state
$\rho^{ABB'}_{2n+1}=(I\otimes
\Lambda_{BB'})|\Psi\rangle\langle\Psi|$. As above, after
discarding $B'$ 1-qubit system the state would become symmetric
extendible that implies $Q_{\rightarrow}(\Lambda_{B})=0$.
Therefore, we obtain $Q_{\rightarrow}(\Lambda_{BB'})\leq 2$.
The power of the above results appears especially in application
of Lemma 3 to any channel reducible to anti-degradable channel
which Choi-Jamiolkowski representation is symmetric extendible
\cite{Lutk1} or channels reducible to degradable channels which
have known capacity \cite{Smith1}.
\textit{\bf Dual picture for one-way distillable entanglement and
private information.} Our results for one-way secret key and
quantum channel capacity lead immediately to similar reduced
formula for private information and one-way distillation
quantities. The private capacity \cite{DevetakW3, DevetakW4}
$\mathcal{P}(\Lambda)$ of a quantum channel is equal to
regularization of private information:
$\mathcal{P}^{(1)}(\Lambda)=\max_{X,\rho_{x}^{A}}(I(X,B)-I(X,E))$
with maximization over classical random variables X and input
quantum states $\rho_{x}^{A}$ depending on the value of X.
Absorbing T into X variable in Theorem 1. leads to definitions for
private information and private capacity \cite{DevetakW4}, thus,
following Lemma 3. we can derive an upper bound on private
information and private capacity via their reduced counterparts:
\textbf{Definition 5.} \textit{For a one-way quantum channel
$\Lambda_{BB'}:B(\cal{H}_{BB'})\rightarrow
B(\cal{H}_{\widetilde{B}\widetilde{B'}})$ the reduced private
information is defined as:
\begin{equation}
\mathcal{P}^{(1)}\downarrow(\Lambda_{BB'}) =
\inf_{\cal{U}}[\mathcal{P}^{(1)}(\cal{U}(\Lambda_{B}))+
\Delta_{P}]
\end{equation}
where $\cal{U}$ denotes unitary operations on Bob's system with a
possible transfer of subsystems from Bob to Eve, i.e.
$\cal{U}(\Lambda_{B}(\rho_{B}))=Tr_{B'}\cal{U}\Lambda_{BB'}(\rho_{BB'})$.
$\Delta_{P} =4 S (\rho_{B'})$ denotes the defect parameter related
to increase of entropy produced by the transfer of B'-subsystem
from Bob's side to Eve.}
\textbf{Theorem 4. }\textit{For a one-way quantum channel
$\Lambda_{BB'}:B(\cal{H}_{BB'})\rightarrow
B(\cal{H}_{\widetilde{B}\widetilde{B'}})$ there holds:
\begin{equation}
\mathcal{P}(\Lambda_{BB'}) \leq
\mathcal{P}\downarrow(\Lambda_{BB'})
\end{equation}
where
$\mathcal{P}\downarrow(\Lambda_{BB'})=\lim_{n}\mathcal{P}^{(1)}\downarrow(\Lambda_{BB'}^{\otimes
n })/n$ denotes the reduced private capacity. Particularly, for
identity operation $\cal{U}=id$ on Bob's side one obtains:
$\mathcal{P}(\Lambda_{BB'}) \leq \mathcal{P}(\Lambda_{B})+
4S(\rho_{B'})$}
The proof can be conducted in analogy to Theorem 2. and Lemma 3,
however, for regularization of reduced private information it is
crucial to derive the below lemma for a one-copy case:
\textbf{Lemma 5. }\textit{For every one-way quantum channel
$\Lambda_{BB'}:B(\cal{H}_{BB'})\rightarrow
B(\cal{H}_{\widetilde{B}\widetilde{B'}})$ there holds:
\begin{equation}
\mathcal{P}^{(1)}(\Lambda_{BB'}) \leq
\mathcal{P}^{(1)}\downarrow(\Lambda_{BB'})
\end{equation}
}
\textit{Proof.} To prove this lemma it suffices to absorb variable
T into X in Theorem 1. for definition of private information and
conduct the proof in analogy to the proof of Lemma 2 for a channel
$\Lambda_{BB'}$ and a chosen state $\rho$ sent through it. $\Box$
We can now propose a new bound on distillation of entanglement by
means of one-way LOCC. This result is based on observation
\cite{DevetakW3, DevetakW4} that one-way distillable entanglement
$D_{\rightarrow}$ of a state $\rho_{AB}$ can be represented as
regularization of one-copy formula:
$D^{(1)}_{\rightarrow}(\rho_{AB})=\max_{\textbf{T}}\sum_{l=1}^{L}\lambda_{l}I_{c}(A\rangle
B)_{\rho_{l}}$ where the maximization is over quantum instruments
$T = (T_{1}, \dots , T_{L})$ on Alice’s system,
$\lambda_{l}=TrT_{l}(\rho_{A})$, $T_{l}$ is assumed to have one
Kraus operator $T_{l}(\rho)=A_{l}\rho A_{l}^{\dag}$ and
$\rho_{l}=\frac{1}{\lambda_{l}}(T_{l}\otimes id)\rho_{AB}$. Basing
on the results of Observation 1. and Lemma 3. we derive a general
formula for the bound on one-way distillable entanglement applying
the reduced quantity:
\textbf{Definition 4.} \textit{For a bipartite state
$\rho_{ABB'}\in B(\cal{H}_{A}\otimes \cal{H}_{B}\otimes
\cal{H}_{B'})$ shared between Alice and Bob (B and B' system) the
reduced one-way distillable entanglement is defined as:
\begin{equation}
D_{\rightarrow}^{(1)}\downarrow(\rho_{ABB'}) =
\inf_{\cal{U}}[D_{\rightarrow}^{(1)}(\cal{U}(\rho_{AB}))+
\Delta_{D_{\rightarrow}}]
\end{equation}
where $\cal{U}$ denotes unitary operations on Bob's system with a
possible transfer of subsystems from Bob to Eve, i.e.
$\cal{U}(\rho_{AB})=Tr_{B'}(I\otimes \cal{U})\rho_{ABB'}$.
$\Delta_{D_{\rightarrow}} =2 S (\rho_{B'})$ denotes the defect
parameter related to increase of entropy produced by the transfer
of B'-subsystem from Bob's side to Eve.}
\textbf{Theorem 5. }\textit{For a bipartite state $\rho_{ABB'}\in
B(\cal{H}_{A}\otimes \cal{H}_{B}\otimes \cal{H}_{B'})$ shared
between Alice and Bob (B and B' system) there holds:
\[
D_{\rightarrow}(\rho_{ABB'}) \leq
D_{\rightarrow}\downarrow(\rho_{ABB'})
\]
where $\Delta_{D_{\rightarrow}}=2S(\rho_{B'})$ and
$D_{\rightarrow}\downarrow(\rho_{ABB'})=\lim_{n}D_{\rightarrow}^{(1)}\downarrow(\rho_{ABB'}^{\otimes
n })/n$ denotes regularized version of reduced one-way distillable
entanglement for one copy. Particularly, for identity operation
$\cal{U}=id$ on Bob's side one obtains:
$D_{\rightarrow}(\rho_{ABB'}) \leq D_{\rightarrow}(\rho_{AB})+
2S(\rho_{B'})$.}
The proof of this theorem can be conducted in analogy to the
previous proofs for bounds on one-way secret key and quantum
channel capacity. The left inequality is an immediate implication
of the following lemma for the one-copy formula:
\textbf{Lemma 6. }\textit{For every bipartite state $\rho_{ABB'}$
there holds:
\begin{equation}
D_{\rightarrow}^{(1)}(\rho_{ABB'}) \leq
D_{\rightarrow}^{(1)}\downarrow(\rho_{ABB'})
\end{equation}
}
\textit{Proof.} It suffices to use results of Observation 1. to
notice that for a chosen set of instruments $\textbf{T}$ on Alice
side for calculation of $D_{\rightarrow}^{(1)}(\rho_{ABB'})$ the
inequality holds as extension of inequality from Observation 1. by
multiplicands $\lambda_{l}$ on the left and right side. However,
if in case of calculating $D_{\rightarrow}^{(1)}(\rho_{AB})$ there
exists a set $\textbf{T'}$ maximizing $D_{\rightarrow}(\rho_{AB})$
better than $\textbf{T}$, then right hand side of the inequality
can be only greater. $\Box$
It is crucial to notice that the 'defect' parameters $\Delta$ for
the reduced quantities are subadditive and hence, can be exploited
in case of composite systems and regularization:
\textit{\bf Corollary. }\textit{For the reduced quantities of
$\{K_{\rightarrow},\mathcal{P}, \mathcal{Q}_{\rightarrow},
D_{\rightarrow}\}$ for composite systems there holds:
$\Delta_{X}(\rho\otimes\sigma)\leq\Delta_{X}(\rho)+\Delta_{X}(\sigma)$
and
$\Delta_{Y}(\Lambda\otimes\Gamma)\leq\Delta_{Y}(\Lambda)+\Delta_{Y}(\Gamma)$
where $X=\{K_{\rightarrow},D_{\rightarrow}\}$ stands for states
and $Y=\{\mathcal{Q}_{\rightarrow},\mathcal{P}\}$ for channels
respectively. }
To prove the above corollary it suffices to use subadditivity of
entropy for composite systems since Bob can act with a unitary
operation before he discard some part of his subsystem. This
property of the parameters enables regularization in the
asymptotic regime of the reduced quantities for large systems
$\rho^{\otimes n}$.
\textit{Example 4. Activable multi-qubit bound entangled states.}
As an example illustrating this bound we consider an activated
bound entangled state $\rho_{II}$ \cite{Dur} which is distillable
if the parties (Alice and Bob) form two groups containing between
$40\%$ and $60\%$ of all parties of the system in the state
$\rho_{II}$. If Alice or Bob posses less than $40\%$ of the system
or system is shared between more than two parties, then the state
becomes undistillable. This state for large amount of particles
can manifest features characteristic for 'macroscopic
entanglement' with no 'microscopic entanglement'. For definition
of the state, let us consider the family $\rho_{N}$ of N-qubit
states:
$\rho=\sum_{\sigma=\pm}\lambda_{0}^{\sigma}|\Psi_{0}^{\sigma}\rangle\langle\Psi_{0}^{\sigma}|+
\sum_{k\neq0}\lambda_{k}(|\Psi_{k}^{+}\rangle\langle\Psi_{k}^{+}|+|\Psi_{k}^{-}\rangle\langle\Psi_{k}^{-}|)$
where $|\Psi_{k}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|k_{1}k_{2}\ldots
k_{N-1}0\rangle\pm|\overline{k}_{1}\overline{k}_{2}\ldots
\overline{k}_{N-1}1\rangle)$ are GHZ-like states with
$k=k_{1}k_{2}\ldots k_{N-1}$ being a chain of $N-1$ bits and
$k_{i}=0, 1$ if $\overline{k}_{i}=1, 0$, thus, the state is
parameterized by $2^{N-1}$ coefficients. Let us consider now a
bipartite splitting $\mathcal{P}$ where Alice takes $0.6N$ of
qubits and Bob takes the other $0.4 N$ qubits. We can immediately
show that $D_{\rightarrow}(\rho_{II})\leq -
2(\lambda_{0}^{\pm}+2\sum_{k}\lambda_{k})\log(\lambda_{0}^{\pm}+2\sum_{k}\lambda_{k})$
since for Bob transferring one qubit to the environment, we obtain
undistillable state $D_{\leftrightarrow}(\rho_{N-1})=0$. It is
noticeable that even for a large macroscopic system with
$N\rightarrow\infty$, $D_{\rightarrow}(\rho_{II})\leq - 2
(\lambda_{0}^{\pm}+2\sum_{k}\lambda_{k})\log(\lambda_{0}^{\pm}+2\sum_{k}\lambda_{k})$.
It can be easily shown that with the same method it is possible to
achieve an upper bound on one-way quantum channel capacity
$Q_{\rightarrow}$.
\textit{\bf Conclusions.} In this paper we proposed new reduced
versions of quantum quantities: reduced one-way quantum key,
distillable entanglement and reduced corresponding capacities. We
show that in some cases they may provide bounds on the non-reduced
versions simplifying drastically their estimations. It is evident
especially in case of states of large systems which is supported
by examples. The open problem is whether they can be applied to
non-additivity problem of quantum channel capacities and quantum
secure key \cite{Smith, Smith1}. Further, it is not known if they
have analogs in general quantum networks and whether the bounds
can be improved by better estimation of defect parameters.
\textit{\bf Acknowledgments.} The authors thank Michal Horodecki
for critical comments on this paper. This work was supported by
Ministry of Science and Higher Education grant No N202 231937.
Part of this work was done in National Quantum Information Center
of Gdansk.
\end{document} |
\begin{document}
\title{Splitting the K\"unneth formula}
\author{Laurence R. Taylor}
\address{Department of Mathematics\newline
\indent University of Notre Dame\newline
\indent Notre Dame, IN 46556\newline
\indent U.S.A.}
\email{[email protected]}
\begin{abstract}
There is a description of the
torsion product of two modules in terms of generators and
relations given by Eilenberg and Mac Lane.
With some additional data on the chain complexes there is
a splitting of the map in the K\"unneth\ formula in terms of
these generators.
Different choices of this additional data determine a natural coset
reminiscent of the indeterminacy in a Massey triple product.
In one class of examples the coset actually is a Massey triple product.
The explicit formulas for a splitting enable proofs of results on the
behavior of the interchange map and the long exact sequence
boundary map on all the terms in the K\"unneth\ formula.
Information on the failure of naturality of the splitting is also obtained.
\end{abstract}
\maketitle
\date{\today}
\section{Introduction}
Fix a principal ideal domain $R$ and let $\complex[1]_\ast$ and
$\complex[3]_\ast$ be two chain complexes of $R$ modules.
The K\"unneth\ formula states that
if $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic then
there is a short exact sequence
\begin{xyMatrixLine}
0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\complex[1]_\ast)
\tensor[R] H_\secondIndex(\complex[3]_\ast)
\ar[r]^-{\cs{cross product}}&
H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)
\ar[r]^-{\cs{to torsion product}}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_\secondIndex(\complex[3]_\ast)\to0
\end{xyMatrixLine}
which is natural for pairs of chain maps and which is split.
For a proof in this generality see
for example Dold \cite{Dold}*{VI, 9.13}.
Let $\cs{to torsion product}_{k}\def\secondInt{\ell}\def\totalInt{n,\secondInt}\colon
H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)
\to
H_{k}\def\secondInt{\ell}\def\totalInt{n}(\complex[1]_\ast)\tor[R] H_{\secondInt}(\complex[3]_\ast)$
denote $\cs{to torsion product}$ followed by projection.
Say that a map
$\sigma\colon
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast) \to
H_{i}\def\secondIndex{j+\secondIndex+1}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)$
\emph{splits the K\"unneth\ formula at $(i}\def\secondIndex{j,\secondIndex)$}
provided
$\cs{to torsion product}_{k}\def\secondInt{\ell}\def\totalInt{n,\secondInt}\circ \sigma =
\identyMap{H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R]
H_{\secondIndex}(\complex[3]_\ast)}$ if
$(k}\def\secondInt{\ell}\def\totalInt{n,\secondInt)=(i}\def\secondIndex{j,\secondIndex)$ and is $0$ otherwise.
\section{The main idea}\sectionLabel{main idea}
Suppose the $R$ modules in the complexes $\complex[1]_\ast$ and
$\complex[3]_\ast$ are free, so the K\"unneth\ formula holds.
The general case is discussed in \S \namedSection{general case}.
In \cite{Eilenberg-Mac Lane}*{\S11}
Eilenberg and Mac Lane gave a generators and relations
description of the torsion product:
$\complex[1]\tor[R]\complex[3]$ is the free $R$ module on
symbols $\cs{elementary tor}{\element[1]}{RElement}{\element[2]}$
where $RElement\in R$,
$\element[1]\in \complex[1]$ with $\element[1]\@ifnextchar_{\LRT@P}{P}DotRElement = 0$ and
$\element[2]\in \complex[3]$ with $RElement\@ifnextchar_{\LRT@P}{P}Dot\element[2] = 0$
modulo four types of relations described below,
(\ref{free cycle gives map}.1) --
(\ref{free cycle gives map}.4).
The symbols
$\cs{elementary tor}{\element[1]}{RElement}{\element[2]}$
will be called \emph{elementary tors}.
In what follows, given any complex $\complex[2]_{\ast}$,
$\complexCycles[2]_\ast$ denotes the cycles and
$\complexBoundaries[2]_\ast$ denotes the boundaries.
Given any cycle $\elementCycle[4]$ of degree $\abs{\element[4]}$
in $\complex[2]_{\ast}$,
write $\homologyClassOf{\elementCycle[4]}\in H_{\abs{\element[4]}}(\complex[2]_\ast)$
for the homology class $\elementCycle[4]$ represents.
Let $\cyclesToHomology[2]\colon \complexCycles[2]_\ast
\to H_\ast(\complex[2]_\ast)$
denote the canonical map.
Mac Lane \cite{Mac Lane}*{Prop.~V.10.6} describes a cycle in
$H_{\totalInt}(\complex[1]_\ast\tensor[R]\complex[3]_\ast)$
representing a given elementary tor in the range of $\cs{to torsion product}$.
Mac Lane's cycle is defined as follows.
Lift $\element[1]$ to a cycle, $\elementCycle[1]$, and $\element[2]$ to a
cycle $\elementCycle[3]$.
Since $\element[1]\@ifnextchar_{\LRT@P}{P}Dot RElement = 0$,
$\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement$ is a boundary.
Choose $\liftToChain[1]\in \complex[1]_{\abs{\element[1]}+1}$ so that
$\boundary[1]_{\abs{\element[1]}+1}(\liftToChain[1]
) =
\elementCycle[1] \@ifnextchar_{\LRT@P}{P}DotRElement$.
Choose $\liftToChain[3]$ so that
$\boundary[3]_{\abs{\element[3]}+1}(\liftToChain[3]) =
RElement\@ifnextchar_{\LRT@P}{P}Dot \elementCycle[3]$.
Up to sign and notation, Mac Lane's cycle is given by
\namedNumber[Formula]{Mac Lane cycle A}
\begin{equation*}\tag{\ref{Mac Lane cycle A}}
\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \liftToChain[1]; \elementCycle[3], \liftToChain[3]\bigr) =
(-1)^{\abs{\element[1]}+1}
\elementCycle[1] \tensor \liftToChain[3] +
\liftToChain[1] \tensor \elementCycle[3]
\end{equation*}
Mac Lane puts the sign in front of the other term but then gets a sign
when evaluating $\cs{to torsion product}$.
Mac Lane also writes (\ref{Mac Lane cycle A}) as a Bockstein.
The short exact sequence $\xyLine[@C10pt]{0\ar[r]&R
\ar[rr]^-{\@ifnextchar_{\LRT@P}{P}DotRElement}&&
R\ar[rr]^-{\rho^{RElement}}&&\ry{RElement}\ar[r]&0}$
gives rise to a long exact sequence whose boundary term is called
the Bockstein associated to the sequence:
\begin{math}
\mathfrak b^{RElement}_{\totalInt}\colon
H_{\totalInt}\bigl({\complex[2]_\ast} {\tensor[R]}\ry{RElement}\bigr)
\to H_{\totalInt-1}({\complex[2]_{\ast}})
\end{math}
In terms of the Bockstein and the pairing
\begin{equation*}
H_{k}\def\secondInt{\ell}\def\totalInt{n}\bigl(\complex[1]_\ast\tensor[R]\ry{RElement}\bigr)
\cs{cross product}
H_{\secondInt}\bigl(\complex[3]_\ast\tensor[R]\ry{RElement}\bigr)
\to
H_{k}\def\secondInt{\ell}\def\totalInt{n+\secondInt}\bigl(\complex[1]_\ast\tensor[R]\complex[3]_\ast
\tensor[R]\ry{RElement}\bigr)
\end{equation*}
\vskip-10pt
\namedNumber[Formula]{Mac Lane cycle B}
\begin{equation*}\tag{\ref{Mac Lane cycle B}}
\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \liftToChain[1]; \elementCycle[3], \liftToChain[3]\bigr) =
(-1)^{\abs{\element[1]}+1}\mathfrak b^{RElement}_{\abs{\element[1]}+\abs{\element[3]}+2}\bigl(
\liftToChain[1] \tensor \liftToChain[3] \bigr)
\end{equation*}
Given a different choice of cycle for $\elementCycle[1]$,
say $\elementCycleB[1]$,
$\elementCycleB[1] =\elementCycle[1] + \boundary[1]_{\abs{\element[1]}+1}
(\liftDelta[1])$.
Take $\liftToChainA[1]{1} = \liftToChain[1] + \liftDelta[1]\@ifnextchar_{\LRT@P}{P}Dot RElement$.
With a similar choice of lift on the right,
$\cs{tor cycles into}\bigl(
{\elementCycleB[1]}, \liftToChainA[1]{1}; \elementCycleB[3], \liftToChainA[3]{1}\bigr)
-
\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \liftToChain[1]; \elementCycle[3], \liftToChain[3]\bigr)$
is a boundary and so different choices of cycles give the same homology class.
\vskip10pt
Indeterminacy comes from the choices of $\liftToChain[1]$ and $\liftToChain[3]$.
With $\elementCycle[1]$ and $\elementCycle[3]$ fixed,
$\liftToChain[1]$ is determined up to a cycle.
Let $\liftToChainA[1]{1} = \liftToChain[1] + \liftCycle[1]$ and
let $\liftToChainA[3]{1} = \liftToChain[3] + \liftCycle[3]$.
Then
\begin{equation*}
{\homologyClassOf[big]{
\cs{tor cycles into}\bigl(
{\element[1]}, \liftToChainA[1]{1}; \element[2], \liftToChainA[3]{1}\bigr)}} =
\homologyClassOf[big]{\cs{tor cycles into}\bigl(
{\element[1]}, \liftToChain[1]; \element[2], \liftToChain[3]\bigr) +
(-1)^{\abs{\liftCycle[1]}}
\bigl(\element[1]\cs{cross product} \homologyClassOf{\liftCycle[3]}\bigr)}
+
\bigl(\homologyClassOf{\liftCycle[1]}\cs{cross product} \element[2]\bigr)
\end{equation*}
Since $\homologyClassOf{\liftCycle[1]}$ and
$\homologyClassOf{\liftCycle[3]}$ can be chosen arbitrarily,
any element in the coset
$\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast}
{\abs{\element[1]}}{\abs{\element[2]}}$
can be realized.
Let
\namedNumber[Formula]{double coset}
\begin{equation*}\tag{\ref{double coset}}
\cosetTor[{\element[1]}]{RElement}{\element[2]} \subset
H_{\abs{\element[1]}+\abs{\element[2]}+1}
(\complex[1]_\ast\tensor[R]\complex[3]_\ast)
\end{equation*}
\noindent
denote the coset determined by any of the
$\homologyClassOf[big]{
\cs{tor cycles into}\bigl(
{\element[1]}, \liftToChainA[1]{1}; \element[2], \liftToChainA[3]{1}\bigr)}$.
The above discussion and Proposition V.10.6 of \cite{Mac Lane}
shows the following.
\begin{ThmS}[Mac Lane main lemma]{Lemma}
For two complexes of free $R$ modules, $R$ a PID, the element
$\homologyClassOf[big]{
\cs{tor cycles into}\bigl(
{\element[1]}, \liftToChain[1]; \element[2], \liftToChain[3]\bigr)}$
determines $\cosetTor[{\element[1]}]{RElement}{\element[2]}
$
a well-defined coset of
$\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast}
{\abs{\element[1]}}{\abs{\element[2]}}$
such that
\begin{equation*}
\cs{to torsion product}_{s,t}
\bigl(\cosetTor[{\element[1]}]{RElement}{\element[2]}\bigr) =
\begin{cases}
\cs{elementary tor}{\element[1]}{RElement}{\element[2]} &
s=\abs{\element[1]}, t = \abs{\element[2]}\\
0&\text{otherwise}\\
\end{cases}
\end{equation*}
\end{ThmS}
To get a splitting requires one more step.
Since $R$ is a PID, the set of boundaries in a free chain complex
is a free submodule and hence there is a splitting of the boundary maps.
Choose splittings for the complexes being considered here:
$\complexSplitting[1]\colon\complexBoundaries[1]\to \complex[1]_{\ast+1}$ and
$\complexSplitting[3]\colon\complexBoundaries[3]\to \complex[3]_{\ast+1}$.
Define
\namedNumber{torsion product cycle}
\newCS{torsion product cycle 1}{{\ref{torsion product cycle}.1}}
\newCS{torsion product cycle 2}{{\ref{torsion product cycle}.2}}
\newCS{env:torsion product cycle 1}{{Formula}}
\newCS{env:torsion product cycle 2}{{Formula}}
\begin{align*}\tag{\cs{torsion product cycle 1}}
\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \complexSplitting[1]; \elementCycle[3],
\complexSplitting[3];RElement\bigr) =&
(-1)^{\abs{\element[1]}+1}
\elementCycle[1]\tensor
\complexSplitting[3]\bigl(RElement \elementCycle[3]\bigr) +
\complexSplitting[1]\bigl(\elementCycle[1] RElement\bigr) \tensor
\elementCycle[3]\\
\tag{\cs{torsion product cycle 2}}
\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \complexSplitting[1];
\elementCycle[3], \complexSplitting[3];RElement\bigr) =&
(-1)^{\abs{\element[1]}+1}\mathfrak b^{RElement}_{\abs{\element[1]}+\abs{\element[3]}+2}
\Bigl(\complexSplitting[1]\bigl(\elementCycle[1] RElement\bigr) \tensor
\complexSplitting[3]\bigl(RElement \elementCycle[3]\bigr)\Bigr)
\end{align*}
\begin{ThmS}[free splitting is independent of cycles]{Lemma}
The homology class
$\homologyClassOf[big]{\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \complexSplitting[1];
\elementCycle[3], \complexSplitting[3];RElement\bigr)}$
is independent of the choice of cycles
$\elementCycle[1]$ and $\elementCycle[3]$.
\end{ThmS}
\begin{proof}
See the paragraph just below (\ref{Mac Lane cycle B}).
\end{proof}
Define
\begin{equation*}
\cs{homology splitting}[{\complexSplitting[1]}]
{\complexSplitting[3]}_{\abs{\element[1]}, \abs{\element[2]}}
\bigl(
\cs{elementary tor}{\element[1]}{RElement}{\element[2]}
\bigr)
=
\homologyClassOf[big]{\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \complexSplitting[1];
\elementCycle[3], \complexSplitting[3]; RElement\bigr)}
\end{equation*}
\begin{ThmS}[free cycle gives map]{Theorem}
For fixed splittings $\complexSplitting[1]$ and $\complexSplitting[3]$, the function
$\cs{homology splitting}[{\complexSplitting[1]}]
{\complexSplitting[1]}_{\abs{\element[1]}, \abs{\element[2]}}$
defined on elementary tors induces an $R$ module map
\begin{equation*}
\cs{homology splitting}[{\complexSplitting[1]}]
{\complexSplitting[3]}_{\abs{\element[1]}, \abs{\element[2]}}\colon
H_{\abs{\element[1]}}(\complex[1]_\ast)\tor[R]
H_{\abs{\element[2]}}(\complex[3]_\ast) \to
H_{\abs{\element[1]} + \abs{\element[2]}+1}(\complex[1]_\ast\tensor[R]
\complex[3]_\ast)
\end{equation*}
which splits the K\"unneth\ formula at $\bigl(\abs{\element[1]},\abs{\element[2]}\bigr)$.
\end{ThmS}
\begin{proof}
The splitting at $\bigl(\abs{\element[1]},\abs{\element[2]}\bigr)$
follows from \namedRef{Mac Lane main lemma}.
Fix splittings and let
$\localName{\element[1]}{RElement}{\element[2]}
=
\homologyClassOf{\cs{tor cycles into}\bigl(
{\elementCycle[1]}, \complexSplitting[1];
\elementCycle[3], \complexSplitting[3];RElement\bigr)}$.
By Eilenberg and Mac Lane \cite{Eilenberg-Mac Lane}*{\S 11},
to prove $\cs{homology splitting}$ is a module map,
it suffices to prove the following
\vskip 10pt
\noindent(\ref{free cycle gives map}.1)
\enumline{$\localName{\element[1]_1}{RElement}{\element[2]} +
\localName
{\element[1]_2}{RElement}{\element[2]} =
\localName
{\element[1]_1+\element[1]_2}{RElement}{\element[2]}$}
{$\element[1]_{i}\def\secondIndex{j}RElement = 0$; $RElement\element[2]=0$}
(\ref{free cycle gives map}.2)
\enumline{$\localName
{\element[1]}{RElement}{\element[2]_1} +
\localName
{\element[1]}{RElement}{\element[2]_2} =
\localName
{\element[1]}{RElement}{\element[2]_1+\element[2]_2}$}
{$\element[1]RElement=0$; $RElement \element[2]_{i}\def\secondIndex{j}=0$}
(\ref{free cycle gives map}.3)
\Enumline{$\localName
{\element[1]}{RElement_1\cdot RElement_2}{\element[2]} =
\localName
{\element[1] RElement_1}{RElement_2}{\element[2]}$}
{$\element[1] RElement_1 RElement_2 = 0$; $RElement_2\element[2]=0$}
(\ref{free cycle gives map}.4)
\Enumline{$\localName
{\element[1]}{RElement_1\cdot RElement_2}{\element[2]} =
\localName
{\element[1]}{RElement_1}{RElement_2\element[2]}$}
{$\element[1]RElement_1=0$;
$RElement_1 RElement_2\element[2]=0$}
These formulas are easily verified at the chain level
using (\cs{torsion product cycle 1}), \namedRef{free splitting is independent of cycles}
and carefully chosen cycles.
\end{proof}
\begin{DefS}{Remark}
Eilenberg and Mac Lane work over $\mathbb Z$ but,
as pointed out explicitly in
\cite{Mac Lane slides}*{about the middle of page 285},
the proof uses nothing more than that submodules of free modules are
free and that finitely generated modules are direct sums of cyclic modules.
Hence the results are valid for PID's.
\end{DefS}
\begin{DefS}{Remark}
The data contained in a splitting is surely related to the structure introduced
by Heller in \cite{Heller}.
See also Section \namedSection{Bocksteins determine}.
\end{DefS}
\section{Free Approximations}\sectionLabel{free approximations}
A result attributed to Dold by Mac Lane \cite{Mac Lane}*{Lemma 10.5}
is that given any chain complex over a PID
there exists a free chain complex with a quasi-isomorphic chain map
to the original complex.
In this paper any such complex and quasi-isomorphism will be called
a \emph{free approximation}.
\begin{DefS*}{Warning}
Some authors also require the chain map to be surjective.
\end{DefS*}
Here is a review of a construction of a free approximation,
mostly to establish notation.
Some lemmas needed later are also proved here.
\def\chainMap[1]+ \chainMap[2]{ }
A \emph{weak splitting} of a chain complex $\complex[1]_\ast$
at an integer $\totalInt$ is a free resolution
$\xyLine[@C20pt]{0\ar[r]&
\freeBoundaries[1]_{\totalInt}
\ar[r]^-{\iota^{\complex[1]}_{\totalInt}}&
\freeCycles[1]_{\totalInt}
\ar[rr]^-{\hat{\freeCyclesMap[0]_{ }}_{H_{\totalInt}(\complex[1]_\ast)}}&&
H_{\totalInt}(\complex[1]_\ast)\ar[r]&0}$
and a pair of maps
$\splitPair[1]_{\totalInt} = \{\freeCyclesMap[1]_{\totalInt}, \freeBoundariesMap[1]_{\totalInt} \}$
of the resolution into $\complex[1]_\ast$
where
$\freeCyclesMap[1]_{\totalInt}\colon \freeCycles[1]_{\totalInt}\to
\complexCycles[1]_{\totalInt}$
and
$\freeBoundariesMap[1]_{\totalInt}\colon
\freeBoundaries[1]_{\totalInt}\to \complex[1]_{\totalInt+1}$.
It is further required that \\
\null\hskip5pt\begin{minipage}{0.30\textwidth}
\begin{xyMatrix}
\freeCycles[1]_{\totalInt}\ar[r]^-{\freeCyclesMap[1]_{\totalInt}}
\ar[dr]_{\hat{\freeCyclesMap[0]_{ }}_{H_{\totalInt}(\complex[1]_\ast)}}&
\ar[d]^-{\cyclesToHomology[1]_{\totalInt}}
\complexCycles[1]_{\totalInt}
\\
&H_{\totalInt}(\complex[1]_\ast)
\end{xyMatrix}
\end{minipage}\hbox to 0.8in{\hfil and\hfil} \begin{minipage}{0.35\textwidth}
\begin{xyMatrix}[@C40pt]
\freeBoundaries[1]_{\totalInt}\ar[r]^-{\iota^{\complex[1]}_{\totalInt}}\ar[d]^-{\freeBoundariesMap[1]_{\totalInt}}
&\freeCycles[1]_{\totalInt}\ar[d]^-{\freeCyclesMap[1]_{\totalInt}}\\
\complex[1]_{\totalInt+1}\ar[r]^-{\boundary[1]_{\totalInt+1}}&
\complexCycles[1]_{\totalInt}\\
\end{xyMatrix}
\end{minipage}
commute.
The complex is said to be \emph{weakly split} if it is weakly split at $\totalInt$
for all integers $\totalInt$.
Any module over a PID has a free resolution and any complex has a weak splitting.
If the complex is free, a splitting as in \S\namedSection{main idea}
is a weak splitting.
\vskip 10pt
Given a weakly split complex, define a complex whose groups are
$\naturalFreeComplex[1]_{\totalInt} =
\freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus}
\freeCycles[1]_{\totalInt}$
and whose boundary maps are the compositions
\noindent
\resizebox{\textwidth}{!}{{$
\xymatrix@C30pt{
\complexFreeBoundaryMap[1]_{\totalInt}\colon
\naturalFreeComplex[1]_{\totalInt} =
\freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt}\ar[r]&
\freeBoundaries[1]_{\totalInt-1}\ar[r]^-{\iota^{\complex[1]}_{\totalInt-1}}&
\freeCycles[1]_{\totalInt-1}\ar[r]&
\freeBoundaries[1]_{\totalInt-2} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt-1}
=
\naturalFreeComplex[1]_{\totalInt-1}
}
$
}}
\noindent
The submodule
$0\displaystyle\mathop{\oplus} \freeBoundaries[1]_{\totalInt-1}\subset
\freeBoundaries[1]_{\totalInt-2}
\displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt-1} =
\naturalFreeComplex[1]_{\totalInt-1}$ is the image of
$\complexFreeBoundaryMap[1]_{\totalInt}$
so one choice of splitting, called the \emph{canonical splitting}, is the composition
\begin{xyMatrix}
\naturalFreeSplitting[1]\colon
\mathbf{B}_{\totalInt-1}(\naturalFreeComplex[1]_\ast)=0 \displaystyle\mathop{\oplus}
\freeBoundaries[1]_{\totalInt-1}\ar[r]&
\freeBoundaries[1]_{\totalInt-1}\ar[r]&
\freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus}
\freeCycles[1]_{\totalInt} =
\naturalFreeComplex[1]_{\totalInt}
\end{xyMatrix}
\begin{ThmS}[free approximations]{Lemma}
The map
\begin{equation*}
\naturalFreeMap[1]_{\totalInt}=\freeBoundariesMap[1]_{\totalInt-1}+
\freeCyclesMap[1]_{\totalInt}\colon
\naturalFreeComplex[1]_{\totalInt} = \freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus}
\freeCycles[1]_{\totalInt} \to \complex[1]_{\totalInt}
\end{equation*}
is a chain map which is a quasi-isomorphism.
If $\freeCyclesMap[1]_\ast\colon\freeCycles[1]_\ast \to
\complexCycles[1]_\ast$ is onto
then $\naturalFreeMap[1]_{\ast}$ is onto.
It is always possible to choose $\freeCyclesMap[1]_\ast$ to be onto.
\end{ThmS}
The proofs of the claimed results are standard.
\begin{ThmS}[cover surjective chain maps]{Lemma}
Let $\chainMap[1]_\ast\colon\complex[1]_\ast\to\complex[3]_\ast$ be a
surjective chain map
and let
$\naturalFreeMap[3]_{\ast}\colon \freeApproximation[3]_\ast\to\complex[3]_\ast$
be a free approximation.
Then there exist free approximations
$\naturalFreeMap[1]_{\ast}\colon \freeApproximation[1]_\ast\to\complex[1]_\ast$
and surjective chain maps $\freeApproximationChainMap[1]_\ast\colon
\freeApproximation[1]_\ast \to \freeApproximation[3]_\ast$ making
\begin{xyMatrix}
\freeApproximation[1]_\ast\ar[r]^-{\freeApproximationChainMap[1]_\ast}
\ar[d]^-{\naturalFreeMap[1]_{\ast}}& \freeApproximation[3]_\ast
\ar[d]^-{\naturalFreeMap[3]_{\ast}}\\
\complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}&
\complex[3]_\ast
\end{xyMatrix}
commute.
\end{ThmS}
\begin{proof}
Let
$\xymatrix{
P_\ast\ar[r]^-{\hat{\chainMap[1]_{ }}_\ast}
\ar[d]^-{\zeta_{\ast}}& \freeApproximation[3]_\ast
\ar[d]^-{\naturalFreeMap[3]_{\ast}}\\
\complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}&
\complex[3]_\ast
}$
be a pull back.
\def\chainMap[1]+ \chainMap[2]{P}
Since $\chainMap[1]_\ast$ is onto, so is $\hat{\chainMap[1]_{ }}_\ast$
and the kernel complexes are isomorphic.
By the 5 Lemma, $\zeta_\ast$ is a quasi-isomorphism.
Let $\naturalFreeMap[10]_\ast\colon
\freeApproximation[1]_\ast\to P_\ast$
be a surjective free approximation.
Then $\naturalFreeMap[1]_{\ast} = \zeta_\ast\circ \naturalFreeMap[10]_{\ast}$
and $\freeApproximationChainMap[1]_\ast =
\hat{\chainMap[1]_{ }}_\ast\circ \naturalFreeMap[10]_\ast$ are
the desired maps.
\end{proof}
\begin{ThmS}[short exact free approximation]{Lemma}
If $
\xymatrix@1@C10pt{
0\ar[r]&
\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&&
\complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&&
\complex[2]_\ast\
\ar[r]& 0}$
is exact, there exist free approximations
making the diagram below commute.
\begin{xyMatrix}[@C10pt]
0\ar[r]&\freeApproximation[1]_\ast\ar[rr]^-{\freeMapApproximation[1]_\ast}
\ar[d]^-{\vertMap{\complex[1]}_\ast}&&
\freeApproximation[3]_\ast\ar[rr]^-{\freeMapApproximation[2]_\ast}
\ar[d]^-{\vertMap{\complex[3]}_\ast}&&
\freeApproximation[2]_\ast
\ar[d]^-{\vertMap{\complex[2]}_\ast}
\ar[r]& 0\\
0\ar[r]&\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&&
\complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&&
\complex[2]_\ast\ar[r]&0
\end{xyMatrix}
\end{ThmS}
\begin{proof}
Use \namedRef{cover surjective chain maps} to get $\freeMapApproximation[2]$.
Let $\freeApproximation[1]_\ast$ be the kernel complex, hence free.
There is a unique map $\vertMap{\complex[1]}_{\ast}$
making the diagram commute.
By the 5 Lemma, $\vertMap{\complex[1]}_{\ast}$ is a quasi-isomorphism.
\end{proof}
\begin{ThmS}[Dold splitting]{Lemma}
Suppose $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic.
Suppose
$\naturalFreeMap[1]_{\ast}\colon \freeApproximation[1]_\ast\to\complex[1]_\ast$
and
$\naturalFreeMap[3]_{\ast}\colon \freeApproximation[3]_\ast\to\complex[3]_\ast$
are free approximations.
Then so is
{\setlength\belowdisplayskip{-10pt}
\begin{equation*}
\naturalFreeMap[1]_{\ast}
\tensor \naturalFreeMap[3]_{\ast}\colon
\freeApproximation[1]_\ast\tensor[R]\freeApproximation[3]_\ast
\to\complex[1]_\ast\tensor[R]\complex[3]_\ast
\end{equation*}
}\end{ThmS}\nointerlineskip
\namedNumber{Dold splitting diagram}
\begin{proof}
The K\"unneth\ formula is natural for chain maps so
\noindent\resizebox{\textwidth}{!}{{$\xymatrix{
0\to
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H_{i}\def\secondIndex{j}(\freeApproximation[1]_\ast)\tensor[R]
H_{\secondIndex}(\freeApproximation[3]_\ast)
\ar[r]^-{\cs{cross product}}
\ar[d]^-{\hbox{\tiny$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
\naturalFreeMap[1]_\ast
\tensor \naturalFreeMap[3]_\ast$}}_-{\hbox to 1in{(\ref{Dold splitting diagram})
}}
&
H_{\totalInt}(\freeApproximation[1]_\ast \tensor[R]
\freeApproximation[3]_\ast)
\ar[r]^-{\cs{to torsion product}}
\ar[d]^-{(\naturalFreeMap[1]\tensor \naturalFreeMap[3])_\ast}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
H_{i}\def\secondIndex{j}(\freeApproximation[1]_\ast)\tor[R]
H_{\secondIndex}(\freeApproximation[3]_\ast)
\to 0
\ar[d]^-{\hbox{\tiny$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
\naturalFreeMap[1]_\ast\tor \naturalFreeMap[3]_\ast$}}
\\
0\to
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tensor[R] H_{\secondIndex}(\complex[3]_\ast)
\ar[r]^-{\cs{cross product}}
&
H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)
\ar[r]^-{\cs{to torsion product}}
&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast)
\to 0
}$}}
\noindent
commutes.
The left and right vertical maps are tensor and torsion products of isomorphisms and
hence isomorphisms.
The middle vertical map is an isomorphism by the 5 Lemma.
\end{proof}
\section{The general case}\sectionLabel{general case}
With notation and hypotheses as in \namedRef{Dold splitting},
applying $(\naturalFreeMap[1]\tensor \naturalFreeMap[3])_\ast$
to the cycle in (\cs{torsion product cycle 1})
gives
\namedNumber{torsion product cycle II}
\newCS{torsion product cycle II 1}{{\ref{torsion product cycle II}.1}}
\newCS{torsion product cycle II 2}{{\ref{torsion product cycle II}.2}}
\newCS{env:torsion product cycle II 2}{{Formula}}
\begin{equation*}\tag{\cs{torsion product cycle II 1}}
\cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)\bigl(\elementCycle[1], RElement, \elementCycle[3]\bigr) =
\epsilon\,
\freeCyclesMap[1]_\ast(\elementCycle[1])
\tensor
\freeBoundariesMap[3]_\ast\bigl(RElement \elementCycle[3]\bigr) +
\freeBoundariesMap[1]_\ast
\bigl(\elementCycle[1] RElement\bigr) \tensor
_\ast(\elementCycle[3])\\
\end{equation*}
\noindent
where $\elementCycle[1]\in \freeCycles[1]_\ast$ satisfies
$\freeHomologyMap[1]_\ast(\elementCycle[1]) =
\element[1]$,
$\elementCycle[3]\in \freeCycles[3]_\ast$ satisfies
$\freeHomologyMap[3]_\ast(\elementCycle[3]) =
\element[2]$ and $\epsilon=(-1)^{\abs{\element[1]}+1}$.
In general there is no analogue to
(\cs{torsion product cycle 2}) because not all complexes have the
necessary Bocksteins.
If $\complex[1]_\ast$ and $\complex[3]_\ast$ are torsion free then
the necessary Bocksteins exist and applying
$(\naturalFreeMap[1]\tensor \naturalFreeMap[3])_\ast$
to (\cs{torsion product cycle 2}) gives
\begin{align*}
\tag{\cs{torsion product cycle II 2}}
\cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)\bigl(\elementCycle[1], RElement, \elementCycle[3]\bigr) =&
\epsilon\,
\mathfrak b^{RElement}_{\abs{\element[1]}+\abs{\element[3]}+2}\Bigl(
\freeBoundariesMap[1]_\ast\bigl(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}DotRElement\bigr) \tensor
\freeBoundariesMap[3]_\ast
\bigl(RElement \@ifnextchar_{\LRT@P}{P}Dot\elementCycle[3]\bigr) \Bigr)
\end{align*}
\begin{ThmS}{Lemma}
The homology class
$\homologyClassOf[big]{\cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)(\elementCycle[1],RElement, \elementCycle[3])}$
is independent of the lifts $\elementCycle[1]$ and $\elementCycle[3]$.
\end{ThmS}
\begin{proof}
The cycles $\elementCycle[1]$ and $\elementCycle[3]$ are cycles in
$\naturalFreeComplex[1]_\ast$ and $\naturalFreeComplex[3]_\ast$
so the result is immediate from
\namedRef{free splitting is independent of cycles}
\end{proof}
\begin{ThmS}[weak splittings give map]{Theorem}
Assume $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic.
For fixed weak splittings $\splitPair[1]_\ast$ and $\splitPair[3]_\ast$
taking the homology class of $\cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)(\elementCycle[1],\elementCycle[3])$
yields a map
\begin{equation*}
\cs{homology splitting}[{\splitPair[1]_\ast}]
{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\colon
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R]
H_{\secondIndex}(\complex[3]_\ast) \to
H_{i}\def\secondIndex{j+\secondIndex+1}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)
\end{equation*}
which splits the K\"unneth\ formula at $(i}\def\secondIndex{j,\secondIndex)$.
\end{ThmS}
\begin{proof}
The cycle \ref{torsion product cycle II}.1 is the image of the cycle
\ref{torsion product cycle}.1 and so
$\cs{homology splitting}$ is a map
by \namedRef{free cycle gives map}.
\namedNumberRef{Dold splitting}
applies and (\ref{Dold splitting diagram}) has exact rows.
The splitting result
follows from \namedRef{free cycle gives map}.
\end{proof}
\begin{ThmS}[weak splitting map in correct coset]{Corollary}
The map $\cs{homology splitting}[{\splitPair[1]_\ast}]
{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}$
will depend on the weak splittings.
For any choices of weak splittings,
$\cs{homology splitting}[{\splitPair[1]_\ast}]
{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl(
\cs{elementary tor}{\element[1]}{RElement}{\element[2]}
\bigr)$ is in the same coset of
$\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast}
{i}\def\secondIndex{j}{\secondIndex}$.
Denote this coset by $\cosetTor[{\element[1]}]{RElement}{\element[2]}$.
\end{ThmS}
\begin{proof}
Suppose given two weak splittings,
$\splitPair[1]_{i}\def\secondIndex{j} =
\{\freeCyclesMap[1]_{i}\def\secondIndex{j}, \freeBoundariesMap[1]_{i}\def\secondIndex{j}\}$ and
$\splitPairA[1]_{i}\def\secondIndex{j} =
\{ \freeCyclesMapA[1]_{i}\def\secondIndex{j}, \freeBoundariesMapA[1]_{i}\def\secondIndex{j}\}$.
Then $\freeCyclesMapA[1]_{i}\def\secondIndex{j} - \freeCyclesMap[1]_{i}\def\secondIndex{j}
\colon\freeCycles[1]_{i}\def\secondIndex{j}
\to \complexCycles[1]_{i}\def\secondIndex{j} \to H_{i}\def\secondIndex{j}(\complex[1]_\ast)$ is trivial so
$\freeCyclesMapA[1]_{i}\def\secondIndex{j} - \freeCyclesMap[1]_{i}\def\secondIndex{j}
\colon\freeCycles[1]_{i}\def\secondIndex{j}
\to \complexBoundaries[1]_{i}\def\secondIndex{j}$.
Since $\freeCycles[1]_{i}\def\secondIndex{j}$ is free, there exists a lift
$\Psi_{i}\def\secondIndex{j}\colon
\freeCycles[1]_{i}\def\secondIndex{j} \to \complex[1]_{i}\def\secondIndex{j+1}$.
Next consider $\xyLine{
\boundary[1]_{i}\def\secondIndex{j+1}\bigl(\Psi_{i}\def\secondIndex{j} -
(\freeBoundariesMapA[1]_{i}\def\secondIndex{j} - \freeBoundariesMap[1]_{i}\def\secondIndex{j})
\bigr)
\colon \freeCycles[1]_{i}\def\secondIndex{j} \to \complex[1]_{i}\def\secondIndex{j+1}
\ar[r]^-{\boundary[1]_{i}\def\secondIndex{j+1}}& \complex[1]_{i}\def\secondIndex{j}}$.
This map is also trivial so there is a unique map
$\freeCycles[1]_{i}\def\secondIndex{j} \to \complexCycles[1]_{i}\def\secondIndex{j+1}$
and hence a unique map
$\Phi_{i}\def\secondIndex{j}\colon \freeCycles[1]_{i}\def\secondIndex{j} \to
\complexCycles[1]_{i}\def\secondIndex{j+1} \to
H_{i}\def\secondIndex{j+1}(\complex[1]_\ast)$.
Then
$\cs{homology splitting}[{\splitPairA[1]_\ast}]
{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl(
\cs{elementary tor}{\element[1]}{RElement}{\element[2]}
\bigr)
-
\cs{homology splitting}[{\splitPair[1]_\ast}]
{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl(
\cs{elementary tor}{\element[1]}{RElement}{\element[2]}
\bigr)
= (-1)^{i}\def\secondIndex{j+1}
\homologyClassOf[big]{\Phi_{i}\def\secondIndex{j}(\elementCycle[1])
\cs{cross product} \elementCycle[3]}
\in
H_{i}\def\secondIndex{j+1}(\complex[1]_\ast)\cs{cross product} \element[3]$.
A similar calculation shows the variation in the other variable lies in
$\element[1]\cs{cross product}H_{\secondIndex+1}(\complex[3]_\ast)$.
\end{proof}
\section{Splitting via Universal Coefficients}\sectionLabel{Bocksteins determine}
In the torsion free case,
\cs{env:torsion product cycle II 2} \cs{torsion product cycle II 2}
suggests another way to produce a splitting.
The Universal Coefficients formula says that for a torsion-free complex
$\complex[2]_\ast$, there exists a natural short exact sequence which
is unnaturally split:
\noindent\resizebox{\textwidth}{!}{{$\xymatrix@C18pt{
0\ar[r]&H_{\totalInt}\bigl(\complex[2]_\ast\bigr)
\tensor[R]\ry{RElement[4]}\ar[rr]&&
H_{\totalInt}\bigl(\complex[2]_\ast
\tensor[R]\ry{RElement[4]}\bigr)\ar[rr]^-{\universalCoefficientsMap{2}{4}{\totalInt}}&&
\rtorsion{RElement[4]}{H_{\totalInt-1}(\complex[2]_{\ast})}\ar[r]&0
}$}}
\noindent
where for a fixed $RElement$ in a PID $R$ and an $R$ module
$\@ifnextchar_{\LRT@P}{P}$, $\rtorsion{RElement}{\@ifnextchar_{\LRT@P}{P}} = P\tor[R]\ry{RElement}$
denotes the submodule
of elements annihilated by $RElement$.
\noindent
The Bockstein $\mathfrak b^{RElement[4]}_{\totalInt}$ is the composition
{\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{-10pt}
\begin{equation*}
\xymatrix@C18pt{
H_{\totalInt}\bigl(\complex[2]_\ast
\tensor[R]\ry{RElement[4]}\bigr)\ar[rr]^-{\universalCoefficientsMap{2}{4}{\totalInt}}&&
\rtorsion{RElement[4]}{H_{\totalInt-1}(\complex[2]_{\ast})} \subset H_{\totalInt-1}(\complex[2]_{\ast})
}
\end{equation*}
}
\begin{ThmS}[Bocksteins in correct coset]{Theorem}
Let $\complex[1]_\ast$ and $\complex[3]_\ast$ be torsion-free complexes.
Given $\element[1]\in H_{i}\def\secondIndex{j}(\complex[1]_\ast)$
pick $\elementR[1]\in
H_{i}\def\secondIndex{j+1}\bigl(\complex[1]_\ast\tensor[R]\ry{RElement}\bigr)$ such that
$\universalCoefficientsMap{1}{0}{i}\def\secondIndex{j+1}(\elementR[1]) = \element[1]$.
Given $\element[3]\in H_{\secondIndex}(\complex[3]_\ast)$
pick $\elementR[3]\in
H_{\secondIndex+1}\bigl(\complex[3]_\ast\tensor[R]\ry{RElement}\bigr)$ such that
$\universalCoefficientsMap{3}{0}{\secondIndex+1}(\elementR[3]) = \element[3]$.
On elementary tors $\cs{elementary tor}
{\element[1]}{RElement}{\element[3]}$
define
\begin{equation*}
\splitByCrossProductOfBocksteins_{i}\def\secondIndex{j,\secondIndex}\bigl(\cs{elementary tor}
{\element[1]}{RElement}{\element[3]}\bigr) =
(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\elementR[1] \tensor \elementR[3]
\bigr)
\end{equation*}
Then
$\splitByCrossProductOfBocksteins_{i}\def\secondIndex{j,\secondIndex}\bigl(\cs{elementary tor}
{\element[1]}{RElement}{\element[3]}\bigr) \in\cosetTor[{\element[1]}]{RElement}{\element[2]}$.
\end{ThmS}
\begin{proof}
From \namedRef{weak splitting map in correct coset},
$(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}(\elementR[1]\tensor
\elementR[3])$ lies in $\cosetTor[{\element[1]}]{RElement}{\element[2]}$
if the splittings used are ones from a weak splitting.
Any other choice of splitting for $\complex[1]_\ast$ is of the form
$\elementR[1] + X_{\element[1]}$ for $X_{\element[1]}\in
H_{i}\def\secondIndex{j+1}(\complex[1]_\ast)$
and
any other choice of splitting for $\complex[3]_\ast$ is of the form
$\elementR[3] + X_{\element[3]}$ for $X_{\element[3]}\in
H_{\secondIndex+1}(\complex[3]_\ast)$.
Then
\begin{equation*}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
(\elementR[1]+ X_{\element[1]})\tensor(
\elementR[3]+X_{\element[3]})\bigr) =
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}(\elementR[1]\tensor
\elementR[3]) + X_{\element[1]}\cs{cross product} \element[3]
+ (-1)^{i}\def\secondIndex{j+1}
\element[1]\cs{cross product} X_{\element[3]}
\end{equation*}
The result follows.
\end{proof}
If the Universal Coefficients splittings are chosen arbitrarily the map on the
elementary tors may not descend to a map on the torsion product.
This problem is overcome as follows.
A family of splittings
\begin{equation*}
\splitBocksteinHomology^{\complex[1],RElement}_{\totalInt}\colon
\rtorsion{RElement}{ H_{\totalInt}(\complex[1]_{\ast})}
\to H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{RElement}\bigr)\\
\end{equation*}
one for each non-zero $RElement\inR$ is \emph{a compatible family
of splittings of $\complex[1]_\ast$ at $\totalInt$} provided,
for all non-zero elements $RElement[4]_1$, $RElement[4]_2\inR$
the diagram
\noindent\resizebox{\textwidth}{!}{{$\xymatrix{
\rtorsion{RElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)}
\ar[r]^-{\subset}\ar[d]^-{\splitBocksteinHomology^{\complex[1],RElement[4]_2}_\totalInt}&
\rtorsion{RElement[4]_1\@ifnextchar_{\LRT@P}{P}DotRElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)}
\ar[r]^-{\@ifnextchar_{\LRT@P}{P}Dot[1]RElement[4]_2}\ar[d]^-{\splitBocksteinHomology^{\complex[1],RElement[4]_1\@ifnextchar_{\LRT@P}{P}DotRElement[4]_2}_\totalInt}&
\rtorsion{RElement[4]_1}{H_{\totalInt}(\complex[1]_\ast)}
\ar[d]^-{\splitBocksteinHomology^{\complex[1],RElement[4]_1}_\totalInt}\\
H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{RElement[4]_2}\bigr)
\ar[r]^-{RElement[4]_1\@ifnextchar_{\LRT@P}{P}Dot[1]}
\ar[d]^-{\universalCoefficientsMapA{1}{RElement[4]_2}{\totalInt+1}}&
H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R]
\ry{RElement[4]_1\@ifnextchar_{\LRT@P}{P}DotRElement[4]_2}\bigr)
\ar[r]^-{\rho^{RElement[4]_1}}
\ar[d]^-{\universalCoefficientsMapA{1}{RElement[4]_1\@ifnextchar_{\LRT@P}{P}Dot[1]RElement[4]_2}{\totalInt+1}}&
H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{RElement[4]_1}\bigr)
\ar[d]^-{\universalCoefficientsMapA{1}{RElement[4]_1}{\totalInt+1}}\\
\rtorsion{RElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)}
\ar[r]^-{\subset}&
\rtorsion{RElement[4]_1\@ifnextchar_{\LRT@P}{P}DotRElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)}
\ar[r]^-{\@ifnextchar_{\LRT@P}{P}Dot[1]RElement[4]_2}&
\rtorsion{RElement[4]_1}{H_{\totalInt}(\complex[1]_\ast)}
\\
}$}}
\noindent
commutes,
where the horizontal maps are induced from the short exact sequence of modules
$\xyLine[@C30pt]{0\to\ry{RElement[4]_2}\ar[r]^-{RElement[4]_1\@ifnextchar_{\LRT@P}{P}Dot[1]}&
\ry{RElement[4]_1\@ifnextchar_{\LRT@P}{P}DotRElement[4]_2}\ar[r]^-{\rho^{RElement[4]_1}}&
\ry{RElement[4]_1}} \to 0$ and the rows are exact.
The diagram consisting of the bottom two rows always commutes and the
vertical maps from the first row to the third are the identity.
If the splittings come from a weak splitting of $\complex[1]_\ast$ then they
are compatible for any $\totalInt$.
\begin{ThmS}{Theorem}
Suppose $\complex[1]_\ast$ and $\complex[3]_\ast$ are torsion-free.
Given a compatible family of splittings of $\complex[1]_\ast$ at $i}\def\secondIndex{j$
and a compatible family of splittings of $\complex[3]_\ast$ at $\secondIndex$,
the formula
\begin{equation*}
\splitByCrossProductOfBocksteinsA{\elementRr[1]{i}\def\secondIndex{j}}{\elementRr[3]{\secondIndex}}_{i}\def\secondIndex{j,\secondIndex}\bigl(\cs{elementary tor}
{\element[1]}{RElement}{\element[3]}\bigr) =
(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\elementRr[1]{i}\def\secondIndex{j} \cs{cross product} \elementRr[3]{\secondIndex}
\bigr)
\end{equation*}
defines a map from $H_{i}\def\secondIndex{j}(\complex[1]_\ast)
\tor[R] H_{{\secondIndex}}(\complex[3]_\ast)$ to
$H_{{i}\def\secondIndex{j}+{\secondIndex}+1}
(\complex[1]_\ast\tensor[R]\complex[3]_\ast)$ splitting the
K\"unneth\ formula at $({i}\def\secondIndex{j},{\secondIndex})$.
\end{ThmS}
\begin{proof}
It follows from \namedRef{Bocksteins in correct coset}
that if $\splitByCrossProductOfBocksteins_{i}\def\secondIndex{j,\secondIndex}$
is a map then it splits the K\"unneth\ formula at
$(i}\def\secondIndex{j,\secondIndex)$.
To show $\splitByCrossProductOfBocksteinsA{\elementRr[1]{i}\def\secondIndex{j}}
{\elementRr[3]{\secondIndex}}_{i}\def\secondIndex{j,\secondIndex}$
is a map, it suffices to show that
(\ref{free cycle gives map}.1-\ref{free cycle gives map}.4) hold.
Equations (\ref{free cycle gives map}.1) and
(\ref{free cycle gives map}.2) hold whether the splittings
are compatible or not since the cross product, and
hence $\splitByCrossProductOfBocksteinsA{\elementRr[1]{i}\def\secondIndex{j}}
{\elementRr[3]{\secondIndex}}_{i}\def\secondIndex{j,\secondIndex}$ is bilinear.
\newcommand{\elementT}[3][0]{\splitBocksteinHomology^{\complex[#1],#3}_{#2}(\element[#1])}
\newcommand{\elementS}[4][0]{\splitBocksteinHomology^{\complex[#1],#3}_{#2}(#4)}
To verify (\ref{free cycle gives map}.3) it suffices to show
\namedNumber{equal Bocksteins}
\begin{equation*}\tag{\ref{equal Bocksteins}}
\mathfrak b^{RElement_1
\@ifnextchar_{\LRT@P}{P}Dot RElement_2}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\elementT[1]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
\cs{cross product}
\elementT[3]{\secondIndex}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
\bigr)
=
\mathfrak b^{RElement_2}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\elementS[1]{i}\def\secondIndex{j}{RElement_2}{\element[1]\@ifnextchar_{\LRT@P}{P}Dot RElement_1}
\cs{cross product} \elementT[3]{\secondIndex}{RElement_2}
\bigr)
\end{equation*}
To compute a Bockstein of a homology class, $\element[4]\in H_{\totalInt}\bigl(
\complex[2]_\ast\tensor[R]\ry{RElement[4]}\bigr)$, first lift to a chain,
$\hat{\element[4]}\in \complex[2]_{\totalInt}$ and then
$\boundary[2]_{\totalInt}(\hat{\element[4]}) = RElement[4] \element[100]$.
The class $\element[100]$ is unique because $\complex[2]_{\totalInt}$ is
torsion-free and $\mathfrak b^{RElement[4]}_{\totalInt}(\element[4])$
is the homology class represented by $\element[100]$.
\newcommand{\elementTC}[3][0]{\splitBocksteinChains^{\complex[#1],#3}_{#2}(\element[#1])}
\newcommand{\elementSC}[4][0]{\splitBocksteinChains^{\complex[#1],#3}_{#2}(#4)}
There are four homology classes in (\ref{equal Bocksteins}).
For uniform notation, given
$\elementS[2]{\totalInt}{RElement[4]}{\element[4]}$, let
$\elementSC[2]{\totalInt}{RElement[4]}{\element[4]}$ be a lift to a representing
chain.
The cross product of homology classes is represented by the tensor product of
chains so
$C_1 =
\elementTC[1]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
\tensor
\elementTC[3]{\secondIndex}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
$ is a chain to compute the left hand side of (\ref{equal Bocksteins})
and
$C_2 =
\elementSC[1]{i}\def\secondIndex{j}{RElement_2}{\element[1]\@ifnextchar_{\LRT@P}{P}Dot RElement_1}
\tensor \elementTC[3]{\secondIndex}{RElement_2}
$
is a chain to compute the right hand side of (\ref{equal Bocksteins}).
Note $\homologyClassOf[Big]{\boundary[1]_{i}\def\secondIndex{j}\bigl(
\elementTC[1]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
\bigr)} =
\elementT[1]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
(RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2)
$
and
$\homologyClassOf[Big]{\boundary[1]_{i}\def\secondIndex{j}\bigl(
\elementSC[1]{i}\def\secondIndex{j}{RElement_2}{\element[1]\@ifnextchar_{\LRT@P}{P}DotRElement_1}
\bigr)} =
\elementS[1]{i}\def\secondIndex{j}{RElement_2}{\element[1]\@ifnextchar_{\LRT@P}{P}DotRElement_1}
(RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2)
$.
If the splittings are compatible,
$\elementT[1]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2} =
\elementS[1]{i}\def\secondIndex{j}{RElement_2}{\element[1]\@ifnextchar_{\LRT@P}{P}DotRElement_1}$
so choose
$\elementTC[1]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2} =
\elementSC[1]{i}\def\secondIndex{j}{RElement_2}{\element[1]\@ifnextchar_{\LRT@P}{P}DotRElement_1}$.
Also
$\homologyClassOf[Big]{\boundary[3]_{i}\def\secondIndex{j}\bigl(
\elementTC[3]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
\bigr)} =
(RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2)
\elementT[3]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2}
$
whereas
$\homologyClassOf[Big]{\boundary[3]_{i}\def\secondIndex{j}\bigl(
\elementSC[3]{i}\def\secondIndex{j}{RElement_2}{\element[3]}
\bigr)} =
RElement_2\elementS[3]{i}\def\secondIndex{j}{RElement_2}{\element[3]}
$.
If the splittings are compatible,
$\elementT[3]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2} =
\elementS[3]{i}\def\secondIndex{j}{RElement_2}{\element[3]}$
so choose
$\elementTC[3]{i}\def\secondIndex{j}{RElement_1\@ifnextchar_{\LRT@P}{P}Dot RElement_2} =
RElement_1
\elementSC[3]{i}\def\secondIndex{j}{RElement_2}{\element[3]}$.
It follows that $C_1 = RElement_1\@ifnextchar_{\LRT@P}{P}Dot C_2$.
Since
\begin{xyMatrix}[@C12pt]
0\ar[r]&R\ar[rr]^-{RElement_2}
\ar[d]_-{\identyMap{R}}
&&R\ar[rr]^-{\rho^{RElement_2}}
\ar[d]^-{RElement_1\@ifnextchar_{\LRT@P}{P}Dot[1]}
&&\ry{RElement_2}\ar[r]
\ar[d]^{RElement_1\@ifnextchar_{\LRT@P}{P}Dot[1]}
&0
\\
0\ar[r]&R\ar[rr]^-{RElement_1\@ifnextchar_{\LRT@P}{P}Dot[1]RElement_2}&&R\ar[rr]^-{\rho^{RElement_1\@ifnextchar_{\LRT@P}{P}Dot[1]RElement_2}}
&&\ry{RElement_1\@ifnextchar_{\LRT@P}{P}DotRElement_2}\ar[r]&0
\end{xyMatrix}
commutes,
$\mathfrak b^{RElement_1
\@ifnextchar_{\LRT@P}{P}Dot RElement_2}_{i}\def\secondIndex{j+\secondIndex+2}(C_1)
=
\mathfrak b^{RElement_2}_{i}\def\secondIndex{j+\secondIndex+2}(RElement_1 C_2)
$ as required.
\end{proof}
\section{Naturality of the splitting}\sectionLabel{Naturality}
\namedNumber{p0}
\namedNumber{p1}
\namedNumber{p2}
Fix a chain map $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$
between two weakly split chain maps.
Pick a map $\freeCyclesChainMap[1]_{\totalInt}\colon
\freeCycles[1]_{\totalInt} \to \freeCycles[2]_{\totalInt}$
satisfying
\begin{equation*}\tag{\ref{p0}}
\cyclesToHomology[2]_{\totalInt}\circ \freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} =
\chainMap[1]_{\totalInt}\circ
\cyclesToHomology[1]_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}\colon
\freeCycles[1]_{\totalInt} \to H_{\totalInt}(\complex[2]_\ast)
\end{equation*}
Since the right hand square in the diagram below commutes
\begin{xyMatrix}[@C1pt]
\freeBoundaries[1]_{\totalInt}\ \subset
\ar@<-12pt>@{.>}[d]^-{\freeBoundariesChainMap[1]_{\totalInt}}
&
\ar@<-2pt>[d]^-{\freeCyclesChainMap[1]_{\totalInt}}
\freeCycles[1]_{\totalInt}
\ar[rrrrrrr]^-{\cyclesToHomology[1]_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}}
&&&&&&&
\ar[d]^-{\chainMap[1]_{\totalInt}}
H_{\totalInt}(\complex[1]_\ast)
\ar[d]^-{\chainMap[1]_{\totalInt}}\\
\freeBoundaries[2]_{\totalInt}\ \subset
&
\freeCycles[2]_{\totalInt}
\ar[rrrrrrr]^-{\cyclesToHomology[2]_{\totalInt}\circ \freeCyclesMap[2]_{\totalInt}}
&&&&&&&
H_{\totalInt}(\complex[2]_\ast)
\\
\end{xyMatrix}
\noindent
there exists a unique map
$\freeBoundariesChainMap[1]_{\totalInt} \colon
\freeBoundaries[1]_{\totalInt} \to \freeBoundaries[2]_{\totalInt}$
making the left hand square commute.
The set of choices for $\freeCyclesChainMap[1]_{\totalInt}$ consists of any one
choice plus any map
$L_{\totalInt}\colon \freeCycles[1]_\ast\to \freeBoundaries[2]_{\totalInt}$.
The restricted map is $\freeBoundariesChainMap[1]_{\totalInt}$ plus the
restriction of $L_{\totalInt}$.
\vskip10pt
The maps
$
\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}$
and
$
\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}
$
have domain $\freeCycles[1]_{\totalInt}$ and range $\complexCycles[2]_{\totalInt}$
and they represent the same homology class.
Hence
$
\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}
-
\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}$
lands in $\complexBoundaries[2]_{\totalInt}$.
Since $\freeCycles[1]_{\totalInt}$ is free, there is a lift of this difference to a map
$\weakMap[1]_{\totalInt}\colon\freeCycles[1]_{\totalInt} \to \complex[2]_{\totalInt+1}$
satisfying
\begin{equation*}\tag{\ref{p1}}
\boundary[2]_{\totalInt+1}\circ \weakMap[1]_{\totalInt} =
\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}
-
\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}
\end{equation*}
If $\freeCyclesChainMap[1]_{\totalInt}$ is replaced by
$\freeCyclesChainMap[1]_{\totalInt} + L_{\totalInt}$,
a choice for the new $\weakMap[1]_{\totalInt}$ is
$\weakMap[1]_{\totalInt} + \freeBoundariesMap[2]\circ L_{\totalInt}$.
The set of solutions to (\ref{p1}) consists of one solution, $\weakMap[1]_{\totalInt}$,
plus any map of the form
$\Lambda_{\totalInt}\colon \freeCycles[1]_{\totalInt} \to
\complexCycles[2]_{\totalInt+1}\subset
\complex[2]_{\totalInt+1}$.
Given a fixed solution to (\ref{p1}) consider
\begin{equation*}
\xi = \weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -
\bigl(
\freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap_{\totalInt+1}\circ \freeBoundariesMap[1]_{\totalInt}
\bigr)\colon
\freeBoundaries[1]_{\totalInt}
\to
\complex[2]_{\totalInt+1}
\end{equation*}
\noindent
Notice if
$\freeCyclesChainMap[1]_{\totalInt}$ is replaced by
$\freeCyclesChainMap[1]_{\totalInt} + L_{\totalInt}$,
the new $\xi$ is the same map as the old $\xi$.
The image of $\xi$ is contained in the cycles of $\complex[2]_{\totalInt+1}$
and so gives a map
\begin{equation*}\tag{\ref{p2}}
\weakHomologyMap[1]_{\totalInt} =
\bigl(
\freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap_{\totalInt+1}\circ \freeBoundariesMap[1]_{\totalInt}
\bigr) -
\weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
\colon \freeBoundaries[1]_{\totalInt}
\to
H_{\totalInt+1}(\complex[2]_{\ast})
\end{equation*}
which does not depend on the choice of $\freeCyclesChainMap[1]_\ast$.
The map $\weakHomologyMap[1]_\ast$ induces a map
\begin{equation*}
\weakTorsionHomologyMap[1]<1>_{\totalInt}\colon
\rtorsion{RElement[1]}{H_{\totalInt}(\complex[1]_\ast) \to
H_{\totalInt+1}(\complex[2]_\ast)}\tensor \ry{r}
\end{equation*}
defined as follows.
Given $\element[1]\in {}\rtorsion{RElement[1]}{H_{\totalInt}(\complex[1]_\ast)}$
pick
$\elementCycle[1]\in \freeCycles[1]_{\totalInt}$
so that $\homologyClassOf{\freeCyclesMap[1]_{\totalInt}(\elementCycle[1])} = \element[1]$.
Then $\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1]
\in\freeBoundaries[1]_{\totalInt}$ so let
$\weakTorsionHomologyMap[1]<1>_{\totalInt}(\element[1])$ be the homology class
represented by
$\weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1])$
reduced mod $RElement[1]$.
\begin{ThmS}{Proposition}
Given a chain map $\chainMap[1]_\ast\colon \complex[1]_\ast\to\complex[2]_\ast$
between two weakly split chain complexes over a PID $R$,
the map
\begin{equation*}
\weakTorsionHomologyMap[1]<1>_{\totalInt}
\colon
\rtorsion{RElement[1]}{H_{\totalInt}(\complex[1]_\ast) \to
H_{\totalInt+1}(\complex[2]_\ast)}\tensor \ry{r}
\end{equation*}
is well-defined regardless of the choices made in (\ref{p0}) and (\ref{p1}).
\end{ThmS}
\begin{proof}
Any other choice of element in $\freeCycles[1]_{\totalInt}$ has the form
$\elementCycle[1] + b$
for $b\in \freeBoundaries[1]_{\totalInt}$.
Then
${
\weakHomologyMap[1]_{\totalInt}\Bigl(\bigl(\elementCycle[1] +
b\bigr)\@ifnextchar_{\LRT@P}{P}Dot RElement[1]\Bigr) =
\weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1] ) +
\weakHomologyMap[1]_{\totalInt}
\bigl(b\@ifnextchar_{\LRT@P}{P}Dot RElement[1] \bigr) =
\weakHomologyMap[1]_{\totalInt}(\elementCycle[1] \@ifnextchar_{\LRT@P}{P}Dot RElement[1] ) +
\weakHomologyMap[1]_{\totalInt}\bigl(b\bigr)
\@ifnextchar_{\LRT@P}{P}Dot RElement[1]
}$
since \penalty-1000 $b\in\freeBoundaries[1]_{\totalInt}$.
Hence $\weakHomologyMap[1]_{\totalInt}\Bigl(\bigl(\elementCycle[1] +
b\bigr)\@ifnextchar_{\LRT@P}{P}Dot RElement[1]\Bigr)$ and
$\weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1])$
represent the same element in
$H_{\totalInt+1}(\complex[2]_\ast)\tensor \ry{r}$
and therefore $\weakTorsionHomologyMap[1]<1>_{\totalInt}$ is well-define.
Since $\weakHomologyMap[1]_{\totalInt}$ is an $R$ module map, so is
$\weakTorsionHomologyMap[1]<1>_{\totalInt}$.
Given a second lift, it has the form $\weakMap[1]_{\totalInt} + \Lambda$
where $\Lambda\colon \freeCycles[1]_{\totalInt} \to \complexCycles[2]_{\totalInt+1}$
and the new $\weakHomologyMap$ is
$\weakHomologyMap[1]_{\totalInt} - \Lambda$.
Compute \begin{math}
\bigl(\weakHomologyMap[1]_{\totalInt} - \Lambda\bigr)
(\elementCycle[1] \@ifnextchar_{\LRT@P}{P}Dot RElement[1] ) =
\weakHomologyMap[1]_{\totalInt}(\elementCycle[1] \@ifnextchar_{\LRT@P}{P}Dot RElement[1] ) -
\Lambda(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1])
\end{math}
But $\Lambda$ is defined on all of $\freeCycles[1]_{\totalInt}$ so
\begin{math}
\bigl(\weakHomologyMap[1]_{\totalInt} - \Lambda\bigr)
(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1] ) =
\weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\@ifnextchar_{\LRT@P}{P}Dot RElement[1]) -
\Lambda(\elementCycle[1])\@ifnextchar_{\LRT@P}{P}Dot RElement[1]
\end{math}
and $\weakTorsionHomologyMap[1]<1>_{\totalInt}$ is independent of the lift.
\end{proof}
\begin{DefS*}{Remark}
A similar result holds for left $R$ modules.
\end{DefS*}
\begin{DefS}[weak split chain map definition]{Definition}
A \emph{weak split chain map} between two weakly split chain complexes
$\{\complex[1]_\ast, \splitPair[1]_\ast\}$ and
$\{\complex[2]_\ast, \splitPair[2]_\ast\}$ consists of a
chain map $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$,
a map $\freeCyclesChainMap[1]_\ast\colon
\freeCycles[1]_\ast \to \freeCycles[2]_\ast$ satisfying (\ref{p0})
and a map
$\weakMap[1]_{\totalInt}\colon\freeCycles[1]_{\totalInt} \to \complex[2]_{\totalInt+1}$
satisfying (\ref{p1}).
From the above discussion, given any two weakly split chain complexes
and a chain map between them, this data can be completed to a weakly
split chain map.
The map $\weakTorsionHomologyMap[1]<1>_{\ast}$ is independent of this
completion.
\end{DefS}
\begin{ThmS}[deviation from naturality in Kunneth formula]{Theorem}
Suppose given four weakly split complexes and weakly split chain maps
$\chainMap[3]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$ and
$\chainMap[2]_\ast\colon \complex[3]_\ast \to \complex[4]_\ast$.
If $\cs{elementary tor}{\element[1]}{RElement}{\element[2]}\in
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast)$ then
\begin{align*}
\cs{homology splitting}[{\splitPair[2]_\ast}]{\splitPair[4]_\ast}_{i}\def\secondIndex{j,\secondIndex}
\bigl(\cs{elementary tor}{\chainMap[1](\element[1])}{RElement}{\chainMap[2](\element[2])}
\bigr)
&=\
\bigl(\chainMap[1]\tensor\chainMap[2]\bigr)_\ast\bigl(
\cs{homology splitting}[{\splitPair[1]_\ast}]{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}
(\cs{elementary tor}{\element[1]}{RElement}{\element[2]})
\bigr) +\\
\noalign{\vskip 10pt}&\hskip-40pt
(-1)^{i}\def\secondIndex{j} \chainMap[1](\element[1])\cs{cross product}
\weakTorsionHomologyMap[2]<1>_{\secondIndex}(\element[2])
+
\weakTorsionHomologyMap[1]<1>_{i}\def\secondIndex{j}(\element[1])\cs{cross product}
\chainMap[2](\element[2])
\end{align*}
\end{ThmS}
\begin{DefS}{Remark}
The $\weakTorsionHomologyMap$ maps take values in
$H_\ast(\,\_\,)\tensor \ry{RElement[1]}$ but since the other factor in the
cross product is $RElement[1]$-torsion, each cross product is well-defined in
$H_{i}\def\secondIndex{j+\secondIndex+1}(\complex[2]_\ast\tensor[R] \complex[4]_\ast)$.
\end{DefS}
\begin{proof}
It suffices to check the formula on elementary tors so fix
$\cs{elementary tor}{\element[1]}{RElement}{\element[3]}$.
The corresponding cycle \ref{torsion product cycle II} is
\begin{equation*}
X_0 = (-1)^{\abs{\element[1]}+1}
\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])
\tensor
\freeBoundariesMap[3]_{\secondIndex}\bigl(RElement \elementCycle[3]\bigr) +
\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr) \tensor
\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])
\end{equation*}
Evaluating $\chainMap[1]\otimes\chainMap[2]$ on $X_0$ gives
\begin{equation*}
X_1=(-1)^{{i}\def\secondIndex{j}+1}
\chainMap[1]_{i}\def\secondIndex{j}\bigl(\freeCyclesMap[1]_{{i}\def\secondIndex{j}}(\elementCycle[1])\bigr)
\tensor
\chainMap[2]_{\secondIndex+1}
\Bigl(\freeBoundariesMap[3]_{\secondIndex}
\bigl(RElement \elementCycle[3]\bigr)\Bigr) +
\chainMap[1]_{i}\def\secondIndex{j+1}\Bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr)\Bigr) \tensor
\chainMap[2]_{\secondIndex}
\bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\bigr)
\end{equation*}
and a chain representing
$\cs{homology splitting}[{\splitPair[2]_\ast}]
{\splitPair[4]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl(
\cs{elementary tor}{\chainMap[1](\element[1])}
{RElement}{\chainMap[2](\element[2])}
\bigr)
$ is
\noindent\resizebox{\textwidth}{!}{{
$X_2 = (-1)^{i}\def\secondIndex{j+1}
\freeCyclesMap[2]_{i}\def\secondIndex{j}\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)
\tensor
\Bigl(\freeBoundariesMap[4]_{\secondIndex}
\bigl(RElement \freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr)\Bigr) +
\Bigl(\freeBoundariesMap[2]_{\secondIndex}
\bigl(\freeCyclesChainMap[1]_{\secondIndex}(\elementCycle[1]) RElement\bigr)
\Bigr)
\tensor
\freeCyclesMap[4]_{\secondIndex}
\bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr)
$}}
It suffices to prove the theorem for $\chainMap[1]_\ast\tensor
\identyMap{\complex[3]_\ast}$
and then for $\identyMap{\complex[2]_\ast} \tensor\chainMap[2]_\ast$
and these calculations are straightforward.
\end{proof}
\begin{math check}
\vskip10pt
It suffices to prove the theorem for $\chainMap[1]_\ast\tensor
\identyMap{\complex[3]_\ast}$
and then for $\identyMap{\complex[2]_\ast} \tensor\chainMap[2]_\ast$.
Here is the proof for $\chainMap[1]_\ast\tensor \identyMap{\complex[3]_\ast}$.
In this special case, $X_1$ and $X_2$ become
\begin{align*}
Y_1=&
(-1)^{i}\def\secondIndex{j+1}
\chainMap[1]_{i}\def\secondIndex{j}\bigl(\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)
\tensor
\freeBoundariesMap[3]_{\secondIndex}\bigl(RElement \elementCycle[3]\bigr) +
\chainMap[1]_{i}\def\secondIndex{j+1}\Bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr)\Bigr) \tensor
\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\\
Y_2=&
(-1)^{i}\def\secondIndex{j+1}
\freeCyclesMap[2]_{i}\def\secondIndex{j}\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)
\tensor
\freeBoundariesMap[3]_{\secondIndex}\bigl(RElement \elementCycle[3]\bigr) +
\freeBoundariesMap[2]_{i}\def\secondIndex{j}
\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) RElement\bigr) \tensor
\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])
\end{align*}
By (\ref{p1})
\begin{equation*}
\freeCyclesMap[2]_{i}\def\secondIndex{j}\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) =
\chainMap[1]_{i}\def\secondIndex{j}\bigl(\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) +
\boundary[2]_{i}\def\secondIndex{j+1}\bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)
\end{equation*}
By (\ref{p2})
\begin{equation*}
\freeBoundariesMap[2]_{i}\def\secondIndex{j}
\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) RElement\bigr) =
\freeBoundariesMap[2]_{i}\def\secondIndex{j}
\bigl(\freeBoundariesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] RElement)\bigr) =
\chainMap[1]_{i}\def\secondIndex{j+1}\Bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr)\Bigr) +
\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] RElement)
+
\weakHomologyMap[1]_{i}\def\secondIndex{j} (\elementCycle[1]\@ifnextchar_{\LRT@P}{P}DotRElement[1])
\end{equation*}
Hence
\alignLine{
Y_2 - Y_1 =& (-1)^{i}\def\secondIndex{j+1}
\Bigl(\boundary[2]_{i}\def\secondIndex{j+1}
\bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)\Bigr)\tensor
\Bigl(\freeBoundariesMap[3]_{\secondIndex}
\bigl(RElement \elementCycle[3]\bigr)\Bigr) +
\Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] RElement) +
\weakHomologyMap[1]_{i}\def\secondIndex{j} (\elementCycle[1]\@ifnextchar_{\LRT@P}{P}DotRElement[1])
\Bigr)\tensor
\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3]) = \\&
(-1)^{i}\def\secondIndex{j+1}\boundary[5]_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor
\freeBoundariesMap[3]_{\secondIndex}\bigl(RElement \elementCycle[3]\bigr)
\bigr) +
\weakTorsionHomologyMap[1]<1>_{i}\def\secondIndex{j}(\element[1])\tensor
\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])
}
since
\begin{align*}
\boundary[5]_{i}\def\secondIndex{j+\secondIndex+2}&
\Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor
\freeBoundariesMap[3]_{\secondIndex}
\bigl(RElement \elementCycle[3]\bigr)\Bigr) =\\&
\Bigl(\boundary[2]_{i}\def\secondIndex{j+1}
\bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)\Bigr)\tensor
\Bigl(\freeBoundariesMap[3]_{\secondIndex}
\bigl(RElement \elementCycle[3]\bigr)\Bigr)+
(-1)^{i}\def\secondIndex{j+1}
\Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] )\Bigr)\tensor
\Bigl(RElement\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\Bigr)=\\&
\Bigl(\boundary[2]_{i}\def\secondIndex{j+1}
\bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)\Bigr)\tensor
\Bigl(\freeBoundariesMap[3]_{\secondIndex}
\bigl(RElement \elementCycle[3]\bigr)\Bigr)+
(-1)^{i}\def\secondIndex{j+1}
\Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] RElement)\Bigr)\tensor
\Bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\Bigr)
\end{align*}
\vskip10pt
For the other case $X_1$ and $X_2$ become
\begin{align*}
Y_1=&(-1)^{i}\def\secondIndex{j+1}
\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])
\tensor
\chainMap[2]_{\secondIndex+1}
\Bigl(\freeBoundariesMap[3]_{\secondIndex}
\bigl(RElement \elementCycle[3]\bigr)\Bigr) +
\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr) \tensor
\chainMap[2]_{\secondIndex}\bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\bigr)\\
Y_2=&(-1)^{i}\def\secondIndex{j+1}
\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])
\tensor
\freeBoundariesMap[4]_{\secondIndex}
\bigl(RElement \freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr) +
\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr) \tensor
\freeCyclesMap[4]_{\secondIndex}
\bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr)
\end{align*}
By (\ref{p1})
\begin{equation*}
\freeCyclesMap[4]_{\secondIndex}
\bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr) =
\chainMap[2]_{\secondIndex}
\bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\bigr) +
\boundary[4]_{\secondIndex+1}
\bigl( \weakMap[2]_{\secondIndex}(\elementCycle[3])\bigr)
\end{equation*}
By (\ref{p2})
\begin{equation*}
\freeBoundariesMap[4]_{\secondIndex}
\bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3]) RElement\bigr) =
\freeBoundariesMap[4]_{\secondIndex}
\bigl(\freeBoundariesChainMap[2]_{\secondIndex}(\elementCycle[3] RElement)\bigr) =
\chainMap[2]_{\secondIndex+1}\Bigl(\freeBoundariesMap[3]_{\secondIndex}
\bigl(\elementCycle[3] RElement\bigr)\Bigr) +
\weakMap[2]_{\secondIndex}(\elementCycle[3] RElement)
-
\weakHomologyMap[2]_{\secondIndex}
(\elementCycle[3]\@ifnextchar_{\LRT@P}{P}DotRElement[1])
\end{equation*}
Hence
\begin{align*}
Y_2-Y_1=& (-1)^{i}\def\secondIndex{j+1}
\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor
\bigl(
\weakMap[2]_{\secondIndex}(\elementCycle[3] RElement)
+
\weakHomologyMap[2]_{\secondIndex}
(\elementCycle[3]\@ifnextchar_{\LRT@P}{P}DotRElement[1])
\bigr) +
\freeBoundariesMap[1]_{i}\def\secondIndex{j}
\bigl(\elementCycle[1] RElement\bigr) \tensor
\boundary[4]_{\secondIndex+1}
\bigl( \weakMap[2]_{\secondIndex}(\elementCycle[3])\bigr) =\\&
\boundary[6]_{i}\def\secondIndex{j+\secondIndex+2}\bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j}
(\elementCycle[1] RElement) \tensor
\weakMap[2]_{\secondIndex}(\elementCycle[3] RElement)\bigr)
+(-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor
\weakHomologyMap[2]_{\secondIndex} (\elementCycle[3]\@ifnextchar_{\LRT@P}{P}DotRElement[1]
\end{align*}
\end{math check}
\begin{ThmS}[naturality of cosets]{Corollary}
Given chain maps
$\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$ and
$\chainMap[2]_\ast\colon \complex[3]_\ast \to \complex[4]_\ast$
\begin{equation*}
\bigl(\chainMap[1]_\ast\tensor \chainMap[2]_\ast\bigr)_\ast\bigl(
\cosetTor[{\element[1]}]{RElement}{\element[2]}
\bigr)\subset
\cosetTor[{\chainMap[1]_\ast(\element[1])}]{RElement}
{\chainMap[2]_\ast(\element[2])}
\end{equation*}
In words, the cosets are natural and do not depend on the
weak splittings of the complexes.
\end{ThmS}
\begin{proof}
First check that the $0$-cosets behave correctly:
\noindent\mathLine{
\bigl(\chainMap[1]_\ast\tensor \chainMap[2]_\ast\bigr)_\ast\Bigl(
\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast}
{i}\def\secondIndex{j}{\secondIndex}\Bigr)
\subset
\fundamentalCoset{\chainMap[1]_\ast(\element[1])}
{\chainMap[2]_\ast(\element[2])}
{\complex[2]_\ast}{\complex[4]_\ast}
{i}\def\secondIndex{j}{\secondIndex}}
By \namedRef{deviation from naturality in Kunneth formula}
$\cs{homology splitting}[{\splitPair[2]_\ast}]{\splitPair[4]_\ast}_{i}\def\secondIndex{j,\secondIndex}
\bigl(\cs{elementary tor}{\chainMap[1](\element[1])}{RElement}
{\chainMap[2](\element[2])}
\bigr)
\subset
\cosetTor[{\chainMap[1]_\ast(\element[1])}]{RElement}
{\chainMap[2]_\ast(\element[2])}$.
One application of \namedRef{deviation from naturality in Kunneth formula} is
to the case in which $\chainMap[1]_\ast$ is the identity but the weak splittings
change.
Hence changing the weak splittings does not change the cosets.
The result follows.
\end{proof}
\section{The interchange map and the K\"unneth\ formula}
There are natural isomorphisms
$I\colon \complex[1]\tensor[R] \complex[3]
\cong
\complex[3]\tensor[R] \complex[1]$
and
$I\colon \complex[1]\tor[R] \complex[3]
\cong
\complex[3]\tor[R] \complex[1]$.
On elementary tensors, $I(\element[1]\tensor \element[2]) =
\element[2]\tensor \element[1]$ and
$I(\cs{elementary tor}
{\element[1]}{RElement}{\element[2]})=\cs{elementary tor}
{\element[2]}{RElement}{\element[1]}$.
Applying $I$ to the tensor product of two chain complexes
is not a chain map: a sign is required.
The usual choice is
\begin{equation*}
T\colon \complex[1]_\ast\tensor[R] \complex[3]_\ast \to
\complex[3]_\ast\tensor[R] \complex[1]_\ast
\end{equation*}
defined on elementary tensors by
$T(\element[1] \tensor \element[2]) =
(-1)^{\abs{\element[1]}\abs{\element[2]}}
\element[2] \tensor \element[1]$.
It follows that the cross product map satisfies
\begin{equation*}
T_\ast(\element[1] \cs{cross product} \element[2]) =
(-1)^{\abs{\element[1]}\abs{\element[2]}}
\element[2] \cs{cross product} \element[1]
\end{equation*}
for all $\element[1]\in H_{\abs{\element[1]}}(\complex[1]_\ast)$
and
$\element[2]\in H_{\abs{\element[2]}}(\complex[3]_\ast)$.
\begin{ThmS}[flip theorem I]{Theorem}
For all $\element[1]\in H_{i}\def\secondIndex{j}(\complex[1]_\ast)$
and
$\element[2]\in H_{\secondIndex}(\complex[3]_\ast)$
{\setlength\belowdisplayskip{-10pt}
\begin{equation*}
T_\ast\Bigl(\cs{homology splitting}[{\splitPair[1]}]
{\splitPair[3]}_{i}\def\secondIndex{j+\secondIndex+1}\bigr(
\cs{elementary tor}{\element[1]}{RElement}{\element[2]}\bigr)\Bigr) =
(-1)^{i}\def\secondIndex{j\cdot\secondIndex+1}
\cs{homology splitting}[{\splitPair[3]}]{\splitPair[1]}_{i}\def\secondIndex{j+\secondIndex+1}
\bigl(
\cs{elementary tor}{\element[2]}{RElement}{\element[1]}\bigr)
\end{equation*}
}
\end{ThmS}
\begin{proof}
Apply $T$ to the cycle in \cs{torsion product cycle II 1}.
\end{proof}
\begin{math check}
$\epsilon\,
\freeCyclesMap[1]_\ast(\elementCycle[1])
\tensor
\freeBoundariesMap[3]_\ast\bigl(RElement \elementCycle[3]\bigr) +
\freeBoundariesMap[1]_\ast
\bigl(\elementCycle[1] RElement\bigr) \tensor
\freeCyclesMap[3]_\ast(\elementCycle[3])
$.
\begin{align*}
T\Bigl(&
\epsilon\,
\freeCyclesMap[1]_\ast(\elementCycle[1])
\tensor
\freeBoundariesMap[3]_\ast\bigl(RElement \elementCycle[3]\bigr) +
\freeBoundariesMap[1]_\ast
\bigl(\elementCycle[1] RElement\bigr) \tensor
\freeCyclesMap[3]_\ast(\elementCycle[3])\Bigr) =
\\&
(-1)^{i}\def\secondIndex{j(\secondIndex+1)}\Bigl(
(-1)^{i}\def\secondIndex{j+1}
\freeBoundariesMap[3]_\ast\bigl(RElement \elementCycle[3]\bigr)
\tensor
\freeCyclesMap[1]_\ast(\elementCycle[1])
\Bigr)
+
(-1)^{(i}\def\secondIndex{j+1)\secondIndex}\Bigl(
\freeCyclesMap[3]_\ast(\elementCycle[3])
\tensor
\freeBoundariesMap[1]_\ast
\bigl(\elementCycle[1] RElement\bigr)
\Bigr)=\\&
(-1)^{i}\def\secondIndex{j\secondIndex+1}\Bigl(
\freeBoundariesMap[3]_\ast\bigl(RElement \elementCycle[3]\bigr)
\tensor
\freeCyclesMap[1]_\ast(\elementCycle[1])
+
(-1)^{i}\def\secondIndex{j\secondIndex+1}\Bigl(
(-1)^{\secondIndex+1}
\freeCyclesMap[3]_\ast(\elementCycle[3])
\tensor
\freeBoundariesMap[1]_\ast
\bigl(\elementCycle[1] RElement\bigr)
\Bigr)=\\&
(-1)^{i}\def\secondIndex{j\secondIndex+1}\Bigl(
(-1)^{\secondIndex+1}
\freeBoundariesMap[3]_\ast\bigl(RElement \elementCycle[3]\bigr)
\tensor
\freeCyclesMap[1]_\ast(\elementCycle[1])
+
\freeCyclesMap[3]_\ast(\elementCycle[3])
\tensor
\freeBoundariesMap[1]_\ast
\bigl(\elementCycle[1] RElement\bigr)
\Bigr)
\end{align*}
This cycle represents
$(-1)^{i}\def\secondIndex{j\cdot\secondIndex+1}
\cs{homology splitting}[{\splitPair[3]}]{\splitPair[1]}_{i}\def\secondIndex{j+\secondIndex+1}
\bigl(
\cs{elementary tor}{\element[2]}{RElement}{\element[1]}\bigr)
$.
\end{math check}
\begin{ThmS}[flip and Kunneth]{Corollary}
If $R$ is a PID and if $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic
\noindent\resizebox{\textwidth}{!}{{$\xymatrix{
0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tensor[R]
H_{\secondIndex}(\complex[3]_\ast)
\ar[r]^-{\cs{cross product}}
\ar[d]^-{\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}(-1)^{i}\def\secondIndex{j \secondIndex} I}
&
H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)
\ar[r]^-{\cs{to torsion product}}
\ar[d]^-{T_\ast}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R]
H_{\secondIndex}(\complex[3]_\ast)\to0
\ar[d]\ar[d]_-{\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
(-1)^{i}\def\secondIndex{j \secondIndex + 1} I}
\\
0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H_{i}\def\secondIndex{j}(\complex[3]_\ast)\tensor[R]
H_{\secondIndex}(\complex[1]_\ast)
\ar[r]^-{\cs{cross product}}&
H_{\totalInt}(\complex[3]_\ast\tensor[R] \complex[1]_\ast)
\ar[r]^-{\cs{to torsion product}}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1}
H_{i}\def\secondIndex{j}(\complex[3]_\ast)\tor[R]
H_{\secondIndex}(\complex[1]_\ast)\to0
}$}}
\noindent
commutes.
The splittings can be chosen to make the diagram commute.
\end{ThmS}
\section{The boundary map and the K\"unneth\ formula}
The boundary map in question is the map associated with the long
exact homology sequence for a short exact sequence of chain complexes.
Before stating the result some preliminaries are needed.
\begin{DefS}{Definition}
A pair of composable chain maps
$\xymatrix@1@C12pt{\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&&
\complex[3]_\ast}$ and
$\xymatrix@1@C12pt{\complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&&
\complex[2]_\ast}$
form \emph{a weak exact sequence} provided there exists a short exact sequence
of free approximations and chain maps making
(\ref{weak exact sequence diagram}) below commute.
\namedNumber{weak exact sequence diagram}
{\setlength\belowdisplayskip{-10pt}
\begin{equation*}\tag{\ref{weak exact sequence diagram}}
\xymatrix@C10pt{
0\ar[r]&\freeApproximation[1]_\ast\ar[rr]^-{\freeApproximationChainMap[1]_\ast}
\ar[d]^-{\vertMap{\complex[1]}_\ast}&&
\freeApproximation[3]_\ast\ar[rr]^-{\freeApproximationChainMap[2]_\ast}
\ar[d]^-{\vertMap{\complex[3]}_\ast}&&
\freeApproximation[2]_\ast
\ar[d]^-{\vertMap{\complex[2]}_\ast}
\ar[r]& 0\\
&
\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&&
\complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&&
\complex[2]_\ast}
\end{equation*}
}
\end{DefS}
Given a weak exact sequence there is a long exact homology sequence
coming from the long exact sequence of the top row of
(\ref{weak exact sequence diagram}):
\begin{equation*}
\xymatrix@C28pt{\cdots\to
H_{i}\def\secondIndex{j+1}(\complex[2]_\ast)\ar[r]^-{\boldsymbol{\partial}_{i}\def\secondIndex{j+1}}&
H_{i}\def\secondIndex{j}(\complex[1]_\ast)\ar[r]^-{\chainMap[1]_\ast}&
H_{i}\def\secondIndex{j}(\complex[3]_\ast)\ar[r]^{\chainMap[2]_\ast}&
H_{i}\def\secondIndex{j}(\complex[2]_\ast)\ar[r]^-{\boldsymbol{\partial}_{i}\def\secondIndex{j}}&\cdots
}\]
The boundary $\boldsymbol{\partial}_{i}\def\secondIndex{j+1} =
\vertMap{\complex[1]}_\ast \circ \partial_{i}\def\secondIndex{j+1}\circ
(\vertMap{\complex[2]}_\ast)^{-1}$ where
$\partial_{i}\def\secondIndex{j+1}$ is the usual boundary in the
long exact homology sequence for the free complexes.
\begin{ThmS}{Lemma}
A short exact sequence of chain complexes
\begin{equation*}
\xyLine[@C10pt]{
0\ar[r]&
\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&&
\complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&&
\complex[2]_\ast\
\ar[r]& 0}
\end{equation*}
is weak exact.
The boundary $\boldsymbol{\partial}_{i}\def\secondIndex{j+1}$ is the usual boundary
map.
\end{ThmS}
\begin{proof}
The commutative diagram of free approximations
(\ref{weak exact sequence diagram}) is given by
\namedRef{short exact free approximation}.
The description of the boundary map is immediate.
\end{proof}
\begin{ThmS}[weak exact is preserved by products]{Lemma}
If $\complex[1]_\ast\tor[R]\complex[4]_\ast$,
$\complex[3]_\ast\tor[R]\complex[4]_\ast$ and
$\complex[2]_\ast\tor[R]\complex[4]_\ast$ are acyclic
and if
$\xymatrix@1{
\complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}&
\complex[3]_\ast\ar[r]^-{\chainMap[2]_\ast}&
\complex[2]_\ast
}$ is weak exact, then so are
{\setlength\abovedisplayskip{0pt}
\setlength\belowdisplayskip{0pt}
\begin{equation*}
\xymatrix@C40pt@R10pt{
\complex[1]_\ast\tensor[R]\complex[4]_\ast
\ar[r]^-{\chainMap[1]_\ast \tensor \identyMap{\complex[4]_\ast}}&
\complex[3]_\ast\tensor[R]\complex[4]_\ast
\ar[r]^-{\chainMap[2]_\ast \tensor \identyMap{\complex[4]_\ast}}&
\complex[2]_\ast\tensor[R]\complex[4]_\ast
\\
\complex[4]_\ast\tensor[R]\complex[1]_\ast
\ar[r]^-{\identyMap{\complex[4]_\ast} \tensor \chainMap[1]_\ast}&
\complex[4]_\ast\tensor[R]\complex[3]_\ast
\ar[r]^-{\identyMap{\complex[4]_\ast}\tensor\chainMap[2]_\ast}&
\complex[4]_\ast\tensor[R]\complex[2]_\ast
}\end{equation*}}
\end{ThmS}
\begin{proof}
Pick free approximations
satisfying (\ref{weak exact sequence diagram}),
$\vertMap{\complex[1]}_\ast$, $\vertMap{\complex[3]}_\ast$,
$\vertMap{\complex[2]}_\ast$
and a free approximation $\vertMap{\complex[4]}_\ast$.
By \namedRef{Dold splitting} the required free approximations are
$\vertMap{\complex[1]}_{\ast} \tensor\vertMap{\complex[4]}_{\ast}$,
$\vertMap{\complex[3]}_{\ast} \tensor\vertMap{\complex[4]}_{\ast}$,
$\vertMap{\complex[2]}_{\ast} \tensor\vertMap{\complex[4]}_{\ast}$,
or
$\vertMap{\complex[4]}_{\ast} \tensor\vertMap{\complex[1]}_{\ast}$,
$\vertMap{\complex[4]}_{\ast} \tensor\vertMap{\complex[3]}_{\ast}$,
$\vertMap{\complex[4]}_{\ast} \tensor\vertMap{\complex[2]}_{\ast}$.
\end{proof}
\begin{DefS*}{Warning}
Even if $\xymatrix@1{
\complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}&
\complex[3]_\ast\ar[r]^-{\chainMap[2]_\ast}&
\, \complex[2]_\ast
}$ is short exact, the pair $\chainMap[1]_\ast\tensor \identyMap{\complex[4]_\ast}$ and
$\chainMap[2]_\ast\tensor \identyMap{\complex[4]_\ast}$
may only be weak exact.
For them to be short exact requires that either
$\complex[2]_\ast$ or $\complex[4]_\ast$
be torsion free.
\end{DefS*}
\begin{ThmS}[boundary of elementary tor]{Theorem}
Suppose $\complex[1]_\ast\tor[R]\complex[4]_\ast$,
$\complex[3]_\ast\tor[R]\complex[4]_\ast$ and
$\complex[2]_\ast\tor[R]\complex[4]_\ast$ are acyclic
and suppose
$\xymatrix@1{
\complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}&
\complex[3]_\ast\ar[r]^-{\chainMap[2]_\ast}&
\,\complex[2]_\ast
}$ is weak exact.
Then for $\element[1]\in H_{i}\def\secondIndex{j}(\complex[2]_\ast)$ and
$\element[2]\in H_{\secondIndex}(\complex[4]_\ast)$
{\setlength\belowdisplayskip{-10pt}
\begin{equation*}
\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}
\bigl(\cosetTor[{\element[1]}]{RElement}{\element[2]}\bigr)
\subset -
\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{RElement}{\element[2]}
\end{equation*}
}
\end{ThmS}\nointerlineskip
\begin{proof}
By \namedRef{weak exact is preserved by products}
it may be assumed that the complexes are all free.
Pick compatible splittings for $\complex[1]_\ast$, $\complex[2]_\ast$ and
$\complex[4]_\ast$.
Recall that Bocksteins and long exact sequence boundary maps anti-commute
and that in short exact sequences of free chain complexes
$\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex}(\elementCycle[1]\otimes\elementCycle[4])
=
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\elementCycle[1])\otimes\elementCycle[4]
$.
A routine calculation completes the proof.
\end{proof}
\begin{math check}
Then
\begin{equation*}
(-1)^{i}\def\secondIndex{j+1}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\bigr)
\in\cosetTor[{\element[1]}]{RElement}{\element[2]}
\end{equation*}
Since $\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1} \circ
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}
=
- \mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}\circ
\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+2}$
\begin{align*}
\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl(
(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(&
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\bigr)\Bigr)
=\\& (-1)^{i}\def\secondIndex{j}
\biggl(\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}\Bigr(
\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\bigr)\Bigr)\biggr)=\\&
(-1)^{i}\def\secondIndex{j}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}\Bigr(
\boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])\bigr)
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\Bigr)\ .
\end{align*}
On the other side
\begin{equation*}
(-1)^{i}\def\secondIndex{j-1+1}
\mathfrak b^{RElement}_{i}\def\secondIndex{j-1+\secondIndex+2}\Bigl(
\splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\Bigr)
\in\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{RElement}{\element[2]}
\end{equation*}
Both
$\splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)$
and
$\boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])\bigr)
$
are chains in $\complex[1]_{i}\def\secondIndex{j}$ which are cycles in
$\complex[1]_{i}\def\secondIndex{j}\tensor[R]\ry{RElement}$.
Applying Bocksteins shows
$\mathfrak b^{RElement}_{i}\def\secondIndex{j}\Bigl(
\splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)\Bigr)
=
RElement \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])$
and
$\mathfrak b^{RElement}_{i}\def\secondIndex{j}\Bigl(\boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])\bigr)
\Bigr)
= -\boldsymbol{\partial}_{i}\def\secondIndex{j}
\Bigl(\mathfrak b^{RElement}_{i}\def\secondIndex{j+1}
\bigl(\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])
\bigr)\Bigr) =
-\boldsymbol{\partial}_{i}\def\secondIndex{j}
(RElement \element[1]) = -RElement \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])
$.
Hence
$Z = \splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)
+
\boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])\bigr)
$
is a cycle.
Hence
\begin{align*}
&\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl(
(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\bigr)\Bigr)
=\\&
\hskip 40pt(-1)^{i}\def\secondIndex{j}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}\biggl(
\Bigl(Z -
\splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)\Bigr)
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\biggr) =\\&
(-1)^{i}\def\secondIndex{j+1}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}
\Bigl(\splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\Bigr)
+(-1)^{i}\def\secondIndex{j+1}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}\bigl(
Z \cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\bigr)=\\&
\hskip 40pt-(-1)^{i}\def\secondIndex{j}
\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+1}
\Bigl(\splitBocksteinHomology^{\complex[1],RElement}_{i}\def\secondIndex{j-1}\bigl(
\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\Bigr)
+(-1)^{i}\def\secondIndex{j+1+i}\def\secondIndex{j}
Z \cs{cross product} \element[2]
\end{align*}
and therefore
\begin{equation*}
\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl(
(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{RElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(
\splitBocksteinHomology^{\complex[2],RElement}_{i}\def\secondIndex{j}(\element[1])
\cs{cross product}
\splitBocksteinHomology^{\complex[4],RElement}_{\secondIndex}(\element[2])
\bigr)\Bigr)
\in -\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{RElement}{\element[2]}
\end{equation*}
Since one element of
$\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}
\bigl(\cosetTor[{\element[1]}]{RElement}{\element[2]}\bigr)
$ is in
$-\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{RElement}{\element[2]}$
and since
\mathLine{\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl(
\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[4]_\ast}
{i}\def\secondIndex{j}{\secondIndex}\Bigr)
\subset
\bigl(\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\cs{cross product}
H_{\secondIndex+1}(\complex[4]_\ast)\bigr) \displaystyle\mathop{\oplus}
\bigl(H_{i}\def\secondIndex{j}(\complex[1]_\ast)\cs{cross product}\element[2]\bigr)
}
the result follows.
\end{math check}
\begin{ThmS}{Corollary}
With assumptions and notation as in \namedRef{boundary of elementary tor}
{\setlength\belowdisplayskip{-10pt}
\begin{equation*}
\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}
\bigl(\cosetTor[{\element[2]}]{RElement}{\element[1]}\bigr)
\subset (-1)^{\secondIndex+1}
\cosetTor[{\element[2]}]{RElement}{{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}}
\end{equation*}}
\end{ThmS}\nointerlineskip
\begin{proof}
Apply the interchange map (\ref{flip theorem I}) to get to the situation of
\namedRef{boundary of elementary tor} and then apply the interchange map
again.
\end{proof}
\begin{ThmS}[boundary and Kunneth]{Corollary}
With assumptions and notation as in \namedRef{boundary of elementary tor}
let
$\boldsymbol{\partial}_{i}\def\secondIndex{j}\tor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}
\colon H_{i}\def\secondIndex{j}(\complex[2]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast)
\to
H_{i}\def\secondIndex{j-1}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast)
$ be the map defined by
$\boldsymbol{\partial}_{i}\def\secondIndex{j}\tor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}\bigl(
\cs{elementary tor}{\element[1]}{RElement}{\element[2]}) =
\cs{elementary tor}{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}{RElement}{\element[2]}
$.
Then
\noindent\resizebox{\textwidth}{!}{{$\xymatrix@R30pt{
0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1}
H_{i}\def\secondIndex{j}(\complex[2]_\ast)\tensor[R] H_{\secondIndex}(\complex[4]_\ast)
\ar[r]^-{\cs{cross product}}
\ar[d]^-{\hbox{\tiny{$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1}
\boldsymbol{\partial}_{i}\def\secondIndex{j}\tensor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}$}}}&
H_{\totalInt+1}(\complex[2]_\ast\tensor[R] \complex[4]_\ast)
\ar[r]^-{\cs{to torsion product}}
\ar[d]_-{\boldsymbol{\partial}_{\totalInt+1}}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H_{i}\def\secondIndex{j}(\complex[2]_\ast)\tor[R]
H_{\secondIndex}(\complex[4]_\ast)\to0
\ar[d]\ar[d]_-{\hbox{\tiny{$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} -\boldsymbol{\partial}_{i}\def\secondIndex{j}\tor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}$}}}
\\
0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1}
H_{i}\def\secondIndex{j-1}(\complex[1]_\ast)\tensor[R] H_{\secondIndex}(\complex[4]_\ast)
\ar[r]^-{\cs{cross product}}&
H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[4]_\ast)
\ar[r]^-{\cs{to torsion product}}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H_{i}\def\secondIndex{j-1}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast)\to0
}$}}
\noindent
commutes.
\end{ThmS}
\begin{proof}The proof is immediate.
\end{proof}
\section{The Massey triple product}
Suppose $X$ and $Y$ are CW complexes with finitely many cells in each
dimension.
Then the cellular cochains are free $\mathbb Z$ modules and the K\"unneth\ formula
plus the Eilenberg-Zilber chain homotopy equivalence yields a K\"unneth\ formula
\mathLine{
\xymatrix{
0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}
H^{i}\def\secondIndex{j}(X)\otimes H^{\secondIndex}(Y)
\ar[r]^-{\cs{cross product}}&
H^{\totalInt}(X\times Y)\ar[r]^-{\cs{to torsion product}}&
\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1}
H^{i}\def\secondIndex{j}(X)\tor H^{\secondIndex}(Y)\to0
}}
Given $u\in H^{i}\def\secondIndex{j}(X)$ define $\secondU{u}\in H^{i}\def\secondIndex{j}(X\times Y)$
by $\secondU{u}=p_X^\ast(u)$ where $p_X\colon X\times Y \to X$
is the projection.
For $v\in H^{\secondIndex}(Y)$ define $\secondU{v}\in H^{\secondIndex}(X\times Y)$
similarly and recall
$u\cs{cross product} v = \secondU{u} \cup \secondU{v}$ where $\cup$
denotes the cup product.
\begin{ThmS}{Theorem}
With notation as above and non-zero $m\in \mathbb Z$
\begin{equation*}
\cosetTor[u]{m}{v} =
\boldsymbol{\langle}
\secondU{u}, \secondU{(m)},\secondU{v}
\boldsymbol{\rangle}
\end{equation*}
where $\boldsymbol{\langle}
\secondU{u}, \secondU{(m)},\secondU{v}
\boldsymbol{\rangle}
$ is the Massey triple product of the indicated cohomology classes
where $\secondU{(m)}$ is $m$ times the
multiplicative identity in $H^0(X\times Y)$.
\end{ThmS}
The proof is immediate from \namedRef{Mac Lane cycle A}
and the definition of the Massey triple product.
\section{Weakly split chain complexes}
Heller's category in \cite{Heller} carries much the same information as
weak splittings.
\gdef\chainMap[1]+ \chainMap[2]{\chainMap[2]\circ\chainMap[1]}
\begin{ThmS}{Proposition}
If $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[3]_\ast$ and
$\chainMap[2]_\ast\colon \complex[3]_\ast \to \complex[2]_\ast$ are
weakly split chain maps, then
$\chainMap[2]\circ\chainMap[1]$ is weakly split by
$\freeCyclesChainMap[10]_{\totalInt}=
\freeCyclesChainMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}$
and
$\weakMap[10]_{\totalInt} = \chainMap[2]_{\totalInt+1}\circ \weakMap[1]_{\totalInt} +
\weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}$.
With these choices
\begin{equation*}
\weakTorsionHomologyMap[10]<1>_{\totalInt} =
(\chainMap[2]_{\totalInt+1}\tensor \identyMap{\ry{RElement}})\circ\weakTorsionHomologyMap[1]<1>_{\totalInt}
+
\weakTorsionHomologyMap[2]<1>_{\totalInt}\circ \chainMap[1]_{\totalInt}
\end{equation*}
\end{ThmS}
\begin{proof}
Formula (\ref{p0}) is immediate.
Formula (\ref{p1}) is a routine calculation.
It is straightforward to check
$\weakHomologyMap[10]_{\totalInt} =
\chainMap[2]_{\totalInt+1}\circ \weakHomologyMap[1]_{\totalInt} +
\weakHomologyMap[2]_{\totalInt} \circ
\freeBoundariesChainMap[1]_{\totalInt}
$ from which the formula for the $\weakTorsionHomologyMap$ follows.
\end{proof}
\begin{DefS}{Remark}
Composition can be checked to be associative.
\def\chainMap[1]+ \chainMap[2]{\identyMap{{\complex[1]_{ }}_{\ast}}}
The pair $\freeCyclesChainMap[10]_{\ast}=\chainMap[1]+ \chainMap[2]$ and
$\weakMap[10]_{\ast}=0$ give the identity
for any weak spitting of $\complex[1]_{\ast}$.
Hence weakly split chain complexes and weakly split chain maps form a category.
\end{DefS}
\begin{math check}
\begin{equation*}
\boundary[2]_{\totalInt+1}\circ \weakMap[10]_{\totalInt} =
\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}
-
\chainMap[2]_{\totalInt}\circ\chainMap[1]_{\totalInt}
\circ \freeCyclesMap[1]_{\totalInt}
\end{equation*}
\begin{align*}
\boundary[2]_{\totalInt+1}\bigl(&\chainMap[4]_{\totalInt+1}\circ
\weakMap[1]_{\totalInt}+
\weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}\bigr) =
\chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) +
\boundary[2]_{\totalInt+1}\bigl(\weakMap[2]_{\totalInt}
\circ \freeCyclesChainMap[1]_{\totalInt}\bigr) =\\&
\chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) +
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[2]_{\totalInt}
-
\chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ
\freeCyclesChainMap[1]_{\totalInt} =\\&
\chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) +
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} \bigr)
-
\bigl(\chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ
\freeCyclesChainMap[1]_{\totalInt} =\\&
\chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) +
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)
-
\bigl(\chainMap[4]_{\totalInt}\circ
\freeCyclesMap[3]_{\totalInt}\bigr)\circ \freeCyclesChainMap[1]_{\totalInt}
=\\&
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)+
\chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr)
-
\bigl(\chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ
\freeCyclesChainMap[1]_{\totalInt} =\\&
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)+
\chainMap[4]\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}
-
\freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}\bigr) =\\&
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)+
\chainMap[4]_{\totalInt}\bigl(
-\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}\bigr) =
\bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)-
\chainMap[4]_{\totalInt}\circ
\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}
\end{align*}
The required formula has been verified.
\begin{equation*}
\weakHomologyMap[10]_{\totalInt} =
\weakMap[10]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -
\bigl(
\freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[10]_{\totalInt}
-
(\chainMap[2]_{\ast}\circ \chainMap[1]_{\ast})_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}
\bigr)\colon
\freeBoundaries[1]_{\totalInt}
\to
\complex[4]_{\totalInt+1}
\end{equation*}
\begin{align*}
\weakMap[10]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -
\bigl( &
\freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[10]_{\totalInt}
-
(\chainMap[2]_{\ast}\circ \chainMap[1]_{\ast})_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}
\bigr) =\\&
\bigl( \chainMap[2]_{\totalInt+1}\circ \weakMap[1]_{\totalInt} +
\weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}\bigr)
\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-\\&\hskip10pt
\bigl(
\freeBoundariesMap[4]_{\totalInt}\circ
\freeBoundariesChainMap[2]_{\totalInt}\circ
\freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap[2]_{\totalInt+1}\circ \chainMap[1]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}
\bigr) =\\&
\chainMap[2]_{\totalInt+1}\Bigl(\weakMap[1]_{\totalInt}
\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-\bigl(
\freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap[1]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}\bigr)\Bigr)
+\\&
\weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}
\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-
\bigl(
\freeBoundariesMap[4]_{\totalInt}\circ
\freeBoundariesChainMap[2]_{\totalInt}\circ
\freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap[2]_{\totalInt+1}\circ \freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
\bigr)=\\&
\chainMap[2]_{\totalInt+1}\Bigl(\weakMap[1]_{\totalInt}
\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-\bigl(
\freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap[1]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}\bigr)\Bigr)
+\\&
\Bigl(\weakMap[2]_{\totalInt}
\big\vert_{_{\scriptstyle\freeBoundaries[2]_{\totalInt}}}
-
\bigl(
\freeBoundariesMap[4]_{\totalInt}\circ
\freeBoundariesChainMap[2]_{\totalInt}
-
\chainMap[2]_{\totalInt+1}\circ \freeBoundariesMap[2]_{\totalInt}\bigr)
\Bigr)
\freeBoundariesChainMap[1]_{\totalInt}=\\&
\chainMap[2]_{\totalInt+1}\circ \weakHomologyMap[1]_{\totalInt} +
\weakHomologyMap[2]_{\totalInt} \circ
\freeBoundariesChainMap[1]_{\totalInt}
\end{align*}
The result follows.
\end{math check}
\begin{ThmS}{Proposition}
Let $\chainMap[3]_\ast\colon \complex[1]_\ast \to \complex[3]_\ast$ be
a weakly split chain map and suppose
$\chainMap[4]_\ast\colon \complex[1]_\ast \to \complex[3]_\ast$ is a chain
map chain homotopic to $\chainMap[3]$.
Let $D_\ast\colon \complex[1]_\ast \to \complex[3]_{\ast+1}$ be a chain
homotopy with
\begin{equation*}
\chainMap[4]_\ast - \chainMap[3]_\ast = \boundary[3]_{\ast+1}\circ D_\ast +
D_{\ast-1}\circ \boundary[1]_\ast
\end{equation*}
Then $\chainMap[4]$ is weakly split by
$\freeCyclesChainMap[2]_{\totalInt}=\freeCyclesChainMap[1]_{\totalInt}$
and
\begin{equation*}
\weakMap[2]_{\totalInt} = \weakMap[1] + D_{\totalInt} \circ
\freeCyclesMap[1]_{\totalInt} +
\boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt}
\end{equation*}
With these choices,
$\weakTorsionHomologyMap[2]<1>_{\totalInt}=\weakTorsionHomologyMap[1]<1>_{\totalInt}$
\end{ThmS}
\begin{proof}
Since chain homotopic maps induce the same map in homology, it is possible
to take
$\freeCyclesChainMap[1]_{\totalInt}=\freeCyclesChainMap[2]_{\totalInt}$ and then
$\freeBoundariesChainMap[1]_{\totalInt}=\freeBoundariesChainMap[2]_{\totalInt}$
The required verifications are straightforward.
\end{proof}
\begin{math check}
\begin{align*}
\boundary[3]_{\totalInt+1}\circ \weakMap[2]_{\totalInt} = &
\boundary[3]_{\totalInt+1}\bigl(\weakMap[1]_{\totalInt} +
D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt}
+
\boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt}
\bigr)=\\&
\bigl(\freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}
-
\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}\bigr)
+
\bigl(
\chainMap[4]_{\totalInt} - \chainMap[3]_{\totalInt} -
D_{\totalInt-1}\boundary[1]_\ast
\bigr)\circ \freeCyclesMap[1]_{\totalInt}=\\&
\freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} -
\chainMap[4]_{\totalInt} \circ\freeCyclesMap[1]_{\totalInt} =
\freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[2]_{\totalInt} -
\chainMap[4]_{\totalInt} \circ\freeCyclesMap[1]_{\totalInt}
\end{align*}
\begin{align*}
&\weakHomologyMap[2]_{\totalInt} =
\weakMap[2]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-\bigl(
\freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt}
-
\chainMap[2]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}\bigr)
\bigr)
= \\&
\weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
+ D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} +
\boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt}
-\bigl(
\freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt}
-
\chainMap[2]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}\bigr)
\bigr)
= \\&
\weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-
\bigl(\freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap[1]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}
\bigr)
+ D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} +
\boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt}
-\\&\hskip30pt
\bigl(
\chainMap[1]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}
-
\chainMap[2]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}\bigr)
\bigr)
=\\&
\weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}}
-
\bigl(\freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt}
-
\chainMap[1]_{\totalInt+1}
\circ \freeBoundariesMap[1]_{\totalInt}
\bigr)
+
D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt}
-
D_{\totalInt}\circ \boundary[1]_{\totalInt+1}\circ \freeBoundariesMap[1]_{\totalInt}
=
\weakHomologyMap[1]_{\totalInt}
\end{align*}
The required formulas have been verified.
\end{math check}
The remaining results are routine verifications.
\begin{ThmS}[weakly split direct sum]{Proposition}
Given two weakly split chain complexes, $\{\complex[1]_\ast$, $\splitPair[1]_\ast\}$
and $\{\complex[2]_\ast$, $\splitPair[2]_\ast\}$, then
$\complex[1]_\ast\displaystyle\mathop{\oplus} \complex[3]_\ast$ is weakly split by
the following data:\\
\def\chainMap[1]+ \chainMap[2]{\complex[1]\oplus \complex[3]}
$\freeCycles[100]_{\totalInt}=
\freeCycles[1]_{\totalInt}\displaystyle\mathop{\oplus}
\freeCycles[2]_{\totalInt}$,
$\freeCyclesMap[100]_{\totalInt} =
\freeCyclesMap[1]_{\totalInt}\displaystyle\mathop{\oplus}\freeCyclesMap[2]_{\totalInt}
$.
Then
$\freeBoundaries[100]_{\totalInt}=
\freeBoundaries[1]_{\totalInt}\displaystyle\mathop{\oplus}
\freeBoundaries[2]_{\totalInt}$ so let
$\freeBoundariesMap[100]_{\totalInt} =
\freeBoundariesMap[1]_{\totalInt}\displaystyle\mathop{\oplus}\freeBoundariesMap[2]_{\totalInt}
$.
\end{ThmS}
\begin{ThmS}{Proposition}
Given weakly split chain maps $\chainMap[1]_\ast\colon\complex[1]_\ast
\to\complex[2]_\ast$
and
$\chainMap[2]_\ast\colon\complex[3]_\ast
\to\complex[4]_\ast$ then $\chainMap[1]_\ast\displaystyle\mathop{\oplus}\chainMap[2]_\ast$ is
weakly split by
\def\chainMap[1]+ \chainMap[2]{\chainMap[1]\oplus \chainMap[2]}
$\freeCyclesChainMap[10]_{\totalInt}=
\freeCyclesChainMap[2]_{\totalInt}\displaystyle\mathop{\oplus} \freeCyclesChainMap[1]_{\totalInt}$
and
$\weakMap[10]_{\totalInt} = \weakMap[1]_{\totalInt} \displaystyle\mathop{\oplus}
\weakMap[2]_{\totalInt}$.
With these choices
\begin{equation*}
\weakTorsionHomologyMap[10]<1>_{\totalInt} = \weakTorsionHomologyMap[1]<1>_{\totalInt}
\displaystyle\mathop{\oplus}
\weakTorsionHomologyMap[2]<1>_{\totalInt}
\end{equation*}
\end{ThmS}
\begin{DefS}{Remark}
The zero complex with its evident splitting
is a zero for the direct sum operation.
The zero chain map between any two weakly split complexes is
weakly split by letting \def\chainMap[1]+ \chainMap[2]{0_\ast}
$\freeCyclesChainMap[10]_{\totalInt}$ and
$\weakMap[10]_{\totalInt}$ be trivial.
Then $\weakTorsionHomologyMap[10]<1>_{\totalInt}$ is also trivial.
\end{DefS}
There is an internal sum result.
\begin{ThmS}{Proposition}
Given weakly split chain maps $\chainMap[1]_\ast\colon\complex[1]_\ast
\to\complex[2]_\ast$
and
$\chainMap[2]_\ast\colon\complex[1]_\ast
\to\complex[2]_\ast$ then $\chainMap[1]_\ast+\chainMap[2]_\ast$ is
weakly split by
\def\chainMap[1]+ \chainMap[2]{\chainMap[1]+ \chainMap[2]}
$\freeCyclesChainMap[10]_{\totalInt}=
\freeCyclesChainMap[2]_{\totalInt}+ \freeCyclesChainMap[1]_{\totalInt}$
and
$\weakMap[10]_{\totalInt} = \weakMap[1]_{\totalInt} +
\weakMap[2]_{\totalInt}$.
With these choices
\begin{equation*}
\weakTorsionHomologyMap[10]<1>_{\totalInt} = \weakTorsionHomologyMap[1]<1>_{\totalInt}
+
\weakTorsionHomologyMap[2]<1>_{\totalInt}
\end{equation*}
\end{ThmS}
\begin{DefS}{Remark}
Unlike the direct sum case (\ref{weakly split direct sum}),
there does not seem to be an easy way to weakly split the tensor product.
\end{DefS}
\begin{references}
\bib{Dold}{book}{
author={Dold, Albrecht},
title={Lectures on algebraic topology},
series={Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences]},
volume={200},
edition={2},
publisher={Springer-Verlag, Berlin-New York},
date={1980},
pages={xi+377},
isbn={3-540-10369-4},
review={\MR{606196 (82c:55001)}},
}
\bib{Eilenberg-Mac Lane}{article}{
author={Eilenberg, Samuel},
author={Mac Lane, Saunders},
title={On the groups $H(\Pi,n)$. II. Methods of computation},
journal={Ann. of Math. (2)},
volume={60},
date={1954},
pages={49--139},
issn={0003-486X},
review={\MR{0065162 (16,391a)}},
}
\bib{Heller}{article}{
author={Heller, Alex},
title={On the K\"unneth theorem},
journal={Trans. Amer. Math. Soc.},
volume={98},
date={1961},
pages={450--458},
issn={0002-9947},
review={\MR{0126479 (23 \#A3775)}},
}
\bib{Mac Lane slides}{article}{
author={MacLane, Saunders},
title={Slide and torsion products for modules},
journal={Univ. e Politec. Torino. Rend. Sem. Mat.},
volume={15},
date={1955--56},
pages={281--309},
review={\MR{0082488 (18,558b)}},
}
\bib{MacLane}{book}{
author={Mac Lane, Saunders},
title={Homology},
edition={1},
note={Die Grundlehren der mathematischen Wissenschaften, Band 114},
publisher={Springer-Verlag, Berlin-New York},
date={1967},
pages={x+422},
review={\MR{0349792 (50 \#2285)}},
}
\end{references}
\end{document} |
\begin{document}
\title[Weak limit of an immersed surface sequence
with bounded Willmore functional]
{\bf Weak limit of an immersed surface sequence
with bounded Willmore functional}
\author[Yuxiang Li]{Yuxiang Li\\
{\,\,\,\,mall\it Department of Mathematical Sciences},\\
{\,\,\,\,mall\it Tsinghua University,}\\
{\,\,\,\,mall\it Beijing 100084, P.R.China.}\\
{\,\,\,\,mall\it Email: [email protected].}}
\date{}
\maketitle
\begin{abstract} This paper is an extension of
\cite{K-L}. In this paper,
we will study the blowup behavior
of a surface sequence $\Sigma_k$ immersed in $\mathbb{R}^n$ with
bounded Willmore functional and fixed genus $g$.
We will prove that, we can
decompose $\Sigma_k$ into finitely many parts:
$$\Sigma_k=\bigcup_{i=1}^m\Sigma_k^i,$$
and find $p_k^i\in \Sigma_k^i$, $\lambda_k^i
\in\mathbb{R}$, such that $\frac{\Sigma_k^i-p_k^i}
{\lambda_k^i}$
converges locally in the sense of
varifolds to a complete branched immersed surface
$\Sigma_\infty^i$ with
$$\,\,\,\,um_i\int_{\Sigma_\infty^i}K_{\Sigma_\infty^i}=2\pi(2-2g).$$
The basic tool we use in this paper is a generalized convergence theorem
of F. H\'elein.
\end{abstract}
{{\bf Keywords}: Willmore functional, Bubble tree.}
{{\bf Mathematics Subject Classification}: Primary 58E20, Secondary
35J35.}
\date{}
\maketitle
\,\,\,\,ection{Introduction}
For an immersed surface $\ f : \Sigma
\rightarrow \mathbb{R}^n\ $ the Willmore functional
is defined by
\begin{displaymath}
W(f) = \frac{1}{4} \int_\Sigma |H_f|^2 d \mu_{f},
\end{displaymath}
where $H_f=\Delta_{g_f}f$ denotes the mean curvature vector of $f$, $g_f = f^*
g_{euc}$ the pull-back metric and $\mu_f$ the induced area measure
on $\Sigma$.
This functional first appeared in the papers of Blaschke \cite{Bl}
and Thomsen \cite{T}, and was reinvented and popularized
by Willmore \cite{W}.
We denote the infimum of Willmore functional of immersed surfaces
of genus $p$ by $\beta_p^n$. We have $\beta_p^n\geq 4\pi$ by
Gauss-Bonnet formula, and $\beta_p^n<8\pi$ as observed by Pinkall
and Kusner \cite{K} independently. Willmore conjectured
that $\beta_1^n$ is attained by Clifford torus. This conjecture is
still open.
Given a surface sequence with bounded Willmore functional
and measure, we are particularly interested to know
what the limit looks like?
In other words, we expect to understand
the blowup behavior of such a surface sequence.
It is very important
as we meet blowup almost
everywhere in the
study of Willmore functional.
For example, if $\Sigma_t$
is a Willmore flow defined on $[0,T)$, then
by $\epsilon$-regularity proved
in \cite{K-S}, $\int_{B_\rho\cap
\Sigma_t}|A_t|^2<\epsilon$ implies
$\|\nabla_{g_t}^mA_t\|_{L^\infty(B_\frac{\rho}{2}
\cap \Sigma_t)}
<C(m,\rho)$. Then
$\Sigma_t$ converges smoothly
in any compact subset of $\mathbb{R}^n$ minus the concentration points set which is defined by
$$\mathcal{S}=\{p\in\mathbb{R}^n:\lim_{r\rightarrow 0}
\liminf_{t\rightarrow T}
\int_{B_r(p)\cap\Sigma_t} |A_t|^2>0\}.$$
So, if we want to have a good knowledge of Willmore flow, we
have to learn the
behavior, especially the structure of the bubble trees
of $\Sigma_t$ near the concentration
points.
Note that $W(f_k)<C$ implies
$\int_{\Sigma}|A_{k}|^2
d\mu_k<C'$.
One expects that
$\|f_k\|_{W^{2,2}}$ is equivalent to
$\int|A_k|^2d\mu_k=\int g_k^{ij}g_k^{km}A_{ik}A_{jm}
\,\,\,\,qrt{|g_k|}dx$. However, it is not always true.
One reason is that the diffeomorphism group
of a surface is extremely big. Therefore, even when an
immersion sequence $f_k$ converges smoothly,
we can easily find a diffeomorphism sequence
$\phi_k$ such that $f_k\circ \phi_k$ will not converge.
Moreover,
the Sobolev embedding
$W^{2,2q}\hookrightarrow C^1$ is invalid when $q=1$,
so that it is impossible to estimate the $L^\infty$
norms of
$g^{-1}_k$ and $g_{k}$ via the Sobolev inequalities
directly.
To overcome these difficulties, an
approximate decomposition lemma was used by L. Simon
when he proved the existence of the minimizer
\cite{S}. He proved that $\beta_p^n$ can be attained if
$p=1$ or
\begin{equation}\label{simon}
p>1,\,\,\,\, and\,\,\,\, \beta_p^n< \omega_p^n=\min\Big\{4\pi+\,\,\,\,um\limits_{i}(\beta_{p_i}^n-4\pi):
\,\,\,\,um\limits_{i} p_i =p,\,1 \leq p_i < p\Big\}.
\end{equation}
Then Bauer and Kuwert proved that \eqref{simon} is
always true,
thus $\beta_p^n$ can be attained for any $p$ and $n$ \cite{B-K}.
Later, such a technique was extended by
W. Minicozzi to get the
minimizer of $W$ on Lagrangian tori \cite{M},
by Kuwert-Sch\"atzle to get
the minimizer of $W$ in a fixed conformal
class \cite{K-S3}, and by Sch\"atzle
to get the minimizer of $W$ with boundary
condition \cite{Sh}.
In a recent paper \cite{K-L}, we presented a new approach.
Given an immersion sequence
$f_k$, we consider each $f_k$ as a conformal immersion
of $(\Sigma,h_k)$ in $\mathbb{R}^n$, where $h_k$ is the
smooth metric with Gaussian curvature $\pm1$ or 0.
On the one hand, the conformal
diffeomorphism group of $(\Sigma,h_k)$ is very small.
On
the other hand, if we set $g_{f_k}=e^{2u_k} g_{euc}$ on an
isothermal coordinate system, then we can estimate
$\|u_k\|_{L^\infty}$ from the compensated compactness property of
$K_{f_k}e^{2u_k}$. Thus we may get the upper boundary of
$\|f_k\|_{W^{2,2}}$ via the equation $\Delta_{h_k}f_k=H_{f_k}$.
However, the compensated compactness only holds when the $L^2$ norm
of the second fundamental form is small locally, thus the blowup
analysis is needed here. Our basic tools are the following 2
results:
\begin{thm}\cite{H}\label{Helein} Let $f_k\in W^{2,2}(D,\mathbb{R}^n)$
be a sequence of conformal immersions with induced metrics
$(g_k)_{ij} = e^{2u_k} \delta_{ij}$, and assume
$$
\int_D |A_{f_k}|^2\,d\mu_{g_k} \leq \gamma <
\gamma_n =
\begin{cases}
8\pi & \mbox{ for } n = 3,\\
4\pi & \mbox{ for }n \geq 4.
\end{cases}
$$
Assume also that $\mu_{g_k}(D) \leq C$ and $f_k(0) = 0$.
Then $f_k$ is bounded in $W^{2,2}_{loc}(D,\mathbb{R}^n)$, and there
is a subsequence such that one of the following two alternatives
holds:
\begin{itemize}
\item[{\rm (a)}] $u_k$ is bounded in $L^\infty_{loc}(D)$ and
$f_k$ converges weakly in $W^{2,2}_{loc}(D,\mathbb{R}^n)$ to a conformal
immersion $f \in W^{2,2}_{loc}(D,\mathbb{R}^n)$.
\item[{\rm (b)}] $u_k \to - \infty$ and $f_k \to 0$ locally uniformly on $D$.
\end{itemize}
\end{thm}
\begin{thm}\label{D.K.}\cite{D-K}
Let $h_k,h_0$ be smooth Riemannian metrics on a surface $M$,
such that $h_k \to h_0$ in $C^{s,\alpha}(M)$, where $s \in \mathbb{N}$,
$\alpha \in (0,1)$. Then for each $p \in M$ there exist
neighborhoods $U_k, U_0$ and smooth conformal diffeomorphisms
$\vartheta_k:D \to U_k$, such that $\vartheta_k \to \vartheta_0$
in $C^{s+1,\alpha}(\overline{D},M)$.
\end{thm}
A $W^{2,2}$-conformal
immersion is defined as follows:
\begin{defi}\label{defconformalimmersion}
Let $(\Sigma,g)$ be a Riemann surface. A map $f\in W^{2,2}(\Sigma,g,\mathbb{R}^n)$
is called a conformal immersion, if the induced metric
$g_{f} = df\otimes df$ is given by
$$
g_{f} = e^{2u} g \quad \mbox{ where } u \in L^\infty(\Sigma).
$$
For a Riemann surface $\Sigma$ the set of all
$W^{2,2}$-conformal immersions is denoted by
$\mathbb{C}I(\Sigma,g,\mathbb{R}^n)$. When $f\in W^{2,2}_{loc}
(\Sigma,g,\mathbb{R}^n)$ and $u\in L^\infty_{loc}(\Sigma)$, we say
$f\in W^{2,2}_{conf,loc}(\Sigma,g,\mathbb{R}^n)$.
\end{defi}
\begin{rem}
F. H\'elein first proved Theorem \ref{Helein} is true for $\gamma<
\frac{8\pi}{3}$ \cite[Theorem 5.1.1]{H}. In \cite{K-L}, we show
that the constant $\gamma_n$ is optimal.
\end{rem}
\noindent Theorem \ref{Helein} together with Theorem \ref{D.K.}
give the convergence of a $W^{2,2}$-conformal sequence
of $(D,h_k)$ in $\mathbb{R}^n$ with $h_k$ converging smoothly
to $h_0$.
Then using the theory of moduli space of Riemann surface,
we proved in \cite{K-L} the following
\begin{thm}\label{KL}\cite{K-L} Let $f\in W^{2,2}_{conf}
(\Sigma,h_k,\mathbb{R}^n)$. If
\begin{equation}\label{omega}
W(f_k)\leq \left\{\begin{array}{ll}
8\pi-\delta&p=1\\
\min\{8\pi,\omega_p\}-\delta&p>1
\end{array}\right.,\,\,\,\, \delta>0,
\end{equation}
then
the conformal class sequence
represented by $h_k$ converges in $\mathcal{M}_p$.
\end{thm}
In other words, $h_k$
converges to a metric $h_0$ smoothly. This was
also proved by T. Rivi\`{e}re \cite{R}. Then up to
M\"obius transformations, $f_k$
will converge weakly in $W^{2,2}_{loc}(\Sigma
\,\,\,\,etminus\{\mbox{finite points}\},h_0)$ to a
$W^{2,2}(\Sigma,h_0)$-conformal immersion. In this way,
we give a new proof of the existence of
minimizer of Willmore functional with fixed genus.
\eqref{omega} also gives us a hint that,
it is the degeneration of complex structure that makes
the trouble for the convergence of an immersion sequence
with
\begin{equation}\label{bmw}
\mu(f_k)+W(f_k)<C.
\end{equation}
In \cite{C-L}, the Hausdorff limit of $\{f_k\}$
with \eqref{bmw} was studied,
using conformal immersion as a tool.
We proved that, the limit of $f_0$ is a conformal
branched immersion from a stratified
surface $\Sigma_\infty$ into $\mathbb{R}^n$.
Briefly speaking,
if $(\Sigma_0,h_0)$ is the limit of $(\Sigma,h_k)$
in $\overline{\mathcal{M}_p}$, then
$f_k$ converges weakly in the $W^{2,2}$ sense
in any component of $\Sigma_0$ away from the blowup points
$$\mathcal{S}(f_k)=\{p\in D:\lim_{r\rightarrow 0}
\liminf_{k\rightarrow+\infty}\int_{B_r(p,h_0)}|A(f_k)|^2d\mu_{f_k}\geq 4\pi\}.$$
Meanwhile, some bubble trees, which consist of $W^{2,2}$ branched conformal
immersions of $S^2$ in $\mathbb{R}^n$ will appear.
As a corollary, we get the following
\begin{pro}\cite{C-L}
Let $f_k:\Sigma\rightarrow \mathbb{R}^n$ be a sequence of smooth
immersions with \eqref{bmw}.
Assume the Hausdorff limit of $f_k(\Sigma)$
is not a union of $W^{2,2}$ branched conformal immersed
spheres. Then the complex structure of $c_k$ induced by $f_k$ diverges in
the moduli space if and only if there are a seqence of closed
curves $\gamma_k$
which are nontrivial in $H^1(\Sigma)$, such that
the length of $f_k(\gamma_k)$ converges to 0.
\end{pro}
Thus, when the conformal class induced by $f_k$
diverges in the moduli space, topology will
be lost. They are two reasons why the topology is lost.
One reason is that Theorem \ref{Helein} does not ensure
the limit is an immersion on each component of $\Sigma_0$.
If $f_k$ converges to a point in some components, then
some topologies are taken away. The other reason is that on each collar which is conformal
to $Q(T_k)=S^1\times[-T_k,T_k]$ with $T_k\rightarrow+\infty$,
there must exist a sequence
$t_k\in[-T_k,T_k]$ such that $f_k(S^1\times\{t_k\})$ will shrink to a point.
It is not easy to calculate how many topologies are lost, but it
is indeed possible to find where $\int_\Sigma K_{f_k}d\mu_{f_k}$
is lost. We have to study those bubbles
which have nontrivial topologies but shrink to points.
For this sake, we should check if those conformal
immersion sequences which converge to points
will converge to immersions after being rescaled:
\begin{thm}\label{convergence}
Let $\Sigma$ be a smooth connected Riemann surface without boundary,
and $\Omega_k\,\,\,\,ubset\,\,\,\,ubset\Sigma$ be domains with
$$\Omega_1\,\,\,\,ubset \Omega_2\,\,\,\,ubset\cdots\Omega_k\,\,\,\,ubset\cdots,\,\,\,\,
\bigcup_{i=1}^\infty\Omega_i=\Sigma.$$
Let $\{h_k\}$ be a smooth metric sequence over $\Sigma$
which converges to $h_0$ in $C^\infty_{loc}(\Sigma)$, and
$\{f_k\}$ be
a conformal immersion sequence of $(\Omega_k,h_k)$
in $\mathbb{R}^n$ satisfying
\begin{itemize}
\item[{\rm 1)}] $\mathcal{S}(f_k):=
\{p\in\Sigma: \lim\limits_{r\rightarrow 0}\liminf\limits_{k\rightarrow+\infty}
\int_{B_r(p,h_0)}|A_{f_k}|^2d\mu_{f_k}\geq 4\pi \}=\emptyset$.
\item[{\rm 2)}] $f_k(\Omega_k)$ can be extended to a closed compact
immersed surface $\Sigma_k$ with
$$\int_{\Sigma_k}(1+|A_{f_k}|^2)d\mu_{f_k}<\Lambda.$$
\end{itemize}
Take a curve $\gamma:[0,1]\rightarrow \Sigma$,
and set $\lambda_k=diam\, f_k(\gamma[0,1])$.
Then
we can find a subsequence of $\frac{f_k
-f_k(\gamma(0))}{\lambda_k}$ which
converges weakly in
$W^{2,2}_{loc}(\Sigma)$ to an
$f_0\in W_{conf,loc}^{2,2}(\Sigma,\mathbb{R}^n)$.
Further, we can find an inverse $I=\frac{y-y_0}{|y-y_0|^2}$
with $y_0\notin f_0(\Sigma)$ such that
$$\int_\Sigma(1+|A_{I(f_0)}|^2)d\mu_{I(f_0)}<+\infty.$$
\end{thm}
When $\Sigma$ is a compact closed surface minus finitely
many points,
$f_0$ may not be compact. However,
by Removability of singularity (see Theorem \ref{removal} in
section 2), $I(f_0)$ is a conformal branched immersion.
Thus $f_0$ is complete.
\begin{defi}
We call $f$ a generalized limit of $f_k$, if we can
find a point $x_0\notin \mathcal{S}(f_k)$ and a positive
sequence $\lambda_k$ which is equivalent to 1 or tends to 0,
such that
$\frac{f_k-f_k(x_0)}{\lambda_k}$ converges to $f$ weakly in
$W^{2,2}_{loc}(\Sigma\,\,\,\,etminus \mathcal{S}(f_k))$.
\end{defi}
Obviously, if $f$ and $f'$ are both generalized limits
of $f_k$, then $f=\lambda f'+b$ for some $\lambda$ and $b$.
We will not distinguish between $f$ and $f'$.
Near the concentration points, we will get
some bubbles. The divergence of complex structure also
gives us some bubbles. In
\cite{C-L}, we only considered the bubbles with
$\lambda_k\equiv1$.
In this paper, we will study the bubbles with $\lambda_k
\rightarrow 0$ which do not appear in the
Hausdorff limit. All the bubbles
can be considered as conformal branched
immersions from $\mathbb{C}$ (or $S^1\times \mathbb{R}$, $S^2$) into
$\mathbb{R}^n$.
However,
the structures of bubble trees here
are much more complicated than those of
harmonic maps. For example, there might
exist infinite many bubbles here, therefore, we should
neglect the bubbles which do not carry
total Gauss curvature.
\begin{defi}
We say a conformal branched immersion of $S^1\times\mathbb{R}$
into $\mathbb{R}^n$ is trivial, if for any $t$,
$$\int_{S^1\times\{t\}}\kappa\neq 2m\pi+\pi,\,\,\,\,
for\,\,\,\, some\,\,\,\, m\in\mathbb{Z}.$$
\end{defi}
The bubble trees constructed in this paper
consist of finitely many branches.
Small branches are on the big branches level by level.
Each branch consists of nontrivial bubbles, bubbles
with concentration, and the first bubble (see definitions in
Section 4).
We can classify the bubbles into four types:
$T_\infty$, $T_0$, $B_\infty$ and $B_0$
(see Definition \ref{typeofbubble}). We will show that a $T_0$ type bubble must follow a $B_\infty$ type bubble,
and a $T_\infty$ type bubble must follow a $B_0$ type
bubble.
Moreover, we have total Gauss curvature identity.
To state the total Gauss curvature identity precisely, we
have to divide it into 3 cases.
{\bf Hyperbolic case (genus$>1$):} Let $\Sigma_0$ be the stable surface in $
\overline{\mathcal{M}}_g$ with nodal points $\mathcal{N}=\{
a_1,\cdots, a_{m}\}$.
$\Sigma_0$ is obtained by pinching
some curves in a surface to points,
thus $\Sigma_0\,\,\,\,etminus\mathcal{N}$ can be divided
into finitely many components $\Sigma_0^1$, $\cdots$,
$\Sigma_0^s$. For each $\Sigma_0^i$, we can
extend $\Sigma_0^i$ to a smooth closed Riemann surface $\overline{\Sigma_0^i}$
by adding a point at each puncture. Moreover, the
complex structure of $\Sigma_0^i$ can be extended
smoothly to a complex structure of $\overline{\Sigma_0^i}$.
We say $h_0$ to be a hyperbolic structure on $\Sigma_0$ if $h_0$ is a smooth complete metric on
$\Sigma_0\,\,\,\,etminus\mathcal{N}$ with finite volume and Gauss curvature $-1$.
We define $\Sigma_{0}(a_j,\delta)$ to be the domain in $\Sigma_0$ which satisfies
$$a_j\in \Sigma_0(a_j,\delta),\,\,\,\, and \,\,\,\, injrad_{\Sigma_0\,\,\,\,etminus\mathcal{N}}^{h_0}(p)<\delta\,\,\,\, \forall p\in\Sigma_0(a_j,\delta)\,\,\,\,etminus\{a_j\}.$$
We set $h_0^i$ to be a
smooth metric over $\overline{\Sigma_0^i}$
which
is conformal to $h_0$ on $\Sigma_0^i$. We may assume $h_0^i$
has curvature $\pm1$ or curvature $0$ and measure 1.
Now, we let $\Sigma_k$ be a sequence of compact Riemann
surfaces of
fixed genus $g$ whose metrics $h_k$ have curvature $-1$,
such that $\Sigma_k
\rightarrow \Sigma_0$ in
$\overline{\mathcal{M}_g}$.
Then, there exist
a maximal collection $\Gamma_k = \{\gamma_k^1,\ldots,\gamma_k^{m}\}$
of pairwise disjoint, simply closed geodesics in $\Sigma_k$
with $\ell^j_k = L(\gamma_k^j) \to 0$, such that after passing
to a subsequence the following hold:
\begin{itemize}
\item[{\rm (1)}] There are maps $\varphi_k \in C^0(\Sigma_k,\Sigma_0)$,
such that $\varphi_k: \Sigma_k \backslash \Gamma_k \to \Sigma_0 \backslash \mathcal{N}$
is diffeomorphic and $\varphi_k(\gamma_k^j) = a_j$ for $j = 1,\ldots,m$.
\item[{\rm (2)}] For the inverse diffeomorphisms
$\psi_k:\Sigma_0 \backslash \mathcal{N} \to \Sigma_k \backslash \Gamma_k$,
we have $\psi_k^\ast (h_k) \to h_0$ in $C^\infty_{loc}(\Sigma_0 \backslash
\mathcal{N})$.
\item[{\rm (3)}] Let $c_k$ be the complex structure over $\Sigma_k$, and $c_0$
be the complex structure over $\Sigma_0\,\,\,\,etminus\mathcal{N}$. Then
$$\psi_{k}^*(c_k)\rightarrow c_0\,\,\,\, in\,\,\,\,
C^\infty_{loc}(\Sigma_0\,\,\,\,etminus\mathcal{N}).$$
\item[{\rm (4)}]For each $\gamma_k^j$, there is a
collar $U_k^j$ containing $\gamma_k^j$, which is
isometric to cylinder
$$Q_k^j=S^1\times(-\frac{\pi^2}{l_k^j},\frac{\pi^2}{l_k^j}),\,\,\,\,
with\,\,\,\, metric\,\,\,\,
h_k^j=\left(\frac{1}{2\pi\,\,\,\,in(\frac{l_k^j}{2\pi}t+\theta_k)}\right)^2(dt^2+d\theta^2),$$
where $\theta_k=\arctan(\,\,\,\,inh(
\frac{l_k^j}{2}))+\frac{\pi}{2}$.
Moreover, for any $(\theta,t)\in S^1\times
(-\frac{\pi^2}{l_k^j},\frac{\pi^2}{l_k^j})$, we have
\begin{equation}\label{injrad}
\,\,\,\,inh(injrad_{\Sigma_k}(t,\theta))\,\,\,\,in(
\frac{l_k^jt}{2\pi}+\theta_k)
=\,\,\,\,inh\frac{l_k^j}{2}.
\end{equation}
Let $\phi_k^j$ be the isometric between $Q_k^j$ and $U_k^j$. Then
$\varphi_k\circ\phi_k^{j}(T_k^j+t,\theta)\cup
\varphi_k\circ\phi_k^{j}(-T_k^j+t,\theta)$ converges in $C^\infty_{loc}((-\infty,0)\cup (0,\infty))$ to an isometric from $S^1\times(-\infty,0)\cup S^1\times(0,+\infty)$ to $\Sigma_0(a_j,1)\,\,\,\,etminus \{a_j\}$.
\end{itemize}
Items 1) and 2) in the above can be found in Proposition 5.1 in \cite{Hum}.
The main part of 3) is just the collar Lemma.
Now, we consider a sequence $f_k\in W^{2,2}_{conf}
(\Sigma,h_k,\mathbb{R}^n)$, with
$$\mu(f_k)+W(f_k)<\Lambda.$$
By Theorem \ref{convergence}, on each component
$\Sigma_k^i$, $f_k\circ \psi_k$ has a generalized limit
$f_0^i\in W^{2,2}_{conf}(\overline{\Sigma_k^i}\,\,\,\,etminus
A^i,h_0^i,\mathbb{R}^n)$,
where $A^i$ is a finite set. We have the following
\begin{thm}\label{main}
Let $f^1$, $f^2$, $\cdots$ be
all of the non-trivial bubbles of $\{f_k\}$. Then
$$\,\,\,\,um_i\int_{\overline{\Sigma_k^i}}K_{f_0^i}d\mu_{f_0^i}+
\,\,\,\,um_i\int_{S^2}K_{f^i}d\mu_{\varphi^i}=2\pi\chi(\Sigma).$$
\end{thm}
{\bf Torus case:} Let $(\Sigma,h_k)=\mathbb{C}/(\pi,z)$, where
$|z|\geq\pi$ and $|\mathbb{R}e{z}|\leq\frac{\pi}{2}$.
We can write
$$(\Sigma,h_k)=S^1\times\mathbb{R}/G_k,$$
where $S^1$ is the circle with perimeter 1 and
$G_k\cong \mathbb{Z}$ is the transformation group generalized by
$$(t,\theta)\rightarrow (t+a_k,\theta+\theta_k),\,\,\,\, where\,\,\,\,
a_k\geq \,\,\,\,qrt{\pi^2-\theta_k^2},\,\,\,\, and\,\,\,\, \theta_k\in [-\frac{\pi}{2},\frac{\pi}{2}].$$
$(\Sigma_k,h_k)$ diverges
in $\mathcal{M}_1$ if and only if $a_k\rightarrow+\infty$.
Then any $f_k\in W^{2,2}_{conf}(\Sigma,h_k,\mathbb{R}^n)$ can be
lifted to a conformal immersion $f_k':S^1\times\mathbb{R}
\rightarrow\mathbb{R}^n$ with
$$f_k'(t,\theta)=f_k'(t+a_k,\theta+\theta_k).$$
After translating, we may assume that
$f_k'(-t+\frac{a_k}{2},\theta)$ and $f_k'(t-\frac{a_k}{2},\theta)$
have no concentrations. We let $\lambda_k=diam
f_k'(S^1\times{\frac{a_k}{2}})$,
then $\frac{f_k'(-t+\frac{a_k}{2},\theta)-f(\frac{a_k}
{2},0)}{\lambda_k}$ and $\frac{f_k'(t-\frac{a_k}{2},\theta)
-f_k'(\frac{a_k}{2},\theta_k)}{\lambda_k}$ will
converge to $f_0^1$ and $f_0^2$ respectively in
$W^{2,2}_{loc}(S^1\times[0,+\infty))$. However, they can be
glued together via
$$f_0=\left\{\begin{array}{ll}
f_0^1(-t,\theta)&t\leq 0\\
f_0^2(t,\theta+\theta_0)&t>0,
\end{array}\right.$$
into a conformal immersion of $S^1\times\mathbb{R}$
in $\mathbb{R}^n$,
where $\theta_0=\lim\limits_{k\rightarrow+\infty}\theta_k$.
Then we have
\begin{thm}\label{main2}
$$\int_{S^1\times\mathbb{R}}K_{f_0}d\mu_{f_0}
+\,\,\,\,um_{i=1}^m\int_{S^1\times\mathbb{R}}K_{f^i}d\mu_i=0,$$
where $f^1$, $\cdots$, $f^m$ are all of the non-trivial bubbles of $f_k'$.
\end{thm}
{\bf Sphere case:} When $\Sigma$ is the sphere, we can
let $h_k\equiv h_0$. There is no bubble from collars.
We have
\begin{thm}\label{sphere} Let $f_0$ be the generalized limit of $f_k$.
Then
$$\int_{S^2}K_{f_0}d\mu_{f_0}+\,\,\,\,um_{i=1}^m\int_{S^1\times\mathbb{R}}
K_{f^i}d\mu_{f^i}=4\pi,$$
where $f^1$, $\cdots$, $f^m$ are all of the non-trivial bubbles.
\end{thm}
Put Theorem \ref{main}--\ref{sphere} together, we get
the main theorem of this paper, which is
a precise version of Theorem \ref{KL}:
\begin{thm}
Let $\Sigma_k$ be a sequence of surfaces
immersed in
$\mathbb{R}^n$ with bounded Willmore functional.
Assume $g(\Sigma_k)=g$. Then we can
decompose $\Sigma_k$ into finite parts:
$$\Sigma_k=\bigcup_{i=1}^m\Sigma_k^i,\,\,\,\, \Sigma_i\cap\Sigma_j
=\emptyset,$$
and find $p_k^i\in \Sigma_k^i$, $\lambda_k^i
\in\mathbb{R}$, such that $\frac{\Sigma_k^i-p_k^i}
{\lambda_k^i}$
converges locally in the sense of
varifolds to a complete branched immersed surface
$\Sigma_\infty^i$ with
$$\,\,\,\,um_i\int_{\Sigma_\infty^i}K_{\Sigma_\infty^i}=2\pi(2-2g),
\,\,\,\, and\,\,\,\, \,\,\,\,um_{i}W(\Sigma_\infty^i)\leq \lim_{k\rightarrow+\infty}
W(\Sigma_k).$$
\end{thm}
\begin{rem}
Parts of Theorem \ref{sphere} have appeared in \cite{L-L},
in which we assumed that
$\{f_k\}\,\,\,\,ubset W^{2,2}_{conf}(D,\mathbb{R}^n)$
and does not converge to a point.
\end{rem}
\,\,\,\,ection{Preliminary}
\,\,\,\,ubsection{Hardy estimate}
Let $f\in W^{2,2}_{conf}(D,\mathbb{R}^n)$ with $g_f=
e^{2u}(dx^1\otimes dx^1+dx^2\otimes dx^2)$ and
$\int_D|A_f|^2<4\pi-\delta$. $f$ induces a
Gauss map
$$G(f)=e^{-2u}(f_1\wedge f_2):D\rightarrow G(2,n)\hookrightarrow
\mathbb{C}\mathbb{P}^{n-1}.$$
Following \cite{M-S}, we define
the map $\Phi(f):\mathbb{C} \to \mathbb{C} P^{n-1}$ by
$$
\Phi(f)(z) =
\left\{\begin{array}{ll}
G(f)(z) & \mbox{ if }z \in D\\
G(f)(\frac{1}{\overline{z}}) & \mbox{ if }z \in \mathbb{C} \backslash \overline{D}.
\end{array}\right.$$
Then $\Phi(f)\in W_0^{1,2}(\mathbb{C},\mathbb{C} P^{n-1})$ and
$\int_{\mathbb{C}}{\Phi}^*(f) (\omega) =0$, where $\omega$
is the K\"ahler form of $\mathbb{C}\mathbb{P}^{n-1}$. Thus by Corollary 3.5.7
in \cite{M-S},
$\Psi(f)=*\Phi^*(f)(\omega)$ is in Hardy space, and
\begin{equation}\label{Psi}
\|\Psi(f)\|_{
\mathcal{H}}<C(\delta)\|A_f\|_{L^2(D)}.
\end{equation}
Note that
\begin{equation}\label{psi-k}
\Psi(f)|_{D}=K_{f}e^{2u}.
\end{equation}
If we set that $v$ solves the equation $-\Delta v=\Psi(f)$,
$v(\infty)=0$, then we have
\begin{equation*}
\|v\|_{L^\infty(\mathbb{R}^n)}+\|\nabla v\|_{L^2(\mathbb{R}^n)}+\|\nabla^2 v\|_{L^1(\mathbb{R}^n)}<
C\|\Psi(f)\|_{\mathcal{H}}.
\end{equation*}
Noting that $u-v$ is harmonic on $D$, we get
\begin{equation}\label{hardy0}
\|u\|_{L^\infty(D_\frac{1}{2})}+\|\nabla u\|_{L^2(D_\frac{1}{2})}+\|\nabla^2 u\|_{L^1(D_\frac{1}{2})}<
C(\|\Psi(f)\|_{\mathcal{H}}+\|u\|_{L^1(D)}).
\end{equation}
\,\,\,\,ubsection{Gauss-Bonnet formula}
Let $f\in W^{2,2}_{conf}(\Sigma,g,\mathbb{R}^n)$ with
$g_f=e^{2u}g$. Let $\gamma$ be
a smooth curve.
On $\gamma$, we define
\begin{equation}\label{geodesic.curvature}
\kappa_{f}=\frac{\partial u}{\partial n}+\kappa_g,
\end{equation}
where $n$ is one of the
unit normal field along $\gamma$ which is compatible to
$\kappa_g$. By \eqref{hardy0}, $\frac{\partial u}{\partial n}$
is well-defined.
In \cite{K-L}, we proved that $u$ satisfies the weak equation
$$-\Delta_g u=K_{f}e^{2u}-K_g.$$
Then, for any domain $\Omega$ with smooth boundary,
we have the Gauss-Bonnet formula:
$$\int_{\partial\Omega}\kappa_f=\chi(\overline{\Omega})+\int_{\Omega}
K_fd\mu_f.$$
\,\,\,\,ubsection{Convergence of $\int K_{f_k}d\mu_{f_k}$}
By \eqref{Psi} \eqref{psi-k} \eqref{hardy0} and
Theorem \ref{Helein}, we have :
\begin{lem}\label{measureconvergence}
Let $f_k$ be a conformal sequence from $D$ into $\mathbb{R}^n$
with $g_{f_k}=e^{2u_k}g_0$ and
$\int_D|A_{f_k}|^2d\mu_f\leq \gamma<4\pi$, which converges
to $f_0$ weakly. We assume $f_0$ is not a point map, and
$g_{f_0}=e^{2u_0}g_0$.
Then we can find a subsequence, such that
\begin{equation}\label{hardy}
K_{f_k}d\mu_{f_k} \rightharpoonup
K_{f_0}d\mu_{f_0}\,\,\,\, over \,\,\,\, D_\frac{1}{2},\,\,\,\, \mbox{in distribution,}
\end{equation}
and
\begin{equation*}
u_k\rightharpoonup u_0,\,\,\,\, in\,\,\,\, W^{1,2}(D_\frac{1}{2}).
\end{equation*}
\end{lem}
We will use the following
\begin{cor}
Let $f_k$ be a conformal sequence of $D\,\,\,\,etminus
D_\frac{1}{2}$ in $\mathbb{R}^n$, which converges to $f_0\in W^{2,2}_{{conf},loc}(D\,\,\,\,etminus D_\frac{1}{2},\mathbb{R}^n)$.
For any $t\in(\frac{1}{2},1)$ with $\partial D_t\cap \mathcal{S}(f_k)
=\emptyset$, we have
$$\lim_{k\rightarrow+\infty}\int_{\partial D_t}\kappa_{f_k} ds_k=
\int_{\partial D_t}\kappa_{f_0}ds_0.$$
\end{cor}
\proof Take $s\in (t,1)$, such that $\mathcal{S}(f_k)\cap
\overline{D_s\,\,\,\,etminus D_t}=\emptyset$.
Let $g_{f_k}=e^{2u_k}g_0$ and $\varphi\in C^\infty_0(D_s)$,
which is 1 on $D_t$.
Then we have
$$-\int_{\partial D_t}\frac{\partial u_k}{\partial r}ds=-
\int_{D_s\,\,\,\,etminus D_t}\nabla u_k\nabla\varphi
d\,\,\,\,igma
+\int_{D_s\,\,\,\,etminus D_t}\varphi K_ke^{2u_k}d\mu_{f_k},$$
and the right-hand side will converge to
$$-
\int_{D_s\,\,\,\,etminus D_t}\nabla u_0\nabla\varphi
d\,\,\,\,igma+\int_{D_s\,\,\,\,etminus D_t}\varphi K_0e^{2u_0}d\mu_{f_0},\,\,\,\,
as\,\,\,\, k\rightarrow+\infty.$$
Then we get
$$-\int_{\partial D_t}\frac{\partial u_k}{\partial r}ds
\rightarrow -\int_{\partial D_t}\frac{\partial u_0}{\partial r}ds.$$
By \eqref{geodesic.curvature} we
get
$$\int_{\partial D_t}\kappa_k\rightarrow
\int_{\partial D_t}\kappa_0.$$
$
\Box$\\
\,\,\,\,ubsection{Removability of singularity}
We have the following
\begin{thm}\label{removal}\cite{K-L}
Suppose that $f\in W^{2,2}_{{conf},loc}(D\backslash \{0\},\mathbb{R}^n)$ satisfies
$$
\int_D |A_f|^2\,d\mu_g < \infty \quad \mbox{ and } \quad \mu_g(D) < \infty,
$$
where $g_{ij} = e^{2u} \delta_{ij}$ is the induced metric. Then
$f \in W^{2,2}(D,\mathbb{R}^n)$ and we have
\begin{eqnarray*}
u(z) & = & m\log |z|+ \omega(z) \quad \mbox{ where }
m\geq 0,\, z\in \mathbb{Z},\,\omega \in C^0 \cap W^{1,2}(D),\\
-\Delta u & = & -2m\pi \delta_0+K_g e^{2u} \quad \mbox{ in }D.
\end{eqnarray*}
The multiplicity of the immersion at $f(0)$ is given by
$$
\theta^2\big(f(\mu_g \llcorner D_\,\,\,\,igma(0)),f(0)\big) = m+1 \quad \mbox{ for any small }
\,\,\,\,igma > 0.
$$
Moreover, we have
\begin{equation}\label{kappa2}
\lim_{t\rightarrow 0}\int_{\partial D_t}\kappa_{f}
ds_f=2\pi (m+1).
\end{equation}
\end{thm}
\proof We only prove \eqref{kappa2}. For the
proof of other part of the theorem, one can refer to
\cite{K-L}.
Observe that
$$|\int_{\partial D_t}\frac{\partial u}
{\partial r}-\int_{\partial D_{t'}}\frac{\partial u}
{\partial r}|=|\int_{D_t\,\,\,\,etminus D_{t'}}
Kd\mu|\rightarrow 0$$
as $t, t'\rightarrow 0$. Then
$\lim\limits_{t\rightarrow 0}\int_{\partial D_t}\frac{\partial u}
{\partial r}$ exists.
Since $\omega\in W^{1,2}(D_r)$,
we can find $t_k\in [2^{-k-1},2^{-k}]$, s.t.
$$(2^{-k}-2^{-k-1})\int_{\partial D_{t_k}}|
\frac{\partial w}{\partial r}|
=\int_{2^{-k-1}}^{2^{-k}}(\int_{\partial D_t}
|\frac{\partial w}{\partial r}|)dt\leq C\|\nabla w\|_{L^2
(D_{2^{-k}})}2^{-k},$$
which implies that
$\int_{\partial D_{t_k}}\frac{\partial w}{\partial r}
\rightarrow 0$. Then we get
$\int_{\partial D_{t_k}}\frac{\partial u}
{\partial r}\rightarrow 2\pi m$, which implies
that
$$\lim_{t\rightarrow 0}
\int_{\partial D_{t}}\frac{\partial u}
{\partial r}\rightarrow 2\pi m.$$
$
\Box$\\
\begin{rem} In the proof of Theorem \ref{removal} in \cite{K-L}, we get that
\begin{equation}\label{isolate}
\lim_{z\rightarrow 0}\frac{|f(z)-f(0)|}{|z|^{m+1}}=\frac{e^{w(0)}}{m+1}.
\end{equation}
\end{rem}
We give the following definition:
\begin{defi} \label{defconformalimmersion}
A map $f\in W^{2,2}(\Sigma,\mathbb{R}^n)$
is called a $W^{2,2}$- branched conformal immersion, if we can find
finitely many points $p_1$, $\cdots$, $p_m$, s.t.
$f\in W^{2,2}_{conf,loc}(\Sigma\,\,\,\,etminus\{p_1,\cdots,p_m\})$, and
$$
\mu(f)<+\infty,\,\,\,\, \int_{\Sigma}|A_f|^2d\mu_f<+\infty.
$$
\end{defi}
For the behavior at infinity of complete conformally
parameterized surfaces, we have the following
\begin{thm}\label{removal2}
Suppose that $f\in W^{2,2}_{{conf},loc}(\mathbb{C}\,\,\,\,etminus D_R,\mathbb{R}^n)$ with
$$
\int_{\mathbb{C}\,\,\,\,etminus D_R} |A_f|^2\,d\mu_{g} < \infty,
$$
where $g_{ij} = e^{2u} \delta_{ij}$ is the induced
metric. We assume $f(\mathbb{C}\,\,\,\,etminus D_{2R})$
is complete. Then we have
\begin{equation*}
u(z) = m\log |z|+ \omega(z) \quad
\mbox{ where }
m\geq 0,\, z\in \mathbb{Z},\,\omega \in W^{1,2}(\mathbb{C}\,\,\,\,etminus D_{2R}).
\end{equation*}
Moreover, we have
\begin{equation}\label{kappa3}
\lim_{t\rightarrow +\infty}\int_{\partial D_t}\kappa_{f}
ds_f=2\pi (m+1).
\end{equation}
\end{thm}
The proof of \eqref{kappa3} is similar to that of
\eqref{kappa2}. Other part of the proof can
be found in \cite{M-S}.
Though Muller-Sverak's result was stated for smooth
surface, it is easy to check that
their proof also holds for a $W^{2,2}$ conformal immersion.
\,\,\,\,ection{Proof of Theorem \ref{convergence}}
We first prove the following
\begin{lem}\label{convergence2} Suppose $(\Sigma,h_k)$ to be smooth Riemann surfaces,
where $h_k$ converges to $h_0$ in $C^\infty_{loc}(\Sigma)$.
Let $\{f_k\}\,\,\,\,ubset W^{2,2}_{{conf},loc}(\Sigma,h_k,\mathbb{R}^n)$ with
$$\mathcal{S}(f_k)=\{p\in\Sigma: \lim_{r\rightarrow 0}
\liminf_{k\rightarrow+\infty}\int_{B_r(p,h_0)}|A_{f_k}|^2d\mu_{f_k}\geq 4\pi\}
=\emptyset.$$
Then $f_k$ converges in $W^{2,2}_{loc}(\Sigma,h_0,\mathbb{R}^n)$
to a point or an $f_0\in W^{2,2}_{conf,loc}(\Sigma,h_0,\mathbb{R}^n)$.
\end{lem}
\proof Let $g_{f_k}=e^{2u_k}h_k$.
We only need to prove the following statement:
for any $p\in\Sigma$, we
can find a neighborhood $V$ which is independent of
$\{f_k\}$, such that $f_k$ converges weakly to $f_0$
in
$W^{2,2}(V,h_0)$. Moreover,
$\|u_k\|_{L^\infty(V)}<C$ if and only if $f_0\in W^{2,2}_{conf}
(V,\mathbb{R}^n)$; $u_k\rightarrow-\infty$ uniformly,
if and only if $f_0$ is a point map.
Now we prove this statement: Given a point $p$, we choose
$U_k$, $U_0$, $\vartheta_k$, $\vartheta_0$
as in the Theorem \ref{D.K.}.
Set $\vartheta_k^*(h_k)=e^{2v_k}g_0$, where
$g_0=(dx^1)^2+(dx)^2$.
We may assume $v_k\rightarrow v_0$ in $C^\infty_{loc}(D)$.
Let $\hat{f}_k=f_k(\vartheta_k)$ which
is a map from $D$ into $\mathbb{R}^n$. It is easy to check that
$\hat{f}_k\in W^{2,2}_{conf}(D,\mathbb{R}^n)$ and
$g_{f_k}=e^{2u_k+2v_k}g_0$. By Theorem \ref{Helein}, we can assume
that $\hat{f}_k$ converges to $\hat{f}_0$
weakly in $W^{2,2}(D_\frac{3}{4})$. Moreover,
$\hat{f}_0$ is a point when $u_k+v_k\
\rightarrow-\infty$ uniformly on $D_\frac{3}{4}$,
and a conformal immersion when $\,\,\,\,up_{k}
\|u_k+v_k\|_{L^\infty(D_\frac{3}{4})}<+\infty$.
Let $V=\vartheta_0(D_\frac{1}{2})$.
Since $\vartheta_k$ converges to $\vartheta_0$,
$\vartheta_k^{-1}(V)\,\,\,\,ubset D_\frac{3}{4}$
for any sufficiently large $k$ and
$f_k=\hat{f}_k(\vartheta_k^{-1})$
converges to $f_0=\hat{f}_0
(\vartheta_0^{-1})$ weakly in $W^{2,2}(V,h_0)$.
Moreover,
$f_0$ is a conformal immersion when $\|u_k\|_{L^\infty(V)}<C$,
and a point when $u_k\rightarrow-\infty$ uniformly in
$V$.
$
\Box$\\
{\it The proof of Theorem \ref{convergence}:} When $f_k$ converges to a conformal immersion
weakly, the result is obvious. Now we assume that
$f_k$ converges to a point. For this case,
$\lambda_k
\rightarrow 0$.
Put $f_k'=\frac{f_k-f_k(\gamma(0))}{\lambda_k}$,
$\Sigma_k'=\frac{\Sigma_k-f_k(\gamma(0))}{\lambda_k}$.
We have two cases:
\noindent Case 1: $diam(f_k')<C$. Letting $\rho$ in
inequality (1.3) in \cite{S} tend to infinity,
we get $\frac{\Sigma_k'\cap B_\,\,\,\,igma(\gamma(0))}{\,\,\,\,igma^2}\leq C$ for any $\,\,\,\,igma>0$,
hence we get $\mu(f_k')<C$ by taking
$\,\,\,\,igma=diam(f_k')$. Then Lemma \ref{convergence2}
shows that $f_k'$ converges weakly
in $W^{2,2}_{loc}(\Sigma,h_0)$. Since
$diam\, f_k'
(\gamma)=1$, the weak limit is not a point.
\noindent Case 2: $diam(f_k')\rightarrow +\infty$. We take a point
$y_0\in\mathbb{R}^n$ and a constant $\delta>0$, s.t.
$$B_\delta(y_0)\cap \Sigma_k'=\emptyset,\,\,\,\, \forall k.$$
Let $I=\frac{y-y_0}{|y-y_0|^2}$, and
$$f_k''=I(f_k'),\,\,\,\, \Sigma_k''=I(\Sigma_k').$$
By conformal invariance of Willmore functional
\cite{C,W}, we have
$$\int_{\Sigma''}|A_{\Sigma''}|^2d\mu_{\Sigma''}
=\int_{\Sigma}|A_\Sigma|^2d\mu_{\Sigma}<\Lambda.$$
Since $\Sigma_k''\,\,\,\,ubset B_\frac{1}{\delta}(0)$, also by (1.3) in \cite{S},
we get $\mu(f_k'')<C$. Thus
$f_k''$ converges weakly in $W^{2,2}_{loc}(\Sigma\,\,\,\,etminus
\mathcal{S}(f_k''),h_0)$.
Next, we prove that $f_k''$ will not converge to a point by assumption.
If $f_k''$ converges to a point in
$W^{2,2}_{loc}(\Sigma\,\,\,\,etminus \mathcal{S}(f_k''))$,
then the limit must be 0, for $diam\,(f_k')$
converges to $+\infty$.
By the
definition of $f_k''$, we can find a $\delta_0>0$,
such that $f_k''(\gamma)\cap
B_{\delta_0}(0,h_0)=\emptyset$. Thus for any $p\in \gamma([0,1])
\,\,\,\,etminus \mathcal{S}(f_k'')$, $f_k''$ will not converge to $0$. A contradiction.
Then we only need to prove that $f_k'$ converges weakly in
$W^{2,2}_{loc}(\Sigma,h_0,\mathbb{R}^n)$.
Let $f_0''$ be the limit of $f_k''$. By Theorem \ref{removal},
$f_0''$ is a branched immersion of $\Sigma$ in $\mathbb{R}^n$.
Let $\mathcal{S}^*=f_0^{''-1}(\{0\})$.
By \eqref{isolate}, $\mathcal{S}^*$ is isolate.
First, we prove that for any $\Omega\,\,\,\,ubset\,\,\,\,ubset\Sigma\,\,\,\,etminus
(\mathcal{S}^*\cup\mathcal{S}(\{f_k''\})$, $f_k'$
converges weakly in $W^{2,2}(\Omega,h_0,\mathbb{R}^n)$:
Since $f_0''$ is continuous
on $\bar{\Omega}$, we may assume
$dist(0,f_0''(\Omega))>\delta>0$. Then $dist(0,f_k''(\Omega))>\frac{\delta}{2}$
when $k$ is sufficiently large. Noting that $f_k'
=\frac{f_k''}{|f_k''|^2}+y_0$, we get that $f_k'$ converges weakly in
$W^{2,2}(\Omega,h_0,\mathbb{R}^n)$.
Next, we prove that for each
$p\in \mathcal{S}^*\cup\mathcal{S}(\{f_k''\})$, $f_k'$ also converges in
a neighborhood of $p$.
We use the denotation $U_k$, $U_0$, $\vartheta_k$ and
$\vartheta_0$ with $\theta_k(0)=p$ again.
We only need to prove that $\hat{f}_k'=f_k'(\vartheta_k)$ converges
weakly in $W^{2,2}(D_\frac{1}{2})$.
Let $g_{\hat{f}_k'}=e^{2\hat{u}_k'}(dx^2+dy^2)$.
Since $\hat{f}_k'\in W^{2,2}_{conf}
(D_{4r})$ with $\int_{D_{4r}}|A_{\hat{f}_k'}|^2d\mu_{\hat{f}_k'}<4\pi$ when $r$ is
sufficiently small and $k$ sufficiently large,
by the arguments in subsection 2.1,
we can find a $v_k$ solving the equation
$$-\Delta v_k=K_{\hat{f}_k'}e^{2\hat{u}_k'},\,\,\,\, z\in D_r\,\,\,\, and\,\,\,\, \|v_k\|_{L^\infty(D_r)}<C.$$
Since $f_k'$ converges to a conformal
immersion in $D_{4r}\,\,\,\,etminus D_{\frac{1}{4}r}$, by Theorem \ref{Helein},
we may assume that
$\|\hat{u}_k'\|_{L^\infty(D_{2r}\,\,\,\,etminus
D_r)}<C$.
Then
$\hat{u}_k'-v_k$ is a harmonic function with
$\|\hat{u}_k'-v_k\|_{L^\infty(\partial D_{2r}(z))}<C$,
then we get $\|\hat{u}_k'(z)-v_k(z)\|_{L^\infty(D_{2r}(z))}<C$
by the Maximum Principle. Thus, $\|\hat{u}_k'\|_{L^\infty(D_{2r})}<C$,
which implies $\|\nabla f_k'\|_{L^\infty(D_{2r})}<C$.
By the equation $\Delta \hat{f}_k'=e^{2\hat{u}_k'}H_{\hat{f}_k'}$, and
the fact that $\|e^{2\hat{u}_k'}H_{\hat{f}_k'}\|_{L^2
(D_{2r})}^2<
e^{\|\hat{u}_k'\|_{L^\infty}}\int_{D_{2r}}|H_{\hat{f}_k'}|^2d\mu_{{\hat{f}_k'}}$,
we get $\|\nabla{\hat{f}_k'}\|_{W^{1,2}(D_{r})}<C$.
Recalling that $\hat{f}_k'$ converges in $C^0(D_r\,\,\,\,etminus
D_\frac{r}{2})$, we complete the proof.
$
\Box$\\
\begin{rem}
In fact, we proved that $\mathcal{S}^*=\emptyset$.
\end{rem}
\,\,\,\,ection{Analysis of the neck}
For a sequence of conformal immersions from
a surface into $\mathbb{R}^n$ with the conformal class divergence,
the blowup comes from concentrations and collars.
Both cases can be changed into a blowup
analysis of a conformal immersion sequence
of $S^1\times[0,T_k]$ in $\mathbb{R}^n$ with $T_k\rightarrow+\infty$. So we first analyze the blow up procedure
on long cylinders without concentrations.
\,\,\,\,ubsection{Classification of bubbles of a simple
sequence over an infinite cylinder}
Let $f_k$ be an immersion sequence
of
$S^1\times [0,T_k]$ in $\mathbb{R}^n$ with $T_k\rightarrow+\infty$.
We say $f_k$ has concentration, if we can find a sequence
$\{(\theta_k,t_k)\}\,\,\,\,ubset S^1\times [0,T_k]$, such that
$$\lim_{r\rightarrow 0}\liminf_{k\rightarrow+\infty}\int_{D_r(\theta_k,t_k)}
|A_{f_k}|^2d\mu_{f_k}\geq 4\pi.$$
We say $\{f_k\}$ is simple if:
\begin{itemize}
\item[{\rm 1)}] $f_k$ has no concentration;
\item[{\rm 2)}] $f_k(S^1\times[0,T_k])$ can be extended to a compact closed
immersed surface $\Sigma_k$ with
$$\int_{\Sigma_k}(1+|A_{f_k}|^2)d\mu_{f_k}<\Lambda.$$
\end{itemize}
When $\{f_k\}$ is simple, we say $f_0$ is a bubble of $f_k$,
if we can find
a sequence $\{t_k\}\,\,\,\,ubset [0,T_k]$ with
$$t_k\rightarrow+\infty,\,\,\,\, and\,\,\,\, T_k-t_k\rightarrow+
\infty,$$
such that $f_0$ is a generalized
limit of $f_k(\theta,t_k+t)$. If $f_0$ is nontrivial,
we call it a nontrivial bubble.
For convenience, we call the generalized limit of $f(\theta,t+T_k)$
and $f(\theta,t)$ the top and the bottom
respectively.
Note that the top and the bottom are in $W^{2,2}_{conf}(S^1\times(-\infty,
0])$ and $W^{2,2}_{conf}(S^1\times[0,+\infty))$
respectively.
\begin{defi}
Let $f^1$ and $f^2$ be two
bubbles which are limits of
$f_k(\theta,t+t_k^1)$
and $f_k(\theta,t+t_k^2)$
respectively.
We say these two bubbles are the same, if
$$\,\,\,\,up_k|t_k^1-t_k^2|<+\infty.$$
When $f^1$ and $f^2$ are not the same,
we say $f^1$ is in front of $f^2$ (or $f^2$ is behind $f^1$)
if $t_k^1<t_k^2$. We say
$f^2$ follows $f^1$, if $f^2$ is behind $f^1$
and there are no non-trivial bubbles
between $f^1$ and $f^2$.
\end{defi}
Obviously, the bubbles in this section must be
in $W^{2,2}_{conf}(S^1\times\mathbb{R})$,
and must be one of the following:
\begin{itemize}
\item[1).] $S^2$-type, i.e. $I(f^0)(S^1\times\{\pm\infty\})
\neq 0$;
\item[2).] Catenoid-type, i.e. $I(f^0)(S^1\times\{\pm\infty\})=0$;
\item[3).] Plain-type, i.e. one and only one
of $I(f^0)(S^1\times\{\infty\})$, $I(f^0)(S^1\times\{-\infty\})$
is 0,
\end{itemize}
where $I=\frac{y-y_0}{|y-y_0|^2}$,
$y_0\notin f^0(S^1\times\mathbb{R})$.
We give another classification of bubbles:
\begin{defi}\label{typeofbubble}
We call a bubble
$f^0$ to be a bubble of
\begin{itemize}
\item[] type $T_{\infty}$ if
$diam f^0(S^1\times\{+\infty\})=+\infty$; type $T_0$ if
$diam f^0(S^1\times\{+\infty\})=0$;
\item[] type $B_{\infty}$ if
$diam f^0(S^1\times\{-\infty\})=+\infty$; type $B_0$ if
$diam f^0(S^1\times\{-\infty\})=0$.
\end{itemize}
\end{defi}
We say $f_k$ has $m$ non-trivial bubbles, if we
can not find the ($m+1$)-th non-trivial bubble for any
subsequence of $f_k$.
\begin{rem}
Let $f_0$ be a bubble. By \eqref{kappa2} and \eqref{kappa3},
$$\lim_{t\rightarrow+\infty}\int_{S^1\times\{t\}}\kappa_{f^0}
=2m^+\pi,\,\,\,\, and\,\,\,\,
\lim_{t\rightarrow+\infty}\int_{S^1\times\{t\}}\kappa_{f^0}
=2m^-\pi$$
for some $m^+$ and $m^-\in\mathbb{Z}$. Then
$f^0$ is trivial implies that
$\int_{S^1\times \mathbb{R}}K_{f^0}d\mu_{f^0}=0$. Thus
both $S^2$ type of bubbles and catenoid type of bubbles
are non-trivial.
\end{rem}
\begin{rem} It is easy to check that
$\mu(f^0)<+\infty$ implies that $f^0$ is
a sphere-type bubble and is of type $(B_0,T_0)$.
\end{rem}
\begin{rem} If $f^{0'}$ is a catenoid-type bubble,
then it is of type $(B_\infty,T_\infty)$;
If $f^{0'}$ is a plain-type bubble,
then it is of type $(B_\infty,T_0)$
or $(B_0,T_\infty)$.
\end{rem}
First, we study the case that
$f_k$ has no bubbles.
Basically, we want to show that after scaling, the
image of $f_k$ will converge to a topological disk.
\begin{lem}
If $f_k$ has no bubbles, then
$$\frac{diam\, f_k(S^1\times \{1\})}{diam\,
f_k(S^1\times\{T_k-1\})}\rightarrow 0\,\,\,\,
or\,\,\,\, +\infty.$$
\end{lem}
\proof Assume this lemma is not true. Then we may assume
$\frac{diam\, f_k(S^1\times \{1\})}{diam\,
f_k(S^1\times\{T_k-1\})}\rightarrow\lambda\in (0,+\infty)$. Let
$\lambda_k=diam f_k(S^1\times\{1\})$. By Theorem
\ref{convergence2}, $\frac{f_k(\theta,t)-f_k(0,1)}{\lambda_k}$
converges to $f^B$ weakly in $W^{2,2}_{loc}
(S^1\times(0,+\infty))$, and
$\frac{f_k(\theta,t+T_k)-f_k(0,T_k-1)}{\lambda_k}$ converges to
$f^T$ weakly in $W^{2,2}_{loc} (S^1\times(-\infty,0))$
respectively.
When $diam f^B(S^1\times \{+\infty\})=0$, we set $\delta_k$ and
$t_k$ to be defined by
$$\delta_k=diam f_k(S^1\times\{t_k\})=
\inf_{t\in [1,T_k-1]}f_k(S^1\times\{t\}).$$
Obviously, $\delta_k\rightarrow 0$, and
$t_k\rightarrow+\infty$, $T_k-t_k
\rightarrow+\infty$. $
\frac{f_k(\theta,t)-f_k(0,t_k)}{\delta_k}$
will converge to a non-trivial bubble. A contradiction.
When $diam f^B(S^1\times \{+\infty\})=+\infty$,
we set $\delta_k'$ and $t_k'$ to be defined by
$$\delta_k'=diam f_k(S^1\times\{t_k'\})=
\,\,\,\,up_{t\in [1,T_k-1]}f_k(S^1\times\{t\}),$$
then we can also get a bubble.
$
\Box$\\
Now we assume $f_k$ has no
bubbles,
and $\frac{diam\, f_k(S^1\times \{1\})}{diam\,
f_k(S^1\times\{T_k-1\})}\rightarrow +\infty$.
Let $\lambda_k=diam\,
f_k(S^1\times\{T_k-1\})$. The bottom $f^B$ is the weak limit of
$f_k'=\frac{f_k(\theta,t)-f_k(0,1)}{\lambda_k}$.
Let $\phi$ be the conformal diffeomorphism
from $D\,\,\,\,etminus\{0\}$ to $S^1\times[0,+\infty)$.
Then $f^B\circ\phi$ is an immersion of $D$
in $\mathbb{R}^n$ perhaps with branch point $0$.
Moreover, by the arguments in \cite{C-L} or in \cite{C},
we have
$$f^B(\phi(0))=\lim_{t\rightarrow+\infty}\lim_{k\rightarrow+\infty}
f_k'(\theta,T_k-t).$$
Since $diam f_k'(S^1\times\{T_k-1\})\rightarrow 0$,
$f_k'(\theta,T_k-t)$ converges to a point, then the Hausdorff
limit of $f_k'((0,T_k))$ is a branched conformal immersion of $D$.
\begin{rem}
In fact, the above results and arguments hold for a sequence
$\{f_k\}$ which has neither $S^2$-type
nor catenoid-type bubbles.\\
\end{rem}
Next, we show when $\{f_k\}$ has bubbles, how we will
find out all of them.
We need the following simple lemma:
\begin{lem}\label{interval} After passing to a subsequence, we can find
$0=d_k^0<d_k^1<\cdots<d_k^l=T_k$,
where $l\leq\frac{\Lambda}{4\pi}$,
such that
$$d_k^i-d_k^{i-1}\rightarrow+\infty,\,\,\,\, i=1,\cdots,l\,\,\,\, and
\int_{S^1\times\{d_k^i\}}\kappa_k=2m_i\pi+\pi,\,\,\,\, m_i\in\mathbb{Z},\,\,\,\,
i=1,\cdots,l-1,
$$
and
$$\lim_{T\rightarrow+\infty}\,\,\,\,up_{t\in [d_k^{i-1}+T,
d_k^i-T]}\left|
\int_{S^1\times \{t\}}\kappa_k-\int_{S^1\times\{d_k^{i-1}+T\}}\kappa_k\right|< \pi.$$
\end{lem}
\proof Let $\Lambda<4m\pi$. We prove the lemma by
induction of $m$.
We first prove it is true for $m=1$. Let
$$\lim_{t\rightarrow+\infty}
\lim_{k\rightarrow+\infty}\int_{S^1\times\{t\}}=2m_1\pi,\,\,\,\,
\lim_{t\rightarrow+\infty}
\lim_{k\rightarrow+\infty}\int_{S^1\times\{T_k-t\}}=2m_2\pi,$$
where $m_1$ and $m_2$ are integers.
Thus, we can find $T$, such that
$$\left|\int_{S^1\times\{T\}}\kappa_k-2m_1\pi\right|<\epsilon,\,\,\,\,
and \,\,\,\, \left|\int_{S^1\times\{T_k-T\}}\kappa_k-2m_2\pi\right|<\epsilon$$
when $k$ is sufficiently large. Take a $t_0\in (T,T_k-T)$,
such that
$$\int_{S^1\times[T,t_0]}|A_{f_k}|^2<2\pi,\,\,\,\,
\int_{S^1\times[t_0,T_k-T]}|A_{f_k}|^2\leq 2\pi.$$
By Gauss-Bonnet,
$$\left|\int_{S^1\times\{t\}}\kappa_k-\int_{S^1\times\{T\}}\kappa_k\right|
\leq \int_{S^1\times[T,t]}|K_{f_k}|d\mu_{f_k}
\leq \frac{1}{2}\int_{S^1\times[T,t_0]}|A_{f_k}|^2d\mu_{f_k}<\pi,
\,\,\,\,\forall t\in(T,t_0),$$
$$\left|\int_{S^1\times\{t\}}\kappa_k-\int_{S^1\times\{T_k-T\}}\kappa_k\right|
\leq
\frac{1}{2}\int_{S^1\times[t_0,T_k-T]}|A_{f_k}|^2d\mu_{f_k}<\pi,
\,\,\,\,\forall t\in(t_0,T_k-T).$$
Thus, we can take $\epsilon$ to be very small so that
$\int_{S^1\times\{t\}}
\neq 2i\pi$ for any $i\in\mathbb{Z}$ and
$t\in (T,T_k-T)$.
Now, we assume the result is true for $m$, and prove it
is also true for $m+1$. We have two cases.
Case 1, there is a sequence $\{t_k\}$, such that
$t_k\rightarrow+\infty$, $T_k-t_k\rightarrow+\infty$,
$\int_{S^1\times\{t_k\}}\kappa_k=2m_k\pi+\pi$
for some $m_k\in\mathbb{Z}$. For this case, we let
$f_k'=\frac{f_k(t+t_k,\theta)-f_k(t_k,0)}{\lambda_k}$
which converges weakly to $f_0'$, where $\lambda_k=diam f_k(S^1\times\{t_k\})$. Then by Gauss-Bonnet
$$\int_{S^1\times\mathbb{R}}|K_{f_0'}|\geq
\left|\int_{S^1\times(0,+\infty)}K_{f_0'}\right|
+\left|\int_{S^1\times(-\infty,0)}K_{f_0'}\right|\geq
2\pi.$$
Thus, $\int_{S^1\times\mathbb{R}}|A_{f_0'}|^2\geq 4\pi$. We can
find $T$, such that
$$\int_{S^1\times[0,t_k-T]}|A_{f_k}|^2<4(m-1)\pi,\,\,\,\,
and\,\,\,\, \int_{S^1\times[t_k+T,T_k]}|A_{f_k}|^2<4(m-1)\pi$$
when $k$ is sufficiently large. Thus, we can use
induction on $[0,t_k-T]$ to get $0=\bar{d}_k^0<
\bar{d}_k^1<\cdots<\bar{d}_k^{\bar{l}}=t_k-T$, and
on $[t_k+T,T_k]$ to get $t_k+T=\tilde{d}_k^0<\cdots<
\tilde{d}_k^{\tilde{l}}=T_k$. We can set
$$d_k^i=\left\{\begin{array}{ll}
\bar{d}_k^i&i<\bar{l}\\
t_k&i=\bar{l}\\
\tilde{d}_k^{i-l}&i>\bar{l}
\end{array}\right.$$
Then, we complete the proof.
$
\Box$\\
Set
$f_k^i=\frac{f_k(t+d_k^i,\theta)-f_k(d_k^i,0)}{
diam\, f_k(S^1\times\{d_k^i\})}$,
and assume $f_k^i\rightharpoonup f^i$. It is easy to check that
$$\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}
\int_{S^1\times\{d_k^i+T\}}\kappa_k
=\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}
\int_{S^1\times\{d_k^{i+1}-T\}}\kappa_k,$$
we get
$$\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}\int_{S^1\times[d_k^i+T,
d_k^{i+1}-T]}K_{f_k}=0.$$
\begin{rem}
In fact, we can get that for any $t_k<t_k'$ with
$$t_k-d_k^i\rightarrow+\infty, \,\,\,\, and\,\,\,\, d_k^{i+1}-t_k'\rightarrow+\infty,$$
we have
$$\lim_{k\rightarrow+\infty}\int_{S^1\times[t_k,
t_k']}K_{f_k}=0.$$
\end{rem}
Hence, we get
\begin{pro}\label{simple}
Let $f_k$ be a simple sequence on $S^1\times[0,T_k]$. Then after
passing to a subsequence,
$f_k$ has finitely many bubbles. Moreover, we have
$$
\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}
\int_{S^1\times [T,T_k-T]}K_{f_k}d\mu_{f_k}=\,\,\,\,um_{i=1}^m
\int_{S^1\times\mathbb{R}}K_{f^i}d\mu_{f^i},$$
where $f^1$, $\cdots$, $f^m$ are all of the bubbles.
\end{pro}
Next, we prove a property of the order of the bubbles.
\begin{thm} Let $f^1$, $f^2$ be two bubbles. Then
1). If $f^1$ and $f^2$ are of type
$T_0$ and $B_0$ respectively, then
there is at least one catenoid-type bubble
between them.
2). If $f^1$ and $f^2$ are of type
$T_\infty$ and $B_\infty$ respectively, then
there is at least one $S^2$-type bubble
between $f^1$ and $f^{2}$.
\end{thm}
\proof 1).
Suppose $\frac{f_k(\theta,t_k^1+t)-f_k(0,t_k^1)}
{diam\, f_k(S^1\times\{t_k^1\})}\rightharpoonup f^1$, and
$\frac{f_k(\theta,t_k^{2}+t)-f_k(0,t_k^{2})}
{diam\, f_k(S^1\times\{t_k^2\})}\rightharpoonup f^{2}$.
Let $t_k'$ be defined by
\begin{equation}\label{infdiam}
diam\, f_k(S^1\times\{t_k'\})=\inf\{diam\,
f_k(S^1\times\{t\}):t\in[t_k^1+T,t_k^{2}-T]
\},
\end{equation}
where $T$ is sufficiently large. Since $f^1$ is
of type $T_0$ and $f^{2}$ of type $B_0$,
we get
$$\lim_{t\rightarrow+\infty}diam\, f^1(S^1\times\{t\})=0,\,\,\,\, and\,\,\,\,
\lim_{t\rightarrow-\infty}diam\, f^{2}(S^1\times\{t\})=0.$$
Then, we have
$$t_k'-t_k^1\rightarrow+\infty,\,\,\,\, t_k^{2}-t_k'
\rightarrow+\infty.$$
If we set $f_k'(t)=\frac{f_k(\theta,t_k'+t)-
f_k(0,t_k')}{diam\, f_k(S^1\times\{t_k'\})}$,
then $f_k'$ will converge to a bubble $f'$
with
$$diam\, f'(S^1\times \{0\})
=\inf \{diam\,
f'(S^1\times\{t\}):t\in \mathbb{R}
\}=1.$$
Thus, $f'$ is a catenoid type bubble.
2). If we replace \eqref{infdiam} with
\begin{equation*}
diam\, f_k(S^1\times\{t_k'\})=\,\,\,\,up\{diam\,
f_k(S^1\times\{t\}):t\in[t_k^1+T,t_k^{2}-T]
\},
\end{equation*}
we will get 2).
$
\Box$\\
The structure of the bubble tree of a simple sequence
is clear now: {\it The $S^2$ type bubbles stand in
a line, with a unique catenoid type bubble between the two neighboring
$S^2$-type bubbles. There might
exist plain-type bubbles between the neighboring $S^2$ type
and catenoid type bubbles. A $T_0$ type bubble must follow a $B_\infty$ type bubble,
and a $T_\infty$ type bubble must follow a $B_0$ type bubble.}
\,\,\,\,ubsection{Bubble trees for a sequence of immersed $D$}
In this subsection, we will consider a conformal
immersion sequence
$f_k: D\rightarrow \mathbb{R}^n$ with $\mathcal{S}(f_k)=\{0\}$.
We assume that
$f_k(D)$ can be extended to a closed
embedded surface $\Sigma_k$
with
$$\int_{\Sigma_k}(1+|A_{\Sigma_k}|^2)d\mu<\Lambda.$$
Take $z_k$ and $r_k$, s.t.
\begin{equation}\label{top}
\int_{D_{r_k}(z_k)}|A_{f_k}|^2d\mu_{f_k}=4\pi-\epsilon,
\end{equation}
and $\int_{D_r(z)}|A_{f_k}|^2d\mu_{f_k}<4\pi-\epsilon$ for any $r<r_k$ and
$D_{r}(z)\,\,\,\,ubset D_\frac{1}{2}$, where $\epsilon$ is sufficiently small.
We set $f_k'=f_k(z_k+r_kz)-f_k(z_k)$. Then
$\mathcal{S}(f_k',D_L)=\emptyset$ for any $L$.
Thus, we can find $\lambda_k$, s.t.
$\frac{f_k'(z)}{\lambda_k}$ converges weakly to
$f^F$ which is a conformal immersion of $\mathbb{C}$
in $\mathbb{R}^n$. We call $f^F$ the first bubble of $f_k$
at the concentration point $0$.
It will be convenient to
make a conformal change of the domain. Let $(r,
\theta)$ be the polar coordinates centered at $z_k$.
Let $\varphi_k:S^1\times\mathbb{R}^1\rightarrow\mathbb{R}^2$ be the mapping
given by
$$r=e^{-t},\theta=\theta.$$
Then
$$\varphi_k^*(dx^1\otimes dx^1+dx^2\otimes dx^2)=
\frac{1}{r^2}(dt^2+d\theta^2).$$
Thus $f_k\circ\varphi_k$ can be considered as a conformal immersion
of $S^1\times [0,+\infty)$ in $\mathbb{R}^n$. For simplicity,
we will also denote $f_k\circ\varphi_k$ by $f_k$.
Set $T_k=-\log r_k$. Similarly to Lemma \ref{interval},
we have
\begin{lem}\label{interval2}
There is
$t_k^0=0<s_k^1<s_k^2< \cdots< s_k^l=T_k$,
such that $l\leq \frac{\Lambda}{4\pi}$ and
1). $\int_{S^1\times(s_k^i-1,s_k^i+1)}|A_{f_k}|^2\geq 4\pi$;
2). $\lim\limits_{T\rightarrow +\infty}\lim\limits_{
k\rightarrow+\infty}\,\,\,\,up\limits_{t\in [d_k^i+T,d_k^{i+1}-T]}
\int_{S^1\times(t-1,t+1)}|A_{f_k}|^2<4\pi$.
\end{lem}
Let
$f_k^i=f_k(\theta,s_k^i+t)$. A generalized limit of $f_k^i$
is called a bubble with concentration (which may be trivial).
There are $W^{2,2}$-conformal immersions
of $S^1\times\mathbb{R}$ with finite branch points and
finite $L^2$ norm of the second fundamental form.
However, if we neglect the concentration points,
we can also define the types of $T_\infty$, $T_0$, $B_{\infty}$,
and $B_0$ for it.
Obviously, we can find a $T'$, such that
$f_k$ is simple on $S^1\times[s_k^{i}+T',s_k^{i+1}-T']$.
Note that the top of $f_k$ on $S^1\times[s_k^i+T',s_k^{i+1}-T']$
is just a part of a generalized limit of $f_k^{i+1}$ and the
bottom of $f_k$ on $S^\times[s_k^i+T',s_k^{i+1}-T']$
is just a part of a generalized limit of $f_k^{i-1}$.
We call the union of nontrivial bubbles of $f_k$ on each $[s_k^i,s_k^{i+1}]$,
the generalized limit of $f_k^i$ and $f^F$
the first level of bubble tree. By Proposition \ref{simple},
we have
$$\begin{array}{lll}
\lim\limits_{r\rightarrow 0}\lim\limits_{
k\rightarrow+\infty}\displaystyle{\int}_{D_r}K_{f_k}&=&
\,\,\,\,um\limits_{i=1}^{l}\lim\limits_{T\rightarrow+\infty}
\lim\limits_{k\rightarrow+\infty}\displaystyle{\int}_{S^1\times
[s_k^i-T'-T,s_k^i+T'+T]} K_{f_k^i}\\[2.0ex]
&&+
\,\,\,\,um\limits_{i=0}^l\lim\limits_{T\rightarrow+\infty}
\lim\limits_{k\rightarrow+\infty}\displaystyle{\int}_{S^1\times
[s_k^i+T'+T,s_k^{i+1}-T'-T]} K_{f_k^i}\\[2.0ex]
&=&\,\,\,\,um\limits_{(r,\theta)\in \mathcal{S}(\{f_k^i\})}\lim\limits_{r\rightarrow 0}\lim\limits_{k\rightarrow+\infty}
\displaystyle{\int}_{B_r(t,\theta)}K_{f_k^i}+
\,\,\,\,um_j\int_{S^1\times\mathbb{R}}K_{f^j},
\end{array}$$
where $\{f^j\}$ are all the bubbles of the first level.
Next,
at each concentration point of $\{f_k^i\}$, we get
the first level of $\{f_k^i\}$. We
usually call them the second level of bubble trees. Such a construction
will stop after finite steps.
\begin{lem}\label{identity1}After
passing to a subsequence,
$f_k$ has finitely many non-trivial bubbles.
Moreover, for any $r<1$
$$\lim_{k\rightarrow+\infty}
\int_{D_r}K_{f_k}d\mu_{f_k}=\int_{D_r}K_{f^0}d\mu_{f^0}+
\,\,\,\,um_{i=1}^m\int_{S^1\times\mathbb{R}}K_{f^i}d\mu_{f^i},$$
where $f^0$ is the generalized limit of $f_k$, and $f^1$, $f^2$,
$\cdots$, $f^m$ are all of the non-trivial
bubbles.
\end{lem}
\,\,\,\,ubsection{Immersion sequence of cylinder which is not simple}
Now we assume
$f_k$ is not simple on $S^1\times[0,T_k]$. We also assume
$f_k(S^1\times[0,T_k])$ can be extended to a closed
immersed surface $\Sigma_k$
with
$$\int_{\Sigma_k}(1+|A_{\Sigma_k}|^2)d\mu<\Lambda.$$
Moreover, we assume $f_k(t,\theta)$ and $f_k(T_k+t,\theta)$
have no concentration.
Then we still have Lemma \ref{interval2}.
The other properties
are the same as those of the immersion of $D$. Moreover, we have
$$\lim_{k\rightarrow+\infty}
\int_{S^1\times[0,T_k]}K_{f_k}d\mu_{f_k}=\int_{S^1\times[0,+\infty)}K_{f^B}
d\mu_{f^B}+
\int_{S^1\times(-\infty,0]}K_{f^T}d\mu_{f^T}+\,\,\,\,um_{i=1}^m\int_{\mathbb{C}}K_{f^i}d\mu_{f^i},$$
where $f^1$, $\cdots$, $f^m$
are all of the nontrivial bubbles.
\,\,\,\,ection{Proof of Theorem \ref{main}}
Since Theorem \ref{main2} can be deduced directly from
subsection 4.3, and Theorem \ref{sphere} can be deduced
directly from subsection 4.2,
we only prove Theorem \ref{main}.
{\it Proof of Theorem \ref{main}:}
Take a curve $\gamma_i\,\,\,\,ubset\Sigma_0^i\,\,\,\,etminus\mathcal{S}(\{f_k\circ
\psi_k\})$ with $\gamma_i(0)=p_i$. We set $\lambda_i=diam\,f_k(\gamma_i)$,
and $\tilde{f}_k^i=\frac{f_k\circ\psi_k-f_k\circ\psi_k(p_i)}{\lambda_k^i}$
which is a mapping from $\Sigma_0^i$
into $\mathbb{R}^n$. It is easy to
check that $\tilde{f}_k^i\in W^{2,2}_{{conf},loc}
(\Sigma_0^i,\psi_k^{\ast}(h_k),\mathbb{R}^n)$.
Given a point $p\in\Sigma_0^i$. We choose
$U_k$, $U_0$, $\vartheta_k$, $\vartheta_0$ as in the
Theorem \ref{D.K.}.
Let $\hat{f}_k^i=\tilde{f}_k^i(\vartheta_k)$ which
is a map from $D$ into $\mathbb{R}^n$.
Let $V=\vartheta(D_\frac{1}{2})$. Since $\vartheta_k$
converges to $\vartheta_0$, $\vartheta_k^{-1}(V)\,\,\,\,ubset
D_\frac{3}{4}$
for any sufficiently large $k$.
When $p$ is not a concentration point, by Lemma \ref{measureconvergence},
for any
$\varphi$ with $supp\varphi\,\,\,\,ubset\,\,\,\,ubset V$,
we have
$$\int_{V}\varphi K_{\tilde{f}_k^i}d\mu_{\tilde{f}_k^i}
=\int_{D_\frac{3}{4}}\varphi(\vartheta_k)K_{\hat{f}_k^i}
d\mu_{\hat{f}_k^i}\rightarrow
\int_{D_\frac{3}{4}}\varphi(\vartheta_0)
K_{\hat{f}_0^i}=\int_V\varphi K_{f_0^i}d\mu_{f_0^i}.$$
When $p$ is a concentration point, by Lemma \ref{identity1}, we get
$$\int_{V}\varphi K_{\tilde{f}_k^i}d\mu_{\tilde{f}_k^i}
\rightarrow
\int_V\varphi K_{f_0^i}d\mu_{f_0^i}+
\varphi(p)\,\,\,\,um_j\int_{S^1\times \mathbb{R}}K_{f^i_j}d\mu_{f^i_j},$$
where $\{f^i_j\}$ is the set of nontrivial bubbles of $\hat{f}_k^i$
at $p$.
Next, we consider the convergence of $f_k$ at
the collars. Let $a^j$ be the intersection of $\overline{\Sigma_0^i}$
and $\overline{\Sigma_0^{i'}}$.
We set $\check{f}_k^j=f_k(\phi_k^j)$, and $T_k^j
=\frac{\pi^2}{l_k^j}-T$.
We may choose $T$ to be sufficiently large such that
$\check{f}_k^j(T_k^j-t,\theta)$ and $\check{f}_k^j
(-T_k^j+t,\theta)$ have no blowup point. Then $\check{f}_k^j$
satisfies the conditions in subsection 2.4.
So the convergence of $\check{f}_k^j$ is clear.
Since
$$\check{f}_k^j=f_k\circ\phi_k^j=f_k\circ\psi_k\circ(\varphi_k\circ\phi_k^j)=
\tilde{f}_k(\varphi_k\circ\phi_k^j).$$
The images of the limit of $\check{f}_k^j(T_k^j-t,\theta)$ and $\check{f}_k^j
(-T_k^j+t,\theta)$
are parts of the images of $\tilde{f}_0^i$ and $\tilde{f}_0^{i'}$.
Then we have
$$\lim_{\delta\rightarrow 0}
\lim_{k\rightarrow+\infty}\int_{\Sigma_0(\delta,a^j)}
K_{f_k}=\,\,\,\,um_i\int_{S^1\times\mathbb{R}}K_{f^{i'}},$$
where all $f^{i'}$ are nontrivial bubbles of $\check{f}_k^j$.
$
\Box$\\
\,\,\,\,ection{A remark about trivial bubbles}
The methods in section 4 can be also used to find all bubbles
with $\|A\|_{L^2}\geq\epsilon_0$ for a fixed $\epsilon_0>0$.
We only consider the simple sequence $f_k$ on
$S^1\times[0,T_k]$ here.
Let $t_k$ be a sequence with $t_k, T_k-t_k\rightarrow \infty$, such that
$\frac{f_k(t+t_k,\theta)-f_k(t_k,0)}{\lambda_k}$
converges to a $f_0\in W^{2,2}(S^1\times\mathbb{R},\mathbb{R}^n)$ with
$\int_{S^1\times\mathbb{R}}|A_{f_0}|^2\geq\epsilon_0^2$.
Take $T$, such that $\int_{S^1\times[-T,T]}
|A_{f_0}|^2\geq\frac{\epsilon_0^2}{2}$. We consider
the convergence on $S^1\times [0,t_k-T]$
and $S^1\times[t_k+T,T_k]$ respectively.
In this way, we can find out all the bubbles.
\end{document} |
\begin{document}
\begin{abstract}
We prove that a finite group acting by birational automorphisms of a non-trivial Severi--Brauer surface
over a field of characteristic zero contains a normal abelian subgroup of index at most~$3$.
Also, we find an explicit bound for orders of such finite groups in the case when the base field contains all roots of~$1$.
\end{abstract}
\title{Birational automorphisms of Severi--Brauer surfaces}
\section{Introduction}
A \emph{Severi--Brauer variety} of dimension $n$ over a field $\mathbb{K}$ is a variety
that becomes isomorphic to the projective space of dimension $n$ over the algebraic closure of~$\mathbb{K}$.
Such varieties are in one-to-one correspondence with central simple algebras of dimension~$(n+1)^2$
over~$\mathbb{K}$. They have many nice geometric properties. For instance, it is known that
a Severi--Brauer variety over $\mathbb{K}$
is isomorphic to the projective space if and only if it has a $\mathbb{K}$-point.
We refer the reader to \cite{Artin} and~\cite{Kollar-SB} for other basic
facts concerning Severi--Brauer varieties.
Automorphism groups of Severi--Brauer varieties can be described in terms
of the corresponding central simple algebras, see Theorem~E on page~266
of~\cite{Chatelet}, or~\mbox{\cite[\S1.6.1]{Artin}}, or \cite[Lemma~4.1]{ShramovVologodsky}.
As for the group of birational automorphisms, something is known in the case of surfaces.
Namely, let
$\mathbb{K}$ be a field of characteristic zero (more generally, one can assume that either $\mathbb{K}$ is perfect, or
its characteristic is different from $2$ and~$3$). Let $S$ be a \emph{non-trivial} Severi--Brauer surface over $\mathbb{K}$,
i.e. one that is not isomorphic to~$\mathbb{P}^2$.
In this case generators for the group $\operatorname{Bir}(S)$ of birational automorphisms of $S$ are known,
see \cite{Weinstein}, or~\cite{Weinstein-new}, or~\cite[Theorem~2.6]{Is-UMN},
or Theorem~\ref{theorem:Weinstein} below.
Moreover, relations between the generators are known as well, see~\mbox{\cite[\S3]{IskovskikhTregub}}.
This may be thought of as an analog of the classical theorem of Noether describing the generators
of the group $\operatorname{Bir}(\mathbb{P}^2)$ over an algebraically closed field,
and the results concerning relations between them (see \cite{Gizatullin}, \cite{Iskovskikh-SimpleGiz}, \cite{IKT}).
Regarding finite groups acting by automorphisms or birational automorphisms on Severi--Brauer surfaces, the following result is known.
\begin{theorem}[{see \cite[Proposition~1.9(ii),(iii)]{ShramovVologodsky}, \cite[Corollary~1.5]{ShramovVologodsky}}]
\label{theorem:ShramovVologodsky}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero.
Suppose that $\mathbb{K}$ contains all roots of $1$.
The following assertions hold.
\begin{itemize}
\item[(i)] If $G\subset\operatorname{Aut}(S)$ is a finite subgroup,
then every non-trivial element of $G$ has order~$3$, and
$G$ is a $3$-group of order at most~$27$.
\item[(ii)] There exists a constant $B=B(S)$ such that for any finite subgroup $G\subset\operatorname{Bir}(S)$
one has $|G|\le B$.
\end{itemize}
\end{theorem}
In this paper we prove the result making Theorem~\ref{theorem:ShramovVologodsky} more precise.
\begin{theorem}\label{theorem:main}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero,
and let $G\subset\operatorname{Bir}(S)$ be a finite group.
The following assertions hold.
\begin{itemize}
\item[(i)] The order of $G$ is odd.
\item[(ii)] The group $G$ is either abelian, or contains a normal abelian subgroup
of index~$3$.
\item[(iii)] If $\mathbb{K}$ contains all roots of $1$, then
$G$ is an abelian $3$-group of order at most~$27$.
\end{itemize}
\end{theorem}
I do not know if the bounds in Theorem~\ref{theorem:main}(ii),(iii) are optimal.
In particular, I am not aware of an example of a finite non-abelian group
acting by birational (or biregular, cf.~Proposition~\ref{proposition:SB-Bir-vs-Aut} below)
automorphisms on a non-trivial Severi--Brauer surface.
Note that
it is easy to construct an example of a non-trivial Severi--Brauer
surface with an action of a group of order~$9$, see Example~\ref{example:cyclic-algebra} below.
In certain cases this is the largest finite subgroup of the automorphism
group of a Severi--Brauer surface; see Lemma~\ref{lemma:27}, which improves the result
of Theorem~\ref{theorem:ShramovVologodsky}(i).
It would be interesting to obtain a complete description of finite groups
acting by biregular and birational automorphisms
on Severi--Brauer surfaces, cf.~\cite{DI}.
Theorem~\ref{theorem:main}(ii) can be reformulated
by saying that the \emph{Jordan constant}
(see e.g.~\mbox{\cite[Definition~1.1]{Yasinsky}} for a definition)
of the birational automorphism group of a
non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero is at most $3$.
This shows one of the amazing differences between birational geometry of
non-trivial Severi--Brauer surfaces and the projective plane,
since in the latter case the corresponding Jordan constant may be much larger.
For instance, if the base field is algebraically closed, the Jordan constant
of the group of birational automorphisms of $\mathbb{P}^2$ equals~$7200$,
see~\mbox{\cite[Theorem~1.9]{Yasinsky}}. Moreover, by the remark made after
Theorem~5.3 in~\cite{Serre2009} the multiplicative analog of this constant
for $\mathbb{P}^2$
equals~\mbox{$2^{10}\cdot 3^4\cdot 5^2\cdot 7$} in the case of the algebraically closed field
of characteristic zero, while by Theorem~\ref{theorem:main}(ii)
a similar constant for a non-trivial Severi--Brauer surface
also equals either~$1$ or~$3$.
To prove Theorem~\ref{theorem:main}, we establish the following intermediate result that
might be of independent interest (see also Proposition~\ref{proposition:summary} below for a more precise
statement).
\begin{proposition}\label{proposition:SB-Bir-vs-Aut}
Let $S$ be a non-trivial Severi--Brauer surface over a field of characteristic zero,
and let $G\subset\operatorname{Bir}(S)$ be a finite non-abelian group.
Then $G$ is conjugate to a subgroup of $\operatorname{Aut}(S)$.
\end{proposition}
It would be interesting to find out if Proposition~\ref{proposition:SB-Bir-vs-Aut}
holds for finite abelian subgroups of birational automorphism groups
of non-trivial Severi--Brauer surfaces.
The plan of the paper is as follows. In~\S\ref{section:birational} we study surfaces that may be birational to
a non-trivial Severi--Brauer surface.
In~\S\ref{section:G-birational} we study finite groups acting on such surfaces.
In~\S\ref{section:bounds} we prove Proposition~\ref{proposition:SB-Bir-vs-Aut} and Theorem~\ref{theorem:main}.
\textbf{Notation and conventions.}
Throughout the paper assume that all varieties are projective.
We denote by ${\boldsymbol{\mu}}_n$ the cyclic group of order $n$,
and by $\mathfrak{S}_n$ the symmetric group on $n$ letters.
Given a field $\mathbb{K}$, we denote by $\bar{\mathbb{K}}$ its algebraic closure.
For a variety $X$ defined over $\mathbb{K}$, we denote by $X_{\bar{\mathbb{K}}}$
its scalar extension to~$\bar{\mathbb{K}}$.
By a point of degree $d$ on a variety defined over some field $\mathbb{K}$ we mean a closed point whose
residue field is an extension of~$\mathbb{K}$ of degree~$d$; a $\mathbb{K}$-point is a point of degree~$1$.
By a linear system on a variety~$X$ over a field $\mathbb{K}$ we mean a twisted linear subvariety
of a linear system on $X_{\bar{\mathbb{K}}}$ defined over~$\mathbb{K}$; thus, a linear system is not
a projective space in general, but a Severi--Brauer variety.
By a degree of a subvariety $Z$ of a Severi--Brauer variety $X$ over a field $\mathbb{K}$
we mean the degree of the subvariety $Z_{\bar{\mathbb{K}}}$ of $X_{\bar{\mathbb{K}}}\cong\mathbb{P}^n_{\bar{\mathbb{K}}}$ with respect to
the hyperplane in~$\mathbb{P}^n_{\bar{\mathbb{K}}}$.
For a Severi--Brauer variety
$X$ corresponding to the central simple algebra $A$,
we denote by~$X^{\mathrm{op}}$ the Severi--Brauer variety corresponding to the algebra opposite to~$A$.
A del Pezzo surface is a smooth surface with an ample anticanonical class.
For a del Pezzo surface $S$, by its degree we mean its (anti)canonical degree~$K_S^2$.
Let $X$ be a variety over an algebraically closed field with an action of
a group $G$, and let $g$ be an element of $G$. By $\operatorname{Fix}_X(G)$ and $\operatorname{Fix}_X(g)$ we denote
the loci of fixed points of the group $G$ and the element $g$ on $X$, respectively.
\textbf{Acknowledgements.}
I am grateful to A.\,Trepalin and V.\,Vologodsky for many useful discussions.
I was partially supported by
the HSE University Basic Research Program,
Russian Academic Excellence Project~\mbox{``5-100''},
by the Young Russian Mathematics award, and by the Foundation for the
Advancement of Theoretical Physics and Mathematics ``BASIS''.
\section{Birational models of Severi--Brauer surfaces}
\label{section:birational}
In this section we study surfaces that may be birational to
a non-trivial Severi--Brauer surface.
The following general result is sometimes referred to as the theorem of Lang and Nishimura.
\begin{theorem}[{see e.g.~\cite[Proposition~IV.6.2]{Kollar-RatCurves}}]
\label{theorem:Lang-Nishimura}
Let $X$ and $Y$ be smooth projective varieties over an arbitrary field $\mathbb{K}$.
Suppose that $X$ is birational to $Y$. Then $X$ has a $\mathbb{K}$-point if and only if $Y$ has a $\mathbb{K}$-point.
\end{theorem}
\begin{corollary}\label{corollary:Lang-Nishimura}
Let $X$ and $Y$ be smooth projective varieties over an arbitrary field $\mathbb{K}$, and let $r$ be a positive integer.
Suppose that $X$ is birational to $Y$, and $X$ has a point of degree not divisible by~$r$.
Then $Y$ has a point of degree not divisible by~$r$.
\end{corollary}
The following result concerning Severi--Brauer surfaces is well-known.
\begin{theorem}[{see e.g.~\cite[Theorem~53(2)]{Kollar-SB}}]
\label{theorem:SB-point-degree}
Let $S$ be a non-trivial Severi--Brauer surface over an arbitrary field. Then $S$ does not contain points of degree $d$ not divisible by $3$.
\end{theorem}
\begin{corollary}\label{corollary:many-dP}
Let $S$ be a non-trivial Severi--Brauer surface over an arbitrary field. Then
$S$ is not birational to any conic bundle, and not birational to any del Pezzo surface of degree
different from $3$, $6$, and $9$.
\end{corollary}
\begin{proof}
Suppose that $S$ is birational to a surface $S'$ with a conic bundle structure~\mbox{$\phi\colon S'\to C$}.
Then $C$ is a conic itself, so that $C$ has a point of degree $2$. This implies that
the surface $S'$ has a point of degree $2$ or $4$, and by Corollary~\ref{corollary:Lang-Nishimura}
the surface~$S$ has a point of degree not divisible by~$3$.
This gives a contradiction with Theorem~\ref{theorem:SB-point-degree}.
Now suppose that $S$ is birational to a del Pezzo surface $S'$
of degree $d$ not divisible by $3$. Then the intersection of two general elements
of the anticanonical linear system~$|-K_{S'}|$ is an effective zero-cycle of degree $d$
defined over the base field. Thus $S'$ has a point of degree not divisible by $3$.
By Corollary~\ref{corollary:Lang-Nishimura} this again gives a contradiction with Theorem~\ref{theorem:SB-point-degree}.
\end{proof}
\begin{corollary}\label{corollary:dP-large-Picard-rank}
Let $S$ be a non-trivial Severi--Brauer surface over an arbitrary field. Then~$S$
is not birational to any del Pezzo surface of degree $6$ of Picard rank greater than~$2$,
and not birational to any del Pezzo surface of degree $3$ of Picard rank greater than~$3$.
\end{corollary}
\begin{proof}
Suppose that $S$ is birational to a del Pezzo surface $S'$ as above.
Then there exists a birational contraction from $S'$ to a del Pezzo surface
$S''$ of degree $K_{S''}^2>K_{S'}^2$ and Picard rank~\mbox{$\operatorname{rk}Pic(S'')=\operatorname{rk}Pic(S')-1$}.
If $K_{S'}^2=6$ and $\operatorname{rk}Pic(S')>2$, then $\operatorname{rk}Pic(S'')>1$, and thus~\mbox{$6<K_{S''}^2<9$}.
This gives a contradiction with Corollary~\ref{corollary:many-dP}.
If $K_{S'}^2=3$ and $\operatorname{rk}Pic(S')>3$, then~$S''$ is either a del Pezzo surface of degree~$6$
with~\mbox{$\operatorname{rk}Pic(S'')>2$}, which is impossible by the above argument, or a del Pezzo surface of degree
not divisible by $3$, which is impossible by Corollary~\ref{corollary:many-dP}.
\end{proof}
To proceed we will need the following general fact about non-trivial Severi--Brauer surfaces.
\begin{lemma}\label{lemma:SB-surface-curves}
Let $S$ be a non-trivial Severi--Brauer surface over an arbitrary field $\mathbb{K}$, and let~$C$ be a curve on $S$.
Then the degree of $C$ is divisible by $3$.
\end{lemma}
\begin{proof}
It is well-known (see for instance \cite[Exercise~3.3.5(iii)]{GS})
that there exists an exact sequence of groups
$$
1\to\operatorname{Pic}(S)\to\operatorname{Pic}(S_{\bar{\mathbb{K}}})^{\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})}\stackrel{b}\to \operatorname{Br}(\mathbb{K})_3,
$$
where $\operatorname{Br}(\mathbb{K})_3$ is the $3$-torsion part of the Brauer group of $\mathbb{K}$. Furthermore, the image
of~$b$ is non-trivial since the Severi--Brauer surface
$S$ is non-trivial, see for instance \cite[Exercise~3.3.4]{GS}.
This means that any line bundle on $S$ has degree divisible by $3$, and the assertion follows.
\end{proof}
\begin{remark}
For an alternative proof of Lemma~\ref{lemma:SB-surface-curves},
one can consider the image $C'$ of~$C$ under a general
automorphism of $S$
(see \cite[Lemma~4.1]{ShramovVologodsky} for a description of the automorphism group of~$S$).
Then the zero-cycle $Z=C\cap C'$ is defined over $\mathbb{K}$ and has degree coprime to~$3$,
so that the assertion follows from Theorem~\ref{theorem:SB-point-degree}.
\end{remark}
Given $d\le 6$ distinct points $P_1,\ldots,P_d$ on $\mathbb{P}^2_{\bar{\mathbb{K}}}$ over an algebraically closed field $\bar{\mathbb{K}}$, we will say that
they are \emph{in general position} if no three of them are contained in a line,
and all six (in the case $d=6$) are not
contained in a conic (cf.~\mbox{\cite[Remark~IV.4.4]{Manin}}). If~$P$ is a point of degree $d$ on a Severi--Brauer
surface $S$ over a perfect field~$\mathbb{K}$, we will say that~$P$ is in general position if the $d$ points
of the set~\mbox{$P_{\bar{\mathbb{K}}}\subset\mathbb{P}^2_{\bar{\mathbb{K}}}$} are in general position. Note that if $P$ is a point
of degree~$d$ in general
position, then the blow up of $S$ at
$P$ is a del Pezzo surface of degree $9-d$, see
for instance~\mbox{\cite[Theorem~IV.2.6]{Manin}}.
\begin{lemma}\label{lemma:SB-general-position}
Let $S$ be a non-trivial Severi--Brauer surface over a perfect field~$\mathbb{K}$,
and let~\mbox{$P\in S$} be a point of degree $d$. Suppose that $d=3$ or $d=6$. Then
$P$ is in general position.
\end{lemma}
\begin{proof}
Suppose that $d=3$ and $P$ is not in general position. Then the three points
of $P_{\bar{\mathbb{K}}}$ are contained in some line $L$ on $\mathbb{P}^2_{\bar{\mathbb{K}}}$.
The line $L$ is $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant
and thus defined over $\mathbb{K}$,
which is impossible by Lemma~\ref{lemma:SB-surface-curves}.
Now suppose that $d=6$ and $P$ is not in general position. Let $P_{\bar{\mathbb{K}}}=\{P_1,\ldots,P_6\}$.
If at least four of the points~\mbox{$P_1,\ldots,P_6$} are contained in a line, then a line with such a property is unique,
and we again get a contradiction with Lemma~\ref{lemma:SB-surface-curves}.
Assume that no four of the points~\mbox{$P_1,\ldots,P_6$} are contained in a line, but some three of them (say,
the points~$P_1$, $P_2$, and $P_3$) are contained in a line which we denote by $L$. The line $L$ is not $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant by Lemma~\ref{lemma:SB-surface-curves},
so that there exists a line $L'\neq L$ that is $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-conjugate to~$L$.
If $L'$ does not pass through any of the points $P_1$, $P_2$, and~$P_3$, then $L$ and $L'$ are the only
lines in $\mathbb{P}^2_{\bar{\mathbb{K}}}$ that contain three of the points $P_1,\ldots,P_6$. This gives a contradiction
with Lemma~\ref{lemma:SB-surface-curves}. Hence, up to relabelling the points, we may assume that $P_3=L\cap L'$,
and the points $P_4$ and $P_5$ are contained in~$L'$.
Let $L_{ij}$ be the line passing through the points $P_i$ and $P_j$, where $i\in\{1,2\}$ and $j\in\{4,5\}$.
Note that $L_{ij}$ are pairwise different, none of them coincides with $L$ or $L'$, and no three of them
intersect at one point.
If the point $P_6$ is not contained in any of the lines $L_{ij}$, then
$L$ and $L'$ are the only
lines that contain three of the points $P_1,\ldots,P_6$, which is impossible by Lemma~\ref{lemma:SB-surface-curves}.
If $P_6$ is an intersection point of two of the lines $L_{ij}$, then
there are exactly four
lines that contain three of the points $P_1,\ldots,P_6$, which is also impossible by Lemma~\ref{lemma:SB-surface-curves}.
Thus, we see that $P_6$ must be contained in a unique line among $L_{ij}$, say, in $L_{15}$. Now there are exactly three lines
that contain three of the points $P_1,\ldots,P_6$, namely, $L$, $L'$, and $L_{15}$.
We see that each of the points $P_1$, $P_3$, and $P_5$ is contained in two of these lines,
while each of the points $P_2$, $P_4$, and $P_6$ is contained in a unique such line. However, the Galois group~\mbox{$\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$} acts
transitively on the points~\mbox{$P_1,\ldots, P_6$}, which gives a contradiction.
Therefore, we may assume that the points $P_1,\ldots, P_6$ are contained in some irreducible conic~$C$.
Obviously, such a conic is unique, and thus $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant. This again gives a contradiction
with Lemma~\ref{lemma:SB-surface-curves}.
\end{proof}
Let $S$ be a non-trivial Severi--Brauer surface over a perfect field~$\mathbb{K}$, and let~$A$
be the corresponding central simple algebra.
Recall that $S^{\mathrm{op}}$ the Severi--Brauer surface corresponding to the algebra opposite to~$A$.
There are two classes of interesting birational maps from $S$ to $S^{\mathrm{op}}$ which we describe below
(cf. \cite[Lemma~3.1]{IskovskikhTregub}, or cases~(c) and~(e) of~\cite[Theorem~2.6(ii)]{Is-UMN}).
Let $P$ be a point of degree $3$ on $S$. Then $P$ is in general position by Lemma~\ref{lemma:SB-general-position}.
Blowing up $P$ and blowing down the proper transforms of the three lines on $S_{\bar{\mathbb{K}}}\cong\mathbb{P}^2_{\bar{\mathbb{K}}}$
passing through the pairs of points of $P_{\bar{\mathbb{K}}}$,
we obtain a birational map $\tau_P$ to another Severi--Brauer surface $S'$.
This map is given by the linear system of conics passing through $P$.
Similarly, let $P$ be a point of degree $6$ on $S$. Then $P$ is in general position by Lemma~\ref{lemma:SB-general-position}.
Blowing up $P$ and blowing down the proper transforms of the six conics on~\mbox{$S_{\bar{\mathbb{K}}}\cong\mathbb{P}^2_{\bar{\mathbb{K}}}$}
passing through the quintuples of points of $P_{\bar{\mathbb{K}}}$,
we obtain a birational map $\eta_P$ to another Severi--Brauer surface $S'$.
This map is given by the linear system of quintics singular at~$P$ (or, in other words, at each point of~$P_{\bar{\mathbb{K}}}$).
In both of the above cases one has $S'\cong S^{\mathrm{op}}$. This follows from a well-known general
fact about the degrees of birational maps between Severi--Brauer varieties with given classes,
see for instance~\cite[Exercise~3.3.7(iii)]{GS}.
For more details on the maps $\tau_P$, see~\mbox{\cite[\S2]{Corn}}.
\begin{remark}\label{remark:Amitsur}
It is known that $S$ and $S^{\mathrm{op}}$ are the only Severi--Brauer surfaces birational to~$S$, see for instance~\cite[Exercise~3.3.6(v)]{GS}.
\end{remark}
The following theorem was first published in~\cite{Weinstein}
(see also~\cite{Weinstein-new}). It can be also obtained
as a particular case of a much more general result, see~\cite[Theorem~2.6]{Is-UMN}.
We provide a sketch of its proof for the reader's convenience.
\begin{theorem}\label{theorem:Weinstein}
Let $S$ be a non-trivial Severi--Brauer surface over a perfect field $\mathbb{K}$, and let $S'$ be
a del Pezzo surface over $\mathbb{K}$ with $\operatorname{rk}Pic(S')=1$.
Suppose that $S$ is birational to~$S'$. Then either
$S'\cong S$, or $S'\cong S^{\mathrm{op}}$. Moreover, any birational map~\mbox{$\theta\colon S\dasharrow S'$}
can be written as a composition
$$
\theta=\theta_1\circ\ldots\circ\theta_k,
$$
where each of the maps $\theta_i$ is either an automorphism, or a map $\tau_P$ or $\eta_P$ for some point~$P$ of
$S$ or~$S^{\mathrm{op}}$.
\end{theorem}
\begin{proof}[Sketch of the proof]
Let $\theta\colon S\dasharrow S'$ be a birational map, and suppose that $\theta$ is not
an isomorphism.
Choose a very ample linear system ${\mathscr{L}}'$ on $S'$, and let ${\mathscr{L}}$ be its proper transform
on $S$. Then ${\mathscr{L}}$ is a mobile non-empty (and in general incomplete)
linear system on $S$. Write
$$
{\mathscr{L}}\sim_{\mathbb{Q}} -\mu K_S
$$
for some positive rational number $\mu$; note that $3\mu$ is an integer.
By the Noether--Fano inequality
(see \cite[Lemma~1.3(i)]{Is-UMN}),
one has $\operatorname{mult}_{P}({\mathscr{L}})>\mu$ for some point $P$ on $S$. Let~$d$ be the degree of $P$, and let
$L_1$ and $L_2$ be two general members of the linear system~${\mathscr{L}}$ (defined over~$\bar{\mathbb{K}}$).
We see that
$$
9\mu^2=L_1\cdot L_2\ge d\operatorname{mult}_{P}({\mathscr{L}})^2>d\mu^2,
$$
and thus $d<9$. Hence one has $d=3$ or $d=6$ by Theorem~\ref{theorem:SB-point-degree},
and the point $P$ is in general position by Lemma~\ref{lemma:SB-general-position}.
Consider a birational map $\theta_P$ defined as follows: if $d=3$, we let $\theta_P=\tau_P$,
and if~\mbox{$d=6$}, we let $\theta_P=\eta_P$. Let $\theta^{(1)}=\theta\circ\theta_P^{-1}$. Let
${\mathscr{L}}_1$ be the proper transform of ${\mathscr{L}}$ (or ${\mathscr{L}}'$) on the
surface~$S^{\mathrm{op}}$, and write
$$
{\mathscr{L}}_1\sim_{\mathbb{Q}} -\mu_1 K_{S^{\mathrm{op}}}
$$
for some positive rational number $\mu_1$ such that~\mbox{$3\mu_1\in\mathbb{Z}$}.
Using the information about~$\theta_P$ provided in~\cite[Lemma~3.1]{IskovskikhTregub}, we see that
$\mu_1<\mu$.
Therefore, applying the same procedure to the surface~\mbox{$S_1=S^{\mathrm{op}}$}, the birational map $\theta^{(1)}$,
and the linear system ${\mathscr{L}}_1$ and arguing by induction, we prove the theorem.
\end{proof}
A particular case of Theorem~\ref{theorem:Weinstein} is the following result that we will need below.
\begin{corollary}\label{corollary:Weinstein}
Let $S$ be a non-trivial Severi--Brauer surface over a perfect field.
Then~$S$ is not birational to any del Pezzo surface $S'$ of degree~$3$ or~$6$
with~\mbox{$\operatorname{rk}Pic(S')=1$}.
\end{corollary}
The part of Corollary~\ref{corollary:Weinstein} concerning del Pezzo surfaces of degree $3$ also follows
from~\mbox{\cite[Chapter~V]{Manin}}.
The part concerning del Pezzo surfaces of degree $6$ can be obtained from
\cite[\S2]{IskovskikhTregub}.
Corollaries~\ref{corollary:dP-large-Picard-rank} and \ref{corollary:Weinstein} show that a del Pezzo surface
of degree $6$ birational to a non-trivial Severi--Brauer
surface must have Picard rank equal to~$2$, and a del Pezzo surface
of degree $3$ birational to a non-trivial Severi--Brauer
surface must have Picard rank equal to~$2$ or~$3$.
In the next section we will obtain further restrictions on such surfaces provided
that they are $G$-minimal with respect to some finite group~$G$.
\section{$G$-birational models of Severi--Brauer surfaces}
\label{section:G-birational}
In this section we study finite groups acting on surfaces birational
to a non-trivial Severi--Brauer surface.
We start with del Pezzo surfaces of degree $6$. Recall that over an algebraically closed field $\bar{\mathbb{K}}$
of characteristic zero
a del Pezzo surface of degree $6$ is unique up to isomorphism, and its
automorphism group is isomorphic to $(\bar{\mathbb{K}}^*)^2\rtimes (\mathfrak{S}_3\times{\boldsymbol{\mu}}_2)$.
More details on this can be found in~\cite[Theorem~8.4.2]{Dolgachev}.
Given a del Pezzo surface $S'$ of degree $6$ over an arbitrary field $\mathbb{K}$ of
characteristic zero, we will call $\mathfrak{S}_3\times{\boldsymbol{\mu}}_2$ its \emph{Weyl group}.
For every element $\theta\in\operatorname{Aut}(S')$ we will refer to the image of $\theta$
under the composition of the embedding $\operatorname{Aut}(S')\hookrightarrow\operatorname{Aut}(S'_{\bar{\mathbb{K}}})$ with the natural homomorphism
$$
\operatorname{Aut}(S'_{\bar{\mathbb{K}}})\to\mathfrak{S}_3\times{\boldsymbol{\mu}}_2
$$
as the image of $\theta$ in the Weyl group. Similarly, we will consider the image of the Galois group
$\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$ in the Weyl group.
Let $\sigma\colon\mathbb{P}^2\dasharrow\mathbb{P}^2$ be the \emph{standard Cremona involution},
that is, a birational involution acting as
$$
(x:y:z)\mapsto\left(\frac{1}{x}:\frac{1}{y}:\frac{1}{z}\right)
$$
in some homogeneous coordinates $x$, $y$, and $z$. This involution becomes regular
on the del Pezzo surface of degree $6$ obtained as a blow up of $\mathbb{P}^2$ at the points
$(1:0:0)$, $(0:1:0)$, and $(0:0:1)$. Let $\hat{\sigma}$ be its image in the Weyl group $\mathfrak{S}_3\times{\boldsymbol{\mu}}_2$.
Then $\hat{\sigma}$ is the generator of the center ${\boldsymbol{\mu}}_2$ of the Weyl group.
If one thinks about~\mbox{$\mathfrak{S}_3\times{\boldsymbol{\mu}}_2$}
as the group of symmetries of a regular hexagon, then $\hat{\sigma}$ is an involution
that interchanges the opposite sides of the hexagon or, in other words, a rotation by~$180^\circ$.
If $S'$ is a del Pezzo surface of degree $6$ over some field of characteristic zero and $\theta$ is its automorphism,
we will say that $\theta$ is \emph{of Cremona type} if its image in the Weyl group
coincides with $\hat{\sigma}$.
\begin{lemma}\label{lemma:Cremona-type}
Let $S'$ be a del Pezzo surface of degree $6$ over a field $\mathbb{K}$ of characteristic zero, and let $\theta$
be its automorphism of Cremona type. Then $\theta$ is an involution that has exactly four fixed points on~$S'_{\bar{\mathbb{K}}}$.
\end{lemma}
\begin{proof}
Using a birational contraction $S'_{\bar{\mathbb{K}}}\to\mathbb{P}^2_{\bar{\mathbb{K}}}$,
consider the automorphism $\theta$
as a birational automorphism of $\mathbb{P}^2_{\bar{\mathbb{K}}}$.
Then $\theta$ can be represented as a composition of the standard Cremona
involution $\sigma$ with some element of the standard torus acting on $\mathbb{P}^2_{\bar{\mathbb{K}}}$.
Thus, one can choose homogeneous coordinates $x$, $y$, and $z$
on $\mathbb{P}^2_{\bar{\mathbb{K}}}$ so that $\theta$ acts as
$$
(x:y:z)\mapsto\left(\frac{\alpha}{x}:\frac{\beta}{y}:\frac{\gamma}{z}\right)
$$
for some non-zero $\alpha,\beta,\gamma\in\bar{\mathbb{K}}$.
Therefore, $\theta$ is conjugate to the involution~$\sigma$ via the automorphism
$$
(x:y:z)\mapsto (\sqrt{\alpha}x:\sqrt{\beta}y:\sqrt{\gamma}z).
$$
This shows that $\theta$ has the same number of fixed points on $\mathbb{P}^2_{\bar{\mathbb{K}}}$ as $\sigma$,
while it is easy to see that the fixed points of the latter are the four points
$$
(1:1:1),\quad (-1:1:1), \quad (1:-1:1),\quad (1:1:-1).
$$
It remains to notice that a fixed point of $\theta$ on $S'_{\bar{\mathbb{K}}}$ cannot be contained
in a $(-1)$-curve, and thus all of them are mapped to fixed points of $\theta$ on~$\mathbb{P}^2_{\bar{\mathbb{K}}}$.
\end{proof}
\begin{lemma}\label{lemma:dP6-rk-1}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero, and let
$S'$ be a del Pezzo surface of degree $6$ over $\mathbb{K}$. Suppose that there exists a finite group
$G\subset\operatorname{Aut}(S')$ such that $\operatorname{rk}Pic(S')^G=1$. Then $S'$ is not birational to $S$.
\end{lemma}
\begin{proof}
Suppose that $S'$ is birational to $S$.
We know from Corollary~\ref{corollary:dP-large-Picard-rank} that~\mbox{$\operatorname{rk}Pic(S')\le 2$}.
Hence by Corollary~\ref{corollary:Weinstein} one has~\mbox{$\operatorname{rk}Pic(S')=2$}.
Since $S'$ does not have a structure of a conic bundle by Corollary~\ref{corollary:many-dP}, we see that
$S'$ has a contraction on a del Pezzo surface of larger degree.
Again by Corollary~\ref{corollary:many-dP},
this means that $S'$ is a blow up of a Severi--Brauer surface at a point of degree~$3$.
Hence the image $\Gamma$ of the Galois group~\mbox{$\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$} in the Weyl group~\mbox{$\mathfrak{S}_3\times{\boldsymbol{\mu}}_2$}
contains an element of order~$3$.
Since the cone of effective curves
on $S'$ has two extremal rays and $\operatorname{rk}Pic(S')^G=1$, we see that $G$ must contain an element
whose image in the Weyl group has order~$2$. On the other hand,
the image of $G$ in the Weyl group commutes with~$\Gamma$. Since $\hat{\sigma}$
is the only element of order $2$ in~\mbox{$\mathfrak{S}_3\times{\boldsymbol{\mu}}_2$} that commutes with an element
of order~$3$, we conclude that $G$
must contain an element $\theta$
of Cremona type. The element~$\theta$ has exactly $4$ fixed points on $S'_{\bar{\mathbb{K}}}$ by Lemma~\ref{lemma:Cremona-type}.
This implies that there is a point of degree not divisible by~$3$ on $S'$.
Thus the assertion follows from Corollary~\ref{corollary:Lang-Nishimura} and Theorem~\ref{theorem:SB-point-degree}.
\end{proof}
Now we deal with del Pezzo surfaces of degree $3$, i.e. smooth cubic surfaces in $\mathbb{P}^3$. Recall from that for a del Pezzo surface
$S'$ of degree $3$ over a field $\mathbb{K}$ the action of the groups~\mbox{$\operatorname{Aut}(S')$} and $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$ on $(-1)$-curves on $S'$ defines homomorphisms
of these groups to the Weyl group~$\mathrm{W}(\mathrm{E}_6)$. Furthermore, the homomorphism
$\operatorname{Aut}(S')\to\mathrm{W}(\mathrm{E}_6)$ is an embedding. The order of the Weyl group~$\mathrm{W}(\mathrm{E}_6)$ equals~$2^7\cdot 3^4\cdot 5$.
We refer the reader to~\cite[Theorem~8.2.40]{Dolgachev}
and~\cite[Chapter~IV]{Manin} for details.
\begin{lemma}\label{lemma:dP3-not-3-group}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero, and let
$S'$ be a del Pezzo surface of degree $3$ over $\mathbb{K}$. Suppose that there exists a non-trivial automorphism
$g\in\operatorname{Aut}(S')$ such that the order of $g$ is not a power of $3$. Then~$S'$ is not birational to $S$.
\end{lemma}
\begin{proof}
We may assume that the order $p=\mathrm{ord}(g)$ is prime. Thus one has $p=2$ or~\mbox{$p=5$}, since the order of the Weyl group
$\mathrm{W}(\mathrm{E}_6)$ is not divisible by primes greater than~$5$.
The action of $g$ on $S'_{\bar{\mathbb{K}}}$ can be of one of the three types listed in
\cite[Table~2]{Trepalin-cubic}; in the notation of \cite[Table~2]{Trepalin-cubic} these are
types~1, 2, and~6. It is straightforward to check
that if $g$ is of type $1$, then the fixed point locus
$\operatorname{Fix}_{S'_{\bar{\mathbb{K}}}}(g)$ consists of a smooth elliptic curve and one isolated point;
if $g$ is of type $2$, then $\operatorname{Fix}_{S'_{\bar{\mathbb{K}}}}(g)$ consists of a $(-1)$-curve and three isolated points;
if $g$ is of type $6$, then $\operatorname{Fix}_{S'_{\bar{\mathbb{K}}}}(g)$ consists of a four isolated points. Note that a $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant
$(-1)$-curve on $S'$ always contains a $\mathbb{K}$-point, since such a curve is a line
in the anticanonical embedding of $S'$. Therefore, in each of the above three cases we
find a $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant set of points on $S'_{\bar{\mathbb{K}}}$ of cardinality coprime to~$3$.
Thus the assertion follows from Corollary~\ref{corollary:Lang-Nishimura} and Theorem~\ref{theorem:SB-point-degree}.
\end{proof}
\begin{corollary}\label{corollary:dP3-rkPic-3}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero, and let
$S'$ be a del Pezzo surface of degree $3$ over $\mathbb{K}$ birational to $S$.
Suppose that there exists a subgroup $G\subset\operatorname{Aut}(S')$ such that
$\operatorname{rk}Pic(S')^G=1$. Then~$\operatorname{rk}Pic(S')=3$.
\end{corollary}
\begin{proof}
We know from Corollaries~\ref{corollary:dP-large-Picard-rank} and
\ref{corollary:Weinstein} that
either $\operatorname{rk}Pic(S')=2$, or $\operatorname{rk}Pic(S')=3$.
Suppose that $\operatorname{rk}Pic(S')=2$, so that the cone of effective curves
on $S'$ has two extremal rays. Since $\operatorname{rk}Pic(S')^G=1$, the group $G$ must contain an element of even order.
Therefore, the assertion follows from Lemma~\ref{lemma:dP3-not-3-group}.
\end{proof}
\begin{corollary}\label{corollary:dP3-3-group}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero, and let
$S'$ be a del Pezzo surface of degree $3$ over $\mathbb{K}$ birational to $S$.
Let~\mbox{$G\subset\operatorname{Aut}(S')$} be a subgroup such that $\operatorname{rk}Pic(S')^G=1$. Then $G$ is isomorphic to
a subgroup of~${\boldsymbol{\mu}}_3^3$.
\end{corollary}
\begin{proof}
We know from Corollary~\ref{corollary:dP3-rkPic-3} that $\operatorname{rk}Pic(S')=3$.
Hence $S'$ is a blow up of a Severi--Brauer surface at two points of degree $3$.
This means that the image $\Gamma$ of the Galois group~\mbox{$\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$} in
the Weyl group $\mathrm{W}(\mathrm{E}_6)$ contains an element conjugate to
$$
\gamma=(123)(456)
$$
in the notation of~\cite[\S4]{Trepalin-cubic}.
We may assume that $\Gamma$ contains $\gamma$ itself, so that
the image of~$G$ in $\mathrm{W}(\mathrm{E}_6)$ is contained in the centralizer $Z(\gamma)$
of $\gamma$. By~\mbox{\cite[Proposition~4.5]{Trepalin-cubic}}
one has
$$
Z(\gamma)\cong({\boldsymbol{\mu}}_3^2\rtimes{\boldsymbol{\mu}}_2)\times \mathfrak{S}_3.
$$
On the other hand, we know from Lemma~\ref{lemma:dP3-not-3-group} that the order of $G$ is a power of $3$.
Since the Sylow $3$-subgroup of $Z(\gamma)$ is isomorphic to ${\boldsymbol{\mu}}_3^3$, the required
assertion follows.
\end{proof}
\begin{remark}
For an alternative proof of Corollary~\ref{corollary:dP3-3-group}, suppose that $G$ is a non-abelian $3$-group.
One can notice that
in this case the image of $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$ in $\mathrm{W}(\mathrm{E}_6)$ is
either trivial, or is generated by an element of type~$\mathrm{A}_2^3$ in the notation of
\cite{Carter}. In the former case $\operatorname{rk}Pic(S')=7$, which is impossible by
Corollary~\ref{corollary:dP-large-Picard-rank}.
In the latter case $\operatorname{rk}Pic(S')=1$, which is impossible by Corollary~\ref{corollary:Weinstein}.
Thus, $G$ is an abelian $3$-group. The rest is more or less straightforward, cf.~\mbox{\cite[Theorem~6.14]{DI}}.
\end{remark}
Let us summarize the results of this section.
\begin{proposition}\label{proposition:summary}
Let $S$ be a non-trivial Severi--Brauer surface over a field $\mathbb{K}$ of characteristic zero, and let
$G\subset\operatorname{Bir}(S)$ be a finite subgroup.
Then $G$ is conjugate either to a subgroup of
$\operatorname{Aut}(S)$, or to a subgroup of
$\operatorname{Aut}(S^{op})$, or to a subgroup of
$\operatorname{Aut}(S')$, where $S'$ is a del Pezzo surface of degree $3$ over $\mathbb{K}$ birational to $S$
such that~\mbox{$\operatorname{rk}Pic(S')=3$} and~\mbox{$\operatorname{rk}Pic(S')^G=1$}.
In the latter case $G$ is isomorphic to
a subgroup of~${\boldsymbol{\mu}}_3^3$.
\end{proposition}
\begin{proof}
Regularizing the action of $G$ and running a $G$-Minimal Model Program (see~\mbox{\cite[Theorem~1G]{Iskovskikh80}}),
we obtain a $G$-surface
$S'$ birational to $S$, such that~$S'$ is either a del Pezzo surface with
$\operatorname{rk}Pic(S')^G=1$, or a conic bundle. The case of a conic bundle is impossible by Corollary~\ref{corollary:many-dP}.
Thus by Corollary~\ref{corollary:many-dP} and Lemma~\ref{lemma:dP6-rk-1}
we conclude that $S'$ is a del Pezzo surface of degree $9$ or $3$.
In the former case $S'$ is a Severi--Brauer surface itself, so
that $S'$ is isomorphic either to $S$ or to $S^{\mathrm{op}}$ by Remark~\ref{remark:Amitsur}.
In the latter case
we have $\operatorname{rk}Pic(S')=3$ by Corollary~\ref{corollary:dP3-rkPic-3}. Furthermore, in this case~$G$ is isomorphic to
a subgroup of~${\boldsymbol{\mu}}_3^3$ by
Corollary~\ref{corollary:dP3-3-group}.
\end{proof}
I do not know the answer to the following question.
\begin{question}
Does there exist an example of a finite abelian group $G$
acting on a smooth cubic surface~$S'$ over a field of characteristic zero,
such that $S'$ is birational to a non-trivial
Severi--Brauer surface and~\mbox{$\operatorname{rk}Pic(S')^G=1$}?
In other words, does there exist a non-trivial Severi--Brauer surface
$S$ over a field of characteristic zero and a finite abelian
group~\mbox{$G\subset\operatorname{Bir}(S)$}, such that
the action of $G$ can be regularized on
some smooth cubic surface, but
$G$ is not conjugate to a subgroup of~\mbox{$\operatorname{Aut}(S)$}?
\end{question}
\section{Automorphisms of Severi--Brauer surfaces}
\label{section:bounds}
In this section we prove
Proposition~\ref{proposition:SB-Bir-vs-Aut}
and Theorem~\ref{theorem:main}.
We start with a couple of simple auxiliary results.
Let ${\mathscr{H}}_3$ denote the Heisenberg group of order $27$; this is the only non-abelian
group of order $27$ and exponent $3$. Its center $\operatorname{z}({\mathscr{H}}_3)$ is isomorphic to ${\boldsymbol{\mu}}_3$,
and there is a non-split exact sequence
$$
1\to\operatorname{z}({\mathscr{H}}_3)\to{\mathscr{H}}_3\to{\boldsymbol{\mu}}_3^2\to 1.
$$
On the other hand, one can also represent ${\mathscr{H}}_3$ as a semi-direct product ${\mathscr{H}}_3\cong{\boldsymbol{\mu}}_3^2\rtimes{\boldsymbol{\mu}}_3$.
\begin{lemma}\label{lemma:P2}
Let $\bar{\mathbb{K}}$ be an algebraically closed field of characteristic zero,
and let
$$
G\subset\operatorname{Aut}(\mathbb{P}^2)\cong\operatorname{PGL}_3(\bar{\mathbb{K}})
$$
be a finite subgroup.
The following assertions hold.
\begin{itemize}
\item[(i)] If the order of $G$ is odd, then $G$ is either abelian, or contains a normal abelian subgroup
of index~$3$.
\item[(ii)] If the order of $G$ is odd and $G$ is non-trivial,
then $G$ contains a subgroup $H$ such that either~\mbox{$|\operatorname{Fix}_{\mathbb{P}^2}(H)|=3$},
or $\operatorname{Fix}_{\mathbb{P}^2}(H)$ has a unique isolated point.
Moreover, one can choose $H$ with such a property so that either $H=G$,
or $H$ is a normal subgroup of index $3$ in~$G$.
\item[(iii)] If $G\cong{\mathscr{H}}_3$, then $G$ contains an element $g$ such that
$\operatorname{Fix}_{\mathbb{P}^2}(g)$ has a unique isolated point.
\item[(iv)] The group $G$ is not isomorphic to ${\boldsymbol{\mu}}_3^3$.
\end{itemize}
\end{lemma}
\begin{proof}
Let $\tilde{G}$ be the preimage of $G$ under the natural projection
$$
\pi\colon\operatorname{SL}_3(\bar{\mathbb{K}})\to\operatorname{PGL}_3(\bar{\mathbb{K}}),
$$
and let $V\cong\bar{\mathbb{K}}^3$ be the corresponding three-dimensional representation of $\tilde{G}$.
We will use the classification of finite subgroups
of $\operatorname{PGL}_3(\bar{\mathbb{K}})$, see for instance~\cite[Chapter~V]{Blichfeldt}.
Suppose that the order of $G$ is odd. Then it follows from the classification that either~$\tilde{G}$ is abelian, so that $V$ splits as a sum of three one-dimensional
$\tilde{G}$-representations such that not all of them are isomorphic to each other; or $\tilde{G}$ is non-abelian and there exists a
surjective homomorphism $\tilde{G}\to{\boldsymbol{\mu}}_3$ whose kernel $\tilde{H}$ is an abelian group,
so that $V$ splits as a sum of three one-dimensional
$\tilde{H}$-representations, and $\tilde{G}/\tilde{H}\cong{\boldsymbol{\mu}}_3$ transitively permutes these
$\tilde{H}$-representations. In other words, the group $G$ cannot be primitive (see~\mbox{\cite[\S60]{Blichfeldt}}
for the terminology), and $V$ cannot split as a sum of a one-dimensional and an irreducible two-dimensional
$\tilde{G}$-representation.
This proves assertion~(i).
If $\tilde{G}$ is abelian, let $\tilde{H}=\tilde{G}$. Thus,
if $|G|$ is odd and $G$ is non-trivial,
in all possible cases we see that the group
$H=\pi(\tilde{H})\subset\operatorname{PGL}_3(\bar{\mathbb{K}})$ is a group with the properties required
in assertion~(ii).
Suppose that $G\cong{\mathscr{H}}_3$.
Then it follows from the classification (cf.~\cite[6.4]{Borel}) that
$$
\tilde{G}\cong{\boldsymbol{\mu}}_3^3\rtimes{\boldsymbol{\mu}}_3,
$$
and $\tilde{H}\cong{\boldsymbol{\mu}}_3^3$.
The elements of $\tilde{H}$ can be simultaneously diagonalized.
Hence the image~\mbox{$H\cong{\boldsymbol{\mu}}_3^2$} of $\tilde{H}$ in $\operatorname{PGL}_3(\bar{\mathbb{K}})$
contains an element $g$
such that $\operatorname{Fix}_{\mathbb{P}^2}(h)$ consists of a line and an isolated point.
This proves assertion~(iii).
Assertion~(iv) directly follows from the classification (cf.~\cite[6.4]{Borel}).
\end{proof}
Most of our remaining arguments are based on the following observation.
\begin{lemma}
\label{lemma:SB-Aut}
Let $S$ be a non-trivial Severi--Brauer surface over a field of characteristic zero, and
let $G\subset\operatorname{Aut}(S)$ be a finite subgroup. Then the order of $G$ is odd.
\end{lemma}
\begin{proof}
Suppose that the order of $G$ is even. Then $G$ contains an element $g$ of order $2$.
Consider the action of $g$ on $S_{\bar{\mathbb{K}}}\cong\mathbb{P}^2_{\bar{\mathbb{K}}}$.
The fixed point locus $\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(g)$ is a union of a line and a unique isolated point $P$. Since the
Galois group $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$ commutes with $g$, the point $P$ is $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant,
which is impossible by assumption.
\end{proof}
\begin{corollary}
\label{corollary:SB-Aut}
Let $S$ be a non-trivial Severi--Brauer surface over a field of characteristic zero, and
let $G\subset\operatorname{Aut}(S)$ be a finite subgroup. Then
\begin{itemize}
\item[(i)] the group $G$ is either abelian, or contains a normal abelian subgroup
of index~$3$;
\item[(ii)] there exists a $G$-invariant point of degree $3$ on $S$.
\end{itemize}
\end{corollary}
\begin{proof}
By Lemma~\ref{lemma:SB-Aut}, the order of $G$ is odd.
The action of $G$ on $S_{\bar{\mathbb{K}}}\cong\mathbb{P}^2_{\bar{\mathbb{K}}}$ gives
an embedding $G\subset\operatorname{PGL}_3(\bar{\mathbb{K}})$.
By Lemma~\ref{lemma:P2}(i)
the group $G$ is either abelian, or contains a normal abelian subgroup
of index~$3$. This proves assertion~(i).
To prove assertion~(ii), we may assume that $G$ is non-trivial.
By Lemma~\ref{lemma:P2}(ii), the group $G$ contains a subgroup $H$ such that either $|\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(H)|=3$,
or $\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(H)$ has a unique isolated point; moreover, one can choose $H$ with such a property so that either~$H$ coincides with~$G$,
or $H$ is a normal subgroup of index $3$ in~$G$.
In any case, $\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(H)$ cannot have a unique isolated point, because this
point would be $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant,
which is impossible by assumption.
Hence $|\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(H)|=3$. Since $H$ is a normal subgroup in $G$, the set $\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(H)$
is $G$-invariant. This gives a $G$-invariant point of degree $3$ on~$S$ and proves assertion~(ii).
\end{proof}
\begin{remark}
It is interesting to note that the analogs of Lemma~\ref{lemma:SB-Aut} and Corollary~\ref{corollary:SB-Aut}
do not hold for Severi--Brauer curves, i.e. for conics. Thus, a conic over the field
$\mathbb{R}$ of real numbers defined by the equation
$$
x^2+y^2+z^2=0
$$
in $\mathbb{P}^2$ with homogeneous coordinates $x$, $y$, and $z$ has no $\mathbb{R}$-points. However, it
is acted on by all finite groups embeddable into $\operatorname{PGL}_2(\mathbb{C})$,
that is, by cyclic groups, dihedral groups, the tetrahedral group~$\mathfrak{A}_4$, the octahedral group~$\mathfrak{S}_4$, and the icosahedral group~$\mathfrak{A}_5$.
From this point of view finite groups acting on non-trivial Severi--Brauer curves
are \emph{more} complicated than those acting on non-trivial Severi--Brauer surfaces.
It would be interesting to obtain a complete classification of finite groups acting on Severi--Brauer surfaces
similarly to what is done for conics in~\cite{Garcia-Armas}.
\end{remark}
Recall from~\S\ref{section:birational} that a Severi--Brauer
surface $S$ is birational to the surface~$S^{op}$. In particular,
the groups $\operatorname{Bir}(S)$ and $\operatorname{Bir}(S^{op})$ are (non-canonically) isomorphic,
and the group $\operatorname{Aut}(S^{op})$ is (non-canonically) realized as a subgroup
of~\mbox{$\operatorname{Bir}(S)$}. Note also that~\mbox{$\operatorname{Aut}(S^{op})\cong \operatorname{Aut}(S)$},
although these groups are not conjugate in~\mbox{$\operatorname{Bir}(S)$}.
Corollary~\ref{corollary:SB-Aut} has the following geometric consequence.
\begin{corollary}\label{corollary:conjugate}
Let $S$ be a non-trivial Severi--Brauer surface over a field of characteristic zero, and
let $G\subset\operatorname{Aut}(S^{op})$ be a finite subgroup. Then $G$ is conjugate to a subgroup of~$\operatorname{Aut}(S)$.
\end{corollary}
\begin{proof}
By Corollary~\ref{corollary:SB-Aut}(ii) the group $G$ has
an invariant point $P$ of degree $3$ on $S'$, and by Lemma~\ref{lemma:SB-general-position} this point is in general position.
Blowing up~$P$ and blowing down the proper transforms of the three lines on $S'_{\bar{\mathbb{K}}}\cong\mathbb{P}^2_{\bar{\mathbb{K}}}$
passing through the pairs of the three points of $P_{\bar{\mathbb{K}}}$,
we obtain a (regular) action of $G$ on the surface~$S$ together with a $G$-equivariant
birational map $\tau_P\colon S'\dasharrow S$, cf.~\S\ref{section:birational}.
This means that $G$ is conjugate to a subgroup of~$\operatorname{Aut}(S)$.
\end{proof}
Now we prove Proposition~\ref{proposition:SB-Bir-vs-Aut}.
\begin{proof}[Proof of Proposition~\ref{proposition:SB-Bir-vs-Aut}]
We know from Proposition~\ref{proposition:summary} that $G$ is conjugate to a subgroup of
$\operatorname{Aut}(S')$, where $S'\cong S$ or $S'\cong S^{op}$. In the former case we are done.
In the latter case~$G$ is conjugate to a subgroup of $\operatorname{Aut}(S)$ by Corollary~\ref{corollary:conjugate}.
\end{proof}
Similarly to Lemma~\ref{lemma:SB-Aut}, we prove the following.
\begin{lemma}
\label{lemma:27}
Let $\mathbb{K}$ be a field of characteristic zero that contains all roots of $1$.
Let $S$ be a non-trivial Severi--Brauer surface over $\mathbb{K}$, and
let $G\subset\operatorname{Aut}(S)$ be a finite subgroup.
Then $G$ is isomorphic to a subgroup of~${\boldsymbol{\mu}}_3^2$.
\end{lemma}
\begin{proof}
We know from Theorem~\ref{theorem:ShramovVologodsky}(i)
that every non-trivial element of $G$ has order $3$, and~\mbox{$|G|\le 27$}.
Assume that $G$ is not isomorphic to a subgroup of~${\boldsymbol{\mu}}_3^2$.
Then either $G\cong{\boldsymbol{\mu}}_3^3$ or~\mbox{$G\cong{\mathscr{H}}_3$}.
The former case is impossible by Lemma~\ref{lemma:P2}(iv).
Thus, we have $G\cong{\mathscr{H}}_3$.
By Lemma~\ref{lemma:P2}(iii)
the group $G$ contains an element $g$ such that $\operatorname{Fix}_{S_{\bar{\mathbb{K}}}}(g)$ has a unique isolated point. This latter point
must be $\operatorname{Gal}(\bar{\mathbb{K}}/\mathbb{K})$-invariant, which is impossible by assumption.
\end{proof}
Finally, we prove our main result.
\begin{proof}[Proof of Theorem~\ref{theorem:main}]
Assertion (i) follows from Proposition~\ref{proposition:summary}
and Lemma~\ref{lemma:SB-Aut}.
Assertion~(ii) follows from Proposition~\ref{proposition:SB-Bir-vs-Aut} and
Corollary~\ref{corollary:SB-Aut}(i).
Assertion (iii) follows
from Proposition~\ref{proposition:summary}
and Lemma~\ref{lemma:27}.
\end{proof}
I do not know if the bound provided by Theorem~\ref{theorem:main}(iii)
(or Lemma~\ref{lemma:27}) is optimal. However, in certain cases it is easy to construct
non-trivial Severi--Brauer surfaces with an action of the group~\mbox{${\boldsymbol{\mu}}_3^2$}.
\begin{example}\label{example:cyclic-algebra}
Let $\mathbb{K}$ be a field of characteristic different from $3$ that contains a primitive
cubic root of unity $\omega$. Let $a, b\in \mathbb{K}$ be the elements such that $b$ is not
a cube in $\mathbb{K}$, and~$a$ is not contained in the image of the Galois norm for the
field extension~\mbox{$\mathbb{K}\subset\mathbb{K}(\sqrt[3]{b})$}.
Consider the algebra~$A$ over~$\mathbb{K}$ generated by variables $u$ and $v$
subject to relations
$$
u^3=a,\quad v^3=b,\quad uv=\omega vu.
$$
Then $A$ is a central division algebra, see for instance~\mbox{\cite[Exercise~3.1.6(ii),(iv)]{GS}}.
One has $\dim A=9$, so that $A$ corresponds to a non-trivial Severi--Brauer
surface $S$. Conjugation by $u$ defines an automorphism of order $3$ of $A$
(sending $u$ to $u$ and $v$ to~$\omega v$). Together with conjugation by
$v$ it generates a group ${\boldsymbol{\mu}}_3^2$ acting by automorphisms of $A$ and $S$.
\end{example}
\end{document} |
\begin{document}
\title{Representations associated to small nilpotent orbits for complex Spin Groups}
\begin{abstract}
This paper
provides a comparison between the
$K$-structure of unipotent representations and regular sections of
bundles on nilpotent orbits for complex groups of type $D$. Precisely,
let $ G_ 0 =Spin(2n,{\mathbb{C}})$ be the Spin complex group
viewed as a real group, and $K\cong G_0$ be the complexification of the maximal compact subgroup of $G_0$.
We compute
$K$-spectra of the regular functions on some small nilpotent orbits ${\mathcal{O}}$ transforming according to characters
$\psi$ of $C_{ K}({\mathcal{O}})$ trivial on the connected component of the
identity $C_{ K}({\mathcal{O}})^0$. We then match them with the
${K}$-types of the genuine (\ie representations which do not factor
to $SO(2n,\mathbb C)$) unipotent representations attached to ${\mathcal{O}}$.
\end{abstract}
\author{Dan Barbasch}
\address[D. Barbasch]{Department of Mathematics\\
Cornell University\\Ithaca, NY 14850, U.S.A.}
\email{[email protected]}
\thanks{D. Barbasch was supported by an NSA grant}
\author{Wan-Yu Tsai}
\address[Wan-Yu Tsai]{Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, TAIWAN}
\email{[email protected]}
\maketitle
\section{Introduction}
Let $G_0\subset G$ be the real points of a complex linear reductive algebraic
group $G$ with Lie algebra $\mathfrak g_0$ and
maximal compact subgroup $K_0$. Let $\mathfrak g_0=\mathfrak k_0+\mathfrak s_0$
be the Cartan decomposition, and $\mathfrak g=\mathfrak k+\mathfrak s$ be the
complexification. Let $K$ be the complexification of $K_0.$
\begin{definition}
Let ${\mathcal{O}}:= K\cdot e\subset \mathfrak s$. We say that an irreducible
admissible representation $\Xi$ is associated to ${\mathcal{O}},$ if ${\mathcal{O}}$
occurs with nonzero multiplicity in the associated cycle in the sense
of \cite{V2}.
An irreducible module $\Xi$ of $G_0$ is called unipotent
associated to a nilpotent orbit ${\mathcal{O}}\subset \mathfrak s$ and
infinitesimal character $\la_{{\mathcal{O}}}$, if it satisfies
\begin{description}
\item[1] It is associated to ${\mathcal{O}}$ and its annihilator
$Ann_{U(\mathfrak g)}\Xi$ is the unique maximal primitive ideal with
infinitesimal character $\la_{{\mathcal{O}}}$,
\item[2] $\Xi$ is unitary.
\end{description}
Denote by ${\mathcal{U}} _{G_0}({\mathcal{O}},\la_{{\mathcal{O}}})$ the set of unipotent
representations of $G_0$ associated to ${\mathcal{O}}$ and $\la_{{\mathcal{O}}}$.
\end{definition}
{Let $C_K({\mathcal{O}}):= C_K(e)$ denote the centralizer of $e$ in $K$, and
let $A_K({\mathcal{O}}):=C_K({\mathcal{O}})/C_K({\mathcal{O}})^0$ be the component group.}
Assume that $G_0$ is connected, and a complex
group viewed as a real Lie group. In this case $G\cong G_0\times G_0,$
and $K\cong G_0$ as complex groups. Furthermore $\mathfrak s\cong\mathfrak g_0$
as complex vector spaces, and the action of $K$ is the adjoint action.
In this case it is conjectured that there exists an infinitesimal character
$\la_{{\mathcal{O}}}$ such that in addition,
\begin{description}
\item[3] There is a 1-1 correspondence $\psi\in
\widehat{ A_K({\mathcal{O}})}\longleftrightarrow \Xi({\mathcal{O}},\psi)\in
{\mathcal{U}}_{G_0}({\mathcal{O}},\la_{{\mathcal{O}}})$ satisfying the additional condition
$$
\Xi({\mathcal{O}},\psi)\bigb_{K}\cong R({\mathcal{O}},\psi),
$$
\end{description}
where
\begin{equation}\label{def-reg-fun}
\begin{aligned}
R({\mathcal{O}}, \psi) &= {\mathrm{Ind}}_{C_{K}(e)} ^{K} (\psi) \\
&= \{f: K\to V_{\psi} \mid f(gx) =\psi(x) f(g) \ \forall g\in K, \ x\in C_K (e)\}
\end{aligned}
\end{equation}
is the ring of regular functions on ${\mathcal{O}}$ transforming according to
$\psi$. Therefore, $R({\mathcal{O}},\psi)$ carries a $K$-representation.
\begin{comment}
{\clrr As pointed out in \cite{BWT}, this conjecture cannot be valid
for real groups. For complex groups the codimension greater than one
of ${\mathcal{O}}$ in ${\overline{\mathcal{O}}}$ is satisfied, so the conjecture is plausible.
}
\end{comment}
Conjectural parameters $\la_{\mathcal{O}}$ satisfying the conditions above
are studied in \cite{B}, along with results establishing the validity of
this conjecture for large classes of nilpotent orbits in the classical
complex groups. Such parameters $\la_{\mathcal{O}}$ are available for the
exceptional groups as well, \cite{B} for $F_4$, and to appear elsewhere
for type $E.$
{
This conjecture cannot be valid for all nilpotent orbits in the case of
real groups; the intersection of a complex nilpotent orbit with $\mathfrak
s$ consists of several components. $R({\mathcal{O}},\psi)$ can be
the same for different components, whereas the representations with
associated variety containing a given component have drastically different
$K$-structures. Examples can be found in \cite{V1}. As explained in
\cite{V1} Chapter 7 and \cite{V2} Theorem 4.11,
if the codimension of the orbit ${\mathcal{O}}$ is $\ge 2,$ then
$\Xi\mid_K=R({\mathcal{O}},\phi)-Y$ with $\phi$ an algebraic representation,
and $Y$ an $S(\mathfrak g/\mathfrak k)$-module supported on orbits of strictly
smaller dimension. The orbits ${\mathcal{O}}$ under consideration in this
paper have codimension $\ge 2.$
Even when $\codim{\mathcal{O}} \ge 2,$ (\eg the case of the minimal orbit in
certain real forms of type $D,$) many examples are known where there
are no representations with associated variety ${\mathcal{O}}$ or any real
form of its complexification. }
In this paper we investigate this conjecture for \textit{small} orbits
in the complex case by different techniques than in \cite{B}; paper
\cite{BTs} investigates the analogue for the real $Spin$ groups.
For the condition of \textit{small} we require that
$$
[\mu : R({\mathcal{O}}, \psi)]\le c_{{\mathcal{O}}}
$$
\ie that the multiplicity of any $\mu\in \widehat{K}$ be uniformly
bounded. This puts a restriction on $\dim{\mathcal{O}}$:
\begin{equation}\label{eq:dim-cond}
\dim {\mathcal{O}} \leq \text{rank}(\mathfrak k) + |\Delta ^+(\mathfrak k,\mathfrak t)|,
\end{equation}
where $\mathfrak t\subset \mathfrak k$ is a Cartan subalgebra, and $\Delta^+(\mathfrak
k,\mathfrak t)$ is a positive system.
The reason for this restriction is as follows. Let $(\Pi,X)$ be an
admissible representation of $G_0$, and $\mu$ be the highest weight of a
representation $(\pi,V)\in \widehat{K}$ which is dominant for
$\Delta^+(\mathfrak k,\mathfrak t)$. Assume that $\dim{\mathrm{Hom}}_K[\pi,\Pi]\le C$, and
$\Pi$ has associated variety {\cf \cite{V2})}. Then
$$
\dim\{ v\ :\ v\in X \text{ belongs to an isotypic component with
} ||\mu||\le t\}\le Ct^{|\Delta^+(\mathfrak k,\mathfrak t)|+\dim \mathfrak t}.
$$
The dimension of $(\pi,V)$ grows like $t^{|\Delta^+(\mathfrak k,\mathfrak t) |}$, the
number of representations with highest weight $||\mu||\le t$
grows like $t^{\dim\mathfrak t},$ and the
multiplicities are assumed uniformly bounded. On the other hand, considerations
involving primitive ideals imply that the dimension of this set grows
like $t^{\dim G\cdot e/2}$ with $e\in{\mathcal{O}},$ and half the
dimension of (the complex orbit) $G\cdot e$ is the
dimension of the ($K$-orbit) $K\cdot e\in\mathfrak s.$ { In the
case of type $D,$ condition (\ref{eq:dim-cond}) coincides with being
spherical, see \cite{P}. Since we only deal with characters of
$C_{ K}({\mathcal{O}})$, multiplicity $\le 1$ is guaranteed}.
In the case of the complex groups of type $D_n$, we
consider $G_0=Spin(2n,{\mathbb{C}})$ viewed as a real group, and hence $K\cong G_0$ is the complexification of the maximal compact subgroup $K_0= Spin(2n)$ of $G$.
In Section 2 we list all small
nilpotent orbits satisfying (\ref{eq:dim-cond}) and describe the (component groups) of their centralizers.
In Section 3, we compute $R({\mathcal{O}},\psi)$ for each ${\mathcal{O}}$ in \ref{ss:orbits} and $\psi\in \widehat{A_{ K}({\mathcal{O}})}$. In Section 4
we associate to each ${\mathcal{O}}$ an infinitesimal character $\la_{{\mathcal{O}}}$ by \cite{B}.
The fact is that ${\mathcal{O}}$ is the minimal orbit which can be the associated variety of
a $(\mathfrak g, K)$-module with infinitesimal character $(\la_L, \la_R)$, with $\la_L$ and $\la_R$ both conjugate to $\la_{{\mathcal{O}}}$.
We make a complete list of irreducible modules ${\overline{X}}(\la_L,\la_R)$ (in terms of Langlands classification) which are attached to ${\mathcal{O}}.$
Then we match the $ K$-structure of these representations with $R({\mathcal{O}},\psi)$.
This demonstrates the conjecture we state in the beginning of the introduction. The following theorem summarizes this.
\begin{theorem}
With notation as above, view $ G_0={Spin}(2n,{\mathbb{C}})$ as a real group. The $ K$-structure
of each representations in ${\mathcal{U}}_{G_0}({\mathcal{O}},\la_{{\mathcal{O}}})$ is calculated explicitly and matches the
$K$-structure of the $R({\mathcal{O}},\psi)$ with $\psi \in \widehat{A_{K}({\mathcal{O}})}$.That is, there is a 1-1 correspondence $\psi\in
\widehat { A_{{K}}({\mathcal{O}})}\longleftrightarrow \Xi({\mathcal{O}},\psi)\in
{\mathcal{U}}_{ G _0 }({\mathcal{O}},\la_{\mathcal{O}})$ satisfying
$$
\Xi({\mathcal{O}},\psi)\mid_{ K}\cong R({\mathcal{O}},\psi).
$$
\end{theorem}
For the case $O(2n,\mathbb C)$ (rather than $Spin(2n,\mathbb C)$), the
$K$-structure of the representations studied in this paper were
considered earlier in \cite{McG} and \cite{BP1}.
\section{Preliminaries}
\subsection{Nilpotent Orbits} \label{ss:orbits}
The complex nilpotent orbits of type $D_n$ are parametrized by
partitions of $2n$, with even blocks occur with even multiplicities, and with $I, II$ in the \textit{very even} case (see \cite{CM}).
The small nilpotent orbits satisfying (\ref{eq:dim-cond}) are those ${\mathcal{O}}$ with $\dim {\mathcal{O}}\le n^2$.
We list them out as the following four cases:
$$
\begin{aligned}
Case\ 1:\ &n=2p&&\quad {\mathcal{O}}=[3\ 2^{n-2} \ 1]&& \dim {\mathcal{O}} = n^2 \\
Case\ 2:\ &n=2p \ \text{ or } \ 2p+1 && \quad {\mathcal{O}} = [ 3\ 2^{2k} \ 1^{2n-4k-3}] \ \ \footnotesize{0\le k \le p -1} &&\dim {\mathcal{O}} =4nk-4k^2+4n-8k-4\\
Case\ 3:\ &a=2p &&\quad{\mathcal{O}}=[2^n]_{I,II} && \dim {\mathcal{O}} = n^2-n \\
Case\ 4:\ &a=2p \ \text{ or } \ 2p+1 &&\quad {\mathcal{O}}=[ 2^{2k} \ 1^{2n-4k}] \ \ 0\le k < n/2&& \dim {\mathcal{O}} = 4nk-4k^2-2k\\
\end{aligned}
$$
Note that these are the orbits listed in \cite{McG}. The proof of the next Proposition, and the details about the nature of the component groups, are
in Section \ref{ss:clifford}.
\begin{prop} (Corollary \ref{c:cgp})
\begin{description}
\item[Case 1] If ${\mathcal{O}}=[3\ 2^{2p-2}\ 1],$ then $A_{{K}}({\mathcal{O}})\cong
\mathbb Z_2\times\mathbb Z_2$.
\item[Case 2] If ${\mathcal{O}}=[3\ 2^{2k}\ 1^{2n-4k-3}]$ with $2n-4k-3>1,$ then $A_{{K}}({\mathcal{O}})\cong
\mathbb Z_2.$
\item[Case 3] If ${\mathcal{O}}=[2^{2p}]_{I,II}$, then $A_{{K}} ({\mathcal{O}}) \cong\mathbb Z_2.$
\item[Case 4] If ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$ with $2k<n,$ then
$A_{{K}} ({\mathcal{O}}) \cong 1.$
\end{description}
In all cases $C_{K}({\mathcal{O}})=Z( K)\cdot C_{ K}({\mathcal{O}})^0.$
\end{prop}
\section{Regular Sections}
We use the notation introduced in Sections 1 and 2.
We compute the centralizers needed for $R({\mathcal{O}},\psi)$ in $\mathfrak k$ and in ${K}$. We use the standard roots
and basis for $\mathfrak{so}(2n,{\mathbb{C}}).$ A basis for the Cartan subalgebra is given by $H({\epsilon}_i)$, the root vectors are
$X(\pm{\epsilon}_i\pm{\epsilon}_j)$. Realizations in terms of the Clifford algebra and explicit calculations are in Section \ref{ss:clifford}.
Let $e$ be a representative the orbit ${\mathcal{O}}$, and let $\{e,h,f\}$ be the corresponding
Lie triple. Let
\begin{itemize}
\item $C_{\mathfrak k} (h)_i$ be the $i$-eigenspace of $ad(h)$ in $\mathfrak k$,
\item $C_{\mathfrak k} (e)_i$ be the $i$-eigenspace of $ad(h)$ in the
centralizer of $e$ in $\mathfrak k$,
\item $C_\mathfrak k (h)^+:= \sum \limits _{i>0} C_\mathfrak k (h) _i$, and $C_\mathfrak k (e)^+:= \sum \limits _{i>0} C_\mathfrak k (e) _i.$
\end{itemize}
\begin{comment}
\begin{table}[h]
\caption{$D_n$, $n=2p$} \label{nilp-orbit1}
\begin{center}
\begin{tabular} {c| c| c| c | c |c}
${\mathcal{O}}$& $[3 \ 2^{n-2} \ 1 ]$& $[ 3\ 2^{2k} \ 1^{2n-4k-3}]$ &
$[2^n]_I $ & $ [2^n]_{II} $ & $[2^{2k} \ 1^{2n-4k}] $ \\
&& $0\le k \le p-1$&&& $0\le k \le p -1$ \\ \hline
$\dim {\mathcal{O}}$& $n^2 $ & $4nk-4k^2+4n-8k-4$& $n^2-n$& $n^2-n$& $4nk-4k^2-2k$\\\hline
$A_K({\mathcal{O}})$& ${\mathbb Z}_2\times {\mathbb Z}_2$&${\mathbb Z}_2$&${\mathbb Z} _2$& ${\mathbb Z} _2$& 1
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{$D_n$, $n=2p+1$} \label{nilp-orbit2}
\begin{center}
\begin{tabular}{c|c|c}
${\mathcal{O}}$& $[3\ 2^{2k} \ 1^{2n-4k-3} ]$ & $[ 2^{2k} \ 1^{2n-4k}]$ \\
& $0\le k \le p-1$& $0\le k\le p$ \\\hline
$\dim {\mathcal{O}}$& $4nk-4k^2+4n-8k-4$ & $4nk-4k^2-2k$\\\hline
$A_K({\mathcal{O}})$& ${\mathbb Z}_2$ & 1
\end{tabular}
\end{center}
\end{table}
\end{comment}
\subsection{} We describe the centralizer for ${\mathcal{O}}= [ 3 \ 2^{2k}\ 1^{2n-4k-3}]$ in detail. These are Cases 1 and 2.
Representatives for $e$ and $h$ are
$$
\begin{aligned}
e&= X({\epsilon}_1-{\epsilon}_{2k+2}) +X({\epsilon}_1+{\epsilon}_{2k+2}) + \sum \limits_{2\le i\le 2k+1} X({\epsilon}_i +{\epsilon} _{k+i})\\
h&= 2 H({\epsilon}_1)+\sum\limits_{2\le i\le 2k+1} H({\epsilon}_i) = H(2, \underset{2k}{\underbrace{1,\dots,1}},
\underset{n-1-2k}{\underbrace{0, \dots, 0}}).
\end{aligned}
$$
Then
\begin{equation}
\begin{aligned}
C_{\mathfrak k} (h)_0 &= \mathfrak{gl} (1)\times \mathfrak{gl} (2k)\times { \mathfrak{so}(2n-2-4k)}, \\
C_{\mathfrak k} (h)_1 &= Span\{ X({\epsilon}_1 - {\epsilon}_i) , \ X({\epsilon}_i \pm {\epsilon} _j ) , \ 2\le i\le 2k+1<j\le n\}, \\
C_{\mathfrak k} (h)_2 &= Span \{ X( {\epsilon} _1\pm {\epsilon} _j), \ X({\epsilon}_i+ {\epsilon} _l), 2\le i \neq l \le 2k+1<j \le n \}, \\
C_{\mathfrak k} (h)_3 &= Span \{ X( {\epsilon}_1 +{\epsilon} _i ), \ 2\le i\le 2k+1\}.
\end{aligned}
\end{equation}
Similarly
\begin{equation}
\begin{aligned}
C_{\mathfrak k}(e)_0 &\cong \mathfrak{sp}(2k)\times{ {\mathfrak{so}(2n-3-4k)}},\\
C_{\mathfrak k} (e)_1 &= Span\{ X( {\epsilon}_1 - {\epsilon}_i ) - X({\epsilon} _{k+i }
\pm {\epsilon}_{2k+2}) , \ X({\epsilon} _1 -{\epsilon} _{k+i})- X({\epsilon} _i \pm{\epsilon} _{2k+2}), \ 2\le i \le k+1, \\
& X({{\epsilon}_j }\pm {\epsilon}_l ) , \ 2\le j\le 2k+1, \ 2k+3\le l \le n\},\\
C_{\mathfrak k} (e)_2 &= C_{\mathfrak k} (h)_2 , \\
C_{\mathfrak k} (e)_3 &= C_{\mathfrak k} (h)_3.
\end{aligned}
\end{equation}
We denote by $\chi$ the trivial character of $C_{\mathfrak k}(e)$. A representation of ${K}$ will be denoted by its highest weight:
$$
V=V(a_1,\dots, a_p), \quad a_1\ge \dots \ge |a_p|,
$$
with all $a_i\in {\mathbb Z}$ or all $a_i\in {\mathbb Z}+1/2$.
We will compute
\begin{equation}
\label{eq:mreal}
{\mathrm{Hom}}_{C_{\mathfrak k}(e)}[V^*, \chi]=
{\mathrm{Hom}}_{C_{\mathfrak k}(e)_0}\left [V^*/(C_{\mathfrak k}(e)^+V^*), \chi \right ]:=
\left (V^*/(C_{\mathfrak k}(e)^+V^*\right )^\chi.
\end{equation}
\subsection{Case 1.} $n=2p$, ${\mathcal{O}}=[3\ 2^{n-2}\ 1]$.
In this case $C_{\mathfrak k}(h)_0 = \mathfrak{gl}(1)\times \mathfrak{gl} (n-2)\times \mathfrak{so}(2), C_{\mathfrak k}(e)_0 =\mathfrak{sp}(n-2)$.
Consider the parabolic $\mathfrak p = \mathfrak l +\mathfrak n$
determined by $h$,
\begin{equation}
\label{eq:ell}
\begin{aligned}
\mathfrak l &= C_{\mathfrak k}(h)_0 \cong \mathfrak{gl}(1)\times \mathfrak{gl}(n-2)\times \mathfrak{so}(2), \\
\mathfrak n &= C_{\mathfrak k}(h)^+.
\end{aligned}
\end{equation}
We denote by $V^*$, the dual of $V$. Since $n=2p,$ $V^*\cong V.$ Then
$V^*$ is a quotient of a generalized Verma module $M(\la)=U(\mathfrak
k)\otimes _{U(\overline{\mathfrak p} )} F(\la)$, where $\la$ is a weight of
$V^*$ which is dominant for $\overline{\mathfrak p}$. This is
$$
\la = (-a_1;-a_{n-1} , \dots, -a_2; -a_n ).
$$
The $;$ denotes the fact that this is a (highest) weight of $\mathfrak l\cong \mathfrak{gl}(1)\times \mathfrak{gl}(n-2)\times \mathfrak{so}(2)$.
We choose the standard positive root
system $\triangle^+ (\mathfrak l)$ for $\mathfrak l$. { As a $C_\mathfrak k(e)_0$-module,
$$
\mathfrak n =C_{\mathfrak
k}(e)^+\oplus \mathfrak n ^{\perp},
$$
where we can choose $\mathfrak n ^{\perp}=Span \{ X({\epsilon}_1-{\epsilon}_{j} ) , \ 2\le j\le n-1
\}$. This complement is $\mathfrak l$-invariant.
It restricts to the standard
module of $C_{\mathfrak k}(e)_0=\mathfrak{sp}(n-2).$ }
The generalized Bernstein-Gelfand-Gelfand resolution is:
\begin{equation}
\label{eq:bgg:cx}
0 \dots \longrightarrow \bigoplus _{w\in W^+, \ \ell(w)=k}M(w\cdot\la) \longrightarrow\dots \longrightarrow \bigoplus _{w\in W^+, \ \ell(w)=1}M(w\cdot\la) \longrightarrow M(\la) \longrightarrow
V^* \longrightarrow 0,
\end{equation}
with $w\cdot \la:= w(\la+{\mathrm{h}}o(\mathfrak k))-{\mathrm{h}}o(\mathfrak k)$, and $w\in W^+$, the $W(\mathfrak l)$-coset representatives that make $w\cdot \la$ dominant for $\Delta^+(\mathfrak l).$ This
is a free $C_{\mathfrak k}(e)^+$-resolution so we can compute cohomology by considering
\begin{equation}
\label{eq:coh}
0 \dots \longrightarrow \bigoplus _{w\in W^+, \ \ell(w)=k} \overline{M(w\cdot\la)} \longrightarrow\dots \longrightarrow \bigoplus _{w\in W^+, \ \ell(w)=1}\overline{M(w\cdot\la)} \longrightarrow \overline{M(\la)} \longrightarrow \overline{V^*} \longrightarrow 0,
\end{equation}
where $\overline{X}$ denotes $X/[C_{\mathfrak k}(e)^+ X]$.
Note that in the sequences,
$M(w\cdot\la)\cong S(\mathfrak n)\otimes_{\mathbb C}
F(w\cdot\la)$ and $\overline{M(w\cdot\la)}\cong S(\mathfrak n^{\perp})\otimes_{\mathbb C}
F(w\cdot\la)$.
As an $\mathfrak l$-module, $\mathfrak n^{\perp}$ has highest weight
$(1;0, \dots,0,-1 ;0)$. Then $S^k(\mathfrak n^\perp)\cong F(k;0,\dots ,0 ,-k;0)$
as an $\mathfrak l$-module.
Let $\mu:=(-\alpha_1 ; -\alpha_{n-1} , \dots, -\alpha_2 ; -\alpha_n)$ be the highest weight of an $\mathfrak l$-module.
By the Littlewood-Richardson rule,
\begin{equation}
\label{eq:tensorproduct}
S^k(\mathfrak n ^{\perp})\otimes F_{\mu}=\sum V(-\alpha_1 +k; -\alpha_{n-1} -k_{n-1} ,
\dots, -\alpha_3-k_3, -\alpha_2-k_2;-\alpha_n ).
\end{equation}
The sum is taken over
$$\{k_i\ | \ k_i\ge 0,\ \sum k_i =k, \ 0\le k_i\le \alpha_{i-1}-\alpha_{i}, \ 3\le i \le n-1\}.$$
\begin{lemma}\label{le:Sn}
Hom$ _{C_{\mathfrak k}(e) _0}[S^k (\mathfrak n ^{\perp}) \otimes F_{\mu} : \chi] \neq 0$ for every $\mu$.
The multiplicity is 1.
\begin{proof}
Since $(\mathfrak{gl}(n-2),\mathfrak{sp}(n-2))$ is a hermitian symmetric pair, Helgason's
theorem implies that a composition factor in $S(\mathfrak n^\perp)\otimes
F_{\mu}$ admits $C_{\mathfrak k}(e)_0$-fixed vectors only if
$$
-\alpha_{n-1}-k_{n-1}=-\alpha_{n-2}-k_{n-2},\
-\alpha_{n-3}- k_{n-3}=-\alpha_{n-4}-k_{n-4},\dots ,-\alpha_3-k_3=-\alpha_2-k_2.
$$
The conditions ${ 0\le k_i\le \alpha_{i-1}-\alpha_{i} } $ imply
\begin{equation}
\begin{aligned}
k_{n-2} =0, &\quad k_{n-1} = \alpha_{n-2}-\alpha_{n-1}, \\
\hspace*{4em} \vdots\\
k_4=0,& \quad k_5 =\alpha_4-\alpha_5, \\
k_2=0, & \quad k_3=\alpha_2-\alpha_3.
\end{aligned}
\end{equation}
Therefore, given $\mu$, the weight of the $C_{\mathfrak k}(e)_0$-fixed vector in $S(\mathfrak n ^{\perp})\otimes F_{\mu}$ is
$$
(-\alpha_1+\alpha_2-\alpha_3+\alpha_4-\alpha_5+\dots+\alpha_{n-2}-\alpha_{n-1} ; -\alpha_{n-2},-\alpha_{n-2}, \dots, -\alpha_2,\alpha_2; -\alpha_n),
$$
and the multiplicity is 1.
\end{proof}
\end{lemma}
\begin{cor}
For every $V(a_1,\dots,a_n)\in \widehat{{K}},$, Hom$_{C_{\mathfrak k}(e)} [V,\chi]= 0$ or 1. The action of $\ad h$ is $-2\sum\limits_{1\le i \le p} a_{2i-1}$.
\begin{proof}
The first statement follows from Lemma \ref{le:Sn} and the surjection
$$
\overline{M(\la)}\cong S(\mathfrak n^{\perp}) \otimes_{{\mathbb{C}}} F(\la)\longrightarrow \overline{V^*}\longrightarrow 0.
$$
The action of $\ad h$ is computed from the module
\begin{eqnarray}\label{eq:fix-vec}
V(-a_1+k; -a_{n-2}, -a_{n-2} , \dots, -a_2,-a_2 ; -a_n)
\end{eqnarray}
with
$k=a_2-a_3+a_4-a_5+\dots+a_{n-2}-a_{n-1}$. The value is $-2\sum\limits_{1\le i \le p} a_{2i-1}$.
\end{proof}
\end{cor}
\subsubsection*{$\mathbf{\ell(w)=1}$}
To show that the weights in (\ref{eq:fix-vec}) actually occur, it is enough to show that these weights
do not occur in the term in the BGG resolution
(\ref{eq:coh}) with $\ell(w)=1$.
We calculate $w\cdot\la:$
$$
{\mathrm{h}}o={\mathrm{h}}o(\mathfrak k)=(-(n-1) ; -1, -2, \dots, -(n-2); 0)
$$
is dominant for $\overline{\mathfrak p}$, and
$$
\la +{\mathrm{h}}o = (-a_1-n+1; -a_{n-1}-1, -a_{n-2} -2 , \dots, -a_2-n+2; -a_n).
$$
There are three elements $w\in W^+$ of length 1. They are the left
$W(\mathfrak l)$-cosets of
$$
w_1=s_{{\epsilon}_1- { {\epsilon}_{n-1} }}, \ w_2= s_{ { {\epsilon}_{2}}-{\epsilon}_n}, w_3=s_{ { {\epsilon}
_{2} } +{\epsilon} _n}.
$$
So
\begin{equation}
\begin{aligned}
w_1\cdot \la &= { (-a_2 +1 ; -a_{n-1}, -a_{n-2}, \dots, -a_4, -a_3,-a_1-1 ; -a_n)},\\
w_2\cdot \la&= { (-a_1; -a_n+1, -a_{n-2}, -a_{n-3}, \dots, -a_3, -a_2; -a_{n-1}-1 )} ,\\
w_3\cdot\la &= { (-a_1; a_n +1, -a_{n-2},-a_{n-3} ,\dots, -a_3,-a_2;a_{n-1}+1)}.
\end{aligned}
\end{equation}
\begin{lemma}
For all $\la$, Hom$_{C_{\mathfrak k}(e)}[\overline{M(w_i\cdot \la)} , \chi ]=1$. The eigenvalues of $\ad h$ are different from
$-2\sum \limits_{1\le i\le p}a_{2i-1}$ for each $w_i$.
\begin{proof}
The $\mathfrak{sp}(n-2)$-fixed weights come from {$S(\mathfrak n^\perp)\otimes
F(w_i\cdot \la), \ i=1, 2, 3, \ $ are }
\begin{equation}\label{eq:spn-2}
{ \begin{aligned}
w_1&&\longleftrightarrow& \mbox{ \footnotesize $( a_1-a_2-a_3 +a_4 -a_5+\dots +a_{n-2}-a_{n-1} +2 ; -a_{n-2},-a_{n-2}, \dots , -a_4,-a_4, -a_2,-a_2; -a_n)$ },\\
w_2 &&\longleftrightarrow & \mbox{ \footnotesize $ (-a_1+a_2-a_3+\dots +a_{n-4}-a_{n-3}+a_{n-2}-a_n+1; -a_{n-2}, -a_{n-2}, \dots,-a_4, -a_4,-a_2,-a_2;-a_{n-1}-1)$ },\\
w_3 &&\longleftrightarrow& \mbox{ \footnotesize $(-a_1+a_2- a_3+\dots + a_{n-4}-a_{n-3}+a_{n-2}+a_n+1 ; -a_{n-2} , -a_{n-2}, \dots,-a_4,-a_4, -a_2, -a_2; a_{n-1}-1)$}.
\end{aligned}}
\end{equation}
The negatives of the weights of $h$ are
{
\begin{equation}
\label{eq:wth}
\begin{aligned}
&w_0=1&&\longleftrightarrow &&2(a_1+a_3+\dots +a_{n-1}),\\
&w_1&&\longleftrightarrow &&2(a_2+a_3+a_5 \dots +a_{n-1}+1),\\
&w_2&&\longleftrightarrow &&2(a_1+a_3+\dots +a_{n-3} +a_{n}-1),\\
&w_3&&\longleftrightarrow &&2(a_1+a_3+\dots +a_{n-3}-a_n -1).
\end{aligned}
\end{equation}
}
The last three weights are not equal to the first one. This completes
the proof.
\end{proof}
\end{lemma}
\begin{theorem}\label{th:reg}
Every representation $V(a_1,\dots,a_n)$ has $C_{\mathfrak k}(e)$ fixed vectors and the multiplicity is 1.
We write $C_K({\mathcal{O}}):= C_K(e)$.
In summary,
$$
\text{Ind}_{C_{{K}}({\mathcal{O}})^0} ^{{K}} (Triv)=\bigoplus _{a \in \widehat{{K}} } V(a_1,\dots,a_n).
$$
\end{theorem}
Theorem \ref{th:reg} can be
interpreted as computing regular functions on the universal cover
$\widetilde{\mathcal{O}}$ of
${\mathcal{O}}$ transforming trivially under $C_{\mathfrak k}(e)_0$.
We decompose it further:
\begin{eqnarray}\label{eq:decomp}
R(\widetilde{{\mathcal{O}}}, Triv):= {\mathrm{Ind}}_{C_{{ K} } ({\mathcal{O}})^0} ^{ K} (Triv) = {\mathrm{Ind}} _{C_{{K}} ({\mathcal{O}})} ^{{K}} \left
[{\mathrm{Ind}}_{C_{ {K}} ({\mathcal{O}})^0} ^{ C_{ {K}} ({\mathcal{O}}) } (Triv)
\right ].
\end{eqnarray}
The inner induced module splits into
\begin{equation} \label{eq:char}
{\mathrm{Ind}}_{C_{ {K}} ({\mathcal{O}})^0} ^{ C_{ {K}} ({\mathcal{O}}) } (Triv)=\sum\psi
\end{equation}
where $\psi $ are the irreducible representations of $C_{ K}({\mathcal{O}})$
trivial on $C_{ K}({\mathcal{O}})^0.$ Thus, the sum in (\ref{eq:char}) is taken over $\widehat{A_K({\mathcal{O}})}$.
Then
\begin{equation}\label{sum-reg-fun}
R(\widetilde{{\mathcal{O}}}, Triv) =\text{Ind}_{C_{ K} ({\mathcal{O}})^0} ^{ K}(Triv)=\sum \limits_{\psi \in \widehat{A_K({\mathcal{O}})} } R({\mathcal{O}},\psi).
\end{equation}
We will decompose $R({\mathcal{O}},\psi)$ explicitly as a representation of
${K}$.
\begin{lemma}
Let $\mu_i$, $1\le i \le 4$, be the following $K$-types parametrized
by their highest weights:
\begin{eqnarray*}
&\mu_1= (0, \dots, 0), \mu _2 = (1,0,\dots ,0), \\
&\mu_3= (\frac{1}{2},\dots,\frac{1}{2} ), \mu_4=
(\frac{1}{2},\dots,\frac{1}{2},-\frac{1}{2} ).
\end{eqnarray*}
Let $\psi_i $ be the restriction of the highest weight of $\mu _i$
to $C_{{K}}({\mathcal{O}})$, respectively. Then
\begin{eqnarray*}
\text{Ind}_{C_{{K}}({\mathcal{O}})^0} ^{ C_{{K}}({\mathcal{O}}) } (Triv)=\sum \limits_{i=1} ^4 \psi _i.
\end{eqnarray*}
\end{lemma}
\begin{prop} \label{p:regfun1}
The induced representation (\ref{sum-reg-fun}) decomposes as
$$
\text{Ind} _{C_{ K} ({\mathcal{O}}) } ^{ K} (Triv) =\sum _{i=1} ^4 R({\mathcal{O}},\psi_i)
$$
where
\begin{eqnarray*}
R({\mathcal{O}}, \psi _1)&= \text{Ind} _{C_{ K} ({\mathcal{O}}) } ^{ K}(\psi_1)=\bigoplus V(a_1,\dots,a_n)&\quad \text{ with } a_i\in {\mathbb Z}, \ \sum a_i \in 2{\mathbb Z}, \\
R({\mathcal{O}}, \psi _2)&= \text{Ind} _{C_{ K} ({\mathcal{O}}) } ^{ K} (\psi_2)=\bigoplus V(a_1,\dots,a_n)&\quad \text{ with } a_i\in {\mathbb Z}, \sum a_i \in 2{\mathbb Z}+1 , \\
R({\mathcal{O}}, \psi _3)&= \text{Ind} _{C_{ K} ({\mathcal{O}}) } ^{ K}(\psi_3)=\bigoplus V(a_1,\dots,a_n) &\quad \text{ with } a_i\in {\mathbb Z}+1/2,\sum a_i \in 2{\mathbb Z}+p,\\
R({\mathcal{O}}, \psi _4)&= \text{Ind} _{C_{ K} ({\mathcal{O}}) } ^{ K} (\psi_4)=\bigoplus V(a_1,\dots,a_n)&\quad \text{ with } a_i\in {\mathbb Z}+1/2, \sum a_i \in 2{\mathbb Z}+p+1.
\end{eqnarray*}
\end{prop}
\subsection{Case 2} ${\mathcal{O}}=[3\ 2^{2k} \ 1^{2n-4k-3}],$ $0\le k \le p-1$.
\subsubsection*{} Consider the parabolic $\mathfrak p = \mathfrak l +\mathfrak n$
determined by $h$:
\begin{equation*}
\begin{aligned}
\mathfrak l &= C_{\mathfrak k}(h)_0 \cong \mathfrak{gl}(1)\times \mathfrak{gl}(2k)\times
{\mathfrak{so}(2n-2-4k)}, \\
\mathfrak n &= C_{\mathfrak k}(h)^+.
\end{aligned}
\end{equation*}
In this section, let ${\epsilon}=-1$ when $n$ is even; ${\epsilon}=1$ when $n$ is
odd. The dual of $V,$ denoted $V^*$, has lowest weight $({\epsilon}
a_n,-a_{n-1},\dots, -a_2, -a_1)$. It is therefore a quotient of a
generalized Verma module
$M(\la)=U(\mathfrak k)\otimes _{U(\overline{\mathfrak p} )} F(\la)$, where $\la$
is dominant for $\overline{\mathfrak p}$, and dominant for the standard
positive system for $\mathfrak l:$
$$
{\la = (-a_1; \underset{2k}{\underbrace{ -a_{2k+1} , \dots, -a_3,
-a_2}}; \underset{n-1-2k}{\underbrace{ a_{2k+2}, \dots, a_{n-1}, {\epsilon} a_n}}).}
$$
$\mathfrak n =C_{\mathfrak k}(e)^+\oplus \mathfrak n ^{\perp}$ as a module for $C_{\mathfrak
k}(e)_0$. A basis for $\mathfrak n^\perp\subset C_\mathfrak k(h)_1$ is given by
$$
{ \{ X({\epsilon}_1{ -} {\epsilon} _{2k+2}) \}},
\quad
2\le i\le 2k+1.
$$
This is the standard representation of
$\mathfrak{sp}(2k),$ trivial for {$\mathfrak{so}(2n-4-4k).$} We write {its highest weight as}
$$
{(1;0,\dots ,0,-1;0,\dots ,0)}.
$$
We can now repeat the argument for the case $k=p;$ there is an added
constraint that ${a_{2k+3}}=\dots =a_n=0$ because the representation with highest weight $(a_{2k+2},\dots ,a_{n-1},{\epsilon} a_n)$ of $\mathfrak{so}(2n-2-4k)$ must have fixed vectors for ${\mathfrak{so}(2n-3-4k)}.$
Then the next theorem follows.
\begin{theorem}
A representation $V(a_1,\dots,a_n)$ has $C_{\mathfrak k }(e)$ fixed vectors if and only if $$a_{2k+3}=\dots=a_n=0,$$ and the multiplicity is 1.
In summary,
$$
\text{Ind}_{C_{ K}({\mathcal{O}})^0}^{ K} (Triv)=\bigoplus V(a_1,\dots, a_{2k+2},0\dots,0), \quad \text{ with } a_1\ge \dots \ge a_{2k+2}\ge 0, \ a_i\in {\mathbb Z}.
$$
\end{theorem}
As in (\ref{sum-reg-fun}), we decompose $\text{Ind}_{C_{ K}({\mathcal{O}})^0}^{ K} (Triv)$ further in to sum of $R({\mathcal{O}},\psi)$ with $\psi\in\widehat{A_K({\mathcal{O}})}$.
\begin{lemma}
Let $\mu_1, \mu_2 $ be the following ${K}$-types parametrized by
their highest weights:
\begin{eqnarray*}
\mu_1= (0, \dots, 0), \mu _2 = (1,0,\dots ,0).
\end{eqnarray*}
Let $\psi_i $ be the restriction of the highest weight of $\mu _i$
to $C_G({\mathcal{O}})$, respectively. Then
\begin{eqnarray*}
\text{Ind}_{C_{ K} ({\mathcal{O}})^0} ^{ C_{ K}({\mathcal{O}}) } (Triv)=\psi_1 +\psi _2.
\end{eqnarray*}
\end{lemma}
\begin{prop}\label{p:regfun2}
The induced representation (\ref{sum-reg-fun}) decomposes as
$$
\text{Ind} _{C_{ K} ({\mathcal{O}}) ^0} ^{ K} (Triv) = R({\mathcal{O}},\psi_1) + R({\mathcal{O}},\psi_2)
$$
where
\begin{eqnarray*}
R({\mathcal{O}}, \psi _1)&= \text{Ind} _{C_{K} ({\mathcal{O}}) } ^{ K}(\psi_1)=\bigoplus V(a_1,\dots,a_{2k+2},0,\dots,0)&\quad \text{ with } a_i\in {\mathbb Z}, \ \sum a_i \in 2{\mathbb Z}, \\
R({\mathcal{O}}, \psi _2)&= \text{Ind} _{C_{ K} ({\mathcal{O}}) } ^{ K} (\psi_2)=\bigoplus V(a_1,\dots,a_{2k+2},0,\dots,0)&\quad \text{ with } a_i\in {\mathbb Z}, \sum a_i \in 2{\mathbb Z}+1. \\
\end{eqnarray*}
\end{prop}
\subsection{}
Now we treat ${\mathcal{O}}=[2^{2k} \ 1^{2n-4k}]$ with $0\le k\le p$. These are Cases 3 and 4. When $k=p$ (and hence $n=2p$), the orbit
is labeled by $I,II$. The computation is similar and easier than the previous two cases. We state the results for $R(\widetilde{O},Triv)$ as follows.
\begin{theorem}\
\begin{description}
\item[Case 3] For $k=p$, so $n=2p,$
$$
\begin{aligned} &{\mathcal{O}} _I =[ 2^n]_I,\qquad &R(\widetilde{{\mathcal{O}}_I},Triv)= \text{Ind}_{C_{ K}({\mathcal{O}}_{I}) ^0} ^{ K}(Triv)=\bigoplus V(a_1, a_1, a_3, a_3,
\dots, a_{n-1}, a_{n-1}),\\
&{\mathcal{O}} _{II}=[ 2^n]_{II},\qquad &R(\widetilde{{\mathcal{O}}_{II}},Triv)=\text{Ind}_{C_{ K}({\mathcal{O}}_{II}) ^0} ^{ K}(Triv)=\bigoplus V(a_1, a_1, a_3, a_3, \dots,
a_{n-1}, -a_{n-1}).
\end{aligned}
$$
\item[Case 4]
{For $k\le p-1$},
$$
{\mathcal{O}}= [2^{2k}\ 1^{2n-4k}],\qquad
R(\widetilde{{\mathcal{O}}},Triv)=\text{Ind}_{C_{ K}({\mathcal{O}}) ^0} ^{ K}(Triv)= {\bigoplus V(a_1, a_1, a_3, a_3, \dots, a_{2k-1}, a_{2k-1}, 0,\dots, 0)},
$$
satisfying $a_1\ge a_3\ge \dots \ge a_{2k-1}\ge 0$.
\end{description}
\begin{proof}
We treat the case $n=2p$ and ${k\le p-1};$ $n=2p+1$ is similar. {A representative of ${\mathcal{O}}$ is}
$e=X({\epsilon}_1+{\epsilon}_{2})+\dots +X({\epsilon}_{2k-1}+{\epsilon}_{2k})$, and the corresponding middle element in the Lie
triple is
$h=H(\underset{2k}{\underbrace{{1,\dots
,1}}},\underset{n-2k}{\underbrace{{0,\dots ,0}}})$.
Thus
\begin{equation}
\label{eq:ch_i}
\begin{aligned}
&C_{\mathfrak k}(h)_0=\mathfrak{gl}({2k})\times \mathfrak{so}(2n-{4k})\\
&C_{\mathfrak k}(h)_1=Span\{ X({\epsilon}_i\pm {\epsilon}_j)\}\qquad {1\le i \le 2k< j\le n},\\
&C_{\mathfrak k}(h)_2=Span\{ X({\epsilon}_l+{\epsilon}_m)\}\qquad 1\le l\ne m\le {2k}.
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
&C_{\mathfrak k}(e)_0=\mathfrak{sp}(2k)\times \mathfrak{so}(2n-4k)\\
&C_{\mathfrak k}(e)_1=C_{\mathfrak k}(h)_1,\\
&C_{\mathfrak k}(e)_2=C_{\mathfrak k}(h)_2.
\end{aligned} \label{eq:cei}
\end{equation}
As before, let $\mathfrak p=\mathfrak l+\mathfrak n$ be the parabolic subalgebra
determined by $h,$ and $V=V(a_1,\dots ,a_n)$ be an irreducible
representation of $K$. Since we assumed $n=2p,$ $V=V^*.$
In this case $C_{\mathfrak k}(e)^+=\mathfrak n,$ so
Kostant's theorem implies $V/[C_{\mathfrak k}(e)^+V]=V_{\mathfrak l}(a_1,\dots
a_{2k};a_{2k+1},\dots ,a_n)$ as a $\mathfrak{gl}(2k)\times \mathfrak{so}(2n-4k)$-module.
Since we want $\mathfrak{sp}(2k)\times
\mathfrak{so}(2n-4k)$-fixed vectors, $a_{2k+1}=\dots =a_n=0,$ and Helgason's
theorem implies $a_1=a_2,a_3=a_4,\dots ,a_{2k-1}=a_{2k}$.
{When $n=2p$, and ${\mathcal{O}}=[2^n]_{I,II}$, the calculations are similar to $k\le p-1.$ The choices $I,II$ are
$$
\begin{aligned}
& e_I=X({\epsilon}_1-{\epsilon}_2)+X({\epsilon}_3-{\epsilon}_4)+\dots+X({\epsilon}_{n-1}-{\epsilon}_n)\quad &&h_I=H(1,\dots,1),\\
&e_{II}=X({\epsilon}_1-{\epsilon}_2)+X({\epsilon}_3-{\epsilon}_4)+\dots +X({\epsilon}_{n-3}-{\epsilon}_{n-2})+X({\epsilon}_{n-1}+{\epsilon}_n),\quad
&&h_{II}=H(1,\dots,1, -1).
\end{aligned}
$$
These orbits are induced from the two nonconjugate maximal parabolic subalgebras with $\mathfrak{gl}(n)$ as Levi components, and $R(\widetilde{{\mathcal{O}}_{I,II}},Triv)$ are just the induced modules from the trivial representation on the Levi component. }
\end{proof}
\end{theorem}
We aim at decomposing $R(\widetilde{{\mathcal{O}}},Triv) =\sum R({\mathcal{O}},\psi)$ with $\psi\in \widehat{A_K({\mathcal{O}})}$ as before.
\begin{lemma}\
\begin{description}
\item[Case 3] { $n=2p$, ${\mathcal{O}} = [2^n]_{I,II}$}.
Let $\mu_1$, $\mu_2$, $\nu_1$, $\nu_2$, be:
\begin{eqnarray*}
&\mu_1= (1, \dots, 1), \mu _2 = (\frac{1}{2},\dots\frac12), \\
&\nu_1= (1,\dots,1,-1), \nu_2=
(\frac{1}{2},\dots,\frac{1}{2},-\frac{1}{2} ).
\end{eqnarray*}
Let $\psi_i $ be the restriction of the highest weight of $\mu _i$
to $C_K(e)$, and $\phi _i$ be the restriction of the highest weight of $\nu_i$, respectively. Then
\begin{eqnarray*}
\text{Ind}_{C_K ({\mathcal{O}}_I)^0} ^{ C_K ({\mathcal{O}}_I) } (Triv)&=&\psi_1 +\psi _2,\\
\text{Ind}_{C_K ({\mathcal{O}}_{II})^0} ^{ C_K ({\mathcal{O}}_{II}) } (Triv)&=&\phi_1 +\phi _2.
\end{eqnarray*}
The $\psi_i,\phi_i$ are viewed as
representations of $\widehat{A_K({\mathcal{O}}_{I,II})}$, and $\psi_1$ and $\phi_1$ are $Triv,$ $\psi_2,\phi_2$ are $Sgn.$
\item[Case 4] { ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$, $0\le k \le p-1$}.
\begin{eqnarray*}
\text{Ind}_{C_K ({\mathcal{O}})^0} ^{ C_K({\mathcal{O}}) } (Triv)=Triv.
\end{eqnarray*}
\end{description}
\end{lemma}
Then we are able to split up $R(\widetilde{{\mathcal{O}}},Triv)$ as a sum of $R({\mathcal{O}},\psi)$ as in (\ref{sum-reg-fun}).
\begin{prop} \label{p:regfun3}\
\begin{description}
\item[Case 3]
{ $n=2p$, ${\mathcal{O}} = [2^n]_{I,II}$}: $R(\widetilde{\mathcal{O}}_{I, II}) = R({\mathcal{O}}_{I,II},Triv) + R({\mathcal{O}}_{I,II}, Sgn)$ with
\begin{eqnarray*}
R({\mathcal{O}}_I, Triv)&=& \text{Ind} _{C_{ K}({\mathcal{O}} _I) } ^{ K} (Triv)=\bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},a_{n-1}), \quad \text{ with } a_i\in{\mathbb Z}, \\
R({\mathcal{O}}_I, Sgn)&= &\text{Ind} _{C_{ K} ({\mathcal{O}} _I) } ^{ K}(Sgn)=\bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},a_{n-1}), \quad \text{ with } a_i\in{\mathbb Z} +1/2, \\
R({\mathcal{O}}_{II}, Triv)&=&\text{Ind} _{C_{ K}({\mathcal{O}} _{II}) } ^{ K}(Triv)=\bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},-a_{n-1}), \quad \text{ with } a_i\in{\mathbb Z}, \\
R({\mathcal{O}}_{II}, Sgn)&=&\text{Ind} _{C_{ K}({\mathcal{O}} _{II}) } ^{ K}(Sgn)=\bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},-a_{n-1}), \quad \text{ with } a_i\in{\mathbb Z} +1/2,
\end{eqnarray*}
satisfying $a_1\ge a_3 \ge \dots\ge a_{n-1}\ge 0$.
\item[Case 4]
{ ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$, $0\le k \le p-1$}:
\begin{eqnarray*}
R(\widetilde{{\mathcal{O}}},Triv)=R({\mathcal{O}},Triv)= \text{Ind} _{C_{ K} ({\mathcal{O}})} ^{ K} (Triv)= \bigoplus V(a_1,a_1,a_3,a_3,\dots, a_{2k-1},a_{2k-1},0,\dots,0), \ \text{ with } a_i\in {\mathbb Z},
\end{eqnarray*}
satisfying $a_1\ge a_3 \ge \dots\ge a_{2k-1}\ge 0$
\end{description}
\end{prop}
\section{Representations with small support}
\subsection{Langlands Classification} Let $G$ be a complex linear
algebraic
reductive group viewed as a real Lie group. Let $\theta$ be a Cartan
involution with fixed points $K.$ Let $G\supset B=HN\supset H=TA$ be a
Borel subgroup containing a fixed $\theta$-stable Cartan subalgebra $H$, with
$$
\begin{aligned}
&T=\{ h\in H\ \mid \ \theta(h)=h\},\\
&A=\{h\in H\ \mid \ \theta(h)=h^{-1}\}.
\end{aligned}
$$
The Langlands classification is as follows. Let
$\chi\in\widehat H.$ Denote by
$$
X(\chi):=Ind_B^G[\chi\otimes \one]_{K\text{-finite}}
$$
the corresponding admissible standard module (Harish-Chandra
induction). Let $(\mu,\nu)$ be the differentials of $\chi\mid_T$ and
$\chi\mid_A$ respectively. Let $\la_L=(\mu+\nu)/2$ and
$\la_R=(\mu-\nu)/2$. We write $X(\mu,\nu)=X(\la_L,\la_R)=X(\chi).$
\begin{theorem}\
\label{t:langlands}
\begin{enumerate}
\item $X(\mu,\nu)$ has a unique irreducible subquotient denoted
$\overline{X}(\mu,\nu)$ which contains the $K$-type with extremal
weight $\mu$ occurring with multiplicity one in $X(\mu,\nu).$
\item $\overline{X}(\mu,\nu)$ is the unique irreducible quotient when
$\langle Re\nu,\al\rangle >0$ for all $\alpha\in\Delta(\mathfrak n,\mathfrak h),$ and
the unique irreducible submodule when $\langle Re\nu,\al\rangle <0$.
\item $\overline{X}(\mu,\nu)\cong\overline{X}(\mu',\nu')$ if and only if there
is $w\in W$ such that $w\mu=\mu', w\nu=\nu'.$ Similarly for
$(\la_L,\la_R).$
\end{enumerate}
\end{theorem}
Assume $\la_L,\ \la_R$ are both dominant integral.
Write $F(\la)$ to be the finite dimensional representation of $G$ with infinitesimal character $\la$.
Then
${\overline{X}}(\la_L,-\la_R)$ is the finite dimensional representation
$F(\la_L)\otimes {F(-w_0\la_R)}$ where $w_0\in W$ is the long Weyl
group element. The lowest $K$-type has extremal weight
$\la_L-\la_R$. Weyl's character formula implies
$$
{\overline{X}}(\la_L,-\la_R)=\sum \limits _{w\in W} {\epsilon}(w)X(\la_L,-w\la_R).
$$
\subsubsection*{}
In the following contents in this section, we use different notation as follows. We write
$(\widetilde{G}, \widetilde{K})=(Spin (2n,{\mathbb{C}}), Spin(2n) )$ and $(G,K)=(SO(2n,{\mathbb{C}}) , SO(2n))$.
\subsection{Infinitesimal characters} \label{s:infchar}
From \cite{B}, we can associate to each ${\mathcal{O}}$ in Section \ref{ss:orbits} an infinitesimal character
$\la_{{\mathcal{O}}}$. The fact is that ${\mathcal{O}}$ is the minimal orbit which can be the associated variety of
a $(\mathfrak g, K)$-module with infinitesimal character $(\la_L, \la_R)$, with $\la_L$ and $\la_R$ both conjugate to $\la_{{\mathcal{O}}}$.
The $\la_{{\mathcal{O}}}$ are listed below.
\begin{description}
\item[Case 1] {$n=2p$, ${\mathcal{O}}=[3 \ 2^{n-2} \ 1]$,}
$$\la_{{\mathcal{O}}} ={\mathrm{h}}o/2= (p-\frac{1}{2}, \dots,\frac{3}{2}, \frac{1}{2}\mid p-1,\dots, 1, 0).$$
\item[Case 2]
{ ${\mathcal{O}}=[3\ 2^{2k} \ 1^{2n-4k-3}]$, $0\le k\le p-1$,}
$$ \la_{{\mathcal{O}}}=( k+\frac{1}{2}, \dots,\frac{3}{2}, \frac{1}{2}\mid n-k-2,\dots, 1, 0).$$
\item[Case 3]
{$n=2p$, ${\mathcal{O}}_{I,II}=[2^n]_{I, II}$,}
\begin{eqnarray*}
\la _{{\mathcal{O}} _I}&=& \left ( \frac{2n-1}{4} ,\frac{2n-5}{4} , \dots, \frac{-(2n-7)}{4} , \frac{ -(2n-3)}{4} \right),\\
\la _{{\mathcal{O}}_{II}} &=& \left ( \frac{2n-1}{4} ,\frac{2n-5}{4} , \dots, \frac{-(2n-7)}{4} , \frac{ (2n-3)}{4} \right).
\end{eqnarray*}
\item[Case 4]
{ ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$, $0\le k \le p-1$,}
$$
\la_{{\mathcal{O}}}= (k,k-1,\dots, 1 ; n-k-1, \dots, 1, 0).
$$
\end{description}
Notice that the infinitesimal characters in Cases 1 and 2 are nonintegral. For instance, in Case 1, $\la_{{\mathcal{O}}}={\mathrm{h}}o/2$, where ${\mathrm{h}}o$ is half
sum of the positive roots of type $D_{2p}$. The integral system is of
type $D_p\times D_p$. The notation $|$ separates the coordinates of the two $D_p$.
\subsection{} \label{ss:rep} We define
the following irreducible modules in terms of Langlands classification:
\begin{description}
\item[Case 1] {$n=2p$, ${\mathcal{O}}=[3 \ 2^{n-1} \ 1]$}.
\begin{enumerate}
\item[(i)] $\Xi_1 = {\overline{X}} (\la_{{\mathcal{O}}},-\la_{{\mathcal{O}}})$;
\item[(ii)] $\Xi_2= {\overline{X}} (\la_{{\mathcal{O}}}, -w_1\la_{{\mathcal{O}}} )$, where $w_1\la_{{\mathcal{O}}}= (p-\frac{1}{2}, \dots,
\frac{3}{2}, -\frac{1}{2} \mid p-1,\dots, 1, 0)$;
\item[(iii)] $\Xi_3= {\overline{X}} (\la_{{\mathcal{O}}}, -w_2\la_{{\mathcal{O}}})$, where $w_2\la_{{\mathcal{O}}}= (p-1,\dots, 1, 0 \mid
p-\frac{1}{2}, \dots, \frac{3}{2}, \frac{1}{2} )$;
\item[(iv)] $\Xi_4= {\overline{X}} (\la_{{\mathcal{O}}},-w_3 \la_{{\mathcal{O}}})$, where $w_3\la_{{\mathcal{O}}}=(p-1,\dots, 1, 0 \mid
p-\frac{1}{2}, \dots, \frac{3}{2}, -\frac{1}{2} )$.
\end{enumerate}
\item[Case 2]
{${\mathcal{O}}=[3\ 2^{2k} \ 1^{2n-4k-3}]$, $0\le k\le p-1$}.
\begin{enumerate}
\item [(i)] $\Xi_1={\overline{X}}(\la_{{\mathcal{O}}},-\la_{{\mathcal{O}}})$;
\item [(ii)]$\Xi_2= {\overline{X}}(\la_{{\mathcal{O}}},-w_1\la_{{\mathcal{O}}}),\ w_1\la_{{\mathcal{O}}}= ( k+\frac{1}{2}, \dots,\frac{3}{2}, \frac{1}{2}\mid n-k-2,\dots, 1, 0 ).$
\end{enumerate}
\item[Case 3]
{$n=2p$, ${\mathcal{O}}_{I,II}=[2^n]_{I, II}$}.
\begin{enumerate}
\item [(i)] $\Xi_{I} ={\overline{X}}(\la_ {{\mathcal{O}}_I} , -\la_{{\mathcal{O}}_{I} } )$;
\item [(i$'$)] $\Xi _{I} ={\overline{X}}(\la _{{\mathcal{O}}_{I}} , -w\la
_{{\mathcal{O}}_{I}})$,\quad { $w\la _{{\mathcal{O}} _I}=\left(\frac{2n-3}{2},\dots ,-\frac{2n-1}{4}\right)$};
\item [(ii)] $\Xi_{II} ={\overline{X}}(\la_ {{\mathcal{O}}_{II}} , -\la_{{\mathcal{O}}_{II} } )$;
\item [(ii$'$)] $\Xi '_{II} ={\overline{X}}(\la _{{\mathcal{O}}_{II}},\quad -w\la_{{\mathcal{O}}_{II}})$, { $w\la _{{\mathcal{O}}
_{II}}=\left(\frac{2n-3}{4},\dots
,-\frac{2n-5}{4},\frac{2n-1}{4}\right)$};
\end{enumerate}
\item[Case 4] {${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$, $0\le k \le p-1$}.
\begin{enumerate}
\item[(i)] $\Xi = {\overline{X}}(\la_{\mathcal{O}}, -\la_{\mathcal{O}}) $.
\end{enumerate}
\end{description}
\begin{remark}
The representations introduced above form the set ${\mathcal{U}}_{\widetilde{G}}({\mathcal{O}},\la_{{\mathcal{O}}})$.
\end{remark}
\subsubsection*{Notation} \label{ss:notation}
We write $F(\la)$ for the finite dimensional representation of the appropriate $SO$ or $Spin$ group with infinitesimal character $\la$;
write $V(\mu)$ for the finite dimensional representation of the appropriate $SO$ or $Spin$ group with highest weight $\mu$.
\subsection{$\widetilde{K}$-structure} We compute the $\widetilde{K}$-types of each representation listed in \ref{ss:rep}.
\subsubsection*{Case 1}
The arguments are refinements of those in
\cite{McG}.
Let $\widetilde{H}$ be the image of $Spin(2p,{\mathbb{C}})\times Spin(2p,\mathbb C)$ in $Spin(4p,\mathbb C)$, and $\widetilde{U}$ the image of the maximal compact subgroup $Spin(2p)\times Spin(2p)$ in $\widetilde{K}$. Irreducible representations of $\widetilde{U}$ can be viewed as $Spin(2p)\times Spin(2p)$-representations such that $\pm(I,I)$ acts trivially.
Cases (i) and (ii) factor to representations of $SO(2n,{\mathbb{C}}),$ (iii) and (iv)
are genuine for $Spin(2n,{\mathbb{C}}).$
The Kazhdan-Lusztig conjectures for
nonintegral infinitesimal character together with Weyl's formula for the character of a finite dimensional module, imply that
\begin{equation}
\label{eq:charfla}
\overline{X}({\mathrm{h}}o/2, -w_i{\mathrm{h}}o/2)=\sum_{w\in W(D_p\times D_p)} {\epsilon}(w) X({\mathrm{h}}o/2,-ww_i{\mathrm{h}}o/2),
\end{equation}
since $W(\la_{{\mathcal{O}}})=W(D_p\times D_p)$.
Restricting (\ref{eq:charfla}) to $\widetilde{K},$
and using Frobenius reciprocity, we get
\begin{equation} \label{eq:ind-K-U}
\overline{X}({\mathrm{h}}o/2, - w_i{\mathrm{h}}o/2)\mid_{\widetilde{K}}=Ind_{\widetilde U }^{\widetilde{K}}
[F_1({\mathrm{h}}o/2) {\otimes} F_2(-w_i{\mathrm{h}}o/2)],
\end{equation}
where $F_{1,2}$ are finite dimensional representations of the two factors $Spin(2p,\mathbb C)\times Spin(2p,\mathbb C)$ with infinitesimal
character ${\mathrm{h}}o/2$ and $-w_i{\mathrm{h}}o/2$, respectively.
The terms $[F_1({\mathrm{h}}o/2) {\otimes } F_2(-w_i{\mathrm{h}}o/2)]$ are
\begin{description}
\item[(i)] $V (1/2,\dots ,1/2)\otimes V(1/2,\dots ,1/2) \boxtimes V(0,\dots ,0)\otimes V(0,\dots
,0)$,
\item[(ii)] $V(1/2,\dots ,-1/2)\otimes V(1/2,\dots ,1/2) \boxtimes V(0,\dots
,0)\otimes V(0,\dots ,0)$,
\item[(iii)] $V(1/2,\dots ,1/2)\otimes V(0,\dots ,0)\boxtimes V(0,\dots
,0)\otimes V(1/2,\dots ,1/2)$,
\item[(iv)] $V(1/2,\dots ,1/2)\otimes V(0,\dots ,0)\boxtimes V(0,\dots
,0)\otimes V(1/2,\dots ,-1/2)$
\end{description}
as $Spin(n)\times Spin(n)$-representations (see \ref{ss:notation} for the notation).
\begin{comment}
\sout{
In the case $n=2p$, when $G= SO(2n,{\mathbb{C}})$, $G_{{\mathbb R}} ^{split}
=SO(n,n), K=S[O(n)\times O(n)]$; when $K=Spin(2n,{\mathbb{C}}), G_{{\mathbb R}}
^{split} = Spin(n,n)$, $K=Spin(n)\times Spin(n) / \{(1,1), (b,b)\}$,
where $b$ is the nontrivial element in the kernel of the covering map
$Spin(n)\to SO(n)$.
It is more convenient to work with $Pin(n,n)$ and $Pin(n)\times
Pin(n).$
A representation of $Pin(2p)$ will be denoted by $F(a_1,\dots
,a_p;{\epsilon})$ where ${\epsilon}=\{\pm 1, 1/2\}$ as follows. If $a_p=0,$ there
are two inequivalent representations with this highest weight,
${\epsilon}=\pm 1$ is according to Weyl's convention. In all other casess there
is a unique representation with this highest weight, ${\epsilon}=1/2$ or
${\epsilon}$ is suppressed altogether.
Representations of $S[Pin(n)\times Pin(n)]$ are
parametrized by restrictions of $F(a,{\epsilon}_1)\boxtimes F(b,{\epsilon}_2)$ with
the following equivalences:
\begin{enumerate}
\item If one of ${\epsilon}_i=\frac{1}{2}$, say, ${\epsilon}_1 =\frac{1}{2}$,
then $F(a;{\epsilon}_1)\boxtimes F(b;{\epsilon}_2)=F(a'; \delta_1)\boxtimes F(b' ;
\delta_2)$ if and only if $a=a', b=b', {\epsilon} _1=\delta _1, {\epsilon}_2= \delta_2$.
\item If ${\epsilon}_1, {\epsilon}_2, \delta_1, \delta_2\in \{\pm 1\}$, then
$F(a;{\epsilon}_1)\boxtimes F(b;{\epsilon}_2)=F(a'; \delta_1)\boxtimes F(b' ;
\delta_2)$ iff $a=a', b=b', {\epsilon}_1{\epsilon}_2=\delta_1\delta_2$.
\end{enumerate}
\end{comment}
\begin{lemma} \label{pinrep}
Let $SPIN_+ =V(\frac{1}{2},\dots,\frac{1}{2}),$ and $SPIN_- = V(\frac{1}{2},\dots,\frac{1}{2}, -\frac{1}{2}) \in \widehat{Spin(n)}$. Then
\begin{equation}
\begin{aligned}
SPIN _+ \otimes SPIN_+ &= \bigoplus \limits _{0\le k\le [\frac{p}{2}]}
V(\underset{2k}{\underbrace{1\dots 1}},
\underset{p-2k}{\underbrace{0\dots 0}} ) , \\
SPIN_+\otimes SPIN_- &= \bigoplus \limits _{0\le k\le [\frac{p-1}{2}]}
V(\underset{2k+1}{\underbrace{1\dots 1}}.
\underset{p-2k-1}{\underbrace{0\dots 0}} )
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
The proof is straightforward.
\end{proof}
Lemma \ref{pinrep} implies that (\ref{eq:ind-K-U}) becomes
\begin{equation}\label{ind-fine1}
\begin{aligned}
(i)\ &{\overline{X}} ({\mathrm{h}}o/2,- {\mathrm{h}}o/2)\mid_{\widetilde{K}} &=\text{Ind} ^{\widetilde{K}} _{\widetilde U} &\left [\bigoplus \limits _{0\le k \le [\frac{p}{2}] } V(\underset{2k}{\underbrace{1,\dots,1}},0,\dots,0 )\boxtimes V(0,\dots,0) \right ]
\\
(ii)\ &{\overline{X}} ({\mathrm{h}}o/2,- w_1{\mathrm{h}}o/2)\mid_{\widetilde{K}} & =\text{Ind} ^{\widetilde{K}} _{\widetilde U} &\left [\bigoplus \limits _{0\le k \le [\frac{p-1}{2}] } V(\underset{2k+1}{\underbrace{1,\dots,1}},0,\dots,0 )\boxtimes V(0,\dots,0) \right ]
\\
(iii)\ &{\overline{X}} ({\mathrm{h}}o/2, -w_2{\mathrm{h}}o/2)\mid_{\widetilde{K}} &=\text{Ind} ^{\widetilde{K}}_{\widetilde U}
&\left [V(1/2,\dots ,1/2)\boxtimes V(1/2,\dots ,1/2)\right] \\
(iv)\ &{\overline{X}} ({\mathrm{h}}o/2, -w_3{\mathrm{h}}o/2)\mid_{\widetilde{K}} &=\text{Ind} ^{\widetilde{K}}_{ \widetilde U}
&\left [V(1/2,\dots ,1/2)\boxtimes V(1/2,\dots ,-1/2)\right].
\end{aligned}
\end{equation}
\begin{prop}\label{p:n2p}
\begin{equation}
\begin{aligned}
&{\overline{X}} ({\mathrm{h}}o/2,-{\mathrm{h}}o/2) |_{\widetilde K}= \bigoplus
V(a_1,\dots ,a_n),\quad \text{ with } a_i\in\mathbb Z, \ \sum a_i\in 2 \mathbb Z, \\
&{\overline{X}} ({\mathrm{h}}o/2, -w_1 {\mathrm{h}}o/2) |_{\widetilde K} = \bigoplus
V(a_1,\dots,a_n),\quad \text{ with } a_i\in\mathbb Z, \ \sum a_i\in 2 \mathbb Z+1,\\
&{\overline{X}} ({\mathrm{h}}o/2,-w_2{\mathrm{h}}o/2) |_{\widetilde K}= \bigoplus
V(a_1,\dots, a_n),\quad \text{ with } a_i\in\mathbb Z+1/2, \ \sum a_i\in 2 \mathbb Z+p,\\
&{\overline{X}} ({\mathrm{h}}o/2,-w_3{\mathrm{h}}o/2) |_{\widetilde K} =\bigoplus
V(a_1,\dots, a_n),\quad \text{ with } a_i\in\mathbb Z+1/2, \ \sum a_i\in 2 \mathbb Z+p+1.
\end{aligned}
\end{equation}
\end{prop}
\begin{proof}
In the first two cases we can substitute $\big({G}^{split},K^{split}):=\big(SO(2p,2p),S[O(2p)\times O(2p)])\big)$ for
$\big(\widetilde{K},\widetilde U\big),$ and $\big(Spin(2p,2p),Spin(2p)\times Spin(2p)/\{\pm (I,I)\}\big)$ for the last two cases.
The problem of computing
the $\widetilde{K}$-structure of $\overline{X}$ reduces to finding the finite
dimensional representations of $\widetilde{G}^{split}$ which contain factors of
$F({\mathrm{h}}o/2)\otimes F(-w_i{\mathrm{h}}o/2).$
Any finite dimensional
representation of $\widetilde{G}^{split}$ is a Langlands quotient of a principal
series. Principal series have fine lowest $K$-types (see \cite{V}). Let
${M}A$ be a split Cartan subgroup of $\widetilde{G}^{split}.$ A principal series is
parametrized by a $(\delta,\nu)\in\widehat{M}A.$ The $\delta$ are called
fine, and each fine ${K^{split}}$-type $\mu$ is a direct sum of a Weyl group
orbit of a $fine$ $\delta.$ This implies
that the multiplicities in (\ref{ind-fine1}) are all one, and all the
finite dimensional representations occur in
$(i),(ii),(iii),(iv)$. The four formulas correspond to the various orbits of the $\delta.$
\end{proof}
\subsubsection*{Case 2: ${\mathcal{O}}=[3\ 2^{2k}\ 1^{2n-4k-3}]$, $0\le k \le p-1$}
Recall that $$\la_{{\mathcal{O}}}= ( k+\frac{1}{2}, \dots,\frac{3}{2}, \frac{1}{2}\mid n-k-2,\dots, 1, 0),$$ and the integral system is $D_k\times D_{n-k}.$
The irreducible modules
are of the form $\overline{X}(\la_L,-w\la_R)$ such that $\la_{{\mathcal{O}}}$ is
dominant, $w_i\la_{{\mathcal{O}}}$ is antidominant for $D_{k}\times D_{n-k},$ and they
factor to $SO(2n,\mathbb C).$ These representations are listed in \ref{ss:rep}.
\subsubsection*{} We need to work with the real form $\big(SO(r,s), S[O(r)\times O(s)]\big)$.
A representation of $O(n)$, $r=2m+\eta$ with $\eta=0$ or $1$, will be denoted by $V(a_1,\dots,a_m; {\epsilon})$, with ${\epsilon}=\pm 1,1/2$ according to Weyl's convention, and $a_1\ge a_2\ge \dots\ge a_m\ge 0.$ If $a_m=0,$ there
are two inequivalent representations with this highest weight, one for ${\epsilon}=1$, one for ${\epsilon}=-1.$ Each restricts irreducibly to $SO(r)$ as the representation $V(a_1,\dots,a_m)\in \widehat{SO(r)}$.
When $a_m\neq 0$, there
is a unique representation with this highest weight, ${\epsilon}=1/2$ or
${\epsilon}$ is suppressed altogether. The restriction of this representation to $SO(r)$ is a sum of two representations $V(a_1,\dots,a_m)$ and $V(a_1,\dots,a_{m-1},-a_m)$.
Representations of $Pin(s)$ are parametrized in the same way, with $a_1\ge \dots \ge a_m\ge 0$ allowed to be nonnegative decreasing half-integers.
Representations of $S[O(r)\times O(s)]$ are
parametrized by restrictions of $V(a;{\epsilon}_1)\boxtimes V(b; {\epsilon}_2)$ with
the following equivalences:
\begin{enumerate}
\item If one of ${\epsilon}_i=\frac{1}{2}$, say, ${\epsilon}_1 =\frac{1}{2}$,
then $V(a;{\epsilon}_1)\boxtimes V(b;{\epsilon}_2)=V(a'; \delta_1)\boxtimes V(b' ;
\delta_2)$ if and only if $a=a', b=b', {\epsilon} _1=\delta _1, {\epsilon}_2= \delta_2$.
\item If ${\epsilon}_1, {\epsilon}_2, \delta_1, \delta_2\in \{\pm 1\}$, then
$V(a;{\epsilon}_1)\boxtimes V(b;{\epsilon}_2)=V(a'; \delta_1)\boxtimes V(b' ;
\delta_2)$ iff $a=a', b=b', {\epsilon}_1{\epsilon}_2=\delta_1\delta_2$.
\end{enumerate}
\begin{lemma} \label{pin-rep-1}
Let $PIN =V(\frac{1}{2}\dots, \frac{1}{2}) \in \widehat{Pin(s)}, s=2m+\eta$ with $\eta=0$ or 1. Then
\begin{equation}
PIN\otimes PIN =\sum _{\ell=0} ^{m-1} V(\underset{k}{\underbrace{1\dots 1}},
\underset{m-\ell}{\underbrace{0\dots 0}} ; {\epsilon} ) + V(1,\dots,1;1/2),
\end{equation}
where the sum in over ${\epsilon}=1$ and $-1$.
\begin{proof}
Omitted.
\end{proof}
\end{lemma}
\subsubsection*{}
We will use the groups $U=S[O(2k)\times O(2n-2k)]\subset K=SO(2n)$.
Again, the representations that we want are in \ref{ss:rep}.
As before,
\begin{equation}
\label{eq:cf2}
\overline{X}(\la_{{\mathcal{O}}}, -w_i\la_{\mathcal{O}})=\sum_{w\in W(D_{k}\times D_{n-k})} {{\epsilon}(w)}X(\la_{\mathcal{O}}, - ww_i\la_{\mathcal{O}}).
\end{equation}
{Restricting to $K,$
and using Frobenius reciprocity, (\ref{eq:cf2}) implies
\begin{equation} \label{ind-K-U-2}
\overline{X}(\la_{\mathcal{O}}, - w_i\la_{\mathcal{O}})\mid_{K }=\text{Ind}_{U }^{K}
[F_1(\la_{{\mathcal{O}}}) {\otimes} F_2(-w_i\la_{\mathcal{O}})].
\end{equation}
}
The terms $[F_1(\la_{\mathcal{O}})\otimes F_2(-w_i\la_{\mathcal{O}})]$ are
\begin{description}
\item[(i)] $V(1/2,\dots ,1/2)\otimes V(0,\dots ,0)\boxtimes V(1/2,\dots
,1/2)\otimes V(0,\dots ,0)$,
\item[(ii)] $V(1/2,\dots ,1/2,-1/2)\otimes V(0,\dots ,0)\boxtimes V(1/2,\dots ,
1/2,-1/2)\otimes V(0,\dots ,0)$.
\end{description}
\
\begin{lemma}
\begin{equation}\label{ind-fine6}
\begin{aligned}
{\overline{X}} (\la_{\mathcal{O}}, -\la_{\mathcal{O}}) =\text{Ind} ^{K} _{U} \Big [ & \sum \limits _{0\le 2\ell \le k }
V(\underset{2\ell}{\underbrace{1,\dots,1}},0,\dots,0 ;1 )\boxtimes
V(0,\dots,0;1)\\
\\ &+ \sum \limits _{0\le 2\ell \le k}
V(\underset{2\ell}{\underbrace{1,\dots,1}},0,\dots,0 ;1 )\boxtimes
V(0,\dots,0;-1)\Big],\\
{\overline{X}} (\la_{\mathcal{O}}, -w_1\la_{\mathcal{O}}) =\text{Ind} ^{K} _U \Big [ & \sum \limits _{0\le 2\ell+1 \le k }
V(\underset{2l+1}{\underbrace{1,\dots,1}},0,\dots,0 ;1 )\boxtimes
V(0,\dots,0;1)
\\ &+ \sum \limits _{0\le 2\ell+1 \le k}
V(\underset{2\ell+1}{\underbrace{1,\dots,1}},0,\dots,0 ;1 )\boxtimes
V(0,\dots,0;-1) \Big].
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
This follows from Lemma \ref{pin-rep-1}.
\end{proof}
\begin{prop}
\begin{equation}\label{eq:ktypeskn}
\begin{aligned}
& {\overline{X}} (\la_{\mathcal{O}},-\la_{\mathcal{O}})|_{\widetilde{K}} = \bigoplus V(a_1,\dots,a_{k}, 0,\dots ,0),\quad \text{ with } \ a_i\in \mathbb Z, \ \sum a_i\in 2{\mathbb Z}\\
&{\overline{X}} (\la_{\mathcal{O}}, -w_1\la_{\mathcal{O}} )|_{\widetilde{K}} = \bigoplus V(a_1,\dots,a_{k},0,\dots ,0),\quad \text{ with } \ a_i\in \mathbb Z.\ \sum a_i\in 2{\mathbb Z} +1.
\end{aligned}
\end{equation}
\end{prop}
\begin{proof}
The proof is almost identical to that of Proposition \ref{p:n2p}.
When $k=p-1$,
the group $\widetilde{G^{split}}$ in the proof of Proposition \ref{p:n2p} is replaced by
$G^{qs}=SO(2p,2p+2)$ and $\widetilde{U}$ is replaced by $U=S[O(2p)\times O(2p+2)].$
When $k<p-1$,
the group $\widetilde{G^{split}}$ is replaced by
$G^{k,n-k}=SO(2k,2n-2k)$ and $\widetilde{U}$ is replaced by $U= S[O(2k)\times O(2n-2k)].$
We follow \cite{V}.
The $K$-types $\mu$ in (\ref{ind-fine6}) have $\mathfrak q(\la_L)$
the $\theta$-stable parabolic $\mathfrak
q=\mathfrak l+\mathfrak u$ determined by $\xi=(0,\dots
,0;\underset{n-2k-2}{\underbrace{1,\dots ,1}},0\dots ,0)$. The Levi
component is $S[O(2k)\times O(2k+2)].$ The resulting
$\mu_L=\mu-2{\mathrm{h}}o(\mathfrak u\cap\mathfrak s)$ are fine $U\cap L$-types. A bottom layer argument
reduces the proof to the quasisplit case $n=2p+1$.
\end{proof}
\subsubsection*{Cases 3,4}
We use the infinitesimal characters in \ref{s:infchar} and the representations are from \ref{ss:rep} again.
In Case 4, ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$ with $k<p$. There is a unique irreducible representation with
associated support ${\mathcal{O}}$, and it is spherical. It is a special
unipotent representation with character given by \cite{BV}.
When {$n=2p$ and $k=p$}, there are two nilpotent orbits ${\mathcal{O}}_{I,II} = [2^n]_{I,II}$.
The representations $\Xi_{I,II}$ in \ref{ss:rep} are
spherical representations, one each for ${\mathcal{O}}_{I,II}$ that are not genuine.
The two representations are induced irreducibly from the
trivial representation of the parabolic subgroups with Levi components
$GL(n)_{I,II}.$
On the other hand, the representations $\Xi'_{I,II}$ are induced irreducibly from the
character $Det ^{1/2}$ of the parabolic subgroups with Levi components
$GL(n)_{I,II}.$
All of these are unitary.
\begin{prop}
\label{p:kstruct3}
The $\widetilde K$-types of these representations are:
\begin{description}
\item[Case 3] ${\mathcal{O}}_{I,II}=[2^{2p}]_{I,II}:\ $
\begin{equation}
\begin{aligned}
\Xi_I |_{\widetilde{K}} & =& \bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},a_{n-1}) & \text{ with } a_i\in {\mathbb Z}, \\
\Xi '_I |_{\widetilde{K}} & =& \bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},a_{n-1}) & \text{ with } a_i\in {\mathbb Z} +1/2,\\
\Xi_{II}|_{\widetilde{K}} & =& \bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},-a_{n-1}) & \text{ with } a_i\in {\mathbb Z}, \\
\Xi '_{II}|_{\widetilde{K}} & =& \bigoplus V(a_1,a_1,a_3,a_3,\dots,a_{n-1},-a_{n-1}) & \text{ with } a_i\in {\mathbb Z} +1/2,
\end{aligned}
\end{equation}
satisfying $a_1\ge a_3\ge \dots \ge a_{n-1}\ge 0$
\item[Case 4] ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}],\ 0\le k< n/2:\ $
$$\Xi | _{\widetilde{K}}= \bigoplus V(a_1,a_1,\dots,
a_k,a_k,0,\dots ,0), \ \text{ with } \ a_i\in {\mathbb Z},$$
satisfying $a_1\ge a_3\ge \dots \ge a_k\ge 0.$
\end{description}
\begin{proof}
These are well known.
The cases $[2^{n}]_{I,II}$ follow by Helgason's theorem since
$(D_{n},A_{n-1})$ is a symmetric pair (for the real form
$SO^*(2n)$). They also follow by the method
outlined below for the other cases.
For $2k<n,$ the methods outlined in \cite{BP2} combined with \cite{B} give the
answer; the representations are $\Theta$-lifts of the trivial
representation of $Sp(2k,{\mathbb{C}}).$ More precisely ${\overline{X}}(\la_{{\mathcal{O}}},-\la_{\mathcal{O}})$
is $\Omega/[\mathfrak{sp}(2k,{\mathbb{C}})\Omega]$ where $\Omega$ is the oscillator
representation for the pair $O(2n,{\mathbb{C}})\times Sp(2k,{\mathbb{C}})$.
The $K$-structure can then be computed using seesaw
pairs, namely $\Omega$ is also the oscillator representation for the
pair $O(2n)\otimes Sp(4k,{\mathbb R})$.
\end{proof}
\end{prop}
\subsection{}
We resume the notation used in Section 3. Let $(G_0, K) = (Spin (2n,{\mathbb{C}}), Spin(2n,{\mathbb{C}}))$.
By comparing Propositions \ref{p:regfun1}, \ref{p:regfun2}, \ref{p:regfun3}
and the $K$-structure of representations listed in this section, we have
the following matchup.
\begin{description}
\item[Case 1] $\Xi_i |_{ K} = R({\mathcal{O}},\psi_i), \ 1\le i \le 4$;
\item[Case 2] $\Xi_i |_{ K} = R({\mathcal{O}},\psi_i), \ i=1,2$;
\item[Case 3] $\Xi_I |_{ K} =R({\mathcal{O}}_I, Triv)$, $\Xi ' _I |_{ K} =R({\mathcal{O}}_I, Sgn)$, \\ \hspace*{2.5em} $\Xi_{II}|_{ K} =R({\mathcal{O}}_{II}, Triv)$, $\Xi '_{II} |_{ K} =R({\mathcal{O}}_{II}, Sgn)$;
\item[Case 4] $\Xi |_{ K} = R({\mathcal{O}} , Triv)$.
\end{description}
Then the following theorem follows.
\begin{theorem}\label{t:main}
Attain the notation above. Let $ G_0={Spin}(2n,{\mathbb{C}})$ be viewed as a real group. The $K$-structure
of each representations in ${\mathcal{U}}_{G_0}({\mathcal{O}},\la_{{\mathcal{O}}})$ is calculated explicitly and matches the
$K$-structure of the $R({\mathcal{O}},\psi)$ with $\psi \in \widehat{A_{ K}({\mathcal{O}})}$.That is, there is a 1-1 correspondence $\psi\in
\widehat { A_{{K}}({\mathcal{O}})}\longleftrightarrow \Xi({\mathcal{O}},\psi)\in
{\mathcal{U}}_{ G_0}({\mathcal{O}},\la_{\mathcal{O}})$ satisfying
$$
\Xi({\mathcal{O}},\psi)\mid_{ K}\cong R({\mathcal{O}},\psi).
$$
\end{theorem}
\section{Clifford algebras and Spin groups} \label{ss:clifford}
Since the main interest is in the case of $Spin(V),$ the simply
connected groups of type $D,$ we realize everything in the context of
the Clifford algebra.
\subsection{} Let $(V,Q)$ be a quadratic space of even dimension $2n$, with a basis
$\{e_i,f_i\}$ with $1\le i\le n,$ satisfying $Q(e_i,f_j)=\delta_{ij},$
$Q(e_i,e_j)=Q(f_i,f_j)=0$. Occasionally we will replace $e_j,f_j$ by
two orthogonal vectors $v_j,w_j$ satisfying
$Q(v_j,v_j)=Q(w_j,w_j)=1,$ and orthogonal to the $e_i,f_i$ for $i\ne
j.$ Precisely they will satisfy $v_j=(e_j+f_j)/\sqrt{2}$ and
$w_j=(e_j-f_j)/(i\sqrt{2})$ (where $i:=\sqrt{-1},$ not an index). Let
$C(V)$ be the Clifford algebra with automorphisms $\al$ defined by
$\al(x_1\cdots x_r)=(-1)^r x_1\cdots x_r$ and $\star$ given by $(x_1\cdots
x_r)^\star=(-1)^r x_r\cdots x_1, $ subject to the relation
$xy+yx=2Q(x,y)$ for $x,y\in V$. The double cover of $O(V)$ is
$$
Pin(V):=\{ x\in C(V)\ \mid\ x\cdot x^\star=1,\ \al(x)Vx^\star\subset V\}.
$$
The double cover $Spin(V)$ of $SO(V)$ is given by the elements in $Pin(V)$ which are in $C(V)^{even},$ \ie $\disp{Spin(V):=Pin(V)\cap C(V)^{even}}.$
For $Spin,$ $\al$ can be suppressed from the notation since it is the identity.
The action of $Pin(V)$ on $V$ is given by ${\mathrm{h}}o(x)v=\al(x)vx^*.$ The
element $-I\in SO(V)$ is covered by
\begin{equation}
\label{eq:-spin}
\pm \Ep_{2n}=\pm i^{n-1}vw{\mathrm{pr}}od_{1\le j\le n-1} [1-e_jf_j]=\pm
i^n{\mathrm{pr}}od_{1\le j\le n} [1-e_jf_j].
\end{equation}
These elements satisfy
$$
\Ep_{2n}^2=
\begin{cases}
+Id &\text{ if } n\in 2\mathbb Z,\\
-Id&\text { otherwise.}
\end{cases}
$$
The center of $Spin(V)$ is
$$
Z(Spin(V))=\{\pm I, \pm\Ep_{2n}\}\cong
\begin{cases}
\mathbb Z_2\times \mathbb
Z_2 &\text{ if }n \text{ is even,}\\
\mathbb Z_4 &\text{ if } n \text{ is odd}.
\end{cases}
$$
The Lie algebra of $Pin(V)$ as well as $Spin(V)$ is formed of elements
of even order $\le 2$ satisfying
$$
x+x^\star=0.
$$
The adjoint action is {$\ad x(y)=xy-yx$}. A
Cartan subalgebra and the root vectors corresponding to the usual
basis in Weyl normal form are formed of the elements
\begin{equation}
\begin{aligned}
\label{eq:liea}
&(1-e_if_i)/2&&\longleftrightarrow &&H({\epsilon}_i)\\
&e_ie_j/2&&\longleftrightarrow &&X(-{\epsilon}_i-{\epsilon}_j),\\
&e_if_j/2&&\longleftrightarrow &&X(-{\epsilon}_i+{\epsilon}_j),\\
&{f_i} f_j/2&&\longleftrightarrow &&X({\epsilon}_i+{\epsilon}_j).
\end{aligned}
\end{equation}
\begin{comment}
\subsubsection*{Root Structure}\ We use $1\le i\le p$ and $1\le j\le
q-1$ consistently. We give a realization of the Lie algebra for
$Spin(2p+1,2q-1)$. The case $Spin(2p,2q)$, is (essentially) obtained
by suppressing the short roots.
\begin{tabular}{ll}
{Compact} &Noncompact\\
&\\
${\mathfrak t}=\{(1-e_if_i), (1-e_{p+j}f_{p+j})\}$ & $\mathfrak a=\{v^+v^-\}$\\
$h({\epsilon}_i),\ h({\epsilon}_{p+j})$ &$h({\epsilon}_{p+q})$\\
$f_iv^+, e_iv^+, f_{p+j}v^-, e_{p+j}v^-$ &$f_iv^-, e_iv^-, v^+f_{p+j}, v^+e_{p+j}$\\
$X({\epsilon}_i)_c, X(-{\epsilon}_{i})_c, X({\epsilon}_{p+j})_c, X(-{\epsilon}_{p+j})_c$&$X({\epsilon}_i)_{n}, X(-{\epsilon}_i)_{n} X({\epsilon}_{p+j})_{n}, X(-{\epsilon}_{-p+j})_{n}$\\
$f_if_l, f_ie_l, e_ie_l, e_if_l $& $f_if_{p+j}, f_ie_{p+j}, e_ie_{p+l}, e_if_{p+l}$\\
$f_{p+j}f_{p+m},f_{p+j}e_{p+m}, e_{p+j}f_{p+m}, e_{p+j}e_{p+m}$& \\
$X({\epsilon}_i+{\epsilon}_l),X({\epsilon}_i-{\epsilon}_l), X(-{\epsilon}_i-{\epsilon}_l), X(-{\epsilon}_i+{\epsilon}_l)$&
$X({\epsilon}_i+{\epsilon}_{p+j}), X({\epsilon}_i-{\epsilon}_{p+j})$,\\
& $X(-{\epsilon}_i-{\epsilon}_{p+j}), X(-{\epsilon}_i+{\epsilon}_{p+j})$,\\
$X({\epsilon}_{p+j}+{\epsilon}_{p+m}), X({\epsilon}_{p+j}-{\epsilon}_{p+m}),$ &\\
$X(-{\epsilon}_{p+j}-{\epsilon}_{p+m}), X(-{\epsilon}_{p+j}+{\epsilon}_{p+m}).$&
\end{tabular}
\end{comment}
\subsection{Nilpotent Orbits}
We write $\widetilde{K}=Spin(V)=Spin (2n,{\mathbb{C}}) $, $K=SO(V)=SO(2n,{\mathbb{C}})$. A nilpotent orbit of an element $e$ will have Jordan blocks denoted by
\begin{equation}
\label{eq:blocks}
\begin{aligned}
&e_1\longrightarrow e_2\longrightarrow\dots \longrightarrow
e_k\longrightarrow v\longrightarrow -f_k\longrightarrow
f_{k-1}\longrightarrow {-f_{k-2} \longrightarrow} \dots \longrightarrow \pm f_1\longrightarrow 0\\
&\begin{matrix}
&e_1\longrightarrow &e_2&\longrightarrow&\dots &\longrightarrow
&e_{2\ell}\longrightarrow 0\\
&f_{2\ell}\longrightarrow &-f_{2\ell-1}&\longrightarrow&\dots &\longrightarrow
&-f_1\longrightarrow 0
\end{matrix}
\end{aligned}
\end{equation}
with the conventions about the $e_i,f_j,v$ as before. There is an even
number of odd sized blocks, and any two blocks of equal odd size $2k+1$
can be replaced by a pair of blocks of the form as the even ones. A realization of the odd block is given by $ \displaystyle\frac{1}{2}\left ( {\sum \limits _{i=1} ^{k-1} }e_{i+1}f_i +vf_k\right ),$ and a realization of the even blocks by $\dpfr \left( {\sum \limits _{i} ^{2l-1}}e_{i+1}f_{i}\right).$ When there are only even blocks, there are two orbits; one block of the form
$\big(\sum_{1\le i< \ell-1} e_{i+1}f_{i}+e_\ell f _{\ell-1}\big)/2$ is replaced by $\big(\sum_{1\le i< \ell-1} e_{i+1}f_{i}+ f_\ell f_{\ell-1}\big)/2.$
The centralizer of $e$ in $\mathfrak{so}(V)$ has Levi component isomorphic to a product
of $\mathfrak{so}(r_{2k+1})$ and $\mathfrak{sp}(2r_{2\ell})$ where $r_j$ is the number of
blocks of size $j.$ The centralizer of $e$ in $SO(V)$ has Levi
component ${\mathrm{pr}}od Sp(2r_{2\ell})\times S[{\mathrm{pr}}od O(r_{2k+1})]$.
For each odd sized block define
\begin{equation}
\label{eq:epsilon}
\Ep_{2k+1}=i^{k}v{\mathrm{pr}}od (1-e_jf_j).
\end{equation}
This is an element in $Pin(V),$ and acts by $-Id$ on the block. Even
products of $\pm \Ep_{2k+1}$ belong to $Spin(V),$ and represent the
connected components of $C_{\widetilde{K}}(e).$
\begin{prop}
Let $m$ be the number of distinct odd blocks. Then
$$A_K({\mathcal{O}}) \cong \begin{cases} {\mathbb Z} _2 ^{m-1} & \mbox{ if } m > 0 \\ 1 & \mbox{ if } m=0. \end{cases} $$ Furthermore,
\begin{enumerate}
\item If $E$ has an odd block of size $2k+1$ with $r_{2k+1}>1,$ then
$A_{\widetilde{K}}({\mathcal{O}})\cong A_K({\mathcal{O}}).$
\item If all $r_{2k+1}\le 1,$ then there is an exact sequence
$$
1\longrightarrow \{\pm I\}\longrightarrow
A_{\widetilde{K}}({\mathcal{O}})\longrightarrow A_K({\mathcal{O}})\longrightarrow
0.
$$
\end{enumerate}
\end{prop}
\begin{proof}
Assume that there is an ${r_{2k+1}>1}.$ Let
$$
\begin{matrix}
&e_1&\rightarrow &\dots &\rightarrow &e_{2k+1}&\rightarrow 0\\
&f_{2k+1}&\rightarrow&\dots &\rightarrow &-f_1 &\rightarrow 0
\end{matrix}
$$
be two of the blocks. In the Clifford algebra this element is
$e=(e_2f_1+\dots +e_{2k+1}f_{2k})/2.$ The element ${\sum \limits _{ j=1} ^{2k+1}}(1-e_{j}f_{j})$ in the Lie
algebra commutes with $e$. So its exponential
\begin{equation}\label{path1}
{\mathrm{pr}}od \exp\big( i\theta(1-e_{j}f_{j})/2\big)=
{\mathrm{pr}}od [\cos\theta/2 + i\sin\theta/2 (1-e_{j}f_{j})]
\end{equation}
also commutes with $e.$ {At $\theta=0$,
the element in (\ref{path1}) is $I$; at $\theta =2\pi$, it is $-I$.} Thus $-I$ is in the connected component of the identity of
$A_{\widetilde{K}}({\mathcal{O}})$ (when $r_{2k+1}>1$), and therefore $A_{\widetilde{K}}({\mathcal{O}})=A_K({\mathcal{O}}).$
Assume there are no blocks of odd size. Then $C_K({\mathcal{O}}) {\cong {\mathrm{pr}}od Sp (r_{2l})}$ is simply
connected, so $C_{\widetilde{K}}({\mathcal{O}})\cong C_K({\mathcal{O}})\times\{\pm I\}.$ { Therefore $A_{\widetilde{K}} ({\mathcal{O}}) \cong {\mathbb Z} _2$.}
Assume there are {$m$ distinct odd blocks with $m\in 2{\mathbb Z} _{>0}$ and $r_{2k_1+1}=\cdots =r_{2k_m +1}=1.$ In this case, $C_K({\mathcal{O}})\cong {\mathrm{pr}}od Sp(r_{2l}) \times S[ \underset{m}{\underbrace{O(1) \times \cdots \times O(1)}} ]$
, and hence $A_{\widetilde{K}} ({\mathcal{O}})\cong {\mathbb Z}_2^{m-1}$.
Even products of $\{ \pm \Ep _{2k_j +1} \}$ are representatives of elements in $A_{\widetilde{K} }({\mathcal{O}})$.} They satisfy
$$
\Ep_{2k+1}\cdot\Ep_{2\ell+1}=
\begin{cases}
-\Ep_{2\ell +1}\cdot\Ep_{2k+1} &k\ne \ell,\\
{ (-1)^kI} &k=\ell.
\end{cases}
$$
\end{proof}
\begin{cor} \
\label{c:cgp}
\begin{enumerate}
\item If ${\mathcal{O}}=[3\ 2^{n-2}\ 1],$ then {$A_{\widetilde{K}}({\mathcal{O}})\cong
\mathbb Z_2\times\mathbb Z_2=\{\pm\Ep_3\cdot\Ep_1,\pm
I\}$}.
\item If ${\mathcal{O}}=[3\ 2^{2k}\ 1^{2n-4k-3}]$ with $2n-4k-3>1,$ then $A_{\widetilde{K}}({\mathcal{O}})\cong
\mathbb Z_2.$
\item If ${\mathcal{O}}=[2^{n}]_{I,II}$ ($n$ even), then $A_{\widetilde{K}} ({\mathcal{O}}) \cong\mathbb Z_2.$
\item If ${\mathcal{O}}=[2^{2k}\ 1^{2n-4k}]$ with $2k<n,$ then
$A_{\widetilde{K}} ({\mathcal{O}}) \cong 1.$
\end{enumerate}
In all cases $C_{\widetilde K}({\mathcal{O}})=Z(\widetilde K)\cdot C_{\widetilde K}({\mathcal{O}})^0.$
\end{cor}
\end{document} |
\begin{document}
\gdef\@thefnmark{}\@footnotetext{\textup{2000} \textit{Mathematics Subject Classification}:
57N05, 20F38, 20F05}
\gdef\@thefnmark{}\@footnotetext{\textit{Keywords}:
Mapping class groups, nonorientable surfaces, involutions}
\newenvironment{prooff}{
\par \noindent {\it Proof}\ }{
$\mathchoice\sqr67\sqr67\sqr{2.1}6\sqr{1.5}6$
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\title[Generating the Mapping Class Group of a Nonorientable Surface] {Generating the Mapping Class Group of a Nonorientable Surface by Two Elements or By Three involutions}
\author[T{\"{u}}l\.{i}n Altun{\"{o}}z, Mehmetc\.{i}k Pamuk, and O\u{g}uz Y{\i}ld{\i}z ]{T{\"{u}}l\.{i}n Altun{\"{o}}z, Mehmetc\.{i}k Pamuk, and O\u{g}uz Y{\i}ld{\i}z}
\address{Department of Mathematics, Middle East Technical University,
Ankara, Turkey}
\email{[email protected]} \email{[email protected]} \email{[email protected]}
\begin{abstract}
We prove that, for $g\geq19$ the mapping class group of a nonorientable surface of genus $g$, ${\rm Mod}(N_g)$, can be generated by two elements, one of which is of order $g$. We also prove that for $g\geq26$, ${\rm Mod}(N_g)$ can be generated by three involutions if $g\geq26$.
\end{abstract}
\maketitle
\setcounter{secnumdepth}{2}
\setcounter{section}{0}
\section{Introduction}
The mapping class group ${\rm Mod}(N_g)$ of closed connected nonorientable surface $N_g$ is defined to be the group of the isotopy classes of all self-diffeomorphisms
of $N_g$. In this paper, we are interested in finding generating sets for ${\rm Mod}(N_g)$ consisting of least possible number of elements. Since this group is not abelian, a generating set must contain at least two elements.
Szepietowski~\cite{sz2} proved that ${\rm Mod}(N_{g})$ is generated by three elements for all $g\geq3$.
Our first result (see Theorem~\ref{thm1}) answers Problem $3.1(a)$ in~\cite[p.$91$]{F} (cf Problem $5.4$ in~\cite{mk44}).
\begin{thma}\label{thma}
For $g\geq 19$, the mapping class group ${\rm Mod}(N_{g})$ is generated by two elements.
\end{thma}
The next aim of the paper is to find an answer Problem $3.1(b)$ in~\cite[p.$91$]{F}. Szepietowski showed that ${\rm Mod}(N_{g})$ can be generated by involutions~\cite{sz1} and later he showed that ${\rm Mod}(N_{g})$ can be generated by four involutions if $g\geq4$~\cite{sz2}. One can deduce that it can be generated by three involutions by the work of Birman and Chillingworth~\cite{bc} if $g=3$. It is known that any group generated by two involutions is isomorphic to a quotient of a dihedral group. Thus the mapping class group ${\rm Mod}(N_{g})$ cannot be generated by two involutions. This implies that any generating set consisting only involutions must contain at least three elements. In this direction, we get the following result (see Theorem~\ref{teven} and Theorem~\ref{todd}):
\begin{thmb}\label{thmb}
For $g\geq 26$, the mapping class group ${\rm Mod}(N_{g})$ can be generated by three involutions.
\end{thmb}
Let us also point out that ${\rm Mod}(N_{g})$ admits an epimorphism onto the automorphism group of $H_1(N_g;\mathbb{Z}_2)$ preserving the $\pmod{2}$ intersection pairing~\cite{mpin} and this group is isomorphic to (see~\cite{mk3} and~\cite{sz3})
\begin{eqnarray}
\begin{cases} Sp(2h;\mathbb{Z}_2)&\text{if $g=2h+1$,} \\
Sp(2h;\mathbb{Z}_2)\ltimes \mathbb{Z}_{2}^{2h+1}&\text{if $g=2h+2$.} \end{cases}
\nonumber
\end{eqnarray}
Hence, the action of mapping classes on $H_1(N_g;\mathbb{Z}_2)$ induces an epimorphism from ${\rm Mod}(N_{g})$ to
$Sp\big(2\lfloor\dfrac{g-1}{2}\rfloor;\mathbb{Z}_2\big)$, which immediately implies the following corollary:
\begin{corc}
The symplectic group $Sp\big( g-1; \mathbb{Z}_2\big)$ can be generated by two elements for every odd $g\geq19$ and also by three involutions for every odd $g\geq27$. Similarly, the group $Sp\big(g-2; \mathbb{Z}_2\big) \ltimes \mathbb{Z}_{2}^{g-1}$ can be generated by two elements for every even $g\geq 20$ and also by three involutions for every even $g\geq26$.
\end{corc}
\noindent
\textit{Acknowledgments.} The first author was partially supported by the Scientific and
Technologic Research Council of Turkey (TUBITAK)[grant number 120F118].
\par
\section{Preliminaries} \label{S2}
Let $N_g$ be a closed connected nonorientable surface of genus $g$.
Note that the {\textit{genus}} for a nonorientable surface is the number
of projective planes in a connected sum decomposition. We use the model for the surface $N_g$ as a sphere with $g$ crosscaps represented shaded disks in all figures of this paper. Note that a crosscap is obtained by deleting the interior of such a disk and identifying the antipodal points on the resulting boundary. The {\textit{mapping class group}}
${\rm Mod}(N_g)$ of the surface $N_g$ is the group of the isotopy classes of self-diffeomorphisms of $N_g$. We
use the functional notation for the composition of two diffeomorphisms; if $f$ and $g$ are two diffeomorphisms,
the composition $fg$ means that $g$ is applied first.
A simple closed curve on a nonorientable surface $N_g$ is
\textit{one-sided} if its regular neighbourhood is
a M\"{o}bius band and \textit{two-sided} if it is an annulus. If $a$ is a two-sided simple closed curve on $N_g$, to define the Dehn twist $t_a$ about the curve $a$, we need to choose one of two possible
orientations of its regular neighbourhood (as we did for the curves in Figure~\ref{G}). Throughout the paper, the right-handed
Dehn twist $t_a$ about the curve $a$ will be denoted by the corresponding capital
letter $A$. In our notation, both the curves on $N_g$ and self-diffeomorphisms of $N_g$ shall be considered up to isotopy. In the following we shall make repeated use of some basic relations in ${\rm Mod}(N_g)$: for two-sided simple closed curves $a$ and $b$
on $N_g$ and for any $f\in {\rm Mod}(N_g)$,
\begin{itemize}
\item \textit{Commutativity:} If $a$ and $b$ are disjoint, then $AB=BA$.
\item \textit{Conjugation:} If $f(a)=b$, then $fAf^{-1}=B^{\varepsilon}$, where $\varepsilon=\pm 1$
depending on the orientation of a regular neighbourhood of $f(a)$ with respect to the chosen orientation.
\end{itemize}
\begin{figure}
\caption{The curves $a_1,a_2,b_i,c_i,\alpha_i,\beta_i$ and $\gamma_i$ on the surface $N_g$, where $g=2r$ or $g=2r+2$. Note that we do not have the curve $c_r$ when $g$ is odd.}
\label{G}
\end{figure}
\begin{figure}
\caption{The homeomorphisms $u$ and $y=Au$.}
\label{Y}
\end{figure}
Consider the Klein bottle $K$ with a hole in Figure~\ref{Y}.
We define a \textit{crosscap transposition} $u$ as the isotopy classes of a diffeomorphism interchanging two consecutive crosscaps as shown on the left hand side of Figure~\ref{Y} and equals to the identity outside the Klein bottle with one hole $K$. The effect of the diffeomorphism $y=Au$ on the interval $c$
as in Figure~\ref{Y} can be also constructed as sliding a M{\"{o}}bius band once along the core of another one and keeping each point of the boundary of $K$ fixed. This is a \textit{$Y$-homeomorphism}~\cite{l1} (also called a \textit{crosscap slide}~\cite{mk4}). Note that $A^{-1}u$ is a $Y$-homemorphism i.e. the other choice of the orientation for a neighbourhood of the curve $a$ also gives a $Y$-homeomorphism. We also note that $y^{2}$ is a Dehn twist about $\partial K$.
It is known that ${\rm Mod}(N_g)$ is generated by Dehn twists and a $Y$-homeomorphism (one crosscap slide)~\cite{l1}. We remark that crosscap transpositions can be used instead of crosscap slides since a crosscap transposition equals to the product of a Dehn twist and a crosscap slide.
Before we finish Preliminaries, let us state a theorem which is used in the proofs of following theorems. We work with the model in Figure~\ref{T} in such a way that the surface is obtained from the $2$-sphere by deleting the interiors of $g$ disjoint disks which are in a circular position and identifying the antipodal points on the boundary. Moreover, note that the rotation $T$ by
$\frac{2\pi}{g}$ about the $x$-axis maps the crosscap $\mathcal{C}_i$ to $\mathcal{C}_{i+1}$ for $i=1,\ldots,g-1$ and $\mathcal{C}_g$ to $\mathcal{C}_1$.
\begin{theorem}\label{thm2.1}
For $g\geq7$, the mapping class group ${\rm Mod}(N_g)$ can be generated by the elements $T$, $A_1A_2^{-1}$, $B_1B_2^{-1}$, and a $Y$-homeomorphism (or a crosscap transposition).
\end{theorem}
\begin{proof}
Let $G$ be the subgroup of ${\rm Mod}(N_g)$ generated by the set
$\lbrace T,A_1A_{2}^{-1},B_1B_{2}^{-1} \rbrace$.
Szepietowski~\cite[Theorem $3$]{sz2} showed that $A_1,A_2,B_i$ and $C_i$ as shown in Figure~\ref{G}, together with a $Y$-homeomorphism generate ${\rm Mod}(N_g)$.
Therefore, it is enough to prove that the elements $A_1,A_2,B_i$ and $C_i$ are contained in $G$ for $i=1,\ldots,r$.
Let $\mathcal{S}$ denote the finite set of isotopy classes of two-sided non-separating
simple closed curves appearing throughout the paper with chosen orientations of neighborhoods. Define a subset $\mathcal{G}$ of $\mathcal{S}\times \mathcal{S}$
as
\[
\mathcal{G} =\lbrace(a,b): AB^{-1}\in G \rbrace.
\]
Using the similar arguments in the proof of ~\cite[Theorem $5$]{mk1}, the set $\mathcal{G}$ satisfies
\begin{itemize}
\item if $(a,b)\in \mathcal{G}$, then $(b,a)\in \mathcal{G}$ (symmetry),
\item if $(a,b) \ \textrm{and} \ (b,c)\in \mathcal{G}$, then $(a,c)\in \mathcal{G}$ (transitivity) and
\item if $(a,b)\in \mathcal{G}$ and $H\in G$ then $(H(a),H(b))\in \mathcal{G}$ ($G$-invariance).
\end{itemize}
Thus, $\mathcal{G}$ defines an equivalence relation on $\mathcal{S}$. \\
\noindent
We begin by showing that $B_iC_{j}^{-1}$ is contained in $G$ for all $i,j$. It follows from the definition of $G$ and from the fact that $T(b_1,b_2)=(c_1,c_2)$, we have $C_1C_{2}^{-1}\in G$ (here, we use the notation $f(a,b)$ to denote $(f(a),f(b))$). Also, by conjugating $C_1C_{2}^{-1}$ with powers of $T$, one can conclude that $G$ contains the elements $B_{i}B_{i+1}^{-1}$ and $C_{i}C_{i+1}^{-1}$. Moreover, the transitivity implies that the elements $B_{i}B_{j}^{-1}$ and $C_{i}C_{j}^{-1}$ are in $G$. To start with, since $B_2B_{3}^{-1}\in G$ and it is easy to verify that
\[
B_2B_{3}^{-1}A_2A_{1}^{-1}(b_2,b_3)=(a_2,b_3),
\]
so that $A_2B_{3}^{-1} \in G$. Then, we have
\[
(A_1A_{2}^{-1})(A_2B_{3}^{-1})(B_3B_{2}^{-1})=A_{1}B_{2}^{-1}\in G,
\]
since $G$ contains each of the factors. Thus, $T(a_1,b_2)=(b_1,c_2)$ implies that $B_1C_{2}^{-1}$ is also in $G$. Moreover, $G$ contains the element
\[
B_1C_{1}^{-1}=(B_1C_{2}^{-1})(C_2C_{1}^{-1}).
\]
Thus, $B_iC_{i}^{-1}\in G$ by conjugating with powers of $T$ for all $i=1,\ldots,r-1$. Again,the transitivity implies that $B_iC_{j}^{-1}\in G$. Note that, we have
\begin{itemize}
\item $(A_1B_{2}^{-1})(B_2C_{1}^{-1})=A_1C_{1}^{-1} \in G$,
\item$(C_1A_{1}^{-1})(A_1A_{2}^{-1})=C_1A_{2}^{-1}\in G$ and
\item$(C_2C_{1}^{-1})(C_1A_{1}^{-1})=C_2A_{1}^{-1}\in G$
\end{itemize}
from which it follows that the elements $A_1C_{1}^{-1}$, $C_1A_{2}^{-1}$ and $C_2A_{1}^{-1}$ are all in $G$.
It can also be verified that
\[
(A_1B_{2}^{-1})(A_1C_{1}^{-1})(A_1C_{2}^{-1})(A_1B_2^{-1})(a_2,a_1)=(d_2,a_1)
\]
so that $D_2A_1^{-1}\in G$. Also, the element $D_2C_2^{-1}=(D_2A_1^{-1})(A_1C_2^{-1})$ is in $G$. It can also be shown that
\[
(C_2B_{1}^{-1})(C_2A_{1}^{-1})(C_2C_{1}^{-1})(C_2B_{1}^{-1})(d_2,c_2)=(d_1,c_2),
\]
which implies that $G$ contains $D_1C_{2}^{-1}$. Thus, $G$ contains the element
\[
D_1A_1^{-1}=(D_1C_2^{-1})(C_2A_1^{-1})
\]
(here, the curves $d_1$ and $d_2$ are shown in \cite[Figure $4$]{bk}).
By similar arguments as in the proof of ~\cite[Lemma $5$]{bk}, for $g\geq7$ the lantern relation implies that
\[
A_3=(A_2C_{1}^{-1})(D_1C_{2}^{-1})(D_2A_{1}^{-1} ).
\]
Since $G$ contains each factor on the right hand side, $A_3\in G$. It follows from
the diffeomorphism $A_3(B_3B_1^{-1})$ maps the curve $a_3$ to $b_3$ that
\[
B_3=A_3(B_3B_{1}^{-1})A_3(B_1B_{3}^{-1})A_{3}^{-1}\in G.
\]
By conjugating $B_3$ with the powers of $T$, we conclude that $A_1,B_1,C_1,\ldots B_{r-1},C_{r-1}$ and $B_r$
are all in $G$. Moreover,
\[
A_2=(A_2A_{1}^{-1})A_1 \in G.
\]
Therefore, the Dehn twist generators are contained in $G$. This finishes the proof.
\end{proof}
\section{A generating set for ${\rm Mod}(N_g)$}\label{S3}
In this section, we work with the model in Figure~\ref{T}. Let us denote by $u_i$ the crosscap transposition supported on the one holed Klein bottle whose boundary is the curve $\alpha_i$ shown in Figure~\ref{G}. Note that the rotation $T$ takes $\alpha_i$ to $\alpha_{i+1}$ and the crosscap $\mathcal{C}_{i}$ to $\mathcal{C}_{i+1}$, which implies that $Tu_iT^{-1}=u_{i+1}$.
\begin{figure}
\caption{The rotation $T$ and the curves $c_2,\gamma_{10}
\label{T}
\end{figure}
\begin{theorem}\label{thm1}
For $g\geq19$, the mapping class group ${\rm Mod}(N_g)$ is generated by $\lbrace T,u_{g-1}\Gamma_{10}C_2^{-1} \rbrace$.
\end{theorem}
\begin{proof}
Let $F_1=u_{g-1}\Gamma_{10}C_2^{-1}$ and let us denote by $G$ the subgroup of ${\rm Mod}(N_g)$ generated by $T$ and $F_1$. It follows from Theorem~\ref{thm2.1} that it suffices to prove that the subgroup $G$ contains the elements $A_1A_2^{-1}$, $B_1B_2^{-1}$ and $u_{g-1}$ to prove that $G={\rm Mod}(N_g)$.
Let $F_2$ denote the conjugation of $F_1$ by $T^{-4}$.
It follows from $T^{-4}$ maps the curves $(\alpha_{g-1},\gamma_{10}, c_2)$ to $(\alpha_{g-5},\gamma_{6},a_1)$ that
\[
F_2=T^{-4}F_1T^4=u_{g-5}\Gamma_{6}A_1^{-1}
\]
is contained in $G$. Let $F_3$ denote the element $(F_2F_1^{-1})F_2(F_2F_1^{-1})^{-1}$ that is contained in $G$. Hence
\[
F_3=(F_2F_1^{-1})F_2(F_2F_1^{-1})^{-1}=u_{g-5}C_2A_1^{-1}.
\]
Since we have similar cases in the remaining parts of the paper, let us give some details before we proceed. It can be verified that the diffeomorphism $F_2F_1^{-1}$ send the curves $(\alpha_{g-5},\gamma_6,a_1)$ to the curves $(\alpha_{g-5},c_2,a_1)$. Then, we get
\begin{eqnarray*}
F_3&=&(F_2F_1^{-1})F_2(F_2F_1^{-1})^{-1}\\
&=&(F_2F_1^{-1})u_{g-5}\Gamma_{6}A_1^{-1}(F_2F_1^{-1})^{-1}\\
&=&u_{g-5}C_2A_1^{-1}.
\end{eqnarray*}
Thus, we have the elements $F_2F_3^{-1}=\Gamma_{6}C_2^{-1}$ and $T^4(\Gamma_{6}C_2^{-1})T^{-4}=\Gamma_{10}C_4^{-1}$, which are both contained in $G$.
Moreover, we have the following elements
\begin{eqnarray*}
F_4&=&(C_4\Gamma_{10}^{-1})F_1=u_{g-1}C_4C_2^{-1},\\
F_5&=&T^{-1}F_4T=u_{g-2}B_4B_2^{-1} \textrm{ and }\\
F_6&=&(F_4F_5)F_3(F_4F_5)^{-1}=u_{g-5}B_2A_1^{-1},
\end{eqnarray*}
all of which are contained in the subgroup $G$. From this, we get the element $F_6F_3^{-1}=B_2C_2^{-1}\in G$. Also, we have $T(B_2C_2^{-1})T^{-1}=C_2B_3^{-1}\in G$, which gives rise to
\[
B_2B_3^{-1}=(B_2C_2^{-1})(C_2B_3^{-1})\in G.
\]
This implies that $T^{-2}(B_2B_3^{-1})T^2=B_1B_2^{-1}$ is in $G$. We also have the elements
\begin{eqnarray*}
T^{2}(C_2B_3^{-1})T^{-2}&=&C_3B_4^{-1}\in G \textrm{ and }\\
T^{-2}(\Gamma_{10}C_4^{-1})T^{2}&=&\Gamma_8C_3^{-1}\in G,
\end{eqnarray*}
implying that $\Gamma_8B_4^{-1}=(\Gamma_8C_3^{-1})(C_3B_4^{-1})\in G$. The conjugation of the element $\Gamma_8B_4^{-1}$ by $T^{-7}$ is the element $\Gamma_1A_1^{-1}=A_2A_1^{-1}$ which is contained in $G$. By the proof of Theorem~\ref{thm2.1}, the subgroup $G$ contains the elements $A_1$, $A_2$, $B_i$ and $C_i$ for $i=1,\ldots,r$. Then, in particular we have the elements $T^9A_2T^{-9}=\Gamma_{10}\in G$ and $C_2\in G$. We conclude that $u_{g-1}=F_1(C_2\Gamma_{10}^{-1})\in G$, which completes the proof.
\end{proof}
\section{Involution generators for ${\rm Mod}(N_g)$}
In the first part of this section, where the genus of the surface $N_g$ is even, we refer to Figure~\ref{RE} for the involution generators $\rho_1$ and $\rho_2$ of $N_g$. The elements $\rho_1$ and $\rho_2$ are reflections about the indicated planes in Figure~\ref{RE} in such a way that the rotation $T$, depicted in Figure~\ref{T}, is given by $T=\rho_2\rho_1$. \begin{figure}
\caption{The reflections $\rho_1$ and $\rho_2$ for $g=2r+2$.}
\label{RE}
\end{figure}
\begin{theorem}\label{teven}
For $g=2r+2\geq26$, the mapping class group ${\rm Mod}(N_g)$ is generated by the involutions $\rho_1$, $\rho_2$ and $\rho_2A_2B_rB_3u_{r+3}$.
\end{theorem}
\begin{proof}
Consider the surface $N_g$ as in Figure~\ref{RE}. It follows from
\[
\rho_2(a_2)=a_2 \textrm{ and } \rho_2(b_r)=b_3
\]
and also $\rho_2$ reverses the given orientation of a neighbourhood of a two-sided simple closed curve that
\[
\rho_2A_2\rho_2=A_2^{-1} \textrm{ and } \rho_2B_r\rho_2=B_3^{-1}.
\]
Since $\rho_2u_{r+3}\rho_2=u_{r+3}^{-1}$, one can verify that the element $\rho_2A_2B_rB_3u_{r+3}$ is an involution. Let $H_1=A_2B_rB_3u_{r+3}$ and let $H$ be the subgroup of ${\rm Mod}(N_g)$ generated by the set
\[
\lbrace \rho_1,\rho_2, \rho_2H_1\rbrace.
\]
It is clear that $H_1$ and $T=\rho_2\rho_1$ are contained in the subgroup $H$. By Theorem~\ref{thm2.1}, we need to prove that the subgroup $H$ contains the elements $A_1A_2^{-1}$, $B_1B_2^{-1}$ and $u_{r+3}$. Let $H_2$ be the conjugation of $H_1$ by $T^7$. Thus
\[
H_2=T^7H_1T^{-7}=\Gamma_8C_2C_6u_{r+10} \in H.
\]
Let
\[
H_3=(H_2H_1)H_2(H_2H_1)^{-1}=\Gamma_8B_3C_6u_{r+10},
\]
which is also in $H$. From this, we get the element $H_2H_3^{-1}=C_2B_3^{-1}\in H$ implying that $T(C_2B_3^{-1})T^{-1}=B_3C_3^{-1} \in H$. One can easily see that $B_iC_i^{-1} \in H$ by conjugating $B_3C_3^{-1}$ with powers of $T$. Also, since $T(B_3C_3^{-1})T^{-1}=C_3B_4^{-1} \in H$, similarly $C_{i}B_{i+1}^{-1}\in H$ by conjugating $C_3B_4^{-1}$ with powers of $T$. Hence, we have the elements
\[
B_iB_{i+1}^{-1}=(B_iC_i^{-1})(C_{i}B_{i+1}^{-1})
\]
which are in $H$ for all $i=1,\ldots,r-1$. Moreover, $B_iB_j^{-1}\in H$ by the transitivity. In particular $B_1B_2^{-1}\in H$. Now, we have the following elements
\begin{eqnarray*}
H_4&=&(B_7B_3^{-1})H_1=A_2B_7B_ru_{r+3} \textrm{ if }r\neq 16,17,18,\\
(H_4&=&(B_9B_3^{-1})H_1=A_2B_9B_ru_{r+3} \textrm{ if }r=16,17,18,)\\
H_5&=&T^{6}H_4T^{-6}=\Gamma_{7} B_{10}B_2u_{r+9}\textrm{ if }r\neq 16,17,18,\\
(H_5&=&T^{6}H_4T^{-6}=\Gamma_{7} B_{12}B_2u_{r+9} \textrm{ if }r=16,17,18,)\\
H_6&=&(H_5H_4)H_5(H_5H_4)^{-1}=\Gamma_{7} B_{10}A_2u_{r+9}\textrm{ if }r\neq 16,17,18,\\
(H_6&=&(H_5H_4)H_5(H_5H_4)^{-1}=\Gamma_{7} B_{12}A_2u_{r+9}\textrm{ if }r=16,17,18,)
\end{eqnarray*}
which are all contained in $H$. Thus, we get the element $H_6H_5^{-1}=A_2B_2^{-1}\in H$. On the other hand, since $C_1B_2^{-1}$ is contained in $H$, the subgroup $H$ contains the following elements
\[
T^{-2}(C_1B_2^{-1})T^2=A_1B_1^{-1},
\]
\[
(A_1B_1^{-1})(B_1B_2^{-1})=A_1B_2^{-1},
\]
\[
(A_2B_2^{-1})(B_2A_1^{-1})=A_2A_1^{-1}.
\]
It follows from $T$, $A_1A_2^{-1}$ and $B_1B_2^{-1}$ are in $H$ that the Dehn twists $A_1$, $A_2$, $B_i$ and $C_i$ are also in $H$ for $i=1,\ldots,r$. This implies that
\[
u_{r+3}=(B_3^{-1}B_r^{-1}A_2^{-1})H_1\in H,
\]
which completes the proof.
\end{proof}
\begin{figure}
\caption{The reflections $\rho_1$ and $\rho_2$ for $g=2r+1$.}
\label{RO}
\end{figure}
In the second part of this section, where the genus of the surface $N_g$ is odd, we refer to Figure~\ref{RO} for the involution generators $\rho_1$ and $\rho_2$ of $N_g$. Similarly, the elements $\rho_1$ and $\rho_2$ are reflections about the indicated planes in Figure~\ref{RO} such that the rotation $T$ in Figure~\ref{T} is given by $T=\rho_2\rho_1$. In the proof of the following theorem, we use the crosscap transposition supported on the one holed Klein bottle whose boundary is the curve $\beta_i$ shown in Figure~\ref{G}. Let us denote this crosscap transposition by $v_i$.
Note that the rotation $T$ sends $\beta_i$ to $\beta_{i+1}$ and the crosscap $\mathcal{C}_i$ to $\mathcal{C}_{i+1}$, which implies that $Tv_iT^{-1}=v_{i+1}$.
\begin{theorem}\label{todd}
For $g=2r+1\geq27$, the mapping class group ${\rm Mod}(N_g)$ is generated by the involutions $\rho_1$, $\rho_2$ and $\rho_2A_2C_{r-1}B_3v_{r+2}$.
\end{theorem}
\begin{proof}
We will follow the proof of Theorem~\ref{teven}, closely. Let us consider the surface $N_g$ as in Figure~\ref{RO}. Since
\[
\rho_2(a_2)=a_2 \textrm{ and } \rho_2(c_{r-1})=b_3
\]
and also since $\rho_2$ reverses the given orientation of a neighbourhood of a two-sided simple closed curve, we get
\[
\rho_2A_2\rho_2=A_2^{-1} \textrm{ and } \rho_2C_{r-1}\rho_2=B_3^{-1}.
\]
By the fact that $\rho_2v_{r+2}\rho_2=v_{r+2}^{-1}$, it can be easy to verify that the element $\rho_2A_2C_{r-1}B_3\phi_{r+2,r+4}$ is an involution. Let $E_1=A_2C_{r-1}B_3v_{r+2}$ and let $K$ denote the subgroup of ${\rm Mod}(N_g)$ generated by the set
\[
\lbrace \rho_1,\rho_2, \rho_2E_1\rbrace.
\]
It is easy to see that $E_1$ and $T=\rho_2\rho_1$ are in $K$. By Theorem~\ref{thm2.1}, we need to show that $K$ contains the elements $A_1A_2^{-1}$, $B_1B_2^{-1}$ and $v_{r+2}$. Let $E_2$ be the following:
\[
E_2=T^7E_1T^{-7}=\Gamma_8C_2C_6v_{r+9} \in K.
\]
Consider the element
\[
E_3=(E_2E_1)E_2(E_2E_1)^{-1}=\Gamma_8B_3C_6v_{r+9},
\]
which belongs to $K$. One can conclude that the element $E_2E_3^{-1}=C_2B_3^{-1}\in K$, which implies that $T(C_2B_3^{-1})T^{-1}=B_3C_3^{-1} \in K$. From this, we get the elements $B_iC_i^{-1} \in H$ by conjugating $B_3C_3^{-1}$ with powers of $T$. Also, since $T(B_3C_3^{-1})T^{-1}=C_3B_4^{-1} \in K$, $C_{i}B_{i+1}^{-1}\in K$ by again conjugating $C_3B_4^{-1}$ with powers of $T$. Thus, we get the elements
\[
B_iB_{i+1}^{-1}=(B_iC_i^{-1})(C_{i}B_{i+1}^{-1}),
\]
which belong to $K$ for all $i=1,\ldots,r-1$. Also, using the transitivity $B_iB_j^{-1}\in K$. In particular $B_1B_2^{-1}\in K$. Moreover, we have the elements
\begin{eqnarray*}
E_4&=&(B_7B_3^{-1})E_1=A_2B_7C_{r-1}v_{r+2} \textrm{ if }r\neq 16,17,18,19,\\
(E_4&=&(B_9B_3^{-1})E_1=A_2B_9C_{r-1}v_{r+2} \textrm{ if }r=16,17,18,19,)\\
E_5&=&T^{6}E_4T^{-6}=\Gamma_{7} B_{10}B_2v_{r+8}\textrm{ if }r\neq 16,17,18,19,\\
(E_5&=&T^{6}E_4T^{-6}=\Gamma_{7} B_{12}B_2v_{r+8} \textrm{ if }r=16,17,18,19,)\\
E_6&=&(E_5E_4)E_5(E_5E_4)^{-1}=\Gamma_{7} B_{10}A_2v_{r+8}\textrm{ if }r\neq 16,17,18,19,\\
(E_6&=&(E_5E_4)E_5(E_5E_4)^{-1}=\Gamma_{7} B_{12}A_2v_{r+8}\textrm{ if }r=16,17,18,19,)
\end{eqnarray*}
which are all contained in the subgroup $K$. Thus, we conclude that the element $E_6E_5^{-1}=A_2B_2^{-1}\in K$.
Since the element $C_1B_2^{-1}\in K$, as in the proof of Theorem~\ref{teven}, one can conclude that the Dehn twists $A_1$, $A_2$, $B_i$ and $C_j$ are in $K$ for $i=1,\ldots,r$ and $j=1,\ldots,r-1$. This implies that $v_{r+2}=(B_3^{-1}C_{r-1}^{-1}A_2^{-1})E_1\in K$, which finishes the proof.
\end{proof}
\end{document} |
\begin{document}
\mathfrak{m}aketitle
\begin{abstract}
We begin by reviewing Zhu's theorem on modular invariance of trace functions associated to a vertex operator algebra, as well as a generalisation by the author to vertex operator superalgebras. This generalisation involves objects that we call `odd trace functions'. We examine the case of the $N=1$ superconformal algebra. In particular we compute an odd trace function in two different ways, and thereby obtain a new representation theoretic interpretation of a well known classical identity due to Jacobi concerning the Dedekind eta function.
\end{abstract}
\section{Introduction}
One of the most significant theorems in the theory of vertex operator algebras (VOAs) is the modular-invariance theorem of Zhu \cite{Zhu}. The theorem states that under favourable circumstances the graded dimensions of certain modules over a VOA are modular forms for the group $SL_2(\mathfrak{m}athbb{Z})$. The favourable circumstances are that the VOA be rational, $C_2$-cofinite, and be graded by integer conformal weights (we define all terms in Section \ref{definitions} and state Zhu's theorem fully in Section \ref{zhusection} below).
Numerous generalisations of Zhu's theorem have appeared in the literature: to twisted modules over VOAs \cite{DLMorbifold}, to vertex operator superalgebras (VOSAs) and their twisted modules \cite{DZ}, \cite{DZ2} (see also \cite{Jordan}), to intertwining operators for VOAs \cite{Huang}, \cite{M2}, to twisted intertwining operators \cite{Y}, and to non rational VOAs \cite{Mnonrational}.
In \cite{meCMP} the present author relaxed the assumption of integer conformal weights of $V$ to allow arbitrary rational conformal weights. This work was carried out in the setting of twisted modules over a rational $C_2$-cofinite VOSA. Actually it is worth noting that in that paper the condition of $C_2$-cofiniteness was also relaxed slightly, allowing applications to some interesting examples such as affine VOAs at admissible level.
One of the features of \cite{meCMP} is the appearance of odd trace functions (see Section \ref{meat} for the definition) which are to be included alongside the more usual (super)trace functions in order to achieve modular invariance. Although similar in many ways, these odd traces differ from (super)traces in that they act nontrivially on odd elements of a vector superspace, whereas the (super)trace must always vanish on such elements. The results of \cite{meCMP} are reviewed in Section \ref{meat} (for simplicity in the special case of Ramond twisted modules).
In the present work, in Section \ref{ex4}, we compute an odd trace function for a particular example: the $N=1$ superconformal minimal model of central charge $c = -21/4$. We evaluate the odd trace function on the superconformal generator (which is an odd element of conformal weight $3/2$), using the strong constraint of its modular invariance. The odd trace function in question equals the weight $3/2$ modular form $\eta(\tau)^3$, where $\eta(\tau)$ is the well known Dedekind eta function.
We then give a different proof of this equality (up to an ambiguity of signs) using a BGG resolution and some simple combinatorics. The result is a representation theoretic interpretation of the classical identity
\begin{align*}
\eta(\tau)^3 = q^{1/8} \sum_{n \in \mathfrak{m}athbb{Z}} (4n+1) q^{n(2n+1)}
\end{align*}
similar in spirit, but a little different, to the celebrated proof coming from the affine Weyl-Kac denominator identity (\cite{IDLA} Chapter 12).
\emph{Acknowledgements:} I would like to warmly thank the organisers of the conference `Lie Superalgebras' at Universit\`{a} di Roma Sapienza where this work was presented. Also to express my gratitude to IMPA and to the IHES where the writing of this paper was completed.
\section{Definitions}\lambdabel{definitions}
For us a \underline{vertex superalgebra} \cite{Kac}, \cite{FBZ} is a quadruple $V, {\left|0\right>}, T, Y$ where $V$ is a vector superspace, ${\left|0\right>} \in V$ an even vector, $T : V \rightarrow V$ an even linear map, and $Y : V \otimes V \rightarrow V((z))$, denoted $u \otimes v \mathfrak{m}apsto Y(u, z)v = \sum_{n \in \mathfrak{m}athbb{Z}} u_{(n)}v z^{-n-1}$, is also even. These data are to satisfy the following axioms.
\begin{itemize}
\item The unit identities $Y({\left|0\right>}, z) = I_V$ and $Y(u, z){\left|0\right>}|_{z=0} = u$.
\item The translation invariance identity $Y(Tu, z) = \partial_z Y(u, z)$.
\item The Cousin property that the three expressions
\[
Y(u, z)Y(v, w) x \quad \quad p(u, v) Y(v, w) Y(u, z) x, \quad \text{and} \quad Y(Y(u, z-w)v, w) x,
\]
which are elements of $V((z))((w))$, $V((w))((z))$, and $V((w))((z-w))$, are images of a single element of $V[[z, w]][z^{-1}, w^{-1}, (z-w)^{-1}]$ under natural inclusions into those three spaces.
\end{itemize}
An equivalent definition, more convenient for some applications, is the following. A vertex superalgebra is a triple $V, {\left|0\right>}, Y$ where these data are as above, but satisfy the following axioms.
\begin{itemize}
\item The unit identities ${\left|0\right>}_{(n)}u = \delta_{n, -1} u$, $u_{(-1)}{\left|0\right>} = u$ and $u_{(n)}{\left|0\right>} = 0$ for $n > 0$.
\item The Borcherds identity (also known, in a different notation, as the Jacobi identity)
\[
B(u, v, x; m, k, n) = 0 \quad \text{for all $u, v, x \in V$, $m, k, n \in \mathfrak{m}athbb{Z}$},
\]
where
\begin{align*}
B(u, v, x; m, k, n) &= \sum_{j \in \mathfrak{m}athbb{Z}_+} \binom{m}{j} (u_{(n+j)}v)_{(m+k-j)} x \\
&- \sum_{j \in \mathfrak{m}athbb{Z}_+} (-1)^j \binom{n}{j} \left[ u_{(m+n-j)} v_{(k+j)} - (-1)^n p(u, v) v_{(k+n-j)} u_{(m+j)} \right] x.
\end{align*}
\end{itemize}
A \underline{vertex algebra} is a purely even vertex superalgebra.
Let $V$ be a vertex superalgebra. A $V$-module is a vector superspace $M$ with a vertex operation $Y^M : V \otimes M \rightarrow M((z))$ such that
\begin{align}\lambdabel{moduleborcherds}
Y^M({\left|0\right>}, z) = I_M, \quad \text{and} \quad B(u, v, x; m, k, n) = 0
\end{align}
for all $u, v \in V$, $x \in M$, $m, k, n \in \mathfrak{m}athbb{Z}$.
For the present theory we require the extra structure of a \underline{conformal vector}. This is a vector $\omega \in V$ such that its modes $L_n = \omega_{(n-1)} \in \en V$ furnish $V$ with a representation of the Virasoro algebra, i.e.,
\[
[L_m, L_n] = (m-n) L_{m+n} + \delta_{m, -n} \frac{m^3-m}{12} c
\]
(here $c \in \mathfrak{m}athbb{C}$ is an invariant of $V$ called the central charge), $L_0$ is diagonal with real eigenvalues bounded below, and $L_{-1} = T$. We call a vertex superalgebra with conformal vector a \underline{vertex operator superalgebra} or VOSA, and we use the term VOA to distinguish the purely even case.
A $V$-module $M$ is called a \underline{positive energy} module if $L_0 \in \en M$ acts diagonally with eigenvalues bounded below. In particular $V$ is a positive energy $V$-module. The eigenvalues of $L_0 \in \en V$ are called \underline{conformal weights}, and if $L_0 u = \Delta u$ we write
\[
Y(u, z) = \sum_{n \in \mathfrak{m}athbb{Z}} u_{(n)} z^{-n-1} = \sum_{n \in -\Delta + \mathfrak{m}athbb{Z}} u_n z^{-n-\Delta}
\]
(so that $u_n = u_{(n-\Delta+1)}$). The \underline{zero mode} $u_0 \in \en M$ attached to $u \in V$ is special because it commutes with $L_0$ and thus preserves the eigenspaces of the latter. A VOSA $V$ is said to be \underline{rational} if its category of positive energy modules is semisimple, i.e., it contains finitely many irreducible objects, and any object is isomorphic to a direct sum of irreducible ones.
The condition of $C_2$-cofiniteness is an important finiteness condition of vertex (super)algebras introduced by Zhu. We say that $V$ is \underline{$C_2$-cofinite} if
\[
\dim \left( V / V_{(-2)}V \right) < \infty.
\]
\section{The Theorem of Zhu}\lambdabel{zhusection}
Now we come to the theorem of Zhu \cite{Zhu}.
\begin{thm}[Zhu] \lambdabel{zhuthm}
Let $V$ be a VOA (i.e., purely even VOSA) such that:
\begin{itemize}
\item $V$ is rational,
\item the conformal weights of $V$ lie in $\mathfrak{m}athbb{Z}_+$,
\item $V$ is $C_2$-cofinite.
\end{itemize}
We associate to each irreducible positive energy module $M$, and $u \in V$, the series
\[
S_M(u, \tau) = \tr_M u_0 q^{L_0 - c/24},
\]
convergent for $q = e^{2\pi i \tau}$ of modulus less than $1$. There is a grading $V = \oplus_{\mathfrak{n}abla \in \mathfrak{m}athbb{Z}_+} V_{[\mathfrak{n}abla]}$ such that for $u \in V_{[\mathfrak{n}abla]}$ the span $\mathfrak{m}athbb{C}C(u)$ of the (finitely many) functions $S_M(u, \tau)$ defined above is modular invariant of weight $\mathfrak{n}abla$, i.e.,
\[
(c\tau + d)^\mathfrak{n}abla f\left(\frac{a\tau+b}{c\tau+d}\right) \in \mathfrak{m}athbb{C}C(u) \quad \text{for all $f(\tau) \in \mathfrak{m}athbb{C}C(u)$ and $\slmat \in SL_2(\mathfrak{m}athbb{Z})$}.
\]
\end{thm}
Here is an outline of the proof of Zhu's theorem.
\begin{enumerate}
\item Introduce a space $\mathfrak{m}athbb{C}C$ of maps $S(u, \tau) : V \times \mathfrak{m}athcal{H} \rightarrow \mathfrak{m}athbb{C}$ linear in $V$ and holomorphic in $\mathfrak{m}athcal{H} = \{\tau \in \mathfrak{m}athbb{C} | \text{Im}{\tau} > 0\}$ satisfying certain axioms, this space is called the `conformal block' of $V$ or the space of conformal blocks of $V$. The definition of $\mathfrak{m}athbb{C}C$ can be understood in terms of elliptic curves and their moduli \cite{FBZ}.
\item It is automatic from its definition that $\mathfrak{m}athbb{C}C$ admits an action of the group $SL_2(\mathfrak{m}athbb{Z})$, namely,
\[
[S \cdot A](u, \tau) = (c\tau + d)^{-\mathfrak{n}abla_u} S(u, A\tau),
\]
where $\mathfrak{n}abla$ is the grading mentioned above.
\item It is proved by direct calculation that $\tr_M u_0^M q^{L_0 - c/24}$ is a conformal block (at least as a formal power series).
\item Using the $C_2$-cofiniteness condition, one shows that any fixed $S \in \mathfrak{m}athbb{C}C$ satisfies some differential equation, and consequently is expressible as a power series in $q$ (whose coefficients are linear maps $V \rightarrow \mathfrak{m}athbb{C}$).
\item \lambdabel{zhustep} The lowest order coefficient $C_0 : V \rightarrow \mathfrak{m}athbb{C}$ in the series expansion factors to a certain quotient $\zhu(V)$ of $V$. This quotient has the structure of a unital associative algebra, and $C_0$ is symmetric, i.e., $C_0(ab) = C_0(ba)$.
\item There is a natural bijection
\[
\text{irreducible positive energy $V$-modules} \longleftrightarrow \text{irreducible $\zhu(V)$-modules},
\]
and so if $V$ is rational, $\zhu(V)$ is finite dimensional semisimple. Thus we can write $C_0 = \sum_N \alpha_N \tr_N$ where the sum is over irreducible $\zhu(V)$-modules and $\alpha_N \in \mathfrak{m}athbb{C}$.
\item Write the corresponding sum $\sum_N \alpha_N S_M$ where $M$ is the $V$-module associated to $N$. Subtract this conformal block from $S$.
\item One can repeat the process and show that $S$ is exhausted by trace functions in a finite number of steps.
\end{enumerate}
\section{Generalisation to the Supersymmetric Case, and to Rational Conformal Weights}\lambdabel{meat}
Results described in this section are drawn from \cite{meCMP}.
Many examples of interest, especially in the supersymmetric case, are graded by noninteger conformal weights. So it is first necessary (and of independent interest) to relax the condition of integer conformal weights in Theorem \ref{zhuthm}. Therefore let $V$ be a VOA whose conformal weights lie in $\mathfrak{m}athbb{Q}$ (and are bounded below) rather than in $\mathfrak{m}athbb{Z}_+$, and which satisfies the other conditions of the theorem. Then the trace functions $S_M(u, \tau)$ and their span $\mathfrak{m}athbb{C}C(u)$ are defined as before. There exists a certain \emph{rational} grading $V = \oplus_{\mathfrak{n}abla \in \mathfrak{m}athbb{Q}} V_{[\mathfrak{n}abla]}$ in place of the usual integer grading. It is then true that for $u \in V_{[\mathfrak{n}abla]}$ the space $\mathfrak{m}athbb{C}C(u)$ is invariant under the weight $\mathfrak{n}abla$ action\footnote{The precise definition of the action involves choices of roots of unity in general. See \cite{meCMP} for details.}, not of $SL_2(\mathfrak{m}athbb{Z})$, but of its congruence subgroup
\begin{align*}
\Gamma_1(N) = \{\slmat \in SL_2(\mathfrak{m}athbb{Z}) | b \equiv 0 \bmod N, \,\text{and}\,\, a \equiv d \equiv 1 \bmod N\}.
\end{align*}
Here $N$ is the least common multiple of the denominators of conformal weights of vectors in $V$. This number is finite because of the condition of $C_2$-cofinitness.
It is possible to achieve invariance under the whole of $SL_2(\mathfrak{m}athbb{Z})$ by altering our definition of $V$-module. Define a \underline{Ramond twisted} $V$-module to be a vector superspace $M$ together with fields
\[
Y^M(u, z) = \sum_{n \in -\Delta_u + \mathfrak{m}athbb{Z}} u_{(n)} z^{-n-1} = \sum_{n \in \mathfrak{m}athbb{Z}} u_n z^{-n-\Delta}
\]
satisfying (\ref{moduleborcherds}) for all $m \in -\Delta_u + \mathfrak{m}athbb{Z}$, $k \in -\Delta_v + \mathfrak{m}athbb{Z}$, $n \in \mathfrak{m}athbb{Z}$. Notice that the ranges of indices of the modes are modified so that $u \in V$ always possesses integrally graded modes $u_n \in \en M$, and in particular always possesses a zero mode. Let us call a VOA \underline{Ramond rational} if its category of positive energy Ramond twisted modules is semisimple.
Let $V$ be a Ramond rational, $C_2$-cofinite VOA with rational conformal weights bounded below. Attach the trace function
\[
S_M(u, \tau) = \tr_M u_0 q^{L_0 - c/24}
\]
to $u \in V$ and $M$ an irreducible positive energy Ramond twisted $V$-module, and let $\mathfrak{m}athbb{C}C(u)$ be the span of all such trace functions. Then for $u \in V_{[\mathfrak{n}abla]}$ the space $\mathfrak{m}athbb{C}C(u)$ is invariant under the weight $\mathfrak{n}abla$ action of the full modular group $SL_2(\mathfrak{m}athbb{Z})$.
The previous paragraph stated a result for VOAs. Upon passage from VOAs to VOSAs, one might expect the claim to hold with trace functions simply replaced by supertrace functions $S_M(u, \tau) = \str_M u_0 q^{L_0 - c/24}$. This would be true, except for an interesting subtlety which can be traced to Step \ref{zhustep} of the proof outline given in Section \ref{zhusection}.
In the present situation it is appropriate to replace the usual Zhu algebra with a certain superalgebra (which we also refer to as the Zhu algebra and denote $\zhu(V)$) introduced in the necessary level of generality in \cite{DK}. If $V$ is Ramond rational then $\zhu(V)$ is finite dimensional and semisimple. The lowest coefficient $C_0$ of the series expansion of a conformal block now descends to a supersymmetric function on $\zhu(V)$, i.e., $C_0(ab) = p(a, b)C_0(ba)$.
The classification of pairs $(A, \varphi)$, where $A$ is a finite dimensional simple superalgebra (over $\mathfrak{m}athbb{C}$) and $\varphi$ is a supersymmetric function on $A$, is as follows.
\begin{itemize}
\item $A = \en(N)$ for some finite dimensional vector superspace $N$, and $\varphi$ is a scalar multiple of $\str_N$.
\item $A = \en(P)[\theta] / (\theta^2 - 1)$ where $P$ is a vector space and $\theta$ is an odd indeterminate, and $\varphi$ is a scalar multiple of $a \mathfrak{m}apsto \tr_P(a \theta)$.
\end{itemize}
The first case is the analogue of the usual Wedderburn theorem. The superalgebra of the second case is known as the \underline{queer superalgebra} and is often denoted $Q_n$ (where $n = \dim P$). Clearly we have
\[
Q_n \cong \left\{ \twobytwo X Y Y X | X, Y \in \text{Mat}_n(\mathfrak{m}athbb{C}) \right\},
\]
and the unique up to a scalar factor supersymmetric function on $Q_n$ is $\left(\begin{smallmatrix}{X}&{Y}\\{Y}&{X}\\ \end{smallmatrix}\right) \mathfrak{m}apsto \tr Y$, which is known as the \underline{odd trace}.
Roughly speaking modular invariance will hold for $\mathfrak{m}athbb{C}C(u)$ defined as the span of supertrace functions together with apropriate analogues of odd trace functions. More precisely:
\begin{defn}\lambdabel{queertracedefn}
Let $V$ be a Ramond rational, $C_2$-cofinite VOSA with rational conformal weights bounded below. Let $A$ be a simple component of $\zhu(V)$, $N$ the corresponding unique $\mathfrak{m}athbb{Z}_2$-graded irreducible module, and $M$ the corresponding irreducible positive energy Ramond twisted $V$-module. If $A \cong \en(P)[\theta]/(\theta^2-1)$ is queer then let $\Theta : M \rightarrow M$ denote the lift to $M$ of the map $\theta : N \rightarrow N$ of multiplication by $\theta$.
In this case we define the \underline{odd trace function}
\[
S_M(u, \tau) = \tr_M u_0 \Theta q^{L_0 - c/24}.
\]
If $A$ is not queer then we define the \underline{supertrace function}
\[
S_M(u, \tau) = \str_M u_0 q^{L_0 - c/24}.
\]
\end{defn}
Now we can state the main theorem: it is Theorem 1.3 of \cite{meCMP} applied to the special case of untwisted characters of Ramond twisted $V$-modules.
\begin{thm}[\cite{meCMP}]\lambdabel{mythm}
Let $V$ be a VOSA as in Definition \ref{queertracedefn}, and let $\mathfrak{m}athbb{C}C(u)$ be the span of the supertrace functions and odd trace functions attached to all irreducible positive energy Ramond twisted $V$-modules. There exists a grading $V = \oplus_{\mathfrak{n}abla \in \mathfrak{m}athbb{Q}} V_{[\mathfrak{n}abla]}$ such that for $u \in V_{[\mathfrak{n}abla]}$ the space $\mathfrak{m}athbb{C}C(u)$ is invariant under the weight $\mathfrak{n}abla$ action of $SL_2(\mathfrak{m}athbb{Z})$.
\end{thm}
In the next two sections we view some examples of odd trace functions.
\section{Example: The Neutral Free Fermion}
This example is considered more fully in \cite{meCMP}, the interested reader may refer there for further details.
We consider the Lie superalgebras $A^\text{tw}$ (resp. $A^\text{untw}$)
\[
(\oplus_{n} \mathfrak{m}athbb{C} \psi_n) \oplus \mathfrak{m}athbb{C} 1
\]
where the direct sum ranges over $n \in 1/2 + \mathfrak{m}athbb{Z}$ (resp. $n \in \mathfrak{m}athbb{Z}$). Here the vector $1$ is even, $\psi_n$ is odd. The commutation relations in both cases are
\[
[\psi_{m}, \psi_{n}] = \delta_{m, -n} 1.
\]
We introduce the Fock module
\[
V = U(A^\text{tw}) \otimes_{U(A^\text{tw}_+)} \mathfrak{m}athbb{C} {\left|0\right>},
\]
where
\[
A^\text{tw}_+ = \mathfrak{m}athbb{C} 1 \oplus (\oplus_{n \mathfrak{m}athfrak{g}eq 1/2} \mathfrak{m}athbb{C} \psi_n)
\]
and $\mathfrak{m}athbb{C} {\left|0\right>}$ is the $A^\text{tw}_+$-module on which $1$ acts as the identity and $\psi_n$ acts trivially.
It is well known \cite{Kac} that $V$ can be given the structure of a VOSA. The Virasoro element is $\omega = \frac{1}{2} \psi_{-3/2}\psi_{-1/2}{\left|0\right>}$ with central charge $c = 1/2$. The vector $\psi = \psi_{-1/2}{\left|0\right>}$ has conformal weight $1/2$ and associated vertex operator
\[
Y(\psi, z) = \sum_{n \in 1/2 + \mathfrak{m}athbb{Z}} \psi_n z^{-n-1/2}.
\]
Ramond twisted $V$-modules are, in particular, modules over the untwisted Lie superalgebra $A^\text{untw}$. In fact the unique irreducible positive energy Ramond twisted $V$-module is
\[
M = U(A^\text{untw}) \otimes_{U(A^\text{untw}_+)} (\mathfrak{m}athbb{C} v + \mathfrak{m}athbb{C} \overline{v})
\]
where
\[
A^\text{untw}_+ = \mathfrak{m}athbb{C} 1 \oplus (\oplus_{n \mathfrak{m}athfrak{g}eq 0} \mathfrak{m}athbb{C} \psi_n)
\]
and $\mathfrak{m}athbb{C} v + \mathfrak{m}athbb{C} \overline{v}$ is the $A^\text{untw}_+$-module on which $1$ acts as the identity, $\psi_n$ acts trivially for $n > 0$, $\psi_0 v = \overline{v}$ and $\psi_0 \overline{v} = v/2$. We note that $V$ is $C_2$-cofinite and Ramond rational (as well as being rational).
The (Ramond) Zhu algebra of $V$ is explicitly isomorphic to the queer superalgebra $Q_1 = \mathfrak{m}athbb{C}[\theta] / (\theta^2 = 1)$ via the map $[{\left|0\right>}] \mathfrak{m}apsto 1$, $[\psi] \mathfrak{m}apsto \sqrt{2} \theta$. Thus the lowest graded piece $M_0 = \mathfrak{m}athbb{C} v + \mathfrak{m}athbb{C} \overline{v}$ of $M$ is a $Q_1$-module.
The corresponding odd trace function is
\[
S_M(u, \tau) = \tr_M u_0 \Theta q^{L_0-c/24}
\]
where $\Theta : M \rightarrow M$ is as in Definition \ref{queertracedefn}. By unwinding that definition we see that $\Theta : \mathfrak{m}athbf{m} w \mathfrak{m}apsto \mathfrak{m}athbf{m} \psi_0 w$, where $w$ is $v$ or $\overline{v}$, and the monomial $\mathfrak{m}athbf{m} \in U(A^\text{untw} / A^\text{untw}_+)$.
The odd trace function $S_M(u, \tau)$ vanishes on $u = {\left|0\right>}$ (indeed on all even vectors of $V$), but it acts nontrivially on the odd vector $\psi$ (which is pure of Zhu weight $1/2$). Therefore $S_M(\psi, \tau)$ must be a modular form on $SL_2(\mathfrak{m}athbb{Z})$ of weight $1/2$ (with possible multiplier system).
Indeed one may verify that $\psi_0 \Theta$ acts as $(-1)^{\text{length}(\mathfrak{m}athbf{m})}$ on the monomial vector $\mathfrak{m}athbf{m} w$, and so
\begin{align*}
S_M(\psi, \tau) &= q^{-c/24} q^{L_0|{M_0}} (1 - q^1) (1 - q^2) \cdots \\
&= q^{1/24} \prod_{n=1}^\infty (1-q^n) = \eta(\tau)
\end{align*}
(here we have used that $c = 1/2$, and that $L_0|_{M_0} = 1/16$). We have recovered the well known Dedekind eta function $\eta(\tau)$ which is indeed a modular form on $SL_2(\mathfrak{m}athbb{Z})$ of weight $1/2$.
\section{Example: The $N=1$ Superconformal Algebra} \lambdabel{ex4}
First we recall the definition of the Neveu-Schwarz Lie superalgebra $\mathfrak{n}stw$, and its Ramond-twisted variant $\mathfrak{n}suntw$ (which is often called the Ramond superalgebra).
\begin{defn}
As vector superspaces the Lie superalgebras $\mathfrak{n}stw$ (resp. $\mathfrak{n}suntw$) are
\[
(\oplus_{n \in \mathfrak{m}athbb{Z}} \mathfrak{m}athbb{C} L_n) \oplus (\oplus_{m} \mathfrak{m}athbb{C} G_n) \oplus \mathfrak{m}athbb{C} C
\]
where the direct sum ranges over $m \in \mathfrak{m}athbb{Z}$ (resp. $m \in 1/2 + \mathfrak{m}athbb{Z}$). Here $C$ and $L_n$ are even, $G_m$ is odd. The commutation relations in both cases are
\begin{align} \lambdabel{NS}
\begin{split}
[L_m, L_n] &= (m-n)L_{m+n} + \frac{m^3-m}{12} \delta_{m, -n} C, \\
[G_m, L_n] &= (m - \frac{n}{2}) G_{m+n}, \\
[G_m, G_n] &= 2L_{m+n} + \frac{1}{3}(m^2 - \frac{1}{4}) \delta_{m, -n} C,
\end{split}
\end{align}
with $C$ central.
\end{defn}
As usual we introduce the Verma $\mathfrak{n}stw$-module
\[
M^{\text{tw}}(c, h) = U(\mathfrak{n}stw) \otimes_{U(\mathfrak{n}stw_+)} \mathfrak{m}athbb{C} v_{c, h}
\]
where
\begin{align*}
\mathfrak{n}stw_+ = \mathfrak{m}athbb{C} C + \oplus_{n \mathfrak{m}athfrak{g}eq 0} \mathfrak{m}athbb{C} L_n + \oplus_{m \mathfrak{m}athfrak{g}eq 1/2} \mathfrak{m}athbb{C} G_m,
\end{align*}
and $\mathfrak{m}athbb{C} v_{c, h}$ is the $\mathfrak{n}stw_+$-module on which $C$ acts by $c \in \mathfrak{m}athbb{C}$, $L_0$ acts by $h \in \mathfrak{m}athbb{C}$, and higher modes act trivially. It is well known \cite{Kac}, \cite{KacWang} that the quotient $\text{NS}^c = M^\text{tw}(c, 0) / U(\mathfrak{n}stw) G_{-1/2} v_{c, 0}$ is a VOSA of central change $c$, as is the irreducible quotient $\text{NS}_c$.
We shall also require the (generalised) Verma $\mathfrak{n}suntw$-modules
\[
M(c, h) = U(\mathfrak{n}suntw) \otimes_{U(\mathfrak{n}suntw_+)} S_{c, h}
\]
and their irreducible quotients $L(c, h)$ (we omit the superscript $\text{untw}$) where
\begin{align*}
\mathfrak{n}suntw_+ = \mathfrak{m}athbb{C} C + \oplus_{n \mathfrak{m}athfrak{g}eq 0} \mathfrak{m}athbb{C} L_n + \oplus_{m \mathfrak{m}athfrak{g}eq 0} \mathfrak{m}athbb{C} G_m,
\end{align*}
and $S_{c, h}$ is the $\mathfrak{n}suntw_+$-module characterised by:
\begin{align*}
\begin{array}{ll}
\text{$S_{c, h} = \mathfrak{m}athbb{C} v_{c, h}$ with $G_0 v_{c, h} = 0$} & \text{if $h = c/24$}, \\
\text{$S_{c, h} = \mathfrak{m}athbb{C} v_{c, h} + \mathfrak{m}athbb{C} G_0 v_{c, h}$} & \text{if $h \mathfrak{n}eq c/24$}, \\
\end{array}
\end{align*}
with $C = c$, $L_0 = h$, and positive modes acting trivially in both cases.
A clear summary of the representations of $\text{NS}^c$ and $\text{NS}_c$ can be found in \cite{supermilas}. Here we focus on the Ramond twisted representations and shall often omit the adjective `Ramond twisted'. Generically $\text{NS}_c = \text{NS}^c$ is irreducible and all the $\mathfrak{n}suntw$-modules $L(c, h)$ acquire the structure of positive energy $\text{NS}_c$-modules. For certain values of $c$ though, $\text{NS}_c$ is a nontrivial quotient of $\text{NS}^c$ and the irreducible positive energy $\text{NS}_c$-modules are finite in number and are all of the form $L(c, h)$. In fact, $\text{NS}_c$ is a (Ramond) rational VOSA when
\begin{align*}
c = c_{p, p'} = \frac{3}{2}\left(1 - \frac{2(p'-p)^2}{pp'}\right)
\end{align*}
for $p, p' \in \mathfrak{m}athbb{Z}_{>0}$ with $p < p'$, $p' - p \in 2\mathfrak{m}athbb{Z}$ and $\mathfrak{m}athfrak{g}cd(\frac{p'-p}{2}, p) = 1$. In this case the irreducible positive energy $\text{NS}_c$-modules are precisely the $\mathfrak{n}suntw$-modules $L(c, h)$ where
\begin{align*}
h = h_{r, s} = \frac{(rp'-sp)^2 - (p'-p)^2}{8pp'} + \frac{1}{16}
\end{align*}
for $1 \leq r \leq p-1$ and $1 \leq s \leq p'-1$ with $r-s$ odd.
Let $c$ be one of these special values from now on. The irreducible $\zhu(\text{NS}_c)$-modules are precisely the lowest graded pieces of the modules $L(c, h)$ introduced above. The lowest graded piece is of dimension $1$ if $h = c/24$ (there is clearly at most one such module for any fixed value of $c$), and is of dimension $1|1$ if $h \mathfrak{n}eq c/24$. It is known that $\zhu(\text{NS}_c)$ is supercommutative (it is a quotient of $\zhu(\text{NS}^c) \cong \mathfrak{m}athbb{C}[x, \theta] / (\theta^2 - x + c/24)$ where $x$ is even and $\theta$ odd). Therefore the simple components of $\zhu(\text{NS}_c)$ with the $1|1$-dimensional modules are all copies of the queer superalgebra $Q_1$, and the component with the $1$-dimensional module (if it exists) is $\mathfrak{m}athbb{C}$.
Let us consider the case $c = -21/4$ (so $p=2$, $p'=8$) for which the two irreducible positive energy modules are $M_i = L(c, h_i)$ where $h_1 = -3/32$ and $h_2 = -7/32 = c/24$. The first of these is the unique queer module. Theorem \ref{mythm} tells us that the supertrace function $S_{M_2}(u, \tau)$ and the odd trace function $S_{M_1}(u, \tau)$ together span an $SL_2(\mathfrak{m}athbb{Z})$-invariant space whose weight is the Zhu weight of $u$. Assume further that $u \in V$ is odd. Then $S_{M_2}(u, \tau)$ vanishes as the supertrace of an odd element. But $S_{M_1}(u, \tau)$ need not vanish, and it will be a modular form (with multiplier system).
Unwinding Definition \ref{queertracedefn} we see that $\Theta : \mathfrak{m}athbf{m} v_{c, h} \mathfrak{m}apsto \mathfrak{m}athbf{m} G_0 v_{c, h}$ where $\mathfrak{m}athbf{m} \in U(\mathfrak{n}suntw / \mathfrak{n}suntw_+)$, and
\[
S_{M_1}(u, \tau) = \tr_{M_1} u_0 \Theta q^{L_0 - c/24}.
\]
The VOSA $\text{NS}_c$ possesses a distinguished element $\mathfrak{n}u = G_{-3/2}{\left|0\right>}$ of conformal weight $3/2$, it satisfies $\mathfrak{n}u_0 = G_0$. It turns out that $\mathfrak{n}u$ is of pure Zhu weight $3/2$ and so by the above remarks
\[
F(\tau) := \tr_{M_1} G_0 \Theta q^{L_0 - c/24}
\]
is a modular form of weight $3/2$. On the top level of $M_1$, $G_0 \Theta = G_0^2 = h - c/24 = 1/8$, so the top level contribution to $F(\tau)$ is $\frac{1}{4}q^{1/8}$. This is already enough information to determine $F(\tau)$ completely. The cube of the Dedekind eta function is $q^{1/8}$ times an ordinary power series in $q$, so the quotient $f(\tau) = F(\tau) / \eta(\tau)^3$ is a holomorphic modular form of weight $0$ for $SL_2(\mathfrak{m}athbb{Z})$, possibly with a multiplier system. Since the $q$-series of $f$ has integer powers of $q$ we have $f(T\tau) = f(\tau)$, and since $S^2 = 1$ we have only the possibilities $f(S\tau) = \pm f(\tau)$. But in $SL_2(\mathfrak{m}athbb{Z})$ we have the relation $(ST)^3 = 1$, so if $S$ acted by $-1$ on $f$ we would have $f(\tau) = f(-T^3\tau) = -f(\tau)$. Hence $f(S\tau) = f(\tau)$ and, since it is a genuine holomorphic modular form on $SL_2(\mathfrak{m}athbb{Z})$, we have $f(\tau) = 1$. Thus
\begin{align} \lambdabel{calcofS}
F(\tau) = \eta(\tau)^3 / 4.
\end{align}
We next compute $F(\tau)$ using representation theory. We obtain (up to some undetermined signs) the following well known classical identity of Jacobi
\begin{align}\lambdabel{jaceta}
\eta(\tau)^3 = q^{1/8} \sum_{n \in \mathfrak{m}athbb{Z}} (4n+1) q^{n(2n+1)}.
\end{align}
We begin by considering the trace of $G_0 \Theta q^{L_0}$ on the Verma module $M(c, h)$. The action of $G_0 \Theta$ on the monomial
\begin{align*}
\mathfrak{m}athbf{m} v = L_{m_1} \cdots L_{m_s} G_{n_1} \cdots G_{n_t} v,
\end{align*}
where $m_1 \leq \ldots \leq m_s \leq -1$, $n_1 < \ldots < n_t \leq -1$, and $v$ is $v_{c, h}$ or $G_0 v_{c, h}$, looks like
\[
\mathfrak{m}athbf{m} G_0 v \mathfrak{m}apsto G_0 \mathfrak{m}athbf{m} G_0 v = (-1)^t \mathfrak{m}athbf{m} G_0^2 v + \text{reduced terms}.
\]
Reduced terms resulting from a single use of the commutation relations are of the same length as $\mathfrak{m}athbf{m}$, but contain different numbers of the symbols $L$ and $G$. Reduced terms resulting from more than one use of the commutation relations are strictly shorter than $\mathfrak{m}athbf{m}$. Therefore none of these terms contribute to the trace. Consider monomials $\mathfrak{m}athbf{m}$ as above with a fixed value of $N = \sum_{i=1}^s m_i + \sum_{j=1}^t n_j$. A simple generating function argument shows that if $N > 0$ then the number of such monomials with $t$ even is the same as the number with $t$ odd. Thus the only nonzero term in $\tr G_0 \Theta q^{L_0}$ is the leading term.
It is known that $L(c = -21/4, h_0 = -3/32)$ has a BGG resolution
\[
0 \leftarrow L(c, h_0) \leftarrow M(c, h_0) \leftarrow M(c, h_1) \oplus M(c, h_{-1}) \leftarrow M(c, h_2) \oplus M(c, h_{-2}) \leftarrow \cdots,
\]
where $h_n = -3/32 - n(2n+1)$ for all $n \in \mathfrak{m}athbb{Z}$, and that all $M(c, h_k)$ are naturally embedded in $M(c, h_0)$ \cite{IoharaKoga}. We therefore identify each Verma module with its image in $M(c, h_0)$. The trace we seek is given as an alternating sum over the terms in the resolution.
From this we see already that the only nonzero coefficients in the $q$-expansion of $\eta(\tau)^3$ must be for powers $q^{1/8 - n(2n+1)}$. We can also easily determine the coefficients up to a sign. Indeed we have $\Theta^2 = h_0 - c/24 = 1/8$, while the operator $G_0|_{M(c, h_k)}$ preserves the top piece $S_k$ of $M(c, h_k)$ and squares to $h_k^2 - c/24$. Therefore the operator $(G_0 \Theta)|_{S_k}$ (which is diagonal on the $1|1$-dimensional space $S_k$) squares to
\[
(h_k-c/24)^2/8 = \left[(4k+1)/8\right]^2.
\]
This matches perfectly with (\ref{jaceta}). To determine the signs of the coefficients directly it seems to be necessary to know some further information about the singular vectors, it would be nice to find a simpler derivation.
Of course similar arguments may be applied to other rational $\text{NS}_c$ and their modules. If $L_0$ happens to take the value $c/24$ on one of the levels of a module then the arguments potentially become more intricate.
\end{document} |
\begin{document}
\title{Quantum Sampling Problems, BosonSampling and Quantum Supremacy}
\newcommand{Centre for Quantum Computation and Communications Technology, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia}{Centre for Quantum Computation and Communications Technology, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia}
\newcommand{Centre for Quantum Computation and Communications Technology, Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, Australia}{Centre for Quantum Computation and Communications Technology, Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, Australia}
\author{A.~P.~Lund}
\affiliation{Centre for Quantum Computation and Communications Technology, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia}
\author{Michael~J.~Bremner}
\affiliation{Centre for Quantum Computation and Communications Technology, Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, Australia}
\author{T.~C.~Ralph}
\affiliation{Centre for Quantum Computation and Communications Technology, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia}
\begin{abstract}
There is a large body of evidence for the potential of greater computational power using information carriers that are quantum mechanical over those governed by the laws of classical mechanics. But the question of the exact nature of the power contributed by quantum mechanics remains only partially answered. Furthermore, there exists doubt over the practicality of achieving a large enough quantum computation that definitively demonstrates quantum supremacy. Recently the study of computational problems that produce samples from probability distributions has added to both our understanding of the power of quantum algorithms and lowered the requirements for demonstration of fast quantum algorithms. The proposed quantum sampling problems do not require a quantum computer capable of universal operations and also permit physically realistic errors in their operation. This is an encouraging step towards an experimental demonstration of quantum algorithmic supremacy. In this paper, we will review sampling problems and the arguments that have been used to deduce when sampling problems are hard for classical computers to simulate. Two classes of quantum sampling problems that demonstrate the supremacy of quantum algorithms are $\BosonSampling$ and $\IQP$ Sampling. We will present the details of these classes and recent experimental progress towards demonstrating quantum supremacy in $\BosonSampling$.
\end{abstract}
\maketitle
\section{Introduction}
There is a growing sense of excitement that in the near future prototype quantum computers might be able to outperform any classical computer. That is, they might demonstrate supremacy over classical devices~\cite{Preskill12}. This excitement has in part been driven by theoretical research into the complexity of intermediate quantum computing models, which over the last 15 years has seen the physical requirements for a quantum speedup lowered while increasing the level of rigour in the argument for the difficulty of classically simulating such systems.
These advances are rooted in the discovery by Terhal and DiVincenzo \cite{Terhal02} that sufficiently accurate classical simulations of even quite simple quantum computations could have significant implications for the interrelationships between computational complexity classes~\cite{Papadimitriou}. Since then the theoretical challenge has been to demonstrate such a result holds for levels of precision commensurate with what is expected from realisable quantum computers. A first step in this direction established that classical computers cannot efficiently mimic the output of ideal quantum circuits to within a reasonable multiplicative (or relative) error in the frequency with which output events occur without similarly disrupting the expected relationships between classical complexity classes~\cite{BJS10, AA}. In a major breakthrough Aaronson and Arkhipov laid out an argument for establishing that efficient classical simulation of linear optical systems was not possible, even if that simulation was only required to be accurate to within a reasonable total variation distance. Their argument revealed a deep connection between the complexity of sampling from quantum computers and conjectures regarding the average-case complexity of a range of combinatorial problems. The linear optical system they proposed was the class of problems called $\BosonSampling$ which is the production of samples from Fock basis measurements of linearly scattering individual Bosons. Using the current state of the art of classical computation an implementation of $\BosonSampling$ using 50 photons would be sufficient to demonstrate quantum supremacy.
Since then many experimental teams have attempted to implement Aaronson and Arkhipov's $\BosonSampling$ problem~\cite{TIL13, SPR13, CRE13, BAR13, BRO13, SPA14} while theorists have extended their arguments to apply to a range of other quantum circuits, most notably commuting quantum gates on qubits, a class known as $\IQP$~\cite{BMS15}. These generalizations give hope that experimental demonstration of quantum supremacy on sufficiently high fidelity systems of just 50 qubits~\cite{Boixo16}.
In this review we will present the theoretical background behind $\BosonSampling$ and its generalizations, while also reviewing recent experimental demonstrations of $\BosonSampling$.
From a theoretical perspective we focus on the connections between the complexity of counting problems and the complexity of sampling from quantum circuits. This is of course not the only route to determining the complexity of quantum circuit sampling, and recent work by Aaronson and Chen explores several interesting alternative pathways \cite{Aaronson16}.
\section{Computational complexity and quantum supremacy}
\begin{table}
\caption{\label{decision-table}Definitions of complexity classes used in this review. The ``Type'' column describes the outputs generated by algorithms within the class. ``{\em D}'' denotes decision problems which output a single bit, whose values are often interpreted as 'accept' and 'reject'. ``{\em C}'' denotes counting problems which output a non-negative integer. ``{\em Z}'' denotes problems that generalise counting problems and allow negative integer outputs. The definitions give the properties algorithms are required to have within each class. }
\begin{ruledtabular}
\begin{tabularx}{60mm}{ccX}
Class & Type & Definition \\ \hline
\P & {\em D} & deterministic with polynomial runtime on a classical computer \\
\EQP & {\em D} & deterministic with polynomial runtime on a quantum computer \\
\BPP & {\em D} & random with classical statistics and an error probability less than $1/3$ \\
\BQP & {\em D} & random with quantum statistics and an error probability less than $1/3$ \\
\NP & {\em D} & outputs can be verified using an algorithm from \P \\
\PP & {\em D} & random with classical statistics and an error probability less than $1/2$ \\
\SharpP & {\em C} & counts the number of 'accept' outputs for circuits from \P \\
\GapP & {\em Z} & difference between the number of 'accept' and 'reject' outputs for circuits from \P \\
\PSPACE & {\em D} & polynomial memory requirements on a classical computer
\end{tabularx}
\end{ruledtabular}
\end{table}
The challenge in rigorously arguing for quantum supremacy is compounded by the difficulty of bounding the ultimate power of classical computers. Many examples of significant quantum speedups over the best-known classical algorithms have been discovered, see \cite{Montanaro15} for a useful review. The most celebrated of these results is Shor's polynomial time quantum algorithm for factorisation~\cite{Shor95}. This was a critically important discovery for the utility of quantum computing, but was not as satisfying in addressing the issue of quantum supremacy due to the unknown nature of the complexity of factoring. The best known classical factoring algorithm, the general number field sieve, is exponential time (growing as $e^{c n^{1/3} \ln^{2/3}{n}}$ where $n$ is the number of bits of the input number). However, in order to prove quantum supremacy, or really any separation between classical and quantum computational models, it must proven for {\em all} possible algorithms and not just those that are known.
The challenge of bounding the power of classical computation is starkly illustrated by the persistent difficulty of resolving the $\P$ versus $\NP$ question, where the extremely powerful non-deterministic Turing machine model cannot be definitively proven to be more powerful than standard computing devices. The study of this question has led to an abundance of nested relationships between classes of computational models, or complexity classes. Some commonly studied classes are shown in TABLE~\ref{decision-table}. Many relationships between the classes can be proven, such as $\P \subseteq \NP$, $\PP \subseteq \PSPACE$ and $\NP \subseteq \PP$, however, strict containments are rare. Questions about the nature of quantum supremacy are then about what relationships one can draw between the complexity classes when introducing quantum mechanical resources.
A commonly used technique in complexity theory is to prove statements relative to an ``oracle''. This is basically an assumption of access to a machine that solves a particular problem instantly. Using this concept one can define a nested structure of oracles called the ``polynomial hierarchy''\cite{Stockmeyer76} of complexity classes. At the bottom of the hierarchy are the classes $\P$ and $\NP$ which are inside levels zero and one respectively. Then there is the second level which contains the class $\NP^{\NP}$ which means problems solvable in $\NP$ with access to an oracle for problems in $\NP$. If $\P \neq \NP$ then this second level is at least as powerful the first level and possibly more powerful due to the ability to access the oracle. Then the third level contains $\NP^{\NP^{\NP}}$, and so on. Higher levels are defined by continuing this nesting. Each level of the hierarchy contains the levels below it. Though not proven, it is widely believed that every level is strictly larger than the next. This belief is primarily due to the relationships of this construction to similar hierarchies such as the arithmetic hierarchy for which higher levels are always strictly larger. If it turns out that two levels are equal, then one can show that higher levels do not increase and this situation is called a polynomial hierarchy collapse. A polynomial hierarchy collapse to the first level would mean that $\P=\NP$. A collapse at a higher level is a similar statement but relative to an oracle. It is the belief that there is no collapse of the polynomial hierarchy at any level that is used in demonstrating the supremacy of quantum sampling algorithms. Effectively one is forced into a choice between believing that the polynomial hierarchy of classical complexity classes collapses or that quantum algorithms are more powerful than classical ones.
\section{Sampling problems}
\label{SecSampling}
\begin{figure}
\caption{\label{SampDemo}
\label{SampDemo}
\end{figure}
Sampling problems are those problems which output random numbers according to a particular probability distribution (see FIG.~\ref{SampDemo}). In the case of a classical algorithm, one can think of this class as being a machine which transforms uniform random bits into non-uniform random bits according to the required distribution. When describing classes of sampling problems the current convention is to prefix ``Samp-'' to the class in which computation takes place. So $\SampP$ is the class described above using an efficient classical algorithm and $\SampBQP$ would be those sampling problems which are efficiently computable using a quantum mechanical algorithm with bounded error.
All quantum computations on $n$ qubits can be expressed as the preparation of an $n$-qubit initial state $|0\rangle^{\otimes n}$, a unitary evolution corresponding to a uniformly generated quantum circuit $C$ followed by a measurement in the computational basis on this system. In this picture the computation outputs a length $n$ bitstring $x\in\{0,1\}^n$ with probability
\begin{equation}
\label{qu-prob}
p_x=|\langle x|C|0\rangle^{\otimes n}|^2.
\end{equation}
In this way quantum computers produce probabilistic samples from a distribution determined by the circuit $C$. Within this model $\BQP$ is those decision problems solved with a bounded error rate by measuring a single output qubit. $\SampBQP$ is the class of problems that can be solved when we are allowed to measure all of the output qubits.
It is known that quantum mechanics produces statistics which cannot be recreated classically as in the case of quantum entanglement and Bell inequalities. However, these scenarios need other physical criterion to be imposed, such as sub-luminal signaling, to rule out classical statistics. Is there an equivalent ``improvement'' in sampling quantum probability distributions when using complexity classes as the deciding criterion? That is, does $\SampBQP$ strictly contain $\SampP$? The answer appears to be yes and there is a (almost) provable separation between the classical and quantum complexity.
Key to these arguments is understanding the complexity of computing the output probability of a quantum circuit from Eq.~(\ref{qu-prob}). In the 1990s it was shown that there are families quantum circuits for which computing $p_x$ is $\SharpP$-hard in the \emph{worst-case} \cite{Fortnow98, Fenner98}. The suffix ``-hard'' is used to indicate that the problem can, with a polynomial time overhead, be transformed into any problem within that class. $\SharpP$-hard includes all problems in $\NP$. Also, every problem inside the Polynomial Hierarchy can be solved inside the class of decision problems within $\SharpP$-hard, which is written $\P^\SharpP$~\cite{Toda}. Importantly this $\SharpP$-hardness does not necessarily emerge only from the most complicated quantum circuits, but rather can be established even for non-universal, or intermediate, families of quantum circuits such as $\IQP$ \cite{BJS10} and those used in $\BosonSampling$ \cite{AA}. This is commonly established by demonstrating that any quantum circuit can be simulated using the (non-physical) resource of postselection alongside the intermediate quantum computing model \cite{Terhal02,BJS10}.
In fact it is possible to show that computing $p_x$ for many, possibly intermediate, quantum circuit families is actually $\GapP$-complete, a property that helps to establish their complexity under approximations. $\GapP$ is a slight generalization of $\SharpP$ that contains all of the problems inside $\SharpP$ (see table \ref{decision-table}). Note that the suffix ``-complete'' indicates that the problem is both -hard and a member of the class itself. An estimate $\tilde{Q}$ of a quantity $Q$ is accurate to within a {\em multiplicative} error $\epsilon^\prime$ when $Q e^{-\epsilon^\prime} \leq \tilde{Q} \leq Q e^{\epsilon^\prime}$ or alternatively, as $\epsilon^\prime$ small is the usual case of interest, $Q (1 - \epsilon^\prime) \leq \tilde{Q} \leq Q (1 + \epsilon^\prime)$. When a problem is $\GapP$-complete it can be shown that multiplicative approximations of the outputs from these problems are still $\GapP$-complete.
It is important to recognize that quantum computers are not expected to be able to calculate multiplicative approximations to $\GapP$-hard problems, such as computing $p_x$, in polynomial time. This would imply that quantum computers could solve any problem in $\NP$ in polynomial time, which is firmly believed to not be possible. However, an important algorithm from Stockmeyer~\cite{Stockmeyer83} gives us the ability to compute good multiplicative approximations to $\SharpP$-complete problems by utilizing an $\NP$ oracle and by sampling from polynomial-sized classical circuits. The stark difference in complexity under approximations between $\SharpP$ and $\GapP$ can be used to establish a separation between the difficulty of sampling from classical and quantum circuits. If there were an efficient classical algorithm for sampling from families of quantum circuits with $\GapP$-hard output probabilities, then we could use Stockmeyer's algorithm to find a multiplicative approximation to these probabilities with complexity that is inside the third level of the Polynomial Hierarchy, however this causes a contradiction because $\P^\GapP$ contains the entire Polynomial Hierarchy (and it is assumed to not collapse). With such arguments it can be shown that it is not possible to even sample from the outputs to within a constant multiplicative error of many intermediate quantum computing models without a collapse in the Polynomial Hierarchy \cite{AA, BJS10, BJS10, Morimae13, Jozsa14, Bouland16}.
Such results suggest quantum supremacy can be established easily, however, quantum computers can only achieve \emph{additive} approximations to their own ideally defined circuits. An estimate $\tilde{Q}$ of a quantity $Q$ is accurate within an additive error $\epsilon$ if $Q - \epsilon \leq \tilde{Q} \leq Q + \epsilon$. Implementations of quantum circuits are approximate in an additive sense because of the form of naturally occurring errors, our limited ability to learn the dynamics of quantum systems, and finally because quantum circuits use only finite gate sets. In order to demonstrate quantum supremacy we need a fair comparison between what a quantum computer can achieve and what can be achieved with classical algorithms. Following the above line of reasoning, we would need to demonstrate that if a classical computer could efficiently produce samples from a distribution which is close in an additive measure, like the total variation distance, from the target distribution then we would also see a collapse in the Polynomial Hierarchy. Being close in total variation distance means, with error budget $\beta$, samples from a probability distribution $q_x$ satisfying $\sum_x |p_x - q_x| \leq \beta$ are permitted. An error of this kind will tend to generate {\em additive} errors in the outputs. The key insight of Aaronson and Arkhipov was that for some special families of randomly chosen quantum circuits an overall additive error budget causes Stockmeyer's algorithm to give an additive estimate $\tilde{Q}$ that might \emph{also} be a good multiplicative approximation.
\section{$\BosonSampling$ problems}
Aaronson and Arkipov~\cite{AA} describe a simple model for producing output probabilities that are $\SharpP$-hard. Their model uses bosons that interact only by linear scattering. The bosons must be prepared in a Fock state and measured in the Fock basis.
Linear bosonic interactions, or linear scattering networks, are defined by dynamics in the Heisenberg picture that generate a linear relationship between of the annihilation operators of each mode. That is, only those unitary operators $\mathcal{U}$ which act on the Fock basis such that
\begin{equation}
\mathcal{U}^\dagger a_i \mathcal{U} = \sum_{j} u_{ij} a_j
\end{equation}
where $a_i$ is the $i$-th mode's annihilation operator and the $u_{ij}$ form a unitary matrix which for $m$ modes is a $m \times m$ matrix. It is important to make a distinction from the unitary operator $\mathcal{U}$ which acts upon the Fock basis and the unitary matrix defined by $u_{ij}$ which describes the linear mixing of modes. For optical systems the matrix $u_{ij}$ is determined by how linear optical elements, such as beam-splitters and phase shifters, are laid out. In fact all unitary networks can be constructed using just beam-splitter and phase shifters~\cite{Reck94}.
The class $\BosonSampling$ is defined as quantum sampling problems where a fiducial $m$-mode $n$-boson Fock state
\begin{equation}
\label{eq:bsinput}
\ket{\underbrace{1,1,1,\ldots,1}_n,\underbrace{0,0,\ldots,0}_{m-n}}
\end{equation}
is evolved through a linear network with the output being samples from the distribution that results after a Fock basis measurement of all modes. The linear interaction is then the input to the algorithm and the output is the sample from the probability distribution. FIG.~\ref{bsamp} shows a schematic representation of this configuration. The set of events which are then output by the algorithm is a tuple of $m$ non-negative integers whose sum is $n$. This set is denoted $\Phi_{m,n}$.
\begin{figure*}
\caption{\label{bsamp}
\label{bsamp}
\end{figure*}
The probability distribution of output events is related to the matrix permanent of sub-matrices of $u_{ij}$. The matrix permanent is defined in a recursive way like the common matrix determinant, but without the alternation of addition and subtraction. For example
\begin{equation}
Per \left(\begin{array}{cc}a & b \\ c & d\end{array}\right) = a d + c b.
\end{equation}
\begin{equation}
Per \left(\begin{array}{ccc}a & b & c \\ d &e &f \\g &h &i \end{array}\right) = a e i + a h f + b d i + d g f + c d h + c g e .
\end{equation}
Or in a more general form
\begin{equation}
Per (A) = \sum_{\sigma \in \mathcal{S}_n} \prod_{i=1}^n a_{i,\sigma(i)}
\end{equation}
where $\mathcal{S}_n$ represents the elements of the symmetric group of permutations of $n$ elements. With this, we can now define the output distribution of the linear network with the input state from Eq.~\ref{eq:bsinput}. For an output event $S=(s_1,s_2,\ldots,s_n)\in \Phi_{m,n}$, the probability of $S$ is then
\begin{equation}
\label{eq:BSprobability}
p_S = \frac{|Per(A_S)|^2}{s_1! s_2! \ldots s_n!}
\end{equation}
where the matrix $A_S$ is a $n \times n$ sub-matrix of $u_{ij}$ where row $i$ is repeated $s_i$ times and only the first $n$ columns are used. One critical observation of this distribution is that all events are proportional to the square of a matrix permanent derived from the original network matrix $u_{ij}$. Also, the fact that each probability is derived from a permanent of a sub-matrix of the same unitary matrix ensures all probabilities are less than $1$ and the distribution is normalised.
The complexity of computing the matrix permanent is known to be $\SharpP$-complete for the case of matrices with entries that are $0$ or $1$~\cite{Val79}. It is also possible to show that for a matrix with real number entries is $\SharpP$-hard to multiplicatively estimate~\cite{AA}. Therefore, using the argument presented above, the case of sampling from this exact probability distribution implies a polynomial hierarchy collapse.
The question is then if sampling from approximations of $\BosonSampling$ distributions also implies the same polynomial hierarchy collapse. The answer that Aarsonson and Arkipov found~\cite{AA} is that the argument does hold because of a feature that is particular to the linear optical scattering probabilities. When performing the estimation of the matrix permanent for exact sampling, the matrix is scaled and embedded in $u_{ij}$. The probability of one particular output event, with $n$ ones in the locations of where the matrix was embedded, is then proportional to the matrix permanent squared. The matrix permanent can then be estimated multiplicatively in the third level of the polynomial hierarchy. But any event containing $n$ ones in $\Phi_{m,n}$ could have been used to determine the location of the embedding. This means that, if the estimation is made on a randomly chosen output event, and that event is hidden from the algorithm implementing approximate $\BosonSampling$, then the expected average error in the estimation will be the overall permitted error divided by the total number of events which could have been used to perform the estimation.
An important consideration of the approximate sampling argument is that the input matrix appears to be drawn from Gaussianly distributed random matrices. This ensures that there is a way of randomly embedding the matrix into $u_{ij}$ so that there is no information accessible to the algorithm about where that embedding has occurred. This is possible when the unitary network matrix is sufficiently large (strictly $m=O(n^5\ln^2 n)$ but $m=O(n^2)$ is likely to be OK). Also, under this condition, the probability of events detected with two or more bosons in a single detector tends to zero for large $n$ (the so-called ``Bosonic Birthday Paradox''). There are $\binom{m}{n}$ events in $\Phi_{m,n}$ with only $n$ ones and so the error budget can be evenly distributed over just these events. There are exponentially many of these events and so the error in the probability of an individual event does not dominate but is as small as the average expected probability itself.
With this assumption about the distribution of input matrices, the proof for hardness of approximate sampling relies on the problem of estimating the permanents of Gaussian random matrices still being in $\SharpP$-hard. Furthermore, as the error allowed to the sampling probabilities is defined in terms of total variation distance, the error in estimation becomes additive rather than multiplicative.
This changes the situation from the hardness proof for exact sampling enough to be concerned that the proof may not apply. Aaronson and Arkipov therefore isolated the requirements for the hardness proof to still apply down to two conjectures that must hold for additive estimation of permanents for Gaussian random matrices to be $\SharpP$-hard. They are the Permenants of Gaussian Conjecture (PGC) and the Permenant Anti-Concentration Conjecture (PACC). The PACC conjecture says that if the matrix permanents of Gaussian random matrices are not too concentrated around zero. If this holds then additive estimation of permanents for Gaussian random matrices is polynomial-time equivalent to multiplicative estimation. The PGC is that multiplicative estimation of permanents from Gaussian random matrices is $\SharpP$-hard. In both of these conjectures there are related proofs that seem close, but do not exactly match the conditions required. Nevertheless, both of these conjectures are highly plausible.
\section{Experimental implementations of $\BosonSampling$}
Several small scale implementations of $\BosonSampling$ have been performed with quantum optics. Implementing $\BosonSampling$ using optics is an ideal choice as the linear network consists of a large multi-path interferometer. Then the inputs are single photon states which are injected into the interferometer and single photon counters are placed at all $m$ output modes and the arrangement of photons at the output, shot-by-shot, is recorded. Due to the suppression of multiple photon counts under the conditions for approximate $\BosonSampling$, single photon counters can be replaced by detectors that detect the presence or absence of photons (e.g. avalanche photo diodes).
Within these optical implementations, the issues of major concern are photon loss, mode-mismatch, network errors and single photon state preparation and detection imperfections. Some of these issues can be dealt with by adjusting the theory and checking that the hardness proof still holds. In the presence of loss one can post-select on events where all $n$ photons make it to the outputs. This provides a mechanism to construct proof of principle devices but does incur an exponential overhead which prevents scaling to large devices. Rohde and Ralph studied bounds on loss in BosonSampling by finding when efficient classical simulation of lossy BosonSampling is possible in two simulation strategies: Gaussian states and distinguishable input photons~\cite{Rohde2012}. Aaronson and Brod~\cite{Aaronson2016} have shown that in the case where the number of photons lost is constant, then hardness can still be shown. However, this is not a realistic model of loss as the number of photons lost will be proportional to the number of photons input. Leverrier and Garcia-Patron have shown that it is a necessary condition for errors in the network to be tolerable provided the error in the individual elements scales as $O(n^{-2})$~\cite{Lev2014}. Later Arkipov showed the sufficient condition is element errors scaling as $o(n^{-2}\log^{-1}m)$~\cite{Arkhipov2015}. Rahimi-Keshari, et al. showed a necessary condition for hardness based on the presence of negativity of phase-space quasiprobability distributions~\cite{Rahimi2016}. This give inequalities constraining the overall loss and noise of a device implementing $\BosonSampling$.
The majority of the initial experiments were carried out with fixed, on-chip interferometers \cite{TIL13,SPR13,CRE13,BAR13}, though one employed a partially tunable arrangement using fibre optics \cite{BRO13}. The largest network so far was demonstrated by N.~Spagnolo, et al, where 3 photons were injected into 5, 7, 9 and 13 mode optical networks~\cite{SPA14}. In this experiment the optical networks were multi-mode integrated interferometers fabricated in glass chips by femtosecond laser writing. The photon source was parametric down-conversion with four photon events identified via post-selection, where 3 of the photons were directed through the on-chip network and the 4th acted as a trigger. Single photon detectors were placed at all outputs, enabling the probability distribution to be sampled.
For the 13 mode experiment there are 286 possible output events from $\Phi_{13,3}$ consisting of just zeros and ones. To obtain the expected probability distributions the permanents for the sub-matrices corresponding to all configurations were calculated. Comparing the experimentally obtained probabilities with the predictions showed excellent agreement for all the chips. Such a direct comparison would become intractable for larger systems -- both because of the exponentially rising complexity of calculating the probabilities, and because of the exponentially rising amount of data needed to experimentally characterise the distribution. N.~Spagnolo, et al demonstrated an alternative approach whereby partial validation of the device can be obtained efficiently by ruling out the possibility that the distribution was simply a uniform one \cite{AAR13}, or that the distribution was generated by sending distinguishable particles through the device \cite{CAR13}. In both cases, only small sub-sets of the data were needed and the tests could be calculated efficiently.
The BosonSampling problem is interesting because, as we have seen, there are very strong arguments to suggest that medium scale systems, such as 50 bosons in 2,500 paths, are intractable for classical computers. Indeed, even for smaller systems, say 20 bosons in 400 paths, no feasible classical algorithms are currently known which can perform this simulation. This suggests that quantum computations can be carried out in this space without fault tolerant error correction that may rival the best current performance on classical computers. In addition, there is a variation of the problem referred to as scatter-shot, or Gaussian BosonSampling which can be solved efficiently by directly using the squeezed states deterministically produced by down converters as the input (rather than single photon states) \cite{LUN14} which has been experimentally demonstrated on a small scale using up to six independent sources for the Gaussian states~\cite{Bentivegna15}. Thus the major challenge to realising an intermediate optical quantum computer of this kind is the ability to efficiently (i.e. with very low loss and noise) implement a reconfigurable, universal linear optical network over hundreds to thousands of modes. On-chip designs such as the 6 mode reconfigurable, universal circuit demonstrated by J.~Carolan, et al~\cite{CAR15} are one of several promising ways forward. Another interesting approach is the reconfigurable time-multiplexed interferometer proposed by Motes et al.~\cite{Motes2014} and recently implemented in free-space by Y.~He, et al~\cite{HE16}. This latter experiment is also distinguished by the use of a quantum dot as the single photon source which have also be utilised in the spatial multiplexed interferometers~\cite{Loredo16}. Finally, it is possible to construct a theory for realistic interferometers including polarisation and temporal degrees of freedom can be considered and also give rise to probabilities proportional to matrix permanents~\cite{Laibacher15, Tamma16}.
\section{Sampling with the circuit model and $\IQP$}
Last year Bremner, Montanaro, and Shepherd extended the $\BosonSampling$ argument to $\IQP$ (Instantaneous Quantum Polynomial-time) circuits, arguing that if such circuits could be classically simulated to within a reasonable additive error, then the Polynomial Hierarchy would collapse to the third level \cite{BMS15}. Crucially, these hardness results rely only on the conjecture that the average-case and worst-case complexities of quantum amplitudes of $\IQP$ circuits coincide. Only the one conjecture is needed as the $\IQP$ analogue of the PACC was proven to be true. As this argument is native to the quantum computing circuit model, any architecture for quantum computation can implement $\IQP$ Sampling. It also means that error correction techniques can be used to correct noise in such implementations. Furthermore, the $\IQP$ Sampling and the related results on Fourier Sampling by Fefferman and Umans \cite{Fefferman15} demonstrate that generalizations of the Aaronson and Arkhipov argument \cite{AA} could potentially be applied to a much wider variety of quantum circuit families, allowing the possibility of sampling arguments that are both better tailored to a particular experimental setup and for their complexity to be dependent on new theoretical conjectures.
$\IQP$ circuits \cite{BS09,BJS10} are an intermediate model of quantum computation where every circuit has the form $C=H^{\otimes n}DH^{\otimes n}$, where $H$ is a hadamard gate and $D$ is an efficiently generated quantum circuit that is diagonal in the computational basis. \emph{\IQP sampling} then simply corresponds to performing measurements in the computational basis on the state $H^{\otimes n}DH^{\otimes n}|0\rangle^{\otimes n}$. In \cite{BMS15} it was argued that classical computers could not efficiently sample from $\IQP$ circuits where $D$ is chosen uniformly at random from circuits composed of: (1) $\sqrt{CZ}$ (square-root of controlled-Z), and $T=\left( \begin{smallmatrix} 1&0\\0 & e^{i\pi/4} \end{smallmatrix} \right)$ gates; or (2) $Z$, $CZ$, and $CCZ$ (doubly controlled-Z) gates. This argument was made assuming that it is $\SharpP$-hard to multiplicatively approximate a constant fraction of instances of (the modulus-squared of): (C1) the complex-temperature partition function of a random 2-local Ising model; or (C2) the (normalized) gap of a degree-3 polynomial (over $\mathbb{F}_2$). These conjectures can be seen as $\IQP$ analogues of Boson Sampling's PGC. In the case of (1) these circuits correspond to random instances of the Ising model drawn from the complete graph, as depicted in FIG.~\ref{IQPfig}.
\begin{figure}
\caption{\label{IQPfig}
\label{IQPfig}
\end{figure}
The worst-case complexity of the problems in both (C1) and (C2) can be seen to be $\SharpP$-hard as these problems are directly proportional to the output probabilities of the $\IQP$ circuit families (1) and (2). These families are examples of sets that become universal under postselection and as a result their output probabilities are $\SharpP$-hard (as mentioned in Section \ref{SecSampling}). This is shown by noting that for either of the gate sets (1) or (2), the only missing ingredient for universality is the ability to perform hadamard gates at any point within the circuit. In \cite{BJS10} it was shown that such gates can be replaced with a ``hadamard gadget", which requires 1 postselected qubit and controlled-phase gate per hadamard gate. It can be shown that the complexity of computing the output probabilities of IQP circuits, $p_x= |\langle x|H^{\otimes n}DH^{\otimes n}|0\rangle^{\otimes n}|^2$, is $\SharpP$-hard in the worst case and this also holds under multiplicative approximation \cite{BMS15, Fujii13, Goldberg14}.
The hardness of $\IQP$-sampling to within additive errors follows from the observation that Stockmeyer's algorithm combined with sufficiently accurate classical additive simulation returns a very precise estimate to the probability $p_0=|\langle 0|^{\otimes n}C_y|0\rangle^{\otimes n}|^2$ for a wide range of randomly chosen circuits $C_y$. A multiplicative approximation to $p_0$ can be delivered on a large fraction of choices of $y$ when both: (a) for a random bitstring $x$, the circuit $\bigotimes_{i=1}^nX^{x_i}$ is a hidden subset of the randomly chosen circuits $C_y$; and (b) $p_0$ anti-concentrates on the random choices of circuits $C_y$. Both of these properties hold for the randomly chosen $\IQP$ circuit families (1) and (2) above, and more generally hold for any random family of circuits that satisfies the Porter-Thomas distribution \cite{Boixo16}. Classical simulations of samples from $C_y$ implies a Polynomial Hierarchy collapse if a large enough fraction of $p_0$ are also $\SharpP$-hard under multiplicative approximations - and definitively proving such a statement remains a significant mathematical challenge. As mentioned above in \cite{BMS15} the authors could only demonstrate sufficient \emph{worst-case} complexity for evaluating $p_0$ for the circuit families (1) and (2), connecting the complexity of these problems to key problems in complexity theory.
The $\IQP$ circuit families discussed above allow for gates to be applied between any qubits in a system. This means that there could be $O(n^2)$ gates in a random circuit for (1) and $O(n^3)$ gates for (2), with many of them long-range. From an experimental perspective this is challenging to implement as most architectures have nearest-neighbour interactions. Clearly these circuits can be implemented with nearest-neighbour gates from a universal gate set, however many SWAP gates would need to be applied. Given that many families of quantum circuits can have $\SharpP$-hard output probabilities this suggests it is worthwhile understanding if more efficient schemes can be found. It is also important to identify new average-case complexity conjectures that might lead to a proof that quantum computers cannot be classically simulated.
The challenge in reducing the resource requirements for sampling arguments is to both maintain the anti-concentration property and the conjectured $\SharpP$-hardness of the average-case complexity of the output probabilities. Recently it was shown that \emph{sparse} $\IQP$-sampling, where $\IQP$ circuits are associated with random sparse graphs, has both of these features \cite{BMS16}. It was proved that anticoncentration can be achieved with only $O(n\log n)$ long-range gates or rather in depth $O(\sqrt{n} \log n)$ with high probability in a universal 2d lattice architecture.
If we take as a guiding principle that in the worst-case output probabilities should not have a straightforward sub-exponential algorithm, then the 2d architecture depth cannot be less than $O(\sqrt{n})$ as there exist classical algorithms for computing any quantum circuit amplitude for a depth $t$ circuit on a 2d-lattice that scale as $O(2^{t\sqrt{n}})$. This suggests that there might be some room still to optimize the results of \cite{BMS16}, and is further evidenced by a recent numerical study suggesting that anti-concentration, and subsequently quantum supremacy, could be achieved in systems where gates are chosen at random from a universal gate set on a square lattice with depth scaling like $O(\sqrt{n})$. Such arguments give hope that a quantum experiment on approximately 50 qubits could be performed, assuming that the rate of error can be kept low enough.
Recently it has also been proposed that sampling from 2d ``brickwork'' states cannot be classically simulated \cite{Gao16}. Such states have depth $O(1)$ and as such their output probabilities are thought not to anticoncentrate and can be classically computed in sub-exponential $2^{O(\sqrt{n})}$ time. However, the authors argue that there are some output probabilities that are $\GapP$-complete, yet might be reliably approximated via Stockmeyer's algorithm without anticoncentration. This is possible under considerably stronger average-case complexity conjectures than those appearing in \cite{BMS15, BMS16}, and also requires polynomially more qubits.
Finally it should be remarked that the level of experimental precision required to definitively demonstrate quantum supremacy, even given generous constant total variation distance bound (such as required in \cite{BMS15}), is very high. Asymptotically this typically requires the precision of each circuit component must improve by an inverse polynomial in the number of qubits. This is likely hard to achieve with growing system size without the use of fault tolerance constructions. More physically reasonable is to assume that each qubit will at least have a constant error rate, which corresponds to a total variation distance scaling like $O(n)$. Recently it was shown that if an $\IQP$ circuit has the anti-concentration property, and it suffers from a constant amount of depolarizing noise on each qubit then there is an classical algorithm that can classically simulated it to within a reasonable total variation distance \cite{BMS16}. However, it should be remarked for a constant number of qubits this algorithm will likely still have a very large run-time. By contrast, $\IQP$ remains classically hard under the error model for multiplicative classical approximations \cite{Fujii16}. Intriguingly, this class of errors can be corrected without the full arsenal of fault tolerance, retrieving supremacy for additive error approximations requiring only operations from $\IQP$ albeit with a cost in terms of gates and qubits \cite{BMS16}. This suggests that unambiguous quantum supremacy may yet require error correction, though the level of error correction required remains a very open question.
\section{Conclusion}
Quantum sampling problems have provided a path towards experimental demonstration of the supremacy of quantum algorithms with significantly lower barriers than previously thought necessary for such a demonstration. The two main classes of sampling problems demonstrating quantum supremacy are $\BosonSampling$ and $\IQP$ which are intermediate models of optical and qubit based quantum information processing architectures. Even reasonable approximations to the outputs from these problems, given some highly plausible conjectures, are hard for classical computers to compute.
Some future directions for research in this area involve a deeper understanding of these classes as well as experimentally addressing the technological challenges towards implementations that outperform the current best known classical algorithms. Theoretical work on addressing what is possible within these classes, such as detecting and correcting with errors within the intermediate models will both aid understanding and benefit experimental implementations. There has been some study of the verification of limited aspects of these devices~\cite{Tichy14,Walschaers16,Aolita15} but more work is required. As $\BosonSampling$ and $\IQP$ are likely outside the Polynomial Hierarchy, an efficient reconstruction of the entire probability distribution which is output from these devices will likely be impossible. However, one can build the components, characterise them and their interactions, build and run such a device to within a known error rate. Beyond this multiplayer games based on sampling problems in $\IQP$ have been proposed to test whether a player is actually running an $\IQP$ computation \cite{BS09}. Recently the complexity of $\IQP$ sampling has been connected to the complexity of quantum algorithms for approximate optimization problems \cite{Farhi16}, suggesting further applications of $\IQP$ and closely related classes. Applications of BosonSampling to molecular simulations~\cite{Huh2015}, metrology~\cite{Motes2015} and decision problems~\cite{Nikolopoulos2016} have been suggested, though more work is needed in this space. Nevertheless, the results from quantum sampling problems have undoubtedly brought us closer to the construction of a quantum device which definitively displays the computational power of quantum mechanics.
\section*{Contributions}
All authors contributed equally to this work.
\section*{Funding}
APL and TCR received financial support from the Australian Research Council Centre of Excellence for Quantum Computation and Communications Technology (Project No. CE110001027). MJB has received financial support from the Australian Research Council via the Future Fellowship scheme (Project No. FT110101044).
\section*{Competing interests}
The authors declare no competing financial interests.
\end{document} |
\begin{document}
\begin{abstract}
We survey recent work on moduli spaces of manifolds with an emphasis
on the role played by (stable and unstable) homotopy theory. The
theory is illustrated with several worked examples.
\end{abstract}
\maketitle
\section{Introduction}
The study of manifolds and invariants of manifolds was begun more
than a century ago. In this entry we shall discuss the parametrised
setting: invariants of \emph{families} of manifolds, parametrised by a
base manifold $X$. The invariants we look for will be cohomology
classes in $X$, characteristic classes.
This article will be structured in the following way.
\begin{enumerate}[(i)]
\item A discussion of the abstract classification theory.
The main content here is the precise definition of the kind of
families we consider, and an outline of how a classifying space may
be constructed. There is one such classifying space for each pair $(W,\rho_W)$ consisting of a closed manifold $W$ and a tangential structure $\rho_W$, see \S\ref{sec:smooth-bundles-their}. The more general case where $W$ is compact with boundary is briefly discussed in \S\ref{sec:boundary}.
\item Definition of Miller--Morita--Mumford classes, the main examples of
characteristic classes. With this definition in place, we state a first version of the main result of \cite{GR-W2, GR-W3, GR-W4}: a formula (see Theorem~\ref{thm:main-cohomological}) for rational cohomology of the classifying spaces in a range of degrees, for many instances of $(W,\rho_W)$.
\item Statement of the main result of \cite{GR-W2, GR-W3, GR-W4} in their general forms. These statements require a bit more homotopy theory to formulate but apply quite generally, at least for even-dimensional manifolds, see the theorems in \S\ref{sec:gener-vers-main}.
\item Examples of calculations and applications. The results surveyed in \S\S\ref{sec:char-class-XXX}--\ref{sec:gener-vers-main} are quite well suited for explicit calculations. We believe this to be an important feature of the theory, and have included a supply of worked examples of various types to illustrate this aspect, many of which have not previously been published. In \S\ref{sec:RatCalc} we will focus on calculations in rational cohomology, and we include a detailed study of some complete intersections. In \S\ref{sec:AbCalc} we carry out some integral homology calculations, focusing on $H_1$.
\end{enumerate}
Our theorems apply in even dimensions $2n \geq 6$, but were inspired by the breakthrough theorem of Madsen and Weiss \cite{MW} in dimension $2$, building on earlier ideas of Madsen and Tillmann \cite{MT} and Tillmann \cite{Tillmann}.
\section{Smooth bundles and their classifying spaces}
\label{sec:smooth-bundles-their}
First, some conventions. By \emph{smooth manifold}\mathrm{ind}ex{manifold} we shall always mean a Hausdorff, second countable topological manifold equipped
with a maximal $C^\infty$ atlas, and \emph{smooth map} shall always mean
$C^\infty$. We shall generally use the letter $\mathcal{M}$
with various decorations for variants of classifying spaces for families of manifolds,
and the letter $\mathcal{F}$ with various decorations for the
functor it classifies.
\subsection{Smooth bundles}
\label{sec:smooth-bundles}
Our notion of ``family'' of manifolds will be \emph{smooth fibre
bundle}, possibly equipped with extra tangent bundle structure, as
follows. If $V$ is a $d$-dimensional real vector bundle we shall
write $\mathrm{Fr}(V)$ for the associated frame bundle, which is a
principal $\mathrm{GL}_d(\bR)$-bundle.
\begin{definition}\label{def:family} Let $d$ be a non-negative integer.
\begin{enumerate}[(i)]
\item A \emph{smooth fibre bundle}\mathrm{ind}ex{fibre bundle} of dimension $d$ consists of smooth manifolds $E$
and $X$ (without boundary) and a smooth proper map $\pi: E\to X$
such that $D\pi: TE \to \pi^* TX$ is surjective and the vector
bundle $T_\pi E = \mathrm{Ker}(D\pi)$ has $d$-dimensional fibres. The bundle $T_\pi E$ is called the \emph{vertical tangent bundle}.\mathrm{ind}ex{vertical tangent bundle}
\item If $\Theta$ is a space with a continuous action of
$\mathrm{GL}_d(\bR)$, a \emph{smooth fibre bundle with $\Theta$-structure}\mathrm{ind}ex{$\Theta$-structure}
consists of a smooth fibre bundle $\pi: E \to X$, together with a continuous
$\mathrm{GL}_d(\bR)$-equivariant map
$\mathrm{st}r: \mathrm{Fr}(T_\pi E) \to \Theta$.
\end{enumerate}
\end{definition}
Typical choices of $\Theta$ include the terminal one $\Theta = \{\ast\}$,
and $\Theta = \mathbbm{Z}^\times = \{\pm 1\}$ on which $\mathrm{GL}_d(\bR)$ acts by multiplication by
the sign of the determinant. In the former case a $\Theta$-structure
is no information, and in the latter it is the data of a continuously
varying family of orientations of the $d$-manifolds $\pi^{-1}(x)$. (The space of equivariant maps $\mathrm{Fr}(T_\pi E) \to \Theta$ may be modelled in other ways, equivalent up to weak equivalence, see \S\ref{sec:gl_dr-spaces-versus}.)
Any smooth map $f: X' \to X$ will be transverse to any smooth fibre bundle
$\pi: E \to X$ as above, and the pullback $(f^* \pi): f^* E \to X'$ is
again a smooth fibre bundle. Given a $\Theta$-structure $\mathrm{st}r$ on
$\pi$, we shall write $f^*\mathrm{st}r$ for the induced structure on
$(f^* \pi)$.
\subsection{Classifying spaces}
\label{sec:classifying-spaces}
\newcommand{\mathcal{F}}{\mathcal{F}}
\newcommand{\mathsf{Man}}{\mathsf{Man}}
\newcommand{\mathsf{sSets}}{\mathsf{sSets}}
\newcommand{\mathrm{op}}{\mathrm{op}}
The natural equivalence relation between the bundles considered above
is \emph{concordance}, which we recall.
\begin{definition} Let $\pi_0: E_0 \to X$ and $\pi_1: E_1 \to X$ be
smooth bundles with $\Theta$-structures
$\mathrm{st}r_0: \mathbf{F}r(T_{\pi_0} E_0) \to \Theta$ and
$\mathrm{st}r_1: \mathbf{F}r(T_{\pi_1} E_1) \to \Theta$.
\begin{enumerate}[(i)]
\item An \emph{isomorphism} between $(\pi_0,\mathrm{st}r_0)$ and
$(\pi_1,\mathrm{st}r_1)$ is a diffeomorphism $\phi: E_0 \to E_1$ over $X$,
such that the induced map
$\mathrm{Fr}(D_\pi \phi): \mathbf{F}r(T_{\pi_0}E_0) \to \mathbf{F}r(T_{\pi_1}E_1)$
is over $\Theta$.
\item A \emph{concordance}\mathrm{ind}ex{concordance} between $(\pi_0,\mathrm{st}r_0)$ and
$(\pi_1,\mathrm{st}r_1)$ is a smooth fibre bundle $\pi: E \to \bR \times X$
with $\Theta$-structure $\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$, together
with isomorphisms from $(\pi_0,\mathrm{st}r_0)$ and $(\pi_1, \mathrm{st}r_1)$ to
the pullbacks of $(\pi,\mathrm{st}r)$ along the two embeddings
$X \cong \{0 \} \times X \subset \bR\times X$ and
$X \cong \{1 \} \times X \subset \bR\times X$.
\end{enumerate}
\end{definition}
Pulling back is functorial up to isomorphism and preserves being concordant.
\begin{definition}
For a smooth manifold $X$ without boundary, let $\mathcal{F}^\Theta[X]$
denote the set of concordance classes of pairs $(\pi,\mathrm{st}r)$ of a
smooth fibre bundle $\pi: E \to X$ with $\Theta$-structure
$\mathrm{st}r: \mathrm{Fr}(T_\pi E) \to \Theta$. (Note that $X$ is fixed,
but $E$ is allowed to vary.)
\end{definition}
\begin{theorem}\label{thm:existence-of-classi}
The functor $X \mathrm{map}sto \mathcal{F}^\Theta[X]$ is representable in the
(weak) sense that there exists a topological space
$\mathcal{M}^\Theta$\mathrm{ind}ex{moduli space!of $\Theta$-manifolds} and a natural bijection
\begin{equation}\label{eq:3}
\mathcal{F}^\Theta[X] \cong [X,\mathcal{M}^\Theta],
\end{equation}
where the codomain denotes homotopy classes of continuous maps.
\end{theorem}
There are various ways to prove this representability statement. In
the series \cite{GR-W2, GR-W3, GR-W4} we did this by constructing explicit point-set
models in terms of submanifolds of $\bR^\infty$. Instead of repeating
what we said there, let us outline an approach based on simplicial sets and the
observation that the functor $\mathcal{F}^\Theta$ may be upgraded to take
values in \emph{spaces}, and the natural bijection~(\ref{eq:3}) may be
upgraded to a natural weak equivalence to the mapping
space.
\begin{definition}\label{defn:represented-functor}
Let $\Delta^\bullet_e$ be the cosimplicial smooth manifold given by
the extended simplices
$\Delta^p_e = \{t \in \bR^{p+1} \mid t_0 + \dots + t_p = 1\}$.
For a smooth manifold $X$ let $\mathcal{F}^\Theta_\bullet(X)$ denote the
simplicial set whose $p$-simplices are all pairs $(\pi,\mathrm{st}r)$
consisting of a smooth fibre bundle $\pi: E \to \Delta^p_e \times X$
with $\Theta$-structure $\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$.
\end{definition}
As they stand, these definitions are not quite rigorous:
$\mathcal{F}^\Theta[X]$ and $\mathcal{F}^\Theta_p(X)$ are not (small) sets for
size reasons, and $\mathcal{F}^\Theta_p(X)$ is not functorial in
$[p] \in \Delta$ because pullback is not strictly associative on the
level of underlying sets. This may be fixed in standard ways, e.g.\
by requiring the underlying set of $E$ to be a subset of
$X \times \Omega$ for a set $\Omega$ of sufficiently large
cardinality. See e.g.\ \cite[Section 2.1]{MW} for more
detail.
\begin{proposition}
Let $\mathsf{Man}$ and $\mathsf{sSets}$ be the categories of smooth manifolds and
simplicial sets, and let
$\mathcal{F}^\Theta_\bullet: \mathsf{Man}^\mathrm{op} \to \mathsf{sSets}$ be the functor defined
above. Then we have a natural bijection
\begin{equation*}
\mathcal{F}^\Theta[X] \cong \pi_0 \mathcal{F}^\Theta_\bullet (X)
\end{equation*}
and a natural weak equivalence of simplicial sets
\begin{equation}\label{eq:4}
\mathcal{F}^\Theta_\bullet(X) \xrightarrow\simeq
\mathrm{Maps}(\mathrm{Sing}(X),\mathcal{F}_\bullet(\{\ast\})).
\end{equation}
Consequently, we may take $\mathcal{M}^\Theta := |\mathcal{F}^\Theta_\bullet(\{\ast\})|$ in Theorem~\ref{thm:existence-of-classi}.
\end{proposition}
In this statement we take $\mathrm{Sing}(X)$ to mean the smooth singular set, i.e.\ the simplicial set $[p] \mathrm{map}sto C^\infty(\Delta^p_e,X)$. It is equivalent to the usual simplicial set made out of all continuous maps $\Delta^p \to X$ by a smooth approximation argument, and in particular the evaluation map $|\mathrm{Sing}(X)| \to X$ is a homotopy equivalence by Milnor's theorem. The codomain of~(\ref{eq:4}) is the simplicial set of maps into the Kan complex $\mathcal{F}_\bullet(\{\ast\})$, homotopy equivalent to the space of maps $X \to |\mathcal{F}_\bullet(\{\ast\})|$.
\begin{proof}[Proof sketch]
The first claim follows by identifying 1-simplices of
$\mathcal{F}^\Theta_\bullet(X)$ with concordances between 0-simplices.
For the second claim, we define the
map~(\ref{eq:4}) by pulling back. To check that it
is a weak equivalence one first verifies that both sides send open
covers $X = U \cup V$ to homotopy pullback squares and countable
increasing unions to homotopy limits. Hence it suffices to check the
case $X = \bR^n$. Then one checks that both sides send
$X \times \bR \to X$ to a weak equivalence, so it suffices to check
$X = \{\ast\}$, which is obvious.
The third claim follows by combining the first and the second claim and the homotopy equivalence $|\mathrm{Sing}(X)| \to X$ given by evaluation. Alternatively it may be quoted directly from \cite[Section 2.4]{MW}.
\end{proof}
The above proof of Theorem~\ref{thm:existence-of-classi} gives an explicit bijection~(\ref{eq:3}). Indeed, an element $(\pi,\mathrm{st}r) \in \mathcal{F}_0(X)$ gives a morphism of simplicial sets $\mathrm{Sing}(X) \to \mathcal{F}^\Theta_\bullet(\{\ast\})$ and hence a canonical zig-zag
\begin{equation}\label{eq:24}
X \overset{\mathrm{ev}}\longleftarrow |\mathrm{Sing}(X)| \overset{(\pi,\mathrm{st}r)}\lra | \mathcal{F}^\Theta_\bullet(\{\ast\})| = \mathcal{M}^\Theta.
\end{equation}
The map $\mathrm{ev}: |\mathrm{Sing}(X)| \to X$ is a homotopy equivalence, and we may choose a homotopy inverse. For example, a smooth triangulation of $X$ gives such an inverse, which is even a section. The resulting homotopy class of map $X \to \mathcal{M}^\Theta$ corresponds to $[(\pi,\mathrm{st}r)]$ under the bijection~(\ref{eq:3}).
For later purposes, let us explain how a universal fibration $\pi^\Theta : \mathcal{E}^\Theta \to \mathcal{M}^\Theta$ modelling the bundle $\pi: E \to X$ may be constructed by the same simplicial method. Let $\widetilde{\mathcal{F}}^\Theta_\bullet(X)$ be the simplicial set whose $p$-simplices are triples $(\pi,\mathrm{st}r,s)$ where $(\pi,\mathrm{st}r)$ is as before and $s: \Delta^p_e \times X \to E$ is a section of $\pi$, and set $\mathcal{E}^\Theta := |\widetilde{\mathcal{F}}^\Theta_\bullet(\{\ast\})|$. The zig-zag~(\ref{eq:24}) above associated to an element $(\pi,\mathrm{st}r) \in \mathcal{F}_0^\Theta(X)$ now extends to a canonical diagram
\begin{equation*}
\begin{tikzcd}
E \dar[']{\pi} & {|\mathrm{Sing}(E)|} \lar[']{\simeq}\dar\rar & \mathcal{E}^\Theta \dar{\pi^\Theta}\\
X & {|\mathrm{Sing}(X)|} \lar[']{\simeq} \rar{(\pi,\mathrm{st}r)} & \mathcal{M}^\Theta
\end{tikzcd}
\end{equation*}
in which both squares are homotopy cartesian, and the top right-hand map is that associated to the data $(\mathrm{pr}_1 : E \times_X E \to E, \mathrm{st}r\circ D\mathrm{pr}_2, \mathrm{diag} : E \to E \times_X E)$.
Finally, a $p$-simplex $(\pi: E \to \Delta^p_e, \mathrm{st}r, s)$ of $\mathcal{E}^\Theta$ gives a map $\ell = (\mathrm{st}r/\mathrm{GL}_d(\mathbb{R})): E \cong (\mathbf{F}r(T_\pi E)/\mathrm{GL}_d(\bR)) \to (\Theta/\mathrm{GL}_d(\bR))$. Composing with the section $s\vert_{\Delta^p}: \Delta^p \to E$ then gives rise to a map of simplicial sets
$$\widetilde{\mathcal{F}}^\Theta_\bullet(\{*\}) \lra \mathrm{Sing}(\Theta/\mathrm{GL}_d(\bR))$$
realising to a map $\mathcal{E}^\Theta \to |\mathrm{Sing}(\Theta/\mathrm{GL}_d(\bR))|$. The orbit space $\Theta/\mathrm{GL}_d(\bR)$ need not be well behaved, and we would like to replace it by the homotopy orbit space $B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\bR) := (E\mathrm{GL}_d(\bR) \times \Theta)/\mathrm{GL}_d(\bR)$. Taking homotopy orbits of the map $\Theta \to \{*\}$ yields a map $\theta : B \to B\mathrm{GL}_d(\bR)$. Repeating the construction above with $E\mathrm{GL}_d(\bR) \times \Theta$ instead of $\Theta$ allows us to construct a zig-zag
\begin{equation*}
\mathcal{E}^\Theta \overset{\simeq}\longleftarrow \mathcal{E}^{E \mathrm{GL}_d(\bR) \times \Theta} \overset{\ell}\lra |\mathrm{Sing}(B)| \overset{\mathrm{ev}}\lra B \overset{\theta}\lra B\mathrm{GL}_d(\bR).
\end{equation*}
Slightly less precisely, we shall summarise this situation as a diagram
\begin{equation}\label{eq:25}
\begin{tikzcd}
E \dar[']{\pi} \rar & \mathcal{E}^\Theta \dar{\pi^\Theta} \rar{\ell} & B \rar{\theta}& B\mathrm{GL}_d(\bR)
\\
X \rar & \mathcal{M}^\Theta
\end{tikzcd}
\end{equation}
where the square is homotopy cartesian. The vector bundle classified by the composition $\theta\circ\ell: \mathcal{E}^\Theta \to B\mathrm{GL}_d(\bR)$ is a universal instance of $T_\pi E$, and shall be denoted $T_\pi \mathcal{E}^\Theta$. The factorisation through $\theta$ gives an equivariant map $\mathrm{st}r^\Theta : \mathbf{F}r(T_\pi \mathcal{E}^\Theta) \to \Theta$, a universal instance of $\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$.
\subsection{Path connected classifying spaces}
The bundles classified so far are inconveniently general. For
example, we have not made any restrictions on the diffeomorphism type
of the fibres of $\pi: E \to X$. Hence, if e.g.\ $\Theta = \{\ast\}$,
the set $\pi_0(\mathcal{M}^\Theta)$ is in bijection with the set of
diffeomorphism classes of compact smooth $d$-manifolds without
boundary, a countably infinite set for any $d \geq 0$. The homotopy
type of $\mathcal{M}^\Theta$ then encodes at once the classification of
smooth manifolds up to diffeomorphism and the classification of smooth
bundles, because it is the ``moduli space'' (or ``$\infty$-groupoid'')
of \emph{all} $d$-manifolds.
We shall often study one path component of
$\mathcal{M}^\Theta$ at a time, which corresponds to fixing the
concordance class of the fibres of the classified bundles. We
introduce the following notation.
\begin{definition}
Let $\Theta$ be a space with $\mathrm{GL}_d(\bR)$ action, $W$ be a
compact $d$-manifold without boundary, and
$\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$ be an equivariant map. Considering this as a family over a point yields a $[(W,\mathrm{st}r_W)] \in \pi_0 \mathcal{M}^\Theta$, and we shall write
$\mathcal{M}^\Theta(W,\mathrm{st}r_W) \subset \mathcal{M}^\Theta$\mathrm{ind}ex{moduli space!of $\Theta$-manifolds} for the
path component containing $(W,\mathrm{st}r_W)$. This path component is a classifying space
for smooth fibre bundles $\pi: E \to X$ with structure
$\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$, whose restriction to any point
$\{x\} \subset X$ is concordant to $(W,\mathrm{st}r_W)$.
\end{definition}
\begin{remark}\label{remark:special-cases}
In the special case $\Theta = \{\ast\}$ the maps $\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$ are irrelevant, and in this case we shall write simply $\mathcal{M}(W)$. This space classifies smooth bundles $\pi: E \to X$ with fibres diffeomorphic to $W$ with no further structure, and we have the weak equivalence
\begin{equation*}
\mathcal{M}(W) \simeq B\Diff(W),
\end{equation*}
where $\Diff(W)$ is the diffeomorphism group\mathrm{ind}ex{diffeomorphism group} of $W$ in the $C^\infty$ topology. Similarly, if $\Theta = \mathbbm{Z}^\times$ with the orientation action and $W$ is given an orientation $\mathrm{st}r_W$, we shall write $\mathcal{M}^\mathrm{or}(W)$ for $\mathcal{M}^\Theta(W,\mathrm{st}r_W)$ and there is a weak equivalence
\begin{equation*}
\mathcal{M}^\mathrm{or}(W) \simeq B\Diff^\mathrm{or}(W),
\end{equation*}
where $\Diff^\mathrm{or}(W) \subset \Diff(W)$ is the subgroup of orientation preserving diffeomorphisms. For general $\Theta$, the homotopy type is described as the Borel construction
\begin{equation*}
\mathcal{M}^\Theta(W,\mathrm{st}r_W) \simeq
\frac{\{\text{$\mathrm{st}r: \mathbf{F}r(TW) \to \Theta$, equivariantly homotopic to
$\mathrm{st}r_W$}\}}
{\left(\substack{\text{topological group of diffeomorphisms $W
\to W$} \\ \text{preserving the
equivariant homotopy
class of $\mathrm{st}r_W$}}\right)}.
\end{equation*}
See \cite[Definition 1.5]{GR-W2}, \cite[Section 1.1]{GR-W2}, or \cite[Section 1.1]{GR-W4} for further discussion of this point of view.
\end{remark}
\section{Characteristic classes}
\label{sec:char-class-XXX}
The main topic of this article is the study of characteristic classes
of the sort of bundles described above, i.e.\ the calculation of the
cohomology ring of the classifying spaces
$\mathcal{M}^\Theta(W,\mathrm{st}r_W)$. We first recall the conclusions in
rational cohomology, which are easier to state and often gives an
explicit {formula} for the ring of characteristic classes.
\subsection{Characteristic classes}
\label{sec:char-class-1}
As before, let $B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$ denote the Borel construction. For a smooth bundle $\pi: E \to X$ with $\Theta$-structure
$\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$, we may form the Borel construction
with $\mathrm{GL}_d(\bR)$ and obtain a correspondence, i.e.\ a diagram
\begin{equation}\label{eq:1}
X \overset{\pi}\longleftarrow E \overset{\simeq}\longleftarrow \mathbf{F}r(T_\pi E)\backslash\!\!\backslashd
\mathrm{GL}_d(\bR) \xrightarrow{\mathrm{st}r\backslash\!\!\backslashd \mathrm{GL}_d(\bR)} B.
\end{equation}
We write $\ell: E \to B$ for the homotopy class of maps associated to $\mathrm{st}r\backslash\!\!\backslashd \mathrm{GL}_d(\bR)$. We shall let $\mathbbm{Z}^{w_1}$ denote the coefficient system on $B$ arising from
the non-trivial action of $\pi_0(\mathrm{GL}_d(\bR)) = \mathbbm{Z}^\times$ on $\mathbbm{Z}$, and
let $A^{w_1} = A \otimes \mathbbm{Z}^{w_1}$ for an abelian group $A$, and we shall use the same notation for these coefficient systems pulled back to $E$ along~(\ref{eq:1}). Then we have a homomorphism
\begin{equation}
\label{eq:20}
\ell^*: H^{k+d}(B;A^{w_1}) \lra H^{k+d}(E;A^{w_1}).
\end{equation}
We also have a
\emph{fibre integration}\mathrm{ind}ex{fibre integration} homomorphism
\begin{equation}
\label{eq:17}
\int_\pi: H^{k+d}(E;A^{w_1}) \lra H^k(X;A),
\end{equation}
where $d$ is the dimension of the fibres of $\pi$. It may be
defined e.g.\ as the composition $H^{k+d}(E;A^{w_1}) \twoheadrightarrow E_\infty^{k,d} \subset E_2^{k,d}$ in the Serre spectral sequence for the fibration $\pi$, or by a Pontryagin--Thom\mathrm{ind}ex{Pontryagin--Thom construction} construction as in \S\ref{sec:mmm-class-gener} below.
Given a class $c \in H^{d + k}(B;A^{w_1})$, we may combine~(\ref{eq:20}) and~(\ref{eq:17}) and define the \emph{Miller--Morita--Mumford classes}\mathrm{ind}ex{Miller--Morita--Mumford class}\mathrm{ind}ex{MMM-class} (or MMM-classes) by the push-pull formula
\begin{equation*}
\kappa_c(\pi) = \int_\pi \ell^*c \in H^k(X;A).
\end{equation*}
This class is easily seen to be natural with respect to pullback along
smooth maps $X' \to X$, and in fact comes from a universal class
\begin{equation*}
\kappa_c \in H^k(\mathcal{M}^\Theta(W,\mathrm{st}r_W);A)
\end{equation*}
for any $(W,\mathrm{st}r_W)$, any $A$, and any $c \in H^{k+d}(B;A^{w_1})$, defined by the analogous push-pull formula in the universal instance~(\ref{eq:25}).
In the special case $k = 0$ and $X = \{\ast\}$, the definition of
$\kappa_c \in H^0(X;A) = A$ simply reproduces the usual
\emph{characteristic numbers},\mathrm{ind}ex{characteristic number} c.f.\ \cite[\S16]{MS}. For example, if $d = 2n$ and $e$ comes from the Euler class in $H^d(B\mathrm{GL}_d(\mathbb{R});\mathbb{Z}^{w_1})$, then $\kappa_e \in H^0(X;\mathbb{Z}) = \mathbb{Z}$ is the Euler characteristic of $W$. Henceforth we shall mostly be interested in the case $k > 0$.
If $A=\mathbbm{k}$ is a field, we may combine with cup product to get a map of graded rings
\begin{equation}\label{eq:2}
\mathbbm{k}[\kappa_c \mid c \in \text{basis of $H^{>d}(B;\mathbbm{k}^{w_1})$}] \lra H^*(\mathcal{M}^\Theta(W,\mathrm{st}r_W);\mathbbm{k}),
\end{equation}
whose domain is the free graded-commutative $\mathbbm{k}$-algebra on symbols $\kappa_c$, one for each element $c$ in a chosen basis (or more invariantly, on a degree-shifted copy of the graded $\mathbbm{k}$-vector space $H^{>d}(B;\mathbbm{k}^{w_1})$).
\subsection{Genus}\label{sec:genus}
All of our results will hold in a range of degrees depending on \emph{genus}, a numerical invariant that we first introduce.
Assume that $d = 2n$ and let a $\mathrm{GL}_{2n}(\bR)$-space $\Theta$ be
given.
We shall assume that $n > 0$ and that the homotopy orbit space
$\Theta\backslash\!\!\backslashd \mathrm{GL}_{2n}(\bR)$ is connected, i.e.\ that
$\pi_0(\mathrm{GL}_{2n}(\bR)) = \mathbbm{Z}^\times$ acts transitively on
$\pi_0(\Theta)$.
The genus will be defined in terms of the manifold obtained from $S^n \times S^n$ by removing a point, which plays a special role in this theory. Up to diffeomorphism this manifold may be obtained
as a pushout
\begin{equation}\label{eq:26}
S^n \times \bR^n \hookleftarrow \bR^n \times \bR^n \hookrightarrow \bR^n
\times S^n,
\end{equation}
where the embeddings are induced by a choice of
coordinate chart $\bR^n \hookrightarrow S^n$. Following
\cite[Definition 1.3]{GR-W3} a $\Theta$-structure on $S^n \times \bR^n$ shall be
called \emph{admissible} if it ``bounds a disk'', i.e.\ is (equivariantly) homotopic
to a structure that extends over some embedding
$S^n \times \bR^n \hookrightarrow \bR^{2n}$. Note that this is automatic if $\pi_n(\Theta) = 0$ for some basepoint.
A structure on
$S^n \times S^n \setminus \{\ast\}$ is admissible if the
restriction to each piece of the gluing~(\ref{eq:26}) is admissible.
\begin{definition}
Assume $d = 2n > 0$ and that $W$ is connected. The \emph{genus}\mathrm{ind}ex{genus} $g(W, \mathrm{st}r_W)$
of a $\Theta$-manifold $(W,\mathrm{st}r_W)$ is the maximal number of disjoint embeddings
$j: S^n \times S^n \setminus \{\ast\} \to W$ such that $j^*\mathrm{st}r_W$ is
admissible.
\end{definition}
This appeared as \cite[Definition 1.3]{GR-W3}. For example, when
$n = 1$ and $\Theta = \pi_0(\mathrm{GL}_2(\bR)) = \mathbb{Z}^\times$, this is
precisely the usual genus of an oriented connected 2-manifold. The admissibility condition may be illustrated by the case $\Theta = \mathrm{GL}_2(\bR)$, corresponding to framings on 2-manifolds. The Lie group framing on $\Sigma = S^1 \times S^1$ satisfies $g(\Sigma,\rho_\mathrm{Lie}) = 0$, but there exist other framings $\rho$ for which $g(\Sigma,\rho) = 1$.
In \S\ref{sec:disc-moduli-spac} below we shall explain how to determine a lower bound on the number $g(W,\mathrm{st}r_W)$ in terms of more accessible invariants.
\subsection{Main theorem in rational cohomology}
\label{sec:main-theor-rati}
The main results of \cite{GR-W2, GR-W3, GR-W4} imply that the ring
homomorphism homomorphism~(\ref{eq:2}) is often an isomorphism in a
range of degrees when $\mathbbm{k} = \mathbb{Q}$.
We explain the statement.
\begin{definition}
Assume $\Theta\backslash\!\!\backslashd \mathrm{GL}_{2n}(\bR)$ is connected. We say $\Theta$ is
\emph{spherical}\mathrm{ind}ex{$\Theta$-structure!spherical} if $S^{2n}$ admits a $\Theta$-structure, i.e.\ if
there exists an equivariant map $\mathbf{F}r(TS^{2n}) \to \Theta$.
\end{definition}
The condition is equivalent to existence of an $\mathrm{O}(2n)$-equivariant map
$\mathrm{O}(2n+1) \to \Theta$. This obviously holds if the $\mathrm{GL}_{2n}(\bR)$-action on
$\Theta$ admits an extension to an action of $\mathrm{GL}_{2n+1}(\bR)$ which
holds in many cases, e.g.\ $\Theta = \{\pm 1\}$ with the orientation
action. See \cite[Section 5.1]{GR-W2} for more information about this
condition.
\begin{theorem}\label{thm:main-cohomological}
Let $d = 2n > 4$, $W$ be a closed simply-connected $d$-manifold, and $\mathrm{st}r_W: \mathbf{F}r(W) \to \Theta$ be a $\Theta$-structure which is $n$-connected (or, equivalently, such that the associated $\ell_W : W \to B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\mathbb{R})$ is $n$-connected). Equip $B$ with the local system $\mathbb{Q}^{w_1}$ as above, and assume that $H^{k+d}(B;\mathbb{Q}^{w_1})$ is finite dimensional for each $k \geq 1$. Then the ring homomorphism
\begin{equation*}
\mathbb{Q}[\kappa_c \mid c \in \text{basis of $H^{>d}(B;\mathbb{Q}^{w_1})$}] \lra H^*(\mathcal{M}^\Theta(W,\mathrm{st}r_W);\mathbb{Q}),
\end{equation*}
as in~(\ref{eq:2}) is an isomorphism in cohomological degrees $\leq (g(W, \mathrm{st}r_W)-4)/3$. If in addition $\Theta$ is spherical, then the range may be improved
to $ \leq (g(W, \mathrm{st}r_W)-3)/2$.
\end{theorem}
\subsection{Estimating genus}
\label{sec:disc-moduli-spac}
To apply Theorem~\ref{thm:main-cohomological} we must calculate the invariant $g(W,\mathrm{st}r_W)$, or at least be able to give useful lower bounds for it. This section explains a general such lower bound, under the assumptions that $d = 2n > 4$, and that the homotopy orbit space $B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\mathbb{R})$ is simply-connected. We may then choose an equivariant map $\Theta \to \mathbbm{Z}^\times$ by which any $\Theta$-structure induces an orientation, and we shall assume given such a map.
There is an obvious upper bound for $g(W,\mathrm{st}r_W)$: it is certainly no
larger than the number of hyperbolic summands in $H_n(W;\mathbbm{Z})$ equipped
with its intersection form. For odd $n$ this is in turn no more than
$b_n/2$ and for even $n$ it is no more than $\min(b_n^+,b_n^-)$, where
we write $b_n$ for the middle Betti number of $W$ and in the even case,
$b_n = b_n^+ + b_n^-$ for its splitting into positive and negative
parts. More usefully, \cite[Remark 7.16]{GR-W3} gives the following \emph{lower}
bound on genus.
\begin{theorem}
Assume $d = 2n > 4$, that the homotopy orbit space
$B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$ is simply-connected, and that
$\ell_W: W \to B$ (or equivalently $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$) is
$n$-connected. Write $g^a(W) = \min(b_n^+,b_n^-)$ for $n$
even and $g^a(W) = b_n/2$ for $n$ odd. Then
\begin{equation}\label{eq:15}
g^a(W) - c \leq g(W,\mathrm{st}r_W) \leq g^a(W),
\end{equation}
with $c = 1+e$, where $e$ is the minimal number of generators of
the abelian group $H_n(B;\mathbbm{Z})$. If $n$ is even or if $n \in \{3,7\}$ then one may take
$c = e$.
\end{theorem}
\begin{remark}
Let us briefly point out that the estimate~(\ref{eq:15}) may be
expressed using characteristic numbers. Indeed, writing $b_i = b_i(B) = b_i(W)$ for $i = 1, \dots, n-1$, we have
\begin{equation}\label{eq:18}
g^a(W) = (-1)^n\big(\chi(W)/2 - \sum_{i = 0}^{n-1} (-1)^i b_i\big) - |\sigma(W)|/2,
\end{equation}
where $\chi(W) = \int_W e(TW)$ is the Euler characteristic and
$\sigma(W) = \int_W \mathcal{L}(TW)$ is the signature (where we
write $\sigma(W) = 0$ when $n$ is odd).
\end{remark}
\section{General versions of main results}
\label{sec:gener-vers-main}
For some purposes, the rational cohomology statement in
Theorem~\ref{thm:main-cohomological} suffices, but there are several homotopy theoretic strengthenings and variations given in \cite{GR-W4}, which we now explain.
\subsection{Stable homotopy enhancement}
To state a more robust version of Theorem~\ref{thm:main-cohomological}
above, we must first introduce a space $\Omega^\infty MT\Theta$
associated to the $\mathrm{GL}_d(\bR)$-space $\Theta$. The map $B = \Theta\backslash\!\!\backslashd \mathrm{GL}_d(\bR) \to B\mathrm{GL}_d(\bR)$ classifies a
$d$-dimensional real vector bundle over $B$, and we shall write
$MT\Theta$ for the Thom spectrum of its virtual inverse and
$\Omega^\infty MT\Theta$ for the corresponding infinite loop
space. The following result is a restatement of \cite[Corollary 1.7]{GR-W4}.
\begin{theorem}\label{thm:homotopical}
Let $d = 2n > 4$, $W$ be a closed simply-connected $d$-manifold, and $\mathrm{st}r_W: \mathbf{F}r(W) \to \Theta$ be an
$n$-connected equivariant map. Then there is a map
\begin{equation}\label{eq:5}
\alpha : \mathcal{M}^\Theta(W,\mathrm{st}r_W) \lra \Omega^\infty MT\Theta,
\end{equation}
inducing an isomorphism in integral homology onto the path component that it hits, in degrees $\leq (g(W, \mathrm{st}r_W) - 4)/3$.
If in addition $\Theta$ is spherical, then~(\ref{eq:5}) induces an
isomorphism in homology in degrees $\leq (g(W, \mathrm{st}r_W)-3)/2$.
\end{theorem}
The statement proved in \cite{GR-W4} is in fact stronger: the map $\alpha$ induces an isomorphism in homology with local coefficients in degrees up to $(g(W, \mathrm{st}r_W) - 4)/3$. We say that the map is \emph{acyclic} in this range of degrees. (The induced map in
$\pi_1$ is of course far from an isomorphism.)
\begin{remark}\label{rem:RatCalc}
Let us also briefly recall why
Theorem~\ref{thm:main-cohomological} is a consequence of
Theorem~\ref{thm:homotopical}: under the Thom isomorphism $H^{k+d}(B;\mathbb{Q}^{w_1}) \cong H^k(MT\Theta;\mathbb{Q})$ each class $c \in H^{k+d}(B;\mathbb{Q}^{w_1})$ may be
represented by a spectrum map $MT\Theta \to \Sigma^k H\mathbb{Q}$. If we
choose a rational basis
$\mathcal{B}_k \subset H^{k+d}(B;\mathbb{Q}^{w_1}) \cong H^k(MT\Theta;\mathbb{Q})$ and represent each basis element by a spectrum map $MT\Theta \to \Sigma^k H\mathbb{Q}$, we obtain
\begin{equation*}
MT\Theta \lra \prod_{k = 1}^\infty \prod_{c \in \mathcal{B}_k} \Sigma^k H\mathbb{Q}
\end{equation*}
which induces isomorphisms in rational homology and hence in
rationalised homotopy in positive degrees, at least if each $\mathcal{B}_k$ is finite in which case the product in the codomain may be replaced by the wedge.
It follows that the induced map of infinite loop spaces
\begin{equation*}
\Omega^\infty MT\Theta \lra \prod_{k = 1}^\infty \prod_{\mathcal{B}_k} K(\mathbb{Q},k)
\end{equation*}
induces an equivalence in rationalised homotopy groups in positive
degrees, hence in rational cohomology when restricted to any path
component of its domain.
\end{remark}
\subsection{MMM-classes in generalised cohomology}\label{sec:mmm-class-gener}
There is a preferred map~(\ref{eq:5}) in Theorem
\ref{thm:homotopical}. Following \cite{MT}, in \cite[Remark 1.11]{GR-W2} we gave an
explicit point-set model. Here we shall explain the map in a conceptual but somewhat
informal way. It is obtained from the following two ingredients.
\begin{enumerate}[(i)]
\item A smooth $d$-manifold $W$ and an equivariant map
$\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$ induces a continuous map
$$\ell_W = \mathrm{st}r_W\backslash\!\!\backslashd\mathrm{GL}_d(\bR): W \simeq \mathbf{F}r(TW)\backslash\!\!\backslashd\mathrm{GL}_d(\bR) \lra B = \Theta
\backslash\!\!\backslashd \mathrm{GL}_d(\bR)$$
under which the canonical bundle $\gamma$ on
$B\mathrm{GL}_d(\bR) = \ast\backslash\!\!\backslashd\mathrm{GL}_d(\bR)$ is pulled back to $TW$. By
passing to Thom spectra of inverse bundles one gets a map
\begin{equation}\label{eq:9}
W^{-TW} \lra B^{-\gamma} = MT\Theta,
\end{equation}
in the stable homotopy category.
\item If $W$ is a closed manifold there is a canonical map
\begin{equation}\label{eq:8}
S^0 \lra W^{-TW}
\end{equation}
which is Spanier--Whitehead dual\mathrm{ind}ex{duality!Spanier--Whitehead} to the canonical map $W \to \{\ast\}$ under Atiyah duality $D(W_+) \simeq W^{-TW}$. The actual map of spectra depends on certain choices, but in the right
setup these choices form a contractible space. (For example, one may choose
an embedding $W \hookrightarrow \bR^\infty$ and get the map~(\ref{eq:8}) by
the Pontryagin--Thom\mathrm{ind}ex{Pontryagin--Thom construction} collapse construction.)
\end{enumerate}
If $W$ is a smooth closed $d$-manifold and $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$
is an equivariant map, we may compose~(\ref{eq:8}) and~(\ref{eq:9}) to
get a map of spectra $S^0 \to MT\Theta$, i.e.\ a point
in $\Omega^\infty MT\Theta$. We shall write $\alpha(W,\mathrm{st}r_W) \in \Omega^\infty MT\Theta$ for this point, and write
\begin{equation*}
\Omega^\infty_{[W,\mathrm{st}r_W]} MT\Theta \subset \Omega^\infty MT\Theta
\end{equation*}
for the path component containing $\alpha(W,\mathrm{st}r_W)$. This is the path component that the map~(\ref{eq:5}) lands in.
The map~(\ref{eq:5}) is given by a parametrised version of this
construction: given a family
$(\pi: E \to X, \mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta)$ as in
Definition~\ref{def:family} over some base manifold $X$, it associates
a continuous map $\alpha: X \to \Omega^\infty MT\Theta$. Said differently, it
comes from a composition of spectrum maps, the parametrised analogues
of~(\ref{eq:8}) and~(\ref{eq:9}) respectively,
\begin{equation}
\label{eq:16}
\begin{aligned}
\Sigma^\infty_+ X & \lra E^{-T_\pi E}\\
E^{-T_\pi E} &\lra MT\Theta.
\end{aligned}
\end{equation}
In any case,~(\ref{eq:5}) is a universal version of this construction.
\begin{remark}
Applying spectrum homology and the Thom isomorphism to the first map in~(\ref{eq:16}), we get precisely the fibre integration homomorphism~(\ref{eq:17}), while the second gives~(\ref{eq:20}). This explains the connection to the characteristic classes in \S\ref{sec:char-class-1}.
\end{remark}
\begin{remark}\label{remark:disconnected}
There is an improvement to Theorem~\ref{thm:homotopical}, in which the domain of
(\ref{eq:5}) is replaced by a disconnected space
$\mathcal{M}^\Theta_n$ containing $\mathcal{M}^\Theta(W,\mathrm{st}r_W)$ as
one of its path components, cf.\ \S\ref{sec:two-fiber-sequences} below and \cite[Section
8]{GR-W4}. In this improved formulation, the number
$g(W,\mathrm{st}r_W) \in \mathbb{N}$ appearing in Theorem~\ref{thm:homotopical} is replaced by a function
\begin{equation*}
\pi_0(\mathcal{M}^\Theta_n) \overset{\alpha_*}\lra \pi_0(MT\Theta) \overset{\bar{g}^\Theta}\lra \mathbb{Z},
\end{equation*}
whose value on the path component $\mathcal{M}^\Theta(W,\mathrm{st}r_W)$ is
the \emph{stable genus}, defined in \cite[Section 5]{GR-W3} and
\cite[Section 1.3]{GR-W4}. It is at least $g(W,\mathrm{st}r_W)$.
If we write $\chi: \pi_0(MT\Theta) \to \mathbbm{Z}$ and
$\sigma: \pi_0(MT\Theta) \to \mathbbm{Z}$ be the homomorphisms arising from
the Euler class and (for even $n$) the Hirzebruch class,
then~(\ref{eq:18}) defines a function
$g^a: \pi_0(MT\Theta) \to \mathbb{N}$. In terms of this function, the
estimate~(\ref{eq:15}) also holds for $\bar{g}^\Theta$.
\end{remark}
\subsection{General tangential structures}
The requirement in Theorems \ref{thm:main-cohomological} and \ref{thm:homotopical} that $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$ be $n$-connected
appears quite restrictive at first sight. For example, it usually
rules out the interesting special cases $\Theta = \{\ast\}$ and
$\Theta = \{\pm 1\}$ from Remark~\ref{remark:special-cases}, so that
the cohomology of $B\Diff(W)$ and $B\Diff^\mathrm{or}(W)$ are not immediately calculated by Theorem~\ref{thm:homotopical}.
A more generally useful version of Theorem~\ref{thm:homotopical},
which holds without the connectivity assumption, may be deduced by a
rather formal homotopy theoretic trick, based on the observation that
the map~(\ref{eq:5}) is functorial in the $\mathrm{GL}_d(\bR)$-space $\Theta$.
In particular, any map $\Theta \to \Theta$ induces a self-map of
$\Omega^\infty MT\Theta$. The following result is \cite[Corollary
1.9]{GR-W4}.
\begin{theorem}\label{thm:general-str}
For $d = 2n > 4$ and $\Lambda$ a $\mathrm{GL}_d(\bR)$-space, let $W$ be a
closed simply-connected smooth $d$-dimensional manifold, and
$\lambda_W: \mathbf{F}r(TW) \to \Lambda$ be an equivariant map. Choose an equivariant Moore--Postnikov\mathrm{ind}ex{Moore--Postnikov stage} $n$-stage
\begin{equation}\label{eq:MPFactorisation}
\lambda_W : \mathbf{F}r(TW) \overset{\mathrm{st}r_W}\lra \Theta \overset{u}\lra \Lambda,
\end{equation}
i.e.\ a factorisation where $u$ is $n$-co-connected equivariant fibration and $\mathrm{st}r_W$ is an $n$-connected equivariant cofibration, and write $\mathrm{hAut}(u)$ for the group-like topological monoid consisting of equivariant weak equivalences $\Theta \to \Theta$ over $\Lambda$.
This topological monoid acts on $\Omega^\infty MT\Theta$, and there
is a continuous map
\begin{equation}\label{eq:11}
\alpha: \mathcal{M}^\Lambda(W,\lambda_M) \lra
(\Omega^\infty MT\Theta)\backslash\!\!\backslashd \mathrm{hAut}(u).
\end{equation}
which, when regarded as a map onto the path component that it hits,
induces an isomorphism in homology with local coefficients in
degrees $\leq (g(W,\lambda_W)-4)/3$, and if $\Theta$ is spherical it
induces an isomorphism in integral homology in degrees
$\leq (g(W,\lambda_W) - 3)/2$.
\end{theorem}
We emphasise that the homotopy orbit space is formed in the category
of spaces, not of infinite loop spaces (there is a comparison map
$(\Omega^\infty MT\Theta)\backslash\!\!\backslashd \mathrm{hAut}(u) \to
\Omega^\infty((MT\Theta) \backslash\!\!\backslashd \mathrm{hAut}(u))$ but it is not a weak
equivalence and we shall not need its codomain). Equivariant factorisations \eqref{eq:MPFactorisation} always exist, and are unique up to contractible choice.
Let us also point out that the group $\pi_0(\mathrm{hAut}(u))$ likely
acts non-trivially on $\pi_0(MT\Theta)$. The construction of the
map~(\ref{eq:11}), outlined in \S\ref{sec:two-fiber-sequences}
below, together with the orbit-stabiliser theorem, lets us re-write
the relevant path component of the codomain of~(\ref{eq:11}) in the
following way, which is how the theorem above is typically used in
practice.
\begin{corollary}\label{cor:general-str}
Let $d$, $\Lambda$, $W$, $\lambda_W$, $\Theta$, $\mathrm{st}r_W$, and $u$ be as
in Theorem~\ref{thm:general-str}, and write
\begin{equation*}
\mathrm{hAut}(u)_{[W,\mathrm{st}r_W]} \subset \mathrm{hAut}(u)
\end{equation*}
for the submonoid stabilising the element
$[W,\mathrm{st}r_W] \in \pi_0(MT\Theta)$ defined in
\S\ref{sec:mmm-class-gener}. The action of this submonoid on
$\Omega^\infty MT\Theta$ restricts to an action on the path
component $\Omega^\infty_{[W,\mathrm{st}r_W]} MT\Theta$ defined in
\S\ref{sec:mmm-class-gener}, and~(\ref{eq:11}) factors through
a map of path connected spaces
\begin{equation*}
\alpha: \mathcal{M}^\Lambda(W,\lambda_W) \lra (\Omega^\infty_{[W,\mathrm{st}r_W]} MT\Theta)\backslash\!\!\backslashd(\mathrm{hAut}(u)_{[W,\mathrm{st}r_W]})
\end{equation*}
which induces an isomorphism on homology in a range, as in
Theorem~\ref{thm:general-str}. \qed
\end{corollary}
\begin{remark}
In both Theorem~\ref{thm:general-str} and Corollary~\ref{cor:general-str} above, the range is expressed in terms of $g(W,\lambda_W)$ but as explained in \cite[Lemma 9.4]{GR-W4} this is equal to $g(W,\mathrm{st}r_W)$ when $\lambda_W$ is factored equivariantly as an $n$-connected map $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$ followed by an $n$-co-connected map $u: \Theta \to \Lambda$. Hence the estimates in \S\ref{sec:disc-moduli-spac} apply, when $e$ is the minimal number of generators for the abelian group $H_n(\Theta \backslash\!\!\backslashd \mathrm{GL}_{2n}(\bR))$. The value of $e$ likely depends on the map $\lambda_W: \mathbf{F}r(TW) \to \Lambda$, even if $W$ and $\Lambda$ are fixed.
\end{remark}
\begin{remark}\label{rem:finite-calculation}
The fact that $u$ is $n$-co-connected implies that
$\mathrm{hAut}(u)$ is an $(n-1)$-type. Hence it is in some sense a finite problem to
determine and describe $\mathrm{hAut}(u)$: finitely many homotopy
groups $\pi_0, \dots, \pi_{n-1}$ and finitely many $k$-invariants.
This finiteness is one of the conceptual advantages of our approach
to $\mathcal{M}(W) \simeq B\Diff(W)$, over the more classical method which
at first gives a formula for the \emph{structure space}
$\mathcal{S}(W) \simeq G(W) / \widetilde{\Diff}(W)$, where $G(W)$ is the monoid of homotopy equivalences from $W$ to itself and
$\widetilde{\Diff}(W)$ is the block diffeomorphism group.\mathrm{ind}ex{diffeomorphism group!block} In that method, one subsequently
has to study the difference between $\Diff(W)$\mathrm{ind}ex{diffeomorphism group} and
$\widetilde{\Diff(W)}$, but also has to take homotopy orbits by the
monoid $G(W)$ of homotopy automorphisms of $W$. While this last
step is in some sense ``purely homotopy theory'', it is in practice
very difficult to get a good handle on $G(W)$, even when $W$ is
relatively simple and even when one is working up to rational
equivalence. See the work of Berglund and Madsen \cite{BerglundMadsen, BerglundMadsenII} for a recent example.
\end{remark}
\subsection{Two fibre sequences}
\label{sec:two-fiber-sequences}
In \cite[Section 9]{GR-W4}, Theorem~\ref{thm:general-str} is deduced
from the special case given in Theorem~\ref{thm:homotopical} by a rather formal
argument: the homology equivalence in Theorem~\ref{thm:homotopical} is natural
in the $\mathrm{GL}_{2n}(\bR)$-space $\Theta$, and hence induces a homology
equivalence by taking homotopy colimit over any diagram in
$\mathrm{GL}_{2n}(\bR)$-spaces; in particular one may form homotopy orbits by
$\mathrm{hAut}(u: \Theta \to \Lambda)$, and
Theorem~\ref{thm:general-str} is deduced by identifying the resulting
map of homotopy orbit spaces with~(\ref{eq:11}). The two fibre sequences
arising from these homotopy orbit constructions, see diagram~(\ref{eq:21}) below, are important for carrying out calculations in concrete examples, and hence we recall
this story in slightly more detail.
\begin{definition}
Let $u: \Theta \to \Lambda$ be an equivariant $n$-co-connected
fibration.
\begin{enumerate}[(i)]
\item Let $\mathcal{M}^\Theta_n \subset \mathcal{M}^\Theta$ be the
union of those path components $\mathcal{M}^\Theta(W,\mathrm{st}r_W)$ for
which $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$ is $n$-connected.
\item Let $\mathcal{M}^\Lambda_u \subset \mathcal{M}^\Lambda$ be the
union of those path components $\mathcal{M}^\Lambda(W,\lambda_W)$ for which $\lambda_W$
\emph{admits} a factorisation through an $n$-connected equivariant
map $\mathrm{st}r_W : \mathbf{F}r(TW) \to \Theta$.
\end{enumerate}
\end{definition}
There is an obvious forgetful map
$\mathcal{M}^\Theta_n \to \mathcal{M}^\Lambda_u$ given by composing
$\mathrm{st}r_W : \mathbf{F}r(TW) \to \Theta$ with $u$. The monoid $\mathrm{hAut}(u)$ has the correct homotopy type when $\Theta$ is equivariantly cofibrant, which we shall assume. It
acts on $\mathcal{M}^\Theta_n$ by postcomposing
$\mathrm{st}r_W : \mathbf{F}r(TW) \to \Theta$ with self-maps of $\Theta$. This action
commutes with the forgetful map, and induces a map
\begin{equation}\label{eq:13}
(\mathcal{M}^\Theta_n)\backslash\!\!\backslashd \mathrm{hAut}(u) \lra
\mathcal{M}^\Lambda_u.
\end{equation}
The following lemma is proved by elementary homotopy theoretic
methods, cf.\ \cite[Section 9]{GR-W4}.
\begin{lemma}
The map~(\ref{eq:13}) is a weak equivalence. Hence there is an
induced fibre sequence of the form
\begin{equation*}
\mathcal{M}^\Theta_n \lra \mathcal{M}^\Lambda_u \lra B
(\mathrm{hAut}(u)).
\end{equation*}
\end{lemma}
The map $\mathcal{M}^\Theta_n \to \Omega^\infty MT\Theta$ explained in
\S\ref{sec:mmm-class-gener} commutes with the actions of
$\mathrm{hAut}(u)$ and induces a map of fibre sequences
\begin{equation}\label{eq:21}
\begin{tikzcd}
\mathcal{M}^\Theta_n \rar \dar &\mathcal{M}^\Lambda_u
\rar \dar & B(\mathrm{hAut}(u)) \arrow[equals]{d}\\
\Omega^\infty MT\Theta \rar & (\Omega^\infty MT\Theta)\backslash\!\!\backslashd
\mathrm{hAut}(u) \rar & B(\mathrm{hAut}(u)).
\end{tikzcd}
\end{equation}
In the setup of Theorem~\ref{thm:general-str},
$\mathcal{M}^\Theta(W,\mathrm{st}r_W)$ is one path component of
$\mathcal{M}^\Theta_n$, and similarly $\mathcal{M}^\Lambda(W,\lambda_W)$
is one path component of $\mathcal{M}^\Lambda_u$. A slightly stronger version of
Theorem~\ref{thm:homotopical}, which is the statement actually proved in \cite[Section 8]{GR-W4}, shows that the left-most vertical map is acyclic in the
range of degrees indicated in Remark~\ref{remark:disconnected}. Theorem~\ref{thm:general-str} is then deduced by a spectral sequence
comparison argument.
\begin{remark}\label{rem:Kreck}
This formulation has content even at the level of path components. Suppose that $W$ is a simply-connected $2n$-manifold for $2n \geq 6$, $\mathrm{st}r_W : \mathbf{F}r(TW) \to \Theta$ is $n$-connected, and $g(W, \mathrm{st}r_W) \geq 3$. If $(W', \mathrm{st}r_{W'})$ is another such $\Theta$-manifold and
$$\alpha(W, \mathrm{st}r_W) = \alpha(W', \mathrm{st}r_{W'}) \in \pi_0(\Omega^\infty MT\Theta) = \pi_0(MT\Theta)$$
then it follows that $(W, \mathrm{st}r_W)$ and $(W', \mathrm{st}r_{W'})$ lie in the same path-component of $\mathcal{M}^\Theta_n$, i.e.\ there is a diffeomorphism from $W$ to $W'$ which pulls back $\mathrm{st}r_{W'}$ to $\mathrm{st}r_W$ up to homotopy. This recovers a theorem of Kreck \cite[Theorem D]{Kreck}, though our requirement on genus is slightly stronger than Kreck's.
\end{remark}
In practice, one usually calculates the cohomology of $\Omega^\infty_{[W,\mathrm{st}r_W]} MT\Theta$
first, and then uses a spectral sequence to calculate the homology or cohomology of the Borel construction in Corollary~\ref{cor:general-str}, or equivalently (for $g(W, \lambda_W) \geq 3$) one calculates the cohomology of
$\mathcal{M}^\Theta(W,\mathrm{st}r_W)$ and then uses the spectral sequence for the fibre sequence
\begin{equation}\label{eq:22}
\mathcal{M}^\Theta(W, \mathrm{st}r_W)\lra \mathcal{M}^\Lambda(W, \lambda_W) \lra B(\mathrm{hAut}(u)_{[W, \mathrm{st}r_W]}).
\end{equation}
We shall see examples of such calculations in \S\ref{sec:WgRat}, \S\ref{sec:VdRat}, and \S\ref{sec:OtherRat}.
\subsection{$\mathrm{GL}_d(\bR)$-spaces versus spaces over $B\mathrm{O}(d)$}
\label{sec:gl_dr-spaces-versus}
It is well known that the homotopy theory of spaces with action of $\mathrm{GL}_d(\bR)$ is equivalent to the homotopy theory of spaces over $B\mathrm{GL}_d(\bR) \simeq B\mathrm{O}(d)$, where the weak equivalences are the equivariant maps that are weak equivalences of underlying spaces, respectively fibrewise maps that are weak equivalences of underlying spaces. The translation goes via the space $E\mathrm{GL}_d(\bR)$ which simultaneously comes with an action of $\mathrm{GL}_d(\bR)$ and a map to $B\mathrm{GL}_d(\bR)$. Explicitly, given a $\mathrm{GL}_d(\bR)$-space $\Theta$, the Borel construction $B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$ comes with a map $B \to B\mathrm{GL}_d(\bR)$; conversely, given a space $B$ and a map $\theta: B \to B\mathrm{GL}_d(\bR)$ the fibre product $\Theta = E\mathrm{GL}_d(\bR) \times_{B\mathrm{GL}_d(\bR)} B$ comes with an action; these processes are inverse up to (equivariant/fibrewise) weak equivalence, as $E\mathrm{GL}_d(\bR)$ is contractible.
Therefore all of the theorems above that depend on a $\mathrm{GL}_d(\bR)$-space $\Theta$ may be stated in equivalent ways taking as input a space $B$ and a map $\theta: B \to B\mathrm{O}(d)$.\mathrm{ind}ex{$\theta$-structure} In the papers
\cite{GR-W2, GR-W3, GR-W4} we have taken the latter point of view.
In this picture, a $\theta$-structure on a manifold $W$ is a (fibrewise linear) vector bundle map $\hat{\ell}_W : TW \to \theta^* \gamma$, where $\gamma$ denotes the universal vector bundle on $B\mathrm{GL}_d(\bR)$. As in those papers, we shall use the notation
\begin{equation*}
MT\theta = MT\Theta,
\end{equation*}
when $\theta: B \to B\mathrm{O}(d)$ is the map corresponding to the
$\mathrm{GL}_d(\bR)$-space $\Theta$; i.e., $MT\theta = B^{-\theta}$ is the Thom
spectrum of the virtual inverse of the vector bundle classified by
$\theta: B \to B\mathrm{O}(d)$.
We have already seen definitions which may be stated
more directly in terms of $(B,\theta)$ than of
$(\Theta,\text{action})$, e.g.\ the characteristic classes $\kappa_c$
from \S\ref{sec:char-class-1}. The constructions in
Theorem~\ref{thm:general-str} form another such example, which we shall now explain. Given $\Lambda$ and $\lambda_W$ as in the theorem, set
$B' = \Lambda \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$. Up to contractible choice, $\lambda_W$
induces a map $W \to B'$, which one then Moore--Postnikov factors as
\begin{equation*}
W \lra B \lra B',
\end{equation*}
into an $n$-connected cofibration followed by an $n$-co-connected
fibration. In this picture, $\mathrm{hAut}(u)$ is simply the
group-like monoid of those self-maps of $B$ over $B'$ that are weak
equivalences. For this to have to the correct homotopy type $B$ should be fibrant and cofibrant in the category of spaces over $B'$.
In \S\ref{sec:RatCalc} and \S\ref{sec:AbCalc} we will exclusively adopt this point of view.
\subsection{Boundary}
\label{sec:boundary}
A further generalisation, also proved in \cite[Section 9]{GR-W4},
allows the compact manifolds $W$ to have non-empty boundary. The boundary should then be a closed
$(2n-1)$-manifold $P$, which should be equipped with an equivariant map
$\mathrm{st}r_P: \mathbf{F}r(\varepsilon^1 \mathrm{op}lus TP) \to \Theta$. The pair $(P,\mathrm{st}r_P)$
should be fixed and every compact $2n$-manifold in sight should come
with a specified diffeomorphism $\partial W \cong P$ compatible with a
structure $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$.
In terms of classified bundles as in \S\ref{sec:smooth-bundles}
and \S\ref{sec:classifying-spaces}, $\mathcal{F}^\Theta_\bullet$ should be
replaced with the functor $\mathcal{F}_\bullet^{\Theta, P,\mathrm{st}r_P}$ whose value on a smooth manifold
$X$ (without boundary, possibly non-compact) has 0-simplices the
smooth proper maps $\pi: E \to X$ equipped with equivariant maps
$\mathrm{st}r: \mathbf{F}r(T_\pi E) \to \Theta$ and a diffeomorphism $\partial E = X \times P$
such that the restriction of $\mathrm{st}r$ to
$\mathbf{F}r(T_\pi E \vert_{\partial E}) = X \times \mathbf{F}r(TP)$ is equal to the map arising
from $\mathrm{st}r_P$.
This kind of bundle also admits a classifying space, denoted
$\mathcal{N}^\Theta(P,\mathrm{st}r_P)$ and called the \emph{moduli space of
null bordisms}\mathrm{ind}ex{moduli space!of null bordisms} of $(P,\mathrm{st}r_P)$. The subspace defined by the
condition that $\mathrm{st}r_W: \mathbf{F}r(TW) \to \Theta$ be $n$-connected is denoted
$\mathcal{N}^\Theta_n(P,\mathrm{st}r_P)$ and is the \emph{moduli space of
highly connected null bordisms}.\mathrm{ind}ex{moduli space!of highly-connected null bordisms} The path component of
$\mathcal{N}^\Theta(P,\mathrm{st}r_P)$ containing $(W,\mathrm{st}r_W)$ shall be
denoted $\mathcal{M}^\Theta(W,\mathrm{st}r_W)$, as before. Notice that
$(P,\mathrm{st}r_P)$ is determined by $P = \partial W$ and $\mathrm{st}r_W$ by
restricting $\mathrm{st}r_W$ to $TW\vert_P \cong \varepsilon^1 \mathrm{op}lus TP$. These classifying spaces are introduced in \cite[Definition 1.1]{GR-W4} using a similar notation.
Theorem~\ref{thm:homotopical} then has the following direct generalisation, also stated as \cite[Corollary 1.8 and Section 8.4]{GR-W4}.
\begin{theorem}
Let $d = 2n > 4$, let $\Theta$ be a $\mathrm{GL}_d(\bR)$-space,
let $P$ be a closed smooth $(d-1)$-manifold and $\mathrm{st}r_P: \mathbf{F}r(\varepsilon^1 \mathrm{op}lus TP) \to \Theta$ be a $\mathrm{GL}_d(\bR)$-equivariant map. Then there is a map (canonical up to homotopy, see below)
\begin{equation}\label{eq:14}
\alpha: \mathcal{N}^\Theta_n(P,\mathrm{st}r_P) \lra
\Omega_{\alpha(P,\mathrm{st}r_P),0}(\Omega^{\infty-1}MT\Theta),
\end{equation}
where $\Omega_{\alpha(P,\mathrm{st}r_P),0}$ denotes the space of paths
starting at a certain point
$\alpha(P,\mathrm{st}r_P) \in \Omega^{\infty-1}MT\Theta$ and ending at the
basepoint, with the following property.
When restricted to the path component containing a
particular $(W,\mathrm{st}r_W)$, it is a homology equivalence onto the path
component it hits, in degrees up to $(g(W,\mathrm{st}r_W) - 4)/3$ and
possibly with twisted coefficients. If in addition $\Theta$ is
spherical, then~(\ref{eq:14}) induces an isomorphism in homology
with constant coefficients in degrees up to $(g(W,\mathrm{st}r_W) - 3)/2$.
\end{theorem}
Both the point $\alpha(P,\mathrm{st}r_P)$ and the map~(\ref{eq:14}) are
constructed by the procedure in \S\ref{sec:mmm-class-gener}. If
the codomain of~(\ref{eq:14}) is non-empty, it is of course non-canonically homotopy
equivalent to $\Omega^\infty MT\Theta$, but the map most naturally
takes values in the path space. In the special case $P = \emptyset$ we have $\mathcal{N}^\Theta_n(\emptyset) = \mathcal{M}^\Theta_n$, and the map \eqref{eq:14} is the same as the map appearing in~(\ref{eq:21}).
As in \S\ref{sec:two-fiber-sequences}, a version for a general tangential structure $\Lambda$ may be deduced by taking homotopy orbits with respect to the monoid
\begin{equation}\label{eq:19}
\mathrm{hAut}(u \ \mathrm{rel}\ P) = \{\phi \in \mathrm{hAut}(u) \mid
\phi \circ \mathrm{st}r_P = \mathrm{st}r_P\},
\end{equation}
provided $\mathrm{st}r_P: \mathbf{F}r(\varepsilon^1 \mathrm{op}lus TP) \to \Theta$ is an
equivariant cofibration and $u: \Theta \to \Lambda$ is an equivariant
$n$-co-connected fibration, as can be arranged. Homotopy equivalently, factor the induced map $W \to B' = \Lambda \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$ as an $n$-connected cofibration
$W \to B$ followed by an $n$-co-connected fibration $B \to B'$, and
define $\mathrm{hAut}(u \ \mathrm{rel}\ P)$ as the homotopy
equivalences of $B$ over $B'$ and under $P$. We formulate the conclusion.
\begin{theorem}
\label{thm:main-with-boundary}
Let $n$, $d$, $\Lambda$ and $\lambda_W: \mathbf{F}r(TW) \to \Lambda$ be as in Theorem~\ref{thm:general-str}, but allow $W$ to be a compact manifold with boundary $P = \partial W$. Let $\Theta$, $\mathrm{st}r_W$, and $u$ be as in Theorem~\ref{thm:general-str}, and $\mathrm{st}r_P$ denote the restriction of $\mathrm{st}r_W$ to $P$. Then there is a map
\begin{equation}
\alpha: \mathcal{M}^\Lambda(W,\lambda_W) \lra \big(\Omega_{\alpha(P,\mathrm{st}r_P),0} \Omega^{\infty - 1} MT\Theta\big) \backslash\!\!\backslashd \mathrm{hAut}(u \ \mathrm{rel}\ P)
\end{equation}
which, when regarded as a map onto the path component that it hits,
induces an isomorphism in homology in a range of degrees, exactly as
in Theorem~\ref{thm:general-str}.
\end{theorem}
This is \cite[Theorem 9.5]{GR-W4}, and the three lines following its proof. The relevant path component may again be re-written using the orbit-stabiliser theorem, as in Corollary~\ref{cor:general-str}.
\begin{remark}
As in Remark \ref{remark:special-cases}, in the special case $\Theta=\{*\}$ a $\Theta$-structure contains no information and we can simply write $\mathcal{M}(W)$ for $\mathcal{M}^\Theta(W,\mathrm{st}r_W)$. This space classifies smooth fibre bundles with fibres diffeomorphic to $W$ and trivialised boundary, and we have a weak equivalence
$$\mathcal{M}(W) \simeq B\Diff_\partial(W)$$
where $\Diff_\partial(W)$ denotes the group of diffeomorphisms of $W$ which fix an open neighbourhood of the boundary, with the $C^\infty$ topology.
\end{remark}
There are no connectivity assumptions imposed on
$\mathrm{st}r_P: \mathbf{F}r(\varepsilon^1 \mathrm{op}lus TP) \to \Theta$, but if it happens to
be $(n-1)$-connected then the monoid $\mathrm{hAut}(u \ \mathrm{rel}\ P)$
is contractible. More generally we have the following.
\begin{lemma}\label{lem:aHutContr}
If the pair $(W, P)$ is $c$-connected for some $c \leq n-1$, then the monoid $\mathrm{hAut}(u \ \mathrm{rel}\ P)$ is a (non-empty) $(n-c-2)$-type. In particular, it is contractible if $(W,P)$ is $(n-1)$-connected.
\end{lemma}
A familiar special case of this observation is the fact that if a
diffeomorphism of an oriented surface with non-empty boundary is the
identity on the boundary, then the diffeomorphism is automatically
orientation preserving.
\begin{proof}
As before, let us write $B = \Theta \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$ and
$B' = \Lambda \backslash\!\!\backslashd \mathrm{GL}_d(\bR)$. We are then asking for homotopy
automorphisms of $B$ over $u: B \to B'$ and under $\ell_P :P \hookrightarrow B$. By adjunction, to give a nullhomotopy of a map
$f : S^k \to \mathrm{hAut}(u \ \mathrm{rel}\ P)$ is to solve the relative
lifting problem
\begin{equation*}
\begin{tikzcd}
(S^k \times B) \cup_{S^k \times P} (D^{k+1} \times P) \dar \arrow{rr}{\tilde{f} \cup (\ell_{P} \circ \text{proj})} & & B \dar{u}\\
D^{k+1} \times B \rar{\text{proj}} & B \rar{u} & B'.
\end{tikzcd}
\end{equation*}
The map $\ell_P: P \to B$ is $c$-connected because both $P \subset W$ and $\ell_W: W \to B$ are (the latter is even $n$-connected). Thus the pair
\begin{equation*}
(D^{k+1} \times B, (S^k \times B) \cup_{S^k \times P} (D^{k+1} \times P))
\end{equation*}
is $(c+k+1)$-connected. But the map $u: B \to B'$ is
$n$-co-connected, so there are no obstructions to solving this
lifting problem if $c+k+1 \geq n$, i.e.\ $k \geq n - c - 1$. This proves that $\mathrm{hAut}(u\ \mathrm{rel}\ P)$ is an $(n-c-2)$-type, and it is non-empty because it contains the identity map.
\end{proof}
\begin{remark}
Formulating a statement which is valid for manifolds with non-empty boundary is not purely for the purpose of added generality: it is essential for the strategy of proof in all three papers \cite{GR-W2}, \cite{GR-W3}, \cite{GR-W4}. For example, the homological stability results in \cite{GR-W3} are proved by a long \emph{handle induction} argument, in which a compact manifold is decomposed into finitely many \emph{handle attachments}; even if one is mainly interested in closed manifolds, this process will create boundary. Similarly, an important role in both \cite{GR-W2} and \cite{GR-W4} is played by \emph{cobordism categories} as studied in \cite{GMTW}, whose morphisms are manifolds with boundary and composition is gluing along common boundary components. For example, given a $\Theta$-cobordism $(K, \mathrm{st}r_K) : (P, \mathrm{st}r_P) \leadsto (Q, \mathrm{st}r_Q)$ there is a continuous map
$$(K, \mathrm{st}r_K) \cup - : \mathcal{N}^\Theta(P,\mathrm{st}r_P) \lra \mathcal{N}^\Theta(Q,\mathrm{st}r_Q)$$
given by gluing on $(K, \mathrm{st}r_K)$.
\end{remark}
\subsection{Fundamental group}\label{sec:pi1}
The main theorem in either of the three forms given above
(Theorems~\ref{thm:main-cohomological}, \ref{thm:homotopical},
and~\ref{thm:general-str}) assumed the manifolds $W$ were simply-connected, but in fact it suffices that the fundamental groups
$\pi = \pi_1(W,w)$ be \emph{virtually polycyclic},\mathrm{ind}ex{virtually polycyclic} i.e.\ has a subnormal series with finite or cyclic quotients. In this case the \emph{Hirsch length} $h(\pi)$ is the number of infinite cyclic quotients in such a series. The only price to pay is that the ranges of homology equivalence
become offset by a constant depending on $h(\pi)$: the homology
isomorphisms Theorems~\ref{thm:homotopical} and~\ref{thm:general-str}
hold in degrees $\leq (g(W,\mathrm{st}r_W) - (h(\pi) + 5))/2$ with integral
coefficients if $\Theta$ is spherical, and in degrees $\leq (g(W,\mathrm{st}r_W) - (h(\pi) + 6))/3$ with
local coefficients. This generalisation was established by
Friedrich \cite{NinaPaper}.
For arbitrary $\pi$ there is a sense in which the theorems
hold in ``infinite genus'': certain maps become acyclic after taking a
colimit over forming connected sum with $S^n \times S^n$ infinitely
many times. In this form the assumption $2n > 4$ is also unnecessary.
See \cite[Sections 1.2 and 7]{GR-W4} for the statement and proof.
\subsection{Outlook}
\label{sec:outlook}
We have attempted to give an overview of the methods developed in
\cite{GR-W2}, \cite{GR-W3}, \cite{GR-W4}, with an emphasis on the main
results from there as they may be applied in calculations in practice.
This is by no means a survey of everything known, let us briefly
mention some recent developments and applications that we have not covered:
\begin{enumerate}[(i)]
\item These results---in the form of the calculation described in Theorem \ref{thm:Wg1Rat} below---have been used by Weiss \cite{WeissDalian} to prove that $p_n \neq e^2 \in H^{4n}(B\mathrm{STop}(2n);\mathbb{Q})$ for large enough $n$. These methods were later used by Kupers \cite{KupersFin} to establish the finite generation of homotopy groups of $\Diff_\partial(D^d)$ for $d \neq 4,5,7$.
\item These results have been used by Botvinnik, Ebert, and Randal-Williams \cite{BER-W}, Ebert and Randal-Williams \cite{ER-Wpsc}, and Botvinnik, Ebert, and Wraith \cite{BEW} to study the topology of spaces of Riemannian metrics of positive scalar, or Ricci, curvature.
\item These results have been used by Krannich \cite{KrannichExotic} to show that if $\Sigma$ is a homotopy $2n$-sphere\mathrm{ind}ex{homotopy sphere} then $B\Diff(M)$ and $B\Diff(M \# \Sigma)$ have the same homology in the stable range with $\mathbb{Z}[\frac{1}{k}]$-coefficients, where $k$ is the order of $\Sigma$ is the group of homotopy spheres.
\item Progress towards a similar understanding for manifolds of \emph{odd dimension} has been made by Perlmutter \cite{PerlmutterProdSpheres, PerlmutterLink, PerlmutterStabHand}, Botvinnik and Perlmutter \cite{BotvinnikPerlmutter} and Hebestreit and Perlmutter \cite{HebestreitPerlmutter}.
\item Progress towards versions for \emph{topological} and \emph{piecewise linear} manifolds has been made by Gomez-Lopez \cite{LopezThesis}, Kupers \cite{KupersTopStab}, and Gomez-Lopez and Kupers \cite{LopezKupers}.
\item Progress towards versions for \emph{equivariant smooth} manifolds has been made by Galatius and Sz\H{u}cs \cite{SzucsGalatius}.
\item There are analogues of many of the theorems above, when the topological group $\Diff(W)$ is replaced by its \emph{underlying discrete group}, by Nariman \cite{Nariman1, Nariman2, Nariman3}.
\item Progress towards understanding the homotopy equivalences of high genus manifolds have been obtained by Berglund and Madsen \cite{BerglundMadsenII}. At present their results seem to be qualitatively quite different from the results described here.
\end{enumerate}
\section{Rational cohomology calculations}\label{sec:RatCalc}
We have already advertised the feature that the general theory surveyed in \S\ref{sec:char-class-XXX} and \S\ref{sec:gener-vers-main} above is amenable to explicit calculations. In this section and the next, we back up this claim with some examples, while simultaneously illustrating how the abstract homotopy theory in \S\ref{sec:gener-vers-main} plays out in concrete examples.
In practice, given a manifold $W$ and a structure $\lambda_W: \mathrm{Fr}(W) \to \Lambda$, one typically first estimates the genus of $(W,\lambda_W)$. This step is trivial for the manifolds considered in \S\ref{sec:RatWgs} and \S\ref{sec:WgRat} below, which are defined to have high genus, but is quite interesting for the complete intersections considered in \S\ref{sec:VdRat}. The next step would typically be to determine the associated highly-connected structure $\mathrm{st}r_W: \mathrm{Fr}(W) \to \Theta$ and the space $B\mathrm{hAut}(u\ \mathrm{rel}\ P)$. This step is mostly resolved by Lemma~\ref{lem:aHutContr} for the example given in \S\ref{sec:RatWgs}, but is again interesting for the examples in \S\ref{sec:WgRat} and especially \S\ref{sec:VdRat}. For calculations in rational cohomology, the last step would then typically be to understand the Serre spectral sequence associated to~(\ref{eq:22}). The complete intersections in \S\ref{sec:VdRat} again provide an interesting and very non-trivial illustration.
\subsection{The manifolds $W_{g,1}$}\label{sec:RatWgs}
Recall that we write $W_g = g(S^n \times S^n)$ for the $g$-fold connected sum, and $W_{g,1} = D^{2n} \# W_g \cong W_g \setminus \mathrm{int}(D^{2n})$. These manifolds play a distinguished role in the theory described above, as they are used to measure the genus of arbitrary $2n$-manifolds: in this sense $W_{g,1}$ is the simplest manifold of genus $g$. In the case $2n=2$ the solution by Madsen and Weiss \cite{MW} of the Mumford Conjecture\mathrm{ind}ex{Mumford Conjecture}\mathrm{ind}ex{Madsen--Weiss theorem} gave a description of $H^*(\mathcal{M}^\mathrm{or}(W_{g,1});\mathbb{Q})$ in terms of Miller--Morita--Mumford classes in a stable range of degrees. In this section we wish to explain how the analogue of Madsen and Weiss' result in dimensions $2n \geq 6$ follows from the theory described above.
The Moore--Postnikov $n$-stage
$$\tau_{W_{g,1}} : W_{g,1} \overset{\ell_{W_{g,1}}}\lra B \overset{\theta}\lra B\mathrm{O}(2n)$$
of a map classifying the tangent bundle of $W_{g,1}$ has the cofibration $\ell_{W_{g,1}}$ $n$-connected and the fibration $\theta$ $n$-co-connected. As $W_{g,1}$ is $(n-1)$-connected and the map $\tau_{W_{g,1}}$ is nullhomotopic---because $W_{g,1}$ admits a framing---we may identify $\theta$ with the $n$-connected cover of $B\mathrm{O}(2n)$, which we write as
$$\theta_n : B\mathrm{O}(2n)\langle n\rangle \overset{\theta_n^\mathrm{or}}\lra B\mathrm{SO}(2n) \overset{\mathrm{or}}\lra B\mathrm{O}(2n).$$
As the pair $(W_{g,1}, \partial W_{g,1})$ is $(n-1)$-connected, by Lemma \ref{lem:aHutContr} we find that $\mathrm{hAut}(\theta_n^\mathrm{or}, \ell_{\partial W_{g,1}})$ is contractible. Thus by Theorem \ref{thm:main-with-boundary} there is a map
$$\alpha : \mathcal{M}^\mathrm{or}(W_{g,1}) \simeq \mathcal{M}^{\theta_n}(W_{g,1},\ell_{W_{g,1}}) \lra \Omega^\infty MT\theta_n$$
which is a homology equivalence onto the path component that it hits, in degrees $* \leq \frac{g-3}{2}$, as long as $2n \geq 6$.
The rational cohomology of a path component of $\Omega^\infty MT\theta_n$ is calculated as described in Remark~\ref{rem:RatCalc}, in terms of $H^*(B\mathrm{O}(2n)\langle n \rangle ;\mathbb{Q})$. To work this out we identify $B\mathrm{O}(2n)\langle n\rangle = B\mathrm{SO}(2n)\langle n\rangle$. As
$$H^*(B\mathrm{SO}(2n);\mathbb{Q}) = \mathbb{Q}[e, p_1, p_2, \ldots, p_{n-1}]$$
is a free graded-commutative algebra the effect of taking the $n$-connected cover on cohomology groups is a simple as possible: it simply eliminates all free generators of degree $ \leq n$. Thus
$$H^*(B\mathrm{SO}(2n)\langle n \rangle ;\mathbb{Q}) = \mathbb{Q}[e, p_{\lceil \frac{n+1}{4} \rceil}, \ldots, p_{n-1}].$$
Combining all the above, we obtain the following.
\begin{theorem}\label{thm:Wg1Rat}
For $2n \geq 6$, let $\mathcal{B}$ denote the set of monomials in the classes $e$, $p_{n-1}$, $p_{n-2}$, \ldots, $p_{\lceil \frac{n+1}{4}\rceil}$. Then the map
$$\mathbb{Q}[\kappa_c \,\,\vert\,\, c \in \mathcal{B}, |c| > 2n] \lra H^*(\mathcal{M}^\mathrm{or}(W_{g,1});\mathbb{Q})$$
is an isomorphism in degrees $* \leq \frac{g-3}{2}$.
\end{theorem}
This is \cite[Corollary 1.8]{GR-W3}. As we mentioned above, for $2n=2$ the same statement (with a slightly different range) was earlier proved by Madsen and Weiss.
\subsection{The manifolds $W_g$}\label{sec:WgRat}
For $2n=2$ it is a theorem of Harer \cite{H} that the map
\begin{equation}\label{eq:CloseBdy}
\mathcal{M}^\mathrm{or}(W_{g,1}) \lra \mathcal{M}^\mathrm{or}(W_{g})
\end{equation}
given by attaching a disk induces an isomorphism on homology in a stable range of degrees. On the other hand $\mathcal{M}^\mathrm{or}(W_{g})$ is the homotopy type of the stack $\mathbf{M}_g$ of genus $g$ Riemann surfaces, and it is in this way that Madsen and Weiss' topological result determines $H^*(\mathbf{M}_g;\mathbb{Q})$ in a stable range.
In dimensions $2n \geq 6$ it is no longer true that the map \eqref{eq:CloseBdy} induces an isomorphism on homology in a stable range of degrees. In this section we shall explain why, by using the theory described above to calculate $H^*(\mathcal{M}^\mathrm{or}(W_{g});\mathbb{Q})$ in a stable range.
Continuing to write
$$\theta_n : B\mathrm{O}(2n)\langle n\rangle \overset{\theta_n^\mathrm{or}}\lra B\mathrm{SO}(2n) \overset{\mathrm{or}}\lra B\mathrm{O}(2n)$$
as in the previous section, this tangential structure is also the Moore--Postnikov $n$-stage of the map classifying the tangent bundle of $W_g$. By Theorem \ref{thm:general-str} there is a map
$$\alpha : \mathcal{M}^\mathrm{or}(W_g) \lra (\Omega^\infty MT\theta_n) /\!\!/ \mathrm{hAut}(\theta^\mathrm{or}_n)$$
that induces an isomorphism on homology, onto the path component that it hits, in degrees $* \leq \tfrac{g-3}{2}$ as long as $2n \geq 6$.
In order to calculate the rational cohomology of $\mathcal{M}^\mathrm{or}(W_g)$ in a range of degrees we could try to calculate the rational cohomology of the relevant path component of $(\Omega^\infty MT\theta_n) /\!\!/ \mathrm{hAut}(\theta^\mathrm{or}_n)$, but instead we shall use the fibre sequence \eqref{eq:22}. Choosing a $\theta_n$-structure $\ell_{W_g}$ on $W_g$, this is a fibre sequence
\begin{equation}\label{eq:WgFibSeq}
\mathcal{M}^{\theta_n}(W_g, \ell_{W_g}) \lra \mathcal{M}^\mathrm{or}(W_g) \overset{\xi}\lra B(\mathrm{hAut}(\theta_n^\mathrm{or})_{[W_g, \ell_{W_g}]}).
\end{equation}
As $\ell_{W_g} : W_g \to B\mathrm{O}(2n)\langle n\rangle$ is $n$-connected, we may apply Theorem \ref{thm:main-cohomological}, giving that
$$\mathbb{Q}[\kappa_c \,\,\vert\,\, c \in \mathcal{B}, |c| > 2n] \lra H^*(\mathcal{M}^{\theta_n}(W_{g},\ell_{W_g});\mathbb{Q})$$
is an isomorphism in degrees $* \leq \frac{g-3}{2}$. All the classes $\kappa_c$ are defined on the total space $\mathcal{M}^\mathrm{or}(W_g)$ of the fibration \eqref{eq:WgFibSeq}, which implies that this fibration satisfies the Leray--Hirsch property in the stable range. Thus the Leray--Hirsch map
\begin{equation}\label{eq:27}
\mathbb{Q}[\kappa_c \,\,\vert\,\, c \in \mathcal{B}, |c| > 2n] \otimes H^*(B(\mathrm{hAut}(\theta_n^\mathrm{or})_{[W_g, \ell_{W_g}]});\mathbb{Q}) \lra H^*(\mathcal{M}^\mathrm{or}(W_{g});\mathbb{Q})
\end{equation}
is an isomorphism in degrees $* \leq \frac{g-3}{2}$.
To complete this calculation we must calculate the rational cohomology of the space $B(\mathrm{hAut}(\theta_n^\mathrm{or})_{[W_g, \ell_{W_g}]}$, and describe the map
$$\xi^* : H^*(B(\mathrm{hAut}(\theta_n^\mathrm{or})_{[W_g, \ell_{W_g}]});\mathbb{Q}) \lra H^*( \mathcal{M}^\mathrm{or}(W_g);\mathbb{Q})$$
in terms that we understand.
\subsubsection{Identifying $\mathrm{hAut}(\theta^\mathrm{or}_n)$}
The map $\theta^\mathrm{or}_n : B\mathrm{O}(2n)\langle n \rangle \to B\mathrm{SO}(2n)$ is a principal fibration for the group-like topological monoid $\mathrm{SO}[0,n-1]$, the truncation of $\mathrm{SO}$, so is classified by the map $B\mathrm{SO}(2n) \to B\mathrm{SO} \to B(\mathrm{SO}[0,n-1])$, and hence there is a homomorphism
$$\iota : \mathrm{SO}[0,n-1] \lra \mathrm{hAut}(\theta^\mathrm{or}_n)$$
given by the principal group action.
\begin{lemma}\label{lem:WgIdhAut}
The map $\iota$ is a weak homotopy equivalence.
\end{lemma}
\begin{proof}
If we fix a basepoint $* \in B\mathrm{O}(2n)\langle n \rangle$, which identifies the fibre through $*$ with $\mathrm{SO}[0,n-1]$, then there is a map
\begin{align*}
ev: \mathrm{hAut}(\theta^\mathrm{or}_n) &\lra \mathrm{SO}[0,n-1]\\
\varphi & \longmapsto \varphi(*),
\end{align*}
and it is clear that $ev \circ \iota$ is the identity. Thus it is enough to show that $ev$ is a weak homotopy equivalence. Suppose we are given a map $(f, g): (D^{k+1}, S^k) \to (\mathrm{SO}[0,n-1], \mathrm{hAut}(\theta_n^\mathrm{or}))$. This determines a relative lifting problem
\begin{equation*}
\begin{tikzcd}
(S^k \times B\mathrm{O}(2n)\langle n \rangle) \cup_{S^k \times \{*\}} (D^{k+1} \times \{*\}) \rar{g \cup f} \dar& B\mathrm{O}(2n)\langle n \rangle \dar{\theta_n^\mathrm{or}}\\
D^{k+1} \times B\mathrm{O}(2n)\langle n \rangle \rar{\theta_n^\mathrm{or} \circ \pi_2} & B\mathrm{SO}(2n)
\end{tikzcd}
\end{equation*}
and finding a nullhomotopy of $(f, g)$ is the same as solving this relative lifting problem. The obstructions for doing so lie in the groups
$$\widetilde{H}^{i-k}(B\mathrm{O}(2n)\langle n \rangle; \pi_{i+1}(B\mathrm{SO}(2n), B\mathrm{O}(2n)\langle n \rangle)),$$
but these groups are zero if $i-k \leq n$ or if $i \geq n$, so always vanish.
\end{proof}
In particular, the submonoid $\mathrm{hAut}(\theta_n^\mathrm{or})_{[W_g, \ell_{W_g}]} \leq \mathrm{hAut}(\theta_n^\mathrm{or})$ is in fact the whole of $\mathrm{hAut}(\theta^\mathrm{or}_n)$, so we may identify the cohomology of its classifying space with
\begin{equation}\label{eq:29}
H^*(B\mathrm{hAut}(\theta^\mathrm{or}_n);\mathbb{Q}) = H^*(B\mathrm{SO}[0,n];\mathbb{Q}) = \mathbb{Q}[p_1, p_2, \ldots, p_{\lfloor \frac{n}{4}\rfloor}].
\end{equation}
\subsubsection{Miller--Morita--Mumford class interpretation}
Combining~(\ref{eq:27}) and~(\ref{eq:29}) gives a formula for $H^*(\mathcal{M}^\mathrm{or}(W_g);\mathbb{Q})$ in a range of degrees. In fact the classes obtained by pulling back $p_1, p_2, \ldots, p_{\lfloor \frac{n}{4}\rfloor}$ along the map
$$\xi : \mathcal{M}^\mathrm{or}(W_g) \lra B\mathrm{hAut}(\theta^\mathrm{or}_n)$$
may be re-interpreted as Miller--Morita--Mumford classes. We shall use the following lemma to explain this.
\begin{lemma}\label{lem:VTangDescends}
Let $\pi^\mathrm{or}: \mathcal{E}^\mathrm{or}(W_g) \to \mathcal{M}^\mathrm{or}(W_g)$ denote the the path component of the fibration \eqref{eq:25} modelling the universal oriented $W_g$-bundle, and $\tau : \mathcal{E}^\mathrm{or}(W_g) \to B\mathrm{SO}(2n)$ denote the map classifying the vertical tangent bundle. Then the square
\begin{equation}\label{eq:VTangDescends}
\begin{gathered}
\begin{tikzcd}
\mathcal{E}^\mathrm{or}(W_g) \arrow[rr,"\tau"] \dar{\pi^\mathrm{or}} & & B\mathrm{SO}(2n) \dar \\
\mathcal{M}^\mathrm{or}(W_g) \rar{\xi} & B\mathrm{SO}[0,n] & B\mathrm{SO}(2n)[0,n] \lar[']{\simeq}
\end{tikzcd}
\end{gathered}
\end{equation}
commutes up to homotopy.
\end{lemma}
\begin{proof}
Let $\pi^{\theta_n} : \mathcal{E}^{\theta_n}(W_g,\ell_{W_g}) \to \mathcal{M}^{\theta_n}(W_g, \ell_{W_g})$ denote the path component of the fibration modelling the universal $W_g$-bundle with $\theta_n$-structure, which as in \eqref{eq:25} comes with maps
$$\tau : \mathcal{E}^{\theta_n}(W_g,\ell_{W_g}) \overset{\ell}\lra B\mathrm{O}(2n)\langle n \rangle \overset{\theta_n^\mathrm{or}}\lra B\mathrm{SO}(2n)$$
whose composition classifies the (oriented) vertical tangent bundle. This gives a commutative square
\[\begin{tikzcd}
\mathcal{E}^{\theta_n}(W_g,\ell_{W_g}) \dar{\pi^{\theta_n}} \rar{\ell}& B\mathrm{O}(2n)\langle n \rangle \dar\\
\mathcal{M}^{\theta_n}(W_g, \ell_{W_g}) \rar& *
\end{tikzcd}\]
of $\mathrm{hAut}(\theta^\mathrm{or}_n)$-spaces and $\mathrm{hAut}(\theta^\mathrm{or}_n)$-equivariant maps. Taking homotopy orbits, and replacing the spaces at each corner with homotopy equivalent models, we obtain the homotopy commutative square
\[\begin{tikzcd}
\mathcal{E}^\mathrm{or}(W_g) \dar{\pi^\mathrm{or}} \rar{\tau}& B\mathrm{SO}(2n) \dar\\
\mathcal{M}^\mathrm{or}(W_g) \rar{\xi}& B\mathrm{SO}[0,n].
\end{tikzcd}\]
Here the right-hand map is the truncation $B\mathrm{SO}(2n) \to B\mathrm{SO}(2n)[0,n]$ followed by the identification $B\mathrm{SO}(2n)[0,n] \overset{\sim}\to B\mathrm{SO}[0,n]$, as required.
\end{proof}
Now we may calculate as follows: by this lemma we have
$$(\pi^\mathrm{or})^* \xi^*(p_i) = \tau^*(p_i),$$
and so by the projection formula (and commutativity of the cup product)
$$\kappa_{ep_i} = \int_{\pi^\mathrm{or}} \tau^*(e \cdot p_i) = \left(\int_{\pi^\mathrm{or}} \tau^*e \right) \cdot \xi^*(p_i) = \chi(W_g) \cdot \xi^*(p_i)$$
and hence, for $\chi(W_g) = 2 + (-1)^n 2g \neq 0$, we have $\xi^*(p_i) = \frac{1}{\chi(W_g)} \kappa_{ep_i}$.
Combined with the previous discussion, we obtain the following.
\begin{theorem}\label{thm:WgRat}
For $2n \geq 6$, let $\mathcal{B}$ denote the set of monomials in the classes $e$, $p_{n-1}$, $p_{n-2}$, \ldots, $p_{\lceil \frac{n+1}{4}\rceil}$, and $\mathcal{C}$ denote the set of the remaining Pontryagin classes $p_1, p_2, \ldots, p_{\lfloor \frac{n}{4}\rfloor}$. Then the map
$$\mathbb{Q}[\kappa_c \,\,\vert\,\, c \in (\mathcal{B} \sqcup e\cdot\mathcal{C}), |c|>2n] \lra H^*(\mathcal{M}^\mathrm{or}(W_g);\mathbb{Q})$$
is an isomorphism in degrees $* \leq (g-3)/2$.
\end{theorem}
For $2n=2$ the same statement (with a slightly different range) holds by the theorem of Madsen and Weiss, and in this case the set $\mathcal{C}$ is empty and the result is the same as that of Theorem \ref{thm:Wg1Rat}. For $2n \geq 6$ we have $\mathrm{hAut}(\theta^\mathrm{or}_n) \simeq \mathrm{SO}[0,n-1] \not\simeq *$ and so it follows from the discussion in this section that the map \eqref{eq:CloseBdy} is \emph{not} an isomorphism on integral cohomology in any range of degrees (though for $2n=6$ it is still an isomorphism on rational cohomology in a stable range, as $\mathrm{SO}[0,2] \simeq K(\mathbb{Z}/2,1)$ is rationally acyclic; in Theorem \ref{thm:WgRat} this corresponds to the fact that $\mathcal{C}$ is empty in this case).
\begin{remark}
In the statement of Theorem \ref{thm:WgRat} we do \emph{not} assert that $\kappa_c=0$ for monomials $c \not\in \mathcal{B} \sqcup e\cdot\mathcal{C}$. Indeed a further consequence of the homotopy commutativity of \eqref{eq:VTangDescends} is the following description of $\kappa_c$ for a general monomial $c=e^i\cdot p_1^{j_1} \cdots p_{n-1}^{j_{n-1}}$ in terms of the generators of Theorem \ref{thm:WgRat}:
$$\kappa_c = \left(\frac{\kappa_{e\cdot p_1}}{\chi(W_g)}\right)^{j_1} \cdot \left(\frac{\kappa_{e\cdot p_2}}{\chi(W_g)}\right)^{j_2} \cdots \left(\frac{\kappa_{e\cdot p_k}}{\chi(W_g)}\right)^{j_k} \cdot \kappa_{\left(e^i \cdot p_{k+1}^{k_{k+1}} \cdots p_{n-1}^{j_{n-1}}\right)},$$
where we write $k = {\lfloor \frac{n}{4}\rfloor}$. This follows immediately from the observation that $p_i(T_\pi) = \pi^*\xi^*(p_i) = \pi^*(\frac{\kappa_{e\cdot p_i}}{\chi(W_g)})$ for $i \leq k$.
\end{remark}
\subsection{Hypersurfaces in $\mathbb{C}\mathbb{P}^4$}\label{sec:VdRat}
If $V \subset \mathbb{C}\mathbb{P}^{r+1}$ is a smooth hypersurface,\mathrm{ind}ex{hypersurface} determined by a homogeneous complex polynomial of degree $d$, then it is an observation of Thom that its diffeomorphism type depends only on the degree $d$, and not on the particular polynomial: we call the resulting $2r$-manifold $V_{d}$. As we shall explain in \S\ref{sec:genus-v_d}, these interesting manifolds tend to have large genus. More generally, for smooth complete intersections\mathrm{ind}ex{complete intersection} of such hypersurfaces the diffeomorphism type depends only on the degrees, and much is understood about the classification up to diffeomorphism of such manifolds in terms of these degrees, by Libgober and Wood \cite{LibgoberWood}, Kreck \cite{Kreck}, and others.
We shall explain how the theory described above applies in the non-trivial example of a hypersurface $V_d \subset \mathbb{C}\mathbb{P}^4$ of degree $d$, and determine a formula for the rational cohomology of $\mathcal{M}^\mathrm{or}(V_d)$ in a range of degrees. Let us start with outlining the steps again, and state the conclusions in this example.
\begin{enumerate}[(i)]
\item Determine the genus of $V_d$: it turns out to be $\frac{1}{2} (d^4-5d^3+10d^2-10d+4)$.
\item Determine the Moore--Postnikov 3-stage $V_d \xrightarrow{\ell_{V_d}} B_d \xrightarrow{\theta_d} B\mathrm{SO}(6)$ of a map classifying the oriented tangent bundle of $V_d$.
\item Calculate the ring $H^*(\mathcal{M}^{\theta_d}(V_d,\ell_{V_d});\mathbb{Q})$ in the stable range. It turns out to be the $\mathbb{Q}$-algebra
\begin{equation}\label{eq:45}
A = \mathbb{Q}[\kappa_{t^n c} \mid \text{$c \in \mathcal{B}$, $n \geq 0$, $|c| + 2n > 6$}],
\end{equation}
where $\mathcal{B}$ is the set of monomials in classes $p_1$, $p_2$, and $e$ of degree $|p_1| = 4$, $|p_2| = 8$, and $|e| = 6$, and $t$ is a class of degree $2$.
\item Use the spectral sequence arising from Corollary~\ref{cor:general-str} to determine the cohomology of $\mathcal{M}^\mathrm{or}(V_d)$ from that of $\mathcal{M}^{\theta_d}(V_d, \ell_{V_d})$ in a stable range. The result is a short exact sequence
\begin{equation}\label{eq:23}
0 \lra H^*(\mathcal{M}^\mathrm{or}(V_d);\mathbb{Q}) \lra A \overset{d_3}\lra A \lra 0,
\end{equation}
where $d_3: A \to A$ is the unique derivation satisfying $d_3(\kappa_{t^n c}) = n \kappa_{t^{n-1} c}$. (The result is a scalar when $|t^{n-1} c| = 6$; this scalar is a characteristic number of $V_d$ and therefore the derivation $d_3$ depends on the degree $d$.)
\end{enumerate}
\subsubsection{Algebraic topology of $V_{d}$}
By the Lefschetz hyperplane theorem the inclusion $i: V_{d} \to \mathbb{C}\mathbb{P}^{4}$ is 3-connected. This first implies that $V_{d}$ is simply-connected. Writing $H^*(\mathbb{C}\mathbb{P}^{4};\mathbb{Z}) = \mathbb{Z}[x]/(x^{5})$ for $x=c_1(\mathcal{O}(1))$ and $t=i^*(x)$, we have
$$H^0(V_{d};\mathbb{Z}) = \mathbb{Z} \quad\quad H^1(V_{d};\mathbb{Z})=0 \quad\quad H^2(V_{d};\mathbb{Z}) = \mathbb{Z}\{t\}$$
and hence by Poincar{\'e} duality we have
$$H^4(V_{d};\mathbb{Z}) = \mathbb{Z}\{s\} \quad\quad H^5(V_{d};\mathbb{Z})=0 \quad\quad H^6(V_{d};\mathbb{Z}) = \mathbb{Z}\{u\}$$
where $\langle [V_{d}], u \rangle=1$ and $s \cdot t = u$. We also have $\langle i_*[V_{d}], x^3 \rangle=d$, obtained by intersecting $V_{d}$ with a generic $\mathbb{C}\mathbb{P}^1 \subset \mathbb{C}\mathbb{P}^{4}$, giving $t^3 = d \cdot u$ and hence $t^2 = d \cdot s$. By definition, $V_{d}$ is the zero locus of a transverse section of $\mathcal{O}(d) \to \mathbb{C}\mathbb{P}^{4}$, so its normal bundle in $\mathbb{C}\mathbb{P}^{4}$ is $i^*(\mathcal{O}(d))$ and hence as complex vector bundles we have
$$TV_{d} \mathrm{op}lus i^*(\mathcal{O}(d)) \mathrm{op}lus \underline{\mathbb{C}} = i^*(T\mathbb{C}\mathbb{P}^{4}) \mathrm{op}lus \underline{\mathbb{C}} = i^*(\mathcal{O}(1))^{\mathrm{op}lus 5}$$
so taking total Chern classes yields $c(TV_{d}) = i^*\left(\frac{(1+x)^{5}}{(1+d x)}\right)$. We can therefore extract
\begin{align*}
c_3(TV_{d}) &= \left(\binom{5}{3} - \binom{5}{2}d + \binom{5}{1}d^2 -d^3\right)t^3
\end{align*}
and so compute the Euler characteristic of $V_{d}$ as $\langle [V_{d}], c_3(TV_{d}) \rangle$ to be
$$\chi(V_{d}) = d \cdot(10-10d+5d^2-d^3).$$
We therefore find that $H^3(V_{d};\mathbb{Z})$ is free of rank $4-\chi(V_{d}) = d^4-5d^3+10d^2-10d+4$, which finishes our calculation of the cohomology of $V_{d}$.
For later use we record two further characteristic classes of $V_{d}$, namely
\begin{align*}
w_2(TV_{d}) &= (5-d)t \mod 2\\
p_1(TV_{d}) &= (5-d^2)t^2,
\end{align*}
obtained from the identities $w_2 \equiv c_1 \mod 2$ and $p_1 = c_1^2-2c_2$ among characteristic classes of complex vector bundles.
\subsubsection{Genus of $V_{d}$}\label{sec:genus-v_d}
The genus of $V_d$ may be estimated from below by the methods of \S\ref{sec:disc-moduli-spac}, but in this case it turns out that an exact formula is possible.
It is a theorem of Wall \cite{WallClassV} that any simply-connected smooth 6-manifold $W$ has a decomposition $W \cong M \# g(S^3 \times S^3)$ with $H_3(M;\mathbb{Z})=0$, and so the genus of such a $W$ is given by half its third Betti number. Thus we have
$$g(V_{d}) = \frac{1}{2} (d^4-5d^3+10d^2-10d+4).$$
Similar formulae are known for higher dimensional smooth complex complete intersection varieties, see e.g.\ \cite{wood1975removing, morita1975kervaire, browder1979complete}.
\begin{remark}
More generally, if $\mathcal{L}$ is an ample line bundle over a smooth projective complex manifold $M$ of complex dimension $n+1$ then for all $d \gg 0$ we may consider the smooth manifolds $U_d$ arising as the zeroes of generic holomorphic sections of $\mathcal{L}^{\otimes d}$. Writing $x := c_1(\mathcal{L})$, as $\mathcal{L}$ is ample we have $N := \int_M x^{n+1} \neq 0$. Writing $i : U_d \hookrightarrow M$ for the inclusion, and using that $i_*[U_d]$ is Poincar{\'e} dual to $e(\mathcal{L}^{\otimes d}) = dx$, it follows that $\int_{U_d} i^*(x)^n = d \cdot N \neq 0$. The analogue of the calculation above gives that $c(TU_d) = i^*(\frac{c(TM)}{(1+dx)})$ and hence we have $\chi(U_d) = (-1)^n d^{n+1} N +O(d^n)$ and so $b_n = d^{n+1} N +O(d^n)$. If $n$ is odd then by the discussion in \S\ref{sec:disc-moduli-spac} we have
$$g(U_d) = \frac{1}{2} d^{n+1} N +O(d^n).$$
If $n$ is even, then the analogous calculation with the total Hirzebruch $\mathcal{L}$-class gives that $\mathcal{L}(TU_d) = i^*(\frac{\mathcal{L}(TM)}{dx/\tanh(dx)})$ so $ \sigma(U_d) = \frac{2^{n+2}(2^{n+2}-1) B_{n+2}}{(n+2)!} d^{n+1} N +O(d^n)$, where $B_i$ denote the Bernoulli numbers. Hence, by the discussion in \S\ref{sec:disc-moduli-spac}, we have
$$g(U_d) = \frac{1}{2}\left(1-\frac{2^{n+2}(2^{n+2}-1) |B_{n+2}|}{(n+2)!}\right) d^{n+1} N +O(d^n).$$
The term $\frac{2^{n+2}(2^{n+2}-1) |B_{n+2}|}{(n+2)!}$ does not matter much for large $n$: the fact that the Taylor series for $\tanh(z)$ has convergence radius $\pi/2$ implies that that term is asymptotically smaller than $(2/\pi)^{n+\varepsilon}$ as $n \to \infty$, for any $\varepsilon > 0$; in particular it quickly becomes much smaller than 1. In the relevant cases $n \geq 4$ it is at most $2/15$.
\end{remark}
\subsubsection{Moore--Postnikov 3-stage of $V_{d}$}
Let us write
$$\tau: V_{d} \overset{\ell_{V_d}}\lra B_{d} \overset{\theta_{d}}\lra B\mathrm{SO}(6)$$
for the Moore--Postnikov 3-stage of a map $\tau$ classifying the oriented tangent bundle of $V_{d}$, so $\ell_{V_d}$ is 3-connected and $\theta_{d}$ is 3-co-connected. From this we easily calculate the homotopy groups of $B_{d}$, as
\begin{align*}
0=\pi_1(V_{d}) \overset{\sim}\lra &\pi_1(B_{d})\\
\mathbb{Z}=\pi_2(V_{d}) \overset{\sim}\lra &\pi_2(B_{d})\\
&\pi_3(B_{d}) \overset{\sim}\lra \pi_3(B\mathrm{SO}(6))=0\\
&\pi_i(B_{d}) \overset{\sim}\lra \pi_i(B\mathrm{SO}(6)) \text{ for all } i \geq 4.
\end{align*}
To understand the map $\theta_{d}$ on homotopy groups, it remains to understand the composition
\begin{equation}\label{eq:2ndSWVd}
\tau_* : \mathbb{Z}=\pi_2(V_{d}) \overset{\sim}\lra \pi_2(B_{d}) \lra \pi_2(B\mathrm{SO}(6)) = \mathbb{Z}/2.
\end{equation}
The latter group is detected by the Stiefel--Whitney class $w_2$, so this map is non-zero if and only if the class $w_2(TV_{d}) \in H^2(V_{d};\mathbb{Z}/2) \cong \mathrm{Hom}(\pi_2(V_{d}), \mathbb{Z}/2)$ is non-zero. We have seen that $w_2(TV_{d}) = (5-d)t$, so \eqref{eq:2ndSWVd} is surjective if and only if $d$ is even.
Let us abuse notation by writing ${t} \in H^2(B_{d};\mathbb{Z})$ for the unique class which pulls back to $t$ along $\ell_{V_d}$. If $d$ is even, then ${t}$ satisfies ${t} \equiv w_2(\theta_{d}^*\gamma) \mod 2$.
Thus there is a $\mathrm{Spin^c}$-structure on the bundle $\theta_d^*\gamma$ with $c_1={t}$, and choosing one provides a commutative diagram
\begin{equation*}
\begin{tikzcd}
B_{d} \arrow[rd, "\theta_{d}"] \arrow[rr, "f"]& & B\mathrm{Spin^c}(6) \arrow[ld]\\
& B\mathrm{SO}(6).
\end{tikzcd}
\end{equation*}
It may be directly checked using the above calculations that the map $f$ induces an isomorphism on all homotopy groups, so is a weak equivalence (over $B\mathrm{SO}(6)$).
If $d$ is odd then we have $w_2(\theta_{d}^*\gamma)=0$, so we may choose a $\mathrm{Spin}$-structure on the bundle $\theta_{d}^*\gamma$, which provides a commutative diagram
\begin{equation*}
\begin{tikzcd}
B_{d} \arrow[rd, "\theta_{d}"] \arrow[rr, "h"]& & B\mathrm{Spin}(6) \arrow[ld]\\
& B\mathrm{SO}(6).
\end{tikzcd}
\end{equation*}
It may be directly checked using the above calculations that the map $h \times {t} : B_{d} \to B\mathrm{Spin}(6) \times K(\mathbb{Z}, 2)$ induces an isomorphism on all homotopy groups, so is a weak equivalence (over $B\mathrm{SO}(6)$).
In either case, the map
$$\theta_{d} \times t : B_{d} \lra B\mathrm{SO}(6) \times K(\mathbb{Z},2)$$
is a rational homotopy equivalence (over $B\mathrm{SO}(6)$), so we have
$$H^*(B_{d};\mathbb{Q}) = \mathbb{Q}[t, p_1, p_2, e].$$
Writing as in the previous examples $\mathcal{B}$ for the set of monomials in $p_1$, $p_2$, and $e$, by Theorem \ref{thm:main-cohomological} the map
$$\mathbb{Q}[\kappa_{t^i c} \, | \, c \in \mathcal{B}, i \geq 0, |c|+2i > 6] \lra H^*(\mathcal{M}^{\theta_{d}}(V_{d},\ell_{V_d});\mathbb{Q})$$
is an isomorphism in degrees $* \leq \frac{d^4-5d^3+10d^2-10d+4}{4}$, establishing \eqref{eq:45}.
\subsubsection{Change of tangential structure}\label{sec:ChangeOfStr}
We wish to use the above to compute the rational cohomology of $\mathcal{M}^\mathrm{or}(V_{d})$ in a range of degrees, so must analyse the forgetful map
$\mathcal{M}^{\theta_{d}}(V_{d}, \ell_{V_d}) \to \mathcal{M}^\mathrm{or}(V_{d})$. We shall do this in two stages, given by the maps of tangential structures
\begin{equation*}
\begin{tikzcd}
B_{d} \arrow[rrd,',"\theta_{d}"] \arrow[rr, "u = \theta_{d} \times t"]& & B\mathrm{SO}(6) \times K(\mathbb{Z},2) \arrow[d, "\mu"] \arrow[rr, "\mu"]& & B\mathrm{SO}(6) \arrow[lld, "\mathrm{Id}"] \\
& & B\mathrm{SO}(6).
\end{tikzcd}
\end{equation*}
The space of $\theta_{d}$-structures on $V_{d}$ refining the $\mu$-structure $u \circ \ell_{V_d}$ is homotopy equivalent to the space of lifts
\begin{equation*}
\begin{tikzcd}
& B_{d} \arrow[d, "u"]\\
V_{d} \arrow{r}[swap]{u \circ \ell_{V_d}} \arrow[ru, dashed]& B\mathrm{SO}(6) \times K(\mathbb{Z},2),
\end{tikzcd}
\end{equation*}
and $\pi_0(\mathrm{hAut}(u))$ acts on the set of homotopy classes of such lifts. If $G \leq \mathrm{hAut}(u)$ is the submonoid of those path components that preserve the $\theta_d$-structure $\ell_{V_d}$ up to diffeomorphisms of $V_d$ preserving the $\mu$-structure $u \circ \ell_{V_d}$, then there is a fibration sequence
$$\mathcal{M}^{\theta_{d}}(V_{d}, \ell_{V_d}) \lra \mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d}) \lra BG.$$
By the discussion in Remark \ref{rem:Kreck}, $G = \mathrm{hAut}(u)_{[V_d, \ell_{V_d}]}$ as long as $g(V_d,\ell_{V_d}) \geq 3$, and this fibration sequence is an instance of \eqref{eq:22}.
As we have seen above, the map $u$ is a rational homotopy equivalence, and it is immediate from this that $\pi_i(\mathrm{hAut}(u)) \otimes \mathbb{Q}=0$ for $i>0$, so $G$ has no higher rational homotopy groups.
We claim that $\pi_0(G)$ is also trivial, and in fact we shall show that $\pi_0(\mathrm{hAut}(u))$ is trivial (so $G=\mathrm{hAut}(u)$). To see this, let $\phi \in \mathrm{hAut}(u)$, and we must then show that the following lifting problem admits a solution:
\begin{equation*}
\begin{tikzcd}
{\partial [0,1]} \times B_{d} \arrow[rr, "\mathrm{Id} \sqcup \phi"] \arrow[d]& & B_{d} \arrow[d, "u"]\\
{[0,1]} \times B_{d} \arrow[r, "proj"] \arrow[rru, dashed] & B_{d} \arrow[r, "u"]& B\mathrm{SO}(6) \times K(\mathbb{Z},2).
\end{tikzcd}
\end{equation*}
By consideration of the cases $B_{d} = B\mathrm{Spin^c}(6)$ and $B_{d} = B\mathrm{Spin}(6) \times K(\mathbb{Z},2)$, we see that the homotopy fibre of $u$ is a $K(\mathbb{Z}/2,1)$, so there is a unique obstruction to finding the required lift, lying in
$$H^2([0,1] \times B_{d}, \partial [0,1] \times B_{d};\mathbb{Z}/2) \cong {H}^1(B_{d};\mathbb{Z}/2)=0.$$
It follows that $BG$ is simply-connected and has trivial higher rational homotopy groups, so $\mathcal{M}^{\theta_{d}}(V_{d}, \ell_{V_d}) \to \mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d})$ is a rational homotopy equivalence.
Analogously to the above, if $H \leq \mathrm{hAut}(\mu)$ is the submonoid of those path components that preserve the $\mu$-structure $u \circ \ell = \tau \times t$ up to orientation-preserving diffeomorphism of $V_d$, then there is a fibration sequence
$$\mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d}) \lra \mathcal{M}^\mathrm{or}(V_{d}) \lra BH.$$
Again, by Remark \ref{rem:Kreck}, $H = \mathrm{hAut}(\mu)_{[V_d, u \circ \ell_{V_d}]}$ as long as $g(V_d, u \circ \ell_{V_d}) \geq 3$, and this fibration sequence is an instance of \eqref{eq:22}. As the fibration $\mu$ is trivial, we have
\begin{align*}
\mathrm{hAut}(\mu) &\simeq \mathrm{map}(B\mathrm{SO}(6), \mathrm{hAut}(K(\mathbb{Z},2)))\\
&\simeq \mathbb{Z}^\times \ltimes \mathrm{map}(B\mathrm{SO}(6), K(\mathbb{Z},2))\\
&\simeq \mathbb{Z}^\times \ltimes K(\mathbb{Z},2).
\end{align*}
The non-trivial path component of this monoid acts on $H^2(B\mathrm{SO}(6) \times K(\mathbb{Z},2);\mathbb{Z}) = \mathbb{Z}\{t\}$ as $-1$, but any orientation-preserving diffeomorphism of $V_d$ fixes $t^3 \in H^6(V_d;\mathbb{Z})$ so acts as $+1$ on $H^2(V_d;\mathbb{Z}) = \mathbb{Z}\{t\}$. Thus the non-trivial path component of $\mathrm{hAut}(\mu)$ does not lie in $H$, so $H \simeq K(\mathbb{Z},2)$. Thus the fibration sequence is of the form
$$\mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d}) \lra \mathcal{M}^\mathrm{or}(V_{d}) \lra K(\mathbb{Z},3).$$
The Serre spectral sequence for this fibration, in rational cohomology, has two columns and so a single possible non-zero differential. In the stable range, using the above, it has the form
$$E_2^{*,*} = \Lambda[\iota_3] \otimes \mathbb{Q}[\kappa_{t^i c} \, | \, c \in \mathcal{B}, i \geq 0, |c|+2i > 6] \Longrightarrow H^*(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q}).$$
It remains to determine the $d_3$-differential, which by the Leibniz rule is done by the following lemma.
\begin{lemma}\label{lem:VdIdentifyDiff}
We have $d_3(\kappa_{t^n c}) = \iota_3 \otimes n \cdot \kappa_{t^{n-1} c}$.
\end{lemma}
\begin{proof}
We have $d_3(\kappa_{t^n c}) = \iota_3 \otimes x$ for some $x$. The action map
$$a: K(\mathbb{Z},2) \times \mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d}) \lra \mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d})$$
classifies the following data: the $V_{d}$-bundle
$$\pi : K(\mathbb{Z},2) \times \mathcal{E}^{\mu}(V_{d}, u \circ \ell_{V_d}) \to K(\mathbb{Z},2) \times \mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d})$$
pulled back by projection to the second factor, equipped with the $\mu$-structure
$$K(\mathbb{Z},2) \times \mathcal{E}^{\mu}(V_{d}, u \circ \ell_{V_d}) \overset{\tau \times \tilde{t}}\lra B\mathrm{SO}(6) \times K(\mathbb{Z},2)$$
where $\tau$ is given by projection to $\mathcal{E}^{\mu}(V_{d}, u \circ \ell_{V_d})$ and its vertical tangent bundle, and $\tilde{t} = \iota_2 \otimes 1 + 1 \otimes t$.
The class $x$ is related to this action by the formula
$$a^*(\kappa_{t^n c}) = 1 \otimes \kappa_{t^n c} + \iota_2 \otimes x + \cdots.$$
Using the description above we calculate $a^*(\kappa_{t^n c})$ as
$$\pi_!((\iota_2 \otimes 1 + 1 \otimes t)^n \cdot \tau^*(c)) = \pi_!\left(\sum_{i=0}^n \binom{n}{i} \iota_2^{i} \otimes (t^{n-i} \cdot \tau^*c)\right)$$
and the K{\"u}nneth factor in $H^2(K(\mathbb{Z},2);\mathbb{Z}) \otimes H^{|\kappa_{t^n c}|-2}(\mathcal{M}^{\mu}(V_{d}, u \circ \ell_{V_d});\mathbb{Z})$ is $\iota_2 \otimes (n \cdot \kappa_{t^{n-1} c})$. It follows that $x = n \cdot \kappa_{t^{n-1} c}$, as required.
\end{proof}
It follows from this lemma that the differential $d_3$ is a surjection from the first column to the third column, so that
$$H^*(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q}) \cong \mathrm{Ker}(d_3 \circlearrowright \mathbb{Q}[\kappa_{t^n c} \, | \, c \in \mathcal{B}, i \geq 0, |c|+2n > 6])$$
in degrees $* \leq \frac{d^4-5d^3+10d^2-10d+4}{4}$, establishing~(\ref{eq:23}).
It may at first appear that this ring does not depend on ${d}$, but this formula is to be understood carefully. If $|\kappa_{t^n c}|=2$ then $d_3(\kappa_{t^n c}) \in \mathbb{Q}$ is a scalar, and must be evaluated: this is a \emph{boundary condition} for the derivation $d_3$, and is a characteristic number of $V_{d}$. The $\kappa_{t^n c}$ of degree 2 are given by the $t^i c$ of degree 8, so are $p_2$, $p_1^2$, $te$, $t^2 p_1$, and $t^4$, and these have
\begin{align*}
d_3(\kappa_{p_2}) &= 0\\
d_3(\kappa_{p_1^2}) &= 0\\
d_3(\kappa_{te}) &= \kappa_e = \chi(V_{d}) = d \cdot(10-10d+5d^2-d^3)\\
d_3(\kappa_{t^2 p_1}) &= 2 \kappa_{tp_1} = 2 d (5-d^2)\\
d_3(\kappa_{t^4}) &= 4 \kappa_{t^3} = 4\cdot d.
\end{align*}
For the penultimate one we use the calculation of the first Pontryagin class of $V_d$. As an example, $H^2(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q})$ is 4-dimensional and is spanned by the classes
\begin{align*}
\kappa_{p_2}, \quad \kappa_{p_1^2}, \quad \kappa_{te} - \frac{10-10d+5d^2-d^3}{4}\kappa_{t^4}, \quad \text{ and } \quad \kappa_{t^2 p_1} - \frac{5-d^2}{2}\kappa_{t^4}.
\end{align*}
\begin{remark}
In the Serre spectral sequence for the fibration $\pi: \mathcal{E}^\mathrm{or}(V_{d}) \to \mathcal{M}^\mathrm{or}(V_{d})$ modelling the universal oriented $V_g$-bundle as in \eqref{eq:25}, the class $t \in H^2(V_{d};\mathbb{Q}) = E_2^{0,2}$ must be a permanent cycle. (This may be seen as the Euler class of the vertical tangent bundle $T_\pi\mathcal{E}^\mathrm{or}(V_{d})$ restricts to $e(TV_{d}) \in H^6(V_g;\mathbb{Q})$, so this must be a permanent cycle, and this is a non-zero multiple of $t^3$. As $d_3(t^3) = 3 t^2 \cdot d_3(t)$, if $d_3(t) \neq 0$ then $t^3$ would not by a permanent cycle, a contradiction.) Thus there exists a class $\bar{t} \in H^2(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q})$ restricting to $t \in H^2(V_{d};\mathbb{Q})$. We may therefore construct the class $\kappa_{\bar{t}^n c} := \pi_!(\bar{t}^n c) \in H^*(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q})$.
However, the class $\bar{t}$ is not uniquely determined by the above discussion: if $\delta t \in H^2(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q})$ is any class then $\bar{\bar{t}} =\bar{t} + \pi^*(\delta t)$ is another possible choice, and we then have
$$\kappa_{\bar{\bar{t}}^n c} = \pi_!((\bar{t} + \pi^*(\delta t))^n c) = \kappa_{\bar{t}^n c} + (\delta t)(n \cdot \kappa_{\bar{t}^{n-1} c}) + (\delta t)^2 \cdots,$$
a potentially different cohomology class.
By the formula in Lemma \ref{lem:VdIdentifyDiff}, we may think of the derivation $d_3$ as being $\frac{\partial}{\partial t}$. From this point of view the polynomials in the classes $\kappa_{t^n c}$ that lie in the kernel of $d_3 = \frac{\partial}{ \partial t}$ are precisely those that are independent of the choice of $\bar{t}$ when evaluated in $H^*(\mathcal{M}^\mathrm{or}(V_{d});\mathbb{Q})$ as described above.
\end{remark}
\subsection{Another $\mathrm{Spin}^c(6)$ example}\label{sec:OtherRat}
For a simply-connected manifold $W$ of dimension $2n \geq 6$, the formula of Theorem \ref{thm:general-str} for the homology of $\mathcal{M}^\mathrm{or}(W)$ in a range of degrees seems at first glance as though it only depends on the equivariant homotopy type of the $\mathrm{GL}_{2n}(\mathbb{R})$-space $\Theta$ having an $n$-co-connected equivariant map $u : \Theta \to \mathbb{Z}^\times$ and an $n$-connected equivariant map $\mathrm{st}r_W : \mathbf{F}r(TW) \to \Theta$. However, the codomain of the map \eqref{eq:11} is the disconnected space $(\Omega^\infty MT\Theta)\backslash\!\!\backslashd \mathrm{hAut}(u)$, and the different path-components of this space can have different cohomology, even rationally. In this section we give an example of this behaviour.
\begin{construction}
Let $V \to S^2$ be the unique non-trivial 5-dimensional real vector bundle, and $M = S(V)$ be its sphere bundle; it is an $S^4$-bundle over $S^2$ with the same homology as $S^4 \times S^2$. If we write $\pi : M \to S^2$ for the bundle projection, then there is an isomorphism $TM \cong \pi^*(V)\mathrm{op}lus \varepsilon^1$. In particular, the $\mathrm{Spin}^c$-structure on $V$ given by a generator of $H^2(S^2;\mathbb{Z})$ gives one on $M$ (which is $\mathrm{Spin}^c$-nullbordant), and the corresponding map $\ell_M : M \to B\mathrm{Spin}^c(6)$ is 3-connected. This induces a $\mathrm{Spin}^c$-structure on $M_g := M \# g (S^3 \times S^3)$ such that $\ell_{M_g} : M_g \to B\mathrm{Spin}^c(6)$ is also 3-connected.
\end{construction}
Let $\theta : B\mathrm{Spin}^c(6) \to B\mathrm{SO}(6)$. As in the last section, if $K \leq \mathrm{hAut}(\theta)$ is the submonoid of those path components that stabilise the $\theta$-structure $\ell_{M_g}$ up to diffeomorphism of $M_g$, then there is a fibration sequence
$$\mathcal{M}^{\theta}(M_g,\ell_{M_g}) \lra \mathcal{M}^\mathrm{or}(M_g) \lra BK.$$
\begin{lemma}
We have $\mathrm{hAut}(\theta) \simeq \mathbb{Z}^\times \ltimes K(\mathbb{Z},2)$.
\end{lemma}
\begin{proof}
The fibration $\theta : B\mathrm{Spin}^c(6) \to B\mathrm{SO}(6)$ has fibre $K(\mathbb{Z},2)$, and is principal, so there is an action of $K(\mathbb{Z},2)$ on $B\mathrm{Spin}^c(6)$ fibrewise over $B\mathrm{SO}(6)$. Furthermore, writing $\mathrm{Spin}^c(6) = \mathrm{Spin}(6) \times_{\mathbb{Z}^\times} \mathrm{U}(1)$ we see that complex conjugation on the $\mathrm{U}(1)$ factor gives an involution $c$ of $B\mathrm{Spin}^c(6)$ over $B\mathrm{SO}(6)$. Together these give a map $\mathbb{Z}^\times \ltimes K(\mathbb{Z},2) \to \mathrm{hAut}(\theta)$ which can be shown to be an equivalence by obstruction theory just as in \S\ref{sec:ChangeOfStr} or the proof of Lemma \ref{lem:WgIdhAut}.
\end{proof}
\begin{lemma}\label{lem:M_ghAut}
We have $K = \mathrm{hAut}(\theta)$.
\end{lemma}
Let us give two proofs of this lemma, one in terms of the manifolds themselves, and one using the infinite loop spaces of the relevant Thom spectrum.
\begin{proof}
The proof of the previous lemma shows that if $\ell$ and $\ell'$ are two $\theta$-structures on $M_g$ then there is a unique obstruction to them being homotopic, namely
$$\ell^*(t) - (\ell')^*(t) \in H^2(M_g;\mathbb{Z}).$$
We therefore see that $\ell_{M_g}$ and $c\circ \ell_{M_g}$, where $c$ is the involution of $B\mathrm{Spin}^c(6)$ over $B\mathrm{SO}(6)$, are not fibrewise homotopic as the obstruction is $2\ell_{M_g}^*(t) \neq 0 \in H^2(M_g;\mathbb{Z})$.
However, pulling back the vector bundle $V \to S^2$ along a diffeomorphism of $S^2$ of degree $-1$ gives an isomorphic oriented vector bundle, as $\pi_2(B\mathrm{SO}(5))= \mathbb{Z}/2$, and so this degree $-1$ diffeomorphism is covered by a diffeomorphism $M \to M$ which acts as $-1$ on $H^2(M;\mathbb{Z})$ and as $+1$ on $H^4(M;\mathbb{Z})$, so is orientation-reversing. Composing this with the fibrewise antipodal map of $\pi : M \to S^2$ gives a diffeomorphism $\varphi: M \to M$ which acts as $-1$ on both $H^2(M;\mathbb{Z})$ and $H^4(M;\mathbb{Z})$, so is orientation-preserving: we may then isotope it to fix a disc, and hence extend it to a diffeomorphism $\varphi_g : M_g \to M_g$ acting as $-1$ on $H^2(M;\mathbb{Z})$ and on $H^4(M;\mathbb{Z})$.
Now the $\theta$-structures $\ell_{M_g} \circ D\varphi_g$ and $c \circ \ell_{M_g}$ on $M_g$ are homotopic, as
$$(\ell_{M_g} \circ D\varphi_g)^*(t) = \varphi_g^*(\ell_{M_g}^*(t)) = - \ell_{M_g}^*(t) = \ell_{M_g}^*(-t) = (c \circ \ell_{M_g})^*(t).$$
This shows that $c \in \mathrm{hAut}(\theta)$ lies in the submonoid $K$, as it preserves the $\theta$-structure $\ell_{M_g}$ up to a diffeomorphism of $M_g$.
\end{proof}
\begin{proof}[Alternative proof]
By the discussion in Remark \ref{rem:Kreck}, the submonoid $K \leq \mathrm{hAut}(\theta)$ agrees with $\mathrm{hAut}(\theta)_{[M_g, \ell_{M_g}]}$ as long as $g \geq 3$, so is the stabiliser of $\alpha(M_g, \ell_{M_g}) \in \pi_0(MT\mathrm{Spin}^c(6))$.
Thomifying the map $B\mathrm{Spin}^c(6) \to B\mathrm{Spin}^c$ gives a fibre sequence of spectra
$$F \lra MT\mathrm{Spin}^c(6) \lra \Sigma^{-6}M\mathrm{Spin}^c$$
and it is easy to check that $F$ is connective and has $\pi_0(F) \cong \mathbb{Z}$. We therefore have an exact sequence
$$\pi_0(F) \cong \mathbb{Z} \lra \pi_0(MT\mathrm{Spin}^c(6)) \lra \pi_6(M\mathrm{Spin}^c) = \Omega_6^{\mathrm{Spin}^c},$$
and the left-hand map can be seen to send a generator to $\alpha(S^6, \ell_{S^6})$, where $\ell_{S^6}$ is the unique $\mathrm{Spin}^c(6)$-structure on $S^6$ compatible with its orientation.
As the $\mathrm{Spin}^c(6)$-manifold $M$ is constructed as the sphere bundle of a $\mathrm{Spin}^c$ vector bundle, its class is trivial in $\Omega_6^{\mathrm{Spin}^c}$ as it bounds the associated disc bundle; similarly for $M_g = M \# g(S^3 \times S^3)$. Thus $\alpha(M_g, \ell_{M_g})$ is a multiple of $\alpha(S^6, \ell_{S^6})$ (by taking Euler characteristic we see that it is $2-g$ times it) and so is fixed by $\mathrm{hAut}(\theta)$, as the $\mathrm{Spin}^c(6)$-structure on $S^6$ is unique given its orientation.
\end{proof}
We may therefore develop the following diagram of fibration sequences
\begin{equation*}
\begin{tikzcd}
\mathcal{M}^{\theta}(M_g, \ell_{M_g}) \ar[r] \ar[equal]{d} & X_g \ar[r] \ar[d] & K(\mathbb{Z},3) \ar[d]\\
\mathcal{M}^{\theta}(M_g, \ell_{M_g}) \ar[r] \ar[d] & \mathcal{M}^\mathrm{or}(M_g) \ar[r] \ar[d] & B(\mathbb{Z}^\times
\ltimes K(\mathbb{Z},2)) \ar[d]\\
* \ar[r] & B\mathbb{Z}^\times \ar[equal]{r} & B\mathbb{Z}^\times,
\end{tikzcd}
\end{equation*}
whose middle row is the fibration sequence, with lower middle arrow defined to make the bottom right-hand square commute, and $X_g$ as its homotopy fibre, and top right-hard square homotopy cartesian.
The calculation of the previous section applies to the top row, showing that
$$H^*(X_g;\mathbb{Q}) = \mathrm{Ker}(d_3 \circlearrowright \mathbb{Q}[\kappa_{t^n c} \, | \, c \in \mathcal{B}, i \geq 0, |c|+2n > 6])$$
in a stable range, this time subject to the boundary conditions
\begin{align*}
d_3(\kappa_{te}) &= \kappa_e = \chi(M_{g}) = 4-2g\\
d_3(\kappa_{t^2 p_1}) &= 2 \kappa_{tp_1} = 0\\
d_3(\kappa_{t^4}) &= 4 \kappa_{t^3} = 0.
\end{align*}
However, now the Serre spectral sequence for the middle column gives the calculation
$$H^*(\mathcal{M}^\mathrm{or}(M_g);\mathbb{Q}) = H^*(X_g;\mathbb{Q})^{\mathbb{Z}^\times} = \mathrm{Ker}(d_3 \circlearrowright \mathbb{Q}[\kappa_{t^n c} \, | \, c \in \mathcal{B}, i \geq 0, |c|+2n > 6])^{\mathbb{Z}^\times}$$
in a stable range, where the invariants are taken with respect to the involution $t \mathrm{map}sto -t$.
Let us explain something of the structure of this ring in low degrees. In particular, we shall see that unlike the previous examples it not a free graded-commutative algebra, even in the stable range where our formulae apply.
Before taking $\mathbbm{Z}^\times$-invariants, in degree 2 the kernel is spanned by $\{\kappa_{p_2}, \kappa_{p_1^2}, \kappa_{t^2 p_1}, \kappa_{t^4}\}$, and these classes are all fixed by the involution, giving
$$\dim_\mathbb{Q} H^2(\mathcal{M}^\mathrm{or}(M_g);\mathbb{Q}) = 4.$$
In degree 4 the kernel of $d_3$ is 16 dimensional, spanned by the 10-dimensional vector space $\mathrm{Sym}^2(\mathbb{Q}\{\kappa_{p_2}, \kappa_{p_1^2}, \kappa_{t^2 p_1}, \kappa_{t^4}\})$ along with the classes
\begin{align*}
& \kappa_{te}\kappa_{p_2} - (4-2g)\kappa_{tp_2}\\
& \kappa_{te}\kappa_{p_1^2} - (4-2g)\kappa_{tp_1^2}\\
& (4-2g)\kappa_{t^3 p_1} - 3\kappa_{t^2p_1}\kappa_{te}\\
& (4-2g)\kappa_{t^5} - 5\kappa_{t^4}\kappa_{te}\\
& \kappa_{p_1e}\\
& \kappa_{te}^2 - (4-2g) \kappa_{t^2e}.
\end{align*}
Of these, the last two classes are invariant under the involution while first four are \emph{anti-}invariant, and hence
$$\dim_\mathbb{Q} H^4(\mathcal{M}^\mathrm{or}(M_g);\mathbb{Q}) = 12.$$
In higher degrees, we find that even though, for example, the class $\kappa_{te}\kappa_{p_2} - (4-2g)\kappa_{tp_2}$ is not invariant under the involution, its square is invariant and therefore defines a class in $H^8(\mathcal{M}^\mathrm{or}(M_g);\mathbb{Q})$. Similarly with products of any two classes that are anti-invariant and in the kernel of $d_3$. In degree 16 we find the relation
\begin{align*}
&((\kappa_{te}\kappa_{p_2} - (4-2g)\kappa_{tp_2})(\kappa_{te}\kappa_{p_1^2} - (4-2g)\kappa_{tp_1^2}))^2\\
&\quad\quad = (\kappa_{te}\kappa_{p_2} - (4-2g)\kappa_{tp_2})^2 (\kappa_{te}\kappa_{p_1^2} - (4-2g)\kappa_{tp_1^2})^2
\end{align*}
among squares of classes of degree 8, showing that the ring is not free.
\section{Abelianisations of mapping class groups}\label{sec:AbCalc}
The theory described above may in principle be used for calculations in integral homology and cohomology, though this is of course far more difficult. In practice such calculations are restricted to low dimensions, and have a different flavour to those described in \S\ref{sec:RatCalc}. Here one must obtain information about the low-dimensional homology of $\Omega^\infty MT\Theta$, which is roughly the same as the low-dimensional homotopy of $\Omega^\infty MT\Theta$, which is the homotopy of the spectrum $MT\Theta$ in small positive degrees. But the spectrum $MT\Theta$ is non-connective, so computing its $\pi_i$ is comparable to computing $\pi_{i+2n}$ of a connective spectrum (in the alternative proof of Lemma \ref{lem:M_ghAut} we have already engaged with this a bit, though we avoided having to actually compute).
As an example of the kinds of calculations that one is required to make, and to give some ideas of the kinds of techniques that can be used to tackle them, in this section we shall survey the calculation in \cite{GR-WAb} of $H_1(\mathcal{M}(W_{g,1});\mathbb{Z})$, and then describe analogous calculations for certain non-simply connected 6-manifolds.
Recall that for a manifold $W$, possibly with boundary, its \emph{mapping class group}\mathrm{ind}ex{mapping class group} is
$$\Gamma_\partial(W) := \pi_0(\Diff_\partial(W)).$$
Equivalently, it is the fundamental group of $B\Diff_\partial(W)$, so by the Hurewicz theorem we may identify its abelianisation as
$$\Gamma_\partial(W)^{ab} \cong H_1(B\Diff_\partial(W);\mathbb{Z}) \cong H_1(\mathcal{M}(W);\mathbb{Z}).$$
\subsection{The manifolds $W_{g,1}$}\label{sec:AbWgs}
We return to the $2n$-manifolds $W_{g,1}$ of \S\ref{sec:RatWgs}. Just as in that section, there is a map
$$\mathcal{M}(W_{g,1}) \simeq \mathcal{M}^{\theta_n}(W_{g,1}, \ell_{W_{g,1}}) \lra \Omega^\infty MT\theta_n$$
that for $2n \geq 6$ is a homology isomorphism in degrees $\leq \frac{g-3}{2}$ onto the path component that it hits. In particular, as long as $g \geq 5$ we have isomorphisms
$$\Gamma_\partial(W_{g,1})^{ab} \cong H_1(\mathcal{M}(W_{g,1});\mathbb{Z}) \cong H_1(\Omega^\infty_0 MT\theta_n ; \mathbb{Z}) \cong \pi_1(MT\theta_n),$$
using that all path components of $\Omega^\infty_0 MT\theta_n$ are homotopy equivalent, that the Hurewicz map is an isomorphism (as this space is a loop space), and that $\pi_1$ of the space $\Omega^\infty_0 MT\theta_n$ is the same as that of the spectrum $MT\theta_n$.
In \cite{GR-WAb} we attempted to calculate this group, at least in terms of other standard groups arising in geometric topology. Here we shall summarise the results and general strategy of that paper, though we refer there for more details.
To state the main result, consider the bordism theory $\Omega_*^{\langle n \rangle}$ associated to the fibration $B\mathrm{O}\langle n \rangle \to B\mathrm{O}$ given by the $n$-connected cover, and represented by the spectrum $M\mathrm{O}\langle n \rangle$ (cf.\ \cite[p.\ 51]{Stong}). The natural map $B\mathrm{O}(2n)\langle n \rangle \to B\mathrm{O}\langle n \rangle$ covering the stabilisation map $B\mathrm{O}(2n) \to B\mathrm{O}$ provides a map of spectra
$$s : MT\theta_n \lra \Sigma^{-2n} M\mathrm{O}\langle n \rangle,$$
and on $\pi_1$ this gives a homomorphism $s_*: \pi_1(MT\theta_n) \to \Omega_{2n+1}^{\langle n \rangle}$. The group $\pi_1(MT\theta_n)$ is determined in terms of this as follows.
\begin{theorem}\label{thm:AbWgsThm}
There is an isomorphism
$$s_* \mathrm{op}lus f : \pi_1(MT\theta_n) \lra \Omega_{2n+1}^{\langle n \rangle} \mathrm{op}lus \begin{cases}
(\mathbb{Z}/2)^2 & \text{ if $n$ is even}\\
0 & \text{ if $n$ is 1, 3, or 7}\\
\mathbb{Z}/4 & \text{ else}
\end{cases}$$
for a certain homomorphism $f$.
\end{theorem}
Furthermore, the groups $\Omega_{2n+1}^{\langle n \rangle}$ are related to the stable homotopy groups of spheres as follows: there is a homomorphism\mathrm{ind}ex{cokernel of $J$}
$$\rho': \mathrm{Cok}(J)_{2n+1} \lra \Omega_{2n+1}^{\langle n \rangle}$$
given by considering a stably framed manifold as a manifold with $B\mathrm{O}\langle n \rangle$-structure, which is surjective and whose kernel is generated by the class of a certain homotopy sphere\mathrm{ind}ex{homotopy sphere} $\Sigma_Q^{2n+1}$. In several cases it follows from work of Stolz that the class of $\Sigma_Q$ in $\mathrm{Cok}(J)$ is trivial---so $\rho'$ is an isomorphism---but this is not known in general. Combining the above with known calculations of $\Omega_{*}^{\langle 2 \rangle} = \Omega_{*}^{\langle 3 \rangle} = \Omega_{*}^{\mathrm{Spin}}$ and $\Omega_{*}^{\langle 4 \rangle} = \Omega_{*}^{\mathrm{String}}$ gives the following.
\begin{table}[h]
\centering
\label{table:1}
\begin{tabular}{c|c c c c c c c}
$n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$\pi_1(MT\theta_n)$ & 0 & $(\mathbb{Z}/2)^2$ & 0 & $(\mathbb{Z}/2)^4$ & $\mathbb{Z}/4$ & $(\mathbb{Z}/2)^2 \mathrm{op}lus \mathbb{Z}/3$ & $\mathbb{Z}/2$ \\
\end{tabular}
\end{table}
Let us outline the proof of Theorem \ref{thm:AbWgsThm} which was given in \cite{GR-WAb}, as we shall need to refer to details of this argument in the following section. The argument combines methods from (stable) homotopy theory with Theorem \ref{thm:homotopical} again. Starting with homotopy theory, we first let $F$ denote the homotopy fibre of the map of spectra $s : MT\theta_n \to \Sigma^{-2n} M\mathrm{O}\langle n \rangle$, and construct a map $\Sigma^{-2n} \mathrm{SO}/\mathrm{SO}(2n) \to F$ which can be shown to be $n$-connected, for example by computing its effect on homology. On the other hand $\mathrm{SO}/\mathrm{SO}(2n)$ is $(2n-1)$-connected, so by Freudenthal's suspension theorem the map
$$\pi_{2n+1}(\mathrm{SO}/\mathrm{SO}(2n)) \lra \pi_{2n+1}^s(\mathrm{SO}/\mathrm{SO}(2n)) \cong \pi_1^s(\Sigma^{-2n} \mathrm{SO}/\mathrm{SO}(2n))$$
is an isomorphism for $n \geq 2$, and similarly for one homotopy group lower. It follows from a calculation of Paechter \cite{Paechter} that $\pi_{2n+1}(\mathrm{SO}/\mathrm{SO}(2n))$ is $(\mathbb{Z}/2)^2$ if $n$ is even and is $\mathbb{Z}/4$ if $n$ is odd, and also that $\pi_{2n}(\mathrm{SO}/\mathrm{SO}(2n)) \cong \mathbb{Z}$. Putting the above together, we find an exact sequence
\begin{equation}\label{eq:SES}
\Omega_{2n+2}^{\langle n \rangle} \overset{\partial} \lra \begin{cases}
(\mathbb{Z}/2)^2 & \text{ if $n$ is even}\\
\mathbb{Z}/4 & \text{ if $n$ is odd}
\end{cases} \lra \pi_1(MT\theta_n) \lra \Omega_{2n+1}^{\langle n \rangle} \lra \mathbb{Z}.
\end{equation}
The rightmost map is zero (as its domain is easily seen to be a torsion group). In the cases $n \in \{1,3,7\}$ it can be shown that the images of $\mathbb{C}\mathbb{P}^2$, $\mathbb{H}\mathbb{P}^2$, and $\mathbb{O}\mathbb{P}^2$ under the leftmost map are non-zero modulo 2, so the leftmost map is surjective. In the remaining cases one must show that the leftmost map is zero, and that the resulting short exact sequence is split, via a homomorphism $f$ as in the statement of Theorem \ref{thm:AbWgsThm}.
At this point is is convenient to use the isomorphism $\Gamma_\partial(W_{g,1})^{ab} \cong \pi_1(MT\theta_n)$ for some $g \gg 0$. The action of $\Gamma_\partial(W_{g,1})$ on $H_n(W_{g,1};\mathbb{Z})$ respects the $(-1)^n$-symmetric intersection form $\lambda$, and if $n \neq 1$, $3$, or $7$ then it also respects a certain quadratic refinement $\mu$ of this bilinear form. This yields a homomorphism
$$\Gamma_\partial(W_{g,1}) \lra \mathrm{Aut}(H_n(W_{g,1};\mathbb{Z}), \lambda, \mu).$$
These automorphism groups have been studied by other authors, and their abelianisations have been identified (for $g \gg 0$) as $(\mathbb{Z}/2)^2$ if $n$ if even or $\mathbb{Z}/4$ if $n$ is odd. A careful analysis of the maps involved shows that the resulting homomorphism
$$f : \pi_1(MT\theta_n) \cong \Gamma_\partial(W_{g,1})^{ab} \lra \mathrm{Aut}(H_n(W_{g,1};\mathbb{Z}), \lambda, \mu)^{ab} \cong \begin{cases}
(\mathbb{Z}/2)^2 & \text{ if $n$ is even}\\
\mathbb{Z}/4 & \text{ if $n$ is odd}
\end{cases}$$
splits the short exact sequence arising from \eqref{eq:SES}, as required.
\begin{remark}
The Pontryagin dual of the finite abelian group $H_1(\mathcal{M}(W_{g,1});\mathbb{Z})$ calculated here is the torsion subgroup of $H^2(\mathcal{M}(W_{g,1});\mathbb{Z})$. The torsion free quotient of the latter group has been analysed in detail by Krannich and Reinhold \cite{KR}. The (unknown, at present) order of the element $[\Sigma_Q] \in \mathrm{Cok}(J)_{2n+1}$ arises there too.
\end{remark}
\subsection{Some non-simply-connected 6-manifolds}
For the example discussed in the previous section the theory described above is not the only way to calculate $\Gamma_\partial(W_{g,1})^{ab}$, because Kreck \cite{KreckAut} has described the groups $\Gamma_\partial(W_{g,1})$ up to two extension problems, and Krannich \cite{KrannichMCG} has recently resolved these extensions completely for $n$ odd, and determined enough about them to calculate $\Gamma_\partial(W_{g,1})^{ab}$ for all $n \geq 3$ and all $g \geq 1$. However, for even slightly more complicated manifolds such an alternative approach is not available, and we suggest that the theory described above is the best way to approach the calculation of $\Gamma_\partial(W)^{ab}$. In this section we illustrate this with an example which seems inaccessible by other means.
Let $G$ be a virtually polycyclic group,\mathrm{ind}ex{virtually polycyclic} and consider a compact 6-manifold $W$ such that
\begin{enumerate}[(i)]
\item a map $\tau_W$ classifying the tangent bundle of $W$ admits a lift $\ell_W$ along
$$\theta : B\mathrm{Spin}(6) \times BG \overset{\mathrm{pr}_1}\lra B\mathrm{Spin}(6) \lra B\mathrm{O}(6)$$
such that $\ell_W : W \to B\mathrm{Spin}(6) \times BG$ is 3-connected, and
\item $(W, \partial W)$ is 2-connected.
\end{enumerate}
Such manifolds exist for any virtually polycyclic $G$: these groups satisfy Wall's \cite{WallFin} finiteness condition ($F$) by \cite[p.183]{Ratcliffe}, and so also ($F_3$), so $W$ may be taken to be a regular neighbourhood of an embedding of a finite 3-skeleton of $BG$ into $\mathbb{R}^6$. As further examples of such manifolds, one may take
\begin{equation}\label{ex:From3Mfld}
W=(M^3 \times D^3) \# g(S^3 \times S^3)
\end{equation}
where $M^3$ is a closed oriented 3-manifold that is irreducible (so that $\pi_2(M)=0$) and has virtually polycyclic fundamental group.
Recall from \S\ref{sec:pi1} that the Hirsch length of a virtually polycyclic group is the number of infinite cyclic quotients in a subnormal series.
\begin{theorem}\label{thm:NinasEx}
Suppose that $G$ is virtually polycyclic of Hirsch length $h$, and $W$ is a 6-manifold satisfying (i) and (ii) above, of genus $g(W) \geq 7 + h$. Then there is a short exact sequence
\begin{equation*}
0 \lra G^{ab} \lra \Gamma_\partial(W)^{ab} \lra \mathrm{ko}_7(BG) \lra 0
\end{equation*}
which is (non-canonically) split.
\end{theorem}
This short exact sequence was first established by Friedrich \cite{NinaThesis}, who also showed that it is split after inverting 2. We shall give a different argument to hers, which gives the splitting at the prime 2 as well.
The following three examples concern the manifolds $W$ of construction \eqref{ex:From3Mfld} with $M$ a 3-manifold having finite fundamental group, and $g \geq 7$ so the hypotheses of Theorem \ref{thm:NinasEx} are satisfied.
\begin{example}\label{ex:Ab1}
Let $M^3 = L_{p,q}^3$ be the $(p,q)$th lens space, with fundamental group $G=\mathbb{Z}/p$ with $p$ prime. Then we have
$$\Gamma_\partial(W)^{ab} \cong \begin{cases}
\mathbb{Z}/2 \mathrm{op}lus \mathbb{Z}/4 & \text{if $p=2$}\\
\mathbb{Z}/3 \mathrm{op}lus \mathbb{Z}/9 & \text{if $p=3$}\\
(\mathbb{Z}/p)^3 & \text{if $p\geq 5$}.
\end{cases}$$
The required calculation of $\mathrm{ko}_7(B\mathbb{Z}/p)$ may be extracted from \cite[Example 7.3.1]{GB2} for $p=2$, and from the Atiyah--Hirzebruch spectral sequence, the fact that $\mathrm{ko}_*(B\mathbb{Z}/p)[\tfrac{1}{2}]$ is a summand of $\mathrm{ku}_*(B\mathbb{Z}/p)[\tfrac{1}{2}]$, and \cite[Remark 3.4.6]{GB1} for odd $p$.
\end{example}
\begin{example}
Let $M^3$ be the spherical 3-manifold with fundamental group $G=Q_8$. Then we have
$$\Gamma_\partial(W)^{ab} \cong (\mathbb{Z}/2)^2 \mathrm{op}lus (\mathbb{Z}/4)^2 \mathrm{op}lus \mathbb{Z}/64.$$
The required calculation of $\mathrm{ko}_7(BQ_8)$ may be extracted from \cite[p.\ 138]{GB2}.
\end{example}
\begin{example}
Let $M^3 = \Sigma^3$ be the Poincar{\'e} homology 3-sphere, with fundamental group $G$ the binary icosahedral group, isomorphic to $SL_2(\mathbb{F}_5)$. Then $g$ we have
$$\Gamma_\partial(W)^{ab} \cong (\mathbb{Z}/5)^2 \mathrm{op}lus \mathbb{Z}/9 \mathrm{op}lus \mathbb{Z}/64.$$
As $\Sigma^3$ is a homology sphere, $H_1(BG;\mathbb{Z}) \cong H_1(\Sigma^3;\mathbb{Z})=0$ so we must just calculate $\mathrm{ko}_7(BG)$. The order of $G \cong SL_2(\mathbb{F}_5)$ is $120 = 2^3 \cdot 3 \cdot 5$, so we shall calculate the localisations $\mathrm{ko}_7(BG)_{(p)}$ for $p \in \{2,3,5\}$. Recall that the cohomology ring of $BG$ is $H^*(BG;\mathbb{Z}) = \mathbb{Z}[z]/(120 \cdot z)$ with $|z|=4$, which may be computed from the fibration sequence $S^3 \to \Sigma^3 \to BG$.
When $p$ is odd, $G$ has cyclic Sylow $p$-subgroup, so by transfer $\mathrm{ko}_7(BG)_{(p)}$ is a summand of $\mathrm{ko}_7(B\mathbb{Z}/p)_{(p)}$, which we have explained in Example \ref{ex:Ab1} is $\mathbb{Z}/9$ for $p=3$ and $(\mathbb{Z}/5)^2$ for $p=5$. Comparing this with the Atiyah--Hirzebruch spectral sequence computing $\mathrm{ko}_*(BG)_{(p)}$ gives the claimed answer.
When $p=2$, $G$ has Sylow 2-subgroup $Q_8$, so by transfer $\mathrm{ko}_7(BG)_{(2)}$ is a summand of $\mathrm{ko}_7(BQ_8)_{(2)}$, which we have explained in Example \ref{ex:Ab1} is $(\mathbb{Z}/4)^2 \mathrm{op}lus \mathbb{Z}/64$. By a theorem of Mitchell and Priddy \cite[Theorem D]{MitchellPriddy} there is a stable splitting of $BQ_8$ as $BG_{(2)} \vee X \vee X$ for some spectrum $X$, from which it follows that $\mathrm{ko}_7(BG)_{(2)}$ is either $\mathbb{Z}/64$ or $(\mathbb{Z}/4)^2 \mathrm{op}lus \mathbb{Z}/64$; we may see that the first case occurs from the Atiyah--Hirzebruch spectral sequence computing $\mathrm{ko}_*(BG)_{(2)}$.
\end{example}
The rest of this section is concerned with the proof of Theorem~\ref{thm:NinasEx}.
\subsubsection{Reduction to homotopy theory}
We have assumed that $(W, \partial W)$ is 2-connected, so by Lemma \ref{lem:aHutContr} we have that $\mathrm{hAut}(\theta, {\partial W}) \simeq *$. Thus by Theorem \ref{thm:homotopical} and the discussion in \S\ref{sec:pi1} there is a map
$$\alpha : \mathcal{M}(W) \simeq \mathcal{M}^{\theta}(W) \lra \Omega^\infty MT\theta = \Omega^\infty( MT\mathrm{Spin}(6) \wedge BG_+)$$
which induces an isomorphism on homology onto the path component that it hits in degrees $* \leq \frac{g(W)-h-5}{2}$, so in particular as long as $g(W) \geq 7 + h$ it induces an isomorphism on first homology. As described above the first homology of $\mathcal{M}(W)$ is the abelianisation of the mapping class group $\Gamma_\partial(W)$, so to establish Theorem \ref{thm:NinasEx} we must establish a short exact sequence
\begin{equation}\label{eq:ToSplit2}
0 \lra G^{ab} \lra \pi_1(MT\mathrm{Spin}(6) \wedge BG_+) \lra \mathrm{ko}_7(BG) \lra 0
\end{equation}
and show that it is split.
\subsubsection{The exact sequence}
Specialising the construction in \S\ref{sec:AbWgs} to $2n=6$, we showed that there is a cofibration sequence of spectra
\begin{equation}\label{eq:CofSeq}
F \lra MT\mathrm{Spin}(6) \lra \Sigma^{-6}M\mathrm{Spin},
\end{equation}
that $F$ is connective, and that $\pi_0(F) \cong \mathbb{Z}$ and $\pi_1(F) \cong \mathbb{Z}/4$. The Atiyah--Hirzebruch spectral sequence for $F \wedge BG_+$ gives isomorphisms
$$\pi_0(F \wedge BG_+) \cong \mathbb{Z} \quad \quad\quad \pi_1(F \wedge BG_+) \cong \mathbb{Z}/4 \mathrm{op}lus H_1(BG;\mathbb{Z})$$
where the splitting in the latter is induced by the retraction $S^0 \to BG_+ \to S^0$ of pointed spaces. Smashing the cofibration sequence \eqref{eq:CofSeq} with $BG_+$, the associated long exact sequence of homotopy groups has the form
$$\Omega_8^{\mathrm{Spin}}(BG) \overset{\partial}\lra \mathbb{Z}/4 \mathrm{op}lus H_1(BG;\mathbb{Z}) \lra \pi_1(MT\mathrm{Spin}(6) \wedge BG_+) \lra \Omega_7^{\mathrm{Spin}}(BG) \lra \mathbb{Z} $$
and by naturality the long exact sequence for $G=\{e\}$ splits off of this one. As $\pi_1(MT\mathrm{Spin}(6))=0$ by Theorem \ref{thm:AbWgsThm}, and $\Omega_7^{\mathrm{Spin}}=0$ by \cite{MilnorSpin}, by the discussion in \S\ref{sec:AbWgs} this leaves an exact sequence
$$\widetilde{\Omega}_8^{\mathrm{Spin}}(BG) \overset{\partial}\lra H_1(BG;\mathbb{Z}) \lra \pi_1(MT\mathrm{Spin}(6) \wedge BG_+) \lra \Omega_7^{\mathrm{Spin}}(BG) \lra 0.$$
The Atiyah--Bott--Shapiro map $M\mathrm{Spin} \to ko$ is 8-connected, so the induced map $\Omega_7^{\mathrm{Spin}}(BG) \to \mathrm{ko}_7(BG)$ is an isomorphism. To obtain the claimed short exact sequence we shall therefore prove the following.
\begin{lemma}\label{lem:ConnectingTrivial}
If $G$ is finitely-generated then the connecting map $\partial : \widetilde{\Omega}_8^{\mathrm{Spin}}(BG) \to H_1(BG;\mathbb{Z})$ is trivial.
\end{lemma}
\begin{proof}
If this connecting map were non-trivial for some finitely-generated $G$, then because every non-trivial element of $H_1(BG;\mathbb{Z}) = G^{ab}$ remains non-trivial under some homomorphism $G^{ab} \to \mathbb{Z}/p^k$ with $p$ prime and $k \geq 1$, by naturality this connecting map would be non-trivial for $G=\mathbb{Z}/p^k$. So it suffices to show that the map is trivial in this case.
As $M\mathrm{Spin} \to ko$ is 8-connected, the map $\widetilde{\Omega}_8^{\mathrm{Spin}}(BG) \to \widetilde{\mathrm{ko}}_8(BG)$ is an isomorphism. The result then follows as $\widetilde{\mathrm{ko}}_8(B\mathbb{Z}/p^k)=0$, by a trivial application of the Atiyah--Hirzebruch spectral sequence for $p$ odd and by \cite[Theorem 2.4]{BGS} for $p=2$.
\end{proof}
\subsubsection{Simplifying the splitting problem}
Let $x : F \to H\mathbb{Z}$ be the 0-th Postnikov truncation of $F$. The composition
$$k: \Sigma^{-6} M\mathrm{Spin} \overset{\partial}\lra \Sigma F \overset{\Sigma x}\lra \Sigma H\mathbb{Z}$$
represents, under the Thom isomorphism, some element $\kappa \in H^7(B\mathrm{Spin};\mathbb{Z})$ which we identify as follows.
\begin{lemma}
We have $\kappa = \beta(w_6)$.
\end{lemma}
\begin{proof}
We have $H^7(B\mathrm{Spin};\mathbb{Z}/2) = \mathbb{Z}/2\{w_7\}$, but $w_7$ of course vanishes when restricted to $B\mathrm{Spin}(6)$. The long exact sequence on $\mathbb{Z}/2$-cohomology for \eqref{eq:CofSeq} contains the portion
$$H^1(MT\mathrm{Spin}(6);\mathbb{Z}/2) \longleftarrow H^1(\Sigma^{-6}M\mathrm{Spin};\mathbb{Z}/2) = \mathbb{Z}/2\{w_7 \cdot u_{-6}\} \longleftarrow H^1(\Sigma F ;\mathbb{Z}/2)$$
and the left-hand map is zero, so the right-hand map is surjective. As the map $x$ is 1-connected, the map $\Sigma x$ is 2-connected, so we have an isomorphism
$$(\Sigma x)^* : \mathbb{Z}/2\{\Sigma \iota\} = H^1(\Sigma H\mathbb{Z};\mathbb{Z}/2) \overset{\sim}\lra H^1(\Sigma F;\mathbb{Z}/2).$$
It follows that $\kappa$ reduces to $w_7 \neq 0$ modulo 2. The integral cohomology of $B\mathrm{Spin}$ is known to only have torsion of order 2 (this may be deduced from \cite{Kono}), so the Bockstein sequence for $B\mathrm{Spin}$ shows that $H^7(B\mathrm{Spin};\mathbb{Z}) = \mathbb{Z}/2$, which must therefore be generated by $\beta w_6$ as this reduces modulo 2 to $\mathrm{Sq}^1(w_6)=w_7$. Thus $\kappa = \beta(w_6)$.
\end{proof}
As $w_6 \cdot u_{-6} = \mathrm{Sq}^6(u_{-6})$, we may write the map $k$ as the composition
$$k: \Sigma^{-6} M\mathrm{Spin} \overset{u_{-6}}\lra \Sigma^{-6} H\mathbb{Z}/2 \overset{\mathrm{Sq}^6}\lra H\mathbb{Z}/2 \overset{\beta}\lra \Sigma H\mathbb{Z}$$
and so we may form a spectrum $E$ fitting into the diagram
\[\begin{tikzcd}
MT\mathrm{Spin}(6) \rar \dar& \Sigma^{-6} M\mathrm{Spin} \rar{\partial} \dar{u_{-6}}& \Sigma F \dar{\Sigma x}\\
E \rar& \Sigma^{-6} H\mathbb{Z}/2 \rar{\beta\mathrm{Sq}^6}& \Sigma H\mathbb{Z}
\end{tikzcd}\]
in which the rows are homotopy cofibre sequences. Smashing with $BG_+$ and considering the map of long exact sequences gives
\[\begin{tikzcd}
\Omega_8^{\mathrm{Spin}}(BG) \rar \dar & H_8(BG;\mathbb{Z}/2) \dar{(\beta\mathrm{Sq}^6)_*=0}\\
\mathbb{Z}/4 \mathrm{op}lus H_1(BG;\mathbb{Z}) \rar{\mathrm{pr}_2} \dar& H_1(BG;\mathbb{Z}) \dar\\
\pi_1(MT\mathrm{Spin}(6)\wedge BG_+) \rar \dar& \pi_1(E\wedge BG_+) \dar \arrow[bend right=40, dashed, swap]{u}{\sigma_G}\\
\Omega_7^{\mathrm{Spin}}(BG) \rar \dar& H_7(BG;\mathbb{Z}/2) \dar{(\beta\mathrm{Sq}^6)_*=0}\\
0 \rar& H_0(BG;\mathbb{Z})
\end{tikzcd}\]
where the columns are exact and the two indicated maps are zero by instability of the (co)homology operation $\beta\mathrm{Sq}^6$ in these degrees. It therefore suffices to show that the right-hand short exact sequence has a dashed splitting $\sigma_G$ as indicated. (Note that the commutativity of the top square gives another proof of Lemma \ref{lem:ConnectingTrivial}.)
\subsubsection{Reducing the splitting problem to cyclic groups}
The abelianisation homomorphism $a: G \to G^{ab}$ induces a map on the short exact sequences of the form
\[\begin{tikzcd}
0 \rar \dar& H_1(BG;\mathbb{Z}) \rar \dar{=} & \pi_1(E \wedge BG_+) \dar{a_*} \rar & H_7(BG;\mathbb{Z}/2) \dar{a_*} \rar& 0 \dar\\
0 \rar & H_1(BG^{ab};\mathbb{Z}) \rar & \arrow[l, bend right=30, dashed, "\sigma_{G^{ab}}"] \pi_1(E \wedge BG^{ab}_+) \rar & H_7(BG^{ab};\mathbb{Z}/2) \rar& 0
\end{tikzcd}\]
so if we can show that the short exact sequence is split for $G^{ab}$, say via the dashed homomorphism $\sigma_{G^{ab}}$, then the sequence for $G$ is also split, via $\sigma_G := \sigma_{G^{ab}} \circ a_*$. This reduces us to studying abelian groups.
On the other hand, if such short exact sequences are split for each cyclic group of prime power order or $\mathbb{Z}$, then we may write $G^{ab} = C_1 \mathrm{op}lus \cdots \mathrm{op}lus C_n$ for cyclic groups $C_i$ of prime power order or $\mathbb{Z}$ and combine their splittings to obtain one for $G^{ab}$ (though of course it depends on the choice of expression for $G^{ab}$ as a sum of cyclic subgroups, so is not canonical). We have therefore reduced the question of splitting the short exact sequences \eqref{eq:ToSplit2} to the case of such cyclic groups.
\subsubsection{The splitting for $G=\mathbb{Z}/p^k$ or $\mathbb{Z}$}
If $G=\mathbb{Z}$ or $G=\mathbb{Z}/p^k$ with $p$ odd then $H_7(BG;\mathbb{Z}/2)=0$ and so the short exact sequence becomes
$$0 \lra G \lra \pi_1(E \wedge BG_+) \lra 0 \lra 0$$
which is certainly split.
For $G=\mathbb{Z}/2^k$ we must go to more trouble: in this case the short exact sequence becomes
\begin{equation}\label{eq:Z2kSeq}
0 \lra \mathbb{Z}/2^k \lra \pi_1(E \wedge B\mathbb{Z}/2^k_+) \lra \mathbb{Z}/2 \lra 0
\end{equation}
so is either split or else $\pi_1(E \wedge B\mathbb{Z}/2^k_+) \cong \mathbb{Z}/2^{k+1}$. Consider first smashing the cofibre sequence defining $E$ with $S/2$, giving the cofibration sequence
$$S/2 \wedge E \lra S/2 \wedge \Sigma^{-6} H\mathbb{Z}/2 \simeq \Sigma^{-6} H\mathbb{Z}/2 \vee \Sigma^{-5} H\mathbb{Z}/2 \overset{\mathrm{Sq}^7 \vee \mathrm{Sq}^6}\lra \Sigma H\mathbb{Z}/2 \simeq S/2 \wedge H\mathbb{Z}.$$
Now further smashing this with $B\mathbb{Z}/2^k_+$ and considering the long exact sequence on homotopy groups gives an exact sequence
\begin{equation*}
\begin{tikzcd}
H_8(B\mathbb{Z}/2^k;\mathbb{Z}/2) \mathrm{op}lus H_7(B\mathbb{Z}/2^k;\mathbb{Z}/2) \rar{\mathrm{Sq}^7_* \mathrm{op}lus \mathrm{Sq}^6_*} \ar[draw=none]{d}[name=X, anchor=center]{} & H_1(B\mathbb{Z}/2^k;\mathbb{Z}/2)
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (X.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]{dl} \\
\pi_1(S/2 \wedge E\wedge B\mathbb{Z}/2^k_+) \rar \ar[draw=none]{d}[name=Y, anchor=center]{}& H_7(B\mathbb{Z}/2^k;\mathbb{Z}/2) \mathrm{op}lus H_6(B\mathbb{Z}/2^k;\mathbb{Z}/2)
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Y.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]{dl}[near end]{\mathrm{Sq}^7_* \mathrm{op}lus \mathrm{Sq}^6_*}\\
H_0(B\mathbb{Z}/2^k;\mathbb{Z}/2)
\end{tikzcd}
\end{equation*}
and so $\pi_1(S/2 \wedge E\wedge B\mathbb{Z}/2^k_+)$ has cardinality 8, because the homology operations $\mathrm{Sq}^7_*$ and $\mathrm{Sq}^6_*$ are zero in these degrees by instability. On the other hand computing with the long exact sequence as above gives an exact sequence
$$0 \lra \mathbb{Z} = H_0(B\mathbb{Z}/2^k ; \mathbb{Z}) \lra \pi_0(E \wedge B\mathbb{Z}/2^k_+) \lra H_6(B\mathbb{Z}/2^k ; \mathbb{Z}/2) = \mathbb{Z}/2 \lra 0$$
which is split via $B\mathbb{Z}/2^k_+ \to S^0$. Thus
$$\mathrm{Ker}(2 \cdot - : \pi_0(E \wedge B\mathbb{Z}/2^k_+) \to \pi_0(E \wedge B\mathbb{Z}/2^k_+)) = \mathbb{Z}/2$$
and so, as $\pi_1(S/2 \wedge E\wedge B\mathbb{Z}/2^k_+)$ has cardinality 8, it follows that
$$\mathrm{Coker}(2 \cdot - : \pi_1(E \wedge B\mathbb{Z}/2^k_+) \to \pi_1(E \wedge B\mathbb{Z}/2^k_+))$$
has cardinality 4. Thus $\pi_1(E \wedge B\mathbb{Z}/2^k_+)$ cannot be cyclic, so \eqref{eq:Z2kSeq} is split.
\noindent \textbf{Acknowledgements.} The authors would like to thank M.\ Krannich for useful comments on a draft of this paper. S.\ Galatius was partially supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No.\ 682922), and by NSF grant DMS-1405001. O.\ Randal-Williams was partially supported by the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement No.\ 756444).
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Complexity of the path avoiding forbidden pairs problem revisited}
\author{Jakub Kov\'a\v c}
\ead{[email protected]}
\address{Department of Computer Science,
Comenius University,
Mlynsk\'a Dolina,\\842~48 Bratislava, Slovakia}
\begin{abstract}
Let $G=(V,E)$ be a directed acyclic graph with two distinguished vertices $s, t$,
and let $F$ be a set of \emph{forbidden pairs} of vertices. We say that a path in $G$
is \emph{safe}, if it contains at most one vertex from each pair $\{u,v\}\in F$.
Given $G$ and $F$, the \emph{path avoiding forbidden pairs} (PAFP) problem is
to find a safe $s$--$t$ path in $G$.
We systematically study the complexity of different special cases of the PAFP problem
defined by the mutual positions of fobidden pairs.
Fix one topological ordering $\prec$ of vertices; we say that pairs $\{u,v\}$ and $\{x,y\}$ are
\emph{disjoint}, if $u\prec v \prec x\prec y$, \emph{nested}, if $u\prec x\prec y \prec v$, and
\emph{halving}, if $u\prec x\prec v\prec y$.
The PAFP problem is known to be NP-hard in general or if no two pairs are disjoint;
we prove that it remains NP-hard even when no two forbidden pairs are nested.
On the other hand, if no two pairs are halving, the problem is known to be solvable in
cubic time. We simplify and improve this result by showing an $O(M(n))$ time algorithm,
where $M(n)$ is the time to multiply two $n\times n$ boolean matrices.
\end{abstract}
\begin{keyword}
path \sep forbidden pair \sep NP-hard \sep dynamic programming
\end{keyword}
\end{frontmatter}
\input intro
\input prelim
\input ordered2
\input paren
\input concl
\subsubsection*{Acknowledgements.}
The autor would like to thank Bro\v na Brejov\' a for many constructive comments.
The research of Jakub Kov\'a\v c is supported by
APVV grant SK-CN-0007-09,
Marie Curie Fellowship IRG-231025 to Dr.\ Bro\v na Brejov\' a,
Comenius University grant UK/121/2011, and
by National Scholarship Programme (SAIA), Slovak Republic.
Preliminary version of this work appeared in \citet{pcr}.
\end{document} |
\begin{document}
\title{Social Cost Guarantees in Smart Route Guidance
}
\author{Paolo Serafino\inst{1} \and
Carmine Ventre\inst{2} \and
Long Tran-Thanh\inst{3} \and
Jie Zhang\inst{3} \and
Bo An\inst{4} \and
Nick Jennings\inst{5}
}
\institute{Gran Sasso Science Institute, Italy
\email{[email protected]}\\
\and
University of Essex, UK, \email{[email protected]}
\and
University of Southampton, UK, \email{\{ltt08r,jie.zhang\}@soton.ac.uk }
\and
Nanyang Technological University, Singapore,
\email{[email protected]}
\and
Imperial College London, UK, \email{[email protected]}}
\maketitle
\begin{abstract}
We model and study the problem of assigning traffic in an urban road network infrastructure.
In our model, each driver submits their intended destination
and is assigned a route to follow
that minimizes the social cost (i.e., travel distance of all the drivers).
We assume drivers are strategic and try to manipulate the system (i.e., misreport their intended destination and/or deviate from the assigned route) if they can reduce their travel distance by doing so.
Such strategic behavior is highly undesirable as it can lead to an overall suboptimal traffic assignment and cause congestion.
To alleviate this problem, we develop moneyless mechanisms that are resilient to
manipulation by the agents and offer provable approximation guarantees on the social cost obtained by the solution.
We then empirically test the mechanisms studied in the paper, showing that they can be effectively used in practice in order to compute manipulation resistant traffic allocations.
\end{abstract}
\section{Introduction}
Recent years have witnessed increasing interest in the development of efficient traffic control systems \cite{osorio2015urban,leontiadis2011effectiveness,djahel2013adaptive}. This is motivated by the significant negative impact on the quality of life of both road users and residents caused by heavy traffic congestion levels in large cities such as London, Beijing, and Los Angeles.
Indeed, heavy congestion is known to be a major cause of air and noise pollution, which are widely recognized as the main cause of many health issues~\cite{Kryzanowski2005,Stanfeld2003}.
Adding to this is the economic cost associated with the large amount of time spent in traffic jams, which reduces the productivity of the economy~\cite{goodwin2004economic}.
Moreover, the situation is expected to become significantly worse in the future when the population, and thus the traffic flow, in large cities will be much bigger than at present.
Unfortunately, conventional traffic control systems have proven unable to efficiently decrease congestion levels, as they are not designed to be adaptive to the dynamics of city traffic, which changes over space and time.
On the other hand, it has been shown~\cite{raphael2015goods,levin2017optimizing} that by putting some sort of intelligence/smartness into traffic control systems, we can make them adapt to the changes of the traffic flow.
A key objective within these smart traffic control systems is to address the so-called \emph{traffic assignment problem} (TAP), in which mobile agents (i.e., typically drivers) declare their intended destination to the system, perhaps via their satellite navigation systems, and are then assigned a route to follow, in such a way that some objective function of the overall traffic flow in the system is optimized (i.e., minimizing the total traveled distance or maintaining an efficient traffic load balance).
As these agents are typically self-interested and strategic (i.e., they try to maximize their own utility, disregarding whether this is detrimental to the global optimization goal), they may manipulate the system whenever they can benefit from doing so~\cite{levin2017optimizing,vasirani2012market}.
This kind of opportunistic behavior is highly undesirable as it will increase the total social cost (i.e., decreasing the total load balance or increasing the total congestion level).
As such, incentivizing agents not to be strategic is a key design objective of these traffic assignment systems~\cite{raphael2015goods,levin2017optimizing,vasirani2012market}.
Given this, we focus on \emph{strategyproof}
TAP mechanisms, which guarantee that it is in the agent's best interest to always report her true destination and follow the assigned route.
Furthermore, we assume that money transfers between the mechanism and the agents are not available.
This is a common assumption in many domains \cite{ProcacciaT13} that will facilitate the likely real-world deployment of the system by lowering set up costs (i.e., avoiding the construction of tolling booths).
The remainder of the paper is organized as follows.
In Section 2 we discuss related works. In Section 3 we introduce our model for TAP and prove that Pareto optimal allocations theoretically guarantee that agents will follow their assigned paths (Theorem \ref{thm:pareto_opt}).
We then move to study deterministic (Section \ref{sec:deterministic_mechs}) and randomized (Section \ref{sec:randomized_mechs}) Pareto optimal mechanisms for our problem.
We show that the approximation ratio of \textit{deterministic} strategyproof mechanisms is lower bounded by 3 (Theorem \ref{thm:deterministic_LB}), while the Serial Dictatorship mechanism can achieve an upper bound of $2^n - 1$ and it is Pareto-optimal and non-bossy (Theorems \ref{thm:apx_SD} and \ref{thm:tightness_SD}), where $n$ is the number of agents (Theorems \ref{thm:apx_SD} and \ref{thm:tightness_SD}).
Furthermore, if we require non-bossiness and Pareto optimality, we are able to close this approximation ratio gap by showing that the Bipolar Serial Dictatorship mechanism is the \emph{only} strategyproof mechanism.
For \textit{randomized} mechanisms, we show that the approximation ratio is lower bounded by $\frac{11}{10}$ (Theorem \ref{thm:randomizedLB}). In addition, the Random Serial Dictatorship mechanism can achieve an $n$-approximation (Theorems 8 and 9), while still preserving the desired properties of Pareto-optimality and non-bossyness.
In addition to these theoretical results, we present an extensive experimental evaluation on traffic networks generated from real road network data, which show how the mechanisms studied in the paper provide good performance in practice, despite the high theoretical worst case approximation guarantee.
\noindent
Full proofs and definitions can be found in the Appendix.
\section{Related Work}\label{sec:related_work}
There is a large body of literature on traffic network modelling and assignment~\cite{beckmann1956studies,skabardonis1997improved,coogan2015compartmental,daganzo1994cell}.
However, these works typically ignore the strategic behaviour of participating agents.
Nevertheless, they can be useful to model the underlying traffic network in our work.
In particular, we follow the widely used traffic model proposed in~\cite{beckmann1956studies}.
To tackle the strategic behaviour of the agents, several researchers have suggested employing mechanism design with money and auction theory for traffic control~\cite{raphael2015goods,levin2017optimizing,vasirani2012market,Brenner2010}.
These works typically rely on the computation of the VCG auction in order to assign vehicles to paths.
However, they require monetary incentives, and typically focus on a local control level, such as intersection management (as VCG is typically computationally hard, and thus, not readily scalable~\cite{conitzer2006failures}).
A number of researchers have focused on mechanism design without money~\cite{ProcacciaT13,resource_augmentation}.
However, none of these mechanisms can be easily applied to the traffic assignment problem, as they do not take into account the features of the underlying traffic network structure.
As we will show, TAP bears some resemblance to the problem of assigning indivisible objects \cite{DBLP:journals/jet/BogomolnaiaDE05,Svensson1999,Filos-RatsikasF014}, although these results are not directly applicable to our scenario. Indeed TAP has a much more complex structure (mainly due to the underlying traffic network topology) which traditional assignment mechanisms fail to address.
\section{Model and Preliminary Definitions}\label{sec:model}
A \emph{traffic assignment problem} (TAP) consists of a set of agents $A=\{a_1, \ldots, a_n \}$ and a road network infrastructure, represented as a directed graph $G=(V,E)$, where: (\emph{i}) $V = \{v_1,\ldots, v_{|V|}\}$ is the set of nodes representing the junctions of the road network infrastructure; and (\emph{ii}) $E\subseteq V\times V$ is the set of directed edges representing one-way road segments.
Each edge $e\in E$ has a \emph{capacity} $c : E \rightarrow \mathbb{N}^+$, which determines the maximum number of agents that can travel through the edge at any given time, and a \emph{weight function} $w: E \rightarrow \mathbb{R}^+$ which represents the cost incurred by the agent traveling through the edge (i.e., travel distance). Furthermore, each edge is associated to a \emph{transit time} $\tau: E \rightarrow \mathbb{Z}^+$ which represents the \emph{free travel time of the edge} (i.e., the minimum travel time needed to travel through the road at maximum allowed speed).
This means that agent $a_i$ setting off at time $t$ from node $v_o$ and heading to node $v_d$ through the edge $(v_o,v_d)$ will reach node $v_d$ at time $t+ \tau(v_o,v_d)$, and will occupy edge $(v_o,v_d)$ only in the time interval $[t, t+\tau(v_o,v_d)]$. Unless stated otherwise, we assume that edges $(u,v)$ and $(v,u)$ are \emph{symmetrical}: for all $(u,v),(v,u)\in E$ $c(u,v)=c(v,u)$, $w(u,v)=w(v,u)$ and $\tau (u,v)= \tau (v,u)$.
As in \cite{Nesterov2003}, we assume that if the flow of traffic through an edge does not exceed its capacity, then no congestion occurs and the traveling time equals the free travel time.
Initially, at time $t=0$, agents reside on a (publicly known) set $O\subseteq V$ of nodes\footnote{Restricting origins/destinations of journeys to road junctions is without loss of generality since fictitious nodes that serve the sole purpose of acting as starting/ending point of a journey can always be created by edge splitting operations.} of the graph, $O_i$ being the initial location of agent $a_i$.
Each agent $a_i\in A$ wants to reach an intended destination $D_i\in V$, which is the agent's private information and will be referred to in the remainder as her \emph{type}.
Agents submit (or \emph{bid}) a destination to an \emph{allocation mechanism}, which then assigns each agent a path in order to optimize a certain objective function.
More formally, let $\mathcal{P}$ be the set of all possible simple paths between any two nodes in $G$.
Let $\mathbf{D}=\left(D_1,\ldots,D_n \right)\in V^n$ be a \emph{vector of declarations} (also referred to as \emph{bids}) by the agents and $\mathbf{D}I$ be the vector of declarations of all agents but $a_i$.
A \emph{mechanism} $M^{G, O} : V^n \rightarrow \mathcal{P}^n$ maps a vector of declarations to \emph{feasible paths} (i.e., not exceeding the capacity of the edges at any given time) on $G$, given the initial locations $O$ of the agents.
We write $M(\mathbf{D})$ instead of $M^{G,O}(\mathbf{D})$ when $G$ and $O$ can be deduced from the context.
The path associated to agent $a_i$ is denoted as $M_i(\mathbf{D})$.
A traffic assignment $S=M(\mathbf{D})$ induces a \emph{flow over time}\footnote{Sometimes also referred to as \emph{dynamic flow} in the literature. We prefer the term \emph{flow over} time as the adjective \emph{dynamic} has often been used in many algorithmic settings to refer to problems where the input data
arrive online or change over time. We assume that all the agents are present at time $t=0$ and the network is cleared after the last agent reaches their destination.} $f_{S} : E\times \mathcal{T} \rightarrow \mathbb{N^+}$, where $\mathcal{T}$ is a suitable discretization of time w.r.t. the transit times of the edges of $G$ (for simplicity we will assume that $\mathcal{T} = \{0,1,\ldots,T\}$, where $T$ is a time horizon sufficient for the network to clear. Thus, $f_{S}(u,v;t) = |\{a_i \in A | (u,v)\in S_i \}|$ is the number of agents that are assigned a path that contains edge $(u,v)$ at time $t\in \mathcal{T}$. Feasibility constraints imply that $f_{S}(u,v;t)\leq c(u,v)$ for all $t\in\mathcal{T}$.
In the remainder, without loss of generality, we will study the problem on the \emph{time-expanded network} \cite{FordFulkerson1,FordFulkersonBook} of $G$ and consider the \emph{static} flow through it (i.e, the transit of an agent over and edge is instantaneous).
A time-expanded network is a properly constructed directed graph with cost and capacity functions on the edges just like $G$, but no transit time
(i.e. travel time is instantaneous through all the edges). For completeness, we give the definition of time expanded networks in the Appendix.
This is without loss of generality from the point of view of SP, Pareto-optimality, non-bossines and approximation guarantee since
it is well known (see
\cite{FordFulkerson1,FordFulkersonBook}) that a flow over time is equivalent to a static flow on the corresponding time-expanded
network.
Let $f^{-i}_S : E \rightarrow \mathbb{N}$ be the flow induced by traffic assignment $S$ generated by agents $A\setminus \{a_i \}$, formally for all $e \in E$, $f^{-i}_{S}(e) = |\{ a_j \in A: e \in S_j, j \neq i\}|$.
The \emph{residual graph} $G_{f}^{-i}$ is a graph such that: (\emph{i}) $G_{f}^{-i}$ has the same nodes and edges as $G$; (\emph{ii}) each edge $e\in E$ of $G_{f}^{-i}$ has capacity $c(e)-{f}^{-i}_S(e)$.
For any two nodes $u,v\in V$, let $\mathcal{P}_{u,v}$ denote the set of simple paths in $G$ connecting $u$ to $v$.
Furthermore, for all traffic assignments $S=M(\mathbf{D})$ and all agents $a_i$, let $\mathcal{P}_{u,v}^i (S) = \{P\in \mathcal{P}_{u,v} | \forall e \in P, c(e)>{f}^{-i}_{S}(e)\}$.
Informally, $\mathcal{P}_{u,v}^{i}(S)$ is the set of paths connecting $u$ and $v$ that have spare capacity from the perspective of agent $a_i$ (i.e., they can be used by agent $a_i$) when the other agents implement $S$.
Then, the set of reactions available to agent $a_i$ having type $D_i$ at allocation $S$ is defined as $R_i (S) = \mathcal{P}^i_{O_i,D_i} (S)$.
Agents are not constrained to follow their assigned path but can choose a different one, subject to capacity constraints\footnote{We do not prevent agents from using edges other than the ones belonging to their assigned paths, as doing so would result in a waste of public resources (i.e., road capacity). To avoid congestion, though, we assume that agents not following their assigned route can be disincentivized from using an edge that, according to the scheduled traffic, is filled to capacity. This can be easily implemented in a smart traffic control system through the use of traffic cameras that check cars' number plates.}.
To model this, as per \cite{Nissim_2012}, we assume that, after the mechanism computes a traffic allocation, the agents can \emph{react} by choosing an action from a set $R_i \subseteq \mathcal{P}$.
Hence, the actual \emph{cost function} of an agent depends on: (\emph{i}) her true type $D_i$; (\emph{ii}) the allocation $S$ chosen by the mechanism on input the bids reported by the agents; and (\emph{iii}) the reactions chosen by the agents.
We can now formally define the cost function of an agent. Given an allocation $S'=M(D'_i,\mathbf{D}I)$, the cost of an agent of type $D_i$ with respect to $S'$ is defined as:
$cost_i(S',D_i) = \min_{P\in R_i(S')}w(P)$
where $w(P)=\sum_{(u,v)\in P}w(u,v)$ denotes the cost of $P$.
We assume that agents are risk-neutral.
In what follows, we define a set of desiderata for our allocation mechanism, namely: (\emph{i}) strategyproofness, (\emph{ii}) Pareto optimality and (\emph{iii}) non-bossiness.
A deterministic mechanism $M$ is \emph{strategyproof} (SP for short) if, for all agents $a_i$, for all declarations $D_i$ and $D'_i$ and all declarations of the other agents $\mathbf{D}I$, agent $a_i$ cannot decrease her cost by misreporting her true type, namely:
\begin{equation}
cost_i(M(\mathbf{D}),D_i)\leq cost_i(M(D'_i,\mathbf{D}I),D_i) \label{eq:sp}
\end{equation}
A randomized mechanism is \emph{strategyproof in expectation} if \eqref{eq:sp} holds in expectation (i.e., over the random choices of the mechanism).
A randomized mechanism is \emph{universally strategyproof} if agents cannot gain by
lying regardless of the random choices made by the mechanism, i.e., the output of the mechanism is a
distribution over strategyproof deterministic allocations.
The \emph{social cost} of an allocation $S$ is defined as $SC(S,\mathbf{D}) = \sum_{a_i\in A}cost_i(S,D_i)$.
A mechanism $OPT$ is \emph{optimal} for TAP if $OPT(\mathbf{D}) \in \arg\min_{S\in \mathcal{P}^n} SC(S,\mathbf{D})$ for all $\mathbf{D}$.
A mechanism $M$ is an $\alpha$--approximation (w.r.t the optimal social cost) with $\alpha\in \mathbb{R}$, $\alpha \geq 1$, being referred to as the \emph{approximation ratio} of $M$, if, for all $\mathbf{D}$, $SC(M(\mathbf{D}),\mathbf{D})\leq \alpha\cdot SC(OPT(\mathbf{D}),\mathbf{D})$.
A traffic allocation $S\in \mathcal{P}^n$ is \emph{Pareto optimal} if there exists no other feasible traffic allocation $S'$ such that $cost_j(S',D_j)\leq cost_j(S,D_j)$ for all $a_j$, and $cost_k(S',D_k)<cost_k(S,D_k)$ for some $a_k$.
Pareto optimal allocations are of particular interest in our scenario, because, as proven in Theorem \ref{thm:pareto_opt}, they are a min-cost response in the available reactions $R_i(S)$ of an agent. This gives us a theoretical guarantee that agents will actually implement Pareto optimal solutions returned by the mechanism.
\begin{theorem}\label{thm:pareto_opt}
Let $S=M(\mathbf{D})$ be a traffic assignment and let $R_i(S)$ be the set of reactions available to $a_i$ at $S$.
If $S$ is Pareto optimal, then $M_i(\mathbf{D})\in \arg\min_{P\in R_i(S)}w(P)$.
\end{theorem}
\iftoggle{proof_thm_1}{
\begin{proof}
Let us suppose by contradiction that there exists a reaction $r'_i\in R_i$ that is feasible and strictly better than $M_i(\mathbf{D})$, i.e. $c(r'_i)<c(M_i(\mathbf{D}))$.
Then, a route assignment $M'$ such that $M'_j = M_j$ for all $j\neq i$ and $M'_i = r_i$ is still feasible. Since $cost_j(M')=cost_j(M')$ for all $j\neq i$, and $cost_i(M')<cost_i(M)$, $M$ is not Pareto optimal.
\end{proof}
}
Finally, a mechanism $M$ is \emph{non-bossy} if $M_i(\mathbf{D}) = M_i(D'_i,\mathbf{D}I)$ implies that $M_j(\mathbf{D}) = M_j(D'_i,\mathbf{D}I)$, for all $a_i,a_j\in N$ and all $\mathbf{D}$ and $D'_i$. In other words, non-bossyness excludes (arguably undesirable) mechanisms that allow one agent to change the allocation of other agents without changing her own too.
In the remainder of this paper, we focus on strategyproof mechanisms for TAP that approximately achieve the optimal social cost. In particular, we are interested in mechanisms that are also Pareto-optimal and non-bossy.
\section{Deterministic Mechanisms}\label{sec:deterministic_mechs}
In this section, we discuss deterministic mechanisms for TAP.
In particular, we first provide a lower bound on the approximation ratio of SP deterministic mechanisms.
\begin{theorem}\label{thm:deterministic_LB}
There is no $\alpha$-approximate deterministic SP mechanism for the traffic assignment problem with $\alpha<3 - \varepsilon$, for any $\varepsilon>0$.
\end{theorem}
\iftoggle{proof_thm_1_sketch}{
\begin{proof}
\emph{(Sketch)}
Consider the graph depicted in Figure \ref{fig:LB_instance_2}, where the labels on the edges represent their capacity (red) and length (black). There are two agents $a_1$ and $a_2$ at node $A$, whose intended destinations are, respectively, $D$ and $G$.
The length of path $(B,C)$ is $K = \min\{2, \frac{10-4\varepsilon}{\varepsilon}\}$ and the length of path $(F,E)$ is $K-1$.
\begin{figure}
\caption{Lower bound instance}
\label{fig:LB_instance_2}
\end{figure}
The instance has two Pareto optimal solutions, depending on which player is allocated the edge $(A,F)$. The optimal allocation $\mathbf{P}^*=OPT(\mathbf{D})$ is $P^*_1 = (A,B,C,D)$ and $P^*_2 = (A,F,E,G)$, $cost_1(\mathbf{P}^*,D) = K+2$ and $cost_2(\mathbf{P}^*,G) = K+1$, and $SC(\mathbf{P}^*,\mathbf{D})=2K+3$. The second best solution is $P_1 = (A,F,E,D)$ and $P_2 = (A,B,C,E,G)$.
$M$ achieves an approximation not better than $3-\varepsilon$, a contradiction.
If $M$ returns the optimal allocation, agent $a_1$ declares $D'_1=F$ instead of her true type. By SP $M$ cannot allocate the edge $(A, F)$ to $a_1$ ($a_1$ can use path $(A,F,E,D)$ and reach her true destination).
Therefore, $M(D_1', D_2)$ must return $P'_1 = (A,B,C,E,F)$ and $P'_2 = (A,F,E,G)$, with $SC((P'_1,P'_2), (D_1', D_2)) = 3K+2$, whilst the optimum for this instance is $OPT_1(D'_1,D_2) = (A,F)$ and $OPT_2(D'_1,D_2) =(A,B,C,E,G)$. The social cost of the optimum is then $SC(OPT(D'_1,D_2),(D'_1,D_2)) = K+4$.
Therefore $M$ has an approximation ratio higher than $3-\varepsilon$.
If $M(\mathbf{D})$ returns $M_1(\mathbf{D}) = P_1$ and $M_2(\mathbf{D}) = P_2$, then agent $a_2$ reports $D'_2=F$ instead of her true type.
The optimal allocation is $OPT_1(D_1,D_2') = (A,B,C,D)$ and $OPT_2(D_1,D_2') = (A,F)$, and $SC(OPT(D'_2,D_1),(D'_2,D_1)) = K+3$
(i.e., agent $a_2$ can use the route $(A,F,E,G)$ to reach her true destination).
In this case the best (in terms of approximation ratio) SP allocation is $P'_1=(A,F,E,D)$ and $P'_2=(A,B,C,E,F)$, with a cost of $SC((P'_1,P'_2),(D_1,D_2')) = 3K+2$.
This solution has an approximation ratio higher than $3-\varepsilon$.
\end{proof}
}{
\iftoggle{proof_thm_1}{
\begin{proof}
Given $\varepsilon>0$, consider the graph depicted in Figure \ref{fig:LB_instance_2}, where the labels on the edges represent their capacity (red) and length (black). The instance we consider has two agents $A= \{a_1,a_2\}$, both initially located at node $A$, whose intended destination is $D$ and $G$, respectively (namely, $\mathbf{D}=(D,G)$).
The length of the path $(B,C)$ is $K = \min\{2, \frac{10-4\varepsilon}{\varepsilon}\}$ and the length of path $(F,E)$ is $K-1$.
\begin{figure}
\caption{Lower bound instance}
\label{fig:LB_instance_2}
\end{figure}
Let us consider a generic $\alpha$-approximate mechanism $M$.
Assume by contradiction that $M$ is strategyproof and $\alpha$-approximate with $\alpha<3-\varepsilon$.
The instance has two Pareto optimal solutions, depending on which player is allocated the edge $(A,F)$ (note that only one agent at a time can use edge $(A,F)$ as its capacity is 1). The optimal allocation $\mathbf{P}^*=OPT(\mathbf{D})$ is $P^*_1 = (A,B,C,D)$ and $P^*_2 = (A,F,E,G)$, $cost_1(\mathbf{P}^*,D) = K+2$ and $cost_2(\mathbf{P}^*,G) = K+1$, and $SC(\mathbf{P}^*,\mathbf{D})=2K+3$. The second best solution is $P_1 = (A,F,E,D)$ and $P_2 = (A,B,C,E,G)$. We note that $cost_1(P_1,D_1)=K+1$ and $cost_2(P_2,D_2)=K+3$, for a social cost of $SC(M(\mathbf{D}),\mathbf{D}) = 2K+4$.
We are going to prove that, regardless of the solution $M$ returns on this instance, there is another instance where to maintain SP, $M$ achieves an approximation not better than $3-\varepsilon$, a contradiction.
Let us assume first that $M$ returns the optimal allocation.
If agent $a_1$ declares $D'_1=F$ instead of her true type, by SP, $M$ cannot allocate the edge $(A, F)$ to $a_1$. In fact, assume for the sake of contradiction that $M(D_1', D_2)$ allocates $(A,F)$ to $a_1$.
Then $a_2$ is allocated path $(A,B,C,E,G)$ and $a_1$ can use path $(A,F,E,D)$ and reach her true destination, thus having:
$$cost_1(M(D'_1,D_2),D_1)=K+1<cost_1(M(\mathbf{D}),D_1)=K+2.$$
Therefore, $M(D_1', D_2)$ must return $P'_1 = (A,B,C,E,F)$ and $P'_2 = (A,F,E,G)$, with $SC((P'_1,P'_2), (D_1', D_2)) = 3K+2$, whilst the optimum for this instance is $OPT_1(D'_1,D_2) = (A,F)$ and $OPT_2(D'_1,D_2) =(A,B,C,E,G)$. The social cost of the optimum is then $SC(OPT(D'_1,D_2),(D'_1,D_2)) = K+4$.
Therefore $M$ has an approximation ratio higher than $3-\varepsilon$.
Let us now suppose that $M(\mathbf{D})$ returns $M_1(\mathbf{D}) = P_1$ and $M_2(\mathbf{D}) = P_2$.
In this case, consider the case that agent $a_2$ reports $D'_2=F$ instead of her true type.
The optimal allocation is $OPT_1(D_1,D_2') = (A,B,C,D)$ and $OPT_2(D_1,D_2') = (A,F)$, and $SC(OPT(D'_2,D_1),(D'_2,D_1)) = K+3$.
As before, this allocation is not strategyproof as:
\begin{eqnarray*}
cost_2(OPT_2(D_1, D'_2),D_2)= K+1 \\
< cost_2(M_2(D_1,D'_2),D_2)=K+3
\end{eqnarray*}
(i.e., agent $a_2$ can use the route $(A,F,E,G)$ to reach her true destination).
As above, one can easily check that in this case the best (in terms of approximation ratio) strategyproof allocation is $P'_1=(A,F,E,D)$ and $P'_2=(A,B,C,E,F)$, with a cost of $SC((P'_1,P'_2),(D_1,D_2')) = 3K+2$.
This solution has an approximation ratio higher than $3-\varepsilon$.
\end{proof}
}
}
The above theorem implies the following corollary:
\begin{corollary}
The optimal allocation is not strategyproof for TAP.
\end{corollary}
These impossibility results suggest that in order to achieve strategyproofness we have to give up on optimality.
This naturally leads to asking to what extent can we approximate the optimal social welfare while satisfy the desired properties.
As a first step to answer this question, we examine the well-known Serial Dictatorship mechanism that is deterministic and notoriously satisfies our three desiderata (i.e., strategyproofness, Pareto optimality and non-bossiness).
\begin{definition}
Mechanism Serial Dictatorship (SD), given an ordering $a_1\prec, \ldots, \prec a_n$ of the agents, allocates paths to agents in $n$ stages such that at stage $i$ agent $a_i$ is allocated her minimum cost path in the residual graph $G_f^{-\{a_1,\ldots,a_{i-1}\}}$.
\end{definition}
The following theorem proves that SD is indeed feasible under some mild conditions:
\begin{theorem}
If $G$ is $K$-edge-connected\footnote{A graph is $K$-edge-connected if it remains connected when strictly fewer than $K$ edges are removed.}, mechanism SD is feasible for $K$ agents.
\end{theorem}
\iftoggle{proof_thm_3}{
\begin{proof}
If the graph is $K$-edge connected, the allocation returned by the Serial Dictator will always be feasible, (i.e. paths assigned to different agents will not overlap and there is always an assignable path for each agents).
This follows from the fact that in a $K$-edge-connected graph there are at least $K$ edge disjoint paths between any pair of nodes.
\end{proof}
}
Next we provide an upper bound on the approximation ratio of SD, and thus, on its worst case performance.
In order to prove our result, we make the following assumption:
\begin{definition}
The \emph{deviation on capacious path assumption (DoCP)} assumes that whenever the SD mechanism allocates to an agent a path that is different from the one that the optimal mechanism would allocate, the assigned path has sufficient capacity to potentially be allocated to all the remaining agents.
\end{definition}
To better understand this assumption, consider the following example.
With reference to Figure \ref{fig:capacious_paths}, let $a_i$ be an agent and $P^*_i$ be the path she is assigned in the optimal allocation (i.e., $OPT_i= P^*_i$).
If agent $a_i$ is not assigned $P^*_i$ by SD, there must be an agent $a_j$, where $j\prec i$ in the ordering used by SD, such that: (\emph{i}) $SD_j= P_j\neq OPT_j$ and (\emph{ii}) $P_j\cap P^*_i \neq \emptyset$ and (\emph{iii}) at least one edge of $P^*_i$ is saturated after $a_j$ is assigned $P_j$.
In such a situation, we say that agent $a_i$ is blocked by agent $a_j$.
Let $\alpha_i \in P_j\cap P^*_i$ ($\beta_i \in P_j\cap P^*_i$, respectively) be the first (last, respectively) node of $P^*_i$ in $P_j$.
The DoCP assumption postulates that if $a_j$ blocks $a_i$, then the \emph{alternative path of blocked agent $a_i$ through blocking agent $a_j$} $\Gamma_{i}^{j} = (O_i,\alpha _i,O_j,D_j,\beta_i,D_i)$ has at least capacity $n-|\{a_k \in A | a_j \prec a_k\}|$ in the residual graph $G_f^{-\{a_1,\ldots
,a_{j}\}}$. That is, all agents yet to be assigned by SD after $a_j$ can be accommodated on this path.
We note that, by construction, if agent $a_i$ is blocked by agent $a_j$ then path $\Gamma^{j}_{i}$ always exists, although unless we assume DoCP, it might not have spare capacity to be assigned to agent $a_i$.
\begin{figure}
\caption{Deviation on capacious paths}
\label{fig:capacious_paths}
\end{figure}
It is not difficult to see that if we relax the DoCP assumption, then the approximation ratio of SD is not bounded by any function of the number of agents on certain pathological TAP instances.
\begin{theorem}\label{thm:apx_SD}
Under the DoCP assumption, SD is at most $(2^n-1)$-approximate.
\end{theorem}
\iftoggle{proof_thm_4_sketch}{
\begin{proof}[Proof sketch]
We prove the claim by induction on the number of players. Let $OPT_i$ denote the cost and solution (with a slight abuse of notation) of the optimal allocation that only considers bids of agents $j \leq i$. Similarly, let $SD_i$ denote the cost and solution of $SD$ on input all the bids of agents $j \leq i$.
Base of the induction ($i=1$): trivially $OPT_1=SD_1$.
Now assume that the claim is true for $i-1$ and, for $j \leq i$, let $P^*_j$ ($P_j$, respectively) be the path assigned to agent $j$ by $OPT_i$ ($SD_i$, respectively). For a path $P$, we let $w(P)$ denote the cost of the path in the given graph G.
We want to prove that under the DoCP assumption, the following holds:
\begin{equation}\label{eq:paths}
w(P_i) \leq OPT_i + SD_{i-1}.
\end{equation}
If $P^*_i=P_i$ then we are done.
Therefore, we can assume that $P^*_i \neq P_i$.
This means that the paths $P_j$ allocated to agents $j < i$ by $SD_i$ saturate some of the edges of $P^*_i$.
Now, for at least one of these agents, say $\bar{j}$, $P^*_{\bar{j}} \neq P_{\bar{j}}$ for otherwise also in $OPT_i$ path $P^*_i$ would be unavailable to $i$. But then $w(P_i) \leq w(\Gamma^{\bar{j}}_i)$, $\Gamma^{\bar{j}}_i$ being the path that connects $O_i$ to $D_i$ through $O_{\bar{j}}$, as per the definition of DoCP.
Note that, under the DoCP assumption, $\Gamma^{\bar{j}}_i$ is always feasible.
Since $\Gamma^{\bar{j}}_i$ uses only edges in $OPT_i \cup SD_{i-1}$ (i.e. $P^*_i$ and $P^*_j$ are in $OPT_i$, paths $(O_i,\alpha_i)$ and $(\beta_i,D_j)$ belong to $SD_{i-1}$), \eqref{eq:paths} is proven.
We finally observe that \eqref{eq:paths} and the inductive hypothesis yield:
\begin{align*}
SD_i & = SD_{i-1} + w(\Gamma^{\bar{j}}_i) \leq 2 SD_{i-1} + OPT_i \\
& \leq 2((2^{i-1}-1) OPT_{i-1}) + OPT_i \leq (2^i-1) OPT_i.
\end{align*}
\end{proof}
}{
\iftoggle{proof_thm_4}{
\begin{proof}
We are going to prove the claim by induction on the number of players. Specifically, let $OPT_i$ denote the cost and solution (with a slight abuse of notation) of the optimal allocation that only considers bids of agents $j \leq i$. Similarly, $SD_i$ denotes the cost and solution of $SD$ on input all the bids of agents $j \leq i$.
For the base of the induction with $i=1$ it is clear that $OPT_1=SD_1$.
Now assume that the claim is true for $i-1$ and, for $j \leq i$, let $P^*_j$ ($P_j$, respectively) be the path assigned to agent $j$ by $OPT_i$ ($SD_i$, respectively). For a path $P$, we let $w(P)$ denote the cost of the path in the given graph G.
We want to prove that under the DoCP assumption, the following holds:
\begin{equation}\label{eq:paths}
w(P_i) \leq OPT_i + SD_{i-1}.
\end{equation}
Let us begin by observing that if $P^*_i=P_i$ then we are done.
Therefore, we can assume that $P^*_i \neq P_i$.
This means that the paths $P_j$ allocated to agents $j < i$ by $SD_i$ saturate some of the edges of $P^*_i$.
Now, for at least one of these agents, say $\bar{j}$, $P^*_{\bar{j}} \neq P_{\bar{j}}$ for otherwise also in $OPT_i$ path $P^*_i$ would be unavailable to $i$. But then $w(P_i) \leq w(\Gamma^{\bar{j}}_i)$, $\Gamma^{\bar{j}}_i$ being the path that connects $O_i$ to $D_i$ through $O_{\bar{j}}$, as per the definition of DoCP.
Note that, under the DoCP assumption, $\Gamma^{\bar{j}}_i$ is always feasible.
Since $\Gamma^{\bar{j}}_i$ uses only edges in $OPT_i \cup SD_{i-1}$ (i.e. $P^*_i$ and $P^*_j$ are in $OPT_i$, paths $(O_i,\alpha_i)$ and $(\beta_i,D_j)$ belong to $SD_{i-1}$), \eqref{eq:paths} is proven.
We can then conclude the proof, by observing that \eqref{eq:paths}, along with the inductive hypothesis, yield:
\begin{align*}
SD_i = SD_{i-1} + w(\Gamma^{\bar{j}}_i) & \leq 2 SD_{i-1} + OPT_i \\
& \leq 2((2^{i-1}-1) OPT_{i-1}) + OPT_i \\ & \leq (2^i-1) OPT_i.
\end{align*}
\end{proof}
}
}
As the $(2^n-1)$-approximation ratio can be prohibitively large for large $n$, we ask ourselves whether we can further improve this upper bound. Unfortunately, the following theorem answers this question in the negative.
\begin{theorem}\label{thm:tightness_SD}
Under the DoCP assumption, the bound of Theorem \ref{thm:apx_SD} is tight.
\end{theorem}
\iftoggle{proof_thm_5}{
\begin{proof}
Let us consider the instance in Figure \ref{fig:sd_tight_instance}, where there are $n$ nodes $v_1,\ldots, v_n$ and $n$ agents $A=\{a_1,\ldots, a_n\}$ such that agent $a_i$ is initially located at node $v_i$. All agents want to reach the same destination $D$.
Each link has capacity $1$.
Agent $a_1$ has two paths to her destination $D$: one direct path that costs $1+\varepsilon$ (where $\varepsilon\ll 1$ is a small constant) and a path costing $1$ that goes through the node agent $a_2$ is initially located on.
Each agent $a_i$, for $i=2,\dots,n-1$ has two paths: one direct path that costs $\varepsilon$ and a path costing $2^{i-1}$ that goes through the node agent $a_{i+1}$ is initially located on.
Agent $a_n$ has two direct paths, costing $\varepsilon$ and $2^{n-1}$ respectively.
The optimal traffic assignment assigns agent $a_1$ to the path that costs $1+\varepsilon$ and the other agents to the path costing $\varepsilon$, and has a cost of $1+\varepsilon n$.
Let us consider ordering $a_1\prec a_2 \prec \ldots \prec a_n$.
On this ordering, mechanism SD assigns agent $a_1$ the path costing $1$, and to each agent $a_i$, for $i=2,\ldots, a_n$ the path costing $2^{i-1}$ for a total cost of $\sum_{i=0}^{n-1} 2^i = 2^n-1$.
For $\varepsilon$ close to $0$, the approximation ratio of SD on the instance depicted in Figure \ref{fig:sd_tight_instance} is hence close to $2^n-1$.
\begin{figure}
\caption{Tight instance for SD}
\label{fig:sd_tight_instance}
\end{figure}
\end{proof}
}
We now provide a characterization of SP, Pareto-optimal, and non-bossy mechanisms for a subset of instances of TAP, named TAP$^+$\, and we prove that the family of all mechanisms satisfying the above properties is comprised by a generalization of SD, namely \emph{Bi-polar Serial Dictatorship} (BSD).
Such a characterization extends naturally to TAP instances.
TAP$^+${} is subset of instances of TAP having a peculiar structure: (\emph{i}) every agent has the same source node $O$; (\emph{ii}) $O$ has outgoing edges with unitary capacity and no ingoing edges, let $E_O = \{(O,v_1),\ldots, (O,v_m)\}$ denote the set of outgoing edges of $O$; and (\emph{iii}) the set of possible destinations that the agents can declare is restricted to a given subset $\mathcal{D}\subset V$.
\begin{definition}
Given an ordering of the agents $\{i_1,i_2\}\prec i_3\prec \ldots\prec i_n$ and a bipartition $\{X_1,X_2\}$ of the set of alternatives $X$ (i.e., paths in the case of TAP) such that $X_1\cap X_2 = \emptyset$ and $X_1 \cup X_2 = X$, a BSD mechanism executes SD with ordering $i_2\prec i_1 \prec \ldots \prec i_n$ if $\min_{x\in X} cost_1(x) = \min_{x\in X} cost_2(x) = x\in X_2$; otherwise SD with ordering $i_1\prec i_2\prec \ldots \prec i_n$ is executed.\end{definition}
\begin{theorem}\label{thm:reduction}
A traffic allocation mechanism for TAP$^+$ is Pareto-optimal, SP and non-bossy if and only if it is a Bi-polar Serially Dictatorial Rule.
\end{theorem}
\iftoggle{proof_thm_6_sketch}{
\begin{proof}[Proof sketch]
We reduce an instance of the problem of \emph{assigning indivisible objects} with general ordinal preferences \cite{DBLP:journals/jet/BogomolnaiaDE05} (AIO for short) to TAP$^+${}.
In an instance of AIO, a set of objects $X = \{x_1,\ldots ,x_m\}$ has to be assigned to a set of agents $A = \{a_1,\ldots ,a_n\}$, such that every agent receives at most one object and no agent is left without an object if there are objects still available.
Agents have ordinal general preferences $\succeq _i$, where $x \succeq_i y$ for $x,y\in X$ means that agent $i$ (weakly) prefers object $x$ to object $y$.
From an instance of AIO, we build an instance of TAP$^+${} as follows.
TAP$^+${} has the same set of agents $A$ as AIO.
Graph $G$ of TAP$^+${} has a node $O$ such that $O_i=O$ for all $a_i \in A$.
For every object $x_j\in X$ we construct in $G$ a node $v_j$ and an edge $(O,v_j)$ such that $c(O,v_j)=1$ and $w(O,v_j) = \varepsilon$ for $0<\varepsilon\ll 1$.
Let $\Psi$ be the set of all possible preference relations over $X$.
We construct $|\Psi|$ destination nodes $D_k$, one for each preference relation $\succeq \in \Psi$ and for each $k\in{1,\ldots, |\Psi|}$.
For each $j \in \{1,\ldots, m \}$ we add an edge $(v_j,D_k)$ having capacity 1 and weight $w(v_j,D_k)$ equal to the \emph{ranking} of $x_j$ according to $\succeq$.
We can now transform an instance of the so-constructed TAP$^+${} problem to an instance of the AIO problem, and vice versa.
In \cite{DBLP:journals/jet/BogomolnaiaDE05} it is proved that BSD is the only Pareto optimal, SP and non-bossy mechanism for AIO.
This characterization transfers to TAP$^+${} due to the reduction sketched above.
\end{proof}
}{
\iftoggle{proof_thm_6}{
\begin{proof}
We will reduce an instance of the problem of \emph{assigning indivisible objects} with general ordinal preferences \cite{DBLP:journals/jet/BogomolnaiaDE05} (AIO for short) to TAP$^+${}.
An instance of AIO is composed of a set of objects $X = \{x_1,\ldots ,x_m\}$ that have to be assigned to a set of agents $A = \{a_1,\ldots ,a_n\}$, such that every agent receives at most one object and no agent is left without an object if there are objects still available.
Agents have ordinal general preferences $\succeq _i$, where $x \succeq_i y$ for $x,y\in X$ means that agent $i$ (weakly) prefers object $x$ to object $y$.
From an instance of AIO, we can build an instance of TAP$^+${} as follows.
TAP$^+${} has the same set of agents $A$ as AIO.
Graph $G$ of TAP$^+${} has a node $O$ such that $O_i=O$ for all $a_i \in A$.
For every object $x_j\in X$ we construct in $G$ a node $v_j$ and an edge $(O,v_j)$ such that $c(O,v_j)=1$ and $w(O,v_j) = \varepsilon$ for $0<\varepsilon\ll 1$.
Let $\Psi$ be the set of all possible preference relations over $X$.
We construct\footnote{We note that, although $|\Psi|$ can be exponential in $m$, it is always finite. We remark that graphs of exponential size are not an issue here since the characterization we are proving in this theorem does not rely on computational efficiency.} $|\Psi|$ destination nodes $D_k$, one for each preference relation $\succeq \in \Psi$ and for each $k\in{1,\ldots, |\Psi|}$.
For each $j \in \{1,\ldots, m \}$ we add an edge $(v_j,D_k)$ having capacity 1 and weight $w(v_j,D_k)$ equal to the \emph{ranking}\footnote{
The alternatives in $X$ can be partitioned in subsets ${X_1,\ldots, X_\ell,\ldots}$ such that any two elements $x_1,x_2\in X_\ell$ are indifferent according to $\succeq$ and, for any $x_1\in X_\ell$ and $x_2 \in X_{\ell+1}$, $x_1$ is strictly preferred to $x_2$ according to $\succeq$. Then $\ell$ is the ranking of $x\in X_\ell$.} of $x_j$ according to $\succeq$.
Figure \ref{fig:reduction_example} gives an example of the reduction applied to an AIO game with $A=\{a_1,a_2,a_3\}$, $X=\{x_1,x_2,x_3\}$ and $\Psi$ being the set of all possible \emph{linear orderings} over $X$. The labels on the edges of the graph of Figure \ref{fig:reduction_example} represent the costs of the edges, whereas all capacities are set to 1.
By construction, the following hold: (\emph{i}) any path allocation on $G$ must include all the edges $(O,v_j)$; (\emph{ii}) any edge $(O,v_j)$ is used by at most one path; and (\emph{iii}) only one agent can be assigned any given edge $(O,v_j)$ due to the capacity constraint.
We can now easily transform an instance of the so-constructed TAP$^+${} problem to an instance of the AIO problem, and vice versa.
In \cite{DBLP:journals/jet/BogomolnaiaDE05} it is proved that BSD is the only Pareto optimal, SP and non-bossy mechanism for AIO.
This characterization trivially transfers to TAP$^+${} due to the reduction sketched above.
Indeed, let us suppose that there exists an SP, Pareto optimal and non-bossy mechanism for TAP.
Such mechanism would be SP, Pareto optimal and non-bossy for the AIO instance as well.
\end{proof}
}
}
Next, we investigate the performance of BSD and show that it does not asymptotically perform better than SD.
In particular, we state that:
\begin{lemma}
BSD cannot achieve an approximation ratio lower than $\Omega(2^n)$ for TAP.
\end{lemma}
\iftoggle{proof_lemma_1}{
\begin{proof}
We are going to show an instance of TAP where BSD has an approximation ratio of $\Omega(2^n)$.
Let us take the instance of Figure \ref{fig:sd_tight_instance} and let us consider the ordering $\{a_1,a_2\}\prec a_3\prec \ldots \prec a_n$.
Let us consider $X_1 = \{v_2,D\}$ and $X_2 = E\setminus X_1$.
The so-defined BSD mechanism, on input the instance of figure \ref{fig:sd_tight_instance} would always execute SD with ordering $a_1 \prec a_2 \prec \ldots, a_n$. We know from Theorem \ref{thm:tightness_SD} that under this ordering the approximation ratio of SD is $\Omega(2^n)$.
\end{proof}
}
\section{Randomized Mechanisms}\label{sec:randomized_mechs}
Given the undesirable approximation guarantees of deterministic mechanisms, we now turn to randomization. Randomized mechanisms can often be interpreted as fractional mechanisms for the deterministic solutions, under mild conditions.
We start by proving the following inapproximability lower bound:
\begin{theorem}\label{thm:randomizedLB}
There is no $\alpha$-approximate universally truthful randomized mechanism for the traffic assignment problem with $\alpha<11/10$.
\end{theorem}
\iftoggle{proof_thm_7}{
\begin{proof}
Our approach is based on Yao's minimax principle \cite{Yao}. In our context, this principle states that the approximation ratio of the best universally truthful randomized mechanism is equal to the approximation ratio of the best deterministic truthful mechanism under a worst-case input distribution. Accordingly, we exhibit a probability distribution over input instances for which any deterministic truthful mechanism cannot attain an approximation guarantee better than $11/10$.
The two instances are taken from the proof of Theorem \ref{thm:deterministic_LB}, where we set $K=2$. Specifically, we consider the instance in Figure \ref{fig:LB_instance_2}, that we name $I$, and the very same instance where agent $a_1$ reports $F$; we call this instance $I'$. We consider a probability distribution over $I$ and $I'$ that returns $I$ with probability $\lambda$ and $I'$ with the remaining probability $1-\lambda$, where $\lambda=2/3$. The expected value of the optimum will then be $(\lambda+1)K+4-\lambda=20/3$.
Let $M$ be a SP deterministic mechanism. From the arguments in the proof of the theorem above, we know that $M$ must assign the edge $(A,F)$ to the same agent in both instances $I$ and $I'$. If $M$ allocates $(A,F)$ to agent $a_1$ in both the instances then its expected social cost will be $(\lambda+1)K+4=22/3$ for an approximation ratio of $11/10$. If instead $M$ allocates $(A,F)$ to agent $a_2$ in both the instances then the expected social cost of the mechanism will be $(3-\lambda)K+\lambda+2=22/3$; the approximation ratio of $M$ would then be $11/10$.
\end{proof}
}
In the remainder of this section, we study the randomized version of SD for TAP, which is universally strategyproof, (ex-post) Pareto optimal and non-bossy.
\begin{definition}
The Randomized Serial Dictatorship (RSD) mechanism computes uniformly at random an ordering $\sigma$ over the agents and returns the output of SD over ordering $\sigma$.
\end{definition}
The following results gives a tight bound on the approximation ratio of RSD.
\begin{theorem}
\label{theorem:RSD_upperbound}
Under the DoCP assumption, RSD is at most $n$-approximate.
\end{theorem}
\iftoggle{proof_thm_8}{
\begin{proof}[Proof sketch]
We are going to prove the claim by induction on the number of agents. As above, let $OPT_i$ denote the cost of the optimal solution with paths assigned only to agents $a_j$, with $j \leq i$. With a slight abuse of notation we also let $OPT_i$ denote the solution itself. Similarly, $RSD_i$ denotes the expected cost of $RSD$ on input all the bids of agents $a_j$, $j \leq i$.
For the base of the induction with $i=1$, it is clear that $RSD_1$ is the optimal solution.
Now assume that the claim is true for $i-1$ and consider an instance with $i$ agents.
Let $I_{-k}(P)$, $P$ being a path from $O_k$ to $D_k$, be the instance of the problem without agent $a_k$ and with the capacity of the directed edges in $P$ diminished by one (i.e., as if the path $P$ were used by $a_k$).
Note that by the DoCP assumption, one of the agents $a_j$, with $j \neq k$, is guaranteed to be able to use the edges of $P$ in the opposite direction than $a_k$.
We now let $OPT_{-k,P}$ and $RSD_{-k,P}$ be the cost of the optimum and expected cost of RSD on $I_{-k}(P)$, respectively. Moreover, let $\pi_j$ be the path minimizing the cost of agent $a_j$ (i.e., the path that $SD$ would assign to $a_j$ if she was the first to choose). We then have
{\scriptsize
\begin{align*}
RSD_i &= \frac1i \sum_{k=1}^i \left(\rule{0ex}{3ex}w(\pi_k) + RSD_{-k,\pi_k}\right)
\leq \frac1i \sum_{k=1}^i \left(\rule{0ex}{3ex}w(\pi_k) + (i-1) OPT_{-k, \pi_k}\right)\\
& \leq \frac1i \sum_{k=1}^i w(\pi_k) + \frac1i \sum_{k=1}^i \left(\rule{0ex}{3ex}(i-1) (OPT_i + w(\pi_k))\right)
\leq \frac1i OPT_i + (i-1) OPT_i + \frac{i-1}{i}OPT_i \\
& = i \cdot OPT_i
\end{align*}
}
\noindent
where the first equality follows from the definition of RSD, i.e., with probability $1/i$ each agent $k$ will have the first choice. As for the inequalities, we note that the first follows from the inductive hypothesis whilst the last from the observation that $OPT_i \geq \sum_{k=1}^i w(\pi_k)$. We are left with the second inequality. That is, we prove that under the DoCP
$
OPT_{-k, \pi_k} \leq OPT_i + w(\pi_k).
$
If $OPT_i$ allocates $\pi_k$ to agent $a_k$ then we are done.
Otherwise, let $P_k$ be the path that $a_k$ gets in $OPT_i$ and note that the paths $P_j$ allocated to agents $a_j$ $j \neq k$ by $OPT_i$ saturates some of the edges of $P_k$; let $a_{\bar{j}}$ be one of these agents.
Consider now the solution $S$ to $I_{-k}(\pi_k)$ where all agents but $a_{\bar{j}}$ are allocated the same path as in $OPT_i$ and agent $a_{\bar{j}}$ is given, instead of $P_{\bar{j}}$, the alternative path $\Gamma_{\bar{j}}^{k}$ through agent $a_k$.
Observe that $\Gamma_{\bar{j}}^{k}$ uses the same directed edges of $P_{\bar{j}}$ and $P_k$ and the edges of $\pi_k$ in opposite direction and, as observed above, under the DoCP assumption, is a feasible path for $a_{\bar{j}}$ and $S$ a feasible solution to $I_k(\pi_k)$, whose social cost is denoted $SC(S)$. But then:
\begin{align*}
OPT_{-k, \pi_k} & \leq SC(S) = OPT_i - w(P_j) - w(P_k) + w(P)
\leq OPT_i + w(\pi_k)
\end{align*}
where the last inequality follows from the fact that the edges in $P \setminus (P_k \cup P_j)$ are a subset of the edges in $\pi_k$.
\end{proof}
}
\begin{theorem}
\label{theorem:RSD_lowerbound}
The approximation ratio of RSD is $\Omega(n)$.
\end{theorem}
\noindent
This means that by allowing randomness in the allocation mechanism, we can improve the exponential approximation ratio of the deterministic case to a linear one.
\iftoggle{proof_thm_9_sketch}{
\begin{proof}[Proof sketch of Theorem~\ref{theorem:RSD_lowerbound}]
The proof uses the same construction as the instance of Figure \ref{fig:sd_tight_instance}, with $k<n$ nodes.
One agent is initially located at node $v_1$, whereas $1+2\cdot3^{i-1}$ agents are initially located at node $v_i$, for $i=2,\ldots, k-1$ .
With a little abuse of notation, let $|v_i|$ denote the number of agents initially located at node $v_i$, and let $n_i = \sum_{\ell=0}^i |v_\ell|$.
Edges $(v_1,D)$ and $(v_1,v_2)$ have capacity $1$, whereas edges $(v_i,D)$ and $(v_i,v_{i+1})$ have capacity $1+2\cdot3^{i-1}$ for $i>1$.
Let $a_1 \prec\ldots \prec a_n$ be an ordering over the agents.
We will be interested in orderings that possess the \emph{chain of levels} property, namely for all $i=1,2,\ldots, k-1$ at least one agent located at node $i$ appears after all agents of levels $0,1,\ldots i-1$.
The property of a chain of levels ordering with respect to the instance of Figure \ref{fig:sd_tight_instance} is that it forces at least one agent located at node $v_i$, for all $i = 1,\ldots, k-1$ to use the path $P = (v_i,v_{i+1},D)$, at a cost of $2^{i-1}$ for the agents, and an overall social cost of $\sum_{i=1}^{k-1}2^{i-1}=2^k-1 > 2^{k-1}$.
We argue that the probability that a chain of levels ordering is chosen by RSD is $\Pi_{i=1}^{k-1} \left(1-\frac{n_{i-1}}{n_i}\right)$.
Indeed, we can look at the process of randomly generating an ordering as follows. First an ordering for the agents located at each node is uniformly generated at random.
Then orderings of agents of consecutive nodes are merged together in lexicographic order.
In particular, we start merging the orderings of nodes $v_1$ and $v_2$.
There are $\binom{|v_2|+|v_1|}{v_1} = \binom{n_2}{n_1}$ such orderings, whereas there are $\binom{n_2-1}{n_1}$ orderings where one agent located at node $v_2$ follows all the agents located at node $v_1$. The partial ordering obtained so far is randomly merged with the ordering of agents at node $v_3$ and the procedure continues until the partial ordering is complete. When merging agents at node $v_i$ with the current partial ordering, we note that there are $\binom{n_i}{n_{i-1}}$ possible orderings, and $\binom{n_i-1}{n_{i-1}}$ where for all $\ell = 1,\ldots, i$, one agent located at node $v_\ell$ follows all the agents located at node $v_{\ell-1}$ in the ordering (i.e., fix one agent from node $v_\ell$ in the last position and compute all possible orderings of the other agents).
Hence, the probability of one agent at node $v_\ell$ appearing after all agents at node $v_{\ell-1}$ is $\binom{n_i-1}{n_{i-1}}/\binom{n_i}{n_{i-1}} = \left(1-\frac{n_{i-1}}{n_i}\right)$.
Since the random orderings generated at each stage are independent, the probability that for all $i = 1, 2, \ldots, k-1$ at least one agent at node $v_i$ appears after all agents at node $v_{i-1}$ in a random ordering is $\Pi_{i=1}^{k-1}\left(1-\frac{n_{i-1}}{n_i}\right)$.
Hence, the probability that a chain of levels ordering is chosen by RSD for the instance of Figure \ref{fig:sd_tight_instance} is $(2/3)^{k-1}$.
Finally, the expected cost of RSD is at least $(4/3)^{k-1} = n^{\log_3(4/3)}\approx n^{0.262}$. Since the optimal allocation costs $1+\epsilon \cdot n$, the approximation ratio is $\Omega(n)$ for $\epsilon$ close to $0$.
\end{proof}
}{
\iftoggle{proof_thm_9}{
\begin{proof}
The proof uses the same construction as the instance of Figure \ref{fig:sd_tight_instance}, with $k<n$ nodes.
One agent is initially located at node $v_1$, whereas $1+2\cdot3^{i-1}$ agents are initially located at node $v_i$, for $i=2,\ldots, k-1$ .
With a little abuse of notation, let $|v_i|$ denote the number of agents initially located at node $v_i$, and let $n_i = \sum_{\ell=0}^i |v_\ell|$.
Edges $(v_1,D)$ and $(v_1,v_2)$ have capacity $1$, whereas edges $(v_i,D)$ and $(v_i,v_{i+1})$ have capacity $1+2\cdot3^{i-1}$ for $i>1$.
Let $a_1 \prec\ldots \prec a_n$ be an ordering over the agents.
We will be interested in orderings that possess the \emph{chain of levels} property, namely for all $i=1,2,\ldots, k-1$ at least one agent located at node $i$ appears after all agents of levels $0,1,\ldots i-1$.
The property of a chain of levels ordering with respect to the instance of Figure \ref{fig:sd_tight_instance} is that it forces at least one agent located at node $v_i$, for all $i = 1,\ldots, k-1$ to use the path $P = (v_i,v_{i+1},D)$, at a cost of $2^{i-1}$ for the agents, and an overall social cost of $\sum_{i=1}^{k-1}2^{i-1}=2^k-1 > 2^{k-1}$.
We argue that the probability that a chain of levels ordering is chosen by RSD is $\Pi_{i=1}^{k-1} \left(1-\frac{n_{i-1}}{n_i}\right)$.
Indeed, we can look at the process of randomly generating an ordering as follows. First an ordering for the agents located at each node is uniformly generated at random.
Then orderings of agents of consecutive nodes are merged together in lexicographic order.
In particular, we start merging the orderings of nodes $v_1$ and $v_2$.
There are $\binom{|v_2|+|v_1|}{v_1} = \binom{n_2}{n_1}$ such orderings, whereas there are $\binom{n_2-1}{n_1}$ orderings where one agent located at node $v_2$ follows all the agents located at node $v_1$. The partial ordering obtained so far is randomly merged with the ordering of agents at node $v_3$ and the procedure continues until the partial ordering is complete. When merging agents at node $v_i$ with the current partial ordering, we note that there are $\binom{n_i}{n_{i-1}}$ possible orderings, and $\binom{n_i-1}{n_{i-1}}$ where for all $\ell = 1,\ldots, i$, one agent located at node $v_\ell$ follows all the agents located at node $v_{\ell-1}$ in the ordering (i.e., fix one agent from node $v_\ell$ in the last position and compute all possible orderings of the other agents).
Hence, the probability of one agent at node $v_\ell$ appearing after all agents at node $v_{\ell-1}$ is $\binom{n_i-1}{n_{i-1}}/\binom{n_i}{n_{i-1}} = \left(1-\frac{n_{i-1}}{n_i}\right)$.
Since the random orderings generated at each stage are independent, the probability that for all $i = 1, 2, \ldots, k-1$ at least one agent at node $v_i$ appears after all agents at node $v_{i-1}$ in a random ordering is $\Pi_{i=1}^{k-1}\left(1-\frac{n_{i-1}}{n_i}\right)$.
Hence, the probability that a chain of levels ordering is chosen by RSD for the instance of Figure \ref{fig:sd_tight_instance} is $(2/3)^{k-1}$.
Finally, the expected cost of RSD is at least $(4/3)^{k-1} = n^{\log_3(4/3)}\approx n^{0.262}$. Since the optimal allocation costs $1+\epsilon \cdot n$, the approximation ratio is $\Omega(n)$ for $\epsilon$ close to $0$.
\end{proof}
}
}
\iftoggle{proof_thm_10}{
\begin{proof}
We reduce the ordinal AIO problem studied in \cite{Filos-RatsikasF014} to TAP$^+${} through the same reduction as Theorem \ref{thm:reduction}.
Note that AIO problem in \cite{Filos-RatsikasF014} requires that the $|X| = |A|$ but this can be easily accommodated.
In \cite{Filos-RatsikasF014} the authors prove that no truthful in expectation mechanism for the AIO problem can achieve an approximation ratio lower than $\Omega{(\sqrt{n})}$. In virtue of the reduction from ordinal AIO to TAP$^+${}, this results holds for TAP$^+${} as well.
\end{proof}
}
\begin{figure*}
\caption{Experimental results on Rome99}
\label{fig:Rome_results}
\caption{Structural characteristics of test graphs}
\label{tab:graphs_stats}
\end{figure*}
\begin{figure*}
\caption{Experimental results on NY-4000}
\label{fig:NY-4000_results}
\caption{Experimental results on NY-10000}
\label{fig:NY-10000_results}
\end{figure*}
\section{Experimental Results}\label{sec:experimental_results}
In this section we present the results of the experimental evaluation we conducted in order to assess
whether the theoretical inapproximability lower bounds impose a high approximation cost on real-life instances. In short, we will show that they do not.
In particular, we have measured the approximation ratio obtained by SD and RSD on three real-life graphs extracted from the DIMCAS 99 shortest path implementation challenge benchmark datasets \cite{DIMACS}.
In particular, Rome99 represents a large portion of the directed road network of the city of Rome, Italy, from 1999. The graph contains 3353 vertices and 8870 edges. Vertices correspond to intersections between roads and edges correspond to roads or road segments.
NY-4000 and NY-10000 are two subgraphs extracted from NY-d, a larger distance graph (with 264,346 nodes and 733,846 edges) representing a large portion the road network infrastrucutre of New York City, USA.
The two graphs were obtained by taking a subset, respectively, of the first 4000 and 10000 nodes of the graph while ensuring that the connectivity was preserved by adding edges representing paths through nodes of the original graph not included in the subgraph.
In Table \ref{tab:graphs_stats} some statistics related to the structural characteristics of our test graphs are reported, where $\delta^+_{AVG}$ represents the average outdegree of a node (i.e. the average number of edges originating from a node) and $c_{AVG}$ is the average capacity of the outgoing edges of a node.
In our experimental assessment, we studied the variation of the approximation ratio of SD and RSD on the test graphs while varying the \emph{resource augmentation factor}.
The resource augmentation factor is the key parameter of the resource augmentation framework \cite{resource_augmentation}, a novel comparison framework where a truthful mechanism that allocates ``scarce resources'' is evaluated by its worst-case performance on an instance where such ``scarce resources'' are augmented, against the optimal mechanism on the same instance with the original amount of resources.
In \cite{resource_augmentation} it is argued that this is a fairer comparison framework than the traditional approximation ratio, which compares the performance of a mechanism that is severely limited by the requirement of truthfulness to that of an omnipotent mechanism that operates under no restrictions and has access to the real inputs of the agents.
An equivalent resource augmentation framework is often also used in the analysis of online algorithms.
In the TAP scenario, the natural resource to be augmented is the capacity of the existing edges, modelled by the augmentation factor $\gamma$, which in our framework is defined as the factor by which the average capacity of the edges departing from a node is multiplied, spreading the excess capacity evenly among the outgoing edges of the node.
More formally, if $c_{AVG}(v)$ is the average capacity of node $v$, then the augmented average capacity $c^{\gamma}_{AVG}(v) = \gamma \cdot c_{AVG}$, and the capacity of each outgoing edge is set as $\frac{c_{AVG}(v)}{\delta^+(v)}$, where $\delta^+(v)$ is the outdegree of $v$.
In our experiments we ranged the augmentation factor $\gamma$ in the interval $[1,2]$, which means increasing the initial capacity until it is doubled.
To run our experiments, we generated three separate populations of agent-origin-destination triplets, one population for each test graph, each comprising a number of triplets roughly equal to $1/3$ of the nodes of the graph. The size of the population of triplets was empirically tailored to let the competition for popular links arise without making the allocation problem unfeasible.
For each agent-origin-destination triplet in the population, both the origin and the destination were independently drawn uniformly at random from the set of the nodes of the graph, with replacement (i.e. the same node can be the origin/destination of multiple triplets).
Figures \ref{fig:Rome_results}, \ref{fig:NY-4000_results} and \ref{fig:NY-10000_results} show the results of our experimental analysis, respectively on graph Rome99, NY-4000 and NY-10000.
In particular, the left hand side plot represents the absolute value of the social cost for the optimal mechanism,
expressed in kilometers, for SD and for RSD, whereas the right hand side plot represents the approximation ratio for SD and RSD.
From our experimental analysis we can see that the actual approximation ratio of both SD and RSD is much lower than the predicted theoretical worst-case approximation.
In particular, our experiments show that the approximation ratios of SD and RSD are quite similar and strongly $o(n)$ on the investigated road networks.
This is due to the fact that such theoretical approximation lower bounds rely on pathological instances that are quite unlikely to occur in real life graphs.
It is also worth noting the beneficial effect that augmenting the capacity of existing roads has on the approximation ratio: increasing the augmentation factor steadily decreases the approximation ratio on both Rome99 and NY-4000.
On the other hand a marked decrease is noticeable only if we increase the augmentation factor to 1.8 in the case of NY-10000.
This phenomenon is due to the already reach topological structure of NY-10000, which necessitates less augmentation to yield good performances.
\section{Conclusions}
\label{Sec:conclusions}
In this paper we investigate the problem of strategyproof traffic assignment without monetary incentives.
We study two SP mechanism for our problem, namely Serial Dictatorship and its randomized counterpart Random Serial Dictatorships.
For deterministic mechanisms we prove that Serial Dictatorship is $2^n-1$ under some mild assumptions, and characterize Bipolar Serial Dictatorship as the only SP, Pareto optimal and non-bossy deterministic mechanism for our problem.
In the randomized case, we prove that Random Serial Dictatorship is $n$-approximate.
Finally we assess the performance of Serial Dictatorship and Random Serial Dictatorship on real road network infrastructure, and show that they exhibit good approximation guarantees.
In particular, RSD is almost indistinguishable from SD, which means that the instances giving rise to the inapproximability results rarely occur in practice.
Note that our work is the first that addresses the problem of moneyless strategyproof traffic assignment. Although it ignores a number of properties that occur in real-world scenarios (e.g., dynamic network behavior, or asynchronous bid submissions), it still serves as a proof of concept for the existence of moneyless strategyproof assignment mechanisms.
\appendix
\section{Appendix}
In the following we give the proofs of the theorems that were omitted in the main body of the paper due to space limitations.
\subsection{Time-Expanded Networks}
Let $G = (V, E)$ be a network with capacities $c$,
non-negative integral transit times $\tau$, and costs $w$ on the edges.
For a given time horizon $T \in \mathbb{Z}>0$, the corresponding time-expanded network $G^T = (V^T, E^T )$ with capacities and costs on the edges
is defined as follows.
For each node $v \in V$ there are $T$ copies $v_0 , v_1 ,\ldots, v_{T -1}$, that is,
$V^{T} = \{v_t | v \in V, t \in \{0, 1, \ldots ,T - 1\}\}$ .
For each edge $e = (v, w) \in E$, there are $T - \tau(e)$ copies $e_0, e_1,\ldots , e_{T -1-\tau(e)}$ where edge $e_t$ connects
node $v_t$ to node $w_{t + \tau(e)}$ . Edge $e_t$ has capacity $c(e_t) = c(e)$ and cost $w(e_t) = w(e)$.
Moreover, $E_T$ contains holdover edges $(v_t , v_{t+1} )$ for $v \in V$ and $t = 0,\ldots, T - 2$.
The capacity of holdover edges is infinite and they have zero cost.
\subsection{Omitted Theorems}
\begin{theorem}
Let $S=M(\mathbf{D})$ be a traffic assignment and let $R_i(S)$ be the set of reaction available to $a_i$ at $S$.
If $S$ is Pareto optimal, then $M_i(\mathbf{D})\in \arg\min_{P\in R_i(S)}w(P)$.
\end{theorem}
\begin{proof}
If $M_i(\mathbf{D})\notin \arg\min_{P\in R_i(S)}w(P)$, then there must exist a reaction $r'_i\in R_i$ that is strictly better than $M_i(\mathbf{D})$ for agent $a_i$, i.e. $c(r'_i)<c(M_i(\mathbf{D}))$.
Then, a route assignment $M'$ such that $M'_j = M_j$ for all $j\neq i$ and $M'_i = r_i$ is still feasible. Since $cost_j(M')=cost_j(M')$ for all $j\neq i$, and $cost_i(M')<cost_i(M)$, $M$ is not Pareto optimal.
\end{proof}
\begin{theorem}\label{thm:deterministic_LB}
There is no $\alpha$-approximate deterministic SP mechanism for the traffic assignment problem with $\alpha<3 - \varepsilon$, for any $\varepsilon>0$.
\end{theorem}
\begin{proof}
Given $\varepsilon>0$, consider the graph depicted in Figure \ref{fig:LB_instance_2}, where the labels on the edges represent their capacity (red) and length (black). The instance we consider has two agents $A= \{a_1,a_2\}$, both initially located at node $A$, whose intended destination is $D$ and $G$, respectively (namely, $\mathbf{D}=(D,G)$).
The length of the path $(B,C)$ is $K = \min\{2, \frac{10-4\varepsilon}{\varepsilon}\}$ and the length of path $(F,E)$ is $K-1$.
\begin{figure}
\caption{Lower bound instance}
\label{fig:LB_instance_2}
\end{figure}
Let us consider a generic $\alpha$-approximate mechanism $M$.
Assume by contradiction that $M$ is strategyproof and $\alpha$-approximate with $\alpha<3-\varepsilon$.
The instance has two Pareto optimal solutions, depending on which player is allocated the edge $(A,F)$ (note that only one agent at a time can use edge $(A,F)$ as its capacity is 1). The optimal allocation $\mathbf{P}^*=OPT(\mathbf{D})$ is $P^*_1 = (A,B,C,D)$ and $P^*_2 = (A,F,E,G)$, $cost_1(\mathbf{P}^*,D) = K+2$ and $cost_2(\mathbf{P}^*,G) = K+1$, and $SC(\mathbf{P}^*,\mathbf{D})=2K+3$. The second best solution is $P_1 = (A,F,E,D)$ and $P_2 = (A,B,C,E,G)$. We note that $cost_1(P_1,D_1)=K+1$ and $cost_2(P_2,D_2)=K+3$, for a social cost of $SC(M(\mathbf{D}),\mathbf{D}) = 2K+4$.
We are going to prove that, regardless of the solution $M$ returns on this instance, there is another instance where to maintain SP, $M$ achieves an approximation not better than $3-\varepsilon$, a contradiction.
Let us assume first that $M$ returns the optimal allocation.
If agent $a_1$ declares $D'_1=F$ instead of her true type, by SP, $M$ cannot allocate the edge $(A, F)$ to $a_1$. In fact, assume for the sake of contradiction that $M(D_1', D_2)$ allocates $(A,F)$ to $a_1$.
Then $a_2$ is allocated path $(A,B,C,E,G)$ and $a_1$ can use path $(A,F,E,D)$ and reach her true destination, thus having:
$$cost_1(M(D'_1,D_2),D_1)=K+1<cost_1(M(\mathbf{D}),D_1)=K+2.$$
Therefore, $M(D_1', D_2)$ must return $P'_1 = (A,B,C,E,F)$ and $P'_2 = (A,F,E,G)$, with $SC((P'_1,P'_2), (D_1', D_2)) = 3K+2$, whilst the optimum for this instance is $OPT_1(D'_1,D_2) = (A,F)$ and $OPT_2(D'_1,D_2) =(A,B,C,E,G)$. The social cost of the optimum is then $SC(OPT(D'_1,D_2),(D'_1,D_2)) = K+4$.
Therefore $M$ has an approximation ratio higher than $3-\varepsilon$.
Let us now suppose that $M(\mathbf{D})$ returns $M_1(\mathbf{D}) = P_1$ and $M_2(\mathbf{D}) = P_2$.
In this case, consider the case that agent $a_2$ reports $D'_2=F$ instead of her true type.
The optimal allocation is $OPT_1(D_1,D_2') = (A,B,C,D)$ and $OPT_2(D_1,D_2') = (A,F)$, and $SC(OPT(D'_2,D_1),(D'_2,D_1)) = K+3$.
As before, this allocation is not strategyproof as:
\begin{eqnarray*}
cost_2(OPT_2(D_1, D'_2),D_2)= K+1 \\
< cost_2(M_2(D_1,D'_2),D_2)=K+3
\end{eqnarray*}
(i.e., agent $a_2$ can use the route $(A,F,E,G)$ to reach her true destination).
As above, one can easily check that in this case the best (in terms of approximation ratio) strategyproof allocation is $P'_1=(A,F,E,D)$ and $P'_2=(A,B,C,E,F)$, with a cost of $SC((P'_1,P'_2),(D_1,D_2')) = 3K+2$.
This solution has an approximation ratio higher than $3-\varepsilon$.
\end{proof}
\begin{theorem}
If $G$ is $K$-edge-connected\footnote{A graph is $K$-edge-connected if removing at most $K$ edges from it does not disconnect the graph.}, mechanism Serial Dictator is feasible for $K$ agents.
\end{theorem}
\begin{proof}
If the graph is $K$-edge connected, the allocation returned by the Serial Dictator will always be feasible, (i.e. paths assigned to different agents will not overlap and there is always an assignable path for each agents).
This follows from the fact that in a $K$-edge-connected graph there are at least $K$ edge disjoint paths between any pair of nodes.
\end{proof}
\setcounter{theorem}{4}
\begin{theorem}\label{thm:tightness_SD}
Under the DoCP assumption, the bound of Theorem 4 is tight.
\end{theorem}
\begin{proof}
Let us consider the instance in Figure \ref{fig:sd_tight_instance}, where there are $n$ nodes $v_1,\ldots, v_n$ and $n$ agents $A=\{a_1,\ldots, a_n\}$ such that agent $a_i$ is initially located at node $v_i$. All agents want to reach the same destination $D$.
Each link has capacity $1$.
Agent $a_1$ has two paths to her destination $D$: one direct path that costs $1+\varepsilon$ (where $\varepsilon\ll 1$ is a small constant) and a path costing $1$ that goes through the node agent $a_2$ is initially located on.
Each agent $a_i$, for $i=2,\dots,n-1$ has two paths: one direct path that costs $\varepsilon$ and a path costing $2^{i-1}$ that goes through the node agent $a_{i+1}$ is initially located on.
Agent $a_n$ has two direct paths, costing $\varepsilon$ and $2^{n-1}$ respectively.
The optimal traffic assignment assigns agent $a_1$ to the path that costs $1+\varepsilon$ and the other agents to the path costing $\varepsilon$, and has a cost of $1+\varepsilon n$.
Let us consider ordering $a_1\prec a_2 \prec \ldots \prec a_n$.
On this ordering, mechanism SD assigns agent $a_1$ the path costing $1$, and to each agent $a_i$, for $i=2,\ldots, a_n$ the path costing $2^{i-1}$ for a total cost of $\sum_{i=0}^{n-1} 2^i = 2^n-1$.
For $\varepsilon$ close to $0$, the approximation ratio of SD on the instance depicted in Figure \ref{fig:sd_tight_instance} is hence close to $2^n-1$.
\begin{figure}
\caption{Tight instance for SD}
\label{fig:sd_tight_instance}
\end{figure}
\end{proof}
\setcounter{theorem}{5}
\begin{theorem}\label{thm:reduction}
A traffic allocation mechanism for TAP$^+$ is Pareto-optimal, SP and non-bossy if and only if it is a Bi-polar Serially Dictatorial Rule.
\end{theorem}
\begin{proof}
We will reduce an instance of the problem of \emph{assigning indivisible objects} with general ordinal preferences
\cite{DBLP:journals/jet/BogomolnaiaDE05} (AIO for short) to TAP$^+${}.
An instance of AIO is composed of a set of objects $X = \{x_1,\ldots ,x_m\}$ that have to be assigned to a set of agents $A = \{a_1,\ldots ,a_n\}$, such that every agent receives at most one object and no agent is left without an object if there are objects still available.
Agents have ordinal general preferences $\succeq _i$, where $x \succeq_i y$ for $x,y\in X$ means that agent $i$ (weakly) prefers object $x$ to object $y$.
From an instance of AIO, we can build an instance of TAP$^+${} as follows.
TAP$^+${} has the same set of agents $A$ as AIO.
Graph $G$ of TAP$^+${} has a node $O$ such that $O_i=O$ for all $a_i \in A$.
For every object $x_j\in X$ we construct in $G$ a node $v_j$ and an edge $(O,v_j)$ such that $c(O,v_j)=1$ and $w(O,v_j) = \varepsilon$ for $0<\varepsilon\ll 1$.
Let $\Psi$ be the set of all possible preference relations over $X$.
We construct\footnote{We note that, although $|\Psi|$ can be exponential in $m$, it is always finite. We remark that graphs of exponential size are not an issue here since the characterization we are proving in this theorem does not rely on computational efficiency.} $|\Psi|$ destination nodes $D_k$, one for each preference relation $\succeq \in \Psi$ and for each $k\in{1,\ldots, |\Psi|}$.
For each $j \in \{1,\ldots, m \}$ we add an edge $(v_j,D_k)$ having capacity 1 and weight $w(v_j,D_k)$ equal to the \emph{ranking}\footnote{
The alternatives in $X$ can be partitioned in subsets ${X_1,\ldots, X_\ell,\ldots}$ such that any two elements $x_1,x_2\in X_\ell$ are indifferent according to $\succeq$ and, for any $x_1\in X_\ell$ and $x_2 \in X_{\ell+1}$, $x_1$ is strictly preferred to $x_2$ according to $\succeq$. Then $\ell$ is the ranking of $x\in X_\ell$.} of $x_j$ according to $\succeq$.
Figure \ref{fig:reduction_example} gives an example of the reduction applied to an AIO game with $A=\{a_1,a_2,a_3\}$, $X=\{x_1,x_2,x_3\}$ and $\Psi$ being the set of all possible \emph{linear orderings} over $X$. The labels on the edges of the graph of Figure \ref{fig:reduction_example} represent the costs of the edges, whereas all capacities are set to 1.
\begin{table}[]
\centering
\caption{Mapping elements of $\Psi$ to destination nodes of $G$}
\label{my-label}
\begin{tabular}{|l|l|}
\hline
$D_1$ & $x_1\prec x_2\prec x_3$ \\ \hline
$D_2$ & $x_1\prec x_3\prec x_2$ \\ \hline
$D_3$ & $x_2\prec x_1\prec x_3$ \\ \hline
$D_4$ & $x_2\prec x_3\prec x_1$ \\ \hline
$D_5$ & $x_3\prec x_1\prec x_2$ \\ \hline
$D_6$ & $x_3\prec x_2\prec x_1$ \\ \hline
\end{tabular}
\end{table}
By construction, the following hold: (\emph{i}) any path allocation on $G$ must include all the edges $(O,v_j)$; (\emph{ii}) any edge $(O,v_j)$ is used by at most one path; and (\emph{iii}) only one agent can be assigned any given edge $(O,v_j)$ due to the capacity constraint.
We can now easily transform a path allocation for the so-constructed TAP$^+${} problem to an allocation of objects to agents in the AIO problem, and vice versa.
Indeed, let $P_i$ be the path assigned to agent $a_i$ in the TAP$^+${} problem. If $P_i$ contains edge $(O,v_j)$ we allocate object $x_j$ to agent $a_i$ in the AIO instance, and vice versa from an allocation for the AOI problem to an allocation for the TAP$^+${} problem.
In \cite{DBLP:journals/jet/BogomolnaiaDE05} it is proved that BSD is the only Pareto optimal, SP and non-bossy mechanism for AIO.
This characterization trivially transfers to TAP$^+${} due to the reduction sketched above.
Indeed, let us suppose that there exists an SP, Pareto optimal and non-bossy algorithm for TAP.
Such algorithm would be SP, Pareto optimal and non-bossy for the AIO instance as well.
\begin{figure}
\caption{Reduction example}
\label{fig:reduction_example}
\end{figure}
\end{proof}
\begin{lemma}
BSD cannot achieve an approximation ratio lower than $\Omega(2^n)$ for TAP.
\end{lemma}
\begin{proof}
We are going to show an instance of TAP where BSD has an approximation ratio of $\Omega(2^n)$.
Let us take the instance of Figure \ref{fig:sd_tight_instance} and let us consider the ordering $\{a_1,a_2\}\prec a_3\prec \ldots \prec a_n$.
Let us consider $X_1 = \{v_2,D\}$ and $X_2 = E\setminus X_1$.
The so-defined BSD mechanism, on input the instance of figure \ref{fig:sd_tight_instance} would always execute SD with ordering $a_1 \prec a_2 \prec \ldots, a_n$. We know from Theorem \ref{thm:tightness_SD} that under this ordering the approximation ratio of SD is $\Omega(2^n)$.
\end{proof}
\begin{theorem}\label{thm:randomizedLB}
There is no $\alpha$-approximate randomized universally truthful mechanism for the traffic assignment problem with $\alpha<11/10$.
\end{theorem}
\begin{proof}
Our approach is based on Yao's minimax principle \cite{Yao}. In our context, this principle states that the approximation ratio of the best universally truthful randomized mechanism is equal to the approximation ratio of the best deterministic truthful mechanism under a worst-case input distribution. Accordingly, we exhibit a probability distribution over input instances for which any deterministic truthful mechanism cannot attain an approximation guarantee better than $11/10$.
The two instances are taken from the proof of Theorem \ref{thm:deterministic_LB}, where we set $K=2$. Specifically, we consider the instance in Figure \ref{fig:LB_instance_2}, that we name $I$, and the very same instance where agent $a_1$ reports $F$; we call this instance $I'$. We consider a probability distribution over $I$ and $I'$ that returns $I$ with probability $\lambda$ and $I'$ with the remaining probability $1-\lambda$, where $\lambda=2/3$. The expected value of the optimum will then be $(\lambda+1)K+4-\lambda=20/3$.
Let $M$ be a SP deterministic mechanism. From the arguments in the proof of the theorem above, we know that $M$ must assign the edge $(A,F)$ to the same agent in both instances $I$ and $I'$. If $M$ allocates $(A,F)$ to agent $a_1$ in both the instances then its expected social cost will be $(\lambda+1)K+4=22/3$ for an approximation ratio of $11/10$. If instead $M$ allocates $(A,F)$ to agent $a_2$ in both the instances then the expected social cost of the mechanism will be $(3-\lambda)K+\lambda+2=22/3$; the approximation ratio of $M$ would then be $11/10$.
\end{proof}
\setcounter{theorem}{8}
\begin{theorem}
The approximation ratio of RSD is $\Omega(n)$.
\end{theorem}
\begin{proof}
The proof uses the same construction as the instance of Figure \ref{fig:sd_tight_instance}, with $k<n$ nodes.
One agent is initially located at node $v_1$, whereas $1+2\cdot3^{i-1}$ agents are initially located at node $v_i$, for $i=2,\ldots, k-1$ .
With a little abuse of notation, let $|v_i|$ denote the number of agents initially located at node $v_i$, and let $n_i = \sum_{\ell=0}^i |v_\ell|$.
Edges $(v_1,D)$ and $(v_1,v_2)$ have capacity $1$, whereas edges $(v_i,D)$ and $(v_i,v_{i+1})$ have capacity $1+2\cdot3^{i-1}$ for $i>1$.
Let $a_1 \prec\ldots \prec a_n$ be an ordering over the agents.
We will be interested in orderings that possess the \emph{chain of levels} property, namely for all $i=1,2,\ldots, k-1$ at least one agent located at node $i$ appears after all agents of levels $0,1,\ldots i-1$.
The property of a chain of levels ordering with respect to the instance of Figure \ref{fig:sd_tight_instance} is that it forces at least one agent located at node $v_i$, for all $i = 1,\ldots, k-1$ to use the path $P = (v_i,v_{i+1},D)$, at a cost of $2^{i-1}$ for the agents, and an overall social cost of $\sum_{i=1}^{k-1}2^{i-1}=2^k-1 > 2^{k-1}$.
We argue that the probability that a chain of levels ordering is chosen by RSD is $\Pi_{i=1}^{k-1} \left(1-\frac{n_{i-1}}{n_i}\right)$.
Indeed, we can look at the process of randomly generating an ordering as follows. First an ordering for the agents located at each node is uniformly generated at random.
Then orderings of agents of consecutive nodes are merged together in lexicographic order.
In particular, we start merging the orderings of nodes $v_1$ and $v_2$.
There are $\binom{|v_2|+|v_1|}{v_1} = \binom{n_2}{n_1}$ such orderings, whereas there are $\binom{n_2-1}{n_1}$ orderings where one agent located at node $v_2$ follows all the agents located at node $v_1$. The partial ordering obtained so far is randomly merged with the ordering of agents at node $v_3$ and the procedure continues until the partial ordering is complete. When merging agents at node $v_i$ with the current partial ordering, we note that there are $\binom{n_i}{n_{i-1}}$ possible orderings, and $\binom{n_i-1}{n_{i-1}}$ where for all $\ell = 1,\ldots, i$, one agent located at node $v_\ell$ follows all the agents located at node $v_{\ell-1}$ in the ordering (i.e., fix one agent from node $v_\ell$ in the last position and compute all possible orderings of the other agents).
Hence, the probability of one agent at node $v_\ell$ appearing after all agents at node $v_{\ell-1}$ is $\binom{n_i-1}{n_{i-1}}/\binom{n_i}{n_{i-1}} = \left(1-\frac{n_{i-1}}{n_i}\right)$.
Since the random orderings generated at each stage are independent, the probability that for all $i = 1, 2, \ldots, k-1$ at least one agent at node $v_i$ appears after all agents at node $v_{i-1}$ in a random ordering is $\Pi_{i=1}^{k-1}\left(1-\frac{n_{i-1}}{n_i}\right)$.
Hence, the probability that a chain of levels ordering is chosen by RSD for the instance of Figure \ref{fig:sd_tight_instance} is $(2/3)^{k-1}$.
Finally, the expected cost of RSD is at least $(4/3)^{k-1} = n^{\log_3(4/3)}\approx n^{0.262}$. Since the optimal allocation costs $1+\epsilon \cdot n$, the approximation ratio is $\Omega(n)$ for $\epsilon$ close to $0$.
\end{proof}
\end{document} |
\begin{document}
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\markboth{Dimitar Mekerov}{Lie groups as $4$-dimensional
nonintegrable almost product manifolds}
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\title{Lie groups as $4$-dimensional Riemannian
or pseudo-Riemannian almost product manifolds
with nonintegrable structure}
\author{Dimitar Mekerov}
\maketitle
{\mathfrak{S}mall
{\it Abstract.} A Lie group as a 4-dimensional pseudo-Riemannian
manifold is considered. This manifold is equipped with an almost
product structure and a Killing metric in two ways. In the first
case à Riemannian almost product manifold with nonintegrable
structure is obtained, and in the second case -- a
pseudo-Riemannian one. Each belongs to a 4-parametric family of
manifolds, which are characterized geometrically.
{\it Mathematics Subject Classification (2000):} 53C15, 53C50 \\
{\it Key words:} almost product manifold, Lie group, Riemannian
metric, pseudo-Riemann\-ian metric, nonintegrable structure,
Killing metric}
\mathfrak{S}ection{Preliminaries}
Let $M$ be a differentiable manifold with a tensor field $P$ of
type $(1,1)$ and a Riemannian metric $g$ such that
\begin{equation}\label{1.1}
P^2=id,\quad g(Px,Py)=g(x,y)
\end{equation}
for arbitrary $x$, $y$ of the algebra $\mathfrak{X}(M)$ of the smooth vector
fields on $M$. The tensor field $P$ is called an \emph{almost
product structure}. The manifold $(M,P,g)$ is called a
\emph{Riemannian} (\emph{pseudo-Riemannian}, resp.) \emph{almost
product manifold}, if $g$ is a Riemannian (pseudo-Riemannian,
resp.) metric. If ${\rm tr}{P}=0$, then $(M,P,g)$ is an
even-dimensional manifold. The classification from
\cite{StGr:connect} of Riemannian almost product manifolds is made
with respect to the tensor field $F$ of type (0,3), defined by
\begin{equation}\label{1.2}
F(x,y,z)=g\left(\left(\nabla_x P\right)y,z\right),
\end{equation}
where $\nabla$ is the Levi-Civita connection of $g$. The tensor
$F$ has the following properties:
\[
F(x,y,z)=F(x,z,y)=-F(x,Py,Pz),\quad F(x,y,Pz)=-F(x,Py,z).
\]
In the case when $g$ is a pseudo-Riemannian metric, the same
classification is valid for pseudo-Riemannian almost product
manifolds, too. In these classifications the condition
\begin{equation}\label{sigma}
F(x,y,z)+F(y,z,x)+F(z,x,y)=0
\end{equation}
defines a class $\mathcal{W}_3$, which is only the class of the three basic
classes $\mathcal{W}_1$, $\mathcal{W}_2$ and $\mathcal{W}_3$ with nonintegrable structure
$P$.
The class $\mathcal{W}_0$, defined by the condition $F(x,y,z)=0$, is
contained in the other classes. For this class $\nabla P=0$ and
therefore it is an analogue of the class of K\"ahlerian manifolds
in the almost Hermitian geometry.
The curvature tensor field $R$ is defined by $R(x,y)z=\nabla_x
\nabla_y z - \nabla_y \nabla_x z - \nabla_{[x,y]}z$ and the
corresponding tensor field of type $(0,4)$ is determined by
$R(x,y,z,w)=g(R(x,y)z,w)$.
Let $\{e_i\}$ be a basis of the tangent space $T_pM$ at a point
$p\in M$ and $g^{ij}$ be the components of the inverse matrix of
$g$ with respect to $\{e_i\}$. Then the Ricci tensor $\rho$ and
the scalar curvature $\tau$ are defined as follows
\begin{equation}\label{1.3}
\rho(y,z)=g^{ij}R(e_i,y,z,e_j),
\end{equation}
\begin{equation}\label{1.4}
\tau=g^{ij}\rho(e_i,e_j).
\end{equation}
The square norm of $\nabla P$ is defined by
\begin{equation}\label{1.5}
\norm{\nabla P}=g^{ij}g^{ks}g\left(\left(\nabla_{e_i}P\right)e_k,\left(\nabla_{e_j}P\right)e_s\right).
\end{equation}
It is clear that $\nabla P=0$ implies $\norm{\nabla P}=0$ but the inverse
implication for the pseudo-Riemannian case is not always true. We
shall call a pseudo-Riemannian almost product manifold
\emph{isotropic $P$-manifold} if $\nabla P=0$.
The Weyl tensor on a $2n$-dimensional pseudo-Riemannian manifold
($n\mathfrak{g}eq 2$) is
\begin{equation}\label{1.6}
W=R-\frac{1}{2n-2}\left(\psi_1(\rho)-\frac{\tau}{2n-1}\pi_1\right),
\end{equation}
where
\[
\begin{array}{l}
\psi_1(\rho)(x,y,z,w)=g(y,z)\rho(x,w)-g(x,z)\rho(y,w) \\
\phantom{\psi_1(\rho)(x,y,z,w)}+\rho(y,z)g(x,w)-\rho(x,z)g(y,w); \\
\pi_1(x,y,z,w)=g(y,z)g(x,w)-g(x,z)g(y,w). \\
\end{array}
\]
Moreover, for $n\mathfrak{g}eq 2$ the Weyl tensor $W$ is zero if and only if
the manifold is \emph{conformally flat}.
If $\alpha$ is a non-degenerate 2-plane spanned by vectors $x, y
\in T_pM, p\in M$, then its sectional curvature is
\begin{equation}\label{1.7}
k(\alpha)=\frac{R(x,y,y,x)}{\pi_1(x,y,y,x)}.
\end{equation}
\mathfrak{S}ection{A Lie group as a 4-dimensional
pseudo-Rie\-mannian manifold with Killing metric}
Let $V$ be a real 4-dimensional vector space with a basis
$\{E_i\}$. Let us consider a structure of a Lie algebra determined
by commutators $[E_i,E_j]=C_{ij}^k E_k$, where $C_{ij}^k$ are
structure constants satisfying the anti-commutativity condition
$C_{ij}^k=-C_{ji}^k$ and the Jacobi identity $C_{ij}^k
C_{ks}^l+C_{js}^k C_{ki}^l+C_{si}^k C_{kj}^l=0$.
Let $G$ be the associated connected Lie group and $\{X_i\}$ be a
global basis of left invariant vector fields which is induced by
the basis $\{E_i\}$ of $V$. Then we have the decomposition
\begin{equation}\label{2.1}
[X_i,X_j]=C_{ij}^k X_k.
\end{equation}
Let us consider the manifold $(G,g)$, where $g$ is a metric
determined by the conditions
\begin{equation}\label{2.2}
\begin{array}{c}
g(X_1,X_1)=g(X_2,X_2)=g(X_3,X_3)=g(X_4,X_4)=1, \\[4pt]
g(X_i,X_j)=0\quad \text{for}\quad i\neq j \\
\end{array}
\end{equation}
or by the conditions
\begin{equation}\label{2.3}
\begin{array}{c}
g(X_1,X_1)=g(X_2,X_2)=-g(X_3,X_3)=-g(X_4,X_4)=1, \\[4pt]
g(X_i,X_j)=0\quad \text{for}\quad i\neq j. \\
\end{array}
\end{equation}
Obviously, $g$ is a Riemannian metric if it is determined by
\eqref{2.2} and $g$ is a pseudo-Riemannian metric of signature
(2,2) if it is determined by \eqref{2.3}.
It is known that the metric $g$ on the group $G$ is called a
\emph{Killing metric} \cite{Hel} if the following condition is
valid
\begin{equation}\label{2.5}
g([X,Y],Z)+g([X,Z],Y)=0.
\end{equation}
where $X$, $Y$, $Z$ are arbitrary vector fields.
If $g$ is a Killing metric, then according to the proof of
Theorem~2.1 in \cite{MaGrMe-4} the manifold $(G,g)$ is
\emph{locally symmetric}, i.e. $\nabla R=0$. Moreover, the
components of $\nabla$ and $R$ are respectively
\begin{equation}\label{2.6}
\nabla_{ij}=\nabla_{X_i} X_j=\frac{1}{2}[X_i,X_j],
\end{equation}
\begin{equation}\label{2.7}
R_{ijks}=R(X_i,X_j,X_k,X_s)=-\frac{1}{4}g\left([X_i,X_j],[X_k,X_s]\right).
\end{equation}
\mathfrak{S}ection{A Lie group as a Riemannian almost product manifold
with Killing metric and nonintegrable structure}
In this section we consider a Riemannian manifold $(G,P,g)$ with a
metric $g$ determined by \eqref{2.2} and a structure $P$ defined
as follows
\begin{equation}\label{3.1}
PX_1=X_3,\quad PX_2=X_4,\quad PX_3=X_1,\quad PX_4=X_2.
\end{equation}
Obviously, $P^2=\mathrm{id}$. Moreover, \eqref{2.2} and \eqref{3.1} imply
\begin{equation}\label{3.2}
g(PX_i,PX_j)=g(X_i,X_j).
\end{equation}
Therefore, $(G,P,g)$ is a Riemannian almost product manifold.
For the manifold $(G,P,g)$ we propose that $g$ be a Killing
metric. Then $(G,P,g)$ is locally symmetric.
From \eqref{2.6} we obtain
\begin{equation}\label{3.3}
\left( \nabla_{X_i} P \right)X_j=\frac{1}{2}\bigl([X_i,PX_j]-P[X_i,X_j]\bigr).
\end{equation}
Then, according to \eqref{1.2}, for the components of $F$ we have
\begin{equation}\label{3.4}
F_{ijk}=\frac{1}{2}g\bigl([X_i,PX_j]-P[X_i,X_j],X_k\bigr).
\end{equation}
Hence, having in mind \eqref{2.2}, \eqref{3.1} and \eqref{3.2}, we
get
\begin{equation}\label{3.5}
F_{ijk}+F_{jki}+F_{kij}=0,
\end{equation}
i.e. $(G,P,g)$ belong to the class $\mathcal{W}_3$.
According to \eqref{2.5}, we have
\begin{equation}\label{3.6}
g\bigl([X_i,X_j],X_i\bigr)=g\bigl([X_i,X_j],X_j\bigr)=0.
\end{equation}
Then the following decomposition is valid
\begin{equation}\label{3.7}
\begin{array}{ll}
[X_1,X_2]= C_{12}^3 X_3 +C_{12}^4 X_4,\qquad & [X_2,X_3]= C_{23}^1 X_1 +C_{23}^4
X_4,\\[4pt]
[X_1,X_3]= C_{13}^2 X_2 +C_{13}^4 X_4,\qquad & [X_2,X_4]= C_{24}^1 X_1
+C_{24}^3 X_3,\\[4pt]
[X_1,X_4]= C_{14}^2 X_2 +C_{14}^3 X_3,\qquad & [X_3,X_4]= C_{34}^1 X_1
+C_{34}^2 X_2.\\[4pt]
\end{array}
\end{equation}
Now we apply again \eqref{2.5} using \eqref{3.7}. So we obtain
\begin{equation}\label{3.8}
\begin{array}{ll}
[X_1,X_2]= \lambda_1 X_3 +\lambda_2 X_4,\qquad & [X_2,X_3]= \lambda_1 X_1
+\lambda_3 X_4,\\[4pt]
[X_1,X_3]= -\lambda_1 X_2 +\lambda_4 X_4,\qquad & [X_2,X_4]= \lambda_2 X_1
-\lambda_3 X_3,\\[4pt]
[X_1,X_4]= -\lambda_2 X_2 -\lambda_4 X_3,\qquad & [X_3,X_4]= \lambda_4 X_1
+\lambda_3 X_2,\\[4pt]
\end{array}
\end{equation}
where $\lambda_1=C_{12}^3$, $\lambda_2=C_{12}^4$,
$\lambda_3=C_{23}^4$, $\lambda_4=C_{13}^4$. We verify immediately
that the Jacobi identity is satisfied in this case.
Let the conditions \eqref{3.8} be satisfied for a Riemannian
almost product manifold $(G,P,g)$ with structure $P$ and metric
$g$, determined by \eqref{3.1} and \eqref{2.2}, respectively. Then
we verify directly that $g$ is a Killing metric.
Therefore, the following theorem is valid.
\begin{thm}\label{thm-3.1}
Let $(G,P,g)$ be a 4-dimensional Riemannian
almost product manifold, where $G$ is the connected Lie group with
an associated Lie algebra, determined by a global basis $\{X_i\}$
of left invariant vector fields, and $P$ and $g$ are the almost
product structure and the Riemannian metric, determined by
\eqref{3.1} and \eqref{2.2}, respectively. Then $(G,P,g)$ is a
$\mathcal{W}_3$-manifold with a Killing metric $g$ iff $G$ belongs to the
4-parametric family of Lie groups, determined by \eqref{3.8}.
\end{thm}
From this point on, until the end of this section we shall
consider the Riemannian almost product manifold $(G,P,g)$
determined by the conditions of \thmref{thm-3.1}.
Using \eqref{3.4}, \eqref{3.8}, \eqref{3.1} and \eqref{3.2}, we
obtain the following nonzero components of the tensor $F$:
\begin{equation}\label{3.9}
\begin{array}{l}
F_{211}=-F_{233}=2F_{134}=2F_{323}=-2F_{112}=-2F_{314}=\lambda_1,\\[4pt]
F_{144}=-F_{122}=2F_{212}=2F_{423}=-2F_{234}=-2F_{414}=\lambda_2,\\[4pt]
F_{322}=-F_{344}=2F_{214}=2F_{434}=-2F_{223}=-2F_{412}=\lambda_3,\\[4pt]
F_{433}=-F_{411}=2F_{141}=2F_{321}=-2F_{132}=-2F_{334}=\lambda_4.\\[4pt]
\end{array}
\end{equation}
The other nonzero components of $F$ are obtained from the property
$F_{ijk}=F_{ikj}$.
Let $F$ be the Nijenhuis tensor on $(G,P,g)$, i.e.
\[
N_{ij}=[X_i,X_j]+P[PX_i,X_j]+P[X_i,PX_j]-[PX_i,PX_j].
\]
According to \eqref{3.1} and \eqref{3.8}, for the square norm
$\norm{N}=N_{ik}N_{js}g^{ij}g^{ks}$ of $N$ we get
\begin{equation}\label{3.10}
\norm{N}=32\left(\lambda_1^2+\lambda_2^2+\lambda_3^2+\lambda_4^2\right).
\end{equation}
For the square norm of $\nabla P$, using \eqref{1.5}, \eqref{2.2}
and \eqref{3.3}, we obtain
\begin{equation}\label{3.11}
\norm{\nabla P}=4\left(\lambda_1^2+\lambda_2^2+\lambda_3^2+\lambda_4^2\right).
\end{equation}
From \eqref{2.7}, having in mind \eqref{2.2} and \eqref{3.8}, we
receive the following nonzero components of the curvature tensor
$R$:
\begin{equation}\label{3.12}
\begin{array}{ll}
R_{1221}=\frac{1}{4}\left(\lambda_1^2+\lambda_2^2\right),\quad
&
R_{1331}=\frac{1}{4}\left(\lambda_1^2+\lambda_4^2\right),\\[4pt]
R_{1441}=\frac{1}{4}\left(\lambda_2^2+\lambda_4^2\right),\quad
&
R_{2332}=\frac{1}{4}\left(\lambda_1^2+\lambda_3^2\right),\\[4pt]
R_{2442}=\frac{1}{4}\left(\lambda_2^2+\lambda_3^2\right),\quad
&
R_{3443}=\frac{1}{4}\left(\lambda_3^2+\lambda_4^2\right),\\[4pt]
R_{1341}=R_{2342}=\frac{1}{4}\lambda_1\lambda_2,\quad
&
R_{3123}=R_{4124}=\frac{1}{4}\lambda_3\lambda_4,\\[4pt]
R_{1231}=R_{4234}=\frac{1}{4}\lambda_2\lambda_4,\quad
&
R_{2142}=R_{3143}=\frac{1}{4}\lambda_1\lambda_3,\\[4pt]
R_{1241}=R_{3243}=-\frac{1}{4}\lambda_1\lambda_4,\quad
&
R_{2132}=R_{4134}=-\frac{1}{4}\lambda_2\lambda_3.\\[4pt]
\end{array}
\end{equation}
The other nonzero components of $R$ are obtained from the
properties $R_{ijks}=R_{ksij}$ and $R_{ijks}=-R_{jiks}=-R_{ijsk}$.
From \eqref{1.3}, having in mind \eqref{2.2}, we receive the
components $\rho_{ij}=\rho(X_i,X_j)$ of the Ricci tensor $\rho$.
The nonzero components of $\rho$ are:
\begin{equation}\label{3.13}
\begin{array}{c}
\begin{array}{ll}
\rho_{11}=\frac{1}{2}\left(\lambda_1^2+\lambda_2^2+\lambda_4^2\right),\quad
&
\rho_{22}=\frac{1}{2}\left(\lambda_1^2+\lambda_2^2+\lambda_3^2\right),\\[4pt]
\rho_{33}=\frac{1}{2}\left(\lambda_1^2+\lambda_3^2+\lambda_4^2\right),\quad
&
\rho_{44}=\frac{1}{2}\left(\lambda_2^2+\lambda_3^2+\lambda_4^2\right),\\[4pt]
\end{array}
\\[4pt]
\begin{array}{lll}
\rho_{12}=\frac{1}{2}\lambda_3\lambda_4,\quad
&
\rho_{13}=-\frac{1}{2}\lambda_2\lambda_3,\quad
&
\rho_{14}=\frac{1}{2}\lambda_1\lambda_3,\\[4pt]
\rho_{23}=\frac{1}{2}\lambda_2\lambda_4,\quad
&
\rho_{24}=-\frac{1}{2}\lambda_1\lambda_4,\quad
&
\rho_{34}=\frac{1}{2}\lambda_1\lambda_2.\\[4pt]
\end{array}
\end{array}
\end{equation}
The other nonzero components of $\rho$ are obtained from the
property $\rho_{ij}=\rho_{ji}$.
For the scalar curvature $\tau$, using \eqref{1.4}, we obtain
\begin{equation}\label{3.14}
\tau=\frac{3}{2}\left(\lambda_1^2+\lambda_2^2+\lambda_3^2+\lambda_4^2\right).
\end{equation}
From \eqref{1.6}, having in mind \eqref{2.2}, \eqref{3.12},
\eqref{3.13} and \eqref{3.14}, we get for the Weyl tensor $W=0$.
Then $(G,P,g)$ is a conformally flat manifold.
For the sectional curvatures $k_{ij}=k(\alpha_{ij})$ of basic
2-planes $\alpha_{ij}=(X_i,X_j)$, according to \eqref{1.7},
\eqref{3.12} and \eqref{2.2}, we have:
\begin{equation}\label{3.15}
\begin{array}{ll}
k_{12}=\frac{1}{4}\left(\lambda_1^2+\lambda_2^2\right),\quad
&
k_{13}=\frac{1}{4}\left(\lambda_1^2+\lambda_4^2\right),\\[4pt]
k_{14}=\frac{1}{4}\left(\lambda_2^2+\lambda_4^2\right),\quad
&
k_{23}=\frac{1}{4}\left(\lambda_1^2+\lambda_3^2\right),\\[4pt]
k_{24}=\frac{1}{4}\left(\lambda_2^2+\lambda_3^2\right),\quad
&
k_{34}=\frac{1}{4}\left(\lambda_3^2+\lambda_4^2\right).\\[4pt]
\end{array}
\end{equation}
The obtained geometric characteristics of the considered manifold
are generalized in the following
\begin{thm}\label{thm-3.2}
Let $(G,P,g)$ be the 4-dimensional Riemannian
almost product manifold where $G$ is the Lie group determined by
\eqref{3.8}, and the structure $P$ and the metric $g$ are
determined by \eqref{3.1} and \eqref{2.2}, respectively. Then
\begin{enumerate}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\item
$(G,P,g)$ is a locally symmetric $\mathcal{W}_3$-manifold with Killing metric $g$ and zero Weyl tensor;
\item
The nonzero components of the basic tensor $F$, the curvature tensor $R$ and the Ricci tensor $\rho$ are
\eqref{3.9}, \eqref{3.12} and \eqref{3.13}, respectively;
\item
The square norms of the Nijenhuis tensor $N$ and $\nabla P$ are \eqref{3.10} and \eqref{3.11}, respectively;
\item
The scalar curvature $\tau$ and the sectional curvatures $k_{ij}$
of the basic 2-planes are \eqref{3.14} and \eqref{3.15}, respectively.
\end{enumerate}
\end{thm}
Let us remark that the 2-planes $\alpha_{13}$ and $\alpha_{24}$
are \emph{$P$-invariant 2-planes}, i.e.
$P\alpha_{13}=\alpha_{13}$, $P\alpha_{24}=\alpha_{24}$. The
2-planes $\alpha_{12}$, $\alpha_{14}$, $\alpha_{23}$,
$\alpha_{34}$ are \emph{totally real 2-planes}, i.e.
$\alpha_{12}\perp P\alpha_{12}$, $\alpha_{14}\perp P\alpha_{14}$,
$\alpha_{23}\perp P\alpha_{23}$, $\alpha_{34}\perp P\alpha_{34}$.
Then the equalities \eqref{3.15} imply the following
\begin{thm}\label{thm-3.3}
Let $(G,P,g)$ be the 4-dimensional Riemannian
almost product manifold where $G$ is the Lie group determined by
\eqref{3.8}, and the structure $P$ and the metric $g$ are
determined by \eqref{3.1} and \eqref{2.2}, respectively. Then
\begin{enumerate}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\item
$(G,P,g)$ is of constant $P$-invariant sectional curvatures
iff
\[\lambda_1^2+\lambda_4^2=\lambda_2^2+\lambda_3^2;\]
\item
$(G,P,g)$ is of constant totally real sectional curvatures
iff
\[\lambda_1^2=\lambda_4^2,\qquad \lambda_2^2=\lambda_3^2\].
\end{enumerate}
\end{thm}
\mathfrak{S}ection{A Lie group as a pseudo-Riemannian almost product manifold
with Killing metric and nonintegrable structure}
In this section we consider a pseudo-Riemannian manifold $(G,P,g)$
with a metric $g$ determined by \eqref{2.3} and a structure $P$
defined as follows
\begin{equation}\label{4.1}
PX_1=X_1,\quad PX_2=X_1,\quad PX_3=-X_3,\quad PX_4=-X_4.
\end{equation}
Obviously, $P^2=\mathrm{id}$. Moreover, \eqref{2.3} and \eqref{4.1} imply
\begin{equation}\label{4.2}
g(PX_i,PX_j)=g(X_i,X_j).
\end{equation}
Therefore, $(G,P,g)$ is a pseudo-Riemannian almost product
manifold.
For the manifold $(G,P,g)$ we propose that $g$ be a Killing
metric. Then $(G,P,g)$ is locally symmetric and the equalities
\eqref{2.6}, \eqref{2.7}, \eqref{3.3} and \eqref{3.4} are valid.
From \eqref{2.3}, \eqref{4.1} and \eqref{4.2} we obtain
\eqref{3.5}, i.e. $(G,P,g)$ is a $\mathcal{W}_3$-manifold.
Now, the equalities \eqref{3.6} and \eqref{3.7} are also
satisfied. According to \eqref{2.5}, from \eqref{3.7} we obtain
\begin{equation}\label{4.3}
\begin{array}{ll}
[X_1,X_2]= \lambda_2 X_3 -\lambda_1 X_4,\qquad & [X_2,X_3]= -\lambda_2 X_1
-\lambda_3 X_4,\\[4pt]
[X_1,X_3]= \lambda_2 X_2 +\lambda_4 X_4,\qquad & [X_2,X_4]= \lambda_1 X_1
+\lambda_3 X_3,\\[4pt]
[X_1,X_4]= -\lambda_1 X_2 -\lambda_4 X_3,\qquad & [X_3,X_4]= -\lambda_4 X_1
+\lambda_3 X_2,\\[4pt]
\end{array}
\end{equation}
where $\lambda_1=C_{24}^1$, $\lambda_2=C_{12}^3$,
$\lambda_3=C_{24}^3$, $\lambda_4=C_{13}^4$. We verify immediately
that the Jacobi identity is satisfied in this case.
Let the conditions \eqref{4.3} be satisfied for a
pseudo-Riemannian almost product manifold $(G,P,g)$ with structure
$P$ and metric $g$ determined by \eqref{4.1} and \eqref{2.3},
respectively. Then we verify directly that $g$ is a Killing
metric.
Therefore, the following theorem is valid.
\begin{thm}\label{thm-4.1}
Let $(G,P,g)$ be a 4-dimensional pseudo-Riemannian
almost product manifold, where $G$ is the connected Lie group with
an associated Lie algebra, determined by a global basis $\{X_i\}$
of left invariant vector fields, and $P$ and $g$ are the almost
product structure and the pseudo-Riemannian metric, determined by
\eqref{4.1} and \eqref{2.3}, respectively. Then $(G,P,g)$ is a
$\mathcal{W}_3$-manifold with a Killing metric $g$ iff $G$ belongs to the
4-parametric family of Lie groups, determined by \eqref{4.3}.
\end{thm}
From this point on, until the end of this section we shall
consider the pseudo-Riemannian almost product manifold $(G,P,g)$
determined by the conditions of \thmref{thm-4.1}.
In an analogous way of the previous section, we get some geometric
characteristics of $(G,P,g)$.
We obtain the following nonzero components of the tensor $F$:
\begin{equation}\label{4.4}
\begin{array}{ll}
F_{124}=-F_{214}=\lambda_1, \qquad &
F_{213}=-F_{123}=\lambda_2,\\[4pt]
F_{423}=-F_{324}=\lambda_3, \qquad &
F_{314}=-F_{413}=\lambda_4.\\[4pt]
\end{array}
\end{equation}
The other nonzero components of $F$ are obtained from the
properties $F_{ijk}=F_{ikj}$.
The square norms of the Nijenhuis tensor $N$ and $\nabla P$ are
respectively:
\begin{equation}\label{4.5}
\norm{N}=24\left(\lambda_1^2+\lambda_2^2-\lambda_3^2-\lambda_4^2\right),
\end{equation}
\begin{equation}\label{4.6}
\norm{\nabla P}=-4\left(\lambda_1^2+\lambda_2^2-\lambda_3^2-\lambda_4^2\right).
\end{equation}
The nonzero components of the curvature tensor $R$ and the Ricci
tensor $\rho$ are respectively:
\begin{equation}\label{4.7}
\begin{array}{ll}
R_{1221}=-\frac{1}{4}\left(\lambda_1^2+\lambda_2^2\right),\quad
&
R_{1331}=\frac{1}{4}\left(\lambda_2^2-\lambda_4^2\right),\\[4pt]
R_{1441}=-\frac{1}{4}\left(\lambda_1^2-\lambda_4^2\right),\quad
&
R_{2332}=\frac{1}{4}\left(\lambda_2^2-\lambda_3^2\right),\\[4pt]
R_{2442}=\frac{1}{4}\left(\lambda_1^2-\lambda_3^2\right),\quad
&
R_{3443}=\frac{1}{4}\left(\lambda_3^2+\lambda_4^2\right),\\[4pt]
R_{1341}=R_{2342}=-\frac{1}{4}\lambda_1\lambda_2,\quad
&
R_{2132}=-R_{4134}=\frac{1}{4}\lambda_1\lambda_3,\\[4pt]
R_{1231}=-R_{4234}=\frac{1}{4}\lambda_1\lambda_4,\quad
&
R_{2142}=-R_{3143}=\frac{1}{4}\lambda_2\lambda_3,\\[4pt]
R_{1241}=-R_{3243}=\frac{1}{4}\lambda_2\lambda_4,\quad
&
R_{3123}=R_{4124}=\frac{1}{4}\lambda_3\lambda_4;\\[4pt]
\end{array}
\end{equation}
\begin{equation}\label{4.8}
\begin{array}{c}
\begin{array}{ll}
\rho_{11}=-\frac{1}{2}\left(\lambda_1^2+\lambda_2^2-\lambda_4^2\right),\quad
&
\rho_{22}=-\frac{1}{2}\left(\lambda_1^2+\lambda_2^2-\lambda_3^2\right),\\[4pt]
\rho_{33}=\frac{1}{2}\left(\lambda_2^2-\lambda_3^2-\lambda_4^2\right),\quad
&
\rho_{44}=\frac{1}{2}\left(\lambda_1^2+\lambda_3^2-\lambda_4^2\right),\\[4pt]
\end{array}
\\[4pt]
\begin{array}{lll}
\rho_{12}=-\frac{1}{2}\lambda_3\lambda_4,\quad
&
\rho_{13}=\frac{1}{2}\lambda_1\lambda_3,\quad
&
\rho_{14}=\frac{1}{2}\lambda_2\lambda_3,\\[4pt]
\rho_{23}=\frac{1}{2}\lambda_1\lambda_4,\quad
&
\rho_{24}=\frac{1}{2}\lambda_2\lambda_4,\quad
&
\rho_{34}=-\frac{1}{2}\lambda_1\lambda_2.\\[4pt]
\end{array}
\end{array}
\end{equation}
The other nonzero components of $R$ and $\rho$ are obtained from
the properties $R_{ijks}=R_{ksij}$, $R_{ijks}=-R_{jiks}=-R_{ijsk}$
and $\rho_{ij}=\rho_{ji}$.
The scalar curvature is
\begin{equation}\label{4.9}
\tau=-\frac{3}{2}\left(\lambda_1^2+\lambda_2^2-\lambda_3^2-\lambda_4^2\right).
\end{equation}
We get for the Weyl tensor that $W=0$. Then $(G,P,g)$ is a
conformally flat manifold.
The sectional curvatures $k_{ij}=k(\alpha_{ij})$ of basic 2-planes
$\alpha_{ij}=(X_i,X_j)$ are:
\begin{equation}\label{4.10}
\begin{array}{ll}
k(\alpha_{13})=-\frac{1}{4}\left(\lambda_2^2-\lambda_4^2\right),\quad
&
k(\alpha_{24})=-\frac{1}{4}\left(\lambda_1^2-\lambda_3^2\right),\\[4pt]
k(\alpha_{12})=-\frac{1}{4}\left(\lambda_1^2+\lambda_2^2\right),\quad
&
k(\alpha_{14})=-\frac{1}{4}\left(\lambda_1^2-\lambda_4^2\right),\\[4pt]
k(\alpha_{23})=-\frac{1}{4}\left(\lambda_2^2-\lambda_3^2\right),\quad
&
k(\alpha_{34})=\frac{1}{4}\left(\lambda_3^2+\lambda_4^2\right).\\[4pt]
\end{array}
\end{equation}
Since $\alpha_{ij}=P\alpha_{ij}$ then all basic 2-planes are
$P$-invariant. It is used to check that now $(G,P,g)$ does not
accept constant $P$-invariant sectional curvatures.
The obtained geometric characteristics of the considered manifold
are generalized in the following
\begin{thm}\label{thm-4.2}
Let $(G,P,g)$ be the 4-dimensional pseudo-Riemannian
almost product manifold where $G$ is the Lie group determined by
\eqref{4.3}, and the structure $P$ and the metric $g$ are
determined by \eqref{4.1} and \eqref{2.3}, respectively. Then
\begin{enumerate}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\item
$(G,P,g)$ is a locally symmetric conformally flat $\mathcal{W}_3$-manifold with Killing metric $g$;
\item
The nonzero components of the basic tensor $F$, the curvature tensor $R$ and the Ricci tensor $\rho$ are
\eqref{4.4}, \eqref{4.7} and \eqref{4.8}, respectively;
\item
The square norms of the Nijenhuis tensor $N$ and $\nabla P$ are \eqref{4.5} and \eqref{4.6}, respectively;
\item
The scalar curvature $\tau$ and the sectional curvatures $k_{ij}$
of the basic 2-planes are \eqref{4.9} and \eqref{4.10}, respectively.
\end{enumerate}
\end{thm}
The last theorem implies immediately the following
\begin{cor}
Let $(G,P,g)$ be the 4-dimensional pseudo-Riemannian
almost product manifold where $G$ is the Lie group determined by
\eqref{4.3}, and the structure $P$ and the metric $g$ are
determined by \eqref{4.1} and \eqref{2.3}, respectively. Then the
following propositions are equivalent:
\begin{enumerate}
\renewcommand{(\roman{enumi})}{(\roman{enumi})}
\item
$(G,P,g)$ is an isotropic $P$-manifold;
\item
$(G,P,g)$ is a scalar flat manifold;
\item
The Nijenhuis tensor is isotopic;
\item
The condition $\lambda_1^2+\lambda_2^2-\lambda_3^2-\lambda_4^2=0$ is
valid.
\end{enumerate}
\end{cor}
\textit{Dimitar Mekerov\\
University of Plovdiv\\
Faculty of Mathematics and Informatics
\\
Department of Geometry\\
236 Bulgaria Blvd.\\
Plovdiv 4003\\
Bulgaria
\\
e-mail: [email protected]}
\end{document} |
\begin{document}
\title{On Hunting for Taxicab Numbers}
\author{P. Emelyanov \\
Institute of Informatics Systems \\
6 avenue Lavrentiev, 630090, Novosibirsk, Russia \\
e-mail: [email protected]}
\maketitle
\begin{abstract}
In this article, we make use of some known method to investigate
some properties of the numbers represented as sums of two equal
odd powers, i.e., the equation $x^n+y^n=N$\/ for $n\ge3$. It was
originated in developing algorithms to search new taxicab numbers
(i.e., naturals that can be represented as a sum of positive cubes
in many different ways) and to verify their minimality. We discuss
properties of diophantine equations that can be used for our
investigations. This techniques is applied to develop an algorithm
allowing us to compute new taxicab numbers (i.e., numbers
represented as sums of two positive cubes in $k$ different ways),
for $k=7\ldots14$\/.
\end{abstract}
\section*{Introduction}
This work was originated in searching new so--called {\em taxicab
numbers}, i.e., naturals $T_k$\/ that can be
represented/decomposed as/into a sum of positive cubes in $k$
different ways, and verifying their minimality. We made use of
some known method to investigate properties of the cubic equation
that could help us to find new taxicab numbers.
Already Fermat proved that numbers expressible as a sum of two
cubes in $n$\/ different ways exist for any $n$. But still finding
taxicab numbers and proving their minimality are hard
computational problems. Whereas the first nontrivial taxicab
number $T_2=1729$\/ became widely--known in 1917 thanks to
Ramanujan and Hardy, next ones were only found with help of
computers: $T_3=87539319$ (J. Leech, 1957), $T_4=6963472309248$
(E. Rosenstiel, J.A. Dardis, and C.R. Rosenstiel, 1991),
$\mbox{\bf\em W}_5=T_5=48988659276962496$ (D. Wilson, 1997,
\cite{Wilson-jis-1999}). It is known that these numbers are
minimal. For $\mbox{\bf\em R}_6=T_6=24153319581254312065344$ (R.L.
Rathbun, 2002) as well as for next discovered taxicab numbers it
is unknown.
In January--September 2006 the author computed
$T_7=139^3\mbox{\bf\em R}_6$, $T_8=727^3T_7$, $T_9=4327^3T_8$,
$T_{10}=38623^3T_9$, and $T_{11}=45294^3T_{10}$. At the end of
2006 the author learned about the results of C. Boyer
\cite{Boyer-2006} who established smaller $T_7,\ldots,T_{11}$\/
and first $T_{12}$\/ in December 2006. At the begin of 2007 the
author computed $T_{13}$\/ and $T_{14}$.
The article is organized as follows. We start with putting the
equation in a new form. Next, we deduce simple properties of the
equation of interest based on this presentation. At the end, we
present a new algorithm to compute taxicab numbers which we used
to find new ones.
\section{Common Properties}
We are interested in the problem of representations (also called
decompositions) of numbers as the sums of two positive odd
$n$-powers; i.e., solvability of the equation
\begin{equation}\label{OriginalEquation}
x^n+y^n=N
\end{equation}
in positive integers. A solution of this equation is also called a
representation or a decomposition of the number $N$. The equation
of interest is too ``smooth'' in its original form. We want to
make it ``uneven''. We are going to consider this equation in the
following $m\pm{}h$-form ($m\neq h>0$)
\begin{equation}\label{TheEquation}
(m-h)^n+(m+h)^n=N
\end{equation}
which is not an infrequent guest in number--theoretical proofs.
Although only even numbers can be directly represented in this
way, there is a simple transformation that allows us to treat this
equation for odd $N$\/ as well. In fact, any pair $(x,y)$\/
consisting of even and odd integers can be represented as
$(t-s-1,t+s)$. If $N$\/ is odd, we write
\[
(t-s-1)^n+(t+s)^n=N.
\]
Multiplying both sides by $2^n$\/ we can put the previous equation
into the form
\begin{equation}\label{TheEqOdd}
((2t-1)-(2s+1))^n+((2t-1)+(2s+1))^n=2^nN
\end{equation}
and, then some extra steps are needed to obtain representations of
$N$\/ itself. For the exponent 3, the least odd number $N$\/ for
which $2^3N$\/ yields a not only proper two cubes representation
is 513:
\[
2^3 513=2^3\left(1^3+8^3\right)=(12-3)^3+(12+3)^3=4104.
\]
Notice that 4104 is the least even number represented as a sum of
two cubes in two different ways. Next, assume $N$\/ to be even if
we do not explicitly state the contrary.
We are interested in any prime powers, although sometimes it is
sufficient that they are odd only. Such representations for odd
powers are closely related to divisors of the numbers of interest.
We shall refer to $m$\/ as a {\em median}\/ of the corresponding
power representation and to $N_d$\/ as an integer quotient $N/d$\/
if it exists. We shall make use the following property (a simple
corollary of Quadratic Reciprocity Law) of odd prime divisors of
binary forms:
\begin{Property}\label{BinaryForm3}
\[
p ~|~ ax^2+by^2 ~\wedge~ \gcd(ax,by)=1 ~~\Longrightarrow~~
\legendre{ab}{p}=(-1)^\frac{p-1}{2}.
\]
In particular, for the binary form $u^2+3v^2$\/ the forbidden
divisors are
\[
\legendre{3}{p}\not=(-1)^\frac{p-1}{2}; \mbox{~~i.e.,~~}
p\equiv5,11\mod{12}.
\]
\end{Property}
Given $N$\/ and its divisors, by solving an $n-1$-order polynomial
equation
\begin{equation}\label{ExpansionEQ}
(m-h)^n+(m+h)^n=
2m\left(
\sum_{k=0}^{\frac{n-1}{2}}\bincoeff{n}{2k}m^{n-2k-1}h^{2k}
\right)=N
\end{equation}
with respect to $h$, we can either ``easily'' find some
representation(s) of this number or prove that it is impossible.
Notice that in this polynomial $m$\/ and $h$\/ occur only in odd
and even powers, respectively.
We start the investigation by establishing the following simple
properties of {\bf Equation (\ref{TheEquation})}.
\begin{Lemma}\label{DivNModulo}
If $m$\/ is a median of some representation of $N$, then
\[
m\equiv N_2\mod{n}.
\]
If $n | N$, then also $n^2 | N$. If $n \mbox{$\hspace*{4pt}|\hspace*{-3.75pt}/~$}{} N$, then
$N=2m(nt+1)$.
\end{Lemma}
\proof
First, rewriting {\bf Equation (\ref{TheEquation})} in the form
\[
m^n+n\sum_{k=1}^{\frac{n-1}{2}}\frac{1}{n}\bincoeff{n}{2k}m^{n-2k}h^{2k}=N_2
\]
we can derive the modular equation $m^n\equiv N_2\mod{n}$. Next:
\begin{itemize}
\item By applying Fermat's Little Theorem we have the first
statement.
\item Because $n | N$, therefore also $n | m$, and this yields the
second statement.
\item Because $n \mbox{$\hspace*{4pt}|\hspace*{-3.75pt}/~$}{} N$, then also $n \mbox{$\hspace*{4pt}|\hspace*{-3.75pt}/~$}{} m$. By
applying Fermat's Little Theorem to $m^{n-1}\equiv
N_{2m}\mod{n}$\/ we have the third statement.
\end{itemize}
\qed
Because $h$\/ is ranged in $(0,m)$\/ it is easy to establish
\begin{Lemma}\label{DivBounds}
A necessary condition for $N$\/ to have a representation as the
sum of $n$-powers is
\[
\exists m|N ~~:~~ \sqrt[n]{\frac{N}{2^n}}~<~m~<~\sqrt[n]{\frac{N}{2}}.
\]
\end{Lemma}
\noindent Obviously, the number of such representations does not
exceed the number of divisors of $N$\/ satisfying this condition
(see also {\bf Lemma \ref{TaxicabLowerBound}}).
{\bf Lemmas 1} and {\bf2} allow us to estimate numbers being the
sum of two odd powers higher than 2 in $k$\/ ways. If a number has
two different representations for the power $n$, then the medians
$m_1, m_2$\/ corresponding to them also satisfy the congruence
$m_1\equiv m_2\mod{n}$. Because
$\sqrt[n]{N/2^n}+n(k-1)\leq\sqrt[n]{N/2}$, we have the following
properties of generalized taxicab numbers
\begin{Lemma}\label{TaxicabLowerBound}
If number $T(n,k)$, $k>1$, represented as the sum of two
$n$-powers in $k$\/ ways is even, then it has at least $k$\/
divisors in the range $(\sqrt[n]{N/2^n},\sqrt[n]{N/2})$\/ and the
following lower bound holds
\[
T(n,k)\geq2\left(\frac{2n}{2-\sqrt[n]{2}}\right)^n(k-1)^n
\]
\end{Lemma}
\noindent This bound is far from optimal due to a quite
conservative assumption about the gaps between medians. This is a
subject of further investigation. Recall that only wide-known
theoretical bound for $T(3,k)=T_k$\/ is Silverman's result
\cite{Silverman-jlms-1983} that describes its logarithmic
behavior:
\[
\log T_k=o(k^{r+2/r}),
\]
where $r$\/ is the highest rank of {\bf Equation
(\ref{OriginalEquation})}. The highest rank known now is 5.
When there are "too many" taxicab medians they cannot be relative
prime because all of them are divisors. Hence they share common
divisors. In particular, for taxicab medians $m_1<\ldots<m_k$\/
the following inequality holds:
\[
\mbox{lcm}(m_1,\ldots,m_k)\leq (2m_1)^n.
\]
The cubic equation in the form $m^2+3h^2=N_{2m}$\/ provides a way
to derive parameterizations of the two cubes representation
problem\footnote{Here we treat independently the median $m$\/ and
its co-factor $N_{2m}$; therefore this does not cover general
cases.}. We mention only those of them that relate to the taxicab
numbers problem. It arises when $N_{2m}$\/ is a cube and this case
is connected to the well-known problem of the decomposition of
numbers into two rational cubes (positive or not) which was
investigated by Fermat, Euler, Sylvester, and other researchers.
G\'erardin proved \cite[Chapter XX]{Dickson-1999} that all
solutions of $u^2+3v^2=w^3$\/ with $\gcd(u,v)=1$\/ are generated
by \label{Gerardin}
\[
(t^3-9ts^2)^2+3(3t^2s-3s^3)^2=(t^2+3s^2)^3.
\]
We have
\[
(t^3-3t^2s-9ts^2+3s^3)^3+(t^3+3t^2s-9ts^2-3s^3)^3=2(t^3-9ts^2)(t^2+3s^2)^3,
\]
and next
\[
\left(\frac{t^3-3t^2s-9ts^2+3s^3}{t^2+3s^2}\right)^3+
\left(\frac{t^3+3t^2s-9ts^2-3s^3}{t^2+3s^2}\right)^3=2t(t-3s)(t+3s).
\]
So, if the diophantine equation
$2t^3-18ts^2\pm{}Nr^3=0$\/ is solvable
\footnote{Euler's solution of the two rational cubes problem is
slightly different.}, then $N$\/ is decomposable.
This can be simplified into one-parametric examples as follows
\[
\left(\frac{w^3+3w^2-6w+1}{3(w^2-w+1)}\right)^3-
\left(\frac{w^3-6w^2+3w+1}{3(w^2-w+1)}\right)^3=w(w-1),
\]
and
\[
\pm{}\left(\frac{8w^9\pm{}24w^6+6w^3\mp{}1}{3w(4w^6\pm{}2w^3+1)}\right)^3\mp{}
\left(\frac{8w^9\mp{}12w^6-12w^3\mp{}1}{3w(4w^6\pm{}2w^3+1)}\right)^3=4w^3\pm{}2.
\]
Also, the substitution $t-3s=u^2v, t+3s=uv^2$\/ gives
\[
\left(\frac{u^3+6u^2v+3uv^2-v^3}{3(u^2+uv+v^2)}\right)^3+
\left(\frac{v^3+6v^2u+3vu^2-u^3}{3(u^2+uv+v^2)}\right)^3=uv(u+v)
\]
which provides the following parametrization of the sum of two
integer powers
\[
\left(\frac{p^9+6p^6q^3+3p^3q^6-q^9}{3pq(p^6+p^3q^3+q^6)}\right)^3+
\left(\frac{q^9+6q^6p^3+3q^3p^6-p^9}{3pq(p^6+p^3q^3+q^6)}\right)^3=p^3+q^3.
\]
Catalan's parametrization
\[
\left({\mbox{\small$\frac12$}}\,(t+s)(t-2s)(s-2t)\right)^2+
3\left({\mbox{\small$\frac32$}}\,ts(t-s)\right)^2=
\left(t^2-ts+s^2\right)^3
\]
leads us to another rational cubes identity
\[
\left(\frac{t^3-3t^2s+s^3}{t^2-ts+s^2}\right)^3+
\left(\frac{t^3-3ts^2+s^3}{t^2-ts+s^2}\right)^3=(t+s)(2s-t)(s-2t).
\]
The substitution $2s-t=u^2v, s-2t=uv^2$\/ gives the following
identity
\[
\left(\frac{u^3+3u^2v-6uv^2+v^3}{3(u^2-uv+v^2)}\right)^3-
\left(\frac{u^3-6u^2v+3uv^2+v^3}{3(u^2-uv+v^2)}\right)^3=uv(u-v)
\]
which provides the following parametrization of the sum of two
integer powers
\[
\left(\frac{p^9+3p^6q^3-6p^3q^6+q^9}{3pq(p^6-p^3q^3+q^6)}\right)^3-
\left(\frac{p^9-6p^6q^3+3p^3q^6+q^9}{3pq(p^6-p^3q^3+q^6)}\right)^3=p^3-q^3.
\]
It is easy to note that these parameterizations of the sum and the
difference of two integer cubes also give parameterizations to the
diophantine equation $X^3+Y^3=S^3+T^3$. Euler's parametric
solution to $X^3+Y^3=S^3+T^3$\/ is
\[
\begin{array}{lll}
X = w (1-(u -3v)(u^2+ 3v^2)) &~~~& Y = w ((u + 3v)(u^2+ 3v^2)-1) \\
S = w ((u + 3v)-(u^2+ 3v^2)^2) & & T = w ((u^2+ 3v^2)^2+ (3v-u))
\end{array}
\]
Finally, we mention some properties of the equation of interest
that can be used to investigate taxicab numbers. Sometimes we can
improve the congruence of {\bf Lemma \ref{DivNModulo}}:
\begin{Lemma}
If $(m-h)^p+(m+h)^p=N$,~ $\gcd(m,h)=1$\/,~ $m\not\equiv
h\mod{2}$,~ then
\[
p=3 ~~\Rightarrow~~ m\equiv N_2\mod{12}
\]
\[
p=5 ~~\Rightarrow~~ m\equiv N_2\mod{20}.
\]
\end{Lemma}
\proof We write down $2m^3+6mh^2=N$\/ as $m^2-h^2+4h^2=N_{2m}$\/
and $2m^5+20m^3h^2+10mh^4=N$\/ as $5m(m^2+h^2)^2-4m^5=N_2$.
Considering these equations by modulo 4 we conclude $m\equiv
N_2\mod{4}$. Combining this congruence with the congruence from
{\bf Lemma \ref{DivNModulo}} we obtain these lemma statements.
\qed
The forbidden divisors condition for two-squares representation is
well known since Fermat's work. For cubic and quintic equations
there are analogies which follow from {\bf Property
\ref{BinaryForm3}}:
\begin{Lemma}\label{ForbiddenDivisors}
Necessary conditions for $N$\/ to have a cubic/quintic
representation with $\gcd(m,h)=1$\/ are the following:
\begin{enumerate}
\item It has no prime divisors of forms $12t+5$\/ and $12t+11$\/
(the cubic case) or of forms $10t\pm{}1$\/ (the quintic case), or
\item If such divisors exist, then all of them are factors of the
median.
\end{enumerate}
\end{Lemma}
\noindent{\bf Remark.}~ In view of the cubic case of {\bf Lemma
\ref{ForbiddenDivisors}}, we can mention the results of Euler et
al for the divisors of numbers in the form $u^2+3v^2$: all prime
divisors have the same form $\alpha^2+3\beta^2$.
\vskip\baselineskip
\section{New Taxicab/Cabtaxi Numbers}
Before we discuss cubic taxicab numbers, we briefly consider the
equation $x^5+y^5=u^5+v^5$. No such number is known within the
range up to $1.05\cdot10^{26}$. We have not yet found any, but we
found some solution in Gaussian integers:
\[
\begin{array}{c}
\left(t^2+s^2-(t^2-2ts-s^2)\imath\right)^5+\left(t^2+s^2+(t^2-2ts-s^2\right)\imath)^5=\\
\left(t^2+s^2-(t^2+2ts-s^2)\imath\right)^5+\left(t^2+s^2+(t^2+2ts-s^2\right)\imath)^5=\\
-8(t^2+s^2)(t^4-2t^3s-6t^2s^2+2ts^3+s^4)(t^4+2t^3s-6t^2s^2-2ts^3+s^4)
\end{array}
\]
The least such positive number is
$3800=(5-\imath)^5+(5+\imath)^5=(5-7\imath)^5+(5+7\imath)^5$.
The observation that $T_6=79^3T_5$\/ stirs up our interest in
searching for new taxicab numbers $T_k$\/ in the same way. The
usual definition of taxicab numbers is equipped with a condition
that they are minimal. But for brevity we designate all multi-ways
representable numbers as taxicab numbers.
Even an open question
\footnote{
C. Calude et al \cite{CaludeCaludeDinneen-jucs-2003} (with an
update \cite{CaludeCaludeDinneen-CDMTCS-2005}) stated that the
minimality of $T_6$\/ can be confirmed with the probability
$>0.99$\/ but G. Martin criticized their considerations in
Mathematical Reviews MR2149410 (2006a:11175).
}
~about the minimality of $T_6$\/ does not matter. To compute some
$k+1$--way representable number we can try any $k$-way
representable number. Our approach can produce non-minimal
numbers, but such numbers can be used to check their minimality or
to search for smaller ones. We believe that this median--based
approach reducing the length of tested numbers in three times
allows us to check the minimality of $T_6$\/ and $T_7$.
Notice that Wilson \cite{Wilson-jis-1999} used similar ideas
(cubic multipliers) to find 5--way representable number
$\mbox{\bf\em W}_5=48988659276962496$\/ in 1997 but his approach
is more expensive even for small numbers. During this search a
six-way example was also detected. Inspired by Wilson's approach
in 2002 R. L. Rathbun \cite{Rathbun-2002} presented the smaller
candidate
\[
\mbox{\bf\em R}_6=79^3\,\mbox{\bf\em W}_5=24153319581254312065344.
\]
Rathbun also mentioned multipliers $139$\/ and $727$\/ giving
other examples of six-way representable numbers. Our approach
demonstrates that they appear in multipliers of $T_9$\/ and
$T_{11}$, respectively.
In the first version of this article (December 2006) we described
a modification of our algorithm that produces some taxicab
numbers. In January--September 2006 with help of this algorithm we
computed $T_7=139^3\mbox{\bf\em R}_6$, $T_8=727^3T_7$,
$T_9=4327^3T_8$, $T_{10}=38623^3T_9$, and $T_{11}=45294^3T_{10}$.
At that moment we learned about results of C. Boyer
\cite{Boyer-2006} who established smaller $T_7,\ldots,T_{11}$\/
and first $T_{12}$\/ in December 2006. Unfortunately he has not
yet published details of his algorithm. Our renewed algorithm,
given later in this article, produces the same numbers. Also, for
the first time we found $T_{13}$\/ and $T_{14}$.
The main idea of our approach is not too surprising. If we know
some $k$--way representable number $T_k$, then we can try to find
$T_{k+1}$\/ in the form $\mu^3\,T_k$. If $m_1,\ldots,m_k$\/ are
medians of the representations of $T_k$, then medians of the
representations of $T_{k+1}$\/ are
$\mu{}m_1,\ldots,\mu{}m_k,d^\prime{}d$\/ where
$d^\prime\in\mbox{\rm divisors}(\mu^3)$\/ (the first version of
the algorithm uses only $d^\prime=1$) and $d\in\mbox{\rm
divisors}(T_k)$. A simple observation is that the multiplier of
interest does not exceed $2T_k^{2/3}$.
The iterative procedure formalizing this idea and using the
properties of the equation is the following:
\begin{description}
\item[$\bullet$~~~] Create an ordered array $D$\/ of all divisors
of $T_k$\/ excluding known too small divisors, i.e., less than
$\sqrt[3]{T_k/4}$.
\item[$\bullet$~~~] For multipliers $M$\/ from 2 to $\lfloor
2T_k^{2/3}\rfloor$\/ do
\begin{description}
\item[$\bullet$~~~] Let $N=M^3T_k$;
\item[$\bullet$~~~] For $\mu\in\mbox{divisors}(M^3)$\/ do
\begin{description}
\item[$\bullet$~~] Using dichotomic search, find a range of $D$\/
where the divisors satisfying {\bf Lemma \ref{DivBounds}} for
$\frac1\mu{}N$\/ are located;
\item[$\bullet$~~] Within this range for divisors $d$\/ such that
$\mu{}d\equiv \frac12N\!\mod{3}$\/ do: if the value
$(\frac{1}{2\mu{}d}N-(\mu{}d)^2)/3$\/ is a perfect square, then
$\mu{}d$\/ is the $k+1$\/ median and therefore $N$\/ is $T_{k+1}$.
Otherwise continue.
\end{description}
\end{description}
\end{description}
A set of all divisors of $T_k$\/ may be space-consuming. To avoid
the explicit computation of this set we used the following trick.
A taxicab number $T_k$\/ is a product $M\!\cdot T_s$\/ where
$T_s$\/ is a ``seed'', i.e., a small taxicab number with an easily
computed set of divisors and $M=(\mu_{s+1}\cdots\mu_k)^3$.
Evidently $M=1$\/ for $T_{k+1}=T_{s+1}$. Thus computing
$T_{k+1}$\/ we split the loop iterating through all divisors of
$T_k$\/ into two nested loops: the outer loop iterating through
all divisors of $M$\/ and the inner one iterating through those
divisors of $T_s$\/ such that product of the first iterator, the
second iterator, and some divisor of the current cubic multiplier
satisfies {\bf Lemma \ref{DivBounds}}.
Choice of the seed $T_s$\/ affects the space used by the
algorithm. We used $\mbox{\bf\em W}_5$\/ to compute new $T_k$\/
for $k=7\ldots12$. But for the next numbers, cardinality of the
divisor set for $M$\/ exceeds one for $T_s$\/ more and more. To
balance the cardinalities of these sets we took greater seeds.
{\begin{center}
\begin{tabular}{||r|r|r|r||}
\hline\hline
Ways & Seed & Multiplier & Time~~~~~ \\
\hline\hline
7 & 5 & 101 & 58 s. \\
8 & 5 & 127 & 5 m. 1 s. \\
9 & 5 & 139 & 18 m. 47 s. \\
10 & 5 & 377 & 4 h. 8 m. \\
11 & 5 & 727 & 123 h. 20 m. \\
12 & 5 & 2971 & 152 d. \\
13 & 6 & 4327 & 21 h. 8 m.$^{a)}$ \\
14 & 6 & 7549 & 23 m. 39 s.$^{b)}$ \\
\hline\hline
\end{tabular}
{\small
\vskip\baselineskip\flushleft{$^{a)}$ To compute this number we
examined only prime multipliers great than 2971.}
\vspace*{-1em}\flushleft{$^{b)}$ To compute this number we
examined only this multiplier.}
}
\vskip\baselineskip {\bf Table 1.} Computational results.
\end{center}}
{\bf Table 1.} represents multipliers producing new taxicab
numbers. In {\bf APPENDIX A} we give these numbers themselves and
their decompositions.
Also, we found that all of our taxicab numbers $T(3,k)$\/ are {\em
cabtaxi} (i.e., without the restriction on the cubes of the
decomposition to be positive) numbers $C(3,k+2)$. Surprisingly the
multiplier 5 gives cabtaxi numbers of higher orders:
$5^3T(3,k)=C(3,k+4)$. We checked this property for $k=6\ldots12$.
\section*{Final Remark}
In September 2007 we learned about new results of C. Boyer who
established new taxicab numbers for $n=13\ldots19$\/ and cabtaxi
numbers for $n=10\ldots30$. Boyer's article is going to be
published in a mathematical magazine.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\section*{APPENDIX A. Taxicab numbers decompositions}
\noindent$T_7=101^3\,\mbox{\bf\em
R}_6=24885189317885898975235988544$:
\[
\begin{array}{rcrc}
58798362^3 & + & 2919526806^3 & = \\
309481473^3 & + & 2918375103^3 & = \\
459531128^3 & + & 2915734948^3 & = \\
860447381^3 & + & 2894406187^3 & = \\
1638024868^3 & + & 2736414008^3 & = \\
1766742096^3 & + & 2685635652^3 & = \\
1847282122^3 & + & 2648660966^3 & \\
\end{array}
\]
\noindent$T_8=127^3\,T_7$=50974398750539071400590819921724352:
\[
\begin{array}{rcrc}
7467391974^3 & + & 370779904362^3 & = \\
39304147071^3 & + & 370633638081^3 & = \\
58360453256^3 & + & 370298338396^3 & = \\
109276817387^3 & + & 367589585749^3 & = \\
208029158236^3 & + & 347524579016^3 & = \\
224376246192^3 & + & 341075727804^3 & = \\
234604829494^3 & + & 336379942682^3 & = \\
288873662876^3 & + & 299512063576^3 &
\end{array}
\]
\noindent$T_9=139^3\,T_8=136897813798023990395783317207361432493888$:
\[
\begin{array}{rcrc}
1037967484386^3 & + & 51538406706318^3 & = \\
4076877805588^3 & + & 51530042142656^3 & = \\
5463276442869^3 & + & 51518075693259^3 & = \\
8112103002584^3 & + & 51471469037044^3 & = \\
15189477616793^3 & + & 51094952419111^3 & = \\
28916052994804^3 & + & 48305916483224^3 & = \\
31188298220688^3 & + & 47409526164756^3 & = \\
32610071299666^3 & + & 46756812032798^3 & = \\
40153439139764^3 & + & 41632176837064^3 & \\
\end{array}
\]
\noindent$T_{10}=377^3\,T_9=7335345315241855602572782233444632535674275447104$:
\[
\begin{array}{rcrc}
391313741613522^3 & + & 19429979328281886^3 & = \\
904069333568884^3 & + & 19429379778270560^3 & = \\
1536982932706676^3 & + & 19426825887781312^3 & = \\
2059655218961613^3 & + & 19422314536358643^3 & = \\
3058262831974168^3 & + & 19404743826965588^3 & = \\
5726433061530961^3 & + & 19262797062004847^3 & = \\
10901351979041108^3 & + & 18211330514175448^3 & = \\
11757988429199376^3 & + & 17873391364113012^3 & = \\
12293996879974082^3 & + & 17627318136364846^3 & = \\
15137846555691028^3 & + & 15695330667573128^3 & \\
\end{array}
\]
\noindent$T_{11}=727^3\,T_{10}=2818537360434849382734382145310807703728251895897826621632$:
\[
\begin{array}{rcrc}
284485090153030494^3 & + & 14125594971660931122^3 & = \\
657258405504578668^3 & + & 14125159098802697120^3 & = \\
1117386592077753452^3 & + & 14123302420417013824^3 & = \\
1497369344185092651^3 & + & 14120022667932733461^3 & = \\
2223357078845220136^3 & + & 14107248762203982476^3 & = \\
4163116835733008647^3 & + & 14004053464077523769^3 & = \\
6716379921779399326^3 & + & 13600192974314732786^3 & = \\
7925282888762885516^3 & + & 13239637283805550696^3& = \\
8548057588027946352^3 & + & 12993955521710159724^3& = \\
8937735731741157614^3 & + & 12815060285137243042^3& = \\
11005214445987377356^3 & + & 11410505395325664056^3& \\
\end{array}
\]
\noindent$T_{12}=2971^3\,T_{11}=\\
73914858746493893996583617733225161086864012865017882136931801625152$:
\[
\begin{array}{rcrc}
845205202844653597674^3 & + & 41967142660804626363462^3 & = \\
1933097542618122241026^3 & + & 41965889731136229476526^3 & = \\
1952714722754103222628^3 & + & 41965847682542813143520^3 & = \\
3319755565063005505892^3 & + & 41960331491058948071104^3 & = \\
4448684321573910266121^3 & + & 41950587346428151112631^3 & = \\
6605593881249149024056^3 & + & 41912636072508031936196^3 & = \\
12368620118962768690237^3 & + & 41606042841774323117699^3 & = \\
19954364747606595397546^3 & + & 40406173326689071107206^3 & = \\
23546015462514532868036^3 & + & 39334962370186291117816^3 & = \\
25396279094031028611792^3 & + & 38605041855000884540004^3 & = \\
26554012859002979271194^3 & + & 38073544107142749077782^3 & = \\
32696492119028498124676^3 & + & 33900611529512547910376^3 & \\
\end{array}
\]
\noindent$T_{13}=4327^3\,T_{12}=\\
5988146776742829080553965820313279739849705084894534523771076163371248442670016$:
\[
\begin{array}{rcrc}
3657202912708816117135398^3 & + & 181591826293301618274700074^3 & = \\
8364513066908614936919502^3 & + & 181586404866626464944928002^3 & = \\
8449396605357004644311356^3 & + & 181586222922362752472011040^3 & = \\
14364582330027624823994684^3 & + & 181562354361812068303667008^3 & = \\
19249457059450309721505567^3 & + & 181520191447994609864354337^3 & = \\
28582404724165067827090312^3 & + & 181355976285742254187920092^3 & = \\
53519019254751900122655499^3 & + & 180029347376357496130283573^3 & = \\
54818831102057750995052604^3 & + & 179911586979069103444414128^3 & = \\
86342536262893738285181542^3 & + & 174837511984583610680880362^3 & = \\
101883608906300383719991772^3 & + & 170202382175796081666789832^3& = \\
109889699639872260803223984^3 & + & 167044016106588827404597308^3& = \\
114899213640905891306456438^3 & + & 164744225351606675259562714^3& = \\
141477721399036311385473052^3 & + & 146687946088200794808196952^3& \\
\end{array}
\]
\noindent$T_{14}=7549^3\,T_{13}=$
\[
\begin{array}{c}
257608810925730001281963766003343299028977072 ~~~\backslash\\
~~~~~~~~~~~~~~~~~~~
5881505682307757452553496715044742867424072384:
\end{array}
\]
\[
\begin{array}{rcrc}
27608224788038852868255119502^3 & + & 1370836696688133916355710858626^3 & = \\
63143709142093134158805320598^3 & + & 1370795770338163183869261487098^3 & = \\
63784494973840028059906426444^3 & + & 1370794396840916418411211340960^3 & = \\
108438232009378539796335869516^3 & + & 1370614213077319303624382243392^3 & = \\
145314151341790388087645525283^3 & + & 1370295925240911309866010890013^3 & = \\
215768573262722097026704765288^3 & + & 1369056264981068276864608774508^3 & = \\
404015076354122094025926361951^3 & + & 1359041543344122738287510692577^3 & = \\
413827355989433962261652107596^3 & + & 1358152570104992661901882252272^3 & = \\
617989830682279948575932296880^3 & + & 1327627770274178602420131034444^3 & = \\
651799806248584830314835460558^3 & + & 1319848377971621677029965852738^3 & = \\
769119363633661596702217886828^3 & + & 1284857783045084620502596441768^3 & = \\
829557342581395696803537855216^3 & + & 1261015277588639058077305078092^3 & = \\
867374163775198573472439650462^3 & + & 1243654157179278791534438927986^3 & = \\
1068015318841325114648936069548^3 & + & 1107347305019827800007078790648^3 & \\
\end{array}
\]
\end{document} |
\begin{document}
\maketitle
\begin{abstract}
We extend the edge-colouring notion of \emph{core} (subgraph induced
by the vertices of maximum degree) to \emph{$t$-core} (subgraph
induced by the vertices $v$ with $d(v)+\mu(v)> \Delta+t$), and find
a sufficient condition for $(\Delta+t)$-edge-colouring. In
particular, we show that for any $t\geq 0$, if the $t$-core of $G$
has multiplicity at most $t+1$, with its edges of multiplicity $t+1$
inducing a multiforest, then $\chi'(G) \leq \Delta+t$. This extends
previous work of Ore, Fournier, and Berge and Fournier. A stronger
version of our result (which replaces the multiforest condition with
a vertex-ordering condition) generalizes a theorem of Hoffman and
Rodger about cores of $\Delta$-edge-colourable simple graphs. In
fact, our bounds hold not only for chromatic index, but for the
\emph{fan number} of a graph, a parameter introduced by Scheide and
Stiebitz as an upper bound on chromatic index. We are able to give
an exact characterization of the graphs $H$ such that
$\Fan(G) \leq \Delta(G)+t$ whenever $G$ has $H$ as its $t$-core.
\end{abstract}
\section{Introduction}\label{sec:introduction}
In this paper a \emph{graph} is permitted to have parallel edges but no loops; we will say \emph{simple graph} when we wish to disallow parallel edges.
A \emph{$k$-edge-colouring} of a graph $G$ is a function that assigns a
colour from $\{1, \ldots, k\}$ to each edge of $G$ so that adjacent
edges receive different colours. The \emph{chromatic index} of $G$,
$\chi'(G)$, is the minimum $k$ such that $G$ is $k$-edge-colourable;
the maximum degree $\Delta(G)$ is an obvious lower bound for
$\chi'(G)$. When the graph $G$ is understood, we sometimes write $\Delta$ for
$\Delta(G)$.
Numerous authors have found sufficient conditions for
$\Delta$-edge-colouring a simple graph $G$ by studying its \emph{core},
that is, the graph induced by the its vertices of degree $\Delta$.
An early such result is due to Fournier~\cite{fournier77, fournier73}:
\begin{theorem}[Fournier~\cite{fournier77, fournier73}]\label{thm:fournier}
If $G$ is a simple graph and the core of $G$ is a forest, then
$\chi'(G) = \Delta(G)$.
\end{theorem}
This result was strengthened by Hoffman and Rodger~\cite{hoffman-rodger} who showed that if $B$ is the core of a graph $G$, and $B$ permits a specific vertex-ordering called a \emph{full B-queue}, then $G$ is $\Delta$-edge-colourable. We defer a precise definition of \emph{full B-queue} to Section~\ref{sec:bqueue} of this paper, but we state their result now, noting that if $B$ is a forest, then it indeed has a full B-queue. Hoffman and Rodger \cite{hoffman-rodger} also provided an efficient algorithm for deciding whether or not a graph $B$ has a full $B$-queue; in fact they showed that the greedy algorithm works.
\begin{theorem}[Hoffman--Rodger~\cite{hoffman-rodger}]\label{thm:hoffman-rodger}
Let $G$ be a simple graph with core $B$. If $B$ has a
full $B$-queue, then $\chi'(G) = \Delta(G)$.
\end{theorem}
Simple graphs can be divided into those of \emph{class I} (having chromatic index $\Delta$) or \emph{class II} (having chromatic index $\Delta+1$), but in general the chromatic index of $G$ can be as high as $\Delta+\mu$, where $\mu=\mu(G)$ is the maximum edge-multiplicity of $G$. This classical bound of Vizing \cite{vizing} also has the following local refinement due to Ore, where $\mu(v)$ denotes the maximum edge multiplicity incident to vertex $v$.
\begin{theorem}[Ore~\cite{ore-fourcolor}]\label{thm:ore}
For every graph $G$, $\chi'(G) \leq \max_{v \in V(G)}[d(v) + \mu(v)]$.
\end{theorem}
We define the \emph{t-core} of $G$ to be the subgraph induced by the vertices $v$ with \[d(v)+\mu(v)> \Delta+t.\] Observe that the $0$-core of a nonempty
simple graph is simply its core. Ore's Theorem can be restated as: ``For any $t\geq 0$, if the $t$-core of a graph $G$ is empty, then $G$ is $(\Delta+t)$-edge-colourable''. We improve this and generalize
Theorem~\ref{thm:fournier} as follows. Here, by \emph{multiforest},
we mean a graph whose underlying simple graph is a forest.
\begin{theorem}\label{thm:forestcore}
Let $G$ be a graph and let $t\geq 0$. If the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of multiplicity $t+1$ inducing a multiforest, then $\chi'(G) \leq \Delta+t$.
\end{theorem}
The $t=0$ case of Theorem~\ref{thm:forestcore} implies
Theorem~\ref{thm:fournier} (and is already slightly stronger, since
Theorem~\ref{thm:forestcore} allows $G$ to be a multigraph even though
$t=0$ forces the $0$-core of $G$ to be simple whenever the hypothesis
is met). When $t=\mu(G)-1$, the hypothesis of
Theorem~\ref{thm:forestcore} is just that the edges of multiplicity
$\mu$ in the $t$-core induce a multiforest; this strengthens a
previous result of Berge and Fournier \cite{berge-fournier}, who
showed that if the $(\mu-1)$-core of $G$ is edgeless, then $G$ is
$(\Delta+\mu-1)$-edge-colourable.
The multiplicity condition in Theorem~\ref{thm:forestcore} is sharp,
and this can already be seen with a fat triangle. Consider the
multigraph $G$ obtained from $K_3$ by giving two edges multiplicity
$t+1$ and the remaining edge multiplicity $t+2$. Now
$\Delta(G) = 2t+3$, and the $t$-core of $G$ is simply the $t+2$
parallel edges (since for each of those endpoints, degree plus
multiplicity is $3t+5>\Delta(G)+t$, while for the other vertex this
sum is only $3t+3$). Hence, the $t$-core of $G$ is a multiforest but
with multiplicity $t+2$; this discrepancy from Theorem
\ref{thm:forestcore} is already enough to cause a problem, as of
course this fat triangle has $\chi'(G)=3t+4 > \Delta(G)+t$.
Theorem \ref{thm:forestcore} is in fact a corollary of a stronger result we prove, which generalizes Theorem \ref{thm:hoffman-rodger}. Theorem \ref{thm:hoffman-rodger} is about a condition on the core (0-core) of a simple graph that guarantees $\Delta$-edge-colourability; here we get a condition on the $t$-core of a graph that guarantees $(\Delta+t)$-edge-colourability (with the same condition when $t=0$).
\begin{theorem}\label{thm:Bqueuecore} Let $G$ be a graph, let $t\geq 0$, and let $H$ be the $t$-core of $G$. If $H$ has multiplicity at most $t+1$, and the underlying simple graph $B$ of those maximum multiplicity edges has a full $B$-queue, then $\chi'(G)\leq \Delta(G)+t$.
\end{theorem}
We can actually state Theorem \ref{thm:Bqueuecore} (and hence Theorem \ref{thm:forestcore}) in an even stronger way, by replacing $\chi'(G)$ with the \emph{fan number} $\Fan(G)$. Scheide and Stiebitz~\cite{SS} introduced $\Fan(G)$ to essentially describe the smallest $k$ for which
Vizing's Fan Inequality (see Section~\ref{sec:fan}) can be used to prove that $G$ is
$k$-edge-colourable, in particular proving the following.
\begin{theorem}[Scheide--Stiebitz~\cite{SS}]\label{thm:SS}
For any graph $G$, $\chi'(G) \leq \Fan(G)$.
\end{theorem}
We are able to give an exact characterization of the graphs $H$ such that $\Fan(G) \leq \Delta(G)+t$ whenever $G$ has $H$ as its $t$-core. In particular, we will define $\corefan(H)$
for a graph $H$ (which we'll think of as being the $t$-core of
$G$), and prove the following pair of theorems.
\begin{theorem}\label{thm:corefan}
Let $G$ be a graph, let $t\geq 0$, and let $H$ be
the $t$-core of $G$. If $\corefan(H) \leq t$, then $\Fan(G)\leq \Delta+t$.
\end{theorem}
\begin{theorem}\label{thm:converse}
Let $H$ be a graph, and let $t$ be a nonnegative integer. If
$\corefan(H) > t$, then there exists a graph $G$ with
$t$-core $H$ such that $\Fan(G)> \Delta(G)+t$.
\end{theorem}
This pair of results can be thought of as a sort of multigraph analog
to the work of Hoffman~\cite{hoffman}, who found a necessary and
sufficient condition for a simple graph $H$ to be the core of a simple
graph $G$ containing a so-called overfull subgraph of the same maximum
degree. Overfull graphs are known to be class II. The graph $G$
constructed in Theorem~\ref{thm:converse} does not necessarily satisfy
$\chi'(G) > \Delta(G) + t$, as one might hope, but the lower bound on
the fan number suggests that fan-recolouring would not suffice to
$(\Delta+t)$-edge-colour these graphs.
Our paper is organized as follows. We'll define $\Fan$ and $\corefan$
in Section~\ref{sec:fan}, spending time to motivate these definitions
according to Vizing's Adjacency Lemma, and conclude the section with a
proof of Theorem \ref{thm:corefan}. In Section~\ref{sec:bqueue} we'll
give a precise definition of $B$-queue and full $B$-queue, and prove
Theorem \ref{thm:Bqueuecore}. In particular, we'll show that when $H$
is the $t$-core of $G$, and $H$ has all the assumptions of Theorem
\ref{thm:Bqueuecore}, then $\corefan(H)\leq t$, and hence Theorems
\ref{thm:SS} and \ref{thm:corefan} imply that
$\chi'(G)\leq Fan(G)\leq \Delta+t$. Our proof of Theorem
\ref{thm:converse} is the subject of Section~\ref{sec:converse}.
\begin{remark}
The word ``core'' has several different meanings in graph theory. In
addition to the usage above, it has a definition in the setting of
graph homomorphisms. Moreover, the term ``$k$-core'' has also been
used in a degeneracy context, to refer to the component of $G$ that
remains after iteratively deleting vertices of degree at most $k$.
\end{remark}
\section{Proof of Theorem \ref{thm:corefan}}\label{sec:fan}
In the introduction we described $\Fan(G)$ as essentially describe the
smallest $k$ for which the following theorem, Vizing's Fan Inequality, can be used to
prove that $G$ is $k$-edge-colourable. Let us now say more about this.
prove that $G$ is $k$-edge-colourable. Let us now say more about this.
\begin{theorem}\label{thm:vizfan}\emph{(Vizing's Fan Inequality
\cite{vizing}, see also \cite{SSFT})} Let $G$ be a graph, let
$k\geq \Delta$, and suppose there is a $k$-edge-colouring of $J-e$
for some $J\subseteqeq G$ and $e=xy \in E(G)$. Then either $J$ is
$k$-edge-colourable, or there exists a vertex-set
$Z\subseteqeq N_J(x)$ such that $|Z|\geq 2$, $y\in Z$, and
\begin{equation}\label{fanineq}
\sum_{z\in Z} \left( d_J(x) +\mu_J(x, z) - k\right) \geq 2.
\end{equation}
\end{theorem}
Vizing's Theorem (and Ore's Theorem) follow immediately from the fan
inequality. To see this, consider an edge-minimal counterexample $G$
(so let $J=G$ in Theorem \ref{thm:vizfan}), and note that setting
$k=\Delta(G)+\mu(G)$ (or $k=\max_{v \in V(G)}[d(v) + \mu(v)]$) makes
inequality (\ref{fanineq}) impossible to satisfy.
In order to apply Theorem \ref{thm:vizfan}, we would certainly need
$k\geq \Delta$. Given this however, if we had a $k$-edge-colouring of
$J-e$ for some $e=xy\in E(J)$ and we knew that for \emph{every}
$Z\subseteqeq N(x)$ with $y\in Z$ and $\sizeof{Z} \geq 2$,
\[\sum_{z \in Z}(d_J(z) + \mu_J(x,z) - k) \leq 1,\]
then we'd get a proof of $k$-edge-colourability of $J$ via Theorem
\ref{thm:vizfan}. On the other hand, if we knew that
\[d_J(x) + d_J(y) - \mu_J(x,y) \leq k,\] for such an $e=xy$, then we'd
get our $k$-edge-colouring extending to $J$ simply because $e$ sees at
most $k-1$ different edges in $G$. With this in mind, Scheide and
Stiebitz~\cite{SS} defined the \emph{fan-degree}, $\deg_J(x,y)$, of
the pair $x, y\in V(J)$ as the smallest nonnegative integer $k$ such
that either:
\begin{enumerate}[(i)]
\item $d_J(x) + d_J(y) - \mu_J(x,y) \leq k$, or
\item $\sum_{z \in Z}(d_J(z) + \mu_J(x,z) - k) \leq 1$ for all $Z \subseteq N_J(x)$ with $y \in Z$ and $\sizeof{Z} \geq 2$.
\end{enumerate}
So, we could extend the $k$-edge-colouring of $J-e$ to $J$ provided we
knew that $\deg_J(x,y)\leq k$. Of course, our goal is to
$k$-edge-colour all of $G$, not just some subgraph $J$. However, if
$G$ is not $k$-edge-colourable, then there exists $J\subseteqeq G$ with
the property that $J-e$ is $k$-edge-colourable for all $e\in E(J)$ but
$J$ is not $k$-edge-colourable. If, for \emph{this} $J$, we knew that
there was a choice of $xy\in E(J)$ with $d_J(x,y)\leq k$, then we'd
know that $J$ is $k$-edge-colourable after all, and hence so is
$G$. If such a choice of $xy$ existed for \emph{every} subgraph $J$ of
$G$ (say with at least one edge), then we would certainly get that $G$
is $k$-edge-colourable. Hence, Scheide and Stiebitz~\cite{SS} defined
the \emph{fan number}, $\fan(G)$, of a graph $G$ by
\[ \fan(G) = \max_{J \subseteq G, E(J)\neq \emptyset} \min\{\deg_J(x,y) \colon\, xy \in E(J)\}, \]
with $\fan(G)$ defined to be 0 for an edgeless graph $G$. Recalling the requirement that $k\geq \Delta$, they finally defined $\Fan(G)=\max\{\Delta, \fan(G)\}$, and established Theorem \ref{thm:SS}.
Now suppose that the graph $G$ has $t$-core $H$. We would like to be
able to look just at $H$ and determine that $\Fan(G)\leq \Delta+
t$. To this end, we would like to describe a condition on $H$ that
would guarantee that for every $J\subseteqeq G$, there exists
$xy\in E(J)$ with $\deg_J(x,y)\leq \Delta+t$. We'll forget about (i)
for this purpose, and try to get a condition on $H$ which guarantees
(ii) for such $J, x, y$. If $K=J\cap H$, then we're trying to get a
guarantee for $J$ by only looking at $K$. The good news here is that
if, for example, some vertex $z\in Z$ is in $J$ but not $K$, then $z$
is not in the $t$-core, so in particular,
\[d_J(z) + \mu_J(x,z) - (\Delta(G)+t)\leq 0,\]
that is, the vertex $z$ is insignificant in terms of establishing (ii). There are more details to handle, but we'll see that the following definition is the right condition to require. Note that while we'll think of $H$ as being the $t$-core of a graph $G$, this definition takes as input any graph $H$.
For any graph $H$, subgraph $K \subseteq H$, and ordered pair of
vertices $(x,y)$ with $xy \in E(K)$, we define the \emph{cfan
degree}, denoted $\cdeg_{H,K}(x,y)$, as the smallest nonnegative integer $l$
such that for all $Z \subseteq N_K(x)$ with $y \in Z$, we have
\[ \sum_{z \in Z}(d_K(z) - d_H(z) + \mu_K(x,z) - l) \leq 1. \]
Note that, in contrast to the fan degree, the cfan degree does
\emph{not} impose the restriction that $\sizeof{Z} \geq 2$ when
determining which sets $Z \subseteq N_K(x)$ must be considered.
The \emph{cfan number} of $H$, written $\corefan(H)$, is then
defined by
\[ \corefan(H) = \max_{K \subseteq H, E(K)\neq\emptyset}\min\{ \cdeg_{H,K}(x,y) \colon\, xy \in E(K) \}, \]
with $\corefan(H)$ defined to be 0 for an edgeless graph $H$. With this definition established, we can now prove Theorem \ref{thm:corefan}.
\begin{proof}[Proof of Theorem~\ref{thm:corefan}]
Suppose that $\corefan(H) \leq t$. We will show that this implies that $\fan(G)\leq \Delta+t$, which in turn implies that $\Fan(G)\leq \Delta+t$, as desired.
If $G$ is an edgeless graph, then $\fan(G)=0$ by definition, so our result is immediate. Now suppose that $G$ has at least one edge, and let any subgraph $J \subseteq G$ with $E(J)\neq \emptyset$ be given. We will show that there exists $xy\in E(G)$ with
$\deg_J(x,y)\leq \Delta(G)+t$; in particular we will show that for all $Z \subseteq N_J(x)$ with $y \in Z$ and $\sizeof{Z} \geq 2$,
\[\sum_{z \in Z}(d_J(z) + \mu_J(x,z) - (\Delta(G)+t)) \leq 1.\]
Let $K = J \cap H$. We consider two cases:
either $K$ contains an edge, or $K$ contains no edges.
\caze{1}{$K$ contains an edge.} In this case, since $K\subseteqeq H$ and we know that $\corefan(H)\leq t$, we know that there exists $xy\in E(K)$ with $\cdeg_{H, K}(x,y)\leq t$, that is, with
\[\sum_{z \in Z}(d_K(z) - d_H(z) + \mu_K(x,z) - t) \leq 1 \]
for all $Z \subseteq N_K(x)$ with $y \in Z$. Note that for any $z\in V(K)$,
\[d_J(z)-d_K(z)+d_H(z)\leq d_G(z)\leq \Delta(G).\]
So we get that for all $Z \subseteq N_K(x)$ with $y \in Z$,
\[\sum_{z \in Z}(d_J(z) + \mu_K(x,z) - (\Delta(G)+t)) \leq 1\]
Now observe that if $w \in N_J(x) - V(H)$, then by the definition of
the $t$-core of $G$, we have $d_G(w) + \mu(w) \leq \Delta(G)+t$. So
in fact we can say that the above sum holds for all
$Z\subseteqeq N_J(z)$ with $y\in Z$ and $\sizeof{Z} \geq 2$, as
desired. (Note that this is the reason we cannot impose the
restriction that $\sizeof{Z} \geq 2$ in the definition of cfan
degree: if we imposed that restriction and had $N_K(x) = \{y\}$ but
$\sizeof{N_J(x)} \geq 2$, we would have no control over the value
of $d_J(y) + \mu_K(x,y) - (\Delta(G)+t)$.)
\caze{2}{$K$ has no edges.} In this case, let $(x,y)$ be any pair such that $xy \in E(J)$,
taking $x \in V(H)$ if possible. Our choice of $x$ implies that for all $z \in N_J(x)$,
we have $z \notin V(H)$, hence $d_G(z) + \mu_G(z) \leq \Delta(G)+t$ by the definition of a $t$-core.
Thus, for every $Z \subseteq N_J(x)$ with $y \in Z$ and $\sizeof{Z} \geq 2$, we have
\[ \sum_{z \in Z}(d_J(z) + \mu_J(x,z) - (\Delta(G)+t)) \leq \sum_{z \in
Z}(d_G(z) + \mu_G(x,z) - (\Delta(G)+t)) \leq 1, \]
as needed.
\end{proof}
\section{Proof of Theorem \ref{thm:Bqueuecore}}\label{sec:bqueue}
We start this section by providing the definition of a \emph{full B-queue}, which is needed for Theorem \ref{thm:Bqueuecore} (and for Theorem \ref{thm:hoffman-rodger}). Hoffman and Rodger \cite{hoffman-rodger} defined a \emph{$B$-queue} of a simple graph
$B$ to be a sequence of vertices $(u_1, \ldots, u_q)$ and a sequence
of vertex subsets $(S_0, S_1, \ldots, S_q)$ such that:
\begin{enumerate}[(i)]
\item $S_0 = \emptyset$, and
\item For all $i \in [q]$:
\begin{itemize}
\item $S_i = N(u_i) \cup \{u_i\} \cup S_{i-1}$,
\item $1 \leq \sizeof{S_i \setminus S_{i-1}} \leq 2$,
\item $u_i \notin \{u_1, \ldots, u_{i-1}\}$, and
\item $\sizeof{S_i \setminus (S_{i-1} \cup \{u_i\})} \leq 1$.
\end{itemize}
\end{enumerate}
If $S_q = V(B)$ then we say the $B$-queue is \emph{full}. We noted
in the introduction that every simple forest $B$ admits a full
$B$-queue. To see this, first suppose that a $B$-queue
$(u_1, \ldots, u_{i-1})$ and $(S_0, \ldots, S_{i-1})$ has already
been defined for $B$, but the $B$-queue is not full,
ie. $S_{i-1}\neq V(B)$. If $B-S_{i-1}$ consists only of isolated
vertices, then they may be chosen in any order as
$u_{i}, u_{i+1}, \ldots$ so as to get a full $B$-queue. If not, then
$B-S_{i-1}$ is a forest, so it contains a leaf vertex which can be
chosen for $u_{i}$. With this choice $|S_{i}-S_{i-1}|=2$ and
$\sizeof{S_{i} \setminus (S_{i} \cup \{u_i\})}=1$ (since
$u_{i}\not\in S_{i-1}$ in this case), so $(u_1, \ldots, u_i)$ and
$(S_0, \ldots, S_i)$ is again a $B$-queue. This process can be
repeated until the $B$-queue is full.
In addition to forests, there are many other simple graphs $B$ that have full $B$-queues. For example, while a cycle itself does not have a full $B$-queue (there is no valid choice for $u_1$), adding any number of pendant edges to a cycle allows the same procedure described above to yield a full $B$-queue. For another, more complicated example, see \cite{hoffman-rodger}.
In this section, we'll prove the following result.
\begin{theorem}\label{thm:Bqueuecorefan} Let $G$ be a graph, let $t\geq 0$, and let $H$ be the $t$-core of $G$. If $H$ has multiplicity at most $t+1$, and the underlying simple graph $B$ of those maximum multiplicity edges has a full $B$-queue, then $\corefan(H)\leq t$.
\end{theorem}
Given the conclusion of Theorem \ref{thm:Bqueuecorefan}, Theorems \ref{thm:SS} and \ref{thm:corefan} immediately tell us that
\[ \chi'(G)\leq \Fan(G)\leq \Delta+t, \]
which in particular implies Theorem \ref{thm:Bqueuecore}.
We'll prove Theorem \ref{thm:Bqueuecorefan} by establishing a sequence of lesser results. The first such lemma, which follows, says that when looking for an upper bound on $\corefan(H)$, it suffices to look at a subgraph of $H$ formed by high-multiplicity edges.
\begin{lemma}\label{lem:lowmult}
Let $H$ be a graph, let $t$ be a nonnegative integer, and let $H_{>t}$ be
the subgraph of $H$ consisting of the edges with multiplicity greater
than $t$. The following are equivalent:
\begin{enumerate}[(i)]
\item $\corefan(H) \leq t$,
\item $\corefan(H_{>t}) \leq t$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $H' = H_{>t}$.
All nonempty subgraphs of $H'$ are also subgraphs of $H$ which must be considered
when computing $\corefan(H)$, so (i)$\implies$(ii) is immediate. To show that (ii)$\implies$(i),
let $K$ be any nonempty subgraph of $H$. We will find a pair $(x,y)$ with $\cdeg_{H,K}(x,y) \leq t$.
If all edges of $K$ have multiplicity at most $t$, then let $(x,y)$
be any pair with $xy \in E(K)$. For any $Z \subseteq N_K(x)$ with
$y \in Z$, all terms of the sum
\[ \sum_{z \in Z}(d_K(z) - d_H(z) + \mu_K(x,z) - t) \] are
nonpositive, so this sum is clearly at most $1$, as desired.
Thus, we may assume that $K$ has some edges of multiplicity at least
$t+1$. Let $K' = K \cap E(H')$; now $K'$ is a nonempty subgraph
of $H'$, so we obtain a pair $(x,y)$ such that
$\cdeg_{H', K'}(x,y) \leq t$. We claim that also
$\cdeg_{H, K}(x,y) \leq t$.
For any $Z \subseteq N_K(x)$ with
$y \in Z$, let $Z' = Z \cap N_{K'}(x)$. For any $z \in Z-Z'$, we
have $\mu_K(x,z) \leq t$, so the contribution of $z$ to the sum
\[ \sum_{z \in Z}[d_K(z) - d_H(z) + \mu_K(x,z) - t] \]
is nonpositive. Moreover, for every $v \in V(K)$, every edge
that is lost when we pass from $K$ to $K'$ is also lost when we
pass from $H$ to $H'$, so that $d_K(v) - d_{K'}(v) \leq d_H(v) - d_{H'}(v)$,
which rearranges to $d_K(v) - d_{H}(v) \leq d_{K'}(v) - d_{H'}(v)$. Since
also $\mu_K(u,v) = \mu_{K'}(u,v)$ for all $uv \in E(K')$, this yields
\begin{align*}
\sum_{z \in Z}[d_K(z) - d_H(z) + \mu_K(x,z) - t] &\leq \sum_{z \in Z'}[d_K(z) - d_H(z) + \mu_K(x,z) - t] \\
&\leq \sum_{z \in Z'}[d_{K'}(z) - d_{H'}(z) + \mu_{K'}(x,z) - t] \leq 1,
\end{align*}
where the last inequality follows from $\cdeg_{H', K'}(x,y) \leq t$.
\end{proof}
When trying to determine $\corefan$ for a given graph, one need only
focus on \emph{full multiplicity} subgraphs, as indicated in the
following lemma, and we'll see this to be a helpful idea. A subgraph
$K$ of a graph $H$ has \emph{full multiplicity} if
$\mu_{K}(e)=\mu_H(e)$ for all $e\in E(K)$. (Note that some edges of
$H$ may be omitted from $K$ entirely.)
\begin{lemma}\label{lem:fullmult}
For any graph $H$,
\[ \corefan(H) = \max_K \min \{ \cdeg_{H,K}(x,y) \colon\, xy \in E(K) \}, \]
where the maximum is taken over all nonempty subgraphs $K \subseteq H$ such that $K$ has full
multiplicity.
\end{lemma}
\begin{proof} Consider a subgraph $K$ of graph $H$, and define $K'$ to be the subgraph of $H$ having the same underlying simple graph as $K$, but having full multiplicity. Note that the truth of the lemma would follow if we could establish
\[ \cdeg_{H,K'}(x,y) \geq \cdeg_{H,K}(x,y) \]
for all pairs $(x,y)$ with $xy \in E(K)$. To this end, note that the definition of cdeg gives us
\begin{equation}
\label{eq:znkprime}
\sum_{z \in Z'}(d_{K'}(z) - d_H(z) + \mu_{K'}(x,y) - \cdeg_{H,K'}(x,y)) \leq 1
\end{equation}
for all $Z' \subseteq N_{K'}(x)$ with $y \in Z$.
Since $d_{K'}(z) \geq d_{K}(z)$ and $\mu_{K'}(x,y) \geq \mu_K(x,y)$ for all $x,y$, we see that
for all such $Z'$, we also have
\begin{equation}
\label{eq:znk}
\sum_{z \in Z'}(d_{K}(z) - d_{H}(z) + \mu_{K}(x,y) - \cdeg_{H,K'}(x,y)) \leq 1.
\end{equation}
Since $N_{K'}(x) = N_{K}(x)$ for all $x$, we see that Inequality~\eqref{eq:znk} holds
(with $Z$ replacing $Z'$) for all $Z \subseteq N_{K}(x)$ with $y \in Z$. This gives us our desired inequality.
\end{proof}
We now turn our focus to computing $\corefan$ in graphs of
\emph{constant multiplicity}, that is, graphs where every edge has the
same multiplicity.
\begin{lemma}\label{lem:tplus1}
Let $H$ be a graph of constant multiplicity $t+1$, and for every
nonempty subgraph $K \subseteq H$, let $Z(K) = \{v \in V(K) \colon\, d_K(v) = d_H(v)\}$.
The following are equivalent:
\begin{enumerate}[(i)]
\item $\corefan(H) \leq t$,
\item For every nonempty full-multiplicity subgraph $K \subseteq H$, there is an edge $xy \in E(K)$
such that $\sizeof{(N_H(x) \cap Z(K)) - y} \leq d_H(y) - d_K(y)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i)$\implies$(ii): Let any nonempty full-multiplicity subgraph $K \subseteq H$ be given,
and let $(x,y)$ be a pair such that $xy \in E(K)$ and $\cdeg_{H,K}(x,y) \leq t$. Let $Z = (N_H(x) \cap Z(K)) \cup \{y\}$.
Observe that $z \in N_H(x) \cap Z(K)$ implies that $xz \in E(K)$, so that $Z \subseteq N_K(x)$;
thus, since $\cdeg_{H,K}(x,y) \leq t$, we have
\[ \sum_{z \in Z}[ d_K(z) - d_H(z) + \mu_K(x,z) - t] \leq 1. \]
Since vertices in $Z - y$ contribute exactly $1$ to this sum, this implies that
\[ \sizeof{Z - y} + (d_K(y) - d_H(y) + \mu_K(x,y) - t) \leq 1, \]
so by the definition of $Z$,
\[ \sizeof{(N_H(x) \cap Z(k)) -y} \leq d_H(y) - d_K(y) - \mu_K(x,y) + t + 1 = d_H(y) - d_K(y), \]
where in the last equation we have $\mu_K(x,y) = t+1$ since $K$ is full-multiplicity and $xy \in E(K)$.
(ii)$\implies$(i): We apply Lemma~\ref{lem:fullmult}. Let any
nonempty full-multiplicity subgraph $K \subseteq H$ be given, and let
$xy$ be an edge such that
$\sizeof{(N_H(x) \cap Z(K)) - y} \leq d_H(y) - d_K(y)$. We claim
that $\cdeg_{H,K}(x,y) \leq t$.
Let $Z$ be any subset of $N_K(x)$ containing $y$. We must show that
$\sum_{z \in Z}[ d_K(z) - d_H(z) + \mu_K(x,z) - t)] \leq 1$. Observe
that any elements of $Z - Z(K)$ contribute
a nonpositive term to this sum, while elements of $Z(K)$ contribute $1$, so that
\begin{align*}
\sum_{z \in Z}[ d_K(z) - d_H(z) + \mu_K(x,z) - t] &\leq \sum_{z \in (N_H(x)\cap Z(K)) \cup \{y\}}[ d_K(z) - d_H(z) + \mu_K(x,z) - t ] \\
&\leq \sizeof{(N_H(x) \cap Z(K)) - y} + (d_K(y) - d_H(y) + 1) \\
&\leq (d_H(y) - d_K(y)) + (d_K(y) - d_H(y) + 1) \\
&= 1.\qedhere
\end{align*}
\end{proof}
\begin{lemma}\label{lem:flatten}
Let $B$ be a simple graph, and let $B_s$ and $B_t$ be graphs of
constant multiplicity $s+1$ and $t+1$ respectively, with underlying
simple graph $B$. If $0\leq s < t$ and $\corefan(B_s) \leq s$, then
$\corefan(B_t) \leq t$.
\end{lemma}
\begin{proof}
First observe that for every vertex $v \in V(B)$, we have
$N_B(v) = N_{B_s}(v) = N_{B_t}(v)$; thus, we suppress the
subscripts and simply write $N(v)$. To avoid double-subscripts, we
will also write $d_s$ and $\mu_s$ as shorthand for $d_{B_s}$ and
$\mu_{B_s}$, and likewise for $t$.
We verify Condition~(ii) of Lemma~\ref{lem:tplus1} for $B_t$. Let
$K$ be any nonempty full-multiplicity subgraph of $B_t$, and let
$K'$ be the full-multiplicity subgraph of $B_s$ having the same
underlying simple graph. Observe that $Z(K') = Z(K)$, and that $K'$
is nonempty since it has the same underlying simple graph as $K$.
Applying Lemma~\ref{lem:tplus1} to $B_s$, there is an edge $xy \in E(K')$ such that
\[
\sizeof{(N(x) \cap Z(K')) - y} \leq d_{s}(y) - d_{K'}(y).
\]
By the definition of $K'$, we have $\mu_K(xy) > 0$, that is, $xy \in E(K)$.
Since $Z(K') = Z(K)$, it therefore suffices to show that $d_{s}(y) - d_{K'}(y) \leq d_{t}(y) - d_{K}(y)$. This follows from observing that if $J$ is the common underlying simple
graph of $K$ and $K'$, then
\[ d_{s}(y) - d_{K'}(y) = (s+1)[d_{B}(y) - d_{J}(y)] \leq (t+1)[d_{B}(y) - d_J(y)] = d_{t}(y) - d_{K}(y). \qedhere \]
\end{proof}
\begin{remark}
The converse of Lemma~\ref{lem:flatten} is not true: for a simple graph $H$, it is possible that
$\corefan(H_t) \leq t+1$ yet $\corefan(H) > 0$, where we consider $H$ itself as the graph $B_s$ for $s=0$.
Consider the simple graph $H$ shown in Figure~\ref{fig:flattenconverse}.
\begin{figure}
\caption{Simple graph $H$ such that $\corefan(H_1) \leq 1$ but $\corefan(H) > 0$.}
\label{fig:flattenconverse}
\end{figure}
To see that $\corefan(H) > 0$, consider the subgraph $K = H-v$. If $\corefan(H)\leq 0$, then there exists $xy\in E(K)$ with
\[ \sum_{z \in Z}(d_K(z) - d_H(z) +1 - 0) \leq 1 \] for all
$Z\subseteqeq N_K(x)$ with $y\in Z$. However, the only vertex in $K$
that does not have the same degree in $K$ as it does in $H$ is $u$,
and this difference in degree is only one. Hence $u$ is the only
vertex that could contribute a nonpositive amount to this sum. If
$x$ is not $u$ or $w$, then $x$ has degree $3$ in $K$, so the sum
cannot be at most 1. On the other hand, if $x$ is $u$ or $w$, then
while $x$ has only degree two in $K$, neither of these neighbours is
$u$.
To see that $\corefan(H_1) \leq 1$, let any full-multiplicity
subgraph $K \subseteq H_1$ with $K\neq \emptyset$ be given. According
to Lemma \ref{lem:tplus1} we need only find $xy \in E(K)$ with
\[\sizeof{(N_{H_1}(x) \cap Z(K)) - y} \leq d_{H_1}(y) - d_K(y),\]
where $Z(K) = \{v \in V(K) \colon\, d_K(v) = d_H(v)\}$. If $uv \notin E(K)$ but $K$ does have at least one edge incident to $u$, then we may take $y=u$ and $x \in N_K(u)$, so that
\[ \sizeof{(N_{H_1}(x) \cap Z(K)) - y} \leq 2 \leq d_{H_1}(y) -
d_{K}(y). \] (Recall that $H_1$ has constant multiplicity $2$,
so at a minimum, the two copies of $uv$ are missing at $y$ in the
subgraph $K$.) Otherwise, we may choose $xy$ so that
$(N_{H_1}(x) \cap Z(K)) - y =\emptyset$ and hence we immediately
get our desired inequality: if $vu\in E(K)$ then we take
$(x,y) = (v,u)$, and if $K$ has no edges incident to $u$ then
$Z(K)$ is either $\emptyset$ or $\{w\}$, and in the latter
case we may choose $y=w$.
\end{remark}
The following is the final result we need in order to write our proof of Theorem \ref{thm:Bqueuecorefan}.
\begin{theorem}\label{thm:qcorefan}
Let $B$ be a simple graph. If $B$ has a full $B$-queue, then $\corefan(H) \leq 0$.
\end{theorem}
\begin{proof}
Consider a full $B$-queue with vertex sequence $(u_1, \ldots, u_q)$ and set sequence $(S_1, \ldots, S_q)$.
Let $K\subseteqeq H$ with $E(K)\neq\emptyset$. According to Lemma \ref{lem:tplus1} we need only find $xy \in E(K)$ with
\[\sizeof{(N_{H}(x) \cap Z(K)) - y} \leq d_B(y) - d_K(y),\]
where $Z(K) = \{v \in V(K) \colon\, d_K(v) = d_H(v)\}$. We consider two cases:
\caze{1}{$K$ contains an edge incident to $u_i$ for some $i$.}
Choose $i$ to be the smallest such index, and let $x = u_i$. If
there is some vertex in $N_K(u_i) \cap (S_i \setminus S_{i-1})$,
let $y$ be such a vertex; otherwise, let $y$ be an arbitrary
element of $N_K(x)$. We claim that
$N_{H}(x) \cap Z(K) \subseteq \{y\}$. To this end, consider any
$z \neq y \in N_K(x)$. The definition of a full $B$-queue implies
that $z \in S_j$ for some $j \leq i$. Take the smallest such
$j$. If $j=i$, then necessarily $z=y$, since
$\sizeof{S_i \setminus (S_{i-1} \cup \{u_i\})} \leq 1$. Otherwise,
$j < i$, and since $z\neq u_j$ (by choice of $i$) we know that
$u_j \in N_B(z)$. But again, by our choice of $i$, the edge $u_jz$
cannot be in $K$, and so $z \notin Z(K)$. It follows
that \[ \sizeof{N_H(x) \cap Z(K) - y} = \sizeof{N_K(x) \cap Z(K) - y} = 0 \leq d_B(y) - d_K(y),\] as
desired.
\caze{2}{$K$ has no edges incident to $u_i$ for any $i$.} In this case, choose any $xy \in E(K)$. Since the $B$-queue is full, every vertex in $B$ is incident to at least one of $u_1, \ldots, u_q$, so $N(x) \cap Z(K) =\emptyset$. Thus again we have
\[\sizeof{(N(x) \cap Z(K)) - y} = 0 \leq d_B(y) - d_K(y)\]
as desired.
\end{proof}
\begin{remark} The converse of Theorem~\ref{thm:qcorefan} does not
hold. In particular, there is an infinite family
$\{H_p\}_{p \geq 4}$ of simple graphs such that for all $p$,
$\corefan(H_p) = 0$ but $B=H_p$ does not have a full $B$-queue. To
see this, let $B=H_p$ be the graph obtained from the complete graph
$K_p$ by designating a special vertex $v$ and attaching $p-2$
pendant edges to $v$. Let $z_1, \ldots, z_{p-2}$ be the vertices of
degree $1$ adajcent to $v$. The graph $H_4$ is shown in
Figure~\ref{fig:bqueueconverse}.
\begin{figure}
\caption{The graph $H_4$, a simple graph with $\corefan(H_4) = 0$ that does not admit a full $B$-queue.}
\label{fig:bqueueconverse}
\end{figure}
If $H_p$ has a full $B$-queue with vertex sequence
$(u_1, u_2, \ldots)$ then at least one vertex of the $K_p$ must
occur as a $u_i$; choose the smallest such $i$. If $u_i \in S_{i-1}$,
then $u_i=v$ and $\sizeof{S_i \setminus S_{i-1}} \geq \sizeof{N(u_i)\setminus S_{i-1}}\geq 3$, a
contradiction. If $u_i\not\in S_{i-1}$, then $\sizeof{S_i\setminus S_{i-1}} \geq 3$,
again a contradiction. Hence $H_p$ does not admit a full $B$-queue.
Now we claim that $\corefan(H_p) \leq 0$. We apply Lemma~\ref{lem:tplus1}.
Let $K$ be any subgraph of $H_p$ with $E(K)\neq \emptyset$. If $K$ does not contain any of the pendant edges incident to $v$, but $v$ does have at least one incident edge in $K$,
then let $y=v$ and take any $x \in N_K(v)$. Since $|N_K(x)|\leq p-1$ and $d_{H_p}(y) - d_K(y) \geq p-2$, we get that
\[ \sizeof{(N_{H_p}(x) \cap Z(K)) - y} \leq p-2 \leq d_B(y) - d_K(y), \]
as desired. Otherwise we can choose $xy\in E(K)$ so that $(N_{H_p}(x) \cap Z(K)) - y=\emptyset$ and hence we immediately get our desired inequality; if we can take $x=z_i$ for some $i$ then this is the case, else there are no edges incident to $v$. Since $v$ is joined to every vertex in $B$, this yields $Z(K) = \emptyset$, so that $N_{H_p}(x) \cap Z(K) = \emptyset$
no matter which $x$ we choose.
\end{remark}
We can now prove Theorem \ref{thm:Bqueuecorefan}.
\begin{proof}[Proof of Theorem \ref{thm:Bqueuecorefan}] By Lemma \ref{lem:lowmult}, it suffices to show
that $\corefan(H_{>t})\leq t$, where $H_{>t}$ is the subgraph of $H$
consisting of the edges with multiplicity greater than $t$. If $H$
has multiplicity at most $t$, then $H_{>t}$ is edgeless, and hence
$\corefan(H_{>t})=0$ by definition. So, we may assume that $H$ has
maximum multiplicity exactly $t+1$, and that $H_{>t}$ is the subgraph $H_t$ of $H$
consisting precisely of all the edges in $H$ of multiplicity
$t+1$. By Lemma \ref{lem:flatten}, we get our desired result of
$\corefan(H_t)\leq t$ provided $\corefan(B)\leq 0$. Since $B$ has a
full $B$-queue, Theorem \ref{thm:qcorefan} indeed tells us that
$\corefan(B)\leq 0$, thus completing our proof.
\end{proof}
\section{Proof of Theorem \ref{thm:converse}}\label{sec:converse}
Before we begin the main proof of Theorem~\ref{thm:converse}, it will
help to record a lemma about constructing regular graphs with perfect
matchings.
\begin{lemma}\label{lem:regmatching}
Let $n$, $k$, and $r$ be positive integers with $r\leq n$ and $k<n$. If $k$ and $r$ are even,
then there is a $k$-regular simple graph $G$ on $n$ vertices containing a
vertex set $S_r$ of size $r$ such that the induced subgraph $G[S_r]$
has a perfect matching.
\end{lemma}
\begin{proof}
For any even $k$ and any $n>k$, the standard circulant graph
construction (see, e.g., Chapter~12 of \cite{handbook-algo}) yields
a $k$-regular simple graph on $n$ vertices with a matching $M$ that covers
at least $n-1$ vertices. In particular, $M$ covers $n$ vertices if
$n$ is even, and $n-1$ vertices if $n$ is odd. Since $r$ is even, we
see that the number of vertices covered by $M$ is at least
$r$. Thus, one may choose any set of $r/2$ edges in $M$, and take
$S_r$ to be the set of vertices covered by those edges.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:converse}] Our goal is to build a graph $G$ whose $t$-core is $H$ and with $\fan(G)>\Delta(G)+t$ (and hence $\Fan(G)>\Delta+t$). Since $\corefan(H)>t$ there exists $K\subseteqeq H$, $E(K)\neq\emptyset$ with $\cdeg_{H,K}(x,y)> t$ for all $xy \in E(K)$, that is, with
\[\sum_{z \in Z}(d_K(z) - d_H(z) + \mu_K(x,z) - t) > 1\]
for some $Z\subseteqeq N_K(x)$ with $y\in Z$. Note that we may choose $K$ so that it has no isolated vertices. Choose positive integers $D$ and $r$ satisfying all of the following conditions:
\begin{enumerate}[(a)]
\item $r\geq \Delta(H) + 6 + t$
\item $r$ is even
\item $D \geq 3r + t$
\item $D \geq \Delta(H) + 2r^2$,
\item $D+t$ is an even multiple of $r-1$, with this multiple being at least 4.
\end{enumerate}
Initialize $G=H$. Our construction proceeds in several
stages; at each stage, we will add vertices and/or edges to $G$. When the construction is complete we will verify that $G$ indeed has $t$-core $H$ and $\fan(G)>\Delta(G)+t$.
\colon\,age{1} In this stage, we will add vertices and edges to our initial $G=H$ in
order to guarantee that $d_G(x) = D$ for all $x \in V(K)$.
Let $p = \sizeof{V(K)}$, and write $V(K) = \{x_1, \ldots,
x_p\}$. Note that since $K$ is not edgeless, we know that $p\geq
2$. For each $x_i \in V(K)$, let $d_i = D - \deg_H(x_i)$. We can
write each $d_i$ as $d_i = \alpha_i (r-1) + \beta_i$ where $\alpha_i$
and $\beta_i$ are integers with $0 \leq \beta_i \leq r-2$. We
rewrite this equation as
\[ d_i = (\alpha_i - \beta_i)(r-1) + \beta_i r. \] Let
$a_{i,r-1} = \alpha_i - \beta_i$ and let $a_{i,r} = \beta_i$. We know that $a_{i,r}\in\{0, \ldots, r-1\}$; note also that by assumption (d),
\begin{align*}
a_{i, r-1}&=\alpha_i-\beta_i
= \left(\tfrac{d_i-\beta_i}{r-1}\right)-\beta_i
=\left(\tfrac{D-\deg_H(x_i)-\beta_i}{r-1}\right)-\beta_i\\
&\geq \left(\tfrac{D-\Delta(H)-\beta_i}{r-1}\right)-\beta_i \geq \left(\tfrac{2r^2-\beta_i}{r-1}\right)-\beta_i \geq r
\end{align*}
If $\sum a_{i,r}$ is odd, then we redefine the first pair
$(a_{1,r}, a_{1,r-1})$ to be $(a_{1,r} +(r-1), a_{1,r-1} -r)$, which
will change the parity of the sum since $r$ is even by assumption
(b). Given the above inequality, we still have that
$a_{i, r}, a_{i, r-1} \geq 0$ for all $i$; in fact we know that
$a_{i, r-1}\geq r$ except possibly when $i=1$. We also still have that
\[d_i=a_{i, r}(r)+ a_{i, r-1}(r-1).\]
For $\ell \in \{r-1, r\}$, let $s_{\ell} = \sum_{i=1}^p a_{i,\ell}$,
and let $S_{\ell}$ be a set of $s_{\ell}$ new vertices added to
$G$. Our definition of the numbers $a_{i, \ell}$ guarantees that
$s_r$ is even, and this will be helpful for us in a later stage of
our construction.
For each $x_i \in V(K)$, choose a disjoint set $T$ of $a_{i,\ell}$
vertices from $S_{\ell}$, and add an edge of multiplicity $\ell$
from $x_i$ to each vertex of $T$. Once we complete this procedure
for all $i$ and both $\ell$, we see that every vertex in $K$ has
degree $D$. Let $S = S_{r-1} \cup S_r$ with $s=|S|$.
\colon\,age{2} In this stage, we will add edges within $S$ to ensure that
every vertex in $S_{\ell}$ ends with degree $D-{\ell}+t$.
Our strategy in this stage will be, roughly, to first paste in a
regular multigraph on the vertex set $S$ to bring the vertices of
$S_{r-1}$ up to degree $D-(r-1)+1$ and the vertices of $S_r$ up to
degree $D-r+t+2$, and then remove parallel copies of a (carefully
planted) perfect matching from $S_r$ so that those vertices end with
degree $D-r+t$.
Let $k = \frac{D - 2(r-1)+t}{r-1} = \frac{D+t}{r-1} - 2$. By
assumption (e), $k$ is an even integer and $k\geq 2$. Since $sk$ is
even, we can construct a $k$-regular simple graph on the vertex set
$S$ provided $s>k$. To verify this, start by observing the
following, where we are using $r \geq \Delta(H)$ (by a weak version
of assumption (a)):
\[ s = \sum_{i=1}^p (a_{i-1} + a_i) \geq \sum_{i=1}^p \frac{d_i}{r} \geq \sum_{i=1}^p \frac{D-r}{r} \geq 2\frac{D-r}{r} = \frac{2D}{r} - 2. \]
To show $s>k$, it remains to prove that $2D/r > (D+t)/(r-1)$, which is true iff $D>t\left(\tfrac{r}{r-2}\right)$. This follows from $D > 2t$ (by assumption (c) coupled with a weak version of assumption (a)) and $r \geq 4$ (by an even weaker version of assumption (a)).
Let $A$ be a $k$-regular simple graph on the vertex set $S$. Since
$s_r$ is even, as established in the previous stage of construction,
Lemma~\ref{lem:regmatching} allows us to choose $A$ so that it
contains a perfect matching $M$ on the vertex set $S_r$.
We now modify $G$ as follows: make $G[S]$ have underlying graph $A$
with edges in $A-M$ having multiplicity $r-1$ and edges in $M$
having multiplicity $r-3$.
Observe that at this point, every vertex in $S_{r-1}$ has degree
\[ (r-1) + (r-1)k = (r-1) + (D+t - 2(r-1)) = D - (r-1)+t, \]
while every vertex in $S_r$ has degree
\[ r + (r-1)k - 2 = r + (D+t - 2(r-1)) - 2 = D-r+t. \]
\colon\,age{3} In this last stage, we will bring each vertex $v \in V(H) - V(K)$
up to degree $D$. We do this by simply adding a single pendant edge
of multiplicity $r-1$ to $v$, followed by enough pendant edges of
multiplicity $1$ to obtain the desired degree. Note that this is possible since, by assumption (d),
$D \geq \Delta(H) +2r^2 \geq \Delta(H)+ r-1$, so the pendant edge of multiplicity
$r-1$ cannot itself make the degree greater than $D$.
This completes our construction of $G$.
\textbf{Verification of Properties.} We begin by verifying that $H$ is the $t$-core of $G$. To this end, note that $d(v) = D$ for
all $v \in V(H)$. For all $v \in S$, we have
$d(v) \leq D-(r-1)+t$, which is less than $D$ since $r-1 > t$ by a weak version of assumption (a). The endpoints of the pendant edges from Stage~3
have $d(v) \leq r-1 < D$ (using assumption (e) weakly). Hence $\Delta(G) = D$, with this degree achieved precisely by the vertices in $H$. Now, note that every vertex
of $H$ is incident to an edge of multiplicity $r-1$ or greater. So, for any vertex $v\in V(H)$,
\[d(v) + \mu(v) \geq D + (r-1) > D+t.\]
For $v \in S_r$ we have
\[d(v) + \mu(v) = (D-r+t) + r = D+t,\]
while for $v \in S_{r-1}$ we
have
\[d(v) + \mu(v) = (D - (r-1)+t) + (r-1) = D+t.\]
The endpoints
of the pendant edges added in Stage~3 have
\[d(v) + \mu(v) \leq (r-1)+(r-1) \leq D+t,\]
where the last inequality is another weak application of (e). Hence $H$ is indeed the $t$-core of $G$.
To verify that $\fan(G) > D+t$, we choose $J=G[K\cup S]$ and show that
$\deg_{J}(x,y) > D+t$ for all $xy \in E(J)$.
We know that $\mu(x,y) \leq r$ for all $xy \in E(J)$, with this coming from a weakened assumption (a) (and the fact that $\Delta(H)\geq \mu(H)$) when $xy\in E(K)$. Using the computations above, we know that
$d_J(v) \geq D-r$ for all $x \in V(J)$, with this coming from $r\geq \Delta(H)$ (again by assumption (a)) when $v\in V(K)$. We thus get
\[ d_J(x) + d_J(y) - \mu(x,y) \geq 2D - 3r > D+t, \]
where the last inequality is by assumption (c). Thus, Condition~(i) of the definition
of $d_J(x,y)$ fails for $k=D+t$.
Next we show that Condition~(ii) fails for $k=D+t$ as well. Consider
any $xy \in E(J)$. We consider two cases,
according to the location of $x$.
\caze{1}{$x \in V(K)$.} If $y \in V(K)$, let $y' = y$. Otherwise,
since $x$ is not isolated in $K$, we can take some $y' \in N_K(x)$. Since
$\corefan(K) > t$, there is a set $Z' \subseteq N_K(x)$ with
$y' \in Z'$ such that
\[ \sum_{z \in Z'}[ d_K(z) - d_H(z) + \mu_K(x,z) - t ] > 1. \]
Since $J$ includes all the edges of $K$, we have $Z' \subseteq N_J(x)$, with
$\mu_K(x,z) = \mu_J(x,z)$ for all $z \in Z'$. Furthermore, $d_H(z) - d_K(z) = D - d_J(z)$
for all $z \in Z'$ since $d_G(z) = D$ and the only $G$-edges incident to $z$ not
included in $J$ are the edges in $E(H) - E(K)$. Thus,
\begin{equation}
\label{ieq:xcase1}
\sum_{z \in Z'}[ d_J(z) + \mu_J(x,z) - (D + t)] = \sum_{z \in Z'}[d_K(z) - d_H(z) + \mu_K(x,z) - t] > 1.
\end{equation}
If $y \in Z'$ then this immediately implies that $\deg_J(x,y) > D + t$. Otherwise,
by our choice of $y'$, we have $y \in S_{\ell}$ for some $\ell$, in which case
\[ d_J(y) + \mu(x,y) - (D+t) = (D-\ell+t) + \ell - (D+t) = 0, \] so letting
$Z = Z' \cup \{y\}$ does not change the sum in
Inequality~\eqref{ieq:xcase1}, and the set $Z$ witnesses
$\deg_J(x,y) > D+t$.
\greg{\caze{2}{$x \in S$.}} Let $y'$ be the unique neighbor of $x$ in $V(K)$.
Observe that
\begin{equation}
\label{ieq:yprime}
d_J(y') + \mu(x,y') - (D + t) \geq (D - \Delta(H)) + (r-1) - (D + t) \geq r-1-\Delta(H) - t \geq 5,
\end{equation}
where the second last inequality comes from assumption (a).
Observe that for any $z \in N_J(x)$, even if $z \in S$ we still have
\begin{equation}
\label{ieq:minus3}
d_J(z) + \mu(x,z) - (D+t) \geq (D-r+t) + (r-3) - (D+t) = -3.
\end{equation}
If $y = y'$, then taking any $z \in N_J(x) - y$ and putting $Z = \{z,y\}$, we see that Inequalities
\eqref{ieq:yprime} and \eqref{ieq:minus3} together imply that $Z$ witnesses $\deg_J(x,y) > D+t$.
(Such a $z$ exists because, by our construction, $d_J(x) \geq (r-1)k \geq 2\mu(x)$.)
If $y \neq y'$, then since $y \in N_J(x)$, taking $Z = \{y, y'\}$
again yields a set witnesing $\deg_J(x,y) > D+t$, via the same
inequalities.
\end{proof}
\end{document} |
\begin{document}
\title[Dicritical singularities of Levi-flat hypersurfaces and foliations]{Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations}
\author{Andr\'es Beltr\'an}
\address[A. Beltr\'an]{Dpto. Ciencias - Secci\'on Matem\'aticas, Pontificia Universidad Cat\'olica del Per\'u.}
\curraddr{Av. Universitaria 1801, San Miguel, Lima 32, Peru}
\email{[email protected]}
\author{Arturo Fern\'andez-P\'erez}
\address[A. Fern\'andez-P\'erez]{Departamento de Matem\'atica - ICEX, Universidade Federal de Minas Gerais, UFMG}
\curraddr{Av. Ant\^onio Carlos 6627, 31270-901, Belo Horizonte-MG, Brasil.}
\email{[email protected]}
\author{Hern\'an Neciosup}
\address[H. Neciosup]{Dpto. Ciencias - Secci\'on Matem\'aticas, Pontificia Universidad Cat\'olica del Per\'u.}
\curraddr{Av. Universitaria 1801, San Miguel, Lima 32, Peru}
\email{[email protected]}
\thanks{This work is supported by the Pontificia Universidad Cat\'olica del Per\'u project VRI-DGI 2016-1-0018. Second author is partially supported by CNPq grant number 301825/2016-5}
\subjclass[2010]{Primary 32V40 - 32S65}
\keywords{Levi-flat hypersurfaces - Holomorphic foliations}
\begin{abstract}
We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in $\mathbb{P}^2$.
\end{abstract}
\maketitle
\section{Introduction}
\par In this paper we study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two with emphasis on the type of singularities of them. Singular Levi-flat hypersurfaces in complex manifolds appear in many contexts, for example the zero set of the real-part of a holomorphic function or as an invariant set of a holomorphic foliation. While singular Levi-flat hypersurfaces have many properties of complex subvarieties, they have a more complicated geometry and inherit many pathologies from general real-analytic subvarieties. The interconnection between singular Levi-flat hypersurfaces
and holomorphic foliations have been studied by many authors, see for example \cite{burns}, \cite{brunella}, \cite{alcides}, \cite{normal}, \cite{arturo}, \cite{generic}, \cite{arnold}, \cite{libro}, \cite{lebl}, \cite{singularlebl} and \cite{shafikov}.
\par Let $M$ be a real-analytic closed subvariety of real dimension 3 in a compact complex manifold $X$ of complex dimension two. Unless specifically stated, subvarieties
are analytic, not necessarily algebraic. Throughout the text, the term \textit{real-analytic hypersurface} will be employed with the
meaning \textit{real-analytic subvariety of real dimension 3}.
Let us denote $M ^{*}$ the set of points of $M$ near which $M$ is a nonsingular real-analytic hypersurface. $M$ is said to be {\textit{Levi-flat}} if the codimension one distribution on $M^{*}$
$$T^{\mathbb{C}}M^{*}=TM^{*}\cap J(TM^{*})\subset TM^{*}$$
is integrable, in Frobenius sense. It follows that $M^{*}$ is foliated locally by immersed one-dimensional complex manifolds, the foliation defined by $T^{\mathbb{C}}M^{*}$ is called the \textit{Levi-foliation} and will be denoted by $\mathcal{L}$.
\par Let $\{U_j\}_{j\in I}$ be an open covering of $X$. A \textit{holomorphic foliation} $\mc{F}$ on $X$ can be described by a collection of holomorphic 1-forms $\omega_j\in\Omega^{1}_{X}(U_{j})$ with isolated zeros such that
\begin{equation*}
\omega_i=g_{ij}\omega_j\,\,\,\,\,\,\,\,\,\text{on}\,\,\,U_i\cap U_j,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g_{ij}\in\mathcal{O}^{*}_{X}(U_i\cap U_j).
\end{equation*}
The cocycle $\{g_{ij}\}$ defines a line bundle $N_{\mc{F}}$ on $X$, called \textit{normal bundle} of $\mc{F}$. The \textit{singular set} $\textsf{Sing}(\mc{F})$ of $\mc{F}$ is the finite subset of $X$ defined by
$$\textsf{Sing}(\mc{F})\cap U_{j}=\text{zeros of}\,\,\,\omega_{j},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall j\in I.$$
A point $q\not\in\textsf{Sing}(\mc{F})$ is said to be \textit{regular}.
We will be denote $c_1(N_{\mc{F}})\in H^2(X,\mathbb{Z})$ the \textit{first Chern class} of $N_{\mc{F}}$ and if $\Omega$ is a smooth closed 2-form on $X$ which represents, in the De Rham sense, the first Chern class of $N_{\mc{F}}$, we will use the following notation
$$c_1^{2}(N_{\mc{F}}):=\int_{X}\Omega\wedge\Omega.$$
We shall say that a holomorphic foliation $\mathcal{F}$ on $X$ is \textit{tangent} to $M$ if locally the leaves of the Levi foliation $\mathcal{L}$ on $M^{*}$ are also
leaves of $\mathcal{F}$. A singular point $p\in M$ is called \textit{dicritical} if for every neighborhood $U$ of $p$, infinitely many leaves of the Levi-foliation on $M^{*}\cap U$ have $p$ in their closure. Analogously, a singular point $p\in\textsf{Sing}(\mc{F})$ of a holomorphic foliation $\mc{F}$ is called \textit{dicritical} if for every neighborhood $U$ of $p$, infinitely many leaves have $p$ in their closure. Otherwise it is called \textit{non-dicritical}.
\par Recently dicritical singularities of singular real-analytic Levi-flat hypersurfaces have been characterized in terms of the \textit{Segre varieties}, see for instance Pinchuk-Shafikov-Sukhov \cite{pinchuk}. Using this characterization, the notion of dicritical singularity in the theory of holomorphic foliations coincides with the notion of Segre-degenerate singularity of a real-analytic Levi-flat hypersurface and therefore, we can use results of residue-type indices associated to singular points of holomorphic foliations \cite{index}, \cite{suwa} to prove the following result.
\begin{maintheorem}\label{main_theorem}
Let $\mc{F}$ be a holomorphic foliation on a compact complex manifold $X$ of complex dimension two tangent to an irreducible real-analytic Levi-flat hypersurface $M \subset X$ such that $\textsf{Sing}(\mc{F})\subset M$. Suppose $c^2_1(N_\mc{F})>0$. Then there exists a dicritical singularity $p \in M$ such that $\mc
{F}$ has a non-constant meromorphic first integral at $p$.
\end{maintheorem}
\par We emphasize that the existence of dicritical singularities of $M$ depends on the condition $c^2_1(N_\mc{F})>0$, because otherwise the result is false. In section \ref{examples_paper} we give some examples that show the importance of the condition $c^2_1(N_\mc{F})>0$. The condition $\textsf{Sing}(\mc{F})\subset M$ is used in the proof of Theorem \ref{main_theorem}, we do not know if this condition can be removed.
\par In the sequel we apply Theorem \ref{main_theorem} to non-dicritical projective foliations, that is, holomorphic foliations on $\mathbb{P}^2$ with only non-dicritical singularities.
\begin{maincorollary}
Let $\mc{F}$ be a holomorphic foliation on $\mathbb{P}^2$ with only non-dicritical singularities. Then there are no singular real-analytic Levi-flat hypersurfaces in $\mathbb{P}^{2}$ tangent to $\mc{F}$ that contain $\textsf{Sing}(\mathcal{F})$.
\end{maincorollary}
Now we apply Theorem \ref{main_theorem} for holomorphic foliations of degree 2 on $\mathbb{P}^2$ with a unique singularity.
\begin{secondcorollary}
Let $\mc{F}$ be a holomorphic foliation of degree 2 on $\mathbb{P}^2$ with a unique singular point $p$. Suppose that $\mc{F}$ is tangent to a singular real-analytic Levi-flat hypersurface $M\subset\mathbb{P}^2$ and $p\in M$. Then, up to automorphism, $\mc{F}$ is given in affine coordinates $(x,y)\in\mathbb{C}^2$ by the 1-form
$$\omega=x^2dx+y^2(xdy-ydx).$$
Moreover, let $[x:y:z]$ be the homogenous coordinates of $\mathbb{P}^2$, then $R=\frac{y^3-3x^2z}{3x^3}$ is a rational first integral for $\mc{F}$.
\end{secondcorollary}
\par We say that a singularity $p$ of a germ of a real-analytic Levi-flat hypersurface $M$ is said to be \textit{semialgebraic}, if the germ of $M$ at $p$ is biholomorphic to a semialgebraic Levi-flat hypersurface. We recall that a real-analytic Levi-flat hypersurface is said to be \textit{semialgebraic}, if it is contained in a codimension one real-analytic subvariety defined by the vanishing of a real polynomial.
\par To continue we apply Theorem \ref{main_theorem} to find \textit{semialgebraic} singularities of singular real-analytic Levi-flat hypersurfaces which are tangent to singular holomorphic foliations on compact complex manifolds of complex dimension two. Similarly results of algebraization of singularities of holomorphic foliations can be found in \cite{genzmer}. Recently in \cite{casale}, Casale considered the algebraization problem for \textit{simple dicritical singularities} of germs of holomorphic foliations. These singularities are those that become nonsingular after one blow-up and such that a unique leaf is tangent to the exceptional divisor with tangency order of one. Motived by \cite{casale}, we state the following result.
\begin{Thirdcorollary}
Let $\mc{F}$ be a holomorphic foliation on a compact complex manifold $X$ of complex dimension two tangent to an irreducible real-analytic Levi-flat hypersurface $M \subset X$ such that $\textsf{Sing}(\mc{F})\subset M$. Suppose that $c^2_1(N_{\mc{F}})>0$ and $\mc{F}$ has only a unique simple dicritical singularity $p\in M$. Then there exists an algebraic surface $V$, a rational
function $H$ on $V$ and a point $q\in V$ such that the germ of $M$ at $p$ is biholomorphic to a semialgebraic Levi-flat hypersurface $M'\subset V$ in a neighborhood of $q$.
\end{Thirdcorollary}
\par Finally we state a result that guarantee the existence of dicritical singularities of $M$ in presence of invariant compact complex curves by $\mc{F}$.
\begin{secondtheorem}\label{second}
Let $\mc{F}$ be a holomorphic foliation on a compact complex manifold $X$ of complex dimension two tangent to an irreducible real-analytic Levi-flat hypersurface $M \subset X$. Suppose that the self-intersection $C\cdot C>0$, where $C\subset M$ is an irreducible compact complex curve invariant by $\mc{F}$, then there exists a dicritical singularity $p \in \textsf{Sing} (\mc{F})\cap C$ such that $\mc
{F}$ has a non-constant meromorphic first integral at $p$.
\end{secondtheorem}
\par To prove Theorem \ref{second} we use the Camacho-Sad index (cf. \cite{CS}) and a result of Cerveau-Lins Neto (see Theorem \ref{lins-cerveau}).
\par We organize the paper as follows: in section \ref{indices} we review some definitions and results about indices of holomorphic foliations at singular points. In section \ref{Existence} we give the proof of Theorem \ref{main_theorem}. Section \ref{application} is devoted to show some applications of Theorem \ref{main_theorem}.
In section \ref{dicritical} we provide the proof of Theorem \ref{second}. Finally, in section \ref{examples_paper} we show some examples which show the importance of the hypotheses in theorems \ref{main_theorem} and \ref{second}.
\section{Indices of holomorphic foliations}\label{indices}
\par In this section we state two important results on indices of holomorphic foliations: Camacho-Sad index \cite{CS} and
the Baum-Bott index \cite{baum}. The first one concerns the computation of $C\cdot C$, where $C\subset X$ is an invariant compact curve by $\mathcal{F}$ and the second one concerns the computation of $c^{2}_1(N_{\mc{F}})$. More references for these index theorems can be found in \cite[Chapter V]{suwa}, see also \cite{birational} and \cite{index}.
\par First, we recall the definition of meromorphic and holomorphic first integral for holomorphic foliations. Let $\mc{F}$ be a singular holomorphic foliation on $X$. Recall that $\mathcal{F}$ admit a \textit{meromorphic} (\textit{holomorphic}) first integral at $p\in X$, if there exists a neighborhood $U$ of $p$ and
a \textit{meromorphic} (\textit{holomorphic}) function $h$ defined in $U$ such that its indeterminacy (zeros)
set is contained in $\textsf{Sing}(\mc{F})\cap U$ and its level curves contain the leaves of $\mathcal{F}$ in $U$.
\subsection{Camacho-Sad index}
\par Let us consider a separatrix $C$ at $p\in X$. Let $f$ be a holomorphic function on a neighborhood of $p$ and defining $C =\{f=0\}$.
We may assume that $f$ is reduced, i.e. $df\neq 0$ outside $p$. Then \cite{lins}, \cite{suwa} there are functions $g$, $k$ and a 1-form $\eta$ on a neighborhood of $p$ such that
$$g\omega=kdf+f\eta$$
and moreover $k$ and $f$ are prime, i.e. $k\neq 0$ on $C^{*}=C\setminus\{p\}$.
The Camacho-Sad index \cite{CS} is defined as
$$\text{CS}(\mc{F},C,p)=-\frac{1}{2\pi i}\int_{\partial{C}}\frac{1}{k}\eta,$$
where $\partial{C}=C\cap S^{3}$ and $S^3$ is a small sphere around $p$; $\partial{C}$ is oriented as a boundary of $S^{3}\cap B^4$, with $B^4$ a ball containing $p$.
\par If $C\subset X$ is a compact complex curve invariant by $\mc{F}$, one has the formula due by Camacho-Sad.
\begin{theorem}\cite[Camacho-Sad]{CS}\label{CS}
$$\sum_{p\in\textsf{Sing}(\mc{F})\cap C}\text{CS}(\mc{F},C,p)=C\cdot C.$$
\end{theorem}
\subsection{Baum-Bott index}
Let $\mc{F}$ be a holomorphic foliation with isolated singularities on $X$. Let $p\in X$ be a singular point of $\mc{F}$; near $p$ the foliation is given by a holomorphic vector field $$v=F(x,y)\frac{\partial}{\partial{x}}+G(x,y)\frac{\partial}{\partial{y}}$$ or by a holomorphic 1-form $$\omega=F(x,y)dy-G(x,y)dx.$$
\par Let $J(x,y)$ be the Jacobian matrix of $(F,G)$ at $(x,y)$, then following \cite{baum} one can define the Baum-Bott index at $p\in\textsf{Sing}(\mc{F})$ as
$$\text{BB}(\mc{F},p)=\text{Res}_{p}\Big\{\frac{(\text{Tr} J)^2}{F\cdot G}dx\wedge dy\Big\}.$$
The Baum-Bott index depended only on the conjugacy class of the germ of $\mc{F}$ at $p$. For example when the singularity $p$ is non-degenerate then $$\text{BB}(\mc{F},p)=\frac{(\text{Tr} J(p))^2}{\det J(p)}=\frac{\lambda_1}{\lambda_2}+\frac{\lambda_2}{\lambda_1}+2,$$ where $\lambda_1$ and $\lambda_2$ are the eigenvalues of the linear part $Dv(p)$ of $v$ at $p$. The set of separatrices $S$ through $p$ is formed by two transversal branches $C_1$ and $C_2$, both of them analytic.
We note also that $$\text{CS}(\mc{F},S,p)=\text{CS}(\mc{F},C_1,p)+\text{CS}(\mc{F},C_2,p)+2[C_1,C_2]_p=\frac{\lambda_1}{\lambda_2}+\frac{\lambda_2} {\lambda_1}+2,$$ where $[C_1,C_2]_{p}$ denotes the intersections number between the curves $C_1$ and $C_2$ at $p$. We remark that $\text{CS}(\mc{F},C_i,p)$ are computed over $\partial{C}_{i}$ for each $i=1,2$ respectively.
Thus $\text{BB}(\mc{F},p)=\text{CS}(\mc{F},S,p)$. Of course this remains valid for generalized curve foliations with non-dicritical singularities.
\begin{theorem}\cite[Brunella]{index}\label{brunella_index}
Let $\mc{F}$ be a non-dicritical germ of holomorphic foliation at $0\in\mathbb{C}^2$ and let $S$ be the union of all its separatrices. If $\mc{F}$ is a generalized curve foliation, then
\begin{eqnarray*}
\text{BB}(\mc{F},0)&=&\text{CS}(\mc{F},S,0).
\end{eqnarray*}
\end{theorem}
\par We recall that the foliation, induced by $v$, is said to be \textit{generalized curve} at $0\in\mathbb{C}^2$ if there are no saddle-nodes in its reduction of singularities. It is easily seen that Theorem \ref{brunella_index} is false for saddle-nodes singularities.
The following formula will used in the proof of Theorem \ref{main_theorem}.
\begin{theorem}\cite[Baum-Bott]{baum}\label{baum_bott}
$$\sum_{p\in\textsf{Sing}(\mc{F})}\text{BB}(\mc{F},p)=c^{2}_1(N_{\mc{F}}).$$
\end{theorem}
\section{Existence of dicritical singularities of Levi-flat hypersurfaces}\label{Existence}
\par In order to prove Theorem \ref{main_theorem} we need the following results.
\begin{theorem}[Cerveau-Lins Neto \cite{alcides}]\label{lins-cerveau}
Let $\mathcal{F}$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq{2}$, of codimension one holomoprhic foliation tangent to a germ of an irreducible real-analytic hypersurface $M$. Then $\mathcal{F}$ has a non-constant meromorphic first integral. In the case of dimension two we can precise more:
\begin{enumerate}
\item If $\mc{F}$ is dicritical then it has a non-constant meromorphic first integral.
\item If $\mc{F}$ is non-dicritical then it has a non-constant holomorphic first integral.
\end{enumerate}
\end{theorem}
Now we prove the following lemma.
\begin{lemma}\label{baum-bott_signo}
Let $\mc{F}$ be a germ of a non-dicritical holomorphic foliation at $0\in\mathbb{C}^2,$ tangent to a germ of an irreducible real-analytic Levi-flat hypersurface $M$ at $0\in\mathbb{C}^2$. Then the Baum-Bott index satisfies
$\text{BB}(\mc{F},0)\leq 0.$
\end{lemma}
\begin{proof}
Since $\mc{F}$ is non-dicritical at $0\in\mathbb{C}^2$, we have $\mc{F}$ has a non-constant holomorphic first integral $g\in\mathcal{O}_2$ by Theorem \ref{lins-cerveau}. In particular, $\mc{F}$ is a generalized curve foliation and Theorem \ref{brunella_index} implies that
$$\text{BB}(\mc{F},0)=\text{CS}(\mc{F},S,0),$$
where $S$ is the union of all separatrices of $\mc{F}$ at $0\in\mathbb{C}^2$. To prove the lemma we need calculate $CS(\mc{F},S,0)$. In fact, as $dg\wedge\omega=0$, then $dg=h\omega$, where $h\in\mathcal{O}_2$. Moreover, if $g=g^{\ell_1}_1\cdots g^{\ell_k}_{k}$, we get $S=\displaystyle\bigcup^{k}_{j=1} C_{j}$ with $C_j=\{g_j=0\}$ and
$$\sum^{k}_{j=1}\ell_jg_1\cdots\widehat{g_j}\cdots g_k dg_j=h_1\omega,$$
where $h_1=\frac{h}{g^{\ell_1-1}_1\cdots g^{\ell_k-1}_k}$. Hence $$\text{BB}(\mc{F},0)=\text{CS}(\mc{F},S,0)=-\sum_{1\leq i< j\leq k}\frac{(\ell_i-\ell_j)^2}{\ell_i\ell_j}[g_i,g_j]_{0}\leq 0,$$
here $[g_i,g_j]_{0}$ denotes the number of intersections between $C_i$ and $C_j$ at $0\in\mathbb{C}^2$.
\end{proof}
To continue we prove Theorem \ref{main_theorem}.
\subsection{Proof of Theorem \ref{main_theorem}}
We use the Theorem \ref{baum_bott} and Theorem \ref{lins-cerveau} to prove that $\mc{F}$ has a dicritical singularity in $X$. In fact, suppose by contradiction that $\textsf{Sing}(\mc{F})$ consists only of non-dicritical singularities. Take any point $p\in\textsf{Sing} (\mc{F})$ and let $U$ be a small neighborhood of $p$ in $X$ such that $\mc{F}$ is represented by a holomorphic 1-form $\omega$ on $U$ and $p$ is an isolated singularity of $\omega$. Since $\mc{F}$ and $M$ are tangent in $U$ and $p\in M$, we have $\mc{F}|_{U}$ admits a holomorphic first integral $g\in\mc{O}(U)$, that is, $\omega\wedge dg=0$ on $U$, by Theorem \ref{lins-cerveau}.
\par Applying Lemma \ref{baum-bott_signo}, we get $\text{BB}(\mc{F},p)\leq 0$, for any $p\in\textsf{Sing}(\mc{F})$. But Baum-Bott's formula implies that
$$\sum_{p\in\textsf{Sing}(\mc{F})}\text{BB}(\mc{F},p)=c^{2}_{1}(N_{\mc{F}})>0,$$
which is absurd. Therefore, there exists a dicritical singularity $p$ of $\mc{F}$. Applying again Theorem \ref{lins-cerveau}, we obtain a non-constant meromorphic first integral for $\mc{F}$ in a neighborhood of $p$.
\section{Applications of Theorem \ref{main_theorem}}\label{application}
\par First we apply Theorem \ref{main_theorem} to holomorphic foliations of $\mathbb{P}^2$ with only non-dicritical singularities.
\begin{corollary}
Let $\mc{F}$ be a holomorphic foliation on $\mathbb{P}^2$ with only non-dicritical singularities. Then there are no singular real-analytic Levi-flat hypersurfaces in $\mathbb{P}^{2}$ tangent to $\mc{F}$ that contain $\textsf{Sing}(\mathcal{F})$.
\end{corollary}
\begin{proof}
Suppose by contradiction that $\mc{F}$ is tangent to a singular real-analytic Levi-flat hypersurface $M\subset\mathbb{P}^2$ such that $\textsf{Sing}(\mc{F})\subset M$. Since $N_{\mc{F}}=\mathcal{O}_{\mathbb{P}^2}(d+2)$, where $d$ is a positive integer, one has $c^{2}_{1}(N_{\mc{F}})=(d+2)^{2}>0$. Therefore $\mc{F}$ has a dicritical singularity by Theorem \ref{main_theorem}. But it is absurd with the assumption.
\end{proof}
\par The Jouanolou foliation $\mathcal{J}_d$ of degree $d$ on $\mathbb{P}^2$, is given in affine coordinates $(x,y)\in\mathbb{C}^2$ by
$$\omega_d=(y^{d}-x^{d+1})dy-(1-x^{d}y)dx.$$
It is well known that $\mathcal{J}_d$ belongs to a holomorphic foliations class of $\mathbb{P}^2$ without algebraic solutions, this means, $\mathcal{J}_d$ does not admit invariant algebraic curves, see for instance \cite{lins}.
\par On the other hand, we know that $\mathcal{J}_d$ is a foliation with only non-dicritical singularities, because $\mathcal{J}_d$ has $d^{2}+d+1$ singularities and for each singularity passing only two analytic separatrices. Hence the above corollary shows that $\mathcal{J}_d$ does not tangent to any singular real-analytic Levi-flat hypersurface of $\mathbb{P}^2$.
\par Now we apply Theorem \ref{main_theorem} to holomorphic foliations of degree 2 on $\mathbb{P}^2$ with only one singularity.
\begin{corollary}
Let $\mc{F}$ be a holomorphic foliation of degree 2 on $\mathbb{P}^2$ with a unique singular point $p$. Suppose that $\mc{F}$ is tangent to a singular real-analytic Levi-flat hypersurface $M\subset\mathbb{P}^2$ and $p\in M$. Then, up to automorphism, $\mc{F}$ is given in affine coordinates $(x,y)\in\mathbb{C}^2$ by the 1-form
$$\omega=x^2dx+y^2(xdy-ydx).$$
Moreover, let $[x:y:z]$ be the homogenous coordinates of $\mathbb{P}^2$, then $R=\frac{y^3-3x^2z}{3x^3}$ is a rational first integral for $\mc{F}$.
\end{corollary}
\begin{proof}
Let us assume that $\mc{F}$ has a unique singularity, say $p$. Applying Theorem \ref{main_theorem} we have $\mc{F}$ has a non-constant meromorphic first integral $f/g$ in a neighborhood of $p$. Now since $\mc{F}$ has degree 2, one can apply the main theorem of \cite{deserti} which asserts that, up automorphism, $\mc{F}$ is given in affine coordinates $(x,y)\in\mathbb{C}^2$ by one of the following types:
\begin{enumerate}
\item $\omega_1=x^2dx+y^2(xdy-ydx)$;
\item $\omega_2=x^2dx+(x+y^2)(xdy-ydx)$;
\item $\omega_3=xydx+(x^2+y^2)(xdy-ydx)$;
\item $\omega_4=(x+y^2-x^2y)dy+x(x+y^2)dx$.
\end{enumerate}
The foliations given in (2) and (3) have first integral respectively
$$\left(2+\frac{1}{x}+2\left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)^2\right)\exp\left(-\frac{y}{x}\right)\,\,\,\text{and}\,\,\,\,\left(\frac{y}{x}\right)\exp\left(\frac{1}{2}\left(\frac{y}{x}\right)^2-\frac{1}{x}\right),$$
and the foliation in (4) has no meromorphic first integral, see \cite{deserti}. Then $\mc{F}$ can only be the foliation induced by
$$\omega_1=x^2dx+y^2(xdy-ydx).$$
It is easy to check that $R(x,y,z)=\frac{y^3-3x^2z}{3x^3}$ is a rational first integral for $\mc{F}$.
\end{proof}
\par An \textit{algebraizable singularity} is a germ of a singular holomorphic foliation which can be defined in some appropriated local chart by a differential equation with algebraic coefficients. It is proved in \cite{genzmer} the existence of countable many classes of saddle-node singularities which are not algebraizable. In \cite{casale}, Casale studied \textit{simple dicritical singularities}, these singularities are those that become nonsingular after one blow-up and such that a unique leaf is tangent to the exceptional divisor with tangency order of one. He show that a simple dicritical singularity with meromorphic first integral is algebraizable (cf. \cite[Theorem 1]{casale}).
\begin{theorem}[Casale]\label{Casale_theorem}
If $\mathcal{F}$ is a simple dicritical foliation at $0\in\mathbb{C}^2$ with a meromorphic first integral then there exist an algebraic surface $S$, a rational function $H$ on $S$ and a point $p\in S$ such that $\mathcal{F}$ is biholomorphic to the foliation given by the level curves of $H$ in a neighborhood of $p$.
\end{theorem}
\par Similarly, our aim is give a result of algebraization of singularities for real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two.
\begin{corollary}
Let $\mc{F}$ be a holomorphic foliation on a compact complex manifold $X$ of complex dimension two tangent to an irreducible real-analytic Levi-flat hypersurface $M \subset X$ such that $\textsf{Sing}(\mc{F})\subset M$. Suppose that $c^2_1(N_{\mc{F}})>0$ and $\mc{F}$ has only a unique simple dicritical singularity $p\in X$. Then there exists an algebraic surface $V$, a rational
function $H$ on $V$ and a point $q\in V$ such that the germ of $M$ at $p$ is biholomorphic to a semialgebraic Levi-flat hypersurface $M'\subset V$ in a neighborhood of $q$.
\end{corollary}
\begin{proof}
Since $p\in X$ is the unique singularity of $\mc{F}$, we have $\mc{F}$ has a non-constant meromorphic first integral in a neighborhood of $p$ by Theorem \ref{main_theorem}. Then it is follows from Theorem \ref{Casale_theorem} that there exist an algebraic surface $V$, a rational function $H$ on $V$ and a point $q\in V$ such that $\mc{F}$ is biholomorphic to the foliation given by level curves of $H$ in a neighborhood of $q$. According to \cite[Lemma 5.2]{lebl}, there exists a real-algebraic subvariety Levi-flat $N\subset V$ (of real dimension three) such that $M$ is biholomorphic to a subset $M'\subset N$. Thus $M'$ is semialgebraic.
\end{proof}
\section{Dicritical singularities of Levi-flat hypersurfaces in presence of compact leaves}\label{dicritical}
We use Camacho-Sad's formula (Theorem \ref{CS}) and Theorem \ref{lins-cerveau} to prove the following result.
\begin{theorem}\label{variotional}
Let $\mc{F}$ be a holomorphic foliation on a compact complex manifold $X$ of complex dimension two tangent to an irreducible real-analytic Levi-flat hypersurface $M \subset X$. Suppose that $C\cdot C>0$, where $C\subset M$ is an irreducible compact complex curve invariant by $\mc{F}$, then there exists a dicritical singularity $p \in \textsf{Sing} (\mc{F})\cap C$ such that $\mc
{F}$ has a non-constant meromorphic first integral at $p$.
\end{theorem}
\begin{proof}
Suppose by contradiction that all the singularities of $\mc{F}$ over $C$ are non-dicritical.
Take any point $q\in\textsf{Sing} (\mc{F})\cap C$ and let $U$ be a neighborhood of $q$ in $X$ such that $C\cap U=\{f=0\}$, $\mc{F}$ is represented by a holomorphic 1-form $\omega$ on $U$ and $q$ is an isolated singularity of $\omega$. Since $\mc{F}$ and $M$ are tangent in $U$, we have $\mc{F}|_{U}$ admits a holomorphic first integral $g\in\mc{O}(U)$, that is, $\omega\wedge dg=0$ on $U$, by Theorem \ref{lins-cerveau}. Then $dg=h\omega$, where $h\in\mc{O}(U)$. If
$g=g^{\ell_1}_1\cdots g^{\ell_k}_{k}$, we get $f=g_i$ for some $i$ and
$$\sum^{k}_{j=1}\ell_jg_1\cdots\widehat{g_j}\cdots g_k dg_j=h_1\omega,$$
where $h_1=\frac{h}{g^{\ell_1-1}_1\cdots g^{\ell_k-1}_k}$. It follows that,
$$CS(\mc{F},C,q)=-\sum_{i\neq j}\frac{\ell_{j}}{\ell_i}[g_i,g_j]_q$$
where $[g_{i},g_j]_{q}$ denotes the intersections number between the curves $\{g_{i}=0\}$ and $\{g_{j}=0\}$ at $q$. In particular,
$$CS(\mc{F},C,q)\leq 0\,\,\,\,\,\text{for any}\,\,q\in\textsf{Sing} (\mc{F})\cap C. $$
It follows from Theorem \ref{CS} that
$$\sum_{q\in\textsf{Sing}{\mc{F}}\cap C}CS(\mc{F},C,q)=C\cdot C.$$
But this is a contradiction with $C\cdot C> 0.$
Now since $p\in M$, we can apply again Theorem \ref{lins-cerveau} to find a meromorphic first integral for $\mc{F}$ in a neighborhood of $p$.
\end{proof}
\section{Examples}\label{examples_paper}
First we give an example where the hypotheses of Theorem \ref{main_theorem} are satisfied.
\begin{example}
The canonical local example of a real-analytic Levi-flat hypersurface in $\mathbb{C}^2$ is given by $\text{Im} z_1=0$. This hypersurface can be extended to all $\mathbb{P}^2$ given by
$$M=\{[Z_1:Z_2:Z_3]\in\mathbb{P}^2: Z_1\bar{Z}_3-\bar{Z}_1Z_3=0\}.$$
Moreover, this hypersurface is tangent to holomorphic foliation $\mathcal{F}$ given by the level of rational function $Z_1/Z_3$. Note also that $\mathcal{F}$ has degree $0$ and therefore $c^2_1(N_\mathcal{F})=c^2_1(\mathcal{O}_{\mathbb{P}^2}(2))=4$. The foliation $\mathcal{F}$ has a dicritical singularity at $[0:1:0]\in M$.
\end{example}
We now give two examples where Theorem \ref{main_theorem} is false.
\begin{example}
Consider the Hopf surface $X=(\mathbb{C}^2-\{0\})/\Gamma_{a,b}$ induced by the infinite cyclic subgroup of $GL(2,\mathbb{C})$ generated by the transformation $(z_1,z_2)\mapsto(az_1,bz_2)$ with $|a|,|b|>1$. Levenberg-Yamaguchi \cite{Levenberg} proved that the domain $$D=\{(z_1,z_2)\cdot\Gamma_{a,b}|\,z_1\in\mathbb{C}, \text{Im}\,\,z_2>0\}$$ in $X$ with $b\in\mathbb{R}$ is Stein. Furthermore it is bounded by a real-analytic Levi-flat hypersurface
$$M=\{(z_1,z_2)\cdot\Gamma_{a,b}|\,z_1\in\mathbb{C}, \text{Im}\,\,z_2=0\}.$$
It is clear that the levels of the holomorphic function $f(z_1,z_2)=z_2$ on $\mathbb{C}^2-\{0\}$ defines a holomorphic foliation $\mc{F}$ on $X$ tangent to $M$. Since any line bundle on $X$ is flat \cite{mall} we have $c^2_1(N_\mc{F})=0$ and $M$ has no singularities in $X$.
\end{example}
\begin{example}
Let $X=\mathbb{P}^1\times\mathbb{P}^1$ and let $\mc{F}$ be the foliation given by the vertical fibration on $X$.
Now let $\pi:X\to\mathbb{P}^1$ be the projection on the first coordinate and let $M=\pi^{-1}(\gamma)$, where $\gamma$ is a real-analytic embedded loop in $\mathbb{P}^1$. Take a fiber $F\subset M$. We have $F$ is isomorphic to $\mathbb{P}^1$ with $F^2=0$.
Clearly $M$ is a Levi-flat hypersurface in $X$ tangent to $\mc{F}$ and it has no dicritical singularities in $F$.
\end{example}
\vskip 0.2 in
\noindent{\it\bf Acknowledgments.--}
The authors wishes to express his thanks to Alcides Lins Neto (IMPA) for several helpful comments during the preparation of the paper. Also, we would like to thank the referee for suggestions and pointing out corrections.
\end{document} |
\begin{document}
\setcounter{page}{1}
\title{Group Variable Selection via a Hierarchical Lasso and Its
Oracle Property}
\author{Nengfeng Zhou\\
Consumer Credit Risk Solutions\\
Bank of America\\
Charlotte, NC 28255\\
\\
Ji Zhu\footnote{Address for correspondence: Ji Zhu, 439 West
Hall, 1085 South University Ave, Ann Arbor, MI
48109-1107. E-mail: [email protected].}\\
Department of Statistics\\
University of Michigan\\
Ann Arbor, MI 48109\\
}
\maketitle
\begin{abstract}
In many engineering and scientific applications, prediction
variables are grouped, for example, in biological applications where
assayed genes or proteins can be grouped by biological roles or
biological pathways. Common statistical analysis methods such as
ANOVA, factor analysis, and functional modeling with basis sets also
exhibit natural variable groupings. Existing successful group
variable selection methods such as Antoniadis and Fan (2001), Yuan
and Lin (2006) and Zhao, Rocha and Yu (2009) have the limitation of
selecting variables in an ``all-in-all-out'' fashion, i.e., when one
variable in a group is selected, all other variables in the same
group are also selected. In many real problems, however, we may want
to keep the flexibility of selecting variables within a group, such
as in gene-set selection. In this paper, we develop a new group
variable selection method that not only removes unimportant groups
effectively, but also keeps the flexibility of selecting variables
within a group. We also show that the new method offers the
potential for achieving the theoretical ``oracle'' property as in
Fan and Li (2001) and Fan and Peng (2004).
\end{abstract}
\begin{quote} \small
{\bf Keywords}: Group selection; Lasso; Oracle property;
Regularization; Variable selection
\end{quote} \normalsize
\setcounter{page}{1}
\section{Introduction} \label{sec:intro}
Consider the usual regression situation: we have training data,
$(\V{x}_1,y_1)$, $\ldots$, $(\V{x}_i,y_i)$, $\ldots$,
$(\V{x}_n,y_n)$, where $\V{x}_i = (x_{i1},\ldots,x_{ip})$ are the
predictors and $y_i$ is the response. To model the response $y$ in
terms of the predictors $x_1,\ldots,x_p$, one may consider the
linear model:
\begin{equation} \label{intro:eq01}
y = \beta_0 + \beta_1 x_{1} + \ldots + \beta_p x_{p} + \varepsilon,
\end{equation}
where $ \varepsilon$ is the error term. In many important practical
problems, however, prediction variables are ``grouped.'' For example,
in ANOVA factor analysis, a factor may have several levels and can
be expressed via several dummy variables, then the dummy variables
corresponding to the same factor form a natural ``group.''
Similarly, in additive models, each original prediction variable may
be expanded into different order polynomials or a set of basis
functions, then these polynomials (or basis functions) corresponding
to the same original prediction variable form a natural ``group.''
Another example is in biological applications, where assayed genes
or proteins can be grouped by biological roles (or biological
pathways).
For the rest of the paper, we assume that the prediction variables
can be divided into $K$ groups and the $k$th group contains $p_k$
variables. Specifically, the linear model (\ref{intro:eq01}) is now
written as
\begin{eqnarray} \label{intro:eq02}
{y}_i &= &
\beta_0 + \sum_{k=1}^{K} \sum_{j=1}^{p_k} \beta_{kj} x_{i,kj} +
\varepsilon_i.
\end{eqnarray}
And we are interested in finding out which variables, especially
which ``groups,'' have an important effect on the response. For
example, $(x_{11},\ldots,x_{1p_1})$, $(x_{21},\ldots,x_{2p_2})$,
$\ldots$, $(x_{K1},\ldots,x_{Kp_K})$ may represent different
biological pathways, $y$ may represent a certain phenotype and we
are interested in deciphering which and how these biological
pathways ``work together'' to affect the phenotype.
There are two important challenges in this problem: prediction
accuracy and interpretation. We would like our model to accurately
predict on future data. Prediction accuracy can often be improved by
shrinking the regression coefficients. Shrinkage sacrifices some
bias to reduce the variance of the predicted value and hence may
improve the overall prediction accuracy. Interpretability is often
realized via variable selection. With a large number of prediction
variables,
we often would like to determine a smaller subset that exhibits the
strongest effects.
Variable selection has been studied extensively in the literature,
for example, see \citeasnoun{GeorgeMcCulloch93},
\citeasnoun{Breiman95}, \citeasnoun{Tibshirani96},
\citeasnoun{GeorgeFoster00}, \citeasnoun{FanLi01},
\citeasnoun{ZouHastie05},
\citeasnoun{LinZhang06} and \citeasnoun{WuEtAl07}.
In particular, lasso \cite{Tibshirani96}
has gained much attention in recent years. The lasso criterion
penalizes the $L_1$-norm of the regression coefficients to achieve a
sparse model:
\begin{equation} \label{eq:crit03}
\max_{\beta_0, \beta_{kj}} - \frac{1}{2} \sum_{i=1}^{n} \left( y_i
-\beta_0 - \sum_{k=1}^{K} \sum_{j=1}^{p_k}
\beta_{kj} x_{i,kj} \right)^2 - \lambda \sum_{k=1}^{K}
\sum_{j=1}^{p_k} |\beta_{kj}|,
\end{equation}
where $\lambda \geq 0$ is a tuning parameter. Note that by location
transformation, we can always assume that the predictors and the
response have mean 0, so we can ignore the intercept in equation
(\ref{eq:crit03}).
Due to the singularity at $\beta_{kj} = 0$, the $L_1$-norm penalty
can shrink some of the fitted coefficients to be {\it exact} zero
when making the tuning parameter sufficiently large. However, lasso
and other methods above are for the case when the candidate
variables can be treated individually or ``flatly.'' When variables
are grouped, ignoring the group structure and directly applying
lasso as in (\ref{eq:crit03}) may be sub-optimal. For example,
suppose the $k$th group is unimportant, then lasso tends to make
individual estimated coefficients in the $k$th group to be zero,
rather than the whole group to be zero, i.e., lasso tends to make
selection based on the strength of individual variables rather than
the strength of the group, often resulting in selecting more groups
than necessary.
\citeasnoun{AntoniadisFan01}, \citeasnoun{YuanLin06} and
\citeasnoun{ZhaoRochaYu06} have addressed the group variable
selection problem in the literature. \citeasnoun{AntoniadisFan01}
proposed to use a
blockwise additive penalty in the setting of wavelet approximations.
To increase the estimation precision, empirical wavelet
coefficients were thresholded or shrunken in blocks (or groups)
rather than individually.
\citeasnoun{YuanLin06} and \citeasnoun{ZhaoRochaYu06}
extended the lasso model (\ref{eq:crit03}) for group variable
selection.
\citeasnoun{YuanLin06} chose to penalize the $L_2$-norm of the
coefficients within each group, i.e., $\sum_{k=1}^K \|
\V{\beta}_{k} \|_2$, where
\begin{equation} \label{intro:eq05}
\| \V{\beta}_{k} \|_2 =
\sqrt{\beta_{k1}^2 + \ldots + \beta_{kp_k}^2}.
\end{equation}
Due to the singularity of $\| \V{\beta}_{k} \|_2$ at $\V{\beta}_k =
\V{0} $, appropriately tuning $\lambda$ can set the whole
coefficient vector $\V{\beta}_k = \V{0}$, hence the $k$th group is
removed from the fitted model. We note that in the setting of
wavelet analysis, this method reduces to
\citeasnoun{AntoniadisFan01}.
Instead of using the $L_2$-norm penalty, \citeasnoun{ZhaoRochaYu06}
suggested using the $L_{\infty}$-norm penalty, i.e.,
$\sum_{k=1}^K \| \V{\beta}_{k} \|_{\infty}$, where
\begin{equation} \label{intro:eq07}
\| \V{\beta}_{k} \|_{\infty} = \max ( {|\beta_{k1}|,
|\beta_{k2}|, \ldots, |\beta_{kp_k}| } ).
\end{equation}
Similar to the $L_2$-norm, the $L_{\infty}$-norm of $\V{\beta}_{k}$
is also singular when $\V{\beta}_k = \V{0} $; hence when $\lambda$
is appropriately tuned, the $L_{\infty}$-norm can also effectively
remove unimportant groups.
However, there are some possible limitations with these methods:
Both the $L_2$-norm penalty and the $L_{\infty}$-norm penalty select
variables in an ``all-in-all-out'' fashion, i.e., when one variable
in a group is selected, {\it all} other variables in the same group
are also selected. The reason is that both $ \| \V{\beta}_{k} \|_2$
and $\| \V{\beta}_{k} \|_{\infty}$ are singular only when the whole
vector $\V{\beta}_k = \V{0}$. Once a component of $\V{\beta}_k$ is
non-zero, the two norm functions are no longer singular. This can
also be heuristically understood as the following: for the
$L_2$-norm (\ref{intro:eq05}), it is the ridge penalty that is under
the square root; since the ridge penalty can not do variable
selection (as in ridge regression), once the $L_2$-norm is non-zero
(or the corresponding group is selected), all components will be
non-zero. For the $L_{\infty}$-norm (\ref{intro:eq07}), if the
``max($\cdot$)'' is non-zero, there is no increase in the penalty
for letting all the individual components move away from zero. Hence
if one variable in a group is selected, all other variables are also
automatically selected.
In many important real problems, however, we may want to keep the
flexibility of selecting variables {\it within} a group. For
example, in the gene-set selection problem, a biological pathway may
be related to a certain biological process, but it does not
necessarily mean all the genes in the pathway are all related to the
biological process. We may want to not only remove unimportant
pathways effectively, but also identify important genes within
important pathways.
For the $L_{\infty}$-norm penalty, another possible limitation is
that the estimated coefficients within a group tend to have the same
magnitude, i.e. $|\beta_{k1}| = |\beta_{k2}| = \ldots =
|\beta_{kp_k}|$; and this may cause some serious bias, which
jeopardizes the prediction accuracy.
In this paper, we propose an extension of lasso for group variable
selection, which we call hierarchical lasso (HLasso). Our method not
only removes unimportant groups effectively, but also keeps the
flexibility of selecting variables within a group. Furthermore,
asymptotic studies motivate us to improve our model and show that
when the tuning parameter is appropriately chosen, the improved
model has the {\it oracle} property \cite{FanLi01,FanPeng04}, i.e., it
performs as well as if the correct underlying model were given in
advance. Such a theoretical property has not been previously studied
for group variable selection at both the group level and the within
group level.
The rest of the paper is organized as follows. In Section
\ref{sec:model}, we introduce our new method: the hierarchical
lasso. We propose an algorithm to compute the solution for the
hierarchical lasso in Section \ref{sec:alg}. In Sections
\ref{sec:theory} and \ref{sec:improve}, we study the asymptotic
behavior of the hierarchical lasso and propose an improvement for
the hierarchical lasso.
Numerical results are in
Sections \ref{sec:resultsimu} and \ref{sec:resultreal}, and we
conclude the paper with Section \ref{sec:summary}.
\section{Hierarchical Lasso}
\label{sec:model}
In this section, we extend the lasso method for group variable
selection so that we can effectively remove unimportant variables at
both the group level and the within group level.
We reparameterize $\beta_{kj}$ as
\begin{equation} \label{intro:eq08}
\beta_{kj} = d_k \alpha_{kj} , ~~~ k = 1,\ldots,K; ~j =
1,\ldots,p_k,
\end{equation}
where $d_k \ge 0 $ (for identifiability reasons). This decomposition
reflects the information that $\beta_{kj}, j = 1, \ldots, p_k$, all
belong to the $k$h group, by treating each $\beta_{kj}$
hierarchically. $d_k$ is at the first level of the hierarchy,
controlling $\beta_{kj}, j = 1, \ldots, p_k$, as a group;
$\alpha_{kj}$'s are at the second level of the hierarchy, reflecting
differences within the $k$th group.
For the purpose of variable selection, we consider the following
penalized least squares criterion:
\begin{eqnarray}\label{model:eq01}
\max_{d_k,\alpha_{kj} } && - \frac{1}{2} \sum_{i=1}^{n} \left(
y_{i} - \sum_{k=1}^K d_k\sum_{j=1}^{p_k} \alpha_{kj} x_{i,kj}
\right)^2 \nonumber \\
&& - \lambda_1 \cdot \sum_{k=1}^K d_k
- \lambda_2 \cdot \sum_{k=1}^K \sum_{j=1}^{p_k} |\alpha_{kj}| \\
\nonumber
\textrm{subject to} && d_k\ge 0, ~k=1, \ldots, K,
\end{eqnarray}
where $\lambda_1 \ge 0 $ and $\lambda_2 \ge 0$ are tuning
parameters. $\lambda_1$ controls the estimates at the group level,
and it can effectively remove unimportant groups: if $d_k$ is
shrunken to zero, all $\beta_{kj}$ in the $k$th group will be equal
to zero. $\lambda_2$ controls the estimates at the variable-specific
level: if $d_k$ is not equal to zero, some of the $\alpha_{kj}$
hence some of the $\beta_{kj},j=1,\ldots,p_k$, still have the
possibility of being zero; in this sense, the hierarchical penalty
keeps the flexibility of the $L_1$-norm penalty.
One may complain that such a hierarchical penalty may be more
complicated to tune in practice, however, it turns out that the two
tuning parameters $\lambda_1$ and $\lambda_2$ in (\ref{model:eq01})
can be simplified into one. Specifically, let $\lambda = \lambda_1
\cdot \lambda_2$, we can show that (\ref{model:eq01}) is equivalent
to
\begin{eqnarray} \label{model:eq02}
\max_{d_k,\alpha_{kj} } && - \frac{1}{2} \sum_{i=1}^{n} \left(
y_{i} - \sum_{k=1}^K d_k\sum_{j=1}^{p_k} \alpha_{kj}
x_{i,kj} \right)^2
- \sum_{k=1}^K d_k - \lambda \sum_{k=1}^K \sum_{j=1}^{p_k} |\alpha_{kj}|
\\ \nonumber
\textrm{subject to} && d_k\ge 0, k=1, \ldots, K.
\end{eqnarray}
\begin{lemma} \label{lemma0}
Let ($\hat{\V{d}}^{\ast}, \hat{\V{\alpha}}^{\ast}$) be a local maximizer of (\ref{model:eq01}),
then there exists a local maximizer ($\hat{\V{d}}^{\star},
\hat{\V{\alpha}}^{\star}$) of (\ref{model:eq02}) such that
$ \hat{d}_k^{\ast} \hat{\alpha}_{kj}^{\ast} = \hat{d}_k^\star
\hat{\alpha}_{kj}^\star. $
Similarly, if ($\hat{\V{d}}^{\star}, \hat{\V{\alpha}}^{\star}$)
is a local maximizer of (\ref{model:eq02}),
there exists a local maximizer ($\hat{\V{d}}^{\ast},
\hat{\V{\alpha}}^{\ast}$) of (\ref{model:eq01})
such that
$ \hat{d}_k^{\ast} \hat{\alpha}_{kj}^{\ast} = \hat{d}_k^\star
\hat{\alpha}_{kj}^\star. $
\end{lemma}
The proof is in the Appendix. This lemma indicates that the final
fitted models from (\ref{model:eq01}) and (\ref{model:eq02}) are the
same, although they may provide different $d_k$ and $\alpha_{kj}$.
This also implies that in practice, we do not need to tune
$\lambda_1$ and $\lambda_2$ separately; we only need to tune one
parameter $\lambda = \lambda_1\cdot\lambda_2$ as in
(\ref{model:eq02}).
\section{Algorithm}
\label{sec:alg}
To estimate the
$d_k$ and $\alpha_{kj}$ in (\ref{model:eq02}), we can use an
iterative approach, i.e., we first fix $d_k$ and estimate
$\alpha_{kj}$, then we fix $\alpha_{kj}$ and estimate $d_k$, and we
iterate between these two steps until the solution converges. Since
at each step, the value of the objective function (\ref{model:eq02})
decreases, the solution is guaranteed to converge.
When $d_k$ is fixed, (\ref{model:eq02}) becomes a lasso problem,
hence we can use either the LAR/LASSO algorithm \cite{Efron04} or a
quadratic programming package to efficiently solve for
$\alpha_{kj}$. When $\alpha_{kj}$ is fixed, (\ref{model:eq02})
becomes a non-negative garrote problem. Again, we can use either an
efficient solution path algorithm or a quadratic programming package
to solve for $d_k$. In summary, the algorithm proceeds as follows:
\begin{itemize}
\item[1.] (Standardization) Center $\V{y}$. Center and normalize $\V{x}_{kj}$.
\item[2.] (Initialization) Initialize $d_k^{(0)}$ and
$\alpha_{kj}^{(0)}$ with some plausible values.
For example, we can set $d_k^{(0)}=1$ and use the least squares
estimates or the simple
regression estimates by regressing the response $\V{y}$ on each
of the $\V{x}_{kj}$ for
$\alpha_{kj}^{(0)}$. Let $ \beta_{kj}^{(0)} = d_{k}^{(0)}
\alpha_{kj}^{(0)}$ and $m$ = 1.
\item[3.] (Update $\alpha_{kj}$) Let
\[ \tilde{x}_{i,kj} = d_k^{(m-1)} x_{i,kj}, ~~ k = 1,\ldots,K;
~j = 1,\ldots, p_k, \]
then
\[ \alpha_{kj}^{(m)} = \arg \max_{\alpha_{kj}}
- \frac{1}{2} \sum_{i=1}^{n} \left( y_{i} - \sum_{k=1}^K
\sum_{j=1}^{p_k} \alpha_{kj} \tilde{x}_{i,kj} \right)^2
- \lambda \sum_{k=1}^K \sum_{j=1}^{p_k} |\alpha_{kj}|. \]
\item[4.] (Update $d_{k}$) Let
\[ \tilde{x}_{i,k} = \sum_{j=1}^{p_k} \alpha_{kj}^{(m)}
x_{i,kj}, ~~ k = 1,\ldots,K, \]
then
\[ d_{k}^{(m)} = \arg \max_{d_{k} \ge 0}
- \frac{1}{2} \sum_{i=1}^{n} \left( y_{i} - \sum_{k=1}^K
d_{k} \tilde{x}_{i,k} \right)^2
- \sum_{k=1}^K d_{k}. \]
\item[5.] (Update $\beta_{kj}$) Let
\[ \beta_{kj}^{(m)} = d_{k}^{(m)} \alpha_{kj}^{(m)}. \]
\item[6.] If $\| \beta_{kj}^{(m)} - \beta_{kj}^{(m-1)} \|$ is
small enough, stop the algorithm. Otherwise, let $m \leftarrow m+1$ and
go back to Step 3.
\end{itemize}
\subsection{Orthogonal Case}
To gain more insight into the hierarchical penalty, we have also
studied the algorithm in the orthogonal design case. This can be
useful, for example, in the wavelet setting, where each $\V{x}_{kj}$
corresponds to a wavelet basis function, different $k$ may
correspond to different ``frequency'' scales, and different $j$ with
the same $k$ correspond to different ``time'' locations.
Specifically, suppose $\T{\V{x}}_{kj} \V{x}_{kj} = 1$ and
$\T{\V{x}}_{kj} \V{x}_{k^{\prime}j^{\prime}} = 0$ if $k\neq
k^{\prime}$ or $j\neq j^{\prime}$, then Step 3 and Step 4 in the
above algorithm have closed form solutions.
Let $\hat{\beta}_{kj}^{\mbox{ols}} = \T{\V{x}}_{kj}\V{y}$ be the ordinary
least squares solution when $\V{x}_{kj}$ are orthonormal to each
other.
\begin{itemize}
\item[Step 3.] When $d_k$ is fixed,
\begin{equation} \label{orth:eq01}
\alpha_{kj}^{(m)} = \mathbb{I}(d_k^{(m-1)}>0) \cdot
\textrm{sgn}( \hat{\beta}_{kj}^{\mbox{ols}} ) \cdot \left( \frac{|
\hat{\beta}_{kj}^{\mbox{ols}}|}{d_k^{(m-1)} } -
\frac{\lambda}{(d_k^{(m-1)})^2} \right)_+ .
\end{equation}
\item[Step 4.] When $\alpha_{kj}$ is fixed,
\begin{equation} \label{orth:eq02}
d_k^{(m)} = \mathbb{I}(\exists j, \alpha_{kj}^{(m)} \neq 0) \cdot
\left( \sum_{j=1}^{p_k}
\frac{(\alpha_{kj}^{(m)})^2}{\sum_{j=1}^{p_k} (\alpha_{kj}^{(m)})^2}
\frac{\hat{\beta}_{kj}^{\mbox{ols}}}{\alpha_{kj}^{(m)}} -
\frac{1}{\sum_{j=1}^{p_k} (\alpha_{kj}^{(m)})^2} \right)_+ .
\end{equation}
\end{itemize}
Equations (\ref{orth:eq01}) and (\ref{orth:eq02}) show that both
$d_k^{(m)}$ and $\alpha_{kj}^{(m)}$ are soft-thresholding estimates.
Here we provide some intuitive explanation.
We first look at $\alpha_{kj}^{(m)}$ in equation (\ref{orth:eq01}).
If $d_{k}^{(m-1)} = 0$, it is natural to estimate all $\alpha_{kj}$ to
be zero because of the penalty on $\alpha_{kj}$. If $d_{k}^{(m-1)} >
0$, then from our reparametrization, we have $\alpha_{kj} =
\beta_{kj} / d_k^{(m-1)} $, $j = 1, \ldots,p_k$. Plugging in
$\hat{\beta}_{kj}^{\mbox{ols}}$ for $\beta_{kj}$, we obtain
$\tilde{\alpha}_{kj} = \hat{\beta}_{kj}^{\mbox{ols}} / d_k^{(m-1)}$.
Equation (\ref{orth:eq01}) shrinks $\tilde{\alpha}_{kj}$, and the
amount of shrinkage is inversely proportional to $(d_k^{(m-1)})^2$.
So when $ d_k^{(m-1)} $ is large, which indicates the $k$th group is
important, the amount of shrinkage is small, while when $
d_k^{(m-1)} $ is small, which indicates the $k$th group is less
important, the amount of shrinkage is large.
Now considering $d_k^{(m)}$ in equation (\ref{orth:eq02}). If all
$\alpha_{kj}^{(m)}$ are zero, it is natural to estimate $d_k^{(m)}$
also to be zero because of the penalty on $d_k$. If not all
$\alpha_{kj}^{(m)}$ are 0, say $\alpha_{kj_1}^{(m)}, \ldots,
\alpha_{kj_r}^{(m)} $ are not zero, then we have $d_k = \beta_{kj_s}
/\alpha_{kj_s}^{(m)}, 1\le s \le r$. Again, plugging in
$\hat{\beta}_{kj_s}^{\mbox{ols}}$ for $\beta_{kj_s}$, we obtain $r$
estimates for $d_k$: $\tilde{d}_k = \hat{\beta}_{kj_s}^{\mbox{ols}}
/\alpha_{kj_s}^{(m)}, 1\le s \le r$. A natural estimate for $d_k$ is
then a weighted average of the $\tilde{d}_k$, and equation
(\ref{orth:eq02}) provides such a (shrunken) average, with weights
proportional to $(\alpha_{kj}^{(m)})^2$.
\section{Asymptotic Theory}
\label{sec:theory}
In this section, we explore the asymptotic behavior of the
hierarchical lasso method.
The hierarchical lasso criterion (\ref{model:eq02}) uses $d_k$ and
$\alpha_{kj}$. We first show that it can also be written in an
equivalent form using the original regression coefficients
$\beta_{kj}$.
\begin{theorem} \label{thm1}
If ($\hat{\V{d}}, \hat{\V{\alpha}}$) is a
local maximizer of (\ref{model:eq02}), then $\hat{\V{\beta}}$, where
$\hat{\beta}_{kj} = \hat{d}_k \hat{\alpha}_{kj} $, is a local
maximizer of
\begin{eqnarray} \label{asym:eq01}
\max_{\beta_{kj} } && - \frac{1}{2}
\sum_{i=1}^{n} \left( y_{i} - \sum_{k=1}^K \sum_{j=1}^{p_k}
x_{i,kj}\beta_{kj} \right)^2 \nonumber \\
& & - 2\sqrt{\lambda} \cdot \sum_{k=1}^K
\sqrt{ |\beta_{k1}|
+ |\beta_{k2}| + \ldots + |\beta_{kp_k}|}.
\end{eqnarray}
On the other hand, if $\hat{\V{\beta}}$ is a local maximizer of
(\ref{asym:eq01}), then
we define ($\hat{\V{d}}, \hat{\V{\alpha}}$), where
$\hat{d}_k = 0, \hat{\V{\alpha}}_k =0 $ if $\|
\hat{\V{\beta}}_{k} \|_1 = 0$,
and $\hat{d}_k = \sqrt{ \lambda \| \hat{\V{\beta}}_{k}
\|_1 }, \hat{\V{\alpha}}_k = \frac{ \hat{\V{\beta}}_k }{
\sqrt{\lambda \| \hat{\V{\beta}}_{k} \|_1 } }$
if $\| \hat{\V{\beta}}_{k} \|_1 \ne 0$.
Then the so-defined ($\hat{\V{d}}, \hat{\V{\alpha}}$) is a local
maximizer of (\ref{model:eq02}).
\end{theorem}
Note that the penalty term in (\ref{asym:eq01}) is similar to the
$L_2$-norm penalty (\ref{intro:eq05}), except that under each
square root, we now penalize the $L_1$-norm of $\V{\beta}_k$,
rather than the sum of squares. However, unlike the $L_2$-norm,
which is singular
only at the point $\V{\beta}_k = \V{0}$, (i.e., the whole vector is
equal to $\V{0}$), the square root of the $L_1$-norm is singular at
$\beta_{kj} = 0$ no matter what are the values of other
$\beta_{kj}$'s. This explains, from a different perspective, why the
hierarchical lasso can remove not only groups, but also variables
within a group even when the group is selected. Equation
(\ref{asym:eq01}) also implies that the hierarchical lasso belongs
to the ``CAP'' family in \citeasnoun{ZhaoRochaYu06}.
{ We study the asymptotic properties allowing the total number of
variables $P_n$, as well as the number of groups $K_n$ and the
number of
variables within each group $p_{nk}$, to go to $\infty$, where
$P_n = \sum_{k=1}^{K_n} p_{nk}$.
Note that we add a subscript ``$n$'' to $K$ and $p_k$ to
denote that these quantities can change with $n$.
Accordingly, $\V{\beta}$, $y_i$ and $x_{i,kj}$ are also changed to
$\V{\beta}_n$, $y_{ni}$ and $x_{ni,kj}$.} We write $2\sqrt{\lambda}$
in (\ref{asym:eq01}) as $n\lambda_n$, and the criterion
(\ref{asym:eq01}) is re-written as
\begin{eqnarray} \label{asym:eq01n}
\max_{\beta_{n,kj} } && - \frac{1}{2}
\sum_{i=1}^{n} \left( y_{ni} - \sum_{k=1}^{K_n} \sum_{j=1}^{p_{nk}}
x_{ni,kj}\beta_{n,kj} \right)^2 \nonumber \\
& & - n{\lambda_n} \cdot \sum_{k=1}^{K_n}
\sqrt{ |\beta_{n,k1}|
+ |\beta_{n,k2}| + \ldots + |\beta_{n,kp_{nk}}|}.
\end{eqnarray}
Our asymptotic analysis in this section is based on the criterion
(\ref{asym:eq01n}).
Let $\V{\beta}_n^0 = \T{(\V{\beta}_{{\cal
A}_n}^0, \V{\beta}_{{\cal B}_n}^0, \V{\beta}_{{\cal C}_n}^0 )}$ be
the underlying true parameters, where
\begin{eqnarray} \label{eq:crit13a}
{\cal{A}}_n &=& \{ (k,j): \beta_{n,kj}^0 \neq 0 \}, \nonumber \\
{\cal{B}}_n &=& \{ (k,j): \beta_{n,kj}^0 = 0, \V{\beta}_{nk}^0 \neq
0 \}, \nonumber \\
{\cal{C}}_n &=& \{ (k,j): \V{\beta}_{nk}^0 = 0 \}, \nonumber \\
{\cal D}_n &=& {\cal B}_n \cup {\cal C}_n.
\end{eqnarray}
Note that ${\cal A}_n$ contains the indices of coefficients which
are truly non-zero, ${\cal C}_n$ contains the indices where the
whole ``groups'' are truly zero, and ${\cal B}_n$ contains the
indices of zero coefficients, but they belong to some non-zero
groups. So ${\cal A}_n$, ${\cal B}_n$ and ${\cal C}_n$ are disjoint
and they partition all the indices. We have the following theorem.
\begin{theorem} \label{thm2}
If $ \sqrt{n} \lambda_n = O(1) $, then there exists a
root-($n/P_n$) consistent local maximizer $\hat{\V{\beta}}_n =
\T{(\hat{\V{\beta}}_{{\cal A}_n}, \hat{\V{\beta}}_{{\cal B}_n},
\hat{\V{\beta}}_{{\cal C}_n})}$ of (\ref{asym:eq01n}), and if
also $P_nn^{-3/4}/{\lambda_n} \rightarrow 0$ as $n \rightarrow \infty$,
then $\F{Pr}(\hat{\V{\beta}}_{{\cal C}_n} = 0) \rightarrow 1$.
\end{theorem}
Theorem \ref{thm2} implies that the hierarchical lasso method can
effectively remove unimportant {\it groups}. For the above
root-($n/P_n$) consistent estimate, however, if ${{\cal B}_n} \neq
\emptyset$ (empty set), then $\F{Pr}(\hat{\V{\beta}}_{{\cal B}_n} =
0) \rightarrow 1$ is not always true. This means that although the
hierarchical lasso method can effectively remove {\it all}
unimportant {\it groups} and {\it some} of the unimportant {\it
variables} within important groups, it cannot effectively remove
{\it all} unimportant {\it variables} within important groups.
In the next section, we improve the hierarchical lasso method to
tackle this limitation.
\section{Adaptive Hierarchical Lasso}
\label{sec:improve}
To improve the hierarchical lasso method, we apply the adaptive idea
which has been used in \citeasnoun{Breiman95},
\citeasnoun{WangLiTsai06},
\citeasnoun{ZhangLu06},
and \citeasnoun{Zou06}, i.e., to penalize different coefficients
differently. Specifically, we consider
\begin{eqnarray} \label{eq:crit13b}
\max_{\beta_{n,kj}} && - \frac{1}{2} \sum_{i=1}^{n} \left( y_{ni} -
\sum_{k=1}^{K_n} \sum_{j=1}^{p_k} x_{ni,kj}\beta_{n,kj} \right)^2
\nonumber \\
&& - n\lambda_n \cdot \sum_{k=1}^{K_n} \sqrt{ w_{n,k1} |\beta_{n,k1}| + w_{n,k2}
|\beta_{n,k2}| + \ldots + w_{n,kp_k} |\beta_{n,kp_{nk}}| },
\end{eqnarray}
where $w_{n,kj}$ are pre-specified weights. The intuition is that if
the effect of a variable is strong, we would like the corresponding
weight to be small, hence the corresponding coefficient is lightly
penalized. If the effect of a variable is not strong, we would like
the corresponding weight to be large, hence the corresponding
coefficient is heavily penalized. In practice, we may consider using
the ordinary least squares estimates or the ridge regression
estimates to help us compute the weights, for example,
\begin{equation} \label{eq:wts}
w_{n,kj} = \frac{1}{|\hat{\beta}_{n,kj}^{\textrm{ols}}|^\gamma}
~~~\textrm{or}~~~ w_{n,kj} =
\frac{1}{|\hat{\beta}_{n,kj}^{\textrm{ridge}}|^\gamma},
\end{equation}
where $\gamma$ is a positive constant.
\subsection{Oracle Property}
\label{sec:theory2}
\subsubsection*{Problem Setup}
Since the theoretical results we develop for (\ref{eq:crit13b}) are
not restricted to the squared error loss, for the rest of the
section, we consider the generalized linear model. For generalized
linear models, statistical inferences are based on underlying
likelihood functions. We assume that the data
$\V{V}_{ni}=(\V{X}_{ni}, Y_{ni}), ~i=1, \ldots, n$ are independent
and identically distributed for every $n$. Conditioning on
$\V{X}_{ni} = \V{x}_{ni}$, $Y_{ni}$ has a density
$f_n(g_n(\T{\V{x}}_{ni} \V{\beta}_n), Y_{ni})$, where $g_n(\cdot)$
is a known link function. We maximize the penalized log-likelihood
\begin{eqnarray} \label{eq:crit131c}
\max_{\beta_{n,kj}}~ Q_n(\V{\beta}_n) &=& L_n(\V{\beta}_n) - J_n(\V{\beta}_n)
\nonumber \\
&=& \sum_{i=1}^{n} \ell_n(g_n(\T{\V{x}}_{ni} \V{\beta}_n), y_{ni}) - n
\sum_{k=1}^K p_{\lambda_n,\V{w}_n}(\V{\beta}_{nk} ),
\end{eqnarray}
where $\ell_n(\cdot,\cdot) = \log f_n(\cdot, \cdot)$ denotes the
conditional log-likelihood of $Y$, and
\[ p_{\lambda_n,\V{w}_n}(\V{\beta}_{nk}) = \lambda_n \sqrt{
w_{n,k1} |\beta_{n,k1}| + \ldots + w_{n,kp_k} |\beta_{n,kp_{nk}}| }. \]
Note that under the normal distribution, $\ell_n(g_n(\T{\V{x}}_{ni}
\V{\beta}_n), y_{ni}) = - \frac{(y_{ni} - \T{\V{x}}_{ni}
\V{\beta}_n)^2}{2C_1} + C_2$, hence (\ref{eq:crit131c}) reduces to
(\ref{eq:crit13b}).
The asymptotic properties of (\ref{eq:crit131c}) are described in
the following theorems, and the proofs are in the Appendix. We note
that the proofs follow the spirit of \citeasnoun{FanLi01} and
\citeasnoun{FanPeng04}, but due to the grouping structure and the
adaptive weights, they are non-trivial extensions of
\citeasnoun{FanLi01} and \citeasnoun{FanPeng04}.
To control the adaptive weights, we define:
\begin{eqnarray*}
a_n &=& \max \{w_{n,kj}: \beta_{n,kj}^0 \neq 0 \}, \\
b_n &=& \min \{w_{n,kj}: \beta_{n,kj}^0 = 0 \}.
\end{eqnarray*}
We assume that
$$ 0 < c_1 < \min \{ \beta_{n,kj}^0 : \beta_{n,kj}^0 \neq 0 \} <
\max \{ \beta_{n,kj}^0 : \beta_{n,kj}^0 \neq 0 \} < c_2 <
\infty. $$
Then we have the following results.
\begin{theorem} \label{thm_b1}
For every $n$, the observations $\{ \V{V}_{ni}, i = 1, 2,
\ldots, n \}$ are independent and identically distributed, each
with a density $f_n(\V{V}_{n1}, \V{\beta}_n)$
that satisfies conditions (A1)-(A3) in the Appendix. If $\frac{
{P^4_n} } {n} \rightarrow 0$ and $P_n^2 \lambda_n \sqrt{a_n} =
o_p(1)$, then there exists a local maximizer $\hat{\V{\beta}}_n$ of
$Q_n(\V{\beta}_n)$ such that $\| \hat{\V{\beta}}_n -\V{\beta}_n^0
\|=O_p( \sqrt {P_n} ( n^{-1/2}+ \lambda_n \sqrt{a_n}))$.
\end{theorem}
Hence by choosing $\lambda_n \sqrt{a_n} = O_p(n^{-1/2} ) $, there
exists a root-$(n/P_n)$ consistent penalized likelihood estimate.
\begin{theorem} \label{thm_b2}
For every $n$, the observations $\{ \V{V}_{ni}, i = 1, 2, \ldots, n \}$
are independent and identically
distributed, each with a density $f_n(\V{V}_{n1}, \V{\beta}_n)$
that satisfies conditions (A1)-(A3) in the Appendix. If $\frac{
{P^4_n} } {n} \rightarrow 0$, $\lambda_n \sqrt{a_n } = O_p(n^{-1/2}
)$ and $ \frac{ {P_n^{2}} }{\lambda_n^2 {b_n}} = o_p(n )$, then
there exists a root-$(n/P_n)$ consistent local maximizer
$\hat{\V{\beta}}_n$ such that:
\begin{enumerate}
\item[(a)] Sparsity: $\F{Pr}(\hat{\V{\beta}}_{n,{{\cal{D}}_n }} = 0)
\rightarrow 1 $, where ${\cal D}_n = {\cal B}_n \cup {\cal C}_n$.
\item[(b)] Asymptotic normality: If $ \lambda_n \sqrt{a_n } =
o_p( {(nP_n)}^{-1/2} )$ and $ \frac{P_n^5}{n}
\rightarrow 0$ as $n\rightarrow \infty$, then we also have:
\begin{equation*}
\sqrt{n} \M{A}_n \M{I}_n^{1/2} ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
\rightarrow {\cal{N}} (\V{0}, \M{G}),
\end{equation*}
\end{enumerate}
where $\M{A}_n$ is a $q \times |{{\cal{A}}_n }| $ matrix such that
$\M{A}_n \T{\M{A}}_n \rightarrow \M{G}$ and $\M{G}$ is a $q\times
q$ nonnegative symmetric matrix. $\M{I}_n(
{\V{\beta}}_{n,{{\cal{A}}_n }}^0 )$ is the Fisher
information matrix knowing $\V{\beta}_{ {\cal{D}}_n }^0 = 0$.
\end{theorem}
The above requirements $\lambda_n \sqrt{a_n } = o_p( {(nP_n)}^{-1/2}
)$ and $\frac{ {P_n^{2}} }{\lambda_n^2 {b_n}} = o_p(n)$
as $n\rightarrow \infty$ can be satisfied by selecting $\lambda_n$
and ${w_{n,kj} }$ appropriately. For example, we may let $\lambda_n
= \frac{(nP_n)^{-1/2}}{\mathrm{log}n} $ and $ {w_{n,kj} } =
\frac{1}{|\hat{\beta}^0_{n,kj}|^2}$, where $\hat{\beta}^0_{n,kj}$ is
the un-penalized likelihood estimate of ${\beta}_{n,kj}^0$, which is
root-($n/P_n$) consistent. Then we have $a_n = O_p(1) $ and
$\frac{1}{b_n} = O_p(\frac{P_n}{n})$. Hence $\lambda_n \sqrt{a_n } =
o_p({(nP_n)}^{-1/2})$ and $ \frac{
{P_n^{2}} }{\lambda_n^2 {b_n}} = o_p(n )$ are satisfied when $ \frac{P_n^5}{n}
\rightarrow 0$.
\subsection{Likelihood Ratio Test}
Similarly as in \citeasnoun{FanPeng04}, we develop a likelihood
ratio test for testing linear hypotheses:
$$ H_0: \M{A}_n {\V{\beta}}_{n,{{\cal{A}}_n }}^0 = 0
\mathrm{~~~vs.~~~}
H_1: \M{A}_n {\V{\beta}}_{n,{{\cal{A}}_n }}^0 \ne 0 , $$
where $\M{A}_n$ is a $ q \times |{{\cal{A}}_n }| $ matrix and $
\M{A}_n \T{\M{A}}_n \rightarrow \M{I}_q$ for a fixed $q$. This
problem includes testing simultaneously the significance of
several covariate variables.
We introduce a natural likelihood ratio test statistic, i.e.
\[ T_n = 2 \left\{ \sup_{\Omega_n} Q_n(\V{\beta}_n|\V{V}) -
\sup_{\Omega_n, \M{A}_n {\V{\beta}}_{n,{{\cal{A}}_n }} = 0 }
Q_n(\V{\beta}_n|\V{V}) \right\} , \]
where $\Omega_n$ is the parameter space for $\V{\beta}_n$. Then
we can obtain the following theorem regarding the asymptotic null
distribution of the test statistic.
\begin{theorem} \label{thm_b3}
When conditions in $(b)$ of Theorem \ref{thm_b2} are satisfied,
under $H_0$ we have
\[ T_n \rightarrow \chi_q^2, ~~~\mathrm{as}~ n \rightarrow
\infty. \]
\end{theorem}
\section{Simulation Study} \label{sec:resultsimu}
In this section, we use simulations to demonstrate the hierarchical
lasso (HLasso) method, and compare the results with those of some
existing methods.
Specifically, we first compare hierarchical lasso with some other
group variable selection methods, i.e., the $L_2$-norm group lasso
(\ref{intro:eq05}) and the $L_\infty$-norm group lasso
(\ref{intro:eq07}). Then we compare the adaptive hierarchical
lasso with some other ``oracle'' (but non-group variable
selection) methods, i.e., the SCAD and the adaptive lasso.
We extended the simulations in \citeasnoun{YuanLin06}.
We considered a model which had both categorical
and continuous prediction variables. We first generated seventeen independent
standard normal variables, $Z_1, \ldots, Z_{16}$ and W. The
covariates were then defined as $X_j=(Z_j+W)/\sqrt{2}$. Then the
last eight covariates $X_{9}, \ldots, X_{16}$ were discretized to
0, 1, 2, and 3 by $\Phi^{-1}(1/4)$, $\Phi^{-1}(1/2)$ and
$\Phi^{-1}(3/4)$. Each of $X_1,
\ldots, X_8$ was expanded through a fourth-order polynomial,
and only the main effects of $X_9, \ldots, X_{16}$ were
considered. This gave us a total of eight continuous groups with
four variables in each group and eight categorical groups with three
variables in each group. We considered two cases.
\begin{description}
\item[Case 1.] ``All-in-all-out''
\begin{eqnarray*}
Y &=& \left[ X_3 + 0.5X^2_3 + 0.1X^3_3 + 0.1X^4_3 \right] +
\left[ X_6 - 0.5X^2_6 + 0.15X^3_6
+ 0.1X^4_6 \right] \\
&& + \left[ \mathbb{I}(X_{9}=0) + \mathbb{I}(X_{9}=1) +
\mathbb{I}(X_{9}=2) \right] + \varepsilon.
\end{eqnarray*}
\item[Case 2.] ``Not all-in-all-out''
\begin{equation*}
Y = \left( X_3 + X^2_3 \right) + \left( 2X_6 - 1.5X^2_6
\right) + \left[ \mathbb{I}(X_{9}=0) +
2~\mathbb{I}(X_{9}=1) \right] + \varepsilon.
\end{equation*}
\end{description}
For all the simulations above, the error term $\varepsilon$ follows
a normal distribution $\F{N}(0, \sigma^2)$, where $\sigma^2$ was set
such that each of the signal to noise ratios, $\F{Var}(\T{\V{X}}
\V{\beta})/\F{Var}(\epsilon)$, was equal to 3. We generated $n=400$
training observations from each of the above models, along with 200
validation observations and 10,000 test observations. The validation
set was used to select the tuning parameters $\lambda$'s that
minimized the validation error. Using these selected $\lambda$'s, we
calculated the mean squared error (MSE) with the test set.
We repeated this 200 times and computed the average MSEs and their
corresponding standard errors. We also recorded how frequently the
important variables were selected and how frequently the unimportant
variables were removed. The results are summarized in Table
\ref{tab:3}.
As we can see, all shrinkage methods perform much better than OLS;
this illustrates that some regularization is crucial for prediction
accuracy. In terms of prediction accuracy, we can also see that when
variables in a group follow the ``all-in-all-out'' pattern, the
$L_2$-norm (group lasso) method performs slightly better than the
hierarchical lasso method (Case 1 of Table \ref{tab:3}). When
variables in a group do not follow the ``all-in-all-out'' pattern,
however, the hierarchical lasso method performs slightly better than
the $L_2$-norm method (Case 2 of Table \ref{tab:3}). For variable
selection, we can see that in terms of identifying important
variables, the four shrinkage methods, the lasso, the
$L_\infty$-norm, the $L_2$-norm, and the hierarchical lasso all
perform similarly (``Non-zero Var.'' of Table \ref{tab:3}). However,
the $L_2$-norm method and the hierarchical lasso method are more
effective at removing unimportant variables than lasso and the
$L_\infty$-norm method (``Zero Var.'' of Table \ref{tab:3}).
\begin{table}[tbh]
\caption{Comparison of several group variable selection methods,
including the $L_2$-norm group lasso, the $L_\infty$-norm group
lasso and the hierarchical lasso. The OLS and the regular
lasso are used as benchmarks.
The upper part is for
Case 1, and the lower part is for Case 2. ``MSE'' is the mean
squared error on the test set. ``Zero Var.'' is the percentage
of correctly removed unimportant variables. ``Non-zero Var.''
is the percentage of correctly identified important variables.
All the numbers outside parentheses are means over 200
repetitions, and the numbers in the parentheses are the
corresponding standard errors.}
\label{tab:3}
\begin{center}
\begin{tabular}{l|c|c|c|c|c}
\hline \hline
\multicolumn{6}{l}{Case 1: ``All-in-all-out''} \\
\hline & OLS & Lasso & $L_{\infty}$ & $L_2$ &HLasso
\\
\hline
MSE & 0.92 (0.018) & 0.47 (0.011) & 0.31 (0.008) & 0.18 (0.009) & 0.24 (0.008) \\
\hline
Zero Var. & - & 57\% (1.6\%) & 29\% (1.4\%) & 96\% (0.8\%) & 94\% (0.7\%) \\
\hline
Non-Zero Var. & - & 92\% (0.6\%) & 100\% (0\%) & 100\% (0\%) & 98\% (0.3\%) \\
\hline \hline
\multicolumn{6}{l}{Case 2: ``Not all-in-all-out''} \\
\hline
& OLS & Lasso & $L_{\infty}$ & $L_2$ &HLasso \\
\hline
MSE& 0.91 (0.018) & 0.26 (0.008) & 0.46 (0.012) & 0.21 (0.01) & 0.15 (0.006) \\
\hline
Zero Var.& - & 70\% (1\%) & 17\% (1.2\%) & 87\% (0.8\%) & 91\% (0.5\%) \\
\hline
Non-zero Var.& - & 99\% (0.3\%) & 100\% (0\%) & 100\% (0.2\%) & 100\% (0.1\%) \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
In the above analysis, we used the criterion (\ref{model:eq02}) or
(\ref{asym:eq01}) for the hierarchical lasso, i.e., we did not use
the adaptive weights $w_{kj}$ to penalize different coefficients
differently. To assess the improved version of the hierarchical
lasso, i.e. criterion (\ref{eq:crit13b}), we also considered using
adaptive weights. Specifically, we applied the OLS weights in
(\ref{eq:wts}) to (\ref{eq:crit13b}) with $\gamma=1$. We compared
the results with those of SCAD and the adaptive lasso, which also
enjoy the oracle property. However, we note that SCAD and the
adaptive lasso do not take advantage of the grouping structure
information. As a benchmark, we also computed the Oracle OLS
results, i.e., OLS using only the important variables. The results
are summarized in Table \ref{tab:4}. We can see that in the
``all-in-all-out'' case, the adaptive hierarchical lasso removes
unimportant variables more effectively than SCAD and adaptive lasso,
and consequently, the adaptive hierarchical lasso outperforms SCAD
and adaptive lasso by a significant margin in terms of prediction
accuracy. In the ``not all-in-all-out'' case, the advantage of
knowing the grouping structure information is reduced, however, the
adaptive hierarchical lasso still performs slightly better than SCAD
and adaptive lasso, especially in terms of removing unimportant
variables.
To assess how the sample size affects different ``oracle'' methods,
we also considered $n$=200, 400, 800, 1600 and 3200. The results
are summarized in Figure \ref{fig:oracle}, where the first row
corresponds to the ``all-in-all-out'' case, and the second row
corresponds to the ``not all-in-all-out'' case. Not surprisingly,
as the sample size increases, the performances of different methods
all improve: in terms of prediction accuracy, the MSE's all decrease
(at about the same rate) and get closer to that of the Oracle OLS;
in terms of variable selection, the probabilities of identifying the
correct model all increase and approach one. However, overall, the
adaptive hierarchical lasso always performs the best among the three
``oracle'' methods, and the gap is especially prominent in terms of
removing unimportant variables when the sample size is moderate.
\begin{table}[tbph]
\caption{Comparison of several ``oracle'' methods, including the
adaptive hierarchical lasso, SCAD and the adaptive lasso. SCAD
and adaptive lasso do not take advantage of the grouping
structure information. The Oracle OLS uses only important
variables. Descriptions for the rows are the same as those in
the caption of Table \ref{tab:3}.}
\label{tab:4}
\begin{center}
\begin{tabular}{l|c|c|c|c}
\hline \hline
\multicolumn{5}{l}{Case 1: ``All-in-all-out''} \\
\hline & Oracle OLS & Ada Lasso & SCAD & Ada HLasso \\
\hline
MSE & 0.16 (0.006) & 0.37 (0.011) & 0.35 (0.011) & 0.24 (0.009) \\
\hline
Zero Var. & - & 77\% (0.7\%) & 79\% (1.1\%) & 98\% (0.3\%) \\
\hline
Non-Zero Var. & - & 94\% (0.5\%) & 91\% (0.6\%) & 96\% (0.5\%) \\
\hline \hline
\multicolumn{5}{l}{Case 2: ``Not all-in-all-out''} \\
\hline
& Oracle OLS & Ada Lasso & SCAD & Ada HLasso \\
\hline
MSE& 0.07 (0.003) & 0.13 (0.005) & 0.11 (0.004) & 0.10 (0.005) \\
\hline
Zero Var.& - & 91\% (0.3\%) & 91\% (0.4\%) & 98\% (0.1\%) \\
\hline
Non-zero Var.& - & 98\% (0.4\%) & 99\% (0.3\%) & 99\% (0.3\%) \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\caption{Comparison of several oracle methods, including the
SCAD, the adaptive lasso and the adaptive hierarchical lasso.
SCAD and adaptive lasso do not take advantage of the grouping
structure information.
The Oracle OLS uses only important variables.
The first row corresponds to the ``all-in-all-out''
case, and the second row corresponds to the
``not all-in-all-out'' case.
``Correct zero ratio'' records the percentage of correctly
removed unimportant variables. ``Correct non-zero ratio''
records the percentage of correctly identified important
variables.}
\label{fig:oracle}
\end{figure}
\section{Real Data Analysis}\label{sec:resultreal}
In this section, we use a gene expression dataset from the NCI-60
collection of cancer cell lines to further illustrate the
hierarchical lasso method. We sought to use this dataset to identify
targets of the transcription factor p53, which regulates gene
expression in response to various signals of cellular stress. The
mutational status of the p53 gene has been reported for 50 of the
NCI-60 cell lines, with 17 being classified as normal and 33 as
carrying mutations \cite{OlivierEtal02}.
Instead of single-gene analysis, gene-set information has recently
been used to analyze gene expression data. For example,
\citeasnoun{SubramanianEtal05} developed the Gene Set Enrichment
Analysis (GSEA), which is found to be more stable and more powerful
than single-gene analysis. \citeasnoun{Efron07} improved the GSEA
method by using new statistics for summarizing gene-sets. Both
methods are based on hypothesis testing. In this analysis, we
consider using the hierarchical lasso method for gene-set selection.
The gene-sets used here are the cytogenetic gene-sets and the
functionals gene-sets from the GSEA package
\cite{SubramanianEtal05}. We considered 391 overlapping
gene-sets with the size of each set greater than 15.
Since the response here is binary (normal vs mutation), following
the result in Section \ref{sec:theory2}, we use the logistic hierarchical
lasso regression, instead of the least square hierarchical lasso.
Note that a gene may belong to multiple gene-sets, we thus extend
the hierarchical lasso to the case of overlapping groups. Suppose
there are $K$ groups and $J$ variables. Let $\mathcal{G}_k$ denote
the set of indices of the variables in the $k$th group. One way to
model the overlapping situation is to extend the criterion
(\ref{model:eq02}) as the following:
\begin{eqnarray}\label{logisticHlasso}
\max_{d_k,\alpha_{j}} && \sum_{i=1}^{n} \ell
\left(\sum_{k=1}^K d_k \sum_{j: j \in \mathcal{G}_k}
\alpha_{j} x_{i,j},~y_{i}\right) \\ \nonumber
&& - \sum_{k=1}^K d_k - \lambda \cdot \sum_{j=1}^J |\alpha_j|
\\ \nonumber
\textrm{subject to} && d_k\ge 0, ~k=1, \ldots, K,
\end{eqnarray}
where $\alpha_j$ can be considered as the ``intrinsic'' effect of a
variable (no matter which group it belongs to), and different group
effects are represented via different $d_k$. In this section,
$\ell(\eta_i,y_i) = y_i\eta_i - \log(1+e^{\eta_i})$ is the logistic
log-likelihood function with $y_i$ being a 0/1 response. Also notice
that if each variable belongs to only one group, the model reduces
to the non-overlapping criterion (\ref{model:eq02}).
We randomly split the 50 samples into the training and test sets 100
times; for each split, 33 samples (22 carrying mutations
and 11 being normal) were used for training and the rest 17 samples
(11 carrying mutations and 6 being normal) were for
testing. For each split, we applied three methods, the logistic
lasso, the logistic $L_2$-norm group lasso \cite{MeierEtAl08}
and the logistic hierarchical lasso.
Tuning parameters were chosen using five-fold cross-validation.
We first compare the prediction accuracy of the three methods. Over
the 100 random splits, the logistic hierarchical lasso has an
average misclassification rate of 14\% with the standard error
1.8\%, which is smaller than 23\%(1.7\%) of the logistic lasso and
32\%(1.2\%) of the logistic group lasso. To assess the stability of
the prediction, we recorded the frequency in which each sample, as a
test observation, was correctly classified. For example, if a sample
appeared in 40 test sets among the 100 random splits, and out of the
40 predictions, the sample was correctly classified 36 times, we
recorded 36/40 for this sample. The results are shown in Figure
\ref{fig:freq}. As we can see, for most samples, the logistic
hierarchical lasso classified them correctly for most of the random
splits, and the predictions seemed to be slightly more stable than
the logistic lasso and the logistic $L_2$-norm group lasso.
Next, we compare gene-set selection of these three methods.
The most notable difference is that both logistic lasso and the
logistic hierarchical lasso selected gene CDKN1A most frequently out
of the 100 random split, while the logistic $L_2$-norm group lasso
rarely selected it. CDKN1A is also named as wild-type p53 activated
fragment-1 (p21), and it is known that the expression of gene CDKN1A
is tightly controlled by the tumor suppressor protein p53, through
which this protein mediates the p53-dependent cell cycle G1 phase
arrest in response to a variety of stress stimuli
\cite{LohEtAl03}.
We also compared the gene-sets selected by the logistic hierarchical
lasso with those selected by the GSEA of
\citeasnoun{SubramanianEtal05} and the GSA of \citeasnoun{Efron07}.
The two most frequently selected gene-sets by the hierarchical lasso
are \emph{atm pathway} and \emph{radiation sensitivity}. The most
frequently selected genes in \emph{atm pathway} by the logistic
hierarchical lasso are CDKN1A, MDM2 and RELA, and the most
frequently selected genes in \emph{radiation sensitivity} are
CDKN1A, MDM2 and BCL2. It is known that MDM2, the second commonly
selected gene, is a target gene of the transcription factor tumor
protein p53, and the encoded protein in MDM2 is a nuclear
phosphoprotein that binds and inhibits transactivation by tumor
protein p53, as part of an autoregulatory negative feedback loop
\cite{KubbutatEtAl97,MollPetrenko03}.
Note that the gene-set \emph{radiation sensitivity} was also
selected by GSEA and GSA. Though the gene-set \emph{atm pathway}
was not selected by GSEA and GSA, it shares 7, 8, 6, and 3 genes
with gene-sets \emph{radiation sensitivity}, \emph{p53 signalling},
\emph{p53 hypoxia pathway} and \emph{p53 Up} respectively, which
were all selected by GSEA and GSA. We also note that GSEA and GSA
are based on the {\it marginal} strength of each gene-set, while the
logistic hierarchical lasso fits an ``additive'' model and uses the
{\it joint} strengths of gene-sets.
\begin{figure}
\caption{The number of samples vs the frequency that a
sample was correctly classified on 100 random splits of the p53
data. }
\label{fig:freq}
\end{figure}
\section{Discussion} \label{sec:summary}
In this paper, we have proposed a hierarchical lasso method for
group variable selection. Different variable selection methods have
their own advantages in different scenarios. The hierarchical lasso
method not only effectively removes unimportant groups, but also
keeps the flexibility of selecting variables within a group. We show
that the improved hierarchical lasso method enjoys an oracle
property, i.e., it performs as well as if the true sub-model were
given in advance. Numerical results indicate that our method works
well, especially when variables in a group are associated with the
response in a ``not all-in-all-out'' fashion.
The grouping idea is also applicable to other regression and
classification settings, for example, the multi-response regression
and multi-class classification problems. In these problems, a
grouping structure may not exist among the prediction variables, but
instead, natural grouping structures exist among {\it parameters}.
We use the multi-response regression problem to illustrate the point
\cite{BreimanFriedman97,TurlachVenablesWright05}. Suppose we observe
$(\V{x}_1, \V{y}_1)$, $\ldots$, $(\V{x}_n, \V{y}_n)$, where each
$\V{y}_i = (y_{i1}, \ldots, y_{iK})$ is a vector containing $K$
responses, and we are interested in selecting a subset of the
prediction variables that predict well for all of the multiple
responses. Standard techniques estimate $K$ prediction functions,
one for each of the $K$ responses, $f_k(\V{x}) = \beta_{k1} x_1 +
\cdots + \beta_{kp} x_p, k=1, \ldots, K$. The prediction variables
$(x_1, \ldots, x_p)$ may not have a grouping structure, however, we
may consider the coefficients corresponding to the same prediction
variable form a natural group, i.e., $(\beta_{1j}, \beta_{2j},
\ldots, \beta_{Kj})$. Using our hierarchical lasso idea, we
reparameterize $\beta_{kj} = d_j \alpha_{kj}$, $d_j \ge 0$, and we
consider
\begin{eqnarray*}
\max_{d_j \ge 0, \alpha_{kj}} && - \frac{1}{2} \sum_{k=1}^K \sum_{i=1}^n
\left( y_{ik} -
\sum_{j=1}^p d_j \alpha_{kj} x_{ij} \right)^2 \\
&& - \lambda_1 \cdot \sum_{j=1}^p d_j
- \lambda_2 \cdot \sum_{j=1}^p \sum_{k=1}^K |\alpha_{kj}|.
\end{eqnarray*}
Note that if $d_j$ is shrunk to zero, all $\beta_{kj}, k=1,\ldots,K$
will be equal to zero, hence the $j$th prediction variable will be
removed from all $K$ predictions. If $d_j$ is not equal to zero,
then some of the $\alpha_{kj}$ and hence some of the $\beta_{kj}$,
$k=1, \ldots, K$, still have the possibility of being zero.
Therefore, the $j$th variable may be predictive for some responses
but non-predictive for others.
One referee pointed out the work by \citeasnoun{HuangEtAl07}, which
we were not aware of when our manuscript was first completed and
submitted in 2007. We acknowledge that the work by
\citeasnoun{HuangEtAl07} is closely related with ours, but there are
also differences. For example:
\begin{itemize}
\item We proved the oracle property for both group selection
and within group selection, while \citeasnoun{HuangEtAl07}
considered the oracle property only for group selection.
\item Our theory applies to the generalized maximum likelihood
estimate, while \citeasnoun{HuangEtAl07} considered the
penalized least squares estimate.
\item Handling overlapping groups. It is
not unusual for a variable to be a member of several
groups. The gene expression date we analyzed in Section
\ref{sec:resultreal} is such an example: given a plethora of
biologically defined gene-sets, not surprisingly, there will be
considerable overlap among these sets.
In \citeasnoun{HuangEtAl07}, a prediction variable that appears
in more than one group gets penalized more heavily than
variables appearing in only one group. Therefore, a prediction
variable belonging to multiple groups is more likely to be
removed than a variable belonging to only one group. We are
not sure whether this is an appealing property.
In our approach, as shown in (\ref{logisticHlasso}),
if a prediction variable
belongs to multiple groups, it does not get penalized more
heavily than other variables that belong to only one group.
\end{itemize}
\section*{Acknowledgments}
Zhou and Zhu were partially supported by grants DMS-0505432,
DMS-0705532 and DMS-0748389 from the National Science Foundation.
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\section*{Appendix}
\subsection*{Proof of Lemma 1}
Let $Q^*(\lambda_1, \lambda_2, \V{d}, \V{\alpha})$ be the criterion
that we would like to maximize in equation (7) and let
$Q^\star(\lambda, \V{d}, \V{\alpha})$ be the corresponding criterion
in equation (8).
Let ($\hat{\V{d}}^*, \hat{\V{\alpha}}^*$) be a local
maximizer of
$Q^*(\lambda_1, \lambda_2, \V{d}, \V{\alpha})$. We would like to
prove ($\hat{\V{d}}^\star = \lambda_1\hat{\V{d}}^*,
\hat{\V{\alpha}}^\star = \hat{\V{\alpha}}^*/\lambda_1$) is a local
maximizer of $Q^\star(\lambda, \V{d}, \V{\alpha})$.
We immediately have $$Q^*(\lambda_1, \lambda_2, {\V{d}},
{\V{\alpha}}) = Q^\star(\lambda, \lambda_1{\V{d}},
{\V{\alpha}}/\lambda_1).$$ Since ($\hat{\V{d}}^*, \hat{\V{\alpha}}^*$) is a local maximizer of
$Q^*(\lambda_1, \lambda_2, \V{d}, \V{\alpha})$, there exists $\delta
> 0$ such that if ${\V{d}}^{\prime}$, ${\V{\alpha}}^{\prime} $ satisfy
$ \| {\V{d}}^{\prime} - \hat{\V{d}}^* \| +
\| {\V{\alpha}}^{\prime} - \hat{\V{\alpha}}^* \| < \delta $ then $
Q^*(\lambda_1, \lambda_2, {\V{d}}^{\prime},
{\V{\alpha}}^{\prime} ) \le Q^*(\lambda_1, \lambda_2, \hat{\V{d}}^*,
\hat{\V{\alpha}}^*).$
Choose $\delta^{\prime} $ such that $
\frac{\delta^{\prime}}{\min\left( \lambda_1,
\frac{1}{\lambda_1}\right)} \le \delta$, for any
(${\V{d}}^{\prime\prime}, {\V{\alpha}}^{\prime\prime}$) satisfying $
\| {\V{d}}^{\prime\prime} - \hat{\V{d}}^\star \| + \|
{\V{\alpha}}^{\prime\prime} - \hat{\V{\alpha}}^\star \| <
\delta^{\prime} $ we have
\begin{eqnarray*}
\left\| \frac{ \V{d}^{\prime\prime} }{\lambda_1} -
\hat{\V{d}}^* \right\| +
\| { \lambda_1 \V{\alpha}}^{\prime\prime} -
\hat{\V{\alpha}}^* \|
& \le & \frac{ \lambda_1 \left\| \frac{ \V{d}^{\prime\prime} }
{\lambda_1} - \hat{\V{d}}^* \right\| +
\frac{1}{\lambda_1} \left\| \lambda_1 \V{\alpha}^{\prime\prime} -
\hat{\V{\alpha}}^* \right\| }
{ \min\left( \lambda_1, \frac{1}{\lambda_1}\right) }
\\
& = & \frac{ \| {\V{d}}^{\prime\prime} - \hat{\V{d}}^\star \| +
\| {\V{\alpha}}^{\prime\prime} - \hat{\V{\alpha}}^\star \| }
{ \min\left( \lambda_1, \frac{1}{\lambda_1}\right) }
\\
& < & \frac{ \delta^{\prime}} { \min\left( \lambda_1,
\frac{1}{\lambda_1}\right) }
\\
& < & \delta.
\end{eqnarray*}
Hence \begin{eqnarray*}
Q^\star(\lambda, \hat{\V{d}}^{\prime\prime}, \hat{\V{\alpha}}^{\prime\prime})
&=& Q^\ast (\lambda_1, \lambda_2, \hat{\V{d}}^{\prime\prime}/\lambda_1,
\lambda_1\hat{\V{\alpha}}^{\prime\prime}) \\
&\le & Q^\ast (\lambda_1, \lambda_2, \hat{\V{d}}^\ast,
\hat{\V{\alpha}}^\ast) \\
&=&
Q^\star(\lambda, \hat{\V{d}}^\star, \hat{\V{\alpha}}^\star).
\end{eqnarray*} Therefore, ($\hat{\V{d}}^\star = \lambda_1\hat{\V{d}}^*,
\hat{\V{\alpha}}^\star = \hat{\V{\alpha}}^*/\lambda_1$) is a local
maximizer of $Q^\star(\lambda, \V{d}, \V{\alpha}).$
Similarly we can prove that for any local maximizer
($\hat{\V{d}}^\star, \hat{\V{\alpha}}^\star$) of $Q^\star(\lambda,
\V{d}, \V{\alpha})$, there is a corresponding local maximizer
($\hat{\V{d}}^*, \hat{\V{\alpha}}^*$) of $Q^*(\lambda_1, \lambda_2,
\V{d}, \V{\alpha})$ such that $ \hat{d}_k^{\ast}
\hat{\alpha}_{kj}^{\ast} = \hat{d}_k^\star
\hat{\alpha}_{kj}^\star. $
\begin{lemma} \label{lemma1}
Suppose ($\hat{\V{d}}, \hat{\V{\alpha}} $) is a local maximizer of
(8). Let $\hat{\V{\beta}} $ be the Hierarchical Lasso estimate
related to ($\hat{\V{d}}, \hat{\V{\alpha}} $), i.e.,
$\hat{\beta}_{kj} = \hat{d}_k \hat{\alpha}_{kj} $. If $\hat{d}_k =
0$, then $\hat{\V{\alpha}}_k =0 $; if $\hat{d}_k \ne 0$, then $\|
\hat{\V{\beta}}_{k} \|_1 \ne 0$ and $\hat{d}_k = \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} \|_1 }, \hat{\V{\alpha}}_k = \frac{
\hat{\V{\beta}}_k }{ \sqrt{\lambda \| \hat{\V{\beta}}_{k} \|_1 } }
$.
\end{lemma}
\subsection*{Proof of Lemma 2}
If $\hat{d}_k = 0$, then $\hat{\V{\alpha}}_k =0 $ is quite obvious.
Similarly, if $\hat{\V{\alpha}}_k =0 $, then $\hat{d}_k = 0$.
Therefore, if $\hat{d}_k \ne 0$, then $\hat{\V{\alpha}}_k \ne 0 $
and $\| \hat{\V{\beta}}_{k} \|_1 \ne 0$.
We prove $\hat{d}_k = \sqrt{ \lambda \| \hat{\V{\beta}}_{k} \|_1 },
\hat{\V{\alpha}}_k = \frac{ \hat{\V{\beta}}_k }{ \sqrt{\lambda \|
\hat{\V{\beta}}_{k} \|_1 } } $ for $\hat{d}_k \ne 0$ by
contradiction. Suppose $\exists k^{\prime}$ such that
$\hat{d}_{k^\prime} \ne 0 $ and $\hat{d}_{k^\prime} \ne \sqrt{
\lambda \| \hat{\V{\beta}}_{k^\prime} \|_1 }$. Let $ \frac{ \sqrt{
\lambda \| \hat{\V{\beta}}_{k^\prime} \|_1 } }{ \hat{d}_{k^\prime} }
= c $. Then $\hat{\V{\alpha}}_k = c \frac{ \hat{\V{\beta}}_k }{
\sqrt{\lambda \| \hat{\V{\beta}}_{k} \|_1 } } $. Suppose $c > 1$.
Let $ \tilde{d}_{k} = \hat{d}_{k} $ and $ \tilde{\V{\alpha}}_{k} =
\hat{\V{\alpha}}_{k} $ for $k\ne k^\prime$ and $
\tilde{d}_{k^\prime} = \delta^\prime \hat{d}_{k^\prime} $ and $
\tilde{\V{\alpha}}_{k^\prime} = \hat{\V{\alpha}}_{k^\prime} \frac{ 1
}{ \delta^\prime }$, where $\delta^\prime$ satisfies $ c >
\delta^\prime > 1 $ and is very close to 1 such that $ \|
\tilde{d}_{k^\prime} - \hat{d}_{k^\prime} \|_1 + \|
\tilde{\V{\alpha}}_{k^\prime} - \hat{\V{\alpha}}_{k^\prime} \|_1 <
\delta $ for some $\delta >0 $.
Then we have
\begin{eqnarray*}
Q^\star(\lambda, \tilde{\V{d}}, \tilde{\V{\alpha}}) -
Q^\star(\lambda, \hat{\V{d}}, \hat{\V{\alpha}})
&=& - \delta^\prime | \hat{d}_{k^\prime} | -
\frac{1}{\delta^\prime} \lambda \| \hat{\V{\alpha}}_{k^\prime} \|_1 +
| \hat{d}_{k^\prime} | +
\lambda \| \hat{\V{\alpha}}_{k^\prime} \|_1 \\
&=& \left( - \frac{\delta^\prime }{ c } - \frac{c}{\delta^\prime} +
\frac{ 1 }{ c } + c \right) \sqrt{ \lambda \| \hat{\V{\beta}}_{k^\prime}\|_1 } \\
&=& \frac{ 1 }{ c }( \delta^\prime - 1 )
\left(\frac{c^2}{\delta^\prime} - 1 \right) \sqrt{ \lambda \|
\hat{\V{\beta}}_{k^\prime} \|_1 } \\
& > & 0.
\end{eqnarray*}
Therefore, for any $\delta > 0 $, we can find
$\tilde{\V{d}},\tilde{\V{\alpha}}$ such that $ \| \tilde{\V{d}} -
\hat{\V{d}} \|_1 + \|\tilde{\V{\alpha}} - \hat{\V{\alpha}} \|_1 <
\delta $ and $Q^\star(\lambda, \tilde{\V{d}}, \tilde{\V{\alpha}}) >
Q^\star(\lambda, \hat{\V{d}}, \hat{\V{\alpha}}) $. These contradict
with $(\hat{\V{d}},\hat{\V{\alpha}})$ being a local maximizer.
Similarly for the case when $ c < 1 $. Hence, we have the result
that if $\hat{d}_k \ne 0$, then $\hat{d}_k = \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} \|_1 }, \hat{\V{\alpha}}_k = \frac{
\hat{\V{\beta}}_k }{ \sqrt{\lambda \| \hat{\V{\beta}}_{k} \|_1 } }
$.
\subsection*{Proof of Theorem 1}
Let $ Q(\lambda, \V{\beta})$ be the corresponding criterion in
equation (11).
Suppose ($\hat{\V{d}}, \hat{\V{\alpha}}$) is a local maximizer of
$Q^\star(\lambda, \V{d}, \V{\alpha})$, we first show that
$\hat{\V{\beta}}$, where $\hat{\beta}_{kj}=
\hat{d}_k\hat{\alpha}_{kj} $, is a local maximizer of $
Q(\lambda,\V{\beta})$, i.e. there exists a $ \delta^\prime $ such
that if $ \| \F{trace}iangle\V{\beta} \|_1 < \delta^\prime $ then
$Q(\lambda, \hat{\V{\beta}} + \F{trace}iangle\V{\beta} ) \le
Q(\lambda,\hat{\V{\beta}})$.
We denote $\F{trace}iangle\V{\beta} = \F{trace}iangle\V{\beta}^{(1)} +
\F{trace}iangle\V{\beta}^{(2)} $, where $\F{trace}iangle\V{\beta}^{(1)}_k = 0$
if $\| \hat{\V{\beta}}_{k} \|_1 = 0 $ and
$\F{trace}iangle\V{\beta}^{(2)}_k = 0$ if $\| \hat{\V{\beta}}_{k} \|_1 \ne
0 $. We have $ \| \F{trace}iangle\V{\beta} \|_1 = \|
\F{trace}iangle\V{\beta}^{(1)} \|_1 + \| \F{trace}iangle\V{\beta}^{(2)} \|_1 $.
Now we show $Q(\lambda, \hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)} )
\le Q(\lambda, \hat{\V{\beta}})$ if $ \delta^\prime $ is small
enough. By Lemma 2, we have $\hat{d}_k = \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} \|_1 }, \hat{\V{\alpha}}_k =
\frac{\hat{\V{\beta}}_k } { \sqrt{\lambda \| \hat{\V{\beta}}_{k}
\|_1} } $ if $\| \hat{d}_{k} \|_1 \ne 0 $ and $\hat{\V{\alpha}}_k =
\V{0} $ if $\| \hat{d}_{k} \|_1 = 0$. Furthermore, let
$\hat{d}_k^\prime =
\sqrt{ \lambda \| \hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k \|_1 },
\hat{\V{\alpha}}_k^\prime = \frac{ \hat{\V{\beta}}_k +
\F{trace}iangle\V{\beta}^{(1)}_k } { \sqrt{\lambda \| \hat{\V{\beta}}_{k}
+ \F{trace}iangle\V{\beta}^{(1)}_k \|_1 } } $ if $\| \hat{d}_{k} \|_1 \ne
0 $. Let $\hat{d}_k^\prime = 0, \hat{\V{\alpha}}_k^\prime = \V{0} $
if $\| \hat{d}_{k} \|_1 = 0$. Then we have $Q^\star(\lambda,
\hat{\V{d}}^\prime, \hat{\V{\alpha}}^\prime ) = Q(\lambda,
\hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)} )$ and $ Q^\star(\lambda,
\hat{\V{d}}, \hat{\V{\alpha}}) = Q(\lambda, \hat{\V{\beta}}) $.
Hence we only need to show that $Q^\star(\lambda,
\hat{\V{d}}^\prime, \hat{\V{\alpha}}^\prime ) \le Q^\star(\lambda,
\hat{\V{d}}, \hat{\V{\alpha}}) $. Note that ($\hat{\V{d}},
\hat{\V{\alpha}}$) ia a local maximizer of $Q^\star(\lambda,
\V{d},\V{\alpha})$. Therefore there exists a $\delta$ such that for
any ${\V{d}}^{\prime}, {\V{\alpha}}^{\prime}$ satisfying
$\|{\V{d}}^{\prime} - \hat{\V{d}} \|_1 +
\| {\V{\alpha}}^{\prime} - \hat{\V{\alpha}} \|_1 <
\delta $, we have $Q^\star(\lambda, {\V{d}}^\prime, {\V{\alpha}}^\prime ) \le
Q^\star(\lambda, \hat{\V{d}}, \hat{\V{\alpha}}) $.
Now since
\begin{eqnarray*}
|\hat{d}_k^\prime - \hat{d}_k| &=& | \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k \|_1 } - \sqrt{
\lambda \| \hat{\V{\beta }}_{k}
\|_1 }| \\
& \le & | \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} \|_1 - \lambda \| \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 } - \sqrt{ \lambda \| \hat{\V{\beta }}_{k}
\|_1 }| \\
& \le & \frac{1}{2} \frac{ \lambda \|
\F{trace}iangle\V{\beta}^{(1)}_k \|_1 }{ \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} \|_1 - \lambda \| \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 }} \\
& \le & \frac{1}{2} \frac{ \lambda \|
\F{trace}iangle\V{\beta}^{(1)}_k
\|_1 }{ \sqrt{ \lambda a - \lambda \delta^\prime }} \\
& \le & \frac{1}{2} \frac{ \lambda \|
\F{trace}iangle\V{\beta}^{(1)}_k \|_1 }{ \sqrt{ \lambda a/2 }},
\end{eqnarray*}
where $ a = \min \{ \| \hat{\V{\beta}}_{k} \|_1 : \|
\hat{\V{\beta}}_{k} \|_1 \ne 0 \} $ and $ \delta^\prime < a/2$.
Furthermore
\begin{eqnarray*}
\| \hat{\V{\alpha}}_k^\prime - \hat{\V{\alpha}}_k \|_1 &=&
\left\| \frac{ \hat{\V{\beta}}_k + \F{trace}iangle\V{\beta}^{(1)}_k }{
\sqrt{\lambda \| \hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 } } - \frac{ \hat{\V{\beta}}_k }{ \sqrt{\lambda \|
\hat{\V{\beta}}_{k}
\|_1 } } \right\|_1 \\
& \le & \left\| \frac{ \hat{\V{\beta}}_k + \F{trace}iangle\V{\beta}^{(1)}_k }{
\sqrt{\lambda \| \hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 } } - \frac{ \hat{\V{\beta}}_k }{ \sqrt{\lambda \|
\hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 } } \right\|_1 \\
& & + \left\| \frac{ \hat{\V{\beta}}_k }{
\sqrt{\lambda \| \hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 } } - \frac{ \hat{\V{\beta}}_k }{ \sqrt{\lambda \|
\hat{\V{\beta}}_{k}
\|_1 } } \right\|_1 \\
& \le & \frac{ \|
\F{trace}iangle\V{\beta}^{(1)}_k
\|_1 }{ \sqrt{ \lambda a/2 }} \\
& & + \frac{ \| \hat{\V{\beta}}_k \|_1 | \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k \|_1 } - \sqrt{
\lambda \| \hat{\V{\beta }}_{k} \|_1 }| }{ \sqrt{ \lambda \|
\hat{\V{\beta}}_{k} + \F{trace}iangle\V{\beta}^{(1)}_k \|_1 }
\sqrt{\lambda \| \hat{\V{\beta}}_{k}
\|_1 } } \\
& \le & \frac{ \|
\F{trace}iangle\V{\beta}^{(1)}_k \|_1 }{ \sqrt{ \lambda a/2 }} + \frac{ b
}{ \sqrt{ \lambda a/2 } \sqrt{ \lambda a} } \left( \frac{1}{2}
\frac{ \lambda \| \F{trace}iangle\V{\beta}^{(1)}_k
\|_1 }{ \sqrt{ \lambda a/2 }} \right) \\
& \le & \|
\F{trace}iangle\V{\beta}^{(1)}_k \|_1 \left( \frac{ 1 }{ \sqrt{ \lambda
a/2 }} + \frac{ b }{ a \sqrt{ \lambda a } } \right),
\end{eqnarray*}
where $ b = \max \{ \| \hat{\V{\beta}}_{k} \|_1 : \|
\hat{\V{\beta}}_{k} \|_1 \ne 0 \} $.
Therefore, there exists a small enough $\delta^\prime$, if $ \|
\F{trace}iangle\V{\beta}^{(1)} \|_1 < \delta^\prime $ we have $\|
\hat{\V{d}}^{\prime} - \hat{\V{d}} \|_1 +
\| \hat{\V{\alpha}}^{\prime} - \hat{\V{\alpha}} \|_1 < \delta $.
Hence $ Q^\star(\lambda, \hat{\V{d}}^\prime, \hat{\V{\alpha}}^\prime
) \le Q^\star(\lambda, \hat{\V{d}}, \hat{\V{\alpha}}) $ (due to
local maximality) and $Q(\lambda, \hat{\V{\beta}} +
\F{trace}iangle\V{\beta}^{(1)} ) \le Q(\lambda, \hat{\V{\beta}})$.
Next we show $ Q(\lambda, \hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)}
+ \F{trace}iangle\V{\beta}^{(2)} ) \le Q(\lambda, \hat{\V{\beta}} +
\F{trace}iangle\V{\beta}^{(1)} ).$ Note that
\[
Q(\lambda, \hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)} +
\F{trace}iangle\V{\beta}^{(2)} ) -
Q(\lambda, \hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)} ) = \T{
\F{trace}iangle\V{\beta}^{(2)} } \nabla L( \hat{\V{\beta}}^* ) -
\sum_{k=1}^{K} \sqrt{ \lambda \| \F{trace}iangle\V{\beta}^{(2)} \|_1 }, \]
where $\V{\beta}^*$ is a vector between $\hat{\V{\beta}} +
\F{trace}iangle\V{\beta}^{(1)} + \F{trace}iangle\V{\beta}^{(2)}$ and
$\hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)} $. Since $ \|
\F{trace}iangle\V{\beta}^{(2)} \|_1 < \delta^\prime $ is small enough, the
second term dominates the first term, hence we have $ Q(\lambda,
\hat{\V{\beta}} + \F{trace}iangle\V{\beta}^{(1)} +
\F{trace}iangle\V{\beta}^{(2)} ) \le Q(\lambda, \hat{\V{\beta}} +
\F{trace}iangle\V{\beta}^{(1)} ) $.
Overall, we have that there exists a small enough $\delta^\prime$,
if $ \| \F{trace}iangle\V{\beta} \|_1 < \delta^\prime $, then $ Q(\lambda,
\hat{\V{\beta}} + \F{trace}iangle\V{\beta} ) \le Q(\lambda,
\hat{\V{\beta}} ) $, which implies that $\hat{\V{\beta}}$ is a local
maximizer of $Q(\lambda, {\V{\beta}})$.
Similarly, we can prove that if $\hat{\V{\beta}}$ is a local
maximizer of $Q(\lambda, {\V{\beta}})$, and if we let
$\hat{d}_k = \sqrt{ \lambda \| \hat{\V{\beta}}_{k} \|_1 },
\hat{\V{\alpha}}_k = \frac{\hat{\V{\beta}}_k }
{ \sqrt{\lambda \| \hat{\V{\beta}}_{k} \|_1} } $
for $\| \hat{\V{\beta}}_{k} \|_1 \ne 0 $ and let $\hat{d}_k = 0,
\hat{\V{\alpha}}_k = \V{0} $ for $\| \hat{\V{\beta}}_{k} \|_1 = 0$,
then ($\hat{\V{d}}, \hat{\V{\alpha}}$) is a local maximizer of
$Q^\star(\lambda, \V{d}, \V{\alpha})$.
\subsection*{Regularity Conditions }
Let $S_n$ be the number of non-zero groups, i.e., $\|
\V{\beta}_{nk}^0 \| \neq 0$. Without loss of generality, we assume
\begin{eqnarray*}
\| \V{\beta}_{nk}^0 \| &\neq& 0, ~\textrm{for}~ k=1, \ldots, S_n, \\
\| \V{\beta}_{nk}^0 \| &=& 0, ~\textrm{for}~ k=S_n+1, \ldots, K_n.
\end{eqnarray*}
Let $s_{nk}$ be the number of non-zero coefficients in group $k,
1\le k \le S_n$; again, without loss of generality, we assume
\begin{eqnarray*}
\beta_{n,kj}^0 &\neq& 0, ~\textrm{for}~ k=1, \ldots, S_n;~ j = 1,
\ldots, s_{nk}, \\
\beta_{n,kj}^0 &=& 0, ~\textrm{for}~ k=1, \ldots, S_n;~ j=s_{nk}+1, \ldots, p_{nk}.
\end{eqnarray*}
For simplicity, we write
$\beta_{n,kj}$, $p_{nk}$ and $s_{nk}$ as $\beta_{ kj}$, $p_k$ and
$s_k$ in the following.
Since we have diverging number of parameters, to keep the uniform
properties of the likelihood function, we need some conditions on
the higher-order moment of the likelihood function, as compared to
the usual condition in the asymptotic theory of the likelihood
estimate under finite parameters (Lehmann and Casella 1998).
\begin{itemize}
\item[(A1)] For every $n$, the observations $\{ \V{V}_{ni}, i = 1, 2,
\ldots, n \}$ are independent and identically distributed, each
with a density $f_n(\V{V}_{n1}, \V{\beta}_n)$. $f_n(\V{V}_{n1},
\V{\beta}_n)$
has a common support and the model is identifiable. Furthermore, the first and second
logarithmic derivatives of $f_n$ satisfy the equations
\begin{eqnarray*}
\F{E}_{{\V{\beta}_n}}\left[\frac{\partial \log f_n(\V{V}_{n1},
\V{\beta}_n) } {\partial\beta_{kj}}\right] &=& 0,
~~~\textrm{for}~ k=1, \ldots, {K_n};~ j=1, \ldots, p_k \\
\M{I}_{k_1j_1k_2j_2} (\V{\beta}_n) &=& \F{E}_{\V{\beta}_n} \left[
\frac{\partial}{\partial\beta_{k_1j_1}} \log f_n(\V{V}_{n1}, \V{\beta}_n)
\frac{\partial}{\partial\beta_{k_2j_2}} \log f_n(\V{V}_{n1},
\V{\beta}_n) \right] \\
&=& \F{E}_{\V{\beta}_n} \left[ -
\frac{\partial^2}{\partial\beta_{k_1j_2} \partial\beta_{k_2j_2}}
\log f_n(\V{V}_{n1}, \V{\beta}_n) \right].
\end{eqnarray*}
\item[(A2)] The Fisher information matrix
\begin{equation}
\M{I}(\V{\beta}_n) =
\F{E}_{{\V{\beta}_n}}\left[ \frac{\partial}{\partial{\V{\beta}_n}}
\log f_n(\V{V}_{n1}, \V{\beta}_n)
\frac{ \T{\partial} }{\partial{\V{\beta}_n}} \log
f_n(\V{V}_{n1}, \V{\beta}_n) \right]
\\ \nonumber
\end{equation}
satisfies the condition
$$ 0 < C_1 < \lambda_{\min} \{ \M{I}(\V{\beta}_n) \} \le
\lambda_{\max} \{ \M{I}(\V{\beta}_n) \} < C_2 < \infty ,$$ and for
any $k_{1}, j_{1}, k_{2}, j_{2}$, we have
\begin{eqnarray*}
\F{E}_{{\V{\beta}_n}}\left[\frac{\partial}{\partial\beta_{k_{1}j_{1}}}
\log f_n(\V{V}_{n1}, \V{\beta}_n) \frac{\partial}
{\partial\beta_{k_{2}j_{2}}} \log f_n(\V{V}_{n1}, \V{\beta}_n)
\right]^2 & < & C_3 < \infty,
\\
\F{E}_{{\V{\beta}_n}}\left[- \frac{\partial^2}
{\partial\beta_{k_{1}j_{1}}\partial\beta_{k_{2}j_{2}}}
\log f_n(\V{V}_{n1}, \V{\beta}_n) \right]^2 & < & C_4 < \infty.
\end{eqnarray*}
\item[(A3)] There exists an open subset $\omega_n$ of $\Omega_n \in R^{P_n}$
that contains the true parameter
point $\V{\beta}_n^0$ such that for almost all $\V{V}_{n1}$,
the density $f_n(\V{V}_{n1}, \V{\beta}_n)$
admits all third derivatives $\partial^3 f_n(\V{V}_{n1}, \V{\beta}_n)/
(\partial\beta_{k_{1}j_{1}}\partial\beta_{k_{2}j_{2}}\partial\beta_{k_{3}j_{3}})$
for all ${\V{\beta}_n}\in\omega_n $. Furthermore, there exist
functions $M_{{nk_{1}j_{1}}{k_{2}j_{2}}{k_{3}j_{3}}}$ such that
\begin{equation}
\left|\frac{\partial^3}{\partial\beta_{k_{1}j_{1}}\partial\beta_{k_{2}j_{2}} \partial\beta_{k_{3}j_{3}}}
\log f_n(\V{V}_{n1}, \V{\beta}_n) \right| \leq
M_{{nk_{1}j_{1}}{k_{2}j_{2}}{k_{3}j_{3}}}(\V{V}_{n1})
~~ \mathrm{for~ all~} {\V{\beta}_n}\in\omega_n,
\\ \nonumber
\end{equation}
and $ \F{E}_{{\V{\beta}_n}}[
M_{{nk_{1}j_{1}}{k_{2}j_{2}}{k_{3}j_{3}}}^2(\V{V}_{n1}) ]
< C_5 < \infty $.
\end{itemize}
These regularity conditions guarantee the asymptotic normality of
the ordinary maximum likelihood estimates for diverging number of
parameters.
For expositional simplicity, we will first prove Theorem 3 and
Theorem 4, then prove Theorem 2.
\subsection*{Proof of Theorem 3}
We will show that for any given $\epsilon > 0$, there exists a
constant $C$ such that
\begin{equation} \label{sec_app_infty072}
\F{Pr}\left\{ \sup_{ \| \V{u}\| = C} Q_n(\V{\beta}_n^0 + \alpha_n
\V{u} ) < Q_n(\V{\beta}_n^0) \right\} \ge 1- \epsilon,
\end{equation}
where $\alpha_n = \sqrt{P_n} ( n^{-1/2} + \lambda_n \sqrt{a_n}
/2\sqrt{c_1} ) $.
This implies that with probability at least $1- \epsilon $, that
there exists a local maximum in the ball $\{ \V{\beta}_n^0 +
\alpha_n \V{u} : {\|\V{u}\| \le C} \} $.
Hence, there exists a local maximizer such that $\| \hat{\V{\beta}}_n -\V{\beta}_n^0
\| = O_p(\alpha_n).$
Since $1/2\sqrt{c_1}$ is a constant, we have $\| \hat{\V{\beta}}_n
-\V{\beta}_n^0
\| = O_p( \sqrt{P_n} (n^{-1/2} + \lambda_n \sqrt{a_n} ) ) $.
Using $p_{\lambda_n,\V{w}_n}(0) = 0 $, we have
\begin{eqnarray} \label{sec_app_infty0072a}
D_n(\V{u}) &=& Q_n(\V{\beta}_n^0 + \alpha_n \V{u} ) - Q_n(\V{\beta}_n^0)
\nonumber \\
&\le& L_n(\V{\beta}_n^0 + \alpha_n \V{u} ) - L_n(\V{\beta}_n^0) \nonumber \\
& & ~~~ - n \sum_{k=1}^{S_n}
( p_{\lambda_n,\V{w}_n}( {\V{\beta}}_{nk}^0+ \alpha_n \V{u}_k )
- p_{\lambda_n,\V{w}_n}({\V{\beta}}_{nk}^0 ) ) \nonumber \\
& \F{trace}iangleq & (I) + (II).
\end{eqnarray}
Using the standard argument on the Taylor
expansion of the likelihood function, we have
\begin{eqnarray} \label{sec_app_infty0073}
(I) & = & \alpha_n \T{\V{u}} {\nabla} L_n (\V{\beta}_n^0) +
\frac{1}{2} \T{\V{u}} \nabla^2 L_n (\V{\beta}_n^0) \V{u} \alpha_n^2
+ \frac{1}{6} \T{\V{u}} {\nabla} \{ \T{\V{u}} \nabla^2 L_n
(\V{\beta}_n^\ast) \V{u} \} \alpha_n^3 \nonumber \\
& \F{trace}iangleq & I_1 + I_2 + I_3,
\end{eqnarray}
where $\V{\beta}_n^\ast$ lies between $\V{\beta}_n^0$ and
$\V{\beta}_n^0 + \alpha_n \V{u}.$ Using the same argument as in the
proof of Theorem 1 of Fan and Peng (2004), we have
\begin{eqnarray}\label{sec_app_infty0073I1}
|I_1| & = & O_p( \alpha_n^2 n) \| \V{u} \|, \\
I_2 & = & -\frac{n \alpha_n^2}{2} \T{\V{u}} \M{I}_n (\V{\beta}_n^0) \V{u} +
o_p(1) n \alpha_n^2 \| \V{u} \|^2,
\end{eqnarray}
and
\begin{eqnarray*}
| I_3 | & = & \left| \frac{1}{6} \sum_{k_1=1}^{K_n} \sum_{j_1=1}^{p_k}
\sum_{k_2=1}^{K_n} \sum_{j_2=1}^{p_k} \sum_{k_3=1}^{K_n}
\sum_{j_3=1}^{p_k} \frac{\partial^3 L_n(\beta_n^\ast)
}{\partial\beta_{k_{1}j_{1}}\partial\beta_{k_{2}j_{2}}
\partial\beta_{k_{3}j_{3}}} u_{k_{1}j_{1}} u_{k_{2}j_{2}} u_{k_{3}j_{3}} \alpha_n^3 \right|
\nonumber \\
& \le & \frac{1}{6} \sum_{i=1}^{n} \left\{ \sum_{k_1=1}^{K_n} \sum_{j_1=1}^{p_k}
\sum_{k_2=1}^{K_n} \sum_{j_2=1}^{p_k} \sum_{k_3=1}^{K_n}
\sum_{j_3=1}^{p_k} M_{{nk_{1}j_{1}}{k_{2}j_{2}}{k_{3}j_{3}}}^{2}
(V_{ni}) \right\}^{1/2}
\| \V{u} \|^3 \alpha_n^3 \nonumber \\
& = & O_p(P_n^{3/2} \alpha_n ) n \alpha_n^2
\| \V{u} \|^3.
\end{eqnarray*}
Since
$\frac{ P_n^4 } {n} \rightarrow 0$ and $ P_n^2 \lambda_n \sqrt{a_n }
\rightarrow 0
$ as $n \rightarrow \infty$, we have
\begin{equation} \label{sec_app_infty0073I3} | I_3 | = o_p( n \alpha_n^2 )
\| \V{u} \|^3. \end{equation}
From (\ref{sec_app_infty0073I1})-(\ref{sec_app_infty0073I3}), we can
see that, by choosing a sufficiently large $C$, the first term in
$I_2$ dominates $I_1$ uniformly on $\|\V{u}\| = C$; when $n$ is
large enough, $I_2$ also dominates $I_3$ uniformly on $\|\V{u}\| =
C$.
Now we consider $(II)$. Since $\alpha_n = \sqrt{P_n} ( n^{-1/2} +
\lambda_n \sqrt{a_n} /2\sqrt{c_1} ) \rightarrow 0 $, for $\| \V{u}
\| \leq C$ we have
\begin{equation} \label{sec_app_infty:thm1_1}
|\beta_{kj}^0 + \alpha_n u_{kj}
| \geq |\beta_{kj}^0| - |\alpha_n u_{kj}
| > 0 \end{equation}
for $n$ large enough and $\beta_{kj}^0 \neq 0$.
Hence, we have
\begin{eqnarray*}
& & p_{\lambda_n,\V{w}_n} ( \V{\beta}_{nk}^0+ \alpha_n \V{u}_k )
- p_{\lambda_n,\V{w}_n}(\V{\beta}_{nk}^0 )
\nonumber \\
& = & \lambda_n ( \sqrt{ w_{n,k1} |\beta_{k1}^0 + \alpha_n u_{k1}
| + \ldots +w_{n,kp_k} |\beta_{kp_k}^0 + \alpha_n u_{kp_k} | }-
\sqrt{ w_{n,k1} |\beta_{k1}^0| + \ldots +w_{n,kp_k} |\beta_{kp_k}^0| } ) \nonumber \\
& \geq & \lambda_n ( \sqrt{ w_{n,k1} |\beta_{k1}^0 + \alpha_n u_{k1}
| + \ldots + w_{n,ks_k} |\beta_{ks_k}^0 + \alpha_n u_{ks_k} | } -
\sqrt{ w_{n,k1} |\beta_{k1}^0| + \ldots + w_{n,ks_k} |\beta_{ks_k}^0| } ) \nonumber \\
&\geq & \lambda_n ( \sqrt{ w_{n,k1} |\beta_{k1}^0| +
\ldots + w_{n,ks_k} |\beta_{ks_k}^0| - \alpha_n( w_{n,k1} | u_{k1}
| + \ldots + w_{n,ks_k} | u_{ks_k} | )} \nonumber \\
&&
~~~~~~~~~~~~~~ - \sqrt{ w_{n,k1} |\beta_{k1}^0| +
\ldots + w_{n,ks_k} |\beta_{ks_k}^0| } )
~~~~~~~~ (\mathrm{for~ n ~large~ enough, ~ by} ~~ (\ref{sec_app_infty:thm1_1}) )\nonumber \\
& = & \lambda_n \sqrt{ w_{n,k1} |\beta_{k1}^0| +
\ldots + w_{n,ks_k} |\beta_{ks_k}^0| } ( \sqrt { 1 - \gamma_{nk} } - 1 ),
\end{eqnarray*}
where $\gamma_{nk}$ is defined as $\gamma_{nk} =
\frac{\alpha_n(w_{n,k1} | u_{k1} | + \ldots +w_{n,ks_k} | u_{ks_k} |
) }{
w_{n,k1} |\beta_{k1}^0| + \ldots +w_{n,ks_k} |\beta_{ks_k}^0| } $.
For $n$ large enough, we have $0 \leq \gamma_{nk} < 1$
and $\gamma_{nk} \leq \frac{\alpha_n \| \V{u}_k \|
(w_{n,k1} + \ldots +w_{n,ks_k} ) }{
c_1 ( w_{n,k1} + \ldots +w_{n,ks_k} ) } =
\frac{\alpha_n \| \V{u}_k \| } { c_1 } \leq \frac{\alpha_nC } { c_1
} \rightarrow 0 $ with probability tending to 1 as $n \rightarrow
\infty$.
Therefore,
\begin{eqnarray*}
& & p_{\lambda_n,\V{w}_n} ( \V{\beta}_{nk}^0+ \alpha_n \V{u}_k )
- p_{\lambda_n,\V{w}_n}(\V{\beta}_{nk}^0 )
\nonumber \\
& \geq & \lambda_n \sqrt{ w_{n,k1} |\beta_{k1}^0| +
\ldots + w_{n,ks_k} |\beta_{ks_k}^0| } ( \sqrt { 1 - \gamma_{nk} } - 1 ) \nonumber \\
& \geq & \lambda_n \sqrt{ w_{n,k1} |\beta_{k1}^0| +
\ldots + w_{n,ks_k} |\beta_{ks_k}^0| }
\left( \frac{1+|o_p(1)|}{2} ( - \gamma_{nk} ) \right) ~~~~~~~~~~~~~~~~~
\nonumber \\
&&~~~~~~ ( \mathrm{Using~} \gamma_{nk} = o_p(1) \mathrm{~and ~Taylor ~expansion } )~~~~~~~~~~~~~~~~~
\nonumber \\
& \geq & - \lambda_n \frac{\alpha_n(w_{n,k1} | u_{k1} | + \ldots
+w_{n,ks_k} | u_{ks_k} | ) }{ \sqrt{
w_{n,k1} |\beta_{k1}^0| + \ldots +w_{n,ks_k} |\beta_{ks_k}^0| } }
\left( \frac{1+|o_p(1)|}{2} \right) \nonumber \\
& \geq & -\alpha_n \lambda_n
\frac{ \| \V{u}_k \| \sqrt{a_n s_k} }{ 2 \sqrt{c_1} }
(1+|o_p(1)| ).
\end{eqnarray*}
Therefore, the term $(II)$ in (\ref{sec_app_infty0072a}) is bounded
by
$$ n \alpha_n \lambda_n \left( \sum_{k=1}^{S_n}
\frac{ \| \V{u}_k \| \sqrt{a_n s_k} }{ 2 \sqrt{c_1} } \right) (1+|o_p(1)|),$$
which is further bounded by
$$ n \alpha_n \lambda_n \sqrt{a_n } (
\| \V{u} \| \cdot
\frac{ \sqrt{P_n} }{ 2 \sqrt{ c_1 } } ) (1+|o_p(1)|).
$$
Note that $\alpha_n = \sqrt{P_n} ( n^{-1/2} + \lambda_n \sqrt{a_n}
/2\sqrt{c_1} ) $, hence the above expression is bounded by
$$ \| \V{u} \| n \alpha_n^2 (1+|o_p(1)|).$$
This term is also dominated by the first term of $I_2$ on $\|\V{u}\|
= C$ uniformly. Therefore, $D_n(\V{u}) < 0$ is satisfied uniformly
on $\|\V{u}\| = C$. This completes the proof of the theorem.
\subsection*{Proof of Theorem 4 }
We have proved that if $ \lambda_n \sqrt{a_n} = O_p(n^{-1/2} )$,
there exists a root-$(n/P_n)$ consistent estimate $\hat{\V{\beta}}_n $.
Now we prove that this root-$(n/P_n)$ consistent estimate has
the oracle sparsity under the condition
$ \frac{P_n^{2} }{\lambda_n^2 {b_n}} = o_p( n )$, i.e.,
$\hat{\beta}_{kj} = 0$ with probability tending to 1 if
${\beta}_{kj}^0 = 0$.
Using Taylor's expansion, we have
\begin{eqnarray} \label{sec_app_infty076}
\frac{\partial Q_n(\V{\beta}_n) }{\partial\beta_{kj}} & = &
\frac{\partial L_n(\V{\beta}_n) }{\partial\beta_{kj}} -
n \frac{\partial p_{\lambda_n,\V{w}_n}(\V{\beta}_{nk} ) }{\partial\beta_{kj}}
\nonumber \\
& = & \frac{\partial L_n(\V{\beta}_n^0) }{\partial\beta_{kj}} +
\sum_{k_{1}=1}^{K_n} \sum_{j_{1}=1}^{p_{k_{1}}}
\frac{\partial^2L_n(\V{\beta}^0)}
{\partial\beta_{kj}\partial\beta_{k_{2}j_{2}} } ( \beta_{k_{1}j_{1}}
- \beta_{k_{1}j_{1}}^0 )
\nonumber \\
& & + \frac{1}{2} \sum_{k_{1}=1}^{K_n} \sum_{j_{1}=1}^{p_{k_{1}}}
\sum_{k_{2}=1}^{K_n} \sum_{j_{2}=1}^{p_{k_{2}}}
\frac{\partial^3L_n({\V{\beta}_n^{\ast}})}
{\partial\beta_{kj}\partial\beta_{k_{1}j_{1}}\partial\beta_{k_{2}j_{2}}}
( \beta_{k_{1}j_{1}} - \beta_{k_{1}j_{1}}^0 )
( \beta_{k_{2}j_{2}} - \beta_{k_{2}j_{2}}^0 )
\nonumber \\
& &
- \frac{ n \lambda_n w_{n,kj} }{ 2\sqrt{w_{n,k1} |\beta_{k1}| +
\ldots + w_{n,kp_k} |\beta_{kp_k}| } } \mathrm{sgn} (\beta_{kj})
\\ \nonumber
& \F{trace}iangleq & I_1 + I_2 + I_3 + I_4,
\end{eqnarray}
where ${\V{\beta}_n^{\ast}}$ lies between ${\V{\beta}_n}$ and
$\V{\beta}_n^0$.
Using the argument in the proof of Lemma 5 of Fan and Peng (2004),
for any $\V{\beta}_n$ satisfying $\| {\V{\beta}}_n -\V{\beta}_n^0
\| = O_p( \sqrt{P_n/n} )$, we have
\begin{eqnarray*}
I_1 & = & O_p(\sqrt{n}) = O_p( \sqrt{nP_n} ), \\
I_2 & = & O_p( \sqrt{nP_n} ), \\
I_3 & = & o_p( \sqrt{nP_n} ).
\end{eqnarray*}
Then, since $\hat{\V{\beta}}_n $ is a root-$(n/P_n)$ consistent estimate
maximizing $Q_n(\V{\beta}_n)$, if $\hat{\beta}_{kj} \neq 0$, we have
\begin{eqnarray} \label{sec_app_infty085}
\left. \frac{\partial Q_n(\V{\beta}_n)
}{\partial\beta_{kj}}\right|_{{\V{\beta}_n} = \hat{\V{\beta}}_n } &
= & O_p( \sqrt{nP_n} ) - \frac{ n \lambda_n w_{n,kj} }{ 2
\sqrt{w_{n,k1} |\hat{\beta}_{k1}| + \ldots + w_{n,kp_k}
|\hat{\beta}_{kp_k}| } } \mathrm{sgn}
(\hat{\beta}_{kj}) \nonumber \\
& = & 0.
\end{eqnarray}
Therefore, $$ \frac{ n \lambda_n w_{n,kj} }{ \sqrt{ w_{n,k1}
|\hat{\beta}_{k1}| + \ldots + w_{n,kp_k} |\hat{\beta}_{kp_k}| } } =
O_p( \sqrt{nP_n} )~~~~~~\mathrm{for}~\hat{\beta}_{kj} \neq 0. $$
This can be extended to
$$ \frac{ n \lambda_n w_{n,kj}
|\hat{\beta}_{kj}|}{ \sqrt{ w_{n,k1} |\hat{\beta}_{k1}| + \ldots +
w_{n,kp_k} |\hat{\beta}_{kp_k}| } } = |\hat{\beta}_{kj}|
O_p(\sqrt{nP_n} ), $$ for any $\hat{\beta}_{kj}$ with
$\hat{\V{\beta}}_{nk} \neq 0 $. If we sum this over all $j$ in the
$k$th group, we have
\begin{equation}\label{sec_app_infty085a}
n \lambda_n \sqrt{w_{n,k1} |\hat{\beta}_{k1}| + \ldots + w_{n,kp_k}
|\hat{\beta}_{kp_k}| } = \sum_{j=1}^{p_k}|\hat{\beta}_{kj}| O_p(
\sqrt{nP_n} ).
\end{equation}
Since $\hat{\V{\beta}}_n $ is a root-$(n/P_n)$ consistent estimate
of ${\V{\beta}}_n ^0$, we have $|\hat{\beta}_{kj}| = O_p(1)$
for $(k,j) \in {{\cal{A}}_n } $
and $|\hat{\beta}_{kj}| = O_p( \sqrt{P_n/n} )$
for $(k,j) \in {\cal{B}}_n \cup {\cal{C}}_n $.
Now for any $k$ and $j$ satisfying ${\beta}_{kj}^0 = 0$ and
$\hat{\beta}_{kj} \neq 0$, equation (\ref{sec_app_infty085}) can be
written as:
\begin{eqnarray} \label{sec_app_infty086} \left.
\frac{\partial Q_n(\V{\beta}_n)
}{\partial\beta_{kj}}\right|_{{\V{\beta}_n}= \hat{\V{\beta}}_n } & =
&
\frac{ 1 }{ 2 \lambda_n \sqrt{w_{n,k1} |\hat{\beta}_{k1}|
+ \ldots + w_{n,kp_k} |\hat{\beta}_{kp_k}| } } \\ \nonumber
& & ( O_p( \sqrt{P_n/n} )
n \lambda_n \sqrt{w_{n,k1} |\hat{\beta}_{k1}| + \ldots + w_{n,kp_k} |\hat{\beta}_{kp_k}| }
\\ \nonumber
& &
- n \lambda_n^2 w_{n,kj} \mathrm{sgn} (\hat{\beta}_{kj}) ) \\ \nonumber
& = & 0.
\end{eqnarray}
Denote $ h_{nk} = O_p( \sqrt{P_n/n} )
n \lambda_n \sqrt{w_{n,k1} |\hat{\beta}_{k1}| +
\ldots + w_{n,kp_k} |\hat{\beta}_{kp_k}| } $.
Let $ h_n = \sum_{k=1}^{K_n} h_{nk} $.
By equation (\ref{sec_app_infty085a}), we have
$ h_n = \sum_{k=1}^{K_n} O_p( \sqrt{P_n/n} ) \sum_{j=1}^{p_k}|\hat{\beta}_{kj}| O_p(
\sqrt{nP_n} ) = O_p(P_n^2) $. Since $ \frac{ {P_n^{2}} }{\lambda_n^2
{b_n}} = o_p(n)$ guarantees that $n \lambda_n^2 b_n $ dominates
$h_n$ with probability tending to 1 as $n \rightarrow \infty$,
the first term in (\ref{sec_app_infty086})
is dominated by the second term as $n \rightarrow \infty$
uniformly for all $k$ and $j$ satisfying ${\beta}_{kj}^0 = 0$
since $w_{n,kj} \ge b_n $ and $h_n > h_{nk}$.
Denote $ g_{nk} = 2\lambda_n \sqrt{w_{n,k1} |\hat{\beta}_{k1}| +
\ldots + w_{n,kp_k} |\hat{\beta}_{kp_k}| } / (n \lambda_n^2 b_n ) $.
Let $ g_n = \sum_{k=1}^{K_n} g_{nk} $. By equation
(\ref{sec_app_infty085a}), we have
$ g_n = 2 \sum_{k=1}^{K_n} (1/n) \sum_{j=1}^{p_k}|\hat{\beta}_{kj}| O_p(
\sqrt{nP_n} ) /(n \lambda_n^2 b_n ) = o_p(1/\sqrt{nP_n}) $. The
absolute value of the second term in (\ref{sec_app_infty086}) is
bounded below by $1/g_n$. So with probability uniformly converging
to 1 the second term in the derivative $\frac{\partial Q(\V{\beta})
}{\partial \beta_{kj}}|_{\V{\beta} = \hat{\V{\beta}}_n}$ will go to
$\infty$ as $n \rightarrow \infty$, which is a contradiction with
equation (\ref{sec_app_infty086}). Therefore, for any $k$ and $j$
satisfying ${\beta}_{kj}^0 = 0$, we have $\hat{\beta}_{kj} = 0$ with
a probability tending to 1 as $n \rightarrow \infty$. We have $
\hat{\V{\beta}}_{{\cal{D}}_n } = 0 $ with probability tending to 1
as well.
Now we prove the second part of Theorem 4. From the above proof, we
know that there exists $ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }},
\V{0}) $
with probability tending to 1, which is a
root-$(n/P_n)$ consistent local maximizer of
$Q( {\V{\beta}_n} )$. With a slight abuse of notation, let
$ Q_n(\V{\beta}_{n,{{\cal{A}}_n }} ) = Q_n(\V{\beta}_{n,{\cal{A}}_n
},\V{0} ) $. Using the Taylor expansion on $ \nabla
Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n }} )$ at point
$\V{\beta}_{n,{{\cal{A}}_n }}^0$, we have
\begin{eqnarray} \label{sec_app_infty087}
& &
\frac{1}{n} ( \nabla^2 L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
- \nabla J_{n } ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }}) )
\\ \nonumber
& = & - \frac{1}{n} \left( \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
+ \frac{1}{2} \T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{\cal{A}}_n
}^0)}
\nabla^2 \{ \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^\ast ) \}
( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \right)
,
\end{eqnarray}
where ${\V{\beta}}_{n,{{\cal{A}}_n }}^\ast$ lies between
$\hat{\V{\beta}}_{n,{{\cal{A}}_n }}$ and ${\V{\beta}}_{n,
{\cal{A}}_n }^0$.
Now we define
$$ {\cal{C}}_n \F{trace}iangleq \frac{1}{2} \T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{\cal{A}}_n
}^0) }
\nabla^2 \{ \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^\ast ) \}
( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ). $$
Using the Cauchy-Schwarz inequality, we have
\begin{eqnarray} \label{sec_app_infty089}
\left\| \frac{1}{n} {\cal{C}}_n \right\|^2 &
\le &
\frac{1}{n^2} \sum_{i=1}^n n \| \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0 \|^4
\sum_{k_1 =1}^{S_n} \sum_{j_1=1}^{p_k}
\sum_{k_2 =1}^{S_n} \sum_{j_2=1}^{p_k} \sum_{k_3=1}^{S_n}
\sum_{j_3=1}^{p_k} M_{{nk_{1}j_{1}}{k_{2}j_{2}}{k_{3}j_{2}}}^{3}
(\V{V}_{ni})
\nonumber \\
& = & O_p( {P_n^2}/{n^2} ) O_p(P_n^3) =
O_p( {P_n^5}/{n^2} ) = o_p ( {1}/{n} )
. \end{eqnarray}
Since $\frac{ {P_n^5} } {n} \rightarrow 0$ as $n \rightarrow
\infty$, by Lemma 8 of Fan and Peng (2004), we have
$$ \left\| \frac{1}{n} \nabla^2 L_n(
{\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) + \M{I}_n(
{\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \right\| = o_p ( {1}/{P_n} )
$$
and
\begin{equation}\label{sec_app_infty090} \left\|
\left(\frac{1}{n} \nabla^2 L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) +
\M{I}_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \right)
( \hat{\V{\beta}}_{n,{\cal{A}}_n } - {\V{\beta}}_{n,{\cal{A}}_n }^0 ) \right\| =
o_p ( {1}/{\sqrt{nP_n}} ) =
o_p ( {1}/{\sqrt{n }} ).\end{equation}
Since
\begin{eqnarray} \label{appB:crit102}
& & \sqrt{w_{n,k1} |\hat{\beta}_{k1}| + \ldots + w_{n,ks_k}
|\hat{\beta}_{ks_k}| } \nonumber \\
& = & \sqrt{w_{n,k1} | {\beta}_{k1}^0|( 1+O_p(\sqrt{P_n/n}) ) +
\ldots + w_{n,ks_k} |{\beta}_{ks_k}^0|(
1+O_p(\sqrt{P_n/n}) ) } \nonumber \\
& = & \sqrt{w_{n,k1} | {\beta}_{k1}^0| + \ldots + w_{n,ks_k}
|{\beta}_{ks_k}^0| } ( 1+O_p(\sqrt{P_n/n}) ) \nonumber,
\end{eqnarray}
we have
$$ \frac{
\lambda_n w_{n,kj} }{ \sqrt{w_{n,k1} |\hat{\beta}_{k1}| + \ldots +
w_{n,ks_k} |\hat{\beta}_{ks_k}| } } = \frac{ \lambda_n w_{n,kj} }{
\sqrt{w_{n,k1} | {\beta}_{k1}^0| + \ldots + w_{n,ks_k}
|{\beta}_{ks_k}^0| } } ( 1+O_p(\sqrt{P_n/n}) ) .$$
Furthermore, since $$ \frac{
\lambda_n w_{n,kj} }{ \sqrt{w_{n,k1} | {\beta}_{k1}^0| + \ldots +
w_{n,ks_k} |{\beta}_{ks_k}^0| } } \le \frac{ \lambda_n w_{n,kj} }{
\sqrt{ w_{n,kj} c_1 } }
\leq \frac{ \lambda_n \sqrt{ a_n }
}{ \sqrt{ c_1 } }
= o_p( {(nP_n)}^{-1/2} ) $$ for $ (k,j) \in {{\cal{A}}_n } $, we
have
$$ \left( \frac{1}{n} \nabla J_{n } (
\hat{\V{\beta}}_{n,{\cal{A}}_n }) \right)_{kj} = \frac{ \lambda_n
w_{n,kj} }{ 2 \sqrt{w_{n,k1} |\hat{\beta}_{k1}| + \ldots +
w_{n,ks_k} |\hat{\beta}_{ks_k}| } } = o_p( {(nP_n)}^{-1/2} ) $$ and
\begin{equation}\label{sec_app_infty091}
\left\| \frac{1}{n} \nabla J_{n } (
\hat{\V{\beta}}_{n,{\cal{A}}_n }) \right\| \le \sqrt{P_n} o_p(
{(nP_n)}^{-1/2} ) = o_p ( {1}/{\sqrt{n }} ).
\end{equation}
Together with (\ref{sec_app_infty089}), (\ref{sec_app_infty090}) and
(\ref{sec_app_infty091}), from (\ref{sec_app_infty087}) we have
$$ \M{I}_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
= \frac{1}{n} \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
+ o_p( {1}/{\sqrt{n }} ).$$
Now using the same argument as in the proof of Theorem 2 of Fan and
Peng (2004), we have
\begin{equation*}
\sqrt{n} \M{A}_n \M{I}_n^{1/2} ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0
) \rightarrow \sqrt{n} \M{A}_n \M{I}_n^{-1/2} ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
\left(\frac{1}{n} \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )\right)
\rightarrow {\cal{N}} (\V{0}, \M{G}),
\end{equation*}
where $ \M{A}_n $ is a $q \times |{{\cal{A}}_n }| $ matrix such that
$ \M{A}_n \T{\M{A}_n} \rightarrow \M{G}$ and $\M{G}$ is a $q\times
q$ nonnegative symmetric matrix.
\subsection*{Proof of Theorem 2}
Note that when $w_{n,kj} = 1 $, we have $a_n = 1$ and $b_n = 1$. The
conditions $ \lambda_n \sqrt{a_n } = O_p(n^{-1/2} )$ and
$ \frac{ {P_n^{2}} }{\lambda_n^2 {b_n}} = o_p(n )$ in Theorem
4 become $\lambda_n \sqrt{n} = O_p(1) $ and $\frac{P_n}{\lambda_n
\sqrt{n} } \rightarrow 0$. These two conditions cannot be satisfied
simultaneously by adjusting $\lambda_n$, which implies that
$\F{Pr}(\hat{\V{\beta}}_{\cal{D}} = 0) \rightarrow 1 $ cannot be
guaranteed.
We will prove that by choosing $\lambda_n$ satisfying
$\sqrt{n}{\lambda_n} = O_p(1) $ and $P_n n^{-3/4}/{\lambda_n}
\rightarrow 0$ as $n \rightarrow \infty$, we can have a root-$n$
consistent local maximizer $\hat{\V{\beta}}_n =
\T{(\hat{\V{\beta}}_{{\cal A}_n}, \hat{\V{\beta}}_{{\cal B}_n},
\hat{\V{\beta}}_{{\cal C}_n})}$ such that
$\F{Pr}(\hat{\V{\beta}}_{{\cal C}_n} = 0) \rightarrow 1$.
Similar as
in the proof of Theorem 4, we let
$ h_n^{\prime} = \sum_{k=S_n+1}^{K_n} h_{nk} $.
By equation (\ref{sec_app_infty085a}), we have
$ h_n^{\prime} = \sum_{k=S_n+1}^{K_n} O_p( \sqrt{P_n/n} )
\sum_{j=1}^{p_k}|\hat{\beta}_{kj}| O_p(
\sqrt{nP_n} ) = O_p(P_n^2/\sqrt{n}) $. Since $P_n
n^{-3/4}/{\lambda_n} \rightarrow 0$ guarantees that $n \lambda_n^2 $
dominates $h_n^{\prime}$ with probability tending to 1 as $n
\rightarrow \infty$,
the first term in (\ref{sec_app_infty086})
is dominated by the second term as $n \rightarrow \infty$
uniformly
for any $k$ satisfying ${\V{\beta}}_{nk}^0 = 0$
since $w_{n,kj} = 1 $ and $h_n^{\prime} > h_{nk}$.
Similar as in the proof of Theorem 4, we have $
\hat{\V{\beta}}_{{\cal{C}}_n } = 0 $ with probability tending to 1.
\subsection*{Proof of Theorem 5}
Let $ N_n = |{{\cal{A}}_n }| $ be the number of nonzero parameters.
Let $\M{B}_n$ be an $( N_n - q ) \times N_n $ matrix which satisfies
$\M{B}_n \T{\M{B}}_n = \M{I}_{N_n - q } $ and $\M{A}_n \T{\M{B}}_n =
0 $. As ${\V{\beta}}_{n,{{\cal{A}}_n }}$ is in the orthogonal
complement to the linear space that is spanned by the rows of
$\M{A}_n$ under the null hypothesis $H_0$, it follows that
$$ {\V{\beta}}_{n,{{\cal{A}}_n }} = \T{\M{B}}_n {\V{\gamma}}_{n} , $$
where ${\V{\gamma}}_{n}$ is an $( N_n - q ) \times 1 $ vector. Then,
under $H_0$ the penalized likelihood estimator is also the local
maximizer $\hat{\V{\gamma}}_{n}$ of the problem
$$ Q_n({\V{\beta}}_{n,{{\cal{A}}_n }}) = \max_{ {\V{\gamma}}_{n} } Q_n( \T{\M{B}}_n {\V{\gamma}}_{n} ). $$
To prove Theorem 5 we need the following two lemmas.
\begin{lemma}
Under condition $(b)$ of Theorem 4 and the null hypothesis $H_0$, we
have
$$ \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - {\V{\beta}}_{n,{{\cal{A}}_n }}^0
= \frac{1}{n} \M{I}_n^{-1} ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
\nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) + o_p(n^{-1/2}), $$
$$ \T{\M{B}}_n ( \hat{\V{\gamma}}_{n} - {\V{\gamma}}_{n}^0
) = \frac{1}{n} \T{\M{B}}_n \{ \M{B}_n \M{I}_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \T{\M{B}}_n \}^{-1}
\M{B}_n \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) + o_p(n^{-1/2}). $$
\end{lemma}
\subsection*{Proof of of Lemma 3}
We need only prove the second equation. The first equation can be
shown in the same manner. Following the proof of Theorem 4, it
follows that under $H_0$,
$$ \M{B}_n \M{I}_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \T{\M{B}}_n ( \hat{\V{\gamma}}_{n} -
{\V{\gamma}}_{n}^0 ) = \frac{1}{n} \M{B}_n \nabla L_n(
{\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) + o_p(n^{-1/2}).
$$
As the eigenvalue $\lambda_i ( \M{B}_n \M{I}_n(
{\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \T{\M{B}}_n ) $ is uniformly
bounded away from 0 and infinity, we have
$$ \T{\M{B}}_n ( \hat{\V{\gamma}}_{n} - {\V{\gamma}}_{n}^0
) = \frac{1}{n} \T{\M{B}}_n \{ \M{B}_n \M{I}_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \T{\M{B}}_n \}^{-1}
\M{B}_n \nabla L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) + o_p(n^{-1/2}). $$
\begin{lemma}
Under condition $(b)$ of Theorem 4 and the null hypothesis $H_0$, we
have
\begin{eqnarray} \label{sec_app_infty095}
& &
Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n }}) - Q_n( \T{\M{B}}_n \hat{\V{\gamma}}_{n} )
\\ \nonumber
& = & \frac{n}{2} \T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\M{I}_n ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) } + o_p(1).
\end{eqnarray}
\end{lemma}
\subsection*{Proof of Lemma 4}
A Taylor's expansion of $ Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n
}}) - Q_n( \T{\M{B}}_n \hat{\V{\gamma}}_{n} )$ at the point $
\hat{\V{\beta}}_{n,{{\cal{A}}_n }} $ yields
$$ Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n }}) - Q_n( \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) = T_1 + T_2 + T_3 + T_4,$$
where
\begin{eqnarray*}
T_1 & = & \T{\nabla} Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n }})
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) } , \\
T_2 & = & - \frac{1}{2} \T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\nabla^2 L_n( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} )
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) } , \\
T_3 & = & \frac{1}{6} \T{\nabla}
\{
\T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\nabla^2 L_n( {\V{\beta}}_{n,{{\cal{A}}_n }}^{\star} )
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\}
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) } , \\
T_4 & = & \frac{1}{2} \T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\nabla^2 J_{n} ( {\V{\beta}}_{n,{{\cal{A}}_n }}^{\ast} )
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} )
} .
\end{eqnarray*}
We have $ T_1 = 0 $ as $ \T{\nabla}
Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n }}) = 0$.
Let $ \M{\Theta}_n = \M{I}_n ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )$
and $ \M{\Phi}_n = \frac{1}{n} \nabla L_n(
{\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) $. By Lemma 2 we have
\begin{eqnarray*}
& & { ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\nonumber \\
& = & \M{\Theta}_n^{-1/2} \{
\M{I}_n - \M{\Theta}_n^{1/2} \T{\M{B}}_n ( \M{B}_n \M{\Theta}_n \T{\M{B}}_n )^{-1} \M{B}_n\M{\Theta}_n^{1/2}
\}
\M{\Theta}_n^{-1/2} \M{\Phi}_n \nonumber \\
& & + o_p(n^{-1/2}).
\end{eqnarray*}
$\M{I}_n - \M{\Theta}_n^{1/2} \T{\M{B}}_n ( \M{B}_n \M{\Theta}_n \T{\M{B}}_n )^{-1} \M{B}_n\M{\Theta}_n^{1/2}$ is an
idempotent matrix with rank $q$. Hence, by a standard argument and condition
(A2),
$$ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) = O_p( \sqrt{\frac{q}{n} } ) .$$
We have \begin{equation} \label{sec_app_infty096} \left( \frac{1}{n}
\nabla^2 J_{n } ( {\V{\beta}}_{n,{\cal{A}}_n }) \right)_{kjk_1j_1} =
0, ~~~~~~\mathrm{for}~k \ne k_1 ~~~~~~
\end{equation} and
\begin{eqnarray} \label{sec_app_infty097}
& & \left( \frac{1}{n} \nabla^2 J_{n } (
{\V{\beta}}_{n,{\cal{A}}_n }^{\ast} ) \right)_{kjkj_1} \nonumber \\
& = & \frac{ \lambda_n
w_{n,kj} w_{n,kj_1} }{ 4 ( {w_{n,k1} | {\beta}_{k1}^{\ast} | +
\ldots + w_{n,ks_k} | {\beta}_{ks_k}^{\ast} | } )^{3/2} } \nonumber \\
& = & \frac{ \lambda_n
w_{n,kj} w_{n,kj_1} }{ 4 ( {w_{n,k1} | {\beta}_{k1}^{0} | + \ldots
+ w_{n,ks_k} | {\beta}_{ks_k}^{0} | } )^{3/2} } (1+o_p(1) ) \nonumber \\
& \le & \frac{ \lambda_n \sqrt{a_n} }{ 4 ( c_1 )^{3/2} } (1+o_p(1) ) \nonumber \\
& = & o_p( (nP_n)^{-1/2} ).
\end{eqnarray}
Combining (\ref{sec_app_infty096}), (\ref{sec_app_infty097}) and
condition $ q < P_n $, following the proof of $I_3$ in Theorem 3, we
have
$$ T_3 = O_p( n P_n^{3/2} n^{-3/2} q ^{3/2} ) = o_p(1) $$
and
\begin{eqnarray*}
T_4 & \le & n \left\| \frac{1}{n} \nabla^2 J_{n } (
{\V{\beta}}_{n,{\cal{A}}_n }^{\ast} ) \right\|
\| \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} \|^2
\nonumber \\
& = & n {P_n} o_p( (nP_n)^{-1/2} ) O_p(\frac{q}{n})
\nonumber \\
& = & o_p(1).
\end{eqnarray*}
Thus,
\begin{equation} \label{sec_app_infty098}
Q_n(\hat{\V{\beta}}_{n,{{\cal{A}}_n }}) - Q_n( \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) =
T_2 + o_p(1).
\end{equation}
It follows from Lemmas 8 and 9 of Fan and Peng (2004) that
$$ \left\| \frac{1}{n} \nabla^2 L_n( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} ) +
\M{I}_n ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 ) \right\| =
o_p\left( \frac{1}{\sqrt{P_n}} \right).$$
Hence, we have
\begin{eqnarray} \label{sec_app_infty099}
& & \frac{1}{2} \T{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) }
\{
\nabla^2 L_n( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} )
+ n \M{I}_n ( {\V{\beta}}_{n,{{\cal{A}}_n }}^0 )
\}
{ ( \hat{\V{\beta}}_{n,{{\cal{A}}_n }} - \T{\M{B}}_n \hat{\V{\gamma}}_{n} ) } \nonumber \\
& \le & o_p\left( n\frac{1}{\sqrt{P_n}} \right) O_p( {\frac{q}{n} } ) =
o_p(1).
\end{eqnarray}
The combination of (\ref{sec_app_infty098}) and (\ref{sec_app_infty099}) yields (\ref{sec_app_infty095}).
\subsection*{Proof of Theorem 5 }
Given Lemmas 3 and 4, the proof of the Theorem is similar to the
proof of Theorem 4 in Fan and Peng (2004).
\end{document} |
\begin{document}
\begin{abstract}
We define and study Cartan--Betti numbers of a graded ideal $J$ in the exterior algebra
over an infinite field which include the usual graded Betti numbers of $J$ as a special case.
Following ideas of Conca regarding Koszul--Betti numbers over the symmetric algebra,
we show that Cartan--Betti numbers increase by passing to the generic initial ideal and
the squarefree lexsegement ideal respectively.
Moreover, we characterize the cases where the inequalities become equalities.
As combinatorial applications
of the first part of this note and some further symmetric algebra methods
we establish results about algebraic shifting of simplicial complexes
and use them to compare different shifting operations. In particular,
we show that each shifting operation does not decrease the number of
facets, and that the exterior shifting is the best among the exterior
shifting operations in the sense that it increases the number of
facets the least.
\end{abstract}
\maketitle
\; | \;ection{Introduction}
Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ of characteristic $0$,
let $I \; | \;ubset S$ be a graded ideal and denote by
$\beta^S_{ij}(S/I)=\dim_K \Tor_i^S(S/I,K)_j$
the graded Betti numbers of $S/I$.
In the last decades
the graded Betti numbers $\beta^S_{ij}(S/I)$ were studied intensively.
To $I$ one associates several important
monomial ideals like the generic initial ideal $\gin(I)$ with respect to the
reverse lexicographic order, or the lexsegment ideal $\lex(I)$.
It is well-known by work of \cite{BI93}, \cite{HU93} and \cite{PA96}
that there are inequalities
$\beta^S_{ij}(S/I) \leq \beta^S_{ij}(S/\gin(I)) \leq \beta^S_{ij}(S/\lex(I))$ for all $i,j$.
The cases where these are equalities for all $i,j$ have been characterized by
Aramova--Herzog--Hibi \cite{ARHEHI00ideal} and \cite{HEHI99}.
The first inequality is an equality for all $i,j$ if and only if $J$ is a componentwise linear ideal,
and we have equalities everywhere for all $i,j$ if and only if $J$ is a Gotzmann ideal.
All these results were generalized to so--called Koszul--Betti numbers in \cite{CO04}.
Let $E=K\langle e_1,\dots,e_n \rangle$ be the
exterior algebra over an infinite field $K$ and $J \; | \;ubset E$ a graded ideal.
Aramova--Herzog--Hibi (\cite{ARHEHI97}) showed that the constructions and results mentioned above
hold similarly for $J$. More precisely, there exists a generic initial ideal $\gin(J)\; | \;ubset E$
with respect to the reverse lexicographic order,
and the unique squarefree lexsegment ideal $\lex(J)\; | \;ubset E$ with the same Hilbert function as $J$.
Let $\beta^E_{ij}(E/J)=\dim_K \Tor_i^E(E/J,K)_j$ be the graded Betti numbers of $E/J$.
Then we also have the inequalities
$\beta^E_{ij}(E/J) \leq \beta^E_{ij}(E/\gin(J)) \leq \beta^E_{ij}(E/\lex(J))$
for all integers $i,j$. The first inequality is again an equality for all $i,j$
if and only if $J$ is a componentwise linear ideal
as was observed in \cite{ARHEHI00ideal}.
The first part of this paper
is devoted to extend the latter results to statements about Cartan--Betti numbers,
similarly as Conca did in the symmetric algebra case for Koszul--Betti numbers.
Over the exterior algebra there exists
the construction of the Cartan complex and Cartan homology (see Section \ref{cartan} for details),
which behave in many ways like the Koszul complex and Koszul homology.
Following Conca's ideas we define the Cartan--Betti numbers as
$
\beta^E_{ijp}(E/J)= \dim_K H_i(f_1,\dots,f_p;E/J)_j
$
where $H_i(f_1,\dots,f_p;E/J)$ is the $i$-th Cartan homology
of $E/J$ with respect to a generic sequence of linear forms
$f_1,\dots,f_p$ for $1\leq p \leq n$. Observe that this definition does not depend on the chosen
generic sequence and that these modules are naturally graded.
We set $H_i(f_1,\dots,f_p;E/J)=0$ for $i>p$. Note that
$\beta^E_{ijn}(E/J)= \beta^E_{ij}(E/J)$ are the graded Betti numbers of $E/J$.
Observe that a generic initial ideal $\gin_\tau(J)$ can be defined with respect to any term
order
$\tau$ on $E$.
Our first main result, Theorem \ref{mainleq}, shows that, for all integers $i,j,p$:
$$
\beta^E_{ijp}(E/J) \leq \beta^E_{ijp}(E/\gin_\tau(J)) \leq
\beta^E_{ijp}(E/\lex(J)) .
$$
Analogously to result in \cite{CO04} we can characterize precisely
the cases where the inequalities become equalities.
At first we have:
\begin{thm}
Let $J \; | \;ubset E$ be a graded ideal.
The following conditions are equivalent:
\begin{enumerate}
\item
$\beta_{ijp}^E(E/J)= \beta_{ijp}^E(E/\gin(J))$ for all $i,j,p$;
\item
$\beta_{1jn}^E(E/J)= \beta_{1jn}^E(E/\gin(J))$ for all $j$;
\item
$J$ is a componentwise linear ideal;
\item
A generic sequence of linear forms $f_1,\dots,f_n$ is a proper sequence of $E/J$.
\end{enumerate}
\end{thm}
We refer to Section \ref{cartan} for the definition of a proper sequence.
Moreover, we show:
\begin{thm}
For each graded ideal $J \; | \;ubset E$, the following statements are
equivalent:
\begin{enumerate}
\item
$\beta^E_{ijp}(E/J)= \beta^E_{ijp}(E/\lex(J))$
for all $i,j,p$;
\item
$\beta^E_{1jn}(E/J)= \beta^E_{1jn}(E/\lex(J))$
for all $j$;
\item
$J$ is a Gotzmann ideal in the exterior algebra;
\item
$\beta^E_{0jp}(E/J) = \beta^E_{0jp}(E/\lex(J))$
for all $j,p$
and
$J$ is componentwise linear.
\end{enumerate}
\end{thm}
Section \ref{cartan} ends with the discussion which generic initial
ideal increases the Cartan--Betti numbers the least.
In Theorem \ref{mainginleq} we show that for all $i,j,p$ and for any term order $\tau$:
$$
\beta_{ijp}^E(E/J)\leq \beta_{ijp}^E(E/\gin(J)) \leq
\beta_{ijp}^E(E/\gin_\tau(J)) .
$$
The second part of this note presents combinatorial applications
of the results mentioned above and some symmetric algebra methods.
More precisely,
we study properties of simplicial complexes under
shifting operations. Let $\mathcal{C}([n])$ be
the set of simplicial complexes on $n$ vertices. Following
\cite{KA84}, \cite{KA01}, a shifting operation is a map
$\; | \;hift \colon \mathcal{C}([n]) \to \mathcal{C}([n])$ satisfying
certain conditions. We refer to Section \ref{sec-def} for precise
definitions. Intuitively, shifting replaces each complex $\Gamma$ with
a combinatorially simpler complex $\Delta (\Gamma)$ that still
captures some properties of $\Gamma$. Shifting has become an important
technique that has been successfully applied in various contexts and
deserves to be investigated in its own right (cf., for example,
\cite{ARHEHI00}, \cite{ARHE}, \cite{BNT}, \cite{BNT2},
\cite{MH1}, \cite{MH2}, \cite{Nevo}).
Several shifting operations can
be interpreted algebraically. Denote by $I_{\Gamma}$ the
Stanley--Reisner ideal of $\Gamma$ in the polynomial ring $S =
K[x_1,\ldots,x_n]$. Then
symmetric shifting $\Delta^s$ can be realized by passing to a kind of
polarization of the generic initial ideal of $I_{\Gamma}$ with respect
to the reverse lexicographic order (cf.\ Example
\ref{exsymmetric}). Similarly, denote by $J_{\Gamma}$ the exterior
Stanley--Reisner ideal in the exterior algebra $E = K\langle
e_1,\dots,e_n\rangle$. The passage from $J_{\Gamma}$ to its generic
initial ideal with respect to any term order $\tau$ on $E$ leads to
the exterior algebraic shifting operation $\Delta^{\tau}$ (Example
\ref{exexterior}). If $\tau$ is the reverse-lexicographic order, we
call $\Delta^e := \Delta^{\tau}$ the exterior algebraic shifting.
While each shifting operation preserves the $f$-vector, the number of
facets may change. However, we show in Theorem \ref{thm-adeg-incr}
that it cannot decrease.
After recalling basic definitions in Section \ref{sec-def},
we first study symmetric algebraic shifting. To this end we use degree
functions. In Section \ref{sec-degree} we show that the smallest
extended degree is preserved under symmetric algebraic shifting whereas the
arithmetic degree $\adeg (\Gamma)$ may increase because it equals the
number of facets. However, the smallest extended degree and the
arithmetic degree of each shifted complex agree.
Furthermore we show that exterior algebraic shifting shift is the best exterior shifting operation in the sense
that it increases the number of facets the least, i.\ e.\ for any term order $\tau$
$$
\adeg \Gamma \leq \adeg \Delta^e(\Gamma) \leq \adeg \Delta^\tau(\Gamma).
$$
Moreover, $\Delta^e$ preserves the arithmetic degree of $\Gamma$ if
and only $\Gamma$ is sequentially Cohen--Macaulay. These results rely
on the full strength of the results in Section \ref{cartan}. The proof
also uses Alexander duality and a reinterpretation of the arithmetic
degree over the exterior algebra.
In Section \ref{ringprop} we use degree functions to give a
short new proof of the fact that $\Delta^s (\Gamma)$ is a pure complex
if and only if $\Gamma$ is Cohen--Macaulay. We then derive a
combinatorial interpretation of the smallest extended degree using
iterated Betti numbers and show that these numbers agree with the
$h$-triangle if and only if $\Gamma$ is sequentially Cohen--Macaulay.
For notions and results related to commutative algebra we refer to \cite{EI}, \cite{BRHE98} and \cite{VA}.
For details on
combinatorics we refer to the book \cite{ST96}
and \cite{KA01}.
\; | \;ection{Cartan--Betti numbers}
\label{cartan}
In this section we introduce and study Cartan--Betti numbers of
graded ideals in the exterior algebra $E=K\langle e_1,\dots,e_n \rangle$ over
an infinite field $K$. Our results rely on techniques from Gr\"obner basis
theory in the exterior algebra. Its basics are treated in \cite{ARHEHI97}. We begin with
establishing some extensions of the theory that are analogous to results over the
polynomial ring.
Recall that a monomial of degree $k$ in $E$ is an element
$e_F=e_{a_1} \wedge \ldots \wedge e_{a_k}$
where $F=\{a_1,\ldots,a_k\}$ is a subset of $[n]$ with $a_1< \ldots < a_k$.
To simplify notation, we write sometimes $f g = f \wedge g$ for any two elements $f, g \in E$.
We will use only term orders $\tau$ on $E$ that satisfy
$
e_1 >_{\tau} e_2 >_{\tau} \ldots >_{\tau} e_n.$
Given a term order $\tau$ on $E$
and a graded ideal $J\; | \;ubset E$
we denote by $\ini_\tau(J)$ and $\gin_\tau(J)$ respectively
the {\em initial ideal} of $J$ and the {\em generic initial ideal} of $J$
with respect to $\tau$.
We also write
$\ini(J)$ and $\gin(J)$ for
the initial ideal of $J$ and the generic initial ideal of $J$
with respect to the reverse lexicographic order on $E$ induced by
$e_1>\cdots>e_n$.
Recall that a monomial ideal
$J \; | \;ubset E$ is called a {\em squarefree strongly stable ideal} with respect to $e_1>\dots>e_n$
if for all $F\; | \;ubseteq [n]$ with $e_F \in J$
and all $i \in F$, $j<i$, $j \not\in F$
we have $e_j\wedge e_{F \; | \;etminus\{i\}} \in J$.
Given an ideal $J\; | \;ubset E$ and a term order $\tau$ on $E$, note that
$\gin_\tau(J)$ is
always squarefree strongly stable and that $\gin_\tau(J)=J$ iff $J$ is
squarefree
strongly stable which can be seen analogously to the case of a
polynomial ring. As
expected, the passage to initial ideals increases the graded Betti
numbers. The next
result generalizes \cite[Proposition 1.8]{ARHEHI97}. It is the
exterior analogue of a well-known fact for the polynomial ring (cf.\
\cite[Lemma 2.1]{CO04}).
\begin{prop}
\label{helpergb}
Let $J, J' \; | \;ubset E$ be graded ideals and let $\tau$ be any term
order on $E$. Then, for all $i, j \in \mathbb{Z}$:
$$
\dim_K [\tor^E_i (E/J, E/J')]_j \leq \dim_K [\tor^E_i (E/\inn_{\tau}
(J), E/\inn_{\tau} (J'))]_j.
$$
\end{prop}
\begin{proof}
This follows from standard deformation arguments. The proof is
verbatim the same as the
one of the symmetric version in \cite[Lemma 2.1]{CO04} after replacing
the polynomial
ring $S$ with the exterior algebra $E$.
\end{proof}
In the following we present exterior versions of results of Conca \cite{CO04}.
We follow the strategy of his proofs and replace symmetric algebra methods and the Koszul
complex by exterior algebra methods and the Cartan complex. We first recall the
construction of the Cartan complex (see, e.g., \cite{ARHEHI97} or \cite{HE01}). For
important results on this complex, we refer to \cite{ARAVHE00}, \cite{ARHEHI97}, \cite{ARHEHI98},
\cite{ARHEHI00ideal}.
For a sequence $\vb=v_1,\ldots, v_m \; | \;ubseteq E_1$, the Cartan complex
$C_{{\hbox{\large\bf.}}}(\vb;E)$ is defined to be the free divided power algebra $E\langle
x_1,\ldots,x_m \rangle$ together with a differential $\delta$. The free divided power
algebra $E\langle x_1,\ldots,x_m \rangle$ is generated over $E$ by the divided powers
$x_{i}^{(j)}$, \ $i=1,\ldots,m$ and $j\geq 0$, that satisfy the relations
$x_{i}^{(j)}x_{i}^{(k)}=\frac{(j+k)!}{j!k!}x_{i}^{(j+k)}$. We set $x_{i}^{(0)}=1$ and
$x_{i}^{(1)}=x_{i}$ for $i=1,\ldots, m$. Hence $C_{{\hbox{\large\bf.}}}(\vb;E)$ is a free $E$-module
with basis $x^{(a)}:=x_{1}^{(a_1)}\ldots x_{m}^{(a_m)}$, $a = (a_1,\ldots,a_m) \in
\mathbb{N}^{m}$. We set $\deg x^{(a)}=i$ if $|a|:=a_1+\ldots+a_m=i$ and
$C_{i}(\vb;E)=\bigoplus_{|a|=i} Ex^{(a)}$. The $E$-linear differential $\partial$ on
$C_{{\hbox{\large\bf.}}}(\vb;E)$ is defined as follows. For $x^{(a)}=x_{1}^{(a_1)}\ldots x_{m}^{(a_m)}$
we set $\partial(x^{(a)})=\; | \;um_{a_i>0} v_{i} \cdot x_{1}^{(a_1)}\ldots x_{i}^{(a_{i}-
1)}\ldots x_{m}^{(a_m)}.$ One easily checks that $\partial\circ\partial=0$, thus
$C_{{\hbox{\large\bf.}}}(\vb;E)$ is indeed a complex.
We denote by $\mathcal{M}$ the category of finitely generated graded left and right
$E$-modules $M$, satisfying $ax=(-1)^{|\deg(a)||\deg(x)|}xa$ for all homogeneous $a\in E$
and $x\in M$. For example, every graded ideal $J\; | \;ubseteq E$ belongs to $\mathcal{M}$.
For $M \in \mathcal{M}$ and $\vb=v_1,\ldots, v_m \; | \;ubseteq E_1$, the complex
$C_{{\hbox{\large\bf.}}}(\vb;M)=C_{{\hbox{\large\bf.}}}(\vb;E)\otimes_E M$ is called the {\em Cartan complex} of $M$
with respect to $\vb$. Its homology is denoted by $H_{{\hbox{\large\bf.}}}(\vb;M)$; it is called {\em
Cartan homology}.
There is a natural grading of this complex and its homology. We set $\deg x_i =1$ and
$C_{j}(\vb;M)_{i}:=\text{span}_{K}\{m_{a}x^{(b)} \; | \; m_a \in M_a, a+|b|=i,|b|=j \}.$
Cartan homology can be computed recursively. For $j=1,\ldots,m-1$, the following sequence
is exact:
\begin{eqnarray} \label{eq-cartan}
\hspace*{.5cm} 0 \to C_{{\hbox{\large\bf.}}}(v_1,\ldots,v_j;M) \overset{\iota}{\to}
C_{{\hbox{\large\bf.}}}(v_1,\ldots,v_{j+1};M)\overset{\varphi}{\to}C_{{\hbox{\large\bf.}}-
1}(v_1,\ldots,v_{j+1};M)(-1)\to 0.
\end{eqnarray}
Here $\iota$ is a natural inclusion map and $\varphi$ is given by
$$
\varphi(g_0 + g_1x_{j+1}+\ldots+g_{k}x_{j+1}^{(k)})=g_1 + g_2x_{j+1}+\ldots+g_{k}x_{j+1}^{(k-1)}
\text{ for }
g_i \in C_{i-k}(v_1,\ldots ,v_{j};M).
$$
The associated long exact homology sequence is
$$
\ldots \to H_{i}(v_1,\ldots,v_j;M) \overset{\alpha_i}{\to}
H_{i}(v_1,\ldots,v_{j+1};M)\overset{\beta_i}{\to} H_{i-
1}(v_1,\ldots,v_{j+1};M)(-1)
$$
$$
\overset{\delta_{i-1}}{\to} H_{i-1}(v_1,\ldots,v_j;M) \overset{\alpha_{i-
1}}{\to} H_{i-1}(v_1,\ldots,v_{j+1};M)\overset{\beta_{i-1}}{\to}\ldots,
$$
where $\alpha_{i}$ and $\beta_{i}$ are induced by $\iota$ and $\varphi$, respectively. For a
cycle $z=g_0 + g_1x_{j+1}+\ldots+g_{i-1}x_{j+1}^{(i-1)}$ in $C_{i-
1}(v_1,\ldots,v_{j+1};M)$ one has $\delta_{i-1}([z])=[g_0v_{j+1}]$.
It is now easy to
see that e.~ g.\ for
$e_t,\dots,e_n$, the Cartan complex $C_{{\hbox{\large\bf.}}}(e_t,\dots,e_n;E)$ is a free
resolution of $E/(e_t,\dots,e_n)$. Hence, for each module $M \in
\mathcal{M}$, there are isomorphisms
\begin{equation}
\label{tor}
\Tor_i^{E}(E/(e_t,\dots,e_n),M)\cong H_{i}(e_t,\dots,e_n;M).
\end{equation}
In particular,
for $\eb=e_1,\dots,e_n$ we have $K\cong E/(\eb)$ and there are isomorphisms of graded $K$-vector spaces $ \Tor_i^{E}(K,M)\cong
H_{i}(\eb;M)$.
Cartan homology is useful to study resolutions of graded ideals in the exterior algebra.
For example, it has been used to show that the Castelnuovo--Mumford regularity $\reg(E/J)=\max\{j \mid \exists $ $i \text{ such that } \Tor_i(K,E/J)_{i+j}\neq 0 \}$
does not change by passing to the generic initial ideal with respect to the reverse
lexicographic order, i.~e.\ $ \reg(E/J)= \reg(E/\gin(J))$ (cf.\ \cite{ARHE}, Theorem 5.3).
There are several other algebraic invariants which behave similarly. See for example
\cite{HETE} for results in this direction. Here, we are interested in the following
numbers which, for $p = n$, include the graded Betti numbers of $E/J$:
\begin{defi} \label{def-CB}
Let $J \; | \;ubset E$ be a graded ideal and
$f_1,\dots,f_p$ be a generic sequence of linear forms.
Let $C(f_1,\dots,f_p;E/J)$ be the Cartan complex
of $E/J$ with respect to that sequence
and $H(f_1,\dots,f_p;E/J)$ the corresponding Cartan homology.
We denote by
$$
\beta^E_{ijp}(E/J)= \dim_K H_i(f_1,\dots,f_p;E/J)_j
$$
the {\em Cartan--Betti numbers} of $E/J$ for $i=0,\dots,p$.
We set $H_i(f_1,\dots,f_p;E/J)=0$ for $i>p$.
\end{defi}
A generic sequence means here that
there exists a non-empty Zariski open set $U$ in $K^{p\times n}$ such that
if one chooses the $p\times n$ coefficients
that express $f_1,\dots,f_p$ as linear combinations of $e_1,\dots,e_n$ inside $U$,
then the ranks of the maps in each degree in the Cartan complex are as big as possible.
In particular, the $K$-vector space dimension of the homology
is the same for each generic sequences and thus the definition of
$\beta^E_{ijp}(E/J)$ does not depend on the chosen generic sequence.
We are going to explain that it is easy to compute the Cartan--Betti numbers for
squarefree strongly stable ideals in the exterior algebra. In fact, given a generic
sequence of linear forms $f_1,\dots,f_p$ there exists an invertible upper triangular
matrix $g$ such that the linear space spanned by $f_1,\dots,f_p$ is mapped by the induced
isomorphism $g \colon E \to E $ to the one generated by
$e_{n-p+1},\dots,e_n$. It follows
that, as graded
$K$-vector spaces, $ H_i(f_{1},\dots,f_p;E/J)\cong
H_i(e_{n-p+1},\dots,e_n;E/g(J))$. Since $J$ is squarefree strongly
stable we have that $g(J)=J$ and thus we get:
\begin{lem}
\label{cartanhelper} Let $J \; | \;ubset E$ be a squarefree strongly stable ideal. Then we
have for all $i,j,p$:
$$
\beta_{ijp}^E(E/J)=\dim_K H_i(e_{n-p+1},\dots,e_n;E/J)_j.
$$
\end{lem}
Aramova, Herzog and Hibi (\cite{ARHEHI97}) computed the Cartan--Betti numbers for squarefree strongly
stable ideals which gives rise to Eliahou--Kervaire type resolutions in the exterior
algebra. To present this result we need some more notation. Recall that for a monomial
ideal $J\; | \;ubset E$ we denote by $G(J)$ the unique minimal system of monomial generators
of $J$. For a monomial $e_F \in E$ where $F \; | \;ubseteq [n]$ we set
$\max(F)=\max\{i \in F\}.$ Given a set $G$ of monomials we define:
\begin{eqnarray*}
m_i(G) & = & |\{e_F \in G : \max(e_F)=i \}|, \\
m_{\leq i}(G) & = & |\{e_F \in G : \max(e_F) \leq i \}|, \\
m_{ij}(G) & = & |\{e_F \in G : \max(e_F)=i,\ |F|=j \}|.
\end{eqnarray*}
For a monomial ideal or a vector space generated by monomials $J$, we denote by $m_i(J)$,
$m_{\leq i}(J)$ and $m_{ij}(J)$ the numbers $m_i(G)$, $m_{\leq i}(G)$ and $m_{ij}(G)$
where $G$ is the set of minimal monomial generators of $J$. Using Lemma
\ref{cartanhelper} we restate the result of Aramova, Herzog and Hibi as follows:
\begin{thm}
\label{cartanbetti}
Let $J \; | \;ubset E$ be a squarefree strongly stable ideal.
Then the Cartan--Betti numbers are given by
the following formulas:
\begin{enumerate}
\item{(Aramova--Herzog--Hibi)}
For all $i>0$, $p \in [n]$ and every $j \in \mathbb{Z}$
we have
$$
\beta_{ijp}^E(E/J)
=
\; | \;um_{k = n-p+1}^n m_{k,j-i+1}(J) \binom{k+p-n+i-2}{i-1}.
$$
\item
For all $p \in [n]$ and every $j \in \mathbb{Z}$
we have
$$
\beta_{0jp}^E(E/J)
=
\binom{n-p}{j}-m_{\leq n-p}(J_j)
$$
where $J_j$ is the degree $j$ component of the ideal $J$.
\end{enumerate}
\end{thm}
\begin{proof}
Only (ii) needs a proof since (i) immediately follows from the proof of Proposition 3.1 of \cite{ARHEHI97}. But since $e_{n-p+1},\dots,e_n$ is generic for $E/J$
we can compute $\beta_{0jp}^E(E/J)$
from the Cartan homology with respect to this sequence. Then the formula
follows from a direct computation.
\end{proof}
Note that for $p=n$ we get the graded Betti numbers of $E/J$.
Using the numbers $m_{\leq i}(J_j)$, it is possible to compare the Cartan--Betti numbers
of squarefree strongly stable ideals. In fact we have the following:
\begin{prop}
\label{maintechnical}
Let $J, J' \; | \;ubset E$
be squarefree strongly stable ideals with the same Hilbert function
such that
$m_{\leq i}(J_j)\leq m_{\leq i}(J'_j)$ for all $i,j$.
Then
$
\beta_{ijp}^E(E/J')
\leq
\beta_{ijp}^E(E/J)
$
for all
$i,j,p$
with equalities everywhere
if and only if
$m_{\leq i}(J_j)= m_{\leq i}(J'_j)$ for all $i,j$.
\end{prop}
\begin{proof}
It follows from
Theorem \ref{cartanbetti} (ii) and the assumption that
\begin{eqnarray*}
\beta_{0jp}^E(E/J) - \beta_{0jp}^E(E/J') &=& \binom{n-p}{j} - m_{\leq n-p}(J_j) -
\binom{n-p}{j} + m_{\leq n-p}(J'_j)\\
&=&
m_{\leq n-p}(J'_j) - m_{\leq n-p}(J_j)\\
&\geq&
0.
\end{eqnarray*}
This shows the desired inequalities for $i=0$.
Next assume that $i>0$ and observe
\begin{eqnarray*}
\lefteqn{ m_{k}(J_{j-i+1})=|\{e_F \in J : \max (e_F) = k, \ |F| = j-i+1\}| }\\
&=&
|\{e_F \in G(J) : \max (e_F) = k, |F| = j-i+1\}| \\
&+&
| \{e_F \in J : \exists e_H \in J_{j-i}, \max (e_H)\leq k-1, e_F=e_H \wedge e_k\} |\\
&=&
m_{k,j-i+1}(J) + m_{\leq k-1}(J_{j-i}).\\
\end{eqnarray*}
We compute
\begin{eqnarray*}
\lefteqn{ \beta_{ijp}^E(E/J') } \\
&=&
\; | \;um_{k = n-p+1}^n m_{k,j-i+1}(J') \binom{k+p-n+i-2}{i-1}\\
&=&
\; | \;um_{k = n-p+1}^n \bigl( m_{k}(J'_{j-i+1}) - m_{\leq k-1}(J'_{j-i}) \bigr) \binom{k+p-n+i-2}{i-1}\\
&=&
\; | \;um_{k = n-p+1}^n \bigl(
m_{\leq k}(J'_{j-i+1})- m_{\leq k-1}(J'_{j-i+1}) - m_{\leq k-1}(J'_{j-i})
\bigr) \binom{k+p-n+i-2}{i-1}\\
&=&
m_{\leq n}(J'_{j-i+1}) \binom{p+i-2}{i-1}
- m_{\leq n-p}(J'_{j-i+1})\\
&+&
\; | \;um_{k = n-p+1}^{n-1}
m_{\leq k}(J'_{j-i+1})
\bigl(
\binom{k+p-n+i-2}{i-1}
-
\binom{k+p-n+i-1}{i-1}
\bigr)\\
&-&
\; | \;um_{k = n-p+1}^n
m_{\leq k-1}(J'_{j-i})
\binom{k+p-n+i-2}{i-1}\\
&=&
\dim_K (J'_{j-i+1}) \binom{p+i-2}{i-1}
- m_{\leq n-p}(J'_{j-i+1})\\
&-&
\; | \;um_{k = n-p+1}^{n-1}
m_{\leq k}(J'_{j-i+1})
\binom{k+p-n+i-2}{i-2}
\\
&-&
\; | \;um_{k = n-p+1}^n
m_{\leq k-1}(J'_{j-i})
\binom{k+p-n+i-2}{i-1}\\
\end{eqnarray*}
and similarly $\beta_{ijp}^E(E/J)$. We know by assumption that $\dim_K
(J'_{j-i+1})=\dim_K (J_{j-i+1})$ and $m_{\leq i}(J_j)\leq m_{\leq i}(J'_j)$ for all
$i,j$. Hence the last equation of the computation implies $\beta_{ijp}^E(E/J)-
\beta_{ijp}^E(E/J') \geq 0$ because the left-hand side is a sum of various $m_{\leq
i}(J'_j)-m_{\leq i}(J_j)$ times some binomial coefficients. Moreover, we have equalities
everywhere if and only if $m_{\leq i}(J_j)= m_{\leq i}(J'_j)$ for all $i,j$.
\end{proof}
Using the above results as well as \cite[Lemma 3.7]{ARHEHI98} instead of Proposition 3.5
in \cite{CO04}, we get the following rigidity statement that is proved analogously to
Proposition 3.7 in \cite{CO04}:
\begin{cor}
\label{rigid}
Let $J, J' \; | \;ubset E$
be a squarefree strongly stable ideals with the same Hilbert function
and $m_{\leq i}(J_j)\leq m_{\leq i}(J'_j)$ for all $i,j$.
Then the following statements are equivalent:
\begin{enumerate}
\item
$\beta_{ijp}^E(E/J')=\beta_{ijp}^E(E/J)$
for all $i,j$ and $p$;
\item
$\beta_{ij}^E(E/J')=\beta_{ij}^E(E/J)$
for all $i$ and $j$;
\item
$\beta_{1j}^E(E/J')=\beta_{1j}^E(E/J)$
for all $j$;
\item
$\beta_{1}^E(E/J')=\beta_{1}^E(E/J)$;
\item
$m_{i}(J'_j)=m_{i}(J_j)$
for all $i,j$;
\item
$m_{\leq i}(J'_j)=m_{\leq i}(J_j)$
for all $i,j$.
\end{enumerate}
\end{cor}
Given a graded ideal $J \; | \;ubset E$ there exists the unique
{\em squarefree lexsegment ideal} $\lex(J) \; | \;ubset E$ with the same Hilbert function as $J$
(see \cite{ARHEHI97} for details).
Next we present a mild variation of a crucial result
in \cite{ARHEHI98} which was an important step to compare Betti numbers
of squarefree strongly stable ideals
and their corresponding squarefree lexsegment ideals.
\begin{thm}{(Aramova--Herzog--Hibi)}
\label{lexext}
Let $J \; | \;ubset E$ be a squarefree strongly stable ideal and let $L \; | \;ubset
E$ be its squarefree lexsegment ideal. Then
$$m_{\leq i}(L_j) \leq m_{\leq i}(J_j) \text{ for all } i,j.$$
\end{thm}
\begin{proof}
It is a consequence of \cite[Theorem 3.9]{ARHEHI98} by observing the following facts.
For $j\in \mathbb{N}$ consider the ideals $J'=(J_j)$ and $L'=(L_j)$ of $E$.
Then $J'$ is a squarefree strongly stable ideal,
$L'$ is a squarefree lexsegment ideal,
and
$\dim_K L'_t \leq \dim_K J'_t$ for all $t$ by \cite[Theorem 4.2]{ARHEHI97}.
\end{proof}
As \cite[Lemma 2]{CO03} one shows for Cartan--Betti numbers:
\begin{lem}
\label{someeq}
Let $J \; | \;ubset E$ be a graded ideal.
Then
$$
\beta_{0jp}^E(E/J) = \beta_{0jp}^E(E/\gin(J)) \text{ for all } j,p.
$$
\end{lem}
\begin{proof}
Consider the graded $K$-algebra homomorphism $g\colon E\to E$ that is induced by a
generic matrix in $\GL_n(K)$. Then the linear forms
$f_i := g^{-1}(e_{n-p+i})$, $i=1,\ldots,p$, are generic too and the Hilbert functions of
$E/(J+L)$ and $E/(g(J)+L')$
coincide, where $L=(f_1,\dots,f_p)$ and $L' = (e_{n-p+1},\ldots,e_n)$. Note that passing
to the initial ideals does not change the Hilbert function and that
$\ini(g(J)+L')=\ini(g(J))+L'$
because the chosen term order is revlex. Furthermore, we
have that $\ini(g(J))=\gin(J)$ because $g$ is generic. Hence, the Hilbert function of
$E/(J+L)$
equals to that of
$E/(\gin(J)+L')$.
Since
$ \beta_{0jp}^E(E/J)= \dim_K (E/(J+L))_j $
and
$ \beta_{0jp}^E(E/\gin(J))= \dim_K (E/(\gin(J)+L'))_j $,
our claim follows.
\end{proof}
The following result generalizes \cite[Prop. 1.8, Theorem 4.4]{ARHEHI97}. The proof is
similar to \cite[Theorem 4.3]{CO04}.
\begin{thm}
\label{mainleq}
Let $J \; | \;ubset E$ be a graded ideal and
$\tau$ an arbitrary term order on $E$.
Then
\begin{enumerate}
\item
$
\beta_{ijp}^E(E/J)
\leq
\beta_{ijp}^E(E/\gin_\tau(J))
$
for all $i,j,p$.
\item
$
\beta_{ijp}^E(E/J)
\leq
\beta_{ijp}^E(E/\lex(J))
$
for all $i,j,p$.
\end{enumerate}
\end{thm}
\begin{proof}
(i): Let $g\in \GL_n(K)$ be a generic matrix and let $f_i$ be the preimage of
$e_{n-p+1}$ under the induced $K$-algebra isomorphism $g \colon E\to E$ where
$i=1,\dots,p$. Then
$$
H_i(f_1,\dots, f_p;E/J)
\cong
H_i(e_{n-p+1},\dots,e_{n};E/g(J))
\cong
\Tor^E_i(E/g(J), E/(e_{n-p+1},\dots,e_{n}))
$$
where the last isomorphism was noted above in (\ref{tor}).
Proposition \ref{helpergb} provides:
\begin{eqnarray*}
\beta^E_{ijp}(E/J)
&\leq&
\dim_K \Tor^E_i(E/\ini_\tau(g(J)), E/(e_{n-p+1},\dots,e_{n}))_j\\
&=&
\dim_K H_i(e_{n-p+1},\dots,e_{n}; E/\ini_\tau(g(J)))_j.
\end{eqnarray*}
Since $g$ is a generic matrix, $\ini_\tau(g(J))=\gin_\tau(J)$ is
squarefree strongly
stable. Thus we can apply Lemma \ref{cartanhelper} and (i) follows.
(ii): By (i) we may replace $J$ by $\gin(J)$, thus we may assume that
$J$ is squarefree
strongly stable. Now (ii) follows from \ref{lexext} (ii) and \ref{maintechnical}.
\end{proof}
The next results answer the natural question, in which cases we have
equalities in Theorem \ref{mainleq}. Denote by $J_{\langle t\rangle}$ the ideal that is generated by all degree
$t$ elements of the graded ideal $J$. Then $J \; | \;ubset E$ is called
{\em componentwise linear} if, for all $t\in \mathbb{N}$, the ideal has an $t$-linear resolution,
i.~e.\ $\Tor_i^E(J_{\langle t\rangle},K)_{i+j}=0$ for $j \neq t$.
Given a module $M \in \mathcal{M}$, we call, in analogy to the symmetric case,
the sequence $v_1,\dots,v_m \in E$ a {\em proper $M$-sequence} if
for all $i\geq 1$ and all $1\leq j< m$ the maps
$$
\delta_i \colon H_i(v_1,\dots,v_{j+1};M)(-1) \to H_i(v_1,\dots,v_{j};M)
$$
are zero maps.
The following result generalizes \cite[Theorem 2.1]{ARHEHI00ideal}.
We follow the idea of the proof of \cite[Theorem 4.5]{CO04}.
\begin{thm}
\label{maincl}
Let $J \; | \;ubset E$ be a graded ideal.
The following conditions are equivalent:
\begin{enumerate}
\item
$\beta_{ijp}^E(E/J)= \beta_{ijp}^E(E/\gin(J))$ for all $i,j,p$;
\item
$\beta_{1jn}^E(E/J)= \beta_{1jn}^E(E/\gin(J))$ for all $j$;
\item
$J$ is a componentwise linear ideal;
\item
A generic sequence of linear forms $f_1,\dots,f_n$ is a proper sequence of $E/J$.
\end{enumerate}
\end{thm}
\begin{proof}
(i) $\mathbb{R}ightarrow$ (ii): This is trivial.
(ii) $\mathbb{R}ightarrow$ (iii): This follows from of \cite[Theorem 1.1 and Theorem
2.1]{ARHEHI00ideal}. In fact, the proof of Aramova, Herzog an Hibi shows this stronger
result.
(iii) $\mathbb{R}ightarrow$ (iv):
Assume that $J$ is componentwise linear.
Let $f_1,\dots,f_n$ be a generic sequence of linear forms.
We have to show that
for all $i\geq 1$ and all $1\leq j< n$
the homomorphisms
$
\delta_i \colon H_i(f_1,\dots,f_{j+1};E/J)(-1) \to H_i(f_1,\dots,f_{j};E/J)$
are zero maps.
Assume first that $J$ is generated in a single degree $d$. Since the regularity does not
change by passing to the generic initial ideal with respect to revlex, it follows that
also $\gin(J)$ is generated in degree $d$. By Theorem \ref{cartanbetti} (i) we get that
for $i\geq 1$ the homology module $H_i(f_1,\dots,f_j;E/\gin(J))$ is concentrated in the
single degree $i+d-1$. Theorem \ref{mainleq} (i) shows that the same is true also for $H_i(f_1,\dots,f_j;E/J)$.
This implies that $H_i(f_1,\dots,f_{j+1};E/J)(-1)$ has non-trivial homogeneous elements only in degree $i+d$
and the module
$H_i(f_1,\dots,f_{j};E/J)$ has only homogeneous elements in degree $i+d-1$.
Since $\delta_i$ is a homogeneous homomorphism of degree zero, this implies that
simply by degree reasons $\delta_i$ is the zero map.
Now we consider the general case. Recall that
for a cycle $c$ in some complex we write $[c]$ for
the corresponding homology class.
For any element $e \in E$ we denote by $\bar e$ the corresponding
residue class in $E/J$ to distinguish notation.
Now we fix some $i$ and $j$.
Let $a \in H_i(f_1,\dots,f_j;E/J)(-1)$ be a homogeneous element of degree $s$.
We have to show that $\delta_i(a)=0$.
There exists an homogeneous element $z \in C_i(f_1,\dots,f_j;E)$ of degree $s-1$
such that $\bar z \in C_i(f_1,\dots,f_j;E/J)$ is a cycle representing $a$,
i.~e. $\partial_i(\bar z)=0$ and $a=[\bar z]$.
Since $\partial_i(\bar z)=0$ we have that
$\partial_i(z) \in JC_{i-1}(f_1,\dots,f_j;E)$ because
$JC_{i-1}(f_1,\dots,f_j;E)=C_{i-1}(f_1,\dots,f_j;J)$
and we have the commutative diagram
$$
\begin{array}{ccccccc}
0 \ \rightarrow & C_i(f_1,\dots,f_j;J) & \longrightarrow & C_i(f_1,\dots,f_j;E) & \longrightarrow & C_i(f_1,\dots,f_j;E/J) & \rightarrow \ 0\\
& \downarrow \partial_i & & \downarrow \partial_i & &
\downarrow \partial_i & \\
0 \ \rightarrow & C_{i-1}(f_1,\dots,f_j;J) & \longrightarrow & C_{i-1}(f_1,\dots,f_j;E) & \longrightarrow & C_{i-1}(f_1,\dots,f_j;E/J) & \rightarrow \ 0\\
\end{array}
$$
\noindent Since $\partial_i (z) $ is homogeneous of degree $s-1$, we obtain that $\partial_i(z)$
is in $J_k C_{i-1}(f_1,\dots,f_j;E)$ for $k=(s-1)-(i-1)=s-i$.
Set $J' := J_{<k>}$.
Then $[\bar z]$ can also be considered as the homology class of $z$ in $H_i(f_1,\dots,f_j;E/J')$.
By construction, $J'$ is generated in a single
degree and, by assumption, it has a linear resolution. Hence we know already that
$\delta_i([\bar z])=0$ as an element of $H_i(f_1,\dots,f_{j-1};E/J')$ (in degree $s$).
The natural homomorphism
$H_i(f_1,\dots,f_{j-1};E/J') \to H_i(f_1,\dots,f_{j-1};E/J)$
induced by
the short exact sequence $0\to J/J' \to E/J' \to E/J \to 0$
implies that
$\delta_i(a)=\delta_i([\bar z])=0$ as an element of $H_i(f_1,\dots,f_j;E/J)$ (in degree $s$)
which we wanted to show.
(iv) $\mathbb{R}ightarrow$ (i):
By Lemma \ref{someeq} we know that
$\beta^E_{0jp}(E/J)=\beta^E_{0jp}(E/\gin(J))$ for all $j,p$.
Observe that
$\beta^E_{ij1}(E/J)=\beta^E_{ij1}(E/\gin(J))$ for $i>0$ and $j\in \mathbb{Z}$
as one easily checks using properties of $\gin$.
We prove (i) by showing
that the numbers $\beta_{ijp}^E(E/J)$ only depend on the numbers
$\beta_{0jp}^E(E/J)$ and $\beta_{ij1}^E(E/J)$.
Since $f_1,\dots,f_n$ is a proper sequence of $E/J$, the long exact Cartan homology
sequence splits into short exact sequences
$$
0 \to H_1(f_1,\dots,f_{p-1};E/J) \to H_1(f_1,\dots,f_{p};E/J) \to
H_0(f_1,\dots,f_{p};E/J)(-1)
$$
$$
\to
H_0(f_1,\dots,f_{p-1};E/J)
\to
H_0(f_1,\dots,f_{p};E/J)
\to
0
$$
and, for $i>1$,
$$
0 \to H_i(f_1,\dots,f_{p-1};E/J) \to H_i(f_1,\dots,f_{p};E/J) \to
H_{i-1}(f_1,\dots,f_{p};E/J)(-1) \to 0.
$$
Thus
$$
\beta^E_{1jp}(E/J) = \beta^E_{1jp-1}(E/J) + \beta^E_{0 j-1 p}(E/J) - \beta^E_{0jp-1}(E/J)
+ \beta^E_{0jp}(E/J)
$$
and, for $i > 1$,
$$
\beta^E_{ijp}(E/J) = \beta^E_{ijp-1}(E/J) + \beta^E_{i-1 j-1 p}(E/J).
$$
Since $\beta^E_{ijp}(E/J)=0$ for all $i>p$, it is easy to see that these equalities imply
(i).
\end{proof}
Recall that a graded ideal $J \; | \;ubset E$ is called a {\em Gotzmann ideal} if the growth
of its Hilbert function is the least possible, i.\ e.\
$\dim_K E_1 \cdot J_j = \dim_K E_1\cdot \lex(J)_j$
for all $j$. The next result is an exterior version of the corresponding result
\cite[Theorem 4.6]{CO04} in the polynomial ring.
\begin{thm}
\label{maingotz} For each graded ideal $J \; | \;ubset E$, the following statements are
equivalent:
\begin{enumerate}
\item
$\beta^E_{ijp}(E/J)= \beta^E_{ijp}(E/\lex(J))$
for all $i,j,p$;
\item
$\beta^E_{1jn}(E/J)= \beta^E_{1jn}(E/\lex(J))$
for all $j$;
\item
$J$ is a Gotzmann ideal in the exterior algebra;
\item
$\beta^E_{0jp}(E/J) = \beta^E_{0jp}(E/\lex(J))$
for all $j,p$
and
$J$ is componentwise linear.
\end{enumerate}
\end{thm}
\begin{proof}
(i) $\mathbb{R}ightarrow$ (ii): This is trivial.
(ii) $\Leftrightarrow$ (iii):
This follows immediately from the definition of Gotzmann ideals.
(ii) $\mathbb{R}ightarrow$ (iv): By Theorems \ref{mainleq} and \ref{maincl}, a Gotzmann ideal is componentwise linear and
$\beta^E_{0jp}(E/J) = \beta^E_{0jp}(E/\lex(J))$ for all $j,p$.
(iv) $\mathbb{R}ightarrow$ (i):
From Theorem \ref{cartanbetti} (ii) we know that $m_{\leq n-p}(\gin(J)_j) = m_{\leq n-p}(\lex(J)_j)$ for all $j, p$. Then it follows from Corollary \ref{rigid} that
$\beta^E_{ijp}(E/\gin(J)) = \beta^E_{ijp}(E/\lex(J))$ for all $i,j,p$. Since $J$ is componentwise linear, the assertion follows from Theorem \ref{maincl}.
\end{proof}
We conclude this section by comparing the Cartan--Betti numbers of
generic initial ideals. The goal is to show that in the family
$$
\gins(J) :=\{\gin_\tau(J) : \tau \text{ a term order of } E\}
$$
of all generic initial ideals of $J$, the revlex-gin has the smallest
Cartan--Betti numbers.
We need a lemma and some more notation. Let $V \; | \;ubset E_i$ be a
$d$-dimensional
subspace. Then $\bigwedge^d V$ is a $1$-dimensional subspace of
$\bigwedge^d E_i$. We
identify it with any of its nonzero elements. An exterior monomial in $\bigwedge^d E_i$
is by definition an element of the form $m_1 \wedge \ldots \wedge m_d$ where
$m_1,\ldots,m_d$ are distinct monomials in $E_i$. It is called a {\em $\tau$-standard}
exterior monomial if $m_1
>_{\tau} \ldots >_{\tau} m_d$. The vector space $\bigwedge^d E_i$ has a $K$-basis of
$\tau$-standard exterior monomials that we order lexicographically by
$
m_1 \wedge \ldots \wedge m_d >_{\tau} n_1 \wedge \ldots \wedge n_d
$
if $m_i >_{\tau} n_i$ for the smallest index $i$ such that $m_i \neq n_i$.
Using this order one defines the initial (exterior) monomial
$\inn_{\tau} (f)$ of any $f \in \bigwedge^d E_i$
as the maximal $\tau$-standard exterior monomial with respect to $>_{\tau}$
which
appears in the unique representation of $f$ as a sum of
$\tau$-standard exterior monomials.
Similarly, the initial subspace $\inn_{\tau}(V)$ of any subspace $V$ of $\bigwedge^d E_i$ is defined as the subspace
generated by all $\inn_{\tau} (f)$ for $f \in V$.
The following result and its proof are analogous to \cite[Corollary 1.6]{CO03}.
\begin{lem}
\label{helperdim} Let $\tau$ and $\; | \;igma$ be term orders on $E$ and let $V \; | \;ubset E_i$
be any $d$-dimensional subspace. If $m_1 \wedge \ldots \wedge m_d$ and $n_1 \wedge
\ldots \wedge n_d$ are $\tau$-standard monomials such that $m_1,\ldots,m_d $ is a $K$-basis of $\gin_{\tau} (V) $ and
$n_1,\ldots,n_d$ is a $K$-basis of $\gin_{\tau} (\inn_{\; | \;igma} (V)) $, then $m_i \geq_{\tau} n_i$ for all $i = 1,\ldots,d$.
\end{lem}
Now we get analogously to the symmetric case \cite[Theorem 5.1]{CO04}:
\begin{thm}
\label{mainginleq}
Let $J \; | \;ubset E$ be a graded ideal and $\tau$ a term order on $E$.
Then
$$
\beta_{ijp}^E(E/\gin(J)) \leq \beta_{ijp}^E(E/\gin_\tau(J)) \quad
\text{ for all } i,j,p.
$$
\end{thm}
\begin{proof}
Set $J'=\gin(J)$ and $J''=\gin_\tau(J)$. Note that $J'$ and $J''$
are squarefree
strongly stable ideals with the same Hilbert function as $J$. Thus by
Proposition \ref{maintechnical},
it is enough to show that $m_{\leq i}(J''_j) \leq m_{\leq i}(J'_j)$
for all $i$ and $j$.
Let $a_1,\dots,a_k$ be the generators of $J'_j$ and let $b_1\dots,b_k$
be those of
$J''_j$. We may assume that the $a_r$'s and the $b_r$'s are listed in the revlex order.
Then we claim:
$$
a_r\geq b_r
\text{ in the revlex order for all } r.
$$
This implies that $\max(a_r)\leq \max(b_r)$ for all $r$, and hence
$m_{\leq i}(J''_j)\leq
m_{\leq i}(J'_j)$. Thus it remains to shows the claim.
Let $V,W \; | \;ubset E_j$ be two monomial vector spaces of the same dimension where $V$ has a
monomial $K$-basis $v_1,\dots,v_k$ and $W$ has a monomial $K$-basis $w_1,\dots,w_k$.
Define $V \geq_{\revlex} W$ if $v_i \geq_{\revlex} w_i$, $v_i>_{\revlex}
v_{i+1}$ and $w_i>_{\revlex} w_{i+1}$ for all $i$. Then Lemma \ref{helperdim} provides:
$$
\gin_{\revlex}(V)
\geq_{\revlex}
\gin_{\revlex}(\ini_\tau(V)).
$$
If $V \; | \;ubset E_j$ is a generic subspace, then $\ini_\tau(V)= \gin_\tau(V)$. Since
$\gin_{\revlex}(\gin_\tau(V))=\gin_\tau(V)$, we then get $ \gin_{\revlex}(V)
\geq_{\revlex} \gin_\tau(V). $ Choosing $V=J_j'$ proves the claim.
\end{proof}
Observe that inequalities similar to the crucial
$m_{\leq i}((\gin_\tau(J))_j) \leq m_{\leq i}(\gin(J)_j)$ above
were shown by \cite[Proposition 2.4]{M05}.
\; | \;ection{Simplicial complexes and algebraic shifting}
\label{sec-def}
In the second part of this paper we give some combinatorial applications
of the exterior algebra methods presented in Section \ref{cartan}
and some symmetric algebra methods which will be presented below.
For this we introduce at first some notation and discuss some basic concepts.
Recall that $\Gamma$ is called a {\em simplicial complex} on the
vertex set $[n]=\{1,\dots,n\}$ if $\Gamma$ is a subset of the power set of $[n]$ which is
closed under inclusion, i.~e.\ if $F \; | \;ubseteq G$ and $G \in \Gamma$, then $F\in \Gamma$.
The elements $F$ of $\Gamma$ are called {\em faces}, and the maximal elements under
inclusion are called {\em facets}. We denote the set of all facets by $\facets(\Gamma)$.
If $F$ consists of $d+1$ elements of $[n]$, then $F$ is called a {\em $d$-dimensional} face,
and we write $\dim F = d$. The empty set is a face of dimension $-1$. Faces of dimension
0, 1 are called {\em vertices} and {\em edges}, respectively. $\Gamma$
is called {\em
pure} if all facets have the same dimension. If $\Gamma$ is non-empty,
then the {\em
dimension} $\dim \Gamma $ is the maximum of the dimensions of the
faces of $\Gamma$. Let
$f_i$ be the total number of $i$-dimensional faces of $\Gamma$ for
$i=-1,\dots,\dim
\Gamma$. The vector $f(\Gamma)=(f_{-1},\dots,f_{\dim \Gamma})$ is
called the {\em
$f$-vector} of $\Gamma$.
Several constructions associate other simplicial complexes to a given
one. For example,
the {\em Alexander dual} of $\Gamma$ is defined as $\Gamma^* =\{F
\; | \;ubseteq [n] : F^c
\not\in \Gamma\}$ where $F^c=[n] \; | \;etminus F$. This gives a duality
operation on
simplicial complexes in the sense that $\Gamma^*$ is indeed a
simplicial complex on the
vertex set $[n]$ and we have that $(\Gamma^*)^*=\Gamma$.
The connection to algebra is due to the following construction. Let $K$ be a field and
$S=K[x_1,\dots,x_n]$ be the polynomial ring in $n$ indeterminates. $S$ is a graded ring
by setting $\deg x_i=1$ for $i=1,\dots,n$. We define the {\em Stanley--Reisner ideal}
$I_\Gamma=(\prod_{i \in F} x_i : F\; | \;ubseteq [n],\ F\not\in \Gamma)$ and the corresponding
{\em Stanley--Reisner ring} $K[\Gamma]=S/I_\Gamma$. For a subset $F \; | \;ubset [n]$, we
write $x_F$ for the squarefree monomial $\prod_{i \in F} x_i$. We will say that $\Gamma$
has an algebraic property like Cohen--Macaulayness if and only if $K[\Gamma]$ has this
property. Note that these definitions may depend on the ground field $K$.
There is an analogous exterior algebra construction due to Aramova, Herzog and Hibi (e.g.\ see \cite{HE01}).
Let $E=K \langle e_1,\dots,e_n\rangle$ be the exterior algebra on $n$ exterior variables.
$E$ is also a graded ring by setting $\deg e_i=1$ for $i=1,\dots,n$.
One defines the {\em exterior Stanley--Reisner ideal}
as $J_\Gamma=(e_F : F \not\in \Gamma)$
and the {\em exterior face ring} $K\{\Gamma\}=E/J_\Gamma$.
We keep these notations throughout the paper. Further information on simplicial complexes
can be found, for example, in the books \cite{BRHE98} and
\cite{ST96}.
Let $\mathcal{C}([n])$ be the set of simplicial complexes on $[n]$. Following
constructions of Kalai, we define axiomatically the concept of algebraic shifting. See
\cite{HE01} and \cite{KA01} for surveys on this subject. We call a map $\; | \;hift \colon
\mathcal{C}([n]) \to \mathcal{C}([n])$ a {\em shifting operation} if the
following conditions are satisfied:
\begin{enumerate}
\item[(S1)]
If $\Gamma \in \mathcal{C}([n])$,
then $\; | \;hift(\Gamma)$ is a {\em shifted complex},
i.~e.\ for all $F \in \; | \;hift(\Gamma)$ and $j > i \in F$
we have that $(F \; | \;etminus \{i\}) \cup \{j\} \in \; | \;hift(\Gamma)$.
\item[(S2)]
If $\Gamma \in \mathcal{C}([n])$ is shifted,
then $\; | \;hift(\Gamma)=\Gamma$.
\item[(S3)]
If $\Gamma \in \mathcal{C}([n])$, then
$f(\Gamma)=f(\; | \;hift(\Gamma))$.
\item[(S4)]
If $\Gamma', \Gamma \in \mathcal{C}([n])$ and $\Gamma' \; | \;ubseteq \Gamma$ is a subcomplex,
then $\; | \;hift(\Gamma')$ is a subcomplex of $ \; | \;hift(\Gamma)$.
\end{enumerate}
See \cite{BNT2} for variations of these axioms.
Note that a simplicial complex $\Gamma$ is shifted
if and only if the Stanley--Reisner ideal
$I_\Gamma \; | \;ubset S=K[x_1,\dots,x_n]$
is a {\em squarefree strongly stable ideal} with respect to $x_1>\dots>x_n$,
i.~e.\ for all $F\; | \;ubseteq [n]$ with $x_F \in I_\Gamma$
and all $i \in F$, $j<i$, $j \not\in F$
we have that $(x_jx_F)/x_i \in I_\Gamma$.
Analogously,
$\Gamma$ is shifted
if and only if the exterior Stanley--Reisner ideal $J_\Gamma \; | \;ubset
E=K\langle e_1,\dots,e_n \rangle$
is a {\em squarefree strongly stable ideal} with respect to $e_1>\dots>e_n$.
Two of the most important examples of such operations are defined as
follows.
\begin{exa}{(Symmetric algebraic shifting)}
\label{exsymmetric} At first we introduce the symmetric algebraic shifting
introduced in \cite{KA84}, \cite{KA91}. Here, we follow the algebraic approach of Aramova,
Herzog and Hibi \cite{ARHEHI00}.
Assume that $K$ is a field of characteristic $0$ and let $S=K[x_1,\dots,x_n]$.
We consider the following operation on monomial ideals of $S$.
For a monomial
$m=x_{i_1}\cdots x_{i_t}$ with $i_1\leq i_2 \leq \dots \leq i_t$
of $S$ we set $m^\; | \;igma=x_{i_1}x_{i_2+1}\dots x_{i_t+ t-1}$.
For a monomial ideal $I$ with unique minimal system of generators $G(I)=\{m_1,\dots,m_s\}$
we set $I^\; | \;igma=(m_1^\; | \;igma,\dots,m_s^\; | \;igma)$ in a suitable polynomial ring with sufficiently many variables.
Let $\Gamma$ be a simplicial complex on the vertex set $[n]$ with
Stanley--Reisner
ideal
$I_{\Gamma}$.
The {\em symmetric algebraic shifted complex} of $\Gamma$ is the unique
simplicial complex $\Delta^s(\Gamma)$ on the vertex set $[n]$ such that
$$
I_{\Delta^s(\Gamma)} = \bigl(\gin(I_\Gamma)\bigr)^\; | \;igma \; | \;ubset S.
$$
It is not obvious that $\Delta^s(\cdot)$ is indeed a shifting operation. The first
difficulty is to show that $I_{\Delta^s(\Gamma)}$ is an ideal of $S$. This and
the proofs of the other properties can be found in \cite{ARHEHI00} or \cite{HE01}.
\end{exa}
\begin{exa}{(Exterior algebraic shifting)}
\label{exexterior} Exterior algebraic shifting was also defined by Kalai in \cite{KA84}.
Let $E=K\langle e_1,\dots,e_n\rangle$ where $K$ is any infinite field. Let
$\Gamma$ be a simplicial complex on the vertex set $[n]$. The {\em exterior algebraic
shifted complex} of $\Gamma$ is the unique simplicial complex $\Delta^e(\Gamma)$ on the
vertex set $[n]$ such that
$$
J_{\Delta^e(\Gamma)}
=
\gin(J_{\Gamma}).
$$
For an introduction to the theory
of Gr\"obner bases in the exterior algebra see \cite{ARHEHI97}. As opposed to symmetric algebraic shifting, it is much easier to see that $\Delta^ e(\cdot)$ is indeed a shifting operation
since generic initial ideals in the exterior algebra are already squarefree strongly
stable. See again \cite{HE01} for details.
\end{exa}
There are several other shifting operations. Since the proof of \cite[Proposition
8.8]{HE01} works for arbitrary term orders $\tau$, one can take a generic initial ideal
$\gin_\tau(\cdot)$ in $E$ with respect to any term order $\tau$ to obtain, analogously to
Example \ref{exexterior}, the {\em exterior algebraic shifting operation} $\Delta^\tau(\cdot)$ with respect to the
term order $\tau$.
\; | \;ection{Degree functions of simplicial complexes I}
\label{sec-degree}
In commutative algebra degree functions are designed to provide measures for the size and
the complexity of the structure of a given module. Given a simplicial complex $\Gamma$ on
the vertex set $[n]$, we recall the definition of several degree functions on the
Stanley--Reisner ring $K[\Gamma]$ in terms of combinatorial data. The first important
invariant is the degree (or multiplicity) $\deg \Gamma$ of $\Gamma$. By definition $\deg
\Gamma=\deg K[\Gamma]$ is the degree of the Stanley--Reisner ring of $\Gamma$. It can be
combinatorially described as
\begin{equation} \label{eq-comb-deg}
\deg \Gamma=f_{\dim \Gamma}(\Gamma), \; \text{ the number of faces of maximal dimension.}
\end{equation}
Since any algebraic shifting operation $\; | \;hift(\cdot)$
preserves the $f$-vector,
we have that
\begin{equation} \label{eq-deg-inv}
\deg \Gamma = \deg \; | \;hift(\Gamma).
\end{equation}
In case $\Gamma$ is a Cohen--Macaulay complex, this invariant gives a lot of information
about $\Gamma$, because $\Gamma$ is pure and thus $\deg \Gamma$ counts all facets. There
are several other degree functions which take into account more information about
$\Gamma$.
Next we study the {\em arithmetic degree}.
Algebraically it is defined as
$$
\adeg \Gamma =
\adeg K[\Gamma]
=
\; | \;um l(H^0_{\p}(K[\Gamma]_{\p})) \cdot \deg \bigl(K[\Gamma]/\p \bigr)
$$
where the sum runs over the associated prime ideals of $K[\Gamma]$
and $l(\cdot)$ denotes the length function of a module.
In \cite{VA-98} and \cite{BNT} it was noted that the results of
\cite{STTRVO} imply that
\begin{equation} \label{eq-comb-adeg}
\adeg \Gamma = |\{F \in \facets(\Gamma)\}|
\end{equation}
is the number of facets of $\Gamma$.
Similarly, one defines for $i=0,\dots,\dim K[\Gamma]$
$$
\adeg_i \Gamma =
\adeg_i K[\Gamma]
=
\; | \;um l(H^0_{\p}(K[\Gamma]_{\p})) \cdot \deg \bigl(K[\Gamma]/\p \bigr)
$$
where the sum runs over the associated prime ideals of $K[\Gamma]$
such that $\dim K[\Gamma]/\p=i$ and gets
\begin{equation} \label{eq-comb-adeg-i}
\adeg_i \Gamma = |\{F \in \facets(\Gamma): \dim F=i-1\}|.
\end{equation}
By definition, shifting preserves the $f$-vector and the number of
facets of maximal
dimension of a simplicial complex. However, the total number of facets
may change under
shifting. But it may only increase, as we show now.
\begin{thm}
\label{thm-adeg-incr}
Let $\Delta$ be a shifting operation and
$\Gamma$ a $(d-1)$-dimensional simplicial complex. Then
$$
\adeg_i \Gamma \leq \adeg_i \Delta (\Gamma)
\text{ for } i=0,\dots,d.
$$
In particular,
$|\facets (\Gamma)|\leq |\facets (\Delta (\Gamma))|$.
\end{thm}
\begin{proof}
Let $m-1$ be the smallest dimension of a facet of
$\Gamma$. We prove the assertion by induction on the number $(d-1)-(m-1)=d-m \geq 0$.
If $d-m = 0$, then $\Gamma$ is a pure complex and only
$\adeg_d \Gamma$ is different from zero.
Using Identity \eqref{eq-deg-inv} we get
$\adeg_d \Gamma = \deg \Gamma = \deg \; | \;hift (\Gamma) = \adeg_d \; | \;hift (\Gamma)$ as claimed.
Let $d > m$. Observe that $\adeg_i \Gamma=0$ for $i<m$.
Denote by $F_1,\ldots,F_s \in \Gamma$ the $m-1$-dimensional
facets of
$\Gamma$ and let $\Gamma'$ be the subcomplex of $\Gamma$ whose facets are the remaining
facets of $\Gamma$. Thus, we have in particular:
$$
f_i (\Gamma') = f_i (\Gamma) \quad \mbox{for all} ~ i > m-1.
$$
Since shifting does not change the $f$-vector and $\; | \;hift (\Gamma')$ is a subcomplex of
$\; | \;hift (\Gamma)$, we conclude that
$$
f_i (\; | \;hift (\Gamma')) = f_i (\; | \;hift (\Gamma)) \quad \mbox{for all} ~ i > m-1,
$$
and, in particular, that each facet of $\; | \;hift (\Gamma')$ of dimension $> m-1$ is also a
facet of $\; | \;hift (\Gamma)$.
It follows from the induction hypothesis that
$$
\adeg_i \Gamma
=
\adeg_i \Gamma'
\leq
\adeg_i \Delta(\Gamma')
=
\adeg_i \Delta(\Gamma)
\text{ for }
i>m.
$$
Hence, it remains to show that
$\; | \;hift (\Gamma)$ has at least $s=\adeg_{m} \Gamma$ facets of dimension $m-1$.
To this end note that, by definition of $\Gamma'$, $f_{m-1} (\Gamma) = f_{m-1} (\Gamma') + s$,
thus we get for the shifted complexes $f_{m-1} (\; | \;hift (\Gamma)) = f_{m-1} (\; | \;hift (\Gamma')) +
s$. Let $G \in \; | \;hift (\Gamma)$ be any $(m-1)$-dimensional face that is not in
$\; | \;hift (\Gamma')$. Assume that $G$ is strictly contained in a face $\widetilde{G}$ of
$\; | \;hift (\Gamma)$. Then $\dim \widetilde{G} > m-1$ and, using the fact that the faces of
$\; | \;hift (\Gamma')$ and $\; | \;hift (\Gamma)$ of dimension $> m-1$ coincide, we conclude that
$\widetilde{G} \in \; | \;hift (\Gamma')$.
Thus $G \in \; | \;hift (\Gamma')$. This contradiction
to the choice of $G$ shows that $G$ is a facet of $\; | \;hift (\Gamma)$ and we are done.
\end{proof}
Observe that we only used axioms (S3) and (S4) of a shifting operation to prove
Theorem \ref{thm-adeg-incr}.
It would be interesting to know if there are further results in this direction like:
\begin{quest}
\label{importantquestion}
Is there a shifting operation $\; | \;hift(\cdot)$ such that $\adeg$ increases the least,
i.~e.\, for every simplicial complex $\Gamma$, we have $\adeg \; | \;hift (\Gamma) \leq \adeg
\; | \;hift' (\Gamma)$ for each algebraic shifting operation $\; | \;hift'(\cdot)$?
\end{quest}
We use the results in Section \ref{cartan} to give a partial answer to the previous question.
We begin by observing that shifting and Alexander duality commute.
\begin{lem}
\label{shifthelper} Let $\Gamma$ be a simplicial complex on the vertex
set $[n]$. Let
$\Delta^{\tau}(\cdot)$ be the exterior shifting operation with respect to a
term order $\tau$ on $E$. Then Alexander duality commutes with shifting, i.\ e.\ we have $
\Delta^\tau(\Gamma)^* = \Delta^\tau(\Gamma^*). $
\end{lem}
\begin{proof}
In \cite{HETE} this was proved for $\tau$ being the revlex order. But the proof works for
any term order $\tau$.
\end{proof}
As a further preparation, we need an interpretation of the arithmetic degree over the
exterior face ring using the socle. The socle $\Soc N$ of a finitely generated $E$-module
$N$ is the set of elements $x \in N$ such that $(e_1,\dots,e_n)x=0$. Observe that $\Soc
N$ is always a finite-dimensional $K$-vector space.
\begin{lem}
\label{arithdesc} Let $\Gamma$ be a $(d-1)$-dimensional
simplicial complex on the vertex set $[n]$.
Then
$$
\adeg_i \Gamma = \dim_K \Soc K\{\Gamma\}_i
\text{ for }
i=0,\dots,d.
$$
In particular, $\adeg \Gamma = \dim_K \Soc K\{\Gamma\}$.
\end{lem}
\begin{proof}
The residue classes of the monomials $e_F$, $F \in \Gamma$, form a $K$-vector space basis
of $K\{\Gamma\}$. Hence, the facets of $\Gamma$ correspond to a $K$-basis of $\Soc
K\{\Gamma\}$.
\end{proof}
We have seen in Theorem \ref{thm-adeg-incr} that each shifting operation increases the
number of facets. Below we show that among the exterior shifting operations, standard
exterior shifting with respect to revlex leads to the least possible increase.
Now we recall the concept of a sequentially Cohen--Macaulay module. Let $M$ be a finitely
generated graded $S$-module. The module $M$ is said to be {\em sequentially
Cohen--Macaulay} (sequentially CM modules for short), if there exists a finite filtration
\begin{equation}
\label{filter}
0=M_0 \; | \;ubset M_1 \; | \;ubset M_2 \; | \;ubset \dots \; | \;ubset M_r=M
\end{equation}
of $M$ by graded submodules of $M$
such that each quotient $M_i/M_{i-1}$ is Cohen--Macaulay and
$\dim M_1/M_0 <\dim M_2/M_1 <\dots<\dim M_r/M_{r-1}$ where $\dim$ denotes
the Krull dimension of $M$.
\begin{thm}
\label{shifted}
Let $\Gamma$ be a $(d-1)$-dimensional simplicial complex on the vertex set $[n]$. For any term order $\tau$ on $E$, we have
$$
\adeg_i \Gamma \leq \adeg_i \Delta^e(\Gamma) \leq \adeg_i \Delta^\tau(\Gamma)
\text{ for }
i=0,\dots,d.
$$
In particular,
$\adeg\Gamma \leq \adeg\Delta^e(\Gamma) \leq \adeg \Delta^\tau(\Gamma)$.
Moreover, $\adeg \Gamma = \adeg \Delta^e(\Gamma)$ if and only if $\Gamma$ is sequentially
Cohen--Macaulay.
\end{thm}
\begin{proof}
Using $\adeg_i \Gamma = \dim_K \Soc (E/J_{\Gamma})_i$ and the definition of Alexander duality,
it is easy to see that $\adeg_i \Gamma$ coincides with the number of minimal generators of
$J_{\Gamma^*}$ of degree $n-i$ which is $\beta_{0n-i}^E(J_{\Gamma^*})$,
i.~e.\ we have
$ \adeg_i \Gamma = \beta_{0n-i}^E(J_{\Gamma^*}) =
\beta_{1n-i}^E(E/J_{\Gamma^*}).$ Hence Lemma \ref{shifthelper} provides
$$
\adeg_i \Gamma
=
\beta_{1n-i}^E(E/J_{\Gamma^*})
\leq
\beta_{1n-i}^E(E/\gin(J_{\Gamma^*}))
=
\beta_{1n-i}^E(E/J_{\Delta^e(\Gamma^*)})
$$
$$
= \adeg_i \Delta^e(\Gamma^*)^* = \adeg_i \Delta^e(\Gamma)
$$
for $i=0,\dots,d$.
Since $ \beta_{1j}^E(E/J_{\Gamma^*}) \leq
\beta_{1j}^E(E/\gin(J_{\Gamma^*})) $ for all
$j$, it follows from Theorem \ref{maincl} that we have equality if and only if
$J_{\Gamma^*}$ is
componentwise linear. By \cite{HEHI99}, this is the case if and only if the
Stanley--Reisner ideal $I_{\Gamma^*} \; | \;ubset S=K[x_1,\dots,x_n]$ is
componentwise linear.
But the latter is equivalent to $\Gamma$ being sequentially
Cohen--Macaulay according to
\cite{HEREWE}.
Combining the above argument and Theorem \ref{mainginleq}, we obtain also
$$
\adeg_i \Delta^e(\Gamma) = \beta_{1n-i}^E(E/\gin(J_{\Gamma^*})) \leq
\beta_{1n-i}^E(E/\gin_\tau(J_{\Gamma^*})) = \adeg_i \Delta^\tau(\Gamma),
$$
and the proof is complete.
\end{proof}
\begin{rem}
\
\begin{enumerate}
\item
Observe that Theorem \ref{mainginleq} is essentially equivalent to the
following inequalities $m_{\leq i}((\gin_\tau(J))_j) \leq m_{\leq i}(\gin(J)_j)$.
Thus
one could argue in the above proof of Theorem \ref{shifted}
by using the latter numbers instead of the corresponding graded Betti-numbers.
(Again see also \cite[Proposition 2.4]{M05} for the needed inequalities.)
\item
The fact that
$\adeg \Gamma = \adeg \Delta^e(\Gamma)$ if and only if $\Gamma$ is sequentially
Cohen--Ma\-caulay can also be shown by refining the arguments in the proof of Theorem \ref{thm-adeg-incr}
and using the following facts:
a simplicial complex $\Gamma$ is sequentially Cohen--Macaulay
if and only if each subcomplex $\Gamma^{[i]}$ generated by the $i$-faces of $\Gamma$
is Cohen--Macaulay (see \cite{DU});
$\Gamma$ is Cohen--Macaulay if and only if $\Delta^e(\Gamma)$ is pure.
\end{enumerate}
\end{rem}
The arithmetic degree has nice properties especially for Cohen--Macaulay $K$-algebras.
However, there are some disadvantages for non-Cohen--Macaulay $K$-algebras. See
\cite{VA} for a discussion. Vasconcelos axiomatically
defined the following
concept. Recall that $S=K[x_1,\dots,x_n]$ where $K$ is a field. A numerical function
$\Deg$ that assigns to every finitely generated graded $S$-module a non-negative integer
is said to be an {\em extended degree function} if it satisfies the following conditions:
\begin{enumerate}
\item
If $L=H^0_{\m}(M)$, then $\Deg M =\Deg M/L +l(L).$
\item
If $y\in S_1$ is sufficiently general and $M$-regular,
then $\Deg M \geq \Deg M/yM.$
\item
If $M$ is a Cohen--Macaulay module, then $\Deg M=\deg M.$
\end{enumerate}
There exists a {\em smallest extended degree function} $\; | \;deg$ in the sense that $\; | \;deg M
\leq \Deg M$ for every other extended degree function.
\begin{rem}
\label{sdeghelper}
We need the following properties of $\; | \;deg$. (For proofs see
\cite{NR}.) For simplicity we state them only for a graded $K$-algebra $S/I$ where $I
\; | \;ubset S$ is a graded ideal.
\begin{enumerate}
\item
$S/I$ is Cohen--Macaulay if and only if $\; | \;deg S/I= \deg S/I$. (This is true for any
extended Degree function.)
\item
$\adeg S/I\leq \; | \;deg S/I$.
\item
If $S/I$ is sequentially Cohen--Macaulay, then $\adeg S/I=\; | \;deg S/I$.
\item
$\; | \;deg S/I = \; | \;deg S/\gin(I)$.
\end{enumerate}
\end{rem}
Then we have:
\begin{thm}
\label{sedgadegshift}
Let $\Gamma$
be a simplicial complex on the vertex set $[n]$.
We have that:
\begin{enumerate}
\item
$\; | \;deg \; | \;hift(\Gamma)=\adeg \; | \;hift(\Gamma)$ for each algebraic shifting operation
$\; | \;hift$.
\item
$\deg \Gamma \leq \adeg \Gamma \leq
\; | \;deg \Gamma = \; | \;deg \; | \;hift^s(\Gamma)=\adeg \; | \;hift^s(\Gamma)$.
\end{enumerate}
\end{thm}
\begin{proof}
In the proof we use that $\; | \;hift(\Gamma)$ is always sequentially Cohen--Macaulay, i.\ e.\
$K[\; | \;hift(\Gamma)]$ is a sequentially CM-ring (cf.\ Proposition \ref{prop-shifted-sCM}
below). This fact and the properties of $\; | \;deg$ observed in Remark \ref{sdeghelper}
imply (i).
The only critical equality in (ii) is
$\; | \;deg \Gamma = \; | \;deg \; | \;hift^s(\Gamma)$.
We compute
\begin{eqnarray*}
\; | \;deg \Gamma
&=& \; | \;deg S/I_\Gamma\\
&=& \; | \;deg S/\gin(I_\Gamma)\\
&=& \adeg S/\gin(I_\Gamma)\\
&=& \adeg S/I_{\; | \;hift^s(\Gamma)}\\
&=& \; | \;deg S/I_{\; | \;hift^s(\Gamma)}
\end{eqnarray*}
where the first equality is the definition of
$\; | \;deg \Gamma$,
the second one was noted above and
the third equality follows from the fact that $S/\gin(I_\Gamma)$
is sequentially Cohen--Macaulay (see \cite[Theorem 2.2]{HESB}).
The forth equality
follows again from
\cite[Theorem 6.6]{BNT}
and the last equality is (i) which we proved already.
\end{proof}
We already mentioned
that $\deg \Gamma$ is the number of facets of maximal dimension of $\Gamma$
and
$\adeg \Gamma$ is the number
of facets of $\Gamma$.
Theorem \ref{sedgadegshift}
provides a combinatorial interpretations of $\; | \;deg \Gamma$.
It is the number of facets of $\Delta^s(\Gamma)$.
\; | \;ection{The Cohen--Macaulay property and iterated Betti numbers}
\label{ringprop}
In this section we use degree functions to relate properties of the shifted complex and
iterated Betti numbers to the original complex.
Our first observation is well-known to specialists. It states that shifted complexes have
a nice algebraic structure.
\begin{prop}
\label{prop-shifted-sCM}
Shifted simplicial complexes are sequentially Cohen--Macaulay.
In particular,
if $\Gamma$ is a simplicial complex on the vertex set $[n]$
and $\; | \;hift(\cdot)$ is an arbitrary algebraic shifting operation,
then $\; | \;hift(\Gamma)$ is sequentially Cohen--Macaulay.
\end{prop}
\begin{proof}
Let $\Gamma$ be a shifted simplicial complex.
Recall that then $I_{\Gamma} \; | \;ubset S$ is a squarefree strongly stable ideal
with respect to $x_1>\dots>x_n$, i.~e.\ for all
$x_F=\prod_{l \in F}x_l \in I_{\Gamma}$
and $i$ with $x_i|x_F$ we have for all $j<i$ with $x_j\nmid x_F$
that $(x_F/x_i)x_j\in I_{\Gamma}$.
It is easy to see that we have that $I_{\Gamma}$ is squarefree strongly stable if and
only if $I_{\Gamma^*}$ is squarefree strongly stable where $\Gamma^*$ is the Alexander
dual of $\Gamma$. It is well-known that in this situation $I_{\Gamma^*}$ is a so-called
componentwise linear ideal (see Section \ref{cartan} for the definition) and thus the
theorem follows now from \cite[Theorem 9]{HEREWE}.
\end{proof}
\begin{rem}
Alternatively one
can prove Proposition \ref{prop-shifted-sCM}
using more combinatorial arguments as follows.
Shifted simplicial complexes are non-pure shellable by \cite{BW97}
and thus sequentially Cohen--Macaulay by \cite{ST96}.
\end{rem}
It is easy to decide whether a shifted simplicial complex is Cohen--Macaulay because of
the following well-known fact:
\begin{prop}
Let $\Gamma$ be a shifted simplicial complex on the vertex set $[n]$. Then the following
statements are equivalent:
\begin{enumerate}
\item
$\Gamma$ is Cohen--Macaulay;
\item
$\Gamma$ is pure.
\end{enumerate}
\end{prop}
\begin{proof}
If $\Gamma$ is Cohen--Macaulay, then it is well-known that
$\Gamma$ is pure.
Assume now that $\Gamma$ is a pure shifted simplicial complex.
We compute
$$
\; | \;deg S/I_{\Gamma}
=
\adeg S/I_{\; | \;hift^s(\Gamma)}
=
\adeg S/I_{\Gamma}
=
\deg S/I_{\Gamma}.
$$
The first equality was shown in Theorem \ref{sedgadegshift}, the second one holds because
$\Gamma$ is shifted, and the third one because $\Gamma$ is pure. Hence $\; | \;deg
S/I_{\Gamma}=\deg S/I_{\Gamma}$ and this implies by Remark \ref{sdeghelper} that $\Gamma$
is Cohen--Macaulay.
\end{proof}
Intuitively, shifting leads to a somewhat simpler complex and one would like to transfer
properties from $\; | \;hift (\Gamma)$ to $\Gamma$. In this respect, we have:
\begin{prop}
\label{justnoted} Let $\Gamma$ be a simplicial complex on the vertex set $[n]$ and let
$\; | \;hift(\cdot)$ be any algebraic shifting operation such that $\; | \;deg \Gamma \leq \; | \;deg
\; | \;hift(\Gamma)$. If $\; | \;hift(\Gamma)$ is Cohen--Macaulay, then $\Gamma$ is
Cohen--Macaulay.
\end{prop}
\begin{proof}
Suppose that $\; | \;hift(\Gamma)$ is Cohen--Macaulay. Then $\; | \;hift(\Gamma)$ is pure and we
have that $\adeg \; | \;hift(\Gamma) = \deg \; | \;hift(\Gamma)$. Thus, we get
$$
\deg \Gamma
\leq
\; | \;deg \Gamma
\leq
\; | \;deg \; | \;hift(\Gamma)
=
\adeg \; | \;hift(\Gamma)
=
\deg \; | \;hift(\Gamma)
=
\deg \Gamma.
$$
The only critical equality is
$\; | \;deg \; | \;hift(\Gamma)
= \adeg \; | \;hift(\Gamma)$ which follows from Remark \ref{sdeghelper} (iii) and Proposition \ref{prop-shifted-sCM}.
Hence $\deg \Gamma=\; | \;deg \Gamma$ and it follows that $\Gamma$ is Cohen--Macaulay.
\end{proof}
For exterior algebraic shifting, the following result was proved in \cite{KA01}.
For symmetric algebraic shifting, this result follows from some general homological
arguments given in \cite{BYCHPO}, as noted in \cite{BNT} and essentially first appeared in \cite{KA91} (Theorem 6.4). Below, we provide a very short proof using only degree functions.
\begin{thm}
\label{cmnice} Let $\Gamma$ be a simplicial complex on the vertex set $[n]$. Then the
following statements are equivalent:
\begin{enumerate}
\item
$\Gamma$ is Cohen--Macaulay;
\item
$\; | \;hift^{s}(\Gamma)$ is Cohen--Macaulay;
\item
$\; | \;hift^{s}(\Gamma)$ is pure.
\end{enumerate}
\end{thm}
\begin{proof}
We already know that (ii) and (iii) are equivalent.
Suppose that (i) holds.
Then
$$
\deg \; | \;hift^s(\Gamma)
=
\deg \Gamma
=
\; | \;deg \Gamma
=
\adeg \; | \;hift^s(\Gamma)
\geq
\adeg \Gamma
\geq
\deg \Gamma
=
\deg \; | \;hift^s(\Gamma).
$$
The first equality is trivial.
The second one holds because $\Gamma$ is Cohen--Macaulay,
and the third one follows from Theorem \ref{sedgadegshift} (ii).
The first inequality is due to
Theorem \ref{thm-adeg-incr} and the second one follows from the definition of $\adeg$.
Hence $\adeg \; | \;hift^s(\Gamma) = \deg \; | \;hift^s(\Gamma)$, and thus $\; | \;hift^s(\Gamma)$ is
pure, as claimed in (iii).
The fact that (iii) implies (i) follows from $\; | \;deg \Gamma=\adeg \; | \;hift^s(\Gamma) \leq
\; | \;deg \; | \;hift^s (\Gamma)$ and Proposition \ref{justnoted}.
\end{proof}
\begin{rem}
Using Theorem \ref{shifted} one can prove in a similar way that Theorem \ref{cmnice} holds also for the exterior shifting $\Delta^e$.
\end{rem}
For the next results, we need some further notation and definitions. At first we recall a
refinement of the $f$- and $h$-vector of a simplicial complex due to \cite{BW96}.
\begin{defi}
Let $\Gamma$ be a $(d-1)$-dimensional simplicial complex on the vertex set $[n]$. We
define
$$
f_{i,r}(\Gamma) = |\{ F \in \Gamma : \deg_{\Gamma} F = i \text{ and } |F| = r \}|
$$
where $\deg_{\Gamma} F= \max \{|G| : F \; | \;ubseteq G \in \Gamma\}$ is the {\em degree} of a
face $F \in \Gamma$. The triangular integer array $f(\Gamma) = (f_{i,r}(\Gamma))_{0\leq r
\leq i\leq d}$ is called the {\em $f$-triangle} of $\Gamma$. Further we define
$$
h_{i,r}(\Gamma) =
\; | \;um_{s=0}^r
(-1)^{r-s}\binom{i-s}{r-s} f_{i,s}(\Gamma).
$$
The triangular array $h(\Gamma)= (h_{i,r}(\Gamma))_{0\leq r\leq i\leq d}$
is called the {\em $h$-triangle} of $\Gamma$.
\end{defi}
It is easy to see that giving the $h$-triangle, one can also compute the $f$-triangle and
thus these triangles determine each other. For $F \; | \;ubseteq [n]$ let $\init(F)=\{k, \dots, n\}$ if $k,\dots,n \in F$, but $k-1 \not\in F$. (Here we set $\init(F)=\emptyset$ if no such $k$ exists.) Duval proved in \cite[Corollary 6.2]{DU} that if $\Gamma$ is sequentially
Cohen--Macaulay, then $h_{i,r} (\Gamma)= |\{F \in \facets(\Gamma) : |\init(F)|=i-r,\ |F|=i
\}|$. In particular, this formula holds for shifted simplicial complexes. Using this fact we define:
\begin{defi}
Let $\Gamma$ be a simplicial complex on the vertex set $[n]$
and $\; | \;hift(\cdot)$ an arbitrary algebraic shifting operation.
We call
$$
b^\Delta_{i,r}(\Gamma)
=
h_{i,r}(\; | \;hift(\Gamma))
=
|\{F \in \facets(\; | \;hift(\Gamma)) : |\init(F)|=i-r,\ |F|=i \}|
$$
the {\em $\Delta$-iterated Betti numbers} of $\Gamma$.
\end{defi}
For symmetric and exterior shifting these
notions were defined and studied in \cite{BNT} and \cite{DURO}.
Since $\; | \;hift(\; | \;hift(\Gamma))=\; | \;hift(\Gamma)$
we have that
$$
b^\Delta_{i,r}(\Gamma)
=
b^\Delta_{i,r}(\; | \;hift(\Gamma))
$$
and therefore $\Delta$-iterated Betti numbers are invariant under each algebraic shifting
operation $\; | \;hift(\cdot)$. As noted in \cite{BNT}, there is the following comparison
result.
\begin{thm} \label{thm-it-Betti}
Let $\Gamma$
be a simplicial complex on the vertex set $[n]$
and $\Delta(\cdot)$ be the symmetric shifting $\Delta^s$ or the exterior shifting $\Delta^e $.
Then the following conditions are equivalent:
\begin{enumerate}
\item
$\Gamma$ is sequentially Cohen--Macaulay;
\item
$b^{\; | \;hift}_{i,r}(\Gamma)=h_{i,r}(\Gamma)$
for all $i,r$.
\end{enumerate}
In particular, if $\Gamma$ is sequentially Cohen--Macaulay,
then
all iterated Betti numbers
of $\Gamma$ with respect to the symmetric shifting and the exterior
algebraic shifting coincide.
\end{thm}
\begin{proof}
Duval proved in \cite[Theorem 5.1]{DU} the equivalence of (i) and (ii) for exterior
algebraic shifting $\; | \;hift^e$. But his proof works also for the symmetric shifting
since he used only axioms (S1)--(S4) of an algebraic shifting operation and the fact that Theorem \ref{cmnice} holds also for the exterior shifting.
\end{proof}
The next result extends Theorem 6.6 of \cite{BNT} where the case of the symmetric
algebraic shifting operation was studied. It follows by combining Identities
\eqref{eq-comb-deg}, \eqref{eq-comb-adeg}, Proposition \ref{prop-shifted-sCM}, and Remark
\ref{sdeghelper} (iii).
\begin{thm}
Let $\Gamma$ be a simplicial complex on the vertex set $[n]$ of dimension $d-1$ and let
$\; | \;hift(\cdot)$ be any algebraic shifting operation. Then we have:
\begin{enumerate}
\item
$
\deg \; | \;hift(\Gamma) = \; | \;um_{r=0}^{d} b^{\; | \;hift}_{d,r}(\Gamma)
= |\{F \in \facets(\; | \;hift(\Gamma)),\ \dim F=d-1 \}|.
$
\item
$
\; | \;deg \; | \;hift(\Gamma)= \adeg \; | \;hift(\Gamma)
= \; | \;um_{r,i} b^{\; | \;hift}_{i,r}(\Gamma)
= |\{F \in \facets(\; | \;hift(\Gamma))\}|.
$
\end{enumerate}
\end{thm}
The last two results suggest the following question:
\begin{quest}
\ What is the relationship between $b^{\; | \;hift}_{i,r}(\Gamma)$ for different shifting
operations? For example, in \cite{BNT} it is conjectured that
$b^{\; | \;hift^s}_{i,r}(\Gamma) \leq b^{\; | \;hift^e}_{i,r}(\Gamma)$ for all $i,r$.
\end{quest}
\; | \;ection*{Acknowledgments}
The authors would like to thank the referee for the careful reading and the very helpful comments.
\end{document} |
\begin{document}
\title{}
\author{}
\date{}
\maketitle
\renewcommand{\arabic{section}.\arabic{equation}}{\arabic{section}.\arabic{equation}}
\begin{center}
{\Large The} ${\LARGE H}^{\infty }${\Large -control problem for parabolic
systems with singular Hardy potentials}
Gabriela Marinoschi
\textquotedblleft Gheorghe Mihoc-Caius Iacob\textquotedblright\ Institute of
Mathematical Statistics and
Applied Mathematics of the Romanian Academy,
Calea 13 Septembrie 13, Bucharest, Romania
\end{center}
\noindent Abstract. We solve the $H^{\infty }$-control problem with state
feedback for infinite dimensional boundary control systems of parabolic type
with distributed disturbances and apply the results to equations with Hardy
potentials with the singularity inside or on the boundary, in the cases of a
distributed control and of a boundary control.
Keywords: $H^{\infty }$-control, feedback control, robust control, abstract
parabolic problems, Hardy potentials
MSC 2020: 93B36, 93B52, 93B35, 35K90
\section{Introduction\label{Intro}}
\setcounter{equation}{0}
The $H^{\infty }$-control is a technique used in control theory to design
robust stabilizing feedback controllers that force a system to achieve
stability with a prescribed performance even if the system output may be
corrupted by perturbations. This method involves a transfer function which
incorporates the effects of the input perturbations towards the output
observation. The aim is to determine the optimal feedback controller which
minimizes the effect of these perturbations on the output, by ensuring that
the $L^{2}$-norm of the transfer function is smaller that the $L^{2}$-norm
of the perturbation with a certain prescribed bound. This turns out in
finding a suboptimal control solution constructed by means of a mathematical
optimization problem. The formal $H^{\infty }$-control theory was initiated
by Zames in \cite{Zames}, as an optimization problem with an operator norm,
in particular, the $H^{\infty }$-norm. State space formulations were
initially developed in \cite{Glover} and \cite{Packard-Doyle} and continued
later by the formulation of the necessary and sufficient conditions for the
existence of an admissible controller in terms of solutions of algebraic
Riccati equations. The state-space approach for linear infinite-dimensional $
H^{\infty }$-control problems was developed in further works and we cite
here e.g., \cite{VB-H-92}, \cite{VB-H-95-hyp}, \cite{VB-H-95}, \cite{Bas-Ber}
, \cite{Opmeer-Staffans-2010}, \cite{Opmeer-Staffans-2019}, \cite{Pri-Tow},
\cite{vanK-93}, \cite{vanK-al-93}, \cite{vanS-91}, \cite{vaS-92}, \cite{VB-S}
, the last for Navier-Stokes equations.
In this paper we discuss the $H^{\infty }$-control problem for linear
infinite dimensional systems of parabolic type and give applications for
equations with singular Hardy potentials, of the type $\frac{\lambda }{
|x|^{2}},$ which as far as we know is a novel approach. Following the papers
\cite{VB-H-92}, \cite{VB-H-95-hyp}, \cite{VB-H-95}, where the $H^{\infty }$
-control abstract problem was solved with assumptions proper for the
hyperbolic case, we prove here a main result stating the formulation of the $
H^{\infty }$-control problem in the parabolic case, relying on appropriate
assumptions for parabolic operators. This is further applied to three
parabolic control systems with Hardy potentials and with distributed or
boundary controls. There is an extensive literature on Hardy-type
inequalities with the singularity located inside the domain or on the
boundary, focusing also on controllability studies (see e.g., \cite{Ervedoza}
, \cite{Fragnelli-Mugnai}, \cite{Vancostenoble-Zuazua}). Besides the high
mathematical interest in such singular equations revealed in the past
decades, a parabolic operator with a Hardy potential term describes a
non-standard growth condition which may affect the behavior of the solutions
to diffusive physical models, as for example of heat transfer or diffusion
of contaminants in fluids. Also, it may represent an equivalent formulation
of a system of two equations in which a state in one equation is represented
as a fundamental solution by the other one. Operators with other similar
potentials can arise for example in quantum mechanics, \cite{Baras-Goldstein}
or in combustion theory, \cite{Bebernes}, \cite{Fragnelli-Mugnai}. Linear
parabolic equations with Hardy potentials have been studied in connection
with stationary Schr\"{o}dinger equations $-\Delta y+V(x)y+E(x)y=f$ with the
singular potential $V\in L^{\infty }(\Omega \backslash x_{0})$ arising from
the uncertainty principle. The robust stabilization of the corresponding
dynamic control system $y_{t}-\Delta y+V(x)y+E(x)y=B_{1}w+B_{2}u$, via the $
H^{\infty }$-control method, with the control $u$ and the exogenous
perturbation $w$ has direct implication for the equilibrium solution to the
above Schr\"{o}dinger equation. The content of the paper is briefly
described below.
In Section \ref{Prel} we present the mathematical formulation of the $
H^{\infty }$-control problem. In Section \ref{Main}, after specifying the
work hypotheses we provide the main result stating the existence of the
feedback controller determined via a Riccati equation. In Sections \ref
{Distributed} and \ref{Boundary} there are given applications for parabolic
equations in the $N$-dimensional case with a distributed control and a
boundary control, respectively, and with Hardy potentials with interior
singularity, while in Section \ref{1D} it is treated the $1$-$D$ case with a
boundary singular Hardy potential.
\section{Problem presentation and preliminaries\label{Prel}}
\setcounter{equation}{0}
In this section we briefly explain the state-space approach of the $
H^{\infty }$-control problem for the linear system
\begin{eqnarray}
&&y^{\prime }(t)=Ay(t)+B_{1}w(t)+B_{2}u(t),\mbox{ \ }t\in \mathbb{R}
_{+}:=(0,+\infty ) \label{1} \\
&&z(t)=C_{1}y(t)+D_{1}u(t),\mbox{ }t\in \mathbb{R}_{+}, \label{2} \\
&&y(0)=y_{0}, \label{1'}
\end{eqnarray}
where $A,$ $B_{1},$ $B_{2},$ $C_{1},$ $D_{1}$ are linear operators
satisfying hypotheses that will be immediately specified. Here, $y$ is the
system state, $u$ is the control input, $w$ is an exogenous input, or an
unknown perturbation and $z$ is the performance output.
At this point we put down a few notation, definitions and results necessary
for explaining the problem. Let $X$ be a real Hilbert space with the scalar
product and norm denoted by $(\cdot ,\cdot )_{X}$ and $\left\Vert \cdot
\right\Vert _{X},$ respectively and $X^{\prime }$ is its dual. The symbol $
\left\langle \cdot ,\cdot \right\rangle _{X^{\prime },X}$ is the pairing
between $X^{\prime }$ and $X.$ Let $A$ be a linear closed operator on $X$
with the domain $D(A):=\{y\in X;$ $Ay\in X\}$ dense in $X.$ By $A^{\ast }$
we denote the the adjoint of $A.$ If $Y$ is another Hilbert space, $L(X,Y)$
represent the space of all linear continuous operators from $X$ to $Y.$
Let $H,$ $U,$ $W,$ $Z$ be real Hilbert spaces identified with their duals.
For the beginning we assume:
\begin{itemize}
\item[$(i_{1})$] $A$ is the infinitesimal generator of an analytic $C_{0}$
-semigroup $e^{At}$ on the Hilbert space $H,$ $e^{At}$ is compact for $t>0,$
and
\begin{equation}
B_{1}\in L(W,H),\mbox{ }B_{2}\in L(U,(D(A^{\ast }))^{\prime }),\mbox{ }
C_{1}\in L(H,Z),\mbox{ }D_{1}\in L(U,Z). \label{1i}
\end{equation}
Here, $(D(A^{\ast }))^{\prime }$ is the dual of the domain of $A^{\ast }$,
where $D(A^{\ast })$ is organized as a Hilbert space with the scalar product
$(y_{1},y_{2})_{D(A^{\ast })}=(A^{\ast }y_{1},A^{\ast
}y_{2})_{H}+(y_{1},y_{2})_{H}$ for $y_{1},$ $y_{2}\in D(A^{\ast }).$
\end{itemize}
\noindent We note that the space $(D(A^{\ast }))^{\prime }$ is the
completion of $H$ in the norm $\left\vert \left\Vert y\right\Vert
\right\vert =\left\Vert (A-\lambda _{0}I)^{-1}y\right\Vert _{H},$ $\lambda
_{0}\in \rho (A).$ Also, we define the extension of the operator $A$ from $H$
to $(D(A^{\ast }))^{\prime },$ denoted for convenience still by $A,$ by
\begin{equation}
\left\langle Ay,\psi \right\rangle _{(D(A^{\ast }))^{\prime },D(A)}=\left(
y,A^{\ast }\psi \right) _{H},\mbox{ for }y\in H,\mbox{ }\psi \in D(A^{\ast
}). \label{401}
\end{equation}
We shall work with both operators and if not seen clearly from the context
which operator is used, we shall specify this.
\noindent Let us consider the uncontrolled system $y^{\prime }(t)=Ay(t),$ $
t\in \mathbb{R}_{+},$ $y(0)=y_{0},$ with $A$ the infinitesimal generator of
a $C_{0}$-semigroup on $H.$
\begin{definition}
\textrm{The operator $A$ generates an exponentially stable semigroup $e^{At}$
if
\begin{equation}
\left\Vert e^{At}\right\Vert _{L(H,H)}\leq Ce^{-\alpha t},\mbox{ for all }
t\geq 0, \label{4}
\end{equation}
where $\alpha $ and $C$ are positive constants. }
\end{definition}
\noindent Relation (\ref{4}) still reads
\begin{equation}
\left\Vert e^{At}y\right\Vert _{H}\leq Ce^{-\alpha t}\left\Vert y\right\Vert
_{H},\mbox{ for all }y\in H\mbox{ and all }t\geq 0. \label{5}
\end{equation}
\noindent Moreover, a result of Datko (see \cite{Datko}) asserts that
relation (\ref{5}) is equivalent to
\begin{equation}
\int_{0}^{\infty }\left\Vert y(t)\right\Vert _{H}^{2}dt<\infty . \label{6}
\end{equation}
\begin{definition}
\textrm{The pair $(A,C_{1})$ in system (\ref{1})-(\ref{2}) is exponentially
detectable if there exists $K\in L(Z,H)$ such that $A+KC_{1}$ generates an
exponentially stable semigroup. }
\end{definition}
In order to state our $H^{\infty }$-control problem, we recall some issues
about such a problem. Assume that under certain conditions system (\ref{1})-(
\ref{1'}) has a mild solution $y\in C([0,T];H)$ for all $T>0$ and $u$ can be
represented as a feedback controller $u=Fy,$ where generally $F:U\rightarrow
H$ is a linear closed and densely defined operator. Then, the solution $
(y(t),z(t))$ becomes dependent only on $w(t)$ and reads
\begin{eqnarray}
&&y(t)=e^{(A+B_{2}F)t}y_{0}+\displaystyle
\int_{0}^{t}e^{(A+B_{2}F)(t-s)}B_{1}w(s)ds,\ t\in \lbrack 0,\infty ),
\label{7} \\
&&z(t)=(C_{1}+D_{1}F)e^{(A+B_{2}F)t}y_{0}+(C_{1}+D_{1}F)\displaystyle
\int_{0}^{t}e^{(A+B_{2}F)(t-s)}B_{1}w(s)ds. \label{8}
\end{eqnarray}
The latter equation can be still written
\begin{equation}
z(t)=f_{0}(t)+(G_{F}w)(t),\mbox{ }t\geq 0 \label{8-1}
\end{equation}
where $f_{0}(t)=(C_{1}+D_{1}F)e^{(A+B_{2}F)t}y_{0}\in Z,$ $t\geq 0,$ and $
G_{F}:L^{2}(\mathbb{R}_{+},W)\rightarrow L^{2}(\mathbb{R}_{+},Z),$ defined by
\begin{equation}
(G_{F}w)(t)=(C_{1}+D_{1}F)\int_{0}^{t}e^{(A+B_{2}F)(t-s)}B_{1}w(s)ds\in Z,
\mbox{ }t\geq 0, \label{8-2}
\end{equation}
shows the transfer of the influence of the perturbation input $w$ to the
output. Roughly speaking, the $H^{\infty }$-control problem means to find a
feedback controller which stabilizes exponentially the system (with $
y_{0}=0) $, with a certain specified performance for the output $G_{F}w$,
depending on a given constant $\gamma .$ Such a feedback control $F$ is
called a suboptimal solution and the $H^{\infty }$ problem can be formulated
as follows: given $\gamma >0,$ find the feedback control $F$ which
exponentially stabilizes system (\ref{1})-(\ref{2}) such that $\left\Vert
G_{F}\right\Vert _{L(L^{2}(\mathbb{R}_{+},W),L^{2}(\mathbb{R}
_{+},Z))}<\gamma .$
To be more precise in what concerns the relation with the Hardy space $
H^{\infty }$, we briefly recall a well-known result property of
vector-valued Hardy classes (see e.g., \cite{Staffans}, \cite{Staffans-Weiss}
, \cite{Curtain-Zwart}, Theorem A6.26). The space $H^{\infty }$ is defined
as the vector space of bounded holomorphic functions on the right half
plane, $\mathbb{C}_{+}=\{z\in \mathbb{C};$ $\mathit{Re}\,z>0\}$, with the
norm $\left\Vert f\right\Vert _{H^{\infty }}=\sup_{\left\vert z\right\vert
<1}\left\vert f(z)\right\vert .$ Let us take the Laplace transform in system
(\ref{1})-(\ref{2}) and get
\begin{equation}
\widehat{z}(\zeta )=C_{1}(\zeta I-A-B_{2}F)^{-1}y_{0}+\widehat{G_{F}}(\zeta )
\widehat{w}(\zeta ). \label{9-0}
\end{equation}
The function $\widehat{G_{F}}:\mathbb{C}_{+}\rightarrow L(W,Z),$
\begin{equation}
\widehat{G_{F}}(\zeta )=(C_{1}+D_{1}F)(\zeta I+A+B_{2}F)^{-1}B_{1}
\label{9-1}
\end{equation}
is the transfer function in the frequency domain, giving a relationship
between the input and output of the system. It plays an important role in
control theory by providing an insight in how disturbances in the system can
affect the output. The results in the papers cited before express the fact
that the $L^{2}$-operator norm of the gain in the time domain is equal to
the Hardy $H^{\infty }(L(W,Z))$-norm of the transfer operator in the
frequency domain, i.e.,
\begin{equation}
\left\Vert G_{F}\right\Vert _{L(L^{2}(\mathbb{R}_{+},W),L^{2}(\mathbb{R}
_{+},Z))}:=\sup_{w\in L^{2}(\mathbb{R}_{+},W)}\frac{\left\Vert
G_{F}w\right\Vert _{L^{2}(\mathbb{R}_{+},Z)}}{\left\Vert w\right\Vert
_{L^{2}(\mathbb{R}_{+},W)}}=\sup_{\zeta \in \mathbb{C}_{+}}\left\Vert
\widehat{G_{F}}(\zeta )\right\Vert _{L(W,Z)}=:\left\Vert \widehat{G_{F}}
\right\Vert _{H^{\infty }}<\gamma . \label{9-2}
\end{equation}
\textit{Notation and some necessary results.} We end this section by
recalling some other notation and results necessary in the paper. We denote
by $H^{m}(\Omega )$ the Sobolev spaces $W^{2,m}(\Omega ),$ for $m\geq 1$ and
by $H_{0}^{1}(\Omega )$ the space $\{y\in H^{1}(\Omega );$ $tr(y)=0$ on $
\Gamma \},$ where $tr(y)$ is the trace operator of $y$ on $\Gamma :=\partial
\Omega .$ Moreover, $H^{-1}(\Omega )$ denotes the dual of $H_{0}^{1}(\Omega
).$ Given a Banach space $X$ and $T\in (0,\infty ]$ we define by $
L^{p}(0,T;X)$ the space of $L^{p}$ $X$-valued functions on $(0,T),$ $p\in
\lbrack 1,\infty ],$ by $C([0,T];X)$ the space of continuous $X$-valued
functions on $(0,T)$ and $W^{1,p}(0,T;X)=\{u\in L^{p}(0,T;X);$ $du/dt\in
L^{p}(0,T;X)\}.$
Let $L:D(L)\subset H\rightarrow H$ be a linear operator defined on the
Hilbert space $H.$ We say that $L$ is $m$-accretive if $L$ is accretive,
meaning that $(Ly,y)_{H}\geq 0,$ $\forall y\in D(L),$ and if $R(I+L)=H,$
where $R$ is the range. The operator $L$ is quasi $m$-accretive or $\omega $-
$m$-accretive if $\omega I+L$ is $m$-accretive for some $\omega >0.$
\textit{Hardy inequalities.} Let $N>3$ and let $\Omega $ be an open bounded
subset of $\mathbb{R}^{N},$\textit{\ }with $0\in \Omega .$ Then we have
\begin{equation}
\int_{\Omega }\left\vert \nabla y(x)\right\vert ^{2}dx\geq H_{N}\int_{\Omega
}\frac{\left\vert y(x)\right\vert ^{2}}{\left\vert x\right\vert ^{2}}dx,
\mbox{ for all }y\in H_{0}^{1}(\Omega ), \label{HN}
\end{equation}
where $H_{N}=\frac{(N-2)^{2}}{4}$ is optimal (see \cite{Brezis-Vazquez}, p.
452, Theorem 4.1).
Let $\Omega =(0,1).$ Then we have
\begin{equation}
\int_{0}^{1}\left\vert y^{\prime }(x)\right\vert ^{2}dx\geq \frac{1}{4}
\int_{0}^{1}\frac{y(x)}{\left\vert x\right\vert ^{2}}dx,\mbox{ }\forall y\in
H^{1}(0,1),\mbox{ }y(0)=0, \label{HN0}
\end{equation}
see \cite{Brezis-Marcus}, p. 217, or Lemma A.1, p. 234.
We recall the Young's inequality for convolutions $(f\ast
g)(t)=\int_{0}^{\infty }f(t-\tau )g(\tau )d\tau ,$
\begin{equation}
\left\Vert f\ast g\right\Vert _{L^{r}(0,\infty )}\leq \left\Vert
f\right\Vert _{L^{p}(0,\infty )}\left\Vert g\right\Vert _{L^{q}(0,\infty )},
\mbox{ where }\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},\mbox{ }1\leq p,\mbox{ }
q,\mbox{ }r\leq \infty . \label{400}
\end{equation}
For simplicity, where there is no risk of confusion, the $L^{p}(\Omega )$
-norm will be denoted by $\left\Vert \cdot \right\Vert _{p},$ $p\in \lbrack
1,\infty ],$ instead of $\left\Vert \cdot \right\Vert _{L^{p}(\Omega )}.$ We
set $\mathbb{R}=(-\infty ,\infty )$ and $\mathbb{R}_{+}=(0,\infty ).$ Also, $
\left\vert \cdot \right\vert $ will represent the Euclidian norm in $\mathbb{
R}^{N},$ for any $N=1,2,...,$ accordingly. In the further calculations $C,$ $
C_{1},...,C_{N},$ $C_{T}$ denote positive constants (which may change from
line to line), $C_{N}$ depending on $N,$ via $\lambda <H_{N}$ and $C_{T}$
depending on $T.$
\section{The main result\label{Main}}
\setcounter{equation}{0}
Besides $(i_{1})$ we assume the following hypotheses:
\begin{itemize}
\item[$(i_{2})$] the next relation takes place:
\begin{equation}
\left\Vert B_{2}^{\ast }e^{A^{\ast }t}\right\Vert _{L(H,U)}\in L^{1}(0,T),
\mbox{ for all }T>0, \label{12}
\end{equation}
\item[$(i_{3})$] the pair $(A,C_{1})$ is exponentially detectable (that is
there exists $K\in L(Z,H)$ such that $A+KC_{1}$ generates an exponentially
stable semigroup) and
\begin{equation}
\int_{0}^{\infty }\left\Vert B_{2}^{\ast }e^{(A^{\ast }+C_{1}^{\ast }K^{\ast
})t}y\right\Vert _{U}dt\leq C\left\Vert y\right\Vert _{H},\mbox{ for all }
y\in H, \label{12-0}
\end{equation}
\item[$(i_{4})$] $\left\Vert D_{1}^{\ast }D_{1}u\right\Vert _{U^{\ast
}}=\left\Vert u\right\Vert _{U}$ and $D_{1}^{\ast }C_{1}=0.$
\end{itemize}
\noindent Let us comment a little these hypotheses. The $L^{1}$
-admissibility hypothesis of the observation operator $B_{2}^{\ast }$ in $
(i_{2})$ is made in order to ensure the existence of a mild solution to (\ref
{1}) in $L^{2}(0,T;H)$ for every $T>0,$ with initial condition $y_{0}$ and
inputs $u\in L^{2}(0,T;U)$ and $w\in L^{2}(0,T;W)$. In an ideal situation
when $B_{2}\in L(U,H),$ eqs. (\ref{1})-(\ref{1'}) have a unique mild
solution $y\in C([0,T];H),$ for every $T>0,$ given by
\begin{equation}
y(t)=e^{At}y_{0}+\int_{0}^{t}e^{A(t-s)}B_{1}w(s)ds+
\int_{0}^{t}e^{A(t-s)}B_{2}u(s)ds,\mbox{ }t\in \lbrack 0,\infty ).
\label{13}
\end{equation}
But generally, $B_{2}$ may be not continuous from $U$ to $H,$ in some
situations its range being in a larger abstract space, indicated before to
be $(D(A^{\ast }))^{\prime }.$ The unique solution to (\ref{1})-(\ref{1'})
is in this case in $C([0,\infty );(D(A^{\ast }))^{\prime }).$ Consequently,
the previous formula should be written in a weak sense, that is for all $
t\geq 0,$ we have
\begin{equation}
(y(t),\varphi )_{H}=(e^{At}y_{0},\varphi )_{H}+\int_{0}^{t}\left(
e^{A(t-s)}(B_{1}w(s),\varphi )_{H}+(u(s),B_{2}^{\ast }e^{A^{\ast
}(t-s)}\varphi )_{U}\right) ds,\mbox{ }\forall \varphi \in H,\mbox{ }
y_{0}\in H. \label{13-1}
\end{equation}
Assumption $(i_{2})$ ensures that $y\in L^{2}(0,T;H),$ and this follows by
proving that $\int_{0}^{T}(y(t),\varphi (t))_{H}dt<C\left\Vert \varphi
\right\Vert _{L^{2}(0,T;H)},$ for $\varphi \in L^{2}(0,T;H).$ Indeed, this
is clearly seen for the first two terms in (\ref{13-1}), since $B_{1}w\in
L^{2}(\mathbb{R}_{+};H)$. For the last term we calculate
\begin{eqnarray}
&&\int_{0}^{T}\int_{0}^{t}\left( u(s),B_{2}^{\ast }e^{A^{\ast }(t-s)}\varphi
(t)\right) _{U}dsdt=\int_{0}^{T}\int_{s}^{T}\left( u(s),B_{2}^{\ast
}e^{A^{\ast }(t-s)}\varphi (t)\right) _{U}dtds \label{16-0} \\
&\leq &\left( \int_{0}^{T}\left\Vert u(s)\right\Vert _{U}^{2}ds\right)
^{1/2}\left( \int_{0}^{T}\left\Vert \int_{0}^{T}B_{2}^{\ast }e^{A^{\ast
}(t-s)}\varphi (t)dt\right\Vert _{U}^{2}ds\right) ^{1/2} \notag \\
&\leq &\left\Vert u\right\Vert _{L^{2}(0,T;U)}\left\{ \left(
\int_{0}^{T}\left\Vert B_{2}^{\ast }e^{A^{\ast }t}\right\Vert
_{L(H,U)}dt\right) \left( \int_{0}^{T}\left\Vert \varphi (t)\right\Vert
_{H}^{2}dt\right) ^{1/2}\right\} \notag \\
&\leq &\left\Vert u\right\Vert _{L^{2}(0,T);U)}\left( \int_{0}^{T}\left\Vert
B_{2}^{\ast }e^{A^{\ast }t}\right\Vert _{L(H,U)}dt\right) \left\Vert \varphi
\right\Vert _{L^{2}(0,T;H)}\leq C\left\Vert \varphi \right\Vert
_{L^{2}(0,T;H)}, \notag
\end{eqnarray}
where we used $(i_{2})$ and the Young's inequality for convolution (\ref{400}
) with $p=1,$ $q=r=2.$ Then, it follows that $y\in L^{2}(0,T;H)$ and the
last term in (\ref{13}) is in $H$.
Regarding (\ref{12-0}) we mention that the corresponding result related to $
L^{2}$ instead of $L^{1}$ is a particular case of Theorem 5.4.2 in \cite
{Tucsnak-Weiss}, so that we expect that (\ref{12}) and the detectability
hypothesis imply (\ref{12-0}), at least in some cases. However, we keep here
relation (\ref{12-0}) as a hypothesis and check it in the applications, by
different proofs according the case. In applications, the first relation in
hypothesis $(i_{4})$ may be weaken to $D_{1}^{\ast }D_{1}\geq \epsilon I$
(see e.g., \cite{VB-S}). However, for certain choices of operators $D_{1}$
and $C_{1},$ relations $(i_{4})$ may be proved as they are.
Theorem 3.1 below is the main result concerning the $H^{\infty }$-control
problem under hypotheses $(i_{1})$-$(i_{4})$ and it gives a representation
for the feedback operator $F$ which is a suboptimal solution to our $
H^{\infty }$-control problem.
This theorem was proved, under some appropriate hypotheses for the
hyperbolic case in \cite{VB-H-92} and \cite{VB-H-95-hyp}. Actually, instead
of (\ref{12}) there it was used the $L^{2}$-admissibility condition
\begin{equation}
\int_{0}^{T}\left\Vert B_{2}^{\ast }e^{A^{\ast }t}y\right\Vert
_{U}^{2}dt\leq C_{T}\left\Vert y\right\Vert _{H}^{2},\mbox{ for every }y\in H
\mbox{ and }T>0. \label{14}
\end{equation}
For the treatment of specific parabolic problems intended to be achieved in
the paper, we have in mind to adapt that approach to the case covered by
assumptions $(i_{1})-(i_{4})$ to obtain the following main result.
\begin{theorem}
\label{Th-main}Let hypotheses $(i_{1})-(i_{4})$ hold and let $\gamma >0.$
Assume that there exists $F\in L(H,U)$ such that $A+B_{2}F$ generates an
analytic exponentially stable $C_{0}$-semigroup on $H$ and
\begin{equation}
\left\Vert G_{F}\right\Vert _{L(L^{2}(\mathbb{R}_{+};W),L^{2}(\mathbb{R}
_{+};Z))}<\gamma . \label{14-1}
\end{equation}
Then, there exists a Hilbert space $\mathcal{X}\subset H$ with dense and
continuous injection and an operator
\begin{equation}
P\in L(H,H)\cap L(\mathcal{X},D(A^{\ast })),\mbox{ }P=P^{\ast }\geq 0,
\label{14-2}
\end{equation}
which satisfies the algebraic Riccati equation
\begin{equation}
A^{\ast }Py+P(A-B_{2}B_{2}^{\ast }P+\gamma ^{-2}B_{1}B_{1}^{\ast
}P)y+C_{1}^{\ast }C_{1}y=0,\mbox{ }\forall y\in \mathcal{X}, \label{15}
\end{equation}
where $B_{2}^{\ast }P\in L(\mathcal{X},U)$ and the operators
\begin{equation}
\Lambda _{P}:=A-B_{2}B_{2}^{\ast }P+\gamma ^{-2}B_{1}B_{1}^{\ast }P,\mbox{ }
\Lambda _{P}^{1}:=A-B_{2}B_{2}^{\ast }P \label{15-prim}
\end{equation}
with the domain $\mathcal{X}$ generate exponentially stable semigroups on $H$
. Moreover, the feedback control
\begin{equation}
\widetilde{F}=-B_{2}^{\ast }P \label{16}
\end{equation}
solves the $H^{\infty }$-problem, that is $\left\Vert G_{\widetilde{F}
}\right\Vert _{L(L^{2}(\mathbb{R}_{+};W),L^{2}(\mathbb{R}_{+};Z))}<\gamma .$
Conversely, assume that there exists a solution $P$ to equation (\ref{15})
with the properties (\ref{14-2}) and such that the corresponding operators $
\Lambda _{P}$ and $\Lambda _{P}^{1}$ generate exponentially stable
semigroups on $H.$ Then, the feedback operator $\widetilde{F}=-B_{2}^{\ast
}P $ solves the $H^{\infty }$-problem (\ref{14-1}).
\end{theorem}
The space $\mathcal{X}$ will be defined in the theorem proof before Lemma
\ref{operators}, in (\ref{3-43-2}). Moreover, we shall show in Lemma \ref
{operators} that if the operator $\Lambda _{P}$ with the domain $D(\Lambda
_{P})=\{y\in H;$ $\Lambda _{P}y=(A-B_{2}B_{2}^{\ast }P+\gamma
^{-2}B_{1}B_{1}^{\ast }P)y\in H\}$ is closed, then $\mathcal{X}=D(\Lambda
_{P}).$ This will happen in all examples given the next sections.
\noindent \textbf{Proof }of Theorem \ref{Th-main}\textbf{. }We assume first
that there exists a solution $F\in L(H,U)$ to the $H^{\infty }$-control
problem such that $A_{F}:=A+B_{2}F$ generates an analytic exponentially
stable $C_{0}$-semigroup and (\ref{14-1}) holds. We must prove that there
exists $P$ satisfying (\ref{14-2})-(\ref{16}).
The state-space approach of the above $H^{\infty }$-control problem comes
back to solve the differential game
\begin{equation}
\sup_{w\in L^{2}(\mathbb{R}_{+},W)}\inf_{u\in L^{2}(\mathbb{R}_{+},U)}\frac{1
}{2}\int_{0}^{\infty }(\left\Vert z(t)\right\Vert _{Z}^{2}-\gamma
^{2}\left\Vert w(t)\right\Vert _{W}^{2})dt, \label{11}
\end{equation}
subject to (\ref{1})-(\ref{1'}), which ensures a prescribed bound on the
Hardy norm $H^{\infty }$ of the transfer operator (see e.g., \cite
{VB-H-95-hyp}).
Let $J:L^{2}(\mathbb{R}_{+};U)\times L^{2}(\mathbb{R}_{+};W)\rightarrow
\lbrack -\infty ,\infty ]$ be defined as
\begin{equation}
J(u,w)=\frac{1}{2}\int_{0}^{\infty }\{\left\Vert
C_{1}y(t)+D_{1}u(t)\right\Vert _{Z}^{2}-\gamma ^{2}\left\Vert
w(t)\right\Vert _{W}^{2}\}dt \label{200}
\end{equation}
and consider first a minimization problem, for a fixed $w\in L^{2}(\mathbb{R}
_{+};W)$,
\begin{equation}
\inf_{u\in L^{2}(\mathbb{R}_{+};U)}J(u,w), \label{201}
\end{equation}
subject to system (\ref{1})-(\ref{2}). By hypothesis $(i_{4})$ we see that
\begin{equation}
J(u,w)=\frac{1}{2}\int_{0}^{\infty }\{\left\Vert C_{1}y(t)\right\Vert
_{Z}^{2}+\left\Vert u(t)\right\Vert _{U}^{2}-\gamma ^{2}\left\Vert
w(t)\right\Vert _{W}^{2}\}dt, \label{202}
\end{equation}
so $u\rightarrow J(u,w)$ is strictly convex, whence it easily can be shown
that (\ref{201}) has a unique solution
\begin{equation}
u^{\ast }=\Gamma w \label{203}
\end{equation}
with $\Gamma :L^{2}(\mathbb{R}_{+};W)\rightarrow L^{2}(\mathbb{R}_{+};U).$
We denote by $y^{u^{\ast }}$ the solution to (\ref{1}) corresponding to $
u^{\ast }$ (realizing the minimum in (\ref{201})) and $w,$ that is, $
y^{u^{\ast }}:=y^{u^{\ast },w}.$
\begin{lemma}
There exists $p\in C(\mathbb{R}_{+};H)\cap L^{2}(\mathbb{R}_{+};H)$
satisfying
\begin{equation}
p^{\prime }(t)=-A_{F}^{\ast }p(t)+C_{1}^{\ast }C_{1}y^{u^{\ast }}(t)+F^{\ast
}u^{\ast }(t),\mbox{ }t\in \mathbb{R}_{+}, \label{204}
\end{equation}
\begin{equation}
u^{\ast }(t)=B_{2}^{\ast }p(t),\mbox{ a.e. }t>0. \label{205}
\end{equation}
\end{lemma}
\begin{proof}
We note first that the solution to the equation $y^{\prime
}(t)=A_{F}y(t)+B_{1}w(t)$ with $y_{0}\in H,$ where $A_{F}=A+B_{2}F$ is
exponentially stable on $H$, is in $L^{2}(\mathbb{R}_{+};H).$ Indeed,
\begin{eqnarray*}
&&\left\Vert y(t)\right\Vert _{H}\leq Ce^{-\alpha t}\left\Vert
y_{0}\right\Vert _{H}+\int_{0}^{t}\left\Vert e^{A_{F}(t-s)}w(s)\right\Vert
_{H}ds \\
&\leq &Ce^{-\alpha t}\left\Vert y_{0}\right\Vert _{H}+\int_{0}^{t}e^{-\alpha
(t-s)}\left\Vert w(s)\right\Vert _{W}ds,\mbox{ }t\geq 0,
\end{eqnarray*}
and by applying the Young's inequality for convolution (\ref{400}) with $
r=2, $ $p=1$ and $q=2$ we obtain
\begin{eqnarray}
&&\int_{0}^{\infty }\left\Vert y(t)\right\Vert _{H}^{2}dt\leq C\left\{
\left\Vert y_{0}\right\Vert _{H}^{2}+\int_{0}^{\infty }\left(
\int_{0}^{t}e^{-\alpha (t-s)}\left\Vert w(s)\right\Vert _{W}ds\right)
^{2}dt\right\} \label{11-2} \\
&\leq &C\left\Vert y_{0}\right\Vert _{H}^{2}+C\left( \int_{0}^{\infty
}e^{-\alpha t}dt\right) ^{2}\int_{0}^{\infty }\left\Vert w(t)\right\Vert
_{W}^{2}dt\leq C(\left\Vert y_{0}\right\Vert _{H}^{2}+\left\Vert
w\right\Vert _{L^{2}(0,\infty ;W)}^{2})<\infty . \notag
\end{eqnarray}
We specify that the solution to (\ref{204}) should be understand in the
following mild sense
\begin{equation*}
p(t)=-\int_{t}^{\infty }e^{-A_{F}^{\ast }(s-t)}(C_{1}^{\ast }C_{1}y^{u^{\ast
}}(s)+F^{\ast }u^{\ast }(s))ds,
\end{equation*}
and so $C([0,\infty );H)$ because $F^{\ast }\in L(U,H),$ $C_{1}^{\ast
}C_{1}\in L(H,H).$ Since $A_{F}^{\ast }$ generates an analytic $C_{0}$
-semigroup it follows by its regularizing effect that $p\in
W^{1,2}(0,T;H)\cap L^{2}(0,T;D(A_{F}^{\ast })),$ for all $T>0.$
We introduce $v:=u-Fy$ and write problem (\ref{201}) as
\begin{equation}
\inf_{v\in L^{2}(\mathbb{R}_{+};U)}\frac{1}{2}\int_{0}^{\infty }\{\left\Vert
C_{1}y(t)\right\Vert _{Z}^{2}+\left\Vert Fy(t)+v(t)\right\Vert
_{U}^{2}-\gamma ^{2}\left\Vert w(t)\right\Vert _{W}^{2}\}dt \label{207}
\end{equation}
subject to $y^{\prime }(t)=A_{F}y(t)+B_{1}w(t)+B_{2}v(t),$ $t\geq 0,$ $
y(0)=y_{0}$. Since the functional is weakly lower semicontinous and convex,
it follows that (\ref{207}) has a unique solution $v^{\ast }=u^{\ast
}-Fy^{u^{\ast }},$ with $u^{\ast }$ the solution to (\ref{201}) and $
y^{u^{\ast }}$ the solution to (\ref{1}) corresponding to $u^{\ast }$ and $
w. $
We set the variation $v^{\lambda }=v^{\ast }+\lambda V,$ where $\lambda >0,$
$V\in L^{2}(\mathbb{R}_{+};U)$ and write the system in variations
\begin{equation}
Y^{\prime }(t)=A_{F}Y(t)+B_{2}V(t),\mbox{ }Y(0)=0, \label{209}
\end{equation}
where $Y(t)=\lim_{\lambda \rightarrow 0}\frac{y^{v^{\lambda }}-y^{v^{\ast }}
}{\lambda }$ weakly in $L^{2}(\mathbb{R}_{+};H).$ Eq. (\ref{209}) can be
still written as $Y^{\prime }(t)=AY(t)+B_{2}(FY(t)+V(T))$ and so it is
easily seen that it has a unique solution $Y$ belonging to $W^{1,2}(\mathbb{R
}_{+};(D(A_{F}^{\ast }))^{\prime })\cap L^{2}(\mathbb{R}_{+};H)$, the latter
following in the same way as shown before for $y(t)$ in (\ref{16-0})$.$
Writing that $v^{\ast }$ realizes the minimum in (\ref{207}), in particular
that $J(v^{\lambda },w)\geq J(v^{\ast },w),$ we deduce
\begin{equation*}
\int_{0}^{\infty }\left\{ (C_{1}y^{u^{\ast }}(t),C_{1}Y(t))_{Z}+(Fy^{u^{\ast
}}(t)+v^{\ast }(t),FY(t)+V(t))_{U}\right\} dt\geq 0.
\end{equation*}
If $\lambda \rightarrow -\lambda $ we obtain the reverse inequality, so that
in conclusion
\begin{equation}
\int_{0}^{\infty }\left\{ (C_{1}^{\ast }C_{1}y^{u^{\ast }}(t)+F^{\ast
}Fy^{u^{\ast }}(t)+F^{\ast }v^{\ast }(t),Y(t))_{H}+(Fy^{u^{\ast
}}(t)+v^{\ast }(t),V(t))_{U}\right\} dt=0, \label{210}
\end{equation}
for all $V\in L^{2}(\mathbb{R}_{+};U).$ By testing the first equation (\ref
{209}) by $p(t)\in D(A_{F}^{\ast }),$ solution to (\ref{204}) and
integrating with respect to $t$ from $0$ to $\infty ,$ we obtain
\begin{equation}
\int_{0}^{\infty }(p^{\prime }(t)+A_{F}^{\ast
}p(t),Y(t))_{H}dt+\int_{0}^{\infty }(B_{2}^{\ast }p(t),V(t))_{U}dt=0,
\label{211}
\end{equation}
which by (\ref{204}) yields
\begin{equation}
\int_{0}^{\infty }\left( C_{1}^{\ast }C_{1}y^{u^{\ast }}(t)+F^{\ast }u^{\ast
}(t),Y(t)\right) _{H}dt=-\int_{0}^{\infty }(B_{2}^{\ast }p(t),V(t))_{U}dt.
\label{212}
\end{equation}
By comparison with (\ref{210}), where we write $v^{\ast }=u^{\ast
}-Fy^{u^{\ast }},$ this yields
\begin{equation}
\int_{0}^{\infty }(-B_{2}^{\ast }p(t)+u^{\ast }(t),V(t))_{U}dt=0,\mbox{ for
all }V\in L^{2}(\mathbb{R}_{+};U). \label{213}
\end{equation}
Therefore, we obtain (\ref{205}) as claimed.
\end{proof}
\noindent Then, the dual system (\ref{204}) can be still written by the
replacement of $v^{\ast }$ as
\begin{equation}
p^{\prime }(t)=-A^{\ast }p(t)+C_{1}^{\ast }C_{1}y^{u^{\ast }}(t),\mbox{ a.e.
}t\in \mathbb{R}_{+}. \label{214-2}
\end{equation}
Now, let us consider the function $\varphi :L^{2}(\mathbb{R}
_{+};W)\rightarrow \mathbb{R}_{+},$ $\varphi (w)=-J(\Gamma w,w),$that is
\begin{equation*}
\varphi (w)=\frac{1}{2}\int_{0}^{\infty }\left( \gamma ^{2}\left\Vert
w(t)\right\Vert _{W}^{2}-\left\Vert C_{1}y^{u^{\ast }}(t)+D_{1}u^{\ast
}(t)\right\Vert _{Z}^{2}\right) dt,
\end{equation*}
where $y^{u^{\ast }}$ is the solution to (\ref{1}) corresponding to $
(u^{\ast },w).$ By (\ref{8-1}) and (\ref{8-2}) we have
\begin{equation*}
C_{1}y^{u^{\ast }}(t)+D_{1}u^{\ast }(t)=G_{F}w(t)-f_{0}(t)
\end{equation*}
and so
\begin{eqnarray*}
&&\left\Vert (C_{1}y^{u^{\ast }}(t)+D_{1}u^{\ast }(t)\right\Vert
_{Z}^{2}=\left\Vert G_{F}w(t)\right\Vert
_{Z}^{2}-2(G_{F}w(t),f_{0}(t))_{Z}+\left\Vert f_{0}(t)\right\Vert _{Z}^{2} \\
&\leq &(1+\delta )\left\Vert G_{F}w(t)\right\Vert _{Z}^{2}+C_{\delta
}\left\Vert f_{0}(t)\right\Vert _{Z}^{2},\mbox{ }\forall t\geq 0.
\end{eqnarray*}
Now, we integrate from $0$ to $\infty ,$ note that $f_{0}\in L^{2}(\mathbb{R}
_{+};H),$ and get,
\begin{equation*}
\int_{0}^{\infty }\left\Vert (C_{1}y^{u^{\ast }}(t)+D_{1}u^{\ast
}(t)\right\Vert _{Z}^{2}dt\leq (1+\delta )(\gamma ^{2}-\varepsilon
)\int_{0}^{\infty }\left\Vert w(t)\right\Vert ^{2}+C_{\delta },
\end{equation*}
where $\varepsilon $ is fixed and the last inequality is implied by (\ref
{14-1}). We can find $\delta $ and $\widetilde{\delta }$ such that $
(1+\delta )(\gamma ^{2}-\varepsilon )\leq \gamma ^{2}-\widetilde{\delta },$
which is verified with the choice $\widetilde{\delta }<\varepsilon -\delta
(\gamma ^{2}-\varepsilon )$ and $\delta <\frac{\varepsilon }{\gamma
^{2}-\varepsilon }.$ Then\thinspace\
\begin{equation*}
\varphi (w)\geq \widetilde{\delta }\int_{0}^{\infty }\left\Vert
w(t)\right\Vert _{W}^{2}dt+C,\mbox{ }
\end{equation*}
and it turns out that $\varphi $ attains its minimum on $L^{2}(\mathbb{R}
_{+};W)$ in a unique point $w^{\ast }.$
\begin{lemma}
We have
\begin{equation}
w^{\ast }(t)=-\gamma ^{-2}B_{1}^{\ast }p(t),\mbox{ a.e. }t>0, \label{215}
\end{equation}
where $p\in W^{1,2}(0,T;H)$ is the solution to (\ref{214-2}).
\end{lemma}
\begin{proof}
Recall that $u^{\ast }=\Gamma w$ and that $y^{u^{\ast }}$ satisfies the
problem
\begin{equation*}
(y^{u^{\ast }})^{\prime }(t)=Ay^{u^{\ast }}(t)+B_{1}w(t)+B_{2}\Gamma w(t),
\mbox{ }t\in \mathbb{R}_{+},\mbox{ }y^{\ast }(0)=y_{0}
\end{equation*}
and proceed by giving variations to $w,$ that is $w^{\lambda }=w^{\ast
}+\lambda \widetilde{w},$ $w\in L^{2}(\mathbb{R}_{+};H).$ Then, the system
in variations is
\begin{equation}
Y^{\prime }(t)=AY(t)+B_{1}\widetilde{w}(t)+B_{2}\Gamma \widetilde{w}(t)\mbox{
, }t\in \mathbb{R}_{+},\mbox{ }Y(0)=0 \label{215-1}
\end{equation}
and the condition of optimality reads
\begin{equation}
\int_{0}^{\infty }(\gamma ^{2}w^{\ast }(t)-\Gamma ^{\ast }\Gamma w^{\ast
}(t),\widetilde{w}(t))_{W}dt-\int_{0}^{\infty }(C_{1}^{\ast }C_{1}y^{u^{\ast
}}(t),Y(t))_{H}dt=0, \label{215-3}
\end{equation}
for all $\widetilde{w}\in L^{2}(\mathbb{R}_{+};W).$ Let us recall the dual
system (\ref{214-2}) and test (\ref{215-1}) by $p(t)$ and integrate for $
t\in (0,\infty ).$ We get
\begin{equation}
\int_{0}^{\infty }(p^{\prime }(t)+A^{\ast }p(t),Y(t))_{H}dt+\int_{0}^{\infty
}(B_{1}^{\ast }p(t)+\Gamma ^{\ast }B_{2}^{\ast }p(t),\widetilde{w}
(t))_{W}dt=0. \label{215-4}
\end{equation}
The latter and (\ref{215-3}) gives
\begin{equation*}
\int_{0}^{\infty }(\gamma ^{2}w^{\ast }(t)-\Gamma ^{\ast }\Gamma w^{\ast
}(t),\widetilde{w}(t))_{W}dt+\int_{0}^{\infty }(B_{1}^{\ast }p(t)+\Gamma
^{\ast }B_{2}^{\ast }p(t),\widetilde{w}(t))_{W}dt=0,
\end{equation*}
so that, since $-\Gamma ^{\ast }\Gamma w^{\ast }(t)+\Gamma ^{\ast
}B_{2}^{\ast }p(t)=-\Gamma ^{\ast }u^{\ast }(t)+\Gamma ^{\ast }u^{\ast
}(t)=0,$ we obtain
\begin{equation*}
\int_{0}^{\infty }(\gamma ^{2}w^{\ast }(t)+B_{1}^{\ast }p(t),\widetilde{w}
(t))_{W}dt=0,
\end{equation*}
for all $\widetilde{w}\in L^{2}(\mathbb{R}_{+};W),$ that implies (\ref{215}
), as claimed.
\end{proof}
Thus, we have proved that (\ref{11}) has a unique solution $(u^{\ast
},w^{\ast })$ with the corresponding state denoted $y^{\ast },$
characterized by the Euler-Lagrange system
\begin{equation}
y^{\ast \prime }(t)=Ay^{\ast }(t)+B_{1}w^{\ast }(t)+B_{2}u^{\ast }(t),\mbox{
}t\in \mathbb{R}_{+},\mbox{ }y^{\ast }(0)=y_{0}, \label{216}
\end{equation}
\begin{equation}
p^{\prime }(t)=-A^{\ast }p(t)+C_{1}^{\ast }C_{1}y^{\ast }(t),\mbox{ }t\in
\mathbb{R}_{+}, \label{217}
\end{equation}
\begin{equation}
u^{\ast }(t)=B_{2}^{\ast }p(t),\mbox{ a.e. }t>0. \label{218}
\end{equation}
\begin{equation}
w^{\ast }(t)=-\gamma ^{-2}B_{1}^{\ast }p(t),\mbox{ a.e. }t>0, \label{219}
\end{equation}
where we already know that
\begin{eqnarray*}
y^{\ast } &\in &C([0,\infty );(D(A^{\ast }))^{\prime })\cap L^{2}(0,T;H),
\mbox{ }\forall T>0, \\
p &\in &C([0,\infty );H)\cap L^{2}(\mathbb{R}_{+};H).
\end{eqnarray*}
\begin{lemma}
Let $y_{0}\in H.$ Then,
\begin{equation}
y^{\ast }\in C([0,\infty );H)\cap W^{1,2}(\delta ,T;H),\mbox{ }\forall
\delta ,\mbox{ }0<\delta \leq T<\infty , \label{3-35-2}
\end{equation}
\begin{equation}
p\in W^{1,2}(0,T;H)\cap L^{2}(0,T;D(A^{\ast })),\mbox{ }\forall T>0.\mbox{ }
\label{3-35-1}
\end{equation}
\end{lemma}
\begin{proof}
Since $A^{\ast }$ generates an analytic $C_{0}$-semigroup and $C_{1}^{\ast
}C_{1}y^{\ast }\in L^{2}(\mathbb{R}_{+};H)$ we see by (\ref{217}) that (\ref
{3-35-1}) holds. Moreover, by (\ref{216}) and (\ref{218}) we have
\begin{eqnarray}
y^{\ast }(t) &=&e^{At}y_{0}+\int_{0}^{t}e^{A(t-s)}B_{1}w^{\ast
}(s)ds+\int_{0}^{t}e^{A(t-s)}B_{2}B_{2}^{\ast }p(s)ds \label{3-35-5} \\
&=&e^{At}y_{0}+g_{1}(t)+g_{2}(t),\mbox{ }\forall t\geq 0. \notag
\end{eqnarray}
The first two terms are in $C([0,\infty );H)\cap W^{1,2}(\delta ,T;H).$ By (
\ref{3-35-1}), $B_{2}B_{2}^{\ast }p\in W^{1,2}(0,T;(D(A^{\ast }))^{\prime })$
and so we may represent it as $B_{2}B_{2}^{\ast }p=(A-\omega I)f,$ with $
f\in W^{1,2}(0,T;H),$ for $\omega $ sufficiently large. This yields
\begin{eqnarray*}
g_{2}(t) &=&\int_{0}^{t}e^{A(t-s)}(A-\omega I)f(s)ds=-\omega
\int_{0}^{t}e^{A(t-s)}f(s)ds-\int_{0}^{t}\left( \frac{d}{ds}
e^{A(t-s)}\right) f(s)ds \\
&=&-\omega
\int_{0}^{t}e^{A(t-s)}f(s)ds-f(t)+e^{At}f(0)+\int_{0}^{t}e^{A(t-s)}f^{\prime
}(s)ds,\mbox{ }\forall t\geq 0.
\end{eqnarray*}
Since $e^{At}$ is an analytic semigroup it follows that $g(t)=
\int_{0}^{t}e^{A(t-s)}f^{\prime }(s)ds$, the solution to $g^{\prime
}(t)=Ag(t)+f(t),$ $g(0)=0\in D(A),$ is in $W^{1,2}(0,T;H),$ as the first two
terms. Though $f(0)\notin D(A)$, the third term is in $C([0,\infty );H)\cap
W^{1,2}(\delta ,T;H),$ $\forall 0<\delta \leq T<\infty $ and so is $g_{2}$
and $y^{\ast },$ too. Moreover, since $A^{\ast }$ is analytic, then (\ref
{3-35-1}) holds.
\end{proof}
\noindent \textit{Proof} (of Theorem \ref{Th-main}, continued). Now we set
\begin{equation}
Py_{0}:=-p(0),\mbox{ for }y_{0}\in H \label{220}
\end{equation}
and note that $P\in L(H,H).$
Moreover, by adding (\ref{216}) multiplied by $p(t)$ with (\ref{217})
multiplied by $y^{\ast }(t)$ and integrating on $(0,\infty )$ we get
\begin{eqnarray}
-2(y_{0},p(0))_{H} &=&\int_{0}^{\infty }\left\{ \left\langle Ay^{\ast
}(t),p(t)\right\rangle _{(D(A^{\ast }))^{\prime },D(A^{\ast })}+(w^{\ast
}(t),B_{1}^{\ast }p(t))_{W}+(u^{\ast }(t),B_{2}^{\ast }p(t))_{U}\right\} dt
\notag \\
&&+\int_{0}^{\infty }\left\{ -\left\langle Ay^{\ast }(t),p(t)\right\rangle
_{(D(A^{\ast }))^{\prime },D(A^{\ast })}+(C_{1}^{\ast }C_{1}y^{\ast
}(t),p(t))_{H}\right\} dt \label{220-1} \\
&=&\int_{0}^{\infty }\left\{ -\gamma ^{2}\left\Vert w^{\ast }(t)\right\Vert
_{W}^{2}+\left\Vert u^{\ast }(t)\right\Vert _{U}^{2}+\left\Vert C_{1}y^{\ast
}(t)\right\Vert _{Z}^{2}\right\} dt \notag
\end{eqnarray}
whence
\begin{eqnarray*}
(Py_{0},y_{0})_{H} &=&-(p(0),y_{0})_{H}=\frac{1}{2}\int_{0}^{\infty }\left(
\left\Vert C_{1}y^{\ast }(t)\right\Vert _{Z}^{2}+\left\Vert u^{\ast
}(t)\right\Vert _{U}^{2}-\gamma ^{2}\left\Vert w^{\ast }(t)\right\Vert
_{W}^{2}\right) dt \\
&=&\sup_{w\in L^{2}(\mathbb{R}_{+};W)}\inf_{u\in L^{2}(\mathbb{R}_{+};U)}
\frac{1}{2}\int_{0}^{\infty }\left( \left\Vert C_{1}y(t)\right\Vert
_{Z}^{2}+\left\Vert u(t)\right\Vert _{U}^{2}-\gamma ^{2}\left\Vert
w(t)\right\Vert _{W}^{2}\right) dt \\
&\geq &\inf_{u\in L^{2}(\mathbb{R}_{+};U)}\frac{1}{2}\int_{0}^{\infty
}\left( \left\Vert C_{1}y(t)\right\Vert _{Z}^{2}+\left\Vert u(t)\right\Vert
_{U}^{2}\right) dt\geq 0,
\end{eqnarray*}
hence $P\geq 0.$
Moreover, $P=P^{\ast }.$ Indeed, let $y_{0},$ $z_{0}\in H$ and $(y^{\ast
},p),$ $(z^{\ast },q)$ be the corresponding solutions to (\ref{216})-(\ref
{219}). Namely, $(z^{\ast },q)$ satisfy
\begin{equation*}
z^{\ast \prime }(t)=Az^{\ast }(t)+B_{1}w^{\ast }(t)+B_{2}u^{\ast }(t),\mbox{
}t\in \mathbb{R}_{+},\mbox{ }y^{\ast }(0)=y_{0},
\end{equation*}
\begin{equation*}
q^{\prime }(t)=-A^{\ast }q(t)+C_{1}^{\ast }C_{1}z^{\ast }(t),\mbox{ }t\in
\mathbb{R}_{+}.
\end{equation*}
We see that
\begin{equation*}
\frac{d}{dt}(p(t),z^{\ast }(t))_{H}=\frac{d}{dt}(q(t),y^{\ast }(t))_{H},
\mbox{ }\forall t\geq 0
\end{equation*}
and this yields $(Py_{0},z_{0})_{H}=(y_{0},Pz_{0})_{H},$ as claimed.
We recall that by the dynamic programming principle (see e.g., \cite
{VB-book-94}, p. 104), the minimization problem (\ref{201}) for $w=w^{\ast
}, $ is equivalent with the following problem
\begin{equation*}
\inf_{u\in L^{2}(\mathbb{R}_{+};U)}\frac{1}{2}\int_{t}^{\infty }\left(
\left\Vert C_{1}y(s)\right\Vert _{Z}^{2}+\left\Vert u(s)\right\Vert
_{U}^{2}-\gamma ^{2}\left\Vert w^{\ast }(s)\right\Vert _{W}^{2}\right) ds
\end{equation*}
subject to (\ref{1})-(\ref{2}) in $S_{t}=\{(t,\infty );$ $y(t)=y^{\ast
}(t)\},$ for every $t\geq 0.$ Since $u^{\ast }$ is the solution to this
problem it follows by (\ref{220}) that
\begin{equation}
p(t)=-Py^{\ast }(t),\mbox{ }\forall t\geq 0. \label{221}
\end{equation}
We denote by $T_{P}(t):H\rightarrow H$ the family of operators
\begin{equation}
T_{P}(t)y_{0}=y^{\ast }(t),\mbox{ }\forall t\geq 0 \label{222}
\end{equation}
where $y^{\ast }(t)$ is the solution to (\ref{216}) with $u^{\ast }$ and $
w^{\ast }$ given by (\ref{217})-(\ref{219}). By (\ref{3-35-2}) it follows
that $T_{P}(t)$ is a $C_{0}$-semigroup on $H.$
Let us denote by $A_{P}$ the infinitesimal generator of $T_{P}(t),$ that is
\begin{equation}
\frac{dy^{\ast }}{dt}(t)=A_{P}y^{\ast }(t),\mbox{ }\forall t\geq 0,\mbox{ }
y^{\ast }(0)=y_{0}, \label{222-1}
\end{equation}
or, equivalently
\begin{equation}
y^{\ast }(t)=e^{A_{P}t}y_{0},\mbox{ }t\geq 0,\mbox{ }\forall y_{0}\in H.
\label{222-2}
\end{equation}
If $y_{0}\in D(A_{P})$ we have
\begin{equation}
y^{\ast }\in C^{1}([0,T];H)\cap C([0,T];D(A_{P})),\mbox{ }\forall T>0.
\label{3-43-1}
\end{equation}
Here, $D(A_{P})=\{y\in H;$ $A_{P}y\in H\}$ is the domain of $A_{P}$. The
space $\mathcal{X}$ in Theorem \ref{Th-main} is actually
\begin{equation}
\mathcal{X}:=D(A_{P}). \label{3-43-2}
\end{equation}
Now, replacing in the right-hand side of (\ref{216}) $u^{\ast }$ and $
w^{\ast }$ by (\ref{218})-(\ref{219}), (\ref{217}) and (\ref{221}) we get
\begin{equation}
y^{\ast \prime }(t)=\widetilde{\Lambda _{P}}y^{\ast }(t) \label{216-1}
\end{equation}
where $\widetilde{\Lambda _{P}}$ is the operator
\begin{equation*}
\widetilde{\Lambda _{P}}:H\rightarrow (D(A^{\ast }))^{\prime },\mbox{ }
\widetilde{\Lambda _{P}}y=Ay-B_{2}B_{2}^{\ast }Py+\gamma
^{-2}B_{1}B_{1}^{\ast }Py\in (D(A^{\ast }))^{\prime }
\end{equation*}
and $A$ is the extension from $H$ to $(D(A^{\ast }))^{\prime }.$
We define by $\Lambda _{P}:D(\Lambda _{P})\subset H\rightarrow H$ the
restriction of the operator $\widetilde{\Lambda _{P}}$ to $H,$ namely
\begin{eqnarray}
\Lambda _{P}y &=&(A-B_{2}B_{2}^{\ast }P+\gamma ^{-2}B_{1}B_{1}^{\ast }P)y,
\mbox{ }y\in D(\Lambda _{P}), \label{222-3} \\
D(\Lambda _{P}) &=&\{y\in H;\mbox{ }(A-B_{2}B_{2}^{\ast }P+\gamma
^{-2}B_{1}B_{1}^{\ast }P)y\in H\}. \notag
\end{eqnarray}
\begin{lemma}
\label{operators}We have
\begin{equation}
P\in L(\mathcal{X},D(A^{\ast })), \label{223}
\end{equation}
\begin{equation}
B_{2}^{\ast }P\in L(\mathcal{X};U), \label{223-1}
\end{equation}
\begin{equation}
A_{P}y=\Lambda _{P}y,\mbox{ for all }y\in \mathcal{X}\subset D(\Lambda _{P})
\label{224}
\end{equation}
and $\Lambda _{P}$ generates a $C_{0}$-semigroup on $H.$
Moreover, if $\Lambda _{P}$ is closed in $H$, then
\begin{equation}
\mathcal{X}=D(A_{P})=D(\Lambda _{P}). \label{224-1}
\end{equation}
\end{lemma}
\begin{proof}
Let $y_{0}\in D(A_{P})$. We know by (\ref{3-43-1}) that $y^{\ast }\in
C^{1}([0,T];H)$ and $A_{P}y^{\ast }\in C([0,T];H)$ for all $T>0,$ and so by (
\ref{217}) it follows therefore that $p^{\prime }\in C([0,T];H)$ and so $
A^{\ast }p\in C([0,T];H).$ Hence, $A^{\ast }p(0)\in H.$ It follows that $
p(0)\in D(A^{\ast })$ and so $Py_{0}\in D(A^{\ast }).$ This implies (\ref
{223}). Since $P\in (\mathcal{X},D(A^{\ast }))$ and $B_{2}^{\ast }\in
L(D(A^{\ast }),U)$ it follows (\ref{223-1}).
We have by (\ref{222-1}) and (\ref{3-43-1}) that
\begin{equation*}
\frac{d}{dt}(y^{\ast }(t),\varphi )_{H}=(A_{P}y^{\ast }(t),\varphi )_{H},
\mbox{ }\forall t\geq 0,\mbox{ }\varphi \in H.
\end{equation*}
On the other hand, by (\ref{216-1}) we have (see the weak form (\ref{13-1})
applied to $\widetilde{\Lambda _{P}}:H\rightarrow (D(A^{\ast }))^{\prime }$)
\begin{equation*}
\frac{d}{dt}(y^{\ast }(t),\varphi )_{H}=\left\langle \frac{dy^{\ast }}{dt}
(t),\varphi )\right\rangle _{(D(A^{\ast }))^{\prime },D(A^{\ast
})}=\left\langle \widetilde{\Lambda _{P}}y^{\ast }(t),\varphi \right\rangle
_{(D(A^{\ast }))^{\prime },D(A^{\ast })},\mbox{ }\forall t\geq 0,\mbox{ }
\forall \varphi \in D(A^{\ast }).
\end{equation*}
Hence,
\begin{equation*}
(A_{P}y^{\ast }(t),\varphi )_{H}=\left\langle \widetilde{\Lambda _{P}}
y^{\ast }(t),\varphi \right\rangle _{(D(A^{\ast }))^{\prime },D(A^{\ast })},
\mbox{ }\forall t\geq 0,\mbox{ }\forall \varphi \in D(A^{\ast }).
\end{equation*}
Recalling that $y^{\ast }\in C^{1}([0,\infty );H)\subset C([0,\infty
);(D(A^{\ast }))^{\prime })$ and letting $t\rightarrow 0$ we get
\begin{equation*}
(A_{P}y_{0},\varphi )_{H}=\left\langle \widetilde{\Lambda _{P}}y_{0},\varphi
\right\rangle _{(D(A^{\ast }))^{\prime },D(A^{\ast })},\mbox{ }\forall
\varphi \in D(A^{\ast }).
\end{equation*}
This implies that $\widetilde{\Lambda _{P}}y_{0}\in H,$ namely $y_{0}\in
D(\Lambda _{P}),$ and $A_{P}y_{0}=\Lambda _{P}y_{0}$ on $D(A_{P})\subset
D(\Lambda _{P}),$ that is (\ref{224}).
Since these two operators coincide on $D(A_{P})$ then $\Lambda _{P}$
generates a $C_{0}$-semigroup on $H.$
Now, $D(A_{P})\subset D(\Lambda _{P})\subset H$ and since $D(A_{P})$ is
dense in $H$ it follows that $D(\Lambda _{P})$ is dense in $H$ and $D(A_{P})$
is dense in $D(\Lambda _{P}).$
Assume that $\Lambda _{P}$ is closed and let $y_{0}\in D(\Lambda _{P}).$
There exists $(y_{0}^{n})_{n}\subset D(A_{P})$, $y_{0}^{n}\rightarrow y_{0}$
in $H$ and by (\ref{224}) we have
\begin{equation*}
(A_{P}y_{0}^{n},\varphi )_{H}=\left( \Lambda _{P}y_{0}^{n},\varphi \right)
_{H},\mbox{ }\varphi \in H,
\end{equation*}
which implies (using the adjoint of $A_{P}^{\ast }$ which is the generator
of a $C_{0}$-semigroup on $H)$ that
\begin{equation*}
\left( y_{0}^{n},A_{P}^{\ast }\varphi \right) _{H}=\left( \Lambda
_{P}y_{0}^{n},\varphi \right) _{H},\mbox{ }\varphi \in D(A_{P}^{\ast
})\subset H.
\end{equation*}
Since $\Lambda _{P}$ is closed, by letting $n\rightarrow \infty $ we obtain
\begin{equation*}
(y_{0},A_{P}^{\ast }\varphi )_{H}=\left( \Lambda _{P}y_{0},\varphi \right)
_{H},\mbox{ }\varphi \in D(A_{P}^{\ast }).
\end{equation*}
Then, $\varphi \rightarrow (y_{0},A_{P}^{\ast }\varphi )_{H}$ is a linear
continuous functional on $H$ and $\left\vert (y_{0},A_{P}^{\ast }\varphi
)_{H}\right\vert \leq C\left\Vert \varphi \right\Vert _{H},$ so that $
y_{0}\in D(A_{P})$ and (\ref{224}) is proved.
\end{proof}
\begin{proof}
(of Theorem \ref{Th-main}, continued). To prove that $P$ is a solution to
the Riccati equation (\ref{15}) we use the relation
\begin{equation*}
\frac{d}{dt}(y^{\ast }(t),p(t))_{H}=\left\langle (y^{\ast })^{\prime
}(t),p(t)\right\rangle _{(D(A^{\ast }))^{\prime },D(A^{\ast })}+(y^{\ast
}(t),p^{\prime }(t))_{H}
\end{equation*}
and calculate by (\ref{216})-(\ref{219}) and (\ref{221}) a relation as done
for (\ref{220-1}) but integrating from $t$ to $\infty $. We get
\begin{eqnarray*}
(Py^{\ast }(t),y^{\ast }(t))_{H} &=&(-p(t),y^{\ast }(t))_{H} \\
&=&\frac{1}{2}\int_{t}^{\infty }\left( \left\Vert C_{1}y^{\ast
}(t)\right\Vert _{Z}^{2}+\left\Vert u^{\ast }(t)\right\Vert _{U}^{2}-\gamma
^{2}\left\Vert w^{\ast }(t)\right\Vert _{W}^{2}\right) dt,\mbox{ }t\geq 0.
\end{eqnarray*}
If $y_{0}\in D(A_{P})$ this implies by differentiating (by using (\ref{222-1}
) and (\ref{3-43-1})) that
\begin{eqnarray*}
&&(Py^{\ast }(t),A_{P}y^{\ast }(t))_{H}+(PA_{P}y^{\ast }(t),y^{\ast
}(t))_{H}+\left\Vert C_{1}y^{\ast }(t)\right\Vert _{Z}^{2} \\
&&+\left\Vert B_{2}^{\ast }Py^{\ast }(t)\right\Vert _{U}^{2}-\gamma
^{2}\left\Vert \gamma ^{-2}B_{1}^{\ast }Py^{\ast }(t)\right\Vert _{W}^{2}=0,
\mbox{ }t\geq 0
\end{eqnarray*}
and since, by (\ref{223}), $B_{2}^{\ast }P\in L(D(A_{P}),D(A^{\ast }))$ we
obtain for $t\rightarrow 0$ the equation
\begin{equation}
2(Py_{0},A_{P}y_{0})_{H}+\left\Vert C_{1}y_{0}\right\Vert
_{Z}^{2}+\left\Vert B_{2}^{\ast }Py_{0}\right\Vert _{U}^{2}-\gamma
^{-2}\left\Vert B_{1}^{\ast }Py_{0}\right\Vert _{W}^{2}=0,\mbox{ }\forall
y_{0}\in D(A_{P}). \label{225}
\end{equation}
By differentiating along $z\in D(A_{P})$ we get
\begin{equation*}
(Py_{0},A_{P}z)_{H}+(Pz,A_{P}y_{0})_{H}+((B_{2}B_{2}^{\ast }-\gamma
^{-2}B_{1}B_{1}^{\ast })Pz,Py_{0})_{H}+(C_{1}^{\ast }C_{1}y_{0},z)_{H}=0,
\end{equation*}
for all $y_{0,}$ $z\in D(A_{P}).$ But here $A_{P}y=\Lambda _{P}y$ for $y\in
D(A_{P})$ and we can replace $A_{P}$ by $\Lambda _{P}$ in the previous
equation obtaining after all calculations
\begin{equation*}
\left( A^{\ast }Py_{0},z\right) _{H}+\left( P(A-B_{2}B_{2}^{\ast }+\gamma
^{-2}B_{1}B_{1}^{\ast })Py_{0},z\right) _{H}+(C_{1}^{\ast
}C_{1}y_{0},z)_{H}=0
\end{equation*}
for all $y_{0,}$ $z\in D(A_{P}),$ namely (\ref{15}).
For proving that the semigroup $e^{\Lambda _{P}t\mbox{ }}$is exponentially
stable we use the detectability assumption $(i_{3}).$ Let us take $K\in
L(Z,H)$ and write eq. (\ref{216}) in the following form
\begin{equation*}
y^{\ast \prime }(t)=(A+KC_{1})y^{\ast }(t)+B_{2}u^{\ast }(t)+B_{1}w^{\ast
}(t)-KC_{1}y^{\ast }(t),\mbox{ }t\geq 0,
\end{equation*}
or equivalently,
\begin{eqnarray*}
y^{\ast }(t)
&=&e^{(A+KC_{1})t}y_{0}+\int_{0}^{t}e^{(A+KC_{1})(t-s)}(B_{2}u^{\ast
}(s)+B_{1}w^{\ast }(s))ds \\
&&-\int_{0}^{t}e^{(A+KC_{1})(t-s)}KC_{1}y^{\ast }(s)ds,\mbox{ for all }t\geq
0.
\end{eqnarray*}
Since $B_{1}w^{\ast },$ $KC_{1}y^{\ast }\in L^{2}(\mathbb{R}_{+};H)$ and $
e^{(A+KC_{1})t}$ is exponentially stable it remains to show that
\begin{equation}
t\rightarrow \int_{0}^{t}e^{(A+KC_{1})(t-s)}B_{2}u^{\ast }(s)ds\in L^{2}(
\mathbb{R}_{+};H). \label{225-1}
\end{equation}
To this end, for each $\psi \in L^{2}(\mathbb{R}_{+};H),$ using the Young's
inequality (\ref{400}) (with $p=1,$ $q=r=2)$ and (\ref{12-0}) we calculate
\begin{eqnarray*}
&&\int_{0}^{\infty }\left( \psi
(t),\int_{0}^{t}e^{(A+KC_{1})(t-s)}B_{2}u^{\ast }(s)ds\right) _{H}dt \\
&=&\int_{0}^{\infty }\left( \int_{s}^{\infty }B_{2}^{\ast }e^{(A^{\ast
}+C_{1}^{\ast }K^{\ast })(t-s)}\psi (t)dt,u^{\ast }(s)\right) _{U}ds \\
&\leq &\left( \int_{0}^{\infty }\left( \int_{s}^{\infty }\left\Vert
B_{2}^{\ast }e^{(A^{\ast }+C_{1}^{\ast }K^{\ast })(t-s)}\psi (t)\right\Vert
_{U}dt\right) ^{2}ds\right) ^{1/2}\left( \int_{0}^{\infty }\left\Vert
u^{\ast }(s)\right\Vert _{U}^{2}ds\right) ^{1/2} \\
&\leq &\left\Vert u^{\ast }\right\Vert _{L^{2}(0,\infty ;U)}\left(
\int_{0}^{\infty }\left( \int_{0}^{\infty }\left\Vert B_{2}^{\ast
}e^{(A^{\ast }+C_{1}^{\ast }K^{\ast })(t-s)}\right\Vert _{L(H,U)}\left\Vert
\psi (t)\right\Vert _{H}dt\right) ^{2}ds\right) ^{1/2} \\
&\leq &\left\Vert u^{\ast }\right\Vert _{L^{2}(0,\infty ;U)}\left(
\int_{0}^{\infty }\left\Vert B_{2}^{\ast }e^{(A^{\ast }+C_{1}^{\ast }K^{\ast
})s}\right\Vert _{L(H,U)}ds\right) \left( \int_{0}^{\infty }\left\Vert \psi
(s)\right\Vert _{H}^{2}ds\right) ^{1/2} \\
&\leq &C\left\Vert u^{\ast }\right\Vert _{L^{2}(0,\infty ;U)}\left\Vert \psi
\right\Vert _{L^{2}(0,\infty ;U)}\leq C_{1}\left\Vert \psi \right\Vert
_{L^{2}(0,\infty ;U)},
\end{eqnarray*}
and this implies (\ref{225-1}), as claimed.
We shall prove now that the operator $\Lambda _{P}^{1}:=A-B_{2}B_{2}^{\ast }P
$ generates an exponentially stable $C_{0}$-semigroup in $H$ with the domain
$\{y\in H;$ $(A-B_{2}B_{2}^{\ast }P)y\in H\}=D(A_{P}).$ The solution $
y^{\ast }(t)$ to (\ref{216}) is in $L^{2}(\mathbb{R}_{+};H)$ can be written
also as
\begin{equation*}
y^{\ast }(t)=e^{\Lambda _{P}^{1}t}y_{0}+\gamma ^{-2}\int_{0}^{t}e^{\Lambda
_{P}^{1}(t-s)}B_{1}B_{1}^{\ast }Py^{\ast }(s)ds
\end{equation*}
and since the second term on the right-hand side is in $L^{2}(\mathbb{R}
_{+};H),$ it follows that $e^{\Lambda _{P}^{1}t}y_{0}\in L^{2}(\mathbb{R}
_{+};H).$
Now, we shall prove (\ref{14-1}). Let us consider the equation
\begin{equation}
y^{\prime }(t)=(A-B_{2}B_{2}^{\ast }P)y(t)+B_{1}w(t),\mbox{ }t\geq 0,\mbox{ }
y(0)=0, \label{225-2}
\end{equation}
with $w\in L^{2}(\mathbb{R}_{+};W).$ As seen earlier, this equation has a
unique mild solution and by (\ref{13-1}) we have
\begin{equation}
\frac{d}{dt}(y(t),\varphi )_{H}=(y(t),(A^{\ast }-B_{2}B_{2}^{\ast }P)\varphi
)_{H}+(B_{1}w(y),\varphi )_{H},\mbox{ }\forall \varphi \in D(A^{\ast }).
\label{3-49-1}
\end{equation}
Let $p(t)=-Py(t),$ $t>0.$ Since by (\ref{223}) $P\in L(D(A_{P}),D(A^{\ast
})) $ it follows that $p(t)\in D(A^{\ast })$ and $p$ is the solution to eq. (
\ref{217}) with $y^{\ast }$ replaced by $y.$ Moreover, as seen earlier by (
\ref{217}) it follows that $A^{\ast }p,$ $p^{\prime }\in L^{2}(\mathbb{R}
_{+};H)$ and we have by (\ref{3-49-1})
\begin{equation*}
\frac{d}{dt}(y(t),p(t))_{H}=(y(t),(A^{\ast }-B_{2}B_{2}^{\ast
}P)p(t))_{H}+(B_{1}w(t),p(t))_{H}+(y^{\prime }(t),p(t))_{H}.
\end{equation*}
Then we calculate using (\ref{221}) and (\ref{15})
\begin{eqnarray*}
&&\frac{d}{dt}(Py(t),y(t))_{H}=2(Py(t),y^{\prime }(t))_{H} \\
&=&2(Py(t),Ay(t))_{H}-2\left\Vert B_{2}^{\ast }Py(t)\right\Vert
_{H}^{2}+2(B_{1}w(t),Py(t))_{H} \\
&=&\left\Vert B_{2}^{\ast }Py(t)\right\Vert _{H}^{2}-\gamma ^{-2}\left\Vert
B_{1}^{\ast }Py(t)\right\Vert _{H}^{2}-\left\Vert C_{1}y(t)\right\Vert
_{H}^{2}-2\left\Vert B_{2}^{\ast }Py(t)\right\Vert
_{H}^{2}+2(B_{1}w(t),Py(t))_{H} \\
&=&-\left\Vert B_{2}^{\ast }Py(t)\right\Vert _{H}^{2}-\left\Vert
C_{1}y(t)\right\Vert _{H}^{2}-\gamma ^{-2}\left\Vert B_{1}^{\ast
}Py(t)\right\Vert _{H}^{2}+2(w(t),B_{1}^{\ast }Py(t))_{W},\mbox{ a.e. }t>0.
\end{eqnarray*}
Integrating this from $0$ to $\infty $ we obtain
\begin{equation*}
0=\int_{0}^{\infty }\left( -\left\Vert B_{2}^{\ast }Py(t)\right\Vert
_{H}^{2}-\left\Vert C_{1}y(t)\right\Vert _{H}^{2}-\gamma ^{-2}\left\Vert
B_{1}^{\ast }Py(t)\right\Vert _{H}^{2}+2(w(t),B_{1}^{\ast }Py(t))_{W}\right)
dt,
\end{equation*}
since $y(0)=0$ and $\lim_{t\rightarrow \infty }(Py(t),y(t))_{H}=0.$
Therefore,
\begin{eqnarray*}
&&\int_{0}^{\infty }\left( \left\Vert C_{1}y(t)\right\Vert
_{H}^{2}+\left\Vert B_{2}^{\ast }Py(t)\right\Vert _{H}^{2}\right) dt \\
&=&\int_{0}^{\infty }\left( -\gamma ^{-2}\left\Vert B_{1}^{\ast
}Py(t)\right\Vert _{H}^{2}+2(w(t),B_{1}^{\ast }Py(t))_{H}-\gamma
^{2}\left\Vert w(t)\right\Vert _{W}^{2}\right) dt+\int_{0}^{\infty }\gamma
^{2}\left\Vert w(t)\right\Vert _{W}^{2}dt \\
&=&\int_{0}^{\infty }\gamma ^{2}\left\Vert w(t)\right\Vert
_{W}^{2}dt-\int_{0}^{\infty }\gamma ^{2}\left\Vert \widetilde{w}
(t)\right\Vert _{W}^{2}dt,
\end{eqnarray*}
where
\begin{equation}
\widetilde{w}(t)=w(t)-\gamma ^{-2}B_{1}^{\ast }Py(t). \label{226}
\end{equation}
If we prove that there exists $\alpha >0$ such that
\begin{equation}
\left\Vert \widetilde{w}\right\Vert _{L^{2}(0,\infty ;W)}\geq \alpha
\left\Vert w\right\Vert _{L^{2}(0,\infty ;W)},\mbox{ }\forall w\in L^{2}(
\mathbb{R}_{+};W), \label{227}
\end{equation}
it follows that
\begin{equation*}
\gamma ^{2}\left( \left\Vert w\right\Vert _{L^{2}(\mathbb{R}
_{+};W)}^{2}-\left\Vert \widetilde{w}\right\Vert _{L^{2}(\mathbb{R}
_{+};W)}^{2}\right) \leq \gamma ^{2}(1-\alpha )\left\Vert w\right\Vert
_{L^{2}(\mathbb{R}_{+};W)}^{2}
\end{equation*}
and therefore
\begin{equation*}
\int_{0}^{\infty }\left( \left\Vert C_{1}y(t)\right\Vert _{H}^{2}+\left\Vert
B_{2}^{\ast }Py(t)\right\Vert _{H}^{2}\right) dt\leq (\gamma ^{2}-\delta
)\left\Vert w\right\Vert _{L^{2}(\mathbb{R}_{+};W)}^{2}
\end{equation*}
with $\delta >0$ independent on $w.$ Therefore,
\begin{equation*}
\int_{0}^{\infty }\left( \left\Vert C_{1}y(t)\right\Vert _{H}^{2}+\left\Vert
B_{2}^{\ast }Py(t)\right\Vert _{H}^{2}\right) dt\leq (\gamma ^{2}-\delta
)\int_{0}^{\infty }\left\Vert w(t)\right\Vert _{W}^{2}dt,
\end{equation*}
which by (\ref{8-2}) implies (\ref{14-1}). We note that once (\ref{227})
proved, $\alpha $ can be chosen smaller such that $\alpha <1.$ It remains to
prove (\ref{227}) and this will be done in Lemma 3.6 given at the end of
this section.
Therefore, $G$ corresponding to $\widetilde{F}:=-B_{2}^{\ast }P$ has the
property $\left\Vert G_{\widetilde{F}}w\right\Vert _{L^{2}(\mathbb{R}
_{+};Z)}<\gamma \left\Vert w\right\Vert _{L^{2}(\mathbb{R}_{+};W)},$ that is
$\widetilde{F}$ is the feedback operator which solves the $H^{\infty }$
-control problem. This ends the proof of the the first part of Theorem \ref
{Th-main}.
Assume now that $P$ is a solution to equation (\ref{15}), satisfying (\ref
{14-2}), such that $\Lambda _{P}=A-(B_{2}B_{2}^{\ast }-\gamma
^{-2}B_{1}B_{1}^{\ast })P$ generates an exponentially stable semigroup on $
\mathcal{X}$. We set $y^{\ast }(t)=e^{\Lambda _{P}t}y_{0},$ for $y_{0}\in H,$
so that $y^{\ast }\in C([0,\infty );H)\cap L^{2}(\mathbb{R}_{+};H).$ Let us
define $p(t)=-Py^{\ast }(t),$ for $t\geq 0.$ Then, $p\in L^{2}(\mathbb{R}
_{+};H)\cap C([0,\infty );H)$ and by replacing $Py^{\ast }(t)$ in (\ref{15})
we get that $p$ satisfies equation (\ref{217}) with the regularity obtained
in Lemma 3.4. The , as before. Finally, we show that the operator $\Lambda
_{P}^{1}$ generates an exponentially stable semigroup and that the
controller $\widetilde{F}y=-B_{2}^{\ast }Py$ stabilizes equation $y^{\prime
}(t)=(A+B_{2}\widetilde{F})y(t)+B_{1}w(t),$ $y(0)=0,$ arguing as before
beginning from (\ref{225-2}). This ends the proof of Theorem \ref{Th-main}.
\end{proof}
It remains to prove (\ref{227}). We set
\begin{equation}
\Phi (w)=\left\Vert \widetilde{w}\right\Vert _{L^{2}(\mathbb{R}_{+};W)}^{2}.
\label{228}
\end{equation}
\begin{lemma}
We have
\begin{equation}
\Phi (w)\geq \alpha \left\Vert w\right\Vert _{L^{2}(0,\infty ;W)}^{2},\mbox{
for all }w\in L^{2}(\mathbb{R}_{+};W), \label{229}
\end{equation}
where $\alpha >0.$
\end{lemma}
\begin{proof}
We proceed by reduction to absurdity. Assume that (\ref{229}) does not hold
and argue from this a contradiction. Thus, let $(w_{n})_{n}\subset L^{2}(
\mathbb{R}_{+};W)$ be such that $\left\Vert w_{n}\right\Vert _{L^{2}(\mathbb{
R}_{+};W)}=1,$ $\forall n\in \mathbb{N}$ and $\Phi (w_{n})\rightarrow 0$ as $
n\rightarrow \infty .$ Hence, by (\ref{228}) and eq. (\ref{225-2}) we have
\begin{equation}
\Phi (w_{n})=\left\Vert w_{n}-\gamma ^{-2}B_{1}^{\ast
}P\int_{0}^{t}e^{(A-B_{2}B_{2}^{\ast }P)t}B_{1}w_{n}(s)ds\right\Vert _{L^{2}(
\mathbb{R}_{+};W)}\rightarrow 0,\mbox{ as }n\rightarrow \infty . \label{230}
\end{equation}
On the other hand, on a subsequence, we have $w_{n}\rightarrow \overline{w}$
weakly in $L^{2}(\mathbb{R}_{+};W),$ and since $\Phi $ is weakly lower
semicontinuous in $L^{2}(\mathbb{R}_{+};W)$ (because it is continuous and
convex) we have by (\ref{230}) that $\Phi (\overline{w})=0$ which implies
that
\begin{equation*}
\overline{w}(t)=\gamma ^{-2}B_{1}^{\ast }P\int_{0}^{t}e^{(A-B_{2}B_{2}^{\ast
}P)t}B_{1}\overline{w}(s)ds,\mbox{ }\forall t\geq 0.
\end{equation*}
By Gronwall's lemma we deduce that $\overline{w}(t)=0.$ Now, if we prove
that
\begin{equation*}
w_{n}\rightarrow \overline{w}\mbox{ strongly in }L^{2}(\mathbb{R}_{+};W)
\mbox{ as }n\rightarrow \infty
\end{equation*}
(namely, that $(w_{n})_{n}$ is compact in $L^{2}(\mathbb{R}_{+};W))$ we
arrive to a contradiction because, the choice $\left\Vert w_{n}\right\Vert
_{L^{2}(\mathbb{R}_{+};W)}=1$ implies $\left\Vert \overline{w}\right\Vert
_{L^{2}(\mathbb{R}_{+};W)}=1,$ which was found before to be $0.$
To prove that $(w_{n})_{n}$ is compact in $L^{2}(\mathbb{R}_{+};W),$ by (\ref
{230}) it suffices to show that the sequence
\begin{equation*}
z_{n}(t)=\gamma ^{-2}B_{1}^{\ast }P\int_{0}^{t}e^{(A-B_{2}B_{2}^{\ast
}P)t}B_{1}w_{n}(s)ds,\mbox{ }t\geq 0
\end{equation*}
is compact in $L^{2}(\mathbb{R}_{+};W),$ that is, it contains a convergent
subsequence. Taking into account that
\begin{equation}
\left\Vert z_{n}\right\Vert _{L^{2}(T,\infty ;W)}\rightarrow 0\mbox{ as }
T\rightarrow \infty ,\mbox{ uniformly in }n, \label{230-0}
\end{equation}
since $A-B_{2}B_{2}^{\ast }P$ generates an exponentially stable semigroup,
it suffices to prove that $(z_{n})_{n}$ is compact in $L^{2}(0,T;W),$ for
each $T>0.$ We set
\begin{equation}
S(t)=e^{(A-B_{2}B_{2}^{\ast }P)t},\mbox{ }t\geq 0 \label{230-1}
\end{equation}
and prove that $\{S(t)\}$ is compact for each $t>0.$ This means that the set~
$\{S(t)y_{0};$ $y_{0}\in H;$ $\left\Vert y_{0}\right\Vert _{H}\leq M\}$ is
relatively compact in $H.$ Since $A-B_{2}B_{2}^{\ast }P=\Lambda _{P}-\gamma
^{-2}B_{1}B_{1}^{\ast }P$ and $B_{1}B_{1}^{\ast }P\in L(H,H)$ and $\Lambda
_{P}=A_{P}$ on $D(A_{P})$ it suffices to show that $T_{P}(t)=e^{A_{P}t}$ is
compact for each $t>0.$ This follows by density by showing first that $
\{T_{P}(t)y_{0};$ $y_{0}\in D(A_{P}),$ $\left\Vert A_{P}y_{0}\right\Vert
_{H}+\left\Vert y_{0}\right\Vert _{H}\leq M\}$ is relatively compact in $H.$
To this end, for $\varepsilon >0,$ we write $T_{P}(t)y_{0}$ in the following
form
\begin{eqnarray}
&&T_{P}(t)y_{0}=e^{At}y_{0}-\int_{0}^{t}e^{A(t-s)}(B_{2}B_{2}^{\ast
}Py(s)-\gamma ^{-2}B_{1}B_{1}^{\ast }Py(s))ds \label{231} \\
&=&e^{At}y_{0}-e^{A\varepsilon }\int_{0}^{t-\varepsilon
}e^{A(t-s-\varepsilon )}(B_{2}B_{2}^{\ast }Py(s)-\gamma
^{-2}B_{1}B_{1}^{\ast }Py(s))ds \notag \\
&&-\int_{t-\varepsilon }^{t}e^{A(t-s)}(B_{2}B_{2}^{\ast }Py(s)-\gamma
^{-2}B_{1}B_{1}^{\ast }Py(s))ds, \notag
\end{eqnarray}
where $y(t)=T_{P}(t)y_{0}.$ If $\mathcal{M}=\{y_{0}\in D(A_{P});$ $
\left\Vert A_{P}y_{0}\right\Vert _{H}+\left\Vert y_{0}\right\Vert _{H}\leq
M\},$ relation (\ref{231}) yields
\begin{eqnarray*}
&&T_{P}(t)\mathcal{M=}\left\{ e^{At}y_{0};\mbox{ }y_{0}\in \mathcal{M}
\right\} \\
&&-\left\{ e^{A\varepsilon }\int_{0}^{t-\varepsilon }e^{A(t-s-\varepsilon
)}(B_{2}B_{2}^{\ast }Py(s)-\gamma ^{-2}B_{1}B_{1}^{\ast }Py(s))ds;\mbox{ }
y_{0}\in \mathcal{M}\right\} \\
&&-\left\{ \int_{t-\varepsilon }^{t}e^{A(t-s)}(B_{2}B_{2}^{\ast
}Py(s)-\gamma ^{-2}B_{1}B_{1}^{\ast }Py(s))ds;\mbox{ }y_{0}\in \mathcal{M}
\right\} =\mathcal{M}_{1}+\mathcal{M}_{2}+\mathcal{M}_{3}.
\end{eqnarray*}
In the sum above, $\mathcal{M}_{1}$ is relatively compact because $e^{At}$
is compact by $(i_{1})$. Next, we write $\mathcal{M}_{2}=\mathcal{M}_{21}+
\mathcal{M}_{22}$ where $\mathcal{M}_{2i}=\left\{ e^{A\varepsilon
}\int_{0}^{t-\varepsilon }e^{A(t-s-\varepsilon )}B_{i}B_{i}^{\ast }Py(s)ds;
\mbox{ }y_{0}\in \mathcal{M}\right\} ,$ $i=1,2.$ $\mathcal{M}_{21}$ is
relatively compact because $e^{A\varepsilon }$ is compact and $
\int_{0}^{t-\varepsilon }e^{A(t-s-\varepsilon )}B_{1}B_{1}^{\ast }Py(s)ds$
is bounded,
\begin{eqnarray*}
\left\Vert \int_{0}^{t-\varepsilon }e^{A(t-s-\varepsilon )}B_{1}B_{1}^{\ast
}Py(s)ds\right\Vert _{H} &\leq &C\int_{0}^{t}\left\Vert B_{1}B_{1}^{\ast
}Py(s)\right\Vert _{U}ds \\
&\leq &C\int_{0}^{t}\left\Vert B_{1}^{\ast }Py(s)\right\Vert _{H}ds\leq
Ct\left\Vert y_{0}\right\Vert _{H}.
\end{eqnarray*}
Then,
\begin{eqnarray*}
&&\left\Vert \int_{0}^{t-\varepsilon }e^{A(t-s-\varepsilon
)}B_{2}B_{2}^{\ast }Py(s)ds\right\Vert _{H}=\sup_{\varphi \in H,\left\Vert
\varphi \right\Vert _{H}\leq 1}\left( \int_{0}^{t-\varepsilon
}e^{A(t-s-\varepsilon )}B_{2}B_{2}^{\ast }Py(s)ds,\varphi \right) _{H} \\
&\leq &\sup_{\left\Vert \varphi \right\Vert _{H}\leq
1}\int_{0}^{t-\varepsilon }\left( B_{2}^{\ast }Py(s),B_{2}^{\ast }e^{A^{\ast
}(t-s-\varepsilon )}\varphi \right) _{U}ds\leq \sup_{\left\Vert \varphi
\right\Vert _{H}\leq 1}\int_{0}^{t-\varepsilon }\left\Vert B_{2}^{\ast
}Py(s)\right\Vert _{U}\left\Vert B_{2}^{\ast }e^{A^{\ast }(t-s-\varepsilon
)}\varphi \right\Vert _{U}ds \\
&\leq &\sup_{\left\Vert \varphi \right\Vert _{H}\leq
1}\int_{0}^{t-\varepsilon }\left\Vert Py(s)\right\Vert _{D(A^{\ast
})}\left\Vert B_{2}^{\ast }e^{A^{\ast }(t-s-\varepsilon )}\right\Vert
_{L(H,U)}\left\Vert \varphi \right\Vert _{H}ds \\
&\leq &\int_{0}^{t-\varepsilon }\left\Vert y(s)\right\Vert
_{D(A_{P})}\left\Vert B_{2}^{\ast }e^{A^{\ast }(t-s-\varepsilon
)}\right\Vert _{L(H,U)}ds \\
&\leq &\int_{0}^{t-\varepsilon }\left\Vert A_{P}y(s)\right\Vert
_{H}\left\Vert B_{2}^{\ast }e^{A^{\ast }(t-s-\varepsilon )}\right\Vert
_{L(H,U)}ds\leq C\left\Vert A_{P}y_{0}\right\Vert
_{H}\int_{0}^{t-\varepsilon }\left\Vert B_{2}^{\ast }e^{A^{\ast
}(t-s-\varepsilon )}\right\Vert _{L(H,U)}ds\leq C_{T},
\end{eqnarray*}
hence $\mathcal{M}_{22}$ is relatively compact, too. We also have
\begin{equation*}
\left\Vert \int_{t-\varepsilon }^{t}e^{A(t-s)}B_{1}B_{1}^{\ast
}Py(s)ds\right\Vert _{H}\leq C\int_{t-\varepsilon }^{t}\left\Vert
B_{2}B_{2}^{\ast }Py(s)\right\Vert _{H}ds\leq C\varepsilon
\end{equation*}
and similarly we estimate that the term corresponding to $B_{2}B_{2}^{\ast }P
$ is bounded by $C\varepsilon .$ Since $\varepsilon $ is arbitrary it
follows that $T_{P}(t)\mathcal{M}$ is compact and, as mentioned earlier, it
follows by density that the set $\left\{ T_{P}(t)y_{0};\mbox{ }\left\Vert
y_{0}\right\Vert _{H}\leq M\right\} $ is compact for each $M$ and $t>0,$
fixed. Now, coming back to $z_{n}$ we write
\begin{equation*}
z_{n}(t)=\gamma ^{-2}B_{1}^{\ast }PS(\varepsilon )\left(
\int_{0}^{t-\varepsilon }S(t-s-\varepsilon )B_{1}w_{n}(s)ds\right) +\gamma
^{-2}B_{1}^{\ast }P\int_{t-\varepsilon }^{t}S(t-s)B_{1}w_{n}(s)ds
\end{equation*}
and get
\begin{eqnarray*}
&&\left\Vert \int_{0}^{t-\varepsilon }S(t-s-\varepsilon
)B_{1}w_{n}(s)ds\right\Vert _{H}\leq C\left\Vert \int_{0}^{t-\varepsilon
}e^{-\beta (t-s-\varepsilon )}B_{1}w(s)ds\right\Vert _{H} \\
&\leq &C\int_{0}^{t-\varepsilon }\left\Vert B_{1}w_{n}(s)\right\Vert
_{H}ds\leq C\left\Vert w\right\Vert _{L^{2}(\mathbb{R}_{+};W)}\leq C,\mbox{ }
\forall t\geq 0,
\end{eqnarray*}
hence, $\left\{ S(\varepsilon )\left( \int_{0}^{t-\varepsilon
}S(t-s-\varepsilon )B_{1}w_{n}(s)ds\right) \right\} $ is compact in $H.$
Taking into account that $\left\Vert \int_{t-\varepsilon
}^{t}S(t-s)B_{1}w(s)ds\right\Vert _{H}\leq C\varepsilon ,$ it follows that $
(z_{n}(t))_{n}$ is compact in $H,$ for every $t>0.$ Also, it is
equi-uniformly continuous, that is $\left\Vert
z_{n}(t+h)-z_{n}(t)\right\Vert _{H}\leq \varepsilon $ if $\left\vert
h\right\vert \leq \delta (\varepsilon ),$ for any $t.$ The latter follows
because the semigroup $S(t)$ is continuous for $t>0$ in the uniform operator
topology (see \cite{Pazy}, p. 48, Theorem 3.2), and this means that $
\left\Vert (S(t+h)-S(t))\theta \right\Vert _{H}\leq \delta _{1}(h)\left\Vert
\theta \right\Vert _{H},$ where $\delta _{1}(h)\rightarrow 0,$ and $\theta
\in H.$ Then,
\begin{eqnarray*}
&&\left\Vert z_{n}(t+h)-z_{n}(t)\right\Vert _{H}\leq
C_{1}\int_{t}^{t+h}\left\Vert S(t+h-s)B_{1}Pw_{n}(s)\right\Vert _{H}ds \\
&&+C_{2}\int_{0}^{t}\left\Vert (S(t+h-s)-S(t-s))B_{1}Pw_{n}(s)\right\Vert
_{H}ds \\
&\leq &C_{1}\int_{t}^{t+h}e^{-\beta (t+h-s)}\left\Vert w_{n}(s)\right\Vert
_{H}ds+C_{2}\delta _{2}(h),
\end{eqnarray*}
where $\delta _{2}(h)\rightarrow 0$ as $h\rightarrow 0.$ Then, by
Ascoli-Arzel\`{a}'s theorem, $(z_{n})_{n}$ is compact in $C([0,T];H),$ for
every $T>0$ and so $z_{n}\rightarrow z$ strongly in $L^{2}(0,T;H),$ for
every $T>0.$ Recalling (\ref{230-0}) we note that
\begin{equation*}
\left\Vert z_{n}-z\right\Vert _{L^{2}(\mathbb{R}_{+};H)}^{2}=\int_{0}^{T}
\left\Vert z_{n}(t)-z(t)\right\Vert _{H}^{2}dt+\int_{T}^{\infty }\left\Vert
z_{n}(t)-z(t)\right\Vert _{H}^{2}dt\rightarrow 0,\mbox{ as }n\rightarrow
\infty
\end{equation*}
because the first term tends to $0$ by the compactness argument developed
before and
\begin{equation*}
\int_{T}^{\infty }\left\Vert z_{n}(t)-z(t)\right\Vert _{H}^{2}dt\leq
2\int_{T}^{\infty }\left\Vert z_{n}(t)\right\Vert
_{H}^{2}dt+\int_{T}^{\infty }\left\Vert z(t)\right\Vert
_{H}^{2}dt\rightarrow 0
\end{equation*}
by (\ref{230-0}) and the fact that $A-B_{2}B_{2}^{\ast }P$ generates an
exponentially stable semigroup.
Going back to (\ref{230}), it follows that $\left\Vert
(w_{n}-z_{n})(t)\right\Vert _{W}\rightarrow 0,$ a.e. $t>0,$ and so $
(w_{n})_{n}$ is compact, as claimed. This ends the proof of Lemma 3.5 and
also of Theorem 3.1.
\end{proof}
\begin{remark}
Theorem \ref{Th-main} reduces the existence of a robust feedback controller $
F$ satisfying (\ref{14-1}) to the existence of a solution $P$ to (\ref{15})
in the same way as for $B_{1}=B_{2}=D_{1}=0,$ $C_{1}=I,$ the Lyapunov
equation $A^{\ast }P+PA=I$ is related to the stability of the semigroup $
e^{At}.$ In the specific examples discussed in the next sections we shall
show that the operatorial equation (\ref{15}) reduces to a nonlinear
integro-differential elliptic equation.
\end{remark}
\section{The case of a $N$-$D$ distributed control\label{Distributed}}
\setcounter{equation}{0}
Let $\Omega $ be an open bounded subset of $\mathbb{R}^{N},$ $N>3$ with the
boundary $\Gamma =\partial \Omega $ sufficiently smooth and assume that $
0\in \Omega .$ We consider the following singular system
\begin{eqnarray}
&&\left. y_{t}-\Delta y-\frac{\lambda y}{|x|^{2}}-a(x)y=B_{1}w+B_{2}u,
\right. \left. \mbox{in }(0,\infty )\times \Omega ,\right. \label{21} \\
&&\left. y=0,\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ on }(0,\infty )\times \Gamma ,\right.
\label{21-1} \\
&&\left. y(0)=y_{0},\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in }\Omega ,\right. \label{21-2} \\
&&\left. z=C_{1}y+D_{1}u,\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ in }(0,\infty )\times \Omega ,\right.
\label{21-3}
\end{eqnarray}
where $\lambda >0,$ $\left\vert \cdot \right\vert $ denotes the Euclidian
norm in $\mathbb{R}^{N},$ for any $N=1,2,...$, according the case and $a$
has the expression
\begin{equation}
a(x)=a_{0}\chi _{\Omega _{0}}(x),\mbox{ }a_{0}>0,\mbox{ }\Omega _{0}\subset
\Omega . \label{21-4}
\end{equation}
In this problem
\begin{equation}
y_{0}\in L^{2}(\Omega ) \label{21-5}
\end{equation}
and we choose
\begin{equation}
H=W=Z=L^{2}(\Omega ),\mbox{ }U=\mathbb{R}, \label{21-6}
\end{equation}
\begin{eqnarray}
B_{1}w &=&\chi _{\omega _{1}}(x)w,\mbox{ \ }B_{2}u=b(x)u,\mbox{ \ }
\label{22} \\
C_{1}y &=&\chi _{\Omega _{C}}(x)y,\mbox{ \ }D_{1}u=d(x)u,\mbox{ }x\in \Omega
,\mbox{ }u\in \mathbb{R}, \notag
\end{eqnarray}
where $\Omega _{0},$ $\Omega _{C},$ $\omega _{1}$ are open sets of $\Omega ,$
$\chi _{\omega }$ is characteristic functions of the set $\omega \subset
\Omega ,$
\begin{equation}
\omega _{1}\sqsubseteq \Omega ,\mbox{ }\Omega _{0}\sqsubseteq \Omega
_{C}\subset \Omega ,\mbox{ } \label{22-5}
\end{equation}
and
\begin{equation}
b\in L^{2}(\Omega ),\mbox{ }d\in L^{2}(\Omega ),\mbox{ }d(x)=\chi _{\Omega
\backslash \Omega _{C}.}. \label{22-1}
\end{equation}
We begin by checking the hypotheses $(i_{1})-(i_{4}).$
$(i_{1})$ By their expressions we see that
\begin{equation*}
B_{1},C_{1}\in L(L^{2}(\Omega ),L^{2}(\Omega )),\mbox{ }B_{2},D_{1}\in L(
\mathbb{R},L^{2}(\Omega ))
\end{equation*}
and $B_{2}^{\ast }:L^{2}(\Omega )\rightarrow \mathbb{R}$ is defined by
\begin{equation}
B_{2}^{\ast }v=\int_{\Omega }b(x)v(x)dx,\mbox{ for }v\in L^{2}(\Omega ).
\label{22-3}
\end{equation}
We recall the Hardy inequality (\ref{HN}) and consider $\lambda <H_{N}.$ We
introduce the self-adjoint operator
\begin{equation}
A:D(A)\subset L^{2}(\Omega )\rightarrow L^{2}(\Omega ),\mbox{ }Ay=\Delta y+
\frac{\lambda y}{|x|^{2}}+ay, \label{23}
\end{equation}
with
\begin{equation}
D(A)=\{y\in H_{0}^{1}(\Omega );\mbox{ }Ay\in L^{2}(\Omega )\}. \label{24}
\end{equation}
It is clear that $\overline{D(A)}=L^{2}(\Omega )$ because $D(A)$ contains $
C_{0}^{\infty }(\Omega \backslash \{0\}).$ Then, equation (\ref{21}) can be
equivalently written
\begin{equation}
y^{\prime }(t)=Ay(t)+B_{1}w(t)+B_{2}u(t),\mbox{ }t\geq 0. \label{24-1}
\end{equation}
In order to show that $A$ generates a $C_{0}$-semigroup on $L^{2}(\Omega ),$
we have to prove that $A$ is $\omega $-$m$-dissipative on $L^{2}(\Omega ),$
or that $-A$ is $\omega $-$m$-accretive on $L^{2}(\Omega )$ (see \cite
{VB-book-2010}, p. 155).
\begin{lemma}
Let $\lambda <H_{N}.$ The operator $-A$ is $\omega $-m-accretive on $
L^{2}(\Omega ),$ for $\omega >a_{0}.$
\end{lemma}
\begin{proof}
This means to show that $-A$ is $\omega $-accretive, that is $((\omega
I-A)y,y)_{2}\geq 0$ for some $\omega >0$ and all $y\in L^{2}(\Omega )$ and
that $\omega I-A$ is surjective. To this end we shall use several times the
Hardy inequality (\ref{HN}) which ensures that $\frac{y}{x}\in L^{2}(\Omega
) $ if $y\in H_{0}^{1}(\Omega ).$ We have
\begin{eqnarray*}
((\omega I-A)y,y)_{2} &=&\omega \int_{\Omega }\left\vert y\right\vert
^{2}dx+\int_{\Omega }\left\vert \nabla y\right\vert ^{2}dx-\lambda
\int_{\Omega }\frac{\left\vert y\right\vert ^{2}}{\left\vert x\right\vert
^{2}}dx-a_{0}\int_{\Omega _{0}}\left\vert y\right\vert ^{2}dx \\
&\geq &\left( 1-\frac{\lambda }{H_{N}}\right) \int_{\Omega }\left\vert
\nabla y\right\vert ^{2}dx+(\omega -a_{0})\int_{\Omega }\left\vert
y\right\vert ^{2}dx \\
&\geq &\frac{1}{2}\left( 1-\frac{\lambda }{H_{N}}\right) \left\Vert \nabla
y\right\Vert _{2}^{2}+\frac{H_{N}}{2}\left( 1-\frac{\lambda }{H_{N}}\right)
\left\Vert \frac{y}{x}\right\Vert _{2}^{2}+(\omega -a_{0})\left\Vert
y\right\Vert _{2}^{2}
\end{eqnarray*}
which shows that $-A$ is $\omega $-accretive on $L^{2}(\Omega )$ for $
\lambda <H_{N}$ and $\omega >a_{0}.$
To prove the surjectivity of $\omega I-A,$ we show that the range $R(\omega
I-A)=L^{2}(\Omega ).$ Thus, let $f\in L^{2}(\Omega )$ and prove that the
equation
\begin{equation}
\omega y-Ay=f \label{24-3}
\end{equation}
has a solution $y\in D(A)$, by the equivalent variational formulation
expressed by the minimization problem
\begin{equation}
\min_{y\in H_{0}^{1}(\Omega )}\left\{ J(y)=\int_{\Omega }\left( \frac{1}{2}
\left\vert \nabla y\right\vert ^{2}-\frac{\lambda }{2}\frac{y^{2}}{
\left\vert x\right\vert ^{2}}-\frac{\omega -a(x)}{2}y^{2}-fy\right)
dx\right\} , \label{24-0}
\end{equation}
subject to (\ref{24-1}) and $y(0)=y_{0}\in L^{2}(\Omega ).$ For $\omega
>a_{0}$ we have
\begin{equation*}
\frac{1}{2}\left( 1-\frac{\lambda }{H_{N}}\right) \int_{\Omega }\left\vert
\nabla y\right\vert ^{2}dx+(\omega -a_{0})\int_{\Omega }y^{2}dx-\frac{1}{
2(\omega -a_{0})}\int_{\Omega }\left\vert f\right\vert ^{2}dx\leq J(\varphi
)<\infty
\end{equation*}
so that $J$ has an infimum $d.$ Taking a minimizing sequence $(y_{n})_{n}$
we have
\begin{equation}
d\leq J(y_{n})\leq d+\frac{1}{n} \label{24-2}
\end{equation}
and so
\begin{equation*}
\left\Vert \nabla y_{n}\right\Vert _{2}+\left\Vert y_{n}\right\Vert
_{2}+\left\Vert \frac{y_{n}}{x}\right\Vert _{2}\leq C_{N}\mbox{ for }\omega
>a_{0}.
\end{equation*}
Further, $C,$ $C_{N},$ $C_{T}$ denote some constants (which may change from
line to line), $C_{N}$ depending on $N,$ via $\lambda <H_{N}$ and $C_{T}$
depending on $T.$
We deduce that on a subsequence denoted still by $n$ it follows that
\begin{equation*}
y_{n}\rightarrow y\mbox{ weakly in }H_{0}^{1}(\Omega ),\mbox{ }\frac{y_{n}}{x
}\rightarrow l\mbox{ weakly in }L^{2}(\Omega )
\end{equation*}
and by compactness $y_{n}\rightarrow y$ strongly in $L^{2}(\Omega ).$ Then $
\frac{y_{n}}{x}\rightarrow \frac{y}{x}$ a.e. on $\Omega $ and $l=\frac{y}{x}$
by the Vitali's theorem. We can now pass to the limit in (\ref{24-2}),
relying on the weakly lower semicontinuity of $J$ and get that $J(y)=d,$
that is $y$ realizes the minimum in (\ref{24-0}).
Next, we give a variation $y^{\sigma }=y+\sigma \eta ,$ for $\sigma >0$ and $
\eta \in H_{0}^{1}(\Omega ),$ and particularize the condition of optimality,
namely $J(\widetilde{y})\geq J(y)$ for any $\widetilde{y}\in
H_{0}^{1}(\Omega )$ for $\widetilde{y}=y^{\sigma }.$ We calculate
\begin{equation*}
\lim_{\sigma \rightarrow 0}\frac{J(y^{\sigma })-J(y)}{\sigma }=\int_{\Omega
}\left( (\omega -a(x))y\eta +\nabla y\cdot \nabla \eta -\frac{\lambda y\eta
}{\left\vert x\right\vert ^{2}}-f\eta \right) dx\geq 0.
\end{equation*}
Repeating the calculus for $\sigma \rightarrow -\sigma $ we get the reverse
inequality, so that finally we can write
\begin{equation*}
\int_{\Omega }\left\langle (\omega -a(x))y-\Delta y-\frac{\lambda y}{
\left\vert x\right\vert ^{2}}-f,\eta \right\rangle _{H^{-1}(\Omega
),H_{0}^{1}(\Omega )}dx=0\mbox{ for all }\eta \in H_{0}^{1}(\Omega ),
\end{equation*}
which implies that $y$ is the weak solution to the equation (\ref{24-3}).
The solution is also unique because $J$ is strictly convex and the system is
linear. By (\ref{24-3}) we see that $Ay\in L^{2}(\Omega ),$ so that $y\in
D(A).$
\end{proof}
In conclusion, $A$ generates an analytic $C_{0}$-semigroup on $L^{2}(\Omega
) $ for $\lambda <H_{N}.$
Moreover, since as earlier seen, the operator $(\omega I-A)^{-1}$ is a
compact operator for $\omega >a_{0},$ it follows that $e^{At}$ is compact
for all $t>0.$
$(i_{2})$ Let $y_{0}\in L^{2}(\Omega ),$ $u\in L^{2}(\mathbb{R}_{+},\mathbb{R
}),$ $w\in L^{2}(\mathbb{R}_{+};L^{2}(\Omega )).$ Since $B_{1}w+B_{2}u\in
L^{2}(0,T;L^{2}(\Omega ))$ and $y_{0}\in \overline{D(A)}=L^{2}(\Omega ),$
eq. (\ref{24-1}) with $y(0)=y_{0}$ has a unique mild solution $y\in
C([0,T],L^{2}(\Omega )),$ given by (\ref{13}) for any $T>0$ (see \cite
{VB-book-2010}, p. 131, Corollary 4.1). The solution also satisfies $y\in
L^{2}(0,T;H_{0}^{1}(\Omega ))\cup W^{1,2}(0,T;H^{-1}(\Omega )).$
In order to prove $(i_{3})$ we provide the following lemma.
\begin{lemma}
Let $\lambda <H_{N}.$ Then, the pair $(A,C_{1})$ is exponentially detectable.
\end{lemma}
\begin{proof}
Let $K\equiv -kI$, with $k\geq a_{0}$ and set $A_{1}=A+KC_{1}.$ This is
still $\omega $-$m$-accretive, so that $A_{1}$ generates a $C_{0}$-semigroup
on $L^{2}(\Omega ),$ $S_{1}(t)=e^{A_{1}t}.$ Hence $y(t)=e^{A_{1}t}y_{0}$
satisfies
\begin{equation}
\frac{dy}{dt}(t)=A_{1}y(t),\mbox{ }t\geq 0,\mbox{ }y(0)=y_{0}. \label{29}
\end{equation}
Recalling the expression of $C_{1},$ multiplying (\ref{29}) by $y(t)$ and
applying again (\ref{HN}) we get
\begin{equation}
\frac{1}{2}\frac{d}{dt}\left\Vert y(t)\right\Vert _{2}^{2}+\left( 1-\frac{
\lambda }{H_{N}}\right) \left\Vert \nabla y(t)\right\Vert
_{2}^{2}+k\int_{\Omega _{C}}\left\vert y(t)\right\vert ^{2}ds\leq
a_{0}\int_{\Omega _{0}}\left\vert y(t)\right\vert ^{2}dx. \label{30}
\end{equation}
We take into account that $\Omega _{0}\sqsubseteq \Omega _{C}$ and $k\geq
a_{0}$, and integrate from $0$ to $t.$ We obtain
\begin{equation*}
\frac{1}{2}\left\Vert y(t)\right\Vert _{2}^{2}+\left( 1-\frac{\lambda }{H_{N}
}\right) \int_{0}^{t}\left\Vert \nabla y(s)\right\Vert
_{2}^{2}ds+(k-a_{0})\int_{\Omega _{0}}\left\vert y(t)\right\vert ^{2}ds\leq
\frac{1}{2}\left\Vert y_{0}\right\Vert _{2}^{2},\mbox{ }\forall t>0.
\end{equation*}
From here and the Poincar\'{e} inequality it follows that
\begin{equation}
\int_{0}^{t}\left\Vert y(s)\right\Vert _{2}^{2}ds\leq C_{N}\left\Vert
y_{0}\right\Vert _{2}^{2},\mbox{ for all }t>0, \label{31}
\end{equation}
with $C_{N}$ a constant depending on $H_{N}.$ Letting $t\rightarrow \infty $
in (\ref{31}) we finally get that
\begin{equation}
\int_{0}^{\infty }\left\Vert y(s)\right\Vert _{2}^{2}ds\leq C_{N}\left\Vert
y_{0}\right\Vert _{2}^{2}. \label{31-0}
\end{equation}
This means by Datko's result, previously recalled, that $e^{A+KC_{1}}$
generates an exponentially stable semigroup, that is there exists $\alpha >0$
such that
\begin{equation*}
\left\Vert e^{(A+KC_{1})t}y\right\Vert _{2}\leq Ce^{-\alpha t}y_{2}\mbox{
for all }y\in L^{2}(\Omega ).
\end{equation*}
Then,
\begin{eqnarray*}
&&\int_{0}^{\infty }\left\Vert B_{2}^{\ast }e^{(A^{\ast }+C_{1}^{\ast
}K^{\ast })t}y\right\Vert _{U}dt\leq C\int_{0}^{\infty }\left\Vert
e^{(A^{\ast }+C_{1}^{\ast }K^{\ast })t}y\right\Vert _{2}dt \\
&\leq &C\left\Vert y\right\Vert _{2}\int_{0}^{\infty }e^{-\alpha
t}dt=C\left\Vert y\right\Vert _{2},\mbox{ }\forall y\in L^{2}(\Omega ),
\end{eqnarray*}
that is (\ref{12-0}) is verified.
\end{proof}
$(i_{4})$ By (\ref{22-1}) we have
\begin{equation*}
\left\Vert D_{1}u\right\Vert _{2}^{2}=u^{2}\left\Vert d\right\Vert
_{L^{2}(\Omega \backslash \Omega _{C})}=u^{2}
\end{equation*}
and
\begin{equation*}
D_{1}^{\ast }C_{1}y=\int_{\Omega }d(x)\chi _{\Omega _{C}}(x)y(x)dx=0.
\end{equation*}
The hypotheses being checked, we can formulate the $H^{\infty }$-control
problem for system (\ref{21})-(\ref{21-3}) as in Theorem \ref{Th-main}.
In order to explicit Theorem \ref{Th-main} and to give a differential
formulation for it, as announced in Remark 3.7, we recall that the linear
continuous operator $P\in L(L^{2}(\Omega ),L^{2}(\Omega ))$ can be
represented by the L. Schwartz kernel theorem (see e.g., \cite{Lions-control}
, p. 166) as an integral operator with a kernel $P_{0}\in L^{2}(\Omega
\times \Omega ),$ namely
\begin{equation}
P\varphi (x)=\int_{\Omega }P_{0}(x,\xi )\varphi (\xi )d\xi ,\mbox{ for all }
\varphi \in C_{0}^{\infty }(\Omega ). \label{36}
\end{equation}
By (\ref{22}) and (\ref{22-3}) we have
\begin{eqnarray}
B_{1}B_{1}^{\ast }\varphi (x) &=&\chi _{\omega _{1}}(x)\varphi (x),\mbox{ }
C_{1}C_{1}^{\ast }\varphi (x)=\chi _{\Omega _{C}}(x)\varphi (x),\mbox{ }
\notag \\
B_{1}B_{1}^{\ast }P\varphi (x) &=&\chi _{\omega _{1}}(x)\int_{\Omega
}P_{0}(x,\xi )\varphi (\xi )d\xi ,\mbox{ } \notag \\
PB_{1}B_{1}^{\ast }P\varphi (x) &=&\int_{\Omega }\int_{\Omega }\chi _{\omega
_{1}}(\overline{\xi })P_{0}(x,\overline{\xi })P_{0}(\overline{\xi },\xi
)\varphi (\xi )d\overline{\xi }d\xi \label{36-0}
\end{eqnarray}
\begin{eqnarray}
B_{2}B_{2}^{\ast }\varphi (x) &=&b(x)\int_{\Omega }b(\overline{x})\varphi (
\overline{x})d\overline{x},\mbox{ }x\in \Omega , \notag \\
B_{2}B_{2}^{\ast }P\varphi (x) &=&b(x)\int_{\Omega }\int_{\Omega }b(
\overline{x})P_{0}(\overline{x},\xi )\varphi (\xi )d\overline{x}d\xi ,\mbox{
}x\in \Omega , \notag \\
PB_{2}B_{2}^{\ast }P\varphi (x) &=&\int_{\Omega }\int_{\Omega }\int_{\Omega
}P_{0}(x,\overline{\xi })P_{0}(\overline{x},\xi )b(\overline{\xi })b(
\overline{x})\varphi (\xi )d\overline{x}d\overline{\xi }d\xi . \label{36-1}
\end{eqnarray}
Moreover, by a straightforward calculation we obtain
\begin{equation}
A^{\ast }P\varphi (x)=\int_{\Omega }\left( \Delta _{x}P_{0}(x,\xi )+\frac{
\lambda P_{0}(x,\xi )}{|x|^{2}}+a(x)P_{0}(x,\xi )\right) \varphi (\xi )d\xi ,
\label{36-2}
\end{equation}
\begin{equation}
PA\varphi (x)=\int_{\Omega }\varphi (\xi )\left( \Delta _{\xi }P_{0}(x,\xi )+
\frac{\lambda P_{0}(x,\xi )}{|\xi |^{2}}+a(\xi )P_{0}(x,\xi )\right) d\xi ,
\label{36-3}
\end{equation}
and by denoting $E:=B_{2}B_{2}^{\ast }-\gamma ^{-2}B_{1}B_{1}^{\ast },$ we
have
\begin{eqnarray*}
PEP\varphi (x) &=&\int_{\Omega }\varphi (\xi )d\xi \int_{\Omega
}\int_{\Omega }P_{0}(x,\overline{\xi })P_{0}(\overline{x},\xi )b(\overline{
\xi })b(\overline{x})d\overline{x}d\overline{\xi } \\
&&-\gamma ^{-2}\int_{\Omega }\varphi (\xi )d\xi \int_{\Omega }\chi _{\omega
_{1}}(\overline{\xi })P_{0}(x,\overline{\xi })P_{0}(\overline{\xi },\xi )d
\overline{\xi }.
\end{eqnarray*}
For $x\in \Omega $ we define the distribution $\mu _{x}\in \mathcal{D}
^{\prime }(\Omega )$ by
\begin{equation*}
\mu _{x}(\varphi )=\chi _{\Omega _{C}}(x)\varphi (x)=\int_{\Omega }\delta
(x-\xi )\chi _{\Omega _{C}}(\xi )\varphi (\xi )d\xi ,\mbox{ }\forall \varphi
\in C_{0}^{\infty }(\Omega ),
\end{equation*}
where $\delta $ is the Dirac distribution. Then, by replacing all these in (
\ref{15}), we deduce the equation
\begin{eqnarray}
&&\Delta _{x}P_{0}(x,\xi )+\Delta _{\xi }P_{0}(x,\xi )+\lambda P_{0}(x,\xi
)\left( \frac{1}{|x|^{2}}+\frac{1}{|\xi |^{2}}\right) +(a(x)+a(\xi
))P_{0}(x,\xi ) \notag \\
&&-\int_{\Omega }\int_{\Omega }P_{0}(x,\overline{\xi })P_{0}(\overline{x}
,\xi )b(\overline{\xi })b(\overline{x})d\overline{x}d\overline{\xi }+\gamma
^{-2}\int_{\Omega }\chi _{\omega _{1}}(\overline{\xi })P_{0}(x,\overline{\xi
})P_{0}(\overline{\xi },\xi )d\overline{\xi } \label{37} \\
&=&-\delta (x-\xi )\chi _{\Omega _{C}}(\xi ),\mbox{ in }\mathcal{D}^{\prime
}(\Omega \times \Omega ). \notag
\end{eqnarray}
This equation is accompanied by the conditions
\begin{equation}
P_{0}(x,\xi )=0,\mbox{ }\forall (x,\xi )\in \Gamma \times \Gamma ,
\label{38}
\end{equation}
\begin{equation}
P_{0}(x,\xi )=P(\xi ,x),\mbox{ }\forall (x,\xi )\in \Omega \times \Omega ,
\label{39}
\end{equation}
\begin{equation}
P_{0}(x,\xi )\geq 0,\mbox{ }\forall (x,\xi )\in \Omega \times \Omega
\label{40}
\end{equation}
and so we can enounce the following
\begin{theorem}
Let $\gamma >0$ and let $A,$ $B_{1},$ $B_{2},$ $C_{1}$ and $D_{1}$ be given
by (\ref{23}) and (\ref{22}), respectively. Then there exists $\widetilde{F}
\in L(L^{2}(\Omega ),\mathbb{R})$ which solves the $H^{\infty }$-control
problem for system (\ref{21})-(\ref{21-3}) if and only if there exists a
solution $P_{0}\in D(A)\times D(A)$ to (\ref{37})-(\ref{38}), satisfying (
\ref{39})-(\ref{40}). Moreover, in this case
\begin{equation}
\widetilde{F}y=-\int_{\Omega }\int_{\Omega }b(x)P_{0}(x,\xi )y(\xi )d\xi dx,
\mbox{ }\forall y\in L^{2}(\Omega ), \label{40-0}
\end{equation}
is a feedback controller which solves the $H^{\infty }$-problem for system (
\ref{21})-(\ref{21-3}).
\end{theorem}
In this case it is easily seen that $\Lambda _{P}=A-B_{2}B_{2}^{\ast
}P+\gamma ^{-2}B_{1}B_{1}^{\ast }P$ has the domain $D(\Lambda _{P})=D(A),$
and since $\Lambda _{P}$ is closed it follows that $\mathcal{X}=D(A).$
Moreover, by (\ref{40-0}) we see that $\widetilde{F}\in L(L^{2}(\Omega ),
\mathbb{R}).$
A\ direct approach of problem (\ref{37})-(\ref{40}) is an interesting
problem by itself but is beyond the objective of this work.
\section{Dirichlet boundary control\label{Boundary}}
As in the previous section let $\Omega $ be an open bounded subset of $
\mathbb{R}^{N},$ $N>3$ with the boundary $\Gamma =\partial \Omega $
sufficiently smooth and such that $0\in \Omega .$ Consider the following
system
\begin{eqnarray}
&&\left. y_{t}-\Delta y-\frac{\lambda y}{|x|^{2}}-a(x)y=B_{1}w,\right.
\left. \mbox{in }(0,\infty )\times \Omega ,\right. \label{41} \\
&&\left. y=\widetilde{u},\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ on }(0,\infty )\times \Gamma ,\right.
\label{42} \\
&&\left. y(0)=y_{0},\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ in }\Omega ,\right. \label{43} \\
&&\left. z=C_{1}y+D_{1}u,\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ in }(0,\infty )\times \Omega ,\right. \label{44}
\end{eqnarray}
where $y_{0}\in L^{2}(\Omega )$, $a$ is again given by (\ref{21-4}) and
\begin{eqnarray}
\widetilde{u}(t,x) &=&\sum\limits_{j=1}^{m}\alpha _{j}(x)u_{j}(t),\mbox{ }
u_{j}(t)\in \mathbb{R}\mbox{ a.e. }t\in (0,\infty ),\mbox{ }j=1,...,m,
\label{44-1} \\
\alpha &=&(\alpha _{1},...,\alpha _{m})\in \left( L^{2}(\Gamma )\right) ^{m},
\mbox{ }\alpha _{j}\geq 0\mbox{ a.e. }x\in \Gamma . \notag
\end{eqnarray}
We assume in addition that
\begin{equation}
\frac{D_{0}\alpha _{j}}{x}\in L^{2}(\Omega ),\mbox{ }j=1,...,m. \label{46}
\end{equation}
The expression (\ref{44-1}) allows the possibility to consider combinations
of conditions on subsets of the boundary for the controls $u_{j}(t)\in
\mathbb{R}.$ The hypothesis (\ref{46}) will be justified later.
$(i_{1})$ For this problem we choose
\begin{equation}
H=W=Z=L^{2}(\Omega ),\mbox{ }U=\mathbb{R}^{m}, \label{46-1}
\end{equation}
\begin{equation}
B_{1}w=\chi _{\omega _{1}}(x)w,\mbox{ }C_{1}y=\chi _{\Omega _{C}}(x)y,\mbox{
\ }D_{1}u=\sum\limits_{j=1}^{m}d_{j}(x)u_{j},\mbox{ }x\in \Omega ,
\label{44-2}
\end{equation}
$u=(u_{1},...,u_{m}),$ with the conditions $\omega _{1}\sqsubseteq \Omega ,$
$\Omega _{0}\sqsubseteq \Omega _{C},$ and
\begin{equation}
d_{j}\in L^{2}(\Omega ),\mbox{ }d_{j}(x)=0\mbox{ on }\Omega _{C},\mbox{ }
\int_{\Omega \backslash \Omega _{C}}d_{j}d_{k}dx=\delta _{jk}. \label{44-4}
\end{equation}
Thus, $B_{1}\in L(L^{2}(\Omega ),L^{2}(\Omega )),$ $C_{1}\in L(L^{2}(\Omega
),L^{2}(\Omega ))$ and $D_{1}:U\rightarrow L^{2}(\Omega ).$ The operator $
B_{2}$ will be further defined. The operator $A$ is the same as before, that
is
\begin{equation}
A:D(A)\subset L^{2}(\Omega )\rightarrow L^{2}(\Omega ),\mbox{ }Ay=\Delta y+
\frac{\lambda y}{|x|^{2}}+a(x)y, \label{45-1}
\end{equation}
\begin{equation}
D(A)=\left\{ y\in H_{0}^{1}(\Omega );\mbox{ }Ay\in L^{2}(\Omega )\right\} .
\label{45-2}
\end{equation}
By Lemma 4.1, for $\lambda <H_{N}$ and $\omega >a_{0},$ it follows that $-A$
is $\omega $-$m$-accretive on $L^{2}(\Omega )$ and self-adjoint, so that $A$
generates a $C_{0}$ compact semigroup $e^{At}$ on $L^{2}(\Omega )$.
Moreover, as we shall see later, if $y\in D(A)$ then $y\in H^{2}(\Omega
\backslash \{0\}).$
In order to write equation\ (\ref{41}) in the operatorial form, we need some
preliminaries. Let us consider the problem
\begin{equation}
\Delta \theta =0\mbox{ in }\Omega ,\mbox{ }\theta =v\mbox{ on }\Gamma ,\mbox{
for }t>0. \label{47}
\end{equation}
The boundary condition is meant in the sense of the trace of $\theta $ on $
\Gamma ,$ generally denoted by $tr(\theta ).$ But, if any confusion is
avoided we shall no longer indicate the trace by the symbol $\mathit{tr}$.
The unique solution to this problem is the well-known Dirichlet map, $
v\rightarrow \theta ,$ here denoted by $D_{0}v.$ If $v\in L^{2}(\Gamma ),$
then $D_{0}:L^{2}(\Gamma )\rightarrow H^{1/2}(\Omega )$ and it satisfies $
\left\Vert D_{0}v\right\Vert _{H^{1/2}(\Omega )}\leq C\left\Vert
v\right\Vert _{L^{2}(\Gamma )}$ (see e.g. \cite{L-T}).
In our case, $v=\widetilde{u}\in L^{2}(\mathbb{R}_{+};L^{2}(\Gamma ))$ and
so $D_{0}\widetilde{u}(t)\in H^{1/2}(\Omega )$ and
\begin{equation*}
\left\Vert D_{0}\widetilde{u}(t)\right\Vert _{H^{1/2}(\Omega )}\leq
C\left\Vert \widetilde{u}(t)\right\Vert _{L^{2}(\Gamma )},\mbox{ a.e. }t>0.
\end{equation*}
Moreover, since $\widetilde{u}$ is given by (\ref{44-1}) and $D_{0}$ is
linear we have
\begin{equation}
D_{0}\widetilde{u}(t)=\sum\limits_{j=1}^{m}u_{j}(t)D_{0}\alpha _{j},\mbox{ }
t>0. \label{47-1}
\end{equation}
Let us introduce the operator
\begin{equation}
A_{0}:D(A_{0})=D(A)\subset L^{2}(\Omega )\rightarrow L^{2}(\Omega ),\mbox{ }
A_{0}y=\Delta y+\frac{\lambda y}{\left\vert x\right\vert ^{2}}. \label{47-2}
\end{equation}
This operator is $m$-dissipative on $L^{2}(\Omega )$ by a similar proof as
in Lemma 4.1. Let us determine the Dirichlet mapping $v\rightarrow Dv$
corresponding to $A_{0},$ that is
\begin{equation}
\Delta Dv+\frac{\lambda Dv}{|x|^{2}}=0\mbox{ in }\Omega ,\mbox{ }Dv=v\mbox{
on }\Gamma . \label{48}
\end{equation}
\begin{lemma}
For $\lambda <H_{N},$ $Dv$ associated to $A_{0}$ exists and it is unique for
$v\in L^{2}(\Gamma )$ satisfying $\frac{D_{0}v}{x}\in L^{2}(\Omega ).$
Moreover, one has
\begin{equation}
Dv\in H^{1/2}(\Omega )\mbox{ and }\left\Vert Dv\right\Vert _{H^{1/2}(\Omega
)}\leq C\left( \left\Vert v\right\Vert _{L^{2}(\Gamma )}+\left\Vert \frac{
D_{0}v}{x}\right\Vert _{L^{2}(\Omega )}\right) . \label{49}
\end{equation}
\end{lemma}
\begin{proof}
Let $t$ be fixed and denote $\varphi =Dv-D_{0}v$ and consider the equation
\begin{equation}
\Delta \varphi +\frac{\lambda \varphi }{\left\vert x\right\vert ^{2}}=-\frac{
\lambda D_{0}v}{\left\vert x\right\vert ^{2}}\mbox{ in }\Omega ,\mbox{ }
\varphi =0\mbox{ on }\Gamma . \label{50}
\end{equation}
We assert that problem (\ref{50}) has a unique solution in $D(A)$ and prove
it via a variational technique, by showing that the solution to (\ref{50})
is given by the minimization of the functional $\Psi (\varphi ),$
\begin{equation}
\min_{\varphi \in H_{0}^{1}(\Omega )}\left\{ \Psi (\varphi )=\int_{\Omega
}\left( \frac{1}{2}\left\vert \nabla \varphi \right\vert ^{2}-\frac{1}{2}
\frac{\lambda \varphi ^{2}}{\left\vert x\right\vert ^{2}}-\frac{\lambda
\varphi D_{0}v}{\left\vert x\right\vert ^{2}}\right) dx\right\} . \label{51}
\end{equation}
It is easily seen that
\begin{equation*}
\left( \frac{1}{2}-\frac{\lambda }{H_{N}}\right) \int_{\Omega }\left\vert
\nabla \varphi \right\vert ^{2}dx-\lambda \int_{\Omega }\left\vert \frac{
D_{0}v}{x}\right\vert ^{2}dx\leq \Psi (\varphi )<\infty ,
\end{equation*}
so that $\Psi $ has an infimum $d.$ We note here the necessity of the
assumption $\frac{D_{0}v}{x}\in L^{2}(\Omega )$. Next, we proceed as in
Lemma 4.1 and show that $\varphi \in H_{0}^{1}(\Omega )$ is the unique weak
solution to the equation (\ref{50}). By (\ref{50}) we note that by
multiplying by $\varphi $ we get
\begin{equation*}
\left\Vert \nabla \varphi \right\Vert _{2}^{2}+\frac{\lambda }{2}\left\Vert
\frac{\varphi }{x}\right\Vert _{2}^{2}\leq \frac{\lambda }{2}\left\Vert
\frac{D_{0}v}{x}\right\Vert _{2}^{2}.
\end{equation*}
Then, it follows that $Dv=\varphi +D_{0}v$ which is the Dirichlet map for (
\ref{48}), has the properties $Dv\in H^{1/2}(\Omega ),$ $\frac{Dv}{x}=\frac{
\varphi }{x}+\frac{D_{0}v}{x}\in L^{2}(\Omega )$ and $\left\Vert
Dv\right\Vert _{H^{1/2}(\Omega )}\leq \left\Vert \varphi +D_{0}v\right\Vert
_{H^{1/2}(\Omega )}\leq C\left( \left\Vert \varphi \right\Vert
_{H_{0}^{1}(\Omega )}+\left\Vert D_{0}v\right\Vert _{H^{1/2}(\Omega
)}\right) ,$ implying (\ref{49}).
\end{proof}
\noindent Lemma 5.1 implies that the operator $D:L^{2}(\Gamma )\rightarrow
L^{2}(\Omega )$ with the domain $\left\{ v\in L^{2}(\Gamma );\frac{D_{0}v}{x}
\in L^{2}(\Omega )\right\} $ is closed and densely defined. We denote by $
D^{\ast }:L^{2}(\Omega )\rightarrow L^{2}(\Gamma )$ its adjoint.
Now, we can write the operatorial form of the system. Let $
u=(u_{1},...,u_{m})$ and assume for the beginning that
\begin{equation*}
u\in W^{1,2}(0,T;\mathbb{R}^{m}),\mbox{ }w\in W^{1,2}(0,T;L^{2}(\Omega )),
\mbox{ }T\geq 0
\end{equation*}
and note that $D\widetilde{u}(t)$ is well defined due to (\ref{46}), $D
\widetilde{u}(t)\in H^{1/2}(\Omega )$ and
\begin{equation*}
\left\Vert D\widetilde{u}(t)\right\Vert _{H^{1/2}(\Omega )}\leq
C\sum\limits_{j=1}^{m}|u_{j}(t)|\left( \left\Vert \alpha _{j}\right\Vert
_{L^{2}(\Gamma )}+\left\Vert \frac{D_{0}\alpha _{j}}{x}\right\Vert
_{L^{2}(\Omega )}\right) ,\mbox{ a.e. }t>0.
\end{equation*}
We and write the difference system (\ref{41}) and (\ref{48}),
\begin{eqnarray*}
&&(y-D\widetilde{u})_{t}-\Delta (y-D\widetilde{u})-\frac{\lambda (y-D
\widetilde{u})}{|x|^{2}}-a(x)(y-D\widetilde{u}) \\
&=&B_{1}w-(D\widetilde{u})_{t}+a(x)D\widetilde{u},\mbox{ in }(0,\infty
)\times \Omega , \\
y-D\widetilde{u} &=&0\mbox{, on }(0,\infty )\times \Gamma ,\mbox{ }(y-D
\widetilde{u})(0)=y_{0}-\widetilde{\theta _{0}}\mbox{ in }\Omega ,
\end{eqnarray*}
where $\widetilde{\theta _{0}}=D\widetilde{u}(0).$ The solution to the
previous system reads
\begin{equation*}
(y-D\widetilde{u})(t)=e^{At}(y_{0}-\widetilde{\theta _{0}}
)+\int_{0}^{t}e^{A(t-s)}(B_{1}w+aD\widetilde{u})(s)ds-
\int_{0}^{t}e^{A(t-s)}(D\widetilde{u})_{t}(s)ds.
\end{equation*}
Integrating by parts the last right-hand side term we obtain
\begin{eqnarray*}
y(t)-D\widetilde{u}(t) &=&e^{At}y_{0}-e^{At}\widetilde{\theta _{0}}
+\int_{0}^{t}e^{A(t-s)}(B_{1}w+a(x)D\widetilde{u})(s)ds \\
&&-D\widetilde{u}(t)+e^{At}\widetilde{\theta _{0}}-\int_{0}^{t}e^{A(t-s)}AD
\widetilde{u}(s)dx
\end{eqnarray*}
which yields
\begin{equation*}
y(t)=e^{At}y_{0}-\int_{0}^{t}e^{A(t-s)}AD\widetilde{u}(s)dx+
\int_{0}^{t}e^{A(t-s)}(B_{1}w+a(x)D\widetilde{u})(s)ds.
\end{equation*}
The formula is preserved by density if $u\in L^{2}(0,T;\mathbb{R}^{m})$ and $
w\in L^{2}(0,T;L^{2}(\Omega ))$ and this represents the solution to the
equation
\begin{equation}
y^{\prime }(t)=Ay(t)+B_{1}w(t)-AD\widetilde{u}(t)+a(x)D\widetilde{u}(t),
\mbox{ }y(0)=y_{0}. \label{53}
\end{equation}
Since $D\widetilde{u}(t)$ is not in $D(A)$ one must interpret $AD\widetilde{u
}(t)$ by using the extension $\widetilde{A}$ of $A$ to the whole space $
L^{2}(\Omega )$ by
\begin{equation}
\widetilde{A}:L^{2}(\Omega )\rightarrow (D(A))^{\prime },\mbox{ }
\left\langle \widetilde{A}y,\psi \right\rangle _{(D(A))^{\prime
},D(A)}=(y,A\psi ),\mbox{ }\forall \psi \in D(A), \label{54}
\end{equation}
see (\ref{401}). Now, we can define $B_{2}:U\rightarrow (D(A))^{\prime },$
\begin{equation}
B_{2}u=-\widetilde{A}\left( \sum\limits_{j=1}^{m}u_{j}D\alpha _{j}\right)
+a(x)\sum\limits_{j=1}^{m}u_{j}D\alpha
_{j}=-\sum\limits_{j=1}^{m}u_{j}A_{0}D\alpha _{j}\mbox{,} \label{55}
\end{equation}
where $u=(u_{1},...,u_{m})\in U=\mathbb{R}^{m}.$ Expression (\ref{55}) is
well defined since $D\alpha _{j}\in H^{1/2}(\Omega )\subset L^{2}(\Omega )$
and $a\in L^{\infty }(\Omega ).$ Eventually, we can express equations (\ref
{41})-(\ref{42}) as
\begin{eqnarray}
y^{\prime }(t) &=&Ay(t)+B_{1}w(t)+B_{2}\widetilde{u}(t),\mbox{ }t\geq 0,
\mbox{ } \label{56} \\
y(0) &=&y_{0} \notag
\end{eqnarray}
with $\widetilde{A}$ defined in (\ref{54}), $B_{2}$ defined in (\ref{55})
and $\widetilde{u}$ defined in (\ref{44-1}).
$(i_{2})$ For verifying (\ref{12}) we need to calculate $B_{2}^{\ast }$ and $
D^{\ast }.$ We denote by $\frac{\partial v}{\partial \nu }$ the normal
derivative of $v$ on the boundary $\Gamma .$ We give the following lemma.
\begin{lemma}
The operator $B_{2}^{\ast }:D(A)\rightarrow \mathbb{R}^{m}$ is given by
\begin{equation}
(B_{2}^{\ast }v)_{j}=-\left( \alpha _{j},\frac{\partial v}{\partial \nu }
\right) _{L^{2}(\Gamma )}\mbox{, for }v\in D(A),\mbox{ }j=1,...,m,
\label{60}
\end{equation}
where $\frac{\partial v}{\partial \nu }\in L^{2}(\Gamma ).$
The operator $D^{\ast }:L^{2}(\Omega )\rightarrow L^{2}(\Gamma ),$ is
defined by
\begin{equation}
D^{\ast }p=\frac{\partial }{\partial \nu }(A_{0}^{-1}p)\mbox{ on }\Gamma ,
\mbox{ for }p\in L^{2}(\Omega ). \label{59-1}
\end{equation}
\end{lemma}
\begin{proof}
We use the definition of $B_{2}$ and for $v\in D(A)$ we calculate
\begin{eqnarray}
&&\left\langle B_{2}u,v\right\rangle _{(D(A))^{\prime },D(A)}=\left\langle -
\widetilde{A}\left( \sum\limits_{j=1}^{m}u_{j}D\alpha _{j}\right)
+a\sum\limits_{j=1}^{m}u_{j}D\alpha _{j},v\right\rangle _{(D(A))^{\prime
},D(A)} \label{57} \\
&=&-\sum\limits_{j=1}^{m}\left( u_{j}D\alpha _{j},Av\right) _{L^{2}(\Omega
)}+\sum\limits_{j=1}^{m}\left( u_{j}D\alpha _{j},av\right) _{L^{2}(\Omega
)}=\sum\limits_{j=1}^{m}u_{j}\left( D\alpha _{j},-Av+av\right)
_{L^{2}(\Omega )} \notag \\
&=&u\cdot (D\alpha ,-Av+av)_{L^{2}(\Omega )}=u\cdot (D\alpha
,-A_{0}v)_{L^{2}(\Omega )}, \notag
\end{eqnarray}
where $(D\alpha ,A_{0}v)_{L^{2}(\Omega )}$ denotes the vector with the
components $(D\alpha _{j},-A_{0}v)_{L^{2}(\Omega )}$ for $v\in D(A).$ Here
we took into account that $-Av+av=-A_{0}v$ with $A_{0}$ defined in (\ref
{47-2}). Hence, we can define the components of $B_{2}^{\ast
}:D(A)\rightarrow U^{\ast }=U=\mathbb{R}^{m}$ by
\begin{equation}
(B_{2}^{\ast }v)_{j}=\left( D\alpha _{j},-A_{0}v\right) _{L^{2}(\Omega )},
\mbox{ }v\in D(A),\mbox{ }j=1,...,m. \label{57-1}
\end{equation}
For the computation of $(D\alpha _{j},-A_{0}v)_{L^{2}(\Omega )}$ let us
consider the generic systems
\begin{equation}
\Delta D\beta +\frac{\lambda D\beta }{\left\vert x\right\vert ^{2}}=0,\mbox{
}D\beta =\beta \mbox{ on }\Gamma ,\mbox{ }\beta \in L^{2}\left( \Gamma
\right) , \label{58}
\end{equation}
\begin{equation}
-\Delta v-\frac{\lambda v}{\left\vert x\right\vert ^{2}}=p,\mbox{ }v=0\mbox{
on }\Gamma ,\mbox{ }p\in L^{2}(\Omega ). \label{58-1}
\end{equation}
The second system has a unique solution $v\in H_{0}^{1}(\Omega ).$ In order
to make a rigorous calculus we assume first that $\beta \in H^{1}\left(
\Gamma \right) $ and $-A_{0}$ is replaced by for $\varepsilon >0$ by
\begin{equation}
-A_{0,\varepsilon }=-\Delta -\frac{\lambda }{\left\vert x\right\vert
^{2}+\varepsilon },\mbox{ }D(A_{0,\varepsilon })=H^{2}(\Omega )\cap
H_{0}^{1}(\Omega ). \label{58-2}
\end{equation}
Thus, the equation $-A_{0,\varepsilon }v=p$ has a unique solution $
v_{\varepsilon }\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )$ and all
operations below make sense. We multiply the approximating equation for $
D\beta $ by the solution $v_{\varepsilon }.$ By applying the Green's formula
we obtain
\begin{equation*}
\int_{\Omega }\left( D\beta \Delta v_{\varepsilon }+\frac{\lambda
v_{\varepsilon }D\alpha _{j}}{\left\vert x\right\vert ^{2}+\varepsilon }
\right) dx+\int_{\Gamma }\left( v_{\varepsilon }\frac{\partial D\beta }{
\partial \nu }-D\beta \frac{\partial v_{\varepsilon }}{\partial \nu }\right)
dx=0
\end{equation*}
which implies, by using (\ref{58-1}) and the boundary condition for $D\alpha
_{j},$ that
\begin{equation*}
-\int_{\Omega }pD\beta dx=\int_{\Gamma }\beta \frac{\partial v_{\varepsilon }
}{\partial \nu }d\sigma ,\mbox{ }\forall \beta \in L^{2}(\Gamma ).
\end{equation*}
Therefore, we have for each $p\in L^{2}(\Omega )$
\begin{equation*}
(D\beta ,p)_{L^{2}(\Omega )}=\left( \beta ,\frac{\partial }{\partial \nu }
(A_{0,\varepsilon }^{-1}p)\right) _{L^{2}(\Gamma )}\mbox{ }\forall \beta \in
H^{1}\left( \Gamma \right) ,
\end{equation*}
which can be written also as
\begin{equation*}
(D\beta ,-A_{0,\varepsilon }v_{\varepsilon })_{L^{2}(\Omega )}=-\left( \beta
,\frac{\partial v_{\varepsilon }}{\partial \nu }\right) _{L^{2}(\Gamma )}
\mbox{ for }\beta \in H^{1}\left( \Gamma \right) ,\mbox{ }v_{\varepsilon
}\in D(A_{0,\varepsilon }).
\end{equation*}
These remain true at limit as $\varepsilon \rightarrow 0,$ hence
\begin{equation}
(D\beta ,p)_{L^{2}(\Omega )}=\left( \beta ,\frac{\partial }{\partial \nu }
(A_{0}^{-1}p)\right) _{L^{2}(\Gamma )}\mbox{ for }\beta \in H^{1}\left(
\Gamma \right) ,\mbox{ }p\in L^{2}(\Omega ), \label{58-0}
\end{equation}
\begin{equation}
(D\beta ,-A_{0}v)_{L^{2}(\Omega )}=-\left( \beta ,\frac{\partial v}{\partial
\nu }\right) _{L^{2}(\Gamma )}\mbox{ for }\beta \in H^{1}\left( \Gamma
\right) ,\mbox{ }v\in D(A_{0})=D(A) \label{59}
\end{equation}
and the latter makes sense since $\frac{\partial v}{\partial \nu }\in
H^{-1/2}(\Gamma ).$ We note that both $A_{0,\varepsilon }$ and $A_{0}$ are
surjective, because they are $m$-accretive and coercive. Then, by (\ref{58-0}
) we can define $D^{\ast }:L^{2}(\Omega )\rightarrow L^{2}(\Gamma ),$ by (
\ref{59-1}).
Going back to (\ref{57-1}) and using (\ref{59}) in which we set $\beta
:=\alpha _{j}$ it turns out that we can define $B_{2}^{\ast
}:D(A)\rightarrow U$ by
\begin{equation}
(B_{2}^{\ast }v)_{j}=(D\alpha _{j},-A_{0}v)_{L^{2}(\Gamma )}=-\left( \alpha
_{j},\frac{\partial v}{\partial \nu }\right) _{L^{2}(\Gamma )}\mbox{, for }
v\in D(A). \label{59-0}
\end{equation}
It remains to show that $\frac{\partial v}{\partial \nu }$ belongs to $
L^{2}(\Gamma )$ if $v\in D(A).$ Indeed, there exists $(v_{\varepsilon
})_{\varepsilon }\subset H^{2}(\Omega )\cap D(A)$ such that $v_{\varepsilon
}\rightarrow v$ strongly in $D(A)$, $\frac{\partial v_{\varepsilon }}{
\partial \nu }\rightarrow \frac{\partial v}{\partial \nu }$ strongly in $
H^{-1/2}(\Gamma )$ as $\varepsilon \rightarrow 0$ and
\begin{equation}
(B_{2}^{\ast }v_{\varepsilon })_{j}=-\left( \alpha _{j},\frac{\partial
v_{\varepsilon }}{\partial \nu }\right) _{L^{2}(\Gamma )}. \label{60-00}
\end{equation}
We recall that $0\in \Omega $. We consider $\varphi \in C^{4}(\overline{
\Omega })$ defined by
\begin{equation*}
\varphi (x)=\left\{
\begin{array}{l}
0,\mbox{ if }x\in \Omega _{\delta } \\
1,\mbox{ if }x\in \Omega \backslash \Omega _{2\delta }
\end{array}
\right.
\end{equation*}
where $\delta >0$ is such that $\Omega _{\delta }=\{x\in \Omega ;$ $
\left\Vert x\right\Vert <\delta \}$ and $0\in \Omega _{\delta }.$ The
function $\varphi v_{\varepsilon }\in H^{2}(\Omega \backslash \Omega
_{2\delta }).$ Indeed, since $v_{\varepsilon }\in H^{2}(\Omega )$ it follows
that there exists $f\in L^{2}(\Omega )$ such that $f=Av_{\varepsilon }$ and
so $\Delta v_{\varepsilon }=f-\frac{\lambda v_{\varepsilon }}{\left\vert
x\right\vert ^{2}}\in L^{2}(\Omega \backslash \Omega _{2\delta }).$ We have
\begin{equation*}
\Delta (\varphi v_{\varepsilon })=\varphi \Delta v_{\varepsilon }+2\nabla
\varphi \cdot \nabla v_{\varepsilon }+v_{\varepsilon }\Delta \varphi \in
L^{2}(\Omega ).
\end{equation*}
This together with the boundary condition $\varphi v_{\varepsilon }=0$ on $
\Gamma $ implies that $\varphi v_{\varepsilon }\in H^{2}(\Omega \backslash
\Omega _{2\delta })$ and so $v_{\varepsilon }\in H^{2}(\Omega \backslash
\Omega _{2\delta }),$ too, because $\varphi =1$ on $\Omega \backslash \Omega
_{2\delta }.$ Consequently, $\frac{\partial v_{\varepsilon }}{\partial \nu }
\in H^{1/2}(\Gamma )\subset L^{2}(\Gamma ).$ This is preserved by density
nearby the boundary. Finally, (\ref{59-0}) remains true by density for $
\alpha _{j}\in L^{2}(\Gamma )$ and so this implies (\ref{60}).
\end{proof}
Now, we pass to the proof of $(i_{2}).$ Such a result is proved for the
Laplace operator in \cite{vbp}, p. 320, Proposition 4.39, but here we give a
complete different proof under our hypotheses.
To this end, we recall that $Ay=A_{0}y+ay$ with $A_{0}$ defined in (\ref
{47-2}) and consider the problem
\begin{equation}
\frac{dy}{dt}(t)+B_{0}y(t)-ay=0,\mbox{ in }(0,T)\times \Omega ,\mbox{ }
y(0)=y_{0}\in L^{2}(\Omega ) \label{60-0}
\end{equation}
where
\begin{equation}
B_{0}=-A_{0},\mbox{ }B_{0}=-\Delta -\frac{\lambda }{\left\vert x\right\vert
^{2}},\mbox{ }B_{0}:D(B_{0})=D(A_{0})\rightarrow L^{2}(\Omega ).
\label{60-B0}
\end{equation}
The operator $B_{0}$ is $m$-accretive, $B_{0}=B_{0}^{\ast }$ and $B_{0}-aI$
is $\omega $-$m$-accretive. The unique solution to problem (\ref{60-0}) has
also the property $y(t)\in D(A)=D(A_{0})$ a.e. $t\in (0,T)$ by the
regularizing effect (see \cite{VB-book-2010}, p. 158 Theorem 4.11).
First, we determine two estimates. We multiply equation (\ref{60-0}) first
by $y(t)$ and integrate over $(0,t).$ We obtain, using Gronwall's lemma
\begin{equation}
\left\Vert y(t)\right\Vert
_{2}^{2}+\int_{0}^{t}(B_{0}y(s),y(s))_{2}ds=C_{T}\left\Vert y_{0}\right\Vert
_{2}^{2},\mbox{ }\forall t\in \lbrack 0,T]. \label{60-1}
\end{equation}
Then, we multiply (\ref{60-0}) by $tB_{0}y(t)$ which yields
\begin{equation}
\frac{1}{2}\frac{d}{dt}\left( tB_{0}y(t),y(t)\right) _{2}+t\left\Vert
B_{0}y(t)\right\Vert _{2}^{2}=\frac{1}{2}\left( B_{0}y(t),y(t)\right)
_{2}+(ay(t),B_{0}y(t))_{2}. \label{60-2}
\end{equation}
We integrate this and by (\ref{60-1}) we get
\begin{equation}
t(B_{0}y(t),y(t))_{2}+\int_{0}^{t}s\left\Vert B_{0}y(s)\right\Vert
_{2}^{2}ds\leq C\int_{0}^{t}(B_{0}y(s),y(s))_{2}ds\leq C_{T}\left\Vert
y_{0}\right\Vert _{2}^{2}. \label{60-3}
\end{equation}
To prove $(i_{2})$ we have to estimate
\begin{eqnarray}
\left\Vert B_{2}^{\ast }e^{At}y_{0}\right\Vert _{\mathbb{R}^{m}}
&=&\left\Vert B_{2}^{\ast }y(t)\right\Vert _{\mathbb{R}^{m}}=\left\Vert
\left( -\left( \alpha _{j},\frac{\partial y(t)}{\partial \nu }\right)
_{L^{2}(\Gamma )}\right) _{j=1}^{m}\right\Vert _{\mathbb{R}^{m}}
\label{60-01} \\
&\leq &\sum\limits_{j=1}^{m}\left\Vert \alpha _{j}\right\Vert _{L^{2}(\Gamma
)}\left\Vert \frac{\partial y(t)}{\partial \nu }\right\Vert _{L^{2}(\Gamma
)}, \notag
\end{eqnarray}
thus, actually we have to estimate $\left\Vert \frac{\partial y(t)}{\partial
\nu }\right\Vert _{L^{2}(\Gamma )}$ for $t>0.$ Since we shall relate this to
the fractional powers of the operator $B_{0},$ for a rigorous computation
involving its fractional powers we shall rely again on the approximation, $
B_{0,\varepsilon }=-A_{0,\varepsilon },$ see (\ref{58-2}). We proceed with
all calculations for the approximating equation (\ref{60-0}) with $
B_{0,\varepsilon }$ instead of $B_{0}$ and pass to the limit at the end.
Thus, $D(B_{0,\varepsilon })=H^{2}(\Omega )\cap H_{0}^{1}(\Omega ),$ $
B_{0,\varepsilon }:D(B_{0,\varepsilon })\subset L^{2}(\Omega )\rightarrow
L^{2}(\Omega )$ and it is $m$-accretive and self-adjoint.
Therefore, we recall that the fractional powers are defined by $
B_{0,\varepsilon }^{s}:D(B_{0,\varepsilon }^{s})\subset L^{2}(\Omega
)\rightarrow L^{2}(\Omega )$, $s\geq 0,$ see \cite{Pazy}$.$ Then, $
D(B_{0,\varepsilon }^{s})\subset H^{2s}(\Omega )$ with equality iff $2s<3/2,$
see e.g., \cite{Fujiwara}. We have the interpolation inequality
\begin{equation}
\left\Vert B_{0,\varepsilon }^{s}w\right\Vert _{2}\leq C\left\Vert
B_{0,\varepsilon }^{s_{1}}w\right\Vert _{2}^{\lambda }\left\Vert
B_{0,\varepsilon }^{s_{2}}w\right\Vert _{2}^{1-\lambda },\mbox{ for }
s=\lambda s_{1}+(1-\lambda )s_{2}, \label{300}
\end{equation}
and the relations
\begin{equation}
\left\Vert B_{0,\varepsilon }^{s}w\right\Vert _{2}\leq C\left\Vert
B_{0,\varepsilon }^{s_{1}}w\right\Vert _{2}\mbox{ if }s<s_{1}, \label{301}
\end{equation}
\begin{equation}
\left\Vert B_{0,\varepsilon }^{s}w\right\Vert _{H^{m}(\Omega )}\leq
C\left\Vert B_{0,\varepsilon }^{s+m/2}w\right\Vert _{2}. \label{302}
\end{equation}
Now, we come back to $\frac{\partial y}{\partial \nu }(t)$ and using the
trace theorem and (\ref{302}) applied to $B_{0,\varepsilon }$ we write for
the approximating solution
\begin{equation}
\left\Vert \frac{\partial y_{\varepsilon }}{\partial \nu }(t)\right\Vert
_{L^{2}(\Gamma )}\leq C\left\Vert y_{\varepsilon }(t)\right\Vert
_{H^{3/2}(\Omega )}\leq C\left\Vert B_{0,\varepsilon }^{3/4}y(t)\right\Vert
_{L^{2}(\Omega )}, \label{60-02}
\end{equation}
so that we must estimate $\left\Vert B_{0,\varepsilon }^{3/4}y(t)\right\Vert
_{H}.$
Next, we use (\ref{300}) and write
\begin{equation}
\left\Vert B_{0,\varepsilon }^{3/4}y_{\varepsilon }(t)\right\Vert _{2}\leq
C\left\Vert B_{0,\varepsilon }y_{\varepsilon }(t)\right\Vert
_{2}^{3/4}\left\Vert y_{\varepsilon }(t)\right\Vert _{2}^{1/4}. \label{60-4}
\end{equation}
Further, we calculate via H\"{o}lder's inequality
\begin{eqnarray}
&&\int_{0}^{t}\left\Vert B_{0,\varepsilon }y_{\varepsilon }(s)\right\Vert
_{2}^{3/4}ds=\int_{0}^{t}s^{p}\left\Vert B_{0,\varepsilon }y_{\varepsilon
}(s)\right\Vert _{2}^{3/4}s^{-p}ds \label{60-5} \\
&\leq &\left( \int_{0}^{t}s^{8p/3}\left\Vert B_{0,\varepsilon
}y_{\varepsilon }(t)\right\Vert _{2}^{2}ds\right) ^{3/8}\left(
\int_{0}^{t}s^{-8p/5}ds\right) ^{5/8} \notag \\
&=&\left( \int_{0}^{t}s\left\Vert B_{0,\varepsilon }y_{\varepsilon
}(s)\right\Vert _{2}^{2}ds\right) ^{3/8}\left( \int_{0}^{t}s^{-3/5}ds\right)
^{5/8} \notag \\
&\leq &C\left( \int_{0}^{t}s\left\Vert B_{0,\varepsilon }y_{\varepsilon
}(s)\right\Vert _{2}^{2}ds\right) ^{3/8}\left( t^{2/5}\right) ^{5/8}, \notag
\end{eqnarray}
where we chose $p=\frac{3}{8}.$ This together with (\ref{60-01}), (\ref
{60-02}), (\ref{60-4}) and (\ref{60-1}) implies
\begin{eqnarray}
&&\int_{0}^{t}\left\Vert B_{2,\varepsilon }^{\ast }e^{As}y_{0}\right\Vert _{
\mathbb{R}^{m}}ds\leq C\int_{0}^{t}\left\Vert \frac{\partial y_{\varepsilon }
}{\partial \nu }(s)\right\Vert _{L^{2}(\Gamma )}ds\leq
C\int_{0}^{t}\left\Vert B_{0,\varepsilon }y_{\varepsilon }(s)\right\Vert
_{L^{2}(\Omega )}^{3/4}ds \label{60-7} \\
&\leq &C\int_{0}^{t}\left\Vert B_{0,\varepsilon }y_{\varepsilon
}(s)\right\Vert _{2}^{3/4}\left\Vert y_{\varepsilon }(s)\right\Vert
_{2}^{1/4}ds\leq C_{T}\left\Vert y_{0}\right\Vert
_{2}^{1/4}\int_{0}^{t}\left\Vert B_{0,\varepsilon }y_{\varepsilon
}(s)\right\Vert _{2}^{3/4}ds \notag \\
&\leq &C_{T}\left\Vert y_{0}\right\Vert _{2}^{1/4}\left\Vert
y_{0}\right\Vert _{2}^{3/4}\left( t^{2/5}\right) ^{5/8}\leq C_{T}\left\Vert
y_{0}\right\Vert _{2},\mbox{ }\forall t\in \lbrack 0,T]. \notag
\end{eqnarray}
Passing to the limit by recalling (\ref{59}) we get $(i_{2})$ as claimed.
This hypothesis has also an important consequence. We note that (\ref{56})
with the initial condition $y(0)=y_{0}\in L^{2}(\Omega )$ has a unique
solution $y\in C([0,T];(D(A))^{\prime }),$
\begin{equation}
y(t)=e^{At}y_{0}+\int_{0}^{t}e^{A(t-s)}(B_{1}w(s)+B_{2}u(s))ds,\mbox{ }t\in
\lbrack 0,\infty ). \label{60-6}
\end{equation}
We are going to show first that $(i_{2})$ ensures in addition that $y\in
L^{2}(0,T;L^{2}(\Omega )).$
Actually, we shall prove the following assertion: if (\ref{12}) takes place
then the solution $y$ to (\ref{56}) belongs to $L^{2}(0,T;L^{2}(\Omega ))$
if $u\in L^{2}(0,T;U).$ Since in (\ref{60-6}) the sum between the first and
the last term corresponding to the contribution of $w$ is already in $
C([0,T];L^{2}(\Omega ))$ we focus only on the term $Y(t):=
\int_{0}^{t}e^{A(t-s)}B_{2}u(s)ds$ and show as in (\ref{16-0}) that $
\left\Vert Y\right\Vert _{L^{2}(0,T;L^{2}(\Omega ))}\leq C\left\Vert
u\right\Vert _{L^{2}(0,T;U)}.$ In conclusion, equation (\ref{56}) with the
initial condition $y_{0}\in L^{2}(\Omega )$ has a mild solution $y\in
L^{2}(0,T;L^{2}(\Omega ))$.
$(i_{3})$ The first part of hypothesis $(i_{3}),$ that is the detectability
of the pair $(A,C_{1})$ follows as in Lemma 4.2. Now we prove (\ref{12-0}).
We recall that
\begin{equation*}
A_{1}y=A_{0}y+a_{0}\chi _{\Omega _{0}}(x)y-k\chi _{\Omega _{C}}(x)y
\end{equation*}
with $A_{0}$ defined in (\ref{47-2}) and consider the problem
\begin{equation}
\frac{dy}{dt}(t)+B_{0}y(t)=a_{0}\chi _{\Omega _{0}}(x)y-k\chi _{\Omega
_{C}}(x)y,\mbox{ in }(0,T)\times \Omega ,\mbox{ }y(0)=y_{0}\in L^{2}(\Omega )
\label{61}
\end{equation}
where $B_{0}=-A_{0}$ is $m$-accretive, $B_{0}=B_{0}^{\ast }$ and $A_{1}$ is $
m$-accretive. Then, problem (\ref{61}) has a unique solution $
y(t)=S_{1}(t)y_{0},$ where $S_{1}(t)$ is the $C_{0}$-semigroup generated by $
A_{1}$. The solution $y\in L^{2}(0,T;H_{0}^{1}(\Omega ))$ and $y(t)\in D(A)$
a.e. $t\in (0,T)$.
Since $A_{1}=A+KC_{1}$ generates an exponentially stable semigroup we have
\begin{equation}
\left\Vert y(t)\right\Vert _{2}\leq e^{-\alpha t}\left\Vert y_{0}\right\Vert
_{2},\mbox{ }\alpha =k-a_{0}. \label{62}
\end{equation}
Moreover, $S_{1}(t)$ is analytic and so
\begin{equation}
\left\Vert A_{1}y(t)\right\Vert _{2}\leq \frac{C_{T}}{t}\left\Vert
y(t)\right\Vert _{2},\mbox{ }\forall t\in (0,T). \label{63-0}
\end{equation}
Since $\left\Vert B_{0}y\right\Vert _{H}\leq \left\Vert A_{1}y\right\Vert
_{H}+C\left\Vert y\right\Vert _{H}$ it follows that
\begin{equation}
\left\Vert B_{0}y(t)\right\Vert _{2}\leq \frac{C_{T}}{t}\left\Vert
y(t)\right\Vert _{2},\mbox{ }\forall t\in (0,T). \label{63}
\end{equation}
The previous calculations for proving point $(i_{2})$ hold here too, and by (
\ref{60-7}) we have
\begin{eqnarray}
&&\left\Vert B_{2}^{\ast }e^{(A+KC_{1})t}y_{0}\right\Vert _{\mathbb{R}
^{m}}=\left\Vert B_{2}^{\ast }y(t)\right\Vert _{\mathbb{R}^{m}}=\left\Vert
\left( -\left( \alpha _{j},\frac{\partial y(t)}{\partial \nu }\right)
_{L^{2}(\Gamma )}\right) _{j=1}^{m}\right\Vert _{\mathbb{R}^{m}} \label{64}
\\
&\leq &\sum\limits_{j=1}^{m}\left\Vert \alpha _{j}\right\Vert _{L^{2}(\Gamma
)}\left\Vert \frac{\partial y(t)}{\partial \nu }\right\Vert _{L^{2}(\Gamma
)}\leq C\left\Vert B_{0}y(t)\right\Vert _{2}^{3/4}\left\Vert
y_{0}\right\Vert _{2}^{1/4}e^{-\alpha t/4}, \notag
\end{eqnarray}
where $y(t)=S_{1}(t)y_{0}$ is the solution to (\ref{61}). Thus,
\begin{equation}
\int_{0}^{T}\left\Vert B_{2}^{\ast }e^{(A+KC_{1})t}y_{0}\right\Vert _{
\mathbb{R}^{m}}dt\leq C_{T}\left\Vert y_{0}\right\Vert _{2},\mbox{ for }
T\geq 0. \label{64-0}
\end{equation}
On the other hand, for $t>T$ we have
\begin{equation*}
\left\Vert A_{1}y(t)\right\Vert _{2}=\left\Vert
A_{1}S_{1}(T)S_{1}(t-T)y(t)\right\Vert _{2}\leq \frac{C_{T}}{T}\left\Vert
S_{1}(t-T)y(t)\right\Vert _{2}\leq \frac{C_{T}}{T}e^{-\alpha
(t-T)}\left\Vert y_{0}\right\Vert _{2}.
\end{equation*}
Then we calculate
\begin{eqnarray*}
&&\left\Vert B_{0}y(t)\right\Vert _{H}^{3/4}\leq \left( \left\Vert
A_{1}y(t)\right\Vert _{H}+C\left\Vert y(t)\right\Vert _{H}\right) ^{3/4}\leq
C\left\Vert A_{1}y(t)\right\Vert _{H}^{3/4}+C\left\Vert y(t)\right\Vert
_{H}^{3/4} \\
&\leq &\frac{C_{T}}{T^{3/4}}e^{-3\alpha (t-T)/4}\left\Vert y_{0}\right\Vert
_{2}^{3/4}+C\left\Vert y_{0}\right\Vert _{2}^{3/4},
\end{eqnarray*}
hence, by (\ref{64})
\begin{eqnarray}
\left\Vert B_{2}^{\ast }e^{(A^{\ast }+KC_{1})t}y_{0}\right\Vert _{U} &\leq
&\left( \frac{C_{T}}{T^{3/4}}e^{-3\alpha (t-T)/4}+1\right) \left\Vert
y_{0}\right\Vert _{2}^{3/4}\left\Vert y_{0}\right\Vert _{2}^{1/4}e^{-\alpha
t/4} \label{64-2} \\
&=&\left( \frac{C_{T}}{T^{3/4}}e^{-3\alpha (t-T)/4}+e^{-\alpha t/4}\right)
\left\Vert y_{0}\right\Vert _{2},\mbox{ for }t>T. \notag
\end{eqnarray}
In particular, let $T=1$ and by (\ref{64-0}) and (\ref{64-2}) we finally get
\begin{eqnarray}
&&\int_{0}^{\infty }\left\Vert B_{2}^{\ast }e^{(A^{\ast
}+KC_{1})t}y_{0}\right\Vert _{U}dt \label{64-1} \\
&=&\int_{0}^{1}\left\Vert B_{2}^{\ast }e^{(A^{\ast
}+KC_{1})t}y_{0}\right\Vert _{U}dt+\int_{1}^{\infty }\left\Vert B_{2}^{\ast
}e^{(A^{\ast }+KC_{1})t}y_{0}\right\Vert _{U}dt \notag \\
&\leq &C_{1}\left\Vert y_{0}\right\Vert _{2}+\left\Vert y_{0}\right\Vert
_{2}\int_{1}^{\infty }\left( C_{1}e^{-3\alpha (t-1)/4}+e^{-\alpha
t/4}\right) dt\leq C\left\Vert y_{0}\right\Vert _{2}\mbox{,} \notag
\end{eqnarray}
for all $y_{0}\in L^{2}(\Omega )$. In conclusion, we have obtained (\ref
{12-0}) as claimed.
$(i_{4})$ The adjoint of $D_{1}$ is $D_{1}^{\ast }:L^{2}(\Omega )\rightarrow
\mathbb{R}^{m}$
\begin{equation*}
D_{1}^{\ast }v=\left( \int_{\Omega }d_{1}(x)v(x)dx,...,\int_{\Omega
}d_{m}(x)v(x)dx\right) .
\end{equation*}
Then, by (\ref{44-4}), $\left\Vert D_{1}u\right\Vert _{L^{2}(\Omega
)}^{2}=\int_{\Omega }\left( \sum\limits_{j=1}^{m}d_{j}(x)\right) ^{2}dx=1,$
and $\int_{\Omega }d_{j}(x)\chi _{\Omega _{C}}(x)ydx=0,$ hence $D_{1}^{\ast
}C_{1}y(\xi )=0.$
Then, calculating the operators in (\ref{15}) we see that formulae (\ref
{36-0}), (\ref{36-2})-(\ref{36-3}) are the same and using (\ref{59-0}) we
get
\begin{equation*}
PB_{2}B_{2}^{\ast }P\varphi (x)=\int_{\Omega }\varphi (\xi )\left(
\sum\limits_{j=1}^{m}A_{j}(\xi )A_{j}(x)\right) d\xi
\end{equation*}
where
\begin{equation*}
A_{j}(\xi )=\int_{\Gamma }\alpha _{j}(\sigma )\frac{\partial P_{0}}{\partial
\nu _{\sigma }}(\sigma ,\xi )d\sigma .
\end{equation*}
Proceedings with all calculations as in Section \ref{Distributed} we have
\begin{theorem}
Let $\gamma >0$ and let $A,$ $B_{1},$ $C_{1}$ and $D_{1}$ be given by (\ref
{45-1}) and (\ref{44-2}), respectively and $B_{2},$ $B_{2}^{\ast }$ be given
by (\ref{55}) and (\ref{60}). Assume that $P_{0}\in D(A)\times D(A)$ is a
solution to equation
\begin{eqnarray}
&&\Delta _{x}P_{0}(x,\xi )+\Delta _{\xi }P_{0}(x,\xi )+\lambda P_{0}(x,\xi
)\left( \frac{1}{|x|^{2}}+\frac{1}{|\xi |^{2}}\right) +(a(x)+a(\xi
))P_{0}(x,\xi ) \notag \\
&&-\sum\limits_{j=1}^{m}A_{j}(x)A_{j}(\xi )+\gamma ^{-2}\int_{\Omega }\chi
_{\omega _{1}}(\overline{\xi })P_{0}(x,\overline{\xi })P_{0}(\overline{\xi }
,\xi )d\overline{\xi } \label{64-4} \\
&=&-\delta (x-\xi )\chi _{\Omega _{C}}(\xi ),\mbox{ in }\mathcal{D}^{\prime
}(\Omega \times \Omega ), \notag
\end{eqnarray}
with conditions (\ref{38})-(\ref{40}). Then, the feedback control $
\widetilde{F}\in L(L^{2}(\Omega ),\mathbb{R}^{m}),$
\begin{equation}
(\widetilde{F}y)_{j}=\int_{\Omega }y(\xi )\left( \alpha _{j},\frac{\partial
P_{0}}{\partial \nu }(\cdot ,\xi )\right) _{L^{2}(\Gamma )}d\xi ,\mbox{ }
j=1,...m,\mbox{ }\forall y\in L^{2}(\Omega ) \label{64-5}
\end{equation}
solves the $H^{\infty }$-problem.
\end{theorem}
In this case, by (\ref{54}), (\ref{55}) and (\ref{57-1}) we have
\begin{equation*}
\Lambda _{P}y=A_{0}\left( y+\int_{\Omega }y(\xi )\sum\limits_{j=1}^{m}\left(
\int_{\Gamma }\alpha _{j}(\sigma )\frac{\partial P_{0}}{\partial \nu
_{\sigma }}(\sigma ,\xi )d\sigma \right) D\alpha _{j}d\xi \right) +ay+\chi
_{\omega _{1}}\int_{\Omega }P_{0}(x,\xi )y(\xi )d\xi
\end{equation*}
and we see that
\begin{equation*}
D(\Lambda _{P})=\left\{ y\in H;\mbox{ }y+\int_{\Omega }y(\xi
)\sum\limits_{j=1}^{m}\left( \int_{\Gamma }\alpha _{j}(\sigma )\frac{
\partial P_{0}}{\partial \nu _{\sigma }}(\sigma ,\xi )d\sigma \right)
D\alpha _{j}d\xi \in D(A)\right\} .
\end{equation*}
Moreover, $\Lambda _{P}$ is closed because if $y_{n}\rightarrow y$ in $H$,
since $A_{0}$ is closed we see that $\Lambda _{P}y_{n}\rightarrow \Lambda
_{P}y$ in $H.$ Then, by Lemma \ref{operators} we deduce that $\mathcal{X}
=D(\Lambda _{P}).$
\section{Dirichlet boundary control in an $1D$ domain with a boundary
singularity\label{1D}}
\setcounter{equation}{0}
We briefly discuss here the $H^{\infty }$-boundary control problem for an
one-dimensional parabolic equation with the singularity on the boundary.
Namely, let $\Omega =(0,1)$ and consider the system
\begin{eqnarray}
&&\left. y_{t}-\Delta y-\frac{\lambda y}{|x|^{2}}-a(x)y=B_{1}w,\right.
\left. \mbox{in }(0,\infty )\times \Omega ,\right. \label{70} \\
&&\left. y(t,0)=0,\mbox{ }y(t,1)=u\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \
\ for }t\geq 0,\right. \label{71} \\
&&\left. y(0)=y_{0},\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ in }\Omega ,\right. \label{72} \\
&&\left. z=C_{1}y+D_{1}u,\right. \left. \mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ in }(0,\infty )\times \Omega ,\right. \label{73}
\end{eqnarray}
where $y_{0}\in L^{2}(\Omega ),$ $u\in \mathbb{R}.$
$
$
$(i_{1})$ For this problem we choose $H=W=Z=L^{2}(\Omega ),$ $U=\mathbb{R},$
\begin{equation}
B_{1}w=\chi _{\omega _{1}}(x)w,\mbox{ }C_{1}y=\chi _{\Omega _{C}}(x)y,\mbox{
\ }D_{1}u=d(x)u,\mbox{ }x\in \Omega , \label{73-2}
\end{equation}
with the conditions $\omega _{1}\sqsubseteq \Omega ,$ $\Omega _{0}\subset
\Omega _{C},$ and
\begin{equation}
d\in L^{2}(\Omega ),\mbox{ }d(x)=0\mbox{ on }\Omega _{C},\mbox{ }
\int_{\Omega \backslash \Omega _{C}}d^{2}(x)dx=1. \label{73-4}
\end{equation}
Thus, $B_{1}\in L(L^{2}(\Omega ),L^{2}(\Omega )),$ $C_{1}\in L(L^{2}(\Omega
),L^{2}(\Omega ))$ and $D_{1}:U\rightarrow L^{2}(\Omega ).$
We deal again with the operator $A:D(A)\subset L^{2}(\Omega )\rightarrow
L^{2}(\Omega ),$ $Ay=\Delta y+\frac{\lambda y}{|x|^{2}},$ which is $\omega $-
$m$-accretive on $L^{2}(\Omega )$ and generates a compact $C_{0}$-semigroup
on $L^{2}(\Omega ).$ The difference here is that in the calculus of the
accretivity of $-A$ we use the Hardy inequality (\ref{HN0}) instead of (\ref
{HN}). Next, we define
\begin{equation}
B:\mathbb{R\rightarrow R\times R}\mbox{, }Bu=(0,u) \label{74}
\end{equation}
and consider problem $\Delta \theta =0,$ $\theta =Bu$ on $\Gamma =\{0,1\}$
which provides the Dirichlet map $D_{0}u,$ associated to $\Delta $ and $Bu,$
expressed in this case by
\begin{equation}
D_{0}u=ux. \label{76}
\end{equation}
Next, the problem
\begin{equation}
\Delta Du+\frac{\lambda Du}{|x|^{2}}=0,\mbox{ }Du=Bu\mbox{ on }\Gamma ,
\label{77}
\end{equation}
provides the Dirichlet map associated to $A_{0}$ defined in (\ref{47-2}).
Making the difference $\varphi =Du-D_{0}u$ we write the equation
\begin{equation*}
\Delta \varphi +\frac{\lambda \varphi }{\left\vert x\right\vert ^{2}}=-\frac{
\lambda u}{x},\mbox{ }\varphi =0\mbox{ on }\Gamma .
\end{equation*}
By a similar calculus as in Lemma 5.1, where we note that in this case while
solving (\ref{51}) we have
\begin{equation*}
\left( \frac{1}{2}-\frac{\lambda }{H_{N}}\right) \int_{\Omega }\left\vert
\nabla \varphi \right\vert ^{2}dx-\left\vert u\right\vert ^{2}\leq \Psi
(\varphi )<\infty ,
\end{equation*}
we deduce that $\Psi $ has a minimum. Thus, we find that $\varphi \in
H_{0}^{1}(\Omega ),$ $\frac{\varphi }{x}\in L^{2}(\Omega )$ and
\begin{equation}
Du=\varphi +ux\in H^{1}(\Omega ),\mbox{ }\frac{Du}{x}\in L^{2}(\Omega ).
\label{78}
\end{equation}
We define
\begin{equation}
B_{2}:U=\mathbb{R}\rightarrow L^{2}(\Omega ),\mbox{ }B_{2}u=-\widetilde{A}
Du+a(x)Du=-uA_{0}D(0,1) \label{79}
\end{equation}
where $\widetilde{A}$ is defined as in (\ref{54}) and $D(0,1)$ is the
Dirichlet map corresponding to the boundary data $y(t,0)=1,$ $y(t,1)=1.$
Then, $B_{2}^{\ast }:D(A)\rightarrow \mathbb{R}$ and Lemma 5.2 implies that
\begin{equation}
B_{2}^{\ast }v=-v^{\prime }(1),\mbox{ }v\in D(A),\mbox{ }D^{\ast
}p=p^{\prime }(1), \label{80}
\end{equation}
where $D^{\ast }:L^{2}(\Omega )\rightarrow \mathbb{R}.$ We recall that $p$
is in $H^{2}$ in the neighborhood of the boundary $x=1.$
Hypotheses $(i_{2}),$ $(i_{3})$ and $(i_{4})$ are proved as in Section \ref
{Boundary}.
Finally, we calculate the term $PB_{2}B_{2}^{\ast }P\varphi (x)$, the other
terms being the same as in the previous sections,
\begin{equation*}
PB_{2}B_{2}^{\ast }P\varphi (x)=\int_{\Omega }\int_{\Omega }\frac{\partial
P_{0}}{\partial x}(1,\xi )\frac{\partial P_{0}}{\partial \xi }(x,1)\varphi
(\xi )d\xi
\end{equation*}
and replacing in (\ref{15}) we get
\begin{theorem}
Let $\gamma >0$ and let $A,$ $B_{1},$ $C_{1}$ and $D_{1}$ be given by (\ref
{45-1}) and (\ref{73-2}), respectively and $B_{2},$ $B_{2}^{\ast }$ be given
by (\ref{74}) and (\ref{80}). Assume that $P_{0}\in D(A)\times D(A)$ is a
solution to equation
\begin{eqnarray}
&&\Delta _{x}P_{0}(x,\xi )+\Delta _{\xi }P_{0}(x,\xi )+\lambda P_{0}(x,\xi
)\left( \frac{1}{|x|^{2}}+\frac{1}{|\xi |^{2}}\right) +(a(x)+a(\xi
))P_{0}(x,\xi ) \notag \\
&&+\int_{\Omega }\frac{\partial P_{0}}{\partial x}(1,\xi )\frac{\partial
P_{0}}{\partial \xi }(x,1)d\xi +\gamma ^{-2}\int_{\Omega }\chi _{\omega
_{1}}(\overline{\xi })P_{0}(x,\overline{\xi })P_{0}(\overline{\xi },\xi )d
\overline{\xi } \label{81} \\
&=&-\delta (x-\xi )\chi _{\Omega _{C}}(\xi ),\mbox{ }(x,\xi )\in \Omega
\times \Omega , \notag
\end{eqnarray}
with the boundary conditions $P_{0}(x,0)=P_{0}(x,1)=0$ for $x\in (0,1)$ and
by symmetry $P_{0}(0,\xi )=P_{0}(1,\xi )=0.$ Then, the feedback control $
\widetilde{F}\in L(D(A),\mathbb{R}),$
\begin{equation}
\widetilde{F}y=\int_{\Omega }y(\xi )\frac{\partial P_{0}}{\partial x}(1,\xi
)d\xi ,\mbox{ }y\in L^{2}(\Omega ) \label{82}
\end{equation}
solves the $H^{\infty }$-problem.
\end{theorem}
In this case
\begin{equation*}
\Lambda _{P}y=A_{0}\left( y+\int_{\Omega }\frac{\partial P_{0}}{\partial x}
(1,\xi )D(0,1)y(\xi )d\xi \right) +ay+\chi _{\omega _{1}}\int_{\Omega
}P_{0}(x,\xi )y(\xi )d\xi
\end{equation*}
which is closed, so that
\begin{equation*}
\mathcal{X}=D(\Lambda _{P})=\left\{ y\in L^{2}(\Omega );\mbox{ }
y+\int_{\Omega }\frac{\partial P_{0}}{\partial x}(1,\xi )D(0,1)y(\xi )d\xi
\in D(A)\right\} .
\end{equation*}
\noindent \textbf{Acknowledgment.} This work was supported by a grant of the Ministry of Research,
Innovation and Digitization, CNCS - UEFISCDI, project number
PN-III-P4-PCE-2021-0006, within PNCDI III.
\end{document} |
\begin{document}
\title[nearest points and DC functions]{nearest points and delta convex functions in Banach spaces}
\author{Jonathan M. Borwein \and Ohad Giladi}
\address{Centre for Computer-assisted Research Mathematics and its Applications (CARMA), School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia}
\email{[email protected], [email protected]}
\begin{abstract}
Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.
\end{abstract}
\subjclass[2010]{46B10. 41A29}
\maketitle
\section{Nearest points in Banach spaces}
\subsection{Background}\label{sec intro}
Let $(X, \|\cdot\|)$ be a real Banach space, and let $C\subseteq X$ be a non-empty closed set. Given $x\in X$, its distance from $C$ is given by
\[d_C(x) = \inf_{y\in C} \|x-y\|.\]
If there exists $z\in C$ with $d_C(x) = \|x-z\|$, we say that $x$ has a \emph{nearest point} in $C$. Let also
\[N(C) = \big\{x\in X: x \text{ has a nearest point in $C$ }\big\}.\]
One can then ask questions about the structure of the set $N(C)$. This question has been studied in \cite{Ste63, Lau78, Kon80, Zaj83, BF89, DMP91, Dud04, RZ11, RZ12} to name just a few. More specifically, the following questions are at the heart of this note:
\begin{center}
\emph{Given a nonempty closed set $C\subseteq X$, how large is the set $N(C)$? When is it non-empty?}
\end{center}
One way to do so is to consider sets which are large in the set theoretic sense, such as dense $G_{\delta}$ sets. We begin with a few definitions.
\begin{defin}
If $N(C) = X$, i.e., every point in $X$ has a nearest point in $C$, then $C$ is said to be proximinal. If $N(C)$ contains a dense $G_{\delta}$ set, then $C$ is said to be almost proximinal.
\end{defin}
In passing we recall that If every point in $X$ is uniquely proximinal then $C$ is said to be a \emph{Chebyshev set} It has been conjectured for over half a century, that in Hilbert space
Chebyshev sets are necessarily convex, but this is only proven for weakly closed sets \cite{BV10}. See also \cite{FM15} for a recent survey on the topic.
For example, closed convex sets in reflexive spaces are proximinal, as well as closed sets in finite dimensional spaces. See \cite{BF89}. One can also consider stronger notions of ``large" sets. See Section \ref{sec porous}. First, we also need the following definition.
\begin{defin}
A Banach space is said to be a (sequentially) Kadec space if for each sequence $\{x_n\}$ that converges weakly to $x$ with $\lim \|x_n\| = \|x\|$, $\{x_n\}$ converges to $x$ in norm, i.e.,
\[\lim_{n\to \infty} \|x-x_n\| =0.\]
\end{defin}
With the above definitions in hand, the following result holds.
\begin{thm}[Lau \cite{Lau78}, Borwein-Fitzpatrick \cite{BF89}]\label{thm Lau BF}
If $X$ is a reflexive Kadec space and $C\subseteq X$ is closed, then $C$ is almost proximinal.
\end{thm}
The assumptions on $X$ are in fact necessary.
\begin{thm}[Konjagin \cite{Kon80}]\label{thm Kon}
If $X$ is not both Kadec and reflexive, then there exist $C\subseteq X$ closed and $U\subseteq X\setminus C$ open such that no $x\in U$ has a nearest point in $C$.
\end{thm}
It is known that under stronger assumption on $X$ one can obtain stronger results on the set $N(C)$. See Section \ref{sec porous}.
\subsection{Fr\'echet sub-differentiability and nearest points}\label{sec diff}
We begin with a definition.
\begin{defin}
Assume that $f: X\to \mathbb R$ is a real valued function with $f(x)$ finite. Then $f$ is said to be Fr\'echet sub-differentiable at $x\in X$ if there exists $x^*\in X^*$ such that
\begin{align}\label{cond subdiff}
\liminf_{y\to 0}\frac{f(x+y)-f(x)-x^*(y)}{\|y\|} \ge 0.
\end{align}
The set of points in $X^*$ that satisfy \eqref{cond subdiff} is denoted by $\mathbb Partial f(x)$.
\end{defin}
Sub-derivatives have been found to have many applications in approximation theory. See for example \cite{BF89, BZ05, BL06, BV10, Pen13}.
One of the connections between sub-differentiability and the nearest point problem was studied in \cite{BF89}. Given $C\subseteq X$ closed, the following modification of a construction of \cite{Lau78} was introduced.
\begin{align*}
L_n(C) = \Big\{x\in X\setminus C : \exists x^*\in \mathbb S_{X^*} \text{ s.t. }\sup_{\delta>0}~\inf_{z\in C\cap B(x,d_C(x)+\delta)}~x^*(x-z) > \big(1-2^{-n}\big)d_C(x)\Big\},
\end{align*}
where $\mathbb S_{X^*}$ denotes the unit sphere of $X^*$. Also, let
\begin{align*}
L(C) = \bigcap_{n=1}^{\infty}L_n(C).
\end{align*}
The following is known.
\begin{prop}[Borwein-Fitzpatrick \cite{BF89}]\label{prop open}
For every $n\in \mathbb N$, $L_n(C)$ is open. In particular, $L(C)$ is $G_{\delta}$.
\end{prop}
Finally, let
\begin{align*}
\Omega(C) = & \Big\{ x\in X\setminus C : \exists x^*\in \mathbb S_{X^*}, \text{ s.t. } \forall \varepsilonilon >0, \exists \delta>0,
\\ & ~~\quad\inf_{z\in C\cap B(x,d_C(x)+\delta)}x^*(x-z) > \big(1-\varepsilonilon\big)d_C(x)\Big\}.
\end{align*}
While $L(C)$ is $G_{\delta}$ by Proposition \ref{prop open}, under the assumption that $X$ is reflexive, the following is known.
\begin{prop}[Borwein-Fitzpatrick \cite{BF89}]
If $X$ is reflexive then $\Omega(C) = L(C)$. In particular, $\Omega(C)$ is $G_{\delta}$.
\end{prop}
The connection to sub-differentiability is given in the following proposition.
\begin{prop}[Borwein-Fitzpatrick \cite{BF89}]
If $x\in X\setminus C$ and $\mathbb Partial d_C(x) \neq \emptyset$, then $x\in \Omega(C)$.
\end{prop}
Also, the following result is known.
\begin{thm}[Borwein-Preiss \cite{BP87}]\label{thm var}
If $f$ is lower semicontiuous on a reflexive Banach space, then $f$ is Fr\'echet sub-differentiable on a dense set.
\end{thm}
In fact, Theorem \ref{thm var} holds under a weaker assumption. See \cite{BP87, BF89}.
Since the distance function is lower semicontinuous, it follows that it is sub-differentiable on a dense subset, and therefore, by the above propositions, $\Omega(C)$ is a dense $G_{\delta}$ set. Thus, in order to prove Theorem \ref{thm Lau BF}, it is only left to show that every $x\in \Omega(C)$ has a nearest point in $C$. Indeed, if $\{z_n\}\subseteq C$ is a minimizing sequence, then by extracting a subsequence, assume that $\{z_n\}$ has a weak limit $z\in C$. By the definition of $\Omega(C)$, there exists $x^* \in \mathbb S_{X^*}$ such that
\[\|x-z\| \ge x^*(x-z) = \lim_{n\to \infty} x^*(x-z_n) \ge d_C(x) = \lim_{n\to \infty}\|x-z_n\|.\]
On the other hand, by weak lower semicontinuity of the norm,
\[\lim_{n\to \infty} \|x-z_n\| \ge \|x-z\|,\]
and so $\|x-z\| = \lim\|x-z_n\|$. Since it is known that $\{z_n\}$ converges weakly to $z$, the Kadec property implies that in fact $\{z_n\}$ converges in norm to $z$. Thus $z$ is a nearest point. This completes the proof of Theorem \ref{thm Lau BF}.
This scheme of proof from \cite{BF89} shows that differentiation arguments can be used to prove that $N(C)$ is large.
\subsection{Nearest points in non-Kadec spaces}\label{sec non-Kadec}
It was previously mentioned that closed convex sets in reflexive spaces are proximinal. It also known that non-empty ``Swiss cheese" sets (sets whose complement is a mutually disjoint union of open convex sets) in reflexive spaces are almost proximinal \cite{BF89}. These two examples show that for some classes of closed sets, the Kadec property can be removed. Moreover, one can consider another, weaker, way to ``measure" whether a set $C\subseteq X$ has ``many" nearest points: ask whether the set of nearest points in $C$ to points in $X\setminus C$ is dense in the boundary of $C$. Note that if $C$ is almost proximinal, then nearest points are dense in the boundary. The converse, however, is not true. In \cite{BF89} an example of a non-Kadec reflexive space was constructed where for every closed set, the set of nearest points is dense in its boundary. The following general question is still open.
\begin{qst}
Let $(X, \|\cdot\|)$ be a reflexive Banach space and $C\subseteq X$ closed. Is the set of nearest points in $C$ to points in $X\setminus C$ dense in its boundary?
\end{qst}
Relatedly, if the set $C$ is norm closed and bounded in a space with the \emph{Radon-Nikodym property} as is the caae of reflexive space, then $N(C)$ is nonempty and is large enough so that $\overline{\rm conv} C = \overline{\rm conv} N(C)$ \cite{BF89}.
\subsection{Porosity and nearest points}\label{sec porous}
As was mentioned in subsection \ref{sec diff}, one can consider stronger notions of ``large" sets. One is the following notion.
\begin{defin}
A set $S\subseteq X$ is said to be porous if there exists $c\in (0,1)$ such that for every $x\in X$ and every $\varepsilonilon>0$, there is a $y \in B(0,\varepsilonilon)\setminus \{0\}$ such that
\[ B(x+y, c\|y\|) \cap S = \emptyset.\]
A set is said to be $\sigma$-porous if it a countable union of porous sets. Here and in what follows, $B(x,r)$ denotes the closed ball around $x$ with radius $r$.
\end{defin}
See \cite{Zaj05, LPT12} for a more detailed discussion on porous sets. It is known that every $\sigma$-porous set is of the first category, i.e., union of nowhere dense set. Moreover, it is known that the class of $\sigma$-porous sets is a proper sub-class of the class of first category sets. When $X=\mathbb R^n$, one can show that every $\sigma$-porous set has Lebesgue measure zero. This is not the case for every first category set: $\mathbb R$ can be written as a disjoint union of a set of the first category and a set of Lebesgue measure zero. Hence, the notion of porosity automatically gives a stronger notion of large sets: every set whose complement is $\sigma$-porous is also a dense $G_{\delta}$ set.
A Banach space $(X, \|\cdot\|)$ is said to be \emph{uniformly convex} if the function
\begin{align}\label{def uni conv}
\delta(\varepsilonilon) = \inf\left\{ 1-\left\|\frac{x+y}2\right\| ~ : ~ x, y\in \mathbb S_X, \|x-y\| \ge \varepsilonilon \right\},
\end{align}
is strictly positive whenever $\varepsilonilon>0$. Here $\mathbb S_X$ denotes the unit sphere of $X$. In \cite{DMP91} the following was shown.
\begin{thm}[De Blasi-Myjak-Papini \cite{DMP91}]\label{thm DMP}
If $X$ is uniformly convex, then $N(C)$ has a $\sigma$-porous compliment.
\end{thm}
In fact, \cite{DMP91} proved a stronger result, namely that for every $x$ outside a $\sigma$-porous set, the minimization problem is \emph{well posed}, i.e., there is unique minimizer to which every minimizing sequence converges. See also \cite{FP91, RZ11, RZ12} for closely related results in this direction.
The proof of Theorem \ref{thm DMP} builds on ideas developed in \cite{Ste63}. However, it would be interesting to know whether one could use differentiation arguments as in Section \ref{sec diff}. This raises the following question:
\begin{qst}
Can differentiation arguments be used to give an alternative proof of Theorem \ref{thm DMP}?
\end{qst}
More specifically, if one can show that $\mathbb Partial d_C \neq \emptyset$ outside a $\sigma$-porous set, then by the arguments presented in Section \ref{sec diff}, it would follow that $N(C)$ has a $\sigma$-porous complement. Next, we mention two important results regarding differentiation in Banach spaces.
\begin{thm}[Preiss-Zaj\'i\v{c}ek \cite{PZ84}]\label{thm PZ}
If $X$ has a separable dual and $f:X\to \mathbb R$ is continuous and convex, then $X$ is Fr\'echet differentiable outside a $\sigma$-porous set.
\end{thm}
See also \cite[Sec. 3.3]{LPT12}. Theorem \ref{thm PZ} implies that if, for example, $d_C$ is a linear combination of convex functions (see more on this in Section \ref{sec DC}), then $N(C)$ has a $\sigma$-porous complement. Also, we have the following.
\begin{thm}[C\'uth-Rmoutil \cite{CR13}]\label{thm CR}
If $X$ has a separable dual and $f:X\to \mathbb R$ is Lipschitz, then the set of points where $f$ is Fr\'echet sub-differentiable but not differentiable is $\sigma$-porous.
\end{thm}
Since $d_C$ is 1-Lipschitz, the questions of seeking points of sub-differentiability or points of differentiability are similar. Theorem \ref{thm PZ} and Theorem \ref{thm CR} remain true if we consider $f:A\to \mathbb R$ where $A\subseteq X$ is open and convex.
\section{DC functions and DC sets}\label{sec DC}
\subsection{Background}
\begin{defin}
A function $f:X\to \mathbb R$ is said to be delta-convex, or DC, if it can be written as a difference of two convex functions on $X$.
\end{defin}
This notion was introduced in \cite{Har59} and was later studied by many authors. See for example \cite{KM90, Cep98, Dud01, VZ01, DVZ03, BZ05, Pav05, BB11}. In particular, \cite{BB11} gives a good introduction to this topic. We will discuss here only the parts that are closely related to the nearest point problem.
The following is an important proposition. See for example \cite{VZ89, HPT00} for a proof.
\begin{prop}\label{prop select}
If $f_1,\dots,f_k$ are DC functions and $f:X\to \mathbb R$ is continuous and $f(x)\in \big\{f_1(x),\dots,f_n(x)\big\}$. Then $f$ is also DC.
\end{prop}
The result is true if we replace the domain $X$ by any convex subset.
\subsection{DC functions and nearest points}
Showing that a given function is in fact DC is a powerful tool, as it allows us to use many known results about convex and DC functions. For example, if a function is DC on a Banach space with a separable dual, then by Theorem \ref{thm PZ}, it is differentiable outside a $\sigma$-porous set. In the context of the nearest point problem, if we know that the distance function is DC, then using the scheme presented in Section \ref{sec diff}, it would follow that $N(C)$ has a $\sigma$-porous complement. The same holds if we have a difference of a convex function and, say, a smooth function.
The simplest and best known example is when $(X,\|\cdot\|)$ is a Hilbert space, where we have the following.
\begin{align*}
d_C^2(x) & = \inf_{y\in C}\|x-y\|^2
\\ & = \inf_{y\in C}\Big[\|x\|^2-2\langle x,y\rangle +\|y\|^2\Big]
\\ & = \|x\|^2 - 2\sup_{y\in C}\Big[\langle x,y\rangle - \|y\|^2/2\Big],
\end{align*}
and the function $x\mapsto \sup_{y\in C}\Big[\langle x,y\rangle - \|y\|^2/2\Big]$ is convex as a supremum of affine functions. Hence $d_C^2$ is DC on $X$.
Moreover, in a Hilbert space we have the following result (see \cite[Sec. 5.3]{BZ05}).
\begin{thm}\label{thm local DC}
If $(X, \|\cdot\|)$ is a Hilbert space, $d_C$ is locally DC on $X\setminus C$.
\end{thm}
\begin{proof}
Fix $y\in C$ and $x_0\in X\setminus C$. It can be shown that if we let $f_y(x) = \|x-y\|$, then $f_y$ satisfies
\begin{align*}
\big\|f_y'(x_1)-f_y'(x_2)\big\|_{X^*} \le L_{x_0}\|x_1-x_2\|, ~~ x_1,x_2 \in B_{x_0},
\end{align*}
where $L_{x_0} = \frac 4 {d_S(x_0)}$ and $B_{x_0} = B\Big(x_0, \frac 1 2 d_C(x_0)\Big)$. In particular,
\begin{align}\label{lip prop}
\big(f_y'(x+tv_1)-f_y'(x+t_2v)\big)(v) \le L_{x_0}(t_2-t_1), ~~ v\in \mathbb S_X, t_2> t_1 \ge 0,
\end{align}
whenever $x+t_1v, x+t_2v \in B_{x_0}$. Next, the convex function $F(x) = \frac {L_{x_0}} 2 \|x\|^2$ satisfies
\begin{align}\label{anti lip hilbert}
\big(F'(x_1)-F'(x_2)\big)(x_1-x_2) \ge L_{x_0}\|x_1-x_2\|^2, ~~ \forall x_1,x_2\in X.
\end{align}
In particular
\begin{align}\label{anti lip}
\big(F'(x+t_2v)-F'(x+t_1v)\big)(v) \ge L_{x_0}(t_2-t_1), ~~ v\in \mathbb S_X, ~t_2>t_1\ge 0.
\end{align}
Altogether, if $g_y(x) = F(x)-f_y(x)$, then
\begin{align*}
\big(g_y'(x+t_2v)-g_y'(x+t_1v)\big)(v) \stackrel{\eqref{lip prop}\wedge \eqref{anti lip}}{\ge} 0, ~~ v\in \mathbb S_X, ~t_2>t_1\ge 0,
\end{align*}
whenever $x+t_1v, x+t_2v \in B_{x_0}$. This implies that $g_y$ is convex on $B_{x_0}$. It then follows that
\begin{align*}
d_C(x) & = \frac {L_{x_0}} 2 \|x\|^2 -\sup_{y\in C}\Bigg[~\frac {L_{x_0}} 2 \|x\|^2-\|x-y\|\Bigg] = h(x) - \sup_{y\in C}g_y(x)
\end{align*}
is DC on $B_{x_0}$.
\end{proof}
\begin{remark}
Even in $\mathbb R^2$ there are sets for which $d_C$ is not DC everywhere (not even locally DC), as was shown in \cite{BB11}. Thus, the most one could hope for is a locally DC function on $X\setminus C$.
\end{remark}
Given $q\in (0,1]$, a norm $\|\cdot\|$ is said to be $q$\emph{-H\"older smooth} at a point $x\in X$ if there exists a constant $K_x\in (0,\infty)$ such that for every $y\in \mathbb S_X$ and every $\tau>0$,
\begin{align*}
\frac{\|x+\tau y\|}{2} + \frac{\|x-\tau y\|}{2} \le 1+K_x\tau^{1+q}.
\end{align*}
If $q=1$ then $(X,\|\cdot\|)$ is said to be \emph{Lipschitz smooth} at $x$.
The spaces $L_p$, $p \ge 2$ are known to be Lipschitz smooth, and in general $L_p$, $p>1$, is $s$-H\"older smooth with $s = \min\{1,p-1\}$.
A Banach space is said to be $p$\emph{-uniformly convex} if for every $x,y\in \mathbb S_X$,
\begin{align*}
1-\left\|\frac{x+y}{2}\right\| \ge L\|x-y\|^p.
\end{align*}
Note that this is similar to assuming that $\delta(\varepsilonilon) = L\varepsilonilon^p$ in \eqref{def uni conv}. The spaces $L_p$, $p>1$, are $r$-uniformly convex with $r = \max\{2,p\}$.
One could ask whether the scheme of proof of Theorem \ref{thm local DC} can be used in a more general setting.
\begin{prop}\label{prop Hilbert}
Let $(X, \|\cdot\|)$ be a Banach space, $C\subseteq X$ a closed set, and fix $x_0\in X\setminus C$ and $y\in C$. Assume that there exists $r_0$ such that $f_y(x) = \|x-y\|$ has a Lipschitz derivative on $B(x_0,r_0)$:
\begin{align}\label{lip of der}
\big\|f_y'(x_1)-f_y'(x_2)\| \le L_{x_0}\|x_1-x_2\|.
\end{align}
Then the norm is Lipschitz smooth on $-y + B_{x_0} = B(x_0-y, r_0)$. If in addition there exists a function $F: X \to \mathbb R$ satisfying
\begin{align}\label{strong conv}
\big(F'(x_1)-F'(x_2)\big)(x_1-x_2) \ge L_{x_0}\|x_1-x_2\|^2, ~~ \forall x_1,x_2\in B(x_0,r_0),
\end{align}
then $(X,\|\cdot\|)$ admits an equivalent norm which is 2-uniformly convex. In particular, if $X=L_p$ then $p=2$.
\end{prop}
\begin{proof}
To prove the first assertion note that \eqref{lip of der} is equivalent to
\begin{align*}
\|x-y+h\|+\|x-y-h\|-2\|x-y\| \le L_{x_0}\|h\|^2, ~~ x\in B_{x_0}.
\end{align*}
See for example \cite[Prop. 2.1]{Fab85}.
To prove the second assertion, note that a function that satisfies \eqref{strong conv} is also known as \emph{strongly convex}: one can show that \eqref{strong conv} is in fact equivalent to the condition
\begin{align*}
f\left(\frac{x_1+x_2}{2}\right) \le \frac 1 2f(x_1)+\frac 1 2 f(x_2) - C\|x_1-x_2\|^2,
\end{align*}
for some constant $C$. See for example \cite[App. A]{SS07}. This implies that there exists an equivalent norm which is 2-uniformly convex (\cite[Thm 5.4.3]{BV10}).
\end{proof}
\begin{remark}
From \cite{Ara88} it is know that if $F:X\to \mathbb R$ satisfies
\begin{align*}
\big(F'(x_1)-F'(x_2)\big)(v) \ge L\|x_1-x_2\|^2,
\end{align*}
for \emph{all} $x_1,x_2\in X$, and also that $F$ is twice (Fr\'echet) differentiable at one point, then $(X,\|\cdot\|)$ is isomorphic to a Hilbert space.
\end{remark}
\begin{remark}
If we replace the Lipschitz condition by a H\"older condition
\begin{align*}
\big\|f_y'(x_1)-f_y'(x_2)\big\| \le \|x_1-x_2\|^{\beta}, ~~ \beta <1,
\end{align*}
then in order to follow the same scheme of proof of Theorem \ref{thm local DC}, instead of \eqref{anti lip hilbert}, we would need a function $F$ satisfying
\begin{align*}
\big(F'(x_1)-F'(x_2)\big)(x_1-x_2) \ge \|x_1-x_2\|^{1+\beta}, ~~ x_1,x_2\in B_{x_0}.
\end{align*}
which implies
\begin{align}\label{anti holder}
\big\|F'(x_1)-F'(x_2)\big\| \ge \|x_1-x_2\|^{\beta}, ~~ x_1,x_2\in B_{x_0}.
\end{align}
If $G= (F')^{-1}$, then we get
\begin{align*}
\|Gx_1-Gx_2\| \le \|x_1-x_2\|^{1/\beta}, ~~ x_1,x_2 \in F'(B_{x_0}),
\end{align*}
which can occur only if $G$ is a constant. Hence \eqref{anti holder} cannot hold and the scheme of proof cannot be used if we replace the Lipschitz condition by a H\"older condition.
\end{remark}
\subsection{DC sets, DC representable sets}
\begin{defin}
A set $C$ is is said to be a DC set if $C=A\setminus B$ where $A,B$ are convex.
\end{defin}
We can also consider the following class of sets.
\begin{defin}
A set $C\subseteq X$ is said to be DC representable if there exists a DC function $f:X\to R$ such that $C = \big\{x\in X: f(x) \le 0\big\}$.
\end{defin}
Note that if $C = A\setminus B$ is a DC set, then we can write $C = \Big\{\mathbbm 1_{B}-\mathbbm 1_{A}+1/2 \le 0\Big\}$, where $\mathbbm 1_A$, $\mathbbm 1_B$ are the indicator functions of $A,B,$ respectively. Therefore, $C$ is DC representable. Moreover, we have the following.
\begin{thm}[Thach \cite{Th93}]
Assume that $X$ and $Y$ are two Banach space, and $T:Y\to X$ is surjective map with $\mathrm{ker}(T) \neq \emptyset$. Then for any set $M\subseteq X$ there exists a DC representable set $D\subseteq Y$, such that $M= T(D)$.
\end{thm}
Also, the following is known. See \cite{HPT00}.
\begin{prop}
If $C$ is a DC representable set, then there exist $A,B \subseteq X\oplus \mathbb R$ convex, such that $x\in C \iff (x,x') \in A\setminus B$.
\end{prop}
\begin{proof}
Define $g_1(x,x') = f_1(x)-x'$, $g_2(x,x') = f_2(x)-x'$. Let $A = \big\{(x,x') : g_1(x,x')\le 0\big\}$, $B = \big\{(x,x'): g_2(x,x') \le 0\big\}$. Then $x\in C \iff (x,x') \in A\setminus B$.
\end{proof}
In particular, every DC representable set in $X$ is a projection of a DC set in $X\oplus \mathbb R$. The following theorem was proved in \cite{TK96}
\begin{thm}[Thach-Konno \cite{TK96}]
If $X$ is a reflexive Banach space and $C\subseteq X$ is closed, then $C$ is DC representable.
\end{thm}
This raises the following question.
\begin{qst}\label{qst DC}
Is it true that for some classes of spaces, e.g. uniformly convex spaces, there exists $\alpha>0$ such that $d_C^{\alpha}$ is locally DC on $X\setminus C$ whenever $C$ is a DC representable set?
\end{qst}
If the answer to Question \ref{qst DC} is positive, then by the discussion in subsection \ref{sec diff} we could conclude that $N(C)$ has a $\sigma$-porous complement, thus giving an alternative proof of Theorem \ref{thm DMP}. One could also ask Question \ref{qst DC} for DC sets instead of DC representable sets.
To end this note, we discuss some simple cases where DC and DC representable sets can be used to study the nearest point problem.
\begin{prop}
Assume that $C = X \setminus \bigcup_{a\in \Lambda}U_a$, where each $U_a$ is an open convex set. Then $d_C$ is locally DC (in fact, locally concave) on $X\setminus C$.
\end{prop}
\begin{proof}
First, it is shown in \cite[Sec. 3]{BF89} that if $a\in \Lambda$, then $d_{X\setminus U_a}$ is concave on $U_a$. Next, it also shown in \cite{BF89} that if $x\in U_a$ then $d_{X\setminus U_a}(x) = d_C(x)$. In particular, $d_C$ is concave on $U_a$.
\end{proof}
\begin{prop}
Assume that $C = A\setminus B$ is a closed DC set, and assume $A$ is closed and $B$ is open, then $d_C$ is convex whenever $d_{C}(x) \le d_{A\cap B}$.
\end{prop}
\begin{proof}
Since $A = \big(A\setminus B\big) \bigcup B$, we have
$$d_A(x) = \min\big\{d_{A\setminus B}(x), d_{A\cap B}(x)\big\} = \min\big\{d_{C}(x), d_{A\cap B}(x)\big\}.$$
Hence, if $d_C(x) \le d_{A\cap B}(x)$ then $d_C(x) = d_A(x)$ is convex.
\end{proof}
\begin{prop}
Assume that $C$ is a DC representable set, i.e., $C = \big\{x\in X: f_1(x)-f_2(x)\le 0\big\}$, and that $f_2(x) =\max_{1\le i \le m}\varphi_i(x)$, where $\varphi_i$ is affine. Then $d_C$ is DC on $X$.
\end{prop}
\begin{proof}
Write
\begin{align*}
C & = \Big\{x : f_1(x)-f_2(x)\le 0\Big\}
\\ &= \Big\{x : f_1(x)-\max_{1\le i \le m}\varphi_i(x) \le 0\Big\}
\\ & = \Big\{x : \min_{1\le i \le m}\big(f_1(x)-\varphi_i(x)\big) \le 0\Big\}
\\ & = \bigcup_{i=1}^n \Big\{x : f_1(x)-\varphi_i(x) \le 0\Big\}.
\end{align*}
where the sets $\Big\{x~ : ~ f_1(x)-\varphi_i(x) \le 0\Big\}$ are convex sets. Hence, we have that
\[d_C(x) = \min_{1\le i \le m}d_{C_i}(x),\]
is a minimum of convex sets and therefore by Proposition \ref{prop select} is a DC function.
\end{proof}
In \cite{Cep98} it was shown that if $X$ is superreflexive, then any Lipschitz map is a uniform limit of DC functions. See also \cite[Sec. 5.1]{BV10}. We have the following simple result.
\begin{prop}
If $X$ is separable, then $d_C$ is a limit (not necessarily uniform) of DC functions.
\end{prop}
\begin{proof}
If $X$ is separable, i.e., there exists a countable $Q = \{q_1,q_2,\dots\}\subseteq X$ with $\bar Q = X$. We have
\begin{align*}
d_C(x) = \inf_{z\in C}\|x-z\| = \inf_{z\in C\cap Q}\|x-z\| = \lim_{n\to \infty} \Big[\min_{z\in C\cap Q_n}\|x-z\|\Big],
\end{align*}
where $Q_n = \{q_1,q_2,\dots,q_n\}$. Again by Proposition \ref{prop select} we have that $\min_{z\in C\cap Q_n}\|x-z\|$ is a DC function as a minimum of convex functions.
\end{proof}
\section{Conclusion}
Despite many decades of study, the core questions addressed in this note are still far from settled. We hope that our analysis will encourage others to take up the quest, and also to reconsider the related \emph{Chebshev problem} \cite{B07,BV10}.
\end{document} |
\begin{document}
\title{The EPR Argument \\
in a Relational Interpretation \\
of Quantum Mechanics}
\author{Federico Laudisa\\
Department of Philosophy, University of Florence, \\
Via Bolognese 52, 50139 Florence, Italy}
\maketitle
\date{}
\begin{abstract}
\noindent
It is shown that in the Rovelli {\it relational} interpretation of
quantum mechanics, in which the notion of absolute
or observer independent state is rejected, the conclusion of the ordinary
EPR argument turns out to be frame-dependent, provided the conditions
of the original argument are suitably adapted to the new interpretation.
The consequences of this result for the `peaceful coexistence' of quantum mechanics and
special relativity are briefly discussed.
\end{abstract}
\section{Introduction}
The controversial nature of observation in quantum mechanics has been at the
heart of the debates on the foundations of the theory since its early days.
Unlike the situation in classical theories, the problem of how and where
we should localize the boundary between systems that observe and systems that
are observed is not merely a practical one, nor has there been a widespread
consensus on whether there is some fundamental difference between the
two classes of systems. The emphasis on the reference to an appropriate
observational context, in order for most properties of quantum mechanical
systems to be meaningful, has been for example the focus of Bohr's reflections
on the foundations of quantum mechanics, and the attention to this
level of description has been inherited to a certain extent even by those
interpretations of the theory that urged to go well beyond the Copenhagen
standpoint.
Although these controversies deal primarily with
the long-studied measurement problem, we are naturally led to ask ourselves
whether a deeper emphasis on the role of the observer might suggest new
directions also about nonlocality.
The celebrated argument of Einstein, Podolsky and Rosen (EPR) for the incompleteness
of quantum mechanics turns essentially on the possibility for an observer of predicting
the result of a measurement performed in a spacetime region that is supposed
to be isolated from the
region where the observer is localized. The quantum mechanical description of a typical
EPR state, however, prevents from conceiving that result as simply revealing a
{\it preexisting} property, so that the upshot of the
argument is the alternative between completeness and locality: by assuming the
former we must give up the latter, and viceversa.
\footnote{Clearly both options are available here since we restrict
our attention to the EPR argument. But after the Bell theorem, it is a
widespread opinion that the only viable option for ordinary quantum mechanics
is in fact the first.}
It then turns out rather natural to ask whether, and to what extent,
taking into due account the fact that quantum predictions are the
predictions {\it of a given observer} affects remarkably the structure and
the significance of the argument.
The ordinary EPR argument is formulated in nonrelativistic quantum mechanics,
whose symmetries constitute a Galilei semigroup.
Therefore, an obvious form of observer dependence that must be taken into account
is the frame dependence that must enter into the description when the whole
framework of the EPR argument is embedded into the spacetime of special
relativity. It can be shown that a relativistic EPR argument still works and,
as a consequence, events pertaining to a quantum system may `depend' on events pertaining
a different quantum system localized in a space-like separated region.
\footnote{Clearly there is no consensus on what a reasonable interpretation of this
`dependence' might be, but a thorough discussion of this point is beyond the scope
of the present paper.}
However, in the relativistic EPR argument, a special attention
must be paid to the limitations placed by this generalization to the attribution of
properties to subsystems of a composite system:
the lesson of a relativistic treatment of the EPR argument lies
in the caution one must take when making assumptions on what an observer
knows about the class of quantum mechanical events taking place in
the absolute elsewhere of
the observer himself. Such observer dependence is essentially linked to
the space-like separation between the regions of the measurements and
prevents from using a property-attributing language without qualifications;
an analysis of these limitations shows that the relativistic EPR argument does not in fact support
the widespread claim that nonlocality involves an {\it instantaneous} creation of properties
at a distance.
\footnote{See (\cite{GG}, \cite{Ghirardi}); for an analysis of the
relevance of this fact to the status of superluminal causation
in quantum mechanics, see \cite{Laudisa}.}
It must be stressed that this argument implies by itself no definite
position about the existence of such influences {\it in the physical
world}: it involves only the {\it logical} compatibility
between the idea of action-at-a-distance and the special relativistic
account of the spacetime structure. More generally, what the argument
is meant to point out is the necessity of a
shift to the relativistic regime, in order
to rigorously assess whether some frequently stated claims about
the metaphysical consequences of
the EPR argument are really consistent with spacetime physics.
But according to a recent {\it relational} interpretation
of quantum mechanics, advanced by Carlo Rovelli (\cite{Rov}), we need not shift to
a fully relativistic quantum theory to find a fundamental form of observer dependence.
In this interpretation, the very notion of state of a physical system
should be considered meaningless unless it is not understood as relative to another
physical system, that plays temporarily the role of observer: when dealing
with concrete physical systems, it is the specification of such observer
that allows the ascription of a state to a system to make sense, so that,
to a certain extent, by a relational point of view the selection of such observers features in
quantum theory as the specification of a frame of reference does in relativity theory.
In the following section the relational interpretation will be
sketched, whereas in section 3 a relational analysis of the
(relativistic) EPR argument will show that whether the ordinary conclusion of the
argument - quantum mechanics is either incomplete or nonlocal - holds or not with respect
to a given observer depends on the frame of reference of the latter.
In the final section we will briefly
discuss the consequences of this analysis for the so called `peaceful
coexistence thesis' concerning the relation between quantum theory and
relativity theory.
\section{Relational quantum mechanics and the observer dependence of states}
In the relational interpretation of quantum mechanics, the relativistic frame dependence
of an observer's predictions is not the only source of observer dependence in quantum mechanics:
a fundamental form of observer dependence is detected already
in {\it non-relativistic} quantum mechanics and that concerns the very
definability of physical state. In the relational
interpretation, the notion of absolute or
observer independent state of a system is rejected: it would make no sense
to talk about a state of a physical system
without referring to an observer, with respect to which that state is defined
(\cite{Rov}). Although the analysis of its assumptions and
its consequences is developed within non-relativistic quantum mechanics,
this claim is somehow reminiscent of the Einsteinian operational critique
of the absolute notion of simultaneity for distant observers, and the main
idea underlying the interpretation is put forward by analyzing the different
accounts that two observers give of the same sequence of events in a typical
quantum measurement process.
Let us consider a system $S$ and a physical quantity
$Q$ that can be measured on $S$. We assume that the possible measurement results are two,
$q_1$ and $q_2$ (for simplicity the spectrum of $Q$ is assumed to be simple
and non-degenerate). The premeasurement state of $S$ at a time $t_1$
can then be written as $\alpha_1\phi_{q_1}+\alpha_2\phi_{q_2},$
with $\alpha ,\beta$ complex numbers
such that $\vert\alpha_1\vert^2+\vert\alpha_2\vert^2=1.$ If we suppose
that a measurement of $Q$ is performed and the measurement result is $q_2,$ according to ordinary
quantum mechanics, at a postmeasurement time $t_2$ the state of $S$ is given
by $\phi_{q_2}.$ Let $O$ be the observer performing
the measurement; the sequence that $O$ observes is then
\begin{equation}
\underbrace{\alpha_1\phi_{q_1}+\alpha_2\phi_{q_2}}_{t_1}\,\,
\Longrightarrow\,\,\,\underbrace{\phi_{q_2}}_{t_2}
\label{eq:sequence-O}
\end{equation}
\noindent
Let us now consider how a second observer $O'$ might describe this same
measurement process, concerning in this case the {\it composite} system
$S+O.$ We denote by $\psi_{init}$ the premeasurement state of $O$ and
by $\psi_{q_1}$ and $\psi_{q_2}$ respectively the
eigenstates of the pointer observable (namely the
states that correspond to recording the $Q$-measurement results
$q_1,q_2$). The premeasurement
state of $S+O$ at $t_1,$ belonging to the tensor product Hilbert space
$\cal{H}_S\otimes\cal{H}_O$
of the Hilbert spaces $\cal{H}_S$ and $\cal{H}_O$ associated to
$S$ and $O$ respectively, is then expressed as
\begin{equation}
\psi_{init}\otimes(\alpha_1\phi_{q_1}+\alpha_2\phi_{q_2});
\label{eq:S+O-t1}
\end{equation}
if $O'$ performs no measurement, by linearity (\ref{eq:S+O-t1}) at $t_2$ becomes
\begin{equation}
\alpha_1\psi_{q_1}\otimes\phi_{q_1}+\alpha_2\psi_{q_2}\otimes\phi_{q_2}.
\label{eq:S+O-t2}
\end{equation}
The measurement process, as described by $O',$ is then given by the sequence
\begin{equation}
\underbrace{\psi_{init}\otimes(\alpha_1\phi_{q_1}+\alpha_2\phi_{q_2})}_{t_1}
\,\,
\Longrightarrow
\,\,
\underbrace{\alpha_1\psi_{q_1}\otimes\phi_{q_1}+\alpha_2\psi_{q_2}\otimes\phi_{q_2}}_{t_2}.
\label{eq:sequence-O'}
\end{equation}
\noindent
So far $O'$ may only claim that the states of $S$ and the states of $O$ are suitably correlated.
But let us suppose now that $O'$, at time $t_3>t_2$ performs on $S$ a measurement of $Q$. Since
we are dealing with the two different observers $O$ and $O'$, it is natural
to ask what general consistency condition we should assume to hold for the different
descriptions that $O$ and $O'$ might give of the measurement process.
In the present situation, such a condition can be the following:
if $S$ at a time $t$ is in an eigenstate $\psi_q$ of an observable $Q$ relative
to $O$, the observer $O'$ that measures $Q$ on $S$ at $t'>t$ (with no
intermediate measurements, between $t$ and $t'$, of observables that are not compatible with
$Q$) will find the eigenvalue belonging to $\psi_q$: therefore
$S$ will be in the state $\psi_q$ also relative to $O'$ (we will return later to the general
form that this consistency condition assumes).
Therefore, the state of $S+O$ at $t_3$ relative to $O'$ will be $\psi_{q_2}\otimes\phi_{q_2}\,,$ since
at $t_2$ the state of $S$ relative to $O$ had been reduced to $\phi_{q_2}.$
If we now compare (\ref{eq:sequence-O}) and (\ref{eq:sequence-O'}), we see that
$O$ and $O'$ give a different account of the same sequence of events
in the above described measurement: at $t_2$ $O$ attributes to $S$
the state $\phi_{q_2},$ whereas $O',$ that views $S$ as a subsystem of
$S+O,$ attributes to $S$ the state $\alpha_1\phi_{q_1}+\alpha_2\phi_{q_2}.$
The most general assumption of the relational interpretation of
quantum mechanics can be then summarized as follows: the
circumstance that there are different ways in which even a simple
quantum mechanical process like the measurement above can be
described by different observers suggests then that this
relational aspect might be not an accident, but a fundamental
property of quantum mechanics. In addition to this, the relational interpretation makes
a pair of further assumptions,
concerning respectively the universality and the completeness of quantum mechanics.
\noindent
1. {\it All physical systems are equivalent.}
\noindent
No specific assumption is made concerning the systems that are supposed to act as observers,
except that they must satisfy the laws of quantum mechanics: being observer is not a
property fixed once and for all for a privileged class of physical systems, permanently
identifiable as `observation systems' and clearly separable from the rest of physical systems
(\cite{Rov}, p.1644), nor is it implicitly assumed that such observation systems are
conscious entities.
\noindent
2. {\it Relational quantum mechanics is a complete physical theory.}
\noindent
The general circumstance that different observers may give different
accounts of the same processes is no sign of any fundamental incompleteness
of quantum mechanics, but is simply the consequence of a relational
meta-assumption, according to which there is no `absolute' or `from-outside'
point of view from which we might evaluate the states of a physical system or
the value of quantities measurable on that system in those states.
``Quantum mechanics can therefore be viewed as a theory
about the states of systems and values of physical quantities relative to
other systems. [...] If the notion of observer-independent description
of the world is unphysical, a complete description of the world is exhausted
by the relevant information that systems have about each other.'' (\cite{Rov}, p. 1650).
One might be tempted to describe the situation above simply by
saying that the difference between $O$ and $O'$ is that $O$ knows at $t_2$ what the state of $S$
is whereas $O'$ does not, and it is for this reason that $O'$ attributes to $S$ a superposition
state, namely an informationally `incomplete' state. The problem with this position, however,
is that it implicitly assumes the `absolute' viewpoint on states of physical systems, namely
just the viewpoint that the relational interpretation urges to reject as implausible.
Moreover, on the basis of the above account, one might draw a general distinction
between
a {\it description} and an {\it observation} of a system $S$ by an observer $O$:
in the former case, a `description' of $S$ involves no interaction between
$O$ and $S$ {\it at the time in which $O$ describes $S,$} although it is
still necessarily based on some {\it prior} interaction between $S$ and
other systems.
On the other hand, we may say that $O$ `observes' $S$ when $O$ actually
measures some relevant physical quantity on $S$: it is clear that,
in this case, there is an interaction between $S$ and $O$ that occurs
exactly when $O$ is said to `observe' $S.$
If we return to our specific example, we might consider the
`description' that $O'$ gives of $S$ {\it at a given $t$} in terms of
correlation properties of the system $O+S$ as the maximal amount of information on
the measurement process involving $S$ and $O$ that is available to $O'$
in absence of interaction {\it at $t$} between $O'$ and the composite system
$O+S.$ So let us recall the sequence (\ref{eq:sequence-O'}) of events.
$O'$ is supposed to perform no measurement in the time interval $[t_1,t_2]$
and thus can `describe' the state of $S$ at $t_2$ only through some observable
$C_{(O,S)}$ (defined on $\cal{H}_S\otimes\cal{H}_O$), that tests whether $O$ has
correctly recorded the result of
the measurement on $S.$ The eigenvalues of $C_{(O,S)}$ are simply $1$ and $0$.
The states $\psi_{q_1}\otimes\phi_{q_1},$ $\psi_{q_2}\otimes\phi_{q_2}$
turn out to be eigenstates belonging to the eigenvalue $1$ - the
record was correct - whereas the states
$\psi_{q_1}\otimes\phi_{q_2},$ $\psi_{q_2}\otimes\phi_{q_1}$
turn out to be eigenstates belonging to the eigenvalue $0$ - the
record was incorrect, namely
\begin{eqnarray*}
C_{(O,S)}\,\psi_{q_1}\otimes\phi_{q_1}=\psi_{q_1}\otimes\phi_{q_1}, & & C_{(O,S)}\psi_{q_1}\otimes\phi_{q_2} = 0\\
C_{(O,S)}\,\psi_{q_2}\otimes\phi_{q_2}=\psi_{q_2}\otimes\phi_{q_2}, & & C_{(O,S)}\psi_{q_2}\otimes\phi_{q_1}=0.\\
\end{eqnarray*}
On the basis of the above account, the consistency
requirement that we mentioned earlier, and that concerns the relation between
different descriptions of the same event given by different
observers, appears rather natural: it can be expressed as the requirement that if
the only information available to $O'$ is that $O$ has measured $Q$ on $S$
but the result is unknown, the results that $O'$ obtains by performing
a $C_{(O,S)}$-measurement and a $Q$-measurement must be correlated (\cite{Rov}, pp. 1650-2).
\section{A relational analysis of the EPR argument}
An ideal place to look at to see how a relational approach might change the
`absolute' view of quantum mechanical states is just
the framework in which the EPR argument is usually
developed (\cite{EPR}, \cite{Bohm}).
The physical framework common to all variants of this argument involves
a two-particles' system, whose subsystems interact for
a short time and then separate.
The original formulation of the EPR argument takes into account
a pair of quantities for each
particle, such in a way that the members of each pair are
mutually incompatible.
We will take into account the usual non-relativistic Bohm version of
the original argument, formulated as a spin correlation experiment,
and in a simplified form that deals with just one quantity for each particle (\cite{Redhead}).
The assumptions of the argument will be slightly rephrased as compared to the widespread
formulation, but this rephrasing does not substantially affect the argument.
\begin{enumerate}
\item {\sc Reality}
\noindent
If, without interacting with a physical system $S,$ we can predict
with certainty - or with probability one - the result $q$ of
the measurement of a quantity $Q$ performed at time $t$ on
$S,$ then at a time $t'$ immediately after $t,$ there exists a
property - associated with $q$ and denoted by $[q]$ - that is actually
satisfied by $S$: any such property is said to be an {\it objective} property
of $S.$
\item {\sc Completeness}
\noindent
Any physical theory $T$ describing a physical system $S$ accounts for
every objective property of $S.$
\item {\sc Locality}
\noindent
No objective property of a physical system $S$ in a state $s$
can be influenced by measurements performed at a distance on a different
physical system.
\item {\sc Adequacy}
\noindent
The statistical predictions of quantum mechanics are correct.
\end{enumerate}
\noindent
A word of comment on the formulation of the Reality condition is in order.
As is expected, the notion of an objective property $[q]$ of
a physical system $S$ is equivalent to the notion of $S$ having $[q]$
no matter whether $Q$ is measured on $S$ or not. Clearly $S$ may
have non-objective properties such as, for instance, `correlation'
properties: certain possible values of a quantity $R$ measurable on $S$ are
correlated to possible values of a quantity pertaining the pointer
of an apparatus that is supposed to measure $R$ on $S.$ Obviously $S$
cannot be said to have such properties independently from any
measurement (actual or not) of $Q.$
The experimental situation considered (often called
an EPR-Bohm correlation experiment) involves a two spin-1/2 particles'
system $S_1+S_2$ prepared at the source in the singlet state. If we focus only on the spin part,
such state of $S_1+S_2$ can be written for any spatial direction $x$ as
\begin{equation}
{1\over\sqrt 2}[\phi_{1,x}(+)\otimes\phi_{2,x}(-)\,-\,
\phi_{1,x}(-)\otimes\phi_{2,x}(+)]\,,
\label{eq:singlet}
\end{equation}
where:
\begin{itemize}
\item
$\phi_{i,x}(\pm)$ is the eigenvector of the
operator $\sigma_{i,x},$ representing spin up or down along the direction
$x$ for the particle $i=1,2;$
\item
$\phi_{1,x}(+)\otimes\phi_{2,x}(-)$ and
$\phi_{1,x}(-)\otimes\phi_{2,x}(+)$ belong to the tensor product
${\cal H}_1\otimes{\cal H}_2$ of the Hilbert spaces ${\cal H}_1$ and
${\cal H}_2$ associated respectively to subsystems $S_1$ and $S_2$.
\end{itemize}
$S_1$ and $S_2$ are
supposed to fly off in opposite directions; spin measurements are
supposed to be performed when $S_1$ and $S_2$ occupy two widely separated
spacetime regions $R_1$ and $R_2,$ respectively.
It follows from (\ref{eq:singlet}) and Adequacy that if on measurement
we find spin up along the direction $x$ for the particle $S_1,$
the probability of finding spin down along
the same direction $x$ for the particle $S_2$
equals $1$: it is usual to say that $S_1$ and $S_2$ are strictly
anticorrelated.
Let us suppose now that, for a given direction $z,$
we measure $\sigma_{1,z}$ at time $t_1$ and we find $-1.$ Adequacy then allows
us - via anticorrelation - to predict with probability one the result of
the measurement of $\sigma_{2,z}$ for any time $t_2$ immediately after
$t_1,$ namely $+1.$ Then, according to Reality,
there exists an objective property $[+1]$ of $S_2$ at
$t_2.$ By Locality, $[+1]$ was an objective property of $S_2$ also
at a time $t_0 < t_1,$ since otherwise it would have been
`created' instantaneously by the act of performing
a spin measurement on $S_1.$ However, at time $t_0$
the state of $S_2$ is a mixture, namely ${1\over 2}(P_{\phi_{2,x}(+)}+ P_{\phi_{2,x}(-)})$,
since the entangled state
(\ref{eq:singlet}), although it is a pure state of $S_1+S_2,$ uniquely
determines the states of the subsystems as mixed states.
Thus $S_2$ is shown
to satisfy an objective property in a state which is not an eigenstate of
$\sigma_{2,z}.$ However all quantum mechanics is able to predict is the
satisfaction of properties such as $[+1]$
only in eigenstates of $\sigma_{2,z}:$ quantum mechanics then
turns out to be incomplete, since there exist objective properties that are
provably satisfied by a system described by quantum mechanics but that cannot
be described in quantum mechanical terms.
By a strictly logical point of view, the conclusion of the argument can
be rephrased
as the statement that the conjunction of Reality, Completeness, Locality
and Adequacy leads to a contradiction.
In the framework of ordinary
quantum mechanics, Reality and Adequacy cannot be called into question:
whereas the latter simply assumes that the probabilistic statements of
quantum mechanics are reliable,
without the former no quantum system could ever satisfy objective
properties, not even such
a property as having a certain value for a given quantity
one is going to measure, when
the system is prepared in an eigenstate belonging to that (eigen)value
of that quantity.
\footnote{See for instance the clear discussion in \cite{BLM}
on objectivity and non-objectivity of properties in quantum mechanics.}
The alternative then reduces to the choice between Completeness and
Locality: by assuming Completeness, we then turn the above EPR argument into
a nonlocality argument. A relativistic formulation of this argument can also be given:
although the different geometry of spacetime must be taken into account, the only generalization
lies in adapting the Locality condition, to the effect that
objective properties of a physical system cannot be influenced by measurements performed
in space-like separated regions on a different physical system (\cite{GG}).
Let us turn now to a relational analysis of the argument.
In a relational approach to the EPR argument, we have to modify accordingly
its main conditions (Adequacy is obvious), basically by relativizing the objectivity of
properties to given observers. The new versions might read as follows:
\begin{enumerate}
\item[$1'.$] {\sc Reality$^*$}
\noindent
If an observer $O$, without interacting with a physical system $S,$
can predict with certainty (or at any rate with probability one)
at time $t$ the value $q$ of a physical quantity $Q$
measurable on $S$ in a state $s,$ then, at a time $t'$ immediately after $t,$
$q$ corresponds to a property of $S$ that is objective relative to $O.$
\item[$2'.$] {\sc Completeness$^*$}
\noindent
Any physical theory $T$ describing a physical system $S$ accounts for
every property of $S$ that is objective relative to some observer.
\item[$3'.$] {\sc R-Locality$^*$}
\noindent
No property of a physical system $S$ that is objective relative to some
observer can be influenced by measurements performed in space-like separated regions
on a different physical system.
\end{enumerate}
\noindent
Once the above weaker sense of objectivity is defined, Completeness$^*$ is little more
than rephrased, whereas R-Locality$^*$ guarantees that no
property that is non-objective (in the weaker sense of objective as relative to a given observer) can be
turned into an objective one (still in the weaker sense) simply by means of operations
performed in space-like separated regions (at this stage, already the relativistic version
of the Locality condition in the ordinary EPR argument is assumed).
So far the relational versions of the original EPR conditions have
been introduced: but are they sufficient in order to derive the
same conclusion drawn by the original argument? We are interested
into the states of the subsystems $S_1$ and $S_2$ and the values
of spin in those states when $S_1$ and $S_2,$ that initially
interact for a short time and then fly off in opposite directions,
are localized in space-like separated regions $R_1$ and $R_2$
respectively. According to the relational interpretation the
reference to the state of a physical system is meaningful only
relative to some observer. So let us suppose to introduce two
observers, $O_1$ for $S_2$ and $O_2$ for $S_2,$ that, after
coexisting in the spacetime region of the source for the short
time of the interaction, follow each its respective subsystem.
Initially, at time $t_0$, $O_1$ and $O_2$ agree on the state
(\ref{eq:singlet}). But when, after leaving the source, $S_1$ and
$S_2$ are subject to measurement, the spacetime regions in which
the measurements are supposed to take place are mutually isolated.
Then we suppose that, for a given direction $z,$ an observer $O_1$
measures $\sigma_{1,z}$ on $S_1$ at time $t_1 > t_0$ and finds
$-1.$ Now the strict spin value anticorrelation built into the
state (\ref{eq:singlet}) allows $O_1$ to predict with certainty
the spin value for $O_2$ without interacting with it. Namely
Adequacy allows $O_1$ to predict with probability one the value of
$\sigma_{2,z}$ on $S_2$ for any time $t_2$ immediately after
$t_1,$ namely $+1.$ Then, according to Reality$^*$, there exists a
property $[+1]$ of $S_2$ that is objective {\it relative to} $O_1$
at $t_2.$ By R-Locality$^*$, $[+1]$ was an objective property of
$S_2$ relative to $O_1$ also at a time $t_0 < t_1,$ since
otherwise it would have been created by the act of
performing a spin measurement on the space-like separated system
$S_1.$ Then $O_1$ may backtrack at $t_0$ the property $[+1]$ or,
equivalently, the {\it pure} state $\phi_{2,z}(+).$ At this point
the non-relational argument proceeds by pointing out that at $t_0$
the state of $S_2$ as determined by (\ref{eq:singlet}) was the
mixture ${1\over 2}(P_{\phi_{2,x}(+)}+ P_{\phi_{2,x}(-)})$: since
this kind of state cannot possibly display a property like $[+1],$
the charge of incompleteness for quantum mechanics follows. By a
relational viewpoint, however, this conclusion does {\it not}
follow: in fact, we are comparing two states relative to {\it
different} observers. Since the regions $R_1$ and $R_2$ of the
measurements are space-like separated, there are frames in which
the measurement performed by $O_1$ precedes the measurement
performed by $O_2$. Let us suppose that in one of these frames
$O_1$ performs a spin measurement on $S_1$ along the direction $z$
at a time $t_1$ and finds $-1$; then, at $t_2$ immediately after
$t_1$, $S_1$ is in the state $\phi_{1,z}(-)$ relative to $O_1$,
and the latter is allowed to attribute to $S_2$ the pure state
$\phi_{2,z}(+)$ at $t_2$, corresponding to a spin value $+1$.
Thus there is a property $[+1]$ of $S_2$ that is objective relative
to $O_1$; hence, according to R-Locality$^*$, $O_1$ should be
allowed to backtrack at $t_0<t_1$ that same property $[+1]$ - satisfied by
$S_2$ in $\phi_{2,z}(+)$ - but in this case he would derive
incompleteness, since at that time the state of $S_2$ is a
mixture. At time $t_0$, however, $O_2$ did not perform yet any
spin measurement: therefore $O_2$ is still allowed to attribute to $S_1$
and $S_2$ only the mixtures ${1\over 2}(P_{\phi_{1,x}(+)}+
P_{\phi_{1,x}(-)})$ and ${1\over 2}(P_{\phi_{2,x}(+)}+
P_{\phi_{2,x}(-)})$ respectively, and there is no matter of fact
as to which observer `is right'. Until $O_2$ does not perform a
measurement, $O_2$ may then describe the measurement by $O_1$
simply as the establishment of a correlation between $O_1$ and
$S_1$.
The conclusion to be drawn is then that the question:
{\it when is an observer allowed to claim, via the EPR argument, that quantum mechanics is either incomplete
or nonlocal?} has a frame-dependent answer.
Let us denote with $M_O^t(\sigma, S,\rho,R)$ the event that an
observer $O$ performs at time $t$ a measurement of a physical quantity $\sigma$ on a system
$S$ in the state $\rho$ in the region $R$.
If we set
\begin{eqnarray*}
M_R & := & M_{O_1}^{t^R}(\sigma_{1,x}, S_1,{1\over 2}(P_{\phi_{1,x}(+)}+
P_{\phi_{1,x}(-)}),R_1)\\
M_L & := & M_{O_2}^{t^L}(\sigma_{2,x}, S_2,{1\over
2}(P_{\phi_{2,x}(+)}+ P_{\phi_{2,x}(-)}),R_2),
\end{eqnarray*}
the three possible cases are:
\noindent
(a) $M_R$ precedes $M_L$;
\noindent
(b) $M_R$ follows $M_L$.
\noindent
(c) $M_R$ and $M_L$ are simultaneous.
In case (a), the EPR argument works for $O_1$ but not for $O_2$, namely
quantum mechanics is either incomplete or nonlocal relative to
$O_1$ but not relative to $O_2$, whereas the situation is reversed in the case (b).
Finally, in the case (c), what the EPR argument implies is nothing but the outcome-outcome
dependence built into the correlations intrinsic to EPR entangled states (\cite{Shimony1}).
However, until $O_1$ and $O_2$ do not interact, they cannot compare their respective state
attributions so that, up to the interaction time, again each observer can claim either
the incompleteness or the nonlocality of quantum mechanics only relative to himself.
\section{Final discussion}
As we recalled above, the EPR incompleteness argument for ordinary quantum
mechanics can be turned into a nonlocality argument, that appears to
threaten the mutual compatibility at a fundamental level of quantum theory
and relativity theory. The received view has been that, relevant that
nonlocality may be by a foundational viewpoint, the conflict engendered by
it is not as deep as it seems. There would be in fact a `peaceful
coexistence' between the two theories, since locality in quantum mechanics
would be recovered at the statistical level and in any case - it is argued -
nonlocal correlations are uncontrollable so no superluminal transmission of information
is allowed.
\footnote{See for instance \cite{Eberhard}, \cite{GRW} and \cite{Shimony1}.
For dissenting views, one can see \cite{CJ} and \cite{Muller}.}
Such view, however, has several drawbacks, first of all the fact that it is based
on highly controversial notions such as the `controllability' (or
`uncontrollability') of information, or on the vagueness of establishing when
it is exactly that an `influence' becomes a bit of information, or
on whether relativity theory prohibits superluminal exchanges of information but not
superluminal travels of influences, and subtleties
like that. On the other hand, the very inapplicability of
the ordinary EPR argument in relational quantum mechanics prevents this kind of
controversies and provides a new way to interpret the peaceful coexistence
thesis, since in the relational interpretation no compelling alternative between completeness
and locality via the EPR argument can be derived in a frame-independent way.
The observer dependence that affects the EPR argument in a relational approach to
quantum mechanics might also imply a different view of the hidden variables program.
Most hidden variable theories do essentially two things. First,
on the basis of the EPR argument's conclusion they {\it assume}
the incompleteness of quantum mechanics as a starting point; second, they introduce
the hypothesis of a set of `complete' states of a classical sort, the averaging
on which gives predictions that are supposed to agree with the quantum mechanical ones
(see for instance \cite{CS}, \cite{Redhead}).
If however, in a relational approach to quantum mechanics, the conditions of Reality$^*$,
Completeness$^*$, R-Locality$^*$ and Adequacy need not clash from
one frame of reference to the other, the attempt of `completing' quantum mechanics by
introducing hidden variables turns out to be unmotivated by a relational point of view,
and so does any nonlocality argument that has this attempt as a premise. Moreover, the
hypothetical complete states of hidden variables theories are conceived themselves as
observer-{\it in}dependent, so that an absolute view of the states of physical system
would be reintroduced, albeit at the level of the hidden variables.
The issue of whether statistical correlations across space-like
separated regions are a real threat to a peaceful coexistence between quantum theory
and relativity theory is being more and more investigated also in the framework of algebraic
quantum field theory (AQFT): in fact a suitable form of Bell inequality -
an inequality that in different formulations has been extensively
studied with reference to the nonlocality issue in
non-relativistic quantum mechanics (\cite{Bell}) - has been shown
also to be violated in AQFT (see for instance \cite{SW} and
\cite{S}). The questions of whether a relational interpretation of
AQFT might be developed, and of what such interpretation might
have to say about the violation of the Bell inequaliity in AQFT,
appear worth investigating for several reasons. First, an
absolute view of quantum states seems in principle precluded in
AQFT. In the algebraic framework for quantum field theory,
local algebras are axiomatically associated with specific (open bounded) regions
of the Minkowski spacetime: the elements of the algebras are represented as the observables
that can be measured in the spacetime region to which the algebra is associated. The states,
represented as suitable expectation functionals on the given algebra, encode the statistics
of all possible measurements of the observables in the algebra and thus inherit from
the latter the feature of being defined not globally but with respect to a particular
spacetime region. Second, the derivation of a suitable
Bell inequality and the analysis of its violation refer neither to
unspecified hidden variables nor to the need of introducing them.
Finally, the locality that the violation of the suitable Bell
inequality might call into question is
directly motivated by relativistic constraints and is
not a hypothetical condition satisfied only by hidden variables,
and imposed over and above a theory that is intrinsically nonlocal
such as ordinary quantum mechanics.
\noindent
{\bf Acknowledgements}
\noindent
I am deeply grateful to Carlo Rovelli for an interesting
correspondence on an earlier version of this paper:
his insightful comments greatly helped me to clarify a number of relevant
points, although I am of course the only responsible of the arguments defended here.
This work was completed during a visit to the Department of History and Philosophy
of Science of the E\"otv\"os University in Budapest. I wish to thank in particular
Mikl\'os R\'edei and L\'aszl\'o Szab\'o for their support and their hospitality.
This work is supported by a NATO CNR Outreach Fellowship.
\end{document} |
\begin{document}
\title{Quantum nondemolition measurement of optical field fluctuations by optomechanical interaction}
\author{A. Pontin \footnote{Present address: Department of Physics and Astronomy, University College London, WC1E 6BT, United Kingdom}}
\affiliation{Dipartimento di Fisica e Astronomia, Universit\`a di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy}
\affiliation{Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy}
\author{M. Bonaldi}
\affiliation{Institute of Materials for Electronics and Magnetism, Nanoscience-Trento-FBK Division,
38123 Povo, Trento, Italy}
\affiliation{INFN, Trento Institute for Fundamental Physics and Application, I-38123 Povo, Trento, Italy }
\author{A. Borrielli}
\affiliation{Institute of Materials for Electronics and Magnetism, Nanoscience-Trento-FBK Division,
38123 Povo, Trento, Italy}
\affiliation{INFN, Trento Institute for Fundamental Physics and Application, I-38123 Povo, Trento, Italy }
\author{L. Marconi}
\affiliation{CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy}
\author{F. Marino}
\affiliation{Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy}
\affiliation{CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy}
\author{G. Pandraud}
\affiliation{Delft University of Technology, Else Kooi Laboratory, 2628 Delft, The Netherlands}
\author{G. A. Prodi}
\affiliation{INFN, Trento Institute for Fundamental Physics and Application, I-38123 Povo, Trento, Italy }
\affiliation{Dipartimento di Fisica, Universit\`a di Trento, I-38123 Povo, Trento, Italy}
\author{P.M. Sarro}
\affiliation{Delft University of Technology, Else Kooi Laboratory, 2628 Delft, The Netherlands}
\author{E. Serra}
\affiliation{INFN, Trento Institute for Fundamental Physics and Application, I-38123 Povo, Trento, Italy }
\affiliation{Delft University of Technology, Else Kooi Laboratory, 2628 Delft, The Netherlands}
\author{F. Marin}
\email[Electronic mail: ]{[email protected]}
\affiliation{Dipartimento di Fisica e Astronomia, Universit\`a di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy}
\affiliation{CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy}
\affiliation{INFN, Sezione di Firenze, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy}
\affiliation{European Laboratory for Non-Linear Spectroscopy (LENS), Via Carrara 1, I-50019 Sesto Fiorentino (FI), Italy}
\begin{abstract}
According to quantum mechanics, if we keep observing a continuous variable we generally disturb its evolution. For a class of observables, however, it is possible to implement a so-called quantum nondemolition measurement: by confining the perturbation to the conjugate variable, the observable is estimated with arbitrary accuracy, or prepared in a well-known state. For instance, when the light bounces on a movable mirror, its intensity is not perturbed (the effect is just seen on the phase of the radiation), but the radiation pressure allows to trace back its fluctuations by observing the mirror motion. In this work, we implement a cavity optomechanical experiment based on an oscillating micro-mirror, and we measure correlations between the output light intensity fluctuations and the mirror motion. We demonstrate that the uncertainty of the former is reduced below the shot noise level determined by the corpuscular nature of light.
\end{abstract}
\maketitle
\section{Introduction}
Quantum mechanics generally prescribes that, as soon as we observe a system, we actually perturb it.
As a paradigmatic example, in the Heisenberg's microscope a measurement of the position of a particle at the time $t$ perturbs its momentum, thus influencing the particle motion, and actually its position at following times. The consequence of the observation of the system (back-action) deteriorates the accuracy of a continuous measurement on the observable considered (the position). On the other hand, there are observables that are not affected by the disturbance caused by their measurement, the effect of which remains confined to their conjugate variable: their measurement can evade the back-action. For such observables it has been introduced the concept of Quantum Non-Demolition (QND) measurement \cite{ref1,ref2,ref3,ref4}. A QND measurement allows to keep observing a variable with arbitrary accuracy. Examples of QND variables are the quadratures of a mechanical oscillator and, similarly, the fluctuations on the quadratures of the electromagnetic field, defined from its bosonic operators, after separation of their average coherent amplitude ($a = \langle a \rangle + \delta a$), as $\delta X = \delta a+\delta a^{\dagger}$ (amplitude quadrature), $\delta Y = -i(\delta a-\delta a^{\dagger})$ (phase quadrature) and $\delta X^{\phi} = \delta X \cos \phi + \delta Y \sin \phi$ (generic quadrature).
The possibility to perform a QND measurement of a field quadrature (in particular, of the amplitude $\delta X$) by exploiting the radiation pressure exerted on a movable mirror was studied in a seminal work by Jacobs {\it et al.} in 1994 \cite{ref5}.
When the light bounces on a mirror, its intensity is not perturbed: the displacement of the mirror changes the phase of the field, and the optomechanical interaction modifies $\delta Y$, but it leaves $\delta X$ unaffected.
In the proposed experiment, a resonant optical cavity amplifies the intensity fluctuations, and eventually the momentum transferred to the mirror by the bouncing photons. Such fluctuations are actually measured by observing the momentum of the mirror, in particular around a mechanical resonance where its susceptibility increases. The measurement of the mirror motion can be performed interferometrically by a meter field \cite{ref6,ref7,ref8}.
The complete measurement apparatus can be viewed as a system with two outputs: the signal field (i.e., a quantum object), and the result of a continuous measurement on one of its quadratures, yielding a (classical) meter variable $Y_{\mathrm{m}}$. An ideal QND measurement is testified by a perfect correlation between the quantum observable to be estimated, i.e. a field quadrature $X_{\mathrm{s}}$ (signal variable), and $Y_{\mathrm{m}}$. The condition to be satisfied can be written as $C_{X_{\mathrm{s}}Y_{\mathrm{m}}} := |S_{X_{\mathrm{s}}Y_{\mathrm{m}}}|^2/(S_{X_{\mathrm{s}}X_{\mathrm{s}}} S_{Y_{\mathrm{m}}Y_{\mathrm{m}}}) = 1$ where $S_{XY}$ is the cross-correlation spectrum between $X$ and $Y$, and $C_{XY}$ is the so-called magnitude-squared coherence (MSC). It is elucidating to compare the QND procedure with a standard, classical intensity measurement where the signal is the quadrature $X_{\mathrm{s}}$ of the field at one output port of a beam-splitter, while the field at the other output port is detected to provide the meter $Y_{\mathrm{m}}$ (Fig. \ref{fig1}a). With a coherent input the cross-correlation is null, and the measurement can just provide information on possible excess noise: the photon noise of the remaining, usable light remains inaccessible. On the contrary, a QND measurement gives access to the quantum fluctuations of the signal field.
\begin{figure}
\caption{Simplified experimental schemes. (a) Scheme of a classical measurement of the field amplitude fluctuations. (b) Simplified experimental setup for our QND measurement. DHD: double homodyne detection; PBS: polarizing beam-splitter; Pol: polarizer. (c) Schematic composition of the fields after the polarizer, in the complex phase plane. The mean field $E_S$ is formed by superposition of the field reflected by the cavity and then transmitted through the polarizer ($E_R$), and a fraction of the reference field ($E_{\mathrm{ref}
\label{fig1}
\end{figure}
For a deeper understanding of the optomechanical QND measurement, we can consider an ideal scheme exploiting a cavity with coupling rate $\kappa$ and no extra losses, and a resonant input field in a coherent state, whose amplitude ($\delta X_{\mathrm{in}}$) and phase ($\delta Y_{\mathrm{in}}$) quadrature fluctuations have spectral densities $S_{\delta X_{\mathrm{in}} \delta X_{\mathrm{in}}} = S_{\delta Y_{\mathrm{in}} \delta Y_{\mathrm{in}}} = 1/4$.
The position $q$ of the mechanical oscillator embedded in the cavity as end mirror, normalized to its zero-point fluctuations, is given in the Fourier space by
$q = q_{\mathrm{th}} + q_{\mathrm{rp}}$,
where
$q_{\mathrm{rp}}=4 \, \chi \, \chi_{\mathrm{opt}} \sqrt{\Gamma_{\mathrm{BA}}} \, \delta X_{\mathrm{in}}$
is the displacement due to the radiation pressure, and
$S_{q_\mathrm{th} q_\mathrm{th}} = 4 \Gamma_{\mathrm{th}} \,|\chi|^2$
is the displacement spectrum due to the oscillator thermal and quantum noise.
In the above expressions,
$\chi = \omega_m/\left(\omega_m^2-\omega^2 - \mathrm{i}\omega\gamma_m\right)$
is the mechanical susceptibility,
$\chi_{\mathrm{opt}} = 1/\left(1-\mathrm{i} \frac{\omega}{\kappa}\right)$ is the optical susceptibility,
$\Gamma_{\mathrm{BA}} = G^2/\kappa$ is the back-action rate \cite{note1},
and
$\Gamma_{\mathrm{th}} = (\omega/Q) (n_T+1/2)$ is the thermal and quantum coupling rate, where $Q$ is the mechanical quality factor \cite{note2} and the average thermal occupancy is
$n_T = \left(\exp{\left(\frac{\hbar \omega}{kT}\right)}-1\right)^{-1}$.
The field quadratures at the output of the cavity are \cite{ref10}
\begin{eqnarray}
\label{Xo}
\delta X_{\mathrm{out}} & = & \exp (2 \mathrm{i} \phi_{\mathrm{opt}}) \, \delta X_{\mathrm{in}} \\
\label{Yo1}
\delta Y_{\mathrm{out}} & = & \exp (2 \mathrm{i} \phi_{\mathrm{opt}}) \,\delta Y_{\mathrm{in}} + \sqrt{\Gamma_{\mathrm{BA}}} \, \chi_{\mathrm{opt}} \, q \\
&= & \exp (2 \mathrm{i} \phi_{\mathrm{opt}}) \,\delta Y_{\mathrm{in}} + 4 \Gamma_{\mathrm{BA}} \, |\chi_{\mathrm{opt}}|^2 \, \chi \, \delta X_{\mathrm{out}} + \sqrt{\Gamma_{\mathrm{BA}}} \, \chi_{\mathrm{opt}} \, q_{\mathrm{th}}
\label{Yo}
\end{eqnarray}
where $\phi_{\mathrm{opt}} = \arg[\chi_{\mathrm{opt}}]$. The relations (\ref{Yo1}-\ref{Yo}) describe the interaction between the field to be measured and the optomechanical system. In order to complete a QND measurement of the field, we need an additional readout channel, measuring the oscillator displacement with the result
\begin{equation}
Y_{\mathrm{m}} = q + \, q_\mathrm{r}
\label{Eq:qm}
\end{equation}
where $q$ is defined above and $q_\mathrm{r}$ is an additional noise term that includes both the readout imprecision and its back-action (i.e., it comprises the overall measurement accuracy).
In case of detection at the standard quantum limit, the spectrum of $q_\mathrm{r}$ is $S^{SQL}_{q_\mathrm{r} q_\mathrm{r}} = 2 |\chi|$ \cite{Caves}, but the fundamental quantum limit is even lower, i.e., $S^{QL}_{q_\mathrm{r} q_\mathrm{r}} = 2 \mathrm{Im}[\chi]$ \cite{Jaekel,Kampel}. At the oscillator resonance frequency, the two limits coincide.
From the above model, we extract three meaningful considerations.
(i) Eq. (\ref{Xo}) shows that the optomechanical interaction does not perturb the amplitude field quadrature $\delta X$, that is transmitted to the output.
(ii) Eq. (\ref{Eq:qm}) and the definitions of $q$ and $q_{\mathrm{rp}}$ show that the output of the readout contains some information on $\delta X_{\mathrm{out}}$.
(iii) Eq. (\ref{Yo}) shows that, in the output field, amplitude and phase quadratures are correlated.
The first two properties form the basis of the Quantum Non Demolition measurement: $Y_{\mathrm{m}}$ is the result of the QND process, that includes the optomechanical interaction (that does not destroy the variable $\delta X$), and a measurement of $q$. The third observation implies instead that the optomechanical interaction is also producing a field in a squeezed state: since $\delta X_{\mathrm{out}}$ and $\delta Y_{\mathrm{out}}$ are correlated, there is an output field quadrature $\delta X^{\phi}$ for which the fluctuations are below $S_{\delta X_{\mathrm{out}} \delta X_{\mathrm{out}}}$, and eventually below the vacuum level.
Once acquired, $Y_{\mathrm{m}}$ can be used to predict the behavior of the quadrature $\delta X_{\mathrm{out}} \equiv X_{\mathrm{s}}$ of the surviving field, that is estimated as $X_{\mathrm{E}} = \alpha(\omega) Y_{\mathrm{m}}$, where $\alpha(\omega)$ is an arbitrary complex function that is chosen with the aim of minimizing the average residual uncertainty
$S_{\Delta X}^{\alpha} := \langle |X_{\mathrm{s}} - X_{\mathrm{E}}|^2 \rangle$.
In a stationary system, the optimal $\alpha$ is $\alpha_{\mathrm{opt}} = (S_{X_{\mathrm{s}} Y_{\mathrm{m}}})^*/S_{Y_{\mathrm{m}} Y_{\mathrm{m}}}$, and the lowest residual uncertainty on the signal is
$S_{\Delta X} := S_{X_{\mathrm{s}} X_{\mathrm{s}}} \left(1- C_{X_{\mathrm{s}} Y_{\mathrm{m}}} \right)$.
For the considered optomechanical system, such residual uncertainty can be written as $S^{QL}_{\Delta X} = \left( 1+\frac{\mathcal{C}}{1+R} \right)^{-1}$ where $\mathcal{C} = \frac{\Gamma_{\mathrm{BA}} |\chi_{\mathrm{opt}}|^2}{\Gamma_{\mathrm{th}}}$ is the cooperativity. The parameter $R=\frac{\gamma_m Q}{2 \omega_m (n_T+1/2)}$ is originated by the readout noise $q_\mathrm{r}$, considered at the quantum limit, and in general $R \ll 1$ except when the mechanical oscillator is cooled close to its ground state \cite{note3}. An example of this residual spectral density is shown in Fig. \ref{newTheo} with a blue dashed line.
A readout imprecision at the quantum limit requires a rapidly varying detection phase, optimized as a function of the frequency, i.e., a so-called variational readout \cite{ref24}. It is more realistic to consider a QND procedure having a constant, frequency-independent readout imprecision. We can assume that the quantum limit is achieved at the mechanical resonance frequency, and thus set the readout imprecision at $|\chi(\omega_m)| = 1/\gamma_m$. With this choice, we can write the total readout noise as $S_{q_\mathrm{r} q_\mathrm{r}} = \left(1/\gamma_m + \gamma_m|\chi|^2\right)$, where the second term within brackets is originated by the readout back-action, and $R$ must be multiplied by $\frac{1}{2 \mathrm{Im}[\chi]}\left(\frac{1}{\gamma_m}+\gamma_m |\chi|^2\right) \simeq 1+2\left(\frac{\omega-\omega_m}{\gamma_m}\right)^2$. The resulting residual uncertainty is shown in Fig. \ref{newTheo} with a blue solid line.
It is interesting to compare $S^{QL}_{\Delta X}$ with the spectral density in the maximally squeezed output quadrature $S^{\mathrm{min}}$, that is calculated by minimizing the spectral density of the output field quadrature $\delta X^{\phi}$ with respect to $\phi$, for each detection frequency: $S^{\mathrm{min}} = \frac{1+\sin^2(\arg[\chi])\,\mathcal{C}}{1+\mathcal{C}}$. Similarly, the residual uncertainty obtained in a QND measurement at fixed readout imprecision can be compared with the noise in a fixed output quadrature $\delta X^\phi$. The two spectra are shown in Fig. \ref{newTheo} with red lines. A significantly better performance is obtainable with the QND approach, in particular for the realistic experiments using a fixed measurement phase (solid lines). We will further discuss it after the description of our experimental results.
\begin{figure}
\caption{Field fluctuations after an optomechanical setup. Blue, long-dashed curve: residual uncertainty $S^{QL}
\label{newTheo}
\end{figure}
Recent advances in cavity optomechanics \cite{ref10} have allowed to discern the quantum component in the effect of radiation pressure \cite{ref11,ref12,ref13}, and in the correlation between signal and meter \cite{ref12} or between field quadratures after optomechanical interaction \cite{ref14a,ref14,ref14b}. As discussed, the latter is actually the basic ingredient of the observed ponderomotive squeezing \cite{ref14a,ref15,ref16,ref17,ref17a}. The ability of a mechanical oscillator to perform a QND measurement of the radiation intensity fluctuations is experimentally demonstrated with a squeezed microwave source by Clark {\it et al.} \cite{ref17b}, who exploit the phase quadrature of the same driving field, detected after the optomechanical interaction, as a meter of the mirror motion. As discussed, a complete QND scheme requires an additional readout of the mirror motion, without destroying (or perturbing too much) the field to be measured, that is therefore preserved after having disclosed its quantum properties. This is in fact what we are showing in the following, where we describe an experiment that actually achieves a measurement of the transmitted light quantum noise by observing the effect of the photons impact on a movable mirror.
\section{The experiment}
\label{experiment}
In a fair QND procedure, $\alpha (\omega)$ is chosen {\it a priori}, e.g. on the basis of a model, or derived from the analysis of an independent set of data (this analysis could include a destructive measurement of $\delta X_{\mathrm{out}}$ to estimate its correlation with $Y_{\mathrm{m}}$). In order to verify that a quantum measurement is indeed performed, the experimentalist has to measure the output intensity fluctuations as well as their correlation with $Y_{\mathrm{m}}$. In a realistic optomechanical system, the achievement of an ideal QND is prevented by thermal fluctuations of the movable mirror, by optical losses and by the imprecision of the readout. Moreover, detuning between input field frequency and cavity resonance, and/or excess classical input noise, can create a strong classical correlation between the signal field quadrature $X_{\mathrm{s}}$, and the meter field $Y_{\mathrm{m}}$ \cite{ref8}. Therefore, a non-null correlation $C_{X_{\mathrm{s}}Y_{\mathrm{m}}}$, as it occurs in a classical measurement of a noisy field, is not sufficient to guarantee that an even non-ideal, yet quantum QND measurement is achieved. The model-independent condition to be verified is that the information carried by $Y_{\mathrm{m}}$ is sufficient to reduce the residual uncertainty of $X_{\mathrm{s}}$ below the standard quantum fluctuations (shot noise), i.e., that $S_{\Delta X} < 1$ \cite{ref9,Roch}.
Our experiment is based on an oscillating micro-mirror, working as end mirror in a high finesse optical cavity.
This oscillator is fabricated by micro-lithography on a silicon-on-insulator wafer. A detailed description of the fabrication process is reported in Ref. \cite{ref25}, while the design of the device is discussed in Refs. \cite{ref18,ref19}. The oscillator has a particular shape, studied to maximize its mechanical quality factor and isolation from the frame (Fig. \ref{fig5}a). A structure made of alternating torsional and flexural springs supports the central mirror and allows its vertical displacement with minimal internal deformations, reducing the mechanical loss in the highly dissipative optical coating. For the oscillation mode exploited for this work, the movement of the central disk is balanced by four counterweights, so that the four joints are nodal points (Fig. \ref{fig5}b). Its effective mass is $m = 2.5 \times 10^{-7}$ kg, deduced form the thermal peak in the displacement spectrum measured at room temperature, its frequency at cryogenic temperature is $\omega_{m}/2\pi = 169334$ Hz. The quality factor of $1.1 \times 10^{6}$ at cryogenic temperature is measured in a ring-down experiment. In a second oscillation mode, with resonance frequency around $\sim208$ kHz, the counterweights move in phase with the central disk (Fig. \ref{fig5}c), therefore a net recoil force is applied on the joints, inducing a larger coupling with the frame and actually a lower quality factor. The design includes an external double wheel, working as mechanical filtering oscillator (Fig. \ref{fig5}d), with a resonance frequency of $\sim 22$ kHz.
\begin{figure}
\caption{The optomechanical oscillator. (a) SEM image of the full device, including the central oscillator and the external isolating wheel. The central dark disk is the $400 \mu$m diameter highly reflective coating. (b-d) FEM simulations of the displacement corresponding to the balanced oscillator mode exploited in this work (b), the second, unbalanced mode (c), and the first wheel oscillator mode (d).
}
\label{fig5}
\end{figure}
The central coated region of the oscillator is the back mirror of a $L_{\mathrm{c}} = 1.455$ mm long Fabry-P\'erot cavity where the input coupler is a 50 mm radius concave mirror, glued on a piezoelectric transducer used to keep a cavity resonance within the laser tuning range. A cavity half-linewith of $\kappa/2\pi = 2.85$ MHz is measured at cryogenic temperature. From the calculated Finesse (18055), the measured resonance depth in the reflected intensity, and the measured mode matching of $90\%$, we deduce an input coupler transmission of 315 ppm (in agreement with the direct measurement, input rate $\kappa_1/2\pi = 2.58$ MHz) and additional cavity losses of 33 ppm (loss rate $\kappa_2/2\pi = 0.27$ MHz). The cavity is strongly overcoupled to optimize the quantum efficiency.
The cavity is suspended inside an helium flux cryostat and thermalized to its cold finger with soft copper foils. The temperature reached by the cavity mount, measured with a diode sensor, is 4.9 K. A finite elements simulation of the heat propagation inside the mount and the silicon device, at the maximum input laser power, suggests that the oscillator temperature should be few tenths of degree higher. The temperature that gives the best agreement between the experimental spectra and the model is indeed 5.6 K.
The experimental setup is sketched in the simplified scheme of Figure \ref{fig1}b and in more details in Fig. \ref{exp_completo} of Appendix A.
A laser beam from a Nd:YAG source is actively amplitude stabilized, down to an amplitude noise (normalized to shot noise) of 1 + $P$/(24 mW), where $P$ is the laser power. An additional, frequency-shifted auxiliary beam (not shown in Figure \ref{fig1}) is used for controlling the detuning from the optical cavity (see Appendix A for details).
The main beam is split by a polarizing beam-splitter (PBS), the outputs of which are sent into the two arms of an interferometer. On one arm, the beam is mode-matched to the optical cavity. The laser power impinging on the cavity is about 50 $\mu$W from the auxiliary beam, and 38 mW from the main beam. The calculated intracavity power is 350 W, corresponding to $n_{\mathrm{c}} = 1.8 \times 10^{10}$ photons. Optical circulators deviate the reflected beams toward the respective detections.
After the recombination of the two interferometer beams, a beam sampler picks up about $3\%$ of the p-polarized light arriving from the cavity, and $\sim 15\%$ of the s-polarized light from the reference arm of the interferometer. The collected radiation is detected in two homodyne setups whose output signals, opportunely combined, allow to actively stabilize the interferometer with the desired phase difference between the arms and, at the same time, to derive a weak measurement of the cavity phase noise and actually of the motion of the oscillating mirror (see Appendix A). The meter variable $Y_{\mathrm{m}}$ is obtained in this way without the necessity of an additional readout field, at the expenses of a slightly reduced efficiency in the transmission of the signal beam. The vacuum noise entering from the unused port of the beam sampler determines the measurement imprecision of the readout.
The spectrum of the meter (Fig. \ref{fig2}a) is dominated by the fluctuations of the cavity length, mainly due to the oscillating mirror. Therefore, the meter provides indeed a readout for the movable mirror, that in turn performs a measurement of a particular field quadrature (namely, the quadrature that gives rise to the intracavity intensity fluctuations).
\begin{figure}
\caption{Meter spectrum and correlations. (a) Spectral density of the meter field (black). The spectrum is calibrated both in terms of meter shot noise (SQL; right axis), and in terms of single-sided power spectral density (PSD) of cavity displacement noise (left axis). The electronic noise (already subtracted from the displayed spectrum) is 10 dB below the SQL. In the model (magenta) we have introduced phenomenologically a $1/\omega^2$ contribution to account for the tails of low frequency modes, and an additional resonance at $\sim 208$ kHz. (b) Experimental magnitude-squared coherence $C_{X_{\mathrm{s}
\label{fig2}
\end{figure}
Due to the weak detuning, the optomechanical interaction shifts the frequency of the main oscillator mode to $\sim 167500$ Hz and broadens its resonance to 430 Hz (corresponding to an effective temperature of $\sim 2$ mK) \cite{ref10}. With the achieved effective susceptibility, a readout achieving the quantum limited sensitivity at resonance implies an imprecision of $1.5\times 10^{-37}$ m$^2$/Hz
. However, the imprecision level is not visible: the mechanical peak emerges from a background of few $10^{-36}$ m$^2$/Hz given by the tails of low frequency mechanical modes and of the unbalanced oscillator mode at $\sim 208$ kHz. Thermal noise, laser amplitude noise (of classical and quantum origin), and intracavity intensity fluctuations due to the background displacement noise, all contribute with comparable importance to the force noise acting on the oscillator (see the simulations reported in the Appendix C for their quantitative estimations). The last contribution (i.e., eventually, the cavity phase noise) is responsible for the deviation from a Lorentzian shape of the peak, that assumes a Fano profile.
The intracavity amplitude quadrature fluctuations are imprinted on the mirror motion. Since the radiation is slightly detuned on the red side of the optical resonance for improving the systems stability, the reflected field quadratures are rotated with respect to the intracavity field, therefore the fluctuations sensed by the oscillator do not exactly correspond to the amplitude quadrature of the reflected field. In order to explore a range of reflected quadratures, we add to the reflected field a small portion of a beam from the reference arm of the interferometer, with a controlled phase (optical path length).
At this purpose,
after the beam sampler the main beam is filtered by an high extinction ratio ($>10^7$) polarizer. The axis of the transmitted polarization is very close to the p-polarization axis (within $\sim 1^{\circ}$), so that $> 99\%$ of the field from the cavity and $\sim 3\%$ of the field from the reference arm (corresponding to about 2 $\mu$W) are transmitted and superimposed to form the signal field. The latter is thus rotated with respect to the radiation reflected by the cavity, as outlined in Fig. \ref{fig1}c, with a tuning range of about $\pm 10^{-2}$ rad.
The radiation transmitted by the polarizer is actually the observed physical system, and in particular its amplitude fluctuations are the signal variable $X_{\mathrm{s}}$. In order to verify that the meter provides a QND measurement of such fluctuations, they are monitored (destructively) with a standard balanced detection, composed of a $50\%$ beam-splitter and a couple of photodiodes: the sum of their signals gives $X_{\mathrm{s}}$, their difference provides an accurate calibration of the radiation standard quantum level (SQL). With respect to a standard homodyne detection, this scheme improves the phase stability and, above all, the accuracy of the shot noise calibration, that is not trivial in a homodyne with high signal power, at the price of weak additional losses.
The measured common mode rejection of the balanced detection is $\sim 40$ dB, and the total quantum efficiency in the detection of the field reflected by the cavity is $69\%$, including the losses in the beam sampler and in the polarizer, and the $\sim90\%$ efficiency of the homodyne photodetectors.
The sum and difference signals are filtered with high order low-pass, anti-aliasing circuits and acquired by an high resolution digital scope. The complete electronics for the readout of the sum and difference signals are calibrated with a relative accuracy of better than $0.1 \%$.
The linearity of the difference signal versus the detected photocurrent (sum of the two detectors photocurrents) has been checked by sending to the photodiodes the laser radiation of the main beam very far from cavity resonance (Figure \ref{fig8}(b-c)). The residuals of the linear fit show no systematic deviation. The noise variance reported in the figure is calculated by considering the spectrum of the photodiodes difference signal in the intervals 154 -- 163 kHz and 176 -- 180 kHz, and calculating the spectral density at 170 kHz with a linear interpolation. Such linear interpolation is sufficient to account for the fact that the spectrum is not flat, due to the filters in the photodetectors circuits. The same procedure is used for evaluating the SQL in the experimental data, where we exclude in this way the region (163 -- 176 kHz) where the strong oscillator peak could percolate into the difference signal in spite of the high rejection. The electronic noise is equal to the shot noise of 0.63 mA, equivalent to an impinging power of 0.8 mW, and it has a day-to-day reproducibility of $\sim 10\%$. Since it is subtracted from the measured spectra, it contributes to the uncertainty with an additional $0.3 \%$. Taking into account all the analyzed sources of systematic error, we evaluate that their total effect in the calibration of the spectra to the SQL is below $0.5 \%$.
\begin{figure}
\caption{Shot noise calibration. (c) Current noise spectral density at 170 kHz measured in the difference signal of the balanced detection, versus total photocurrent, measured by varying the optical power impinging on the detectors with the laser far from resonance. The cyan straight line is a linear fit to the data. The red circle indicates a typical measurement taken with the fully working experiment (with the laser locked to the cavity), used to calibrate the SQL for the spectra reported in Figures \ref{fig3}
\label{fig8}
\end{figure}
\section{Results and discussion}
\label{discussion}
To verify the claim that the amplitude quadrature of the signal field is measured in a QND way by the mechanical oscillator \cite{ref17b}, and actually that the meter variable well reports the result of this measurement, we have to calculate the residual spectrum $S_{\Delta X}$ and show that it falls below the standard quantum level in a proper frequency range. We observe indeed that the coherence between the meter and the signal (Fig. \ref{fig2}b) reaches values close to unit around the peak frequency, but it mainly reflects classical fluctuations. Only the following comparison with the signal spectrum allows to assess that a QND measurement is indeed performed.
\begin{figure}
\caption{Signal and its residual uncertainty. (a) Spectral density of the signal field $S_{X_{\mathrm{s}
\label{fig3}
\end{figure}
In Fig. \ref{fig3} we show the spectrum of $X_{\mathrm{s}}$, i.e., of the amplitude fluctuations of the output field that is determined by a particular choice of cavity detuning and interferometer reference phase. It displays a typical Fano profile, due to the interference between amplitude fluctuations of the intracavity field, which act on the mirror via radiation pressure, and the field fluctuations induced by the consequent mirror motion. For the chosen reference phase, such interference is constructive on the right of the resonance, and destructive on the left, due to the change in the sign of the real part of the mechanical susceptibility \cite{ref20}. As a consequence, depending on the frequency, the spectral density can be higher or lower than the input amplitude power spectrum, but we always find it above the SQL do to the excess input amplitude noise. This behavior is indeed predicted by the theory (outlined in the Appendix C), and typically occurs for most of the values of the reference phase.
If, on the other hand, we exploit the information carried by the meter and calculate the spectral density $S_{\Delta X}$ of the residual fluctuations, we verify that it falls below the shot noise level in a $\sim 1.5$ kHz broad region, on the high frequency side of the resonance. Its lowest value, normalized to the SQL, is $0.921 \pm 0.012$ (uncertainty corresponding to one standard error) when the spectrum is integrated over 150 Hz (Fig. \ref{fig3}b). By averaging over a 600 Hz band we obtain $0.942 \pm 0.006$, demonstrating a QND measurement with strong statistical significance. The systematic error due to calibration uncertainties is $\pm 0.005$. To fully exploit the information carried by the meter, we have not just used the correlation between $X_{\mathrm{s}}$ and $Y_{\mathrm{m}}$, but also the one between $X_{\mathrm{s}}$ and the square of $Y_{\mathrm{m}}$ (see Appendix B).
We have fitted to the spectrum a complete optomechanical model (described in the Appendix C) where all the system parameters are independently measured, except for the amplitude of the background displacement noise, the detuning and the reference phase. The fact that $S_{\Delta X}$ is below the SQL on just the right hand side of the peak, predicted by the complete model, but in contrast with our simplified introductory description (see Fig. \ref{newTheo}), is due to the favorable frequency/background noise cancellation occurring around the resonance frequency of the free oscillator \cite{ref21}, an effect that can also be interpreted in terms of Opto-Mechanically-Induced Transparency \cite{omit}.
\begin{figure}
\caption{Ponderomotive squeezing and QND measurement. With respect to Fig. \ref{fig3}
\label{fig4}
\end{figure}
For a deeper exploration of the QND measurement, it is interesting to vary the choice of the signal quadrature $X_{\mathrm{s}}$. Such variation has strong effects on the intensity spectrum $S_{X_{\mathrm{s}}X_{\mathrm{s}}}$. In particular, if the phase $\phi_s$ is changed to the opposite side with respect to the reference given by the reflected field (see Fig. \ref{fig1}c), the destructive interference in $X_{\mathrm{s}}$ occurs on the right of the resonance. By accurately tuning the phase, we can now observe an intensity spectrum $S_{X_{\mathrm{s}}X_{\mathrm{s}}}$ falling below the shot noise level (Fig. \ref{fig4}). It is the signature of ponderomotive squeezing \cite{ref15,ref16,ref17,ref22,ref23}. On the other hand, the residual spectrum obtained after exploiting the information carried by $Y_{\mathrm{m}}$ is weakly phase-dependent, as indicated by the fact that the sub-SQL region is now very similar to the one previously shown in Fig. \ref{fig3}.
Both the dependence of $S^{\mathrm{min}}$ from $\arg[\chi]$ and the limited squeezing bandwidth in a given output quadrature are the consequence of the physical origin of the squeezing, that is due to the negative interference (cancellation) between the terms $\delta X_{\mathrm{out}} \cos \phi$ and $\propto \chi \, \delta X_{\mathrm{out}} \sin \phi$ (see Eq. (\ref{Yo})) in the output field quadrature $\delta X^{\phi}$. At a given phase $\phi$, such interference is optimal for a particular value of $\chi$, i.e., for a particular frequency, while it degrades as soon as $\chi$ varies. Moreover, the cancellation is limited by the imaginary part of the second term (actually, by the imaginary part of $\chi$), and it is completely inefficient at resonance, where $\chi$ is purely imaginary (see the dashed red curve in Fig. \ref{newTheo}). Such limiting features are absent in the QND measurement, where an appropriate weighting function $\alpha(\omega)$ can compensate for the frequency dependence of $\chi$ and for its argument. On the other hand, as we have seen, the QND needs an additional measurement (on $q$) besides the optomechanical interaction, that is not necessary to produce the squeezed field. The QND performance depends on the quality of such measurement.
\section{Conclusions}
In conclusion, we have experimentally demonstrated that a QND measurement is performed by means of the mechanical interaction of light with a moving mirror. More specifically, our optomechanical apparatus produces a radiation field whose amplitude fluctuations (including those of quantum origin) are continuously observed. The result of such measurement is available through a meter channel that actually monitors the mirror motion. The back-action of the measurement is almost completely confined to the signal field phase fluctuations, and its weak percolation in the amplitude quadrature is efficiently detected by the meter. As a consequence, the residual fluctuations of the signal amplitude, that remain unknown after exploiting the information brought about by the meter, are below the shot noise. In a measurement process, the SQL is a crucial threshold: one can reduce the noise down to the SQL by just using, in a noise eater, a beam sampler to measure the intensity fluctuations. On the contrary, in a classical apparatus a noise level below the shot can just be obtained inside the close loop containing the detector, i.e., in a destructive measurement. In other words, the quantum photon noise remains elusive in classical experiments, and it can just be catch by a QND measurement \cite{ref9}. This technique is therefore very promising for the application to sub-SQL sensors, including integrated micro-devices and future gravitational wave detectors, where it can either be used to produce sub-shot noise light in a quantum noise eater \cite{Yamamoto}, or directly integrated in the complete measurement procedure by performing a preliminary estimation of the field quantum fluctuations.
This work has been supported by MIUR (”PRIN 2010-2011” and ”QUANTOM”) and by INFN (”HUMOR” project). A.B. acknowledges support from the MIUR under the ”FIRB Futuro in ricerca 2013” funding program, project code RBFR13QUVI.
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\section*{Appendix A: The Pound-Drever-Hall and the double homodyne detections.}
The experimental setup is shown in Fig. \ref{exp_completo}.
On the laser bench, the laser radiation is split into two beams. The first one (auxiliary beam) is frequency shifted by means of two acousto-optic modulators (AOM) operating on opposite diffraction orders. A resonant electro-optic modulator (EOM) provides phase modulation at 13.3~MHz used for a Pound-Drever-Hall (PDH) detection scheme. The PDH signal allows to stabilize the laser frequency to the cavity resonance. The locking bandwidth is about 20 kHz and additional notch filters assure that the servo loop does not influence the system dynamics in the frequency region around the oscillator frequency.
The PDH signal is also bandpass filtered around 22 kHz, and added to the signal driving the intensity modulator of the noise eater acting on the main beam. We so implement a feedback cooling \cite{ref26} on the wheel oscillator, with two purposes: firstly, we improve its dynamic stability, that is otherwise critical due to the combined effect of optomechanical interaction and frequency locking servo loop \cite{ref27}. Secondly, we depress the fluctuations of the wheel oscillator, that would otherwise provide a major contribution to the overall cavity phase noise. We remind that the rms value of such phase noise is large enough that a simple linear expansion of the cavity optical response in not sufficient to account for the reflected field fluctuations. Therefore, even if feedback cooling is just effective on the peak of the wheel resonator, it reduces the contribution brought into the frequency range of interest by nonlinear mixing.
The second beam (main beam) is actively amplitude stabilized, reducing the noise by about 30 dB in the band 100kHz - 200 kHz. Both beams are sent to the experiment bench by means of single-mode, polarization maintaining optical fibers. The main beam is split by a polarizing beam-splitter (PBS), the outputs of which are sent into the two arms of a Michelson interferometer. The length of the reference arm is finely controlled by shifting its end mirror with an inductive transducer. On the other arm, the beam is overlapped to the auxiliary beam, with orthogonal polarizations, in a further PBS and then mode-matched to the optical cavity.
On the path of the radiation exiting from the Michelson interferometer, the two faces of a wedge window, with the bisector plane at the Brewster angle, pick up $1.5\%$ each of the p-polarized light arriving from the cavity, and respectively $6\%$ and $23\%$ of the s-polarized light from the reference beam. On the path of one of these reflections, a quarter-wave plate with the axes parallel to the polarizations adds an additional delay between the fields arriving from the cavity and the reference arms. The fields reflected by the two window faces are analyzed by homodyne setups, each composed of a half-wave plate that rotates the polarizations by $45^{\circ}$, a PBS, and a couple of photodiodes at the two outputs of the PBS. The difference signals of the two couples of photodiodes can be written respectively as $\,V_A \sin \phi\,$ and $\,V_B \cos \phi$, where $\phi$ is the phase difference between the fields coming from the two arms of the Michelson interferometer. The transimpedence gains of the detectors are set to compensate for the different collected powers, in order to have $V_A \simeq V_B$. We electronically derive a weighted average of the two signals $\,V_\mathrm{m} = \beta_1 V_A +\beta_2 (1-\beta_1) V_B \propto \sin(\phi + \phi_0(\beta_1,\beta_2))$, where $\beta_1$ can be chosen between 0 and 1, and $\beta_2 = \pm 1$, so that $-\pi/2 < \phi_0 < \pi/2$. The low frequency component of $V_\mathrm{m}$ is integrated and fed back to the position control of the reference arm mirror, so that the phase $\phi$ is locked to $-\phi_0$ with a servo bandwidth of about 1 kHz. Moreover, the fluctuations of $\delta V_\mathrm{m}$ are now proportional to the fluctuations of the phase quadrature of the field reflected by the cavity, plus a contribution that, due to the low detected power, can be considered as originated by additional vacuum fluctuations. In summary, such combined homodyne detection is equivalent to a standard homodyne detection of the phase quadrature of the radiation reflected from the cavity, and it additionally allows to choose and stabilize the phase difference between the two, orthogonally polarized fields that compose the transmitted main beam. We identify $\delta V_\mathrm{m}$ with our meter variable $Y_\mathrm{m}$.
\begin{figure}
\caption{Detailed scheme of the experimental setup. EOM: electro-optic modulator. AOM: acousto-optic modulator. Pol: polarizer. PBS: polarizing beam-splitter. LPF: low-pass filter. OI: optic isolator. PD: photodiode. FR: Faraday rotator.}
\label{exp_completo}
\end{figure}
\section*{Appendix B: Data acquisition and analysis}
We have acquired simultaneous data streams from three channels: the sum and difference outputs from the final balanced detection, and the meter $Y_{\mathrm{m}}$. The signals are sampled at 5 MHz and several 10 seconds data streams are acquired, separated by lapses of few seconds necessary for data storage. Such delays improve the randomness of the complete data sets, reducing the effect of long term relaxations. The stability of the mean beam power is better than $1 \%$ during the whole measurement period. The data elaborated to obtain the results shown in Figures \ref{fig3} and \ref{fig4} are taken respectively from 5 and 4 consecutive 10 s time series.
The 10 seconds temporal series are divided into 100 ms long intervals. A preliminary selection on the intervals is performed by setting upper limits on the peak and rms values of the sum signal. This selection procedure is useful to reject datasets plagued by strong noise spikes, mainly due to low ($\sim$kHz) frequency modes, generated by instabilities of the helium flux in the cryostat. We keep $\sim90 \%$ of the data intervals.
For each $n$-th interval, we calculate the discrete Fourier transform of the difference signal $\tilde{X}_{-}^{(n)}$, of the sum signal $\tilde{X}_{+}^{(n)}$, of the meter signal $\tilde{Y}_{\mathrm{m}}^{(n)}$, of the square of the meter signal $\tilde{Y}_{\mathrm{sqm}}^{(n)}$ (we distinguish in the following the experimental signal $\tilde{X}_{+}$ from the signal variable $X_\mathrm{s}$ that is obtained from $\tilde{X}_{+}$ after subtraction of the electronic noise and normalization to the SQL).
The spectra to be evaluated are the sum and difference power spectra $S_{X_+X_+}$ and $S_{X_-X_-}$, and the cross-correlation contributions. For all such spectra, we use correct estimators as discussed below in the sub-section ``The statistical estimators".
The final steps of the analysis are the subtraction of the detection electronic noise, and the normalization of the sum and the residual spectra to the SQL. The obtained $S_{X_\mathrm{s}X_\mathrm{s}}$ is plotted in Figures \ref{fig3} and \ref{fig4} (wine traces).
\subsubsection*{Correlation with the square of the meter}
Due to the relatively large rms value of the cavity phase noise, mainly due to several mechanical resonances, a simple linear expansion of the cavity reflection function is not sufficient to account for the whole effect of such fluctuations on the reflected field quadratures. As a consequence, the best estimate of $X_{\mathrm{s}}$ would be a function $f(Y_{\mathrm{m}})$. If we consider its second order expansion, we deduce that a non-null correlation can also exist between $X_{+}$ and the square of $Y_{\mathrm{m}}$, and a more accurate estimate of the signal state can be performed by exploiting all the information provided by the meter signal, i.e., using an appropriate linear combination of $Y_{\mathrm{m}}$ and of its square. This residual uncertainty is found by subtracting from $S_{X_+X_+}$ also the correlation between $X_{+}$ and $Y_{\mathrm{sqm}}$. This is indeed the spectrum of the residual fluctuations that we have plotted in Figures \ref{fig3} and \ref{fig4} (dark green traces). It is compared with the theoretical calculation of $S_{\Delta X}$, that is based on linear expansions of the equations of motion. We remark however that even without the use of the correlation with $Y_{\mathrm{sqm}}$, the normalized spectrum of the residual fluctuations falls below the unit. No further improvements have been obtained by considering correlation with higher order in $Y_{\mathrm{m}}$.
Even the subtraction of just the correlation with $Y_{\mathrm{sqm}}$ from the spectrum $S_{X_+X_+}$ is interesting, as shown in Fig. \ref{fig9}, since it removes some peaks originating from the nonlinearity of the system, improving the agreement with the model. We point out that this is a confirmation of the existence of a quadratic nonlinearity. The correlation with $Y_{\mathrm{sqm}}$ is particularly meaningful around two peaks at $\sim163$ kHz and $\sim187$ kHz, but it also slightly improves the residual around the minimum.
\begin{figure}
\caption{(a) Spectra of the signal (experimental spectrum of the sum signal from the homodyne detection, with the electronic noise subtracted, and normalized to SQL) (purple), the same after subtraction of the correlation with the square of the meter (dark green), and theoretical model (cyan), for the reference phase corresponding to Figure \ref{fig3}
\label{fig9}
\end{figure}
\subsubsection*{The statistical estimators}
We have to estimate power spectra (such as $S_{X_{+}X_{+}}$ and $S_{X_{-}X_{-}}$), as well as cross-correlation contributions (such as $|S_{X_+Y_\mathrm{m}}|^2/S_{Y_\mathrm{m}Y_\mathrm{m}}$), starting from a finite number $N$ of experimental, Fourier transformed time series. For the power spectrum of a variable $X$, a good estimator is straightforwardly $\hat{S}_{XX} = \sum_{n=1,N} |\tilde{X}^{(n)}|^2/N\,$. On the other hand, finding a correct, unbiased indicator for the cross-correlation contribution is not obvious. We have therefore chosen a different point of view.
We are willing to estimate the residual fluctuations of $X$ that remains once the information brought by $Y$ is optimally used (the subscripts of $X$ and $Y$ are omitted in this discussion for the sake of clarity). In a linear system, the information that can be extracted from $Y$ can be written as $\alpha(\omega) \tilde{Y}$, where $\alpha(\omega)$ is a complex function. Therefore, we have to find the function $\alpha(\omega)$ that minimizes the spectral density of $S_{\Delta X}^{\alpha} := \langle |\tilde{X} - \alpha \tilde{Y}|^2 \rangle = S_{XX} + |\alpha|^2 S_{YY} - 2 \mathrm{Re} (\alpha S_{XY})$. By deriving with respect to $\alpha$, we find that its optimal value is
$\alpha_{\mathrm{opt}} = (S_{XY})^*/S_{YY}$
and the lowest residual spectrum is indeed
$S_{\Delta X}^{\mathrm{opt}} = S_{XX}-|S_{XY}|^2/S_{YY}$.
Any different $\alpha$ gives an overestimation of the optimal residual spectrum. On the other hand, for a given $\alpha$, we have a correct, unbiased estimator of the residual spectrum, that is
\begin{equation}
\hat{S}_{\Delta X}^{\alpha} = 1/N \left(\sum |\tilde{X}^{(n)}|^2 + |\alpha|^2 \sum|\tilde{Y}^{(n)}|^2 - 2 \mathrm{Re}\left(\alpha \sum (\tilde{X}^{(n)})^* \tilde{Y}^{(n)}\right) \right) \,.
\label{stima}
\end{equation}
The function $\alpha$ could be chosen {\it a priori}, e.g. on the basis of a model, but for a more realistic analysis we have derived it from the experimental data using the definition of $\alpha_{\mathrm{opt}}$ as guideline, as described in the following. We separate the $N$ intervals into two independent half-sets, according to the parity of the index $n$. From the first half-set we calculate $\alpha$ as
$\alpha_{\mathrm{odd}} = \sum_{\mathrm{odd}\, n} \tilde{X}^{(n)} (\tilde{Y}^{(n)})^*/\sum_{\mathrm{odd}\, n}|\tilde{Y}^{(n)}|^2$,
and from the second half-set we calculate the residual spectrum following Eq. (\ref{stima}), where the sums are taken over the even indexes. We then repeat the procedure by exchanging the two half-sets, and we finally take the average over the two resulting residual spectra. If we calculate the expectation value of our final spectrum $S_{\mathrm{\Delta X}}^{\mathrm{exp}}$, we find:
\begin{displaymath}
\begin{array}{lcl}
E\left[ S_{\Delta X}^{\mathrm{exp}} \right] &
= & \frac{1}{N} \langle \, \sum_{\mathrm{even}\, n} |\tilde{X}^{(n)}|^2 + |\alpha_{\mathrm{odd}}|^2 \sum_{\mathrm{even}\, n}|\tilde{Y}^{(n)}|^2 - 2 \mathrm{Re}(\alpha_{\mathrm{odd}} \sum_{\mathrm{even}\, n} (\tilde{X}^{(n)})^* \tilde{Y}^{(n)}) \\
& & \quad + \, (\mathrm{even} \leftrightarrow \mathrm{odd}) \, \rangle
\\
& = & S_{XX} + \langle |\alpha_{\mathrm{e/o}}|^2 \rangle S_{YY} - 2 \mathrm{Re} \left( \langle \alpha_{\mathrm{e/o}} \rangle S_{XY} \right)
\end{array}
\end{displaymath}
where we have used the independence of the two half-data sets, so that, e.g., $\,\langle \alpha_{\mathrm{odd}} \sum_{\mathrm{even}\, n} f^{(n)} \rangle = \langle \alpha_{\mathrm{odd}} \rangle \langle \sum_{\mathrm{even}\, n} f^{(n)} \rangle \, $ and $\, \langle \alpha_{\mathrm{odd}} \rangle = \langle \alpha_{\mathrm{even}} \rangle := \langle \alpha_{\mathrm{e/o}} \rangle$. Since $ \, \langle |\alpha_{\mathrm{e/o}}|^2 \rangle \ge |\langle \alpha_{\mathrm{e/o}} \rangle|^2 \,$ (a relation valid for any stochastic variable), we can write
$E\left[ S_{\Delta X}^{\mathrm{exp}} \right] \,\ge\, S_{\Delta X}^{\langle\alpha_{\mathrm{e/o}}\rangle} \,\ge\, S_{\Delta X}^{\mathrm{opt}}$.
Therefore, our experimental evaluation of the residual spectrum provides an unbiased, conservative estimator of the residual spectrum.
\begin{figure}
\caption{Difference between the ``odd" and ``even" estimates of the residual fluctuations, normalized to its statistical uncertainty.}
\label{chk}
\end{figure}
We have tested the compatibility of the two independent ``odd"/``even" estimates by calculating their difference normalized to its statistical uncertainty (i.e., to twice the standard deviation of their average). The result is shown in Fig. \ref{chk} for the QND frequency region of Fig. \ref{fig4}b. The normalized differences have an average value of $-0.024\pm0.21$ and a standard deviation of $0.86\pm0.21$, figures compatible with a normal distribution.
The above discussion can be extended to the case of two information channels $Y_1$ and $Y_2$, as follows. We have to find the functions $\alpha_1$ and $\alpha_2$ that minimizes the spectrum of $\,(X - \alpha_1 Y_1 - \alpha_2 Y_2)\,$. The residual spectrum is
\begin{equation}
S_{\Delta X} = S_{XX}+|\alpha_1|^2 S_{Y_1Y_1}+ |\alpha_2|^2 S_{Y_2Y_2}-2 \mathrm{Re}\left(\alpha_1 S_{XY_1} \right)-2 \mathrm{Re}\left(\alpha_2 S_{XY_2} \right)+2 \mathrm{Re}\left(\alpha_1^{*} \alpha_2 S_{Y_1Y_2} \right)
\label{Sxyz}
\end{equation}
and the optimal weight functions are
\begin{equation}
\alpha_{1,\mathrm{opt}} = \frac{S_{Y_2Y_2}S^*_{XY_1}-S^*_{XY_2} S_{Y_1Y_2}}{S_{Y_1Y_1} S_{Y_2Y_2}-|S_{Y_1Y_2}|^2}
\end{equation}
\begin{equation}
\alpha_{2,\mathrm{opt}} = \frac{S_{Y_1Y_1}S^*_{XY_2}-S^*_{XY_1} S_{Y_2Y_1}}{S_{Y_1Y_1} S_{Y_2Y_2}-|S_{Y_1Y_2}|^2} \, .
\end{equation}
As in the case of the single correlation, in order to derive a correct estimator one can separate the data streams into two interlaced subsets, calculate the weight functions from the above expression using, in place of the spectra, averages on half-sets of the correspondent discrete Fourier transforms, calculate the residual spectra $S_{\Delta X}$ according to Eq. (\ref{Sxyz}), exchange the two subsets, and finally average the two results.
In our experiment, we have one single meter $Y_{\mathrm{m}}$. However, as we have already mentioned, a linear approximation is not sufficient to fully exploit it. The best estimate of $X_{\mathrm{s}}$ would be a function $f(Y_{\mathrm{m}})$, that we can ideally expand to the second order, as $f(Y_{\mathrm{m}}) \simeq \alpha_1(\omega) Y_{\mathrm{m}} + \alpha_2 (\omega) Y_{\mathrm{sqm}}$, thus returning to the previous, two channels case.
In the ideal case of an infinite number of measurements in a stationary system, the addition of further orders in $Y_{\mathrm{m}}$ can just improve the estimate. However, in the case of $N$ measurements each further channel adds statistical uncertainty. Moreover, the non-optimal estimator can even increase the residual spectrum if the correlation is not sufficiently strong. As already mentioned, in our case we have indeed verified that the residual spectrum is not further improved by considering higher order terms in $Y_{\mathrm{m}}$.
\section*{Appendix C: The model}
The Hamiltonian of the optomechanical system can be written as
\begin{equation}
H=\hbar\omega_{\mathrm{c}}a^{\dagger}a+\frac{1}{2}\hbar\omega_{m}(p^{2}+q^{2})
-\hbar G_{0}a^{\dagger}a q
\label{hamiltonian}
\end{equation}
where $a$ is the annihilation operator of the cavity mode at frequency $\omega_{\mathrm{c}}$, $p$ and $q$ are the momentum and position operators of the mechanical oscillator, the single-photon coupling strength is $G_0 =-(\omega_{\mathrm{c}}/L_{\mathrm{c}})\sqrt{\hbar/m \omega_m}$.
The evolution equations for the system are derived from the Hamiltonian with the inclusion of an intense laser field at frequency $\omega_{0}$, input vacuum field operators $a_1^{\mathrm{in}}$ (from the input mirror) and $a_2^{\mathrm{in}}$ (from cavity losses), and additional noise terms that will be listed below. They can be written in the frame rotating at the laser frequency, that is detuned by $\Delta_0 = \omega_0 - \omega_{\mathrm{c}}$ with respect to the cavity resonance, as
\begin{eqnarray}
\dot{q}&=&\omega_{m} p, \label{equazioniq}\\
\dot{p}&=&-\omega_{m} q - \gamma_{m} p + G_0 a^{\dagger}a + \xi, \\
\dot{a}&=&-\kappa a +\mathrm{i} \left(\Delta_0+\zeta+G_0 q \right)a +E_0 \nonumber \\
&& +\sqrt{2\kappa_1}\left( a_1^{\mathrm{in}}+\epsilon\right)+\sqrt{2\kappa_2} a_2^{\mathrm{in}} \, .\label{equazionia}
\end{eqnarray}
Here $E_0=\sqrt{2 \kappa_1 P/\hbar \omega_0}$ where $P$ is the input laser power and we take $E_0$ as real, which means that we use the driving laser as phase reference for the optical field.
The mechanical mode is affected by a viscous force with damping rate $\gamma_{m}$ and by a Brownian stochastic force $\xi(t)$.
We have included the laser excess amplitude noise with the real stochastic variable $\epsilon$. The additional cavity phase fluctuations are introduced by a stochastic term $\zeta$ in the detuning. The input fields correlations are
\begin{eqnarray}
&& \bigl\langle a_j^{\mathrm{in}}(t)a_j^{\mathrm{in}}(t')\bigr\rangle = \bigl\langle a_j^{\mathrm{in}, \dagger}(t)a_j^{\mathrm{in}, \dagger}(t')\bigr\rangle = \bigl\langle a_j^{\mathrm{in},\dagger}(t)a_j^{\mathrm{in}}(t')\bigr\rangle= 0,\quad \label{corr1}\\
&& \bigl\langle a_j^{\mathrm{in}}(t)a_j^{\mathrm{in},\dagger}(t')\bigr\rangle = \delta (t-t'), \,\,\, j=1,2.\label{corr2}
\end{eqnarray}
We consider the motion of the system around a steady state characterized by the intracavity electromagnetic field of amplitude $\alpha_s$, and the oscillator at a new position $q_s$, by writing:
\begin{eqnarray}
q&=&q_s+\delta q,\label{eq:ss+fluct_q}\\
p&=&p_s+\delta p,\label{eq:ss+fluct_p}\\
a&=&\alpha_s+\delta a.\label{eq:ss+fluct_a}
\end{eqnarray}
Substituting Eqs.~(\ref{eq:ss+fluct_q})-(\ref{eq:ss+fluct_a}) into Eqs.~(\ref{equazioniq})-(\ref{equazionia}), we obtain the stationary solutions:
\begin{eqnarray}\label{eq:ss_0th}
q_s&=&\frac{G_0}{\omega_m}|\alpha_s|^{2},\\
p_s&=&0,\\
\alpha_s&=&\frac{E_{0}}{\kappa-\mathrm{i}\Delta}, \label{eq:ss_1th}
\end{eqnarray}
where $\Delta = \Delta_0 + G_0 q_s$, and the first order linearized equations for the fluctuations operators
\begin{eqnarray}
\delta \dot{q}&=&\omega_m \delta p,\label{linearq}\\
\delta \dot{p}&=&-\omega_m \delta q-\gamma_m \delta p+G_0\left(\alpha_s\delta a^{\dagger}+\alpha_s^{*} \delta a \right)+\xi,\\
\delta \dot{a}&=&-\left(\kappa-\mathrm{i}\Delta\right)\delta a+\mathrm{i} G_0\alpha_s\delta q+\sqrt{2\kappa_1}(a_1^{\mathrm{in}}+\epsilon) + \mathrm{i} \alpha_s \zeta + \sqrt{2\kappa_2}a_2^{\mathrm{in}}.
\label{lineara}
\end{eqnarray}
The Fourier transformed of Eqs. (\ref{linearq})-(\ref{lineara}), are solved for $a(\omega)$ (we call $a(\omega)$ the Fourier transformed of $\delta a(t)$ and $a^{\dagger}(\omega)$ the Fourier transformed of $\delta a^{\dagger}(t)$, with the same notation for the other fields).
Using the input/output relations
\begin{eqnarray}\label{eq:output}
E_\mathrm{R}&=&\sqrt{2\kappa_1}\alpha_s-\frac{E_0}{\sqrt{2 \kappa_1}} , \\
a^{\mathrm{out}}&=&\sqrt{2\kappa_1}\delta a-\left(a_1^{\mathrm{in}}+\epsilon\right) ,
\end{eqnarray}
we can write the output field, with average value
\begin{equation}
E_\mathrm{R}=\sqrt{\frac{P}{\hbar \omega_0}}\frac{\kappa-2\kappa_2+\mathrm{i}\Delta}{\kappa-\mathrm{i}\Delta}
\label{ER}
\end{equation}
and fluctuation operator
\begin{eqnarray}\label{eq:aout}
a^{\mathrm{out}}(\omega) &=& \nu_1(\omega) a_1^{\mathrm{in}}(\omega) + \nu_2(\omega)a_1^{\mathrm{in},\dagger}(\omega) + \nu_3(\omega)a_2^{\mathrm{in}}(\omega)+\nu_4(\omega)a_2^{\mathrm{in},\dagger}(\omega)\nonumber \\
&&+ \nu_5(\omega) \zeta(\omega)+\nu_6(\omega)\epsilon (\omega) + \nu_{7}(\omega) \xi (\omega) \, ,
\end{eqnarray}
where
\begin{eqnarray*}\label{eq:nu1}
\nu_1(\omega) &=& \frac{\kappa-2\kappa_2+\mathrm{i}\bigl(\Delta+\omega\bigr)}{\kappa-\mathrm{i}\bigl(\Delta+\omega\bigr)}+\frac{\mathrm{i} |G|^2 \kappa_1 \chi_\mathrm{eff}(\omega)}{\bigl[\kappa-\mathrm{i}\bigl(\Delta+\omega\bigr)\bigr]^2}, \\
\label{eq:nu2}
\nu_2(\omega) &=& \frac{\mathrm{i} G^2 \kappa_1 \chi_\mathrm{eff}(\omega)}{\bigl[\kappa-\mathrm{i}\bigl(\Delta+\omega\bigr)\bigr]\bigl[\kappa+\mathrm{i}\bigl(\Delta-\omega\bigr)\bigr]}, \\
\nu_3(\omega) &=&\sqrt{\frac{\kappa_2}{\kappa_1}}\left(\nu_1(\omega)+1\right), \\
\nu_4(\omega) &=&\sqrt{\frac{\kappa_2}{\kappa_1}}\nu_2(\omega), \nonumber\\
\label{eq:nu5}
\nu_5(\omega)&=&\frac{\mathrm{i}\alpha_s}{\sqrt{2\kappa_1}}\left(\nu_1(\omega)-\nu_2(\omega)+1 \right), \\
\nu_6(\omega)&=&\nu_1(\omega)+\nu_2(\omega), \\
\nu_{7}(\omega) &=& \frac{i G \sqrt{\kappa_1} \chi_\mathrm{eff}(\omega)}{\kappa-\mathrm{i}\bigl(\Delta+\omega\bigr)}\,.
\end{eqnarray*}
Here $G=G_0 \sqrt{2} \alpha_s$ is the effective coupling strength, and
\begin{equation}
\chi _\mathrm{ eff}(\omega )=\omega _{m}\Biggl[\omega_{m}^{2}-\omega^{2}-\mathrm{i}\omega \gamma _{m}+\frac{|G|^2\Delta\omega _{m}}{\bigl(\kappa -\mathrm{i}\omega \bigr)^{2}+\Delta^{2}}\Biggr]^{-1} \label{chieffD}
\end{equation}
is the effective mechanical susceptibility modified by the optomechanical coupling.
In the experiment, we split the output field into a weak meter and a signal. They have different optical losses, that are considered in the model using the beam-splitter relations
\begin{eqnarray}
a_{\mathrm{m}} = \sqrt{\eta_{\mathrm{m}}} \,a^{\mathrm{out}} + \sqrt{1-\eta_{\mathrm{m}}} \,a_{\mathrm{3}} \\
a_{\mathrm{s}} = \sqrt{\eta_{\mathrm{s}}} \,a^{\mathrm{out}} + \sqrt{1-\eta_{\mathrm{s}}} \,a_{\mathrm{4}}
\end{eqnarray}
where $a_{3,4}$ are vacuum input fields and $\eta_{\mathrm{m,s}}$ are the efficiencies respectively for the meter and the signal. The correlation between $a_3$ and $a_4$ could be considered by introducing in the model the beam-splitter that separates the meter and signal fields, followed by further beam-splitters modeling the optical losses. However, due to the low efficiency $\eta_{\mathrm{m}}$, to reproduce the results we can safely neglect such correlation and consider vacuum fields $a_{3,4}$ satisfying the relations (\ref{corr1})-(\ref{corr2}) with $j$ extended to (3,4).
We can similarly consider the reference field as contributing to the meter and the signal with independent effective vacuum fields, already included phenomenologically in $a_{3,4}$. To account for the non perfect mode matching we must consider that the field in the non-resonant modes is reflected by the cavity input mirror, and impinges on the detectors where, in first approximation, it does not interfere with the main mode. Therefore, we do not sum the fields, but the fluctuating intensities. The above relations are modified by replacing $\,P \rightarrow \eta_{mm} P, \,$ $\,(a_1^{\mathrm{in}}+\epsilon) \rightarrow \sqrt{\eta_{mm}}\left(a_1^{\mathrm{in}}+\epsilon\right)+\sqrt{1-\eta_{mm}}\,a_5\,$ and $\,a^{\mathrm{out}} \rightarrow \sqrt{\eta_{mm}} a^{\mathrm{out}} + \sqrt{1 - \eta_{mm}} \left(\sqrt{1 - \eta_{mm}} (a_1^{\mathrm{in}}+\epsilon) - \sqrt{\eta_{mm}} a_5\right)$ where $\eta_{mm}$ is the mode matching coefficient and $a_5$ is a further vacuum input field.
The general quadrature of a field $a$ is defined as $\, a \mathrm{e}^{-i \phi}+a^{\dagger}\mathrm{e}^{i \phi}$. For the meter field, we measure the phase quadrature with respect to the field reflected by the cavity. The latter, according to Eq. (\ref{ER}), is dephased by $\,\phi_\mathrm{R} = \arctan \Delta/(\kappa-2\kappa_2) + \arctan \Delta/\kappa \,$ with respect to our reference (i.e., the field at the cavity input). The measured quadrature of the meter is therefore defined by $\,\phi_{\mathrm{m}} = \phi_\mathrm{R} +\pi/2. \,$ Concerning the signal, we are defining as $X_\mathrm{s}$ the amplitude quadrature at the output of the polarizer, i.e., the quadrature defined by the superposition of main and reference fields: $\,\phi_{\mathrm{s}} = \phi_{\mathrm{R}} - \arcsin \left(\sin \phi_0/\sqrt{1+P_\mathrm{R}/P_{\mathrm{ref}}+2\sqrt{P_\mathrm{R}/P_{\mathrm{ref}}} \cos \phi_0}\right) \,$ where $P_\mathrm{R}\,$ ($P_{\mathrm{ref}}$) is the power transmitted by the polarizer and coming from the cavity (reference) arm (Fig. 1c of the main text).
Theoretical curves are obtained by calculating symmetrized power spectra and cross-correlation spectra of the variables $\,Y_{\mathrm{m}}= a_{\mathrm{m}}(\omega) \mathrm{e}^{-\mathrm{i} \phi_{\mathrm{m}}}+a_{\mathrm{m}}^{\dagger}(\omega)\mathrm{e}^{\mathrm{i} \phi_{\mathrm{m}}}\,$ and $\,X_{\mathrm{s}} = a_{\mathrm{s}}(\omega) \mathrm{e}^{-\mathrm{i} \phi_{\mathrm{s}}}+a_{\mathrm{s}}^{\dagger}(\omega)\mathrm{e}^{\mathrm{i} \phi_{\mathrm{s}}}.\,$ The oscillator and cavity parameters, quoted in main text, are all measured independently and fixed in the theoretical calculations. The calculated coupling strengths are $G_0/2\pi = -3.85$ Hz and $G/2\pi = -740$ kHz (at resonance). The input power is $P = 38$ mW. The spectrum $S_{\epsilon\epsilon}$ is 1/4 of the excess intensity noise, normalized to SQL. In our case, we set $S_{\epsilon\epsilon}=0.25\times P/(24 \mathrm{mW})$. The stochastic term in the detuning is linked to the cavity length fluctuations $\delta l$ by $\zeta = \delta l \times \omega_{\mathrm{c}}/L_{\mathrm{c}}$. Its spectrum is modeled with a Lorentzian peak at 208 kHz that roughly reproduces the resonance of the second oscillator mode, plus a $1/\omega^2$ background. The total background amplitude is left as free fitting parameters. The mode-matching parameter and the efficiencies, both for the signal and for the meter, are measured independently. The detuning and the signal phase $\phi_{\mathrm{s}}$ are free fitting parameters.
In addition to the curves already compared with the experimental results in the main text, we show in Figure \ref{fig11} the contributions of the different noise sources to the power spectrum $S_{X_\mathrm{s}X_\mathrm{s}}$ plotted in Fig. 7 of the main text.
\begin{figure}
\caption{Theoretical calculation of the spectrum reported in Figure 7 of the main text (black solid line), normalized to SQL (grey dashed line), together with its contributions: input laser noise (blue dashed-double dotted line), thermal noise (red solid line), vacuum noise entering through cavity losses (dark yellow, long dash-double dotted line), cavity phase noise (greed dashed line).}
\label{fig11}
\end{figure}
The contribution of the cavity phase noise cancels at the bare oscillator frequency, as discussed in Ref. [34] of the main text. The contribution of the laser noise (quantum noise and classical amplitude noise) reaches a minimum at a frequency determined by the best destructive interference between the fluctuations of the laser field (modified by the optical cavity) and those mediated by the optomechanical interaction (originated by the term proportional to $\delta q$ in Eq. (\ref{lineara})). With the parameters used for this spectrum, even this interference occurs close to $\omega_m$. This coincidence allows to observe ponderomotive squeezing, that would otherwise be hidden by the cavity phase noise. The squeezing depth is eventually limited by thermal noise.
\begin{figure}
\caption{Theoretical calculation of the spectrum of the residual fluctuations of $X_\mathrm{s}
\label{fig12}
\end{figure}
In Figure \ref{fig12} we show, for the same parameters, the spectrum of the residual fluctuations of $X_\mathrm{s}$ after the subtraction of the correlation with the meter. To put into evidence the different noise contributions, we start by the residual spectrum where just the laser noise is present, than we add cavity losses, thermal noise and cavity phase noise. Before the last contribution, the interference effect above described is no more necessary to fall below the SQL, and the region where it happens is potentially much larger. However, in agreement with the comment to the previous figure, we see that eventually the cavity phase noise strongly limits the width of this QND region. Its cancellation at $\omega_m$ is crucial, while the minimum of the spectrum is again limited by thermal noise.
\end{document} |
\begin{document}
\title[Barycentric subdivisions of cubical complexes]
{Face numbers of barycentric subdivisions of cubical
complexes}
\author{Christos~A.~Athanasiadis}
\address{Department of Mathematics\\
National and Kapodistrian University of Athens\\
Panepistimioupolis\\
15784 Athens, Greece}
\email{[email protected]}
\date{December 18, 2020}
\thanks{ 2010 \textit{Mathematics Subject Classification.}
Primary 05E45; \, Secondary 26C10, 52B12.}
\thanks{ \textit{Key words and phrases}.
Barycentric subdivision, cubical complex, $h$-polynomial,
Eulerian polynomial, real-rootedness.}
\begin{abstract}
The $h$-polynomial of the barycentric subdivision
of any $n$-dimensional cubical complex with nonnegative
cubical $h$-vector is shown to have only real roots and
to be interlaced by the Eulerian polynomial of type
$B_n$. This result applies to barycentric subdivisions
of shellable cubical complexes and, in particular, to
barycentric subdivisions of cubical convex polytopes
and answers affirmatively a question of Brenti,
Mohammadi and Welker.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:intro}
A fundamental problem in algebraic and geometric
combinatorics is to characterize, or at least obtain
significant information about, the face enumerating
vectors of triangulations of various topological
spaces, such as balls and spheres \cite{StaCCA}.
Face enumerating vectors are often presented in
the form of the $h$-polynomial (see
Section~\ref{sec:enu} for definitions). Properties
such as unimodality, log-concavity,
$\gamma$-positivity and real-rootedness have been of
primary interest \cite{Ath18, Bra15, Bre94b, Sta89}.
One expects that the `nicer' the triangulation is
combinatorially and the space being triangulated is
topologically, the better the behavior of the
$h$-polynomial is.
Following this line of thought, Brenti and
Welker~\cite{BW08} considered an important and well
studied triangulation in mathematics, namely the
barycentric subdivision. They studied the
transformation of the $h$-polynomial of a simplicial
complex $\Delta$ under barycentric subdivision and
showed that the resulting $h$-polynomial has only
real roots (a property with strong implications) for
every simplicial complex $\Delta$ with nonnegative
$h$-polynomial. They asked \cite[Question~3.10]{BW08}
whether the $h$-polynomial of the barycentric
subdivision of any convex polytope has only real
roots, suspecting an affirmative answer (see
\cite[p.~105]{MW17}). This question was
raised again by Mohammadi and
Welker~\cite[Question~35]{MW17} and, as is
typically the case in face enumeration,
it is far more interesting and more challenging for
general polytopes and polyhedral complexes, than it
is for simplicial polytopes
and simplicial complexes. Somewhat surprisingly, no
strong evidence has been provided in the
literature that such a result may (or may not) hold
beyond the simplicial setting. One should also note
that barycentric subdivisions of boundary complexes
of polytopes form a special class of flag
triangulations of spheres and that the real-rootedness
property fails for the $h$-polynomials of this more
general class of triangulations in dimensions higher
that four \cite{Ga05}. At present, it is unclear
where the borderline between positive and negative
results lies.
Mohammadi and Welker (based on earlier discussions
with Brenti) suggested the class of cubical polytopes
as another good test case; see \cite[p.~105]{MW17}. Cubical
complexes and polytopes are important and mysterious
objects with highly nontrivial combinatorial properties
(see, for instance, \cite{Ad96, BBC97, Jo93, JZ00}).
They have been studied both for their own independent
interest, and for the role they play in other
areas of mathematics. Given the intricacy of their
combinatorics, it comes as no surprise that the
question of Brenti and Welker turns out to be more
difficult for them than for simplicial complexes. The
following theorem provides
the first general positive result on this question,
since \cite{BW08} appeared, and suggests that an
affirmative answer should be expected at least for
broad classes of nonsimplicial convex polytopes (or
even more general cell complexes and posets).
\begin{theorem} \label{thm:main}
The $h$-polynomial of the barycentric subdivision
of any shellable cubical complex has only real roots.
In particular, barycentric subdivisions of cubical
polytopes have this property.
\end{theorem}
The case of cubical polytopes was also studied recently
by Hlavacex and Solus~\cite{HS20+}. Using the concept
of shellability and the theory of interlacing
polynomials, they gave an affirmative answer
for cubical complexes which admit a special type of
shelling and applied their result to certain
families of cubical polytopes, such as cuboids,
capped cubical polytopes and neighborly cubical
polytopes.
The proof of the aforementioned result of~\cite{BW08}
applies a theorem of Br\"and\'en~\cite{Bra06} on the
subdivision operator \cite[Section~3.3]{Bra15} to a
formula for the $h$-polynomial of the barycentric
subdivision of a simplicial complex (see
Remark~\ref{rem:BW-formula}). The proof of
Theorem~\ref{thm:main} is motivated by the proof of
the result of~\cite{BW08}, given and extended to the
setting of uniform triangulations of simplicial
complexes in~\cite{Ath20+} (the latter was partially
motivated by \cite[Example~8.1]{Bra15}). To explain
further, we let $h(\Delta, x) = \sum_{i=0}^{n+1}
h_i(\Delta) x^i$ denote the $h$-polynomial and
${\rm sd}(\Delta)$ denote
the barycentric subdivision of an $n$-dimensional
simplicial complex $\Delta$. As already shown
in~\cite{BW08}, there exist polynomials with
nonnegative coefficients $p_{n,k}(x)$ for $k \in
\{0, 1,\dots,n+1\}$, which depend only on $n$ and $k$,
such that
\begin{equation} \label{eq:BW}
h({\rm sd}(\Delta), x) \ = \ \sum_{k=0}^{n+1} h_k(\Delta)
p_{n,k}(x)
\end{equation}
for every $n$-dimensional simplicial complex $\Delta$.
For every $n \in {\mathbb N}$, the polynomials $p_{n,k}(x)$ can
be shown \cite[Example~8.1]{Bra15} to have only real
roots and to form an interlacing sequence. This implies
that their nonnegative linear combination
$h({\rm sd}(\Delta), x)$ also has only real (negative)
roots and that it is interlaced by $p_{n,0}(x)$, which
equals the classical $(n+1)$st Eulerian polynomial
$A_{n+1}(x)$ \cite[Section~1.4]{StaEC1}. The interlacing
condition implies that the roots of $h({\rm sd}(\Delta), x)$
are not arbitrary, but rather that they lie in certain
intervals that depend only on the dimension $n$, formed
by zero and the roots of $A_{n+1}(x)$. The polynomial
$p_{n,k}(x)$ can be interpreted as the $h$-polynomial
of the relative simplicial complex obtained from the
barycentric subdivision of the $n$-dimensional simplex
by removing all faces lying on $k$ facets of the
simplex \cite[Section~5]{Ath20+}
\cite[Section~4.2]{HS20+}.
This paper presents a similar picture for cubical
complexes. We define (see Definition~\ref{def:pBnk})
polynomials $p^B_{n,k}(x)$ for $k \in \{0, 1,\dots,n+1\}$
as the $h$-polynomials of relative simplicial complexes
obtained from the barycentric subdivision of the
$n$-dimensional cube by removing all faces lying on
certain facets of the cube and prove (see
Theorem~\ref{thm:h-trans}) that Equation~(\ref{eq:BW})
continues to hold when $\Delta$ is replaced by an
$n$-dimensional cubical complex ${\mathcal L}$, $p_{n,k}(x)$ is
replaced by $p^B_{n,k}(x)$ and the $h_k(\Delta)$ are
replaced by the entries of the (normalized) cubical
$h$-vector of ${\mathcal L}$, introduced and studied by
Adin~\cite{Ad96}. We provide recurrences (see
Proposition~\ref{prop:pBnk}) for the polynomials
$p^B_{n,k}(x)$ which guarantee that they form an
interlacing
sequence for every $n \in {\mathbb N}$ and conclude that
$h({\rm sd}({\mathcal L}), x)$ has only real (negative) roots and
that it is interlaced by the $n$th Eulerian polynomial
$B_n(x)$ of type $B$ for every $n$-dimensional cubical
complex ${\mathcal L}$ with nonnegative cubical $h$-vector (see
Corollary~\ref{cor:main}). This implies
Theorem~\ref{thm:main}, since shellable cubical
complexes were shown~\cite{Ad96} to have nonnegative
cubical $h$-vector and boundary complexes of convex
polytopes are shellable~\cite{BM71}.
The main results of this paper apply to cubical
regular cell complexes (equivalently, to cubical
posets) and will be stated at this level of
generality. What comes perhaps unexpectedly
is the fact that the transformation of a cubical
$h$-polynomial into a simplicial one can be so well
behaved. Corollary~\ref{cor:main} has nontrivial
applications to triangulations of simplicial
complexes as well; see Remark~\ref{rem:simplicial}.
\section{Face enumeration of simplicial and cubical
complexes} \label{sec:enu}
This section recalls some definitions and background
on the face enumeration of simplicial and cubical
complexes and their triangulations, and shellability.
For more information and any undefined terminology,
we recommend the books \cite{HiAC, StaCCA}. All
cell complexes considered here are assumed to be
finite. Throughout this paper, we set ${\mathbb N} =
\{0, 1, 2,\dots\}$ and denote by $|S|$ the cardinality
of a finite set $S$.
\subsection{Simplicial complexes}
\label{sec:simplicial}
An $n$-dimensional \emph{relative simplicial complex}
\cite[Section~III.7]{StaCCA} is a pair $(\Delta,
\Gamma)$, denoted $\Delta / \Gamma$, where $\Delta$
is an (abstract) $n$-dimensional simplicial complex
and $\Gamma$ is a subcomplex of $\Delta$. The
\emph{$f$-polynomial} of $\Delta / \Gamma$ is defined
as
\[ f(\Delta / \Gamma, x) \ = \ \sum_{i=0}^{n+1} f_{i-1}
(\Delta / \Gamma) x^i, \]
where $f_j(\Delta / \Gamma)$ is the number of
$j$-dimensional faces of $\Delta$ which do not belong
to $\Gamma$. The \emph{$h$-polynomial} is defined as
\begin{eqnarray*} \label{def:simpl-h}
h(\Delta/\Gamma, x) & = & (1-x)^{n+1} f(\Delta/\Gamma,
\frac{x}{1-x}) \ = \ \sum_{i=0}^{n+1} f_{i-1} (\Delta/\Gamma)
\, x^i (1-x)^{n+1-i} \\ & = & \sum_{F \in \Delta/\Gamma}
x^{|F|} (1-x)^{n+1-|F|} \ := \ \sum_{k=0}^{n+1} \, h_k
(\Delta/\Gamma) x^k \nonumber
\end{eqnarray*}
\noindent
and the sequence $h(\Delta/\Gamma) := (h_0(\Delta/\Gamma),
h_1(\Delta/\Gamma),\dots,h_{n+1}(\Delta/\Gamma))$ is called
the \emph{$h$-vector} of $\Delta/\Gamma$. Note that
$f(\Delta / \Gamma, x)$ has only real roots if and only
if so does $h(\Delta/\Gamma, x)$. When $\Gamma$ is empty,
we get the corresponding invariants of $\Delta$ and drop
$\Gamma$ from the notation. Thus, for example,
$h(\Delta, x) = \sum_{k=0}^{n+1} h_k (\Delta) x^k$ is
the (usual) $h$-polynomial of $\Delta$.
{\smallsetminus}allskip
Suppose now that $\Delta$ triangulates an
$n$-dimensional ball. Then, the boundary complex
$\partial \Delta$ is a triangulation of an
$(n-1)$-dimensional sphere and the \emph{interior
$h$-polynomial} of $\Delta$ is defined as
$h^\circ(\Delta, x) = h(\Delta / \partial \Delta,
x)$. The following statement is a special case of
\cite[Lemma~6.2]{Sta87}.
\begin{proposition} \label{prop:hsymmetry}
{\rm (\cite{Sta87})}
Let $\Delta$ be a triangulation of an
$n$-dimensional ball. Let $\Gamma$ be a
subcomplex of $\partial\Delta$ which is homeomorphic
to an $(n-1)$-dimensional ball and $\bar{\Gamma}$ be
the subcomplex of $\partial\Delta$ whose facets are
those of $\partial\Delta$ which do not belong to
$\Gamma$. Then,
\[ h(\Delta / \bar{\Gamma}, x) \ = \ x^{n+1}
h(\Delta / \Gamma, 1/x) . \]
Moreover, $h^\circ(\Delta, x) = x^{n+1} h(\Delta,
1/x)$.
\end{proposition}
\subsection{Cubical complexes}
\label{sec:cubical}
A \emph{regular cell complex} \cite[Section 4.7]{OM}
is a (finite) collection ${\mathcal L}$ of subspaces of a
Hausdorff space $X$, called \emph{cells} or
\emph{faces}, each homeomorphic to a closed unit
ball in some finite-dimensional Euclidean space,
such that: (a) $\varnothing \in {\mathcal L}$; (b) the relative
interiors of the nonempty cells partition $X$;
and (c) the boundary of any cell in ${\mathcal L}$ is a
union of cells in ${\mathcal L}$. The \emph{boundary complex}
of $\sigma \in {\mathcal L}$, denoted by $\partial \sigma$,
is defined as the regular cell complex consisting
of all faces of ${\mathcal L}$ properly contained in $\sigma$.
A regular cell complex ${\mathcal L}$ is called \emph{cubical}
if every nonempty face of ${\mathcal L}$ is combinatorially
isomorphic to a cube. A convex polytope is called
\emph{cubical} if so is its boundary complex.
Given a cubical complex ${\mathcal L}$ of
dimension $n$, we denote by $f_k({\mathcal L})$ the number of
$k$-dimensional faces of ${\mathcal L}$. The cubical
$h$-polynomial was introduced and studied by
Adin~\cite{Ad96} as a (well behaved) analogue of the
(simplicial) $h$-polynomial of a simplicial complex.
Following \cite[Section~4]{EH00}, we define the
(normalized) \emph{cubical $h$-polynomial} of ${\mathcal L}$ as
\begin{equation} \label{def:cub-h}
(1+x) h({\mathcal L}, x) \ = \ 1 \, + \, \sum_{k=0}^n \,
f_k({\mathcal L}) \, x^{k+1} \left( \frac{1-x}{2} \right)^{n-k}
+ \ (-1)^n \, \widetilde{{\rm ch}i} ({\mathcal L}) x^{n+2} ,
\end{equation}
where $\widetilde{{\rm ch}i} ({\mathcal L}) = -1 + \sum_{k=0}^n
(-1)^k f_k({\mathcal L})$ is the reduced Euler characteristic
of ${\mathcal L}$ (the only difference from Adin's definition
is that all coefficients of $h({\mathcal L}, x)$ have been
divided by $2^n$ and, therefore, are not necessarily
integers). We note that $h({\mathcal L}, x)$ is indeed a
polynomial in $x$ of degree at most $n+1$. The
(normalized) \emph{cubical $h$-vector} of ${\mathcal L}$ is the
sequence $h({\mathcal L}) = (h_0({\mathcal L}), h_1({\mathcal L}),\dots,h_{n+1}
({\mathcal L}))$, where $h({\mathcal L}, x) = \sum_{k=0}^{n+1} h_k({\mathcal L})
x^k$.
{\smallsetminus}allskip
Adin showed that $h({\mathcal L}, x)$ has nonnegative
coefficients for every shellable cubical complex ${\mathcal L}$
\cite[Theorem~5~(iii)]{Ad96} (his result is stated for
abstract cubical complexes with the intersection
property, but the proof is valid without assuming the
later). He asked whether the same holds whenever ${\mathcal L}$
is Cohen--Macaulay
\cite[Question~1]{Ad96}. The coefficient $h_k({\mathcal L})$ is
known to be nonnegative for every Cohen--Macaulay ${\mathcal L}$
for $k \in \{0, 1\}$, since $h_0({\mathcal L}) = 1$ and
$h_1({\mathcal L}) = (f_0({\mathcal L}) - 2^n)/2^n$, for $k=n$
\cite[Corollary~1.2]{Ath12} and for $k = n+1$, since
$h_{n+1}({\mathcal L}) = (-1)^n \widetilde{{\rm ch}i} ({\mathcal L})$,
and for every $k$ in the special case that ${\mathcal L}$ is the
cubical barycentric subdivision of a Cohen--Macaulay
simplicial complex \cite{Het96} (see also
Remark~\ref{rem:simplicial}).
\subsection{Barycentric subdivision and shellability}
\label{sec:shell}
The \emph{barycentric subdivision} of a regular cell
complex ${\mathcal L}$ is denoted by ${\rm sd}({\mathcal L})$
and defined as the abstract simplicial complex whose
faces are the chains $\sigma_0 \subset \sigma_1
\subset \cdots \subset \sigma_k$ of nonempty faces
of ${\mathcal L}$. The natural restriction of ${\rm sd}({\mathcal L})$ to a
nonempty face $\sigma \in {\mathcal L}$ is exactly the
barycentric subdivision ${\rm sd}(\sigma)$ of (the
complex of faces of) $\sigma$.
Similarly, by the barycentric subdivision ${\rm sd}(Q)$
of a convex polytope $Q$ we mean that of the complex
of faces
of $Q$. Since ${\rm sd}(Q)$ is a cone over ${\rm sd}(\partial
Q)$, we have $h({\rm sd}(Q), x) = h({\rm sd}(\partial Q), x)$.
For the $n$-dimensional cube $Q$ we have $h({\rm sd}(Q),
x) = B_n(x)$, where $B_n(x)$ is the Eulerian polynomial
which counts signed permutations of $\{1, 2,\dots,n\}$
by the number of descents of type $B$; see, for
instance, \cite[Chapter~11]{Pet15}. The following
well known type $B$ analogue of Worpitzky's identity
\cite[Equation~(13.3)]{Pet15}
\begin{equation}
\label{eq:Bn-gen}
\frac{B_n(x)}{(1-x)^{n+1}} \ = \ \sum_{m \ge 0} \,
(2m+1)^n x^m
\end{equation}
will make computations in the following section
easier.
A regular cell complex ${\mathcal L}$ is called \emph{pure} if
all its facets (faces which are maximal with respect
to inclusion) have the same dimension. Such a
complex ${\mathcal L}$ is called \emph{shellable} if either
it is zero-dimensional, or else there exists a
linear ordering $\tau_1, \tau_2,\dots,\tau_m$ of its
facets, called a \emph{shelling}, such that (a)
$\partial \tau_1$ is shellable; and (b) for
$2 \le j \le m$, the complex of faces of $\partial
\tau_j$ which are contained in $\tau_1 \cup
\tau_2 \cup \cdots \cup \tau_{j-1}$ is pure, of the
same dimension as $\partial \tau_j$, and there exists
a shelling of $\partial \tau_j$ for which the facets
of $\partial \tau_j$ contained in $\tau_1 \cup \tau_2
\cup \cdots \cup \tau_{j-1}$ form an initial segment.
A fundamental result of Bruggesser and Mani
\cite{BM71} states that $\partial Q$ is shellable
for every convex polytope $Q$. For the shellability
of cubical complexes in particular, see
\cite[Section~3]{EH00} \cite[Section~3]{HS20+}.
\section{The $h$-vector transformation}
\label{sec:h-trans}
This section studies the transformation which maps
the cubical $h$-vector of a cubical complex ${\mathcal L}$ to
the (simplicial) $h$-vector of the barycentric
subdivision ${\rm sd}({\mathcal L})$ and deduces
Theorem~\ref{thm:main} from its properties. We begin
with an important definition.
\begin{definition} \label{def:pBnk}
For $n \in {\mathbb N}$ and $k \in \{0, 1,\dots,n+1\}$ we
denote by ${\mathcal C}_{n,k}$ the relative simplicial complex
which is obtained from the barycentric subdivision of
the $n$-dimensional cube by removing
\begin{itemize}
\item[$\bullet$]
no face, if $k=0$,
\item[$\bullet$]
all faces which lie in one facet and $k-1$ pairs
of antipodal facets of the cube (making a total of
$2k-1$ facets), if $k \in \{1, 2,\dots,n\}$,
\item[$\bullet$]
all faces on the boundary of the cube, if $k=n+1$.
\end{itemize}
We define $p^B_{n,k}(x) = h({\mathcal C}_{n,k}, x)$ for $k
\in \{0, n+1\}$, and $p^B_{n,k}(x) =
2 h({\mathcal C}_{n,k}, x)$ for $k \in \{1, 2,\dots,n\}$.
\end{definition}
The polynomials $p^B_{n,k}(x)$ are shown on
Table~\ref{tab:pBnk} for $n \le 3$. For $n=4$,
\[ p^B_{4,k}(x) \ = \ \begin{cases}
1 + 76x + 230x^2 + 76x^3 + x^4,
& \text{if $k = 0$,} \\
108x + 460x^2 + 196x^3 + 4x^4,
& \text{if $k = 1$,} \\
36x + 420x^2 + 300x^3 + 12x^4,
& \text{if $k=2$,} \\
12x + 300x^2 + 420x^3 + 36x^4,
& \text{if $k=3$,} \\
4x + 196x^2 + 460x^3 + 108x^4,
& \text{if $k=4$,} \\
x + 76x^2 + 230x^3 + 76x^4 + x^5,
& \text{if $k=5$}. \end{cases} \]
\noindent
Their significance stems from the following theorem.
\begin{theorem} \label{thm:h-trans}
For every $n$-dimensional cubical complex ${\mathcal L}$,
\[ h({\rm sd}({\mathcal L}), x) \ = \ \sum_{k=0}^{n+1} h_k({\mathcal L})
p^B_{n,k}(x) . \]
\end{theorem}
{\scriptsize
\begin{table}[hptb]
\begin{center}
\begin{tabular}{| l || l | l | l | l | l | l ||} \hline
& $k=0$ & $k=1$ & $k=2$ & $k=3$ & $k=4$ \\ \hline \hline
$n=0$ & 1 & $x$ & & & \\
\hline
$n=1$ & $1+x$ & $4x$ & $x+x^2$ & & \\ \hline
$n=2$ & $1+6x+x^2$ & $12x+4x^2$ &
$4x+12x^2$ & $x+6x^2+x^3$ & \\ \hline
$n=3$ & $1+23x+23x^2+x^3$ & $36x+56x^2+4x^3$
& $12x+72x^2+12x^3$ & $4x+56x^2+36x^3$ &
$x+23x^2+23x^3+x^4$ \\ \hline
\end{tabular}
\caption{The polynomials $p^B_{n,k}(x)$ for $n \le 3$.}
\label{tab:pBnk}
\end{center}
\end{table}}
The proof requires a few preliminary results. We
first summarize some of the main properties of
$p^B_{n,k}(x)$.
\begin{proposition} \label{prop:pBnk}
For every $n \in {\mathbb N}$:
\begin{itemize}
\itemsep=0pt
\item[{\rm (a)}]
The polynomial $p^B_{n,k}(x)$ has nonnegative
coefficients for every $k \in \{0, 1,\dots,n+1\}$;
its degree is equal to $n+1$, if $k = n+1$, and to
$n$ otherwise.
\item[{\rm (b)}]
$p^B_{n,n+1-k}(x) = x^{n+1}p^B_{n,k}(1/x)$ for
every $k \in \{0, 1,\dots,n+1\}$.
\item[{\rm (c)}]
$p^B_{n,0}(x) = B_n(x)$, $p^B_{n,n+1}(x) = xB_n(x)$
and $\sum_{k=0}^{n+1} p^B_{n,k}(x) = B_{n+1}(x)$.
\item[{\rm (d)}]
We have
\[ p^B_{n+1,k+1}(x) \ = \ \begin{cases}
2 p^B_{n+1,0}(x) + 2 (x-1) p^B_{n,0}(x),
& \text{if $k = 0$,} \\
p^B_{n+1,k}(x) + 2(x-1) p^B_{n,k}(x),
& \text{if $1 \le k \le n$,} \\
(1/2) \cdot p^B_{n+1,n+1}(x) + (x-1) p^B_{n,n+1}(x),
& \text{if $k=n+1$} . \end{cases} \]
\item[{\rm (e)}]
The recurrence
\[ p^B_{n+1,k}(x) \ = \ \begin{cases}
{\displaystyle \sum_{i=0}^{n+1} p^B_{n,i}(x)},
& \text{if $k = 0$}, \\
{\displaystyle 2x \sum_{i=0}^{k-1} p^B_{n,i}(x)
\, + \, 2 \sum_{i=k}^{n+1} p^B_{n,i}(x)},
& \text{if $1 \le k \le n+1$}, \\
{\displaystyle x \sum_{i=0}^{n+1} p^B_{n,i}(x)},
& \text{if $k=n+2$} \end{cases} \]
holds for $k \in \{0, 1,\dots,n+1\}$.
\item[{\rm (f)}]
We have
\[ \frac{p^B_{n,k}(x)}{(1-x)^{n+1}} \ = \ \begin{cases}
{\displaystyle \sum_{m \ge 0} \, (2m+1)^n x^m},
& \text{if $k = 0$,} \\
{\displaystyle \sum_{m \ge 0} \, (4m) (2m-1)^{k-1}
(2m+1)^{n-k} x^m},
& \text{if $1 \le k \le n$,} \\
{\displaystyle \sum_{m \ge 1} \, (2m-1)^n x^m},
& \text{if $k=n+1$}. \end{cases} \]
\end{itemize}
\end{proposition}
\begin{proof}
We first note that, as discussed in
Section~\ref{sec:shell}, $p^B_{n,0}(x) = h({\mathcal C}_{n,0},x) =
B_n(x)$. Part (d) follows from Definition~\ref{def:pBnk}
and the definition of the $h$-polynomial of a relative
simplicial complex. Indeed, for $1 \le k \le n$, we have
$f({\mathcal C}_{n+1,k+1}, x) = f({\mathcal C}_{n+1,k}, x) - 2 f({\mathcal C}_{n,k},
x)$. Hence,
\begin{eqnarray*}
h({\mathcal C}_{n+1,k+1}, x) & = & (1-x)^{n+2} f({\mathcal C}_{n+1,k+1},
\frac{x}{1-x}) \\ & & \\ & = &
(1-x)^{n+2} f({\mathcal C}_{n+1,k}, \frac{x}{1-x}) \, - \,
2 (1-x) \cdot (1-x)^{n+1} f({\mathcal C}_{n,k}, \frac{x}{1-x})
\\ & & \\ & = &
h({\mathcal C}_{n+1,k}, x) \, + \, 2(x-1) h({\mathcal C}_{n,k}, x)
\end{eqnarray*}
\noindent
and
\begin{eqnarray*}
p^B_{n+1,k+1}(x) & = & 2 h({\mathcal C}_{n+1,k+1}, x)
\ = \ 2 h({\mathcal C}_{n+1,k}, x) \, + \, 4 (x-1)
h({\mathcal C}_{n,k}, x) \\ & = & p^B_{n+1,k}(x) \, +
\, 2 (x-1) p^B_{n,k}(x).
\end{eqnarray*}
\noindent
The same argument, similar to that in the proof of
\cite[Corollary~5.6]{Ath20+}, works for $k \in \{0,
n+1\}$. Part (f) follows from part (d) by
straightforward induction on $k$ (for fixed $n$),
where the base $k=0$ of the induction holds because
of Equation~(\ref{eq:Bn-gen}).
For part (c) we first note that $p^B_{n,n+1}(x) =
h({\mathcal C}_{n,n+1},x) = h^\circ({\mathcal C}_{n,0}, x) = x^{n+1}
h({\mathcal C}_{n,0}, 1/x) = x^{n+1} B_n(1/x) = xB_n(x)$. The
identity for the sum of the $p^B_{n,k}(x)$ can be
verified directly by summing that of part (f). For
a more conceptual proof, one can use an obvious
shelling of the boundary complex of
the $(n+1)$-dimensional cube to write, as explained
in \cite[Section~3]{HS20+}, the $h$-polynomial
$B_{n+1}(x)$ of its barycentric subdivision as a
sum of $h$-polynomials of relative simplicial
complexes, each one combinatorially isomorphic to
one of the ${\mathcal C}_{n,k}$. The details are left to
the interested reader.
Given (c), the recursion of part (e) follows easily
by induction on $k$ from part (d) (this parallels
the proof of \cite[Lemma~6.3]{Ath20+}).
Part (b) is a consequence of Definition~\ref{def:pBnk}
and Proposition~\ref{prop:hsymmetry}. Alternatively,
it follows from part (f) and standard facts
about rational generating functions; see
\cite[Proposition~4.2.3]{StaEC1}. The nonnegativity
of the coefficients of $p^B_{n,k}(x)$, claimed in part
(a), follows from the recursion of part (e), as well
as from general results \cite[Corollary~III.7.3]{StaCCA}
on the nonnegativity of $h$-vectors of Cohen--Macaulay
relative simplicial complexes. The statement about the
degree of $p^B_{n,k}(x)$, claimed there, follows from
either of parts (d), (e) or (f).
\end{proof}
We leave the problem to find a combinatorial
interpretation of $p^B_{n,k}(x)$ open. Given part
(c) of the proposition, one naturally expects that
there is such an interpretation which refines one
of the known combinatorial interpretations of
$B_{n+1}(x)$.
The following statement is a consequence of a more
general result \cite[Proposition~7.6]{Sta92} of
Stanley on subdivisions of CW-posets. To keep this
paper self-contained, we include a proof.
\begin{proposition} \label{prop:sdc-h}
For every $n$-dimensional cubical complex ${\mathcal L}$,
\[ h({\rm sd}({\mathcal L}), x) \ = \ (1-x)^{n+1} \, + \, x \,
\sum_{k=0}^n f_k({\mathcal L}) (1-x)^{n-k} B_k(x) . \]
\end{proposition}
\begin{proof}
Since every face of ${\rm sd}({\mathcal L})$ is an interior face
of the restriction ${\rm sd}(\sigma)$ of ${\rm sd}({\mathcal L})$ to
a unique face $\sigma \in {\mathcal L}$, we have
\[ f({\rm sd}({\mathcal L}), x) \ = \ \sum_{\sigma \in {\mathcal L}} f^\circ
({\rm sd}(\sigma), x) \ = \ 1 \, + \sum_{\sigma \in {\mathcal L}
{\smallsetminus} \{\varnothing\}} f^\circ({\rm sd}(\sigma), x) . \]
Transforming $f$-polynomials to $h$-polynomials in
this equation and recalling from Section~\ref{sec:enu}
that $h^\circ({\rm sd}(\sigma), x) = x^{k+1} h({\rm sd}(\sigma),
1/x) = x^{k+1} B_k(1/x) = x B_k(x)$ for every nonempty
$k$-dimensional face $\sigma \in {\mathcal L}$, we get
\begin{eqnarray*}
h({\rm sd}({\mathcal L}), x) & = & (1-x)^{n+1} f({\rm sd}({\mathcal L}),
\frac{x}{1-x}) \\ & = & (1-x)^{n+1} \ +
\sum_{\sigma \in {\mathcal L} {\smallsetminus} \{\varnothing\}} (1-x)^{n+1}
f^\circ({\rm sd}(\sigma), \frac{x}{1-x}) \\ & = &
(1-x)^{n+1} \ + \sum_{\sigma \in {\mathcal L} {\smallsetminus} \{\varnothing\}}
(1-x)^{n-\dim(\sigma)} \, h^\circ({\rm sd}(\sigma), x) \\ & = &
(1-x)^{n+1} \, + \ \sum_{k=0}^n f_k({\mathcal L}) (1-x)^{n-k} x
B_k(x)
\end{eqnarray*}
and the proof follows.
\end{proof}
\noindent
\emph{Proof of Theorem~\ref{thm:h-trans}}. Let us
denote by $p({\mathcal L}, x)$ the right-hand side of the desired
equality. Clearly, it suffices to show that $h({\rm sd}({\mathcal L}),
x)/(1-x)^{n+1} = p({\mathcal L}, x)/(1-x)^{n+1}$. From
Proposition~\ref{prop:sdc-h} and
Equation~(\ref{eq:Bn-gen}) we deduce that
\begin{eqnarray*}
\frac{h({\rm sd}({\mathcal L}), x)}{(1-x)^{n+1}} & = & 1 \, + \, x \,
\sum_{k=0}^n f_k({\mathcal L}) \, \frac{B_k(x)}{(1-x)^{k+1}} \ =
\ 1 \, + \, \sum_{m \ge 0} \left( \, \sum_{k=0}^n
f_k({\mathcal L}) (2m+1)^k \right) x^{m+1} \nonumber \\ & = &
1 \, + \, \sum_{m \ge 1} \left( \, \sum_{k=0}^n
f_k({\mathcal L}) (2m-1)^k \right) x^m .
\end{eqnarray*}
Similarly, from part (f) of Proposition~\ref{prop:pBnk}
we get
\[ \frac{p({\mathcal L}, x)}{(1-x)^{n+1}} \ = \ \sum_{k=0}^{n+1}
h_k({\mathcal L}) \, \frac{p^B_{n,k}(x)}{(1-x)^{n+1}} \ = \ 1 \, +
\, \sum_{m \ge 1} a_{\mathcal L}(m) x^m, \]
where
\[ a_{\mathcal L}(y) \ := \ h_0({\mathcal L}) (2y+1)^n \, + \, \sum_{k=1}^n
h_k({\mathcal L}) (4y) (2y-1)^{k-1} (2y+1)^{n-k} \, + \, h_{n+1}({\mathcal L})
(2y-1)^n. \]
Thus, it remains to show that
\begin{equation} \label{eq:final}
\sum_{k=0}^n f_k({\mathcal L}) (2y-1)^k \ = \ a_{\mathcal L}(y) .
\end{equation}
We claim that this is, essentially, the defining
Equation~(\ref{def:cub-h}) of the cubical $h$-polynomial
of ${\mathcal L}$ in disguised form. Indeed, cancelling first
the summand $1 + h_{n+1}({\mathcal L}) x^{n+2} = 1 + (-1)^n
\widetilde{{\rm ch}i} ({\mathcal L}) x^{n+2}$, and then a factor of
$x$, from both sides of (\ref{def:cub-h}) gives that
\[ \sum_{k=0}^n (h_k({\mathcal L}) + h_{k+1}({\mathcal L})) x^k \ = \
\left( \frac{1-x}{2} \right)^n \, \sum_{k=0}^n
f_k({\mathcal L}) \left( \frac{2x}{1-x} \right)^ k . \]
Setting $2x/(1-x) = 2y-1$, so that $x = (2y-1)/(2y+1)$
and $(1-x)/2 = 1/(2y+1)$, the previous identity can
be rewritten as
\begin{equation} \label{def:cub-h2}
\sum_{k=0}^n f_k({\mathcal L}) (2y-1)^k \ = \ \sum_{k=0}^n
\, (h_k({\mathcal L}) + h_{k+1}({\mathcal L})) (2y-1)^k (2y+1)^{n-k} .
\end{equation}
Since the right-hand side is readily equal to
$a_{\mathcal L}(y)$, this proves Equation~(\ref{eq:final})
and the theorem as well.
\qed
To deduce Theorem~\ref{thm:main} from
Theorem~\ref{thm:h-trans} and
Proposition~\ref{prop:pBnk}, we need to recall a few
definitions and facts from the theory of
interlacing polynomials; for more information, see
\cite[Section~8]{Bra15} and references therein. A
polynomial $p(x) \in {\mathbb R}[x]$ is called
\emph{real-rooted} if either it is the zero
polynomial, or every complex
root of $p(x)$ is real. Given two real-rooted
polynomials $p(x), q(x) \in {\mathbb R}[x]$, we say that
$p(x)$ \emph{interlaces} $q(x)$ if the roots
$\{\alpha_i\}$ of $p(x)$ interlace (or alternate
to the left of) the roots $\{\beta_j\}$ of $q(x)$,
in the sense that they can be listed as
\[ \cdots \le \beta_3 \le \alpha_2 \le \beta_2 \le
\alpha_1 \le \beta_1 \le 0. \]
A sequence $(p_0(x), p_1(x),\dots,p_n(x))$ of
real-rooted polynomials is called \emph{interlacing}
if $p_i(x)$ interlaces $p_j(x)$ for all $0 \le i < j
\le n$. Assuming also that these polynomials have
positive leading coefficients, every nonnegative
linear combination of $p_0(x), p_1(x),\dots,p_n(x)$
is real-rooted and interlaced by $p_0(x)$. A standard
way to produce interlacing sequences in combinatorics
is the following. Suppose that $p_0(x),
p_1(x),\dots,p_n(x)$ are real-rooted polynomials
with nonnegative coefficients and set
\[ q_k(x) \ = \ x \sum_{i=0}^{k-1} p_i(x) \, + \,
\sum_{i=k}^n p_i(x) \]
for $k \in \{0, 1,\dots,n+1\}$. Then, if the sequence
$(p_0(x), p_1(x),\dots,p_n(x))$ is interlacing, so is
$(q_0(x), q_1(x),\dots,q_{n+1}(x))$; see
\cite[Corollary~8.7]{Bra15} for a more general
statement.
{\smallsetminus}allskip
The following result is a stronger version of
Theorem~\ref{thm:main}.
\begin{corollary} \label{cor:main}
The polynomial $h({\rm sd}({\mathcal L}), x)$ is real-rooted and
interlaced by the Eulerian polynomial $B_n(x)$ for
every $n$-dimensional cubical complex ${\mathcal L}$ with
nonnegative cubical $h$-vector.
In particular, $h({\rm sd}({\mathcal L}), x)$ and $h({\rm sd}(Q), x)$
are real-rooted and interlaced by $B_n(x)$ for every
shellable, $n$-dimensional cubical complex ${\mathcal L}$ and
every cubical polytope $Q$ of dimension $n+1$,
respectively.
\end{corollary}
\begin{proof}
By an application of the lemma on interlacing
sequences just discussed, the recurrence of part
(e) of Proposition~\ref{prop:pBnk} implies that
$(p^B_{n,0}(x), p^B_{n,1}(x),\dots,p^B_{n,n+1}(x))$
is interlacing for every $n \in {\mathbb N}$ by induction
on $n$. Therefore, being a nonnegative linear
combination of the elements of the sequence by
Theorem~\ref{thm:h-trans}, $h({\rm sd}({\mathcal L}), x)$ is
real-rooted and interlaced by $p^B_{n,0}(x) =
B_n(x)$ for every $n$-dimensional cubical complex
${\mathcal L}$ with nonnegative cubical $h$-vector. This
proves the first statement.
The second statement follows from the first since
shellable cubical complexes are known to have
nonnegative cubical $h$-vector
\cite[Theorem~5~(iii)]{Ad96}, $h({\rm sd}(Q), x) =
h({\rm sd}(\partial Q), x)$ for every convex polytope
$Q$ and because boundary complexes of polytopes
are shellable.
\end{proof}
\begin{remark} \label{rem:simplicial} \rm
Let $\Delta$ be a simplicial complex with nonnegative
$h$-vector and ${\mathcal L}$ be a cubical complex which is
obtained from $\Delta$ by any operation which
preserves nonnegativity of $h$-vectors.
Corollary~\ref{cor:main} implies that $h({\rm sd}({\mathcal L}),
x)$ is real-rooted.
By a result of
Hetyei~\cite{Het96}, such an operation is the
cubical barycentric subdivision ${\mathcal L} = {\rm sd}_c(\Delta)$
(see \cite[p.~44]{Ath18}), also known as barycentric
cover \cite[Section~2.3]{BBC97}, of $\Delta$. Then,
${\rm sd}({\mathcal L})$ becomes the interval triangulation
of $\Delta$ \cite[Section~3.3]{MW17}. This argument
shows that the interval triangulation of $\Delta$
has a real-rooted $h$-polynomial for every simplicial
complex $\Delta$ with nonnegative $h$-vector and
answers in the affirmative the question
of \cite[Problem~33]{MW17}. Although there are
other proofs of this fact in the literature (see
\cite{Ath20+} and references therein), the approach
via Corollary~\ref{cor:main} allows for
more general results, e.g., by applying further
cubical subdivisions of ${\rm sd}_c(\Delta)$ which
preserve the nonnegativity of the cubical
$h$-vector.
\end{remark}
\begin{remark} \label{rem:BW-formula} \rm
Applying the reasoning of the proof of
Proposition~\ref{prop:sdc-h} and of the first few lines
of the proof of Theorem~\ref{thm:h-trans} to an
$n$-dimensional simplicial complex $\Delta$ gives
that
\[ h({\rm sd}(\Delta), x) \ = \ (1-x)^{n+1} \, + \, x \,
\sum_{k=0}^n f_k(\Delta) (1-x)^{n-k} A_{k+1}(x) \]
and
\[
\frac{h({\rm sd}(\Delta), x)}{(1-x)^{n+2}} \ = \
\sum_{m \ge 0} \left( \, \sum_{k=0}^{n+1} f_{k-1}
(\Delta) m^k \right) x^m \ = \ \sum_{m \ge 0}
\left( \, \sum_{k=0}^{n+1} h_k(\Delta) m^k
(m+1)^{n+1-k} \right) x^m . \]
This is the expression at which Brenti and Welker
arrived \cite[Equation~(5)]{BW08} via a different
route and which they used to show that $h({\rm sd}(\Delta),
x)$ has only real roots, provided that $h_k(\Delta)
\ge 0$ for all $k$.
\end{remark}
\begin{remark} \rm
Replacing $2y-1$ by $x$ in (\ref{def:cub-h2})
shows that the equation
\[ \sum_{k=0}^n f_k({\mathcal L}) x^k \ = \ \sum_{k=0}^n
\, (h_k({\mathcal L}) + h_{k+1}({\mathcal L})) \, x^k (x+2)^{n-k} , \]
together with the condition $h_0({\mathcal L}) = 1$, gives
an equivalent way to define the normalized cubical
$h$-vector of an $n$-dimensional cubical complex ${\mathcal L}$.
\end{remark}
\section{Closing remarks}
\label{sec:rem}
There is a large literature on the barycentric
subdivision of simplicial complexes which relates
to the work \cite{BW08}. Many of the questions
addressed there make sense for cubical complexes.
We only consider a couple of them here.
\textbf{a}. Being real-rooted, $h({\rm sd}(\Delta), x)$
is unimodal for every $n$-dimensional simplicial
complex $\Delta$ with nonnegative $h$-vector.
Kubitzke and Nevo showed~\cite[Corolalry~4.7]{KN09}
that the corresponding $h$-vector
$(h_i({\rm sd}(\Delta))_{0 \le i \le n+1}$ has a peak
at $i = (n+1)/2$, if $n$ is odd, and at $i = n/2$
or $i = n/2 + 1$, if $n$ is even. The analogous
statement for cubical complexes follows from
Theorem~\ref{thm:h-trans} since, as in the
simplicial setting, the
unimodal polynomial $p^B_{n,k}(x)$ has a peak at
$i = (n+1)/2$, if $n$ is odd, at $i = n/2$ if $n$
is even and $k \le n/2$, and at $i = n/2 + 1$, if
$n$ is even and $k \ge n/2 + 1$. The latter claim
can be deduced from the recursion
of part (e) of Proposition~\ref{prop:pBnk} by
mimicking the argument given in the simplicial
setting in~\cite[Section~2]{Mur10}. For general
results on the unimodality of $h$-vectors of
barycentric subdivisions of Cohen--Macaulay
regular cell complexes, proven by algebraic methods,
see Corollaries~1.2 and~5.12 in \cite{MY14}.
\textbf{b}. The main result of~\cite{BS20} implies
(see~\cite[Section~8]{Ath20+}) that
$h({\rm sd}(\Delta), x)$ has a nonnegative real-rooted
symmetric decomposition with respect to $n$ for
every triangulation $\Delta$ of an $n$-dimensional
ball. Does this hold if $\Delta$ is replaced by
any cubical subdivision of the $n$-dimensional ball?
Are these symmetric decompositions interlacing? Do
the polynomials $p^B_{n,k}(x)$ have such
properties?
\textbf{c}. The subdivision operator (see
\cite[Section~3.3]{Bra15}) has a natural generalization
in the context of uniform triangulations of simplicial
complexes \cite[Section~5]{Ath20+} which plays a role
in that theory. It may be worth studying the cubical
analogue of this operator further.
\end{document} |
\begin{document}
\title{GIPA: A General Information Propagation Algorithm for Graph Learning}
\author{
Houyi Li\inst{1}\thanks{Contributed to this work when the author worked in Ant and Alibaba Group.} \and
Zhihong Chen\inst{2} \and
Zhao Li\inst{3, 6} \and
Qinkai Zheng \inst{4} \and
Peng Zhang \inst{5}\and
Shuigeng Zhou\inst{1}\thanks{Shuigeng Zhou is the corresponding author.}}
\authorrunning{Li, H. et al.}
\institute{School of computer science, Fudan University, Shanghai, China \and
Alibaba Group, Hangzhou, China \and
Zhejiang University, Hangzhou, China \and
Tsinghua University, Beijing, China \and
Cyberspace Institute of Advanced Technology, Guangzhou University, China \and
Link2Do Technology, Hangzhou, China\\
\email{[email protected], [email protected], [email protected], [email protected], [email protected], [email protected]}
}
\maketitle
\begin{abstract}
Graph neural networks (GNNs) have been widely used in graph-structured data computation, showing promising performance in various applications such as node classification, link prediction, and network recommendation.
Existing works mainly focus on node-wise correlation when doing weighted aggregation of neighboring nodes based on attention, such as dot product by the dense vectors of two nodes. This may cause conflicting noise in nodes to be propagated when doing information propagation. To solve this problem, we propose a General Information Propagation Algorithm (GIPA\xspace), which exploits more fine-grained information fusion including bit-wise and feature-wise correlations based on edge features in their propagation. Specifically, the bit-wise correlation calculates the element-wise attention weights through a multi-layer perceptron (MLP) based on the dense representations of two nodes and their edge; The feature-wise correlation is based on the one-hot representations of node attribute features for feature selection. We evaluate the performance of GIPA\xspace on the Open Graph Benchmark proteins (OGBN-proteins) dataset and the Alipay dataset of Alibaba Group.
Experimental results reveal that GIPA\xspace outperforms the state-of-the-art models in terms of prediction accuracy, e.g., GIPA\xspace achieves an average ROC-AUC of $0.8917\pm 0.0007$, which is better than that of all the existing methods listed in the OGBN-proteins leaderboard.
\keywords{Graph neural networks, Fine-grained information fusion, Bit-wise and feature-wise attention.}
\end{abstract}
\section{Introduction}
Graph representation learning typically aims to learn an informative embedding for each graph node based on the graph topology (link) information.
Generally, the embedding of a node is represented as a low-dimensional feature vector, which can be used to facilitate downstream applications.
This research focuses on homogeneous graphs that have only one type of nodes and one type of edges. The purpose is to learn node representations from the graph topology~\cite{grover2016node2vec,perozzi2014deepwalk,dai2016discriminative}.
Specifically, given a node $u$, either breadth-first search, depth-first search or random walk is used to identify a set of neighboring nodes. Then, $u$'s embedding is learnt by maximizing the co-occurrence probability of $u$ and its neighbors.
Early studies on graph embedding have limited capability to capture neighboring information from a graph because they are based on shallow learning models such as SkipGram~\cite{mikolov2013distributed}.
Moreover, transductive learning is used in these graph embedding methods, which cannot be generalized to new nodes that are absent in the training graph.
Graph neural networks~\cite{kipf2016semi,hamilton2017inductive,velivckovic2017graph} are proposed to overcome the limitations of traditional graph embedding models.
GNNs employ deep neural networks to aggregate feature information from neighboring nodes and thereby have the potential to gain better aggregated embeddings.
GNNs can support inductive learning and infer the class labels of unseen nodes during prediction~\cite{hamilton2017inductive,velivckovic2017graph}.
The success of GNNs is mainly due to the neighborhood information aggregation. However,
GNNs face two challenges: \textit{which neighboring nodes of a target node are involved in message passing? and how much contribution each neighboring node makes to the aggregated embedding?}.
For the former question, neighborhood sampling~\cite{hamilton2017inductive,ying2018graph,chen2018fastgcn,huang2018adaptive,zou2019layer,ji2020accelerating} is proposed for large dense or power-law graphs.
For the latter, neighbor importance estimation is used to attach different weights to different neighboring nodes during feature propagation.
Importance sampling~\cite{chen2018fastgcn,zou2019layer,ji2020accelerating} and attention~\cite{velivckovic2017graph,liu2019geniepath,wang2019heterogeneous,yun2019graph,hu2020heterogeneous} are two popular techniques.
Importance sampling is a special case of neighborhood sampling, where the importance weight of a neighboring node is drawn from a distribution over nodes.
This distribution can be derived from normalized Laplacian matrices~\cite{chen2018fastgcn,zou2019layer} or jointly learned with GNNs~\cite{ji2020accelerating}.
With this distribution, at each step a subset of neighbors is sampled, and aggregated with the importance weights.
Similar to importance sampling, attention also attaches importance weights to neighbors.
Nevertheless, attention differs from importance sampling.
Attention is represented as a neural network and is always learned as a part of a GNN model.
In contrast, importance sampling algorithms use statistical models without trainable parameters.
Existing attention mechanisms consider only the correlation of node-wise, ignoring the suppression of noise information in transmission, and the information of edge features. In real world applications, only partial users authorize the system to collect theirs profiles. The model cannot learn the node-wise correlation between a profiled user and a user we know nothing about. Therefore, existing models will spread noise information, resulting in inaccurate node representations. However, two users who often transfer money to each other and two users who only have a few conversations have different correlation.
In this paper, to solve the problems mentioned above, we present a new graph neural network attention model, namely \underline{G}eneral \underline{I}nformation \underline{P}ropagation \underline{A}lgorithm (GIPA\xspace). We design a bit-wise correlation module and a feature-wise correlation module. Specifically, we believe that each dimension of the dense vector represents a feature of the node. Therefore, the bit-wise correlation module filters at the dense representation level. The dimension of attention weights is equal to that of density vector. In addition, we represent each attribute feature of the node as a one-hot vector. The feature-wise correlation module performs feature selection by outputting the attention weights of similar dimensionality and attribute features. It is worth mentioning that to enable the model to extract better attention weights, edge features that measure the correlation between nodes are also included in the calculation of attention. Finally, GIPA\xspace inputs sparse embedding and dense embedding into the wide end and deep end of the deep neural network for learning specific tasks, respectively.
Our contributions are summarized as follows:
\begin{itemize}
\item[1)]
We design the bit-wise correlation module and the feature-wise correlation module to perform more refined information weighted aggregation from the element level and the feature level, and utilize the edge information.
\item[2)]
Based on the wide \& deep architecture~\cite{WideDeep}, we use dense feature representation and sparse feature representation to extract deep information and retain shallow original information respectively, which provides more comprehensive information for the downstream tasks.
\item[3)]
Experiments on the Open Graph Benchmark (OGB)~\cite{hu2020open} proteins dataset (OGBN-proteins) demonstrate that GIPA\xspace achieves better accuracy with an average ROC-AUC of $0.8917\pm 0.0007$~\footnote{The reproducible code is open source: https://github.com/houyili/gipa\_wide\_deep} than the state-of-the-art methods listed in the OGBN-proteins leaderboard~\footnote{https://ogb.stanford.edu/docs/leader\_nodeprop/\#ogbn-proteins}.
In addition, GIPA\xspace has been tested on billion-scale industrial Alipay dataset.
\end{itemize}
\section{Related Work}
In this section, we review existing attention primitive implementations in brief.
\cite{bahdanau2014neural} proposes an additive attention that calculates the attention alignment score using a simple feed-forward neural network with only one hidden layer.
The alignment score $score(q,k)$ between two vectors $q$ and $k$ is defined as
$score(q,k) = u^T\tanh(W[q;k])$,
where $u$ is an attention vector and the attention weight $\alpha_{q,k}$ is computed by normalizing $scor\tilde{e}_{q,k}$ over all ${q,k}$ values with the softmax function.
The core of the additive attention lies in the use of attention vector $u$.
This idea has been widely adopted by several algorithms~\cite{yang2016hierarchical,pavlopoulos2017deeper} for natural language processing.
\cite{luong2015effective} introduces a global attention and a local attention.
In global attention, the alignment score can be computed by three alternatives: dot-product ($q^Tk$), general ($q^TWk$) and concat ($W[q;k]$).
In contrast, local attention computes the alignment score solely from a vector ($Wq$).
Likewise, both global and local attention normalize the alignment scores with the \textit{softmax} function.
\cite{vaswani2017attention} proposes a self-attention mechanism based on scaled dot-product.
This self-attention computes the alignment score between any $q$ and $k$ as follows:
$scor\tilde{e}_{q,k} = {q^Tk}\big/ {\sqrt{d_k}}$.
This attention differs from the dot-product attention~\cite{luong2015effective} by only a scaling factor of $\frac{1}{\sqrt{d_k}}$.
The scaling factor is used because the authors of~\cite{vaswani2017attention} suspected that for large values of $d_k$, dot-product is large in magnitude, which thereby pushes the \textit{softmax} function into regions where it has extremely small gradients.
More attention mechanisms, like feature-wise attention~\cite{FeatureAtt} and multi-head attention~\cite{vaswani2017attention}, can be referred to some surveys~\cite{attsurvey,NIU202148}.
In addition, these aforementioned attention primitives have been extended to heterogeneous graphs.
HAN~\cite{wang2019heterogeneous} uses a two-level hierarchical attention that consists of a node-level attention and a semantic-level attention.
In HAN, the node-level attention learns the importance of a node to any other node in a meta-path, while the semantic-level one weighs all meta-paths.
HGT~\cite{zhang2019heterogeneous} weakens the dependency on meta-paths and instead uses meta-relation triplets as basic units.
HGT uses node-type-aware feature transformation functions and edge-type-aware multi-head attention to compute the importance of each edge to a target node.
It is worthy of mentioning that heterogeneous models must not always be superior to homogeneous ones, and vice versa. In this paper, unlike the node-wise attention in existing methods that ignores noise propagation, our proposed GIPA\xspace introduces two MLP-based correlation modules, the bit-wise correlation module and the feature-wise correlation module, to achieve fine-grained selective information propagation and utilize the edge information.
\section{Methodology}
\subsection{Preliminaries}
\paragraph{Graph Neural Networks.} Consider an attributed graph $\mathcal{G}=\{\mathcal{V}, \mathcal{E}\}$, where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ is the set of edges. GNNs use the same model framework as follows~\cite{hamilton2017inductive,xu2018powerful}:
\begin{figure*}
\caption{The architecture of GIPA\xspace, which consists of \textit{attention}
\label{fig:gipa_process}
\end{figure*}
\begin{equation}
\label{eq:gnn}
\tilde{h}_i^k=\phi(\tilde{h}_i^{k-1}, \mathcal{F}_{agg}(\{\tilde{h}_j^{k-1}, j\in\mathcal{N}(i)\}))
\end{equation}
and \noindent where $\mathcal{F}_{agg}$ represents an aggregation function, $\phi$ represents an update function, $\tilde{h}_i^k$ represents the node embedding based on node feature $x_i^k$, and $\mathcal{N}(i)$ represents the neighbor node set. The objective of GNNs is to update the embedding of each node by aggregating the information from its neighbor nodes and the connections between them.
\subsection{Model Architecture}
In this section, we present the architecture of GIPA\xspace in~\figurename~\ref{fig:gipa_process}, which extracts node features and edge features in a more general way. Consider a node $i$ with feature embedding $\tilde{h}_i$ and its neighbor nodes $j\in\mathcal{N}(i)$ with feature embedding $\tilde{h}_j$. $\tilde{e}_{i, j}$ represents the edge feature between node $i$ and $j$. The problem is how to generate an expected embedding for node $i$ from its own node feature embedding $\tilde{h}_i$, its neighbors' node features embedding $\{\tilde{h}_j\}$ and the related edge features $\{\tilde{e}_{i, j}\}$.
The workflow of GIPA\xspace consists of three major modules (or processes): \textit{attention}, \textit{propagation} and \textit{aggregation}.
First, GIPA\xspace computes the dense embedding $\tilde{h}_i^d$, $\tilde{h}_j^d$, and $\tilde{e}_{i, j}$, and sparse embedding $\tilde{h}_i^s$, $\tilde{h}_j^s$ from raw features.
Then, the \textit{attention} process calculates the bit-wise attention weights by using fully-connected layers on $\tilde{h}_i^d$, $\tilde{h}_j^d$, and $\tilde{e}_{i, j}$ and feature-wise by using fully-connected layers on $\tilde{h}_i^s$, $\tilde{h}_j^s$, and $\tilde{e}_{i, j}$.
Following that, the \textit{propagation} process focuses on propagating information of each neighbor node $j$ by combining $j$'s node embedding $\tilde{h}_j$ with the associated edge embedding $\tilde{e}_{i, j}$.
Finally, the \textit{aggregation} process aggregates all messages from neighbors to update the embedding of $i$. The following subsections introduce the details of each process in one layer.
\subsubsection{Embedding Layer.}
The GIPA\xspace (wide \& deep) is more suitable for these scenarios: each node in GNN is not composed of text or images, but represents objects such as users, products, and proteins, whose features are composed of category features and statistical features.
For example, the category feature in \textit{ogbn-proteins} dataset is what species the proteins come from.
And the statistical features in \textit{Alipay} dataset are similar to the total consumption of users in a year.
For dense embedding, each integer number can express a category, each floating-point number can express a statistical value, thus a one-dimensional embedding can represent one feature.
However, the sparse embedding of one feature requires more dimensions.
For each category feature, one-hot encoding is required.
For example, a certain ``category feature" has a total of $K$ possible categories, and an $K+1$ dimensional vector is required to represent the feature.
And each "statistical feature" is cut into $K$ categories (using equal-frequency or equal-width cutting method), and then $K+1$ dimensional one-hot encoding is performed.
Thus, the concatenations of dense and sparse embeddings are the inputs of deep part and wide part respectively.
\begin{algorithm}[t]
\caption{The optimization strategy of GIPA}
\label{alg::algorithm1}
\begin{algorithmic}[1]
\State \textbf{Input:} Graph $\mathcal{G}=\{\mathcal{V}, \mathcal{E}\}$; input features $\{x_{v}, \forall v \in V\}$; Number of layer $K$
\State \textbf{Note:} $\tilde{h}_{v}^* \in \{\tilde{h}_{v}^s, \tilde{h}_{v}^d\}$
\State $\tilde{h}_v^{d^0} \leftarrow DenseEmb(x_{v}), \forall v \in V$ \State $\tilde{h}_v^{s^0} \leftarrow SparseEmb(x_{v}), \forall v \in V$
\State\textbf{while} not end of epoch \textbf{do}
\State \qquad Select a subgraph $\mathcal{G}_t=\{\mathcal{V}_t, \mathcal{E}_t\} \in \mathcal{G}$
\State \qquad \textbf{for} each $k \in [1, K]$ \textbf{do}
\State \qquad \qquad \textbf{if} $k > 1$
\State \qquad \qquad \qquad $\tilde{h}_i^{*k-1} \leftarrow o_{i}^{*k-1}, \forall v_i \in V_t$
\State \qquad \qquad $\tilde{\bm{a}}_{i, j}^{*k} \leftarrow \mathcal{F}_{act}(\mathcal{F}_{att}^*(\tilde{h}_i^{*k-1}, \tilde{h}_j^{*k-1}, \tilde{e}_{i, j}|W_{att}^*)), \forall v_i \in V_t$
\State \qquad \qquad $m_{i, j}^{*k} \leftarrow \tilde{\bm{a}}_{i, j}^{*k} * \mathcal{F}_{prop}^*(\tilde{h}_j^{*k-1}, \tilde{e}_{i,j}|W_{prop}^*), \forall v_i \in V_t$
\State \qquad \qquad $o_{i}^{*k} \leftarrow \mathcal{F}_{agg}^*(\sum_{j\in\mathcal{N}(i)}m_{i, j}^{*k}, \hat{h}_i^{*k-1} | W_{agg}^*), \forall v_i \in V_t$
\State \qquad $\hat{y}_i = Deep(o_{i}^{dk})+ W_{wide}(o_{i}^{sk})$
\State \qquad $L \leftarrow \sum_{\forall v_i \in V_t}l(y_i, \hat{y}_i)$
\State \qquad Update all parameters by gradient of $L$
\end{algorithmic}
\end{algorithm}
\subsubsection{Attention Process.}
Different from the existing attention mechanisms like self-attention or scaled dot-product attention, we use MLP to realize a bit-wise attention mechanism and a feature-wise attention mechanism.
The bit-wise and feature-wise \textit{attention} process of GIPA\xspace can be formulated as follows:
\begin{equation}
\label{eq:att1}
\bm{a}_{i, j}^d = \mathcal{F}_{att}^d(\tilde{h}_i^d, \tilde{h}_j^d, \tilde{e}_{i, j}|W_{att}^d) = \text{MLP}([\tilde{h}_i^d || \tilde{h}_j^d || \tilde{e}_{i, j}]|W_{att}^d)
\end{equation}
\begin{equation}
\label{eq:att2}
\bm{a}_{i, j}^s = \mathcal{F}_{att}^s(\tilde{h}_i^s, \tilde{h}_j^s, \tilde{e}_{i, j}|W_{att}^s) = \text{MLP}([\tilde{h}_i^s || \tilde{h}_j^s || \tilde{e}_{i, j}]|W_{att}^s)
\end{equation}
\noindent where $\bm{a}_{i, j}^d \in \mathcal{R}^n$ is bit-wise attention weight and its dimension is the same as that of $\tilde{h}_i^d$, $\bm{a}_{i, j}^s \in \mathcal{R}^m$ is feature-wise attention weight and its dimension is the same as the number of node features, the attention function $\mathcal{F}_{att}^*$ is realized by an \textit{MLP} with learnable weights $W_{att}^*$ (without bias). Its input is the concatenation of the node embeddings $\tilde{h}_i^*$ and $\tilde{h}_j^*$ as well as the edge embedding $\tilde{e}_{i, j}$. As the edge features $\tilde{e}_{i, j}$ measuring the correlation between nodes are input into \textit{MLP}, this attention mechanism could be more representative than previous ones simply based on dot-product. The final attention weight is calculated by an activation function for bit-wise part and feature-wise part:
\begin{equation}
\label{eq:softmax1}
\tilde{\bm{a}}_{i, j}^* = \mathcal{F}_{act}(\bm{a}_{i, j}^*)
\end{equation}
\noindent where $ \mathcal{F}_{act}$ represents the activation function, such as \textit{softmax}, \textit{leaky-relu}, \textit{softplus}, etc. Based on the experimental results, we finally define the activation function as \textit{softplus}. Details can be seen in Section \ref{Hyperparameter}.
\subsubsection{Propagation Process.}
Unlike GAT~\cite{velivckovic2017graph} that considers only the node feature of neighbors, GIPA\xspace incorporates both node and edge embeddings during the \textit{propagation} process:
\begin{equation}
\label{eq:prop1}
p_{i, j}^d = \mathcal{F}_{prop}^d(\tilde{h}_j^d, \tilde{e}_{i,j}|W_{prop}^d) = \text{MLP}([\tilde{h}_j^d || \tilde{e}_{i, j}]|W_{prop}^d),
\end{equation}
\begin{equation}
\label{eq:prop1}
p_{i, j}^s = \mathcal{F}_{prop}^s(\tilde{h}_j^s, \tilde{e}_{i,j}|W_{prop}^s) = \text{MLP}([\tilde{h}_j^s || \tilde{e}_{i, j}]|W_{prop}^s),
\end{equation}
\noindent where the propagation function $\mathcal{F}_{prop}^*$ is also realized by an \text{MLP} with learnable weights $W_{prop}^*$. Its input is the concatenation of a neighbor node dense and sparse embeddings $\tilde{h}_j^*$ and the related edge embedding $\tilde{e}_{i, j}$. Thus, the $propagation$ is done bit-wise and feature-wise rather than node-wise.
Combining the results by \textit{attention} and \textit{propagation} by bit-wise and feature-wise multiplication, GIPA\xspace gets the message $m_{i, j}^d$ and $m_{i, j}^s$ of node $i$ from $j$:
\begin{equation}
\label{eq:msg}
m_{i, j}^d = \tilde{\bm{a}}_{i, j}^d * p_{i, j}^d \ \ \ m_{i, j}^s = \tilde{\bm{a}}_{i, j}^s \otimes p_{i, j}^s
\end{equation}
\subsubsection{Aggregation Process.}
For each node $i$, GIPA\xspace repeats previous processes to get messages from its neighbors. The \textit{aggregation} process first gathers all these messages by a reduce function, summation for example:
\begin{equation}
\label{eq:sum}
m_{i}^{*} = \sum_{j\in\mathcal{N}(i)}m_{i, j}^{*}
\end{equation}
\noindent Then, a residual connection between the linear projection $\hat{h}_i^*$ and the message of $m_i$ is added through concatenation:
\begin{equation}
\label{eq:proj}
\hat{h}_i^d=W_{proj}^d\tilde{h}_i^d \ \ \ \hat{h}_i^s=W_{proj}^s\tilde{h}_i^s
\end{equation}
\begin{equation}
\label{eq:agg}
o_{i}^d = \mathcal{F}_{agg}^d(m_{i}^d, \tilde{h}_i^d | W_{agg}^d) = \text{MLP}([m_{i}^d || \tilde{h}_i^d]|W_{agg}^d) \oplus \hat{h}_i^d
\end{equation}
\begin{equation}
\label{eq:agg}
o_{i}^s = \mathcal{F}_{agg}^s(m_{i}^s, \tilde{h}_i^s | W_{agg}^s) = \text{MLP}([m_{i}^s || \tilde{h}_i^s]|W_{agg}^s) \oplus \hat{h}_i^s
\end{equation}
\noindent where an \textit{MLP} with learnable weights $W_{agg}^*$ is applied to get the final dense output $o_{i}^d$ and sparse output $o_{i}^s$ .
Finally, we would like to emphasize that the process of GIPA\xspace can be easily extended to multi-layer variants by stacking the process multiple times. After we get the aggregated output of the node, $o_{i}^d$ and $o_{i}^s$ respectively input the depth side and wide side of the Deep\&Wide architecture for downstream tasks. See algorithm \ref{alg::algorithm1} for details.
\section{Experiments}
\subsection{Datasets and Settings}
\noindent \textbf{Datasets.} In our experiments, we choose two edge-attribute dataset: the \textit{ogbn-proteins} dataset from OGB~\cite{hu2020open} and \textit{Alipay} dataset\cite{graphtheta}. The \textit{ogbn-proteins} dataset is an undirected and weighted graph, containing 132,534 nodes of 8 different species and 79,122,504 edges with 8-dimensional features. The task is a multi-label binary classification problem with 112 classes representing different protein functions. The \textit{Alipay} dataset is an edge attributed graph, containing 1.40 billion nodes with 575 features and 4.14 billion edges with 57 features. The task is a multi-label binary classification problem. It is worth noting that due to the
high cost of training on \textit{Alipay} data set, we only conduct ablation experiments on the
input features of $\mathcal{F}_{att}$ and $\mathcal{F}_{prop}$ in the industrial data set, as Table \ref{tab:ali_performance}.
\noindent \textbf{Baselines.} Several representative GNNs including SOTA GNNs are used as baselines. For semi-supervised node classification, we utilize GCN \cite{kipf2016semi}, GraphSAGE \cite{hamilton2017inductive}, GAT \cite{velivckovic2017graph}, MixHop \cite{abu2019mixhop}, JKNet \cite{xu2018representation}, DeeperGCN \cite{li2020deepergcn}, GCNII \cite{chen2020simple}, DAGNN \cite{liu2020towards}, MAGNA \cite{wang2020direct}, UniMP \cite{shi2020masked}, GAT+BoT \cite{wang2021bag}, RevGNN \cite{li2021training} and AGDN \cite{sun2020adaptive}. Note that DeeperGCN, UniMP, RevGNN, and AGDN are implemented with random partition. GAT is implemented with neighbor sampling. Except for our GIPA\xspace, results of other methods are from their papers or the OGB leaderboard.
\noindent \textbf{Evaluation metric.} The performance is measured by the average ROC-AUC scores. We follow the dataset splitting settings as recommended in OGB and test the performance of 10 different trained models with different random seeds.
\noindent \textbf{Hyperparameters.} For the number of layers $K$, we search the best value from 1 to 6. As for the activation function of attention process, we consider common activation functions. For details, please refer to Section~\ref{Hyperparameter}.
\noindent \textbf{Running environment.} For \textit{ogbn-proteins} dataset, GIPA\xspace is implemented in Deep Graph Library (DGL)~\cite{wang2019deep} with Pytorch~\cite{paszke2019pytorch} as the backend. Experiments are done in a platform with Tesla V100 (32G RAM). For \textit{Alipay} dataset, GIPA\xspace is implemented in \textit{GraphTheta}~\cite{graphtheta}, and runs on private cloud of Alibaba Group.
\begin{table}[!h]
\centering
\caption{Test and validation performance results (ROC-AUC) on the \textit{ogbn-proteins} dataset. The improvements over comparison methods are statistically significant at 0.05 level.}
\begin{tabular}{lcc}
\hline
Method & Test ROC-AUC & Validation ROC-AUC \\ \hline
GCN & $0.7251\pm0.0035$ & $0.7921\pm0.0018$ \\
GraphSAGE & $0.7768\pm0.0020$ & $0.8334\pm0.0013$ \\
DeeperGCN & $0.8580\pm0.0028$ & $0.9106\pm0.0011$ \\
GAT & $0.8682\pm0.0018$ & $0.9194\pm0.0003$ \\
UniMP & $0.8642\pm0.0008$ & $0.9175\pm0.0006$ \\
GAT+BoT & $0.8765\pm0.0008$ & $0.9280\pm0.0008$ \\
RevGNN-deep & $0.8774\pm0.0013$ & $0.9326\pm0.0006$ \\
RevGNN-wide & $0.8824\pm0.0015$ & $0.9450\pm0.0008$ \\
AGDN & $0.8865\pm0.0013$ & $0.9418\pm0.0005$ \\
\hline
GIPA\xspace-3Layer & $0.8877\pm 0.0011$ & $0.9415\pm0.0023 $ \\
GIPA\xspace-6Layer & $\mathbf{0.8917\pm 0.0007}$ & $\mathbf{0.9472\pm0.0020}$ \\ \hline
\label{tab:performance}
\end{tabular}
\end{table}
\begin{table}[!h]
\centering
\caption{Ablation study on the \textit{ogbn-proteins} dataset.}
\begin{tabular}{lcc}
\hline
Method & Test ROC-AUC & Validation ROC-AUC \\ \hline
GIPA\xspace w/o bit-wise module & $0.8813\pm0.0011$ & $0.9332\pm0.0009$ \\
GIPA\xspace w/o feature-wise module & $0.8701\pm0.0021$ & $0.9320\pm0.0007$ \\
GIPA\xspace w/o edge feature & $0.8599\pm0.0047$ & $0.9204\pm0.0038$ \\
\hline
GIPA\xspace & $\mathbf{0.8917\pm 0.0007}$ & $\mathbf{0.9472\pm0.0020}$ \\ \hline
\label{tab:ab_performance}
\end{tabular}
\end{table}
\subsection{Performance Comparison}
\tablename~\ref{tab:performance} shows the average ROC-AUC and the standard deviation for the test and validation set.
The results of the baselines are retrieved from the ogbn-proteins leaderboard\footnotemark[8].
Our GIPA\xspace outperforms all previous methods in the leaderboard and reaches an average ROC-AUC higher than 0.89 for the first time.
Furthermore, GIPA\xspace only with 3 layer achieved the \textit{SOTA} performance on \textit{ogbn-proteins} dataset.
This result shows the effectiveness of our proposed bit-wise and feature-wise correlation modules, which can leverage the edge features to improve the performance by fine-grained information fusion and noise suppression.
To further investigate the impact of each component in our proposed GIPA\xspace, we conduct the ablation study on the ogbn-proteins dataset.
As shown in \tablename~\ref{tab:ab_performance}, compare with GIPA\xspace w/o bit-wise module, GIPA\xspace w/o feature-wise module. And the combination of these components (i.e., GIPA\xspace) yields the best performance, which indicates the necessity of bit-wise and feature-wise correlation modules.
The average training times per epoch on \textit{ogbn-proteins} with GIPA\xspace, 3-Layer GIPA\xspace, AGDN are 8.2 seconds (s), 5.9s, and 4.9s, respectively.
The average inference times on whole graph of \textit{ogbn-proteins} with GIPA\xspace, 3-Layer GIPA\xspace, AGDN are 11.1s, 9.3s, and 10.7s, respectively.
Compared with ADGN, GIPA\xspace is slower in training speed, but has advantages in inference speed.
\subsection{Hyperparameter Analysis and Ablation Study}\label{Hyperparameter}
\noindent \textbf{Effect of the number of layers $K$.} To study the impact of the number of layers $K$ on performance, we vary its value to $\{1, 2, 3, 4, 5 ,6\}$. As shown in Fig.~\ref{fig:layer}, with the increase of layers, the performance of the model on the test set gradually converges. However, in the comparison between the five layers and the six layers, the increase in the performance of the model on the verification set is far greater than that on the test set. This result shows that with the increase of complexity, the performance of the model can be improved, but there will be an over-fitting problem at a bottleneck. Therefore, we finally set $K$ to 6.
\noindent \textbf{Analysis on Attention Process with Correlation Module.} Because the bit-wise correlation module and the feature-wise correlation module are the most important parts of GIPA\xspace, we make a detailed analysis of their architecture, including activation function and edge feature.
\\
1) Activation function: as shown in Fig.~\ref{fig:activation}, by comparing these activation functions: the performance of \textit{tanh} and \textit{relu} are even worse than the one without \textit{activator}.
The model using \textit{softmax} as the activation function achieves the best performance, but the performance of model using \textit{softplus} is close to it.
This phenomenon indicates that the attention module of GIPA\xspace can adaptively learn the correlation weight between nodes.
Cause $\sum_{j\in N(i)}a_{ij}$ is needed to be calculated firstly by aggregation from all neighbors, the normalization operation of \textit{softmax} is a costly operation in GNN.
On the other hand, GIPA\xspace with \textit{softplus} trains faster than that with \textit{softmax} on \textit{ogbn-proteins}.
Considering the trade-off between cost and performance, \textit{softplus} is adopted on billion-scale \textit{Alipay} dataset.
\\
2) Edge feature: As shown in \tablename~\ref{tab:ab_performance}, we find that the performance of GIPA\xspace is better than that of GIPA\xspace w/o edge feature, which means that the edge feature contains the correlation information between two nodes. Therefore, it is an indispensable feature in attention processing and can help the two correlation modules get more realistic attention weights.
\begin{table}[t]
\centering
\caption{Ablation on \textit{Alipay} dataset. `S' and `D' are primary and neighbor nodes.}
\begin{tabular}{ccccc}
\hline
$\mathcal{F}_{att}$ input & $\mathcal{F}_{prop}$ input& Feature-wise module & ROC-AUC & F1 \\
\hline
Edge & D. Node & None & 0.8784 & 0.1399 \\
Edge \& D. Node & Edge \& D. Node & None & 0.8916 & 0.1571 \\
Edge \& D.+ S. Nodes & Edge \& D. Node & None &0.8943 &0.1611 \\
Edge \& D.+ S. Nodes & Edge \& D. Node & Yes & 0.8961 &0.1623 \\
\hline
\label{tab:ali_performance}
\end{tabular}
\end{table}
\begin{figure*}
\caption{ }
\label{layer}
\caption{ }
\label{fig:layer}
\caption{Effect of hyperparameters. (a) ROC-AUC vs. the number of layers $K$. (b) Convergence on test set vs. activation function of attention.}
\label{fig:activation}
\end{figure*}
\section{Conclusion}
We have presented GIPA\xspace, a new graph attention network architecture for graph data learning.
GIPA\xspace consists of a bit-wise correlation module and a feature-wise correlation module, to leverage edge information and realize the fine granularity information propagation and noise filtering.
Performance evaluation on the ogbn-proteins dataset has shown that our method outperforms the state-of-the-art methods listed in the ogbn-proteins leaderboard.
And it has been tested on the billion-scale industrial dataset.
\end{document} |
\begin{document}
\title[positive $(p,p)$-forms]{\bf Sharp differential estimates of Li-Yau-Hamilton type
for positive $(p,p)$-forms on K\"ahler manifolds}
\alphauthor{ Lei Ni and Yanyan Niu}
\date{}
\maketitle
\begin{abstract} In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a K\"ahler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the K\"ahler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the K\"ahler-Ricci flow and the Ricci flow.
\epsilonnd{abstract}
\section{Introduction}
In this paper, we study the deformation of positive $(p,p)$-forms on a K\"ahler manifold via the ${\bar{\alpha}}r{\partialartial}$-Laplacian heat equation. One of our main goals of this paper is to prove differential Harnack estimates for positive solutions. The Harnack estimate for positive solutions of linear parabolic PDEs of divergence form goes back to the fundamental work of Moser \cite{Moser}. In another fundamental work \cite{LY}, Li and Yau, on Riemannian manifolds with nonnegative Ricci curvature, proved a sharp differential estimate which implies a sharp form of Harnack inequality for the positive solutions. Later, Hamilton \cite{richard-harnack} proved the miraculous matrix differential estimates for the curvature operator of solutions to Ricci flow assuming that the curvature operator is nonnegative. Since the curvature operator of a Ricci flow solution satisfies a nonlinear diffusion-reaction equation, this result of Hamilton is as surprising as it is important. Due to this development people also call this type of sharp estimates Li-Yau-Hamilton (abbreviated as LYH) type estimates. There are many further works \cite{Andrews, brendle, Cao, CN, chow-Gauss, chow-yamabe, Hlinear, Hmcf, N-jams, Ni-JDG07, NT-ajm} in this direction since the foundational estimate of Li and Yau for linear heat equation and Hamilton's one for the Ricci flow, which cover various different geometric evolution equations, including the mean curvature flow, the Gauss curvature flow, the K\"ahler-Ricci flow, the Hermitian-Einstein flow, the Yamabe flow etc.
Since the Harnack estimate for the linear equation implies the regularity of the weak solution, it has been an interesting question that if the celebrated De Giorgi-Nash-Moser theory for the linear equation has its analogue for linear systems. This unfortunately has been known to be false in the most general setting. As a geometric interesting system, the Hodge-Laplacian operator on forms has been extensively studied since the original works of Hodge and Kodaira (see for example Morrey's classics \cite{Morrey} and references therein). It is a natural candidate on which one would like to investigate whether or not the differential Harnack estimates of LYH type still hold. One of the main results of this paper is to prove such LYH type estimates for this system. The positivity (really meaning non-negativity) of the $(p, p)$-form is in the sense of Lelong \cite{L}. In fact, in \cite{N-jams} the first author proved a LYH type estimate for positive semi-definite Hermitian symmetric tensors satisfying the so-called Lichnerowicz-Laplacian heat equation. This in particular applies to solutions of $(1,1)$-forms to the Hodge-Laplacian heat equation. The first main result of this paper is to generalize this result for $(1, 1)$-forms to solutions of $(p, p)$-forms to the Hodge-Laplacian heat equation. The result is proved under a new curvature condition $\mathcal{C}_p$. We say that the curvature operator $\Bbb Rm$ of a K\"ahler manifold $(M, g)$ satisfies $\mathcal{C}_p$ (or lies inside the cone $\mathcal{C}_p$) if
$
\operatorname{l}angle \Bbb Rm(\alphalpha), {\bar{\alpha}}r{\alphalpha}\rangle \gammae 0
$
for any $\alphalpha\sqrt {-1}n \wedge ^{1, 1}(\Bbb C^m)$ such that $\alphalpha =\sum_{k=1}^p X_k \wedge \overline{Y}_k$ with $X_k, Y_k \sqrt {-1}n T'M$. Here $TM\otimes \Bbb C =T'M\oplus T''M$, $\operatorname{l}angle \cdot, \cdot \rangle$ is the bilinear extension of the Riemannian product, and we identify $T'M$ with $\Bbb C^m$. Under the condition $\mathcal{C}_p$ we prove the following result.
\begin{theorem} \operatorname{l}abel{thm11} For $\partialhi(x, t)$, a positive $(p, p)$-form satisfying $\operatorname{l}eft(\frac{\partialartial}{\partialartial t} +\Delta_{{\bar{\alpha}}r\partial}\rightght)\partialhi(x, t) =0$, then
$$\frac{1}{\sqrt{-1}}{\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi+\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{V}\cdot {\bar{\alpha}}r{\partialartial}^* \partialhi-\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{\overline{V}}\cdot \partialartial^* \partialhi + \sqrt{-1} \sqrt {-1}ota_V\cdot \sqrt {-1}ota_{\overline{V}}\cdot \partialhi+\frac{\Lambda \partialhi}{t}\gammae 0
$$
as a $(p-1, p-1)$-form, for any $(1, 0)$ type vector field $V$. Here $\Lambda$ is the adjoint of the operator $L\doteqdot \omega \wedge (\cdot)$ with $\omega$ being the K\"ahler form.
\epsilonnd{theorem}
The above estimate is compatible with the Hodge $*$ operator (see Section 4 for the detailed discussions). Also note the easy fact that $*$-operator maps a positive $(p, p)$-form to a positive $(m-p, m-p)$-form. It then implies that if $\partialsi=*\partialhi$,
$$
\sqrt{-1}\partialartial {\bar{\alpha}}r{\partialartial} \partialsi +\sqrt{-1}V^*\wedge {\bar{\alpha}}r{\partialartial}\partialsi-\sqrt{-1}\overline{V}^* \wedge \partialartial \partialsi +\sqrt{-1}V^*\wedge \overline{V}^* \wedge \partialsi+\frac{L (\partialsi)}{t}\gammae 0
$$
as a $(m-p+1, m-p+1)$-form, with $V^*$ being a $(1,0)$-type $1$-form. This generalizes the matrix estimate for positive solutions to the heat equation proved in \cite{CN}, which asserts $\sqrt{-1}\partialartial {\bar{\alpha}}r{\partialartial}\operatorname{l}og \partialsi+\frac{\omega}{t}\gammae 0$. (Note that this is the matrix version of Li-Yau's estimate: $\Delta \operatorname{l}og \partialsi+\frac{m}{t}\gammae 0$. See also \cite{yau-harmonic} for the earlier work for harmonic functions.)
For the proof, it is a combination of techniques of \cite{Cao}, \cite{N-jams}, \cite{NT-jdg} and Hamilton's argument in \cite{richard-harnack}. Applying to the static solution the above result asserts a differential estimate:
\begin{equation}\operatorname{l}abel{eq:har1}
\frac{1}{\sqrt{-1}}{\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi+\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{V}\cdot {\bar{\alpha}}r{\partialartial}^* \partialhi-\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{\overline{V}}\cdot \partialartial^* \partialhi +\sqrt{-1} \sqrt {-1}ota_V\cdot \sqrt {-1}ota_{\overline{V}}\cdot \partialhi\gammae 0
\epsilonnd{equation}
for any ${\bar{\alpha}}r{\partialartial}$-harmonic positive $(p, p)$-form $\partialhi$ and any vector field $V$ of $(1, 0)$-type. Note here on a noncompact manifold, being harmonic does not imply that $\partialartial^*\partialhi =0$ (or ${\bar{\alpha}}r{\partialartial}^* \partialhi=0$).
As a result of independent interest we also observe that $\mathcal{C}_p$ is an invariant condition under the K\"ahler-Ricci flow, thanks to a general invariant cone result of Wilking, whose proof we include here in the Appendix (see also a recent preprint \cite{CT}). Note that this result of Wilking includes almost all the known invariant cones such as the nonnegativity of bisectional curvature, the nonnegativity of isotropic curvature, etc.
After establishing the invariance of $\mathcal{C}_p$, it is natural to study the heat equation for the Hodge-Laplacian coupled with the K\"ahler-Ricci flow. For this we proved the following nonlinear version of the above estimate.
\begin{theorem} \operatorname{l}abel{thm12}Assume that $\partialhi(x, t)\gammae 0$ is a solution to heat equation of the Hodge-Laplacian coupled with the K\"ahler-Ricci flow: $\frac{\partialartial}{\partialartial t}g_{i{\bar{\alpha}}r{j}}=-R_{i{\bar{\alpha}}r{j}}$. Then
$$
\frac{1}{\sqrt{-1}}{\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi+\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{V}\cdot {\bar{\alpha}}r{\partialartial}^* \partialhi-\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{\overline{V}}\cdot \partialartial^* \partialhi +\sqrt{-1}\sqrt {-1}ota_V\cdot \sqrt {-1}ota_{\overline{V}}\cdot \partialhi+\Lambda_{\operatorname{Ric}}(\partialhi)+\frac{\Lambda \partialhi}{t}\gammae 0
$$as a $(p-1, p-1)$-form for any $(1, 0)$ type vector field $V$. Here $\Lambda_{\operatorname{Ric}}\partialhi $ is the adjoint of $\operatorname{Ric}\wedge (\cdot)$ with $\operatorname{Ric}=\sqrt{-1}R_{i{\bar{\alpha}}r{j}}dz^i\wedge dz^{{\bar{\alpha}}r{j}}$ being the Ricci form.
\epsilonnd{theorem}
To prove the above result it is necessary to prove the following family of matrix differential estimates which interpolate between Hamilton's matrix estimate and Cao's estimate for the K\"ahler-Ricci flow.
\begin{theorem} \operatorname{l}abel{thm13} Let $(M, g(t))$ be a complete solution to the K\"ahler-Ricci flow satisfying the condition $\mathcal{C}_p$ on $M \times [0, T]$. When $M$ is noncompact we assume that the curvature of $(M, g(t))$ is bounded on $M \times [0, T]$. Then
for any $\wedge^{1, 1}$-vector $U$ which can be written as
$U=\sum_{i=1}^{p-1} X_i\wedge {\bar{\alpha}}r{Y}_i+W\wedge {\bar{\alpha}}r{V}$, for $(1, 0)$-type vectors $X_i, Y_i, W, V$, the Hermitian bilinear form
$$
\mathcal{Q}(U\oplus W)\doteqdot
\mathcal{M}_{\alphabb}W^{\alpha}W^{{\bar{\beta}}}+P_{\alphabb\gamma}{\bar{\alpha}}r{U}^{{\bar{\beta}}\gamma}W^{\alpha}
+P_{\alphabb{\bar{\alpha}}r{\gamma}}U^{\alpha{\bar{\alpha}}r{\gamma}}W^{{\bar{\beta}}}
+R_{\alphabb\gammabd}U^{\alpha{\bar{\beta}}}{\bar{\alpha}}r{U}^{{{\bar{\alpha}}r{\delta}}\gamma}
$$
satisfies that $\mathcal{Q}\gammae 0$ for any $t>0$. Moreover, if the equality ever occurs for some $t>0$, the universal cover of $(M, g(t))$ must be a gradient expanding K\"ahler-Ricci soliton.
\epsilonnd{theorem}
Recall that the tensors $\mathcal{M}$ and $P$ are defined as $\mathcal{M}_{\alphabb}=\D R_{\alphabb}+R_{\alphabb\gammabd}R_{\delta{\bar{\alpha}}r{\gamma}}+\frac{R_{\alphabb}}{t}, \quad
P_{\alphabb\gamma}=\nabla_{\gamma}R_{\alphabb}, \quad P_{\alphabb{\bar{\alpha}}r{\gamma}}=\nabla_{{\bar{\alpha}}r{\gamma}}R_{\alphabb}.$
There exists a similar condition $\widetilde{\mathcal{C}}_p$ for Riemannian manifold which can be formulated similarly. Precisely we call the curvature operator satisfies $\widetilde{\mathcal{C}}_p$ if
$
\operatorname{l}angle \Bbb Rm (v), {\bar{\alpha}}r{v} \rangle >0
$
for any nonzero $v\sqrt {-1}n \Lambda^2(\Bbb C^n)$ which can be written as $v=\sum_{i=1}^k{Z_i\wedge W_i}$
for some complex vectors $Z_i$ and $W_i \sqrt {-1}n TM \otimes \Bbb C$. For K\"ahler manifolds it can be shown that $\widetilde{\mathcal{C}}_p=\mathcal{C}_{2p}$ and $\mathcal{C}_{2}$ amounts to the nonnegativity of the complex sectional curvature, a notion goes back at least to the work of Sampson \cite{Sa} on harmonic maps. This leads us to discover another family of matrix differential estimates for the Ricci flow which interpolate the result of Hamilton and a recent result of Brendle.
\begin{theorem} \operatorname{l}abel{thm14} Assume that $(M, g(t))$ on $M \times [0, T]$ satisfies $\widetilde{\mathcal{C}}_p$. When $M$ is noncompact we also assume that the curvature of $(M, g(t))$ is uniformly bounded on $M \times [0, T]$. Then for any $t>0$, the quadratic form
$$
\widetilde{\mathcal{Q}}(W\oplus U)\doteqdot \operatorname{l}angle \mathcal{M}(W), W\rangle +2\operatorname{l}angle P(W), U\rangle +\operatorname{l}angle \Bbb Rm(U), U\rangle
$$
satisfies that $\text{\rm Hess}Q \gammae 0$ for any $(x, t)\sqrt {-1}n M\times [0, T]$, $W\sqrt {-1}n T_x M \otimes \Bbb C$ and $U\sqrt {-1}n \wedge^2(T_xM\otimes \Bbb C) $ such that $U=\sum_{\mu=1}^p W_\mu \wedge Z_\mu$ with $W_p=W$. Furthermore, the equality holds for some $t>0$ implies that the universal cover of $(M, g(t))$ is a gradient expanding Ricci soliton.
\epsilonnd{theorem}
Here $\mathcal{M}$ and $P$ are defined similarly. In fact for $p=1$, our result is slightly stronger than Brendle's estimate. After we finished our paper, we were brought the attention to a recent preprint \cite{CT}, where a similar, but seemly more general result, was formulated in terms of the space-time consideration of Chow and Chu \cite{CC}. In the Spring of 2009 Wilking informed us that he has obtained a differential Harnack estimate for the Ricci flow with positive isotropic curvature, whose precise statement however is not known to us. It is very possible that the above result is a special case of his. Nevertheless our statement and proof here are direct/explicit without involving the space-time formulation. The proof is also rather short (see Section 9), can be easily checked and is motivated by the K\"ahler case.
Here is how we organize the paper. In Section 2 we prove that under the condition $\mathcal{C}_p$ the positivity of the $(p,p)$-forms is preserved under the Hodge-Laplacian heat equation. In Section 3 we derive the invariance of $\mathcal{C}_p$ by refining an argument of Wilking which is detailed in the Appendix. In Section 4 we collect and prove some preliminary formulae needed for the proof of the Theorem \ref{thm11}. The rigidity on the equality case as well as a monotonicity formula implied by Theorem \ref{thm11} was also included in Section 4.
Section 5 is devoted to the proof of Theorem \ref{thm11}. Sections 6 and 8 are devoted to the proof of Theorem \ref{thm12} as Section 7 is on the proof Theorem \ref{thm13}, which is needed in Section 8. Section 9 is on the proof of Theorem \ref{thm14}. Since up to Section 9 we present only the argument for the compact manifolds, Section 10 is the noncompact version of Sections 3, 7, 10, where the metric is assumed to have bounded curvature, while Section 11 supplies the argument for the noncompact version of Section 2, 6, where no upper bound on the curvature is assumed. Due to the length of the paper we shall study the applications of the estimates in a forth coming article.
\section{Heat equation deformation of $(p, p)$-forms}
Let $(M^m, g)$ be a complex Hermitian manifold of complex dimension $m$. Recall that a $(p,p)$-form $\partialhi$ is called positive if for any $x\sqrt {-1}n M$ and for any vectors $v_1, v_2, \cdot\cdot\cdot , v_p\sqrt {-1}n T^{1,0}_xM$, $\operatorname{l}angle \partialhi, \frac{1}{\sqrt{-1}}v_1\wedge \overlineerline {v}_1\wedge \cdot\cdot\cdot \wedge \frac{1}{\sqrt{-1}}v_p\wedge \overlineerline{v}_p\rangle \gammae 0$. By linear algebra (see also \cite{Siu-74}) it is equivalent to the condition that the nonnegativity holds for $v_1, \cdots, v_p$ satisfying that $\operatorname{l}angle v_i, v_j\rangle=\delta_{ij}$. We also denote $\operatorname{l}angle \partialhi, \frac{1}{\sqrt{-1}}v_1\wedge \overlineerline {v}_1\wedge \cdot\cdot\cdot \wedge \frac{1}{\sqrt{-1}}v_p\wedge \overlineerline{v}_p\rangle $ by $\partialhi(v_1, v_2, \cdots, v_p; {\bar{\alpha}}r{v}_1, {\bar{\alpha}}r{v}_2, \cdots, {\bar{\alpha}}r{v}_p)$, or even $\partialhi_{v_1v_2\cdots v_p, {\bar{\alpha}}r{v}_1 {\bar{\alpha}}r{v}_2\cdots {\bar{\alpha}}r{v}_p}$. We say that $\partialhi$ is strictly positive if $\partialhi_{v_1v_2\cdots v_p, {\bar{\alpha}}r{v}_1 {\bar{\alpha}}r{v}_2\cdots {\bar{\alpha}}r{v}_p}$ is positive for any linearly independent $\{v_i\}_{i=1}^p$.
Let $\Delta_{{\bar{\alpha}}r\partial}={\bar{\alpha}}r\partial {\bar{\alpha}}r\partial^* +{\bar{\alpha}}r\partial^* {\bar{\alpha}}r\partial$ be the ${\bar{\alpha}}r\partial$-Hodge Laplacian operator. There also exists a Laplacian operator $\Delta$ defined by
$$
\Delta =\frac{1}{2}\operatorname{l}eft(\nabla_i \nabla_{{\bar{\alpha}}r{i}}+\nabla_{{\bar{\alpha}}r{i}}\nabla_i\rightght).
$$
where $\nabla$ is the induced co-variant derivative on $(p, p)$-forms.
Since the complex geometry, analysis and Riemannian geometry fit better when the manifold is K\"ahler, we assume that $(M, g)$ is a K\"ahler manifold for our discussion. Let $\omega=\sqrt{-1} g_{i{\bar{\alpha}}r{j}}dz^i\wedge dz^{{\bar{\alpha}}r{j}}$ be the K\"ahler form. Clearly $\omega^p$ is a strictly positive $(p, p)$-form.
For a $(p, p)$-form $\partialhi_0$, consider the evolution equation:
\begin{equation}\operatorname{l}abel{eq:11}
\operatorname{l}eft(\frac{\partialartial}{\partialartial t} +\Delta_{{\bar{\alpha}}r\partial}\rightght)\partialhi(x, t) =0
\epsilonnd{equation}
with initial value $\partialhi(x, 0)=\partialhi_0(x)$.
Our first concern is when the positivity of the $(p, p)$-forms is preserved under the above evolution equation. If we denote by $\mathcal{P}_p$ the closed cone consisting all positive $(p, p)$-forms, an equivalent question is whether or not $\mathcal{P}_p$ is preserved under the heat equation (\ref{eq:11}).
The answer is well known for the cases $p=0$ and $p=m$ since the equation is nothing but the regular heat equation.
When $p=1$, this question was studied in \cite{NT-ajm} as well as \cite{NT-jdg} and it was proved that when $(M, g)$ is a complete K\"ahler manifold with nonnegative bisectional curvature, then the positivity is preserved for the solutions satisfying certain reasonable growth conditions, which is needed for the uniqueness of the solution with the given initial data.
It turns out, to prove the invariance of $\mathcal{P}_p$ for $m-1\gammae p\gammae 2$, we need to introduce a new curvature condition which we shall formulate below. We say that the curvature operator $\Bbb Rm$ of a K\"ahler manifold $(M, g)$ satisfies $\mathcal{C}_p$ (or lies inside the cone $\mathcal{C}_p$) if
$$
\operatorname{l}angle \Bbb Rm(\alphalpha), {\bar{\alpha}}r{\alphalpha}\rangle \gammae 0
$$
for any $\alphalpha\sqrt {-1}n \wedge ^{1, 1}(\Bbb C^m)$ (we use $\wedge^{1,1}_{\Bbb R} (\Bbb C^m)$ to denote the space of real wedge-$2$ vectors of $(1, 1)$ type), such that it can be written as $\alphalpha =\sum_{k=1}^p X_k \wedge {\bar{\alpha}}r{Y}_k$. Here $TM\otimes \Bbb C =T'M\oplus T''M$, $\operatorname{l}angle \cdot, \cdot \rangle$ is the bilinear extension of the Riemannian product, and we identify $T'M$ with $\Bbb C^m$. Note that $\operatorname{l}angle \Bbb Rm(X\wedge {\bar{\alpha}}r{Y}), \overlineerline{X\wedge {\bar{\alpha}}r{Y}}\rangle =R_{X{\bar{\alpha}}r{X} Y{\bar{\alpha}}r{Y}}$, the bisectional curvature of the complex plane spanned by $\{ X, Y\}$. Here the cones $\mathcal{C}_p$ interpolate between the cone of nonnegative bisectional curvature and that of the nonnegative curvature operator. In the next section we shall show that in fact $\mathcal{C}_p$ is an invariant condition under the K\"ahler-Ricci flow, which generalizes an earlier result of Bando-Mok \cite{Bando, Mok} on the invariance of the nonnegative bisectional curvature cone. Let us first recall the following well-known computational lemma of Kodaira \cite{M-K}.
\begin{lemma} \operatorname{l}abel{lemma11}Let $\partialhi$ be a $(p, p)$-form, which can be locally expressed as
$$
\partialhi=\frac{1}{(p!)^2}\sum \partialhi_{I_p,{\bar{\alpha}}r{J}_p}\operatorname{l}eft(\sqrt{-1}dz^{i_1}\wedge dz^{{\bar{\alpha}}r{j_1}}\rightght)\wedge \cdot\cdot\cdot\wedge \operatorname{l}eft(\sqrt{-1}dz^{i_p}\wedge dz^{{\bar{\alpha}}r{j_p}}\rightght)
$$
where $I_p=(i_1, \cdot\cdot \cdot i_p)$ and ${\bar{\alpha}}r{J}_p=({\bar{\alpha}}r{j_1}, \cdot\cdot\cdot, {\bar{\alpha}}r{j_p})$. Then
\begin{eqnarray}
\operatorname{l}eft(\Delta_{{\bar{\alpha}}r\partial} \partialhi\rightght)_{I_p, {\bar{\alpha}}r{J}_p}&=&-\frac{1}{2}\operatorname{l}eft(\sum_{ij}g^{i{\bar{\alpha}}r{j}}\nabla_{{\bar{\alpha}}r{j}}\nabla_{i}
\partialhi_{I_p,{\bar{\alpha}}r{J}_p}+\sum_{ij}g^{{\bar{\alpha}}r{j}i}\nabla_i\nabla_{{\bar{\alpha}}r{j}}\partialhi_{I_p,{\bar{\alpha}}r{J}_p}\rightght)
\nonumber\\
&\quad& -\sum_{\mu = 1}^{p}\sum_{\nu =1 }^{p}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j}_\nu}\partialhi_{i_1\cdots(k)_\mu\cdots
i_p,{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j_p}} \operatorname{l}abel{eq:Kodaira}\\
&\quad&+\frac{1}{2}\operatorname{l}eft(\sum_{\nu =1}^{p}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}\partialhi_{I_p,{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})\cdots{\bar{\alpha}}r{j}_p}+\sum_{\mu =1 }^{p}R_{i_\mu}^{\,\,
k}\partialhi_{i_1\cdots(k)_\mu\cdots i_p, {\bar{\alpha}}r{J}_p} \rightght)\nonumber.
\epsilonnd{eqnarray}
Here $R_{i{\bar{\alpha}}r{j}k{\bar{\alpha}}r{l}}$, $R_{i{\bar{\alpha}}r{j}}$, $R^{\,\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j_\nu}}$, $R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j_\nu}}$ are the curvature tensor, Ricci tensor and the index raising of them via the K\"ahler metric, $(k)_{\mu}$ means that the index in the $\mu$-th position is replaced by $k$. Here the repeated index is summed from $1$ to $m$.
\epsilonnd{lemma}
An immediate consequence of (\ref{eq:Kodaira}) is that if $\partialhi$ is a solution of (\ref{eq:11}), then it satisfies that
\begin{equation}\operatorname{l}abel{KB-heat}
\operatorname{l}eft(\frac{\partialartial}{\partialartial t} -\Delta\rightght)\partialhi(x, t) =\mathcal{KB}(\partialhi)
\epsilonnd{equation}
where
\begin{eqnarray*}
\operatorname{l}eft(\mathcal{KB}(\partialhi)\rightght)_{I_p,{\bar{\alpha}}r{J_p}}&=&\sum_{\mu = 1}^{p}\sum_{\nu =1 }^{p}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{j_\nu}}\partialhi_{i_1\cdots(k)_\mu\cdots
i_p,{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_p} \\
&\quad&
-\frac{1}{2}\operatorname{l}eft(\sum_{\nu =1}^{p}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}\partialhi_{I_p,{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})\cdots{\bar{\alpha}}r{j_p}}+\sum_{\mu =1 }^{p}R_{i_\mu}^{\,\,
k}\partialhi_{i_1\cdots(k)_\mu\cdots i_p, {\bar{\alpha}}r{J}_p}\rightght).
\epsilonnd{eqnarray*}
\begin{proposition}\operatorname{l}abel{max-pp} Let $(M, g)$ be a K\"ahler manifold whose curvature operator $\Bbb Rm \sqrt {-1}n \mathcal{C}_p$. Assume that $\partialhi(x, t)$ is a solution of (\ref{eq:11}) such that $\partialhi(x, 0)$ is positive. Then $\partialhi(x, t)$ is positive for $t>0$.
\epsilonnd{proposition}
\begin{proof} When $M$ is a compact manifold, applying Hamilton's tensor maximum principle, it suffices to show that if at $(x_0, t_0)$ there exist $v_1, \cdots v_p$ such that $\partialhi_{v_1 \cdots v_p, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_p}=0$ and for any $(x, t)$ with $t\operatorname{l}e t_0$, $\partialhi \gammae 0$,
$$
\mathcal{KB}(\partialhi)_{v_1 \cdots v_p, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_p}\gammae 0.
$$
This holds obviously if $\{v_i\}_{i=1}^p$ is linearly dependent since $\mathcal{KB}(\partialhi)$ is a $(p, p)$-form. Hence we assume that $\{v_i\}_{i=1}^p$ is linearly independent. By Gramm-Schmidt process, which does not change the sign (or being zero) of $\partialhi_{v_1 \cdots v_p, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_p}$, we can assume that $v_1, \cdots, v_p$ can be extended to a unitary frame. Hence we may assume that $(v_1, \cdots, v_p)$=$(\frac{\partialartial\,\,}{\partialartial z_1}, \cdots, \frac{\partialartial\,\,}{\partialartial z_p})$ with $(z^1,\cdots, z^m)$ being a normal coordinate centered at $x_0$. Hence what we need to verify is that $(\mathcal{KB}(\partialhi))_{1\,\cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}\gammae 0$.
Since we have that
$\partialhi_{1\, 2\, \cdots\, p,\, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{p}}=0$ and
$$
I(t)\doteqdot \partialhi\operatorname{l}eft(\frac{1}{\sqrt{-1}}(v_1+tw_1)\wedge \overlineerline{v_1+tw_1}\wedge \cdots \wedge \frac{1}{\sqrt{-1}}(v_p+tw_p)\wedge \overlineerline{v_p+tw_p}\rightght)\gammae 0
$$
for any $t\gammae 0$ and any vectors $w_1, \cdots, w_p$. The equation $I'(0)=0$ implies that
\begin{equation}\operatorname{l}abel{1st-var}
\sum_{1\operatorname{l}e k, l\operatorname{l}e p} \partialhi_{1\,\cdots\, (w_k)_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}+\partialhi_{1\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, (\overlineerline{w_l})_l\, \cdots\, {\bar{\alpha}}r{p}}=0.
\epsilonnd{equation}
Here $\partialhi_{1\,\cdots\, (w_k)_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}$ the $k$-the holomorphic position is filled by vector $w_k$.
For the simplicity of the notation we write $\partialhi_{1\,\cdots\, (w_k)_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}$ as $\partialhi_{1\,\cdots\, w_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}$.
Since this holds for any $p$-vectors $w_1, \cdots, w_p$, if we replace $t$ by $\sqrt{-1}t$, one can deduce from (\ref{1st-var}) that
\begin{equation}\operatorname{l}abel{1st-var-2}
\sum_{1\operatorname{l}e k\operatorname{l}e p}\partialhi_{1\,\cdots\, w_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}=\sum_{1\operatorname{l}e l\operatorname{l}e p}\partialhi_{1\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, \overlineerline{w_l}\, \cdots\, {\bar{\alpha}}r{p}}=0.
\epsilonnd{equation}
This implies that
$$\operatorname{l}eft(\sum_{1\operatorname{l}e l\operatorname{l}e m} \sum_{\nu =1}^{p}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{\nu}}\partialhi_{1\, \cdots\, p, \, {\bar{\alpha}}r{1}\, \cdots\, ({\bar{\alpha}}r{l})_\nu\, \cdots\,{\bar{\alpha}}r{p}}+\sum_{1\operatorname{l}e k\operatorname{l}e m}\sum_{\mu =1 }^{p}R_{\mu}^{\,\,
k}\partialhi_{1\, \cdots\, (k)_\mu\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}\rightght)=0.
$$
Now the fact $I''(0)\gammae 0$ implies that
\begin{eqnarray}\operatorname{l}abel{2nd-var}
&\quad&\sum_{1\operatorname{l}e k,\, l\operatorname{l}e p} \partialhi_{1\,\cdots\, w_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, \overlineerline{w_l}\, \cdots\, {\bar{\alpha}}r{p}}+\sum_{1\operatorname{l}e k\ne l \operatorname{l}e p}\partialhi_{1\, \cdots\, w_k\, \cdots\, w_l\, \cdots\,p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}\\
&\quad&\quad \quad + \sum_{1\operatorname{l}e k\ne l \operatorname{l}e p}\partialhi_{1\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, \overlineerline{w_k}\, \cdots\, \overlineerline{w_l}\, \cdots\, {\bar{\alpha}}r{p}}\gammae 0. \nonumber
\epsilonnd{eqnarray}
Replacing $t$ by $\sqrt{-1} t$ in $I(t)$, the fact that $I''(0)\gammae 0$ will yield
\begin{eqnarray}\operatorname{l}abel{2nd-var-2}
&\quad&\sum_{1\operatorname{l}e k,\, l\operatorname{l}e p} \partialhi_{1\,\cdots\, w_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, \overlineerline{w_l}\, \cdots\, {\bar{\alpha}}r{p}}-\sum_{1\operatorname{l}e k\ne l \operatorname{l}e p}\partialhi_{1\, \cdots\, w_k\, \cdots\, w_l\, \cdots\,p,\, {\bar{\alpha}}r{1}\, \cdots\, {\bar{\alpha}}r{p}}\\
&\quad&\quad \quad - \sum_{1\operatorname{l}e k\ne l \operatorname{l}e p}\partialhi_{1\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, \overlineerline{w_k}\, \cdots\, \overlineerline{w_l}\, \cdots\, {\bar{\alpha}}r{p}}\gammae 0. \nonumber
\epsilonnd{eqnarray}
Adding them up we have that for any $\mathfrak{w}=\operatorname{l}eft(\begin{array}{l} w_1\\ \vdots\\
w_p\epsilonnd{array}\rightght)\sqrt {-1}n \oplus_{1}^p T^{1, 0}_{x_0}M$, the Hermitian form
\begin{equation}\operatorname{l}abel{2nd-var-3}
\mathcal{J}(\mathfrak{w}, \overlineerline{\mathfrak{w}})\doteqdot \sum_{1\operatorname{l}e k,\, l\operatorname{l}e p} \partialhi_{1\,\cdots\, w_k\, \cdots\, p,\, {\bar{\alpha}}r{1}\, \cdots\, \overlineerline{w_l}\, \cdots\, {\bar{\alpha}}r{p}}
\epsilonnd{equation}
is semi-positive definite. A Hermitian-bilinear form $\mathcal{J}(\mathfrak{w}, \overlineerline{\mathfrak{z}}) $
can be obtained via the polarization. In matrix form, the nonnegativity of $\mathcal{J}(\cdot, \cdot)$ is equivalent to that
\begin{eqnarray*}
A= \operatorname{l}eft(\begin{array}{l}
\partialhi_{(\cdot)\, 2\cdots\, p, \, {\bar{\alpha}}r{(\cdot)}\, {\bar{\alpha}}r{2}\,\cdots\, {\bar{\alpha}}r{p}}\quad
\partialhi_{1\, (\cdot)\, \cdots\, p, \, {\bar{\alpha}}r{(\cdot)}\, {\bar{\alpha}}r{2}\,\cdots\, {\bar{\alpha}}r{p}}\quad \quad \quad
\cdots \quad \quad \quad
\partialhi_{1\, 2\, \cdots\, (\cdot)_p, \, {\bar{\alpha}}r{(\cdot)}\, {\bar{\alpha}}r{2}\,\cdots\, {\bar{\alpha}}r{p}}\\
\partialhi_{(\cdot)\, 2\cdots\, p, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{ (\cdot)}\, \cdots\, {\bar{\alpha}}r{p}}
\quad \partialhi_{1\, (\cdot)\, \cdots\, p, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{(\cdot)}\, \cdots\, {\bar{\alpha}}r{p}}
\quad \quad \quad
\cdots
\quad \quad \quad
\partialhi_{1\, 2\, \cdots\, (\cdot)_p, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{(\cdot)}\, \cdots\, {\bar{\alpha}}r{p}}\\
\quad \quad \quad \cdots\quad \quad \quad \quad \quad \quad \cdots \quad \quad \quad \quad \quad \, \,\, \cdots \quad \quad \quad \quad \quad \quad \cdots \quad \quad \quad\\
\partialhi_{(\cdot)\, 2\cdots\, p, \, {\bar{\alpha}}r{1}\,{\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{ (\cdot)}_p}\quad
\partialhi_{1\, (\cdot)\, \cdots\, p, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{ (\cdot)}_p}\quad \quad \,
\cdots \quad \quad \quad
\partialhi_{1\, 2\, \cdots\, (\cdot)_p, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{ (\cdot)}_p}
\epsilonnd{array}\rightght)
\epsilonnd{eqnarray*}
is a semi-positive definite Hermitian symmetric matrix. Namely $\overline{\mathfrak{w}}^{tr} A \mathfrak{w}\gammae 0$. Here we view $\partialhi_{1\,\cdots\,(\cdot)_\mu\, \cdots\,
p, \, {\bar{\alpha}}r{1}\, \cdots\, ({\bar{\alpha}}r{\cdot})_\nu\, \cdots\, {\bar{\alpha}}r{p}}$ as a matrix such that for any vectors $w, z$, $\overline{z}^{tr} \partialhi_{1\,\cdots\,(\cdot)_\mu\, \cdots\,
p, \, {\bar{\alpha}}r{1}\, \cdots\, ({\bar{\alpha}}r{\cdot})_\nu\, \cdots\, {\bar{\alpha}}r{p}} \cdot w$ is the
Hermitian-bilinear form $\partialhi_{1\,\cdots\,(w)_\mu\, \cdots\,
p, \, {\bar{\alpha}}r{1}\, \cdots\, ({\bar{\alpha}}r{z})_\nu\, \cdots\, {\bar{\alpha}}r{p}}$.
The equivalence can be made via the identity
$
\mathcal{J}(\mathfrak{w}, \overlineerline{\mathfrak{z}})=\operatorname{l}angle A (\mathfrak{w}), \overlineerline{\mathfrak{z}}\rangle.
$
Here $\mathfrak{w}=w_1\oplus\cdots \oplus w_p$, $\mathfrak{z}=z_1\oplus \cdots \oplus z_p$.
If we define $\partialhi^{\mu{\bar{\alpha}}r{v}}$ by
$$
\operatorname{l}angle \partialhi^{\mu{\bar{\alpha}}r{\nu}}(X), \overline{Y}\rangle =\partialhi_{1\cdots (X)_{\mu}\cdots p, {\bar{\alpha}}r{1}\cdots (\overline{Y})_\nu\cdots {\bar{\alpha}}r{p}}
$$
it is easy to see that $\overline{(\partialhi^{\mu{\bar{\alpha}}r{\nu}})^{\operatorname{tr}}}=\partialhi^{\nu {\bar{\alpha}}r{\mu}}$. Using this notation
$\mathcal{J}(\mathfrak{w}, \overlineerline{\mathfrak{z}})$ or $\operatorname{l}angle \mathcal{J}(\mathfrak{w}), \overlineerline{\mathfrak{z}}\rangle$ can be expressed as $\sum \operatorname{l}angle \partialhi^{\mu {\bar{\alpha}}r{\nu}}(w_\mu), \overline{z}_\nu\rangle$. It is easy to check that $\mathcal{J}$ is Hermitian symmetric.
What to be checked is that
$$
\mathcal{KB}(\partialhi)_{1\, 2\, \cdots\, p,\, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{2}\, \cdots\,
{\bar{\alpha}}r{p}}=\sum_{\mu = 1}^{p}\sum_{\nu =1 }^{p}R^{\,
k{\bar{\alpha}}r{l}}_{\mu\,\,{\bar{\alpha}}r{\nu}}\partialhi_{1\,\cdots\,(k)_\mu\, \cdots\,
p, \, {\bar{\alpha}}r{1}\, \cdots\, ({\bar{\alpha}}r{l})_\nu\, \cdots\, {\bar{\alpha}}r{p}} \gammae0.
$$
Under the unitary frame it is equivalent to
\begin{equation}\operatorname{l}abel{goal1}\sum_{\mu = 1}^{p}\sum_{\nu =1 }^{p}R_{\mu\,{\bar{\alpha}}r{\nu}\, l\,{\bar{\alpha}}r{k}}\partialhi_{1\,\cdots\,(k)_\mu\, \cdots\,
p, \, {\bar{\alpha}}r{1}\, \cdots\, ({\bar{\alpha}}r{l})_\nu\, \cdots\, {\bar{\alpha}}r{p}} \gammae0.
\epsilonnd{equation}
Here we have used the 1st-Bianchi identity. If we can show that the Hermitian matrix
\begin{eqnarray*}
B=\operatorname{l}eft(\begin{array}{l}
R_{1{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
R_{1{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{1{\bar{\alpha}}r{p}(\cdot){\bar{\alpha}}r{(\cdot)}}\\
R_{2{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
R_{2{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{2{\bar{\alpha}}r{p}(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\quad\cdots\quad\quad\quad\cdots\quad\quad\quad\cdots\quad\quad\quad\cdots\\
R_{p{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
R_{p{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{p{\bar{\alpha}}r{p}(\cdot){\bar{\alpha}}r{(\cdot)}}
\epsilonnd{array}\rightght)
\epsilonnd{eqnarray*}
is nonnegative, then the inequality (\ref{goal1}) holds since the left hand side of (\ref{goal1}) is just the trace of the product matrix
$B\cdot A$ of the two nonnegative Hermitian symmetric matrices.
On the other hand the nonnegativity of $B$ is equivalent to for any $(1,0)$-vectors $w_1, \cdots, w_p$,
$\operatorname{l}angle B(\mathfrak{w}), \overline{\mathfrak{w}}\rangle \gammae 0$ with $\mathfrak{w}=w_1\oplus\cdots \oplus w_p$. This is equivalent to
$$
\sum R_{i{\bar{\alpha}}r{j} w_j \overline{w}_i}\gammae 0.
$$
Let $\alphalpha =\sum_{i=1}^p \frac{\partialartial}{\partialartial z^i}\wedge \overline{w}_i$ (we later simply denote as $\sum i\wedge \overline{w}_i$). Then $\operatorname{l}angle \Bbb Rm (\alphalpha), \overline{\alphalpha}\rangle \gammae 0$ is equivalent to the above inequality. This proves (\ref{goal1}), hence the proposition, at least for the case when $M$ is compact.
Another way to look at this is to define the transformation $R^{\mu {\bar{\alpha}}r{\nu}}$ as $\operatorname{l}angle R^{\mu {\bar{\alpha}}r{\nu}}(X), \overline{Y}\rangle =R_{\mu{\bar{\alpha}}r{\nu}X \overline{Y}}$. Similarly one can easily check that
$\overline{(R^{\mu {\bar{\alpha}}r{\nu}})^{\operatorname{tr}}}=R^{\nu {\bar{\alpha}}r{\mu}}$. Define transformation $\mathcal{K}$ on $\oplus_{i=1}^p T'M$ by
$\operatorname{l}angle \mathcal{K}(\mathfrak{w}), \overline{\mathfrak{z}}\rangle $ as $\sum \operatorname{l}angle R^{\mu {\bar{\alpha}}r{\nu}}(w_\nu), \overline{z}_\mu\rangle $. It is easy to check that $\mathcal{K}$ is Hermitian symmetric and that $\Bbb Rm \sqrt {-1}n \mathcal{C}_p$ implies $\mathcal{K}\gammae 0$. Simple algebraic manipulation shows that:
\begin{eqnarray*}
\mbox{LHS of (\ref{goal1})} &=& \sum_{\mu, \nu=1}^{p} \operatorname{l}angle R^{\mu{\bar{\alpha}}r{\nu}}(e_l), \overline{e}_k\rangle \operatorname{l}angle \partialhi^{\mu {\bar{\alpha}}r{\nu}}(e_k), \overline {e}_l\rangle\\
&=& \sum_{\mu, \nu=1}^p \operatorname{l}angle R^{\mu{\bar{\alpha}}r{\nu}}(\partialhi^{\mu {\bar{\alpha}}r{\nu}}(e_k)), \overline{e}_k\rangle,
\epsilonnd{eqnarray*}
if $\{e_k\}_{k=1\cdots m}$ is a unitary frame. One can see that this is nothing but the trace of $\mathcal{K} \cdot \mathcal{J}$ since a natural unitary base for $\oplus_{\mu=1}^p T'M$ is $\{E_{\mu k}\}$, where $1\operatorname{l}e \mu \operatorname{l}e p, 1\operatorname{l}e k \operatorname{l}e m$, $E_{\mu k} =\vec{0} \oplus \cdots\oplus (e_k)_{\mu}\oplus \cdots \oplus \vec{0}$. Then $\mathcal{J}(E_{\mu k})=\oplus_{\nu} \partialhi^{\mu {\bar{\alpha}}r{\nu}}(e_k)$. Hence $\mathcal{K}(\mathcal{J}(E_{\mu k}))=\oplus_{\sigma}R^{\sigma {\bar{\alpha}}r{\nu}}(\partialhi^{\mu {\bar{\alpha}}r{\nu}}(e_k))$. This shows that $\operatorname{l}angle \mathcal{K}(\mathcal{J}(E_{\mu k})), \overline{E_{\mu k}}\rangle = \sum \operatorname{l}angle R^{\mu{\bar{\alpha}}r{\nu}}(\partialhi^{\mu {\bar{\alpha}}r{\nu}}(e_k)), \overline{e}_k\rangle$. Hence the left hand side of (\ref{goal1}) can be written as $\operatorname{trace} (\mathcal{K}\cdot \mathcal{J})$. Similarly for any $X_1, \cdots, X_p$ we can define $R^{\mu{\bar{\alpha}}r{\nu}}$ and $\partialhi^{\mu {\bar{\alpha}}r{\nu}}$ and $\mathcal{K}$ and $\mathcal{J}$. The above argument shows that
\begin{equation}
\mathcal{KB}_{X_1\cdots X_p, \overline{X}_1\cdots \overline{X}_p}=\operatorname{trace} (\mathcal{K}\cdot \mathcal{J}).
\epsilonnd{equation}
For noncompact complete manifolds we postpone it to Section 11 (Theorem \ref{max-pp-noncom}).
\epsilonnd{proof}
\section{Invariance of $\mathcal{C}_p$ under the K\"ahler-Ricci flow}
Recently \cite{Wilking}, Wilking proved a very general result on invariance of cones of curvature operators under Ricci flow. The result is formulated for any Riemannian manifold $(M, g)$ of real dimension $n$. Since our proof is a modification of his we first state his result. Identify $TM $ with $\Bbb R^n$ and its complexification $TM \otimes \Bbb C$ with $\Bbb C^n$. Also identify $\wedge^2(\Bbb R^n)$ with the Lie algebra $\mathfrak{so}(n)$. The complexified Lie algebra $\mathfrak{so}(n, \Bbb C)$ can be identified with $\wedge^2(\Bbb C^n)$. Its associated Lie group is ${\mathsf{SO}}(n, \Bbb C)$, namely all complex matrices $A$ satisfying $A\cdot A^{\operatorname{tr}} =A^{\operatorname{tr}} \cdot A =\sqrt {-1}d$. Recall that there exists the natural action of ${\mathsf{SO}}(n, \Bbb C)$ on $\wedge ^2(\Bbb C^n)$ by extending the action $g\sqrt {-1}n {\mathsf{SO}}(n)$ on $x\otimes y$ ($g(x\otimes y) =gx\otimes gy$). Let $\Sigma \subset \wedge^2(\Bbb C^m)$ be a set which is invariant under the action of ${\mathsf{SO}}(n, \Bbb C)$.
Let $\widetilde{\mathcal{C}}_{\Sigma}$ be the cone of curvature operators satisfying that $\operatorname{l}angle R(v), {\bar{\alpha}}r{v} \rangle \gammae 0$ for any $v\sqrt {-1}n \Sigma$. Here we view the space of algebraic curvature operators as a subspace of $S^2(\wedge^2(\Bbb R^n))$ satisfying the first Bianchi identity. Recently (May of 2008) \cite{Wilking}, Wilking proved the following result.
\begin{theorem}[Wilking]\operatorname{l}abel{ww}
Assume that $(M, g(t))$, for $0\operatorname{l}e t\operatorname{l}e T$, is a solution of Ricci flow on a compact manifold. Assume that $\Bbb Rm(g(0))\sqrt {-1}n \widetilde{\mathcal{C}}_{\Sigma}$. Then $\Bbb Rm(g(t))\sqrt {-1}n \widetilde{\mathcal{C}}_{\Sigma}$ for all $t\sqrt {-1}n [0, T]$.
\epsilonnd{theorem}
It is not hard to see that this result contains the previous result of Brendle-Schoen \cite{BS} and Nguyen \cite{Ng} on the invariance of the cone of nonnegative isotropic curvature under the Ricci flow. In particular it implies the invariance of the cone of nonnegative complex sectional curvature, a useful consequence first observed in \cite{BS} (see also \cite{NW} for an alternative proof). By modifying the argument of Wilking one can prove the following result.
\begin{corollary}\operatorname{l}abel{thm:p-NBC}
The K\"ahler-Ricci flow on a compact K\"ahler manifold preserves the cone $\mathcal{C}_p$ for any $p$.
\epsilonnd{corollary}
For Riemannian manifolds, there exists another family of invariant cones which is analogous to $\mathcal{C}_p$.
We say that the complex sectional curvature of $(M, g)$ is $k$-positive if
$$
\operatorname{l}angle \Bbb Rm (v), {\bar{\alpha}}r{v} \rangle >0
$$
for any nonzero $v\sqrt {-1}n \Lambda^2(\Bbb C^n)$ which can be written as $v=\sum_{i=1}^k{Z_i\wedge W_i}$
for some complex vectors $Z_i$ and $W_i \sqrt {-1}n TM \otimes \Bbb C$. Clearly the complex sectional curvature is $1$-positive is the same as positive complex sectional curvature. The $k$-positivity for $k\gammae \frac{n(n-1)}{2}$ is the same as positive curvature operator.
Similarly one has the notion that the complex sectional curvature is $k$-nonnegative. In the space of the algebraic curvature operators $S_B(\mathfrak{so}(n))$, the ones with $k$-nonnegative complex sectional curvature form a cone $\widetilde{\mathcal{C}}_k$. Clearly $\widetilde{\mathcal{C}}_k \subset\widetilde{\mathcal{C}}_{k-1}$. The argument in \cite{NW} (this in fact was proved in an updated version of \cite{NW}) proves that
\begin{theorem}\operatorname{l}abel{thm:PkCSC}
The Ricci flow on a compact manifold preserves $\widetilde{\mathcal{C}}_k$.
\epsilonnd{theorem}
Of course, this result is now also included in the previous mentioned general theorem of Wilking. In fact almost all the known invariant cones of nonnegativity type can be formulated as a special case of Wilking's above theorem.
Note that even for a K\"ahler manifold $\mathcal{C}_p$ is a bigger cone than $\widetilde{\mathcal{C}}_p$. For example, for K\"ahler manifold $(M ,g)$, the nonnegativity of the complex sectional curvature implies that
$$
R_{s{\bar{\alpha}}r{t}j{\bar{\alpha}}r{i}}(a_i {\bar{\alpha}}r{b}_j-c_i {\bar{\alpha}}r{d}_j) \overlineerline{(a_s{\bar{\alpha}}r{b}_t -c_s{\bar{\alpha}}r{d}_t)}\gammae 0
$$
for any complex vector $\vec{a}$($=(a_1, \cdots, a_m)$), $\vec{b}$, $\vec{c}$ and $\vec{d}$.
Namely $(M, g)$ has {\sqrt {-1}t strongly nonnegative sectional curvature in the sense of Siu}, which is in general stronger than the nonnegativity of the sectional curvature (or bisectional curvature). On a K\"ahler manifold, if $\{E_i\}$ is a unitary basis of $T'M$, and letting $X=a_i E_i$, $Y=b_i E_i$, $Z=c_i E_i$ and $W=d_i E_i$, then the above is equivalent to $$\operatorname{l}angle \Bbb Rm( (X+{\bar{\alpha}}r{Z})\wedge ({\bar{\alpha}}r{Y}+W)), \overlineerline{ (X+{\bar{\alpha}}r{Z})\wedge ({\bar{\alpha}}r{Y}+W)}\rangle\gammae 0.$$
In fact a simple computation as the above proves that $\widetilde{\mathcal{C}}_p= \mathcal{C}_{2p}$: For $1\operatorname{l}e i\operatorname{l}e p$, let $Z_i=X_{2i-1}+{\bar{\alpha}}r{Y}_{2i}$ and $W_i={\bar{\alpha}}r{Y}_{2i-1}-X_{2i}$. Then $\Bbb Rm(\sum_{i=1}^p Z_i \wedge W_i)=\Bbb Rm (\sum_{j=1}^{2p} X_j\wedge {\bar{\alpha}}r{Y}_{j})$. Thus by the fact that $\Bbb Rm$ is self-adjoint
$$
\operatorname{l}angle \Bbb Rm (\sum_{i=1}^p Z_i \wedge W_i), \sum_{i=1}^p\overlineerline {Z_i \wedge W_i}\rangle =\operatorname{l}angle \Bbb Rm (\sum_{j=1}^{2p} X_j\wedge {\bar{\alpha}}r{Y}_{j}),\sum_{j=1}^{2p} \overlineerline{X_j\wedge {\bar{\alpha}}r{Y}_{j}}\rangle.
$$
In order to prove Corollary \ref{thm:p-NBC}, first let $G$ be the subgroup of ${\mathsf{SO}}(n, \Bbb C)$ consisting of matrices $A \sqrt {-1}n {\mathsf{SO}}(n, \Bbb C)$ such that $A$ commutes with the almost complex structure $J=\operatorname{l}eft(\begin{matrix} 0 &\sqrt {-1}d\\ -\sqrt {-1}d &0\epsilonnd{matrix}\rightght)$, noting here that $n=2m$. Let $\mathfrak{g}$ be the Lie algebra of $G$. A key observation is that $\mathfrak{g}$ consisting of $c\sqrt {-1}n \mathfrak{so}(n, \Bbb C)$ which commutes with $J$. It is easy to show that $\mathfrak{g}$ is the same as $\wedge^{1, 1}(\Bbb C^m)$ under the identification of $\wedge^2(\Bbb C^n)$ with $\mathfrak{so}(n, \Bbb C)$. More precisely $c=(c_{ij})$ (which is identified with $c_{ij}X_i \wedge {\bar{\alpha}}r{X}_j$ for a unitary basis $\{X_i\}$) is identified with $\operatorname{l}eft(\begin{matrix} a & -b\\ b & a\epsilonnd{matrix}\rightght)$ with $a=c-c^{\operatorname{tr}}$ and $b=-\sqrt{-1}(c+c^{\operatorname{tr}})$. Now the argument of Wilking can be adapted to show the following result for the K\"ahler-Ricci flow.
\begin{theorem} \operatorname{l}abel{kaehler} Let $\Sigma\subset \wedge^{1,1}(\Bbb C^m)$ be a set invariant under the adjoint action of $G$. Let $\mathcal{C}_\Sigma=\{ \Bbb Rm |\, \operatorname{l}angle \Bbb Rm(v), {\bar{\alpha}}r{v}\rangle \gammae 0\}$ for any $v\sqrt {-1}n \Sigma$.
Assume that $(M, g(t))$ (with $t\sqrt {-1}n [0, T]$) is a solution to K\"ahler-Ricci flow on a compact K\"ahler manifold such that $\Bbb Rm(g(0))\sqrt {-1}n \mathcal{C}_\Sigma$. Then for any $t\sqrt {-1}n [0, T]$, $\Bbb Rm(g(t))\sqrt {-1}n \mathcal{C}_\Sigma$.
\epsilonnd{theorem}
If $C\sqrt {-1}n G$ and $v=\sum_{k=1}^p X_k \wedge {\bar{\alpha}}r{Y}_k$ with $X_k\sqrt {-1}n T'M$ and ${\bar{\alpha}}r{Y}_k\sqrt {-1}n T''M$, $C(v)=\sum_{k=1}^p C(X_k) \wedge C({\bar{\alpha}}r{Y}_k)$. Since $C$ is commutative with $J$, $X_k'=C(X_k)\sqrt {-1}n T'M$ and ${\bar{\alpha}}r{Y}_k'=C({\bar{\alpha}}r{Y}_k)\sqrt {-1}n T''M$. Hence the set consisting of all such $v$ is an invariant set $\Sigma_k$ under the adjoint action of $G$. The invariance of cone $\mathcal{C}_k$ follows from the above theorem by applying to $\Sigma=\Sigma_k$.
In fact, one can easily generalize Wilking's result to manifolds with special holonomy group. When the manifold $(M, g)$ has a special holonomy group $G$ with holonomy algebra $\mathfrak{g}\subset \mathfrak{so}(n)$, since for any $v\sqrt {-1}n \mathfrak{so}(n)$, $\Bbb Rm(v)\sqrt {-1}n \mathfrak{g}$ by Ambrose-Singer theorem, Wilking's proof, in particular (\ref{A1}) remains the same even if $\{b^\alphalpha\}$ being an orthonormal basis of $\mathfrak{g}$ instead of an orthonormal basis of $\mathfrak{so}(n)$. Note that in this case $\Bbb Rm(b)=0$ for any $b\sqrt {-1}n \mathfrak{g}^{\partialerp}$. Let $\mathfrak{g}^\Bbb C\subset \mathfrak{so}(n, \Bbb C)$ denote the complexified Lie algebra. It is easy to see then that for any $b\sqrt {-1}n \operatorname{l}eft({\mathfrak{g}^{\Bbb C}}\rightght)^{\partialerp}$, $\operatorname{l}angle \Bbb Rm (b), \overlineerline{w}\rangle =\operatorname{l}angle b, \overlineerline{\Bbb Rm(w)}\rangle =0$ for any $w\sqrt {-1}n \mathfrak{so}(n, \Bbb C)$. Hence $\Bbb Rm(b)=0$. This implies that $\operatorname{l}angle \Bbb Rm^{\#} (v), \overlineerline{w}\rangle=\frac{1}{2}\operatorname{trace}(-\operatorname{ad}_{\overlineerline{w}} \cdot \Bbb Rm \cdot \operatorname{ad}_{v} \cdot \Bbb Rm )$ with the trace taken for transformations of $\mathfrak{g}^\Bbb C$.
\begin{theorem}\operatorname{l}abel{general-holo} Assume that $(M, g_0)$ is a compact manifold with special holonomy group $G$ (and corresponding Lie algebra $\mathfrak{g}$). Let $\Sigma$ be a subset of $\mathfrak{g}^\Bbb C$ satisfying the assumption that it is invariant under the adjoint action of $G^\Bbb C$, the complexification of $G$. Then if the curvature operator $\Bbb Rm$ of $g_0$ lies inside the cone $\mathcal{C}_\Sigma$, the curvature operator $\Bbb Rm$ of $g(t)$, the solution to Ricci flow with initial value $g(0)=g_0$, also lies inside $\mathcal{C}_\Sigma$.
\epsilonnd{theorem}
When $G= \mathsf{U}(m)$ (with $n=2m$), the unitary group, the above result implies Theorem \ref{kaehler}.
All above results in this section remain true on noncompact manifolds if we assume that the solution $g(t)$ has bounded curvature. This shall be proved in Section 10.
\section{A LYH type estimate for positive $(p, p)$-forms.}
First we recall some known computations of Kodaira \cite{M-K}. Let $\partialhi$ be a $(p, q)$-form valued in a holomorphic Hermitian vector bundle $E$ with local frame $\{E_\alpha\}$ and locally $\partialhi=\sum \partialhi^\alpha E_{\alpha}$.
$$
\partialhi^\alpha = \frac{1}{p!q!}\sum \partialhi^\alpha_{I_p{\bar{\alpha}}r{J}_q}dz^{I_p}\wedge dz^{{\bar{\alpha}}r{J}_q}.
$$
Here $I_p=(i_1,\cdots,i_p), {\bar{\alpha}}r{J}_q=({\bar{\alpha}}r{j}_1, \cdots,{\bar{\alpha}}r{j}_q)$ and $dz^{I_p}=dz^{i_1}\wedge
\cdots\wedge dz^{i_p}$, $dz^{{\bar{\alpha}}r{J}_q}=dz^{\overlineerline{j_1}}\wedge\cdots \wedge dz^{\overlineerline{j_q}}.$
For $(p, p)$ forms, $\partialhi_{1\cdots p, {\bar{\alpha}}r{1}\cdots {\bar{\alpha}}r{p}}$ differs from $\partialhi_{1\cdots p {\bar{\alpha}}r{1}\cdots {\bar{\alpha}}r{p}}$ by a factor $\operatorname{l}eft(\frac{1}{\sqrt{-1}}\rightght)^p(-1)^{\frac{p(p-1)}{2}}$.
The following two formulae are well known
\begin{equation}\operatorname{l}abel{eq:41}
({\bar{\alpha}}r{\partial} \partialhi)^\alpha_{I_p{\bar{\alpha}}r{j}_0\cdots{\bar{\alpha}}r{j}_q}=(-1)^p\sum_{\nu =
0}^{q} (-1)^\nu \nabla_{{\bar{\alpha}}r{j}_\nu}
\partialhi^{\alpha}_{I_p{\bar{\alpha}}r{j}_0\cdots \hat{{\bar{\alpha}}r{j}}_\nu\cdots {\bar{\alpha}}r{j}_q},
\epsilonnd{equation}
\begin{equation}\operatorname{l}abel{eq:42}
({\bar{\alpha}}r{\partial}^*\partialhi)^\alpha_{I_p{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{q-1}}=(-1)^{p+1}\sum _{ij}g^{{\bar{\alpha}}r{j}i}\nabla_{i}
\partialhi^{\alpha}_{I_p\overline{jj_1}\cdots{\bar{\alpha}}r{j}_{q-1}}.
\epsilonnd{equation}
Here $\hat{{\bar{\alpha}}r{j_\nu}}$ means that the index ${\bar{\alpha}}r{j}_\nu$ is
removed. From (\ref{eq:41}) and (\ref{eq:42}) we have
\begin{eqnarray}
(\D_{{\bar{\alpha}}r{\partialartial}}\partialhi)^{\alpha}_{I_p {\bar{\alpha}}r{J}_q} & = & -\sum_{ij}g^{{\bar{\alpha}}r{j}i}\nabla_i\nabla_{{\bar{\alpha}}r{j}}\partialhi^{\alpha}_{I_p {\bar{\alpha}}r{J}_q}+
\sum_{\nu = 1}^q \Omega^{\alpha{\bar{\alpha}}r{l}}_{\b\, {\bar{\alpha}}r{j}_\nu}\partialhi^{\b}_{I_p {\bar{\alpha}}r{j}_1\cdots ({\bar{\alpha}}r{l})_\nu \cdots{\bar{\alpha}}r{j}_q}
\nonumber \\
& \,& +\sum_{\nu =1}^{q}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}\partialhi^{\alpha}_{I_p {\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})\cdots{\bar{\alpha}}r{j}_q}
-\sum_{\mu = 1}^{p}\sum_{\nu =1 }^{q}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j}_\nu}\partialhi^{\alpha}_{i_1\cdots(k)_\mu\cdots
i_p {\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_q}\operatorname{l}abel{eq:43}
\epsilonnd{eqnarray}
and
\begin{eqnarray}
(\D_{{\bar{\alpha}}r{\partialartial}}\partialhi)^{\alpha}_{I_p{\bar{\alpha}}r{J}_q} & =&
-\sum_{ij}g^{i{\bar{\alpha}}r{j}}\nabla_{{\bar{\alpha}}r{j}}\nabla_{i}
\partialhi^\alpha_{I_p{\bar{\alpha}}r{J}_q} + \sum_{\nu = 1}^q \Omega^{\alpha{\bar{\alpha}}r{l}}_{\b
{\bar{\alpha}}r{j}_\nu} \partialhi^{\b}_{I_p {\bar{\alpha}}r{j}_1\cdots ({\bar{\alpha}}r{l})_\nu
\cdots{\bar{\alpha}}r{j}_q} -\sum_{\b}\Omega_{\b}^{\alpha}
\partialhi^{\b}_{I_p {\bar{\alpha}}r{J}_q} \nonumber \\
& \, & +\sum_{\mu =1 }^{p}R_{i_\mu}^{\,\,
k}\partialhi^\alpha_{i_1\cdots(k)_\mu\cdots i_p {\bar{\alpha}}r{J}_q} -\sum_{\mu =
1}^{p}\sum_{\nu =1 }^{q}R^{\, \, k{\bar{\alpha}}r{l}}_{i_\mu\, \, \,
{\bar{\alpha}}r{j}_\nu}\partialhi^{\alpha}_{i_1\cdots(k)_\mu\cdots
i_p {\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_q}, \operatorname{l}abel{eq:44}
\epsilonnd{eqnarray}
where $(k)_{\mu}$ means that the index in the $\mu$-th position is replaced by $k$. Here
$$
\Theta^\alpha_{\beta}=\frac{\sqrt{-1}}{2\partiali}\sum \Omega^\alpha_{\beta \, i{\bar{\alpha}}r{j}}dz^i\wedge d{\bar{\alpha}}r{z}^j
$$
is the curvature of $E$ and $\Omega^\alpha_\beta$ is the mean curvature. Here we shall focus on the case that $E$ is the trivial bundle. Recall the contraction $\Lambda$ operator, $\Lambda: \wedge^{p, q}\to \wedge^{p-1, q-1}$, defined by
$$
(\Lambda \partialhi)_{i_1 \cdots i_{p-1} {\bar{\alpha}}r{j}_1 \cdots {\bar{\alpha}}r{j}_{q-1}}=\frac{1}{\sqrt{-1}}(-1)^{p-1} g^{i{\bar{\alpha}}r{j}}\partialhi_{i i_1 \cdots i_{p-1} {\bar{\alpha}}r{j} {\bar{\alpha}}r{j}_1 \cdots {\bar{\alpha}}r{j}_{p-1}}.
$$
Note our definition of $\Lambda$ differs from \cite{M-K} by a sign. With the above notations the K\"ahler identities assert
\begin{equation}\operatorname{l}abel{id1}
\partialartial \Lambda -\Lambda \partialartial =-\sqrt{-1}{\bar{\alpha}}r{\partialartial} ^*, \quad
{\bar{\alpha}}r{\partialartial}\Lambda -\Lambda {\bar{\alpha}}r{\partialartial}=\sqrt{-1}\partialartial^*.
\epsilonnd{equation}
In \cite{Ni-SDG}, the first author speculated that for $\partialhi$, a nonnegative $(p, p)$-form satisfying the Hodge-Laplacian heat equation, the $(p-1, p-1)$-form
\begin{equation}
Q(\partialhi, V)\doteqdot \frac{1}{2\sqrt{-1}}\operatorname{l}eft( {\bar{\alpha}}r{\partialartial}^*\partialartial^*-\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \rightght)\partialhi
+\frac{1}{\sqrt{-1}}({\bar{\alpha}}r{\partialartial}^* \partialhi)_V -\frac{1}{\sqrt{-1}}(\partialartial ^*\partialhi)_{{\bar{\alpha}}r{V}} +\partialhi_{V,
{\bar{\alpha}}r{V}}+\frac{\Lambda \partialhi}{t}\gammae 0 \operatorname{l}abel{qn1}
\epsilonnd{equation}
for any $(1,0)$ type vector field $V$. Here $\partialhi_{V, \overline{V}}$ is a $(p-1,p-1)$-form defined as
$$(\partialhi_{V, \overline{V}})_{I_{p-1},\overline{J}_{p-1}}\doteqdot \partialhi_{V I_{p-1}, \overline{V} J_{p-1}}$$
or equivalently
$$
(\partialhi_{V, \overline{V}})_{I_{p-1}\overline{J}_{p-1}}\doteqdot \frac{1}{\sqrt{-1}}(-1)^{p-1}\partialhi_{V I_{p-1} \overline{V} J_{p-1}},
$$
and for $\partialsi$ and $\partialsi'$, $\partialsi_V$ and $\partialsi'_{{\bar{\alpha}}r{V}}$ are defined as
$$
(\partialsi_V)_{I_{p-1} \overline{J}_q}\doteqdot\partialsi_{V I_{p-1} \overline{J}_q}, \quad \quad (\partialsi'_{{\bar{\alpha}}r{V}})_{I_p \overline{J}_{q-1}}\doteqdot (-1)^p\partialsi'_{I_p {\bar{\alpha}}r{V}\overline{J}_{q-1}}.
$$
When the meaning is clear we abbreviate $Q(\partialhi, V)$ as $Q$.
The expression of $Q$ coincides with the quantity $Z$ in Theorem 1.1 of \cite{N-jams} for $(1,1)$-forms due to (\ref{eq:41}), (\ref{eq:42}) as well as their cousins
\begin{eqnarray}
(\partialartial \partialhi)^\alpha_{i_0i_1\cdots i_p \overline{J}_q}&=&\sum_{\mu=0}^p (-1)^\mu \nabla_{i_\mu} \partialhi^\alpha_{i_0\cdots \hat{i}_\mu \cdots i_p \overline{J}_q},\operatorname{l}abel{eq:47}\\
(\partialartial^* \partialhi)^\alpha_{i_1\cdots i_{p-1} \overline{J}_q}&=& -\sum_{ij}g^{i{\bar{\alpha}}r{j}}\nabla_{{\bar{\alpha}}r{j}}\partialhi^\alpha_{ii_1\cdots i_{p-1} \overline{J}_q}\operatorname{l}abel{eq:48}
\epsilonnd{eqnarray}
since the operators (defined in \cite{N-jams} for the case $p=1$) $\operatorname{div}(\partialhi)_{i_1\cdots i_{p-1}, {\bar{\alpha}}r{J}_p}$, $\operatorname{div}(\partialhi)_{I_p, {\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}$ and $g^{i{\bar{\alpha}}r{j}}\nabla_i \operatorname{div}(\partialhi)_{I_{p-1}, {\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}$ can be identified with $\partialartial^*$, ${\bar{\alpha}}r{\partialartial}^*$ and ${\bar{\alpha}}r{\partialartial}^*\partialartial^*$ etc. To make it precise, first note that in our discussion the bundle is trivial and we can forget about the upper index in $\partialhi^\alphalpha$. It is easy to check that $\Lambda(\partialhi)_{I_{p-1}, \overline{J}_{p-1}}=g^{i{\bar{\alpha}}r{j}}\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}$. In (\ref{qn1}), $({\bar{\alpha}}r{\partialartial}^* \partialhi)_V$ is a $(p-1, p-1)$-form which by the definition can be written as
$$
({\bar{\alpha}}r{\partialartial}^* \partialhi)_V=\sum_{i=1}^m V^i \sqrt {-1}ota_i ({\bar{\alpha}}r{\partialartial}^* \partialhi)
$$
where $\sqrt {-1}ota_i$ is the adjoint of the operator $dz^i \wedge(\cdot)$. Hence $({\bar{\alpha}}r{\partialartial}^* \partialhi)_V=\sqrt {-1}ota_V \cdot {\bar{\alpha}}r{\partialartial}^*\partialhi$.
A direct calculation then shows that
$$
\frac{1}{\sqrt{-1}}\operatorname{l}eft(({\bar{\alpha}}r{\partialartial}^* \partialhi)_V\rightght)_{I_{p-1},\overline{J}_{p-1}}=V^{i} g^{{\bar{\alpha}}r{j}k}\nabla_k \partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}.$$
Similarly, $(\partialartial^* \partialhi)_{\overline{V}}=\sum_{j=1}^m V^{{\bar{\alpha}}r{j}}\sqrt {-1}ota_{{\bar{\alpha}}r{j}}(\partialartial^* \partialhi)=\sqrt {-1}ota_{\overline{V}} \cdot \partialartial^*\partialhi $ and
another direct computation implies that
$$ \frac{1}{\sqrt{-1}}\operatorname{l}eft((\partialartial^*\partialhi)_{\overline{V}}\rightght)_{I_{p-1}, \overline{J}_{p-1}}=-\overline{V^j}g^{i{\bar{\alpha}}r{k}}\nabla_{{\bar{\alpha}}r{k}}\partialhi_{iI_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}.$$
If we define
\begin{eqnarray*}
(\operatorname{div}''(\partialhi))_{I_{p},\overline{J}_{p-1}} \doteqdot\sum _{ij}g^{{\bar{\alpha}}r{j}i}\nabla_{i}
\partialhi_{I_p,\overline{j} \overline{J}_{p-1}}, &\,& (\operatorname{div}'(\partialhi))_{I_{p-1},\overline{J}_{p}} \doteqdot \sum_{ij}g^{i{\bar{\alpha}}r{j}}\nabla_{{\bar{\alpha}}r{j}}\partialhi_{iI_{p-1}, \overline{J}_p}, \\(\operatorname{div}''_{V}(\partialhi))_{I_{p-1}, \overline{J}_{p-1}}\doteqdot(\operatorname{div}''(\partialhi))_{VI_{p-1}, {\bar{\alpha}}r{J}_{p-1}}
,\quad &\, &\quad (\operatorname{div}'_{\overline{V}}(\partialhi))_{I_{p-1}, \overline{J}_{p-1}}\doteqdot (\operatorname{div}'(\partialhi))_{I_{p-1}, \overline{V} {\bar{\alpha}}r{J}_{p-1}}
\epsilonnd{eqnarray*}
then simple calculation shows that
\begin{eqnarray*}
\frac{1}{\sqrt{-1}}({\bar{\alpha}}r{\partialartial}^* \partialhi)_V =\operatorname{div}''_{V}(\partialhi), \quad &\, & \quad \frac{1}{\sqrt{-1}}(\partialartial^* \partialhi)(\overline{V})=-\operatorname{div}'_{\overline{V}}(\partialhi),\\
({\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi)_{I_{p-1}, \overline{J}_{p-1}} &=& \sqrt{-1} \operatorname{div}'' (\operatorname{div}'(\partialhi))_{I_{p-1}, \overline{J}_{p-1}},\\
(\partialartial^*{\bar{\alpha}}r{\partialartial}^* \partialhi)_{I_{p-1}, \overline{J}_{p-1}} &=& -\sqrt{-1} \operatorname{div}' (\operatorname{div}''(\partialhi))_{I_{p-1}, \overline{J}_{p-1}}.
\epsilonnd{eqnarray*}
Hence $Q$ also has the following equivalent form:
\begin{eqnarray*}
Q_{I_{p-1}, \overline{J}_{p-1}}&=&\frac{1}{2}\operatorname{l}eft[\operatorname{div}'' (\operatorname{div}'(\partialhi))+ \operatorname{div}' (\operatorname{div}''(\partialhi))\rightght]_{I_{p-1}, \overline{J}_{p-1}}+(\operatorname{div}'(\partialhi))_{I_{p-1},{\bar{\alpha}}r{V} {\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}\\
&\quad&+(\operatorname{div}''(\partialhi))_{Vi_1\cdots i_{p-1},\overline{J}_{p-1}}+\partialhi_{V I_{p-1}, {\bar{\alpha}}r{V} \overline{J}_{p-1}}+\frac{1}{t}(\Lambda \partialhi)_{I_{p-1}, \overline{J}_{p-1}}.
\epsilonnd{eqnarray*}
Recall that $d=\partialartial +{\bar{\alpha}}r{\partialartial}$ and $d_c=-\sqrt{-1}(\partialartial -{\bar{\alpha}}r{\partialartial})$. One can also write $Q$ as
\begin{equation}\operatorname{l}abel{lyh-ddc}
Q=\frac{1}{2}d_c^* d^* \partialhi -\Pi_{p-1, p-1}\cdot \sqrt {-1}ota_{V+\overline{V}}\cdot d_c^*\partialhi +\partialhi_{V, {\bar{\alpha}}r{V}}+\frac{\Lambda \partialhi}{t}
\epsilonnd{equation}
as well as
\begin{equation}\operatorname{l}abel{lyh-ddpar}
Q=\frac{1}{\sqrt{-1}}{\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi+\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{V}\cdot {\bar{\alpha}}r{\partialartial}^* \partialhi-\frac{1}{\sqrt{-1}}\sqrt {-1}ota_{\overline{V}}\cdot \partialartial^* \partialhi +\partialhi_{V, {\bar{\alpha}}r{V}}+\frac{\Lambda \partialhi}{t}
\epsilonnd{equation}
Here $\Pi_{p-1, p-1}$ is the projection to the $\wedge^{p-1, p-1}$-space.
When $\partialhi$ is $d$-closed and write $\partialsi=\Lambda \partialhi$, the K\"ahler identities and its consequence
$\Delta_{{\bar{\alpha}}r{\partialartial}}=\Delta_\partialartial$ imply that
$$
Q=\partialsi_t +\frac{1}{2}\operatorname{l}eft({\bar{\alpha}}r{\partialartial}{\bar{\alpha}}r{\partialartial}^*+\partialartial \partialartial^*\rightght) \partialsi +\sqrt {-1}ota_{V}\cdot \partialartial \partialsi+\sqrt {-1}ota_{\overline{V}}\cdot {\bar{\alpha}}r{\partialartial}\partialsi +\partialhi_{V, \overline{V}}+\frac{\partialsi}{t}.
$$
Sometimes we also abbreviate $\sqrt {-1}ota_{V} \partialartial \partialsi, \sqrt {-1}ota_{\overline{V}}{\bar{\alpha}}r{\partialartial}\partialsi$ as $\partialartial_V \partialsi, {\bar{\alpha}}r{\partialartial}_{\overline{V}} \partialsi$.
\begin{theorem}\operatorname{l}abel{main-lyh} Let $(M, g)$ be a complete K\"ahler manifold. Assume that $\partialhi(x, t)$ is a positive solution to (\ref{eq:11}). Assume further that the curvature of $(M, g)$ satisfies $\mathcal{C}_p$. Then $Q\gammae 0$ for any $t>0$. Furthermore, if the equality holds somewhere for $t>0$, then $(M, g)$ must be flat.
\epsilonnd{theorem}
\begin{proof} We postpone the proof on the part that $Q\gammae 0$ to Section 5 and 11. Here we show the rigidity result implied by the equality case, which can be reduced to the $p=1$ case treated in \cite{N-jams}. The observation is that $\Lambda (Q(\partialhi, V))=Q(\Lambda \partialhi, V)$. This can be seen via the well-known facts (cf. Corollary 4.10 of Chapter 5 of \cite{wells}) that
$$
[\Lambda, \partialartial^*]=[\Lambda, {\bar{\alpha}}r{\partialartial}^*]=0
$$
as well as $\partialhi_{V, {\bar{\alpha}}r{V}}=\sqrt{-1}\sum_{ij=1}^m V^{i} V^{{\bar{\alpha}}r{j}}\sqrt {-1}ota_{i}\sqrt {-1}ota_{{\bar{\alpha}}r{j}}\partialhi=\sqrt{-1}\sqrt {-1}ota_V\cdot \sqrt {-1}ota_{\overline{V}}\cdot \partialhi$ and the equalities
$$
[\Lambda, \sqrt {-1}ota_{V}]=[\Lambda, \sqrt {-1}ota_{\overline{V}}]=0.
$$
One can refer to (3.19) of Chapter 5 in \cite{wells} for a proof of the above identities.
Hence if $Q(\partialhi, V)=0$, it implies that $Q(\Lambda^{p-1}\partialhi, V)=\Lambda^{p-1}(Q(\partialhi, V))=0$. Now the result follows from Theorem 1.1 of \cite{N-jams} applying to $\Lambda^{p-1}\partialhi$, which is a positive $(1,1)$-form.
\epsilonnd{proof}
Note that in the statement of the theorem, $Q(\partialhi, V)=0$ means it equals to the zero as a $(p-1, p-1)$-form.
\begin{corollary}\operatorname{l}abel{main-con1}
Let $(M, g)$ and $\partialhi$ be as above. Let $\partialsi=\Lambda \partialhi$ and assume that $\partialhi$ is $d$-closed. Then
\begin{equation}\operatorname{l}abel{con1-lyh}
\frac{1}{t}\frac{\partialartial}{\partialartial t}\operatorname{l}eft(t \partialsi\rightght)+\frac{1}{2}\operatorname{l}eft({\bar{\alpha}}r{\partialartial}{\bar{\alpha}}r{\partialartial}^* +\partialartial \partialartial^*\rightght)\partialsi \gammae -\min_{V}\operatorname{l}eft(\partialartial_V \partialsi+{\bar{\alpha}}r{\partialartial}_{\overline{V}}\partialsi +\partialhi(V, \overline{V})\rightght)\gammae 0.
\epsilonnd{equation}
In particular, for any $\partialhi\gammae 0$, if $M$ is compact,
\begin{equation}\operatorname{l}abel{mono}
\frac{d}{dt}\sqrt {-1}nt_M t \partialsi\wedge \omega^{m-p+1} \gammae 0.
\epsilonnd{equation}
\epsilonnd{corollary}
\begin{proof} By adding ${\varepsilon}ilon \omega^p$ and then letting ${\varepsilon}ilon\to 0$, we can assume that $\partialhi$ is strictly positive. Then for any $I_{p-1}=(i_1, \cdots, i_{p-1})$, the Hermitian bilinear form ( in $V$),
$(\partialsi_t+\frac{\partialsi}{t})_{I_{p-1}, \overline{I}_{p-1}}+(\partialartial_V \partialsi+{\bar{\alpha}}r{\partialartial}_{\overline{V}}\partialsi)_{I_{p-1}, \overline{I}_{p-1}}
+\partialhi(V, \overline{V})_{I_{p-1}, \overline{I}_{p-1}}
$ has a minimum. It is then easy to see that for the minimizing vector $V$,
$$
(\partialartial_V \partialsi+{\bar{\alpha}}r{\partialartial}_{\overline{V}}\partialsi)_{I_{p-1}, \overline{I}_{p-1}}
+\partialhi(V, \overline{V})_{I_{p-1}, \overline{I}_{p-1}}=-\partialhi(V, \overline{V})_{I_{p-1}, \overline{I}_{p-1}}.
$$
The first result then follows. For the second one, just notice that
$$
\sqrt {-1}nt_M ({\bar{\alpha}}r{\partialartial}{\bar{\alpha}}r{\partialartial}^*+\partialartial \partialartial^*)\partialsi\wedge \omega^{m-p+1}=0.
$$
\epsilonnd{proof}
\begin{remark} Clearly, if one can perform the integration by parts and control the boundary terms, the monotonicity (\ref{mono}) still holds on noncompact case.
\epsilonnd{remark}
One can define a formal dual operator of $Q(\partialhi, V)$ as
\begin{equation}\operatorname{l}abel{lyh-dual}
Q^*(\partialsi, V^*)\doteqdot\sqrt{-1}\partialartial {\bar{\alpha}}r{\partialartial} \partialsi +\sqrt{-1}V^*\wedge {\bar{\alpha}}r{\partialartial}\partialsi-\sqrt{-1}\overline{V}^* \wedge \partialartial \partialsi +\sqrt{-1}V^*\wedge \overline{V}^* \wedge \partialsi+\frac{\omega\wedge \partialsi}{t}
\epsilonnd{equation}
which maps a $(m-p, m-p)$-form $\partialsi$ to a $(m-p+1, m-p+1)$-form. Here $V^*$ is a $(1,0)$ type co-vector. The following duality can be checked by direct calculations, making use of the following well known identities on $(p,q)$-forms (cf. Proposition 2.4, (1.13) and (3.14) of Chapter 5 of \cite{wells} repectively):
\begin{eqnarray*}
{\bar{\alpha}}r{\partialartial}^* =-* \cdot \partialartial \cdot *\, , \quad &\quad& \quad \partialartial^*=-* \cdot {\bar{\alpha}}r{\partialartial} \cdot *\, ;\\
\Lambda&=& (-)^{p+q} * \cdot (\omega \wedge)\cdot *\, ; \\
\sqrt {-1}ota_{V} =*\cdot (\overline{V}^* \wedge)\cdot *\, ,\quad &\quad&\quad \sqrt {-1}ota_{\overline{V}} =*\cdot (V^* \wedge)\cdot *\, .
\epsilonnd{eqnarray*}
Here $*$ is the Hodge-star operator.
\begin{proposition} Let $\partialhi$, $V$ be as the above discussion. Let $V^*$ be the dual of $V$. Let $*$ be the Hodge star operator. Then
\begin{equation}\operatorname{l}abel{dual1}
Q(\partialhi, V)=* \cdot Q^* (* \cdot \partialhi, V^*).
\epsilonnd{equation}
\epsilonnd{proposition}
By this duality, one can identify the result for the $(m, m)$-forms with that of \cite{CN}. In the rest of this section we derive some preliminary results useful for the proof of Theorem \ref{main-lyh}.
The following lemma follows from (\ref{eq:43}), (\ref{eq:44}) and the fact that $[\Delta_{{\bar{\alpha}}r{\partialartial}}, \partialartial^*]=[\Delta_{{\bar{\alpha}}r{\partialartial}}, {\bar{\alpha}}r{\partialartial}^*]=0$.
\begin{lemma} \operatorname{l}abel{help41} Let $\partialhi$ be a $(p, p)$-form satisfying (\ref{eq:11}). Then $(\frac{\partialartial}{\partialartial t}+\Delta_{{\bar{\alpha}}r{\partialartial}}){\bar{\alpha}}r{\partialartial}^* \partialhi =(\frac{\partialartial}{\partialartial t}+\Delta_{{\bar{\alpha}}r{\partialartial}})\partialartial^* \partialhi=0$. Hence by (\ref{eq:43}), (\ref{eq:44})
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) ({\bar{\alpha}}r{\partialartial}^*\partialhi)_{i i_1\cdots i_{p-1}, {\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}= \sum_{\mu = 1}^{p-1}\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j_\nu}}({\bar{\alpha}}r{\partialartial}^*\partialhi)_{ii_1\cdots(k)_\mu\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}} \operatorname{l}abel{eq:lem41}\\
\quad \quad &
+& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i\,\,{\bar{\alpha}}r{j_\nu}}({\bar{\alpha}}r{\partialartial}^*\partialhi)_{ki_1\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}R_{i}^{\,\,
k}({\bar{\alpha}}r{\partialartial}^*\partialhi)_{ki_1\cdots i_{p-1},{\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}} \nonumber\\
\quad \quad &
-&\frac{1}{2}\operatorname{l}eft(\sum_{\nu =1}^{p-1}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}({\bar{\alpha}}r{\partialartial}^*\partialhi)_{ii_1\cdots i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})\cdots{\bar{\alpha}}r{j}_{p-1}}+\sum_{\mu =1 }^{p-1}R_{i_\mu}^{\,\,
k}({\bar{\alpha}}r{\partialartial}^*\partialhi)_{ii_1\cdots(k)_\mu\cdots i_{p-1},{\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}\rightght),\nonumber\\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\partialartial^*\partialhi)_{ i_1\cdots i_{p-1}, {\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}= \sum_{\mu = 1}^{p-1}\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j_\nu}}(\partialartial^*\partialhi)_{i_1\cdots(k)_\mu\cdots
i_{p-1},{\bar{\alpha}}r{j}{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}} \operatorname{l}abel{eq:lem42}\\
\quad \quad &
+& \sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j}}(\partialartial^*\partialhi)_{i_1\cdots(k)_\mu\cdots
i_{p-1}, {\bar{\alpha}}r{l}{\bar{\alpha}}r{j_1}\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}}(\partialartial^*\partialhi)_{i_1\cdots i_{p-1},{\bar{\alpha}}r{l}{\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}} \nonumber \\
\quad \quad&
-&\frac{1}{2}\operatorname{l}eft(\sum_{\nu =1}^{p-1}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}(\partialartial^*\partialhi)_{i_1\cdots i_{p-1},{\bar{\alpha}}r{j}{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})\cdots{\bar{\alpha}}r{j}_{p-1}}+\sum_{\mu =1 }^{p-1}R_{i_\mu}^{\,\,
k}(\partialartial^*\partialhi)_{i_1\cdots(k)_\mu\cdots i_{p-1},{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}\rightght).\nonumber
\epsilonnd{eqnarray}
\epsilonnd{lemma}
Similarly, one can write the following lemma.
\begin{lemma} \operatorname{l}abel{help42}Let $\partialhi$ be a solution to (\ref{eq:11}). Then $(\frac{\partialartial}{\partialartial t}+\Delta_{{\bar{\alpha}}r{\partialartial}})(\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)=(\frac{\partialartial}{\partialartial t}+\Delta_{{\bar{\alpha}}r{\partialartial}})({\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi)=0$. Hence (\ref{eq:43}), (\ref{eq:44}) imply similar equations for
$\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)$ and $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) ({\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi)$. Namely
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)_{i_1\cdots i_{p-1}, {\bar{\alpha}}r{j}_1\cdots {\bar{\alpha}}r{j}_{p-1}}=\sum_{\mu = 1}^{p-1}\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j_\nu}}(\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)_{i_1\cdots(k)_\mu\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}} \operatorname{l}abel{eq:lm43}\\
&\quad&
-\frac{1}{2}\operatorname{l}eft(\sum_{\nu =1}^{p-1}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}(\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)_{i_1\cdots i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})\cdots{\bar{\alpha}}r{j}_{p-1}}+\sum_{\mu =1 }^{p-1}R_{i_\mu}^{\,\,
k}(\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)_{i_1\cdots(k)_\mu\cdots i_{p-1}, {\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}\rightght).\nonumber
\epsilonnd{eqnarray}
Simply put $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)=\mathcal{KB}(\partialartial^*
{\bar{\alpha}}r{\partialartial}^* \partialhi)$ and $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) ({\bar{\alpha}}r{\partialartial}^*\partialartial^* \partialhi)=\mathcal{KB}(
{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi)$.
\epsilonnd{lemma}
Lemma \ref{help41} implies that
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}''_{V}(\partialhi))_{I_{p-1}, \overline{J}_{p-1}}=\operatorname{div}''_{\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) V}(\partialhi) +\mathcal{KB}(\operatorname{div}''_{V}(\partialhi)) \operatorname{l}abel{eq:lem412}\\
\quad \quad &+& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}}(\operatorname{div}''(\partialhi))_{ki_1\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}\operatorname{div}''_{\operatorname{Ric}(V)}(\partialhi)
\nonumber\\
\quad \quad &-& g^{i{\bar{\alpha}}r{j}}\operatorname{l}eft((\nabla_{i} \operatorname{div}''(\partialhi))_{\nabla_{{\bar{\alpha}}r{j}}V}+(\nabla_{{\bar{\alpha}}r{j}} \operatorname{div}''(\partialhi))_{\nabla_{i}V}\rightght); \nonumber \\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}'_{\overline{V}}(\partialhi))_{I_{p-1}, {\bar{\alpha}}r{J}_{p-1}}= \operatorname{div}'_{\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \overline{V}}(\partialhi)+\mathcal{KB}(\operatorname{div}'_{\overline{V}}(\partialhi)) \operatorname{l}abel{eq:lem413}\\
\quad \quad &
+& \sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{V}}(\operatorname{div}'(\partialhi))_{i_1\cdots(k)_\mu\cdots
i_{p-1}, {\bar{\alpha}}r{l}{\bar{\alpha}}r{j_1}\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}\operatorname{div}'_{\overline{\operatorname{Ric}(V)}}(\partialhi) \nonumber\\
\quad \quad &-& g^{i{\bar{\alpha}}r{j}}\operatorname{l}eft((\nabla_{i} \operatorname{div}'(\partialhi))_{\nabla_{{\bar{\alpha}}r{j}}\overline{V}}+(\nabla_{{\bar{\alpha}}r{j}} \operatorname{div}'(\partialhi))_{\nabla_{i}\overline{V}}\rightght). \nonumber
\epsilonnd{eqnarray}
since for any $r$-tensor $T$, the $r-1$-tensor $T_V(X_1, \cdots X_{r-1})\doteqdot T(V, X_1, \cdots X_{r-1})$ satisfies $\nabla_X T_V =(\nabla_X T)_V +T_{\nabla_X V}$. To compute the evolution equation of $Q$, since
$ \operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \Lambda \partialhi =\mathcal{KB}(\Lambda \partialhi)$, the only term left is the evolution equation on $\partialhi(V, \overline{V})$ which we also abbreviate as $\partialhi_{V,{\bar{\alpha}}r{V}}$. Since $\nabla_X \partialhi_{V,{\bar{\alpha}}r{V}}= (\nabla_X \partialhi)_{V,{\bar{\alpha}}r{V}}+\partialhi_{\nabla_X V, {\bar{\alpha}}r{V}}+\partialhi_{V, \nabla_X {\bar{\alpha}}r{V}}$, Lemma \ref{lemma11} implies that
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \partialhi_{V,{\bar{\alpha}}r{V}}=\mathcal{KB}(\partialhi_{V,{\bar{\alpha}}r{V}})+ R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{V}} \partialhi_{k, {\bar{\alpha}}r{l}}+\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}} \partialhi_{k I_{p-1}, {\bar{\alpha}}r{V} {\bar{\alpha}}r{j}_1\cdots ({\bar{\alpha}}r{l})_\nu\cdots {\bar{\alpha}}r{j}_{p-1}}\operatorname{l}abel{eq:414}\\
\quad \quad &+&\sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{V}} \partialhi_{V i_1\cdots (k)_\mu\cdots i_{p-1}, {\bar{\alpha}}r{l}{\bar{\alpha}}r{J}_{p-1}}-\frac{1}{2}\operatorname{l}eft(\partialhi_{V, \overline{\operatorname{Ric}(V)}}+\partialhi_{\operatorname{Ric}(V), {\bar{\alpha}}r{V}}\rightght) \nonumber\\
\quad \quad &+& \partialhi_{\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) V, {\bar{\alpha}}r{V}}+\partialhi_{V, \operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) {\bar{\alpha}}r{V}}-g^{i{\bar{\alpha}}r{j}}\operatorname{l}eft(\partialhi_{\nabla_i V, \nabla_{{\bar{\alpha}}r{j}}{\bar{\alpha}}r{V}}+\partialhi_{\nabla_{{\bar{\alpha}}r{j}}V, \nabla_i {\bar{\alpha}}r{V}}\rightght)\nonumber\\
\quad \quad &-& g^{i{\bar{\alpha}}r{j}}\operatorname{l}eft((\nabla_i \partialhi)_{\nabla_{{\bar{\alpha}}r{j}} V, {\bar{\alpha}}r{V}}+(\nabla_{i}\partialhi)_{V, \nabla_{{\bar{\alpha}}r{j}}{\bar{\alpha}}r{V}}+(\nabla_{{\bar{\alpha}}r{j}}\partialhi )_{\nabla_i V, {\bar{\alpha}}r{V}}+(\nabla_{{\bar{\alpha}}r{j}}\partialhi)_{ V, \nabla_i{\bar{\alpha}}r{V}}\rightght). \nonumber
\epsilonnd{eqnarray}
\section{The proof of Theorem \ref{main-lyh}.}
Now $Q$ is viewed as a $(p-1, p-1)$-form. For $p=1 $ and $p=m$, the result has been proven earlier. Using the notations introduced in the last section the LYH quantity $Q$, $(p-1, p-1)$-form depending on vector field $V$, can be written as
$$
Q=\frac{1}{2}\operatorname{l}eft[\operatorname{div}'' (\operatorname{div}'(\partialhi))+ \operatorname{div}' (\operatorname{div}''(\partialhi))\rightght]+\operatorname{div}'_{\overline{V}}(\partialhi)+\operatorname{div}''_{V}(\partialhi) +\partialhi_{V, \overline{V}}+\frac{\Lambda \partialhi}{t}.
$$
As before if we assume that at $(x_0, t_0)$, for the first time, for some $V$, $Q_{v_1 v_2\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots{\bar{\alpha}}r{v}_{p-1}}=0$ for some linearly independent vectors $\{v_i\}_{i=1}^{p-1}$. By a perturbation argument as in \cite{N-jams} we can assume without the loss of the generality that $\partialhi$ is strictly positive. As in \cite{richard-harnack}, it suffices to check that at the point $(x_0, t_0)$,
$\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) Q\gammae 0$. Since the complex function (in terms of the variable $z$)
\begin{eqnarray*}
I(z)&\doteqdot &\frac{1}{2}\operatorname{l}eft[\operatorname{div}'' (\operatorname{div}'(\partialhi))+ \operatorname{div}' (\operatorname{div}''(\partialhi))\rightght]_{v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{v}_1(z)\cdots {\bar{\alpha}}r{v}_{p-1}(z)}\\
&\, & +\operatorname{l}eft[\operatorname{div}'_{\overline{V}(z)}(\partialhi)+\operatorname{div}''_{V(z)}(\partialhi) \rightght]_{v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{v}_1(z)\cdots {\bar{\alpha}}r{v}_{p-1}(z)}\\
&\,& +\partialhi_{V(z)v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{V}(z){\bar{\alpha}}r{v}_1(z)\cdots {\bar{\alpha}}r{v}_{p-1}(z)}+\frac{\Lambda \partialhi_{v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{v}_1(z)\cdots {\bar{\alpha}}r{v}_{p-1}(z)}}{t}
\epsilonnd{eqnarray*}
satisfies $I(0)=0$ and $I(z)\gammae 0$ for any variational vectors $v_{\mu}(z), V(z)$, holomorphic in $z$, with $v_{\mu}(0)=v_{\mu}$ and $V(0)=V$. In particular, letting $v_{\mu}(z)=v_{\mu}$ and $V'(0)=X$ we have that
\begin{equation}\operatorname{l}abel{eq:51}
\operatorname{div}'_{{\bar{\alpha}}r{X}}(\partialhi)+\partialhi_{V, {\bar{\alpha}}r{X}}=0= \operatorname{div}''_{X}(\partialhi)+\partialhi_{X, {\bar{\alpha}}r{V}}.
\epsilonnd{equation}
Similarly by fixing $V(z)=V$ and varying $v_{\mu}(z)$, we deduce for any $X$,
\begin{equation}\operatorname{l}abel{eq:52}
Q_{v_1\cdots (X)_\mu\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_{p-1}}=0=Q_{v_1\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots ({\bar{\alpha}}r{X})_{\nu}\cdots v_{p-1}}.
\epsilonnd{equation}
As before, after a changing of variables we may assume that $\{ v_i\}_{i=1}^{p-1} =\{\frac{\partialartial}{\partialartial z^i}\}_{i=1}^{p-1}$.
Since $z=0$ is the minimizing point we have that $\Delta I(0)\gammae 0$. If $v'_\mu(0)=X_\mu$ and $V'(0)=X$, where $v'(z)=\frac{\partialartial v}{\partialartial z}$, this implies that
\begin{eqnarray*}
&\, &\sum_{\mu, \nu=1}^{p-1} Q_{v_1\cdots X_\mu \cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu \cdots {\bar{\alpha}}r{v}_{p-1}}+ \partialhi_{X v_1\cdots v_{p-1}, {\bar{\alpha}}r{X} {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_{p-1}}\\
\quad \quad \quad &+& \sum_{\mu=1}^{p-1}\operatorname{div}'_{{\bar{\alpha}}r{X}}(\partialhi)_{v_1\cdots X_\mu\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_{p-1}}+\partialhi_{V v_1\cdots X_\mu\cdots v_{p-1}, {\bar{\alpha}}r{X} {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_{p-1}}\\
\quad \quad \quad &+& \sum_{\nu=1}^{p-1}\operatorname{div}''_{X}(\partialhi)_{v_1\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu\cdots {\bar{\alpha}}r{v}_{p-1}}+\partialhi_{X v_1\cdots v_{p-1}, {\bar{\alpha}}r{V} {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu \cdots {\bar{\alpha}}r{v}_{p-1}}\\
\quad \quad \quad &\gammae& 0.
\epsilonnd{eqnarray*}
This amounts to that the block matrix
\begin{eqnarray*}
\mathcal{M}_1=\operatorname{l}eft(\begin{array}{l} A \quad \quad\quad \quad S \quad \quad \\
\overlineerline{S}^{tr}\quad \partialhi_{(\cdot )1\cdots p-1, ({\bar{\alpha}}r{\cdot}){\bar{\alpha}}r{1}, \cdots \overline{p-1}}
\epsilonnd{array}\rightght)\gammae 0
\epsilonnd{eqnarray*}
where
\begin{eqnarray*}
A= \operatorname{l}eft(\begin{array}{l}
Q_{(\cdot)\, 2\cdots\, p-1, \, {\bar{\alpha}}r{(\cdot)}\, {\bar{\alpha}}r{2}\,\cdots\, \overline{p-1}}\quad
Q_{1\, (\cdot)\, \cdots\, p-1, \, {\bar{\alpha}}r{(\cdot)}\, {\bar{\alpha}}r{2}\,\cdots\, \overline{p-1}}\quad \quad \quad
\cdots \quad \quad \quad
Q_{1\, 2\, \cdots\, (\cdot)_{p-1}, \, {\bar{\alpha}}r{(\cdot)}\, {\bar{\alpha}}r{2}\,\cdots\, \overline{p-1}}\\
Q_{(\cdot)\, 2\cdots\, p-1, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{ (\cdot)}\, \cdots\, \overline{p-1}}
\quad Q_{1\, (\cdot)\, \cdots\, p-1, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{(\cdot)}\, \cdots\, \overline{p-1}}
\quad \quad \quad
\cdots
\quad \quad \quad
Q_{1\, 2\, \cdots\, (\cdot)_{p-1}, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{(\cdot)}\, \cdots\, \overline{p-1}}\\
\quad \quad \quad \cdots\quad \quad \quad \quad \quad \quad \quad \quad \quad \cdots \quad \quad \quad \quad \quad \, \quad \quad \cdots \quad \quad \quad \quad \quad \quad \cdots \quad \quad \quad\\
Q_{(\cdot)\, 2\cdots\, p-1, \, {\bar{\alpha}}r{1}\,{\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{ (\cdot)}_{p-1}}\quad
Q_{1\, (\cdot)\, \cdots\, p-1, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{ (\cdot)}_{p-1}}\quad \quad \,
\cdots \quad \quad \quad
Q_{1\, 2\, \cdots\, (\cdot)_{p-1}, \, {\bar{\alpha}}r{1}\, {\bar{\alpha}}r{2}\, \cdots\, {\bar{\alpha}}r{ (\cdot)}_{p-1}}
\epsilonnd{array}\rightght)
\epsilonnd{eqnarray*}
and $S$ satisfies that for vectors $X_1,\cdots, X_{p-1}, X$
$$
(\overline{X}_1^{tr}, \cdots, \overline{X}_{p-1}^{tr})\cdot S\cdot X=\sum_{\nu=1}^{p-1}\operatorname{div}''_{X}(\partialhi)_{v_1\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu\cdots {\bar{\alpha}}r{v}_{p-1}}+\partialhi_{X v_1\cdots v_{p-1}, {\bar{\alpha}}r{V} {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu \cdots {\bar{\alpha}}r{v}_{p-1}}.
$$
To check that $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) Q_{1\cdots p-1, {\bar{\alpha}}r{1}\cdots \overline{p-1}}\gammae 0$ we may extend $V$ such that the following holds:
\begin{eqnarray*}
\nabla_i V =\frac{1}{t}\frac{\partialartial}{\partialartial z^i}, \quad &\, & \quad
\nabla_{{\bar{\alpha}}r{i}} V = 0,\\
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) V &=& -\frac{1}{t} V.
\epsilonnd{eqnarray*}
Using these set of equations, (\ref{eq:lem412}), (\ref{eq:lem413}) and (\ref{eq:414}) can be simplified to
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}''_{V}(\partialhi))_{I_{p-1}, \overline{J}_{p-1}}=-\frac{1}{t}\operatorname{div}''_{ V}(\partialhi) +\mathcal{KB}(\operatorname{div}''_{V}(\partialhi)) \operatorname{l}abel{eq:53}\\
\quad \quad &+& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}}(\operatorname{div}''(\partialhi))_{ki_1\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}\operatorname{div}''_{\operatorname{Ric}(V)}(\partialhi)
-\frac{1}{t} \operatorname{div}'( \operatorname{div}''(\partialhi)); \nonumber \\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}'_{\overline{V}}(\partialhi))_{I_{p-1}, {\bar{\alpha}}r{J}_{p-1}}= -\frac{1}{t}\operatorname{div}'_{\overline{V}}(\partialhi)+\mathcal{KB}(\operatorname{div}'_{\overline{V}}(\partialhi)) \operatorname{l}abel{eq:54}\\
\quad \quad &
+& \sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{V}}(\operatorname{div}'(\partialhi))_{i_1\cdots(k)_\mu\cdots
i_{p-1}, {\bar{\alpha}}r{l}{\bar{\alpha}}r{j_1}\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}\operatorname{div}'_{\overline{\operatorname{Ric}(V)}}(\partialhi)
-\frac{1}{t}\operatorname{div}''( \operatorname{div}'(\partialhi)); \nonumber\\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \partialhi_{V,{\bar{\alpha}}r{V}}=\mathcal{KB}(\partialhi_{V,{\bar{\alpha}}r{V}})+ R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{V}} \partialhi_{k, {\bar{\alpha}}r{l}}+\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}} \partialhi_{k I_{p-1}, {\bar{\alpha}}r{V} {\bar{\alpha}}r{j}_1\cdots ({\bar{\alpha}}r{l})_\nu\cdots {\bar{\alpha}}r{j}_{p-1}}\operatorname{l}abel{eq:55}\\
\quad \quad &+&\sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{V}} \partialhi_{V i_1\cdots (k)_\mu\cdots i_{p-1}, {\bar{\alpha}}r{l}{\bar{\alpha}}r{J}_{p-1}}-\frac{1}{2}\operatorname{l}eft(\partialhi_{V, \overline{\operatorname{Ric}(V)}}+\partialhi_{\operatorname{Ric}(V), {\bar{\alpha}}r{V}}\rightght) \nonumber\\
\quad \quad &-& \frac{2}{t}\partialhi_{ V, {\bar{\alpha}}r{V}}-\frac{\Lambda \partialhi}{t^2}-\frac{1}{t}\operatorname{div}'_{\overline{V}}(\partialhi)-\frac{1}{t}\operatorname{div}''_{V}(\partialhi).\nonumber \nonumber
\epsilonnd{eqnarray}
Adding them up with that
$$\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \operatorname{l}eft[\operatorname{div}'' (\operatorname{div}'(\partialhi))+ \operatorname{div}' (\operatorname{div}''(\partialhi))\rightght] =\mathcal{KB}(\operatorname{l}eft[\operatorname{div}'' (\operatorname{div}'(\partialhi))+ \operatorname{div}' (\operatorname{div}''(\partialhi))\rightght] )$$ and $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \Lambda \partialhi=\mathcal{KB}(\Lambda \partialhi)$ , using (\ref{eq:51}) we have that
\begin{eqnarray*}
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) Q_{I_{p-1}, \overline{J}_{p-1}} &=& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}} \operatorname{l}eft(\partialhi_{k I_{p-1}, {\bar{\alpha}}r{V} {\bar{\alpha}}r{j}_1\cdots ({\bar{\alpha}}r{l})_\nu\cdots {\bar{\alpha}}r{j}_{p-1}}+(\operatorname{div}''(\partialhi))_{kI_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}\rightght)\\
\quad \quad &+&\sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{V}} \operatorname{l}eft(\partialhi_{V i_1\cdots (k)_\mu\cdots i_{p-1}, {\bar{\alpha}}r{l}{\bar{\alpha}}r{J}_{p-1}}+(\operatorname{div}'(\partialhi))_{i_1\cdots(k)_\mu\cdots
i_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}\rightght)\\
\quad \quad&+&R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{V}} \partialhi_{kI_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}+\mathcal{KB}(Q)_{I_{p-1}, \overline{J}_{p-1}}-\frac{2 Q_{I_{p-1}, \overline{J}_{p-1}}}{t}.
\epsilonnd{eqnarray*}
Now the nonnegativity of $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) Q_{1\cdots(p-1), {\bar{\alpha}}r{1}\cdots (\overline{p-1})}$ at $(x_0, t_0)$ can be proved in a similar way as the argument in Section 2. First observe that the part of $\mathcal{KB}(Q)_{1\cdots(p-1), {\bar{\alpha}}r{1}\cdots (\overline{p-1})}$ involving only $\operatorname{Ric}$ is
$$
-\frac{1}{2}\sum_{i=1}^{p-1}\operatorname{l}eft(Q_{1\cdots \operatorname{Ric}(i)\cdots (p-1), {\bar{\alpha}}r{1}\cdots (\overline{p-1})}+Q_{1\cdots (p-1), {\bar{\alpha}}r{1}\cdots\overline{\operatorname{Ric}(i)}\cdots (\overline{p-1})}\rightght)
$$
which vanishes due to (\ref{eq:52}).
Hence we only need to establish the nonnegativity of
\begin{eqnarray*}J&\doteqdot &
\sum_{\mu = 1}^{p-1}\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{\mu\,\,{\bar{\alpha}}r{\nu}}Q_{1\cdots(k)_\mu\cdots
(p-1), {\bar{\alpha}}r{1}\cdots({\bar{\alpha}}r{l})_\nu\cdots\overline{p-1}}\\
\quad \quad &+& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,\,{\bar{\alpha}}r{\nu}} \operatorname{l}eft(\partialhi_{k 1 \cdots (p-1), {\bar{\alpha}}r{V} {\bar{\alpha}}r{1}\cdots ({\bar{\alpha}}r{l})_\nu\cdots (\overline{p-1})}+(\operatorname{div}''(\partialhi))_{k1\cdots (p-1),{\bar{\alpha}}r{1}\cdots({\bar{\alpha}}r{l})_\nu\cdots(\overline{p-1})}\rightght)\\
\quad \quad &+&\sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{\mu\,\,{\bar{\alpha}}r{V}} \operatorname{l}eft(\partialhi_{V 1\cdots (k)_\mu\cdots (p-1), {\bar{\alpha}}r{l}\cdots (\overline{p-1})}+(\operatorname{div}'(\partialhi))_{1\cdots(k)_\mu\cdots
(p-1), {\bar{\alpha}}r{l}{\bar{\alpha}}r{1}\cdots (\overline{p-1})}\rightght)\\
\quad \quad&+&R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{V}} \partialhi_{k1\cdots (p-1), {\bar{\alpha}}r{l}{\bar{\alpha}}r{1}\cdots (\overline{p-1})}.
\epsilonnd{eqnarray*}
The curvature operator is in $\mathcal{C}_p$ implies that the matrix
$$
\mathcal{M}_2=\operatorname{l}eft(\begin{array}{l}
R_{1{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
R_{1{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad\quad
\cdots\quad\quad
R_{1\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}} \quad \quad \quad R_{1\overline{V}(\cdot){\bar{\alpha}}r{(\cdot)}}\\
R_{2{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad
R_{2{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad\quad
\cdots\quad\quad
R_{2\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad \quad R_{2\overline{V}(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\quad\cdots\quad\quad\quad\quad \cdots\quad\quad\quad\quad \cdots\quad\quad\quad\cdots\quad \quad \quad \quad \quad \quad \cdots \\
R_{p-1{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
R_{p-1{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{p-1\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad R_{p-1\overline{V}(\cdot){\bar{\alpha}}r{(\cdot)}}\\
R_{V{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
R_{V{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad\,\,
\cdots\quad\quad
R_{V\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad \quad R_{V\overline{V}(\cdot){\bar{\alpha}}r{(\cdot)}}
\epsilonnd{array}\rightght)\gammae 0.
$$
The nonnegativity of $J$ follows from $\operatorname{trace}(\mathcal{M}_1 \cdot \mathcal{M}_2)\gammae 0$.
We define transformations on $T'M$, $(\operatorname{div}''(\partialhi))^{{\bar{\alpha}}r{\nu}}$, $(\operatorname{div}'(\partialhi))^{\mu}$, $\partialhi_{\overline{V}}^{{\bar{\alpha}}r{\nu}}$ and $\partialhi_{V}^{\mu}$ by
\begin{eqnarray*}
\operatorname{l}angle (\operatorname{div}''(\partialhi))^{{\bar{\alpha}}r{\nu}}(X), \overline{Y}\rangle &\doteqdot& (\operatorname{div}''(\partialhi))_{X 1\cdots (p-1), {\bar{\alpha}}r{1}\cdots (\overline{Y})_{\nu}\cdots (\overline{p-1})}, \\
\operatorname{l}angle (\operatorname{div}'(\partialhi))^\mu(X), \overline{Y}\rangle &\doteqdot& (\operatorname{div}'(\partialhi))_{ 1\cdots (X)_\mu\cdots (p-1), \overline{Y} {\bar{\alpha}}r{1}\cdots (\overline{p-1})},\\
\operatorname{l}angle \partialhi_{\overline{V}}^{{\bar{\alpha}}r{\nu}}(X), \overline{Y}\rangle &\doteqdot& \partialhi_{X 1\cdots (p-1), \overline{V} {\bar{\alpha}}r{1}\cdots (\overline{Y})_\nu \cdots (\overline{p-1})},\\
\operatorname{l}angle \partialhi_{V}^{\mu}(X), \overline{Y}\rangle &\doteqdot& \partialhi_{ V1\cdots (X)_\mu \cdots (p-1), \overline{Y}{\bar{\alpha}}r{1}\cdots (\overline{p-1})}.
\epsilonnd{eqnarray*}
Then the operator $S$ defined previously can be written as $S=\oplus_{\nu =1}^{p-1} \operatorname{l}eft[(\operatorname{div}''(\partialhi))^{{\bar{\alpha}}r{\nu}}+\partialhi_{\overline{V}}^{{\bar{\alpha}}r{\nu}}\rightght]$.
If we define $Q^{\mu{\bar{\alpha}}r{\nu}}$ in a similar way as $\partialhi^{\mu{\bar{\alpha}}r{\nu}}$ then the quantity $J$ above can be expressed as
\begin{eqnarray*}
J&=&\sum_{\mu, \nu =1}^{p-1} \operatorname{trace}\operatorname{l}eft(R^{\mu{\bar{\alpha}}r{\nu}} Q^{\mu{\bar{\alpha}}r{\nu}}\rightght)
+\sum_{\nu=1}^{p-1}\operatorname{trace} \operatorname{l}eft( R^{V {\bar{\alpha}}r{\nu}}\cdot ((\operatorname{div}''(\partialhi))^{{\bar{\alpha}}r{\nu}}+\partialhi_{\overline{V}}^{{\bar{\alpha}}r{\nu}})\rightght)\\
&\quad&+\sum_{\nu=1}^{p-1}\overline{\operatorname{trace} \operatorname{l}eft( R^{V {\bar{\alpha}}r{\nu}}\cdot ((\operatorname{div}''(\partialhi))^{{\bar{\alpha}}r{\nu}}+\partialhi_{\overline{V}}^{{\bar{\alpha}}r{\nu}})\rightght)}+\operatorname {trace}(R^{V, {\bar{\alpha}}r{V}}\cdot \partialhi^{p, {\bar{\alpha}}r{p}}).
\epsilonnd{eqnarray*}
Hence one can modify the definitions of transformations $\mathcal{K}$ and $\mathcal{J}$ on $\oplus_{\mu=1}^p T'M$ in Section 2 so that $J=\operatorname{trace}(\mathcal{K}\cdot \mathcal{J})$, $\mathcal{J}$ and $\mathcal{K}$ correspond to $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
\begin{remark}We suspect that the theorem (and later results) still holds even under the weaker assumption $\mathcal{C}_1$, even though the techniques employed here seem not be able to prove such a claim.
\epsilonnd{remark}
\section{Coupled with the K\"ahler-Ricci flow.}
Now we consider $(M^m, g(t))$ a complete solution of the K\"ahler-Ricci flow
\begin{equation}\operatorname{l}abel{eq:KRF}
\frac{\partial}{\partial t}g_{i{\bar{\alpha}}r{j}}=-R_{i{\bar{\alpha}}r{j}}.
\epsilonnd{equation}
Corollary \ref{thm:p-NBC} asserts that $\mathcal{C}_p$ is an invariant
curvature condition under the K\"ahler-Ricci flow. Now we generalize the LYH estimate to the solution of (\ref{eq:11}). Again the result is proved for $p=1$ and $p=m$ in \cite{NT-ajm} and \cite{Ni-JDG07} respectively.
\begin{theorem}\operatorname{l}abel{main-KRFlyh}
Let $(M, g(t))$ be a complete solution to the K\"ahler-Ricci flow
(\ref{eq:KRF}). When $M$ is noncompact we assume that the curvature of $(M, g)$ is uniformly bounded. Assume that $\partialhi$ is a solution to (\ref{eq:11})
with $\partialhi(x, 0)$ being a positive $(p, p)$-form. Assume further that
the curvature of $(M, g(t))$ satisfies $\mathcal{C}_p$. Then for any vector field $V$ of $(1,0)$ type $\widetilde{Q}\gammae 0$
for any $t>0$, where
$$
\widetilde Q=Q+\operatorname{Ric} (\partialhi)
$$
Here $Q$ is the LYH quantity defined in Section 4 and 5, which is a $(p-1,p-1)$-form valued (Hermitian) quadratic form of $V$, $\operatorname{Ric}(\partialhi)$ is a $(p-1, p-1)$-form defined by
$$
\operatorname{Ric}(\partialhi)_{I_{p-1}, \overline{J}_{p-1}}\doteqdot g^{i{\bar{\alpha}}r{l}}g^{k{\bar{\alpha}}r{j}}R_{i{\bar{\alpha}}r{j}}\partialhi_{k I_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}.
$$
\epsilonnd{theorem}
Note that the difference between the above result and Theorem \ref{main-lyh} is that the Laplacian operator $\Delta_{{\bar{\alpha}}r{\partialartial}}$ is time-dependent, namely the $g_{i{\bar{\alpha}}r{j}}$ and the connection used in the definition ${\bar{\alpha}}r{\partialartial}^*$ are evolved by the K\"ahler-Ricci flow equation. Moreover since $\partialartial^*$ and ${\bar{\alpha}}r{\partialartial}^*$ depend on changing metrics now, the quantity $Q$ is different from the static case even though they are defined by the same expression. Amazingly, the theorem asserts that the result still holds if we add a correction term $\operatorname{Ric}(\partialhi)$.
\begin{corollary}\operatorname{l}abel{main-con2}
Let $(M , g)$, $\partialhi$ be as in Theorem \ref{main-KRFlyh}. Assume that $\partialhi$ is $d$-closed and $M$ is compact. Let $\partialsi=\Lambda \partialhi$. Then
\begin{equation}\operatorname{l}abel{mono-krf1}
\frac{d}{d t} \operatorname{l}eft(t\sqrt {-1}nt_M \partialsi\wedge \omega_0^{m-p+1}\rightght)\gammae 0.
\epsilonnd{equation}
Here $\omega_0$ is the K\"ahler form of the initial metric.
\epsilonnd{corollary}
\begin{proof}
Note that $\frac{\partialartial}{\partialartial t} \partialsi +\Delta_{{\bar{\alpha}}r{\partialartial}}\partialsi =\operatorname{Ric}(\partialhi)$, the operators $\partialartial$ and ${\bar{\alpha}}r{\partialartial}$ can be commuted with $\frac{\partialartial}{\partialartial t}$ and $\Delta_{{\bar{\alpha}}r{\partialartial}}$. The rest is the same as the proof of Corollary \ref{main-con1}.
\epsilonnd{proof}
We first start with some lemmas which are the time dependent version of Lemma \ref{help41}, \ref{help42}.
\begin{lemma} \operatorname{l}abel{helpKRF61} Let $\partialhi$ be a $(p, p)$-form satisfying
(\ref{eq:11}). Then under a normal coordinate,
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}''(\partialhi))_{i I_{p-1},\overline{J}_{p-1}}
=\mathcal{KB}(\operatorname{div}''_{i}(\partialhi)) _{I_{p-1}, \overline{J}_{p-1}}\nonumber\\
&
+& \sum_{\nu =1 }^{p-1}R_{i\,{\bar{\alpha}}r{j_\nu} l{\bar{\alpha}}r{k}}
(\operatorname{div}''(\partialhi))_{k I_{p-1}, {\bar{\alpha}}r{j_1}
\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}
-\frac{1}{2}R_{i{\bar{\alpha}}r{k}}(\operatorname{div}''(\partialhi))_{k I_{p-1}, \overline{J}_{p-1}}
\operatorname{l}abel{eq:lem61}\\
&
+&R_{j{\bar{\alpha}}r{k}}\nabla_k \partialhi_{i I_{p-1},{\bar{\alpha}}r{j} \overline{J}_{p-1}}
+\nabla_i R_{j{\bar{\alpha}}r{k}}\partialhi_{k I_{p-1},{\bar{\alpha}}r{j}
\overline{J}_{p-1}}
+\sum_{\mu =1}^{p-1}
\nabla_{i_\mu}R_{l{\bar{\alpha}}r{k}}\partialhi_{i i_1\cdots(k)_\mu\cdots i_{p-1}
{\bar{\alpha}}r{l} \overline{J}_{p-1}};\nonumber\\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}'(\partialhi))_{ I_{p-1},
{\bar{\alpha}}r{j}\overline{J}_{p-1}}= \mathcal{KB}(\operatorname{div}'_{{\bar{\alpha}}r{j}}(\partialhi)) _{I_{p-1}, \overline{J}_{p-1}}\nonumber
\\
\quad \quad &
+& \sum_{\mu =1 }^{p-1}R_{i_\mu{\bar{\alpha}}r{j}\, l{\bar{\alpha}}r{k} }(\operatorname{div}'(
\partialhi))_{i_1\cdots(k)_\mu\cdots i_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}
-\frac{1}{2}R_{l
{\bar{\alpha}}r{j}}(\operatorname{div}'(\partialhi))_{ I_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}
\operatorname{l}abel{eq:lem62}\\
\quad \quad &
+&R_{l{\bar{\alpha}}r{k}}\nabla_{{\bar{\alpha}}r{l}}\partialhi_{k I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\nabla_{{\bar{\alpha}}r{j}}R_{l{\bar{\alpha}}r{k}}\partialhi_{k I_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}
+\sum_{\nu =1}^{p-1}\nabla_{{\bar{\alpha}}r{j}_\mu}R_{l{\bar{\alpha}}r{k}}\partialhi_{k I_{p-1},
{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}.\nonumber
\epsilonnd{eqnarray}
\epsilonnd{lemma}
\begin{proof}
Since $\frac{\partialartial}{\partialartial t} \Gamma^h_{jl}=-g^{h{\bar{\alpha}}r{q}}\nabla_{j}R_{l{\bar{\alpha}}r{q}}$,
\begin{eqnarray}
&\,&\frac{\partialartial}{\partialartial t} (\sum g^{{\bar{\alpha}}r{l}k}\nabla_{k}\partialhi_{i i_1\cdots
i_{p-1},{\bar{\alpha}}r{l}{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}})\nonumber \\
&=& R_{l{\bar{\alpha}}r{k}}\nabla_{k}\partialhi_{i I_{p-1},
{\bar{\alpha}}r{l}\overline{J}_{p-1}}+\nabla_i R_{l{\bar{\alpha}}r{k}}\partialhi_{k I_{p-1}, {\bar{\alpha}}r{l}
\overline{J}_{p-1}}+\sum_{\mu =1}^{p-1}\nabla_{i_{\mu}}R_{l{\bar{\alpha}}r{k}}\partialhi_{
i i_1\cdots(k)_{\mu}
\cdots i_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}}\operatorname{l}abel{eq:help61}\\
\quad\quad &+&\operatorname{l}eft(\operatorname{div}''\operatorname{l}eft(\frac{\partialartial}{\partialartial t} \partialhi\rightght)\rightght)_{i I_{p-1}, \overline{J}_{p-1}},\nonumber
\epsilonnd{eqnarray}
then the first equation follows from the fact that $\Delta_{{\bar{\alpha}}r{\partialartial}}$ is commutative with $\operatorname{div}''$ and (\ref{eq:lem41}).
The second evolution equation follows from taking the conjugation of
the first one.
\epsilonnd{proof}
\begin{lemma} \operatorname{l}abel{helpKRF62}Let $\partialhi$ be a solution to (\ref{eq:11}). Then under the normal coordinate
\begin{eqnarray*}
&\, &\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}'
(\operatorname{div}''(\partialhi)))_{ I_{p-1}, \overline{J}_{p-1}}
= \operatorname{l}eft(\mathcal{KB}(\operatorname{div}'
(\operatorname{div}''(\partialhi)))\rightght)_{ I_{p-1}, \overline{J}_{p-1}}
+\mathcal{E}(\partialhi)_{I_{p-1},\overline{J}_{p-1}}; \\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}''
(\operatorname{div}'(\partialhi)))_{ I_{p-1}, \overline{J}_{p-1}}= \operatorname{l}eft(\mathcal{KB}(
\operatorname{div}''(\operatorname{div}'(\partialhi)))\rightght)_{ I_{p-1}, \overline{J}_{p-1}}+\mathcal{E}(\partialhi)_{I_{p-1},\overline{J}_{p-1}},
\epsilonnd{eqnarray*}
where
\begin{eqnarray}
&\, &\mathcal{E}(\partialhi)_{I_{p-1}, \overline{J}_{p-1}}\doteqdot
R_{j{\bar{\alpha}}r{i}}
\operatorname{l}eft(\nabla_{i}(\operatorname{div}'(\partialhi))_{I_{p-1},{\bar{\alpha}}r{j}\overline{J}_
{p-1}}+\nabla_{{\bar{\alpha}}r{j}}
(\operatorname{div}''(\partialhi))_{i I_{p-1},\overline{J}_{p-1}}
\rightght)\nonumber\\
\quad\quad
&+&\sum_{\mu=1}^{p-1}\nabla_{j}R_{i_{\mu}{\bar{\alpha}}r{k}}
(\operatorname{div}'(\partialhi))_{i_1\cdots(k)_{\mu}\cdots
i_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\sum_{\nu=1}^{p-1}\nabla_{{\bar{\alpha}}r{i}}R_{l{\bar{\alpha}}r{j}_{\nu}}
(\operatorname{div}''(\partialhi))_{i I_{p-1},
{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}}\nonumber\\
\quad\quad
&+&\sum_{\mu=1}^{p-1}R_{l{\bar{\alpha}}r{k}}R_{j{\bar{\alpha}}r{i}i_{\mu}{\bar{\alpha}}r{l}}
\partialhi_{ii_1\cdots(k)_{\mu}\cdots i_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\sum_{\nu=1}^{p-1}R_{l{\bar{\alpha}}r{k}}R_{j{\bar{\alpha}}r{i}k{\bar{\alpha}}r{j}_{\nu}}
\partialhi_{i I_{p-1},
{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}}\nonumber \\
\quad\quad
&+&\D R_{j{\bar{\alpha}}r{i}}
\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\nabla_{k}R_{j{\bar{\alpha}}r{i}}\nabla_{{\bar{\alpha}}r{k}}
\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\nabla_{{\bar{\alpha}}r{k}}R_{j{\bar{\alpha}}r{i}}\nabla_{k}
\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}\nonumber\\
\quad\quad
&+&R_{j{\bar{\alpha}}r{k}}R_{k{\bar{\alpha}}r{i}}
\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
-R_{k{\bar{\alpha}}r{l}}R_{l{\bar{\alpha}}r{k}j{\bar{\alpha}}r{i}}
\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}.\nonumber
\epsilonnd{eqnarray}
\epsilonnd{lemma}
\begin{proof}
\begin{eqnarray*}
&\, &\frac{\partialartial}{\partialartial t} (g^{i{\bar{\alpha}}r{j}}\nabla_{{\bar{\alpha}}r{j}}
(\operatorname{div}''(\partialhi))_{i I_{p-1}, \overline{J}_{p-1}})\\
\quad\quad
&=&R_{j{\bar{\alpha}}r{i}}\nabla_{{\bar{\alpha}}r{j}}
(\operatorname{div}''(\partialhi))_{i I_{p-1}, \overline{J}_{p-1}}
+\sum_{\nu=1}^{p-1}\nabla_{{\bar{\alpha}}r{i}}R_{l{\bar{\alpha}}r{j}_{\nu}}
(\operatorname{div}''(\partialhi))_{i I_{p-1},
{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}}\\
\quad\quad
&+&\nabla_{{\bar{\alpha}}r{i}}(\frac{\partial}{\partial t}(
(\operatorname{div}''(\partialhi))_{i I_{p-1}, \overline{J}_{p-1}})
.
\epsilonnd{eqnarray*}
Now we plug in (\ref{eq:help61}). Applying the commutator formula, the 2nd-Bianchi identity and Lemma
\ref{help42}, we get the first evolution equation. The second one
follows from the first by taking the conjugation.
\epsilonnd{proof}
The next lemma is on $\operatorname{Ric}(\partialhi)$. The proof is via straight forward computation.
\begin{lemma} For $\partialhi$, a solution (\ref{eq:11}), under a normal coordinate,
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (R_{j{\bar{\alpha}}r{i}}\partialhi_{i I_{p-1},
{\bar{\alpha}}r{j}\overline{J}_{p-1}})
=\sum_{\mu = 1}^{p-1}\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,\,{\bar{\alpha}}r{j_\nu}}R_{j{\bar{\alpha}}r{i}}
\partialhi_{ii_1\cdots(k)_{\mu}\cdots i_{p-1}{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}
\cdots{\bar{\alpha}}r{j}_{p-1}}\operatorname{l}abel{eq:lem65}\\
&\quad&
-\frac{1}{2}\operatorname{l}eft(\sum_{\nu =1}^{p-1}R^{{\bar{\alpha}}r{l}}_{\,\,
{\bar{\alpha}}r{j}_\nu}(R_{j{\bar{\alpha}}r{i}}\partialhi_{i I_{p-1},{\bar{\alpha}}r{j}{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_{\nu}
\cdots{\bar{\alpha}}r{j}_{p-1}}+\sum_{\mu =1
}^{p-1}R_{i_\mu}^{\,\,
k}R_{j{\bar{\alpha}}r{i}}\partialhi_{ii_1\cdots(k)_\mu\cdots i_{p-1},
{\bar{\alpha}}r{j}\overline{J}_{p-1}}\rightght)\nonumber\\
&\quad&-\operatorname{l}eft(\sum_{\mu=1}^{p-1}R_{l{\bar{\alpha}}r{k}}R_{j{\bar{\alpha}}r{i}i_{\mu}{\bar{\alpha}}r{l}}
\partialhi_{ii_1\cdots(k)_{\mu}\cdots i_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\sum_{\nu=1}^{p-1}R_{l{\bar{\alpha}}r{k}}R_{j{\bar{\alpha}}r{i}k{\bar{\alpha}}r{j}_{\nu}}
\partialhi_{i I_{p-1},
{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}}\rightght)\nonumber \\
&\quad&-\operatorname{l}eft(\nabla_{k}R_{j{\bar{\alpha}}r{i}}\nabla_{{\bar{\alpha}}r{k}}
\partialhi_{i I_{p-1}， {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\nabla_{{\bar{\alpha}}r{k}}R_{j{\bar{\alpha}}r{i}}\nabla_{k}
\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}\rightght)
+2R_{k{\bar{\alpha}}r{l}}R_{l{\bar{\alpha}}r{k}j{\bar{\alpha}}r{i}}\partialhi_{i I_{p-1},
{\bar{\alpha}}r{j}\overline{J}_{p-1}}\nonumber.
\epsilonnd{eqnarray}
\epsilonnd{lemma}
Adapting the notation from Section 4, Lemma \ref{helpKRF61} implies the following set of formulae.
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}''_{V}(\partialhi))_{I_{p-1},
\overline{J}_{p-1}}=\operatorname{div}''_{\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) V}(\partialhi)
+\mathcal{KB}(\operatorname{div}''_{V}(\partialhi)) \operatorname{l}abel{eq:68}\\
\quad \quad &+& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}}(\operatorname{div}''(\partialhi))_{ki_1\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}
\operatorname{div}''_{\operatorname{Ric}(V)}(\partialhi)
\nonumber\\
\quad\quad&+&
R_{j{\bar{\alpha}}r{k}}\nabla_{k}\partialhi_{Vi_1\cdots i_{p-1},{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1
\cdots{\bar{\alpha}}r{j}_{p-1}}+\nabla_VR_{j{\bar{\alpha}}r{i}}\partialhi_{ii_1\cdots i_{p-1},
{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}\nonumber\\
\quad \quad &+& \sum_{\mu=1}^{p-1}\nabla_{i_{\mu}}R_{j{\bar{\alpha}}r{k}}\partialhi_{Vi_1\cdots
(k)_{\mu}\cdots i_{p-1},{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}-
g^{i{\bar{\alpha}}r{j}}\operatorname{l}eft((\nabla_{i}
\operatorname{div}''(\partialhi))_{\nabla_{{\bar{\alpha}}r{j}}V}+(\nabla_{{\bar{\alpha}}r{j}}
\operatorname{div}''(\partialhi))_{\nabla_{i}V})\rightght); \nonumber \\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}'_{\overline{V}}(\partialhi))_{I_{p-1}, {\bar{\alpha}}r{J}_{p-1}}=
\operatorname{div}'_{\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right)
\overline{V}}(\partialhi)+\mathcal{KB}(\operatorname{div}'_{\overline{V}}(\partialhi))
\operatorname{l}abel{eq:69}\\
\quad \quad &
+& \sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu \,{\bar{\alpha}}r{V}}(\operatorname{div}'(\partialhi))_{i_1\cdots(k)_\mu\cdots
i_{p-1},
{\bar{\alpha}}r{l}{\bar{\alpha}}r{j_1}\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}\operatorname{div}'_{\overline{\operatorname{Ric}(V)
}}(\partialhi) \nonumber\\
\quad\quad &+&
R_{k{\bar{\alpha}}r{i}}\nabla_{{\bar{\alpha}}r{k}}\partialhi_{ii_1\cdots i_{p-1},
\overline{V}{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}
+\nabla_{\overline{V}}R_{j{\bar{\alpha}}r{i}}\partialhi_{ii_1\cdots i_{p-1}, {\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots
{\bar{\alpha}}r{j}_{p-1}}\nonumber\\
\quad\quad &+& \sum_{\nu=1}^{p-1}\nabla_{{\bar{\alpha}}r{j}_{\nu}}R_{j{\bar{\alpha}}r{i}}
\partialhi_{ii_1\cdots i_{p-1},
\overline{V}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{j})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}}
-g^{i{\bar{\alpha}}r{j}}\operatorname{l}eft((\nabla_{i}
\operatorname{div}'(\partialhi))_{\nabla_{{\bar{\alpha}}r{j}}\overline{V}}+(\nabla_{{\bar{\alpha}}r{j}}
\operatorname{div}'(\partialhi))_{\nabla_{i}\overline{V}}\rightght). \nonumber
\epsilonnd{eqnarray}
For $\Lambda\partialhi$, we have the following evolution equation.
\begin{equation}\operatorname{l}abel{eq:610}
(\frac{\partial}{\partial t}-\D)(\Lambda\partialhi)_{i_1\cdots i_{p-1},
{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}=\mathcal{KB}(\Lambda\partialhi)_{i_1\cdots i_{p-1},
{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}+R_{j{\bar{\alpha}}r{i}}\partialhi_{ii_1\cdots i_{p-1},
{\bar{\alpha}}r{j}{\bar{\alpha}}r{j}_1\cdots{\bar{\alpha}}r{j}_{p-1}}.
\epsilonnd{equation}
\section{ A family of LYH estimates for the K\"ahler-Ricci flow under the condition $\mathcal{C}_p$}
Let $(M, g(t))$ be a complete solution to the K\"ahler-Ricci flow. Assume
further that the curvature operator satisfies $\mathcal{C}_p$.
Let
\begin{eqnarray*}
\mathcal{M}_{\alphabb}=\D R_{\alphabb}+R_{\alphabb\gammabd}R_{\delta{\bar{\alpha}}r{\gamma}}+\frac{R_{\alphabb}}{t}, \quad
P_{\alphabb\gamma}=\nabla_{\gamma}R_{\alphabb}, \quad P_{\alphabb{\bar{\alpha}}r{\gamma}}=\nabla_{{\bar{\alpha}}r{\gamma}}R_{\alphabb}.
\epsilonnd{eqnarray*}
Also let $P_{{\bar{\alpha}}r{\beta} \alphalpha \gammaamma}=\nabla_\gammaamma R_{{\bar{\alpha}}r{\beta}\alphalpha}$, $P_{{\bar{\alpha}}r{\beta}\alphalpha {\bar{\alpha}}r{\gammaamma}}=\nabla_{{\bar{\alpha}}r{\gammaamma}}R_{{\bar{\alpha}}r{\beta}\alphalpha}$. Clearly $P_{\alphabb\gamma}=P_{{\bar{\alpha}}r{\beta} \alphalpha \gammaamma}$ and $P_{\alphabb{\bar{\alpha}}r{\gamma}}=P_{{\bar{\alpha}}r{\beta}\alphalpha {\bar{\alpha}}r{\gammaamma}}$.
The second Bianchi identity implies that
\begin{equation*}
P_{\alphabb\gamma}=P_{\gamma{\bar{\beta}}\alpha}, \quad \overline{P_{\alphabb\gamma}}=P_{\b{\bar{\alpha}}{\bar{\alpha}}r{\gamma}}=P_{{\bar{\alpha}}\b{\bar{\alpha}}r{\gamma}}.
\epsilonnd{equation*}
\begin{theorem}\operatorname{l}abel{LYH} Let $(M, g(t))$ be a complete solution to the K\"ahler-Ricci flow satisfying the condition $\mathcal{C}_p$ on $M \times [0, T]$. When $M$ is noncompact we assume that the curvature of $(M, g(t))$ is bounded on $M \times [0, T]$. Then
for any $\wedge^{1, 1}$-vector $U$ which can be written as
$U=\sum_{i=1}^{p-1} X_i\wedge {\bar{\alpha}}r{Y}_i+W\wedge {\bar{\alpha}}r{V}$, for $(1, 0)$-type vectors $X_i, Y_i, W, V$, the Hermitian bilinear form $\mathcal{Q}$ defined as
\begin{equation}\operatorname{l}abel{eq:71}
\mathcal{Q}(U\oplus W)\doteqdot
\mathcal{M}_{\alphabb}W^{\alpha}W^{{\bar{\beta}}}+P_{\alphabb\gamma}{\bar{\alpha}}r{U}^{{\bar{\beta}}\gamma}W^{\alpha}
+P_{\alphabb{\bar{\alpha}}r{\gamma}}U^{\alpha{\bar{\alpha}}r{\gamma}}W^{{\bar{\beta}}}
+R_{\alphabb\gammabd}U^{\alpha{\bar{\beta}}}{\bar{\alpha}}r{U}^{{{\bar{\alpha}}r{\delta}}\gamma}
\epsilonnd{equation}
satisfies that $\mathcal{Q}\gammae 0$ for any $t>0$. Moreover, if the equality ever occurs for some $t>0$, the universal cover of $(M, g(t))$ must be a gradient expanding K\"ahler-Ricci soliton.
\epsilonnd{theorem}
The theorem says that for any vector $W$, viewing $\mathcal{Q}$ as a Hermitian quadratic/bilinear form on $\wedge^{1,1}$ space, it also satisfies $\mathcal{C}_p$, but only for the $\wedge^{1,1}$ vector $U$ with the form $U=\sum_{i=1}^p X_i\wedge \overline{Y}_i$ with $X_p=W$.
If we define $P: T'M \to \wedge^{1,1}$ by the equation $\operatorname{l}angle P(W), \overline{U}\rangle =P_{\alphabb\gamma}\overline{U}^{{\bar{\beta}}\gamma}W^{\alpha}$, the LYH expression can be written as, by abusing the notation with $\mathcal{Q}$ denoting also the Hermitian symmetric transformation,
\begin{equation}\operatorname{l}abel{def-krf}
\operatorname{l}angle \mathcal{Q}(U), \overline{U}\rangle =\operatorname{l}angle \mathcal{M}(W), \overline{W}\rangle +2 Re(\operatorname{l}angle P(W), \overline{U}\rangle )+\operatorname{l}angle \Bbb Rm(U), \overline{U}\rangle.
\epsilonnd{equation}
\begin{remark}
When $p=1$, The inequality (\ref{eq:71}) recovers the LYH inequality of Cao \cite{Cao}. When $p>1$, $\mathcal{Q}$ can be written as
\begin{equation*}
\mathcal{Q}=Z_{\alphabb}W^{\alpha}W^{{\bar{\beta}}}
+(P_{\alphabb\gamma}+R_{\alpha{\bar{\alpha}}r{V}\gamma{\bar{\beta}}})\overline{\tilde{U}}^{{\bar{\beta}}\gamma}
W^{\alpha}+(P_{\alphabb{\bar{\alpha}}r{\gamma}}+R_{V{\bar{\beta}}\alpha{\bar{\alpha}}r{\gamma}})\tilde{U}^{\alpha{\bar{\alpha}}r{\gamma}}W^{{\bar{\beta}}}
+R_{\alphabb\gammabd}\tilde{U}^{\alphabb}\overline{\tilde{U}}^{{{\bar{\alpha}}r{\delta}}\gamma},
\epsilonnd{equation*}
with $\tilde{U}=\sum_{i=1}^{p-1}X_i\wedge {\bar{\alpha}}r{Y}_i$ and
\begin{equation}\operatorname{l}abel{defZ}
Z_{\alphabb}\doteqdot \mathcal{M}_{\alphabb}+P_{\alphabb\gamma}V^{\gamma}+P_{\alphabb{\bar{\alpha}}r{\gamma}}V^{{\bar{\alpha}}r{\gamma}}
+R_{\alphabb\gammabd}V^{\gamma}V^{{{\bar{\alpha}}r{\delta}}}.
\epsilonnd{equation}
Equivalently, if we write the above as $\operatorname{l}angle Z(W\wedge \overline{V}), \overline{W\wedge \overline{V}}\rangle$,
$$
\mathcal{Q}=\operatorname{l}angle Z(W\wedge \overline{V}), \overline{W\wedge \overline{V}}\rangle +2Re\operatorname{l}eft( \widetilde{P}(W\wedge \overline{V}), \overline{\tilde{U}}\rangle\rightght)+\operatorname{l}angle \Bbb Rm (\tilde U), \overline{\tilde{U}}\rangle.
$$
Here $\widetilde{P}$ is defined as $ \widetilde{P}(W\wedge \overline{V})=P(W)+\Bbb Rm(W\wedge \overline{V})$.
Note that Hamilton in \cite{richard-harnack} proved that under the stronger assumption that the curvature operator $\Bbb Rm\gammae0$, $Q(U\oplus W)\gammae 0$ for any $\wedge^2$-vector $U$. For $p$ sufficiently large $\mathcal{C}_p$ is equivalent to $\Bbb Rm \gammae 0$ and by taking $U=U_1+W\wedge \overline{Y}_p$ with $U_1=U-W\wedge \overline{Y}_p$, one can see that the above result implies Hamilton's result. Hence Theorem \ref{LYH} interpolates between Cao's result and Hamilton's result for the K\"ahler-Ricci flow. Please also see \cite{CC} for an interpretation via the space time consideration. In a later section we shall prove another set of estimates which generalize Hamilton's estimate for the Ricci flow on Riemannian manifolds.
\epsilonnd{remark}
One can easily get the following lemma through direct calculation, which can also be derived from Lemma 4.3, 4.4 of \cite{richard-harnack}.
\begin{lemma}\operatorname{l}abel{help71}
\begin{eqnarray}
&\, &\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \mathcal{M}_{\alpha{\bar{\beta}}}=R_{\alphabb\gammabd}\mathcal{M}_{\delta{\bar{\alpha}}r{\gamma}}
-\frac{1}{2}(R_{\alpha{\bar{\alpha}}r{\epsilonta}}\mathcal{M}_{\epsilonta{\bar{\beta}}}+R_{\epsilonta{\bar{\beta}}}\mathcal{M}_{\alpha{\bar{\alpha}}r{\epsilonta}})
+R_{\alphabb\gammabd}R_{\delta{\bar{\alpha}}r{\xi}}R_{\xi{\bar{\alpha}}r{\gamma}}\operatorname{l}abel{lem71}\\
&\quad&+R_{\delta{\bar{\alpha}}r{\gamma}}(\nabla_{\gamma}P_{\alphabb{{\bar{\alpha}}r{\delta}}}
+\nabla_{{{\bar{\alpha}}r{\delta}}}P_{\alphabb\gamma})+P_{\alpha{\bar{\alpha}}r{\xi}\gamma}P_{\xi{\bar{\beta}}{\bar{\alpha}}r{\gamma}}
-P_{\alpha{\bar{\alpha}}r{\xi}{\bar{\alpha}}r{\gamma}}P_{\xi{\bar{\beta}}\gamma}
-\frac{R_{\alphabb}}{t^2},
\nonumber\\
&\, &\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) P_{\alphabb\gamma}=R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}P_{\epsilonta{\bar{\alpha}}r{\xi}\gamma}
+R_{\xi{\bar{\beta}}\gammabd}P_{\alpha{\bar{\alpha}}r{\xi}\delta}
-R_{\alpha{\bar{\alpha}}r{\xi}\gammabd}P_{\xi{\bar{\beta}}\delta}\operatorname{l}abel{lem72}\\
&\quad&
-\frac{1}{2}\operatorname{l}eft(R_{\alpha{\bar{\alpha}}r{\xi}}P_{\xi{\bar{\beta}}\gamma}
+R_{\xi{\bar{\beta}}}P_{\alpha{\bar{\alpha}}r{\xi}\gamma}+R_{\gamma{\bar{\alpha}}r{\xi}}P_{\alphabb\xi}\rightght)
+\nabla_{\gamma}R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}R_{\epsilonta{\bar{\alpha}}r{\xi}}.\nonumber
\epsilonnd{eqnarray}
\epsilonnd{lemma}
By taking the conjugation of (\ref{lem72}), we have
\begin{eqnarray}\operatorname{l}abel{eq:74}
&\, &\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) P_{\alphabb{\bar{\alpha}}r{\gamma}}
=R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}P_{\epsilonta{\bar{\alpha}}r{\xi}{\bar{\alpha}}r{\gamma}}
+R_{\alpha{\bar{\alpha}}r{\xi}\delta{\bar{\alpha}}r{\gamma}}P_{\xi{\bar{\beta}}{{\bar{\alpha}}r{\delta}}}
-R_{\xi{\bar{\beta}}\delta{\bar{\alpha}}r{\gamma}}P_{\alpha{\bar{\alpha}}r{\xi}{{\bar{\alpha}}r{\delta}}}\operatorname{l}abel{lem}\\
&\quad&-\frac{1}{2}\operatorname{l}eft(R_{\alpha{\bar{\alpha}}r{\xi}}P_{\xi{\bar{\beta}}{\bar{\alpha}}r{\gamma}}
+R_{\xi{\bar{\beta}}}P_{\alpha{\bar{\alpha}}r{\xi}{\bar{\alpha}}r{\gamma}}+R_{\xi{\bar{\alpha}}r{\gamma}}P_{\alphabb{\bar{\alpha}}r{\xi}}\rightght)
+\nabla_{{\bar{\alpha}}r{\gamma}}R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}R_{\epsilonta{\bar{\alpha}}r{\xi}}\nonumber.
\epsilonnd{eqnarray}
The evolution equation for the curvature tensor is (see for example \cite{Bando})
\begin{eqnarray}\operatorname{l}abel{eq:75}
&\, &\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) R_{\alphabb\gammabd}
=R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}R_{\epsilonta{\bar{\alpha}}r{\xi}\gammabd}
-R_{\alpha{\bar{\alpha}}r{\xi}\gamma{\bar{\alpha}}r{\epsilonta}}R_{\xi{\bar{\beta}}\epsilonta{{\bar{\alpha}}r{\delta}}}
+R_{\alpha{{\bar{\alpha}}r{\delta}}\xi{\bar{\alpha}}r{\epsilonta}}R_{\epsilonta{\bar{\alpha}}r{\xi}\gamma{\bar{\beta}}}\operatorname{l}abel{eq:}\\
&\quad&-\frac{1}{2}\operatorname{l}eft(
R_{\alpha{\bar{\alpha}}r{\xi}}R_{\xi{\bar{\beta}}\gammabd}+R_{\xi{\bar{\beta}}}R_{\alpha{\bar{\alpha}}r{\xi}\gammabd}
+R_{\gamma{\bar{\alpha}}r{\xi}}R_{\alphabb\xi{{\bar{\alpha}}r{\delta}}}+R_{\xi{{\bar{\alpha}}r{\delta}}}R_{\alphabb\gamma{\bar{\alpha}}r{\xi}}
\rightght)\nonumber.
\epsilonnd{eqnarray}
Now we begin to prove the theorem. We
assume that the curvature of $(M, g(t))$ satisfies $\mathcal{C}_p$. One can adapt the perturbation argument as \cite{richard-harnack} if $\Bbb Rm$ does not have strictly $p$-positive bisectional curvature. Hence when manifold is compact without the loss of generality we may assume that $\Bbb Rm$ has strictly $p$-positive bisectional curvature. Then it is clear that
when $t$ is small $\mathcal{Q}$ is positive, since
the bisectional curvature is strictly $p$-positive and $\mathcal{M}_{\alphabb}$ has a term
$\frac{R_{\alphabb}}{t}$. We claim $\mathcal{Q}\gammae 0$ for all time. If it fails to hold, there
is a first time $t_0$, a point $x_0$, and
$U\sqrt {-1}n \Lambda^{1, 1}T_{x_0}M, W\sqrt {-1}n \Lambda^{1,0}$ such that
$\mathcal{Q}(U\oplus W)=0$, and for any $t\operatorname{l}e t_0, x\sqrt {-1}n M$, $(1, 1)$-vector
$\hat{U}\sqrt {-1}n \Lambda^{1, 1}T_{x}M$ and $(1, 0)$-vector
$\hat{W}\sqrt {-1}n T_xM$, $\mathcal{Q}(\hat{U}\oplus\hat{W})\gammae 0$.
We extend $U$ and $W$ in space-time
at $(x_0, t_0)$ in the following way:
\begin{eqnarray*}
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) U^{\gammabd}
=\frac{1}{2}\operatorname{l}eft(R^{\gamma}_{\alpha}U^{\alpha{{\bar{\alpha}}r{\delta}}}
+R^{{{\bar{\alpha}}r{\delta}}}_{{\bar{\alpha}}r{\alpha}}U^{\gamma{\bar{\alpha}}r{\alpha}}\rightght),\quad &\, & \quad \operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) W^{\alpha}=\frac{1}{2}R_{\beta}^{\alpha}W^{\beta}
+\frac{1}{t}W^{\alpha}
,\\
\nabla_{s}U^{\gammabd}
=R_{s}^{\gamma}W^{{{\bar{\alpha}}r{\delta}}}+\frac{1}{t}g^{\gamma}_{s}W^{{{\bar{\alpha}}r{\delta}}},
\quad \quad \nabla_{{\bar{\alpha}}r{s}}U^{\gammabd}=0,\quad &\,& \quad \nabla_{\gamma}W^{\alpha}=\nabla_{{\bar{\alpha}}r{\gamma}}W^{\alpha}=0
.
\epsilonnd{eqnarray*}
Here $R^{\alpha}_\beta, R^{{\bar{\alpha}}r{\gammaamma}}_{{\bar{\alpha}}r{\delta}}$ are the associated tensors obtained by raising the indices on the Ricci tensor. These sets of equations are the same as those of \cite{richard-harnack} in disguise.
As in \cite{richard-harnack}, it suffices
to check that at the point $(x_0, t_0)$, $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \mathcal{Q}\gammae 0$.
Using the above equations and equations (\ref{lem71}),
(\ref{lem72}), (\ref{eq:74}), (\ref{eq:75}), a lengthy but straight-forward computation shows that
\begin{eqnarray*}
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \mathcal{Q}&=&R_{\alphabb\gammabd}M_{\delta{\bar{\alpha}}r{\gamma}}W^{\alpha}W^{{\bar{\beta}}}
+R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}P_{\epsilonta{\bar{\alpha}}r{\xi}\gamma}{\bar{\alpha}}r{U}^{{\bar{\beta}}\gamma}W^{\alpha}
+R_{\alphabb\xi{\bar{\alpha}}r{\epsilonta}}P_{\epsilonta{\bar{\alpha}}r{\xi}{\bar{\alpha}}r{\gamma}}
U^{\alpha{\bar{\alpha}}r{\gamma}}W^{{\bar{\beta}}}
\\
&\quad&+R_{\alpha{{\bar{\alpha}}r{\delta}}\xi{\bar{\alpha}}r{\epsilonta}}R_{\epsilonta{\bar{\alpha}}r{\xi}\gamma{\bar{\beta}}}U^{\alphabb}
{\bar{\alpha}}r{U}^{{{\bar{\alpha}}r{\delta}}\gamma}\\
&
-&\operatorname{l}eft(P_{\alpha{\bar{\alpha}}r{\xi}{\bar{\alpha}}r{\gamma}}P_{\xi{\bar{\beta}}\gamma}W^{\alpha}W^{{\bar{\beta}}}
+R_{\alpha{\bar{\alpha}}r{\xi}\gammabd}P_{\xi{\bar{\beta}}\delta}{\bar{\alpha}}r{U}^{{\bar{\beta}}\gamma}W^{\alpha}\rightght.\\
&\quad& \operatorname{l}eft.
+R_{\xi{\bar{\beta}}\delta{\bar{\alpha}}r{\gamma}}P_{\alpha{\bar{\alpha}}r{\xi}{{\bar{\alpha}}r{\delta}}}U^{\alpha{\bar{\alpha}}r{\gamma}}W^{{\bar{\beta}}}
+R_{\alpha{\bar{\alpha}}r{\xi}\gamma{\bar{\alpha}}r{\epsilonta}}R_{\xi{\bar{\beta}}\epsilonta{{\bar{\alpha}}r{\delta}}}U^{\alphabb}
{\bar{\alpha}}r{U}^{{{\bar{\alpha}}r{\delta}}\gamma}\rightght)\\
&\quad& +(P_{\alpha{\bar{\alpha}}r{\xi}\gamma}W^{\alpha}+R_{\alpha{\bar{\alpha}}r{\xi}\gammabd}U^{\alpha{{\bar{\alpha}}r{\delta}}})
(P_{\xi{\bar{\beta}}{\bar{\alpha}}r{\gamma}}W^{{\bar{\beta}}}+R_{\xi{\bar{\beta}}\delta{\bar{\alpha}}r{\gamma}}{\bar{\alpha}}r{U}^{{\bar{\beta}}\delta}).
\epsilonnd{eqnarray*}
The above computation can also be derived using Lemma 4.5 of \cite{richard-harnack}.
In the following, $X_p=W, Y_p=V$. To prove
$\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \mathcal{Q}\gammae 0$ it is enough to show that the nonnegativity of
\begin{eqnarray*}
\mathcal{J}
&\doteqdot& R_{X_p{\bar{\alpha}}r{X}_p\gammabd}Z_{\delta{\bar{\alpha}}r{\gamma}}
+\sum_{\nu=1}^{p-1}R_{X_p{\bar{\alpha}}r{X}_{\nu}\gammabd}(P_{\delta{\bar{\alpha}}r{\gamma} Y_{\nu}}
+R_{\delta{\bar{\alpha}}r{\gamma}Y_{\nu}{\bar{\alpha}}r{Y}_p})
+\sum_{\mu=1}^{p-1}R_{X_{\mu}{\bar{\alpha}}r{X}_p\gammabd}(P_{\delta{\bar{\alpha}}r{\gamma}{\bar{\alpha}}r{Y}_{\mu}}
+R_{\delta{\bar{\alpha}}r{\gamma}Y_p{\bar{\alpha}}r{Y}_{\mu}})\\
\quad\quad&
+&\sum_{\mu, \nu=1}^{p-1}R_{X_{\mu}{\bar{\alpha}}r{X}_{\nu}\gammabd}R_{\delta{\bar{\alpha}}r{\gamma}
Y_{\nu}{\bar{\alpha}}r{Y}_{\mu}}
+|P_{X_p{\bar{\beta}}\alpha}+\sum_{\mu=1}^pR_{X_{\mu}{\bar{\alpha}}r{Y}_{\mu}\alphabb}|^2\\
\quad\quad&
-&\operatorname{l}eft(|P_{X_p{\bar{\alpha}}r{\gamma}{{\bar{\alpha}}r{\delta}}}|^2+\sum_{\nu=1}^p
R_{X_p{\bar{\alpha}}r{\gamma}Y_{\nu}{{\bar{\alpha}}r{\delta}}}P_{\gamma{\bar{\alpha}}r{X}_{\nu}\delta}
+\sum_{\mu=1}^p R_{\gamma{\bar{\alpha}}r{X}_p\delta{\bar{\alpha}}r{Y}_{\mu}}
P_{X_{\mu}{\bar{\alpha}}r{\gamma}{{\bar{\alpha}}r{\delta}}}+
\sum_{\mu, \nu=1}^p R_{X_{\mu}{\bar{\alpha}}r{\gamma}Y_{\nu}{{\bar{\alpha}}r{\delta}}}
R_{\gamma{\bar{\alpha}}r{Y}_{\mu}\delta{\bar{\alpha}}r{X}_{\nu}}\rightght),
\epsilonnd{eqnarray*}
where we have respectively replaced $U$ and $W$ by
$\sum_{i=1}^p X_{i}\wedge {\bar{\alpha}}r{Y}_i$ and $X_p$.
Let
\begin{eqnarray*}
A_1=\operatorname{l}eft(\begin{array}{l}
R_{X_1{\bar{\alpha}}r{X}_1(\cdot){\bar{\alpha}}r{(\cdot)}}\qquad
R_{X_1{\bar{\alpha}}r{X}_2(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{X_1{\bar{\alpha}}r{X}_{p}(\cdot){\bar{\alpha}}r{(\cdot)}}
\\
R_{X_2{\bar{\alpha}}r{X}_1(\cdot){\bar{\alpha}}r{(\cdot)}}\qquad
R_{X_2{\bar{\alpha}}r{X}_2(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{X_2{\bar{\alpha}}r{X}_{p}(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\quad\quad\cdots\quad\quad\quad\quad\cdots\qquad\qquad\quad\cdots\quad\quad
\quad\cdots\\
R_{X_p{\bar{\alpha}}r{X}_1(\cdot){\bar{\alpha}}r{(\cdot)}}\qquad
R_{X_p{\bar{\alpha}}r{X}_2(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{X_p{\bar{\alpha}}r{X}_p(\cdot){\bar{\alpha}}r{(\cdot)}}
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
\begin{eqnarray*}
A_2=\operatorname{l}eft(\begin{array}{l}
R_{Y_1{\bar{\alpha}}r{Y}_1(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
R_{Y_1{\bar{\alpha}}r{Y}_2(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad
R_{Y_1{\bar{\alpha}}r{Y}_{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
E^{1{\bar{\alpha}}r{p}}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\\
R_{Y_2{\bar{\alpha}}r{Y}_1(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
R_{Y_2{\bar{\alpha}}r{Y}_2(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad
R_{Y_2{\bar{\alpha}}r{Y}_{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
E^{2{\bar{\alpha}}r{p}}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\quad\quad\cdots\qquad\qquad\cdots\qquad\qquad\cdots\quad\quad
\quad\cdots\quad\quad\quad\cdots\\
R_{Y_{p-1}{\bar{\alpha}}r{Y}_1(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
R_{Y_{p-1}{\bar{\alpha}}r{Y}_2(\cdot){\bar{\alpha}}r{(\cdot)}}\ \quad\cdots\
R_{Y_{p-1}{\bar{\alpha}}r{Y}_{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\
E^{p-1{\bar{\alpha}}r{p}}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\ \overline{E^{1{\bar{\alpha}}r{p}}}^{tr}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\qquad\quad
\overline{E^{2{\bar{\alpha}}r{p}}}^{tr}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad\cdots\quad\quad
\overline{E^{(p-1){\bar{\alpha}}r{p}}}^{tr}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
Z_{(\cdot){\bar{\alpha}}r{(\cdot)}}
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
and
\begin{eqnarray*}
A_3=\operatorname{l}eft(\begin{array}{l}
R_{X_1(\cdot)Y_1(\cdot)}\quad
R_{X_1(\cdot)Y_2(\cdot)}\quad\quad\cdots\quad
R_{X_1(\cdot)Y_{p-1}(\cdot)}\quad
R_{X_1(\cdot)Y_p(\cdot)}+P_{X_1(\cdot)(\cdot)}\\
R_{X_2(\cdot)Y_1(\cdot)}\quad
R_{X_2(\cdot)Y_2(\cdot)}\quad\quad\cdots\quad
R_{X_2(\cdot)Y_{p-1}(\cdot)}\quad
R_{X_2(\cdot)Y_p(\cdot)}+P_{X_2(\cdot)(\cdot)}\\
\quad\quad\cdots\quad\quad\qquad\cdots\qquad\quad\ \cdots\quad
\quad\quad\cdots\qquad\qquad\qquad\cdots\\
R_{X_p(\cdot)Y_1(\cdot)}\quad
R_{X_p(\cdot)Y_2(\cdot)}\quad\quad\cdots\quad
R_{X_p(\cdot)Y_{p-1}(\cdot)}\quad
R_{X_p(\cdot)Y_p(\cdot)}+P_{X_p(\cdot)(\cdot)}
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
where the tensor $Z_{\gammabd}$ is defined as in (\ref{defZ}),
\begin{equation*}
E^{\mu{\bar{\alpha}}r{p}}_{\ \gammabd}=R_{Y_\mu{\bar{\alpha}}r{Y}_p\gammabd}+P_{\gammabd Y_{\mu}}
, \quad \quad \overline{E^{\mu{\bar{\alpha}}r{p}}}^{tr}_{\ \gammabd}
=R_{Y_p{\bar{\alpha}}r{Y}_{\mu}\gammabd}+P_{\gammabd{\bar{\alpha}}r{Y}_{\mu}},
1\operatorname{l}e \mu\operatorname{l}e p-1.
\epsilonnd{equation*}
Note $A_1\gammae 0$ since $\Bbb Rm \sqrt {-1}n \mathcal{C}_p$, $A_2\gammae 0$ since $\mathcal{Q}\gammae 0$.
Now $\mathcal{J}$ can be written as
\begin{equation*}
\mathcal{J}=\operatorname{trace}(A_1\cdot A_2)+|P_{X_p{\bar{\beta}}\alpha}+\sum_{\mu=1}^p
R_{X_{\mu}{\bar{\alpha}}r{Y}_{\mu}\alphabb}|^2-\operatorname{trace}(A_3\cdot {\bar{\alpha}}r{A}_3).
\epsilonnd{equation*}
Since $\mathcal{Q}(U\oplus W)$ achieves the minimum at $(U, W)$ at time
$(x_0, t_0)$, then the second variation
\begin{equation*}
\frac{\partial^2}{\partial s^2}|_{s=0}\mathcal{Q}(U(s)\oplus W(s))\gammae 0,
\epsilonnd{equation*}
where $W(s)=W+sW_p, \
U(s)=\sum_{\mu=1}^{p}(X_{\mu}+sW_{\mu})\wedge
\overline{(Y_{\mu}+sV_{\mu})}$ for any $(1, 0)$-type vectors
$W_{\mu}, V_{\mu}\sqrt {-1}n T^{1, 0}_{x_0}M$.
Through calculation, $\frac{\partial^2}{\partial s^2}|_{s=0}\mathcal{Q}(U(s)\oplus W(s))\gammae 0$ implies that
\begin{eqnarray}
&\, &\sum_{\mu, \nu=1}^p
R_{Y_{\mu}{\bar{\alpha}}r{Y}_{\nu}\alphabb}W_{\nu}^{\alpha}W_{\mu}^{{\bar{\beta}}}
+\sum_{\mu=1}^p (P_{\alphabb Y_{\mu}}W_p^{\alpha}W_{\mu}^{{\bar{\beta}}}
+P_{\alphabb{\bar{\alpha}}r{Y}_{\mu}}W_{\mu}^{\alpha}W_p^{{\bar{\beta}}})+M_{\alphabb}W_p^{\alpha}W_{p}^{{\bar{\beta}}}
\operatorname{l}abel{eq:77}\\
\quad
&+&\sum_{\mu=1}^p (P_{X_p\alphabb}V_{\mu}^{\alpha}W_{\mu}^{{\bar{\beta}}}
+P_{\alpha{\bar{\alpha}}r{X}_p{\bar{\beta}}}W_{\mu}^{\alpha}V_{\mu}^{{\bar{\beta}}})
+\sum_{\mu, \nu=1}^p(R_{Y_{\mu}{\bar{\alpha}}r{X}_{\mu}\alphabb}W_{\nu}^{\alpha}V_{\nu}^{{\bar{\beta}}}
+R_{X_{\mu}{\bar{\alpha}}r{Y}_{\mu}\alphabb}V_{\nu}^{\alpha}W_{\nu}^{{\bar{\beta}}})\nonumber\\
\quad
&+&\sum_{\mu, \nu=1}^p (
R_{X_{\mu}{\bar{\alpha}} Y_{\nu}{\bar{\beta}}}W_{\nu}^{{\bar{\alpha}}}V_{\mu}^{{\bar{\beta}}}
+R_{\alpha{\bar{\alpha}}r{X}_{\mu}\b{\bar{\alpha}}r{Y}_{\nu}}W_{\nu}^{\alpha}V_{\mu}^{\b})
+\sum_{\mu=1}^p(P_{X_{\mu}{\bar{\alpha}}{\bar{\beta}}}W_{p}^{{\bar{\alpha}}}V_{\mu}^{{\bar{\beta}}}+
P_{\alpha{\bar{\alpha}}r{X}_{\mu}\b}W_p^{\alpha}V_{\mu}^{\b})\nonumber\\
\quad
&+&\sum_{\mu, \nu=1}^p
R_{X_{\mu}{\bar{\alpha}}r{X}_{\nu}\alphabb}V_{\nu}^{\alpha}V_{\mu}^{{\bar{\beta}}}\gammae 0.\nonumber
\epsilonnd{eqnarray}
By letting $\mathcal{X}=\operatorname{l}eft(\begin{array}{l} W_1\\ \vdots\\
W_p\epsilonnd{array}\rightght),\, \mathcal{Y}=\operatorname{l}eft(\begin{array}{l} V_1\\ \vdots\\
V_p\epsilonnd{array}\rightght)$, one can deduce from (\ref{eq:77}) that
\begin{equation*}
\overline{\mathcal{X}}^{tr}A_2\mathcal{X}+\overline{\mathcal{Y}}^{tr}A_1\mathcal{Y}+2Re(\mathcal{Y}^{tr}{\bar{\alpha}}r{A}_3\mathcal{X}
+\overline{\mathcal{Y}}^{tr}A_4\mathcal{X})\gammae
0,
\epsilonnd{equation*}
where
\begin{eqnarray*}
A_4=\operatorname{l}eft(\begin{array}{l}
G\quad
0\quad\quad
\cdots\quad\quad
0
\\
0\quad
G\quad\quad
\cdots\quad\quad
0\\
0\quad
0\quad\quad\cdots\quad\quad
0\\
0\quad
0\quad\quad
\cdots\quad\quad
G
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
where $G_{\alphabb}=
P_{\alpha{\bar{\alpha}}r{X}_p{\bar{\beta}}}
+\sum_{\mu=1}^p
R_{Y_{\mu}{\bar{\alpha}}r{X}_{\mu}\alphabb}$.
If we regard $T^{1, 0}_{x_0}M$ as $\mathbb{C}^m$, then $\mathcal{X}, \mathcal{Y} \sqrt {-1}n
\mathbb{C}^{pm}$. By Lemma \ref{lmlyh} below, which is due to Mok according to \cite{Cao} (see also Lemma 2.86 of \cite{Chowetc}), we have
\begin{equation}\operatorname{l}abel{eq:79}
\operatorname{trace}(A_2\cdot A_1)\gammae \operatorname{trace}(A_3\cdot {\bar{\alpha}}r{A}_3).
\epsilonnd{equation}
The inequality (\ref{eq:79}) implies that $\mathcal{J}\gammae 0.$ We then complete the proof of
Theorem \ref{LYH} for the case that $M$ is compact. The case that $M$ is noncompact will be treated in Section 10.
\begin{lemma}\operatorname{l}abel{lmlyh}
Let $S(\mathcal{X}, \mathcal{Y})$ be a Hermitian symmetric quadratic form defined by
\begin{equation*}
S(\mathcal{X}, \mathcal{Y})=A_{i{\bar{\alpha}}r{j}}\mathcal{X}^i\overline{\mathcal{X}^j}+2Re(B_{ij}\mathcal{X}^i\mathcal{Y}^j
+D_{i{\bar{\alpha}}r{j}}\mathcal{X}^i\overline{\mathcal{Y}^j})
+C_{i{\bar{\alpha}}r{j}}\mathcal{Y}^i\overline{\mathcal{Y}^j}.
\epsilonnd{equation*}
If $S$ is semi-positive definite, then
\begin{equation*}
\sum_{i, j=1}^NA_{i{\bar{\alpha}}r{j}}C_{j{\bar{\alpha}}r{i}}\gammae \max\{
\sum_{i, j=1}^N|B_{ij}|^2, \sum_{i, j=1}^N|D_{i{\bar{\alpha}}r{j}}|^2\}.
\epsilonnd{equation*}
\epsilonnd{lemma}
If one prefers notations without indices the first three terms of (\ref{eq:77}) can be written as
$\operatorname{l}angle \mathcal{Q}(\sum_{\mu=1}^p Y_\mu\wedge \overline{W}_\mu), \overline{\sum_{\nu=1}^p Y_\nu\wedge \overline{W}_\nu}\rangle.
$ The last term can be written as $\operatorname{l}angle \Bbb Rm( \sum_{\mu=1}^p X_\mu\wedge \overline{V}_\mu), \overline{\sum_{\nu=1}^p X_\nu\wedge \overline{V}_\nu}\rangle.$
\section{The proof of Theorem \ref{main-KRFlyh}}
Before we start, we remark that for the cases $p=1$ and $p=m$ the result has been previously proved in \cite{N-jams} and \cite{Ni-JDG07}. As before we deal with the compact case first. By an perturbation argument we also consider that $\partialhi$ is strictly positive. Then $\widetilde{Q}>0$ for small $t$. Assume that at some point $(x_0, t_0)$, $\widetilde{Q}=0$ for the first time for some linearly independent vectors $v_1, v_2, \cdots,v_{p-1}$. As in Section 5, let $v_i(z)$ and $V(z)$ be variational vectors such that they depend on $z$ holomorphically.
Now consider the following function in $z$,
\begin{eqnarray*}
\mathcal{I}(z)&\doteqdot &\frac{1}{2}\operatorname{l}eft[\operatorname{div}''
(\operatorname{div}'(\partialhi))+ \operatorname{div}'
(\operatorname{div}''(\partialhi))\rightght]_{v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{v}_1(z)\cdots
{\bar{\alpha}}r{v}_{p-1}(z)}\\
&\, &
+R_{j{\bar{\alpha}}r{i}}\partialhi_{iv_1(z)\cdots v_{p-1}(z),
{\bar{\alpha}}r{j}{\bar{\alpha}}r{v}_1(z)\cdots{\bar{\alpha}}r{v}_{p-1}(z)}\\
&\, &
+\operatorname{l}eft[\operatorname{div}'_{\overline{V}(z)}(\partialhi)+\operatorname{div}''_{V(z)}(\partialhi)
\rightght]_{v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{v}_1(z)\cdots {\bar{\alpha}}r{v}_{p-1}(z)}\\
&\,& +\partialhi_{V(z)v_1(z)\cdots v_{p-1}(z), {\bar{\alpha}}r{V}(z){\bar{\alpha}}r{v}_1(z)\cdots
{\bar{\alpha}}r{v}_{p-1}(z)}+\frac{\Lambda \partialhi_{v_1(z)\cdots v_{p-1}(z),
{\bar{\alpha}}r{v}_1(z)\cdots {\bar{\alpha}}r{v}_{p-1}(z)}}{t}
\epsilonnd{eqnarray*}
which satisfies $\mathcal{I}(0)=0$ and $\mathcal{I}(z)\gammae 0$ for any variational
vectors $v_{\mu}(z), V(z)$
with $v_{\mu}(0)=v_{\mu}$ and $V(0)=V$.
As before, without the loss of generality we may assume that
$\{ v_1, \cdots, v_{p-1}\}=\{\frac{\partialartial}{\partialartial z^1},\cdots,
\frac{\partialartial}{\partialartial z^{p-1}}\}$. By the first and the second variation consideration as in Section 5,
we have that
\begin{equation}\operatorname{l}abel{eq:81}
\operatorname{div}'_{{\bar{\alpha}}r{X}}(\partialhi)+\partialhi_{V, {\bar{\alpha}}r{X}}=0=
\operatorname{div}''_{X}(\partialhi)+\partialhi_{X, {\bar{\alpha}}r{V}},
\epsilonnd{equation}
\begin{equation}\operatorname{l}abel{eq:82}
\widetilde{Q}_{v_1\cdots (X)_\mu\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots
{\bar{\alpha}}r{v}_{p-1}}=0=\widetilde{Q}_{v_1\cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots ({\bar{\alpha}}r{X})_{\nu}\cdots
v_{p-1}},
\epsilonnd{equation}
and for any $(1, 0)$-type vectors $X, X_i$,
\begin{eqnarray*}
&\, &\sum_{\mu, \nu=1}^{p-1}\widetilde{Q}_{v_1\cdots X_\mu \cdots v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu \cdots
{\bar{\alpha}}r{v}_{p-1}}+ \partialhi_{X v_1\cdots v_{p-1}, {\bar{\alpha}}r{X} {\bar{\alpha}}r{v}_1\cdots
{\bar{\alpha}}r{v}_{p-1}}\\
\quad \quad \quad &+& \sum_{\mu=1}^{p-1}\operatorname{div}'_{{\bar{\alpha}}r{X}}(\partialhi)_{v_1\cdots X_\mu\cdots
v_{p-1}, {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_{p-1}}+\partialhi_{V v_1\cdots X_\mu\cdots v_{p-1},
{\bar{\alpha}}r{X} {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{v}_{p-1}}\\
\quad \quad \quad &+& \sum_{\nu=1}^{p-1}\operatorname{div}''_{X}(\partialhi)_{v_1\cdots v_{p-1},
{\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu\cdots {\bar{\alpha}}r{v}_{p-1}}+\partialhi_{X v_1\cdots v_{p-1},
{\bar{\alpha}}r{V} {\bar{\alpha}}r{v}_1\cdots {\bar{\alpha}}r{X}_\nu \cdots {\bar{\alpha}}r{v}_{p-1}}\\
\quad \quad \quad &\gammae& 0.
\epsilonnd{eqnarray*}
This amounts to that the block matrix (as defined in Section 5) $\mathcal{M}_1\gammae 0$. However here $Q$ is replaced by $\widetilde{Q}$.
To check that $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \widetilde{Q}_{1\cdots p-1, {\bar{\alpha}}r{1}\cdots \overline{p-1}}\gammae 0$ we may
extend $V$ such that the following holds:
\begin{eqnarray*}
&\quad& (\frac{\partial}{\partial t}-\D)V^{i}=-\frac{1}{t}V^{i}
\\
&\quad& \nabla_{i}V=R^{j}_i\frac{\partial}{\partial z^j}+\frac{1}{t}\frac{\partial}{\partial z^i}, \quad \
\nabla_{{\bar{\alpha}}r{i}}V^{j}=0.
\epsilonnd{eqnarray*}
Using these set of equations, (\ref{eq:68}), (\ref{eq:69}) and
(\ref{eq:414}) can be simplified to
\begin{eqnarray}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}''_{V}(\partialhi))_{I_{p-1},
\overline{J}_{p-1}}=-\frac{1}{t}\operatorname{div}''_{ V}(\partialhi)
+\mathcal{KB}(\operatorname{div}''_{V}(\partialhi)) \operatorname{l}abel{eq:83}\\
\quad \quad &+& \sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V,\,{\bar{\alpha}}r{j_\nu}}(\operatorname{div}''(\partialhi))_{ki_1\cdots
i_{p-1},{\bar{\alpha}}r{j_1}\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}
\operatorname{div}''_{\operatorname{Ric}(V)}(\partialhi)
-\frac{1}{t} \operatorname{div}'( \operatorname{div}''(\partialhi)) \nonumber \\
\quad\quad &+&R_{j{\bar{\alpha}}r{k}}\nabla_k\partialhi_{V I_{p-1},
{\bar{\alpha}}r{j}\overline{J}_{p-1}}+\nabla_VR_{j{\bar{\alpha}}r{i}}\partialhi_{i I_{p-1},
{\bar{\alpha}}r{j}\overline{J}_{p-1}}+\sum_{\mu =1}^{p-1}\nabla_{i_\mu}R_{j{\bar{\alpha}}r{k}}
\partialhi_{V i_1\cdots(k)_\mu\cdots i_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}\nonumber\\
\quad\quad&-&
R_{l{\bar{\alpha}}r{k}}\nabla_{{\bar{\alpha}}r{l}}(\operatorname{div}''(\partialhi))_{k I_{p-1},
\overline{J}_{p-1}};\nonumber\\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) (\operatorname{div}'_{\overline{V}}(\partialhi))_{I_{p-1}, {\bar{\alpha}}r{J}_{p-1}}=
-\frac{1}{t}\operatorname{div}'_{\overline{V}}(\partialhi)+\mathcal{KB}(\operatorname{div}'_
{\overline{V}}(\partialhi)) \operatorname{l}abel{eq:84}\\
\quad \quad &
+& \sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu,\,{\bar{\alpha}}r{V}}(\operatorname{div}'(\partialhi))_{i_1\cdots(k)_\mu\cdots
i_{p-1},
{\bar{\alpha}}r{l}{\bar{\alpha}}r{j_1}\cdots{\bar{\alpha}}r{j}_{p-1}}-\frac{1}{2}\operatorname{div}'_{\overline{\operatorname{Ric}(V)
}}(\partialhi)
-\frac{1}{t}\operatorname{div}''( \operatorname{div}'(\partialhi)) \nonumber\\
\quad\quad&+&
R_{k{\bar{\alpha}}r{i}}\nabla_{{\bar{\alpha}}r{k}}\partialhi_{i I_{p-1}, \overline{V}\overline{J}_{p-1}}
+\nabla_{\overline{V}}R_{j{\bar{\alpha}}r{i}}\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+\sum_{\nu =1}^{p-1}\nabla_{{\bar{\alpha}}r{j}_\nu}R_{l{\bar{\alpha}}r{i}}
\partialhi_{i I_{p-1},
\overline{V}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_\nu\cdots{\bar{\alpha}}r{j}_{p-1}}\nonumber\\
\quad\quad&-&
R_{l{\bar{\alpha}}r{k}}\nabla_k(\operatorname{div}'(\partialhi))_{I_{p-1},
{\bar{\alpha}}r{l}\overline{J}_{p-1}};\nonumber\\
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \partialhi_{V,{\bar{\alpha}}r{V}}=\mathcal{KB}(\partialhi_{V,{\bar{\alpha}}r{V}})+ R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{V}} \partialhi_{k, {\bar{\alpha}}r{l}}+\sum_{\nu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{V\,{\bar{\alpha}}r{j_\nu}} \partialhi_{k I_{p-1}, {\bar{\alpha}}r{V} {\bar{\alpha}}r{j}_1\cdots
({\bar{\alpha}}r{l})_\nu\cdots {\bar{\alpha}}r{j}_{p-1}}\operatorname{l}abel{eq:85}\\
\quad \quad &+&\sum_{\mu =1 }^{p-1}R^{\,
k{\bar{\alpha}}r{l}}_{i_\mu\,{\bar{\alpha}}r{V}} \partialhi_{V i_1\cdots (k)_\mu\cdots i_{p-1},
{\bar{\alpha}}r{l}{\bar{\alpha}}r{J}_{p-1}}-\frac{1}{2}\operatorname{l}eft(\partialhi_{V, \overline{\operatorname{Ric}(V)}}+\partialhi_{\operatorname{Ric}(V),
{\bar{\alpha}}r{V}}\rightght) \nonumber\\
\quad \quad &-& \frac{2}{t}\partialhi_{ V, {\bar{\alpha}}r{V}}-\frac{\Lambda
\partialhi}{t^2}-\frac{2}{t}R_{j{\bar{\alpha}}r{i}}\partialhi_{i I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}-
\frac{1}{t}\operatorname{div}'_{\overline{V}}(\partialhi)
-\frac {1}{t}\operatorname{div}''_{V}(\partialhi)\nonumber \\
\quad\quad&-&
\operatorname{l}eft(R_{j{\bar{\alpha}}r{k}}\nabla_k\partialhi_{V I_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
+R_{k{\bar{\alpha}}r{i}}\nabla_{{\bar{\alpha}}r{k}}\partialhi_{i I_{p-1}, {\bar{\alpha}}r{V}\overline{J}_{p-1}}
\rightght)-R_{j{\bar{\alpha}}r{k}}R_{k{\bar{\alpha}}r{i}}\partialhi_{iI_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}.\nonumber
\epsilonnd{eqnarray}
Adding them up with the two evolution equations in Lemma \ref{helpKRF62} and
(\ref{eq:lem65}), (\ref{eq:610}), using (\ref{eq:81}) we have that
\begin{eqnarray*}
&\,&\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \widetilde{Q}_{I_{p-1}, \overline{J}_{p-1}}
=\mathcal{KB}(\widetilde{Q})_{I_{p-1}, \overline{J}_{p-1}}
+\sum_{i\, j=1}^m Z_{j{\bar{\alpha}}r{i}}\partialhi_{iI_{p-1}, {\bar{\alpha}}r{j}\overline{J}_{p-1}}
\\
&\quad&+
\sum_{\mu=1}^{p-1}(R_{i_\mu{\bar{\alpha}}r{V}l{\bar{\alpha}}r{k}}
+P_{l{\bar{\alpha}}r{k}i_\mu})(\operatorname{div}'(\partialhi)_{i_1\cdots(k)_{\mu}\cdots i_{p-1},
{\bar{\alpha}}r{l}\overline{J}_{p-1}}+\partialhi_{Vi_1\cdots(k)_{\mu}\cdots i_{p-1}, {\bar{\alpha}}r{l}\overline{J}_{p-1}})\\
&\quad&+\sum_{\nu=1}^{p-1}(
R_{V{\bar{\alpha}}r{j}_{\nu}l{\bar{\alpha}}r{k}}+P_{l{\bar{\alpha}}r{k}{\bar{\alpha}}r{j}_\nu})
(\operatorname{div}''(\partialhi)_{kI_{p-1}, {\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}}
+\partialhi_{kI_{p-1}, {\bar{\alpha}}r{V}{\bar{\alpha}}r{j}_1\cdots({\bar{\alpha}}r{l})_{\nu}\cdots{\bar{\alpha}}r{j}_{p-1}})\\
&\quad&-\frac{2}{t}\widetilde{Q}_{I_{p-1}\overline{J}_{p-1}}.
\epsilonnd{eqnarray*}
Now the nonnegativity of $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \widetilde{Q}_{1\cdots(p-1), {\bar{\alpha}}r{1}\cdots (\overline{p-1})}$ at
$(x_0, t_0)$ can be proved in a similar way as the argument in Section 2. First
observe that the part of $\mathcal{KB}(\widetilde{Q})_{1\cdots(p-1), {\bar{\alpha}}r{1}\cdots
(\overline{p-1})}$ involving only $\operatorname{Ric}$ is
$$
-\frac{1}{2}\sum_{i=1}^{p-1}\operatorname{l}eft(\widetilde{Q}_{1\cdots \operatorname{Ric}(i)\cdots (p-1), {\bar{\alpha}}r{1}\cdots
(\overline{p-1})}+\widetilde{Q}_{1\cdots (p-1), {\bar{\alpha}}r{1}\cdots\overline{\operatorname{Ric}(i)}\cdots (\overline{p-1})}\rightght)
$$
which vanished due to (\ref{eq:82}).
Hence we only need to establish the nonnegativity of
\begin{eqnarray*}\tilde{J}&\doteqdot &
\sum_{\mu = 1}^{p-1}\sum_{\nu =1
}^{p-1}R_{\mu{\bar{\alpha}}r{\nu}l{\bar{\alpha}}r{k}}\widetilde{Q}_{1\cdots(k)_\mu\cdots
(p-1), {\bar{\alpha}}r{1}\cdots({\bar{\alpha}}r{l})_\nu\cdots(\overline{p-1})}\\
\quad \quad &+&
\sum_{\mu=1}^{p-1}(R_{\mu{\bar{\alpha}}r{V}l{\bar{\alpha}}r{k}}
+P_{l{\bar{\alpha}}r{k}\mu})(\operatorname{div}'(\partialhi)_{1\cdots(k)_{\mu}\cdots (p-1),
{\bar{\alpha}}r{l}\overline{J}_{p-1}}+\partialhi_{V1\cdots(k)_{\mu}\cdots (p-1), {\bar{\alpha}}r{l}\overline{J}_{p-1}})\\
\quad\quad
&+&\sum_{\nu=1}^{p-1}(
R_{V{\bar{\alpha}}r{\nu}l{\bar{\alpha}}r{k}}+P_{l{\bar{\alpha}}r{k}{\bar{\alpha}}r{\nu}})
(\operatorname{div}''(\partialhi)_{kI_{p-1}, {\bar{\alpha}}r{1}\cdots({\bar{\alpha}}r{l})_{\nu}\cdots\overline{p-1}}
+\partialhi_{kI_{p-1}, {\bar{\alpha}}r{V}{\bar{\alpha}}r{1}\cdots({\bar{\alpha}}r{l})_{\nu}\cdots\overline{p-1}})\\
\quad\quad &+& \sum_{i, j=1}^m Z_{j{\bar{\alpha}}r{i}}\partialhi_{i1\cdots (p-1), {\bar{\alpha}}r{j}{\bar{\alpha}}r{1}\cdots
(\overline{p-1})}.
\epsilonnd{eqnarray*}
By Theorem \ref{LYH}, the assumption that the curvature operator $\Bbb Rm$ is in $\mathcal{C}_p$ implies that
the matrix
$$
\mathcal{M}_3=\operatorname{l}eft(\begin{array}{l}
R_{1{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
R_{1{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad\quad
\cdots\quad\quad
R_{1\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}} \quad \quad \quad
D^1_{(\cdot){\bar{\alpha}}r{(\cdot)}}\\
R_{2{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad
R_{2{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad\quad
\cdots\quad\quad
R_{2\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad \quad
D^2_{(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\quad\cdots\quad\quad\quad\quad \cdots\quad\quad\quad\quad
\cdots\quad\quad\quad\cdots\quad \quad \quad \quad \quad \quad \cdots \\
R_{p-1{\bar{\alpha}}r{1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad
R_{p-1{\bar{\alpha}}r{2}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad\quad
\cdots\quad\quad
R_{p-1\overline{p-1}(\cdot){\bar{\alpha}}r{(\cdot)}}\quad \quad
D^{p-1}_{(\cdot){\bar{\alpha}}r{(\cdot)}}\\
\overline{D^1}^{tr}_{(\cdot){\bar{\alpha}}r{(\cdot)}}
\quad\qquad
\overline{D^2}^{tr}_{(\cdot){\bar{\alpha}}r{(\cdot)}} \quad\quad\,\,
\cdots\quad\qquad
\overline{D^{p-1}}^{tr}_{(\cdot){\bar{\alpha}}r{(\cdot)}}
\quad \quad \quad Z_{(\cdot){\bar{\alpha}}r{(\cdot)}}
\epsilonnd{array}\rightght)\gammae 0,
$$
where $D^i_{\mu{\bar{\alpha}}r{\nu}}=R_{i{\bar{\alpha}}r{V}\mu{\bar{\alpha}}r{\nu}}
+P_{\mu{\bar{\alpha}}r{\nu}i}$.
The nonnegativity of $\tilde{J}$ follows from
$\operatorname{trace}(\mathcal{M}_1 \cdot
\mathcal{M}_3)\gammae 0$. Here $\mathcal{M}_1$ is the block matrix in Section 5. This proves Theorem \ref{main-KRFlyh} for the case that $M$ is compact. We postpone the proof of the noncompact case to a later section.
\section{LYH type estimates for the Ricci Flow under the condition $\widetilde{\mathcal{C}}_p$}
In this section we prove another set of LYH type estimates for the Ricci flow. Let $(M, g(t))$ be a complete solution to
\begin{equation}\operatorname{l}abel{rfeq}
\frac{\partialartial}{\partialartial t} g_{ij}=-2R_{ij}.
\epsilonnd{equation}
Recall that Hamilton proved that if $\Bbb Rm\gammae 0$ and bounded then the quadratic form
$$
\widetilde{\mathcal{Q}}(W\oplus U)\doteqdot \operatorname{l}angle \mathcal{M}(W), W\rangle +2\operatorname{l}angle P(W), U\rangle +\operatorname{l}angle \Bbb Rm(U), U\rangle\gammae 0
$$
where the $\mathcal{M}$ and $P$ are defined in a normal frame by
\begin{eqnarray*}
\mathcal{M}_{ij}&\doteqdot&\Delta R_{ij}-\frac{1}{2}\nabla_i\nabla_j R +2R_{ikjl}R_{kl} -R_{ik}R_{jk}+\frac{1}{2t}R_{ij},\\
P_{ijk}&\doteqdot& \nabla_i R_{jk}-\nabla_j R_{ik}
\epsilonnd{eqnarray*}
with $\operatorname{l}angle P(W), U\rangle =P_{ijk} W^k U^{ij}$.
One can view $\text{\rm Hess}Q$ as the restriction of a Hermitian quadratic form
$$
\operatorname{l}angle \widetilde{\mathcal{Q}}(W\oplus U), \overline{W\oplus U}\rangle \doteqdot \operatorname{l}angle \mathcal{M}(W), \overline{W}\rangle +2Re\operatorname{l}angle P(W), \overline{U}\rangle +\operatorname{l}angle \Bbb Rm(U), \overline{U}\rangle
$$
which is defined on $\wedge^2(\Bbb C^n)\oplus \Bbb C^n$. We also denote by
$$
\operatorname{l}angle \mathcal{Z}(W\wedge Z), \overline{W\wedge Z}\rangle\doteqdot \text{\rm Hess}Q (W\oplus (W\wedge Z)).
$$
Fixing a $Z$, $\mathcal{Z}$ can be viewed as a Hermitian bilinear form of $W$, which we denote by $\mathcal{Z}_Z$, or still by $\mathcal{Z}$ when the meaning is clear.
In terms of local frame, it can be written as
$$
\operatorname{l}eft(\mathcal{Z}_Z\rightght)_{cd}=\mathcal{M}_{cd}+P_{dac}\overline{Z}^a +P_{cad}Z^a +R_{Zc \overline{Z} d}.
$$
\begin{theorem}\operatorname{l}abel{rf-lyh} Assume that $(M, g(t))$ on $M \times [0, T]$ satisfies $\widetilde{\mathcal{C}}_p$. When $M$ is noncompact we also assume that the curvature of $(M, g(t))$ is uniformly bounded on $M \times [0, T]$. Then for any $t>0$,
$\text{\rm Hess}Q \gammae 0$ for any $(x, t)\sqrt {-1}n M\times [0, T]$, $W\sqrt {-1}n T_x M \otimes \Bbb C$ and $U\sqrt {-1}n \wedge^2(T_xM\otimes \Bbb C) $ such that $U=\sum_{\mu=1}^p W_\mu \wedge Z_\mu$ with $W_p=W$. Furthermore, the equality holds for some $t>0$ implies that the universal cover of $(M, g(t))$ is a gradient expanding Ricci soliton.
\epsilonnd{theorem}
\begin{remark}
In \cite{brendle}, a slightly weaker result was proved for the $p=1$ case. As before, for $p$ large enough the condition $\widetilde{\mathcal{C}}_p$ is equivalent to that $\Bbb Rm\gammae 0$ and the above result is equivalent to Hamilton's theorem. Hence our result gives a family of estimates interpolating between those of \cite{brendle} and \cite{richard-harnack}.
\epsilonnd{remark}
In Theorem 4.1 of \cite{richard-harnack}, the following result was proved by brutal force computations.
\begin{lemma}\operatorname{l}abel{ham-com} At $(x_0, t_0)$, if $W$ and $U$ are extended by the equations:
\begin{eqnarray*}
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) W&=&\frac{W}{t}+\operatorname{Ric}(W), \quad \quad \nabla W=0;\\
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) U^{ab}&=&R^a_c U^{cb}+R^{b}_cU^{ac},\\
\nabla_a U^{bc}&=&\frac{1}{2}(R_{a}^{b} W^c -R_{a}^{c}W^b)+\frac{1}{4t}(g_{a}^bW^c-g_{a}^cW^b),
\epsilonnd{eqnarray*}
then under an orthonormal frame
\begin{eqnarray}\operatorname{l}abel{hamilton-help1}
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \text{\rm Hess}Q &=& 2R_{acbd}\mathcal{M}_{cd}W^a \overline{W}^b -2P_{acd}P_{bdc}W^a\overline{W}^b
\\&\quad&+ 8Re\operatorname{l}eft( R_{adce} P_{dbe} W^c \overline{U}^{ab}\rightght)+4R_{aecf}R_{bedf}U^{ab}\overline{U}^{cd} \nonumber\\
&\quad& +|P(W)+\Bbb Rm(U)|^2.\nonumber
\epsilonnd{eqnarray}
Here $R^a_b$ denotes the $\operatorname{Ric}$ transformation in terms of the local frame.
\epsilonnd{lemma}
Using the notation of \cite{H86}, the term $4R_{aecf}R_{bedf}U^{ab}\overline{U}^{cd}$ can be expressed as $8\operatorname{l}angle \Bbb Rm^{\#} (U), \overline{U}\rangle$.
Assume that $\text{\rm Hess}Q \gammae 0$ for $M\times[0, t_0]$ and at $(x_0, t_0)$ it vanished for $W\oplus U$, with $U=\sum_{\mu=1}^p W_\mu \wedge Z_{\mu}$, and $W_p=W$. Now let $W_\mu(z)$ and $Z_\mu(z)$ be a variation of $W_\mu$ and $Z_\mu$ with $W_\mu(z)=W_\mu+z X_\mu$ and $Z_\mu(z)=Z_\mu+zY_\mu$. Let $\widetilde{\mathcal{I}}(z)\doteqdot \text{\rm Hess}Q (W(z)\oplus U(z))$ with $U(z)=\sum_{\mu=1}^p W_\mu(z)\wedge Z_\mu(z)$. Using $\Delta \widetilde{\mathcal{I}}(0)\gammae 0$, we deduce the following estimate:
\begin{eqnarray}
&\, &\sum_{\mu, \nu=1}^p
R_{X_\mu Z_\mu \overline{X}_\nu \overline{Z}_\nu}
+2Re \operatorname{l}eft(\operatorname{l}angle P(X_p), \overline{\sum_{\mu=1}^p X_\mu \wedge Z_\mu}\rangle\rightght) +\operatorname{l}angle \mathcal{M}(X_p), \overline{X}_{p}\rangle
\operatorname{l}abel{eq:91}\\
\quad
&+&
2Re \operatorname{l}eft(\sum_{\mu, \nu=1}^p R_{X_\mu Z_\mu \overline{W}_\nu \overline{Y}_\nu}\rightght)+
2Re \operatorname{l}eft(\operatorname{l}angle P(X_p), \overline{\sum_{\mu=1}^p W_\mu \wedge Y_\mu}\rangle \rightght)\nonumber\\
\quad
&+&\sum_{\mu, \nu=1}^p
R_{W_{\mu}Y_\mu \overline{W}_\nu \overline{Y}_\nu}\gammae 0.\nonumber
\epsilonnd{eqnarray}
To prove Theorem \ref{rf-lyh} for the compact case, it suffices to show that the right hand side of (\ref{hamilton-help1}) is nonnegative for a null vector $W\oplus U$ with $U=\sum_{\mu=1}^p W_\mu \wedge Z_\mu$ and $W_p=W$. Denote the first four terms in the right hand side of (\ref{hamilton-help1}) by $\text{\rm Hess}J$. Expand it and let $\hat{P}_{dc}(Z_p)=P_{dac}Z^a_p$. We then obtain that
\begin{eqnarray*}
\text{\rm Hess}J &=& 2 R_{W_p c \overline{W}_p d} \mathcal{Z}_{cd}
-2R_{W_p c\overline{W}_pd}\operatorname{l}eft(\hat{P}_{dc}(\overline{Z}_p)+\hat{P}_{cd}(Z_p)\rightght)-2R_{W_p c\overline{W}_p d}
R_{Z_pc\overline{Z}_p d}\\
&\quad& + 2\sum_{\mu, \nu=1}^p\operatorname{l}eft(R_{W_\mu e \overline{W}_\nu f}R_{Z_\mu e \overline{Z}_\nu f}-R_{W_\mu e \overline{Z}_\nu f}R_{Z_\mu e\overline{W}_\nu f}\rightght)\\
&\quad& +4Re \operatorname{l}eft(\sum_{\mu=1}^{p-1}R_{\overline{W}_\mu d W_p e}\hat{P}_{de}(\overline{Z}_\mu)\rightght)+4Re\operatorname{l}eft(R_{\overline{W}_p d W_pe}\hat{P}_{de}(\overline{Z}_p)\rightght)\\
&\quad& -4Re\operatorname{l}eft( \sum_{\mu=1}^{p-1}R_{\overline{Z}_\mu d W_p e}\hat{P}_{de}(\overline{W}_\mu)\rightght)-4Re\operatorname{l}eft(R_{\overline{Z}_p d W_pe}\hat{P}_{de}(\overline{W}_p)\rightght)\\
&\quad& -2\hat{P}_{de}(W_p)\hat{P}_{ed}(\overline{W}_p).
\epsilonnd{eqnarray*}
After some cancelations (the 2nd term and the 7th term on the right hand side above cancel each other) the nonnegativity of $\text{\rm Hess}J=2\operatorname{l}eft(\operatorname{trace}(B_1 B_2)-\operatorname{trace}(B_3\cdot \overline{B}_3)\rightght)$
where
\begin{eqnarray*}
B_1=\operatorname{l}eft(\begin{array}{l}
R_{W_1(\cdot)\overline{W}_1(\cdot)}\qquad
R_{W_1(\cdot)\overline{W}_2(\cdot)}\quad\quad
\cdots\quad\quad
R_{W_1(\cdot)\overline{W}_{p}(\cdot)}
\\
R_{W_2(\cdot)\overline{W}_1(\cdot)}\qquad
R_{W_2(\cdot)\overline{W}_2(\cdot)}\quad\quad
\cdots\quad\quad
R_{W_2(\cdot)\overline{W}_{p}(\cdot)}\\
\quad\quad\cdots\quad\quad\quad\quad\cdots\qquad\qquad\quad\cdots\quad\quad
\quad\cdots\\
R_{W_p(\cdot)\overline{W}_1(\cdot)}\qquad
R_{W_p(\cdot)\overline{W}_2(\cdot)}\quad\quad
\cdots\quad\quad
R_{W_p(\cdot)\overline{W}_p(\cdot)}
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
\begin{eqnarray*}
B_2=\operatorname{l}eft(\begin{array}{l}
R_{\overline{Z}_1 (\cdot) Z_1(\cdot)}\quad\quad\quad
R_{\overline{Z}_1(\cdot) Z_2(\cdot)}\quad\quad\quad
\cdots\quad
R_{\overline{Z}_1(\cdot)Z_{p-1}(\cdot)}\quad \quad
F^{1p}\\
R_{\overline{Z}_2 (\cdot) Z_1(\cdot)}\quad\quad\quad
R_{\overline{Z}_2(\cdot) Z_2(\cdot)}\quad\quad\quad
\cdots\quad
R_{\overline{Z}_2(\cdot)Z_{p-1}(\cdot)}\quad\quad
F^{2p}\\
\quad\quad\cdots\qquad\qquad\quad \cdots\qquad\qquad\quad \cdots\quad\quad\quad
\quad\cdots\quad\quad\quad\cdots\\
R_{\overline{Z}_{p-1} (\cdot) Z_1(\cdot)}\quad\quad
R_{\overline{Z}_{p-1}(\cdot) Z_2(\cdot)}\quad\quad
\cdots\quad
R_{\overline{Z}_{p-1}(\cdot)Z_{p-1}(\cdot)}\quad F^{p-1 p}
\\
\quad \quad \quad \overline{F^{1p}}^{tr}\qquad\quad\quad
\overline{F^{2p}}^{tr}\quad\quad\quad \cdots\quad\quad\quad
\overline{F^{(p-1)p}}^{tr}\quad \quad
\mathcal{Z}_{(\cdot)(\cdot)}
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
and
\begin{eqnarray*}
B_3=\operatorname{l}eft(\begin{array}{l}
R_{W_1(\cdot)\overline{Z}_1(\cdot)}\quad
R_{W_1(\cdot)\overline{Z}_2(\cdot)}\quad\quad\cdots\quad
R_{W_1(\cdot)\overline{Z}_{p-1}(\cdot)}\quad
R_{W_1(\cdot)\overline{Z}_p(\cdot)}+\hat{P}(W_1)\\
R_{W_2(\cdot)\overline{Z}_1(\cdot)}\quad
R_{W_2(\cdot)\overline{Z}_2(\cdot)}\quad\quad\cdots\quad
R_{W_2(\cdot)\overline{Z}_{p-1}(\cdot)}\quad
R_{W_2(\cdot)\overline{Z}_p(\cdot)}+\hat{P}(W_2)\\
\quad\quad\cdots\quad\quad\qquad\cdots\qquad\quad\ \cdots\quad
\quad\quad\cdots\qquad\qquad\qquad\cdots\\
R_{W_p(\cdot)\overline{Z}_1(\cdot)}\quad
R_{W_p(\cdot)\overline{Z}_2(\cdot)}\quad\quad\cdots\quad
R_{W_p(\cdot)\overline{Z}_{p-1}(\cdot)}\quad
R_{W_p(\cdot)\overline{Z}_p(\cdot)}+\hat{P}(W_p)
\epsilonnd{array}\rightght),
\epsilonnd{eqnarray*}
with $F^{\mu p}=R_{\overline{Z}_{\mu}(\cdot)Z_{p}(\cdot)}+\hat{P}_{(\cdot) (\cdot)}(\overline{Z}_\mu)$.
On the other hand using the above notation (\ref{eq:91}) can be re-written as
\begin{eqnarray*}
&\,&\mathcal{Z}_{X_p \overline{X}_p}+2\sum_{\mu=1}^{p-1}Re \operatorname{l}eft( R_{\overline{Z}_\mu \overline{X}_\mu Z_p X_p}\rightght)+2\sum_{\mu=1}^{p-1}Re \operatorname{l}eft( \hat{P}_{\overline{X}_\mu X_p}(\overline{Z}_\mu)\rightght)+\sum_{\mu, \nu =1}^{p-1}R_{\overline{Z}_\mu \overline{X}_\mu Z_\nu X_\nu}\\
&\, & -2\sum_{\mu=1}^{p}Re\operatorname{l}eft( \hat{P}_{\overline{Y}_\mu X_p}(\overline{W}_\mu)+\sum_{\nu=1}^p R_{\overline{W}_\mu \overline{Y}_\mu Z_{\nu} X_\nu}\rightght)+\sum_{\mu, \nu=1}^p R_{W_\mu Y_\mu \overline{W}_\nu \overline{Y}_\nu}\\
&\,& \gammae 0
\epsilonnd{eqnarray*}
which amounts to $S(\mathcal{X}, \mathcal{Y})$
being nonnegative, where
$$
S(\mathcal{X}, \mathcal{Y})=(B_2)_{ij}\mathcal{X}^i\overline{\mathcal{X}^j}-2 Re \operatorname{l}eft( (B_3)_{ij}\mathcal{Y}^i \overline{\mathcal{X}^j}\rightght)+(B_1)_{ij}\mathcal{Y}^i\overline{\mathcal{Y}^j}
$$
with $\mathcal{X}=\operatorname{l}eft(\begin{array}{l} X_1\\ \vdots\\
X_p\epsilonnd{array}\rightght),\, \mathcal{Y}=\operatorname{l}eft(\begin{array}{l} Y_1\\ \vdots\\
Y_p\epsilonnd{array}\rightght)$. Hence by Lemma \ref{lmlyh} we can conclude that $\text{\rm Hess}J\gammae 0$ and complete the proof of Theorem \ref{rf-lyh} for the compact case. The case that $M$ is noncompact will be proved in the next section together with Theorem \ref{LYH}.
\section{Complete noncompact manifolds with bounded curvature}
In this section we first show that under the condition that the curvature tensor of $(M, g(t))$, a solution to the Ricci flow or K\"ahler-Ricci flow, is uniformly bounded on $M \times [0, T]$, the maximum principle can still apply and conclude the invariance of the cone $\mathcal{C}_p$, $\widetilde{\mathcal{C}}_p$ from Section 3. Moreover, the LYH type estimates for the K\"ahler-Ricci flow and the Ricci flow in Section 7 and 9 remain valid.
First we show the invariance of the $\mathcal{C}_p$ and $\widetilde{\mathcal{C}}_p$. In fact the following maximum principle holds on noncompact manifolds. Consider $V$, a vector bundle over $M$, with a fixed metric $\widetilde{h}$, a time-dependent metric connection
$D^{(t)}$. On $M$ there are time-dependent metrics $g(t)$ and $\nabla^{(t)}$, the Levi-Civita connection of $g(t)$. When the meaning is clear we often omit the sup-script $^{(t)}$.
The main concern of this subsection is the diffusion-reaction equation:
\begin{eqnarray}\operatorname{l}abel{pde-ode}
\operatorname{l}eft\{\begin{array}{ll}
\quad &\frac{\partialartial }{\partialartial t} f(x, t)-\Delta f(x, t) =\Phi(f)(x, t),\\
\quad& f(x,0 )=f_0(x).\epsilonnd{array}
\rightght.
\epsilonnd{eqnarray}
Here $\Delta =g^{ij}(x, t)D_iD_j$. We know that after applying the Ulenbeck's trick \cite{H86} the study of the curvature operator under the Ricci flow equation is a subcase of this general formulation. One can modify the proof of Theorem 1.1 in \cite{B-W} to obtain the following result.
\begin{theorem} Assume that $M$ is a complete noncompact manifold and $\Phi$ is locally Lipschitz. Let $(M, g(t))$ be a solution to Ricci flow such that $|\Bbb Rm|(x, t)\operatorname{l}e A$ for some $A>0$ for any $(x, t)\sqrt {-1}n M\times[0, T]$. Let $C(t)\subset V$, $t\sqrt {-1}n [0, T]$, be a family of closed full dimensional cones, depending continuously on $t$. Suppose that each of the cones $C(t)$ is invariant under parallel transport, fiberwise convex and that the family $\{C(t)\}$ is preserved by the ODE $\frac{d}{dt} f(t)=\Phi(f)$. Moreover assume that there exists a smooth section $I$ which is invariant under the parallel transport and $I\sqrt {-1}n C(t)$ for all $t$. If $f(x, t)$ satisfies (\ref{pde-ode}) with $f(x, 0)\sqrt {-1}n C(0)$, $|f|(x, t)\operatorname{l}e B$ on $M\times[0, T]$ for some $B>0$, then $f(x, t)\sqrt {-1}n C(t)$ for $(x, t)\sqrt {-1}n M \times [0, T]$ .
\epsilonnd{theorem}
\begin{proof} The key is Lemma \ref{lemma101} below, which ensures the existence of a smooth function $\varphi$ such that $\varphi(x, t)\to +\sqrt {-1}nfty$ uniformly on $[0, \epsilonta]$ for some $\epsilonta>0$ and $\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \varphi \gammae C\varphi.$ Clearly once we can prove the result for $[0, \epsilonta]$ we can iterate the procedure and get the result on $[0, T]$.
For any ${\varepsilon}ilon>0$, we can fix a compact region $K$ such that $\tilde{f}(x, t)\doteqdot f(x, t)+{\varepsilon}ilon\varphi \operatorname{I}}\newcommand{\scal}{{\mathrm{scal}}\sqrt {-1}n C(t)$ for all $(x, t)$ with $x\sqrt {-1}n M\setminus K$. In fact one can choose $K=\overlineerline{B^0(p, R_0)}$, a closed ball of a certain radius $R_0$ with respect to the initial metric. Now for every $t$, $\rho(x, t)=\operatorname{dist}^2(\tilde f(x, t), C_x(t))$ with $C_x(t)=C(t)\cap V_x$ achieves a maximum somewhere. The argument of \cite{B-W} can be applied and we only need to restrict ourselves over $K\times [0, \epsilonta]$. In particular we let $\rho(t)=\rho(x_0, t)=\max \rho(\cdot, t)$. Since $\Phi$ is locally Lipschitz it is easy to infer that there exists $A'$ such that $|\tilde{f}|+|\Phi(\tilde{f})|\operatorname{l}e A'$ for some constant $A'$, on $K\times[0, T]$. Since $\varphi>0$, we can choose $C$ large enough such that ${\varepsilon}ilon C \varphi \operatorname{I}}\newcommand{\scal}{{\mathrm{scal}} +\Phi(f)-\Phi(\tilde{f}) \sqrt {-1}n C(t)$ for all $(x, t)\sqrt {-1}n K\times [0, \epsilonta]$. Now the rest of the argument in \cite{B-W} can be evoked to conclude that $D_{-} \rho(t)\operatorname{l}e L\rho(t)$ with $L$ depending on the local Lipschitz constant of $\Phi$. Here $D_{-}$ is the lower Dini's derivative from the left. Precisely we have
\begin{eqnarray*}
D_{-}\rho(t)&\operatorname{l}e & \operatorname{l}angle \frac{\partialartial }{\partialartial t}\tilde{f}, \tilde{f}-v_\sqrt {-1}nfty\rangle|_{(x_0, t)}-2\operatorname{l}angle \Phi(v_\sqrt {-1}nfty), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle\\
&=& 2\operatorname{l}angle (\Delta \tilde{f})(x_0, t), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle\\
&\quad& +2\operatorname{l}angle \Phi(f)+{\varepsilon}ilon \operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \varphi \operatorname{I}}\newcommand{\scal}{{\mathrm{scal}}|_{(x_0, t)} -\Phi(v_\sqrt {-1}nfty), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle.
\epsilonnd{eqnarray*}
Here $v_\sqrt {-1}nfty$ is a vector in $V_{x_0}$ such that $\operatorname{dist}(\tilde{f}(x_0, t), v_\sqrt {-1}nfty)=\operatorname{dist}(\tilde{f}(x_0, t), C_{x_0}(t)).$ By Lemma 1.2 of \cite{B-W}
$$
\operatorname{l}angle (\Delta \tilde{f})(x_0, t), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle\operatorname{l}e 0.
$$
For sufficient large $C$,
${\varepsilon}ilon C \varphi \operatorname{I}}\newcommand{\scal}{{\mathrm{scal}} +\Phi(f)-\Phi(\tilde{f}) \sqrt {-1}n C(t)$, which implies that
$\operatorname{l}angle {\varepsilon}ilon C \varphi \operatorname{I}}\newcommand{\scal}{{\mathrm{scal}} +\Phi(f)-\Phi(\tilde{f}), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle\operatorname{l}e 0$. Hence by the convexity of $C(t)$,
$$
\operatorname{l}angle \Phi(f)+{\varepsilon}ilon \operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \varphi \operatorname{I}}\newcommand{\scal}{{\mathrm{scal}}|_{(x_0, t)} +\Phi(\tilde{f}), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle\operatorname{l}e 0.
$$
Combining the above we conclude that
$$
D_{-}\rho(t)\operatorname{l}e 2\operatorname{l}angle \Phi(\tilde{f}(x_0, t))-\Phi(v_\sqrt {-1}nfty), \tilde{f}(x_0, t)-v_\sqrt {-1}nfty\rangle \operatorname{l}e L\rho(t).
$$
The rest of the proof follows from \cite{B-W} verbatim.
\epsilonnd{proof}
\begin{lemma}\operatorname{l}abel{lemma101} Assume that $M$ is a complete noncompact manifold. Let $(M, g(t))$ be a solution to Ricci flow such that $|\Bbb Rm|(x, t)\operatorname{l}e A$ for some $A>0$ for any $(x, t)\sqrt {-1}n M\times[0, T]$. Then there exist $C_1>0$ and a positive function $\varphi(x, t)$ such that for any given $C>0$, there exists $\epsilonta>0$ such that on $M\times[0, \epsilonta]$
\begin{eqnarray*}
\epsilonxp(C_1^{-1}(r_0(x)+1))&\operatorname{l}e& \varphi(x, t)\operatorname{l}e \epsilonxp(C_1(r_0(x)+1)+1), \\
\operatorname{l}eft(\frac{\partial}{\partial t}-\Delta\right) \varphi &\gammae & C\varphi.
\epsilonnd{eqnarray*}
Here $r_0(x)$ is the distance to a fixed point with respect to the initial metric.
\epsilonnd{lemma}
\begin{proof} First by Lemma 5.1 of \cite{richard-harnack}, there exist $f(x)$ such that
\begin{eqnarray*}
C_1^{-1} (1+r_0(x)) &\operatorname{l}e& f(x)\operatorname{l}e C_1(1+r_0(x)),\\
|\nabla f|^2+|\nabla^2 f|&\operatorname{l}e& C_1.
\epsilonnd{eqnarray*}
Now we let $\varphi=\epsilonxp(\alphalpha t+f(x))$. The claimed result follows easily.
\epsilonnd{proof}
\begin{corollary} Let $(M, g(t))$ be a solution to Ricci flow (or K\"ahler-Ricci flow) such that $|\Bbb Rm|(x, t)\operatorname{l}e A$ for some $A>0$ for any $(x, t)\sqrt {-1}n M\times[0, T]$. Then $\widetilde{\mathcal{C}}_p$ is invariant under the Ricci flow. (Respectively, $\mathcal{C}_p$ is invariant under the K\"ahler-Ricci flow.)
\epsilonnd{corollary}
Concerning the LYH type estimates for the Ricci flow and K\"ahler-Ricci flow, we can evoke the perturbation argument of Hamilton \cite{richard-harnack}. Note that by passing to $[{\varepsilon}ilon, T-{\varepsilon}ilon]$, the curvature
bound, due to Shi's derivative estimates, implies that all the derivatives of the curvature are uniformly bounded. Now consider the perturbed quantity
$$
\text{\rm Hess}Q'(W\oplus U)=\operatorname{l}angle \mathcal{M}(W), \overline{W}\rangle+\frac{\varphi}{t} \operatorname{l}angle W, \overline{W}\rangle +2Re \operatorname{l}eft( P(W), \overline{U}\rangle \rightght)+\operatorname{l}angle \Bbb Rm(U),\overline{U}\rangle +\partialsi |U|^2
$$
where $\varphi$ and $\partialsi$ are the functions from Lemma 5.2 of \cite{richard-harnack}. Following the argument of Section 5 in \cite{richard-harnack} verbatim we can show the following result.
\begin{corollary} Assume that $(M, g(t))$ a solution to the Ricci flow on $M \times [0, T]$ such that $|\Bbb Rm|(x, t)\operatorname{l}e A$. Assume that $ \Bbb Rm(g(x, 0))\sqrt {-1}n \widetilde{\mathcal{C}}_p$. Then for any $t>0$,
$\text{\rm Hess}Q \gammae 0$ for any $(x, t)\sqrt {-1}n M\times [0, T]$, $W\sqrt {-1}n T_x M \otimes \Bbb C$ and $U\sqrt {-1}n \wedge^2(T_xM\otimes \Bbb C) $ such that $U=\sum_{\mu=1}^p W_\mu \wedge Z_\mu$ with $W_p=W$. (Respectively, if $(M, g(t))$ is a solution to the K\"ahler-Ricci flow with $\Bbb Rm(g(x, 0))\sqrt {-1}n \mathcal{C}_p$, then $\mathcal{Q}\gammae 0$).
\epsilonnd{corollary}
\section{Complete noncompact manifolds without curvature bound}
We first discuss the existence of the Cauchy problem for (\ref{eq:11}). First we observe that the maximum principle of Section 2 holds for the Dirichlet boundary problem by a perturbation argument adding ${\varepsilon}ilon \omega^p$. Precisely we have the following proposition.
\begin{proposition}
Let $(M, g)$ be a K\"ahler manifold whose curvature operator $\Bbb Rm \sqrt {-1}n \mathcal{C}_p$. Let $\Omega$ be a bounded domain in $M$. Assume that $\partialhi(x, t)$ satisfies that
\begin{eqnarray}\operatorname{l}abel{boundary1}
\operatorname{l}eft\{\begin{array}{ll}
\quad &\frac{\partialartial }{\partialartial t} \partialhi(x, t)+\Delta_{{\bar{\alpha}}r{\partialartial}} \partialhi(x, t) =0,\\
\quad &\partialhi(x, t)|_{\partialartial \Omega}\gammae 0, \\
\quad& \partialhi(x,0 )=\partialhi_0(x)\gammae 0.\epsilonnd{array}
\rightght.
\epsilonnd{eqnarray}
Then $\partialhi(x, t)\gammae 0$ for $t>0$.
\epsilonnd{proposition}
Note that the boundary condition $\partialhi(x, t)|_{\partialartial \Omega}\gammae 0$ means that at any $x\sqrt {-1}n \partialartial \Omega$ and for any $v_1, \cdots, v_p$, $\partialhi(v_1, \cdots v_p; {\bar{\alpha}}r{v}_1, \cdots, {\bar{\alpha}}r{v}_p)\gammae 0$ at $x$. Namely $\partialhi(x, t)|_{\partialartial \Omega}$ does not mean the restriction of the $(p, p)$-form to the boundary.
This will help us to obtain needed estimate to obtain a global solution in the case that $M$ is a noncompact complete manifold via some a priori estimates. A basic assumption is needed to ensure even the short time existence of the Cauchy problem on open manifolds. Here we assume that there exists a positive constant $a$ such that
\begin{equation}\operatorname{l}abel{ass-1}
\mathcal{B}\doteqdot\sqrt {-1}nt_M |\partialhi_0(y)|\epsilonxp(-ar^2(y))\, d\mu(y)< \sqrt {-1}nfty,
\epsilonnd{equation}
where $r(x)$ is the distance function to some fixed point $o\sqrt {-1}n M$. The pointwise norm $|\cdot|$ for $\partialhi$ is defined as $$|\partialhi|^2=\frac{1}{(p!)^2}\sum \partialhi_{I_p, \overline{J}_p}\overline{\partialhi_{K_p, \overline{L}_p}} g^{i_1 {\bar{\alpha}}r{k}_1}\cdots g^{i_p {\bar{\alpha}}r{k}_p} g^{l_1 {\bar{\alpha}}r{j}_1}\cdots g^{l_p{\bar{\alpha}}r{j}_p}.$$
By basic linear algebra, for example, Lemma 2.4 of \cite{Siu-74}, it is easy to see that for
positive $(p, p)$-forms, there exists $C_{p, m}$ such that
\begin{equation}\operatorname{l}abel{basic-1}
|\partialhi|(x)\operatorname{l}e C_{p, m} |\Lambda \partialhi|(x).
\epsilonnd{equation}
Now the existence of the solution to the Cauchy problem can be proved for any continuous positive $(p, p)$-form $\partialhi_0(x)$ satisfying (\ref{ass-1}).
\begin{proposition}\operatorname{l}abel{short-ext}
Let $(M, g)$ be a K\"ahler manifold whose curvature operator $\Bbb Rm \sqrt {-1}n \mathcal{C}_p$. Assume that $\partialhi_0(x)$ satisfies (\ref{ass-1}), then there exists $T_0$ such that the Cauchy problem
\begin{eqnarray}\operatorname{l}abel{cauchy1}
\operatorname{l}eft\{\begin{array}{ll}
\quad &\frac{\partialartial }{\partialartial t} \partialhi(x, t)+\Delta_{{\bar{\alpha}}r{\partialartial}} \partialhi(x, t) =0,\\
\quad& \partialhi(x,0 )=\partialhi_0(x)\gammae 0\epsilonnd{array}
\rightght.
\epsilonnd{eqnarray}
has a solution $\partialhi(x, t)$ on $M\times [0, T_0]$. Moreover, $\partialhi(x, t)\gammae 0$ on $M \times [0, T_0]$ and satisfies the estimate
\begin{equation}\operatorname{l}abel{con-sol}
|\partialhi|(x, t)\operatorname{l}e \mathcal{B}\cdot \frac{C(m, p)}{V_x(\sqrt{t})}\epsilonxp\operatorname{l}eft(2a\, r^2(x)\rightght).
\epsilonnd{equation}
Here $V_x(r)$ is the volume of ball $B(x, r)$.
\epsilonnd{proposition}
\begin{proof} Let $\Omega_\mu$ be a sequence of exhaustion bounded domains. By the standard theory on the linear parabolic system \cite{Friedman, Lady1}, there exist solutions $\partialhi_\mu(x, t)$ on $\Omega_\mu \times [0, \sqrt {-1}nfty)$ such that $\partialhi_\mu(x, t)=0$ on $\partialartial \Omega \times [0, \sqrt {-1}nfty)$. Note that in terms of the language of \cite{Morrey-har}, $\partialhi_\mu(x, t)=0$ on $ \partialartial \Omega$ means that both the tangential part $\mathsf{t} \partialhi_\mu$ and the normal part $\mathsf{n}\partialhi_\mu$ vanish on $\partialartial \Omega$. Hence this is different from the more traditional {\sqrt {-1}t relative} or {\sqrt {-1}t absolute} boundary value problem for differential forms which requires $\mathsf{t} \partialhi=\mathsf{t}\delta \partialhi=0$ and $\mathsf{n} \partialhi=\mathsf{n}d \partialhi=0$ respectively. Nevertheless it is a boundary condition (which was studied in \cite{Morrey-har}) such that together with $\Delta_{{\bar{\alpha}}r{\partialartial}}\partialhi=0$ it is hypo-elliptic and the Schauder estimate of \cite{Simon} applies. To get a global solution we shall prove that there exist uniform (in terms of $\mu$) estimates so that we can extract a convergent sub-sequence. Note that $\Lambda^p \partialhi_\mu$ is a solution to the heat equation and $|\partialhi_\mu|\operatorname{l}e C_{m, p}\Lambda^p \partialhi_\mu$. Let
$$
u(x, t)=\sqrt {-1}nt_M H(x, y, t) |\partialhi_0|(y)\, d\mu(y)
$$
where $H(x, y, t)$ is the positive heat kernel of $M$. By the fundamental heat kernel estimate of Li-Yau \cite{LY}, it is easy to see that, under the assumption (\ref{ass-1}), there exists $T_0$ such that $u(x, t)$ is finite on $K\times[0, T_0]$ for any compact subset $K$. It is easy to see that $|\partialhi_\mu|(x, t)\operatorname{l}e C_{m, p}\Lambda^p \partialhi_\mu\operatorname{l}e C'_{p, m} u(x, t)$ by (\ref{basic-1}) and the maximum principle for the scalar heat equation. Now the interior Schauder estimates \cite{Simon} (see also \cite{Morrey}, Theorem 5.5.3 for the corresponding estimates in the elliptic cases) imply that for any $0< \alphalpha< 1$, $K$, a compact subset of $M$,
$$
\|\partialhi_\mu\|_{2, \alphalpha, \frac{\alphalpha}{2}, K\times [0, T_0]} \operatorname{l}e C(K, p, m, \|\partialhi_\mu\|_{\sqrt {-1}nfty,K\times [0, T_0]} ).
$$
Here $\|\cdot\|_{2, \alphalpha, \frac{\alphalpha}{2}}$ is the $C^{2, \alphalpha}$-H\"odler norm on the parabolic region.
Since $\|\partialhi_\mu\|_{\sqrt {-1}nfty,K\times [0, T_0]}$ is estimated by $u(x, t)$ uniformly, we have established the uniform estimates so that, after passing to a subsequence, $\{\partialhi_\mu(x, t)\}$ converges to a solution $\partialhi(x, t)$ on $M\times [0, T_0]$.
It is obvious from the construction that $\partialhi(x, t)\gammae 0$. To prove the estimate (\ref{con-sol}), appealing Li-Yau's upper estimate
$$
H(x, y, t)\operatorname{l}e \frac{C(n)}{V_x(\sqrt{t})}\epsilonxp\operatorname{l}eft(-\frac{r^2(x, y)}{5t}\rightght)
$$
we can derive that for $0\operatorname{l}e t\operatorname{l}e T_0\operatorname{l}e \frac{1}{10a}$,
\begin{eqnarray*}
u(x, t)&\operatorname{l}e& \frac{C(m)}{V_x(\sqrt{t})}\sqrt {-1}nt_M \epsilonxp\operatorname{l}eft(-\frac{r^2(x, y)}{5t}+ar^2(y)\rightght) |\partialhi_0|(y)\epsilonxp\operatorname{l}eft(-ar^2(y)\rightght)\, d\mu(y)\\
&\operatorname{l}e & \mathcal{B}\cdot \frac{C(m)}{V_x(\sqrt{t})}\epsilonxp\operatorname{l}eft(2a\, r^2(x)\rightght).
\epsilonnd{eqnarray*}
In the second inequality above we used the estimate that for $0\operatorname{l}e t\operatorname{l}e T_0\operatorname{l}e \frac{1}{10a}$
$$
-\frac{r^2(x, y)}{5t}+ar^2(o, y) \operatorname{l}e -\frac{r^2(x, y)}{5t}+2a r^2(o, x)+2a r^2(x, y)\operatorname{l}e 2a r^2(x).
$$
\epsilonnd{proof}
It is clear from the proof that if $\partialhi_0(x)$ satisfies stronger assumption that
\begin{equation}\operatorname{l}abel{ass-2}
\sqrt {-1}nt_M |\partialhi_0|(y)\epsilonxp( -a r^{2-\delta}(y))\, d\mu(y) < \sqrt {-1}nfty
\epsilonnd{equation}
for some positive constants $\delta$ and $a$ then the Cauchy problem has a global solution on $M \times [0, \sqrt {-1}nfty)$.
To deform a general $(p, p)$-form, we need the following generalization on a well-known lemma of Bishop-Goldberg concerning $(1,1)$-forms on manifolds with $\mathcal{C}_1$. This also holds the key to extending Proposition \ref{max-pp} to the noncompact manifolds.
\begin{lemma}\operatorname{l}abel{bg} Assume that $(M, g)$ satisfies $\mathcal{C}_2$. Then for any $(p, q)$-form $\partialhi$,
\begin{equation}\operatorname{l}abel{eq:117}
\operatorname{l}angle \mathcal{KB}(\partialhi), \overline{\partialhi}\rangle \operatorname{l}e 0.
\epsilonnd{equation}
\epsilonnd{lemma}
\begin{proof} We shall check for the $(p, p)$-forms since the argument is the same for $(p, q)$-forms. For $\partialhi=\frac{1}{(p!)^2}\sum \partialhi_{I_p, \overline{J}_p} \operatorname{l}eft(\sqrt{-1}dz^{i_1}\wedge dz^{{\bar{\alpha}}r{j_1}}\rightght)\wedge \cdot\cdot\cdot\wedge \operatorname{l}eft(\sqrt{-1}dz^{i_p}\wedge dz^{{\bar{\alpha}}r{j_p}}\rightght)$, where the summation is for $1\operatorname{l}e i_1,\cdots, i_p, j_1, \cdots, j_p\operatorname{l}e m$.
Under a normal coordinate,
$$
\operatorname{l}angle \partialhi, \overline{\partialsi}\rangle =\frac{1}{(p!)^2} \sum \partialhi_{I_p, \overline{J}_p}\overline{\partialsi_{I_p, \overline{J}_p}}.
$$
Recall that also under the normal coordinate, $\operatorname{l}angle \Bbb Rm( dz^k\wedge dz^{{\bar{\alpha}}r{l}}), \overline{dz^s\wedge dz^{{\bar{\alpha}}r{t}}}\rangle =R_{l{\bar{\alpha}}r{k}s{\bar{\alpha}}r{t}}$. It is easy to check that $\mathcal{C}_2$ implies that
$\operatorname{l}angle \Bbb Rm(Z^*_1\wedge \overline{W}^*_1+Z^*_2\wedge \overline{W}^*_2), \overline{Z^*_1\wedge \overline{W}^*_1+Z^*_2\wedge \overline{W}^*_2}\rangle \gammae 0$, for any $(1,0)$-forms $Z^*_1, Z^*_2, W^*_1, W^*_2$. We shall prove the claim by computing the expression under a normal coordinate.
For any fixed $I_p, J_p$ and $\mu, \nu$ with $1\operatorname{l}e \mu, \nu \operatorname{l}e p$ we can define
$$
\overline{\epsilonta}_\mu \doteqdot\sum_{k_\mu=1}^m \partialhi_{i_1\cdots (k_\mu)_\mu\cdots i_p, \overline{J}_p} dz^{{\bar{\alpha}}r{k}_\mu}, \quad \xi_\nu\doteqdot \sum_{l_\nu=1}^m \partialhi_{I_p, {\bar{\alpha}}r{j}_1\cdots \overline{(l_\nu)_\nu}\cdots {\bar{\alpha}}r{j}_p}dz^{l_\nu}.
$$
Now $\Bbb Rm$ being in $\mathcal{C}_2$ implies that
\begin{equation}\operatorname{l}abel{eq:118}
\operatorname{l}angle \Bbb Rm( dz^{i_\mu}\wedge \overline{\epsilonta}_\mu -\xi_\nu \wedge dz^{{\bar{\alpha}}r{j}_\nu}), \overline{dz^{i_\mu}\wedge \overline{\epsilonta}_\mu -\xi_\nu \wedge dz^{{\bar{\alpha}}r{j}_\nu}} \rangle \gammae 0.
\epsilonnd{equation}
Now using that
\begin{eqnarray*}\Bbb Rm(dz^{i_\mu}\wedge \overline{\epsilonta}_\mu -\xi_\nu \wedge dz^{{\bar{\alpha}}r{j}_\nu})&=&\sum_{k_\mu, st}\partialhi_{i_1\cdots (k_\mu)_\mu\cdots i_p, \overline{J}_p} R_{k_\mu {\bar{\alpha}}r{i}_\mu s{\bar{\alpha}}r{t}}dz^s \wedge dz^{{\bar{\alpha}}r{t}}\\
&\quad& -\sum_{l_\nu, s, t} \partialhi_{I_p, {\bar{\alpha}}r{j}_1\cdots \overline{(l_\nu)_\nu}\cdots {\bar{\alpha}}r{j}_p} R_{j_\nu {\bar{\alpha}}r{l}_\nu s{\bar{\alpha}}r{t}}dz^s\wedge dz^{{\bar{\alpha}}r{t}}
\epsilonnd{eqnarray*}
we can expand the left hand side of (\ref{eq:118}) and obtain that
\begin{eqnarray}
0&\operatorname{l}e& \sum_{k_\mu, k'_\mu}\partialhi_{i_1\cdots (k_\mu)_\mu\cdots i_p, \overline{J}_p}R_{k_\mu {\bar{\alpha}}r{i}_\mu i_\mu {\bar{\alpha}}r{k'}_\mu}\overline{\partialhi_{i_1\cdots (k'_\mu)_\mu\cdots i_p, \overline{J}_p}}\nonumber \\
&-& \sum_{l_\nu, k'_\mu}\partialhi_{I_p, {\bar{\alpha}}r{j}_1\cdots \overline{(l_\nu)_\nu}\cdots {\bar{\alpha}}r{j}_p}R_{j_\nu {\bar{\alpha}}r{l}_\nu i_\mu {\bar{\alpha}}r{k'}_\mu} \overline{\partialhi_{i_1\cdots (k'_\mu)_\mu\cdots i_p, \overline{J}_p}}\\
&-& \sum_{ k_\mu, l'_\nu}\partialhi_{i_1\cdots (k_\mu)_\mu\cdots i_p, \overline{J}_p}R_{k_\mu {\bar{\alpha}}r{i}_\mu l'_\nu {\bar{\alpha}}r{j}_\nu} \overline{ \partialhi_{I_p, {\bar{\alpha}}r{j}_1\cdots \overline{(l'_\nu)_\nu}\cdots {\bar{\alpha}}r{j}_p}}\nonumber\\
&+&\sum_{l_\nu, l'_\nu} \partialhi_{I_p, {\bar{\alpha}}r{j}_1\cdots \overline{(l_\nu)_\nu}\cdots {\bar{\alpha}}r{j}_p} R_{j_\nu {\bar{\alpha}}r{l}_\nu l'_\nu {\bar{\alpha}}r{j}_\nu}\overline{\partialhi_{I_p, {\bar{\alpha}}r{j}_1\cdots \overline{(l'_\nu)_\nu}\cdots {\bar{\alpha}}r{j}_p}}. \nonumber
\epsilonnd{eqnarray}
Now summing for all $1\operatorname{l}e i_1, \cdots, i_p, j_1, \cdots, j_p\operatorname{l}e m$ and $1\operatorname{l}e \mu, \nu\operatorname{l}e p$, a tedious, but straight forward checking shows that the total sum of the right hand
side above is
$-2\operatorname{l}angle \mathcal{KB}(\partialhi), \overline{\partialhi}\rangle$.
\epsilonnd{proof}
An immediate consequence of the above lemma is that {\sqrt {-1}t any harmonic $(p, q)$-form on a compact K\"ahler
manifold with $\mathcal{C}_2$ must be parallel}. This fact was known for harmonic $(p, 0)$-forms under the weaker assumption that $\operatorname{Ric}\gammae 0$ and for harmonic $(1,1)$-forms under the nonnegativity of bisectional curvature. In fact, using the full power of the uniformization result of Mori-Siu-Yau-Mok, the result holds even under $\mathcal{C}_1$. Hence it does not give any new information for compact K\"ahler manifolds.
Another consequence of Lemma \ref{bg} is the following result, which generalizes Lemma 2.1 of \cite{NT-jdg}, by virtually the same argument.
\begin{corollary}\operatorname{l}abel{sub-com} Let $M^m$ be a complete K\"ahler manifold
with $\mathcal{C}_2$. Let $\partialhi(x, t)$ be
a $(p,p)$-form satisfying (\ref{eq:11}) on $M\times
[0, T]$. Then $|\partialhi|(x,t)$ is a sub-solution of the heat equation.
\epsilonnd{corollary}
This together with the proof to Proposition \ref{short-ext} gives the following improvement on the existence of the Cauchy problem for initial $(p, p$-forms not necessarily positive.
\begin{proposition}\operatorname{l}abel{short2}
Let $(M, g)$ be a K\"ahler manifold whose curvature operator $\Bbb Rm \sqrt {-1}n \mathcal{C}_2$. Assume that $\partialhi_0(x)$ satisfies (\ref{ass-1}), then there exists $T_0$ such that the Cauchy problem (\ref{cauchy1})
has a solution $\partialhi(x, t)$ on $M\times [0, T_0]$. Moreover, (\ref{con-sol}) holds.
\epsilonnd{proposition}
\begin{proof} Observe that $\partialhi_\mu(x, t)$ in the proof of Proposition \ref{short-ext} satisfies that
$|\partialhi_\mu|(x, t)$ is a sub-solution to the heat equation, hence $|\partialhi_\mu|(x, t)\operatorname{l}e u(x, t)$. The rest proof of Proposition \ref{short-ext} applies here.
\epsilonnd{proof}
A more important application of the lemma is the following extension of Proposition \ref{max-pp}.
This also extends Theorem 2.1 of \cite{NT-jdg}.
\begin{theorem}\operatorname{l}abel{max-pp-noncom}
Let $(M, g)$ be a complete noncompact K\"ahler manifold with $\mathcal{C}_p$. Let $\partialhi(x, t)$ be a $(p,p)$-form satisfying (\ref{eq:11}) on $M\times
[0, T]$. Assume that $\partialhi(x, 0)\gammae0$ and satisfies (\ref{ass-1}). Assume further that for some $a>0$,
\begin{equation}\operatorname{l}abel{ass-max}
\operatorname{l}iminf_{r\to \sqrt {-1}nfty} \sqrt {-1}nt_0^T \sqrt {-1}nt_{B_o(r)}|\partialhi|^2(x, t) \epsilonxp(-a r^2(x))\, d\mu(x) dt < \sqrt {-1}nfty.
\epsilonnd{equation}
Then $\partialhi(x, t)\gammae 0$. Moreover (\ref{con-sol}) holds.
\epsilonnd{theorem}
Before we prove the theorem, we should remark that even though Proposition \ref{short-ext} provides a solution to the Cauchy problem which is a positive $(p, p)$-form, it is also useful to be able to assert that certain solutions, which are not constructed by Proposition \ref{short-ext}, preserve the positivity. For example, if $\partialhi=\sqrt{-1}\partialartial {\bar{\alpha}}r{\partialartial} \epsilonta$ and $\epsilonta$ satisfies (\ref{eq:11}). It is easy to see that $\partialhi$ satisfies (\ref{eq:11}) since $\Delta_{{\bar{\alpha}}r{\partialartial}}$ is commutable with $\partialartial$ and ${\bar{\alpha}}r{\partialartial}$. If we know that $\sqrt{-1}\partialartial {\bar{\alpha}}r{\partialartial} \epsilonta\gammae 0$ at $t=0$, it is desirable to know when we have $\partialhi(x, t)\gammae 0$.
\begin{proof} We employ the localization technique of \cite{NT-jdg}. Let $\sigma_R$ be a cut-off function between $0$ and $1$ being $1$ in the annulus $A(\frac{R}{4}, 4R)=B(o, 4R)\setminus B(o, \frac{R}{4})$ and supported in the annulus $A(\frac{R}{8}, 8R)$. Let $$u_R(x, t)=\sqrt {-1}nt_M H(x, y, t) |\partialhi|(y, 0) \sigma_R(y)\, d\mu(y)\quad \quad u(x, t)=\sqrt {-1}nt_M H(x, y, t)|\partialhi|(y, 0)d\mu(y).$$ Clearly $u_R(x, t)\operatorname{l}e u(x, t)$. However the following result is proved in \cite{NT-jdg}, Lemma 2.2.
\begin{lemma}[Ni-Tam] Assume that $\partialhi(x, 0)$ satisfies (\ref{ass-1}).
Then there exists $T_0>0$ depending only on
$a$ such that for $R\gammae \max\{ \sqrt{T_0}, 1\},$
the following are true.
\begin{enumerate}
\sqrt {-1}tem There exists a function $\tau=\tau(r)$ with
$\operatorname{l}im_{r\to \sqrt {-1}nfty}\tau(r)=0$ such that for all $(x,t)\sqrt {-1}n
A_o(\frac R2, 2R)\times[0,T_0]$,
$$
u(x,t)\operatorname{l}e u_R(x,t)+\tau(R).
$$
\sqrt {-1}tem For any $r>0$,
$$
\operatorname{l}im_{R\to\sqrt {-1}nfty}\sup_{B_o(r)\times [0,T_0]}u_R=0.
$$
\epsilonnd{enumerate}
\epsilonnd{lemma}
Lemma \ref{bg} above implies that $(u_R(x, t)+\tau(R))\omega^p$ can be used as a barrier on $ \partialartial B_o(R)\times [0, T_0]$ since by Corollary \ref{sub-com} and the maximum principle on complete noncompact Riemannian manifolds, that is where the assumption (\ref{ass-max}) is needed, $|\partialhi|(x, t)\operatorname{l}e u(x, t)\operatorname{l}e u_R(x, t)+\tau(R)$ on $ \partialartial B_o(R)\times [0, T_0]$. In fact $$\operatorname{l}eft((u_R(x, t)+\tau(R))\omega^p+\partialhi\rightght)(v_1, \cdots, v_p; {\bar{\alpha}}r{v}_1, \cdots, {\bar{\alpha}}r{v}_p)\gammae u_R(x, t)+\tau(R)-|\partialhi|(x, t)\gammae 0$$ for any $(v_1, \cdots, v_p)$ which can be extended into a unitary basis of $T'_xM$. Now apply the argument of Proposition \ref{max-pp} on $B_0(R)\times [0, T]$ we can conclude that
$$(u_R(x, t)+\tau(R))\omega^p+\partialhi\gammae 0
$$
on $B_o(R)\times [0, T_0]$
as a $(p, p)$-form since $\partialhi(x, 0)\gammae 0$. Now the result follows by letting $R\to \sqrt {-1}nfty$ and the facts that $\operatorname{l}im_{R\to\sqrt {-1}nfty}\sup_{B_o(r)\times [0,T_0]}u_R=0$ proved in the lemma, and $\tau(R)\to 0$. Since $|\partialhi|(x, t)\operatorname{l}e u(x, t)$, the estimate (\ref{con-sol}) follows as before.
\epsilonnd{proof}
We devote the rest to the proof of Theorem \ref{main-lyh} and Theorem \ref{main-KRFlyh} for the case that $M$ is noncompact complete. Since one can pick a small $\delta>0$ and shift the time $t\to t-\delta$ and multiply the expression $Q$ by $t-\delta$, we may assume without the loss of the generality that $\partialhi, \partialartial^* \partialhi, {\bar{\alpha}}r{\partialartial}^* \partialhi, {\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi$ are all smooth up to $t=0$.
First we observe that if $\partialhi$ is a positive $(p, p)$-form, then $\Lambda^p\partialhi(x, t)$ is a nonnegative solution to the heat equation, hence satisfies (\ref{ass-max}) by the estimate of Li and Yau.
Precisely,
$$
\Lambda^p \partialhi(x, t)\operatorname{l}e \Lambda^p \partialhi (o, 1) \cdot \frac{1}{t^{m}}\cdot \epsilonxp\operatorname{l}eft(\frac{r^2(x)}{4(1-t)}\rightght).
$$
In particular, for $\frac{\delta}{2}<T<1-\delta$, one can find $a>0$ such that
$$
\sqrt {-1}nt_M \operatorname{l}eft(\Lambda^p \partialhi\rightght)^2(x,\frac{\delta}{2}) \epsilonxp(-ar^2(x))\, d\mu(x)+\sqrt {-1}nt_{\frac{\delta}{2}}^T \sqrt {-1}nt_M (\Lambda^p \partialhi )^2 \epsilonxp(-ar^2(x))\, d\mu(x)\, dt <\sqrt {-1}nfty.
$$
Since $|\partialhi|\operatorname{l}e C_{m, p} \Lambda^p \partialhi$ we can conclude that $|\partialhi|$ satisfies the above estimate. Applying Lemma \ref{bg} to $(p-1, p)$ and $(p, p-1)$ forms implies the following estimates: There exists $c_{m, p}>0$ such that
\begin{eqnarray}
\operatorname{l}eft(\Delta -\frac{\partialartial}{\partialartial t}\rightght)|\partialhi|^2 &\gammae& c_{m, p}\operatorname{l}eft(|\partialartial^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialhi|^2\rightght),\operatorname{l}abel{11-help1}\\
\operatorname{l}eft(\Delta -\frac{\partialartial}{\partialartial t}\rightght) \operatorname{l}eft(|\partialartial^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialhi|^2\rightght) &\gammae& c_{m, p} |{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi|^2, \operatorname{l}abel{11-help2}\\
\operatorname{l}eft(\Delta -\frac{\partialartial}{\partialartial t}\rightght) |{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi|^2 &\gammae 0. \operatorname{l}abel{11-help3}
\epsilonnd{eqnarray}
By the same argument of the proof to Lemma 1.4 in \cite{N-jams} we can conclude that there exists $a'>0$ such that
\begin{equation}
\sqrt {-1}nt_{\frac{\delta}{2}}^T \sqrt {-1}nt_M \operatorname{l}eft(|\partialhi|^2 +|\partialartial^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi|^2\rightght)\epsilonxp(-a'r^2(x))< \sqrt {-1}nfty.
\epsilonnd{equation}
Note that by the mean value theorem for the subsolution to the heat equation \cite{LT}, one can obtain pointwise estimates for $|\partialhi|^2 +|\partialartial^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi|^2$ at $t=\delta$.
Now with the help of the argument for the compact case, the same proof as Theorem \ref{max-pp-noncom} via the barrier argument, applying to $Q$ which is viewed a $(p-1, p-1)$-form valued Hermitian symmetric tensor, proves Theorem \ref{main-lyh} on complete noncompact manifolds. The corresponding result with the K\"ahler-Ricci flow, namely Theorem \ref{main-KRFlyh}, is very similar. Hence we keep it brief. Due to the bound on the curvature there exists a positive constant $\alphalpha_{m, A}$, depending on the upper bound $A$ of the curvature tensor so that
\begin{eqnarray}&\,&
\operatorname{l}eft(\Delta -\frac{\partialartial}{\partialartial t}\rightght)\operatorname{l}eft(e^{-\alphalpha_{m, A} t}\cdot |\partialhi|^2 \rightght)\gammae c_{m, p}e^{-\alphalpha_{m, A} t}\cdot \operatorname{l}eft(|\partialartial^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialhi|^2\rightght),\operatorname{l}abel{11-help1-1}\\
&\, &\operatorname{l}eft(\Delta -\frac{\partialartial}{\partialartial t}\rightght)\operatorname{l}eft(e^{-\alphalpha_{m, A} t}\cdot \operatorname{l}eft(|\partialartial^* \partialhi|^2+|{\bar{\alpha}}r{\partialartial}^* \partialhi|^2\rightght)\rightght) \gammae c_{m, p} e^{-\alphalpha_{m, A} t}\cdot |{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi|^2, \operatorname{l}abel{11-help2-2}\\
&\, & \operatorname{l}eft(\Delta -\frac{\partialartial}{\partialartial t}\rightght)\operatorname{l}eft(e^{-\alphalpha_{m, A} t}\cdot |{\bar{\alpha}}r{\partialartial}^* \partialartial^* \partialhi|^2 \rightght) \gammae 0. \operatorname{l}abel{11-help3-3}
\epsilonnd{eqnarray}
There are modified point-wise estimates for the positive solutions coupled with the Ricci flow to replace the Li-Yau's estimate. See for example Theorem 2.7 of \cite{Ni-JDG07} (a result of Guenther \cite{Gu}). There is a corresponding mean value theorem for the nonnegative sub-solutions to the heat equation. See for example Theorem 1.1 of \cite{N-MRL05}. Putting them together the similar argument as the above applies to Theorem \ref{main-KRFlyh}.
\begin{remark} The argument here in fact proves Theorem 1.1 of \cite{N-jams} without the assumption (1.5) there since that assumption is a consequence of the semi-positivity of $h$ and Li-Yau's estimate for positive solutions to the heat equation.
\epsilonnd{remark}
\section{Appendix.}
Here we include Wilking's proof to Theorem \ref{ww} (also included in \cite{CT}) following the notations of Section 3, which was explained to the first author by Wilking in May of 2008.
Recall that $\operatorname{ad}_v: \mathfrak{so}(n, \Bbb C)\to \mathfrak{so}(n, \Bbb C)$ mapping $w$ to $[v, w]$. The operator $\operatorname{ad}_{(\cdot)}$ can be viewed as a map from $\mathfrak{so}(n, \Bbb C)$ to the space of endmorphisms of $\mathfrak{so}(n, \Bbb C)$. It is the derivative of $\operatorname{Ad}$, the adjoint action of $\mathsf{SO}(n, \Bbb C)$ on $\mathfrak{so}(n, \Bbb C)$, which maps $\mathsf{SO}(n, \Bbb C)$ to automorphisms of $\mathfrak{so}(n, \Bbb C)$. This is a basic fact in Lie group theory. Another basic fact from Lie group theory asserts that
$e^{\operatorname{ad}_v}=\operatorname{Ad}(\operatorname{Exp}(v))$.
For the proof of Wilking's theorem we need to recall the following identity for $\Bbb Rm^\#$.
\begin{equation}\operatorname{l}abel{A1}
\operatorname{l}angle \Bbb Rm^{\#} (v), w\rangle =\frac{1}{2}\sum_{\alphalpha, \beta}\operatorname{l}angle [ \Bbb Rm(b^\alphalpha), \Bbb Rm(b^\beta)], v\rangle \operatorname{l}angle [b^\alphalpha, b^\beta], w\rangle.
\epsilonnd{equation}
Here $\{b^\alphalpha\}$ is a basis for $\frak{so}(n, \Bbb C)$. It suffices to show that if $\operatorname{l}angle \Bbb Rm(v_0), \overline{v}_0\rangle =0$, $\operatorname{l}angle \Bbb Rm^{\#}(v_0), \overline{v}_0\rangle\gammae 0$. Here we identify $\wedge^2(\Bbb C^n)$ with $\mathfrak{so}(n, \Bbb C)$ and and observe that the action of $\mathsf{SO}(n, \Bbb C)$ on $\wedge^2 (\Bbb C^n)$ is the same as the adjoint action under the identification.
Given above basic facts from Lie group theory, for any $b\sqrt {-1}n \mathfrak{so}(n, \Bbb C)$, and $z\sqrt {-1}n \Bbb C$, consider $\operatorname{l}eft(\operatorname{Ad}(\operatorname{Exp}(zb))\rightght)(v_0)=\operatorname{Exp}(zb) \cdot v_0 \cdot \operatorname{Exp}(-zb)$. Since $\operatorname{Exp}(zb)\sqrt {-1}n \mathsf{SO}(n, \Bbb C)$, we conclude that $\operatorname{l}eft(\operatorname{Ad}(\operatorname{Exp}(zb))\rightght)(v_0)\sqrt {-1}n \Sigma$. Hence $v(z)=e^{z \, \operatorname{ad}_b}\cdot v_0\sqrt {-1}n \Sigma$. Thus if we define
$$
I(z):=\operatorname{l}angle \Bbb Rm(v(z)), \overlineerline{v(z)}\rangle
$$
it is clear that $I(z)\gammae 0$ and $I(0)=0$, which implies that $\frac{\partialartial^2}{\partialartial z\partialartial \overlineerline{z}} I(z)|_{z=0} \gammae 0$. Hence for any $b\sqrt {-1}n \mathfrak{so}(n, \Bbb C)$,
$\operatorname{l}angle \Bbb Rm ( \operatorname{ad}_b(v_0)), \overlineerline{\operatorname{ad}_b(v_0)}\rangle \gammae 0 $, which can be equivalently written as
\begin{equation}\operatorname{l}abel{A2}
\operatorname{l}angle \Bbb Rm \cdot \operatorname{ad}_{v_0}(b), \operatorname{ad}_{\overlineerline{v}_0} (\overlineerline{b})\rangle \gammae 0.
\epsilonnd{equation}
This is equivalent to $-\operatorname{ad}_{\overlineerline{v}_0}\cdot \Bbb Rm \cdot \operatorname{ad}_{v_0}\gammae 0$, as a Hermitian symmetric tensor.
By equation (\ref{A1}) $ \operatorname{l}angle \Bbb Rm^{\#}(v_0), \overline{v}_0\rangle \gammae 0$ is the same as
$\frac{1}{2}\operatorname{trace}(-\operatorname{ad}_{\overlineerline{v}_0} \cdot \Bbb Rm \cdot \operatorname{ad}_{v_0} \cdot \Bbb Rm )\gammae 0$. This last fact is implied by (\ref{A2}) as follows. Let $\operatorname{l}ambda_{\alphalpha}$ be the eigenvalues of $-\operatorname{ad}_{\overlineerline{v}_0}\cdot \Bbb Rm \cdot \operatorname{ad}_{v_0}$ with eigenvectors $b^{\alphalpha}$. Then for $\operatorname{l}ambda_\alphalpha>0$,
$
b^{\alphalpha}=\frac{1}{\operatorname{l}ambda_\alphalpha} \operatorname{ad}_{\overlineerline{v}_0}(w^{\alphalpha}),
$
where $w^{\alphalpha}=\Bbb Rm \cdot \operatorname{ad}_{v_0}(b^\alphalpha)$. At the mean time
\begin{eqnarray*}
\operatorname{trace}(-\operatorname{ad}_{\overlineerline{v}_0} \cdot \Bbb Rm \cdot \operatorname{ad}_{v_0}
\cdot \Bbb Rm )&=&\sum \operatorname{l}angle \Bbb Rm \cdot (-\operatorname{ad}_{\overlineerline{v}_0} \cdot \Bbb Rm \cdot \operatorname{ad}_{v_0}) (b^\alphalpha), \overlineerline{b^{\alphalpha}}\rangle \\
&=&\sum_{\operatorname{l}ambda_\alphalpha >0}\operatorname{l}ambda_\alphalpha \operatorname{l}angle \Bbb Rm (b^{\alphalpha} ), \overlineerline{b^{\alphalpha}}\rangle\cr
&=&\sum_{\operatorname{l}ambda_\alphalpha >0}\frac{1}{\operatorname{l}ambda_\alphalpha} \operatorname{l}angle \Bbb Rm \cdot \operatorname{ad}_{\overlineerline{v}_0}(w^{\alphalpha}), \operatorname{ad}_{v_0} (\overlineerline{w^\alphalpha})\rangle\\
&=&\sum_{\operatorname{l}ambda_\alphalpha >0}\frac{1}{\operatorname{l}ambda_\alphalpha} \operatorname{l}angle \Bbb Rm\cdot \operatorname{ad}_{v_0} (\overlineerline{w^\alphalpha}), \operatorname{ad}_{\overlineerline{v}_0}(w^{\alphalpha}) \rangle.
\epsilonnd{eqnarray*}
The last expression is nonnegative by (\ref{A2}).
\section*{Acknowledgments.} { } We thank Brett Kotschwar for bringing our attention to \cite{CT}. We also thank Nolan Wallach for helpful discussions, Shing-Tung Yau for his interests. The first author's research is partially supported by a NSF grant DMS-0805834.
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{\sc Addresses:}
{\sc Lei Ni},
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA
email: [email protected]
{\sc Yanyan Niu}, Department of Mathematics and Institute of Mathematics and
Interdisciplinary Science, Capital Normal University, Beijing, China
email: [email protected]
\epsilonnd{document} |
\betaegin{document}
\title{Spectral determinant of the two-photon quantum Rabi model}
\alphauthor{Daniel Braak}
\alphaddress{EP VI and
Center for Electronic Correlations and Magnetism,\\
Institute of Physics,
University of Augsburg, 86135 Augsburg, Germany}
\ead{[email protected]}
\betaegin{abstract}
The various generalized spectral determinants ($G$-functions) of the two-photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the $G$-function proposed by Chen {\it et al.} in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. We derive this function rigorously employing the $\mathbb{Z}_4$-symmetry of the model and show that its zeros correspond to the complete regular spectrum.
\end{abstract}
\maketitle
\sigmaection{Introduction}
The quantum Rabi model (QRM) \cite{jaynes_63} is a well-known minimalistic way to describe the interaction of matter (fermions, discrete degrees of freedom) with light (bosons, continuous degrees of freedom). It consists of a spin 1/2 coupled linearly to a single mode of the radiation field,
\betaegin{equation}
H_R=\omega\alphad a +\Delta\sigma_z + g(a+\alphad)\sigma_x.
\lambdaabel{Hrabi}
\end{equation}
The Pauli matrices $\sigma_{x,z}$ describe the spin 1/2 and $a$ ($\alphad$) are the annihilation (creation) operators of the bosonic mode. The Hilbert space is thus
${\cal H}=L^2(\mathbb{R})\otimes\mathbb{C}^2$. This model is integrable due to its manifest $\mathbb{Z}_2$-symmetry for all values of frequency $\omega$ and coupling $g$ \cite{braak_11}. The small and large coupling regions are also accessible via perturbation theory.For small $g$ the rotating wave approximation is feasible \cite{jaynes_63}, for large $g$ the adiabatic approximation \cite{li_21}. The validity of the adiabatic approximation for large coupling is due to the boundedness of ${1\!\!1}\otimes\sigma_z$ and the fact that the operator $\alphad a +g(a+\alphad) +g^2$ is unitarily equivalent to $\alphad a$, the Hamiltonian of the uncoupled field mode. Therefore, the qualitative features of the spectral graph as function of $g$ do not change much in going from small to large coupling \cite{rossatto_17}, as long as neither $\omega$ nor $\Deltaelta$ become singular \cite{hwang_15,felicetti_20}. Especially the average distance between adjacent levels with the same parity (the eigenvalue of the operator $\hat{P}_R=-\exp(i\phantomii\alphad a)\otimes\sigma_z$ generating the $\mathbb{Z}_2$-symmetry) is always $\omegaega$, independent of $g$ and $\Deltaelta$ \cite{braak_19}.
There are many generalizations of the QRM which describe various implementations within cavity and circuit QED as well as quantum simulation platforms, e.g. the anisotropic \cite{xie_14} and the asymmetric QRM \cite{wakayama_17}, the Rabi-Stark model \cite{mac_15,eckle_17} and the QRM with non-linear coupling \cite{ng_99}, the so-called two-photon quantum Rabi model (2pQRM) which is the subject of the present investigation. The 2pQRM has received lot of attention recently due to the possibilities to realize it in superconducting circuits \cite{felicetti_18,felicetti_19} or via quantum simulation \cite{felicetti_15}.
After a trivial unitary transformation exchanging $\sigma_x$ and $\sigma_z$, the Hamiltonian of the 2pQRM reads
\betaegin{equation}
{\cal H}p = \omega\alphad a + (a^2 + \alphadq)\sigma_z +\Delta\sigma_x.
\lambdaabel{H2p}
\end{equation}
For convenience, we measure $\omega$ and $\Delta$ in units of $g$ which is set to 1.
In contrast to ({\tilde{\rho}}ef{Hrabi}), the coupling term in ({\tilde{\rho}}ef{H2p}) is not relatively bounded with respect to $\alphad a$ \cite{reed_75} and the spectrum undergoes a dramatic change when $\omega$ approaches the critical value $\omega_c=2$ from above. For $\omega$ less than 2, ${\cal H}p$ is no longer bounded from below \cite{ng_99}.
Exactly at $\omega=2$, the spectrum contains a discrete and a continuous part while it is pure point for $\omega>2$. The change from discrete to continuous spectrum at $\omega_c$ has been termed ``spectral collapse'' \cite{felicetti_15}, although this is a misnomer because the discrete spectrum does not collapse to a single point at $\omega_c$.
However, exactly this behavior is seen in all numerical evaluations of the spectrum, simply because the Hilbert space used in these calculations has finite dimension. The continuous part of the spectrum at $\omega_c$ covers the interval $[E_0,\infty[$ with $E_0=-\omega_c/2=-1$ which follows from a direct treatment of this case \cite{lo_20,chan_20}. The critical point $\omega_c=2$ allows for a mapping of the eigenvalue problem to a 1D Schr\"odinger equation in an energy-dependent potential. Similar simplifications at criticality appear also in the Rabi-Stark model \cite{mac_15,chen_20}.
On the other hand, it is not clear from this calculation whether the Hamiltonian exactly at $\omega_c$ is the limit of ${\cal H}p(\omega)$ for $\omega {\tilde{\rho}}ightarrow \omega_c^+$ or a singular special case. To answer this question one has to compute the full spectrum for all $\omega>\omega_c$ and deduce its qualitative behavior in approaching the ``collapse'' point. This can be done without much effort for $\Delta=0$, where the collapse phenomenon already happens. In this case ${\cal H}p$ is equivalent to a linear combination of generators of $\mathfrak{su}(1,1)$ and can be diagonalized by standard methods \cite{ng_99, penna_18}. The general situation is much more complicated. One reason is the fact that the spectral problem for the 2pQRM and the linear QRM cannot be solved by polynomial ansatz functions which yield f.e. the discrete spectrum of the harmonic oscillator and the hydrogen atom. The simple harmonic oscillator with Hamiltonian
\betaegin{equation}
H_{{\tilde{\rho}}m osc}=-\frac{1}{2}\frac{{\tilde{\rho}}d^2}{{\tilde{\rho}}d x^2}+\frac{\omega}{2}x^2-\frac{\omega}{2},
\lambdaabel{hosc}
\end{equation}
has a spectrum of pure point type. The eigenfunctions can be obtained with the ansatz $\phantomisi_n(x)=P_n(x)e^{-\omega x^2/2}$, where $P_n(x)$ is a polynomial of degree $n$, $n\in\rm{I\!N}_0$. Functions of this type are clearly normalizable in $L^2(\mathbb{R})$, so each $\phantomisi_n(x)$ is an eigenfunction of $H_{{\tilde{\rho}}m osc}$ with eigenvalue $E_n=n\omega$. But it is not easy to prove that the discrete spectrum of $H_{{\tilde{\rho}}m osc}$ consists only of the $E_n$, meaning that the set $\{\phantomisi_n\}$ is complete in $L^2(\mathbb{R})$ (supposing in addition the absence of a continuous spectrum). Moreover, the fact that a polynomial ansatz for the eigenfunctions indeed works is due to the simplicity of the differential equation $(H_{{\tilde{\rho}}m osc}-E)\phantomisi=0$ which belongs to the hypergeometric class \cite{slavianov_00}.
In most cases, the eigenfunctions cannot be obtained by simple ansatz functions and the spectral problem becomes difficult because the normalizability condition in $L^2(\mathbb{R})$,
\betaegin{equation}
\lambdaangle\phantomisi|\phantomisi{\tilde{\rho}}angle = \int {\tilde{\rho}}d x\ \phantomisi^\alphast(x)\phantomisi(x) < \infty,
\end{equation}
for the eigenunction $\phantomisi(x)$ requires knowledge of $\phantomisi(x)$ for all $x$, it is \emph{non-local}. Fortunately, there exists a Hilbert space ${\cal B}$, isomorphic to $L^2(\mathbb{R})$, which consists of analytic functions $f(z)$ of the complex variable $z$ \cite{bargmann_61}. In ${\cal B}$, the multiplication operator $z$ is adjoint to the derivative ${\tilde{\rho}}d/{\tilde{\rho}}d z$. All elements $f(z)$ in ${\cal B}$ satisfy two conditions: The first Bargmann condition means they are holomorphic in all of $\mathbb{C}$, which is a local property of $f(z)$. The second Bargmann condition requires finiteness of $|f|$, the Bargmann norm of $f(z)$, obtained form the scalar product,
\betaegin{equation}
\lambdaangle f|g{\tilde{\rho}}angle = \frac{1}{\phantomii}\int {\tilde{\rho}}d z{\tilde{\rho}}d \betaz \ e^{-|z|^2}\betaar{f}(\betaz)g(z).
\lambdaabel{barg-norm}
\end{equation}
This condition is again non-local. But in the case of the harmonic oscillator (and the QRM) the first, local Bargmann condition is sufficient to determine the whole discrete spectrum and prove its completeness. Writing $H_{{\tilde{\rho}}m osc}$ in terms of $a$ and $\alphad$, we have
\betaegin{equation}
H_{{\tilde{\rho}}m osc}=\omega\alphad a = \omega z\deltaz
\end{equation}
in ${\cal B}$, because $a=\deltaz$ and $\alphad=z$. All solutions of $(H_{{\tilde{\rho}}m osc}-E)\phantomisi(z)=0$ read $\phantomisi_E(z)=z^{E/\omega}$. They have finite Bargmann norm for all real $E\gammae 0$.
So, the second Bargmann condition does not determine the discrete spectrum of $H_{{\tilde{\rho}}m osc}$. But the first Bargmann condition requires $E/\omega$ to be a non-negative integer, yielding the spectrum of the harmonic oscillator. Moreover, the set $\{z^n\}$ is clearly a basis for the entire functions in $\mathbb{C}$, so it is complete in ${\cal B}$. The absence of a continuous spectrum follows at once. Note that the $\phantomisi_{E_n}(z)$ are not obtained by a special ansatz because the Schr\"odinger equation has no other solutions than $z^{E/\omega}$.
We show in the following that also the second Bargmann condition, despite its non-locality, can be used to determine the exact spectrum of Hamiltonians with a single continuous degree of freedom, in this case ${\cal H}p$, by analyzing the singularity structure of the corresponding ordinary differential equation in the complex domain. This is done for all coupling regimes in section {\tilde{\rho}}ef{general}. In section {\tilde{\rho}}ef{recur}, we discuss generalized spectral determinants proposed in \cite{zhang_17} and \cite{mac_17} for the case $\omega>2$. In section {\tilde{\rho}}ef{chen} we derive the $G$-function proposed in \cite{chen_12, duan_16}. Conclusions are given in section {\tilde{\rho}}ef{concl}.
\sigmaection{Singularities of the eigenvalue equation}
\lambdaabel{general}
We write an eigenfunction of ${\cal H}p$, $|\phantomisi{\tilde{\rho}}angle\in {\cal B}\otimes\mathbb{C}^2$ as the column vector $|\phantomisi{\tilde{\rho}}angle=(\phantomi_1(z),\phantomi_2(z))^T$.
The eigenvalue equation ${\cal H}p|\phantomisi{\tilde{\rho}}angle=E|\phantomisi{\tilde{\rho}}angle$ is the following coupled system
\betaegin{eqnarray}
\phantomi_1^{(2)} +\omega z\phantomi_1^{(1)} +(z^2-E)\phantomi_1 +\Delta\phantomi_2 &=0,
\lambdaabel{orig1}\\
\phantomi_2^{(2)} -\omega z\phantomi_2^{(1)} +(z^2+E)\phantomi_2 -\Delta\phantomi_1 &=0,
\lambdaabel{orig2}
\end{eqnarray}
where $\phantomi^{(n)}$ denotes the $n$-th derivative of $\phantomi(z)$ with respect to $z$. The system ({\tilde{\rho}}ef{orig1},{\tilde{\rho}}ef{orig2}) has no singular points for $|z|<\infty$ but a strong irregular singularity at $z=\infty$. Upon elimination of $\phantomi_2$, the equation for $\phantomi_1$ is of fourth order,
\betaegin{eqnarray}
\phantomi_1^{(4)} +((2-\omega^2)z^2+2\omega)\phantomi_1^{(2)} \nonumber\\
+ (4+2\omega E-\omega^2)z\phantomi_1^{(1)})
+(z^4-2\omega z^2+2-E+\Delta^2)\phantomi_1=0.
\lambdaabel{orig4}
\end{eqnarray}
The analysis of the irregular singularity at $z=\infty$ according to \cite{ince_12} shows that the singularity is unramified with rank two, or, following the notation of \cite{slavianov_00}, ``$s$-rank'' three. The class of the singularity is four and therefore maximal \cite{ince_12}. This means that for $z{\tilde{\rho}}a \infty$ all solutions are entire and have the asymptotic form
\betaegin{equation}
\phantomi_1(z)=\exp\lambdaeft(\frac{\gamma}{2}z^2+\beta z{\tilde{\rho}}ight)z^{\tilde{\rho}}ho\sigmaum_{n=0}^\infty c_nz^{-n},
\lambdaabel{origasym}
\end{equation}
where $c_0\neq 0$ and $\gamma$ is a solution of the equation
\betaegin{equation}
\gamma^4+(2-\omega)\gamma^2+1=0.
\lambdaabel{origamma}
\end{equation}
The four solutions of ({\tilde{\rho}}ef{origamma}),
\betaegin{equation}
\gamma_{1,4}=\frac{1}{2}(-\omega\phantomim\sigmaqrt{\omega^2-4}),\qquad
\gamma_{2,3}=\frac{1}{2}(\omega\mp\sigmaqrt{\omega^2-4}),
\lambdaabel{origammas}
\end{equation}
called characteristic exponents of second kind with order two, determine the growth type of
the entire function $\phantomi_1(z)$ which also has order two \cite{rubel_96}. The coefficient $\beta$ of the linear term in the exponential factor of ({\tilde{\rho}}ef{origasym}) is an exponent of second kind with order one \cite{slavianov_00}.
In general, the formal expansion ({\tilde{\rho}}ef{origasym}) of a solution $\phantomi_1(z)$ with a fixed $\gamma_j$ is only valid in a certain sector $S_l$
(``Stokes sector'') of the complex plane, while in the complementary sectors the asymptotic expansion requires $\gamma_k\neq\gamma_j$ \cite{slavianov_00}. The growth type of $\phantomi_1(z)$ is then given by the sector and associated $\gamma$ with maximal Re$(\gamma z^2)$.
Now the spectral problem consists in the task to find those solutions of ({\tilde{\rho}}ef{orig4}) which have finite Bargmann norm because the first Bargmann condition \cite{bargmann_61} is satisfied by all of them. The opposite case is realized in the linear QRM: The generic solutions are not entire in $\mathbb{C}$ but their Bargmann norm is always finite as for the harmonic oscillator \cite{braak_11}.
If $|\phantomi_1|<\infty$, $\phantomi_1(z)$ is an element of ${\cal B}$, which is a necessary condition for $|\phantomisi{\tilde{\rho}}angle$ to be an eigenvector of ${\cal H}p$.
To decide whether $|\phantomi_1|$ is finite, it is sufficient to know the asymptotic behavior of $\phantomi_1(z)$ in each Stokes sector $S_l$,
\betaegin{equation}
\fl
\int_{S_l}{\tilde{\rho}}d z{\tilde{\rho}}d\betaz \ e^{-|z|^2}|\phantomi_1(z)|^2 < \infty \Leftrightarrow
\int_{S_l}{\tilde{\rho}}d z{\tilde{\rho}}d\betaz \ e^{-|z|^2} \theta(|z|-R)|\phantomi_1(z)|^2 <\infty,
\lambdaabel{asymapprox}
\end{equation}
where $\theta(x)$ denotes the Heaviside step function.
Inequality ({\tilde{\rho}}ef{asymapprox}) holds for arbitrary $R<\infty$ because $\phantomi_1(z)$ is entire. Now, for $z\in S_l$ and $|z|>R$,
\betaegin{equation}
\phantomi_1(z)=c_0\exp\lambdaeft(\frac{\gamma(S_l)}{2}z^2+\beta(S_l) z{\tilde{\rho}}ight)z^{{\tilde{\rho}}ho(S_l)}\betaig(1+{\cal O}(R^{-1})\betaig)
\lambdaabel{asymstokes}
\end{equation}
for sufficient large but finite $R$
which means that the inequality on the right of ({\tilde{\rho}}ef{asymapprox}) will be satisfied if
\betaegin{equation}
\int_{S_l}{\tilde{\rho}}d z{\tilde{\rho}}d\betaz \ \theta(|z|-R)e^{-|z|^2}\exp\betaig({\rm Re}(\gamma(S_l) z^2 +2\beta(S_l) z)\betaig)|z^{{\tilde{\rho}}ho(S_l)}|^2 <\infty.
\lambdaabel{condasym}
\end{equation}
If ({\tilde{\rho}}ef{condasym}) is satisfied for all Stokes sectors $S_l$, $\phantomi_1$ will be normalizable. It is easy to see that the integral in ({\tilde{\rho}}ef{condasym}) is finite for $|\gamma|<1$ in all of $\mathbb{C}$ and diverges for $|\gamma|>1$ in the Stokes sectors with ${\rm Re}(\gamma z^2)>0$, independent of $\betaeta$ and ${\tilde{\rho}}ho$. If $|\gamma|=1$, the convergence of the integral is determined by $\betaeta\neq0$. If $\gamma$ lies on the unit circle and $\betaeta$ vanishes, ({\tilde{\rho}}ef{condasym}) is determined by ${\tilde{\rho}}ho$.
From ({\tilde{\rho}}ef{origammas}) we can infer three regions in the range of $\omega$ with different properties of ({\tilde{\rho}}ef{condasym}).
\sigmaubsection{$\omega>2$}
\lambdaabel{sec-greater}
In this case, all $\gamma_j$ are real and non-degenerate. $|\gamma_{3,4}|>1$ and $|\gamma_{1,2}|<1$.
The four Stokes sectors $S_l$ are
\betaegin{equation}
z\in S_l \Leftrightarrow (2l - 1)\phantomii/4 < {{\tilde{\rho}}m arg}(z) < (2l+1)\phantomii/4,
\qquad l=0\lambdadots 3.
\lambdaabel{sectors}
\end{equation}
Solutions with $\gamma_{1,2}=-\gamma_{2,1}$ have finite Bargmann norm in all sectors. Solutions with $\gamma_4<-1$ converge in sectors $S_0$ and $S_2$ and diverge in sectors $S_1$ and $S_3$ while solutions with $\gamma_3=-\gamma_4>1$ diverge in sectors $S_0$ and $S_2$ and converge in sectors $S_1$ and $S_3$.
Because the $\gamma_j$ are non-degenerate, we have $\betaeta_j=0$ for all $j$, but this has no bearing on the normalizability properties of $\phantomi_1(z)$.
The fact that all asymptotics have $|\gamma|\neq 1$ entails that in this region the spectrum has a ``pure point'' characteristic \cite{reed_81} and no continuous part if the continuous spectrum requires asymptotics with $|\gamma|=1$. That this is likely correct follows from consideration of the known ``generalized eigenstates'' which are associated with the continuous spectrum \cite{bargmann_61},
\betaegin{equation}
\phantomihi^\vartheta_{x}(z)=\phantomii^{-1/4}e^{-\frac{x^2}{2}}\exp\lambdaeft( -e^{2i\vartheta}\frac{z^2}{2} +\sigmaqrt{2}e^{i\vartheta}xz{\tilde{\rho}}ight),
\lambdaabel{genstates}
\end{equation}
with the real phase angle $0\lambdae\vartheta<2\phantomii$ and $x\in\mathbb{R}$. These states satisfy the orthogonality relation
\betaegin{equation}
\lambdaangle \phantomihi^\vartheta_x|\phantomihi^\vartheta_y{\tilde{\rho}}angle=\delta(x-y),
\lambdaabel{ortho}
\end{equation}
for fixed $\vartheta$, needed to constitute a spectrum
continuous in the parameter $x$. In fact, the image of $\phantomihi_{x_0}^0$ in $L^2(\mathbb{R})$ is $\phantomisi_{x_0}(x)=\delta(x-x_0)$, a generalized eigenstate of the position operator $\hat{x}$, while the image of $\phantomi_p^{\phantomii/2}$ reads $\phantomisi_p(x)=(2\phantomii)^{-1/2}\exp(ipx)$, a plane wave associated with the eigenbasis of the momentum operator $\hat{p}$. Fourier transformation in $L^2(\mathbb{R})$ corresponds to the unitary transform $z{\tilde{\rho}}ightarrow iz$ in ${\cal B}$, embedded in the continuous set of isometries of ${\cal B}$, $z {\tilde{\rho}}ightarrow e^{i\vartheta}z$, each corresponding to a family of generalized eigenfunctions parameterized by the angle $\vartheta$. Each of them is characterized by its exponent of the second kind $\gamma=-e^{2i\vartheta}$ of order two, the generalized eigenvalue is proportional to the exponent of order one. The position basis ($\vartheta=0$) has $\gamma =-1$ and the momentum basis ($\vartheta=\phantomii/2$) $\gamma=1$. It is quite probable, although not yet rigorously demonstrated, that all generalized eigenfunctions constituting the continuous part of the spectrum of a self-adjoint operator in ${\cal B}$ must have an asymptotic expansion of the form ({\tilde{\rho}}ef{genstates}) in each of their Stokes sectors, in general with $\vartheta$ and $x$ varying from sector to sector.
The standard ``scattering problem'' in $L^2(\mathbb{R})$ is thus generalized to a lateral connection problem in ${\cal B}$.
If the conjecture above is correct, the spectrum of ${\cal H}p$ in the region $\omega>2$ has pure point characteristic.
\sigmaubsection{$\omega<2$}
\lambdaabel{sec-less}
All $\gamma_j$ are located on the unit circle, $|\gamma_j|=1$ and are non-degenerate.
\betaegin{equation}
\gamma_{1,4}=\frac{1}{2}(-\omega\phantomim i\sigmaqrt{4-\omega^2}),\qquad
\gamma_{2,3}=\frac{1}{2}(\omega\mp i\sigmaqrt{4-\omega^2}).
\lambdaabel{imorigammas}
\end{equation}
Because the $\gamma_j$ are all different, the $\beta_j$ vanish as in case I. As the arguments of the $\gamma_j$ differ, they do not have common Stokes sectors. The convergence of the integral ({\tilde{\rho}}ef{condasym}) in a certain sector $S_l$ of the complex plane where ({\tilde{\rho}}ef{origasym}) is asymptotically valid depends on whether $S_l$ contains the line $\{z|\alpharg(\phantomim z)=-\alpharg(\gamma)/2\}$. If not, the integral converges.
If the line is contained in $S_l$, the convergence depends on the value of the exponent of the first kind, ${\tilde{\rho}}ho(S_l)$. As example, lets consider the sector
$S_\gamma = \{z| \alpharg(z)\in[-\alpharg(\gamma)/2- \phantomii/4, -\alpharg(\gamma)/2 +\phantomii/4]\}$.
The integral in ({\tilde{\rho}}ef{condasym}) reads then ($\gamma=e^{i\varphi}$)
\betaegin{equation}
\fl
I(R) = e^{\varphi{\rm Im}({\tilde{\rho}}ho)}\int_R^\infty{\tilde{\rho}}d x\int_{-x}^x{\tilde{\rho}}d y\
(x^2+y^2)^{{\rm Re}({\tilde{\rho}}ho)}e^{-2{\rm Im}({\tilde{\rho}}ho)\alpharctan(y/x)} e^{-2y^2},
\lambdaabel{intsg}
\end{equation}
where the arc from $x-ix$ to $x+ix$ has been replaced by a straight line.
We have
\betaegin{equation}
I(R) < e^{\varphi{\rm Im}({\tilde{\rho}}ho)+\mu}2^{{\rm Re}({\tilde{\rho}}ho)}\sigmaqrt{\frac{\phantomii}{2}}\int_R^\infty{\tilde{\rho}}d x \ x^{2{\rm Re}({\tilde{\rho}}ho)},
\end{equation}
with $\mu=\max(0,-{\rm Im}({\tilde{\rho}}ho)\phantomii/2)$.
It follows that ({\tilde{\rho}}ef{condasym}) will be satisfied if ${\rm Re}({\tilde{\rho}}ho)<-1/2$.
In the present case, ${\tilde{\rho}}ho$ is a function of $\gamma$, $\omega$ and $E$,
\betaegin{equation}
{\tilde{\rho}}ho=-\frac{6\gamma^3+2\omega\gamma^2+(2-\omega^2)\gamma +(4+2\omega E-\omega^2)\gamma-2\omega}{4\gamma^3
+2(2-\omega^2)\gamma}.
\lambdaabel{rho}
\end{equation}
A direct calculation shows that for any $\gamma$ in the set ({\tilde{\rho}}ef{imorigammas}) and arbitrary real $E$, ${\rm Re}({\tilde{\rho}}ho)=-3/2$. Therefore all solutions of
({\tilde{\rho}}ef{orig4}) have finite Bargmann norm for all real $E$ which entails that the symmetric operator ${\cal H}p$ is not self-adjoint in ${\cal B}$ for $\omega<2$ and does not have self-adjoint extensions, because ${\cal B}$ is separable \cite{reed_81} and does not contain an uncountable set of mutually orthogonal vectors. We have thus proven the conjecture made by Ng {\it et al.} in \cite{ng_99} about the behavior of ${\cal H}p$ ``beyond the collapse point'', i.e. in the region $\omega < 2$.
\sigmaubsection{$\omega=2$}
\lambdaabel{sec-critical}
Here, $\gamma$ takes the values $+1$ and $-1$, both of them doubly degenerate. It follows from ({\tilde{\rho}}ef{orig4}) together with ({\tilde{\rho}}ef{origasym}) that now $\beta$ is determined in terms of $\gamma$,
\betaegin{equation}
6\beta^2\gamma^2+(2-\omega^2)\beta^2 +6\gamma^3+2\omega\gamma^2+(6-2\omega^2+2\omega E)\gamma -2\omega=0.
\lambdaabel{beta}
\end{equation}
For $\gamma=1$, that means
\betaegin{equation}
\beta^2=-(E+1) \quad \Rightarrow \quad \lambdaeft\{
\betaegin{array}{lcl}
E> -1 & \Rightarrow & \beta\in i\mathbb{R},\\
E<-1 & \Rightarrow & \beta \in \mathbb{R},
\end{array}
{\tilde{\rho}}ight.
\lambdaabel{gammaplus}
\end{equation}
and for $\gamma=-1$,
\betaegin{equation}
\beta^2=(E+1) \quad \Rightarrow \quad \lambdaeft\{
\betaegin{array}{lcl}
E> -1 & \Rightarrow & \beta\in \mathbb{R},\\
E<-1 & \Rightarrow & \beta \in i\mathbb{R},
\end{array}
{\tilde{\rho}}ight.
\lambdaabel{gammaminus}
\end{equation}
In both cases, the two-fold degeneracy of $\gamma$ is lifted by the two possible values for $\beta(\gamma)$, making all four asymptotic solutions linearly independent.
There are now two different parts of the spectrum, $E>-1$ and $E<-1$.
The threshold energy is $E_0=-1$, in accord with \cite{lo_20,chan_20}.
The four Stokes sectors are given in ({\tilde{\rho}}ef{sectors}). The ``critical'' line with $\alpharg(\phantomim z)=-\alpharg(\gamma)/2$ is the real axis for $\gamma=1$ and the imaginary axis for $\gamma=-1$. Let's first look at $\gamma=1$ and $E<-1$. The integral in ({\tilde{\rho}}ef{condasym}) reads in sector $S_0$,
\betaegin{equation}
I_0(R)=\int_R^\infty {\tilde{\rho}}d x\ e^{2\beta x}\int_{-x}^x {\tilde{\rho}}d y\ e^{-2y^2}(x^2+y^2)^{{\rm Re}({\tilde{\rho}}ho)}e^{-2{\rm Im}({\tilde{\rho}}ho)\alpharctan(y/x)},
\end{equation}
as $\beta$ is real and will only converge if $\beta<0$, independent of ${\tilde{\rho}}ho$. In sector $S_1$, we have instead
\betaegin{equation}
\fl
I_1(R)=\int_R^\infty {\tilde{\rho}}d y\ e^{-2y^2}\int_{-y}^y{\tilde{\rho}}d x\ e^{2\beta x}(x^2+y^2)^{{\rm Re}({\tilde{\rho}}ho)}e^{-2{\rm Im}({\tilde{\rho}}ho)[\phantomii/2-\alpharctan(x/y)]}
\end{equation}
which converges unconditionally. Analogously, the integral converges in sector $S_2$ if $\beta > 0$ and independent of $\beta$ in $S_3$. For $\gamma=-1$ and $E<-1$, the same description obtains with the real and imaginary axes interchanged ($\beta\in i\mathbb{R}$). It follows that the spectral problem is equivalent to a lateral connection problem with non-trivial Stokes multipliers \cite{slavianov_00}, because asymptotic solutions in sectors $S_0$ and $S_2$ ($S_1$ and $S_3$) must have different $\beta$ in case $\gamma=1$ ($\gamma=-1$) for a state with $E<-1$ to be normalizable. There is seemingly also a second possibility, namely that $\gamma=1$ in sectors $S_{1,3}$ and $\gamma=-1$ in sectors $S_{0,2}$ which also corresponds to off-diagonal Stokes matrices but involving the exponents of second kind with order two. We will see later that this case cannot occur, but the first one leads to a normalizable eigenstate if the parameter $E$ is determined such that the lateral connection problem is solved by $\phantomi_1(z)$. Indeed we know from the direct calculation \cite{lo_20,chan_20} (see also \cite{duan_16}) that there are normalizable states below $E<-1$, contrary to the statement made in \cite{mac_17}, where it is claimed that such states cannot exist for any $E$.
For $E>-1$ and $\gamma=1$, $\beta$ is imaginary and we have in sector $S_0$,
\betaegin{equation}
\fl
I_0(R) = \int_R^\infty{\tilde{\rho}}d x\int_{-x}^x{\tilde{\rho}}d y\
(x^2+y^2)^{{\rm Re}({\tilde{\rho}}ho)}e^{-2{\rm Im}({\tilde{\rho}}ho)\alpharctan(y/x)} e^{-2y^2-2{\rm Im}(\beta)y}.
\end{equation}
An estimate similar to ({\tilde{\rho}}ef{intsg}) yields now that $I_0(R)$ will be finite if ${\rm Re}({\tilde{\rho}}ho)<-1/2$. The same condition applies to $S_2$, while the integral converges in sectors $S_{1,3}$. For $g=-1$ (real $\beta$) normalizability of a state with $E>-1$ imposes the same condition on ${\tilde{\rho}}ho$ and $S_{0,2}$ replaced by $S_{1,3}$.
We find for $\gamma=1$, ${\tilde{\rho}}ho(1)=-8/5$ and for $\gamma=-1$, ${\tilde{\rho}}ho(-1)=0$. According to the argument for the case $\omega<2$, one would conclude that there is a continuum of normalizable solutions with $\gamma=1$ and $E>-1$, as the integral in the critical sector converges in this case without further condition on $E$. Does this mean that ${\cal H}p$ is not self-adjoint also at the critical point? No -- because no eigenstate of ${\cal H}p$ at $\omega=2$ can have $\phantomi_1$-asymptotics with $\gamma=1$. Only $\gamma=-1$ is a valid exponent of second kind with order two for $\phantomi_1(z)$ satisfying the system ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}). To see this, let's use the simplification of the system at $\omega=2$ mentioned in the introduction, namely
\betaegin{eqnarray}
(\phantomiartial_z^2+2z\phantomiartial_z +z^2)&=(\phantomiartial_z+z)^2 -{1\!\!1}&=2\hat{x}^2 -{1\!\!1},\\
(\phantomiartial_z^2-2z\phantomiartial_z +z^2)&=(\phantomiartial_z-z)^2 +{1\!\!1}&=-2\hat{p}^2 +{1\!\!1}.
\end{eqnarray}
The system ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) becomes
\betaegin{eqnarray}
2\hat{x}^2\phantomi_1 &= (E+1)\phantomi_1-\Delta\phantomi_2,
\lambdaabel{phi1eq}\\
2\hat{p}^2\phantomi_2 &= (E+1)\phantomi_2-\Delta\phantomi_1.
\lambdaabel{phi2eq}
\end{eqnarray}
It can be easily analyzed in the space $L^2(\mathbb{R})$ \cite{lo_20,chan_20}, but the asymptotics of the eigenstates can be inferred already from the special case $\Delta=0$. We find that $\phantomi_1(z)$ is an eigenstate of the position operator $\hat{x}$ with eigenvalue $x_0=\sigmaqrt{E+1}/\sigmaqrt{2}$ which agrees with the value of $\beta$ in ({\tilde{\rho}}ef{gammaminus}), the value ${\tilde{\rho}}ho=0$ found above for $\gamma=-1$ and the form of the corresponding generalized eigenstate of $\hat{x}$ in ({\tilde{\rho}}ef{genstates}). Therefore, only $\gamma=-1$ may occur in the asymptotic form of $\phantomi_1$. This is not changed for $\Delta\neq 0$ because $\phantomi_1$ satisfies then a Schr\"odinger equation on the real line with a potential approaching zero asymptotically \cite{lo_20,chan_20}, therefore the asymptotic form of the scattering states are the same as for $\Delta=0$. Likewise, for $\phantomi_2$, which is the image of $\phantomi_1$ under Fourier transformation (or $z {\tilde{\rho}}ightarrow iz$), its asymptotic form must have $\gamma=1$.
Because the formal solutions $\phantomi_1(z)$ with $\gamma=-1$, real $\beta$
and ${\tilde{\rho}}ho=0$ cannot be made normalizable by adjusting $E$ but are asymptotically generalized eigenstates of $\hat{x}$ for all $E>-1$, they must correspond to the continuous spectrum of the self-adjoint operator ${\cal H}p$ in this range of energy. Moreover, as $\gamma$ is fixed to $-1$ for $\phantomi_1(z)$ in each Stokes sector, the option to have normalizable states for $E>-1$, namely $\gamma=-1$ in sectors $S_{0,2}$ and $\gamma=1$ in sectors $S_{1,3}$, is ruled out. It follows that no bound states are embedded into the continuum as it appears in the Rabi-Stark model \cite{chen_20}. The same argument applies for $E<-1$, however, here we must assume $\Delta\neq 0$ to provide a potential allowing for bound states. The asymptotic form $\tilde{\phantomihi}_2(x)$ of a solution to ({\tilde{\rho}}ef{phi2eq}) in $L^2(\mathbb{R})$ ($\hat{p}=-i\phantomiartial_x$) is obviously $\tilde{\phantomihi}_2(x) \sigmaim \exp(-\sigmaqrt{-(E+1)/2}|x|)$ which corresponds to ({\tilde{\rho}}ef{gammaplus}) with two different $\beta$ in sectors $S_0$ and $S_2$, the first case mentioned above. The second case, different $\gamma$ in sectors $S_{0,2}$ and $S_{1.3}$ is not realized. Therefore, the exponent of second kind with order two is unique throughout the complex plane and the Stokes phenomenon manifests only in the subleading exponent of order one.
\sigmaection{Pure point spectrum for $\omega>2$}
\lambdaabel{purepoint}
Whereas for $\omega\lambdae 2$ the spectrum of ${\cal H}p$ can be understood completely by analyzing the singularity at $z=\infty$, because ${\cal H}p$ is either not self-adjoint ($\omega<2$) or reducible to the Schr\"odinger equation ({\tilde{\rho}}ef{phi2eq}) ($\omega=2$), the spectral problem is non-trivial for $\omega>2$. In this case, ${\cal H}p$ has only discrete eigenvalues and normalizable eigenfunctions, because the exponents of second kind with order two are non-degenerate solutions of ({\tilde{\rho}}ef{origamma}). The spectral condition amounts to the requirement that the formal solution of ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) has only normalizable asymptotics in all four Stokes sectors ({\tilde{\rho}}ef{sectors}).
A sufficient condition for this to happen is clearly that only exponents $\gamma_{1,2}$
appear in the asymptotic expansion of $\phantomi_1(z)$ which occurs only for a discrete set $E_n$, $n=0,1,2 \deltaots$, of the parameter $E$.
We call any function $G_H(E)$ a ``$G$-function'' for $H$, if it has only zeros on the real axis, coinciding with all (respectively the regular subset of) the discrete eigenvalues $E_n < \infty$ of the self-adjoint operator $H$. $G_H(E)$ is thus a generalized spectral determinant,
\betaegin{equation}
G_H(E)=e^{h(E)}\phantomirod_{n=0}^\infty\lambdaeft(1-\frac{E}{E_n}{\tilde{\rho}}ight),
\lambdaabel{spec-det}
\end{equation}
where $h(E)$ may be any function bounded from below such that the right hand side of ({\tilde{\rho}}ef{spec-det}) converges for all finite $E$. The spectral problem can be considered ``solved'' if one finds a $G$-function for $H$, which is, of course, not unique. Note that $h(E)$ does not need to be bounded from above -- which will turn out to be rather useful in section {\tilde{\rho}}ef{chen}.
It is often convenient to define $G$-functions for different sectors of the spectrum related to invariant subspaces under the symmetries of $H$. In the case of ${\cal H}p$, we have a $\mathbb{Z}_4$-symmetry generated by $\hat{P}=\exp(i(\phantomii/2)\alphad a)\sigma_x$. The four eigenvalues of $\hat{P}$ are $\phantomim 1, \phantomim i$ and label four invariant subspaces of $\cal H$. The $\mathbb{Z}_2$-subgroup generated by $\hat{P}^2$ acts only in the bosonic part of $\cal H$ with eigenvalues $\phantomim 1$ corresponding to even and odd functions of $z$. We denote each invariant subspace as ${\cal H}_{ab}$ with $a,b\in\{+,-\}$ corresponding to the eigenvalues of $\hat{P}^2$ and $\hat{P}$: $a= +$ [$-$] for $\phantomi_{1,2}(z)$ even [odd] and $b=\phantomim$ if $\phantomi_1(iz) =\phantomim\phantomi_2(z)$ for even $\phantomi_{1,2}(z)$, resp. $\phantomi_1(iz) =\phantomim i\phantomi_2(z)$ for odd $\phantomi_{1,2}(z)$. These relations between $\phantomi_1(z)$ and $\phantomi_2(z)$ apply to their asymptotic forms ({\tilde{\rho}}ef{asymstokes}) and therefore relate the asymptotics of $\phantomi_1(z)$ in Stokes sectors $S_0$ and $S_2$ to the asymptotics of $\phantomi_2(z)$ in sectors $S_1$ and $S_3$ and vice versa. The asymptotic relations among $\mathbb{Z}_4$-eigenstates will be of importance in the following.
The eigenstates of ${\cal H}p$ belonging to different subspaces may have degenerate eigenvalues at certain points in the parameter space and manifest as level crossings in the spectral graph. In the 2pQRM the level crossings are of two different types: The first, the so-called Juddian case, describes level crossings between ${\cal H}_{++}$ and ${\cal H}_{+-}$, respectively between
${\cal H}_{-+}$ and ${\cal H}_{--}$. It has been investigated in \cite{emary_02,zhang_13}. The two-fold degeneracy of the Juddian solutions corresponds to a two-dimensional representation of $\mathbb{Z}_4/\mathbb{Z}_2 \sigmaim \mathbb{Z}_2$. The second type describes also two-fold degeneracies but between subspaces with different eigenvalues of $\hat{P}^2$. They have been studied in \cite{mac_19}. In both cases the eigenfunctions can be given in terms of either elementary functions or functions belonging to the hypergeometric class. The degenerate eigenvalues belong to the exceptional spectrum \cite{braak_11} and are not the subject of the present paper which concerns the regular spectrum comprising the non-degenerate eigenvalues.It has to be noted that the non-degenerate spectrum may contain certain eigenvalues at isolated points in parameter space (the so-called non-degenerate exceptional spectrum) which have to be dealt with separately, see \cite{braak_19} and section {\tilde{\rho}}ef{chen}. We confine the following discussion to the subspace ${\cal H}_{++}$, containing only even functions of $z$.
\sigmaubsection{The $G$-functions based on recurrence relations}
\lambdaabel{recur}
The eigenvalue $+1$ of $\mathbb{Z}_4/\mathbb{Z}_2$ means that a solution of ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) belonging to ${\cal H}_{++}$ satisfies $\phantomi_1(iz)=\phantomi_2(z)$. If $\phantomi_{1,2}(z)$ are even, their expansion in powers of $z$ around $z=0$ reads
\betaegin{equation}
\phantomi_1(z)=\sigmaum_{n=0}^\infty a_nz^{2n}, \quad \phantomi_2(z)=\sigmaum_{n=0}^\infty (-1)^na_nz^{2n},
\lambdaabel{z4rel}
\end{equation}
with arbitrary $a_0\neq 0$.
From ({\tilde{\rho}}ef{orig1}) we deduce the following three-term recurrence relation for the
$a_n$,
\betaegin{equation}
(2n+2)(2n+1)a_{n+1}+(2n\omega-E+(-1)^n\Delta)a_n+a_{n-1}=0,
\lambdaabel{origrecur}
\end{equation}
and $a_n=0$ for $n<0$.
This recurrence relation is the starting point for the $G$-functions proposed in \cite{zhang_17} and \cite{mac_17}.
A Poincar\'e analysis of ({\tilde{\rho}}ef{origrecur})
with the asymptotic ansatz
\betaegin{equation}
a_n \sigmaim \frac{x^n}{n!} \quad \textrm{for} \quad n{\tilde{\rho}}a \infty,
\lambdaabel{asympa}
\end{equation}
leads to the equation for $x$,
\betaegin{equation}
x^2+\frac{\omega}{2}x+\frac{1}{4}=0,
\lambdaabel{Z-equ}
\end{equation}
with solutions
\betaegin{equation}
x_+=\frac{1}{4}\lambdaeft(\sigmaqrt{\omega^2-4}-\omega{\tilde{\rho}}ight) = \frac{\gamma_1}{2}, \quad
x_-=-\frac{1}{4}\lambdaeft(\sigmaqrt{\omega^2-4}+\omega{\tilde{\rho}}ight) = \frac{\gamma_4}{2}.
\lambdaabel{asympx}
\end{equation}
As $|x_-|>|x_+|$, the Perron-Kreuser theorem \cite{gautschi_67} states that there exist a minimal solution to the recurrence ({\tilde{\rho}}ef{origrecur}) with
\betaegin{equation}
\frac{a_{n+1}}{a_n} \sigmaim \frac{x_+}{n} \quad \textrm{for} \quad n{\tilde{\rho}}a \infty,
\end{equation}
and ({\tilde{\rho}}ef{asympa}) suggests that the asymptotic form of $\phantomi_1(z)$ for $|z|{\tilde{\rho}}a\infty$ reads in this case
\betaegin{equation}
\phantomi_1(z) = \exp\lambdaeft(\frac{\gamma_1}{2}z^2 +r(z){\tilde{\rho}}ight)\lambdaeft[c_0+ \textrm{less divergent terms}{\tilde{\rho}}ight],
\lambdaabel{asympZ}
\end{equation}
where $r(z)$ grows at most linear in $z$ but is undetermined by the asymptotics of $a_n$.
A function with these leading asymptotics in all Stokes sectors corresponds to a normalizable solution of ({\tilde{\rho}}ef{orig4}). Because Pincherle's theorem states that the minimal solution of a three-term recurrence relation, if it exists, is determined by a convergent continued fraction \cite{gautschi_67} (see also \cite{braak_cont}), one may write down a $G$-function for the spectral problem in ${\cal H}_{++}$ as follows \cite{zhang_17},
\betaegin{equation}
G_Z(E)=\frac{E-\Delta}{2} -V_1(E),
\lambdaabel{zhangG}
\end{equation}
with
\betaegin{equation}
\fl
V_n(E)= \frac{a_n^{{\textrm{\tiny min}}}}{a_{n-1}^{{\textrm{\tiny min}}}}=\frac{-1}{2n\omega +(-1)\Delta-E+(2n+2)(2n+1)V_{n+1}}, \quad n=1,2,\lambdadots,
\end{equation}
and $(E-\Delta)/2=a_1/a_0$, as determined from the initial condition for ({\tilde{\rho}}ef{origrecur}). The vanishing of $G_Z(E)$ at some point $E_s$ means that the minimal solution of ({\tilde{\rho}}ef{origrecur}) whose first terms are given by the
continued fraction $V_1(E_s)=a_1^{{\textrm{\tiny min}}}/a_{0}^{{\textrm{\tiny min}}}$ matches the initial condition $a_n=0$ for negative $n$, which is dictated by the analyticity of $\phantomi_1(z)$ at $z=0$. The state is therefore normalizable because the asymptotic behavior of $\phantomi_1(z)$ apparently contains only the exponent $\gamma_1$, as guaranteed by the minimality of the solution to ({\tilde{\rho}}ef{origrecur}). It follows that $E_s$ belongs to the pure point spectrum of ${\cal H}p$ if $G_Z(E_s)=0$.
Zhang's approach is certainly the simplest and most elegant way to obtain a $G$-function for ${\cal H}p$ in the regime $\omega>2$. Following Okubo \cite{okubo_63}, Maciejewski and Stachowiak \cite{mac_17} have proposed a method which is based as well on the determination of those solutions of a three-term (matrix) recurrence relation which correspond to normalizable asymptotics for $\phantomi_{1,2}$ because they contain only exponents $\gamma_{1,2}$. After ``gluing'' them to the initial conditions as above, they obtain another $G$-function ((43) in \cite{mac_17}) which is given in terms of a contour integral over the solution of an auxiliary differential equation.
Coming back to ({\tilde{\rho}}ef{Z-equ}), we have $|x_+|=|x_-|$ for $\omega\lambdae 2$, a minimal solution does not exist and the continued fraction does not converge. One would thus conclude with the authors of \cite{mac_17} that in this regime there are no normalizable solutions of ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) satisfying in addition $\phantomi_1(iz)=\phantomi_2(z)$. However, as we have seen in section {\tilde{\rho}}ef{general}, the pure point spectrum is not empty for $\omega=2$ and \emph{all} formal solutions of ({\tilde{\rho}}ef{orig4}) are normalizable for $\omega < 2$. The discrepancy is resolved by noting that the Poincar\'e analysis of the asymptotic behavior of the recurrence relation only yields the leading, most divergent term in the asymptotic expansion and says nothing about the subleading terms which determine normalizability in the case $|\gamma|=1$. The formal solution ({\tilde{\rho}}ef{z4rel}) may or may not be normalizable even if no minimal solution of ({\tilde{\rho}}ef{origrecur}) exists.
Another consequence is the missing Stokes phenomenon: The minimal solution of the recurrence relation for $\phantomi_1(z)$ fixes the allowed exponent to $\gamma_1$, independent of the argument of $z$. $\phantomi_1(z)$ should approach zero in sectors $S_{0,2}$ whereas $\phantomi_2(z)$ diverges there, obviously contradicting ({\tilde{\rho}}ef{orig1}) if $\Delta\neq 0$. In fact, the form ({\tilde{\rho}}ef{asympZ}) of the asymptotic behavior of $\phantomi_1(z)$ is only valid in sectors $S_{1,3}$ where the term $\exp(\gamma_1z^2/2)$ diverges for $|z|{\tilde{\rho}}a\infty$. In the sectors $S_{0,2}$, where it is recessive, subdominant contributions in the recurrence ({\tilde{\rho}}ef{origrecur}) determine the asymptotic behavior of $\phantomi_1(z)$, given in these sectors as $\phantomi_1(z)\sigmaim \exp(\gamma_2z^2/2)$.
Because the asymptotics of the recurrence relation does not provide the correct behavior of $\phantomi_1(z)$ in all Stokes sectors, one may ask whether indeed all normalizable states satisfying ({\tilde{\rho}}ef{orig1}), ({\tilde{\rho}}ef{orig2}) are related to the minimal solution of ({\tilde{\rho}}ef{origrecur}). For certain values of $E$, there could be an entire function $\phantomi(z)$ behaving asymptotically like $\exp(\gamma_4z^2/2)$ in sectors $S_{0,2}$ and as $\exp(\gamma_3z^2/2)$ in $S_{1,3}$. This function would be normalizable although it belongs to a dominant solution of ({\tilde{\rho}}ef{origrecur}). If this happens, there would be elements of the spectrum not given by zeros of $G_Z(E)$. The same argument applies to the $G$-function of \cite{mac_17}. The phenomenon occurs (for different reasons) in the QRM with linear coupling, where the non-degenerate exceptional spectrum cannot be obtained by Schweber's $G$-function based on continued fractions \cite{braak_16}.
In the present case of the 2pQRM, such an exceptional eigenstate cannot occur, at least not in the non-degenerate part of the spectrum, due to the relation $\phantomi_1(z)=\alpha\phantomi_2(iz)$ with $\alpha\in\{\phantomim1,\phantomim i\}$. It entails that if $\phantomi_1(z)$ has strict recessive behavior in a given sector with exponent $\gamma$, $\phantomi_2(z)$ will have exponent $-\gamma$ in the same sector and would not be normalizable if $\gamma$ is either $\gamma_3$ or $\gamma_4$. Indeed, the $\mathbb{Z}_4$-symmetry of the solution shows that exponents $\gamma_1$ and $\gamma_2$ appear in the asymptotics of both $\phantomi_1$ and $\phantomi_2$ in all sectors but not $\gamma_3$ or $\gamma_4$ if the state is normalizable.
Thus we may conclude that $G_Z(E)$ yields the complete spectrum for $\omega >2$. The argument is less clear for the $G$-function based on Okubo's method \cite{mac_17}. Because the factorial power series in $x_+$ is actually a double series (see (34) in \cite{mac_17}, $x_+$ is called $u_0$), it is not obvious that the asymptotics have the form $\phantomi_1(z)\sigmaim \exp(\phantomim x_+z^2)$ in all sectors.
\betaegin{figure}
\includegraphics[width=0.6\lambdainewidth]{fig1.eps}
\caption{Zhang's $G$-function in sector ${\cal H}_{++}$ for parameters $\omega=2.5$ and $\Delta=0.7$. The poles of $G_Z(E)$ move closer to its zeros $E_n$ for larger $E$.}
\lambdaabel{zgfunction}
\end{figure}
Zhang's $G$-function is shown in \textbf{Figure {\tilde{\rho}}ef{zgfunction}}. It has poles and zeros
which move closer together for higher values of $E$, as typical for a continued fraction. The resolution of higher zeros poses thus known numerical problems but the major drawback of $G_Z(E)$ is the lack of qualitative knowledge about the distribution of its zeros, constituting the regular spectrum of ${\cal H}p$ in ${\cal H}_{++}$. The poles are all of first order and $G_Z(E)$ is monotonous, but the location of the poles is just as unknown as the zeros -- they do not allow to restrict the distribution of the zeros as the $G$-functions for the QRM which possess poles at known positions, namely $E=n\omega$, with $n$ a non-negative integer \cite{braak_11,zhong_13,mac_14}. The $G$-function proposed in \cite{mac_17} has the same problem: The exact eigenvalues of ${\cal H}p$ can be extracted by a numerical procedure but this has no advantage compared with direct diagonalization in a truncated Hilbert space because the graph of the $G$-function has no properties, say the position of maxima or minima, which are known in closed form (see Figures 1 and 2 in \cite{mac_17}). On the other hand, exact diagonalization in a state space of finite dimension suggests the collapse of the complete spectrum to a single point at $\omega_c$, a numerical artifact of the truncation. But the $G$-functions based only on the recurrence relation ({\tilde{\rho}}ef{origrecur}) offer no option to study the pure point spectrum arbitrarily close to $\omega_c$ in a qualitative way.
\sigmaubsection{The $G$-functions based on the $\mathbb{Z}_4$-symmetry}
\lambdaabel{chen}
The recurrence ({\tilde{\rho}}ef{origrecur}) implements the $\mathbb{Z}_4$-symmetry directly via the relation ({\tilde{\rho}}ef{z4rel}) between the series expansions of $\phantomi_1(z)$ and $\phantomi_2(z)$. It is therefore automatically satisfied by all solutions of ({\tilde{\rho}}ef{origrecur}), independent of their normalizability. The question arises whether the $\mathbb{Z}_4$-symmetry can be used to discern normalizable from non-normalizable solutions as in the linear QRM. In the latter case, the two regular singular points of the eigenvalue equation are located at $z=\phantomim g$ and mapped onto each other by the $\mathbb{Z}_2$-symmetry $z{\tilde{\rho}}a -z$ of that model. This can be used to construct a $G$-function which has only zeros at those energies corresponding to solutions $\phantomi_1(z)$,$\phantomi_2(z)$ which are analytic in the whole complex plane by using Frobenius expansions which are analytic in the vicinity of only one of the singular points -- the symmetry requirement $\phantomi_2(z)=\phantomi_1(-z)$ is then equivalent to analyticity at the other singular point \cite{braak_11}.
The situation is different for the 2pQRM: Here there is only a single singular point at $z=\infty$ but with higher rank, preventing all formal solutions to be normalizable, although all are entire. The invariant points of the map $z{\tilde{\rho}}a iz$ are $z=0,\infty$ as in the QRM and $z=0$ is an ordinary point of the system ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}). The symmetry generator acts trivially on the point $z=\infty$ and maps the asymptotics of $\phantomi_{1,2}(z)$ onto each other: $\gamma_1\lambdaeftrightarrow\gamma_2$ and $\gamma_3\lambdaeftrightarrow\gamma_4$. Clearly, because $z{\tilde{\rho}}a iz$ is an isometry of ${\cal B}$, the normalizable forms are mapped onto themselves, as well as the non-normalizable ones.
Trav{\v e}nec \cite{travenec_12} has attempted to use the $\mathbb{Z}_4$-symmetry to derive a $G$-function for ${\cal H}p$ in analogy with the linear QRM as follows. First, separate the asymptotic factor $\exp(\gamma_1z^2/2)$ from the expansions of $\phantomi_{1,2}$,
\betaegin{equation}
\phantomi_j(z)=e^{\frac{\gamma_1}{2}z^2}\betaar{\phantomi}_j(z), \quad
\betaar{\phantomi}_j(z)=\sigmaum_{n=0}^\infty a^j_n z^{2n}, \quad j=1,2.
\end{equation}
The coefficients $a^j_n$ satisfy the coupled recurrence relations
\betaegin{eqnarray}
\fl
(2n+2)(2n+1)a^1_{n+1} &=& (E-\gamma_1-2n(2\gamma_1+\omega))a_n^1 -\Delta a_n^2,
\lambdaabel{trav1}\\
\fl
(2n+2)(2n+1)a^2_{n+1} &=& (-E-\gamma_1-2n(2\gamma_1-\omega))a_n^2 + \Delta a_n^1 +2\omega\gamma_1a^2_{n-1},
\lambdaabel{trav2}
\end{eqnarray}
with initial condition $a_0^1=a_0^2=1$.
The symmetry $\phantomi_1(iz)=\phantomi_2(z)$ is not assumed beforehand in the system ({\tilde{\rho}}ef{trav1}),({\tilde{\rho}}ef{trav2}).
The $G$-function is then defined as
\betaegin{equation}
G_T(E) = \phantomi_2(-iz_0)-\phantomi_1(z_0),
\lambdaabel{Gtrav}
\end{equation}
with $z_0\neq 0$ an arbitrary point in the complex plane. The zeros of $G_T(E)$ should not depend on the choice of $z_0$. The problem with this approach is that the implementation of the symmetry condition in ({\tilde{\rho}}ef{Gtrav}) cannot discern between normalizable and non-normalizable solutions. (Non-)normalizability is an invariant property of any function under $z{\tilde{\rho}}a iz$, as we see from the behavior of the possible asymptotic forms.
Moreover, the initial condition $\phantomi_1(0)=\phantomi_2(0)$ enforces the symmetry for any solution of ({\tilde{\rho}}ef{trav1}),({\tilde{\rho}}ef{trav2}). The function $G_T(E)$ vanishes therefore identically \cite{mac_15-2}. Nevertheless, a straightforward numerical implementation of ({\tilde{\rho}}ef{Gtrav}) yields a function which does not vanish identically and possesses zeros at the exact eigenenergies of ${\cal H}p$ in ${\cal H}_{++}$. Figure~{\tilde{\rho}}ef{tgfunction} shows a plot of $G_T(E)$ for the same parameters as in Figure~{\tilde{\rho}}ef{zgfunction}. The zeros of $G_Z(E)$ and $G_T(E)$ coincide, but the non-zero values of $G_T(E)$ are extremely large.
\betaegin{figure}
\includegraphics[width=0.6\lambdainewidth]{fig2.eps}
\caption{Trav{\v e}nec's $G$-function for $\omega=2.5$ and $\Delta=0.7$ with $z_0=5+5i$. The real (magenta) and imaginary (green) parts of $G_T(E)$ share the same zeros which are located at the eigenenergies of ${\cal H}p$.}
\lambdaabel{tgfunction}
\end{figure}
Both observations, the numerical validity of $G_T(E)$ and its large values, have the same origin, namely the fact that a three-term recurrence relation has a unique minimal solution. The numerical evaluation of ({\tilde{\rho}}ef{trav1}),({\tilde{\rho}}ef{trav2}) will give valid results only if the initial conditions including the parameter $E$ are fine-tuned to yield the minimal solution \cite{gautschi_67, braak_13}. The evaluation of dominant solutions is always numerically unstable, leading to the exponential accumulation of rounding errors which produce the huge non-zero values of $G_T(E)$ for any value of $E$ not belonging to the spectrum. Only if $E$ coincides with some $E_n$, the recurrence yields the (normalizable) minimal solution and the numerics will reproduce $G_T(E)=0$, the value of $G_T(E)$ for \emph{any} $E$ if evaluated exactly. The same mechanism appears also in the $G$-function for the oscillator with quartic anharmonicity proposed by Lay \cite{lay_97}.
Trav{\v e}nec's $G$-function is thus numerically exact in the same sense as Lay's method to compute the spectrum of the quartic oscillator. But as it is only non-zero due to numerical instabilities, $G_T(E)$ cannot be used to draw qualitative inferences about the spectrum in an analytical way because it has no exactly known properties. In this respect, there is no difference between Trav{\v e}nec's \cite{travenec_12}, Zhang's \cite{zhang_17} or Maciejewski's $G$-functions \cite{mac_17}.
The root of the problem is the invariance of the singularity of ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) and its asymptotic solutions under the symmetry operation $z{\tilde{\rho}}a iz$. Therefore, this system of differential equations has to be transformed in such a way that the symmetry
acts non-trivially on its singularity structure.
The bosonic Bogoliubov transformation is well established in quantum optics to describe squeezed light \cite{klimov_09} and was used by Chen {\it et al.} in \cite{chen_12} to derive a $G$-function for the 2pQRM. The derivation was heuristic in the sense that it did not establish a proof that the zeros of this function correspond to normalizable eigenstates of ${\cal H}p$. We shall show that this is indeed the case. In $L^2(\mathbb{R})$, the Bogoliubov transformation amounts to the isometry
\betaegin{equation}
I_\theta[\phantomisi](x)=e^{\theta/2}\phantomisi(e^\theta x) =
\exp\lambdaeft(\frac{\theta}{2}\lambdaeft(a^2-\alphadq{\tilde{\rho}}ight){\tilde{\rho}}ight)\phantomisi(x),
\end{equation}
for $\theta\in\mathbb{R}$. Because $\theta$ is real, the operator acting on $\phantomisi(x)$ is a unitary representation of an element of $SU(1,1)$ with generator
\betaegin{equation}
\frac{1}{2}\lambdaeft(a^2-\alphadq{\tilde{\rho}}ight)= \hat{Y}-\hat{X},
\end{equation}
where $\hat{X}=(1/2)\alphadq$, $\hat{Y}=(1/2)a^2$ and $\hat{Z}=\alphad a +(1/2){1\!\!1}$ fulfill the commutation relations of $\mathfrak{sl}_2(\mathbb{R})$,
\betaegin{equation}
[\hat{Z},\hat{X}]=2\hat{X}, \quad [\hat{Z},\hat{Y}]=-2\hat{Y}, \quad [\hat{X},\hat{Y}]=-\hat{Z}.
\end{equation}
The creation and annihilation operators transform under $I_\theta$ as
\betaegin{eqnarray}
a_\theta &= I_\theta a I^{-1}_\theta &=\textrm{ch}(\theta) a +\sigmah(\theta)\alphad,\nonumber\\
\alphad_\theta &= I_\theta \alphad I^{-1}_\theta &=\textrm{ch}(\theta)\alphad +\sigmah(\theta) a.\nonumber
\end{eqnarray}
Applying $I_\theta$ in the Bargmann space ${\cal B}$ to the system ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) leads to
\betaegin{eqnarray}
\omega_1z\varphi_1^{(1)} - E_1\varphi_1 +\Delta\varphi_2 &=& 0,
\lambdaabel{eq1}\\
\Gamma\varphi_2^{(2)} + \omega_2 z\varphi_2^{(1)} +(\Gamma z^2+E_2)\varphi_2 - \Delta\varphi_1 &=& 0,
\lambdaabel{eq2}
\end{eqnarray}
if $\textrm{th}(2\theta)=-2/\omega$ which requires $\omega>2$.
We have set $\varphi_j(z)=I_\theta[\phantomi_j](z)$.
The parameters are
\betaegin{eqnarray}
\Gamma &=& 2\textrm{ch}(2\theta),\nonumber\\
E_1 &=& E - \sigmah(2\theta) -\omega\sigmah^2(\theta),\nonumber\\
E_2 &=& E + \sigmah(2\theta) -\omega\sigmah^2(\theta),
\lambdaabel{params}\\
\omega_1 &=& -\sigmah(2\theta)\lambdaeft(\frac{\omega^2}{2}-2{\tilde{\rho}}ight) > 0,\nonumber\\
\omega_2 &=& \sigmah(2\theta)\lambdaeft(\frac{\omega^2}{2}+2{\tilde{\rho}}ight) < 0\nonumber.
\end{eqnarray}
The transformed system ({\tilde{\rho}}ef{eq1}),({\tilde{\rho}}ef{eq2}) has a singularity structure different from ({\tilde{\rho}}ef{orig1}),({\tilde{\rho}}ef{orig2}) but it is still invariant under $z{\tilde{\rho}}a -z$, so the solutions are even or odd functions of $z$. Upon elimination of $\varphi_2(z)$, we obtain an equation of third order for $\varphi_1(z)$,
\betaegin{equation}
\varphi_1^{(3)}+\lambdaeft(b_1z+\frac{b_2}{z}{\tilde{\rho}}ight)\varphi_1^{(2)}
+(z^2+b_3)\varphi_1^{(1)} +\lambdaeft(-\frac{E_1}{\omega_1}z+\frac{b_4}{z}{\tilde{\rho}}ight)\varphi_1=0,
\lambdaabel{eq3}
\end{equation}
with abbreviations
\betaegin{eqnarray}
b_1 &=& \frac{\omega_2}{\Gamma},\nonumber\\
b_2 &=& 2-\frac{E_1}{\omega_1},
\lambdaabel{abbrev}\\
b_3 &=& \frac{E_2\omega_1 - E_1\omega_2 +\omega_1\omega_2}{\Gamma\omega_1},\nonumber\\
b_4 &=& \frac{\Delta^2-E_1E_2}{\Gamma\omega_1}\nonumber.
\end{eqnarray}
The equation ({\tilde{\rho}}ef{eq3}) has a regular singular point at $z=0$ and an unramified irregular singular point at $z=\infty$ of rank two and class two. The indicial equation at $z=0$ has three solutions corresponding to exponents ${\tilde{\rho}}ho=0,1, E_1/\omega_1$ in the Frobenius expansion
\betaegin{equation}
\varphi_1(z)=z^{\tilde{\rho}}ho\sigmaum_{n=0}^\infty a_nz^n,
\end{equation}
around $z=0$. The first two correspond to even and odd solutions analytic at $z=0$. The third solution with ${\tilde{\rho}}ho=E_1/\omega_1$ is only entire if $E_1/\omega_1$ is a non-negative integer. This is in close analogy to the linear QRM, where the exceptional spectrum is characterized by $E/\omega\in \rm{I\!N}_0$ \cite{braak_19}.
For now we assume $E_1/\omega_1\noindenttin \mathbb{Z}$. The other cases will be discussed later. The irregular singular point is $z=\infty$, as in ({\tilde{\rho}}ef{orig4}). Rank two entails that the asymptotics of $\varphi_1(z)$ have again the form ({\tilde{\rho}}ef{origasym}). The equation for $\gamma$ reads now
\betaegin{equation}
\gamma^3+\frac{\omega_2}{\Gamma}\gamma^2 +\gamma=0,
\lambdaabel{gamma}
\end{equation}
with solutions $0$, $\omega/2$ and $2/\omega$. The first solution $\gamma=0$ corresponds to the fact that the class of the singularity is not maximal \cite{ince_12}. The indicial equation has degree one, meaning that there is a solution $\varphi_1(z)$ where $z=\infty$ could be a regular singular point. The asymptotic expansion has then the form
\betaegin{equation}
\varphi_1(z)=z^{\tilde{\rho}} \sigmaum_{n=0}^\infty c_nz^{-n}.
\lambdaabel{regasym}
\end{equation}
Plugging this expansion into ({\tilde{\rho}}ef{eq3}), we find that all odd coefficients vanish, $c_{2m+1}=0$, for $m\in\rm{I\!N}_0$, and the exponent of first kind ${\tilde{\rho}}$ has the unique value
${\tilde{\rho}}=E_1/\omega_1$. The even coefficients $c_{2m}$ are determined recursively
\betaegin{eqnarray}
c_n&=\frac{1}{n}{\cal B}ig(\betaig[b_1({\tilde{\rho}}-n+1)({\tilde{\rho}}-n+2)+b_3({\tilde{\rho}}-n+2)+b_4\betaig]c_{n-2}
\lambdaabel{recur-c}\\
&+(4-n)({\tilde{\rho}}-n+3)({\tilde{\rho}}-n+4)c_{n-4}{\cal B}ig),\nonumber
\end{eqnarray}
for $n\gammae 2$ and $c_{-2}=0$. The coefficients in the three-term recurrence relation ({\tilde{\rho}}ef{recur-c}) are diverging for $n{\tilde{\rho}}a\infty$, therefore the convergence radius of ({\tilde{\rho}}ef{regasym}) is zero. There is, as expected, no solution of ({\tilde{\rho}}ef{eq3}) regular at $z=\infty$ if ${\tilde{\rho}}\noindenttin\mathbb{Z}$. Nevertheless, there is a certain Stokes sector $S^\alphast$ where ({\tilde{\rho}}ef{regasym}) is asymptotically valid for $z\in S^\alphast$ and $|z|{\tilde{\rho}}a\infty$. This means that in $S^\alphast$ the branching behavior of $\varphi_1(z)$ is correctly described by the exponent ${\tilde{\rho}}\noindenttin \mathbb{Z}$ which is the \emph{same} as the non-integer exponent ${\tilde{\rho}}ho=E_1/\omega_1$ of the third Frobenius solution in the vicinity of $z=0$. The global behavior of this solution can be described by a branch-cut running from $z=0$ to $z=\infty$ as shown in Figure~{\tilde{\rho}}ef{branch}.
\betaegin{figure}
\includegraphics[width=0.6\lambdainewidth]{fig3.eps}
\caption{The global behavior of the third solution of ({\tilde{\rho}}ef{eq3}) for $E_1/\omega_1\noindenttin\mathbb{Z}$. The branch-cut runs from zero to $\infty$ within the sector $S^\alphast$.}
\lambdaabel{branch}
\end{figure}
The two other solutions have non-zero exponents of second kind $\gamma$ and $\beta=0$. One of them ($\gamma=\omega/2$) has not normalizable asymptotics and the other ($\gamma=2/\omega$) is normalizable if it is asymptotically valid in $S_{0,2}$. For generic values of $E$, both solutions analytic at $z=0$ will exhibit the exponent $\omega/2$ in sectors $S_{0,2}$ and are thus not elements of ${\cal B}$. If $E$ belongs to the spectrum of ${\cal H}p$ in ${\cal H}_{++}$, the even solution of ({\tilde{\rho}}ef{eq3}) will be entire and normalizable with asymptotics described by $\gamma=2/\omega$ in $S_{0,2}$. The asymptotic exponent ${\tilde{\rho}}$ for $\gamma\neq 0$ depends on $E$ and $\gamma$ but has no influence on the normalizability of $\varphi_1$. Because $\gamma\neq0$, its behavior close to $z=\infty$ is determined by the essential singularity of the factor $\exp((\gamma/2)z^2)$ which is isolated if $\varphi_1$ is entire.
Up to now, we have not employed the $\mathbb{Z}_4$-symmetry which is implemented in ${\cal B}$ by the operator $T$ as follows
\betaegin{equation}
T[\phantomisi](z)=\phantomisi(iz)=\exp\lambdaeft(i\frac{\phantomii}{2}z\deltaz{\tilde{\rho}}ight)\phantomisi(z)=
e^{-i\phantomii/4}\exp\lambdaeft(i\frac{\phantomii}{2}\hat{Z}{\tilde{\rho}}ight)\phantomisi(z).
\lambdaabel{origt}
\end{equation}
$T$ is therefore an element of $SL_2(\mathbb{C})$ like the Bogoliubov transformation $I_\theta$. To compute $T_\theta=I_\theta TI_\theta^{-1}$ and to find the normal ordered form
\betaegin{equation}
g = \exp(\alpha_x\hat{X})\exp(\alpha_z\hat{Z})\exp(\alpha_y\hat{Y}), \quad \alpha_{x,y,z}\in\mathbb{C},
\end{equation}
of any group element $g$ in $SL_2(\mathbb{C})$, it is convenient to use the two-dimensional representation with
\betaegin{equation}
\hat{Z}=\lambdaeft(\!\!
\betaegin{array}{cc}
1&0\\
0&-1
\end{array}
\!\!{\tilde{\rho}}ight), \quad
\hat{X}=\lambdaeft(\!\!
\betaegin{array}{cc}
0&i\\
0&0
\end{array}
\!\!{\tilde{\rho}}ight), \quad
\hat{Y}=\lambdaeft(\!\!
\betaegin{array}{cc}
0&0\\
i&0
\end{array}
\!\!{\tilde{\rho}}ight).
\lambdaabel{sl2}
\end{equation}
We find then
\betaegin{equation}
\fl
I_\theta= \frac{1}{\sigmaqrt{\textrm{ch}(\theta)}}
\exp\lambdaeft(-\textrm{th}(\theta)\frac{z^2}{2}{\tilde{\rho}}ight)
\exp\lambdaeft(-\lambdan\textrm{ch}(\theta)z\deltaz{\tilde{\rho}}ight)
\exp\lambdaeft(\textrm{th}(\theta)\frac{1}{2}\deltadz{\tilde{\rho}}ight),
\lambdaabel{Ibarg}
\end{equation}
and
\betaegin{equation}
\fl
T_\theta= \frac{1}{\sigmaqrt{\textrm{ch}(2\theta)}}
\exp\lambdaeft(\frac{2}{\omega}\frac{z^2}{2}{\tilde{\rho}}ight)
\exp\lambdaeft(\lambdaeft[i\frac{\phantomii}{2}-\lambdan\textrm{ch}(2\theta){\tilde{\rho}}ight]z\deltaz{\tilde{\rho}}ight)
\exp\lambdaeft(\frac{2}{\omega}\frac{1}{2}\deltadz{\tilde{\rho}}ight).
\lambdaabel{Tbarg}
\end{equation}
To compute the image of $\varphi_1(z)$ under $T_\theta$, we write
\betaegin{equation}
\varphi_1(z) =\exp\lambdaeft(\frac{\gamma}{2}z^2{\tilde{\rho}}ight)f(z) + g(z),
\lambdaabel{decomp}
\end{equation}
such that $f(z){\tilde{\rho}}a c_0\neq0$ and $g(z)e^{-(\gamma/2)z^2}{\tilde{\rho}}a 0$ for $|z|{\tilde{\rho}}a\infty$ in $S_{0,2}$. The asymptotic behavior of the image
$T_\theta[\varphi_1](z)$ is determined by the diverging factor in ({\tilde{\rho}}ef{decomp}), which can be expressed through an element of $SL_2(\mathbb{C})$ acting on the constant function,
\betaegin{equation}
\exp\lambdaeft(\gamma\frac{z^2}{2}{\tilde{\rho}}ight)=\exp(\gamma\hat{X})[1](z).
\end{equation}
Using now the two-dimensional representation of $T_\theta$ and $\hat{X}$,
\betaegin{equation}
T_\theta=e^{-i\phantomii/4}
\lambdaeft(\!\!
\betaegin{array}{cc}
i\textrm{ch}(2\theta) & -\sigmah(2\theta)\\
-\sigmah(2\theta) & -i\textrm{ch}(2\theta)
\end{array}
\!\!{\tilde{\rho}}ight), \quad
\exp(\gamma\hat{X})=
\lambdaeft(\!\!
\betaegin{array}{cc}
1 & i\gamma\\
0 & 1
\end{array}
\!\!{\tilde{\rho}}ight),
\end{equation}
we obtain
\betaegin{equation}
T_\theta\exp(\gamma\hat{X})=e^{-i\phantomii/4}\exp(\eta_x\hat{X})\exp(\eta_z\hat{Z})\exp(\eta_y\hat{Y}),
\end{equation}
with
\betaegin{equation}
\eta_x(\gamma)=-\frac{\gamma+\textrm{th}(2\theta)}{\textrm{th}(2\theta)\gamma+1}
=\frac{\frac{2}{\omega}-\gamma}{1-\frac{2}{\omega}\gamma}.
\lambdaabel{eta}
\end{equation}
It follows
\betaegin{equation}
T_\theta\exp\lambdaeft(\gamma\frac{z^2}{2}{\tilde{\rho}}ight)=
e^{-i\phantomii/4}\exp\lambdaeft(\eta_x(\gamma)\frac{z^2}{2}{\tilde{\rho}}ight).
\lambdaabel{image}
\end{equation}
Solutions of ({\tilde{\rho}}ef{eq3}) are mapped by $T_\theta$ onto solutions of
\betaegin{equation}
\fl
\varphi_2^{(3)}+\lambdaeft(b_1z+\frac{b_2-2}{z}{\tilde{\rho}}ight)\varphi_2^{(2)}
+(z^2+b_3)\varphi_2^{(1)} +\lambdaeft(\lambdaeft[2-\frac{E_1}{\omega_1}{\tilde{\rho}}ight]z
+\frac{b_4}{z}{\tilde{\rho}}ight)\varphi_2=0,
\lambdaabel{eq4}
\end{equation}
which is obtained from ({\tilde{\rho}}ef{eq1}),({\tilde{\rho}}ef{eq2}) after elimination of $\varphi_1$.
It has the same singular points as ({\tilde{\rho}}ef{eq3}), the exponents of first kind at $z=0$ are $0$, $1$ and $E_1/\omega_1+2$. As usual, the non-integer exponent differs from the one for $\varphi_1$ by an integer, so that these solutions for $\varphi_1$ and $\varphi_2$ fulfill together the system ({\tilde{\rho}}ef{eq1}),({\tilde{\rho}}ef{eq2}). Clearly, the exponents of second kind at $z=\infty$ are the same as in ({\tilde{\rho}}ef{eq3}), with ${\tilde{\rho}}=E_1/\omega_1-2$. Therefore, the branching structure of the third solution of ({\tilde{\rho}}ef{eq4}) with $\gamma=0$ has the form shown in Fig.{\tilde{\rho}}ef{branch} for the associated solution of ({\tilde{\rho}}ef{eq3}).
Lets assume now that the exponent of $\phantomi_1(z)$ in ({\tilde{\rho}}ef{decomp}) is $\gamma=2/\omega$.
The exponent of its image under $T_\theta$ given by ({\tilde{\rho}}ef{eta}) is zero which means that the dominant part of $\phantomi_1(z)$ is mapped onto the recessive part $g(z)$ of $\varphi_2(z)$ in the decomposition ({\tilde{\rho}}ef{decomp}). Because $T_\theta$ is an isometry, normalizable solutions are mapped onto normalizable ones, so the part diverging as $\exp(z^2/\omega)$ is mapped onto a less diverging part of a normalizable solution of ({\tilde{\rho}}ef{eq4}) which must be analytic at $z=0$. On the other hand, if the exponent of $\varphi_1(z)$ is $\gamma=\omega/2$, the exponent of the image $T_\theta[\varphi_1](z)$ would be infinite according to ({\tilde{\rho}}ef{eta}). This indicates that the corresponding solution of ({\tilde{\rho}}ef{eq4}) cannot be entire, otherwise it would be described by either $\gamma=\omega/2$ or $\gamma=2/\omega$ in sectors $S_{0,2}$. The only possibility is that non-normalizable solutions of ({\tilde{\rho}}ef{eq3}) which are analytic at $z=0$ are mapped by $T_\theta$ onto solutions of ({\tilde{\rho}}ef{eq4}) which are not analytic at $z=0$, with non-integer exponent ${\tilde{\rho}}ho=E_1/\omega_1+2$. Recall that those solutions possess an essential but non-isolated singularity at $z=\infty$, because the asymptotic expansion ({\tilde{\rho}}ef{regasym}) is valid in the Stokes sector $S^\alphast$. Their behavior at infinity cannot be captured by an entire function of order two, indicated by the diverging $\eta_x(\gamma)$. They are not elements of ${\cal B}$ already due to their non-analyticity at $z=0$.
It follows that $T_\theta$ maps the set of normalizable solutions of ({\tilde{\rho}}ef{eq3}) onto the corresponding set of solutions of ({\tilde{\rho}}ef{eq4}). These solutions are all analytic at $z=0$ and have exponent $\gamma=2/\omega$ in sectors $S_{0,2}$. The non-normalizable solutions which are analytic at $z=0$ but have exponent $\gamma=\omega/2$ are mapped onto solutions which are \emph{not} analytic at $z=0$ but have there a branch-cut with exponent $E_1/\omega_1+2$. We can now follow the same strategy as for the linear QRM and construct a pair of even functions $\varphi_1(z)$, $\varphi_2(z)$ satisfying the system ({\tilde{\rho}}ef{eq1}),({\tilde{\rho}}ef{eq2}) and both analytic at $z=0$.
If $\varphi_2(z)$ and $\varphi_1(z)$ have the expansions
\betaegin{equation}
\varphi_2(z)=\sigmaum_{m=0}^\infty a_{2m}z^{2m}, \quad
\varphi_1(z)=\sigmaum_{m=0}^\infty \betaa_{2m}z^{2m},
\lambdaabel{expans}
\end{equation}
it follows from ({\tilde{\rho}}ef{eq1}) that the coefficients are related by
\betaegin{equation}
\betaa_n=\frac{\Delta}{E_1-n\omega_1}a_n,
\lambdaabel{p1p2}
\end{equation}
if $E_1/\omega_1$ is not an integer. The coefficients of $\varphi_1$ are uniquely fixed in terms of the coefficients of $\varphi_2$ if both are analytic at $z=0$ without invoking the symmetry. From ({\tilde{\rho}}ef{eq2}) one deduces the three-term recurrence relation for the $a_n$,
\betaegin{equation}
\fl
a_{n+2}=\frac{1}{(n+2)(n+1)\Gamma}\lambdaeft(\lambdaeft[\frac{\Delta^2}{E_1-n\omega_1}-(n\omega_2+E_2){\tilde{\rho}}ight]a_n-\Gamma a_{n-2}{\tilde{\rho}}ight).
\lambdaabel{p2-recur}
\end{equation}
The initial conditions can be chosen as $a_{-2}=0$, $a_0=1$. A Poincar\'e analysis of ({\tilde{\rho}}ef{p2-recur}) shows that the convergence radius of the expansions in ({\tilde{\rho}}ef{expans}) is infinite, as it should be. Both $\varphi_1$ and $\varphi_2$ are entire and their asymptotics in $S_{0,2}$ have either the exponent $\gamma=2/\omega$ if they are normalizable or the exponent $\omega/2$ if not. The symmetry imposes now the additional relation between $\varphi_1$ and $\varphi_2$,
\betaegin{equation}
\varphi_2(z) = T_\theta[\varphi_1](z)
\lambdaabel{symmetry}
\end{equation}
for all $z\in\mathbb{C}$. If $\phantomi_1(z)$ has asymptotics with exponent $\gamma=\omega/2$ in $S_{0,2}$, rendering it non-normalizable, it will be mapped onto a function which is not analytic at $z=0$ and ({\tilde{\rho}}ef{symmetry}) cannot be satisfied, because $\varphi_2(z)$ is analytic by construction. It follows that the solutions of ({\tilde{\rho}}ef{p1p2}) and ({\tilde{\rho}}ef{p2-recur}) cannot satisfy ({\tilde{\rho}}ef{symmetry}) for generic values of $E$ because they both have asymptotics with $\gamma=\omega/2$. If $E$ is element of the regular spectrum of ${\cal H}p$ and corresponds thus to a certain eigenvalue of the $\mathbb{Z}_4$-symmetry, condition ({\tilde{\rho}}ef{symmetry}) will be satisfied because the image of $\varphi_1$ under $T_\theta$ is a normalizable solution of ({\tilde{\rho}}ef{eq4}) which must be analytic at $z=0$. Therefore, $E$ is an element of the spectrum with eigenstate in ${\cal H}_{++}$ if and only if $\varphi_{1,2}$ as determined by ({\tilde{\rho}}ef{p1p2}),({\tilde{\rho}}ef{p2-recur}) satisfy ({\tilde{\rho}}ef{symmetry}). This argument is independent from the presence of the Stokes phenomenon because only sectors $S_{0,2}$ determine the normalizability as $\omega/2>0$ and $2/\omega >0$. The asymptotics in sectors $S_{1,3}$ play no role, in contrast to the original situation described by ({\tilde{\rho}}ef{orig4}) where the exponents of second kind take positive and negative values.
To derive the $G$-function explicitly, we must compute the action of $T_\theta$ given in ({\tilde{\rho}}ef{Tbarg}) on the series expansion of $\varphi_1(z)$ in ({\tilde{\rho}}ef{expans}), i.e. on even powers of $z$,
\betaegin{equation}
\fl
T_\theta[z^{2m}](z)=\frac{1}{\sigmaqrt{\textrm{ch}(2\theta)}}
\exp\lambdaeft(\frac{2}{\omega}\frac{z^2}{2}{\tilde{\rho}}ight)
\exp\lambdaeft(\lambdaeft[i\frac{\phantomii}{2}-\lambdan\textrm{ch}(2\theta){\tilde{\rho}}ight]z\deltaz{\tilde{\rho}}ight)
P^m_{\omega}(z),
\lambdaabel{t-zm}
\end{equation}
where $P^m_\omega(z)$ is a polynomial in $z$ of degree $2m$,
\betaegin{equation}
P_{\omega}^m(z)=\sigmaum_{l=0}^m\omega^{l-m}\frac{(2m)!}{(m-l)!(2l)!}z^{2l}.
\end{equation}
It follows
\betaegin{equation}
T_\theta[z^{2m}](z)=\frac{1}{\sigmaqrt{\textrm{ch}(2\theta)}}e^{z^2/\omega}
P_{\omega}^m\lambdaeft(\frac{e^{i\phantomii/2}z}{\textrm{ch}(2\theta)}{\tilde{\rho}}ight).
\lambdaabel{t-zm2}
\end{equation}
It is evident that expression ({\tilde{\rho}}ef{t-zm2}) becomes most simple at $z=0$,
\betaegin{equation}
T_\theta[z^{2m}](0)=\frac{1}{\sigmaqrt{\textrm{ch}(2\theta)}}\frac{(2m)!}{\omega^m m!},
\end{equation}
and condition ({\tilde{\rho}}ef{symmetry}) would read
\betaegin{equation}
\varphi_2(0) = 1 =\frac{(\omega^2-4)^{1/4}}{\sigmaqrt{\omega}} \sigmaum_{m=0}^\infty \frac{a_{2m}\Delta(2m)!}{(E_1-2m\omega_1)\omega^m m!},
\lambdaabel{first}
\end{equation}
where the $a_{2m}$ are recursively determined by ({\tilde{\rho}}ef{p2-recur}).
However, a Poincar\'e analysis of the series
\betaegin{equation}
f(w) = \sigmaum_{m=0}^\infty \frac{a_{2m}(2m)!}{(E_1-2m\omega_1)\omega^m m!} w^m
\lambdaabel{w-series}
\end{equation}
shows that the convergence radius $R(f)=1$, meaning that $f(w)$ is not defined at $w=1$, which is supposed to yield $T_\theta[\varphi_1](0)$. This is easy to understand because the generic form of $T_\theta[\varphi_1](z)$ has a branch-cut at $z=0$ and each series expansion of it diverges there. The analysis of ({\tilde{\rho}}ef{w-series}) gives thus another independent proof of the fact that a non-normalizable but entire solution of ({\tilde{\rho}}ef{eq3}) is mapped onto a solution of ({\tilde{\rho}}ef{eq4}) which is not analytic at $z=0$.
To derive a $G$-function whose series expansion does converge, we apply $I_\theta^{-1}$ to both sides of ({\tilde{\rho}}ef{symmetry}),
\betaegin{equation}
\phantomi_2(z)=I^{-1}_\theta[\varphi_2](z)=T[I_\theta^{-1}[\varphi_1]](z)=I_\theta^{-1}[\varphi_1](iz).
\lambdaabel{Gc1}
\end{equation}
The expressions in ({\tilde{\rho}}ef{Gc1}) are well defined at $z=0$, so it is possible to evaluate the $G$-function at the invariant point $z_0=0$ of the map $T$. This has the advantage that $\phantomi_2(z_0)-\phantomi_1(iz_0)=0$ is sufficient to determine the regular spectrum in ${\cal H}_{++}$. If $z_0\neq 0$, one would have two independent conditions $\phantomi_2(z_0)=\phantomi_1(iz_0)$ and $\phantomi_2(-iz_0)=\phantomi_1(z_0)$ \cite{braak_13}.
Using ({\tilde{\rho}}ef{Ibarg}), we find for $I_\theta^{-1}[\varphi_2](0)$,
\betaegin{equation}
I_\theta^{-1}[\varphi_2](0) = \frac{1}{\sigmaqrt{\textrm{ch}(\theta)}}\sigmaum_{m=0}^\infty a_{2m}\frac{\textrm{th}(|\theta|)^m(2m)!}{2^m m!}.
\lambdaabel{p2zero}
\end{equation}
The Poincar\'e analysis of the series
\betaegin{equation}
\tilde{f}(w)=\sigmaum_{m=0}^\infty a_{2m}\frac{\textrm{th}(|\theta|)^m(2m)!}{2^m m!} w^m
\end{equation}
yields the convergence radius $R(\tilde{f})=1+\sigmaqrt{\omega^2-4}/\omega>1$, therefore the expression ({\tilde{\rho}}ef{p2zero}) is given by an absolutely converging series. {\it A fortiori} the same applies to $I^{-1}_\theta[\varphi_1](0)$ and we can define a $G$-function as $\Gammac(E)=\phantomi_2(0)-\phantomi_1(0)$, that is
\betaegin{equation}
\Gammac(E)=\sigmaum_{m=0}^\infty\lambdaeft(1-\frac{\Delta}{E_1-2m\omega_1}{\tilde{\rho}}ight)a_{2m}\frac{\textrm{th}(|\theta|)^m(2m)!}{2^m m!}.
\lambdaabel{Gc}
\end{equation}
After adjusting notation, we find $\Gammac(E)=G^{1/4}_+(x(E))$ which is the $G$-function for sector ${\cal H}_{++}$ given in (16) of \cite{duan_16} and has been originally proposed ten years ago by Chen {\it et al.} in \cite{chen_12}.
By examination of the normalizability properties of formal solutions to the Schr\"odinger equation in the Bargmann space, we have demonstrated that Chen's $G$-function yields the complete regular spectrum in ${\cal H}_{++}$ (the other sectors require minor modifications of the derivation). It is shown in Figure~{\tilde{\rho}}ef{cgfunction}.
\betaegin{figure}
\includegraphics[width=0.6\lambdainewidth]{fig4.eps}
\caption{Chen's $G$-function $\Gammac(E)$ for the same parameters as in Figs.{\tilde{\rho}}ef{zgfunction} and {\tilde{\rho}}ef{tgfunction}. This function is characterized by poles at even integer values of $E_1/\omega_1$. The regular zeros are located between the poles which determine the average distance between adjacent zeros.}
\lambdaabel{cgfunction}
\end{figure}
The major qualitative difference of this $G$-function compared with the others proposed by Zhang \cite{zhang_17}, Trav{\v e}nec \cite{travenec_12} and Maciejewski/Stachowiak \cite{mac_17} is the presence of simple poles at even integer values of $E_1/\omega_1$, that means at energies
\betaegin{equation}
E^{(m)}=\lambdaeft(2m+\frac{1}{2}{\tilde{\rho}}ight)\sigmaqrt{\omega^2-4}-\frac{\omega}{2}, \qquad m\in\rm{I\!N}_0.
\lambdaabel{poles}
\end{equation}
For most values of the parameters $\omega$ and $\Delta$ all eigenenergies are regular and located between the poles given by ({\tilde{\rho}}ef{poles}). According to the conjecture formulated in \cite{braak_11}, the number of zeros between adjacent poles is restricted to be 0, 1 or 2. This conjecture has been numerically verified in the linear and 2pQRM and can be derived analytically for small $\Delta$. It plays here a similar role as the string hypothesis in systems solvable by the Bethe ansatz \cite{eckle_19}. As discussed in \cite{duan_16}, the conjecture entails that the average distance between eigenenergies in ${\cal H}_{++}$ is given by the distance between the poles, $2\sigmaqrt{\omega^2-4}$. The poles are equally spaced along the whole real axis for $E \gammae E^{(0)}=(1/2)(\sigmaqrt{\omega^2-4}-\omega)$ and become more dense as $\omega$ approaches the critical value $\omega_c=2$ from above -- the distance between adjacent energy levels goes to zero in this limit. At $\omega_c$, the real axis above the threshold value $E_0=-1$ becomes densely covered with levels, indicating the transition to a continuous spectrum as derived in section {\tilde{\rho}}ef{general} for the critical point itself. In this way the qualitative properties of the spectrum as the system approaches $\omega_c$ can be deduced from the known pole structure of $\Gammac(E)$. There is no ``collapse'' of the spectrum at the point $\omega_c$ to the single energy $E_0$ as falsely inferred from exact diagonalization in a truncated state space but instead the transition to a continuous spectrum at $\omega_c$, spanning the whole real axis from $E_0$ to infinity. The exceptional spectrum in ${\cal H}_{++}$ consists of those energy levels which coincide with a pole energy $E^{(m)}$ in full analogy with the linear QRM. In this case the pole in $\Gammac(E)$ is lifted by a zero of the expansion coefficients in ({\tilde{\rho}}ef{Gc}) and $\Gammac(E^{(m)})$ is finite. It is easy to see that those eigenstates are exactly the special solutions found in \cite{emary_02}. They appear if the asymptotic expansion ({\tilde{\rho}}ef{regasym}) of the third solution to ({\tilde{\rho}}ef{eq3}) is convergent because the recurrence ({\tilde{\rho}}ef{recur-c}) breaks off after $m$ steps, leading to another normalizable entire solution, a polynomial. The exceptional level is thus doubly degenerate. We have seen theat the presence of poles in $\Gammac(E)$ is instrumental to determine the approach to the collapse point without numerical evaluation of the $g$-function. On the other hand, it may seem detrimental to have access only to the regualr part of the spectrum. As shown in \cite{li_add_16,kimoto_21}, it is possible to obtain a $G$-function for the linear QRM whose zeros give the complete spectrum including the degenerate and non-degenerate exceptional parts by multiplying $G_\phantomim(E)$ with appropriate Gamma-functions. The same is possible for $\Gammac(E)$. This may be of importance to make connections with the spectral zeta-functions associated with $H_R$ and ${\cal H}p$ \cite{wakayama_17,kimoto_21}.
The degeneracies of Juddian type, between states in ${\cal H}_{++}$ and $H_{+-}$, are connected to the poles of $\Gammac$ and $G_{\sigmacriptscriptstyle{+-}}$, where $E_1/\omega_1$ is a positive even integer. Likewise the degeneracies between ${\cal H}_{-+}$ and $H_{--}$ correspond to lifting of a pole in $G_{\sigmacriptscriptstyle{-\phantomim}}$. The poles correspond to $E_1/\omega_1$ being a positive odd integer. The degeneracies appearing between even and odd eigenstates with $E_1/\omega_1$ a half-integer are not reflected in these $G$-functions. They correspond to a factorization property of ({\tilde{\rho}}ef{orig4}) for special values of the parameters $\omega$, $\Delta$ which does not follow from the $\mathbb{Z}_4$-symmetry alone \cite{mac_19}.
\sigmaection{Conclusions}
\lambdaabel{concl}
Many qualitative aspects of the spectrum of the two-photon quantum Rabi model which define the three different parameter regimes, $\omega>2$, $\omega=2$ and $\omega<2$, can be inferred without numerical computation by examining the singularity structure of the Schr\"odinger operator in the Bargmann space of entire functions. In these regimes, the spectrum is either pure point ($\omega>2$) or comprises a continuous and a discrete part ($\omega=2$). We have proved that the Hamiltonian ${\cal H}p$ is not self-adjoint for nonzero $\Delta$ and $\omega<2$, but self-adjoint for $\omega\gammae 2$. In the regime $\omega<2$, the Hamiltonian is not only unbounded from below as was already noted in \cite{ng_99}, but has no physical interpretation because it is not self-adjoint. Being unbounded from below alone does not render an operator unphysical as the momentum and position operators demonstrate. One may argue that an operator without lower bound cannot correspond to the energy of the system, but ${\cal H}p$ does not even fulfill the basic requirement for physical interpretation if $\omega < 2$, although it is interpretable at the critical point $\omega=2$. Here the system resembles somewhat a particle moving in a flat potential, an interpretation suggested by the case $\Delta=0$, but the presence of proper bound states for $\Delta\neq 0$ shows that this is not correct. On the other hand, the effective potential following from ({\tilde{\rho}}ef{phi1eq}), ({\tilde{\rho}}ef{phi2eq}) depends on the energy eigenvalue so that no direct mapping to a one-dimensional problem without spin degree of freedom is possible.
The various generalized spectral determinants proposed in the regime $\omega>2$ have different qualitative properties depending on their derivation. While Zhang's $G$-function \cite{zhang_17}, based on a continued fraction, and Chen's $G$-function \cite{chen_12} based on a Bogoliubov (squeezing) transformation, both possess poles on the real axis located between the zeros (eigenvalues of ${\cal H}p$), their position is known analytically only in the latter case. Neither Trav{\v e}nec's \cite{travenec_12} nor Maciejewski's $G$-functions \cite{mac_17} exhibit poles or other known qualitative features, like the position of maxima or minima, to infer the distribution of the zeros without explicit numerical computation. The average distance of energy levels, while independent from the coupling in the linear QRM, changes dramatically in the 2pQRM if the critical coupling $g=\omega/2$ is approached from below (or $\omega {\tilde{\rho}}a 2$ from above for fixed $g=1$). The levels coalesce on the real axis above the threshold energy $E_0=-1$ to form the continuous part of the spectrum at the critical point $\omega=2$. Because the average distance of levels is given by the pole distance in Chen's $G$-function, $\Delta E=2\sigmaqrt{\omega^2-4}$, the exponents of this critical behavior can be computed exactly \cite{duan_16}. Moreover, Chen's $G$-function yields the quasi-exact Juddian points of the spectrum in the same way as the corresponding $G$-function of the linear QRM. While the validity of this $G$-function has been confirmed numerically in great detail, a rigorous derivation based on the normalizability of the wave functions was missing. As the numerical checks fail close to the critical point due to truncation of the Hilbert space, the rigorous proof of validity provided here seems not only of formal but also of practical importance.
\textbf{Acknowledgements} \phantomiar
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant no.: 439943572.
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\end{document} |
\begin{document}
\title{Categorical Torelli theorem for hypersurfaces}
\author{Dmitrii Pirozhkov}
\begin{abstract}
Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree~$d$. The derived category of coherent sheaves on~$X$ contains an interesting subcategory called the Kuznetsov component~$\mathcal{A}_X$. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines $X$ uniquely if~$d > 3$ or if~$d = 3$ and~$n > 3$. This generalizes a result by D.~Huybrechts and J.~Rennemo, who proved the same statement under the additional assumption that $d$ divides $n+2$.
\end{abstract}
\maketitle
\section{Introduction}
Reconstruction of an algebraic variety from some of its invariants is a classical endeavor that started with Torelli proving that a smooth projective curve is determined by its Jacobian as a polarized abelian variety. In 1997 A.~Bondal and D.~Orlov proved \cite{bondal-orlov} that Fano varieties and varieties with ample canonical class can be reconstructed from their derived categories of coherent sheaves. In this note we are interested in some developments arising from Bondal--Orlov's theorem.
To discuss our setting, we first need some notation. Let $k$ be a field of characteristic zero, and let~$V$ be an \mbox{$(n+2)$-dimensional} vector space. We work with an~$n$-dimensional smooth Fano hypersurface~$X \subset \mathbb{P}(V)$ of degree~$d$. The bounded derived category of coherent sheaves on $X$ admits a semiorthogonal decomposition:
\begin{equation}
\label{eq:intro sod}
D^b_{\!\mathrm{coh}}(X) = \langle \mathcal{A}_X, \mathcal{O}_X, \mathcal{O}_X(1), \ldots, \mathcal{O}_X(n-d+1) \rangle,
\end{equation}
where the category $\mathcal{A}_X$ is called a \emph{Kuznetsov component} or a \emph{residual category} of $D^b_{\!\mathrm{coh}}(X)$. The decomposition \eqref{eq:intro sod} implies that, according to \cite[Prop.~3.8]{orlov-glueing}, the category $D^b_{\!\mathrm{coh}}(X)$, and hence by Bondal--Orlov's theorem the variety $X$ itself, can be reconstructed from the following pieces of data:
\begin{itemize}
\item The category $\langle \mathcal{O}_X, \mathcal{O}_X(1), \ldots, \mathcal{O}_X(n-d+1) \rangle$ that depends only on $n$ and $d$, not on the choice of the hypersurface $X$;
\item The Kuznetsov component $\mathcal{A}_X$;
\item A certain glueing data between the two categories above.
\end{itemize}
In general, it is impossible to determine $X$ just from the Kuznetsov component $\mathcal{A}_X$, without the glueing data. An example with cubic fourfolds is described in \cite{pertusi}. In the paper~\cite{huybrechts-rennemo} D.~Huybrechts and J.~Rennemo suggested a different approach to this problem. They used a particular autoequivalence~$\mathbb{P}hi_{\mathcal{A}_X}$ of $\mathcal{A}_X$ called the \emph{rotation functor} (see Definition~\ref{def:rotation functor}). They proved the following reconstruction theorem:
\begin{theorem}[{\cite[Cor.~1.2]{huybrechts-rennemo}}]
A smooth Fano hypersurface $X \subset \mathbb{P}(V)$ of degree $d$ and dimension $n$ with $d | (n+2)$ is determined by the pair $(\mathcal{A}_X, \mathbb{P}hi_{\mathcal{A}_X})$ composed of the Kuznetsov component and the rotation functor (as a dg-category and a dg-endofunctor).
\end{theorem}
In this paper we generalize their approach to any Fano hypersurface, not necessarily satisfying the divisibility condition on $d$ and $n$:
\begin{theorem}[{ = Theorem~\ref{thm:main theorem}}]
\label{thm:main theorem intro}
A smooth Fano hypersurface $X \subset \mathbb{P}(V)$ of degree $d$ and dimension $n$ satisfying~$d > 3$ or~$(d = 3, n > 3)$ is determined by the pair $(\mathcal{A}_X, \mathbb{P}hi_{\mathcal{A}_X})$ composed of the Kuznetsov component and the rotation functor (as a dg-category and a dg-endofunctor).
\end{theorem}
\begin{remark}
The case of cubic threefolds, not included in this theorem, has been studied in~\cite{bmms}. For this case the category $\mathcal{A}_X$ alone suffices to determine $X$.
\end{remark}
The strategy of the proof is similar to the one in \cite{huybrechts-rennemo}. The vector space of natural transformations from the identity functor of $\mathcal{A}_X$ to the rotation functor is isomorphic to~$V^{\scriptstyle\vee}$~(Lemma~\ref{lem:multiplication on kuznetsov component}). The~$d$'th power of the rotation functor is a shift-by-two functor on~$\mathcal{A}_X$~(Theorem~\ref{thm:kuznetsov periodicity}), and thus the~$d$-fold composition of the natural transformations defines a morphism
\begin{equation}
\label{eq:intro map}
S^d V^{\scriptstyle\vee} \to \mathrm{HH}^2(\mathcal{A}_X).
\end{equation}
We show that the kernel of this morphism is exactly the $d$'th graded component of the Jacobian ideal of $X \subset \mathbb{P}(V)$, which determines the hypersurface~$X$ uniquely. A subtle point of the argument is that we do not need an explicit computation of~$\mathrm{HH}^2(\mathcal{A}_X)$. Instead, we prove that the map~\eqref{eq:intro map} factors through $\mathrm{HH}^2(X)$ and use an easy observation from Lemma~\ref{lem:action is nondegenerate if non-diagonal} that shows that the restriction map $\mathrm{HH}^2(X) \to \mathrm{HH}^2(\mathcal{A}_X)$ is an injection on a large subspace of $\mathrm{HH}^2(X)$.
There are many classes of varieties which admit semiorthogonal decompositions similar to the one in \eqref{eq:intro sod} in the sense that one of the components of the decomposition is the "interesting" one, and the others are very simple. In these situations one could investigate some refined version of the Bondal--Orlov's theorem. For a review of known results along this direction, see \cite{pertusi-stellari}. We especially remark the Fano threefold case studied in \cite{infinitesimal-torelli}, due to some similarities with our approach.
While preparing the work I learned of an upcoming paper by J.~Rennemo~\cite{rennemo-reconstruction}, who showed that if $d$ does not divide $n+2$ and the pair $(d, n)$ is not of the form $(4, 4k)$, then the Kuznetsov component alone suffices to reconstruct $X$, with no dependency on the rotation functor. I believe that Theorem~\ref{thm:main theorem intro}, though weaker outside of the case $(d = 4, n = 4k)$, is still of interest, in particular due to the uniform handling of all cases.
\textbf{Structure of the paper}. In Section~\ref{sec:jacobian ring} we recall the notion of the Jacobian ring of a hypersurface and its connection with Hochschild cohomology. In Section~\ref{sec:orthogonal to structure sheaf}, following \cite{kuznetsov-v14}, we perform some computations related to the orthogonal to the structure sheaf in the derived category of a hypersurface. We use those results in Section~\ref{sec:kuznetsov component} to study the $d$'th power of the rotation functor on the Kuznetsov component. Finally, in Section~\ref{sec:injectivity on hochschild} we prove the main Theorem~\ref{thm:main theorem intro}.
\textbf{Notation}. Let $K$ be a field of characteristic zero. In this paper all categories are assumed to be triangulated and $K$-linear, all pullbacks and pushforwards are assumed to be derived, and all varieties are assumed to be smooth.
Let $V$ be an $(n+2)$-dimensional vector space over $K$.
We work with a smooth $n$-dimensional Fano hypersurface $X \subset \mathbb{P}(V)$ with $\mathrm{deg}(X) = d$, defined by an equation $f \in S^d V^{\scriptstyle\vee}$. In particular, $d < n+2$.
We use the notation $\mathbb{P}$ for the projective space $\mathbb{P}(V)$. Since we mostly work with objects in the derived categories $D^b_{\!\mathrm{coh}}(X)$ and $D^b_{\!\mathrm{coh}}(X \times X)$, we sometimes omit the subscript $X$ on objects like the structure sheaf of the diagonal $\mathcal{O}_\Delta \in D^b_{\!\mathrm{coh}}(X \times X)$ to avoid the symbol clutter in formulas when the risk of confusion is small.
\textbf{Acknowledgements}. I thank Emanuele Macr\`i for many helpful conversations. I thank Alexander Kuznetsov for advice and suggestions. I also thank Daniel Huybrechts and J\o rgen Rennemo for their explanations concerning Lemma~\ref{lem:multiplication on kuznetsov component}, and for letting me know of \cite{rennemo-reconstruction}.
\section{Hochschild cohomology and Jacobian ring}
\label{sec:jacobian ring}
In \cite{donagi} Ron Donagi proved a Hodge-theoretic Torelli theorem for a (very general) hypersurface satisfying some conditions on the degree and the dimension. The proof relied on the notion of the Jacobian ring of a hypersurface. We also need this notion for the categorical version of Torelli theorem.
\begin{definition}
\label{def:jacobian ring}
The \emph{Jacobian ring} of a hypersurface $X \subset \mathbb{P}(V)$ defined by an equation~$f \in S^d V^{\scriptstyle\vee}$ is the graded ring given by the quotient
\[
J^\bullet(f) := S^\bullet V^{\scriptstyle\vee}/\langle \tfrac{\partial f}{\partial v} \rangle_{v \in V}.
\]
The graded ideal generated by the partial derivatives of $f$ is called the \emph{Jacobian ideal} of $X$.
\end{definition}
What most interests us is a relation between the Jacobian ring and the Hochschild cohomology of the hypersurface. One aspect of this relation is demonstrated in the combination of Lemma~\ref{lem:h1t and jacobian ideal} and Proposition~\ref{prop:hh2 and jacobian ideal}. For general information on Hochschild cohomology see, for example, \cite{kuznetsov-oldhochschild}. We only recall that Hochschild cohomology of a variety can be computed by the formula
\[
\mathrm{HH}^\bullet(X) = \mathrm{Ext}^\bullet_{X \times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta),
\]
and the Hochschild cohomology of an (admissible) subcategory $\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$ can be computed by the formula
\begin{equation}
\label{eq:hochschild definition subcategory}
\mathrm{HH}^\bullet(\mathcal{A}_X) = \mathrm{Ext}^\bullet_{X \times X}(P, P),
\end{equation}
where the object~$P \in D^b_{\!\mathrm{coh}}(X \times X)$ is a Fourier--Mukai kernel of the (left or right) projection functor to~$\mathcal{A}_X$.
\begin{lemma}
\label{lem:h1t and jacobian ideal}
Consider the normal bundle short exact sequence:
\[
0 \to T_X \to T_{\mathbb{P}}|_X \to \mathcal{O}_X(d) \to 0.
\]
The connecting homomorphism in the long exact sequence of cohomology groups induces a morphism
\[
S^d V^{\scriptstyle\vee} \simeq H^0(\mathbb{P}, \mathcal{O}_{\mathbb{P}}(d)) \to H^0(X, \mathcal{O}_X(d)) \to H^1(X, T_X).
\]
Then this map is surjective and it identifies $H^1(X, T_X)$ with the $d$'th graded component of the Jacobian ring $J^d(f)$.
\end{lemma}
\begin{proof}
The long exact sequence of cohomology groups of the normal short exact sequence contains the following fragment:
\begin{equation}
\label{eq:cohomology of normal exact sequence}
H^0(T_{\mathbb{P}}|_X) \to H^0(\mathcal{O}_X(d)) \to H^1(T_X) \to H^1(T_\mathbb{P}|_X).
\end{equation}
Note that the tangent bundle of $\mathbb{P}$ fits into the Euler short exact sequence
\[
0 \to \mathcal{O}_\mathbb{P} \to V \otimes \mathcal{O}_\mathbb{P}(1) \to T_\mathbb{P} \to 0.
\]
Since $d < n+2$ it is easy to compute that $H^1(X, T_\mathbb{P}|_X) = 0$ and $H^0(X, T_\mathbb{P}|_X) = V^{\scriptstyle\vee} \otimes V/\langle \mathrm{id}_V \rangle$. Thus the sequence \eqref{eq:cohomology of normal exact sequence} simplifies:
\[
V^{\scriptstyle\vee} \otimes V/\langle \mathrm{id}_V \rangle
\to
S^d V^{\scriptstyle\vee} / \langle f \rangle
\to
H^1(T_X) \to 0.
\]
Here the first map sends a decomposable tensor~$\xi \otimes v \in V^{\scriptstyle\vee} \otimes V$ to the element~$\xi \cdot \frac{\partial f}{\partial v} \in S^d V^{\scriptstyle\vee}$. The image of this map in $S^d V^{\scriptstyle\vee}$ is thus the $d$'th component of the Jacobian ideal. Since $f$ lies in its own Jacobian ideal, we conclude that $H^1(X, T_X) \simeq J^d(f)$.
\end{proof}
Recall the notion of the \emph{universal Atiyah class} $\mathrm{At} \in \mathrm{Ext}^1(\mathcal{O}_\Delta, \Delta_*\mathcal{O}mega^1_X)$ (see, e.g., \cite{kuznetsov-markushevich}).
Abusing the notation, we denote the following composition also by $\mathrm{At}$:
\[
H^1(X, T_X) \simeq \mathrm{Ext}_X^1(\mathcal{O}mega^1_X, \mathcal{O}_X) \xrightarrow{\Delta_*} \mathrm{Ext}^1_{X \times X}(\Delta_*\mathcal{O}mega^1_X, \mathcal{O}_\Delta) \xrightarrow{- \circ \mathrm{At}} \mathrm{Ext}^2_{X \times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \cong \mathrm{HH}^2(X).
\]
By the Hochschild--Kostant--Rosenberg theorem (e.g., \cite[Cor.~2.6]{swan-hkr}) this morphism is an injection.
Recall the (universal) linkage class $\varepsilonilon_X$ for a hypersurface $X \subset \mathbb{P}(V)$ \cite[Sec.~3]{kuznetsov-markushevich}: the derived restriction of $\mathcal{O}_{\Delta_\mathbb{P}}$ to $X \times X \subset \mathbb{P} \times \mathbb{P}$ is a complex with two adjacent cohomology sheaves. Thus it fits into a distinguished triangle
\begin{equation}
\label{eq:universal linkage}
\mathcal{O}_{\Delta_\mathbb{P}}|_{X \times X} \to \mathcal{O}_\Delta \xrightarrow{\varepsilonilon_X} \mathcal{O}_\Delta(-d)[2].
\end{equation}
The gluing morphism~$\varepsilonilon_X \in \mathrm{Ext}^2_{X \times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta(-d))$ between the cohomology sheaves is called the \emph{universal linkage class}.
\begin{proposition}
\label{prop:hh2 and jacobian ideal}
The composition with the universal linkage class defines a morphism:
\[
S^d V^{\scriptstyle\vee} \twoheadrightarrow H^0(X, \mathcal{O}_X(d)) \xrightarrow{\Delta_*} \mathrm{Hom}_{X \times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta(d)) \xrightarrow{\varepsilonilon_X \circ -} \mathrm{Ext}^2_{X \times X}(\mathcal{O}_\Delta, \mathcal{O}_\Delta) \cong \mathrm{HH}^2(X).
\]
such that the following triangle commutes:
\begin{equation}\begin{tikzcd}
\label{eq:hh2 and jacobian ideal}
{S^d V^{\scriptstyle\vee}} & {H^1(X, T_X)} \\
& {\mathrm{HH}^2(X)}
\arrow["{\mathrm{At}}", hook', from=1-2, to=2-2]
\arrow[from=1-1, to=1-2]
\arrow[from=1-1, to=2-2]
\end{tikzcd}\end{equation}
where the horizontal arrow is the map from Lemma~\textup{\ref{lem:h1t and jacobian ideal}}.
\end{proposition}
\begin{proof}
Let $\nu\colon \mathcal{O}mega^1 \to \mathcal{O}(-d)[1]$ be the extension class of the conormal exact sequence
\[
0 \to \mathcal{O}(-d) \to \mathcal{O}mega^1_\mathbb{P}|_X \to \mathcal{O}mega^1_X \to 0.
\]
By \cite[Th.~3.2]{kuznetsov-markushevich} the universal linkage class is equal to the composition
\[
\mathcal{O}_\Delta \xrightarrow{\mathrm{At}} \Delta_*\mathcal{O}mega^1[1] \xrightarrow{\Delta_*\nu} \mathcal{O}_\Delta(-d)[2]
\]
of the universal Atiyah class and the pushforward of $\nu$ along the diagonal.
Unwinding the definitions, we see that the diagram~\eqref{eq:hh2 and jacobian ideal} commutes on an element $g \in S^d V^{\scriptstyle\vee}$ if and only if the diagram below commutes:
\[\begin{tikzcd}
{\mathcal{O}_\Delta} & {\Delta_* \mathcal{O}mega^1[1]} & {\mathcal{O}_\Delta(-d)[2]} \\
{\mathcal{O}_\Delta(d)} & {\Delta_*\mathcal{O}mega^1(d)[1]} & {\mathcal{O}_\Delta[2]}
\arrow["{\mathrm{At}}", from=1-1, to=1-2]
\arrow["{\Delta_*\nu}", from=1-2, to=1-3]
\arrow["{\mathrm{At}(d)}", from=2-1, to=2-2]
\arrow["{\Delta_*\nu(d)}", from=2-2, to=2-3]
\arrow["{\cdot g}", from=1-3, to=2-3]
\arrow["{\cdot g}", from=1-1, to=2-1]
\end{tikzcd}\]
This commutativity is clear since multiplication by $g$ is a natural transformation.
\end{proof}
\section{The orthogonal to the structure sheaf}
\label{sec:orthogonal to structure sheaf}
Since $X \subset \mathbb{P}(V)$ is by assumption a Fano hypersurface, the structure sheaf $\mathcal{O}_X \in D^b_{\!\mathrm{coh}}(X)$ is an exceptional object. In this section we study some properties of the right-orthogonal subcategory $\mathcal{O}_X^\perp \subset D^b_{\!\mathrm{coh}}(X)$. The goal is to perform some explicit computations involving the projection functor to $\mathcal{O}_X^\perp$. We will rely on them in Section~\ref{sec:kuznetsov component} where we study the Kuznetsov component $\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$. Nothing in this section is new, all results are already in \cite{kuznetsov-v14}.
We begin with a brief reminder on Fourier--Mukai transforms to fix the notation. For details, see \cite{HuybFM}.
Given an object $K \in D^b_{\!\mathrm{coh}}(X \times X)$ we define the Fourier--Mukai functor~$\mathbb{P}hi_K\colon D^b_{\!\mathrm{coh}}(X) \to D^b_{\!\mathrm{coh}}(X)$ by the formula $\pi_{2 *}(\pi_1^*(-) \otimes K)$. For any $d \in \mathbb{Z}$ the Fourier--Mukai transform $\mathbb{P}hi_{\mathcal{O}_{\Delta}(d)}$ is the functor given by the twist by $\mathcal{O}_X(d)$. For any pair of objects~$F, G \in D^b_{\!\mathrm{coh}}(X)$ the Fourier--Mukai transform along the exterior product $F \boxtimes G$ is the functor
\[
\mathbb{P}hi_{F \boxtimes G}(-) := \mathrm{R}\mathbb{G}_amma\,(- \otimes F) \otimes G.
\]
The \emph{convolution} of two kernels $K_1, K_2 \in D^b_{\!\mathrm{coh}}(X \times X)$ is defined by the formula
\[
K_1 \circ K_2 := \pi_{1 3 *}(\pi_{1 2}^*(K_1) \otimes \pi_{2 3}^*(K_2)),
\]
and it satisfies $\mathbb{P}hi_{K_1 \circ K_2} = \mathbb{P}hi_{K_1} \circ \mathbb{P}hi_{K_2}$. For line bundles of the form~$\mathcal{O}_X(a, b) := \mathcal{O}_X(a) \boxtimes \mathcal{O}_X(b)$ the convolution equals
\begin{equation}
\label{eq:convolution of line bundles}
\mathcal{O}_X(a, b) \circ \mathcal{O}_X(a^\operatorname{pr}ime, b^\operatorname{pr}ime) \cong \mathrm{R}\mathbb{G}_amma\,(\mathcal{O}_X(b + a^\operatorname{pr}ime)) \otimes \mathcal{O}_X(a, b^\operatorname{pr}ime).
\end{equation}
Now we return to the main object of this section.
\begin{definition}
\label{def:kernels for rotations}
We define the object $Q_0 \in D^b_{\!\mathrm{coh}}(X \times X)$ as the complex
\[
Q_0 := [ \mathcal{O}_X \boxtimes \mathcal{O}_X \to \mathcal{O}_\Delta ],
\]
in degrees $-1$ and $0$, representing the left projection functor to the subcategory $\mathcal{O}_X^\perp \subset D^b_{\!\mathrm{coh}}(X)$. We define the objects $Q_i$ for $i > 0$ recursively:
\[
Q_i := Q_{i-1} \circ \mathcal{O}_\Delta(1) \circ Q_0,
\]
where the symbol $\circ$ denotes the convolution of Fourier--Mukai kernels in $D^b_{\!\mathrm{coh}}(X \times X)$.
\end{definition}
\begin{remark}
The functor $D^b_{\!\mathrm{coh}}(X) \to D^b_{\!\mathrm{coh}}(X)$ represented by the object $Q_1 \in D^b_{\!\mathrm{coh}}(X \times X)$ is called a \emph{rotation functor} in \cite{kuznetsov-v14}. It can alternatively be described as a composition
\[
D^b_{\!\mathrm{coh}}(X) \xrightarrow{Q_0} D^b_{\!\mathrm{coh}}(X) \xrightarrow{- \otimes \mathcal{O}_X(1)} D^b_{\!\mathrm{coh}}(X) \xrightarrow{Q_0} D^b_{\!\mathrm{coh}}(X),
\]
where the Fourier--Mukai transform along the object~$Q_0$ is the left projection to the subcategory~$\mathcal{O}_X^\perp$.
\end{remark}
\begin{lemma}
\label{lem:multiplication on the subcategory}
For any $i \geq 0$ there is a natural morphism $\mathcal{O}_\Delta(i) \to Q_i$ which induces a map
\[
m_{Q_i}\colon S^i V^{\scriptstyle\vee} \to \mathrm{Hom}(\mathcal{O}_\Delta, \mathcal{O}_\Delta(i)) \to \mathrm{Hom}(\mathcal{O}_\Delta, Q_i) \to \mathrm{Hom}(Q_0, Q_i).
\]
Furthermore, for $i = 1$ the map $m_{Q_1}$ is an isomorphism of vector spaces.
\end{lemma}
\begin{proof}
By the definition of the left projection functor there is a morphism
\[
\mathcal{O}_\Delta \to Q_0.
\]
We define the map $\mathcal{O}_\Delta(i) \to Q_i$ by repeatedly using the map $\mathcal{O}_\Delta \to Q_0$ as follows:
\[\begin{tikzcd}
{\mathcal{O}_\Delta(i)} & {\mathcal{O}_\Delta \circ \mathcal{O}_\Delta(1) \circ \mathcal{O}_\Delta \circ \cdots \circ \mathcal{O}_\Delta(1) \circ \mathcal{O}_\Delta} \\
{Q_i} & {Q_0 \circ \mathcal{O}_\Delta(1) \circ Q_0 \circ \cdots \circ \mathcal{O}_\Delta(1) \circ Q_0 }
\arrow["{=}", from=1-1, to=1-2]
\arrow["{=}", from=2-1, to=2-2]
\arrow[from=1-2, to=2-2]
\end{tikzcd}\]
The fact that $m_{Q_1}$ is an isomorphism can be checked by a straightforward computation using the resolution
\[
Q_1 \cong [ V^{\scriptstyle\vee} \otimes \mathcal{O}_{X \times X} \to \mathcal{O}(1,0) \oplus \mathcal{O}(0, 1) \to \mathcal{O}_\Delta(1) ]
\]
obtained by using Definition~\ref{def:kernels for rotations} and the formula~\eqref{eq:convolution of line bundles}.
\end{proof}
\begin{definition}
\label{def:beilinson resolution}
We denote by $B_\Delta$ the Beilinson's resolution of the diagonal on $\mathbb{P}(V) \times \mathbb{P}(V)$:
\[
B_\Delta := [ \mathcal{O}_{\mathbb{P}(V)}(-n-1) \boxtimes \mathcal{O}mega_{\mathbb{P}(V)}^{n+1}(n+1) \to \ldots \to \mathcal{O}_{\mathbb{P}(V)}(-1) \boxtimes \mathcal{O}mega_{\mathbb{P}(V)}^1(1) \to \mathcal{O}_{\mathbb{P}(V)} \boxtimes \mathcal{O}_{\mathbb{P}(V)} ].
\]
For any $0 \leq i \leq n+1$ we denote by $s_{\geq -i}(B_\Delta)$ the stupid truncation of this resolution:
\[
s_{\geq -i}(B_\Delta) := [ \mathcal{O}_{\mathbb{P}(V)}(-i) \boxtimes \mathcal{O}mega_{\mathbb{P}(V)}^{i}(i) \to \ldots \to \mathcal{O}_{\mathbb{P}(V)} \boxtimes \mathcal{O}_{\mathbb{P}(V)} ].
\]
\end{definition}
\begin{theorem}[{\cite{kuznetsov-v14}}]
\label{thm:kernels for rotation powers}
For $0 \leq i < d$ the morphism $\mathcal{O}_\Delta(i) \to Q_i$ from Lemma~\textup{\ref{lem:multiplication on the subcategory}} fits into an exact triangle
\[
s_{\geq -i}(B_\Delta)|_{X \times X} \otimes (\mathcal{O}_X(i) \boxtimes \mathcal{O}_X) \xrightarrow{\psi_i} \mathcal{O}_\Delta(i) \to Q_i,
\]
and the map~$\psi_i$ is a twist by~$\mathcal{O}_X(i) \boxtimes \mathcal{O}_X$ of the composition
\[
s_{\geq -i}(B_\Delta)|_{X \times X} \to B_\Delta|_{X \times X} \simeqarrow (\mathcal{O}_{\Delta_{\mathbb{P}(V)}})|_{X \times X} \to \mathcal{O}_\Delta.
\]
\end{theorem}
\begin{proof}
The statement is implicitly contained in the proof of \cite[Lem.~4.2]{kuznetsov-v14}, and can be proved by an inductive computation. For the sake of clarity, we sketch the argument. The base case $i = 0$ is true by definition of $Q_0$. Suppose the statement holds for $Q_i$ with $i < d-1$ and we want to prove it for the object
\[
Q_{i+1} \cong Q_i \circ \mathcal{O}_\Delta(1) \circ [ \mathcal{O}_X(0 ,0) \to \mathcal{O}_{\Delta} ] \cong \mathrm{Cone}(Q_i \circ \mathcal{O}_X(0, 1) \to Q_i \circ \mathcal{O}_\Delta(1)).
\]
This description as a cone, together with the formula for $Q_i$ that we know by induction, shows that $Q_{i+1}$ can be represented by the following complex:
\begin{equation}\begin{tikzcd}
\label{eq:inductive q}
{\mathcal{O}_X \boxtimes (H^0(\mathcal{O}_X(1)) \otimes \mathcal{O}mega^i_{\mathbb{P}}(i)|_X)} &[-6mm] \ldots &[-5mm] {\mathcal{O}_X \boxtimes (H^0(\mathcal{O}_X(i+1)) \otimes \mathcal{O}_X)} &[-6mm] {\mathcal{O}_X \boxtimes \mathcal{O}_X(i+1)} \\
{\mathcal{O}_X(1) \boxtimes \mathcal{O}mega^i_\mathbb{P}(i)|_X} & \ldots & {\mathcal{O}_X(i+1) \boxtimes \mathcal{O}_X} & {\mathcal{O}_\Delta(i+1)}
\arrow[from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\arrow[from=1-2, to=1-3]
\arrow[from=2-2, to=2-3]
\arrow[from=1-3, to=1-4]
\arrow[from=2-3, to=2-4]
\arrow[from=1-1, to=2-1]
\arrow[from=1-2, to=2-2]
\arrow[from=1-3, to=2-3]
\arrow[from=1-4, to=2-4]
\end{tikzcd}
\end{equation}
By induction hypothesis the differentials in the bottom row are given by the contraction with the restriction of the tautological section of $\mathcal{O}_{\mathbb{P}(V)}(-1) \boxtimes T_{\mathbb{P}(V)}(1)$, and the differentials in the top row are induced from that section.
Note that $H^0(\mathcal{O}_X(j)) \cong S^j V^{\scriptstyle\vee}$ for any $j < d$.
Recall that on the projective space we have a resolution for $\mathcal{O}mega^{i+1}_\mathbb{P}(i+1)$ given by a Koszul complex:
\[
0 \to \mathcal{O}mega_\mathbb{P}^{i+1}(i+1) \to V^{\scriptstyle\vee} \otimes \mathcal{O}mega_P^i(i) \to \ldots \to \mathcal{O} \otimes S^{i+1} V^{\scriptstyle\vee} \to \mathcal{O}_\mathbb{P}(i+1) \to 0.
\]
Thus, if $i+1 < d$, we recognize the upper row in the diagram \eqref{eq:inductive q} to be a complex quasiisomorphic to a single vector bundle $\mathcal{O}_X \boxtimes \mathcal{O}mega^{i+1}_\mathbb{P}(i+1)$, put to the leftmost degree. This finishes the inductive argument.
\end{proof}
\begin{theorem}[{\cite{kuznetsov-v14}}]
\label{thm:rotation periodicity}
There exists a morphism $\varphi_{Q_d}\colon Q_d \to Q_0[2]$ such that the following diagram commutes:
\begin{equation}
\label{eq:rotation periodicity commutes}
\begin{tikzcd}
{S^d V^{\scriptstyle\vee}} & {\mathrm{Hom}(Q_0, Q_d)} \\
{\mathrm{HH}^2(X)} & {\mathrm{Ext}^2(Q_0, Q_0)}
\arrow["{\varphi_{Q_d} \circ -}", from=1-2, to=2-2]
\arrow["{m_{Q_d}}", from=1-1, to=1-2]
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\end{tikzcd}
\end{equation}
Here the left vertical map is from Proposition~\textup{\ref{prop:hh2 and jacobian ideal}}, the top horizontal map is from Lemma~\textup{\ref{lem:multiplication on the subcategory}}, and the bottom horizontal arrow comes from the identification of $\mathrm{Ext}^2(Q_0, Q_0)$ with $\mathrm{HH}^2(\mathcal{O}_X^\perp)$.
\end{theorem}
\begin{proof}
This is, again, implicitly contained in the proof of \cite[Lem.~4.2]{kuznetsov-v14}. To explain the commutativity of the diagram, we repeat the argument.
Denote by $\widetilde{Q_d}$ the cone
\begin{equation}
\label{eq:widetilde qd}
\widetilde{Q_d} := \mathrm{Cone}(s_{\geq -d}(B_\Delta|_{X \times X}) \otimes \mathcal{O}(d, 0) \to \mathcal{O}_\Delta(d))
\end{equation}
as in Theorem~\ref{thm:kernels for rotation powers}. The diagram~\eqref{eq:inductive q} for~$i = d-1$ shows that the difference between~$\widetilde{Q_d}$ and~$Q_d$ arises from the fact that $H^0(X, \mathcal{O}_X(d))$ is isomorphic not to~$S^d V^{\scriptstyle\vee}$, but to the quotient~$S^d V^{\scriptstyle\vee} / \langle f \rangle$, where~$f$ is the equation of the hypersurface~$X \subset \mathbb{P}(V)$. More precisely, there is a distinguished triangle
\[
\langle f \rangle \cdot \mathcal{O}_X \boxtimes \mathcal{O}_X[2] \to \widetilde{Q_d} \to Q_d.
\]
Note that the convolution $(\mathcal{O}_X \boxtimes \mathcal{O}_X) \circ Q_0$ is a zero object since the Fourier--Mukai transform along $\mathcal{O}_X \boxtimes \mathcal{O}_X$ vanishes on $\mathcal{O}_X^\perp$, and $Q_0$ is exactly the projector to $\mathcal{O}_X^\perp$. Hence the convolution on the right with $Q_0$ produces an isomorphism
\[
\widetilde{Q_d} \circ Q_0 \to Q_d \circ Q_0 = Q_d.
\]
Consider now the commutative diagram of distinguished triangles arising from the stupid truncation of $B_\Delta|_{X \times X} \simeq \mathcal{O}_{\Delta_\mathbb{P}}|_{X \times X}$:
\begin{equation}
\label{eq:rotation periodicity comparison}
\begin{tikzcd}
{s_{\geq -d}(B_\Delta|_{X \times X}) \otimes \mathcal{O}(d, 0)} & {\mathcal{O}_\Delta(d)} & {\widetilde{Q_d}} \\
{B_\Delta|_{X \times X} \otimes \mathcal{O}(d, 0)} & {\mathcal{O}_\Delta(d)} & {\mathcal{O}_\Delta[2]}
\arrow[from=1-1, to=1-2]
\arrow[from=1-2, to=1-3]
\arrow[from=1-3, to=2-3]
\arrow["", from=2-1, to=2-2]
\arrow["{\varepsilonilon_X}", from=2-2, to=2-3]
\arrow["{=}", from=1-2, to=2-2]
\arrow[from=1-1, to=2-1]
\end{tikzcd}
\end{equation}
Since $B_\Delta$ is a resolution of the structure sheaf of the diagonal $\mathcal{O}_{\Delta_{\mathbb{P}}} \in D^b_{\!\mathrm{coh}}(\mathbb{P} \times \mathbb{P})$, the bottom horizontal triangle is the universal linkage class as defined in \eqref{eq:universal linkage}.
Using the rightmost vertical map, we define the morphism $Q_d \to Q_0[2]$ as the composition:
\[
Q_d \simeq \widetilde{Q_d} \circ Q_0 \to \mathcal{O}_\Delta[2] \circ Q_0 \cong Q_0[2].
\]
It only remains to show the commutativity of the diagram~\eqref{eq:rotation periodicity commutes}.
Recall the natural morphism~$\mathcal{O}_\Delta \to Q_d$ defined in Lemma~\ref{lem:multiplication on the subcategory}. It is easy to see from the definition~\eqref{eq:widetilde qd} that this morphism lifts to the map $\mathcal{O}_\Delta(d) \to \widetilde{Q_d}$, and thus the map $m_{Q_d}$ factors through the map
\[
S^d V^{\scriptstyle\vee} \to \mathrm{Hom}(\mathcal{O}_\Delta, \mathcal{O}_\Delta(d)) \to \mathrm{Hom}(\mathcal{O}_\Delta, \widetilde{Q_d}).
\]
Since the rightmost square in the diagram~\eqref{eq:rotation periodicity comparison} commutes, we additionally see that the map $m_{Q_d}$ factors through the composition with the universal linkage class:
\[
S^d V^{\scriptstyle\vee} \to \mathrm{Hom}(\mathcal{O}_\Delta, \mathcal{O}_\Delta(d)) \xrightarrow{\varepsilonilon_X \circ -} \mathrm{Hom}(\mathcal{O}_\Delta, \mathcal{O}_\Delta[2]),
\]
and the commutativity of the diagram \eqref{eq:rotation periodicity commutes} follows from Proposition~\ref{prop:hh2 and jacobian ideal}.
\end{proof}
\section{Kuznetsov components}
\label{sec:kuznetsov component}
We begin by discussing the basic properties of Kuznetsov components and their rotation functors.
\begin{definition}
\label{def:kuznetsov component}
The \emph{Kuznetsov component} of the hypersurface $X \subset \mathbb{P}(V)$ of degree $d < n+2$ is the category $\mathcal{A}_X$ defined as the left orthogonal to the exceptional sequence
\[
\langle \mathcal{O}_X, \mathcal{O}_X(1), \ldots, \mathcal{O}_X(n-d+1) \rangle
\]
in the category $D^b_{\!\mathrm{coh}}(X)$.
\end{definition}
\begin{definition}
\label{def:functors p}
We define the object $P_0 \in D^b_{\!\mathrm{coh}}(X \times X)$ to be the Fourier--Mukai kernel of the left projector to the subcategory $\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$. Recall the fundamental triangle of projector objects:
\begin{equation}
\label{eq:kuznetsov projectors triangle}
P_0^\operatorname{pr}ime \to \mathcal{O}_\Delta \to P_0.
\end{equation}
where $P_0^\operatorname{pr}ime$ is the right projector to the subcategory $\langle \mathcal{O}_X, \ldots, \mathcal{O}_X(n-d+1)\rangle$.
We define the objects $P_i$ for $i > 0$ recursively:
\[
P_i := P_{i-1} \circ \mathcal{O}_\Delta(1) \circ P_0,
\]
where the symbol $\circ$ denotes the convolution of Fourier--Mukai kernels in $D^b_{\!\mathrm{coh}}(X \times X)$.
\end{definition}
\begin{definition}
\label{def:rotation functor}
The \emph{rotation functor} $\mathbb{P}hi_{\mathcal{A}_X}\colon \mathcal{A}_X \to \mathcal{A}_X$ of $\mathcal{A}_X$ is defined as the composition
\[
\mathcal{A}_X \hookrightarrow D^b_{\!\mathrm{coh}}(X) \xrightarrow{- \otimes \mathcal{O}_X(1)} D^b_{\!\mathrm{coh}}(X) \twoheadrightarrow \mathcal{A}_X.
\]
It can alternatively be described as the Fourier--Mukai transform along the kernel $P_1$.
\end{definition}
The following result by Huybrechts and Rennemo computes the space of natural transformations from the identity functor on $\mathcal{A}_X$ to the rotation functor.
\begin{lemma}[{\cite[Lem.~3.1]{huybrechts-rennemo}}]
\label{lem:multiplication on kuznetsov component}
For any $i \geq 0$ there is a natural morphism $\mathcal{O}_\Delta(i) \to P_i$, which induces a map
\[
m_{P_i}\colon S^i V^{\scriptstyle\vee} \to \mathrm{Hom}(\mathcal{O}_\Delta, \mathcal{O}_\Delta(i)) \to \mathrm{Hom}(\mathcal{O}_\Delta, P_i) \to \mathrm{Hom}(P_0, P_i).
\]
Furthermore, if $d > 3$ or if $d = 3$ and $n > 3$, the map $m_{P_1}\colon V^{\scriptstyle\vee} \to \mathrm{Hom}(P_0, P_1)$ is an isomorphism of vector spaces.
\end{lemma}
For the proof of the main Theorem~\ref{thm:main theorem intro} we need some information about natural transformations from the identity functor of $\mathcal{A}_X$ to the $d$'th power of the rotation functor. The following two results are sufficient for our purposes.
\begin{lemma}
\label{lem:compatibility for rotations}
There exists a natural morphism $P_0 \to Q_0$, which induces a map $P_i \to Q_i$ for any $i \geq 0$. The precomposition with $P_0$ transforms this map into an isomorphism:
\[
P_i \cong P_i \circ P_0 \simeqarrow Q_i \circ P_0.
\]
\end{lemma}
\begin{remark}
Since $P_0$ is the projector to the subcategory $\mathcal{A}_X$, the last claim of Lemma~\ref{lem:compatibility for rotations} essentially means that the Fourier--Mukai transforms $D^b_{\!\mathrm{coh}}(X) \to D^b_{\!\mathrm{coh}}(X)$ along the two kernels~$P_i$ and~$Q_i$ agree on the subcategory $\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$.
\end{remark}
\begin{proof}
Note that $P_0$ and $Q_0$ are projector functors to subcategories $\mathcal{A}_X$ and $\mathcal{O}_X^\perp$, respectively. Since $\mathcal{A}_X \subset \mathcal{O}_X^\perp$, the statement is true for $i = 0$.
Since $P_i$ and $Q_i$ are defined inductively in terms of $P_0$ and $Q_0$, it is enough to show that the convolution with the map $P_0 \to Q_0$ induces an isomorphism
\[
P_0 \circ \mathcal{O}_\Delta(1) \circ P_0 \simeqarrow Q_0 \circ \mathcal{O}_\Delta(1) \circ P_0.
\]
Since $Q_0$ and $P_0$ are projectors to the subcategories $\mathcal{O}_X^\perp$ and $\mathcal{A}_X = \langle \mathcal{O}_X, \ldots, \mathcal{O}_X(n-d+1)\rangle^\perp$, respectively, it is enough to show that the image of the Fourier--Mukai transform along the object $\mathcal{O}_\Delta(1) \circ P_0$ lies in the orthogonal to the exceptional sequence $\langle \mathcal{O}_X(1), \ldots, \mathcal{O}_X(n-d+1)\rangle$. This is clear since $P_0$ is the projector to $\mathcal{A}_X$.
\end{proof}
\begin{theorem}[{\cite{kuznetsov-v14}}]
\label{thm:kuznetsov periodicity}
There is an isomorphism $\varphi_{P_d}\colon P_d \simeq P_0[2]$ such that the following diagram commutes:
\begin{equation}
\label{eq:kuznetsov periodicity}
\begin{tikzcd}
{S^d V^{\scriptstyle\vee}} & {\mathrm{Hom}(P_0, P_d)} \\
{\mathrm{HH}^2(X)} & {\mathrm{Ext}^2(P_0, P_0)}
\arrow["{\varphi_{P_d} \circ -}", from=1-2, to=2-2]
\arrow["{m_{P_d}}", from=1-1, to=1-2]
\arrow[from=1-1, to=2-1]
\arrow[from=2-1, to=2-2]
\end{tikzcd}
\end{equation}
Here the left vertical map is from Proposition~\textup{\ref{prop:hh2 and jacobian ideal}}, and the bottom horizontal arrow comes from the identification of $\mathrm{Ext}^2(P_0, P_0)$ with $\mathrm{HH}^2(\mathcal{A}_X)$.
\end{theorem}
\begin{proof}
This is proved in \cite[Lem.~4.2]{kuznetsov-v14}. We repeat the argument for the convenience of the reader.
By Lemma~\ref{lem:compatibility for rotations} the convolution $Q_d \circ P_0$ is naturally isomorphic to $P_d$.
Recall the morphism $\varphi_{Q_d}\colon Q_d \to Q_0[2]$ from Theorem~\ref{thm:rotation periodicity}. Let $\varphi_{P_d}\colon P_d \to P_0[2]$ be the convolution~$\varphi_{Q_d} \circ P_0$. Then by Theorem~\ref{thm:rotation periodicity} the following diagram commutes
\[\begin{tikzcd}
{S^d V^{\scriptstyle\vee}} & {\mathrm{Hom}(Q_0, Q_d)} & {\mathrm{Hom}(P_0, P_d)} \\
{\mathrm{HH}^2(X)} & {\mathrm{Ext}^2(Q_0, Q_0)} & {\mathrm{Ext}^2(P_0, P_0)}
\arrow["{\varphi_{Q_d} \circ -}", from=1-2, to=2-2]
\arrow["{- \circ P_0}", from=1-2, to=1-3]
\arrow["{- \circ P_0}", from=2-2, to=2-3]
\arrow["{\varphi_{P_d} \circ -}", from=1-3, to=2-3]
\arrow[from=1-1, to=2-1]
\arrow[from=1-1, to=1-2]
\arrow[from=2-1, to=2-2]
\end{tikzcd}\]
The composition of the maps in the upper row is equal to $m_{P_d}$ by definition. Thus the commutativity of the diagram~\eqref{eq:kuznetsov periodicity} is proved. It remains only to show that the map $\varphi_{P_d}$ is an isomorphism. To do this, consider the convolution of the diagram~\eqref{eq:rotation periodicity comparison} used in the proof of Theorem~\ref{thm:rotation periodicity} with the object~$P_0$. Note that $\widetilde{Q_d} \circ P_0 \cong Q_d \circ P_0 \cong P_d$. Thus the rightmost vertical map is exactly the morphism $\varphi_{P_d}\colon P_d \to P_0[2]$, and its cone is isomorphic to the object
\begin{equation}
\label{eq:cone of the isomorphism}
(s_{\leq -d-1}(B_\Delta|_{X \times X}) \otimes \mathcal{O}(d, 0)) \circ P_0[1].
\end{equation}
To show that this cone is zero, by Definition~\ref{def:beilinson resolution} it is enough to check that the convolution
\[
(\mathcal{O}_X(-k + d) \boxtimes \mathcal{O}mega_\mathbb{P}^k(k)|_X) \circ P_0
\]
vanishes for any $k$ satisfying $n+1 \geq k \geq d+1$. Since $P_0$ is the projector to the Kuznetsov component $\mathcal{A}_X$, any object in the image of $P_0$ is right-orthogonal to $\mathcal{O}_X(-k+d)$ for $k \in [d+1; n+1]$, and hence the object~\eqref{eq:cone of the isomorphism} is zero.
\end{proof}
\section{Rotation functors and Hochschild cohomology}
\label{sec:injectivity on hochschild}
Recall that we work with a Fano hypersurface $X \subset \mathbb{P}(V)$ of dimension $n$ and degree $d$. In particular, $d < n+2$.
\begin{lemma}
\label{lem:nondiagonal diamonds for hypersurfaces}
If $d > 3$ or if $d = 3$ and $n \geq 3$, then the Hodge diamond of $X$ is not diagonal.
\end{lemma}
\begin{proof}
By Griffiths' theorem \cite[Th.~8.3]{griffiths} for any $0 \leq p \leq n$ there exists an isomorphism
\[
H^{p, n - p}_{\mathrm{prim}} \simeq J^{t_p}(f)
\]
between the primitive part of the cohomology of $X$ and a particular graded component of the Jacobian ring, where $t_p = (n - p + 1)d - (n+2)$. Since $X$ is smooth, the Jacobian ring is a finite-dimensional graded ring such that any graded component in degrees between~$0$ and~$(d-2)(n+2)$ is nonzero (see, e.g., \cite{donagi}). Thus it is enough to find some $p \neq n/2$ such that $t_p$ lies between $0$ and $(d-2)(n+2)$.
If~$n = 2m+1$ is an odd number, the condition $d \geq 3$ implies that we can take $p = m$, so we get that~$H^{m, m+1}(X) \neq 0$. If~$n = 2m$ is an even number, the condition~$d \geq 3, n \geq 4$ implies that we can take~$p = m-1$, i.e.,~$H^{m-1, m+1}(X) \neq 0$.
\end{proof}
\begin{lemma}
\label{lem:action is nondegenerate if non-diagonal}
Let $X \subset \mathbb{P}(V)$ be a hypersurface. Let $\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$ be an admissible subcategory. Assume that the orthogonal subcategory~$\mathcal{A}_X^\perp \subset D^b_{\!\mathrm{coh}}(X)$ has a full exceptional collection and the Hodge diamond of $X$ is not diagonal. Then the composition
\[
H^1(X, T_X) \hookrightarrow \mathrm{HH}^2(X) \to \mathrm{HH}^2(\mathcal{A}_X)
\]
is an injection.
\end{lemma}
\begin{remark}
A similar idea in a different situation has been recently used in \cite[Thm.~1.2]{infinitesimal-torelli}.
\end{remark}
\begin{proof}
Since $\mathcal{A}_X^\perp$ is generated by an exceptional collection, by the additivity of Hochschild homology \cite[Cor.~7.5]{kuznetsov-oldhochschild} we have
\begin{equation}
\label{eq:hochschild additivity}
\mathrm{HH}_*(X) \cong \mathrm{HH}_*(\mathcal{A}_X) \oplus \mathrm{HH}_*(\mathrm{pt})^{\oplus k},
\end{equation}
where $k$ is the length of the exceptional collection in $\mathcal{A}_X^\perp$. In particular, for any $i \neq 0$ we have an equality~$\mathrm{HH}_i(X) = \mathrm{HH}_i(\mathcal{A}_X)$. There is an action of Hochschild cohomology on Hochschild homology, and this action is compatible with the decomposition~\eqref{eq:hochschild additivity} by construction (see, e.g., \cite[Prop.~6.1]{addington-thomas} for the case where $\mathcal{A}_X$ is isomorphic to $D^b_{\!\mathrm{coh}}(S)$ for some smooth projective variety $S$; the proof works in general).
Let $\xi \in H^1(X, T_X)$ be a nonzero element. We want to show that its image in $\mathrm{HH}^2(\mathcal{A}_X)$ is nonzero. It is enough to show that the class of $\xi$ in $\mathrm{HH}^2(X)$ acts on $\mathrm{HH}_*(\mathcal{A}_X)$ nontrivially. By~\eqref{eq:hochschild additivity} it is enough to show that $\xi$ acts nontrivially on the non-zero degree part of the Hochschild homology of $X$, i.e., to find a complementary class~$\widetilde{\xi} \in H^1(X, T_X)^{\otimes N-1}$ and some integer~$a$ so that the action map
\begin{equation}
\label{eq:action on hochschild cohomology}
\mathrm{HH}_a(X) \xrightarrow{\xi \cdot \widetilde{\xi} \cdot -} \mathrm{HH}_{a+2N}(X)
\end{equation}
is nontrivial, $a \neq 0$ and $a+2N \neq 0$.
Using the (Todd-twisted) Hochschild--Kostant--Rosenberg isomorphism~\cite[Th.~1.4]{hkr-action-compatibility} the action of Hochschild cohomology on Hochschild homology can be reinterpreted in terms of the Hodge structure. Namely, under the isomorphisms
\[
\mathrm{HH}_a(X) \simeq \bigoplus_{i \geq 0} H^{i}(\mathcal{O}mega^{a+i}_X), \qquad \mathrm{HH}^2(X) \simeq \bigoplus_{i \geq 0} H^i(\Lambda^{2-i}(T_X))
\]
the Hochschild-homological action of $H^1(T_X) \subset \mathrm{HH}^2(X)$ on $\mathrm{HH}_\bullet(X)$ differs from the one induced from the contraction morphism $T_X \otimes \mathcal{O}mega^1_X \to \mathcal{O}_X$ only by the multiplication with the Todd class. Since in~\eqref{eq:action on hochschild cohomology} we have $a \neq 0$ and $a+2N \neq 0$, we are only interested in what happens in the middle cohomology of the hypersurface, $H^n(X)$, and thus the twist by the Todd class does not matter for our purposes since it only changes the result by corrections in other cohomological degrees. Hence it is enough to find two integers,~$p < q$, none of which is equal to~$n/2$, and a complementary class~$\widetilde{\xi} \in H^1(X, T_X)^{\otimes q-p-1}$ such that the multiplication map
\begin{equation}
\label{eq:polyvector multiplication}
H^{q, n-q}(X) \xrightarrow{\xi \cdot -} H^{q-1, n-q+1}(X) \xrightarrow{\widetilde{\xi} \cdot -} H^{p, n-p}(X)
\end{equation}
is nonzero.
By Lemma~\ref{lem:nondiagonal diamonds for hypersurfaces} the Hodge diamond of $X$ is not diagonal. By symmetry there are at least two integers $p < q$, none of which are equal to $n/2$, such that $H^{p, n-p}(X)$ and $H^{q, n-q}(X)$ are both nonzero. A refinement of Griffiths' theorem (see, e.g., \cite[Th.~2.2]{donagi}) shows that not only those cohomology groups are isomorphic to the components of the Jacobian ring, but also the action of $H^1(X, T_X) \simeq J^d(f)$ is given by the multiplication in the ring. The multiplication in the Jacobian ring is nondegenerate (see, e.g, \cite[Th.~2.6]{donagi}), and hence it is possible to choose a complementary class $\widetilde{\xi}$ such that the multipilcation~\eqref{eq:polyvector multiplication} is nonzero. Thus the injectivity is proved.
\end{proof}
We are now ready to prove the main theorem of this paper.
\begin{theorem}[{ = Theorem~\ref{thm:main theorem intro}}]
\label{thm:main theorem}
Let $X \subset \mathbb{P}(V)$ be a smooth $n$-dimensional hypersurface of degree~$d < n+2$ given by the equation~$f \in S^d V^{\scriptstyle\vee}$. Assume that~$d \geq 4$, or~$d \geq 3$ and~$n > 3$. Let~$\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$ be the Kuznetsov component of $X$ (Definition~\textup{\ref{def:kuznetsov component}}), and let $\mathbb{P}hi_{\mathcal{A}_X}$ be the rotation functor (Definition~\textup{\ref{def:rotation functor}}).
Then the pair $(\mathcal{A}_X, \mathbb{P}hi_{\mathcal{A}_X})$, as a dg-category with a dg-endofunctor, determines $X$ up to an isomorphism.
\end{theorem}
\begin{proof}
For any $k \geq 0$ the vector space of dg-natural transformations $\mathrm{id}_{\mathcal{A}_X} \Rightarrow \mathbb{P}hi^{\circ k}$, i.e., the homotopy classes of maps in the dg-category of functors from $\mathcal{A}_X$ to itself, is naturally isomorphic to $\mathrm{Hom}_{X \times X}(P_0, P_k)$ \cite{toen}. By Lemma~\ref{lem:multiplication on kuznetsov component} the vector space $\mathrm{Hom}(P_0, P_1)$ is isomorphic to $V^{\scriptstyle\vee}$. The composition of the maps defines for any $k \geq 0$ a morphism
\[
(V^{\scriptstyle\vee})^{\otimes k} \to \mathrm{Hom}(P_0, P_k),
\]
and it factors through the map $m_{P_k}\colon S^k V^{\scriptstyle\vee} \to \mathrm{Hom}(P_0, P_k)$ defined in Lemma~\ref{lem:multiplication on kuznetsov component} by construction. Thus for $k = d$ we get a commutative diagram:
\[\begin{tikzcd}
{S^d V^{\scriptstyle\vee}} & {\mathrm{dgNat}(\mathrm{id}_{\mathcal{A}_X}, \mathbb{P}hi_{\mathcal{A}_X}^{\circ d})} \\
& {\mathrm{Hom}_{X \times X}(P_0, P_d)}
\arrow[from=1-1, to=1-2]
\arrow["{m_{P_d}}"', from=1-1, to=2-2]
\arrow["\cong", from=1-2, to=2-2]
\end{tikzcd}\]
By Theorem~\ref{thm:kuznetsov periodicity} the vector space $\mathrm{Hom}(P_0, P_d)$ is isomorphic to $\mathrm{Ext}^2(P_0, P_0)$, and the kernel of the diagonal arrow is equal to the kernel of the composition
\[
S^d V^{\scriptstyle\vee} \to H^1(X, T_X) \hookrightarrow \mathrm{HH}^2(X) \to \mathrm{HH}^2(\mathcal{A}_X)
\]
By Lemma~\ref{lem:action is nondegenerate if non-diagonal} the composition of the last two arrow is injective. By Proposition~\ref{prop:hh2 and jacobian ideal} the kernel of the first morphism is equal to the $d$'th component of the Jacobian ideal of $f$.
Thus, up to an automorphism of $V$, we reconstructed the $d$'th component of the Jacobian ideal of $f$ as a subspace in $S^d V^{\scriptstyle\vee}$ from the pair $(\mathcal{A}_X, \mathbb{P}hi)$. This subspace, in turn, recovers the hypersurface $X$ up to an automorphism of $V$ by Mather--Yau theorem \cite[Prop.~1.1]{donagi}.
\end{proof}
\begin{corollary}
Let $X \subset \mathbb{P}(V)$ be an $n$-dimensional hypersurface of degree $n+1$. Assume that $n \geq 3$. Then the Kuznetsov component $\mathcal{A}_X \subset D^b_{\!\mathrm{coh}}(X)$, considered as a dg-category, determines $X$ up to an isomorphism.
\end{corollary}
\begin{proof}
If $d = n+1$, the endofunctor $\mathbb{P}hi$ defined in the statement of Theorem~\ref{thm:main theorem} is, up to a shift, the inverse Serre functor of $\mathcal{A}_X$ \cite[Lem.~4.1]{kuznetsov-v14}. Thus it can be canonically recovered as a dg-endofunctor from the dg-structure on $\mathcal{A}_X$, and the result follows from Theorem~\ref{thm:main theorem}.
\end{proof}
\operatorname{pr}intbibliography
\end{document} |
\begin{equation}gin{document}
\title{Quantum illumination for enhanced detection of Rayleigh-fading targets}
\author{Quntao Zhuang$^{1,2}$}
\email{[email protected]}
\author{Zheshen Zhang$^1$}
\author{Jeffrey H. Shapiro$^1$}
\affiliation{$^1$Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\\
$^2$Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA}
\date{\today}
\begin{equation}gin{abstract}
Quantum illumination (QI) is an entanglement-enhanced sensing system whose performance advantage over a comparable classical system survives its usage in an entanglement-breaking scenario plagued by loss and noise. In particular, QI's error-probability exponent for discriminating between equally-likely hypotheses of target absence or presence is 6\,dB higher than that of the optimum classical system using the same transmitted power. This performance advantage, however, presumes that the target return, when present, has known amplitude and phase, a situation that seldom occurs in lidar applications. At lidar wavelengths, most target surfaces are sufficiently rough that their returns are speckled, i.e., they have Rayleigh-distributed amplitudes and uniformly-distributed phases. QI's optical parametric amplifier receiver---which affords a 3\,dB better-than-classical error-probability exponent for a return with known amplitude and phase---fails to offer any performance gain for Rayleigh-fading targets. We show that the sum-frequency generation receiver [Phys. Rev. Lett. {\bf 118}, 040801 (2017)]---whose error-probability exponent for a nonfading target achieves QI's full 6\,dB advantage over optimum classical operation---outperforms the classical system for Rayleigh-fading targets. In this case, QI's advantage is subexponential: its error probability is lower than the classical system's by a factor of $1/\ln(M\begin{eqnarray}r{\kappa}N_S/N_B)$, when $M\begin{eqnarray}r{\kappa}N_S/N_B \gg 1$, with $M\gg 1$ being the QI transmitter's time-bandwidth product, $N_S \ll 1$ its brightness, $\begin{eqnarray}r{\kappa}$ the target return's average intensity, and $N_B$ the background light's brightness.
\end{abstract}
\maketitle
Quantum illumination (QI)~\cite{Sacchi_2005_1,Sacchi_2005_2,Lloyd2008,Tan2008,Lopaeva_2013,Guha2009,Ragy2014,Zheshen_15,Barzanjeh_2015} uses entanglement to
outperform the optimum classical-illumination (CI) system for detecting the presence of a weakly-reflecting target that is embedded in a very noisy background, despite that environment's destroying the initial entanglement~\cite{footnote0}. With optimum quantum reception, QI's error-probability exponent---set by the quantum Chernoff bound (QCB)~\cite{Audenaert2007}---is 6\,dB higher~\cite{Tan2008} than that of the optimum CI system, i.e., a coherent-state transmitter and a homodyne receiver. Until recently, the sole structured receiver for QI that outperformed CI---Guha and Erkmen's optical parametric amplifier (OPA) receiver~\cite{Guha2009}---offered only a 3\,dB increase in error-probability exponent. In Ref.~\cite{Zhuang_2017}, we showed that the sum-frequency generation (SFG) receiver's error-probability exponent reached QI's QCB. Moreover, augmenting that receiver with feed-forward (FF) operations yielded the FF-SFG receiver~\cite{Zhuang_2017}, whose performance, for a low-brightness transmitter, matched QI's Helstrom limit for both the target-detection error probability and the Neyman-Pearson criterion's receiver operating characteristic (ROC)~\cite{zhuang2017entanglement}.
Prior QI performance analyses~\cite{Tan2008,Guha2009,Zhuang_2017,zhuang2017entanglement} have all assumed that the target return has known amplitude and phase, something that seldom occurs in lidar applications. At lidar wavelengths, most target surfaces are sufficiently rough that their returns are speckled, i.e., they have Rayleigh-distributed amplitudes and uniformly-distributed phases~\cite{Goodman1965,Goodman1976,Shapiro1981,Shapiro1982}. It is crucial, therefore, to show that QI maintains a target-detection performance advantage over CI for a target return with random amplitude and phase.
In this paper, we compare QI and CI target detection for Rayleigh-fading targets in the flat-fading limit, when the complex-field envelope of the target return from a single transmitted pulse suffers multiplication by a time-independent Rayleigh-distributed random amplitude and a time-independent uniformly-distributed random phase shift. We show that QI with OPA reception fails to offer any performance advantage over CI in this case. QI with SFG reception does provide an advantage over CI: when $M\begin{eqnarray}r{\kappa}N_S/N_B \gg 1$, its error probability is a factor of $1/\ln(M\begin{eqnarray}r{\kappa}N_S/N_B)$ lower than that of optimum CI, which transmits a coherent state and uses heterodyne reception. Here, $M\gg 1$ is the QI transmitter's time-bandwidth product, $N_S$ is its brightness, $\begin{eqnarray}r{\kappa}$ is the target return's average intensity, and $N_B$ is the background light's brightness.
{\em QI target detection}---.
In QI, the transmitter illuminates the region of interest with a single-spatial-mode, $T$-s-long pulse of signal light produced by pulse carving the continuous-wave output of a spontaneous parametric downconverter (SPDC). The SPDC source is taken to have a $W$-Hz-bandwidth, flat-spectrum phase-matching function with $W \gg 1/T$. The resulting signal pulse is maximally entangled with a corresponding single-spatial-mode, $T$-s-long pulse of idler light that the transmitter retains for subsequent joint measurement with the light returned from the region of interest. The $M = TW \gg 1$ signal-idler mode pairs that comprise the transmitted signal and retained idler pulses are thus in independent, identically-distributed (iid), two-mode squeezed-vacuum states with average photon number $N_S \ll 1$ in each signal and idler mode. Let $\{\hat{a}_{S_m},\hat{a}_{I_m}\}$ be the photon-annihilation operators for the transmitter's $M$ signal and idler modes, and $\{\hat{a}_{R_m}\}$ the photon-annihilation operators of the $M$ modes returned from the region of interest. The target-detection hypothesis test is to determine whether $h=0$ (target absent) or $h=1$ (target present) is true when: $\hat{a}_{R_m} = \hat{a}_{B_m}$, for $h =0$, and $\hat{a}_{R_m} = \sqrt{\kappa}\,e^{i\phi}\hat{a}_{S_m} + \sqrt{1-\kappa}\,\hat{a}_{B_m}$, for $h=1$. Here: the $\{\hat{a}_{B_m}\}$ are photon-annihilation operators for iid background-noise modes that are in the thermal state with average photon number $N_B \gg 1$ when $h=0$ and in the thermal state with average photon number $N_B/(1-\kappa)$ when $h=1$~\cite{footnote1}; $\kappa > 0$ is the target-return's reflectivity; and $\phi$ is the target-return's phase.
Previous theoretical work on QI target detection~\cite{Tan2008,Guha2009,Barzanjeh_2015,Zhuang_2017} has assumed known $\kappa$, $\phi = 0$~\cite{footnote2},
and lossless idler storage. For equally-likely target absence or presence, QI with optimum quantum reception---realizable with FF-SFG~\cite{Zhuang_2017}---has error probability $\Pr(e)_{\rm opt} \simeq e^{-M\kappa N_S/N_B}/2$, QI with OPA reception has error probability $\Pr(e)_{\rm OPA} \simeq e^{-M\kappa N_S/2N_B}/2$, and optimum CI has error probability $\Pr(e)_{\rm CI} \simeq e^{-M\kappa N_S/4N_B}/2$.
Lidar targets are almost always speckle targets, viz., $\sqrt{\kappa}$ and $\phi$ are statistically independent random variables whose respective probability density functions (pdfs) are
$f_{\sqrt{\kappa}}(x) = 2xe^{-x^2/\begin{eqnarray}r{\kappa}}/\begin{eqnarray}r{\kappa}$, for $x >0$, and
$f_\phi(y) = 1/2\pi$, for $0\le y\le 2\pi$,
where $\begin{eqnarray}r{\kappa}$ is the target return's average intensity. These statistics invalidate \emph{all} of the error-probability expressions from the preceding paragraph. Worse, as will soon be seen, they preclude \emph{any} QI receiver from obtaining a single-pulse error probability that decreases exponentially with increasing $M\begin{eqnarray}r{\kappa}N_S/N_B$. For that demonstration we will employ the QCB, an exponentially-tight upper bound on the error probability of optimum quantum reception for multiple-copy quantum state discrimination~\cite{Audenaert2007}.
{\em The QCB applied to QI with Rayleigh fading}---.
Conditioned on knowledge of $h$, $\sqrt{\kappa}$, and $\phi$, the $\{\hat{a}_{R_m},\hat{a}_{I_m}\}$ mode pairs at the QI receiver are in the state $\hat{\boldsymbol \rho}_h(\sqrt{\kappa},\phi) = \otimes_{m=1}^M\hat{\rho}^{(m)}_h(\sqrt{\kappa},\phi)$, with $\hat{\rho}_h^{(m)}(\sqrt{\kappa},\phi)$ being the two-mode, zero-mean, Gaussian state whose Wigner covariance matrix is
\begin{eqnarray}
&
{\mathbf{\Lambda}}_h =
\frac{1}{4}
\left[
\begin{equation}gin{array}{cccc}
(2N_B+1) {\mathbf I}&2C_p{\mathbf R}_h\\
2C_p{\mathbf R}_h&(2N_S+1){\mathbf I}
\end{array}
\right],
\label{hk}
&
\end{eqnarray}
where $N_B \gg 1\gg N_S$ has been used. In this covariance matrix: ${\bf I}$ is the $2\times 2$ identity matrix, and ${\mathbf R}_h={\rm Re}\!\left[e^{i\phi} \left({\mathbf Z}-i{\mathbf X}\right)\right]\delta_{h1}$, where $\delta_{hk}$ is the Kronecker delta function, and ${\mathbf Z}$ and ${\mathbf X}$ are $2\times 2$ Pauli matrices. It follows that the signature of target presence is the nonzero phase-sensitive cross correlation, $C_p=\sqrt{\kappa N_S\left(N_S+1\right)}$, between the returned signal and the retained idler modes.
Erroneous target-detection decisions can be either false-alarm errors, when target presence is declared but no target is present, or miss errors, when target absence is declared but a target is present. For a given target-detection system, the conditional probabilities for these errors to occur are the false-alarm probability $P_F$, and the miss probability $P_M = 1-P_D$, where $P_D$ is the detection probability, i.e., the probability that target presence is declared when a target is present. Almost all QI target detection analyses~\cite{Tan2008,Guha2009,Barzanjeh_2015,Zhuang_2017} have been Bayesian: assign prior probabilities, $\{\pi_h\}$, to $h=0$ and $h=1$, and minimize the error probability, $\Pr(e) = \pi_0P_F + \pi_1P_M$, typically for equiprobable hypotheses, $\pi_0 = \pi_1 = 1/2$. Owing to the difficulty of accurately assigning priors to target absence and presence, a better approach to optimizing target-detection performance is to apply the Neyman-Pearson performance criterion: maximize $P_D$ subject to a constraint on $P_F$. Only recently has this criterion been applied to QI target detection~\cite{zhuang2017entanglement}, and that work assumed knowledge of the target return's amplitude and phase. In this paper, we will consider both performance criteria---minimizing $\Pr(e)$ and maximizing $P_D$ for a given $P_F$---for our Rayleigh-fading QI scenario.
In the Bayesian approach, the minimum error probability for QI target detection is set by the Helstrom limit~\cite{Helstrom1969} for discriminating between the \emph{unconditional} $h=0$ and $h=1$ states,
\begin{equation}gin{equation}
\hat{\begin{eqnarray}r{\boldsymbol \rho}}_h = \int\!{\rm d}x\int\!{\rm d}y\,f_{\sqrt{\kappa}}(x)f_\phi(y)\hat{\boldsymbol \rho}_h(x,y).
\end{equation}
This limit's calculation requires diagonalizing $\pi_1\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1-\pi_0\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0$, so it is intractable for QI with Rayleigh fading, because $\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1$ is not an $M$-fold product state. Nevertheless, applying the QCB will yield an informative result.
Let $D_{\pi_0}(\hat{\boldsymbol \rho}_0(x,y),\hat{\boldsymbol \rho}_1(x,y))$ denote the Helstrom limit for discriminating between $\hat{\boldsymbol \rho}_0(x,y)$ and $\hat{\boldsymbol \rho}_1(x,y)$ that occur with priors $\pi_0$ and $\pi_1$, and let $\xi_{\rm QCB}(\hat{\boldsymbol \rho}_0(x,y),\hat{\boldsymbol \rho}_1(x,y)) \equiv -\lim_{M\rightarrow \infty} {\ln[D_{\pi_0}(\hat{\boldsymbol \rho}_0(x,y),\hat{\boldsymbol \rho}_1(x,y))]}/M$ be the QCB on its error-probability exponent. Then, using the Helstrom limit's being concave in quantum states (see Lemma~1 in the Appendix), we can show (see Lemma~2 in the Appendix) that the Helstrom limit's error-probability exponent for QI target detection, $\xi_{\rm QI}\equiv-\lim_{M\to\infty}{\ln[D_{\pi_0}(\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0,\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1)]}/M$, vanishes, i.e., $\xi_{\rm QI} = 0$, for all $\pi_0\pi_1 \neq 0$. Having $\xi_{\rm QI} = 0$ implies that optimum quantum reception for QI target detection with Rayleigh fading has an error probability that decreases subexponentially with the number of signal-idler mode pairs that are employed. This subexponential error-probability behavior applies to \emph{all} QI receivers, including the FF-SFG, SFG, and OPA receivers. Because OPA receivers are relatively easy to build~\cite{Zheshen_15}---as opposed to the far more complicated SFG and FF-SFG receivers~\cite{Zhuang_2017}---one might hope that QI with OPA reception would offer a performance advantage over optimum CI for the Rayleigh-fading scenario. We next show that such is not the case.
{\em OPA reception for QI with Rayleigh fading}---.
It is difficult to get an analytic error-probability approximation for QI with OPA reception in the Rayleigh-fading scenario, so we will content ourselves with finding its SNR and comparing that result to the SNR for the optimum Rayleigh-fading CI system.
The OPA receiver's essence is converting QI's phase-sensitive cross-correlation signature of target presence to an average photon-number signature that can be sensed with direct detection.
In particular, the OPA receiver measures
$\hat{N} \equiv\sum_{m=1}^M \hat{a}_m^\dagger\hat{a}_m$, where
$\hat{a}_m = \sqrt{G}\,\hat{a}_{I_m} + \sqrt{G-1}\,\hat{a}_{R_m}^\dagger$
is the idler-port output of a low-gain ($\max(N_S/N_B,N_S/\kappa N_B^2) \ll G-1 \sim \sqrt{N_S}/N_B \ll 1$) OPA. Hence, we define its SNR to be
${\rm SNR}_{\rm OPA} \equiv [(\sum_{j=0}^1(-1)^j\langle\hat{N}\rangle_j) /(\sum_{j=0}^1\sqrt{{\rm Var}_j(\hat{N})})]^2$,
where $\langle \hat{N}\rangle_j$ and ${\rm Var}_j(\hat{N})$ for $j=0,1$ are the conditional means and conditional variances of the $\hat{N}$ measurement given $h=j$.
For known $\kappa$ and $\phi =0$, we get
$\langle\hat{N}\rangle_1 - \langle\hat{N}\rangle_0 \approx 2M\sqrt{G(G-1)\kappa N_S(N_S+1)}$. Combining this result with ${\rm Var}_j(\hat{N}) \approx \langle\hat{N}\rangle_j$ for the $\hat{N}$ measurement's conditional variances, gives ${\rm SNR}_{\rm OPA} \approx M\kappa N_S/N_B$ when $N_S \ll 1$, $\kappa\ll 1$ is known, $\phi=0$, and $N_B \gg 1$. In the Rayleigh-fading case, the uniformly-distributed random phase destroys the phase-sensitive cross-correlation signature in $\langle \hat{N}\rangle_1$, leading to $\langle\hat{N}\rangle_1 - \langle\hat{N}\rangle_0 = M(G-1)\begin{eqnarray}r{\kappa}N_S$, and it adds $2M^2(G-1)\begin{eqnarray}r{\kappa}N_S$ to ${\rm Var}_1(\hat{N})$, hence giving us
\begin{equation}gin{equation}
{\rm SNR}_{\rm OPA} \approx \frac{M(G-1)(\begin{eqnarray}r{\kappa}N_S)^2/N_B}{(1+\sqrt{1+ 2M\begin{eqnarray}r{\kappa}N_S/N_B})^2},
\end{equation}
which is much smaller than $M\begin{eqnarray}r{\kappa}N_S/N_B$, the ${\rm SNR}_{\rm OPA}$ for a known $\kappa = \begin{eqnarray}r{\kappa}$ and $\phi=0$~\cite{footnote3}.
Optimum CI for Rayleigh fading does matched filtering of its heterodyne detector's output followed by square-law envelope detection that yields an output, $R$, which is exponentially distributed under both $h=0$ and $h=1$~\cite{VanTrees1}. The SNR for this system, ${\rm SNR}_{\rm CI} \equiv [(\sum_{j=0}^1(-1)^j\langle R\rangle_j)/(\sum_{j=0}^1\sqrt{{\rm Var}_j(R)})]^2$, satisfies
\begin{equation}gin{equation}
{\rm SN}_{\rm CI} = (M\begin{eqnarray}r{\kappa} N_S/2N_B)/\left(1+M\begin{eqnarray}r{\kappa}N_S/2N_B\right)^2,
\end{equation}
which is orders of magnitude greater than ${\rm SNR}_{\rm OPA}$ for Rayleigh fading in the interesting $M\begin{eqnarray}r{\kappa}N_S/N_B \gg 1$ operating regime.
{\em SFG Reception for QI with Rayleigh Fading}---.
The SFG receiver~\cite{Zhuang_2017} uses a succession of $K$ SFG stages. At the input to each such stage a beam splitter taps off a small fraction of the light returned from the region of interest to undergo SFG with the retained idler light. The returned-light output from that SFG process is then recombined with the portion remaining from that stage's input beam-splitter and applied, along with the retained-idler output, to the next stage. Photon-counting measurements are performed on the SFG's sum-frequency output and on the auxiliary output from the return-light beam splitter at the output of each SFG stage. These measurements are used to decide on target absence or presence. Figure~\ref{SFGrcvr} shows a schematic representation of the SFG receiver's $k$th stage, for more details see Ref.~\cite{Zhuang_2017}.
\begin{equation}gin{figure}[th]
\includegraphics[width=0.4\textwidth]{Zhuang_QI_with_fading_Fig1.pdf}
\caption{Schematic representation of the sum-frequency generation (SFG) receiver's $k$th stage, showing only the $m$th mode pair, although all $M$ mode pairs are processed simultaneously. The $m$th mode pair of the returned light ($\hat{a}_{R_m}^{(k)}$) and the retained idler ($\hat{a}_{I_m}^{(k)}$) at the input to the $k$th stage is transformed into the corresponding mode pair at that stage's output by means of SFG. Photon-counting measurements are made on the single-mode sum-frequency output ($\hat{b}^{(k)}$) and the auxiliary output modes ($\{\hat{a}_{E_m}^{(k)} : 1 \le m \le M\}$). The SFG receiver's decision as to target absence or presence is based on the total of all the photon-counting measurements, i.e., $N_T \equiv \sum_{k=1}^K(N_b^{(k)} + N_E^{(k)}),$ where $N_b^{(k)}$ is the outcome of the $\hat{b}^{(k)\dagger}\hat{b}^{(k)}$ measurement, and $N_E^{(k)}$ is the outcome of the $\sum_{m=1}^M\hat{a}_{E_m}^{(k)\dagger}\hat{a}_{E_m}^{(k)}$ measurement.}
\label{SFGrcvr}
\end{figure}
For known $\kappa$ and $\phi = 0$, SFG reception's error probability achieves the QCB. The FF-SFG receiver~\cite{Zhuang_2017} augments the SFG receiver with pre-SFG and post-SFG squeezers, whose parameters are chosen in accordance with a Bayesian update rule that is controlled by feed-forward information from the prior stages. FF-SFG reception reaches the Helstrom limit for QI target detection---in both the Bayesian and Neyman-Pearson settings---for known $\kappa$ and $\phi = 0$~\cite{Zhuang_2017,zhuang2017entanglement}. Because its feed-forward operations exploit $\phi=0$, FF-SFG reception ceases to function effectively when $\phi$ is uniformly distributed. SFG reception, which eschews the use of feed-forward, \emph{does} cope with random amplitude and phase, as we now show.
When $h=0$, the SFG receiver's total photon count---i.e., $N_T\equiv \sum_{k=1}^K(N_b^{(k)} + N_E^{(k)})$ from Fig.~\ref{SFGrcvr}---is the sum of $M$ iid Bose-Einstein random variables, and has mean value $N_0 \simeq -N_S \ln(\epsilon)/2$ for $N_S \ll 1$. When $h=1$, and conditioned on the values of $\kappa$ and $\phi$, the statistics of the SFG receiver's total photon count equal those for direct detection of the coherent state $|\sqrt{(1-\epsilon)M\kappa N_S/N_B}\,e^{i\phi}\rangle$ embedded in a weak thermal-noise background of average photon number $N_0 \ll 1$. In these expressions, $\epsilon \ll 1$ is chosen to obtain good performance, see~\cite{Zhuang_2017} for details. When $M\kappa N_S/N_B \gg N_0$, the thermal contribution to the $h=1$ statistics can be neglected. Then, averaging the $h=1$ conditional state over the $\sqrt{\kappa}$ and $\phi$ statistics results in a thermal state with average photon number $N_1 = (1-\epsilon)M\begin{eqnarray}r{\kappa}N_S/N_B$, implying that the SFG receiver has reduced Rayleigh-fading QI target detection to discriminating between two thermal states,
$\hat{\sigma}_{0} = \sum_{n=0}^\infty [N_0^n/(N_0+1)^{(n+1)}]\ket{n}\bra{n}$ and
$\hat{\sigma}_{1} = \sum_{n=0}^\infty [N_1^n/(N_1+1)^{(n+1)}]\ket{n}\bra{n}$, using photon-counting measurements.
SFG reception's minimum error-probability decision, $\tilde{h} = 0$ or 1, is therefore
$\tilde{h}=\argmax_h \pi_h \!\left[N_h^n/(N_h+1)^{(n+1)}\right]$, where $n$ is the observed photon count.
The preceding rule can be implemented as a threshold test: $\tilde{h} = 1$ if and only if $n > n_t$, where the threshold $n_t$ satisfies $\pi_0 N_0^{n_t}/(N_0+1)^{(n_t+1)}\ge \pi_1 N_1^{n_t}/(N_1+1)^{(n_t+1)}$ and $\pi_0 N_0^{n_t+1}/(N_0+1)^{(n_t+2)}< \pi_1 N_1^{n_t+1}/(N_1+1)^{(n_t+2)}$. SFG reception's ROC---its $P_D$ versus $P_F$ behavior---can now be obtained analytically. For integer $n_t$, we have $P_F^{\rm SFG}= [N_0/(N_0+1)]^{n_t+1}$ and $P_D^{\rm SFG}=[N_1/(N_1+1)]^{n_t+1}$. ROC points intermediate between those generated with integer thresholds are then obtained from randomized tests~\cite{VanTrees2}.
The Bayesian approach's error probability is easily found once its decision rule's threshold $n_t$ is determined. Evaluating the false-alarm and detection probabilities for that threshold value, SFG reception's error probability then follows from
$
\Pr(e)_{\rm SFG}=\pi_0P_F^{\rm SFG}+\pi_1 (1-P_D^{\rm SFG}).
$
For $N_S\to 0$ with $\epsilon\ll1$, we find that $n_t=0$ and hence
\begin{equation}
\Pr(e)_{\rm SFG}\simeq \Pr(e)_{\rm SFG}^{N_S\to0}\equiv \pi_1/(1+M\begin{eqnarray}r{\kappa}N_S/N_B).
\label{NS0}
\end{equation}
This result's algebraic scaling with $M$ is consistent with our earlier finding that optimum quantum reception for Rayleigh-fading QI target detection has an error probability that decreases subexponentially with increasing $M$.
{\em QI versus CI for Rayleigh Fading}---.
We are now prepared to demonstrate that QI target detection with SFG reception enjoys a significant performance advantage over CI target detection in the Rayleigh-fading scenario. We start with the Neyman-Pearson criterion, for which we already have the ROC for QI with SFG reception. The ROC for CI target detection with a coherent-state transmitter and heterodyne detection is~\cite{VanTrees1}
$P_D^{\rm CI}=\left(P_F^{\rm CI}\right)^{1/{\left(1+M\begin{eqnarray}r{\kappa}N_S/N_B\right)}}.$
Figure~\ref{Fig_comparisons_ROC} compares two QI and CI ROCs. Similar to what was assumed in Refs.~\cite{Tan2008,Zhuang_2017}, we took $\begin{eqnarray}r{\kappa} = 0.01$, $N_B=20$, and $\epsilon=0.01$ for both comparisons. In one case we assumed $N_S = 10^{-4}$ and $M = 10^{8.5}$, while in the other we chose $N_S = 10^{-2}$ and $M = 10^{6.5}$. Figure~\ref{Fig_comparisons_ROC} shows that QI target detection with SFG reception has a much higher detection probability than optimum CI target detection at low false-alarm probabilities.
\begin{equation}gin{figure}
\includegraphics[width=0.225\textwidth]{Zhuang_QI_with_fading_Fig2a.pdf}\hspace*{.1in}
\includegraphics[width=0.225\textwidth]{Zhuang_QI_with_fading_Fig2b.pdf}
\caption{QI and CI ROCs for Rayleigh-fading target detection with $\begin{eqnarray}r{\kappa}=0.01$, $N_B = 20$, and $\epsilon=0.01$. (a) $N_S=10^{-4}$ and $M=10^{8.5}$. (b) $N_S=10^{-2}$ and $M=10^{6.5}$.
\label{Fig_comparisons_ROC}
}
\end{figure}
Turning now to the Bayesian approach, we again have the QI result in hand, and we find optimum CI's error probability from $\Pr(e)_{\rm CI}=\min_{P_F^{\rm CI}}[\pi_0 P_F^{\rm CI}+\pi_1(1-P_D^{\rm CI})]$. Figure~\ref{Fig_comparisons} plots $\Pr(e)_{\rm SFG}$ and $\Pr(e)_{\rm CI}$ versus $\log_{10}(M)$ for equally-likely target absence or presence assuming $\begin{eqnarray}r{\kappa} = 0.01$, $N_B = 20$, and $\epsilon = 0.01$ for $N_S = 10^{-4}$ and $N_S = 10^{-2}$. Here we see that QI target detection with SFG reception offers a significantly lower error probability than optimum CI target detection. Indeed, for $MN_S \gg 1$ we obtain the asymptotic result
\begin{equation}
\Pr(e)_{\rm CI}\simeq \frac{\pi_1\ln(M\begin{eqnarray}r{\kappa}N_S/N_B)}{M\begin{eqnarray}r{\kappa}N_S/N_B}+O\!\left(\frac{1}{MN_S}\right),
\end{equation}
which is a factor of $\ln(M\begin{eqnarray}r{\kappa}N_S/N_B)$ higher than the corresponding result for $\Pr(e)_{\rm SFG}^{N_S\to0}$ when $M\begin{eqnarray}r{\kappa}N_S/N_B \gg 1$. Moreover, Fig.~\ref{Fig_comparisons}a shows that $N_S = 10^{-4}$ is small enough to ensure $\Pr(e)_{\rm SFG} \approx \Pr(e)_{\rm SFG}^{N_S\to0}$ for the parameter values employed therein. At high enough $M$ values, however, the effect of background noise in the SFG process becomes significant and $\Pr(e)_{\rm SFG}$ begins to deviate from the ideal $N_S\to0$ result. The onset of this deviation occurs at lower $M$ values when $N_S=10^{-2}$, as seen in Fig.~\ref{Fig_comparisons}b, because the background-noise effect on the SFG process is proportional to $N_S$~\cite{Zhuang_2017}. Nevertheless, QI's advantage over CI persists. We also see that QI target detection's robustness to noise is worse for Rayleigh fading than what our previous results~\cite{Zhuang_2017} showed for known $\kappa$. This reduced robustness arises from noise having greater impact on Rayleigh-fading error probability---because $\kappa \ll \begin{eqnarray}r{\kappa}$ can occur---as opposed to its effect in a nonfading environment with $\kappa = \begin{eqnarray}r{\kappa}$.
\begin{equation}gin{figure}
\includegraphics[width=0.225\textwidth]{Zhuang_QI_with_fading_Fig3a.pdf}
\hspace*{.1in}
\includegraphics[width=0.225\textwidth]{Zhuang_QI_with_fading_Fig3b.pdf}
\caption{QI and CI error probabilities for Rayleigh-fading target detection with $\pi_0=\pi_1=1/2$, $\begin{eqnarray}r{\kappa} = 0.01$, $N_B=20$, and $\epsilon=0.01$. (a) $N_S=10^{-4}$. (b) $N_S=10^{-2}$. The slope discontinuity in $\Pr(e)_{\rm SFG}$ for $N_S = 10^{-2}$ is due to the its receiver's photon-number threshold increasing from $n_t = 0$ to $n_t = 1$ at that point.
\label{Fig_comparisons}
}
\end{figure}
{\em Conclusions}---.
QI target detection is remarkable because it uses entanglement to outperform CI despite environmental loss and noise's destroying that entanglement. Previously, both theory and experiment have demonstrated QI's having an advantage over CI, but \emph{only} for a target return with known amplitude and known phase. Yet lidar targets are generally speckle targets, so their target returns have Rayleigh-distributed amplitudes and uniformly-distributed phases. We have shown that SFG reception affords a target-detection performance advantage over optimum CI for this scenario, but its magnitude is much smaller than what QI provides for the nonfading situation. Nevertheless, our result brings QI target detection closer to practical application, although two major problems remain to be solved: implementing near-lossless idler-storage and near-unity efficiency SFG for low-brightness, broadband light.
Two final points now deserve mention. First, although we have limited our treatment to the Rayleigh-fading scenario, the SFG receiver's immunity to a uniformly-distributed random phase means that it will also be effective against other fading distributions, e.g., the Rician fading that models a target return with both specular and diffuse components~\cite{VanTrees1,swerling1997radar}. Finally, because $N_B\gg 1$ most naturally occurs at microwave, rather than optical, wavelengths~\cite{Barzanjeh_2015}, SFG reception's applicability to a variety of flat-fading scenarios makes it relevant for microwave as well as optical QI.
Q.~Z. acknowledges support from the Claude E. Shannon Research Assistantship. Z.~Z. and J.~H.~S. acknowledge support from Air Force Office of Scientific Research Grant No.~FA9550-14-1-0052.
\setcounter{equation}{0}
\makeatletter
\renewcommand{A\arabic{equation}}{A\arabic{equation}}
{\em Appendix}---. Here we prove the two lemmas that were used earlier.
\noindent{\bf Lemma 1}
\emph{(Concavity of the Helstrom limit)
Consider the problem of discriminating between states $\hat{\sigma}_0=\int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x}) \hat{\rho}_0({\bm x})$ and $\hat{\sigma}_1=\int\! {\rm d} {\bm x}\, f_{\bm X}({\bm x}) \hat{\rho}_1({\bm x})$, where ${\bm X}$ is a random vector, that occur with prior probabilities $\pi_0$ and $\pi_1$. The Helstrom limit for this binary state-discrimination task satisfies
$
D_{\pi_0}(\hat{\sigma}_0,\hat{\sigma}_1)
\ge \int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x}) D_{\pi_0}(\hat{\rho}_0({\bm x}),\hat{\rho}_1({\bm x})).$}
\noindent{\bf Proof.}
Let $\hat{M}_0$ and $\hat{M}_1 = \hat{I}-\hat{M}_0$ be the Helstrom-limit positive operator-valued measurement for discriminating between $\hat{\sigma}_0$ and $\hat{\sigma}_1$ when those states' prior probabilities are $\pi_0$ and $\pi_1$. Then we have that
\begin{equation}gin{eqnarray}
\lefteqn{D_{\pi_0}(\hat{\sigma}_0,\hat{\sigma}_1) = \pi_0 {\rm tr}(\hat{M}_1\hat{\sigma}_0)+\pi_1{\rm tr}(\hat{M}_0\hat{\sigma}_1)} \nonumber \\
&=& \int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x})
\{
\pi_0 {\rm tr}[\hat{M}_1\hat{\rho}_0({\bm x})] + \pi_1{\rm tr}[\hat{M}_0\hat{\rho}_1({\bm x})]\} \nonumber
\\
&\ge& \int\!{\rm d} {\bm x}\, f_{\bm X}({\bm x}) D_{\pi_0}(\hat{\rho}_0({\bm x}),\hat{\rho}_1({\bm x})), \nonumber
\end{eqnarray}
and the proof is complete.
\noindent{\bf Lemma 2}
\emph{(Error-probability exponent for QI with Rayleigh fading) For $h=0,1$, let $\hat{\boldsymbol \rho}_h(\sqrt{\kappa},\phi) = \otimes_{m=1}^M\hat{\rho}_h^{(m)}(\sqrt{\kappa},\phi)$, where $\hat{\rho}_h^{(m)}(\sqrt{\kappa},\phi)$ is the two-mode, zero-mean, Gaussian state whose Wigner covariance matrix is given by Eq.~(\ref{hk}), and let $\hat{\begin{eqnarray}r{\boldsymbol \rho}}_h$ be the unconditional density operators obtained by averaging $\hat{\boldsymbol \rho}_h(\sqrt{\kappa},\phi)$ over Rayleigh and uniform probability density functions for $\sqrt{\kappa}$ and $\phi$, respectively.
Then, for all $\pi_0\pi_1 \neq 0$ we have $\xi_{\rm QI}\equiv-\lim_{M\to\infty}{\ln[D_{\pi_0}(\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0,\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1)]}/M = 0$.}
\noindent{\bf Proof.}
Because $\kappa \le 1$ is required for a passive target, i.e., one that only reflects, the Rayleigh pdf is really an approximation to $f_{\sqrt{\kappa}}(x) = 2xe^{-x^2/\begin{eqnarray}r{\kappa}}/\begin{eqnarray}r{\kappa}(1-e^{-1/\begin{eqnarray}r{\kappa}})$ for $0\le x\le 1$ that is very accurate in QI target detection's $\begin{eqnarray}r{\kappa} \ll 1$ scenario. For proving Lemma~2, however, we need to employ the truncated pdf, so that Lemma~1 and the QCB's exponential tightness for $M$-copy state discrimination gives us
\begin{equation}gin{eqnarray}
\lefteqn{D_{\pi_0}(\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0,\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1)} \nonumber \\
&\ge& \int_0^1\!{\rm d}x\!\int_0^{2\pi}\!{\rm d}y\, \frac{2xe^{-x^2/\begin{eqnarray}r{\kappa}}}{2\pi \begin{eqnarray}r{\kappa}(1-e^{-1/\begin{eqnarray}r{\kappa}})}D_{\pi_0}(\hat{\boldsymbol \rho}_0(x,y),\hat{\boldsymbol \rho}_1(x,y)) \nonumber\\
&\ge& \int_0^1\!{\rm d}x\!\int_0^{2\pi}\!{\rm d}y\, \frac{2xe^{-x^2/\begin{eqnarray}r{\kappa}}}{2\pi \begin{eqnarray}r{\kappa}(1-e^{-1/\begin{eqnarray}r{\kappa}})} \nonumber \\
&&\hspace{.2in}\times\,\,C_{x,y}(M) e^{-M\xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y))}, \nonumber
\end{eqnarray}
where the subunity prefactor, $C_{x,y}(M)$, is an algebraic function of $M$. Specifically, for all $0\le x\le 1$ and $0\le y\le 2\pi$, we have $\lim_{M\to\infty}\ln[C_{x,y}(M)]/M=0$. It follows that for every $\epsilon>0$ there is a finite $M_\epsilon(x,y)$ such that $C_{x,y}(M)\ge e^{-\epsilon M_\epsilon(x,y)}$ for all $M > M_\epsilon(x,y)$.
Because $\Omega \equiv \{0\le x\le 1, 0\le y\le 2\pi\}$ is a compact region, there is a \emph{finite} $M^\star_\epsilon = \max_{(x,y)\in \Omega} M_\epsilon(x,y)$. So, for all $M > M^\star_\epsilon$ we have
\begin{equation}gin{eqnarray}
D_{\pi_0}(\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0,\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1) &\ge& e^{-\epsilon M}\int_0^1\!{\rm d}x\!\int_0^{2\pi}\!{\rm d}y\, \frac{2xe^{-x^2/\begin{eqnarray}r{\kappa}}}{2\pi \begin{eqnarray}r{\kappa}(1-e^{-1/\begin{eqnarray}r{\kappa}})} \nonumber \\
&& \hspace{.2in}\times\,\,e^{-M\xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y))} \nonumber
\end{eqnarray}
But $\min_{(x,y)\in \Omega}\xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y))$ occurs at $x=0$, where $\xi_{\rm QCB}(\hat{\rho}_0(0,y),\hat{\rho}_1(0,y)) = 0$, because $\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0 = \hat{\begin{eqnarray}r{\boldsymbol \rho}}_1$ when the target return's intensity vanishes. Thus, for any $0< \epsilon'<1$ we can define $\Omega_{\epsilon'} = \{(\sqrt{\kappa},\phi) : \xi_{\rm QCB}(\hat{\rho}_0(x,y),\hat{\rho}_1(x,y)) \le \epsilon'\}$, and then weaken our previous lower bound on the Helstrom limit to
\begin{equation}gin{equation}
D_{\pi_0}(\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0,\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1) \ge e^{-(\epsilon+\epsilon') M}\Pr[(\sqrt{\kappa},\phi)\in \Omega_{\epsilon'}] > 0,\nonumber
\end{equation}
where the last inequality follows from $\pi_0\pi_1 \neq 0$.
Applying this bound to the error-probability exponent then leads to
\begin{equation}gin{equation}
\xi_{\rm QI}(\hat{\sigma}_0,\hat{\sigma}_1)
\equiv-\lim_{M\to\infty}{\ln[D_{\pi_0}(\hat{\begin{eqnarray}r{\boldsymbol \rho}}_0,\hat{\begin{eqnarray}r{\boldsymbol \rho}}_1)]}/M\nonumber \\
\le \epsilon + \epsilon'
\nonumber
\end{equation}
Because this upper bound holds for all $\epsilon, \epsilon' > 0$, by continuity our proof is now complete.
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\end{document} |
\begin{document}
\title{When do skew-products exist?}
\author[S.N. Evans]{Steven N. Evans}
\thanks{S.N.E. was supported in part by NSF grant DMS-0907639 and NIH grant 1R01GM109454-01}
\address{Department of Statistics\\
367 Evans Hall \#3860\\
University of California \\
Berkeley, CA 94720-3860 \\
USA}
\email{[email protected]}
\author[A. Hening]{Alexandru Hening }
\thanks{A.H. was supported by EPSRC grant EP/K034316/1}
\address{Department of Statistics \\
University of Oxford \\
1 South Parks Road \\
Oxford OX1 3TG \\
United Kingdom}
\email{[email protected]}
\author{Eric Wayman}
\address{Department of Mathematics\\
University of California\\
970 Evans Hall \#3840\\
Berkeley, CA 94720-3840\\
U.S.A.}
\email{[email protected]}
\date{\today}
\begin{abstract}
The classical skew-product decomposition of planar Brownian motion
represents the process in polar coordinates as an autonomously
Markovian radial part and an angular part that is an independent
Brownian motion on the unit circle time-changed according to
the radial part. Theorem~4 of \cite{L09} gives a broad generalization
of this fact to a setting where there is a diffusion on a manifold $X$
with a distribution that is equivariant under the smooth action of
a Lie group $K$. Under appropriate conditions, there is a decomposition
into an autonomously Markovian ``radial'' part that lives on the space
of orbits of $K$ and an ``angular'' part that is an independent
Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy
subgroup of a point of $x$, that is time-changed with a time-change that
is adapted to the filtration of the radial part.
We present two apparent counterexamples to \cite[Theorem~4]{L09}. In the
first counterexample the angular part is not a time-change of any Brownian motion
on $K/M$, whereas in the second counterexample the angular part is the time-change
of a Brownian motion on $K/M$ but this Brownian motion is not independent of the radial part.
In both of these examples $K/M$ has dimension $1$. The statement and proof of \cite[Theorem~4]{L09}
remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question
of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$
and what conditions lead to there being no decomposition at all or one in which the angular part is a time-changed
Brownian motion but this Brownian motion is not independent of the radial part.
\end{abstract}
\maketitle
\section{Introduction}
The archetypal skew-product decomposition of a Markov process is the decomposition of a Brownian motion in the plane $(B_t)_{t \geq 0}$ into its radial and angular part
\begin{equation}\label{e:BM_skew}
B_t = |B_t| \exp(i \theta_t).
\end{equation}
Here the radial part $(|B_t|)_{t \geq 0}$ is a two-dimensional Bessel process and $\theta_t = y_{\tau_t}$, where $(y_t)_{t \geq 0}$ is a one-dimensional Brownian motion that is independent of the radial part $(|B_t|)_{t \geq 0}$ and $\tau$ is a time-change that is adapted to the filtration generated by the process $|B|$. Specifically, $\tau_t = \int_0^t \frac{1}{|B_s|^2}ds$. See Corollary 18.7 from \cite{K01} for more details.
The most obvious generalization of this result is obtained in \cite{G63}. The
process considered is any time-homogeneous diffusion $(x_t)_{t \geq 0}$ with state space $\mathbb{R}^3$ that satisfies the additional assumptions that almost surely every path does not pass through the origin at positive times and that $(x_t)_{t \geq 0}$ is isotropic in the sense that the law of $(x_t)_{t \geq 0}$ is equivariant under the group of
orthogonal transformations; that is, if we consider a point $(r, \theta) \in \mathbb{R}^3$ in spherical coordinates, where $r \in \mathbb{R}_+$ is the radial coordinate and $\theta$ is a point on the unit sphere $S^2$, and if we take $k \in O(3)$, the orthogonal group on $\mathbb{R}^3$, then
\begin{equation*}
P_{(r, k \theta)}\left( kA \right) = P_{(r, \theta)} \left( A \right)
\end{equation*}
for any Borel set $A$ in path space $C(\mathbb{R}_+, \mathbb{R}^3)$. Here $P_{x}(A)$ is the probability a path started at $x$ belongs to the Borel set $A$ \cite[(2.2)]{G63}.
Theorem 1.2 of \cite{G63} states that we can decompose $(x_t)_{t \geq 0}$ as $x_{t} = r_{t} \theta_{t}$ where the radial motion $(r_t)_{t \geq 0}$ is a time-homogeneous Markov process on $\mathbb{R}_+$ and the angular process $(\theta_{t})_{t \geq 0}$ can be written as $\theta_{t} = B_{\tau_{t}}$, with $(B_t)_{t \geq 0}$ a spherical Brownian motion independent of the radial part and with the time-change $(\tau_{t})_{t \geq 0}$ adapted to the filtration generated by the radial part.
More generally, one can consider a group $G$ acting on $\mathbb{R}^n$ and $(x_t)_{t \geq 0}$ a Markov process on $\mathbb{R}^n$ such that the distribution of $(x_t)_{t\geq 0}$ satisfies the equivariance condition
\[
P_{gx}(gA) = P_x(A)
\]
for any Borel set $A$ in path space. The existence of a skew-product decomposition for this setting
is explored in \cite{Chy08} when $(x_t)_{t \geq 0}$ is a Dunkl process and $G$ is the group of distance preserving transformations of $\mathbb{R}^n$.
The paper \cite{PR88} investigates the skew-product decomposition of a Brownian motion on a $C^{\infty}$ Riemannian manifold $(M,g)$ which can be written as a product of a radial manifold $R$ and an angular manifold $\Theta$, both of which are assumed to be smooth and connected. Provided the Riemannian metric respects the product structure of the manifold in a suitable manner, \cite[Theorem~4]{PR88} establishes the existence of a skew-product decomposition such that the radial motion is a Brownian motion with drift on $R$ and the angular motion is a time-change of a
Brownian motion on $\Theta$ that is independent of the radial motion.
A broadly applicable skew-product decomposition result is obtained in \cite{L09} for a general continuous Markov process $(x_t)_{t \geq 0}$ with state space a smooth manifold $X$ and distribution that is equivariant under the smooth action of a Lie group $K$ on $X$. Here the decomposition of $(x_t)_{t \geq 0}$ is into a radial part $(y_t)_{t \geq 0}$ that is a Markov process on the submanifold $Y$ which is transversal to the orbits of $K$ and an angular part $(z_t)_{t \geq 0}$ that is a process on a general $K$-orbit which can be identified with the homogeneous space $K/M$, where $M$ is the isotropy subgroup of $K$ that is assumed to be the same for all elements $x \in X$. Theorem 4 of \cite{L09} asserts that under suitable conditions the process $(x_t)_{t \geq 0}$ has the same distribution as $(B(a_t)y_t)_{t \geq 0}$, where the radial part $(y_t)_{t \geq 0}$ is a diffusion on $Y$, $(B_t)_{t \geq 0}$ is a Brownian motion on $K/M$ that is independent of $(x_t)_{t \geq 0}$, and $(a_t)_{t \geq 0}$ a
time-change that is adapted to the filtration generated by $(y_t)_{t \geq 0}$.
The present paper was motivated by our desire to understand better the structural features that give rise to skew-product decompositions of diffusions that are equivariant under the action of a group and what it is about the absence of these features which cause such a decomposition not to hold. In attempting to do so, we read the paper \cite{L09}. We found an apparent counterexample to the main result, Theorem 4 of that paper in which there is a decomposition of
the process into an autonomously Markov radial process on $Y$ and an angular part that is a Brownian motion on $K/M$ time-changed according to the radial process, but this Brownian motion is {\bf not}, contrary to the claim of \cite{L09}, independent of the radial process, see Section~\ref{s_counter} for an exposition of the counterexample. This seeming contradiction appears because the assumption from \cite{L09} that $K/M$ is irreducible is not strong enough to ensure the nonexistence of a nonzero $M$-invariant tangent vector in the special case when, as in our construction, $K/M$ has dimension $1$.
It is
the nonexistence of such a tangent vector that is used in the
proof in \cite{L09} to deduce the independence of the radial process and the Brownian motion. Professor Liao pointed out to us that \cite[Theorem~4]{L09} holds under the conditions in \cite{L09}
for $\text{dim}(K/M)> 1$
and that result also holds when $K/M$ has dimension
$1$ if we further assume that there is no $M$-invariant tangent vector.
An anonymous referee pointed out an even simpler counterexample to \cite[Theorem~4]{L09} which we present in Section~\ref{s:rotated}. Namely, one takes
\[
x_t= \Theta_t \begin{pmatrix}U_t\\V_t\end{pmatrix}
\]
where $\begin{pmatrix}U_t\\V_t\end{pmatrix}$ is a planar Brownian motion and $\Theta_t\in SO(2)$ is the matrix that represents rotation about the origin through an angle $t$. We show that in this case that there is no skew-product decomposition for a somewhat different (and perhaps less interesting) reason: the angular part of $(x_t)_{t\geq 0}$ cannot be written as a time-changed Brownian motion on the unit circle in the plane.
The apparent contradiction to \cite[Theorem~4]{L09}is again due to the irreducibility of $K/M$ being inadequate to ensure the non-existence of an $M$ invariant tangent vector when $K/M$ has dimension $1$.
We present both of these counterexamples here because
they illustrate two rather different ways in which
things can go wrong. The latter counterexample
shows that under what look like reasonable conditions one might fail to have a skew-product decomposition because the angular part can't be time-changed to be Brownian, whereas the former counterexample does involve an angular part that is a time-changed Brownian motion, but it is just that this Brownian motion isn't independent of the radial process. We hope that by presenting these two examples we will
prompt further investigation into what general conditions lead to the subtle failure of the usual skew-product decomposition in the first counterexample and what ones lead to the grosser failure in the second counterexample.
The outline of the remainder of the paper is the following.
In Section \ref{s:BM} we check that the classical skew-product decomposition of planar Brownian motion fits in the setting from \cite{L09}, even though the proof of \cite[Theorem~4]{L09} does not, as we have noted, apply to ensure the existence of the skew-product decomposition when, as here, the dimension
of $K/M$ is $1$.
In Section~\ref{s:rotated} we describe the counterexample mentioned above of a planar Brownian motion that is rotated at a constant rate for which
the angular part is not a time-changed Brownian motion on the unit circle in the plane.
In Section \ref{s_counter} we construct the counterexample of a diffusion for which the angular part is a time-changed Brownian motion on the appropriate homogeneous space, but this Brownian motion is not independent of the radial part. Here the diffusion $(x_t)_{t \geq 0}$ has state space the manifold of $2 \times 2$ matrices that have a positive determinant. This diffusion can be represented via the well-known QR decomposition as the product of an
autonomously Markov
``radial'' process $(T_t)_{t\geq 0}$ on the manifold of $2 \times 2$ upper-triangular matrices with positive diagonal entries
and a time-changed ``angular'' process $(U_{R_t})_{t \geq 0}$, where $(U_t)_{t \ge 0}$ is a Brownian motion on the group $SO(2)$ of $2 \times 2$ orthogonal matrices with determinant one and the time-change $(R_t)_{t \ge 0}$ is adapted to the
filtration of the radial process. However, the processes $(U_t)_{t \geq 0}$ and $(T_t)_{t \geq 0}$ are {\bf not} independent.
We end this introduction by noting that analogous skew-product decompositions of superprocesses have been studied in \cite{P91, EM91, H00}. The continuous Dawson-Watanabe (DW) superprocess is a rescaling limit of a system of branching Markov processes while the Fleming-Viot (FV) superprocess is a rescaling limit of the empirical distribution of a system of particles undergoing Markovian motion and multinomial resampling. It
is shown in \cite{EM91} that a FV process is a DW process conditioned to have total mass one. More generally, it is demonstrated in \cite{P91} that the distribution of the DW process conditioned on the path of its total mass process is equal to the distribution of a time-change of a FV process that has a suitable underlying time-inhomogeneous Markov motion. The latter result is extended
to measure-valued processes that may have jumps in \cite{H00}.
A sampling of other results involving skew-products can be found
in \cite{Tay92,La09,El10, Ba06}.
\section{Example 1: Planar Brownian motion}\label{s:BM}
Let $(x_t)_{t \ge 0}$ be a planar Brownian motion.
Following the notation of \cite{L09}, we consider the following set-up.
\begin{enumerate}
\item Let $X=\R^2 \setminus \{(0,0)^T\}$.
\item Let $K$ be the Lie group $SO(2)$ of $2\times2$ orthogonal matrices
with determinant $1$. This group acts on $X$ by $A \mapsto Q^{-1} A$ for
$Q \in K$ and $A \in X$.
\item The quotient of $X$ with respect to the action of $K$ can be identified with the positive $x$ axis. Note that the orbits of $K$ are just circles centered at the origin.
\item The isotropy subgroup of $K$ for an element $x \in X$ is, as usual, the
subgroup $\{k \in K : kx = x\}$. Since every element of $X$ is an invertible
matrix, this subgroup is always the trivial group
consisting of just the identity.
In particular, this subgroup is the same for every
$y$ in the interior of $Y$, as required in \cite[pg~168]{L09}.
We denote this subgroup by $M$.
\end{enumerate}
It is straightforward to check that $(x_t)_{t \ge 0}$ satisfies
all the assumptions of \cite[Theorem~4]{L09}. We refer the reader to Sections \ref{s:rotated} and \ref{s_counter} for details of how to verify these assumptions in more complicated examples.
\begin{remark}\label{r:BM}
In this example, $\text{dim}(K/M)=1$ and there is the skew-product decomposition \eqref{e:BM_skew}.
\end{remark}
\section{Example 2: Rotated planar Brownian motion}\label{s:rotated}
Write $((U_t,V_t)^T)_{t \ge 0}$
for a planar Brownian started from $(x,y)^T$ (where $T$ denotes transpose, so we are
thinking of column vectors). The process $(x_t)_{t\geq 0}:=\left((x_t^1,x_t^2)^T\right)_{t\geq 0}$ started from
$(x,y)^T$ is defined by
\begin{equation}\label{e:rotatedBM}
\begin{pmatrix}x_t^1\\x_t^2\end{pmatrix} = \Theta_t \begin{pmatrix}U_t\\V_t\end{pmatrix},
\end{equation}
where $\Theta_t$ is the matrix that represents rotating though an angle $t$. Thus,
\begin{equation}
\begin{split}
x^1_t &= \mathcal{O}s(t) U_t - \sin(t) V_t\\
x^2_t &= \sin(t) U_t + \mathcal{O}s(t) V_t.
\end{split}
\end{equation}
Then,
\begin{equation*}
\begin{split}
dx^1_t &= \mathcal{O}s(t) dU_t - U_t \sin(t) dt - \sin(t) dV_t - V_t \mathcal{O}s(t) dt\\
dx^2_t &= \sin(t) dU_t + U_t \mathcal{O}s(t) dt + \mathcal{O}s(t) dV_t - V_t \sin(t) dt,
\end{split}
\end{equation*}
which becomes
\begin{equation*}
\begin{split}
dx^1_t &= \mathcal{O}s(t) dU_t - \sin(t) dV_t - Y_t dt\\
dx^2_t &= \sin(t) dU_t + \mathcal{O}s(t) dV_t + X_t dt.
\end{split}
\end{equation*}
If we define martingales $(B_t)_{t \ge 0}$
and $(C_t)_{t \ge 0}$ by
\[
dB_t = \mathcal{O}s(t) dU_t - \sin(t) dV_t
\]
and
\[
dC_t = \sin(t) dU_t + \mathcal{O}s(t) dV_t,
\]
then $[B]_t = t$, $[C]_t = t$ and $[B,C]_t = 0$, so
the process $((B_t,C_t)^T)_{t \ge 0}$ is a
planar Brownian motion and the process $\left((x_t^1,x_t^2)^T\right)_{t\geq 0}$ satisfies
the SDE
\begin{equation}\label{e:SDE_rotatedBM}
\begin{split}
dx^1_t &= dB_t - Y_t dt\\
dx^2_t &= dC_t + X_t dt.
\end{split}
\end{equation}
Following the notation of \cite{L09}, we consider the following set-up.
\begin{enumerate}
\item Let $X=\R^2 \setminus \{(0,0)^T\}$.
\item Let $K$ be the Lie group $SO(2)$ of $2\times2$ orthogonal matrices
with determinant $1$. This group acts on $X$ by $A \mapsto Q^{-1} A$ for
$Q \in K$ and $A \in X$.
\item The quotient of $X$ with respect to the action of $K$ can be identified with the positive $x$ axis. Note that the orbits of $K$ are just circles centered at the origin.
\item The isotropy subgroup of $K$ for an element $x \in X$ is, as usual, the
subgroup $\{k \in K : kx = x\}$. Since every element of $X$ is an invertible
matrix, this subgroup is always the trivial group
consisting of just the identity.
In particular, this subgroup is the same for every
$y$ in the interior of $Y$, as required in \cite[pg~168]{L09}.
We denote this subgroup by $M$.
\item Let $(x_t)_{t \ge 0}$ be the $X$-valued process that is defined in \eqref{e:rotatedBM}.
\end{enumerate}
We now check that $(x_t)_{t \ge 0}$ satisfies
all the assumptions of \cite[Theorem~4]{L09}.
These are as follows:
\begin{enumerate}
\item The process $(x_t)_{t \ge 0}$ is a Feller process with continuous
sample paths.
\item The distribution of $(x_t)_{t \ge 0}$ is equivariant under
the action of $K$. That is, for $k \in K$ the distribution of $(k x_t)_{t \ge 0}$
when $x_0 = x_*$ is the same as the distribution of
$(x_t)_{t \ge 0}$ when $x_0 = k x_*$ \cite[(2)]{L09}.
\item The set $Y$ is a submanifold of $X$ that is transversal to the action of $K$
\cite[(3)]{L09}.
\item For any $y \in Y^0$ (that is, the relative interior of $Y$ -- which
in this case is just $Y$ itself) $T_y X$,
the tangent space of $X$ at $y$, is the direct sum of tangent spaces
$T_y(Ky) \bigoplus T_y Y$ \cite[(5)]{L09}.
\item The homogeneous space $K/M$ is irreducible; that is, the action of $M$
on $T_o(K/M)$ (the tangent space at the coset $o$ containing the identity)
has no nontrivial invariant subspace \cite[pg~177]{L09}.
\end{enumerate}
These assumptions are verified as follows:
\begin{enumerate}
\item This follows from the representation \eqref{e:SDE_rotatedBM}.
\item Since $\Theta_t\in SO(2)$ we have by \eqref{e:rotatedBM} that for any $Q\in SO(2)$
\[
Q x_t = Q \Theta_t\begin{pmatrix}U_t\\V_t\end{pmatrix}.
\]
Since $ Q \Theta_t \in SO(2)$ the condition holds because planar Brownian motion is equivariant under the action of $SO(2)$.
\item This is immediate.
\item $T_y(Ky) = \text{Span}\left\{ (0,1)^T\right\}$ and $T_y(Y) = \text{Span}\left\{(1,0)^T\right\}$ so that
\[
\R^2 = T_y X = T_y(Ky) \oplus T_y(Y)
\]
\item The tangent space of $Ky$ is one-dimensional so $K/M$ is irreducible.
\end{enumerate}
Consequently, $(x_t)_{t \ge 0}$ satisfies all the hypotheses of \cite[Theorem~4]{L09}.
Write $(R_t)_{t\geq 0}$ for the radial process
\[
R_t := |(x_t^1, x_t^2)^T| = |(U_t, V_t)^T|,
\]
and let $(L_t)_{t\geq 0}$ be the angular part of
$((U_t, V_t)^T)_{t \ge 0}$. We can think of $(L_t)_{t\geq 0}$ as living on the unit circle in the complex plane. In
polar coordinates, we have
\[
x_t = (R_t, L_t \exp(it)).
\]
By the usual skew-product for planar Brownian motion recalled in \eqref{e:BM_skew} we have that $L_t =
\exp(i W_{T_t})$, where $W$ is a standard Brownian motion on the line
independent of $R$ and $T$ is a time-change defined
from $R$. Therefore
\[
x_t = (R_t, \exp(i (W_{T_t}+t))).
\]
\begin{proposition}\label{p_mainprop_rotatedBM}
The process $(x_t)_{t\geq 0}$ cannot be written as
\[
x_t = (R_t, \exp(i Z_{S_t})),
\]
where $Z$ is a Brownian motion (possibly with drift) on the line
independent of $R $ and $S$ is an increasing process adapted to the filtration generated by $R$.
\end{proposition}
\begin{proof}
If such a representation was possible, then we would have $Z_t = \tilde Z_t + a t$ for some constant $a \in \R$, where $\tilde Z_t$ is a standard Brownian motion. This would imply that
\begin{equation*}
\begin{split}
\tilde Z &= W\\
S &= T\\
\exp(i a S_t) &= \exp(i t).
\end{split}
\end{equation*}
However, this is not possible: it would mean that
\[
\exp(i t) = \exp(i a T_t),
\]
but $T_t$ is certainly not a constant multiple of $t$
for all $t \ge 0$.
\end{proof}
\begin{remark}\label{r:rotated}
In this example $K/M$ is the unit circle, which has dimension $1$, and there is no skew-product decomposition. The angular part cannot be written as the time-change of any Brownian motion on the unit circle.
\end{remark}
\section{Example 3: A matrix valued process}\label{s_counter}
Recall the well-known QR decomposition which says that any square matrix
can be written as the product of an orthogonal matrix and an
upper triangular matrix, and that this decomposition is unique for invertible matrices if we require the diagonal entries in the
upper triangular matrix to be positive (see, for example, \cite{horn}).
This decomposition is essentially a special case of the Iwasawa decomposition
for semisimple Lie groups.
In the $2 \times 2$ case, uniqueness also holds for QR decomposition of invertible matrices if we require the orthogonal matrix to have determinant one and there are simple
explicit formulae for the factors. Indeed,
if
\begin{equation}\label{eq_qrfact}
A = \left(\begin{matrix}a&b\\ c&d\end{matrix}\right)
\end{equation}
and $\det A = ad - bc \ne 0$,
then $A = \tilde Q \tilde R$, where
\begin{equation}\label{e:Q1}
\tilde{Q}=\frac{1}{\sqrt{a^2+c^2}} \left(\begin{matrix}a&-c\\ c&a\end{matrix}\right) \in SO(2)
\end{equation}
and
\begin{equation}\label{e:R1}
\tilde{R}=\left(\begin{matrix}\sqrt{a^2 + c^2}&\frac{ab + cd}{\sqrt{a^2 + c^2}}\\ 0& \frac{ad-bc}{\sqrt{a^2 + c^2}}\end{matrix}\right).
\end{equation}
In this setting, we consider a $2 \times 2$ matrix of independent Brownian motions and time-change it to produce a Markov process with the property that if the determinant is positive at time $0$, then it stays positive at all times. This ensures that uniqueness of the $QR$-factorization holds at all times and also that the time-changed process falls into the setting of \cite{L09}.
Following the notation of \cite{L09}, we consider the following set-up.
\begin{enumerate}
\item Let $X$ be the manifold of $2\times2$ matrices over $\R$
with strictly positive determinant equipped with the topology it inherits
as an open subset of $\R^{2 \times 2} \mathcal{O}ng \R^4$.
\item Let $K$ be the Lie group $SO(2)$ of $2\times2$ orthogonal matrices
with determinant $1$. This group acts on $X$ by $A \mapsto Q^{-1} A$ for
$Q \in K$ and $A \in X$.
\item The quotient of $X$ with respect to the action of $K$ can,
via the QR decomposition, be identified with
the set $Y$ of upper triangular $2\times2$ matrices with
strictly positive diagonal entries.
\item The isotropy subgroup of $K$ for an element $x \in X$ is, as usual, the
subgroup $\{k \in K : kx = x\}$. Since every element of $X$ is an invertible
matrix, this subgroup is always the trivial group
consisting of just the identity.
In particular, this subgroup is the same for every
$y$ in the interior of $Y$, as required in \cite[pg~168]{L09}.
We denote this subgroup by $M$.
\item Let $(x_t)_{t \ge 0}$ be the $X$-valued process that satisfies
the stochastic differential equation (SDE)
\begin{equation}\label{e_SDE}
dx_t =\left(\begin{matrix} dx^{1,1}_t&dx^{1,2}_t\\dx^{2,1}_t&dx^{2,2}_t\end{matrix} \right) = \left(\begin{matrix} f(x_t) \, dA^{1,1}_t& f(x_t) \, dA^{1,2}_t\\f(x_t) \, dA^{2,1}_t & f(x_t) \, d A^{2,2}_t \end{matrix} \right), \quad x_0 \in X,
\end{equation}
where $A^{1,1}_t$, $A^{1,2}_t$, $A^{2,1}_t$, and $ A^{2,2}_t$ are independent standard one-dimensional Brownian motions, and $f(x):= \frac{\det(x)}{\text{tr}(x'x)+1}$ with $\det$ and $\text{tr}$ denoting the determinant and the trace.
We establish below that \eqref{e_SDE} has a unique strong
solution and that this
solution does indeed take values in $X$.
\end{enumerate}
It follows from the QR decomposition
that $x_t = Q_t T_t$, where, in the terminology of \cite{L09},
the ``angular part'' $Q_t$ belongs to $K$ and the ``radial part''
$T_t$ belongs to $Y$.
We will show that $(T_t)_{t \ge 0}$ is an autonomous diffusion
on $Y$ and that $Q_t = U_{R_t}$, where $(U_t)_{t \ge 0}$
is a Brownian motion on $K$ and $(R_t)_{t \ge 0}$ is an increasing process adapted to
the filtration generated by $(T_t)_{t \ge 0}$.
However, we will establish that \textbf{it is not possible} to take the
Brownian motion $(U_t)_{t \ge 0}$ to be independent of
the process $(T_t)_{t \ge 0}$.
This will contradict the claim of \cite[Theorem~4]{L09} once we have
also checked that the conditions of that result hold.
Note that if we consider $f$ as a function on the space
$\R^{2 \times 2} \mathcal{O}ng \R^4$ of all $2 \times 2$ matrices, then it
has bounded partial derivatives, and hence it is globally
Lipschitz continuous. Consequently, if we allow the initial
condition in $\eqref{e_SDE}$ to be an arbitrary element of $\R^{2 \times 2}$,
then the resulting SDE has a unique strong solution (see, for example, \cite[Ch~5,~Thm~11.2]{RW00}). Moreover, the resulting process
is a Feller process on $\mathbb{R}^{2 \times 2}$
(see, for example, \cite[Ch~5, Thm~22.5]{RW00}).
We now check that $(x_t)_{t \ge 0}$ actually takes values in $X$.
That is, we show that if $x_0$ has positive determinant, then $x_t$ also has positive determinant for all $t \ge 0$. It follows from It\^o's Lemma that
\begin{equation*}
[\det(x_\cdot)]_t = \int_0^t \mathrm{tr}(x_s'x_s)f^2(x_s)\,ds,
\end{equation*}
\begin{equation*}
[\mathrm{tr}(x_\cdot'x_\cdot)]_t = \int_0^t4\mathrm{tr}(x_s'x_s)f^2(x_s)\,ds,
\end{equation*}
and
\begin{equation*}
[\det(x_\cdot),\mathrm{tr}(x_\cdot'x_\cdot)] = \int_0^t4\det(x_s)f^2(x_s)\,ds.
\end{equation*}
Thus, $((\det(x_t), \mathrm{tr}(x_t'x_t)))_{t \ge 0}$ is
a Markov process and there exist independent standard one-dimensional Brownian motions
$(B^1_t)_{t \ge 0}$ and $(B^2_t)_{t \ge 0}$ such that
\[
d \, \det(x_t) = \sqrt{\mathrm{tr}(x_t'x_t)} f(x_t) \, dB^1_t
\]
and
\[
\begin{split}
d \, \mathrm{tr}(x_t'x_t) & = \frac{4\det(x_t) f(x_t)}{\sqrt{\mathrm{tr}(x_t'x_t)}} \, dB^1_t + \sqrt{\frac{4 \mathrm{tr}^2(x_t'x_t) - 16\det(x_t)^2} {\mathrm{tr}(x_t'x_t)}}f(x_t) \, dB^2_t \\
& \quad + 4f^2(x_t) \, dt.
\end{split}
\]
When we substitute for $f$, the above equations transform into
\[
d \, \det(x_t) = \frac{\det(x_t) \sqrt{\mathrm{tr}(x_t'x_t)}}{\mathrm{tr}(x_t'x_t)+1} \, dB^1_t
\]
and
\[
\begin{split}
d \, \mathrm{tr}(x_t'x_t) &=
\frac{4(\det(x_t))^{2}}{\sqrt{\mathrm{tr}(x_t'x_t)} (\mathrm{tr}(x'x)+1)} \, dB^1_t + \sqrt{\frac{4 \mathrm{tr}^2(x_t'x_t) - 16\det(x_t)^2} {\mathrm{tr}(x_t'x_t)}}\frac{\det(x_t)}{\mathrm{tr}(x_t'x_t)+1} \, dB^2_t \\
& \quad + 4 \left( \frac{\det(x_t)}{\mathrm{tr}(x_t'x_t)+1} \right)^2 \, dt. \\
\end{split}
\]
In particular, the process
$(\det(x_t))_{t \ge 0}$ is the stochastic exponential
of the local martingale $(M_t)_{t \ge 0}$, where
\[
M_t = \int^t_0 \frac{\sqrt{\mathrm{tr}(x_s'x_s)}}{\mathrm{tr}(x_s'x_s)+1} \, dB^1_s.
\]
Since $x_0 \in X$, we have $\det(x_0) > 0$, and hence
\[
\det(x_t) = \det (x_0) \exp\left(M_t-M_0-\frac{1}{2}[M]_t\right)
\]
is strictly positive for all $t \ge 0$. This shows that
$(x_t)_{t \ge 0}$ takes values in $X$.
We now check that $(x_t)_{t \ge 0}$ satisfies
all the assumptions of \cite[Theorem~4]{L09}.
These are as follows:
\begin{enumerate}
\item The process $(x_t)_{t \ge 0}$ is a Feller process with continuous
sample paths.
\item The distribution of $(x_t)_{t \ge 0}$ is equivariant under
the action of $K$. That is, for $k \in K$ the distribution of $(k x_t)_{t \ge 0}$
when $x_0 = x_*$ is the same as the distribution of
$(x_t)_{t \ge 0}$ when $x_0 = k x_*$ \cite[(2)]{L09}.
\item The set $Y$ is a submanifold of $X$ that is transversal to the action of $K$
\cite[(3)]{L09}.
\item For any $y \in Y^0$ (that is, the relative interior of $Y$ -- which
in this case is just $Y$ itself) $T_y X$,
the tangent space of $X$ at $y$, is the direct sum of tangent spaces
$T_y(Ky) \bigoplus T_y Y$ \cite[(5)]{L09}.
\item The homogeneous space $K/M$ is irreducible; that is, the action of $M$
on $T_o(K/M)$ (the tangent space at the coset $o$ containing the identity)
has no nontrivial invariant subspace \cite[pg~177]{L09}.
\end{enumerate}
The verifications of (1)--(5) proceed as follows:
\begin{enumerate}
\item We have already observed that solutions of \eqref{e_SDE}
with initial conditions in $\R^{2 \times 2}$ form a Feller process
and that this process stays in the open set $X$ if it starts in $X$,
and so $(x_t)_{t \ge 0}$ is a Feller process on $X$.
\item Suppose that $(x_t)_{t \ge 0}$ is a solution of \eqref{e_SDE}
with $x_0 = x_*$ and $(\hat x_t)_{t \ge 0}$ is a solution of \eqref{e_SDE}
with $\hat x_0 = k x_*$ for some $k \in K$. We have to show that if
we set $\tilde x_t = k^{-1} \hat x_t$, then $(\tilde x_t)_{t \ge 0}$
has the same distribution as $(x_t)_{t \ge 0}$. Note that
$\det \tilde x_t = \det \hat x_t$ and
$\tilde x'_t \tilde x_t = \hat x_t' \hat x_t$, so that
$f(\tilde x_t) = f(\hat x_t)$. Thus,
\[
d \tilde x_t =
f(\tilde x_t)
k^{-1}
\begin{pmatrix}
dA^{1,1}_t & dA^{1,2}_t\\
dA^{2,1}_t & dA^{2,2}_t
\end{pmatrix}, \quad \tilde x_0=x_*.
\]
Now the columns of the matrix
\[
\begin{pmatrix} A^{1,1}_t & A^{1,2}_t \\ A^{2,1}_t& A^{2,2}_t \end{pmatrix}
\]
are independent standard two-dimensional Brownian motions, and so the
same is true of the columns of the matrix
\[
k^{-1}
\begin{pmatrix} A^{1,1}_t & A^{1,2}_t \\ A^{2,1}_t& A^{2,2}_t \end{pmatrix}
\]
by the equivariance of standard two-dimensional Brownian motion under the action of $SO(2)$.
Hence,
\[
k^{-1}
\begin{pmatrix} A^{1,1}_t & A^{1,2}_t \\ A^{2,1}_t & A^{2,2}_t \end{pmatrix} =
\begin{pmatrix} \alpha^{1,1}_t & \alpha^{1,2}_t \\ \alpha^{2,1}_t & \alpha^{2,2}_t \end{pmatrix},
\]
where
$(\alpha^{1,1}_t)_{t \ge 0}$, $(\alpha^{1,2}_t)_{t \ge 0}$,
$(\alpha^{2,1}_t)_{t \ge 0}$, and $(\alpha^{2,2}_t)_{t \ge 0}$
are independent standard Brownian motions.
Since,
\begin{equation*}
d\tilde{x}_t =
f(\tilde{x}_t) \begin{pmatrix} d\alpha^{1,1}_t & d\alpha^{1,2}_t \\ d\alpha^{2,1}_t & d\alpha^{2,2}_t \end{pmatrix}, \quad \tilde{x_0} = x_0,
\end{equation*}
the existence and uniqueness of strong solutions to \eqref{e_SDE}
establishes that the distributions of $(x_t)_{t \ge 0}$ and
$(\tilde{x}_t)_{t \ge 0}$ are equal.
\item It follows from the existence of the
$QR$ decomposition for invertible matrices that $X$
is the union of the orbits $Ky$ for $y \in Y$, and it follows
from the uniqueness of the decomposition for such matrices
that the orbit $Ky$ intersects $Y$ only at $y$.
\item Since the tangent space of $K=SO(2)$ at the identity is the
vector space of $2 \times 2$ skew-symmetric matrices and the tangent space of
$Y$ at the identity is the vector space of $2 \times 2$ upper-triangular matrices,
we have to show that if $W$ is a fixed invertible
upper-triangular $2 \times 2$ matrix and $M$ is a fixed $2 \times 2$ matrix, then
\begin{equation*}
M = S W + V
\end{equation*}
for a unique skew-symmetric $2 \times 2$ matrix $S$ and unique upper-triangular $2 \times 2$ matrix $V$.
Let
\begin{equation*}
M:=\begin{pmatrix} m_{11}&m_{12}\\m_{21}&m_{22}\end{pmatrix}
\quad \text{and} \quad
W:=
\begin{pmatrix} w_{11}&w_{12}\\0&w_{22}\end{pmatrix}.
\end{equation*}
It is immediate that
\begin{equation*}
S =
\begin{pmatrix} 0 &-\frac{m_{21}}{w_{11}}\\\frac{m_{21}}{w_{11}}&0 \end{pmatrix}
\end{equation*}
and
\begin{equation*}
V = \begin{pmatrix} m_{11} &\frac{m_{12} w_{11} + m_{21} w_{22}}{w_{11}}\\0&\frac{m_{22} w_{11} - m_{21} w_{12}}{w_{11}}\end{pmatrix}.
\end{equation*}
\item We have already noted that the
tangent space of $K$ at the identity is the vector space of
skew-symmetric $2\times2$ matrices. This vector space
is one-dimensional and
so this condition holds trivially.
\end {enumerate}
We have now shown that $(x_t)_{t \ge 0}$ satisfies all the hypotheses of \cite[Theorem~4]{L09}. However, we have the following result.
\begin{proposition}\label{p_mainprop}
In the decomposition $x_t= Q_t T_t$ the
$Y$-valued process $(T_t)_{t \ge 0}$ is Markov and
the $K$-valued process $(Q_t)_{t \ge 0}$
may be written as
$Q_t = U_{R_t}$,
where $(U_t)_{t \ge 0}$ is a $K$-valued Brownian motion and
$(R_t)_{t \ge 0}$ is an increasing continuous process
such that $R_0 = 0$ and $R_t - R_s$ is
$\sigma\{T_u : s \le u \le t\}$-measurable for
$0 \le s < t < \infty$. However, there is no such representation
in which $(T_t)_{t \ge 0}$ and $(U_t)_{t \ge 0}$ are independent.
\end{proposition}
\begin{proof}
For all $t \ge 0$ we have $x_t=Q_tT_t$, where
\begin{equation*}
Q_t=\frac{1}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}}
\begin{pmatrix} x^{11}_t &-x^{21}_t \\ x^{21}_t&x^{11}_t\end{pmatrix}
\in K
\end{equation*}
and
\begin{equation*}
T_t=\begin{pmatrix}\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}&\frac{x^{11}_t x^{12}_t + x^{21}_t x^{22}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}}\\ 0& \frac{\det (x_t)}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}}\end{pmatrix}
\in Y.
\end{equation*}
Note that $\det(x_t) = \det(T_t)$ and $\mathrm{tr}(x_t'x_t) = \mathrm{tr}(T_t'T_t)$, and so $f(x_t) = f(T_t)$.
Note also that the complex-valued process
$(x^{11}_t + i x^{21}_t)_{t \ge 0}$ is an isotropic complex
local martingale
in the sense of \cite[Ch 18]{K01}, that is
\[
[x^{11}]=[x^{21}]
\]
and
\[
[x^{22},x^{21}]=0.
\]
In our case
\[
d[x^{11}]_t = d[x^{21}]_t = f^2(T_t) \, dt.
\]
By \cite[Thm~18.5]{K01}, $(\log(x^{11}_t + i x^{21}_t)_{t \ge 0}$
is a well-defined isotropic complex local martingale that can
be written as
\[
\log(x^{11}_t + i x^{21}_t) = \log \left( T^{11}_t \right) + i \theta_t,
\]
where
\begin{equation*}
d[\theta]_t = d[\log(T^{11})]_t= \frac{1}{(T^{11}_t)^2}d[x^{11}]_t = \left( \frac{f(T_t)}{T^{11}_t} \right)^2 \, dt.
\end{equation*}
By the classical result of Dambis, Dubins and Schwarz
(see, for example, \cite[Thm~18.4]{K01}),
there exists a standard complex Brownian motion
$(\tilde{B}_t + iB_t)_{t \ge 0}$ such that $\log(x^{11}_t + i x^{21}_t) = \tilde{B}_{R_t} + iB_{R_t}$, where
\[
R_t = \int^t_0 \left( \frac{f(T_s)}{T_s^{11}} \right)^2 \, ds, \quad t \ge 0.
\]
So, $\theta_t = B_{R_t}$ and $\log(T^{11}_t) = \tilde{B}_{R_t}$. Hence,
\[
\frac{x^{11}_t + i x^{21}_t}{\sqrt{(x^{11}_t)^2 + (x^{21}_t)^2}} = \left( \mathcal{O}s(\theta_t) + i \sin(\theta_t) \right)
\]
and
\begin{equation*}
Q_t = \begin{pmatrix} \mathcal{O}s(B_{R_t} )& -\sin(B_{R_t})\\ \sin(B_{R_t})& \mathcal{O}s(B_{R_t})\end{pmatrix}.
\end{equation*}
Consequently, $Q_t = U_{R_t}$, where
\begin{equation*}
U_t = \begin{pmatrix} \mathcal{O}s(B_t )& -\sin(B_t)\\ \sin(B_t)& \mathcal{O}s(B_t)\end{pmatrix},
\end{equation*}
and $(B_t)_{t \ge 0}$ is a standard one-dimensional Brownian motion.
Note that $(U_t)_{t \ge 0}$ is certainly a Brownian motion on $K=SO(2)$,
and so we have uniquely identified the $K$-valued Brownian motion
$(U_t)_{t \ge 0}$ and the increasing process $(R_t)_{t \ge 0}$
that appear in the claimed decomposition of $(x_t)_{t \ge 0}$.
To complete the proof, it suffices to suppose that
$(U_t)_{t \ge 0}$ is independent of $(T_t)_{t \ge 0}$
and obtain a contradiction.
An application of It\^o's Lemma shows that the entries of
$(U_t)_{t \ge 0}$ satisfy the system of SDEs
\begin{eqnarray*}\label{e_USDE}
dU^{1,1}_t &=& - U^{2,1}_t \, dB_t - \frac{1}{2}U^{1,1}_t \, dt\\
dU^{2,1}_t &=& U^{1,1}_t \, dB_t - \frac{1}{2}U^{2,1}_t \, dt\\
dU^{1,2}_t &=& -U^{1,1}_t \, dB_t + \frac{1}{2}U^{2,1}_t \, dt=-dU^{2,1}_t\\
dU^{2,2}_t &=& - U^{2,1}_t \, dB_t - \frac{1}{2}U^{1,1}_t \, dt = dU^{1,1}_t.
\end{eqnarray*}
We apply Proposition \ref{p_timechange} below to each of the four SDEs in the system describing $(U_t)_{t \ge 0}$, with, in the notation of that result, $(\zeta_t, H_t, K_t)$ being the respective triples
$(U^{1,1}_t ,U^{2,1}_t ,U^{1,1}_t)$,
$(U^{2,1}_t ,U^{1,1}_t ,U^{2,1}_t)$,
$(U^{1,2}_t ,U^{1,1}_t ,U^{2,1}_t)$,
and $(U^{2,2}_t ,U^{2,1}_t ,U^{1,1}_t)$.
In each of the four applications, we let
\begin{itemize}
\item $(\mathcal{F}_t)_{t \ge 0}$ be the filtration generated by $(U_t)_{t \ge 0}$,
\item $(\mathcal{G}_t)_{t \ge 0}$ be the filtration generated by $(T_t)_{t \ge 0}$,
\item $\beta_t = B_t$,
\item $\rho_t = R_t$,
\item $J_t = \left( \frac{f(T_t)}{T_t^{11}} \right)^2$,
\item $\gamma_t = W_t = \int^t_0 \sqrt{\frac{1}{R^{\prime}_s}}dB_{R_s}$.
\end{itemize}
Let $\mathcal{H}_t = \mathcal{F}_{\rho_t} \vee \mathcal{G}_t$, $t \ge 0$,
as in the Proposition \ref{p_timechange}.
It follows by the assumed independence of
$(U_t)_{t \ge 0}$ and $(T_t)_{t \ge 0}$,
part (iii) of Proposition \ref{p_timechange},
and equation \eqref{e_USDE} that the entries of the
time-changed process $Q_t = U_{R_t}$ satisfy the system of SDEs
\begin{eqnarray*}
dQ^{1,1}_t &=& - Q^{2,1}_t \sqrt{R'_t} \, dW_t - \frac{1}{2}Q^{1,1}_t R'_t \, dt = - Q^{2,1}_t \frac{f(T_t)}{T^{11}_t} \, dW_t - \frac{1}{2}Q^{1,1}_t \left(\frac{f(T_t)}{T^{11}_t}\right)^2 \, dt\\
dQ^{2,1}_t &=& Q^{1,1}_t \sqrt{R'_t} \, dW_t - \frac{1}{2}Q^{2,1}_t R'_t \, dt = Q^{1,1}_t \frac{f(T_t)}{T^{11}_t} \, dW_t - \frac{1}{2}Q^{2,1}_t \left(\frac{f(T_t)}{T^{11}_t}\right)^2 \, dt\\
dQ^{1,2}_t &=& -dQ^{2,1}_t = Q^{1,1}_t \sqrt{R'_t} \, dW_t - \frac{1}{2}Q^{2,1}_t R'_t \, dt = Q^{1,1}_t \frac{f(T_t)}{T^{11}_t} \, dW_t - \frac{1}{2}Q^{2,1}_t \left(\frac{f(T_t)}{T^{11}_t}\right)^2 \, dt\\
dQ^{2,2}_t &=& dQ^{1,1}_t = - Q^{2,1}_t \sqrt{R'_t} \, dW_t - \frac{1}{2}Q^{1,1}_t R'_t = - Q^{2,1}_t \frac{f(T_t)}{T^{11}_t} \, dW_t - \frac{1}{2}Q^{1,1}_t \left(\frac{f(T_t)}{T^{11}_t}\right)^2 \, dt.
\end{eqnarray*}
Set
\begin{eqnarray*}
dw^1_t &=& \frac{x^{11}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{11}_t + \frac{x^{21}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{21}_t\\
dw^2_t &=& \frac{-x^{21}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{11}_t + \frac{x^{11}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{21}_t\\
dw^3_t &=& \frac{x^{11}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{12}_t + \frac{x^{21}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{22}_t\\
dw^4_t &=& \frac{-x^{21}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{12}_t + \frac{x^{11}_t}{\sqrt{(x^{11}_t)^2+(x^{21}_t)^2}} \, dA^{22}_t.
\end{eqnarray*}
The processes $(w^i_t)_{t \ge 0}$ are local martingales with $[w^i_t,w^j_t]_t = \delta_{ij}t$, and thus they are independent standard Brownian motions. An application of It\^o's Lemma shows that $(T_t)_{t \ge 0}$ is a diffusion satisfying the following system of SDEs.
\begin{eqnarray*}
dT^{11}_t &=& f(T_t) \, dw^1_t + \frac{f^2(T_t)}{T^{11}_t} \, dt\\
dT^{12}_t &=& \frac{T^{22}_t f(T_t)}{T^{11}_t} \, dw^2_t + f(T_t)dw^3_t - \frac{T^{12}_t f^2(T_t)}{2 (T^{11}_t)^2} \, dt\\
dT^{22}_t &=& \frac{T^{12}_t f(T_t)}{T^{11}_t} \, dw^2_t + f(T_t)dw^4_t - \frac{T^{22}_t f^2(T_t)}{2 (T^{11}_t)^2} \, dt.
\end{eqnarray*}
The assumed independence of the processes
$(U_t)_{t \ge 0}$ and $(T_t)_{t \ge 0}$ and part (iv) of Proposition \ref{p_timechange} give that $[Q^{i,j},T^{k,l}] \equiv 0$ for all $i,j,k$ and $l$.
It follows from It\^o's Lemma that
\begin{eqnarray*}
d(Q_t T_t)^{1,1} &=& d N_t +\frac{Q_t^{1,1}f^2(T_t)}{T^{1,1}_t}\left(1-\frac{1}{2T_t^{1,1}}\right) \, dt,\\
\end{eqnarray*}
where $(N_t)_{t \ge 0}$ is a continuous local martingale
for the filtration $(\mathcal{H}_t)_{t \ge 0}$. This, however,
is not possible because
$(Q_t T_t)^{1,1} = x_t^{1,1}$
and the process $(x^{1,1}_t)_{t \ge 0}$ is a continuous local martingale
for the filtration $(\mathcal{H}_t)_{t \ge 0}$.
\end{proof}
We required the following proposition that collects together some simple
facts about time-changes.
\begin{proposition}\label{p_timechange}
Consider two filtrations
$(\mathcal{F}_t)_{t \ge 0}$ and $(\mathcal{G}_t)_{t \ge 0}$
on an underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
Set $\mathcal{F}_\infty = \bigvee_{t \ge 0} \mathcal{F}_t$
and $\mathcal{G}_\infty = \bigvee_{t \ge 0} \mathcal{G}_t$.
Assume that the sub-$\sigma$-fields $\mathcal{F}_\infty$
and $\mathcal{G}_\infty$ are independent. Suppose that
\[
\zeta_t = \zeta_0 + \int_0^t H_s \, d\beta_s + \int_0^t K_s \, ds,
\]
where $\zeta_0$ is $\mathcal{F}_0$-measurable, the integrands
$(H_t)_{t \ge 0}$
and $(K_t)_{t \ge 0}$ are $(\mathcal{F}_t)_{t \ge 0}$-adapted,
and $(\beta_t)_{t \ge 0}$ is an
$(\mathcal{F}_t)_{t \ge 0}$-Brownian motion. Suppose further that $\rho_t = \int_0^t J_s \, ds$,
where $(J_t)_{t \ge 0}$ is a nonnegative, $(\mathcal{G}_t)_{t \ge 0}$-adapted process such that $\rho_t$
is finite for all $t \ge 0$ almost surely.
For $t \ge 0$ put
\[
\mathcal{F}_{\rho_t}
=
\sigma\{L_{s \wedge \rho_t} : \text{$s \ge 0$ and $L$ is
$(\mathcal{F}_t)_{t \ge 0}$-optional}\}.
\]
Set $\mathcal{H}_t = \mathcal{F}_{\rho_t} \vee \mathcal{G}_t$, $t \ge 0$.
Then the following hold.
\begin{itemize}
\item[(i)]
The process $(\beta_{\rho_t})_{t \ge 0}$ is a continuous local martingale
for the filtration $(\mathcal{H}_t)_{t \ge 0}$ with quadratic variation
$[\beta_{\rho_\cdot}]_t = \rho_t$.
\item[(ii)]
The process $(\gamma_t)_{t \ge 0}$, where
\[
\gamma_t = \int_0^t \sqrt{\frac{1}{J_s}} \, d \beta_{\rho_s},
\]
is a Brownian motion for the filtration $(\mathcal{H}_t)_{t \ge 0}$.
\item[(iii)]
If $\xi_t = \zeta_{\rho_t}$, $t \ge 0$, then
\[
\xi_t = \xi_0 + \int_0^t H_{\rho_s} \sqrt{J_s}\, d \gamma_s + \int_0^t K_{\rho_s} J_s \, ds.
\]
\item[(iv)]
If $(\eta_t)_{t \ge 0}$ is a continuous local martingale
for the filtration $(\mathcal{G}_t)_{t \ge 0}$, then
$(\eta_t)_{t \ge 0}$ is also a continuous local martingale
for the filtration $(\mathcal{H}_t)_{t \ge 0}$
and $[\eta,\gamma] \equiv 0$.
\end{itemize}
\end{proposition}
\begin{remark}\label{r:counter}
In this example $K/M = SO(2)$ has dimension $1$ and there is a type of skew-product decomposition. The angular part can indeed be written as a time-change depending on the radial part of a Brownian motion on $SO(2)$.
However, we cannot take this Brownian motion to be independent of the radial part.
\end{remark}
\section{Open problem}
The apparent counterexamples to \cite[Theorem~4]{L09} arise in Sections \ref{s:rotated} and \ref{s_counter} because $K/M$ is one-dimensional and hence trivially irreducible. When $K/M$ has dimension greater than $1$, irreducibility implies the nonexistence of a nonzero $M$-invariant tangent vector and it is this latter property that is actually used in the proof of \cite[Theorem~4]{L09}. In the examples in Sections \ref{s:BM}, \ref{s:rotated} and \ref{s_counter} the group $M$ is the trivial group consisting of just the identity and there certainly are nonzero $M$-invariant tangent vector.
Therefore, in view of the three examples we presented and Remarks \ref{r:BM}, \ref{r:rotated}, \ref{r:counter} we propose the following open problem.
\begin{question}
Suppose that $(x_t)_{t\geq 0}$ is a continuous Markov process with state space a smooth manifold $X$ and distribution that is equivariant under the smooth action of a Lie group $K$ on $X$ so that we can decompose $(x_t)_{t\geq 0}$ into a radial part $(y_t)_{t\geq 0}$ that is a Markov process on the submanifold $Y$ which is transversal to the orbits of $K$ and an angular part $(z_t)_{t\geq 0}$ that is a process on the homogeneous space $K/M$. Suppose further that $\text{dim}(K/M)=1$.
\begin{enumerate}
\item When can we write $z_t=B_{a_t}$ where $(B_t)_{t\geq 0}$ is a Brownian motion on $K/M$ and $(a_t)_{t\geq 0}$ is a time-change that is adapted to the filtration generated by $(y_t)_{t\geq 0}$.
\item Under which conditions can we take the Brownian motion $(B_t)_{t\geq 0}$ to be independent of $(x_t)_{t\geq 0}$?
\end{enumerate}
\end{question}
\subsection*{Acknowledgment}
We thank Prof. M. Liao for kindly explaining to us the role played by the assumption of irreducibility in \cite[Theorem~4]{L09}. We thank an anonymous referee for comments that improved this manuscript and for the example described in Section \ref{s:rotated}.
\end{document} |
\begin{equation}gin{document}
\title[Generation and evolution of quantum vortex states]{Entanglement by linear SU(2) transformations: generation and evolution of quantum vortex states}
\author{G S Agarwal\footnote{On leave from Physical Research Laboratory, Navrangpura, Ahmedabad 380 009,
India}$^1$ and J Banerji$^2$}
\address {$^1$ Department of Physics, Oklahoma State University,
Stillwater, OK-74078, USA}
\address {$2$ Quantum Optics and Quantum Information Group, Physical Research Laboratory, Navrangpura, Ahmedabad 380 009,
India} \ead{[email protected]}
\begin{equation}gin{abstract}
We consider the evolution of a two-mode system of bosons under the
action of a Hamiltonian that generates linear SU(2)
transformations. The Hamiltonian is generic in that it represents
a host of entanglement mechanisms, which can thus be treated in a
unified way. We start by solving the quantum dynamics analytically
when the system is initially in a Fock state. We show how the two
modes get entangled by evolution to produce a coherent
superposition of vortex states in general, and a single vortex
state under certain conditions. The degree of entanglement between
the modes is measured by finding the explicit analytical
dependence of the Von Neumann entropy on the system parameters.
The reduced state of each mode is analyzed by means of its
correlation function and spatial coherence function. Remarkably,
our analysis is shown to be equally as valid for a variety of
initial states that can be prepared from a two-mode Fock state via
a unitary transformation and for which the results can be obtained
by mere inspection of the corresponding results for an initial
Fock state. As an example, we consider a quantum vortex as the
initial state and also find conditions for its revival and charge
conjugation. While studying the evolution of the initial vortex
state, we have encountered and explained an interesting situation
in which the entropy of the system does not evolve whereas its
wave function does. Although the modal concept has been used
throughout the paper, it is important to note that the theory is
equally applicable for a two-particle system in which each
particle is represented by its bosonic creation and annihilation
operators.\\\\
(Figures in this article are in colour only in the electronic
version)
\end{abstract}
\pacs{03.67.-a, 03.75.Gg, 03.75.Lm, 03.67.Mn, 05.30.Jp, 42.50.Dv}
\section{Introduction\label{1}}
Nonclassical properties of quantum states are actively being
studied for their relevance in quantum computation. It is known
that quantum entanglement is the key to performing communication
and information processing tasks that cannot be realized
classically. For this reason, there has been a surge of activity
towards preparing, identifying and quantifying entangled
systems\cite{review}.
An important source of quantum entanglement has been the
polarization-entangled two-photon states generated from type-II
phase-matched parametric down conversion \cite{pdc}. A variety of
other entangled states can be produced by using various polarizing
components. More recently, the subject of quantum information
processing has been given a new direction with the realization
that a number of quantum logic operations can be performed using
single photons and methods of linear optics \cite{milburn}. Even a
method for quantum teleportation was proposed and implemented
\cite{demartini}. Clearly one needs to examine, in full
generality, the question of transformation of an arbitrary input
state by a device which can mix different states.
We note that a number of special cases for the generation of
entanglement using linear optical devices have been investigated.
Huang and Agarwal \cite{huang} considered multimode systems
described by a Hamiltonian that is quadratic in the mode
operators. They derived conditions for the generation of an
entangled state when the input state was represented by a Gaussian
density matrix. Their treatment covered a large class of states
including squeezed coherent states and even states with thermal
noise. However they did not consider the case of input fields in
Fock states. More recently and more specifically, Kim et al
\cite{knight} examined the question of the generation of entangled
states by a beam splitter using Fock states as input fields.
In this paper we specialise to intensity- or number-preserving
linear transformations belonging to the SU(2) group. For two-mode
states characterized by the annihilation operators $a$ and $b$,
such transformations can be generated by evolution under a
Hamiltonian of the form
\begin{equation}gin{equation} H=g(a^\dagger
b e^{i\phi} +h.c)+\Omega (a^\dagger a -b^\dagger b)\label{eq1}
\end{equation}
where $g$ and $\Omega$ are real constants. Introducing the
generators of the SU(2) group as \begin{equation} \label{eq62}
\fl J_1 = (a^\dagger b + ab^\dagger)/2,\qquad
J_2 = (a^\dagger b - ab^\dagger)/2i,\qquad
J_3 = (a^\dagger a - b^\dagger b)/2,
\end{equation}
the Hamiltonian (\ref{eq1}) can be rewritten in the form
\begin{equation}gin{equation}\label{eq63}
H=v_1 J_1+v_2J_2+v_3J_3\end{equation}
with
\begin{equation}gin{equation}\label{eq64}
v_1=2 g \cos\phi,\qquad v_2=-2g\sin\phi,\qquad v_3=2\Omega.\end{equation}
Motivation for the present work comes from the realization that a
Hamiltonian of the form (\ref{eq1}) can represent a host of
entanglement mechanisms which can thus be treated in a unified
way. Several examples are given as follows.
The beam splitter, used by many authors as an entangler \cite{bs}
can be described by (\ref{eq1}) for $\Omega=0$ if one defines its
amplitude reflection and transmission coefficients by $\cos g$ and
$\sin g$ respectively while $\phi$ denotes the phase difference
between the reflected and transmitted fields .
The parametric frequency conversion by a strong pump field (of
frequency $\omega$) in $\chi^{(2)}$ material can also be represented
by an interaction Hamiltonian of the form (\ref{eq1})
\cite{louisell} with $\Omega=0$. Here $ a$ and $b$ are the
annihilation operators for the signal (of frequency $\omega_a$) and
the idler (of frequency $\omega_b$) respectively, $g$ is a coupling
constant that depends on the amplitude of the pump mode and
$\phi=\Delta\omega t$ where $\Delta\omega=\omega
+\omega_b-\omega_a$. We should note, however, that this Hamiltonian
does not support parametric down conversion.
Polarizing elements such as half- and quarter wave plates also can
act as entangling devices. Quantum mechanically, polarized light
is represented by a pair of orthogonal polarization modes
(described by boson mode operators $a$, $b$), or as points on the
Poincar\'{e} sphere. The effect of a polarizing element on the
field is a SU(2) transformation of the mode operators which
corresponds to rotations on the Poincar\'{e} sphere. The
transformations are generated by Hamiltonians of the form
(\ref{eq1}).
Finally, following the work of Wineland et al \cite{wineland}, we
consider a single laser cooled ion confined in a two dimensional
harmonic trap. The internal and motional degrees of freedom of the
ion can be coupled by applying two classical laser beams. If $a$
and $b$ represent the two oscillatory modes of the ion's quantized
motion and $\phi$ denotes the difference in phase between the two
applied fields, then, under certain conditions \cite{brif} the
Hamiltonian for the ion's motion will be of the form (\ref{eq1})
in the interaction picture.
The present work is also relevant in the context of parallel
developments in the field of optical vortices. An optical vortex
of order $l$ centered at the origin ($r=0$) has a field
distribution of the form $F(r) \exp(il\phi)$. The distribution is
such that the field intensity tends to zero as $r\to 0$ whereas
the phase shift in one cycle around the origin is $2\pi l$ where
$l$ is an integer. The azimuthal mode index $l$ has a physical
meaning in that the vortex carries an orbital angular momentum of
$l \hbar$ per photon \cite{pra92_8185}. This angular momentum can
be imparted to microscopic particles in order to manipulate them
optically \cite{OL96_827,prl95_826}. In recent years, this
understanding has led to considerable interest in the generation
and study of optical vortices both in free space
\cite{allenreview} and in guided media\cite{science,jayvortex}.
A physically realizable field distribution that contains optical
vortices is a higher-order Laguerre-Gaussian ($LG$) beam whose
waist-plane field amplitude is given by
\cite{oc93_123}\begin{equation}gin{equation} \fl u^{LG}_{mn}(x,y,\omega)
=\sqrt{{2\over \pi \omega^2}} {(-1)^p p! \over \sqrt{m!n!}}
e^{-i\theta(m-n)} (r\sqrt{2}/\omega)^{|m-n|}L_p^{|m-n|}
(2r^2/\omega^2) e^{-r^2/\omega^2}\label{eq0} \end{equation} where
$r^2=x^2+y^2$, $\theta=\arctan (y/x)$, $\omega$ is the beam waist,
$p = \min (n,m)$ and $L_p^l(x)$ is a generalized Laguerre
polynomial. $LG$ beams can be produced directly from a laser
\cite{oc94_161}. In fact, in a hydrodynamic formulation of laser
beam dynamics in terms of LG modes, vortices were found to occur in
transverse laser patterns \cite{brambilla1,brambilla2}. Usually
however, $LG$ beams are produced by the conversion or combination of
Hermite-Gaussian ($HG$) beams that are emitted by most laser
cavities. This is made possible because of the fact that any LG mode
can be expressed in terms of HG modes. The waist-plane amplitude of
the HG modes has the form
\begin{equation}gin{equation}\label{three} u^{HG}_{n,m}(x,y,
\omega)=\Phi_{n}(x,\omega)\Phi_{m}(y,\omega)\end{equation} where
\begin{equation}gin{equation}\label{four} \Phi_{n}(x,\omega)= \left({\sqrt{2}\over
\sqrt{\pi} 2^n w n!}\right)^{1/2} H_n(\sqrt{2} x/w) \exp
(-x^2/w^2) \end{equation} and $H_n(x)$ is a Hermite polynomial.
The decomposition of a LG mode in terms of HG modes is given as
\cite{allenreview} \numparts
\begin{equation}gin{eqnarray}
\label{six} u^{LG}_{n,m}(x,y,\omega) & = & \sum_{k=0}^{m+n} i^k
b(n,m,k) u^{HG}_{m+n-k,k}(x,y,\omega)\\
b(n,m,k) & = & \sqrt{{(n+m)! k!\over 2^{n+m} n! m!}}{1\over k!}
{d^k\over dt^k}[(1-t)^n (1+t)^m]\rfloor_{t=0}
\end{eqnarray}
\endnumparts
The vortices as discussed above appear on the transverse amplitude
profile of {\it classical} wave fields. Vortices can also occur in
the configuration space representation of quantum systems of
matter or radiation. Since the HG modes are also the energy
eigenfunctions of a quantum oscillator, quantum vortices should
arise in the study of wave packets of a quantum system that could
be a two-dimensional harmonic oscillator like an ion in a
two-dimensional trap. For a two-mode radiation field characterized
by the annihilation operators $a$, $b$, and represented by a
state vector $\vert\psi\rangle$, the quantum vortex will appear in
the quadrature distribution $\vert\langle
x,y\vert\psi\rangle\vert^2$ where $\vert x,y\rangle$ is the
eigenvector of $(a+a^\dagger)/\sqrt{2}$ and
$(b+b^\dagger)/\sqrt{2}$. Quadrature distributions can be measured
by a homodyne method\cite{leonhardt}. Vortices of matter will
appear in the configuration space probability distribution.
Recently it has been shown that the HG and LG modes are unitarily
related\cite{simon} and the Poincare sphere\cite{poincare_cl}
representing LG beams has an underlying SU(2)
structure\cite{poincare_qm}. The Hamiltonian (1) is therefore
ideally suited to explore the possibility of generating quantum
vortices.
The objective and the plan of the paper are as follows. In section
\ref{2} we obtain the state vector and the wave function of a
two-mode system which is initially in a Fock state and is acted
upon by the Hamiltonian (\ref{eq1}). We show how the two modes get
entangled by evolution and under certain conditions evolve into a
vortex state. The degree of entanglement between the modes is
measured by finding the dependence of the von Neumann entropy on
the system parameters. In section \ref{3} the above analysis is
carried out when the two-mode system is initially in a state that
can be obtained from a Fock state via a unitary transformation. As
an example, a quantum vortex is used as the initial state. We also
find conditions for the revival and the charge conjugation of the
vortex. In section \ref{4} we consider the structure of the
reduced state of each mode. The paper ends with concluding remarks
in section \ref{5}.
\section{Generation of quantum entanglement and creation of a quantum vortex using an initial two-mode Fock state\label{2}}
\subsection{Evolution of the state vector\label{2.1}}
Let us consider the evolution of a two-mode Fock state $\vert
N-j,j\rangle$ when the Hamiltonian is given by (\ref{eq1}) and the
total number ($N$) of photons in the two modes is constant. The
resulting state $|\psi_{Nj}(t)\rangle=U(t)\vert N-j,j\rangle$ can
be obtained by the use of the disentangling theorem. In what
follows, we use a different method. We write $|N-j,j\rangle$ as
\begin{equation}gin{equation}\label{eq2} |N-j,j\rangle={(\hat{a}^\dagger)^{N-j}
(\hat{b}^\dagger)^{j}\over \sqrt{N-j! j!}}|0,0\rangle
\end{equation} and define a new pair of operators
\begin{equation}gin{equation}\label{eq3} {\hat{a}(t)\choose \hat{b}(t)}
=U^\dagger (t) {\hat{a}\choose \hat{b}} U(t) \end{equation} where
$U(t)=\exp(-iHt)$ is the time evolution operator. Then
$|\psi_{Nj}(t)\rangle$ can be written in a compact form as
\begin{equation}gin{equation}\label{eq4}
|\psi_{Nj}(t)\rangle={[\hat{a}^\dagger(-t)]^{N-j}
[\hat{b}^\dagger(-t)]^{j}\over \sqrt{N-j! j!}}|0,0\rangle
\end{equation} Note that the state at time $t$ is obtained by using the operators evaluated at time $-t$. The explicit expressions for $\hat{a}(t)$ and
$\hat{b}(t)$ can be obtained by solving the Heisenberg equations
for the operators. We get \begin{equation}gin{equation}\label{eq5}
{\hat{a}(t)\choose \hat{b}(t)}= {\bf
V}{\hat{a}(0)\choose\hat{b}(0)} ={\bf V}{\hat{a}\choose\hat{b}}
\end{equation} where ${\bf V}=\{v_{ij}\}$ is a $2\times 2$ unitary matrix.
Setting $\sigma=\sqrt{\Omega^2+g^2}$ and $\Omega=\sigma
\cos\Theta$, the matrix elements are written as
\begin{equation}gin{eqnarray}\label{eq6}
v_{11} = \cos \sigma t - i \cos\Theta \sin \sigma t, &&
\qquad v_{12} = - ie^{i\phi}\sin\Theta\sin \sigma t,\nonumber\\
v_{21} = - ie^{-i\phi}\sin\Theta\sin \sigma t, &&\qquad v_{22} =
\cos \sigma t + i \cos\Theta\sin \sigma t. \end{eqnarray} Note
that
\begin{equation}gin{equation}\label{eq7} v_{11}=v^*_{22},\quad
v_{12}=-v^*_{21}\quad\hbox{and}\quad \vert v_{21}\vert^2+\vert
v_{22}\vert^2=1. \end{equation} Substitution in (\ref{eq4})
followed by binomial expansion and the use of (\ref{eq2}) yields
\begin{equation}gin{equation}\label{eq8} |\psi_{Nj}(t)\rangle=\sum_{m=0}^{N-j}\sum_{n=0}^{j}
b_{mn}|N-(m+n), m+n\rangle \end{equation} where
\begin{equation}gin{equation}\label{eq9} \fl b_{mn}={N-j\choose m} {j\choose
n} {N\choose N-j}^{1/2} {N\choose
m+n}^{-1/2}(v_{11})^{N-j-m}(v_{21})^m (v_{12})^{j-n} (v_{22})^n.
\end{equation} {\it The two modes in the state $|\psi_{Nj}\rangle$ are
entangled in the sense that the above double sum cannot be reduced
to the product of two single-mode summations}.
It is instructive to briefly mention the case when the two modes are
initially in a Glauber coherent state $\vert \alpha, \begin{equation}ta\rangle$.
Since the Hamiltonian (\ref{eq1}) conserves photon numbers, the
state at time $t$ will also be a coherent state:
\begin{equation}gin{equation}\label{eq10} U(t)\vert \alpha, \begin{equation}ta\rangle=\vert \alpha (t),\begin{equation}ta
(t)\rangle. \end{equation} Applying (\ref{eq5}) on $\vert \alpha,
\begin{equation}ta\rangle$, we immediately obtain
\begin{equation}gin{equation}\label{eq11} {\alpha(t)\choose \begin{equation}ta (t)}= {\bf
V}{\alpha\choose \begin{equation}ta}.\end{equation} Furthermore, the unitarity
of ${\bf V}$ ensures that
\begin{equation}gin{equation}\label{eq12}\vert\alpha(t)\vert^2+\vert\begin{equation}ta(t)\vert^2=
\vert\alpha\vert^2+\vert\begin{equation}ta\vert^2.\end{equation} Thus {\it no
entanglement occurs if each input mode is in a coherent state}.
In what follows, we exploit coherent states as generating
functions of number states to reduce the double sum in (\ref{eq8})
to a single sum. Expanding both sides of (\ref{eq10}) in number
states and recalling that $\vert \alpha(t)\vert^2 +\vert
\begin{equation}ta(t)\vert^2=\vert \alpha\vert^2 +\vert \begin{equation}ta\vert^2$, we get
the relation \begin{equation}gin{equation}\label{eq13} \sum_m\sum_n {\alpha^m
\begin{equation}ta^n\over \sqrt{m!n!}} U(t)\vert m,n\rangle=\sum_p\sum_q
{\alpha^{p+q}\over \sqrt{p!q!}} \xi_{pq}(\tau)\vert p,q\rangle
\end{equation} where $\tau=\begin{equation}ta/\alpha$ and \begin{equation}\label{eq14}
\xi_{pq}(\tau) = (v_{11}+v_{12}\tau)^p (v_{21}+v_{22}\tau)^q =
\sum_{k=0}^{p+q} {\tau^k \over k!}
\partial_\tau^{(k)} \xi_{pq}(\tau)\rfloor_{\tau\to 0}.\end{equation} Substituting
in (\ref{eq13}) and equating the coefficient of $\alpha^{N-j}
\begin{equation}ta^j$, one gets \begin{equation}gin{equation}\label{eq15}
|\psi_{Nj}(t)\rangle=U(t)\vert N-j,j\rangle=\sum_{q=0}^N
C^{(q)}_{Nj} \vert N-q,q\rangle \end{equation} where
\begin{equation}gin{equation}\label{eq16} C^{(q)}_{Nj}={1\over j!}\left[{(N-j)!
j! \over (N-q)!q!}\right]^{1/2}
\partial_\tau^{(j)} \xi_{N-q,q}(\tau)\rfloor_{\tau\to 0}.\end{equation}
Some useful properties of $\vert C^{(q)}_{Nj}\vert^2$ are derived
in appendix A.
\subsection{ The wave function-- a coherent superposition of vortex states\label{2.2}}
The corresponding wave function in configuration space is obtained
as follows. Using the relation
\begin{equation}gin{equation}\label{eq22}
\langle y\vert q\rangle = {e^{-y^2/2} H_q(y)\over \sqrt{2^q q!
\sqrt{\pi}}},\qquad H_q(y) = (-1)^q e^{y^2}
\partial_y^{(q)} e^{-y^2},\end{equation} and the
corresponding expression for $\langle x\vert N-q\rangle$, we
obtain
\begin{equation}gin{eqnarray}\label{eq23} \psi_{Nj}(x,y,t) & = & \langle x,y\vert U(t)\vert
N-j,j\rangle\nonumber\\& = & {e^{-(x^2+y^2)/2}\over \sqrt{\pi
2^N}}\sum_{q=0}^N C^{(q)}_{Nj}{H_{N-q}(x) H_q(y)
\over\sqrt{(N-q)!q!}}\nonumber\\& = & {(-1)^N e^{(x^2+y^2)/2}\over
\sqrt{\pi 2^N}}\sum_{q=0}^N C^{(q)}_{Nj}{\partial_x^{N-q}
\partial_y^q e^{-(x^2+y^2)}\over\sqrt{(N-q)!q!}
}\end{eqnarray} The wave function has a more appealing form in
polar coordinates as shown below. Writing $x=r\cos\theta$,
$y=r\sin\theta$, and defining
\begin{equation}gin{equation}\label{eq26}
\gamma_{\pm}(\tau)=v_{11}+v_{12}\tau \pm
i(v_{21}+v_{22}\tau),\end{equation} we get (see appendix B)
\begin{equation}gin{equation}\label{eq28} \psi_{Nj}(x,y,t)= \sum_{n=0}^N
b_{Nj}^{(n)} u_{N-n,n}(r,\theta)\end{equation} where
\begin{equation}gin{eqnarray}\label{eq29} b_{Nj}^{(n)} & = & {1\over j!} \sqrt{{(N-j)!
j!\over (N-n)! n! 2^N}}\zeta_{Nn}^{(j)}(0)\nonumber\\
\zeta_{Nn}(\tau)& = & \gamma_+(\tau)^{N-n}
\gamma_-(\tau)^n\nonumber\\ \zeta_{Nn}^{(j)}(0)& = &
\partial_\tau^j \zeta_{Nn}(\tau)\rfloor_{\tau\to 0}\nonumber\\
u_{mn}(r,\theta)& = & u^{LG}_{mn}(x,y,\sqrt{2}).\end{eqnarray}
Recall that for $m\neq n$, $u_{mn}(r,\theta)$ represents a vortex of
order $\vert m-n\vert$ and charge $m-n$ embedded in a Gaussian host
beam of waist $\omega=\sqrt{2}$. Thus {\it for odd values of $N$,
the wave function $\psi_{Nj}(x,y,t)$ becomes a coherent
superposition of vortex states, whereas for even values of $N$, the
superposition will also contain a state (corresponding to $n=N/2$)
that does not have a vortex character.}\cite{zeilinger}.
\subsection{Creation of a single quantum vortex\label{2.3}}
In this section we will derive conditions for the creation of a
single quantum vortex. We reiterate that for light fields, the
vortex will appear in the quadrature distribution whereas for
other systems it will be in the probability distribution in
configuration space.
The initial two-mode Fock state can evolve into a single vortex
state when the summation in (\ref{eq28}) collapses into a single
term. This happens whenever $\gamma_+(0)$ or $\gamma_-(0)$ is
zero. It is easy to show that $\vert v_{21}\vert^2=1/2$ for both
these cases.
If $\gamma_+(0)=0$, then $v_{11}+i v_{21}=0$ and taking the
complex conjugate of this equation, $v_{22}+i v_{12}=0$. Then
$\gamma_+(\tau)=2 v_{12}\tau$ and $\gamma_-(\tau)=-2 i v_{21}$ so
that $\zeta_{Nn}^{(j)}(0)=(2 v_{12})^{N-n}(-2 i v_{21})^n j!
\end{equation*}lta_{N-j,n}$ and, finally \begin{equation}gin{equation}\label{eq30}
\psi_{Nj}(x,y,t)\rfloor_{\gamma_+(0)=0}=2^{N/2} i^{j-N}
v_{21}^{N-j} v_{12}^j u_{j,N-j} (r,\theta).\end{equation} The
condition $\gamma_+(0)=0$ implies that $\Omega=-g\sin \phi$ and
$\sigma \cos \sigma t=-g \cos\phi \sin\sigma t$. We give two
examples for which these conditions are
satisfied.\begin{equation}gin{enumerate}
\item Setting $\Omega=0$, $\phi=\pi$ and $\sigma t=\pi/4$, we get
\begin{equation}gin{equation}\label{eq31} \psi_{Nj}(x,y,t)=i^j u_{j,N-j}(r,\theta).\end{equation} From
(\ref{eq15}), one obtains the corresponding state vector
\begin{equation}gin{equation}\label{eq32} U_0\vert N-j,j\rangle=\sum_{q=0}^N D^{(q)}_{Nj}
\vert N-q,q\rangle\end{equation} where
\begin{equation}gin{equation}\label{eq33} U_0=\exp[{i \pi\over 4} (a^\dagger b
+ ab^\dagger)]\end{equation} and \begin{equation}gin{eqnarray}\label{eq34}
D^{(q)}_{Nj}& = & C^{(q)}_{Nj}\rfloor_{\Omega=0, \phi=\pi,\sigma t
=\pi/4}\nonumber\\ & = & \sqrt{{(N-j)! j!\over 2^N (N-q)! q!}}
{i^q\over j!}\left[
\partial_\tau^j (1+i\tau)^{N-q} (1-i\tau)^q\right]_{\tau\to 0}.\end{eqnarray}
\item Setting $\Omega=g$, $\phi=-\pi/2$ and $\sigma t=\pi/2$, we get
\begin{equation}gin{equation}\label{eq35} \psi_{Nj}(x,y,t)=(-i)^{j+N}
u_{j,N-j}(r,\theta).\end{equation} The operator form of the
corresponding state vector is given by \cite{simonnote}
\begin{equation}gin{equation}\label{eq36} \exp[{i \pi\over 2\sqrt{2}}
\{i(a^\dagger b - ab^\dagger)-(a^\dagger a-b^\dagger b)\}] \vert
N-j,j\rangle.\end{equation}
\end{enumerate}
Following a similar
analysis for $\gamma_-(0)=0$, one obtains
\begin{equation}gin{equation}\label{eq37}
\psi_{Nj}(x,y,t)\rfloor_{\gamma_-(0)=0}=2^{N/2} i^{N-j}
v_{21}^{N-j} v_{12}^j u_{N-j,j} (r,\theta).\end{equation} The
condition $\gamma_-(0)=0$ yields $\Omega=g\sin \phi$ and $\sigma
\cos \sigma t=g \cos\phi \sin\sigma t$. These two conditions are
satisfied , for example, when $\Omega=\phi=0$ and $\sigma
t=\pi/4$. The corresponding wave function is the complex conjugate
of (\ref{eq31}).
We end this section by noting that the above conditions can be
physically realized for a given entangling device. We give an
example in the context of a frequency converter. Suppose the
signal (of frequency $\omega_a$) and the idler (of frequency
$\omega_b$) are initially in Fock states and the converter is
pumped at the difference frequency $\omega_a-\omega_b$. Replacing
$t$ by $L/c$, where $L$ is the length of the non-linear medium and
$c$ is the speed of light, one can adjust the pump amplitude such
that $gL/c=\pi/4$. This setup corresponds to $\Omega=\phi=0$ and
$gt=\pi/4$. In this case the quadrature distribution of the output
state will be a single quantum vortex as mentioned above.
\subsection{Entanglement of the two modes\label{2.4}}
Initially the two modes are not entangled as the state vector
$\vert N-j,j\rangle$ is the direct product of the state vectors
for each mode. In configuration space, this would imply that
$\psi_{Nj}(x,y,0)$ is separable in $x$ and $y$ as indeed it is.
Furthermore, as the time dependence arises solely in $v_{ij}$
which vary as $\cos \sigma t$ or $\sin \sigma t$, the initial
state is revived whenever $\sigma t=k\pi$ where $k$ is an integer.
For even values of $k$, the revival is exact whereas for odd
values of $k$, it is within an overall factor of $(-)^N$. At other
times, the two modes are entangled as is evident in the expression
((\ref{eq8}) or (\ref{eq15})) for the state vector and the
expression ((\ref{eq23}) or (\ref{eq28})) for the corresponding
wave function.
\subsection{Degree of entanglement\label{2.5}}
Note
that the two-mode system $|\psi_{Nj}(t)\rangle$ is in a pure state
whereas the reduced state of each mode, determined by a partial
trace operation, will be a mixed state. The reduced density
operators of modes `a' and `b' are given respectively by \numparts
\begin{equation}gin{eqnarray}\label{eq38} \rho^{(a)}_{Nj} & =Tr_b |\psi_{Nj}\rangle\langle
\psi_{Nj}|& =\sum_{q=0}^N |C^{(q)}_{Nj} |^2 |q\rangle\langle
q|\\\rho^{(b)}_{Nj} & =Tr_a |\psi_{Nj}\rangle\langle \psi_{Nj}|&
=\sum_{q=0}^N |C^{(N-q)}_{Nj}|^2 |q\rangle\langle q|
\end{eqnarray}
\endnumparts
The corresponding von Neumann entropies $S^{(a)}_{Nj}$ and
$S^{(b)}_{Nj}$ provide a measure of the degree of entanglement
between the two modes:
\numparts
\begin{equation}gin{eqnarray}\label{eq39}
S^{(a)}_{Nj}& = & -
\sum_{q=0}^N |C^{(q)}_{Nj} |^2 \log |C^{(q)}_{Nj} |^2\\
\rule{0mm}{6mm}
S^{(b)}_{Nj}& = & - \sum_{q=0}^N |C^{(N-q)}_{Nj}|^2 \log
|C^{(N-q)}_{Nj}|^2 \end{eqnarray}
\endnumparts
By virtue of relations (\ref{eq20}), we get \begin{equation} \label{eq40}
S^{(a)}_{Nj}\rfloor_{|v_{21}|^2\to 1-R} =
S^{(a)}_{Nj}\rfloor_{|v_{21}|^2\to R} =
S^{(a)}_{N,N-j}\rfloor_{|v_{21}|^2\to R}.\end{equation}
Changing the
summation index from $q$ to $N-q$ in the expression for
$S^{(b)}_{Nj}$, one obtains $S^{(a)}_{Nj}=S^{(b)}_{Nj}$. Thus the
symmetry relations (\ref{eq40}) hold good for $S^{(b)}_{Nj}$ as
well. These observations hold for any bipartite system in a pure
state.
It is remarkable that for a given value of $N$, $j$ and $q$, the
dynamics of $\vert C^{(q)}_{Nj}\vert^2$ depends on $\vert
v_{21}\vert^2=\sin^2\Theta\sin^2\sigma t$ only (see appendix A).
This important observation implies that {\it (a) the entropy
$S^{(a)}_{Nj}$ and the reduced density operator $\rho^{(a)}_{Nj}$
are independent of $\phi$ and (b) are symmetric with respect to
the interchange of $\Theta$ and $\sigma t$.} In Figure 1, we plot
$S^{(a)}_{Nj}$ as a function of $\vert v_{21}\vert^2$ for $N=4$
and $j=0, 1, 2$.
\begin{equation}gin{figure}
\setcaptionwidth{4.0in}
\centering\includegraphics[width=4.0in]{fig1jb}
\caption{\label{f1} Plot of $S^{(a)}_{Nj}$ as a function of $\vert
v_{21}\vert^2$ for $N=4$ and $j=0, 1, 2$.}
\end{figure}
Trivially, for $\vert v_{21}\vert^2=0$, the initial pure state
$|N-j,j\rangle$ either does not evolve or is fully revived and the
entropy of the reduced state is zero. For $\vert v_{21}\vert^2=1$,
the initial state swaps the photon numbers in the two modes and
becomes $|j,N-j\rangle$ which is also a pure state. For all other
values of $\vert v_{21}\vert^2$, the initially pure state becomes
a mixed state and the entropy of the reduced state becomes
non-zero. Recall that for $\vert v_{21}\vert^2=1/2$ and $N-j\neq
j$, the quantum state becomes a vortex. Thus {\it a quantum vortex
is indeed an entangled state}. To quantify the degree of
entanglement for a vortex state, we plot $S^{(a)}_{Nj}$ as a
function of $j$ for $\vert v_{21}\vert^2=1/2$ and a given total
number of photons $N$ (see Fig. 2).
\begin{equation}gin{figure}
\setcaptionwidth{5.0in}
\centering
\includegraphics[width=5.0in]{fig2jb}
\caption{\label{f2} Plot of $S^{(a)}_{Nj}$ as a function of $j$
for $\vert v_{21}\vert^2=1/2$ and a given total number of photons
$N$.}
\end{figure}
It is clear that the entropy of the state without a vortex
($j=N/2$) is less than the entropy of the neighboring ($ j\sim N/2$)
vortex states ($N-j\neq j$). This reduction in entropy can be
attributed to the symmetry of the $j=N/2$ state and traced to the
highly oscillatory nature of the Jacobi polynomial appearing in Eq.
(\ref{eq21}). For a given value of $N$, the minimum in the entropy
of a {\it vortex state} occurs for $j=0,N$ in which case $\vert
C^{(q)}_{Nj}\vert^2$ is a binomial distribution (see appendix A).
Interestingly, for $j=0,N$, the vortex state will have the maximum
allowed order ($N$). Thus {\it the vortex state of maximum order
will have minimum entropy} which is counter-intuitive. One would
have expected that the more twists the phase of the state has, more
energetic and more entropic it would be. Note that the symmetry of
$S^{(a)}_{Nj}$ about $\vert v_{21}\vert^2=1/2$ in Fig. 1 and about
$j=N/2$ in Fig. 2 is contained in the relations (\ref{eq40}). Note
also that the vorticity or non-vorticity of the state of lowest
entropy depends on the value of $N$ (see Fig. 3).
\begin{equation}gin{figure}
\setcaptionwidth{4.0in}
\centering\includegraphics[width=4.0in]{fig3jb}
\caption{\label{f3}Plot of $S^{(a)}_{Nj}$ as a function of $N$ for
$\vert v_{21}\vert^2=1/2$ and $j=0, 1, N/2$.}
\end{figure}
We end this section by comparing the entropy values in Figs. 1-3
with $\log_2 (N+1)$, the maximum entropy possible for a given $N$
with an entirely mixed state. For $N=4$, $10$ and $100$, $\log_2
(N+1)$ has the values $2.32193$, $3.45943$ and $6.65821$
respectively.
\section{Evolution of an initial vortex state\label{3}}
\subsection{Evolution of the state vector\label{3.1}}
Let us assume that the two modes are initially in a quantum vortex
state as in (\ref{eq32}). Then the state vector at time t will be
given by \begin{equation}gin{equation}\label{eq41} \vert \tilde{\psi}_{Nj}
(t)\rangle=U(t)U_0\vert N-j,j\rangle\end{equation} Proceeding as
in section \ref{2.1}, we define
\begin{equation}gin{equation}\label{eq42} {\hat{a}(t)\choose \hat{b}(t)}
=[U(t)U_0]^\dagger {\hat{a}\choose \hat{b}} [U(t)U_0]
\end{equation} and obtain \begin{equation}gin{equation}\label{eq43}
{\hat{a}(t)\choose \hat{b}(t)}= {\bf
V}{\hat{a}(0)\choose\hat{b}(0)}. \end{equation} Note, however,
that in this case, \begin{equation}gin{equation}\label{eq44} {\hat{a}(0)\choose
\hat{b}(0)} =U_0^\dagger {\hat{a}\choose \hat{b}}U_0= {\bf
W}{\hat{a}\choose\hat{b}}, \qquad {\bf W}={1\over
\sqrt{2}}\left(\begin{equation}gin{array}{cc}
1 & i \\
i & 1
\end{array}\right). \end{equation}
Thus \begin{equation}gin{equation}\label{eq46} {\hat{a}(t)\choose \hat{b}(t)}=
{\bf \tilde{V}}{\hat{a}\choose\hat{b}}; \quad {\bf \tilde{V}}=
{\bf V}{\bf W}.\end{equation} The action and the effect of the
unitary operator $U_0$ are now clear. $U_0$ transforms the
two-mode Fock state into a different initial state {\it before}
its time evolution begins and thus $U_0$ can be regarded as the
operator for {\it initial state preparation}. The effect of $U_0$
is contained in the unitary matrix ${\bf W}$. As a result, the
overall unitary evolution matrix changes from ${\bf V}$ to ${\bf
\tilde{V}}= {\bf V}{\bf W}$.
In the present case, $U_0$, as given by (\ref{eq33}), prepares a quantum
vortex state as the initial state and the corresponding expression
for ${\bf W}$ is given as in (\ref{eq44}). A different expression
for $U_0$
will generate an initial state that is different from a quantum
vortex. Yet for all these initial states the dynamics is
essentially solved once the corresponding dynamics for a two-mode
Fock state is worked out as in section \ref{2.1}. In each case, one need only calculate
the matrix ${\bf W}$ and replace ${\bf V}$ by ${\bf \tilde{V}}$.
{\it In this sense, our theory not only provides a unified
approach to entanglement through a generic Hamiltonian but also
promises wide applicability to a variety of initial states}.
In the present case, the matrix elements of ${\bf
\tilde{V}}=\{\tilde{v}_{ij}\}$ are obtained easily as
\begin{equation}gin{eqnarray}\label{eq47} \tilde{v}_{11} =
(v_{11}+iv_{12})/\sqrt{2}, &&\qquad \tilde{v}_{12} = (v_{12}+i
v_{11})/\sqrt{2},\nonumber\\ \tilde{v}_{21} = (v_{21}+i
v_{22})/\sqrt{2}, && \qquad \tilde{v}_{22} = (v_{22}+i
v_{21})/\sqrt{2}.\end{eqnarray} Using (\ref{eq7}), one can also
show that
\begin{equation}gin{equation}\label{eq48}\tilde{v}_{11}=\tilde{v}^*_{22},\quad
\tilde{v}_{12}=-\tilde{v}^*_{21}\quad\hbox{and}\quad \vert
\tilde{v}_{21}\vert^2+\vert \tilde{v}_{22}\vert^2=1.
\end{equation}
It is now trivial to obtain the wave vector and the wave function
by borrowing the corresponding results from the previous section.
We simply replace $v_{ij}$ by $\tilde{v}_{ij}$ for $i,j=1,2$ and
for the sake of clarity and comparison, use the same nomenclature
for the new expressions except for a $\tilde{ }$ (tilde) over
them. Thus \begin{equation}gin{equation}\label{eq49} \vert \tilde{\psi}_{Nj}
(t)\rangle=\sum_{q=0}^N \tilde{C}^{(q)}_{Nj} \vert N-q,q\rangle
\end{equation} where \numparts \begin{equation}gin{eqnarray}\label{eq50}
\tilde{C}^{(q)}_{Nj} & = &{1\over j!}\left[{(N-j)! j! \over
(N-q)!q!}\right]^{1/2}
\partial_\tau^{(j)} \tilde{\xi}_{N-q,q}(\tau)\rfloor_{\tau\to 0}\\
\tilde{\xi}_{pq}(\tau) & = & (\tilde{v}_{11}+\tilde{v}_{12}\tau)^p
(\tilde{v}_{21}+\tilde{v}_{22}\tau)^q\end{eqnarray}\endnumparts
Furthermore, the results of appendix A can be used to write
\begin{equation}gin{eqnarray}\label{eq52} \fl \vert \tilde{C}^{(q)}_{Nj}\vert^2 & =
& (N-j)! (N-q)! q! (j!)^{-1} (1-|\tilde{v}_{21}|^2)^N
\left({|\tilde{v}_{21}|^2\over 1-|\tilde{v}_{21}|^2}\right)^{q-j}\vert
f^{(q)}_{Nj}(|\tilde{v}_{21}|^2)\vert^2\\\fl
&=& \label{eq53}\left\{\begin{equation}gin{array}{ll}
\end{equation*}lta_{q,j}, & \mbox{$|\tilde{v}_{21}|^2\to 0$,}\\\rule{0mm}{6mm}
\end{equation*}lta_{q,N-j}, & \mbox{$|\tilde{v}_{21}|^2\to 1$.}
\end{array}
\right.\end{eqnarray} and \begin{equation} \label{eq54}\vert
\tilde{C}^{(q)}_{Nj}\vert ^2\rfloor_{|\tilde{v}_{21}|^2\to 1-R}=
\vert \tilde{C}^{(N-q)}_{Nj}\vert
^2\rfloor_{|\tilde{v}\tilde{}_{21}|^2\to R} = \vert
\tilde{C}^{(q)}_{N,N-j}\vert ^2\rfloor_{|\tilde{v}_{21}|^2\to
R}.\end{equation}
\subsection{The wave function\label{3.2}} The corresponding wave function in
configuration space can be read off from Eq (\ref{eq28}). We get
\begin{equation}gin{equation}\label{eq55} \tilde{\psi}_{Nj}(x,y,t)= \sum_{n=0}^N
\tilde{b}_{Nj}^{(n)} u_{N-n,n}(r,\theta)\end{equation} where
\numparts\begin{equation}gin{eqnarray}\label{eq56} \tilde{b}_{Nj}^{(n)} & = &
{1\over j!} \sqrt{{(N-j)! j!\over
(N-n)! n! 2^N}}\tilde{\zeta}_{Nn}^{(j)}(0)\\
\tilde{\zeta}_{Nn}(\tau)& = & \tilde{\gamma}_+(\tau)^{N-n}
\tilde{\gamma}_-(\tau)^n\\ \tilde{\gamma}_{\pm}(\tau) & = &
\tilde{v}_{11}+\tilde{v}_{12}\tau \pm
i(\tilde{v}_{21}+\tilde{v}_{22}\tau).\end{eqnarray}\endnumparts
Thus {\it a quantum vortex state evolves into a superposition of
vortex states} under the action of the Hamiltonian (\ref{eq1}).
\subsection{Revival and charge conjugation\label{3.3}}
It can be shown that if $\tilde{\gamma}_+(0)$ or
$\tilde{\gamma}_-(0)$ is zero, then the summation in (\ref{eq55})
reduces to a single term. Specifically, if
$\tilde{\gamma}_+(0)=0$, then ${\rm Im}\,v_{21}= {\rm Im
}\,v_{22}=0$ and \begin{equation}gin{equation}\label{eq57}
\tilde{\psi}_{Nj}(x,y,t)= (iv)^j {v^*}^{N-j}
u_{j,N-j}(r,\theta)\end{equation} with $v={\rm Re}\,v_{22}+i {\rm
Re }\,v_{21}$. The above conditions are satisfied for the
following cases: (a) $\sin \sigma t=0$ for arbitrary values of
$\Theta$ and $\phi$. This includes the initial state ($t=0$) and
the state upon revival ($\sigma t = \pi$). (b)
$\sin\Theta=\sin\phi=1$ for arbitrary time. In this case the
initial vortex state becomes an eigenstate of the corresponding
Hamiltonian.
On the other hand, if $\tilde{\gamma}_-(0)=0$, then ${\rm Re
}\,v_{21}= {\rm Re }\,v_{22}=0$ and \begin{equation}gin{equation}\label{eq58}
\tilde{\psi}_{Nj}(x,y,t)= (v)^j (-iv^*)^{N-j}
u_{N-j,j}(r,\theta)\end{equation} with $v={\rm Im }\,v_{22}+i {\rm
Im }\,v_{21}$. Note that
$u_{N-j,j}(r,\theta)=u^*_{j,N-j}(r,\theta)$ and thus
$\tilde{\gamma}_-(0)=0$ is the condition for {\it `charge
conjugation'} or {\it `helicity reversal'} of the initial vortex
state. This condition is fulfilled whenever $\sin\Theta\sin\phi=0$
and $\cos\sigma t=0$.
\subsection{Degree of Entanglement in the superposition state
(\ref{eq55})\label{3.4}}
The reduced density operator of mode 'a' and the
corresponding von Neumann entropy are given respectively by
\numparts\begin{equation}gin{eqnarray}\label{eq59} \tilde{\rho}^{(a)}_{Nj} & =
& Tr_b |\tilde{\psi}_{Nj}\rangle\langle \tilde{\psi}_{Nj}|=
\sum_{q=0}^N |\tilde{C}^{(q)}_{Nj} |^2 |q\rangle\langle
q|\\\label{eq60}\tilde{S}^{(a)}_{Nj}& = & - \sum_{q=0}^N
|\tilde{C}^{(q)}_{Nj} |^2 \log |\tilde{C}^{(q)}_{Nj}
|^2\end{eqnarray}\endnumparts
It is clear that
$\tilde{S}^{(a)}_{Nj}$ will depend on $\vert
\tilde{v}_{21}\vert^2$ in exactly the same way as $S^{(a)}_{Nj}$
does on $\vert v_{21}\vert^2$ except that the form of $\vert
\tilde{v}_{21}\vert^2$ as a function of $\Theta$, $\phi$ and
$\sigma t$ is quite different from $\vert v_{21}\vert^2$. Notably,
$\vert \tilde{v}_{21}\vert^2$ depends on $\phi$ while $\vert
v_{21}\vert^2$ does not. Explicitly, \begin{equation}gin{equation}\label{eq61}
\vert \tilde{v}_{21}\vert^2={1\over 2}-\sin\Theta \sin\sigma t
(\cos\phi \cos \sigma
t-\sin\phi\cos\Theta\sin\sigma t).\end{equation}
\subsection{A case of constant entropy\label{3.5}}
Note that if $\Theta=\phi=\pi/2$ or $\Theta=0$, then $\vert
\tilde{v}_{21}\vert^2=1/2$ so that $ \vert
\tilde{C}^{(q)}_{Nj}\vert^2$ and the entropy
$\tilde{S}^{(a)}_{Nj}$ will not evolve with time. The underlying
reason is as follows.
Using the expressions (\ref{eq62}) for the SU(2) generators, we
obtain,
\begin{equation}gin{equation}\label{eq65} J_3\vert
N-q,q\rangle ={N-2 q\over 2}\vert
N-q,q\rangle\end{equation}
Noting that $U_0$, as given by \ref{eq33}), can be written as $U_0=\exp(i\pi J_1/2)$ and using the
relation $J_2=U_0 J_3 U_0^\dagger$, we also get
\begin{equation}gin{equation}\label{eq66}
J_2U_0\vert
N-j,j\rangle={N-2j\over 2}U_0\vert
N-j,j\rangle\end{equation}
For $\Theta=\phi=\pi/2$, the Hamiltonian reduces to $H=-2g J_2$
for which $U_0\vert N-j,j\rangle$ becomes an eigenstate by virtue
of (\ref{eq66}).
The condition $\Theta=0$ corresponds to $g=0$ and the Hamiltonian
reduces to $2\Omega J_3$. From (\ref{eq32}), (\ref{eq41}) and (\ref{eq49}), one then immediately obtains
$\tilde{C}^{(q)}_{Nj}=\exp(-i\Omega t[N-2q]) D^{(q)}_{Nj}$ so that $ \vert
\tilde{C}^{(q)}_{Nj}\vert^2$, and consequently, the entropy
$\tilde{S}^{(a)}_{Nj}$ become independent of time. The
corresponding
wave function is given by (\ref{eq55}) where the coefficients $\tilde{b}^{(n)}_{Nj}$ have
the value
\begin{equation}\label{eq67} \fl \tilde{b}^{(n)}_{Nj} = {N!\over j!} {N\choose j}^{-1/2} {N\choose n}^{-1/2}
(-i)^{N-n} (\sin\Omega t)^{N-n+j} (\cos\Omega t)^{n-j} f^{(n)}_{Nj}(\cos^2\Omega
t).\end{equation}
It is interesting that although $\vert \tilde{C}^{(q)}_{Nj}\vert^2$ and the
entropy $\tilde{S}^{(a)}_{Nj}$ remain constant for $\Theta=0$,
the initial
vortex state will continue to evolve with time as shown in Figure
4 \cite{oe}.
\begin{equation}gin{figure}
\setcaptionwidth{5.0in}
\centering
\includegraphics[width=5.0in]{fig4jb}
\caption{\label{f4}Time evolution of an initial vortex state for
$\Theta=0$ even though the entropy $\tilde{S}^{(a)}_{Nj}$ remains
constant. Shown here are the contour plots of the absolute square
(top row) and the phase (bottom row) of $\tilde{\psi}_{Nj}(x,y,t)$
as functions of $x$ and $y$ at different times with $b^{(n)}_{Nj}$
as given in (\ref{eq67}). Here, $N=4$, $j=0$ and $\cos^2\Omega t$
has the values (a) 1.0, (b) 0.9, (c) 0.5 and (d) 0.0. The
horizontal and the vertical axes refer to the $x$ and $y$
coordinates respectively. For the phase plots, we have used the
convention that the phase ranges from $-\pi$ to $\pi$. Note that
for $\cos^2\Omega t=0$ the state becomes the complex conjugate of
the initial state as the direction of phase change is reversed.}
\end{figure}
For $N=4$ and $j=0$, the initial state at $t=0$ (Figure 4(a))
corresponding to $\cos^2\Omega t=1.0$ is a vortex of order 4 and
charge -4 as given by equations (\ref{eq31}) and (\ref{eq32}).
Recall that $\Omega=\sigma \cos\Theta$. Thus $\Theta=0$
corresponds to $\Omega=\sigma$. Furthermore, if $\cos^2\Omega
t=0.0$, then, with $\Theta=0$, the conditions for charge
conjugation as given below equation (\ref{eq58}) are satisfied and
we obtain the complex conjugate of the initial vortex. For
$\cos^2\Omega t=0.5$, it is more convenient to use cartesian
co-ordinates. Using the expression for $\tilde{C}^{(q)}_{Nj}$ as
given above and the configuration space representation of number
states as given by (\ref{eq22}), one can use the summation theorem
for Hermite polynomials \cite{gradshteyn} to obtain \numparts
\begin{equation}gin{eqnarray}\label{eq681}\fl \tilde{\psi}_{40}(x,y,t)\rfloor_{\Omega
t=\frac{\pi}{4}} & = & - {e^{-(x^2+y^2)/2}\over \sqrt{\pi 2^4 4!}}
H_4\left(\frac{x-y}{\sqrt{2}}\right)\\& = & {e^{-(x^2+y^2)/2}\over
\sqrt{ 24 \pi}} [ -(x-y)^4+6 (x-y)^2 -3]\end{eqnarray}
\endnumparts
Thus, the wave function is a Gaussian modulated by a Hermite
polynomial. Clearly, its value is real and, therefore, its phase
is either zero or $\pi$ depending respectively on whether the wave
function is $\geq 0$ or negative. It is easy to show that the wave
function vanishes whenever $(x-y)^2=3\pm \sqrt{6}$. Finally, it
may be of some interest to realize that the wave function
corresponds to a SU(2) coherent state $-\vert \tau, N\rangle$ in
the Schwinger representation \cite{oe} with $\tau=-1$ and $N=4$.
\section{Structure of the reduced state\label{4}}
In order to determine the structure of the reduced state
for each mode, we first consider the correlation function in the
$x$-space of mode `a' given by $\langle x\vert
\rho^{(a)}_{Nj}\vert y\rangle$. A classical analog of this
function is the mutual coherence function of a partially coherent
source \cite{classvortex}. Thus the process of reduction of a pure
two-mode state into a mixed state by a partial trace operation
over one mode amounts to loss of coherence and information. We can
also define the spatial coherence function $\gamma^{(a)}_{Nj}(l)$
for the reduced state by \begin{equation}gin{equation}\label{eq68}
\gamma^{(a)}_{Nj}(l)= \int \langle x\vert \rho^{(a)}_{Nj}\vert
x+l\rangle \,dx\end{equation}
When the system is initially in a two-mode Fock state, one obtains
\begin{equation}gin{equation}\label{eq69} \langle x\vert \rho^{(a)}_{Nj}\vert
y\rangle=\sum_{q=0}^N {\vert C^{(q)}_{Nj}\vert^2 \over 2^q q!
\sqrt{\pi}} e^{-(x^2+y^2)/2} H_q(x) H_q(y).\end{equation} The
corresponding expression for $\gamma^{(a)}_{Nj}(l)$ is obtained by
evaluating the standard integral \cite{gradshteyn} in
(\ref{eq68}). We get
\begin{equation}gin{equation}\label{eq70} \gamma^{(a)}_{Nj}(l)= \sum_{q=0}^N \vert
C^{(q)}_{Nj}\vert^2 e^{-l^2/4} L_q(l^2/2)\end{equation} where
$L_q(x)=L_q^0(x)$ is a Laguerre polynomial. Note that only one
term survives in the summations over $q$ by virtue of (\ref{eq19})
whenever $\vert v_{21}\vert^2=0$ or $1$. Thus
\begin{equation}gin{equation}\label{eq71} \langle x\vert \rho^{(a)}_{Nj}\vert
y\rangle ={e^{-(x^2+y^2)/2}\over \sqrt{\pi}}
\left\{\begin{equation}gin{array}{ll}
{H_j(x) H_j(y)\over 2^j j!}, & \mbox{$|v_{21}|^2\to 0$,}\\
\rule{0mm}{6mm}
{H_{N-j}(x) H_{N-j}(y)\over 2^{N-j} (N-j)!}, &
\mbox{$|v_{21}|^2\to 1$.}
\end{array}
\right. \end{equation} and \begin{equation}gin{equation}\label{eq72}
\gamma^{(a)}_{Nj}(l)=e^{-l^2/4}\left\{\begin{equation}gin{array}{ll}
L_j(l^2/2), & \mbox{$|v_{21}|^2\to 0$,}\\
\rule{0mm}{6mm}
L_{N-j}(l^2/2), & \mbox{$|v_{21}|^2\to 1$.}
\end{array}
\right. \end{equation} Furthermore, one can use Eq. (\ref{eq20})
and the definition (\ref{eq38}) to get \numparts \begin{equation}gin{eqnarray}
\label{eq73} \langle x\vert \rho^{(a)}_{Nj}\vert
y\rangle\rfloor_{|v_{21}|^2\to 1-R} & = & \langle
x\vert\rho^{(a)}_{N,N-j}\vert y\rangle\rfloor_{|v_{21}|^2\to R}
\\\label{eq74} \gamma^{(a)}_{Nj}(l)\rfloor_{|v_{21}|^2\to 1-R} & = &
\gamma^{(a)}_{N,N-j}\rfloor_{|v_{21}|^2\to
R}\end{eqnarray}\endnumparts
When the system is initially in a vortex state, Eqs.
(\ref{eq69}-\ref{eq74}) are still valid provided that $|v_{21}|^2$
is replaced by $|\tilde{v}_{21}|^2$.
\begin{equation}gin{figure}
\setcaptionwidth{5.0in}
\centering
\includegraphics[width=5.0in]{fig5jb}
\caption{\label{f5}Contour plots of the correlation function as a
function of $x$ and $y$ for different values of $|v_{21}|^2$. The
parameters are as follows: $N=8$; $j=0$ (top row), $j=4$ (bottom
row); $|v_{21}|^2$ has the values (a) $1.0$, (b) $0.9$, (c) $0.5$,
(d) $0.1$ and (e) $0.0$. The horizontal and the vertical axes
refer to the $x$ and $y$ coordinates respectively.}
\end{figure}
In Fig. 5 we present contour plots of the correlation function as
a function of $x$ and $y$ for a set of values of $|v_{21}|^2$
when $N=8$ and $j=0$ (top
row), $j=4$ (bottom row). The intricate patterns for
$|v_{21}|^2=0$ and $1$ can be explained by using Eq. (\ref{eq71}).
Furthermore, the identical nature of patterns for $N=8$, $j=4$ on
either side of $|v_{21}|^2=1/2$ can be attributed to the property
(\ref{eq73}). Finally in Fig. 6 we plot $\gamma^{(a)}_{Nj}(l)$ as
a function of $l$ and $|v_{21}|^2$ when $N=4$ and $j=0$, $2$. The
patterns for $|v_{21}|^2=0$ and $1$ follow from Eq. (\ref{eq72})
and the symmetry of the plot for $j=2$ about $|v_{21}|^2=1/2$
follows from (\ref{eq74}).
\begin{equation}gin{figure}
\setcaptionwidth{5.0in}
\centering
\includegraphics[width=5.0in]{fig6jb}
\caption{\label{f6} Contour plot of $\gamma^{(a)}_{Nj}(l)$ as a
function of $l$ and $|v_{21}|^2$ when $N=4$ and $j=0$ (left),
$j=2$ (right).}
\end{figure}
\section{Conclusion\label{5}}
In conclusion, we have studied, in a general way, entanglement
produced in a two-mode bosonic system by linear SU(2)
transformations leading to the generation and evolution of quantum
vortex states. The linear SU(2) transformations are generated by
evolving the system under the action of a generic Hamiltonian that
mimics a variety of entanglement mechanisms. We have demonstrated
that these transformations produce a coherent superposition of
quantum vortices in general, and a single quantum vortex under
certain conditions. Furthermore, as one would expect, a vortex state
is found to be an entagled state. When the system is a light field,
the vortex will appear in the quadrature distribution that can be
measured by a homodyne method \cite{leonhardt}. Explicit analytical
results were obtained when the system was initially either in a Fock
state or in a quantum vortex state. In the latter case, we have also
found conditions for its revival and charge conjugation. A simple
recipe was provided to accommodate all other cases for which the
initial state can be reached from a Fock state by a unitary
transformation. Thus we not only provide a unified approach to
entanglement through a generic Hamiltonian but also predict wide
applicability of our results to a variety of initial states.
The ideas developed in this paper can be applied not only to light
fields but also to matter waves such as the Bose Einstein
condensates (BEC). In recent years, the BEC has proved to be an
excellent laboratory for studying (both bipartite and many-particle)
entanglement \cite{bec1,bec2,bec3,deb2,deb3}. The entanglement of
the modes as well as the entanglement of the atoms in a BEC have
been considered. We mention parenthetically that our work is
relevant in the former case. It is also well known that several
mechanisms exist for the generation of vortices in a BEC
\cite{becv,schleich1,schleich2,kapale}. Additionally, Whyte et al.
\cite{whyte} have used the similarity between BECs and laser light
to propose a method for generating Hermite-Gaussian type modes in a
so-called {\it light pulse resonator}. Thus it should indeed be
possible to generate vortices in a two-component BEC by entangling
the two modes of the BEC by linear SU(2) transformations via an
entangling device such as a beam splitter.
\ack
One of us (GSA) would like to thank NSF for supporting this
work under grant no. CCF-0524673.
\appendix
\section{Some useful properties of $\vert C^{(q)}_{Nj}\vert^2$ }
Using Leibniz' rule for the $j$-th derivative of
a product and the relations (\ref{eq7}), one obtains
\begin{equation}\label{eq17} \fl \vert C^{(q)}_{Nj}\vert^2 =
(N-j)! (N-q)! q! (j!)^{-1} (1-|v_{21}|^2)^N
\left({|v_{21}|^2\over 1-|v_{21}|^2}\right)^{q-j}\vert
f^{(q)}_{Nj}(|v_{21}|^2)\vert^2\end{equation} with
\begin{equation}gin{equation}\label{eq18} f^{(q)}_{Nj}(|v_{21}|^2)=\sum_{k=0}^j {(-1)^k
{j\choose k} \over (N-q-k)! (q-j+k)!}\left({|v_{21}|^2\over
1-|v_{21}|^2}\right)^k .\end{equation} It is easy to show that
\begin{equation}gin{equation}\label{eq19} \vert C^{(q)}_{Nj}\vert^2 =
\left\{\begin{equation}gin{array}{ll}
\end{equation*}lta_{q,j}, & \mbox{$|v_{21}|^2\to 0$,}\\ \rule{0mm}{6mm}
\end{equation*}lta_{q,N-j}, & \mbox{$|v_{21}|^2\to 1$.}
\end{array}
\right. \end{equation} Next we derive some important symmetry
properties of $\vert C^{(q)}_{Nj}\vert^2$. First we show that
\begin{equation}gin{eqnarray*} f^{(q)}_{Nj}(1-R) & = & (-1)^j\left({1-R\over
R}\right)^j f^{(N-q)}_{Nj}(R)\\ f^{(q)}_{N,N-j}(R) & = &
(-1)^{N-q}{(N-j)!\over j!}\left({R\over
1-R}\right)^{N-q}f^{(q)}_{Nj}(1-R)\end{eqnarray*} where $0\leq
R\leq 1$. The first relation is obtained from (\ref{eq18}) by
changing the summation index from $k$ to $j-k$ and the second
relation is proved by exploiting the non-negativity of the
factorials in (\ref{eq18}) and changing the summation range
accordingly. Using these two relations we immediately get \begin{equation}
\label{eq20}\vert C^{(q)}_{Nj}\vert ^2\rfloor_{|v_{21}|^2\to 1-R}
= \vert C^{(N-q)}_{Nj}\vert ^2\rfloor_{|v_{21}|^2\to
R}= \vert C^{(q)}_{N,N-j}\vert ^2\rfloor_{|v_{21}|^2\to R}.\end{equation}
Note that if $|v_{21}|^2=1/2$, then
$|v_{11}|^2=|v_{22}|^2=|v_{12}|^2=1/2$ as well. Additionally, if
$j=0$ or $N$, then $\vert C^{(q)}_{Nj}\vert^2=2^{-N} {N\choose
q}$ is a binomial distribution whereas if $j=N/2$, then
\begin{equation}gin{equation}\label{eq21} \vert
C^{(q)}_{N/2,N/2}\vert^2 = (N!)^{-1}[(N/2)! P_{N/2}^{(N/2
-q,q-N/2)}(0)]^2 {N\choose q} \end{equation} where
$P_n^{(\alpha,\begin{equation}ta)}(x)$ is a Jacobi polynomial.
\section{Derivation of equation (\ref{eq28})}
Substituting the expression (\ref{eq16}) for $C^{(q)}_{Nj}$ in
(\ref{eq23}) and performing the summation over $q$ {\it before}
differentiation with respect to $\tau$, we get \begin{equation}\label{eq24} \fl
\psi_{Nj}(x,y,t)= {(-1)^N\over N!} \sqrt{{(N-j)!\over j! 2^N
\pi}}e^{(x^2+y^2)/2} [\partial_\tau^{(j)}
\hat{A}^N(\tau)e^{-(x^2+y^2)}]_{\tau\to 0}\end{equation} where
\begin{equation}gin{equation}\label{eq25}\hat{A}(\tau)=(v_{11}
+v_{12}\tau)\partial_x+(v_{21}
+v_{22}\tau)\partial_y.\end{equation} We introduce $z=x+iy$ so
that $x^2+y^2=zz^*$ and
$\hat{A}(\tau)=\gamma_{+}(\tau)\partial_z+\gamma_{-}(\tau)\partial_{z^*}$
with $\gamma_{\pm}$ given by (\ref{eq26}). Next we expand
$\hat{A}(\tau)^N$ binomially and then use the relation
\begin{equation}gin{equation}\label{eq27}
\partial^m_z\partial^n_{z^*} e^{-zz^*}=\left\{\begin{equation}gin{array}{ll}
(-1)^n m! e^{-zz^*} z^{n-m}L_m^{n-m}(zz^*), & \mbox{$m\leq
n$,}\\
\rule{0mm}{6mm}
(-1)^m n! e^{-zz^*} {z^*}^{m-n}L_n^{m-n}(zz^*), & \mbox{$n\leq
m$.}
\end{array}
\right. \end{equation} to evaluate $\hat{A}(\tau)^N
e^{-(x^2+y^2)}$. Collecting all the terms we finally obtain
(\ref{eq28}).
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\end{document} |
\begin{document}
\title{General mapping of multi-qu$d$it entanglement
conditions to non-separability indicators for quantum optical fields}
\author{Junghee Ryu}
\affiliation{Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore}
\author{Bianka Woloncewicz}
\affiliation{International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdansk, Poland}
\author{Marcin Marciniak}
\affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'{n}sk, 80-308 Gda\'{n}sk, Poland}
\author{Marcin Wie\'{s}niak}
\affiliation{Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gda\'{n}sk, 80-308 Gda\'{n}sk, Poland}
\affiliation{International Centre for Theory of Quantum Technologies (ICTQT),
University of Gdansk, 80-308 Gdansk, Poland}
\author{Marek \.{Z}ukowski}
\affiliation{International Centre for Theory of Quantum Technologies (ICTQT),
University of Gdansk, 80-308 Gdansk, Poland}
\date{\today}
\begin{abstract}
We show that any multi-qudit entanglement witness leads to a non-separability indicator for quantum optical fields, which involves intensity correlations. We get, e.g., {\it necessary and sufficient} conditions for intensity or intensity-rate correlations to reveal polarization entanglement. We also derive separability conditions for experiments involving multiport interferometers, now feasible with integrated optics. We show advantages of using intensity rates rather than intensities, e.g., a mapping of Bell inequalities to ones for optical fields. The results have implication for studies of non-classicality of ``macroscopic" systems of undefined or uncontrollable number of ``particles".
\end{abstract}
\maketitle
\newcommand{\avg}[1]{\langle #1\rangle}
\newcommand{\cre}[1]{a^{\dagger}_{#1}}
\newcommand{\ani}[1]{a_{#1}}
\newcommand{\inner}[2]{\langle#1 | #2 \rangle}
Non-classicality
due to entanglement initially was studied using quantum optical
multiphoton interferometry, see e.g.,~\cite{PAN}.
The experiments were constrained to defined photon number
states, e.g., the two-photon polarization singlet~\cite{ASPECT}.
This includes Greenberger-Horne-Zeilinger (GHZ)~\cite{GHZ} inspired multiphoton interference,
with an interpretation that each detection event signals one photon.
Spurious events of higher photon number counts contributed to a lower interferometric
contrast.
Still, states of undefined photon numbers, e.g., the
squeezed vacuum, can be entangled~\cite{BANASZEK, BOUW, MASZA-REVIEW}.
This form of entanglement of quantum optical fields served e.g., to show that a strongly pumped two-mode (bright) squeezed state allows one
to directly refute the ideas of EPR \cite{EPR}, as it approximates their state, and a form of Bell's Theorem can be shown for it \cite{BANASZEK}. The trick was to use displaced parity observables.
Recently it has been shown that this is also possible for four-mode bright squeezed vacuum \cite{ROSOLEK}, which can be produced via type II parametric down-conversion, see e.g \cite{BOUW, MASZA-REVIEW}. In this case the state approximates a tensor product of two EPR states, and interestingly can also be thought of as a polarization ``super-singlet" of undefined photon numbers \cite{DURKIN}. The approach of Ref. \cite{ROSOLEK} used (effectively) intensity observables, which are less experimentally cumbersome.
With the birth of quantum information science and technology, entanglement became a resource. We have an extended literature on detection of entanglement for systems of finite dimensions, essentially ``particles", see e.g., \cite{HORODECKIS}. It is well known that not all entangled states violate Bell inequalities.
Still there is theory of entanglement indicators, called usually witnesses, which allow to detect entanglement, even if a given state for finite dimensional systems (essentially, qu$d$its) does not violate any known Bell inequalities. The case of two-mode entanglement for optical fields was studied in trailblazing papers of
\cite{SIMON, DUAN}, which discussed ``two-party continuous variable systems", and with a direct quantum optical formalism in \cite{HILLERY}. The entanglement conditions reached in the papers did not involve intensity correlations.
An entanglement condition for four-mode fields, which was borrowing ideas from two spin-1/2 (two-qubit) correlations, involved correlations Stokes operators and was first discussed in \cite{BOUW}. The resulting indicator was used to measure efficiency of an ``entanglement laser".
The output of the ``laser" was bright four-mode vacuum.
We shall present here the most extensive generalization of such an approach, i.e., entanglement indicators for optical fields which are derivatives of multi-qudit entanglement witnesses involving intensity correlations.
In Supplementary Material~\cite{supp} we give examples of entanglement conditions based on such an approach. Some of them are more tight versions of the entanglement conditions mentioned above.
As a growing part of the experimental effort is now directed at non-classical
features of bright (intensive, ``macroscopic") beams of light, e.g., \cite{Lamas01, BOUW-2, Eckstein11, Iskhakov12, Iskhakov13, Kanseri13, Spasibko17} so the time is ripe for a comprehensive study of such entanglement conditions. All that may lead to some new schemes in quantum communication and quantum cryptography, perhaps on the lines of Ref. \cite{DURKIN}.
The emergence of integrated optics allows now to construct stable multiport interferometers \cite{Mattle95, Weihs96, Peruzzo11, Meany12, Metcalf13, Spagnolo13, Carolan15, Schaeff15}, and is our motivation of going beyond two times mode case.
We present a theory of
mapping multi-qu$d$it entanglement witnesses~\cite{HORODECKIS}
into entanglement indicators for quantum optical fields,
which employ intensity
correlations or correlations of intensity rates. By intensity rates we mean the ratio of intensity at a given local detector and the sum of intensities at all local detectors (in some case the second approach leads to better entanglement detection). The method may find applications also in studies of non-classicality of correlations in ``macroscopic" many-body quantum systems of undefined or uncontrollable
number of constituents, e.g., Bose-Einstein condensates \cite{SORENSEN}, other specific states of cold atoms \cite{POLZIK,sorensenPRL}.
The essential ideas are presented
for polarization measurements by two observers and the most simple model of
intensity observable: photon-number in the observed mode.
Next, we present further generalization of our approach, and examples employing
specific indicators involving intensity correlations
for unbiased multiport interferometers.
We discuss generalizations
to multi-party
entanglement
indicators.
We show that the use of rates leads to a modification of quantum optical
Glauber correlation functions, which gives a new
tool for studying non-classicality, and that it also gives a
general method of mapping
standard Bell inequalities into ones
for optical fields.
\begin{figure}
\caption{The experiments (two parties). Two multi-mode beams propagate to two spatially separated measurement stations.
Each station consists of a $d$ input $d$ output tunable multi-port beamsplitter-interferometer (MPBS) and detectors at its outputs.
For polarization measurements put $d_A=d_B=2$, and treat the paths as polarization modes.}
\label{fig:exp}
\end{figure}
We discuss spatially
separated stations, $X=A,B,...$
with
(passive) interferometers
of $d_X$ input and output ports, FIG.~\ref{fig:exp}.
In each output there is a detector which measures
intensity.
One can assume either a pulsed source,
sources acting synchronously~\cite{ZUK-ZEIL-WEIN, KALTENBAEK}
or that the measurement is performed within a short time gate.
Each time gate, or pulsed emission,
is treated as a repetition of the experiment building up averages of observables.
{\it Stokes parameters.}---For the
description of polarization of light,
the standard approach uses Stokes parameters.
Using the photon numbers they read
$
\langle \hat \Theta_{j}\rangle = \langle \hat{a}_{j}^\dagger \hat a_{j} - \hat a_{{j}_{\perp}}^\dagger \hat a_{{j}_{\perp}}\rangle,
$
where $j, j_{\perp}$
denote a pair of orthogonal
polarizations of one of three mutually unbiased
polarization bases $j = 1,2,3 $, e.g.,
$\{ {H}, {V}\}, \{ {45^{\circ}}, {-45^{\circ}}\}, \{ {R}, {L}\}$.
The zeroth parameter $
\langle \hat \Theta_0 \rangle $ is the total intensity:
$\langle \hat N\rangle = \langle \hat a_{j}^\dagger \hat a_{j} + \hat a_{{j}_\perp}^\dagger \hat a_{{j}_\perp} \rangle.$
Alternative {\it normalized} Stokes observables were studied
by some of us~\cite{zuko, zuko1, zuko2}. They were first intorduced in \cite{HE}, however a different technical approach was used.
Following \cite{zuko} one can put
$
\langle \hat S_{j} \rangle = \langle \hat \Pi\frac{(\hat a_{j}^\dagger \hat a_{j} - \hat a_{{j}_\perp}^\dagger \hat a_{{j}_\perp})}{\hat N} \hat \Pi\rangle,
$
and $\langle \hat S_0 \rangle = \langle \hat \Pi\rangle$,
where $\hat \Pi = \mathbb{1} - \ketbra{\Omega}$ and $|\Omega\rangle$
is the vacuum
state for the considered modes,
$\hat{a}_j|\Omega\rangle = \hat{a}_{j_{\perp}}|\Omega\rangle =0$.
Operationally, in the $r$-th run of an experiment, we register photon numbers in the two exits of a polarization analyzer, $n^r_j$ and $n^r_{j_\perp}$, and divide their difference by their sum. If $n^r_j + n^r_{j_\perp}=0$ , the value is put as zero. This does not require any additional measurements, only the data are differently processed than in the standard approach. In~\cite{zuko, zuko1, zuko2} examples of two-party entanglement conditions and Bell inequalities using normalized Stokes operators were given. Here we present a general approach.
{\it Map from two-qubit entanglement witnesses
to entanglement indicators for fields involving Stokes parameters.}---Pauli operators
$ \vec{\sigma}
= (\hat \sigma_1, \hat \sigma_2, \hat \sigma_3)$ and
$\hat \sigma_0 = \mathbb{1}$
form a basis in the real space of one-qubit observables.
Thus, any two-qubit entanglement
witness, $\hat{W}$,
has the following expansion:
$\hat W = \sum_{\mu,\nu}w_{\mu\nu}\hat \sigma_{\mu}^A\otimes \hat \sigma_{\nu}^B$, where $\mu,\nu = 0,1,2,3$ and $w_{\mu\nu}$ are real coefficients. We have $\langle \hat{W}\rangle_{sep}\geq 0$, where $\langle \cdot \rangle_{sep}$ denotes an average for a separable state.
We will show that with each witness $\hat W$
one can
associate entanglement indicators
for polarization measurements involving correlations of Stokes observables for quantum optical fields.
The maps are
$
\hat \sigma_{\mu}^A \otimes \hat{\sigma}_{\nu}^B \to \hat S_{\mu}^A\hat S_{\nu}^B
$
and
$
\hat \sigma_{\mu}^A \otimes \hat{\sigma}_{\nu}^B \to \hat \Theta_{\mu}^A\hat \Theta_{\nu}^B
$,
and they link $\hat{W}$ with its quantum optical analogues
$
\hat{\mathcal{W}}_{S} = \sum_{\mu,\nu}w_{\mu\nu}\hat S_{\mu}^A\hat S_{\nu}^B,
$
and
$
\hat{\mathcal{W}}_ {\Theta} = \sum_{\mu,\nu}w_{\mu\nu}\hat \Theta_{\mu}^A\hat \Theta_{\nu}^B,
$
which fulfill
$\langle \hat{\mathcal{W}}_{S} \rangle_{sep}\geq 0$
and $\langle \hat{\mathcal{W}}_{\Theta} \rangle_{sep}\geq 0$. The proof goes as follows.
{\it Normalized Stokes operators case.}---It is enough to prove that for any mixed state
$\varrho$
one can find a $4 \times 4$
density matrix $\mathbf{\mathfrak{\hat R}}^{AB}_\varrho$
for a pair of qubits, such that:
\begin{equation}
\label{EQUALIZO1}
\frac{\langle{\hat{\mathcal{W}}_S}\rangle _\varrho}{\langle{\hat \Pi^A\hat \Pi^B}\rangle_\varrho}=\Tr \hat W \mathbf{\mathfrak{\hat R}}^{AB}_\varrho.
\end{equation}
First, we show that (\ref{EQUALIZO1}) holds for any pure state $\ket{\psi^{AB}}$.
Let us denote the polarization basis $H$ and $V$ as
$\hat x_H = \hat x_1$ and $\hat x_V = \hat x_2$.
Normalized Stokes operators in arbitrary direction can be put as
$\vec{m}\cdot \vec{S}^{X}$, where
$\vec{m}$ is an arbitrary unit real vector, or in the matrix form
$
\sum_{kl}\hat \Pi^X \frac{{ {\hat {x}}_k^{\dagger}}
(\vec{m}\cdot \vec{\sigma})_{kl}{{ \hat {x}_l}}}{\hat N^X}\hat \Pi^X,
$ with $\hat{x}=\hat{a}$ or $\hat{b}$ depending on the beam $X$, whereas
$\hat{S}_0^X$ reads
$
\sum_{kl}\hat \Pi^X \frac{{ {\hat {x}}_k^{\dagger}} \delta_{kl}
{{ \hat {x}_l}}}{\hat N^X}\hat \Pi^X.
$
We introduce a set of states
\begin{eqnarray}
\label{super_psi}
\ket{\mathbf{\Psi}_{km}^{AB}} =
\hat{a}_k\hat{b}_m\frac{1}{\sqrt{\hat N^A\hat N^B}} \hat \Pi^A\hat \Pi^B \ket{\psi^{AB}},
\end{eqnarray}
where $k, m \in \{1,2 \}$. This allows us to put
\begin{eqnarray}
\label{homomulti}
\expval{\hat S_{\mu}^{A} \hat S_{\nu}^B}{\psi^{AB}}
&=& \sum_{k,l=1}^2\sum_{m,n=1}^2\sigma^{kl}_{\mu }
\sigma^{mn}_{\nu }\bra{\mathbf{\Psi}^{AB}_{km}}\ket{
\mathbf{\Psi}^{AB}_{ln}} \nonumber \\
&=& \Tr\hat
\sigma_{{\mu}}^A \otimes
\hat \sigma_{{\nu}}^B
\hat R^{AB}_\psi,
\end{eqnarray}
where the matrix elements of
$\hat R^{AB}_\psi $ are $\bra{\mathbf{\Psi}^{AB}_{km}}\ket{\mathbf{\Psi}^{AB}_{ln}}$.
As a Gramian matrix, ${\hat {R}}^{AB}_\psi$ is positive. Except for $|\psi^{AB}\rangle$ describing
vacuum at one or both sides, we have
$0<\Tr\hat R^{AB}_\psi= \langle \hat \Pi^A \hat \Pi^B \rangle \leq 1$.
Thus, $\mathbf{\mathfrak{\hat R}}^{AB}_\psi=
\hat R^{AB}_\psi/\langle \hat \Pi^A \hat \Pi^B\rangle$ is an admissible
density matrix of two qubits.
For mixed states $\varrho$, i.e., convex combinations of $\ket{\psi^{AB}_{\lambda}}$'s with weights $p_\lambda$, one gets
$\hat{R}^{AB}_\varrho= \sum_{\lambda}p_{\lambda}{\hat{R}}^{AB}_{\lambda}$ which is positive definite, and its trace is
$ \sum_{\lambda}p_{\lambda}\Tr \hat{R}^{AB}_{\lambda}\leq 1$. Thus after the re-normalization one gets a proper
two-qubit density matrix $ \mathbf{\mathfrak{\hat R}}^{AB}_\varrho$.
As purity of a field state $\ket{\psi^{AB}_{\lambda}}$ does not warrant that the corresponding $\hat{R}^{AB}_{\lambda}$ is a projector, $ \mathbf{\mathfrak{\hat R}}^{AB}_\varrho$ does not have to have the same convex expansion coefficients in terms of pure two-qubit states, as $\varrho$ in terms of $\ket{\psi^{AB}_{\lambda}}$'s.
For any separable pure state of two optical beams
$|\psi^{AB}\rangle_{prod}$, defined
as $F^\dagger_A F^\dagger_B|\Omega\rangle$,
where $F^\dagger_X$ is a polynomial function of
creation operators
for beam (modes) $X$, and $|\Omega\rangle$ is the vacuum state of both beams,
the matrix $\hat R^{AB}$ factorizes:
$\hat R^{AB} = \hat R^{A}
\hat R^{B}$.
Simply, ${}_{prod}\bra{\mathbf{\Psi}^{AB}_{km}}\ket{\mathbf{\Psi}^{AB}_{ln}}_{prod}$ factorizes to
$
\bra{\mathbf{\Psi}^{A}_{k}}\ket{\mathbf{\Psi}^{A}_{l}} \bra{\mathbf{\Psi}^{B}_{m}}\ket{\mathbf{\Psi}^{B}_{n}},
$
where
$\bra{\mathbf{\Psi}^{X}_{k}}\ket{\mathbf{\Psi}^{X}_{l}}$
are elements of matrix $\hat R^{X}$
and $
|\mathbf{\Psi}_{l}^{X}\rangle =
\hat{x}_l\frac{1}{\sqrt{\hat N^X}} \hat{\Pi}^X F^{\dagger}_X|\Omega\rangle.
$
As
$\langle \Omega|F_X \hat{\Pi}^XF^{\dagger}_X|\Omega\rangle^{-1}\hat R^{X}$
can be shown to be
a qubit density matrix and $\langle\hat{W}\rangle_{sep}\geq 0$,
therefore for pure separable states of the optical
beams $\langle\hat{\mathcal{W}}_S\rangle_{prod}\geq 0 $. Obviously,
$\langle \mathcal{\hat W}_S \rangle_{sep}\geq 0 $ also for all mixed separable states.
{\it Standard Stokes operators case.}--- Any standard Stokes operator can be put as
$\vec{m} \cdot \vec{\Theta}^X =
\sum_{kl}
\hat{x}_k^{\dagger}
(\vec{m}\cdot \vec{\sigma})_{kl}
\hat{x}_l.
$
We introduce state vectors
$\ket{\mathbf{\Phi}_{jk}^{AB}} = \hat a_{j}\hat b_k \ket{\psi^{AB}}$.
One has
\begin{eqnarray}
\expval{\hat \Theta_{\mu}^A \hat \Theta_{\nu}^B}{\psi^{AB}} = \Tr \hat \sigma^A_{\mu}
\hat \sigma^B_{\nu} \hat P^{AB},
\end{eqnarray}
where the matrix $\hat P^{AB}$
has entries $\bra{\mathbf{\Phi}_{km}^{AB}}\ket{\mathbf{\Phi}_{ln}^{AB}}$,
it
is {positive definite},
and its trace is $ \langle \hat N_A \hat N_B \rangle.$
Thus, $\mathbf{\mathfrak{\hat P}}^{AB} = \hat P^{AB}/\langle \hat N^A \hat N^B\rangle$ is an admissible two-qubit density matrix, and one has
$
\langle \hat{\mathcal{W}}_\Theta\rangle_\varrho/\langle \hat N ^A\hat N^B\rangle_\varrho = \Tr \hat W
\mathbf{\mathfrak{\hat P}}^{AB}_\varrho.
$
All that leads to $\langle \hat{\mathcal{W}}_\Theta\rangle_{sep}\geq 0$.
Note that, for a general
state $\mathbf{\mathfrak{\hat R}}^{AB}_\varrho$
does not have to be equal to $ \mathbf{\mathfrak{\hat P}}^{AB}_\varrho$. Still,
$\mathbf{\mathfrak{\hat R}}^{AB} = \mathbf{\mathfrak{\hat P}}^{AB}$ for states of defined photon numbers in both beams.
{\it Reverse map.}---
Any linear separability condition expressible in terms of correlation functions of normalized Stokes Parameters
reads:
$
\sum_{\mu\nu} \omega_{\mu\nu}\langle \hat{S}^A_\mu \hat{S}^B_\nu\rangle_{sep} \geq 0.
$
As two-photon states, with one at A and the other at B, are possible field states, thus for any separable such state we must have
$
\sum_{\mu\nu} \omega_{\mu\nu}\langle \hat{S}^A_\mu \hat{S}^B_\nu\rangle_{sep-2-ph} \geq 0.
$
This is algebraically equivalent to
$
\sum_{\mu\nu} \omega_{\mu\nu}\langle \hat{\sigma}_\mu \otimes \hat{\sigma}_\nu\rangle_{sep} \geq 0,
$
for any two-qubit state. We get an entanglement witness. Therefore, we have an isomorphism. Similar proof applies to standard Stokes observables.
{\it Examples.}---In the Supplemental Material~\cite{supp},
we show some examples of entanglement indicators which can be derived with the above method.
This includes a necessary and sufficient conditions for detection of entanglement of two optical beams with correlations of Stokes parameters of the two considered kinds.
{\it Detection losses.}---
Consider the usual model of losses:
a perfect detector in front of which is a beamsplitter of transmission amplitude ${\eta}$, with the reflection channel describing the losses.
Then,
$\langle \hat \Theta_{\mu}^A\hat \Theta_{\nu}^B\rangle$
scales down as $\eta^A\eta^B$ (see Sec. II in the Supplemental Material~\cite{supp}), where
$\eta^X$ for $X=A,B$ is the local detection efficiency.
We have a full resistance of entanglement
detection, using any $\hat{\mathcal{W}}_\Theta$, with respect to such losses.
A different character of
losses may lead to threshold efficiencies.
For the normalized Stokes parameters, it is enough to consider only pure states,
because mixed ones, as convex combinations of such,
cannot introduce anything new in
entanglement conditions linear with respect to the density matrix.
Any pure state is a superposition of Fock states $|F\rangle= |n_{i}^{A},n_{i_\perp}^{A}, n_{j}^{B},n_{j_\perp}^{B} \rangle$, where $n_{i}^{X}$ denotes the number of $i$ polarized photons in beam $X$, and $\hat{S}_{\mu}^A\hat S_{\nu}^B$ are diagonal with respect to the Fock basis related with them. Thus, the dependence on efficiencies of the value of an entanglement indicator, in the case of a pure state, depends on the behavior of its Fock components. One can show, see Sec. II in the Supplemental Material~\cite{supp},
that
$
\label{EFFICIENCY}
{\langle F_\eta|\hat {S}_{\mu}^A\hat{ S}_{\nu}^B |F_\eta \rangle} = H_F
{\langle F|\hat {S}_{\mu}^A\hat{ S}_{\nu}^B |F\rangle},
$
where $|F_\eta\rangle$ is the state $|F\rangle$ after the above described losses in both channels, and
$H_F=\langle F_\eta| \hat{S}_{0}^A\hat S_{0}^B |F_\eta \rangle $, which reads $\prod_{X=A,B}[1-(1-\eta^X)^{m^X}]$, where $m^X$
is the total number of photons in channel $X$, before the losses.
Expanding $|F\rangle$ in terms of Fock states with respect to different polarizations than $i,i_\perp$
and $j,j_\perp$, does not change the values of $m^X$, and
thus the formula stays put for any indices.
Again we have a strong resistance of the
entanglement indicators with respect to losses. Especially for states with high photon numbers, the entanglement conditions based on normalized Stokes
parameters, may be more resistant to losses, because $0<\eta < 1$ one has $\eta<1-(1-\eta)^n$.
{\it Multi-party case.}--- Consider three parties, and the case of indicators of genuine three-beam entanglement. Any genuine three-qubit entanglement witness $\hat{W}^{(3)}$ has the property that it is positive for pure product three-qubit states $|\xi\rangle_{AB,C}=|\psi\rangle_{AB}|\phi\rangle_C$, for similar ones with qubits permuted, and for all convex combinations of such states. With any pure partial product state of the optical beams, e.g. $|\Xi\rangle_{AB,C}=F^\dagger _{AB} F^\dagger_C|\Omega\rangle $, where $F^\dagger_{AB}$ is an operator built of creation operators for beams $A$ and $B$, etc., one can associate, in a similar way as above,
a partially factorizable three-qubit density matrix $\mathfrak{\hat{R}}^{AB}_\psi\mathfrak{\hat{R}}^{C}_\phi$.
Thus, the homomorphism works. Generalizations are obvious.
{\it General Theory.}---Consider a beam of $d_A$ quantum
optical modes propagating toward a
measuring station $A$, and a beam of $d_B$ modes toward station $B$. We associate with
the situation a $d_A \times d_B$ dimensional Hilbert Space, $\mathbb{C}^{d_A} \otimes \mathbb{C}^{d_B}$, which contains pure states of a pair of qudits
of dimensions $d_A$ and $d_B$.
For $X=A,B$, let $\hat V_i^X$, with $i=1,...,d_X^2$, be an orthonormal, i.e. $\Tr \hat V_i^X \hat V_j^X=\delta_{ij}$, Hermitian basis of the space of Hermitian operators acting on $\mathbb{C}^{d_X}$.
Therefore, products $\hat V_i^A\otimes \hat V_j^B$ form an orthonormal basis of the space of Hermitian operators acting on $\mathbb{C}^{ d_A}\otimes \mathbb{C}^{d_B}$.
Thus, any entanglement witness for the pair of qudits, $\hat W$, can be expanded into
\begin{equation}\label{EXPANSION}
\hat W=\sum_{j=1}^{d_A^2}\sum_{k=1}^{d_B^2}w_{jk}\hat V_j^A\otimes
\hat V_k^B,
\end{equation}
with real $w_{jk}$. The optimal
expansion (with the minimal number of terms) is to use a Schmidt basis for $\hat{W}$.
Each $\hat V^X_j$
can be decomposed to a linear combination of
its spectral projections linked with their respective eigenbases, $|x^{(j)}_l\rangle$,
where $x=a$ or $b$
consistently with $X$ and $l=1,..., d_X$. If one fixes a certain pair of bases in
$\mathbb{C}^{d_A}$ and $\mathbb{C}^{d_B}$ as ``computational ones", i.e., starting ones, denoted as $|l_x\rangle$, one can always find local unitary matrices $U^X(j)$ such that $U^X(j)|l_x\rangle= |x^{(j)}_l\rangle$. The construction of Reck et al.~\cite{RECK} fixes (phases in) a local multiport interferometer, which performs such a transformation. We shall call such interferometers $U^X(j)$ ones.
In the case of field modes a passive interferometer performs the following mode transformation:
$\sum_{k} U^X(j)_{lk}\hat x^\dagger_k= \hat x^\dagger_{l}(j)$, where $\hat x^\dagger_{l}(j)$ is the photon creation operator in the $l$-th exit mode of interferometer $U^X(j)$.
A two-party entanglement witness $\hat {\mathcal{W}}_R$
for optical fields,
which uses correlations of intensity {\it rates}
behind pairs of $U^X(j)$ interferometers can be constructed as follows.
For the output $l_x$ of an interferometer, one defines rate observables
as
$\hat{r}_{l_x}=\hat{\Pi}^X\frac{\hat n_{l_x}}{\hat{N}^X} \hat{\Pi}^X$, where $\hat{N}^X=\sum_{l_x=1}^{d_X}\hat{n}_{l_x}$.
The witness $\hat W$
expanded in terms of the computational basis:
\begin{equation} \label{KETBRA}
\hat W=\sum_{k,m}^{{d}_{A}}\sum_{l,n}^{{d}_{B}} w_{klmn}|k_a,l_b\rangle\langle m_a,n_b|,
\end{equation}
allows us to form an entanglement witness for fields:
\begin{equation}
\label{RATE_WITNESS}
\hat {\mathcal{W}}_R=\sum_{k,m}\sum_{l,n}w_{klmn}
\hat \Pi^A\hat \Pi^B\frac { \hat a^\dagger_k \hat b^\dagger_l \hat a_m \hat b_n}
{\hat N^A \hat N^B}\hat \Pi^A\hat \Pi^B.
\end{equation}
For any pure state of the quantum beams $|\Psi\rangle $
\begin{equation}
\frac{\langle \Psi |\hat{\mathcal{W}}_R|\Psi\rangle}{ \langle \Psi|\hat \Pi^A\hat \Pi^B|\Psi\rangle} =
\Tr \hat W \hat{ \mathcal{R}},
\label{eq:avg_rate}
\end{equation}
where the matrix $\hat{\mathcal R}$ has elements $r_{klmn}$
\begin{equation}
r_{klmn} = \frac{1}{\langle \Psi|\hat \Pi^A\hat \Pi^B|\Psi\rangle}
\langle \Psi |\hat \Pi^A\hat \Pi^B\frac { \hat a^\dagger_k \hat b^\dagger_l \hat a_m \hat b_n}
{\hat N^A \hat N^B}\hat \Pi^A\hat \Pi^B|\Psi\rangle.
\end{equation}
Using a
generalization of the earlier derivations
one can show that $\hat{\mathcal R}$
is a two-qudit density matrix, and so on.
The actual measurements,
to be correlations of local ones, should
be performed using the
sequence of pairs of $U^X(j)$ interferometers,
which enter the expansion of the two-qudit entanglement witness (\ref{EXPANSION}). In the entanglement
indicator the rates at
output $x_l(j)$ of the given local
interferometer $U^X(j)$ are multiplied by the respective
eigenvalue of $\hat V_j^X$ related
with the eigenstate $|x^{(j)}_l\rangle$.
To get an entanglement witness for intensities $\mathcal{\hat{W}}_I$ we take $\hat W$ and replace the computational basis kets and bras by suitable creation and annihilation operators:
\begin{equation}
\label{INTENSE_WITNESS}
\hat {\mathcal{W}}_{I}=\sum_{k,m}^{{d}_{A}}\sum_{l,n}^{{d}_B}w_{klmn}
\hat a^\dagger_k \hat b^\dagger_l \hat a_m \hat b_n.
\end{equation}
For any pure state of the quantum beams $|\Psi\rangle$ one has
$
\frac{\langle \Psi|\hat {\mathcal{W}}_{I}|\Psi\rangle}{ \langle \Psi|\hat N_A \hat N_B|\Psi\rangle} = \Tr \hat W {\hat{\mathcal P}},
$
where the matrix $\hat{\mathcal P}$ has elements
$
\frac{1}{\langle \Psi| \hat N_A\hat N_B|\Psi\rangle}\langle \Psi | \hat a^\dagger_k \hat b^\dagger_l \hat a_m \hat b_n | \Psi\rangle,
$
and has all properties of a two-qudit density matrix.
{\it Example showing further extension to unitary operator bases.}---Let $d$ be a power of a prime number.
Consider $d_A=d_B=d$ beams experiment (see Fig.~\ref{fig:exp}), with families of $U^X(m)$
interferometers which link the computational basis of a qudit
with an unbiased basis $m$, belonging to the full set of $d+1$ mutually unbiased ones~\cite{WOOTERS, MUBs}.
We introduce a set of unitary observables
for a qudit:
$
\hat{q}_k(m)= \sum_{j=1}^d\omega^{jk}|j(m)\rangle \langle j(m)|,
\label{EQ:UNI_OBS}
$
with $ | j(m)\rangle =U(m)|j\rangle$ and it is the $j$-th member of $m$-th mutually unbiased basis, and $\omega=\exp(2\pi i/d)$.
Operators $\hat{q}_k(m)/\sqrt{d}$ with $k = 1,...,d-1$
and $m=0,...,d$ and $\hat{q}_0(0)/\sqrt{d}$ form an orthonormal basis in the
Hilbert-Schmidt space of all $d\times d$ matrices (see Sec. III in the Supplemental Material~\cite{supp}).
Thus, we can expand any qudit density matrix as
\begin{equation}\label{COMPLETE}
{\varrho} =\frac{1}{\sqrt{d}} \left[
c_{0,0}\hat{q}_0(0) +
\sum_{m=0}^d\sum_{k=1}^{d-1}c_{m,k}\hat{q}_k(m) \right],
\end{equation}
where
$
c_{m,k} = \Tr \hat{q}_k^{\dagger}(m) \varrho / \sqrt{d},
$
and $c_{0,0}=1/\sqrt{d}$.
As the basis observables
are unitary the expansion coefficients of an entanglement witness operator in terms of such tensor products of such bases
are in general complex.
This is no problem for theory,
but renders useless a direct application in experiments,
as one cannot expect the experimental averages to be real, and thus one has to introduce modifications. Below we present one.
The condition $\Tr \varrho^2\leq 1$
can be put as
\begin{equation}
\frac{1}{d}+\frac{1}{d}\sum_{m=0}^d\sum_{k=1}^{d-1}
\left| \Tr \varrho \hat{q}_{k}(m) \right|^2 \leq 1.
\label{SINGLE2}
\end{equation}
Thus, applying Cauchy-Schwartz estimate,
we get immediately a separability condition for two qudits:
\begin{equation}
\sum_{m=0}^d\sum_{k=1}^{d-1}
\left| \Tr \varrho^{AB}_{sep}
\hat{q}^{A}_{k}(m)\hat{q}^{B\dagger}_{k}(m) \right| \leq (d-1).
\label{PAIR2}
\end{equation}
Our general method defines
a Cauchy-Schwartz-like separability condition homomorphic with (\ref{PAIR2}) as
\begin{equation}
\label{MULTIWARUNEKnew}
\sum_{m=0}^d \sum_{k=1}^{d-1}|\langle \hat Q_k^A(m)\hat
Q_k^{B{\dagger}}(m)\rangle_{sep}| \leq
(d-1)\langle
\hat \Pi^A \hat\Pi^B\rangle_{sep},
\end{equation}
where
\begin{equation}
\label{MULTIOBS}
\hat Q_{k}^X(m)
= \sum_{j=1}^d \hat \Pi^{X}\frac{\omega^{jk}\hat n^X_j(m)}{\hat N^{X}}\hat \Pi^{X}.
\end{equation}
Here $\hat n^X_j(m) = \hat x^{\dagger}_j(m)\hat x_j(m)$ is a photon number operator for output mode $j$ of a multiport $m$, at station $X$.
For generalized observables based on intensity, one can introduce
$
\label{MULTIOLD}
{\hat \chi}_{k}(m) = \sum_{j=1}^d\omega^{jk} \hat n_j(m)
$
to get the following separability condition:
\begin{equation}
\label{MULTIWARUNEKold}
\sum_{m=0}^d \sum_{k=1}^{d-1}
|\langle{ \hat \chi }_k^A(m){\hat
\chi}_k^{B{\dagger}}(m)\rangle_{sep}| \leq
(d-1)\langle
\hat N^A \hat N^B\rangle_{sep}.
\end{equation}
Supplemental Material presents other examples~\cite{supp}.
{\it Implications for optical coherence theory.}---The approach can be generalized further.
Let us take as an example Glauber's correlation functions for optical fields, say $G^{(4)}$
in the form of
$
\langle \hat{I}_A(\vec{x},t) \hat{I}_B(\vec{x}',t')\rangle
$,
where the intensity operator has the usual form of $I_X(\vec{x}, t)=\hat{F}_X^\dagger(\vec x, t) \hat{F}_X(\vec x, t)$,
with normal ordering requiring that operator $\hat{F}_X(\vec x, t)$ is built out of local annihilation operators. The idea of normalized Stokes operators
suggests the following alternative correlation function $\Gamma^{4}(\vec{x},t; \vec{x}', t')$ given by
\begin{eqnarray}
\label{GLAUBER}
\langle \Pi^A\Pi^B \frac{\hat{I}_A(\vec{x},t) \hat{I}_B(\vec{x}',t')}{\int_{a(A)} d\sigma(\vec{x}) \hat{I}_A(\vec{x},t)\int_{a(B)} d\sigma(\vec{x}') \hat{I}_B(\vec{x}',t')}\Pi^A\Pi^B\rangle,&& \nonumber \\
\end{eqnarray}
where $a(X)$ denotes the overall aperture of the detectors in location $X$. Obviously one has $\int_{a(A)} d\sigma(\vec{x})\int_{a(B)} d\sigma(\vec{x}') \Gamma^4(\vec{x},t; \vec{x}', t')=\langle\Pi^A\Pi^B\rangle$, and for fixed $t$ and $t'$ one can define
\begin{eqnarray*}
&&\varrho(\vec{x},\vec{y},\vec{x}' \vec{y}')_{t,t'}
= {\langle\Pi^A\Pi^B\rangle}^{-1} \nonumber \\
&&\times \langle \Pi^A\Pi^B
\frac{\hat{F}^\dagger_A(\vec{y},t)\hat{F}_A(\vec{x},t)
\hat{F}^\dagger_B(\vec{y}',t') \hat{F}_B(\vec{x}',t')}
{\int_{a(A)} d\sigma(\vec{x}) \hat{I}_A(\vec{x},t)\int_{a(B)} d\sigma(\vec{x}') \hat{I}_B(\vec{x}',t')}\Pi^A\Pi^B\rangle,
\end{eqnarray*}
which behaves like a proper two-particle density matrix, provided one constrains the range of $\vec{x}, \vec{y}, \vec{x}', \vec{y}'$ to appropriate sets of apertures. As our earlier considerations use simplified forms of (\ref{GLAUBER}), it is evident that such correlation functions may help us to unveil non-classicality in situations
in which the standard ones fail, see e.g.~\cite{ROSOLEK}.
{\it Bell inequalities.}---The above ideas allow one to introduce a general mapping of qudit Bell inequalities
to the ones for optical fields.
A two-qudit Bell inequality for a final number of local measurement settings $\alpha$ and $\beta$ has the following form:
\begin{eqnarray}
\label{BELL}
&&\sum_{\alpha\beta}\sum_{i=1}^{d_A}\sum_{j=1}^{d_B} K^{ij}_{\alpha\beta} P_{ij}(\alpha, \beta) \nonumber \\
&&+ \sum_{i=1}^{d_A}\sum_\alpha N^i_\alpha P_i(\alpha)
+ \sum_{j=1}^{d_B}\sum _\beta M^j_\beta P_j(\beta)\leq L_R,
\end{eqnarray}
where $P_{ij}(\alpha, \beta)$ denotes the probability of the qudits ending up respectively at detectors $i$ and $j$, when the local setting are as indicated, and $\sum_jP_{ij}(\alpha, \beta)=P_{i}(\alpha)$ and $P_{j}(\beta)=\sum_iP_{ij}(\alpha, \beta)$.
The coefficient matrices $K, N, M$ are real, and $L_R$ is the maximum value allowed by local realism. The bound is calculated by putting $P_{ij}(\alpha,\beta)= D^i(\alpha)D^j(\beta)$ and $P_{i}(\alpha)= D^i(\alpha)$,
$P_{j}(\beta)= D^j(\beta)$, with constraints $0\leq D^i(\alpha/\beta )\leq1$, and $\sum_{i=1}^{d_{A/B}}D^i(\alpha/\beta)=1$. As for a given run of a quantum optical experiment local measured photon intensity rates $r_i(\alpha)$ and $r_j(\beta)$ satisfy exactly the same constraints.
We can replace $P_{ij}(\alpha, \beta)\rightarrow \langle r_i(\alpha)r_j(\beta)\rangle_{LR}$, and $P_{i}(\alpha)\rightarrow \langle r_i(\alpha)\rangle_{LR}$, etc., where $\langle.\rangle_{LR}$ is an average in the case of local realism. The bound $L_R$ stays put. To get a Bell operator we further replace the above by rate observables
$\hat{r}_i(\alpha) \hat{r}_j(\beta)$, etc. Thus any (multiparty) Bell inequality, see e.g.~\cite{BELL-RMP}, can be useful in quantum optical intensity (rates) correlation experiments.
The presented methods for entanglement indicators and Bell inequalities allow also to get steering inequalities for quantum optics.
{\it Conclusions.}---We present tools for a construction of entanglement indicators for optical fields, inspired by the vast literature \cite{HORODECKIS} on entanglement witnesses for finite dimensional quantum systems.
The indicators would be handy for more intense light beams in states of undefined photon numbers, especially in the emerging field of integrated optics multi-spatial mode interferometry (see Supplemental Material~\cite{supp} for examples). One may expect applications in the case of many-body systems, e.g. for an analysis of non-classicality of correlations in Bose-Einstein condensates, like in the ones reported in \cite{SCHMIED}.
\begin{acknowledgments}
{\it Acknowledgments.}---The work is part of the ICTQT IRAP project of FNP, financed by structural funds of EU. MZ acknowledges COPERNICUS grant-award, and discussions with profs. Maria Chekhova and Harald Weinfurter. JR acknowledges the National Research Foundation, Prime Minister’s Office, Singapore and the Ministry of Education, Singapore under the Research Centres of Excellence programme, and discussions with prof. Dagomir Kaszlikowski. MW acknowledges NCN grants number 2015/19/B/ST2/01999 and 2017/26/E/ST2/01008.
\end{acknowledgments}
\section*{Supplemental material}
We give here several examples, and more details concerning some derivations. All separability conditions are generalizations or tighter versions of conditions
presented in \cite{BOUW-2, BOUW, zuko, MULTIPORT, MULTIPORT2}, which were derived using various less general approaches.
\section{Necessary and Sufficient conditions for intensity and rate correlations to reveal entanglement}
\label{apx:exp_iff_qubit}
Two-qubit states are separable
if and only if their partial transposes are positive.
Yu {\it et al.} derived an equivalent family of conditions for two-qubit states~\cite{CHINY} in a form of an inequality, which reads
\begin{eqnarray}
\label{CHINSKIPAULI}
\langle \hat \sigma_{1}^{A}\hat \sigma_{1}^{B} + \hat \sigma_{2}^{A}\hat \sigma_{2}^{B} \rangle^2 &+& \langle \hat \sigma_{3}^{A}\hat{\sigma}_0^{{B}} + \hat{\sigma}_0^{{A}} \hat \sigma_{3}^{B}\rangle^2 \nonumber \\
&\leq& \langle \hat{\sigma}_0^{A}
\hat{\sigma}_0^{B} + \hat \sigma_{3}^{A}\hat \sigma_{3}^{B} \rangle^2,
\end{eqnarray}
where $\hat{\sigma}^X_j=\vec{n}^X_j\cdot \vec{\sigma}^X$ for $X=A, B$, and the unit vectors $\vec{n}^X_j$ form a right-handed Cartesian basis triad.
If a two-qubit state is entangled,
then there exists at least one pair
of such triads
for which the inequality is violated.
The conditions can be put in a form of a
family of entanglement witnesses:
\begin{eqnarray}
\label{ENT-WIT-YU}
W(\alpha) =\hat{\sigma}_0^{A}\hat{\sigma}_0^{B} + \hat \sigma_{3}^{A}\hat \sigma_{3}^{B} &+& \sin{\alpha}( \hat \sigma_{1}^{A}\hat \sigma_{1}^{B} + \hat \sigma_{2}^{A}\hat \sigma_{2}^{B}) \nonumber \\
&+& \cos{\alpha}( \hat \sigma_{3}^{A}\hat{\sigma}_0^{{B}} + \hat{\sigma}_0^{{A}} \hat \sigma_{3}^{B}).
\end{eqnarray}
Our homomorphisms can be used to get the following~\cite{zuko1}: for normalized Stokes operators
\begin{eqnarray}
\label{CHINSKISTOKES1}
\langle \hat S_{1}^{A}\hat S_{1}^{B} + \hat S_{2}^{A}\hat S_{2}^{B} \rangle^2 &+& \langle \hat S_{3}^{A}\hat \Pi^{{B}} +
\hat \Pi^{{A}}\hat S_{3}^{B}\rangle^2 \nonumber \\
&\leq& \langle \hat \Pi^{A}\hat \Pi^{B} + \hat S_{3}^{A}\hat S_{3}^{B}\rangle^2,
\end{eqnarray}
and for standard ones
\begin{eqnarray}
\label{CHINSKISTOKES2}
\langle \hat \Theta_{1}^{A}\hat \Theta_{1}^{B} + \hat \Theta_{2}^{A}\hat \Theta_{2}^{B} \rangle^2 &+& \langle \hat \Theta_{3}^{A}\hat N^{{B}} + \hat N^{{A}}\hat \Theta_{3}^{B}\rangle^2 \nonumber \\
&\leq& \langle \hat N^{A}\hat N^{B} + \hat \Theta_{3}^{A}\hat \Theta_{3}^{B}\rangle^2.
\end{eqnarray}
The homomorphisms warrant that the violations of conditions~(\ref{CHINSKISTOKES1}) and~(\ref{CHINSKISTOKES2}) are {\it necessary and sufficient} to detect entanglement via measurements of correlations of the Stokes observables.
That is, any other condition is sub-optimal, including the ones presented in \cite{BOUW}, \cite{Iskhakov12} and \cite{BOUW-2} for standard Stokes observables.
From the necessary and sufficient condition (\ref{CHINSKISTOKES1}) one can derive its corollary, which is a necessary condition for separability:
\begin{equation}
\label{WARUJNOWYauto}
\sum_{j=1}^{3} |\langle{\hat S}_{{j}}^{A}
{ \hat S}_{{j}}^{B} \rangle_{sep}| \leq \langle \hat
\Pi^A
\hat \Pi^B \rangle_{sep}.
\end{equation}
The condition can be thought as a more tight refinement of the result in \cite{BOUW-2}.
It can be derived using the fact that for two qubits any of the observables
$\sum_k s_k \sigma^A_k \sigma^B_k+ \sigma^A_0 \sigma^B_0$,
for arbitrary $s_k=\pm1$ is non-negative for separable states. This can be reached via an application of the Cauchy inequality for a product pure states of a pair of qubits. Next we apply the homomorphism.
One can also see that (\ref{WARUJNOWYauto}) is the separability condition (14) in the main text for $d=2$.
For the standard Stokes operators the associated separability condition (\ref{WARUJNOWYauto}) reads
\begin{equation}
\label{WARUJSTARYauto}
\sum_{j = 1}^{3} |\langle\hat \Theta_{j}^{A} \hat \Theta_{j}^{B} \rangle_{sep}| \leq \langle
{\hat N}^{A} {\hat N}^{B} \rangle_{sep}.
\end{equation}
This is a tighter version of the condition given in \cite{BOUW-2}.
For states, which locally lead to vanishing averages of local Stokes parameters, here $\langle \hat{S}^A_i \hat{\Pi}^B\rangle =0 $, etc.,
(e.g., for an ideal four-mode bright squeezed vacuum, see below),
the conditions (\ref{WARUJNOWYauto}) and (\ref{CHINSKISTOKES1}) are equivalent.
Thus, in such a case the Cauchy inequality based
condition is necessary and sufficient for detection of entanglement with normalized Stokes operators.
A similar statement can be produced for the analog condition involving
traditional Stokes parameters $\Theta_j$, given by (\ref{WARUJSTARYauto}).
{\it Cauchy-like inequality condition vs. EPR inspired approach}.---Consider four-mode (bright) squeezed vacuum represented by
\begin{equation}
\label{BOH_PROJ_0}
\ket{\Psi^-} =
\frac{1}{\cosh^2\Gamma} \sum_{n=0}^{\infty}\sqrt{n+1} \tanh^n\Gamma
\textcolor{black}{\ket{\psi^{n}_-}},
\end{equation}
where $\Gamma$ describes a gain which is proportional to the pump power, and
$\ket{\psi^{n}_{-}}$ reads
\begin{equation}
\label{PSINONOISE}
\ket{\psi^n_-} = \frac{1}{n! \sqrt{n+1}}
\bigg(\hat a_i^{\dagger}\hat
b_{i_{\perp}}^{\dagger} -\hat
a_{i_{\perp}}^{\dagger}
b_i^{\dagger} \bigg)^n
\ket{\Omega},
\end{equation}
where $\ket{\Omega}$ is the vacuum state.
Perfect EPR-type anti-correlations of which are the main trait of the state allow one
to formulate the following appealing separability
condition (Simon and Bouwmeester, \cite{BOUW}):
\begin{equation}
\label{OLDANDWRONG}
\sum_{j=1}^3 \langle (\hat \Theta_{j}^A
+\hat \Theta_{j}^B )^2\rangle_{sep} \geq
2(\langle \hat N^{A}\rangle + \langle \hat
N^{B}\rangle)_{sep}.
\end{equation}
Note, that for $|\Psi^-\rangle$ and each $|\psi^n_-\rangle$ the left-hand side (LHS) of the above is vanishing.
The underlying inequality beyond the condition (\ref{OLDANDWRONG}) can be extracted with the use of
well-known operator identity (see e.g. \cite{KLYSHKO}):
\begin{equation}
\label{RELOLD}
\sum_{j=1}^3 \hat \Theta_j^2 = \hat
N(\hat N+2).
\end{equation}
Using this the (\ref{OLDANDWRONG}) boils down to
\begin{equation}
\label{WARUJ}
- \sum_{j=1}^3 \langle \hat \Theta_{j}^A
\hat \Theta_{j}^B\rangle_{sep} \leq \frac{1}{2}\langle \hat
N_{A}^2\rangle_{sep}
+ \frac{1}{2}\langle \hat N_{B}^2\rangle_{sep},
\end{equation}
which by the way can be generalized to
$
2 \sum_{j=1}^3| \langle \hat \Theta_{j}^A
\hat \Theta_{j}^B\rangle_{sep}| \leq\langle \hat
N_{A}^2\rangle_{sep}
+ \langle \hat N_{B}^2\rangle_{sep}.
$
Simon-Bouwmeester EPR-like condition (\ref{OLDANDWRONG}), or equivalently (\ref{WARUJ}), cannot be
considered as an entanglement indicator for fields
$\hat{\mathcal{W}}_{\Theta}$
homomorphic in the way proposed here, with a two-qubit (linear) entanglement witness $\hat{W}$.
Detection of entanglement with (\ref{OLDANDWRONG}) depends on a detector efficiency.
The threshold efficiency for entanglement detection, in the case of $2 \times 2$ mode squeezed vacuum $|\Psi^-\rangle$ in (\ref{BOH_PROJ_0}),
considered in \cite{BOUW}
is given by $\eta_{crit} = 1/3$.
It does not depend on the gain parameter $\Gamma$.
Obviously, as $\langle \hat{N}_A \hat{N}_B\rangle \leq \frac{1}{2}\langle\hat
N_{A}^2\rangle
+ \frac{1}{2}\langle \hat N_{B}^2\rangle,$ the inequality (\ref{WARUJ}) is not
optimal.
A more optimal option is to estimate from below the LHS of (\ref{WARUJ}) using a corollary of the
Cauchy-like inequality
$
\label{WARUJCS}
-\sum_{j=1}^3 \langle \hat \Theta_{j}^A
\hat \Theta_{j}^B\rangle \leq \langle \hat N^A\hat N^B\rangle
$, which is tighter than (\ref{WARUJ}).
By combining (\ref{WARUJSTARYauto}) with (\ref{RELOLD}) we get
\begin{equation}
\begin{multlined}
\label{OLDANDNAJS}
\sum_{j=1}^3 \langle (\hat \Theta_{j}^A
+\hat \Theta_{j}^B )^2\rangle_{sep} \\ \geq
2(\langle \hat
N^{A} \rangle + \langle \hat N^{B}\rangle)_{sep} +
\langle (\hat N^{A} - \hat
N^{B})^2\rangle_{sep}.
\end{multlined}
\end{equation}
The new EPR-like necessary condition for separability differs from the one of Simon and Bouwmeester by the second term on the RHS of (\ref{OLDANDNAJS}). As the term is always non-negative, this is a stronger condition.
For the standard quantum optical model of inefficient detection (see the main text, or Sec.~\ref{LOSSES}) the new condition holds for any efficiency.
Note, that (\ref{RELOLD}) does not contribute anything
to the relation (\ref{OLDANDNAJS}), because it is an operator identity. That is, the condition (\ref{OLDANDNAJS}) reduces to (\ref{WARUJSTARYauto}).
For normalized Stokes parameters the
EPR-like separability condition, which is an analog of (\ref{OLDANDWRONG}), reads
\begin{equation}
\begin{multlined}
\label{NEWANDWRONG}
\sum_{j=1}^3 \left \langle \left ({\hat S}_{j}^A
+{\hat S}_{j}^B \right)^2\right \rangle_{sep} \\ \geq
\left \langle \hat \Pi^A\frac{2}{\hat N^A}\hat \Pi^A
+\hat \Pi^B\frac{2}{\hat N^B}\hat \Pi^B \right \rangle_{sep}.
\end{multlined}
\end{equation}
For a derivation, see \cite{zuko} (and see also \cite{MULTIPORT, MULTIPORT2} for its generalizations to $d$ modes).
Entanglement detection with (\ref{NEWANDWRONG}) also depends on the detector efficiency, but for the considered bright squeezed vacuum state the threshold efficiency $\eta_{crit}$ decreases with growing $\Gamma$. The $\eta_{crit}$ is lower than $1/3$ for any finite $\Gamma$.
If one uses the Cauchy-like inequality~(\ref{WARUJNOWYauto}) and the identity $\sum_{i=1}^3\hat{S}_i^2= \hat{\Pi} + \hat{\Pi} \frac{2}{\hat{N}}\hat{\Pi}$ (see \cite{zuko}), then the following tighter EPR-like separability condition emerges
\begin{eqnarray}
\label{NEWANDNAJS}
\sum_{j=1}^3 \left \langle \left ({\hat S}_{j}^A
+{\hat S}_{j}^B \right)^2\right \rangle_{sep} \geq
\langle \hat \Pi^A\frac{2}{\hat N^A}\hat \Pi^A\rangle_{sep} \nonumber \\ +
\langle \hat \Pi^B\frac{2}{\hat N^B} \hat \Pi^B \rangle_{sep}
+ \langle (\hat \Pi^A
-\hat \Pi^B)^2\rangle_{sep}.
\end{eqnarray}
It is equivalent with the much
simpler linear condition (\ref{WARUJNOWYauto}).
The condition presented here has much more resistant to losses that the one derived in \cite{zuko}, and generalized in \cite{MULTIPORT2}, here formula (\ref{NEWANDWRONG}).
\section{Resistance with respect to losses}
\label{LOSSES}
Here we derive the dependence on a detector efficiency
of average values of entanglement indicators for optical fields
$\hat{\mathcal{W}}_{\Theta}$ and $\hat{\mathcal{W}}_{S}$.
Our reasoning can be extended
to an arbitrary number of quantum optical modes and multi-party cases.
The loss model (an ideal detector and a beamsplitter of
transmission amplitude $\sqrt{\eta}$ in front of it) is described by a beamsplitter transformation for the creation operators, see e.g. \cite{KLYSHKO}, which reads
\begin{equation}
\begin{multlined}
\label{ETAKREACJA}
\hat a^{\dagger}_j(\eta)
= \sqrt{\eta} \hat
a^{\dagger}_j+ \sqrt{1-\eta}
\hat c^{\dagger}_j,
\end{multlined}
\end{equation}
where $\hat{a}^{\dagger}_j$ refers to the detection channel in $j$-th mode and
$\hat{c}^{\dagger}_j$ refers to the loss channel linked with the mode.
First, we shall analyze the problem for standard Stokes operators. Let $\ket{\psi^{AB}}$ be a pure state of the modes,
before the photon losses. The unitary transformation $\hat{\mathcal{U}}(\eta)$ describing losses in all channels leads to $\hat{\mathcal U}(\eta)|\psi^{AB}\rangle = |\psi^{AB}(\eta)\rangle$, and we have
\begin{equation}
\expval{\mathcal{\hat W}_{{\Theta}}}{\psi^{AB}(\eta)}
= \expval{\mathcal{\hat W}_{{\Theta}}(\eta)}{\psi^{AB}},
\end{equation}
where
$\mathcal{\hat{W}}_{{\Theta}}(\eta) = \hat{\mathcal{U}}^{\dagger}(\eta)\mathcal{\hat{W}}_{\Theta}\hat{\mathcal{U}}(\eta)$.
A transformed photon number operator $\hat n_j (\eta) = \hat a_j^{\dagger}(\eta)\hat a_j(\eta)$ reads
\begin{eqnarray}
\label{nOPER}
\hat n_{j}(\eta) &=& (\sqrt{\eta} \hat
a_{j}^{\dagger}+ \sqrt{1-\eta}
\hat c_{j}^{\dagger}) (\sqrt{\eta} \hat
a_j + \sqrt{1-\eta} \hat c_{j}) \nonumber \\
&=& \eta \hat n_{j} + \sqrt{\eta(1-\eta)}(\hat c_{j}^{\dagger} \hat a_{j} +
\hat a_{j}^{\dagger} \hat c_{j})
+(1-\eta) \hat c_{j}^{\dagger} \hat c_{j}. \nonumber \\
\end{eqnarray}
Notice that as the original state $\ket{\psi^{AB}}$ does not contain photons in the
loss channels, thus in $ \expval{\hat n_{i}^A(\eta^{A})\hat n_{j}^B(\eta^{B})}{\psi^{AB}}
$ only the first term of the second line of~(\ref{nOPER}) survives. For the transmission amplitudes $\eta^{A}$ and $\eta^{B}$ of beams $A$ and $B$, we have
\begin{equation}
\expval{\hat n_{i}^A(\eta^A)\hat n_{j}^B(\eta^B)}{\psi^{AB}} = \eta^A \eta^B
\expval{\hat n_{i}^A\hat n_{j}^B}{\psi^{AB}}.
\end{equation}
From this we get the dependence of correlations of Stokes operators on detection efficiency in the form of $\langle\hat{ \Theta}_i^A(\eta^A) \hat{ \Theta}_j^B(\eta^B)\rangle=\eta^A\eta^B\langle\hat{ \Theta}_i^A \hat{ \Theta}_j^B\rangle$.
For normalized Stokes operators, the reasoning is as follows.
For Fock states $|F\rangle=| n_{A_i}, n_{A_{i_\perp}}, m_{B_i}, m_{B_{i_\perp}}\rangle$, it is enough to consider only the average value of $\hat{S}^A_3$ for state $|F_A\rangle=|n_{A_H}, m_{A_{V}}\rangle$, which we shall denote for simplicity as $|n, m\rangle$. Obviously for such a state the intensity rate at the detector measuring output $H$, with the detection efficiency $\eta$ for each of the detectors in the station, reads
\begin{eqnarray*}
r_1(\eta)&=&\langle (n,m)_\eta| \hat{\Pi}_A \frac{\hat{n}_H}{\hat{n}_H+\hat{n}_V} \hat{\Pi}_A|(n,m)_\eta\rangle \nonumber \\
&=&\sum_{k=1}^{n}\sum_{l=0}^{m}{{n}\choose{k}}{{m}\choose{l}}\frac{k}{k+l}\eta^{k+l}(1-\eta)^{n+m-k-l}.
\end{eqnarray*}
First we notice that $k{{n}\choose{k}}={n} {{n-1}\choose{k-1}}$, and rewrite the first summation as from $k=0$ to $k=n-1$.
Next, let us consider a function $f(\gamma, \eta)$ of the form
\begin{eqnarray}
f(\gamma, \eta)
&=&n\sum_{k=0}^{n-1}\sum_{l=0}^{m}{{n-1}\choose{k}}{{m}\choose{l}}\frac{1}{k+1+l} \nonumber \\
&&\times \gamma^{k+1+l}(1-\eta)^{n-1+m-k-l},
\end{eqnarray}
which for $\gamma=\eta$ gives $r_1(\eta)$. Its derivative with respect to $\gamma$
reads
\begin{eqnarray}
&&\frac{d}{d\gamma}f(\gamma, \eta)\nonumber \\
&=&n\sum_{k=0}^{n-1}\sum_{l=0}^{m}{{n-1}\choose{k}}{{m}\choose{l}}\gamma^{k+l}(1-\eta)^{n-1+m-k-l}\nonumber\\
&=&n(\gamma+1-\eta)^{n+m-1}.
\end{eqnarray}
This upon integration with respect to $\gamma$, with the initial condition $f(\gamma=0,\eta)=0$, gives for $\gamma=\eta$
the required result:
\begin{equation}
\label{RES}
r_1(\eta)=\frac{n}{n+m}\big(1-(1-\eta)^{n+m}\big).
\end{equation}
It is easy to see that this result has a straightforward generalization
to the case of more than two local detectors (e.g., see Fig. 1 in the main text).
To calculate the dependence on $\eta$ of the rate at detector $i$,
when we have altogether $d$ detectors at the station, we simply replace in the above formulas $\hat{n}_H$ by
$\hat{n}_i$ and $\hat{n}_V$ by $\sum_{j\neq i} \hat{n}_j$, to get $ r_i(\eta)=\frac{n}{n_{tot}}(1-(1-\eta)^{n_{tot}})$, where $n$ is the number of photons in a Fock state in mode $i$
and $n_{tot}$ is the total number of photons.
Note that for four-mode bright squeezed vacuum state (\ref{BOH_PROJ_0}) our entanglement condition for normalized Stokes parameters (\ref{WARUJNOWYauto}) is fully resilient with respect to losses of the kind described above. This is due to the fact that squeezed vacuum is a superposition entangled states (\ref{PSINONOISE}), and each of them violates the separability criterion. As the Stokes operators do not change overall photon numbers on each of the sides of the experiments which we consider here, and states $\ket{\psi^{n}_{-}}$ contain $n$ photons in both beams $A$ and $B$, an inefficient detection in the case of $\ket{\psi^{n}_{-}}$ introduces the same reduction factor on both sides of condition (\ref{WARUJNOWYauto}). The violation of it holds for whatever value of $\eta$. The expectation values for the full squeezed state are simply weighted sum of expectation values for its components $|\psi^n_-\rangle$. The same can be shown for all other squeezed states, and linear separability conditions considered here, including the cases of $d>2$.
\section{Entanglement experiments involving multiport beamsplitters:
homomorphism of single qudit observables and field operators}
\label{apx:qudit_density}
{\it Proof of relation (11) of the main text for qu$d$it states.}---We consider a set of unitary qudit observables of the following form in the main text
\begin{eqnarray}
\label{SMALLq}
\hat{q}_k(m)= \sum_{j=1}^d\omega^{jk}|j(m)\rangle \langle j(m)|,
\end{eqnarray}
where $k=0,1,..., d-1$ and $\omega=\exp(2\pi i/d)$, and $\hat{U}(m) | j\rangle = | j(m)\rangle$ is a unitary transformation of a computational basis ($m=0$) to a vector of a different unbiased basis $m$. We assume that the bases $m\neq m'$ are all mutually unbiased, and consider only dimensions in which we have $d+1$ mutually unbiased bases.
We show that the operators $\hat{q}_k(m)/\sqrt{d}$ with $k = 1,...,d-1$
and $m=0,...,d$, and $\hat{q}_0(0) = \openone$ form an orthonormal basis in the
Hilbert-Schmidt space of (all) $d\times d$ matrices.
The orthonormality of the operators can be established as follows.
We are to prove that
\begin{eqnarray}
\frac{1}{d} \Tr
\hat{q}_k^{\dagger}(m)\hat{q}_{k'}(m') =
\delta_{mm'}\delta_{kk'}.
\label{ORTHO}
\end{eqnarray}
\begin{itemize}
\item For $k'=0$, this is trivial because all $k\neq 0$
operators are traceless (as $\sum_{j=1}^{d}\omega^{jk}=d\delta_{k0}$).
\item For $m\neq m'$, with $k\neq 0$ and $k'\neq 0$, one has
\begin{eqnarray}
&&\frac{1}{d}\sum_{l,j,j'} \omega^{-jk+j' k'} \langle l(m) |j(m)\rangle\langle j(m)| j'(m')\rangle\langle j'(m')|l(m)\rangle \nonumber \\
&=& \frac{1}{d} \sum_{l,j'}\omega^{-lk +j'k'}\langle l(m)| j'(m')\rangle\langle j'(m')|l(m)\rangle \nonumber \\
&=&\frac{1}{d^2}\sum_{l,j'} \omega^{-lk +j'k'} =0,
\label{EQ:ORTHO2}
\end{eqnarray}
where we use the fact that for mutually unbiased bases $|\langle j'(m')|j(m)\rangle|^2=1/d$.
\item For $m=m'$, in the second line of
(\ref{EQ:ORTHO2}) we have \\ $\langle
j'(m')|l(m)\rangle
=\delta_{lj'}$, and we get in the last line
$\frac{1}{d}\sum_l\omega^{l(k-k')}= \delta_{kk'}.$
\end{itemize}
As we have $(d-1)(d+1)+1=d^2$ such orthonormal operators, the basis is complete. QED.
{\it Remarks on the homomorphism.}---We shall now show that for any pure
state of a $d$-mode optical field $\ket{\psi}$,
one can always find a $d \times d$
one qudit density matrix $\mathfrak{M}$ for which the following holds
\begin{equation}
\frac{\expval{\hat Q_{k}(m)}{\psi}}{\expval{\hat \Pi}{\psi}} =
\Tr \hat{q}_{k}(m)\mathfrak{M},
\label{eq:qudit_homo_rate}
\end{equation}
where $ \hat{Q}_{k}(m) $ is defined by (15) in the main text.
For the expectation value, which reads
\begin{eqnarray}\label{Q-OBS}
\bra{\psi} \hat{Q}_{k}(m)\ket{\psi}
= \bra{\psi} \sum_{j=1}^{d}
\hat \Pi \frac{a_{j}^{\dagger} (m)
a_{j}(m) }{\hat{N}} \hat \Pi \, \omega^{jk}
\ket{\psi},
\end{eqnarray}
we introduce a set of states
\begin{equation}
\ket{\phi_{j}(m)} = \ani{j}(m)
\frac{1}{\sqrt{\hat{ N}}} \hat \Pi \ket{\psi},
\label{EQ:STATE}
\end{equation}
which for $m=0$ gives
\begin{equation}
\ket{\phi_{j}(0)} = \ani{j} \frac{1}{\sqrt{\hat{N}}}
\hat \Pi \ket{\psi}.
\end{equation}
Then, one can transform (\ref{Q-OBS}) into
\begin{equation}
\avg{\hat{Q}_{k}(m)} = \sum_{j=1}^d
\inner{\phi_{j}(m)}{\phi_{j}(m)} \omega^{jk}.
\label{EQ:EXP_obs}
\end{equation}
As it was mentioned in the main text, the unitary transformation of the creation operators between input and output beams is $\hat a_{l}^{\dagger} (m) = \sum_{r} {U}_{l
r} (m) \hat a_{r}^{\dagger}$,
where $\hat a^{\dagger}_r =\hat a^{\dagger}_r(m=0)$ is a reference operator and $ U(m=0) = \mathbb{1}$. Thanks to this the state~(\ref{EQ:STATE}) can be put as
\begin{eqnarray}
\ket{\phi_{j}(m)} &=& \sum_{s=1}^d U_{js}^{*} (m) \ani{s}
\frac{1}{\sqrt{\hat{N}}} \hat \Pi \ket{\psi} \nonumber \\
&=& \sum_{s} U_{js}^{*} (m) \ket{\phi_{s}(0)}.
\end{eqnarray}
Therefore, (\ref{EQ:EXP_obs}) can be put as
\begin{eqnarray*}
\avg{\hat{Q}_{k} (m)} &=& \sum_{j,s,r=1}^d
\omega^{jk} \bra{\phi_{r}(0)} {U}_{jr} (m)
U_{js}^* (m) \ket{\phi_{s}(0)}.
\end{eqnarray*}
Let us introduce a matrix, denoted by $M$, whose elements are $M_{sr} = \inner{\phi_{r}(0)}{\phi_{s}(0)}$. Then
\begin{eqnarray}
\sum_{r,s=1}^d \bra{\phi_{r}(0)}
U_{jr} (m) U^*_{js} (m) \ket{\phi_{s}(0)}
\label{MATRIXORI}
\end{eqnarray}
becomes
\begin{eqnarray}
\sum_{r,s=1}^d {U}_{jr} (m)
M_{sr} U_{js}^* (m)
=\left[ U (m) M^T U^{\dagger} (m) \right]_{jj}.
\end{eqnarray}
Finally we arrive at
\begin{eqnarray}
\avg{\hat{Q}_{k}(m)} =\sum_{j} \omega^{jk} \left[ U (m) M^T U^{\dagger} (m) \right]_{jj},
\end{eqnarray}
where $M$ is a (positive
definite) Gramian matrix.
Its trace
is given by $\Tr M = \langle \hat \Pi \rangle \leq 1$.
We can normalize it to get $\mathfrak{M} =M/\langle\hat\Pi\rangle$, which is an admissible qudit density matrix.
Let us now turn back to qudits, and analyze the structure an expectation of the unitary observable~(\ref{SMALLq}).
First, consider a pure state $\ket{\xi}$. The expectation value reads
\begin{eqnarray}
\bra{\xi} \hat{q}_{k}(m) \ket{\xi}&=&
\sum_{j,r,s} \omega^{jk} U_{js}(m) {U}^{*}_{jr} (m)\inner{r}{\xi} \inner{\xi}{s} \nonumber \\
&=&\sum_{j,r,s} \omega^{jk} U_{js} (m) M_{rs}^{\xi} U^{*}_{jr} (m) \nonumber \\
&=&\sum_{j} \omega^{jk} \left[ U(m) M^{\xi T} U^{\dagger} (m) \right]_{jj},
\label{eq:exp_chi}
\end{eqnarray}
where we use $\ket{j (m)} = \sum_{r} U_{jr} (m) \ket{r}$ and introduce a density matrix $M^{\xi}$ for the state $\ket{\xi}$ of elements $M_{rs}^{\xi} = \inner{r}{\xi}\inner{\xi}{s}$.
If we replace $\ket{\xi}$ by a density matrix given by ${\varrho} = \sum_{\lambda} p_\lambda \ket{\xi_\lambda} \bra{\xi_\lambda}$, then the expectation~(\ref{eq:exp_chi}) becomes
\begin{eqnarray}
\Tr\varrho \hat{q}_{k}(m)
&=& \sum_{\lambda} p_\lambda \bra{\xi_\lambda} \hat{q}_{k}(m)
\ket{\xi_\lambda} \nonumber \\
&=&\sum_{j} \omega^{jk} \left[ U(m) M^{\varrho T} U^{\dagger} (m) \right]_{jj},
\end{eqnarray}
where matrix $M^{\varrho}$ has elements given by $M^{\varrho}_{rs}=\sum_{\lambda} p_\lambda \inner{r}{\xi_\lambda} \inner{\xi_\lambda}{s}$.
Therefore, (\ref{eq:qudit_homo_rate}) holds.
Obviously, such reasoning can be generalized to the case of (mixed) states describing correlated beams $A$ and $B$, in the way it is done in the main text.
For intensity-based observables, we have a similar relation
\begin{equation}
\frac{ \expval{{\hat \chi}_{k}(m)}{\psi}}
{\expval{\hat N}{\psi} } = \Tr
\hat{q}_{k}(m)\mathfrak{N},
\end{equation}
where $\mathfrak{N}$ is a possible two-qudit density matrix. Note that in general $\mathfrak{M} \neq
\mathfrak{N}$.
\section{Noise resistance of
Cauchy-Schwartz-like separability condition for Bright Squeezed Vacuum}
\label{NOISE}
Observables based on rates can in some cases allow a more noise resistant
entanglement detection than the ones based directly on intensities.
{\it Distortion noise.}---We take as our working example a $d\times d$ mode bright squeezed vacuum
in the presence of a specific type of noise, which can be treated
as distortion of the state,
which lowers the correlations between the beams.
\subsection{$2\times2$ mode bright squeezed vacuum plus noise}
We build our noise model in following steps. Let us introduce four squeezed vacuum states which are related with the Bell state basis for two qubits.
To make our notation concise let us denote by $k=0$ the polarization $H$ and by $k=1$ polarization $V$, and let us define that the index values follow modulo 2 algebra.
Then one can write down the following
\begin{equation}
\label{BSV-psi}
\ket{\Xi(m,l)} =\frac{1}{\cosh^2 \Gamma} \sum_{n=0}^{\infty} \frac{\tanh^n{\Gamma}}{n!}\left(\sum_k (-1)^{km}a^\dagger_kb^\dagger_{k+l}\right)^n|\Omega\rangle
\end{equation}
and define squeezed vacua related with the Bell states as $\ket{\Xi(0,0)} = \ket{\Phi^+}$, $\ket{\Xi(0,1)}=\ket{\Psi^+}$, $\ket{\Xi(1,0)}=\ket{\Phi^-}$, and $\ket{\Xi(1,1)}=\ket{\Psi^-}$. This notation may look too dense here, but it will help us further on.
Our noise model, which is an analog of the ``white noise" in the case of two qubits, can be defined as
\begin{equation}
\label{NOISE-1}
\varrho^{noise} = \frac{1}{4}(\ketbra{\Psi^-} + \ketbra{\Psi^+} + \ketbra{\Phi^-} + \ketbra{\Phi^+}).
\end{equation}
The following properties of the noise are essential.
For each $i$ and $j$,
\begin{equation}
\Tr
\hat S_i^{A}
\hat S_j^{B}\varrho^{noise} =0.
\end{equation}
That is the noise itself such that it leads to vanishing correlations between components of the Stokes parameters. This is easy to see when one recalls the local unitary transformations, say on side $A$, (replaced here by mode transformations) which link the three other two-qubit Bell states with the singlet. Simply they are equivalent to $\pi$ rotations of Bloch sphere of side $A$ with respect to axes $z$, $x$, and $y$.
The second property is
\begin{equation}
\expval{\hat \Pi^{A}\hat \Pi^B}{\Psi^-}
= \Tr \hat \Pi^{A}
\hat \Pi^{B} \varrho^{noise}.
\end{equation}
{\it For normalized Stokes operators.}---Let us start with the analysis of noise in terms
of the rate observables.
Let $v$ be the
\textit{visibility}, which determines the following
noisy state:
\begin{equation}
\varrho^{AB} = v\ketbra{\Psi^-} + (1-v)\varrho^{noise},
\end{equation}
where $0\leq v \leq1$.
We have to find the threshold $v$ above which our separability condition
$\sum_{i=1}^3|\langle \hat S_i^A\hat S_i^B\rangle|_{sep}
\leq \langle \hat\Pi^A\hat\Pi^B\rangle_{sep}$
fails to hold. It happens when
\begin{equation}
\label{MULTINOISE}
v\sum_{i=1}^{3}
\left| \expval{{\hat S}_i^{A} { \hat S}_i^{B}}{\Psi^-}\right| >
\expval{\hat \Pi^{A}\hat \Pi^B}{\Psi^-}.
\end{equation}
This will be our measure of the resilience with respect to the noise.
Applying the technical facts that for $\ket{\Psi^-}$ one has $\expval{{\hat S}_i^{A}
{ \hat S}_i^{B}}{\Psi^-}=-\expval{({\hat S}_i^{A})^2}{\Psi^-}$ and $ \expval{
\hat \Pi^{A}\hat \Pi^B
}{\Psi^-} = \expval{
\hat \Pi^{A}
}{\Psi^-}$,
one gets
$$\sum_{i = 1}^3
\left| \expval{\hat S_{{i}}^{A}
\hat S_{{i}}^{B}}{\Psi^-} \right| = \expval{\hat \Pi^A +\hat \Pi^A
\frac{2}{\hat N^A}\hat \Pi^A}{\Psi^-} $$ and
the condition for detection of entanglement reads
\begin{equation}
v
\expval{\hat \Pi^A + \hat \Pi^A
\frac{2}{\hat N^A}\hat \Pi^A}{\Psi^-} > \expval{
\hat\Pi^{A}
}{\Psi^-}.
\end{equation}
The threshold visibility $v_{crit}$ is given by
\begin{equation}
\label{NOISEcond}
v_{crit} = \frac{\expval{\hat \Pi^A}{\Psi^-}}{ \expval{\hat \Pi^A
+ \hat{\Pi}^A\frac{2}{\hat N^A}\hat \Pi^A}{\Psi^-}}.
\end{equation}
The respective terms of (\ref{NOISEcond}) are given by
\begin{equation}
\label{NONVAC}
\expval{\hat \Pi^A}{\Psi^-} = 1 -\frac{1}{\cosh^{4}\Gamma}
= 1- \sech^{4}\Gamma
\end{equation}
that follows from the definition of $\langle \hat{\Pi}^A\rangle $ and
\begin{eqnarray}
&& \expval{ \hat{\Pi}^A\frac{1}{\hat N^A} \hat{\Pi}^A}{\Psi^-} \nonumber \\
&&= \frac{2 \tanh^2\Gamma }{\cosh^{4}\Gamma}
{}_3F_{2}(1,1,3;2,2;\tanh^2 \Gamma),
\end{eqnarray}
where ${}_3F_2(1,1,3;2,2;\tanh^2 \Gamma)$ is generalized hypergeometric function.
{\it For standard Stokes operators.}---Following the same reasoning for
observables based rates
the threshold visibility $v_{crit}^{old}$ for observables
based on intensities
is given by
\begin{equation}
\label{VIZold}
v_{crit}^{old} = \frac{\expval{(\hat N^{A})^{2}}{\Psi^-}}{
\expval{\hat N^A(\hat N^A + 2)}{\Psi^-}
}.
\end{equation}
We have
\begin{equation}
\expval{\hat N^A}{\Psi^-} =
2\sinh^2 \Gamma
\end{equation}
and
\begin{eqnarray}
\expval{(\hat{N}^{A})^{2}}{\Psi^-} &=& \frac{2 \tanh^2 \Gamma}{\cosh^4 \Gamma} \frac{2\tanh^2 \Gamma+1}{(1-2\tanh^2 \Gamma)^4} \nonumber \\
&=& \sinh^2 \Gamma ( 3 \cosh 2\Gamma -1 ).
\label{NEWTERM}
\end{eqnarray}
The form of (\ref{NEWTERM}) was
obtained as follows. Let us put $x =\tanh^2{\Gamma}$, and $c=\cosh^4{\Gamma}$. We have
\begin{multline}
\langle (\hat{N}^{A})^{2}\rangle =
\frac{1}{c} \sum_{n=0}^{\infty}x^n(n+1)n^2 =
\frac{x}{c}\frac{d^2}{d x^2} \bigg(\sum_{n=0}^{\infty}nx^{n+1}
\bigg) \\=
\frac{x}{c}\frac{d^2}{dx^2}\bigg(x^2\frac{d}{dx}\sum_{n=0}^{\infty}x^n \bigg) =
\frac{x}{c}\frac{d^2}{dx^2}\bigg(x^2\frac{d}{dx}\bigg(\frac{1}{1-x}\bigg) \bigg) \\= \frac{2x(2x+1)}{c(1-x)^4}.
\end{multline}
Thus, the threshold visibility in function of the
amplification gain $v_{crit}^{old}(\Gamma)$ for the ``macroscopic singlet" $\ket{\Psi^-}$
is
\begin{equation}
v_{crit}^{old}(\Gamma)= \frac{3\cosh{2\Gamma} -1}{3\cosh{2\Gamma} +3}.
\end{equation}
We compare the critical visibilities obtained with the two approaches (normalized vs. standard Stokes parameters) in Fig. \ref{fig:viz_viZ}.
\begin{figure}
\caption{Comparison of critical visibilities
to detect entanglement via the Cauchy-like condition, for four-mode squeezed vacuum $\ket{\Psi^-}
\label{fig:viz_viZ}
\end{figure}
\subsection{Unitary observables for $d$-mode}
{\it Multimode bright squeezed vacuum.}---The bright squeezed vacuum is a state of light
of
undefined photon number which has, due to entanglement, perfect EPR correlations
of numbers of photons between specific modes reaching $A$ and $B$.
Such an entanglement can be observed in multimode
parametric down-conversion emission. The
interaction
Hamiltonian of the process, for a classical pump, is essentially
$\hat H =i \gamma \sum_{j=0}^{d-1}\hat a_j^{\dagger}\hat b_j^{\dagger} + h.c.$
where $\gamma$ is the coupling constant proportional to a pump power. Thus,
$d\times d$ mode (bright) squeezed vacuum state is given by
\begin{equation}
\label{MULTIVAC}
\ket{\Psi_{BSV}^{d}} =\frac{1}{\cosh^d \Gamma} \sum_{n=0}^{\infty}
\sqrt{\frac{(n+d-1)!}{n!(d-1)!}} \tanh^n\Gamma \ket{\psi^{n}_d},
\end{equation}
where $\Gamma = \gamma t$ and $t$ is the interaction time, and
\begin{eqnarray}\label{PSIN}
\ket{\psi^n_d}= \sqrt{\frac{n!(d-1)!}{(n+d-1)!}} \frac{1}{n!} \left(\sum_{j=0}^{d-1}\hat a_j^{\dagger}\hat
b_j^{\dagger}\right)^n\ket{\Omega}.
\end{eqnarray}
{\it Noise model.}--- If we consider the unitary observables, our noise model can look as follows: we build our noise model in following similar steps as for the $d=2$ case.
Let us now index $k$ stand for local modes $k=0,1,...,d-1$ and we shall the modulo $d $ algebra for it.
Then one can write down the following
\begin{equation}
\label{BSV-psi_d}
\ket{\Xi^d(m,l)} =\frac{1}{\cosh^d \Gamma} \sum_{n=0}^{\infty} \frac{\tanh^n{\Gamma}}{n!}\left(\sum_k \omega^{km}a^\dagger_kb^\dagger_{k+l}\right)^n|\Omega\rangle
\end{equation}
with $m$ and $l$ taking values $0,1,...,d-1$. Note that these squeezed $d$-mode vacua are analogs of the following Bell basis for a pair of qudits:
$\frac{1}{\sqrt{d}}\sum_k \omega^{km}|k\rangle\otimes|k+l\rangle$.
Our noise model is defined as
\begin{equation} \label{NOISE-2}
\varrho^{noise} = \frac{1}{d^2} \sum_{m,l} \ketbra{\Xi^d(m,l)}.
\end{equation}
The following properties of the noise are essential for us.
For each $i$ and $j$
\begin{equation}
\label{NOISE-PROPERTY}
\Tr
\hat{Q}_i^{A}(m)
\hat Q_j^{B\dagger}(m')\varrho^{noise} =0,
\end{equation}
and the second property is
\begin{equation}
\Tr (\hat \Pi^{A} \hat \Pi^{B} \varrho^{noise})=\expval{\hat \Pi^{A}\hat \Pi^B}{\Psi^d_{BSV}}.
\end{equation}
We have the same relation for observables based on intensities.
{\it Noise resistance.}---Applying this model we get that
entanglement detection is possible with the Cauchy-like condition for observables based on rates, in the case of $\ket{\Psi^d_{BSV}}$ mixed with the noise, if the threshold
visibility $v_{crit}$ fulfills
\begin{equation}
\label{VIZnewUNI}
v_{crit} = \frac{\expval{\hat
\Pi^A}{\Psi^d_{BSV}}}{
\expval{\hat \Pi^A + \hat
\Pi^A\frac{d}{\hat N^A}\hat \Pi^A}{\Psi^d_{BSV}}}.
\end{equation}
In case of observables based on intensities, we get
\begin{equation}
\label{VIZoldUNI}
v_{crit}^{old} = \frac{\expval{(\hat{N}^{A})^{2}}{\Psi^d_{BSV}}}{
\expval{\hat N^A(\hat N^A + d)}{\Psi^d_{BSV}}
}.
\end{equation}
\subsubsection{$3\times 3$ mode bright squeezed vacuum}
In case of observables based on rates, the respective terms in (\ref{VIZnewUNI}) are as follows:
\begin{eqnarray}
&&\expval{\hat \Pi^A\frac{3}{\hat N^A}\hat \Pi^A}{\Psi^3_{BSV}} \nonumber \\
&&= \frac{1}{\cosh^{6}\Gamma} 9 \tanh^2\Gamma
_3F_2(1,1,4;2,2;\tanh^2
\Gamma)
\end{eqnarray}
and
\begin{equation}
\expval{\hat \Pi^A}{\Psi^3_{BSV}}
= 1- \sech^{6} \Gamma.
\end{equation}
For observables based on intensities in (\ref{VIZoldUNI}) we have
\begin{equation}
\expval{\hat N^A}{\Psi^3_{BSV}} =
3\sinh^2 \Gamma
\end{equation}
and
\begin{eqnarray}
\label{expN2UNI}
\expval{(\hat{N}^{A})^{2}}{\Psi^3_{BSV}}
&=&\frac{1}{\cosh^{6}\Gamma}
\frac{3\tanh^2\Gamma(3\tanh^2\Gamma+1)}{(1-\tanh^2\Gamma)^5} \nonumber \\
&=& 3\sinh^2\Gamma + 12\sinh^4\Gamma
\end{eqnarray}
The first equality of (\ref{expN2UNI}) can be obtained as (here, $x = \tanh^2\Gamma$ and $c =\cosh^6\Gamma$):
\begin{eqnarray}
\langle (\hat {N}^{A})^{2}\rangle &=& \frac{1}{c} \sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{2}n^2x^n \nonumber \\
&=& \frac{x}{2c}\frac{d^2}{dx^2} \bigg(\sum_{n=0}^{\infty} n(n+2)x^{n+1} \bigg) \nonumber \\
&=& \frac{x}{2c}\frac{d^2}{dx^2}\bigg(\frac{d}{dx}\bigg(\sum_{n=0}^{\infty} nx^{n+2}\bigg) \bigg) \nonumber \\
&=& \frac{x}{2c}\frac{d^3}{dx^3}\bigg(x^3 \frac{d}{dx}\sum_{n=0}^{\infty}x^n \bigg) \nonumber \\
&=& \frac{x}{2c}\frac{d^3}{dx^3}\bigg(x^3\frac{d}{dx}\bigg(\frac{1}{1-x}\bigg) \bigg) \nonumber \\
&=& \frac{1}{c}\frac{3x(3x+1)}{(1-x)^5}.
\end{eqnarray}
The threshold visibility in function of the
amplification gain, $v_{crit}(\Gamma)$, for the macroscopic
singlet $\ket{\Psi^3_{BSV}}$
is presented in Fig.~\ref{FIG:suppFigS2}.
\begin{figure}
\caption{Noise resistance for $\ket{\Psi^3_{BSV}
\label{FIG:suppFigS2}
\end{figure}
\section{Derivation of some formulas used in Sec. \ref{NOISE}, and to obtain the general Cauchy-like separability condition}
\subsection{Formula 1}
We shall show the following:
\begin{eqnarray}
\sum_{m=0}^{d} \sum_{k=1}^{d-1} |\hat{Q}_{k} (m)|^2
= (d-1) \left( \hat \Pi + \hat \Pi \frac{d}{\hat N}\hat \Pi \right).
\label{apx:abssquare}
\end{eqnarray}
Note that this is a generalization of the identity $\sum_{i=1}^3\hat{S}_i^2= \hat{\Pi} + \hat{\Pi} \frac{2}{\hat{N}}\hat{\Pi}$.
The field operators involving the unbiased interferometers, within the approach with rates (15) in the main text can be put as
\begin{eqnarray*}
\hat{Q}_{k} (m) = \sum_{l,l'=1}^{d} \left( \sum_{j=1}^{d} \omega^{jk} U_{j l'}(m) U^{*}_{j l}(m) \right) \hat{\Pi}
\frac{\cre{l'} \ani{l}}{\hat{N}} \hat \Pi,
\end{eqnarray*}
and the formula for $\hat{Q}_{k}^{\dagger}$ is the Hermitian conjugate of the above.
The following relations
\begin{eqnarray}
\sum_{j=1}^d {U}_{j l'}(m) U^*_{j l}(m) \omega^{jk} =
[ \hat{q}_{k} (m) ]_{l l'}
\end{eqnarray}
and
\begin{eqnarray}
\sum_{j=1}^d U^*_{j l'}(m) {U}_{j l}(m) \omega^{-jk} =
[ {\hat{q}^{\dagger}_{k} (m)}]_{l l'}
\end{eqnarray}
lead to
\begin{eqnarray}
\hat{Q}_{k} (m) = \sum_{l,l'=1}^{d} [ \hat{q}_{k} (m) ]_{l l'} \hat{\Pi} \frac{\cre{l'} \ani{l}}{\hat{N}} \hat \Pi,
\end{eqnarray}
where $\hat{q}_k(m)$ are the qudit operators (\ref{SMALLq}).
Therefore, we have
\begin{eqnarray}
&&\sum_{m=0}^{d} \sum_{k=1}^{d-1}|\hat{Q}_{k} (m)|^2 = \hat \Pi \frac{1}{\hat{N}} \sum_{l l' n n'=1}^d \left( \sum_{m=0}^{d}\sum_{k=1}^{d-1} \right. \nonumber \\
&&\times [ \hat{q}_{k} (m) ]_{l l'} [{\hat{q}^{\dagger}}_{k} (m) ]_{n n'} \bigg) \cre{l'} \ani{l} \cre{n} \ani{n'} \frac{1}{\hat{N}}\hat \Pi.
\end{eqnarray}
As the operators $\frac{1}{\sqrt{d}}\hat{q}_k(m)$ and $\hat{q}_0(0)=\openone$ form an orthonormal basis in the Hilbert-Schmidt space of $d \times d$ matrix, we have
$$
\delta_{l l'} \delta_{n n'} +
\sum_{m=0}^{d} \sum_{k=1}^{d-1} [ \hat{q}_{k} (m)]_{l
l'}
[ {\hat{q}^{\dagger}}_{k} (m) ]_{n n'} = d
\delta_{ln}\delta_{l'n'}.
$$
All that, and $[a_{i}, a_{j}^{\dagger}] = \delta_{ij}$, allow one to perform the following calculation:
\begin{eqnarray} \label{MULT}
&&\sum_{m=0}^{d} \sum_{k=1}^{d-1} |\hat{Q}_{k} (m)|^2 \nonumber \\
&=& \hat \Pi \frac{1}{\hat{N}} \sum_{l l' n n'=1}^d [ d\delta_{ln}\delta_{l'n'} - \delta_{ll'}\delta_{nn'}]
\cre{l'} \ani{l} \cre{n} \ani{n'} \frac{1}{\hat{N}}\hat \Pi \nonumber \\
&=& \hat \Pi \frac{1}{\hat{N}} \left[ -\sum_{l n} \cre{l} \ani{l} \cre{n} \ani{n} + d\sum_{l l'} \cre{l'} \ani{l} \cre{l} \ani{l'} \right]
\frac{1}{\hat{N}}\hat \Pi \nonumber \\
&=& \hat \Pi \frac{1}{\hat{N}} \left[ -\hat{N}^2 + d \sum_{l l'} \cre{l'} \ani{l'} ( \cre{l} \ani{l} +1 ) - d \sum_{l} \cre{l} \ani{l} \right]
\frac{1}{\hat{N}}\hat \Pi \nonumber \\
&=& \Pi \frac{1}{\hat{N}} \left[ -\hat{N}^2 + d\hat{N}^2 + d^2\hat{N} - d\hat{N} \right] \frac{1}{\hat{N}}\hat \Pi \nonumber \\
&=& \hat \Pi \left[ (d-1) + \frac{d(d-1)}{\hat{N}} \right] \hat \Pi.
\end{eqnarray}
Thus, (\ref{apx:abssquare}) holds.
An analogue relation for the observables involving intensities, which reads
\begin{eqnarray}
\sum_{m=0}^{d} \sum_{k=1}^{d-1} | \hat{\chi}_{k} (m)|^2 = (d-1)\hat N(\hat N+ d),
\label{apx:abssquareint}
\end{eqnarray}
can be obtained by similar steps. It is a generalization of (\ref{RELOLD}).
\subsection{Formula 2}
\label{TECHNICAL_LEMA}
We here calculate the expressions which enter
of Cauchy-Schwartz-like separability conditions
based on rates~(14) and intensities~(16) in the main text
for a $d\times d$ mode bright squeezed vacuum. Some of the formulas are also used in the discussion of noise resistance.
Let us consider first the condition~(16) in the main text: its LHS and RHS read
\begin{eqnarray}
\label{WARUJSINGLET}
\rm{LHS} &=& \sum_{m=0}^d \sum_{k=1}^{d-1}
\left|\expval{\hat \chi_{{k}}^A(m)\hat
\chi_{{k}}^{{B\dagger}}(m)}{\Psi^d_{BSV}} \right|, \nonumber \\
\rm{RHS} &=& (d-1)\expval{ (\hat N^{A})^{2}}{\Psi^d_{BSV}}.
\end{eqnarray}
To get the formula for RHS we used
\begin{equation}
\expval{\hat N^A \hat N^B}{\Psi^d_{BSV}} =
\expval{ (\hat N^{A})^{2}}{\Psi^d_{BSV}}.
\end{equation}
The action of
$\hat \chi_k^B(m = 0)$
on an unnormalized
$\ket{\psi^n_d}$ of (\ref{PSIN}), which we put as $\ket{\phi^n} = \left(\sum_{j=1}^d\hat
a_j^{\dagger}\hat
b_j^{\dagger}\right)^n\ket{\Omega}$, is as follows
\begin{equation}
\label{ROBO}
\hat \chi_k^B
\left(\sum_{j=1}^d\hat a_j^{\dagger}\hat
b_j^{\dagger}\right)^n\ket{\Omega}
= \left(\sum_{j=1}^d\omega^{jk}
\hat b_j^{\dagger}\hat b_j\right)
\left(\sum_{j=1}^d\hat a_j^{\dagger}\hat
b_j^{\dagger}\right)^n\ket{\Omega}.
\end{equation}
Let us denote as
$\hat X \equiv \hat{\chi}_k^B =\sum_{j=1}^d \omega^{jk}\hat b_j^{\dagger}\hat b_j$ and $\hat Y \equiv
\sum_{j=1}^d\hat a_j^{\dagger}\hat b_j^{\dagger}$.
Then, we have
\begin{equation}
[\hat X, \hat Y] =
\left[\sum_{j=1}^d \omega^{jk}\hat b_j^{\dagger}\hat b_j,
\sum_{j=1}^d\hat a_j^{\dagger}\hat b_j^{\dagger} \right] =
\sum_{j=1}^d\omega^{jk}\hat a_j^{\dagger}\hat
b_j^{\dagger}.
\end{equation}
Next, we use the algebraic fact that if
$[[\hat X,\hat Y],\hat Y] =0$, then the following holds $[\hat X,\hat
Y^n] = n[\hat X, \hat Y]\hat Y^{n-1}$ and
$ \hat X\hat Y^n = \hat Y^n\hat X + n[\hat X, \hat Y]\hat Y^{n-1}$.
Applying this relation to (\ref{ROBO}) we get
\begin{eqnarray}
&&{\hat \chi}_k^B
\left(\sum_{j=1}^d\hat a_j^{\dagger}\hat b_j^{\dagger}\right)^n\ket{\Omega}
\nonumber
= \\
&&n \left(\sum_{j=1}^d \omega^{jk}\hat a_j^{\dagger}\hat b_j^{\dagger}\right)
\left(\sum_{j=1}^d \hat a_j^{\dagger} \hat b_j^{\dagger}\right)^{n-1}\ket{\Omega},
\label{apx:relation}
\end{eqnarray}
where we use $\sum_{j=1}^d \omega^{jk}\hat b_j^{\dagger}\hat b_j \ket{\Omega} =0$.
We have the same relation if we replace $\hat \chi_k^B$ by $\hat \chi_k^A$ in (\ref{apx:relation}), i.e.,
\begin{equation}
\label{RESULT}
{\hat \chi}_k^A \left(\sum_{j=1}^d
\hat a_j^{\dagger}\hat b_j^{\dagger}\right)^n\ket{\Omega} =
{\hat \chi }_k^B \left(\sum_{j=1}^d \hat a_j^{\dagger}
\hat b_j^{\dagger}\right)^n\ket{\Omega}.
\end{equation}
The identity (\ref{RESULT}) holds for all $m=0,1,\dots,d$. In the case of $m\neq 0$ the formulas look the same if one employs creation and annihilation operators related with the interferometers $U(m)$ for $A$ and ${U}^\dagger (m)$ for $B$, and the fact that $\sum_{j=1}^d \hat a_j^{\dagger}
\hat b_j^{\dagger}= \sum_{j=1}^d \hat a_j^{\dagger}(m)
\hat b_j^{\dagger}(m)$, which is at the root of EPR correlations of the state. All that, and the identity (\ref{apx:abssquareint}), lead to
\begin{eqnarray}
\label{RESULT1}
\text{LHS} &=& \sum_{m=0}^d \sum_{k=1}^{d-1} \expval{|{\hat \chi}_{{k}}^A(m)|^2}{\Psi^d_{BSV}} \nonumber \\
&=& (d-1)\expval{\hat N^A(\hat N^A+d)}{\Psi^d_{BSV}}.
\end{eqnarray}
Thus, we get
\begin{eqnarray}
\label{WARUNEK}
\expval{\hat N^A(\hat N^A+d)}{\Psi^d_{BSV}}
> \expval{ ({\hat N^{{A}}})^2 }{\Psi^d_{BSV}}, \nonumber \\
\end{eqnarray}
for every $\Gamma$.
A reasoning following similar steps leads to a violation of the Cauchy-Schwartz-like separability condition (14) in the main text for observables involving rates, as for the bright squeezed vacuum we have in this case:
\begin{eqnarray}
\label{TECHNICAL_NEW}
\expval{(\hat \Pi^A + \hat{\Pi}^A\frac{d}{\hat N^A}\hat \Pi^A)}{\Psi^d_{BSV}} >\expval{\hat \Pi^A }{\Psi^d_{BSV}}, \nonumber \\
\end{eqnarray}
where $\hat \Pi^A = (\hat \Pi^A)^2$ was used.
\subsection{Property (\ref{NOISE-PROPERTY}) of the noise model}
The essential property of our noise is that for each $i$ and $j$ we get
\begin{equation}
\label{COOLPROP}
\Tr
\hat{\chi}_i^{A}(m)
\hat{\chi_j}^{B\dagger}(m')\varrho^{noise} =0,
\end{equation}
and we have the same relation for observables based on rates. We shall prove (\ref{COOLPROP}) for $d>2$. For simplicity we will use the intensity approach. The proof for the rate observables is similar.
For an arbitrary $d$ all Bell-like maximally entangled states $ \ket{\Xi^d(k,l)}$ are linked by a unitary transformation that acts on one subsystem. The transformation is as follows:
\begin{eqnarray}
\label{TRANSSS}
\hat{ \mathcal U}^{\dagger}(k,l)\hat b^{\dagger}_n\hat{\mathcal U}(k,l) = \sum_{i=0}^{d-1}U(k,l)_{ni}\hat b^{\dagger}_i = \omega^{nk}\hat b^{\dagger}_{n+l},
\end{eqnarray}
where $\hat b^{\dagger}_n$ stands for $k=0$ and $l=0$. Respectively, for annihilation operators we have:
$
\hat{\mathcal U}^{\dagger}(k,l)\hat b_n\hat {\mathcal U}(k,l) = \sum_{i=0}^{d-1}\bar{U}(k,l)_{ni}\hat b_i =
\omega^{-nk} \hat b_{n+l}.
$
Using transformation (\ref{TRANSSS}) we can present any $ \ket{\Xi^d(k,l)} $ as follows:
\begin{eqnarray}
\label{BSV-psi_d2}
&&\ket{\Xi^d(k,l)} \nonumber \\
&=& \frac{1}{\cosh^d \Gamma} \sum_{n=0}^{\infty} \frac{\tanh^n{\Gamma}}{n!}\left(\sum_m a^\dagger_m \hat{\mathcal U^{\dagger}}(k,l) b^\dagger_{m}\hat{\mathcal U}(k,l)\right)^n|\Omega\rangle \nonumber \\
&=& \hat{\mathcal U^{\dagger}}(k,l) |\Psi^d_{BSV}\rangle.
\end{eqnarray}
Because this transformation is unitary we can replace the action of (\ref{TRANSSS}) on the state by its action on the observables.
Thus, in order to prove (\ref{COOLPROP}) we shall show that for any $i,j\neq 0$
\begin{equation}
\label{NOISE_1} \langle{\Psi_{BSV}^d}|
\sum_{k,l=0}^{d-1} \hat{\chi}_i^{A{}}(m)
\hat{\mathcal U}(k,l)\hat{\chi}_j^{B\dagger}(m')\hat{\mathcal U}^{\dagger}(k,l)|{\Psi_{BSV}^d}\rangle = 0.
\end{equation}
It turns out that the above holds because of the following operator identity
\begin{equation}
\label{NOISE_MZ}
\sum_{k,l=0}^{d-1}
\hat{\mathcal{ U}}(k,l)\hat{\chi}_j^{B \dagger}(m')\hat{\mathcal{ U}}^{\dagger}(k,l) = 0.
\end{equation}
The reverse of transformation
(\ref{TRANSSS}) can be expressed in the following way:
\begin{equation}
\hat{ \mathcal U}(k,l)\hat b^{\dagger}_n\hat{\mathcal U}^\dagger (k,l) = \sum_{i=0}^{d-1}U^{-1}(k,l)_{ni}\hat b^{\dagger}_i = \omega^{-nk}\hat b^{\dagger}_{n-l}.
\end{equation}
Note that $U^{-1}$ can be decomposed as follows $
U^{-1}= Z^kX^l$,
where $(Z^k)_{rn} = \delta_{rn}\omega^{-nk}$, $(X^l)_{ni} = \delta_{(n-l)i} $.
Using the notation introduced above we get
\begin{eqnarray}
&&\hat{\mathcal{U}}(k,l)\hat{\chi}^{B \dagger}_j \hat{\mathcal{U}^{\dagger}}(k,l) \nonumber \\
&=& \hat{\mathcal{U}}(k,l) \sum_{r=0}^{d-1}\omega^{-rj}\hat{b}^{\dagger}_r(m^{\prime}) \hat{b}_r(m^{\prime})\hat{\mathcal U} ^{\dagger}(k,l) \nonumber \\
&=& \sum_{r=0}^{d-1}\omega^{rj}\sum_{s=0}^{d-1} U_{rs}(m^{\prime})\sum_{t=0}^{d-1}(Z^kX^l)_{st}\hat b^{\dagger}_t \nonumber \\ &\times& \sum_{s^{\prime} =0}^{d-1} U^*_{rs^{\prime}}(m^{\prime})\sum_{t^{\prime} =0}^{d-1}(Z^kX^l)^*_{s^{\prime}t^{\prime}} \hat b_{t^{\prime}}.
\label{in_prog1}
\end{eqnarray}
We have
\begin{eqnarray}
\label{in_prog2}
& &\sum_{k,l=0}^{d-1}(Z^kX^l)_{st} (\bar Z^kX^l)_{s^{\prime}t^{\prime}} \nonumber \\
&=& \sum_{k,l=0}^{d-1} \omega^{-sk}\delta_{(s-l)t} \omega^{s^{\prime}k} \delta_{(s^{\prime}-l) t^{\prime}} \nonumber \\
&=& \sum_{l=0}^{d-1}\delta_{(s-l)t} \delta_{(s^{\prime}-l) t^{\prime}}\sum_{k=0}^{d-1} \omega^{-k(s-s^{\prime})} \nonumber \\
&=& \delta_{s,s^{\prime}} \delta_{t,t^{\prime}}.
\end{eqnarray}
Combining (\ref{in_prog1}) and (\ref{in_prog2}) we get
\begin{eqnarray}
&&\sum_{k,l=0}^{d-1}\hat{\mathcal{U}}(k,l)\hat{\chi}^{B \dagger}_j \hat{\mathcal{U}^{\dagger}}(k,l) \nonumber \\
&=& \sum_{r=0}^{d-1}\omega^{-rj} \sum_{s,t=0}^{d-1} U_{r s}(m^{\prime}) U^*_{rs}(m^{\prime}) \hat b_{t}^{\dagger}\hat b_{t} \nonumber \\
&=& \sum_{r=0}^{d-1}\omega^{-rj}\hat N^B =0.
\end{eqnarray}
Thus the identity (\ref{NOISE_MZ}) holds.
\end{document} |
\begin{document}
\frontmatter
\title[Cohomology of the moduli of cubic threefolds]{Cohomology of the moduli space of cubic threefolds and its smooth models}
\author[S. Casalaina-Martin]{Sebastian Casalaina-Martin}
\address{University of Colorado, Department of Mathematics, Boulder, CO 80309}
\email{[email protected]}
\author[S. Grushevsky]{Samuel Grushevsky}
\address{Stony Brook University, Department of Mathematics, Stony Brook, NY 11794-3651}
\email{[email protected]}
\author[K. Hulek]{Klaus Hulek}
\address{Institut f\"ur Algebraische Geometrie, Leibniz Universit\"at Hannover, 30060 Hannover, Germany}
\email{[email protected]}
\author[R. Laza]{Radu Laza}
\address{Stony Brook University, Department of Mathematics, Stony Brook, NY 11794-3651}
\email{[email protected]}
\begin{abstract}
We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily--Borel and toroidal compactifications of the ball quotient model, due to Allcock--Carlson--Toledo. Our starting point is Kirwan's method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli of cubic surfaces is discussed in an appendix.
\end{abstract}
\date{\today}
\maketitle
\tableofcontents
\mainmatter
\chapter{Introduction}\label{sec:intro}
\
\blfootnote{
2010 \emph{Mathematics Subject Classification}. 14J30, 14J10, 14L24, 14F25, 55N33, 55N25.
Research of the first author is supported in part by grants from the Simons Foundation (317572) and the NSA (H98230-16-1-0053). Research of the second author is supported in part by NSF grants DMS-15-01265 and DMS-18-02116. Research of the third author is supported in part by DFG grant Hu-337/7-1. Research of the fourth author is supported in part by NSF grants DMS-12-54812 and DMS-18-02128. The first author would like to thank the Institut f\"ur Algebraische Geometrie at Leibniz Universit\"at for support during the Fall Semester 2017. The first and third authors are also grateful to MSRI Berkeley, which is supported by NSF Grant DMS-14-40140, for providing excellent working conditions in the Spring Semester 2019.
}
Cubic threefolds and their moduli are one of the most studied objects in algebraic geometry. In previous work we have investigated the relationship among various compactifications of the moduli space $\mathcal{M}$
of smooth cubic threefolds, and the purpose of this paper is now to determine the cohomology of these moduli spaces.
The first compactification which one naturally encounters is, as for all hypersurfaces, the GIT compactification $\calM^{\operatorname{GIT}}$ (as studied by Allcock~\cite{allcock} and Yokoyama~\cite{yokoyama}).
It is interesting to note that recently Liu--Xu~\cite{LiuXu} showed that for cubic threefolds (and also for cubic surfaces) $\calM^{\operatorname{GIT}}$ is equal to the moduli space of $K$-stable cubics, thus
providing a differential-geometric perspective on the GIT moduli of cubics.
What makes the case of cubic threefolds especially interesting is the presence of two period maps which lead to further natural compactifications.
The first of these period maps is given by the intermediate Jacobian and was already studied by Clemens--Griffiths~\cite{cg}. The Torelli theorem holds for this period map for cubic threefolds, and one obtains an immersion $\mathcal{M}\hookrightarrow \mathcal{A}_5$ into the moduli space $\mathcal{A}_5$ of principally polarized abelian varieties of dimension $5$. Taking the closure $\overline {IJ}\subset \overline\mathcal{A}_5$ of the locus $IJ$ of intermediate Jacobians in suitable compactifications $\overline\mathcal{A}_5$ of $\mathcal{A}_5$, one obtains geometrically meaningful compactifications of $\mathcal{M}$ (see~\cite{cubics}).
Perhaps even more surprising is that one can construct a $10$-dimensional ball quotient model $\mathcal{B}/\Gamma$ of $\mathcal{M}$, by using the periods of cubic fourfolds (cf.~Allcock--Carlson--Toledo~\cite{act}).
This ball quotient admits naturally the Baily--Borel compactification $(\calB/\Gamma)^*$ and the (unique) toroidal compactification~$\overline{\calB/\Gamma}$, which thus provide two further compactifications of the moduli of smooth cubic threefolds.
It is, in particular, these models which we will study in this paper.
The spaces $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$ are closely related, as explained in~\cite{act} and~\cite{ls}. Briefly, there exists a space $\widehat\mathcal{M}$ dominating both $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$. In fact $\widehat\mathcal{M}$ plays two roles: on the
one hand it is the partial Kirwan blowup of the point $\Xi\in\calM^{\operatorname{GIT}}$
corresponding to the chordal cubic, and on the other hand it is the Looijenga $\mathbb{Q}$-factorialization (cf.~\cite{l1}) associated to the hyperelliptic divisor $\mathcal{H}_h^*\subset(\calB/\Gamma)^*$.
Both compactifications $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$ are singular. The toroidal compactification~$\overline{\calB/\Gamma}$ is a natural (partial) desingularization of~$(\calB/\Gamma)^*$, while a natural (partial) desingularization of $\calM^{\operatorname{GIT}}$ is provided by Kirwan's blowup $\calM^{\operatorname{K}}$, which is smooth up to finite quotient singularities. By construction there is a factorization $\mathcal{M}^K\to\widehat \mathcal{M}\to \calM^{\operatorname{GIT}}$, as $\widehat \mathcal{M}$ is nothing but an intermediary step in the construction of the Kirwan blowup $\mathcal{M}^K$.
The relationship among these compactifications and $\overline{IJ}$ was the subject of our previous works~\cite{cml,cubics, prym}.
In this paper we investigate this relationship further by turning our attention to the cohomology of these moduli spaces. More precisely, we determine the (intersection) cohomology of the compactifications $\calM^{\operatorname{GIT}}$, $\widehat\mathcal{M}$, $\calM^{\operatorname{K}}$, $(\calB/\Gamma)^*$ and $\overline{\calB/\Gamma}$:
\begin{teo}\label{teo:betti}
The Betti numbers of $\calM^{\operatorname{K}}$ and $\overline{\calB/\Gamma}$, and the intersection Betti numbers of $\calM^{\operatorname{GIT}}$, $\widehat\mathcal{M}$, and $(\calB/\Gamma)^*$ are as follows:
\begin{equation}
\renewcommand*{\arraystretch}{1.2}
\begin{array}{l|ccccccccccc}
\hskip2cm j&0&2&4&6&8&10&12&14&16&18&20\\\hline
\dim H^j(\calM^{\operatorname{K}})&1&4&6&10&13&15&13&10&6&4&1\\
\dim IH^j(\calM^{\operatorname{GIT}})&1&1&2&3&4&5&4&3&2&1&1\\
\dim IH^j(\widehat\mathcal{M})&1&2&3&5&6&8&6&5&3&2&1\\
\dim IH^j((\calB/\Gamma)^*)&1&2&3&5&6&7&6&5&3&2&1\\
\dim H^j(\overline{\calB/\Gamma})&1&4&6&10&13&15&13&10&6&4&1
\end{array}
\end{equation}
while all the odd degree (intersection) cohomology vanishes.
\end{teo}
\begin{conv}
As usual with these type of cohomological computations, the cohomology is always with $\mathbb{Q}$ coefficients. This will be our convention throughout the paper.
\end{conv}
\begin{rem}
The easier related case of the moduli space of cubic surfaces is discussed in Appendix~\ref{sec:surfaces}. Specifically, as in the case of cubic threefolds, there exists both a GIT model (one of the standard examples in classical Invariant Theory) and a ball quotient model for the moduli space of cubic surfaces (due to Allcock--Carlson--Toledo~\cite{ACTsurf}; see also~\cite{DvGK}). However, in this lower-dimensional case, the two models are isomorphic (cf.~\cite{ACTsurf}). The cohomology of the GIT model and of its partial Kirwan desingularization were worked out by Kirwan as illustrations of her general theory (see~\cite{kirwanhyp}, and also~\cite{ZhangCubic}). On the other hand, to our knowledge, Theorem~\ref{T:CubSurfH}, which computes the cohomology of the associated toroidal compactification of the ball quotient model for cubic surfaces, and Theorem~\ref{T:Naruki}, which computes the cohomology of the Naruki compactification for the surface case, are new, and possibly of independent interest (see also Remark~\ref{rem_possible_iso} below).
\end{rem}
\begin{rem}\label{rem:sing}
Let us briefly comment on the singularities of the various compactifications that occur in our paper. First, by construction, $\calM^{\operatorname{K}}$ and $\overline{\calB/\Gamma}$ have only finite quotient singularities. In particular, their cohomology coincides with their intersection cohomology (with $\mathbb{Q}$ coefficients). The intermediate space $\widehat\mathcal{M}$, which resolves the birational map $\calM^{\operatorname{GIT}}\dashrightarrow (\calB/\Gamma)^*$, has only toric singularities. In contrast, the two starting points of our analysis, the GIT quotient $\calM^{\operatorname{GIT}}$ and the Baily--Borel compactification $(\calB/\Gamma)^*$, have worse singularities. Specifically, the GIT quotient $\calM^{\operatorname{GIT}}$ has at worst finite quotient singularities along the stable locus $\mathcal{M}^s$, and toric singularities along the GIT boundary, except for the point $\Xi$ that corresponds to the chordal cubic threefold. Finally, $(\calB/\Gamma)^*$ has at worst finite quotient singularities in the interior $\mathcal{B}/\Gamma$, but the singularities at the two isolated boundary points (the cusps) of $(\calB/\Gamma)^*$ are fairly complicated. The precise description of the (partial) resolutions $\calM^{\operatorname{K}}\to \calM^{\operatorname{GIT}}$ and $\overline{\calB/\Gamma}\to (\calB/\Gamma)^*$ constitutes an important part of our paper (see esp. Sections~\ref{sec:prelim} and~\ref{sec:toroidal} respectively).
\end{rem}
\begin{rem}\label{rem_possible_iso}
We note that the first and the last row of this table are identical. This is to say, the Betti numbers of the two compactifications that are smooth up to finite quotient singularities --- the Kirwan blowup $\calM^{\operatorname{K}}$ and the toroidal compactification $\overline{\calB/\Gamma}$ of the ball quotient --- coincide. This leads to the natural question of whether these two compactifications are in fact isomorphic. Geometrically, both $\calM^{\operatorname{K}}$ and $\overline{\calB/\Gamma}$ are blowups of $(\calB/\Gamma)^*$ at the same two points which are the two cusps of $(\calB/\Gamma)^*$. However, it is unclear whether the blowup ideals are the same in both cases (see~\cite[\S5.1]{LOG2} for some related computations).
Similarly, in Appendix~\ref{sec:surfaces} we show that the Betti numbers of the Kirwan blowup and of the toroidal compactification for the moduli of cubic surfaces (see~\cite{ACTsurf}) are also equal (Theorem~\ref{T:CubSurfH}).
Even in this easier case, while we expect that the Kirwan blowup and the toroidal compactification for the moduli of cubic surfaces are in fact isomorphic, some subtle details remain to be settled.
Answering this question will require methods very different from what we use in this paper, and we plan to return to this question in the future.
\end{rem}
In addition to the fact that the (intersection) Betti numbers of a moduli space are a basic invariant of interest, there are several further reasons for our interest in these numbers.
In particular, our work here provides a better understanding of the geometry of the birational maps among the various compactifications of
the moduli space of cubic threefolds. In general it is a natural question to ask how different compactifications of a given moduli space, each often arising as the result of a natural compactification process, relate to each other. One way of understanding
such relations can be via the log-MMP with respect to a suitable linear combination of boundary divisors.
This is a very active subject of research, widely known as {\it the Hassett--Keel program}, in the case of the moduli space of curves ${\mathfrak M}_g$ (see eg.~\cite{HH1,HH2}). The motivation is that the log-MMP allows one to interpolate between a known compactification (such as the Deligne--Mumford compactification $\overline{\mathfrak M}_g$) and a target compactification (such as the canonical model $\mathfrak M_g^{can}$ for $g\ge23$). More recently, Laza and O'Grady~\cite{LOG2,LOG1} have used a variation of log-models to understand the relationship between the GIT and Baily--Borel compactifications for low degree (esp. quartic) $K3$ surfaces. It is natural to ask whether a similar picture arises for moduli spaces of cubics (see~\cite[Sect. 7]{cml} for some further discussion).
In particular, in this context the question raised by the remark above, of whether $\calM^{\operatorname{K}}$ and $\overline{\calB/\Gamma}$ are in fact isomorphic, seems to be the natural starting point, and resolving it might give some indication of the properties of the log-MMP in this case.
In another direction, our results provide a geometric approach to computing the cohomology of an interesting ball quotient (the Allcock--Carlson--Toledo model $\mathcal{B}/\Gamma$ for the moduli space of cubic threefolds) and its compactifications. First, since $\mathcal{B}/\Gamma$ is a locally symmetric variety, there are several interesting questions related to its topology. One natural question is whether its cohomology is generated by arithmetic cycles, i.e., Shimura subvarieties, which in this case will be sub-ball quotients $\mathcal{B}'/\Gamma'$. Our results provide a starting point for identifying some geometrically meaningful candidates for such subvarieties (e.g., loci corresponding to cubic threefolds with specified singularities, or cubics with specified automorphisms), although we are far from being able to answer this question completely. Analogous questions were considered in the case of orthogonal modular varieties (also known as type IV or $K3$ type) under the heading of the Noether--Lefschetz conjecture. This was verified by Bergeron et al.~\cite{bergeron} who show that the cohomology of locally symmetric varieties of type IV is generated at least up to middle dimension by Shimura subvarieties.
Second, we note that one can also approach the computation of the intersection cohomology of Baily--Borel compactifications via automorphic representations and trace formulae. This has been advanced very successfully in the case of the Satake compactification $\Sat$ of the moduli space of principally polarized abelian varieties, where the intersection cohomology is completely known for $g \leq 7$ (also for intersection cohomology with coefficients in any local system), see~\cite{hulektommasi}. This, as well as the work by Bergeron et al., relies on Arthur's endoscopic classification of automorphic representations of the symplectic group. In principle, Arthur's method can also be applied to the unitary group (i.e., the case of ball quotients) as was shown by Mok~\cite{mok}, but to the best of our knowledge the $10$-dimensional case which we treat here has not yet been approached by representation-theoretic methods.
Finally, while there has been some previous work computing the intersection cohomology of Baily--Borel compactifications of ball quotient models, in this paper we work out the cohomology of the \emph{toroidal} compactification.
To our knowledge, this is the first nontrivial example where the
intersection cohomology of the toroidal compactification of an arithmetic ball quotient model of a moduli space has been computed.
The techniques should be applicable to other examples of interest.
In fact, as our ten-dimensional ball quotient is the largest of the ball quotient models related to natural moduli problems, the results should be immediately applicable in these other situations. As mentioned above, in Appendix~\ref{sec:surfaces} we for instance apply our techniques to the ball quotient model of the moduli space of cubic surfaces.
Our approach takes as its starting point Kirwan's general theory (see~\cite{kirwan84,kirwanblowup}) of computing the (intersection) cohomology of GIT quotient spaces.
In her paper~~\cite{kirwanhyp} Kirwan uses her techniques to perform the computations for the cases of cubic and quartic surfaces.
Furthermore, Kirwan and her collaborators have done such computations for Baily--Borel compactifications of the moduli space of $K3$ surfaces of degree $2$ (see~\cite{KL1,KL2}) and the Deligne--Mostow ball quotients (see~\cite{KLW}).
Indeed, the largest Deligne--Mostow ball quotient, corresponding to $12$ points in $\mathbb{P}^1$, is directly related to our analysis, as it corresponds to the hyperelliptic divisor $\mathcal{H}_h^*$ in $(\calB/\Gamma)^*$.
However, our situation is that of the Baily--Borel compactification of the ball quotient $(\calB/\Gamma)^*$, which is of dimension $10$, and goes beyond the Deligne--Mostow examples.
While our basic setup is similar to these works, we encounter various new phenomena and complications, which make our computations considerably more intricate, and in particular require a careful analysis of the geometry of our situation. Combining Kirwan's machinery and geometric descriptions of various unstable and polystable loci (some available in the literature, but with further information deduced in this paper) allows us to compute the cohomology of $\calM^{\operatorname{GIT}}$, $\widehat \mathcal{M}$, and $\calM^{\operatorname{K}}$.
Next we compute the cohomology of the Baily--Borel compactification $(\calB/\Gamma)^*$ by applying the decomposition theorem to the natural morphism $\calM^{\operatorname{K}}\to(\calB/\Gamma)^*$.
We finally compute the cohomology of the toroidal compactification~$\overline{\calB/\Gamma}$ by applying the decomposition theorem to the natural morphism $\overline{\calB/\Gamma}\to (\calB/\Gamma)^*$, which is the blowup of the two points which are the cusps of~$(\calB/\Gamma)^*$.
We note that, as for all ball quotients, there are no choices involved in the construction of the toroidal compactification.
The computation of the cohomology of the toroidal boundary divisors requires a careful analysis of the arithmetic and the geometry of the two cusps. This involves the theory of Eisenstein lattices and leads to a wealth of new geometric insights.
In particular, we are led to generalize a Chevalley type result due to Looijenga and Bernstein--Schwarzman~\cite{Lroot, FMW, BS} to the case of Eisenstein lattices (see \S\ref{S:Eies-Chev}).
Furthermore, as an immediate and easy application of our techniques, we can for instance compute the cohomology of the toroidal compactification of the ball quotient model of the moduli space of cubic surfaces (Theorem~\ref{T:CubSurfH}).
Let us briefly go over the content of our paper. We start in Chapter~\ref{sec:prelim} with some preliminaries. Specifically, we first briefly review (\S\ref{subsec:modulicubic}) the work of Allcock~\cite{allcock} and Allcock--Carlson--Toledo~\cite{act} (see also Looijenga--Swierstra~\cite{ls}) on the moduli space of cubic threefolds and its two compact models $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$. We then review (\S\ref{subsec:kirwanth}) the basic framework of Kirwan's method (and fix the necessary notation). In particular, we introduce the space $\mathcal{M}^K$, the Kirwan (orbifold) desingularization of the GIT model $\calM^{\operatorname{GIT}}$, that plays a key role in our analysis.
In Sections~\ref{sec:HXss} and~\ref{S:CohKirBl-II} we compute the cohomology of the Kirwan resolution $\mathcal{M}^K$.
There are two main steps in the computation.
First is the computation of the equivariant cohomology of the semi-stable locus $X^{ss}$ in the Hilbert scheme of cubic threefolds (\S\ref{sec:HXss}). This is done by computing the usual Kempf stratification of the unstable locus, followed by an excision type argument. A key simplifying observation of Kirwan is that, for the purposes of eventually computing the intersection cohomology (or equivalently, cf. Remark \ref{rem:sing}, the cohomology) of $\calM^{\operatorname{K}}$, one can safely ignore unstable strata of high codimension. In fact, for the analogous computation in the case of quartic surfaces quartic surfaces discussed in~\cite{kirwanhyp}, all unstable strata can be ignored. To our surprise, this is no longer the case for the strata of unstable cubic threefolds, leading to some additional complications in the computation of $H^\bullet(\calM^{\operatorname{K}})$, since the locus of unstable cubic threefolds with a $D_5$ singularity plays a role. The next step, after computing the equivariant cohomology of the semi-stable locus $X^{ss}$, is to compute some correction terms (\S~\ref{S:CohKirBl-II}) that arise from blowing up the loci of strictly polystable points in $X^{ss}$ in the construction of $\calM^{\operatorname{K}}$.
Once the computation of the cohomology of $\calM^{\operatorname{K}}$ is completed, Kirwan's setup allows one to in principle approach the computation of the {\em intersection} cohomology of the GIT compactification~$\calM^{\operatorname{GIT}}$. To do this, Kirwan sets up an appropriate application of a suitable equivariant version of the decomposition theorem. In order to apply this, one needs to solve separate GIT problems for actions on the tangent space of suitable normalizers of stabilizers of strictly semi-stable points. We perform this computation, and turn out to be lucky in that the suitable quotients of strictly semi-stable loci are two points and a $\mathbb{P}^1$ in our case, which allows the computation of relevant intersection local systems. Along the way, we also determine the intersection cohomology of $\widehat \mathcal{M}$ as an intermediate step. This is discussed in Chapter~\ref{sec:IHGIT}.
We then further descend the computations from $\calM^{\operatorname{K}}$ to $(\calB/\Gamma)^*$.
To do this, we apply the decomposition theorem directly to the map $\calM^{\operatorname{K}}\to (\calB/\Gamma)^*$. The crucial point here is that the
Kirwan blowup $\calM^{\operatorname{K}}$ is smooth up to finite quotient singularities and that the map $\calM^{\operatorname{K}}\to (\calB/\Gamma)^*$ is a blowup in two points whose preimages are divisors in $\calM^{\operatorname{K}}$.
The decomposition theorem then has a simple description in terms of the cohomology of these exceptional divisors.
Since most of the work in computing the cohomology of those exceptional divisors was already done in the computation of the intersection cohomology of $\calM^{\operatorname{GIT}}$, the computation becomes feasible.
This is discussed in Chapter~\ref{sec:IHball}.
Finally, in Chapter~\ref{sec:toroidal} we compute the intersection cohomology of the toroidal compactification $\overline{\calB/\Gamma}$. Since $\overline{\calB/\Gamma}$ is a smooth up to finite quotient singularities, this computation is also done by applying directly the decomposition theorem, this time to the morphism $\overline{\calB/\Gamma}\to(\calB/\Gamma)^*$, which is also a blowup of the two cusps in $(\calB/\Gamma)^*$, with the total space smooth (also up to finite quotient singularities). This requires computing the cohomology of the two exceptional toroidal divisors of $\overline{\calB/\Gamma}$, which get contracted to the two cusps of $(\calB/\Gamma)^*$.
This turns out to be an interesting question in its own right, whose solution involves the theory of Eisenstein lattices as well as an equivariant version (Proposition~\ref{thm_L_Eis}) of a Chevalley type theorem of Looijenga~\cite{Lroot} and Bernstein--Schwarzman~\cite{BS}.
As Kirwan's machinery involves computations with equivariant cohomology, for convenience we have summarized in Appendix~\ref{sec:equivcoh} the properties of equivariant cohomology that we will use. To apply this general machinery, one still needs to determine various stabilizers, normalizers, their fixed point sets, etc. Such computations, though elementary, are quite lengthy and laborious. To streamline the flow of the text, we have gathered all such results in Appendix~\ref{S:Elem}. Finally, Appendix~\ref{sec:surfaces} discusses the easier case of the moduli space of cubic surfaces, where we prove that the cohomology of the Kirwan blowup, toroidal, and the Naruki compactifications are all equal.
\section{Notation and conventions}
\subsection{The general setting}\label{SSS:KirSetUp} In order to keep our presentation consistent with that of~\cite{kirwanblowup, kirwanhyp, GIT}, and in order to discuss some of the details of Kirwan's construction, we first recall the general framework.
We start with a complex projective manifold $X\subseteq \mathbb{P}^N$, a complex reductive group $G$ acting algebraically on $X$, and a $G$-linearization of the action on
the very ample line bundle $L=\mathcal{O}_{\mathbb{P}^N}(1)|_X$.
A complex Lie group $G$ is reductive if and only if it is the complexification of a maximal compact subgroup,
and we fix one such subgroup $K$.
We assume that the action and the linearization are induced by a faithful representation
$$
\rho:G\longrightarrow \operatornameeratorname{GL}(N+1,\mathbb{C})
$$
such that $\rho(K)\subset\operatornameeratorname{U}(N+1)$. We fix a maximal algebraic torus $\mathbb{T}\cong (\mathbb{C}^*)^{N+1}$ in $G$, and a corresponding maximal compact torus $T$ in $K$, so that $T$ is a maximal compact subgroup of $\mathbb{T}$. Let $\alpha_0,\dots,\alpha_{N}\in \mathfrak t^\vee$ be the weights of the representation of $K$, lying in the dual to the Lie algebra~$\mathfrak t$ of~$T$; if $(x_0:\dots :x_N)$ are the coordinates on $\mathbb{P}^N=\mathbb{P}\mathbb{C}^{N+1}$ diagonalizing the action of $T$, then we associate to $x_i$ the weight $\alpha_i$.
We fix an inner product on the Lie algebra $\mathfrak k$ of $K$ that is invariant under the adjoint action of $K$ (for example the Killing form), and use its restriction to $\mathfrak t$ to identify $\mathfrak t=\mathfrak t^\vee$.
We also fix once and for all a positive Weyl chamber $\mathfrak t_+$.
\subsection{The case of hypersurfaces}\label{S:hypeKirConv}
In this paper we will be specializing to the case of hypersurfaces of degree $d$ in $\mathbb{P}^n$, and eventually to cubic threefolds. To keep the notation consistent with the previous subsection, and in particular consistent with~\cite{kirwanhyp}, we take $X=\mathbb{P} H^0(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d))=\mathbb{P}\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee$, i.e., $X=\mathbb{P}^N$ with $N=\binom{n+d}{d}-1$, and we take $G=\operatornameeratorname{SL}(n+1,\mathbb{C})$ acting via the natural representation on $\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee$ induced by the canonical matrix action on $(\mathbb{C}^{n+1})^\vee=\mathbb C^{n+1}$.
This induces a linearization of the action for $\mathcal{O}_{\mathbb{P}^N}(1)$.
We note that the action of $G$ on $X$ is not faithful: the center of $\operatornameeratorname{SL}(n+1,\mathbb{C})$, which is isomorphic to $\mathbb{Z}/(n+1)\mathbb{Z}$, consisting of diagonal matrices with the same $(n+1)$-st root of unity along the diagonal, acts trivially on~$X$.
\begin{rem}
As is typical in this situation, there is some choice involved in picking the group $G$. The choice of $\operatornameeratorname{SL}(n+1,\mathbb C)$ is preferable from the perspective of linearizations and GIT (see~\cite[p.33 and Prop.~1.4]{GIT}). On the other hand, since the action of $\mathbb{P}\operatornameeratorname{GL}(n+1,\mathbb C)$ on $X$ is faithful, and automorphisms of a hypersurface are identified with the stabilizer of the corresponding point under this action, it can frequently be convenient to work with $\mathbb{P}\operatornameeratorname{GL}(n+1,\mathbb C)$ when computing stabilizers. Finally, it turns out that sometimes the stabilizers (and related groups) are easier to describe from the group theoretic perspective as subgroups of $\operatornameeratorname{GL}(n+1,\mathbb C)$. Since we can easily go back and forth among the various groups, we take $G=\operatornameeratorname{SL}(n+1,\mathbb C)$, so as to work well in the GIT setting, and be consistent with Kirwan's conventions.
\end{rem}
In this case $K=\operatornameeratorname{SU}(n+1)$, and $T\cong (S^1)^{n}$ is the subgroup of diagonal unitary matrices with determinant $1$.
The root system for $\operatornameeratorname{SU}(n+1)$ is of type $A_n$, with Weyl group the symmetric group $S_{n+1}$, and we fix a positive Weyl chamber $\mathfrak t_+$. The Killing form on $\mathfrak {su}(n+1)$ is given by $A.B=2n\operatornameeratorname{tr}(AB)$; thus when restricted to the diagonal traceless matrices of $\mathfrak t$, identified as the hyperplane $\{(a_0,\dots,a_n)\in \mathbb{R}^{n+1}: \sum a_i=0\} \subseteq \mathbb{R}^{n+1}$, the inner product on $\mathfrak t$ is $2n$ times the standard inner product.
For simplicity, we will always use the standard inner product.
To describe the weights of the representation of $\operatornameeratorname{SU}(n+1)$ concretely,
we take as a basis for $\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee$ the monomials of degree $d$. As usual, we use the notation $x^I:=x_0^{i_0}\dots x_n^{i_n}$, where $I=(i_0,\dots,i_n)$ is a partition of $d$, to index our monomials. A diagonal matrix $\operatornameeratorname{diag}(\lambda_0,\dots,\lambda_n)$ acts on $x^I$ by scaling by $\lambda_0^{i_0}\dots \lambda_n^{i_n}$,
and thus the index $I$ also gives the weight $\alpha_I$ associated to the coordinate $x^I$.
More precisely, the monomials
naturally sit as lattice points in the non-negative quadrant of $\mathbb{Z}^{n+1}$, and the monomials of fixed degree $d$ can be thought of as the lattice points of a simplex in the affine $n$-space whose defining equation is that the sum of coordinates is $d$. We make the identification of monomials of degree $d$ with weights in $\mathfrak t\subseteq \mathbb{R}^{n+1}$ explicit with the assignment $x^I\mapsto \alpha_I:= (i_0-d/(n+1),\dots,i_n-d/(n+1))$.
\subsection{The case of cubic threefolds}
The particular case of interest in this paper is the case of cubic threefolds.
As in the previous subsection, to fix the notation to match~\cite{kirwanhyp}, we set throughout the paper $d=3$ for the degree of the hypersurfaces, $n=4$ for dimension of the ambient $\mathbb{P}^4$, $X=\mathbb{P}^{34}=\mathbb{P}\operatornameeratorname{Sym}^3(\mathbb{C}^5)^\vee$ for the parameter space for cubic threefolds, and $G=\operatornameeratorname{SL}(5,\mathbb{C})$ for the reductive group acting on $X$ via change of coordinates, with the canonical linearization on $\mathcal{O}_{\mathbb{P}^{34}}(1)$.
\subsection{Strictly polystable points}
As before, let $G$ be a reductive group acting on a projective variety $X$ with a $G$-linearized ample line bundle $L$.
A point $x\in X$ is {\it semi-stable} if there exists an invariant section $\sigma\in H^0(X,L^m)^G$ (for some $m\in \mathbb{Z}_+$) such that $\sigma(x)\neq 0$.
We denote by $X^{ss}(L)$, or simply $X^{ss}$ if no confusion on $L$ is possible, the set of semi-stable points. A point $x\in X^{ss}(L)$ is {\it polystable} if the orbit $G\cdot x$ is closed in the locus of semi-stable points $X^{ss}(L)$. The stabilizer of a polystable point is a reductive group. We recall that the points of the GIT quotient $X/\!\!/_L G(=X^{ss}(L)/G)$ are in one-to-one correspondence with the orbits of the polystable points. Finally, a point $x\in X^{ss}(L)$ is {\it stable} if it is polystable with finite stabilizer. We denote by $X^s(L)\subset X^{ss}(L)$ (or simply $X^s$) the open subset of stable points. The quotient $X^s/G$ is a geometric quotient, in particular the points of $X^s/G$ are in one-to-one correspondence with the $G$-orbits in $X^s(L)$.
We will use the terminology of {\it strictly polystable} points for polystable points that are strictly semi-stable (i.e., the point is polystable, and semi-stable, but is not stable).
The main tool for determining the semi-stable/stable points is Mumford's numerical criterion (e.g.~\cite[\S2.1]{GIT}). For the case relevant here, the cubic threefolds, a complete description of the semi-stable/polystable/stable points was done by Allcock~\cite{allcock} and Yokoyama~\cite{yokoyama}, as reviewed below.
\section[Standard compactifications $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$]{Moduli space of cubic threefolds and its standard compactifications $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$}\label{subsec:modulicubic}
\subsection{The GIT compactification $\calM^{\operatorname{GIT}}$}
With $X=\mathbb{P}^{34}=\mathbb{P}\operatornameeratorname{Sym}^3(\mathbb{C}^5)^\vee$, the parameter space for cubic threefolds, and $G=\operatornameeratorname{SL}(5,\mathbb{C})$ acting via change of coordinates, as above,
the natural GIT compactification for the moduli space of cubic threefolds is denoted
$$\calM^{\operatorname{GIT}}:=X/\!\!/ G\,\,.$$
Note that since projective space has Picard rank $1$, and $G=\operatornameeratorname{SL}(5,\mathbb{C})$, there is essentially a unique choice of linearization for defining the GIT quotient~\cite[Prop.~1.4, p.33]{GIT}.
The open subset parameterizing smooth cubics will be denoted throughout by $\mathcal{M}$, and the stable locus will be denoted by $\mathcal{M}^s=X^s/G$. Clearly, one has
$$\mathcal{M}\subset \mathcal{M}^s\subset \calM^{\operatorname{GIT}}\,,$$
and $\mathcal{M}^s$ has at worst finite quotient singularities.
The GIT compactification $\calM^{\operatorname{GIT}}$ for cubic threefolds was analyzed by Allcock in~\cite{allcock} and Yokoyama in~\cite{yokoyama}.
They showed that semi-stability of a cubic threefold is determined by its singularities (with almost no global information needed; this is quite special to this case).
In particular, all the semi-stable cubics have isolated singularities, with a single exception, the {\it chordal cubic}, i.e. the secant variety of a rational normal curve in $\mathbb{P}^4$ (see~\eqref{eq:ch} below for an explicit equation). The chordal cubic is polystable, and we denote by $\Xi$ its orbit, which we view as a special point $\Xi\in \calM^{\operatorname{GIT}}$ of the GIT quotient.
For further reference, we summarize the GIT analysis for cubic threefolds (cf.~\cite[Thms.~1.1 -- 1.4]{allcock}) as follows:
\begin{teo}[{GIT compactification for cubic threefolds,~\cite{allcock}}] \label{T:GITcub}
The following hold:
\begin{itemize}
\item[(1)] A cubic threefold is GIT stable if and only if it has at worst $A_1, \dots ,A_4$-singularities.
\item[(2)] The GIT boundary $\calM^{\operatorname{GIT}} - \mathcal{M}^s$ consists of a rational curve $\mathcal{T}$ and an isolated point $\Delta$.
\item[(3a)] The polystable orbit parameterized by $\Delta$ corresponds to a cubic with $3D_4$-singularities, given by equation~\eqref{eq:3D4}.
\item[(3b)] Under a suitable identification $\mathcal{T}\cong \mathbb{P}^1$, the polystable orbits parameterized by $\mathcal{T} -\{0,1\}$ correspond to cubics with precisely $2A_5$-singularities (see~\eqref{eq:2A5} below for an explicit parameterization).
\item[(3b')] The special point $0\in \mathcal{T}$ corresponds to a cubic with $2A_5+A_1$-singularities (i.e. the cubics with $2A_5$ singularities parameterized by $\mathcal{T}$ can acquire an additional node for a special value of the parameter in $\mathcal{T}\cong \mathbb{P}^1$).
\item[(3b'')] The special point $1\in \mathcal{T}$ corresponds to the chordal cubic (in this situation, the $2A_5$ cubics specialize to a cubic with non-isolated singularities), i.e. the point $\Xi \in \calM^{\operatorname{GIT}}$ identified above.
\end{itemize}
\end{teo}
\begin{rem}\label{R:Alck-poly-form} In what follows, we will need explicit equations for the cubics in strictly polystable orbits. Specifically, we have the following (cf. ~\cite[Thm.~1.2]{allcock}):
\begin{enumerate}
\item The polystable orbit corresponding to the isolated boundary point $\Delta\in\calM^{\operatorname{GIT}}$ is the orbit consisting of cubics with three isolated $D_4$ singularities (a geometric condition that characterizes it, cf.~\cite[Thm. 5.4]{allcock}); one such cubic is given explicitly by the polynomial
\begin{equation}\label{eq:3D4}
F_{3D_4}:=x_0x_1x_2+x_3^3+x_4^3,
\end{equation}
with zero set $V(F_{3D_4})$, which we will call the $3D_4$-cubic.
\item The curve $\mathcal{T}\subset (\calM^{\operatorname{GIT}} -\mathcal{M}^s)$ parameterizes strictly polystable orbits given by polynomials of the form
\begin{equation}\label{eq:2A5}
F_{A,B}=Ax_2^3+x_0x_3^2+x_1^2x_4-x_0x_2x_4+Bx_1x_2x_3,
\end{equation}
with $A,B$ not simultaneously vanishing.
Specifically, one notes that the zero set $V(F_{A,B})$ is projectively equivalent to $V(F_{\lambda^2A,\lambda B})$ for any $\lambda\in\mathbb{C}^*$. In fact, $V(F_{A,B})$ is projectively equivalent to $V(F_{A',B'})$ if and only if $A/B^2=A'/B'^2$. Thus, $C:=4A/B^2$ can be taken as an affine parameter for the rational curve $\mathcal{T}$. The factor of $4$ is taken for numerical convenience: if $C\notin\{0,1\}$, then the cubic $V(F_{A,B})$ has exactly two isolated $A_5$ singularities (a geometric condition that characterizes the cubics $V(F_{A,B})$, cf.~\cite[Thm. 5.7]{allcock}). If $C=0$ (equivalently $A=0$), then the cubic $V(F_{0,B})$ has in addition to the two $A_5$ singularities, an isolated $A_1$ singularity. Finally, if $C=1$ (e.g., $(A,B)=(1,-2)$), then the associated cubic $V(F_{1,-2})$ is the {\it chordal cubic}, i.e., the secant variety of the standard rational normal curve in $ \mathbb{P}^4$ (which is singular precisely along the rational normal curve). Note that
\begin{equation}\label{eq:ch}
F_{1,-2}=\det\begin{pmatrix} x_0&x_1&x_2\\ x_1&x_2&x_3\\ x_2&x_3&x_4\end{pmatrix},
\end{equation}
which makes the relationship to the standard rational normal curve in $\mathbb{P}^4$ more transparent.
\end{enumerate}
\end{rem}
\subsection{The ball quotient model $(\calB/\Gamma)^*$} Looij\-enga--Swierstra~\cite{ls} and independently Allcock--Carlson--Toledo~\cite{act} have constructed a ball quotient model $\mathcal{B}/\Gamma$, where $\mathcal{B}$ is a $10$-dimensional complex ball, and $\Gamma$ is an arithmetic group acting on $\mathcal{B}$, via the period map for cubic fourfolds. The following summarizes the essential aspects of the ball quotient model.
\begin{teo}[{The ball quotient model,~\cite{act} and~\cite{ls}}]
Let $\mathcal{B}/\Gamma$ be the ball quotient model of~\cite{act}. The following hold:
\begin{itemize}
\item[(1)] The period map (defined via eigenperiods of cubic fourfolds)
$$P:\mathcal{M}\to \mathcal{B}/\Gamma$$
is an open embedding with the complement of the image being the union of two irreducible Heegner divisors $D_n:={\calD}_{n}/\Gamma$ (called the nodal divisor) and $D_h:={\calD}_{h}/\Gamma$ (called the hyperelliptic divisor), where ${\calD}_{n}$ and ${\calD}_{h}$ are $\Gamma$-invariant hyperplane arrangements.
\item[(2)] The boundary of the Baily--Borel compactification $(\calB/\Gamma)^*$ consists of two cusps (i.e., $0$-dimensional boundary components), which we will call $c_{3D4}$ and $c_{2A_5}$.
\end{itemize}
\end{teo}
The Baily--Borel compactification $(\calB/\Gamma)^*$ of the ball quotient model discussed above gives a projective compactification for the moduli space of cubic threefolds $\mathcal{M}$. The main result of~\cite{act} and~\cite{ls} is that there is a simple birational relationship between the GIT and Baily--Borel models -- this is an essential result for our analysis. We summarize their results below:
\begin{teo}[{GIT to ball quotient comparison,~\cite{act} and~\cite{ls}}]\label{resgitball}
As above, let $\calM^{\operatorname{GIT}}$ be the GIT compactification of the moduli space of cubic threefolds. Let $(\calB/\Gamma)^*$ be the Baily--Borel compactification of the ball quotient model of~\cite{act}. Then there exists a diagram
$$
\xymatrix{
&\widehat{\mathcal{M}} \ar[ld]_{p} \ar[rd]^{q}&\\
\ \ \calM^{\operatorname{GIT}}\ar@{-->}[rr]^{\overline P} && (\calB/\Gamma)^* \\
}
$$
resolving the birational map between $\calM^{\operatorname{GIT}}$ and $(\calB/\Gamma)^*$ such that:
\begin{itemize}
\item[(1)] $p:\widehat \mathcal{M}\to \calM^{\operatorname{GIT}}$ is the Kirwan blowup of the point $\Xi\in \calM^{\operatorname{GIT}}$, corresponding to the chordal cubic (see \S\ref{S:KirBlUpDef} below, esp.~\eqref{diag_kirwanblowup}). The exceptional divisor $E:=p^{-1}(\Xi)$
of this blowup is naturally identified with the moduli space of $12$ unordered points in $\mathbb{P}^1$.
\item[(2)] $q:\widehat{\calM} \to (\calB/\Gamma)^*$ is a small semi-toric modification as constructed by Looijenga~\cite{l1}. The morphism $q$ is an isomorphism over the interior $\mathcal{B}/\Gamma$ and one of the two cusps of $(\calB/\Gamma)^*$, namely $c_{3D4}$. The preimage under $q$ of the other cusp, $c_{2A5}$, is a curve, which is identified with the strict transform $\widehat {\mathcal{T}}$ of $\mathcal{T}\subset \calM^{\operatorname{GIT}}$ under $p$.
\end{itemize}
In particular note that the period map $P:\mathcal{M}\to \mathcal{B}/\Gamma$ extends to a morphism $\overline P:\calM^{\operatorname{GIT}} -\lbrace \Xi\rbrace\to(\calB/\Gamma)^*$. Furthermore, the following hold:
\begin{itemize}
\item[(3)] Let $E\subset \widehat{\calM}$ be the exceptional divisor of the map $p$. Then the image $q(E)$ is the closure $D_h^*$ in $(\calB/\Gamma)^*$ of the hyperelliptic divisor $D_h\subset \mathcal B/\Gamma$,
while $q$ is an isomorphism over $D_h$ (i.e., $q_{\mid q^{-1}(D_h)}:q^{-1}(D_h)\simeq D_h$).
\item[(4)] $q$ is an isomorphism over the stable locus $\mathcal{M}^s$ and in a neighborhood of the point $\Delta$, corresponding to the $3D_4$ cubic. The image under~$q$ of the locus of cubics with $A_1,\dots,A_4$-singularities is $({\calD}_{n}-{\calD}_{h})/\Gamma$ (equivalently,~$\overline P$ extends over~$\mathcal{M}^s$ and $\overline P(\mathcal{M}^s)=(\mathcal{B} -{\calD}_{h})/\Gamma$).
\item[(5)] $q$ maps $\Delta$ to the cusp $c_{3D4}$ of $(\calB/\Gamma)^*$, and the strict transform $\widehat {\mathcal{T}}$ of the curve $\mathcal{T}$ to the cusp $c_{2A5}$.
\end{itemize}
\end{teo}
\section{The Kirwan blowup $\calM^{\operatorname{K}}$ of the moduli space of cubic threefolds}\label{subsec:kirwanth}
\subsection{Introduction}
The first step towards understanding the cohomology of the GIT and ball quotient models for the moduli of cubic threefolds is to produce a common resolution (with at worst finite quotient singularities). For GIT quotients, Kirwan~\cite{kirwanblowup} gives a general algorithm that achieves this resolution. Roughly speaking, one considers the GIT boundary $\calM^{\operatorname{GIT}}-\mathcal{M}^s$($={\mathcal{T}}\cup\{\Delta\}$ in our situation) and stratifies it in terms of the connected components $R$ of the stabilizers of the associated polystable orbits.
Then, one proceeds by blowing up these strata, starting with the deepest one, in a way that will be explained in detail below.
In our situation, we will see that there are three strata: $\Xi$, $\Delta$ (which are points) and ${\mathcal{T}}-\{\Xi\}$ (which is a curve), with associated connected components of the stabilizers being $\operatornameeratorname{SL}(2,\mathbb{C})$, $(\mathbb{C}^*)^2$, and $\mathbb{C}^*$, respectively.
\subsection{The Kirwan blowup in general}\label{SSS:kirBlUp}
We start with $X$, $G$, and $L$ as in the general setup of \S~\ref{SSS:KirSetUp}.
Let $\mathcal{R}$ be a set of representatives for the (finite) set of conjugacy classes of connected components of stabilizers of strictly polystable points in $X^{ss}$. Denote then $r$ the maximal dimension of the groups in $\mathcal{R}$, and let then $\mathcal{R}(r)\subseteq \mathcal{R}$ be the representative of those subgroups that have dimension $r$. For a given $R\in\mathcal{R}(r)$, we proceed as follows. If $r=0$, then there is nothing to do. Otherwise, set
\begin{equation}\label{E:ZRss}
Z_R^{ss}:=\{x\in X^{ss}\mid R \textrm{ fixes } x\} \subset X^{ss}.
\end{equation}
Kirwan shows that for all $R\in \mathcal{R}(r)$, the loci $G\cdot Z^{ss}_R$ are smooth and closed in $X^{ss}$~\cite[Lem.~5.11, Cor.~5.10]{kirwanblowup}. Now let $\hat \pi:\hat X\to X^{ss}$ be the blowup of $X^{ss}$ along $G\cdot Z^{ss}_R$.
Note that since $G\cdot Z^{ss}_R$ only depends on the conjugacy class of $R$, the same is true for the blowup.
As $G$ acts on the center of the blowup, there is an induced action of $G$ on $\hat X$. Taking $E$ to be the exceptional divisor of the blowup $\hat\pi$, there is a choice of $d\gg 0$ such that $\hat L:=\hat \pi^*L^{\otimes d}\otimes \mathcal{O}(-E)$ is ample and admits a $G$-linearization that makes the following statements true~\cite[Lem.~3.11, Lem.~6.11]{kirwanblowup} (see also~\cite{reichstein}). Let $\hat {\mathcal{R}}$ be a set of representatives for the set of conjugacy classes of connected components of stabilizers of polystable points in the semi-stable locus $\hat X^{ss}$. Then, up to replacing elements of $\hat{\mathcal{R}}$ with conjugates, we have $\hat{\mathcal{R}}\subsetneq \mathcal{R}$~\cite[Lem.~6.1]{kirwanblowup}.
Thus, by induction on the cardinality of the set $\mathcal{R}$, we obtain the desired space $\pi:\widetilde X^{ss}\to X^{ss}$ by iteratively blowing up with respect to a smooth center, and then restricting to the semi-stable locus. Moreover, $\widetilde X^{ss}$ is equipped with a $G$-linearized ample line bundle $\widetilde L$, such that $G$ acts with finite stabilizers. We define the Kirwan blowup to be the space $\widetilde X^{ss}/\!\!/_{\widetilde L}G$ ($=\widetilde X^{ss}/G$); up to isomorphism, this is independent of the choices~\cite[Rem.~6.8 and p.64]{kirwanblowup}. The Kirwan blowup has at worst finite quotient singularities, and there is a birational morphism~\cite[Cor.~6.7]{kirwanblowup}:
$$
\widetilde X^{ss} /\!\!/_{\widetilde L}G\longrightarrow X^{ss}/\!\!/_{ L}G.
$$
\begin{rem}\label{R:Stab+Strict}
For later reference, we recall two further facts regarding the map $\hat \pi:\hat X\to X^{ss}$, and the chosen linearization.
First, if $\hat x\in \hat X - E$, then $\hat x\in \hat X^{ss}$ if and only if $\overline {G\cdot \hat \pi(\hat x)}\cap G\cdot Z_R^{ss}=\emptyset$~\cite{reichstein}.
In other words, outside of the exceptional divisor, the effect of the blowup is to destabilize exactly those strictly semi-stable points that have orbit closure meeting the center of the blowup.
Second, for any $\hat R\in \hat {\mathcal{R}}$ the locus $\hat Z^{ss}_{\hat R}\subseteq \hat X^{ss}$ is the strict transform of the locus $ Z^{ss}_{\hat R}\subseteq X^{ss}$ defined by viewing~$\hat R$ as an element of~$\mathcal{R}$~\cite[Rem.~6.8]{kirwanblowup}.
\end{rem}
\begin{rem}\label{R:pi-r-Def}
We also recall the following fact~\cite[Lem.~8.2]{kirwanblowup}: If $R_1, R_2\in \mathcal{R}(r)$ are different groups of maximal dimension among elements of~$\mathcal{R}$, then $G\cdot Z^{ss}_{R_1}\cap G Z^{ss}_{R_2}=\emptyset$, and any $x$ in $G\cdot Z^{ss}_{R_2}$ remains semi-stable after $X^{ss}$ is blown up along $G\cdot Z^{ss}_{R_1}$. In particular we have~\cite[Cor.~8.3]{kirwanblowup}: the result of successively blowing up $X^{ss}$ along $G\cdot Z^{ss}_R$ for each $R \in \mathcal{R}(r)$ is the same as the blowup of $X^{ss}$ along
$\bigcup _{R\in \mathcal{R}(r)}G\cdot Z^{ss}_R $. Following the notation in~\cite[Cor.~8.3]{kirwanblowup}, we will denote this blowup by $\pi_r:X_r\to X^{ss}$.
Repeating the above process we obtain a sequence of blowups
$$\widetilde X^{ss}:=X_1^{ss}\stackrel{\pi_1}{\longrightarrow} X_{2}^{ss}\stackrel{\pi_{2}}{\longrightarrow}\dots\stackrel{\pi_{r-1}}{\longrightarrow} X^{ss}_r\stackrel{\pi_r}{\longrightarrow} X^{ss}=:X_{r+1}^{ss}\,\,.$$
Note that we allow some of these blowups to be the identity if there are no relevant subgroups in a given dimension. In short, $\pi_{j}$ is the blowup of the locus determined by the subgroups $R\in \mathcal{R}$ of dimension $j$; i.e., by all $R\in\mathcal{R}(j)$. Note that in contrast, if $R_1\in \mathcal{R}(r_1)$ and $R_2\in\mathcal{R}(r_2)$ for $r_1\ne r_2$, then it may happen that $G\cdot Z^{ss}_{R_1}\cap G Z^{ss}_{R_2}\ne \emptyset$.
\end{rem}
\subsection{The Kirwan blowup of the moduli space of cubic threefolds}\label{S:KirBlUpDef}
We now implement the steps outlined in the previous subsection to construct the Kirwan blowup of the moduli space of cubic threefolds.
The first step is to enumerate the connected components of the stabilizers of polystable points. In our situation, this is answered by the following proposition, where as is standard, we write $1$-PS for one-parameter subgroups:
\begin{pro}[The connected components of stabilizers~$R$] \label{pro:stabilizer0}
Let $V$ be a strictly polystable cubic threefold. Then the connected component $\operatornameeratorname{Stab}^0(V)$ of the identity in the stabilizer $\operatornameeratorname{Stab}(V)\subseteq \operatornameeratorname{SL}(5,\mathbb{C})$ is one of the following (up to conjugation):
\begin{itemize}
\item[(1)] The $1$-PS with weights $(2,1,0,-1,-2)$:
\begin{equation}\label{E:R2A5}
R_{2A5}:=\operatornameeratorname{Stab}^0(V(F_{A,B}))=\operatornameeratorname{diag}(\lambda^2,\lambda,1,\lambda^{-1},\lambda^{-2})\cong \mathbb{C}^*,
\end{equation}
for $4A/B^2\ne 1$.
We have $\operatornameeratorname{Stab}^0(V)=R_{2A5}$ (up to conjugation) if and only if $V$ is in the orbit of $V(F_{A,B})$ with $4A/B^2\ne 1$; i.e., if and only if the cubic has exactly two $A_5$ singularities, or exactly two $A_5$ singularities and one $A_1$ singularity. These are the cubic threefolds corresponding to points on the curve $(\mathcal{T}-\lbrace \Xi\rbrace)\, \subseteq \calM^{\operatorname{GIT}}$.
\item[(2)] The three-dimensional group
\begin{equation}\label{E:Rc}
R_c:=\operatornameeratorname{Stab}^0(V(F_{-1,2}))\cong\operatornameeratorname{PGL}(2,\mathbb{C}),
\end{equation}
given as the copy of $\operatornameeratorname{PGL}(2,\mathbb{C})$ embedded into $\operatornameeratorname{SL}(5,\mathbb{C})$ as the image of the $\operatornameeratorname{SL}(2,\mathbb{C})$ representation $\operatornameeratorname{Sym}^4(\mathbb{C}^2)\cong \mathbb{C}^5$ (see Appendix~\ref{sec:equivcoh} for more details on dealing with equivariant cohomology of $\operatornameeratorname{GL}(n+1,\mathbb{C})$ versus $\operatornameeratorname{SL}(n+1,\mathbb{C})$, and related issues).
We have $\operatornameeratorname{Stab}^0(V)=R_{c}$ (up to conjugation) if and only if $V$ is in the orbit of $V(F_{A,B})$ with $4A/B^2= 1$; i.e., if and only if the cubic is projectively equivalent to the chordal cubic. These are the cubic threefolds corresponding to the point $\,\Xi \in \calM^{\operatorname{GIT}}$.
\item[(3)] The two-dimensional torus:
\begin{equation}\label{E:R3D4}
R_{3D_4}:=\operatornameeratorname{Stab}^0(V(F_{3D_4}))=\operatornameeratorname{diag}(s,t,(st)^{-1},1,1)\cong (\mathbb{C}^*)^2.
\end{equation}
We have $\operatornameeratorname{Stab}^0(V)=R_{3D_4}$ (up to conjugation) if and only if $V$ is in the orbit of $V(F_{3D_4})$; i.e., if and only if the cubic has exactly $3D_4$ singularities. These are the cubic threefolds corresponding to the point $\,\Delta \in \calM^{\operatorname{GIT}}$.
\end{itemize}
Moreover, we have
\begin{equation}\label{E:Rcont}
R_{2A_5}\subset R_c, \ \ R_{c}\cap R_{3D_4}=1,
\end{equation}
with the inclusion on the left corresponding to the fact that $\Xi\in \mathcal{T}\subset \calM^{\operatorname{GIT}}$.
\end{pro}
\begin{proof}
From the results of~\cite{allcock} describing polystable cubic threefolds (see Theorem~\ref{T:GITcub} and Remark~\ref{R:Alck-poly-form}, above), it suffices to consider the cubic threefolds of the form $V(F_{A,B})$~\eqref{eq:2A5}, for $A$ and $B$ not simultaneously zero, and $V(F_{3D_4})$~\eqref{eq:3D4}.
It is obvious that each of the groups listed above is connected and stabilizes the corresponding polystable orbit. For instance, $\operatornameeratorname{PGL}(2)$ acting on $\mathbb{P}^4$ via the $\operatornameeratorname{Sym}^4$ representation fixes the standard rational normal curve. Obviously, it will also fix the secant variety of that curve, which is precisely the chordal cubic.
The converse (i.e., the fact that $\operatornameeratorname{Stab}^0(V)$ is precisely as listed, and not larger) follows by a routine calculation. Many straightforward computations with matrices will be relegated to Appendix~\ref{S:Elem}. For the results here, see in particular Proposition~\ref{P:App-R=SL2}, and Propositions~\ref{P:App-R=C*p1},\ref{P:App-R=C*p2},~\ref{P:App-R=C*2}. The relationships~\eqref{E:Rcont} among the $R$ are straightforward from the descriptions of the groups.
\end{proof}
Utilizing the notation from~\eqref{E:R2A5},~\eqref{E:Rc}, and~\eqref{E:R3D4}, it follows that for cubic threefolds we may take
\begin{equation}\label{def_calr}
\mathcal{R}:=\{R_{2A_5},R_{3D_4},R_c\}\longleftrightarrow \{\mathbb{C}^*, (\mathbb{C}^*)^2,\operatornameeratorname{PGL}(2,\mathbb{C})\}
\end{equation}
as a set of representatives for the set of conjugacy classes of connected components of stabilizers of strictly polystable cubic threefolds. For each $R\in \mathcal{R}$, we have the corresponding fixed locus $Z^{ss}_R$, defined in~\eqref{E:ZRss}.
These loci can be described more explicitly:
\begin{pro}[The strata $Z_{R}^{ss}$]\label{P:ZRss}
For cubic threefolds, the fixed loci $Z_R^{ss}$~\eqref{E:ZRss} can be described as follows:
\begin{enumerate}
\item $Z^{ss}_{R_{2A_5}}$ is the set of cubic threefolds defined by the cubic forms:
\begin{equation}\label{eq:ZR2A5}
F=a_0x_2^3+a_1x_0x_3^2+a_2x_1^2x_4+a_3x_0x_2x_4+a_4x_1x_2x_3,
\end{equation}
with $a_1,a_2,a_3\ne 0$, $(a_0,a_4)\ne (0,0)$. For $(A,B)\ne (0,0)$ we have $V(F_{A,B})\in Z^{ss}_{R_{2A_5}}$, and conversely every cubic in $Z^{ss}_{R_{2A_5}}$ is projectively equivalent to a cubic of the form $V(F_{A,B})$ with $(A,B)\ne (0,0)$.
\item $Z^{ss}_{R_c}=\{V(F_{1,-2})\}$, the chordal cubic in standard coordinates.
\item $Z^{ss}_{R_{3D_4}}$ is the set of cubics defined by equations of the form
$$
x_0x_1x_2+P_3(x_3,x_4)
$$
where $P_3(x_3,x_4)$ is an arbitrary homogeneous cubic with three distinct roots.
\end{enumerate}
Moreover, we have the following relationships among the fixed loci:
\begin{equation}\label{E:ZssR-Rel}
Z^{ss}_{R_c}\subset Z^{ss}_{R_{2A5}}, \ \ \ \ Z^{ss}_{R_{2A_5}} \cap Z^{ss}_{R_{3D4}}=\emptyset.
\end{equation}
\end{pro}
\begin{proof}
It is immediate to check that the groups $R_{2A_5}$, $R_{3D_4}$, and $R_{c}$ fix the corresponding loci $Z^{ss}_{R_{2A_5}}$, $Z^{ss}_{R_{3D_4}}$, and $Z^{ss}_{R_{c}}$, respectively. It is a straightforward check that these are in fact the full fixed loci; see also Propositions~\ref{P:App-R=C*p1},\ref{P:App-R=C*p2},\ref{P:App-R=SL2},~\ref{P:App-R=C*2}.
The relationships~\eqref{E:ZssR-Rel} among the $Z^{ss}_R$ are a straightforward consequence of the descriptions above. See also Corollary~\ref{C:App-ZssR-rel}.
\end{proof}
For the Kirwan blowup, we are actually interested in the orbits
$$
G\cdot Z^{ss}_R;
$$
in other words the loci of cubic threefolds that are projectively equivalent to the cubics in a given stratum.
\begin{cor}[The orbits $G\cdot Z^{ss}_R$]\label{C:ZRss}
For cubic threefolds, the orbits of the fixed loci $Z_R^{ss}$ can be described as follows:
\begin{enumerate}
\item $G\cdot Z^{ss}_{R_{2A_5}}$ is the set of polystable cubics projectively equivalent to a $2A_5$ cubic, a $2A_5+A_1$ cubic, or a chordal cubic; i.e., projectively equivalent to a cubic of the form $V(F_{A,B})$ with $(A,B)\ne (0,0)$.
\item $G\cdot Z^{ss}_{R_c}$ is the set of polystable cubics projectively equivalent to the chordal cubic; i.e., projectively equivalent to $V(F_{1,-2})$.
\item $G\cdot Z^{ss}_{R_{3D_4}}$ is the set of polystable cubics with $3D_4$ singularities; i.e., projectively equivalent to $V(F_{3D_4})$.
\end{enumerate}
Moreover,
we have the following relationships among the orbits:
\begin{equation}\label{E:GZssR-Rel}
G\cdot Z^{ss}_{R_c}\subset G\cdot Z^{ss}_{R_{2A5}}, \ \ \ \ G\cdot Z^{ss}_{R_{2A_5}} \cap G\cdot Z^{ss}_{R_{3D4}}=\emptyset.
\end{equation}
\end{cor}
\begin{proof} (1)--(3) follow directly from Proposition~\ref{P:ZRss}(1)--(3).
The first inclusion of~\eqref{E:GZssR-Rel} follows directly from that of~\eqref{E:ZssR-Rel}. The equality on the right follows from (1)--(3), since the cubics in question are not projectively equivalent.
\end{proof}
Now recall that the Kirwan desingularization process consists of successively blowing up $X^{ss}$ along the (strict transforms of the) loci $G\cdot Z_R^{ss}$ in order of $\dim R$, to obtain a smooth space $\widetilde X^{ss}$, and then taking the induced GIT quotient $\widetilde{X}^{ss}/\!\!/_{\widetilde L} G$ with respect to a particular linearization. We denote the resulting desingularization~$\calM^{\operatorname{K}}$ and refer to it as {\it the Kirwan blowup} of $\calM^{\operatorname{GIT}}$.
Concretely, in our situation, this translates into a diagram:
\begin{equation}\label{diag_kirwanblowup}
\resizebox{\textwidth}{!}{
\xymatrix@C=1em{
\widetilde X^{ss}\ar@{=}[d]\\ (\operatornameeratorname{Bl}_{ {G\cdot Z^{ss}_{R_{2A_5},2}}}(X_2^{ss}))^{ss}\ar[r]\ar@{->}[d] & X_2^{ss}=(\operatornameeratorname{Bl}_{{G\cdot Z^{ss}_{R_{3D_4}}}}( X_3^{ss}))^{ss}\ar[r]\ar@{->}[d]& X^{ss}_3=(\operatornameeratorname{Bl}_{G\cdot Z^{ss}_{R_{c}}}(X^{ss}))^{ss}\ar[r] \ar@{->}[d]& X^{ss}\ar[d]\\
\calM^{\operatorname{K}} \ar[r]& \widehat {\widehat {\mathcal{M}}\,\,} \ar[r]& \widehat{\mathcal{M}} \ar[r]&\calM^{\operatorname{GIT}}
}
}
\end{equation}
Here ${{G\cdot Z^{ss}_{R_{2A_5},2}}}$ is the strict transform of the orbit ${{G\cdot Z^{ss}_{R_{2A_5}}}}$.
The Kirwan blowup $\calM^{\operatorname{K}}$ is obtained by first blowing up the point $\Xi\in \calM^{\operatorname{GIT}}$ corresponding to the chordal cubic, followed by blowing up the point $\Delta$ (which is not affected by the first blowup), and then finally blowing up the strict transform $\widehat {\mathcal{T}}$ of ${\mathcal{T}}\subset \calM^{\operatorname{GIT}}$.
To be precise, we must specify the blowup ideals corresponding to the blowups on the lower line of \eqref{diag_kirwanblowup}. These are obtained by descent modulo the action of $G$ from $X^{ss}$ of the reduced ideals defining the blowup $\widetilde {X}^{ss}\to X^{ss}$.
Note that the last two blowups commute (thus their order is irrelevant). Also, the blowup of $\Xi$ (i.e., the first blowup) coincides
with the blowup $\widehat \mathcal{M}$ constructed by Allcock--Carlson--Toledo~\cite{act} in order to resolve the birational period map $\overline P:\calM^{\operatorname{GIT}}\dashrightarrow (\calB/\Gamma)^*$ (i.e., the space discussed above in Theorem~\ref{resgitball}). Indeed, in the Kirwan blowup, in light of Corollary~\ref{C:ZRss}(2), the first step is to blowup $X^{ss}$ along the orbit of the chordal cubic, and then take the GIT quotient with respect to a particular linearization, which is exactly the construction in~\cite[\S 3]{act}.
The space $\widehat {\widehat {\mathcal{M}}\,\,}$ is an auxiliary space from our perspective.
\section{The toroidal compactification}
As with any locally symmetric space, the ball quotient $\mathcal{B}/\Gamma$ has not only the Baily--Borel compactification $(\calB/\Gamma)^*$, but also a toroidal compactification, which is thus another
natural birational model of $\mathcal{M}$. While typically
the construction of toroidal compactifications depends on certain choices, this is not the case for ball quotients. Recall that the cusps are in $1:1$ correspondence
with $\Gamma$-orbits of rational isotropic subspaces of the vector space on which the group $\Gamma$ acts. Since ball quotients are related to hermitian forms
of signature $(1,n)$, the only possibility is given by isotropic lines. This means on the one hand that the Baily--Borel compactification $(\calB/\Gamma)^*$ is obtained from the ball quotient $\mathcal{B}/\Gamma$
by adding finitely many (in our case -- two) points, that is $0$-dimensional cusps, as we have discussed above. On the other hand, from a toric point of view, we are in a $1$-dimensional situation, which
allows no choices. We shall denote the (unique) toroidal compactification by $\overline{\calB/\Gamma}$. It comes with a natural morphism $\overline{\calB/\Gamma} \to (\calB/\Gamma)^*$.
We shall discuss this in more detail in Chapter~\ref{sec:toroidal}.
In summary we have the following diagram illustrating the relationships among all the models of the moduli space of cubic threefolds we have discussed so far:
\begin{equation}\label{E:BirDiagMod}
\xymatrix{
&\calM^{\operatorname{K}}\ar[ldd]_\pi\ar[d]^f\ar[rdd]^g\ar@{<-->}[r]^{}&\overline{\calB/\Gamma}\ar[dd]\\
&\widehat\mathcal{M}\ar[ld]^{p}\ar[rd]_q\\
\calM^{\operatorname{GIT}} \ar@{-->}[rr]^{\overline P} &&(\calB/\Gamma)^*.
}
\end{equation}
While $\calM^{\operatorname{K}}$ and $\overline{\calB/\Gamma}$ can both be viewed as blowups of the two points in $(\calB/\Gamma)^*$ corresponding to the two cusps of the Baily--Borel compactification,
we do not know whether the Kirwan blowup $\calM^{\operatorname{K}}$ and the toroidal compactification $\overline{\calB/\Gamma}$ are isomorphic (see Remark~\ref{rem_possible_iso}). This seems to us an interesting question in its own right, which we plan to revisit in the future.
\chapter[Equivariant cohomology of the semi-stable locus]{The cohomology of the Kirwan blowup, part I: \\ equivariant cohomology of the semi-stable locus}\label{sec:HXss}
Following Kirwan, we will compute the intersection cohomology of the GIT quotient $\calM^{\operatorname{GIT}}$ by first computing the cohomology of the Kirwan blowup $\calM^{\operatorname{K}}$. The first step in computing the cohomology of the Kirwan blowup is to compute the equivariant cohomology of the semi-stable locus. This is accomplished by constructing an equivariantly perfect stratification ~\cite[p.17]{kirwan84} of the unstable locus, and then using the Thom--Gysin sequence. We review the precise setup in this section, and perform this step for the case of cubic threefolds.
\section[The equivariantly perfect stratification]{The equivariantly perfect stratification and the equivariant cohomology of the semi-stable locus in general}\label{S:PXss}
\subsection{Defining the equivariantly perfect stratification $S_\beta$} \label{S:E-P-S-def}
We return to the general setup of \S~\ref{SSS:KirSetUp}, and
recall Kirwan's equivariantly perfect stratification of the unstable locus in $X$, which will allow us to compute the equivariant cohomology of the semi-stable locus.
Our presentation follows~\cite[Ch.8 \S 7]{GIT}, and serves primarily to fix notation.
In addition, one of the main points of the review in this section is that it is difficult to explain the terms in Kirwan's formulas in the case of cubic threefolds without describing the construction, and partially explaining the proofs.
To define the stratification we first define an indexing set $\mathcal{B}$. This consists of the points in the closure $\overline{\mathfrak t}_+$ of the positive Weyl chamber that can be characterized as
follows: they are the closest point to the origin of the convex hull of a nonempty set of the weights $\alpha_0,\dots,\alpha_N$~\cite[Def.~3.13, and \S8 p.59]{kirwan84}.
Using the inner product on $\mathfrak t$ (fixed in \S~\ref{SSS:KirSetUp}), and the corresponding norm $||\cdot ||$, we define for each $\beta \in \mathcal{B}$~\cite[p.173]{GIT},~\cite[Exa.~3.11, Thm.~12.26]{kirwan84}:
\begin{align}
\label{E:Zbeta} Z_\beta&:=\{(x_0:\dots :x_N)\in X\subseteq \mathbb{P}^N: x_j=0 \text { if } \alpha_j.\beta \ne ||\beta ||^2\}\\
\label{E:Ybeta}Y_\beta &:=\{(x_0:\dots :x_N)\in X\subseteq \mathbb{P}^N: x_j=0 \text { if } \alpha_j.\beta < ||\beta ||^2,\\ \nonumber
&\ \ \ \ \ \ \ \ \ \ \text{ and } \exists \ x_i \ne 0 \text{ s.t. } \alpha_i.\beta =||\beta ||^2\}.
\end{align}
Since $Z_\beta$ sits in projective space, for any point $(x_0:\dots :x_N)\in Z_\beta$ there exists some $x_i\ne 0$ with $\alpha_i.\beta =||\beta ||^2$. Thus we have $Z_\beta\subseteq Y_\beta$,
and in fact there is a retraction $$p_\beta :Y_\beta \to Z_\beta$$ that sends $x_i$ to $0$ if $\alpha_i.\beta >||\beta ||^2$ (see~\cite[p.42, Def.~12.18]{kirwan84} and~\cite[p.173]{GIT}).
\begin{rem} To get a geometric sense of the spaces $Z_\beta$ and $Y_\beta$, it can be helpful to consider the special case of hypersurfaces of degree $d$ in $\mathbb P^n$.
This case is described in detail in~\S~\ref{Sec-Hyp-2}.
\end{rem}
For each $\beta\in \mathcal{B}$ we set $K_\beta$ to be the stabilizer of $\beta$ under the adjoint action of the maximal compact subgroup $K$ on its Lie algebra $\mathfrak k$ (recall $\beta \in \mathfrak t\subseteq \mathfrak k$)~\cite[Def.~4.8]{kirwan84},~\cite[p.169]{GIT}. There is an action of $K_\beta$ on $Z_\beta$~\cite[p.25]{kirwan84}, and a particular linearization of the action of the complexification of $K_\beta$ on $Z_\beta$ that is defined in~\cite[\S 8.11]{kirwan84}, and with respect to which we obtain a semi-stable locus $Z^{ss}_\beta$. One defines~\cite[p.173]{GIT},~\cite[(11.2), Def.~12.20]{kirwan84}:
\begin{align}
Y_\beta^{ss}&:=p_\beta ^{-1}(Z_\beta^{ss})\\
\label{E:SbetaDef} S_\beta &:= G\cdot Y_\beta ^{ss}.
\end{align}
It is a fact that
\begin{equation}\label{E:GbPbYb}
S_\beta \cong G\times_{P_\beta }Y_{\beta}^{ss}
\end{equation}
where $P_\beta$ is the parabolic subgroup of $G$ that is the product of the stabilizer $K_\beta$ and the Borel subgroup $B$ associated to the choice of $T$ and $\mathfrak t^+$~\cite[p.173]{GIT},~\cite[Lem.~6.9 and \S 12]{kirwan84}.
In fact, the parabolic subgroup $P_\beta$ can also be described as the subgroup of $G$ that preserves $Y_\beta^{ss}$~\cite[Lem.~13.4]{kirwan84}.
An equivalent algebraic definition of $Z_\beta^{ss}$, and hence of $Y_\beta^{ss}$ and $S_\beta$, is given in~\cite[Def.~12.8, Def.~12.14, Def.~12.20]{kirwan84}. For any $x=(x_0:\dots:x_N)\in X\subseteq \mathbb{P}^N$, we denote by $C(x)\subseteq \mathfrak t$ the convex hull of the collection of weights $\alpha_i$ such that $x_i\ne 0$; we define $\beta(x)$ to be the closest point to the origin in $C(x)$. Then for $\beta\ne 0$ we have the following description, summarizing and slightly rephrasing the discussion of~\cite[\S 12]{kirwan84}:
\begin{align}\label{E:HesseZss}
Z_\beta^{ss}&=\left\{x\in Z_\beta : \beta(x)=\beta, \text{ and for all } g \in G, \ ||\beta(gx)||\le ||\beta|| \right\}.
\end{align}
We will also use the fact that~\cite[Lem.~12.13]{kirwan84}:
\begin{equation}\label{E:S0P0}
S_0=X^{ss}
\text{ and }
P_0=G.
\end{equation}
Finally it is shown in~\cite[Lem.~12.15, 12.16]{kirwan84} that the $S_\beta$ define a $G$-equivariant stratification
\begin{equation}\label{E:SbStrat}
X=\bigsqcup_{\beta\in\mathcal{B}}S_\beta=X^{ss}\sqcup \bigsqcup_{0\ne \beta\in\mathcal{B}}S_\beta.
\end{equation}
We end by observing that one can use~\eqref{E:GbPbYb} to conclude that, if nonempty, $S_\beta$ has dimension
\begin{equation}\label{E:DimSb}
\dim S_\beta = \dim G/P_\beta+\dim Y_\beta.
\end{equation}
We call the right hand side of~\eqref{E:DimSb} the expected dimension of $S_\beta$, and denote this as $\dim_{\operatornameeratorname{exp}} S_\beta$.
\begin{rem}\label{R:SbPOSET}
We order the strata $S_\beta$ as a POSET in the usual way, via inclusions of closures; i.e., $S_{\beta'}\le S_{\beta }$ if $\overline S_{\beta'} \subseteq \overline S_{\beta}$.
The maximal stratum is $S_0=X^{ss}$, if it is nonempty. More generally, we can make a POSET out of $\mathcal{B}$ by setting $\beta'\le \beta$ if $\overline Y_{\beta'}\subseteq \overline Y_{\beta}$, and then if $S_{\beta }$ is nonempty, the inequality $\beta'<\beta$ implies $S_{\beta'} <S_{\beta }$. Indeed, $S_{\beta}$ nonempty implies that $Y^{ss}_{\beta }$ is a dense open subset of $\overline Y_{\beta}$, and consequently $\overline Y_{\beta'}\subseteq \overline {Y_{\beta}^{ss}}$, so that $\overline S_{\beta'} \subseteq \overline S_{\beta}$. Note also that if $\beta '<\beta$, then since $S_{\beta '}\subseteq G \overline Y_{\beta '}\subseteq G\cdot \overline Y_\beta$, we have $\dim S_{\beta'}\le \dim G/P_\beta+\dim Y_\beta$. In other words, we can say that if $\beta '<\beta$, then $\dim_{\operatornameeratorname{exp}}S_{\beta '} \le \dim_{\operatornameeratorname{exp}}S_{\beta}$, and if $S_{\beta}$ is nonempty, the inequality is strict.
\end{rem}
\subsection{Equivariant cohomology of the semi-stable locus}
The Thom--Gysin sequence relating the cohomology of a manifold $Y$, a closed submanifold $Z$ of $Y$, and its complement $Y-Z$, is a long exact sequence of the form
$$
\dots \to H^{i-d}(Z;\mathbb{Q})\to H^i(Y;\mathbb{Q})\to H^i(Y-Z;\mathbb{Q})\to H^{i+1-d}(Z;\mathbb{Q})\to \dots
$$
where $d$ is the codimension of $Z$ in $Y$. The existence of such a sequence implies the following identity for Poincar\'e polynomials:
$$
t^dP_t(Z)-P_t(Y)+P_t(Y-Z)=(1+t)Q(t)
$$
where $Q(t)\in \mathbb{Q}[t]$ has nonnegative coefficients.
Applying this to the stratification~\eqref{E:SbStrat} we obtain the following identities for Poincar\'e polynomials and equivariant Poincar\'e polynomials (i.e., the Poincar\'e polynomials for equivariant cohomology):
\begin{align*}
P_t(X)&=\sum_{\beta}t^{2d(\beta)}P_t(S_{\beta})-(1+t)Q(t),\\
P^G_t(X)&=\sum_{\beta}t^{2d(\beta)}P_t^G(S_{\beta})-(1+t)Q^G(t),
\end{align*}
where the polynomials $Q(t),Q^G(t)\in \mathbb{Q}[t]$ have nonnegative coefficients, and
\begin{equation}\label{E:d(beta)Def}
d(\beta):=\operatornameeratorname{codim}_{\mathbb{C}}S_\beta =\dim X-(\dim G-\dim P_\beta+\dim Y_{\beta}),
\end{equation}
where the equality on the right holds provided $S_\beta$ is nonempty.
One then shows that the stratification is $G$-equivariantly perfect~\cite[p.17]{kirwan84}, implying that $Q^G(t)=0$, so that we have:
\begin{align}\label{E:KD-(3.1)-0}
P^G_t(X)&=\sum_{\beta}t^{2d(\beta)}P_t^G(S_{\beta}).
\end{align}
The key point in showing that the stratification is equivariantly perfect is to consider a degenerate Morse function $f:X\to \mathbb{R}$ given as the composition of the induced moment map $\mu:X\to \mathfrak k^\vee$~\cite[(2.7)]{kirwan84}, with the modulus $||-||:\mathfrak k^\vee \to \mathbb{R}$, induced by the Killing form. The strata $S_\beta$, $Y_\beta$, and $Z_\beta$ then have interpretations with respect to the gradient flow to the critical sets for $f$~\cite[Thm.~12.26]{kirwan84}, and one then uses techniques from Morse theory and symplectic geometry to establish that the stratification is equivariantly perfect~\cite[Thm.~6.18]{kirwan84}.
Finally we observe that equation~\eqref{E:KD-(3.1)-0} can be rewritten as~\cite[Eq.~3.1]{kirwanhyp}:
\begin{equation}\label{E:KD-(3.1)}
P_t^G(X^{ss})=P_t(X)P_t(BG)-\sum_{0\neq \beta \in \mathcal{B}} t^{2d(\beta)}P_t^G(S_{\beta}),
\end{equation}
using~\eqref{E:S0P0}, and a result of Kirwan~\cite[Prop.~5.8]{kirwan84} on equivariant cohomology with respect to compact Lie groups acting on symplectic manifolds (see formulas~\eqref{E:K-EC-1} and~\eqref{E:AS-EC-1}), to write $P_t^G(X)=P_t(X)P_t(BG)$. Note that if $S_\beta$ is empty, our convention is that $t^{2d(\beta)}P_t^G(S_\beta)=0$.
\begin{rem}\label{R:PD-deg} The computation of $P_t^G(X^{ss})$ is
an intermediate step in computing the intersection cohomology of the GIT quotient $X/\!\!/_{\mathcal{O}(1)}G$, and the cohomology of the Kirwan blowup. Both of these cohomology theories satisfy Poincar\'e duality, and therefore in these applications it suffices to compute $P_t^G(X^{ss})$ up to degree equal to the complex dimension of the GIT quotient.
Thus, estimating the dimensions via~\eqref{E:DimSb}, one may in some cases ignore many if not all of the strata $S_\beta$ in~\eqref{E:KD-(3.1)}.
\end{rem}
\section[The equivariant cohomology of the semi-stable locus]{The equivariant cohomology of the locus of semi-stable cubic threefolds} \label{S:ECSScubics}
\subsection{Some observations for hypersurfaces}
\label{Sec-Hyp-2}
Before moving to the case of cubic threefolds, we start by making a few observations that hold for all hypersurfaces.
We continue using the notation from \S \ref{S:hypeKirConv}. Recall that $X=\mathbb{P}\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee$ is the Hilbert space of hypersurfaces of degree $d$ in $\mathbb P^n$, we have identified the Lie algebra of the maximal torus of $\operatornameeratorname{SU}(n+1)$ as $\mathfrak t=\{(a_0,\dots,a_n)\in \mathbb{R}^{n+1}: \sum a_i=0\} \subseteq \mathbb{R}^{n+1}$, the inner product on $\mathfrak t$ is taken to be the standard inner product, and we make the identification of monomials of degree $d$ with weights in $\mathfrak t\subseteq \mathbb{R}^{n+1}$ via the assignment $x^I\mapsto \alpha_I:= (i_0-d/(n+1),\dots,i_n-d/(n+1))$. The Weyl group of $\operatornameeratorname{SU}(n+1)$ is the symmetric group $S_{n+1}$ acting on $\mathfrak t$ by its generators, the reflections in the coordinate hyperplanes.
The indexing set
$\mathcal{B}\subseteq \overline{\mathfrak t}_+$ consists of the points in $\overline{\mathfrak t}_+$ that can be described as the closest point to the origin of the convex hull of a nonempty set of the weights $\alpha_0,\dots,\alpha_N$, which themselves can be viewed as lattice points in a simplex.
The sets $Z_\beta$ and $Y_\beta$ are defined as sets of polynomials (up to scaling) where only certain monomials are allowed to appear with non-zero coefficients. More precisely, $Z_\beta$ is the linear subspace of $\mathbb{P}\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee$ determined by the vanishing of the coefficients of the monomials $x^I$ whose weight $\alpha_I$ does not lie in the affine space orthogonal to $\beta$ (i.e., the coefficient of $x^I$ is zero if $\alpha_I.\beta\ne ||\beta^2||$), and $Y_\beta$ is an open subset of the linear subspace of $\mathbb{P}^N$ determined by the vanishing of the coefficients of the monomials $x^I$ whose weight does not lie on the positive side of the affine space orthogonal to $\beta$.
Said another way, $Z_\beta$ is the linear span of the monomials $x^I$ with weights $\alpha_I$ lying in the affine space orthogonal to $\beta$ (i.e., the span of the monomials $x^I$ with $\alpha_I.\beta= ||\beta^2||$), and $Y_\beta$ is the set of polynomials that are linear combinations of the $x^I$ with weights $\alpha_I$ lying on the non-negative side of the affine space orthogonal to $\beta$, and have at least one monomial $x^I$ appearing with non-zero coefficient that has weight $\alpha_I$ lying in the affine space orthogonal to $\beta$. A similar description of $Z^{ss}_\beta$ follows from~\eqref{E:HesseZss}.
We observe also that for hypersurfaces, from the definition of a parabolic subgroup,
it follows that the parabolic subgroup $P_\beta$ can be described as the subgroup of $G$ that preserves the linear subspace of $\mathbb{P}\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee$ that is the closure of $Y_\beta$; this can make the explicit computation of $P_\beta$ easier.
We define $d(\beta):=\operatornameeratorname{codim}_{\mathbb{C}}S_\beta$, so that if $S_\beta$ is nonempty, we obtain the convenient combinatorial dimension count from~\eqref{E:DimSb}:
\begin{align}\label{E:Sbcodim}
d(\beta) &=n(\beta)-\dim G/P_\beta
\end{align}
where
$n(\beta)=\dim \mathbb{P}\operatornameeratorname{Sym}^d(\mathbb{C}^{n+1})^\vee-\dim Y_\beta$ is the number of weights $\alpha_I$ such that
$\beta.\alpha_I <||\beta ||^2$~\cite[p.47]{kirwanhyp}; i.e., the number of weights lying on the negative side of the affine space orthogonal to $\beta$. In other words, the expected codimension of $S_\beta$ is $d_{\operatornameeratorname{exp}}(\beta)=n(\beta)-\dim G/P_\beta $.
\begin{rem}[Estimating $\dim P_\beta$]\label{R:dimPbbest}
Clearly a key point is to estimate the dimension of $P_\beta$.
To this end, recall that if there is a decomposition of the vector space $\mathbb{C}^{n+1}=W\operatornamelus W'$, and a parabolic subgroup $P$ of $\operatornameeratorname{SL}(n+1,\mathbb{C})$ contains the subgroup $\operatornameeratorname{SL}(W)\operatornamelus \operatornameeratorname{Id}_{W'}$,
then the flag associated to $P$ has as its smallest vector space a vector space of dimension at least $\dim W$. In other words, in appropriate bases, $P$ must be block upper-diagonal with a block of size at least $\dim W$ so that the dimension of $P$ must be at least $\binom{\dim W}{2}$ more than the dimension of the Borel subgroup of upper triangular matrices in $\operatornameeratorname{SL}(n+1,\mathbb{C})$ (which has dimension $\binom{n+2}{2}-1$).
\end{rem}
\subsection{The case of cubic threefolds}
We now implement all this in the case of cubic threefolds. For dimension estimates as in Remark~\ref{R:dimPbbest}, note that
the Borel subgroup of upper triangular matrices has dimension $14$ in this case.
\begin{pro}[Equivariant cohomology of the semi-stable locus]\label{P:PtGXss}
For the moduli of cubic threefolds, the only unstable stratum $S_\beta$ that contributes to formula~\eqref{E:KD-(3.1)}, modulo $t^{11}$, is the complex codimension $5$ stratum corresponding to general $D_5$ cubics
(corresponding to the case (b) in~\cite[Lem.~3.1]{allcock}), which only contributes its equivariant $H^0$, so that finally
\begin{equation}\label{eq:cohxss_our}\begin{aligned}
P_t^G(X^{ss})&\equiv 1+t^2+2t^4+3t^6+5t^8+6t^{10} &\mod t^{11}.
\end{aligned}
\end{equation}
\end{pro}
\begin{proof}
We are claiming that for the moduli of cubic threefolds, the only unstable stratum $S_\beta$ that contributes to formula~\eqref{E:KD-(3.1)}, modulo $t^{11}$, is the complex codimension $5$ stratum corresponding to general $D_5$ cubics
(corresponding to the case (b) in~\cite[Lem.~3.1]{allcock}), which only contributes its equivariant $H^0$, so that we have
\begin{equation*}
\begin{aligned}
P_t^G(X^{ss})&\equiv P_t(X)P_t(B\operatornameeratorname{SL}(5,\mathbb{C}))-t^{10}&\mod t^{11}\, \\
&\equiv (1-t^2)^{-1}(1-t^4)^{-1}(1-t^6)^{-1} (1-t^{8})^{-1}(1-t^{10})^{-1}-t^{10} &\mod t^{11}\,\\
&\equiv 1+t^2+2t^4+3t^6+5t^8+6t^{10} &\mod t^{11}.
\end{aligned}
\end{equation*}
We now explain this. To begin, recalling that
$P_t(X)=P_t(\mathbb{P}^{34})\equiv (1-t^2)^{-1}\mod t^{11}$ and that $ P_t(B\operatornameeratorname{SL}(5,\mathbb{C})) = (1-t^4)^{-1}(1-t^6)^{-1} (1-t^{8})^{-1}(1-t^{10})^{-1} $ (e.g., Example~\eqref{E:BSUn}),
we can write~\eqref{E:KD-(3.1)} as
\begin{equation}\label{eq:cohxssProp}
P_t^G(X^{ss})\equiv (1-t^2)^{-1}(1-t^4)^{-1} \dots (1-t^{10})^{-1} -\sum_{0\neq \beta \in \mathcal{B}} t^{2d(\beta)}P_t^G(S_{\beta})
\ \ \mod t^{11}.
\end{equation}
We will show that the strata $S_\beta$, $\beta\ne 0$, have complex codimension $d(\beta)$ at least $5$, and that there is exactly one stratum of complex codimension $5$. This stratum can then only contribute its equivariant $H^0$, which we will see is $1$-dimensional, completing the proof.
The basic tool we will use is the dimension count for the $S_\beta$ given in~\eqref{E:Sbcodim}, and for convenience we rewrite this with the specific numerics we have here. If we set $r(\beta)$ to be the number of weights $\alpha$ such that $\beta .\alpha \ge ||\beta||^2$, and set $p(\beta )=\dim P_\beta$, then it follows from~\eqref{E:Sbcodim} that if $S_\beta$ is nonempty, then
\begin{align}\label{E:dbetacubic}
d(\beta)=(35 -r(\beta))-\dim G+p(\beta)&=11+p(\beta)-r(\beta)\\
&\ge 25-r(\beta). \label{E:dbetacubic>}
\end{align}
From~\eqref{E:dbetacubic>}, if $r(\beta)< 20$, then $d(\beta)>5$, so that $S_\beta$ cannot contribute to~\eqref{eq:cohxssProp}.
As before, we call the right hand side of~\eqref{E:dbetacubic} the expected codimension of $S_\beta$, i.e., the codimension of $S_\beta$, provided it is nonempty, and denote it by $d_{\operatornameeratorname{exp}}(\beta)$.
For our analysis, we will proceed to estimate $d_{\operatornameeratorname{exp}}(\beta)$, starting from the maximal $\beta$; i.e., we partially order the elements of $\mathcal{B}$ by setting $\beta'\le \beta $ if $\overline Y_{\beta'}\subseteq \overline Y_\beta$, and start with the (possibly empty) strata $S_\beta$, $\beta \ne 0$, such that the associated linear spaces $\overline Y_\beta$ are maximal (among the $\overline Y_\beta$ with $\beta \ne 0$); see also Remark~\ref{R:SbPOSET}. These maximal $\overline Y_\beta$ can be described as maximal linear spaces spanned by monomials destabilized by some $1$-PS, and are classified by
Allcock in~\cite[Lem.~3.1]{allcock}. In terms of Allcock's notation, $r(\beta)$ is the number of black dots in the corresponding diagram in~\cite[Lem.~3.1]{allcock}, and the linear space $\overline Y_{\beta}$ is given by the span of the monomials corresponding to those black dots. We now compute the expected codimension $d_{\operatornameeratorname{exp}}(\beta)$ for all the cases in~\cite[Lem.~3.1]{allcock}, enumerating in the same way as in the reference:
\begin{enumerate}
\item[(a)] Let $S_\beta$ correspond to~\cite[Fig.~3.1(a)]{allcock}. One computes the number of black dots in~\cite[Fig.~3.1(a)]{allcock} to be $r(\beta) = 21$. For the dimension of the parabolic subgroup, it is also easy to see from
\cite[Fig.~3.1(a)]{allcock} that
if we write $(\mathbb{C}^5)^\vee=\mathbb{C}\langle x_1,x_2,x_3\rangle\operatornamelus \mathbb{C}\langle x_0,x_4\rangle$, then
the parabolic subgroup $P_\beta$ contains the subgroup $\operatornameeratorname{SL}(\mathbb{C}\langle x_1,x_2,x_3\rangle)\operatornamelus \operatornameeratorname{Id}_{\mathbb{C}\langle x_0,x_4\rangle}$, so that $\dim P_\beta$ is at least $14+\binom{3}{2}=17$ (Remark~\ref{R:dimPbbest}).
Thus from~\eqref{E:dbetacubic} we have that $d_{\operatornameeratorname{exp}}(\beta)\ge 11+17-21=7$.
\item[(b)] Now let $S_\beta$ correspond to~\cite[Fig.~3.1(b)]{allcock}. Here one computes $r = 21$, while the parabolic subgroup can permute $x_3$ and $x_4$, and thus has a $2\times 2$ block on the diagonal, so that $p\ge 14+1=15$. Thus $d_{\operatornameeratorname{exp}}(\beta)\ge 5$.
\item[(c)] Here $r = 20$, while the parabolic subgroup $P_\beta$ contains the subgroup $\operatornameeratorname{SL}(\mathbb{C}\langle x_2,x_3\rangle) \operatornamelus \operatornameeratorname{Id}_{\mathbb{C}\langle x_0,x_1,x_4\rangle}$. Thus we have $p\ge 15$,
so that from~\eqref{E:dbetacubic} we have $d_{\operatornameeratorname{exp}}(\beta)\ge 6$.
\item[(d)] Here $r= 18$, so that as pointed out above in~\eqref{E:dbetacubic>}, the minimal estimate $p\ge 14$ suffices to give $d_{\operatornameeratorname{exp}}(\beta)\ge 7$.
\item[(e)] Here $r=22$, while the parabolic subgroup can permute $x_2,x_3,x_4$, so that $p\ge 17$, and thus $d_{\operatornameeratorname{exp}}(\beta)\ge 6$.
\item[(f)] Here $r = 19$, so again we can use the minimal estimate $d_{\operatornameeratorname{exp}}(\beta)\ge 6$ in~\eqref{E:dbetacubic>}. (Considering the parabolic subgroup more carefully, one can see that one can permute $x_1$ and $x_2$, so that $p\ge 15$ and $d_{\operatornameeratorname{exp}}(\beta )\ge 7$.)
\end{enumerate}
We now turn our attention to the expected codimension of the strata $S_{\beta '}$, $\beta'\ne 0$, that do not arise in the list above. The point is that for any $0\ne \beta'\in \mathcal{B}$, we have $\beta '\le \beta $ for one of the $\beta$ in the list above, and
if $\beta'<\beta$, then $d(\beta' )\ge d_{\operatornameeratorname{exp}}(\beta')\ge d_{\operatornameeratorname{exp}}(\beta)$ (see, e.g., Remark~\ref{R:SbPOSET}).
For clarity, we summarize what we have shown, and what we will prove to establish the proposition:
\begin{enumerate}
\item For any $0\ne \beta'\in \mathcal{B}$ we have shown that $d(\beta')\ge 6$, unless $\beta'\le \beta$ with $\beta$ as in case (b) above, in which case we have $d(\beta')\ge 5$.
\item We claim that there is exactly one $\beta '\le \beta$ with $\beta$ as in case (b) above such that $d(\beta')=5$.
\item For the stratum $S_{\beta'}$ in (2), we claim that the general point corresponds to a $D_5$ cubic, and that $\dim H^0_G(S_{\beta'})=1$.
\end{enumerate}
If $S_\beta$ were known to be non-empty in case (b), then (2) would follow trivially from (1). We find it is easier to establish the weaker statement (2) directly and argue geometrically.
Let $\beta$ be as in~\cite[Fig.~3.1(b)]{allcock}. This corresponds to the case \cite[Thm.~3.3(iii)]{allcock}: the cubic threefold contains a singularity of nullity $2$ and Milnor number $\ge 5$. In others words, the cubic has a double point whose projectivized tangent cone (a quadric) has corank $2$. This excludes the possibility of $A_k$ singularities (as they have corank $1$). Since the Milnor number is at least $5$, we also exclude the $D_4$ case, leading to $D_5$ singularities or worse. In fact, it is immediate to see\footnote{Namely, this case corresponds to singularities of cubic threefolds with an affine equation $x_3^2+x_4^2+x_1x_2^2+(h.o.t.)$, where higher order terms is with respect to weights $(1/3,1/3,1/2,1/2)$ (see \cite[p. 216]{allcock}). We then note that $x_3^2+x_4^2+x_1x_2^2+2x_3x_1^2$ is analytically equivalent to $(x_3')^2+x_4^2+x_1x_2^2-x_1^4$, which is precisely the normal form for $D_5$ in $4$ variables.} that a generic point of $\overline Y_\beta$ (corresponding to a generic linear combination of the given set of monomials) gives a cubic threefold with an isolated $D_5$ singularity. Moreover, $G \cdot\overline Y_\beta$ contains an open subset of the cubics with an isolated $D_5$ singularity; i.e., a general cubic with an isolated $D_5$ singularity is projectively equivalent to one in $\overline Y_\beta$. Using the versal deformation space of a $D_5$ singularity and checking that locally around a $D_5$ cubic the space of cubics maps surjectively onto the versal deformation space (see e.g.,~\cite[Fact 3.12]{cubics}),
it follows that the locus $G\cdot \overline Y_\beta$ has complex codimension $5$ in the space of cubics. Thus there is a codimension $5$ locus in the non-semi-stable locus $X-X^{ss}$.
As the $S_\beta$ with $\beta\ne 0$, stratify the non-semi-stable locus,
and have codimension at most $5$, a Zariski open subset of $G\cdot\overline Y_\beta$ must be an open subset in
some stratum, say $S_{\beta'}$. For dimension reasons, the only possibility is $\beta'\le \beta$ with $\beta$ as in (b) in the list above. For this stratum we have $d(\beta')=5$, and clearly since $G\cdot\overline Y_\beta$ is connected of codimension $5$ there can be no other stratum $S_{\beta ''}$, $\beta''\ne \beta'$, with $d(\beta'')=5$.
The only thing left to show is (3), that $\dim H^0_G(S_{\beta'})=1$.
We have that $Y_{\beta'}^{ss}$ is connected, being a Zariski open subset of a projective space. Consequently, $S_{\beta'} =G\cdot Y_{\beta'}^{ss}$ is connected. Setting $EG\to BG$ to be the universal principal bundle,
since
$S_{\beta'} \times _G EG $ is the quotient of the connected space $S_{\beta'} \times EG\sim_{\text{hom}}S_{\beta'}$, and is therefore itself connected, it follows finally that
$\dim H^0_G(S_{\beta'})=1$.
\end{proof}
\chapter{The cohomology of the Kirwan blowup, part II}\label{S:CohKirBl-II}
The next, and final, step in computing the cohomology of the Kirwan blowup is to compute some ``correction'' terms arising from the blowups. The key point is that there is an equivariant version of the formula for the cohomology of a blowup, which inductively reduces the problem to the setup of the previous section, namely, computing the equivariant cohomology of semi-stable loci. The subtle point is that these computations all reduce to computations on the exceptional divisors, and then with some more work, to computations on a general normal fiber to each stratum that is blown up. This allows one to compute everything essentially on the original space, making the process feasible in examples. We start by reviewing the general setup, and then specialize to the case of cubic threefolds. One of the main points of the review in this section is that it is difficult to explain the terms in Kirwan's formulas in the case of cubic threefolds without describing the construction, and partially explaining the proofs.
\section{The correction terms in general}
\subsection{The correction terms for a single blowup}\label{SSS:CorTermsI}
It is notationally much easier to explain the correction terms after a single blowup. We start with this case, and then in the next subsection explain what the formulas are for multiple blowups.
We start here in the situation of \S~\ref{SSS:kirBlUp}, where we have fixed a maximal dimensional connected component $R\in \mathcal{R}$ of the stabilizer of a strictly polystable point, taken the blowup
\begin{equation}\label{E:PiHatDef-4.1}
\hat \pi:\hat X\to X^{ss}
\end{equation}
along the locus $G\cdot Z^{ss}_R$~\eqref{E:ZRss}, and chosen a linearization of the action on an ample line bundle $\hat L$ on $\hat X$, as described in \S~\ref{SSS:kirBlUp}.
For simplicity, we further assume that $Z_R^{ss}$ is connected.
The first observation is that from the standard argument about cohomology of blowups of smooth loci~\cite[p.605]{GrHa78}, adapted to the $G$-equivariant setting, one has~\cite[p.67]{kirwanblowup}:
\begin{equation}\label{E:KirYXss}
P_t^G(\hat X)=P^G_t(X^{ss})+P^G_t(\mathbb{P}\mathcal{N})-P^G_t(G\cdot Z_R^{ss})
\end{equation}
where $\mathbb{P}\mathcal{N}$ is the projectivization of
\begin{equation}\label{E:NNbundDef}
\mathcal{N}:=\text{the normal bundle to the orbit } G\cdot Z^{ss}_R
\end{equation}
(i.e., $\mathbb{P}\mathcal{N}$ is the exceptional divisor).
Next, the standard Leray spectral sequence argument for the cohomology of a projective bundle~\cite[Prop.~p.606]{GrHa78}, adapted to the $G$-equivariant setting, gives
$P_t^G(\mathbb{P} \mathcal{N})=P^G_t(G\cdot Z_R^{ss})P_t(\mathbb{P}^{\operatornameeratorname{rk}(\mathcal{N})-1})$~\cite[p.67]{kirwanblowup}, so that we have
~\cite[Lem.~7.2]{kirwanblowup}
\footnote{There is a typo in~\cite[Lem.~7.2]{kirwanblowup}: $P_t^G(Z_R^{ss})$ should be $P_t^G(G\cdot Z_R^{ss})$.}
\begin{align}\label{E:KirL7.2}
P_t^G(\hat X)=P^G_t(X^{ss})+P_t^G(G\cdot Z_R^{ss})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}-1)})
\end{align}
where $\operatornameeratorname{rk}\mathcal{N}$ is the complex rank of the vector bundle, which is equal to the codimension of $G\cdot Z_R^{ss}$.
Using~\eqref{E:KD-(3.1)} applied to $\hat X$, with indexing set $\mathcal{B}_{\hat X}$, and strata $S_{\hat X,\hat \beta}$,
and substituting
into~\eqref{E:KirL7.2}, we obtain
\begin{align}
P_t^G(\hat X^{ss})&=P^G_t(X^{ss})+P_t^G(G\cdot Z_R^{ss})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}-1)})-\!\!\! \sum_{0\ne \hat \beta \in \mathcal{B}_{\hat X}}\!\!t^{2d(\hat X,\hat \beta)}P^G_t(S_{\hat X,\hat \beta}),
\end{align}
where the complex codimension of the stratum $S_{\hat X,\hat \beta}$ in $\hat X$
is given by $d(\hat X,\hat \beta)$ as defined in~\eqref{E:d(beta)Def}, but now applied to the blowup $\hat X$.
Now let
\begin{align}
\label{E:NDef} N&:=N(R)\\
\label{E:StabbDef} \operatornameeratorname{Stab}_G\hat \beta&:=G_{\hat \beta}
\end{align}
be the normalizer of $R$ in $G$, and the stabilizer of $\hat \beta$ in $G$ under the adjoint action, respectively.
Then $G\cdot Z^{ss}_R=G\times_N Z^{ss}_R$, and $S_{\hat X,\beta}=G\times_{N\cap \operatornameeratorname{Stab}_G\hat \beta }(Z_{\hat \beta}^{ss}\cap \hat \pi^{-1}Z^{ss}_R)$~\cite[p.72]{kirwanblowup}, where $Z_{\hat \beta}^{ss}\subseteq \hat X$ is defined as in~\eqref{E:HesseZss}. Therefore we obtain (see~\eqref{E:AB-EC-4})
\cite[(7.13)]{kirwanblowup}
\footnote{There are two typos in \cite[(7.13)]{kirwanblowup}: the formula is missing a $-P_t^N(Z^{ss}_R)$, and the sign before the sum in the formula should be negative. The former is noted in \cite[p.50]{kirwanhyp}, while the latter is essentially noted in \cite[(3.4)]{kirwanhyp}.}
\begin{align}
P_t^G(\hat X^{ss})&=P^G_t(X^{ss})\\
&+P_t^N(Z_R^{ss})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}-1)})\\
\label{E:ExtraTerm00} &- \sum_{0\ne \hat \beta \in \mathcal{B}_{\hat X}}t^{2d(\hat X,\hat \beta)}P^{N\cap \operatornameeratorname{Stab}_G\hat \beta }_t(Z_{\hat \beta}^{ss}\cap \hat \pi^{-1}Z^{ss}_R).
\end{align}
We now come to the more subtle point of relating the final term~\eqref{E:ExtraTerm00} to the representation on the normal slice to the orbit.
Let
\begin{equation}\label{E:xinZssR}
x\in Z^{ss}_R
\end{equation}
be a generic point, and let $\mathcal{N}_x$ be the normal space to $G\cdot Z^{ss}_R$ in $X^{ss}$ at $x\in G\cdot Z^{ss}_R$; i.e., the fiber of $\mathcal{N}$ at $x$. Then we obtain a representation
\begin{equation}\label{E:rhoDef}
\rho:R\to \operatornameeratorname{GL}(\mathcal{N}_x).
\end{equation}
We take $\mathbb{T}_R$ to be the restriction of the maximal torus $\mathbb{T}$ of $G$ under the inclusion $R\subseteq G$, with maximal compact tori $T_R$ and $T$, respectively. This gives an inclusion of (real) Lie algebras $\mathfrak t_R\subseteq \mathfrak t$, and we use the metric on $\mathfrak t_R$ \emph{induced from that of $\mathfrak t$}. (Recall that originally in the setup we were allowed to take any $\operatornameeratorname{Ad}$-invariant metric on $\mathfrak t$, but here we \emph{must} take the induced metric on $\mathfrak t_R$.)
Let
\begin{equation}\label{E:BrhoDef}
\mathcal B(\rho)
\end{equation}
be the indexing set for the induced stratification of $\mathbb P\mathcal N_x$: that is, $\mathcal B(\rho)$ is the set of all $\beta$ in a fixed positive Weyl chamber in $\mathfrak t_R$ such that $\beta$ is the closest point to $0$ of the convex hull of a nonempty set of weights of the representation $\rho$.
Let $S_{\beta'}(\rho)$ for
\begin{equation}\label{E:BBrhoDef}
\beta'\in \mathcal{B}(\rho)
\end{equation}
be the associated $R$-stratification of $\mathbb{P}\mathcal{N}_x$ (as defined in \S~\ref{S:E-P-S-def} and~\eqref{E:SbetaDef}).
Kirwan shows
that for $\hat \beta \in \mathcal{B}_{\hat X}$, naturally in $\mathfrak t$, we may actually take $\hat \beta \in \mathfrak t_R\subseteq \mathfrak t$~\cite[Proof of Lem.~7.9, p.70]{kirwanblowup}, and that there is a surjective
\footnote{
For surjectivity of the map \eqref{E:Brho-Bhat}, we are assuming that $\hat X$ is the full Kirwan blow-up. Otherwise, \eqref{E:Brho-Bhat} will not be surjective, and later, in \eqref{E:finExtraT1}, we would find that there was a further sum of the form in \eqref{E:ExtraTerm00}, corresponding to those $\hat \beta\in \mathcal B_{\hat X}$ that are not in the image of \eqref{E:Brho-Bhat}. This is further clarified in \eqref{eq:ARcontribution} and \eqref{E:finExtraT}.
}
map~\cite[p.73, Lem.~7.6, Lem.~7.9]{kirwanblowup}
\begin{equation}\label{E:Brho-Bhat}
\mathcal{B}(\rho)\to \mathcal{B}_{\hat X}
\end{equation}
taking $\beta'\in \mathcal{B}(\rho)$ to the unique element $\hat \beta\in \mathcal{B}_{\hat X}$ with $\beta'$ in its Weyl group orbit $W(G)\cdot \hat \beta$; here we are identifying $W(G)=W(K)$, where $K$ is the maximal compact subgroup. Note that we prefer to work with Weyl group orbits in the Lie algebra $\mathfrak t$ of the (real) maximal torus $T$, whereas Kirwan prefers to work with the equivalent notion of adjoint orbits in the Lie algebra $\mathfrak k$ of $K$.
Given $\hat \beta\in \mathcal{B}_{\hat X}$, the Weyl group orbit $W(G)$ of $\hat \beta$ decomposes into a finite number of $W(R)$ orbits. There is a unique $\beta'\in \mathcal{B}(\rho)$ in each $W(R)$ orbit contained in $W(G)\cdot \hat \beta$~\cite[Proof of Lem.~7.9, p.71]{kirwanblowup}.
We let
\begin{equation}\label{E:wbRGDef}
w( \beta',R,G)
\end{equation} be the number of $\beta' \in \mathcal{B} (\rho)$ that lie in $W(G)\cdot \hat \beta$, i.e., the number of elements in the fiber of~\eqref{E:Brho-Bhat} containing $\beta'$, which is also equal to the number of $W(R)$ orbits contained in the $W(G)$ orbit of~$\hat \beta$~\cite[p.68]{kirwanblowup}\footnote{There is a typo in~\cite[\S 3, p.49]{kirwanhyp}; the definition there is meant to read: $w(\beta,R,G)$ is the number of $R$-adjoint orbits contained in the $G$-adjoint orbit of $\beta$.}.
The fiber $\hat \pi^{-1}(x)$ of $\hat \pi: \hat X \to X^{ss}$ can be naturally identified with the projective space
$\mathbb{P}\mathcal{N}_x$. For $\hat \beta\in \mathcal{B}_{\hat X}$, we have~\cite[Lem.~7.9]{kirwanblowup}
$$
S_{\hat X,\hat \beta}\cap \mathbb{P}\mathcal{N}_x=\bigcup_{\beta '\in \mathcal{B}(\rho)\cap \operatornameeratorname{Ad}(G)\hat \beta} S_{\beta'}(\rho);
$$
i.e., the union is over the $\beta'$ in the fiber of~\eqref{E:Brho-Bhat} over $\hat \beta$.
Kirwan proves in~\cite[Lem.~7.11]{kirwanblowup} that the codimensions $d(\hat X,\hat \beta)$ and $d(\mathbb{P}\mathcal{N}_x,\beta')$ of the associated strata $S_{\hat X,\hat \beta}$ and $S_{\beta'}(\rho)$ are equal.
Now given $\beta'\in \mathcal{B}(\rho)$, let
$\hat \beta$ be the unique element in $\mathcal{B}_{\hat X}$ with $\beta'$ in its $W(G)$ orbit; i.e., the image of $\beta '$ under~\eqref{E:Brho-Bhat}. Let
\begin{equation}\label{E:StabGb'}
\operatornameeratorname{Stab}_G\beta':=G_{\beta'}
\end{equation}
be the stabilizer of $\beta'$ under the \emph{$G$-adjoint action}.
In general $N\cap \operatornameeratorname{Stab}_G\hat \beta \ne N\cap \operatornameeratorname{Stab}_G\beta '$; they may differ by conjugation by an element of the Weyl group $W(G)$. However, as both groups are conjugate subgroups of $N=N(R)$, they have well-defined actions on $Z^{ss}_R$ (since $N(R)$ preserves $Z^{ss}_R$). Moreover, replacing $Z^{ss}_{\hat \beta}$ ($=:Z^{ss}_{\hat X,\hat \beta}$) with the isomorphic locus
\begin{equation}\label{E:ZsshatXbb'}
Z^{ss}_{\hat X,\beta'}
\end{equation}
defined by the element $\beta'\in W(G)\cdot\hat \beta$ (i.e., via~\eqref{E:HesseZss}), we obtain a well-defined action of $N\cap \operatornameeratorname{Stab}_G\beta'$ on $Z^{ss}_{\hat X,\beta'}\cap \hat \pi^{-1}Z^{ss}_R$, such that
$$
P^{N\cap \operatornameeratorname{Stab}\hat \beta }_t(Z_{\hat \beta}^{ss}\cap \hat \pi^{-1}Z^{ss}_R)=P^{N\cap \operatornameeratorname{Stab}_G \beta' }_t(Z_{\hat X, \beta'}^{ss}\cap \hat \pi^{-1}Z^{ss}_R).
$$
In summary, we have~\cite[(7.15)]{kirwanblowup},~\cite[(3.2), (3.4)]{kirwanhyp}:
\begin{flalign}
P_t^G(\hat X^{ss})
\label{E:finHss}&=P^G_t(X^{ss})& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{``semi-stable locus''}\\
\label{E:finMainT1} &+P_t^N(Z_R^{ss})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}-1)}) & \text{``main term"}\\
\label{E:finExtraT1} &-\!\!\!\!\sum_{0\ne \beta'\in \mathcal{B}(\rho)}\!\!\frac{1}{w(\beta',R,G)}t^{2d(\mathbb{P}\mathcal{N}_x,\beta')}P^{N\cap \operatornameeratorname{Stab}_G \beta' }_t(Z_{\hat X , \beta'}^{ss}\cap \hat \pi^{-1}Z^{ss}_R)\!\!\!\!\!\!
&\text{``extra term"}
\end{flalign}
For the formula above, recall that $Z^{ss}_R$ is defined in~\eqref{E:ZRss}, and the rest of the terms are defined above in this subsection: $\hat X^{ss}$~\eqref{E:PiHatDef-4.1}, $N$
\eqref{E:NDef}, $\mathcal{N}$~\eqref{E:NNbundDef}, $\rho$~\eqref{E:rhoDef}, $\beta '$~\eqref{E:BBrhoDef}, $\mathcal{B}(\rho)$~\eqref{E:BrhoDef}, $w(\beta ',R,G)$~\eqref{E:wbRGDef}, $x\in Z^{ss}_R$~\eqref{E:xinZssR}, $\operatornameeratorname{Stab}_G{\beta'}$~\eqref{E:StabGb'}, and $Z^{ss}_{\hat X,\beta'}$~\eqref{E:ZsshatXbb'}.
The goal in deriving the formulas~\eqref{E:finHss},~\eqref{E:finMainT1},~\eqref{E:finExtraT1} above is to try to reduce the computation of the equivariant cohomology of the blowup $\hat X^{ss}$ to certain computations on $X$. The remark below can be quite helpful in this regard.
\begin{rem} \label{R:PiFibration}
The restriction $\hat \pi : (Z_{\hat X, \beta'}^{ss}\cap \hat \pi^{-1}Z^{ss}_R) \to Z^{ss}_R$ is an $(N\cap \operatornameeratorname{Stab}_G \beta')$-equivariant
fibration with fibers isomorphic to $Z^{ss}_{\beta'}(\rho)$ (as defined in~\eqref{E:Zbeta},~\eqref{E:HesseZss} for the representation $\rho$)~\cite[8.11]{kirwan84}~\cite[p.50]{kirwanhyp}.
Moreover, if for instance $N\cap \operatornameeratorname{Stab}_G \beta'$ acts transitively on $Z^{ss}_R$,
then letting $(N\cap \operatornameeratorname{Stab}_G \beta')_x$ be the stabilizer of the general point $x\in Z^{ss}_R$~\eqref{E:xinZssR}
in $N\cap \operatornameeratorname{Stab}_G \beta'$, we have (e.g.,~\eqref{E:PGfib})
$$
P_t^{N\cap \operatornameeratorname{Stab}_G \beta'}(Z^{ss}_{\hat X, \beta'}\cap \hat \pi^{-1}Z^{ss}_R)=P^{(N\cap \operatornameeratorname{Stab}_G \beta')_x}(\hat \pi^{-1}(x))=P^{(N\cap \operatornameeratorname{Stab}_G \beta')_x}(Z^{ss}_{\beta'}(\rho)).
$$
Note that with the transitive group action on $Z^{ss}_R$, we may take any point $x\in Z^{ss}_R$ to make our computation, since this will only change the computations up to conjugate groups, which will not affect the final outcome.
\end{rem}
\subsection{The correction terms in general} Having reviewed the case of a single blowup, we now give the formulas for the cohomology of the full inductive blowup, $\widetilde X^{ss}$.
We use the notation from \S~\ref{SSS:kirBlUp} and especially Remark~\ref{R:pi-r-Def}.
The relevant formula for computing the cohomology of $\widetilde X^{ss}$, generalizing~\eqref{E:finHss} to the full blowup, is now~\cite[Eq.~3.2]{kirwanhyp}
\begin{equation}\label{eq:ARcontribution}
P_t^G(\widetilde X^{ss})=P_t^G(X^{ss})+\sum_{R\in \mathcal{R}} A_R(t).
\end{equation}
For any $R\in\mathcal{R}$ the term $A_R(t)$ in~\eqref{eq:ARcontribution} records the change of the Betti numbers under the blowup $\pi_{\dim R}:X_{\dim R}\to X_{\dim R+1}$, as defined in Remark~\ref{R:pi-r-Def}; more precisely, one decomposes $\pi_{\dim R}$ into individual blowups of the loci $G\cdot Z^{ss}_{R,\dim R+1}$, where $Z^{ss}_{R,\dim R+1}$ is the strict transform of $Z^{ss}_R$ in $X^{ss}_{\dim R+1}$, and these are the correction terms for that blowup.
The explicit formula for these terms, generalizing~\eqref{E:finMainT1} and~\eqref{E:finExtraT1}, is given by~\cite[Eq.~3.4]{kirwanhyp}
\begin{align}
&A_R(t):=\\
\label{E:finMainT} &P_t^{N(R)}(Z_{R,\dim R+1}^{ss})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}_R-1)}) & \text{``main term"}\\
\label{E:finExtraT} &- \sum_{0\ne \beta'\in \mathcal{B}_R(\rho)}\frac{1}{w(\beta',R,G)}t^{2d(\mathbb{P}\mathcal{N}_x,\beta')}P^{N(R)\cap \operatornameeratorname{Stab}_G \beta' }_t(Z^{ss}_{\beta ',R})& \text{``extra term"}
\end{align}
where $Z^{ss}_{\beta ',R}:=Z_{X_{\dim R}, \beta'}^{ss}\cap \pi_{\dim R}^{-1}Z^{ss}_{R,\dim R+1}$.
These terms, as well as all of the other terms above, are described in \S~\ref{SSS:CorTermsI} (see especially the references after~\eqref{E:finExtraT1}, and also Remark~\ref{R:PiFibration}, and Remark~\ref{R:pi-r-Def}).
\section[The main correction terms for cubic threefolds]{The \emph{main correction} terms for cubic threefolds}
We now compute the main terms~\eqref{E:finMainT} for the case of cubic threefolds. We have taken $\mathcal{R}=\{R_{2A_5}\cong \mathbb{C}^*, R_{3D_4}\cong (\mathbb{C}^*)^2, R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})\}$.
Since we have also already worked out the loci $Z^{ss}_R$ in Proposition~\ref{P:ZRss}, the main point is to understand the normalizers $N(R)$, and their action on the $Z^{ss}_R$.
We will work in the order of descending dimension of $R$, following the Kirwan blowup process.
\subsection{The main correction term for $R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})$, the chordal cubic case}
As we have seen, the first step in the Kirwan blowup process is to blow up the locus corresponding to chordal cubics. We start by describing the main term~\eqref{E:finMainT} for this blowup.
\begin{pro}[Main term for the chordal cubic]\label{P:MT-ChC} For the group $R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})$, the main term~\eqref{E:finMainT} is given by
\begin{align*}
P^{N(R_c)}_t(Z^{ss}_{R_c})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}_{R_c}-1)}) &=(1-t^4)^{-1}(t^2+\dots+t^{24})\\
&= t^2+t^4+2t^6+2t^8+3t^{10}\mod t^{11}.
\end{align*}
\end{pro}
This will follow directly from the following lemma:
\begin{lem}[Proposition~\ref{P:App-R=SL2}]\label{L:RcNorm}
For $R_c$, the group $\operatornameeratorname{PGL}(2,\mathbb{C})$ embedded in $\operatornameeratorname{SL}(5,\mathbb{C})$ via its $\operatornameeratorname{Sym}^4\mathbb C^2$ ($\cong \mathbb C^5$) representation,
the normalizer $N(R_c)$ of $R_c$ in $\operatornameeratorname{SL}(5,\mathbb{C})$ is a split central extension
\begin{equation}\label{E:NRcCent}
1\to\mu_5\to N(R_c)\to \operatornameeratorname{PGL}(2,\mathbb{C})\to 1,
\end{equation}
where $\mu_5$ is the group of $5$-th roots of unity. \qed
\end{lem}
The proof of the lemma is elementary, with all necessary computations given in Proposition~\ref{P:App-R=SL2}.
\begin{proof}[Proof of Proposition~\ref{P:MT-ChC}]
We saw in Proposition~\ref{P:ZRss} that $Z^{ss}_{R_c}$ consists of a single point, $V(F_{-1,2})$; i.e., the chordal cubic.
The stabilizer of $V(F_{-1,2})$ has connected component equal to $R_c$, so that the dimension of $G\cdot Z^{ss}_{R_c}=\dim G-3=21$.
Thus the rank of the normal bundle to the orbit $G\cdot Z^{ss}_{R_c}$ is $\operatornameeratorname{rk}\mathcal{N}_{R_c}=34-21=13$.
Next we compute $P^{N(R_c)}_t(Z^{ss}_{R_c})$.
Since $Z^{ss}_{R_c}$ is a point, we have
\begin{align*}
H^\bullet_{N(R_c)}(Z^{ss}_{R_c})&=H^\bullet(B(N(R_c)))\\
&=H^\bullet (B\mu_5)\otimes H^\bullet (B\operatornameeratorname{PGL}(2,\mathbb{C}))& (\eqref{E:NRcCent}, \eqref{E:K-AS-EC-cent})\\
&= H^\bullet (B\operatornameeratorname{PGL}(2,\mathbb{C}))\\
&=\mathbb{Q}[c] & (\text{Example~\ref{E:BSUn}, \ref{E:H-PGL}})
\end{align*}
where $\deg c=4$. In other words $P^{N(R_c)}_t(Z^{ss}_{R_c})=(1-t^4)^{-1}=1+t^4+t^8+\dots$.
\end{proof}
\subsection{The main correction term for $R_{3D_4}\cong (\mathbb{C}^*)^2$, the $3D_4$ case}
\begin{pro}[Main term for the $3D_4$ cubic]\label{P:MT-3D4}
For the group $R_{3D_4}\cong (\mathbb{C}^*)^2$, the main term~\eqref{E:finMainT} is given by
\begin{align*}
P^{N(R_{3D_4})}_t(Z^{ss}_{R_{3D_4,3}})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}_{R_{3D_4}}-1)})&= (1-t^4)^{-1}(1-t^6)^{-1}(t^2+\dots +t^{22})\\
&=t^2+t^4+2t^6+3t^8+4t^{10} \mod t^{11}.
\end{align*}
\end{pro}
Similarly to the chordal cubic case, this will follow directly from the elementary, but laborious, computations leading to the descriptions of the geometry involved. We will record the results here, while the full proofs are given in Proposition~\ref{P:App-R=C*2} of the Appendix. We first recall the notation $\mathbb{S}_n$ for the ``generalized permutation matrices of size $n$'', which explicitly are the matrices one obtains in $\operatornameeratorname{GL}(n,\mathbb{C})$ by permuting the columns of some diagonal matrix. Moreover, we adopt the convention that when we write an explicit form of a collection of matrices, and then write that it lies in a certain group, that this may impose an extra condition (eg.,~for a $5\times 5$ matrix, we may write $\in\operatornameeratorname{SL}$ to impose that it has determinant one, if it is not automatic from the form of the matrix). We finally record that $R_{3D_4}$ is isomorphic to $(\mathbb{C}^*)^2$, and given in coordinates by
\begin{equation}\label{E:R3D4main}
R_{3D_4}=\operatornameeratorname{diag}(s,t,s^{-1}t^{-1},1,1)\cong (\mathbb{C}^*)^2\,.
\end{equation}
\begin{lem}[{Proposition~\ref{P:App-R=C*2}}]\label{L:R3D4Norm1}
In the notation above:
\begin{enumerate}
\item The normalizer $N(R_{3D_4})$ of $R_{3D_4}$ in $\operatornameeratorname{SL}(5,\mathbb{C})$ is
$$
N(R_{3D_4})=
\left\{\left(
\begin{array}{c|c}
\mathbb{S}_3&\\ \hline
&\operatornameeratorname{GL}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\}\,.
$$
\item The fixed locus $Z_{R_{3D_4}}^{ss}$ is unchanged under the first blowup:
$Z_{R_{3D_4,3}}^{ss}=Z_{R_{3D_4}}^{ss}$.
\item The normalizer acts on $Z_{R_{3D_4}}^{ss}$ transitively: $Z^{ss}_{R_{3D_4}}=N(R_{3D_4}) \cdot \{V(F_{3D_4})\}$. \qed
\end{enumerate}
\end{lem}
We will moreover need to know various other stabilizer groups. We denote by $\operatornameeratorname{Stab}(V(F_{3D4}))\subset\operatornameeratorname{SL}(5,\mathbb{C})$
the stabilizer of the cubic with equation $F_{3D4}$, denote $\operatornameeratorname{Aut}(V(F_{3D4}))$ its stabilizer in $\operatornameeratorname{PGL}(5,\mathbb{C})$, and let $\operatornameeratorname{GL}_{V(F_{3D4})}$ be the stabilizer in $\operatornameeratorname{GL}(5,\mathbb{C})$. We furthermore denote $D:=\{\operatornameeratorname{diag}(\lambda _0,\lambda _1,\lambda_2,\lambda_3,\lambda_4): \lambda_0\lambda_1\lambda_2=\lambda_3^3=\lambda_4^3\}$ an auxiliary group for these computations, which can be explicitly written as the direct product
\begin{equation}\label{E:D=Txmu}
D=\mathbb{T}^3\times \mu_3
\end{equation}
of the torus $\mathbb{T}^3=\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_0^{-1}\lambda_1^{-1}\lambda_3^3,\lambda_3,\lambda_3)\cong (\mathbb{C}^*)^3$ and the group $\mu_3=\operatornameeratorname{diag}(1,1,1,1,\zeta^i)\cong \mathbb{Z}/3\mathbb{Z}$ where $\zeta$ is a primitive $3$-rd root of unity.
\begin{lem}[{Proposition~\ref{P:App-R=C*2}}]\label{L:R3D4Norm2}
The groups defined above are as follows.
\begin{enumerate}
\item The group $\operatornameeratorname{Stab}(V(F_{3D4}))$ is equal to
\begin{equation}\label{E:App-GF3D4}
\operatornameeratorname{Stab}(V(F_{3D4}))=
\left\{\left(
\begin{array}{c|c}
\mathbb{S}_3&\\ \hline
&\mathbb{S}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C}): \lambda_0\lambda_1\lambda_2=\lambda_3^3=\lambda_4^3\right\},
\end{equation}
where the $\lambda_i$ is the unique non-zero element in the $i$-th row.
\item There are central extensions
\begin{equation}\label{E:St3D4Ct}
1\to \mu_5\to \operatornameeratorname{Stab}(V(F_{3D_4})) \to \operatornameeratorname{Aut}(V(F_{3D_4}))\to 1\,,
\end{equation}
\begin{equation}\label{E:GLSt3D4Ct}
1\to \mathbb{C}^*\to \operatornameeratorname{GL}_{V(F_{3D_4})}\to \operatornameeratorname{Aut}(V(F_{3D_4}))\to 1\,.
\end{equation}
\item There is an isomorphism
\begin{equation}\label{E:GLStDs3s2}
\operatornameeratorname{GL}_{V(F_{3D_4})}\cong D\rtimes (S_3\times S_2)\,,
\end{equation}
where the action of $S_3\times S_2$ on $D$ is to permute the entries; $S_3$ permutes the first three entries $\lambda_0,\lambda_1,\lambda_2$, and $S_2$, the last two, $\lambda_3,\lambda_4$. \qed
\end{enumerate}
\end{lem}
The proofs of the two lemma above are by (long) direct computations, given in Proposition~\ref{P:App-R=C*2} in the Appendix.
\begin{rem}
The reason for introducing $\operatornameeratorname{GL}_{V(F_{3D_4})}$ is that while there is a short exact sequence $1\to D\to \operatornameeratorname{Stab}(V(F_{3D_4}))\to S_3\times S_2\to 1$, this sequence does not split. For the purposes of computing equivariant cohomology, it is just as easy to work with central extensions, see~\eqref{E:K-AS-EC-cent}, and so we work with $\operatornameeratorname{GL}_{V(F_{3D_4})}$, where the surjection splits, giving an easy semi-direct product with which to work.
\end{rem}
\begin{proof}[Proof of Proposition~\ref{P:MT-3D4}]
For brevity, we write $R=R_{3D_4}$ and $N=N(R)$. Since by Lemma~\ref{L:R3D4Norm1}(1) the group~$N$ acts transitively on $Z^{ss}_R$, we have $\dim G\cdot Z^{ss}_R=\dim G\cdot \{V(F_{3D_4})\}=24-2=22$.
Thus the rank of the normal bundle to the orbit $G\cdot Z^{ss}_{R}$ is $\operatornameeratorname{rk}\mathcal{N}_{R}=34-22=12$.
Next we compute $P^N_t(Z^{ss}_{R})$.
From Lemma~\ref{L:R3D4Norm1}(3), we have
\begin{align*}
H^\bullet_N(Z^{ss}_{R})&=H^\bullet (B\operatornameeratorname{Stab}(V(F_{3D_4})))\,.
\end{align*}
At the same time we have
\begin{align*}
H^\bullet(B\mathbb{C}^*)\otimes H^\bullet (B&\operatornameeratorname{Stab}(V(F_{3D_4})))\\
&=H^\bullet(B\mathbb{C}^*)\otimes H^\bullet (B\operatornameeratorname{Aut}(V(F_{3D_4})))&(\eqref{E:St3D4Ct},~\eqref{E:K-AS-EC-cent})\\
&=H^\bullet (B \operatornameeratorname{GL}_{V(F_{3D_4})})&(\eqref{E:GLSt3D4Ct})\\
&=H^\bullet(BD)^{S_3\times S_2}& (\eqref{E:GLStDs3s2}, \ \eqref{E:AS-EC-2})\\
&=(H^\bullet(B(\mathbb{T}^3\times \mu_3)))^{S_3\times S_2} & (\eqref{E:D=Txmu})\\
&=H^\bullet(B(\mathbb{T}^3))^{S_3\times S_2} & (\eqref{Exa:SemDirExa})\\
&=\mathbb{Q}[c_1^{(1)},c_1^{(2)},c_1^{(3)}]^{S_3\times S_2},&
\end{align*}
with degree $c_1^{(i)}=2$.
The action of $S_3\times S_2$ is given as follows. First, we observe that the action is obtained from the action of $S_3\times S_2$ on the torus $\mathbb{T}^3$ (e.g., Example~\ref{Exa:SemDirExa}), via the identifications~\eqref{E:D=Txmu} and~\eqref{E:GLStDs3s2}.
Concretely, $\mathbb{T}^3=\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_0^{-1}\lambda_1^{-1}\lambda_3^3,\lambda_3,\lambda_3)\cong (\mathbb{C}^*)^3=\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_3)$.
The action of $S_3\times S_2$ on $\mathbb T^3$ is to permute the entries; $S_3$ permutes the first three entries, and $S_2$, the last two.
Consequently, the $S_2$ factor acts trivially on $\mathbb T^3$.
To describe the action of the $S_3$ factor on $\mathbb T^3\cong (\mathbb{C}^*)^3=\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_3)$, let us denote $S_3=\langle \delta,\gamma : \delta^2=\sigma^3=1,\ \delta \sigma =\sigma^2\delta\rangle$ the standard presentation of $S_3$.
Then
$\delta(\lambda_0,\lambda_1,\lambda_3)=(\lambda_1,\lambda_0,\lambda_3)$, and $\sigma(\lambda_0,\lambda_1,\lambda_3)=(\lambda_0^{-1}\lambda_1^{-1}\lambda_3^3,\lambda_0,\lambda_3)$.
The action of $S_3$ on the symmetric algebra $\mathbb{Q}[c_1^{(1)},c_1^{(2)},c_1^{(3)}]$ is induced by the action of $S_3$ on the vector space $\mathbb{Q}\langle c_1^{(1)},c_1^{(2)},c_1^{(3)}\rangle$, and so we see that $\delta$ and $\sigma$ act by
$$
\delta =
\left(
\begin{array}{ccc}
0&1&0\\
1&0&0\\
0&0&1
\end{array}
\right)\ \ \hbox{and}
\ \ \sigma =
\left(
\begin{array}{ccc}
-1&1&0\\
-1&0&0\\
3&0&1
\end{array}
\right).
$$
At this point, one may use Molien's formula, or simply observe via the characters that the representation of $S_3$ given by the matrices above is isomorphic to the standard representation, which decomposes as the direct sum of the trivial representation and the representation of $S_3$ as the dihedral group acting on the plane.
In any case,
we obtain the generating function for $\mathbb{Q}[c_1^{(1)},c_1^{(2)},c_1^{(3)}]^{S_3\times S_2}$ to be
$$
(1-t^2)^{-1}(1-t^4)^{-1}(1-t^6)^{-1}
$$
Thus putting everything together, we have
\begin{align*}
P^N_t(Z^{ss}_{R})&=(1-t^2)\cdot (1-t^2)^{-1}(1-t^4)^{-1}(1-t^6)^{-1} =(1-t^4)^{-1}(1-t^6)^{-1}\\
&\equiv 1+ t^4+t^6+t^8+t^{10} \mod t^{11}.
\end{align*}
\end{proof}
\subsection{The main correction term for $R_{2A_5}\cong \mathbb{C}^*$, the $2A_5$ case}
\begin{pro}[Main term for $2A_5$ cubics]\label{P:MT-2A5} For the group $R_{2A_5}\cong \mathbb{C}^*$, the main term~\eqref{E:finMainT} is given by
\begin{align*}
P^{N(R_{2A_5})}_t(Z^{ss}_{R_{2A_5,2}})(t^2+\dots +t^{2(\operatornameeratorname{rk}\mathcal{N}_{R_{2A_5}}-1)})&=(1-t^4)^{-1}(1+t^2)(t^2+\dots+t^{18})\\
&\equiv t^2+2t^4+3t^6+4t^8+5t^{10} \mod t^{11}.
\end{align*}
\end{pro}
This will follow directly from the following lemma, which will be proven by direct elementary computations, given in Propositions~\ref{P:App-R=C*p1} and~\ref{P:App-R=C*p2} in the Appendix. We recall from Proposition~\ref{P:ZRss} that $Z^{ss}_{R_{2A_5}}$ is the set of semi-stable cubics defined by equations of the form $a_0x_2^3 + a_1x_0x_3^2 + a_2x_1^2x_4 +a_3 x_0x_2x_4 + a_4x_1x_2x_3=0$.
\begin{lem}[{Propositions~\ref{P:App-R=C*p1} and~\ref{P:App-R=C*p2}}]\label{L:R2A5Norm} For $R_{2A_5}=\operatornameeratorname{diag}(\lambda^{2},\lambda ,1,\lambda ^{-1},\lambda ^{-2})\cong \mathbb{C}^*$:
\begin{enumerate}
\item The normalizer $N(R_{2A_5})$ of $R_{2A_5}$ in $\operatornameeratorname{SL}(5,\mathbb{C})$ is equal to
the subgroup of $\operatornameeratorname{SL}(5,\mathbb{C})$ that is the semi-direct product
\begin{equation}\label{E:LR2A5Norm}
N(R_{2A_5})\cong \mathbb{T}^4\rtimes \mathbb{Z}/2\mathbb{Z}
\end{equation}
of the maximal torus $\mathbb{T}^4$, and the involution $\tau:x_i\mapsto x_{4-i}$, with the semi-direct product given by the homomorphism
$$ \tau \mapsto \left(\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_2,\lambda_3,\lambda_4)\mapsto \operatornameeratorname{diag}(\lambda_4,\lambda_3,\lambda_2,\lambda_1,\lambda_0)\right)\,\,.$$
\item The orbit of the chordal cubic $G\cdot Z^{ss}_{R_c}$ meets $Z^{ss}_{R_{2A_5}}$
precisely in the divisor defined by the equation $4a_0a_1a_2-a_3a_4^2=0$.
Thus the strict transform $Z_{R_{2A_5},2}^{ss}$ is isomorphic to $Z^{ss}_{R_{2A_5}}$.
\item The quotient $Z_{R_{2A_5}}^{ss}/\mathbb{T}^4$ is isomorphic to $\mathbb{P}^1$. \qed
\end{enumerate}
\end{lem}
For the last item we note that $\mathbb{T}^4/\mathbb{C}^*$ acts on $Z_{R_{2A_5}}^{ss}$ with finite stabilizers, and so the quotient $Z_{R_{2A_5}}^{ss}/\mathbb{T}^4=Z_{R_{2A_5}}^{ss}/(\mathbb{T}^4/\mathbb{C}^*)$ is a well-defined variety.
\begin{proof}[Proof of Proposition~\ref{P:MT-3D4}]
For brevity, write $R=R_{2A_5}$ and $N=N(R)$. In Lemma~\ref{L:R2A5Norm}(2), we saw that $Z^{ss}_{R_{},2}=Z^{ss}_{R_{}}$, and that $Z^{ss}_{R_{}}$ has dimension $4$. Now consider the subgroup $G'\subseteq \operatornameeratorname{SL}(5,\mathbb{C})$ consisting of those $g$ such that $g\cdot Z^{ss}_R\subseteq Z^{ss}_R$. This group $G'$ has dimension $4$. Indeed, we have $N\subseteq G'$, so that $\dim G'\ge 4$. On the other hand, the stabilizer of a general point of $Z^{ss}_R$ is $1$-dimensional (it has connected component equal to $R$), so that
if $\dim G'\ge 5$, then the dimension of the orbit of a general point would be $\ge 5-1=\dim Z^{ss}_R$. But then there would be a Zariski dense subset of $Z^{ss}_R$ corresponding to projectively equivalent cubics.
It follows that $\dim G\cdot Z^{ss}_R=\dim Z^{ss}_R+\dim G -\dim G' =4+24-4=24$.
Thus the rank of the normal bundle to the orbit $G\cdot Z^{ss}_{R}$ is $\operatornameeratorname{rk}\mathcal{N}_{R}=34-24=10$.
Next we compute $P^N_t(Z^{ss}_{R})$.
From Lemma~\ref{L:R2A5Norm}, we have
\begin{align*}
H^\bullet_N(Z^{ss}_{R})&=\left(H^\bullet _{\mathbb{T}^4}(Z^{ss}_R)\right)^{\mathbb{Z}/2\mathbb{Z}}& (\eqref{E:LR2A5Norm}, \ \eqref{E:AS-EC-2})\\
&=(H^\bullet(BR)\otimes H^\bullet_{\mathbb{T}^4/R}(Z^{ss}_R))^{\mathbb{Z}/2\mathbb{Z}} & (\eqref{E:K-AS-EC-cent}) \\
&=(H^\bullet(BR)\otimes H^\bullet(\mathbb{P}^1))^{\mathbb{Z}/2\mathbb{Z}} & (\text{Lemma~\ref{L:R2A5Norm}}(3),\ \eqref{E:AS-EC-FinG})\\
&=(\mathbb{Q}[c_1] \otimes \mathbb{Q}[h]/h^2)^{\mathbb{Z}/2\mathbb{Z}} &
\end{align*}
where $\deg c_1=\deg h=2$.
Now one must trace through the constructions to find the action of $\mathbb{Z}/2\mathbb{Z}=\langle \tau\rangle$ on the polynomial ring. The action on the cohomology of $BR$ is induced by the action on $R$, which can easily be seen to be given by $\lambda\mapsto \lambda^{-1}$. Thus the action of $\tau$ on $c_1$ is given by $\tau c_1=-c_1$. The action of $\tau$ on the cohomology of $\mathbb{P}^1$ is induced by the action on $\mathbb{P}^1$. The action of $\tau$ on $Z^{ss}_R$ is given by $\tau (a_0:\dots :a_4)=(a_0:a_2:a_1:a_3:a_4)$. Using the locus $\{V(F_{A,B}):(A,B)\ne (0,0)\}\subseteq Z^{ss}_R$ (i.e., $a_1=a_2=-a_3=1$), one sees that the action on the quotient $\mathbb{P}^1=Z^{ss}_R/\mathbb{T}^4$ is trivial. Thus the action of $\tau$ on $h$ is trivial. Thus we have $ (\mathbb{Q}[c_1] \otimes \mathbb{Q}[h]/h^2)^{\mathbb{Z}/2\mathbb{Z}} =\mathbb{Q}[c_1^2]\otimes \mathbb{Q}[h]/(h^2)$. Thus $P_t^N(Z^{ss}_R)=(1-t^4)^{-1}(1+t^2)$.
\end{proof}
\section[The extra correction terms for cubic threefolds]{The \emph{extra correction terms} for cubic threefolds}
Having computed the {\em main terms} of the contributions $A_R(t)$ given by~\eqref{E:finMainT}, to finish the computation of $H^\bullet (\calM^{\operatorname{K}})$ following Kirwan's method it thus remains to compute the \emph{extra terms} given by~\eqref{E:finExtraT}. A key point is to describe for each $R$ the representation $\rho:R\to \operatornameeratorname{Aut}(\mathcal{N}_x)$ on the normal slice to the orbit $G\cdot Z^{ss}_R$ at a generic point $x\in Z^{ss}_R$. We start in \S~\ref{S:TanOrb} by reviewing a general approach to computing the tangent space to an orbit for the case of hypersurfaces. We then utilize this in the case of cubic threefolds, and consequently obtain the extra terms.
\subsection{Tangent spaces to orbits for hypersurfaces}\label{S:TanOrb}
Let $F\in H^0(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d))$ the form defining a hypersurface $V(F)\subseteq \mathbb{P}^n$. We wish to describe the tangent space to the orbit $\operatornameeratorname{GL}(n+1,\mathbb{C})\cdot F$.
\begin{rem} We will ultimately be interested in the normal space to the orbit $\operatornameeratorname{SL}(n+1,\mathbb{C})\cdot \{V(F)\}$ in $\mathbb{P} H^0(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d))$. However, since the normal space of any submanifold $Y$ in projective space $\mathbb{P}(V)$ can, via the
Euler sequence, be identified with the normal space to its cone $C(Y)$ in $V$, we may instead consider the $\operatornameeratorname{GL}(n+1,\mathbb{C})$ orbit of $F$ in $H^0(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d))$, rather than the $\operatornameeratorname{SL}(n+1,\mathbb{C})$ orbit of $V(F)$ in $\mathbb{P} H^0(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d))$.
\end{rem}
To compute the tangent space to the $\operatornameeratorname{GL}(n+1,\mathbb{C})$ orbit of $F$ we work with the Lie algebra $\mathfrak{gl}(n+1,\mathbb{C})$ and use the exponential map $\exp: \mathfrak{gl}(n+1,\mathbb{C}) \to \operatornameeratorname{GL}(n+1,\mathbb{C})$.
If $e \in \mathfrak{gl}(n+1,\mathbb{C})$ then taking the derivative $\frac{d}{dt}\left.(\exp(te)F)\right|_{t=0}$ gives a tangent vector $t_e$ in the tangent space to the orbit $\operatornameeratorname{GL}(n+1,\mathbb{C})\cdot F$. Taking a basis of $\mathfrak{gl}(n+1,\mathbb{C})$ we then obtain generators for the tangent space
to the orbit $\operatornameeratorname{GL}(n+1,\mathbb{C})\cdot F$.
Concretely, with respect to the coordinates $(x_0:\dots:x_n)$, numbering the rows and columns of matrices from $0$ to $n$, we then take as generators for $\mathfrak{gl}(n+1,\mathbb{C})$ the elementary matrices $e_{ij}$ for all $0\le i, j\le n$, where $e_{ij}$ is the matrix with all zero entries except for the $ij$-the entry which is one. Given a form $F$, we then denote
$$
(DF)_{ij}:=\frac{d}{dt}\left.\left(\exp(te_{ij})F\right)\right|_{t=0}.
$$
We denote by $DF$ the associated matrix with entries $(DF)_{ij}$. Finally, we conclude that the tangent space to the orbit $\operatornameeratorname{GL}(n+1,\mathbb{C})\cdot F$ is given by the span of the entries of the matrix $DF$.
We implement this now in the case of polystable cubic threefolds:
\begin{exa}[Tangent spaces to the orbits of strictly polystable cubic threefolds]\label{Exa:D}
For a strictly polystable cubic threefold defined by a cubic form $F$, the tangent space to the orbit $\operatornameeratorname{GL}(5,\mathbb{C})\cdot F$ is given by the span of the entries of the matrix $DF$. In particular we have:
\begin{enumerate}
\item For $F=F_{A,B}$, $(A,B)\ne (0,0)$, the matrix $DF_{A,B}$ is given by
$$
DF_{A,B}=
$$
$$
\scriptstyle
\hskip-1cm \left(\begin{smallmatrix}
x_0x_3^2-x_0x_2x_4&x_1x_3^2-x_1x_2x_4&x_2x_3^2-x_2^2x_4&x_3^3-x_2x_3x_4&x_3^2x_4-x_2x_4^2\\
2x_0x_1x_4+Bx_0x_2x_3&2x_1^2x_4+Bx_1x_2x_3&2x_1x_2x_4+Bx_2^2x_3&2x_1x_3x_4+Bx_2x_3^2&2x_1x_4^2+Bx_2x_3x_4\\
3Ax_0x_2^2-x_0^2x_4+Bx_0x_1x_3&3Ax_1x_2^2-x_0x_1x_4+Bx_1^2x_3&3Ax_2^3-x_0x_2x_4+Bx_1x_2x_3&3Ax_2^2x_3-x_0x_3x_4+Bx_1x_3^2&3Ax_2^2x_4-x_0x_4^2+Bx_1x_3x_4\\
2x_0^2x_3+Bx_0x_1x_2&2x_0x_1x_3+Bx_1^2x_2&2x_0x_2x_3+Bx_1x_2^2&2x_0x_3^2+Bx_1x_2x_3&2x_0x_3x_4+Bx_1x_2x_4\\
x_0x_1^2-x_0^2x_2&x_1^3-x_0x_1x_2&x_1^2x_2-x_0x_2^2&x_1^2x_3-x_0x_2x_3&x_1^2x_4-x_0x_2x_4
\end{smallmatrix}\right)
$$
To quickly determine all linear equations satisfied by the entries of the matrix $DF$, we note that since $F_{A,B}$ is preserved by the action of $\mathbb{C}^*=R_{2A_5}$, any relation decomposes under the action, given by~\eqref{E:R2A5}, into linear equations among monomials of the same weight. By inspection the weights in the above matrix under the above action range from $+4$ in the bottom left to corner to $-4$ in the top right corner, with entries along each diagonal going down-and-right having the same weight. Then within each diagonal to determine possible linear relations among the entries one first checks if any monomial is repeated. One easily sees then that for $4A/B^2\ne 1$ and $(A,B)\ne (0,0)$ the only possible relation could be in weight $0$, i.e.~among the five entries on the main diagonal. By looking at which monomials repeat in which entries, one finally sees that for $4A/B^2\ne 1$ and $(A,B)\ne (0,0)$ the only linear relation satisfied by the entries of $DF_{A,B}$ is
\begin{equation}\label{eq:rel*}
2(DF_{A,B})_{00}+(DF_{A,B})_{11}-(DF_{A,B})_{33}-2(DF_{A,B})_{44}=0\,.
\end{equation}
\item For $F_{1,-2}$, i.e., $A=1$ and $B=-2$, in addition to the linear relation~\eqref{eq:rel*}, one sees that the entries of $DF_{1,-2}$ satisfies two additional linear relations in weights $1$ and $-1$:
\begin{equation}\label{eq:relSL}\begin{aligned}&(DF_{1,-2})_{10}+2(DF_{1,-2})_{21}+3(DF_{1,-2})_{32}+4(DF_{1,-2})_{43}=0; \\ &(DF_{1,-2})_{34}+2(DF_{1,-2})_{23}+3(DF_{1,-2})_{12}+4(DF_{1,-2})_{01}=0\,,
\end{aligned}
\end{equation}
and no further relations.
\item For $F=F_{3D_4}$, the matrix $DF_{3D_4}$ is given by
$$
DF_{3D_4}=\begin{pmatrix}
x_0x_1x_2&x_1^2x_2&x_1x_2^2&x_1x_2x_3&x_1x_2x_4\\
x_0^2x_2&x_0x_1x_2&x_0x_2^2&x_0x_2x_3&x_0x_2x_4\\
x_0^2x_1&x_0x_1^2&x_0x_1x_2&x_0x_1x_3&x_0x_1x_4\\
3x_0x_3^2&3x_1x_3^2&3x_2x_3^2&3x_3^3&3x_3^2x_4\\
3x_0x_4^2&3x_1x_4^2&3x_2x_4^2&3x_3x_4^2&3x_4^3
\end{pmatrix}\,.
$$
Since all entries of this matrix are monomial, the only possible linear relations are pairwise equalities, up to a constant factor. One sees that the only monomial that repeats more than once is $x_0x_1x_2$, and thus the set of linear relations satisfied by the entries of $DF_{3D_4}$ is
$$
(DF_{3D_4})_{00}=(DF_{3D_4})_{11}=(DF_{3D_4})_{22}.
$$
\end{enumerate}
\end{exa}
\subsection{The extra correction term for $R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})$, the chordal cubic case}
\begin{pro}[Extra term for the chordal cubic] \label{P:ET-ChC}
For the group $R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})$, the extra term~\eqref{E:finExtraT} is given by
\begin{align}\label{E:PET-ChC}
\sum_{0\ne \beta'\in \mathcal{B}(\rho)}\frac{1}{w(\beta',R_c,G)}t^{2d(\mathbb{P}\mathcal{N}_x,\beta')}P^{N(R_c)\cap \operatornameeratorname{Stab} \beta' }_t(Z_{\beta ',R_c}^{ss})&=0 \mod t^{11}.
\end{align}
\end{pro}
Recall that in the formula above $x$ is a general point of $Z^{ss}_{R_c}$ (in this case, since $Z^{ss}_{R_c}$ consists of a single point, $x$ corresponds to the chordal cubic), $\mathcal{N}_x$ is the fiber of the normal bundle to the orbit $G\cdot Z^{ss}_{R_c}$ at $x$, and $\rho:R_c\to \operatornameeratorname{Aut}(\mathcal{N}_x)$ is the induced representation. The proof of the proposition will consist of showing that the codimension $d(\mathbb{P}\mathcal{N}_x,\beta')$ of any stratum $S_{\beta'}(\rho)$ for $0\ne \beta'\in \mathcal{B}(\rho)$ is at least $6$. This will follow from the following lemma, describing the representation $\rho$.
\begin{lem}\label{L:Rc-Nx-Rep}
For $R_c=\operatornameeratorname{PGL}(2,\mathbb{C})$, $\dim\mathcal{N}_x=13$, and the representation $\rho$ of $R_c$ on $\mathcal{N}_x$ is the one induced by the $\operatornameeratorname{SL}(2,\mathbb{C})$-representation $\operatornameeratorname{Sym}^{12}\mathbb{C}^2$, where $\mathbb{C}^2$ is the standard two-dimensional representation. Consequently the weights of the action of the maximal torus $T\cong \mathbb{C}^*$ in $R_c$ are
$$
-12,-10,-8,-6,-4,-2,0,2,4,6,8,10,12.
$$
\end{lem}
\begin{proof}
It suffices to determine the restriction of $\rho$ to the maximal torus $\mathbb{T}$ in $\operatornameeratorname{SL}(2,\mathbb{C})$ (induced by the homomorphism $\operatornameeratorname{SL}(2,\mathbb{C})\to \operatornameeratorname{PGL}(2,\mathbb{C})$).
Recall that $Z^{ss}_{R_c}=\{V(F_{1,-2})\}$, and so to describe $\mathcal{N}_x$, we must simply describe the normal space to the orbit $G\cdot V(F_{1,-2})$ at $V(F_{1,-2})$.
The maximal torus $\mathbb{T}=\operatornameeratorname{diag}(t,t^{-1})$ in $\operatornameeratorname{SL}(2,\mathbb{C})$ acts on coordinates $(x_0:\dots:x_4)$ diagonally by $(t^4:t^2:1:t^{-2}:t^{-4})$. Thus it multiplies each cubic monomial by some power of $t$, so that each monomial is thus an eigenspace for the action of $\mathbb{T}$.
Thus $T_x\mathbb{C}^{35}=\mathbb{C}^{35}$
decomposes as a sum of one-dimensional representations of $\mathbb{T}$ with the following multiplicities of weights
$$
(\pm 12) \times 1, (\pm 10) \times 1, (\pm 8) \times 2, (\pm 6) \times 3, (\pm4) \times 4, (\pm 2) \times 4, (0) \times 5.
$$
The tangent space to the orbit $G\cdot V(F_{1,-2}) $ is generated by the entries of the matrix in Example~\ref{Exa:D}(1).
Each binomial spans an eigenspace for the action of $\mathbb{T}$, and weights of the action of $\mathbb{T}$ on these generators can be computed directly to be equal to
\begin{equation*}
(\pm 8) \times 1, (\pm 6) \times 2, (\pm4) \times 3, (\pm 2) \times 4, ( 0) \times 5.
\end{equation*}
Now the relation $(\ref{eq:rel*})$ is among the weight $0$ generators, and thus we may drop one of them in forming a basis of the tangent space. The two relations~\eqref{eq:relSL} are among generators of weights $2$ and $-2$, respectively, so we can also drop one generator of weight $2$ and $-2$.
In summary, the weights for $\mathbb{T}$ on the tangent space to the orbit are given by
\begin{equation}\label{equ:w2}
(\pm 8) \times 1, (\pm 6) \times 2, (\pm4) \times 3, (\pm 2) \times 3, ( 0) \times 4.
\end{equation}
Taking the complement of the set of weights of the representation on the tangent space to the orbit in the set of weights of the representation on $\mathbb{C}^{35}$ gives the weights of the action on the normal space, proving the lemma.
\end{proof}
\begin{rem}
We note that in fact this result already follows from the geometry as described in~\cite{act}, where it was shown that the exceptional divisor in $\calM^{\operatorname{K}}$ corresponding to the chordal cubic is in fact the locus of Jacobians of hyperelliptic curves of genus five, which is thus the moduli space of twelve points on $\mathbb{P}^1$, which is exactly the GIT quotient for the $12$-th symmetric power of the standard representation of $\operatornameeratorname{SL}(2,\mathbb{C})$ on $\mathbb{C}^2$.
\end{rem}
\begin{proof}[Proof of Proposition~\ref{P:ET-ChC}]
From the description of the weights of $\rho$ in Lemma~\ref{L:Rc-Nx-Rep}, we see that we can take $\mathcal{B}(\rho)=\{0,2,4,6,8,10,12\}$.
We can estimate the codimension $d(\beta')$ for $\beta'\in \mathcal{B}(\rho)$ using~\eqref{E:Sbcodim}; i.e., $d(\beta')=n(\beta')-\dim ( R_{c}/P_{\beta'})$, where $n(\beta')$ is the number of weights less than $\beta'$, namely $6+\beta'/2$, and $P_{\beta'}$ is the associated parabolic subgroup. One can check that $P_{\beta'}$ is equal to the $2$-dimensional Borel subgroup consisting of upper triangular matrices; however, it suffices for our purposes to observe that $P_{\beta'}$ contains the Borel. Thus $d(\beta')\ge (6+\beta'/2)-3+2\ge 6$.
Thus the terms in~\eqref{E:PET-ChC} begin in degree $\ge 2d(\beta')=12$, and are zero modulo $t^{11}$.
\end{proof}
\subsection{The extra correction term for $R_{3D_4}\cong (\mathbb{C}^*)^2$, the $3D_4$ case}
\begin{pro}[Extra term for the $3D_4$ cubic] \label{P:ET-3D4}
For the group $R_{3D_4}\cong (\mathbb{C}^*)^2$, the extra term~\eqref{E:finExtraT} is given by
\begin{align*}
\scriptstyle
-\sum_{0\ne \beta'\in \mathcal{B}(\rho)}\frac{1}{w(\beta',R_{3D_4},G)}t^{2d(\mathbb{P}\mathcal{N}_x,\beta')}P^{N(R_{3D_4})\cap \operatornameeratorname{Stab} \beta' }_t(Z_{\beta', R_{3D_4}}^{ss})&=-t^8-2t^{10} \mod t^{11}.
\end{align*}
\end{pro}
Recall that in the formula above, $x$ is a general point of $Z^{ss}_{R_{3D_4}}$ (in this case, since $Z^{ss}_{R_{3D_4}}$ consists of a single $N(R_{3D_4})$ orbit, so we may take $x=V(F_{3D_4})$), $\mathcal{N}_x$ is the fiber of the normal bundle to the orbit $G\cdot Z^{ss}_{R_c}$ at $x$, and $\rho:R_c\to \operatornameeratorname{Aut}(\mathcal{N}_x)$ is the induced representation. We start with the following lemma, describing the representation $\rho$.
\begin{lem}\label{L:R3D4-Nx-Rep}
For $R_{3D_4}=\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_2,1,1)\cap \operatornameeratorname{SL}(5,\mathbb{C})\cong (\mathbb{C}^*)^2$, and $x=V(F_{3D_4})$, we have $\dim\mathcal{N}_x=12$, with an explicit basis given by
\begin{equation}\label{E:3D4-Nx-Bas}
x_0^3,\ x_1^3,\ x_2^3,\
x_0^2x_3,\ x_1^2x_3, \ x_2^2x_3,\
x_0^2x_4, \ x_1^2x_4, \ x_2^2x_4,\
x_0x_3x_4, \ x_1x_3x_4, \ x_2x_3x_4.
\end{equation}
Each element of this basis is an eigenvector for the action of $R$, so that under the inclusion $\mathfrak t_R=\{(\alpha_0,\alpha_1,\alpha_2,0,0): \sum \alpha_i=0\}\subseteq \mathfrak t=\{(\alpha_0,\dots ,\alpha_4): \sum \alpha_i=0\}\subseteq \mathbb{R}^5$, and the identification $\mathfrak t_R=\mathfrak t_R^\vee$ induced by the metric from $\mathfrak t$, the weights of the representation $$\rho:R_{3D_4}\to \operatornameeratorname{Aut}(\mathcal{N}_x)$$ in the order of the basis elements above, are equal to (see also Figure~\ref{F:curve-weights}):
\begin{equation}
\begin{array}{ccc}
(2,-1,-1,0,0),& (-1,2,-1,0,0), & (-1,-1,2,0,0),\\
(4/3,-2/3,-2/3,0,0),& (-2/3,4/3,-2/3,0,0),& (-2/3,-2/3,4/3,0,0) \\
(4/3,-2/3,-2/3,0,0),& (-2/3,4/3,-2/3,0,0),&
(-2/3,-2/3,4/3,0,0), \\
(2/3,-1/3,-1/3,0,0), & (-1/3,2/3,-1/3,0,0),& (-1/3,-1/3,2/3,0,0).
\end{array}
\end{equation}
\end{lem}
\begin{proof}
The basis for $\mathcal{N}_x$ comes directly from Example~\ref{Exa:D}(3). The identification of $\mathfrak t_R=\mathfrak t_R^\vee$ is given by the composition $\mathfrak t_R^\vee \hookrightarrow \mathfrak t^\vee \stackrel{\sim}{\to}\mathfrak t \twoheadrightarrow \mathfrak t_R$, where the last map is the orthogonal projection, and the rest is immediate. This gives the weights above. Indeed, for a basis monomial $x^I$, writing the group as $\operatornameeratorname{diag}(e^{i\alpha_0},e^{i\alpha_1},e^{i\alpha_2},1,1)$, the associated weight as a linear map (viewed as either a linear map in $\mathfrak t_R^\vee$ or $\mathfrak t^\vee$) is given by $I.\alpha$. The orthogonal projection is given by $(\alpha_0,\dots,\alpha_4)\mapsto (\alpha_0-\frac{1}{3}\sum_{i=0}^2\alpha_i,\alpha_1-\frac{1}{3}\sum_{i=0}^2\alpha_i,\alpha_2-\frac{1}{3}\sum_{i=0}^2\alpha_i,0,0)$. For instance, the monomial $x_0^2x_3$ has index $I=(2,0,0,1,0)$, and the orthogonal projection is then $(\frac{4}{3},-\frac{2}{3},-\frac{2}{3},0,0)$.
\end{proof}
\begin{figure}
\caption{
A sample codimension~$4$ element is given above in blue as $\textcolor{blue}
\label{F:curve-weights}
\end{figure}
We now move to describe the indexing set $\mathcal{B}(\rho)$ associated to the representation $\rho$, as well as various groups and loci associated to the elements $\beta'\in \mathcal{B}(\rho)$. First we note that since $R$ is the two-dimensional torus, the Weyl chamber is all of $\mathbb{R}^2$. By construction, the indexing set $\mathcal{B}(\rho)$ associated to the representation $\rho:R_{3D_4}\to \operatornameeratorname{Aut}(\mathcal{N}_x)$ is then the set of points $\beta'\in\mathfrak t_R=\{(\alpha_0,\alpha_1,\alpha_2,0,0): \sum \alpha_i=0\}$
that can be described as the closest point to the origin (with respect to the standard metric in $\mathbb{R}^5$) in a convex hull of the weights. The codimension of the associated stratum $S_{\beta'}$ is then equal to the number of weights lying on the same side as the origin from the orthogonal complement to $\beta'$. The situation is described by the following lemma, and the corresponding sets and weights are depicted in Figure~\ref{F:curve-weights}.
\begin{lem}\label{L:R(rho)3D4p1}
For the group $R=R_{3D_4}\cong (\mathbb{C}^*)^2$, all codimension $4$ and $5$ strata are as follows.
\begin{enumerate}
\item[(a)] There are $3$ codimension~$4$ elements $\beta'\in \mathcal{B}$: $\frac{1}{2}(-\frac{2}{3},\frac{1}{3},\frac{1}{3})$ (in \color{blue}{blue }\color{black} in Figure~\ref{F:curve-weights});
$\frac{1}{2}(\frac{1}{3},-\frac{2}{3},\frac{1}{3})$ (in \color{green}green \color{black} in the figure); and
$\frac{1}{2}(\frac{1}{3}, \frac{1}{3},-\frac{2}{3})$ (not shown in the figure). For each of these, $w(\beta ',R,G)=3$;
\item[(b)] There are $6$ codimension~$5$ elements $\beta'\in \mathcal{B}'$: $\frac{1}{7}(1,2,-3)$; $\frac{1}{7}(2,-3,1)$; $ \frac{1}{7}(-3,1,2)$; $\frac{1}{7}(-3,2,1)$; $ \frac{1}{7}(1,-3,2)$;
$\frac{1}{7}(2,1,-3)$ (the last of them shown in \color{red}{red }\color{black} in Figure~\ref{F:curve-weights}). For each of these, $w(\beta',R,G)=6$.
\end{enumerate}
Moreover, in each of these two cases, all the elements $\beta'$ are in the same orbit of the Weyl group of $G=\operatornameeratorname{SL}(5,\mathbb{C})$.
\end{lem}
\begin{proof}
To find $\mathcal{B}(\rho)$ one observes that since $R$ is a torus, the Weyl chamber is all of $\mathbb{R}^2$. It is easy to check from~\eqref{E:HesseZss} that since $R$ is a torus, the strata $S_{\beta'}$ are non-empty for the weights as given and shown in the picture. The fact that all 3 elements in each case lie in the same orbit of the Weyl group is also immediate, since $W(G)=S_5$ acts by permuting the entries, which preserves $\mathfrak t_R$ only for the subgroup $S_3$ permuting the first three entries. The weights also then easily follow.
\end{proof}
We now further describe the relevant fixed point sets and the action of the stabilizers. Since all $\beta'$ in the case (a) or in case (b) lie in the same orbit of the Weyl group, it is enough to work with one representative for each case.
\begin{lem}[{Lemmas~\ref{L:App-R(rho)3D4a} and \ref{L:App-R(rho)3D4b}}]\label{L:R(rho)3D4p2} In the notation above:
\begin{enumerate}
\item For $\beta'=\frac{1}{2}(-\frac{2}{3},\frac{1}{3},\frac{1}{3})$ (case (a)), we have
$$
Z^{ss}_{\beta'}=\{[a:b]\in \mathbb{P}\mathbb{C}\langle x_1x_3x_4,x_2x_3x_4\rangle: a\ne 0, b\ne 0 \}\cong \mathbb{C}^*.
$$
\item For $\beta'= \frac{1}{7}(2,1,-3)$ (case (b)), we have
$$
Z^{ss}_{\beta'}=\{[a:b:c]\in \mathbb{P}\mathbb{C}\langle x_0x_3x_4,x_1^2x_3, x_1^2x_4\rangle: a\ne 0,\ \text{and}\ (b,c)\ne (0,0) \}\cong \mathbb{A}^2-\{0\}.
$$
\item For either $\beta'$, the group $N\cap \operatornameeratorname{Stab}_G\beta'$ acts transitively on $Z^{ss}_R$.
\item The action of $(N\cap \operatornameeratorname{Stab}_G\beta')_x$ on $Z^{ss}_{\beta'}$ is induced by change of coordinates, via the inclusion $(N\cap \operatornameeratorname{Stab}_G\beta')_x\subseteq \operatornameeratorname{SL}(5,\mathbb{C})$ and the description of the loci above in terms of cubic forms.
For $\beta '=\frac{1}{2}(-\frac{2}{3},\frac{1}{3},\frac{1}{3})$ (case (a)), the group $(N\cap \operatornameeratorname{Stab}_G\beta')_x$ acts transitively on $Z^{ss}_{\beta '}$, and the stabilizer of the point $(1:1)\in Z^{ss}_{\beta'}$ is given explicitly
as $(\mathbb{C}^*\times \mu_{15})\times(S_2\times S_2)$. \qed
\end{enumerate}
\end{lem}
We note that for case (b), more details on the action of $(N\cap \operatornameeratorname{Stab}_G\beta')_x$ on $Z^{ss}_{\beta'}$ will not be needed.
The proof of the lemma is elementary, with all necessary computations given in Lemmas~\ref{L:App-R(rho)3D4a} and \ref{L:App-R(rho)3D4b} in the Appendix, for the cases (a) and (b), respectively.
\begin{proof}[Proof of Proposition~\ref{P:ET-3D4}]
Recall that we are trying to compute the extra terms~\eqref{E:finExtraT} contributed by the $3D_4$ locus:
\begin{align*}
&- \sum_{0\ne \beta'\in \mathcal{B}(\rho)}\frac{1}{w(\beta',R,G)}t^{2d(\mathbb{P}\mathcal{N}_x,\beta')}P^{N\cap \operatornameeratorname{Stab}_G \beta' }_t(Z^{ss}_{\beta',R}).
\end{align*}
As we saw in Lemma~\ref{L:R(rho)3D4p1}, there are $3$ elements $\beta'\in \mathcal{B}(\rho)$ with $d(\mathbb{P}\mathcal{N}_x,\beta')=4$, and $6$ with $d(\mathbb{P}\mathcal{N}_x,\beta')=5$. The other $\beta'\ne 0$ have $d(\mathbb{P}\mathcal{N}_x,\beta')\ge 6$, and will not contribute modulo $t^{11}$, and so we ignore them. We also saw that the $3$ codimension~$4$ (resp.~$6$ codimension~$5$) elements $\beta'$ were all in the same Weyl group of $G=\operatornameeratorname{SL}(5,\mathbb{C})$ orbit, and so we are free to work with one representative from each orbit. Finally, we showed for the $\beta'$ of codimension~$4$ and $5$ that $N\cap \operatornameeratorname{Stab}_G\beta'$ acts transitively on $Z^{ss}_R$, so that by Remark~\ref{R:PiFibration},
$$
P^{N\cap \operatornameeratorname{Stab}_B\beta'}_r(Z^{ss}_{\beta',R})=P_t^{(N\cap \operatornameeratorname{Stab}_B\beta')_x}(Z^{ss}_{\beta'}).
$$
Let us consider first $P_t^{(N\cap \operatornameeratorname{Stab}_B\beta')_x}(Z^{ss}_{\beta'})$ for the codimension~$5$ loci. Since we have from Lemma~\ref{L:R(rho)3D4p2}(2) that $Z^{ss}_{\beta'}$ is connected, it follows that $P_t^{(N\cap \operatornameeratorname{Stab}_B\beta')_x}(Z^{ss}_{\beta'})=1+O(t)$.
Thus the codimension $5$ extra term is $-\sum_{i=1}^6 \frac{1}{6}t^{10} (1+\dots)$.
Now let us consider $P_t^{(N\cap \operatornameeratorname{Stab}_B\beta')_x}(Z^{ss}_{\beta'})$ for the codimension~$4$ loci. Take the representative $\beta'=\frac{1}{2}(-\frac{2}{3},\frac{1}{3},\frac{1}{3})$. By Lemma~\ref{L:R(rho)3D4p2}(4), the action of $(N\cap \operatornameeratorname{Stab}_B\beta')_x$ on $Z^{ss}_{\beta'}$ is transitive, with the stabilizer equal to $(\mathbb{C}^*\times \mu_{15})\times(S_2\times S_2)$. Thus $P_t^{(N\cap \operatornameeratorname{Stab}_B\beta')_x}(Z^{ss}_{\beta'})=P_t(B\mathbb{C}^*)=(1-t^2)^{-1}$.
In summary, we have
\begin{align*}
(\text{``extra term'' codim 4})\ \ \ \ \ &-\sum_{i=1}^3 \frac{1}{3}t^{8} (1+t^2+\dots) \\
(\text{``extra term'' codim 5})\ \ \ \ \ &-\sum_{i=1}^6 \frac{1}{6}t^{10} (1+\dots)
\end{align*}
This completes the proof of the proposition.
\end{proof}
\subsection{The extra correction term for $R_{2A_5}\cong \mathbb{C}^*$, the $2A_5$ case}
\begin{pro}[Extra term for the $2A_5$ cubics] \label{P:ET-2A5}
For the group $R_{2A_5}\cong \mathbb{C}^*$, the extra term~\eqref{E:finExtraT} is given by
\begin{align*}
-\sum_{0\ne \beta'\in \mathcal{B}(\rho)}\frac{1}{w(\beta',R_{2A_5},G)}t^{2d(\mathbb{P}\mathcal{N}_x,\beta')}P^{N(R_{2A_5})\cap \operatornameeratorname{Stab} \beta' }_t(Z_{\beta',R_{2A_5}}^{ss})&\equiv -t^{10} \mod t^{11}.
\end{align*}
\end{pro}
This will follow from the next lemma.
\begin{lem}\label{L:R2A5-Nx-Rep}
For $R_{2A_5}=\operatornameeratorname{diag}(\lambda^2,\lambda,1,\lambda^{-1},\lambda^{-2})\cong \mathbb{C}^*$, $\dim\mathcal{N}_x=10$, and the weights of the representation $\rho$ of $R_{2A_5}$ on $\mathcal{N}_x$ are
$$
-6,-5,-4,-3,-2,2,3,4,5,6.
$$
\end{lem}
\begin{proof}
The proof is essentially the same as that of Lemma~\ref{L:Rc-Nx-Rep}.
The vector space $T_x\mathbb{C}^{35}=\mathbb{C}^{35}$
decomposes as a sum of one-dimensional representations of $R=R_{2A_5}$ with the following multiplicities of weights
$$
(\pm 6) \times 1, (\pm 5) \times 1, (\pm 4) \times 2, (\pm 3) \times 3, (\pm2) \times 4, (\pm 1) \times 4, (0) \times 5.
$$
The tangent space to the orbit $\operatornameeratorname{GL}(5,\mathbb{C})\cdot F_{A,B} $ of a general $F_{A,B}$ is generated by the entries of the matrix in Example~\ref{Exa:D}(1).
Each binomial spans an eigenspace for the action of $R$, and weights of the action of $R$ on these generators can be computed directly to be equal to
\begin{equation*}
(\pm 4) \times 1, (\pm 3) \times 2, (\pm 2) \times 3, (\pm 1) \times 4, ( 0) \times 5.
\end{equation*}
Now the relation $(\ref{eq:rel*})$ is among the weight $0$ generators, and thus we may drop one of them in forming a basis of the tangent space.
In summary, the weights for $R$ on the tangent space to the orbit $\operatornameeratorname{GL}(5,\mathbb{C}) \cdot F_{A,B}$ are given by
$$
(\pm 4) \times 1, (\pm 3) \times 2, (\pm 2) \times 3, (\pm 1) \times 4, ( 0) \times 4.
$$
Recall, however, that the relevant normal space $\mathcal{N}_x$ is the normal space in $X$ to the orbit $G\cdot Z_R^{ss}$ of the
fixed set $Z_R^{ss}$. We know
that $Z_R^{ss}/G$ is one-dimensional (corresponding to the curve $\mathcal{T}$ of cubics). Thus the tangent space $T_x(G\cdot Z_R^{ss})$, when lifted to $\mathbb{C}^{35}$, is the sum of $T_{F_{A,B}}(\operatornameeratorname{GL}(5,\mathbb{C})\cdot F_{A,B})$ together with a tangent vector representing the direction along $Z_R^{ss}/G$; in other words a tangent vector which comes from varying $4A/B^2$. As such a tangent vector we can take the deformation which simply deforms the cubic $F_{A,B}$ by changing the coefficient $A$. Clearly the derivative in this direction is equal to $\frac{d}{dA}F_{A,B}=x_2^3$.
This is weight $0$ (and, as expected, does not lie in the span of the weight~$0$ space of the orbit). Thus the lift to $\mathbb{C}^{35}$ of the tangent space to the orbit $G\cdot Z^{ss}_R$ is given by a space with weights
$$
(\pm 4) \times 1, (\pm 3) \times 2, (\pm 2) \times 3, (\pm 1) \times 4, ( 0) \times 5.
$$
Taking the complement of the set of weights of the representation on the tangent space in the set of weights of the representation on $\mathbb{C}^{35}$ gives the weights of the representation on the normal space.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{P:ET-2A5}]
From the description of the weights of $\rho$ in Lemma~\ref{L:R2A5-Nx-Rep}, we see that we can take $\mathcal{B}(\rho)= \{\pm 2,\pm 3,\pm 4,\pm 5,\pm 6\}$.
We can compute $d(\beta'):=\operatornameeratorname{codim}_{\mathbb C}\dim S_{\beta'}$ for $\beta'\in \mathcal{B}(\rho)$ using the definition~\eqref{E:SbetaDef};
i.e., $d(\beta')=n(|\beta'|)$, where $n(|\beta'|)$ is the number of weights less than $|\beta'|$, namely $5+|\beta'|-2\ge 5$.
Thus the terms in the formula in Proposition~\eqref{P:ET-2A5} begin in degree $\ge 2d(\beta')=10$, and we must only compute for $\beta'=\pm 2$.
One immediately obtains $w(\beta',R_{2A_5},G)=2$.
Finally, it is easy to see that $Z^{ss}_{\beta',R}$ is connected, since $Z^{ss}_R$ is connected, and one can check that $Z^{ss}_{\beta'}\cong \mathbb{C}$ is connected. Thus we have $P^{N(R_{2A_5})\cap \operatornameeratorname{Stab}_G\beta'}_t(Z^{ss}_{\beta',R})=1+\dots$, completing the proof.
\end{proof}
\section{Putting the terms together to compute the cohomology of $\mathcal{M}^K$}\label{S:MK-Coh-sum}
We now put together the results in the previous sections to complete the proof of Theorem~\ref{teo:betti} for $\calM^{\operatorname{K}}$. Recall that $\calM^{\operatorname{K}}$ has only finite quotient singularities, so the cohomology satisfies Poincar\'e duality. Consequently, as $\dim \calM^{\operatorname{K}}=10$, it suffices to compute $P_t(\calM^{\operatorname{K}})$ modulo $t^{11}$. We have:
\begin{align*}
&P_t(\mathcal{M}^K)=P_t^G(\widetilde X^{ss}) \equiv\\
& 1+t^2+2t^4+3t^6+5t^8+6t^{10}&\text{(Semi-stable locus, Prop.~\ref{P:PtGXss})}\\
&\ +t^2+t^4+2t^6+2t^8+3t^{10}&\text{(Main term, chordal
cubic, Prop.~\ref{P:MT-ChC})}\\
&\ +t^2+t^4+2t^6+3t^8+4t^{10} &\text{(Main term, $3D_4$ cubic, Prop.~\ref{P:MT-3D4})}\\
&\ +t^2+2t^4+3t^6+4t^8+5t^{10}&\text{(Main term, $2A_5$ cubics, Prop.~\ref{P:MT-2A5})}\\
&\ -0&\text{(Extra term, chordal cubic, Prop.~\ref{P:ET-ChC})}\\
&\ -t^8-2t^{10}&\text{(Extra term, $3D_4$ cubic, Prop.~\ref{P:ET-3D4})}\\
&\ -t^{10}&\text{(Extra term, $2A_5$ cubics, Prop.~\ref{P:ET-2A5})}\\
\equiv& 1+4t^2+6t^4+10t^6+13t^8+15t^{10} \mod t^{11}.\hskip-1cm
\end{align*}
This completes the proof of Theorem~\ref{teo:betti} for $\calM^{\operatorname{K}}$.
\chapter{The intersection cohomology of the GIT moduli space $\calM^{\operatorname{GIT}}$}\label{sec:IHGIT}
Our next goal is to compute the intersection cohomology of the GIT moduli space $\calM^{\operatorname{GIT}}$ by comparing this with the (intersection) cohomology of the Kirwan blowup $\calM^{\operatorname{K}}$. We recall that $\calM^{\operatorname{K}}$ is smooth up to finite quotient singularities, which implies that
cohomology and intersection cohomology coincide. The starting point lies in Kirwan's techniques~\cite{kirwanrational1,kirwanhyp}, which in turn use the decomposition theorem in a subtle way. To carry this out
requires a thorough understanding of the geometric situation, and it turns out that this analysis is rather involved.
However, these geometric details will come in handy also in Chapter~\ref{sec:IHball} where we compute the
intersection cohomology of the Baily--Borel compactification $(\calB/\Gamma)^*$.
\section[The Kirwan blowup to the GIT quotient, in general]{Obtaining the intersection cohomology of the GIT quotient from the cohomology of the Kirwan blowup, in general}
\subsection{Intersection cohomology for a single blowup}\label{SSS:IC1}
As before, it is notationally easier to explain the formulas after a single blowup. We start with this case, and then in the next subsection explain what the formulas are for a sequence of blowups.
We start again in the situation of \S~\ref{SSS:kirBlUp}, where we have fixed a maximal dimensional connected component $R\in \mathcal{R}$ of the stabilizer of a strictly polystable point, taken the blowup
\begin{equation}\label{E:piHatDef}
\hat \pi:\hat X\to X^{ss}
\end{equation}
along the locus $G\cdot Z^{ss}_R$~\eqref{E:ZRss}, and chosen a linearization of the action on an ample line bundle $\hat L$ on $\hat X$, as described in \S~\ref{SSS:kirBlUp}.
For simplicity, we further assume that $Z_R^{ss}$ is connected (which is the case for all $R$ for the moduli of cubic threefolds, as computed in the previous section); see~\cite[Rem.~1.19]{kirwanrational1} for the necessary modifications if $Z_R^{ss}$ is disconnected.
Considering GIT quotients, we have the following diagram~\cite[Diag.~1]{kirwanrational1} summarizing the situation:
$$
\xymatrix@C=1em@R=1em{
\hat X^{ss}\ar@{->>}[rrrrr] \ar@{->>}[ddddd]_{\hat \pi}&&&&&\hat X/\!\!/_{\hat L}G \ar@{->>}[ddddd]^{\hat \pi_G}\\
&E^{ss}\ar@{^(->}[lu] \ar@{->>}[rrr] \ar@{->>}[ddd]_{\hat \pi}&&&E/\!\!/_{\hat L}G \ar@{^(->}[ru] \ar@{->>}[ddd]^{\hat \pi_G}&\\
&&\hat {\mathcal{N}}\ar@{->>}[lu] \ar@{->>}[r] \ar@{->>}[d]_{T(\hat \pi)}&\hat{\mathcal{N}}/\!\!/_{\hat L}G \ar@{->>}[ru] \ar@{->>}[d]^{T(\hat \pi)_G} &&\\
&&\mathcal{N}\ar@{->>}[ld] \ar@{->>}[r]&\mathcal{N}/\!\!/_LG\ar@{->>}[rd]&&\\
&G\cdot Z^{ss}_R \ar@{^(->}[ld] \ar@{->>}[rrr]&&&Z_R/\!\!/_{L}N \ar@{^(->}[rd]&\\
X^{ss}\ar@{->>}[rrrrr]&&&&&X/\!\!/_LG\\
}
$$
Most of the notation in the diagram has been introduced before, but here we recall some of the definitions, and explain the remaining notation. We have set $\mathcal{N}$ to be the normal bundle to $G\cdot Z^{ss}_R$ in $X^{ss}$, we defined $E$ to be the exceptional divisor of the blowup $\hat \pi:\hat X\to X^{ss}$, $E^{ss}$ to be the intersection of $E$ with $\hat X^{ss}$, and set $\hat {\mathcal{N}}$ to be the normal bundle to $E^{ss}$ in $\hat X^{ss}$. The morphism $T(\hat \pi):\hat{\mathcal{N}}\to \mathcal{N}$ is induced by the differential of $\hat \pi$. The $G$-actions extend naturally to all of the spaces in the diagram, and the linearizations are induced via pull-back along the respective morphisms.
The group $N$ is defined to be the normalizer of $R$, and we have identified $G\cdot Z^{ss}_R/\!\!/_LG=Z^{ss}_R/\!\!/_LN$ via the identification $G\cdot Z^{ss}_R=G\times_ N Z^{ss}_R$~\cite[p.~72]{kirwanblowup}.
The goal is to use the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber~\cite{bbdg} to compare the intersection cohomology groups $IH^\bullet(\hat X/\!\!/_{\hat L}G)$ and $IH^\bullet(X/\!\!/_LG)$. The interaction between the general theory and Kirwan's results is outlined in~\cite[Rem.~2.2, p.~484]{kirwanrational1}. Here we will review the outline of Kirwan's argument, as this helps clarify the meaning of terms appearing in the formulas, particularly when we move to the specific case of cubic threefolds.
Immediately from the decomposition theorem, since $\hat \pi_G$ is a birational morphism, one has that $IH^\bullet(X/\!\!/_LG)$ is a direct summand of $IH^\bullet(\hat X/\!\!/_{\hat L} G)$, and thus the goal is to determine the extra summands precisely. In other words, denoting $IP_t$ the intersection Poincar\'e polynomial, we can write
\begin{equation}\label{E:BR(t)-intro}
IP_t(X/\!\!/_LG)=IP_t(\hat X/\!\!/_{\hat L}G)-B_R(t),
\end{equation}
for some polynomial $B_R(t)$ with non-negative integral coefficients, and our aim is to compute this polynomial.
The first observation is the following. Given a fibred product diagram
\begin{equation}\label{E:MV-setup}
\xymatrix{
E\ar@{^(->}[r] \ar@{->}[d]&\hat U \ar@{^(->}[r] \ar[d]&\hat V \ar[d]^f\\
C\ar@{^(->}[r]&U\ar@{^(->}[r]&V\\
}
\end{equation}
where $f:\hat V\to V$ is a birational morphism of projective varieties that is an isomorphism on the complement of a closed subvariety $C\subseteq V$, and $U\subseteq V$ is an open neighborhood of $C$, then~\cite[Lem.~2.8]{kirwanrational1}:
\begin{equation}\label{E:IH-Lem1}
\dim IH^i(V)=\dim IH^i(\hat V)-\dim IH^i(\hat U)+\dim IH^i(U).
\end{equation}
We note that this is a slightly more general statement than \cite[Lem.~2.8]{kirwanrational1}, but that proof goes through unchanged to prove \eqref{E:IH-Lem1}.
In our situation, we apply this for $U$ being an open neighborhood of $Z_R/\!\!/_LN$ in $X/\!\!/_LG$, so that $\hat U:=\hat \pi_G^{-1}(U)$ is its inverse image in $\hat X/\!\!/_{\hat L}G$, which is an open neighborhood of $E/\!\!/_{\hat L}G$
in $\hat X/\!\!/_{\hat L}G$, so that we obtain
\begin{equation}\label{E:KR1-L2.8}
\dim IH^i(X/\!\!/_LG)=\dim IH^i(\hat X/\!\!/_{\hat L}G)-\dim IH^i(\hat U)+\dim IH^i(U).
\end{equation}
Kirwan then shows that there is an open neighborhood $U$ as above that is homeomorphic to $\mathcal{N}/\!\!/_LG$, and furthermore such that its preimage $\hat U$ is homeomorphic to $\hat{\mathcal{N}}/\!\!/_{\hat L}G$~\cite[Lem.~2.9, and p.~487]{kirwanrational1}. This establishes~\cite[Cor.~2.11]{kirwanrational1}:
\begin{equation}\label{E:KR1-C2.11}
\dim IH^i(X/\!\!/_LG)=\dim IH^i(\hat X/\!\!/_{\hat L}G)-\dim IH^i(\hat{\mathcal{N}}/\!\!/_{\hat L}G)+\dim IH^i(\mathcal{N}/\!\!/_LG).
\end{equation}
Now we use the fact that $\mathcal{N}/\!\!/_LG\cong (\mathcal{N}|_{Z^{ss}_R})/\!\!/_LN$ and $\hat {\mathcal{N}}/\!\!/_{\hat L}G\cong (\hat{\mathcal{N}}|_{\hat \pi^{-1}Z^{ss}_R})/\!\!/_{\hat L}N$
\cite[p.~493, and 1.7, p.~476]{kirwanrational1}, and the fact that the intersection cohomology of the quotient by a finite group is the subset of intersection cohomology that is invariant under the finite group~\cite[Lem.~2.12]{kirwanrational1}, to conclude that
\begin{align}
\nonumber IH^\bullet(\mathcal{N}/\!\!/_LG)&\cong [IH^\bullet((\mathcal{N}|_{Z^{ss}_R})/\!\!/_LN_0)]^{\pi_0N}\\
\label{E:KR1-Ep493(2)}
IH^\bullet(\hat{\mathcal{N}}/\!\!/_{\hat L}G)&\cong [IH^\bullet((\hat{\mathcal{N}}|_{\hat \pi^{-1}Z^{ss}_R})/\!\!/_{\hat L}N_0)]^{\pi_0N}
\end{align}
where $N_0\subset N$ is the connected component of the identity, and $\pi_0N=N/N_0$ is the group of connected components of~$N$.
A Leray spectral sequence argument for the morphisms $(\mathcal{N}|_{Z^{ss}_R})/\!\!/_LN_0\to Z_R/\!\!/_LN_0$ and $(\hat{\mathcal{N}}|_{\hat \pi^{-1}Z^{ss}_R})/\!\!/_{\hat L}N_0\to Z_R/\!\!/_LN_0$ yields
\cite[p.~493, Lem.~2.15, Prop.~2.13]{kirwanrational1}:
\begin{align}
IH^\bullet((\mathcal{N}|_{Z^{ss}_R})/\!\!/_LN_0)&\cong IH^\bullet(\mathcal{N}_x/\!\!/R)\otimes H^\bullet(Z_R/\!\!/_LN_0)\\
\label{E:KR1-Lem2.15}
IH^\bullet((\hat{\mathcal{N}}|_{\hat \pi^{-1}Z^{ss}_R})/\!\!/_{\hat L}N_0)&\cong IH^\bullet( \hat {\mathcal{N}_x} /\!\!/R)\otimes H^\bullet(Z_R/\!\!/_LN_0)
\end{align}
We emphasize here that we do not projectivize $\mathcal N_x$ or $\hat {\mathcal N}_x$.
Here, as in~\eqref{E:xinZssR}, $x$ is a general point of $Z^{ss}_R$, and the fiber $\mathcal{N}_x$ has an action $\rho:R\to \operatornameeratorname{GL}(\mathcal{N}_x)$ as described in~\eqref{E:rhoDef}, which is used as the linearization. The quotient $\mathcal{N}_x/\!\!/R$ is defined to be $\operatornameeratorname{Spec}(\operatornameeratorname{Sym}^\bullet \mathcal{N}_x^\vee)^R$, the spectrum of the invariant ring (which is why we are not indicating a linearization in the notation), and similarly for $\hat {\mathcal{N}_x}/\!\!/R$.
We are also using the fact established in the proof of~\cite[Prop.~2.13]{kirwanrational1} that $Z_R/\!\!/_LN_0$ has at worst finite quotient singularities, so that its cohomology and intersection cohomology are equal.
Combining this with~\eqref{E:KR1-Ep493(2)} yields~\cite[2.20, p.~494, Lem.~2.15]{kirwanrational1}:
\begin{align}
\label{E:KR1-E2.22(1)}
IH^\bullet(\mathcal{N}/\!\!/_LG)&\cong [IH^\bullet(\mathcal{N}_x/\!\!/R)\otimes H^\bullet(Z_R/\!\!/_LN_0)]^{\pi_0 N}\\
\label{E:KR1-E2.22(2)}
IH^\bullet(\hat {\mathcal{N}}/\!\!/_{\hat L}G)&\cong [IH^\bullet( \hat {\mathcal{N}_x} /\!\!/R)\otimes H^\bullet(Z_R/\!\!/_LN_0)]^{\pi_0 N}.
\end{align}
Finally one shows that $IH^\bullet(\hat{\mathcal{N}}_x/\!\!/R)\cong IH^\bullet(\mathbb{P}(\mathcal{N}_x)/\!\!/R)$~\cite[Lem.~2.15]{kirwanrational1}, and that there is a natural surjection
$$
IH^i(\mathbb{P}(\mathcal{N}_x)/\!\!/R)\to IH^i(\mathcal{N}_x/\!\!/R)
$$
whose kernel is isomorphic to $IH^{i-2}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)$ if $i\le \dim \mathbb{P}(\mathcal{N}_x)/\!\!/R$ and to $IH^i(\mathbb{P}(\mathcal{N}_x)/\!\!/R)$ otherwise
\cite[Cor.~2.17]{kirwanrational1}. Putting this all together we have~\cite[Prop.~2.1]{kirwanrational1}:
\begin{align}
\nonumber \dim IH^i(X/\!\!/_LG)&=\dim IH^i(\hat X/\!\!/_{\hat L}G)\\
\label{E:IH-CorTerm1}
&-\sum_{p+q=i}\dim\left[H^p(Z_R/\!\!/_LN_0)\otimes IH^{\hat q}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)\right]^{\pi_0N}
\end{align}
where $\hat q=q-2$ for $q\le \dim \mathbb{P}(\mathcal{N}_x)/\!\!/R$ and $\hat q=q$ otherwise.
Reiterating from above, $x$ is a general point of $Z^{ss}_R$, the fiber $\mathcal{N}_x$ has an action $\rho:R\to \operatornameeratorname{GL}(\mathcal{N}_x)$ as described in~\eqref{E:rhoDef}, which is used as the linearization, $N_0\subset N$ is the connected component of the identity, and $\pi_0N:=N/N_0$.
The action of $\pi_0N$ on $H^\bullet(Z_{R}/\!\!/_LN_0)$ is induced from the given action of $N$ on $Z^{ss}_R$, and the action on the tensor product is induced via a Leray spectral sequence (see~\eqref{E:KR1-Ep493(2)} and~\eqref{E:KR1-E2.22(2)}).
\begin{rem}\label{R:pi0TrivB}
If the action of $\pi_0N$ on the tensor product $H^p(Z_R/\!\!/_LN_0)\otimes IH^{\hat q}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)$ is trivial on the second factor, then we can conclude from~\eqref{E:IH-CorTerm1} that~\cite[Cor.~2.28]{kirwanrational1}:
\begin{align*}
\dim IH^i(X/\!\!/_LG)&=\dim IH^i(\hat X/\!\!/_{\hat L}G)\\
&-\sum_{p+q=i}\dim H^p(Z_R/\!\!/_LN)\cdot \dim IH^{\hat q}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)
\end{align*}
so that
\begin{align}\label{E:IP-BR(t)}
IP_t(X/\!\!/_LG)&=IP_t(\hat X/\!\!/_{\hat L}G)-\underbrace{P_t(Z_R/\!\!/_L N)\,b_R(t)}_{\text{``}B_R(t)\text{''}}
\end{align}
where $B_R(t)$ is the product indicated above, and the coefficients of the polynomial $b_R(t)$ are essentially the shifted intersection Betti numbers of the GIT quotient $\mathbb{P}(\mathcal{N}_x)/\!\!/R$ defined as follows. Denoting $c=\dim \mathbb{P}(\mathcal{N}_x)/\!\!/R$, and denoting these intersection Betti numbers as
\begin{equation}\label{E:IHPNR}
IP_t(\mathbb{P}(\mathcal{N}_x)/\!\!/R)=1+b_1t+b_2t^2+b_3t^3+\dots+b_{c-1}t^{c-1}+b_{c}t^{c}+b_{c+1}t^{c+1}+b_{c+2}t^{c+2}+\dots+t^{2c},
\end{equation}
the polynomial $b_R(t)$ is defined as
\begin{equation}\label{E:bR(t)-def}
\qquad \quad \qquad \qquad b_R(t):= t^2+b_1t^3+\dots+b_{c-3}t^{c-1}+b_{c-2}t^{c}+b_{c+1}t^{c+1}+b_{c+2}t^{c+2}+\dots+t^{2c}\\
\end{equation}
We recall here that the intersection cohomology always satisfies Poincar\'e duality, and thus we could have written $b_j=b_{2c-j}$ instead.
\end{rem}
\begin{rem}\label{R:pi0TrivExpl}
We remark that the spectral sequence argument for the morphism $(\hat{\mathcal{N}}|_{\pi^{-1}Z^{ss}_R})/\!\!/_{\hat L}N_0\to Z_R/\!\!/_LN_0$
that yielded~\eqref{E:KR1-Lem2.15} also shows that if $Z_R/\!\!/_LN_0$ is simply connected, then the action of $\pi_0N$ on the tensor product splits, and we can conclude from~\eqref{E:IH-CorTerm1} that
\begin{align*}
\dim IH^i(X/\!\!/_LG)&=\dim IH^i(\hat X/\!\!/_{\hat L}G)\\
&-\sum_{p+q=i}\dim H^p(Z_R/\!\!/_LN)\cdot \dim [IH^{\hat q}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)]^{\pi_0N}
\end{align*}
so that
\begin{align}\label{E:IP-BR(t)-1}
IP_t(X/\!\!/_LG)&=IP_t(\hat X/\!\!/_{\hat L}G)-\underbrace{P_t(Z_R/\!\!/_L N)\,b_R(t)}_{\text{``}B_R(t)\text{''}}
\end{align}
where $B_R(t)$ is the product indicated above, and the coefficients $b_i$ of the polynomial $b_R(t)$ are the dimensions of the $\pi_0N$-invariant subspace of the intersection cohomology: $b_i:=\dim [IH^{\hat i}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)]^{\pi_0N}$, shifted as before in~\eqref{E:bR(t)-def}. In other words, we replace~\eqref{E:IHPNR} with $IP_t^{\pi_0N}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)$, and then $b_R(t)$ is obtained from this as in~\eqref{E:bR(t)-def}.
\end{rem}
\subsection{The correction terms in general}
Having reviewed the case of a single blowup, we now give the formulas for the cohomology utilizing the full Kirwan blowup.
We use the notation from \S~\ref{SSS:kirBlUp} and especially Remark~\ref{R:pi-r-Def}; recall in particular that $\widetilde X/\!\!/_{\tilde L}G$ denotes the Kirwan blowup, and $\widetilde X$ and $\tilde L$ are the iterated blowup and linearization, respectively, playing the roles of $\hat X$ and $\hat L$ in the discussion of the single blowup above.
From~\eqref{E:BR(t)-intro}, we have
\begin{equation}\label{E:BR(t)-gen}
IP_t(X/\!\!/_LG)=P_t(\widetilde X/\!\!/_{\tilde L}G)-\sum_{R\in \mathcal{R}}B_R(t),
\end{equation}
and our goal is to describe the polynomials $B_R(t)$ more precisely.
The relevant formula for computing the intersection cohomology of $X/\!\!/_LG$ from that of the full Kirwan blowup $\widetilde X^{ss}/\!\!/_{\tilde L}G$,
generalizing~\eqref{E:IH-CorTerm1}, is~\cite[Thm.~3.1]{kirwanrational1}:
\begin{align}
\nonumber \dim IH^i(X/\!\!/_LG)&=\dim H^i(\widetilde X/\!\!/_{\tilde L}G)\\
\label{eq:IH-FullBU}
&-\sum_{R\in \mathcal{R}} \sum_{p+q=i}\dim[H^p(Z_{R,\dim R+1}/\!\!/_LN_0^R)\otimes IH^{\hat q^R}(\mathbb{P}(\mathcal{N}_x^R)/\!\!/R)]^{\pi_0N^R}.
\end{align}
The notation is explained after~\eqref{E:IH-CorTerm1}, where now the superscript $R$ indicates the corresponding object with respect to the given group $R$. For instance,
$\hat q^R=q-2$ for $q\le \dim \mathbb{P}(\mathcal{N}_x^R)/\!\!/R$ and $\hat q^R=q$ otherwise.
The notation $Z_{R,\dim R+1}$ indicates the strict transform of $Z_R$ in $X_{\dim R+1}$ under the appropriate sequence of blowups in the inductive process (see \S~\ref{SSS:kirBlUp} and especially Remark~\ref{R:pi-r-Def}).
\begin{rem}\label{R:pi0TrivBR}
If for some $R\in \mathcal{R}$ the action of $\pi_0N^R$ on the tensor product $H^p(Z_{R,\dim R+1}/\!\!/_LN_0)\otimes IH^{\hat q}(\mathbb{P}(\mathcal{N}_x^R)/\!\!/R)$ is trivial on the second factor, one can simplify the corresponding term in~\eqref{eq:IH-FullBU} using Remark~\ref{R:pi0TrivB}.
In particular, if for all $R\in \mathcal{R}$ the action of $\pi_0N^R$ on the tensor product $H^p(Z_{R,\dim R+1}/\!\!/_LN_0^R)\otimes IH^{\hat q}(\mathbb{P}(\mathcal{N}_x^R)/\!\!/R)$ is trivial on the second factor, then we have
\begin{align*}
\dim IH^i(X/\!\!/_LG)&=\dim H^i(\widetilde X/\!\!/_{\tilde L}G)\\
&-\sum_{R\in \mathcal{R}}\sum_{p+q=i}\dim H^p(Z_{R,\dim R+1}/\!\!/_LN^R) \dim IH^{\hat q^R}(\mathbb{P}(\mathcal{N}_x^R)/\!\!/R)
\end{align*}
so that
\begin{align}\label{E:IP-BR(t)gen}
IP_t(X/\!\!/_LG)&=P_t(\widetilde X/\!\!/_{\tilde L}G)-\sum_{R\in \mathcal{R}}\underbrace{P_t(Z_{R,\dim R+1}/\!\!/_L N^R)b_R(t)}_{\text{``}B_R(t)\text{''}}
\end{align}
where $B_R(t)$ is the product indicated above, and $b_R(t)$ is defined as in~\eqref{E:bR(t)-def}. If each of the $Z_{R,\dim R+1}/\!\!/_LN_0^R$ in
\eqref{eq:IH-FullBU} is simply connected, then the direct generalization of Remark~\ref{R:pi0TrivExpl} holds.
\end{rem}
\section[The GIT quotient for cubic threefolds]{The intersection cohomology of the GIT quotient for cubic threefolds}
We now apply this in the case of the GIT moduli space of cubic threefolds $\calM^{\operatorname{GIT}}$, for which~\eqref{E:BR(t)-gen} gives
\begin{equation}\label{E:GIT-MK-B}
IP_t(\calM^{\operatorname{GIT}})=P_t(\calM^{\operatorname{K}})-\sum_{R\in \mathcal{R}}B_R(t).
\end{equation}
In our case $\mathcal{R}=\{R_{2A_5}\cong \mathbb{C}^*, R_{3D_4}\cong (\mathbb{C}^*)^2, R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})\}$, and to compute the terms $B_R(t)$, we utilize~\eqref{eq:IH-FullBU}, for most of which the computations have in fact already been done. Indeed, as we will see, we have already checked in the previous section that the quotients $Z_{R,\dim R+1}/\!\!/_LN_0$ are simply connected, so that applying Remark~\ref{R:pi0TrivBR}, we can utilize~\eqref{E:IP-BR(t)gen}, and thus all that remains is to compute the intersection cohomology of the GIT quotients $\mathbb{P}(\mathcal{N}_x^R)/\!\!/R$. We will work out the terms $B_R(t)$ in the order of descending dimension of $R$, following the Kirwan blowup process.
\subsection{The correction term $B_R(t)$ for $R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})$, the chordal cubic case}
\begin{pro}[The $B_R(t)$ term for the chordal cubic]\label{P:BRt-ChC}
For the group $R_c\cong \operatornameeratorname{PGL}(2,\mathbb{C})$, we have
\begin{enumerate}
\item $Z_{R_c}/\!\!/_{\mathcal{O}(1)} N^{R_c}$ is a point.
\item $IP_t(\mathbb{P}(\mathcal{N}_x^{R_c})/\!\!/R_c)=1 +t^2 + 2t^4 +2t^6 + 3t^8 + 3t^{10} + 2t^{12}+ 2t^{14} + t^{16} + t^{18}$.
\item The action of ${\pi_0N^{R_c}}$ on $IH^\bullet(\mathbb{P}(\mathcal{N}_x^{R_c})/\!\!/R_c))$ is trivial.
\end{enumerate}
The term $B_{R_c}(t)$ is equal to
\begin{align*}
P_t(Z_{R_c}/\!\!/_{\mathcal{O}(1)} N^{R_c}) b_{R_c}(t)&\equiv t^2+t^4+2t^6+2t^8+3t^{10} \mod t^{11}.
\end{align*}
\end{pro}
\begin{proof}
For brevity, let us write $R=R_c$, $N=N^{R_c}$, and $\mathcal{N}_x^{R_c}=\mathcal{N}_x$.
By Proposition~\ref{P:ZRss}(2), $Z_R$ is a point, and thus~$Z_R/\!\!/_{\mathcal{O}(1)} N$ is a point, proving (1).
Now the representation $\rho:R\to \mathcal{N}_x $ was worked out in
Lemma~\ref{L:Rc-Nx-Rep} to be the representation of $\operatornameeratorname{PGL}(2,\mathbb{C})$ induced by the $\operatornameeratorname{SL}(2,\mathbb{C})$-representation $\operatornameeratorname{Sym}^{12}\mathbb{C}^2$, where $\mathbb{C}^2$ is the standard two-dimensional representation.
Thus $\mathbb{P}(\mathcal{N}_x )/\!\!/R$ is the GIT moduli space of $12$ \emph{unordered} points in $\mathbb{P}^1$, the intersection cohomology of which was worked out in~\cite[Table, p.~40]{kirwanhyp} (see also \cite{Bri} and \cite{LSa}), giving (2).
Now, since $Z_{R}/\!\!/_{\mathcal{O}(1)} N$ is a point, we obtain from~\eqref{eq:IH-FullBU} that the correction term is equal to $B_R(t)=\sum_{i}t^i[\dim IH^{\hat i}(\mathbb{P}(\mathcal{N}_x )/\!\!/R)]^{\pi_0N}$, where the action is induced by an action of $\pi_0N$ on $\mathbb{P}(\mathcal{N}_x )/\!\!/R$.
It follows from Lemma~\ref{L:RcNorm} that $\pi_0N$ consists of scalar matrices of the form $\zeta^i\cdot \operatornameeratorname{Id}$, where $\zeta$ is a primitive fifth root of unity. Their action is evidently trivial, proving (3).
Finally, by Remark~\ref{R:pi0TrivBR} we conclude that $B_{R}(t)=b_{R}(t)$, where $b_{R}(t)$ is worked out from (2) via~\eqref{E:bR(t)-def} to be
$$
b_{R}(t)=t^2+t^4+2t^6+2t^8+3t^{10} + 2t^{12}+ 2t^{14} + t^{16} + t^{18},
$$
completing the proof.
\end{proof}
\subsection{The correction term $B_R(t)$ for $R_{3D_4}\cong (\mathbb{C}^*)^2$, the $3D_4$ case}
\begin{pro}[The $B_R(t)$ term for the $3D_4$ cubic]\label{P:BRt-3D4} For the group $R_{3D_4}\cong (\mathbb{C}^*)^2$,
we have
\begin{enumerate}
\item $Z_{R_{3D_4}}/\!\!/_{\mathcal{O}(1)} N^{R_{3D_4}}$ is a point.
\item $IP_t^{\pi_0N^{R_{3D_4}}}(\mathbb{P}(\mathcal{N}_x^{R_{3D_4}})/\!\!/R_{3D_4})=1 +t^2 + 2t^4 +3t^6 + 3t^8 + 3t^{10} + 3t^{12}+ 2t^{14} + t^{16} + t^{18}$.
\end{enumerate}
The
term $B_{R_{3D_4}}(t)$ is given by
\begin{align*}
B_{R_{3D_4}}(t)&\equiv t^2+t^4+2t^6+3t^8+3t^{10} \mod t^{11}.
\end{align*}
\end{pro}
\begin{proof}
For brevity, let us write $R=R_{3D_4}$, $N=N^{R_{3D_4}}$, and $\mathcal{N}_x^{R_{3D_4}}=\mathcal{N}_x$.
We have seen in Lemma~\ref{L:R3D4Norm1}(3) that $N^{ }$ acts transitively, so that $Z_{R_{ }}/\!\!/_{\mathcal{O}(1)} N^{ }$ is a point,
while the representation $\rho:R_{ }\to \mathcal{N}_x^{ }$ was worked out in
Lemma~\ref{L:R3D4-Nx-Rep}. The quotient $\mathbb{P}(\mathcal{N}_x^{ })/\!\!/R_{ }$ is a projective toric variety; the intersection cohomology can be worked out either torically, or via the general Kirwan process described in \S~\ref{S:PXss}.
The latter approach is quite elementary in this case, and so we sketch that here. First, from the description of the weights of the action in Lemma~\ref{L:R3D4-Nx-Rep}, it is clear that there are no strictly semi-stable points in $\mathbb{P}(\mathcal{N}_x^{ })$ (this could also be deduced from the fact that we have locally already arrived at the full Kirwan blowup). Thus the GIT quotient has at worst finite quotient singularities. In any case, we have $IP_t(\mathbb{P}(\mathcal{N}_x^{ })/\!\!/R_{ })=P_t(\mathbb{P}(\mathcal{N}_x^{ })/\!\!/R_{ })=P_t^{R_{ }}(\mathbb{P}(\mathcal{N}_x^{ })^{ss})$.
But then using~\eqref{E:KD-(3.1)}, we have
\begin{align}
\nonumber IP_t(\mathbb{P}(\mathcal{N}_x^{ })/\!\!/R_{ })&=P_t^{R_{ }}(\mathbb{P}(\mathcal{N}_x^{ })^{ss})=P_t(\mathbb{P}(\mathcal{N}_x^{ }))P_t(BR_{ })-\sum_{0\neq \beta' \in \mathcal{B}(\rho)} t^{2d(\beta')}P_t^{R_{ }}(S_{\beta'})\\
\nonumber&=P_t(\mathbb{P}^{11})P_t(B(\mathbb{C}^*)^2)-\sum_{0\neq \beta' \in \mathcal{B}(\rho)} t^{2d(\beta')}P_t^{R_{ }}(S_{\beta'})\\
\label{E:IP3D4}&\equiv P_t(\mathbb{P}^{11})P_t(B(\mathbb{C}^*)^2) -3t^8 -3\dim H^1_{(\mathbb{C}^*)^2}(S_{\beta '})t^9\mod t^{10}
\end{align}
where in \eqref{E:IP3D4}, in the notation of Lemma~\ref{L:R(rho)3D4p1}, $\beta'$ is as in case (a), one of the exactly three $0\ne \beta'\in \mathcal{B}(\rho)$ with $d(\beta)\le 4$, with isomorphic corresponding strata; since $\mathbb{P}(\mathcal{N}_x^{ })/\!\!/R_{ }$ has dimension $9$, it suffices by Poincar\'e duality to compute up to $t^{9}$.
We can in fact conclude that the coefficient $-3\dim H^1_{(\mathbb{C}^*)^2}(S_{\beta '})$ of $t^9$ is zero, since it must be non-negative (but it turns out this coefficient does not contribute to the final answer anyway).
Now, since $Z_{R}/\!\!/_{\mathcal{O}(1)} N $ is a point, we obtain from~\eqref{eq:IH-FullBU} that the correction term is equal to $B_R(t)=\sum_{q}t^q[\dim IH^{\hat q}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)]^{\pi_0N}$, where the action is induced by an action of $\pi_0N$ on $\mathbb{P}(\mathcal{N}_x)/\!\!/R$. It follows from Lemma~\ref{L:R3D4Norm1}(1) (see also the computations in Proposition~\ref{P:App-R=C*2}(2)) that $\pi_0N\cong S_3$ acts on $\mathbb{P}(\mathcal{N}_x )/\!\!/R$ via permutation of the coordinates $x_0,x_1,x_2$. In the context of~\eqref{E:IP3D4}, the action on the cohomology of $\mathbb{P}^{11}$ is trivial, the action permutes the $S_{\beta '}$, and acts on the torus $(\mathbb{C}^*)^2$ in the following way. The involution $\delta$ given by $x_0\leftrightarrow x_1$ acts by $(\lambda_0,\lambda_1)\mapsto (\lambda_1,\lambda_0)$, and the cyclic permutation $\sigma$ given by $x_0\mapsto x_1\mapsto x_2\mapsto x_0$ acts by $(\lambda_0,\lambda_1)\mapsto (\lambda_0^{-1}\lambda_1^{-1},\lambda_0)$. Thus the action of $S_3$ on the cohomology of $B(\mathbb{C}^*)^2$ is induced by the action of $S_3$ on the vector space $\mathbb{Q}\langle c_1^{(1)},c_1^{(2)}\rangle$ by the matrices
$$
\delta = \left (
\begin{array}{cc}
0&1\\
1&0\\
\end{array}
\right), \ \ \
\sigma = \left (
\begin{array}{cc}
-1&1\\
-1&0\\
\end{array}
\right).
$$
Looking at the characters, we see that this is the dihedral representation of $S_3$,
which has generating function $(1-t^4)^{-1}(1-t^6)^{-1}$ (the standard representation of $S_3$ has generating function $(1-t^2)^{-1}(1-t^4)^{-1}(1-t^6)^{-1}$, and is the direct sum of the trivial representation and the dihedral representation).
Putting this back together with~\eqref{E:IP3D4}, we obtain
\begin{align*}
\scriptstyle
IP_t^{\pi_0N}(\mathbb{P}(\mathcal{N}_x )/\!\!/R_{})&\equiv (1-t^2)^{-1}\cdot (1-t^4)^{-1}(1-t^6)^{-1}-\frac{1}{3}\left(3t^8-3\dim H^1_{(\mathbb{C}^*)^2}(S_{\beta '}) t^9\right) \\
&\equiv1+t^2+2t^4+3t^6+3t^8-\dim H^1_{(\mathbb{C}^*)^2}(S_{\beta '}) t^9 \mod t^{10}.
\end{align*}
As noted above, $\dim H^1_{(\mathbb{C}^*)^2}(S_{\beta '})=0$, but we have included it here again for clarity with respect to deducing the invariants from~\eqref{E:IP3D4}. This concludes the proof of (2).
Now, computing the numerics as in~\eqref{E:bR(t)-def}, we find finally that
$$
B_R(t)\equiv t^2+t^4+2t^6+3t^8+3t^{10} \mod t^{11}$$
completing the proof (and explaining the claim above that the coefficient of $t^9$ in~\eqref{E:IP3D4} is irrelevant).
\end{proof}
\subsection{The correction term $B_R(t)$ for $R_{2A_5}\cong \mathbb{C}^*$, the $2A_5$ case} \label{S: BR-2A5}
In this section we denote by $\widehat Z_{R_{2A_5}}$ the strict transform of $Z_{R_{2A_5}}$ in the blowup along the chordal cubic locus.
\begin{pro}[The $B_R(t)$ term for $2A_5$ cubics]\label{P:BRt-2A5} For the group $R_{2A_5}\cong \mathbb{C}^*$,
we have
\begin{enumerate}
\item $\widehat Z_{R_{2A_5}}/\!\!/_{\mathcal{O}(1)} N^{R_{2A_5}}\cong \mathbb{P}^1$.
\item $IP_t^{\pi_0N^{R_{2A_5}}}(\mathbb{P}(\mathcal{N}_x^{R_{2A_5}})/\!\!/R_{2A_5})=1 +t^2 + 2t^4 +2t^6 + 3t^8 + 2t^{10}+ 2t^{12} + t^{14} + t^{16}$.
\end{enumerate}
The term $B_{R_{2A_5}}(t)$ is given by
\begin{align*}
B_{R_{2A_5}}(t)&= t^2+2t^4+3t^6+4t^8+4t^{10} \mod t^{11}
\end{align*}
\end{pro}
\begin{proof}
For brevity, let us write $R=R_{2A_5}$, $N=N^{R_{2A_5}}$, and $\mathcal{N}_x^{R_{2A_5}}=\mathcal{N}_x$.
The same argument as used in Proposition~\ref{P:App-R=C*p2} for the proof of Lemma~\ref{L:R2A5Norm}(3) shows that $\widehat Z_{R}/\!\!/_{\mathcal{O}(1)} N $ is a rational normal projective variety of dimension $1$; i.e., $\mathbb{P}^1$.
Now the representation $\rho:R\to \mathcal{N}_x$ was worked out in
Lemma~\ref{L:R2A5-Nx-Rep}. The quotient $\mathbb{P}(\mathcal{N}_x^{R})/\!\!/R$ is a projective toric variety, and as in the previous case it is quite elementary to use our prior computations to compute its intersection cohomology via the general Kirwan process described in \S~\ref{S:PXss}.
Indeed, from the description of the weights of the action in Lemma~\ref{L:R2A5-Nx-Rep} (or from the fact that we have arrived at the Kirwan blowup) it follows that there are no strictly semi-stable points in $\mathbb{P}(\mathcal{N}_x)$, so that the GIT quotient has at worst finite quotient singularities. Thus $IP_t(\mathbb{P}(\mathcal{N}_x)/\!\!/R)=P_t(\mathbb{P}(\mathcal{N}_x )/\!\!/R)=P_t^{R}(\mathbb{P}(\mathcal{N}_x)^{ss})$, and using~\eqref{E:KD-(3.1)} yields
\begin{align}
\nonumber IP_t(\mathbb{P}(\mathcal{N}_x)/\!\!/R) =P_t^{R}(\mathbb{P}(\mathcal{N}_x )^{ss})&=P_t(\mathbb{P}(\mathcal{N}_x ))P_t(BR)-\sum_{0\neq \beta' \in \mathcal{B}(\rho)} t^{2d(\beta')}P_t^{R}(S_{\beta'})\\
\nonumber &=P_t(\mathbb{P}^{9})P_t(B\mathbb{C}^*)-\sum_{0\neq \beta' \in \mathcal{B}(\rho)} t^{2d(\beta')}P_t^{R}(S_{\beta'})\\
\label{E:IP2A5}&\equiv P_t(\mathbb{P}^{9})P_t(B\mathbb{C}^*) \mod t^{9}
\end{align}
since from Lemma~\ref{L:R2A5-Nx-Rep} there are no $0\ne \beta'\in \mathcal{B}(\rho)$ with $d(\beta)\le 4$; since $\mathbb{P}(\mathcal{N}_x )/\!\!/R$ has complex dimension $8$, it suffices by Poincar\'e duality to compute modulo $t^9$.
Now, since $\widehat Z_{R}/\!\!/_{\mathcal{O}(1)} N^{R}$ is simply connected, as described in Remark~\ref{R:pi0TrivBR} the action of $\pi_0N$ splits into an action on the base and an action on the fiber,
so that to compute $B_R(t)$, it suffices to compute $IP_t^{\pi_0N}(\mathbb{P}(\mathcal{N}_x)/\!\!/R)$.
We have seen in Lemma~\ref{L:R2A5Norm} (see Proposition~\ref{P:App-R=C*p2} for more details) that $\pi_0N\cong \mathbb{Z}_2$, and acts on $\mathbb{P}(\mathcal{N}_x )/\!\!/R$ via permutation of the coordinates $x_0,x_1,x_2,x_3,x_4\leftrightarrow x_4,x_3,x_2,x_1,x_0$. In the context of~\eqref{E:IP2A5}, the action of this involution on the cohomology of $\mathbb{P}^{9}$ is trivial, and acts on the torus $\mathbb{C}^*$ by $\lambda\mapsto \lambda^{-1}$. Thus the action of $\mathbb{Z}_2$ on $H^\bullet(B\mathbb{C}^*)=\mathbb Q[c_1]$ ($\deg c_1=2$) is induced by the action of $\mathbb{Z}_2$ on the vector space $\mathbb{Q}\langle c_1\rangle$ by $c_1\mapsto -c_1$ (see Example \ref{Exa:SemDirExa}). Thus $H^\bullet (B\mathbb C^*)^{\mathbb Z_2}=\mathbb Q[c_1^2]$, with generating function $(1-t^4)^{-1}$.
Putting this together with~\eqref{E:IP2A5} we obtain
\begin{align}\label{IPquot2A5}
\nonumber IP_t^{\pi_0N}(\mathbb{P}(\mathcal{N}_x )/\!\!/R_{})&\equiv (1-t^2)^{-1}\cdot (1-t^4)^{-1}\\
&\equiv 1+t^2+2t^4+2t^6+3t^8\mod t^{9}
\end{align}
completing the proof of (2).
From this we obtain the polynomial $b_R(t)=t^2+t^4+2t^6+2t^8+3t^{10}+2t^{12}+2t^{14}+t^{16}+t^{18}$, as in~\eqref{E:bR(t)-def} (with $c=9$).
We find finally that
\begin{align*}
B_R(t)&\equiv (1+t^2)\cdot( t^2+t^4+2t^6+2t^8+2t^{10})\\
&\equiv t^2+2t^4+3t^6+4t^8+4t^{10}\mod t^{11}.
\end{align*}
\end{proof}
\section{Putting the terms together to compute the cohomology of $\calM^{\operatorname{GIT}}$}
We now put together the results in the previous sections to complete the proof of Theorem~\ref{teo:betti} for $\calM^{\operatorname{GIT}}$. Recall that the intersection cohomology of $\calM^{\operatorname{GIT}}$ satisfies Poincar\'e duality. Consequently, as $\dim \calM^{\operatorname{GIT}}=10$, it suffices to compute $IP_t(\calM^{\operatorname{GIT}})$ up to $t^{10}$. We have:
\begin{align*}
&IP_t(\calM^{\operatorname{GIT}}) \equiv\\
&\equiv 1+4t^2+6t^4+10t^6+13t^8+15t^{10} &\text{(Kirwan blowup, Theorem~\ref{teo:betti})}\\
&\ \ \ \ -\ (t^2+t^4+2t^6+2t^8+3t^{10}) &\!\!\!\!\!\text{(Correction term, chordal cubic, Prop.~\ref{P:BRt-ChC})}\\
&\ \ \ \ -\ ( t^2+t^4+2t^6+3t^8+3t^{10}) &\text{(Correction term, $3D_4$ cubic, Prop.~\ref{P:BRt-3D4})}\\
&\ \ \ \ -\ (t^2+2t^4+3t^6+4t^8+4t^{10}) &\text{(Correction term, $2A_5$ cubic, Prop.~\ref{P:BRt-2A5})}\\
&\equiv 1+t^2+2t^4+3t^6+4t^8+5t^{10} \mod t^{11}.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\end{align*}
This completes the proof of Theorem~\ref{teo:betti} for $\calM^{\operatorname{GIT}}$.
\begin{rem}
Recall from~\eqref{eq:ARcontribution} that $P_t(\calM^{\operatorname{K}})=P^G_t(X^{ss})+\sum_{R\in \mathcal{R}}A_R(t))$, where $A_R(t)$ is the difference of the ``main term'' and the ``extra term'' (see~\eqref{E:finMainT} and~\eqref{E:finExtraT}). From~\eqref{E:GIT-MK-B} we have $IP_t(\calM^{\operatorname{GIT}})=P_t(\calM^{\operatorname{K}})-\sum_{R\in \mathcal{R}}B_R(t)$, so that finally
$$
IP_t(\calM^{\operatorname{GIT}})=P_t^G(X^{ss})+\sum_{R\in \mathcal{R}}(A_R(t)-B_R(t)).
$$
From~\cite[Thm.~2.5]{kirwanrational1}, one knows that all the coefficients of the polynomial $\sum_{R\in \mathcal{R}}(A_R(t)-B_R(t))$ are non-positive (note that in examples, the individual terms $A_R(t)-B_R(t)$ may have positive coefficients, e.g.,~\cite[6.5]{kirwanhyp}).
In our situation we have
\begin{align*}
A_{R_c}(t)-B_{R_c}(t)&\equiv \ \ \ 0 \mod t^{11}\\
A_{R_{3D_4}}(t)-B_{R_{3D_4}}(t)&\equiv -t^8 -t^{10}\mod t^{11}\\
A_{R_{2A_5}}(t)-B_{R_{2A_5}}(t)&\equiv \ \ \ 0 \mod t^{11}.
\end{align*}
In other words, up to $t^{10}$ (which is all that matters due to Poincar\'e duality), the intersection cohomology of $\calM^{\operatorname{GIT}}$ agrees with the equivariant cohomology of the semi-stable locus, except for a correction in real codimensions~$8$ and $10$, coming from the blowup of the $3D_4$ locus.
\end{rem}
\section{The intersection cohomology of $\widehat{\mathcal{M}}$}
Since the space $\widehat\mathcal{M}$ is an intermediate step in the Kirwan blowup, obtained after one blows up only the chordal cubic point $\Xi\in\calM^{\operatorname{GIT}}$, its intersection cohomology appears as an intermediate stage in our computations above, simply by taking only the blowup step corresponding to $R_c$, dealt with in Propositions~\ref{P:ET-ChC} and~\ref{P:BRt-ChC}. Thus we have
\begin{align*}
IP_t(\widehat\mathcal{M})\equiv &1+t^2+2t^4+3t^6+4t^8+5t^{10}& \text{($IP_t(\calM^{\operatorname{GIT}})$, Theorem~\ref{teo:betti})}\\
&\ + t^2+t^4+2t^6+2t^8+3t^{10} &\text{(Correction term $B_{R_c}$, Proposition~\ref{P:BRt-ChC})} \\
\equiv &1+2t^2+3t^4+5t^6+6t^8+8t^{10} \mod t^{11}.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\end{align*}
This completes the proof of Theorem~\ref{teo:betti} for $\widehat {\mathcal{M}}$.
\chapter{The intersection cohomology of the ball quotient}\label{sec:IHball}
In this section we use the decomposition theorem in a different way to compute the intersection cohomology of the Baily--Borel compactification $(\calB/\Gamma)^*$ of the ball quotient model $\mathcal{B}/\Gamma$ of $\mathcal{M}$.
For a paper addressing similar situations computing intersection cohomology of arithmetic quotients via the Kirwan blowup, see~\cite{KLW}.
\section{A special case of the decomposition theorem} \label{S:Decomp-Thm}
There is a special case of the decomposition theorem that will be quite useful for us in computing cohomology of the Baily--Borel and toroidal compactifications of the ball quotient.
We start by recalling that if $f:\hat V\to V$ is a map from a variety $\hat V$ of dimension~$n$, smooth up to finite quotient singularities, to a possibly singular variety $V$ that is the blowup of a (not necessarily smooth) point $p\in V$ to an exceptional divisor $E\subset \hat V$, smooth up to finite quotient singularities (cf.~Diagram~\eqref{E:MV-setup}, with $C=p$), then the decomposition theorem gives the following. Writing $P_t(E)=\sum e_jt^j$, the decomposition theorem gives (e.g.,~\cite[Lem.~9.1]{GH-IHAg-17}):
\begin{equation}\label{eq:IHblowup}
P_t(\hat V)=IP_t(V)+e_{2n-2}t^2+e_{2n-3}t^3+\dots+e_{n+1}t^{n-1}+e_nt^n+e_{n+1}t^{n+1}+\dots +e_{2n-2}t^{2n-2}.
\end{equation}
In other words, the correction terms for $P_t(\hat V)$ in degree $\ge n=\dim \hat V$ are the Betti numbers of $E$, and in degree $\le n$ are set up so that Poincar\'e duality holds for $\hat V$. We note that by Poincar\'e duality on $E$, we have $e_{2n-2-i}=e_i$.
\subsection{Comparing to Kirwan's computation}
We now observe that we have already seen another approach to computing $P_t(\hat V)$ in~\eqref{eq:IHblowup}.
Indeed, let $U\subseteq V$ be any open neighborhood of $p$ and let ${\hat U}:=f^{-1}(U)$ (cf.~Diagram~\eqref{E:MV-setup}, with $C=p$). Then from~\eqref{E:IH-Lem1}, we have $IP_t(\hat V)=IP_t(V)+IP_t({\hat U})-IP_t(U)$. Using the fact that $\hat V$ is smooth up to finite quotient singularities, we immediately get
$
P_t(\hat V)=IP_t(V)+P_t({\hat U})-IP_t(U)
$.
If we assume now that
${\hat U}$ retracts onto $E$, then we have
\begin{equation}\label{eq:IHblowup-1}
P_t(\hat V)=IP_t(V)+P_t(E)-IP_t(U).
\end{equation}
\begin{rem}
Combining~\eqref{eq:IHblowup} and~\eqref{eq:IHblowup-1} we find
\begin{equation}\label{E:BR-E-comp}
P_t(E)-IP_t(U)=e_{2n-2}t^2+e_{2n-3}t^3+\dots+e_{n+1}t^{n-1}+e_nt^n+\dots +e_{2n-2}t^{2n-2}.
\end{equation}
\end{rem}
We now compare this description of the decomposition theorem for the blowup of a point to the general setup of the Kirwan blowup machinery, which will provide us with an alternative viewpoint, and enable us to use some of the previous computations to deal with $(\calB/\Gamma)^*$.
To make this comparison, let us return to Kirwan's general situation, for a single blowup, as in Chapter~\ref{SSS:IC1}. We are further assuming that the morphism $\hat \pi_G:\hat X/\!\!/_{\hat L}G\to X/\!\!/_LG$ is such that $\hat X/\!\!/_{\hat L}G$ is smooth up to finite quotient singularities, that the center of the blowup $Z_R/\!\!/_{L}G$ is a point, and that the exceptional divisor $E/\!\!/_{\hat L}G$ is smooth up to finite quotient singularities. In this situation,~\eqref{eq:IHblowup-1} translates to
\begin{equation}\label{E:KR1-C2.11-1}
P_t(\hat X/\!\!/_{\hat L}G)=IP_t(X/\!\!/_LG)+ \underbrace{P_t(E/\!\!/_{\hat L}G)-IP_t(\mathcal{N}/\!\!/G)}_{B_R(t)}\,,
\end{equation}
where we are using the fact mentioned in deducing~\eqref{E:KR1-C2.11} from~\eqref{E:IH-Lem1}, that there is an appropriate open neighborhood $U\subseteq X/\!\!/_{L}G$ of $Z_R/\!\!/_{L}G$ with $U$ homeomorphic to $\mathcal{N}/\!\!/G$. Alternatively, this is~\eqref{E:KR1-C2.11} together with the fact
that
$E/\!\!/_{\hat L}G\cong \hat{ \mathcal{N}}/\!\!/G$~\cite[Lem.~2.15]{kirwanrational1}
(see also~\cite[p.494]{kirwanrational1}).
We note here that in this special case, the formula~\eqref{eq:IHblowup} follows from~\eqref{E:KR1-C2.11-1} using~\eqref{E:KR1-E2.22(1)} and~\eqref{E:KR1-E2.22(2)}, and the shift in degrees mentioned after those equations.
The main point for us, however, is the converse statement, that~\eqref{eq:IHblowup} gives an alternative approach to computing $B_R(t)$, assuming one knows $P_t(E/\!\!/_{\hat L}G)$.
Indeed, if $P_t(E/\!\!/_{\hat L}G)=\sum e_jt^j$, then combining \eqref{E:KR1-C2.11-1} and \eqref{E:BR-E-comp}, one sees that $B_R(t)= e_{2n-2}t^2+e_{2n-3}t^3+\dots+e_{n+1}t^{n-1}+e_nt^n+\dots +e_{2n-2}t^{2n-2}$, where here $n=\dim X/\!\!/_LG$.
Moreover,~\eqref{E:KR1-E2.22(2)} asserts that the cohomology of the exceptional divisor is given in our special situation as
\begin{equation}\label{E:DT-ExcD}
H^i(E/\!\!/_{\hat L}G)=\sum_{p+q=i}\left[H^p(Z_R/\!\!/_LN_0)\otimes IH^{q}(\mathbb{P}(\hat{\mathcal{N}}_x)/\!\!/R)\right]^{\pi_0N}.
\end{equation}
We also note that Remarks~\ref{R:pi0TrivB} and~\ref{R:pi0TrivExpl} give analogous statements for $H^i(E/\!\!/_{\hat L}G)$ in this situation.
\section{The intersection cohomology of the ball quotient}
We start by recalling the birational maps relating the Kirwan blowup, GIT, and Baily--Borel compactifications, and introduce the notation for them (see also \S~\ref{sec:prelim}):
\begin{equation}\label{eq:diagramofmaps}
\xymatrix{
&\calM^{\operatorname{K}}\ar[ldd]_\pi\ar[d]^f\ar[rdd]^g\\
&\widehat\mathcal{M}\ar[ld]^{p}\ar[rd]_q\\
\calM^{\operatorname{GIT}}&&(\calB/\Gamma)^*
}
\end{equation}
Here $\pi$ is the Kirwan blowup, the geometry of which has been the focus of the paper up to this point. The space $\widehat\mathcal{M}$ resolves the birational map from $\calM^{\operatorname{GIT}}$ to $(\calB/\Gamma)^*$, and has been described above in \S\ref{subsec:modulicubic}, following~\cite{act}. The map $p$ is the blowup of the point $\Xi\in\calM^{\operatorname{GIT}}$ that corresponds to the chordal cubic, to a divisor $D_h\subset\widehat\mathcal{M}$.
Recall that~$\mathcal{T}\subset \calM^{\operatorname{GIT}}$ is the rational curve of cubics with $2A_5$ singularities, which contains the point $\Xi$ corresponding to the chordal cubic. Let then $\widehat {\mathcal{T}}=Z_{T,1}^{ss}\subset\widehat\mathcal{M}$ be the strict transform of the curve $\mathcal{T}$ under the map $p$. The map $q$ then consists simply of blowing down the curve $\widehat {\mathcal{T}}$ to a point $c_{2A_5}$, this point being one of the two cusps of $(\calB/\Gamma)^*$. The other cusp $c_{3D_4}$ of $(\calB/\Gamma)^*$ corresponds to the $3D_4$ cubic. Thus the exceptional divisor of the map $g$ consists of two disjoint irreducible components $D_{3D_4}$ and $D_{2A_5}$, contracted to the two points that are the corresponding cusps of $(\calB/\Gamma)^*$.
To compute the intersection cohomology of~$(\calB/\Gamma)^*$, we apply the decomposition theorem to the map $g:\calM^{\operatorname{K}}\to(\calB/\Gamma)^*$. The advantage of working with this map instead of with $q$ is that the domain $\calM^{\operatorname{K}}$ is smooth up to finite quotient singularities, thus its intersection cohomology is equal to its cohomology, and the intersection complex is trivial. We will show that both exceptional divisors $D_{2A_5},D_{3D_4}\subset\calM^{\operatorname{K}}$ of the map $g:\calM^{\operatorname{K}}\to(\calB/\Gamma)^*$ are smooth up to finite quotient singularities, and thus we will be able to use~\eqref{eq:IHblowup}.
The divisor $D_{3D_4}$ is smooth up to finite quotient singularities, since it is obtained as an exceptional divisor in the Kirwan blowup process $\pi$, which is not modified after it is introduced (one can also check directly that in the divisor's description as a GIT quotient, it has no strictly semi-stable points at the stage it is first introduced, and is not modified by the subsequent blowups).
The divisor $D_{2A_5}$ is similarly seen to be smooth up to finite quotient singularities, since it is obtained as the last step of the Kirwan blowup process (one can also check directly that in the divisor's description as a GIT quotient, it has no strictly semi-stable points at the stage it is first introduced).
We now compute the contribution to $P_t(\calM^{\operatorname{K}})$ due to the divisor $D_{3D_4}$. In fact, in this case, rather than using~\eqref{eq:IHblowup}, and computing $P_t(D_{3D_4})$, we will use a slightly different approach.
Since $\pi(D_{3D_4})\in \mathcal{M}^{GIT}$ and $f(D_{3D_4})\in \widehat {\mathcal{M}}$ are both points and $p$ is locally an isomorphism near those points, and since $q$ is locally an isomorphism near the points $f(D_{3D_4})\in \widehat {\mathcal{M}}$ and $c_{3D_4}\in(\calB/\Gamma)^*$, the term $B_{R_{3D_4}}(t)$ in Proposition~\ref{P:BRt-3D4} is precisely the contribution for the blowup $g$, over the cusp $c_{3D_4}$, namely
\begin{equation}\label{E:g3D4-contr}
t^2+t^4 +2t^6 +3t^8 +3t^{10} \mod t^{11}
\end{equation}
For the contribution of the divisor $D_{2A_5}$ the situation is a little trickier, so that we will have to use~\eqref{eq:IHblowup}, and compute $P_t(D_{2A_5})$. The situation is more complicated because $f(D_{2A_5})=\widehat {\mathcal{T}}$ is a curve, contracted by $q$ to the point $c_{2A_5}\in (\calB/\Gamma)^*$, and moreover, since the map $p$ is not an isomorphism in a neighborhood of $\mathcal{T}$ and $\widehat {\mathcal{T}}$ (one must take into account the blowup of the point $\Xi$). Nevertheless, we have essentially already computed $P_t(D_{2A_5})$. Indeed, from~\eqref{E:DT-ExcD}, Proposition~\ref{P:BRt-2A5}, and Remark~\ref{R:pi0TrivExpl} we see that
\begin{align}
\label{E:KirAlt2A5} P_t(D_{2A_5})&=P_t(\mathbb{P}^1)\, IP_t^{\pi_0N}(\mathbb{P}(\mathcal{N}_x)/\!\!/R) & \\
\nonumber &\equiv (1+t^2)(1+t^2+2t^4+2t^6+3t^8)\ \mod t^{9} &\ \text{(From~\eqref{IPquot2A5})}\\
\label{E:g2A5-contr} &\equiv 1+2t^2+3t^4+4t^6+5t^8\ \mod t^9&.
\end{align}
Summarizing, we obtain
\begin{align*}
&IP_t((\calB/\Gamma)^*) \equiv\\
&\equiv 1+4t^2+6t^4+10t^6+13t^8+15t^{10} &\text{(Kirwan blowup $\calM^{\operatorname{K}}$, Theorem~\ref{teo:betti})}\\
&\ \ \ \ -( t^2+t^4 +2t^6 +3t^8 +3t^{10})&\text{($D_{3D_4}$ contribution,~\eqref{E:g3D4-contr})}\\
&\ \ \ \ -( t^2+2t^4 +3t^6 +4t^8 +5t^{10})&\text{($D_{2A_5}$ contribution,~\eqref{E:g2A5-contr},~\eqref{eq:IHblowup})}\\
&\equiv 1+2t^2+3t^4 +5t^6 +6t^8 +7t^{10}\mod t^{11}.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\end{align*}
\begin{rem}
We remark that thus $IP_t(\widehat\mathcal{M})-IP_t((\calB/\Gamma)^*)=t^{10}$; in particular, this difference is not zero. Even though the map $q$ is small (being a contraction of a curve in a 10-fold to a point), it does not induce an isomorphism in intersection cohomology, which is possible for a small map whose domain is not smooth.
\end{rem}
\begin{rem}\label{R:Kirwan2A5}
Frances Kirwan suggested a small variation on how to establish~\eqref{E:KirAlt2A5}.
The main claim is that the map $f|_{D_{2A_5}}:D_{2A_5}\to\widehat{\mathcal{T}} =\mathbb{P}^1$ is a fibration. To see this,
one interprets this map as a GIT quotient of the locus in $\mathbb{P}^{34}$ of all $2A_5$ cubics by the normalizer $N_{2A_5}$. The extra involution $\sigma$ that appears in the stabilizer of the special point (as explained in the appendix) is contained in $N_{2A_5}$ (recall that $\sigma$ acts diagonally, while the normalizer is a $\mathbb{Z}/2\mathbb{Z}$ extension of the maximal diagonal torus). Thus in thinking about the GIT quotient by $N_{2A_5}$, this extra involution plays no role. Thus we have a fibration, with fibers isomorphic to $\mathbb{P}(\mathcal{N}_x)/\!\!/R)/\pi_0N$.
In~\eqref{IPquot2A5} we have computed the invariant part of the cohomology of these fibers. Notice, crucially, that this cohomology is zero in all odd degrees. Thus the spectral sequence computing the cohomology of $D_{2A_5}$, as a fibration over $\mathbb{P}^1$, is completely degenerate, and we obtain~\eqref{E:KirAlt2A5}.
\end{rem}
\chapter{The cohomology of the toroidal compactification}\label{sec:toroidal}
In this section we will compute the cohomology of the toroidal compactification of the ball quotient, completing the proof of Theorem~\ref{teo:betti}.
The starting point of our discussion is the~\cite{act} ball quotient model $\mathcal{B}/\Gamma$ for the moduli of cubic threefolds (see Chapter~\ref{sec:prelim}). We recall that this locally symmetric variety $\mathcal{B}/\Gamma$ is associated to an Eisenstein lattice $\Lambda$ (see \S\ref{subsec:Eisenstein}, esp.~\eqref{eq_def_lambda}) of signature $(1,10)$, that we will review below. Similar to the better known case of $K3$ surfaces, the cusps of the Baily--Borel compactification $(\mathcal{B}/\Gamma)^*$ correspond to $\Gamma$-conjugacy classes of primitive isotropic subspaces in $\Lambda$. By~\cite{act}, there are exactly two cusps, that we label $c_{2A_5}$ and $c_{3D_4}$ respectively, in accordance with Theorem~\ref{resgitball}. The toroidal compactification, which we denote here by~$\overline{\calB/\Gamma}$, is a partial resolution $\overline{\calB/\Gamma}\to(\mathcal{B}/\Gamma)^*$ of the two cusps. Unlike in the Siegel or the orthogonal case,
there are no choices involved, and one can thus speak about {\em the} toroidal compactification $\overline{\calB/\Gamma}$. The reason for the uniqueness is that all cones which are used in the toroidal compactification have
dimension $1$. For this reason it also holds that $\overline{\calB/\Gamma}$ only has finite quotient singularities and, therefore, its singular cohomology and intersection cohomology coincide. From the general theory (see~\cite{AMRT}; see also~\cite{beh} for the ball quotient case), the two exceptional divisors of $\overline{\calB/\Gamma}\to(\mathcal{B}/\Gamma)^*$ are quotients of $9$-dimensional abelian varieties by finite groups (Proposition~\ref{prop_structure_tor}). In each of the two cases occurring here, the relevant abelian variety is in fact $(E_\omega)^9$, where $E_\omega$ is the elliptic curve with $j$-invariant equal to $0$, namely the quotient of the complex numbers by the Eisenstein integers. The content of the first subsection of this section is the identification of the two finite groups $\Gamma_{2A_5}$ and $\Gamma_{3D_4}$ acting on $(E_\omega)^9$ for the two cusps $c_{2A_5}$ and $c_{3D_4}$ respectively. In the second subsection, by adapting a well-known theorem of Looijenga (see~\cite{Lroot},~\cite[Thm. 2.7]{FMW}), we are able to compute the cohomology of these two exceptional divisors $(E_\omega)^9/\Gamma_{2A_5}$ and $(E_\omega)^9/\Gamma_{3D_4}$. Finally, combining this with an application of the decomposition theorem, we conclude the proof of Theorem~\ref{teo:betti}.
\section{The arithmetic of the two cusps of $\mathcal{B}/\Gamma$}\label{S:ArithmeticCusp}
In this subsection, we discuss the structure of the toroidal compactification for the ball quotient model $\mathcal{B}/\Gamma$ for cubic threefolds. To start, we briefly recall the notion of Eisenstein lattice, and the relationship to the even $\mathbb{Z}$-lattices endowed with an order $3$ isometry. We follow with the classification of the cusps of $(\mathcal{B}/\Gamma)^*$, which is closely related to the classification of the Type II boundary components for the Baily--Borel compactification for cubic fourfolds (see~\cite[\S6.1]{laza}). Using some ideas from~\cite{beh}, we can describe the two exceptional divisors of $\overline{\calB/\Gamma}\to(\mathcal{B}/\Gamma)^*$ (Proposition~\ref{prop_structure_tor}).
\subsection{Eisenstein Lattices}\label{subsec:Eisenstein} Let $\mathcal{E}$ be the ring of Eisenstein integers
$$
\mathcal{E}:=\mathbb{Z}[\omega], \ \ \ \omega= e^{\frac{2 \pi i}{3}}.
$$
By an {\it Eisenstein lattice} $\mathcal{G}$ we understand a free $\mathcal{E}$ module, endowed with an hermitian form taking values in $\mathcal E$.
Throughout, we will make the \emph{additional convention that the Hermitian form takes values in $\theta \mathcal{E}$}, where $\theta=\omega-\omega^2(=\sqrt{-3})$, i.e.,
$$\langle -,-\rangle:\mathcal G\times \mathcal G\to \theta \mathcal E;$$
this should be understood as an analogue of even lattices over $\mathbb{Z}$. In particular, note that then $\|x\|^2=\langle x,x\rangle\in 3\mathbb{Z}$.
Associated to an Eisenstein lattice $\mathcal{G}$, there is a usual $\mathbb{Z}$-lattice, that we denote $\mathcal{G}_\mathbb{Z}$. Simply, $\mathcal{G}_\mathbb{Z}$ is the underlying free $\mathbb{Z}$-module. On $\mathcal{G}_\mathbb{Z}$, we define a bilinear symmetric form
$$
(-,-):= - \frac{2}{3} \operatornameeratorname{Re} \langle -,- \rangle: \mathcal{G}_{\mathbb{Z}} \times \mathcal{G}_{\mathbb{Z}} \to \mathbb{Z}.
$$
Under our convention on the hermitian form, $\mathcal{G}_\mathbb{Z}$ is an even lattice. Note that $\mathcal{G}_\mathbb{Z}$ comes endowed with an order $3$ isometry $\rho\in \operatornameeratorname{O}(\mathcal{G}_\mathbb{Z})$,
namely
\begin{equation}\label{eqrho}
\rho(x):=\omega\cdot x,
\end{equation}
where the multiplication is the multiplication by scalars in the Eisenstein module $\mathcal{G}$. Clearly, $\rho$ acts on $\mathcal{G}_\mathbb{Z}$ fixing only the origin (i.e., $\rho(x)\neq x$ for any $x\neq0$).
Conversely, given an even lattice $\mathcal{G}_\mathbb{Z}$ together with an order $3$ isometry $\rho$ (fixing only the origin), we can define an Eisenstein lattice $\mathcal{G}$ reversing the process above. More precisely, the Eisenstein structure on $\mathcal{G}_{\mathbb{Z}}$ is determined by~\eqref{eqrho}. Then, the hermitian form is given by
\begin{equation}\label{arr:hermitianform}
\operatornameeratorname{Re} \langle x,y \rangle=-\frac{3}{2}\cdot(x,y),\quad
i\cdot\operatornameeratorname{Im} \langle x,y \rangle=\frac{\theta}{2}\cdot((\rho-\rho^2)x,y).
\end{equation}
Finally, we note that an isometry $\phi$ of $\mathcal{G}$ induces an isometry of $\mathcal{G}_\mathbb{Z}$, commuting with $\rho$, and conversely. In other words,
\begin{equation}\label{eq_rel_iso}
\operatornameeratorname{Aut}(\mathcal{G},\langle-,-\rangle)=\{\phi\in \operatornameeratorname{O}(\mathcal{G}_\mathbb{Z})\mid \phi\rho=\rho\phi\}.
\end{equation}
By abuse of notation, we will denote by
$$\operatornameeratorname{O}(\mathcal{G}):=\operatornameeratorname{Aut}(\mathcal{G},\langle-,-\rangle)$$ the group of isometries (N.B. $\operatornameeratorname{O}(\mathcal{G})$ is a unitary group, and not an orthogonal group).
We now introduce the Eisenstein lattices relevant to our discussion. First, we consider the following definite lattices (defined in terms of Gram matrices):
$$\mathcal{E}_1: (3),\ \ \mathcal{E}_2: \begin{pmatrix}
3&\theta\\
\bar{\theta}&3\\
\end{pmatrix}, \ \ \mathcal{E}_3: \begin{pmatrix}
3&\theta&0\\
\bar{\theta}&3&\theta\\
0&\bar\theta&3
\end{pmatrix}, \ \ \mathcal{E}_4: \begin{pmatrix}
3&\theta&0&0\\
\bar{\theta}&3&\theta&0\\
0&\bar{\theta}&3&\theta\\
0&0&\bar\theta&3
\end{pmatrix}\,\,.$$
The underlying $\mathbb{Z}$ lattices $(\mathcal{E}_i)_\mathbb{Z}$ are $A_2(-1)$, $D_4(-1)$, $E_6(-1)$, and $E_8(-1)$ respectively. Conversely, we note that the lattices $A_2(-1)$, $D_4(-1)$, $E_6(-1)$ and $E_8(-1)$ admit (up to conjugacy) a unique order $3$ isometry fixing only the origin, and thus they admit a unique Eisenstein structure (e.g.~\cite[Lem.~3]{HKN}).
We also consider the indefinite (signature $(1,1)$) lattice $\mathcal{H}$ defined by
$$\mathcal{H}: \begin{pmatrix}
0&\theta\\
\bar{\theta}&0\\
\end{pmatrix}\,,$$
whose underlying $\mathbb{Z}$ lattice is $2U$ (two copies of the hyperbolic plane).
The Eisenstein lattice used by Allcock--Carlson--Toledo~\cite{act} to define the ball quotient model $\mathcal{B}/\Gamma$ for the moduli of cubic threefolds is
\begin{equation}\label{eq_def_lambda}
\Lambda:= \mathcal{E}_1 + 2\mathcal{E}_4 + \mathcal{H},
\end{equation}
with associated $\mathbb{Z}$ lattice
$$
\Lambda_{\mathbb{Z}} \cong A_2(-1) + 2E_8(-1) + 2U,
$$
which is precisely (up to a sign) the lattice of the primitive middle cohomology of a smooth cubic fourfold. Returning to the construction of $\mathcal{B}/\Gamma$, we recall
$$
\mathcal{B}:=\mathcal{B}_{10}:=\{[z ]: z^2>0\}^+\subset \mathbb{P}(\Lambda\otimes_{\mathcal{E}}\mathbb{C}),
$$
and $\Gamma=\operatornameeratorname{O}(\Lambda)$ acts naturally (properly discontinuously) on $\mathcal{B}$.
Finally, let us recall some basic terminology from Nikulin's theory for even $\mathbb{Z}$-lattices that will be needed later. Let $M$ be an even non-degenerate $\mathbb{Z}$-lattice. The {\it dual lattice}
is $M^\vee=\operatornameeratorname{Hom}_\mathbb{Z}(M,\mathbb{Z})$. Using the quadratic form, the dual $M^\vee$ has the following
description
$$M^\vee=\{w\in M\otimes_\mathbb{Z} \mathbb{Q}\mid (v,w)\in \mathbb{Z} \textrm{ for all } v\in M\}\,,$$
in particular $M\subset M^\vee\subset M\otimes_\mathbb{Z} \mathbb{Q}$. The {\it discriminant group} is the finite group $A_M:=M^\vee/M$. A key insight of Nikulin is that the quadratic form on $M$ induces a finite quadratic form
$$q_M:A_M\to \mathbb{Q}/2\mathbb{Z}\,\,.$$
For example, if $M=E_6(-1)$, then $A_{M}\cong \mathbb{Z}/3$ and $q_M(\xi)=-\frac{4}{3}\in \mathbb{Q}/2\mathbb{Z}$ for $\xi$ a generator of $A_M$.
We also recall that for $v\in M$, the {\it divisibility} $\operatornameeratorname{div} v$ is the positive generator of the ideal $(v,M)\subset \mathbb{Z}$, i.e., the biggest natural number by which all integers $(v,m)$ for $m\in M$ are
divisible. Note that $\frac{v}{\operatornameeratorname{div}{v}}\in M^\vee$, and then via the projection $M^\vee\to A_M=M^\vee/M$ we obtain an element in $A_M$. In fact, every element of $A_M$ arises in this way. If $v\in M$ is primitive, then the order of (the class of) $\frac{v}{\operatornameeratorname{div}{v}}$ in $A_M$ is precisely $\operatornameeratorname{div}{v}$. Returning to the $M=E_6(-1)$ example, we see that $\operatornameeratorname{div}{v}\in\{1,3\}$ for $v\in M$ primitive. Furthermore, if $v$ is primitive with $\operatornameeratorname{div}{v}=3$, then (the class of) $\frac{v}{3}$ is a generator of $A_M$. Using $q_M\left(\frac{v}{3}\right)=-\frac{4}{3}\in \mathbb{Q}/2\mathbb{Z}$, one concludes $v^2=-12 \pmod{18}$. For $M=E_6(-1)$, there exists indeed an element $v$ of norm $-12$ and divisibility $3$.
\begin{rem}\label{R:DualEisen}
Most of the above discussion can be adapted to the case of Eisenstein lattices. Here, for $v$ in an Eisenstein lattice $\mathcal{G}$, we define $\operatornameeratorname{div} v$ as the generator of the ideal $\langle \mathcal{G},v\rangle\subset \theta \mathcal{E}$.
Clearly $\theta$ divides $\operatornameeratorname{div} v$, and $\operatornameeratorname{div} v$ divides $ \|v\|^2$. Similarly, following the conventions in~\cite[p.285]{ALeech},~\cite[p.8645]{Ma} we define $\mathcal G^*=\operatornameeratorname{Hom}_{\mathcal E}(\mathcal G,\mathcal E)$, and under the identification $\mathcal G^*=\{\nu\in \mathcal G\otimes_{\mathcal E}\mathbb Q(\omega): \langle \lambda ,\nu\rangle \in \mathcal E\ \forall\ \lambda \in \mathcal G\}$, we naturally obtain a Hermitian form on $\mathcal G^*$ making it an Eisenstein lattice (with our conventions).
Under the $\theta$-value assumption on the Hermitian form,
it holds that
$$\frac{1}{\theta}\mathcal{G}\subset \mathcal{G}^*\subset \mathcal{G}\otimes_\mathcal{E}\mathbb{Q}(\omega)\,.$$
Thus, the natural ``unimodularity'' condition in this setup is $\theta\mathcal{G}^*\cong \mathcal{G}$. For the lattices considered here, $\mathcal{E}_4$ and $\mathcal{H}$ satisfy this condition, while $\mathcal{E}_1$, $\mathcal{E}_2$, and $\mathcal{E}_3$ do not. Note that under the natural identification $\mathcal G\otimes_{\mathcal E}\mathbb Q(\omega)=\mathcal G\otimes_{\mathcal E}\mathcal E\otimes_{\mathbb Z}\mathbb Q=\mathcal G\otimes_{\mathbb Z}\mathbb Q$, we have $(\theta\mathcal G^*)_{\mathbb Z}=(\mathcal G_{\mathbb Z})^\vee$~\cite[p.8645]{Ma}, so that the notions of unimodularity in the Eisenstein, and underlying integral case, agree. Finally, we observe that under our $\theta$-value assumption on the Hermitian form, it is typically more convenient to work with $\mathcal G':= \operatornameeratorname{Hom}_{\mathcal E}(\mathcal G,\theta \mathcal E)= \{\nu\in \mathcal G\otimes_{\mathcal E}\mathbb Q(\omega): \langle \lambda ,\nu\rangle \in \theta \mathcal E\ \forall\ \lambda \in \mathcal G\}$. Clearly $\mathcal G'=\theta\mathcal G^*$, the unimodularity condition becomes $\mathcal G'\cong \mathcal G$, and we have $(\mathcal G')_{\mathbb Z}=(\mathcal G_{\mathbb Z})^\vee$.
\end{rem}
\subsection{Identification of the two cusps of $(\calB/\Gamma)^*$}
As mentioned above, a cusp of the Baily--Borel compactification corresponds to a primitive isotropic subspace (automatically of rank $1$) $\mathcal{F}\subset \Lambda$, considered up to the action of $\Gamma$. As a consequence of~\cite{act} (see Theorem~\ref{resgitball}), we know that there are precisely two cusps, and thus two possible $\mathcal{F}$. Our goal here is to describe these two cases explicitly. First, we note that a standard invariant that in many cases suffices to distinguish the Baily--Borel cusps is the definite lattice $\mathcal{F}^\perp/\mathcal{F}$ of rank $9$, where $\mathcal{F}^\perp$ denotes the orthogonal complement of $\mathcal{F}$ in $\Lambda$ (N.B. since $\mathcal{F}$ is isotropic, $\mathcal{F}\subset \mathcal{F}^\perp$).
A weaker invariant is the associated $\mathbb{Z}$-lattice $(\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}$ (negative definite of rank $18$). An even weaker invariant is $R\left((\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}\right)$, i.e. the sublattice of $(\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}$ spanned by roots (i.e., $-2$ classes). In our situation this weak invariant suffices to distinguish the cusps, as there are only two of them.
\begin{lem}\label{lem_inv_iso}
With notation as above (e.g., $\mathcal{F}$ is the isotropic subspace associated to the corresponding cusp), the following hold:
\begin{itemize}
\item[i)] for the cusp $c_{2A_5}$ of $(\mathcal{B}/\Gamma)^*$, $R\left((\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}\right)\cong 2E_8(-1)+A_2(-1)$;
\item[ii)] for the cusp $c_{3D_4}$ of $(\mathcal{B}/\Gamma)^*$, $R\left((\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}\right)\cong 3E_6(-1)$.
\end{itemize}
\end{lem}
\begin{proof}
By Theorem~\ref{resgitball}, we know that $c_{2A_5}$ and $c_{3D_4}$ correspond to semi-stable cubic threefolds with $2A_5$ singularities and $3D_4$ singularities respectively. The ball quotient model $\mathcal{B}/\Gamma$ for cubic threefolds is obtained by considering the eigenperiods (see~\cite{DK}) of cubic fourfolds with a $\mu_3$ action (namely, to a cubic threefold $V(f_3(x_0,\dots,x_4))$ one associates the cubic fourfold $V(f_3(x_0,\dots,x_4)+x_5^3)$). This construction is compatible with GIT and Baily--Borel compactifications. One immediately checks that the construction associates to a cubic threefold with $2A_5$ (resp. $3D_4$) singularities a semi-stable cubic fourfold with $2\widetilde E_8$ (resp. $3\widetilde E_6$) singularities. Now the claim follows from the classification of Type II boundary components for cubic fourfolds (and the discussion of their geometric meaning) in~\cite[\S6.1]{laza}).
\end{proof}
\begin{rem}\label{rem_comparebb}
For further reference, let us note the following. Let $(\mathcal{D}/\Gamma')^*$ be the Baily--Borel compactification for the moduli of cubic fourfolds (as discussed, this is associated to the lattice $\Lambda_\mathbb{Z}(-1)=2E_8+A_2+2U$). By construction, there exists a natural morphism $$A:(\mathcal{B}/\Gamma)^*\to (\mathcal{D}/\Gamma')^*\,,$$
which is generically an embedding (in fact, a normalization of the image). The two cusps of the Baily--Borel compactification $(\mathcal{B}/\Gamma)^*$ map to points on the Type II components of $(\mathcal{D}/\Gamma')^*$ (corresponding to the fact that $\mathcal{F}_\mathbb{Z}$ is an isotropic rank $2$ subspace of $\Lambda_\mathbb{Z}$). The lemma above says that $A(c_{2A_5})\in II_{2E_8+A_2}$ and $A(c_{2A_5})\in II_{3E_6}$ respectively (where $II$ indexed by a root lattice denotes a Type II boundary component in $(\mathcal{D}/\Gamma')^*$). It is well known (in full generality) that
the Type II boundary components of $(\mathcal{D}/\Gamma')^*$ are modular curves, while here in fact $II_{2E_8+A_2}, II_{3E_6}\cong \mathfrak h/\operatornameeratorname{SL}(2,\mathbb{Z})$. It is then clear
(by construction) that the $A(c_{2A_5})$ and $A(c_{3D_4})$ map to the special points on $II_{2E_8+A_2}$ and $II_{3E_6}$ respectively corresponding to $j$-invariant equal to $0$.
\end{rem}
It remains now to identify two possibilities of primitive isotropic subspaces $\mathcal{F}\subset \Lambda$ such that the associated invariant $R((\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z})$ is as in Lemma~\ref{lem_inv_iso}.
The first case is immediate.
\begin{lem}[$2A_5$ cusp]
If $\mathcal{F}$ is an isotropic subspace in the summand $\mathcal{H}$ of $2\mathcal{E}_4+\mathcal{E}_1+\mathcal{H}= \Lambda$, then $\mathcal{F}^\perp/\mathcal{F}\cong 2\mathcal{E}_4+\mathcal{E}_1$. Hence $\mathcal{F}$ defines the cusp $c_{2A_5}$.
\end{lem}
\begin{proof}
This is clear.
\end{proof}
For the second case (cusp $c_{3D_4}$), the argument is more lengthy, and we begin with some preliminary discussion. We know that once we have found $\mathcal F$, then we will have $R((\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z})\cong 3E_6$, and in fact $(\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}$ is a lattice in the genus of $2E_8+A_2$ (see~\cite[Ch. 5]{scattone} and~\cite[\S6.1]{laza}). In other words, $(\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}$ is an index $3$ overlattice of $3E_6$. As noted in Lemma~\ref{lem_inv_iso}, a semi-stable cubic threefold with $3D_4$ singularities leads (via the Allcock--Carlson--Toledo construction of adding a new monomial $x_5^3$ to the defining equation) to a semi-stable cubic fourfold with $3\widetilde E_6$ singularities, which in turns leads to the $3E_6$ sublattice in the vanishing cohomology. Thus, we see that the order $3$ isometry $\rho$ on $(\mathcal{F}^\perp/\mathcal{F})_\mathbb{Z}$ defining the Eisenstein lattice $\mathcal{F}^\perp/\mathcal{F}$ is compatible with order $3$ isometries on each of the $E_6$ factors (giving $\mathcal{E}_3$ lattices). In other words, $\mathcal{F}^\perp/\mathcal{F}$ is an index $3$ overlattice of $3\mathcal{E}_3$. We will now define an index $3$ overlattice $\widetilde{3\mathcal{E}_3}$ of $3\mathcal{E}_3$ (a posteriori, indeed $\mathcal{F}^\perp/\mathcal{F}\cong \widetilde{3\mathcal{E}_3}$). We start with three copies of $\mathcal{E}_3$, or equivalently with three copies of $E_6(-1)$ each endowed with an isometry $\rho$ of order $3$ (fixing only the origin). By Nikulin theory, see~\cite[\S1.4]{nikulin}
there exists an index $3$ overlattice $\widetilde{3E_6(-1)}$ of $3E_6(-1)$.
Indeed, overlattices of $3E_6(-1)$ correspond to isotropic subgroups $H \subset A_{3E_6(-1)}$ of the discriminant group by taking the inverse image of $H$ under the projection $(3E_6(-1))^{\vee} \to A_{3E_6(-1)}$.
Here we take the subgroup $H$ generated by the diagonal embedding of $A_{E_6(-1)}$ into $A_{(3E_6(-1))}$. In fact, up to isometries of $3E_6(-1)$ this is the only isotropic subgroup.
Explicitly, we can find elements $z_i\in E_6(-1)^{(i)}$ (the $i^{th}$ copy) with $z_i^2=-12$ and $\operatornameeratorname{div} z_i=3$.
Then $\widetilde{3E_6(-1)}$ is the lattice generated by $3E_6(-1)$ and $\frac{z_1+z_2+z_3}{3}$ (inside $3E_6(-1)\otimes \mathbb{Q}$). Note also that for each of the $E_6(-1)$ components, the isometry $\rho$ acts trivially on the discriminant (simply, the automorphism group of the discriminant $A_{E_6(-1)}\cong \mathbb{Z}/3$ has order $2$, while $\rho$ has order $3$).
This means that $\rho$ (defined component-wise) on $3E_6(-1)$ extends to an isometry of $\widetilde{3E_6(-1)}$ whose only fixed point is the origin, thus giving the Eisenstein lattice $\widetilde{3\mathcal{E}_3}$ (recall the hermitian form is determined as in~\eqref{arr:hermitianform})
and in fact this
is the only such overlattice.
\begin{rem}\label{rem_discr_3e6}
Note that $z_i$ as above are chosen such that $\frac{z_i}{3}$ generate the discriminant of the respective copy of $E_6(-1)$.
The condition $\operatornameeratorname{div} z_i=3$
guarantees that $\frac{z_i}{3}\in E_6(-1)^\vee$, and then its projection into $A_{E_6(-1)}\cong \mathbb{Z}/3\mathbb{Z}$ is a generator. Note also that $z_i$ is divisible by $\theta$
when we view $E_6(-1)(=\mathcal{E}_3)$ as an Eisenstein lattice; indeed, we compute
$$\frac{1}{3} (\theta\cdot z_i) = \frac{1}{3}(\rho(z_i)-\rho^2(z_i))=\frac{\rho(z_i)}{3}-\frac{\rho^2(z_i)}{3}=0\in A_{E_6(-1)}\,,$$
or equivalently $\theta \cdot z_i=3v_i$ for some $v_i\in E_6(-1)^{(i)}$, and then $z_i=-\theta v_i$ (this relation makes sense even over $\mathbb{Z}$, by interpreting $\theta$ as the endomorphism $\rho-\rho^2$; over $\mathcal{E}$, $\rho$ is the multiplication $\omega$, and thus $\rho-\rho^2$ is the multiplication by $\theta\in \mathcal{E}$). Returning to $\widetilde{3E_6(-1)}$ and the companion Eisenstein lattice $\widetilde{3\mathcal{E}_3}$, we note that the discriminant of $\widetilde{3E_6(-1)}$ is $\mathbb{Z}/3$ and it is generated by the class of $\frac{z_1-z_2}{3}$. Furthermore, the following hold
\begin{itemize}
\item $(z_1-z_2)^2=z_1^2+z_2^2=-24$.
\item $z_1-z_2\in \widetilde {3E_6(-1)}$ is primitive and $\operatornameeratorname{div}(z_1-z_2)=3$ (even in $\widetilde{3E_6(-1)}$).
\item $(z_1-z_2)=\theta\cdot (v_1-v_2)$.
\end{itemize}
In terms of the Hermitian norm, note that $\|z_1-z_2\|^2=36$, and then $\|v_1-v_2\|^2=12$.
\end{rem}
The lattice $\widetilde{3\mathcal{E}_3}$ satisfies the following key property.
\begin{pro}\label{prop_3e6tilde}
There is an isomorphism of indefinite Eisenstein lattices
\begin{equation}
\Lambda(\cong \mathcal{E}_1 + 2\mathcal{E}_4 + \mathcal{H}) \cong \widetilde{3\mathcal{E}_3} + \mathcal{H}.
\end{equation}
\end{pro}
\begin{proof}
Let us first note that the underlying $\mathbb{Z}$-lattices are indeed isomorphic, i.e., forgetting the Eisenstein structure, it holds that
$$
A_2(-1) + 2E_8(-1) +2U \cong \widetilde{3E_6(-1)} +2U\,.$$
Indeed, the two lattices have the same signature and isomorphic discriminant groups (together with the quadratic from on it), thus they are in the same genus (see~\cite[Cor.~1.9.4]{nikulin}). Since the signature is indefinite, this genus contains only one element (see~\cite[Cor.~1.13.3]{nikulin}).
To lift this isometry to an isometry of Eisenstein lattices, we would need to know that (up to the action of the orthogonal group) there exists a unique Eisenstein structure on the $\mathbb{Z}$-lattice $A_2(-1) + 2E_8(-1) +2U$. We were not able to find such a result in the literature, but a related result is known: {\it an even indefinite unimodular lattice (e.g., $2E_8(-1) +2U$) admits at most one Eisenstein lattice structure} (see~\cite[Lem.~2.6]{basak}). Since $\mathcal{E}_1 + 2\mathcal{E}_4 + \mathcal{H}$ is the direct sum of a unimodular Eisenstein lattice $2\mathcal{E}_4 + \mathcal{H}$ (with underlying unimodular $\mathbb{Z}$-lattice $2E_8(-1)+2U$) and a rank $1$ lattice $\mathcal{E}_1$ spanned by a norm $3$ vector $v$, it suffices to find a vector $\widetilde{3\mathcal{E}_3} + \mathcal{H}$ such that $\|w\|^2=3$ and $(w)^\perp_{\widetilde{3\mathcal{E}_3} + \mathcal{H}}$ is unimodular. This in turn is equivalent to $\|w\|^2=3$ and $\operatornameeratorname{div} w=3$.
By the discussion of Remark~\ref{rem_discr_3e6}, one sees that $w'=v_1-v_2$ (with the notations of the remark) satisfies the right divisibility condition, but not the norm condition. However, we can correct the norm by taking $w=w'+\theta u$ with $u\in \mathcal{H}$ and $\|u\|^2=-3$. This completes the proof.
\end{proof}
As a consequence of the above proposition, we conclude:
\begin{cor}[$3D_4$ cusp]
If $\mathcal{F}$ is an isotropic subspace of the summand $\mathcal{H}$ of the sum
$\widetilde{3\mathcal{E}_3}+\mathcal{H}\cong \Lambda$, then $\mathcal{F}^\perp/\mathcal{F}\cong \widetilde{3\mathcal{E}_3}$. Hence $\mathcal{F}$ defines the cusp $c_{3D_4}$.
\end{cor}
\begin{proof}
This is clear.
\end{proof}
\subsection{Structure of the two boundary divisors of $\overline{\calB/\Gamma}$}
We denote the two boundary divisors of $\overline{\calB/\Gamma}$ corresponding to the cusps $c_{2A_5}$ and $c_{3D_4}$ by $T_{2A_5}$ and $T_{3D_4}$, respectively. To be able to treat both cusps simultaneously we write
\begin{equation}\label{equ:decomp}
\Lambda = \mathcal{G}+ \mathcal{H}
\end{equation}
where $\mathcal{G} = \mathcal{E}_1 + 2\mathcal{E}_4$ or $\mathcal{G}= \widetilde{3\mathcal{E}_3}$, respectively. As before, we can choose a rank $1$ primitive isotropic subspace $\mathcal{F}\subset \mathcal{H}$, and then $\mathcal{G} \cong \mathcal{F}^{\perp}/\mathcal{F}$.
We denote by $E_{\omega}$ the elliptic curve with an order $3$ automorphism and note that
$$
E_{\omega}= \mathbb{C}/\mathcal{E}.
$$
We can write
$$
(E_{\omega})^9= E_{\omega} \otimes_{\mathcal{E}} \mathcal{G}=\mathbb{C}^9/\mathcal{G}.
$$
This description defines a natural action of $\operatorname{O}(\mathcal{G})$ on the $9$-dimensional abelian variety $(E_{\omega})^9$. The aim of this subsection is to prove the following
\begin{pro}\label{prop_structure_tor}
The following holds:
\begin{itemize}
\item[(1)] $T_{2A_5} \cong (E_{\omega}\otimes_{\mathcal E}(\mathcal E_1+2\mathcal E_4))/\operatorname{O}( \mathcal{E}_1 + 2\mathcal{E}_4)\ \ (\cong (E_{\omega})^9/\operatorname{O}( \mathcal{E}_1 + 2\mathcal{E}_4))$;
\item[(2)] $T_{3D_4} \cong (E_{\omega}\otimes_{\mathcal E}\widetilde{3\mathcal E_3})/\operatorname{O}(\widetilde{3\mathcal{E}_3}) \ \ (\cong (E_{\omega})^9/\operatorname{O}(\widetilde{3\mathcal{E}_3}))$.
\end{itemize}
\end{pro}
\begin{proof}
We will give the proof for both cusps simultaneously. For this we pick an isomorphism as in (\ref{equ:decomp}) and an isotropic vector $h$
in $\mathcal{H}$.
As a matter of notation, by $F$ we will denote the cusp given by the isotropic line $\mathcal F=\mathcal Eh$.
Now we choose $b_1:=h$ and extend this to a basis of $\Lambda$ such that the
hermitian form with respect to this basis has the Gram matrix
$$
Q=
\left(
\begin{array}{c|c|c}
0 & 0 & \theta \\ \hline
0 & B & 0\\ \hline
\bar \theta & 0 & 0\rule{0pt}{2.6ex}
\end{array}
\right)
$$
Here $b_2, \dots, b_{10}$ form a basis of $\mathcal{G}$, and $B$ is the Gram matrix of $\mathcal{G}$ with respect to this basis. In order to understand the boundary we first have to determine
certain subgroups of $\operatorname{O}(\Lambda)$ related to the cusp $F$ (here we will only be dealing with the integral groups).
The first is the stabilizer subgroup $N(F)$ corresponding to $F$, i.e.~the subgroup of $\operatorname{O}(\Lambda)$ fixing the line spanned by $h$. A straightforward calculation, see~\cite[Sec.~4]{beh}, gives
\begin{equation}\label{pro:structureboundarycomp}
N(F)= \left\{ g \in \operatorname{O}(\Lambda): g= \left( \begin{array}{c|c|c}
u & v & w \\ \hline
0 & X & y \\ \hline
0 & 0 & s
\end{array}
\right)\right\}.
\end{equation}
Note that, in particular, this implies that $X\in \operatorname{O}(\mathcal{G})$. Its unipotent radical is given by
\begin{equation}
W(F)= \left\{g \in N(F): g= \left(
\begin{array}{c|c|c}
1 & v & w \\ \hline
0 & 1 & y \\ \hline
0 & 0 & 1
\end{array}
\right) \right\}
\end{equation}
and finally the center of the unipotent radical is
\begin{equation}
U(F)= \left\{g\in W(F): g= \left(
\begin{array}{c|c|c}
1 & 0 & w \\ \hline
0 & 1 & 0 \\ \hline
0 & 0 & 1
\end{array}
\right), w \in \mathbb{Z} \right\} \cong \mathbb{Z}.
\end{equation}
We have already introduced coordinates $(z_0:z_1: \dots : z_{10})$ on $\mathcal{B} \subset \mathbb{P}(\Lambda\otimes_{\mathcal{E}}\mathbb{C})$ and we can assume that $z_{10}=1$.
The first step in the toroidal compactification is to take the partial quotient of $\mathcal{B}$ by $U(F)$. This is given by
\begin{equation}
\begin{aligned} \mathcal{B} &\to \mathbb{C}^* \times \mathbb{C}^9 \\ (z_0, \dots, z_{9}) &\mapsto (t_0=e^{2 \pi i z_0}, z_1, \dots, z_9).
\end{aligned}
\end{equation}
Adding the toroidal boundary means adding the divisor $\{0\} \times \mathbb{C}^9$, and we will use $z_1, \dots, z_9$ as coordinates on this boundary divisor. The quotient $N(F)/U(F)$ then acts on $\mathcal{B}/U(F)$ and this quotient gives the
toroidal compactification of $\mathcal{B}$ near the cusp $F$. Here we are only interested in the structure of the boundary divisor and hence in the action of $N(F)/U(F)$ on $\{0\} \times \mathbb{C}^9$. A straightforward calculation shows that
\begin{equation}\label{equ:action}
g=\left(
\begin{array}{c|c|c}
u & v & w \\ \hline
0 & X & y \\ \hline
0 & 0 & s
\end{array}
\right): \underline{z} \mapsto \frac{1}{s}(X\underline{z} + y)
\end{equation}
where $\underline{z}=(z_1, \dots, z_9)$. We first look at the normal subgroup $W(F)$, matrices whose elements act as follows
$$
g=\left(
\begin{array}{c|c|c}
1 & v & w \\ \hline
0 & 1 & y \\ \hline
0 & 0 & 1
\end{array}
\right): \underline{z} \mapsto \underline{z} + y.
$$
Since $g\in \operatorname{O}(\Lambda)$ we have $y \in \mathcal{E}^9$ and we claim that all vectors in $\mathcal{E}^9$ appear as entries in matrices $g\in W(F)$. Indeed a straightforward calculation, see~\cite[Sec.~4]{beh}, shows that the condition that $g\in \operatorname{O}(\Lambda)$ is
$$
By+\bar{v}^t\theta=0, \quad \bar{y}^tBy+\bar{\theta}w + \theta \bar w=0.
$$
Given $y$ we define $v$ by $\bar{v}^t=-\frac{1}{\theta}By$. This is in $\mathcal{E}^9$ since the coefficients of $By$ have values in $\theta \mathcal{E}$.
Finally, we must check that we can find a suitable element $w \in \mathcal{E}$.
We know that $\bar{y}^tBy \in 3\mathbb{Z}$
and hence we can write $\bar{y}^tBy=3n$ for some integer $n$.
Hence we can take $w= -\frac12 + \frac{i}{2} \sqrt{3}n \in \mathcal{E}$ if $n$ is odd and $w= \frac{i}{2} \sqrt{3}n \in \mathcal{E}$ if $n$ is even, respectively . This shows that
$$
\mathbb{C}^9/W(F)\cong (E_{\omega})^9.
$$
Next we consider the action of the subgroup
$$
\left\{ g \in \operatorname{O}(\Lambda): g= \left( \begin{array}{c|c|c}
1 & 0 & 0 \\ \hline
0 & X & 0 \\ \hline
0 & 0 & 1
\end{array}
\right)\right\}.
$$
which acts on $(E_{\omega})^9$ as claimed in the proposition.
It remains to consider elements of the form
$$
g=\left(
\begin{array}{c|c|c}
u & 0 & 0 \\ \hline
0 & 1 & 0 \\ \hline
0 & 0 & s
\end{array}
\right) \in N(F).
$$
The condition that such a matrix lies in $\operatorname{O}(\Lambda)$ is that $s\bar{u}=1$ with $s\in \mathcal{E}$. Hence $s$ is a power of $\omega$ and these elements act on $(E_{\omega})^9$ by multiplication with powers of $\omega$. But by
(\ref{equ:action}) these elements are already in
$\operatorname{O}(\mathcal{G})$ and hence we do not get a further quotient, and the claim follows.
\end{proof}
\subsection{Isometry groups associated to the two cusps} We now discuss the isometry groups $\operatornameeratorname{O}(2\mathcal{E}_4+\mathcal{E}_1)$ and $\operatorname{O}(\widetilde{3\mathcal{E}_3})$ associated to the two cusps $c_{2A_5}$ and $c_{3D_4}$ respectively (see Proposition~\ref{prop_structure_tor}). As we will discuss below, these groups are easily determined once the isometry groups of the basic lattices $\mathcal{E}_3$ and $\mathcal{E}_4$ are understood.
It turns out that the lattices $\mathcal{E}_3$ and $\mathcal{E}_4$ are special lattices, they are ``root lattices'' in the sense of Eisenstein lattices. Consequently, the associated isometry groups $\operatorname{O}(\mathcal{E}_i)$ are essentially the complex reflections $W(\mathcal{E}_i)$ generated by the roots ($W(\mathcal{E}_i)$ is the analogue for Eisenstein lattices of the usual Weyl group).
To start our discussion of the isometry groups, let us recall that the role of reflections is taken by {\em triflections}. First, an {\it Eisenstein root} is an element $r\in \mathcal{G}$ with $\|r\|^2=\langle r, r \rangle =3$. Note that $r$, when viewed as an element of
$\mathcal{G}_\mathbb{Z}$, is then a root in the usual sense, i.e. $r^2=(r,r)=-2$.
The role of the $(-2)$-reflections is taken by {\em triflections}. To explain these, let $r$ be an Eisenstein root in the above sense and
define
$$
R_r: x \mapsto x - (1-\omega) \frac{\langle r,x \rangle }{\langle r,r \rangle }x.
$$
This defines an isometry of order $3$ (called a triflection). For an Eisenstein lattice $\mathcal{G}$, we define
$$W(\mathcal{G})\subset \operatorname{O}(\mathcal{G})$$
to be the subgroup of isometries generated by triflections $R_r$ in all roots $r$ defined above. Note that it follows, for instance, that $W(\mathcal{E}_3)$ is the subgroup of $W(E_6)$ consisting of the elements that commute with $\rho$ (similar statements hold for all $\mathcal{E}_n$ for $n=1,\dots,4$).
We will now discuss the isometry group of the relevant Eisenstein lattices.
Clearly
\begin{equation}\label{E:OOE1}
\operatorname{O}(\mathcal{E}_1) = W(\mathcal{E}_1) \times \mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}
\end{equation}
where the factors are generated by multiplication by $\omega$ and $-1$, respectively. The (complex) Weyl groups $W(\mathcal{E}_3)$ and $W(\mathcal{E}_4)$ are well known complex reflection groups, typically denoted by $L_3$ and $L_4$ (see~\cite[Table 2]{Dolgachev-ref}), and described as follows.
\begin{pro}\label{propwe3}
The following holds:
\begin{itemize}
\item[(1)]
The Weyl group $W(\mathcal{E}_3)$ is a group of order $2^3 \cdot 3^4=648$. It is isomorphic to $\operatornameeratorname{U}(3,\mathbb{F}_4)$, respectively a semidirect product of an extra special group of order $27$ and exponent $3$ with
$\operatornameeratorname{SL}(2,\mathbb{F}_3)$.
\item[(2)] $\operatorname{O}(\mathcal{E}_3) \cong W(\mathcal{E}_3) \rtimes \mathbb{Z}/2\mathbb{Z}$.
\end{itemize}
\end{pro}
\begin{proof}
For the first item see for example~\cite[Thm.~8.42]{LT}. For the second item, we recall that the underlying $\mathbb{Z}$-lattice is $E_6(-1)$. It is well known that $\operatorname{O}(E_6(-1))=W(E_6(-1))\rtimes \mathbb{Z}/2\mathbb{Z}$, with the $\mathbb{Z}/2\mathbb{Z}$ factor corresponding to the outer automorphism $\tau$ given by the symmetry of the Dynkin diagram. To recover the groups $W(\mathcal{E}_3)$ and $\operatorname{O}(\mathcal{E}_3)$, we note that the Eisenstein structure on $E_6(-1)(=\mathcal{E}_3)$ determines an order $3$ element $\rho\in W(E_6(-1))$. Then, $W(\mathcal{E}_3)$ can be obtained as the centralizer of $\rho$ in $W(E_6(-1))$ (e.g.~\cite[Ex. 9.5]{Dolgachev-ref}). Similarly, $\operatorname{O}(\mathcal{E}_3)$ is the centralizer of $\rho$ in $\operatorname{O}(E_6(-1))$ (see~\eqref{eq_rel_iso}). Finally, a direct calculation shows that
$\tau$ commutes with the order $3$ automorphism $\rho$ which defines the Eisenstein structure on $E_6(-1)$. Item (2) follows.
\end{proof}
\begin{pro}\label{propwe4}
The following holds:
\begin{itemize}
\item[(1)] The Weyl group $W(\mathcal{E}_4)$ has order $2^7 \cdot 3^5 \cdot 5= 155,520$. It is isomorphic to $\mathbb{Z}/3\mathbb{Z} \times \operatornameeratorname{Sp}(4,\mathbb{F}_3)$.
\item[(2)] $W(\mathcal{E}_4)=\operatorname{O}(\mathcal{E}_4)$.
\end{itemize}
\end{pro}
\begin{proof} This follows from~\cite[Thm. 5.2]{ALeech} (see also~\cite[Thm.~8.43]{LT}).
\end{proof}
With these preliminaries, we can now conclude the computation of the groups of isometries occurring for the two cusps of the ball quotient model.
\begin{pro} \label{lem:autoEisenstein}
The following holds:
\begin{itemize}
\item[(1)] $\operatorname{O}(\mathcal{E}_1 + 2 \mathcal{E}_4) \cong W(\mathcal{E}_1) \times (W(\mathcal{E}_4)^{\times 2} \rtimes S_2)$
\item[(2)] $\operatorname{O}(\widetilde{3\mathcal{E}_3}) \cong (W(\mathcal E_3)^{\times 3} \rtimes S_3) \rtimes \mathbb{Z}/2\mathbb{Z}$.
\end{itemize}
Here $S_n$ denotes the symmetric group in $n$ elements.
\end{pro}
\begin{proof}
Let $\mathcal{G}$ be a definite Eisenstein lattice. We start with two basic remarks. Firstly, the set of Eisenstein roots $\mathcal{R}(\mathcal{G})$ is finite, and in fact (under our scaling assumptions) coincides (set-theoretically) with the set of $-2$ roots for the
$\mathbb{Z}$-lattice $\mathcal{G}_\mathbb{Z}$.
Indeed, this follows from~\eqref{arr:hermitianform}.
Secondly, any isometry $\phi\in \operatorname{O}(\mathcal{G})$ preserves the set of roots ($\phi(\mathcal{R}(\mathcal{G}))=\mathcal{R}(\mathcal{G})$). Furthermore, if the set of roots $\mathcal{R}(\mathcal{G})$ generates $\mathcal{G}$ (over $\mathbb{Q}(\omega)$), then $\phi$ is determined uniquely by the action of $\phi$ on the finite set $\mathcal{R}(\mathcal{G})$. In our situation, $\mathcal{G}=2\mathcal{E}_4+\mathcal{E}_1$ or $\mathcal{G}=\widetilde{3\mathcal{E}_3}$, and we know $R(\mathcal{G}_\mathbb{Z})=2E_8(-1)+ A_2(-1)$ and $R(\mathcal{G}_\mathbb{Z})=3E_6(-1)$ respectively (see Lemma~\ref{lem_inv_iso}), where we denote by $R(\mathcal{G}_\mathbb{Z})$ the sublattice of $\mathcal{G}_\mathbb{Z}$ spanned by the roots $\mathcal{R}(\mathcal{G}_\mathbb{Z})$. Thus, given $\phi\in \operatorname{O}(2\mathcal{E}_4+\mathcal{E}_1)$ (or $\phi\in \operatorname{O}(\widetilde{3\mathcal{E}_3})$ respectively), after a permutation (giving the $S_2$ and $S_3$ factors above), we can assume that $\phi$ preserves the irreducible root summands $2\mathcal{E}_4+\mathcal{E}_1$ (and respectively each of the three $\mathcal{E}_3$ in $\widetilde{3\mathcal{E}_3}$). Then the isometry $\phi$ is determined by the action of $\phi$ on the summands $\mathcal{E}_1$, $\mathcal{E}_4$ and $\mathcal{E}_3$ respectively (which were described in Propositions~\ref{propwe4} and~\ref{propwe3}); see also~\cite[Lem.~1]{HKN} for a related argument.
This shows immediately that the isometries of $\mathcal{E}_1 + 2 \mathcal{E}_4$ are as claimed. We have also seen that any isometry of $\widetilde{3\mathcal{E}_3}$ is the extension of an isometry of $3\mathcal{E}_3$. Now, an isometry of $3\mathcal{E}_3$ lifts to
$\widetilde{3\mathcal{E}_3}$ if and only if the induced isometry on the discriminant preserves the defining subgroup $H\subset A_{3E_6(-1)}\cong (\mathbb{Z}/3)^3$. This immediately shows that $W(\mathcal E_3)^{\times 3} \rtimes S_3 \subset \operatorname{O}(\widetilde{3\mathcal{E}_3})$.
To complete the proof we recall from Proposition~\ref{propwe3} that $\operatorname{O}(\mathcal{E}_3) \cong W(\mathcal{E}_3) \rtimes \mathbb{Z}/2\mathbb{Z}$ where the $\mathbb{Z}/2\mathbb{Z}$-factor is generated by the involution $\tau$ given by the symmetry of the Dynkin diagram. Since $\tau$ acts by $-1$ on
$D(E_6(-1))\cong \mathbb{Z}/3\mathbb{Z}$ it follows that only the identity and the diagonal element $(\tau,\tau,\tau)$ extend to $\widetilde{3\mathcal{E}_3}$. This gives the extra factor of $\mathbb{Z}/2\mathbb{Z}$ and completes the proof.
\end{proof}
\section{The cohomology of the toroidal boundary divisors} We will compute the cohomology of the toroidal compactification $\overline{\calB/\Gamma}$ by applying the decomposition theorem to the map $\overline{\calB/\Gamma} \to (\mathcal{B}/\Gamma)^*$ and using our knowledge of the cohomology of $(\mathcal{B}/\Gamma)^*$ (see \S~\ref{sec:IHball}).
For this we require the knowledge of the cohomology of the two toroidal boundary divisors $T_{2A_5}$ and $T_{3D_4}$.
Since the lattices involved in the definition of the two exceptional divisors (see Proposition~\ref{prop_structure_tor}) are essentially direct sums of $\mathcal{E}_3$ and $\mathcal{E}_4$ lattices, the main ingredient needed to compute the cohomology of $T_{2A_5}$ and $T_{3D_4}$ respectively is the cohomology of the spaces $(E_\omega\otimes_{\mathcal{E}} \mathcal{E}_k)/W(\mathcal{E}_k)$ for $k=3,4$. These are quotients of abelian varieties by finite groups, and it turns out that these spaces are equivariantly isogenous to weighted projective spaces of dimension $k$ (see Proposition~\ref{thm_L_Eis} and~\eqref{E:Eisen-Looij}, below, for a precise statement), and thus they have simple cohomology, making the remaining computations routine.
\subsection{An Eisenstein analogue of Chevalley's theorem} \label{S:Eies-Chev}
The fact that $(E_\omega\otimes_{\mathcal{E}} \mathcal{E}_k^*)/W(\mathcal{E}_k)$ for $k=3,4$ have the same cohomology as weighted projective spaces of dimension $k$ is an analogue over $\mathcal{E}$ of a Chevalley type Theorem due to Looijenga~\cite{Lroot}. Specifically, we recall that if $R$ is an irreducible ADE root lattice (we make this assumption for simplicity) and $E$ is an elliptic curve, then $(E\otimes_\mathbb{Z} R^\vee)/W(R)\cong W\mathbb{P}^r$, where $W\mathbb{P}^r$ denotes a weighted projective space of dimension $r$ equal to the rank of $R$ (see~\cite[Thm. 2.7]{FMW}). In the Eisenstein case, we get that the quotients are equivariantly isogenous to weighted projective spaces; in particular we obtain:
\begin{pro}\label{thm_L_Eis}
Let $E_\omega$ be the elliptic curve with $j$-invariant $0$. Then:
\begin{itemize}
\item[(1)] $H^\bullet((E_\omega\otimes_\mathcal{E} \mathcal{E}_3)/W(\mathcal{E}_3))\cong H^\bullet(W\mathbb{P}(1,2,2,3))$
\item[(2)] $H^\bullet((E_\omega\otimes_\mathcal{E} \mathcal{E}_4)/W(\mathcal{E}_4))\cong H^\bullet(W\mathbb{P}(2,3,4,5,6))$.
\end{itemize}
\end{pro}
\begin{proof}
Chevalley type theorems for complex reflection groups acting on projective varieties were obtained by Bernstein and Schwarzman~\cite{BS}. In particular, the fact that the cohomology of $(E_\omega\otimes \mathcal{E}_k)/W(\mathcal{E}_k)$ agrees with that of weighted projective space (in much more generality) follow from~\cite[\S2.3]{BS}.
For the reader's convenience, we sketch a geometric proof of the two cases that are needed in our paper, following the outline of~\cite{FMW}. For simplicity, we will discuss only the $\mathcal{E}_3$ case, the other case being obtained by minor changes. Let us discuss first the situation over $\mathbb{Z}$ (i.e., the classical setup of Looijenga), namely the statement that $(E\otimes_\mathbb{Z} E_6^\vee)/W(E_6)\cong W\mathbb{P}(1,1,1,2,2,2,3)$ (N.B.~for the moment, $E$ is any (fixed) elliptic curve).
We consider the moduli of anticanonical pairs\footnote{In general, {\it an anticanonical pair} $(S,D)$ is a rational surface $S$ together with a reduced anticanonical cycle $D\in |-K_S|$. Clearly, $D$ is either a smooth elliptic curve (as in our case) or a cycle of rational curves. The latter case is sometimes known also as {\it Looijenga pair}. The moduli of anticanonical pairs is well understood: the case when $D$ is smooth is essentially reviewed here, while the harder case when $D$ is singular is treated in~\cite{GHK},~\cite{F13}.}, \emph{with $E$ fixed},
$$\mathcal{P}_E=\{(S,E)\mid S \textrm{ is a degree $3$ del Pezzo surface, and }E\in|-K_S|\}\,.$$
This moduli space has two different descriptions. On the one hand, as an instance of Pinkham's general theory of deformations of singularities with $\mathbb{C}^*$-action (see~\cite{pinkham}), we obtain
a GIT description as $\mathcal{P}_E=B_{-}/\!\!/ \mathbb{C}^*$, where $B_{-}\cong \mathbb{C}^7$ is the negative weight deformation space for the versal deformation
of the singularity of type $\widetilde E_6$ (the affine cone over the elliptic curve $E\subset \mathbb{P}^2$). (In general, the versal deformation space of a singularity is a germ. However, for singularities with $\mathbb{C}^*$-action, there is an induced $\mathbb{C}^*$-action, which allows one to globalize the negative weight subspace. Thus, in the case of quasi-homogeneous hypersurface singularities, one can take $B_{-}$ to be an affine space.)
The versal deformations of $\widetilde E_6$ are easily described explicitly, and as a consequence one gets
\begin{equation}\label{wp_iso}
\mathcal{P}_E\cong W\mathbb{P}(1,1,1,2,2,2,3).
\end{equation}
(We refer to~\cite{rthesis} for further related discussion.) On the other hand, one gets a period map
\begin{eqnarray}\label{pe_quot}
\Phi_E:\mathcal{P}_E&\to&\operatornameeratorname{Hom}_\mathbb{Z}(E_6(-1),E)/W(E_6)\\
(S,E)&\to& H^2(S,E)\notag
\end{eqnarray}
which can be explained as follows: Firstly, $W(E_6)$ is the monodromy group acting on the primitive cohomology $E_6(-1)\cong H^2(S,\mathbb{Z})_{0}$ of the del Pezzo surface $S$. Secondly, from the exact sequence of a pair one gets
$$
0\to H^1(E)\to H^2(S,E)\to H^2(S)_0\to 0,
$$
where we are using that $H^2(S)_0=\ker (H^2(S)\to H^2(E))$. This shows that the mixed Hodge structure on $H^2(S,E)$ is an extension of a trivial weight $2$ Hodge structure (type $(1,1)$) by the elliptic curve $E$. Carlson~\cite{carlson} showed that these type of extensions are classified by $\operatornameeratorname{Hom}_\mathbb{Z}(H^2(S)_0,E)$. Since the elliptic curve $E$ is fixed, the monodromy $W(E_6)$ acts naturally on $H^2(S)_0\cong E_6(-1)$ and on $\operatornameeratorname{Hom}_\mathbb{Z}(H^2(S)_0,E)$. Thus, the period space (i.e., the period domain modulo monodromy) for $\Phi_E$ is $\operatornameeratorname{Hom}_\mathbb{Z}(E_6(-1),E)/W(E_6)$,
showing that the period map above is well defined. Let us then note that in fact the period map $\Phi_E$ has an easy geometric description. Namely, by identifying $H^2(S)_0$ with $\operatornameeratorname{Pic}(S)_0$, i.e., degree $0$ line bundles on $S$ with respect to the polarization $-K_S$, and $E$ with $\operatornameeratorname{Pic}^0(E)$, one sees that the ``period point''
$$\Psi:=\Phi_E(S,E)\in \operatornameeratorname{Hom}_\mathbb{Z}(H^2(S)_0,E)$$ is just the natural restriction morphism
\begin{eqnarray}
\Psi:\operatornameeratorname{Pic}(S)_0&\to& \operatornameeratorname{Pic}^0(E)\notag\\
\mathcal L&\to&\mathcal L_{\mid E}.\label{eq_psi}
\end{eqnarray}
Finally, an easy Torelli type theorem (essentially, the surface $S$ is the blow-up of $6$ points in $\mathbb{P}^2$ which lie on a smooth cubic curve $C\cong E$) establishes that $\Phi_E$ is an isomorphism. (We refer to~\cite{carlson} and~\cite{Fannals} for details of the period map construction and the Torelli theorem. In particular, we note that our case is one of the main examples in~\cite{carlson}.) Comparing the GIT~\eqref{wp_iso} and Hodge theoretic~\eqref{pe_quot} descriptions of $\mathcal{P}_E$, one obtains the claimed result $(E\otimes_\mathbb{Z} E_6^\vee)/W(E_6)\cong W\mathbb{P}(1,1,1,2,2,2,3)$.
Returning to our situation, i.e., the Eisenstein analogue of the above argument, we proceed as follows. We specialize to the case $E=E_\omega$, and we define $\mathcal{P}_{E_\omega}^\omega$ to be the subspace of $\mathcal{P}_{E_\omega}$ corresponding to pairs $(S,E_\omega)$ for which the $\mu_3$-action on $E_{\omega}$ extends (linearly) to $S$. It is easy to see that we can choose a normal form for $S$ as follows
$$S=V((x^3+y^3+z^3)+a_1txy+a_2 t^2x+a_3t^2y+a_4t^3)\subset \mathbb{P}^3\,,$$
with the elliptic curve $E_\omega$ being the hyperplane at infinity $(t=0)$.
Since $E_\omega$ is fixed, the only transformation allowed is the rescaling of $t$. This shows that $\mathcal{P}_{E_\omega}^\omega\cong W\mathbb{P}(1,2,2,3)$ (and this is a natural subspace of $\mathcal{P}_{E_\omega}\cong W\mathbb{P}(1,1,1,2,2,2,3)$).
Let us now discuss the restriction of the period map $\Phi_{E_\omega}$ to the subspace $\mathcal{P}_{E_\omega}^\omega\subset \mathcal{P}_{E_\omega}$. By definition $\mathcal{P}_{E_\omega}^\omega$ is the locus of pairs $(S,E_\omega)$ that admit an order $3$ automorphism $f$. Since $f$ preserves $K_S$, we see that $f^*$ acts as an order $3$ isometry, call it $\rho$, on $H^2(S)_0\cong E_6(-1)$. On the other hand, the restriction of $f$ to $E_\omega$ acts as multiplication by $\omega$. Since $f$ acts compatibly on the pair $(S,E)$, we get $(f^*\mathcal L)_{\mid E}=f^*(\mathcal L_{\mid E})$, which in turn is equivalent to saying (compare~\eqref{eq_psi})
$$\Psi(\rho(\mathcal L))=\omega\cdot \Psi(\mathcal L)\,.$$
We conclude that the period domain for the restricted period map $\Phi_{E_\omega \mid \mathcal{P}_{E_\omega}^\omega}$ is the $\omega$-eigenspace in $(E_\omega\otimes_\mathbb{Z} E_6^\vee)$ (w.r.t.~the action induced by $\rho$ on the second factor).
In short we have
\begin{equation}\label{E:Eisen-Looij}
W\mathbb{P}(1,2,2,3)\cong \mathcal{P}_{E_\omega}^\omega\cong (E_\omega\otimes_\mathbb{Z} E_6^\vee)_\omega/W(\mathcal{E}_3),
\end{equation}
where the subscript indicates the $\omega$-eigenspace.
We now recall the identification $E_6^\vee=((\mathcal E_3)_{\mathbb Z})^\vee = (\mathcal E_3')_{\mathbb Z}$ from Remark~\ref{R:DualEisen}. By considering the eigenspaces for the $\omega$ action on the right factor of $\mathcal E \otimes_{\mathbb Z}(\mathcal E_3')_{\mathbb Z}$, we obtain inclusions $\mathcal E_3 \hookrightarrow \mathcal E_3'\hookrightarrow (\mathcal E\otimes_\mathbb{Z} (\mathcal E_3')_{\mathbb Z})_\omega= (\mathcal E\otimes_\mathbb{Z} E_6^\vee)_\omega$, with torsion co-kernels. Tensoring with $E_\omega \otimes_{\mathcal E}-$, we obtain isogenies
$E_\omega\otimes_{\mathcal E}\mathcal E_3 \twoheadrightarrow E_\omega \otimes_{\mathcal E}\mathcal E_3'\twoheadrightarrow (E_\omega\otimes_{\mathbb Z} E_6^\vee)_\omega$, which are $W(\mathcal E_3)$-equivariant. Since isogenies give isomorphisms on the cohomology of abelian varieties with rational coefficients, we have the identifications $H^\bullet (E_\omega\otimes_{\mathcal E}\mathcal E_3) /W(\mathcal E_3))=H^\bullet (E_\omega\otimes_{\mathcal E}\mathcal E_3))^{W(\mathcal E_3)}=H^\bullet ((E_\omega\otimes _{\mathbb Z}E_6^\vee )_\omega)^{W(\mathcal E_3)}=H^\bullet ((E_\omega\otimes _{\mathbb Z}E_6^\vee )_\omega/{W(\mathcal E_3)})=H^\bullet (W\mathbb{P}(1,2,2,3))$.
\end{proof}
\subsection{The cohomology of the divisors $T_{2A_5}$ and $T_{3D_4}$}
We are now ready to compute the topology of the toroidal boundary components.
The result is the following
\begin{pro}\label{pro:cohomoloytorbound}
The cohomology of the toroidal boundary divisors $T_{2A_5}$ and $T_{3D_4}$ is given by the following table:
\begin{equation}
\renewcommand*{\arraystretch}{1.3}
\begin{array}{r|ccccccccccc}
j&0&2&4&6&8&10&12&14&16&18\\\hline
\dim H^j(T_{2A_5})&1&2&3&4&5&5&4&3&2&1\\
\dim H^j(T_{3D_4})&1&1&2&3&3&3&3&2&1&1\\
\end{array}
\end{equation}
All odd cohomology vanishes.
\end{pro}
\begin{proof}
As discussed in Proposition~\ref{thm_L_Eis}, the quotients $(E_{\omega}\otimes_\mathcal{E} \mathcal{E}_i)/W(\mathcal{E}_i)\cong(E_\omega)^i/W(\mathcal{E}_i)$ (for $i=3,4$) have the cohomology of weighted projective spaces. Hence, as graded vector spaces, we have
\begin{equation}
H^\bullet((E_{\omega})^i/W(\mathcal{E}_i),\mathbb{Q}) \cong \mathbb{Q}[x]/(x^{i+1}).
\end{equation}
We shall first treat the case $T_{2A_5}$. It follows from Proposition~\ref{pro:structureboundarycomp} and Lemma~\ref{lem:autoEisenstein} that
\begin{equation}
T_{2A_5} \cong E_{\omega}/W(\mathcal{E}_1) \times ((E_{\omega})^4/W(\mathcal{E}_4))^{\times 2}/S_2.
\end{equation}
We shall first compute the cohomology of the second factor. For this we have to consider the $S_2$ invariant parts of a tensor product
$\mathbb{Q}[x]/(x^{i+1}) \otimes \mathbb{Q}[y]/(y^{i+1})$. The invariants in each degree are given by $1$ in degree $0$, $x+y$ in degree $1$, $x^2 + y^2,xy$ in degree $2$, $x^3+y^3, x^2y + xy^2$ in degree $3$, and
$x^4+y^4,x^3y + xy^3, x^2y^2$ in degree $4$. Hence, using Poincar\'e duality we see that all the odd cohomology vanishes, and that the entire cohomology is equal to
\begin{equation}
P_t\left(\left((E_{\omega})^4/W(\mathcal{E}_4)\right)^{\times 2}/S_2\right)=1+t^2+2t^4+2t^6+3t^8+2t^{10}+2t^{12}+t^{14}+t^{16}.
\end{equation}
The cohomology of the first factor is that of $\mathbb{P}^1$, equal to $1+t^2$, and an application of the K\"unneth formula therefore gives
\begin{equation}
\begin{aligned}
P_t(T_{2A_5})&=(1+t^2)\cdot(1+t^2+2t^4+2t^6+3t^8+2t^{10}+2t^{12}+t^{14}+t^{16})\\
&=1+2t^2+3t^4+4t^6+5t^8+5t^{10}+4t^{12}+3t^{14}+2t^{16}+t^{18}.
\end{aligned}
\end{equation}
We shall now treat the second boundary component $T_{3D_4}$. We first note that the inclusion $3\mathcal{E}_3 \subset \widetilde{3\mathcal{E}_3}$ gives us an \'etale $3:1$ map
\begin{equation}\label{equ:covering}
\mathbb{C}^9/3\mathcal{E}_3 \cong (E_{\omega})^9 \to \mathbb{C}^9/ \widetilde{3\mathcal{E}_3}.
\end{equation}
To compute the cohomology of $T_{3D_4}$ is equivalent to computing the invariant cohomology of $\mathbb{C}^9/ \widetilde{3\mathcal{E}_3}$ under the group
$\operatorname{O}(\widetilde{3\mathcal{E}_3}) \cong (W(\mathcal E_3)^{\times 3} \rtimes S_3) \rtimes \mathbb{Z}/2\mathbb{Z}$. Since the covering group of the \'etale $3:1$ map (\ref{equ:covering})
acts by translation on the product of elliptic curves, and hence trivially on cohomology,
this is equivalent
to computing the invariant cohomology of $\mathbb{C}^9/3\mathcal{E}_3 \cong (E_{\omega})^9$. We will first restrict to the subgroup $W(\mathcal E_3)^{\times 3} \rtimes S_3$.
Again using the fact that each factor $(E_{\omega})^3/W(\mathcal{E}_3)$ has the cohomology of a weighted projective space, and counting invariants under the symmetry group $S_3$ as above, we obtain for the invariant cohomology
\begin{equation}
P_t\left((E_\omega)^9\right)^{W(\mathcal E_3)^{\times 3} \rtimes S_3}=1+t^2+2t^4+3t^6+3t^8+3t^{10}+3t^{12}+2t^{14}+t^{16}+t^{18}.
\end{equation}
It remains to consider the action of the outer automorphism $\tau$
(see the proof of Proposition~\ref{lem:autoEisenstein}),
which acts diagonally on the triple product. Note, however, that $(E_{\omega})^3/W(\mathcal{E}_3)$ has $1$-dimensional cohomology
in even degree and no odd cohomology. Hence $\tau$ acts trivially on the cohomology of $(E_{\omega})^3/W(\mathcal{E}_3)$, and this finishes the proof.
\end{proof}
\begin{rem}
Note that the Betti numbers of $T_{3D_4}$ and $T_{2A_5}$ agree with those of $D_{3D_4}$ and $D_{2A_5}$, given by formulas~\eqref{E:g3D4-contr} and~\eqref{E:g2A5-contr}, respectively.
\end{rem}
\section{The cohomology of the toroidal compactification}
At this point, we can conclude the computation of the cohomology of the toroidal compactification $\overline{\calB/\Gamma}$. This completes the proof of our main Theorem~\ref{teo:betti}:
\begin{teo}\label{teo:coh_obg}
The cohomology of the toroidal compactification $\overline{\calB/\Gamma}$ of the ball quotient is given by
\begin{equation}
\renewcommand*{\arraystretch}{1.5}
\begin{array}{r|ccccccccccc}
j&0&2&4&6&8&10&12&14&16&18 &20\\\hline
\dim H^j(\overline{\calB/\Gamma})&1&4&6&10&13&15&13&10&6&4&1
\end{array}
\end{equation}
All odd cohomology vanishes.
\end{teo}
\begin{proof}
We recall that the toroidal compactification $\overline{\calB/\Gamma}$ is smooth up to finite quotient singularities, and that the morphism $\overline{\calB/\Gamma}\to(\calB/\Gamma)^*$ is the blowup of two points. Hence we can
apply the decomposition theorem in the form of~\cite[Lem.~9.1]{GH-IHAg-17}, i.e.~we are in the special case of \S\ref{S:Decomp-Thm}, to the morphism $\overline{\calB/\Gamma} \to (\calB/\Gamma)^*$. We thus compute
\begin{align*}
&P_t(\overline{\calB/\Gamma}) \equiv\\
&\equiv 1+2t^2+3t^4 +5t^6 +6t^8 +7t^{10}&\text{($IP_t((\calB/\Gamma)^*)$, from the previous section)}\\
&\ \ \ \ + t^2 +t^4 +2t^6 +3t^8 +3t^{10}&\text{($T_{3D_4}$ contribution, from Proposition~\ref{pro:cohomoloytorbound})}\\
&\ \ \ \ + t^2 +2t^4 +3t^6 +4t^8 +5t^{10}&\text{($T_{2A_5}$ contribution, from Proposition~\ref{pro:cohomoloytorbound})}\\
&\equiv 1+4t^2+6t^4 +10t^6 +13t^8 +15t^{10}\mod t^{11}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\end{align*}
by applying equation~\eqref{eq:IHblowup} to determine the contribution to the cohomology of $\overline{\calB/\Gamma}$ from each of the two exceptional divisors.
\end{proof}
\appendix
\chapter{Equivariant cohomology}\label{sec:equivcoh}
In this appendix we review a few basic facts from the theory of equivariant cohomology. The first subsection, \S\ref{S:A-AB83}, is a review of~\cite[\S 13]{atiyahbott83}.
In \S\ref{S:A-G/K}, we review some results concerning the equivariant cohomology of Lie groups.
In \S\ref{S:A-KEC} we recall~\cite[Prop.~5.8]{kirwan84} concerning equivariant cohomology for quotients of symplectic manifolds by compact Lie groups.
These results are all standard by now, but unfortunately, we are not aware of a reference where the results are all stated. As is the case throughout the paper, for a topological space $X$, we use the convention $H^\bullet(X)=H^\bullet (X,\mathbb Q)$.
\section{Review of Atiyah--Bott}\label{S:A-AB83}
For any topological group $G$, a classifying space $BG$ is defined as the base of a left principal $G$-bundle $EG\to BG$ whose total space $EG$ is contractible. A classifying space is unique up to homotopy, so that in particular $H^\bullet (BG)$ depends only on $G$. For every topological group $G$ a classifying space exists~\cite{milnor56}.
\begin{exa}\label{E:A-BGL}
The principal $\operatornameeratorname{GL}(n,\mathbb{C})$-bundle induced by the universal vector bundle $E_n $ over the Grassmannian $\operatornameeratorname{Gr}(n,\mathbb{C}^\infty)$ makes $\operatornameeratorname{Gr}(n,\mathbb{C}^\infty)$ into a classifying space for $\operatornameeratorname{GL}(n,\mathbb{C})$. The cohomology can be described as $H^\bullet(B\operatornameeratorname{GL}(n,\mathbb{C}),\mathbb{Z})\cong \mathbb{Z}[c_1,\dots,c_n]$ with $c_i$ taken to have degree $2i$. From this one can deduce that $P_t(B\operatornameeratorname{GL}(n,\mathbb{C})) = (1-t^2)^{-1}(1-t^4)^{-1}\dots (1-t^{2n})^{-1}$.
\end{exa}
More generally if $G$ acts on a topological space $X$ on the right, and a choice of classifying space $BG$ has been made, then
we define $X_G:=X\times_G EG:=(X\times EG)/G$, which is
a locally trivial fibration over $BG$ with fiber $X$ and structure group $G$. The $G$-equivariant cohomology of $X$ is defined to be the ordinary cohomology of $X_G$:
\begin{equation}\label{E:AB-EC-1}
H^\bullet_G(X):=H^\bullet(X_G).
\end{equation}
In particular we have $H^\bullet(BG)=H^\bullet_G(\ast)$, where $\ast$ is a topological space with one point. Moreover, it is well known (e.g.,~\cite[Thm.~6.10.5]{weibel94}) that $H^\bullet(BG)\cong H^\bullet_{\mathsf {gp}}(G,\mathbb{Q})$; i.e., that the cohomology of the classifying space is given by the group cohomology.
If the quotient map $X\to X/G$ is a right principal $G$-bundle, for instance if $G$ is a compact Lie group acting freely on a manifold $X$, then
\begin{equation}\label{E:AB-EC-2}
H^\bullet_G(X)\cong H^\bullet (X/G).
\end{equation}
Indeed, since $X\to X/G$ is a right principal $G$-bundle, applying $X\times_G -$ to the canonical morphism $EG\to \ast$, we obtain that the morphism $X_G=X\times_G EG \to X\times_G \ast \cong X/G$ is a locally trivial fibration with contractible fiber $EG$.
Thus there is a homotopy equivalence $X_G\simeq X/G$.
Note that if $G$ is a compact Lie group acting properly on a manifold $X$, with finite stabilizers, then~\eqref{E:AB-EC-2} still holds. In this case, the fiber of $X_G\to X/G$ over a point $[x]\in X/G$ is isomorphic to $E_G/G_x$, which satisfies $H^i(E_G/G_x)=0$, $i\ge 1$; thus one may conclude via the Leray spectral sequence.
Another useful observation is the following. If $G$ is a subgroup of a topological group $G'$, and $G'\to G'/G$ is a principal $G$-bundle, for instance if $G$ is a closed subgroup of a Lie group $G'$,
then
\begin{equation}\label{E:AB-EC-4}
H^\bullet_G(X)\cong H^\bullet_{G'}(X\times_G G').
\end{equation}
The proof is as follows: Since $G'\to G'/G$ is a principal $G$-bundle, we have that $EG'\to EG'/G$ is a principal $G$-bundle as well, so that we may take $EG=EG'$. Thus we have
$
(X\times_G G')_{G'}:= (X\times_G G' ) \times_{G'} EG' \cong X\times _G EG '=X_G$.
As an immediate application, if a Lie group $G$ acts transitively on $X$ with $X=x\cdot G$ for some $x\in X$, and $G_x$ is the stabilizer of $x$, then $X\cong G_x\backslash G \cong x \times_{G_x}G$, so that~\eqref{E:AB-EC-4} gives
\begin{equation}\label{E:AB-EC-5}
H_{G_x}(x)\cong H_G(X).
\end{equation}
\section{Compact and complex Lie groups} \label{S:A-G/K}
Here we focus on the situation where $K$ is a subgroup of a topological group $G$ such that the quotient map $G\to G/K$ is a principal $K$-bundle; for instance $K$ is a closed subgroup of a Lie group $G$. In this situation $EG\to EG/K$ is also a principal $K$-bundle, so we may take $EK=EG$. Since $X\times EG\to X\times BG$ is a principal $G$-bundle, we have that
$$
X_K=(X\times EG)/K \longrightarrow (X\times EG)/G=X_G
$$
is a locally trivial fibration with fiber $G/K$.
Under various assumptions on $G$ and $K$ we can deduce some further consequences.
For instance, if $G$ is a connected Lie group and $K$ is a maximal compact subgroup, then
\begin{equation}\label{E:AS-EC-1}
H^\bullet_G(X)\cong H^\bullet_K(X).
\end{equation}
Indeed in this case $X_K\to X_G$ is a homotopy equivalence, since $G/K$ is homeomorphic to $\mathbb{R}^n $ for some $n$.
\begin{exa}\label{E:BUn}
Since $\operatornameeratorname{U}(n)$ is a maximal compact subgroup of $\operatornameeratorname{GL}(n,\mathbb{C})$, we have from~\eqref{E:AS-EC-1} and Example~\ref{E:A-BGL} that $H^\bullet(B\operatornameeratorname{U}(n),\mathbb{Z})\cong \mathbb{Z}[c_1,\dots,c_n]$ with $c_i$ taken to have degree $2i$. From this one can deduce that $P_t(B\operatornameeratorname{U}(n)) = (1-t^2)^{-1}(1-t^4)^{-1}\dots (1-t^{2n})^{-1}$.
\end{exa}
\begin{exa}\label{E:BSUn} Identifying $S^1=\operatornameeratorname{U}(1)$ with the group of $n\times n$ diagonal matrices with all entries equal, the surjective multiplication homomorphism $\operatornameeratorname{SU}(n)\times S^1\to \operatornameeratorname{U}(n )$ has kernel isomorphic to the group $\mu_n$ of $n$-th roots of unity. The Lyndon/Hochschild--Serre spectral sequence (e.g.,~\cite[Thm.~6.8.2]{weibel94}) for the normal subgroup $\mu_n$ of $\operatornameeratorname{SU}(n)\times S^1$ then degenerates, since the higher group cohomology for $\mu_n$, being torsion, vanishes with $\mathbb{Q}$-coefficients (e.g.,~\cite[Cor.~6.3.5]{weibel94}), giving an isomorphism $H^\bullet_{\mathsf {gp}}(\operatornameeratorname{U}(n))\cong H_{\mathsf {gp}}^\bullet(\operatornameeratorname{SU}(n)\times S^1)$. Finally, as $H_{\mathsf {gp}}^\bullet(\operatornameeratorname{SU}(n)\times S^1)\cong H_{\mathsf {gp}}^\bullet(\operatornameeratorname{SU}(n) )\otimes H_{\mathsf {gp}}^\bullet( S^1)$ (e.g.,~\cite[Exe.~6.1.10]{weibel94}),
we obtain
\begin{equation}\label{E:SU-EC-1}
H^\bullet (B{\operatornameeratorname{U}(n)}) =H^\bullet (B{\operatornameeratorname{SU}(n)})\otimes H^\bullet(BS^1).
\end{equation}
Since $H^\bullet(BS^1)\cong H^\bullet(B\operatornameeratorname{U}(1))\cong \mathbb{Q}[c_1]$,
we have $P_t(B\operatornameeratorname{SU}(n))=(1-t^4)^{-1}\dots (1-t^{2n})^{-1}$. As $\operatornameeratorname{SU}(n)$ is a maximal compact subgroup of $\operatornameeratorname{SL}(n)$, one has $P_t(B\operatornameeratorname{SL}(n))=P_t(B\operatornameeratorname{SU}(n))$ and $H^\bullet (B{\operatornameeratorname{GL}(n,\mathbb{C})}) =H^\bullet (B{\operatornameeratorname{SL}(n,\mathbb{C})})\otimes H^\bullet(B\mathbb{C}^*)$. Using the short exact sequence $1\to \mu_n\to \operatornameeratorname{SL}(n,\mathbb C)\to \operatornameeratorname{PGL}(n,\mathbb C)\to 1$, similar arguments show that $H^\bullet_{\mathsf {gp}}(\mathbb \operatornameeratorname{PGL}(n,\mathbb C))=H^\bullet_{\mathsf {gp}}(\operatornameeratorname{SL}(n,\mathbb C))$, so that $H^\bullet(B\operatornameeratorname{PGL}(n,\mathbb C))=H^\bullet(B\operatornameeratorname{SL}(n,\mathbb C))$.
\end{exa}
If $K$ is a closed normal subgroup of a Lie group $G$, and $G/K$ is a finite group, then
\begin{equation}\label{E:AS-EC-2}
H^\bullet_G(X)=(H^\bullet_K(X))^{(G/K)}.
\end{equation}
Indeed in this case $X_K\to X_G$ is a principal bundle for the finite group $G/K$.
Note that the action of $G/K$ on $H_K^\bullet(X)$ is induced by an action of $G/K$ on $X_K$. As a particular example, if $G$ is finite, one obtains as a special case
\begin{equation}\label{E:AS-EC-FinG}
H_G^\bullet(X)=H^\bullet(X)^G=H^\bullet(X/G).
\end{equation}
\begin{exa}\label{Exa:SemDirExa} Suppose we have $G=K\rtimes F$, where $K$ is a compact Lie group and $F$ is a finite group. Let $\phi:F\to \operatornameeratorname{Aut}(K)$ be the homomorphism associated to the semidirect product. This induces a homomorphism $\Phi:F\to \operatornameeratorname{Aut}(BK)$, giving the action of $F$ on $H^\bullet(BK)$ such that $H^\bullet(BG)=H^\bullet(BK)^F$.
When $K=T=(S^1)^r$ is a compact torus, this can be made more explicit. We have $\operatornameeratorname{Aut}(T)=\operatornameeratorname{GL}(r,\mathbb{Z})$, and $BT=\prod^r BS^1=\prod^r\mathbb{P}^\infty_{\mathbb{C}}$.
The canonical action of $\operatornameeratorname{Aut}(T)$ on $BT$ is given, for each $\phi\in \operatornameeratorname{Aut}(T)$, by sending a right principal $H$-bundle $P\to B$ to $P\times_{H,\phi}H$. More concretely, $H^\bullet(BT)=\operatornameeratorname{Sym}^\bullet H^2(BT)=\operatornameeratorname{Sym}^\bullet \mathbb{Q}\langle c_1^{(1)},\dots,c^{(r)}_1\rangle = \mathbb{Q}[c_1^{(1)},\dots,c^{(r)}_1]$, $\deg c_1^{(i)}=2$, $i=1,\dots,r$. Viewing $\phi\in \operatornameeratorname{Aut}(T)=\operatornameeratorname{GL}(r,\mathbb{Z})\subseteq \operatornameeratorname{GL}(r,\mathbb{Q})$ as a matrix, we obtain an action of $\phi$ on $\mathbb Q^r= \mathbb{Q}\langle c_1^{(1)},\dots,c^{(r)}_1\rangle=H^2(BT)$ by matrix multiplication.
This induces an action of
$\phi$ on $H^\bullet(BT)=\operatornameeratorname{Sym}^\bullet H^2(BT)$,
which one can check agrees with the canonical action under these identifications.
Similarly, if $K=T\times \Gamma$ for a finite abelian group $\Gamma$, and $T$ a compact torus as above, then $H^\bullet(BK)^F=H^\bullet(BT)^F$, where the action of $F$ on $BT$ is induced by the action of $F$ on $T$, viewing $T$ as the connected component of the identity.
\end{exa}
If $K$ is a compact connected Lie group and $T$ is a maximal torus in $K$,
\begin{equation}\label{E:AS-EC-3}
H^\bullet_T(X)=H^\bullet_K(X)\otimes H^\bullet(K/T).
\end{equation}
Indeed, in this case the fibration $X_T\to X_K$ has fiber given by the flag variety
$K/T$. A direct computation
(see e.g.,~\cite[p.35]{kirwan84})
shows that the associated Leray spectral sequence degenerates, giving~\eqref{E:AS-EC-3}.
We focus again on the situation where $K$ is a subgroup of a topological group $G$ such that the quotient map $G\to G/K$ is a principal $K$-bundle; for instance $K$ is a closed subgroup of a Lie group $G$.
If $K$ is central and contained in the kernel of the map $G\to \operatornameeratorname{Aut}(X)$, then
\begin{equation}\label{E:K-AS-EC-cent}
H^\bullet_G(X)= H^\bullet(BK) \otimes H^\bullet_{G/K}(X).
\end{equation}
Indeed, we start with the observation that, with $G$ acting on $E(G/K)$ via the quotient map to $G/K$, we have that $EG\times E(G/K)$ is contractible with a free $G$-action. Thus we have
$$X\times_G EG\simeq X\times_G (EG\times E(G/K))=((X\times EG\times E(G/K))/K)/(G/K)$$
$$
=((X\times_K EG) \times E(G/K))/(G/K)=(X\times_KEG)\times_{G/K}E(G/K)
$$
$$
\simeq (X\times_KEK)\times_{G/K}E(G/K),
$$
where in the last step we are using that $EG\simeq EK$ (\S~\ref{S:A-G/K}). Now using the fact that $K$ acts trivially on $X$, we obtain that this is equal to
$(X\times BK)\times_{G/K}E(G/K)$. Considering $BK$ as the universal base for principal $K$-bundles, and that the action of $G/K$ on a $K$-principal bundle is given via conjugation, then the fact that $K$ is central implies that the action of $G/K$ on $BK$ is homotopic to the trivial action. Thus we finally arrive at
$BK\times (X\times_{G/K}E(G/K))$, completing the proof.
\footnote{
Alternatively, as suggested to us by Frances Kirwan, one can consider the Leray spectral sequence for the fibration $X \times_G( EG \times E(G/K)) \to EG/K = BK$ with fiber $X \times_{G/K} E(G/K)$, and use Deligne's argument as in~\cite[p.35]{kirwan84} to show the spectral sequence degenerates.
}
\begin{exa}\label{E:H-PGL}
There is a central extension $1\to \mu_n\to \operatornameeratorname{SL}(n,\mathbb{C})\to \operatornameeratorname{PGL}(n,\mathbb{C})\to 1$. Consequently, since $H^\bullet (B\mu_n)=\mathbb Q$, we have that if $\operatornameeratorname{PGL}(n,\mathbb{C})$ acts on $X$, then $H^\bullet _{\operatornameeratorname{PGL}(n,\mathbb{C})}(X)=H^\bullet_{\operatornameeratorname{SL}(n,\mathbb{C})}(X)$, for the induced $\operatornameeratorname{SL}(n,\mathbb{C})$ action. Note, in particular, that applying this to the case where $X$ is a point gives $H^\bullet (B\operatornameeratorname{PGL}(n,\mathbb C))=H^\bullet(B\operatornameeratorname{SL}(n,\mathbb C))$ (Example \ref{E:BSUn}).
\end{exa}
\section{Kirwan's result for compact groups acting on symplectic manifolds}\label{S:A-KEC}
For a compact symplectic manifold $X$ acted on by a compact connected Lie group $K$ such that the moment map exists,
it is shown in~\cite[Prop.~5.8]{kirwan84} that the Leray spectral sequence for the fibration $X_K\to BK$ degenerates, giving
\begin{equation}\label{E:K-EC-1}
H^\bullet_K(X)\cong H^\bullet(BK)\otimes H^\bullet(X).
\end{equation}
If $K$ is disconnected, setting $K_0$ to be its identity component, it follows from~\eqref{E:K-EC-1} and~\eqref{E:AS-EC-2} that $H_K^\bullet (X)$ is the invariant part of
$ H^\bullet (BK_0)\otimes H^\bullet (X)$
under the action of the finite group $K/K_0$.
\section{Fibrations}
Suppose we have a right $G$-equivariant fibration
$$
\xymatrix@R=1em{
F\ar[r] \ar[d]& X \ar[d]^\pi\\
y\ar[r]& Y
}
$$
with $G$ acting transitively on $Y$, and with stabilizer $G_y$ of a point $y\in Y$. Then we have the following equality of equivariant Poincar\'e polynomials:
\begin{equation}\label{E:PGfib}
P^G(X)=P^{G_y}(F).
\end{equation}
This is straightforward from the definitions.
\chapter{Stabilizers, normalizers, and fixed loci for cubic threefolds}\label{S:Elem}
In this section we compute some stabilizers, normalizers, and fixed loci for cubic threefolds, which have appeared in the main body of the paper. While the computations are fairly elementary, they are nevertheless somewhat lengthy, and we have included the details here for the convenience of the reader.
\section{Connected component $\mathbb{C}^*$}
We recall that $2A_5$ cubics of the form $V(F_{A,B})$ are given by equations~\eqref{eq:2A5}, and that for $4A/B^2\ne 1$ such a cubic has either exactly two $A_5$ singularities, or two $A_5$ singularities and an $A_1$ singularity. We continue to denote by $\operatornameeratorname{Aut}(V(F_{A,B}))\subseteq \operatornameeratorname{PGL}(5,\mathbb{C})$ and by $\operatornameeratorname{Stab}(V(F_{A,B}))\subset\operatornameeratorname{SL}(5,\mathbb{C})$ the stabilizers of such a cubic, and recall that by definition $R_{2A5}:=\operatornameeratorname{Stab}^0(V(F_{A,B}))$ is the connected component of the stabilizer. All these are computed by the following proposition, which enhances the statement of Lemma~\ref{L:R2A5Norm}(1) with more computations.
\begin{pro}\label{P:App-R=C*p1}
For a cubic of the form $V(F_{A,B})$ with $4A/B^2\ne 1$:
\begin{enumerate}
\item
The connected component of the stabilizer is the $1$-PS \begin{equation}\label{E:App-R2A5}
R_{2A5}=\operatornameeratorname{diag}(\lambda^2,\lambda,1,\lambda^{-1},\lambda^{-2})\cong \mathbb{C}^*
\end{equation}
(i.e. the $1$-PS with weights $(2,1,0,-1,-2)$).
For a polystable cubic $V$, we have $\operatornameeratorname{Stab}^0(V)=R_{2A5}$ (up to conjugation) if and only if $V$ is in the orbit of $V(F_{A,B})$ with $4A/B^2\ne 1$. These are the cubics corresponding to points on the curve $(\mathcal T-\lbrace\Xi\rbrace) \subseteq \calM^{\operatorname{GIT}}$.
\item If $4A/B^2\ne 0,1,\infty$, then the stabilizer $\operatornameeratorname{Aut}(V(F_{A,B}))\subseteq \operatornameeratorname{PGL}(5,\mathbb{C})$ is
$$
\operatornameeratorname{Aut}(V(F_{A,B}))\cong R_{2A_5}\rtimes \mathbb{Z}/2\mathbb{Z}\cong \mathbb{C}^*\rtimes \mathbb{Z}/2\mathbb{Z},
$$
where the involution is $\tau:x_i \mapsto x_{4-i}$, and the semi-direct product is given by the homomorphism $\mathbb{Z}/2\mathbb{Z} \to \operatornameeratorname{Aut}(\mathbb{C}^*)$ defined by $ \tau \mapsto (\lambda \mapsto \lambda^{-1})$.
Furthermore, we have
$$
1\to \mu_5\to \operatornameeratorname{Stab}(V(F_{A,B}))\to \operatornameeratorname{Aut}(V(F_{A,B}))\to 1.
$$
\item If $4A/B^2=\infty$, then
$$
\operatornameeratorname{Aut}(V_{F_{1,0}}) \cong (\mathbb{C}^*\times \mathbb{Z}/2\mathbb{Z})\rtimes\mathbb{Z}/2\mathbb{Z},
$$
where the second $\mathbb{Z}/2\mathbb{Z}$ factor corresponds to the automorphism $\tau$ that exists for a generic $C$ (and thus also for $C=\infty$), while the first $\mathbb{Z}/2\mathbb{Z}$ factor is given by the involution $\sigma:(x_0:x_1:x_2:x_3:x_4)\mapsto (x_0:-x_1:x_2:x_3:x_4)$, which commutes with the diagonal action of $\mathbb{C}^*$.
\item The normalizer $N(R_{2A_5})$ is equal to
$$N(R_{2A_5})\cong \mathbb{T}^4\rtimes \mathbb{Z}/2\mathbb{Z}\,,$$
where $\mathbb{T}^4$ is the maximal torus, and the $\mathbb{Z}/2\mathbb{Z}$ factor corresponds to the involution $\tau:x_i \mapsto x_{4-i}$.
\end{enumerate}
\end{pro}
\begin{rem}
Before proceeding with the proof, we note that the above does not cover the case of automorphisms for $C=4A/B^2=1$, i.e. the case of the chordal cubic $F_{1,-2}$. Indeed, in Kirwan's machinery this is a separate blowup, and will be treated separately in the next proposition, Proposition~\ref{P:App-R=SL2} --- the proof of which uses this proposition.
\end{rem}
\begin{proof}
Allcock~\cite[Thm.~5.4]{allcock} states that the automorphism group of a general $F_{A,B}$ is as stated in (2), and that the automorphism group of $F_{1,0}$ is as stated in (3) --- we have just included an explicit description.
For completeness, we give a determination of the automorphism group for the case $C=0$, i.e.,~for the $2A_5+A_1$ case:
$$
F_{0,1}=x_0x_3^2+x_1^2x_4-x_0x_2x_4+x_1x_2x_3.
$$
We make the following geometric observation, as mentioned in~\cite{allcock}. Any automorphism must map singularities of the cubic to singularities, and so must its inverse. One easily checks that $F_{0,1}$ has $A_5$ singularities at the points $(1:0:0:0:0)$ and $(0:0:0:0:1)$, and an $A_1$ singularity at $(0:0:1:0:0)$ (which is not there for cubics $F_{A,B}$ with $A\ne 0$). Thus for any automorphism $\gamma\in\operatornameeratorname{Aut}(V_{F_{0,1}})$, either $\gamma$ or $\tau\circ\gamma$ must fix each of these three points. Thus, after possibly composing with $\tau$, such an automorphism must have the form
$$
g:=\left(\begin{smallmatrix} *&*&0&*&0\\ 0&*&0&*&0\\ 0&*&*&*&0\\ 0&*&0&*&0\\ 0&*&0&*&*\end{smallmatrix}\right).
$$
Denoting coefficients of this matrix by $a_{ij}$ for $0\le i,j\le 4$, we see for example that the coefficient of the monomial $x_0x_1^2$ in $F_{0,1}(gx)$
would be equal to $a_{00}a_{13}^2$. Since this coefficient must be zero, while $a_{00}$ cannot be zero in such an invertible matrix, it implies that $a_{13}=0$. Similarly from the coefficient of $x_4x_3^2$ in $ F_{0,1} (gx)$ being zero we deduce that $a_{31}=0$. Continuing in this way, one sees finally that the matrix $a$ must be diagonal.
Denoting this diagonal matrix then by $\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_2,\lambda_3,\lambda_4)$, and requiring the matrix to act on $F_{0,1}$ by scaling it by some $a$, we get the equations
$$
\lambda_0\lambda_3^2=a;\ \lambda_4\lambda_1^2=a;\ \lambda_0\lambda_2\lambda_4=a;\ \lambda_1\lambda_2\lambda_3=a.
$$
As we are interested in the automorphisms in $\operatornameeratorname{PGL}(5,\mathbb{C})$, all $\lambda_i$ are non-zero, and we can always rescale to make $\lambda_2=1$. We then express everything in terms of $\lambda_1$. From the last equation one gets $\lambda_3=a\lambda_1^{-1}$, from the second equation one gets $\lambda_4=a\lambda_1^{-2}$, substituting $\lambda_3$ in the first equation yields $\lambda_0=a\lambda_3^{-2}=a^{-1}\lambda_1^2$, and thus the third equation finally yields $a=\lambda_0\lambda_4=a^{-1}\lambda_1^2a\lambda_1^{-2}=1$, so that the matrix is diagonal of the form $\operatornameeratorname{diag}(\lambda^2,\lambda,1,\lambda^{-1},\lambda^{-2})$, i.e.,~lies in the generic $\mathbb{C}^*$ stabilizer.
We finally prove (4), that is determine the normalizer $N=N(R_{2A_5})$. For this we do a direct computation. Indeed, a matrix $n=(n_{ij})_{0\le i\le j\le 4}$ lies in $N$ if and only if for any $s\in \mathbb{T}$ there exists an $s'\in \mathbb{T}$ such that $nsn^{-1}=s'$, where we think of $s\in \mathbb{T}$ as the diagonal matrix $\operatornameeratorname{diag}(s^2,s,1,s^{-1},s^{-2})$. If this is the case, the map $s\mapsto s'$ gives a homomorphism of the torus. Since conjugating by $n^{-1}$ gives an inverse, this homomorphism must be an isomorphism, and thus we must have either $s'=s$ or $s'=s^{-1}$. Furthermore, note that the involution ${\mathfrak{j}}$ that gives the permutation of coordinates $x_i\mapsto x_{4-i}$ satisfies ${\mathfrak{j}} s{\mathfrak{j}}=s^{-1}$, and thus for any $n$ such that $nsn^{-1}=s^{-1}$ we have $(n{\mathfrak{j}})s(n{\mathfrak{j}})^{-1}=s$. This implies that the normalizer $N$ is a semidirect product of its subgroup $N_0$ consisting of $n$ such that $nsn^{-1}=s$ for all $s\in \mathbb{T}$, and the $\mathbb{Z}/2\mathbb{Z}$ generated by ${\mathfrak{j}}$. Finally, the matrix equality $ns=sn$ for any $n\in N_0$ translates into the equalities $n_{ij}s^{2-i}=n_{ij}s^{2-j}$ for all $0\le i\le j\le 4$ for the entries of the matrix, which must be valid for arbitrary $s$. Thus for $i=j$ there is no restriction on $n_{ij}$, while for $i\ne j$ we must have $n_{ij}=0$. This implies that $N_0\subset G$ consists of diagonal matrices, and is thus the maximal torus $\mathbb{T}^4$, and $N=\mathbb{T}^4\rtimes \mathbb{Z}/2\mathbb{Z}$, as claimed.
\end{proof}
We now describe the fixed locus, and the action of the normalizer on it, supplementing the statement of Lemma~\ref{L:R2A5Norm}(2), (3) with more details.
\begin{pro}\label{P:App-R=C*p2}
\begin{enumerate}
\item
The fixed locus $Z^{ss}_{R_{2A_5}}$~\eqref{E:ZRss} is the set of cubics defined by equations of the form
\begin{equation}\label{E:App-ZR2A5}
F=a_0x_2^3+a_1x_0x_3^2+a_2x_1^2x_4+a_3x_0x_2x_4+a_4x_1x_2x_3,
\end{equation}
with $a_1,a_2,a_3\ne 0$, $(a_0,a_4)\ne (0,0)$. For $(A,B)\ne (0,0)$ we have $V(F_{A,B})\in Z^{ss}_{R_{2A_5}}$, and conversely every cubic in $Z^{ss}_{R_{2A_5}}$ is projectively equivalent to a cubic of the form $V(F_{A,B})$ with $(A,B)\ne (0,0)$.
\item The orbit of the chordal cubic meets $Z^{ss}_{R_{2A_5}}$
in the divisor defined by the equation
$$
4a_0a_1a_2+a_3a_4^2=0.
$$
\item $Z_{R_{2A_5}}^{ss}/N(R_{2A_5}) \cong \mathbb{P}^1$. We also have $Z_{R_{2A_5}}^{ss}/\mathbb{T}^4\cong \mathbb{P}^1$.
\end{enumerate}
\end{pro}
\begin{proof}
We prove (1) by describing all semi-stable cubics that are stabilized by $R=R_{2A_5}=\operatornameeratorname{diag}(\lambda^{2},\lambda,1,\lambda^{-1},\lambda^{-2})$.
To be stabilized by this torus, the monomials must all be of the same weight with respect to that torus. If they all have the same non-zero weight, then they would be unstable with respect to the $1$-PS $R$, and therefore unstable. So we are reduced to looking for the monomials of weight $0$ with respect to that torus. We obtain the projective space of weight $0$ monomials for $R$:
$$
Z_R=\mathbb{P}^4=\{a_0x_2^3+a_1x_0x_3^2+a_2x_1^2x_4+a_3x_0x_2x_4+a_4x_1x_2x_3=0\}.
$$
We next use Allcock's description of the unstable locus. For this, we note that conveniently, the monomials in question are indicated in black squares in~\cite[Fig.~3.2(c)]{allcock}. Now, returning to~\cite[Fig.~3.1]{allcock}, describing unstable cubics, we have that~\cite[Fig.~3.1(a)]{allcock} implies the cubic is unstable if $(a_1,a_3)= (0,0)$,~\cite[Fig.~3.1(b)]{allcock} implies the same if $a_3= 0$,~\cite[Fig.~3.1(c)]{allcock} implies the same if $a_1=0$,~\cite[Fig.~3.1(d)]{allcock} implies the same if $(a_0,a_4)=(0,0)$,~\cite[Fig.~3.1(e)]{allcock} implies the same if $a_2=0$, and~\cite[Fig.~3.1(f)]{allcock} implies the same if $(a_0,a_4)= (0,0)$. Thus in summary, the cubic is unstable if at least one of $a_1,a_2,a_3$ is $0$, or if
$(a_0,a_4)= (0,0)$. Therefore, conversely, let us assume that $a_1,a_2,a_3\ne 0$, and $(a_0,a_4)\ne (0,0)$. Then using the maximal torus $\mathbb{T}^4$, we can easily put the equation for the cubic in the form $F_{A,B}$, with $(A,B)\ne (0,0)$. As Allcock has shown these are all polystable, we see that every point in $Z^{ss}_R$ is semi-stable (in fact, polystable). Moreover, we see that every cubic in $Z^{ss}_R$ can be taken to a cubic of the form $V(F_{A,B})$ by the action of the maximal torus $\mathbb{T}^4$.
(2) We now want to identify the orbit of the chordal cubic inside of $Z_R^{ss}$. The claim is that
$$
G\cdot \{V(F_{-1,2})\} \cap Z_R^{ss}=\{4a_0a_1a_2+a_3a_4^2=0\}\subseteq Z^{ss}_R.
$$
Given a cubic in $Z_R^{ss}$, we saw in the proof of (1) that we could take it into a cubic of the form $V(F_{A,B})$ using just the maximal torus.
So to determine if $(a_0:\dots :a_4)$ defines a cubic in the orbit of the chordal cubic, it suffices to consider the maximal torus orbit, and see whether one can take the cubic into one defined by $F_{A,B}$ with $4A/B^2=1$; in other words, to see when the torus takes $(a_0:\dots :a_4)$ into $(A:1:1:-1:B)$ with $4A/B^2=1$.
The torus $\operatornameeratorname{diag}(s_0,\dots,s_4)$ acts on $(a_0:\dots:a_4)$ by
\begin{align*}
a_0&\mapsto a_0s_2^3\\
a_1&\mapsto a_1s_0s_3^2\\
a_2&\mapsto a_2s_1^2s_4\\
a_3&\mapsto a_3s_0s_2s_4\\
a_4&\mapsto a_4s_1s_2s_3.
\end{align*}
It is immediate to check that if $(a_0:\dots:a_4)=(A:1:1:-1:B)$ with $4A/B^2=1$, then the full orbit satisfies the given equation (a $2A_5$ cubic is chordal if and only if $4A/B^2=1$). Conversely, let us show that if $(a_0:\dots:a_4)$ satisfies the given equations, then we can find $\operatornameeratorname{diag}(s_0,\dots,s_4)$ taking $(a_0:\dots:a_4)$ into the form $(A:1:1:-1:B)$. The first thing to note is that if $a_4$ or $a_0$ is zero, then the equation $4a_0a_1a_2+a_3a_4^2=0$ implies both are zero (since the other $a_i$ are assumed non-zero), so we can assume none of the $a_i$ are zero.
We want $s_0,\dots,s_4$ such that:
\begin{align*}
a_1s_0s_3^2&=1\\
a_2s_1^2s_4&=1\\
a_3s_0s_2s_4&=-1\\
4a_0s_2^3-a_4^2s_1^2s_2^2s_3^2&=0.
\end{align*}
Canceling $s_2^2$, we can take the last equation as $4a_0s_2-a_4^2s_1^2s_3^2=0$.
In other words, we have
\begin{align*}
s_0&=\frac{1}{a_1s_3^2}\\
s_4&=\frac{1}{a_2s_1^2}\\
s_2&=-\frac{1}{a_3s_0s_4}\\
s_2&=\frac{a_4^2s_1^2s_3^2}{4a_0}.
\end{align*}
Taking $s_1$ and $s_3$ arbitrary defines $s_0, s_4,s_2$ via the first three equations. Then one can check that the last equation holds, since by assumption $4a_0a_1a_2+a_3a_4^2=0$.
(3)
Since $N$ is 4-dimensional, and the stabilizer of a generic point (which is contained in $N$) is $1$-dimensional, it follows that the quotient $Z^{ss}_R/N$ is $1$-dimensional. As this quotient is clearly unirational and normal (it is the quotient of a normal space by a reductive group action),
it must be an open subset of $\mathbb{P}^1$. Since the copy of $\mathbb{P}^1\subseteq Z^{ss}_{R}$ given by $V(F_{A,B})$ for $(A,B)\ne (0,0)$ surjects onto the quotient, the quotient is also compact, and is therefore isomorphic to $\mathbb{P}^1$. The identical proof works for the quotient $Z_{R_{2A_5}}^{ss}/\mathbb{T}^4\cong \mathbb{P}^1$.
\end{proof}
\section{Connected component $\operatornameeratorname{PGL}(2,\mathbb{C})$}
\begin{pro}\label{P:App-R=SL2}
For the cubic of the form $V(F_{1,-2})$~\eqref{eq:2A5}; i.e., the chordal cubic, the connected component of the stabilizer is
\begin{equation}\label{E:App-Rc}
R_{c}:=\operatornameeratorname{Stab}^0(V(F_{1,-2}))\cong\mathbb{P} \operatornameeratorname{GL}(2,\mathbb{C})
\end{equation}
given as the copy of $\operatornameeratorname{PGL}(2,\mathbb{C})$ embedded into $\operatornameeratorname{SL}(5,\mathbb{C})$ via the representation $\operatornameeratorname{Sym}^4(\mathbb{C}^2)$ ($\cong \mathbb{C}^5$).
For a polystable cubic $V$, we have $\operatornameeratorname{Stab}^0(V)=R_{c}$ (up to conjugation) if and only if $V$ is in the orbit of $V(F_{A,B})$ with $4A/B^2= 1$; i.e., if and only if the cubic is projectively equivalent to the chordal cubic. These are the cubics corresponding to the point $\Xi \in \calM^{\operatorname{GIT}}$.
Moreover, we have
\begin{enumerate}
\item The full stabilizer group of $V(F_{1,-2})$ in $\operatornameeratorname{PGL}(5,\mathbb{C})$ is $\operatornameeratorname{PGL}(2,\mathbb{C})$, and thus there is a split central extension
\begin{equation}\label{E:App-StabCh1}
1\to \mu_5\to \operatornameeratorname{Stab}(V(F_{1,-2}))\to \operatornameeratorname{PGL}(2,\mathbb{C})\to 1.
\end{equation}
\item The normalizer $N(R_c)$ is equal to the stabilizer $\operatornameeratorname{Stab}(V(F_{1,-2}))$.
\item The fixed locus is $Z^{ss}_{R_c}=\{V(F_{1,-2})\}$; i.e., it is the point corresponding to the chordal cubic.
\end{enumerate}
\end{pro}
\begin{proof}
The fact
\eqref{E:App-Rc} follows from~\cite{allcock}. Indeed, the stabilizer in $\operatornameeratorname{PGL}(5,\mathbb{C})$ of the cubic $V(F_{1,-2})$ is computed in~\cite[Thm.~5.4]{allcock} to be $\operatornameeratorname{PGL}(2,\mathbb{C})$ embedded via the $\operatornameeratorname{Sym}^4$-representation. This immediately gives~\eqref{E:App-StabCh1}: to show that the connected component of the identity is $\operatornameeratorname{PGL}(2,\mathbb{C})$, it suffices to construct a section of~\eqref{E:App-StabCh1}. For this, observe that the standard representation of $\operatornameeratorname{SL}(2,\mathbb{C})$ on $\mathbb{C}^2$ induces a homomorphism $\operatornameeratorname{SL}(2, \mathbb{C})\to \operatornameeratorname{SL}(\operatornameeratorname{Sym}^4\mathbb{C}^2)$, with kernel equal to $\mu_2$; in other words, the image is $\operatornameeratorname{PGL}(2,\mathbb{C})$, providing the section.
For (2) it is convenient to recall the $\operatornameeratorname{Sym}^4\mathbb{C}^2$ representation of $\operatornameeratorname{SL}(2,\mathbb{C})$ explicitly. The matrix
$$
\left(
\begin{array}{cc}
a&b\\
c&d
\end{array}
\right)\in \operatornameeratorname{SL}(2,\mathbb{C})
$$
acts on $\mathbb{C}^2$ by sending homogeneous coordinates $(t_0:t_1)$ to $(at_0+bt_1: ct_0+dt_1)$. Then, in terms of the standard basis for $\operatornameeratorname{Sym}^4\mathbb{C}^2$:
$$
(t_0^4:t_0^3t_1:t_0^2t_1^2:t_0t_1^3:t_1^4),
$$
the action of $
\left(
\begin{array}{cc}
a&b\\
c&d
\end{array}
\right)$ is given by the rule:
\begin{align*}
t_0^4&\mapsto (at_0+bt_1)^4=a^4t_0^4+4a^3bt_0^3t_1+6a^2b^2t_0^2t_1^2+4ab^3t_0t_1^3+b^4t_1^4\\
t_0^3t_1&\mapsto (at_0+bt_1)^3(ct_0+dt_1)=\dots\\
\vdots &
\end{align*}
Thus the induced homomorphism $\operatornameeratorname{SL}(2,\mathbb{C}) \to \operatornameeratorname{SL}(5,\mathbb{C})$ is given explicitly by:
\begin{equation}\label{E:SL2RepCh}
\scriptstyle
\left(
\begin{array}{cc}
a&b\\
c&d
\end{array}
\right)\mapsto
\left(
\begin{array}{ccccc}
a^4&a^3c &a^2c^2 &ac^3&c^4\\
4a^3b&3a^2bc+a^3d&2abc^2+2a^2cd&bc^3+3ac^2d&4c^4d\\
6a^2b^2&3ab^2c+3a^2bd&b^2c^2+4abcd+a^2d^2&3bc^2d+3acd^2&6c^2d^2\\
4ab^3&b^3c+3ab^2d&2b^2cd+2abd^2&3bcd^2+ad^3&4cd^3\\
b^4&b^3d&b^2d^2&bd^3&d^4\\
\end{array}
\right).
\end{equation}
The kernel is given by the matrix $\left(
\begin{array}{cc}
-1&0\\
0&-1
\end{array}
\right)$ confirming that the image of $\operatornameeratorname{SL}(2,\mathbb{C})$ in $\operatornameeratorname{SL}(5,\mathbb{C})$ is $\operatornameeratorname{PGL}(2,\mathbb{C})$.
We now move on to the proof of (2). We first introduce the matrix
$\tau$ which is the matrix associated to the involution sending $x_i$ to $x_{4-i}$; note that $\tau$ is the image of the matrix $\left(
\begin{array}{cc}
0&i\\
i&0
\end{array}
\right)$ in $\operatornameeratorname{SL}(2,\mathbb{C})$. This will be needed in the proof of the next claim.
Since $R_c$ being the connected component is clearly normal in $\operatornameeratorname{Stab}(V(F_{1,-2}))$, we have $\operatornameeratorname{Stab}(V(F_{1,-2}))\subseteq N(R_c)=N$. For the converse, we argue with two claims:
\vskip .2 cm \emph{Claim 1: For any $n\in N$, there is a $g\in R_c$ with $ng\in \mathbb{T}^4\cap N$, where $\mathbb{T}^4$ is the maximal torus.}
\vskip .2cm \noindent
Indeed, any element $n\in N$ must conjugate the standard maximal torus $\mathbb{T}\subset\operatornameeratorname{PGL}(2,\mathbb{C})$, which is embedded into $\operatornameeratorname{SL}(5,\mathbb{C})$, into some torus $\mathbb{T}'\subset\operatornameeratorname{SL}(5,\mathbb{C})$. Since all such tori are conjugate under the action of $\operatornameeratorname{PGL}(2,\mathbb{C})$, this means there must exist some $g'\in\operatornameeratorname{PGL}(2,\mathbb{C})$ such that $n':=ng'$ fixes the maximal torus $\mathbb{T}$ as a set, which is simply to say that $n'$ lies in the normalizer of $R_{2A_5}$, computed in Proposition~\ref{P:App-R=C*p1} to be the subgroup generated by $\mathbb{T}^4$ and $\tau$. Thus for $i\in \{0,1\}$, we have $n'':=ng'\tau^i\in \mathbb{T}^4$. We may as well replace $g'$ with $g=g'\tau\in \operatornameeratorname{PGL}(2,\mathbb{C})$.
\vskip .2 cm \emph{Claim 2: $\mathbb{T}^4\cap N\subseteq \langle \mu_5,\operatornameeratorname{PGL}(2,\mathbb{C})\rangle=\operatornameeratorname{Stab}(V(F_{1,-2}))$.}
\vskip .2cm \noindent
This will suffice to prove (2), since then for any $n\in N$, there is a $g\in R_c$ such that $ng=s\in \operatornameeratorname{Stab}(V(F_{1,-2}))$. Since $R_c\subseteq \operatornameeratorname{Stab}(V(F_{1,-2}))$, we have $n\in \operatornameeratorname{Stab}(V(F_{1,-2}))$.
Thus we just need to show the claim.
For this we consider the special case of upper triangular matrices $\left(\begin{smallmatrix} 1&t\\0&1\end{smallmatrix}\right)\in\operatornameeratorname{SL}(2,\mathbb{C})$ for arbitrary $t\in\mathbb{C}$.
The fourth symmetric power of such a matrix gives its action as an element $M_t$ of $\operatornameeratorname{SL}(5,\mathbb{C})$:
\begin{equation}\label{eq:sym4t}
\scriptstyle
M_t\circ \begin{pmatrix}x_0\\ x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\begin{pmatrix} 1&4t&6t^2&4t^3&t^4\\ 0&1&3t&3t^2&t^3\\ 0&0&1&2t&t^2\\ 0&0&0&1&t\\ 0&0&0&0&1\end{pmatrix}\circ \begin{pmatrix}x_0\\ x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\left(\begin{array}{r} x_0+4t\ x_1+6t^2x_2+4t^3x_3+t^4x_4\\ x_1+3t\ x_2+3t^2x_3+t^3x_4\\ x_2+2t
\ x_3+t^2x_4\\ x_3+t\ x_4\\ x_4\end{array}\right).
\end{equation}
and we need to check whether a diagonal matrix $d\in \mathbb{T}^4$ (where $ \mathbb{T}^4$ is the maximal torus of $\operatornameeratorname{SL}(5,\mathbb{C})$), can conjugate $M_t$ to the action of some element of $\operatornameeratorname{SL}(2,\mathbb{C})$. Since conjugating an upper triangular matrix with $1$'s on the diagonal by a diagonal matrix leaves it upper-triangular with $1$'s on the diagonal, we need to check when for any $t\in\mathbb{C}$ there exists a $t'\in\mathbb{C}$ such that $dM_t d^{-1}=M_{t'}$. Again, $t\mapsto t'$ is then an isomorphism of the additive group, so that it is either the identity or $t\mapsto -t$.
If the map on $t$ is the identity, i.e.,~if for any $t$ the identity $dM_td^{-1}=M_t$, holds, then the equality of the last columns of these matrices yields that each $d_i/d_4$ must be equal to one, so that all $d_i$ are equal, and thus $d$ is scalar multiplication by an arbitrary $5$th root of unity. We note that such a scalar multiplication is not an element of $\operatornameeratorname{SL}(2,\mathbb{C})$ because it is easily checked not to be an element of the diagonal maximal torus $\mathbb{T}$ as above. On the other hand, for the case when for any $t$ the identity $dM_{t}d^{-1}=M_{-t}$ holds, looking again at the last column of these matrices shows that $d_0=d_2=d_4=-d_3=-d_1$, so that $d$ is the product of $\operatornameeratorname{diag}(1,-1,1,-1,1)$ and an arbitrary scalar fifth root of unity. However, the diagonal matrix $\operatornameeratorname{diag}(i,-i)\in\operatornameeratorname{SL}(2,\mathbb{C})$ gives rise precisely to the matrix $\operatornameeratorname{diag}(1,-1,1,-1,1)$ under the fourth symmetric power map, and thus this diagonal matrix is already accounted for by the $\operatornameeratorname{SL}(2,\mathbb{C})$.
(3) We now determine the set of cubics fixed by the action of $\operatornameeratorname{PGL}(2,\mathbb{C})$. Let $V$ be such a cubic. Since $R_{2A_5}\subseteq \operatornameeratorname{PGL}(2,\mathbb{C})$, we must have that $V\in Z^{ss}_{R_{2A_5}}$. If $V$ were not in the orbit of the chordal cubic, then we have seen in Proposition~\ref{P:App-R=C*p1} that the stabilizer would have dimension $1$, which would be a contradiction. Thus $V$ is in the orbit of the chordal cubic, say $V=g\cdot V(F_{1,-2})$. But then the connected component $R_V$ of the stabilizer of $V$ is equal to $gR_cg^{-1}$. If $V$ is fixed by $R_c$, then for dimension reasons, we must have $R_V=R_c$, so that $g$ is in the normalizer of $R_c$. But we saw in (1) that $N(R_c)=\operatornameeratorname{Stab}(V(F_{1,-2}))$, so that $V=V(F_{1,-2})$.
\end{proof}
\begin{rem}
Recall that in the construction of the Kirwan blowup $\calM^{\operatorname{K}}$, one first blows up the point $\Xi \in \mathcal{M}^{GIT}$ corresponding to the chordal cubic, followed by a blowup of the strict transform of the rational curve $\mathcal T$ parameterizing $2A_5$ cubics (the point $\Delta$, corresponding to the $3D_4$ cubic, can be dealt with separately). To fix notation, let $\widehat D_c$ be the exceptional divisor of the blowup of $\Xi$, and let $\widehat {\mathcal T}$ be the strict transform of $\mathcal T$ in this blowup. We explain here that $\widehat {\mathcal T}$ meets $\widehat D_c$ in a single point.
On the one hand, by investigating the proof of Proposition~\ref{P:App-R=C*p2}(3), describing $\mathcal T$ as the quotient $Z^{ss}_{R_{2A_5}}/N(R_{2A_5})$, one can show that $\mathcal T$ is locally unibranched near $\Xi$, and thus
that $\widehat {\mathcal T}$ meets $\widehat D_c$ in a single point.
On the other hand, this can alternatively be seen via the identification of $\widehat D_c$ with the GIT of 12 points on $\mathbb P^1$.
More precisely, this one point of intersection of $\widehat{\mathcal T}$ and $\widehat{D_c}$ can be identified as follows. By construction, and smoothness of the Kirwan blowup up to finite quotient singularities, every point of intersection of $\widehat{\mathcal T}$ with the exceptional divisor $\widehat D_c$ must have a stabilizer containing $\mathbb{C}^*$. On the other hand, since the exceptional divisor does not intersect the locus of $3D_4$ cubics, and there are no further blowups in constructing $\calM^{\operatorname{K}}$, any point on $\widehat D_c$ with a $\mathbb{C}^*$ contained in its stabilizer must be contained in $\widehat {\mathcal T}$. Now, since $\widehat D_c$ is isomorphic to the GIT quotient of 12 points in $\mathbb{P}^1$, the only strictly semi-stable points are where precisely 6 of the 12 points have come together; moreover, one can see immediately that for such a point to have an infinite stabilizer requires the remaining 6 points to also have come together. Thus the only strictly semi-stable points are where the 12 points were separated in two groups of 6. In other words, $\widehat{\mathcal T}\cap \widehat{D_c}$ is the strictly semi-stable point of $\widehat D_c$ corresponding to the case where the 12 points were separated in two groups of 6.
Having explained that $\widehat{\mathcal T}\cap \widehat{D_c}$ consists of a single point, we now point out further that with the identification of $\widehat D_c$ as the GIT of 12 points in $\mathbb P^1$, the stabilizers of all of the points of $\widehat{\mathcal T}$ can be described uniformly (as extension of $\mathbb C^*$ as in Proposition~\ref{P:App-R=C*p1}). Indeed, consider this point $\widehat{\mathcal T}\cap \widehat{D_c}$.
By acting by $\operatornameeratorname{PGL}(2,\mathbb{C})$, we can move the two underlying points (of the pairs of 6 points) to $0$ and $\infty\in\mathbb{P}^1$, respectively, so that $\mathbb{C}^*$ acts by rescaling the coordinate~$z$, and there is an extra involution $z\mapsto 1/z$, so that the stabilizer of these two 6-tuples of points (recall that the points are unlabeled) is $\mathbb{C}^*\rtimes\mathbb{Z}/2\mathbb{Z}$.
Recalling that the automorphisms of the chordal cubic were identified with the automorphisms of the rational normal curve, we can describe this group as follows.
One can identify the rational normal curve explicitly in coordinates as $(t_0^4:t_0^3t_1:t_0^2t_1^2:t_0t_1^3:t_1^4)$ as done in the proof of Proposition~\ref{P:App-R=SL2}. Then the action of the involution $\tau$ is induced by the involution $(t_0:t_1)\mapsto (t_1:t_0)$ on $\mathbb{P}^1$, and thus the stabilizer of the point $\widehat{\mathcal T}\cap \widehat{D_c}$ can be identified concretely as a subgroup of $\operatornameeratorname{PGL}(5,\mathbb C)$, and is the same as for points of $\mathcal T$ with $C\ne 0,1,\infty$ (as described in Proposition~\ref{P:App-R=C*p1}(1)).
\end{rem}
\section{Connected component $(\mathbb{C}^*)^2$}
We now give the computations for the $3D_4$ case, proving Lemmas~\ref{L:R3D4Norm1} and~\ref{L:R3D4Norm2} and providing some more information. We use the notation for the groups involved in the statements of these lemmas.
\begin{pro}\label{P:App-R=C*2}
\begin{enumerate}
\item For the cubic of the form $V(F_{3D_4})$~\eqref{eq:3D4}, i.e., with $3D_4$ singularities, the connected component $R_{3D4}$ of the stabilizer in $\operatornameeratorname{SL}(5,\mathbb{C})$ is given by equation~\eqref{E:R3D4main}. For a polystable cubic $V$, we have $\operatornameeratorname{Stab}^0(V)=R_{3D_4}$ (up to conjugation) if and only if $V$ is in the orbit of $V(F_{3D_4})$; i.e., if and only if the cubic has exactly $3D_4$ singularities. These are the cubics corresponding to the point $\Delta \in \calM^{\operatorname{GIT}}$.
\item The normalizers and stabilizers in $\operatornameeratorname{SL}(5,\mathbb{C})$, $\operatornameeratorname{PGL}(5,\mathbb{C})$, $\operatornameeratorname{GL}(5,\mathbb{C})$ are as given in Lemma~\ref{L:R3D4Norm2}, described as certain central extensions in terms of the group $D$ defined there.
\item The fixed locus $Z^{ss}_{R_{3D_4}}$ is the set of cubics defined by equations of the form
$$
x_0x_1x_2+P_3(x_3,x_4)
$$
where $P_3(x_3,x_4)$ is an arbitrary homogeneous cubic with three distinct roots, and the normalizer $N(R_{3D_4})$ acts on it transitively, as stated in Lemma~\ref{L:R3D4Norm1} (3).
\end{enumerate}
\end{pro}
\begin{proof} (1) and (2): We compute explicitly all the groups involved.
We first derive the stabilizer group $\operatornameeratorname{GL}_{V(F_{3D_4})}$ of $V(F_{3D_4})$ in $\operatornameeratorname{GL}(5,\mathbb{C})$.
To begin, it is clear that the group
\begin{equation}\label{E:App-GF3D4-1}
\left\{\left(
\begin{array}{c|c}
\mathbb{S}_3&\\ \hline
&\mathbb{S}_2\\
\end{array}
\right): \lambda_0\lambda_1\lambda_2=\lambda_3^3=\lambda_4^3\right\}\subseteq \operatornameeratorname{GL}(5,\mathbb{C})
\end{equation}
stabilizes $V(F_{3D_4})$. We wish to show that this is all of the matrices in the stabilizer. For this, we observe that any symmetry must permute the $3$ singularities of the cubic, and thus permute the points $(1:0:0:0:0)$, $(0:1:0:0:0)$ and $(0:0:1:0:0)$. This forces a matrix stabilizing $V(F_{3D_4})$ to be of the form:
$$
\left(
\begin{array}{c|c}
\mathbb{S}_3&*\\ \hline
0&\operatornameeratorname{GL}_2\\
\end{array}
\right).
$$
Such a transformation sends the monomial $x_0x_1x_2$ to $(\lambda_0 x_0+*x_3+*x_4)\cdot (\lambda_1 x_1+*x_3+*x_4)\cdot (\lambda_2 x_2+*x_3+*x_4)$, where all the $\lambda$'s are non-zero, and $*$ are the entries of the unknown $2\times 3$ block of the matrix. Furthermore, $x_3$ and $x_4$ are sent to linear combinations of only $x_3$ and $x_4$. Thus all entries $*$ must be equal to zero, or otherwise applying this transformation to $F_{3D_4}$ would give a cubic with non-zero coefficient of some monomial $x_ax_b x_c$ with $0\le a<b\le 2$ and $3\le c\le 4$. Thus we have deduced that the matrix stabilizing $V(F_{3D_4})$ must actually be of the form
$$
\left(
\begin{array}{c|c}
\mathbb{S}_3&0\\ \hline
0&\operatornameeratorname{GL}_2\\
\end{array}
\right).
$$
However, for a matrix in $\operatornameeratorname{GL}_2$ acting on the span of $x_3$ and $x_4$ to stabilize $x_3^3+x_4^3$, it must lie in $\mathbb{S}_2$, or some cross terms would appear, and thus the stabilizer can only contain matrices of the form
$$
\left(
\begin{array}{c|c}
\mathbb{S}_3&0\\ \hline
0&\mathbb{S}_2\\
\end{array}
\right).
$$
Finally the conditions $\lambda_0\lambda_1\lambda_2=\lambda_3^3=\lambda_4^3$ are obvious. This completes the proof that the stabilizer group is as claimed.
We now want to describe the structure of the stabilizer group $\operatornameeratorname{GL}_{V(F_{3D_4})}$ in $\operatornameeratorname{GL}(5,\mathbb{C})$ more precisely.
There is clearly a left exact sequence
$$
1\to D\to \operatornameeratorname{GL}_{V(F_{3D_4})} \to S_3\times S_2\to 1
$$
where $D$ is the subgroup of diagonal matrices in $\operatornameeratorname{GL}_{V(F_{3D_4})}$, and the map to $S_3\times S_2$ is the one taking a generalized permutation matrix to the associated permutation matrix. There is an obvious section $S_3\times S_2\to \operatornameeratorname{GL}_{V(F_{3D_4})}$, viewing $S_3\times S_2$ as block diagonal permutation matrices. This means
$$
\operatornameeratorname{GL}_{V(F_{3D4})}\cong D\rtimes (S_3\times S_2)
$$
where the action of $S_3\times S_2$ on $D$ is to permute the entries.
We now wish to describe $D$.
Concretely, $D=\{\operatornameeratorname{diag}(\lambda _0,\lambda _1,\lambda_2,\lambda_3,\lambda_4): \lambda_0\lambda_1\lambda_2=\lambda_3^3=\lambda_4^3\}$.
Fixing the torus $\mathbb{T}^3=\operatornameeratorname{diag}(\lambda_0,\lambda_1,\lambda_0^{-1}\lambda_1^{-1}\lambda_3^3,\lambda_3,\lambda_3)\cong (\mathbb{C}^*)^3$, we have $\mathbb{T}^3\subseteq D$, and we now describe the quotient. Given an element of $D$, then up to elements of $\mathbb{T}^3$, we may assume it is of the form
$\operatornameeratorname{diag}(1,1,\lambda_2,1,\lambda_4)$. But then we must have $1\cdot 1\cdot \lambda_2= 1^3=\lambda_4^3$, so that $\lambda_2=1$ and $\lambda_4$ is a $3$-rd root of unity.
Fixing the group $\mu_3=\operatornameeratorname{diag}(1,1,1,1,\zeta^i)\cong \mathbb{Z}/3\mathbb{Z}$ where $\zeta$ is a primitive $3$-rd root of unity, we have
$$
D=\mathbb T^3\times \mu_3.
$$
We determine the normalizer $N(R_{3D_4})$ by an explicit computation. Indeed, if a matrix $n=(n_{ij})_{0\le i\le j\le 4}$ lies in $N$, then for any $(s_1,s_2)\in \mathbb T^2$ we have
\begin{equation}\label{eq:normT2}
\operatornameeratorname{diag}(s_1,s_2,s_1^{-1}s_2^{-1},1,1)\cdot n=n\cdot \operatornameeratorname{diag}(t_1,t_2,t_1^{-1}t_2^{-1},1,1)
\end{equation}
for some $(t_1,t_2)\in \mathbb T^2$. We first observe that~\eqref{eq:normT2} immediately implies that for any $0\le i\le 2$ and $3\le j\le 4$ we must have $n_{ij}=0$. Furthermore, we note that this equality implies no restrictions whatsoever on the entries $n_{33},n_{34},n_{43},n_{44}$, which can thus be arbitrary. The map $f:(s_1,s_2)\mapsto (t_1,t_2)$ is an automorphism of $\mathbb T^2$, which is to say that $t_1=s_1^as_2^b$ and $t_2=s_1^cs_2^d$ for some matrix $\left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)\in\operatornameeratorname{SL}(2,\mathbb{Z})$. By writing down the conditions for the entries $n_{ij}$ with $0\le i\le j\le 2$ of the matrix, we see that these elements can be non-zero only if the map $f$ permutes the three diagonal entries $s_1,s_2,s_1^{-1}s_2^{-1}$. Conversely, any such permutation lies in the normalizer with respect to $\operatornameeratorname{GL}(5,\mathbb{C})$. If this permutation, as an element of $S_3$, is even, we compose $n$ with this permutation of coordinates $x_0,x_1,x_2$; if such a permutation is odd, we compose $n$ with this permutation of $x_0,x_1,x_2$, together with changing the signs of $x_0,x_1,x_2$ (so that the resulting transformation is still in $\operatornameeratorname{SL}(5,\mathbb{C})$). Thus $N$ is a semidirect product of $S_3$ and of the normal subgroup $N_0\subset N$ for which $f$ is the identity map. Finally, if $f$ is the identity map, so that $t_1=s_1$ and $t_2=s_2$, then clearly~\eqref{eq:normT2} implies that the submatrix $(n_{ij})_{0\le i\le j\le 2}$ is diagonal. Thus finally $N_0$ is the intersection of $\mathbb T^3\times \operatornameeratorname{GL}(2,\mathbb{C})$ with $\operatornameeratorname{SL}(5,\mathbb{C})$, and thus $N$ is as claimed.
(3) We now describe the fixed locus $Z^{ss}_{R_{3D_4}}$. As usual, to be semi-stable, and fixed by $R_{3D_4}\cong (\mathbb{C}^*)^2$, the cubic must be defined by monomials of weight $0$ with respect to any $1$-PS in $R_{3D_4}$. It is easy to see that the only such monomials are
$$
x_0x_1x_2+P_3(x_3,x_4)
$$
where $P_3(x_3,x_4)$ is a homogeneous cubic.
Allcock has shown that these are semi-stable if and only if $P_3(x_3,x_4)$ has $3$ distinct roots; i.e., the cubic has exactly $3D_4$ singularities. More precisely, as mentioned earlier,~\cite[Thm.~4.1]{allcock} shows that the orbit of $V(F_{3D_4})$ is closed in the semi-stable locus. But any cubic as above with $P_3(x_3,x_4)$ having multiple roots is in the closure of the orbit of $V(F_{3D_4})$, but does not have $3D_4$ singularities, which is a contradiction.
Finally, the matrices of the form
$$
\left(
\begin{array}{c|c}
\operatornameeratorname{Id}_3&0\\ \hline
0&\operatornameeratorname{SL}_2\\
\end{array}
\right)
$$
that lie in the normalizer clearly act transitively on $Z^{ss}_{R_{3D_4}}$.
\end{proof}
\begin{cor}\label{C:App-ZssR-rel}
We have the following relationships among the fixed loci:
\begin{equation}\label{E:App-ZssR-Rel}
Z^{ss}_{R_c}\subset Z^{ss}_{R_{2A5}}, \ \ \ \ Z^{ss}_{R_{2A_5}} \cap Z^{ss}_{R_{3D4}}=\emptyset.
\end{equation}
\end{cor}
\begin{proof}
The inclusion on the left follows immediately from the first inclusion in~\eqref{E:Rcont}. For the equation on the right in~\eqref{E:App-ZssR-Rel}, suppose that $x\in Z^{ss}_{R_{2A_5}}\cap Z^{ss}_{R_{3D_4}}$, and let $V$ be the corresponding cubic. Then $\operatornameeratorname{Aut}^0(V)\supseteq R_{2A_5}\cup R_{3D_4}$, and one can see this implies it contains a $3$-torus isomorphic to $(\mathbb{C}^*)^3$. On the other hand, $V$ degenerates to a polystable cubic, and consequently we have that $\operatornameeratorname{Aut}^0(V)$ is contained in a conjugate of $ R$ for some $R\in \mathcal{R}$. For dimension reasons, it would have to be contained in a conjugate of $R_c=\operatornameeratorname{SL}(2,\mathbb{C})$, but this does not contain a $3$-torus.
\end{proof}
We recall from Lemma~\ref{L:R3D4Norm1} the normalizer
$$
N=N(R_{3D_4})=
\left\{\left(
\begin{array}{c|c}
\mathbb{S}_3&\\ \hline
&\operatornameeratorname{GL}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\}\,,
$$
and define a subgroup $N_0$:
$$
N_0:=
\left\{\left(
\begin{array}{c|c}
\mathbb{T}^3&\\ \hline
&\operatornameeratorname{GL}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\}.
$$
Recall also that the stabilizer $G_x=G_{3D_4}$ of $x=V(F_{3D_4})$ in $G=\operatornameeratorname{SL}(5,\mathbb{C})$ is:
\begin{equation*}
G_{F_{3D4}}=
\left\{\left(
\begin{array}{c|c}
\mathbb{S}_3&\\ \hline
&\mathbb{S}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C}): \lambda_1\lambda_2\lambda_3=\lambda_4^3=\lambda_5^3\right\}.
\end{equation*}
Here $\lambda_i$ is the unique non-zero entry in column $i$. We now compute the relevant stabilizers and their action, proving Lemma~\ref{L:R(rho)3D4p2} and providing more details. We record two propositions, separately for the cases when $\beta'$ correspond to the codimension 4 and codimension 5 strata, as in the cases (a) and (b) of Lemma~\ref{L:R(rho)3D4p1}, respectively.
\begin{lem}\label{L:App-R(rho)3D4a}
For $\beta'=\frac{1}{2}(-\frac{2}{3},\frac{1}{3},\frac{1}{3})$ (case (a) of Lemma~\ref{L:R(rho)3D4p1}),
we have
\begin{enumerate}
\item
$$
\operatornameeratorname{Stab}_G\beta'=
\left\{\left(
\begin{array}{c|c|c}
\mathbb{C}^*&&\\ \hline
&\operatornameeratorname{GL}_2&\\ \hline
&&\operatornameeratorname{GL}_2
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\},
$$
$$
N\cap \operatornameeratorname{Stab}_G\beta' =
\left\{\left(
\begin{array}{c|c|c}
\mathbb{C}^*&&\\ \hline
&\mathbb{S}_2&\\ \hline
&&\operatornameeratorname{GL}_2
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\}\,.
$$
\item The group $N\cap \operatornameeratorname{Stab}_G\beta'$ acts transitively on $Z^{ss}_R$.
\item The stabilizer of a point is
$$\scriptstyle{
(N\cap \operatornameeratorname{Stab}_G\beta')_x=G_{F_{3D_4}}\cap N\cap \operatornameeratorname{Stab}_G\beta' =
\left\{\left(
\begin{array}{c|c|c}
\mathbb{C}^*&&\\ \hline
&\mathbb{S}_2&\\ \hline
&&\mathbb{S}_2
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C}):\lambda_1\lambda_2\lambda_3=\lambda_4^3=\lambda_5^3\right\}.}
$$
\item The locus $Z^{ss}_{\beta'}$ is
$$
Z^{ss}_{\beta'}=\{[a:b]\in \mathbb{P}\mathbb{C}\langle x_1x_3x_4,x_2x_3x_4\rangle: a\ne 0, b\ne 0 \}\cong \mathbb{C}^*.
$$
\item The action of $(N\cap \operatornameeratorname{Stab}_G\beta')_x$ on $Z^{ss}_{\beta'}$ is induced by change of coordinates, via the inclusion $(N\cap \operatornameeratorname{Stab}_G\beta')_x\subseteq \operatornameeratorname{SL}(5,\mathbb{C})$, and the description of the loci above in terms of cubic forms. In fact
that $(N\cap \operatornameeratorname{Stab}_G\beta')_x$ acts transitively on $Z^{ss}_{\beta '}$, and the stabilizer of the point $(1:1)\in Z^{ss}_{\beta'}$ is given by
$$\scriptstyle{
((N\cap \operatornameeratorname{Stab}_G\beta')_x)_{(1:1)}
=\left\{\left(
\begin{array}{c|c|c}
\mathbb{C}^*&&\\ \hline
&\mathbb{S}_2&\\ \hline
&&\mathbb{S}_2
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C}):\lambda_0\lambda_1\lambda_2=\lambda_3^3=\lambda_4^3,\ \lambda_1=\lambda_2\right\}}.
$$
Here $\lambda_i$ is the unique non-zero entry in column $i$.
In fact we have $$((N\cap \operatornameeratorname{Stab}_G\beta')_x)_{(1:1)}\cong (\mathbb{C}^*\times \mu_{15})\times (S_2\times S_2)\,,$$ where $\mathbb{C}^*=\operatornameeratorname{diag}(\lambda^{-2},\lambda,\lambda,1,1)$, $\mu_{15}=\operatornameeratorname{diag}(\zeta^{3i},1,1,\zeta^i,\zeta^{-4i})$ for $\zeta$ a primitive $15$-th root of unity, and the first (resp.~second) copy of $S_2$ is the subgroup of $((N\cap \operatornameeratorname{Stab}_G\beta')_x)_{(1:1)}$ generated by the
matrix
$$
\left(
\begin{array}{ccccc}
-1&&&&\\
&0&1&&\\
&1&0&&\\
&&&-1&\\
&&&&-1\\
\end{array}
\right)
,\ \
\left(resp.~
\left(
\begin{array}{ccccc}
-1&&&&\\
&1&&&\\
&&1&&\\
&&&0&-1\\
&&&-1&0\\
\end{array}
\right)
\right).
$$
\end{enumerate}
\end{lem}
The results for the codimension 5 orbits are as follows.
\begin{lem}\label{L:App-R(rho)3D4b}
For $\beta'= \frac{1}{7}(2,1,-3)$ (case (b) of Lemma~\ref{L:R(rho)3D4p1}), we have
\begin{enumerate}
\item
$$
\operatornameeratorname{Stab}_G\beta'
= N_0=
\left\{\left(
\begin{array}{c|c}
\mathbb{T}^3&\\ \hline
&\operatornameeratorname{GL}_2
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\},
$$
$$
N\cap \operatornameeratorname{Stab}_G\beta' = N_0=
\left\{\left(
\begin{array}{c|c}
\mathbb{T}^3&\\ \hline
&\operatornameeratorname{GL}_2
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C})\right\}.
$$
\item The group $N\cap \operatornameeratorname{Stab}_G\beta'$ acts transitively on $Z^{ss}_R$.
\item The stabilizer of a point is
$$\scriptstyle{
(N\cap \operatornameeratorname{Stab}_G\beta')_x=G_{F_{3D_4}}\cap N\cap \operatornameeratorname{Stab}_G\beta' =
\left\{\left(
\begin{array}{c|c}
\mathbb{T}^3&\\ \hline
&\mathbb{S}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C}):\lambda_1\lambda_2\lambda_3=\lambda_4^3=\lambda_5^3\right\}}.
$$
\item The locus $Z^{ss}_{\beta'}$ is
$$
Z^{ss}_{\beta'}=\{[a:b:c]\in \mathbb{P}\mathbb{C}\langle x_0x_3x_4,x_1^2x_3, x_1^2x_4\rangle: a\ne 0,\ \text{and}\ (b,c)\ne (0,0) \}\cong \mathbb{A}^2-\{0\}.
$$
\end{enumerate}
\end{lem}
We prove both lemmas in parallel.
\begin{proof}
(1) Given $\beta'$, we are first looking at computing the stabilizer $\operatornameeratorname{Stab}_G\beta'$ for the group $G=\operatornameeratorname{SL}(5,\mathbb{C})$ acting by the adjoint representation; i.e., conjugation. Since all of our $\beta'$ are given explicitly as diagonal matrices, this is quite easy. Indeed, given any diagonal matrix $D=\operatornameeratorname{diag}(d_1,\dots,d_n)$ and any $n\times n$ matrix $A$, the condition that $AD=DA$ is given by $d_ia_{ij}=d_ja_{ij}$. In other words: if $d_i=d_j$, then $a_{ij}$ may be arbitrary; if $d_i\ne d_j$, then $a_{ij}=0$. The rest is an elementary computation.
(2) It is immediate that $N_0$ acts transitively on $Z^{ss}_R$. Thus, since for each $\beta'$ in either case (a) or (b) we have $N_0\subseteq N\cap \operatornameeratorname{Stab}_G\beta'$, and we are done.
(3) Recall that we computed
\begin{equation*}
G_{F_{3D4}}=
\left\{\left(
\begin{array}{c|c}
\mathbb{S}_3&\\ \hline
&\mathbb{S}_2\\
\end{array}
\right)\in \operatornameeratorname{SL}(5,\mathbb{C}): \lambda_1\lambda_2\lambda_3=\lambda_4^3=\lambda_5^3\right\}.
\end{equation*}
The rest follows immediately from the previous parts.
(4) This follows immediately from the definitions, by inspection of the previous computations.
(5) (for case (a) only) We have $\mathbb{T}^2=\operatornameeratorname{diag} (\lambda_0,\lambda_1,\lambda_0^{-1}\lambda_1^{-1},1,1)\subseteq (N\cap \operatornameeratorname{Stab}_G\beta')_x$.
The action of $\mathbb{T}^2$ on $Z^{ss}_{\beta'}$ is given by
$\operatornameeratorname{diag} (\lambda_0,\lambda_1,\lambda_0^{-1}\lambda_1^{-1},1,1) \cdot (a:b)=(\lambda_1a:\lambda_0^{-1}\lambda_1^{-1}b)$, thus the action of $\mathbb{T}^2$ on $Z^{ss}_{\beta'}$ is transitive, and therefore the same is true of $(N\cap \operatornameeratorname{Stab}_G\beta')_x$. The stabilizer $((N\cap \operatornameeratorname{Stab}_G\beta')_x)_{(1:1)}$ is easily worked out to be as claimed, from the previous description of $(N\cap \operatornameeratorname{Stab}_G\beta')_x$. The direct product decomposition can be deduced as follows. First, let $D'$ be the diagonal matrices in $((N\cap \operatornameeratorname{Stab}_G\beta')_x)_{(1:1)}$. There is a short exact sequence $$1\to D'\to ((N\cap \operatornameeratorname{Stab}_G\beta')_x)_{(1:1)}\to S_2\times S_2\to 1\,,$$ and the matrices given above clearly define a section. Those matrices commute, and commute with the diagonal matrices, and so we obtain a direct product $D'\times (S_2\times S_2)$.
Now we have essentially already analyzed the diagonal matrices $D'$; indeed we described a group $D\subseteq \operatornameeratorname{GL}(5,\mathbb{C})$ of diagonal matrices in Proposition~\ref{P:App-R=C*2}(1), with $D'\subseteq D\cong (\mathbb{C}^*)^3\times \mu_3$. One can easily deduce the structure of $D'$ from this.
For clarity, we reproduce the argument in this special case.
Assume we have a diagonal matrix $\operatornameeratorname{diag}(\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5)\in D'$. Since we are only interested up to the torus $\mathbb{T}=\operatornameeratorname{diag}(\lambda^{-2},\lambda,\lambda,1,1)$, we may scale so that $\lambda_1=\lambda_2=1$. We now have that $\lambda_0=\lambda_3^3=\lambda_4^3$, and $\lambda_0\lambda_3\lambda_4=1$. Together these imply that $\lambda_3^4\lambda_4=1$.
This implies that $\lambda_4=\lambda_3^{-4}$.
This implies
$\lambda_3^3=\lambda_4^3=(\lambda_3^{-4})^3=\lambda_3^{-12}$, so that $\lambda_3^{15}=1$;
i.e., $\lambda_3$ is a $15$-th root of unity. In other words, up to scaling by the torus, any diagonal matrix in $D'$ is of the form $\operatornameeratorname{diag}(\lambda_4^3,1,1,\lambda_4,\lambda_4^{-4})$
where $\lambda_4$ is a $15$-th root of unity. We may as well write:
$$
D'=
\left\{
\left(
\begin{array}{ccccc}
\lambda^{-2}\zeta^{3j}&&&&\\
&\lambda&&&\\
&&\lambda &&\\
&&&\zeta^{j}&\\
&&&&\zeta^{-4j}\\
\end{array}
\right): \lambda \in \mathbb{C}^*,\ \zeta =e^{2\pi i/15},\ j=0,\dots,14
\right\}.
$$
This completes the proof.
\end{proof}
\chapter{The moduli space of cubic surfaces}
\label{sec:surfaces}
As a demonstration of the techniques developed in the paper, we briefly outline how one obtains analogous results for the moduli space of cubic curves and surfaces. The new results in this appendix are the computations of the Betti numbers of the toroidal and Naruki compactifications of the moduli space of cubic surfaces (Theorem~\ref{T:CubSurfH}).
\section{The moduli space of cubic curves}
The case of cubic curves is trivial, but nevertheless we review this situation, for completeness. The GIT moduli space $\calM^{\operatorname{GIT}}_{\operatornameeratorname{curve}}$ has stable points corresponding to smooth cubic curves, and strictly semi-stable points corresponding to cubic curves with nodes. There is a unique strictly polystable orbit, corresponding to the cubic curve $V(x_0x_1x_2)$, the so-called $3A_1$ cubic curve.
Being a normal rational projective variety of dimension~$1$, we have $\calM^{\operatorname{GIT}}_{\operatornameeratorname{curve}}\cong \mathbb{P}^1$.
The natural period map is $\mathcal{M}_{\operatornameeratorname{curve}}\to \mathfrak{H}/\Gamma_1$, taking a cubic curve to its Jacobian, with $\Gamma_1=\operatornameeratorname{SL}(2,\mathbb{Z})$; here $\mathcal{M}_{\operatornameeratorname{curve}}$ is the locus of smooth cubic curves. As the Baily--Borel compactification $(\mathfrak{H}/\Gamma_1)^*$ is also a normal rational projective variety of dimension~$1$, it is also isomorphic to $\mathbb{P}^1$, and the period map extends to an isomorphism $\calM^{\operatorname{GIT}}_{\operatornameeratorname{curve}}\cong (\mathfrak{H}/\Gamma_1)^*$. Note also that
the boundary of the Baily--Borel compactification is already a divisor (it is simply a point on a curve), and since $(\mathfrak{H}/\Gamma_1)^*$ is smooth, it is its own canonical toroidal compactification $\overline{\mathfrak{H}/\Gamma_1}=(\mathfrak{H}/\Gamma_1)^*$. Finally, since $\calM^{\operatorname{GIT}}_{\operatornameeratorname{curve}}$ has a strictly polystable point, the Kirwan blowup is not just the identity map; however, since the Kirwan blowup $\calM^{\operatorname{K}}_{\operatornameeratorname{curve}}$ is smooth, projective and of dimension~$1$, it is also isomorphic to $\mathbb{P}^1$, so that $\calM^{\operatorname{K}}_{\operatornameeratorname{curve}}\to \calM^{\operatorname{GIT}}_{\operatornameeratorname{curve}}$ is an isomorphism. In other words, all of the compactifications in question are isomorphic to $\mathbb{P}^1$, and the cohomology is obvious.
\section{The moduli space of cubic surfaces}
The moduli of cubic surfaces has a number of compactifications constructed in a similar way to those of cubic threefolds. To begin with, the GIT compactification $\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}$
can be described as follows (see e.g.,~\cite[\S 7.2(b)]{mukai}). A cubic surface $V$ is:
\begin{itemize}
\item stable if and only if it has at worst $A_1$ singularities,
\item semi-stable if and only if it is stable, or has at worst $A_2$ singularities, and does not contain the axes of the $A_2$ singularities,
\item strictly polystable if and only if it is projectively equivalent to $V(x_0x_1x_2+x_3^3)$ (the so-called $3A_2$ cubic).
\end{itemize}
Note that it is a classical result that $\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}\cong W\mathbb{P}(1,2,3,4,5)$ (see~\cite[(2.4)]{DvGK}).
By considering the triple cover of $\mathbb{P}^3$ branched along a cubic surface, one obtains a cubic threefold, and via the period map for cubic threefolds, one obtains a period map to a $4$-dimensional ball quotient $\mathcal{M}_{\operatornameeratorname{surf}}\to \mathcal{B}_4/\Gamma_4$ (see~\cite{ACTsurf}); here $\mathcal{M}_{\operatornameeratorname{surf}}$ is the locus of smooth cubic surfaces. This is an open embedding, and the complement of the image is the Heegner divisor $D_n=\mathcal{D}_n/\Gamma_4$. The rational period map $\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}\dashrightarrow (\mathcal{B}_4/\Gamma_4)^*$ to the Baily--Borel compactification extends to an isomorphism, taking the discriminant $D_{A_1}\subset \calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}$ to the divisor $D_n$. Under this isomorphism, the unique strictly polystable point $\Delta\in \calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}$ corresponding to the $3A_2$ cubic is identified with the sole cusp of $(\mathcal{B}_4/\Gamma_4)^*$, which we thus denote $c_{3A_2}$. The Kirwan blowup $\calM^{\operatorname{K}}_{\operatornameeratorname{surf}}\to \calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}$ is a blowup with center supported at $\Delta$.
In a different direction, Naruki~\cite{naruki} has constructed a modular compactification $\widetilde \mathcal{N}$ of the moduli space of marked cubic surfaces (this was subsequently reworked by~\cite{HKT09} from a different perspective). There is a natural action by $W(E_6)$ on $\widetilde\mathcal{N}$, and denoting by $\overline\mathcal{N}=\widetilde\mathcal{N}/W(E_6)$, we get another smooth (as always, up to finite quotient singularities) compactification for the moduli space of cubic surfaces. As discussed in~\cite{DvGK}, $\overline\mathcal{N}$ maps to $\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}\cong (\mathcal{B}_4/\Gamma_4)^*$, and this map contracts a divisor to the boundary point $\Delta$ (resp.~$c_{3A_2}$);
denoting this divisor $D_{\overline \mathcal{N}_{3A_2}}$, this contraction induces an isomorphism $\overline\mathcal{N}-D_{\overline \mathcal{N}_{3A_2}}\cong \calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}-\Delta$~\cite[\S2.10]{DvGK}.
In summary, we have a diagram (compare~\eqref{E:BirDiagMod})
\begin{equation}\label{eq_diag_surf}
\xymatrix{
&\calM^{\operatorname{K}}_{\operatornameeratorname{surf}} \ar[ld] \ar[rd]\ar@{<-->}[r]&\overline{\mathcal{B}_4/\Gamma_4} \ar[d]\ar@{<-->}[r]&\overline{\mathcal{N}}\ar[ld]\\ \calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}} \ar[rr]^{\sim}&&(\mathcal{B}_4/\Gamma_4)^*
}
\end{equation}
where $\overline{\mathcal{B}_4/\Gamma_4} $ is the (again, unique) toroidal compactification. The purpose of this section is to establish that these three compactifications ($\calM^{\operatorname{K}}_{\operatornameeratorname{surf}}$, $\overline{\mathcal{B}_4/\Gamma_4}$, and $\overline{\mathcal{N}}$) have the same cohomology.
Note that all three spaces are blowups of the point $c_{3A_2}\in (\mathcal{B}_4/\Gamma_4)^*$; we expect that they are all isomorphic, but this is not yet known (compare Remark~\ref{rem_possible_iso}).
In~\cite{kirwanhyp} and~\cite{ZhangCubic}
the (intersection) Betti numbers of the spaces $\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}\cong (\mathcal{B}_4/\Gamma_4)^*$ and $\calM^{\operatorname{K}}_{\operatornameeratorname{surf}}$ were computed\footnote{
Note there is an error in~\cite[Thm.~1.6, p.50, and 5.2]{kirwanhyp} regarding the Betti numbers of $\calM^{\operatorname{K}}_{\operatornameeratorname{surf}}$, corrected in~\cite{ZhangCubic}. Specifically, the set $\mathcal{R}$ of connected components of stabilizers consists only of $\mathbb{T}^2$, and does not also include $\operatornameeratorname{SO}(3,\mathbb{C})$, as claimed in~\cite[p.59]{kirwanhyp}: the only strictly polystable orbit is the orbit of the $3A_2$ cubic surface, with connected component of the stabilizer given by $\mathbb{T}^2$. The rest of the computations in~\cite{kirwanhyp} go through unchanged, and yield $P_t(\calM^{\operatorname{K}}_{\operatornameeratorname{surf}})=P^G_t(X_{\operatornameeratorname{surf}}^{ss})+A_{\mathbb{T}^2}(t)\equiv (1+t^2+2t^4)+t^2\equiv 1+2t^2+2t^4\mod t^5$; i.e., one simply does not add the $A_{\operatornameeratorname{SO}(3,\mathbb{C})}(t)\equiv t^2+t^4 \mod t^5$ contribution from the erroneous group $R=\operatornameeratorname{SO}(3,\mathbb{C})$. The computation of $IP_t(\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}})$ is then also corrected by omitting the terms corresponding to $R=\operatornameeratorname{SO}(3,\mathbb{C})$, so that one obtains $IP_t(\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}})=P_t(\calM^{\operatorname{K}}_{\operatornameeratorname{surf}})-B_{\mathbb{T}^2}(t)\equiv (1+2t^2+2t^4)-(t^2+t^4)\equiv 1+t^2+t^4\mod t^5$; i.e., the formula for $IP_t(\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}})$ in \cite[Thm.~1.6]{kirwanhyp} is correct.
}
:
\begin{equation}\label{E:KirThmCubSurf}
\begin{array}{r|ccccc}
j&0&2&4&6&8\\\hline
\dim H^j(\calM^{\operatorname{K}}_{\operatornameeratorname{surf}})&1&2&2&2&1 \rule{0pt}{2.6ex}\\
\dim IH^j(\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}})=\dim IH^j((\mathcal{B}_4/\Gamma_4)^*)&1&1&1&1&1 \rule{0pt}{2.6ex}
\end{array}
\end{equation}
with all odd degree (intersection) cohomology vanishing.
Note that the bottom row is immediate, since $\calM^{\operatorname{GIT}}_{\operatornameeratorname{surf}}$ is a weighted projective space, as recalled above.
For the cohomology of the toroidal compactification of the ball quotient model $\mathcal{B}_4/\Gamma_4$ of the moduli of cubic surfaces, we apply the same approach (but, of course, with easier computational details) as for cubic threefolds (see Chapter~\ref{sec:toroidal}). As announced, we obtain that the cohomology of the toroidal compactification coincides with the cohomology of the Kirwan blowup $\calM^{\operatorname{K}}_{\operatornameeratorname{surf}}$.
\begin{teo}\label{T:CubSurfH}
The Betti numbers of the toroidal compactification of the ball quotient model $\overline{\mathcal{B}_4/\Gamma_4}$ of the moduli space of cubic surfaces are as follows:
\begin{equation}\label{E:CubSurfH}
\begin{array}{r|ccccc}
\hskip2cm j&0&2&4&6&8\\\hline
\dim H^j(\overline{\mathcal{B}_4/\Gamma_4})&1&2&2&2&1 \rule{0pt}{2.8ex}
\end{array}
\end{equation}
while all the odd degree cohomology vanishes.
\end{teo}
\section{The proof of Theorem~C.1}
In this section, following the setup of \S\ref{S:ArithmeticCusp}, we discuss the structure of the toroidal compactification $\overline{\mathcal{B}_4/\Gamma_4}$ of the ball quotient model for surfaces, and prove Theorem~\ref{T:CubSurfH}.
\subsection{The Eisenstein lattice for cubic surfaces}
The Eisenstein lattice used by Allcock--Carlson--Toledo~\cite[(2.7.1)]{ACTsurf} to define the ball quotient model $\mathcal{B}_4/\Gamma_4$ for the moduli of cubic surfaces is
\begin{equation}
\Lambda = \mathcal{E}_1(-1)+4 \mathcal{E}_1
\end{equation}
with the associated $\mathbb{Z}$-lattice
$$
\Lambda_\mathbb{Z}=A_2+4A_2(-1)
$$
(see~\cite[\S5, \S6]{DvGK} for a discussion of the lattice $\Lambda_\mathbb{Z}$ and its relevance to the ball quotient construction).
Returning to the construction of $\mathcal{B}_4/\Gamma_4$, we recall
$$
\mathcal{B}_4:=\{[z]: z^2>0\}^+\subseteq \mathbb{P}(\Lambda \otimes_{\mathcal{E}}\mathbb{C}),
$$
and $\Gamma_4:=\operatornameeratorname{O}(\Lambda)$ acts naturally (properly discontinuously) on $\mathcal{B}_4$. Let us note that one can construct a natural $W(E_6)$-cover
\begin{equation}\label{markedball}
\mathcal{B}_4/\Gamma_4^m\to \mathcal{B}_4/\Gamma_4
\end{equation} of the ball quotient model parameterizing marked cubic surfaces (i.e., cubic surfaces with the $27$ lines labeled). This corresponds to an arithmetically defined normal subgroup $\Gamma_4^m\subset \Gamma_4$ with $\Gamma_4/\Gamma_4^m\cong \pm1\times W(E_6)$ (with $\pm 1$ acting trivially on $\mathcal{B}_4$); we refer to~\cite[\S6.10]{DvGK} and~\cite[\S3]{ACTsurf} for details.
\subsection{Identifying the cusp of $(\mathcal{B}_4/\Gamma_4)^*$}
From the description above, it is elementary to find a representative isotropic line $\mathcal{F}\subseteq \Lambda$ defining the cusp $c_{3A_2}$, namely the one generated by
$$
h=(1,1,0,0,0).
$$
One then sees immediately that
$$
h^\perp/h=3\mathcal{E}_1,
$$
and we recall then that $(3\mathcal{E}_1)_\mathbb{Z}=3A_2(-1)$.
\subsection{The isometry group of the cusp}
Clearly, $\operatornameeratorname{O}(\mathcal{E}_1)=\mathbb{Z}_3\times \mathbb{Z}_2$ (compare~\eqref{E:OOE1}), with $\mathbb{Z}_3$ acting by $\omega$ and $\mathbb{Z}_2$ acting by $-1$.
It is easy to see that
\begin{equation}\label{E:OO3EE1}
\operatornameeratorname{O}(3\mathcal{E}_1)=\operatornameeratorname{O}(\mathcal{E}_1)^{\times 3}\rtimes S_3=(\mathbb{Z}_3\times \mathbb{Z}_2)^{\times 3}\rtimes S_3
\end{equation}
where the semi-direct product is given by the action of $S_3$ on the three copies of $\operatornameeratorname{O}(\mathcal{E}_1)$.
\subsection{The structure of the toroidal boundary divisor}
We denote the boundary divisor of $\overline{\mathcal{B}_4/\Gamma_4}$
corresponding to the cusp $c_{3A_2}$ by $T_{3A_2}$.
\begin{lem}\label{lem_struc_tor_surf}
The following holds:
$$
T_{3A_2}\cong (E_\omega\otimes_{\mathcal{E}}3\mathcal{E}_1)/\operatornameeratorname{O}(3\mathcal{E}_1) \ \ (\cong (E_\omega^3)/\operatornameeratorname{O}(3\mathcal{E}_1)).
$$
\end{lem}
\begin{proof}
The proof is analogous to that of Proposition~\ref{prop_structure_tor}, with a minor difference.
To make this appendix accessible to readers who are primarily interested in cubic surfaces we will give a self-contained proof here, but also comment on the differences to the previous case.
We start with $\Lambda= \mathcal{E}_1(-1) + 4\mathcal{E}_1$ and
the isotropic vector $b_1:=h=(1,1,0,0,0)$.
We will denote the corresponding cusp given by the isotropic line $\mathcal{F}=\mathcal{E} h$ by $F$.
We then add $b_2,b_3,b_4$ where each $b_i$ is a generator of a copy of $3\mathcal{E}_1=h^{\perp}/h$, and complement this by
$b_5=(1,-1,0,0,0)$. The difference to Proposition~\ref{prop_structure_tor} is that this is a $\mathbb{Q}(\sqrt{-3})$-basis of $\mathcal{E}_1(-1) + 4\mathcal{E}_1$, and not an $\mathcal{E}$-basis. With respect to this
basis the hermitian form is given by
$$
Q=
\left(
\begin{array}{c|c|c}
0 & 0 & 6 \\ \hline
0 & B & 0\\ \hline
6 & 0 & 0
\end{array}
\right)
$$
where
$$
B=
\left(
\begin{array}{c|c|c}
3 & 0 & 0 \\ \hline
0 & 3 & 0\\ \hline
0 & 0 & 3
\end{array}
\right).
$$
In order to determine the structure of the boundary one first has to understand the structure of the
stabilizer subgroup $N(F)$ corresponding to $F$, i.e.~the subgroup of $\operatorname{O}(\Lambda)$ fixing the line spanned by $h$. A straightforward calculation, see~\cite[Sec.~4]{beh}, gives
\begin{equation}\label{pro:structureboundarycompA}
N(F)= \left\{ g \in \operatorname{O}(\Lambda): g= \left( \begin{array}{c|c|c}
u & v & w \\ \hline
0 & X & y \\ \hline
0 & 0 & s
\end{array}
\right)\right\}.
\end{equation}
Note that, in particular, this implies that $X\in \operatorname{O}(3\mathcal{E})$. Its unipotent radical is given by
\begin{equation}
W(F)= \left\{g \in N(F): g= \left(
\begin{array}{c|c|c}
1 & v & w \\ \hline
0 & 1 & y \\ \hline
0 & 0 & 1
\end{array}
\right) \right\}
\end{equation}
and finally the center of the unipotent radical is
\begin{equation}
U(F)= \left\{g\in W(F): g= \left(
\begin{array}{c|c|c}
1 & 0 & w \\ \hline
0 & 1 & 0 \\ \hline
0 & 0 & 1
\end{array}
\right), w \in \mathbb{Z} \right\} \cong \mathbb{Z}.
\end{equation}
We have natural coordinates coordinates $(z_0:z_1: z_2:z_3:z_{4})$ on $\mathcal{B} \subset \mathbb{P}(\Lambda\otimes_{\mathcal{E}}\mathbb{C})$ and we can assume that $z_4=1$. Then we obtain a map
\begin{equation}
\begin{aligned} \mathcal{B} &\to \mathbb{C}^* \times \mathbb{C}^3 \\ (z_0, z_1, z_2, z_{3}) &\mapsto (t_0=e^{2 \pi i z_0}, z_1, z_2, z_3)
\end{aligned}
\end{equation}
and adding the toroidal boundary amounts to adding $\{0\} \times \mathbb{C}^3$.
The quotient $N(F)/U(F)$ then acts on $\mathcal{B}/U(F)$ and this quotient gives the
toroidal compactification of $\mathcal{B}$ near the cusp $F$. Here we are only interested in the structure of the boundary divisor and hence in the action of $N(F)/U(F)$ on $\{0\} \times \mathbb{C}^3$.
By a straightforward calculation
\begin{equation}\label{equ:action2}
g=\left(
\begin{array}{c|c|c}
u & v & w \\ \hline
0 & X & y \\ \hline
0 & 0 & s
\end{array}
\right): \underline{z} \mapsto \frac{1}{s}(X\underline{z} + y)
\end{equation}
where $\underline{z}=(z_1, z_2, z_3)$. We first look at the normal subgroup $W(F)$, matrices whose elements act as follows
$$
g=\left(
\begin{array}{c|c|c}
1 & v & w \\ \hline
0 & 1 & y \\ \hline
0 & 0 & 1
\end{array}
\right): \underline{z} \mapsto \underline{z} + y.
$$
Since $g\in \operatorname{O}(\Lambda)$, we must necessarily have $y \in \mathcal{E}^3$ (where we now use the notation $\mathcal{E}^3$ rather than $3\mathcal{E}$ since we want to
emphasize the vector space structure rather than the lattice). This is where there is a difference to the case of cubic fourfolds: it is no longer true that all vectors in $\mathcal{E}^3$ appear as entries $y$ in matrices $g\in W(F)$.
Indeed by a straightforward calculation, see~\cite[Sec.~4]{beh}, the condition that $g\in \operatorname{O}(\Lambda)$ is
$$
By+6\bar{v}^t=0, \quad \bar{y}^tBy+6w + 6 \bar w=0.
$$
Given $y$ we want to define $v$ by $\bar{v}^t=-\frac{1}{6}By$. Since $By \in 3 \cdot \mathcal{E}^3$ and $v$ must be in $\mathcal{E}^3$ this requires that $y \in 2 \cdot \mathcal{E}^3$. Note that we can then also find a suitable $w\in \mathcal{E}$.
However, scaling the lattice by a factor $2$ gives isomorphic quotients showing
$$
\mathbb{C}^3/W(F)\cong (E_{\omega})^3.
$$
The rest of the argument is now again very close to Proposition~\ref{prop_structure_tor}.
Clearly, the subgroup
$$
\left\{ g \in \operatorname{O}(\Lambda): g= \left( \begin{array}{c|c|c}
1 & 0 & 0 \\ \hline
0 & X & 0 \\ \hline
0 & 0 & 1
\end{array}
\right)\right\}.
$$
acts on $(E_{\omega})^3$ as claimed in the proposition.
It remains to consider elements of the form
$$
g=\left(
\begin{array}{c|c|c}
u & 0 & 0 \\ \hline
0 & 1 & 0 \\ \hline
0 & 0 & s
\end{array}
\right) \in N(F).
$$
The condition that such a matrix lies in $\operatorname{O}(\Lambda)$ is that $s\bar{u}=1$ with $s\in \mathcal{E}$. Hence $s$ is a power of $\omega$ and these elements act on $(E_{\omega})^3$ by multiplication with powers of $\omega$. But by
(\ref{equ:action2}) these elements are already accounted for by matrices with $u=s=1$ and
$X \in \operatorname{O}(3\mathcal{E})$ and hence we do not get a further quotient. Thus the claim follows.
\end{proof}
\subsection{The cohomology of the toroidal boundary divisor}
It is elementary to see from the descriptions above that
$$
(\mathcal{E}_1\otimes_{\mathcal{E}}E_\omega)/\operatornameeratorname{O}(\mathcal{E}_1)\cong \mathbb{P}^1.
$$
It follows that
$$
T_{3A_2}=(3\mathcal{E}_1\otimes_{\mathcal{E}}E_\omega)/\operatornameeratorname{O}(3\mathcal{E}_1)=(\mathbb{P}^1)^3/S_3=\mathbb{P}^3.
$$
In particular, we get:
\begin{cor}\label{lem_coho_torbound_ball}
The Betti numbers of the toroidal boundary divisor $T_{3A_2}$ of $\overline{\mathcal{B}_4/\Gamma_4}$ are given by
$b_0( T_{3A_2})=b_2( T_{3A_2})=b_4( T_{3A_2})=b_6( T_{3A_2})=1$ and $b_1( T_{3A_2})=b_3( T_{3A_2})=b_5( T_{3A_2})=0$.
\end{cor}
\subsection{The cohomology of the toroidal compactification}\label{completeproof}
We can now complete the proof of Theorem~\ref{T:CubSurfH} using the Decomposition Theorem for the morphism $\overline {\mathcal{B}_4/\Gamma_4}\to (\mathcal{B}_4/\Gamma_4)^*$.
We have
\begin{align*}
P_t(\overline {\mathcal{B}_4/\Gamma_4}) \equiv\ &1+t^2+t^4& \text{($IP_t((\mathcal{B}_4/\Gamma_4)^*)$, from~\eqref{E:KirThmCubSurf})}\\
&\ +( t^2 +t^4)&\text{($T_{3A_2}=\mathbb{P}^3$ contribution, from Corollary \ref{lem_coho_torbound_ball})}\\
\equiv\ & 1+2t^2+2t^4 \mod t^{5}\!\!\!\!\!\!\!\!
\end{align*}
by applying equation~\eqref{eq:IHblowup} to determine the contribution to the cohomology of $\overline {\mathcal{B}_4/\Gamma_4}$ from the exceptional divisor.
\section{The cohomology of the Naruki compactification} For completeness, let us note that the cohomology of the Naruki compactification $\overline \mathcal{N}$ coincides with the cohomology of toroidal and Kirwan compactifications for the moduli of cubic surfaces.
\begin{pro}\label{T:Naruki}
The Betti numbers of the Naruki compactification $\overline \mathcal{N}=\widetilde \mathcal{N}/W(E_6)$ of the moduli space of cubic surfaces are as follows:
\begin{equation}\label{E:CubSurfHNar}
\begin{array}{r|ccccc}
\hskip2cm j&0&2&4&6&8\\\hline
\dim H^j(\overline{\mathcal{N}})&1&2&2&2&1 \rule{0pt}{2.8ex}
\end{array}
\end{equation}
while all the odd degree cohomology vanishes.
\end{pro}
\begin{proof}
The Naruki compactification $\widetilde \mathcal{N}$ is a modular compactification for the moduli of marked cubic surfaces. Clearly, $W(E_6)$ acts on $\widetilde \mathcal{N}$, and we have defined $\overline \mathcal{N}=\widetilde \mathcal{N}/W(E_6)$. On the other hand, as discussed above we recall that there exists a marked ball quotient model $\mathcal{B}_4/\Gamma_4^m$, which is a $W(E_6)$ cover of $\mathcal{B}_4/\Gamma_4$ (see~\eqref{markedball}).
Then, there exists a ($W(E_6)$-equivariant) period map $\widetilde \mathcal{N}\to (\mathcal{B}_4/\Gamma_4^m)^*$ contracting $40$ divisors $D_i$ in $\widetilde \mathcal{N}$ to the $40$ cusps of the Baily--Borel compactification $(\mathcal{B}_4/\Gamma_4^m)^*$ (see~\cite[\S2.9]{DvGK}\footnote{In~\cite{DvGK}, the image of the (extended) period map $\widetilde \mathcal{N}\to (\mathcal{B}_4/\Gamma_4^m)^*$ is denoted by $\mathcal{N}$. For consistency with our notations, a better notation would be $\mathcal{N}^*(=\mathcal{N})$. Of course, $\mathcal{N}^*=(\mathcal{B}_4/\Gamma_4^m)^*$ as the period map is surjective.}). Furthermore (cf. loc. cit.), $D_i\cong (\mathbb{P}^1)^3$ (and the singularities of $(\mathcal{B}_4/\Gamma_4^m)^*$ at the $40$ cusps are cones over the Segre embedding of $(\mathbb{P}^1)^3$). The $40$ exceptional divisors $D_i$ are conjugated under the action of $W(E_6)$. Thus, taking the quotient by $W(E_6)$, we obtain $\overline \mathcal{N}\to (\mathcal{B}_4/\Gamma_4)^*$, which contracts a divisor $D$ to the unique cusp of $(\mathcal{B}_4/\Gamma_4)^*$. Since $D$ is a quotient of $D_i\cong (\mathbb{P}^1)^3$ by a finite group that contains $S_3$ permuting the three $\mathbb{P}^1$ factors, it is immediate to see that $D$ has the rational cohomology of $\mathbb{P}^3$. The claim now follows as before (see \S\ref{completeproof}).
\end{proof}
\backmatter
\printindex
\end{document} |
\begin{document}
\title[Pontryagin space structure in RKHS's]{Pontryagin space structure in
reproducing kernel Hilbert spaces over $*$--semigroups}
\author[F.H. Szafraniec, M. Wojtylak]{ Franciszek Hugon Szafraniec \and Micha\l{} Wojtylak }
\address{F. H. Szafraniec, Instytut Matematyki, Uniwersytet
Jagiello\'nski, ul. \L ojasiewicza 6, 30 348 Krak\'ow, Poland
M. Wojtylak, Instytut Matematyki, Uniwersytet
Jagiello\'nski, ul. \L ojasiewicza 6, 30 348 Krak\'ow, Poland\\ VU University Amsterdam,
Department of Mathematics,
Faculty of Exact Sciences,
De Boelelaan 1081 a, 1081 HV Amsterdam\\
}
\textrm{e}mail{[email protected]}
\textrm{e}mail{[email protected]}
\thanks{The first author was supported by the MNiSzW grant
N201 026 32/1350. He also would like to acknowledge an assistance of
the EU Sixth Framework Programme for the Transfer of Knowledge
``Operator theory methods for differential equations'' (TODEQ) \#
MTKD-CT-2005-030042.}
\subjclass[2000]{primary: 43A35 \and 46C20 \and 47B32}
\keywords{$*$-semigroup \and shift operator \and Pontryagin space \and fundamental symmetry}
\begin{abstract}
The geometry of spaces with indefinite inner product, known also as
Krein spaces, is a basic tool for developing Operator Theory
therein. In the present paper we establish a link between this
geometry and the algebraic theory of $*$-semigroups. It goes via
the positive definite functions and related to them reproducing
kernel Hilbert spaces. Our concern is in describing properties of
elements of the semigroup which determine shift operators which
serve as Pontryagin fundamental symmetries.
\textrm{e}nd{abstract}
\maketitle
\section*{Introduction}
There are two ways of looking at $*$--semigroups and positive
definite functions defined on them. The first consists in
intense analysis of the algebraic structure of a semigroup so as
to establish conditions on it, which ensure prescribed properties to hold for \underbar{any}
positive definite functions. One of the
properties frequently considered is representing \underbar{each}
of the positive definite functions as moments of a positive
measure. This attitude has been successfully undertaken by
Bisgaard resulting in a considerable number of papers, see in
particular
\cite{biss-scmat,biss-two,biss-sep,biss-ext,biss-CC,biss-fact} and
references therein, some of them we are going to exploit here.
The other way is to determine \underbar{which} of the positive
definite functions posses desired properties; a typical example
within this category is to detect multidimensional moment
sequences.
In the present paper we are going to develop the first thread
putting forward the following problem. Impose necessary and sufficient conditions on a
distinguished element $u$ of a $*$--semigroup $S$ to generate a Pontryagin fundamental
symmetry of \underbar{any} reproducing kernel Hilbert space over the semigroup in question;
the precise formulation is exposed as ({\tt P}), p. \pageref{a}. Surprisingly, our problem
has found a simple algebraic solution. In the context of $*$-separative commutative
semigroups the condition on $u$ is: $u=u^*$, $u+u=0$ and $u+s\neq s$ for only a finite
number of $s\in S$ (see Proposition \ref{krein}, Theorem \ref{main}).
Let us mention that $*$-semigroups and positive definite functions on them have been
originated by Sz.-Nagy in his famous Appendix \cite{nagy}. He uses the reproducing kernel
Hilbert space factorization to prove his "general dilation theorem" for operator valued
functions. Since then the RKHS technique has been used from time to time for proving results
with dilation flavour behind the screen. We are going to follow suit here.
\section{Shift operators connected with positive definite
functions -- formulation of the problem}
By a $*$-semigroup we understand a commutative semigroup with an involution, not necessarily
having the neutral element. The involution is always denoted by the symbol ``$*$'' and the
semigroup operation is always written in an additive way. In the case when the
$*$--semigroup $S$ has a neutral element $0$ we say that $\phi:S\to\mathbb{C}$ is {\it positive
definite} (we write $\phi\in\Pp S$) if for every $N\in\mathbb{N}:=\set{0,1,\dots}$ and every
$s_0,\dots, s_N\in S$, $\xi_0,\dots, \xi_N\in\mathbb{C}$ we have $\sum_{i,j=0}^N\xi_i\bar\xi_j
\phi(s_j^*+s_i)\geq 0$. With such $\phi$ we link a reproducing kernel Hilbert space
$\mathcal{H}^\phi\subseteq \mathbb{C}^S$ with a reproducing kernel defined by $K^\phi(s,t):=\phi(t^*+s)$,
$t,s\in S$ (see e.g. ~\cite[p.81]{BChR}). For $s\in S$ we set $K^\phi_s:=K(\cdot,s)$ ($s\in
S$), it is known that $K^\phi_s\in\mathcal{H}^\phi$ and $f(s)=\seq{f,K^\phi_s}$ for every
$f\in\mathcal{H}^\phi$. The set $\mathcal{D}^\phi:=\textrm{lin}\set{K^\phi_s:s\in S}$ is dense in $\mathcal{H}^\phi$.
For an element $u\in S$ we define {\it the shift operator}
(sometimes called also the translation operator) $
A(u,\phi):\mathcal{D}^\phi\to\mathcal{D}^\phi$, by
$$
A(u,\phi)\left(\sum_j\xi_jK_{s_j}^\phi\right):=\sum_j\xi_jK_{s_j+u}^\phi.
$$
It can be shown that $A( u,\phi)$ is well defined, linear and
closable (see e.g. \cite[Proposition, p.253]{szaf-bdd},
\cite[p.90]{BChR}). Our main object will be the closure of the
above operator, denoted below by $u_\phi$. It is a matter of
simple verification that $\mathcal{D}^\phi$ is contained in the domain of
$(u_\phi)^*$ (the adjoint of the operator $u_\phi$) and
$(u_\phi)^*f=(u^*)_\phi f$ for $f\in\mathcal{D}^\phi$.
Under the above circumstances we can state our problem as follows.
\begin{enumerate} \label{a}
\item[({\tt P})] Provide necessary and sufficient conditions on
the element $u\in S$ for the operator $u_\phi$ to be a
fundamental symmetry of a Pontryagin space, i.e. to satisfy
$$
u_\phi=u_\phi^*,\quad u_\phi^2=I_{\mathcal{H}^\phi},\quad
\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty
$$
\underbar{for every} $\phi\in\Pp S$.
\textrm{e}nd{enumerate}
Obviously, the
condition $u_\phi=u_\phi^*$ together with $u_\phi^2=I_{\mathcal{H}^\phi}$ imply that $u_\phi$ must be bounded
on $\mathcal{H}$.
We continue with some basic definitions and notations concerning $*$-semigroups. Let $S$ and
$T$ be $*$-semigroups, a mapping $\chi:S\to T$ is called $*$-{\it homomorphism} if
$\chi(s+t)=\chi(s)+\chi(t)$ for all $s,t\in S$, $\chi(s^*)=(\chi(s))^*$ for all $s\in S$. A
{\it character} on $S$ is a nonzero $*$-homomorphism $\chi:S\to\mathbb{C}$ where the latter set
is understood as a semigroup with multiplication as the operation and the conjugation as
involution. It is obvious that if $S$ has a neutral element $0$ then $\chi(0)=1$ for all
$\chi\in S^*$. Let $\mathcal{A}(S^*)$ be the least $\sigma$-algebra of subsets of $S^*$ rendering
measurable all the functions
\begin{equation}\label{shat}
\mathcal{H}at s:S^*\ni\chi\mapsto\chi(s)\in\mathbb{C},\quad s\in S.
\textrm{e}nd{equation}
If $\mu$ is a
positive measure on $S^*$ such that all the functions listed in
(\ref{shat}) are square-integrable we define a function
$\mathcal{L}(\mu):S\to\mathbb{C}$ by
$\mathcal{L}(\mu)(s):=\int_{S^*}\sigma(s)d\mu(\sigma)$ ($s\in S$). We call
$\phi:S\to\mathbb{C}$ a {\it moment function} ($\phi\in\M S $) if
$\phi=\mathcal{L}(\mu)$ for some measure $\mu$ on $S^*$. It is easy to
verify that $\M S \subseteq\Pp S$, if the latter inclusion is an equality
then we call $S$ {\it semiperfect}. For examples of semiperfect and
non-semiperfect $*$-semigroups and more general concepts of
semiperfectness see ~\cite{biss-scmat}.
We call $S$ {\it $*$-separative} if the characters separate points in $S$. The {\it greatest
$*$-homomorphic $*$-separative image of $S$} is the semigroup $S/_\sim$, where the equivalence
relation $\sim$ on $S$ is defined by the condition that $s\sim t$ if and only if
$\sigma(s)=\sigma(t)$ for all $\sigma\in S^*$; addition and involution in $S/_\sim$ are those
that make the quotient mapping a $*$-homomorphism. The elements of $S/_\sim$ will be denoted
as equivalence classes $[s]$ ($s\in S$). If $S$ has a zero, then we will use the symbol $0$
for the neutral element of both $S$ and $S/_\sim$, instead of using $[0]$. Since every
character on $S$ generates a character on $S/_\sim$, the latter semigroup is in fact
$*$-separative. The following simple proposition gives answer to the question when the
operator $ u_\phi$ defined above is a fundamental symmetry of a Krein space, i.e. when $(
u_\phi)^2= u_\phi$ and $ u_\phi=( u_\phi)^*$.
\begin{prop}\label{krein}
Let $S$ be a commutative $*$-semigroup with zero. For each element $u\in S$ the following
conditions are equivalent:
\begin{itemize}
\item[(i)]{$[2u]= 0$ and $[u]=[u^*]$;}
\item[(ii)]{for every $\phi\in\M S $ we have $(u_\phi)^2=
I_{\mathcal{H}^\phi}$ and $(u_\phi)^*= u_\phi$;} \item[(iii)]{for every
$\phi\in\Pp S$ we have $(u_\phi)^2 = I_{\mathcal{H}^\phi}$ and $(u_\phi)^*=
u_\phi$;}
\textrm{e}nd{itemize}
Moreover, {\rm (i)} implies that $ u_\phi$ is a bounded, selfadjoint operator on $\mathcal{H}^\phi$ for every
$\phi\in\Pp S$.
\textrm{e}nd{prop}
\begin{proof}
(i)$ \Rightarrow$(iii) Suppose that (i) holds and let $\phi\in\Pp S$. For every $\sigma\in S^*$ we have
$\sigma(u)=\sigma(u^*)$ and $\sigma(2u)=\sigma(0)=1$. Consequently, for every $\sigma\in S^*$ and every $s\in
S$
\begin{equation}\label{sigma2}
\sigma(s+u)=\sigma(s+u^*),\quad\sigma(t+2u)=\sigma(t),\quad \sigma(s^*+u^*+u+s)=\sigma(s^*+s).
\textrm{e}nd{equation}
By \cite[Thm.2]{biss-sep} we have
\begin{equation}\label{sigma3-1}
\phi(s+u)=\phi(s+u^*),\quad s\in S,
\textrm{e}nd{equation}
\begin{equation}\label{sigma3-2}
\phi(s+2u)=\phi(s),\quad s\in S,
\textrm{e}nd{equation}
\begin{equation}\label{sigma3-3}
\phi(s^*+u^*+u+s)=\phi(s^*+s),\quad s\in S,
\textrm{e}nd{equation}
The last of these three equalities, together with \cite[Cor.1]{szaf-bdd}, implies that the
operator $u_\phi$ is in $\bold{B}(\mathcal{H}^\phi)$. It is also selfadjoint, since for $f=\sum_i \xi_i
K_{s_i}^\phi\in\mathcal{D}^\phi$ we have
$$
\seq{u_\phi f,f}=
\sum_{i,j}\xi_i\bar\xi_j\seq{K_{s_i+u}^\phi,K_{s_j}^\phi}=\sum_{i,j}\xi_i\bar\xi_j\phi(s_i+u+s_j^*)
\stackrel{(\ref{sigma3-1})}=
$$ $$
\sum_{i,j}\xi_i\bar\xi_j\phi(s_i+u^*+s_j^*)=
\seq{f, u_\phi f}.
$$
The fact that $(u_\phi)^2=I_\mathcal{H}^\phi$ can be obtained
similarly as selfadjointness of $u_\phi$, with the use of
(\ref{sigma3-2}) instead of (\ref{sigma3-1}). This finishes the
proof of (iii) and of the `Moreover' part of the proposition.
The implication (iii)$ \Rightarrow$(ii) is trivial. The proof (ii)$ \Rightarrow$(i) goes by contraposition.
Suppose first that $[2u]\neq 0$, i.e. there exists a character $\sigma$ such that
$\sigma(u)^2=\sigma(2u)\neq\sigma(0)=1$. We put $\phi:=\mathcal{L}(\delta_\sigma)$, where
$\delta_\sigma$ stands for the Dirack measure on $S^*$ concentrated in $\sigma$. Let
$\seq{\,\cdot\,,-}$ denote the scalar product on $\mathcal{H}^\phi$. Observe that
$$
\seq{K_0^\phi,K_0^\phi}=\phi(0)=\sigma(0)\neq\sigma(2u)=\phi(2u)=\seq{K_{2u}^\phi,K_0^\phi}=\seq{(
u_\phi)^2K_0^\phi,K_0^\phi}.
$$
In consequence, $I_{\mathcal{H}^\phi}\neq ( u_\phi)^2$. Similarly, if $\tau(u)\neq\tau(u^*)$ for some
$\tau\in S^*$ then for $\psi:=\mathcal{L}(\delta_\tau)$ the operator $u_\psi$ is not symmetric in
$\mathcal{H}^\psi$.
\textrm{e}nd{proof}
Let us consider now a situation when $S$ and $T$ are $*$-semigroups
with zeros and $h$ is a $*$-homomorphism from $S$ into $T$
satisfying $h(0)=0$. Note that if an element $u\in S$ is such that
$[2u]=0$, $[u]=[u^*]$ then $[2h(u)]=0$, $[h(u)]=[h(u)^*]$. This
comes from the fact that for every character $\sigma$ on $T$ the
function $\sigma\circ h$ is a character on $S$. Observe also that
$\phi\circ h\in\Pp S$ for every $\phi\in\Pp T$.
\begin{prop}\label{ST}
Assume that $S$, $T$ and $h$ are as above and that $h$ is
additionally onto. Let $u\in S$ be such that $[2u]=0$, $[u]=[u^*]$
and let $\phi\in\Pp T$. Then the operators $u_{\phi\circ h}$ in
$\mathcal{H}^{\phi\circ h}$ and $h(u)_{\phi}$ in $\mathcal{H}^\phi$ are unitarily
equivalent.
\textrm{e}nd{prop}
\begin{proof}
Let $\seq{\cdot,-}_\phi$ and
$\seq{\cdot,-}_{\phi\circ h}$ denote the scalar products on $\mathcal{H}^\phi$ and $\mathcal{H}^{\phi\circ h}$
respectively. Since
\begin{eqnarray*}
\seq{\sum_{j=1}^N \xi_j K^{\phi\circ h} _{s_j} ,\sum_{j=1}^N
\xi_j K^{\phi\circ h} _{s_j}}_{\!\!\phi\circ h}
&=& \sum_{i,j=1}^N \xi_i\bar\xi_j (\phi\circ h)(s_i+s_j^*) \\ =
\sum_{i,j=1}^N \xi_i\bar\xi_j \phi(h(s_i)+h(s_j^*))&=&
\seq{\sum_{j=1}^N \xi_j K^{\phi} _{h(s_j)},\sum_{j=1}^N \xi_j
K^{\phi} _{h(s_j)}}_{\!\!\phi},
\textrm{e}nd{eqnarray*}
the condition $ V( K^{\phi\circ h}_{s}):= K^{\phi}_{h(s)} $ ($s\in
S$) properly defines an isometry between $\mathcal{H}^{\phi\circ h}$ and
$\mathcal{H}^{\phi}$. Since $h$ is onto, the range of $V$ is dense in
$\mathcal{H}^\phi$, and so $V$ is a unitary operator. To finish the proof we
need to show that
\begin{equation}\label{VuuV}
h(u)_\phi Vf=V u_{\phi\circ h}f, \quad f\in\mathcal{H}^{\phi\circ h}.
\textrm{e}nd{equation}
This can be easily verified for $f\in\mathcal{D}^{\phi\circ h}$. Since all
the operators appearing in (\ref{VuuV}) are bounded, the proof is
finished.
\textrm{e}nd{proof}
Applying the above to the quotient semigroup $T=S/_\sim$ and $h$ as
the quotient mapping, together with the fact from \cite{biss-fact}
that every $\phi\in\Pp S$ ($\phi\in\M S$) is of the form
$\phi=\psi\circ h$ for some $\psi\in\Pp{S/_\sim}$ ($\psi\in\MS/_\sim$)
gives the following.
\begin{cor}\label{Nowyjeszcze}
Assume that $S$ is a $*$-semigroup with zero and $u\in S$ is such
that $[2u]=0$, $[u]=[u^*]$. Then
\begin{eqnarray*}
\dim\ker(u_\phi+I)&<&+\infty \textrm{ for every }\phi\in\Pp S\,\, (\phi\in\M S)\\
\iff \dim\ker([u]_{\psi}+I)&<&+\infty\textrm{ for every
}\psi\in\PpS/_\sim\,\, (\psi\in\MS/_\sim)
\textrm{e}nd{eqnarray*}
\textrm{e}nd{cor}
\section{Examples}
\begin{exm}\label{exm1}
Consider the semigroup $S={\mathbb{Z}}_2\times\mathbb{N}$ with standard addition
and the identical involution. As usually (cf. \cite{BChR}) we
identify $S^*$ with $\set{-1,1}\times\mathbb{R}$, note that $S$ is
$*$-separative. The only nonzero element satisfying $u=u^*$, $2u=0$
is $u=(1,0)$. Let $\phi$ be a positive definite mapping, we will now
compute the eigenspaces of $ u_\phi$. Since $S$ is semiperfect
(\cite{biss-two}), there exists a Borel measure $\mu$ on $S^*$ such
that $\phi=\mathcal{L}(\mu)$. Having our interpretation of characters in
mind we get
$$
\phi(x,n)=\int_\mathbb{R} (-1)^xt^nd\mu_-(t)+\int_\mathbb{R} t^nd\mu_+ (t),\quad
x\in{\mathbb{Z}}_2,\,n\in\mathbb{N},
$$
with $\mu_\pm:=\mu\rest{{\set{\pm1}}\times\mathbb{R}}$. Let us define the functions
$f_{x,n}:S^*\to S^*$ ($x\in{\mathbb{Z}}_2$, $n\in\mathbb{N}$) by
$$
f_{x,n}(\textrm{e}ps,t):=\textrm{e}ps^xt^x ,\quad \textrm{e}ps\in\set{-1,1},\,t\in\mathbb{R},\,x\in{\mathbb{Z}}_2,\,n\in\mathbb{N},
$$
and note that they all are square integrable. By $\mathcal P^\mu$ we define the closure in
$L^2(\mu)$ of the linear span of the functions $f_{x,n}$ ($x\in{\mathbb{Z}}_2$, $n\in\mathbb{N}$). The
formula
\begin{equation}\label{Vdef}
V(K^\phi_{x,n}):=f_{x,n},\qquad x\in{\mathbb{Z}}_2,\,n\in\mathbb{N},
\textrm{e}nd{equation}
constitutes a unitary isomorphism between $\mathcal{H}^\phi$ and $\mathcal P^\mu$. The shift operator
$u_\phi$ is unitary equivalent (via $V$) to the following operator $M$
$$
(M f)(\textrm{e}ps,t):=\textrm{e}ps f (\textrm{e}ps,t),\qquad (\textrm{e}ps,t)\in\set{-1,1}\times\mathbb{R},\, f\in\mathcal{P}^\mu.
$$
It is not hard to see that
$$
\ker(M\pm I)=\set{f\in \mathcal{P}^\mu: f(\pm1,\cdot)=0\,\,\, \,
\mu_\pm\textrm{--a.e}}.
$$
Note that $\dim\ker(u_\phi\pm I)=\dim\ker(M\pm I)$ and the latter is finite dimensional if
and only if the support of $\mu_{\mp}$ is a finite set. In particular there exists a mapping
$\phi\in\Pp S=\M S $ such that $\dim\ker( u_\phi+I)=\infty$.
\textrm{e}nd{exm}
At this point we present a useful for our purposes construction. Let $S$ and $T$ be two
disjoint $*$-semigroups (which only a formal restriction) and let $h:S\mapsto T$ be a
$*$-homomorphism. We endow the set $S\cup T$ with the $*$-semigroup structure in the
following way. The addition on $S\cup T$, denoted by the same symbol `$+$', is defined by
$$
s+t:=\left\{\begin{array}{rcl}
s+t & : & s,t\in S\textrm{ or } s,t\in T\\
s+h(t) & : & s\in S,\, t\in T\\
t+h(s) & : & t\in S,\, s\in T\\
\textrm{e}nd{array}\right.
$$
The involution on $S\cup T$ (still denoted by `$*$') is such that its restriction to both
$S$ and $T$ is the original involution on $S$ and $T$, respectively. We denote the above
constructed semigroup by $\mathrm{U}(S,T,h)$. The reader can easily check a general fact,
that if $S$ and $T$ are $*$-separative then $\mathrm{U}(S,T,h)$ is $*$-separative as well.
\begin{exm}\label{exm2} Consider a semigroup $S=\mathrm{U}({\mathbb{Z}}_2,\mathbb{N},h_0)$
where $h_0(x)=0$ ($x\in {\mathbb{Z}}_2$).
The element $0_{{\mathbb{Z}}_2}$ is the neutral element of $S$. Take $u:=1_{{\mathbb{Z}}_2}$, clearly
$u=u^*$ and $2u=0_{{\mathbb{Z}}_2}$. Let $\phi$ be any positive definite function on $S$ and suppose
that $f\in\ker( u_\phi+I)$. This means that for $n\in\mathbb{N}$
$$
f(n)=f(n+u)=\seq{f,K^\phi_{n+u}}=\seq{f,u_\phi K^\phi_n}=\seq{u_\phi f, K^\phi_n}=-f(n),
$$
hence $f\rest\mathbb{N}=0$. A similar calculation shows that $f(u)=-f(0_{{\mathbb{Z}}_2})$. Hence,
the eigenspace $\ker( e_\phi+I)$ is spanned by the single function
$$
f(s)=\left\{\begin{array}{rcl}
0 &:& s\in\mathbb{N}\\
1 & : & s=0_{{\mathbb{Z}}_2}\\
-1 & : & s=1_{{\mathbb{Z}}_2}
\textrm{e}nd{array} \right.
$$
if $f\in\mathcal{H}^\phi$ or is trivial otherwise. Resuming, $\dim\ker(u_\phi+I)\leq 1$ for all
positive definite $\phi$.
\textrm{e}nd{exm}
\section{Main result}
\begin{thm}\label{main}
Let $S$ be a commutative $*$-semigroup with zero and let $u\in S$ be such that $[2u]=0$ and
$[u]=[u^*]$. Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] the set $\set{[s]\in S/_\sim\colon [u+s]\neq[s]}$ is
finite\,\footnote{\ Cf. Remark \ref{stupidstyle}.};
\item[(ii)]{ $\sigma(u)=-1$ for only finitely many $\sigma\in S^*$;}
\item[(iii)]{$\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty$ for all $\phi\in\M S $;}
\item[(iv)]{$\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty$ for all $\phi\in\Pp S$.}
\textrm{e}nd{itemize}
Moreover, if {\rm (i)} holds then $\dim\ker(u_\phi+I_{\mathcal{H}^\phi})$ is
less or equal to the half of the number elements of the set
mentioned in {\rm (i)}.
\textrm{e}nd{thm}
Let us stress here the solution of the main problem of the paper,
which follows from Proposition~\ref{krein} and Theorem~\ref{main} :
{\it the operator $ u_\phi$ is a fundamental symmetry of a
Pontryagin space for every $\phi\in\Pp S$ $(\!\!$ equivalently:
$\phi\in\M S$$)$ if and only if $[2u]=0$, $[u]=[u^*]$ and the set
$\set{[s]\inS/_\sim:[u+s]\neq[s]}$ is finite.}
\begin{rem}\label{stupidstyle}
Condition (ii) is equivalent to
\begin{itemize}
\item[(ii')]{\it $\sigma([u])=-1$ for only finitely many characters $\sigma$ on $S/_\sim$,}
\textrm{e}nd{itemize}
since the characters on $S$ and $S/_\sim$ are in one-to-one
correspondence. This, together with Corollary \ref{Nowyjeszcze} (and
the remarks above it) allows us to reduce the proof of Theorem
\ref{main} to the case when $S$ is $*$-separative. However, note
that the conditions (i)--(iv) are \underbar{not} equivalent to the
following:
\begin{itemize}
\item[]{\it the set $\set{s\in S: u+s\neq s}$ is finite.}
\textrm{e}nd{itemize}
Indeed, consider the following example. Let $S={\mathbb{Z}}_4\times\mathbb{N}$
with the natural operation $+$ and the identical involution. The
greatest $*$-separative homomorphic image of $S$ is ($*$-isomorphic
with) ${\mathbb{Z}}_2\times\mathbb{N}$ and the quotient homomorphism maps
$u:=(2,0)$ to $(0,0)$. We have that $u+s\neq s$ for all $s\in S$ but
the set mentioned in (i) is empty. It remains on open problem if the
condition
\begin{itemize}
\item[]{\it the set $\set{s\in S: [u+s]\neq [s]}$ is finite.}
\textrm{e}nd{itemize}
is equivalent to (i).
\textrm{e}nd{rem}
Before the proof we introduce the notion of a $*$-archimedean
component of a semigroup. We call a $*$-semigroup $H$ (not
necessarily with 0) {\it $*$-archimedean} if for all $s,t\in H$
there exists $m\in\mathbb{N}\setminus\set0$ such that $m(s+s^*)\in t+ H$.
An {\it $*$-archimedean component} of a $*$-semigroup $S$ is a
maximal $*$-archimedean $*$-subsemigroup. Though $*$-archimedean
component is $*$-semigroup for itself it is possible for it not to
have the neutral element even if $S$ does. It can be shown that two
elements $s,t$ belong to the same $*$-archimedean component of $S$
if and only if $m(s+s^*)\in t+S$ and $n(t+t^*)\in s+S$ for some
$m,n\in\mathbb{N}\setminus\set0$. Furthermore, $S$ is the disjoint union
of its $*$-archimedean components, see \cite[Section 4.3]{clipres}
for the case of identical involution. The following Lemma was proven
in \cite{biss-ext} (Lemma 2), the proof for an arbitrary involution
requires minimal effort.
\begin{lem}\label{extchar}
If $H$ is a $*$-archimedean component of a $*$-semigroup $S$ then every character on $H$ is
everywhere nonzero and extends to a character on $S$.
\textrm{e}nd{lem}
If $H$ and $K$ are two
$*$-archimedean components of $S$ then $H+K$ is contained in one single $*$-archimedean
component of $S$. If $(S_i)_{i\in I}$ is the family of all $*$-archimedean components of $S$
then we define the operation $+$ on $I$ by:
$$
i+j=k\textrm{ if and only if } S_i+S_j\subseteq S_k,\quad i,j,k\in I.
$$
Since $S_i+S_i\subseteq S_i$ for all $i\in I$ we have that $i+i=i$. Therefore $I$ is a
semilattice with the natural partial order given by the condition that $i\leq j$ if and only
if $i+j=j$. The following easy lemma is left as an exercise for the reader.
\begin{lem}\label{elat}
Let $S$ be a $*$-semigroup with zero and let $(S_i)_{i\in I}$ be the family of all
$*$-archimedean components $S$. If $u\in S$ be such that $2u=0$, then $u$ belongs to the
same $*$-archimedean component as 0. In particular, $u+S_i\subseteq S_i$ for all $i\in I$.
\textrm{e}nd{lem}
\begin{proof}[Proof of Theorem ~\ref{main}]
As it was said in Remark \ref{stupidstyle} we may assume that $S$ is
$*$-separative. To prove (i)$ \Rightarrow$(iv) let us put
$$
U:=\set{ s\in S: u+ s= s}
$$
and suppose that the set $ S\setminus U$ contains only a
finite number $M$ of elements. Take $\phi\in\Pp S$. We show that
$$
\dim\ker( u_\phi+I)\leq M/2,
$$
this will also prove the last statement of Theorem \ref{main}. Let us fix an arbitrary
$f\in\ker( u_\phi+I)$. We have
$$
f( s+ u)=\seq{f,u_\phi K_{s}^\phi}=\seq{u_\phi f,K_s^\phi}=-f( s),\quad s\in S.
$$
This means
that $f\rest{ U}\textrm{e}quiv 0$. Observe that the relation
$$
aRb \Longleftrightarrow (a=u+b\textrm{ or }a=b)
$$
is an equivalence relation on $ S\setminus U$ and that the each equivalence class contains
exactly two elements. Take any representees $s_1,\dots, s_{M/2}$ of the equivalence classes of
$R$. It is clear, that
$$
\ker( u_\phi+I)\subseteq\textrm{lin}\set{\delta_{s_i}-\delta_{s_i+u}:i=1,\dots, M/2}.
$$
Consequently $\dim\ker( u_\phi+I)\leq M/2$.\\
(iv)$ \Rightarrow$(iii) is obvious. (iii)$ \Rightarrow$(ii) Suppose that $\set{\sigma_n:n\in\mathbb{N}}$ is an
infinite set of characters satisfying $\sigma_n(u)=-1$ ($n\in\mathbb{N}$). We define a measure
$\mu$ on $S^*$ by $\mu:=\sum_{n=0}^\infty 2^{-n} \delta_{\sigma_n}$ and we take a mapping
$\phi=\mathcal{L}(\mu)$. Note that for every $N\in\mathbb{N}$ and for every $s_0,\dots, s_N\in S$,
$\xi_0,\dots,\xi_N\in\mathbb{C}$ we have
$$
\bigg|\sum_{j=0}^N\xi_j\sigma_n(s_j)\bigg|^2\leq
2^n\int_{S^*}\bigg|\sum_{j=0}^N\xi_j\sigma(s_j)\bigg|^2d\mu(\sigma)=
2^n\sum_{i,j=0}^N\xi_i\bar\xi_j\int_{S^*}\sigma(s_i+s_j^*)d\mu(\sigma)=
$$ $$
=2^n\sum_{i,j=0}^N\xi_i\bar\xi_j\phi(s_i+s_j^*),\quad n\in\mathbb{N}.
$$
By the RKHS Test (\cite{test} and also \cite{test2}) we get that $\sigma_n\in\mathcal{H}^\phi$ for
$n=1,2,\dots$. Now observe that
$$
u_\phi(\sigma_n)(s)=\sigma_n(u+s)=-\sigma_n(s),\quad s\in S,n=1,2\dots.
$$
Therefore $\sigma_n\in\ker( u_\phi+I)$, $n=1,2,\dots$. It remains to
show that the functions $\sigma_n$, $n\in\mathbb{N}$, are linearly
independent. But this results from the well known fact that all
characters are linearly independent\,\footnote{\ We can use the
following argument: For every $s\in S$ the function $\mathcal{H}at s$ is a
character on the dual semigroup $S^*$ and it is trivial that the
family $(\mathcal{H}at s)_{s\in S}$ separates elements of $S^*$. Proposition
2 of \cite{lacunae} (see also \cite[Proposition 6.1.8]{BChR}) says
that if $T$ is a semigroup and $C\subseteq T^*$ separates points, then
the functions $\mathcal{H}at t\rest C$, $t\in T$ are linearly independent in
$\mathbb{C}^C$. We use this result for $T=S^*$ and $C=\set{\mathcal{H}at s:s\in
S}$, the functions $\mathcal{H}at\sigma\rest C$ can
be identified with characters on $S$.}.\\
(ii)$ \Rightarrow$(i) Suppose that the number of elements $ s\in S$ satisfying $ u+ s\neq s$ is
infinite. We show that there exists infinitely many characters $\sigma$ on $ S$ such that
$\sigma( u)=-1$.
Let $( S_i)_{i\in I}$ be the family of all $*$-archimedean components of $ S$. Set
$$
I_0:=\set{i\in I: u+ s\neq s\textrm{ for some } s\in S_i}.
$$
Our assumption implies that\,\footnote{\ Remark \ref{eitheror} shows
that it is even equivalent to} either
\begin{equation}\label{Case1}
\textrm{ $ S_j$ is infinite for some $j\in I_0$}
\textrm{e}nd{equation}
or
\begin{equation}\label{Case2}
I_0\textrm{ is infinite}.
\textrm{e}nd{equation}
Let us first assume (\ref{Case1}). Take $s_0\in S_j$ such that
$u+s_0\neq s_0$. By Lemma \ref{elat} we have $ u+ s_0\in S_j$. The
$*$-semigroup $S_j$ is $*$-separative as a subsemigroup of a
$*$-separative semigroup $S$. Therefore, there exists a character
$\sigma_0$ on $ S_j$ such that $\sigma_0( u+ s_0)\neq\sigma_0( s_0).
$
By
Lemma \ref{extchar} $\sigma_0$ extends to some character $\tilde{\sigma_0}$ on $ S$. Since $
u= u^*$ and $2 u=0$ we have $\tilde{\sigma_0}( u)\in\set{-1,1}$. But
$$
\tilde{\sigma_0}( u)\sigma_0( s_0)=\sigma_0( u+ s_0)\neq\sigma_0( s_0).
$$
Hence, $\tilde{\sigma_0}( u)=-1$, which means that $\sigma_0( u+ s_0)=-\sigma_0( s_0)$.
Denote by $A$ the set of all those characters $\sigma$ on $ S_j$ satisfying $\sigma( u+
s_0)=-\sigma( s_0)$. Since $\sigma_0$ is everywhere nonzero on $ S_j$ (Lemma \ref{extchar}),
the mapping
$$
S_j^*\ni\sigma\longmapsto \sigma_0\sigma\in S_j^*
$$
is bijective. Moreover, it maps $A$ onto $ S_j^*\setminus A$. Since $ S_j$ is infinite and
$*$-separative, $ S_j^*$ is infinite as well. Hence, $A$ is infinite. By Lemma
\ref{extchar}, there is an infinite number of characters $\sigma$ on $ S$ satisfying
$\sigma( u+ s_0)=-\sigma( s_0)$ and consequently $\sigma( u)=-1$.\\
Let us assume now (\ref{Case2}). For each $i\in I_0$ we take a
character $\sigma_i$ on $ S$ satisfying $\sigma_{i}( u)=-1$,
$\sigma_{i}( s)\neq 0$ for $s\in S_{i}$ (such a character exist by
repeating the proof from the previous case). We also define a family
of characters $\chi_{i}\in S^*$ (${i\in I_0}$) by
$$
\chi_{i}( s)=\left\{\begin{array}{rl} 1 & \textrm{ if } s\in S_j\textrm{ and }j\leq i\\
0 & \textrm{
otherwise}
\textrm{e}nd{array}\right.,\qquad s\in S,\,i\in I_0.
$$
Finally, we put $\rho_i:=\sigma_i\chi_i$ ($i\in I_0$). By Lemma \ref{elat} we have that $ u$
is in the same $*$-archimedean component as 0, denote this component by $ S_{j_0}$. It is
clear that $j_0\leq i$ for all $i\in I$, therefore $\chi_i( u)=1$ and $\rho_i( u)=-1$ for
all $i\in I_0$. The only thing that lasts is to show that $\rho_i\neq\rho_j$ for $i\neq j$.
If $i\neq j$ then, by symmetry, we can assume that $j\nleq i$. Thus $\chi_i\rest{ S_j}=0$
and $\rho_i\rest{ S_j}=0$. But $\chi_j\rest{ S_j}\textrm{e}quiv1$ and $\sigma_j$ is, by definition,
everywhere nonzero on $ S_j$. Therefore
$\rho_i\rest{ S_j}\textrm{e}quiv0\neq\rho_j\rest{ S_j}$.\\
\textrm{e}nd{proof}
\begin{rem}\label{eitheror}
The alternative of (\ref{Case1}) and (\ref{Case2}) in the proof of
(ii)$ \Rightarrow$(i) becomes more clear if we observe that $ u+ s\neq s$
for \underbar{all} $ s\in S_i$, provided that $i\leq j\in I_0$.
Indeed, suppose that $ u+ s= s$ for some $ s\in S_i$, $ u+ s_0\neq
s_0$ for some $ s_0\in S_j$ and $i\leq j$. This gives us
$$
( u+ s_0)+( s+ s_0)= u+ s+ s_0+ s_0= s_0+( s+ s_0).
$$
By Lemma \ref{elat} we have
$ u+ s_0\in S_j$. Moreover, $ s+ s_0\in S_j$ because $i\leq j$. The
semigroup $ S_j$ is cancellative as a $*$-archimedean component of a
$*$-separable group (see \cite[p.63]{biss-CC}). This gives us $ u+
s_0= s_0$, contradiction. The example below shows that both
(\ref{Case1}) and (\ref{Case2}) are possible.
\textrm{e}nd{rem}
\begin{exm}
Let $S={\mathbb{Z}}_2\times\mathbb{N}$, with the natural addition on ${\mathbb{Z}}_2$ and
$\mathbb{N}$ and the identical involution. The element $u=(1,0)$ is like
in (\ref{Case1}).
Let us now consider the semigroup $T={\mathbb{Z}}_2\times\mathbb{N}$, with the
natural addition on ${\mathbb{Z}}_2$ and maximum as the operation on $\mathbb{N}$,
the involution is again set to identity. It is easy to see that $T$
is $*$--separable. The element $u=(1,0)$ is such that (\ref{Case2})
is satisfied. This example shows one more thing. Namely, the
condition
\begin{itemize}
\item[]{ \it $\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty$ for all $\phi\in\M S $ of compact support}
\textrm{e}nd{itemize}
is \underbar{not} equivalent to any of the conditions of Theorem \ref{main}. Indeed, the
characters on $T$ form a discrete, enumerable set. If the mapping $\phi\in\M S $ is
compactly supported then it is finitely supported and consequently the space $\mathcal{H}^\phi$ is
finite dimensional. Hence, $u=(1,0)$ satisfies the above condition, but does not satisfy
(i).
Note that in Proposition \ref{krein} restricting to compactly
supported moment functions is possible because the function $\phi$
constructed in the proof (ii)$ \Rightarrow$(i) is supported by only one
character.
\textrm{e}nd{exm}
\section{Functions with a finite number of negative squares}
The condition $\dim\ker( u_\phi+I_{\mathcal{H}^\phi})<\infty$ can be written also in the language of
negative squares. Precisely speaking, by {\it the number of negative squares of a mapping }
$\psi:S\to \mathbb{C}$ satisfying
\begin{equation}\label{symmetric}
\psi(s)=\overline{\psi(s^*)}, \qquad s\in S,
\textrm{e}nd{equation}
we understand the maximum, taken over all numbers $N\in\mathbb{N}$ and
all sequences $s_0,\dots, s_N$, of the number of negative eigenvalues
of the symmetric matrix $\left(\psi(s_i+s_j^*)\right)_{i,j=0}^N$.
Note that if $[u]=[u^*]$ and $\phi\in\Pp S$ then the mapping
$\psi:=\phi(\cdot+u)$ satisfies (\ref{symmetric}). Indeed, take any
character $\sigma$. Then
$\sigma(s+u)=\sigma(s)\sigma(u)=\sigma(s)\sigma(u^*)=\sigma(s+u^*)$.
By the result of \cite{biss-sep} we get $\phi(s+u)=\phi(s+u^*)$.
Combining this with $\phi(t)=\overline{\phi(t^*)}$ ($t\in S$)
\cite[4.1.6]{BChR} proofs the claim.
\begin{prop}\label{(v)}
Let $S$ be a $*$--semigroup with zero and let $u\in S$ be such that $[2u]=0$ and $[u]=[u^*]$
and let $\phi\in\Pp S$. Then the number of negative squares of the mapping $\phi(\cdot+u)$
equals $\dim\ker(u_\phi+I)$. Consequently, conditions {\rm (i)--(iv)} of Theorem \ref{main}
are equivalent to each of the following:
\begin{itemize}
\item[(v)] the mapping $\phi(\cdot+u)$ has a finite number of negative squares for every $\phi\in\Pp S$;
\item[(vi)] the mapping $\phi(\cdot+u)$ has a finite number of negative squares for every $\phi\in\M S $.
\textrm{e}nd{itemize}
\textrm{e}nd{prop}
\begin{proof} (cf. \cite{berg,langerio,sasvari} for similar arguments) First let
us assume that $\dim\ker(u_\phi+I)=m\in\mathbb{N}$. Consider the indefinite inner product space
$(\mathcal{H}^\phi, \seq{u_\phi\,\cdot\,,\,\cdot\,})$. Since $\mathcal{D}^\phi$ is dense in $\mathcal{H}^\phi$ we can
find elements $s_1,\dots, s_k\in S$ and vectors $\alpha^i=(\alpha^i_1,\dots,
\alpha^i_k)\in\mathbb{C}^k$ ($i=1,\dots, m$) such that the elements
\begin{equation}\label{fi}
f^i:=\sum_{j=1}^k\alpha^i_j K^\phi_{s_j}\qquad (i=1,\dots, m)
\textrm{e}nd{equation}
span an $m$-dimensional negative subspace (\cite[Theorem IX.1.4]{bognar}). Let
\begin{equation}\label{A}
A:=\left(\phi(s_j+s_{j'}^*+u)\right)_{j,j'=1}^k\in\mathbb{C}^{k\times k}.
\textrm{e}nd{equation}
Note that for $i,l=1,\dots, m$ we have
\begin{equation}\label{fitransf}
\seq{A \alpha^l,\alpha^i}=
\sum_{j,j'=1}^k\alpha^{l}_j\overline{\alpha^{i}_{j'}}K^\phi(s_j+u,s_{j'})
=\seq{u_\phi f^l,f^i}.
\textrm{e}nd{equation}
Hence, the subspace $\textrm{lin}\set{\alpha^1,\dots,\alpha^k}$ is a negative subspace of the
indefinite inner product space $(\mathbb{C}^m,\seq{A\cdot,\cdot})$. Since $f^1,\dots, f^m$ are
linearly independent, the vectors $\alpha^1,\dots,\alpha^m$ are linearly independent as well.
Therefore, the matrix $A$ has at least $m$ negative eigenvalues.
Now let us assume that for some choice of $s_1,\dots, s_k\in S$ the
matrix $A$ defined as in (\ref{A}) has $m$ negative eigenvalues.
Then there exists linearly independent vectors
$\alpha^i=(\alpha^i_1,\dots, \alpha^i_k)\in\mathbb{C}^k$ ($i=1,\dots, m$) such
that
\begin{equation}
\seq{A \alpha^l,\alpha^i}=\delta_{il}\lambda_i\qquad
i,l=1,\dots, m,
\textrm{e}nd{equation}
with some $\lambda_1,\dots,\lambda_m<0$. We define $f_1,\dots, f_m$ as in
(\ref{fi}) (with the new meaning of $s_1,\dots, s_m$). Using the
calculation in (\ref{fitransf}) we get that the space
$\textrm{lin}\set{f_1,\dots, f_m}$ is a negative subspace of $(\mathcal{H}^\phi,
\seq{u_\phi\cdot,\cdot})$. We show now that $f_1,\dots, f_m$ are
linearly independent. If $\sum_{i=1}^m \beta_i f_i=0$ for some
$\beta_1,\dots,\beta_m\in\mathbb{C}$ then, by (\ref{fitransf}),
$$
\seq{A \sum_{i=1}^m\beta_i\alpha^i,\sum_{i=1}^m\beta_i\alpha^i}=\seq{u_\phi \sum_{i=1}^m
\beta_i f_i, \sum_{i=1}^m \beta_i f_i}=0.
$$
But $A$ is strictly negative on $\textrm{lin}\set{\alpha^1,\dots,\alpha^m}$ and $\alpha_1,\dots,\alpha_m$
are linearly independent. Hence, $\beta_1=\cdots=\beta_m=0$.
\textrm{e}nd{proof}
\section{More examples}
The reader can easily check that Theorem ~\ref{main} can be applied to Examples ~\ref{exm1}
and~\ref{exm2}. The next example concerns the estimation of the dimension of the eigenspace
in Theorem ~\ref{main}. We will show that this dimension can be any number between $0$ and
$M/2$, where $M$ is defined as in the proof of Theorem 2.
\begin{exm}
Let $S={\mathbb{Z}}_2^m$ with identical involution and let $u=(1,0,0,\dots,0)$.
We have $M=2^m$. The dual semigroup $S^*$ can be identified with
$\set{-1,1}^m$. There are $2^{m-1}$ characters $\sigma$ on $S$
satisfying $\sigma(u)=-1$ and $2^{m-1}$ characters $\sigma$
satisfying $\sigma(u)=1$. We denote those characters by
$\sigma_1,\dots,\sigma_{2^{m-1}}$ and $\rho_1,\dots,\rho_{2^{m-1}}$,
respectively. For fixed $k,l\in\set{0,\dots, 2^{m-1}}$ we
put\,\footnote{\ We use the convention $\sum_{i=1}^0 a_i:=0$ }
$\mu:=\sum_{i=1}^k \delta_{\sigma_i}+\sum_{j=1}^{l}\delta_{\rho_j}$
and $\phi:=\mathcal{L}(\mu)$. Since the support of the measure is consists
of $k+l$ points, the space $\mathcal{H}^\phi$ is $k+l$ dimensional. To see
this one can use the interpretation of $\mathcal{H}^\phi$ as $\mathcal
P^\mu$, as in Example \ref{exm1}. Now let us observe that
\begin{equation}\label{sigmaker}
\sigma_1,\dots,\sigma_k\in\ker( u_\phi+I),\quad
\rho_1,\dots,\rho_{l}\in\ker( u_\phi-I),
\textrm{e}nd{equation}
by the same
argument as in the proof of Theorem \ref{main} (iii)$ \Rightarrow$(ii). Since the characters are
always linearly independent we get that $\dim\ker( u_\phi+I)\geq k$ and $\dim\ker(
u_\phi-I)\geq l$. But the eigenspaces corresponding to $-1$ and $1$ are orthogonal, thus
$\dim\ker( u_\phi+I)=k$ and $\dim\ker( u_\phi-I)=l$.
Let us put $e=(0,1,0,\dots, 0)$ and take two numbers $l_1\in\set{0,\dots, 2^{m-1}}$ and
$l_2\in\set{0,\dots, 2^{m-2}}$. Using the same technique we can construct a mapping $\phi\in\Pp
S$ such that $\dim\ker(u_\phi+I)=l_1$ and $\dim\ker(e_\phi+I)=l_2$.
\textrm{e}nd{exm}
In the following example there are three elements satisfying
$2u=0$ and $u=u^*$, with three different upper bounds for the
dimensions of the kernel.
\begin{exm}
Let us consider the semigroup $S=\mathrm{U}({\mathbb{Z}}_2^2,{\mathbb{Z}}_2,\pi)$ where
$\pi(x,y)=x$ for $x,y\in{\mathbb{Z}}_2$. The involution on $S$ is identity. Note that $(1,0)+s\neq
s$ for $s\in S$, but $(0,1)+s\neq s$ only for $s\in{\mathbb{Z}}_2^2$. Hence, the upper bounds for
the dimensions of the kernels are three and two, respectively. The dimension of the kernel
for $(0,0)$ is obviously zero.
\textrm{e}nd{exm}
\begin{rem}
Let us take two $*$-separative semigroups $S$ and $T$, both having neutral elements ($0_S$
and $0_T$ respectively) and a $*$-homomorphism $h:S\to T$ satisfying $h(0_S)=0_T$. Take an
element $u\in T$ satisfying $2u=0_T$ and $u=u^*$. The $*$-semigroup $\mathrm{U}(S,T,h)$ has
a zero, namely $0_S$. However, the element $u$, understood as an element of
$\mathrm{U}(S,T,h)$, does not satisfy the condition $2u=0_{\mathrm{U}(S,T,h)}$.
Nevertheless, we have $3u=u$ and $u^*=u$, which by \cite{szaf-bdd} guaranties boundedness
(and hence selfadjointness) of $u_\phi$ for any $\phi\in\Pp{\mathrm{U}(S,T,h)}$. The
indefinite inner product space $(\mathcal{H}^\phi,\seq{u_\phi\cdot,-})$ is then degenerate, i.e.
$u_\phi$ has a non-trivial kernel.
\textrm{e}nd{rem}
We could investigate, instead of positive definite mappings on
$S$, the set of positive definite forms on $S$. Namely, we say that $\phi:S\times
\mathcal{E}\times \mathcal{E}\to\mathbb{C}$ is a { form over ($S,\mathcal{E}$)} if for every
$s\in S$ the mapping $\phi(s,\,\cdot\,,-)$ is a hermitian bilinear form on the linear space
$\mathcal{E}$. We say that a form is positive definite if for every finite sequences
$(s_k)_k\subseteq S$, $(f_k)_k\subseteq \mathcal{E}$ we have $\sum_{i,j} \phi(s_j^*+s_i;f_i,f_j))\geq
0$. For a positive definite form $\phi$ we can construct a Hilbert space $\mathcal{H}^\phi$ which
together with the functions $K_{s,f}^\phi$ ($s\in S$, $f\in\mathcal E$) constitute a RKHS.
Like in the case of $\mathcal{E}=\mathbb{C}$, cf. \cite{szaf-bdd} and also \cite{general}, we can
define the (closed) shift operator associated with an element $u\in S$ by
$u_\phi(K_{s,f}^\phi)=K_{s+u,f}^\phi$.
The following example shows, that in this setting the equivalence
in Theorem \ref{main} no longer holds.
\begin{exm}
Let $S={\mathbb{Z}}_2$ (with the identical involution) and let
$\mathcal{E}$ be an \underbar{infinite} dimensional {Hilbert} space.
Consider the following form
$$
\phi(x,f,g)=\left\{\begin{array}{rcl}
\seq{f,g}_\mathcal{E} & : & x=0\\
\seq{-f,g}_\mathcal{E} & : & x=1
\textrm{e}nd{array}\right. .
$$
Note that
$$
\sum_{x,y=0,1}
\phi(x+y,f_x,f_y)=\seq{f_0,f_0}+\seq{f_1,f_1}-2\Re\seq{f_1,f_0}=\norm{f_1-f_0}^2
$$
which is greater or equal to zero for any choice of $f_0,f_1\in\mathcal{H}$.
The space $\mathcal{H}^\phi$ can be realized as $\mathcal{H}^\phi=\mathcal{E}$ so as $K_{0,f}^\phi=f$ and
$K_{1,g}=-g$, $f,g\in \mathcal{E}$.
Take $u=1$. The condition (i) of Theorem \ref{main} is satisfied
because the semigroup is of finite cardinality. On the other hand
$u_\phi K_{0,f}^\phi = K_{1,f}^\phi=-f$ for $f\in\mathcal{H}$. Hence,
$\dim\ker(u_\phi+I)=\dim\mathcal{H}=\infty$.
\textrm{e}nd{exm}
\section{Final remarks}
Our work is connected in a
way with many other papers and books. Let us mention some of
them.
\begin{itemize}
\item The transformation $\phi\mapsto\phi(\cdot+u)$ has been
investigated by Bisgaard. He showed in \cite{biss-fact} that it
is always a sum of four positive definite mappings. \item
Functions with finite number of negative spaces on topological
groups has been considered in the book of Sasv\'ari
\cite{sasvari}. \item In \cite{berg} sequences on
$\mathbb{N}$ with a finite number of negative squares are considered.
\item In \cite{BChR} the authors consider negative definite
sequences, which is a subclass of mappings with a finite number
of negative squares. \item Finally, in \cite{biss-cor}
definitizing ideals are investigated.
\textrm{e}nd{itemize}
\begin{thebibliography}{99}
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squares, {\it J. Funct. Anal.} {\bf79} (1988), 260--287.}
\bibitem{BChR}{Ch. Berg, J.P.R. Christensen, P. Ressel, {\it Harmonic Analysis on Semigroups,}
Springer-Verlag, New-York Berlin Heidelberg Tokyo {\bf1984}.}
\bibitem{bognar}{ J. Bogn\'ar, {\it Indefinite Inner Product
Spaces}, Springer-Verlag {\bf1974}.}
\bibitem{biss-scmat}{T.M. Bisgaard, On the Relation between the
Scalar Moment Problem and the Matrix Moment Problem on $*$-semigroups, \it Semigroup Forum
\rm {\bf68} (2005), 25-46. }
\bibitem{biss-two}{T.M. Bisgaard, Two sided complex moment problem,
{\it Ark. Mat.} {\bf27} (1989), 23--28.}
\bibitem{biss-sep}{T.M. Bisgaard, Separation by characters or positive definite functions, {\it Semigroup Forum} {\bf53} (1996), 317-320.}
\bibitem{biss-ext}{T.M. Bisgaard, Extensions of Hamburger's Theorem, {\it Semigroup Forum} {\bf57} (1998), 397-429.}
\bibitem{biss-CC}{T.M. Bisgaard, Semiperfect countable $\mathbb{C}$-separative $C$-finite semigroups, {\it Collect. Math.} {\bf52} (2001), 55-73}
\bibitem{biss-fact}{T.M. Bisgaard, Factoring of Positive Definite Functions
on Semigroups, {\it Semigroup Forum} {\bf64} (2002), 243-264.}
\bibitem{biss-cor}{T.M. Bisgaard, H. Cornean, Nonexistence in General of a Definitizing Ideal of the
Desired Codimension, {\it Positivity } {\bf7} (2003), 297–302.}
\bibitem{lacunae}{D. Cicho\'n, J. Stochel, F.H. Szafraniec, Extending positive
definiteness, preprint}
\bibitem{clipres}{A.H. Clifford, G.B. Preston, \it The Algebraic Theory of semigroups \rm
2 vols., AMS, Providence, {\bf1961, 1967}.}
\bibitem{langerio}{I.S. Iohvidov, M.G. Krein, H. Langer, {\it Introduction to the Spectral
Theory of operators in Spaces with an Indefinite Metric},
Akademie-Verlag, Berlin {\bf1982}.}
\bibitem{sasvari}{Z. Sasv\'ari, {\it Positive Definite and definitizable functions},
Akademie Verlag, Berlin {\bf1994}.}
\bibitem{szaf-bdd}{F.H. Szafraniec, Boundness of the shift operator
related to positive definite forms: an application to moment problems, {\it Ark. Mat.} {\bf
19} (1981), 251-259.}
\bibitem{test} F.H. Szafraniec, Interpolation and domination by positive
definite kernels, in {\textrm{e}m Complex Analysis - Fifth Romanian-Finish Seminar}, Part 2,
Proceedings, Bucarest (Romania), 1981, eds. C. Andrean Cazacu, N. Boboc, M. Jurchescu and I.
Suciu, Lecture Notes in Math., {\bf 1014}, 291-295, Springer, Berlin-Heidelberg, 1893.
\bibitem{general} F.H. Szafraniec, The Sz.-Nagy "th\'eor\`eme principal" extended. Application to subnormality,
{\it Acta Sci. Math. {\rm(}Szeged{\rm)} }, {\bf57}(1993), 249-262.
\bibitem{test2} F.H. Szafraniec, The reproducing kernel Hilbert space and its multiplication operators,
{\textrm{e}m Oper. Theory Adv. Appl.}, {\bf114}(2000), 253-263.
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linear transformations in Hilbert space which extend beyond this space}, Appendix to F.
Riesz, B. Sz.--Nagy, {\it Functional Analysis}, Ungar, New York, {\bf1960}.
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297-307.}
\textrm{e}nd{thebibliography}
\textrm{e}nd{document} |
\begin{document}
\begin{frontmatter}
\title{Lower bounds for the smallest singular value of structured random matrices}
\runtitle{Invertibility of structured random matrices}
\author{\fnms{Nicholas} \snm{Cook}\ead[label=e1]{[email protected]}\thanksref{t1}}
\thankstext{t1}{Partially supported by NSF postdoctoral fellowship DMS-1266164.}
\address{Department of Mathematics\\
University of California\\
Los Angeles, CA 90095-1555\\
\printead{e1}}
\affiliation{University of California, Los Angeles}
\runauthor{N. Cook}
\begin{abstract}
We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries.
Specifically, we consider $n\times n$ matrices with complex entries of the form
\[
M = A\perp\!\!\!\perprc X + B = (a_{ij}\xi_{ij} + b_{ij})
\]
where $X=(\xi_{ij})$ has iid centered entries of unit variance and $A$ and $B$ are fixed matrices.
In our main result we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B= Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero.
As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph.
In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses.
Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemer\'edi's Regularity Lemma, and a version of the Restricted Invertibility Theorem due to Spielman and Srivastava.
\end{abstract}
\begin{keyword}[class=MSC]
\kwd[Primary ]{60B20}
\kwd[; secondary ]{15B52}
\end{keyword}
\begin{keyword}
\kwd{Random matrices}
\kwd{condition number}
\kwd{regularity lemma}
\kwd{metric entropy}
\end{keyword}
\end{frontmatter}
\section{Introduction}
Throughout the article we make use of the following standard asymptotic notation:
$f=O(g)$, $f\ll g$, $g\gg f$ all mean that $|f|\le Cg$ for some absolute constant $C<\infty$.
We indicate dependence of the implied constant on parameters with subscripts, e.g.\ $f\ll_\alpha g$.
$C,c, c', c_0$, etc.\ denote unspecified constants whose value may be different at each occurence, and are understood to be absolute if no dependence on parameters is indicated.
\subsection{Background}
Recall that the singular values of an $n\times n$ matrix $M$ with complex entries are the eigenvalues of $\sqrt{M^*M}$, which we arrange in non-increasing order:
\[
\|M\|=s_1(M)\ge \cdots\ge s_n(M)\ge0.
\]
(throughout we write $\|\cdot \|$ for the $\ell_2^n\to \ell_2^n$ operator norm).
$M$ is invertible if and only if $s_n(M)>0$, in which case $s_n(M)= \|M^{-1}\|^{-1}$.
We (informally) say that $M$ is ``well-invertible" if $s_n(M)$ is well-separated from zero.
The largest and smallest singular values of random matrices with independent entries have been intensely studied, in part due to applications in theoretical computer science.
Motivated by their work on the first electronic computers, von Neumann and Goldstine sought upper bounds on the condition number $\kappa(M)= s_1(M)/s_n(M) $ of a large matrix $M$ with iid entries \perp\!\!\!\perptep{vNG:numerical}.
More recently, bounds on the condition number of non-centered random matrices have been important in the theory of \emph{smoothed analysis of algorithms} developed by Spielman and Teng \perp\!\!\!\perptep{SST:smoothed}.
The smallest singular value
has also received attention due to its connection with proving convergence of the empirical spectral distribution -- see
\perp\!\!\!\perptep{TaVu:circ, BoCh:survey}.
Much is known about the largest singular value for random matrices with independent entries.
First we review the iid case: we denote by $X=X_n$ an $n\times n$ matrix whose entries $\xi_{ij}$ are iid copies of a centered complex random variable with unit variance, and refer to such $X$ as an ``iid matrix".
From the works \perp\!\!\!\perptep{BSY:largestsv, BKY:largestsv} it is known that $\frac{1}{\sqrt{n}}s_1(X_n)\in (2-\eps,2+\eps)$ with probability tending to one as $n\to \infty$ for any fixed $\eps>0$.
In connection with problems in computer science and the theory of Banach spaces there has been considerable interest in obtaining non-asymptotic bounds for matrices with independent but non-identically distributed entries; see the recent works \perp\!\!\!\perptep{BaHa} and \perp\!\!\!\perptep{vanHandel:norm} and references therein for an overview.
The picture is far less complete for the smallest singular value of random matrices; however, recent years have seen much progress for the case of the iid matrix $X$.
The limiting distribution of $\sqrt{n}s_n(X)$ was obtained by Edelman for the case of Gaussian entries \perp\!\!\!\perptep{Edelman:condition}, and this law was shown by Tao and Vu to hold for all iid matrices with entries $\xi_{ij}$ having a sufficiently large finite moment \perp\!\!\!\perptep{TaVu:ssv}.
Quantitative lower tail estimates for $s_n(X)$ proved to be considerably more challenging than bounding the operator norm.
The first breakthrough was made by Rudelson \perp\!\!\!\perptep{Rudelson:inv}, who showed that if $X$ has iid real-valued sub-Gaussian entries, that is
\begin{equation} \label{subgaussian}
\e \expo{ |\xi|^2/K_0}\le 2
\end{equation}
for some $K_0<\infty$,
then
\begin{equation} \label{bd:rudelson}
\pro{ s_n(X) \le tn^{-3/2}} \ll_{K_0} t + n^{-1/2}\qquad \text{for all $t\ge 0$}.
\end{equation}
Around the same time, in \perp\!\!\!\perpte{TaVu:cond} Tao and Vu used methods from additive combinatorics to obtain bounds of the form
\begin{equation} \label{tavu:arb}
\pro{ s_n(X) \le n^{-\beta}} \ll n^{-\alpha}
\end{equation}
for any fixed $\alpha>0$ and $\beta$ sufficiently large depending on $\alpha$, for the case that the entries of $X$ take values in $\{-1,0,1\}$.
Roughly speaking, their approach was to classify potential almost-null vectors $v$ according to the amount of additive structure present in the multi-set of coordinate values $\{v_j\}_{j=1}^n$.
They extended \eqref{tavu:arb} to uncentered matrices with general entry distributions having finite second moment in \perp\!\!\!\perpte{TaVu:circ} (see Theorem \ref{thm:tavu} below), which was instrumental for their proof of
the celebrated circular law for the limiting spectral distribution of $\frac{1}{\sqrt{n}}X$.
Motivated by these developments, in \perp\!\!\!\perpte{RuVe:ilo} Rudelson and Vershynin found a different way to quantify the additive structure of a vector $v$ called the \emph{essential least common denominator}, and obtained the following improvement of \eqref{bd:rudelson}, \eqref{tavu:arb} for matrices with sub-Gaussian entries:
\begin{equation} \label{bd:ruve_exp}
\pro{ s_n(X) \le tn^{-1/2}} \ll_{K_0} t + e^{-cn}.
\end{equation}
This estimate is optimal up to the implied constant and $c=c(K_0)>0$ (with $K_0$ as in \eqref{subgaussian}).
Finally, we mention that there has also been work on \emph{upper tail bounds} for the smallest singular value -- see in particular \perp\!\!\!\perpte{RuVe:uppertail, NgVu:normal} -- but we do not consider this problem further in the present work.
\subsection{A general class of non-iid matrices}
In this paper we are concerned with bounds for the smallest singular value of random matrices with independent but non-identically distributed entries.
The following definition allows us to quantify the dependence of our bounds on the distribution of the matrix entries.
\begin{definition}[Spread random variable] \label{def:spread}
Let $\xi$ be a complex random variable and let $\kappa\ge 1$.
We say that $\xi$ is \emph{$\kappa$-spread} if
\begin{equation} \label{def:kappa0}
\var \big[\,\xi\un(|\xi-\e\xi|\le \kappa)\,\big] \ge \frac1\kappa.
\end{equation}
\end{definition}
\begin{remark} \label{rmk:kappap}
It follows from the monotone convergence theorem that any random variable $\xi$ with non-zero second moment is $\kappa$-spread for some $\kappa<\infty$.
Furthermore, if $\xi$ is centered with unit variance and finite $p$th moment $\mu_p$ for some $p>2$, then it is routine to verify that $\xi$ is $\kappa$-spread with $\kappa =3 (3\mu_p^p)^{1/(p-2)}$, say.
\end{remark}
Our results concern the following general class of matrices:
\begin{definition}[Structured random matrix] \label{def:profile}
Let $A=(\sig_{ij})$ and $B=(\be_{ij})$ be deterministic $n\times m$ matrices with $\sig_{ij}\in [0,1]$ and $\be_{ij}\in \C$ for all $i,j$.
Let $X=(\xi_{ij})$ be an $n\times m$ matrix with independent entries, all identically distributed to a complex random variable $\xi$ with mean zero and variance one.
Put
\begin{equation} \label{Mdef}
\M=A\perp\!\!\!\perprc X+B = (\sig_{ij}\xi_{ij}+ \be_{ij})_{i,j=1}^n
\end{equation}
where $\perp\!\!\!\perprc$ denotes the matrix Hadamard product.
We refer to $A$, $B$ and $\xi$ as the \emph{standard deviation profile}, \emph{mean profile} and \emph{atom variable}, respectively.
We denote the $L^p$ norm of the atom variable by
\begin{equation} \label{Lp}
\mu_p:= (\e|\xi|^p)^{1/p}.
\end{equation}
Without loss of generality, we assume throughout that $\xi$ is $\kappa_0$-spread for some fixed $\kappa_0\ge1$.
\end{definition}
(While all of our results are for square matrices, we give the definition for the general rectangular case as we will often need to consider rectangular submatrices in the proofs.)
\begin{remark} \label{rmk:shiftscale}
The assumption that the entries of $\M$ are shifted scalings of random variables $\xi_{ij}$ having a common distribution is made for convenience, as it allows us to access some standard anti-concentration estimates (see Section \ref{sec:fourier}).
We expect the proofs can be modified to cover general matrices with independent entries having specified means and variances (possibly with additional moment hypotheses), but we do not pursue this here.
\end{remark}
As a concrete example one can consider a centered non-Hermitian band matrix, where one sets $a_{ij}\equiv 0$ for $|i-j|$ exceeding some bandwidth parameter $w\in [n-1]$ -- see Corollary \ref{cor:band}.
The singular value distributions for structured random matrices have been studied in connection with wireless MIMO networks \perp\!\!\!\perptep{TuVe:rmt_wireless,HLN:detequiv}.
The limiting spectral distributions and spectral radius for certain structured random matrices have been used to model the dynamical properties of neural networks \perp\!\!\!\perptep{RaAb:neural, ARS:block}.
In the recent work \perp\!\!\!\perptep{CHNR} with Hachem, Najim and Renfrew, the limiting spectral distribution was determined for a general class of centered structured random matrices.
That work required bounds on the smallest singular value for shifts of centered matrices by scalar multiples of the identity, which was the original motivation for the results in this paper (in particular, Corollary \ref{cor:scalarshift} below is a key input for the proofs in \perp\!\!\!\perptep{CHNR}).
The picture for the smallest singular value of structured random matrices is far less complete than for the largest singular value.
Here we content ourselves with identifying sufficient conditions on the matrices $A,B$ and the distribution of $\xi$ for a structured random matrix $M$ to be well-invertible with high probability. Specifically, we seek to address the following:
\begin{question} \label{prob:main}
Let $\M$ be an $n\times n$ random matrix as in Definition \ref{def:profile}.
Under what assumptions on the standard deviation and mean profiles $A,B$
and the distribution of the atom variable $\xi$
do we have
\begin{equation} \label{lowertail}
\pr\big( s_n(\M) \le n^{-\beta}\big) = O(n^{-\alpha})
\end{equation}
for some constants $\alpha,\beta>0$?
\end{question}
The case that $B=-z\sqrt{n} I $ for some fixed $z\in \C$ (where $I$ denotes the $n\times n$ identity matrix) is of particular interest for applications to the limiting spectral distribution of centered random matrices.
As we shall see in the next subsection, existing results in the literature give lower tail bounds for $s_n(M)$ that are uniform in the shift $B$ under the size constraint $\|B\|=n^{O(1)}$, i.e.
\begin{equation} \label{lowertail-uniform}
\sup_{B\in \mM_n(\C): \;\|B\| \le n^{C}} \pr\big( s_n(A\perp\!\!\!\perprc X + B) \le n^{-\beta}\,\big) = O(n^{-\alpha}).
\end{equation}
for some constant $C>0$ (results stated for centered matrices generally extend in a routine manner to allow a perturbation of size $\|B\|=O(\sqrt{n})$).
Such bounds can be viewed as matrix analogues of classical anti-concentration (or ``small ball") bounds of the form
\begin{equation} \label{anticonc:general}
\sup_{z\in \C} \pro{ |S_n - z|\le r} \le f(r) + o(1)
\end{equation}
for a sequence of scalar random variables $S_n$ (such as the normalized partial sums of an infinite sequence of iid variables), where $f:\R_+\to \R_+$ is some continuous function such that $f(r)\to 0$ as $r\to 0$.
In fact, bounds of the form \eqref{anticonc:general} are a central ingredient in the proofs of estimates \eqref{lowertail-uniform}.
Roughly speaking, the translation invariance of \eqref{anticonc:general} causes the uniformity in the shift $B$ in \eqref{lowertail-uniform} to come for free once one can handle the centered case $B=0$ (the assumption $\|B\|=n^{O(1)}$ is needed to have some continuity of the map $u\mapsto \|Mu\|$ on the unit sphere in order to apply a discretization argument).
In light of this we may pose the following:
\begin{question} \label{prob:sub}
Let $\M$ be an $n\times n$ random matrix as in Definition \ref{def:profile}, and let $\gamma>0$.
Under what assumptions on the standard deviation profile $A$
and the distribution of the atom variable $\xi$ do we have
\begin{equation} \label{lowertail:sub}
\sup_{B\in \mM_n(\C): \; \|B\|\le n^\gamma}\pr\big( s_n(\M) \le n^{-\beta}\,\big) = O(n^{-\alpha})
\end{equation}
for some constants $\alpha,\beta>0$?
\end{question}
The following simple observation puts a clear limitation on the standard deviation profiles $A$ for which we can expect to have \eqref{lowertail:sub}.
\begin{observation} \label{obs:submatrix}
Suppose that $A=(\sig_{ij})$ has a $k\times m$ submatrix of zeros for some $k,m$ with $k+m>n$.
Then $A\perp\!\!\!\perprc X$ is singular with probability 1.
Thus, \eqref{lowertail:sub} fails (by taking $B=0$) for any fixed $\alpha,\beta>0$.
\end{observation}
Theorem \ref{thm:broad} below (see also Theorem \ref{thm:ruze} for the Gaussian case) shows that the above is in some sense the only obstruction to obtaining \eqref{lowertail:sub}.
\subsection{Previous results} \label{sec:previous}
Before stating our main results on Questions \ref{prob:main} and \ref{prob:sub} we give an overview of what is currently in the literature.
For the case of a constant standard deviation profile $A$ and essentially arbitrary mean profile $B$ we have the following result of Tao and Vu:
\begin{theorem}[Shifted iid matrix \perp\!\!\!\perptep{TaVu:circ}] \label{thm:tavu}
Let $X$ be an $n\times n$ matrix with iid entries $\xi_{ij}\in \C$ having mean zero and variance one.
For any $\alpha,\gamma>0$ there exists $\beta>0$ such that for any fixed (deterministic) $n\times n$ matrix $B$ with $\|B\|\le n^\gamma$,
\begin{equation} \label{tavu:bound}
\pr\big( s_n(X+B) \le n^{-\beta} \,\big) = O_{\alpha,\gamma}(n^{-\alpha}).
\end{equation}
\end{theorem}
A stronger version of the above bound was established earlier by Sankar, Spielman and Teng for the case that $X$ has iid standard Gaussian entries \perp\!\!\!\perptep{SST:smoothed}.
For the case that $B=0$, the bound \eqref{bd:ruve_exp} of Rudelson and Vershynin gives the optimal dependence $\beta = \alpha +1/2$ for the exponents, but requires the stronger assumption that the entries are real-valued and sub-Gaussian (we remark that their proof extends in a routine manner to allow an arbitrary shift $B$ with $\|B\|=O(\sqrt{n})$).
Recently, the sub-Gaussian assumption for \eqref{bd:ruve_exp} was relaxed by Rebrova and Tikhomirov to only assume a finite second moment \perp\!\!\!\perptep{ReTi}.
When the entries of $M$ have bounded density the problem is much simpler.
The following is easily obtained by the argument in \perp\!\!\!\perpte[Section 4.4]{BoCh:survey}.
\begin{proposition}[Matrix with entries having bounded density \perp\!\!\!\perptep{BoCh:survey}] \label{prop:bdd}
Let $\M$ be an $n\times n$ random matrix with independent entries having density on $\C$ or $\R$ uniformly bounded by $\varphi> 1$.
For every $\alpha>0$ there is a $\beta=\beta(\alpha,\varphi)>0$ such that
\begin{equation}
\pr\big( s_n(\M) \le n^{-\beta}\,\big) = O(n^{-\alpha}).
\end{equation}
\end{proposition}
Note that above we make no assumptions on the moments of the entries of $\M$ -- in particular, they may have heavy tails.
The following result of Bordenave and Chafa\"i (Lemma A.1 in \perp\!\!\!\perptep{BoCh:survey}) relaxes the hypothesis of continuous distributions from Proposition \ref{prop:bdd} while still allowing for heavy tails, but comes at the cost of a worse probability bound.
\begin{proposition}[Heavy-tailed matrix with non-degenerate entries {\perp\!\!\!\perptep{BoCh:survey}}] \label{prop:BoCh}
Let $Y$ be an $n\times n$ random matrix with independent entries $\eta_{ij}\in \C$.
Suppose that for some $p,r,\ha>0$ we have that for all $i,j\in [n]$,
\begin{equation} \label{BoCh:nondeg}
\pro{ |\eta_{ij}|\le r} \ge p, \quad\quad \var( \eta_{ij}\un(|\eta_{ij}|\le r)) \ge \ha^2.
\end{equation}
For any $s\ge1$, $t\ge 0$, and any fixed $n\times n$ matrix $B$ we have
\begin{equation}
\pro{ s_n(Y+B) \le \frac{t}{\sqrt{n}}, \; \|Y+B\| \le s} \ll_{p,r,\ha} \sqrt{\log s}\bigg(ts^2 + \frac{1}{\sqrt{n}}\bigg).
\end{equation}
\end{proposition}
The non-degeneracy conditions \eqref{BoCh:nondeg} do not allow for some entries to be deterministic.
Litvak and Rivasplata \perp\!\!\!\perptep{LiRi} obtained a lower tail estimate of the form \eqref{lowertail} for centered random matrices having a sufficiently small constant proportion of entries equal to zero deterministically.
Below we give new results (Theorems \ref{thm:broad} and \ref{thm:super}) allowing all but an arbitrarily small (fixed) proportion of entries to be deterministic.
Finally, we recall a theorem of Rudelson and Zeitouni \perp\!\!\!\perptep{RuZe} for Gaussian matrices, showing that Observation \ref{obs:submatrix} is essentially the only obstruction to obtaining \eqref{lowertail:sub}.
To state their result we need to set up some graph theoretic notation, which will be used repeatedly throughout the paper.
To a non-negative $n\times m$ matrix $A=(\sig_{ij})$ we associate a bipartite graph $\Gamma_A = ([n],[m], E_A)$, with $(i,j)\in E_A$ if and only if $\sig_{ij}>0$.
For a row index $i\in [n]$ we denote by
\begin{equation} \label{def:nbhd}
\mN_A(i) = \set{j\in [m]: \sig_{ij}>0}
\end{equation}
its neighborhood in $\Gamma_A$.
Thus, the neighborhood of a column index $j\in [m]$ is denoted $\mN_{A^\tran}(j)$.
Given sets of row and column indices $I\subset [n], J\subset[m]$, we define the associated \emph{edge count}
\begin{equation} \label{def:edges}
e_A(I,J) := |\{(i,j)\in [n]\times [m]: \sig_{ij}>0\}|.
\end{equation}
We will generally work with the graph that only puts an edge $(i,j)$ when $\sig_{ij}$ exceeds some fixed cutoff parameter $\ha>0$.
Thus, we denote by
\begin{equation} \label{Aha}
A(\ha) = (\sig_{ij}1_{\sig_{ij}\ge \ha})
\end{equation}
the matrix which thresholds out entries smaller than $\ha$.
Rudelson and Zeitouni work with Gaussian matrices whose matrix of standard deviations $A=(\sig_{ij})$ satisfies the following expansion-type condition.
\begin{definition}[Broad connectivity] \label{def:broad}
Let $A=(\sig_{ij})$ be an $n\times m$ matrix with non-negative entries.
For $I\subset [n]$ and $\delta\in (0,1)$, define the set of \emph{$\delta$-broadly connected} neighbors of $I$ as
\begin{equation} \label{def:broadnbr}
\mN_{A}^{(\delta)}(I)= \{ j\in [m]: |\mN_{A^\tran}(j)\cap I|\ge \delta |I|\}.
\end{equation}
For $\delta,\nu\in (0,1)$, we say that $A$ is \emph{$(\delta,\nu)$-broadly connected} if
\begin{enumerate}[(1)]
\item $|\mN_A(i)| \ge \delta m$ for all $i\in [n]$;
\item $|\mN_{A^\tran}(j)|\ge \delta n$ for all $j\in [m]$;
\item $|\mN_{A^\tran}^{(\delta)}(J)| \ge \min(n,(1+\nu)|J|)$ for all $J\subset [m]$.
\end{enumerate}
\end{definition}
\begin{theorem}[Gaussian matrix with broadly connected profile \perp\!\!\!\perptep{RuZe}] \label{thm:ruze}
Let $G$ be an $n\times n$ matrix with iid standard real Gaussian entries, and let $A$ be an $n\times n$ matrix with entries $\sig_{ij}\in [0,1]$ for all $i,j$.
With notation as in \eqref{Aha}, assume that $A(\ha)$ is $(\delta,\nu)$-broadly connected for some $\ha,\delta,\nu\in (0,1)$.
Let $K\ge1$, and let $B$ be a fixed $n\times n$ matrix with $\|B\|\le K\sqrt{n}$.
Then for any $t\ge0$,
\begin{equation} \label{bound:ruze}
\pr\big( s_n(A\perp\!\!\!\perprc G+ B) \le tn^{-1/2}\,\big) \ll_{\delta,\nu,\ha} K^{O(1)}t+ e^{-cn}
\end{equation}
for some $c=c(\delta,\nu,\ha)>0$.
\end{theorem}
Note that the assumption of broad connectivity gives us an ``epsilon of separation" from the bad example of Observation \ref{obs:submatrix}.
Thus, Theorem \ref{thm:ruze} provides a near-optimal answer to Question \ref{prob:sub} for Gaussian matrices.
\begin{remark} \label{rmk:ruze_dense}
Since the dependence of the bound \eqref{bound:ruze} on the parameters $\delta$ and $\nu$ is not quantified, Theorem \ref{thm:ruze} only addresses Question \ref{prob:sub} for \emph{dense} standard deviation profiles, i.e. when $A$ has a non-vanishing proportion of large entries.
While it would not be difficult to quantify the steps in \perp\!\!\!\perptep{RuZe}, the resulting dependence on parameters is not likely to be optimal.
\end{remark}
\subsection{New results} \label{sec:results}
Our first result removes the Gaussian assumption from Theorem \ref{thm:ruze}, though at the cost of a worse probability bound.
Recall the parameter $\kappa_0$ from Definition \ref{def:profile}.
\begin{theorem}[General matrix with broadly connected profile] \label{thm:broad}
Let $\M=A\perp\!\!\!\perprc X+B$ be an $n\times n$ matrix as in Definition \ref{def:profile}, and assume that $A(\ha)$ is $(\delta,\nu)$-broadly connected for some $\ha,\delta,\nu\in (0,1)$.
Let $K\ge 1$.
For any $t\ge 0$,
\begin{equation} \label{broad:poly}
\pro{ s_n(\M) \le \frac{t}{\sqrt{n}}, \, \|\M\| \le K\sqrt{n} } \ll_{K,\delta,\nu,\ha,\kappa_0} t + \frac{1}{\sqrt{n}}.
\end{equation}
\end{theorem}
\begin{remark} \label{rmk:distribution_broad}
While we have stated no moment assumptions on the atom variable $\xi$ over the standing assumption of unit variance, the restriction to the event $\{\|M\|\le K\sqrt{n}\}$ requires us to assume at least four finite moments to deduce $\pr(s_n(M) \le t/\sqrt{n}) \ll t + o(1)$.
Here we give a lower tail estimate at the optimal scale $s_n(M)\sim n^{-1/2}$; however, the arguments in this paper can be used to establish a polynomial lower bound on $s_n(M)$ of non-optimal order under larger perturbations $B$ (similar to \eqref{bd:super} below).
\end{remark}
\begin{remark}[Improving the probability bound]
We expect that the probability bound in \eqref{broad:poly} can be improved by making use of more advanced tools of Littlewood--Offord theory introduced in \perp\!\!\!\perptep{TaVu:circ, RuVe:ilo},
though it appears these tools cannot be applied in a straightforward manner.
In the interest of keeping the paper of reasonable length we do not pursue this here.
\end{remark}
\begin{remark}[Bounds on moderately small singular values]
The methods used to prove Theorem \ref{thm:broad} together with an idea of Tao and Vu from \perp\!\!\!\perptep{TaVu:esd} can be used to give lower bounds of optimal order on $s_{n-k}(M)$ with $n^{\eps}\le k \le cn$ for any $\eps>0$ and a sufficiently small constant $c=c(\kappa_0,\ha,\delta,\nu,K)>0$; see \perp\!\!\!\perptep[Theorem 4.5.1]{Cook:thesis}.
Such bounds are of interest for proving convergence of the empirical spectral distribution; see \perp\!\!\!\perptep{TaVu:esd, BoCh:survey}.
\end{remark}
In light of Observation \ref{obs:submatrix}, Theorem \ref{thm:broad} gives an essentially optimal answer to Question \ref{prob:sub} for \emph{dense} random matrices (see Remark \ref{rmk:ruze_dense}).
It would be interesting to establish a version of this result that allows for only a proportion $o(1)$ of the entries to be random.
Indeed, we expect a version of the above theorem to hold when $A$ has density as small $(\log^{O(1)}n)/n$. (Quantifying the dependence on $\delta,\nu$ in \eqref{broad:poly} would only allow a slight polynomial decay in the density.)
We note that they broad connectivity hypothesis includes many standard deviation profiles of interest, such as band matrices:
\begin{corollary}[Shifted non-Hermitian band matrices] \label{cor:band}
Let $M=A\perp\!\!\!\perprc X+ B$ be an $n\times n$ matrix as in Definition \ref{def:profile}, and assume that for some fixed $\ha,\eps\in (0,1)$, $\sig_{ij} \ge \ha$ for all $i,j$ with $\min(|i-j|,n-|i-j|)\le \eps n$.
Let $K\ge 1$.
Then \eqref{broad:poly} holds for any $t\ge 0$ (with implied constant depending on $K,\ha,\eps$ and $\kappa_0$).
\end{corollary}
We defer the proof to Appendix \ref{app:band}.
\begin{remark}
It is possible to modify our argument for the above corollary to treat a band profile that does not ``wrap around", i.e.\ only enforcing $a_{ij}\ge \ha$ for $i,j$ with $|i-j|\le \eps n$.
\end{remark}
Having addressed Question \ref{prob:sub}, we now ask whether we can further relax the assumptions on the standard deviation profile $A$ by assuming more about the mean profile $B$.
In particular, can we make assumptions on $B$ that give \eqref{lowertail} while allowing $A\perp\!\!\!\perprc X$ to be singular deterministically?
Of course, a trivial example is to take $A=0$ and $B$ any invertible matrix.
Another easy example is to take take $B$ to be \emph{very well-invertible}, with $s_n(B) \ge K\sqrt{n}$ for a large constant $K>0$ (for instance, take $B=K\sqrt{n}I$, where $I$ is the identity matrix).
Indeed, standard estimates for the operator norm of random matrices with centered entries (cf.\ Section \ref{sec:op}) give $\|A\perp\!\!\!\perprc X\| = O(\sqrt{n})$ with high probability provided the atom variable $\xi$ satisfies some additional moment hypotheses.
From the triangle inequality
\begin{align*}
s_n(M) &= \inf_{u\in S^{n-1}} \|(A\perp\!\!\!\perprc X+B)u\| \ge s_n(B) - \|A\perp\!\!\!\perprc X\|,
\end{align*}
so $s_n(M)\gg \sqrt{n}$ with high probability if $K$ is sufficiently large.
The problem becomes non-trivial when we allow $B$ to have singular values of size $\eps \sqrt{n}$ for small $\eps>0$ and $A$ as in Observation \ref{obs:submatrix}.
In this case any proof of a lower tail estimate of the form \eqref{lowertail} must depart significantly from the proofs of the results in the previous section by making use of arguments which are not translation invariant.
Our main result shows that when the mean profile $B$ is a \emph{diagonal} matrix with smallest entry at least an arbitrarily small (fixed) multiple of $\sqrt{n}$, then we do not need to assume anything further about the standard deviation profile $A$.
\begin{theorem}[Main result]
\label{thm:main}
Fix arbitrary $r_0\in (0,\frac12]$, $K_0\ge1$, and let $Z$ be a (deterministic) diagonal matrix with diagonal entries $z_1,\dots,z_n\in \C$ satisfying
\begin{equation} \label{constraint:zi}
|z_i|\in [r_0,K_0]\qquad \forall i\in[n].
\end{equation}
Let $M$ be an $n\times n$ random matrix as in Definition \ref{def:profile} with $B=Z\sqrt{n}$, and assume $\mu_{4+\eta}<\infty$ for some fixed $\eta>0$.
There are $\alpha(\eta)>0$ and $\beta(r_0,\eta,\mu_{4+\eta})>0$ such that
\begin{equation} \label{bd:gen}
\pr\big( s_n(M)\le n^{-\beta} \, \big) = O_{r_0,K_0,\eta,\mu_{4+\eta}}(n^{-\alpha}).
\end{equation}
\end{theorem}
\begin{remark}[Moment assumption]
The assumption of $4+\eta$ moments is due to our use of a result of Vershynin, Theorem \ref{thm:vershynin} below, on the operator norm of products of random matrices.
Apart from this, at many points in our argument we use that an $m\times m$ submatrix of $M$ has operator norm $O(\sqrt{m})$ with high probability (assuming $m$ grows with $n$), which requires at least four finite moments.
Under certain additional assumptions on the standard deviation profile we only need to assume two moments -- see Remark \ref{rmk:relaxmom}.
\end{remark}
\begin{remark}[Dependence of $\alpha,\beta$ on parameters] \label{rmk:rdep}
The proof gives $\alpha(\eta)=\frac19\min(1,\eta)$. If we were to assume $\xi$ has finite $p$th moment for a sufficiently large constant $p$ then we could take any fixed $\alpha<1/2$ in \eqref{bd:gen}.
The dependence of $\beta$ on
$\mu_{4+\eta}$
and $r_0$ given by our proof is very bad, of the form
\begin{equation}
\beta = \text{twr}\big(O_\eta(1)\exp((\mu_{4+\eta}/r_0)^{O(1)})\big)
\end{equation}
where $\text{twr}(x)$ is a tower exponential $2^{2^{\iddots^2}}$ of height $x$.
(The factor $O_\eta(1)$ comes from Vershynin's bound mentioned in the previous remark -- we do not know the precise dependence on $\eta$, but we expect it is relatively mild.)
This is due to our use of Szemer\'edi's regularity lemma (specifically, a version for directed graphs due to Alon and Shapira -- see Lemma \ref{lem:regularity}).
It would be interesting to obtain a version of Theorem \ref{thm:main} with a better dependence of $\beta$ on the parameters.
\end{remark}
As we remarked above, the case of a diagonal mean profile is of special interest for the problem of proving convergence of the empirical spectral distribution of centered random matrices with a variance profile.
\begin{corollary}[Scalar shift of a centered random matrix] \label{cor:scalarshift}
Let $X=(\xi_{ij})$ be an $n\times n$ matrix whose entries are iid copies of a centered complex random variable $\xi$ having unit variance and $(4+\eta)$-th moment $\mu_{4+\eta}<\infty$ for some fixed $\eta>0$.
Let $A=(\sig_{ij})$ be a fixed $n\times n$ non-negative matrix with entries uniformly bounded by $\sigma_{\max}<\infty$.
Put $Y= \frac1{\sqrt{n}}A\perp\!\!\!\perprc X$, and fix an arbitrary $z\in \C\setminus\{0\}$.
There are constants $\alpha=\alpha(\eta)>0$ and $\beta=\beta(|z|,\eta,\mu_{4+\eta},\sigma_{\max})>0$ such that
\begin{equation} \label{bd:scalarshift}
\pr\big( s_n(Y-zI)\le n^{-\beta} \, \big) =O_{|z|,\sigma_{\max},\mu_{4+\eta}}(n^{-\alpha}).
\end{equation}
\end{corollary}
While our main motivation was to handle diagonal perturbations of centered random matrices, we conjecture that Theorem \ref{thm:main} extends to matrices as in Definition \ref{def:profile} with more general mean profiles $B$:
\begin{conjecture} \label{conj:main}
Theorem \ref{thm:main} continues to hold for $B\in \mM_n(\C)$ not necessarily diagonal, where the constraint \eqref{constraint:zi} is replaced with $\frac1{\sqrt{n}}s_i(B)\in [r_0,K_0]$ for all $1\le i\le n$.
\end{conjecture}
\subsection{Ideas of the proof} \label{sec:ideas}
Here we give an informal discussion of the main ideas in the proof of Theorem \ref{thm:main}.
\subsubsection*{Regular partitions of graphs}
As with Theorem \ref{thm:broad}, the key is to associate the standard deviation profile $A$ with a graph.
Since we want the diagonal of $M$ to be preserved under relabeling of vertices will will associate $A$ with a directed graph (digraph) which puts an edge $i\to j$ whenever $a_{ij}$ exceeds some small threshold $\ha>0$.
Since $A$ has no special connectivity structure \emph{a priori}, we will apply a version of Szemer\'edi's regularity lemma for digraphs (Lemma \ref{lem:regularity}) to partition the vertex set $[n]$ into a bounded number of parts of equal size $I_1,\dots, I_m$, together with a small set of ``bad" vertices $I_{\badd}$, such that for most $(k,l)\in [m]^2$ the subgraph on $I_k\cup I_l$ enjoys certain ``pseudorandomness" properties.
These properties will not be quite strong enough to control the smallest singular value of the corresponding submatrix $M_{I_k,I_l}$ of $M$, but we can apply a ``cleaning" procedure (as it is called in the extremal combinatorics literature) to remove a small number of bad vertices from each part in the partition (which we add to $I_{\badd}$), after which we will be able to control $s_{\min}(M_{I_k,I_l})$ for most $(k,l)\in [m]^2$.
We defer the precise formulation of the pseudorandomness properties and corresponding bound on the smallest singular value to Definition \ref{def:super} and Theorem \ref{thm:super} below.
\subsubsection*{Schur complement formula}
The task will then be to lift this control on the invertibility of submatrices to the whole matrix $M$.
The key tool here is the \emph{Schur complement formula} (see Lemma \ref{lem:schur}) which allows us to control the smallest singular value of a block matrix
\begin{equation} \label{sketch:partition}
\begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix}
\end{equation}
assuming some control on the smallest singular values of (perturbations of) the diagonal block submatrices $M_{11},M_{22}$ and on the operator norm of the off-diagonal submatrices $M_{12}, M_{21}$.
The control on the smallest singular value of the whole matrix is somewhat degraded, but this is acceptable as we will only apply Lemma \ref{lem:schur} a bounded number of times.
If we can find a generalized diagonal of ``good" block submatrices that are well-invertible under additive perturbations,
then after permuting the blocks to lie on the main diagonal we can apply the Schur complement bound along a nested sequence of submatrices partitioned as in \eqref{sketch:partition}, where $M_{11}$ is a ``good" matrix and $M_{22}$ is well-invertible by the induction hypothesis.
We remark that the strategy of leveraging properties of a small submatrix using the Schur complement formula was recently applied in a somewhat different manner in \perp\!\!\!\perptep{BEYY:band} to prove the universality of spectral statistics of random Hermitian band matrices.
\subsubsection*{Decomposition of the reduced digraph}
At this point it is best to think of the regular partition $I_1,\dots, I_m$ as inducing a ``macroscopic scale" digraph $\mR = ([m],E)$ (often called the \emph{reduced digraph} in extremal combinatorics) that puts an edge $(k,l)\in E$ whenever the corresponding submatrix $A_{I_k,I_l}$ is pseudorandom and sufficiently dense.
If we can cover the vertices of $\mR$ with vertex-disjoint directed cycles, then we will have found a generalized diagonal of submatrices of $M$ with the desired properties, and we can finish with a bounded number of applications of the Schur complement formula as described above.
Of course, it may be the case that $\mR$ cannot be covered by disjoint cycles.
For instance, if $A$ were to have all ones in the first $n/2$ columns and all zeros in the last $n/2$ columns then roughly half of the vertices of $\mR$ would have no incoming edges.
This is where we make crucial use of the diagonal perturbation $Z\sqrt{n}$ (indeed, without this perturbation $M$ would be singular in this example).
The top left $n/2\times n/2$ submatrix of $M$ is dense, and we can apply Theorem \ref{thm:super} to control its smallest singular vale.
The bottom right $n/2\times n/2$ submatrix is a diagonal matrix with diagonal entries of size at least $r_0\sqrt{n}$, and hence its smallest singular value is at least $r_0\sqrt{n}$.
This argument even allows for the bottom right submatrix of $A$ to be nonzero but sufficiently sparse: we can use the triangle inequality and standard bounds on the operator norm of sparse random matrices to argue that the smallest singular value of the bottom right submatrix is still of order $\gg r_0\sqrt{n}$.
We handle the general case as follows.
We greedily cover as many of the vertices of $\mR$ as we can with disjoint cycles -- call this set of vertices $U_{\cyc}\subset[m]$.
At this point we have either covered the whole graph (and we are done) or the graph on the remaining vertices $U_{\free}$ is cycle-free. This means that the vertices of $\mR$ can be relabeled so that its adjacency matrix is upper-triangular on $U_{\free}\times U_{\free}$.
Write $J_{\cyc} = \bigcup_{k\in U_{\cyc}} I_k$, $J_{\free}=\bigcup_{k\in U_{\free}}I_k$ and denote the corresponding submatrices of $A$ on the diagonal by $A_{\cyc}, A_{\free}$, and likewise for $M$.
We thus have a relabeling of $[n]$ under which $A_{\free}$ is close to upper triangular (there may be some entries of $A_{\free}$ below the diagonal of size less than $\ha$, or which are contained in a small number of exceptional pairs from the regular partition).
Crucially, this relabeling has preserved the diagonal, so the submatrix $M_{\free}$ is a diagonal perturbation of an (almost) upper-triangular random matrix.
We then show that such a matrix has smallest singular value of order $\gg_{r_0}\sqrt{n}$ with high probability.
With another application of the Schur complement bound we can combine the control on the submatrices $M_{\cyc},M_{\free}$ (along with standard bounds on the operator norm for the off-diagonal blocks) to conclude the proof.
(Actually, the bad set $I_{\badd}$ of rows and columns requires some additional arguments, but we do not discuss these here.)
This concludes the high level description of the proof of Theorem \ref{thm:main}.
We only remark that the above partitioning and cleaning procedures will generate various error terms and residual submatrices (such as the vertices in $I_{\badd}$, or the small proportion of pairs $(I_k,I_l)$ which are not sufficiently pseudorandom).
As the smallest singular value is notoriously sensitive to perturbations, it will take some care to control these terms.
We will use some high-powered tools such as bounds on the operator norm of sparse random matrices and products of random matrices due to Lata\l a and Vershynin -- see Section \ref{sec:op}.
\subsubsection*{Invertibility from connectivity assumptions}
Now we state the specific pseudorandomness condition on a standard deviation profile under which we have good control on the smallest singular value.
While ``pseudorandom" generally means that the edge distribution in a graph is close to uniform on a range of scales, we will only need control from below on the edge densities (morally speaking, we want the matrix $A$ to be as far as possible from the zero matrix, the most poorly invertible matrix).
The following one-sided condition is taken from the combinatorics literature (see \perp\!\!\!\perpte[Definition 1.6]{KoSi:survey}). The reader should recall the notation introduced in \eqref{def:nbhd}--\eqref{Aha}.
\begin{definition}[Super-regularity] \label{def:super}
Let $A$ be an $n\times m$ matrix with non-negative entries.
For $\delta,\eps\in (0,1)$, we say that $A$ is \emph{$(\delta,\eps)$-super-regular} if the following hold:
\begin{enumerate}[(1)]
\item $|\mN_A(i)| \ge \delta m$ for all $i\in [n]$;
\item $|\mN_{A^\tran}(j)|\ge \delta n$ for all $j\in [m]$;
\item $e_A(I,J)\ge \delta |I||J|$ for all $I\subset[n],J\subset[m]$ with $|I|\ge \eps n$ and $|J|\ge \eps m$.
\end{enumerate}
\end{definition}
The reader should compare this condition with Definition \ref{def:broad}.
Conditions (1) and (2) are are the same in both definitions, while it is not hard to see that condition (3) above implies
\begin{equation}
|\mN_{A^\tran}^{(\delta)}(J)| \ge (1-\eps)n
\end{equation}
whenever $|J|\ge \eps n$ (with notation as in \eqref{def:broadnbr}), which is stronger than condition (3) in Definition \ref{def:broad} for such $J$.
On the other hand, conditions (1) and (2) imply that $|\mN_{A^\tran}^{(\sqrt{\delta}/2)}(J)|\ge \frac12\delta n$ for any $J\subset[m]$ (see Lemma \ref{lem:goodrows0}), so super-regularity is stronger than broad connectivity for $\eps,\eta$ sufficiently small depending on $\delta$.
\begin{theorem}[Matrix with super-regular profile] \label{thm:super}
Let $M=A\perp\!\!\!\perprc X+B$ be an $n\times n$ matrix as in Definition \ref{def:profile}.
Assume that $A(\ha)$ (as defined in \eqref{Aha}) is $(\delta,\eps)$-super-regular for some $\delta,\ha\in (0,1)$ and $0<\eps<c_1\delta \ha^2$ with $c_1>0$ a sufficiently small constant.
For any $\gamma\ge1/2$ there exists $\beta = O(\gamma^2)$ such that
\begin{equation} \label{bd:super}
\pr\big( s_n(M) \le n^{-\beta},\, \|M\|\le n^\gamma\,\big) \ll_{\gamma,\delta,\ha,\kappa_0} \sqrt{\frac{\log n}{n}}.
\end{equation}
\end{theorem}
Note that Theorem \ref{thm:super} allows for a mean profile $B$ of arbitrary polynomial size in operator norm, whereas in Theorem \ref{thm:broad} we only allowed $\|B\|=O(\sqrt{n})$.
The ability to handle such large perturbations will be crucial in the proof of Theorem \ref{thm:main}, as the iterative application of the Schur complement bound discussed above will lead to perturbations of increasingly large polynomial order.
We defer discussion of the key technical ideas for Theorem \ref{thm:broad} and Theorem \ref{thm:super} to Sections \ref{sec:comp} and \ref{sec:incomp}.
We only mention here that our proof of Theorem \ref{thm:super} makes crucial use of a new ``entropy reduction" argument, which allows us to control the event that $\|Mu\|$ is small for some $u$ in certain portions of the sphere $S^{n-1}$ by the event that this holds for some $u$ in a random net of relatively low cardinality.
The argument uses an improvement by Spielman and Srivastava \perp\!\!\!\perptep{SpSr:rit} of the classic Restricted Invertibility Theorem due to Bourgain and Tzafriri \perp\!\!\!\perptep{BoTz:rit} -- see Section \ref{sec:comp} for details.
\subsection{Organization of the paper}
The rest of the paper is organized as follows.
Sections \ref{sec:anti}, \ref{sec:comp} and \ref{sec:incomp} are devoted to the proofs of Theorems \ref{thm:broad} and \ref{thm:super}.
We prove these theorems in parallel as they involve many similar ideas.
In Section \ref{sec:anti} we collect some standard lemmas on anti-concentration for random walks and products of random matrices with fixed vectors, along with some facts about nets in Euclidean space.
In Section \ref{sec:comp} we show that random matrices as in Theorems \ref{thm:broad} and \ref{thm:super} are well-invertible over sets of ``compressible" vectors in the unit sphere, and in Section \ref{sec:incomp} we establish control over the complementary set of ``incompressible" vectors.
Theorem \ref{thm:main} is proved in Section \ref{sec:diag}.
\subsection{Notation} \label{sec:notation}
In addition to the asymptotic notation defined at the beginning of the article,
we will occasionally use the notation $f=o(g)$ to mean that $f/g\rightarrow 0$ as $n\rightarrow \infty$, where the parameter $n$ will be the size of the matrix under consideration (this will only be for the sake of brevity, as all of our arguments are quantitative).
$\mM_{n,m}(\C)$ denotes the set of $n\times m$ matrices with complex entries. When $m=n$ we will write $\mM_n(\C)$.
For a matrix $A=(a_{ij})\in \mM_{n,m}(\C)$ we will sometimes use the notation $A(i,j)= a_{ij}$.
For $I\subset[n],J\subset[m]$, $A_{I,J}$ denotes the $|I|\times |J|$ submatrix with entries indexed by $I\times J$.
We abbreviate $A_{J}:= A_{J,J}$.
$\|\cdot\|$ denotes the Euclidean norm when applied to vectors, and the $\ell_2^m\to\ell_2^n$ operator norm when applied to an $n\times m$ matrix.
$\|A\|_{\HS}$ denotes the Hilbert--Schmidt (or Frobenius) norm of a matrix $A$.
We will sometimes denote the smallest singular value of a square matrix $M$ by $s_{\min}(M)$ (in situations where $M$ is a submatrix of a larger matrix this will often be clearer than writing the dimension).
We denote the unit sphere in $\C^n$ by $S^{n-1}$.
For $J\subset [n]$, we denote by $\C^J\subset\C^n$ (resp. $S^J\subset S^{n-1}$) the set of vectors (resp. unit vectors) in $\C^n$ supported on $J$.
Given a vector $v\in \C^n$, we denote by $v_J\in \C^n$ the projection of $v$ to the coordinate subspace $\C^J$.
For $m\in \N$, $x\in \R$, ${[m]\choose x}$ denotes the family of subsets of $[m]$ of size $\lf x\rf$.
When considering a random matrix $M$ as in Definition \ref{def:profile},
we use $R_i$ to denote the $i$th row of $M$, and write
\begin{equation} \label{def:sigmaalgebra}
\mF_{I,J}:= \langle \{\xi_{ij}\}_{i\in I,j\in J}\rangle
\end{equation}
for the sigma algebra of events generated by the entries $\{\xi_{ij}\}_{i\in I,j\in J}$ of $X$.
For $I\subset[n]$ we write $\pr_I(\cdot)$ for probability conditional on $\mF_{[n]\setminus I,[n]}$.\\
\noindent{\bf Acknowledgements.}
The author thanks David Renfrew and Terence Tao for useful conversations, and also thanks David Renfrew for providing helpful comments on a preliminary version of the manuscript.
\section{Preliminaries} \label{sec:anti}
\subsection{Partitioning and discretizing the sphere} \label{sec:net}
For the proofs of Theorems \ref{thm:broad} and \ref{thm:super} we make heavy use of ideas and notation developed in \perp\!\!\!\perptep{LPRT,LPRTV2,Rudelson:inv, RuVe:ilo} and related ideas from geometric functional analysis.
In particular, in order to lower bound
\[
s_n(M) = \inf_{u\in S^{n-1}} \|Mu\|
\]
we partition the sphere into sets of vectors of different levels of ``compressibility", which we presently define, and separately obtain control on the infimum of $\|Mu\|$ over each set.
Recall from Section \ref{sec:notation} our notation $\C^J\subset \C^m$ for the set of vectors supported on $J\subset[m]$.
For a set $T\subset \C^n$ and $\rho>0$ we write $T_\rho$ for the set of points within Euclidean distance $\rho$ of $T$.
We recall also the following definitions from \perp\!\!\!\perptep{RuZe}.
For $\theta,\rho\in (0,1)$, we define the set of \emph{compressible vectors}
\begin{equation} \label{def:compr}
\Comp(\theta,\rho) := S^{m-1}\cap\bigcup_{J\in {[m]\choose \theta m}} (\C^J)_\rho
\end{equation}
and the complementary set of \emph{incompressible vectors}
\begin{equation} \label{def:incompr}
\Incomp(\theta,\rho) := S^{m-1}\setminus \Comp(\theta,\rho).
\end{equation}
That is, $\Comp(\theta,\rho)$ is the set of unit vectors within (Euclidean) distance $\rho$ of a vector supported on at most $\theta m$ coordinates.
On the other hand, incompressible vectors enjoy the following property which will lead to good anti-concentration properties for an associated random walk.
\begin{lemma}[Incompressible vectors are spread, cf.\ {\perp\!\!\!\perpte[Lemma 3.4]{RuVe:ilo}}] \label{lem:spread}
Fix $\theta,\rho\in (0,1)$ and let $v\in \Incomp(\theta,\rho)$.
There is a set $L^+\subset[m]$ with $|L^+|\ge \theta m$ such that $|v_j|\ge \rho/\sqrt{m}$ for all $j\in L^+$.
Moreover, for all $\lambda\ge1$ there is a set $L\subset [m]$ with $|L| \ge (1-\frac1{\lambda^2})\theta m$ such that for all $j\in L$,
\[
\frac{\rho}{\sqrt{m}} \le |v_j| \le \frac{\lambda}{\sqrt{\theta m}}.
\]
\end{lemma}
\begin{proof}
Take $L^+=\{j:|v_j|\ge \rho/\sqrt{m}\}$ and denote $L^-= \{j:|v_j|\le \lambda/\sqrt{\theta m}\}$.
Since $v$ lies a distance at least $\rho$ from any vector supported on at most $\theta m$ coordinates we must have $|L^+|\ge \theta m$, which gives the first claim.
On the other hand, since $v\in S^{m-1}$, by Markov's inequality we have
$|(L^-)^c|\le \theta m/\lambda^2$,
so taking $L=L^+\cap L^-$ we have $|L|\ge (1-\frac1{\lambda^2})\theta m$.
\end{proof}
For fixed choices of $\theta,\rho$ we informally refer to the coordinates of $v\in \Incomp(\theta,\rho)$ where $|v_j|\ge \rho/\sqrt{n}$ as the \emph{essential support of $v$}.
Now we recall a standard fact about nets of the sphere of controlled cardinality.
For $\rho>0$, recall that a $\rho$-net of a set $T\subset \C^m$ is a finite subset $\Sigma\subset T$ such that for all $v\in T$ there exists $v'\in \Sigma$ with $\|v-v'\|\le \rho$.
\begin{lemma}[Metric entropy of the sphere] \label{lem:net}
Let $V\subset \C^m$ be a subspace of (complex) dimension $k$, let $T\subset V\cap S^{m-1}$, and let $\rho\in(0,1)$.
Then $T$ has a $\rho$-net $\Sigma\subset T$ of cardinality $|\Sigma|\le (3/\rho)^{2k}$.
\end{lemma}
\begin{proof}
Let $\Sigma\subset T$ be a $\rho$-separated (in Euclidean distance) subset that is maximal under set inclusion.
It follows from maximality that $\Sigma$ is a $\rho$-net of $T$.
Let $\Sigma_{\rho/2}$ denote the $\rho/2$ neighborhood of $\Sigma$ in $V$.
Noting that $\Sigma_{\rho/2}$ is a disjoint union of $k$-dimensional Euclidean balls of radius $\rho/2$, we have
$$|\Sigma| c_{k} (\rho/2)^{2k} \le \text{vol}_k(\Sigma_{\rho/2}) \le c_{k} (1+\rho/2)^{2k}$$
where $\text{vol}_k$ denotes the $k$-dimensional Lebesgue measure on $V$ and $c_{k}$ is the volume of the Euclidean unit ball in $\C^k$.
The desired bound follows by rearranging.
\end{proof}
\begin{comment}
The existence of $\rho$-nets of controlled size allows us to obtain uniform control from below on $\|M u\|$ for all $u$ drawn from a low-dimensional subset of the sphere, off a small event.
The following is essentially Lemma 3.3 from \perp\!\!\!\perptep{RuZe}, adjusted to the current setting of matrices over $\C$ rather than $\R$.
\Red{Remove this, as it's only applied once, and give the proof in-line.}
\begin{corollary}[Control by a net] \label{cor:net}
Let $\M$ be an $n\times m$ random matrix with complex entries.
Let $V\subset \C^m$ be a $k$-dimensional subspace, and let $T\subset V\cap S^{m-1}$.
Let $K\ge1$, and assume that for some $\rho,p_0\in (0,1)$, for any $u\in T$ we have
\begin{equation} \label{pbound}
\pr\big( \|\M u\|\le \rho K\sqrt{n}\big) \le p_0.
\end{equation}
Then
\begin{equation} \label{net:unionbd}
\pro{ \mB(K) \wedge \big\{ \, \exists u\in T_{\rho/4}: \|\M u\| \le (\rho/2) K\sqrt{n}\,\big\}} \le \left(\frac{12}{\rho}\right)^{2k}\cdot p_0.
\end{equation}
\end{corollary}
\begin{proof}
By Lemma \ref{lem:net}, $T$ has a $\rho/4$-net $\Sigma$ of size at most $(12/\rho)^{2k}$.
On the event that $\|M\|\le K\sqrt{n}$ and there exists $u\in T_{\rho/4}$ such that $\|Mu\|\le (\rho/2)K\sqrt{n}$, we fix such a $u$ and let $u'\in \Sigma$ be such that $\|u-u'\|<\rho/2$ (which exists by the triangle inequality).
It follows that
$$\|Mu'\|\le \|Mu\|+ \|M(u-u')\|\le (\rho/2)K\sqrt{n} + K\sqrt{n}\cdot (\rho/2) = \rho K\sqrt{n}.$$
We have shown that
$$
\big\{\|M\|\le K\sqrt{n}\; \mbox{ and }\;\exists u\in T_{\rho/4}: \|Mu\|\le (\rho/2)K\sqrt{n} \big\} \subset \big\{ \exists u \in \Sigma: \|Mu\|\le \rho K\sqrt{n}\big\}.
$$
The desired result now follows from taking a union bound over the at most $(12K/\rho)^{2k}$ choices of $u\in \Sigma$ and \eqref{pbound}.
\end{proof}
\end{comment}
\subsection{Anti-concentration for scalar random walks} \label{sec:fourier}
In this subsection we collect some standard anti-concentration estimates for scalar random walks, which are perhaps the most central tool for proving that random matrices are (well-)invertible with high probability.
\begin{definition}[Concentration probability]
Let $\xi$ be a complex-valued random variable.
For $v\in \C^n$ we let
\begin{equation}
S_\xi(v) = \sum_{j=1}^n \xi_jv_j
\end{equation}
where $\xi_1,\dots,\xi_n$ are iid copies of $\xi$.
For $r\ge0$ we define the \emph{concentration probability}
\begin{equation}
p_{\xi,v}(r)= \sup_{z\in \C}\pr\big( |S_\xi(v)-z|\le r\big).
\end{equation}
\end{definition}
Throughout this section we operate under the following distributional assumption on $\xi$.
\begin{definition}[Controlled second moment, cf.\ {\perp\!\!\!\perpte[Definition 2.2]{TaVu:circ}}] \label{def:kappa}
Let $\kappa\ge 1$. A complex random variable $\xi$ is said to have \emph{$\kappa$-controlled second moment} if one has the upper bound
\begin{equation} \label{cond.kappa1}
\e|\xi|^2\le \kappa
\end{equation}
(in particular, $|\e \xi|\le \kappa^{1/2}$), and the lower bound
\begin{equation} \label{cond.kappa2}
\e [\re(z\xi-w)]^2 \un(|\xi|\le \kappa) \ge \frac1\kappa [\re(z)]^2
\end{equation}
for all $z\in \C, a\in \R$.
\end{definition}
Roughly speaking, a complex random variable $\xi$ has controlled second moment if its distribution has a one-(real-)dimensional marginal with fairly large variance on some compact set.
The following is a quantitative version of \perp\!\!\!\perpte[Lemma 2.4]{TaVu:circ}, and shows that by multiplying the matrices $X$ and $B$ in Definition \ref{def:profile} by a scalar phase (amounting to multiplying $M$ by a phase, which does not affect its singular values) we can assume the atom variable $\xi$ has $O(\kappa_0)$-controlled second moment in all of our proofs with no loss of generality.
The proof is
deferred to Appendix \ref{app:anti}.
\begin{lemma} \label{lem:wlog.kappa}
Let $\xi$ be a centered complex random variable with unit variance, and assume $\xi$ is $\kappa_0$-spread for some $\kappa_0 \ge 1$ (see Definition \ref{def:spread}).
Then there exists $\theta \in \R$ such that $e^{i\theta}\xi$ has $\kappa$-controlled second moment for some $\kappa=O(\kappa_0)$.
\end{lemma}
Below we give two standard bounds on the concentration function $p_{\xi,v}(r)$ when $\xi$ is a $\kappa$-controlled random variable and $v\in S^{n-1}$.
The first gives a crude constant order bound that is uniform in $v\in S^{n-1}$:
\begin{lemma}[Crude anti-concentration, cf.\ {\perp\!\!\!\perpte[Corollary 6.3]{TaVu:smooth}}] \label{lem:anti_crude}
Let $\xi$ be a complex random variable with $\kappa$-controlled second moment.
There exists $r_0>0$ depending only on $\kappa$ such that $p_{\xi,v}(r_0) \le 1-r_0$ for all $v\in S^{n-1}$.
\end{lemma}
Note that Lemma \ref{lem:anti_crude} is sharp for the case that $v$ is a standard basis vector.
The following gives an improved bound when $v$ has small $\ell_\infty$ norm.
\begin{lemma}[Improved anti-concentration] \label{lem:anti_improved}
Let $\xi$ be a complex random variable that is $\kappa$-controlled for some $\kappa>0$, and let $v\in S^{n-1}$.
For all $r\ge0$,
\begin{equation} \label{be:1d}
p_{\xi,v}(r) \ll_\kappa r + \|v\|_\infty.
\end{equation}
\end{lemma}
\begin{comment}
\Blue{May omit.}
\begin{remark}
Since $\xi$ and the coefficients of $v$ are complex one would expect a bound with $r^2$ in place of $r$ on the right hand side of \eqref{be:1d}, provided the set of phases of the components of $v$ is in some sense ``genuinely two dimensional".
Indeed, by a modification of the proof below one can show
\begin{equation}
p_{\xi,v}(r) \ll_\kappa \frac1{\alpha^2} (r^2 + \|v\|_\infty^2)
\end{equation}
for all $r\ge 0$, provided $v$ satisfies
\begin{equation} \label{disaligned}
\inf_{\theta\in \R}\|\re(e^{i\theta}v)\| \ge \alpha\|v\|
\end{equation}
for some $\alpha>0$.
The condition \eqref{disaligned} was used in a more sophisticated anti-concentration bound of Rudelson and Vershynin -- see \perp\!\!\!\perpte[Theorem 3.3]{RuVe:rectangular}.
We will not need such improvements in the present work.
\end{remark}
\end{comment}
Lemma \ref{lem:anti_improved} can be deduced from the Berry--Ess\'een theorem (which is the approach taken in \perp\!\!\!\perptep{LPRT}, for instance), but this would require $\xi$ to have finite third moment, which we do not assume.
(Generally speaking, higher moment assumptions should only be necessary to prove concentration bounds as opposed to anti-concentration.)
Since we could not locate a proof in the literature for the case that $\xi$
and
the coefficients of $v$ take values in $\C$, we provide a proof in
Appendix \ref{app:anti}.
\subsection{Anti-concentration for the image of a fixed vector} \label{sec:image}
In this subsection we boost the anti-concentration bounds for scalar random variables from the previous sections to anti-concentration for the image of a fixed vector under a random matrix.
The following lemma of Rudelson and Vershynin is convenient for this task.
\begin{lemma}[Tensorization, cf.\ {\perp\!\!\!\perpte[Lemma 2.2]{RuVe:ilo}}] \label{lem:tensorize}
Let $\zeta_1,\dots, \zeta_n$ be independent non-negative random variables.
\begin{enumerate}[(a)]
\item Suppose that for some $\eps_0,p_0>0$ and all $j\in [n]$, $\pro{\zeta_j\le \eps_0}\le p_0$. There are $c_1,p_1\in (0,1)$ depending only on $p_0$ such that
\begin{equation}
\pr\bigg( \sum_{j=1}^n \zeta_j^2 \le c_1\eps_0^2n\bigg) \le p_1^n.
\end{equation}
\item Suppose that for some $K,\eps_0\ge 0$ and all $j\in [n]$, $\pro{\zeta_j\le\eps} \le K\eps$ for all $\eps\ge \eps_0$.
Then for all $\eps\ge \eps_0$,
\begin{equation}
\pr\bigg( \sum_{j=1}^n \zeta_j^2 \le\eps^2n \bigg) \le (CK\eps)^n.
\end{equation}
\end{enumerate}
\end{lemma}
Note that in part (a) we have given more specific dependencies on the parameters than in \perp\!\!\!\perptep{RuVe:ilo}. For completeness we provide the proof of this modified version in Appendix \ref{app:anti}.
Let $\M=A\perp\!\!\!\perprc X+B$ be as in Definition \ref{def:profile}.
Recall that we denote by $R_i$ the $i$th row of $\M$.
In the following lemmas we assume that the atom variable $\xi$ has $\kappa$-controlled second moment for some fixed $\kappa\ge1$.
For $v\in \C^m$ and $i\in [n]$ we write
\begin{equation} \label{def:vi}
v^i :=(v_j\sig_{ij})_{j=1}^m
\end{equation}
For $\alpha>0$ we denote
\begin{equation} \label{def:Ialpha}
I_\alpha(v):= \{i\in [n]: \|v^i\|\ge \alpha\}.
\end{equation}
\begin{lemma}[Crude anti-concentration for the image of a fixed vector] \label{lem:fixed_crude}
Fix $v\in \C^m$ and let $\alpha>0$ such that $I_\alpha(v)\ne \varnothing$.
For all $I_0\subset I_\alpha(v)$,
\begin{equation}
\sup_{w\in \C^n} \pr_{I_0}\Big( \|\M v -w\| \le c_0 \alpha|I_0|^{1/2} \Big) \le e^{-c_0|I_0|}
\end{equation}
where $c_0>0$ is a constant depending only on $\kappa$ (recall our notation $\pr_{I_0}(\,\cdot\,)$ from Section \ref{sec:notation}).
\end{lemma}
\begin{proof}
Fix $w\in \C^n$ arbitrarily.
For any $i\in I_\alpha(v)$ and any $t\ge0$ we have
\begin{align*}
\pro{ |R_i\cdot v -w_i|\le t} &\;\le\; p_{\xi,v^i}(t) \,=\,p_{\xi,v^i/\|v^i\|}(t/\|v^i\|) \,\le\, p_{\xi,v^i/\|v^i\|}(t/\alpha).
\end{align*}
Taking $t=\alpha r_0$, by Lemma \ref{lem:anti_crude} we have
\begin{equation}
\pro{ |R_i\cdot v -w_i|\le \alpha r_0} \le 1-r_0
\end{equation}
where $r_0>0$ depends only on $\kappa$.
Fix $I_0\subset I_\alpha(v)$ arbitrarily. We may assume without loss of generality that $I_0$ is non-empty.
By Lemma \ref{lem:tensorize}(a) there exists $c_1>0$ depending only on $\kappa$ such that
\begin{equation} \label{eq:crude1}
\pr_{I_0}\bigg( \sum_{i\in I_0} |R_i\cdot v - w_i|^2 \le c_1 r_0^2\alpha^2 |I_0|\bigg) \le e^{-c_1|I_0|}.
\end{equation}
Now for any $\tau\ge 0$,
\begin{align*}
\pr_{I_0}\Big( \|\M v -w\| \le \tau |I_0|^{1/2}\Big) &= \pr_{I_0}\bigg( \sum_{i=1}^n |R_i\cdot v -w_i|^2 \le \tau^2 |I_0|\bigg)\\
&\le \pr_{I_0}\bigg( \sum_{i\in I_0} |R_i\cdot v -w_i|^2 \le \tau^2 |I_0|\bigg)
\end{align*}
and the claim follows by taking $\tau=c_1^{1/2}r_0\alpha =: c_0\alpha$ and applying \eqref{eq:crude1}.
\end{proof}
By similar lines, using Lemmas \ref{lem:tensorize}(b) and \ref{lem:anti_improved} in place of Lemmas \ref{lem:tensorize}(a) and \ref{lem:anti_crude}, respectively, one obtains the following, which is superior to Lemma \ref{lem:fixed_crude} for vectors $v$ with small $\ell_\infty$ norm.
The details are omitted.
\begin{lemma}[Improved anti-concentration for the image of a fixed vector] \label{lem:fixed_improved}
Fix $v\in \C^m$.
Let $\alpha>0$ such that $I_\alpha(v)\ne \varnothing$ and fix $I_0\subset I_\alpha(v)$ nonempty.
For all $t\ge 0$,
\begin{equation}
\sup_{w\in \C^n} \pr_{I_0}\Big( \|\M v -w\| \le t|I_0|^{1/2}\Big) \le O_\kappa\bigg(\frac{1}{\alpha}\big( t+\|v\|_\infty\big) \bigg)^{|I_0|}.
\end{equation}
\end{lemma}
\section{Invertibility from connectivity: Compressible vectors} \label{sec:comp}
In this section we combine the anti-concentration estimates from Section \ref{sec:anti} with union bounds over $\eps$-nets (as obtained for instance from Lemma \ref{lem:net}) to prove that with high probability, a random matrix $M$ as in Theorem \ref{thm:broad} or Theorem \ref{thm:super} is well-invertible on the set of compressible vectors $\Comp(\theta,\rho)$ (as defined in \eqref{def:compr}) for appropriate choices of $\theta,\rho$.
Hence, there will be a competition between the quality of the anti-concentration estimates and the cardinality of the $\eps$-nets.
For small values of $\theta$ we can use $\eps$-nets of small cardinality, but only have poor anti-concentration bounds (namely, Lemma \ref{lem:fixed_crude}), while for large $\theta$ the nets are very large, but we have access to the improved anti-concentration of Lemma \ref{lem:fixed_improved}.
In both cases we start with a crude result, Lemma \ref{lem:high}, giving control for the vectors in $\Comp(\theta_0,\rho_0)$ for some small value of $\theta_0$ (possibly depending on $n$).
We then use an iterative argument argument to obtain control on $\Comp(\theta,\rho)$ for larger values of $\theta$ while lowering the parameter $\rho$.
For Theorem \ref{thm:broad} we want to take $\theta$ close to 1, while for Theorem \ref{thm:super} a constant order value of $\theta$ will suffice.
It turns out that that while the standard $\eps$-net from Lemma \ref{lem:net} suffices to prove Lemma \ref{lem:high}, it is insufficient to obtain control on $\Comp(\theta,\rho)$ for the desired values of $\theta$.
For the broadly connected case this is essentially due to working in $\C^n$ rather than $\R^n$, which causes a factor $2$ increase in metric entropies (this difficulty was not present in the proof of Theorem \ref{thm:ruze} in \perp\!\!\!\perptep{RuZe} as they worked in $\R^n$).
The situation is worse for the case of Theorem \ref{thm:super}, the main source of difficulty being that $\|B\|$ can be of arbitrary polynomial order. As a consequence, the starting point $\theta_0$ for our iterative argument will be of size $o(1)$.
This prevents us from using the third condition of the super-regularity hypothesis (see Definition \ref{def:super}), which only ``sees" vectors that are essentially supported on more than $\eps n$ coordinates.
We deal with this by \emph{reducing the entropy cost} of the nets over which we take union bounds.
In Section \ref{sec:entropy} we prove Lemma \ref{lem:entropy} which shows, roughly speaking, that if we have already established control on vectors in $\Comp(\theta,\rho)$ for some $\theta,\rho$, then we can control the vectors in $\Comp(\theta+\Delta,\rho')$ for some small $\Delta, \rho'$ using a \emph{random} net of significantly smaller cardinality than the net provided by Lemma \ref{lem:net}.
We can then increment $\theta$ from $\theta_0$ up to size $\gg 1$, taking steps of size $\Delta$.
For the broadly connected case we can continue and take $\theta$ as close to $1$ as desired.
The entropy reduction argument for Lemma \ref{lem:entropy} makes use of a strong version of the well-known Restricted Invertibility Theorem due to Spielman and Srivastava -- see Theorem \ref{thm:rit}.
We now state the main results of this section.
For $K\ge1$ we denote the \emph{boundedness event}
\begin{equation} \label{event:bdd}
\mB(K):= \big\{\|\M\|\le K\sqrt{n}\,\big\}.
\end{equation}
With a fixed choice of $K$ we write
\begin{equation} \label{def:eventcomp}
\event(\theta,\rho):= \mB(K) \wedge \big\{\, \exists u\in \Comp(\theta,\rho): \|\M u\|\le \rho K \sqrt{n}\,\big\}.
\end{equation}
\begin{comment}
\begin{proposition}[Compressible vectors: broadly connected profile] \label{prop:slight}
Let $\M=A\perp\!\!\!\perprc X+B$ be as in Definition \ref{def:profile} with $n/2\le m\le 2n$.
Assume that $\xi$ has $\kappa$-controlled second moment for some $\kappa\ge1$ (see Definition \ref{def:kappa}), and that for some $\ha,\delta,\nu\in (0,1)$ we have
\begin{enumerate}[(1)]
\item $|\mN_{A(\ha)^\tran}(j)|\ge \delta n$ for all $j\in [m]$;
\item $|\mN_{A(\ha)^\tran}^{(\delta)}(J)| \ge \min((1+\nu)|J|,n)$ for all $J\subset[m]$.
\end{enumerate}
Let $K\ge1$.
There is a constant $\rho=\rho(\kappa,\ha,\delta,\nu,K)>0$ such that for any $0<\theta\le (1-\frac\delta4)\min(\frac{n}m,1)$,
\begin{equation}
\pro{\event(\theta,\rho)}=O(e^{-c_\kappa\delta\ha^2n})
\end{equation}
where $c_\kappa>0$ depends only on $\kappa$ and the implied constant depends only on $\kappa,\ha,\delta,\nu$ and $K$.
\end{proposition}
\end{comment}
\begin{proposition}[Compressible vectors: broadly connected profile] \label{prop:slight}
Let $\M=A\perp\!\!\!\perprc X+B$ be as in Definition \ref{def:profile} with $n/2\le m\le 2n$, and assume that $\xi$ has $\kappa$-controlled second moment for some $\kappa\ge1$ (see Definition \ref{def:kappa}).
Let $K\ge1$ and $\ha,\delta,\nu\in (0,1)$.
There exist $\theta_0(\kappa,\ha,\delta,K)>0$ and $\rho(\kappa,\ha,\delta,\nu,K)>0$ such that the following holds.
Assume
\begin{enumerate}[(1)]
\item $|\mN_{A(\ha)^\tran}(j)|\ge \delta n$ for all $j\in [m]$;
\item $|\mN_{A(\ha)^\tran}^{(\delta)}(J)| \ge \min((1+\nu)|J|,n)$ for all $J\subset[m]$ with $|J|\ge \theta_0m$.
\end{enumerate}
Then for any $0<\theta\le (1-\frac\delta4)\min(\frac{n}m,1)$,
\begin{equation}
\pro{\event(\theta,\rho)}\ll_{\kappa,\ha,\delta,\nu,K} \expo{-c_\kappa\delta\ha^2n}
\end{equation}
where $c_\kappa>0$ depends only on $\kappa$.
\end{proposition}
The following gives control of compressible vectors for more general profiles than in Proposition \ref{prop:slight} (essentially removing the condition (2)).
However, we have to take the parameter $\rho$ much smaller, and we only cover vectors that are essentially supported on a small (linear) proportion of the coordinates, rather than a proportion close to one.
\begin{proposition}[Compressible vectors: general profile with large perturbation] \label{prop:comp}
Let $\M=A\perp\!\!\!\perprc X+B$ be as in Definition \ref{def:profile} with $n/2\le m\le 2n$.
Assume $\xi$ has $\kappa$-controlled second moment for some $\kappa\ge1$, and that for some $\sig_0>0$ we have
\begin{equation} \label{LB:columnvar}
\sum_{i=1}^n \sig_{ij}^2\ge \sig_0^2n\quad\text{ for all }j\in [m].
\end{equation}
Fix $\gamma\ge 1/2$ and let $1\le K=O(n^{\gamma-1/2})$.
Then for some $\rho= \rho(\gamma,\sig_0,\kappa,n) \gg_{\gamma,\sig_0,\kappa} n^{-O(\gamma^2)}$ and a sufficiently small constant $c_0>0$ we have
\begin{equation}
\pro{\event(c_0\sig_0^2,\rho)}\ll_{\gamma,\sig_0,\kappa}\expo{-c_\kappa\sig_0^2n}
\end{equation}
where $c_\kappa>0$ depends only on $\kappa$.
\end{proposition}
\begin{comment}
\Red{Either shorten this or the intro to the section}
The remainder of this section is organized as follows.
In Section \ref{sec:high} we apply Corollary \ref{cor:net} and the crude anti-concentration bound of Lemma \ref{lem:anti_crude}
to obtain a crude form of Proposition \ref{prop:comp} (which also applies to matrices as in Proposition \ref{prop:slight}).
In Section \ref{sec:entropy} we present a key technical result, Lemma \ref{lem:entropy}, which allows one to control the invertibility of a random matrix over a portion of the sphere using a random net of small cardinality, assuming one already has some control over compressible vectors.
In the remaining sections Lemma \ref{lem:entropy} is applied with the improved anti-concentration estimate of Lemma \ref{lem:anti_improved} in an iterative argument to upgrade the crude control from Lemma \ref{lem:high} to obtain Propositions \ref{prop:slight} and \ref{prop:comp}.
\end{comment}
\subsection{Highly compressible vectors} \label{sec:high}
In this subsection we establish the following crude version of Proposition \ref{prop:comp}, giving control on vectors in $\Comp(\theta_0,\rho_0)$ with $\theta_0$ sufficiently small depending on $\sig_0$ and $K$.
\begin{lemma}[Highly compressible vectors] \label{lem:high}
Let $\M=A\perp\!\!\!\perprc X+B$ be as in Definition \ref{def:profile} with $m\le 2n$.
Assume that $\xi$ has $\kappa$-controlled second moment for some $\kappa\ge1$.
Suppose also that there is a constant $\sig_0>0$ such that for all $j\in [m]$, $\sum_{i=1}^n \sig_{ij}^2 \ge \sig_0^2n$.
Let $K\ge 1$.
Then with notation as in \eqref{def:eventcomp} we have
\begin{equation}
\pro{ \event(\theta_0,\rho_0)} \le e^{-c_\kappa\sig_0^2n}
\end{equation}
where $\theta_0= c_\kappa\sig_0^2 /\log(K/\sig_0^2)$ and $\rho_0=c_\kappa\sig_0^2/K$ for a sufficiently small $c_\kappa>0$ depending only on $\kappa$.
\end{lemma}
We will need the following lemma, which ensures that the set $I_\alpha(v)$ from \eqref{def:Ialpha} is reasonably large when the columns of $A$ have large $\ell_2$ norm.
A similar argument has been used in \perp\!\!\!\perptep{LiRi} and \perp\!\!\!\perptep{RuZe}.
\begin{lemma}[Many good rows] \label{lem:goodrows0}
Let $A$ be an $n\times m$ matrix as in Definition \ref{def:profile}, and assume that for some $\sig_0>0$ we have $\sum_{i=1}^n \sig_{ij}^2\ge \sig_0^2 n$ for all $j\in [m]$.
Then for any $v\in S^{m-1}$ we have $|I_{\sig_0/2}(v)|\ge \frac12\sig_0^2n$.
\end{lemma}
\begin{proof}
Writing $\alpha=\sig_0/\sqrt{2}$, we have
\begin{align*}
\sig_0^2 n &\le \sum_{i=1}^n\sum_{j=1}^m |v_j|^2 \sig_{ij}^2\\
&=\sum_{i\in I_{\alpha}(v)}\sum_{j=1}^m |v_j|^2 \sig_{ij}^2 + \sum_{i\notin I_{\alpha}(v)}\sum_{j=1}^m |v_j|^2 \sig_{ij}^2\\
&\le \sum_{i\in I_{\alpha}(v)}\sum_{j=1}^m |v_j|^2 + \sum_{i\notin I_{\alpha}(v)} \frac{1}2\sig_0^2 \\
&\le |I_{\alpha}(v)| + \frac12 \sig_0^2n
\end{align*}
and rearranging gives the claim.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:high}]
Fix $J\subset [m]$ of size $\lf \theta_0m\rf$ and let $v\in S^J$ be arbitrary.
Writing $\alpha=\sig_0/\sqrt{2}$, by Lemma \ref{lem:fixed_crude} and our choice of $\rho_0$ (with $c_\kappa>0$ sufficiently small depending on $\kappa$),
\[
\pro{ \|\M v\|\le \rho_0K \sqrt{n}} \le \pro{ \|\M v\|\le c_\kappa\sig_0|I_{\alpha}(v)|^{1/2}}
\le e^{-c_\kappa|I_{\alpha}(v)|}.
\]
Applying Lemma \ref{lem:goodrows0}, we obtain
\begin{equation} \label{high:forallv}
\pro{ \|\M v\|\le \rho_0K\sqrt{n}} \le e^{-c_\kappa\sig_0^2 n}\qquad \forall v\in S^J
\end{equation}
(adjusting $c_\kappa$).
By Lemma \ref{lem:net} we may fix $\Sigma_J\subset S^J$ a $\rho_0/4$-net for $S^J$ such that $|\Sigma_J|\le (12/\rho_0)^{2k}$.
Suppose that $\|M\|\le K\sqrt{n}$ and that $\|Mu\|\le \rho_0 K\sqrt{n}$ for some $u\in S^{m-1}\cap (\C^J)_{\rho_0/4}$.
Let $u'\in \C^J$ with $\|u-u'\|\le \rho_0/4$, and let $u''\in \Sigma_J$ with $\|u''-\frac{u'}{\|u'\|}\|\le \rho_0/4$.
By the triangle inequality,
\[
\| u-u''\| \le \|u-u'\| + \left\| u'-\frac{u'}{\|u'\|}\right\| + \left\| \frac{u'}{\|u'\|} - u''\right\| \le 3\rho_0/4
\]
where the bound on the middle term follows from $|\|u'\|-1|\le \rho_0/4$ (also by the triangle inequality).
We have
\[
\|Mu''\|\le \|Mu\| + \|M(u-u'')\| \le \rho_0K\sqrt{n} + K\sqrt{n}\cdot (3\rho_0/4) \le 2\rho_0 K\sqrt{n}.
\]
Applying the union bound and \eqref{high:forallv} (adjusting $c_\kappa$ to replace $\rho_0$ by $2\rho_0$),
\begin{align*}
&\pro{ \exists u\in S^{m-1}\cap (\C^J)_{\rho_0/8}: \|Mu\|\le \rho_0K\sqrt{n}} \\
&\qquad\qquad\qquad\qquad\qquad\qquad\le \pro{ \exists u''\in \Sigma_J: \|Mu''\|\le 2\rho_0K\sqrt{n}}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\le O(1/\rho_0)^{2\theta_0 m} e^{-c_\kappa a_0^2n}
\end{align*}
From \eqref{def:compr} and applying the union bound over all choice of $J\in {[m]\choose \theta_0 m}$,
\begin{align*}
\pro{ \event(\theta_0,\rho_0/4)} &\le O(1/\theta_0)^{\theta_0m} O(1/\rho_0)^{2\theta_0m} e^{-c_\kappa\sig_0^2n}\le O\left(\frac{1}{\theta_0\rho_0^2}\right)^{2\theta_0 n} e^{-c_\kappa\sig_0^2n},
\end{align*}
where we used our assumption $m\le 2n$.
The desired bound now follows from substituting our choices of $\theta_0,\rho_0$, and again adjusting the constant $c_\kappa$ to replace $\rho_0/4$ by $\rho_0$ in the above.
\end{proof}
\subsection{An entropy reduction lemma} \label{sec:entropy}
The aim of this subsection is to establish the following:
\begin{lemma}[Control by a random net of small cardinality] \label{lem:entropy}
For every $I\subset[n], J\subset [m]$, $\eps>0$ there is a random finite set $\Sigma_{I,J}(\eps)\subset S^J$, measurable with respect to $\mF_{I.J} =\langle \{\xi_{ij}\}_{i\in I,j\in J}\rangle$, such that the following holds.
Let $\rho\in (0,1)$, $K>0$ and $0<\theta< \frac nm$.
On $\mB(K)\wedge \event(\theta,\rho)^c$, for all $J\subset [m]$ with $|J|>\theta m$ and all $\beta,\rho'\in (0,1)$,
putting
\begin{equation} \label{def:rhodub}
\rho''=\frac{6\rho'}{\beta \rho}\left(\frac{n}{\lf \theta m\rf}\right)^{1/2}
\end{equation}
there exists $I\subset[n]$ with $|I|=\lf (1-\beta)^2\lf \theta m\rf\rf$ such that
\begin{enumerate}[(1)]
\item $|\Sigma_{I,J}(\rho'')| \le (C/\rho'')^{2(|J|-|I|)}$ for an absolute constant $C>0$, and
\item for any $u\in S^{m-1}\cap (\C^J)_{\rho'}$ such that $\|\M u\|\le \rho'K\sqrt{n}$, we have $\dist(u,\Sigma_{I,J}(\rho'')) \le 3\rho''$.
\end{enumerate}
Furthermore, writing
\begin{equation}
\good_{I,J}(\rho'') := \bigg\{ \big| \Sigma_{I,J}(\rho'')\big|\le \bigg(\frac{C}{\rho''}\bigg)^{2(|J|-|I|)} \bigg\}
\end{equation}
we have that for any $\theta'\in (\theta,1]$,
\begin{align}
&\event(\theta,\rho)^c \wedge \event(\theta',\rho') \subset \notag\\
&\quad\quad
\bigvee_{J\in {[m]\choose \theta' m}} \bigvee_{I\in {[n]\choose (1-\beta)^2\lf \theta m\rf}} \bigg( \good_{I,J}(\rho'')
\wedge\Big\{ \exists u \in \Sigma_{I,J}(\rho''): \|\M u\|\le 4\rho''K\sqrt{n} \Big\}\bigg).\label{cont:entropy}
\end{align}
\end{lemma}
\begin{remark}
We obtain the random set $\Sigma_{I,J}(\eps)$ as the intersection of the sphere $S^J$ with an $\eps$-net of the kernel of the submatrix $\M_{I, J}$.
However, for our purposes it only matters that it is fixed by conditioning on the rows $\{R_i\}_{i\in I}$, has small cardinality, and serves as a net for almost-null vectors of $\M$ that are supported on $J$.
\end{remark}
To prove Lemma \ref{lem:entropy} we use the following version of the Restricted Invertibility Theorem \perp\!\!\!\perptep{SpSr:rit} (the version below is taken from \perp\!\!\!\perpte[Theorem 3.1]{MSS:icm}).
\begin{theorem}[Restricted Invertibility Theorem]
\label{thm:rit}
Suppose $v_1,\dots, v_n\in \C^m$ are such that $\sum_{i=1}^n v_iv_i^* = I_m$.
For any $\beta\in (0,1)$, there is a subset $I\subset [n]$ of size $|I|= \lf (1-\beta)^2m\rf$ for which
\begin{equation}
\lambda_{|I|}\bigg(\sum_{i\in I} v_iv_i^*\bigg) \ge \beta^2m/n
\end{equation}
where $\lambda_k(A)$ denotes the $k$th largest eigenvalue of a Hermitian matrix $A$.
\end{theorem}
This has the following consequence, which can be seen as a robust quantitative version of the basic fact from linear algebra that the row rank of a matrix is equal to its column rank.
\begin{corollary} \label{cor:rit}
Let $M$ be an $n\times m$ matrix with $n\ge m$, and assume $s_m(M)\ge \eps_0\sqrt{n}$ for some $\eps_0>0$.
For any $\beta\in (0,1)$ there exists $I\subset [n]$ with $|I|=\lf (1-\beta)^2 m\rf$ such that
\[
s_{|I|}(M_{I, [m]}) \ge \beta \eps_0\sqrt{m}.
\]
\end{corollary}
\begin{remark}
The original Restricted Invertibility Theorem of Bourgain and Tzafriri \perp\!\!\!\perptep{BoTz:rit} only gives $|I|\ge cm$ and $s_{|I|}(M_{I, [m]}) \ge c \eps_0\sqrt{m}$ for some (small) absolute constant $c>0$, while it will be important for our purposes to be able to take $I$ of size close to $m$.
\end{remark}
\begin{proof}[Proof of Corollary \ref{cor:rit}]
By the singular value decomposition it suffices to consider $M$ of the form
$
M=U\Sigma
$
where $U$ is an $n\times m$ matrix with orthonormal columns and $\Sigma$ is an $m\times m$ diagonal matrix with entries bounded below by $\eps_0\sqrt{n}$.
Fix $\alpha \in (0,1)$.
Letting $v_1^*,\dots, v_n^*\in \C^m$ denote the rows of $U$, it follows from orthonormality that
\[
I_m = U^*U = \sum_{i=1}^n v_iv_i^*.
\]
Hence, we can apply Theorem \ref{thm:rit} to obtain a subset $I\subset[n]$ with $|I|= \lf (1-\beta)^2m\rf$
such that
\[
s_{|I|}(U_{I, [m]})^2 = \lambda_{|I|}\bigg( \sum_{i\in I} v_iv_i^*\bigg) \ge \beta^2m/n.
\]
Now we have
\[
s_{|I|}(M_{I, m}) \ge s_{|I|}(U_{I, m}) s_m(\Sigma) \ge \beta \sqrt{\frac{m}{n}} \eps_0\sqrt{n} = \beta\eps_0\sqrt{m}.\qedhere
\]
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:entropy}]
Let $I\subset [n], J\subset[m]$, and write $V_{I,J} = \C^J\cap \ker(\M_{I,J})$.
Conditional on $\mF_{I,J}$, for $\eps>0$ we let $\Sigma_{I,J}(\eps)$ be an $\eps$-net of $S^{m-1}\cap V_{I,J}$.
By Lemma \ref{lem:net} we may take
\begin{equation} \label{sigmacard1}
|\Sigma_{I,J}(\eps)| = O(1/\eps)^{2\dim(V_{I,J})}.
\end{equation}
Let $\rho,\rho'\in (0,1)$, $K>0$ and $0<\theta< \frac nm$.
Fix $\beta\in (0,1)$ and $J\subset [m]$ with $|J|>\theta m$.
On $\event(\theta,\rho)^c$, for all $J_0\subset J$ with $|J_0|=\lf \theta m\rf$ we have
\[
s_{\lf \theta m\rf}(\M_{[n], J_0})\ge \rho K\sqrt{n}.
\]
By Corollary \ref{cor:rit} there exists $I\subset [n]$ with $|I|=\lf (1-\beta)^2 \lf \theta m\rf\rf$ such that
\[
s_{|I|}(\M_{I, J_0})\ge \beta \rho K\sqrt{\lf \theta m\rf}.
\]
By the Cauchy interlacing law,
\begin{equation} \label{sklb}
s_{|I|}(\M_{I, J}) \ge \beta \rho K\sqrt{\lf \theta m\rf}.
\end{equation}
In particular, the submatrix $(y_{ij})_{i\in I,j\in J}$ has full row-rank, which implies $\dim(V_{I,J}) = |J| - |I|$.
From \eqref{sigmacard1} we conclude
\begin{equation} \label{sigmacard2}
|\Sigma_{I,J}(\eps)| = O(1/\eps)^{2(|J|-|I|)}
\end{equation}
for any $\eps>0$.
Now suppose there exists $u\in S^{m-1}\cap (\C^J)_{\rho'}$ such that
\begin{equation}
\|\M u\|\le \rho'K\sqrt{n}.
\end{equation}
Letting $v'\in \C^J$ such that $\|u-v'\|\le \rho'$, and putting $v:=v'/\|v'\|\in S^J$, by the triangle inequality we have $\|u-v\|\le 2\rho'$ and
\begin{equation} \label{Muub}
\|\M v\|\le \|\M u\|+\|\M \|\|u-v\| \le 3\rho'K\sqrt{n}.
\end{equation}
On the other hand,
\[
\|\M v\| \ge \|\M_{I, [m]}v\| = \|\M_{I, [m]}(\id -P_{V_{I,J}})v\|
\]
where $P_{V_{I,J}}$ is the matrix for orthogonal projection to the subspace $V_{I,J}$.
Applying \eqref{sklb},
\[
\|\M v\| \ge \|(\id - P_{V_{I,J}})v\| \beta \rho K\sqrt{\lf \theta m\rf}.
\]
Together with \eqref{Muub} this implies that $v$ lies within distance
\begin{equation}
\frac{3\rho' \sqrt{n}}{\beta \rho \sqrt{\lf \theta m\rf}} =\rho''/2
\end{equation}
of the subspace $V_{I,J}$. Since $v$ is a unit vector we have $\dist(v,S^{m-1}\cap V_{I,J})\le \rho''$ by the triangle inequality, and
\begin{align*}
\dist(u,\Sigma_{I,J}(\rho'')) &\le \|u-v\|+\rho''+ \dist(v, S^{m-1}\cap V_{I,J}) \le 2\rho' + 2\rho'' \le 3\rho''
\end{align*}
as desired (that $2\rho'\le \rho''$ follows from inspection of \eqref{def:rhodub}).
Now to prove \eqref{cont:entropy}, let $\theta'\in (\theta,1]$.
Intersecting with $\event(\theta,\rho)^c$ and applying the first part of the lemma,
\begin{align}
&\event(\theta,\rho)^c\wedge\event(\theta',\rho') \notag\\
&\quad = \mB(K)\wedge\event(\theta,\rho)^c\wedge\bigvee_{J\in {[m]\choose \theta' m}} \Big\{\exists v\in (S^J)_{\rho'}: \|\M v\|\le \rho' K\sqrt{n}\Big\}\notag\\
&\quad\subset
\bigvee_{J\in {[m]\choose \theta' m}} \bigvee_{I\in {[n]\choose (1-\beta)^2\lf \theta m\rf}} \bigg( \good_{I,J}(\rho'')
\wedge\Big\{ \exists u \in \Sigma_{I,J}(\rho''): \|\M u\|\le 4\rho''K\sqrt{n} \Big\}\bigg)
\end{align}
where in the last line we noted that for $v\in (S^J)_{\rho'}, u\in \Sigma_{I,J}(\rho'')$ such that $\|u-v\|\le 3\rho''$, we have
\[
\|\M u\|\le \|\M v\|+3\rho'' K\sqrt{n} \le (\rho'+3\rho'')K\sqrt{n}\le 4\rho'' K\sqrt{n}.\qedhere
\]
\end{proof}
\subsection{Broadly connected profile: Proof of Proposition \ref{prop:slight}}
We will obtain Proposition \ref{prop:slight} from an iterative application of the following lemma:
\begin{lemma}[Incrementing compressibility: broadly connected profile] \label{lem:increment_broad}
Let $\M=A\perp\!\!\!\perprc X+B$ be as in Definition \ref{def:profile} with $m\ge n/2$.
Assume $\xi$ has $\kappa$-controlled second moment for some $\kappa\ge1$, and that for some $\ha,\delta,\nu,\theta_1\in (0,1)$ we have
\begin{enumerate}[(1)]
\item $|\mN_{A(\ha)}(j)|\ge \delta n$ for all $j\in [m]$;
\item $|\mN_{A(\ha)}^{(\delta)}(J)|\ge \min((1+\nu) |J|,n)$ for all $J\subset[m]$ with $|J|\ge (\theta_1/2)m$.
\end{enumerate}
Let $K\ge1$, $\rho\in (0,1)$, and $\theta\in [\theta_1,1)$ such that $(1+\frac\nu2)\theta m<n$.
There exists $\rho'=\rho'(\kappa,\ha,\delta,\nu,\rho,\theta,K)>0$ such that
\begin{equation}
\pro{ \event(\theta,\rho)^c \wedge \event\Big( \Big(1+\frac{\nu}{10}\Big)\theta,\rho'\Big)}
=O_{\kappa,\ha,\delta,\nu,\rho,\theta,K}(e^{-n}).
\end{equation}
\end{lemma}
\begin{proof}
We may assume $n$ is sufficiently large depending on $\kappa,\ha,\delta,\nu,\rho,\theta,K$.
Write $\theta'=\big(1+\frac{\nu}{10}\big)\theta$ and take $\beta=\frac{\nu}{10}$.
Let $\rho'>0$ to be taken sufficiently small depending on $\kappa,\ha,\delta,\nu,\rho,\theta,K$, and let $\rho''$ be as in \eqref{def:rhodub}.
Intersecting the right hand side of \eqref{cont:entropy} with $\event(\theta,\rho)^c$, we have
\begin{align}
&\event(\theta,\rho)^c \wedge \event(\theta',\rho') \subset \notag\\
&\quad
\bigvee_{J\in {[m]\choose \theta' m }} \bigvee_{I\in {[n]\choose (1-\beta)^2\lf \theta m\rf}} \good_{I,J}(\rho'')
\wedge\event(\theta,\rho)^c\wedge\Big\{ \exists u \in \Sigma_{I,J}(\rho''): \|\M u\|\le 4\rho''K\sqrt{n} \Big\}\notag\\
&
\subset\bigvee_{J\in {[m]\choose \theta' m}} \bigvee_{I\in {[n]\choose (1-\beta)^2\lf \theta m\rf}} \good_{I,J}(\rho'')
\wedge\Big\{ \exists u \in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta,\rho): \|\M u\|\le 4\rho''K\sqrt{n} \Big\}\label{unionIJ}
\end{align}
where the second line follows by taking $\rho'$ small enough that $4\rho''<\rho$.
Fix $J\subset [m]$ and $I\subset[n]$ of sizes $\lf \theta'm\rf,\lf(1-\beta)^2\lf \theta m\rf\rf$, respectively, and condition on $\mF_{I,[n]}$ (recall the notation \eqref{def:sigmaalgebra}) to fix $\Sigma_{I,J}(\rho'')$.
Consider an arbitrary element $u\in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta,\rho)$.
By Lemma \ref{lem:spread}, there is a set $L\subset[m]$ with $|L|\ge (1-\frac{\nu}{C_0^2})\theta m$
and
\begin{equation} \label{flat:broad}
\frac{\rho}{\sqrt{m}}\le |u_j|\le \frac{C_0}{\sqrt{\nu\theta m}}
\end{equation}
for all $j\in L$, where $C_0>0$ is an absolute constant to be taken sufficiently large.
For any $i\in \mN^{(\delta)}(L)$, we have
\begin{equation}
\|(u_L)^i\|^2 \ge \sum_{i\in L:\sig_{ij}\ge \ha} |u_j|^2\sig_{ij}^2\ge \frac{\rho^2}{m}\ha^2\delta |L| \ge \frac12\rho^2\ha^2\delta\theta=:\alpha^2
\end{equation}
where in the last inequality we took $C_0$ sufficiently large.
Hence,
\begin{equation}
|I_\alpha(u_L)|\ge |\mN^{(\delta)}(L)| \ge \min\big(n,(1+\nu)(1-\nu/C_0^2)\theta m\big) \ge \Big(1+\frac{\nu}2\Big)\theta m
\end{equation}
taking $C_0$ larger if necessary, where in the second inequality we used our assumption $\theta\ge \theta_1$, and in the third inequality we used our assumption $(1+\frac\nu2)\theta m<n$.
Fix $I_0\subset I_\alpha(u_L)\setminus I$ of size $n_0:=\lf (1+\frac\nu2)\theta m\rf -|I|$.
In particular,
\begin{align}
\frac{\nu}{2}\theta m\le n_0 &\le \Big(1+\frac\nu2\Big)\theta m -(1-2\beta)\theta m \le \nu\theta m \label{broad:i0}
\end{align}
and
\begin{align}
n_0+2|I|-2|J| &\ge \Big(1+\frac\nu2\Big)\theta m + (1-2\beta)\theta m- 2\Big(1+\frac{\nu}{10}\Big)\theta m -O(1) \notag\\
&=\frac1{10}\nu\theta m-O(1). \label{i0i2j2}
\end{align}
by our choice of $\beta$.
By Lemma \ref{lem:fixed_improved},
\begin{equation} \label{broad:fixedbd}
\pr_{I_0}\big( \|\M u\| \le 4\rho''K\sqrt{n}\big) \le O_\kappa\bigg(\frac1\alpha\bigg( \frac{\rho''K\sqrt{n}}{\sqrt{|I_0|}} + \frac{1}{\sqrt{\nu\theta m}}\bigg)\bigg)^{n_0}\le O_\kappa\bigg(\frac{\rho''K}{\alpha\theta^{1/2}}\bigg)^{n_0}
\end{equation}
where in the second inequality we applied the assumption $m\ge n/2$ and assumed that $n$ is sufficiently large that $\rho''\gg 1/K\sqrt{n}$ (it follows from \eqref{def:rhodub} and our assumption that $\rho'$ is independent of $n$ that $\rho''$ is bounded below independent of $n$).
Suppose that $\good_{I,J}(\rho'')$ holds.
Since the bound \eqref{broad:fixedbd} is uniform in the choice of $I_0$, we can undo the conditioning and apply the union bound over elements of $\Sigma_{I,J}(\rho'')\setminus \Comp(\theta,\rho)$ to find
\begin{align*}
&\pr\Big( \exists u\in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta,\rho): \|\M u\|\le 4\rho''K\sqrt{n}\Big) \\
&\qquad\qquad\qquad\qquad\qquad\le O\bigg(\frac{1}{\rho''}\bigg)^{2(|J|-|I|)} O_\kappa\bigg(\frac{\rho''K}{\alpha\theta^{1/2}}\bigg)^{n_0}\\
&\qquad\qquad\qquad\qquad\qquad=O_\kappa\bigg(\frac{K}{\alpha\theta^{1/2}}\bigg)^{n_0}O(\rho'')^{n_0+2|I|-2|J|}\\
&\qquad\qquad\qquad\qquad\qquad=O_\kappa\bigg(\frac{K}{\alpha\theta^{1/2}}\bigg)^{\nu \theta m}O(\rho'')^{\frac1{10}\nu\theta m - O(1)}
\end{align*}
where in the last line we applied the bounds \eqref{broad:i0} and \eqref{i0i2j2}.
Since this is uniform in $I,J$, we can undo the conditioning on $\mF_{I,[n]}$ and apply \eqref{unionIJ} with another union bound over the choices of $I,J$ to obtain
\begin{equation} \label{incr:pen}
\pro{\event(\theta,\rho)^c\wedge\event(\theta',\rho')}
\le 2^{m+n} O_\kappa\bigg(\frac{K}{\alpha\theta^{1/2}}\bigg)^{\nu \theta m}O\bigg(\frac{\rho'}{\nu\rho\theta^{1/2}}\bigg)^{\frac1{10}\nu\theta m - O(1)}
\end{equation}
where we have substituted the definition of $\rho''$.
The result now follows by taking $\rho'$ sufficiently small.
\end{proof}
Now we conclude the proof of Proposition \ref{prop:slight}.
From our assumptions it follows that for all $j\in[m]$ we have $\sum_{i=1}^n \sig_{ij}^2 \ge \delta\ha^2n$.
Together with our assumption $m\le 2n$, this means we can apply Lemma \ref{lem:high} to find that
\begin{equation} \label{fromhigh}
\pr(\event(\theta_0,\rho_0)) \le e^{-c_\kappa\delta\ha^2n}
\end{equation}
where $\theta_0= c_\kappa\delta\ha^2/\log(K/\delta\ha^2)$ and $\rho_0=c_\kappa\delta\ha^2/K$.
We may assume without loss of generality that $\nu\le \delta/2$.
For $l\ge1$ set $\theta_l=(1+\frac{\nu}{10})^l\theta_0$, and let $k$ be the smallest $l$ such that $\theta_l\ge \theta$.
We have
\[
\Big(1+\frac{\nu}{2}\Big)\theta_{k-1}m \le \Big(1+\frac{\nu}{2}\Big)\theta m \le \Big(1-\frac{\delta^2}{16}\Big)\min(m,n)<n.
\]
In particular, $(1+\nu/10)^k\theta_0\le (1+\nu/10)\theta\le 1$, so
\begin{equation}
k\le \frac{\log\frac1{\theta_0}}{\log\big(1+\frac{\nu}{10}\big)} \ll_{\kappa,\ha,\delta,\nu,K}1.
\end{equation}
Applying Lemma \ref{lem:increment_broad} inductively, we have that for every $1\le l\le k$ there is $\rho_l>0$ depending only on $\kappa,\ha,\delta,\nu$ and $K$ such that
\begin{equation}
\pro{ \event(\theta_l,\rho_l)\setminus \event(\theta_{l-1},\rho_{l-1})} = O_{\kappa,\ha,\delta,\nu,K}(e^{-n}).
\end{equation}
Together with \eqref{fromhigh} and the union bound,
\begin{align*}
\pro{ \event(\theta,\rho)} &\le \pro{\event(\theta_0,\rho_0)} + \sum_{l=1}^k \pro{\event(\theta_l,\rho_l)\setminus \event(\theta_{l-1},\rho_{l-1})}\\
&\le e^{-c_\kappa\delta\ha^2n} + O_{\kappa,\ha,\delta,\nu,K}(e^{-n}) = O_{\kappa,\ha,\delta,\nu,K}(e^{-c_\kappa\delta\ha^2n}).
\end{align*}
\subsection{General profile: Proof of Proposition \ref{prop:comp}}
For technical reasons (essentially due to the fact that we want to allow the operator norm to have arbitrary polynomial size) the anti-concentration argument from the previous section will not suffice here, and we will need the following substitute.
Roughly speaking, while previously we argued by isolating a large set of coordinates on which the vector $u$ is ``flat" (see \eqref{flat:broad}), here we will need to locate a set on which $u$ is \emph{very flat}, only fluctuating by a constant factor.
This is done by a simple dyadic decomposition of the range of $u$, which is responsible for the loss of a logarithmic factor in the probability bound.
A similar argument will be used in Section \ref{sec:super}.
\begin{lemma}[Anti-concentration for the image of an incompressible vector] \label{lem:anti_incomp}
Let $M$ be as in Proposition \ref{prop:comp}.
Let $v\in \Incomp(\theta,\rho)$ for some $\theta,\rho\in (0,1)$, and fix $I_0\subset [n]$ with $|I_0|\le \frac14\sig_0^2n$.
Then for all $t\ge a_0\rho/\sqrt{m}$,
\begin{equation}
\sup_{w\in \C^n} \pr_{[n]\setminus I_0} \Big( \|Mv-w\| \le t \sqrt{n} \Big) = O_\kappa\left(\frac{t\log^{1/2}(\frac{\sqrt{m}}{\rho})}{\sig_0^2\rho \theta^{1/2}}\right)^{\frac14\sig_0^2n}.
\end{equation}
\end{lemma}
\begin{remark}
Proceeding as in the proof of Lemma \ref{lem:increment_broad} would yield
\begin{equation}
\sup_{w\in \C^n} \pr_{[n]\setminus I_0} \Big( \|Mv-w\| \le t \sqrt{n} \Big) = O_\kappa\left(\frac{t}{\sig_0^2\rho \theta^{1/2}}\right)^{\frac14\sig_0^2n} \quad\quad \text{for all }\; t\ge \frac{a_0}{\sqrt{\theta m}}.
\end{equation}
The ability to take $t$ down to the scale $\sim \rho/\sqrt{m}$ will be crucial in the proof of Lemma \ref{lem:increment_gen} below.
\end{remark}
\begin{proof}
We begin by finding a set of indices on which $v$ varies by at most a factor of 2.
For $k\ge 0$ let $L_k = \{j\in [m]: 2^{-(k+1)}<|v_j|\le 2^{-k}\}$.
Since $v\in \Incomp(\theta,\rho)$, we have
\[
|L^+| := |\{j\in [m]: |v_j|\ge \rho/\sqrt{m}\}| \ge \theta m.
\]
Indeed, were this not the case then $v$ would be within distance $\rho$ of the vector $v_{L^+}$ whose support is smaller than $\theta m$, implying $v\in \Comp(\theta,\rho)$.
Thus,
$
L^+\subset \bigcup_{k=0}^\ell L_k
$
for some $\ell \ll \log(\frac{\sqrt{m}}{\rho})$.
By the pigeonhole principle there exists $k^*\le \ell$ such that $L^*:= L_{k^*}$ satisfies
\begin{equation} \label{LB:Lstar}
|L^*|\ge \frac{\theta n}{\ell} \gg \frac{\theta m}{ \log(\frac{\sqrt{m}}{\rho})}.
\end{equation}
Denote $I^*:= I_{\frac{a_0}{2}\|v_{L^*}\|}(v_{L^*})$.
By Lemma \ref{lem:goodrows0},
\begin{equation} \label{LB:istar}
|I^*|\ge \frac12\sig_0^2 n.
\end{equation}
Fix $i\in I^*$.
By definition of $I^*$,
\begin{equation} \label{vi:2}
\| (v^i)_{L^*}\| \ge \frac{1}{2}a_0 \|v_{L^*}\|
\end{equation}
and since $|v_j|\gg \rho/\sqrt{m}$ on $L^*$,
\begin{equation} \label{vstar:2}
\|v_{L^*}\| \gg \frac{\rho}{\sqrt{m}}|L^*|^{1/2}.
\end{equation}
Furthermore, since $a_{ij}\le 1$ for all $j\in [m]$ and $v$ varies by a factor at most 2 on $L^*$,
\begin{equation} \label{vi:infty}
\|(v^i)_{L^*}\|_\infty \le \|v_{L^*}\|_{\infty} \le 2\frac{\|v_{L^*}\|}{|L^*|^{1/2}}.
\end{equation}
Fix $w\in \C^n$ arbitrarily, and recall that $R_i$ denotes the $i$th row of $M$.
By Lemma \ref{lem:anti_improved} and the above estimates,
for all $t\ge 0$ we have
\begin{align*}
\pr(|R_i\cdot v-w_i|\le t)
&\ll_\kappa \frac{t+ \|(v^i)_{L^*}\|_\infty }{\|(v^i)_{L^*}\|}\\
&\ll \frac{1}{a_0} \left( \frac{t}{\|v_{L^*}\|} + \frac{\|(v^i)_{L^*}\|_\infty}{\|v_{L^*}\|}\right)\\
&\ll \frac1{a_0} \left( \frac{t}{\rho} \left(\frac{m}{|L^*|}\right)^{1/2} + \frac1{|L^*|^{1/2}}\right)\\
&= \frac1{a_0} \left( \frac{m}{|L^*|}\right)^{1/2} \left( \frac{t}{\rho} + \frac1{\sqrt{m}}\right).
\end{align*}
By Lemma \ref{lem:tensorize},
\begin{align*}
\pr_{I^*\setminus I_0} \Big( \|Mv-w\| \le t |I^*\setminus I_0|^{1/2}\Big)
&\le \pr_{I^*\setminus I_0} \Big( \sum_{i\in I^*\setminus I_0}|R_i\cdot v - w_i|^2 \le t^2 |I^*\setminus I_0|\Big) \\
&= O_\kappa\left(\frac{t\sqrt{m}}{a_0\rho|L^*|^{1/2}}\right)^{|I^*\setminus I_0|}
\end{align*}
for all $t\ge \rho/\sqrt{m}$.
Substituting the lower bounds \eqref{LB:Lstar}, \eqref{LB:istar} on $|L^*|$ and $|I^*|$ and our assumption $|I_0| \le \frac14\sig_0^2n$,
\[
\pr_{I^*\setminus I_0} \bigg( \|Mv-w\| \le \frac12ta_0\sqrt{n} \bigg) = O_\kappa\left(\frac{t\log^{1/2}(\frac{\sqrt{m}}{\rho})}{a_0\rho \theta^{1/2}}\right)^{\frac14\sig_0^2n}
\]
for all $t\ge \rho/\sqrt{m}$.
The result now follows by replacing $t$ with $2t/a_0$ as undoing the conditioning on the remaining rows in $[n]\setminus I_0$.
\end{proof}
Now we are ready to prove the analogue of Lemma \ref{lem:increment_broad} for general profiles.
Whereas in the broadly connected case we obtained control on vectors in $\Comp((1+\beta)\theta,\rho')$ after restricting to the event that we have control on $\Comp(\theta,\rho)$, for small $\beta>0$, here we will also need to assume control on $\Comp(\theta_0,\rho_0)$ for a fixed small $\theta_0$ at each step.
The control on $\Comp(\theta,\rho)$ will be used to obtain a net of low cardinality using Lemma \ref{lem:entropy}, while the control on $\Comp(\theta_0,\rho_0)$ will be used to obtain good anti-concentration estimates using Lemma \ref{lem:anti_incomp}.
(In the broadly connected case the control on $\Comp(\theta,\rho)$ was sufficient for both purposes.)
\begin{lemma}[Incrementing compressibility: general profile] \label{lem:increment_gen}
Let $\M$ be as in Proposition \ref{prop:comp}, fix $\gamma>1/2$ and put $K=n^{\gamma-1/2}$. Let $\theta_0,\rho_0$ be as in Lemma \ref{lem:high}, and fix $\theta\in [\theta_0,c_0\sig_0^2]$, where $c_0$ is a sufficiently small constant (we may assume the constant $c$ in Lemma \ref{lem:high} is sufficiently small so that this interval is non-empty).
We have
\begin{equation} \label{bound:increment_gen}
\pro{ \event(\theta_0,\rho_0)^c\wedge\event(\theta,\rho)^c \wedge \event( \theta+\beta a_0^2,\rho'\Big)}
=O_{\gamma,\sig_0,\kappa}(e^{-n})
\end{equation}
for some $\rho' \gg_{\gamma,\sig_0,\kappa} n^{-O(\gamma)}\rho$,
where we set
\begin{equation} \label{set:beta}
\beta= c_1\min\left(1,\frac1{\gamma-1/2}\right)
\end{equation}
for a sufficiently small constant $c_1>0$.
\end{lemma}
\begin{proof}
Let $\rho'>0$ to be taken sufficiently small, and let $\rho''$ be as in \eqref{def:rhodub}.
We denote $\theta'=\theta +\beta a_0^2$.
Intersecting both sides of \eqref{cont:entropy} with $\event(\theta_0,\rho_0)^c$, we have
\begin{align}
&\event(\theta_0,\rho_0)^c\wedge\event(\theta,\rho)^c \wedge \event(\theta',\rho') \subset \notag\\
&
\bigvee_{J\in {[m]\choose \theta' m }} \bigvee_{I\in {[n]\choose (1-\beta)^2\lf \theta m\rf}} \good_{I,J}(\rho'')
\wedge\Big\{ \exists u \in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta_0,\rho_0): \|\M u\|\le 4\rho''K\sqrt{n} \Big\} \label{unionIJ:gen}
\end{align}
where we have assumed $\rho'$ is small enough that $4\rho''<\rho_0$.
Fix $J\subset[m]$ and $I\subset[n]$ of size $\lf \theta'm\rf$, $\lf (1-\beta)^2\lf \theta m\rf\rf$, respectively, and condition on $\mF_{I,[n]}$ to fix $\Sigma_{I,J}(\rho'')$.
Fix an arbitrary $u\in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta_0,\rho_0)$.
From Lemma \ref{lem:anti_incomp} we have
\begin{equation}
\pr_{[n]\setminus I} \Big( \|Mu\| \le 4\rho'' K\sqrt{n} \Big) = O_\kappa\left(\frac{\rho'' K\log^{1/2}(\frac{\sqrt{n}}{\rho_0})}{\sig_0^2\rho_0 \theta_0^{1/2}}\right)^{\frac14\sig_0^2n}
\end{equation}
provided
\begin{equation} \label{assume:rhodub}
\rho'' \ge \frac{c a_0\rho_0}{K\sqrt{n}}
\end{equation}
for some small constant $c>0$ (note that we used our assumption $n/2\le m\le 2n$).
\begin{comment}
By Lemma \ref{lem:spread} there is a set $L\subset[m]$ with $|L| \ge \frac12\theta_0m$ and
\begin{equation}
\frac{\rho_0}{\sqrt{m}}\le |u_j|\le \frac{2}{\sqrt{\theta m}} \quad \forall j\in L.
\end{equation}
Let $u_L$ denote the restriction of $u$ to $L$, and $\hat{u}= u_L/\|u_L\|$.
By Lemma \ref{lem:goodrows0}, $|I_{\sig_0/2}(\hat{u})|\ge \frac12\sig_0^2n$.
Since $\|u_L\|\ge |L|\rho_0^2/m\ge \frac12\theta_0\rho_0^2$ we have $I_\alpha(u_L)\supset I_{a_0/2}(\hat{u})$ with $\alpha^2 :=\frac18\sig_0^2\theta_0\rho_0^2$, so
\begin{equation}
|I_\alpha(u_L)| \ge \frac12\sig_0^2n.
\end{equation}
By our assumptions we have $|I|\le \theta m\le \frac14 \sig_0^2 n$, so we may choose a set $I_0\subset I_\alpha(u_L)\setminus I$ with
\begin{equation} \label{comp:i0lb}
|I_0|\ge \frac14\sig_0^2n.
\end{equation}
Applying Lemma \ref{lem:fixed_improved} (conditioning on the columns of $M$ outside $L$) we have
\begin{align*}
\pr_{I_0}\big( \|Mu\|\le 4\rho'' K\sqrt{n}\big)
&\le O\bigg( \frac1\alpha \bigg( \frac{\rho''K\sqrt{n}}{\sqrt{|I_0|}} + \frac1{\sqrt{\theta_0 m}} \bigg)\bigg)^{|I_0|}\\
&= O\bigg(\frac{\rho'' K}{\sig_0^2\rho_0\sqrt{\theta_0}}\bigg)^{\frac14\sig_0^2n}
\end{align*}
where in the second line we have applied \eqref{comp:i0lb} and the assumption $m\ge n/2$, and we have assumed
\begin{equation} \label{assume:rhodub}
\rho'' \ge \frac{a_0}{K\sqrt{\theta_0m}}.
\end{equation}
\end{comment}
Applying the union bound over the choices of $u\in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta_0,\rho_0)$, on the event $\good_{I,J}(\rho'')$ we have
\begin{align*}
&\pr\Big( \exists u\in \Sigma_{I,J}(\rho'')\setminus \Comp(\theta_0,\rho_0): \, \|Mu\|\le 4\rho'' K\sqrt{n} \Big) \\
&\qquad\qquad\qquad \le O\left(\frac1{\rho''}\right)^{2(|J|-|I|)} O_\kappa\left(\frac{\rho'' K\log^{1/2}(\frac{\sqrt{n}}{\rho_0})}{\sig_0^2\rho_0 \theta_0^{1/2}}\right)^{\frac14\sig_0^2n}\\
&\qquad\qquad\qquad= O\left(\frac1{\rho''}\right)^{2(|J|-|I|)} O_{\kappa,a_0}\left(\rho''K^2 \log(K\sqrt{n}) \right)^{\frac14\sig_0^2n}
\end{align*}
where in the second line we substituted the expressions for $\rho_0,\theta_0$ from Lemma \ref{lem:high}.
Denoting $\eps = \rho'' K^2$, the above bound rearranges to
\begin{equation} \label{comp:rearrange}
O_{\kappa,a_0}(\log n)^n n^{O(\gamma)} n^{O(\gamma-1/2)(|J|-|I|)} \eps^{\frac14\sig_0^2 n - 2(|J| - |I|)}.
\end{equation}
We can bound
\begin{align*}
|J|-|I|
&= \theta m + \beta \sig_0^2m - (1-\beta)^2 \theta m + O(1) \le \beta \sig_0^2m +2\beta \theta m + O(1)\\
& = O(\beta \sig_0^2 m) + O(1)
\end{align*}
where we used our assumption that $\theta\le c_0\sig_0^2$.
In particular, $|J|-|I| \le \frac18 \sig_0^2 n+O(1)$ if the constant $c_1$ in \eqref{set:beta} is sufficiently small, and \eqref{comp:rearrange} is bounded by
\begin{equation}
O_{\kappa,a_0}(\log n)^n n^{O(\gamma)}n^{O(\gamma-1/2) \beta a_0^2 m} \eps^{\frac18\sig_0^2n - O(1) }.
\end{equation}
Applying the union bound over the choices of $I,J$ in \eqref{unionIJ:gen}, which incurs a harmless factor of $2^{m+n} = O(1)^n$, and substituting the expression \eqref{set:beta} for $\beta$ we have
\begin{equation}
\pro{ \event(\theta_0,\rho_0)^c\wedge\event(\theta,\rho)^c \wedge \event(\theta+\beta a_0^2,\rho'\Big)} = O_{\kappa,a_0}(\log n)^n n^{O(\gamma)}\eps^{-O(1)}(n^{O(c_1)} \eps^{1/8})^{a_0^2n}.
\end{equation}
It only remains to check that we can take $\eps$ sufficiently small to obtain \eqref{bound:increment_gen}.
From \eqref{assume:rhodub} we are constrained to take
\[
\eps = \rho'' K^2 \ge \frac{c \sig_0\rho_0K}{\sqrt{n}} = \frac{c' a_0^3 }{\sqrt{n}}
\]
for some constant $c'\in (0,1)$ sufficiently small.
Taking $\eps = a_0^3/\sqrt{n}$ and $c_1$ sufficiently small we have
\begin{equation}
\pro{ \event(\theta_0,\rho_0)^c\wedge\event(\theta,\rho)^c \wedge \event(\theta+\beta a_0^2,\rho'\Big)} \le O_{\kappa,\sig_0}(1)^n n^{O(\gamma)} n^{-.01 \sig_0^2 n}
\end{equation}
which yields \eqref{bound:increment_gen} as desired.
With this choice of $\eps$,
\[
\rho' \gg \rho'' \beta \rho \theta\ge \rho'' \beta \rho \theta_0 \gg_{\kappa,\sig_0,\gamma} \rho n^{-2\gamma + 1/2-o(1)}
\]
as desired (recall that $\theta_0\gg_\kappa \sig_0^2/\log(K/a_0) \gg_{\gamma,\sig_0,\kappa} 1/\log n$).
\end{proof}
Now we conclude the proof of Proposition \ref{prop:comp}.
Since the event $\mB(K)$ is monotone under increasing $K$, by perturbing $\gamma$ and assuming $n$ is sufficiently large we may take $K=n^{\gamma-1/2}$ with $\gamma>1/2$.
Let $\rho_0,\theta_0$ be as in Lemma \ref{lem:high}, and for $l\ge 1$ we let $\theta_l = \theta_0+ l\beta a_0^2$ with $\beta=\beta(\gamma)$ as in \eqref{set:beta}.
By Lemma \ref{lem:increment_gen} we can inductively define a sequence $\rho_l$ such that for each $l\ge 1$ such that $\theta_l \le c_0\sig_0^2$,
\[
\rho_l \gg_{\gamma,\sig_0,\kappa} n^{-O(\gamma)}\rho_{l-1}
\]
and
\[
\pro{ \event(\theta_0,\rho_0)^c\wedge \event(\theta_{l-1},\rho_{l-1})^c\wedge \event(\theta_l,\rho_l)} = O_{\gamma,\sig_0,\kappa}(e^{-n}).
\]
Applying the union bound, for some $k=O(\gamma)$ we have
\begin{align*}
\pro{ \event(c_0\sig_0^2,\rho) }
&\le \pro{ \event(\theta_0,\rho_0)} + \sum_{l=1}^{k}
\pro{ \event(\theta_0,\rho_0)^c\wedge \event(\theta_{l-1},\rho_{l-1})^c\wedge \event(\theta_l,\rho_l)}\\
& \le e^{-c_\kappa\sig_0^2n} + O_{\gamma,\sig_0,\kappa} (e^{-n})\\
& = O_{\gamma,\sig_0,\kappa} (e^{-c_\kappa\sig_0^2n})
\end{align*}
and
$
\rho \gg_{\gamma,\sig_0,\kappa} n^{-O(\gamma^2)}.
$
This concludes the proof of Proposition \ref{prop:comp}.
\section{Invertibility from connectivity: Incompressible vectors} \label{sec:incomp}
In this section we conclude the proofs of Theorems \ref{thm:broad} and \ref{thm:super} by bounding the event that $\|Mu\|$ is small for some incompressible vector $u$ (recall the terminology from Section \ref{sec:net}).
We follow the (by now standard) approach of reducing to the event that a fixed row $R_i$ of $M$ lies close to the span of the remaining rows, an idea which goes back to the work of Koml\'os on the singularity probability for Bernoulli matrices \perp\!\!\!\perptep{Komlos67,Komlos68,Komlos77}.
This can in turn be controlled by the event that a random walk $R_i\cdot v$ concentrates near a particular point, where $v$ is a fixed unit vector in the orthocomplement of the remaining rows.
Independence of the rows allows us to condition on $v$, and our results from the previous section allow us to argue that $v$ is incompressible.
For the case that the entries of $R_i$ have variances uniformly bounded below, we could then complete the proof by applying the anti-concentration estimate of Lemma \ref{lem:anti_improved}.
In the present setting, however, a proportion $1-\delta$ of the entries of $R_i$ may have zero variance.
For the case of broadly connected profile we follow the argument of Rudelson and Zeitouni \perp\!\!\!\perptep{RuZe} and use Proposition \ref{prop:slight} to show $v$ has essential support of size $(1-\delta/2)n$, and hence has non-trivial overlap with the support of $R_i$.
For the case of a super-regular profile, Proposition \ref{prop:comp} only gives that $v$ has essential support of size $\gg \delta \ha^2$.
In Lemma \ref{lem:overlap} we make use of a double counting argument to show that if we choose the row $R_i$ at random, on average it will have good overlap with the corresponding normal vector $v^{(i)}$ (which also depends on $i$).
Here is where we make crucial use of the super-regularity hypothesis on $A$.
Lemma \ref{lem:overlap} is a natural extension of a double counting argument used by Koml\'os in his work on the singularity probability for Bernoulli matrices, and which was applied to bound the smallest singular value of iid matrices by Rudelson and Vershynin in \perp\!\!\!\perptep{RuVe:ilo}.
We were also inspired by
a similar refinement of the double counting argument from the recent paper \perp\!\!\!\perptep{LLTTY} on the singularity probability for adjacency matrices of random regular digraphs.
\subsection{Proof of Theorem \ref{thm:broad}} \label{sec:broad}
By Lemma \ref{lem:wlog.kappa} and multiplying $X$ and $B$ by a phase (which does not affect our hypotheses) we may assume that $\xi$ has $O(\kappa_0)$-controlled second moment.
Fix $K\ge 1$, and let $\rho=\rho(\kappa,\ha,\delta,\nu,K)$ be as in Proposition \ref{prop:slight}.
We may assume $n$ is sufficiently large depending on $\kappa,\ha,\delta,\nu,K$.
For the remainder of the proof we restrict to the event $\mB(K) = \{\|M\|\le K\sqrt{n}\}$.
For $j\in [n]$ let $M^{(i)}$ denote the $n-1\times n$ matrix obtained by removing the $i$th row from $M$.
Define the good event
\begin{equation}
\good = \Big\{ \forall i\in [n], \forall u\in \Comp(1-\delta/2,\rho), \; \|u^*M\|,\|M^{(i)}u\| >\rho K\sqrt{n}\Big\}.
\end{equation}
Applying Proposition \ref{prop:slight} to $M^*$ and $M^{(i)}$ for each $i\in[n]$ (using our restriction to $\mB(K)$) and the union bound we have
\begin{equation}
\pr(\good) = 1- O_{\kappa,\ha,\delta,\nu,K}(ne^{-c_\kappa\delta \ha^2 n}) = 1- O_{\ha,\delta,\nu,K}(e^{-c_\kappa\delta \ha^2 n})
\end{equation}
adjusting $c_\kappa$ slightly.
Let $t\le 1$, and define the event
\begin{equation}
\event(t) = \good\wedge \big\{ \exists u\in \Incomp(1/10,\rho): \|u^*M\|\le t/\sqrt{n}\big\}.
\end{equation}
For $n$ sufficiently large (larger than $1/\rho K$) it suffices to show
\begin{equation} \label{goal:broad1}
\pr(\event(t)) \ll_{\kappa,\ha,\delta,\nu,K} t + n^{-1/2}.
\end{equation}
Recalling that $R_i$ denotes the $i$th row of $M$, we denote
\begin{equation}
R_{-i} = \Span(R_j: j\in [n]\setminus \{i\})
\end{equation}
and let \begin{equation}
\event_i(t) = \good \wedge \{\dist(R_i,R_{-i}) \le t/\rho\}.
\end{equation}
We now use a double counting argument of Rudelson and Vershynin from \perp\!\!\!\perptep{RuVe:ilo} to control $\event(t)$ in terms of the events $\event_i(t)$.
Suppose that $\event(t)$ holds, and let $u\in \Incomp(1/10, \rho)$ such that $\|u^*M\|\le t/\sqrt{n}$.
Then we must have $|u_i| \ge \rho/\sqrt{n}$ for at least $n/10$ elements $i\in[n]$.
For each such $i$ we have
\[
\frac{t}{\sqrt{n}} \ge \|u^*M\| = \bigg\| \sum_{j=1}^n \overline{u_j} R_j\bigg\|
\ge \bigg\| P_{R_{-i}^\perp} \sum_{j=1}^n \overline{u_j} R_j\bigg\|
= |u_i| \left\| P_{R_{-i}^\perp} R_i\right\|
\ge \frac{\rho}{\sqrt{n}} \dist(R_i,R_{-i})
\]
where we denote by $P_W$ the orthogonal projection to a subspace $W$.
Thus, on $\event(t)$ we have that $\event_i(t)$ holds for at least $n/10$ values of $i\in[n]$, so by double counting,
\begin{equation}
\pro{ \event(t)} \le \frac{10}{n} \sum_{i=1}^n \pro{\event_i(t)}.
\end{equation}
Now it suffices to show that for arbitrary fixed $i\in [n]$,
\begin{equation} \label{goal:broad2}
\pr(\event_i(t)) \ll_{\kappa,\ha,\delta,\nu,K} t + n^{-1/2}.
\end{equation}
Fix $i\in [n]$ and condition on $\{R_j: j\in [n]\setminus \{i\}\}$.
Draw a unit vector $u\in R_{-i}^\perp$ independent of $R_i$, according to Haar measure (say).
Since $\dist(R_i,R_{-i}) \le |R_i\cdot u|$, it suffices to show
\begin{equation} \label{goal:broad3}
\pro{|R_i\cdot u|\le t/\rho } \ll_{\kappa,\ha,\delta,\nu,K} t + n^{-1/2}.
\end{equation}
Since $u\in \ker(M^{(i)})$, on $\good$ we have that $u\in \Incomp(1-\frac\delta2,\rho)$.
By Lemma \ref{lem:spread} there exists $L\subset [n]$ of size $|L| \ge (1-\frac34\delta)n$ such that
\[
\frac{\rho}{\sqrt{n}} \le |u_j| \le \frac{10}{\sqrt{\delta n}}
\]
for all $j\in L$.
By assumption we have $|\mN_{A(\ha)}(i)| = |\{ j\in [n]: \sig_{ij} \ge \ha\}| \ge \delta n$,
so letting $J=\mN_{A(\ha)}(i)\cap L$ we have $|J|\ge \delta n/4$.
Denoting $v= (u^i)_J = (\sig_{ij} u_j 1_{j\in J})_j$, we have
\[
\|v\|^2 = \sum_{j\in J} \sig_{ij}^2 |u_j|^2 \ge |J|\ha^2 \rho^2/n \ge \delta\ha^2\rho^2/4
\]
and
\[
\|v\|_\infty \le \|u_J\|_{\infty} \le \frac{10}{\sqrt{\delta n}}
\]
(recall that $a_{ij}\le 1$ for all $i,j\in [n]$).
Conditioning on $u$ and $\{\xi_{ij}\}_{j\notin J}$, we apply Lemma \ref{lem:anti_improved} to conclude
\begin{align*}
\pro{ |R_i\cdot u| \le t/\rho} \ll_\kappa \frac{1}{\|v\|}\left( \frac{t}{\rho} + \|v\|_\infty\right) \ll \frac{1}{\rho\ha\delta^{1/2}} \left(\frac{t}\rho + \frac{1}{\sqrt{\delta n}}\right)
\end{align*}
which gives \eqref{goal:broad3} as desired.
\subsection{Proof of Theorem \ref{thm:super}} \label{sec:super}
By Lemma \ref{lem:wlog.kappa} and multiplying $X$ and $B$ by a phase (which does not affect our hypotheses) we may assume that $\xi$ has $\kappa=O(\kappa_0)$-controlled second moment.
Fix $\gamma\ge1/2$ and let $K=O(n^{\gamma-1/2})$.
We will show that for all $\tau\ge 0$,
\begin{equation} \label{bound:super2}
\pro{ s_n(M) \le \frac{\tau}{\sqrt{n}}\,, \; \|M\|\le K\sqrt{n}} \ll_{\gamma,\ha,\delta,\kappa} n^{O(\gamma^2)}\tau + \sqrt{\frac{\log n}{n}}.
\end{equation}
For the remainder of the proof we restrict to the boundedness event
\begin{equation}
\mB(K) = \{\|M\|\le K\sqrt{n}\}.
\end{equation}
By the assumption that $A(\ha)$ is $(\delta,\eps)$-super-regular we have
\[
\sum_{i=1}^n \sig_{ij}^2 \ge \delta \ha^2n
\]
for all $j\in [n]$.
Let $\sig_0 = \delta^{1/2}\ha$, and let $\rho=\rho(\gamma,\sig_0,\kappa n)$ and $c_0$ be as in Proposition \ref{prop:comp}.
In particular,
\begin{equation} \label{incomp:rholb}
\rho \gg_{\gamma,\delta,\ha}n^{-O(\gamma^2)}.
\end{equation}
Denoting $\theta=c_0\delta\ha^2$, for $\tau>0$ we define the good event
\begin{equation}
\good(\tau) = \Big\{ \forall u\in \Comp(\theta,\rho), \, \|Mu\|,\|u^*M\| > \tau/\sqrt{n}\Big\}.
\end{equation}
Applying Proposition \ref{prop:comp} to $M$ and $M^*$, along with the union bound, we have
\begin{equation} \label{LB:goodtau}
\pro{\good(\tau)} = 1-O_{\gamma,\delta,\ha,\kappa}(e^{-c_\kappa\delta\ha^2n})
\end{equation}
as long as $\tau\le \rho K n$.
Let $0<\tau\le 1$ to be chosen later.
Recalling our notation $M^{(i)}$ from Section \ref{sec:broad}, we define the sets
\begin{equation}
S_i(\tau) = \left\{ u\in S^{n-1}: \|M^{(i)}u\|\le \frac{\tau}{\sqrt{n}}\right\}.
\end{equation}
Informally, for small $\tau$ this is the set of unit almost-normal vectors to the subspace $R_{-i}$ spanned by the rows of $M^{(i)}$.
In Lemma \ref{lem:overlap} below we reduce our task to bounding the probability that a row $R_i$ is nearly orthogonal to a vector $u^{(i)}\in S_i(\tau)$ that is independent of $R_i$, and also has many large coordinates in the support of $R_i$.
The reduction uses the super-regularity hypothesis together with a careful averaging argument.
It turns out that for this argument to work it is important to consider almost-normal vectors rather than normal vectors (as in the proof of Theorem \ref{thm:broad}).
Writing $\mN(i)= \mN_{A(\ha)}(i)$, we define the \emph{good overlap events}
\begin{equation}
\mO_i(\tau) = \big\{ \exists u\in S_i(\tau): |\mN(i) \cap L^+(u,\rho)|\ge \delta \theta n\big\}
\end{equation}
where
\begin{equation} \label{super:Lplus}
L^+(u)= \{j\in [n]: |u_j|\ge \rho/\sqrt{n}\}.
\end{equation}
On $\mO_i(\tau)$ we fix a vector $u^{(i)} = u^{(i)}(M^{(i)},\tau)\in S_i(\tau)$, chosen measurably with respect to $M^{(i)}$, satisfying $|\mN(i)\cap L^+(u,\rho)|\ge \delta \theta n$.
\begin{lemma}[Good overlap on average] \label{lem:overlap}
Recall the parameter $\eps$ from our super-regularity hypothesis (cf.\ Definition \ref{def:super}), and assume $\eps\le \theta/2$.
Then
\begin{equation}
\pro{\good(\tau)\wedge\Big\{ s_n(M)\le \frac{\tau}{\sqrt{n}}\Big\}} \le \frac{2}{\theta n} \sum_{i=1}^n \pro{ \mO_i(\tau)\wedge \bigg\{ |R_i\cdot u^{(i)}|\le \frac{2\tau}{\rho}\bigg\}}.
\end{equation}
\end{lemma}
\begin{proof}
Suppose $\good(\tau)\wedge\{s_n(M)\le \tau/\sqrt{n}\}$ holds.
Then there exist $u,v\in S^{n-1}$ such that $\|Mu\|, \|M^*v\|\le \tau/\sqrt{n}$.
By our restriction to $\good(\tau)$ we must have $u,v\in \Incomp(\theta,\rho)$.
With notation as in \eqref{super:Lplus} we have $|L^+(u)|, |L^+(v)|\ge \theta n$.
In particular, $|L^+(u)|\ge \eps n$, so
\begin{equation}
|\mN(i) \cap L^+(u)| \ge \delta |L^+(u)| \ge \delta \theta n
\end{equation}
for at least $(1-\eps)n$ elements $i\in [n]$.
Indeed, otherwise we would have
\[
e_{A(\ha)}(I,L^+(u)) = \sum_{i\in I} |\mN(i)\cap L^+(u)| <\delta |I||L^+(u)|
\]
for some $I\subset[n]$ with $|I|>\eps n$, which contradicts our assumption that $A(\ha)$ is $(\delta,\eps)$-super-regular.
Since
$
\|M^{(i)}u\|\le \|Mu\|\le \frac{\tau}{\sqrt{n}}
$
for all $i\in [n]$, we have that $u\in S_i(\tau)$ for all $i\in [n]$.
Thus,
\begin{equation} \label{super:goodvi}
\left| \big\{ i\in L^+(v): \mO_i(\tau) \mbox{ holds}\big\}\right| \ge \theta n -\eps n\ge \theta n/2.
\end{equation}
Fix $i\in L^+(v)$ such that $\mO_i(\tau)$ holds.
We have
\begin{align*}
\frac{\tau}{\sqrt{n}} \ge \|v^*M\|\ge |v^*Mu^{(i)}| \ge |v_i| |R_i\cdot u^{(i)}| - \bigg|\sum_{j\ne i} \overline{v_j} R_j\cdot u^{(i)}\bigg|.
\end{align*}
The first term on the right hand side is bounded below by $\frac{\rho}{\sqrt{n}}|R_i\cdot u^{(i)}|$ since $i\in L^+(v)$.
By Cauchy--Schwarz the second term is bounded above by $\|M^{(i)} u^{(i)}\|\le \tau/\sqrt{n}$, since $u^{(i)}\in S_i(\tau)$.
Rearranging we conclude
$
|R_i\cdot u^{(i)}|\le 2\tau/\rho
$
for all $i\in L^+(v)$ such that $\mO_i(\tau)$ holds.
Letting $\mE_i(t)=\{|R_i\cdot u^{(i)}|\le t\}$,
we have shown that on the event $\good(\tau)\wedge\{s_n(M)\le \tau/\sqrt{n}\}$, the event
$\mO_i(\tau)\wedge \mE_i(2\tau/\rho)$ holds for at least $\theta n/2$ values of $i\in [n]$ (from \eqref{super:goodvi}). It follows that
\[
\sum_{i=1}^n \un(\mO_i(\tau)\wedge \mE_i(2\tau/\rho)) \ge \frac{\theta n}{2} \un(\good(\tau)\wedge\{s_n(M)\le \tau/\sqrt{n}\}).
\]
Taking expectations on each side and rearranging yields the claim.
\end{proof}
Fix $i\in [n]$ arbitrarily, and suppose that $\mO_i(\tau)$ holds.
We condition on the rows $\{R_{j}\}_{j\in [n]\setminus \{i\}}$ to fix $u^{(i)}$.
We begin by finding a large set on which $u^{(i)}$ is flat, following a similar dyadic pigeonholing argument as in the proof of Lemma \ref{lem:anti_incomp}.
Letting $L_k = \{j\in [n]: 2^{-(k+1)} < |u^{(i)}_j| \le 2^{-k}$, since
\[
\delta \theta n \le |\mN(i) \cap L^+(u^{(i)})| \le \bigg| \bigcup_{k=0}^{\ell} \mN(i)\cap L_k \bigg|
\]
for some $\ell \ll \log(\sqrt{n}/\rho)$, by the pigeonhole principle there exists $k^*\le \ell$ such that $J:= \mN(i)\cap L_{k^*}$ satisfies
\begin{equation} \label{incomp:Jbound}
|J| \ge \delta \theta n/\ell \gg \frac{\delta \theta n}{\log(\sqrt{n}/\rho)}.
\end{equation}
Let us denote $v= (\sig_{ij} u^{(i)}_j 1_{j\in J})_j$.
Since $\sig_{ij} \ge \ha$ for $j\in \mN(i)$ and $|u_j^{(i)}|\gg \rho/\sqrt{n}$ for $j\in L_{k^*}$,
\begin{equation} \label{v:2}
\|v\| \ge \ha \|(u^{(i)})_J\| \gg \ha \rho (|J|/n)^{1/2}
\end{equation}
and since $u^{(i)}$ varies by at most a factor of $2$ on $J$,
\begin{equation} \label{v:infty}
\|v\|_\infty \le \|u^{(i)}1_{J}\|_\infty \le 2\|u^{(i)}\|/|J|^{1/2}.
\end{equation}
By further conditioning on the variables $\{\xi_{ij}\}_{j\notin J}$ and applying Lemma \ref{lem:anti_improved} along with the estimates \eqref{v:2}, \eqref{v:infty} we have
\begin{align*}
\pro{ |R_i\cdot u^{(i)}|\le 2\tau/\rho}
&\ll_\kappa \frac{\tau/\rho + \|v\|_{\infty}}{\|v\|}\\
&\ll \frac{1}{\ha}\left(\frac{\tau/\rho}{\rho(|J|/n)^{1/2}} + \frac{1}{|J|^{1/2}}\right)\\
&= \frac{1}{\ha} \left(\frac{n}{|J|}\right)^{1/2} \left(\frac{\tau}{\rho^2} + \frac1{\sqrt{n}}\right).
\end{align*}
Inserting the bound \eqref{incomp:Jbound} and undoing all of the conditioning, we have shown
\[
\pro{ \mO_i(\tau)\wedge \bigg\{ |R_i\cdot u^{(i)}|\le \frac{2\tau}{\rho}\bigg\}}
\ll_\kappa \frac{1}{\ha\sqrt{\delta \theta}} \left(\frac{\tau}{\rho^2} + \frac1{\sqrt{n}}\right) \log^{1/2}(\sqrt{n}/\rho).
\]
Since the right hand side is uniform in $i$, applying Lemma \ref{lem:overlap} (taking $c_1=c_0/2$) and substituting the expression for $\theta$ we have
\begin{equation}
\pro{\good(\tau)\wedge\Big\{ s_n(M)\le \frac{\tau}{\sqrt{n}}\Big\}}
\ll_\kappa \frac{1}{\ha^4\delta^2} \left(\frac{\tau}{\rho^2} + \frac1{\sqrt{n}}\right) \log^{1/2}(\sqrt{n}/\rho)
\end{equation}
for all $\tau\ge0$ (note that this bound is only nontrivial when $\tau\le \rho^2$, in which case our constraint $\tau\le \rho Kn$ from \eqref{LB:goodtau} holds).
The bound \eqref{bound:super2} now follows by substituting the lower bound \eqref{incomp:rholb} on $\rho$ and the bound \eqref{LB:goodtau} on $\good(\tau)^c$ (which is dominated by the $O(n^{-1/2}\log^{1/2}n)$ term).
This concludes the proof of Theorem \ref{thm:super}.
\section{Invertibility under diagonal perturbation: Proof of main theorem} \label{sec:diag}
In this final section we prove Theorem \ref{thm:main}. See Section \ref{sec:ideas} for a high level discussion of the main ideas.
In Sections \ref{sec:tools} and \ref{sec:op} we collect the main tools of the proof: the regularity lemma, the Schur complement bound, and bounds on the operator norm of random matrices.
In Section \ref{sec:decomp} we apply the regularity lemma to decompose the standard deviation profile $A$ into a bounded number of submatrices enjoying various properties.
In Section \ref{sec:highlevel} we apply the decomposition to prove Theorem \ref{thm:main}, on two technical lemmas, and in the final sections we prove these lemmas.
\subsection{Preliminary Tools} \label{sec:tools}
We begin by stating a version of the regularity lemma suitable for our purposes.
Recall that in Theorem \ref{thm:broad} we associated the standard deviation profile $A$ with a bipartite graph.
Here it will be more convenient to associate $A$ with a directed graph.
That is, to a non-negative square matrix $A=(\sig_{ij})_{1\le i,j\le n}$ we associate a directed graph $\Gamma_A$ on vertex set $[n]$ having an edge $i\rightarrow j$ when $\sig_{ij}>0$ (note that we allow $\Gamma_A$ to have self-loops, though the diagonal of $A$ will have a negligible effect on our arguments).
The notation \eqref{def:nbhd}--\eqref{def:edges} extends to this setting.
Additionally, we denote the \emph{density} of the pair $(I,J)$
\[
\rho_A(I,J):= \frac{e_A(I,J)}{|I||J|}.
\]
\begin{definition}[Regular pair] \label{def:regular.pair}
Let $A$ be an $n\times n$ matrix with non-negative entries.
For $\eps>0$, we say that a pair of vertex subsets $I,J\subset [n]$ is \emph{$\eps$-regular for $A$} if for every $I'\subset I, J'\subset J$ satisfying
\[
|I'|> \eps |I|, \quad |J'|> \eps|J|
\]
we have
\[
|\rho_A(I',J')-\rho_A(I,J)|<\eps.
\]
\end{definition}
The following is a version of the regularity lemma for directed graphs which follows quickly from a stronger result of Alon and Shapira \perp\!\!\!\perpte[Lemma 3.1]{AlSh:testing}.
Note that \perp\!\!\!\perpte[Lemma 3.1]{AlSh:testing} is stated for directed graphs without loops, which in the present setting means that it only applies to matrices $A$ with diagonal entries equal to zero.
However, Lemma \ref{lem:regularity} follows from applying \perp\!\!\!\perpte[Lemma 3.1]{AlSh:testing} to the matrix $A'$ formed be setting the diagonal entries of $A$ to zero, and noting that the diagonal has a negligible impact on the edge densities $\rho_A(I,J)$ when $|I|,|J|\gg n$.
\begin{lemma}[Regularity Lemma] \label{lem:regularity}
Let $\eps>0$.
There exists $m_0\in \N$ with $\eps^{-1}\le m_0\ll_{\eps}1$ such that for all $n$ sufficiently large depending on $\eps$, for every $n\times n$ non-negative matrix $A$ there is a partition of $[n]$ into $m_0+1$ sets $I_0,I_1,\dots, I_{m_0}$ with the following properties:
\begin{enumerate}[(1)]
\item $|I_0|<\eps n$;
\item $|I_1|=|I_2|=\cdots =|I_{m_0}|$;
\item all but at most $\eps m_0^2$ of the pairs $(I_k,I_l)$ are $\eps$-regular for $A$.
\end{enumerate}
\end{lemma}
\begin{remark}
The dependence on $\eps$ of the bound $m_0\le O_{\eps}(1)$ is very bad: a tower of exponentials of height $O(\eps^{-C})$.
Indeed, as in Szemer\'edi's proof for the setting of bipartite graphs \perp\!\!\!\perptep{Szemeredi:lemma}, the proof in \perp\!\!\!\perptep{AlSh:testing} gives such a bound with $C=5$.
It was shown by Gowers that for undirected graphs one cannot do better than $C=1/16$ in general \perp\!\!\!\perptep{Gowers:towers}.
As remarked in \perp\!\!\!\perptep{AlSh:testing}, his argument carries over to give a similar result for directed graphs.
\end{remark}
We will apply this in Section \ref{sec:decomp} to partition the standard deviation profile into a bounded number of manageable submatrices.
The following elementary fact from linear algebra will be used to lift the invertibility properties obtained for these submatrices back to the whole matrix.
\begin{lemma}[Schur complement bound] \label{lem:schur}
Let $M\in \mM_{N+n}(\C)$, which we write in block form as
$$M=\begin{pmatrix} A & B\\ C&D\end{pmatrix}$$
for $A\in \mM_{N}(\C), B\in \mM_{N,n}(\C), C\in \mM_{n,N}(\C), D\in \mM_n(\C)$.
Assume that $D$ is invertible.
Then
\begin{equation} \label{bound:schur}
s_{N+n}(M) \ge \bigg(1+ \frac{\|B\|}{s_n(D)}\bigg)^{-1} \bigg(1+ \frac{\|C\|}{s_n(D)}\bigg)^{-1} \min\Big( s_n(D), \;s_N(A-BD^{-1}C)\Big) .
\end{equation}
\end{lemma}
\begin{proof}
From the identity
\[
\begin{pmatrix} A & B\\ C&D\end{pmatrix}=
\begin{pmatrix} I_N & BD^{-1} \\ 0&I_n\end{pmatrix}
\begin{pmatrix} A-BD^{-1}C & 0\\ 0&D\end{pmatrix}
\begin{pmatrix} I_N & 0\\ D^{-1}C&I_n\end{pmatrix}
\]
we have
\begin{align*}
\begin{pmatrix} A & B\\ C&D\end{pmatrix}^{-1} &=
\begin{pmatrix} I_N & 0\\ -D^{-1}C&I_n\end{pmatrix}
\begin{pmatrix} (A-BD^{-1}C)^{-1} & 0\\ 0&D^{-1}\end{pmatrix}
\begin{pmatrix} I_N & -BD^{-1} \\ 0&I_n\end{pmatrix}.
\end{align*}
We can use the triangle inequality to bound the operator norm of the first and third matrices on the right hand side by $1+\|BD^{-1}\|$ and $1+\|CD^{-1}\|$, respectively.
Now by sub-multiplicativity of the operator norm,
\begin{align*}
\|M^{-1}\| &\le (1+\|BD^{-1}\|)(1+\|D^{-1}C\|) \max(\|(A-BD^{-1}C)^{-1}\|,\|D^{-1}\|) \\
&\le \bigg( 1+ \frac{\|B\|}{s_n(D)}\bigg)\bigg( 1+ \frac{\|C\|}{s_n(D)}\bigg) \max(\|(A-BD^{-1}C)^{-1}\|,\|D^{-1}\|).
\end{align*}
The bound \eqref{bound:schur} follows after taking reciprocals.
\end{proof}
\subsection{Control on the operator norm} \label{sec:op}
The following lemma summarizes the control we will need on the operator norm of submatrices and products of submatrices of $M$.
\begin{lemma}[Control on the operator norm] \label{lem:opcontrol}
Let $\xi\in \C$ be a centered random variable with $\e|\xi|^{4+\eta}\le 1$ for some $\eta\in (0,1)$.
Let $\asp \in (0,1)$.
Then the following hold for all $n\ge 1$:
\begin{enumerate}[(a)]
\item (Control for sparse matrices)
If $A\in \mM_n([0,1])$ is a fixed matrix and $X=(\xi_{ij})$ is an $n\times n$ matrix of iid copies of $\xi$, then
\begin{equation}
\|A \perp\!\!\!\perprc X\| \ll \tau \sqrt{n}
\end{equation}
except with probability $O_\tau(n^{-\eta/8})$, where $\tau=\tau(A)\in [0,1]$ is any number such that
\begin{equation}
\sum_{k=1}^n a_{ik}^2, \, \sum_{k=1}^n a_{kj}^2 \le \tau^2 n
\end{equation}
for all $i,j\in [n]$, and
\begin{equation} \label{4th.mom.bd}
\sum_{i,j=1}^n a_{ij}^4 \le \tau^4 n^2.
\end{equation}
\item (Control for matrix products)
Let $m\in [\asp n,n]$.
If $A\in \mM_{n,m}([0,1])$ and $D\in \mM_{m, n}(\C)$ are fixed matrices with $\|D\|\le 1$, and $X=(\xi_{ij})$ is an $n\times m$ matrix of iid copies of $\xi$, then
\begin{equation} \label{mbp:2}
\|D(A\perp\!\!\!\perprc X)\| \ll_\eta
\sqrt{m}
\end{equation}
except with probability $O_{\asp }(n^{-\eta/8})$.
\end{enumerate}
\end{lemma}
\begin{remark}
The probability bounds in the above lemma can be improved under higher moment assumptions on $\xi$, and improve to exponential bounds under the assumption that $\xi$ is sub-Gaussian (see \eqref{subgaussian}).
\end{remark}
We will use standard truncation arguments to deduce Lemma \ref{lem:opcontrol} from the following bounds on the expected operator norm of random matrices due to Lata\l a and Vershynin.
\begin{theorem}[Lata\l a \perp\!\!\!\perptep{Latala}] \label{thm:latala}
Let $n,m$ be sufficiently large and let $Y$ be an $n\times m$ random matrix with independent, centered entries $Y_{ij}\in \R$ having finite fourth moment. Then
\begin{equation}
\e \|Y\| \ll \max_{i\in [n]} \Bigg( \sum_{j=1}^m \e Y_{ij}^2 \Bigg)^{1/2} + \max_{j\in [m]} \Bigg(\sum_{i=1}^n \e Y_{ij}^2 \Bigg)^{1/2} + \Bigg( \sum_{i=1}^n\sum_{j=1}^m \e Y_{ij}^4\Bigg)^{1/4}.
\end{equation}
\end{theorem}
\begin{theorem}[Vershynin \perp\!\!\!\perptep{Vershynin:product}] \label{thm:vershynin}
Let $\eta\in (0,1)$ and $n,m,N$ sufficiently large natural numbers.
Let $D\in \mM_{m,N}(\R)$ be a deterministic matrix satisfying $\|D\|\le 1$ and $Y\in \mM_{N,n}(\R)$ be a random matrix with independent centered entries $Y_{ij}$ satisfying $\e |Y_{ij}|^{4+\eta}\le 1$.
Then
\begin{equation}
\e \|DY\| \ll_\eta \sqrt{n} + \sqrt{m}.
\end{equation}
\end{theorem}
\begin{proof}[Proof of Lemma \ref{lem:opcontrol}]
We begin with (a).
By splitting $X$ into real and imaginary parts and applying the triangle inequality we may assume $\xi$ is a real-valued random variable.
Set $\eta_0= \min(1/4,\eta/32)$ and define the product event
\begin{equation} \label{op1:eij}
\event = \bigwedge_{i,j=1}^n \event_{ij}; \quad\quad \event_{ij} = \big\{ |\xi_{ij}| \le n^{1/2-\eta_0}\big\}.
\end{equation}
By Markov's inequality,
\begin{equation} \label{op1:pij}
\pr( \event_{ij}^c) \le n^{-(4+\eta)(1/2-\eta_0)} \le n^{-1}
\end{equation}
for all $i,j\in [n]$.
By the union bound,
\begin{equation} \label{op1:event}
\pr(\event^c) \le n^2 n^{-(4+\eta)(1/2-\eta_0)} \le n^{-\eta/8}.
\end{equation}
We denote
\[
X' = (\xi_{ij}') = (\xi_{ij} - \e\xi_{ij}\un_{\event_{ij}}) = X - \e (X\un_{\event}).
\]
First we show
\begin{equation} \label{op1:AeX}
\|A\perp\!\!\!\perprc \e(X\un_{\event})\| \ll \tau \sqrt{n}.
\end{equation}
Since the variables $\xi_{ij}$ are centered,
$
|\e(\xi_{ij}\un_{\event_{ij}})| = | \e (\xi_{ij} \un_{\event_{ij}^c})|.
$
By two applications of H\"older's inequality and \eqref{op1:pij},
\[
|\e (\xi_{ij} \un_{\event_{ij}^c})| \le (\e |\xi_{ij}|^4)^{1/4} \pr(\event_{ij}^c)^{3/4} \le n^{-3/4}.
\]
Thus,
\begin{equation}
\|A\perp\!\!\!\perprc \e(X\un_{\event})\| \le \|A\perp\!\!\!\perprc \e(X\un_{\event})\|_{\HS} \le n^{-3/4} \|A\|_{\HS} \le \tau n^{1/4}
\end{equation}
which yields \eqref{op1:AeX} with room to spare.
Now from \eqref{op1:event}, \eqref{op1:AeX} and the triangle inequality it is enough to show
\begin{equation} \label{op1:goal1}
\pro{\event \wedge\big\{ \|A\perp\!\!\!\perprc X'\| \ge C\tau \sqrt{n} \big\}} = O_\tau(n^{-\eta/8})
\end{equation}
for a sufficiently large constant $C>0$ (we will actually show an exponential bound).
First note that the variables $\xi_{ij}'\un_{\event_{ij}}$ are centered and satisfy $\e |\xi_{ij}'\un_{\event_{ij}}|^4 = O(1)$.
It follows from Theorem \ref{thm:latala} that
\begin{align*}
\e\un_\event \|A\perp\!\!\!\perprc X'\|
&\ll
\max_{i\in [n]} \Bigg( \sum_{j=1}^n a_{ij}^2 \Bigg)^{1/2} + \max_{j\in [n]} \Bigg(\sum_{i=1}^n a_{ij}^2 \Bigg)^{1/2} + \Bigg( \sum_{i,j=1}^n a_{ij}^4\Bigg)^{1/4}\\
&\ll \tau \sqrt{n}.
\end{align*}
Thus, \eqref{op1:goal1} will follow if we can show
\begin{equation} \label{op1:goal2}
\pro{ \|A\perp\!\!\!\perprc X'\|\un_\event - \e \|A\perp\!\!\!\perprc X'\| \un_\event\ge \tau \sqrt{n} } = O_\tau(n^{-\eta/8}).
\end{equation}
This in turn follows in a routine manner from Talagrand's inequality \perp\!\!\!\perpte[Theorem 6.6]{Talagrand:newlook} (see also \perp\!\!\!\perpte[Corollary 4.4.11]{AGZ:book}):
Observe that $X\mapsto \|A\perp\!\!\!\perprc X\|$ is a convex and 1-Lipschitz function on the space $\mM_n(\R)$ equipped with the (Euclidean) Hilbert--Schmidt metric.
Since the matrix $X'\un_\event$ has centered entries that are bounded by $O(n^{1/2-\eta_0})$, Talagrand's inequality gives that the left hand side of \eqref{op1:goal2} is bounded by
\begin{equation}
O\big(\exp(-c\tau^2 n/(n^{1/2-\eta_0})^2)\big) = O\big( \exp(-c \tau^2 n^{2\eta_0})\big)
\end{equation}
which gives \eqref{op1:goal2} with plenty of room.
Now we turn to part (b).
The proof follows a very similar truncation argument to the one in part (a), so we only indicate the necessary modifications.
As before, by splitting $D$ and $X$ into real and imaginary parts and applying the triangle inequality we may assume $D$ and $X$ are real matrices.
We define $\event$ as in \eqref{op1:eij}, with
$
\event_{ij}= \big\{ |\xi_{ij}|\le (n\sqrt{m})^{1/3-\eta_1}\big\}
$
and
\begin{equation}
\eta_1= \frac{1}{4}\frac{\eta}{4+\eta}.
\end{equation}
With this choice of $\eta_1$, Markov's inequality and the union bound give $\pr(\event^c) = O_{\asp} (n^{-\eta/8})$.
Taking $X'= X-\e(X\un_\event)$ as before, we can bound
$
\|D(A\perp\!\!\!\perprc \e(X\un_\event))\| \le \|A\perp\!\!\!\perprc \e(X\un_\event)\|
$
by submultiplicativity of the operator norm, and the same argument as before gives
\begin{equation} \label{op2:det}
\|A\perp\!\!\!\perprc \e(X\un_\event)\| \le nm(n\sqrt{m})^{-\frac34(4+\eta)(1/3-\eta_1)} = m^{1/2-\eta/32} = o(\sqrt{m}).
\end{equation}
Since $X'\un_{\event}$ has centered entries with finite moments of order $4+\eta$, by Theorem \ref{thm:vershynin} we have
\begin{equation} \label{op2:e}
\e \|D(A\perp\!\!\!\perprc X'\un_\event)\| \ll_\eta \sqrt{m}.
\end{equation}
The mapping $X\mapsto \|D(A\perp\!\!\!\perprc X)\|$ is convex and 1-Lipschitz with respect to the Hilbert--Schmidt metric on $\mM_{n}(\R)$ (since $\|D\|\le 1$) so using Talagrand's inequality as in part (a) we find that
\begin{align*}
\pro{ \|D(A\perp\!\!\!\perprc X'\un_\event)\| - \e \|D(A\perp\!\!\!\perprc X'\un_\event)\| \ge \sqrt{m}}
&\ll \expo{ -c m / (n\sqrt{m})^{2/3-2\eta_1}}\\
&\le \expo{-c'(\asp) n^{c\eta } }
\end{align*}
for some constant $c>0$ and $c'(\asp)>0$ sufficiently small depending on $\asp$.
As the last line is bounded by $O_{\asp}(n^{-\eta/8})$, the result follows from the above, \eqref{op2:det}, \eqref{op2:e} and the triangle inequality by the same argument as for part (a).
\end{proof}
\subsection{Decomposition of the standard deviation profile} \label{sec:decomp}
We now begin the proof of Theorem \ref{thm:main}, which occupies the remainder of the paper.
In the present subsection we prove Lemma \ref{lem:decomp} below, which shows that the standard deviation profile $A$ can be partitioned into a bounded collection of submatrices with certain nice properties.
For the motivation behind this lemma (and the notation $J_{\free},J_{\cyc}$) see Section \ref{sec:ideas}.
\begin{lemma} \label{lem:decomp}
Let $A$ be an $n\times n$ matrix with entries $a_{ij}\in [0,1]$.
Let $\eps,\delta ,\ha\in (0,1)$, and assume $\eps$ is sufficiently small depending on $\delta $.
There exists $0\le \me \ll_\eps1$, a partition
\begin{align}
[n]&=J_{\badd}\cup J_{\free}\cup J_{\cyc} \notag\\
&= J_{\badd}\cup J_{\free}\cup J_1\cup\cdots \cup J_{\me } \label{decomp:bf1}
\end{align}
and a set $F\subset[n]^2$ satisfying the following properties:
\begin{enumerate}[(1)]
\item\label{decomp:1} $\eps n\ll |J_{\badd}|\ll \delta^{1/2} n$.
\item\label{decomp:F} $|F| \ll \delta n^2$, and for all $i\in J_{\free}$,
\begin{equation} \label{decomp:Frows}
|\{j\in J_{\free}: (i,j) \in F\}|, \; |\{j\in J_{\free}: (j,i) \in F\}| \le \delta^{1/2} n.
\end{equation}
\item\label{decomp:2} If $J_{\free}\ne\varnothing$ then there is a permutation $\tau:J_{\free}\to J_{\free}$ such that for all $(i,j)\in J_{\free}\times J_{\free} \setminus F$ with $\tau(i)\ge \tau(j)$, $\sig_{ij} <\ha$.
\item\label{decomp:3} If $\me \ge 1$ then
\begin{equation}
|J_1|=\cdots = |J_{\me }| \gg_\eps n
\end{equation}
and there is a permutation $\pi:[\me ]\to [\me ]$ such that for all $1\le k\le \me $, $A(\ha)_{J_k,J_{\pi(k)}}$ is $(2\delta,2\eps)$-super-regular
(see Definition \ref{def:super}).
\end{enumerate}
\end{lemma}
\begin{proof}
We begin by applying Lemma \ref{lem:regularity} to $A(\ha)$ to obtain $m_0\in \N$ with $\eps^{-1}\le m_0=O_\eps(1)$ and a partition $[n]=I_0\cup\cdots \cup I_{m_0}$ satisfying the properties in that lemma.
The partition $I_0,\dots,I_{m_0}$ is almost what we need.
In the remainder of the proof we perform a ``cleaning" procedure (as it is commonly referred to in the extremal combinatorics literature) to obtain a partition $J_0,\dots, J_{m_0}$ with improved properties, where $J_k\subset I_k$ for each $1\le k\le m_0$, and $J_0\supset I_0$ collects the leftover elements.
We start by forming a \emph{reduced digraph} $\mR=([m_0],E)$ on the vertex set $[m_0]$ with directed edge set
\begin{equation}
E:= \Big\{ (k,l)\in [m_0]^2: (I_k,I_l) \mbox{ is $\eps$-regular and } \rho_{A(\ha)}(I_k,I_l) >5\delta\Big\}.
\end{equation}
Next we find a (possibly empty) set $T\subset [m_0]$ such that the induced subgraph $\mR(T)$ is covered by vertex-disjoint directed cycles, and the induced subgraph $\mR([m_0]\setminus T)$ is cycle-free.
Such a set can be obtained by greedily removing cycles and the associated vertices from $\mR$ until the remaining graph has no more directed cycles.
By relabeling $I_1,\dots, I_{m_0}$ we may take $T=[\me]$, where $\me\in [0, m_0]$.
Assuming $\me \ne 0$, the fact that $\mR([\me ])$ is
covered by vertex-disjoint cycles
is equivalent to the existence of a permutation $\pi:[\me ]\to [\me ]$ such that $(k,\pi(k))\in E$ for all $1\le k\le \me $.
Now we will obtain the sets $J_1,\dots, J_{\me }$ obeying the properties in part (\ref{decomp:3}) of the lemma.
Let $1\le k\le \me $.
We have that $(I_k,I_{\pi(k)})$ is $\eps$-regular with density $\rho_k:= \rho_{A(\ha)} (I_k,I_{\pi(k)})>5\delta$, so if we assume $\eps\le \delta$ then for every $I\subset I_k, J\subset I_{\pi(k)}$ with $|I|,|J|\ge \eps |I_k|$,
\begin{equation} \label{decomp:edge}
e_{A(\ha)}(I,J) \ge (\rho_k-\eps)|I||J| \ge 4\delta |I||J|.
\end{equation}
It remains to ensure that conditions (1) and (2) from Definition \ref{def:super} also hold, which we will do by removing a small number of rows and columns.
Letting
\[
I_k' = \big\{ i\in I_k: |\mN_{A(\ha)}(i)\cap I_{\pi(k)}|<4\delta|I_k|\big\}
\]
we have $e_{A(\ha)}(I_k',I_{\pi(k)})<4\delta|I_k'||I_{\pi(k)}|$, and it follows that $|I_k'|\le \eps |I_k|$.
Similarly, letting
\[
I_k'' = \big\{ i\in I_k: |\mN_{A(\ha)^\tran}(i)\cap I_{\pi^{-1}(k)}|<4\delta|I_k|\big\}
\]
we have $|I_k''|\le \eps |I_k|$.
Letting $I_k^*\subset I_k$ be a set of size $\lf 2\eps|I_k|\rf$ containing $I_k'\cup I_k''$, we take
\begin{equation} \label{decomp:badcyc}
J_k= I_k\setminus I_k^*.
\end{equation}
With this definition we have $|J_1|=\cdots |J_{\me }|$, and for each $1\le k\le \me , i\in J_k$,
\begin{equation}
|\mN_{A(\ha)}(i)\cup J_{\pi(k)}|, \, |\mN_{A(\ha)^\tran}(i)\cap J_{\pi^{-1}(k)}|\ge (4\delta-2\eps)|I_k|\ge 2\delta|J_k|.
\end{equation}
Furthermore, for each $1\le k\le \me $ and $I\subset J_k, J\subset J_{\pi(k)}$ with $|I|,|J|\ge 2\eps |J_k|$,
if we assume $\eps\le 1/4$ then $|I|,|J|\ge \eps |I_k|$, so by \eqref{decomp:edge}
\begin{equation}
e_{A(\ha)}(I,J) \ge 4\delta|I||J|.
\end{equation}
It follows that for every $1\le k\le \me $ the submatrix $A(\ha)_{J_k,J_{\pi(k)}}$ is $(2\delta,2\eps)$-super-regular, which concludes the proof of part (\ref{decomp:3}) of the lemma.
Now we prove parts (\ref{decomp:F}) and (\ref{decomp:2}).
We will obtain $J_{\free}$ by removing a small number of bad elements from $I_{\free}:=\bigcup_{k=\me +1}^{m_0}I_k$.
Since the induced subgraph $\mR([\me +1,m_0])$ is cycle-free we may relabel $I_{\me +1},\dots, I_{m_0}$ so that
\begin{equation} \label{decomp:relabel}
(k,l) \notin E\; \mbox{ for all $\me < l\le k\le m_0$}.
\end{equation}
We take
\begin{equation}
F= \big\{ (i,j)\in [n]^2: (i,j)\in I_k\times I_l \mbox{ for some $(k,l)\notin E$}\big\}.
\end{equation}
The contribution to $F$ from irregular pairs $(I_k,I_l)$ is at most $\eps n^2$ by the regularity of the partition $I_0,\dots, I_{m_0}$, and the contribution from pairs $(I_k,I_l)$ with density less than $5\delta$ is at most $5\delta n^2$.
Hence,
\begin{equation} \label{decomp:Fbound}
|F| \le \eps n^2+5\delta n^2 \le 6\delta n^2
\end{equation}
giving the first estimate in (\ref{decomp:F}) (recall that we assumed $\eps\le \delta$).
Setting
\begin{equation} \label{decomp:Frows1}
I_{\free}'= \left\{ i\in I_{\free}: \max\big( |\{j\in [n]: (i,j)\in F\}|, |\{j\in [n]: (j,i)\in F\}|\big) \ge \delta^{1/2}n\right\}
\end{equation}
it follows from \eqref{decomp:Fbound} that
\begin{equation}
|I_{\free}'| \le 12\delta^{1/2}n.
\end{equation}
Let $I_{\free}^*\subset I_{\free}$ be any set containing $I_{\free}'$ of size $\min(|I_{\free}|, \lf 12 \delta^{1/2}n\rf)$ and take $J_{\free}= I_{\free}\setminus I_{\free}^*$.
The bounds \eqref{decomp:Frows} now follow immediately from \eqref{decomp:Frows1}.
For part (\ref{decomp:2}), from \eqref{decomp:relabel} we may take for $\tau$ any ordering of the elements of $J_{\free}$ that respects the order of the sets $J_k:=I_k\setminus I_{\free}^*$, i.e. so that $\tau(j)\ge \tau(i)$ for all $i\in J_k, j\in J_l$ and all $\me <l\le k\le m_0$.
Finally, taking
\begin{equation}
J_{\badd} = I_0 \cup I_{\free}^* \cup \bigcup_{k=1}^{\me } I_k^*.
\end{equation}
we have
\[
|J_{\badd}| \le \eps n + 12\delta^{1/2}n + 2\eps n \le 15 \delta^{1/2}n
\]
giving the upper bound in part (\ref{decomp:1}).
Now recalling that we took
$|I_{\free}^*|= \min(|I_{\free}|, \lf 12 \delta^{1/2}n\rf)$
and $|I_k^*| = \lf 2\eps |I_k|\rf$ for all $1\le k\le \me $, we also have the lower bound
\begin{align*}
|J_{\badd}|
&\ge \min\bigg(|I_{\free}^*|, \; \bigg|\bigcup_{k=1}^{\me } I_k^*\bigg| \bigg) \\
&\ge \min\bigg( \lf 12\delta^{1/2}n\rf ,\, |I_{\free}|,\, 2\eps \bigg| \bigcup_{k=1}^{\me } I_k \bigg| - \me \bigg)\\
&= \min \bigg( \lf 12 \delta^{1/2}n\rf, \, \bigg| \bigcup_{k=\me +1}^{m_0} I_k \bigg| , \,
2\eps \bigg| \bigcup_{k=1}^{\me } I_k \bigg| - \me \bigg) \\
&\gg \eps n
\end{align*}
where we used that at least one of the sets $I_{\free} = \bigcup_{k=\me +1}^{m_0} I_k$, $I_{\cyc} = \bigcup_{k=1}^{\me } I_k$ must be of size at least $n/4$, say.
This gives the lower bound in part (\ref{decomp:1}) and completes the proof.
\end{proof}
\subsection{High level proof of Theorem \ref{thm:main}} \label{sec:highlevel}
In this subsection we prove Theorem \ref{thm:main} on two lemmas (Lemmas \ref{lem:nil} and \ref{lem:cyc}) which give control on the smallest singular values of the submatrices $M_{J_{\free}}$ and (perturbations of) $M_{J_{\cyc}}$, with $J_{\free}, J_{\cyc}$ as in Lemma \ref{lem:decomp}.
The proofs of these lemmas are deferred to the remaining subsections.
By our moment assumptions on $\xi$ it follows that $\xi$ is $\kappa_0$-spread for some $\kappa_0=O(\mu_{4+\eta}^2)$ (see Remark \ref{rmk:kappap}).
By Lemma \ref{lem:wlog.kappa} and multiplying $X$ and $B$ by a phase we may assume $\xi$ has $O(\mu_{4+\eta}^2)$-controlled second moment.
Without loss of generality we may assume $\eta<1$.
We introduce parameters $\ha,\delta,\eps\in (0,1)$ to be chosen sufficiently small depending on $r_0,\eta$, and $\mu_{4+\eta}$; specifically we will have the following dependencies:
\begin{equation} \label{depends}
\ha=\ha(r_0,\mu_{4+\eta}), \quad \delta=\delta(r_0,\eta,\mu_{4+\eta}), \quad \eps=\eps(\ha,\delta).
\end{equation}
For the remainder of the proof we assume that $n$ is sufficiently large depending on all parameters (which will only depend on $r_0,K_0, \eta$ and $\mu_{4+\eta}$).
We begin by summarizing the control we have on the operator norm of submatrices of $A\perp\!\!\!\perprc X$.
From Lemma \ref{lem:opcontrol}(a) we have that for any fixed $B=(b_{ij})\in \mM_n([0,1])$ and any $I,J\subset[n]$ with $|I|\le |J|$,
\begin{equation} \label{opIJ:improved}
\pro{ \|(B\perp\!\!\!\perprc X)_{I,J}\| \le \tau K \sqrt{|J|}} = 1-O_\tau(|J|^{-\eta/8})
\end{equation}
for some $K=O(\mu_{4+\eta})$, and any $\tau\le 1$ satisfying
\begin{equation}
\tau \ge \frac1{|J|^{1/2}}\max\left( \max_{i \in I}\left( \sum_{j\in J} b_{ij}^2\right)^{1/2}, \;\max_{j\in J}\left( \sum_{i\in I} b_{ij}^2 \right)^{1/2}, \left( \sum_{i,j=1}^n b_{ij}^4\right)^{1/4} \right),
\end{equation}
and similarly with $|J|$ replaced by $|I|$ if $|J|\le |I|$.
In particular, taking $\tau=1$ and $B=A$ we have
\begin{align}
\|(A\perp\!\!\!\perprc X)_{I,J}\| &\ll_{\mu_{4+\eta}} \sqrt{\max(|I|,|J|)} \notag \\
&\qquad \text{with probability } 1-O(\max(|I|,|J|)^{-\eta/8}). \label{opIJ}
\end{align}
(We state \eqref{opIJ:improved} for general $B\in \mM_n([0,1])$ as at one point we will apply this to a residual matrix obtained by subtracting off a collection of ``bad" entries from $A$.)
We now apply Lemma \ref{lem:decomp} (assuming $\eps$ is sufficiently small depending on $\delta$) to obtain a partition $[n] = J_{\badd}\cup J_{\free} \cup J_{\cyc}$ and a set $F\subset[n]^2$ satisfying the properties (1)--(4) in the lemma.
In the following we abbreviate $M_{\free}:= M_{J_{\free}}$ and $M_{\cyc}:= M_{J_{\cyc}}$.
\begin{lemma} \label{lem:nil}
Assume $n_1:=|J_{\free}|\ge \delta^{1/2}n$.
If $\ha,\delta$ are sufficiently small depending on $r_0$ and
$\mu_{4+\eta}$,
then
\begin{equation}
s_{n_1}(M_{\free}) \gg_{\mu_{4+\eta},r_0} \sqrt{n}
\end{equation}
except with probability $O_{\mu_{4+\eta},r_0,\delta}(n^{-\eta/9})$.
\end{lemma}
(Note that while the definition of $M_{\free}$ depends on $\eps$, the bounds in the above lemma are independent of $\eps$.)
\begin{lemma} \label{lem:cyc}
Assume $n_2:=|J_{\cyc}| \ge \delta^{1/2}n$.
Fix $\gamma\ge1$ and let $W\in \mM_{n_2}(\C)$ be a deterministic matrix with $\|W\|\le n^{\gamma}$.
There exists $\beta= \beta(\gamma,\ha,\delta)$ such that if $\eps=\eps(\ha,\delta)$ is sufficiently small,
\begin{equation}
\pro{ s_{n_2}(M_{\cyc} + W) \le n^{-\beta}} \ll_{K_0,\gamma,\delta,\ha,\mu_{4+\eta}} \sqrt{\frac{\log n}{n}}.
\end{equation}
\end{lemma}
\begin{remark} \label{rmk:relaxmom}
We note that in the proof of Lemma \ref{lem:cyc} we do not make use of the fact that the atom variable $\xi$ has more than two finite moments (the dependence on $\mu_{4+\eta}$ is only through the parameter $\kappa_0=O(\mu_{4+\eta}^2)$).
In particular, we can remove the extra moment hypotheses in Theorem \ref{thm:main} under the additional assumption that the standard deviation profile $A$ contains a generalized diagonal of block submatrices which are super-regular and of dimension linear in $n$ (that is, if we can take $J_{\badd}=J_{\free}=\varnothing$ in \eqref{decomp:bf1}).
\end{remark}
We defer the proofs of Lemmas \ref{lem:nil} and \ref{lem:cyc} to subsequent sections, and conclude the proof of Theorem \ref{thm:main}.
Note that at this stage (before we have applied Lemma \ref{lem:nil} or \ref{lem:cyc}) the only constraint we have put on the parameters in \eqref{depends} is to assume $\eps$ is sufficiently small depending on $\delta$ for the application of Lemma \ref{lem:decomp}.
We proceed in the following steps:\\
\begin{itemize}
\item []
\begin{itemize}
\item [ {\bf Step 1:}] Bound the smallest singular value of $M_{\free}$ using Lemma \ref{lem:nil}. In this step we fix $\sigma(r_0,\mu_{4+\eta})$, while $\delta$ is assumed to be sufficiently small depending on $r_0,\mu_{4+\eta}$ but is otherwise left free.
\item [ {\bf Step 2:}] Bound the smallest singular value of
\begin{equation} \label{decompM1}
M_1 := M_{J_{\free}\cup J_{\badd},\,J_{\free}\cup J_{\badd}} = \begin{pmatrix} M_{\free} & B_1 \\ C_1 & M_0 \end{pmatrix}.
\end{equation}
using the result of Step 1, the Schur complement bound of Lemma \ref{lem:schur},
\eqref{opIJ:improved} and Lemma \ref{lem:opcontrol}(b).
In this step we fix $\delta(r_0,\eta,\mu_{4+\eta})$.
\item [ {\bf Step 3:}] Bound the smallest singular value of
\begin{equation} \label{decompM}
M= \begin{pmatrix} M_{\cyc} & B_2\\ C_2 & M_1\end{pmatrix}.
\end{equation}
using the result of Step 2, the Schur complement bound of Lemma \ref{lem:schur}, and Lemma \ref{lem:cyc}.
In this step we fix $\eps(\ha,\delta)$.
\end{itemize}
\end{itemize}
The case that one of $J_{\free}$ or $J_{\cyc}$ is small (or empty) can be handled essentially by skipping either Step 1 or Step 3.
We will begin by assuming
\begin{equation} \label{LB:nilcyc}
|J_{\free}|,\, |J_{\cyc}| \ge \delta^{1/2}n
\end{equation}
and address the case that this does not hold at the end.
\subsubsection*{Step 1}
By Lemma \ref{lem:nil} and the assumption \eqref{LB:nilcyc}, we can take $\ha$ and $\delta$ sufficiently small depending on $r_0$ and
$\mu_{4+\eta}$
such that
\begin{equation} \label{LB:Mnil}
s_{\min}(M_{\free}) \gg_{\mu_{4+\eta},r_0} \sqrt{n}
\end{equation}
except with probability $O_{\mu_{4+\eta},r_0,\delta}(n^{-\eta/9})$.
We now fix $\ha=\ha(r_0,\mu_{4+\eta})$ once and for all, but leave $\delta$ free to be taken smaller if necessary.
By independence of the entries of $M$ we may now condition on a realization of $M_{\free}$ such that \eqref{LB:Mnil} holds.
\subsubsection*{Step 2}
By \eqref{opIJ} and \eqref{LB:nilcyc} we have $\|C_1\|=O_{\mu_{4+\eta}}(\sqrt{n})$ except with probability $O_\delta(n^{-\eta/8})$. We henceforth condition on a realization of $C_1$ satisfying this bound.
Together with \eqref{LB:Mnil} this gives
\begin{equation}
\|C_1M_{\free}^{-1}\| \le \frac{\|C_1\|}{s_{\min}(M_{\free})} \ll_{\mu_{4+\eta},r_0}1.
\end{equation}
Since $B_1$ is independent of $C_1$ and $M_{\free}$ we can apply
Lemma \ref{lem:opcontrol}(b)
to conclude
\begin{equation} \label{C1MB1}
\|C_1M_{\free}^{-1}B_1\| \ll_{\eta,\mu_{4+\eta}}\|C_1M_{\free}^{-1}\| |J_{\badd}|^{1/2} \ll_{\eta,\mu_{4+\eta},r_0} |J_{\badd}|^{1/2}
\end{equation}
except with probability
$O_{\eps}(n_1^{-\eta/8}) = O_{\delta,\eps}(n^{-\eta/9})$,
where we have used the lower bound $|J_{\badd}|\gg \eps n$ from Lemma \ref{lem:decomp}(1).
On the other hand, by the triangle inequality and \eqref{opIJ},
\begin{equation} \label{step2:M0}
s_{\min}(M_0) = s_{\min}(Z_{J_{\badd}} \sqrt{n} + (A\perp\!\!\!\perprc X)_{J_{\badd}}) \ge r_0\sqrt{n} - O_{\mu_{4+\eta}}(|J_{\badd}|^{1/2})
\end{equation}
except with probability $O(|J_{\badd}|^{-\eta/8}) = O_\eps(n^{-\eta/9})$.
Again by the triangle inequality and the previous two displays,
\begin{equation}
s_{\min}(M_0-C_1M_{\free}^{-1} B_1) \ge r_0\sqrt{n} - O_{\eta,\mu_{4+\eta},r_0}(|J_{\badd}|^{1/2})
\end{equation}
except with probability $O_{\delta,\eps}(n^{-\eta/9})$.
Since $|J_{\badd}|\ll \delta^{1/2}n$ we can take $\delta$ smaller, if necessary, depending on $r_0,\eta,\mu_{4+\eta}$ to conclude that
\begin{equation} \label{step2:condition}
s_{\min}(M_0-C_1M_{\free}^{-1} B_1)\ge (r_0/2) \sqrt{n}
\end{equation}
except with probability $O_{\delta,\eps}(n^{-\eta/9})$.
We may henceforth condition on the event that \eqref{step2:condition} holds.
Of an event with probability $O_\delta(n^{-\eta/8})$ we may also assume $\|B_1\|=O_{\mu_{4+\eta}}(\sqrt{n})$.
From Lemma \ref{lem:schur} and the preceding estimates we have
\begin{align}
s_{\min}(M_1)
&\gg \left( 1+ \frac{O_{\mu_{4+\eta}}(\sqrt{n})}{s_{\min}(M_{\free})}\right)^{-2}
\min\big[ s_{\min}(M_{\free}), s_{\min}(M_0- C_1 M_{\free}^{-1} B_1)\big] \notag\\
&\gg_{\mu_{4+\eta},r_0}\min\big[ \sqrt{n}, s_{\min}(M_0-C_1M_{\free}^{-1}B_1)\big] \notag\\
&\gg_{\mu_{4+\eta},r_0} \sqrt{n}. \label{step2:final}
\end{align}
At this point we fix $\delta=\delta(r_0, \eta,\mu_{4+\eta})$.
\subsubsection*{Step 3}
Condition on a realization of $M_1$ such that \eqref{step2:final} holds.
By \eqref{opIJ} we may also condition on realizations of the matrices $B_2,C_2$ in \eqref{decompM} such that $\|B_2\|,\|C_2\| \ll_{\mu_{4+\eta}}\sqrt{n}$.
Applying Lemma \ref{lem:schur},
\begin{align}
s_n(M)
&\gg \left( 1+ \frac{O_{\mu_{4+\eta}}(\sqrt{n})}{s_{\min}(M_1)}\right)^{-2} \min \big[ s_{\min}(M_1), \,
s_{\min}(M_{\cyc} - B_2 M_1^{-1} C_2) \big] \notag\\
& \gg_{\mu_{4+\eta},r_0} \min \big[ \sqrt{n}, s_{\min}(M_{\cyc} - B_2 M_1^{-1} C_2) \big]. \label{step3:start}
\end{align}
By our estimates on $\|B_2\|,\|C_2\|$ and $s_{\min}(M_1)$ we have
\begin{equation}
\|B_2M_1^{-1}C_2\| \ll_{\mu_{4+\eta}} \frac{ n}{s_{\min}(M_1)} \ll_{\mu_{4+\eta},r_0} \sqrt{n}
\end{equation}
(unlike in Step 2, here we did not need the stronger control on matrix products provided by \eqref{mbp:2}).
Now since $M_2$ is independent of $M_1,B_2,C_2$, we can apply Lemma \ref{lem:cyc} with $\gamma=0.51$ (say), fixing $\eps$ sufficiently small depending on $\ha(r_0,\mu_{4+\eta})$ and $\delta(r_0,\eta,\mu_{4+\eta})$, to obtain
\begin{equation}
\pro{ s_{\min}(M_{\cyc} - B_2 M_1^{-1} C_2) \le n^{-\beta}} \ll_{K_0, r_0,\eta,\mu_{4+\eta}} \sqrt{\frac{\log n}{n}}
\end{equation}
for some $\beta = \beta(r_0,\eta,\mu_{4+\eta})>0$.
The result now follows from the above and \eqref{step3:start}, taking $\alpha=\min(\eta/9,1/4)$, say.
It only remains to address the case that the assumption \eqref{LB:nilcyc} fails.
We may assume that $\delta$ is small enough that only one of these bounds fails.
In this case we simply redefine $J_{\badd}$ to include the smaller of $J_{\cyc}, J_{\free}$.
Note that we still have $|J_{\badd}|= O(\delta^{1/2}n)$.
If $|J_{\cyc}|<\delta^{1/2}n$, then with this new definition of $J_{\badd}$ we have $M=M_1$, and the desired bound on $s_n(M)$ follows from \eqref{step2:final} (with plenty of room).
If $|J_{\free}|< \delta^{1/2}n$ then we skip Step 2, proceeding with Step 3 using $M_0$ in place of $M_1$. The bound \eqref{step2:final} in this case follows from \eqref{step2:M0} and the bound $|J_{\badd}|\ll \delta^{1/2}n$, taking $\delta$ sufficiently small depending on $\mu_{4+\eta},r_0$.
This concludes the proof of Theorem \ref{thm:main}.
\subsection{Proof of Lemma \ref{lem:nil}} \label{sec:nil}
We denote
\begin{equation}
A_{F}= (\sig_{ij} 1_{(i,j)\in F}).
\end{equation}
By the estimates on $F$ in Lemma \ref{lem:decomp} we can apply \eqref{opIJ:improved} with $\tau=O(\delta^{1/4})$ to obtain
\begin{equation}
\|(A_{F}(\ha)\perp\!\!\!\perprc X)_{J_{\free}}\| \ll_{\mu_{4+\eta}} \delta^{1/4}\sqrt{n}
\end{equation}
except with probability at most $O_{\delta}(n_1^{-\eta/8}) = O_{\delta}(n^{-\eta/9})$.
By another application of \eqref{opIJ:improved} with $\tau=1$,
\begin{equation}
\big\|\big((A-A(\ha))\perp\!\!\!\perprc X\big)_{J_{\free}}\big\| \ll_{\mu_{4+\eta}} \ha\sqrt{n}
\end{equation}
except with probability at most $O_{\delta}(n^{-\eta/9})$.
Let
\begin{equation}
\tM_{\free} := (\tA\perp\!\!\!\perprc X)_{J_{\free}} + Z_{J_{\free}}\sqrt{n}, \quad\quad \tA := A(\ha)-A_{F}(\ha).
\end{equation}
By the above estimates and the triangle inequality,
\begin{align}
s_{\min}(M_{\free}) &\ge s_{\min}(\tM_{\free}) - \|((A-\tA)\perp\!\!\!\perprc X)_{J_{\free}}\| \notag\\
&\ge s_{\min}(\tM_{\free})- O_{\mu_{4+\eta}}(\delta^{1/4}+ \ha)\sqrt{n} \label{nil:redux}
\end{align}
except with probability $O_{\delta}(n^{-\eta/9})$.
Thus, it suffices to show
\begin{equation} \label{nil:goal1}
s_{\min}(\tM_{\free})\gg_{\mu_{4+\eta},r_0}\sqrt{n}.
\end{equation}
except with probability $O_{\mu_{4+\eta},r_0,\delta}(n^{-\eta/9})$ -- the result will then follow from \eqref{nil:goal1} and \eqref{nil:redux} by taking $\delta,\ha$ sufficiently small depending on $\mu_{4+\eta},r_0$.
Furthermore, by Lemma \ref{lem:decomp}(3) and conjugating $M_{\free}$ by a permutation matrix we may assume that $\tA$ is (strictly) upper triangular.
Now it suffices to prove the following:
\begin{lemma}
Let $M= A\perp\!\!\!\perprc X + B$ be an $n\times n$ matrix as in Definition \ref{def:profile}, and further assume that for some $r_0>0, K\ge 1, \alpha>0$,
\begin{itemize}
\item $A$ is upper triangular;
\item $B=Z\sqrt{n} = \diag(z_i\sqrt{n})_{i=1}^n$ with $|z_i|\ge r_0$ for all $1\le i\le n$;
\item $\xi$ is such that for all $n'\ge 1$ and any fixed $A'\in \mM_{n'}([0,1])$, $\|A'\perp\!\!\!\perprc X'\|\le K\sqrt{n'}$ except with probability $O((n')^{-\alpha})$.
\end{itemize}
Then $s_n(M) \gg_{K,r_0}\sqrt{n}$ except with probability $O_{K,r_0}(1)^\alpha n^{-\alpha}$.
\end{lemma}
\begin{remark}
The proof gives an implied constant of order $\exp(-O(K/r_0)^{O(1)})$ in the lower bound on $s_n(M)$.
\end{remark}
To deduce Lemma \ref{lem:nil} we apply the above lemma with $M=\tM_{\free}$, $\alpha=\eta/8$, $K=O(\mu_{4+\eta})$ (by \eqref{opIJ}) and $n_1\gg_{\delta}n$ in place of $n$, which gives that \eqref{nil:goal1} holds with probability
\begin{equation}
1-O_{\mu_{4+\eta},r_0}(n_1^{-\eta/8}) = 1-O_{\mu_{4+\eta},r_0,\delta}(n^{-\eta/9})
\end{equation}
where in the first bound we applied our assumption that $\eta<1$.
\begin{proof}
First we note that we may take $n$ to be a dyadic integer, i.e. $n=2^q$ for some $q\in \N$.
Indeed, if this is not the case, then letting $2^q$ be the smallest dyadic integer larger than $n$ we can increase the dimension of $M$ to $2^q$ by padding $A$ out with rows and columns of zeros, adding additional rows and columns of iid copies of $\xi$ to $X$, and extending the diagonal of $Z$ with entries $z_i\equiv r_0$ for $n<i\le 2^q$.
The hypotheses on $A$ and $Z$ in the lemma are still satisfied, and the smallest singular value of the new matrix is a lower bound for that of the original matrix (since the original matrix is a submatrix of the new matrix).
Now fix an arbitrary dyadic filtration $\mF= \bigcup_{p\ge 0}\{J_s: s\in \{0,1\}^p\}$ of $[n]$, where we view $\{0,1\}^0$ as labeling the trivial partition of $[n]$, consisting only of the empty string $\varnothing$, so that $J_\varnothing = [n]$.
Thus, for every $0\le p< q$ and every binary string $s\in \{0,1\}^{p}$, $J_s$ has cardinality $n2^{-p}$ and is evenly partitioned by $J_{s0},J_{s1}$.
For a binary string $s$ we abbreviate $M_s:= M_{J_s}$ and similarly define $A_s,X_s,Z_s$.
We also write $B_s=M_{J_{s0},J_{s1}}$, so that we have the block decomposition
\begin{equation} \label{Ms:block}
M_s= \begin{pmatrix} M_{s0} & B_s\\ 0 & M_{s1} \end{pmatrix}.
\end{equation}
For $p\ge1$ define the boundedness event
\begin{equation}
\mB^*(p) = \big\{ \|A\perp\!\!\!\perprc X\| \le K\sqrt{n}\} \wedge \big\{ \forall s\in \{0,1\}^{p}, \; \|A_s\perp\!\!\!\perprc X_s\| \le K\sqrt{n 2^{-p}}\big\}.
\end{equation}
By our assumption on $\xi$ we have
\begin{equation} \label{mBstar:lb}
\pr(\mB^*(p))\ge 1- O(n^{-\alpha}) - 2^{p} O((n2^{-p})^{-\alpha}) = 1- O(2^{(1+\alpha)p}n^{-\alpha}).
\end{equation}
For arbitrary $s\in \{0,1\}^{p}$, by the triangle inequality we have that on $\mB^*(p)$,
\begin{align*}
s_{\min}(M_s) &\ge s_{\min}(Z_s) - \|A_s\perp\!\!\!\perprc X_s\| \ge (r_0 - K2^{-p/2})\sqrt{n}.
\end{align*}
Setting $p_0= \lf 2\log (2K/r_0)\rf +1$ we have that on $\mB^*(p_0)$,
\begin{equation} \label{lambdap:00}
s_{\min}(M_s) \ge (r_0/2)\sqrt{n}
\end{equation}
for all $s\in \{0,1\}^{p_0}$.
For the remainder of the proof we restrict the sample space to the event $\mB^*(p_0)$ and will use the Schur complement bound (Lemma \ref{lem:schur}) to show that the desired lower bound on $s_{\min}(M)$ holds deterministically (note that by \eqref{mBstar:lb} and our choice of $p_0$, $\mB^*(p_0)$ holds with probability
$1-O_{K,r_0}(n^{-\alpha})$).
For $0\le p\le p_0$ let
\begin{equation}
\lambda_p= \min_{s\in \{0,1\}^p} \frac{1}{\sqrt{n}} s_{\min}(M_s).
\end{equation}
From \eqref{lambdap:00} we have
\begin{equation} \label{lambdap:0}
\lambda_{p_0}\ge r_0/2
\end{equation}
Now let $1\le p\le p_0$ and $s\in \{0,1\}^{p-1}$. By the block decomposition \eqref{Ms:block} and Lemma \ref{lem:schur},
\begin{align*}
s_{\min}(M_s)
&\gg \left(1+ \frac{\|B_s\|}{s_{\min}(M_{s0})}\right)^{-1} \min\big( s_{\min}(M_{s0}), s_{\min}(M_{s1})\big)\\
&\ge (1+K/\lambda_p)^{-1}\lambda_p\sqrt{n}
\end{align*}
so $\lambda_{p-1} \gg (1+K/\lambda_p)^{-1}\lambda_p\sqrt{n}$ for all $0\le p\le p_0$.
Applying this iteratively along with \eqref{lambdap:0} we conclude $\lambda_0\gg_{K,r_0} 1$, i.e.
\begin{equation}
s_{\min}(M) \gg_{K,r_0} \sqrt{n}
\end{equation}
as desired.
\end{proof}
\subsection{Proof of Lemma \ref{lem:cyc}} \label{sec:cyc}
We may assume throughout that $n$ is sufficiently large depending on the parameters $K_0,\gamma,\delta,\ha$, and $\mu_{4+\eta}$. Note we may also assume $\gamma> 2$ without loss of generality.
We will apply only the following crude control on the operator norm of submatrices:
\begin{equation} \label{crude.op}
\pr(\|(A\perp\!\!\!\perprc X)_{I,J}\|\ge n^2 ) \le n^{-2} \quad \forall I,J\subset[n].
\end{equation}
Indeed, for any $I,J\subset[n]$,
\begin{align*}
\pr(\|(A\perp\!\!\!\perprc X)_{I,J}\|\ge n^2 ) \le \pr(\|A\perp\!\!\!\perprc X\|_{\HS}\ge n^2) .
\end{align*}
Furthermore, $\e\|A\perp\!\!\!\perprc X\|_{\HS}^2 \le \e \|X\|_{\HS}^2 = n^2$,
and \eqref{crude.op} follows from the above display and Markov's inequality.
By multiplying $M_{\cyc}$ by a permutation matrix we may assume that $A_k:= A_{J_k}$ is $(2\delta,2\eps)$-super-regular for $1\le k\le \me $ (unlike in the proof of Lemma \ref{lem:nil} the diagonal matrix $Z\sqrt{n}$ plays no special role here).
We denote $J_{\le k}= J_1\cup\cdots\cup J_k$, and for any matrix $W$ of dimension at least $|J_{\le k}|$ we abbreviate
\begin{equation}
W_k=W_{J_k},\quad W_{\le k} = W_{J_{\le k}},\quad W_{\le k-1, k} = W_{J_{\le k-1}, J_{k}}, \quad W_{k,\le k-1}= W_{J_{k}, J_{\le k-1}}
\end{equation}
so that for $2\le k\le \me $ we have the block decomposition
\begin{equation}
W_{\le k} = \begin{pmatrix} W_{\le k-1} & W_{\le k-1,k}\\ W_{k,\le k-1} & W_{k}\end{pmatrix}.
\end{equation}
Let us denote
\begin{equation} \label{nprime.eps}
n'= |J_1|=\cdots=|J_{\me }| \gg_\eps n.
\end{equation}
For $1\le k\le \me -1$, $\beta>0$ and a fixed $kn'\times kn'$ matrix $W$, we denote the event
\begin{equation}
\event_k(\beta,W) := \big\{ s_{kn'}(M_{\le k} + W) > n^{-\beta}\big\}.
\end{equation}
Let $\gamma>2$ and fix an arbitrary matrix $W\in \mM_{n',n'}(\C)$ with $\|W\|\le n^\gamma$.
By \eqref{crude.op} we have
\begin{equation}
\|M_1+W\| \le K_0\sqrt{n} + n^2 + n^\gamma \le 2n^\gamma
\end{equation}
with probability $1-O(n^{-2})$ if $n$ is sufficiently large depending on $K_0$ and $\gamma$.
By Theorem \ref{thm:super} there exists $\beta_1(\gamma) = O(\gamma^2)$ such that if $\eps$ is sufficiently small depending on $\ha,\delta$, then
\begin{align}
&\pr\big(\event_1(\beta_1,W)^c\big) \notag\\
&\le \pro{ \|M_1+W\|>2n^\gamma} + \pro{ \event_1(\beta_1,W)^c\wedge \{ \|M_1+W\|\le 2n^\gamma\}}\notag\\
&\ll_{\gamma,\delta,\ha,\eps,\mu_{4+\eta}} \sqrt{\frac{\log n}{n}}, \label{cyc:event1}
\end{align}
where we have used \eqref{nprime.eps} to write $n$ in $n^{-\beta_1}$ rather than $n'$, and the fact that the atom variable is $O(\mu_{4+\eta}^2)$-spread.
Now let $2\le k\le \me $, and suppose we have found a function $\beta_{k-1}(\gamma)$ such that for any $\gamma>2$ and any fixed $(k-1)n'\times (k-1)n'$ matrix $W$ with $\|W\|\le n^\gamma$,
\begin{equation}
\pro{ \event_{k-1}(\beta_{k-1}(\gamma),W)^c} \ll_{\gamma,\delta,\ha,\eps,\mu_{4+\eta}} \sqrt{\frac{\log n}{n}}.
\end{equation}
Fix a $kn'\times kn'$ matrix $W$ with $\|W\|\le n^\gamma$.
By Lemma \ref{lem:schur} we have
\begin{align}
s_{kn'}( M_{\le k} + W)
&\gg \left( 1+ \frac{\|(M+W)_{\le k-1, k}\|}{s_{(k-1)n'}(M_{\le k-1}+W_{\le k-1})}\right)^{-1}
\left( 1+ \frac{\|(M+W)_{k,\le k-1}\|}{s_{(k-1)n'}(M_{\le k-1}+W_{\le k-1})}\right)^{-1} \notag\\
&\quad\quad\quad\quad \times \min\Big[ s_{(k-1)n'}(M_{\le k-1}+W_{\le k-1}) , s_{n'}\big(M_k + B_k\big)\Big] \label{Mk:schur}
\end{align}
where we have abbreviated
\begin{equation}
B_k:=W_k - (M+W)_{k,\le k-1}( M_{\le k-1} + W_{\le k-1})^{-1} (M+W)_{\le k-1,k} .
\end{equation}
Suppose that the event $\event_{k-1}(\beta_{k-1}(\gamma),W_{\le k-1})$ holds.
We condition on a realization of the submatrix $M_{\le k-1}$ satisfying
\begin{equation}
s_{(k-1)n'}(M_{\le k-1}+W_{\le k-1}) \ge n^{-\beta_{k-1}(\gamma)}.
\end{equation}
Moreover, from \eqref{crude.op} we have
\begin{equation}
\|(M+W)_{\le k-1, k}\|, \|(M+W)_{k,\le k-1}\| \le K_0\sqrt{n}+n^2+n^\gamma\le 2n^\gamma
\end{equation}
with probability $1-O(n^{-2})$. Conditioning on the event that the above holds, from the previous two displays we have
$
\|B_k\| \le n^\gamma+ 4n^{\gamma + \beta_{k-1}(\gamma)}.
$
Again by \eqref{crude.op},
\begin{equation}
\|M_k+ B_k\|\le K_0\sqrt{n}+n^2 + 4n^{\gamma+\beta_{k-1}(\gamma)} \le 5n^{\gamma+\beta_{k-1}(\gamma)}
\end{equation}
with probability $1-O(n^{-2})$ in the randomness of $M_k$.
By Theorem \ref{thm:super} and independence of $M_k$ from $M_{\le k-1},M_{k,\le k-1},M_{k,\le k-1}$, there exists $\beta_k' = O(\gamma^2 + \beta_{k-1}(\gamma)^2)$ such that
\begin{equation} \label{MkBk:bound}
\pro{s_{n'}(M_k+B_k) \le n^{-\beta_k'}} \ll_{\gamma,\delta,\ha,\eps,\mu_{4+\eta}} \sqrt{\frac{\log n}{n}}.
\end{equation}
Restricting further to the event that $s_{n'}(M_k+B_k) > n^{-\beta_k'}$ and substituting the above estimates into \eqref{Mk:schur}, we have
\begin{equation}
s_{kn'}(M_{\le k}+W) \gg n^{-2\gamma-2\beta_{k-1}(\gamma)}\min(n^{-\beta_{k-1}(\gamma)}, n^{-\beta_k'}) \ge n^{-\beta_k(\gamma)}
\end{equation}
for some $\beta_k(\gamma) = O(\gamma^2 + \beta_{k-1}(\gamma)^2)$.
With this choice of $\beta_k(\gamma)$ we have shown
\begin{equation}
\pro{ \event_k(\beta_{k}(\gamma), W_{\le k})^c \wedge \event_{k-1}(\beta_{k-1}(\gamma),W_{\le k-1})} \ll_{\gamma,\delta,\ha,\eps,\mu_{4+\eta}} \sqrt{\frac{\log n}{n}}.
\end{equation}
Applying this bound for all $2\le k'\le k$ together with \eqref{cyc:event1} and Bayes' rule we conclude that for any fixed $k$ and any square matrix $W$ of dimension at least $kn'$ and operator norm at most $n^\gamma$,
\begin{equation}
\pro{ \event_k(\beta_{k}(\gamma), W_{\le k})^c } \ll_{\gamma,\delta,\ha,\eps,\mu_{4+\eta}} k \sqrt{\frac{\log n}{n}}.
\end{equation}
The result now follows by taking $k=\me $ and recalling that $\me =O_\eps(1)$.
\appendix
\section{Invertibility for perturbed non-Hermitian band matrices} \label{app:band}
In this appendix we prove Corollary \ref{cor:band}.
By conditioning on the entries $\xi_{ij}$ with $\min(|i-j|,n-|i-j|)> \eps n$ and absorbing the corresponding entries of $A\perp\!\!\!\perprc X$ into $B$ we may assume the entries of $A(\ha)$ are zero outside the band.
By Theorem \ref{thm:broad} it suffices to show that $A(\ha)$ is $(\delta,\nu)$-broadly connected for $\delta,\nu\in (0,1)$ sufficiently small depending on $\eps$.
Throughout the proof we may assume that $n$ is sufficiently large depending on $\eps$, i.e.\ $n\ge n_0$ for any $n_0(\eps)\in \N$.
Let $\delta,\nu\in (0,1)$ to be chosen sufficiently small depending on $\eps$.
For all $i\in [n]$ we have $|\mN_{A(\ha)}(i)|,|\mN_{A^\tran(\ha)}(i)|\ge 2\eps n$, so taking $\delta<2\eps$, it only remains to verify the third condition in Definition \ref{def:broad}.
Note that if $|J|>(1-\eps)n$ we trivially have $|J(i)| \ge |\mN_{A(\ha)}(i)| - \eps n \ge \eps n$ for every $i\in [n]$, and the condition holds in this case.
Fix a set $J\subset[n]$ with $1\le |J|\le (1-\eps)n$. For the remainder of the proof we abbreviate $J(i):= J\cap \mN_{A(\ha)}(i)$ and
\[
I_\delta:= \mN_{A^\tran(\ha)}^{(\delta)}(J) = \{i: |J(i)| \ge \delta |J|\}.
\]
It will be convenient to view $i\mapsto |J(i)|$ as a function on the torus $\Z/n\Z$ (which we identify with $[n]$ in the natural way). From double counting we have
\begin{equation} \label{ex:doublect}
\sum_{i\in \Z/n\Z} |J(i)| = (1+\lf 2\eps n\rf)|J| \ge 2\eps |J|.
\end{equation}
On the other hand, we have the discrete derivative bound
\begin{equation} \label{discderiv}
||J(i)|-|J(i-1)||\le 1 \quad \forall i\in \Z/n\Z.
\end{equation}
Suppose towards a contradiction that
\begin{equation} \label{Ji:suppose}
|I_\delta| <(1+\nu) |J|.
\end{equation}
Since we took $\delta<2\eps$, from \eqref{ex:doublect} and the pigeonhole principle it follows that $|I_\delta|\ge 1$.
We decompose $I_\delta=\cup_{l\in L} I_l$ as a disjoint union of interval subsets $I_l=[a_l,b_l]\subset\Z/n\Z$ that are pairwise separated by a distance at least 2.
We further split $L=L_>\cup L_\le$, where $L_>= \{l\in L: |I_l|\ge 4\eps n\}$ and $L_\le = L\setminus L_>$.
Note that for each $l\in L$ we have
\begin{equation} \label{endpoints}
|J(a_l)|=|J(b_l)| = \lf \delta |J|\rf + 1.
\end{equation}
From the bound \eqref{discderiv} and the endpoint conditions \eqref{endpoints} we see that within $I_l$,
\begin{equation} \label{Ji.pointwise}
|J(i)|\le \min\big[ \lf \delta |J|\rf + 1+ \min(i-a_l,b_l-i) , \, 2\eps n+1\big],
\end{equation}
where the second argument in the outer minimum comes from the bound $|J(i)| \le \mN_{A(\ha)}(i)\le 2\eps n+1$.
For $l\in L_\le$ we ignore the second argument in the outer minimum (which only increases the bound), and sum to obtain
\[
\sum_{i\in I_l} |J(i)| \le (\delta|J| +1) |I_l| + \frac14 |I_l|^2 \le (1+\delta |J| + \eps n)|I_l|,\quad l\in L_\le.
\]
For $l\in L_>$ we have
\begin{align*}
\sum_{i\in I_l} |J(i)|
&=\sum_{i\in I_l: \min(i-a_l,b_l-i)\le 2\eps n} \lf\delta |J|\rf + 1+ \min(i-a_l,b_l-i)\\
&\qquad+ (2\eps n+1) |\{i\in I_l: i-a_l, b_l-i\ge 2\eps n+1\}|\\
&\le 4\eps n (\lf \delta |J|\rf + 1) + 4\eps^2 n^2 + (2\eps n+1) (|I_l| -4\eps n)\\
&\le (2\eps n+1)|I_l| + 4\eps n\delta |J| - 4\eps^2n^2.
\end{align*}
From the previous two displays we obtain
\begin{align*}
\sum_{i\in \Z/n\Z} |J(i)|
&\le \delta |J| n + \sum_{i\in I_\delta} |J(i)|\\
&\le \delta |J| n + \sum_{l\in L_\le} (1+\delta |J| + \eps n)|I_l| \\
&\qquad \qquad+ \sum_{l\in L_>} \Big[(2\eps n+1)|I_l| + 4\eps n\delta |J| - 4\eps^2 n^2\Big]\\
&= \delta|J|n + 4\eps n(\delta |J|-\eps n)|L_>|\\
&\qquad\qquad+(1+\delta|J|+ \eps n) \sum_{l\in L_\le }|I_l| + (2\eps n+1)\sum_{l\in L_>} |I_l|.
\end{align*}
If $|L_>|=0$ then
\begin{align*}
\sum_{i\in \Z/n\Z} |J(i)| \le \delta |J| n + (1+\delta |J|+\eps n) |I_\delta|.
\end{align*}
Combining with \eqref{ex:doublect} and rearranging we obtain
\[
|I_\delta| \ge \frac{(2\eps - \delta)|J| n}{1+ \eps n + \delta |J|} \ge \frac{2\eps-\delta}{\eps + \delta}|J|,
\]
and we contradict \eqref{Ji:suppose} taking $\nu<1/2$, say, and $\delta<c\eps$ for a sufficiently small constant $c>0$.
If $|L_>|\ge 1$, from our assumption $\delta<2\eps$ we have
\begin{align*}
\sum_{i\in \Z/n\Z} |J(i)|
&\le \delta |J| n -2\eps n|L_>| + (2\eps n+1) \sum_{i\in L} |I_l|\\
&\le \delta |J| n - 2\eps^2 n^2+ (2\eps n+1) |I_\delta|.
\end{align*}
Together with \eqref{ex:doublect} this gives
\[
|I_\delta| \ge \frac{2\eps n}{2\eps n+1} |J| + \frac{2\eps^2n^2-\delta n|J|}{2\eps n+1} \ge \frac{2\eps n}{2\eps n+1} |J| + \frac14\eps n
\]
where in the last bound we took $\delta<\eps^2$ and assumed $n\ge 1/\eps$.
Taking $\nu<\eps/8$, say, we contradict \eqref{Ji:suppose} if $n$ is sufficiently large.
The claim follows.
\section{Proofs of anti-concentration lemmas}
\label{app:anti}
In this appendix we prove Lemmas \ref{lem:wlog.kappa}, \ref{lem:anti_improved} and \ref{lem:tensorize}.
All three are established by modification of existing arguments from the literature.
\subsection{Proof of Lemma \ref{lem:wlog.kappa}} \label{app:kappa}
\eqref{cond.kappa1} is immediate by our assumptions.
It remains to show
\begin{equation} \label{kappa.goal}
\e | \re(z\xi - w)|^2\un(|\xi|\le {\kappa_0}) \gg \frac1{\kappa_0} |\re(z)|^2
\end{equation}
for all $z,w\in \C$ after rotating $\xi$ by a phase if necessary.
We may assume $\kappa_0$ is larger than any fixed constant.
Let $\event$ denote the event $\{|\xi|\le \kappa_0\}$.
By Chebyshev's inequality,
\begin{equation} \label{kappa.cheb}
\pr(\event) \ge 1-\frac1{\kappa_0^2}.
\end{equation}
Fix $z,w\in \C$.
Write $\widetilde{\e}:= \e(\cdot|\event)$.
By \eqref{kappa.cheb} and assuming $\kappa_0$ is sufficiently large we have that the left hand side of \eqref{kappa.goal} is $\gg \widetilde{\e}| \re(z\xi - w)|^2$, so it suffices to show
\begin{equation}
\widetilde{\e}| \re(z\xi - w)|^2 \gg \frac{1}{\kappa_0}|\re(z)|^2
\end{equation}
after rotating $\xi$ by a phase.
Denoting $\eta:= \xi -\widetilde{\e}\xi$, we have
\[
\widetilde{\e}| \re(z\xi - w)|^2 = \widetilde{\e}| \re(z\eta + (\widetilde{\e} \xi - w))|^2
= \widetilde{\e}| \re(z\eta)|^2 + |\widetilde{\e} \xi - w|^2
\]
so it suffices to show that after rotating $\xi$ by a phase,
\begin{equation}
\widetilde{\e}| \re(z\eta)|^2 \gg \frac{1}{\kappa_0} |\re(z)|^2.
\end{equation}
We first estimate the conditional variance of $\eta$.
We have
\begin{align*}
\widetilde{\e}|\eta|^2
& = \widetilde{\e}|\xi|^2 - |\widetilde{\e}\xi|^2 \\
&= \frac1{\pr(\event)} \e |\xi|^2 \un_{\event} - \frac1{\pr(\event)^2} |\e \xi \un_{\event}|^2\\
&= \frac1{\pr(\event)^2} \var(\xi\un_{\event}) + \frac1{\pr(\event)}\left(1-\frac1{\pr(\event)}\right) \e |\xi|^2\un_\event\\
&= \frac{1}{\pr(\event)^2} \left( \var(\xi\un_\event) - \pr(\event^c)\e |\xi|^2\un_{\event}\right)\\
&\gg \var(\xi\un_\event) - O(1/\kappa_0^2)
\end{align*}
where in the final line we applied \eqref{kappa.cheb}, the assumption $\e|\xi|^2=1$, and assumed $\kappa_0$ is sufficiently large.
Now by our assumption that $\xi$ is $\kappa_0$-spread we have $\var(\xi\un_\event) \gg 1/\kappa_0$, so
\begin{equation}
\widetilde{\e}|\eta|^2 \gg 1/\kappa_0
\end{equation}
taking $\kappa_0$ larger if necessary.
Now consider the covariance matrix
\begin{equation}
\Sigma_{\kappa_0} := \begin{pmatrix} \widetilde{\e} |\re(\eta)|^2 & \widetilde{\e} (\re(\eta)\im(\eta))\\
\widetilde{\e} (\re(\eta)\im(\eta)) & \widetilde{\e} |\im(\eta)|^2
\end{pmatrix}.
\end{equation}
Writing $z=a-ib$ and letting $x=(a\quad b)^\tran$ be the associated column vector, we have
\begin{equation}
\widetilde{\e}| \re(z\eta)|^2 = \widetilde{\e} |a\re(\eta) + b \im(\eta)|^2 = x^\tran \Sigma_{\kappa_0} x.
\end{equation}
Since $\Sigma_{\kappa_0}$ has two non-negative eigenvalues $\sigma^2_1\ge \sigma^2_2\ge 0$ summing to $\widetilde{\e}|\eta|^2 \gg 1/{\kappa_0}$, it follows that $\sigma_1^2 \gg 1/{\kappa_0}$.
We may rotate $\xi$ by an appropriate phase to assume the corresponding eigenspace is spanned by $(1\quad 0)^\tran$.
This gives
\[\widetilde{\e}| \re(z\eta)|^2 \gg \sigma_1^2 |\re(z)|^2 \gg \frac1{{\kappa_0}} |\re(z)|^2\] as desired.
\subsection{Proof of Lemma \ref{lem:anti_improved}} \label{app:improved}
We first need to recall a couple of lemmas from \perp\!\!\!\perptep{TaVu:smooth, TaVu:circ}.
\begin{lemma}[Fourier-analytic bound, cf.\ {\perp\!\!\!\perpte[Lemma 6.1]{TaVu:smooth}}] \label{lem:fourier}
Let $\xi$ be a complex-valued random variable.
For all $r> 0$ and any $v\in S^{n-1}$ we have
\begin{equation}
p_{\xi,v}(r) \ll r^2\int_{w\in \C: |w|\le 1/r} \exp\bigg( -c\sum_{j=1}^n \|wv_j\|_\xi^2\bigg) \dd w
\end{equation}
where
\begin{equation}
\|z\|_\xi^2:= \e \|\re(z(\xi-\xi'))\|_{\R/\Z}^2,
\end{equation}
$\xi'$ is an independent copy of $\xi$, and $\|x\|_{\R/\Z}$ denotes the distance from $x$ to the nearest integer.
\end{lemma}
The next lemma gives an important property enjoyed by the ``norm" $\|\cdot\|_\xi$ from Lemma \ref{lem:fourier} under the assumption that $\xi$ has $\kappa$-controlled second moment.
\begin{lemma}[cf.\ {\perp\!\!\!\perpte[Lemma 5.3]{TaVu:circ}}] \label{lem:modbound}
For any $\kappa>0$ there are constants $c_1,c_2>0$ such that if $\xi$ is $\kappa$-controlled, then $\|z\|_\xi \ge c_1|\re(z)|$ whenever $|z|\le c_2$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:anti_improved}]
Let $r\ge0$.
We may assume $r\ge C_0\|v\|_\infty$ for any fixed constant $C_0>0$ depending only on $\kappa$.
From Lemma \ref{lem:fourier},
\[
p_{\xi,v}(r) \ll r^2\int_{|w|\le 1/r} \exp\bigg(-c\sum_{j=1}^n \|wv_j\|_\xi^2\bigg)\dd w.
\]
If $C_0$ is sufficiently large depending on $\kappa$, it follows from Lemma \ref{lem:modbound} that whenever $|w|\le 1/r$, $\|wv_j\|_\xi \ge c_1|\re(wv_j)|$, giving
\[
p_{\xi,v}(r) \ll r^2\int_{|w|\le 1/r} \exp\bigg(-c'\sum_{j=1}^n (\re(wv_j))^2\bigg)\dd w
\]
where $c'$ depends only on $\kappa$.
By change of variable,
\begin{equation} \label{be:cov}
p_{\xi,v}(r) \ll \int_{|w|\le 1} \exp\bigg(-\frac{c'}{r^2}\sum_{j=1}^n (\re(wv_j))^2\bigg)\dd w.
\end{equation}
Write $v_j=r_je^{i\theta_j}$ for each $j\in [n]$.
Since $v\in S^{n-1}$ we have $\sum_{j=1}^n r_j^2=1$.
By Jensen's inequality,
\begin{align*}
p_{\xi,v}(r) &\ll \int_{|w|\le 1} \exp\bigg( -\frac{c'}{r^2} \sum_{j=1}^n r_j^2\big( \re(we^{i\theta_j})\big)^2\bigg)\dd w \\
&\le \int_{|w|\le 1}\sum_{j=1}^n r_j^2 \exp\bigg( -\frac{c'}{r^2} \big(\re(we^{i\theta_j})\big)^2\bigg)\dd w.
\end{align*}
By rotational invariance the last expression is equal to
\[
\sum_{j=1}^n r_j^2\int_{|w|\le 1} \exp\bigg( -\frac{c'}{r^2} (\re(w))^2\bigg)\dd w = \int_{|w|\le 1} \exp\bigg( -\frac{c'}{r^2} (\re(w))^2\bigg)\dd w
\]
which by direct computation is seen to be of size $O(r)$ (with implied constant depending on $\kappa$).
Together with our assumption that $r\ge C_0\|v\|_\infty$ this gives \eqref{be:1d}.
\end{proof}
\subsection{Proof of Lemma \ref{lem:tensorize}}
We only prove part (a) as part (b) is given in \perp\!\!\!\perpte[Lemma 2.2]{RuVe:ilo}.
Let $c_1>0$ to be taken sufficiently small depending on $p_0$, and let $\alpha>0$ a sufficiently small constant to be chosen later.
We have
\begin{align}
\pro{ \sum_{j=1}^n |\zeta_j|^2 \le c_1\eps_0^2 n}
&= \pro{ n- \frac1{c_1\eps_0^2} \sum_{j=1}^n |\zeta_j|^2 \ge 0} \notag\\
&\le \e \expo{ c_1\alpha n - \frac{\alpha}{\eps_0^2} \sum_{j=1}^n |\zeta_j|^2 } \notag\\
&= e^{c_1\alpha n} \prod_{j=1}^n \e \expo{ -\alpha |\zeta_j|^2/\eps_0^2}. \label{tensor:above}
\end{align}
For arbitrary $j\in [n]$ we have
\begin{align*}
\e \expo{ -\alpha |\zeta_j|^2/\eps_0^2}
&= \int_0^1 \pro{ \expo{-\alpha|\zeta_j|^2/\eps_0^2 } \ge u} \dd u\\
&= \int_0^\infty \pro{ |\zeta_j| \le s\eps_0/\sqrt{\alpha}} \dd (e^{-s^2})\\
&\le p_0\int_0^{\sqrt{\alpha}} \dd(e^{-s^2}) + \int_{\sqrt{\alpha}}^\infty \dd(e^{-s^2})\\
&= p_0(1-e^{-\alpha}) + e^{-\alpha}\\
&= 1- (1-p_0)(1-e^{-\alpha}).
\end{align*}
Inserting this in \eqref{tensor:above}, we obtain
\begin{align*}
\pro{ \sum_{j=1}^n |\zeta_j|^2 \le c_1\eps_0^2 n}
&\le e^{c_1\alpha n} \big[ 1- (1-p_0)(1-e^{-\alpha}) \big]^n\\
&\le \expo{ n\big(c_1\alpha - (1-p_0)(1-e^{-\alpha})\big)}.
\end{align*}
The claim now follows by setting $c_1=(1-p_0)/2$ (for instance) and taking $\alpha$ a sufficiently small constant.
\end{document} |
\begin{document}
\title{Microscopic biasing of discrete-time quantum trajectories}
\author{Dario Cilluffo}
\email{Corresponding author:[email protected]}
\affiliation{Universit$\grave{a}$ degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr\`e, via Archirafi 36, I-90123 Palermo, Italy}
\affiliation{NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa, Italy}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at T\"ubingen, Auf der Morgenstelle 14, 72076 T\"ubingen, Germany}
\author{Giuseppe Buonaiuto}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at T\"ubingen, Auf der Morgenstelle 14, 72076 T\"ubingen, Germany}
\author{Igor Lesanovsky}
\affiliation{School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom}
\affiliation{Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at T\"ubingen, Auf der Morgenstelle 14, 72076 T\"ubingen, Germany}
\author{Angelo Carollo}
\affiliation{Universit$\grave{a}$ degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr\`e, via Archirafi 36, I-90123 Palermo, Italy}
\affiliation{Radiophysics Department, National Research Lobachevsky State University of Nizhni Novgorod, 23 Gagarin Avenue, Nizhni Novgorod 603950, Russia}
\author{Salvatore Lorenzo}
\affiliation{Universit$\grave{a}$ degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr\`e, via Archirafi 36, I-90123 Palermo, Italy}
\author{G. Massimo Palma}
\affiliation{Universit$\grave{a}$ degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr\`e, via Archirafi 36, I-90123 Palermo, Italy}
\affiliation{NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa, Italy}
\author{Francesco Ciccarello}
\affiliation{Universit$\grave{a}$ degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segr\`e, via Archirafi 36, I-90123 Palermo, Italy}
\affiliation{NEST, Istituto Nanoscienze-CNR, Piazza S. Silvestro 12, 56127 Pisa, Italy}
\author{Federico Carollo}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at T\"ubingen, Auf der Morgenstelle 14, 72076 T\"ubingen, Germany}
\begin{abstract}
We develop a microscopic theory for biasing the quantum trajectories of an open quantum system,
which renders rare trajectories typical.
To this end we consider a discrete-time quantum dynamics, where the open system collides sequentially with qubit probes which are then measured. A theoretical framework is built in terms of thermodynamic functionals in order to characterize its quantum trajectories (each embodied by a sequence of measurement outcomes). We show that the desired biasing is achieved by suitably modifying the Kraus operators describing the discrete open system dynamics. From a microscopical viewpoint and for short collision times, this corresponds to adding extra collisions which enforce the system to follow a desired rare trajectory. The above extends the theory of biased quantum trajectories from Lindblad-like dynamics to sequences of arbitrary dynamical maps, providing at once a transparent physical interpretation.
\end{abstract}
\date{\today}
\maketitle
\sigmaection{Introduction}
Controlling quantum systems typically requires coping with dissipative (open) nonequilibrium dynamics. As dissipation is due to the ineliminable effect of an environment, the system-environment coupling needs to be controlled, or even explicitly harnessed, as in dissipative quantum computing/state engineering \cite{KempePRA01, DFLidar, DFPalma} or preparation of decoherence-free steady states \cite{PhysRevLett.89.277901,BeigePRL00,GonzalezTudelaPRL15}.
At a fundamental level, open quantum dynamics result from single stochastic realizations called quantum trajectories \cite{Gardiner2004,breuerpetruccione}. In each of these, the open system, which is effectively continuously monitored by the “environment”, undergoes an overall non-unitary time-evolution interrupted, at random times, by quantum jumps. Each jump, which for atom-photon systems is in one-to-one correspondence with the irreversible emission and detection of a photon, causes a sudden change of the system state \cite{ZollerPRA1987,BrunPRA2000}.
\begin{figure}
\caption{{\bf Quantum collision model.}
\label{fig1}
\end{figure}
In analogy with equilibrium thermodynamic ensembles \cite{touchette2009large,greiner2012thermodynamics}, the collection of quantum trajectories and their probability can be treated as a macroscopic (non-equilibrium) state. Each single realization can be thought of as a microstate, characterized by the number of occurred jumps, typically extensive with the observation time. The average properties of the system are determined by typical trajectories of the macroscopic state, while rare ones govern deviations from such behaviour \cite{10.1143/PTPS.184.304,PhysRevLett.111.120601,garaspects,Jack:2020aa}.
Given the above scenario, {controlling the statistics of trajectories} is crucial: a major benefit would, e.g., be the possibility of engineering devices with the desired emission properties. This is yet a challenging task since changing the jumps statistics in fact entails turning rare trajectories into typical \cite{garrahan2010thermodynamics,carollo2018making,PhysRevLett.122.130605}. First progress along this line was recently made by showing that a preselected set of rare trajectories of a Markovian open quantum system described by a Lindblad master equation can always be seen as the typical realizations of an alternative (still Markovian) system \cite{carollo2018making}. Yet, the physical connection between the two systems (which may be radically different) is not straightforwardly interpreted.
This work approaches the problem of tailoring trajectories statistics from a much wider viewpoint in two main respects. On the one hand, we go beyond the master equation approach addressing the question: how should we modify the way system and environment interact at a {\it microscopic} level in order to turn rare trajectories into typical as desired? On the other hand we go beyond continuous-time processes and address {\it discrete}-time quantum dynamics corresponding to a sequence of stochastic quantum maps on the open system.
To achieve the above, we use a quantum collision model (CM): in fact the natural and simplest microscopic framework for describing quantum trajectories and weak measurements \cite{brunSimple2002, altamirano_unitarity_2017, Gross2018, Ciccarello,Cilluffo2020Timebin, Seah_2019}.
In a CM [see Fig.~1], the system of interest unitarily interacts, in a sequential way, with a large collection of environmental subunits (or probes), which constitute a thermal bath. After the collision, each probe undergoes a projective measurement, whose result is recorded. The sequence of measurement outcomes (see Fig.~1) defines a quantum trajectory. Since the unitary collision correlates the system and the probe, measuring the latter changes the state of the system as well. This change is tiny in most cases but occasionally can be dramatic and culminate in a quantum jump.
Exploiting thermodynamic functionals, we characterize the ensemble of trajectories in CMs and show how the system-probe interaction can be modified so as to bias the statistics of measurement outcomes on the probes. Notably, this unveils the physical mechanism turning rare trajectories into typical. As will be shown, for short collision times, the modified dynamics is obtained by adding extra collisions which enforce the system dynamics far from the average one so as to sustain a trajectory with desired measurement outcomes.
\sigmaection{Collision model}
The environmental probes [see Fig.~\ref{fig1}] are labeled by $n=1,2,...,N$ and assumed to be non-interacting. Each is modeled as a qubit with states $\{\ket{0}, \ket{1}\}$ (note that quantum optics master equations and photo detection schemes are always describable in terms of qubit probes \cite{Wiseman, Gross2018}).
Each system-probe collision is described by the pairwise unitary
\begin{align}
U(H_S,V) =\exp[-i (H_S\otimes \mathbb{1}{+}V) \Delta t)],
\label{U-coll}
\end{align}
with $H_S$ the free Hamiltonian of system $S$ (generally including a drive) and $V$ the $S$-probe interaction Hamiltonian. Note that $U$ can be seen as a gate acting on system and probe \cite{scarani_thermalizing_2002, Ciccarello} according to an associated quantum-circuit representation (see Fig.~\ref{fig2}).
In the following, we assume that initially $S$ and the probes are in the uncorrelated state $\varrho_0 = \rho_0 \bigotimes_n \eta_n$ with $\rho_0$ ($\eta_n$) the initial state of $S$ (probe $n$). We will set $\eta_n=\ket{0}\!_n\langle 0|$ (the generalization to mixed state is straightforward).
Right after colliding with $S$, under the action of the unitary $U(H_S,V)$, each probe is measured in the basis $\{\ket{k_n}\}$ with $k=0,1$ [see Fig.~\ref{fig2}(a)]. In an atom-field setup
outcome $\ket{0}$ means no emission while $\ket{1}$ signals one photon emitted by $S$ and detected. The state of $S$ after $n$ steps, $\rho_n$, is the average over all possible discrete trajectories (unconditional dynamics). Between two subsequent steps, it evolves as $\rho_{n+1}=\mathcal{E}[\rho_{n}]$, where the map
\begin{equation}
\mathcal{E}[\rho]:=\sigmaum_{k=0}^1 K_k \rho K_k^\dag\, ,\quad \mbox{with} \quad K_k=\langle k|U(H_S,V)|0\rangle\, ,
\label{K-map}
\end{equation}
is completely positive and trace preserving (CPT). $K_k$ are the so called Kraus operators acting on $S$. In particular, trace preservation (equivalent to probability conservation) holds due to $\sigmaum_{k=0,1} K^\dagger_k K_k=\mathbb{1}$.
We take the linear system-probe coupling
\begin{align}
V=\tfrac{1}{\sigmaqrt{\Delta t}} (J \otimes \sigmaigma_{+}+J^\dag \otimes \sigmaigma_{-})\,,
\label{int_lin}
\end{align}
where $J$ is an operator on $S$ having the units of the square root of a frequency, and
$\sigmaigma_{-}=\sigmaigma_{+}^\dag=|0\rangle\!\langle 1|$.
In spite of its simplicity, this model of interaction describes a wide variety of representative physical situations \cite{Gross2018, Doherty_1999}.
Also, note that Eq.~ref{K-map} is independent of the probe label since so are $U$ and $\eta_n$.
\begin{figure}
\caption{{\bf Quantum circuits.}
\label{fig2}
\end{figure}
Being a sequence of identical CPT maps, the overall discrete dynamics of $S$ is Markovian and, in the limit of vanishing collision time, it reduces to the continuous-time Markovian open quantum dynamics described by the Lindblad master equation $\dot \rho=\mathcal{L}[\rho]$ with \cite{Lindblad1976,Gorini1976}
\begin{equation}
\mathcal{L}[\rho]=-i\left[H_S,\rho\right]+J\rho J^\dagger -\tfrac{1}{2}\left\{J^\dagger J, \rho \right\}\, .
\label{Lindblad}
\end{equation}
\sigmaection{Biased collisional trajectories}
In contrast to the average (deterministic) dynamics generated by Eq.~ref{K-map}, each specific quantum trajectory is composed of the specific measurement outcomes on the probes and is thus stochastic. At each step, the state of $S$ evolves as \cite{BrunPRA2000}
\begin{align}
\ket{\psi_{n+1}} = K_k \ket{\psi_n} / \|K_k \ket{\psi_n}\|
\end{align}
with $p_k =\| K_k \ket{\psi_n} \|^2$ being the probability to measure the $n$th probe in state $\ket{k}$ (we have assumed an initial pure state for the system, $\rho_0=|\psi_0\rangle\langle \psi_0|$, for the sake of argument).
Each $K_k$ is in one-to-one correspondence with a particular measurement outcome.
Here we focus on measurement outcomes described by operator $K_1$. Let then $P_N(M)$ be the probability of observing $M$ times the action of $K_1$ in a realization of the collision dynamics up to the discrete time $N$. For large $N$, this has the form \cite{touchette2009large,garrahan2010thermodynamics,StegmannPRB2018,Chetrite_2015}
\begin{align}
P_N(M)\sigmaim e^{-N\varphi(m)}\, ,
\end{align}
with $m=M/N$ being the frequency with which the probe has been measured in state $\ket{k=1}$. The (positive, semi-definite) function $\varphi$ is the so-called \textit{large deviation function}. It only vanishes when $m$ is equal to its typical value $\langle m\rangle$, i.e.,~the most likely to observe. This function fully characterizes the statistics of the random variable $M$. To obtain it, it is convenient to define the moment generating function (MGF) of the observable
\begin{align}
Z_N(s):=\sigmaum_{M=0}^\infty P_N(M)e^{-s\, M}\xrightarrow[N \gg 1]{} e^{N \theta(s)} \, ,
\label{mgf}
\end{align}
where the real variable $s$ is called ``counting field", and
$\theta(s)$ is the scaled cumulant generating function (SCGF) valid at stationarity for the observable $M$ \cite{touchette2009large}
\begin{align}
\theta(s):=\lim_{N\to\infty}\frac{1}{N} \log Z_N(s),
\label{scgf}
\end{align}
In line with the arguments of \cite{touchette2009large,garrahan2010thermodynamics}, the SCGF can be calculated as the logarithm of the largest real eigenvalue of a \textit{tilted} Kraus map [\cfEq.~ref{K-map}] (see the Appendix \cite{SM} for further details).
\begin{align}
\mathcal{E}_s [X] = K_0 X K_0 + e^{-s} K_1 X K_1 \, .\label{Kraus-s}
\end{align}
The map $\mathcal{E}_s$ does not represent a physical process, but is rather a mathematical tool that is of help to recover $\theta(s)$.
The (physical) process is retrieved for $s=0$.
The probability distribution $P_N(M)$ is determined by the behavior of $\theta(s)$ through derivatives with respect to $s$, taken at the ``physical point" $s=0$.
Yet, looking at Eq.~ref{mgf}, after normalizing by $Z_N(s)$, one can define a set of {\it biased} probabilities
\begin{align}
P_N^s(M)=\frac{e^{-s\, M}P_N(M)}{Z_N(s)}\, .\label{PNMs}
\end{align}
For $s>0$, these probabilities enhance occurrence of trajectories featuring smaller-than-typical values of $M$, while for $s<0$, instead, larger values of $M$ are favored \cite{garrahan2011quantum}.
Remarkably, these apparently fictitious probabilities in fact describe rare ensembles of trajectories of the original collision model \cite{PhysRevLett.111.120601}.
Cumulants of the biased probability distribution $P_N^s(M)$ can be determined through derivatives of $\theta(s)$ for values of $s$ different by zero; for instance, the rate of the measurement of probes in state $\ket{1}$ is, for $P_N^s(M)$, $\langle m \rangle_s=\tfrac{1}{N}\langle M \rangle_s = -\theta'(s)$.
\begin{figure}
\caption{\textbf{Turning biased trajectories into typical.}
\label{fig3bis}
\end{figure}
\sigmaection{Turning biased trajectories into typical}
So far we have constructed the probability distribution Eq.~ref{PNMs} by hand and noted how these actually describe rare dynamical events. Here, we show how to modify the system-probe collision in a way that $P_N^s(M)$ become instead physical probabilities. In other words we will show how, by tuning the interaction between system and probes, the rare behavior of
the original process can become the typical one of the
new dynamics (see Fig.~\ref{fig3bis}).
\begin{figure*}
\caption{{\bf Discrete-time quantum trajectories of a three-level system.}
\label{fig3}
\end{figure*}
As mentioned earlier $P_N^{{s}}(M)$ is generated by the tilted map $\mathcal{E}_{{s}}$ [\cfEq.~ref{Kraus-s}] which is {\it not} CPT (i.e., it does not represent a legitimate physical process) since probability is not preserved. The task is thus to turn $\mathcal{E}_{{s}}$ into a well-defined CPT map. This is achieved by introducing a \emph{Doob} transform of the dynamics \cite{10.1143/PTPS.184.304,garrahan2010thermodynamics, carollo2018making} for {\it discrete-time} quantum processes, embodied by the auxiliary CPT map (see the Appendix \cite{SM})
\begin{equation}
\tilde{\mathcal{E}}[X]=\tilde{K}_0X\tilde{K}_0^\dagger +\tilde{K}_1X\tilde{K}_1^\dagger\, .\label{doob-discrete}
\end{equation}
The explicit expression of the modified Kraus operators $\tilde{K}$ is given in \cite{SM}.
The probability distribution associated with the map $\tilde{\mathcal{E}}$ is exactly the desired one $P_N^{{s}}$ for long times.
Notably, for short collision times $\Delta t$, the replacement $\mathcal{E}\rightarrow \tilde{\mathcal{E}}$ [\cfEq.~ref{K-map}] is equivalent to changing the system-probe collision unitary as
\begin{equation}
U({H}_S,{V})\rightarrow U(\tilde{H}_S,\tilde{V})\,.
\label{change}
\end{equation}
Here the new Hamiltonian $\tilde{H}_S$ and jump operator $\tilde{J}$ match those obtained via the Doob transform for continuous-time Lindblad processes \cite{garrahan2010thermodynamics,carollo2018making,PhysRevLett.122.130605}. As a consequence, the new Kraus operators are
\begin{align}
\tilde{K}_k=\langle k |U(\tilde{H}_S,\tilde{V})|0\rangle\,.
\label{Kraus_D}
\end{align}
The new system-probe collision unitary Eq.~ref{change} can be decomposed as (see the Appendix)
\begin{equation}
U(\tilde{H}_S,\tilde{V})= U(H'_S,V')U(H_S,V)U(H'_S,V')\,,\label{3coll}
\end{equation}
where $H'_S=\tfrac{1}{2}(\tilde{H}_S{-}H_S)$ and $V'=\tfrac{1}{2}(\tilde{V}{-}V)$. The associated quantum circuit is shown in Fig.~\ref{fig2}(b). This decomposition makes apparent the mechanism by which rare events can be sustained so as to make them typical: {\it extra} unitary collisions, added to the original one $U(H_S,V)$, drive the system away from typicality, pinning its dynamical behavior to the fluctuations of interest.
Note also that the same task can be accomplished by a single additional collision according to
\begin{equation}
U(\tilde{H}_S,\tilde{V})= U(H''_S,V'')\,U(H_S,V)\,
\label{2-coll1}
\end{equation}
with
\begin{equation}
H''_S=2H'_S\,,\,\,\,V''=2V'+i\tfrac{\Delta t}{2}\,[\tilde{V},V]\,.
\label{2-coll}
\end{equation}
This is obtained from Eq.~ref{3coll} by swapping the last two unitaries and applying the Baker-Campbell-Hausdorff formula \cite{greiner2013field} to leading order.
Note that the second term in $V''$ (cf. Eq.~ref{2-coll}) is of order $\mathcal{O}(1)$ in $\Delta t$, and represents an extra system-prob coupling. Eq.~s Eq.~ref{2-coll1} and Eq.~ref{2-coll} hold for any collision time $\Delta t$.
\sigmaection{Driven three-level system.}
As an example, we discuss here a simple system, with rich dynamical behaviour which illustrates how our ideas can be exploited to investigate reduced-system discrete-time dynamics in metastable or prethermal regimes as well as to bias and drive such interesting dynamics.
Let $S$ be a coherently driven three-level system [see Fig.~\ref{fig3} (a)].
Each transition $|g\rangle \leftrightarrow|e_k\rangle$, with $k=1,2$, is driven with a Rabi frequency $\Omega_k$ according to the Hamiltonian
\begin{equation}
{H}_{S} = \sigmaum_{k} \Omega_k (\sigma^{(k)}_+ + \sigma^{(k)}_- ) \,,
\end{equation}
where $\sigma^{(k)}_{-} = |g\rangle_S \langle e_k| = \sigma^{(k)\,^\dag}_{+}$. For the sake of argument we assume the lasers to be in resonance with the atomic transitions. Additionally, we set $J=\sigmaqrt{\gamma}\,\sigma^{(1)}_-$ [\cfEq.~ref{int_lin}], meaning that only state $\ket{e_1}$ can decay with rate $\gamma$ by emitting an excitation into the environment (corresponding to outcome $\ket{1_n}$).
For short collision times, intermittent emission is known to occur \cite{PlenioJumpRMP,CookPRAIntermittency}, which can been explained as the coexistence of two deeply different phases of emission much like a first-order phase transition \cite{garaspects}.
Notably, the developed framework allows to investigate such transition-like behaviour away from the Lindblad dynamical regime, i.e., for {\it finite} collision times $\Delta t$.
The collision Hamiltonian reads
\begin{align}
V=\sigmaqrt{\tfrac{\gamma}{\Delta t}} (\sigma^{(1)}_- \otimes \sigmaigma_{+} + \sigma^{(1)}_+ \otimes \sigmaigma_{-})\,,
\label{int_ex}
\end{align}
thus through the biased map $\mathcal{E}_s$ (\cfEq.~ Eq.~ref{Kraus-s}) we work out the auxiliary map $\tilde{\mathcal{E}}$ and study the statistics of the quantum trajectories generated by quantum jump MonteCarlo.
To this end, we plot in Fig.~\ref{fig3}(b) the time-averaged rate of probe measurements in state $\ket{1}$, $\langle m\rangle/\Delta t = -\partial_s (\theta(s,\Delta t))/\Delta t$, as a function of $s$ and $\Delta t$ for $\Omega_1/\gamma=1$ and $\Omega_1/\Omega_2=1/10$.
This dynamical order parameter allows us to distinguish active (bright) and inactive (dark) trajectory regimes [some representative samples of quantum trajectories are shown in Fig.~ \ref{fig3}(c)].
The boundary line --clearly visible in Fig.~\ref{fig3}(b)-- represents a sharp crossover between the two dynamical regimes. Along this boundary, trajectories feature intermittent emission of excitations from the system.
As $\Delta t$ grows up, the crossover occurs at a different value of $s$ and its sharpness changes. Thus, away from the short-$\Delta t$ (Lindblad) regime, both typical and atypical emission rates are modified.
\sigmaection{Conclusions}
We presented a microscopic framework for the statistical characterization of quantum trajectories in discrete-time processes.
This provides a quantitative tool for studying dynamical fluctuations beyond the standard continuous-time regime corresponding to the Lindblad master equation.
A recipe was given allowing to turn rare quantum trajectories into typical upon addition of extra collisions between the system and each probe.
It is worth noting that this is reminiscent of a giant-atom dynamics (a giant atom couples to the field at two or more points \cite{Kockum5years}), which can indeed be described as cascaded collisions \cite{giovannettiMaster2012a,giovannettiMaster2012,LorenzoPRAflux} yet involving the same system $S$ \cite{Carollo2020CCC}. \\
While we have focussed on collisions of the form described in Eq.~ref{U-coll} and Eq.~ref{int_lin}, our results for discrete-time collision models do not depend on the specific form of the collision unitary. We note that also the interpretation of the Doob dynamics as a collision model with an additional collision should extend straightforwadly to such more general cases \cite{SM}.
We note that it is also possible to use this formalism to obtain the finite-time statistics of emissions for discrete-time quantum maps as well as their finite-time Doob transform. This can be done by following essentially the same steps used for the continuous-time case, as for instance done in \cite{carollo2018making} where, in order to obtain the finite-Doob dynamics, the continuous-time dynamics has been first discretized.
The method introduced here shows how to engineer open quantum dynamics in order to produce desired emission patterns, without the need for changing the detection/ post-selection scheme \cite{BudiniPRE2011}.
Moreover, the presented qubit-based protocol can be implemented with experimental quantum simulator platforms based on trapped ions \cite{Schindler_2013} or Rydberg atoms \cite{Browaeys_2020,Weimer_2010}.
\sigmaection{Acknowledgments}
F. Carollo~acknowledges support through a Teach@T\"ubingen Fellowship. IL~acknowledges support from EPSRC [Grant No.~EP/R04421X/1], from The Leverhulme Trust [Grant No.~RPG-2018-181] and from the ``Wissenschaftler-R\"uckkehrprogramm GSO/CZS" of the Carl-Zeiss-Stiftung and the German Scholars Organization e.V..
A. Carollo acknowledges support from the Government of the Russian Federation through Agreement No.~074-02-2018-330 (2).
We acknowledge support from MIUR through project PRIN Project 2017SRN-BRK QUSHIP.
The research leading to these results has received funding from the European Union’s H2020 research and innovation programme [Grant Agreement No. 800942 (ErBeStA)].
\onecolumngrid
\renewcommand\thesection{A\arabic{section}}
\renewcommand\theequation{A\arabic{equation}}
\renewcommand\thefigure{S\arabic{figure}}
\sigmaetcounter{equation}{0}
\sigmaetcounter{figure}{0}
\appendix
\sigmaection*{Appendix}
\sigmaubsection{Doob transform of the discrete process.}
Although the large deviation function $\varphi(m)$ encompasses full information about the asymptotic behavior of the probability distribution $P_N(M)$, it is not, in general, easy to access by direct calculation.
The diagonalization of tilted map $\mathcal{E}_{s}$ (\cf Eq.~ref{Kraus-s}) does the job, providing the SCGF $\theta(s)$ (\cf Eq.~ref{scgf}) as the logarithm of its maximum eigenvalue that we name $\Lambda_s := e^{\theta(s)}$.
$\theta(s)$ captures the asymptotic behavior of cumulant generating function $\log Z_N(s)$, and is linked to $\varphi(m)$ through the Legendre-Fenchel transform \cite{touchette2009large}
\begin{align}
\varphi(m)=-\max_{\forall s}\left[m\, s + \theta(s)\right]\, .
\end{align}
Furthermore $\mathcal{E}_{s}$ biases the original probabilities $P_N(M)$ through an exponential factor.
By exploiting the partition function (MGF) $Z_N(s)$ we can thus define the tilted probability
\begin{align}
P_N^{s}(M)=\frac{e^{-s M}P_N(M)}{Z_N(s)}\, .
\end{align}
This ensemble --so-called $s$-ensemble-- contains information about the properties and the dynamical features associated with a rare event of the originial process. However, $\mathcal{E}_{s}$ is not a well-defined quantum discrete dynamics since the dual map does not preserve the identity, $\mathcal{E}_{s}^*[{\mathbb 1}]\neq{\mathbb 1}$.
Nonetheless, as we now demonstrate, it is possible to transform this tilted into a proper dynamics, which reproduces as typical the rare outcomes of the original collision model $\mathcal{E}$. We can obtain this dynamics as follows. Suppose we are interested in the behaviour of the system associated with probabilities $P_N^{s}(M)$. Then, we can define the (Doob) quantum discrete quantum
\begin{equation}
\tilde{\mathcal{E}}[\rho]=\sigmaum_{k=0,1}\tilde{K}_k \rho \tilde{K}_k^\dagger \, , \quad \mbox{ where }\quad \tilde{K}_0 =\frac{1}{\Lambda_s^{1/2}}\ell^{1/2}K_0\ell^{-1/2} \qquad \mbox{ and } \quad \tilde{K}_1 =\frac{e^{-s/2}}{\Lambda_s^{1/2}}\ell^{1/2}K_1\ell^{-1/2}\, .
\label{discrete-Doob}
\end{equation}
Here we have that $\ell$ is the left eigen-operator of the tilted map $\mathcal{E}_s$ associated with its eigenvalue with largest real part $\Lambda_s = e^{\theta(s)}$. Namely, $\ell$ is the operator such that
\begin{align}
\mathcal{E}_s^*[\ell]=\Lambda_s \, \ell\, .
\end{align}
The map $\tilde{\mathcal{E}}$ is completely positive and we also have that $\tilde{\mathcal{E}}^*[{\mathbb 1}]={\mathbb 1}$. The latter equality follows from
\begin{align}
\tilde{\mathcal{E}}^*[{\mathbb 1}]=\frac{1}{\Lambda_{s}}\ell^{-1/2}\, \mathcal{E}_s^* [\ell]\ell^{-1/2}=\frac{1}{\Lambda_s}\ell^{-1/2}\left(\Lambda_{s}\ell\right)\ell^{-1/2}={\mathbb 1}\, .
\end{align}
Because of these properties, the map in Eq.~ref{discrete-Doob} is a proper discrete quantum dynamics and, as we have discussed, reproduces as typical the rare event of the original processes $P_N^{s}(M)$.
\sigmaubsection{Doob transform in the collision model.}
In the above section, we have demonstrate how to obtain the Doob dynamics of a discrete-time quantum process. Here, instead, we want to consider that our initial dynamics describes a collision model, meaning that $\Delta t\to 0$. Using this fact, we show how the Doob dynamics is in fact a new collision model with effective Hamiltonian and jump operator which coincides with those of the Doob transform for the continuous-time Lindblad case \cite{carollo2018making}.
First of all, we notice that in the collision model limit, $\Delta t\ll1$, the tilted Kraus map is approximately given by
\begin{align}
\mathcal{E}_s[\rho]\approx e^{\Delta t \, \mathcal{L}_s} [\rho]\, ,
\end{align}
where $\mathcal{L}_s$ is the tilted Lindblad operator \cite{garrahan2010thermodynamics,carollo2018making}
\begin{align}
\mathcal{L}_{s}[\rho]=-i[H_S,\rho]+e^{-s}J\rho J^\dagger -\frac{1}{2}\left\{\rho,J^\dagger J\right\}\, .
\end{align}
As such, the left eigen-operator of $\mathcal{L}_s$ is approximately also the eigen-operator of $\mathcal{E}_{ s}$, $\ell$, at first-order in $\Delta t$. This also implies that the largest real eigenvalue of the tilted map can be written as
\begin{align}
\Lambda_s\approx e^{\Delta t\, \chi(s)}\, ,
\end{align}
where $\chi(s)$ is given by the largest real eigenvalue of the tilted Lindbladian map $\mathcal{L}_s$.
We can now focus on the Doob transform in Eq.~ref{discrete-Doob} and consider the small collision-time limit. The second term on the right hand side is thus equivalent to
\begin{align}
\tilde{K}_1\rho \tilde{K}_1^\dagger \approx \frac{e^{-s}}{e^{\Delta t\, \chi(s)}}\Delta t\, \ell^{1/2} J \ell^{-1/2}\, \rho \, \ell^{-1/2} \tilde J^\dagger \ell^{1/2} \approx \Delta t\, \tilde J \rho \tilde J^\dagger\, ;
\end{align}
here $\tilde{J}=e^{-s/2}\ell^{1/2}J\ell^{-1/2}$ (which corresponds to the jump operator of the continuous time Doob dynamics \cite{carollo2018making}) and the last term ${e^{\Delta t\, \chi(s)}}$ only contributes at the zero-th order in $\Delta t$.
Considering the first term on the right hand side of Eq.~ref{discrete-Doob}, up to first order in $\Delta t$, we obtain
\begin{align}
\tilde{K}_0\rho\tilde{K}_0^\dagger \approx 1+\left[-i\ell^{1/2}H_{\rm eff}\ell^{-1/2}\rho+i\rho\ell^{-1/2}H_{\rm eff}^\dagger \ell^{1/2}-\chi(s)\rho\right]\Delta t\, ,
\end{align}
and this, with similar computation as those done in Ref.~\cite{carollo2018making} gives
\begin{align}
\tilde{K}_0\rho\tilde{K}_0^\dagger \approx 1 -i\left(\tilde{H}_S-\frac{i}{2}\tilde{J}^\dagger \tilde{J} \right)\rho\Delta t+i \rho \left(\tilde{H}_S+\frac{i}{2}\tilde{J}^\dagger \tilde{J}\right)\Delta t\, ,
\end{align}
where $\tilde{H}_S$ coincide with the Hamiltonian of the continuous time Doob dynamics
\begin{align}
\tilde{H}_S &= \frac{1}{2} \ell^{1/2} \left(
H - \frac{i}{2} J^\dag J
\right) \ell^{-1/2} + {\rm H.c.}\,.
\end{align}
In light of this result, we can write the unitary interaction between system and probe as a new collision model with
\begin{align}
U(\tilde{H}_S,\tilde{V})=\exp[-i (\tilde{H}_S\otimes {\mathbb 1}{+}\tilde{V}) \Delta t)]\, ,
\end{align}
and
\begin{align}
\tilde{V}=\frac{1}{\sigmaqrt{\Delta t}} (\tilde{J}\otimes \sigmaigma_{+}+\tilde{J}^\dag \otimes \sigmaigma_{-})\, .
\end{align}
\sigmaubsection{The Doob collision dynamics as a three-collision model.}
In this section we show how it is possible to write the Doob dynamics as a collision model where system and probe collide three times with one collision being exactly the original one.
To show this we start by writing the Doob unitary collision with the $n$-th ancilla as
\begin{align}
U(\tilde{H}_S,\tilde{V})=\exp\left[-i(A+B)\right]\, ,
\end{align}
where we have
\begin{align}
A=\Delta t\left(H_S\otimes {\mathbb 1}+V\right)\, , \qquad \mbox{and}\qquad B=\Delta t\left(\tilde{H}_S\otimes{\mathbb 1}-H_S\otimes{\mathbb 1}+\tilde{V}-V\right)\, .
\end{align}
Due to the fact that we are interested in the regime $\Delta t\ll1$, we just need to preserve the terms of the unitary operator $U(\tilde{H}_S,\tilde{V})$ only up to first order in $\Delta t$. Recalling that terms $V \Delta t$ are actually of order $\sigmaqrt{\Delta t}$ this means it is sufficient to guarantee that the unitary $U(\tilde{H}_S,\tilde{V})$ is preserved up to the second order products of $A,B$. A possible decomposition is thus given by the second-order Trotter decomposition
\begin{align}
U(\tilde{H}_S,\tilde{V})=\exp(-iB/2)\exp(-iA)\exp(-iB/2)+o(\Delta t)\, .
\end{align}
Noticing that $\exp(-iA)=U(H_S,V)$, and since we can define
\begin{align}
\exp(-iB/2)=U(H'_S,V')\, , \quad \mbox{ with } \quad H'_S=(\tilde{H}_S-H_S)/2\, \quad \mbox{ and }\quad V'=(\tilde{V}-V)/2\, ,
\end{align}
we are allowed to write the Doob collision as
\begin{align}
U(\tilde{H}_S,\tilde{V})\approx U(H'_S,V')U(H_S,V)U(H'_S,V')\, .
\end{align}
\sigmaubsection{Doob dynamics as a two-collision model for finite collision time}
In this section we show that, also for the case of finite collision time, it is possible to have an interpretation of the Doob dynamics in Eq.~ref{doob-discrete}, as a collision dynamics with one additional unitary collision.
As shown in the main text, the Doob dynamics for generic discrete-time processes is given by
$$
\tilde{\mathcal{E}}[X]=\sigmaum_{k=0,1}\tilde{K}_kX\tilde{K}_k\, .
$$
By Stinespring dilation theorem, it is possible to interpret these operators as
$$
\tilde{K}_k=\bra{k}\tilde{U}\ket{0}\, ,
$$
where $\tilde{U}$ is a suitable unitary collision between system and bath. With $U$ being the original collision of the process, one can always write
$$
\tilde{U}=W_1U=UW_2\, ,
$$
where $W_1=\tilde{U}U^\dagger$ and $W_2=U^\dagger\tilde{U}$. This means that we can interpret the unitary interaction of the Doob process $\tilde{U}$, as a the sequence of two collisions involving the original one and an extra one, which is sustaining as typical the rare behaviour of the original dynamics.
\end{document} |
\begin{document}
\title{Local Gradient Estimates for Second-Order Nonlinear Elliptic and Parabolic Equations by the Weak Bernstein's Method }
\author{G.Barles\thanks{Institut Denis Poisson (UMR CNRS 7013)
Université de Tours, Université d'Orléans, CNRS. Parc de Grandmont
37200 Tours, France. Email: [email protected] \newline
\indent This work was partially supported by the project ANR MFG (ANR-16-CE40-0015-01) funded by the French National Research Agency } }
\maketitle
\noindent {\bf Key-words}: Second-order elliptic and parabolic equations, gradient bounds, weak Bernstein's method, viscosity solutions.
\\
{\bf MSC}:
35D10
35D40,
35J15
35K10
\begin{abstract}
{\footnotesize In the theory of second-order, nonlinear elliptic and parabolic equations, obtaining local or global gradient bounds is often a key step for proving the existence of solutions but it may be even more useful in many applications, for example to singular perturbations problems. The classical Bernstein's method is a well-known tool to obtain these bounds but, in most cases, it has the defect of providing only a priori estimates. The ``weak Bernstein's method'', based on viscosity solutions' theory, is an alternative way to prove the global Lipschitz regularity of solutions together with some estimates but it is not so easy to perform in the case of local bounds. The aim of this paper is to provide an extension of the ``weak Bernstein's method'' which allows to prove local gradient bounds with reasonnable technicalities.}
\end{abstract}
\maketitle
The classical Bernstein's method is a well-known tool for obtaining gradient estimates for solutions of second-order, elliptic and parabolic equations (cf. Caffarelli and Cabr\'e \cite{CClivre} Gilbarg and Trudinger\cite{GT} (Chap. 15) and Lions\cite{LB}). The underlying idea is very simple: if $\Omega$ is a domain in $\mathbb R^N$ and $u : \Omega \to \mathbb R$ is a smooth solution of
$$ -\Delta u = 0 \quad\hbox{in }\Omega\; ,$$
where $\Delta$ denotes the Laplacian in $\mathbb R^N$, then $w:=|Du|^2$ satisfies
$$ -\Delta w \leq 0 \quad\hbox{in }\Omega\; .$$
The gradient bounded is deduced from this property by using the Maximum Principle if one knows that $Du$ is bounded on $\partial \Omega$ and this bound on the boundary is usually the consequence of the existence of barriers functions.
Of course this strategy, consisting in showing that $w:=|Du|^2$ is a {\em subsolution} of an elliptic equation and then using the
Maximum Principle, can be applied to far more general equations but it has a clear defect: in order to justify the above
computations, the solution has to be $C^3$ and, since it is rare that the solution has such a regularity, the classical Bernstein's method provides, in general, only {\em a priori estimates}; then one has to find a suitable approximation of the equation, with smooth enough solutions, to actually obtain the gradient bound.
In 1990, this difficulty was partially overcomed by the weak Bernstein's method whose idea is even simpler: if one looks at the maximum of the function
$$(x,y) \mapsto u(x)-u(y)-L|x-y| \quad\hbox{in }\overline \Omega \times \overline \Omega\; ,$$
and if one can prove that it is achieved only for $x=y$ for $L$ large enough, then $|Du|\leq L$. Surprisingly, as it is explained in the introduction of \cite{B-wb}, the computations and structure conditions which are needed to obtain this bound are the same (or almost the same with tiny differences) as for the classical Bernstein's method. Of course, the main advantage of the weak Bernstein's method is that it does not require $u$ to be smooth since there is no differentiation of $u$ and it can even be used in the framework of viscosity solutions.
Problem solved? Not completely because the weak Bernstein's method is not of an easy use if one looks for local bounds instead of global bounds. In fact, in order to get such local gradient bounds, the only possible way seems to multiply the solution by a cut-off function and to look for a gradient bound for this new function. Unfortunately, this new function satisfies a rather complicated equation where the derivatives of the cut-off function appear at different places and the computations become rather technical. The classical Bernstein's method also faces similar difficulties but, at least in some cases, succeeds in providing these local bounds in a not too complicated way.
The aim of this article is to describe a slight improvement of the weak Bernstein's method which allows to obtain local gradient bounds in a simpler way, ``simpler'' meaning that the technicalities are as reduced as possible, although some are unavoidable. This improvement is based on an idea of P.~Cardaliaguet~\cite{C1} which dramatically simplifies a matrix analysis which is keystone in \cite{B-wb} but also allows this extension to local bounds.
To present our result, we consider second-order, possibly degenerate, elliptic equations which we write in the general form
\begin{equation}\label{GFNLE}
F(x, u, D u, D^2u) = 0 \quad\hbox{in }\Omega\; ,
\end{equation}
where $\Omega$ is a domain of $\mathbb R^N$ and $F :\Omega \times \mathbb R \times \mathbb R^N \times {\mathcal S}^N \to \mathbb R $ is a locally Lipschitz continuous function, ${\mathcal S}^N$ denotes the space of $N \times N$ symmetric matrices, the solution $u$ is a real-valued function defined on $\Omega$, $Du, D^2u$ denote respectively its gradient and Hessian matrix. We assume that $F$ satisfies the (degenerate) ellipticity condition : for any $(x,r,p)\in\Omega \times \mathbb R \times \mathbb R^N$ and for any $X,Y\in{\mathcal S}^N$,
$$
F(x,r,p,X) \leq F(x,r,p,Y)\quad\hbox{if }X\geq Y.
$$
Our results consist in providing several general ``structure conditions'' on $F$ under which one has a local gradient bound
depending or not on the local oscillation of $u$ and the uniform ellipticity of the equation. We also consider the parabolic case
for which we give a structure condition on the equation allowing to prove a local gradient bound, depending on the local oscillation of
$u$, where ``local'' means both in space and time.
In the stationary framework, we focus in particular on the following example
\begin{equation}\label{PartEqn}
-\Delta u + |Du|^m = f(x) \quad\hbox{in }\Omega\; ,
\end{equation}
where $m>1$ and $f \in W^{1,\infty}_{loc}(\Omega)$, which is a particular case for which the classical Bernstein's method
provides local bound (independent of the oscillation of $u$) in a rather easy way, while it is not the case for the weak Bernstein's method.
We conclude this introduction by two remarks: the first one concerns the ``structure conditions'' on $F$ on which our results are based.
In \cite{B-wb}, it is pointed out that, in general, the equation we consider does not satisfy these structure conditions and we have to make a
change of unknown function $v=\psi(u)$, choosing $\psi$ in order that the new equation for $v$ satisfies them. Obviously,
the same remark is true here and we provide an example where such a change allows to obtain the desired gradient bound. But, contrarily to
\cite{B-wb}, we are not going to study the effect of such changes in a more systematic way.
The second remark concerns the method we are going to present: the results we obtain are based on several choices we made at several
places and, in particular, in the estimates of the terms we have to handle. Clearly, many variants are possible and we have just tried to
convince the reader that,
actually, the technicalities are really ``reasonnable'' as we pretend it in the abstract.\\
\noindent{\bf Acknowledgement:} the author would like to thank the anonymous referees whose remarks led to significant improvements of the readability of this article.
\section{Some preliminary results}\label{prelim}
In this section, we are going to construct the functions we use in the proof of our main result. To do so, we introduce $\mathcal{K}$ which is the class of continuous functions $\chi:[0,+\infty)\to [0,+\infty)$ such that $\chi(t)=0$ if $t\leq 1$, $\chi$ is increasing on $[1,+\infty[$, $\chi(t)\leq {\tilde K}(\chi)t^\beta$ for $t\geq 1$, for some $0<\beta < 1/2$ and some constant ${\tilde K}(\chi)>0$, and
$$ \int_1^{+\infty}\frac{dt}{t\chi(t)}<+\infty .$$
The first ingredient we use below is a smooth function $\varphi : [0,1[ \to \mathbb R$ such that $\varphi(0)=0$, $\varphi'(0)=1 \leq \varphi'(t)$ for any $t\in [0,1[$ with $\varphi(t) \to +\infty$ as $t\to 1^-$ and which solves the ode $\varphi''(t)= K_1\varphi'(t) \chi(\varphi'(t))$ for some constant $K_1>0$. In fact the existence of such function is classical using that
$$ \int_1^{\varphi'(t)} \frac{ds}{s\chi(s)} = K_1 t\, ,$$
and by choosing $K_1=\int_1^{+\infty} \frac{ds}{s\chi(s)}$ we already see that $\varphi' (t) \to +\infty$ as $t\to 1^-$. Moreover
$$ \int_{\varphi'(t)}^{+\infty} \frac{ds}{s\chi(s)} = K_1 (1-t) \; ,$$
and therefore, for $t$ close enough to $1$
$$ K_1 (1-t) \geq [{\tilde K}(\chi)]^{-1}\int_{\varphi'(t)}^{+\infty} \frac{ds}{s^{1+\beta}}= [{\tilde K}(\chi)\beta]^{-1}\varphi'(t)^{-\beta}\; .$$
This means that
$$\varphi'(t) \geq \left(\frac{K_1 (1-t)}{[{\tilde K}(\chi)\beta]^{-1}}\right)^{-1/\beta} \; ,$$
and therefore $\varphi'(t)$ is not integrable at $1$ since $1/\beta>2$. Hence we have $\varphi(t) \to +\infty$ as $t\to 1^-$.
On the other hand, given $x_0 \in \mathbb R^N$ and $R>0$, we use below a smooth function $C: B(x_0,3R/4) \to \mathbb R$ is a smooth function such that $C(z)= 1$ on $B(x_0,R/4)$, $C(z) \geq 1$ in $ B(x_0,3R/4)$ and $C(z)\to +\infty$ when $z\to \partial B(x_0,3R/4)$ and with
$$ \frac{|D^2C(x)|}{C(x)} , \frac{|DC(x)|^2}{[C(x)]^2} \leq K_2(R) [\chi(C(x))]^2\; ,$$ where $\chi$ is a function in the class $\mathcal{K}$. If $C_1$ is a function which satisfies the above properties for $x_0=0$ and $R=1$, we see that we can choose $C$ as
$$ C(x)=C_1\left(\frac{x-x_0} R\right)\; ,$$
and therefore $K_2(R)$ behaves like $R^{-2}K_2(1)$.
To build $C_1$, we first solve
$$ \psi'' (t) = K_3 \psi (t)[\chi(\psi (t))]^2, \; \psi(0)=1,\; \psi'(0)=0\; ,$$
for some constant $K_3$ to be chosen later on.
Multiplying the equation by $2 \psi'(t)$, we obtain that
$$ \psi'(t) = F(\psi(t))\; ,$$
where
$$ [F(\tau)]^2= 2K_3 \int_1^\tau s[\chi(s)]^2 ds\; .$$
Again we look for a function $\psi$ such that $\psi(t) \to +\infty$ as $t\to 1^{-}$ and to do so, the following condition should hold
$$ \int_1^{+\infty} \frac{d\tau}{F(\tau)} < +\infty\; .$$
But, since $\chi$ is increasing,
$$ [F(\tau)]^2 \geq 2K_3 \int_{\tau/2}^\tau s[\chi(s)]^2 ds\geq \; 2K_3 [\tau/2 \chi(\tau/2)]^2 ,$$
and since $\tau \mapsto \chi(\tau/2)$ is in $\mathcal{K}$, we have the result for $F$, and then for $\psi$ by choosing appropriately the constant $K_3$.
Moreover
$$ [F(\tau)]^2 \leq 2K_3 (\tau-1) \tau[\chi(\tau)]^2 \leq 2K_3 [\tau \chi(\tau)]^2 \; ,$$
and therefore
$$ \psi'(t) \leq (2K_3)^{1/2} \psi(t) \chi(\psi(t))\; .$$
Finally, we can extend $\psi$ by setting $\psi(t)=1$ for $t\leq 0$ and the equations satisfied by $\psi$ show that we define in that way a $C^2$-function on $(-\infty,1)$.
With such a $\psi$, the construction of $C_1$ is easy, we may choose
$$ C_1(x):= \psi \bigl(4(|x| - 1/2)\bigr)\quad \hbox{for }x\in B(0,3/4) ,$$
and define $C$ from $C_1$ as above. We notice that, because of the properties of $\psi$, $\dfrac{|DC(x)|}{[C(x)]^2}$ remains bounded on
$B(x_0,3R/4)$ and is a $O(R^{-1})$, a property that we will use later on.
\section{The Main Result}
In the statement of our main result below, for the sake of clarity, we are going to drop the arguments of the partial derivatives of $F$ and
to simply denote by $F_s$ the quantity $\dfrac{\partial F}{\partial s} (x,r,p,M)$ for $s=x,r,p,M$. Actually these arguments are $(x,r,p,M)$ everywhere.
Our result is the following
\begin{theorem} \label{main}Assume that $F$ is a locally Lipschitz function in $\Omega \times \mathbb R \times \mathbb R^N \times {\mathcal S}^N \to \mathbb R$ which satisfies : $F(x,r,p,M)$ is Lipschitz continuous in $M$ and
$$
F_M(x,r,p,M) \leq 0 \;\hbox{and}\; F_r(x,r,p,M) \geq 0\quad\hbox{a.e. in }\Omega \times \mathbb R \times \mathbb R^N \times {\mathcal S}^N\; ,$$
and let $u\in C(\Omega)$ be a solution of (\ref{GFNLE}).\\
(i) {\bf (Uniformly elliptic equation with coercive gradient dependence: estimates which are independant of the oscillation of $u$)} Assume that there exist a function $\chi \in \mathcal{K}$ and $0<\eta\leq1$ such that, for any $K>0$, there exists $L= L(F,K)$ large enough such that
$$ -(1+\eta)|F_x| |p| (1+K\chi(\eta |p|)) - K |F_p| |p|^2 \left(1+K\chi(\eta |p| )\right) \chi(\eta |p| ) - \dfrac1{1+\eta}F_M\cdot M^2 $$
$$ \geq \eta + K \bigl( |p| \left(1+K\chi(\eta |p| )\right) \chi(\eta |p| )\bigr)^2 \; \hbox{a.e.},
$$
in the set $$\{(x,r,p,M);\ |F(x,r,p,M))| \leq K \eta |p|[1 + K\chi(\eta|p|)]+\eta\; ,\; |p|\geq L\}\; .$$
If $\overline{B(x_0,R)} \subset \Omega$ then $u$ is Lipschitz continuous in $B(x_0,R/2)$ and $|Du| \leq {\bar L} $ in $B(x_0,R/2)$ where $\bar L$ depends only on $F$ and $R$.\\
(ii) {\bf (Uniformly elliptic equation with coercive gradient dependence: estimates depending the oscillation of $u$)} Assume that there exist a function $\chi \in \mathcal{K}$ and $0<\eta\leq1$ small enough such that, for any $K>0$, there exists $L= L(F,K)$ large enough such that
$$-(1+\eta) |F_x||p| - K|F_p| |p|^2\chi(\eta |p|) - \frac{1}{1+\eta} F_M\cdot M^2 \geq \eta
+ K |p|^2\chi(\eta |p|)^{2}\; \hbox{a.e.},$$
in the set $\{(x,r,p,M);\ |F(x,r,p,M))| \leq K |p|+\eta \; ,\; |p|\geq L\}$. If $\overline{B(x_0,R)} \subset \Omega$ then $u$ is Lipschitz continuous in
$B(x_0,R/2)$ and $|Du| \leq {\bar L} $ in $B(x_0,R/2)$ where $\bar L$ depends on $F$, $R$ and $osc_R (u)$, the oscillation of $u$ on $
\overline{B(x_0,R)}$.\\
(iii) {\bf (Non-uniformly elliptic equation : estimates depending the oscillation of $u$)} Assume that there exist a function $\chi \in \mathcal{K}$
and $0< \eta \leq 1$ small enough such that, for any $K>0$, there exists $L=L(F,K)$ large enough such that
$$ -(1+\eta) |F_x| |p| +(1-\eta)^2 F_r|p|^2 - K|F_p| |p|^2\chi(\eta |p|)- \frac{1}{1+\eta} F_M\cdot M^2$$
$$ \geq \eta + K |p|^2\chi(\eta |p|)^{2}\; \hbox{a.e.},$$
in the set $\{(x,r,p,M);\ |F(x,r,p,M))| \leq K |p|+\eta \; ,\; |p|\geq L\}$. If $\overline{B(x_0,R)} \subset \Omega$ then $u$ is Lipschitz continuous in $B(x_0,R/2)$ and $|Du| \leq {\bar L} $ in $B(x_0,R/2)$ where $\bar L$ depends on $F$, $R$ and $osc_R (u)$.
\end{theorem}
{ As an application we consider Equation (\ref{PartEqn}): in order to have a gradient estimate which is independant of the oscillation of $u$, i.e. Result (i) in Theorem~\ref{main},
the idea is to choose $\chi(t)=(t-1)^\beta$ for $t\geq 1$ with $0<\beta <1/2$ and $\gamma:=1+2\beta < m$. The most important point is that, for large $|p|$, the constraint on $F$ reads
$$|F(x,r,p,M))| \leq K\eta |p|(1+ K(\eta |p|)^{\beta})+\eta$$ and therefore $|F(x,r,p,M))|$ behaves as $K^2(\eta |p|)^{1+\beta}$ if $|p|$ is
large enough. Since $1+\beta<m$, this implies that, for such $(x,r,p,M)$,
$$ {\rm Tr}(M)\geq \frac12 |p|^m - ||f||_{L^{\infty}(B(x_0,R)}\; .$$
But, by Cauchy-Schwarz inequality
$$ {\rm Tr}(M)\leq C(N)[{\rm Tr}(M^2)]^{1/2}\; .$$
Therefore the term $-F_M\cdot M^2$ behaves like $|p|^{2m}$. For the other terms, we have, for large $|p|$
\begin{enumerate}
\item the term $|F_x| |p| (1+K\chi(\eta |p|)) $ behaves like $|p|^{1+\beta}=|p|^{\gamma-\beta}$;
\item the term $|F_p| |p|^2 \left(1+K\chi(\eta |p| )\right) \chi(\eta |p| )$ behaves like $|p|^{m+1+2\beta}=|p|^{m+\gamma}$;
\item the term $K ||F_M||_\infty \bigl( |p| \left(1+K\chi(\eta |p| )\right) \chi(\eta |p| )\bigr) ^2$ behaves like $|p|^{2(1+2\beta)}=|p|^{2\gamma}$.
\end{enumerate}
Since $\gamma < m$, the term $-F_M\cdot M^2$ clearly dominates all the other terms as $|p|$ tends to $+\infty$; therefore we have the gradient bound since the assumption holds for any $0<\eta \leq 1$. Moreover the classical case ($m=1$) can be also treated under the assumptions of Result~(ii).
In this example, it is also clear that we can replace the term $|Du|^m$ by a term $H(Du)$ where $H$ satisfies: there exists $\chi \in \mathcal{K}$ such that
$$\frac{|p|\chi(|p|)}{H(p)}\to 0 \quad\hbox{as } |p|\to +\infty\; , $$
and
$$ \frac{|H_p|(|p|\chi(|p|))^2}{[H(p)]^2}\to 0 \quad\hbox{as } |p|\to +\infty\; .$$}
\ \\
In the case of non-uniformly elliptic equation, the gradient bound comes necessarely from the $F_r|p|^2$-term. We consider the equation
\begin{equation}\label{PartEqnNUN}
-{\rm Tr}(A(x)D^2 u) + |Du|^m = f(x) \quad\hbox{in }\Omega\; ,
\end{equation}
where $m>1$ and $f $ is locally bounded and Lipschitz continuous; concerning $A$, we use the classical assumption: $A(x)=\sigma(x)\cdot \sigma^T(x)$ for some bounded, Lipschitz continuous function $\sigma$, where $\sigma^T(x)$ denotes the transpose matrix of $\sigma(x)$.
In order to obtain a local gradient bound for $u$, a change of variable is necessary: assuming (without loss of generality) that $u\geq 1$ at least in the ball $\overline{B(x_0,R)}$, we can use the change $u=\exp(v)$. The equation satisfied by $v$ is
$$
-{\rm Tr}(A(x)D^2 v) +A(x)Dv\cdot Dv+ \exp((m-1)v)|Dv|^m = \exp(-v)f(x) \quad\hbox{in }\Omega\; ,
$$
And the aim is now to apply Theorem~\ref{main}-(iii) to get the gradient bound for $v$ (hence for $u$).
The computation of the different terms gives
$$ F_r(x,r,p,M)= (m-1)\exp((m-1)r)|p|^m + \exp(-r)f(x)\; ,$$
$$ F_x(x,r,p,M)= -{\rm Tr}(A_x(x)M)+A_x(x)p\cdot p-\exp(-r)f_x (x)$$
$$F_p(x,r,p,M)= 2A(x)p +\exp((m-1)r)|p|^{m-2}p\; ,$$
$$ - F_M(x,r,p,M)M^2= {\rm Tr}(A(x)M^2)\; .$$
We first use Cauchy-Schwarz inequality and the assumption on $A$ to deduce that, for any $\eta>0$
$$ |{\rm Tr}(A_x(x) M)| |p|\leq \frac{1}{1+\eta} {\rm Tr}(A(x)M^2)+ O((|\sigma_x||p|)^2)\; ;$$
This control of the first term in $F_x(x,v,p,M)$ is the only use of the term $- F_M(x,v,p,M)M^2$ .
Therefore the $F_r(x,r,p,M)|p|^2$-term which behaves like $|p|^{m+2}$ if $m>1$, has to control the terms
$$ (A_x(x)\cdot p)(p\cdot p)=O(|p|^3)\; ,\; -\exp(-v)f_x (x) |p|=O(|p|)\; ,\; 2A(x)p\cdot p =O(|p|^2)\; .$$
We have now to consider the $F_p$-term and the term $K \bigl(|p|\chi (\eta |p|)\bigr)^{2}$ in the right-hand side. Notice that, for the time being, we have not chosen $\chi$ nor $\eta$.
The $F_p$-term behaves as $|p|^{\max(1,m-1)}$ and therefore $|F_p| |p|^2\chi(\eta |p|)$ behaves as $|p|^{\max(3,m+1)}\chi(\eta |p|)$.
On the other hand, $K \bigl(|p|\chi (\eta |p|)\bigr)^{2}$ behaves as $|p|^2[\chi(\eta |p|)]^2$. If we choose any $\chi \in \mathcal{K}$, because of the growth of such $\chi$ at infinity, these two terms are controlled by the $F_r|p|^2$-one. Therefore Theorem~\ref{main} (iii) applies.
It is worth pointing out that, in this last example, we do not use the fact that the assumption has to hold only
in the set $\{(x,r,p,M);\ |F(x,r,p,M))| \leq K |p|+\eta \; ,\; |p|\geq \bar L\}$, a fact which is going to be (almost) the general case in the parabolic setting.
{
\section{Proof of Theorem~\ref{main}}
We start by proving (i) : the aim is to prove that, for any $x\in B(x_0,R/4)$, $D^+u(x)$ is bounded with an explicit bound. This will provide the desired gradient bound. We recall that $$ D^+u(x)=\{p\in\mathbb R^n:\ u(x+h)\leq u(x)+p\cdot h+o(|h|) \ \text{ as } h\to 0\}.$$
To do so, we consider on
$$\Gamma_L :=\{(x,y) \in B(x_0,3R/4) \times B(x_0,R) : LC(x)(|x-y| +\alpha)<1\}$$
the following function
$$ \Phi(x,y)= u(x)-u(y) - \varphi\left (LC(x)(|x-y| +\alpha)\right )\; ,$$
where
\begin{itemize}
\item $L\geq \max(1,4/R)$ is a constant which is our future gradient bound (and therefore which has to be choosen large enough),
\item the functions $\varphi$ and $C$ are built in Section~\ref{prelim},
\item $\alpha >0$ is a small constant devoted to tend to $0$.
\end{itemize}
We remark that the above function achieves its maximum in the open set $\Gamma_L$: indeed, if $(x,y) \in \Gamma_L$, we have $ LC(x)\alpha<1$ and therefore $x\in \overline{B(x_0,R')}$ for some $R'<3R/4$. Moreover $LC(x)|x-y|<1$ implies $|x-y|<L^{-1}$ and, since $L> 4/R$, this implies $y\in \overline{B(x_0,R'+R/4)}$ and $R'+R/4<R$. Therefore, clearly $\Phi(x,y) \to - \infty$ if $(x,y)\to \partial \Gamma_L$.
Next we argue by contradiction: if, for some $L$, this maximum is achieved for any $\alpha$ at $({\bar x}_\alpha,{\bar y}_\alpha)$ with ${\bar x}_\alpha={\bar y}_\alpha$, then $\Phi({\bar x}_\alpha,{\bar x}_\alpha)=- \varphi(LC({\bar x}_\alpha)\alpha)$ and therefore necessarely ${\bar x}_\alpha \in B(x_0,R/4)$ by the maximality property and the form of $C$. Moreover, for any $x,y$
$$u(x)-u(y) - \varphi(LC(x)(|x-y| +\alpha))\leq -\varphi(L\alpha)\; ,$$
and if this is true, for a fixed $L$, this implies that, for any $x,y$
$$u(x)-u(y) - \varphi(LC(x)|x-y| )\leq 0\; .$$
Choosing $x\in B(x_0,R/4)$, we have
$$ u(y)-u(x) \geq - \varphi(L|x-y| )\; ,$$
and this inequality implies that any element in $D^+u(x)$ has a norm which is less than $L$, which we wanted to prove.
Notice that, by using slightly more complicated arguments, the same conclusion is true if, for some $L$, we have ${\bar x}_\alpha-{\bar y}_\alpha \to 0$ when $\alpha \to 0$.
Therefore, we may assume without loss of generality that, for any fixed $L$, the maximum points $({\bar x}_\alpha,{\bar y}_\alpha)$ of $\Phi$, satisfies not only ${\bar x}_\alpha\neq {\bar y}_\alpha$ for $\alpha$ small enough but ${\bar x}_\alpha-{\bar y}_\alpha$ is bounded away from $0$ when $\alpha \to 0$. We are going to prove that this is a contradiction for $L$ large enough.
For the sake of simplicity of notations, we omit the indice $\alpha$ in all the quantities which depends on $\alpha$ (actually they also depend on $L$). In particular, we denote by $(x,y)$ a maximum point of $\Phi$ and we set $t=LC(x)(|x-y| +\alpha)$ and
$$ p= \varphi'(t)LC(x) \frac{(x-y)}{|x-y|} \; ,\; q= \varphi'(t)LDC(x) (|x-y|+\alpha)\; .$$
By a classical result of the User's guide (cf. Crandall, Ishii and Lions \cite{users}), there exist matrices $X,Y \in {\mathcal S}^N$ such that $(p+q,X)\in \overline{D^{2,+}}u(x)$, $(p,Y)\in \overline{D^{2,-}}u(y)$, for which the following viscosity inequalities hold
$$ F(x,u(x), p+q,X) \leq 0\; ,\; F(y,u(y), p,Y) \geq 0\; .$$
Moreover the matrices $X,Y$ satisfy, for any $\varepsilon >0$
$$ \left(-\frac1\varepsilon + ||A||\right) I_{2N} \leq \left(\begin{array}{cc}X & 0 \\0 & -Y\end{array}\right)\leq A + \varepsilon A^2$$
and where, if $\psi(x,y)= \varphi(LC(x)(|x-y| +\alpha))$, $A=D^2\psi(x,y)$ and $||A||=\max\{|\lambda|:\ \hbox{$\lambda $ is an eigenvalue of $A$}\}$.
Since $\varepsilon>0$ is arbitrary and since we are going to use only the second above inequality, we may choose a sufficiently
small $\varepsilon$ in order that the term $\varepsilon A^2$ becomes negligible. Using this remark, we argue
below assuming that $\varepsilon=0$ in order to simplify the exposure.
With this convention, the matrices $X,Y$ satisfy, for any $r,s \in \mathbb R^N$
\begin{equation}\label{fu}
Xr\cdot r - Ys \cdot s \leq \gamma_1|r-s|^2+2\gamma_2 |r-s||r|+\gamma_3|r|^2\; ,
\end{equation}
where
$$ \gamma_1= \frac{\varphi'(t)LC(x)}{|x-y|}+ \varphi''(t)(LC(x))^2\; ,$$
$$ \gamma_2= \varphi'(t)L|DC(x)|+ \varphi''(t)L^2|DC(x)|C(x)(|x-y|+\alpha)\; ,$$
$$ \gamma_3= \varphi'(t)\frac{|D^2C(x)|}{C(x)}t+ \varphi''(t)\frac{|DC(x)|^2}{[C(x)]^2}t^2\; ,$$
By easy manipulations, it is easy to see that
$$ \gamma_2 \leq \gamma_1\frac{|DC(x)|}{C(x)}(|x-y|+\alpha) + o_\alpha(1)\leq \gamma_1 K_2^{1/2}\chi(C(x))(|x-y|+\alpha)+ o_\alpha(1)\; ,$$
$$ \gamma_3 \leq \gamma_1 K_2[\chi(C(x))]^2(|x-y|+\alpha)^2+ o_\alpha(1)\; ,$$
where the $o_\alpha(1)$ comes from terms of the form $\alpha/|x-y|$. Again, for the sake of clarity, we are going to drop these terms which play no role at the end.
By Cauchy-Schwarz inequality, we deduce that, using $\eta$ appearing in the assumption,
\begin{equation}\label{si}
Xr\cdot r - Ys \cdot s \leq (1+\eta)\gamma_1|r-s|^2+ B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2|r|^2\; ,
\end{equation}
where $B(R,\eta)=(1+\eta^{-1})K_2$ depends on $R$ through $K_2$ and therefore is a $O(R^{-2})$ if $\eta$ is fixed.
Coming back to $p$ and $q$, we also have
$$ |q| = |p|\frac{|DC(x)|}{C(x)} (|x-y|+\alpha) \leq |p|\frac{|DC(x)|}{L [C(x)]^2} \leq O((RL)^{-1})|p|\; ,$$
since $LC(x)(|x-y|+\alpha)\leq 1$, $C\geq 1$ everywhere and since $\dfrac{|DC(x)|}{ [C(x)]^2}$ is a $O(R^{-1})$.
In order to have simpler formulas, we denote below by $\varpi_1$ any quantity which is a $O((RL)^{-1})$.
Now we arrive at the key point of the proof: by \eqref{fu}, choosing $r=0$, we have $-Y\leq \gamma_1I_N$ where $I_N$ is the identity matrix in $\mathbb R^N$. Therefore the matrix $\displaystyle I_N+[(1+\eta)\gamma_1]^{-1}Y$ is invertible and rewriting \eqref{si} as
$$ Xr\cdot r \leq Ys \cdot s + (1+\eta)\gamma_1|r-s|^2+ B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2|r|^2 \; ,$$
we can take the infimum in $s$ in the right-hand side and we end up with
$$X \leq Y(I_N+\frac{1}{(1+\eta)\gamma_1}Y)^{-1}+B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2I_N\; .$$
Setting $\tilde Y:= Y(I_N+\frac{1}{(1+\eta)\gamma_1}Y)^{-1}$, this implies that we have $(p+q,\tilde Y+3\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2I_N)\in \overline{D^{2,+}}u(x)$, $(p,Y)\in \overline{D^{2,-}}u(y)$ and then, using the Lipschitz continuity of $F$ in $M$, we have
the viscosity inequalities
$$ F(x,u(x), p+q,\tilde Y) \leq ||F_M||_\infty B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2\; ,$$
$$ F(y,u(y), p,Y) \geq 0\;.$$
Next we introduce the function
$$
g(\tau):= F(X(\tau), U(\tau), P(\tau), Z(\tau))-\tau||F_M||_\infty B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2 \; ,$$
where
$$ X(\tau) = \tau x+(1-\tau)y\; ,\; U(\tau)= \tau u(x)+(1-\tau)u(y)\; ,\; P(\tau)=p+\tau q\; ,$$
$$ Z(\tau) = Y(I_N+\frac{\tau}{(1+\eta)\gamma_1}Y)^{-1}\; .$$
From now on, in order to simplify the exposure, we are going to argue as if $F$ were $C^1$: the case when $F$ is just locally Lipschitz continuous follows from tedious but standard approximation arguments.
The above viscosity inequalities read $g(0)\geq 0$ and $g(1)\leq 0$ : if we can show that the $C^1$-function $g$ satisfies $g'(\tau)>0$ if $g(\tau)=0$, we would have a contradiction. Therefore we compute
\begin{eqnarray*}
g'(\tau) &=& F_x\cdot(x-y)+F_r (u(x)-u(y)) + F_p\cdot q + F_M\cdot Z'(\tau)\\
&&-||F_M||_\infty B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2\; ,
\end{eqnarray*}
and using that $F_r \geq 0$ and $Z'(\tau)=-((1+\eta)\gamma_1)^{-1}[Z(\tau)]^2$, we are lead to
\begin{eqnarray*}
g'(\tau) &\geq & (\gamma_1)^{-1}\biggl\{-|F_x| \gamma_1 |x-y|- \gamma_1 |F_p| |q| - \dfrac1{1+\eta}F_M\cdot [Z(\tau)]^2\\
&& -B(R,\eta) ||F_M||_\infty (\gamma_1)^2 [\chi(C(x))]^2(|x-y|+\alpha)^2\biggr\}.
\end{eqnarray*}
Before estimating the different terms inside the brackets, we point out that, contrarily to \cite{B-wb} where $Z(\tau)$ was given by
$\tau X + (1-\tau)Y$ and where we had to prove an inequality between $X-Y$ and $-[Z(\tau)]^2$, here this inequality comes for free
because of the form of $Z(\tau)$: this is the key idea of Cardaliaguet \cite{C1}.
Now we estimate the terms $\gamma_1 |x-y|$, $\gamma_1 |q|$ and $\gamma_1 \chi(C(x))(|x-y|+\alpha)$ in terms of $|P(\tau)|$ in
order to be able to use the assumptions on $F$.
Using that $LC(x)(|x-y|+\alpha) \leq 1$ and the properties of $\varphi$, we have
\begin{eqnarray*}
\gamma_1|x-y| & \leq & \varphi'(t)LC(x)+ \varphi''(t)(LC(x))^2|x-y|\\
& \leq & \varphi'(t)LC(x) + K_1\varphi'(t)\chi(\varphi'(t)) LC(x)\\
& \leq & |P(\tau)| (1+\varpi_1)\left(1 + K_1\chi(\varphi'(t)) \right)\\
& \leq & |P(\tau)| (1+\varpi_1)\left(1+K_1 \chi(L^{-1}|P(\tau)| (1+\varpi_1))\right)\; .
\end{eqnarray*}
Indeed, recalling the estimate on $|q|$, $ \varphi'(t)LC(x)=|p|=|P(\tau)|(1+\varpi_1\tau)$ and, on an other hand, since $C\geq 1$, we have
$$\chi(\varphi'(t))\leq \chi(L^{-1}|p|)\leq \chi(L^{-1}|P(\tau)| (1+\varpi_1)).$$
From now on, we are going to assume that $L$ is chosen large enough in order to have $L^{-1} (1+\varpi_1)\leq \eta$ and, since $R$ is fixed, $|\varpi_1| \leq \eta$. Notice that these constraints on $L$ depend only on $R$ and $\eta$, hence on $R$ and $F$.
Using this choice, the above estimate of $\chi(\varphi'(t))$ -- and we can argue in the same way for $\chi(C(x))$-- takes the simple form
\begin{equation}\label{kest}
\chi(\varphi'(t)), \chi(C(x)) \leq \chi(\eta |P(\tau)|).
\end{equation}
This leads to the
simpler estimate
$$ \gamma_1|x-y| \leq |P(\tau)| (1+\eta)(1+K_1\chi(\eta |P(\tau)|))\; .$$
In the same way, since we can take $\alpha$ as small as we want and $|x-y|$ is bounded away from $0$, one has
$$ \gamma_1 (|x-y| +\alpha) \leq |P(\tau)| (1+\eta)\left(1+K_1\chi(\eta |P(\tau)|)\right) +o_\alpha(1)\; .$$
This allows to estimate the $F_p$-term, namely
\begin{align*}
\gamma_1 |q| & \leq \gamma_1 |p| \frac{|DC(x)|}{C(x)}(|x-y| +\alpha)\\
&\leq |P(\tau)|^2 (1+\eta)^2 \left(1+K_1\chi(\eta |P(\tau)| )\right)K_2^{1/2} \chi(C(x))+o_\alpha(1),\\
& \leq K_2^{1/2} |P(\tau)|^2 (1+\eta)^2 \left(1+K_1\chi(\eta |P(\tau)| )\right) \chi(\eta |P(\tau)| )+o_\alpha(1)\; .
\end{align*}
Finally, by the same estimates
$$\gamma_1 \chi(C(x)) (|x-y|+\alpha) \leq |P(\tau)| (1+\eta)\left(1+K_1\chi(\eta |P(\tau)| )\right)\chi(\eta |P(\tau)| ) +o_\alpha(1)\; .$$
We end up with
\begin{eqnarray*}
g'(\tau) &\geq & (\gamma_1)^{-1}\biggl\{-|F_x| |P(\tau)| (1+\eta)(1+K_1\chi(\eta |P(\tau)|) ) \\
&& \phantom{(\gamma_1)^{-1}\biggl\{} - |F_p| K_2^{1/2} |P(\tau)|^2 (1+\eta)^2 \left(1+K_1\chi(\eta |P(\tau)| )\right) \chi(\eta |P(\tau)| ) \\
&& \phantom{(\gamma_1)^{-1}\biggl\{} - \dfrac1{1+\eta}F_M\cdot [Z(\tau)]^2\\
&& -B(R,\eta) ||F_M||_\infty \bigl( |P(\tau)| (1+\eta)\left(1+K_1\chi(\eta |P(\tau)| )\right) \chi(\eta |P(\tau)| )\bigr) ^2\biggr\} \\
&& + o_\alpha(1).
\end{eqnarray*}
On the other hand, in order to take into account the constraint $g(\tau)= 0$, we have to estimate $\gamma_1 [\chi(C(x))]^2 (|x-y|+\alpha)^2$.
Since $|x-y|$ is bounded away from $0$ and $LC(x)(|x-y|+\alpha)\leq 1$, we have
\begin{eqnarray*}
\gamma_1(|x-y|+\alpha)^2 & \leq & \varphi'(t)+ \varphi''(t) +o_\alpha (1)\\
& \leq & \varphi'(t)[1 + K_1\chi(\varphi'(t))]+ o_\alpha (1)\\
& \leq & (1+\eta) \frac{|P(\tau)|}{LC(x)}[1 + K_1\chi(\varphi'(t))]+o_\alpha (1)\\
& \leq & (1+\eta) \frac{|P(\tau)|}{LC(x)}[1 + K_1\chi(\eta|P(\tau)| )]+o_\alpha (1).
\end{eqnarray*}
But $\dfrac{[\chi(C(x))]^2}{C(x)} \leq \tilde K(\chi)$ and therefore
$$ \gamma_1 [\chi(C(x))]^2 (|x-y|+\alpha)^2 \leq \eta(1+\eta)\tilde K(\chi) |P(\tau)|[1 + K_1\chi(\eta |P(\tau)| )]+o_\alpha (1).$$
This implies
$$ |F(X(\tau), U(\tau), P(\tau), Z(\tau))|\leq \eta(1+\eta)\tilde K(\chi) |P(\tau)|[1 + K_1\chi(\eta|P(\tau)|)]+o_\alpha (1)\; ,
$$
while
$$ |P(\tau)| \geq (1-\eta)L\; .$$
The conclusion follows by applying the assumption on $F$ for $L$ large enough and $\alpha$ small enough in order that the $o_\alpha(1)$-terms are controlled by the $\eta$-terms. Taking $L$ large enough depending on $\eta$ and $R$, we have a contradiction and the proof of (i) is complete.
Now we turn to the proof of (ii) where we choose $\varphi(t)=t$ and
$$\Gamma'_L :=\{(x,y) \in B(x_0,3R/4) \times B(x_0,R) : LC(x)(|x-y| +\alpha)\leq osc_R (u)\}\; .$$
The proof follows the same arguments, except that the fact that $\varphi''(t)\equiv 0$ allows different estimates on the $\gamma_i$, $i=1,2,3$ because several terms do not exist anymore. We denote by $\varpi_2$ any quantity of the form $O(osc_R (u)(RL)^{-1})$ and we choose $L$
large enough in order to have $|\varpi_2|\leq \eta$ for any of these terms and $L^{-1} \leq \eta/(1+\eta)$. We notice that, here, the constraints on $L$ depend not only on $R$ and $\eta$ but also on $osc_R (u)$.
We have $p= LC(x) \dfrac{(x-y)}{|x-y|}$ and therefore
$$ |q|= L.|DC(x)| (|x-y|+\alpha)= |p| \frac{|DC(x)|}{C^2}\frac{LC(x)(|x-y|+\alpha)}{L}=\varpi_2|p|\leq \eta |p|\; ,$$
since $\dfrac{|DC(x)|}{C^2}\leq O(R^{-1})$. Using this inequality and taking into account our choice of $L$, it is easy to check that (\ref{kest}) still holds.
Moreover we have
$$ \gamma_1= \frac{LC(x)}{|x-y|}\; ,\; \gamma_2= L.|DC(x)|\; ,
\; \gamma_3= L |D^2C(x)|(|x-y|+\alpha) \; .$$
And we still have the same estimates on $\gamma_1, \gamma_2,\gamma_3$
$$ \gamma_2 = \gamma_1\frac{|DC(x)|}{C(x)}|x-y| \leq \gamma_1 \chi(C(x))|x-y|\; ,$$
$$ \gamma_3 \leq \gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2\; .$$
The proof is then done in the same way as in the first case with the computation of $ g'(\tau)$ and then with the estimates of the different terms
\begin{eqnarray*}
g'(\tau) &\geq & (\gamma_1)^{-1}\left\{-|F_x| \gamma_1 |x-y|-\gamma_1 |F_p| |q| - \frac{1}{1+\eta}F_M\cdot [Z(\tau)]^2\right.\\
&& \left . -B(R,\eta) ||F_M||_\infty (\gamma_1)^2 [\chi(C(x))]^2(|x-y|+\alpha)^2\right\}\; .
\end{eqnarray*}
But here
$$\gamma_1 |x-y|=|p| \leq |P(\tau)| (1+\eta)\; ,$$
and in the same way,
\begin{align*}
\gamma_1 |q| & = \frac{LC}{|x-y|} L |DC(x)|(|x-y| +\alpha)\\
&\leq |p|^2 \dfrac{|DC(x)|}{C(x)}(1+o_\alpha(1))\\
& \leq K_2^{1/2} (1+\eta)^2 |P(\tau)|^2 \chi(\eta |P(\tau)| )+o_\alpha(1)\; ,
\end{align*}
and
$$\gamma_1 \chi(C(x)) (|x-y|+\alpha) \leq (1+\eta)|P(\tau)|\chi(\eta|P(\tau)|)+o_\alpha (1)\; .$$
We end up with
\begin{eqnarray*}
g'(\tau) &\geq & (\gamma_1)^{-1}\biggl\{-|F_x| (1+\eta)|P(\tau)| - K_2^{1/2} (1+\eta)^2|F_p| |P(\tau)|^2 \chi(\eta |P(\tau)| ) \\
&& -\frac{1}{1+\eta} F_M\cdot [Z(\tau)]^2 - B(R,\eta) ||F_M||_\infty (1+\eta)^2 |P(\tau)|^2[\chi(\eta|P(\tau)|)]^2 \\
&& +o_\alpha (1) \biggr\}\; .
\end{eqnarray*}
On the other hand, for the constraint $g(\tau)= 0$, we have
\begin{align*}
\gamma_1[\chi(C(x))]^2(|x-y|+\alpha)^2 &= |p|\frac{[\chi(C(x))]^2}{C(x)}\dfrac{LC(|x-y|+\alpha)^2}{|x-y|}\\
&\leq (1+\eta)[\tilde K(\chi)]^2 |P(\tau)|(1+\varpi_2)(1+o_\alpha(1))\\
&\leq (1+\eta)^2[\tilde K(\chi)]^2|P(\tau)|+o_\alpha(1)\; ,
\end{align*}
and
$$ |P(\tau)|\geq LC(x) (1-\eta)\geq L(1-\eta)\; .$$
Hence
\begin{equation}\label{F-prop}
|F(X(\tau), U(\tau), P(\tau), Z(\tau))| \leq B(R,\eta)||F_M||_\infty (1+\eta)^2 |P(\tau)| + o_\alpha(1)\; .
\end{equation}
The conclusion follows as in the first case by applying the assumption on $F$ for $L$ large enough and $\alpha$ small enough for which we have a contradiction.
For the proof of (iii), we keep the same test-function and the same set $\Gamma'_L$ but since we are not expecting the gradient bound to come from the same term in $g'(\tau)$, we are going to change the strategy in our computation of $g'(\tau)$ by keeping the $F_r$-term. Using that
$F_r \geq 0$ and
$$ u(x)-u(y)\geq LC(x)(|x-y|+\alpha)= \frac{|p |^2}{\gamma_1}(1+o_\alpha(1))\; ,$$
we obtain
\begin{eqnarray*}
g'(\tau)& = & F_x\cdot(x-y)+F_r (u(x)-u(y)) + F_p\cdot q + F_M\cdot Z'(\tau)\\
&&-||F_M||_\infty B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2\; ,\\
&\geq& (\gamma_1)^{-1}\left\{F_x\cdot p+ F_r |p|^2 - \gamma_1 |F_p| |q| - \frac{1}{1+\eta}F_M\cdot [Z(\tau)]^2\right.\\
&& \left . -B(R,\eta) ||F_M||_\infty (\gamma_1)^2 [\chi(C(x))]^2(|x-y|+\alpha)^2+o_\alpha(1)\right\}\; .
\end{eqnarray*}
This computation is close to the one given in \cite{B-wb} if there is no localization term ($C\equiv1$).
Since $|P(\tau)| (1-\eta)\leq |p| \leq |P(\tau)| (1+\eta)$ and using anagolous estimates as above, we are lead to
\begin{eqnarray*}
g'(\tau) &\geq & (\gamma_1)^{-1}\biggl\{-(1+\eta) |F_x| |P(\tau)| +(1-\eta)^2 F_r |P(\tau)|^2\\
&& - K_2^{1/2} (1+\eta)^2|F_p| |P(\tau)|^2 \chi(\eta |P(\tau)| )- \frac{1}{1+\eta}F_M\cdot [Z(\tau)]^2 \\
\\
&& -B(R,\eta) ||F_M||_\infty (1+\eta)^2 [\tilde K(\chi)]^2 |P(\tau)|^2[\chi(\eta|P(\tau)|)]^2 \biggr\}+o_\alpha (1) \; .
\end{eqnarray*}
On the other hand, the constraint $g(\tau)= 0$ still implies (\ref{F-prop}) and we also conclude by choosing $L$ large enough and $\alpha$ small enough.}
\section{The parabolic case}
In this section, we consider evolution equations under the general form
\begin{equation}\label{GFNLP}
u_t + F(x, t, u, D u, D^2u) = 0 \quad\hbox{in }\Omega \times (0,T)\; ,
\end{equation}
and the aim is to provide a local gradient bound where ``local'' means both local in space and time. As a consequence, we will have to provide a localization also in time and a second main difference is that we will not be able to use that the equation holds since the $u_t$-term has no property in general and therefore the assumptions on $F$ have to hold for any $x, t, r, p, M$ and not only those for which $F(x, t, r, p, M)$ is close to $0$.
\begin{theorem} \label{mainP}{\bf (Estimates for non-uniformly parabolic equations : estimates depending the oscillation of $u$)}\\
Assume that $F$ is a locally Lipschitz function in $\Omega \times (0,T) \times \mathbb R \times \mathbb R^N \times {\mathcal S}^N$ which satisfies : $F(x,t,r,p,M)$ is Lipschitz continuous in $M$ and
$$
F_M(x,t,r,p,M) \leq 0 \quad\hbox{a.e. in }\Omega \times (0,T)\times \mathbb R \times \mathbb R^N \times {\mathcal S}^N \; ,$$
and let $u\in C(\Omega\times (0,T))$ be a solution of (\ref{GFNLP}). Assume that there exists a function $\chi \in \mathcal{K}$, $0<\eta\leq 1$
such that, for any $K>0$, there exists $L=L(\eta,K)$ large enough such that, for $|p|\geq L$,
we have $F_r(x,t,r,p,M)\geq 0$ and
$$ -(1+\eta) |F_x| |p| (1+\chi(\eta |p|)) - K |F_p| |p|^2 \left(1+\chi(\eta |p| )\right) \chi(\eta |p| )- \frac{1}{1+\eta} F_M\cdot M^2$$
$$ \geq \eta + K |p|^2\biggl( \chi((1+\eta) |p|)+ \chi(\eta |p|)^{2}\biggr)\; \hbox{a.e.},$$
If $\overline{B(x_0,R)} \subset \Omega$ and $\delta>0$, then $u$ is Lipschitz continuous in $x$ in $B(x_0,R/2)\times [\delta, T-\delta]$ and $|Du| \leq {\bar L} $ in $B(x_0,R/2)\times [\delta, T-\delta]$ where $\bar L$ depends on $F$, $R$, $\delta$ and the oscillation of $u$ in
$B(x_0,R)\times (\delta/2,T-\delta]$.
\end{theorem}
It is worth pointing out that the assumptions of Theorem~\ref{mainP} are rather close to the one of Theorem~\ref{main} (iii) and the same computations provide a gradient bound for the evolution equation
\begin{equation}\label{PartEqnNUN-P}
u_t-{\rm Tr}(A(x)D^2 u) + |Du|^m = f(x) \quad\hbox{in }\Omega\times (0,T)\; ,
\end{equation}
if $m>1$.
\noindent{\bf Proof of Theorem~\ref{mainP} :} We argue as in the proof of Theorem~\ref{main} (iii), except that here $L=L(t)$ with $L(t) \to +\infty$ as $t \to (\delta/2)^+$. We still choose $\varphi(t)=t$ and we denote by $\Gamma'_L$, the subset of points $(x,y,t) \in B(x_0,3R/4) \times B(x_0,R)\times (\delta/2,T-\delta]$ such that
$$ L(t)C(x)(|x-y| +\alpha)\leq osc_{R,\delta} (u)\},$$
where $osc_{R,\delta} (u)$ denotes the oscillation of $u$ in
$B(x_0,R)\times (\delta/2,T-\delta]$.
We consider maximum points $(x,y,t) \in \Gamma'_L$ of the function
$$(x,y,t)\mapsto u(x,t)-u(y,t) - L(t)C(x)(|x-y| +\alpha)\; ,$$
and, if $x\neq y$, we are lead to the viscosity inequalities
$$ a+ F(x,t, u(x,t), p+q,X) \leq 0\; ,\; b+F(y,t,u(y,t), p,Y) \geq 0\; ,$$
where $(a,p+q,X)\in D^{2,+}u(x,t)$, $(p,Y)\in D^{2,-}u(y,t)$ and $$a-b \geq L'(t)C(x)(|x-y|+\alpha).$$
As in the proof of Theorem~\ref{main}, the second inequality holds for $\tilde Y$ as well and subtracting these inequalities, we have
$$ L'(t)C(x)(|x-y|+\alpha)+ F(x,u(x), p+q,X) -F(y,u(y), p,\tilde Y)\leq 0\; .$$
Then, with the notations of the proof of Theorem~\ref{main}, we introduce
$$
g(\tau) := F(X(\tau), U(\tau), P(\tau), Z(\tau))-\tau ||F_M||_\infty B(R,\eta)\gamma_1 [\chi(C(x))]^2(|x-y|+\alpha)^2]$$
$$ +\tau L'(t)C(x)(|x-y|+\alpha) \; .$$
Here we have no information on the signs of $g(0)$ and $g(1)$, we only know that $g(1)-g(0)\leq 0$; therefore, in order to have the contradiction, we have to show that $g'(\tau) >0$ for any $0\leq\tau\leq1$ if we choose a function $L(\cdot)$ such that $L(t)$ is large enough for any $t\in (\delta/2,T-\delta]$.
The computation of $g'(\tau)$ and the estimates are done as above; we have just to estimate the new term $L'(t)C(x)(|x-y|+\alpha)$ which is
multiplied by $\gamma_1$ when we put it inside the bracket. We have
$$ \gamma_1 L'(t)C(x)(|x-y|+\alpha) = L(t) L'(t)[C(x)]^2 (1+o_\alpha(1)) \; ,$$
and if we choose $L$ as the solution of the ode
$$ L'(t)=-k_T L(t)\chi(L(t))\; , L(T-\delta) = L_T \;\hbox{(large enough)}\; .$$
By choosing properly $k_T>0$, we have $L((\delta/2)^+)=+\infty$ (notice that $k_T$ decreases when $L_T$ increases). Since $L(t)\leq |p|\leq (1+\eta) |P(\tau)|$, we have
$$L(t) L'(t)[C(x)]^2 \geq -k_T |P(\tau)|^2 \chi((1+\eta)|P(\tau)|)\; .$$
Using this estimate, the conclusion follows as above by applying the assumption on $F$ for $K$ large enough and $\alpha$ small enough for which we have a contradiction by taking $L_T $ large enough.
\thebibliography{99}
\bibitem{B-wb} Barles, G., (1991), A weak Bernstein method for fully nonlinear elliptic equations. J.
Diff. and Int. Equations, vol 4, n${}^\circ$ 2, pp 241-262.
\bibitem{CClivre}
L. A. Caffarelli and X. Cabr\'e,
{\em Fully non-linear elliptic equations.}
American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, 1995.
\bibitem{C1} Cardaliaguet, P. : Personal communication.
\bibitem{users} Crandall, M.G., Ishii, H., Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27(1) 1-67.
\bibitem{GT}
Gilbarg D., Trudinger N.-S., {\em Elliptic Partial Differential Equations of
Second Order}, Second edition, Springer, 2001.
\bibitem{LB}
Lions P.-L., {\em Generalized solutions of {H}amilton-{J}acobi
equations}, vol.~69 of Research Notes in Mathematics, Pitman (Advanced
Publishing Program), Boston, Mass., 1982.
\end{document} |
\begin{document}
\title{Hashing Learning with Hyper-Class Representation}
\author{Shichao~Zhang,~\IEEEmembership{Senior~Member,~IEEE, Jiaye Li}
\IEEEcompsocitemizethanks{\IEEEcompsocthanksitem School of Computer Science and Engineering,
Central South University, Changsha 410083, PR China. \protect\\
E-mail: [email protected] and [email protected]
Corresponding author: ****.
}
\thanks{Manuscript received August **, 2022; revised ** **, 2022.}}
\markboth{}
{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for Computer Society Journals}
\IEEEtitleabstractindextext{
\begin{abstract}
Existing unsupervised hash learning is a kind of attribute-centered calculation. It may not accurately preserve the similarity between data. This leads to low down the performance of hash function learning. In this paper, a hash algorithm is proposed with a hyper-class representation. It is a two-steps approach. The first step finds potential decision features and establish hyper-class. The second step constructs hash learning based on the hyper-class information in the first step, so that the hash codes of the data within the hyper-class are as similar as possible, as well as the hash codes of the data between the hyper-classes are as different as possible. To evaluate the efficiency, a series of experiments are conducted on four public datasets. The experimental results show that the proposed hash algorithm is more efficient than the compared algorithms, in terms of mean average precision (MAP), average precision (AP) and Hamming radius 2 (HAM2).
\end{abstract}
\begin{IEEEkeywords}
data representation; hyper-class representation; hash learning
\end{IEEEkeywords}}
\maketitle
\IEEEdisplaynontitleabstractindextext
\IEEEpeerreviewmaketitle
\IEEEraisesectionheading{\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{O}{n} the road to artificial intelligence, a major open problem is learning representation of data, which can be more efficiently applied to various data mining algorithms.
A good data representation can make the follow-up learning task simple and efficient\cite{ray2022teaching}\cite{al2022fuzzy}. Traditional data representation only retains the information of data value. In order for AI to understand our world, it must be able to distinguish and separate the potential information hidden under the observed data. Therefore, how to construct an appropriate and efficient data representation is very important.
Most previous data representations convert text and image data into tables or matrices, which store discrete or continuous values of features\cite{xiao2022survey}\cite{nebli2022quantifying}. This often loses some potential information in the data. For example, in Fig. \ref{fig1}, we show an unlabeled data set (it has only three-dimensional features). The x-axis, y-axis and z-axis represent its three features, namely, height, weight and age. Obviously, according to their age, they can be divided into four classes, namely, childhood, teenager, middle-aged and elderly. This part of information is lost in Fig. \ref{fig1}. In this paper, we call this part of information hyper-class. As shown in Fig. \ref{fig2}, the information of hyper-classes 1-4 is added to the data. It is different from clustering. Clustering generates the class information of the data by calculating the similarity between all the features of the data. Hyper-class is the similarity calculation for a certain feature of all data. The hyper-class representation of data can add some potential information to the data, {\em i.e.,~} hyper-class. In addition, it also simplifies the representation of data to a certain extent. For example, the original value of age may be a continuous value of 1-100, but now there are only four values: childhood, teenager, middle-aged and elderly.
\begin{figure}
\caption{An unlabeled dataset.}
\label{fig1}
\end{figure}
\begin{figure}
\caption{Dataset under hyper-class representation}
\label{fig2}
\end{figure}
Hash learning is an effective measure to solve the approximate nearest neighbor retrieval of large-scale data. Its core idea is to keep the similarity between data as short as possible. In unsupervised hash algorithm, the learned hash code can not perform approximate nearest neighbor retrieval well due to the lack of data labels. In addition, the missing label information can not guide the generation of hash code, which will lead to the high similarity of hash code in data of different classes, thus reducing the retrieval accuracy.
To solve the above problems, this paper proposes a novel hash algorithm under hyper-class representation. Specifically, we first find the potential decision features from the unlabeled data and establish the hyper-class representation of the data (the search decision features and feature selection algorithms are different. Feature selection is to find the feature subset that can best represent the overall information of the whole data, while we are looking for the feature that can best be used as the decision feature of the data). Then we construct a hash method according to the hyper-class representation, which can make the hash codes of the data within the hyper-class as similar as possible and the hash codes of the data between the hyper-classes as different as possible. On this basis, we can learn a more suitable hash code for approximate nearest neighbor retrieval.
The main contributions of this paper are as follows:
\begin{itemize}
\item We improve the hyper-class representation of data and apply it to hash learning. It can represent the potential hyper-class information in the data and simplify the representation of the data.
\item Based on the hyper-class representation, an effective hash algorithm is proposed, which can learn the more appropriate hash code for approximate nearest neighbor retrieval. At the same time, an alternating iterative algorithm is proposed to optimize it.
\end{itemize}
In addition, we conducted a series of experiments to show that the proposed algorithm exceeds the state-of-the-art methods in terms of MAP, AP and HAM2.
The rest of this paper is organized as follows. Section \ref{Preliminary} briefly reviews previous related work. Section \ref{Method} describes our proposed method in detail, the optimization process and time complexity analysis. Section \ref{experiments} shows the results of all algorithms on real datasets. Section \ref{conclusion} summarizes the full paper.
\section{Related work}\label{Preliminary}
In this section, we briefly introduce some data representation methods and hash methods.
\subsection{Data representation}
Data representation is the basis of data mining algorithm\cite{xiao2022survey}\cite{jiang2021novel}. An appropriate data representation can make the performance of data mining algorithm get twice the result with half the effort \cite{zhang2022hyper}. The traditional data representation methods include list method\cite{choi2021relative}, drawing method\cite{you2022novel} and equation method\cite{rasool2022introduction}. List method is the most commonly used data representation method. It puts the values of data into a table. Usually, each row of the table represents a sample and each column represents a feature. It only saves the value information of the data. Drawing method is to represent the data in the form of image. This kind of representation is generally not directly applicable to the data mining task of computer\cite{zhu2020unsupervised}. It also needs to further represent the data in a form that is easy to be processed by computer\cite{zhu2019efficient}. Equation method is to express the data with function according to some linear or nonlinear relationship in the data\cite{tang2018robust}. This method is rarely used because it is difficult to represent most data with only one equation.
In addition to the traditional data representation methods, with the rapid development of artificial intelligence, many data representation methods have been proposed. For example, entity relationship representation \cite{al2021conceptual}. It uses entities to represent each sample or event, each entity has its corresponding features, and the relationship between entities is represented by edges. It is widely used in the field of natural language processing. On this basis, it is often used together with one-hot encoding\cite{yu2022missing}. For example, one-hot encoding is used to represent some concepts in medical data. It represents medical concepts as a binary vector, and the number of elements in the vector is the number of classes of medical concepts. The position corresponding to medical concepts in this vector is 1, otherwise it is 0. Although one-hot encoding is simple, it still has some shortcomings. For example, it will increase the dimension of data. High dimensional data will bring trouble to the algorithm\cite{zhu2018low}\cite{zhu2018one}. At this time, the sparse representation of data just solves this problem. Sparse representation is usually carried out by using sparse constraints, which can be expressed as the following convex optimization problem:
\begin{eqnarray}
\label{eq1}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf W,\mathbf E} {\left\| \mathbf W \right\|_1} + \alpha {\left\| \mathbf E \right\|_{2,1}}\\
s.t.,\mathbf X = \mathbf X\mathbf W,diag(\mathbf W) = 0
\end{array}
\end{eqnarray}
where ${\left\| \mathbf W \right\|_1}$ is $l_1-$norm of $\mathbf W$ matrix and ${\left\| \mathbf E \right\|_{2,1}}$ is $l_{2,1}-$norm of noise matrix $\mathbf E$. $\mathbf X$ is the data matrix. In order to avoid obtaining the trivial solution $\mathbf W = \mathbf I$, the restriction of $diag(\mathbf W) = 0$ is added. Sparse representation can make the values in the data more sparse without affecting the information contained in the data itself. Based on the sparse representation of data, it is further proposed to use it for feature selection of data. For example, $l_1-$norm restriction is applied to the feature weight vector, which can remove the redundant features, so as to retain only the features that can better represent the data information, and further obtain the low dimensional representation of the data. Or we can obtain the low dimensional representation of the data by limiting the $l_{2,1}-$norm of the feature relation matrix. Of course, some other subspace learning algorithms also belong to the low-dimensional representation of data, such as principal component analysis (PCA)\cite{hasan2021review}, linear discriminant analysis (LDA)\cite{guo2021reverse} and local preserving projects (LPP)\cite{qiang2021robust}.
In recent years, with the advent of alphago, artificial intelligence has ushered in a new upsurge\cite{9566737}. Many data representation methods based on deep learning have been proposed. Zhou {\em et al.~} used deep learning and sparse representation to diagnose weak fault of bearing \cite{zhou2022hybrid}. Liu {\em et al.~} proposed a data representation method based on knowledge map \cite{liu2022knowledge}. Specifically, it first constructs a multi-layer knowledge map for industrial Internet of things data. Then it uses the cognitive driven knowledge map to realize the automatic data fusion. Finally, it embeds the graph representation into the knowledge map and makes reasoning. Ahmadian {\em et al.~} proposed a reliable depth representation method \cite{ahmadian2022reliable}. Specifically, it first proposes a probability model and reliability measure to score users implicitly. Then these implicit scores are input into deep sparse coding to generate a new representation of user characteristics. Finally, it proposes a new similarity measure function to calculate the similarity between users and recommend projects to target users. Prusa {\em et al.~} proposed a new data representation method for deep neural network and text classification \cite{prusa2016designing}. It can reduce the dimension of data, {\em i.e.,~} from \emph m characters to ${\log _2}(m)$ characters. Based on this, it can greatly reduce the memory consumption of the computer and reduce the amount of calculation. In addition, this method can reduce the memory use of the computer by 16 times, so as to reduce the number of neurons in deep learning. Bethge {\em et al.~} proposed a representation learning method based on private encoder \cite{bethge2022domain}. Specifically, it first uses a neural network architecture for private coding from the far end of the data. Then the learned feature representation is shared to the global classifier. Finally, it uses domain alignment to learn the data representation independent of the data source. Jokanovic {\em et al.~} studied the impact of data representation on deep learning \cite{jokanovic2016effect}. It mainly aims at the impact of falls related injuries on the elderly. Specifically, it studies the influence of different time-frequency representations on the performance of deep learning detector. Finally, they found that the appropriate time-frequency representation method is very important to the performance of the depth detector. Yi {\em et al.~} studied the method, trend and application of graph representation \cite{yi2022graph}. They analyzed the application of graph representation in biological information research in detail. Rebuffi {\em et al.~} proposed incremental data representation, {\em i.e.,~} it is not a fixed data representation, but a representation that changes continuously with the increase of data \cite{rebuffi2017icarl}. Prabhakar and Lee proposed an improved sparse representation method \cite{prabhakar2022improved}. Specifically, it first preprocesses the data, and then thins the preprocessed data. Finally, it develops six different sparse representation optimization combinations. Wang {\em et al.~} proposed a data representation method and applied it to network intrusion detection system \cite{wang2022representation}. Specifically, it first learns the explicit and implicit representation of data from the two spaces of samples and features, so as to establish the model of network behavior. Then, it establishes an unsupervised eigenvalue representation module to learn the relationship between features. Finally, it establishes a supervised neural network module to represent the object and learn the potential implicit relationship in the data.
\subsection{Hashing learning}
Hash learning is one of the effective measures for approximate nearest neighbor retrieval of large-scale high-dimensional data \cite{zhu2017graph}. Hash learning can be divided into data independent hash and data dependent hash according to whether there is a training process\cite{shen2022learning}. According to the label information, the data dependent hash can be divided into supervised hash and unsupervised hash\cite{wang2017survey}. Next, we will introduce them.
\textbf{Data independent hash}: Data independent hash has no training process. Its core is to preset the hash function and learn the distribution information in the data. The traditional data independent hash includes random hash, local sensitive hash and structure projection hash \cite{hayashi2016more} \cite{liu2015structure}. The core idea of random hash is first to reduce the dimension of data, and then randomly select the hash function in a specific function set. The prediction time complexity of random hash can be constant. Local sensitive hash (LSH) makes the hash codes of two similar points in the original space similar by maintaining the local structure of the data \cite{andoni2017optimal}. Although it ensures the structure information of data, its efficiency is relatively low and requires a long hash code. The core of structure projection hash is to divide or map the data space through some data structure (such as tree) or some projection (such as Hilbert curve).
Data independent hashes do not need training time\cite{kafai2014discrete}. They are widely used in the fields of approximate duplicate image detection and large-scale retrieval. However, they often need a long hash code to ensure the performance of the algorithm, which will lead to the memory burden of the computer and reduce the scope of their application.
\textbf{Data dependent hash}: The data dependent hash function obtains the hash function according to the training data\cite{jose2022deep}. Its core idea is to learn hash function by mining the information of linear relationship, nonlinear relationship, local structure and global structure in data. Unlike the data independent hash, the hash codes it learns are relatively compact, which can reduce the internal consumption of computer storage. Because of its high efficiency, data dependent hash is widely used in approximate nearest neighbor retrieval of large-scale image data\cite{yang2021deep}.
According to whether the data has labels, the data dependent hash can be divided into supervised hash and unsupervised hash. In supervised hash learning, it usually uses the relationship between label information and data to establish hash function. Lin {\em et al.~} proposed a supervised hash algorithm ({\em i.e.,~} FastH)\cite{lin2014fast}. Specifically, it uses the block search method of graph cutting to solve large-scale reasoning, and uses the training enhanced decision tree to learn the nonlinear relationship in the data and construct the hash function. In unsupervised hash learning, it usually does not use data label information. For example, the iterative quantization method (ITQ)\cite{gong2012iterative} proposed by Gong {\em et al.~} It uses the rotation problem of zero center data to solve the binary code, and uses the alternating minimization algorithm to calculate the quantization error of data mapping to the vertex of zero center binary hypercube. In addition, according to the form of data, data dependent hash can be divided into single-mode hash and cross mode hash. Monomodal hash only works on one form of data, such as images or text, and only constructs hash codes in monomodal data to approximate neighbor search. For example, k-nearest neighbors hashing (KNNH)\cite{he2019k} and concatenation hashing (CH)\cite{weng2020concatenation}. Transmembrane hash refers to establishing the association between multiple modes, so as to learn their hash code\cite{li2021adaptive}. It uses the hash code of one mode to retrieve the data of another mode, such as the hash code of text to retrieve the image or the hash code of image to retrieve the text.
\section{Our Approach}\label{Method}
In this section, we will introduce the constructed hyper-class representation, the proposed hash algorithm and the optimization process of the algorithm in detail.
\subsection{Notation}
In this paper, we use uppercase bold letters to represent matrices and lowercase bold letters to represent vectors. Given a matrix $\mathbf X = [{x_{ij}}]$, its \emph i-th row and \emph j-th column are represented as $\mathbf X_i$ and $\mathbf X^j$ respectively. Frobenius norm, $l_{2,1}-$norm and $l_1-$norm of matrix $\mathbf X$ are ${\left\| \mathbf X \right\|_F} = {(\sum\nolimits_i {\sum\nolimits_j {x_{ij}^2} } )^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}}}$, ${\left\| \mathbf X \right\|_{2,1}} = \sum\nolimits_i {{{(\sum\nolimits_j {x_{ij}^2} )}^{{1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}}}} $ and ${\left\| \mathbf X \right\|_1} = \sum\nolimits_i {\sum\nolimits_j {\left| {{x_{ij}}} \right|} } $ respectively. The transpose, inverse and trace of matrix $\mathbf X$ are expressed as ${\mathbf X^T}$, ${\mathbf X^{ - 1}}$ and $tr(\mathbf X)$, respectively.
We summarize these notations used in our paper in Table \ref{tab1}.
\begin{table}[!tb]
\centering
\caption{\footnotesize The detail of the notations used in this paper.}
\centering
{\footnotesize
\begin{tabular}[c]{|c|c|} \hline
$\mathbf X$ & training data \\ \hline
$\mathbf y$ & test data \\ \hline
$\mathbf X_i$ & the \emph i-th row of $\mathbf X$ \\ \hline
$\mathbf X^j$ & the \emph j-th column of $\mathbf X$ \\ \hline
$\mathbf X_{^\neg i}$ & All remaining rows after \emph i-th row is removed from $\mathbf X$ \\ \hline
${\left\| \mathbf X \right\|_F}$ & the frobenius norm of $\mathbf{X}$, {\em i.e.,~} $||\mathbf{X}|{|_F} = \sqrt {\sum\nolimits_{i,j} \mathbf{x}_{i,j} ^2}$ \\ \hline
${\left\|\mathbf x \right\|_1}$ &the $l_1$ -norm of $\mathbf X$, {\em i.e.,~} ${\left\|\mathbf x \right\|_1} = \sum\nolimits_{i = 1}^n {\left| {{x_i}} \right|}$ \\ \hline
${\left\| \mathbf X \right\|_{2,1}}$ &$l_{2,1}-$norm of matrix {\em i.e.,~} ${\left\| \mathbf X \right\|_{2,1}} = \sum\nolimits_{i = 1}^n {{{\left\| {{\mathbf x_i}} \right\|}_2}} $ \\ \hline
$\mathbf X^T$ & the transpose of $\mathbf X$ \\ \hline
$\mathbf X^{-1}$ & the inverse of $\mathbf X$ \\ \hline
$tr(\mathbf X)$ & the trace of $\mathbf X$ \\ \hline
$\mathbf H$ & coding matrix \\ \hline
$\mathbf E$ & identity matrix \\ \hline
\end{tabular}}
\label{tab1}
\end{table}
\subsection{Find decision features and establish hyper-class}
Hyper-class representation is a form of data representation. In this paper, the core process of establishing hyper-class representation is as follows: 1. Find potential decision feature. 2. Use the found decision features to classify the data into hyper-classes, so as to generate the hyper-class information of the data. The first step is very important, which is different from the traditional unsupervised feature selection algorithm. Unsupervised feature selection is to select the feature subset that can best represent the overall information of the data. The first step here is to find the most likely decision features as class labels. Next, we first carry out the first step. Specifically, we use each feature to fit all other features, and use the least square loss to get the relationship between each feature and other features. In order to maintain the nonlinear relationship in the data, we use the kernel function to map the samples. In addition, in order to obtain the closed form solution of the method, we introduce the $l_{2,1}-$norm limit of the relationship matrix, as shown in the following formula:
\begin{eqnarray}
\label{eq2}
\begin{array}{l}
\mathop {\min }\limits_\mathbf W \sum\limits_{i = 1}^d {\left\| {{\mathbf X_i} - k({\mathbf X_{{}^\neg i}})\mathbf W} \right\|_F^2} + \varsigma {\left\| \mathbf W \right\|_{2,1}}
\end{array}
\end{eqnarray}
where ${\mathbf X_i} \in {\mathbb R^{1 \times n}}$ represents the \emph i-th feature and ${\mathbf X_{{}^\neg i}} \in {\mathbb R^{(d - 1) \times n}}$ represents all features except the \emph i-th feature. $\mathbf W \in {\mathbb R^{n \times n}}$ is the relation matrix. $k({\mathbf X_{{}^\neg i}}) \in {\mathbb R^{1 \times n}}$ is the vector after ${\mathbf X_{{}^\neg i}}$ is mapped to the kernel space. The specific mapping process is as follows:
\begin{eqnarray}
\label{eq3}
\begin{array}{l}
k({\mathbf X_{{}^\neg i}}) = [\varphi ({({\mathbf X_{{}^\neg i}})^{{1^T}}}{({\mathbf X_{{}^\neg i}})^1}),\varphi ({({\mathbf X_{{}^\neg i}})^{{2^T}}}{({\mathbf X_{{}^\neg i}})^2})\\
, \ldots ,\varphi ({({\mathbf X_{{}^\neg i}})^{{n^T}}}{({\mathbf X_{{}^\neg i}})^n})]
\end{array}
\end{eqnarray}
In addition, in order to obtain the possibility of each feature as a decision feature, we apply a weight $v_i$ to each feature (there are \emph d features in total). The greater the weight, the more it can be used as a decision feature. As follows:
\begin{eqnarray}
\label{eq4}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf v, \mathbf W} \sum\limits_{i = 1}^d {{v_i}\left\| {{\mathbf X_i} - k({\mathbf X_{{}^\neg i}})\mathbf W} \right\|_F^2} + \varsigma {\left\| \mathbf W \right\|_{2,1}}\\ - \sigma (\frac{1}{2}\left\| \mathbf v \right\|_2^2 - {\left\| \mathbf v \right\|_1})
\end{array}
\end{eqnarray}
where $\mathbf v \in {\mathbb R^{d \times 1}}$ represents the weight that each feature can be used as a decision feature. $\sigma $ and $\varsigma $ are adjustable parameters. $\frac{1}{2}\left\| \mathbf v \right\|_2^2 - {\left\| \mathbf v \right\|_1}$ is the ``soft" weight regular term, because it can obtain the actual value of the weight of each feature. Next, we solve Eq. (\ref{eq4}) by alternating iteration.
(1) Fix $\mathbf v$ and solve $\mathbf W$.
When $\mathbf v$ is fixed, Eq. (\ref{eq4}) can be written as follows:
\begin{eqnarray}
\label{eq5}
\begin{array}{l}
\mathop {\min }\limits_\mathbf W \sum\limits_{i = 1}^d {{v_i}\left\| {{\mathbf X_i} - k({\mathbf X_{{}^\neg i}})\mathbf W} \right\|_F^2} + \varsigma {\left\| \mathbf W \right\|_{2,1}}
\end{array}
\end{eqnarray}
To facilitate the solution, Eq. (\ref{eq5}) can be further written as:
\begin{eqnarray}
\label{eq6}
\begin{array}{l}
\mathop {\min }\limits_\mathbf W \left\| {\mathbf Q - \mathbf G\mathbf W} \right\|_F^2 + \varsigma {\left\| \mathbf W \right\|_{2,1}}
\end{array}
\end{eqnarray}
where $\mathbf Q = \mathbf V\mathbf X$, $\mathbf G = \mathbf V \tilde {\mathbf X}$, $\mathbf V = diag(\sqrt \mathbf v )$ and $\tilde {\mathbf X} = [k({\mathbf X_{{}^\neg 1}});k({\mathbf X_{{}^\neg 2}}); \ldots ;k({\mathbf X_{{}^\neg d}})]$. Further, we can get that Eq. (\ref{eq6}) is equivalent to the following equation:
\begin{eqnarray}
\label{eq7}
\begin{array}{l}
\mathop {\min }\limits_\mathbf W \left\| {\mathbf Q - \mathbf G\mathbf W} \right\|_F^2 + \varsigma tr({\mathbf W^T}\mathbf F\mathbf W)
\end{array}
\end{eqnarray}
We use Eq. (\ref{eq7}) to derive $\mathbf W$, and we can get:
\begin{eqnarray}
\label{eq8}
\begin{array}{l}
- {\mathbf Q^T}\mathbf G + {\mathbf G^T}\mathbf G\mathbf W + \varsigma \mathbf F\mathbf W
\end{array}
\end{eqnarray}
where
\begin{eqnarray}
\label{eq9}
\begin{array}{l}
{F_{ii}} = \frac{1}{{2\sqrt {\left\| {{\mathbf W^i}} \right\|_2^2 + \varepsilon } }}(\varepsilon \to 0,i = 1,2, \ldots ,n)
\end{array}
\end{eqnarray}
If we make Eq. (\ref{eq8}) equal to zero, we can get the solution of $\mathbf W$ as follows:
\begin{eqnarray}
\label{eq10}
\begin{array}{l}
\mathbf W = {({\mathbf G^T}\mathbf G + \varsigma F)^{ - 1}}{\mathbf G^T}\mathbf Q
\end{array}
\end{eqnarray}
(2) Fix $\mathbf W$ and solve $\mathbf v$.
When $\mathbf W$ is fixed, Eq. (\ref{eq4}) can be written as follows:
\begin{eqnarray}
\label{eq11}
\begin{array}{l}
\mathop {\min }\limits_\mathbf v \sum\limits_{i = 1}^d {{v_i}\left\| {{\mathbf X_i} - k({\mathbf X_{{}^\neg i}})\mathbf W} \right\|_F^2} - \sigma (\frac{1}{2}\left\| \mathbf v \right\|_2^2 - {\left\| \mathbf v \right\|_1})
\end{array}
\end{eqnarray}
We let $L({\mathbf X_i},{\mathbf X_{{}^\neg i}}, \mathbf W) = \left\| {{\mathbf X_i} - k({\mathbf X_{{}^\neg i}})\mathbf W} \right\|_F^2$, then Eq. (\ref{eq11}) can be further written as follows:
\begin{eqnarray}
\label{eq12}
\begin{array}{l}
\mathop {\min }\limits_\mathbf v \sum\limits_{j = 1}^d {{v_i}L({\mathbf X_i},{\mathbf X_{{}^\neg i}},\mathbf W)} - \frac{1}{2}\sigma \sum\limits_{i = 1}^d {(v_i^2 - 2{v_i})}
\end{array}
\end{eqnarray}
We use Eq. (\ref{eq12}) to derive $v_i$ and make the derivative zero, and we can get the solution of $\mathbf v$ as follows:
\begin{eqnarray}
\label{eq13}
\begin{array}{l}
{v_i} = \left\{ {\begin{array}{*{20}{c}}
{1 - \frac{{L({\mathbf X_i},{\mathbf X_{{}^\neg i}},\mathbf W)}}{\sigma },}&{L({\mathbf X_i},{\mathbf X_{{}^\neg i}},\mathbf W) < \sigma }\\
{0,}&{L({\mathbf X_i},{\mathbf X_{{}^\neg i}},\mathbf W) \ge \sigma }
\end{array}} \right.
\end{array}
\end{eqnarray}
After the optimal solution of $\mathbf v$ is obtained, the weight of each feature as a decision feature is obtained. We can obtain the most likely decision feature by the following formula:
\begin{eqnarray}
\label{eq14}
\begin{array}{l}
df = \max \{ {v_1},{v_2}, \ldots ,{v_d}\}
\end{array}
\end{eqnarray}
Through Eq. (\ref{eq14}), after obtaining the most likely decision feature $df$, we use the method in literature \cite{makarychev2022performance} to divide the data under decision feature $df$.
\subsection{Hash method based on hyper-class representation}
After the data is represented by hyper-class, its original data still exists. Hyper-class representation only adds some potential hyper-class information ({\em i.e.,~} class $\{ 1,2,3, \ldots ,c\}$, assuming that the established hyper-class has \emph c classes) on the basis of the original data. When the hyper-class has only one class, {\em i.e.,~} all data are of the same class, and their class labels are the same ($c=1$). At this time, our idea is to make the similarity between the hash codes corresponding to each sample as high as possible, because they belong to the same class.
When $c=1$, {\em i.e.,~} there is only one class. At this time, the objective function is as follows:
\begin{eqnarray}
\label{eq15}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf H,\mathbf U,\mathbf S} \sum\nolimits_{i,j}^n {\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{ij}}} + \alpha \sum\nolimits_i^n {\left\| {{\mathbf s_i}} \right\|_2^2} \\
+ \beta \sum\nolimits_{i = 1}^n {\left\| {{\mathbf H^{(i)}} - \bm \mu } \right\|_2^2} \\
s.t.{\rm{ }}\mathbf s_i^T1 = 1,~{s_{ii}} = 0,\\
{s_{ij}} \ge 0, ~if~{\rm{ }}j \in N(i),~{\rm{ }}otherwise{\rm{ }}~0
\end{array}
\end{eqnarray}
where $\mathbf S \in {\mathbb R^{n \times n}}$ is the similarity matrix of hash codes, which records the similarity between the hash codes of each sample, where ${s_{ii}} = 0$, because the similarity between the hash code of each sample and itself is 0. ${\mathbf H^{(i)}}$ is the hash code of the \emph i-th sample, and ${\mathbf H^{(j)}}$ is the hash code of the \emph j-th sample. $U$ is the mapping matrix. $\bm \mu$ is the average of all sample hash codes. In Eq. (\ref{eq15}), $\sum\nolimits_{i,j}^n {\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{ij}}} $ calculates the similarity between the hash codes of all samples, and $\sum\nolimits_i^n {\left\| {{\mathbf s_i}} \right\|_2^2} $ is the regular term limit of variable $\mathbf s$. The main function of $\sum\nolimits_{i = 1}^n {\left\| {{\mathbf H^{(i)}} - \bm \mu } \right\|}$ is to reduce the deviation between all hash codes and the mean, so as to reduce the influence of outliers. Eq. (\ref{eq15}) only considers the relationship between hash codes and does not consider the relationship between hash codes and samples. Therefore, we further obtain the following formula:
\begin{eqnarray}
\label{eq16}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf H,\mathbf U,\mathbf S} \sum\nolimits_{i,j}^n {\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{ij}}} + \alpha \sum\nolimits_i^n {\left\| {\mathbf s_i} \right\|_2^2} \\
+ \beta \sum\nolimits_{i = 1}^n {\left\| {{\mathbf H^{(i)}} - \bm \mu } \right\|_2^2} + \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2 + \eta \left\| \mathbf U \right\|_2^2\\
s.t.~{\rm{ }}\mathbf s_i^T1 = 1,~{s_{ii}} = 0,\\
{s_{ij}} \ge 0,~if{\rm{ }}j \in N(i),~{\rm{ }}otherwise{\rm{ }}0,~{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{eqnarray}
where $\left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2$ considers the relationship between the hash code and the sample, and $\left\| \mathbf U \right\|_2^2$ is the regular term containing $\mathbf U$, which can maintain the closed solution of the function. $\mathbf U{\mathbf U^T} = \mathbf I$ is the orthogonal limit of $\mathbf U$, which can make the hyperplane of hash function irrelevant ({\em i.e.,~} orthogonal to each other). In practice, it is almost impossible for the hyper-class representation we established to have only one class ({\em i.e.,~} in most cases, $c > 1$). We further construct the objective function in the following two cases:
When $c=2$, {\em i.e.,~} there are only two hyper-class to be constructed. At this time, the objective function is as follows:
\begin{eqnarray}
\label{eq17}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf H,\mathbf U,\mathbf S} \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2{s_1^{(ij)}}} + \alpha \sum\nolimits_i^{{n_1}} {\left\| {{s_1^{(i)}}} \right\|_2^2} \\
- \beta \sum\nolimits_{i = 1}^{{n_1}} {\left\| {\mathbf H_1^{(i)} - {\bm \mu _2}} \right\|_2^2} + \sum\nolimits_{i,j}^{{n_2}} {\left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2{s_2^{(ij)}}} \\
+ \alpha \sum\nolimits_i^{{n_2}} {\left\| {{\mathbf s_2^{(i)}}} \right\|_2^2} - \beta \sum\nolimits_{i = 1}^{{n_2}} {\left\| {\mathbf H_2^{(i)} - {\bm \mu _1}} \right\|_2^2} \\
+ \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2 + \eta \left\| \mathbf U \right\|_2^2\\
s.t.~{\rm{ }}\mathbf s_1^T1 = \mathbf 1,~{s_1^{ii}} = 0, \mathbf s_2^T1 = \mathbf 1,~{s_2^{ii}} = 0,\\
{s_{ij}} \ge 0,~if{\rm{ }}j \in N(i),~{\rm{ }}otherwise{\rm{ }}~0,~{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{eqnarray}
where $n_1$ represents the number of samples in the first hyper-class and $n_2$ represents the number of samples in the second hyper-class. $\bm \mu_1$ is the center of the hash code of the first hyper-class data, and $\bm \mu_2$ is the center of the hash code of the second hyper-class data. In Eq. (\ref{eq17}), it should be noted that we use $\sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2{s_{ij}}} $ and $\sum\nolimits_{i,j}^{{n_2}} {\left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2{s_{ij}}} $ to calculate the similarity of the internal hash codes of the first hyper-class and the second hyper-class respectively. $\sum\nolimits_{i = 1}^{{n_1}} {\left\| {\mathbf H_1^{(i)} - {\bm \mu _2}} \right\|}$ is used to calculate the similarity between the hash code of the internal data of the first hyper-class and the hash code of the second hyper-class. $\sum\nolimits_{i = 1}^{{n_2}} {\left\| {\mathbf H_2^{(i)} - {\bm \mu _1}} \right\|} $ is used to calculate the similarity between the hash code of the internal data of the second hyper-class and the hash code of the first hyper-class. Different from the case of $c=1$, we use a minus sign for the three terms of the objective function, because $\sum\nolimits_{i = 1}^{{n_1}} {\left\| {\mathbf H_1^{(i)} - {\bm \mu _2}} \right\|}$ and $\sum\nolimits_{i = 1}^{{n_2}} {\left\| {\mathbf H_2^{(i)} - {\bm \mu _1}} \right\|}$ represent the similarity between different hyper-classes hash codes, which can reduce the similarity between data hash codes between hyper-classes. {\em i.e.,~} it can make the hash code similarity of the internal data of each hyper-class high and the hash code similarity between different hyper-class data low. Similarly, further, we can get the case of $c > 2$.
When $c > 2$, the objective function is as follows:
\begin{equation}
\label{eq18}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf H,\mathbf U,\mathbf S} \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2s_1^{(ij)}} + \alpha \sum\nolimits_i^{{n_1}} {\left\| {\mathbf s_1^{(i)}} \right\|_2^2} \\
- \beta \sum\nolimits_{i = 1}^{{n_1}} {\left\| {{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)} - {\mathbf V_{{}^\neg 1}}} \right\|_2^2} + \sum\nolimits_{i,j}^{{n_2}} {\left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2s_2^{(ij)}} \\
+ \alpha \sum\nolimits_i^{{n_2}} {\left\| {\mathbf s_2^{(i)}} \right\|_2^2} - \beta \sum\nolimits_{i = 1}^{{n_2}} {\left\| {{\mathbf I_{{}^\neg 2}}\mathbf H_2^{(i)} - {\mathbf V_{{}^\neg 2}}} \right\|_2^2} \\
+ \cdots + \sum\nolimits_{i,j}^{{n_c}} {\left\| {\mathbf H_c^{(i)}\mathbf U - \mathbf H_c^{(j)}\mathbf U} \right\|_2^2s_c^{(ij)}} + \alpha \sum\nolimits_i^{{n_c}} {\left\| {\mathbf s_c^{(i)}} \right\|_2^2} \\
- \beta \sum\nolimits_{i = 1}^{{n_c}} {\left\| {{\mathbf I_{{}^\neg c}}\mathbf H_c^{(i)} - {\mathbf V_{{}^\neg c}}} \right\|_2^2} + \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2 + \eta \left\| \mathbf U \right\|_2^2\\
s.t.{\rm{ }}\mathbf s_1^{{{(i)}^T}}1 = 1,\mathbf s_2^{{{(i)}^T}}1 = 1, \ldots ,\mathbf s_c^{{{(i)}^T}}1 = 1,\\
s_1^{(ii)} = 0,s_2^{(ii)} = 0, \ldots ,s_c^{(ii)} = 0\\
s_1^{(ij)},s_2^{(ij)}, \ldots s_c^{(ij)} \ge 0,if{\rm{ }}j \in N(i),{\rm{ }}otherwise{\rm{ }}~0,{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{equation}
In Eq. (\ref{eq18}), $n_c$ represents the number of samples in the \emph c-th hyper-class. ${\mathbf I_{{}^\neg 1}}{\mathbf I_{{}^\neg 2}} \ldots {\mathbf I_{{}^\neg c}}$ is all 1 vectors of $(c - 1) \times 1$. ${\mathbf V_{{}^\neg 1}} \in {\mathbb R^{(c - 1) \times l}}$ represents the mean matrix composed of the mean values of hash codes in each hyper-class except the first hyper-class. Similarly, ${\mathbf V_{{}^\neg 2}} \in {\mathbb R^{(c - 1) \times l}}$ represents the mean matrix composed of the mean values of hash codes in each hyper-class except the second hyper-class. ${\mathbf V_{{}^\neg c}} \in {\mathbb R^{(c - 1) \times l}}$ represents the mean matrix composed of the mean values of hash codes in each hyper-class except the \emph c hyper-class. ${\mathbf H_1} \in {\mathbb R^{{n_1} \times l}}$, ${\mathbf H_2} \in {\mathbb R^{{n_2} \times l}}$ and ${\mathbf H_c} \in {\mathbb R^{{n_c} \times l}}$ represent the hash code matrix of the samples in the first, second and \emph c hyper-class, respectively. Its idea is the same as in the case of $c=2$. Based on the hyper-class representation, the similarity of hash codes of data within hyper-class is as large as possible, and the similarity of hash codes between hyper-classes is as small as possible.
\subsection{Hash method optimization process}\label{Hash method optimization process}
In this section, we optimize the proposed objective function in two cases ({\em i.e.,~} $c=1$ and $c>1$). We still use the optimization method of alternating iteration ({\em i.e.,~} solving one variable and fixing other variables) to optimize them.
When $c=1$, the optimization process of the objective function ({\em i.e.,~} Eq. (\ref{eq16}) is as follows:
When $\mathbf U$ and $\mathbf S$ are fixed to solve $\mathbf H$, Eq. (\ref{eq16}) can be written as follows:
\begin{eqnarray}
\label{eq19}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H \sum\nolimits_{i,j}^n {\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{ij}}} + \beta \sum\nolimits_{i = 1}^n {\left\| {{\mathbf H^{(i)}} - \bm \mu } \right\|} _2^2\\
+ \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2
\end{array}
\end{eqnarray}
Further, Eq. (\ref{eq19}) can be written as follows:
\begin{eqnarray}
\label{eq20}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H \beta \sum\nolimits_{i = 1}^n {\left( {\left\langle {{\mathbf H^{(i)}},{\mathbf H^{(i)}}} \right\rangle - 2\left\langle {{\mathbf H^{(i)}},\bm \mu } \right\rangle + \left\langle {\bm \mu ,\bm \mu } \right\rangle } \right)} \\
+ \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2 + tr({\mathbf U^T}{\mathbf H^T}\mathbf L\mathbf H\mathbf U)
\end{array}
\end{eqnarray}
Because $tr({\mathbf U^T}{\mathbf H^T}\mathbf L\mathbf H\mathbf U) = tr(\mathbf U{\mathbf U^T}{\mathbf H^T}\mathbf L\mathbf H)$ and $\mathbf U{\mathbf U^T} = \mathbf I$. Therefore, the first term in Eq. (\ref{eq20}) can be written as $tr({\mathbf H^T}\mathbf L\mathbf H)$. We use Eq. (\ref{eq20}) to find the derivative of ${\mathbf H^{(i)}}$, and let the derivative is 0, so we can get the following formula:
\begin{eqnarray}
\label{eq21}
\begin{array}{l}
{\mathbf L^{(i)}}[{({\mathbf E^{(i)}})^T}{\mathbf H^{(i)}} + \sum\limits_{j \ne i}^{n - 1} {{{({\mathbf E^{(i)}})}^T}{\mathbf H^{(j)}}} ]\\
+ (\beta + \lambda ){\mathbf H^{(i)}} - \lambda {({\mathbf X^T}{\mathbf U^T})^i} = \beta \frac{1}{n}\sum\limits_{i = 1}^n {{{({\mathbf X^T}{\mathbf U^T})}^i}}
\end{array}
\end{eqnarray}
where $\mathbf E$ is the identity matrix, the solution of ${\mathbf H^{(i)}}$ can be further obtained as follows:
\begin{equation}
\label{eq22}
\begin{array}{l}
{\mathbf H^{(i)}} = {({\mathbf L^{(i)}}{({\mathbf E^{(i)}})^T} + \beta + \lambda )^{ - 1}}\\
(\lambda {({\mathbf X^T}{\mathbf U^T})^i} - {\mathbf L^{(i)}}\sum\limits_{j \ne i}^{n - 1} {{{({\mathbf E^{(i)}})}^T}{\mathbf H^{(j)}}} + \beta \frac{1}{n}\sum\limits_{i = 1}^n {{{({\mathbf X^T}{\mathbf U^T})}^i}} )
\end{array}
\end{equation}
When $\mathbf H$ and $\mathbf S$ are fixed to solve $\mathbf U$, Eq. (\ref{eq16}) can be written as follows:
\begin{eqnarray}
\label{eq23}
\begin{array}{l}
\mathop {\min }\limits_\mathbf U \sum\nolimits_{i,j}^n {\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{ij}}} \\
+ \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2 + \eta \left\| \mathbf U \right\|_2^2\\
s.t.~{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{eqnarray}
Further, Eq. (\ref{eq23}) can be rewritten as follows:
\begin{eqnarray}
\label{eq24}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H tr({\mathbf U^T}{\mathbf H^T}\mathbf L\mathbf H\mathbf U)\\
+ \lambda (({\mathbf H^T} - \mathbf U\mathbf X)(\mathbf H - {\mathbf X^T}{\mathbf U^T})) + \eta \left\| \mathbf U \right\|_2^2\\
s.t.{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{eqnarray}
Let's take $f(\mathbf U) = tr({\mathbf U^T}{\mathbf H^T}\mathbf L\mathbf H\mathbf U) + \lambda (({\mathbf H^T} - \mathbf U\mathbf X)(\mathbf H - {\mathbf X^T}{\mathbf U^T})) + \eta \left\| \mathbf U \right\|_2^2$ and take the derivative of $\mathbf U$, and we can get:
\begin{eqnarray}
\label{eq25}
\begin{array}{l}
\frac{{\partial f(\mathbf U)}}{{\partial \mathbf U}} = 2{\mathbf H^T}\mathbf L\mathbf H\mathbf U\\
+ \lambda (2\mathbf U\mathbf X{\mathbf X^T} - 2{\mathbf H^T}{\mathbf X^T}) + 2\eta \mathbf I\mathbf U
\end{array}
\end{eqnarray}
Because Eq. (\ref{eq24}) has an orthogonal restriction on $\mathbf U$, {\em i.e.,~} $\mathbf U{\mathbf U^T} = \mathbf I$, we use the method in literature \cite{wen2013feasible} to solve it.
When $\mathbf H$ and $\mathbf U$ are fixed to solve $\mathbf S$, Eq. (\ref{eq16}) can be written as follows:
\begin{eqnarray}
\label{eq26}
\begin{array}{l}
\mathop {\min }\limits_\mathbf U \sum\nolimits_{i,j}^n {\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{ij}}} + \alpha \sum\nolimits_i^n {\left\| {{\mathbf s_i}} \right\|_2^2} \\
s.t.~{\rm{ }}\mathbf s_i^T\mathbf 1 = 1,~{s_{ii}} = 0,\\
{s_{ij}} \ge 0,~if{\rm{ }}j \in N(i),~{\rm{ }}otherwise{\rm{ }}0
\end{array}
\end{eqnarray}
Optimizing $\mathbf s$ is equivalent to optimizing each ${\mathbf s_i}(i = 1,...,n)$ separately, so we further convert the optimization problem into the following formula:
\begin{eqnarray}
\label{eq27}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf s_i^T\mathbf 1 = 1,{s_{i,i}} = 0,{s_{i,j}} \ge 0} \sum\nolimits_{i,j}^n {(\left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2{s_{i,j}}} \\ + \alpha s_{i,j}^2)
\end{array}
\end{eqnarray}
Here, let $\mathbf Z \in {\mathbb R^{n \times n}}$, where ${Z_{i,j}} = \left\| {{\mathbf H^{(i)}}\mathbf U - {\mathbf H^{(j)}}\mathbf U} \right\|_2^2$, so equation (26) further becomes:
\begin{eqnarray}
\label{eq28}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf s_i^T\mathbf 1 = 1,~{s_{i,i}} = 0,~{s_{i,j}} \ge 0} \left\| {{\mathbf s_i} + \frac{1}{{2\alpha }}{\mathbf Z_i}} \right\|_2^2
\end{array}
\end{eqnarray}
Under KKT conditions, we can get the following results:
\begin{eqnarray}
\label{eq29}
\begin{array}{l}
{s_{i,j}} = {( - \frac{1}{{2\alpha }}{Z_{i,j}} + \tau )_ + }
\end{array}
\end{eqnarray}
Since each hash code has close neighbors, we arrange each ${\mathbf Z_i}(i = 1,...,n)$ in descending order, {\em i.e.,~} ${\hat Z_i} = \{ {\hat Z_{i,1}},...,{\hat Z_{i,n}}\}$, then we can know that ${s_{i,k + 1}} = 0$ and ${s_{i,k}} > 0$. Available:
\begin{eqnarray}
\label{eq30}
\begin{array}{l}
- \frac{1}{{2\alpha }}{\hat Z_{i,k + 1}} + \tau \le 0
\end{array}
\end{eqnarray}
Under the condition ${\mathbf s_i}^T\mathbf 1 = 1$, we can get:
\begin{eqnarray}
\label{eq31}
\begin{array}{l}
\sum\nolimits_{j = 1}^k {(\frac{1}{{2\alpha }}{{\hat Z}_{i,k}} + \tau )} = 1 \Rightarrow \tau = \frac{1}{k} + \frac{1}{{2k\alpha }}\sum\nolimits_{j = 1}^k {{{\hat Z}_{i,k}}}
\end{array}
\end{eqnarray}
When $c>1$, the optimization process of the objective function ({\em i.e.,~} Eq. (\ref{eq18}) is as follows:
When $\mathbf U$ and $\mathbf S$ are fixed to solve $\mathbf H$, Eq. (\ref{eq18}) can be written as follows:
\begin{eqnarray}
\label{eq32}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2s_1^{(ij)}} \\
- \beta \sum\nolimits_{i = 1}^{{n_1}} {\left\| {{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)} - {\mathbf V_{{}^\neg 1}}} \right\|_2^2} \\
+ \sum\nolimits_{i,j}^{{n_2}} {\left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2s_2^{(ij)}} \\
- \beta \sum\nolimits_{i = 1}^{{n_2}} {\left\| {{\mathbf I_{{}^\neg 2}}\mathbf H_2^{(i)} - {\mathbf V_{{}^\neg 2}}} \right\|_2^2} \\
+ \cdots + \sum\nolimits_{i,j}^{{n_c}} {\left\| {\mathbf H_c^{(i)}\mathbf U - \mathbf H_c^{(j)}\mathbf U} \right\|_2^2s_c^{(ij)}} \\
- \beta \sum\nolimits_{i = 1}^{{n_c}} {\left\| {{\mathbf I_{{}^\neg c}}\mathbf H_c^{(i)} - {\mathbf V_{{}^\neg c}}} \right\|_2^2} + \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2
\end{array}
\end{eqnarray}
In Eq. (\ref{eq32}), we can find that $\mathbf H = [{\mathbf H_1};{\mathbf H_2}; \ldots ;{\mathbf H_c}] \in {\mathbb R^{n \times l}}$. Therefore, we still use the alternating iteration method to solve it. {\em i.e.,~} when fixing ${\mathbf H_2}, \ldots ,{\mathbf H_c}$ to solve $\mathbf H_1$, Eq. (\ref{eq32}) can be written as follows:
\begin{eqnarray}
\label{eq33}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2s_1^{(ij)}} \\
- \beta \sum\nolimits_{i = 1}^{{n_1}} {\left\| {{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)} - {\mathbf V_{{}^\neg 1}}} \right\|_2^2} + \lambda \left\| {{\mathbf H_1} - {\mathbf X_1}^T{\mathbf U^T}} \right\|_F^2
\end{array}
\end{eqnarray}
where ${\mathbf V_{{}^\neg 1}} \in {\mathbb R^{(c - 1) \times l}}$ represents the mean matrix composed of the mean values of hash codes in each hyper-class except the first hyper-class. Eq. (\ref{eq33}) can be further written as follows:
\begin{eqnarray}
\label{eq34}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H - \beta \sum\nolimits_{i = 1}^{{n_1}} {\left( \begin{array}{l}
\left\langle {{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)},{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)}} \right\rangle - 2\left\langle {{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)},{\mathbf V_{\neg 1}}} \right\rangle \\
+ \left\langle {{\mathbf V_{\neg 1}},{\mathbf V_{\neg 1}}} \right\rangle
\end{array} \right)} \\
+ \lambda \left\| {{\mathbf H_1} - {\mathbf X_1}^T{\mathbf U^T}} \right\|_F^2 + tr({\mathbf U^T}{\mathbf H_1}^T{\mathbf L_1}{\mathbf H_1}\mathbf U)
\end{array}
\end{eqnarray}
Because $tr({\mathbf U^T}{\mathbf H_1}^T{\mathbf L_1}{\mathbf H_1}\mathbf U) = tr(\mathbf U{\mathbf U^T}{\mathbf H_1}^T{\mathbf L_1}{\mathbf H_1})$ and $\mathbf U{\mathbf U^T} = \mathbf I$. Therefore, the first term in Eq. (\ref{eq34}) can be written as $tr({\mathbf H_1}^T{\mathbf L_1}{\mathbf H_1})$. We use Eq. (\ref{eq34}) to find the derivative of $\mathbf H_1^{(i)}$, and make derivative is 0, so we can get the following formula:
\begin{eqnarray}
\label{eq35}
\begin{array}{l}
\mathbf L_1^{(i)}[{({\mathbf E^{(i)}})^T}\mathbf H_1^{(i)} + \sum\limits_{j \ne i}^{{n_1} - 1} {{{({\mathbf E^{(j)}})}^T}\mathbf H_1^{(j)}} ] \\+ (\beta + \lambda )\hat H_1^{(i)} - \lambda {({\mathbf X_1}^T{\mathbf U^T})^i} = - \beta \mathbf I_{{}^\neg 1}^T{\mathbf V_{{}^\neg 1}}
\end{array}
\end{eqnarray}
According to Eq. (\ref{eq35}), we can get the solution of $\mathbf H_1$ as follows:
\begin{eqnarray}
\label{eq36}
\begin{array}{l}
\hat H_1^{(i)} = {(\mathbf L_1^{(i)}{({\mathbf E^{(i)}})^T} - \beta + \lambda )^{ - 1}}\\(\lambda {({\mathbf X_1}^T{\mathbf U^T})^i} - \mathbf L_1^{(i)}\sum\limits_{j \ne i}^{{n_1} - 1} {{{({\mathbf E^{(j)}})}^T}\mathbf H_1^{(j)}} - \beta \mathbf I_{{}^\neg 1}^T{\mathbf V_{{}^\neg 1}})
\end{array}
\end{eqnarray}
Similarly, the solution of ${\mathbf H_2},{\mathbf H_3}, \ldots ,{\mathbf H_c}$ can be obtained as follows:
\begin{eqnarray}
\label{eq37}
\begin{array}{l}
\left\{ {\begin{array}{*{20}{c}}
\begin{array}{l}
\hat H_2^{(i)} = {(\mathbf L_2^{(i)}{({\mathbf E^{(i)}})^T} - \beta + \lambda )^{ - 1}}\\
(\lambda {({\mathbf X_2}^T{\mathbf U^T})^i} - \mathbf L_2^{(i)}\sum\limits_{j \ne i}^{{n_2} - 1} {{{({\mathbf E^{(j)}})}^T}\mathbf H_2^{(j)}} - \beta \mathbf I_{{}^\neg 2}^T{\mathbf V_{{}^\neg 2}})
\end{array}\\
\begin{array}{l}
\hat H_3^{(i)} = {(\mathbf L_3^{(i)}{({\mathbf E^{(i)}})^T} - \beta + \lambda )^{ - 1}}\\
(\lambda {({\mathbf X_3}^T{\mathbf U^T})^i} - \mathbf L_3^{(i)}\sum\limits_{j \ne i}^{{n_3} - 1} {{{({\mathbf E^{(j)}})}^T}\mathbf H_3^{(j)}} - \beta \mathbf I_{{}^\neg 3}^T{\mathbf V_{{}^\neg 3}})
\end{array}\\
\vdots \\
\begin{array}{l}
\hat H_c^{(i)} = {(\mathbf L_c^{(i)}{({\mathbf E^{(i)}})^T} - \beta + \lambda )^{ - 1}}\\
(\lambda {({\mathbf X_c}^T{\mathbf U^T})^i} - \mathbf L_c^{(i)}\sum\limits_{j \ne i}^{{n_c} - 1} {{{({\mathbf E^{(j)}})}^T}\mathbf H_c^{(j)}} - \beta \mathbf I_{{}^\neg c}^T{\mathbf V_{{}^\neg c}})
\end{array}
\end{array}} \right.
\end{array}
\end{eqnarray}
After we get the solution of ${\mathbf H_1},{\mathbf H_2}, \ldots ,{\mathbf H_c}$, we can splice them together and get the solution of $\mathbf H$.
When $\mathbf H$ and $\mathbf S$ are fixed to solve $\mathbf U$, Eq. (\ref{eq18}) can be written as follows:
\begin{eqnarray}
\label{eq38}
\begin{array}{l}
\mathop {\min }\limits_U \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2s_1^{(ij)}} \\
+ \sum\nolimits_{i,j}^{{n_2}} {\left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2s_2^{(ij)}} \\
+ \cdots + \sum\nolimits_{i,j}^{{n_c}} {\left\| {\mathbf H_c^{(i)}\mathbf U - \mathbf H_c^{(j)}\mathbf U} \right\|_2^2s_c^{(ij)}} \\
+ \lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2 + \eta \left\| \mathbf U \right\|_2^2\\
s.t.~{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{eqnarray}
Further, Eq. (\ref{eq38}) can be written as follows:
\begin{eqnarray}
\label{eq39}
\begin{array}{l}
\mathop {\min }\limits_\mathbf H tr({\mathbf U^T}{\mathbf H_1}^T{\mathbf L_1}{\mathbf H_1}\mathbf U) + tr({\mathbf U^T}{\mathbf H_2}^T{\mathbf L_2}{\mathbf H_2}\mathbf U)\\
+ \cdots + tr({\mathbf U^T}{\mathbf H_c}^T{\mathbf L_c}{\mathbf H_c}\mathbf U)\\
+ \lambda (({\mathbf H^T} - \mathbf U\mathbf X)(\mathbf H - {\mathbf X^T}{\mathbf U^T})) + \eta \left\| \mathbf U \right\|_2^2\\
s.t.~{\rm{ }}\mathbf U{\mathbf U^T} = \mathbf I
\end{array}
\end{eqnarray}
We use Eq. (\ref{eq39}) to derive $\mathbf U$, and we can get:
\begin{eqnarray}
\label{eq40}
\begin{array}{l}
\frac{{\partial f(\mathbf U)}}{{\partial \mathbf U}} = 2{\mathbf H_1}^T{\mathbf L_1}{\mathbf H_1}\mathbf U + 2{\mathbf H_2}^T{\mathbf L_2}{\mathbf H_2}\mathbf U\\
+ \cdots + 2{\mathbf H_c}^T{\mathbf L_c}{\mathbf H_c}\mathbf U\\
+ \lambda (2\mathbf U\mathbf X{\mathbf X^T} - 2{\mathbf H^T}{\mathbf X^T}) + 2\eta \mathbf I \mathbf U
\end{array}
\end{eqnarray}
Because Eq. (\ref{eq39}) has an orthogonal restriction on $\mathbf U$, {\em i.e.,~} $\mathbf U{\mathbf U^T} = \mathbf I$, we still use the method in literature \cite{wen2013feasible} to solve it.
When $\mathbf H$ and $\mathbf U$ are fixed to solve $\mathbf S$, Eq. (\ref{eq18}) can be written as follows:
\begin{eqnarray}
\label{eq41}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf H,\mathbf U,\mathbf S} \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2s_1^{(ij)}} + \alpha \sum\nolimits_i^{{n_1}} {\left\| {\mathbf s_1^{(i)}} \right\|_2^2} \\
+ \sum\nolimits_{i,j}^{{n_2}} {\left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2s_2^{(ij)}} + \alpha \sum\nolimits_i^{{n_2}} {\left\| {\mathbf s_2^{(i)}} \right\|_2^2} \\
+ \cdots + \sum\nolimits_{i,j}^{{n_c}} {\left\| {\mathbf H_c^{(i)}\mathbf U - \mathbf H_c^{(j)}\mathbf U} \right\|_2^2s_c^{(ij)}} + \alpha \sum\nolimits_i^{{n_c}} {\left\| {\mathbf s_c^{(i)}} \right\|_2^2} \\
s.t.~{\rm{ }}\mathbf s_1^{{{(i)}^T}}\mathbf 1 = 1,~\mathbf s_2^{{{(i)}^T}}\mathbf 1 = 1, \ldots ,~\mathbf s_c^{{{(i)}^T}}1 = 1,\\
s_1^{(ii)} = 0,~s_2^{(ii)} = 0, \ldots ,~s_c^{(ii)} = 0\\
s_1^{(ij)},~s_2^{(ij)}, \ldots s_c^{(ij)} \ge 0,if{\rm{ }}j \in N(i),~{\rm{ }}otherwise{\rm{ }}0
\end{array}
\end{eqnarray}
Since $\mathbf S = {[{\mathbf s_1},{\mathbf s_2}, \ldots ,{\mathbf s_c}]^T}$, we still adopt the alternating iterative method to solve it. {\em i.e.,~} when fixing ${\mathbf s_2}, \ldots ,{\mathbf s_c}$ to solve $\mathbf s_1$, Eq. (\ref{eq41}) can be written as follows:
\begin{eqnarray}
\label{eq42}
\begin{array}{l}
\mathop {\min }\limits_{\mathbf H,\mathbf U,\mathbf S} \sum\nolimits_{i,j}^{{n_1}} {\left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2s_1^{(ij)}} + \alpha \sum\nolimits_i^{{n_1}} {\left\| {\mathbf s_1^{(i)}} \right\|_2^2} \\
s.t.~{\rm{ }}\mathbf s_1^{{{(i)}^T}}1 = 1,~s_1^{(ii)} = 0\\
s_1^{(ij)} \ge 0,~if{\rm{ }}j \in N(i),~{\rm{ }}otherwise{\rm{ }}0
\end{array}
\end{eqnarray}
According to the method of solving $\mathbf S$ when $c=1$, {\em i.e.,~} Eq. (\ref{eq29}), we can get the solution of $\mathbf s_1$ as follows:
\begin{eqnarray}
\label{eq43}
\begin{array}{l}
s_1^{(i,j)} = {( - \frac{1}{{2\alpha }}Z_1^{(i,j)} + {\tau _1})_ + }
\end{array}
\end{eqnarray}
where $\mathbf Z \in {R^{{n_1} \times {n_1}}}$, $Z_1^{(i,j)} = \left\| {\mathbf H_1^{(i)}\mathbf U - \mathbf H_1^{(j)}\mathbf U} \right\|_2^2$, ${\tau _1} = \frac{1}{k} + \frac{1}{{2k\alpha }}\sum\nolimits_{j = 1}^k {\hat Z_1^{i,k}}$. Similarly, we can get the solution of ${\mathbf s_2}, \ldots ,{\mathbf s_c}$ as follows:
\begin{eqnarray}
\label{eq44}
\begin{array}{l}
\left\{ {\begin{array}{*{20}{c}}
{s_2^{(i,j)} = {{( - \frac{1}{{2\alpha }}Z_2^{(i,j)} + {\tau _2})}_ + }}\\
{s_3^{(i,j)} = {{( - \frac{1}{{2\alpha }}Z_3^{(i,j)} + {\tau _3})}_ + }}\\
\vdots \\
{s_c^{(i,j)} = {{( - \frac{1}{{2\alpha }}Z_c^{(i,j)} + {\tau _c})}_ + }}
\end{array}} \right.
\end{array}
\end{eqnarray}
where $Z_2^{(i,j)},Z_3^{(i,j)}, \ldots ,Z_c^{(i,j)}$ and ${\tau _2},{\tau _3}, \ldots ,{\tau _c}$ are respectively as follows:
\begin{eqnarray}
\label{eq45}
\begin{array}{l}
\left\{ {\begin{array}{*{20}{c}}
{Z_2^{(i,j)} = \left\| {\mathbf H_2^{(i)}\mathbf U - \mathbf H_2^{(j)}\mathbf U} \right\|_2^2}\\
{Z_3^{(i,j)} = \left\| {\mathbf H_3^{(i)}\mathbf U - \mathbf H_3^{(j)}\mathbf U} \right\|_2^2}\\
\vdots \\
{Z_c^{(i,j)} = \left\| {\mathbf H_c^{(i)}\mathbf U - \mathbf H_c^{(j)}\mathbf U} \right\|_2^2}
\end{array}} \right.
\end{array}
\end{eqnarray}
\begin{eqnarray}
\label{eq46}
\begin{array}{l}
\left\{ {\begin{array}{*{20}{c}}
{{\tau _2} = \frac{1}{k} + \frac{1}{{2k\alpha }}\sum\nolimits_{j = 1}^k {\hat Z_2^{i,k}} }\\
{{\tau _3} = \frac{1}{k} + \frac{1}{{2k\alpha }}\sum\nolimits_{j = 1}^k {\hat Z_3^{i,k}} }\\
\vdots \\
{{\tau _c} = \frac{1}{k} + \frac{1}{{2k\alpha }}\sum\nolimits_{j = 1}^k {\hat Z_c^{i,k}} }
\end{array}} \right.
\end{array}
\end{eqnarray}
So far, the optimization process of the proposed algorithm is completed.
\begin{algorithm}
\setlength{\algomargin}{0.5em}
\SetAlgoLined
\caption{\label{alg1} Training stage of HCH.}
\IncMargin{0.5em}
\KwIn{Training set $\mathbf{X} \in \mathbb{R}^{d \times n}$, $c$ (the number of clusters), $l$ (the number of bits), adjustable parameters $\varsigma$, $\sigma$, $\alpha$, $\beta$, $\lambda$ and $\eta$ \;}
\KwOut{$\mathbf v \in {\mathbb R^{d \times 1}}$, $\mathbf B \in {\mathbb R^{n \times l}}$, $
\mathbf U \in {\mathbb R^{l \times d}}$;}
Initialize \emph t=0\;
Randomly initialize $\mathbf W^{(0)}$ and $\mathbf v^{(0)}$\;
\For{$i=1 \to d $}
{
Calculate $k({\mathbf X_{{}^\neg i}})$ via Eq. (\ref{eq3})\;
}
\Repeat{converge}{
Update $\mathbf F^{(t)}$ via
${F_{ii}} = \frac{1}{{2\sqrt {\left\| {{\mathbf W^i}} \right\|_2^2 + \varepsilon } }}$\;
Update $\mathbf W^{(t+1)}$ via Eq. (\ref{eq10})\;
Updata $\mathbf v^{(t+1)}$ via Eq. (\ref{eq13})\;
\emph{t} = \emph{t}+1 \;
}
Get the potential decision feature $df$ via Eq. (\ref{eq14})\;
The method in reference \cite{makarychev2022performance} is used to cluster or divide the potential decision feature, so as to construct hyper-class\;
Initialize \emph t=0, initialize $\mathbf S$ to all zero matrix, randomly initialize $\mathbf U^{(0)}$ and $\mathbf H^{(0)}$\;
\Repeat{converge}{
Update $\mathbf H^{(t+1)}$ via Eqs. (\ref{eq36}) and (\ref{eq37}) \;
Update $\mathbf U^{(t+1)}$ via Eq. (\ref{eq40}) and Reference \cite{wen2013feasible}\;
Updata $\mathbf S^{(t+1)}$ via Eqs. (\ref{eq43}) and (\ref{eq44})\;
\emph{t} = \emph{t}+1 \;
}
Calculate $\mathbf B$ via $sgn(\mathbf X^T\mathbf U^T)$ for all the data points $\mathbf X$;
\end{algorithm}
\begin{algorithm}
\setlength{\algomargin}{0.5em}
\SetAlgoLined
\caption{\label{alg2} Testing stage of HCH.}
\IncMargin{0.5em}
\KwIn{$\mathbf y \in {\mathbb R^d}$ and $\mathbf U \in {\mathbb R^{l \times d}}$\;}
\KwOut{${\mathbf b_y} \in {\mathbb R^l}$;}
Calculate $\mathbf b_y$ via $sgn(\mathbf y^T\mathbf U^T)$\;
Calculate the Hamming distance between $\mathbf b_y$ and $\mathbf B$;
\end{algorithm}
\subsection{Summary of algorithm}
As shown in algorithm \ref{alg1} and algorithm \ref{alg2}, the proposed HCH algorithm can be divided into four steps, {\em i.e.,~} 1. Use the relationship between each feature and all other features to learn the potential decision features in the data, as shown in steps 1-12. 2. establish hyper-class according to the obtained decision features and clustering algorithm, as shown in step 13. 3. construct the hash learning objective function according to the hyper-class in the data, {\em i.e.,~} Eq. (\ref{eq18}). And optimize the objective function, as shown in step 14-20. 4. perform binary hash coding on the data according to the learned $\mathbf U$ matrix.
In addition, in the proposed HCH algorithm, we use the alternative iterative optimization method to solve the objective function, which can make the algorithm converge quickly\cite{chen2019extended}. In section 4.4, we also verify its convergence.
\subsection{Time complexity analysis}
In the proposed HCH algorithm, in the training phase, the time complexity of finding decision features, clustering method, hash function learning and binarization are $O(t({n^2}d + d))$, $O(dcn)$, $O(t(n{l^2} + ndl + {n^2}l))$ and $O(nl)$ respectively. Where, $t$ represents the number of iterations, $n$ represents the number of samples, $d$ represents the number of features, $l$ represents the number of bits, and $c$ represents the number of clusters. In the test phase, we need to spend $O(dl)$ time to binarize the test data and $O(1)$ time to perform reverse lookup in the hash table. Since $l$ is usually less than $d$ and $t$ is a small number, the time complexity of HCH is $O({n^2}d + dcn)$ in the training phase and $O(dl)$ in the testing phase. In addition, we also list the time complexity comparison of some popular hash algorithms, such as ITQ, FastH, KNNH, CH, LSH, MDSH and ALECH, as shown in Table \ref{tab2}.
In the training phase, the spatial complexity of HCH is $O(d(c + n))$, and that in the test phase is $O((d + m)l)$.
\begin{table}[!tb]
\centering
\caption{\footnotesize Time complexity of comparison algorithm.}
\centering
{\footnotesize
\begin{tabular}[c]{|c|c|c|} \hline
&Training phase &Test phase \\ \hline
ITQ &$O({d^2}n + {l^3})$ &$O(dl)$ \\ \hline
FastH &$O({d^2}n)$ &$O(dl)$ \\ \hline
KNNH &$O({n^2}(d + \log n))$ &$O(1)$ \\ \hline
CH &$O(2t{d^2}n + tncd)$ & $O(dl)$ \\ \hline
LSH &$O(1)$ &$O(1)$ \\ \hline
MDSH &$O({d^2}n)$ & $O(dl)$ \\ \hline
ALECH &$O(nld + {d^3} + n{d^2})$ & $O(dl)$ \\ \hline
HCH &$O({n^2}d + dcn)$ & $O(dl)$ \\ \hline
\end{tabular}}
\begin{tablenotes}
\footnotesize
\item where $t$ represents the number of iterations, $n$ represents the number of samples, $d$ represents the number of features, $l$ represents the number of bits, and $c$ represents the number of clusters.
\end{tablenotes}
\label{tab2}
\end{table}
\section{Experiments} \label{experiments}
In this section, we compare the performance of the proposed algorithm with seven comparison algorithms on four real datasets.
\begin{figure*}
\caption{The value of each iteration of the objective function on all datasets}
\label{fig3}
\end{figure*}
\subsection{Datasets}
In our experiment, we used four real datasets, namely, Mnist, Cifar, Labelme and Place. Their details are as follows:
The Mnist dataset contains 70000 images and their labels. Each image is represented by a 784 dimensional vector. 60000 of them are training sets and 10000 are testing sets. In the testing sets, the first 5000 data are more regular than the last 5000 data. The reason is that they come from different sources.
The Cifar dataset was collected by Alex krizhevsky, Vinod Nair, and Geoffrey Hinton. It contains a total of 60000 color images, each of which is 32$\times$32. It has a total of 10 classes, each containing 6000 images. In this paper, we use 50000 images as the training set and 10000 images as the testing set.
The Labelme dataset contains a total of 50000 images, and each image is a 256$\times$256 jpeg image. Among them, 45007 images are used as the training set and 4993 images are used as the testing set.
There are 2448872 images in the Place dataset. They can be divided into 205 classes, and each sample has 128 features.
\begin{figure*}
\caption{The HAM2 of our proposed method with different parameters' setting {\em i.e.,~}
\label{fig4}
\end{figure*}
\subsection{Comparied Algorithms}
ITQ\cite{gong2012iterative}: this method is a classic data dependent hash algorithm. It learns the binary hash code of the data by minimizing the quantization error between the data and the vertex of the zero center binary hypercube. In addition, it can be used for unsupervised PCA embedding and supervised canonical correlation analysis (CCA) embedding.
FastH\cite{lin2014fast}: it is a supervised hash algorithm. Different from other nonlinear hash algorithms, it uses decision tree to mine nonlinear relationships in data rather than kernel functions. Specifically, it first proposes a sub module formula for hash code binary reasoning. Then it trains the enhanced decision tree to adapt it to binary hash codes. Finally, it verifies the effect on high-dimensional data in experiments.
LSH\cite{andoni2017optimal}: this method is a locally sensitive hash algorithm. Its core idea is to divide the random space of data. Given a query sample, it first searches in the same divided space as the query sample. In other words, it improves query speed by shortening the retrieval range of data.
MDSH\cite{weiss2012multidimensional}: it is a multi-dimensional hash algorithm. It aims to establish the affinity relationship between data rather than the distance between data. Specifically, it first establishes a formula for learning binary coding to obtain the affinity matrix, and then it uses the threshold eigenvector of the affinity matrix to obtain bits. Finally, in the experiment, it also shows some results with the increase of the number of bits.
KNNH\cite{he2019k}: it is a KNN hash algorithm. Specifically, it first uses PCA algorithm to reduce the dimension of the original data. Then KNN algorithm is used to find k nearest neighbors of each data. Finally, it learns the binary representation in each subspace of the data.
CH\cite{weng2020concatenation}: this method is a hash algorithm based on clustering technology. Different from other hash algorithms based on clustering technology, it mainly focuses on reducing the influence of clustering boundary samples. It uses alternating iterative algorithm to carry out hash learning and clustering at the same time, so as to maintain the relative position of each data to the cluster center. In addition, it also splices the hash function learning in each cluster to obtain the binary code corresponding to all data.
ALECH\cite{li2021adaptive}: this method is a hash algorithm based on adaptive label correlation. Specifically, it first uses the least square loss to obtain the relationship between semantic labels and hash codes. Then it uses the alternating iterative optimization method to optimize and solve the proposed objective function. Finally, it maps the data to the kernel space to obtain the hash function. This method is a supervised hash algorithm.
\begin{figure*}
\caption{The HAM2 results of our proposed method with different parameters' setting {\em i.e.,~}
\label{fig5}
\end{figure*}
\begin{figure*}
\caption{Precision-recall curves of all hashing methods, on four data sets at 16 hash bits}
\label{fig6}
\end{figure*}
\begin{figure*}
\caption{Precision-recall curves of all hashing methods, on four data sets at 32 hash bits}
\label{fig7}
\end{figure*}
\begin{figure*}
\caption{Precision-recall curves of all hashing methods, on four data sets at 48 hash bits}
\label{fig8}
\end{figure*}
\begin{figure*}
\caption{Precision-recall curves of all hashing methods, on four data sets at 64 hash bits}
\label{fig9}
\end{figure*}
\begin{figure*}
\caption{HAM2 results of all hashing methods on four data sets at different number of hash bits, {\em i.e.,~}
\label{fig10}
\end{figure*}
\begin{table*}[!ht]
\centering
\caption{ Map results of all hash algorithms on Mnist and Cifar datasets.}
\centering
{
\centering
{
\begin{tabular}
{|p{1.3cm}|p{0.9cm}p{0.9cm}p{0.9cm}p{0.9cm}|p{0.9cm}p{0.9cm}p{0.9cm}p{0.9cm}|} \hline
\multirow{1}{*}{Method} &\multicolumn{4}{c|}{Mnist} &\multicolumn{4}{c|}{Cifar-10} \\ \hline
&16 &32 &48 &64 &16 &32 &48 &64 \\\hline
\multirow{1}{*}{ITQ} &0.3929 &0.4102 &0.4054 &0.3985 &0.1725 &0.1770 &0.1774 &0.1747 \\ \hline
\multirow{1}{*}{FastH} &0.2745 &0.2574 &0.2538 &0.2442 &0.1318 &0.1366 &0.1316 &0.1309 \\ \hline
\multirow{1}{*}{KNNH} &0.3092 &0.3046 &0.3049 & 0.3096 &0.1570 &0.1852 &0.1734 &0.1717 \\ \hline
\multirow{1}{*}{CH} &0.4135 &0.3768 &0.3279 &0.3388 &0.1580 &0.1525 &0.1594 & 0.1482 \\ \hline
\multirow{1}{*}{LSH} &0.2103 &0.2710 &0.2885 &0.2974 &0.1239 &0.1470 &0.1462 & 0.1520 \\ \hline
\multirow{1}{*}{MDSH} &0.3270 &0.3953 &0.3483 &0.3050 &0.1685 &0.1445 &0.1508 & 0.1407 \\ \hline
\multirow{1}{*}{ALECH} &0.2419 &0.2225 &0.2133 &0.2067 &0.1374 &0.1273 &0.1246 &0.1202 \\ \hline
\multirow{1}{*}{HCH} &0.3758 &0.4104 &0.4101 &0.4182 &0.1736 &0.1798 &0.1791 &0.1752 \\ \hline
\end{tabular}
}
}
\label{tab3}
\end{table*}
\begin{table*}[!ht]
\centering
\caption{ Map results of all hash algorithms on Labelme and Place datasets.}
\centering
{
\centering
{
\begin{tabular}
{|p{1.3cm}|p{0.9cm}p{0.9cm}p{0.9cm}p{0.9cm}|p{0.9cm}p{0.9cm}p{0.9cm}p{0.9cm}|} \hline
\multirow{1}{*}{Method} &\multicolumn{4}{c|}{Labelme} &\multicolumn{4}{c|}{Place} \\ \hline
&16 &32 &48 &64 &16 &32 &48 &64 \\\hline
\multirow{1}{*}{ITQ} &0.1971 &0.2121 &0.2162 &0.2377 &0.0706 &0.1145 &0.1482 &0.1654 \\ \hline
\multirow{1}{*}{FastH} &0.1942 &0.1894 &0.1949 &0.1950 &0.0854 &0.1131 &0.1200 & 0.1400 \\ \hline
\multirow{1}{*}{KNNH} &0.2125 &0.1957 &0.2048 &0.1831 &0.1029 &0.1256 &0.1376 &0.1385 \\ \hline
\multirow{1}{*}{CH} &0.1904 &0.1745 &0.1775 &0.1731 &0.1022 &0.1211 &0.1381 &0.1312 \\ \hline
\multirow{1}{*}{LSH} &0.1582 &0.1659 &0.1732 &0.1819 &0.0528 &0.0860 &0.1135 &0.1218 \\ \hline
\multirow{1}{*}{MDSH} &0.1779 &0.1772 &0.1714 & 0.1816 &0.0713 &0.1037 &0.1226 & 0.1427 \\ \hline
\multirow{1}{*}{ALECH} &0.1389 &0.1275 &0.1251 & 0.1225 &0.0524 &0.0906 &0.1114 & 0.1285 \\ \hline
\multirow{1}{*}{HCH} &0.1984 &0.2192 &0.2187 &0.2401 &0.1154 &0.1283 &0.1529 &0.1863 \\ \hline
\end{tabular}
}
}
\label{tab4}
\end{table*}
\subsection{Experimental setting}
In this paper, we mainly carry out two part experiments: 1 The convergence and parameter sensitivity of the proposed algorithm are analyzed. 2. The performance comparison between the proposed algorithm and other outstanding hash algorithms. In the second part of the experiment, we compared the precision recall performance of all algorithms on four data sets, The mean average accuracy ($MAP$) of different bits and the average precision of Hamming radius 2 corresponding to different bits. In addition, in the proposed algorithm, the value range of parameters $\alpha$, $\beta$, $\lambda$ and $\eta$ is: $[{10^{ - 3}},{10^{ - 2}},10,1,10,{10^2},{10^3}]$. $MAP$ is the mean value of the average precision ($AP$) returned by the retrieval of all query samples. The calculation formula of $AP$ is as follows:
\begin{eqnarray}
\label{eq47}
\begin{array}{l}
AP = \frac{1}{l}\sum\limits_{r = 1}^K {Precision(r)\sigma (r)}
\end{array}
\end{eqnarray}
where $l$ is the number of samples of real relevant nearest neighbors, and $Precision(r)$ is the accuracy of the training data retrieved by top r. If the r-th instance is related to the sample of the query, then $\sigma (r) = 1$, otherwise $\sigma (r) = 0$. $MAP$ is defined as follows:
\begin{eqnarray}
\label{eq48}
\begin{array}{l}
MAP = \frac{1}{m}\sum\limits_{r = 1}^m {AP(i)}
\end{array}
\end{eqnarray}
where $m$ represents the number of query samples, and $AP(i)$ is the average accuracy of the \emph i-th sample.
\subsection{Convergence and parameter sensitivity analysis of the algorithm}
As shown in Fig. \ref{fig3}, it shows the objective function value of each iteration of the proposed algorithm on four data sets. We can find that the proposed algorithm converges within 5 iterations on four data sets. This shows that the proposed algorithm has fast convergence speed and can greatly reduce the time cost of training. The proposed objective function has four parameters, namely, $\alpha$, $\beta$, $\lambda$ and $\eta$, as shown in Eq. (\ref{eq18}). We have done the HAM2 result experiment of the proposed method under different parameter settings, as shown in Fig. \ref{fig4} and Fig. \ref{fig5}. In Fig. \ref{fig4}, we set the values of $\lambda$ and $\eta$ to 1, and then adjust the values of $\alpha$ and $\beta$ to conduct the experiment. From Fig. \ref{fig4}, we can see that on the data sets Mnist and Cifar, when $\alpha = 10^2$ or $10^3$, the proposed algorithm achieves the best effect. On the dataset of Labelme and Place, when $\alpha =10^{-3}$ and $\beta = 10^3$, the proposed algorithm has the best effect. In addition, we can also find that different $\alpha$ and $\beta$ values will affect the effect of the proposed algorithm. Because $\alpha$ controls the similarity between hash codes in the established hyper-class, {\em i.e.,~} $\alpha \sum\nolimits_i^{{n_1}} {\left\| {\mathbf s_1^{(i)}} \right\|_2^2} ,\alpha \sum\nolimits_i^{{n_2}} {\left\| {\mathbf s_2^{(i)}} \right\|_2^2} , \ldots ,\alpha \sum\nolimits_i^{{n_c}} {\left\| {\mathbf s_c^{(i)}} \right\|_2^2} $, and ${\mathbf s_1},{\mathbf s_2}, \ldots ,{\mathbf s_c}$ stores the similarity relationship between hash codes in each hyper-class. $\beta$ controls the similarity of hash codes between hyper-classes, {\em i.e.,~} $\beta \sum\nolimits_{i = 1}^{{n_1}} {\left\| {{\mathbf I_{{}^\neg 1}}\mathbf H_1^{(i)} - {\mathbf V_{{}^\neg 1}}} \right\|_2^2} ,\beta \sum\nolimits_{i = 1}^{{n_2}} {\left\| {{\mathbf I_{{}^\neg 2}}\mathbf H_2^{(i)} - {\mathbf V_{{}^\neg 2}}} \right\|_2^2} , \ldots ,\\\beta \sum\nolimits_{i = 1}^{{n_c}} {\left\| {{\mathbf I_{{}^\neg c}}\mathbf H_c^{(i)} - {\mathbf V_{{}^\neg c}}} \right\|_2^2} $, which indicates the similarity between each hyper-class and other hyper-classes. Therefore, we need to carefully adjust the values of the parameters $\alpha$ and $\beta$.
In Fig. \ref{fig5}, we set the values of $\alpha$ and $\beta$ to 1, and then adjust the values of $\lambda$ and $\eta$ to conduct the experiment. From Fig. \ref{fig5}, we can see that the optimal parameters are different on each data set. For example, on Mnist dataset, when $\lambda = 10^0$ and $\eta = 10^2$, the proposed algorithm achieves the best results. Similarly, when $\lambda = [{10^3},{10^{ - 1}},{10^{ - 3}}]$ and $\eta = [{10^3},{10^2}]$, the proposed algorithm achieves the best results on datasets Cifar, Labelme and Place respectively. Because they affect the values of $\lambda \left\| {\mathbf H - {\mathbf X^T}{\mathbf U^T}} \right\|_F^2$ and $\eta \left\| \mathbf U \right\|_2^2$ respectively, these two terms control the value of the relationship matrix between the hash code and the sample. Therefore, we also need to carefully adjust the values of the parameters $\lambda$ and $\eta$.
\subsection{Performance comparison with other hash algorithms}
We show the precision recall curves of all algorithms on four data sets under different bit numbers ({\em i.e.,~} 16, 32, 48, 64), as shown in Figs. \ref{fig6}-\ref{fig9}. In addition, we also show the HAM2 results of all algorithms under different bit numbers and $MAP$ results under different bits, as shown in Fig. \ref{fig10}, table \ref{tab3} and table \ref{tab4}.
From figs. \ref{fig6}-\ref{fig9}, we can find that the proposed algorithm achieves the best performance. The reason is that the proposed algorithm is a hash algorithm based on hyper-class representation, which can make the hash code similarity within each hyper-class as high as possible and the hash code similarity between hyper-classes as low as possible. So it can learn more suitable hash codes. Specifically, on Mnist and Cifar datasets, the proposed HCH algorithm achieves the best results on each number of bits, compared with other comparison algorithms. When the number of bits is only 16, the proposed algorithm does not achieve the best performance on the dataset Labelme and Place. The reason is that different datasets have different characteristics. In the datasets Labelme and Place, 16 bits are not enough to represent the similarity of data within the hyper-class and between hyper-classes. However, when the hash code is larger than 16 bits, {\em i.e.,~} when the hash code is 32 bits, the proposed HCH algorithm still achieves the best results.
From Fig. \ref{fig10}, we can see that the proposed HCH algorithm achieves the best performance in the HAM2 results under different bit numbers. Specifically, on Mnist and Place datasets, when the bit number is 64, the effect of HCH algorithm is most obvious on HAM2. On data sets Cifar and Labelme, when the number of bits is 48, the effect of HCH algorithm is most obvious on HAM2. Therefore, the proposed HCH algorithm has the best hash learning ability compared with other comparison algorithms.
From table \ref{tab3} and table \ref{tab4}, we can see that the proposed algorithm achieves the best results in the $MAP$ results. Specifically, on Mnist data, when the number of bits is 64, HCH is improved by 1.97\% compared with ITQ algorithm. Compared with CH algorithm, HCH algorithm is improved by 7.94\%. On the Place dataset, HCH is improved by 2.09\% compared with ITQ algorithm. Compared with CH algorithm, HCH algorithm is improved by 5.11\%. The reason is that the proposed HCH algorithm is different from the previous data independent hash algorithm (LSH) and data dependent hash algorithm (FastH, ITQ, KNNH, CH, etc.). HCH depends on the hyper-class representation of data. It considers the potential class information in data from the level of data characteristics, {\em i.e.,~} hyper-class information. According to the hyper-class information, we can further construct a hash code that can better represent the relationship between data features, so as to improve the hash learning ability of the algorithm.
\section{Conclusion} \label{conclusion}
In this paper, we have proposed a hash algorithm based on hyper-class representation. Specifically, we first calculate the relationship between each feature and all other features, and apply a weight to each feature, so as to select the feature that is most likely to be used as the decision feature. Then we use the selected decision features to construct the hyper-class representation of data. Finally, we propose a new hash algorithm based on the principle of ``the similarity of hash codes withinhyper-class is as high as possible, and the similarity between hash codes of data between hyper-classes is as low as possible". In the experiment, the proposed algorithm shows better performance on four data sets, compared with other comparison algorithms.
The proposed hash algorithm is based on the hyper-class representation of data. Therefore, in the future work, we plan to apply the proposed representation of data to other fields. {\em i.e.,~} different data representations are proposed for different data mining algorithms.
\section*{Acknowledgment}
This work has been supported in part by the Natural Science Foundation of China under grant 61836016.
\ifCLASSOPTIONcaptionsoff
\fi
{}
\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{Zhang.jpg}}]{Shichao Zhang}
is a China National Distinguished Professor with the Central South University, China. He holds a PhD degree from the Deakin University, Australia. His research interests include data mining and big data. He has published 90 international journal papers and over 70 international conference papers. He is a CI for 18 competitive national grants. He serves/served as an associate editor for four journals.
\end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{Li.jpg}}]{Jiaye Li}
is currently working toward the PhD degree at Central South University, China. His research interests include machine learning, data mining and deep learning.
\end{IEEEbiography}
\end{document} |
\begin{document}
\title{Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form}
\begin {abstract} We construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two-fluid flows in 1-D when the data are $L^1$ in densities and $L^\infty$ in velocities. To this end, we use a method that reduces this system of PDEs to a family of systems of four ODEs in Banach spaces whose smooth solutions are these approximate solutions. This method is constructive: using standard numerical methods for ODEs one can easily and accurately compute these approximate solutions which, therefore, from the mathematical proof, can serve for comparison with numerical schemes. One observes agreement with previously known solutions from scientific computing [S. Evje, T. Flatten. Hybrid Flux-splitting Schemes for a common two fluid model. J. Comput. Physics 192, 2003, p. 175-210]. We show that one recovers the solutions of these authors (exactly in one case, with a slight difference in another case). Then we propose an efficient numerical scheme for the original system of two-fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above. \\
\end{abstract}
AMS classification: 35D30, 35F25, 65M06, 76-XX.\\
Keywords: partial differential equations, approximate solutions, weak asymptotic methods, fluid dynamics.\\
\\
\textit{*this research has been done thanks to financial support of FAPESP, processo 2012/15780-9.}\\
\textbf{1. Introduction}.\\
We study a basic model used to describe mathematically a mixture of two immiscible fluids in the isothermal case and without transfer of momentum between the two fluids, \cite{EvjeFlatten} p. 179, \cite{Cortes} p. 465,
\begin{equation}\frac{\partial}{\partial t}(\rho_1 \alpha_1)+\frac{\partial}{\partial x}(\rho_1\alpha_1 u_1)=0,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(\rho_2 \alpha_2)+\frac{\partial}{\partial x}(\rho_2\alpha_2 u_2)=0,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(\rho_1 \alpha_1u_1)+\frac{\partial}{\partial x}(\rho_1\alpha_1( u_1)^2)+\frac{\partial}{\partial x}((p_1-p_1^{int})\alpha_1)+\alpha_1\frac{\partial}{\partial x}(p_1^{int})=g\alpha_1\rho_1,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(\rho_2 \alpha_2u_2)+\frac{\partial}{\partial x}(\rho_2\alpha_2( u_2)^2)+\frac{\partial}{\partial x}((p_2-p_2^{int})\alpha_2)+\alpha_2\frac{\partial}{\partial x}(p_2^{int})=g\alpha_2\rho_2,\end{equation}
\begin{equation}\alpha_1+\alpha_2=1,\end{equation}
\begin{equation}p_1=K_1\rho_1-b_1, \ \ p_2=K_2\rho_2-b_2,\end{equation}
\\
where the two fluids are denoted by the indices $1$ and $2$, for instance mixture of oil and natural gas in extraction tubes of oil exploitation \cite{Avelar}. The physical variables are
the densities $\rho_i(x,t)$, the velocities $u_i(x,t)$, the volumic proportions
$\alpha_i(x,t)$, the pressures
$p_i(x,t)$, the phasic pressures $p_i^{int}(x,t)$ at the interface, $i=1,2$, and
$g$ is the component of the gravitational acceleration in the direction of the tube.
Equations (6) are the state laws stated in \cite{EvjeFlatten} p. 179; it is assumed $b_1-b_2>0, K_1>0$ and $ K_2>0$.
Equations (1) and (2) are the continuity equations for each fluid: they express mass conservation. Equations (3) and (4) are the Euler equations for each fluid: they express momentum conservation. A natural assumption is to state the equality of the four pressures $p_i$ and $ p_i^{int}, i=1,2$. This simplest assumption of equal pressure leads to a nonhyperbolic model, called the equal pressure model \cite{EvjeFjelde} p.677, \cite{Munk} p. 2589, \cite{Toumi} p. 287, \cite{Wendroff} p. 372-373 that we study in this paper. \\
We construct families of differentiable functions $S(x,t,\epsilon)$ that, when plugged into the equal pressure model, tend asymptotically to satisfy it when $\epsilon\rightarrow 0$. We prove that these families of functions are weak asymptotic methods. The concept of weak asymptotic method and its relevance has been put in evidence by many authors \cite{Albeverio, Danilov1, Mitrovic, Panov, Shelkovichmat} by explicit calculations and by reduction of the problem of description of nonlinear waves interaction to the resolution of systems of ordinary differential equations, as a continuation of Maslov' s theory. In other words our families of functions tend to satisfy the system modulo a remainder that tends to 0 when $\epsilon\rightarrow 0$. To construct these families we use a method which consists in solving a system of four ordinary differential equations in a Banach space whose solutions are the approximate solutions of the one pressure model. This method allows us to compute the solutions with standard convergent numerical schemes for ODEs, thus permitting comparaison with existing numerical solutions of the equal pressure model obtained in scientific computing. We observe the approximate solutions we obtain agree with the results presented in \cite{EvjeFlatten}, with a small difference in one case which diminishes in presence of the pressure correction, which can be considered as a mathematical justification of these numerical results. The system (1-6) is in nondivergence form, i.e. the derivatives cannot be transfered to test functions because of the terms $\alpha_i\frac{\partial p_i^{int}}{\partial x} $ in (3, 4). Therefore the study of the solutions of this system in presence of shock waves is problematic and we use a family of approximate solutions that are classical differentiable functions which permits at the limit to obtain "exact solutions" that are irregular functions such as discontinuous functions. In this way the weak asymptotic methods presented here can be a tool for mathematical and numerical investigations of discontinuous solutions despite this system is in nondivergence form. Various systems in divergence form have been obtained by replacing (10,11) below by their sum, and then by introducing a new equation \cite{Evje1, Evje2, Evje3, EvjeFlattenFriis, EvjeFriis, EvjeKarlsen1, EvjeKarlsen2}.\\
From a physical viewpoint the equations of fluid dynamics are mared with some imprecision since they do not take into account some minor effects and the molecular structure of matter. It is natural to expect these equations and their imprecision should be stated in the sense of distributions in the space variables. Weak asymptotic methods provide approximate solutions that enter into this imprecision for $\epsilon>0$ small enough. Therefore they could be considered as some convenient way to approximate possible solutions to the equations of physics. In the absence of a uniqueness result of a privileged family of weak asymptotic methods (all giving same results) that should represent physics in a given physical situation, we have to content to check numerically that the weak asymptotic methods we present give the known solutions at the limit $\epsilon\rightarrow 0$.\\
\textbf{2. Simplified statement of the system.}\\ In order to simplify the study of the system (1-6) with the equal pressure assumption we transform it into a system of four equations with four unknown functions by changes of unknown and algebraic calculations. We set
\begin{equation} r_1=\rho_1 \alpha_1, r_2=\rho_2 \alpha_2, \ \alpha=\alpha_1. \end{equation}
Then (1-5) with equal pressures has the form
\begin{equation}\frac{\partial}{\partial t}(r_1)+\frac{\partial}{\partial x}(r_1u_1)=0,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(r_2)+\frac{\partial}{\partial x}(r_2u_2)=0\end{equation}
\begin{equation}\frac{\partial}{\partial t}(r_1u_1)+\frac{\partial}{\partial x}(r_1(u_1)^2)+\alpha \frac{\partial}{\partial x}p=gr_1,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(r_2u_2)+\frac{\partial}{\partial x}(r_2(u_2)^2)+(1-\alpha)\frac{\partial}{\partial x}p=gr_2,\end{equation}
and the two state laws (6) are left unchanged. The 6 unknowns are now $r_1, r_2, u_1, u_2, \alpha$ and $ p$. Then we transform the equations in a way which will be more convenient to construct the weak asymptotic method since we will have only the four unknown functions $r_1, r_2, u_1$ and $u_2$.\\
$\bullet$ From (6), $p=K_1\rho_1-b_1=K_2\rho_2-b_2 $ implies
\begin{equation}\rho_2=\frac{-b_1+b_2+K_1\rho_1}{K_2}.\end{equation}
Note that this calculation is linear so it can be done rigorously even in presence of shock waves.\\
$\bullet$\textit{ Calculation of $\alpha$ in function of $r_1$ and $ r_2$}. We multiply the equality $p=K_1\frac{r_1}{\alpha}-b_1=K_2\frac{r_2}{1-\alpha}-b_2$ by $\alpha(1-\alpha)$ to obtain (13) below:
this is a nonlinear calculation. In the case of discontinuous solutions it is well known such nonlinear calculations usually change the solutions. This formal calculation is usual for this system and the observation of the numerical results in section 6, observation 3, shows a posteriori that this nonlinear calculation giving the formula (13) is justified.
No unjustified nonlinear calculations are done after (13). This calculation gives
\begin{equation}(1-\alpha)K_1r_1-b_1\alpha(1-\alpha)=\alpha K_2r_2-b_2\alpha(1-\alpha),\end{equation}
i.e. $F(\alpha)=0$,
setting
\begin{equation}F(X)=X^2(b_1-b_2)+X(-K_1r_1-b_1-K_2r_2+b_2)+K_1r_1.\end{equation}
One has $F(0)=K_1r_1> 0$ and $ F(1)=-K_2r_2<0$ which implies that $F$ has one and only one root $\alpha\in ]0,1[$ in the case $r_1>0$ and $r_2>0$ i.e. in absence of void regions in each fluid. In this case, since $F(1)<0$ and since it is assumed $b_1-b_2>0$, the second root is $>1$. Therefore the discriminant $\Delta=(K_1r_1+K_2r_2+b_1-b_2)^2-4(b_1-b_2)K_1r_1$ is $>0$ and the solution $\alpha\in ]0,1[$ is given by
\begin{equation} \alpha=\frac{K_1r_1+K_2r_2+b_1-b_2-(\Delta)^{\frac{1}{2}}}{2(b_1-b_2)}.\end{equation}
$\bullet$ The following result will be used below
\begin{equation}0\leq\frac{K_1r_1}{b_1-b_2+K_1r_1+K_2r_2}\leq \alpha\leq 1.\end{equation}
Proof. From (14), $F(X)\geq -X(b_1-b_2+K_1r_1+K_2r_2)+K_1r_1$ since $b_1-b_2>0$; therefore $F(\frac{K_1r_1}{b_1-b_2+K_1r_1+K_2r_2})\geq 0$, hence the result since $F(\alpha)=0$ and $F(1)\leq 0$.$\Box$\\
Finally we can eliminate $\alpha$ from (10, 11) and we obtain the following statement of the system: first the continuity equations
\begin{equation}\frac{\partial}{\partial t}(r_1)+\frac{\partial}{\partial x}(r_1u_1)=0,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(r_2)+\frac{\partial}{\partial x}(r_2u_2)=0,\end{equation}
then the Euler equations in the form
\begin{equation}\frac{\partial}{\partial t}(r_1u_1)+\frac{\partial}{\partial x}(r_1(u_1)^2)+r_1\frac{\partial}{\partial x}\Phi_1=gr_1,\end{equation}
\begin{equation}\frac{\partial}{\partial t}(r_2u_2)+\frac{\partial}{\partial x}(r_2(u_2)^2)+r_2\frac{\partial}{\partial x}\Phi_2=gr_2,\end{equation}
where
\begin{equation} \Phi_1=K_1 log \rho_1, \ \rho_1=\frac{r_1}{\alpha}, \ \ \Phi_2=K_2 log \rho_2, \
\rho_2=\frac{r_2}{1-\alpha}=\frac{-b_1+b_2+K_1\rho_1}{K_2},\end{equation}
with $\alpha$ given by (15).
The system is now a system of four scalar PDEs with the four unknowns $r_1, r_2, u_1$ and $ u_2$.\\
\textbf{3. Statement of the weak asymptotic method.}\\
Setting
\begin{equation}u_i^+=\frac{|u_i|+u_i}{2}, \ \ u_i^-=\frac{|u_i|-u_i}{2},\end{equation}
one has
\begin{equation} u_i^+-u_i^-=u_i, \ \ u_i^++u_i^-=|u_i|. \end{equation}
The two continuity equations and the two Euler equations are replaced by the following ODEs, $i=1,2$
\begin{equation} \frac{d}{dt}r_i(x,t,\epsilon)=\frac{1}{\epsilon}[(r_i u_i^+)(x-\epsilon,t,\epsilon)-(r_i |u_i|)(x,t,\epsilon)+(r_i u_i^-)(x+\epsilon,t,\epsilon)]+\epsilon^\beta,\end{equation} with $\beta>0$ to be defined later,\\
$ \frac{d}{dt}(r_iu_i)(x,t,\epsilon)=\frac{1}{\epsilon}[(r_i u_iu_i^+)(x-\epsilon,t,\epsilon)-$\begin{equation}(r_iu_i |u_i|)(x,t,\epsilon)+(r_i u_iu_i^-)(x+\epsilon,t,\epsilon)]-r_i(x,t,\epsilon)\frac{\partial}{\partial x}\Phi_i
(x,t,\epsilon)+gr_i(x,t,\epsilon).\end{equation}
The potentials $\Phi_i$, $i=1,2$, are defined by
\begin{equation}\Phi_i(x,t,\epsilon)=K_i[log(\rho_i(.,t,\epsilon)+\epsilon^N)*\phi_{\epsilon^\gamma}](x),
\end{equation} with $N$ and $\gamma$ to be defined later, $\phi\in \mathcal{C}_c^\infty(\mathbb{R}), \ \phi\geq 0$ and $ \int\phi(\mu)d\mu=1$. The convolution in (26) permits that the derivative $\frac{\partial}{\partial x}\Phi_i$ in (25) makes sense: thus the fact the equations (19, 20) are not in divergence form does not cause any trouble for the approximating sequences.
We recall $\alpha$ is defined in (15), \ $\rho_1=\frac{r_1}{\alpha},$ and one will prove $\alpha_i(x,t,\epsilon)>0 \ \forall \epsilon>0$;
$\rho_2$ is given in (12, 21), $u_i=\frac{r_iu_i}{r_i}$ and one will prove $
r_i(x,t,\epsilon)>0 \ \forall \epsilon>0$. This will follow from (33) below, which, from (13), implies $\alpha\not=0$ and $\alpha\not=1$.\\
We assume $r_{i,0}$ and $u_{i,0}, i=1,2$ are given initial conditions on the 1-D torus $\mathbb{T}=\mathbb{R}/(2\pi \mathbb{Z})$ with the properties $r_{i,0} \in L^1(\mathbb{T})$ and $u_{i,0}\in L^\infty(\mathbb{T})$ and that $r_{i,0}^\epsilon$ and $ u_{i,0}^\epsilon$ are regularizations of $r_{i,0}$ and $u_{i,0}$ respectively, with uniform $L^1$ and $L^\infty$ bounds respectively (independent on $\epsilon$), and $r_{i,0}^\epsilon (x)>0 \ \ \forall x$.\\
\textbf{Theorem.} \textit{If $0<\gamma<\frac{1}{6}$ and $N-1-\beta-3\gamma>0$ the system of four ODEs (24, 25) complemented by the relations (6, 7, 26) provides a weak asymptotic method for the system (17, 18, 19, 20, 21).}\\
Sections 4 and 5 are devoted to the proof of the Theorem.\\
\textbf{4. A priori inequalities for fixed $\epsilon$.} \\
We seek solutions on the 1-D torus $\mathbb{T}=\mathbb{R}/(2\pi \mathbb{Z})$. Families $(r_{i,0}^\epsilon)_\epsilon$ and $ (r_{i,0}^\epsilon u_{i,0}^\epsilon)_\epsilon$ of approximations of initial conditions are given on $\mathbb{T}$. For fixed $\epsilon>0$ we assume existence and uniqueness of a solution of (24, 25) of class $\mathcal{C}^1$ $$[0,\delta(\epsilon)[\longmapsto \mathcal{C}_b(\mathbb{R})^4,$$
$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\longmapsto (r_i(.,t,\epsilon), r_iu_i(.,t,\epsilon))$$
such that
\begin{equation}\exists m>0 \ / \ r_i(x,t,\epsilon)\geq m \ \forall x\in \mathbb{R} \ \forall t\in [0,\delta(\epsilon)[,\end{equation}
\begin{equation}\exists M>0 \ / \| u_i(.,t,\epsilon)\|_\infty\leq M, \| r_i(.,t,\epsilon)\|_\infty\leq M \ \forall t\in [0,\delta(\epsilon)[.\end{equation}
\\
\textbf {Proposition 1 (a priori inequalities).}\\
$ \bullet \ \forall \epsilon>0, \ \forall t\in [0,\delta(\epsilon)[ \ \ r_i(.,t,\epsilon)\in L^1(\mathbb{T}),$ and \begin{equation}\int_{-\pi}^\pi r_i(x,t,\epsilon)dx = \int_{-\pi}^\pi r_i(x,0,\epsilon)dx+2\pi\epsilon^\beta t,\end{equation}
\begin{equation} \bullet \ \exists C>0 \ / \ \ \|\frac{\partial}{\partial x}\Phi_i(.,t,\epsilon)\|_\infty \leq \frac{C}{\epsilon^{3\gamma}} \ \forall t\in [0,\delta(\epsilon)[,\forall \epsilon>0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{equation}
\begin{equation} \bullet \|u_i(.,t,\epsilon)\|_\infty \leq \|u_i(.,0,\epsilon)\|_\infty +\frac{2(C+g)} {\epsilon^{3\gamma}}\delta(\epsilon) \ \forall t\in [0,\delta(\epsilon)[,\forall \epsilon>0. \ \ \ \end{equation}
Setting
\begin{equation} k(\epsilon)=max_{i=1,2}\|u_i(.,0,\epsilon)\|_\infty+ \frac{2(C+g) \delta(\epsilon)}{\epsilon^{3\gamma}},\end{equation}
then $\forall t\in [0,\delta(\epsilon)[,\forall \epsilon>0, $
\begin{equation} \bullet \ r_i(x,0,\epsilon)exp(\frac{-k(\epsilon)t}{\epsilon}) \leq r_i(x,t,\epsilon) \leq 2\|r_i(.,0,\epsilon)\|_\infty exp(\frac{2k(\epsilon)t}{\epsilon}) \ \forall x\in \mathbb{R}.\end{equation}
Proof of Proposition 1.\\
$\bullet$ From (23, 24),\\
\\
$\frac{d}{dt}\int_{-\pi}^{+\pi}r_i(x,t,\epsilon)dx=
\frac{1}{\epsilon}[\int_{-\pi}^{+\pi}(r_i u_i^+)(x-\epsilon,t,\epsilon)dx-\int_{-\pi}^{+\pi}(r_i u_i^+)(x,t,\epsilon)dx-\\
\\
\int_{-\pi}^{+\pi}(r_i u_i^-)(x,t,\epsilon)dx+\int_{-\pi}^{+\pi}(r_i u_i^-)(x+\epsilon,t,\epsilon)dx]+2\pi\epsilon^\beta
=0+2\pi\epsilon^\beta$\\
\\
by periodicity of $r_i$ and $u_i$. \\
$\bullet$ From (26),
$$\frac{\partial}{\partial x}(\Phi_i)(x,t,\epsilon)=K_i\int log[\rho_i(x-y,t,\epsilon)+\epsilon^N]\frac{1}{\epsilon^{2\gamma}} \phi'(\frac{y}{\epsilon^\gamma})dy.$$
If $\rho_i(x-y,t,\epsilon)\leq 1$, one uses the bound $|log(\epsilon^N)|\leq \frac{const}{\epsilon^\gamma}.$
If $\rho_i(x-y,t,\epsilon)> 1$, one uses the fact that $\rho_i(.,t,\epsilon)\in L^1(\mathbb{T})$ with $L^1$ norm independent on $\epsilon$ and $t\in[0,\delta(\epsilon)[$. The result that $\rho_i(.,t,\epsilon)\in L^1(\mathbb{T})$ with $L^1$ norm independent on $\epsilon$ and $t$ follows from formula (16) that implies $\rho_1=\frac{r_1}{\alpha}\leq r_1\frac{b_1-b_2+K_1r_1+K_2r_2}{K_1r_1}$. Then one notices that $b_1-b_2>0, K_i>0,r_i>0$ and the result follows from (29). For $\rho_2$ one uses (12). \\
$\bullet$Now we proceed to the proof of (31).
From (24) and the assumption that the solution of the ODEs is of class $\mathcal{C}^1$ on $[0,\delta(\epsilon)[$ valued in the Banach space $\mathcal{C}(\mathbb{T})$, one obtains, for fixed $\epsilon>0$ and for $dt>0$ small enough with $t+dt<\delta(\epsilon)$, that\\
$r_i(x,t+dt,\epsilon)=r_i(x,t,\epsilon)+$\\
$$\frac{dt}{\epsilon}[(r_i u_i^+)(x-\epsilon,t,\epsilon)-(r_i |u_i|)(x,t,\epsilon)+(r_i u_i^-)(x+\epsilon,t,\epsilon)]+dt.o(x,t,\epsilon)(dt)+\epsilon^\beta dt=$$ \begin{equation}\frac{dt}{\epsilon}(r_i u_i^+)(x-\epsilon,t,\epsilon)+(1-\frac{dt}{\epsilon}|u_i|(x,t,\epsilon))r_i(x,t,\epsilon)+\frac{dt}{\epsilon}(r_i u_i^-)(x+\epsilon,t,\epsilon)+dt.o(x,t,\epsilon)(dt)+\epsilon^\beta dt\end{equation}
\\
where $\|o(.,t,\epsilon)(dt)\|_\infty \rightarrow 0$ when $dt\rightarrow 0$ uniformly for $t$ in a compact set of $[0,\delta(\epsilon)[$, from the mean value theorem in the form $f(t+dt)=f(t)+dt f'(t) +dt. r(t,dt),$ with $ \|r(t,dt)\|\leq sup_{0<\theta<1}\|f'(t+\theta dt)-f'(t)\|$. Notice that there is no uniformness in $\epsilon$. For $dt>0$ small enough (depending on $\epsilon$) the single term $ (1-\frac{dt}{\epsilon}|u_i|(x,t,\epsilon))r_i(x,t,\epsilon)$ dominates the term $dt.o(x,t,\epsilon)(dt)$ from (27, 28). Since, further, $r_i u_i^\pm\geq0$, one can invert (34). Dropping the useless term $\epsilon^\beta dt$ one obtains\\
$\frac{1}{r_i(x,t+dt,\epsilon)}\leq$\\
\\
$[\frac{dt}{\epsilon}(r_i u_i^+)(x-\epsilon,t,\epsilon)+[1-\frac{dt}{\epsilon}|u_i|(x,t,\epsilon)]r_i(x,t,\epsilon)+\frac{dt}{\epsilon}(r_i u_i^-)(x+\epsilon,t,\epsilon)]^{-1}+\\
\\
dt.o(x,t,\epsilon)(dt)$\\
\\
where the new $o$ has still the property that $\|o(.,t,\epsilon)(dt)\|_\infty \rightarrow 0$ when $dt\rightarrow 0$ uniformly for $t\in[0,\delta'] $ if $\delta'<\delta(\epsilon)$.\\
Applying the analog of (34) for $r_i u_i$ in place of $r_i$, with the supplementary terms $r_i\frac{\partial}{\partial x}(\Phi_i)$ and $gr_i$ from (25), one obtains, using (27, 28)
$$u_i(x,t+dt,\epsilon)=\frac{(r_i u_i)(x,t+dt,\epsilon)}{r_i(x,t+dt,\epsilon)}\leq$$
$$ \frac
{\frac{dt}{\epsilon}(r_i u_i u_i^+)(x-\epsilon,t,\epsilon)+[1-\frac{dt}{\epsilon}|u_i|(x,t,\epsilon)](r_i u_i)(x,t,\epsilon)+\frac{dt}{\epsilon}(r_i u_i u_i^-)(x+\epsilon,t,\epsilon)}
{\frac{dt}{\epsilon}(r_i u_i^+)(x-\epsilon,t,\epsilon)+[1-\frac{dt}{\epsilon}|u_i|(x,t,\epsilon)]r_i(x,t,\epsilon)+\frac{dt}{\epsilon}(r_i u_i^-)(x+\epsilon,t,\epsilon)}$$
\begin{equation} +dt\frac{r_i(x,t,\epsilon)}{r_i(x,t+dt,\epsilon)}[|\frac{\partial}{\partial x}(\Phi_i)(x,t,\epsilon)|+g]+ dt.o(x,t,\epsilon)(dt)\end{equation}
where the new $o$ has the same property as in (34) for fixed $\epsilon$.
For $dt>0$ small enough the first term in the second member is a barycentric combination of $u_i(x-\epsilon,t,\epsilon), u_i(x,t,\epsilon)$ and $ u_i(x+\epsilon,t,\epsilon)$. The quotient $\frac{r_i(x,t+dt,\epsilon)}{r_i(x,t,\epsilon)} $ tends to $1$ when $dt\rightarrow 0$ (for fixed $\epsilon$). Finally one obtains, using also (30), that
\begin{equation} \|u_i(.,t+dt,\epsilon)\|_\infty\leq \|u_i(.,t,\epsilon)\|_\infty+dt\frac{const}{\epsilon^{3\gamma}}+dt.\|o(.,t,\epsilon)(dt)\|_\infty\end{equation}
with uniform bound of $o$ when $t$ ranges in a compact set in $[0,\delta(\epsilon)[$, for fixed $\epsilon$. One obtains the bound (31) as in \cite{Colombeaugravitation} by dividing the interval $[0,t]$ into $n$ small intervals $[\frac{it}{n},\frac{(i+1)t}{n}], 0\leq i \leq n-1$, applying (36) in each small interval, which gives
$$\|u_i(.,(i+1)\frac{t}{n},\epsilon)\|_\infty \leq \|u_i(.,i\frac{t}{n},\epsilon)\|_\infty+\frac{t}{n}\frac{const}{\epsilon^{3\gamma}}+\frac{t}{n}o(\frac{t}{n}),$$
summing on $i$ and using that $o(\frac{t}{n})\rightarrow 0$ when $n\rightarrow \infty$.\\
$\bullet$The proofs of the two inequalities (33) follows from (24) that gives the inequalities $\frac{d}{dt}r_i(x,t,\epsilon)\geq-\frac{\|u_i\|_\infty}{\epsilon}r_i(x,t,\epsilon)$ and $\frac{d}{dt}r_i(x,t,\epsilon)\leq \frac{2\|u_i\|_\infty\|r_i\|_\infty}{\epsilon}$ using (31) to evaluate $\|u_i\|_\infty$. They are given in detail in section 2 of \cite{Colombeaugravitation}. \\
The existence of a unique global solution to (24, 25) for fixed $\epsilon$ is obtained from the a priori estimates (29-33) from classical ODEs arguments of the theory of ODEs in Banach spaces in the Lipschitz case. Indeed for fixed $\epsilon>0$ , if $0<\lambda<1$ and $\Omega_\lambda:=\{(X_i,Y_i)\in \mathcal{C}(\mathbb{T})^4 / \ \forall x\in \mathbb{T} \ \lambda<X_i(x)<\frac{1}{\lambda}, |Y_i(x)|<\frac{1}{\lambda}\}$ the four equations (24, 25) with variables $X_i=r_i$ and $ Y_i=r_i u_i$ have the Lipschitz property on $\Omega_\lambda$ with values in $\mathcal{C}(\mathbb{T})^4$, with Lipschitz constants $\leq \frac{1}{\lambda^3}$. We refer to \cite {Colombeaugravitation} section 4 for details.\\
\textbf{5. Proof of the weak asymptotic method.} \\
It remains to prove that the solution of the system of ODEs (24, 25) and the formula (26) provide a weak asymptotic method for system (17, 18, 19, 20, 21) when $\epsilon\rightarrow 0$. To this end one has to prove that $\forall \psi\in \mathcal{C}_c^\infty(\mathbb{R})$, (37-39) below hold when $\epsilon\rightarrow 0$
\begin{equation} \int \frac{d}{dt}r_i(x,t,\epsilon)\psi(x)dx-\int (r_i u_i)(x,t,\epsilon)\psi'(x)dx \rightarrow 0,\end{equation}
$ \int \frac{d}{dt}(r_i u_i)(x,t,\epsilon)\psi(x)dx-$\begin{equation}\int (r_i (u_i)^2)(x,t,\epsilon)\psi'(x)dx +\int r_i(x,t,\epsilon)\frac{\partial}{\partial x}(\Phi_i)(x,t,\epsilon)\psi(x)dx-g\int r_i(x,t,\epsilon)\psi(x)dx\rightarrow 0,\end{equation}
\begin{equation} \int\Phi_i(x,t,\epsilon)\psi(x)dx-K_i\int \log[\rho_i(x,t,\epsilon)]\psi(x)dx\rightarrow 0 \end{equation}
where (37) means satisfaction of (17, 18), (38) satisfaction of (19, 20) and (39) satisfaction of the two state laws in (21) in the sense of distributions at the limit $\epsilon\rightarrow 0$.\\
The proof of (37) is as follows: from (23, 24, 29, 30, 31), a change of variable and $\frac{\psi(x+\epsilon)-\psi(x)}{\epsilon}=\psi'(x)+O_x(\epsilon)$,\\
$ \int \frac{d}{dt}r_i(x,t,\epsilon)\psi(x)dx=
\frac{1}{\epsilon}\int (r_i u_i^+)(x,t,\epsilon)[\psi(x+\epsilon)-\psi(x)]dx -\frac{1}{\epsilon}\int (r_i u_i^-)(x,t,\epsilon)$\\
\\
$[\psi(x)-\psi(x-\epsilon)]dx+
\int\epsilon^\beta \psi(x)dx =
\int (r_i u_i)(x,t,\epsilon)\psi'(x)dx+\int_{compact} (r_i u_i^+)(x,t,\epsilon)$ \\
\\
$O_x(\epsilon)dx+
\int_{compact} (r_i u_i^-)(x,t,\epsilon) O_x(\epsilon)dx+O(\epsilon^\beta)= \int(r_iu_i)(x,t,\epsilon)\psi'(x) dx+$\\
\\
$(const+\frac{const}{\epsilon^{3\gamma}}) O(\epsilon)+O(\epsilon^\beta)= \int(\rho u)(x,t,\epsilon)\psi'(x) dx+O(\epsilon^{1-3\gamma})+O(\epsilon^\beta)$.\\
\\
This gives (37) if $0<\gamma<\frac{1}{3}$. The proof of (38) is similar since the additional terms $\int r_i(x,t,\epsilon)\frac{\partial}{\partial x}\Phi_i(x,t,\epsilon)\psi(x)dx$ and $g\int r_i(x,t,\epsilon)\psi(x) dx$ are the same in (25) and (38): one obtains a remainder $\frac{const}{\epsilon^{6\gamma}} O(\epsilon)$ because of one more factor $u_i$ and the bound (31). Finally one chooses $0<\gamma<\frac{1}{6}$.\\
To check (39) one has to prove from (26) that $\forall \psi \in \mathcal{C}_c^\infty(\mathbb{R})$
\begin{equation}\int\{[(log(\rho_i(.,t,\epsilon)+\epsilon^N)*\phi_{\epsilon^\gamma}](x)-log[\rho_i(x,t,\epsilon)]\}\psi(x)dx \rightarrow 0 \end{equation}
when $\epsilon\rightarrow 0$. To this end we share the integral (40) into two parts (41, 42) below and we prove that each tends to 0 when $\epsilon\rightarrow 0$. Let
\begin{equation}I= \int\{[(log(\rho_i(.,t,\epsilon)+\epsilon^N)*\phi_{\epsilon^\gamma}](x)-log[\rho_i(x,t,\epsilon)+\epsilon^N]\}\psi(x)dx \end{equation}
and
\begin{equation} J=\int\{(log[\rho_i(x,t,\epsilon)+\epsilon^N]-log[\rho_i(x,t,\epsilon)]\}\psi(x)dx.\end{equation}
Now\\
$I=\int\{(log[\rho_i(x-\epsilon^\gamma\mu,t,\epsilon)+\epsilon^N]-log[\rho_i(x,t,\epsilon)+\epsilon^N]\}\phi(\mu)\psi(x)d\mu dx=$\\
\\
$\int log[\rho_i(x,t,\epsilon)+\epsilon^N]\phi(\mu)[\psi(x+\epsilon^\gamma\mu)-\psi(x)]d\mu dx.$\\
\\
Since $\rho_i(x,t,\epsilon)\geq 0$ from (16, 21, 33), using its $L^1$ property (29) in the case $\rho_i(x,t,\epsilon)>1$ and using the term $\epsilon^N$ in the case $\rho_i(x,t,\epsilon)\leq1$, as in the proof of (30), one has $|I|\leq const.log(\frac{1}{\epsilon})\epsilon^\gamma$. Therefore $I\rightarrow 0$ when $\epsilon\rightarrow 0$.\\
Now, (42) and the mean value theorem give
\begin{equation}|J|\leq \epsilon^N\frac{1}{min (\rho_i)} const\end{equation}
if $min(\rho_i)$ denotes the inf of $\rho_i(x,t,\epsilon)$ for fixed $t,\epsilon$ when $x$ ranges in a compact set containing the support of $\psi$.
The problem is to obtain an inf. bound of $min(\rho_i)$; this is the purpose of the term $\epsilon^\beta$ in (24). From (24),
$\frac{d r_i}{dt}(x,t,\epsilon)\geq-\frac{1}{\epsilon}r_i(x,t,\epsilon)\|u_i(.,t,\epsilon)\|_\infty +\epsilon^\beta \geq - \frac{1}{\epsilon}r_i(x,t,\epsilon)\frac{const}{\epsilon^{3\gamma}}T +\epsilon^\beta$ if $t\in [0,T[$, from (31) applied with $\delta(\epsilon)=T$.\\
Setting $A:=const \frac{T}{\epsilon^{1+3\gamma}}$ and $B:=\epsilon^\beta$, one has
$\frac{dr_i}{dt}\geq -Ar_i+B. $ Comparing with the exact solution of
the ODE $\frac{dX}{dt}(x,t)=-AX(x,t)+B$ with initial condition $X(x,0)=r_{i,0}(x,\epsilon)$, namely
$X(x,t)=r_{i,0}(x,\epsilon) e^{-At}+\frac{B}{A}(1-e^{-At})\geq \frac{B}{A}(1-e^{-At})$, we obtain the bound
\begin{equation}r_i(x,t,\epsilon)\geq const(t).\epsilon^{1+\beta+3\gamma}\end{equation}
for $\epsilon>0$ small enough and fixed $t$.
Now using (16) we can obtain a lower bound of $min\rho_i$\\
\begin{equation}\rho_1(x,t,\epsilon) =\frac{r_1(x,t,\epsilon)}{\alpha(x,t,\epsilon)}\geq r_1(x,t,\epsilon)\geq const. \epsilon^{1+\beta+3\gamma}. \end{equation}
Similarly, from (12)\begin{equation}\rho_2(x,t,\epsilon) \geq const.\epsilon^{1+\beta+3\gamma}.\end{equation}
From (43),
$|J|\leq const(t).\epsilon^{N-1-\beta-3\gamma}$ and it suffices to choose $N-1-\beta-3\gamma>0$
to obtain that $J\rightarrow 0$ when $\epsilon \rightarrow 0.$ $\Box$\\
\textbf{6. Numerical observations from the weak asymptotic method.} \\
We will present two shock tube problems selected from \cite{EvjeFlatten}. The pressure laws are those in \cite{EvjeFlatten} p. 179-180: $K_1=10^6, K_2=10^5, b_i=K_i\rho_{0,i}-p_{0,i}, \rho_{0,1}=1000, p_{0,1}=10^5, \rho_{0,2}=0$ and $p_{0,2}=0$. The final time is $T=0.001$, with 1000 cells on [0,1] (or $T=0.01$ on $[0,10]$), therefore $\Delta x=\epsilon=(1000)^{-1} $ and the CFL number is $r=\frac{\Delta t}{\Delta x}= 10^{-6}$. We use the explicit Euler order one method for the ODEs (24, 25). We choose $\delta=1, \beta=100$ and $ N=100$ ($\beta$ and $N$ do not matter since there is no void region in any fluid). We regularize the initial conditions $\omega_{i,0}=r_{i,0}, r_{i,0}u_{i,0}$ by an averaging
\begin{equation}\nu \omega_{i,0}(x-\epsilon)+(1-2\nu)\omega_{i,0}(x)+\nu\omega_{i,0}(x+\epsilon), \
\nu=0.1.\end{equation}
We represent the convolution in (26) by a similar averaging of $\Phi_i$ on 5 cells instead of 3 with coefficient $\nu$=0.15. We use a small averaging as (47) at each step in $r_i$ and $r_i u_i, i=1,2$. Concerning this last averaging one observes that the minimal needed values of $\nu$ tend to 0 when $r\rightarrow 0$: $\nu=10^{-2}, 10^{-3}$ and $ 10^{-4}$ when $r=10^{-4}, 10^{-5}$ and $ 10^{-6}$ respectively. Therefore this regularization can be considered as a numerical artefact absent in the ODE formulation which corresponds to $r=0$.
The Riemann conditions are $\alpha=0.71,0.7, p=265000,265000, u_1=1,1$ and $u_2=65,50$ for test 1 and $\alpha=0.7,0.1, p=265000,265000, u_1=10,15$ and $ u_2=65,50$ for test 2.\\
\textit{Observation 1}. For shock tube problem 1 (figure 1) one observes the same results as those depicted in $\cite{EvjeFlatten}$. For shock tube problem 2 (figure 2) one observes a slight difference for the second step value in the right panels: 2.46 $10^5$ instead of 2.50 $10^5$ (top panel) and 89 instead of 84 (bottom panel). These values do not change with discretizations ranging from 100 to 20000 cells, with different values of $r$ and the other parameters, and are also exactly those obtained from the direct adaptation of the scheme in section 7 below. With the pressure correction adopted in \cite{EvjeFlatten} one observes from the scheme in section 7 that this difference tends to disappear, figures 4 and 5, therefore it is presumably a consequence of the pressure correction adopted in \cite{EvjeFlatten}. Modulo this difference one observes that the results we obtain without pressure correction agree with the results obtained in \cite{EvjeFlatten} even with pressure correction, which appear therefore as depictions of approximate solutions. \\
\textit{Observation 2}. In both tests it has been observed that the left and right discontinuities satisfy with great precision the 3 standard jump conditions of system (8-11): the two ones from (8, 9) and the third one from the equation obtained by adding (10) and (11). They satisfy also with great precision the two formal jump conditions (62) one can calculate from nonlinear algebraic calculations with the nonconservative equations as done in the appendix.
The arrays below give on a line the values of the wave velocities computed from the two equations (8, 9) i.e. $c=\frac{[r_1u_1]}{[r_1]}$ and $c=\frac{[r_2u_2]}{[r_2]},$ the value computed adding the equations (10, 11) without gravitation, i.e. $c=\frac{[r_1(u_1)^2+r_2(u_2)^2+p]}{[r_1u_1+r_2u_2]}$, and the two formal results (62). They are calculated from the numerical step values in figures 1 and 2. We first give the results for the shock tube problem 1, then for the shock tube problem 2. When the jump conditions are satisfied all values on a line should be equal since they are the value of the velocity of a shock wave obtained from the 5 different formulas.
\vskip13cm
\begin{displaymath}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{shock tube problem 1 }\\
\hline
&$ c_1$&$c_2$&$c_3$&$c_4$&$c_5$\\
\hline
left&-255.95 & -255.90 & -255.82 &-255.84 &-255.75\\
\hline
middle& -1.52&-1.49&17.94&10.23&-0.95\\
\hline
right&370.35&370.23& 370.24& 370.28&370.15\\
\hline
\end{tabular}
\end{displaymath}
\begin{displaymath}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{shock tube problem 2}\\
\hline
&$ c_1$&$c_2$&$c_3$&$c_4$&$c_5$\\
\hline
left& -240.23 & -240.85 & -240.81 &-240.65 &-240.91\\
\hline
middle& 9.34&9.30&8.30&13.78&10.52\\
\hline
right&358.32&358.70& 358.84& 358.53&358.77\\
\hline
\end{tabular}
\end{displaymath}
These jump formulas are very well satisfied by the left and right discontinuities but are not satisfied by the middle discontinuity except the two jump conditions from the two continuity equations. Since it is proved the results depict approximate solutions (from the theorem and from a careful numerical solution of the ODEs) an explanation could be that the middle discontinuity is not a classical shock wave as suggested by the singularities often observed on top or bottom of this discontinuity, which could denote it is some kind of more complicated wave, possibly not a shock wave. To test this hypothesis we did numerical tests for different volumic compositions of the fluids. One observes that in the case of equal volume fractions on both sides in the Riemann problem there appears a very neat singularity in the middle discontinuity which is present in the other cases but far clearly visible on the volumic fraction when both sides of the volumic fraction are equal (figure 3). With the values of pressure and velocities of shock tube problem 1 the observed singularity in volumic fraction is small, while it is quite large with the values of shock wave problem 2, figure 3. This explains why the middle discontinuity does not satisfy well, or does not satisfy at all in some cases, the expected conservative jump conditions: it is not a classical shock wave i.e. a mere moving discontinuity. It is natural that something else than a mere shock wave occurs: if the Riemann problem were solved by three standard shock waves we would have 8+3=11 unknown values (the 3 velocities and the 8 step values) for 12 equations (the 4 jump conditions at each discontinuity supposing one has solved the ambiguity in the 2 nonconservative equations). The values of wave velocities in case of figure 3, computed from the 5 algebraic formulas as in the above arrays are
\begin{displaymath}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
\multicolumn{6}{|c|}{shock tube problem 3}\\
\hline
&$ c_1$&$c_2$&$c_3$&$c_4$&$c_5$\\
\hline
left& -253.35 & -253.34 & -253.33 &-253.33 &-253.31\\
\hline
middle& -8473&-1099&25.3&12.8&-16.9\\
\hline
right& 368.99&368.96& 368.96& 368.97&368.92\\
\hline
\end{tabular} \end{displaymath}
\\
Since we have an approximate solution that can be computed with arbitrary precision it is possible to observe this singular part of the solution. Numerical investigation on the "object" that appears in the liquid fraction for $\alpha=0.60$ (top-left panel in figure 3) shows that \\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
$\bullet$ The area of the region located below the line $\alpha=0.60$ and above the curve $\alpha$ is constant (independent on $\epsilon$) for fixed time when $\epsilon$ varies and it is proportional to time even up to very large values of time (tests were done up to 100 times the value of time used in figures 1, 2 and 3 with the scheme in section 7).
$\bullet$ For rather small values of the time such as those in figures 1, 2 and 3 the object travels with constant speed and its width on $\alpha=0.60$ tends to 0 when $\epsilon\rightarrow 0$, roughly as $\sqrt\epsilon$ for fixed $t$ and as $\sqrt t$ for fixed $\epsilon$; its minimum value diminishes when the time increases. For large values of the time this decrease of the minimum is stopped because one always has $\alpha(x)>0 \ \forall x$ and one observes the width of the object then increases proportionally to $t$ so as to maintain an area proportional to time.
\textit{Observation 3.} One could state different values of $\gamma$ for the spreading of the state laws of the two fluids when they are very different, for instance a liquid and a gas: we have observed that modifications representing the convolution are unefficient to produce significative differences in the solution because in the present case the discontinuities take place on a large number of cells thus making the results rather unsensitive to modifications that would not be important enough to modify significatively the aspect of the jumps of $\alpha$ and $p$ (the nonconservative terms in (10, 11)): indeed the great sensibility on the slight modifications of the schemes for systems in nondivergence form has been observed in the case the discontinuities take place on a very small number of cells. The numerical schemes observed in the case of the multifluid system are robust in the sense that small modifications of the scheme do not affect significantly the result precisely because the discontinuities are spread over a large number of cells. Because of this fact one observes in the three arrays corresponding to figures 1, 2 and 3 that not only the conservative jump conditions (the three values $c_1,c_2$ and $c_3$) but also the two formal jump conditions (the two values $c_4$ and $c_5$) are satisfied, showing the evidence that, to some extent, one can compute formally on the system, thus allowing the formal nonlinear calculation done to obtain formula (13).\\
\textbf{7. A transport-correction scheme.}\\
We propose here a natural numerical scheme for the numerical solution of system (17-21). The scheme is an adaptation of the Le Roux et al numerical method of splitting into transport and pressure correction as described in \cite{Baraille}, extending to two fluids the scheme done in \cite{ColombeauNMPDE} for one fluid. The space $\mathbb{R}\times [0,+\infty[$ is divided into rectangular cells $[ih-\frac{h}{2},ih+\frac{h}{2}]\times [n\Delta t, (n+1)\Delta t[, i\in\mathbb{Z}, n\in\mathbb{N}$.\\
Given the family $\{(r_1)_i^n,(r_2)_i^n,(r_1u_1)_i^n,(r_2u_2)_i^n\}_{i\in\mathbb{Z}}$ of values of these variables on the interval $[ih-\frac{h}{2},ih+\frac{h}{2}]$ at time $n\Delta t$ we seek the family of values $\{(r_1)_i^{n+1},(r_2)_i^{n+1},(r_1u_1)_i^{n+1},(r_2u_2)_i^{n+1}\}_{i\in\mathbb{Z}}$ at time $(n+1)\Delta t$.\\
$\bullet$ \textit{First step: transport}. For k=1, 2
\begin{equation}(u_k)_i^n:=\frac{(r_ku_k)_i^n}{(r_k)_i^n} \end{equation} if $ (r_k)_i^n\not=0$, any value if $(r_k)_i^n=0,$
\begin{equation} (u_k)_i^{n,+}:=\frac{|(u_k)_i^n|+(u_k)_i^n}{2}, (u_k)_i^{n,-}:=\frac{|(u_k)_i^n|-(u_k)_i^n}{2}.\end{equation}
The CFL condition is $r|(u_k)_i^n|<1 \ \forall k,i,n$.
Then if $r=\frac{\Delta t}{h}$ we set
\begin{equation} (\overline{r_k})_i:=r (r_k)_{i-1}^n(u_k)_{i-1}^{n,+}+(1 -r |(u_k)_i^n|) (r_k)_i^n + r (r_k)_{i+1}^n(u_k)_{i+1}^{n,-},\end{equation}
\begin{equation} (\overline{r_ku_k})_i:=r (r_ku_k)_{i-1}^n(u_k)_{i-1}^{n,+} +(1 -r |(u_k)_i^n|) (r_ku_k)_i^n + r (r_ku_k)_{i+1}^n(u_k)_{i+1}^{n,-}.\end{equation}\\
$\bullet$ \textit{Second step: averaging}. We choose a value $\mu, 0<\mu<0.5$,
\begin{equation} (r_k)_i^{n+1}:=\mu(\overline{r_k})_{i-1}+(1-2\mu)(\overline{r_k})_{i}+\mu(\overline{r_k})_{i+1},\end{equation}
\begin{equation}\widetilde{ (r_ku_k)}_i:=\mu(\overline{r_ku_k})_{i-1}+(1-2\mu)(\overline{r_ku_k})_{i}+\mu(\overline{r_ku_k})_{i+1}.\end{equation}\\
$\bullet$ \textit{Third step: pressure correction}.
\begin{equation} \Delta_i:=(K_1\overline{ (r_1)_i}+K_2\overline{ (r_2)_i}+b_1-b_2)^2-4(b_1-b_2)K_1\overline{ (r_1)_i},\end{equation}
\begin{equation} \alpha_i:=\frac{K_1\overline{ (r_1)_i}+K_2\overline{ (r_2)_i}+b_1-b_2-\sqrt(\Delta)}{2(b_1-b_2)},\end{equation}
\begin{equation} p_i:=K_1\frac{ \overline{(r_1)_i} } { \alpha_i }-b_1 \ \ if \ \alpha_i\not=0,\end{equation}
\\
\begin{equation} (r_1u_1)_i^{n+1}:=\widetilde{ (r_1u_1)}_i-\frac{r}{2} \alpha_i(p_{i+1}-p_{i-1}),\end{equation}
\\
\begin{equation} (r_2u_2)_i^{n+1}:=\widetilde{ (r_2u_2)}_i-\frac{r}{2} (1-\alpha_i)(p_{i+1}-p_{i-1}).\end{equation}
\ \ \\
In (52, 58) we have obtained the family $\{(r_1)_i^{n+1},(r_2)_i^{n+1},(r_1u_1)_i^{n+1},(r_2u_2)_i^{n+1}\}_{i\in\mathbb{Z}}$.\\
\\
Now we justify the choice of an arbitrary value in density when a denominator in (48) is null.\\
\textbf{Proposition.}
\textit{When $(r_k)_i^{n+1}=0, k=1$ or $2$, then $(r_ku_k)_i^{n+1}=0$. }\\
proof.
Assume $(r_k)_i^{n+1}=0$. Then from (52), the strict inequality in $\mu$ and the positiveness of $r_k$ imply
\begin{equation}(\overline{r_k})_{i-1}=0=(\overline{r_k})_{i}=(\overline{r_k})_{i+1}.\end{equation}
Now notice that $(\overline{ r_k})_i=0$ implies $(r_k)_i^n=0$ from (50) and the strict inequality in the CFL condition. Further since $r\not=0$ it also implies from (50) that $ (r_k)_{i-1}^n(u_k)_{i-1}^{n,+}=0$, which implies $ (r_k u_k)_{i-1}^n(u_k)_{i-1}^{n,+}=0$, and similarly
$ (r_k u_k)_{i+1}^n(u_k)_{i+1}^{n,-}=0$. Therefore $(\overline{r_k})_i=0$ implies $(\overline{r_ku_k})_i=0$. Therefore from (59)
\vskip10cm
\textit{Figure 4. The shock tube problem 2 without correction (continuous curve) and with correction (+). One observes a small difference in two step values in the right panels.}
\begin{equation}(\overline{r_ku_k})_{i-1}=0=(\overline{r_ku_k})_{i}=(\overline{r_ku_k})_{i+1}. \end{equation}
Therefore from (53) $ (\widetilde{r_ku_k})_i=0$. From (56, 59) one has also $p_{i-1}=b_1=p_{i}=p_{i+1}$ if $k=1$. Finally, from (57, 58), we obtain $(r_ku_k)_i^{n+1}=0$. $\Box$\\
Following calculations in \cite{ColombeauSiam, ColombeauNMPDE} one can prove, under assumptions to be checked, such as boundedness of the velocity field when $h\rightarrow 0$, that the scheme tends to satisfy the equations when $h\rightarrow 0$.
\textit{Numerical observations.} First it has been observed that the scheme has always given the same result as the weak asymptotic method. It has the advantage to be more efficient and of a very easy use since one has only to fix the value of the CFL number $r$ and then the value of the averaging parameter $\mu$ in (52, 53).\\
The scheme in this section has been used with the interface pressure modelling (11) in \cite{EvjeFlatten} p. 180 which ensures the hyperbolicity of the system. In the case of shock tube problem 2 one can observe a slight difference relatively to the absence of correction (figure 4: 1000 space steps, $r=0.002, \mu=0.1$):
\vskip9cm
\textit{Figure 5. Quality of the transport-correction scheme: +++ results with pressure correction and (continuous line) without pressure correction. The curves are obtained with 100 space steps only.}\\
\\
the second step values from the left in pressure and gas velocity are 248000 and 86.5 respectively instead of 246000 and 88.5. Since the tests in \cite{EvjeFlatten} have been done in presence of this pressure correction and are close to the values we obtain with this correction, this explains the small disagreement observed when comparing the results in \cite{EvjeFlatten} figures 4 and 5 p. 197 and 198 with those in figure 2 for these two step values. Besides this difference the results in figure 1 and 3 are unchanged in absence or presence of the pressure correction, in particular the presence of the middle "singular wave" is independent of the presence of pressure correction.\\
The numerical quality of the scheme is tested in figure 5, both in absence and presence of pressure correction: a dicretization in 100 space cells suffices to obtain the step values and the jump formulas (as in the above arrays corresponding to figures 1, 2 and 3) with precision.\\
\textbf{9. Conclusion.}\\
The approximate solutions we have constructed with full proof and rather arbitrary initial data provide a mathematical tool that permits theorical and numerical investigations of the initial value problem for the equal pressure model of multifluid flows in the isothermal case. Since numerical calculations of these approximate solutions can be done easily and accurately with standard ODEs methods these approximate solutions can play the role of explicit solutions for mathematical and numerical investigations. They show that numerical schemes from scientific computing give an approximate solution besides the mathematical peculiarities of the model. They can show that supplementary terms such as pressure corrections do not modify (shock tube problem 1) or modify only slightly (shock tube problem 2) the solution. They permit to investigate the nature of the "solutions" put in evidence by these approximate solutions and by scientific computing although this system is in nondivergence form. \\
\\
Acknowledgements. The author is very grateful to members of the Instituto de Matematica of The Universidade de S\~ao Paulo, of the Instituto de Matematica, Estatistica e Computa\c c\~ao Cientifica of the Universidade Estadual de Campinas and of the Instituto de Matematica of the Universidade Federal do Rio de Janeiro for their attention, encouragements and suggestions while doing this work.\\
\textbf{Appendix. Formal calculations on the system.}\\
We obtain jump formulas from formal calculations. We observe in section 6 that these jump formulas are satisfied by the left shock waves and by the right shock waves in figures 1, 2 and 3. Developping (10) with $g=0$ and simplifying from (8), then dividing by $r_1$ one obtains\\
\begin{equation}\frac{\partial}{\partial t}(u_1)+u_1\frac{\partial}{\partial x}(u_1)+\frac{\alpha\frac{\partial}{\partial x} p}{r_1}=0.\end{equation}
Using the state law (6) $p=K_1\rho_1-b_1$ and $\rho_1=\frac{r_1}{\alpha}$ one obtains
$$\frac{\partial}{\partial t} (u_1)=\frac{\partial}{\partial x}(-K_1log(\rho_1)-\frac{(u_1)^2}{2}),$$
which gives the jump condition
\begin{equation} c=K_1\frac{log(\rho_{1,r})-log(\rho_{1,l})}{[u_1]}+\frac{u_{1,r}+u_{1,l}}{2}
\end{equation}
where $c$ denotes the velocity of the shock wave. The same calculation holds from (10) and (9) and gives (62) with index 2 and the same fomula with index 2.\\
\end{document} |
\begin{document}
\maketitle
\begin{abstract} In the present note we prove a multiplicity result for a Kirchhoff type problem involving a critical term, giving a partial positive answer to a problem raised by Ricceri.
\end{abstract}
\section{Introduction}
Nonlocal boundary value problems of the type
$$
\left\{
\begin{array}{ll}
- \left( a+b\displaystyle\int_\Omega |\nabla u|^2 dx\right)\Delta u=
f(x,u), & \hbox{ in } \Omega \\ \\
u=0, & \hbox{on } \partial \Omega
\end{array}
\right.
$$
are related to the stationary version of the Kirchhoff equation
$$\frac{\partial^2 u}{\partial t^2}- \left( a+b\displaystyle\int_\Omega |\nabla u|^2 dx\right)\Delta u=f_1(t,x,u),$$ first proposed by Kirchhoff to describe the transversal oscillations of a stretched string. Here $\Omega$ is a bounded domain of ${\mathbb R}^N$, $u$ denotes the displacement, $f_1$ is the
external force, $b$ is the initial tension and $a$ is related to the intrinsic properties
of the string.
Note that, this type of nonlocal equations appears in other fields like biological systems, where $u$ describes a process depending on the average of itself, like population density (see for instance \cite{CL}).
The first attempt to find solutions for subcritical nonlinearities, by means of variational methods, is due to Ma and Rivera \cite{MR} and Alves, Corr\^{e}a and Ma \cite{ACM} who combined minimization arguments with truncation techniques and a priori estimates. Using Yang index and critical group arguments or the theory of invariant sets of descent flows, Perera and Zhang (see \cite{PZ,ZP}) proved existence results for the above problem. Multiplicity theorems can be found for instance in \cite{CKW,MZ,R0}.
The existence or multiplicity of solutions of the Kirchhoff type problem with critical exponents in a bounded domain (or even in the whole space) has been studied by using different techniques as variational methods, genus theory, the Nehari manifold, the Ljusternik--Schnirelmann category theory (see for instance \cite{CF,Fan,F,FS}).
It is worth mentioning that Mountain Pass arguments combined with the Lions' Concentration Compactness principle \cite{L} are still the most popular tools to deal with such problems in the presence of a critical term. Applications to the lower dimensional case ($N<4$) can be found in \cite{ACF,LLG,N}, while for higher dimensions ($N\geq4$) we refer to \cite{H1,H2,N0,YM}. Notice that in order to employ the Concentration Compactness principle, $a$ and $b$ need to satisfy suitable constraints.
In order to state our main result we introduce the following notations: we endow the Sobolev space $H^1_0(\Omega)$ with the
classical norm $\|u\|=\left( \int_{\Omega }|\nabla u|^2 \ dx\right)^{\frac{1}{2}}$ and denote by $\|u\|_{q}$ the Lebesgue norm in $L^{q}(\Omega)$ for $1\leq q \leq 2^\star$, i.e. $\|u\|_{q}=\left(\int_{\Omega} |u|^{q} \ dx\right)^{\frac{1}{q}}$.
Let $S_N$ be the embedding constant of $H^1_0(\Omega)\hookrightarrow L^{2^\star}(\Omega)$, i.e.
\[\|u\|^2_{2^\star}\leq S_N^{-1} \|u\|^2 \qquad \mbox{for every } \ u\in H^1_0(\Omega). \]
Let us recall that (see Talenti \cite{Talenti} and Hebey \cite{H1} for an explicit espression)
\begin{equation}\label{2*}
S_N=\frac{N(N-2)}{4}\omega_N^{\frac{2}{N}},
\end{equation}
where $\omega_N$ is the volume of the unit ball in ${\mathbb R}^N$.
For $N\geq4$ denote by $C_1(N)$ and $C_2(N)$ the constants
\[
C_1(N)=
\begin{cases}\displaystyle
\frac{4(N-4)^{\frac{N-4}{2}}}{N^{\frac{N-2}{2}}S_{N}^{\frac{N}{2}}} & N>4\\ \\
\displaystyle \frac{1}{S_{4}^{2}}, & N=4,
\end{cases}
\qquad \mbox{ and } \qquad
C_2(N)=\begin{cases}
\displaystyle\frac{2(N-4)^{\frac{N-4}{2}}}{(N-2)^{\frac{N-2}{2}}S_{N}^{\frac{N}{2}}} & N>4\\ \\
\displaystyle \frac{1}{S_{4}^{2}}, & N=4.
\end{cases}.
\]
Notice that $C_1(N)\leq C_2(N)$.
Our result reads as follows:
\begin{theor}\label{our theorem} Let $a, b$ be positive numbers, $N\ge4$. \\
\noindent (A) If $ a^{\frac{N-4}{2}} b\geq C_1(N)$, then, for each $\lambda>0$ large enough and for each convex set $C\subseteq L^2(\Omega)$ whose closure in $L^2(\Omega)$ contains $H^1_0(\Omega)$, there exists $v^*\in C$ such that the functional
\[u\to \frac{a}{2} \int_{\Omega}|\nabla u|^2 dx +\frac{b}{4} \left( \int_{\Omega}|\nabla u|^2 dx\right)^2-\frac{1}{2^\star}\int_{\Omega}|u|^{2^\star} dx-\frac{\lambda}{2}\int_{\Omega}|u(x)-v^*(x)|^2 dx\] has two global minima.
\noindent (B) If $a^{\frac{N-4}{2}} b> C_2(N)$,
then,
for each $\lambda>0$ large enough and for each convex set $C\subseteq L^2(\Omega)$ whose closure in $L^2(\Omega)$ contains $H^1_0(\Omega)$, there exists $v^*\in C$ such that the problem
$$
\left\{
\begin{array}{ll}
- \left( a+b\displaystyle\int_\Omega |\nabla u|^2 dx\right)\Delta u=
|u|^{2^\star-2}u+\lambda (u-v^*(x)), & \hbox{ in } \Omega \\ \\
u=0, & \hbox{on } \partial \Omega
\end{array}
\right.\eqno{(\mathcal{P}_{\lambda})}
$$
has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)$ of the energy functional defined in (A).
\end{theor}
The paper is motivated by a recent work of Ricceri where the author studied problem $(\mathcal P_\lambda)$ in the subcritical case, i.e. when $|u|^{2^\star-2}u$ is replaced by $|u|^{p-2}u$ with $p<2^\star$. In \cite[Proposition 1]{R}, the existence of two global minima for the energy functional (and three solutions for the associated Kirchhoff problem) is obtained for every $a\geq 0$ and $b>0$. In the same paper, the following challenging question was raised (see \cite[Problem 1]{R}):
\begin{question}
Does the conclusion of Proposition 1 hold if $N>4$ and $p=2^\star$?
\end{question}
Notice that, for $N>4$ and $p=2^\star$ the energy functional associated to $(\mathcal{P}_{\lambda})$ is bounded from below while if $N=4(=2^\star)$ this is not true any more for arbitrary $b$. Moreover, when $p=2^\star$ the embedding of $H^1_0(\Omega)$ into $L^{p}(\Omega)$ fails to be compact and one can not apply directly the abstract tool which leads to \cite[Theorem 1 \& Proposition 1]{R}.
The main result of the present note gives a partial positive answer to the above question and prove that Proposition 1 of \cite{R} holds for $p=2^\star$ and $N\geq 4$ provided that $a$ and $b$ satisfies a suitable crucial inequality.
Namely, we prove that the interaction between the Kirchhoff type operator and the critical nonlinearity ensures the sequentially weakly lower semicontinuity of the energy functional, a key property which allows to apply the minimax theory developed in \cite[Theorem 3.2]{R1} (see also Theorem \ref{minimax theorem} below).
\section{Proofs}
The proof of Theorem \ref{our theorem} relies on the following key lemma (see \cite{FFK} for a deeper study on this topic).
\begin{lem}\label{semicontinuity}
Let $N\geq 4$ and $a, b$ be positive numbers such that $ a^{\frac{N-4}{2}} b\geq C_1(N)$. Denote by $\mathcal F:H^1_0(\Omega)\to{\mathbb R}$ the functional
\[\mathcal F(u)=\frac{a}{2}\|u\|^2+\frac{b}{4} \|u\|^4-\frac{1}{2^\star}\|u\|^{2^\star}_{2^\star} \qquad \mbox{for every }\ u \in H^1_0(\Omega).\]
Then, $\mathcal F$ is sequentially weakly lower semicontinuous in $H^1_0(\Omega)$.
\end{lem}
\begin{proof}
Fix $u \in H^1_0(\Omega)$ and let $\{u_n\} \subset H^1_0(\Omega)$ such that $u_n\rightharpoonup u$ in $H^1_0(\Omega)$. Thus,
\begin{align*}
\mathcal{F}(u_n)-\mathcal{F}(u) =&\frac{a}{2}(\|u_n\|^2-\|u\|^2)+\frac{b}{4}(\|u_n\|^4-\|u\|^4)\\ &-\frac{1}{2^\star}\left(\|u_n\|_{2^\star}^{2^\star}-\|u\|_{2^\star}^{2^\star}\right).
\end{align*}
It is clear that \begin{align*}\|u_n\|^2-\|u\|^2&=\|u_n-u\|^2+2\int_{\Omega}\nabla(u_n-u)\nabla u \\ &= \|u_n-u\|^2+o(1),
\end{align*}
and
\begin{align*}
\|u_n\|^4-\|u\|^4&=\left(\|u_n-u\|^2+o(1)\right)\left(\|u_n-u\|^2+2\int_{\Omega}\nabla u_n\nabla u\right)\\&=\left(\|u_n-u\|^2+o(1)\right)\left(\|u_n-u\|^2+2\int_{\Omega}\nabla (u_n-u)\nabla u+2\|u\|^2\right)\\
&=\left(\|u_n-u\|^2+o(1)\right)\left(\|u_n-u\|^2+2\|u\|^2+o(1)\right).
\end{align*}
Moreover, from the Br\'ezis-Lieb lemma, one has
$$\|u_n\|_{2^\star}^{2^\star}-\|u\|_{2^\star}^{2^\star}=\|u_n-u\|_{2^\star}^{2^\star}+o(1).$$
Putting together the above outcomes,
\begin{align*}
\mathcal{F}(u_n)-\mathcal{F}(u)=&\frac{a}{2}\|u_n-u\|^2+\frac{b}{4}\left(\|u_n-u\|^4+2\|u\|^2\|u_n-u\|^2\right)-\frac{1}{2^\star}\|u_n-u\|_{2^\star}^{2^\star}+o(1) \\{\geq}& \frac{a}{2}\|u_n-u\|^2+\frac{b}{4}\left(\|u_n-u\|^4+2\|u\|^2\|u_n-u\|^2\right)-\frac{{S}_N^{-\frac{2^\star}{2}}}{2^\star}\|u_n-u\|^{2^\star}+o(1)
\\ \geq& \frac{a}{2}\|u_n-u\|^2 +\frac{b}{4}\|u_n-u\|^4-\frac{{S}_N^{-\frac{2^\star}{2}}}{2^\star}\|u_n-u\|^{2^\star}+o(1)\\=& \|u_n-u\|^2 \left(\frac{a}{2}+\frac{b}{4}\|u_n-u\|^2-\frac{{S}_N^{-\frac{2^\star}{2}}}{2^\star}\|u_n-u\|^{2^\star-2}\right)+o(1).
\end{align*}
Denote by $f:[0,+\infty[\to{\mathbb R}$ the function $\displaystyle f(x)=\frac{a}{2}+\frac{b}{4}x^2-\frac{{S}_N^{-\frac{2^\star}{2}}}{2^\star}x^{2^\star-2}$. We claim that $f(x)\geq 0$ for all $x\geq 0$.
Indeed, when $N=4$, and $b{S}_4^2\geq 1$,
\[f(x)=\frac{a}{2}+\frac{b}{4}x^2-\frac{{S}_4^{-2}}{4}x^2=\frac{a}{2}+\frac{1}{4}\left(b-\frac{1}{{S}_4^2}\right)x^2\geq \frac{a}{2}.\]
If $N>4$, it is immediately seen that $f$ attains its minimum at $$x_0=\left(\frac{2^\star }{2(2^\star-2)}{S}_N^{\frac{2^\star}{2}}b\right)^{\frac{1}{2^\star-4}}$$ and the claim is a consequence of the assumption $\displaystyle a^\frac{N-4}{2}b\geq C_1(N)$.
Thus, $$\liminf_{n\to \infty}(\mathcal{F}(u_n)-\mathcal{F}(u))\geq \liminf_{n \to \infty}\|u_n-u\|^2 f(\|u_n-u\|)\geq 0,$$ and the thesis follows.
\end{proof}
\begin{rem}We point out that the constant $C_1(N)$ in Lemma \ref{semicontinuity} is optimal, i.e. if
$ a^{\frac{N-4}{2}} b< C_1(N)$ the functional $\mathcal F$ is no longer sequentially weakly lower semicontinuous (see \cite{FFK}).
\end{rem}
In the next lemma we prove the Palais Smale property for our energy functional. Notice that the same constraints on $a$ and $b$ appear in \cite{H1} where such property was investigated for the critical Kirchhoff equation on closed manifolds by employing the $H^1$ (which is the underlying Sobolev space) decomposition.
\begin{lem}\label{Palais Smale}
Let $N \ge 4$ and $a,b$ be positive numbers such that $a^{\frac{N-4}{2}}b>C_{2}(N)$. For $\lambda>0, v^*\in H_{0}^{1}(\Omega)$ denote by $\mathcal{E}:H_{0}^{1}(\Omega)\to\mathbb{R}$ the functional
defined by
\[
\mathcal{E}(u)=\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}-\frac{1}{2^{\star}}\|u\|_{2^{\star}}^{2^{\star}}-\frac{\lambda}{2}\|u-v^{\star}\|_{2}^{2}
\qquad \mbox{for every }\ u \in H^1_0(\Omega).\] Then, $\mathcal E$
satisfies the Palais-Smale (shortly (PS)) condition.
\end{lem}
\begin{proof}
Let $\{u_{n}\}$ be a (PS) sequence for $\mathcal E$, that is
\[
\begin{cases}
\mathcal{E}(u_{n})\to c\\
\mathcal{E}'(u_{n})\to0
\end{cases}\mbox{as }n\to\infty.
\]
Since $\mathcal E$ is coercive, $\{u_{n}\}$ is bounded and there exists $u\in H_{0}^{1}(\Omega)$ such that (up to a subsequence)
\begin{align*}
u_{n} & \rightharpoonup u\mbox{ in }H_{0}^{1}(\Omega),\\
u_{n} & \to u\mbox{ in }L^{p}(\Omega),\ p\in[1,2^{\star}),\\
u_{n} & \to u\mbox{ a.e. in }\Omega.
\end{align*}
Using the second concentration compactness lemma of Lions \cite{L}, there exist an at most countable index set $J$,
a set of points $\{x_{j}\}_{j\in J}\subset\overline\Omega$ and two families of positive
numbers $\{\eta_{j}\}_{j\in J}$, $\{\nu_{j}\}_{j\in J}$ such that
\begin{align*}
|\nabla u_{n}|^{2} & \rightharpoonup d\eta\geq|\nabla u|^{2}+\sum_{j\in J}\eta_{j}\delta_{x_{j}},\\
|u_{n}|^{2^\star} & \rightharpoonup d\nu=|u|^{2^\star}+\sum_{j\in J}\nu_{j}\delta_{x_{j}},
\end{align*}
(weak star convergence in the sense of measures), where $\delta_{x_{j}}$ is the Dirac mass concentrated at
$x_{j}$ and such that
$$ S_{N} \nu_{j}^{\frac{2}{2^\star}}\leq\eta_{j} \qquad \mbox{for every $j\in J$}.$$
Next, we will prove that the index set $J$ is empty. Arguing
by contradiction, we may assume that there exists a $j_{0}$ such
that $\nu_{j_{0}}\neq0$. Consider now, for $\varepsilon>0$ a non negative cut-off function $\phi_\varepsilon$ such that
\begin{align*}
&\phi_{\varepsilon} =1\mbox{ on }B(x_{0},\varepsilon),\\
&\phi_{\varepsilon} =0\mbox{ on } \Omega\setminus B(x_{0},2\varepsilon),\\
&|\nabla\phi_{\varepsilon}| \leq\frac{2}{\varepsilon}.
\end{align*}
It is clear that the sequence $\{u_{n}\phi_{\varepsilon}\}_{n}$ is
bounded in $H_{0}^{1}(\Omega)$, so that
\[
\lim_{n\to\infty}\mathcal{E}'(u_{n})(u_{n}\phi_{\varepsilon})=0.
\]
Thus
\begin{align}\label{calc 1}
o(1) & =(a+b\|u_{n}\|^{2})\int_{\Omega}\nabla u_{n}\nabla(u_{n}\phi_{\varepsilon})-\int_{\Omega}|u_{n}|^{2^\star}\phi_{\varepsilon}-\lambda\int_{\Omega}(u_{n}-v^{*})(u_{n}\phi_{\varepsilon}) \nonumber \\
& =(a+b\|u_{n}\|^{2})\left(\int_{\Omega}|\nabla u_{n}|^{2}\phi_{\varepsilon}+\int_{\Omega}u_{n}\nabla u_{n}\nabla\phi_{\varepsilon}\right)-\int_{\Omega}|u_{n}|^{2^\star}\phi_{\varepsilon}-\lambda\int_{\Omega}(u_{n}-v^{*})(u_{n}\phi_{\varepsilon}).
\end{align}
Moreover, using H\"{o}lder inequality, one has
\[
\left|\int_{\Omega}(u_{n}-v^{*})(u_{n}\phi_{\varepsilon})\right|\leq \left(\int_{B(x_{0},2\varepsilon)}(u_{n}-v^{*})^2\right)^\frac{1}{2} \left(\int_{B(x_{0},2\varepsilon)}u_n^2\right)^\frac{1}{2},
\] so that
\[\lim_{\varepsilon\to0}\lim_{n\to\infty}\int_{\Omega}(u_{n}-v^{*})(u_{n}\phi_{\varepsilon})=0.\]
Also,
\begin{eqnarray*}
\left|\int_\Omega u_{n}\nabla u_{n}\nabla\phi_{\varepsilon}\right|&=&\left|\int_{B(x_{0},2\varepsilon)}u_{n}\nabla u_{n}\nabla\phi_{\varepsilon}\right|\leq \left(\int_{B(x_{0},2\varepsilon)}|\nabla u_n|^2\right)^\frac{1}{2}
\left(\int_{B(x_{0},2\varepsilon)}|u_n\nabla \phi_\varepsilon|^2\right)^\frac{1}{2}\\
&\leq& C \left(\int_{B(x_{0},2\varepsilon)}|u_n\nabla \phi_\varepsilon|^2\right)^\frac{1}{2}.
\end{eqnarray*}
Since $$\lim_{n\to\infty}\int_{B(x_{0},2\varepsilon)}|u_n\nabla \phi_\varepsilon|^2=\int_{B(x_{0},2\varepsilon)}|u\nabla \phi_\varepsilon|^2,$$ and
\begin{eqnarray*}
\left(\int_{B(x_{0},2\varepsilon)}|u\nabla \phi_\varepsilon|^2\right)^\frac{1}{2}&\leq &
\left(\int_{B(x_{0},2\varepsilon)} |u|^{2^\star}\right)^\frac{1}{2^\star}
\left(\int_{B(x_{0},2\varepsilon)}|\nabla \phi_\varepsilon|^N \right)^\frac{1}{N}\\
&\leq& C \left(\int_{B(x_{0},2\varepsilon)} |u|^{2^\star}\right)^\frac{1}{2^\star}
\end{eqnarray*}
we get
\[
\lim_{\varepsilon\to0}\lim_{n\to\infty}(a+b\|u_{n}\|^{2})\left|\int_\Omega u_{n}\nabla u_{n}\nabla\phi_{\varepsilon}\right|=0.
\]
Moreover, as $0\leq \phi_\varepsilon\leq 1$,
\begin{eqnarray*}
\lim_{n\to\infty}(a+b\|u_{n}\|^{2})\int_{\Omega}|\nabla u_{n}|^{2}\phi_{\varepsilon}&\geq&
\lim_{n\to\infty}\left[a\int_{B(x_{0},2\varepsilon)}|\nabla u_{n}|^{2}\phi_{\varepsilon}+b\left(\int_{\Omega}|\nabla u_{n}|^{2}\phi_{\varepsilon}\right)^{2}\right]\\&\geq&
a\int_{B(x_{0},2\varepsilon)}|\nabla u|^{2}\phi_{\varepsilon}+b\left(\int_{\Omega}|\nabla u|^{2}\phi_{\varepsilon}\right)^{2}+a\eta_{j_{0}}+b\eta_{j_{0}}^{2}.
\end{eqnarray*}
So, as $\int_{B(x_{0},2\varepsilon)}|\nabla u|^{2}\phi_{\varepsilon}\to 0$ as $\varepsilon\to 0$,
\[
\lim_{\varepsilon\to0}\lim_{n\to\infty}(a+b\|u_{n}\|^{2})\int_{\Omega}|\nabla u_{n}|^{2}\phi_{\varepsilon} \geq a\eta_{j_{0}}+b\eta_{j_{0}}^{2}.\]
Finally,
\begin{align*}
\lim_{\varepsilon\to0}\lim_{n\to\infty}\int_\Omega|u_{n}|^{2^\star}\phi_{\varepsilon} & =\lim_{\varepsilon\to0}\int_\Omega |u|^{2^\star}\phi_{\varepsilon}+\nu_{j_{0}}=\lim_{\varepsilon\to0}\int_{B(x_{0},2\varepsilon)} |u|^{2^\star}\phi_{\varepsilon}+\nu_{j_{0}}=\nu_{j_{0}}.
\end{align*}
Summing up the above outcomes, from \eqref{calc 1}
one obtains
\begin{align*}
0 & \geq a\eta_{j_{0}}+b\eta_{j_{0}}^{2}-\nu_{j_0}\geq a\eta_{j_{0}}+b\eta_{j_{0}}^{2}-S_{N}^{-\frac{2^\star}{2}}\eta_{j_{0}}^{\frac{2^\star}{2}}\\
& =\eta_{j_{0}}\left(a+b\eta_{j_{0}}-S_{N}^{-\frac{2^\star}{2}}\eta_{j_{0}}^{\frac{2^\star-2}{2}}\right).
\end{align*}
Denote by $f_{1}:[0,+\infty[\to\mathbb{R}$ the function ${\displaystyle f_{1}(x)=a+bx-S_{N}^{-\frac{2^\star}{2}}x^{\frac{2^\star-2}{2}}}$.
As before, assumptions on $a$ and $b$ imply that $f_{1}(x)>0$ for all $x\geq0$. Thus
\[
a+b\eta_{j_{0}}-S_{N}^{-\frac{2^\star}{2}}\eta_{j_{0}}^{\frac{2^\star-2}{2}}>0,
\]
therefore $\eta_{j_{0}}=0,$ which is a contradiction. Such conclusion implies
that $J$ is empty, that is
\[\lim_{n\to\infty}\int_{\Omega}|u_n|^{2^\star}= \int_{\Omega}|u|^{2^\star}\]
and the uniform convexity of $L^{2^\star}(\Omega)$ implies that \[
u_{n}\to u\mbox{ in }L^{2^\star}(\Omega).
\]
Now, recalling that the derivative of the function $$u\to \frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}$$ satisfies the $(S_+)$ property, in a standard way one can see that $u_{n}\to u\mbox{ in }H_{0}^{1}(\Omega)$, which proves
our lemma.
\end{proof}
In the proof of our result, the main tool is the following theorem:
\begin{theor}[Ricceri \cite{R1}, Theorem 3.2]\label{minimax theorem}
Let $X$ be a topological space, $E$ a real Hausdorff
topological vector space, $C\subseteq E$ a convex set,
$f : X\times C \to {\mathbb R}$ a function which is lower semicontinuous,
inf--compact in $X$, and upper semicontinuous and concave in $C$. Assume also that
\begin{equation}\label{minimax}
\sup_{v\in C}\inf_{x\in X}f(x,v)<\inf_{x\in X}\sup_{v\in C} f(x,v).
\end{equation}
Then, there exists $v^*\in C$ such that the function $f(\cdot, v^*)$ has at least two global
minima.
\end{theor}
\noindent {\bf Proof of Theorem \ref{our theorem}}
We apply Theorem \ref{minimax theorem} with $X=H^1_0(\Omega)$ endowed with the weak topology, $E=L^2(\Omega)$ with the strong topology, $C$ as in the assumptions.
Let $\mathcal F$ as in Lemma \ref{semicontinuity}, i.e.
\[\mathcal F(u)=\frac{a}{2}\|u\|^2+\frac{b}{4} \|u\|^4-\frac{1}{2^\star}\|u\|^{2^\star}_{2^\star} \qquad \mbox{for every }\ u \in H^1_0(\Omega).\]
From Lemma \ref{semicontinuity}, $\mathcal F$ is sequentially weakly lower semicontinuous, and coercive, thus, the set $M_\mathcal F$ of its global minima is non empty.
Denote by
\begin{equation}\label{lambdastar}\lambda^\star=\inf\left\{\frac{\mathcal F(u)-\mathcal F(v)}{\|u-v\|_2^2} \ : \ (v, u)\in M_{\mathcal F}\times H^1_0(\Omega), \ v\neq u \right\}
\end{equation} and fix $\lambda>\lambda^\star$.
Let $f: H^1_0(\Omega)\times C\to{\mathbb R} $ be the function
\[f(u,v)=\mathcal F(u)-\lambda \|u-v\|_2^2.\]
From the Eberlein Smulyan theorem it follows that $f(\cdot, v)$ has weakly compact sublevel sets in $H^1_0(\Omega)$. It is also clear that $f(u, \cdot)$ is continuous and concave in $L^2(\Omega)$. Let us prove \eqref{minimax}.
Recalling that the closure of $C$ in $L^2(\Omega)$ (denoted by ${\overline C}$) contains $H^1_0(\Omega)$, one has
\begin{align}\label{first}
\inf_{u\in H^1_0(\Omega)}\sup_{v\in C } f(u,v)&=\inf_{u\in H^1_0(\Omega)}\sup_{v\in \overline {C}}f(u,v)\nonumber \\&\geq
\inf_{u\in H^1_0(\Omega)}\sup_{v\in H^1_0(\Omega) } f(u,v)\nonumber\\&=
\inf_{u\in H^1_0(\Omega)}\sup_{v\in H^1_0(\Omega) } (\mathcal F(u)-\lambda \|u-v\|_2^2)\nonumber \\\nonumber&=\inf_{u\in H^1_0(\Omega)}(\mathcal F(u)-\lambda \inf_{v\in H^1_0(\Omega)}\|u-v\|_2^2)\\&=
\min_ {H^1_0(\Omega)}\mathcal F
\end{align}
Since $\lambda>\lambda^\star$, there exist $u_0, v_0\in H^1_0(\Omega), u_0\neq v_0$ and $\varepsilon>0 $ such that
\begin{align*}
&\mathcal F(u_0)-\lambda \|u_0-v_0\|_2^2<\mathcal F(v_0)-\varepsilon,\\
& \mathcal F(v_0)= \min_ {H^1_0(\Omega)}\mathcal F.
\end{align*}
Thus, if $h:L^2(\Omega)\to{\mathbb R}$ is the function defined by $h(v)=\inf_{u\in H^1_0(\Omega)}(\mathcal F(u)-\lambda \|u-v\|_2^2)$,
then, $h$ is upper semicontinuous in $L^2(\Omega)$ and \[h(v_0)\leq \mathcal F(u_0)-\lambda \|u_0-v_0\|_2^2<\mathcal F(v_0)-\varepsilon.\]
So, there exists $\delta>0$ such that $h(v)<\mathcal F(v_0)-\varepsilon$ for all $\|v-v_0\|_2\leq \delta.$
Therefore,
\[\sup_{\|v-v_0\|_2\leq \delta }\inf_{u\in H^1_0(\Omega)}(\mathcal F(u)-\lambda \|u-v\|_2^2)\leq \mathcal F(v_0)-\varepsilon.\]
On the other hand,
\[\sup_{\|v-v_0\|_2\geq \delta }\inf_{u\in H^1_0(\Omega)}(\mathcal F(u)-\lambda \|u-v\|_2^2)\leq
\sup_{\|v-v_0\|_2\geq \delta } (\mathcal F(v_0)-\lambda \|v_0-v\|_2^2)\leq \mathcal F(v_0)-\lambda\delta^2.\]
Summing up the above outcomes, we obtain
\begin{align}\label{second}
\sup_{v\in C }\inf_{u\in H^1_0(\Omega)} f(u,v)&\leq \sup_{v\in L^2(\Omega) }\inf_{u\in H^1_0(\Omega)}f(u,v)\nonumber \\&=
\sup_{v\in L^2(\Omega) }\inf_{u\in H^1_0(\Omega)}(\mathcal F(u)-\lambda \|u-v\|_2^2)\nonumber\\&<\mathcal F(v_0)=\min_{H^1_0(\Omega)}\mathcal F.
\end{align}
From \eqref{first} and \eqref{second}, claim \eqref{minimax} follows.
Applying Theorem \ref{minimax theorem}, we deduce the existence of $v^*\in C$ such that the energy functional
\[\mathcal E(u)=\mathcal F(u)-\frac{\lambda}{2}\|u-v^*\|_2^2\] associated to our problem has two global minima, which is claim $(A)$. In order to prove $(B)$ we observe that, since the functional is of class $C^1$, such global minima turns out to be weak solutions of our problem. The third solution follows by Lemma \ref{Palais Smale} (recall that $C_2(N)\geq C_1(N)$) and a classical version of the Mountain Pass theorem by Pucci and Serrin \cite{PS}.\qed
\begin{rem} For sake of clarity, we calculate the approximate
values of the constants $C_1(N)$ and $C_2(N)$ for some $N$:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$N$ & $C_1(N)$ & $C_{2}(N)$\tabularnewline
\hline
\hline
5 & 0.002495906672 & 0.002685168050\tabularnewline
\hline
6 & 0.0001990835458 & 0.0002239689890\tabularnewline
\hline
7 & 0.00001712333233 & 0.00001985538802\tabularnewline
\hline
9 & 1.269275934$\cdot10^{-7}$ & 1.529437355$\cdot10^{-7}$\tabularnewline
\hline
\end{tabular}
\end{center}
\end{rem}
{\begin{question}
Notice that if $N=4$ then, for $b S_N^2< 1$, $\mathcal E$ is unbounded from below. Indeed, if $\{u_n\}$ is such that $\frac{\|u_n\|^2}{\|u_n\|_4^2}\to S_N$, then we can fix $c$ and $\bar n$ such that $\frac{\|u_{\bar n}\|^2}{\|u_{\bar n}\|_4^2}<c<b^{-\frac{1}{2}}$. Thus
\[
\mathcal{E}(\tau u_{\bar n})<\frac{a\tau^2}{2}\|u_{\bar n}\|^{2}+\frac{\tau^4}{4}\left(b-\frac{1}{c^2}\right)\|u_{\bar n}\|^{4}-\frac{\lambda}{2}\|\tau u_{\bar n}-v^{*}\|_{2}^{2}\to-\infty, \mbox{as} \ \tau\to+\infty.
\] It remains an open question if, when $N>4$, Theorem \ref{our theorem} holds for every $a\geq 0, b>0$ with $ a^{\frac{N-4}{2}} b< C_1(N)$.
\end{question}
{\bf Acknowledgment} This work was initiated when Cs.
Farkas visited the Department of Mathematics of the University of
Catania, Italy. He thanks the financial support of Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
\end{document} |
\begin{document}
\begin{abstract}
We study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this family by using Dirichlet polynomial approximations of $L^k(s,\chi)$ of length $X$, with $Q<X<Q^2$. Assuming the Generalized Lindel\"{o}f Hypothesis, we prove an asymptotic formula for these approximations of the twisted moments. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith for this family of $L$-functions, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family.
\end{abstract}
\maketitle
\begingroup
\hypersetup{hidelinks}
\tableofcontents
\endgroup
\section{Historical overview and motivation}\label{sec: HistoricalOverview}
In recent decades, there has been much interest and measured progress in the study of moments of $L$-functions. The program has its beginnings in the study of the $2k$th moment
\[
M_k(T):=\int_{0}^{T}\left|\zeta\left(\tfrac{1}{2}+it\right) \right|^{2k}\,dt
\]
of the Riemann zeta-function $\zeta(s)$, where $k$ is any positive real number. A great deal of effort has been made to understand $M_k(T)$ for different values of $k$ as $T\to \infty$, yet asymptotic formulas for $M_k(T)$ have remained stubbornly out of reach in all but a few cases. In 1918, Hardy and Littlewood \cite{HL} showed that $M_1(T)\sim T\log T$ as $T\to\infty$, and in 1926 Ingham \cite{Ingham} showed that $M_2(T) \sim (2\pi^2)^{-1}T\log^4T$ as $T\to\infty$. To date, an asymptotic formula is not known to hold for any other $M_k(T)$. Historically, the original motivation for studying $M_k(T)$ has been to prove the Lindel\"{o}f Hypothesis (LH), which asserts that\footnote{Here and throughout this paper, we employ Vinogradov notation and use $f \ll g$ to mean $f=O(g)$.} for any $\varepsilon>0$, $\zeta(1/2+it)\ll t^\varepsilon$ as $t\rightarrow \infty$. In fact, if one could show that $M_k(T)\ll T^{1+\varepsilon}$ for all positive integers $k$ and arbitrarily small $\varepsilon>0$, then LH would follow~\cite[Theorem~13.2]{Titchmarsh}. Proving an asymptotic formula for $M_k(T)$ for any integer $k\ge 3$ is now considered an important problem in its own right.
A folklore conjecture predicts that if $k$ is a positive real number, then, for some unspecified constant $c_k$, we have $M_k(T) \sim c_kT(\log T)^{k^2}$ as $T \to \infty$. In support of this conjecture, it is now known due to the work of many authors that
\[
T(\log T)^{k^2} \ll M_k(T) \ll T(\log T)^{k^2} ,
\]
where the lower bound holds for any real $k\ge 0$, and the upper bound holds unconditionally for $0\le k \le 2$ and conditionally on the Riemann Hypothesis for $k>2$ (see \cite{Ram2}, \cite{Ram1}, \cite{HBlower}, \cite{Soundzeta}, \cite{RadSound}, \cite{Harperzeta}, \cite{BettinChandeeRadziwill}, \cite{BettinChandeeRadziwill}, \cite{HeapRadziwillSound}), and \cite{SoundHeap}). The problem of finding an asymptotic formula for $M_k(T)$ for $k\geq 3$ is so intractable that, up until recently, there had been no viable guess for the exact value of the coefficient $c_k$ in the conjecture $M_k(T) \sim c_kT(\log T)^{k^2}$ for any integer $k\geq 3$. In 1993, Conrey and Ghosh~\cite{ConreyGhoshtalk,ConreyGhosh} predicted the exact value of $c_3$. Later, Conrey and Gonek~\cite{ConreyGonek} used a different approach to conjecture the exact values of both $c_3$ and $c_4$. Both approaches involve heuristic number-theoretic arguments, and the predicted values of $c_3$ agree. Recently, Ng~\cite{Ng} has made the heuristic argument of Conrey and Gonek rigorous, and used it to prove an asymptotic formula for $M_3(T)$ under the assumption of an additive divisor conjecture.
A breakthrough was made in the late 90's when Keating and Snaith \cite{KeatingSnaithRMTzeta} modeled $M_k(T)$ via characteristic polynomials of large random matrices. Doing so allowed them to conjecture the exact value of $c_k$ for all complex $k$ with $\re(k)\ge-1/2$. Remarkably, their predictions agree with the Conrey-Ghosh-Gonek conjectures for $c_3$ and $c_4$ . Later, Diaconu, Goldfeld, and Hoffstein~\cite{DGH} used the theory of multiple Dirichlet series to conjecture the value of $c_k$ for all natural numbers $k$. Despite the differences between these approaches, all the conjectures agree.
Keating and Snaith \cite{KeatingSnaithRMTLfunctions2, KeatingSnaithRMTLfunctions} have made analogous predictions for various families of $L$-functions. One family that has received much attention in the literature is the family of all primitive Dirichlet $L$-functions of modulus $q$. Precisely, let $\chi\bmod q$ be a primitive Dirichlet character, and let
\[
L(s,\chi) = \sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}=\prod_{p}\left(1-\frac{\chi(p)}{p^s}\right)^{-1}, \quad \re(s)>1
\]
be its associated Dirichlet $L$-function. In 1931, Paley \cite{Paley} showed that $\sum_{\chi}|L(1/2,\chi)|^2\sim (\phi^2(q)/q)\log q $ as $q\to \infty$, where the sum is over all characters modulo $q$. The work of Heath-Brown \cite{HB4th} shows
$$
\sideset{}{^*}\sum_{\chi \bmod q}|L(\tfrac{1}{2},\chi)|^4\sim \frac{\phi^*(q)}{2\pi^2}\prod_{p|q}\frac{(1-\tfrac{1}{p})^3}{(1+\tfrac{1}{p})}(\log q)^4, \quad q \to \infty
$$
with some restrictions on $q$, where $*$ is used to indicate that the sum is over primitive characters and $\phi^*(q)$ is the number of primitive characters modulo $q$. Soundararajan \cite{Sound4th} improved the result to hold for all $q$. Young \cite{Young4th} showed that this asymptotic formula holds with a power savings error term when the modulus $q$ is prime. Progress for this family is at the same level as that of the zeta-function, and asymptotic expressions have only been obtained for the second and fourth moments. Likewise, sharp lower and upper bounds for the $2k$th moments can be computed; see \cite{RSlower}, \cite{Soundzeta}, \cite{HBupper}, \cite{Harperzeta}, and \cite{SoundHeap}.
By averaging over all $q\le Q$, Huxley \cite{Huxley} used the large sieve inequality to obtain upper bounds of the predicted order of magnitude for $\sum_{q\le Q}\sum_{\chi\bmod q}^*|L(1/2,\chi)|^{2k}$ with $k=3,4$. A recent innovation of Conrey, Iwaniec, and Soundararajan \cite{CISAsymptoticLargeSieve} allowed them to prove an asymptotic formula for the sixth moment averaged over all $q$, albeit with an additional small averaging over the critical line \cite{CIS6th}. Their method, called the \textit{asymptotic large sieve}, was later refined by Chandee and Li \cite{ChandeeLi8Dirichlet} in the context of the eighth moment with the same additional averaging. The asymptotic large sieve has also been used to study the zeros of primitive Dirichlet $L$-functions (see \cite{CISGaps}, \cite{CISCriticalZeros}, \cite{ChandeeLeeLiuRadziwill}) and the twisted second moment \cite{CIS}. (See Section~\ref{sec: outline} for a more detailed discussion on the asymptotic large sieve.)
Inspired by the discovery of Keating and Snaith, Conrey, Farmer, Keating, Rubinstein, and Snaith \cite{CFKRS} used random matrix theory as a guide to formulate a heuristic, which we refer to as ``the CFKRS recipe" or simply ``the recipe," that predicts precise asymptotic formulas for integral moments of various families of $L$-functions. For the family of primitive Dirichlet $L$-functions, the CFKRS recipe leads to the conjecture
\[
\sum_{q\le Q}\,\sideset{}{^*}\sum_{\chi \bmod q}\left|L\left(\tfrac{1}{2},\chi \right)\right|^{2k} \sim c_k\sum_{q\le Q} \,\sideset{}{^*}\sum_{\chi \bmod q}\prod_{p| q}\Bigg(\sum_{m=0}^{\infty}\frac{\binom{m+k-1}{k-1}^2}{p^m}\Bigg)^{-1}(\log q)^{k^2}, \qquad Q \to \infty
\]
for all positive integers $k$, with an explicit value of $c_k$. More generally, the CFKRS recipe predicts an asymptotic formula for
\begin{equation}\label{eqn: moment}
\sum_{q\le Q}\,\sideset{}{^*}\sum_{\chi \bmod q}\prod_{\alpha \in A}L\left(\tfrac{1}{2}+\alpha,\chi \right)\prod_{\beta \in B}L\left(\tfrac{1}{2}+\beta,\overline{\chi} \right),
\end{equation}
where $A,B$ are finite multisets of small complex numbers, which we refer to as ``shifts." These shifts allowed Conrey et al.~\cite{CFKRS} to write the conjecture as a combinatorial sum that reveals some underlying structure in the asymptotic formula. Within each term in the sum, the shifts appear in an arrangement that involves element exchanges between the multisets $A$ and $B$. Thus each term in the conjectured asymptotic formula can be described as having $\ell$ ``swaps," where $\ell$ is the number of elements exchanged by each multiset with the other. Each $\ell$-swap term may contain leading order terms, lower order terms, or both. We precisely state the conjecture in the context of our main theorem in Conjecture \ref{con: conjecture} below.
The CFKRS recipe arrives at the conjecture by assuming that certain terms are negligible in the calculation of the moment. While this leads to the ``final simple answer that should emerge" \cite[page 35]{CFKRS}, the heuristic does not indicate how or why those terms can be ignored. Recently, Conrey and Keating~\cite{CK1}, \cite{CK2}, \cite{CK3}, \cite{CK4}, \cite{CK5} have developed a new approach to this problem for $\zeta(s)$ using Dirichlet polynomial approximations. They estimate the moments
\begin{equation*}
\int_{T}^{2T}\prod_{\alpha \in A} \zeta(\tfrac{1}{2}+\alpha+it)\prod_{\beta \in B} \zeta(\tfrac{1}{2}+\beta-it)\, dt
\end{equation*}
by approximating the product over $\alpha \in A$ by a Dirichlet polynomial of length $X$ and doing the same for the product over $\beta \in B$. One of their early observations suggests that the size of $X$ determines the values of $\ell$ for which the $\ell$-swap terms contribute at most $o(T)$ to the conjectured asymptotic formula. In particular, they predict that if $X < T/(2\pi)$ then all but the zero-swap term contribute $o(T)$. Similarly, if $T/\pi < X< T^2/(4\pi^2)$ then all but the zero- and one-swap terms should contribute $o(T)$, if $T^2/\pi^2 < X < T^3/(8\pi^3)$ then all but the zero-, one-, and two-swap terms should contribute $o(T)$, and so on.
This prediction reveals the difficulty in obtaining asymptotic formulas for higher moments of $L$-functions. Historically, the approach to calculating moments has been to use the approximate functional equation, and this is in fact the approach used in the CFKRS recipe. For low moments (with $k=1,2$, say), only the so-called ``diagonal" terms from the approximate functional equation contribute to the main term. On the other hand, the previously mentioned conjectures of Conrey et al.~ and Conrey and Keating indicate that high moments have the more delicate and challenging feature that some of the ``off-diagonal" terms actually contribute to the main term. In order to extract these contributions, more sophisticated techniques are needed.
\section{Main result}
We are interested in understanding the twisted $2k$th moment of all primitive Dirichlet $L$-functions of modulus $q$, averaged over all moduli $q\le Q$. To state the result precisely, we must introduce a bit of notation. In Section \ref{sec: notation}, we give a more comprehensive overview of the notation used in this article, with clarifying examples.
For a finite multiset $A=\{\alpha_1,\alpha_2,\dots,\alpha_r\}$ of complex numbers $\alpha_i$, we define $\tau_A(m)$ for positive integers $m$ by
\begin{equation*}
\tau_A(m) := \sum_{m_1\cdots m_r =m}m_1^{-\alpha_1}\cdots m_r^{-\alpha_r},
\end{equation*}
where the sum is over all positive integers $m_1,\dots,m_r$ such that $m_1\cdots m_r =m$. Thus, if $\chi$ is a Dirichlet character, then
$$
\sum_{m=1}^{\infty} \frac{\tau_A(m)\chi(m)}{m^s} = \prod_{\alpha\in A }L(s+\alpha,\chi)
$$
for all $s$ such that the left-hand side converges absolutely, where the product on the right-hand side is over all $\alpha \in A$, counted with multiplicity. For any multiset $A$ and $s\in \mathbb{C}$, we define $A_s$ to be the multiset $A$ with $s$ added to each element. In other words, if $A=\{\alpha_1,\alpha_2,\dots,\alpha_r\}$, then
\begin{equation*}
A_s := \{\alpha_1+s,\alpha_2+s,\dots,\alpha_r+s\}.
\end{equation*}
If $A$ and $B$ are multisets, then we let $A\cup B$ denote the multiset sum of $A$ and $B$ and $A\smallsetminus B$ denote the multiset difference. We write $A^{-}$ to denote the multiset $A$ with each element multiplied by $-1$.
In this paper, we study the moments \eqref{eqn: moment} with twists $\chi(h)\overline{\chi}(k)$ using Dirichlet polynomial approximations. Thus the main object that we are interested in is
\begin{equation}\label{eqn: S(h,k)def}
\begin{split}
\mathcal{S}(h,k):=\sum_{q=1}^{\infty}W\left(\frac{q}{Q}\right) \sideset{}{^\flat}\sum_{\chi \bmod q} \chi(h)\overline{\chi}(k) \sum_{m=1}^{\infty}\frac{\tau_A(m)\chi(m)}{\sqrt{m}}V\left(\frac{m}{X}\right) \sum_{n=1}^{\infty}\frac{\tau_B(n)\overline{\chi}(n)}{\sqrt{n}}V\left(\frac{n}{X}\right),
\end{split}
\end{equation}
where $W$ is a smooth, nonnegative function that is compactly supported on $(0,\infty)$, the symbol $\flat$ denotes that the sum is over all even, primitive characters modulo $q$, and $V$ is a smooth, nonnegative function that is compactly supported on $[0,\infty)$ and satisfies $V(0)>0$. Note that the length of the $m$-sum, as well as the $n$-sum, is of the same order of magnitude as $X$. Note also that we use the symbol $k$ in \eqref{eqn: S(h,k)def} for the twist $\overline{\chi}(k)$. This $k$ should not be interpreted as the same $k$ we use when we refer to the $2k$th moment.
In order to state the asymptotic formula for $\mathcal{S}(h,k)$ that is predicted by the CFKRS recipe, we define
\begin{equation}\label{eqn: I_l(h,k)def}
\begin{split}
\mathcal{I}_\ell(h,k) &:= \sum_{\substack{q=1 \\ (q,hk)=1} }^{\infty} W\left( \frac{q}{Q}\right)\sideset{}{^\flat}\sum_{\chi \bmod q} \frac{1}{(2\pi i )^2} \int_{(\varepsilon)} \int_{(\varepsilon)} X^{s_1+s_2} \widetilde{V}(s_1) \widetilde{V}(s_2) \\
&\hspace{.5in}\times \sum_{\substack{U\subseteq A, V\subseteq B \\ |U|=|V|=\ell}} \prod_{\alpha\in U} \frac{\mathscr{X} (\tfrac{1}{2}+\alpha +s_1 )}{ q^{\alpha+s_1} } \prod_{\beta\in V} \frac{\mathscr{X} (\tfrac{1}{2}+\beta+s_2 )}{ q^{\beta+s_2} } \\
&\hspace{.5in}\times \sum_{\substack{1\leq m,n<\infty \\ mh=nk\\ (mn,q)=1}} \frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (m) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (n) }{\sqrt{mn}} \,ds_2\,ds_1 ,
\end{split}
\end{equation}
where $\varepsilon>0$ is an arbitrarily small constant,
\begin{equation*}
\widetilde{V}(s) : = \int_0^{\infty} V(x) x^{s-1}\,dx
\end{equation*}
is the Mellin transform of $V$, and
\begin{equation*}
\mathscr{X} (s) := {\pi}^{s-\frac{1}{2}} \frac{\Gamma(\frac{1}{2}-\frac{1}{2}s)}{\Gamma(\frac{1}{2}s)}.
\end{equation*}
Here, the sum over $U,V$ should be interpreted as taking into account the multiplicity of the elements in $A$ and $B$. The sum $\mathcal{I}_\ell(h,k)$ is precisely the sum of all the $\ell$-swap terms from the recipe prediction. We call these terms the ``$\ell$-swap terms" because the multiset $A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}$ results from taking the set $A_{s_1}$ and replacing the $\ell$ elements of $U_{s_1}$ with the negatives of the $\ell$ elements in $V_{s_2}$. Similarly, $B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-}$ results from taking the multiset $B_{s_2}$ and replacing the $\ell$ elements of $V_{s_2}$ with the negatives of the $\ell$ elements in $U_{s_1}$. Thus, we are swapping $\ell$ elements from $A_{s_1}$ with $\ell$ elements from $(B_{s_2})^-$. In particular, $\mathcal{I}_0(h,k)$ is the zero-swap term, $\mathcal{I}_1(h,k)$ is the sum of the one-swap terms, and so on. We remark that the $m,n$-sum should be interpreted as its analytic continuation, which we write explicitly in \eqref{eqn: I_l(h,k)def2} below.
In Section \ref{sec: recipe}, we show how to derive the following conjecture for the asymptotic behavior of $\mathcal{S}(h,k)$ using the CFKRS recipe.
\begin{conjecture}\label{con: conjecture}
Let $A$ and $B$ be finite multisets of complex numbers $\ll 1/\log Q$, where $Q$ is a large parameter. Define $\mathcal{S}(h,k)$ by \eqref{eqn: S(h,k)def}. Then, for all $X>0$,
\[
\mathcal{S}(h,k) \sim \sum_{\ell=0}^{\min\{|A|,|B|\}}\mathcal{I}_\ell(h,k), \qquad \text{as } Q \to \infty.
\]
\end{conjecture}
Towards this conjecture, we prove the following theorem.
\begin{theorem}\label{thm: main}
Let $Q$ be a large parameter and $X=Q^{\eta}$ with $1<\eta<2$. Let $A$ and $B$ be finite multisets of complex numbers $\ll 1/\log Q$, and define $\mathcal{S}(h,k)$ by \eqref{eqn: S(h,k)def}. Then, assuming the Generalized Lindel\"{o}f Hypothesis, we have
\begin{equation}\label{eqn: main}
\mathcal{S}(h,k) = \mathcal{I}_0(h,k) + \mathcal{I}_1(h,k) + \mathcal{E}(h,k),
\end{equation}
where the error term $\mathcal{E}(h,k)$ satisfies, for arbitrarily small $\epsilon>0$,
\begin{equation}\label{eqn: mainbound}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda}_k}{\sqrt{hk}} \mathcal{E}(h,k) \ll_{\epsilon,|A|,|B|,V,W} Q^{1+\frac{\vartheta}{2}+\frac{\eta}{2}+\epsilon} + Q^{\frac{5}{2}-\frac{\eta}{2}+\vartheta+\epsilon}
\end{equation}
uniformly for $0<\vartheta < 2-\eta$ and arbitrary complex numbers $\lambda_h$ such that $\lambda_h\ll_{\varepsilon} h^{\varepsilon}$ for arbitrarily small $\varepsilon>0$.
\end{theorem}
Theorem \ref{thm: main} proves that, under GLH, the zero- and one-swap terms conjectured by the CFKRS recipe are correct. This provides the first rigorous evidence beyond the diagonal terms for the conjecture of Conrey et al.~\cite{CFKRS} for the general $2k$th moment of this family.
While the recipe provides a detailed prediction for the asymptotic formula satisfied by \eqref{eqn: S(h,k)def}, at present it seems difficult to rigorously prove all the steps involved. We thus approach the problem in a different way using the asymptotic large sieve, which in recent years has become one of the primary tools for studying moments of primitive Dirichlet $L$-functions. Our general strategy in proving Theorem~\ref{thm: main} is based on the approach of Conrey, Iwaniec, and Soundararajan~\cite{CIS}, who applied the asymptotic large sieve to study the twisted second moment. Thus, our work is similar to theirs in many respects. However, there are crucial differences due to the generality of our situation and the intricacy of the predicted asymptotic formula that we aim to prove.
The crux of the proof is to uncover the one-swap terms and then show that they match the prediction in Conjecture~\ref{con: conjecture}. The difficulty here is that while Conjecture~\ref{con: conjecture} tells us what the one-swap terms should look like, and the asymptotic large sieve gives us a general idea of where we might find them, neither gives any indication on how to extract the one-swap terms from the asymptotic formula that results from using the asymptotic large sieve. We achieve this through delicate and deliberate contour integration by breaking the predicted one-swap terms into several residues (Section~\ref{subsec: one-swap1}), doing the same for one of the main terms brought about by the use of the asymptotic large sieve (Section~\ref{subsec: one-swap2}), and then matching these residues to show that they are asymptotically equal via Euler product identities (Section~\ref{sec: matchresidues}).
\
\noindent{\bf Remarks}
\begin{itemize}
\item The main terms in \eqref{eqn: main} are of size about $Q^2$. If we also assume that $\vartheta < (\eta-1)/2$, then the right hand side of \eqref{eqn: mainbound} is $\ll Q^{2-\delta}$ for some $\delta>0$.
\item It can be shown using \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Stirlingchi}, and \eqref{eqn: I_l(h,k)def2} below that, if $A,B$ are as in Theorem~\ref{thm: main}, then $\mathcal{I}_{\ell}(h,k)\ll Q^{2-2\ell\varepsilon+\delta} X^{2\varepsilon} (hk)^{\delta}$ for arbitrarily small $\delta>0$. Hence, if $X=Q^{\eta}$ with $\eta<\ell$, then $\mathcal{I}_{\ell}(h,k) \ll Q^{2-\delta}(hk)^{\delta}$ for some $\delta>0$. Thus Theorem~\ref{thm: main} is consistent with Conjecture~\ref{con: conjecture} when $X=Q^{\eta}$ with $1<\eta<2$.
\item We assume the Generalized Lindel\"{o}f Hypothesis (GLH) in a few key places, which we identify throughout the course of the proof. In each of these instances, there may be a large number of zeta-functions or $L$-functions that we need to bound. If the cardinalities of $A$ and $B$ are not too large, then it may be possible to carry out these estimations unconditionally.
\item For convenience, we have only considered even primitive characters. For odd characters, some parts of the arguments are simpler, while in other parts only small changes are needed. The conclusion of the theorem for odd primitive characters is the same except that we must replace the function $\mathscr{X}(s)$ with $\pi^{s-\frac{1}{2}} \Gamma(\frac{2-s}{2})/\Gamma(\frac{s+1}{2})$ in the definition of $\mathcal{I}_\ell(h,k)$. We describe the changes to the proof carefully in Section~\ref{sec: outline}.
\item The terms $\mathcal{I}_0(h,k)$ and $\mathcal{I}_1(h,k)$ are both holomorphic functions of the shifts $\alpha \in A$ and $\beta \in B$. We prove this fact at the end of Section \ref{sec: proofofthm}. We may use \eqref{eqn: vandermonde1swap} as a convenient way to evaluate $\mathcal{I}_1(h,k)$ when some of the elements in $A\cup B$ have multiplicity greater than 1. In particular, we can use \eqref{eqn: vandermonde1swap} to evaluate $\mathcal{I}_1(h,k)$ when all the shifts $\alpha \in A$ and $\beta \in B$ are 0.
\end{itemize}
The one-swap terms have also been found for other families of $L$-functions. Hamieh and Ng \cite{HamiehNg} do this for the $2k$th moments of $\zeta(s)$ under the assumption of an additive divisor conjecture by making some of the arguments in the work of Conrey and Keating \cite{CK3} rigorous. In our situation, we do not need to assume an analogous divisor conjecture because we are able to leverage the asymptotic large sieve. On the other hand, we must assume GLH because the factors $\tau_A$ and $\tau_B$ are unchanged when applying the asymptotic large sieve and thus give rise to a potentially large number of $L$-functions. Conrey and Rodgers \cite{ConreyRodgers} have found the one-swap terms for the family of quadratic Dirichlet $L$-functions. They also do not need to assume any divisor conjecture because they are able to use the Poisson summation method of Soundararajan \cite{SoundPoisson}. As in our situation, they also need to assume GLH to bound large numbers of $L$-function factors.
Analogous results have been proved unconditionally in the function field setting. Andrade and Keating~\cite{AndradeKeating} used the CFKRS recipe to predict the asymptotic formulas for moments of $L$-functions associated with hyperelliptic curves of genus $g$ over a fixed finite field, where $g$ is a parameter going to infinity. Florea~\cite{FloreaThesis} has recovered the one-swap terms for this family. Moreover, Bui, Florea, and Keating~\cite{BuiFloreaKeating} have found the one-swap terms for the 2-level density of zeros of this family. In this setting, the Poisson summation method is the primary tool for studying moments of $L$-functions (see also \cite{Florea4th}, \cite{Florea23}, \cite{BuiFloreaKeatingRoditty-Gershon}, and \cite{BuiFloreaKeating2}). For a different family over function fields, Sawin~\cite{Sawin} has formulated a heuristic that recovers the CFKRS prediction, which he then confirms under the assumption of a conjecture on the vanishing of certain cohomology groups.
In order to extract the two-swap terms predicted by Conjecture~\ref{con: conjecture}, the discussion at the end of Section~\ref{sec: HistoricalOverview} suggests that we must work with a Dirichlet polynomial approximation of length $X>Q^2$. In this situation, the predicted two-swap terms are of size about $Q^2$. Without any additional input, the asymptotic large sieve does not seem effective when $X>Q^2$ because it no longer reduces the moduli of the character sums for such $X$ (see Section \ref{sec: outline} for more details). In fact, the predicted two-swap terms should be hidden inside the term $\mathcal{E}(h,k)$ in \eqref{eqn: main}, and thus we no longer expect the left-hand side of \eqref{eqn: mainbound} to be $\ll Q^{2-\delta}$ when $X>Q^2$. This limitation of the asymptotic large sieve is analogous to the limitation of the Poisson summation method in evaluating high moments of the family of quadratic Dirichlet $L$-functions.
With some additional work, we may be able to use our result to study the sixth moment of primitive Dirichlet $L$-functions. There could also be potential applications to studying gaps between zeros of Dirichlet $L$-functions.
\
\noindent\emph{Outline of the article.} In Section \ref{sec: notation}, we give a comprehensive list of all the notation used in the article. In Section \ref{sec: recipe}, we use the CFKRS recipe to derive Conjecture~\ref{con: conjecture}. We give a detailed outline of the proof of Theorem~\ref{thm: main} in Section \ref{sec: outline}. The remaining sections are devoted to proving the theorem. In Section~\ref{sec: diagonal}, we examine the diagonal terms to extract the zero-swap term. We study the off-diagonal terms in Sections~\ref{sec: Lsum}-\ref{sec: Ur}, where we extract the one-swap terms. Finally, in Section \ref{sec: proofofthm}, we complete the proof of Theorem \ref{thm: main} and prove the holomorphy of $\mathcal{I}_0(h,k)$ and $\mathcal{I}_1(h,k)$.
\
\noindent{\bf Acknowledgments.}\, Work on this project began in the summer of 2020 at the American Institute of Mathematics as part of the NSF Focused Research Group ``Averages of $L$-functions and Arithmetic Stratification" supported by NSF DMS-1854398 FRG. We are grateful to Brian Conrey for suggesting this problem, for many helpful discussions, and for all the support and encouragement. We also thank David Farmer, Alexandra Florea, and Brad Rodgers for a number of useful comments that improved the exposition. The second author thanks the American Institute of Mathematics for providing a focused research environment in February and March 2022, during which the manuscript was prepared. The first author is supported by NSF DMS-1854398 FRG, and the second author is partially supported by NSF DMS-1902193 and NSF DMS-1854398 FRG.
\section{Notation, conventions, and preliminaries}\label{sec: notation}
In this section, we collect our commonly used notation for the reader's convenience. We also list a number of technical assumptions and basic facts that we use throughout the paper. The reader may choose to skip this section and only refer to it when needed.
We employ standard notation in analytic number theory and use $\int_{(c)}$ to denote integrals along the line from $c-i\infty$ to $c+i\infty$. We let $\varepsilon>0$ denote an arbitrarily small constant whose value may change from one line to the next. We also sometimes use $\epsilon>0$ to denote an arbitrarily small constant, except that the value of $\epsilon$ remains the same all throughout. This distinction between $\varepsilon$ and $\epsilon$ will often be harmless, and we will use $\epsilon$ only when the situation requires more concreteness, such as when dealing with integrals like
$$
\int_{(\epsilon)}\int_{(\epsilon/2)} \Gamma(w)\Gamma(z)\Gamma(w-z)\,dz\,dw.
$$
The symbol $\varepsilon$ may sometimes depend on $\epsilon$, but only when the concreteness of $\epsilon$ is no longer required. When at least one of $\varepsilon$ or $\epsilon$ is present, in some fashion, in an inequality or error term, we allow implied constants to depend on $\varepsilon$ or $\epsilon$ without necessarily indicating so in the notation. We sometimes indicate the dependence of implied constants on variables by the use of subscripts: for example, $Y\ll_b Z$ or $Y=O_b(Z)$ means that the implied constant may depend on $b$.
The symbol $p$ always denotes a prime number. We use $\text{ord}_p(m)$ to denote the exponent of $p$ in the prime factorization of $m$. For example, ord$_3(72)=2$ and ord$_5(84)=0$. We let $\phi$ be the Euler totient function, and $\mu$ the M\"{o}bius function. If $h$ and $k$ are positive integers that are present in some form in an equation or inequality, then we use $H$ to denote $h/(h,k)$ and $K$ to denote $k/(h,k)$.
For a multiset $E=\{\xi_1,\xi_2,\dots,\xi_j\}$ of complex numbers, we define $\tau_E(m)$ for positive integers $m$ by
\begin{equation}\label{eqn: taudef}
\tau_E(m) := \sum_{m_1\cdots m_j =m}m_1^{-\xi_1}\cdots m_j^{-\xi_j},
\end{equation}
where the sum is over all positive integers $m_1,\dots,m_j$ such that $m_1\cdots m_j =m$. Thus, for example, if $\xi_1=\cdots=\xi_j=0$, then $\tau_E(m)$ is the $j$-fold divisor function. If $E$ is empty, then we define $\tau_E(1)=1$ and $\tau_E(m)=0$ for all other $m$. It follows that if $E$ is a finite multiset of complex numbers, then
$$
\sum_{m=1}^{\infty} \frac{\tau_E(m)}{m^s} = \prod_{\xi\in E }\zeta(s+\xi)
$$
for all $s$ such that the left-hand side converges absolutely, where $\zeta(s)$ is the Riemann zeta-function and the product on the right-hand side is over all $\xi\in E$, counted with multiplicity. We define $\tau_E(p^{-1})$ to be zero for every multiset $E$. If $r$ is a real number such that each element of $E$ has real part $\geq r$, then \eqref{eqn: taudef} and the divisor bound imply
\begin{equation}\label{eqn: divisorbound}
\tau_E(m)\ll_{\varepsilon} m^{-r+\varepsilon}.
\end{equation}
If $E$ is a multiset of complex numbers and $s\in \mathbb{C}$, then we define $E_s$ to be the multiset $E$ with $s$ added to each element. In other words, if $E=\{\xi_1,\xi_2,\dots,\xi_j\}$, then
\begin{equation*}
E_s := \{\xi_1+s,\xi_2+s,\dots,\xi_j+s\}.
\end{equation*}
It follows immediately from this definition and \eqref{eqn: taudef} that
\begin{equation}\label{eqn: taufactoringidentity}
\tau_{E_s}(m) = m^{-s}\tau_E(m).
\end{equation}
If $E$ is a multiset, then we let $|E|$ denote its cardinality, counting multiplicity. If $D$ and $E$ are multisets, then we let $D\cup E$ denote the multiset sum of $D$ and $E$, which means that the multiplicity of each element in $D\cup E$ is exactly the sum of the multiplicity of the element in $D$ and its multiplicity in $E$. Similarly, we define $D\smallsetminus E$ to be the multiset difference, which is the multiset with each element having multiplicity equal to its multiplicity as an element of $D$ minus its multiplicity as an element of $E$ if this difference is nonnegative, and equal to zero otherwise. Thus, for example, if $A=\{\alpha_1,\alpha_2,\dots,\alpha_j\}$ is a multiset of complex numbers, $\alpha=\alpha_1$, and $\beta$ and $s$ are complex numbers, then \eqref{eqn: taudef} implies
\begin{equation*}
\tau_{A_s \smallsetminus \{\alpha+s\} \cup \{-\beta-s\}} (m) = \sum_{m_1\cdots m_j=m} m_1^{\beta+s}m_2^{-\alpha_2-s}m_3^{-\alpha_3-s}\cdots m_j^{-\alpha_j-s}
\end{equation*}
for every positive integer $m$. For most of our proofs, we will be dealing with sets instead of multisets, and in most cases $D\smallsetminus E$ and $D\cup E$ reduce to ordinary set difference and set union, respectively.
The letter $Q$ denotes a parameter tending to $\infty$, and $\vartheta \in (0,1)$ is a parameter. We define $X=Q^{\eta}$ with $\eta$ a parameter satisfying $1<\eta<2$. The quantities $C$ and $Y$, which satisfy $C\geq 1$ and $Y\geq XQ^{\vartheta}$ and are introduced in Sections~\ref{sec: outline} and \ref{sec: Ur}, respectively, are positive parameters that we will choose to be powers of $Q$ at the end of the proof of Theorem~\ref{thm: main}. The sequence $\lambda_1,\lambda_2,\dots$ is an arbitrary sequence of complex numbers such that $\lambda_h \ll_{\varepsilon} h^{\varepsilon}$ for all positive integers $h$. We use this sequence only to prove the property \eqref{eqn: mainbound} of $\mathcal{E}(h,k)$. In Section~\ref{sec: U2}, we use the symbol $\delta$ to denote the reciprocal of an arbitrarily large power of $Q$, say
\begin{equation}\label{eqn: deltadef}
\delta=Q^{-99}.
\end{equation}
In many places in the same section and in other sections, we also use the symbol $\delta$ as an index of a product, but this will not cause confusion.
We let $A$ and $B$ be arbitrary fixed finite multisets of complex numbers. We usually denote elements of $A$ by $\alpha$ and elements of $B$ by $\beta$. We assume that $\alpha,\beta\ll 1/\log Q$ for all $\alpha\in A$ and $\beta\in B$, with the implied constant arbitrary but fixed. For convenience, we let $C_0>0$ be a fixed arbitrary constant and assume all throughout our proof of Theorem~\ref{thm: main} that if $A=\{\alpha_1,\alpha_2,\dots,\alpha_j\}$ and $B=\{\beta_1,\beta_2,\dots,\beta_{\ell}\}$, then
\begin{equation}\label{eqn: orbitals}
\begin{split}
|\alpha_{\nu}| &= \frac{2^{\nu}C_0}{\log Q} \ \ \ \text{for }\nu=1,2,\dots,j, \text{ and}\\
|\beta_{\nu}| &= \frac{2^{j+\nu}C_0}{\log Q} \ \ \ \text{for }\nu=1,2,\dots,\ell.
\end{split}
\end{equation}
This ensures that we do not encounter double poles when dealing with expressions such as $\prod_{\alpha\in A,\beta\in B}\zeta(\alpha+\beta+s)$. A consequence of \eqref{eqn: orbitals} is that if $J_1,J_2$ are subsets of $\{1,2,\dots,j\}$ and $L_1,L_2$ are subsets of $\{1,2,\dots,\ell\}$ such that either $J_1\neq J_2$ or $L_1\neq L_2$, then
\begin{equation}\label{eqn: zetaalphaalphabound}
\zeta\bigg( 1 + \sum_{\nu\in J_1} \alpha_{\nu} +\sum_{\nu\in L_1}\beta_{\nu} - \sum_{\nu\in J_2} \alpha_{\nu} - \sum_{\nu\in L_2}\beta_{\nu}\bigg) \ll \log Q.
\end{equation}
We will eliminate the assumption \eqref{eqn: orbitals} in Section~\ref{sec: proofofthm} and show that Theorem~\ref{thm: main} holds for arbitrary finite multisets $A$ and $B$ such that $\alpha,\beta\ll 1/\log Q$ for all $\alpha\in A$ and $\beta\in B$. The assumption \eqref{eqn: orbitals} is unnecessary in carrying out the Euler product evaluations in Lemmas~\ref{lem: 1swapeulerbound}, \ref{lem: Kfunctionaleqn}, and \ref{lem: U2eulerbound} and Subsection~\ref{sec: matchresidues}. For those calculations, we only need the elements of $A$ and $B$ to be arbitrarily small, and so the assumption that $\alpha,\beta\ll 1/\log Q$ for all $\alpha\in A$ and $\beta\in B$ suffices.
We define the Mellin transform of a function $f$ by
\begin{equation}\label{eqn: mellindef}
\widetilde{f}(s) : = \int_0^{\infty} f(x) x^{s-1}\,dx.
\end{equation}
We assume that $V$ is a fixed smooth function from $[0,\infty)$ to $[0,\infty)$ that has compact support. We suppose that $V(0)>0$, since otherwise the $m$-sum (or $n$-sum) in \eqref{eqn: S(h,k)def} tends to $0$ as $X\rightarrow \infty$ and is thus an invalid approximation of the product of $L$-functions. Without loss of generality, we may assume that $V(0)=1$ since we may normalize by dividing $V(x)$ by $V(0)$. Integrating by parts, we see from the definition \eqref{eqn: mellindef} of $\widetilde{V}$ that if $\re(s)>0$, then
\begin{equation}\label{eqn: mellinVIBP}
\widetilde{V}(s) = -\frac{1}{s}\int_0^{\infty}V'(x) x^s\,dx.
\end{equation}
The latter integral is holomorphic for $\re(s)>-1$ since $V'$ is bounded and compactly supported. It thus follows from \eqref{eqn: mellinVIBP} that $s=0$ is a simple pole of $\widetilde{V}$ and
\begin{equation}\label{eqn: mellinVresidue}
\underset{s=0}{\text{Res}}\ \widetilde{V}(s) = \lim_{s\rightarrow 0} s\widetilde{V}(s) = 1
\end{equation}
because $V(0)=1$. We may apply integration by parts again to the right-hand side of \eqref{eqn: mellinVIBP} to analytically continue $\widetilde{V}(s)$ to $\re(s)>-2$. Repeating this process indefinitely, we see that $\widetilde{V}(s)$ is meromorphic on all of $\mathbb{C}$ with possible poles only at the non-positive integers.
We assume that $W$ is a fixed smooth function from $(0,\infty)$ to $[0,\infty)$ that has compact support. This means that the support of $W$ is bounded away from $0$, and it follows immediately from \eqref{eqn: mellindef} and Morera's theorem that $\widetilde{W}(s)$ is an entire function. The definition \eqref{eqn: mellindef} and a repeated application of integration by parts shows that if $n$ is a positive integer, then
\begin{equation}\label{eqn: mellinrapiddecay}
\widetilde{V}(s),\widetilde{W}(s) \ll_n \frac{1}{|s|^n}
\end{equation}
as $s\rightarrow \infty$. We will repeatedly use this fact without mention to justify moving lines of integration.
We allow implied constants to depend on $\varepsilon$, $\epsilon$, the cardinalities $|A|$ and $|B|$, the implied constant in the assumption $\alpha,\beta\ll 1/{\log Q}$, or the functions $V$ and $W$ without necessarily indicating so in the notation. The implied constants never depend on the actual values of $\alpha,\beta$ nor on any of $Q,X,C,Y,h,k,\lambda_h,\lambda_k, \vartheta,\eta$.
We define $\mathscr{X}(s)$ by
\begin{equation}\label{eqn: scriptXdef}
\mathscr{X} (s) = {\pi}^{s-\frac{1}{2}} \frac{\Gamma(\frac{1}{2}-\frac{1}{2}s)}{\Gamma(\frac{1}{2}s)}.
\end{equation}
In other words, we write the functional equation of $\zeta(s)$ as $\zeta(s)=\mathscr{X}(s)\zeta(1-s)$. The poles of $\mathscr{X}$ are at the odd positive integers, and Stirling's formula implies \cite[(4.12.3)]{Titchmarsh}
\begin{equation}\label{eqn: Stirlingchi}
\mathscr{X}(s) \asymp (1+|s|)^{\frac{1}{2}-\re(s)}
\end{equation}
for $s$ in any fixed vertical strip such that $s$ is bounded away from the poles of $\mathscr{X}$. The relation $f\asymp g$ means $f \ll g$ and $f \gg g$. We will use \eqref{eqn: Stirlingchi} repeatedly without mention. We define $\mathcal{H}(z,w)$ by
\begin{equation}\label{eqn: Hdef}
\mathcal{H}(z,w)=\sqrt{\pi}\frac{\Gamma(\tfrac{1-w}{2})\Gamma(\tfrac{z}{2})\Gamma(\tfrac{w-z}{2})}{\Gamma(\tfrac{w}{2})\Gamma(\tfrac{1-z}{2})\Gamma(\tfrac{1-w+z}{2})}.
\end{equation}
It follows from this and the definition \eqref{eqn: scriptXdef} of $\mathscr{X}$ that
\begin{equation}\label{eqn: Hintermsofchifactor}
\mathcal{H}(z,w)=\mathscr{X}(w)\mathscr{X}(1-z)\mathscr{X}(1-w+z).
\end{equation}
This and \eqref{eqn: Stirlingchi} imply
\begin{equation}\label{eqn: Hbound}
\mathcal{H}(z,w) \asymp |w|^{\frac{1}{2}-\re(w)}|z|^{\re(z)-\frac{1}{2}} |w-z|^{\re(w-z)-\frac{1}{2}}
\end{equation}
for $w,z$ in any fixed vertical strip such that $w$, $z$, and $w-z$ are bounded away from the integers.
We will repeatedly use without mention the well-known fact that $\zeta(s)$ and the Dirichlet $L$-functions each have at most polynomial growth in fixed vertical strips. Oftentimes, this polynomial growth is offset by the rapid decay \eqref{eqn: mellinrapiddecay} of the Mellin transforms. However, there are certain points in our argument, particularly when estimating integrals involving a large number of zeta or $L(s,\chi)$ factors, where we will need to assume the following.
\begin{glh}
The Lindel\"{o}f Hypothesis for $\zeta(s)$ holds and
$$
L(\tfrac{1}{2}+it,\psi) \ll_{\varepsilon} (q(1+|t|))^{\varepsilon}
$$
for all real $t$ and all non-principal Dirichlet characters $\psi$ modulo $q$, where the implied constant depends only on $\varepsilon$.
\end{glh}
The Generalized Riemann Hypothesis implies GLH~\cite{ConreyGhoshLindeloff}. We will explicitly mention our assumption of GLH each time we use it.
For conciseness, we adopt the convention that any expression of a sum in $\Sigma$-notation that contains the symbol $\pm$ means a sum of two copies of that expression: one with the symbol $\pm$ replaced by $+$, the other with $\pm$ replaced by $-$, and both with $\mp$ replaced by the sign opposite that replacing $\pm$. For example,
$$
\sum_{\substack{d|q \\ d|(m\pm n)}} \psi(\mp d) f(\pm d) g(d)
$$
means the same as
$$
\sum_{\substack{d|q \\ d|(m+ n)}} \psi(- d)f( d) g(d) + \sum_{\substack{d|q \\ d|(m- n)}} \psi( d)f(-d) g(d).
$$
and
$\sum_{a} h(\pm a)$ means the same as $\sum_{a} h(a) +\sum_{a} h(-a)$. On the other hand, we use the typical interpretation of $\pm$ in expressions like
\[
\int_{0}^{\infty } \frac{c |mh\pm e^{\xi}nk|}{gx Q} W\left( \frac{c |mh\pm e^{\xi}nk|}{gx Q}\right) x^{w-1} \,dx
\]
and in definitions such as
\[
\ell := \frac{|mh\pm nk|}{d}.
\]
We end this section with two lemmas that we will apply in various sections.
\begin{lemma}\label{lem: Lemma2ofCIS}\cite[Lemma 2]{CIS}
If $(mn,q)=1$, then
\[
\sideset{}{^{\flat}}\sum_{\chi \bmod q} \chi(m)\overline{\chi(n)} = \frac{1}{2}\Bigg(\sum_{\substack{d|q\\d|(m\pm n)}}\phi(d)\mu\left(\frac{q}{d}\right) \Bigg),
\]
where the $\flat$ indicates that the sum is over all the even primitive characters. Here, we have adopted the previously mentioned convention that the right-hand side means a sum of two copies of itself: one with $\pm$ replaced by $+$, and the other with $\pm$ replaced by $-$.
\end{lemma}
\begin{lemma}\label{lem: sumstoEulerproducts}
If $f(m_1,m_2,\dots,m_j;p)$ is a complex-valued function such that
$$
f(m_1,m_2,\dots,m_j;p) = f(p^{\text{ord}_p(m_1)},p^{\text{ord}_p(m_2)},\dots,p^{\text{ord}_p(m_j)};p)
$$
for all positive integers $m_1,m_2,\dots,m_j$ and primes $p$, then
$$
\sum_{1 \leq m_1 ,m_2 ,\dots,m_j<\infty} \prod_p f(m_1,m_2,\dots,m_j;p) = \prod_p \sum_{0 \leq b_1 , b_2 ,\dots,b_j<\infty} f(p^{b_1},p^{b_2},\dots,p^{b_j};p)
$$
if absolute convergence holds for both sides.
\end{lemma}
\begin{proof}[Proof sketch]
This can be proved using a standard argument (see, for example, \cite[Theorem~11.7]{Apostol}) together with the fact that $\prod_{p>y} f(1,\dots,1;p)\rightarrow 1$ as $y\rightarrow \infty$.
\end{proof}
\section{The CFKRS recipe for conjecturing asymptotic formulas for moments}\label{sec: recipe}
In this section, we apply the heuristic of Conrey et~al.~\cite{CFKRS} to conjecture the asymptotic formula for the sum $\mathcal{S}(h,k)$ defined by \eqref{eqn: S(h,k)def}. We also make the definition \eqref{eqn: I_l(h,k)def} of $\mathcal{I}_{\ell}(h,k)$ more explicit by writing out the analytic continuation of the $m,n$-sum. Furthermore, we write the $q$-sum in \eqref{eqn: I_l(h,k)def} in terms of an integral in order to facilitate subsequent calculations. For a more detailed discussion on the CFKRS recipe and its applications to other families of $L$-functions, see \cite{CFKRS}.
We first apply Mellin inversion, interchange the order of summation, and observe that
\begin{equation*}
\sum_{m=1}^{\infty}\frac{\tau_A(m)\chi(m)}{m^{\frac{1}{2}+s_1}} \sum_{n=1}^{\infty}\frac{\tau_B(n)\bar\chi(n)}{n^{\frac{1}{2}+s_2}} = \prod_{\alpha\in A} L(\tfrac{1}{2}+\alpha+s_1,\chi) \prod_{\beta\in B} L(\tfrac{1}{2}+\beta+s_2,\overline{\chi})
\end{equation*}
by the definition \eqref{eqn: taudef} of $\tau_E$ to deduce from \eqref{eqn: S(h,k)def} that
\begin{equation}\label{eqn: towardstherecipe1}
\begin{split}
\mathcal{S}(h,k) = \frac{1}{(2\pi i )^2} \int_{(2)} \int_{(2)} X^{s_1+s_2} \widetilde{V}(s_1) \widetilde{V}(s_2) \sum_{q=1}^{\infty} W\left( \frac{q}{Q}\right) \sideset{}{^\flat}\sum_{\chi \bmod q} \chi(h) \overline{\chi}(k)\\
\times \prod_{\alpha\in A} L(\tfrac{1}{2}+\alpha+s_1,\chi) \prod_{\beta\in B} L(\tfrac{1}{2}+\beta+s_2,\overline{\chi}) \,ds_2\,ds_1,
\end{split}
\end{equation}
where $\widetilde{V}$ is defined by~\eqref{eqn: mellindef}. We may move the lines of integration to $\re(s_1)=\re(s_2)=\varepsilon$ because of the rapid decay of $\widetilde{V}$ and the fact that $L(s,\chi)$ is entire for non-principal $\chi$. Now recall that if $\chi$ is an even primitive character of conductor $q$, then $L(s,\chi)$ satisfies the functional equation \cite[\S9]{Davenport}
\begin{equation*}
L(s,\chi) = G(\chi)q^{-s} \mathscr{X} (s) L(1-s,\overline{\chi}),
\end{equation*}
where $G(\chi)= \sum_{n\bmod{q}}\chi(n) \exp(2\pi i n/q)$ is the Gauss sum and $\mathscr{X} (s)$ is defined by \eqref{eqn: scriptXdef}. Then we have the approximate functional equation
\begin{equation*}
L(s,\chi) \approx \sum_n \frac{\chi(n)}{n^s} + G(\chi)q^{-s} \mathscr{X} (s) \sum_n \frac{\overline{\chi}(n)}{n^{1-s}}.
\end{equation*}
We replace each $L(s,\chi)$ factor in \eqref{eqn: towardstherecipe1} with the right-hand side of its approximate functional equation, and then multiply out the resulting product. We formally discard all the resulting terms except for those that have the same number of $G(\chi)$ factors as $G(\overline{\chi})$ factors. For the remaining terms, we use the fact that $G(\chi)G(\overline{\chi})=q$ \cite[\S9]{Davenport}, and formally extend the sums from the approximate functional equations to $\infty$. We then write the sums in terms of the function $\tau_E$ defined by \eqref{eqn: taudef}, and use the approximation \cite[(4.3.4)]{CFKRS}
\begin{equation*}
\sideset{}{^\flat}\sum_{\chi \bmod q} \chi(hm) \overline{\chi}(kn) \approx \left\{ \begin{array}{cl} \displaystyle \sideset{}{^\flat}\sum_{\chi \bmod q} 1 & \text{if }hm=kn \text{ and } (hkmn,q)=1\\ \\ 0 & \text{else}, \end{array}\right.
\end{equation*}
which we expect to follow from the orthogonality of Dirichlet characters (see also Lemma~\ref{lem: Lemma2ofCIS}). This leads us to conjecture Conjecture~\ref{con: conjecture}.
We may put Conjecture \ref{con: conjecture} into a more explicit form by writing out the analytic continuation of the $m,n$-sum in \eqref{eqn: I_l(h,k)def}. We do this by formally writing it as an Euler product, multiplying it by
\begin{equation}\label{eqn: CFKRSzetafactors}
\prod_{\substack{ \gamma\in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } } \zeta(1+\gamma+\delta),
\end{equation}
and then dividing it by the Euler product of \eqref{eqn: CFKRSzetafactors}. In other words, we claim that the definition \eqref{eqn: I_l(h,k)def} of $\mathcal{I}_{\ell}(h,k)$ with the $m,n$-sum written explicitly as its analytic continuation is
\begin{align}
\mathcal{I}_{\ell}(h,k) &= \sum_{\substack{U\subseteq A, V\subseteq B \\ |U|=|V|=\ell}} \sum_{\substack{q=1 \\ (q,hk)=1} }^{\infty} W\left( \frac{q}{Q}\right)\sideset{}{^\flat}\sum_{\chi \bmod q} \frac{1}{(2\pi i )^2} \int_{(\varepsilon)} \int_{(\varepsilon)} X^{s_1+s_2} \widetilde{V}(s_1) \widetilde{V}(s_2) \notag\\
&\hspace{.25in}\times \prod_{\alpha\in U} \frac{\mathscr{X} (\tfrac{1}{2}+\alpha +s_1 )}{ q^{\alpha+s_1} } \prod_{\beta\in V} \frac{\mathscr{X} (\tfrac{1}{2}+\beta+s_2 )}{ q^{\beta+s_2} } \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }} \zeta(1+\gamma+\delta) \notag\\
&\hspace{.25in}\times \prod_{p|q}\Bigg\{ \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \Bigg\} \prod_{p | hk}\Bigg\{ \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \notag\\
&\hspace{.5in}\times\sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}}\frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p^m) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p^n) }{p^{m/2}p^{n/2} }\Bigg\} \notag\\
&\hspace{.25in}\times \prod_{p\nmid qhk}\Bigg\{ \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \notag\\
&\hspace{.5in}\times\sum_{m=0}^{\infty}\frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p^m) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p^m) }{p^{m} }\Bigg\} \,ds_2\,ds_1. \label{eqn: I_l(h,k)def2}
\end{align}
We now prove our claim by showing that the Euler product in \eqref{eqn: I_l(h,k)def2} converges absolutely for $A,B$ satisfying $\alpha,\beta\ll 1/\log Q$ for all $\alpha\in A$ and $\beta\in B$. To do this, we make the following observations for such $A,B$. If Re$(s_1)=$Re$(s_2)=\varepsilon$, then
\begin{equation}\label{eqn: CFKRSeuler1}
\begin{split}
\prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right)
& = 1- \sum_{\substack{ \gamma\in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } } \frac{1}{p^{1+\gamma+\delta}} + O\left( \frac{1}{p^{1+\varepsilon}}\right) \\
& = 1 -\frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p ) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p ) }{ p } + O\left( \frac{1}{p^{1+\varepsilon}}\right),
\end{split}
\end{equation}
where the last equality follows from the definition \eqref{eqn: taudef} of $\tau_E$. Furthermore, \eqref{eqn: divisorbound} implies
\begin{equation}\label{eqn: CFKRSeuler2}
\sum_{m=2}^{\infty} \frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p^{m}) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p^{m}) }{p^{m}} \ll \frac{1}{p^{2-\varepsilon}}
\end{equation}
for Re$(s_1)=$Re$(s_2)=\varepsilon$. From this and \eqref{eqn: CFKRSeuler1}, we deduce that if $p\nmid qhk$, then the local factor in \eqref{eqn: I_l(h,k)def2} corresponding to $p$ is $1+O(p^{-1-\varepsilon})$. Hence the Euler product in \eqref{eqn: I_l(h,k)def2} converges absolutely.
We next prove an integral expression for the $q$-sum in \eqref{eqn: I_l(h,k)def2} in order to facilitate the proof of Theorem~\ref{thm: main}. We first observe that if Re$(s_1)=$Re$(s_2)=\varepsilon$ and $\alpha,\beta\ll 1/\log Q$ for all $\alpha\in A$ and $\beta\in B$, then
\begin{align}
& \prod_{p|q}\Bigg\{ \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \Bigg\} \prod_{p | hk}\Bigg\{ \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right)\notag \\
&\hspace{.5in}\times\sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}}\frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p^m) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p^n) }{p^{m/2}p^{n/2} }\Bigg\} \notag \\
& = \prod_{p|qhk} O(1) \ll (qhk)^{\varepsilon}. \label{eqn: CFKRSeuler3}
\end{align}
Now Lemma~\ref{lem: Lemma2ofCIS} with $m=n=1$ implies
\begin{equation*}
\sideset{}{^\flat}\sum_{\chi \bmod q}1 = \frac{1}{2}\sum_{d|q} \phi(d) \mu \left( \frac{q}{d}\right) +O(1).
\end{equation*}
We insert this into \eqref{eqn: I_l(h,k)def2}. The total contribution of the $O(1)$ error term is at most $\ll_{\varepsilon} X^{\varepsilon}Q^{1+\varepsilon}(hk)^{\varepsilon}$ if we assume \eqref{eqn: orbitals}, since we have \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: CFKRSeuler1}, \eqref{eqn: CFKRSeuler2}, and \eqref{eqn: CFKRSeuler3}. We then write $W(q/Q)$ as an integral using its Mellin transform. We take this integral to be along Re$(w)=2+\varepsilon$ to keep the $q$-sum absolutely convergent. Expressing the $q$-sum as an Euler product using Lemma~\ref{lem: sumstoEulerproducts}, we then deduce from \eqref{eqn: I_l(h,k)def2} that, if \eqref{eqn: orbitals} holds, then
\begin{equation}\label{eqn: IandIstar}
\mathcal{I}_{\ell}(h,k) = \mathcal{I}_{\ell}^*(h,k) + O(X^{\varepsilon}Q^{1+\varepsilon}(hk)^{\varepsilon}),
\end{equation}
where $\mathcal{I}_{\ell}^*(h,k)$ is defined by
\begin{align}
\mathcal{I}_{\ell}^*(h,k)
& = \sum_{\substack{U\subseteq A, V\subseteq B \\ |U|=|V|={\ell}}} \frac{1}{2(2\pi i )^3} \int_{(\varepsilon)} \int_{(\varepsilon)} \int_{(2+\varepsilon)} X^{s_1+s_2} Q^w \widetilde{V}(s_1) \widetilde{V}(s_2)\widetilde{W}(w) \notag\\
&\hspace{.25in} \times \prod_{\alpha\in U} \mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \prod_{\beta\in V} \mathscr{X} (\tfrac{1}{2}+\beta+s_2 ) \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }} \zeta(1+\gamma+\delta) \notag \\
&\hspace{.25in} \times \prod_{p|hk} \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \notag\\
&\hspace{.5in}\times \sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}}\frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p^m) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p^n) }{p^{m/2}p^{n/2} } \notag\\
&\hspace{.25in} \times \prod_{p\nmid hk} \Bigg\{ \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \times \bigg( 1 + \frac{p-2}{p^{w+\sum_{\alpha\in U} (\alpha+s_1) +\sum_{\beta\in V} (\beta + s_2)} } \notag\\
&\hspace{.5in} + \left( 1-\frac{1}{p}\right)^2 \frac{p^{2(1-w-\sum_{\alpha\in U} (\alpha+s_1) -\sum_{\beta\in V} (\beta + s_2))}}{1-p^{1-w-\sum_{\alpha\in U} (\alpha+s_1) -\sum_{\beta\in V} (\beta + s_2)}} \bigg) \notag\\
&\hspace{.5in} + \prod_{\substack{\gamma \in A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-} \\ \delta \in B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} }} \left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \notag\\
&\hspace{.75in}\times \sum_{m=1}^{\infty} \frac{\tau_{A_{s_1}\smallsetminus U_{s_1} \cup (V_{s_2})^{-}} (p^m) \tau_{B_{s_2}\smallsetminus V_{s_2} \cup (U_{s_1})^{-} } (p^m) }{p^m} \Bigg\} \,dw \,ds_2\,ds_1. \label{eqn: Istardef}
\end{align}
\section{Initial setup and outline of the proof of Theorem~\ref{thm: main}}\label{sec: outline}
We may assume that $(q,mnhk)=1$ in the definition \eqref{eqn: S(h,k)def} of $\mathcal{S}(h,k)$ since otherwise the summand is zero. We may thus apply Lemma~\ref{lem: Lemma2ofCIS} to deduce from \eqref{eqn: S(h,k)def} that
\begin{equation}\label{eqn: applyLemma2ofCIS}
\mathcal{S}(h,k) = \frac{1}{2} \sum_{\substack{1\leq q<\infty \\ (q,hk)=1}}W\left(\frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{c,d\geq 1 \\ cd=q \\ d|mh\pm nk }} \phi(d)\mu(c).
\end{equation}
Let $C>0$ be a parameter that we will choose to be some power of $Q$ at the end of our proof of Theorem~\ref{thm: main}. We use the notation of \cite{CIS} and split the right-hand side of \eqref{eqn: applyLemma2ofCIS} to write
\begin{equation}\label{eqn: Ssplit}
\mathcal{S}(h,k)=\mathcal{L}(h,k)+\mathcal{D}(h,k)+\mathcal{U}(h,k),
\end{equation}
where $\mathcal{L}(h,k)$ is the sum of the terms with $c>C$, $\mathcal{D}(h,k)$ is the sum of the ``diagonal'' terms with $c\leq C$ and $mh=nk$, and $\mathcal{U}(h,k)$ is the sum of the ``off-diagonal'' terms with $c\leq C$ and $mh\neq nk$. In other words, $\mathcal{L}(h,k)$, $\mathcal{D}(h,k)$, and $\mathcal{U}(h,k)$ are defined by
\begin{equation}\label{eqn: Lsum}
\mathcal{L}(h,k):= \frac{1}{2}\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{c>C, d \geq 1 \\ cd=q \\ d|mh\pm nk}}\phi(d)\mu(c),
\end{equation}
\begin{equation}\label{eqn: Dsum}
\mathcal{D}(h,k):= \frac{1}{2}\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1\\ mh=nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}}V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{1\leq c\leq C, d \geq 1 \\ cd=q \\ d|mh\pm nk}}\phi(d)\mu(c),
\end{equation}
and
\begin{equation}\label{eqn: Usum}
\mathcal{U}(h,k):= \frac{1}{2}\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{1\leq c\leq C, d \geq 1 \\ cd=q \\ d|mh\pm nk}}\phi(d)\mu(c),
\end{equation}
respectively. The purpose of splitting the $c$-sum this way is that we need the $c$-sum to be finite when we apply the asymptotic large sieve.
For the rest of this section, we outline our strategy for estimating each of $\mathcal{L}(h,k)$, $\mathcal{D}(h,k)$, and $\mathcal{U}(h,k)$. The presentation in this section will be terse in comparison to the actual arguments.
We treat $\mathcal{D}(h,k)$ in Section~\ref{sec: diagonal}. There, we extend the $c$-sum in \eqref{eqn: Dsum} to $\infty$, apply Mellin inversion, and then write sums in terms of an Euler product to show that, up to an admissible error term, $\mathcal{D}(h,k)$ equals the zero-swap term $\mathcal{I}_0(h,k)$, which is defined by \eqref{eqn: I_l(h,k)def} with $\ell=0$.
We evaluate $\mathcal{L}(h,k)$ in Section~\ref{sec: Lsum}. As in the approach of \cite{CIS}, we detect the divisibility condition $d|mh\pm nk$ using character sums and split $\mathcal{L}(h,k)$ into
$$
\mathcal{L}^0(h,k) + \mathcal{L}^r(h,k),
$$
where $\mathcal{L}^0(h,k)$ is the contribution of the principal characters while $\mathcal{L}^r(h,k)$ is the rest of the sum. We use M\"{o}bius inversion to convert $\mathcal{L}^0(h,k)$ into a sum over $c\leq C$ and show later that it cancels with a term from our analysis of $\mathcal{U}(h,k)$. We bound
$$
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h\overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k)
$$
by applying Mellin inversion and writing the $m,n$-sum in terms of Dirichlet $L$-functions. We use GLH to bound these $L$-functions, and then apply the large sieve. The role of $C$ here is to make the bound from applying the large sieve $\ll Q^{2-\varepsilon}$. Our use of GLH differs from the approach in \cite{CIS}, where they are able to apply the bound for the fourth moment because they have only a few $L$-functions in their setting.
The analysis of $\mathcal{U}(h,k)$ forms the most difficult part of the proof, and is done in Sections \ref{sec: U(h,k)split}, \ref{sec: U2}, and \ref{sec: Ur}. The first step in our analysis of $\mathcal{U}(h,k)$ is to make a change of variables and switch from the divisor $d$ of $mh\pm nk$ to the ``complementary modulus'' $\ell$ given by
\begin{equation}\label{eqn: switchtocomplementary}
\ell = \frac{|mh\pm nk|}{d}.
\end{equation}
We then use character sums to detect the condition $\ell | mh\pm nk$ and arrive at (essentially)
\begin{equation*}
\begin{split}
\mathcal{U}(h,k)&\approx \frac{1}{2} \sum_{c=1}^C \mu(c) \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\\
&\hspace{.25in} \times\sum_{\ell=1}^{\infty} \frac{1}{ \ell} \sum_{\psi \bmod \ell} \psi (mh) \overline{\psi}( \mp nk) \frac{|mh\pm nk|}{ \ell} W\left( \frac{c|mh\pm nk|}{\ell Q}\right)
\end{split}
\end{equation*}
(the unabridged version of this is \eqref{eqn: Uafterswitch} in Section~\ref{sec: U(h,k)split}). This technique of switching to the complementary modulus is at the heart of the \textit{asymptotic large sieve} due to Conrey, Iwaniec, and Soundararajan \cite{CISAsymptoticLargeSieve}; see also \cite{CIS6th} and \cite{ChandeeLi8Dirichlet}. The purpose of switching from the divisor $d$ to the complementary modulus \eqref{eqn: switchtocomplementary} is to reduce the moduli of the characters we use to detect the divisibility condition. This, in turn, leads to a tighter upper bound when applying the large sieve inequality. Indeed, the variable $d$ in \eqref{eqn: Usum} satisfies $d\asymp Q/c$ because $cd=q$ and $q\asymp Q$ by the support of $W$. Thus, $d$ can be of size $\asymp Q$ since $c$ may be $1$. On the other hand, the variable $\ell$ in \eqref{eqn: switchtocomplementary} can only be at most $\ll XCQ^{\vartheta-1}$ for $h,k\leq Q^{\vartheta}$ since $d\asymp Q/c$, $c\leq C$, and $m,n\ll X$ in \eqref{eqn: Usum} by the support of $V$. If $X\ll Q^{2-\varepsilon}$, then $XCQ^{\vartheta-1}$ is a factor of $Q^{\varepsilon}$ smaller than $Q$ for suitably small $C$ and $\vartheta$. This technique and the asymptotic large sieve have proven to be extremely useful in the study of the family of primitive Dirichlet $L$-functions (see, for example, \cite{CIS6th}, \cite{CISCriticalZeros}, \cite{ChandeeLeeLiuRadziwill}, and \cite{ChandeeLi8Dirichlet}).
After expressing $\mathcal{U}(h,k)$ in terms of character sums, we may split $\mathcal{U}(h,k)$ into
$$
\mathcal{U}^0(h,k) + \mathcal{U}^r(h,k),
$$
where $\mathcal{U}^0(h,k)$ is the contribution of the principal characters while $\mathcal{U}^r(h,k)$ is the rest of the sum. We bound
\begin{equation}\label{eqn: Ursumtobound}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h\overline{\lambda_k}}{\sqrt{hk}} \mathcal{U}^r(h,k)
\end{equation}
in Section~\ref{sec: Ur} through a procedure similar to that in \cite{CIS}. In this method, we first make a change of variables to remove some of the dependencies of the summation variables $m,n,h,k$ on each other. We then apply Mellin inversion, write the sum in terms of an Euler product, and then move the lines of integration closer to zero so that the resulting exponent of $X$ in the integrand has small real part. The Euler product contains a potentially large number of $L$-function factors, and we use GLH to bound these $L$-functions. We split the integrals into dyadic parts, and bound the Mellin transforms carefully by treating each dyadic part differently. This technical step, which we carry out explicitly in \eqref{eqn: beforelargesieve}, is a bit more delicate than the estimations in \cite{CIS} because there are more variables of integration after we apply Mellin inversion. Finally, we apply the large sieve inequality to estimate the character sums. It is at this point that we see the effectiveness of using the complementary modulus \eqref{eqn: switchtocomplementary}. If the character sums involve characters of modulus $Q$, then the large sieve inequality alone may not be enough to show that \eqref{eqn: Ursumtobound} has order of magnitude smaller than that of the main term in the predicted asymptotic formula for $\mathcal{S}(h,k)$.
To evaluate the contribution $\mathcal{U}^0(h,k)$ of the principal characters, we first apply Mellin inversion on the function $W$ and write the $\ell$-sum as an Euler product using Lemma 6 of \cite{CIS} (Lemma~\ref{lem: Lemma6ofCIS} in Section~\ref{sec: U(h,k)split}). We then move the line of integration to write
$$
\mathcal{U}^0(h,k) = \mathcal{U}^1(h,k) + \mathcal{U}^2(h,k),
$$
where $\mathcal{U}^1(h,k)$ is the residue from the pole of the (analytic continuation of the) Euler product, while $\mathcal{U}^2(h,k)$ is the integral along the new line. The residue $\mathcal{U}^1(h,k)$ is equal to the negative of $\mathcal{L}^0(h,k)$ plus an admissible error term, and thus cancels $\mathcal{L}^0(h,k)$.
We analyze the integral $\mathcal{U}^2(h,k)$ in Section~\ref{sec: U2} to uncover the predicted one-swap terms. This is where we carry out the delicate contour integration mentioned below Theorem~\ref{thm: main}. To begin, we apply Proposition~2 of \cite{CIS} (stated as Proposition~\ref{prop: CISProp2} in Section~\ref{sec: U2}) and separate the variables $m$ and $n$ in $|mh\pm nk|$ by writing $|mh\pm nk|^w$ in terms of an integral of a meromorphic function. We then apply Mellin inversion on the function $V$ and express the sum as an Euler product. We determine the analytic continuation of this Euler product, and then move the lines of integration to suitable locations to express $\mathcal{U}^2(h,k)$ as a sum of several residues and error terms. We use the Lindel\"{o}f Hypothesis for $\zeta(s)$ to justify moving some of the lines of integration and to bound one of the error terms. We also carry out a similar analysis of the sum $\mathcal{I}_1(h,k)$ of the one-swap terms from Conjecture~\ref{con: conjecture}. We then find that each residue in the expression for $\mathcal{U}^2(h,k)$ can be matched with a residue in the expression for $\mathcal{I}_1(h,k)$ in such a way that corresponding residues are equal up to a negligible error term. This step requires proving identities involving several Euler products. These Euler product identities, in turn, are consequences of certain properties of the function $\tau_E$, the chief one being
\begin{equation*}
\begin{split}
\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell}) - \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) \\
= \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{\ell}) -p^{\alpha+\beta} \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}}(p^{j-1}) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{\ell-1})
\end{split}
\end{equation*}
(Lemma~\ref{lem: CK3identity} in Section~\ref{sec: U2}), which stems from the work of Conrey and Keating \cite{CK3} on moments of zeta. Conjecture \ref{con: conjecture}, predicted by the CFKRS recipe, plays a crucial role in the analysis of $\mathcal{U}^2(h,k)$, as it provides a clear answer to aim for in untangling $\mathcal{U}^2(h,k)$.
\
\noindent{\it Changes in the proof for the odd case.} We now describe the changes we need to make in our proof in order to handle the odd primitive characters. The version of Lemma~\ref{lem: Lemma2ofCIS} for odd primitive characters states that if $(mn,q)=1$, then
\begin{equation*}
\sideset{}{^{\text{odd}}}\sum_{\chi \bmod q}\chi(m)\overline{\chi(n)} = \frac{1}{2}\sum_{\substack{d|q \\ d|m-n}} \phi(d)\mu\left( \frac{q}{d}\right) - \frac{1}{2}\sum_{\substack{d|q \\ d|m+n}} \phi(d)\mu\left( \frac{q}{d}\right),
\end{equation*}
where the superscript ``odd'' indicates that the sum is over all the odd primitive characters. Thus, to handle the sum over the odd primitive characters, we change our convention about the symbol $\pm$ and have $-1$ multiplied to the copy that has $\pm$ replaced by $+$. A consequence of this sign change is that the analogues of $\mathcal{L}^0(h,k)$ and $\mathcal{U}^0(h,k)$ for odd primitive characters are zero. The main term in the asymptotic formula for the analogue of $\mathcal{D}(h,k)$ is unaffected by the sign change, and so \eqref{eqn: Dis0swap} still holds with $\mathcal{D}(h,k)$ replaced by its analogue. The sign change does not affect the other bounds in our proof. In evaluating the analogue of $\mathcal{U}^2(h,k)$, instead of using Proposition~\ref{prop: CISProp2}, we use the version of it for
$$
|1-r|^{-\omega} - |1+r|^{-\omega}.
$$
This version has the function
$$
\mathscr{X}(\omega)\mathscr{Y}(1-z)\mathscr{Y}(1-\omega+z)
$$
in place of $\mathcal{H}(z,\omega)$, where $\mathscr{Y}(s)$ is defined by
$$
\mathscr{Y}(s) = \pi^{s-\frac{1}{2}} \frac{\Gamma( 1-\frac{1}{2}s)}{\Gamma(\frac{1}{2}+\frac{1}{2}s)}.
$$
\section{The diagonal terms \texorpdfstring{$\mathcal{D}(h,k)$}{D(h,k)}}\label{sec: diagonal}
In this section, we focus on the sum $\mathcal{D}(h,k)$ of the diagonal terms, defined by \eqref{eqn: Dsum}. We first perform a short analysis of the main contribution $\mathcal{I}^*_0(h,k)$ of the zero-swap term. We will then see that $\mathcal{I}^*_0(h,k)$ coincides exactly with the main contribution of $\mathcal{D}(h,k)$.
\subsection{The prediction for the zero-swap term}
We may simplify $\mathcal{I}^*_0(h,k)$, defined by \eqref{eqn: Istardef} with $\ell=0$, by cancelling the zeta-function factors $\zeta(1+\alpha+\beta+s_1+s_2)$ with the convergent products of the corresponding local factors. We also apply \eqref{eqn: taufactoringidentity}. The result is
\begin{align*}
\begin{split}
\mathcal{I}^*_0(h,k)= \frac{1}{2(2\pi i )^3} &\int_{(\varepsilon)} \int_{(\varepsilon)} \int_{(2+\varepsilon)} X^{s_1+s_2} Q^w \widetilde{V}(s_1) \widetilde{V}(s_2)\widetilde{W}(w)\\
&\hspace{.25in}\times\prod_{p|hk} \sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}}\frac{\tau_{A}(p^m)\tau_{B} (p^n)}{p^{m(1/2+s_1)}p^{n(1/2+s_2)} } \\
&\hspace{.75in}\times \prod_{p \nmid hk} \left( \frac{(1-p^{-w})^2}{1-p^{1-w}}+ \sum_{\ell=1}^{\infty} \frac{\tau_{A} (p^\ell) \tau_{B } (p^{\ell} ) }{p^{\ell(1+s_1+s_2)}} \right) \,dw \,ds_2\,ds_1.
\end{split}
\end{align*}
To simplify the latter $m,n$-sum, define $H:=h/(h,k)$ and $K:=k/(h,k)$. A given pair $m,n$ is a pair of nonnegative integers with $m+\text{ord}_p(h) = n+\text{ord}_p(k)$ if and only if there is a nonnegative integer $\ell$ such that $m=\ell + \text{ord}_p(K)$ and $n= \ell + \text{ord}_p(H)$. Hence we may write the $m,n$ sum as
\begin{align*}
\frac{1}{p^{\text{ord}_p(K)(1/2+s_1)+\text{ord}_p(H)(1/2+s_2)}} \sum_{\ell=0}^{\infty}\frac{\tau_A(p^{\text{ord}_p(K)+\ell})\tau_B(p^{\text{ord}_p(H)+\ell})}{p^{\ell(1+s_1+s_2)}}.
\end{align*}
Thus we predict that
\begin{equation}\label{eqn: zero-swap}
\begin{split}
\mathcal{I}^*_0(h,k)=\frac{1}{2(2\pi i )^3} &\int_{(\varepsilon)} \int_{(\varepsilon)} \int_{(2+\varepsilon)} \frac{X^{s_1+s_2}}{H^{1/2+s_2}K^{1/2+s_1}} Q^w \widetilde{V}(s_1) \widetilde{V}(s_2)\widetilde{W}(w)\\
&\times \prod_{p|hk} \sum_{\ell=0}^{\infty}\frac{\tau_A(p^{\text{ord}_p(K)+\ell})\tau_B(p^{\text{ord}_p(H)+\ell})}{p^{\ell(1+s_1+s_2)}}\\
&\hspace{.5in}\times\prod_{p\nmid hk} \left( \frac{(1-p^{-w})^2}{1-p^{1-w}}+ \sum_{\ell=1}^{\infty} \frac{\tau_{A} (p^\ell) \tau_{B } (p^{\ell} ) }{p^{\ell(1+s_1+s_2)}} \right) \,dw \,ds_2\,ds_1.
\end{split}
\end{equation}
\subsection{\texorpdfstring{$\mathcal{D}(h,k)$}{D(h,k)} coincides with the prediction for the zero-swap term}
In this subsection, we show that $\mathcal{D}(h,k)$, defined by \eqref{eqn: Dsum}, is equal to the right-hand side of \eqref{eqn: zero-swap} plus an admissible error term. To this end, we first make a change of variables in the $m,n$ sum. Since $H:=h/(h,k)$ and $K:=k/(h,k),$ the condition $mh=nk$ is equivalent to the condition that $m=K\ell \text{ and } n=H\ell$ for some positive integer $\ell$. We thus arrive at
\begin{equation*}
\begin{split}
\mathcal{D}(h,k)= \frac{1}{2}\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}}& W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq \ell <\infty \\ (\ell,q)=1}} \frac{\tau_A(K\ell) \tau_B(H\ell) }{\ell\sqrt{HK}} V\left( \frac{K\ell}{X} \right)
V\left( \frac{H\ell}{X} \right) \sum_{\substack{1\leq c\leq C, d \geq 1 \\ cd=q \\ d|K\ell h\pm H\ell k}}\phi(d)\mu(c).
\end{split}
\end{equation*}
Recall that we use the notation $d|K\ell h\pm H\ell k$ to signify that we are adding two copies of the sum: one with $d|K\ell h- H\ell k$ and the other with $d|K\ell h+ H\ell k$. In the first copy, we are summing over all $d$ because $K h= H k$. In the second copy, the condition that $d$ divides $K\ell h+ H\ell k$ is equivalent to the condition that $d|2$ because $K h= H k$ and $(q,hk\ell)=1$. Thus, the $c,d$-sum in the second copy has at most two terms, and so the second copy is bounded by
\[
\ll Q\sum_{\ell \ll X} \frac{(HK\ell)^\varepsilon}{\ell \sqrt{HK}} \ll Q\frac{(XHK)^\varepsilon}{\sqrt{HK}}.
\]
Hence
\begin{align*}
\mathcal{D}(h,k)&=\frac{1}{2}\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\le \ell < \infty\\(\ell,q)=1}}\frac{\tau_A(K\ell)\tau_B(H\ell)}{\ell\sqrt{HK}}V\left(\frac{K\ell}{X}\right)V\left(\frac{H\ell}{X}\right)\\
&\hspace{.25in}\times\sum_{\substack{1\leq c\leq C, d \geq 1 \\ cd=q}}\phi(d)\mu(c)+ O\left( Q\frac{(XHK)^\varepsilon}{\sqrt{HK}} \right).
\end{align*}
We next extend the $c$-sum to $\infty$. The error introduced in doing so is
\begin{align*}
&\ll \sum_{q \ll Q}\sum_{\ell \ll X} \frac{(HK\ell)^\varepsilon}{\ell \sqrt{HK}}\sum_{\substack{c>C , d \geq 1 \\ cd=q}}\phi(d) \ \ll \frac{Q^2}{C} \frac{(XHK)^{\varepsilon}}{\sqrt{HK}}.
\end{align*}
Note that we are careful to estimate the $c$-sum in terms of $C$, which is necessary because the main term in Theorem~\ref{thm: main} is of size about $Q^2$. Later, we will choose $C$ as a specific positive power of $Q$ to control this error term. Setting $\phi^\star(q):=\sum_{cd=q}\phi(d)\mu(c)$, we now have
\begin{align*}
\mathcal{D}(h,k)&=\frac{1}{2}\sum_{\substack{1\le q <\infty \\ (q,hk)=1}}W\left(\frac{q}{Q}\right)\phi^\star(q)\sum_{\substack{1\le \ell < \infty \\ (\ell,q)=1}}\frac{\tau_A(K\ell)\tau_B(H\ell)}{\ell \sqrt{HK}}V\left(\frac{K\ell}{X}\right)V\left(\frac{H\ell}{X}\right)\\
&\hspace{.25in}+ O\left(\left(Q+\frac{Q^2}{C}\right)\frac{(XHK)^\varepsilon}{\sqrt{HK}}\right).
\end{align*}
Next, write $V,W$ in terms of their Mellin transforms using Mellin inversion to find
\begin{align}\label{eqn: diagonalapplyMellin}
\begin{split}
\mathcal{D}(h,k)&=\frac{1}{2(2\pi i)^3}\int_{(\varepsilon)}\int_{(\varepsilon)}\frac{X^{s_1+s_2}}{H^{1/2+s_1}K^{1/2+s_2}}\widetilde{V}(s_1)\widetilde{V}(s_2)\int_{(2+\varepsilon)}Q^w\widetilde{W}(w)\\
&\hspace{.25in}\times\sum_{\substack{1\le q <\infty \\ (q,hk)=1}}q^{-w}\phi^\star(q)\sum_{\substack{1\le \ell < \infty \\ (\ell,q)=1}}\frac{\tau_A(K\ell)\tau_B(H\ell)}{\ell^{1+s_1+s_2}}\,dw\,ds_2\,ds_1\\
&\hspace{.5in}+O\left(\left(Q+\frac{Q^2}{C}\right)\frac{(XHK)^\varepsilon}{\sqrt{HK}}\right),
\end{split}
\end{align}
where we have chosen the location of the $w$-line to be along Re$(w)=2 + \varepsilon$ to ensure that the $q$-sum is absolutely convergent. We may then rewrite the $q,\ell$-sum in \eqref{eqn: diagonalapplyMellin} as the Euler product
\begin{equation}\label{eqn: diagonalqlsumeuler}
\prod_{p}\Bigg(\sum_{\substack{0\le q < \infty\\ \min\{ q,\text{ord}_p(h)+\text{ord}_p(k)\}=0}}\!\!\!\!\!\!\!\!p^{-qw}\phi^{\star}(p^q)\sum_{\substack{0\le \ell < \infty\\\min\{\ell,q\}=0}}\frac{\tau_A(p^{\text{ord}_p(K)+\ell})\tau_B(p^{\text{ord}_p(H)+\ell})}{p^{\ell(1+s_1+s_2)}}\Bigg).
\end{equation}
If $p|hk$, then $\text{ord}_p(h)+\text{ord}_p(k)\ge 1$. In this case, for the condition $\min\{ q,\text{ord}_p(h)+\text{ord}_p(k)\}=0$ to hold, we must have $q=0$. Since $\phi^{\star}(p^0) =1$, it follows that the contribution to the Euler product from the primes dividing $hk$ is
\begin{align*}
\prod_{p|hk} \Bigg(\sum_{\ell = 0}^{\infty}\frac{\tau_A(p^{\text{ord}_p(K)+\ell})\tau_B(p^{\text{ord}_p(H)+\ell})}{p^{\ell(1+s_1+s_2)}} \Bigg),
\end{align*}
which we note has no dependence on $w$. Now suppose that $p\nmid hk$. Then $\text{ord}_p(h)+\text{ord}_p(k)=0$, which means we may drop the condition that $\min\{ q,\text{ord}_p(h)+\text{ord}_p(k)\}=0$. The contribution to the Euler product from primes not dividing $hk$ is thus
\begin{align*}
\prod_{p\nmid hk}\sum_{\substack{0\le q,\ell < \infty\\ \min(\ell,q)=0}}p^{-qw}\phi^{\star}(p^q) \frac{\tau_A(p^{\ell})\tau_B(p^{\ell})}{p^{\ell(1+s_1+s_2)}} = \prod_{p\nmid hk}\left( 1 + \sum_{q=1}^{\infty}p^{-qw}\phi^{\star}(p^q)+ \sum_{\ell=1}^{\infty}\frac{\tau_A(p^{\ell})\tau_B(p^{\ell})}{p^{\ell(1+s_1+s_2)}}\right)
\end{align*}
Inserting the definition of $\phi^{\star}$ into the $q$-sum, we directly calculate the $q=1$ term and realize the sum of the terms with $q>1$ as a geometric series to find, after a short calculation, that
\begin{align*}
1 + \sum_{q=1}^{\infty}p^{-qw}\phi^{\star}(p^q) &=1 + \sum_{q=1}^{\infty}p^{-qw} \sum_{cd=p^q}\phi(d)\mu(c)
=\left(\frac{1}{1-p^{1-w}}\right)\left( 1-p^{-w}\right)^2.
\end{align*}
Hence, writing the $q,\ell$-sum in \eqref{eqn: diagonalapplyMellin} as the Euler product \eqref{eqn: diagonalqlsumeuler} and applying the above simplifications, we arrive at
\begin{align*}
\mathcal{D}(h,k)&=\frac{1}{2(2\pi i)^3}\int_{(\varepsilon)}\int_{(\varepsilon)}\int_{(2+\varepsilon)}\frac{X^{s_1+s_2}}{H^{1/2+s_1}K^{1/2+s_2}}Q^w\widetilde{V}(s_1)\widetilde{V}(s_2)\widetilde{W}(w)\\
&\hspace{.25in}\times\prod_{p|hk} \left(\sum_{\ell = 0}^{\infty}\frac{\tau_A(p^{\text{ord}_p(K)+\ell})\tau_B(p^{\text{ord}_p(H)+\ell})}{p^{\ell(1+s_1+s_2)}} \right)\\
&\hspace{1in}\times\prod_{p\nmid hk}\left( \frac{(1-p^{-w})^2}{1-p^{1-w}} + \sum_{\ell=1}^{\infty}\frac{\tau_A(p^{\ell})\tau_B(p^{\ell})}{p^{\ell(1+s_1+s_2)}}\right)\,dw\,ds_1\,ds_2\\
&\hspace{1.5in}+ O\left(\left(Q+\frac{Q^2}{C}\right)\frac{(XHK)^\varepsilon}{\sqrt{HK}}\right).
\end{align*}
After relabeling $s_1$ as $s_2$ and vice versa, we see that the integral above exactly matches the right-hand side of \eqref{eqn: zero-swap}. In other words,
\begin{equation}\label{eqn: Dis0swap}
\mathcal{D}(h,k)=\mathcal{I}^*_0(h,k) + O\left(\left(Q+\frac{Q^2}{C}\right)\frac{(XHK)^\varepsilon}{\sqrt{HK}}\right).
\end{equation}
\section{The term \texorpdfstring{$\mathcal{L}(h,k)$}{L(h,k)}}\label{sec: Lsum}
Recall the definition \eqref{eqn: Lsum} of $\mathcal{L}(h,k)$, and recall that we interpret the $d$-sum therein as two sums: one with the condition $d|mh-nk$ and the other with the condition $d|mh+nk$. We first show how to re-express $\mathcal{L}(h,k)$ in terms of characters modulo $d$. For $(mnhk,d)=1$, the orthogonality of character sums implies
\begin{align*}
\frac{1}{\phi(d)}\sum_{\psi \bmod d} \psi(mh)\overline{\psi}(nk) = \left\{\begin{array}{cl} 1 & \text{if } d|mh-nk \\ \\ 0 & \text{else} \end{array}\right.
\end{align*}
and
\begin{align*}
\frac{1}{\phi(d)}\sum_{\psi \bmod d} \psi(mh)\overline{\psi}(-nk) = \left\{\begin{array}{cl} 1 & \text{if } d|mh+nk \\ \\ 0 & \text{else}. \end{array}\right.
\end{align*}
Since $\overline{\psi}(1)+\overline{\psi}(-1)=2$ if $\psi$ is even and $0$ if $\psi$ is odd, it follows that the sum of these two character sums is
$$
\frac{2}{\phi(d)}\sum_{\substack{\psi \bmod d \\ \psi \text{ even}}} \psi(mh)\overline{\psi}( nk).
$$
Therefore, we may recast $\mathcal{L}(h,k)$ as
\begin{align*}
\mathcal{L}(h,k) &= \sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\\
&\hspace{.5in}\times\sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)\sum_{\substack{\psi \bmod d \\ \psi \text{ even}}} \psi(mh)\overline{\psi}( nk).
\end{align*}
Split the right-hand side to write
\begin{equation}\label{eqn: Lsplit}
\mathcal{L}(h,k) = \mathcal{L}^0(h,k)+\mathcal{L}^r(h,k),
\end{equation}
where $\mathcal{L}^0(h,k)$ is the contribution of the principal character modulo $d$ and $\mathcal{L}^r(h,k)$ is the rest. In other words,
\begin{equation}\label{eqn: L0}
\begin{split}
\mathcal{L}^0(h,k) := \sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)
\sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
\mathcal{L}^r(h,k) :&= \sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\\
&\hspace{.5in}\times \sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)\sum_{\substack{\psi \bmod d \\ \psi \text{ even} \\ \psi\neq \psi_0}} \psi(mh)\overline{\psi}( nk),
\end{split}
\end{equation*}
where $\psi_0$ denotes the principal character modulo $d$.
In this section, we have two goals. First, we will bound the contribution of $\mathcal{L}^r(h,k)$ and show, on average over $h,k$, that it is an acceptable error term. Second, we will rework $\mathcal{L}^0(h,k)$ in preparation to show (later, in Section \ref{sec: U1}) that $\mathcal{L}^0(h,k)$ cancels with a term arising during the analysis of $\mathcal{U}(h,k)$.
\subsection{Bounding the contribution of \texorpdfstring{$\mathcal{L}^r(h,k)$}{Lr(h,k)} }
We may freely interchange the order of summation because each of $W$ and $V$ has compact support, forcing the sums to be finite. We bring the $m,n$-sum inside and then use Mellin inversion to write
\begin{equation*}
\begin{split}
\mathcal{L}^r(h,k) &= \sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)\sum_{\substack{\psi \bmod d \\ \psi \text{ even} \\ \psi\neq \psi_0}} \psi( h)\overline{\psi}( k) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) \psi(m )\overline{\psi}( n ) }{\sqrt{mn}}\\
&\hspace{.5in}\times\frac{1}{(2\pi i)^2}\int_{(\frac{1}{2}+\varepsilon)} \int_{(\frac{1}{2}+\varepsilon)} \frac{X^{s_1+s_2}}{m^{s_1}n^{s_2}}\widetilde{V}(s_1)\widetilde{V}(s_2) \,ds_2 \,ds_1,
\end{split}
\end{equation*}
where we have chosen the lines of integration to be at $\re(s_1)=\re(s_2)=\frac{1}{2}+\varepsilon$ so that in the next step we can interchange the $m,n$-sum and the integrals. Since $q=cd$ and $\psi(\nu)=0$ for $(\nu,d)>1$, the $m,n$-sum is the same as
\begin{equation*}
\begin{split}
\sum_{\substack{1\leq m,n<\infty \\ (mn,c)=1}} \frac{\tau_A(m) \tau_B(n) \psi(m )\overline{\psi}( n ) }{m^{\frac{1}{2}+s_1}n^{\frac{1}{2}+s_2}} &= \prod_{\alpha\in A} L(\tfrac{1}{2}+s_1+\alpha,\psi) \prod_{\beta\in B} L(\tfrac{1}{2}+s_2+\beta,\overline{\psi})\\
&\hspace{.25in}\times \prod_{\alpha\in A}\Bigg(\prod_{p|c}\bigg(1- \frac{\psi(p)}{p^{\frac{1}{2}+s_1+\alpha}} \bigg) \Bigg) \prod_{\beta\in B}\Bigg(\prod_{p|c}\bigg(1- \frac{\overline{\psi}(p)}{p^{\frac{1}{2}+s_2+\beta}} \bigg) \Bigg).
\end{split}
\end{equation*}
Therefore, we have
\begin{equation*}
\begin{split}
\mathcal{L}^r(h,k)=&\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)\sum_{\substack{\psi \bmod d \\ \psi \text{ even} \\ \psi\neq \psi_0}} \psi( h)\overline{\psi}( k)\frac{1}{(2\pi i)^2}\int_{(\frac{1}{2}+\varepsilon)} \int_{(\frac{1}{2}+\varepsilon)} X^{s_1+s_2} \\
&\hspace{.25in}\times \widetilde{V}(s_1)\widetilde{V}(s_2)
\prod_{\alpha\in A} L(\tfrac{1}{2}+s_1+\alpha,\psi) \prod_{\beta\in B} L(\tfrac{1}{2}+s_2+\beta,\overline{\psi})\\
&\hspace{.5in} \times \prod_{\alpha\in A}\Bigg(\prod_{p|c}\bigg(1- \frac{\psi(p)}{p^{\frac{1}{2}+s_1+\alpha}} \bigg) \Bigg) \prod_{\beta\in B}\Bigg(\prod_{p|c}\bigg(1- \frac{\overline{\psi}(p)}{p^{\frac{1}{2}+s_2+\beta}} \bigg) \Bigg) \,ds_2 \,ds_1.
\end{split}
\end{equation*}
We may now move the lines of integration to $\re(s_1)=\re(s_2)=\varepsilon$ by the rapid decay of $\widetilde{V}(s_1)$ and $\widetilde{V}(s_2)$ and the fact that $L(s,\psi)$ has no pole whenever $\psi$ is non-principal. We multiply both sides of the above equation by $ \lambda_h \overline{\lambda_k}(hk)^{-1/2}$, and then sum over all positive integers $h,k\leq Q^{\vartheta}$ to arrive at the quantity we aim to bound:
\begin{equation}\label{eqn: Lrweaimtobound}
\begin{split}
&\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k) = \sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}}\sum_{\substack{1\leq q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)\sum_{\substack{\psi \bmod d \\ \psi \text{ even} \\ \psi\neq \psi_0}} \psi( h)\overline{\psi}( k)\\
&\hspace{.25in}\times \frac{1}{(2\pi i)^2}\int_{( \varepsilon)} \int_{( \varepsilon)} X^{s_1+s_2} \widetilde{V}(s_1)\widetilde{V}(s_2)\prod_{\alpha\in A} L(\tfrac{1}{2}+s_1+\alpha,\psi) \prod_{\beta\in B} L(\tfrac{1}{2}+s_2+\beta,\overline{\psi}) \\
&\hspace{.5in}\times \prod_{\alpha\in A}\Bigg(\prod_{p|c}\bigg(1- \frac{\psi(p)}{p^{\frac{1}{2}+s_1+\alpha}} \bigg) \Bigg) \prod_{\beta\in B}\Bigg(\prod_{p|c}\bigg(1- \frac{\overline{\psi}(p)}{p^{\frac{1}{2}+s_2+\beta}} \bigg) \Bigg) \,ds_2 \,ds_1.
\end{split}
\end{equation}
Now observe that
$$
\prod_{p|c}\bigg|1- \frac{\psi(p)}{p^{\frac{1}{2}+z}} \bigg| \leq \prod_{p|c}(2) \ll_{\varepsilon} c^{\varepsilon}
$$
for any complex number $z$ with $|z|<1/2$. Moreover, it holds that
$$
\sum_{\substack{h,k\leq Q^{\vartheta} \\ (hk,q)=1} } \frac{\lambda_h \overline{\lambda_k} \psi( h)\overline{\psi}( k) }{\sqrt{hk}} = \Bigg|\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{\lambda_h \psi( h) }{\sqrt{h }} \Bigg|^2.
$$
We bound the $L$-functions in \eqref{eqn: Lrweaimtobound} by assuming GLH\footnote{We must assume GLH in this step because of the potentially large number of $L(s,\psi)$ factors. This differs from the argument in Conrey et~al.~\cite{CIS}, where they bound the size of the square of the $L$-function using the large sieve and the approximate functional equation (see the argument following equation (4.6) in \cite{CIS}).}. It follows from these and the triangle inequality that
\begin{equation*}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k) &\ll_{\varepsilon} X^{\varepsilon} \sum_{ 1\leq q<\infty } W\left( \frac{q}{Q}\right) \sum_{\substack{c>C, d \geq 1 \\ cd=q }} (cd)^{\varepsilon} \sum_{\substack{\psi \bmod d \\ \psi \text{ even} \\ \psi\neq \psi_0}} \Bigg|\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{\lambda_h \psi( h) }{\sqrt{h }} \Bigg|^2 \\
&\hspace{.5in}\times \int_{( \varepsilon)} \int_{( \varepsilon)} |s_1s_2|^{\varepsilon} |\widetilde{V}(s_1)||\widetilde{V}(s_2)| \,|ds_2 \,ds_1|.
\end{split}
\end{equation*}
The rapid decay of $\widetilde{V}$ implies that the latter double integral is $\ll 1$. We substitute $q=cd$ and write the $q$-sum as a double sum over $c$ and $d$. Furthermore, in preparation to use the large sieve, we express each $\psi$ in terms of the primitive character it is induced by to deduce the upper bound
\begin{equation*}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k) \ll_{\varepsilon} (XQ)^{\varepsilon} \sum_{c>C} \sum_{d=1}^{\infty} W\left( \frac{cd}{Q}\right) \sum_{u|d} \,\sideset{}{^\flat}\sum_{\substack{\psi \bmod u \\ \psi\neq \psi_0}}\Bigg|\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{\lambda_h \psi( h) }{\sqrt{h }} \Bigg|^2,
\end{split}
\end{equation*}
where we again use $\flat$ to denote that the sum is over even primitive characters. We substitute $d=ru$ and write the $d$-sum as a double sum over $r$ and $u$ to arrive at
\begin{equation*}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k) \ll_{\varepsilon} (XQ)^{\varepsilon} \sum_{c>C} \sum_{r=1}^{\infty} \sum_{u=1}^{\infty} W\left( \frac{cru}{Q}\right) \,\sideset{}{^\flat}\sum_{\substack{\psi \bmod u \\ \psi\neq \psi_0}} \Bigg|\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{\lambda_h \psi( h) }{\sqrt{h }} \Bigg|^2 .
\end{split}
\end{equation*}
Since $W$ is bounded and compactly supported, it follows that
\begin{equation*}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k) \ll_{\varepsilon} (XQ)^{\varepsilon} \sum_{C<c\ll Q} \sum_{r\ll \frac{Q}{c}} \sum_{u \ll \frac{Q}{cr}} \,\sideset{}{^\flat}\sum_{\substack{\psi \bmod u \\ \psi\neq \psi_0}} \Bigg|\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{\lambda_h \psi( h) }{\sqrt{h }} \Bigg|^2 .
\end{split}
\end{equation*}
The large sieve (see, for example, \cite[\S27, Theorem 4]{Davenport}) implies that
\begin{equation*}
\sum_{u \ll \frac{Q}{cr}} \,\sideset{}{^\flat}\sum_{\substack{\psi \bmod u \\ \psi\neq \psi_0}} \Bigg|\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{\lambda_h \psi( h) }{\sqrt{h }} \Bigg|^2 \ll \bigg(Q^{\vartheta} + \frac{Q^2}{c^2r^2} \bigg)\sum_{\substack{h \leq Q^{\vartheta} \\ (h ,q)=1} } \frac{|\lambda_h|^2 }{h}.
\end{equation*}
Hence, since $\lambda_h \ll_{\varepsilon} h^{\varepsilon}$, it follows that
\begin{equation}\label{eqn: Lrbound}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}} \mathcal{L}^r(h,k)
& \ll_{\varepsilon} (XQ)^{\varepsilon} \sum_{C<c\ll Q} \sum_{r\ll \frac{Q}{c}} \bigg(Q^{\vartheta} + \frac{Q^2}{c^2r^2} \bigg)Q^{\varepsilon} \\
& \ll_{\varepsilon} (XQ)^{\varepsilon} \sum_{C<c\ll Q} \bigg(\frac{Q^{1+\vartheta}}{c} + \frac{Q^2}{c^2} \bigg) \\
& \ll_{\varepsilon} (XQ)^{\varepsilon} \bigg(Q^{1+\vartheta+\varepsilon} + \frac{Q^2}{C} \bigg).
\end{split}
\end{equation}
As mentioned in Section~\ref{sec: outline}, we will eventually choose $C$ as a specific positive power of $Q$ to control this error term.
\subsection{Preparing \texorpdfstring{$\mathcal{L}^0(h,k)$}{L0(h,k)} for eventual cancellation}
The goal of this subsection is to put $\mathcal{L}^0(h,k)$ into a form that, as we will eventually see in Section \ref{sec: U1}, cancels with a term arising from our analysis of $\mathcal{U}(h,k)$. To this end, let us first focus on the $c,d$-sum in the definition \eqref{eqn: L0} of $\mathcal{L}^0(h,k)$. We complete the $c$-sum by writing
\[
\sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c) = \sum_{c|q}\mu(c) - \sum_{\substack{c\le C, d \geq 1 \\ cd=q }} \mu(c) = \left\lfloor \frac{1}{q} \right\rfloor - \sum_{\substack{c\le C, d \geq 1 \\ cd=q }} \mu(c).
\]
The latter $c,d$-sum equals $1$ if $q=1$, and so it follows that
\begin{align*}
\sum_{\substack{c>C, d \geq 1 \\ cd=q }} \mu(c)
= \begin{cases}
\displaystyle -\sum_{\substack{c\leq C, d \geq 1 \\ cd=q }} \mu(c) & \text{if } q>1\\ \\
\hphantom{---} 0 &\text{if } q=1.
\end{cases}
\end{align*}
From this and the definition \eqref{eqn: L0} of $\mathcal{L}^0(h,k)$, we arrive at
\begin{equation*}
\mathcal{L}^0(h,k) = -\sum_{\substack{1< q<\infty \\ (q,hk)=1}} W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty \\ (mn,q)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{c\leq C, d \geq 1 \\ cd=q }} \mu(c).
\end{equation*}
Without loss of generality, we may ignore the condition $q>1$ and simply sum over all $1\leq q<\infty$ because the $q=1$ term is zero for large enough $Q$, as $W$ is supported away from $0$. We substitute $q=cd$ and interchange the order of summation to deduce that
\begin{equation}\label{eqn: L0cancelformalmost}
\mathcal{L}^0(h,k) = - \sum_{\substack{1\leq c\le C\\ (c,hk)=1}} \mu(c)\sum_{\substack{1\leq m,n<\infty \\ (mn,c)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{1\le d < \infty \\ (d,mhnk)=1}}W\left( \frac{cd}{Q}\right).
\end{equation}
To evaluate the latter $d$-sum, we use Stieltjes integration and the fact that
$$
\sum_{\substack{d\le x \\ (d,m)=1}}1 = x\frac{\phi(m)}{m} + E(x,m)
$$
for some function $E(x,m)$ such that $E(x,m)=O(m^\varepsilon)$ uniformly for all $x>0$ and positive integers $m$. This results to
\begin{align*}
\sum_{\substack{1\le d < \infty\\ (d,mnhk)=1}}W\left( \frac{cd}{Q}\right) = W\left(\frac{c}{Q}\right) +\frac{\phi(mnhk)}{mnhk}\int_{1}^{\infty}W\left( \frac{cx}{Q}\right)\,dx + \int_{1}^{\infty}W\left(\frac{cx}{Q}\right)\,dE.
\end{align*}
Note that $W(c/Q)\ll 1$. Moreover, we may integrate by parts to see that the last integral is $O((mnhk)^\varepsilon)$ by the bound on $E(x,m)$ and the fact that $W$ is compactly supported. By a change of variables, we have
\[
\frac{c}{Q}\int_{1}^{\infty}W\left( \frac{cx}{Q}\right)\,dx = \int_{0}^{\infty}W(x)\,dx - \int_{0}^{c/Q}W(x)\,dx = \int_{0}^{\infty}W(x)\,dx + O\left(\frac{c}{Q} \right).
\]
Combining these estimates with \eqref{eqn: L0cancelformalmost}, we find that
\begin{equation}\label{eqn: L0cancelform}
\begin{split}
\mathcal{L}^0(h,k) &=-Q\sum_{\substack{1\le c\le C\\ (c,hk)=1}} \frac{\mu(c)}{c}\sum_{\substack{1\leq m,n<\infty \\ (mn,c)=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\frac{\phi(mnhk)}{mnhk}
\int_{0}^{\infty}W(x)\,dx\\
&\hspace{1.5in}+ O\big( (Xhk)^\varepsilon XC \big).
\end{split}
\end{equation}
In Section \ref{sec: U1}, we will show that a part of $\mathcal{U}(h,k)$ cancels with the main term above.
\section{Preparing the term \texorpdfstring{$\mathcal{U}(h,k)$}{U(h,k)} for analysis}\label{sec: U(h,k)split}
There are two goals for this section. The first is to switch to the complementary modulus by making a change of variables in the definition \eqref{eqn: Usum} of $\mathcal{U}(h,k)$ and then express the divisibility condition in terms of character sums. The second goal is to dissect the contribution of the principal characters in order to isolate the part of it containing the predicted one-swap terms.
\subsection{\texorpdfstring{$\mathcal{U}(h,k)$}{U(h,k)}: Switching to the complementary modulus}
Recall the definition \eqref{eqn: Usum} of $\mathcal{U}(h,k)$. We substitute $q=cd$ and rearrange the sum to deduce that
\begin{equation}\label{eqn: Ubeforedsum}
\mathcal{U}(h,k)= \frac{1}{2}\sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \mu(c) \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{ 1\leq d<\infty \\ ( d,mhnk)=1\\ d|mh\pm nk}}\phi(d) W\left( \frac{cd}{Q}\right).
\end{equation}
Let $g=(mh,nk)$. Then the condition that $d|mh\pm nk$ and $( d,mhnk)=1$ is equivalent to the condition that $d|\frac{mh}{g}\pm \frac{nk}{g}$ and $(d,g)=1$. From this and the fact that $\phi(d)= \sum_{ef=d} \mu(e)f$, we see that the $d$-sum in \eqref{eqn: Ubeforedsum} equals
\begin{equation*}
\sum_{\substack{ 1\leq d<\infty \\ ( d,g)=1\\ d|\frac{mh}{g}\pm \frac{nk}{g}}}\phi(d) W\left( \frac{cd}{Q}\right) = \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \mu(e)\sum_{\substack{ 1\leq f<\infty \\ ( f,g)=1\\ ef|\frac{mh}{g}\pm \frac{nk}{g}}} f W\left( \frac{cef}{Q}\right).
\end{equation*}
Use M\"{o}bius inversion to detect the condition $(f,g)=1$ and write the above as
\begin{equation*}
\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \mu(e)\sum_{\substack{ 1\leq f<\infty \\ ef|\frac{mh}{g}\pm \frac{nk}{g}}} \sum_{\substack{a|f \\ a|g}} \mu(a)f W\left( \frac{cef}{Q}\right) =\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \mu(e) \sum_{a|g} \mu(a) \sum_{\substack{ 1\leq f<\infty \\ a|f \\ ef|\frac{mh}{g}\pm \frac{nk}{g}}} f W\left( \frac{cef}{Q}\right).
\end{equation*}
Make a change of variables $f=ab$ in the $f$-sum to see that this equals
\begin{equation}\label{eqn: Udsum}
\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \mu(e) \sum_{a|g} a\mu(a) \sum_{\substack{ 1\leq b<\infty \\ eab|\frac{mh}{g}\pm \frac{nk}{g}}} b W\left( \frac{ceab}{Q}\right).
\end{equation}
Now define the ``complementary modulus'' $\ell$ by
\begin{equation*}
|mh\pm nk| =geab\ell,
\end{equation*}
and use it to make a change of variables in the $b$-sum to write \eqref{eqn: Udsum} as
\begin{equation}\label{eqn: Udsum2}
\begin{split}
\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }}& \mu(e) \sum_{a|g} a\mu(a) \sum_{\substack{ 1\leq \ell <\infty \\ ea\ell |\frac{mh}{g}\pm \frac{nk}{g}}} \frac{|mh\pm nk|}{gea\ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right)\\
&\hspace{.25in} = \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{a|g} \mu(a) \sum_{\substack{ 1\leq \ell <\infty \\ ea\ell |\frac{mh}{g}\pm \frac{nk}{g}}} \frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right).
\end{split}
\end{equation}
Since $g$ is defined by $g=(mh,nk)$, we must have that $ea\ell$ is coprime to each of $mh/g$ and $nk/g$, because if not then the condition $ea\ell |(mh\pm nk)/g$ would imply that $mh/g$ and $nk/g$ are not coprime, contradicting the definition of $g$. Thus the orthogonality of character sums implies
\begin{equation*}
\frac{1}{\phi(ea \ell)} \sum_{\psi \bmod ea\ell} \psi \left( \frac{mh}{g} \right) \overline{\psi}\left( \mp \frac{nk}{g} \right) = \left\{ \begin{array}{cl} 1 & \text{if } ea\ell |\frac{mh}{g}\pm \frac{nk}{g} \\ \\ 0 & \text{else}. \end{array} \right.
\end{equation*}
Hence, we may replace the condition $ea\ell |(mh\pm nk)/g$ in \eqref{eqn: Udsum2} with the above multiplier to conclude that the $d$-sum appearing in \eqref{eqn: Ubeforedsum} is equal to
\begin{equation*}
\begin{split}
\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }}& \frac{\mu(e)}{e} \sum_{a|g} \mu(a) \sum_{\substack{ 1\leq \ell <\infty \\ (ea\ell, \frac{mh}{g}\cdot\frac{nk}{g})=1 }} \frac{1}{\phi(ea \ell)} \sum_{\psi \bmod ea\ell} \psi \left( \frac{mh}{g} \right) \overline{\psi}\left( \mp \frac{nk}{g} \right)\\
&\hspace{.5in}\times\frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right).
\end{split}
\end{equation*}
It follows that
\begin{equation}\label{eqn: Uafterswitch}
\begin{split}
\mathcal{U}(h,k)&= \frac{1}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \mu(c) \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{a|g} \mu(a)\\
&\hspace{.25in} \times\sum_{\substack{ 1\leq \ell <\infty \\ (ea\ell, \frac{mh}{g}\cdot\frac{nk}{g})=1 }} \frac{1}{\phi(ea \ell)} \sum_{\psi \bmod ea\ell} \psi \left( \frac{mh}{g} \right) \overline{\psi}\left( \mp \frac{nk}{g} \right)\frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right).
\end{split}
\end{equation}
Write this as
\begin{equation}\label{eqn: Usplit00}
\mathcal{U}(h,k)= \mathcal{U}^0(h,k) + \mathcal{U}^r(h,k),
\end{equation}
where $\mathcal{U}^0(h,k)$ is the contribution of the principal character in the $\psi$-sum, and $\mathcal{U}^r(h,k)$ is the contribution of the non-principal characters. In other words, $\mathcal{U}^0(h,k)$ and $\mathcal{U}^r(h,k)$ are defined by
\begin{equation}\label{eqn: U0}
\begin{split}
\mathcal{U}^0(h,k)&:= \frac{1}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \mu(c) \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left(\frac{n}{X}\right)\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{a|g} \mu(a) \\
&\hspace{.25in} \times \sum_{\substack{ 1\leq \ell <\infty \\ (ea\ell, \frac{mh}{g}\cdot\frac{nk}{g})=1 }} \frac{|mh\pm nk|}{g \ell\phi(ea \ell)} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right)
\end{split}
\end{equation}
and
\begin{equation}\label{eqn: Ur}
\begin{split}
\mathcal{U}^r(h,k)&:= \frac{1}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \mu(c) \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{a|g} \mu(a) \\
&\hspace{.25in} \times \sum_{\substack{ 1\leq \ell <\infty \\ (ea\ell, \frac{mh}{g}\cdot\frac{nk}{g})=1 }} \frac{1}{\phi(ea \ell)} \sum_{\substack{\psi \bmod ea\ell\\ \psi\neq \psi_0}} \psi \left( \frac{mh}{g} \right) \overline{\psi}\left( \mp \frac{nk}{g} \right)\frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right),
\end{split}
\end{equation}
respectively, where $\psi_0$ denotes the principal character mod $ea\ell$.
\subsection{The principal contribution \texorpdfstring{$\mathcal{U}^0(h,k)$}{U0(h,k)}}
Our goal in this subsection is to separate out a part of $\mathcal{U}^0(h,k)$ that we will eventually prove contains the one-swap terms that are predicted by the recipe. We apply Mellin inversion to write
\begin{equation}\label{eqn: mellinW}
\frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right) = \frac{Q}{2\pi i c} \int_{(\varepsilon)} \ell^{-w} \Upsilon_{\pm}(w;mh,nk) \,dw,
\end{equation}
where
\begin{equation}\label{eqn: Upsilondef}
\Upsilon_{\pm}(w;mh,nk)= \Upsilon_{\pm}(w;mh,nk;c,Q):=\int_{0}^{\infty}\frac{c|mh\pm nk|}{g x Q}W\left(\frac{c|mh\pm nk|}{g x Q} \right)x^{w-1}\,dx.
\end{equation}
We insert \eqref{eqn: mellinW} into \eqref{eqn: U0}, then interchange the order of summation and write the $\ell$-sum as an Euler product using the following lemma.
\begin{lemma}\label{lem: Lemma6ofCIS}\cite[Lemma 6]{CIS}
Let $s$ be a complex number with $\re(s)>0$, and let $u$ and $v$ be coprime natural numbers. Then
\[
\sum_{\substack{\ell=1 \\ (\ell, v)=1}}^{\infty}\frac{1}{\phi(u\ell) \ell^s} = \frac{1}{\phi(u)}\zeta(1+s)R(s;u,v),
\]
where
\begin{equation}\label{eqn: Rdef}
R(s;u,v) = \prod_{p|v}\left(1-\frac{1}{p^{s+1}} \right)\prod_{p\nmid uv}\left(1+\frac{1}{p^{s+1}(p-1)}\right)
\end{equation}
converges absolutely in $\re(s)>-1$.
\end{lemma}
The result is
\begin{equation}\label{eqn: U0beforesplit}
\begin{split}
\mathcal{U}^0(h,k)&= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{\substack{ a|g \\ (ea, \frac{mh}{g}\cdot\frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)}\\
&\hspace{.25in} \times \frac{1}{2\pi i}\int_{(\varepsilon)} \Upsilon_\pm(w;mh,nk) \zeta(1+w) R(w;ea, mhnk/g^2)\,dw.
\end{split}
\end{equation}
Note that $\Upsilon_\pm(w;mh,nk)$ has rapid decay as $|w|\rightarrow \infty$ by \eqref{eqn: Upsilondef} and a repeated application of integration by parts. Hence, we may move the line of integration in \eqref{eqn: U0beforesplit} to Re$(w)=-\epsilon$. Doing so leaves a residue at $w=0$ from the pole of $\zeta(1+w)$, and we arrive at
\begin{equation}\label{eqn: U0split}
\mathcal{U}^0(h,k) = \mathcal{U}^1(h,k)+\mathcal{U}^2(h,k),
\end{equation}
where $\mathcal{U}^1(h,k)$ is the residue, i.e.,
\begin{equation}\label{eqn: U1}
\begin{split}
\mathcal{U}^1(h,k):=
& \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \\
& \times \sum_{\substack{ a|g \\ (ea, \frac{mh}{g}\cdot\frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)} \Upsilon_\pm(0;mh,nk) R(0;ea, mhnk/g^2),
\end{split}
\end{equation}
and $\mathcal{U}^2(h,k)$ is defined by
\begin{equation}\label{eqn: U2}
\begin{split}
& \mathcal{U}^2(h,k):= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \\
& \times \sum_{\substack{ a|g \\ (ea, \frac{mh}{g}\cdot\frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)} \cdot \frac{1}{2\pi i}\int_{(-\epsilon)} \Upsilon_\pm(w;mh,nk) \zeta(1+w) R(w;ea, mhnk/g^2)\,dw.
\end{split}
\end{equation}
\subsection{The term \texorpdfstring{$\mathcal{U}^1(h,k)$}{U1(h,k)} approximately cancels with \texorpdfstring{$\mathcal{L}^0(h,k)$}{L0(h,k)}}\label{sec: U1}
In this subsection, we show that the term $\mathcal{U}^1(h,k)$ defined by \eqref{eqn: U1} cancels with the main contribution of $\mathcal{L}^0(h,k)$, which we have evaluated in \eqref{eqn: L0cancelform}. We first focus on the $e,a$-sum in $\eqref{eqn: U1}$. To express it as an Euler product, we observe that Lemma~\ref{lem: sumstoEulerproducts} and the definition \eqref{eqn: Rdef} of $R$ implies for $\re(w)>-1$ that
\begin{equation}\label{eqn: easum}
\begin{split}
&\sum_{\substack{1\leq e<\infty \\ (e, g)=1}}\frac{\mu(e)}{e} \sum_{\substack{a|g \\ (ea,\frac{mh}{g}\cdot \frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)}R(w;ae,mnhk/g^2)\\
&\hspace{1in}=\prod_{p|mnhk/g^2}\left(1-\frac{1}{p^{1+w}} \right)\prod_{\substack{p|g \\ p\nmid mnhk/g^2}}\left(1+\frac{1}{p^{w+1}(p-1)} -\frac{1}{p-1} \right)\\
&\hspace{1.5in}\times \prod_{\substack{p\nmid g \\ p\nmid mnhk/g^2}}\left(1+\frac{p^{-w}-1}{p(p-1)}\right)
\end{split}
\end{equation}
(this is the same as (7.7) of \cite{CIS}). It follows from this with $w=0$ that
\begin{equation}\label{eqn: easumw0}
\sum_{\substack{1\leq e<\infty \\ (e, g)=1}}\frac{\mu(e)}{e} \sum_{\substack{a|g \\ (ea,\frac{mh}{g}\cdot \frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)}R(0;ae,mnhk/g^2) = \frac{\phi(mnhk)}{mnhk}.
\end{equation}
Now the definition \eqref{eqn: Upsilondef} of $\Upsilon_{\pm}$ and a change of variables gives
\begin{equation*}
\Upsilon_{+}(0;mh,nk) + \Upsilon_{-}(0;mh,nk) = 2\int_0^{\infty} W(u)\,du.
\end{equation*}
From this, \eqref{eqn: U1}, and \eqref{eqn: easumw0}, we deduce that
\begin{equation}\label{eqn: U1simplify}
\mathcal{U}^1(h,k)=
Q \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \frac{\phi(mnhk)}{mnhk} \int_0^{\infty} W(u)\,du.
\end{equation}
In order to show that $\mathcal{U}^1(h,k)$ cancels with the main term of $\mathcal{L}^0(h,k)$ given in \eqref{eqn: L0cancelform}, we must complete the sum above to include the terms $mh=nk$. In order to do this successfully, we must show that the total contribution of the terms with $mh=nk$ is small. By \eqref{eqn: divisorbound} and our assumption that $V$ and $W$ have compact support, the sum of the terms with $mh=nk$ is at most
\begin{equation}\label{eqn: U1boundstep1}
\ll Q\sum_{1\leq c \leq C} \frac{1}{c} \sum_{\substack{1\leq m,n \ll X \\ mh=nk}} \frac{(mn)^\varepsilon }{\sqrt{mn}}.
\end{equation}
Observe that $mh=nk$ if and only if there is an integer $\ell$ such that $m=\ell K$ and $n=\ell H$, where, as before, $H$ and $K$ are defined by $H:=h/(h,k)$ and $K:=k/(h,k)$. Thus \eqref{eqn: U1boundstep1} is
\begin{align*}
\ll (HK)^{-1/2+\varepsilon}Q(\log C) \sum_{1\le \ell \ll X}\frac{1}{\ell^{1-\varepsilon}}
\ll X^{\varepsilon} (HK)^{-1/2+\varepsilon}Q\log C.
\end{align*}
Hence, including the $mh=nk$ terms in \eqref{eqn: U1simplify} gives
\begin{equation}\label{eqn: U1ready}
\begin{split}
\mathcal{U}^1(h,k)&=
Q \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \frac{\phi(mnhk)}{mnhk} \int_0^{\infty} W(u)\,du\\
&\hspace{.5in}+O\bigg(Q\frac{(XCHK)^{\varepsilon}}{\sqrt{HK}} \bigg).
\end{split}
\end{equation}
The main term here cancels with the main term from our analysis of $\mathcal{L}^0(h,k)$, given in \eqref{eqn: L0cancelform}. More precisely, it follows from \eqref{eqn: L0cancelform} and \eqref{eqn: U1ready} that
\begin{equation}\label{eqn: U1ready2}
\mathcal{U}^1(h,k) = -\mathcal{L}^0(h,k) + O \big( (Xhk)^\varepsilon XC \big) + O\bigg(Q\frac{(XCHK)^{\varepsilon}}{\sqrt{HK}} \bigg).
\end{equation}
Summarizing this section, we deduce from \eqref{eqn: Usplit00}, \eqref{eqn: U0split}, and \eqref{eqn: U1ready2} that
\begin{equation}\label{eqn: Uready}
\mathcal{U}(h,k) = -\mathcal{L}^0(h,k)+\mathcal{U}^2(h,k)+\mathcal{U}^r(h,k)+ O \big( (Xhk)^\varepsilon XC \big) + O\bigg(Q\frac{(XCHK)^{\varepsilon}}{\sqrt{HK}} \bigg).
\end{equation}
Looking forward, we show in Section~\ref{sec: U2} that $\mathcal{U}^2(h,k)$ is, up to an admissible error term, equal to the one-swap terms $\mathcal{I}_1(h,k)$ predicted by the recipe. In Section~\ref{sec: Ur}, we bound the average of $(hk)^{-1/2}\mathcal{U}^r(h,k)$ over $h,k$ and show that $\mathcal{U}^r(h,k)$ is an acceptable error term.
\section{The term \texorpdfstring{$\mathcal{U}^2(h,k)$}{U2(h,k)}: extracting the one-swap terms}\label{sec: U2}
Recall that $\mathcal{U}^2(h,k)$, defined by \eqref{eqn: U2}, does not include the diagonal terms $mh=nk$. As in the analysis of $\mathcal{U}^1(h,k)$, we will find it advantageous to add these terms back in, and so we must show that the total contribution of these terms is acceptably small. The analysis that follows is similar to that of $\mathcal{U}^1(h,k)$ in Subsection~\ref{sec: U1}. However, the treatment of $\Upsilon_{\pm}(w;mk,nk)$ is more delicate because the variables $m$ and $n$ are entangled in the factor $|mh\pm nk|$. To ameliorate this challenge, we first introduce a bit of averaging as in Section~7 of \cite{CIS}. This averaging will lead to expressions with absolutely convergent integrals after separating the variables $m$ and $n$ in $\Upsilon_{\pm}(w;mk,nk)$ (Proposition~\ref{prop: CISProp2} below). The absolute convergence, in turn, will allow us to interchange the order of summation in our analysis of $\mathcal{U}^2(h,k)$ and extract the predicted one-swap terms in the subsections that follow.
To begin, we state and prove the averaging result that we will apply as just described.
\begin{lemma}\label{lem: smoothing}
Let $f:[0,\infty)\rightarrow \mathbb{C}$ be a continuously differentiable function of compact support such that $f$ is zero in a neighborhood of zero. Let $x,y,v\in \mathbb{R}$, with $v>0$. Then the function
$$
t\longmapsto f(v|x-t y|)
$$
is continuously differentiable on $\mathbb{R}$. Moreover, if $\,0<\delta<1$, then
\begin{equation*}
f(v|x-y|) = \frac{1}{2\delta}\int_{-\delta}^{\delta} f(v| x-e^{\xi} y|)\,d\xi +O(|vy|\delta),
\end{equation*}
where the implied constant depends only on $f$.
\end{lemma}
\begin{proof}
That the function $t\longmapsto f(v|x-t y|)$ is continuously differentiable on $\mathbb{R}$ follows by the chain rule and the assumption that $f$ is zero in a neighborhood of zero. Moreover, $f'(x)=O(1)$ uniformly on $\mathbb{R}$ because $f$ has compact support, and so
$$
\frac{d}{dt} f(v|x-t y|)=\pm vy f'(v|x-t y|) \ll |vy|.
$$
It follows from this and the fundamental theorem of calculus that, for $0<\delta<1$,
\begin{align*}
\int_{-\delta}^{\delta} f(v| x-e^{\xi} y|)\,d\xi -\int_{-\delta}^{\delta} f(v|x-y|)\,d\xi
& = \int_{-\delta}^{\delta} \int_1^{e^{\xi}} \frac{d}{dt} f(v|x-t y|)\,dt\,d\xi \\
& \ll |vy| \int_{-\delta}^{\delta} |\xi|\,d\xi
\ll |vy|\delta^2.
\end{align*}
Rearranging the terms gives the lemma.
\end{proof}
Before we apply Lemma~\ref{lem: smoothing} to the sum $\mathcal{U}^2(h,k)$ defined by \eqref{eqn: U2}, we first truncate the $w$-integral in \eqref{eqn: U2}. Doing so will enable us to easily deal with the error term arising from the application of Lemma~\ref{lem: smoothing}. To this end, observe that if $\xi\in \mathbb{R}$, then a change of variables implies
\begin{equation}\label{eqn: Upsilonchangeofvar}
\int_{0}^{\infty } \frac{c |mh\pm e^{\xi}nk|}{gx Q} W\left( \frac{c |mh\pm e^{\xi}nk|}{gx Q}\right) x^{w-1} \,dx = \left( \frac{c |mh\pm e^{\xi}nk|}{g Q}\right)^{w} \widetilde{W}(1-w).
\end{equation}
If $w,c,m,h,n,k$ are as in \eqref{eqn: U2}, then $|mh\pm nk|\geq 1$ since $mh\neq nk$, and so the definition \eqref{eqn: Upsilondef} of $\Upsilon_{\pm}(w;mh,nk)$, \eqref{eqn: Upsilonchangeofvar} with $\xi=0$, and \eqref{eqn: mellinrapiddecay} imply that
\begin{equation}\label{eqn: Upsilonrapiddecay}
\Upsilon_{\pm}(w;mh,nk) \ll_{\nu} \frac{(gQ)^{\varepsilon}}{|w|^{\nu}}
\end{equation}
for any positive integer $\nu$. Now the definition \eqref{eqn: Rdef} of $R$ implies that if $\re(w)=-\varepsilon$, then
\begin{equation}\label{eqn: Rbound}
R(w;ea, mhnk/g^2) \ll (mhnk)^{\varepsilon}.
\end{equation}
From this and \eqref{eqn: Upsilonrapiddecay}, we see that the part of the integral in \eqref{eqn: U2} that has $|\text{Im}(w)|\geq (XQ)^{\varepsilon}$ is negligible. Thus, using also \eqref{eqn: divisorbound}, the definition $g=(mh,nk)$, and the assumption that $V$ has compact support, we deduce that
\begin{equation}\label{eqn: U2truncated}
\begin{split}
& \mathcal{U}^2(h,k)= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right)\sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \\
& \times \sum_{\substack{ a|g \\ (ea, \frac{mh}{g}\cdot\frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)} \cdot \frac{1}{2\pi i}\int_{-\epsilon -i(XQ)^{\varepsilon}}^{-\epsilon +i(XQ)^{\varepsilon}} \Upsilon_\pm(w;mh,nk) \zeta(1+w) R(w;ea, mhnk/g^2)\,dw \\
& \hspace{1.5in} + O\big( (Chk)^{\varepsilon} Q^{-99}\big).
\end{split}
\end{equation}
Having truncated the integral in \eqref{eqn: U2}, we now apply Lemma~\ref{lem: smoothing}. Recall that the support of $W$ is a compact subset of $(0,\infty)$. Use Lemma~\ref{lem: smoothing} with $f(u)=uW(u)$ and $\delta$ defined by \eqref{eqn: deltadef} to deduce that the integrand in \eqref{eqn: Upsilondef} satisfies
\begin{align*}
\frac{c |mh\pm nk|}{gx Q} W\left( \frac{c |mh\pm nk|}{gx Q}\right) = \frac{1}{2\delta}\int_{-\delta}^{\delta} \frac{c |mh\pm e^{\xi}nk|}{gx Q} W\left( \frac{c |mh\pm e^{\xi}nk|}{gx Q}\right)\,d\xi +O\left(\frac{c nk \delta}{gx Q}\right).
\end{align*}
We insert this into the definition \eqref{eqn: Upsilondef} of $\Upsilon_{\pm}(w;mh,nk)$. The contribution of the error term is
\begin{align*}
\ll \frac{c nk \delta}{g Q} \left( \frac{c |mh\pm nk|}{g Q}\right)^{-\varepsilon-1} \ll X k \delta (g Q)^{\varepsilon}
\end{align*}
for $w,c,m,h,n,k$ satisfying the conditions in \eqref{eqn: U2truncated}, because $|mh\pm nk|\geq 1$, $c\geq 1$, $n\ll X$, and, by the support of $W$, the integrand in \eqref{eqn: Upsilondef} is zero unless $x\asymp c|mh\pm nk| /(gQ)$. We arrive at
\begin{equation*}
\Upsilon_{\pm}(w;mh,nk) = \frac{1}{2\delta}\int_0^{\infty}\int_{-\delta}^{\delta} \frac{c |mh\pm e^{\xi}nk|}{gx Q} W\left( \frac{c |mh\pm e^{\xi}nk|}{gx Q}\right)x^{w-1}\,d\xi\,dx +O\big( X k \delta (g Q)^{\varepsilon}\big).
\end{equation*}
This and \eqref{eqn: Upsilonchangeofvar} imply
\begin{equation*}
\Upsilon_{\pm}(w;mh,nk) = \frac{1}{2\delta} \int_{-\delta}^{\delta} \left( \frac{c |mh\pm e^{\xi}nk|}{g Q}\right)^{w} \widetilde{W}(1-w)\,d\xi +O\big( X k \delta (g Q)^{\varepsilon}\big).
\end{equation*}
We insert this into \eqref{eqn: U2truncated} to deduce that
\begin{equation}\label{eqn: U2before2}
\begin{split}
\mathcal{U}^2(h,k) &= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{\substack{ a|g \\ (ea, \frac{mh}{g}\cdot\frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)} \\
&\hspace{.25in} \times \frac{1}{2\pi i}\int_{-\epsilon -i(XQ)^{\varepsilon}}^{-\epsilon +i(XQ)^{\varepsilon}} \zeta(1+w) R(w;ea, mhnk/g^2) \left( \frac{c }{g Q}\right)^{w} \widetilde{W}(1-w) \\
&\hspace{.5in}\times \frac{1}{2\delta}\int_{-\delta}^{\delta} |mh\pm e^{\xi}nk|^w \,d\xi\,dw + O\big((XChk)^{\varepsilon} k X^2 Q^{-97}\big),
\end{split}
\end{equation}
where, to bound the error term, we have used \eqref{eqn: divisorbound}, \eqref{eqn: Rbound}, the definition $g=(mh,nk)$, the definition \eqref{eqn: deltadef} of $\delta$, and the assumption that $V$ has compact support.
The following proposition, which is Proposition 2 in \cite{CIS}, enables us to separate the variables $m$ and $n$ in the expression $|mh\pm e^{\xi} nk|$ and thus write the $m,n,e,a$-sum in \eqref{eqn: U2before2} in terms of an Euler product.
\begin{prop}[Proposition 2 of \cite{CIS}]\label{prop: CISProp2}
Let $\omega$ be a complex number with $\re(\omega)>0$. Then for any $0<c<\re(\omega)$, and $r>0$ with $r\ne 1$, we have
\[
|1+r|^{-\omega}+|1-r|^{-\omega}=\frac{1}{2\pi i}\int_{(c)}\mathcal{H}(z,\omega)r^{-z}\,dz.
\]
Therefore, for any $\delta>0$,
\begin{equation}\label{eqn: Prop2Integral}
\frac{1}{2\delta}\int_{-\delta}^{\delta}|1+ e^{\xi}r|^{-\omega} + |1- e^{\xi}r|^{-\omega}\,d\xi = \frac{1}{2\pi i}\int_{(c)}\mathcal{H}(z,\omega)r^{-z}\frac{e^{\delta z}-e^{-\delta z}}{2\delta z}\, dz,
\end{equation}
where $\mathcal{H}(z,\omega)$ is defined by \eqref{eqn: Hdef}. The $z$-integral in \eqref{eqn: Prop2Integral} converges absolutely for $\re(\omega)<1$.
\end{prop}
We apply Proposition~\ref{prop: CISProp2} with $\omega=-w$, $\re(w)=-\epsilon$, $c=\epsilon/2$, and $r=nk/(mh)$, which is $\neq 1$ in \eqref{eqn: U2before2}, to deduce that
$$
\frac{1}{2\delta}\int_{-\delta}^{\delta} \left|1+ e^{\xi} \frac{nk}{mh}\right|^{w} + \left|1- e^{\xi} \frac{nk}{mh}\right|^{w}\,d\xi =\frac{1}{2\pi i} \int_{(\epsilon/2)} \mathcal{H}(z,-w) \left(\frac{nk}{mh}\right)^{-z} \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\,dz.
$$
Multiply both sides by $(mh)^w$ to find that
$$
\frac{1}{2\delta}\int_{-\delta}^{\delta} |mh + e^{\xi} nk|^{w} + |mh - e^{\xi} nk|^{w} \,d\xi =\frac{1}{2\pi i} \int_{(\epsilon/2)} \mathcal{H}(z,-w) (mh)^{w+z}(nk)^{-z} \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\,dz.
$$
We insert this into \eqref{eqn: U2before2} and arrive at
\begin{equation}\label{eqn: U2before}
\begin{split}
\mathcal{U}^2(h,k)&= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }} \frac{\mu(e)}{e} \sum_{\substack{ a|g \\ (ea, \frac{mh}{g}\cdot\frac{nk}{g})=1}} \frac{\mu(a)}{\phi(ea)}\\
&\hspace{.15in}\times \frac{1}{2\pi i}\int_{-\epsilon -i(XQ)^{\varepsilon}}^{ -\epsilon +i(XQ)^{\varepsilon} } \zeta(1+w)R(w;ea, mhnk/g^2) \left( \frac{c}{gQ}\right)^w \widetilde{W}(1-w)\\
&\hspace{.25in}\times\frac{1}{2\pi i} \int_{(\epsilon/2)} \mathcal{H}(z,-w) (mh)^{w+z}(nk)^{-z} \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\,dz \,dw + O\big((XChk)^{\varepsilon} k X^2 Q^{-97}\big).
\end{split}
\end{equation}
By \eqref{eqn: divisorbound}, \eqref{eqn: Hbound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Rbound}, and the assumption that $V$ has compact support, we may extend the $w$-integral in \eqref{eqn: U2before} to infinity by introducing a negligible error. We then insert \eqref{eqn: easum} to deduce that
\begin{align}
\mathcal{U}^2(h,k) & = \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }}\frac{\mu(c)}{c} \sum_{\substack{1\le m,n<\infty\\ (mn,c)=1\\mh\neq nk}}\frac{\tau_A(m)\tau_B(n)}{\sqrt{mn}}V\left(\frac{m}{X}\right)V\left(\frac{n}{X}\right)\frac{1}{2\pi i}\int_{(-\epsilon)}\zeta(1+w) \notag\\
&\hspace{.25in}\times\widetilde{W}(1-w)\left(\frac{c}{gQ}\right)^w \frac{1}{2\pi i}\int_{(\epsilon/2)}\mathcal{H}(z,-w)(mh)^{w+z}(nk)^{-z}\frac{e^{\delta z}-e^{-\delta z}}{2\delta z} \notag\\
&\hspace{.5in}\times \prod_{p|mnhk/g^2}\left(1-\frac{1}{p^{1+w}}\right) \prod_{\substack{p|g\\p\nmid mnhk/g^2}}\left(1+\frac{1}{p^{w+1}(p-1)}-\frac{1}{p-1} \right) \notag\\
&\hspace{.75in}\times \prod_{\substack{p\nmid g\\p\nmid mnhk/g^2}}\left(1+\frac{p^{-w}-1}{p(p-1)}\right)\, dz\, dw + O\big((XChk)^{\varepsilon} k X^2 Q^{-97}\big). \label{eqn: U2before3}
\end{align}
We next add the $mh=nk$ terms to complete the $m,n$-sum. Let us first show that their total, which is the above main term expression with the condition $mh\ne nk$ replaced with $mh=nk$, is acceptably small. As we have seen in the discussion below \eqref{eqn: U1boundstep1}, $mh=nk$ if and only if $m=\ell K$ and $n=\ell H$ for some integer $\ell$. For such an $\ell$, the condition $(mn,c)=1$ is equivalent to $(\ell,c)=1$ because $(c,hk)=1$. Moreover, if $mh=nk$, then the definition $g=(mh,nk)$ implies $g=mh=nk$. Thus the total contribution of the $mh=nk$ terms is
\begin{align}
\frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} & \sum_{\substack{1\le \ell <\infty\\ (\ell,c)=1}}\frac{\tau_A(\ell K)\tau_B(\ell H)}{\ell\sqrt{HK}}V\left(\frac{\ell K}{X}\right)V\left(\frac{\ell H}{X}\right) \frac{1}{2\pi i}\int_{(-\epsilon)}\zeta(1+w) \notag\\
&\times \widetilde{W}(1-w)\left(\frac{c}{Q}\right)^w\frac{1}{2\pi i}\int_{(\epsilon/2)}\mathcal{H}(z,-w)\frac{e^{\delta z}-e^{-\delta z}}{2\delta z} \notag\\
&\hspace{.5in}\times\prod_{p|\ell hk}\left(1+\frac{1}{p^{w+1}(p-1)}-\frac{1}{p-1} \right)\prod_{p\nmid \ell hk}\left(1+\frac{p^{-w}-1}{p(p-1)}\right)\, dz\, dw. \label{eqn: U2diagonalHK}
\end{align}
We may restrict the $\ell$ sum to $1\le \ell \ll X$ because $V$ is compactly supported. The product over $p|\ell hk$ is bounded by $(hk\ell)^\varepsilon$, and the infinite product over $p\nmid \ell hk$ is absolutely convergent since $\re(w)=-\epsilon$. Thus \eqref{eqn: U2diagonalHK} is
\begin{equation}\label{eqn: U2diagonalHK2}
\begin{split}
&\ll Q^{1+\varepsilon}\sum_{1\leq c \leq C}\frac{1}{c^{1+\varepsilon}}\sum_{1\le \ell \ll X}\frac{(hk\ell)^{\varepsilon}}{\ell\sqrt{HK}}\\
& \hspace{.5in} \times \int_{(-\epsilon)}\int_{(\epsilon/2)}\left|\zeta(1+w)\widetilde{W}(1-w)\mathcal{H}(z,-w)\frac{e^{\delta z}-e^{-\delta z}}{2\delta z}\right|\,|dz|\,|dw|
\end{split}
\end{equation}
To bound the latter $w,z$-integral, observe that if $\re(w)=-\epsilon$ and $\re(z)=\epsilon/2$, then \eqref{eqn: mellinrapiddecay} and \eqref{eqn: Hbound} imply that $\widetilde{W}(1-w)\mathcal{H}(z,-w)$ is $O(|w|^{-99}|z|^{\varepsilon-1})$ for $|w-z|\geq |z|/2$, and is $O(|w|^{-99}|z|^{-99})$ for $|w-z|\leq |z|/2$ since $|w|\asymp |z|$ and $|w-z|\geq \epsilon/2$ in this case. Hence
\begin{equation}\label{eqn: zwbound}
\begin{split}
\int_{(-\epsilon)} & \int_{(\epsilon/2)}\left|\zeta(1+w)\widetilde{W}(1-w)\mathcal{H}(z,-w)\frac{e^{\delta z}-e^{-\delta z}}{2\delta z}\right|\,|dz|\,|dw| \\
& \ll \int_{(-\epsilon)}\int_{(\epsilon/2)} |w|^{-98}|z|^{\varepsilon-1}\min\left\{ 1, \frac{1}{\delta|z|}\right\} \,|dz|\,|dw| \ll \left( \frac{1}{\delta}\right)^{\varepsilon}.
\end{split}
\end{equation}
From this, \eqref{eqn: U2diagonalHK2}, and the definition \eqref{eqn: deltadef} of $\delta$, we deduce that the total contribution of the $mh=nk$ terms is
\begin{equation}\label{eqn: U2diagonalbound}
\ll Q^{1+\varepsilon}\sum_{1\leq c \leq C}\frac{1}{c^{1+\varepsilon}}\sum_{1\le \ell \ll X}\frac{(hk\ell)^{\varepsilon}}{\ell\sqrt{HK}} \ll X^{\varepsilon} Q^{1+\varepsilon}\frac{(hk)^\varepsilon(h,k)}{(hk)^{1/2}}.
\end{equation}
We now complete the $m,n$-sum in \eqref{eqn: U2before3} by including the $mh=nk$ terms. As we have just shown, this introduces an error of size \eqref{eqn: U2diagonalbound}. Then, we apply Mellin inversion to $V(m/X)$ and $V(n/X)$ and arrive at
\begin{equation}\label{eqn: U2before4}
\begin{split}
\mathcal{U}^2(h,k)&= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \cdot \frac{1}{(2\pi i)^2}\int_{(2)}\int_{(2)} X^{s_1+s_2}
\sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1}} \frac{\tau_A(m) \tau_B(n) }{m^{\frac{1}{2}+s_1}n^{\frac{1}{2}+s_2}} \widetilde{V}(s_1)\widetilde{V}(s_2)\\
&\hspace{.25in}\times \frac{1}{2\pi i}\int_{(-\epsilon)} \zeta(1+w) \widetilde{W}(1-w) \left( \frac{c}{gQ}\right)^w \frac{1}{2\pi i} \int_{(\epsilon/2)} \mathcal{H}(z,-w) (mh)^{w+z}(nk)^{-z} \\
&\hspace{.5in}\times \frac{e^{\delta z} - e^{-\delta z}}{2\delta z} \prod_{p|mnhk/g^2}\left(1-\frac{1}{p^{1+w}}\right) \prod_{\substack{p|g\\p\nmid mnhk/g^2}}\left(1+\frac{1}{p^{w+1}(p-1)}-\frac{1}{p-1} \right) \\
&\hspace{.75in}\times \prod_{\substack{p\nmid g\\p\nmid mnhk/g^2}}\left(1+\frac{p^{-w}-1}{p(p-1)}\right) \,dz \,dw \,ds_2 \,ds_1 \\
& \hspace{1in} + O\left(X^{\varepsilon}Q^{1+\varepsilon}\frac{(hk)^\varepsilon(h,k)}{(hk)^{1/2}} + (XChk)^{\varepsilon} k X^2 Q^{-97}\right).
\end{split}
\end{equation}
We have chosen the $s_1$- and $s_2$-lines to be at $\re(s_1)=\re(s_2)=2$ to ensure the absolute convergence of the $m,n$-sum.
Our next task is to express the $m,n$-sum in \eqref{eqn: U2before4} as an Euler product. This sum is
\begin{equation}\label{eqn: U2mnea}
\begin{split}
& \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1 }} \frac{\tau_A(m) \tau_B(n) }{ m^{\frac{1}{2}+s_1}n^{\frac{1}{2}+s_2}} g^{-w} m^{w+z}n^{-z} \prod_{p| mnhk/g^2} \left( 1-\frac{1}{p^{1+w}}\right)\\
&\hspace{.5in}\times \prod_{\substack{ p|g \\ p\nmid mnhk/g^2 }} \left( 1+\frac{1}{p^{1+w}(p-1)}-\frac{1}{p-1}\right) \prod_{\substack{ p\nmid g \\ p\nmid mnhk/g^2} } \left(1+ \frac{p^{-w}-1}{p(p-1)} \right) \\
& = \sum_{ 1\leq m,n<\infty} \prod_p f(m,n,p),
\end{split}
\end{equation}
where $f(m,n,p)$ is defined by
\begin{equation*}
f(m,n,p) := F_1(m,n,p)F_2(m,n,p)F_3(m,n,p)
\end{equation*}
with $F_1,F_2,F_3$ defined by
\begin{equation*}
F_1(m,n,p) := \begin{cases} 1 & \text{if } p\nmid c \\ \\ 1 & \text{if } p|c \text{ and } \text{ord}_p(m)=\text{ord}_p(n)=0 \\ \\ 0 & \text{if } p|c \text{ and } \text{ord}_p(mn)>0,
\end{cases}
\end{equation*}
\begin{equation*}
F_2(m,n,p) := \frac{\tau_A(p^{\text{ord}_p(m)}) \tau_B(p^{\text{ord}_p(n)}) }{ p^{(\frac{1}{2}+s_1-w-z)\text{ord}_p(m)} p^{(\frac{1}{2}+s_2+z)\text{ord}_p(n)} p^{w \min\{\text{ord}_p(m)+\text{ord}_p(h), \text{ord}_p(n) + \text{ord}_p(k) \} }},
\end{equation*}
and
\begin{equation*}
F_3(m,n,p) := \begin{cases} \displaystyle 1-\frac{1}{p^{1+w}} & \text{if } p|\frac{mnhk}{g^2} \\ \\ \displaystyle 1+\frac{1}{p^{1+w}(p-1)}-\frac{1}{p-1} & \text{if } p|g \text{ and } p\nmid \frac{mnhk}{g^2} \\ \\ \displaystyle 1+ \frac{p^{-w}-1}{p(p-1)} & \text{if } p\nmid \frac{mnhk}{g} ,\end{cases}
\end{equation*}
respectively. We can rewrite the conditions in $F_3(m,n,p)$ in terms of $\text{ord}_p(m)$, $\text{ord}_p(n)$, $\text{ord}_p(h)$, and $\text{ord}_p(k)$, as follows. Since $g=(mh,nk)$, a prime $p$ divides $mnhk/g^2$ if and only if
$$
\text{ord}_p(m)+\text{ord}_p(h)+\text{ord}_p(n)+\text{ord}_p(k)-2\min\{\text{ord}_p(m)+\text{ord}_p(h), \text{ord}_p(n)+\text{ord}_p(k)\}>0.
$$
Since two real numbers $x,y$ satisfy $x+y-2\min\{x,y\}>0$ if and only if $x\neq y$, it follows that $p| mnhk/g^2$ if and only if $\text{ord}_p(m)+\text{ord}_p(h)\neq \text{ord}_p(n)+\text{ord}_p(k)$. A similar argument shows that a prime $p$ satisfies $p\nmid mnhk/g$ if and only if $p\nmid mhnk$. Thus the definition of $F_3(m,n,p)$ is equivalent to
\begin{equation*}
F_3(m,n,p) = \begin{cases} \displaystyle 1-\frac{1}{p^{1+w}} & \text{if } \text{ord}_p(m)+\text{ord}_p(h)\neq \text{ord}_p(n)+\text{ord}_p(k) \\ \\ \displaystyle 1+\frac{1}{p^{1+w}(p-1)}-\frac{1}{p-1} & \text{if } \text{ord}_p(m)+\text{ord}_p(h)= \text{ord}_p(n)+\text{ord}_p(k) >0 \\ \\ \displaystyle 1+ \frac{p^{-w}-1}{p(p-1)} & \text{if } \text{ord}_p(m)+\text{ord}_p(h)= \text{ord}_p(n)+\text{ord}_p(k) =0 .\end{cases}
\end{equation*}
If $p|c$, then $F_1(m,n,p)=0$ unless $m=n=1$, in which case
\begin{equation*}
f(1,1,p) = F_1(1,1,p)F_2(1,1,p)F_3(1,1,p) = 1+ \frac{p^{-w}-1}{p(p-1)}
\end{equation*}
because $(c,hk)=1$. Thus, from \eqref{eqn: U2mnea} and Lemma~\ref{lem: sumstoEulerproducts}, we deduce that the $m,n$-sum in \eqref{eqn: U2before4} equals
\begin{align*}
& \prod_p \sum_{0\leq m,n<\infty} f(p^m,p^n,p) \\
& = \prod_{p|c} \left( 1+ \frac{p^{-w}-1}{p(p-1)} \right) \prod_{p\nmid c} \sum_{0\leq m,n<\infty} F_2(m,n,p) F_3(m,n,p) \\
& = \prod_{p|c} \left( 1+ \frac{p^{-w}-1}{p(p-1)} \right)\\
& \hspace{.25in}\times \prod_{\substack{p\nmid c \\ p|hk}} \Bigg( \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{1}{p^{1+w}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(\frac{1}{2}+s_1-w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{w \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \\
& \hspace{.75in} + \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)\neq n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{1}{p^{1+w}} \right)}{ p^{m(\frac{1}{2}+s_1-w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{w \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \Bigg) \\
& \hspace{.25in} \times \prod_{\substack{p\nmid c \\ p \nmid hk}} \Bigg( 1+ \frac{p^{-w}-1}{p(p-1)} + \sum_{m=1 }^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{1}{p^{1+w}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1+s_1+s_2)} } \\
& \hspace{.75in} + \sum_{\substack{0\leq m,n<\infty \\ m\neq n } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{1}{p^{1+w}} \right)}{ p^{m(\frac{1}{2}+s_1-w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{w \min\{m, n \} }} \Bigg).
\end{align*}
We substitute this for the $m,n$-sum in \eqref{eqn: U2before4}. For convenience, we also make a change of variables $w\mapsto 1-w$. The result is
\begin{align}
\mathcal{U}^2(h,k)&= \frac{Q}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{c} \cdot \frac{1}{(2\pi i)^4 } \int_{(2)} \int_{(2)} X^{s_1+s_2} \widetilde{V}(s_1)\widetilde{V}(s_2) \int_{(1+\epsilon)} \zeta(2-w) \widetilde{W}(w) \left( \frac{c}{ Q}\right)^{1-w} \notag\\
&\hspace{.25in}\times \int_{(\epsilon/2)} \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}h^{1-w+z}k^{-z} \prod_{p|c} \left( 1+ \frac{p^{w-1}-1}{p(p-1)} \right) \notag\\
&\times \prod_{\substack{p\nmid c \\ p|hk}} \Bigg( \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \notag\\
&\hspace{.5in}+ \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)\neq n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \Bigg) \notag\\
&\times \prod_{\substack{p\nmid c \\ p \nmid hk}} \Bigg(
1+ \frac{p^{w-1}-1}{p(p-1)} + \sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1+s_1+s_2 )} } \notag\\
&\hspace{.5in}+ \sum_{\substack{0\leq m,n<\infty \\ m\neq n } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m, n \} }} \Bigg)\,dz \,dw \,ds_2\,ds_1 \notag\\
&+ O\left(X^{\varepsilon}Q^{1+\varepsilon}\frac{(hk)^\varepsilon(h,k)}{(hk)^{1/2}} + (XChk)^{\varepsilon} k X^2 Q^{-97}\right). \label{eqn: U2ready}
\end{align}
\subsection{Analysis of the predicted one-swap terms from the recipe}\label{subsec: one-swap1}
Before we continue our treatment of $\mathcal{U}^2(h,k)$, we first break down the predicted one-swap terms into several parts via the residue theorem. Afterward, we will show that $\mathcal{U}^2(h,k)$ is equal to the sum of the same parts plus admissible error terms.
Recall that the definition of $\mathcal{I}_1^*(h,k)$ is given by \eqref{eqn: Istardef} with $\ell=1$. For each term in the definition of $\mathcal{I}_1^*(h,k)$, we denote the element of $U$ by $\alpha$ and the element of $V$ by $\beta$, and we multiply the integrand by $ \zeta(w-1+\alpha+s_1+\beta+s_2) $ and divide it by the Euler product of $\zeta(w-1+\alpha+s_1+\beta+s_2)$. This ``factoring out'' of the zeta-function gives us a further analytic continuation of the integrand and allows us to evaluate its residues when shifting contours. With these notations and factorization, we thus write $\mathcal{I}_1^*(h,k)$ as
\begin{equation}\label{eqn: 1swapterms0}
\begin{split}
\mathcal{I}_1^*(h,k) = \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{2(2\pi i )^3} \int_{(\varepsilon)} \int_{(\varepsilon)} \int_{(2+\varepsilon)} X^{s_1+s_2} Q^w \widetilde{V}(s_1) \widetilde{V}(s_2)\widetilde{W}(w) \\
\times \mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \mathscr{X} (\tfrac{1}{2}+\beta+s_2 ) \prod_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\} \\ \delta \in B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} }} \zeta(1+\gamma+\delta) \\
\times \zeta(w-1+\alpha+s_1+\beta+s_2) \mathcal{K}(s_1,s_2,w) \,dw \,ds_2\,ds_1,
\end{split}
\end{equation}
where $\mathcal{K}(s_1,s_2,w)$ is defined by
\begin{align}
\mathcal{K}(s_1,s_2,w) =
& \mathcal{K}(s_1,s_2,w;A,B,\alpha,\beta,h,k) \notag\\
: = & \prod_{p|hk} \Bigg\{ \left( 1 - \frac{1}{p^{w-1+\alpha+s_1+\beta+s_2}}\right) \prod_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\} \\ \delta \in B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \notag\\
& \hspace{.5in} \times \sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}}\frac{\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} } (p^n) }{p^{m/2}p^{n/2} }\Bigg\} \notag\\
& \times \prod_{p\nmid hk} \Bigg\{ \left( 1 - \frac{1}{p^{w-1+\alpha+s_1+\beta+s_2}}\right)\prod_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\} \\ \delta \in B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right) \notag\\
& \hspace{.5in} \times \Bigg(1 + \frac{p-2}{p^{w+ \alpha+s_1 + \beta + s_2 } } + \left( 1-\frac{1}{p}\right)^2 \frac{p^{2(1-w- \alpha-s_1 - \beta - s_2 )}}{1-p^{1-w- \alpha-s_1 - \beta - s_2 }} \notag\\
& \hspace{.75in} + \sum_{m=1}^{\infty} \frac{\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\}} (p^m) }{p^m} \Bigg)\Bigg\}. \label{eqn: 1swapeulerKdef}
\end{align}
To facilitate our estimations, we first prove the following lemma, which will allow us to move lines of integration and bound the integrals that remain after applying the residue theorem.
\begin{lemma}\label{lem: 1swapeulerbound}
Suppose that $\epsilon>0$ is arbitrarily small. Let $\alpha\in A$ and $\beta\in B$, and let $h$ and $k$ be positive integers. If $s_1,s_2,w$ are complex numbers such that
\begin{enumerate}
\item[\upshape{(i)}] $\re(w-1+\alpha+s_1+\beta+s_2)\geq \frac{1}{2}+\varepsilon$, \item[\upshape{(ii)}] $-\frac{1}{2}+5\epsilon\leq \re(s_1+s_2) \leq 2\epsilon$, and
\item[\upshape{(iii)}] either $|\re(s_1)|\leq \epsilon$ or $|\re(s_2)|\leq \epsilon$,
\end{enumerate}
then the product \eqref{eqn: 1swapeulerKdef} defining $\mathcal{K}(s_1,s_2,w;A,B,\alpha,\beta,h,k)$ converges absolutely and we have
$$
\mathcal{K}(s_1,s_2,w;A,B,\alpha,\beta,h,k) \ll_{\varepsilon} (hk)^{\varepsilon}.
$$
\end{lemma}
\begin{proof}
Since $\re(w-1+\alpha+s_1+\beta+s_2)\geq \frac{1}{2}+\varepsilon$, we have
\begin{equation}\label{eqn: 1swapeulerbound1}
\frac{1}{p^{w-1+\alpha+s_1+\beta+s_2} }\ll \frac{1}{p^{\frac{1}{2}+\varepsilon}}.
\end{equation}
Moreover, each term of the form $p^{-1-\gamma-\delta}$ in the definition \eqref{eqn: 1swapeulerKdef} of $\mathcal{K}(s_1,s_2,w)$ satisfies $p^{-1-\gamma-\delta}\ll p^{-\frac{1}{2}-\varepsilon}$ because $-\frac{1}{2}+5\epsilon\leq \re(s_1+s_2) \leq 2\epsilon$ and each element of $A\cup B$ is $\ll 1/\log Q$. We may thus multiply out the product and apply the definition \eqref{eqn: taudef} of $\tau_E$ to deduce that
\begin{equation}\label{eqn: 1swapeulerbound2}
\begin{split}
& \left( 1 - \frac{1}{p^{w-1+\alpha+s_1+\beta+s_2}}\right) \prod_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\} \\ \delta \in B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} }}\left( 1- \frac{1}{p^{1+\gamma+\delta}}\right)\\
& = 1 - \frac{1}{p^{w-1+\alpha+s_1+\beta+s_2}} - \sum_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\} \\ \delta \in B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} }}\frac{1}{p^{1+\gamma+\delta}} + O\left( \frac{1}{p^{1+\varepsilon}}\right) \\
& = 1 - \frac{1}{p^{w-1+\alpha+s_1+\beta+s_2}} - \frac{\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p ) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\}} (p ) }{p } + O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{split}
\end{equation}
We may assume that $|\re(s_2)|\leq \epsilon$ as the proof for the case with $|\re(s_1)|\leq \epsilon$ is similar. Since $ -\frac{1}{2}+5\epsilon \leq \re(s_1+s_2) \leq 2\epsilon$, it then follows that $-\frac{1}{2}+4\epsilon\leq \re(s_1) \leq 3\epsilon $. This, the inequality $|\re(s_2)|\leq \epsilon$, and the bound \eqref{eqn: divisorbound} imply
\begin{equation}\label{eqn: 1swapeulerbound3}
\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) \ll_{\varepsilon} p^{m(\frac{1}{2}-4\epsilon+\varepsilon)},
\end{equation}
and
\begin{equation}\label{eqn: 1swapeulerbound4}
\tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\}} (p^n)\ll_{\varepsilon} p^{n(3\epsilon+\varepsilon)}.
\end{equation}
Therefore
\begin{equation}\label{eqn: 1swapeulerbound6}
\sum_{m=2}^{\infty} \frac{\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\}} (p^m) }{p^m} \ll \sum_{m=2}^{\infty} \frac{p^{m(\frac{1}{2}- \epsilon+\varepsilon) } }{p^m} \ll \frac{1}{p^{1+\varepsilon}}
\end{equation}
and
\begin{equation}\label{eqn: 1swapeulerbound5}
\begin{split}
\sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}}&\frac{\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} } (p^n) }{p^{m/2}p^{n/2} }\\
&\hspace{.5in}\ll \sum_{0\leq m,n<\infty} \frac{p^{m(\frac{1}{2}-4\epsilon+\varepsilon)} p^{n(3\epsilon+\varepsilon)} }{ p^{m/2}p^{n/2} }\\
&\hspace{.5in}\ll 1.
\end{split}
\end{equation}
From \eqref{eqn: 1swapeulerbound1}, \eqref{eqn: 1swapeulerbound2}, \eqref{eqn: 1swapeulerbound3} with $m=1$, \eqref{eqn: 1swapeulerbound4} with $n=1$, and \eqref{eqn: 1swapeulerbound5}, we deduce that if $p|hk$ then the local factor in \eqref{eqn: 1swapeulerKdef} corresponding to $p$ is $O(1)$. On the other hand, from \eqref{eqn: 1swapeulerbound1}, \eqref{eqn: 1swapeulerbound2}, and \eqref{eqn: 1swapeulerbound6}, we deduce that if $p\nmid hk$ then the local factor in \eqref{eqn: 1swapeulerKdef} corresponding to $p$ is $1+O(p^{-1-\varepsilon})$. It follows that the right-hand side of \eqref{eqn: 1swapeulerKdef} converges absolutely, and is $\ll (hk)^{\varepsilon}$ since $\prod_{p|\nu}O(1) \ll {\nu}^{\varepsilon}$ for any positive integer $\nu$.
\end{proof}
We now move the $w$-line in \eqref{eqn: 1swapterms0} to $\re(w)=\frac{3}{2}+\varepsilon$. This leaves a residue from the pole at $w=2-\alpha-s_1-\beta-s_2$. To bound the new integral that has $\re(w)=\frac{3}{2}+\varepsilon$, we use Lemma~\ref{lem: 1swapeulerbound}, \eqref{eqn: zetaalphaalphabound}, and \eqref{eqn: mellinrapiddecay}. Since the residue of $\zeta(s)$ at $s=1$ is $1$, we arrive at
\begin{equation}\label{eqn: 1swapterms1}
\mathcal{I}_1^*(h,k) = \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{2(2\pi i )^2} \int_{(\epsilon)} \int_{(\epsilon)} \mathcal{J} \,ds_2\,ds_1 + O\big( X^{\varepsilon} Q^{\frac{3}{2}+\varepsilon} (hk)^{\varepsilon}\big),
\end{equation}
where, for brevity, we define $\mathcal{J}$ by
\begin{equation}\label{eqn: integrandJdef}
\begin{split}
\mathcal{J} := X^{s_1+s_2} Q^{2-\alpha-s_1-\beta-s_2} \widetilde{V}(s_1) \widetilde{V}(s_2)\widetilde{W}(2-\alpha-s_1-\beta-s_2) \\
\times \mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \mathscr{X} (\tfrac{1}{2}+\beta+s_2 ) \prod_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\} \\ \delta \in B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\} }} \zeta(1+\gamma+\delta) \\
\times \mathcal{K}(s_1,s_2,2-\alpha-s_1-\beta-s_2).
\end{split}
\end{equation}
Notice that we have now specified the lines of integration in \eqref{eqn: 1swapterms1} to be $\re(s_1)=\epsilon$ and $\re(s_2)=\epsilon$, with $\epsilon$ fixed and arbitrarily small. The purpose of this is to make the succeeding estimations more explicit.
Next, we move the $s_2$-line in \eqref{eqn: 1swapterms1} to $\re(s_2)=-\frac{1}{2}+5\epsilon$. This leaves residues from the pole at $s_2=0$ due to the factor $\widetilde{V}(s_2)$, the pole at $s_2=-s_1-\alpha-\beta$ due to the factor $\zeta(1-\alpha-s_1-\beta-s_2)$, and the poles at $s_2=-s_1-\alpha'-\beta'$ due to the factors $\zeta(1+\alpha'+s_1+\beta'+s_2)$, where $\alpha'$ runs through the elements of $A\smallsetminus \{\alpha\}$ and $\beta'$ runs through the elements of $B\smallsetminus \{\beta\}$. To bound the new integral that has $\re(s_2)=-\frac{1}{2}+5\epsilon$, we use Lemma~\ref{lem: 1swapeulerbound}, \eqref{eqn: zetaalphaalphabound}, and \eqref{eqn: mellinrapiddecay}. We arrive at
\begin{equation*}
\begin{split}
\mathcal{I}_1^*(h,k) &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{4\pi i } \int_{(\epsilon)} \underset{s_2=0}{\text{Res}}\ \mathcal{J} \,ds_1 + \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{4\pi i } \int_{(\epsilon)} \underset{s_2=-s_1-\alpha-\beta}{\text{Res}}\ \mathcal{J} \,ds_1\\
&\hspace{.25in}+ \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i } \int_{(\epsilon)} \underset{s_2=-s_1-\alpha'-\beta'}{\text{Res}}\ \mathcal{J} \,ds_1\\
&\hspace{.5in}+ O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}\big) + O\big( X^{\varepsilon} Q^{\frac{3}{2}+\varepsilon} (hk)^{\varepsilon}\big).
\end{split}
\end{equation*}
For brevity, write this as
\begin{equation}\label{eqn: 1swapterms2}
\mathcal{I}_1^*(h,k) = J_1+J_2+J_3 + O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}\big) + O\big( X^{\varepsilon} Q^{\frac{3}{2}+\varepsilon} (hk)^{\varepsilon}\big).
\end{equation}
We first evaluate the contribution $J_1$ of the residue at $s_2=0$. By \eqref{eqn: mellinVresidue} and the definition \eqref{eqn: integrandJdef} of $\mathcal{J}$, we have
\begin{equation}\label{eqn: Ress20J}
\begin{split}
\underset{s_2=0}{\text{Res}}\ \mathcal{J} &= X^{s_1 } Q^{2-\alpha-s_1-\beta } \widetilde{V}(s_1) \widetilde{W}(2-\alpha-s_1-\beta )\mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \mathscr{X} (\tfrac{1}{2}+\beta )\\
&\hspace{.25in}\times\prod_{\substack{\gamma \in A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta \} \\ \delta \in B \smallsetminus \{\beta \} \cup \{-\alpha-s_1\} }} \zeta(1+\gamma+\delta)\mathcal{K}(s_1,0,2-\alpha-s_1-\beta).
\end{split}
\end{equation}
We move the line of integration in the definition
\begin{equation*}
J_1 := \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{4\pi i } \int_{(\epsilon)} \underset{s_2=0}{\text{Res}}\ \mathcal{J} \,ds_1
\end{equation*}
to $\re(s_1)=-\frac{1}{2}+5\epsilon$. We find residues from the pole at $s_1=0$ due to the factor $\widetilde{V}(s_1)$ in \eqref{eqn: Ress20J}, the pole at $s_1=-\alpha-\beta$ due to the factor $\zeta(1-\alpha-s_1-\beta)$, and the poles at $s_1=-\alpha'-\beta'$ due to the factors $\zeta(1+\alpha'+s_1+\beta')$, where $\alpha'$ runs through the elements of $A\smallsetminus \{\alpha\}$ and $\beta'$ runs through the elements of $B\smallsetminus \{\beta\}$. To bound the new integral that has $\re(s_1)=-\frac{1}{2}+5\epsilon$, we use Lemma~\ref{lem: 1swapeulerbound}, \eqref{eqn: zetaalphaalphabound}, and \eqref{eqn: mellinrapiddecay}. We deduce that
\begin{equation*}
\begin{split}
J_1 &= \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \underset{s_1=0}{\text{Res}}\ \underset{s_2=0}{\text{Res}}\ \mathcal{J} + \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \underset{s_1=-\alpha-\beta}{\text{Res}}\ \underset{s_2=0}{\text{Res}}\ \mathcal{J}\\
&\hspace{.25in}+ \frac{1}{2 }\sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \underset{s_1=-\alpha'-\beta'}{\text{Res}}\ \underset{s_2=0}{\text{Res}}\ \mathcal{J} + O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}\big).
\end{split}
\end{equation*}
For brevity, we write this as
\begin{equation}\label{eqn: J1split}
J_1 = J_{11}+J_{12}+J_{13} + O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}\big).
\end{equation}
We deduce from \eqref{eqn: mellinVresidue} and \eqref{eqn: Ress20J} that
\begin{equation}\label{eqn: J11evaluated}
\begin{split}
J_{11} &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{2} Q^{2-\alpha-\beta } \widetilde{W}(2-\alpha-\beta )\mathscr{X} (\tfrac{1}{2}+\alpha ) \mathscr{X} (\tfrac{1}{2}+\beta )\\
&\hspace{.25in}\times\prod_{\substack{\gamma \in A\smallsetminus \{\alpha\} \cup \{-\beta \} \\ \delta \in B \smallsetminus \{\beta \} \cup \{-\alpha\} }} \zeta(1+\gamma+\delta) \mathcal{K}(0,0,2-\alpha-\beta).
\end{split}
\end{equation}
Since $\mathscr{X} (\tfrac{1}{2}-\beta ) \mathscr{X} (\tfrac{1}{2}+\beta ) =1$ by the definition \eqref{eqn: scriptXdef} of $\mathscr{X}$ and the residue of $\zeta(1-\alpha-s_1-\beta)$ at $s_1=-\alpha-\beta$ is $-1$, it follows from \eqref{eqn: Ress20J} that term $J_{12}$ in \eqref{eqn: J1split} equals
\begin{equation}\label{eqn: J12evaluated}
\begin{split}
J_{12} &= - \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} X^{-\alpha-\beta } Q^{2} \widetilde{V}(-\alpha-\beta) \widetilde{W}(2) \prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta}} \zeta(1-\alpha-\beta+\hat{\alpha} +\hat{\beta} )\\
&\times\prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \mathcal{K}(-\alpha-\beta,0,2).
\end{split}
\end{equation}
Next, since the residue of $\zeta(1+\alpha'+s_1+\beta')$ at $s_1=-\alpha'-\beta'$ is $1$, it follows from \eqref{eqn: Ress20J} that the term $J_{13}$ in \eqref{eqn: J1split} equals
\begin{equation}\label{eqn: J13evaluated}
\begin{split}
& J_{13} = \frac{1}{2} \sum_{\substack{\alpha\in A \\ \beta \in B }} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} X^{-\alpha'-\beta' } Q^{2-\alpha -\beta +\alpha'+\beta' } \widetilde{V}(-\alpha'-\beta') \widetilde{W}(2-\alpha-\beta + \alpha'+\beta' )\\
& \hspace{.1in}\times \mathscr{X} (\tfrac{1}{2}+\alpha -\alpha'-\beta' ) \mathscr{X} (\tfrac{1}{2}+\beta ) \zeta(1-\alpha-\beta+\alpha'+\beta' ) \prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta(1-\alpha'-\beta'+\hat{\alpha} +\hat{\beta} )\\
&\hspace{.25in}\times \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \mathcal{K}(-\alpha'-\beta',0,2-\alpha-\beta+\alpha'+\beta').
\end{split}
\end{equation}
This, \eqref{eqn: J1split}, \eqref{eqn: J11evaluated}, and \eqref{eqn: J12evaluated} complete our evaluation of $J_1$.
Having estimated $J_1$, we next turn to the integral $J_2$ from \eqref{eqn: 1swapterms2}. Recall its definition
\begin{equation}\label{eqn: J2def}
J_2 := \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{4\pi i } \int_{(\epsilon)} \underset{s_2=-s_1-\alpha-\beta}{\text{Res}}\ \mathcal{J} \,ds_1.
\end{equation}
Since $\mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \mathscr{X} (\tfrac{1}{2}-\alpha-s_1 )=1$ by the definition \eqref{eqn: scriptXdef} of $\mathscr{X}$ and the residue of $\zeta(1-\alpha-s_1-\beta-s_2)$ at $s_2=-s_1-\alpha-\beta$ is $-1$, we see from the definition \eqref{eqn: integrandJdef} of $\mathcal{J}$ that
\begin{equation}\label{eqn: Ress2s1abJ}
\begin{split}
\underset{s_2=-s_1-\alpha-\beta}{\text{Res}}\ \mathcal{J} &= -X^{-\alpha-\beta} Q^{2} \widetilde{V}(s_1) \widetilde{V}(-s_1-\alpha-\beta)\widetilde{W}(2) \\
&\hspace{.25in}\times \prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta}} \zeta(1+\hat{\alpha} -\alpha+\hat{\beta}-\beta) \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \\
&\hspace{.25in}\times \mathcal{K}(s_1,-s_1-\alpha-\beta,2).
\end{split}
\end{equation}
We move the line of integration in \eqref{eqn: J2def} to $\re(s_1)=-\epsilon-\re(\alpha)-\re(\beta)$. This leaves residues from the poles at $s_1=0$ and $s_1=-\alpha-\beta$ due to the factors $\widetilde{V}(s_1)$ and $\widetilde{V}(-s_1-\alpha-\beta)$ in \eqref{eqn: Ress2s1abJ}, and we arrive at
\begin{equation*}
\begin{split}
J_2 = \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \underset{s_1=0}{\text{Res}}\ \underset{s_2=-s_1-\alpha-\beta}{\text{Res}}\ \mathcal{J} + \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \underset{s_1=-\alpha-\beta}{\text{Res}}\ \underset{s_2=-s_1-\alpha-\beta}{\text{Res}}\ \mathcal{J}\\
+ \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{4\pi i } \int_{(-\epsilon-\re(\alpha)-\re(\beta))} \underset{s_2=-s_1-\alpha-\beta}{\text{Res}}\ \mathcal{J} \,ds_1.
\end{split}
\end{equation*}
For brevity, we write this as
\begin{equation}\label{eqn: J2split}
J_{2} =J_{21} +J_{22} +J_{23}.
\end{equation}
Since the residue of $\widetilde{V}(s)$ at $s=0$ is $1$ by \eqref{eqn: mellinVresidue}, it follows from \eqref{eqn: Ress2s1abJ} that
\begin{equation}\label{eqn: J21evaluated}
\begin{split}
J_{21}&= - \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} X^{-\alpha-\beta} Q^{2} \widetilde{V}(-\alpha-\beta)\widetilde{W}(2) \\
&\hspace{.25in}\times \prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta}} \zeta(1+\hat{\alpha} -\alpha+\hat{\beta}-\beta) \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta)\mathcal{K}(0,-\alpha-\beta,2).
\end{split}
\end{equation}
Similarly, since the residue of $\widetilde{V}(-s_1-\alpha-\beta)$ at $s_1=-\alpha-\beta$ is $-1$ by \eqref{eqn: mellinVresidue}, we see from \eqref{eqn: Ress2s1abJ} that the term $J_{22}$ in \eqref{eqn: J2split} equals
\begin{equation}\label{eqn: J22evaluated}
\begin{split}
J_{22} &= \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} X^{-\alpha-\beta} Q^{2} \widetilde{V}(-\alpha-\beta)\widetilde{W}(2) \\
&\hspace{.25in}\times \prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta}} \zeta(1+\hat{\alpha} -\alpha+\hat{\beta}-\beta) \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta)\mathcal{K}(-\alpha-\beta,0,2).
\end{split}
\end{equation}
To evaluate the integral $J_{23}$ in \eqref{eqn: J2split}, we first prove the following lemma that gives a functional equation for $\mathcal{K}$.
\begin{lemma}\label{lem: Kfunctionaleqn}
Suppose that $\epsilon>0$ is arbitrarily small. Let $\alpha\in A$ and $\beta\in B$, and let $h$ and $k$ be positive integers. If $s_1,s_2,w$ are complex numbers satisfying the conditions (i)--(iii) in Lemma~\ref{lem: 1swapeulerbound},
then
\begin{equation*}
\mathcal{K}(s_1,s_2,w;A,B,\alpha,\beta,h,k) = \left( \frac{h}{k}\right)^{s_1}\mathcal{K}(0,s_1+s_2,w;A,B,\alpha,\beta,h,k).
\end{equation*}
\end{lemma}
\begin{proof}
Lemma~\ref{lem: 1swapeulerbound} guarantees that the product \eqref{eqn: 1swapeulerKdef} defining $\mathcal{K}(s_1,s_2,w)$ converges absolutely and thus $\mathcal{K}(s_1,s_2,w)$ is well-defined. Now \eqref{eqn: taufactoringidentity} implies that
\begin{equation}\label{eqn: Kfunctional1}
\begin{split}
& \tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\}} (p^m) \\
& = \tau_{A \smallsetminus \{\alpha \} \cup \{-\beta-s_1-s_2\}} (p^m) \tau_{B_{s_1+s_2}\smallsetminus \{\beta+s_1+s_2\} \cup \{-\alpha\}} (p^m).
\end{split}
\end{equation}
Similarly, if $m+\text{ord}_p(h)=n+\text{ord}_p(k)$, then \eqref{eqn: taufactoringidentity} implies
\begin{equation}\label{eqn: Kfunctional2}
\begin{split}
\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\} \cup \{-\beta-s_2\}} (p^m) &\tau_{B_{s_2}\smallsetminus \{\beta+s_2\} \cup \{-\alpha-s_1\}} (p^n) \\
&= p^{s_1(n-m)}\tau_{A \smallsetminus \{\alpha \} \cup \{-\beta-s_1-s_2\}} (p^m) \tau_{B_{s_1+s_2}\smallsetminus \{\beta+s_1+s_2\} \cup \{-\alpha\}} (p^n) \\
&= p^{s_1(\text{ord}_p(h)-\text{ord}_p(k))}\tau_{A \smallsetminus \{\alpha \} \cup \{-\beta-s_1-s_2\}} (p^m) \tau_{B_{s_1+s_2}\smallsetminus \{\beta+s_1+s_2\} \cup \{-\alpha\}} (p^n).
\end{split}
\end{equation}
Also, we have
$$
\gamma+\delta = (\gamma-s_1) + (\delta+s_1).
$$
Lemma~\ref{lem: Kfunctionaleqn} follows from this, \eqref{eqn: Kfunctional1}, \eqref{eqn: Kfunctional2}, the definition \eqref{eqn: 1swapeulerKdef} of $\mathcal{K}$, and the fact that
$$
\prod_{p| hk} p^{s_1(\text{ord}_p(h)-\text{ord}_p(k))} = \left( \frac{h}{k}\right)^{s_1}.
$$
\end{proof}
We now evaluate the integral $J_{23}$ in \eqref{eqn: J2split}. Lemma~\ref{lem: Kfunctionaleqn} implies for $\re(s_1)=-\epsilon-\re(\alpha)-\re(\beta)$ that
\begin{equation}\label{eqn: KfunctionaleqnJ23}
\mathcal{K}(s_1,-s_1-\alpha-\beta,2) = \left( \frac{h}{k}\right)^{s_1}\mathcal{K}(0,-\alpha-\beta,2).
\end{equation}
Moreover, a change of variables $s_1\mapsto -s-\alpha-\beta$ gives
$$
\int_{(-\epsilon-\re(\alpha)-\re(\beta))} \widetilde{V}(s_1)\widetilde{V}(-s_1-\alpha-\beta) \left( \frac{h}{k}\right)^{s_1}\,ds_1 = \int_{(\epsilon)} \widetilde{V}(-s-\alpha-\beta) \widetilde{V}(s) \left( \frac{h}{k}\right)^{-s-\alpha-\beta} \,ds.
$$
From this, \eqref{eqn: KfunctionaleqnJ23}, and \eqref{eqn: Ress2s1abJ}, we deduce that the integral $J_{23}$ in \eqref{eqn: J2split} equals
\begin{equation}\label{eqn: J23evaluated}
\begin{split}
J_{23} & = -\sum_{\substack{\alpha\in A \\ \beta\in B}}\frac{1}{4\pi i} \int_{(\epsilon)} X^{-\alpha-\beta} Q^{2} \widetilde{V}(s) \widetilde{V}(-s-\alpha-\beta)\widetilde{W}(2) \\
& \hspace{.25in}\times \prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta}} \zeta(1+\hat{\alpha} -\alpha+\hat{\beta}-\beta) \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \\
& \hspace{.25in}\times \left( \frac{h}{k}\right)^{-s-\alpha-\beta} \mathcal{K}(0,-\alpha-\beta,2)\,ds.
\end{split}
\end{equation}
This, \eqref{eqn: J2split}, \eqref{eqn: J21evaluated}, and \eqref{eqn: J22evaluated} complete our calculation of $J_2$.
Now that we have evaluated $J_2$, we next turn our attention to the term $J_3$ in \eqref{eqn: 1swapterms2}. Recall its definition
\begin{equation}\label{eqn: J3def}
J_3 := \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i } \int_{(\epsilon)} \underset{s_2=-s_1-\alpha'-\beta'}{\text{Res}}\ \mathcal{J} \,ds_1.
\end{equation}
Since the residue of $\zeta(1+\alpha'+s_1+\beta'+s_2)$ at $s_2=-s_1-\alpha'-\beta'$ is $1$, it follows from the definition \eqref{eqn: integrandJdef} of $\mathcal{J}$ that if $\alpha'\in A\smallsetminus\{\alpha\}$ and $\beta'\in B\smallsetminus\{\beta\}$, then
\begin{equation}\label{eqn: Ress2s1a'b'J}
\begin{split}
\underset{s_2=-s_1-\alpha'-\beta'}{\text{Res}}\ \mathcal{J} = X^{-\alpha'-\beta'} Q^{2-\alpha-\beta+\alpha'+\beta'} \widetilde{V}(s_1) \widetilde{V}(-s_1-\alpha'-\beta')\widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \\
\times \mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \mathscr{X} (\tfrac{1}{2}+\beta-s_1-\alpha'-\beta' ) \zeta(1-\alpha-\beta+\alpha'+\beta') \\
\prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta(1+\hat{\alpha} +\hat{\beta} -\alpha'-\beta') \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha) \prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \\
\times \mathcal{K}(s_1,-s_1-\alpha'-\beta',2-\alpha-\beta+\alpha'+\beta').
\end{split}
\end{equation}
We move the line of integration in \eqref{eqn: J3def} to $\re(s_1)=-\epsilon$. We find residues from the poles at $s_1=0$ and $s_1=-\alpha'-\beta'$ due to the factors $\widetilde{V}(s_1)$ and $\widetilde{V}(-s_1-\alpha'-\beta')$ in \eqref{eqn: Ress2s1a'b'J}, and thus deduce that
\begin{equation*}
\begin{split}
J_3 &= \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \underset{s_1=0}{\text{Res}}\ \underset{s_2=-s_1-\alpha'-\beta'}{\text{Res}}\ \mathcal{J} + \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \underset{s_1=-\alpha'-\beta'}{\text{Res}}\ \underset{s_2=-s_1-\alpha'-\beta'}{\text{Res}}\ \mathcal{J}\\
&\hspace{.25in}+ \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i } \int_{(-\epsilon)} \underset{s_2=-s_1-\alpha'-\beta'}{\text{Res}}\ \mathcal{J} \,ds_1.
\end{split}
\end{equation*}
For brevity, write this as
\begin{equation}\label{eqn: J3split}
J_{3} =J_{31} +J_{32} +J_{33}.
\end{equation}
We see from \eqref{eqn: mellinVresidue} and \eqref{eqn: Ress2s1a'b'J} that
\begin{equation}\label{eqn: J31evaluated}
\begin{split}
J_{31} &= \frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} X^{-\alpha'-\beta'} Q^{2-\alpha-\beta+\alpha'+\beta'} \widetilde{V}(-\alpha'-\beta')\widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}\times \mathscr{X} (\tfrac{1}{2}+\alpha ) \mathscr{X} (\tfrac{1}{2}+\beta-\alpha'-\beta' ) \zeta(1-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}\times\prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta(1+\hat{\alpha} +\hat{\beta} -\alpha'-\beta') \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha) \prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \\
&\hspace{.25in}\times \mathcal{K}(0,-\alpha'-\beta',2-\alpha-\beta+\alpha'+\beta').
\end{split}
\end{equation}
Similarly, since the residue of $\widetilde{V}(-s_1-\alpha'-\beta')$ at $s_1=-\alpha'-\beta'$ is $-1$ by \eqref{eqn: mellinVresidue}, we deduce from \eqref{eqn: Ress2s1a'b'J} that the term $J_{32}$ in \eqref{eqn: J3split} equals
\begin{align}
J_{32} &= -\frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} X^{-\alpha'-\beta'} Q^{2-\alpha-\beta+\alpha'+\beta'} \widetilde{V}(-\alpha'-\beta')\widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \notag\\
&\hspace{.25in}\times \mathscr{X} (\tfrac{1}{2}+\alpha -\alpha'-\beta' ) \mathscr{X} (\tfrac{1}{2}+\beta ) \zeta(1-\alpha-\beta+\alpha'+\beta') \notag\\
&\hspace{.25in}\times\prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta(1+\hat{\alpha} +\hat{\beta} -\alpha'-\beta') \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha) \prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \notag \\
&\hspace{.25in}\times \mathcal{K}(-\alpha'-\beta',0,2-\alpha-\beta+\alpha'+\beta'). \label{eqn: J32evaluated}
\end{align}
To simplify the integral $J_{33}$ in \eqref{eqn: J3split}, we apply Lemma~\ref{lem: Kfunctionaleqn} to deduce that
\begin{equation*}
\mathcal{K}(s_1,-s_1-\alpha'-\beta',2-\alpha-\beta+\alpha'+\beta') = \left(\frac{h}{k}\right)^{s_1} \mathcal{K}(0,-\alpha'-\beta',2-\alpha-\beta+\alpha'+\beta').
\end{equation*}
It follows from this and \eqref{eqn: Ress2s1a'b'J} that the integral $J_{33}$ in \eqref{eqn: J3split} equals
\begin{equation}\label{eqn: J33evaluated}
\begin{split}
J_{33} &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i } \int_{(-\epsilon)}
X^{-\alpha'-\beta'} Q^{2-\alpha-\beta+\alpha'+\beta'} \widetilde{V}(s_1) \widetilde{V}(-s_1-\alpha'-\beta')\widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}\times \mathscr{X} (\tfrac{1}{2}+\alpha +s_1 ) \mathscr{X} (\tfrac{1}{2}+\beta-s_1-\alpha'-\beta' ) \zeta(1-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}\times\prod_{ \substack{\hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta(1+\hat{\alpha} +\hat{\beta} -\alpha'-\beta') \prod_{ \hat{\alpha} \neq \alpha} \zeta(1+\hat{\alpha}-\alpha) \prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta) \\
&\hspace{.25in}\times \left( \frac{h}{k}\right)^{s_1} \mathcal{K}(0,-\alpha'-\beta',2-\alpha-\beta+\alpha'+\beta') \,ds_1.
\end{split}
\end{equation}
This, \eqref{eqn: J3split}, \eqref{eqn: J31evaluated}, and \eqref{eqn: J32evaluated} complete our evaluation of $J_3$.
Putting together our calculations, we deduce from \eqref{eqn: 1swapterms2}, \eqref{eqn: J1split}, \eqref{eqn: J2split}, and \eqref{eqn: J3split} that
\begin{equation*}
\begin{split}
\mathcal{I}_1^*(h,k)
& = J_{11}+J_{12}+J_{13} + J_{21}+J_{22}+J_{23} + J_{31}+J_{32}+J_{33} \\
& \hspace{.25in}+ O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}\big) + O\big( X^{\varepsilon} Q^{\frac{3}{2}+\varepsilon} (hk)^{\varepsilon}\big).
\end{split}
\end{equation*}
The terms $J_{12}$ and $J_{22}$ cancel each other by \eqref{eqn: J12evaluated} and \eqref{eqn: J22evaluated}, while $J_{13}$ cancels with $J_{32}$ by \eqref{eqn: J13evaluated} and \eqref{eqn: J32evaluated}. Therefore
\begin{equation}\label{eqn: 1swapsready}
\mathcal{I}_1^*(h,k) = J_{11} + J_{21} +J_{23} + J_{31} +J_{33} + O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}\big) + O\big( X^{\varepsilon} Q^{\frac{3}{2}+\varepsilon} (hk)^{\varepsilon}\big),
\end{equation}
and we have evaluated $J_{11}$ in \eqref{eqn: J11evaluated}, $J_{21}$ in \eqref{eqn: J21evaluated}, $J_{23}$ in \eqref{eqn: J23evaluated}, $J_{31}$ in \eqref{eqn: J31evaluated}, and $J_{33}$ in \eqref{eqn: J33evaluated}. Our goal for the rest of this section is to show that $\mathcal{U}^2(h,k)$ is equal to the right-hand side of \eqref{eqn: 1swapsready} up to an admissible error term.
\subsection{Analysis of \texorpdfstring{$\mathcal{U}^2(h,k)$}{U2(h,k)}}\label{subsec: one-swap2}
We now continue our analysis of $\mathcal{U}^2(h,k)$. Our goal for this subsection and the next is to show that $\mathcal{U}^2(h,k)$ is equal to the right-hand side of \eqref{eqn: 1swapsready} up to an admissible error term. We multiply the integrand in \eqref{eqn: U2ready} by
\begin{equation}\label{eqn: U2factoroutzetas}
\prod_{\alpha\in A} \zeta(-\tfrac{1}{2}+\alpha+s_1+w-z)\prod_{\beta\in B} \zeta(\tfrac{1}{2}+\beta+s_2+z)
\end{equation}
and divide it by the Euler product of \eqref{eqn: U2factoroutzetas}. The result is
\begin{equation}\label{eqn: U2analysis0}
\begin{split}
\mathcal{U}^2(h,k) &= \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^4 } \int_{(2)} \int_{(2)} \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{s_1+s_2}Q^{w} c^{ -w} \\
&\hspace{.25in} \times \widetilde{V}(s_1)\widetilde{V}(s_2) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\alpha\in A} \zeta(-\tfrac{1}{2}+\alpha+s_1+w-z)\prod_{\beta\in B} \zeta(\tfrac{1}{2}+\beta+s_2+z)\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(s_1,s_2,w,z) \,dz \,dw \,ds_2\,ds_1\\
&\hspace{.25in}+ O\left(X^{\varepsilon}Q^{1+\varepsilon}\frac{(hk)^\varepsilon(h,k)}{(hk)^{1/2}} + (XChk)^{\varepsilon} k X^2 Q^{-97}\right),
\end{split}
\end{equation}
where $\mathcal{P}(s_1,s_2,w,z)$ is defined by
\begin{equation}\label{eqn: U2eulerPdef}
\begin{split}
&\mathcal{P}(s_1,s_2,w,z) = \mathcal{P}(s_1,s_2,w,z;A,B,h,k,c) \\
&\hspace{.25in}: = \prod_{p|c} \Bigg\{ \left( 1+\frac{p^{w-1}-1}{p(p-1)} \right) \prod_{\alpha\in A}\left(1-\frac{1}{p^{-\frac{1}{2}+\alpha+s_1+w-z}} \right) \prod_{\beta\in B} \left(1-\frac{1}{p^{\frac{1}{2}+\beta+s_2+z}} \right) \Bigg\}\\
&\hspace{.25in}\times \prod_{\substack{p\nmid c\\ p|hk}} \Bigg\{ \prod_{\alpha\in A}\left(1-\frac{1}{p^{-\frac{1}{2}+\alpha+s_1+w-z}} \right) \prod_{\beta\in B} \left(1-\frac{1}{p^{\frac{1}{2}+\beta+s_2+z}} \right) \\
&\hspace{.25in}\times \Bigg( \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \\
&\hspace{.25in}+ \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)\neq n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \Bigg)\Bigg\} \\
&\hspace{.25in}\times \prod_{\substack{p\nmid c\\ p\nmid hk}} \Bigg\{ \prod_{\alpha\in A}\left(1-\frac{1}{p^{-\frac{1}{2}+\alpha+s_1+w-z}} \right) \prod_{\beta\in B} \left(1-\frac{1}{p^{\frac{1}{2}+\beta+s_2+z}} \right) \\
&\hspace{.25in}\times \Bigg( 1+ \frac{p^{w-1}-1}{p(p-1)} + \sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1+s_1+s_2 )} } \\
&\hspace{.25in}+ \sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(\frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} } + \sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{3}{2}+s_2-w+z)} } \Bigg)\Bigg\}.
\end{split}
\end{equation}
As in our analysis of $\mathcal{I}_1^*(h,k)$ in the previous subsection, this ``factoring out'' of the zeta-functions \eqref{eqn: U2factoroutzetas} gives us the analytic continuation of the integrand and allows us to evaluate its residues when shifting contours.
To facilitate our estimations, we first prove the following lemma, which will allow us to move some lines of integration and bound integrals that remain after applying the residue theorem.
\begin{lemma}\label{lem: U2eulerbound}
Suppose that $\epsilon>0$ is arbitrarily small. Let $h,k,c$ be positive integers with $(c,hk)=1$. If $s_1,s_2,w$ are complex numbers such that
\begin{enumerate}
\item[\upshape{(i)}] $\re(w)\leq 2-\epsilon$,
\item[\upshape{(ii)}] $\re(-\frac{1}{2}+s_1+w-z) \geq \frac{1}{2} +\epsilon$,
\item[\upshape{(iii)}] $\re(\frac{1}{2}+s_2+z) \geq \frac{1}{2}+\epsilon $
\item[\upshape{(iv)}] $\re(1+s_1+s_2)\geq 1+\epsilon$
\item[\upshape{(v)}] $\re(\frac{1}{2}+s_1-z)\geq \epsilon$, and
\item[\upshape{(vi)}] $\re(\frac{3}{2}+ s_2 - w + z)\geq \epsilon$.
\end{enumerate}
then the product \eqref{eqn: U2eulerPdef} defining $\mathcal{P}(s_1,s_2,w,z;A,B,h,k,c)$ converges absolutely and we have
$$
\mathcal{P}(s_1,s_2,w,z;A,B,h,k,c) \ll_{\varepsilon} (chk)^{\varepsilon} (h,k)^{\re(w)-1}.
$$
\end{lemma}
\begin{proof}
For brevity, in this proof we will refer to the conditions in the hypothesis by their respective labels (i), (ii), $\dots$, (vi). We will also repeatedly apply without mention the bounds $\tau_A(m)\ll m^{\varepsilon}$ and $\tau_B(n)\ll n^{\varepsilon}$, which follow from \eqref{eqn: divisorbound} and our assumption that $\alpha, \beta \ll 1/\log Q$ for all $\alpha \in A$ and $\beta \in B$. The condition (i) implies
\begin{equation}\label{eqn: U2eulerbound1}
1+\frac{p^{w-1}-1}{p(p-1)} = 1 + O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
\end{equation}
From (ii), (iii), and our assumption that $\alpha, \beta \ll 1/\log Q$ for all $\alpha \in A$ and $\beta \in B$, we see that
$$
\frac{1}{p^{-\frac{1}{2}+\alpha+s_1+w-z}} \ll \frac{1}{p^{\frac{1}{2}+\varepsilon}}
$$
for all $\alpha\in A$ and
$$
\frac{1}{p^{\frac{1}{2}+\beta+s_2+z}} \ll \frac{1}{p^{\frac{1}{2}+\varepsilon}}
$$
for all $\beta\in B$. Thus, multiplying out the product and applying the definition \eqref{eqn: taudef} gives
\begin{equation}\label{eqn: U2eulerbound2}
\begin{split}
& \prod_{\alpha\in A}\left(1-\frac{1}{p^{-\frac{1}{2}+\alpha+s_1+w-z}} \right) \prod_{\beta\in B} \left(1-\frac{1}{p^{\frac{1}{2}+\beta+s_2+z}} \right) \\
& = 1- \sum_{\alpha\in A } \frac{1}{p^{-\frac{1}{2}+\alpha+s_1+w-z}} -\sum_{\beta\in B} \frac{1}{p^{\frac{1}{2}+\beta+s_2+z}} + O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg) \\
& = 1 - \frac{\tau_A(p)}{p^{-\frac{1}{2}+s_1+w-z}} - \frac{\tau_B(p)}{p^{\frac{1}{2}+s_2+z}} + O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
\end{split}
\end{equation}
By (i) and (iv), we have
\begin{equation}\label{eqn: U2eulerbound6}
\sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1+s_1+s_2 )} } \ll \sum_{m=1}^{\infty} \frac{1}{p^{m(1+\varepsilon)}} \ll \frac{1}{p^{1+\varepsilon}}.
\end{equation}
Next, to estimate the sum
\begin{equation*}
\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(\frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} },
\end{equation*}
we separate it into three parts: the term with $m=0$ and $n=1$, the sum of the terms with $m=0$ and $n\geq 2$, and the sum of the terms with $m\geq 1$. The part with $m=0$ and $n\geq 2$ is at most $O(p^{-1-\varepsilon})$ by (i) and (iii). To bound the part with $m\geq 1$, we evaluate the $n$-sum first and use (i), (iii), and (iv) to write
\begin{equation*}
\begin{split}
\sum_{ 1\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(\frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} }
& \ll \sum_{m=1}^{\infty} \frac{p^{(m+1)\varepsilon}}{p^{m(\re(\frac{1}{2}+s_1 -z))} p^{(m+1)(\re(\frac{1}{2}+s_2+z))} }\\
& = \sum_{m=1}^{\infty} \frac{p^{(m+1)\varepsilon}}{p^{m(\re(1+s_1 +s_2))+\re(\frac{1}{2}+s_2+z)} } \ll \frac{1}{p^{\frac{3}{2}+\varepsilon}}.
\end{split}
\end{equation*}
We thus arrive at
\begin{equation*}
\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(\frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} } = \frac{ \tau_B(p ) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{ \frac{1}{2}+s_2+z } } +O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
\end{equation*}
It follows from this and (vi) that
\begin{equation}\label{eqn: U2eulerbound7}
\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(\frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} } = \frac{ \tau_B(p ) }{ p^{ \frac{1}{2}+s_2+z } } +O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
\end{equation}
A similar argument using (i), (ii), (iv), and (v) leads to
\begin{equation}\label{eqn: U2eulerbound8}
\sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{3}{2}+s_2-w+z)} } = \frac{\tau_A(p)}{p^{ -\frac{1}{2}+s_1+w-z }} + O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
\end{equation}
We next bound the sum
\begin{equation}\label{eqn: U2eulerboundfinite1}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }}.
\end{equation}
For brevity, we denote $h_p:=\text{ord}_p(h)$ and $k_p:=\text{ord}_p(k)$ for the rest of this proof. We make the change of variable $m\mapsto \nu +k_p$ in \eqref{eqn: U2eulerboundfinite1}, so that $n=\nu+h_p$, to write \eqref{eqn: U2eulerboundfinite1} as
\begin{equation*}
\frac{1}{p^{k_p(\frac{1}{2}+s_1-z)} p^{h_p(\frac{3}{2}+s_2-w+z)} } \sum_{\nu=-\min\{h_p,k_p\} }^{\infty} \frac{\tau_A(p^{\nu+k_p}) \tau_B(p^{\nu+h_p}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{\nu(1+s_1+s_2)} }.
\end{equation*}
Hence, by (i) and (iv), we see that $\eqref{eqn: U2eulerboundfinite1}$ is at most
\begin{equation*}
\ll \frac{p^{\varepsilon h_p +\varepsilon k_p + \min\{h_p,k_p\} \re(1+s_1+s_2)} }{p^{k_p(\re(\frac{1}{2}+s_1-z))} p^{h_p(\re(\frac{3}{2}+s_2-w+z))} }.
\end{equation*}
The denominator of this bound is $\geq p^{\min\{h_p,k_p\} \re(\frac{1}{2}+s_1-z )} p^{\min\{h_p,k_p\} \re(\frac{3}{2}+s_2-w+z )} $ by (v) and (vi). It follows that
\begin{equation}\label{eqn: U2eulerbound3}
\begin{split}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }&\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }}\\
&\hspace{.5in}\ll p^{\varepsilon h_p +\varepsilon k_p + \min\{h_p,k_p\} (\re(w)-1)}.
\end{split}
\end{equation}
Next, to bound the sum
\begin{equation*}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)< n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }},
\end{equation*}
we split it into the part with $m<k_p-h_p$ and the part with $m\geq k_p-h_p$ to deduce that
\begin{equation}\label{eqn: U2eulerboundfinite2}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)< n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} =\Sigma_1 + \Sigma_2,
\end{equation}
where
\begin{equation}\label{eqn: U2eulerboundSigma1def}
\Sigma_1 : = p^{(w-1) h_p } \sum_{m=0 }^{k_p-h_p-1}\sum_{n=0}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m( \frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} }
\end{equation}
and
\begin{equation}\label{eqn: U2eulerboundSigma2def}
\Sigma_2 : = p^{(w-1) h_p } \sum_{m= \max\{0,k_p-h_p\} }^{\infty}\sum_{n=m+h_p-k_p+1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m( \frac{1}{2}+s_1 -z)} p^{n(\frac{1}{2}+s_2+z)} }.
\end{equation}
We use (i) to bound $p^w/p^2$ and apply (iii) to estimate the $n$-sums in \eqref{eqn: U2eulerboundSigma1def} and \eqref{eqn: U2eulerboundSigma2def} to see that
\begin{equation}\label{eqn: U2eulerboundSigma1bound1}
\Sigma_1 \ll p^{(\re(w)-1) h_p } \sum_{m=0 }^{k_p-h_p-1} \frac{p^{m\varepsilon}}{ p^{m( \re(\frac{1}{2}+s_1 -z))} }
\end{equation}
and
\begin{equation}\label{eqn: U2eulerboundSigma2bound1}
\Sigma_2 \ll p^{(\re(w)-1) h_p } \sum_{m= \max\{0,k_p-h_p\} }^{\infty} \frac{ 1 }{ p^{m( \re(1+s_1 +s_2)-\varepsilon)} p^{( h_p-k_p+1)(\re(\frac{1}{2}+s_2+z)-\varepsilon)} }.
\end{equation}
If $h_p\geq k_p$, then the $m$-sum on the right-hand side of \eqref{eqn: U2eulerboundSigma1bound1} is zero. Otherwise, it is $O(1)$ by (v). In either case, we have
\begin{equation}\label{eqn: U2eulerboundSigma1bound2}
\Sigma_1 \ll p^{(\re(w)-1) \min\{h_p,k_p\} }.
\end{equation}
If $h_p\geq k_p$, then the $m$-sum in \eqref{eqn: U2eulerboundSigma2bound1} starts at $m=0$ and thus (iii), (iv), and (vi) imply
\begin{equation*}
\Sigma_2 \ll \frac{ p^{(\re(w)-1) h_p } }{ p^{( h_p-k_p+1)(\re(\frac{1}{2}+s_2+z)-\varepsilon)} }= \frac{ p^{(\re(w)-1) k_p } }{ p^{( h_p-k_p )(\re(\frac{3}{2}+s_2-w+z)-\varepsilon)} p^{ \re(\frac{1}{2}+s_2+z)-\varepsilon } } \ll \frac{ p^{(\re(w)-1) k_p } }{p^{\frac{1}{2}+\varepsilon}}
\end{equation*}
On the other hand, if $h_p< k_p$, then the $m$-sum in \eqref{eqn: U2eulerboundSigma2bound1} starts at $m=k_p-h_p$ and it follows from (iii), (iv), and (v) that
\begin{equation*}
\begin{split}
\Sigma_2
& \ll \frac{ p^{(\re(w)-1) h_p } }{ p^{(k_p-h_p) (\re(1+s_1 +s_2)-\varepsilon) } p^{( h_p-k_p+1)(\re(\frac{1}{2}+s_2+z)-\varepsilon)} } \\
& \leq \frac{ p^{(\re(w)-1) h_p } p^{( k_p-h_p+1)\varepsilon} }{ p^{(k_p-h_p) (\re(\frac{1}{2}+s_1 -z)) } p^{ \re(\frac{1}{2}+s_2+z) } } \leq \frac{p^{(\re(w)-1) h_p } p^{( k_p-h_p+1)\varepsilon}. }{p^{\frac{1}{2}+\varepsilon}}
\end{split}
\end{equation*}
In either case, we have
\begin{equation*}
\Sigma_2 \ll p^{(\re(w)-1) \min\{h_p,k_p\} + \varepsilon h_p + \varepsilon k_p + \varepsilon -\frac{1}{2} } .
\end{equation*}
From this, \eqref{eqn: U2eulerboundSigma1bound2}, and \eqref{eqn: U2eulerboundfinite2}, we arrive at
\begin{equation}\label{eqn: U2eulerbound4}
\begin{split}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)< n + \text{ord}_p(k) } } &\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }}\\
&\ll p^{ \varepsilon h_p + \varepsilon k_p + \varepsilon + (\re(w)-1) \min\{h_p,k_p\} }.
\end{split}
\end{equation}
A similar argument using (i), (ii), (iv), (v), and (vi) gives
\begin{equation}\label{eqn: U2eulerbound5}
\begin{split}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h) > n + \text{ord}_p(k) } } &\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(-\frac{1}{2}+s_1+w-z)} p^{n(\frac{1}{2}+s_2+z)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }}\\
&\hspace{.25in}\ll p^{ \varepsilon h_p + \varepsilon k_p + \varepsilon + (\re(w)-1) \min\{h_p,k_p\} }.
\end{split}
\end{equation}
From \eqref{eqn: U2eulerbound1}, \eqref{eqn: U2eulerbound2}, (ii), and (iii), we see that if $p|c$ then the local factor in \eqref{eqn: U2eulerPdef} corresponding to $p$ is $O(1)$. Moreover, from \eqref{eqn: U2eulerbound2}, (ii), (iii), \eqref{eqn: U2eulerbound3}, \eqref{eqn: U2eulerbound4}, and \eqref{eqn: U2eulerbound5}, we deduce that if $p\nmid c$ and $p|hk$ then the local factor in \eqref{eqn: U2eulerPdef} corresponding to $p$ is
$$
\ll p^{(\re(w)-1) \min\{ \text{ord}_p(h),\text{ord}_p(k)\} + \varepsilon \text{ord}_p(h) + \varepsilon \text{ord}_p(k) + \varepsilon }.
$$
Finally, from \eqref{eqn: U2eulerbound1}, \eqref{eqn: U2eulerbound2}, \eqref{eqn: U2eulerbound6}, \eqref{eqn: U2eulerbound7}, and \eqref{eqn: U2eulerbound8}, we see that if $p\nmid chk$ then the local factor in \eqref{eqn: U2eulerPdef} corresponding to $p$ is $1+O(p^{-1-\varepsilon})$. We conclude that the right-hand side of \eqref{eqn: U2eulerPdef} converges absolutely, and is
$$
\ll (chk)^{\varepsilon} (h,k)^{\re(w)-1}
$$
because $c$ and $hk$ are coprime, $(h,k)=\prod_{p|hk} p^{ \min\{ \text{ord}_p(h),\text{ord}_p(k)\}}$, and $\prod_{p|\nu}O(1) \ll {\nu}^{\varepsilon}$ for any positive integer $\nu$.
\end{proof}
We move the $s_2$-line in \eqref{eqn: U2analysis0} to $\re(s_2)=\epsilon$. This leaves a residue from the pole at $s_2=\frac{1}{2}-\beta-z$ for each $\beta\in B$ because of the factors \eqref{eqn: U2factoroutzetas}. Note that we need to assume the Lindel\"{o}f Hypothesis to maintain the absolute convergence of the $z$-integral, as there is an arbitrary number of zeta-functions that depend on $z$ and $\mathcal{H}(z,w-1)$ only decays slowly by \eqref{eqn: Hbound}. The result is
\begin{equation}\label{eqn: U2split}
\mathcal{U}^2(h,k) = I_1 + I_2 + O\left(X^{\varepsilon}Q^{1+\varepsilon}\frac{(hk)^\varepsilon(h,k)}{(hk)^{1/2}} + (XChk)^{\varepsilon} k X^2 Q^{-97}\right),
\end{equation}
where $I_1$ is the integral of the residues at the poles $s_2=\frac{1}{2}-\beta-z$ and $I_2$ is the new integral with $\re(s_2)=\epsilon$. More precisely,
\begin{equation}\label{eqn: U2I1def}
\begin{split}
I_1 &:= \sum_{\beta\in B} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^3 } \int_{(2)} \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{s_1+\frac{1}{2}-\beta-z}Q^{w} c^{ -w} \\
&\hspace{.25in} \times \widetilde{V}(s_1)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\alpha\in A} \zeta(-\tfrac{1}{2}+\alpha+s_1+w-z)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(s_1,\tfrac{1}{2}-\beta-z,w,z) \,dz \,dw \,ds_1
\end{split}
\end{equation}
and
\begin{equation}\label{eqn: U2I2def}
\begin{split}
I_2 &:= \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^4 } \int_{(2)} \int_{(\epsilon)} \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{s_1+s_2}Q^{w} c^{ -w} \\
&\hspace{.25in}\times \widetilde{V}(s_1)\widetilde{V}(s_2) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\alpha\in A} \zeta(-\tfrac{1}{2}+\alpha+s_1+w-z)\prod_{\beta\in B} \zeta(\tfrac{1}{2}+\beta+s_2+z)\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(s_1,s_2,w,z) \,dz \,dw \,ds_2\,ds_1.
\end{split}
\end{equation}
We first bound $I_2$. We move the $s_1$-line in \eqref{eqn: U2I2def} to $\re(s_1)=\epsilon$ to deduce that
\begin{equation}\label{eqn: U2I2split}
I_2 = I_{21} + I_{22},
\end{equation}
where $I_{21}$ is the integral of the residues at the poles $s_1=\frac{3}{2}-\alpha-w+z$, where $\alpha$ runs through the elements of $A$, and $I_{22}$ is the new integral with $\re(s_1)=\epsilon$. In other words,
\begin{equation}\label{eqn: U2I21def}
\begin{split}
I_{21} &:= \sum_{\alpha\in A} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^3 } \int_{(\epsilon)} \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{\frac{3}{2}-\alpha-w+z +s_2}Q^{w} c^{ -w} \\
&\hspace{.25in} \times \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(s_2) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\beta\in B} \zeta(\tfrac{1}{2}+\beta+s_2+z)\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(\tfrac{3}{2}-\alpha-w+z,s_2,w,z) \,dz \,dw \,ds_2
\end{split}
\end{equation}
and
\begin{align}
I_{22} &:= \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^4 } \int_{(\epsilon)} \int_{(\epsilon)} \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{s_1+s_2}Q^{w} c^{ -w} \notag\\
&\hspace{.25in} \times \widetilde{V}(s_1)\widetilde{V}(s_2) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z} \notag\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\alpha\in A} \zeta(-\tfrac{1}{2}+\alpha+s_1+w-z)\prod_{\beta\in B} \zeta(\tfrac{1}{2}+\beta+s_2+z) \notag\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(s_1,s_2,w,z) \,dz \,dw \,ds_2\,ds_1.\label{eqn: U2I22def}
\end{align}
Note that we need to assume the Lindel\"{o}f Hypothesis to justify \eqref{eqn: U2I2split} like we did to validate \eqref{eqn: U2split}. To estimate $I_{22}$, we again assume the Lindel\"{o}f Hypothesis in order to bound the arbitrary number of zeta-functions in \eqref{eqn: U2I22def} that depend on the variable $z$. We apply \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: U2eulerbound}, and argue as in \eqref{eqn: zwbound} to deduce from \eqref{eqn: U2I22def} and the definition \eqref{eqn: deltadef} of $\delta$ that
\begin{equation}\label{eqn: U2I22bound}
I_{22} \ll X^{\varepsilon} Q^{1+\varepsilon} h^{\varepsilon}k^{\varepsilon}.
\end{equation}
Next, to bound $I_{21}$, we move the $w$-line in \eqref{eqn: U2I21def} to $\re(w)= \frac{3}{2}-\epsilon$. We traverse no poles in doing so. We then bound the resulting expression by applying \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: U2eulerbound}. The result is
\begin{equation*}
I_{21} \ll X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon}.
\end{equation*}
From this, \eqref{eqn: U2I22bound}, and \eqref{eqn: U2I2split}, we arrive at
\begin{equation}\label{eqn: U2I2bound}
I_{2} \ll X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon}.
\end{equation}
Having bounded $I_2$, we now turn our attention to the integral $I_1$ defined by \eqref{eqn: U2I1def}. We move the $s_1$-line in \eqref{eqn: U2I1def} to $\re(s_1)=\epsilon$. This leaves a residue from the pole at $s_1=\frac{3}{2}-\alpha-w+z$ for each $\alpha\in A$, and leads to
\begin{equation}\label{eqn: U2I1split}
I_1=I_{11}+I_{12},
\end{equation}
where
\begin{equation}\label{eqn: U2I11def}
\begin{split}
I_{11} &:= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^2 } \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{2-\alpha-\beta-w}Q^{w} c^{ -w} \\
&\hspace{.25in} \times \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(\tfrac{3}{2}-\alpha-w+z,\tfrac{1}{2}-\beta-z,w,z) \,dz \,dw
\end{split}
\end{equation}
and
\begin{equation}\label{eqn: U2I12def}
\begin{split}
I_{12} &:= \sum_{\beta\in B} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^3 } \int_{(\epsilon)} \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{s_1+\frac{1}{2}-\beta-z}Q^{w} c^{ -w} \\
&\hspace{.25in}\times \widetilde{V}(s_1)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\alpha\in A} \zeta(-\tfrac{1}{2}+\alpha+s_1+w-z)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(s_1,\tfrac{1}{2}-\beta-z,w,z) \,dz \,dw \,ds_1.
\end{split}
\end{equation}
To bound $I_{12}$, we move the $w$-line in \eqref{eqn: U2I12def} to the right by a distance of at most $\epsilon/2$, and then move the $z$-line to the right by a distance of at most $\epsilon/2$. We do this in such a way as to maintain the inequality $1+\frac{\epsilon}{2} \leq \re(w-z) \leq 1+\epsilon$, so as to not traverse any pole of $\mathcal{H}(z,w-1)$. We repeat this process until the $w$-line is at $\re(w)=\frac{3}{2}-\epsilon$ and the $z$-line is at $\frac{1}{2}-\frac{3\epsilon}{2}$. This leaves no residues because we do not cross any poles of the integrand. We then bound the resulting integral by applying \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: U2eulerbound}. We arrive at
\begin{equation}\label{eqn: U2I12bound}
I_{12} \ll X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon}.
\end{equation}
To estimate the integral $I_{11}$ defined by \eqref{eqn: U2I11def}, our first task is to extend the $c$-sum in \eqref{eqn: U2I11def} to infinity. To do this, we need to bound the sum
\begin{equation}\label{eqn: U2E11def}
\begin{split}
\sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{c>C \\ (c ,hk)=1 }}& \frac{\mu(c)}{2(2\pi i)^2 } \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{2-\alpha-\beta-w}Q^{w} c^{ -w} \\
&\hspace{.25in} \times \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(\tfrac{3}{2}-\alpha-w+z,\tfrac{1}{2}-\beta-z,w,z) \,dz \,dw.
\end{split}
\end{equation}
We first move the $w$-line in \eqref{eqn: U2E11def} to $\re(w)=\frac{3}{2}$, crossing no poles. Then, we move the $z$-line to $\re(z)=\frac{1}{2}-\epsilon$, again traversing no poles. Afterward, we further move the $w$-line to $\re(w)=2-2\epsilon$. This does not cross any poles since now $\re(z)=\frac{1}{2}-\epsilon$. We bound the new integral that has $\re(w)=2-2\epsilon$ and $\re(z)=\frac{1}{2}-\epsilon$ using \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: U2eulerbound}, and deduce that \eqref{eqn: U2E11def} is at most
\begin{equation*}
\ll \frac{ (XChk)^{\varepsilon} Q^{2} (h,k) }{C\sqrt{hk}}.
\end{equation*}
It follows from this and \eqref{eqn: U2I11def} that
\begin{equation}\label{eqn: I11toR0}
I_{11} = R_0 + O\bigg(\frac{ (XChk)^{\varepsilon} Q^{2} (h,k) }{C\sqrt{hk}}\bigg) ,
\end{equation}
where $R_0$ is defined by
\begin{equation}\label{eqn: R0def}
\begin{split}
R_0 &:= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{c\geq 1 \\ (c ,hk)=1 }} \frac{\mu(c)}{2(2\pi i)^2 } \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{2-\alpha-\beta-w}Q^{w} c^{ -w} \\
&\hspace{.25in}\times \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{P}(\tfrac{3}{2}-\alpha-w+z,\tfrac{1}{2}-\beta-z,w,z) \,dz \,dw.
\end{split}
\end{equation}
We may evaluate the sum of $\mu(c)c^{ -w} \mathcal{P}(\tfrac{3}{2}-\alpha-w+z,\tfrac{1}{2}-\beta-z,w,z)$ over all $c\geq 1$ with $(c ,hk)=1$ by using the definition \eqref{eqn: U2eulerPdef} of $\mathcal{P}$ and Lemma~\ref{lem: sumstoEulerproducts}, with absolute convergence ensured by Lemma~\ref{lem: U2eulerbound}. This and \eqref{eqn: R0def} lead to
\begin{equation}\label{eqn: R0factored}
\begin{split}
R_0 &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{2(2\pi i)^2 } \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{2-\alpha-\beta-w}Q^{w} \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1)\\
&\hspace{.25in}\times\frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\zeta(2-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \prod_{ p|hk } \Bigg\{ \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.5in}\times \Bigg( \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \\
&\hspace{.5in}+ \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)\neq n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \Bigg)\Bigg\} \\
&\hspace{.25in}\times \prod_{ p\nmid hk } \Bigg\{ \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.5in}\times \Bigg( \left(1-\frac{1}{p^w}\right)\left( 1+ \frac{p^{w-1}-1}{p(p-1)} \right) + \sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(3-\alpha-\beta-w )} } \\
&\hspace{.5in}+ \sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(2-\alpha-w)} p^{n(1-\beta)} } + \sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(2-\beta-w)} } \Bigg)\Bigg\}\,dz \,dw .
\end{split}
\end{equation}
From \eqref{eqn: U2I1split}, \eqref{eqn: U2I12bound}, \eqref{eqn: I11toR0}, we deduce that
\begin{equation}\label{eqn: I1toR0}
I_1 = R_0 + O\bigg(\frac{ (XChk)^{\varepsilon} Q^{2} (h,k) }{C\sqrt{hk}}\bigg) + O\Big( X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon} \Big),
\end{equation}
where $R_0$ is expressed as a finite sum of contour integrals in \eqref{eqn: R0factored}.
To be able to shift the contours and evaluate residues, we analytically continue the integrand in \eqref{eqn: R0factored} by multiplying it by
\begin{equation}\label{eqn: U2I11factoroutzetas}
\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w)
\end{equation}
and dividing it by the Euler product of \eqref{eqn: U2I11factoroutzetas}. The result is
\begin{align}
R_0 &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{2(2\pi i)^2 } \int_{(1+\epsilon)}\int_{(\epsilon/2)} X^{2-\alpha-\beta-w}Q^{w} \notag\\
&\hspace{.25in} \times \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z} \notag\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta ) \notag\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{G} (w,\alpha,\beta)\,dz \,dw, \label{eqn: R0def2}
\end{align}
where $\mathcal{G} (w,\alpha,\beta)$ is defined by
\begin{equation}\label{eqn: R0eulerGdef}
\begin{split}
\mathcal{G}(w
& ,\alpha,\beta) = \mathcal{G}(w,\alpha,\beta; A,B,h,k)\\
&:= \prod_{ p|hk } \Bigg\{ \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.75in}\times \Bigg( \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \\
&\hspace{.75in}+ \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)\neq n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \Bigg)\Bigg\} \\
&\hspace{.5in}\times \prod_{ p\nmid hk } \Bigg\{ \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.75in}\times \Bigg( \left( 1-\frac{1}{p^w}\right)\left(1+ \frac{p^{w-1}-1}{p(p-1)}\right) + \sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(3-\alpha-\beta-w )} } \\
&\hspace{.75in}+ \sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(2-\alpha-w)} p^{n(1-\beta)} } + \sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(2-\beta-w)} } \Bigg)\Bigg\}.
\end{split}
\end{equation}
We next prove the following lemma, which we will use to justify moving the lines of integration and bound some of the integrals that remain after applying the residue theorem.
\begin{lemma}\label{lem: R0eulerbound}
Suppose that $\epsilon>0$ is arbitrarily small. Let $\alpha\in A$ and $\beta\in B$, and let $h$ and $k$ be positive integers. If $w$ is a complex number such that
$$
1+\epsilon \leq \re(w) \leq \frac{5}{2}-\epsilon,
$$
then the product \eqref{eqn: R0eulerGdef} defining $\mathcal{G}(w,\alpha,\beta; A,B,h,k)$ converges absolutely and we have
$$
\mathcal{G}(w,\alpha,\beta; A,B,h,k) \ll_{\varepsilon} h^{\frac{1}{2}+\varepsilon} k^{\frac{1}{2}+\varepsilon} (h,k)^{\frac{1}{2}+\varepsilon}.
$$
\end{lemma}
\begin{proof}
In this proof, we will repeatedly apply without mention the bounds $\tau_A(m)\ll m^{\varepsilon}$, and $\tau_B(n)\ll n^{\varepsilon}$, which follow from \eqref{eqn: divisorbound} and the assumption that $\alpha,\beta \ll 1/\log Q$ for all $\alpha \in A$ and $\beta \in B$. Since $\re(w)\leq \frac{5}{2}-\epsilon$, we have
$$
\frac{1}{p^{3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w}} \ll \frac{1}{p^{\frac{1}{2}+\varepsilon}}
$$
for all $\hat{\alpha}\in A$ and $\hat{\beta}\in B$. Also, it holds that
$$
\frac{1}{p^{1+\hat{\alpha}-\alpha}} \ll \frac{1}{p^{1-\varepsilon}}
$$
and
$$
\frac{1}{p^{1+\hat{\beta}-\beta}} \ll \frac{1}{p^{1-\varepsilon}}
$$
for all $\hat{\alpha}\in A$ and $\hat{\beta}\in B$. Hence, multiplying out the product and applying the definition \eqref{eqn: taudef} gives
\begin{equation}\label{eqn: R0eulerbound1}
\begin{split}
\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}&\left(1-\frac{1}{p^{3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&= 1 -\sum_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \frac{1}{p^{3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w}} - \sum_{\hat{\alpha}\in A} \frac{1}{p^{1+\hat{\alpha}-\alpha}} - \sum_{\hat{\beta}\in B} \frac{1}{p^{1+\hat{\beta}-\beta}} + O\left( \frac{1}{p^{1+\varepsilon}}\right) \\
&= 1 - \frac{\tau_A(p)\tau_B(p)}{p^{3-\alpha-\beta-w}} + \frac{\tau_A(p)}{p^{3-\alpha-w}} + \frac{\tau_B(p)}{p^{3-\beta-w}} - \frac{1}{p^{3-w}} - \frac{ \tau_A(p)}{p^{1-\alpha}} - \frac{ \tau_B(p)}{p^{1-\beta}} + O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{split}
\end{equation}
Since $1+\epsilon \leq \re(w)\leq \frac{5}{2}-\epsilon$, we have
\begin{equation}\label{eqn: R0eulerbound2}
\left( 1-\frac{1}{p^w}\right)\left(1+ \frac{p^{w-1}-1}{p(p-1)}\right) = 1+\frac{1}{p^{3-w}} + O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{equation}
The assumption $\re(w)\leq \frac{5}{2}-\epsilon$ also implies $\re(3-\alpha-\beta-w)\geq \frac{1}{2}+\varepsilon$ and thus
\begin{equation}\label{eqn: R0eulerbound3}
\sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(3-\alpha-\beta-w )} } = \frac{\tau_A(p)\tau_B(p)}{p^{3-\alpha-\beta-w}} + O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{equation}
Next, since $p^w/p^2 \ll p^{\frac{1}{2}-\epsilon}$, the terms with $m\geq 1$ in the sum
$$
\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(2-\alpha-w)} p^{n(1-\beta)} }
$$
add up to at most
\begin{equation*}
\ll \sum_{ 1\leq m<n<\infty } \frac{p^{\frac{1}{2}-\epsilon}}{p^{m(-\frac{1}{2}-\varepsilon+\epsilon)} p^{n(1-\varepsilon)} } \ll \sum_{m=1}^{\infty} \frac{p^{\frac{1}{2}-\epsilon} }{p^{m(-\frac{1}{2}-\varepsilon+\epsilon)}p^{(m+1)(1-\varepsilon)}} \ll \frac{1}{p^{1+\varepsilon}},
\end{equation*}
while the terms with $m=0$ and $n\geq 2$ add up to at most
\begin{equation*}
\ll \sum_{ n=2 }^{\infty} \frac{p^{\frac{1}{2}-\epsilon}}{p^{n(1-\varepsilon)} } \ll \frac{1}{p^{\frac{3}{2}+\varepsilon}}.
\end{equation*}
Hence
\begin{equation}\label{eqn: R0eulerbound4}
\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(2-\alpha-w)} p^{n(1-\beta)} } = \frac{ \tau_B(p ) }{ p^{ 1-\beta } } - \frac{ \tau_B(p ) }{ p^{ 3-\beta-w } } + O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{equation}
Similarly, or by symmetry, we have
\begin{equation}\label{eqn: R0eulerbound5}
\sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(2-\beta-w)} } = \frac{\tau_A(p )}{ p^{ 1-\alpha } } - \frac{\tau_A(p ) }{ p^{ 3-\alpha-w } }+ O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{equation}
We next bound the sum
\begin{equation}\label{eqn: R0eulerboundfinite1}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }}.
\end{equation}
For brevity, we denote $h_p:=\text{ord}_p(h)$ and $k_p:=\text{ord}_p(k)$ for the rest of this proof. We make the change of variable $m\mapsto \nu +k_p$ in \eqref{eqn: R0eulerboundfinite1}, so that $n=\nu+h_p$, to see that \eqref{eqn: R0eulerboundfinite1} equals
\begin{equation*}
p^{k_p(w+\alpha-2)+h_p(w+\beta-2)} \sum_{\nu=-\min\{h_p,k_p\} }^{\infty} \frac{\tau_A(p^{\nu+k_p}) \tau_B(p^{\nu+h_p}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{\nu(3-\alpha-\beta-w)} }.
\end{equation*}
This and the inequality $(k_p+h_p -\min\{h_p,k_p\})\re(w)\leq \frac{5}{2}(k_p+h_p -\min\{h_p,k_p\})$ imply
\begin{equation}\label{eqn: R0eulerbound6}
\begin{split}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }&\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^w}{p^{2}(p-1)}-\frac{1}{p-1} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \\
&\hspace{.25in}\ll p^{k_p(\re(w)-2+\varepsilon)+h_p(\re(w)-2+\varepsilon) + \min\{h_p,k_p\} \re( 3-w) }\\
&\hspace{.25in}\ll p^{(\frac{1}{2}+\varepsilon)( h_p +k_p + \min\{h_p,k_p\}) } .
\end{split}
\end{equation}
Next, to bound the sum
\begin{equation*}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h) < n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }},
\end{equation*}
we split it into the part with $m<k_p-h_p$ and the part with $m\geq k_p-h_p$ to deduce that
\begin{equation}\label{eqn: R0eulerboundfinite2}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h) < n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} =\Sigma_1 + \Sigma_2,
\end{equation}
where
\begin{equation}\label{eqn: R0eulerboundSigma1def}
\Sigma_1 : = p^{(w-1) h_p } \sum_{m=0 }^{k_p-h_p-1}\sum_{n=0}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(2-\alpha-w)} p^{n(1-\beta)} }
\end{equation}
and
\begin{equation}\label{eqn: R0eulerboundSigma2def}
\Sigma_2 : = p^{(w-1) h_p } \sum_{m= \max\{0,k_p-h_p\} }^{\infty}\sum_{n=m+h_p-k_p+1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(2-\alpha-w)} p^{n(1-\beta)} }.
\end{equation}
We apply $\re(w)\leq \frac{5}{2}-\epsilon$ and bound the $n$-sums in \eqref{eqn: R0eulerboundSigma1def} and \eqref{eqn: R0eulerboundSigma2def} to deduce that
\begin{equation}\label{eqn: R0eulerboundSigma1bound1}
\Sigma_1 \ll p^{(\frac{3}{2}-\epsilon) h_p } \sum_{m=0 }^{k_p-h_p-1} \frac{p^{\frac{1}{2}-\epsilon}}{ p^{m(-\frac{1}{2}-\varepsilon+\epsilon)}}
\end{equation}
and
\begin{equation}\label{eqn: R0eulerboundSigma2bound1}
\Sigma_2 \ll p^{(\frac{3}{2}-\epsilon) h_p } \sum_{m= \max\{0,k_p-h_p\} }^{\infty} \frac{ p^{\frac{1}{2}-\epsilon} }{ p^{m( \frac{1}{2}-\varepsilon+\epsilon)} p^{( h_p-k_p+1)(1-\varepsilon)} }.
\end{equation}
The right-hand side of \eqref{eqn: R0eulerboundSigma1bound1} is zero if $h_p\geq k_p$, and otherwise it is $ \ll p^{(\frac{3}{2}-\epsilon) h_p + (k_p-h_p) (\frac{1}{2} +\varepsilon)}$. In either case, we have
\begin{equation}\label{eqn: R0eulerboundSigma1bound2}
\Sigma_1 \ll p^{(\frac{1}{2}+\varepsilon)( h_p +k_p + \min\{h_p,k_p\}) }.
\end{equation}
If $h_p\geq k_p$, then the $m$-sum in \eqref{eqn: R0eulerboundSigma2bound1} starts at $m=0$ and thus
\begin{equation*}
\Sigma_2 \ll \frac{ p^{(\frac{3}{2}-\epsilon) h_p + \frac{1}{2}-\epsilon} }{ p^{( h_p-k_p+1)(1-\varepsilon)} } \ll p^{(\frac{1}{2}+\varepsilon) h_p +k_p - \frac{1}{2} + \varepsilon}.
\end{equation*}
On the other hand, if $h_p< k_p$, then the $m$-sum in \eqref{eqn: R0eulerboundSigma2bound1} starts at $m=k_p-h_p$ and hence
\begin{equation*}
\Sigma_2 \ll \frac{ p^{(\frac{3}{2}-\epsilon) h_p + \frac{1}{2}-\epsilon} }{ p^{(k_p-h_p) (\frac{1}{2}-\varepsilon+\epsilon) } p^{( h_p-k_p+1)(1-\varepsilon)} } \ll p^{ (1+\varepsilon) h_p + (\frac{1}{2}+\varepsilon) k_p -\frac{1}{2}+\varepsilon }.
\end{equation*}
In either case, we have
\begin{equation*}
\Sigma_2 \ll p^{(\frac{1}{2}+\varepsilon)( h_p +k_p + \min\{h_p,k_p\}) -\frac{1}{2}+\varepsilon} .
\end{equation*}
From this, \eqref{eqn: R0eulerboundSigma1bound2}, and \eqref{eqn: R0eulerboundfinite2}, we arrive at
\begin{equation}\label{eqn: R0eulerbound7}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h) < n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \ll p^{(\frac{1}{2}+\varepsilon)( h_p +k_p + \min\{h_p,k_p\}) }.
\end{equation}
Similarly, or by symmetry, it holds that
\begin{equation}\label{eqn: R0eulerbound8}
\sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h) > n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(1-w) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \ll p^{(\frac{1}{2}+\varepsilon)( h_p +k_p + \min\{h_p,k_p\}) }.
\end{equation}
From \eqref{eqn: R0eulerbound1}, \eqref{eqn: R0eulerbound2}, \eqref{eqn: R0eulerbound3}, \eqref{eqn: R0eulerbound4}, and \eqref{eqn: R0eulerbound5}, we deduce that if $p\nmid hk$ then the local factor in \eqref{eqn: R0eulerGdef} corresponding to $p$ is $1+O(p^{-1-\varepsilon})$. To bound the local factors corresponding to the primes $p|hk$, observe that \eqref{eqn: R0eulerbound1} is $O(1)$ because $\re(w)\leq \frac{5}{2}-\epsilon$. This, \eqref{eqn: R0eulerbound6}, \eqref{eqn: R0eulerbound7}, and \eqref{eqn: R0eulerbound8} imply that if $p|hk$ then the local factor corresponding to $p$ is
$$
\ll p^{(\frac{1}{2}+\varepsilon)( h_p +k_p + \min\{h_p,k_p\}) }.
$$
We conclude that the right-hand side of \eqref{eqn: R0eulerGdef} converges absolutely, and is
$$
\ll h^{\frac{1}{2}+\varepsilon} k^{\frac{1}{2}+\varepsilon} (h,k)^{\frac{1}{2}+\varepsilon}
$$
because $hk(h,k)=\prod_{p|hk} p^{ h_p+k_p+ \min\{h_p,k_p\}}$ and $\prod_{p|\nu}O(1) \ll {\nu}^{\varepsilon}$ for any positive integer $\nu$.
\end{proof}
We now move the $w$-line in \eqref{eqn: R0def2} rightward to $\re(w)= \frac{5}{2}-\epsilon$ to deduce that
\begin{equation}\label{eqn: R0split}
R_0 = R_1 + R_2 + R_3 + R_4,
\end{equation}
where $R_1$ is the integral of the residue at $w=2$, $R_2$ is the integral of the residue at $w=\frac{3}{2}-\alpha+z$, $R_3$ is the integral of the residues at the poles of \eqref{eqn: U2I11factoroutzetas}, and $R_4$ is the new integral with $\re(w)= \frac{5}{2}-\epsilon$.
We first bound $R_4$, which is defined by
\begin{equation*}
\begin{split}
R_4 &:= \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{2(2\pi i)^2 } \int_{(\frac{5}{2}-\epsilon)}\int_{(\epsilon/2)} X^{2-\alpha-\beta-w}Q^{w} \\
&\hspace{.25in} \times \widetilde{V}(\tfrac{3}{2}-\alpha-w+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(w) \mathcal{H}(z,w-1) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(2-w) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 3+\hat{\alpha}+\hat{\beta} -\alpha-\beta-w) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{1-w+z}k^{-z} \mathcal{G} (w,\alpha,\beta)\,dz \,dw.
\end{split}
\end{equation*}
We move the $z$-line to $\re(z)=\frac{1}{2}-\epsilon$, traversing no poles in the process. We then bound the resulting integral using \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: R0eulerbound}. The result is
\begin{equation}\label{eqn: R4bound}
R_4\ll X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}.
\end{equation}
We next evaluate the integral $R_1$ defined in \eqref{eqn: R0split}. To do this, observe that the winding number in the application of the residue theorem in \eqref{eqn: R0split} is $-1$. Also, the definition \eqref{eqn: Hdef} implies that
\begin{equation*}
\underset{w=2}{\text{Res}}\ \mathcal{H}(z,w-1) = -2
\end{equation*}
because Res$_{s=0}\Gamma(s)=1$ and $\Gamma(1/2)=\sqrt{\pi}$. Furthermore, $\zeta(0)=-1/2$. Hence
\begin{equation}\label{eqn: R1def}
\begin{split}
R_1 &= -\sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{ 4\pi i } \int_{(\epsilon/2)} X^{-\alpha-\beta}Q^{2} \widetilde{V}(-\tfrac{1}{2}-\alpha+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(2) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha-\beta) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-1+z}k^{-z} \mathcal{G} (2,\alpha,\beta)\,dz .
\end{split}
\end{equation}
Some factors here do not depend on $z$, and we only need to evaluate
\begin{equation*}
\int_{(\epsilon/2)} \widetilde{V}(-\tfrac{1}{2}-\alpha+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z} h^{z}k^{-z}\,dz.
\end{equation*}
The part of this with $|\text{Im}z|\geq 1/\delta$ is negligible because of \eqref{eqn: mellinrapiddecay} and the definition \eqref{eqn: deltadef} of $\delta$. In the complementary part with $|\text{Im}z|\leq 1/\delta$, we have
\begin{equation}\label{eqn: exppowerseriesapprox}
\frac{e^{\delta z} - e^{-\delta z}}{2\delta z} = 1 +O(\delta|z|).
\end{equation}
Thus
\begin{equation*}
\begin{split}
\int_{(\epsilon/2)} \widetilde{V}(-\tfrac{1}{2}-\alpha+z)&\widetilde{V}(\tfrac{1}{2}-\beta-z) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z} h^{z}k^{-z}\,dz \\
&\hspace{.25in}= \int_{\frac{\epsilon}{2}-\frac{i}{\delta}}^{\frac{\epsilon}{2}+\frac{i}{\delta}} \widetilde{V}(-\tfrac{1}{2}-\alpha+z)\widetilde{V}(\tfrac{1}{2}-\beta-z)h^{z}k^{-z}\,dz +O\big((hk)^{\varepsilon} \delta \big).
\end{split}
\end{equation*}
By \eqref{eqn: deltadef} and \eqref{eqn: mellinrapiddecay}, we may extend the range of Im$(z)$ in the latter integral to $(-\infty,\infty)$ by adding a negligible quantity. We then make the change of variables $s\mapsto \frac{1}{2}-\beta-z$, and afterward move the line of integration to $\re(s)=\epsilon$. We traverse no poles in doing so, and we arrive at
\begin{equation*}
\begin{split}
\int_{(\epsilon/2)} \widetilde{V}(-\tfrac{1}{2}-\alpha+z)&\widetilde{V}(\tfrac{1}{2}-\beta-z) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z} h^{z}k^{-z}\,dz \\
&\hspace{.25in}= \int_{(\epsilon)} \widetilde{V}(-\alpha-\beta-s)\widetilde{V}(s)\left( \frac{h}{k}\right)^{\frac{1}{2}-\beta-s}\,ds +O\big((hk)^{\varepsilon} \delta \big).
\end{split}
\end{equation*}
From this and \eqref{eqn: R1def}, we deduce that
\begin{equation}\label{eqn: R1evaluated}
\begin{split}
R_1 &= -\sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{ 4\pi i } \int_{(\epsilon)} X^{-\alpha-\beta}Q^{2} \widetilde{V}(-\alpha-\beta-s)\widetilde{V}(s) \widetilde{W}(2)\\
&\hspace{.25in}\times \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha-\beta) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}-\beta-s} k^{-\frac{1}{2}+\beta+s} \mathcal{G} (2,\alpha,\beta)\,ds + O\big( (Xhk)^{\varepsilon}k^{1/2} Q^{-96}\big),
\end{split}
\end{equation}
where we have applied \eqref{eqn: deltadef}, \eqref{eqn: zetaalphaalphabound}, and Lemma~\ref{lem: R0eulerbound} to bound the error term.
Having evaluated $R_1$, we next turn our attention to the integral $R_2$ defined in \eqref{eqn: R0split}. By \eqref{eqn: mellinVresidue}, the residue of $\widetilde{V}(\frac{3}{2}-\alpha-w+z)$ at $w=\frac{3}{2}-\alpha+z$ is $-1$. From this and the fact that the winding number in the application of the residue theorem in \eqref{eqn: R0split} is $-1$, we deduce that
\begin{equation*}
\begin{split}
R_2 &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \frac{1}{4\pi i } \int_{(\epsilon/2)} X^{\frac{1}{2}-\beta-z}Q^{\frac{3}{2}-\alpha+z} \\
&\hspace{.25in}\times \widetilde{V}(\tfrac{1}{2}-\beta-z) \widetilde{W}(\tfrac{3}{2}-\alpha+z) \mathcal{H}(z,\tfrac{1}{2}-\alpha+z) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(\tfrac{1}{2}+\alpha-z) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( \tfrac{3}{2}+\hat{\alpha}+\hat{\beta} -\beta-z) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-z} \mathcal{G} (\tfrac{3}{2}-\alpha+z,\alpha,\beta)\,dz.
\end{split}
\end{equation*}
We move the line of integration to $\re(z)=1-2\epsilon$ to deduce that
\begin{equation}\label{eqn: R2split}
R_2 = R_{21} + R_{22} + R_{23} + R_{24},
\end{equation}
where $R_{21}$ is the residue at $z=\frac{1}{2}-\beta$, $R_{22}$ is the residue at $z=\frac{1}{2}+\alpha$, $R_{23}$ is the sum of the residues at the poles $z=\frac{1}{2}+\alpha'+\beta'-\beta$, where $\alpha'$ runs through the elements of $A\smallsetminus \{\alpha\}$ and $\beta'$ runs through the elements of $B\smallsetminus \{\beta\}$, and $R_{24}$ is the new integral with $\re(z)=1-2\epsilon$. To bound $R_{24}$, we apply \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: R0eulerbound}. The result is
\begin{equation}\label{eqn: R24bound}
R_{24} \ll X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon}.
\end{equation}
We next estimate the residue $R_{21}$ defined in \eqref{eqn: R2split}. By \eqref{eqn: mellinVresidue}, the residue of $\widetilde{V}(\frac{1}{2}-\beta-z)$ at $z=\frac{1}{2}-\beta$ is $-1$. From this and the fact that the winding number in the application of the residue theorem in \eqref{eqn: R2split} is $-1$, we have
\begin{equation}\label{eqn: R21explicit}
\begin{split}
R_{21} &= \frac{1}{2 }\sum_{\substack{\alpha\in A \\ \beta\in B}} Q^{2-\alpha-\beta} \widetilde{W}(2-\alpha-\beta) \mathcal{H}(\tfrac{1}{2}-\beta,1-\alpha-\beta) \frac{e^{\delta (\frac{1}{2}-\beta)} - e^{-\delta (\frac{1}{2}-\beta)}}{\delta (1-2\beta)}\\
&\hspace{.25in}\times \zeta(\alpha+\beta) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} ) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}+\beta} \mathcal{G} (2-\alpha-\beta,\alpha,\beta).
\end{split}
\end{equation}
Now \eqref{eqn: Hintermsofchifactor} gives
\begin{equation*}
\mathcal{H}(\tfrac{1}{2}-\beta,1-\alpha-\beta)=\mathscr{X}(1-\alpha-\beta)\mathscr{X}(\tfrac{1}{2}+\beta)\mathscr{X}(\tfrac{1}{2}+\alpha),
\end{equation*}
and thus the functional equation of $\zeta(s)$ implies
$$
\zeta(\alpha+\beta)\mathcal{H}(\tfrac{1}{2}-\beta,1-\alpha-\beta) = \zeta(1-\alpha-\beta) \mathscr{X}(\tfrac{1}{2}+\beta)\mathscr{X}(\tfrac{1}{2}+\alpha).
$$
It follows from this and \eqref{eqn: R21explicit} that
\begin{equation}\label{eqn: R21explicit2}
\begin{split}
R_{21} &= \frac{1}{2 }\sum_{\substack{\alpha\in A \\ \beta\in B}} Q^{2-\alpha-\beta} \widetilde{W}(2-\alpha-\beta) \mathscr{X}(\tfrac{1}{2}+\alpha)\mathscr{X}(\tfrac{1}{2}+\beta) \frac{e^{\delta (\frac{1}{2}-\beta)} - e^{-\delta (\frac{1}{2}-\beta)}}{\delta (1-2\beta)}\\
&\hspace{.25in}\times \zeta(1-\alpha-\beta) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} ) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}+\beta} \mathcal{G} (2-\alpha-\beta,\alpha,\beta).
\end{split}
\end{equation}
By \eqref{eqn: deltadef} and the assumption that $\alpha,\beta \ll 1/\log Q$ for all $\alpha \in A$ and $\beta \in B$, we have
\begin{equation*}
\frac{e^{\delta (\frac{1}{2}-\beta)} - e^{-\delta (\frac{1}{2}-\beta)}}{\delta (1-2\beta)} = 1 + O\big( Q^{-99}\big).
\end{equation*}
We insert this into \eqref{eqn: R21explicit2} and apply Lemma~\ref{lem: R0eulerbound} and \eqref{eqn: zetaalphaalphabound} to bound the contribution of the error term. The result is
\begin{equation}\label{eqn: R21evaluated}
\begin{split}
R_{21} &= \frac{1}{2 }\sum_{\substack{\alpha\in A \\ \beta\in B}} Q^{2-\alpha-\beta} \widetilde{W}(2-\alpha-\beta) \mathscr{X}(\tfrac{1}{2}+\alpha)\mathscr{X}(\tfrac{1}{2}+\beta) \\
&\hspace{.25in}\times \zeta(1-\alpha-\beta) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} ) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}+\beta} \mathcal{G} (2-\alpha-\beta,\alpha,\beta) + O\big((hk)^{\varepsilon}(h,k)^{1/2}Q^{-96}\big).
\end{split}
\end{equation}
Our next task is to evaluate the residue $R_{22}$ defined in \eqref{eqn: R2split}. To do this, observe that the winding number in the application of the residue theorem in \eqref{eqn: R2split} is $-1$. Moreover, the definition \eqref{eqn: Hdef} implies that
\begin{equation*}
\underset{z=\frac{1}{2}+\alpha}{\text{Res}}\ \mathcal{H}(z,\tfrac{1}{2}-\alpha+z) = -2
\end{equation*}
because Res$_{s=0}\Gamma(s)=1$ and $\Gamma(1/2)=\sqrt{\pi}$. Furthermore, $\zeta(0)=-1/2$. Hence
\begin{equation}\label{eqn: R22explicit}
\begin{split}
R_{22} &= -\frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} X^{-\alpha-\beta}Q^{2}\widetilde{V}(-\alpha-\beta) \widetilde{W}(2) \frac{e^{\delta (\frac{1}{2}+\alpha)} - e^{-\delta (\frac{1}{2}+\alpha) }}{ \delta (1+2\alpha)}\\
&\hspace{.25in}\times \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha-\beta) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}-\alpha} \mathcal{G} (2,\alpha,\beta).
\end{split}
\end{equation}
By \eqref{eqn: deltadef} and the assumption that $\alpha,\beta \ll 1/\log Q$ for all $\alpha \in A$ and $\beta \in B$, we have
\begin{equation*}
\frac{e^{\delta (\frac{1}{2}+\alpha)} - e^{-\delta (\frac{1}{2}+\alpha)}}{\delta (1+2\alpha)} = 1 + O\big( Q^{-99}\big).
\end{equation*}
We insert this into \eqref{eqn: R22explicit} and apply Lemma~\ref{lem: R0eulerbound} and \eqref{eqn: zetaalphaalphabound} to bound the contribution of the error term. The result is
\begin{equation}\label{eqn: R22evaluated}
\begin{split}
R_{22} &= -\frac{1}{2}\sum_{\substack{\alpha\in A \\ \beta\in B}} X^{-\alpha-\beta}Q^{2} \widetilde{V}(-\alpha-\beta) \widetilde{W}(2) \\
&\hspace{.25in}\times \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha-\beta) \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}-\alpha} \mathcal{G} (2,\alpha,\beta) + O\big( (Xhk)^{\varepsilon}(h,k)^{1/2} Q^{-96}\big).
\end{split}
\end{equation}
We next estimate the sum $R_{23}$ defined in \eqref{eqn: R2split}. Since the winding number in the application of the residue theorem in \eqref{eqn: R2split} is $-1$ and
\begin{equation*}
\underset{z=\frac{1}{2}+\alpha'+\beta'-\beta}{\text{Res}}\ \zeta( \tfrac{3}{2}+\alpha'+\beta' -\beta-z)=-1,
\end{equation*}
it follows that
\begin{align}
R_{23} &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta }} \frac{1}{2 } X^{-\alpha'-\beta'}Q^{2-\alpha-\beta+\alpha'+\beta'} \notag \\
&\hspace{.25in}\times \widetilde{V}(-\alpha'-\beta') \widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \mathcal{H}(\tfrac{1}{2}+\alpha'+\beta'-\beta,1-\alpha-\beta+\alpha'+\beta') \notag \\
&\hspace{.25in}\times \frac{e^{\delta (\frac{1}{2}+\alpha'+\beta'-\beta) } - e^{-\delta (\frac{1}{2}+\alpha'+\beta'-\beta)}}{2\delta (\frac{1}{2}+\alpha'+\beta'-\beta)}\zeta(\alpha+\beta-\alpha'-\beta') \notag\\
&\hspace{.25in}\times\prod_{\substack{ \hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha}, \hat{\beta})\neq (\alpha',\beta') }} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha'-\beta') \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta ) \notag\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}-\alpha'-\beta'+\beta} \mathcal{G} (2-\alpha-\beta+\alpha'+\beta',\alpha,\beta). \label{eqn: R23explicit}
\end{align}
Now \eqref{eqn: Hintermsofchifactor} gives
\begin{equation*}
\mathcal{H}(\tfrac{1}{2}+\alpha'+\beta'-\beta,1-\alpha-\beta+\alpha'+\beta')=\mathscr{X}(1-\alpha-\beta+\alpha'+\beta')\mathscr{X}(\tfrac{1}{2}+\beta-\alpha'-\beta')\mathscr{X}(\tfrac{1}{2}+\alpha),
\end{equation*}
and thus the functional equation of $\zeta(s)$ implies
\begin{equation*}
\begin{split}
&\zeta(\alpha+\beta-\alpha'-\beta')\mathcal{H}(\tfrac{1}{2}+\alpha'+\beta'-\beta,1-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}= \zeta(1-\alpha-\beta+\alpha'+\beta') \mathscr{X}(\tfrac{1}{2}+\beta-\alpha'-\beta')\mathscr{X}(\tfrac{1}{2}+\alpha).
\end{split}
\end{equation*}
It follows from this and \eqref{eqn: R23explicit} that
\begin{equation}\label{eqn: R23explicit2}
\begin{split}
R_{23} &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta }} \frac{1}{2 } X^{-\alpha'-\beta'}Q^{2-\alpha-\beta+\alpha'+\beta'} \\
&\hspace{.25in} \times \widetilde{V}(-\alpha'-\beta') \widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \mathscr{X}(\tfrac{1}{2}+\beta-\alpha'-\beta')\mathscr{X}(\tfrac{1}{2}+\alpha) \\
&\hspace{.25in} \times \frac{e^{\delta (\frac{1}{2}+\alpha'+\beta'-\beta) } - e^{-\delta (\frac{1}{2}+\alpha'+\beta'-\beta)}}{2\delta (\frac{1}{2}+\alpha'+\beta'-\beta)}\zeta(1-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}\times\prod_{\substack{ \hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha}, \hat{\beta})\neq (\alpha',\beta') }} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha'-\beta') \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}-\alpha'-\beta'+\beta} \mathcal{G} (2-\alpha-\beta+\alpha'+\beta',\alpha,\beta).
\end{split}
\end{equation}
By \eqref{eqn: deltadef} and the assumption that $\alpha,\beta \ll 1/\log Q$ for all $\alpha \in A$ and $\beta \in B$, we have
\begin{equation*}
\frac{e^{\delta (\frac{1}{2}+\alpha'+\beta'-\beta) } - e^{-\delta (\frac{1}{2}+\alpha'+\beta'-\beta)}}{2\delta (\frac{1}{2}+\alpha'+\beta'-\beta)} = 1 + O\big( Q^{-99}\big).
\end{equation*}
We insert this into \eqref{eqn: R23explicit2} and apply Lemma~\ref{lem: R0eulerbound} and \eqref{eqn: zetaalphaalphabound} to bound the contribution of the error term. The result is
\begin{equation}\label{eqn: R23evaluated}
\begin{split}
R_{23} &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta }} \frac{1}{2 } X^{-\alpha'-\beta'}Q^{2-\alpha-\beta+\alpha'+\beta'} \\
&\hspace{.25in}\times \widetilde{V}(-\alpha'-\beta') \widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \mathscr{X}(\tfrac{1}{2}+\beta-\alpha'-\beta')\mathscr{X}(\tfrac{1}{2}+\alpha) \\
&\hspace{.25in}\times \zeta(1-\alpha-\beta+\alpha'+\beta') \prod_{\substack{ \hat{\alpha} \neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha}, \hat{\beta})\neq (\alpha',\beta') }} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha'-\beta') \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\\
&\hspace{.25in}\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta ) h^{-\frac{1}{2}+\alpha}k^{-\frac{1}{2}-\alpha'-\beta'+\beta} \mathcal{G} (2-\alpha-\beta+\alpha'+\beta',\alpha,\beta) \\
&\hspace{.25in}+ O\big( (Xhk)^{\varepsilon}(h,k)^{1/2} Q^{-96}\big).
\end{split}
\end{equation}
This, \eqref{eqn: R2split}, \eqref{eqn: R24bound}, \eqref{eqn: R21evaluated}, and \eqref{eqn: R22evaluated} complete our evaluation of $R_2$.
Having estimated $R_2$, we next turn our attention to the integral $R_3$ defined in \eqref{eqn: R0split}. Since
\begin{equation*}
\underset{w=2-\alpha-\beta+\alpha'+\beta'}{\text{Res}}\ \zeta( 3+\alpha'+\beta' -\alpha-\beta-w)=-1
\end{equation*}
and the winding number in the application of the residue theorem in \eqref{eqn: R0split} is $-1$, we may write
\begin{equation}\label{eqn: R3def}
\begin{split}
R_3 &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i} \int_{(\epsilon/2)} X^{-\alpha'-\beta'}Q^{2-\alpha-\beta+\alpha'+\beta'} \widetilde{V}(-\tfrac{1}{2}+\beta-\alpha'-\beta'+z)\widetilde{V}(\tfrac{1}{2}-\beta-z)\\
&\hspace{.25in}\widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \mathcal{H}(z,1-\alpha-\beta+\alpha'+\beta') \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(\alpha+\beta-\alpha'-\beta') \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha'-\beta') \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\\
&\hspace{.25in}\times\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )h^{-1+\alpha+\beta-\alpha'-\beta'+z}k^{-z} \mathcal{G} (2-\alpha-\beta+\alpha'+\beta',\alpha,\beta)\,dz .
\end{split}
\end{equation}
By \eqref{eqn: Hintermsofchifactor}, we have
\begin{equation*}
\mathcal{H}(z,1-\alpha-\beta+\alpha'+\beta')=\mathscr{X}(1-\alpha-\beta+\alpha'+\beta')\mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z),
\end{equation*}
and thus the functional equation of $\zeta(s)$ implies
$$
\zeta(\alpha+\beta-\alpha'-\beta')\mathcal{H}(z,1-\alpha-\beta+\alpha'+\beta') = \zeta(1-\alpha-\beta+\alpha'+\beta') \mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z).
$$
It follows from this and \eqref{eqn: R3def} that
\begin{equation}\label{eqn: R3def2}
\begin{split}
R_3 &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i} \int_{(\epsilon/2)} X^{-\alpha'-\beta'}Q^{2-\alpha-\beta+\alpha'+\beta'} \widetilde{V}(-\tfrac{1}{2}+\beta-\alpha'-\beta'+z)\widetilde{V}(\tfrac{1}{2}-\beta-z)\\
&\hspace{.25in}\times\widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z) \frac{e^{\delta z} - e^{-\delta z}}{2\delta z}\\
&\hspace{.25in}\times \zeta(1-\alpha-\beta+\alpha'+\beta') \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha'-\beta') \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\\
&\hspace{.25in}\times\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )h^{-1+\alpha+\beta-\alpha'-\beta'+z}k^{-z} \mathcal{G} (2-\alpha-\beta+\alpha'+\beta',\alpha,\beta)\,dz .
\end{split}
\end{equation}
Some factors here do not depend on $z$, and we only need to evaluate
$$
\int_{(\epsilon/2)} \widetilde{V}(-\tfrac{1}{2}+\beta-\alpha'-\beta'+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z)\frac{e^{\delta z} - e^{-\delta z}}{2\delta z} h^{z}k^{-z}\,dz.
$$
The part of this with $|\text{Im}z|\geq 1/\delta$ is negligible because of \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Stirlingchi}, and the definition \eqref{eqn: deltadef} of $\delta$. In the complementary part with $|\text{Im}z|\leq 1/\delta$, we have \eqref{eqn: exppowerseriesapprox} and thus
\begin{equation*}
\begin{split}
\int_{(\epsilon/2)}& \widetilde{V}(-\tfrac{1}{2}+\beta-\alpha'-\beta'+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \\
&\hspace{.25in}\times\mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z)\frac{e^{\delta z} - e^{-\delta z}}{2\delta z} h^{z}k^{-z}\,dz \\
&= \int_{\frac{\epsilon}{2}-\frac{i}{\delta}}^{\frac{\epsilon}{2}+\frac{i}{\delta}} \widetilde{V}(-\tfrac{1}{2}+\beta-\alpha'-\beta'+z)\widetilde{V}(\tfrac{1}{2}-\beta-z) \\ &\hspace{.25in}\times \mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z) h^{z}k^{-z}\,dz+O\big((hk)^{\varepsilon} \delta \big).
\end{split}
\end{equation*}
By \eqref{eqn: deltadef} and \eqref{eqn: mellinrapiddecay}, we may extend the range of Im$(z)$ in the latter integral to $(-\infty,\infty)$ by adding a negligible quantity. We then make the change of variables
$$
s\longmapsto -\frac{1}{2}+\beta-\alpha'-\beta'+z,
$$
and afterward move the line of integration to $\re(s)=-\epsilon$. We traverse no poles in doing so, and we arrive at
\begin{equation*}
\begin{split}
\int_{(\epsilon/2)}& \widetilde{V}(-\tfrac{1}{2}+\beta-\alpha'-\beta'+z)\widetilde{V}(\tfrac{1}{2}-\beta-z)\\
&\hspace{.25in}\times\mathscr{X}(1-z)\mathscr{X}(\alpha+\beta-\alpha'-\beta'+z)\frac{e^{\delta z} - e^{-\delta z}}{2\delta z} h^{z}k^{-z}\,dz \\
&= \int_{(-\epsilon)} \widetilde{V}(s) \widetilde{V}(-\alpha'-\beta'-s)\\
&\hspace{.25in}\times\mathscr{X}(\tfrac{1}{2}+\beta-\alpha'-\beta'-s) \mathscr{X}(\tfrac{1}{2}+\alpha+s)\left( \frac{h}{k}\right)^{\frac{1}{2}-\beta+\alpha'+\beta'+s}\,ds +O\big((hk)^{\varepsilon} \delta \big).
\end{split}
\end{equation*}
From this and \eqref{eqn: R3def2}, we deduce that
\begin{equation}\label{eqn: R3evaluated}
\begin{split}
R_3 &= \sum_{\substack{\alpha\in A \\ \beta\in B}} \sum_{\substack{\alpha'\neq \alpha \\ \beta'\neq \beta}} \frac{1}{4\pi i} \int_{(-\epsilon)} X^{-\alpha'-\beta'}Q^{2-\alpha-\beta+\alpha'+\beta'} \\
&\hspace{.25in}\times \widetilde{V}(s) \widetilde{V}(-\alpha'-\beta'-s) \widetilde{W}(2-\alpha-\beta+\alpha'+\beta') \\
&\hspace{.25in}\times \mathscr{X}(\tfrac{1}{2}+\beta-\alpha'-\beta'-s) \mathscr{X}(\tfrac{1}{2}+\alpha+s)\zeta(1-\alpha-\beta+\alpha'+\beta')\\
&\hspace{.25in}\times\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta \\ (\hat{\alpha},\hat{\beta})\neq (\alpha',\beta')}} \zeta( 1+\hat{\alpha}+\hat{\beta} -\alpha'-\beta') \prod_{\hat{\alpha}\neq \alpha} \zeta(1+\hat{\alpha}-\alpha)\prod_{\hat{\beta}\neq \beta} \zeta(1+\hat{\beta}-\beta )\\
&\hspace{.25in}\times h^{-\frac{1}{2}+\alpha+ s}k^{-\frac{1}{2}+\beta-\alpha'-\beta'- s} \mathcal{G} (2-\alpha-\beta+\alpha'+\beta',\alpha,\beta)\,ds \\
&\hspace{.25in}+ O\big( (Xhk)^{\varepsilon}k^{1/2} Q^{-96}\big),
\end{split}
\end{equation}
where we have applied \eqref{eqn: deltadef}, \eqref{eqn: zetaalphaalphabound}, \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Hbound}, and Lemma~\ref{lem: R0eulerbound} to bound the error term.
Putting together our calculations, we see from \eqref{eqn: U2split}, \eqref{eqn: U2I2bound}, and \eqref{eqn: I1toR0} that
\begin{equation*}
\mathcal{U}^2(h,k) = R_0 + O \bigg( \bigg( Q^{1+\varepsilon} + \frac{Q^2}{C^{1-\varepsilon}}\bigg) \frac{(Xhk)^{\varepsilon}(h,k)}{\sqrt{hk}} \bigg) + O\Big( X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon} + (XChk)^{\varepsilon} k X^2 Q^{-97}\Big).
\end{equation*}
From this, \eqref{eqn: R0split}, \eqref{eqn: R4bound}, \eqref{eqn: R2split}, and \eqref{eqn: R24bound}, we arrive at
\begin{equation}\label{eqn: U2residues}
\begin{split}
\mathcal{U}^2(h,k) &= R_1+R_{21}+R_{22}+R_{23}+R_{3} + O \bigg( \bigg( Q^{1+\varepsilon} + \frac{Q^2}{C^{1-\varepsilon}}\bigg) \frac{(Xhk)^{\varepsilon}(h,k)}{\sqrt{hk}} \bigg)\\
&\hspace{.25in} + O\Big( X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon} + X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon} + (XChk)^{\varepsilon} k X^2 Q^{-97}\Big),
\end{split}
\end{equation}
where we have evaluated the residue $R_1$ in \eqref{eqn: R1evaluated}, $R_{21}$ in \eqref{eqn: R21evaluated}, $R_{22}$ in \eqref{eqn: R22evaluated}, $R_{23}$ in \eqref{eqn: R23evaluated}, and $R_{3}$ in \eqref{eqn: R3evaluated}. In the next subsection, we will match these five residues with the five residues on the right-hand side of \eqref{eqn: 1swapsready} in such a way that corresponding residues are equal, thus showing that $\mathcal{U}^2(h,k)$ is equal to $\mathcal{I}_1^*(h,k)$ up to an admissible error term.
\subsection{Matching the residues: Euler product evaluations}\label{sec: matchresidues}
To be able to show that each of the residues on the right-hand side of \eqref{eqn: U2residues} is equal to some term on the right-hand side of in \eqref{eqn: 1swapsready}, we will prove the following identity involving the Euler products $\mathcal{G}$ and $\mathcal{K}$.
\begin{lemma}\label{lem: GKeuleridentity}
Let $\alpha\in A$ and $\beta\in B$. Suppose that $h$ and $k$ are positive integers. If $\mathcal{G}$ is defined by \eqref{eqn: R0eulerGdef} and $\mathcal{K}$ by \eqref{eqn: 1swapeulerKdef}, then
\begin{equation}\label{eqn: GKeuleriden}
h^{-\frac{1}{2}+\alpha} k^{-\frac{1}{2}+\beta} \mathcal{G}(2-\alpha-\beta,\alpha,\beta;A,B,h,k) = \mathcal{K}(0,0,2-\alpha-\beta;A,B,\alpha,\beta,h,k).
\end{equation}
\end{lemma}
Our proof of Lemma~\ref{lem: GKeuleridentity} will depend on the following three lemmas. The first is a slight generalization of an identity due to Conrey and Keating~\cite{CK3}
\begin{lemma}\label{lem: CK3identity}
Let $\alpha\in A$ and $\beta\in B$. Suppose that $j$ and $\ell$ are nonnegative integers and $p$ is a prime. Then
\begin{equation*}
\begin{split}
\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell}) - \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) \\
= \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{\ell}) -p^{\alpha+\beta} \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}}(p^{j-1}) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{\ell-1}),
\end{split}
\end{equation*}
where $\tau_E(p^{-1})$ is defined to be zero for any multiset $E$.
\end{lemma}
\begin{proof}
We argue as in \cite{CK3}. Observe that the definition \eqref{eqn: taudef} implies that if $m$ is any nonnegative integer, $E$ is any finite multiset, and $\gamma\in E$, then
\begin{equation}\label{eqn: tauremoveelement}
\tau_E(p^m)=\tau_{E\smallsetminus \{\gamma\}}(p^m) + p^{-\gamma}\tau_E(p^{m-1}).
\end{equation}
We apply this, multiply out the resulting products, and then cancel one $\tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell})$ with its negative to deduce that
\begin{equation*}
\begin{split}
&\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell}) - \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) \\
&= \Big( \tau_{A\smallsetminus\{\alpha\}} (p^j) + p^{\beta}\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^{j-1}) \Big) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) \\
&\hspace{.5in}+ \tau_{A\smallsetminus\{\alpha\}} (p^j) \Big( \tau_{B\smallsetminus\{\beta\} } (p^{\ell}) + p^{\alpha} \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell-1}) \Big)- \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) \\
&= \tau_{A\smallsetminus\{\alpha\}} (p^j)\tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + p^{\beta}\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^{j-1})\tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + p^{\alpha}\tau_{A\smallsetminus\{\alpha\}}(p^{j }) \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell-1}).
\end{split}
\end{equation*}
We add and subtract $p^{\alpha+\beta}\tau_{A\smallsetminus\{\alpha\}\cup\{-\beta\}} (p^{j-1}) \tau_{B\smallsetminus\{\beta\}\cup\{-\alpha\}} (p^{\ell-1})$, and then factor part of the resulting expression to arrive at
\begin{equation*}
\begin{split}
\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell}) - \tau_{A\smallsetminus\{\alpha\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) \\
= \Big( \tau_{A\smallsetminus\{\alpha\}} (p^j) + p^{\beta}\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^{j-1})\Big) \Big( \tau_{B\smallsetminus\{\beta\}} (p^{\ell}) + p^{\alpha} \tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^{\ell-1})\Big) \\
- p^{\alpha+\beta}\tau_{A\smallsetminus\{\alpha\}\cup\{-\beta\}} (p^{j-1}) \tau_{B\smallsetminus\{\beta\}\cup\{-\alpha\}} (p^{\ell-1}).
\end{split}
\end{equation*}
The lemma now follows from this and \eqref{eqn: tauremoveelement}.
\end{proof}
\begin{lemma}\label{lem: taulongidentity}
Let $\alpha\in A$ and $\beta\in B$. Suppose that $j$ and $\ell$ are nonnegative integers and $p$ is a prime. Then
\begin{equation}\label{eqn: taulongiden}
\begin{split}
\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell})&=
(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^j)\tau_{B } (p^{\ell})+p^{-\alpha-\beta} \tau_A(p^j)\tau_{B } (p^{\ell}) \\
&-p^{-\beta} \tau_A(p^j) \tau_{B } (p^{\ell-1})-(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^{j-1})\tau_{B } (p^{\ell-1})
\end{split}
\end{equation}
where $\tau_E(p^{-1})$ is defined to be zero for any multiset $E$.
\end{lemma}
\begin{proof}
We apply \eqref{eqn: tauremoveelement} and multiply out the resulting expression to deduce that
\begin{equation*}
\begin{split}
(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^j)\tau_{B } (p^{\ell})
&= (1-p^{-\alpha-\beta})\Big( \tau_{A} (p^j)+p^{\beta} \tau_{A\cup\{-\beta\}} (p^{j-1}) \Big)\tau_{B } (p^{\ell}) \\
&= \tau_{A} (p^j)\tau_{B } (p^{\ell}) -p^{-\alpha-\beta} \tau_{A} (p^j)\tau_{B } (p^{\ell})\\
&\hspace{.5in}+ (p^{\beta}-p^{-\alpha }) \tau_{A\cup\{-\beta\}} (p^{j-1}) \tau_{B } (p^{\ell})
\end{split}
\end{equation*}
The term $-p^{-\alpha-\beta} \tau_{A} (p^j)\tau_{B } (p^{\ell})$ cancels with its negative on the left-hand side of \eqref{eqn: taulongiden}, and it follows that
\begin{equation*}
\begin{split}
(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^j)\tau_{B } (p^{\ell}) &+p^{-\alpha-\beta} \tau_A(p^j)\tau_{B } (p^{\ell}) \\
&-p^{-\beta} \tau_A(p^j) \tau_{B } (p^{\ell-1}) -(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^{j-1})\tau_{B } (p^{\ell-1}) \\
&\hspace{-.5in}= \tau_{A} (p^j)\tau_{B } (p^{\ell}) + (p^{\beta}-p^{-\alpha }) \tau_{A\cup\{-\beta\}} (p^{j-1}) \tau_{B } (p^{\ell}) \\
&\hspace{.25in}-p^{-\beta} \tau_A(p^j) \tau_{B } (p^{\ell-1}) -(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^{j-1})\tau_{B } (p^{\ell-1}).
\end{split}
\end{equation*}
The right-hand side factors as
\begin{equation*}
\begin{split}
\Big( \tau_{A} (p^j) + (p^{\beta}-p^{-\alpha }) \tau_{A\cup\{-\beta\}} (p^{j-1}) \Big)\Big( \tau_{B } (p^{\ell})-p^{-\beta} \tau_{B } (p^{\ell-1})\Big),
\end{split}
\end{equation*}
which, by \eqref{eqn: tauremoveelement}, equals $\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^j) \tau_{B\smallsetminus\{\beta\}} (p^{\ell})$.
\end{proof}
\begin{lemma}\label{lem: tauseriesidentity}
Let $\beta\in B$. Suppose that $j$ and $\ell$ are nonnegative integers and $p$ is a prime. Then
\begin{equation*}
p^{(\frac{1}{2}-\beta)(j-\ell)} \sum_{ \substack{ 0\leq m,n<\infty \\ m+ j< n+\ell} } \frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m\beta } p^{n(1-\beta)} } = \sum_{\substack{ 0\leq m,n<\infty \\ m+j =n+\ell}} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) - \tau_{A}(p^m) \tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}}
\end{equation*}
\end{lemma}
\begin{proof}
The definition \eqref{eqn: taudef} of $\tau_E$ implies that if $D$ and $E$ are finite multisets, then the Dirichlet convolution $\tau_D*\tau_E$ of $\tau_D$ and $\tau_E$ is $\tau_{D\cup E}$. It follows from this and the definition of Dirichlet convolution that, for each nonnegative integer $m$,
\begin{equation*}
\tau_{A\cup\{-\beta\}} (p^{m-1}) = (\tau_A*\tau_{\{-\beta\}})(p^{m-1}) = \sum_{\nu=0}^{m-1} \tau_A(p^{\nu}) \tau_{\{-\beta\}} (p^{m-1-\nu}) = \sum_{\nu=0}^{m-1} \tau_A(p^{\nu}) p^{\beta(m-1-\nu)}.
\end{equation*}
This and the identity \eqref{eqn: tauremoveelement} imply
\begin{equation*}
\tau_{A\cup\{-\beta\}}(p^m) - \tau_{A}(p^m) = p^{\beta} \tau_{A\cup\{-\beta\}}(p^{m-1})= \sum_{\nu=0}^{m-1} \tau_A(p^{\nu}) p^{\beta(m-\nu)}.
\end{equation*}
Therefore
\begin{equation*}
\sum_{\substack{ 0\leq m,n<\infty \\ m+j =n+\ell}} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) - \tau_{A}(p^m) \tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}} = \sum_{\substack{ 0\leq m,n<\infty \\ m+j =n+\ell}} \frac{ \tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}}\sum_{\nu=0}^{m-1} \tau_A(p^{\nu}) p^{\beta(m-\nu)}.
\end{equation*}
In the latter sum, we may replace $m$ with $n+\ell-j$ to write the sum as
\begin{equation*}
\sum_{n=0}^{\infty} \frac{ \tau_B(p^n) }{p^{n + \frac{1}{2}(\ell-j)} }\sum_{\nu=0}^{n+\ell-j-1} \tau_A(p^{\nu}) p^{\beta(n+\ell-j-\nu)}= p^{(\frac{1}{2}-\beta)(j-\ell)} \sum_{ \substack{ 0\leq \nu,n<\infty \\ \nu+ j< n+\ell} } \frac{\tau_A(p^{\nu}) \tau_B(p^{n}) }{ p^{\nu\beta } p^{n(1-\beta)} }.
\end{equation*}
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem: GKeuleridentity}]
We may write each side of \eqref{eqn: GKeuleriden} as an Euler product by the definitions \eqref{eqn: 1swapeulerKdef} of $\mathcal{K}$ and \eqref{eqn: R0eulerGdef} of $\mathcal{G}$. The Euler products converge absolutely by Lemmas~\ref{lem: 1swapeulerbound} and \ref{lem: R0eulerbound}. To prove Lemma~\ref{lem: GKeuleridentity}, it suffices to show for each $p$ that the local factors corresponding to $p$ in these Euler products agree.
We first examine the local factors corresponding to a given prime $p\nmid hk$. For brevity, let $\mathfrak{F}_p$ denote the local factor corresponding to this $p$ in the Euler product expression for the left-hand side of \eqref{eqn: GKeuleriden}. Thus, from the definition \eqref{eqn: R0eulerGdef} of $\mathcal{G}$, we see that $\mathfrak{F}_p$ is defined by
\begin{equation}\label{eqn: mathfrakFdef}
\begin{split}
\mathfrak{F}_p &:= \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta}}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.25in}\times \Bigg( \left( 1-\frac{1}{p^{2-\alpha-\beta}}\right)\left(1+ \frac{p^{1-\alpha-\beta}-1}{p(p-1)}\right) + \left( 1+\frac{p^{-\alpha-\beta}}{p-1}-\frac{1}{p-1} \right) \sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) }{ p^{m} } \\
&\hspace{.25in}+ ( 1-p^{-\alpha-\beta} )\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m\beta} p^{n(1-\beta)} } + ( 1-p^{-\alpha-\beta} ) \sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n\alpha} } \Bigg).
\end{split}
\end{equation}
Lemma~\ref{lem: tauseriesidentity} with $j=\ell=0$ implies
\begin{equation}\label{eqn: tauseriesidenappl1}
\sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m\beta} p^{n(1-\beta)} } = \sum_{m=0}^{\infty} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^m) }{p^{m}} - \sum_{m=0}^{\infty} \frac{\tau_{A}(p^m) \tau_B(p^m) }{p^{m}}.
\end{equation}
Similarly, Lemma~\ref{lem: tauseriesidentity} with $A$ and $B$ interchanged, $\beta$ replaced by $\alpha$, and $j=\ell=0$ implies
\begin{equation}\label{eqn: tauseriesidenappl2}
\sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n\alpha} } = \sum_{m=0}^{\infty} \frac{ \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^m) }{p^{m}} - \sum_{m=0}^{\infty} \frac{\tau_{A}(p^m) \tau_B(p^m) }{p^{m}}.
\end{equation}
We complete the first $m$-sum in \eqref{eqn: mathfrakFdef} by adding and subtracting its $m=0$ term, and then insert \eqref{eqn: tauseriesidenappl1} and \eqref{eqn: tauseriesidenappl2} to deduce that
\begin{align}
\mathfrak{F}_p &= \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta}}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \notag\\
&\hspace{.25in}\times \Bigg( \left( 1-\frac{1}{p^{2-\alpha-\beta}}\right)\left(1+ \frac{p^{1-\alpha-\beta}-1}{p(p-1)}\right) - \left( 1+\frac{p^{-\alpha-\beta}}{p-1}-\frac{1}{p-1} \right) \notag\\
&\hspace{.5in}+ \left( 2p^{-\alpha-\beta}-1+\frac{p^{-\alpha-\beta}}{p-1}-\frac{1}{p-1} \right) \sum_{m=0}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) }{ p^{m} } \notag\\
&\hspace{.5in}+ ( 1-p^{-\alpha-\beta} )\sum_{m=0}^{\infty} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^m) }{p^{m}} + ( 1-p^{-\alpha-\beta} ) \sum_{m=0}^{\infty} \frac{ \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^m) }{p^{m}} \Bigg). \label{eqn: mathfrakF2}
\end{align}
Observe that there is the factor $(1-1/p)^2$ in \eqref{eqn: mathfrakF2}. This factor is the product of the factor corresponding to $\hat{\alpha}=\alpha$ in the product over $\hat{\alpha}\in A$ and the factor corresponding to $\hat{\beta}=\beta$ in the product over $\hat{\beta}\in B$. We distribute $(1-1/p)$ among the terms in \eqref{eqn: mathfrakF2} and arrive at
\begin{equation}\label{eqn: mathfrakFtoSigma0}
\begin{split}
\mathfrak{F}_p &= \left( 1-\frac{1}{p}\right)\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta}}} \right) \prod_{\hat{\alpha}\neq \alpha }\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\neq \beta} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.25in}\times \Bigg( \left( 1-\frac{1}{p^{2-\alpha-\beta}}\right)\left(1-\frac{1}{p}+ \frac{1}{p^{1+\alpha+\beta}}-\frac{1}{p^2}\right) - \left( 1-\frac{2}{p}+\frac{1}{p^{1+\alpha+\beta}}\right) +\Sigma_0\Bigg),
\end{split}
\end{equation}
where $\Sigma_0$ is defined by
\begin{equation*}
\begin{split}
\Sigma_0 := & \left( 2p^{-\alpha-\beta}-1-\frac{p^{-\alpha-\beta}}{p} \right) \sum_{m=0}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) }{ p^{m} } \\
& + \bigg( 1-p^{-\alpha-\beta} -\frac{1}{p} +\frac{p^{-\alpha-\beta}}{p} \bigg)\Bigg( \sum_{m=0}^{\infty} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^m) }{p^{m}} + \sum_{m=0}^{\infty} \frac{ \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^m) }{p^{m}}\Bigg).
\end{split}
\end{equation*}
Multiply out the products in the latter expression and rearrange the terms to write
\begin{equation*}
\begin{split}
\Sigma_0 &= \sum_{m=0}^{\infty} \frac{(1- p^{-\alpha-\beta})\tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^m) +p^{-\alpha-\beta} \tau_A(p^m) \tau_B(p^m) }{p^m} \\
&\hspace{.25in}+ \sum_{m=0}^{\infty} \frac{(1- p^{-\alpha-\beta})\tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^m) +p^{-\alpha-\beta} \tau_A(p^m) \tau_B(p^m) - \tau_A(p^m)\tau_B(p^m) }{p^m} \\
&\hspace{.25in}- \sum_{m=0}^{\infty} \frac{(1- p^{-\alpha-\beta})\big( \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^m) + \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^m)\big) + p^{-\alpha-\beta} \tau_A(p^m)\tau_B(p^m)}{p^{m+1}}.
\end{split}
\end{equation*}
We make a change of variables in the last $m$-sum on the right-hand side by replacing each instance of $m$ with $m-1$. To the resulting expression for $\Sigma_0$, we add
\begin{equation*}
\begin{split}
0 &= \sum_{m=0}^{\infty} \frac{p^{-\alpha} \tau_A(p^{m-1})\tau_B(p^m) + p^{-\beta} \tau_A(p^{m})\tau_B(p^{m-1}) }{p^m} \\
&\hspace{.25in}- \sum_{m=0}^{\infty} \frac{p^{-\alpha} \tau_A(p^{m-1})\tau_B(p^m) + p^{-\beta} \tau_A(p^{m})\tau_B(p^{m-1}) }{p^m}
\end{split}
\end{equation*}
and rearrange the terms to deduce that
\begin{equation}\label{eqn: Sigma0split}
\Sigma_0 = \sum_{m=0}^{\infty} \Big( D_{1,m}+D_{2,m} + D_{3,m}\Big) \frac{1}{p^m},
\end{equation}
where $D_{1,m}$, $D_{2,m}$, and $D_{3,m}$ are defined by
\begin{equation*}
\begin{split}
D_{1,m}&:= (1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^m)\tau_{B } (p^m) +p^{-\alpha-\beta} \tau_A(p^m)\tau_{B } (p^m) \\
&\hspace{.25in}-p^{-\beta} \tau_A(p^m) \tau_{B } (p^{m-1}) -(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^{m-1})\tau_{B } (p^{m-1}),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
D_{2,m} &:= (1-p^{-\alpha-\beta}) \tau_A (p^m) \tau_{B \cup \{-\alpha \}} (p^m) +p^{-\alpha-\beta} \tau_A (p^m)\tau_B(p^m)\\
&\hspace{.25in}-p^{-\alpha} \tau_A (p^{m-1}) \tau_B(p^m) -(1-p^{-\alpha-\beta}) \tau_A (p^{m-1}) \tau_{B \cup \{-\alpha \}} (p^{m-1}),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
D_{3,m} &:= -\tau_A (p^m)\tau_{B } (p^m) + p^{-\alpha}\tau_A (p^{m-1})\tau_{B } (p^m) \\
&\hspace{.25in}+ p^{-\beta}\tau_A (p^m)\tau_{B } (p^{m-1}) -p^{-\alpha-\beta} \tau_A (p^{m-1})\tau_{B } (p^{m-1}),
\end{split}
\end{equation*}
where we recall that $\tau_E(p^{-1})$ is defined to be zero for any multiset $E$. Now Lemma~\ref{lem: taulongidentity} with $j=\ell=m$ implies
\begin{equation}\label{eqn: D1msimplify}
D_{1,m} = \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\}} (p^{m}).
\end{equation}
Moreover, Lemma~\ref{lem: taulongidentity} with $A$ and $B$ interchanged and $j=\ell=m$ implies
\begin{equation}\label{eqn: D2msimplify}
D_{2,m} = \tau_{A\smallsetminus\{\alpha\}} (p^{m})\tau_{B\smallsetminus\{\beta\}\cup \{-\alpha\}} (p^m) .
\end{equation}
As for $D_{3,m}$, we may factor it and apply \eqref{eqn: tauremoveelement} to deduce that
\begin{equation*}
\begin{split}
D_{3,m}
& = -\big(\tau_A(p^m) -p^{-\alpha}\tau_A(p^{m-1}) \big) \big( \tau_B(p^m) -p^{-\beta}\tau_B(p^{m-1}) \big) \\
& = -\tau_{A\smallsetminus\{\alpha\}}(p^m) \tau_{B\smallsetminus\{\beta\}}(p^m).
\end{split}
\end{equation*}
From this, \eqref{eqn: D1msimplify}, \eqref{eqn: D2msimplify}, and Lemma~\ref{lem: CK3identity} with $j=\ell=m$, we arrive at
\begin{equation*}
\begin{split}
D_{1,m} &+ D_{2,m} + D_{3,m} \\
&= \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m}) -p^{\alpha+\beta} \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}}(p^{m-1}) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m-1}).
\end{split}
\end{equation*}
This and \eqref{eqn: Sigma0split} imply
\begin{equation*}
\begin{split}
\Sigma_0 &= \sum_{m=0}^{\infty} \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m})}{p^m} \\
&\hspace{.25in}-p^{\alpha+\beta} \sum_{m=0}^{\infty} \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^{m-1}) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m-1})}{p^m}.
\end{split}
\end{equation*}
We make a change of variables in the latter $m$-sum by replacing each instance of $m$ with $m+1$. The result is
\begin{equation*}
\begin{split}
\Sigma_0 = \bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg)\sum_{m=0}^{\infty} \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m})}{p^m}.
\end{split}
\end{equation*}
We insert this into \eqref{eqn: mathfrakFtoSigma0} and arrive at
\begin{equation}\label{eqn: Sigma0evaluated}
\begin{split}
\mathfrak{F}_p &= \left( 1-\frac{1}{p}\right)\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta}}} \right) \prod_{\hat{\alpha}\neq \alpha }\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\neq \beta} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.25in}\times \Bigg( \left( 1-\frac{1}{p^{2-\alpha-\beta}}\right)\left(1-\frac{1}{p}+ \frac{1}{p^{1+\alpha+\beta}}-\frac{1}{p^2}\right) - \left( 1-\frac{2}{p}+\frac{1}{p^{1+\alpha+\beta}}\right) \\
&\hspace{.25in}+ \bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg) + \bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg)\sum_{m=1}^{\infty} \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m})}{p^m}\Bigg),
\end{split}
\end{equation}
where we have separated the $m=0$ term from the $m$-sum. A direct calculation gives
\begin{equation*}
\begin{split}
\left( 1-\frac{1}{p^{2-\alpha-\beta}}\right)\left(1-\frac{1}{p}+ \frac{1}{p^{1+\alpha+\beta}}-\frac{1}{p^2}\right) - \left( 1-\frac{2}{p}+\frac{1}{p^{1+\alpha+\beta}}\right) + \bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg) \\
= \left( 1- \frac{1}{p^{1-\alpha-\beta}}\right)\left(1 +\frac{1}{p} \right)\left(1-\frac{1}{p^2} \right).
\end{split}
\end{equation*}
We insert this into \eqref{eqn: Sigma0evaluated} and then factor out $(1-p^{-1+\alpha+\beta})$ to deduce that
\begin{equation*}
\begin{split}
\mathfrak{F}_p = \left( 1-\frac{1}{p}\right)\bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg)\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta}}} \right) \prod_{\hat{\alpha}\neq \alpha }\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\neq \beta} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
\times \Bigg( \left(1 +\frac{1}{p} \right)\left(1-\frac{1}{p^2} \right) + \sum_{m=1}^{\infty} \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{m})}{p^m}\Bigg).
\end{split}
\end{equation*}
The right-hand side is exactly the local factor corresponding to $p$ in the Euler product expression for $\mathcal{K}(0,0,2-\alpha-\beta)$ by the definition \eqref{eqn: 1swapeulerKdef}, because we are assuming that $p\nmid hk$.
We have now shown for each $p\nmid hk$ that the local factors corresponding to $p$ in the Euler product expressions of both sides of \eqref{eqn: GKeuleriden} agree. Our next task is to do the same for each $p|hk$. To this end, let $p|hk$ be given, and let $\mathfrak{G}_p$ denote the local factor corresponding to this $p$ in the Euler product expression for the left-hand side of \eqref{eqn: GKeuleriden}. Also, for brevity, for the rest of this proof we denote $h_p:=\text{ord}_p(h)$ and $k_p:=\text{ord}_p(k)$. With these notations, we see from the definition \eqref{eqn: R0eulerGdef} of $\mathcal{G}$ that $\mathfrak{G}_p$ is defined by
\begin{equation}\label{eqn: mathfrakGdef}
\begin{split}
\mathfrak{G}_p := p^{-(\frac{1}{2}-\alpha)h_p -(\frac{1}{2}-\beta)k_p }\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta} }} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
\times \Bigg( \left( 1+\frac{p^{-\alpha-\beta}}{p-1}-\frac{1}{p-1} \right) \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta) \min\{m+h_p, n + k_p \} }} \\
+ \left( 1-p^{-\alpha-\beta} \right) \sum_{\substack{0\leq m,n<\infty \\ m+h_p\neq n + k_p } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta) \min\{m+h_p, n + k_p \} }} \Bigg).
\end{split}
\end{equation}
If $m+h_p=n+k_p$, then
$$
(-1+\alpha+\beta) \min\{m+h_p,n+k_p \} = \left(-\frac{1}{2}+\alpha\right) (m+h_p) +\left( -\frac{1}{2}+\beta\right)(n+k_p),
$$
and so
\begin{equation}\label{eqn: tauhkseriessimplify}
\begin{split}
p^{-(\frac{1}{2}-\alpha)h_p -(\frac{1}{2}-\beta)k_p }
& \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta) \min\{m+h_p, n + k_p \} }} \\
& = \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{\frac{m}{2}+\frac{n}{2}}}.
\end{split}
\end{equation}
If $m+h_p<n+k_p$, then $\min\{m+h_p,n+k_p \}=m+h_p$ and it follows from Lemma~\ref{lem: tauseriesidentity} with $j=h_p$ and $\ell=k_p$ that
\begin{equation}\label{eqn: tauseriesidenappl3}
\begin{split}
p^{-(\frac{1}{2}-\alpha)h_p -(\frac{1}{2}-\beta)k_p } \sum_{\substack{0\leq m,n<\infty \\ m+h_p< n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta) \min\{m+h_p, n + k_p \} }} \\
= \sum_{\substack{ 0\leq m,n<\infty \\ m+h_p =n+k_p}} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) - \tau_{A}(p^m) \tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}}.
\end{split}
\end{equation}
Similarly, Lemma~\ref{lem: tauseriesidentity} with $A$ and $B$ interchanged, $\beta$ replaced by $\alpha$, $j=k_p$, and $\ell=h_p$ implies
\begin{equation*}
\begin{split}
p^{-(\frac{1}{2}-\alpha)h_p -(\frac{1}{2}-\beta)k_p } \sum_{\substack{0\leq m,n<\infty \\ m+h_p > n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta) \min\{m+h_p, n + k_p \} }}\\
= \sum_{\substack{ 0\leq m,n<\infty \\ m+h_p =n+k_p}} \frac{ \tau_A(p^m) \tau_{B\cup\{-\alpha\}}(p^n) - \tau_{A}(p^m) \tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}}.
\end{split}
\end{equation*}
It follows from this, \eqref{eqn: mathfrakGdef}, \eqref{eqn: tauhkseriessimplify}, and \eqref{eqn: tauseriesidenappl3} that
\begin{equation}\label{eqn: mathfrakG2}
\begin{split}
\mathfrak{G}_p &= \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta} }} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.25in}\times \Bigg( \left( 2p^{-\alpha-\beta}- 1+\frac{p^{-\alpha-\beta}}{p-1}-\frac{1}{p-1} \right) \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{\frac{m}{2} + \frac{n}{2} }} \\
&\hspace{.25in}+ \left( 1-p^{-\alpha-\beta} \right) \sum_{\substack{ 0\leq m,n<\infty \\ m+h_p =n+k_p}} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) + \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}} \Bigg).
\end{split}
\end{equation}
There is the factor $(1-1/p)^2$ in \eqref{eqn: mathfrakG2} by the same reason mentioned below \eqref{eqn: mathfrakF2}. We distribute $(1-1/p)$ among the terms in \eqref{eqn: mathfrakG2} and deduce that
\begin{equation}\label{eqn: mathfrakGtoSigma1}
\mathfrak{G}_p = \Sigma_1\times \left( 1-\frac{1}{p}\right)\prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta} }} \right) \prod_{\hat{\alpha}\neq \alpha}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\neq \beta} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right),
\end{equation}
where $\Sigma_1$ is defined by
\begin{equation*}
\begin{split}
\Sigma_1 &:= \left( 2p^{-\alpha-\beta}- 1-\frac{p^{-\alpha-\beta}}{p} \right) \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) }{ p^{\frac{m}{2} + \frac{n}{2} }} \\
&\hspace{.25in}+ \left( 1-p^{-\alpha-\beta} -\frac{1}{p} + \frac{ p^{-\alpha-\beta} }{p} \right) \sum_{\substack{ 0\leq m,n<\infty \\ m+h_p =n+k_p}} \frac{ \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) + \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^n) }{p^{\frac{m}{2}+\frac{n}{2}}}.
\end{split}
\end{equation*}
Multiply out the products and rearrange the terms to write $\Sigma_1$ as
\begin{equation*}
\begin{split}
&\Sigma_1 = \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{(1- p^{-\alpha-\beta})\tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) +p^{-\alpha-\beta} \tau_A(p^m) \tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2} }} \\
&+ \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{(1- p^{-\alpha-\beta})\tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^n) +p^{-\alpha-\beta} \tau_A(p^m) \tau_B(p^n) - \tau_A(p^m)\tau_B(p^n) }{p^{\frac{m}{2}+\frac{n}{2} }} \\
&- \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{(1- p^{-\alpha-\beta})\big( \tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^n) + \tau_{A}(p^m) \tau_{B\cup\{-\alpha\}}(p^n)\big) + p^{-\alpha-\beta} \tau_A(p^m)\tau_B(p^n)}{p^{1 + \frac{m}{2}+\frac{n}{2} }}.
\end{split}
\end{equation*}
We make changes of variables in the last $m,n$-sum on the right-hand side by replacing each instance of $m$ with $m-1$ and each instance of $n$ with $n-1$. To the resulting expression for $\Sigma_1$, we add
\begin{equation*}
\begin{split}
0 &= \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{p^{-\alpha} \tau_A(p^{m-1})\tau_B(p^n) + p^{-\beta} \tau_A(p^{m})\tau_B(p^{n-1}) }{p^{\frac{m}{2}+\frac{n}{2}}} \\
&\hspace{.25in}- \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{p^{-\alpha} \tau_A(p^{m-1})\tau_B(p^n) + p^{-\beta} \tau_A(p^{m})\tau_B(p^{n-1}) }{p^{\frac{m}{2}+\frac{n}{2}}}
\end{split}
\end{equation*}
and rearrange the terms to deduce that
\begin{equation}\label{eqn: Sigma1split}
\Sigma_1 = \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \Big( D_{1,m,n}+D_{2,m,n} + D_{3,m,n}\Big) \frac{1}{p^{\frac{m}{2}+\frac{n}{2}}},
\end{equation}
where $D_{1,m,n}$, $D_{2,m,n}$, and $D_{3,m,n}$ are defined by
\begin{equation*}
\begin{split}
D_{1,m,n}&:= (1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^m)\tau_{B } (p^n) +p^{-\alpha-\beta} \tau_A(p^m)\tau_{B } (p^n) \\
&\hspace{.25in}-p^{-\beta} \tau_A(p^m) \tau_{B } (p^{n-1}) -(1-p^{-\alpha-\beta}) \tau_{A\cup\{-\beta\}} (p^{m-1})\tau_{B } (p^{n-1}),
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
D_{2,m,n} &:= (1-p^{-\alpha-\beta}) \tau_A (p^m) \tau_{B \cup \{-\alpha \}} (p^n) +p^{-\alpha-\beta} \tau_A (p^m)\tau_B(p^n)\\
&\hspace{.25in}-p^{-\alpha} \tau_A (p^{m-1}) \tau_B(p^n) -(1-p^{-\alpha-\beta}) \tau_A (p^{m-1}) \tau_{B \cup \{-\alpha \}} (p^{n-1}),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
D_{3,m,n} &:= -\tau_A (p^m)\tau_{B } (p^n) + p^{-\alpha}\tau_A (p^{m-1})\tau_{B } (p^n) \\
&\hspace{.25in}+ p^{-\beta}\tau_A (p^m)\tau_{B } (p^{n-1}) -p^{-\alpha-\beta} \tau_A (p^{m-1})\tau_{B } (p^{n-1}),
\end{split}
\end{equation*}
where we recall that $\tau_E(p^{-1})$ is defined to be zero for any multiset $E$. Now Lemma~\ref{lem: taulongidentity} with $j=m$ and $\ell=n$ implies
\begin{equation}\label{eqn: D1mnsimplify}
D_{1,m,n} = \tau_{A\smallsetminus \{\alpha\}\cup\{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\}} (p^n).
\end{equation}
Moreover, Lemma~\ref{lem: taulongidentity} with $A$ and $B$ interchanged, $j=n$, and $\ell=m$ implies
\begin{equation}\label{eqn: D2mnsimplify}
D_{2,m,n} = \tau_{A\smallsetminus\{\alpha\}} (p^m) \tau_{B\smallsetminus \{\beta\}\cup\{-\alpha\}} (p^n).
\end{equation}
As for $D_{3,m,n}$, we may factor it and apply \eqref{eqn: tauremoveelement} to deduce that
\begin{equation*}
\begin{split}
D_{3,m,n}
& = -\big(\tau_A(p^m) -p^{-\alpha}\tau_A(p^{m-1}) \big) \big( \tau_B(p^n) -p^{-\beta}\tau_B(p^{n-1}) \big) \\
& = -\tau_{A\smallsetminus\{\alpha\}}(p^m) \tau_{B\smallsetminus\{\beta\}}(p^n).
\end{split}
\end{equation*}
From this, \eqref{eqn: D1mnsimplify}, \eqref{eqn: D2mnsimplify}, and Lemma~\ref{lem: CK3identity} with $j=m$ and $\ell=n$, we arrive at
\begin{equation*}
\begin{split}
D_{1,m,n} &+ D_{2,m,n} + D_{3,m,n} \\
&= \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{n}) -p^{\alpha+\beta} \tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}}(p^{m-1}) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{n-1}).
\end{split}
\end{equation*}
This and \eqref{eqn: Sigma1split} imply
\begin{equation*}
\begin{split}
\Sigma_1 &= \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{n})}{p^{\frac{m}{2}+\frac{n}{2}}} \\
&\hspace{.25in}- p^{\alpha+\beta} \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}}(p^{m-1}) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{n-1}) }{p^{\frac{m}{2}+\frac{n}{2}}}.
\end{split}
\end{equation*}
We make a change of variables in the latter $m,n$-sum by replacing each instance of $m$ with $m+1$ and each instance of $n$ with $n+1$. The result is
\begin{equation*}
\Sigma_1 = \bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg)\sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{n})}{p^{\frac{m}{2}+\frac{n}{2}}}.
\end{equation*}
We insert this into \eqref{eqn: mathfrakGtoSigma1} and arrive at
\begin{equation*}
\begin{split}
\mathfrak{G}_p &= \left( 1-\frac{1}{p}\right)\bigg( 1-\frac{1}{p^{1-\alpha-\beta}}\bigg) \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta} }} \right) \prod_{\hat{\alpha}\neq \alpha}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\neq \beta} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \\
&\hspace{.25in}\times \sum_{\substack{0\leq m,n<\infty \\ m+h_p= n + k_p } } \frac{\tau_{A\smallsetminus\{\alpha\}\cup \{-\beta\}} (p^m) \tau_{B\smallsetminus\{\beta\} \cup\{-\alpha\}} (p^{n})}{p^{\frac{m}{2}+\frac{n}{2}}}.
\end{split}
\end{equation*}
The right-hand side is exactly the local factor corresponding to $p$ in the Euler product expression for $\mathcal{K}(0,0,2-\alpha-\beta)$ by the definition \eqref{eqn: 1swapeulerKdef}, because we are assuming that $p|hk$.
We have now shown for each $p$ that the local factors corresponding to $p$ in the Euler product expressions of both sides of \eqref{eqn: GKeuleriden} agree. This completes the proof of Lemma~\ref{lem: GKeuleridentity}.
\end{proof}
We will also use the following variant and consequence of Lemma~\ref{lem: GKeuleridentity}.
\begin{lemma}\label{lem: GKeuleridentity2}
Let $\alpha,\alpha^*\in A$ and $\beta,\beta^*\in B$. Suppose that $h$ and $k$ are positive integers. If $\mathcal{G}$ is defined by \eqref{eqn: R0eulerGdef} and $\mathcal{K}$ by \eqref{eqn: 1swapeulerKdef}, then
\begin{equation*}
\begin{split}
h^{-\frac{1}{2}+\alpha} k^{-\frac{1}{2}+\beta-\alpha^*-\beta^*} &\mathcal{G}(2-\alpha-\beta+\alpha^*+\beta^*,\alpha,\beta;A,B,h,k) \\
&\hspace{.25in}= \mathcal{K}(0,-\alpha^*-\beta^*,2-\alpha-\beta+\alpha^*+\beta^*;A,B,\alpha,\beta,h,k).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
The definition \eqref{eqn: R0eulerGdef} implies
\begin{align}
\mathcal{G}(2
& -\alpha-\beta+\alpha^*+\beta^*,\alpha,\beta;A,B,h,k) \notag\\
& = \prod_{ p|hk } \Bigg\{ \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta} -\alpha^*-\beta^*}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \notag\\
&\hspace{.25in} \times \Bigg( \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)= n + \text{ord}_p(k) } }\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1+\frac{p^{-\alpha-\beta+\alpha^*+\beta^*}}{p-1}-\frac{1}{p-1} \right)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta-\alpha^*-\beta^*) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \notag\\
&\hspace{.25in} + \sum_{\substack{0\leq m,n<\infty \\ m+\text{ord}_p(h)\neq n + \text{ord}_p(k) } } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \big( 1-p^{-\alpha-\beta+\alpha^*+\beta^*} \big)}{ p^{m(1-\alpha)} p^{n(1-\beta)} p^{(-1+\alpha+\beta-\alpha^*-\beta^*) \min\{m+\text{ord}_p(h), n + \text{ord}_p(k) \} }} \Bigg)\Bigg\} \notag\\
& \times \prod_{ p\nmid hk } \Bigg\{ \prod_{\substack{ \hat{\alpha}\neq \alpha \\ \hat{\beta}\neq \beta}}\left(1-\frac{1}{p^{1+\hat{\alpha}+\hat{\beta} -\alpha^*-\beta^*}} \right) \prod_{\hat{\alpha}\in A}\left(1-\frac{1}{p^{1+\hat{\alpha}-\alpha}} \right) \prod_{\hat{\beta}\in B} \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) \notag\\
&\hspace{.25in} \times \Bigg( \left( 1-\frac{1}{p^{2-\alpha-\beta+\alpha^*+\beta^*}}\right)\left(1+ \frac{p^{1-\alpha-\beta+\alpha^*+\beta^*}-1}{(p-1)}\right) \notag\\
&\hspace{.5in} + \sum_{m=1}^{\infty} \frac{\tau_A(p^{m}) \tau_B(p^{m}) \left( 1+\frac{p^{-\alpha-\beta+\alpha^*+\beta^*}}{ p-1 }-\frac{1}{p-1} \right)}{ p^{m(1-\alpha^*-\beta^* )} } \notag\\
&\hspace{.5in} + \sum_{ 0\leq m<n<\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \big( 1-p^{-\alpha-\beta+\alpha^*+\beta^*} \big)}{ p^{m(\beta-\alpha^*-\beta^*)} p^{n(1-\beta)} } \notag\\
&\hspace{.5in} + \sum_{ 0\leq n<m <\infty } \frac{\tau_A(p^{m}) \tau_B(p^{n}) \big( 1-p^{-\alpha-\beta+\alpha^*+\beta^*} \big)}{ p^{m(1-\alpha)} p^{n(\alpha-\alpha^*-\beta^*)} } \Bigg)\Bigg\}, \label{eqn: G2aba'b'}
\end{align}
with the product absolutely convergent by Lemma~\ref{lem: R0eulerbound}. Now
\begin{equation*}
\prod_{\hat{\beta}\in B } \left(1-\frac{1}{p^{1+\hat{\beta}-\beta}} \right) = \prod_{\gamma\in B_{-\alpha^*-\beta^*}} \left(1-\frac{1}{p^{1+\gamma-\beta+\alpha^*+\beta^*}} \right),
\end{equation*}
while \eqref{eqn: taufactoringidentity} implies
\begin{equation*}
\frac{ \tau_B(p^{n}) }{ p^{n(1-\beta)} } = \frac{ \tau_{B_{-\alpha^*-\beta^*}}(p^{n}) }{ p^{n(1-\beta+\alpha^*+\beta^*)} }
\end{equation*}
and
\begin{equation*}
\frac{ \tau_B(p^{m}) }{ p^{m(1-\alpha^*-\beta^*)} } = \frac{ \tau_{B_{-\alpha^*-\beta^*}}(p^{m}) }{ p^{m}}.
\end{equation*}
It follows from these, \eqref{eqn: G2aba'b'}, and the definition \eqref{eqn: R0eulerGdef} of $\mathcal{G}$ that
\begin{equation}\label{eqn: GKeuleriden2b}
\begin{split}
\mathcal{G}&(2-\alpha-\beta+\alpha^*+\beta^*,\alpha,\beta;A,B,h,k) \\
&\hspace{.25in}= \mathcal{G}(2-\alpha-\beta+\alpha^*+\beta^*,\alpha,\beta-\alpha^*-\beta^*;A,B_{-\alpha^*-\beta^*},h,k).
\end{split}
\end{equation}
Lemma~\ref{lem: GKeuleridentity} with $B$ replaced by $B_{-\alpha^*-\beta^*}$ and $\beta$ replaced by $\beta-\alpha^*-\beta^*$ implies
\begin{equation}\label{eqn: GKeuleriden2a}
\begin{split}
h^{-\frac{1}{2}+\alpha} k^{-\frac{1}{2}+\beta-\alpha^*-\beta^*} \mathcal{G}(2-\alpha-\beta+\alpha^*+\beta^*,\alpha,\beta-\alpha^*-\beta^*;A,B_{-\alpha^*-\beta^*},h,k) \\
= \mathcal{K}(0,0,2-\alpha-\beta+\alpha^*+\beta^*;A,B_{-\alpha^*-\beta^*},\alpha,\beta-\alpha^*-\beta^*,h,k).
\end{split}
\end{equation}
To see that the right-hand side is the same as
$$
\mathcal{K}(0,-\alpha^*-\beta^*,2-\alpha-\beta+\alpha^*+\beta^*;A,B,\alpha,\beta,h,k),
$$
we make the following observations. If $w=2-\alpha-\beta+\alpha^*+\beta^*$ and $s_1=s_2=0$, then
$$
w-1+\alpha+s_1+(\beta-\alpha^*-\beta^*)+s_2= 1,
$$
$$
A_{s_1} \smallsetminus\{\alpha+s_1\} \cup\{-\beta+\alpha^*+\beta^*-s_2\} = A \smallsetminus\{\alpha \} \cup\{-\beta+\alpha^*+\beta^* \},
$$
and
$$
\big(B_{-\alpha^*-\beta^*}\big)_{s_2} \smallsetminus \{\beta-\alpha^*-\beta^*+s_2\} \cup\{-\alpha-s_1\} = B_{-\alpha^*-\beta^*}\smallsetminus \{\beta-\alpha^*-\beta^*\} \cup\{-\alpha\}.
$$
On the other hand, if $w=2-\alpha-\beta+\alpha^*+\beta^*$, $s_1=0$, and $s_2=-\alpha^*-\beta^*$, then
$$
w-1+\alpha+s_1+\beta+s_2= 1,
$$
$$
A_{s_1} \smallsetminus\{\alpha+s_1\} \cup\{-\beta-s_2\} = A \smallsetminus\{\alpha \} \cup\{-\beta+\alpha^*+\beta^* \},
$$
and
$$
B_{s_2} \smallsetminus \{\beta+s_2\} \cup\{-\alpha-s_1\} = B_{-\alpha^*-\beta^*}\smallsetminus \{\beta-\alpha^*-\beta^*\} \cup\{-\alpha\}.
$$
These observations and the definition \eqref{eqn: 1swapeulerKdef} of $\mathcal{K}$ imply that
\begin{equation*}
\begin{split}
\mathcal{K}(0,0,2-\alpha-\beta+\alpha^*+\beta^*;A,B_{-\alpha^*-\beta^*},\alpha,\beta-\alpha^*-\beta^*,h,k) \\
= \mathcal{K}(0,-\alpha^*-\beta^*,2-\alpha-\beta+\alpha^*+\beta^*;A,B,\alpha,\beta,h,k).
\end{split}
\end{equation*}
From this, \eqref{eqn: GKeuleriden2b}, and \eqref{eqn: GKeuleriden2a}, we arrive at Lemma~\ref{lem: GKeuleridentity2}.
\end{proof}
The special case of Lemma~\ref{lem: GKeuleridentity2} with $\alpha^*=\alpha$ and $\beta^*=\beta$ implies that
\begin{equation}\label{eqn: GKeuleriden3}
\begin{split}
h^{-\frac{1}{2}+\alpha} k^{-\frac{1}{2}-\alpha} \mathcal{G}(2,\alpha,\beta)
= \mathcal{K}(0,-\alpha -\beta ,2).
\end{split}
\end{equation}
As a side note, we mention that \eqref{eqn: GKeuleriden3} may be proved directly from the definitions \eqref{eqn: R0eulerGdef} of $\mathcal{G}$ and \eqref{eqn: 1swapeulerKdef} of $\mathcal{K}$ by using the identity
$$
\frac{\tau_B(p^n)}{p^{n(-\alpha-\beta)}} = \tau_{B_{-\alpha-\beta}}(p^n),
$$
which follows from \eqref{eqn: taufactoringidentity}, and observing that if $m+\text{ord}_p(h)=n+\text{ord}_p(k)$ then
\begin{equation*}
\begin{split}
p^{m(1-\alpha)}p^{n(1-\beta)}p^{-\min\{m+\text{ord}_p(h),n+\text{ord}_p(k)\}}
& = p^{m(1-\alpha)}p^{n(1-\beta)}p^{-\frac{1}{2}(m+\text{ord}_p(h))-\frac{1}{2}(n+\text{ord}_p(k))} \\
& = p^{-(\frac{1}{2}-\alpha)\text{ord}_p(h) -(\frac{1}{2}+\alpha)\text{ord}_p(k)+\frac{m}{2}+n(\frac{1}{2}-\alpha-\beta)}
\end{split}
\end{equation*}
because $\alpha(\text{ord}_p(h)-\text{ord}_p(k))=\alpha(n-m)$.
We are now ready to match each residue on the right-hand side of \eqref{eqn: U2residues} with a residue on the right-hand side of \eqref{eqn: 1swapsready} in such a way that corresponding residues are equal. The identity \eqref{eqn: GKeuleriden3} implies that
\begin{equation*}
\begin{split}
h^{-\frac{1}{2}-\beta-s} k^{-\frac{1}{2}+\beta+s} \mathcal{G}(2,\alpha,\beta)
&= \left( \frac{h}{k}\right)^{-s-\alpha-\beta}h^{-\frac{1}{2}+\alpha} k^{-\frac{1}{2}-\alpha} \mathcal{G}(2,\alpha,\beta)\\
& = \left( \frac{h}{k}\right)^{-s-\alpha-\beta}\mathcal{K}(0,-\alpha -\beta ,2).
\end{split}
\end{equation*}
From this, \eqref{eqn: J23evaluated}, and \eqref{eqn: R1evaluated}, we deduce that
\begin{equation}\label{eqn: R1andJ23}
R_1 = J_{23}+ O\big( (Xhk)^{\varepsilon}k^{1/2} Q^{-96}\big).
\end{equation}
Now from \eqref{eqn: J11evaluated}, \eqref{eqn: R21evaluated}, and Lemma~\ref{lem: GKeuleridentity}, we immediately see that
\begin{equation}\label{eqn: R21andJ11}
R_{21} = J_{11}+ O\big( (hk)^{\varepsilon}(h,k)^{1/2} Q^{-96}\big).
\end{equation}
Next, \eqref{eqn: J21evaluated}, \eqref{eqn: R22evaluated}, and \eqref{eqn: GKeuleriden3} imply
\begin{equation}\label{eqn: R22andJ21}
R_{22} = J_{21}+ O\big( (Xhk)^{\varepsilon}(h,k)^{1/2} Q^{-96}\big).
\end{equation}
From \eqref{eqn: J31evaluated}, \eqref{eqn: R23evaluated}, and Lemma~\ref{lem: GKeuleridentity2} with $\alpha^*=\alpha'$ and $\beta^*=\beta'$, we deduce that
\begin{equation}\label{eqn: R23andJ31}
R_{23} = J_{31}+ O\big( (Xhk)^{\varepsilon}(h,k)^{1/2} Q^{-96}\big).
\end{equation}
Finally, \eqref{eqn: J33evaluated}, \eqref{eqn: R3evaluated}, and Lemma~\ref{lem: GKeuleridentity2} with $\alpha^*=\alpha'$ and $\beta^*=\beta'$ imply
\begin{equation*}
R_{3} = J_{33}+ O\big( (Xhk)^{\varepsilon}k^{1/2} Q^{-96}\big).
\end{equation*}
From this, \eqref{eqn: R1andJ23}, \eqref{eqn: R21andJ11}, \eqref{eqn: R22andJ21}, \eqref{eqn: R23andJ31}, and \eqref{eqn: U2residues}, we arrive at
\begin{equation*}
\begin{split}
\mathcal{U}^2(h,k) &= J_{23}+J_{11}+J_{21}+J_{31}+J_{33} + O \bigg( \bigg( Q^{1+\varepsilon} + \frac{Q^2}{C^{1-\varepsilon}}\bigg) \frac{(Xhk)^{\varepsilon}(h,k)}{\sqrt{hk}} \bigg) \\
&\hspace{.25in}+ O\Big( X^{\varepsilon} Q^{\frac{3}{2}} h^{\varepsilon}k^{\varepsilon} + X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon} + (XChk)^{\varepsilon} hk X^2 Q^{-96}\Big).
\end{split}
\end{equation*}
From this and \eqref{eqn: 1swapsready}, we conclude that
\begin{equation}\label{eqn: U2is1swap}
\begin{split}
\mathcal{U}^2(h,k) &= \mathcal{I}_1^*(h,k) + O \bigg( \bigg( Q^{1+\varepsilon} + \frac{Q^2}{C^{1-\varepsilon}}\bigg) \frac{(Xhk)^{\varepsilon}(h,k)}{\sqrt{hk}} \bigg) \\
&\hspace{.25in}+ O\big( X^{-\frac{1}{2}+\varepsilon} Q^{\frac{5}{2}} (hk)^{\varepsilon} + X^{\varepsilon} Q^{\frac{3}{2}+\varepsilon} (hk)^{\varepsilon} + (XChk)^{\varepsilon} hk X^2 Q^{-96}\big).
\end{split}
\end{equation}
\section{The error term \texorpdfstring{$\mathcal{U}^r(h,k)$}{Ur(h,k)}}\label{sec: Ur}
Recall that $\lambda_1,\lambda_2,\dots$ are arbitrary complex numbers such that $\lambda_h \ll_{\varepsilon} h^{\varepsilon}$ for all $\varepsilon>0$. In this section, we bound the sum
$$
\sum_{h,k\leq Q^{\vartheta}} \frac{\lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k),
$$
where $\mathcal{U}^r(h,k)$ is defined by \eqref{eqn: Ur}. The majority of the work that follows consists of preparing the above sum for an eventual application of the large sieve.
We begin by showing that the terms in \eqref{eqn: Ur} that have sufficiently large $a\ell$ are zero. Since the support of $W$ is compact and contained in $(0,\infty)$, the summand in the definition \eqref{eqn: Ur} of $\mathcal{U}^r(h,k)$ is zero unless $|mh\pm nk|\asymp g\ell Q/c$, which implies that either $mh \gg g\ell Q/c$ or $nk \gg g\ell Q/c$. Since $a|g$, $c\leq C$, and $h,k\leq Q^{\vartheta}$, this means that the summand in \eqref{eqn: Ur} is zero unless
$$
m \gg \frac{Qg \ell}{hc} \geq \frac{Qa\ell}{CQ^{\vartheta}} \ \ \ \ \text{or} \ \ \ \ n \gg \frac{Qg\ell}{kc}\geq \frac{Qa\ell}{CQ^{\vartheta}} .
$$
Now $V(m/X)V(n/X)=0$ except if $m,n\ll X$. Thus the summand in \eqref{eqn: Ur} is zero unless
\begin{equation}\label{eqn: largeal}
X \gg \frac{Qa\ell}{CQ^{\vartheta}}.
\end{equation}
In other words, the terms in the definition \eqref{eqn: Ur} of $\mathcal{U}^r(h,k)$ are zero unless $a\ell \ll XCQ^{\vartheta-1}$.
We next show that the terms in \eqref{eqn: Ur} that have sufficiently large $ae\ell$ are negligible. We first consider the terms that have $mh/{g} \equiv \mp {nk}/{g}$ (mod~$ae\ell$). In this case, $|mh \pm nk|/g$ is a multiple of $ae\ell$ that is not zero because $mh\neq nk$. Thus $|mh \pm nk|/g \geq ae\ell$, and the triangle inequality implies that either $mh/g \geq ae\ell/2$ or $nk/g \geq ae\ell/2$. Since $h,k\leq Q^{\vartheta}$ and $g=(mh,nk)\geq 1$, these lower bounds imply that either $ae\ell \ll m Q^{\vartheta}$ or $ae\ell \ll n Q^{\vartheta}$. Hence, using the support of $V$ in the same way we deduced \eqref{eqn: largeal}, we see that the terms in \eqref{eqn: Ur} that have $mh/{g} \equiv \mp {nk}/{g}$ (mod~$ae\ell$) are zero unless
$ae\ell \ll XQ^{\vartheta}$.
Next, we consider the terms in \eqref{eqn: Ur} that have $mh/{g} \not\equiv \mp {nk}/{g}$ (mod~$ae\ell$) and $ae\ell \gg Y$, where $Y$ is a large parameter that we will choose later (in Section~\ref{sec: proofofthm}). For these terms, the orthogonality of Dirichlet characters implies that the $\psi$-sum in \eqref{eqn: Ur} is $O(1)$. Moreover, we have shown that these terms are zero unless \eqref{eqn: largeal} holds, and thus we may assume that $e\gg YQ^{\vartheta-1}/(XC)$. It follows from these and \eqref{eqn: divisorbound} that the sum of the terms in \eqref{eqn: Ur} that have $mh/{g} \not\equiv \mp {nk}/{g}$ (mod~$ae\ell$) and $ae\ell \gg Y$ is bounded by
\begin{equation}\label{eqn: aelggYterms}
\ll \sum_{1\leq c\leq C} \sum_{1\leq m,n\ll X} \frac{(mn)^{\varepsilon}}{\sqrt{mn}} \sum_{ \frac{Y Q^{1-\vartheta}}{XC} \ll e<\infty} \frac{1}{e} \sum_{a|g} \sum_{\substack{1\leq \ell <\infty \\ a\ell \ll XCQ^{\vartheta -1} } } \frac{(ae\ell)^{\varepsilon}}{ae\ell}\cdot \frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right).
\end{equation}
Since the support of $W$ is compact and contained in $(0,\infty)$, we have $|mh\pm nk|/(g\ell) \ll Q/c$ in \eqref{eqn: aelggYterms}, and so \eqref{eqn: aelggYterms} is
\begin{align*}
\ll & (XCQ)^{\varepsilon} Q \sum_{1\leq c\leq C}\frac{1}{c} \sum_{1\leq m,n\ll X} \frac{(mn)^{\varepsilon}}{\sqrt{mn}} \left(\frac{Y Q^{1-\vartheta}}{XC} \right)^{-1+\varepsilon} \ll (XCQY)^{\varepsilon} \frac{X^2 CQ^{\vartheta}}{Y}.
\end{align*}
This bound is small if $Y$ is, say, a large power of $Q$. We have thus shown that the terms in \eqref{eqn: Ur} that have $mh/{g} \not\equiv \mp {nk}/{g}$ (mod~$ae\ell$) and $ae\ell \gg Y$ are negligible for large enough $Y$.
From all these observations, we deduce for $h,k\leq Q^{\vartheta}$ and $Y\geq XQ^{\vartheta}$ that the total contribution of the terms in the definition \eqref{eqn: Ur} of $\mathcal{U}^r(h,k)$ that have $a\ell \gg XCQ^{\vartheta-1}$ or $ae\ell \gg Y$ is
$$
\ll (XCQY)^{\varepsilon} \frac{X^2 CQ^{\vartheta}}{Y}.
$$
Thus
\begin{align*}
\mathcal{U}^r(h,k)&= \frac{1}{2} \sum_{\substack{1\leq c \leq C \\ (c ,hk)=1 }} \mu(c) \sum_{\substack{1\leq m,n<\infty \\ (mn,c )=1\\ mh\neq nk}} \frac{\tau_A(m) \tau_B(n) }{\sqrt{mn}} V\left( \frac{m}{X} \right) V\left( \frac{n}{X} \right) \sum_{\substack{ 1\leq e <\infty \\ ( e ,g)=1 }}\frac{\mu(e)}{e}\notag\\
&\hspace{.5in} \times \sum_{a|g} \mu(a)\sum_{\substack{ 1\leq \ell <\infty \\ (ea\ell, \frac{mh}{g}\cdot\frac{nk}{g})=1 \\ a\ell \ll XCQ^{\vartheta-1} \\ ae\ell \ll Y }} \frac{1}{\phi(ea \ell)}\sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \psi \left( \frac{mh}{g} \right) \overline{\psi}\left( \mp \frac{nk}{g} \right) \notag\\
&\hspace{.75in} \times\frac{|mh\pm nk|}{g \ell} W\left( \frac{c|mh\pm nk|}{g\ell Q}\right) \ + \ O\left( (XCQY)^{\varepsilon} \frac{X^2 CQ^{\vartheta}}{Y}\right).
\end{align*}
We multiply both sides by $\lambda_h\overline{\lambda_k} (hk)^{-1/2}$ and then sum over all $h,k\leq Q^{\vartheta}$ to arrive at
\begin{equation}\label{eqn: Urshort}
\begin{split}
&\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k)= \frac{1}{2}\sum_{1\leq c \leq C} \mu(c) \sum_{1\leq e<\infty} \frac{\mu(e)}{e} \sum_{1\leq \ell<\infty} \sum_{\substack{1\leq a<\infty \\ a\ell \ll XCQ^{\vartheta-1} \\ ae\ell \ll Y }} \frac{\mu(a)}{\phi(ae\ell) \ell}\\
&\hspace{.25in}\times \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \Big\{ \mathcal{U}^+ (c,a,e,\ell,\psi) + \mathcal{U}^- (c,a,e,\ell,\psi)\Big\}
\ + \ O\left( (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y}\right),
\end{split}
\end{equation}
where $\mathcal{U}^{\pm} (c,a,e,\ell,\psi)$ is defined by
\begin{equation}\label{eqn: Upmdef}
\begin{split}
\mathcal{U}^{\pm} (c,a,e,\ell,\psi) &:= \sum_{\substack{h,k\leq Q^{\vartheta} \\ (c,hk)=1}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}} \sideset{}{'}\sum_{m,n} \frac{\tau_A(m)\tau_B(n)}{\sqrt{mn}} V\left( \frac{m}{X}\right)V\left( \frac{n}{X}\right)\\
&\hspace{.25in}\times\psi \left( \frac{mh}{g} \right) \overline{\psi}\left( \mp \frac{nk}{g} \right) \frac{|mh\pm nk|}{g } W\left( \frac{c|mh\pm nk|}{g\ell Q}\right),
\end{split}
\end{equation}
with the symbol $\sum'$ denoting summation over all positive integers $m,n$ such that $(mn,c)=1$, $mh\neq nk$, $(e,g)=1$, $a|g$, and $(ea\ell, mhnk/g^2)=1$, where $g=(mh,nk)$. We split the $a,e,\ell$-sum in \eqref{eqn: Urshort} into dyadic blocks and deduce that
\begin{equation}\label{eqn: Urshortbound}
\begin{split}
& \sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k) \ll
\sum_{1\leq c \leq C} \sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} \sum_{A<a\leq 2A} \sum_{E<e\leq 2E} \sum_{L<\ell\leq 2L} \frac{(ae\ell)^{\varepsilon}}{ ae^2\ell^2} \\
& \hspace{.25in} \times \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \Big\{ |\mathcal{U}^+ (c,a,e,\ell,\psi)| + |\mathcal{U}^- (c,a,e,\ell,\psi)|\Big\} \ + \ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y},
\end{split}
\end{equation}
where each of the summation variables $A,E,L$ runs through the set $\{2^{\nu}: \nu\in \mathbb{Z}, \nu \geq -1\}$. Note that we are abusing notation here and using the symbol $A$ to denote both the summation variable in \eqref{eqn: Urshortbound} and the set in $\tau_A$ in \eqref{eqn: Upmdef}. However, this will not cause confusion.
To remove the interdependence of the summation variables in \eqref{eqn: Upmdef}, we let $g_1=(h,k)$, $g_2=(m,n)$, $g_3=(m/g_2,k/g_1)$, and $g_4=(n/g_2,h/g_1)$, and make the change of variables $h=g_1g_4H$, $k=g_1g_3K$, $m=g_2g_3M$, and $n=g_2g_4N$. Recalling the definition $g=(mh,nk)$, we note that $g=g_1g_2g_3g_4$. By their definitions, the new variables satisfy the coprimality conditions $(g_3,g_4)=1$, $(H,g_3)=1$, $(K,g_4)=1$, $(H,K)=1$, $(M,g_4)=1$, $(N,g_3)=1$, $(M,N)=1$, $(M,K)=1$, and $(N,H)=1$. Furthermore, the properties of $m,h,n,k$ in \eqref{eqn: Upmdef} are equivalent to $(c,MNHKg_1g_2g_3g_4)=1$, $MH\neq NK$, $(e,g_1g_2g_3g_4)=1$, $a|g_1g_2g_3g_4$, $(ea\ell, MNHK)=1$, $g_1g_4 H\leq Q^{\vartheta}$, $g_1g_3 K\leq Q^{\vartheta}$, and $1\leq M,N<\infty$. Since $V$ has compact support, we may assume that $m,n\ll X$ in \eqref{eqn: Upmdef} and hence $g_2\ll X$. Thus, the result of this change of variables is
\begin{equation}\label{eqn: Upm}
\begin{split}
\mathcal{U}^{\pm} (c,a,e,\ell,\psi) = \sum_{\substack{g_1,g_2,g_3,g_4 \\ M, N, H,K}}^* \frac{ \lambda_{g_1g_4H} \overline{\lambda_{g_1g_3K}}}{g_1\sqrt{g_3g_4HK}} \frac{\tau_A(g_2g_3M)\tau_B(g_2g_4N)}{g_2\sqrt{g_3g_4MN}} V\left( \frac{g_2g_3M}{X}\right)V\left( \frac{g_2g_4N}{X}\right)\\
\times \psi (MH) \overline{\psi}( \mp NK) |MH\pm NK| W\left( \frac{c|MH\pm NK|}{\ell Q}\right),
\end{split}
\end{equation}
where $*$ denotes the conditions for $g_1,g_2,g_3,g_4, M, N, H,K$ listed above.
Our next task is to write \eqref{eqn: Upm} in terms of an Euler product. To this end, we apply Mellin inversion twice to write
\begin{equation}\label{eqn: mellintwice}
\begin{split}
V\left( \frac{g_2g_3x}{X}\right)
& V\left( \frac{g_2g_4y}{X}\right) |xH\pm yK| W\left( \frac{c|xH\pm yK|}{\ell Q}\right) \\
& = \frac{1}{(2\pi i)^2} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\frac{1}{2}+\varepsilon)} (xH)^{-s_1}(yK)^{-s_2} \int_0^{\infty}\int_0^{\infty} u^{s_1-1} v^{s_2-1}\\
&\hspace{.25in}\times V\left( \frac{g_2g_3u}{HX}\right)V\left( \frac{g_2g_4v}{KX}\right)|u\pm v| W\left( \frac{c|u\pm v|}{\ell Q}\right) \,dv \,du\,ds_2\,ds_1.
\end{split}
\end{equation}
We have chosen the lines of integration to be at $\re(s_1)=\re(s_2)=\frac{1}{2}+\varepsilon$ to facilitate later estimations. We let $\Psi:[0,\infty)\rightarrow \mathbb{R}$ be a smooth nonnegative function of compact support such that $\Psi(\xi)=1$ for all $\xi$ in the support of $V$. Then
\begin{align*}
V\left( \frac{g_2g_3u}{HX}\right) = \Psi\Big( \frac{ u}{XQ^{\vartheta}}\Big)V\left( \frac{g_2g_3u}{HX}\right)
\end{align*}
for all $u\geq 0$, and applying Mellin inversion on the right-hand side gives
\begin{align*}
V\left( \frac{g_2g_3u}{HX}\right) = \frac{1}{2\pi i } \Psi\Big( \frac{ u}{XQ^{\vartheta}}\Big) \int_{(\varepsilon)} \left(\frac{XH}{g_2g_3 u} \right)^{s_3} \widetilde{V}(s_3) \,ds_3.
\end{align*}
Similarly,
\begin{align*}
V\left( \frac{g_2g_4v}{KX}\right) =\frac{1}{2\pi i } \Psi\Big( \frac{ v}{XQ^{\vartheta}}\Big)\int_{(\varepsilon)} \left(\frac{XK}{g_2g_4 v} \right)^{s_4} \widetilde{V}(s_4) \,ds_4.
\end{align*}
It follows from these and \eqref{eqn: mellintwice} that
\begin{align*}
V\left( \frac{g_2g_3x}{X}\right)&V\left( \frac{g_2g_4y}{X}\right) |xH\pm yK| W\left( \frac{c|xH\pm yK|}{\ell Q}\right) \\
& = \frac{1}{(2\pi i)^4} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\varepsilon)} \int_{(\varepsilon)} H^{-s_1+s_3}K^{-s_2+s_4} \left(\frac{X}{g_2g_3} \right)^{s_3} \left(\frac{X }{g_2g_4} \right)^{s_4} x^{-s_1}y^{-s_2} \\
&\hspace{.25in} \times\int_0^{\infty} \int_0^{\infty} u^{s_1-s_3-1} v^{s_2-s_4-1} \Psi\Big( \frac{ u}{XQ^{\vartheta}}\Big)\Psi\Big( \frac{ v}{XQ^{\vartheta}}\Big)\widetilde{V}(s_3)\widetilde{V}(s_4) |u\pm v|\\
&\hspace{.5in} \times W\left( \frac{c|u\pm v|}{\ell Q}\right) \,dv \,du\,ds_4\,ds_3\,ds_2\,ds_1.
\end{align*}
Now take $x=M$ and $y=N$, and insert the result into \eqref{eqn: Upm} to deduce that
\begin{equation}\label{eqn: UpmMN}
\begin{split}
\mathcal{U}^{\pm} (c,a,e,\ell,\psi) &= \sum_{\substack{g_1,g_2,g_3,g_4 \\ M, N, H,K}}^* \frac{ \lambda_{g_1g_4H} \overline{\lambda_{g_1g_3K}}}{g_1\sqrt{g_3g_4HK}} \frac{\tau_A(g_2g_3M)\tau_B(g_2g_4N)}{g_2\sqrt{g_3g_4MN}} \psi (MH) \overline{\psi}( \mp NK)\\
&\hspace{.25in}\times \frac{1}{(2\pi i)^4} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\epsilon)} \int_{(\epsilon)} H^{-s_1+s_3}K^{-s_2+s_4} \left(\frac{X}{g_2g_3} \right)^{s_3} \left(\frac{X }{g_2g_4} \right)^{s_4}\\
&\hspace{.5in}\times M^{-s_1}N^{-s_2}\mathcal{V}(s_1,s_2,s_3,s_4 )\,ds_4\,ds_3\,ds_2\,ds_1,
\end{split}
\end{equation}
where $\mathcal{V}(s_1,s_2,s_3,s_4 )$ is defined by
\begin{equation}\label{eqn: Vs}
\begin{split}
\mathcal{V}(s_1,s_2,s_3,s_4 )
&= \mathcal{V}(s_1,s_2,s_3,s_4;X,Q, \vartheta,c,\ell) \\
& : = \widetilde{V}(s_3)\widetilde{V}(s_4)\int_0^{\infty} \int_0^{\infty} u^{s_1-s_3-1} v^{s_2-s_4-1} \Psi\Big( \frac{ u}{XQ^{\vartheta}}\Big)\Psi\Big( \frac{ v}{XQ^{\vartheta}}\Big)\\
&\hspace{.5in}\times |u\pm v| W\left( \frac{c|u\pm v|}{\ell Q}\right) \,dv \,du.
\end{split}
\end{equation}
The following lemma gives a bound for $\mathcal{V}(s_1,s_2,s_3,s_4 )$, and is analogous to \eqref{eqn: mellinrapiddecay}.
\begin{lemma}\label{lem: uvintegralbound}
If $j_1,j_2$ are nonnegative integers and $s_1,s_2,s_3,s_4$ are complex numbers such that $j_1+j_2\geq 1$, $\re(s_1-s_3)>0$, and $\re(s_2-s_4)>0$, then
\begin{align*}
\int_0^{\infty} \int_0^{\infty} u^{s_1-s_3-1} v^{s_2-s_4-1} \Psi\Big( \frac{ u}{XQ^{\vartheta}}\Big)\Psi\Big( \frac{ v}{XQ^{\vartheta}}\Big) |u\pm v| W\left( \frac{c|u\pm v|}{\ell Q}\right) \,dv \,du \\
\ll \frac{ (X Q^{\vartheta } )^{\re(s_1+s_2-s_3-s_4) } }{|s_1-s_3|^{j_1} |s_2-s_4|^{j_2}} \left( \frac{ \ell Q}{c}\right) \bigg(1+\frac{ XcQ^{\vartheta-1}}{\ell} \bigg)^{j_1+j_2-1},
\end{align*}
where the implied constant may depend only on $\Psi$, $W$, $\re(s_1-s_3)$, $\re(s_2-s_4)$, $j_1$, or $j_2$.
\end{lemma}
\begin{proof}
For brevity, let $\mathcal{D}$ denote the double integral in question, and let $W_0$ denote the function $W_0(\xi):=\xi W(\xi)$. Make the change of variables $u\mapsto {u\ell Q}/{c}$ and $v\mapsto {v\ell Q}/{c}$, then integrate by parts with respect to $u$ $j_1$ times and with respect to $v$ $j_2$ times to deduce that
\begin{align*}
\mathcal{D} &= (-1)^{j_1+j_2} \left( \frac{ \ell Q}{c}\right)^{s_1+s_2-s_3-s_4+1}\int_0^{\infty} \int_0^{\infty} \frac{u^{s_1-s_3+j_1-1}}{(s_1-s_3)\cdots (s_1-s_3+j_1-1)}\\
&\times \frac{v^{s_2-s_4+j_2-1}}{(s_2-s_4)\cdots (s_2-s_4+j_2-1)}\frac{\partial^{j_1}}{\partial u^{j_1}}\frac{\partial^{j_2}}{\partial v^{j_2}} \bigg\{ \Psi\Big( \frac{ u \ell }{XcQ^{\vartheta-1}}\Big)\Psi\Big( \frac{ v \ell }{XcQ^{\vartheta-1}}\Big) W_0(|u\pm v|) \bigg\} \,dv \,du.
\end{align*}
We may use the product rule and chain rule to bound the derivatives in the integrand. We also observe that the integrand is zero unless $u,v\ll XcQ^{\vartheta-1}/\ell$ and $|u\pm v|\asymp 1$, because $\Psi$ is supported on a compact subset of $[0,\infty)$ and $W$ is supported on a compact subset of $(0,\infty)$. Thus
\begin{equation}\label{eqn: uvintegralboundD}
\begin{split}
\mathcal{D} \ll \left( \frac{ \ell Q}{c}\right)^{\re(s_1+s_2-s_3-s_4)+1}\bigg(1+ \frac{ \ell }{XcQ^{\vartheta-1}} \bigg)^{j_1+j_2}\frac{1}{|s_1-s_3|^{j_1} |s_2-s_4|^{j_2}} \\
\times \mathop{\iint}_{\substack{0\leq u,v\ll { XcQ^{\vartheta-1}}/{\ell} \\ |u \pm v| \asymp 1 }} u^{\re(s_1-s_3)+j_1-1} v^{\re(s_2-s_4)+j_2-1} \,dv \,du.
\end{split}
\end{equation}
Since $j_1,j_2$ are nonnegative integers with $j_1+j_2\geq 1$, it holds that either $j_1\geq 1$ or $j_2\geq 1$. By renaming the variables $u$ and $v$ if necessary, we may suppose, without loss of generality, that $j_2\geq 1$. Then $v^{j_2-1}\ll (XcQ^{\vartheta-1}/{\ell})^{j_2-1}$. Moreover, for each $u$, the $v$-integral is over an interval of length $\ll \min\{1,XcQ^{\vartheta-1}/{\ell}\}$. Hence the $u,v$-integral in \eqref{eqn: uvintegralboundD} is at most
\begin{equation*}
\ll \bigg( \frac{ XcQ^{\vartheta-1}}{\ell} \bigg)^{\re(s_1+s_2-s_3-s_4) + j_1+j_2-1} \min\left\{ 1, \frac{ XcQ^{\vartheta-1}}{\ell}\right\}.
\end{equation*}
Since $\min\{1,1/x\}\asymp 1/(1+x)$ for $x>0$, this proves the lemma.
\end{proof}
Now \eqref{eqn: mellinrapiddecay}, \eqref{eqn: Vs}, and Lemma~\ref{lem: uvintegralbound} imply that
\begin{equation}\label{eqn: Vbound}
\begin{split}
\mathcal{V}(s_1,s_2,s_3,s_4 ) & \ll_{\varepsilon,j_1,j_2,j_3,j_4} \frac{(X Q^{\vartheta } )^{\re(s_1+s_2-s_3-s_4) }}{|s_1-s_3|^{j_1}|s_2-s_4|^{j_2}|s_3|^{j_3} |s_4|^{j_4}} \left( \frac{ \ell Q}{c}\right) \bigg(1+\frac{ XcQ^{\vartheta-1}}{\ell} \bigg)^{j_1+j_2-1}
\end{split}
\end{equation}
for any nonnegative integers $j_1,j_2,j_3,j_4$ with $j_1+j_2\geq 1$ and any complex numbers $s_1,s_2,s_3,s_4$ such that each of Re$(s_1-s_3)$, Re$(s_2-s_4)$, Re$(s_3)$, and Re$(s_4)$ is $\geq \varepsilon$. It follows that \eqref{eqn: UpmMN} is absolutely convergent, and we may interchange the order of summation to deduce that, recalling the conditions indicated by $*$ and listed before \eqref{eqn: Upm}, we have
\begin{equation}\label{eqn: Upm2}
\begin{split}
\mathcal{U}^{\pm} (c,a,e,\ell,\psi) =
& \frac{1}{(2\pi i)^4} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\frac{1}{2}+\varepsilon)} \int_{(\epsilon)} \int_{(\epsilon)} X^{s_3+s_4}\\
&\times \sum_{\substack{1\leq g_1,g_2,g_3,g_4, H,K \ll \max\{Q^{\vartheta},X\}\\ g_1g_4H \leq Q^{\vartheta}, \ g_1g_3K \leq Q^{\vartheta}, \ g_2\ll X \\ (g_3,g_4)=(H,g_3)=(K,g_4)=(H,K)=1 \\ (ec,g_1g_2g_3g_4)=(cae\ell,HK)=1 \\ a|g_1g_2g_3g_4 }} \frac{ \lambda_{g_1g_4H} \overline{\lambda_{g_1g_3K}} \, \psi ( H) \overline{\psi}( \mp K) }{g_1 g_2^{1+s_3+s_4} g_3^{1+s_3}g_4^{1+s_4} H^{\frac{1}{2}+s_1-s_3}K^{\frac{1}{2}+s_2-s_4}} \\
& \times \sum_{\substack{1\leq M,N<\infty \\ (M,g_4)=(N,g_3)=(M,K)=(N,H)=1 \\ (M,N)=(MN,cae\ell)=1 \\ MH\neq NK }} \frac{\tau_A(g_2g_3M)\tau_B(g_2g_4N)\psi (M ) \overline{\psi}( N) }{M^{\frac{1}{2}+s_1}N^{\frac{1}{2}+s_2}}\\
&\times \mathcal{V}(s_1,s_2,s_3,s_4 )\,ds_4\,ds_3\,ds_2\,ds_1.
\end{split}
\end{equation}
We next write the $M,N$-sum in terms of an Euler product. To do this, we first add and subtract the terms with $MH=NK$ and write
\begin{align}\label{eqn: MNdifference}
\sum_{\substack{1\leq M,N<\infty \\ (M,g_4)=(N,g_3)=(M,K)=(N,H)=1 \\ (M,N)=(MN,cae\ell)=1 \\ MH\neq NK }} \frac{\tau_A(g_2g_3M)\tau_B(g_2g_4N)\psi (M ) \overline{\psi}( N) }{M^{\frac{1}{2}+s_1}N^{\frac{1}{2}+s_2}} =\mathscr{P}_1 - \mathscr{P}_2,
\end{align}
where $\mathscr{P}_1$ is the sum on the left-hand side, except without the condition $MH\neq NK$, and $\mathscr{P}_2$ is the sum with the condition $MH =NK$ instead of $MH\neq NK$. To evaluate $\mathscr{P}_2$, observe that the conditions $(H,K)=1$ and $(M,N)=1$ imply that $MH=NK$ if and only if $M=K$ and $N=H$. Since $(M,K)=(N,H)=1$, this is only possible if $M=N=H=K=1$. Thus
\begin{equation}\label{eqn: P2evaluated}
\mathscr{P}_2=\tau_A(g_2g_3)\tau_B(g_2g_4).
\end{equation}
Next, we express the sum $\mathscr{P}_1$ defined in \eqref{eqn: MNdifference} as an Euler product and write
\begin{equation}\label{eqn: P1euler}
\mathscr{P}_1 = \prod_{\alpha\in A} L(\tfrac{1}{2}+s_1+\alpha,\psi) \prod_{\beta\in B} L(\tfrac{1}{2}+s_2+\beta,\overline{\psi}) \mathcal{R}(s_1,s_2),
\end{equation}
where $\mathcal{R}(s_1,s_2)$ is defined by
\begin{equation}\label{eqn: mathcalRdef}
\begin{split}
\mathcal{R}(s_1,s_2) =
& \mathcal{R}(s_1,s_2; g_2,g_3,g_4,H,K,cae\ell) \\
:= & \prod_p \Bigg\{ \prod_{\alpha\in A}\left( 1-\frac{\psi(p)}{p^{\frac{1}{2}+s_1+\alpha}} \right) \prod_{\beta\in B}\left( 1-\frac{\overline{\psi}(p)}{p^{\frac{1}{2}+s_2+\beta}} \right)\\
& \times \sum_{\substack{0\leq m,n<\infty \\ \min\{m,\text{ord}_p(g_4K) \} = \min\{n,\text{ord}_p(g_3H) \} =0 \\ \min\{m,n\}=\min\{mn,\text{ord}_p(cae\ell) \}=0}} \frac{\tau_A(p^{m+\text{ord}_p(g_2g_3) }) \tau_B(p^{n+\text{ord}_p(g_2g_4) })\psi(p^m)\overline{\psi}(p^n)}{p^{m(\frac{1}{2}+s_1)+n(\frac{1}{2}+s_2)}}\Bigg\}.
\end{split}
\end{equation}
If $\re(s_1),\re(s_2)\geq \varepsilon$ and $p | g_2g_3g_4 HKcae\ell$, then the local factor in \eqref{eqn: mathcalRdef} corresponding to $p$ is $O(p^{\varepsilon \text{ord}_p(g_2g_3g_4)})$ by \eqref{eqn: divisorbound}. Moreover, if $\re(s_1),\re(s_2)\geq \varepsilon$ and $p\nmid g_2g_3g_4 HKcae\ell$, then it follows from \eqref{eqn: taudef} and \eqref{eqn: divisorbound} that the local factor in \eqref{eqn: mathcalRdef} corresponding to $p$ is
\begin{align*}
\prod_{\alpha\in A}
& \left( 1-\frac{\psi(p)}{p^{\frac{1}{2}+s_1+\alpha}} \right) \prod_{\beta\in B}\left( 1-\frac{\overline{\psi}(p)}{p^{\frac{1}{2}+s_2+\beta}} \right) \sum_{\substack{0\leq m,n<\infty \\ \min\{m,n\}=0}} \frac{\tau_A(p^m) \tau_B(p^n)\psi(p^m)\overline{\psi}(p^n)}{p^{m(\frac{1}{2}+s_1)+n(\frac{1}{2}+s_2)}} \\
& = \left( 1-\frac{\tau_A(p)\psi(p)}{p^{\frac{1}{2}+s_1}} + O\left( \frac{1}{p^{1+\varepsilon}}\right)\right) \left( 1-\frac{\tau_B(p)\overline{\psi}(p)}{p^{\frac{1}{2}+s_2}} + O\left( \frac{1}{p^{1+\varepsilon}}\right)\right)\\
& \hspace{.25in} \times \left( 1+\frac{\tau_A(p)\psi(p)}{p^{\frac{1}{2}+s_1}} + \frac{\tau_B(p)\overline{\psi}(p)}{p^{\frac{1}{2}+s_2}} + O\left( \frac{1}{p^{1+\varepsilon}}\right)\right) \\
& = 1 + O\left( \frac{1}{p^{1+\varepsilon}}\right).
\end{align*}
Thus, if $\re(s_1),\re(s_2)\geq \varepsilon$, then the product in \eqref{eqn: mathcalRdef} converges absolutely and we have
\begin{equation}\label{eqn: mathcalRbound}
\mathcal{R}(s_1,s_2) \ll (g_2g_3g_4HKcae\ell)^{\varepsilon}
\end{equation}
because $\prod_{p|\nu}O(1) \ll \nu^{\varepsilon}$ for any positive integer $\nu$. Hence, \eqref{eqn: MNdifference} with \eqref{eqn: P2evaluated} and \eqref{eqn: P1euler} gives an analytic continuation of the $M,N$-sum in \eqref{eqn: Upm2} to the region with $\re(s_1),\re(s_2)\geq \varepsilon$. If $\psi$ is non-principal, then this analytic continuation has no poles in the region, and \eqref{eqn: divisorbound} and \eqref{eqn: mathcalRbound} imply that it is bounded by
$$
\ll (g_2g_3g_4HKcae\ell)^{\varepsilon}\Bigg\{ 1+ \prod_{\alpha\in A} |L(\tfrac{1}{2}+s_1+\alpha,\psi) | \prod_{\beta\in B} |L(\tfrac{1}{2}+s_2+\beta,\overline{\psi})| \Bigg\}
$$
for $\re(s_1),\re(s_2)\geq \varepsilon$. This fact together with \eqref{eqn: Vbound} implies that if $\psi$ is non-principal, then we may move the $s_1$- and $s_2$-lines in \eqref{eqn: Upm2} to $\re(s_1)=\re(s_2)=2\epsilon$ and deduce that
\begin{equation}\label{eqn: Upm3}
\begin{split}
\mathcal{U}^{\pm} (c,a,e,\ell,\psi) \ll
& (XQcae\ell)^{\varepsilon}\int_{(2\epsilon)} \int_{(2\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)}\\
&\times\Bigg| \sum_{\substack{1\leq g_1,g_2,g_3,g_4, H,K \ll \max\{Q^{\vartheta},X\}\\ g_1g_4H \leq Q^{\vartheta}, \ g_1g_3K \leq Q^{\vartheta}, \ g_2\ll X \\ (g_3,g_4)=(H,g_3)=(K,g_4)=(H,K)=1 \\ (ec,g_1g_2g_3g_4)=(cae\ell,HK)=1 \\ a|g_1g_2g_3g_4 }} \frac{ \lambda_{g_1g_4H} \overline{\lambda_{g_1g_3K}} \, \psi ( H) \overline{\psi}( \mp K) }{g_1 g_2^{1+s_3+s_4} g_3^{1+s_3}g_4^{1+s_4} H^{\frac{1}{2}+s_1-s_3}K^{\frac{1}{2}+s_2-s_4}} \Bigg| \\
& \times \Bigg\{ 1+\prod_{\alpha\in A} |L(\tfrac{1}{2}+s_1+\alpha,\psi)| \prod_{\beta\in B} |L(\tfrac{1}{2}+s_2+\beta,\overline{\psi})|\Bigg\}\\
&\times |\mathcal{V}(s_1,s_2,s_3,s_4 )|\,|ds_4\,ds_3\,ds_2\,ds_1|.
\end{split}
\end{equation}
We apply M\"{o}bius inversion to remove the interdependence of the variables $H$ and $K$ and write
\begin{align*}
& \sum_{\substack{ 1\leq H,K \ll \max\{Q^{\vartheta},X\}\\ g_1g_4H \leq Q^{\vartheta}, \ g_1g_3K \leq Q^{\vartheta} \\ (H,g_3)=(K,g_4)=(H,K)=1 \\ (cae\ell,HK)=1 }} \frac{ \lambda_{g_1g_4H} \overline{\lambda_{g_1g_3K}} \, \psi ( H) \overline{\psi}( \mp K) }{ H^{\frac{1}{2}+s_1-s_3}K^{\frac{1}{2}+s_2-s_4}}\\
&\hspace{.5in} = \sum_{\substack{ 1\leq H,K \ll \max\{Q^{\vartheta},X\} \\ g_1g_4H \leq Q^{\vartheta}, \ g_1g_3K \leq Q^{\vartheta} \\ (H,g_3)=(K,g_4)= 1 \\ (cae\ell,HK)=1 }} \sum_{\substack{d|H \\ d|K}} \mu(d) \frac{ \lambda_{g_1g_4H} \overline{\lambda_{g_1g_3K}} \, \psi ( H) \overline{\psi}( \mp K) }{ H^{\frac{1}{2}+s_1-s_3}K^{\frac{1}{2}+s_2-s_4}} \\
&\hspace{.5in} = \sum_{\substack{d\leq Q^{\vartheta} \\ (d,g_3g_4cae\ell)=1}} \frac{\mu(d)|\psi(d)|^2}{d^{1+s_1+s_2-s_3-s_4} } \sum_{\substack{ H\leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_1-s_3} } \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( \mp K) }{ K^{\frac{1}{2}+s_2-s_4}},
\end{align*}
where in the last line we have made the change of variables $H\mapsto dH$ and $K\mapsto dK$. From this, \eqref{eqn: Upm3}, the triangle inequality, and the fact that $\psi(\mp K) = \psi(\mp 1) \psi(K)$, we deduce that
\begin{align*}
\mathcal{U}^{\pm} (c,a,e,\ell,\psi) \ll
& (XQcae\ell)^{\varepsilon}\sum_{\substack{1\leq g_1,g_2,g_3,g_4 \ll \max\{Q^{\vartheta},X\}\\ g_1g_4 \leq Q^{\vartheta}, \ g_1g_3 \leq Q^{\vartheta}, \ g_2\ll X \\ (g_3,g_4)=(ec,g_1 g_2 g_3 g_4)=1 \\ a|g_1g_2g_3g_4 }} \frac{1}{ g_1 g_2^{1+\varepsilon} g_3^{1+\varepsilon}g_4^{1+\varepsilon} } \sum_{\substack{d\leq Q^{\vartheta} \\ (d,g_3g_4cae\ell)=1}} \frac{1}{d^{1+\varepsilon}}\\
& \times \int_{(2\epsilon)} \int_{(2\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)} \Bigg| \sum_{\substack{ H \leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_1-s_3} } \Bigg|\Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}(K) }{ K^{\frac{1}{2}+s_2-s_4}} \Bigg| \\
& \times \Bigg\{ 1+\prod_{\alpha\in A} |L(\tfrac{1}{2}+s_1+\alpha,\psi)| \prod_{\beta\in B} |L(\tfrac{1}{2}+s_2+\beta,\overline{\psi})|\Bigg\}\\
& \times |\mathcal{V}(s_1,s_2,s_3,s_4 )| \,|ds_4\,ds_3\,ds_2\,ds_1|.
\end{align*}
From this and \eqref{eqn: Urshortbound}, we arrive at
\begin{equation}\label{eqn: Urbound}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k) &\ll \sum_{1\leq c \leq C} \sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} \sum_{A<a\leq 2A} \sum_{E<e\leq 2E} \frac{( CXQY)^{\varepsilon}}{ AE^2L^2}\\
& \times \sum_{\substack{1\leq g_1,g_2,g_3,g_4\ll \max\{Q^{\vartheta},X\}\\ g_1g_4 \leq Q^{\vartheta}, \ g_1g_3 \leq Q^{\vartheta}, \ g_2\ll X \\ (g_3,g_4)=(ec,g_1 g_2 g_3 g_4)=1 \\ a|g_1g_2g_3g_4 }} \frac{1}{ g_1 g_2^{1+\varepsilon} g_3^{1+\varepsilon}g_4^{1+\varepsilon} } \sum_{\substack{d\leq Q^{\vartheta} \\ (d,g_3g_4cae )=1}} \frac{1}{d^{1+\varepsilon}}\\
& \times \Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ + \ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y},
\end{split}
\end{equation}
where $\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4}$ is defined by
\begin{align*}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4}
:=& \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}}\sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \int_{(2\epsilon)} \int_{(2\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)}\\
&\Bigg| \sum_{\substack{ H \leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_1-s_3} } \Bigg|\Bigg| \sum_{\substack{ K \leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}(K) }{ K^{\frac{1}{2}+s_2-s_4}} \Bigg| \\
&\Bigg\{ 1+\prod_{\alpha\in A} |L(\tfrac{1}{2}+s_1+\alpha,\psi)| \prod_{\beta\in B} |L(\tfrac{1}{2}+s_2+\beta,\overline{\psi})|\Bigg\}\\
&\times |\mathcal{V}(s_1,s_2,s_3,s_4 )| \,|ds_4\,ds_3\,ds_2\,ds_1|.
\end{align*}
We interchange the order of integration and then make the change of variables $s_5=s_1-s_3$ and $s_6=s_2-s_4$ to write
\begin{equation}\label{eqn: ellpsisum2}
\begin{split}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} = & \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}}\sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \int_{(\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)} |\mathcal{V}(s_3+s_5,s_4+s_6,s_3,s_4)|\\
& \times \Bigg| \sum_{\substack{ H \leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_5} } \Bigg| \Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( K) }{ K^{\frac{1}{2}+s_6}} \Bigg| \\
& \times \Bigg\{ 1 + \prod_{\alpha\in A} |L(\tfrac{1}{2}+s_3+s_5+\alpha,\psi)| \prod_{\beta\in B} |L(\tfrac{1}{2}+s_4+s_6+\beta,\overline{\psi})|\Bigg\}\\
&\times\,|ds_6\,ds_5\,ds_4\,ds_3|.
\end{split}
\end{equation}
Now GLH and the Phragm\'{e}n-Lindel\"{o}f principle together imply that if $\varepsilon>0$ then
$$
L(s,\psi) \ll_{\varepsilon} (q(1+|t|))^{\varepsilon}
$$
for all $s=\sigma+it$ with $\frac{1}{2}\leq \sigma\leq 1$ and real $t$ and all non-principal Dirichlet characters $\psi$ modulo $q$, where the implied constant depends only on $\varepsilon$. It follows from this and \eqref{eqn: ellpsisum2} that
\begin{equation}\label{eqn: ellpsisum3}
\begin{split}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ll & \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}} (ae\ell)^{\varepsilon} \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \int_{(\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)} \int_{(\epsilon)} |\mathcal{V}(s_3+s_5,s_4+s_6,s_3,s_4)| \\
&\times |s_3s_4s_5s_6|^{ \varepsilon} \Bigg| \sum_{\substack{ H \leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_5} } \Bigg| \Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( K) }{ K^{\frac{1}{2}+s_6}} \Bigg|\\
&\times \,|ds_6\,ds_5\,ds_4\,ds_3|.
\end{split}
\end{equation}
Our next task is to apply the bound \eqref{eqn: Vbound} for $\mathcal{V}$. We will facilitate later estimations by choosing particular values of $j_1,j_2,j_3,j_4$ in \eqref{eqn: Vbound} for specific ranges of $s_5$ and $s_6$. To this end, we split the range of integration of the $s_5$- and $s_6$-integrals in \eqref{eqn: ellpsisum3} into dyadic segments to write
\begin{equation}\label{eqn: ellpsisum4}
\begin{split}
&\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4}\\
&\hspace{.25in}\ll \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}} (ae\ell)^{\varepsilon} \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \sum_{S_5} \sum_{S_6} \mathop{\int}_{\substack{S_5\le |s_5| \le 2S_5\\ \re(s_5)=\epsilon} }\mathop{\int}_{\substack{S_6\le |s_6| \le 2S_6\\ \re(s_6)=\epsilon}}\\
&\hspace{.25in}\times \int_{(\epsilon)} \int_{(\epsilon)} |s_3 s_4 s_5 s_6|^{ \varepsilon} |\mathcal{V}(s_3+s_5,s_4+s_6,s_3,s_4)| \\
&\hspace{.25in} \times \Bigg| \sum_{\substack{ H\leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_5} } \Bigg| \Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( K) }{ K^{\frac{1}{2}+s_6}} \Bigg| \,|ds_4\,ds_3\,ds_6\,ds_5|,
\end{split}
\end{equation}
where each of $S_5$ and $S_6$ runs through the set $\{0\} \cup \{2^\nu : \nu\in\mathbb{Z}, \nu\geq 0\}$. Here, we make an abuse of notation and interpret the condition $S_5\le |s_5| \le 2S_5$ to mean $\epsilon\le |s_5|\le 1$ when $S_5=0$, and similarly for $S_6$. We now apply \eqref{eqn: Vbound}. We choose $j_3=j_4=2$ in every situation, while we choose $j_1$ and $j_2$ depending on $S_5$ and $S_6$, as specified in the following table.
\begin{center}
\begin{longtable}{| C | C || C| C |}
\hline
\multicolumn{2}{|c||}{conditions on} & \multicolumn{2}{c|}{\,\,\,\,\,choices of\,\,\,\,\,} \\
\multicolumn{1}{|c}{$S_5$} & \multicolumn{1}{c||}{$S_6$} & \multicolumn{1}{c} {\,\,\,\,\,$j_1$\,\,\,\,\,} & \multicolumn{1}{c|} {$j_2$} \\
\Xcline{1-4}{5\arrayrulewidth}
\endfirsthead
\multicolumn{4}{l} {{(table continued from previous page)}} \\
\hline
\multicolumn{2}{|c||}{conditions on} & \multicolumn{2}{c|}{\,\,\,\,\,choices of\,\,\,\,\,} \\
\multicolumn{1}{|c}{$S_5$} & \multicolumn{1}{c||}{$S_6$} & \multicolumn{1}{c} {\,\,\,\,\,$j_1$\,\,\,\,\,} & \multicolumn{1}{c|} {$j_2$} \\
\Xcline{1-4}{5\arrayrulewidth}
\endhead
\hline \multicolumn{4}{r}{{(table continued on next page)}} \\
\endfoot
\endlastfoot
S_5 = 0 & \vphantom{ \displaystyle\sum_{1}^{2}}S_6 = 0 & 1 & 0\\
\hline
S_5 = 0 & \vphantom{ \displaystyle\sum_{1}^{2}}0 < S_6\leq 1+\displaystyle \frac{XcQ^{\vartheta-1}}{L} & 0 & 1\\
\hline
S_5 = 0 & \vphantom{ \displaystyle\sum_{1}^{2}} S_6>1+\displaystyle \frac{XcQ^{\vartheta-1}}{L} & 0 & 3\\
\hline
\vphantom{ \displaystyle\sum_{1}^{2}} 0<S_5\leq 1+\displaystyle \frac{XcQ^{\vartheta-1}}{L} & S_6=0 & 1 & 0\\
\hline
\vphantom{ \displaystyle\sum_{1}^{2}} S_5> 1+\displaystyle\frac{XcQ^{\vartheta-1}}{L} & S_6=0 & 3 & 0\\
\hline
0<S_5< S_6 & \vphantom{ \displaystyle\sum_{1}^{2}} 0<S_6\leq 1+\displaystyle\frac{XcQ^{\vartheta-1}}{L} & 0 & 1\\
\hline
0<S_5< S_6 & \vphantom{ \displaystyle\sum_{1}^{2}} S_6> 1+\displaystyle\frac{XcQ^{\vartheta-1}}{L} & 0 & 3\\
\hline
\vphantom{ \displaystyle\sum_{1}^{2}} 0<S_5\leq 1+\displaystyle\frac{XcQ^{\vartheta-1}}{L} & 0<S_6\leq S_5 & 1 & 0\\
\hline
\vphantom{ \displaystyle\sum_{1}^{2}} S_5> 1+\displaystyle\displaystyle\frac{XcQ^{\vartheta-1}}{L} & 0<S_6 \leq S_5 & 3 &0\\
\hline
\caption{\label{tab: S5S6j1j2}Our choices of the values of $j_1$ and $j_2$ depend on the ranges of the variables of integration $s_5$ and $s_6$. }
\end{longtable}
\end{center}
We arrive at
\begin{equation}\label{eqn: beforelargesieve}
\begin{split}
& \Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ll (XQ)^{\varepsilon}\left( \frac{ L Q}{c}\right) \sum_{S_5>0} \sum_{S_6>0} \frac{1}{ S_5^{j_1-\varepsilon} S_6^{j_2-\varepsilon} } \bigg(1+\frac{ XcQ^{\vartheta-1}}{L} \bigg)^{j_1+j_2-1} \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}} (ae\ell)^{\varepsilon} \\
& \times \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \int_{\epsilon-i2S_5}^{\epsilon+i2S_5} \int_{\epsilon-i2S_6}^{\epsilon+i2S_6} \Bigg| \sum_{\substack{ H \leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_5} } \Bigg| \Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( K) }{ K^{\frac{1}{2}+s_6}} \Bigg| \,|ds_6\,ds_5|,
\end{split}
\end{equation}
where the values of $j_1$ and $j_2$ depend on $S_5$ and $S_6$ as described in Table~\ref{tab: S5S6j1j2}. Note that, for conciseness, we have bounded the term with $S_5=S_6=0$ in \eqref{eqn: ellpsisum4} by the term with $S_5=S_6=1$ in \eqref{eqn: beforelargesieve}. We may do this because both terms have the same value of $j_1+j_2$ by Table~\ref{tab: S5S6j1j2}. Similarly, we have bounded the sum of the terms with $S_5=0$ and $S_6>0$ in \eqref{eqn: ellpsisum4} by the sum of the terms with $S_5=1$ and $S_6>0$ in \eqref{eqn: beforelargesieve}, and we have bounded the sum of the terms with $S_5>0$ and $S_6=0$ in \eqref{eqn: ellpsisum4} by the sum of the terms with $S_5>0$ and $S_6=1$ in \eqref{eqn: beforelargesieve}.
In order to be able to apply the large sieve inequality, we use the Cauchy-Schwarz inequality to deduce from \eqref{eqn: beforelargesieve} that
\begin{align}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ll
& (X Q )^{\varepsilon} \left( \frac{ L Q}{c}\right) \sum_{S_5>0} \sum_{S_6>0} \frac{(aeLS_5S_6)^{\varepsilon}}{ S_5^{j_1 } S_6^{j_2 } } \bigg(1+\frac{ XcQ^{\vartheta-1}}{L} \bigg)^{j_1+j_2-1} \notag\\
& \times \Bigg( \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}}\sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \Bigg\{ \int_{\varepsilon-i2S_5}^{\varepsilon+i2S_5} \Bigg| \sum_{\substack{ H\leq Q^{\vartheta}/(dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_5} } \Bigg| \,|ds_5|\Bigg\}^2 \Bigg)^{1/2}\notag\\
& \times \Bigg( \sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}}\sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \Bigg\{ \int_{\varepsilon-i2S_6}^{\varepsilon+i2S_6} \Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( K) }{ K^{\frac{1}{2}+s_6}} \Bigg| \,|ds_6| \Bigg\}^2 \Bigg)^{1/2} \label{eqn: SigmaCauchySchwarz}.
\end{align}
We now apply the hybrid large sieve inequality in the form of the following lemma.
\begin{lemma}\label{hybridlargesieve}
Let $R,T,N,\sigma$ be real numbers with $T\geq 3$, $R,N\geq 1$, and $\sigma\geq 1/2$, and let $j$ be a positive integer. If $\{a_n\}$ is any sequence of complex numbers, then
$$
\sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}} \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2 \ll_{\varepsilon} (jRNT)^{\varepsilon} (RNT+jR^2T^2 ) \sum_{ n\leq N } \frac{|a_n|^2}{n^{2\sigma}},
$$
where the $\chi$-sum is over all non-principal Dirichlet characters $\chi$ mod~$qj$.
\end{lemma}
\begin{proof}
The proof of the lemma is contained within the proof of Proposition~1 of \cite{CIS}. For full details, see Appendix \ref{hybridlargesievedetails}.
\end{proof}
From Lemma~\ref{hybridlargesieve} with $R=2L$, $T=2S_5$, $N=Q^{\vartheta}/dg_1g_4$, $\sigma=\frac{1}{2}+\varepsilon$, and $j=ae$, we deduce that
\begin{equation}\label{eqn: applyhybridlargesieve}
\begin{split}
\sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}}
& \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}} \Bigg\{ \int_{\varepsilon-i2S_5}^{\varepsilon+i2S_5} \Bigg| \sum_{\substack{ H \leq Q^{\vartheta} / (dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ \lambda_{dg_1g_4H} \, \psi ( H) }{ H^{\frac{1}{2}+s_5} } \Bigg| \,|ds_5|\Bigg\}^2 \\
& \ll (aeLQS_5)^{\varepsilon} \bigg( \frac{Q^{\vartheta}LS_5}{dg_1g_4} +aeL^2S_5^2 \bigg)\sum_{\substack{ H \leq Q^{\vartheta} / (dg_1g_4) \\ (H,g_3cae\ell) = 1 }} \frac{ |\lambda_{dg_1g_4H}|^2 }{ H^{1+\varepsilon} } \\
& \ll (dg_1g_4aeL QS_5)^{\varepsilon} ( Q^{\vartheta} LS_5 +aeL^2S_5^2 ),
\end{split}
\end{equation}
where the last line follows from the assumption $\lambda_h \ll_{\varepsilon} h^{\varepsilon}$. Note that, in using Lemma~\ref{hybridlargesieve} here, we may assume without loss of generality that $2S_5\geq 3$ since if not, then we may extend the interval of integration because the integrand is nonnegative. Similarly, Lemma~\ref{hybridlargesieve} implies
\begin{align*}
\sum_{\substack{L<\ell\leq 2L \\ (d,\ell)=1}}
& \sum_{\substack{\psi \bmod ae\ell\\ \psi\neq \psi_0}}\Bigg\{ \int_{\varepsilon-i2S_6}^{\varepsilon+i2S_6} \Bigg| \sum_{\substack{ K\leq Q^{\vartheta}/(dg_1g_3) \\ (K,g_4cae\ell)= 1 }} \frac{ \overline{\lambda_{dg_1g_3K}} \, \overline{\psi}( K) }{ K^{\frac{1}{2}+s_6}} \Bigg| \,|ds_6| \Bigg\}^2 \\
& \ll (dg_1g_3aeLQS_6)^{\varepsilon} ( Q^{\vartheta} L S_6 +aeL^2S_6^2 ).
\end{align*}
From this, \eqref{eqn: applyhybridlargesieve}, and \eqref{eqn: SigmaCauchySchwarz}, we arrive at
\begin{equation}\label{eqn: applyhybridlargesieve2}
\begin{split}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ll
& (X Q L aedg_1g_3g_4 )^{\varepsilon}\left( \frac{ L Q}{c}\right) \sum_{S_5>0} \sum_{S_6>0} \frac{1}{ S_5^{j_1-\varepsilon} S_6^{j_2-\varepsilon} } \bigg(1+\frac{ XcQ^{\vartheta-1}}{L} \bigg)^{j_1+j_2-1} \\
& \times \Big( Q^{\vartheta} LS_5 +aeL^2S_5^2 \Big)^{1/2} \Big( Q^{\vartheta} LS_6 +aeL^2S_6^2 \Big)^{1/2}.
\end{split}
\end{equation}
By our choices of the values of $j_1$ and $j_2$ described in Table~\ref{tab: S5S6j1j2}, if $M,N\in \{2^{\nu}: \nu\in \mathbb{Z},\nu\geq 0\}$ are given, then the term on the right-hand side of \eqref{eqn: applyhybridlargesieve2} that corresponds to the pair $(S_5,S_6)=(M,N)$ is equal to the term that corresponds to the pair $(S_5,S_6)=(N,M)$. Thus, the part of the right-hand side of \eqref{eqn: applyhybridlargesieve2} that has $S_6\leq S_5$ is a bound for the left-hand side. In that part, we have $j_2=0$ by Table~\ref{tab: S5S6j1j2}. Hence
\begin{equation}\label{eqn: applyhybridlargesieve3}
\begin{split}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ll (X Q L aedg_1g_3g_4 )^{\varepsilon}\left( \frac{ L Q}{c}\right) \mathop{\sum_{S_5>0} \sum_{S_6>0}}_{S_6\leq S_5} \frac{S_6^{\varepsilon}}{ S_5^{j_1-\varepsilon} } \bigg(1+\frac{ XcQ^{\vartheta-1}}{L} \bigg)^{j_1-1} \\
\times \Big( Q^{\vartheta} LS_5 +aeL^2S_5^2 \Big).
\end{split}
\end{equation}
Recall that, as stated below \eqref{eqn: ellpsisum4}, the variables $S_5$ and $S_6$ in \eqref{eqn: applyhybridlargesieve3} each run through the set $\{2^{\nu}: \nu\in\mathbb{Z},\nu \geq 0\}$. Moreover, as described in Table~\ref{tab: S5S6j1j2}, we have $j_1=1$ for the terms in \eqref{eqn: applyhybridlargesieve3} that have $S_5 \leq 1+ XcQ^{\vartheta-1}/L$ and $j_1=3$ for the terms with $S_5 > 1+ XcQ^{\vartheta-1}/L$. We may thus evaluate the $S_5$- and $S_6$-sums in \eqref{eqn: applyhybridlargesieve3} by writing
\begin{equation}\label{eqn: geometricsum1}
\sum_{0<S_6 \leq S_5} S_6^{\varepsilon} \ll S_5^{\varepsilon}
\end{equation}
for each $S_5$,
\begin{equation}\label{eqn: geometricsum2}
\sum_{0<S_5 \leq 1+ XcQ^{\vartheta-1}/L} \frac{ Q^{\vartheta} LS_5 +aeL^2S_5^2 }{S_5^{1-\varepsilon}} \ll (XcQ)^{\varepsilon} \bigg(Q^{\vartheta} L + aeL^2 \bigg(1 + \frac{ XcQ^{\vartheta-1} }{L} \bigg) \bigg),
\end{equation}
and
\begin{align*}
\sum_{S_5 > 1+ XcQ^{\vartheta-1}/L}
& \frac{ Q^{\vartheta} LS_5 +aeL^2S_5^2 }{S_5^{3-\varepsilon}}\bigg(1 + \frac{ XcQ^{\vartheta-1} }{L} \bigg)^2 \\
& \ll (XcQ)^{\varepsilon} \bigg(Q^{\vartheta} L + aeL^2 \bigg(1 + \frac{ XcQ^{\vartheta-1} }{L} \bigg) \bigg).
\end{align*}
From this, \eqref{eqn: applyhybridlargesieve3}, \eqref{eqn: geometricsum1}, and \eqref{eqn: geometricsum2}, we deduce that
\begin{equation*}
\Sigma_{c,a,e,d,g_1,g_2,g_3,g_4} \ll (X QLcaedg_1g_3g_4)^{\varepsilon}\left( \frac{ L Q}{c}\right)\big( Q^{\vartheta}L + aeL^2 +aeL XcQ^{\vartheta-1} \big) .
\end{equation*}
From this and \eqref{eqn: Urbound}, we arrive at
\begin{align}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k) \ll
& \sum_{1\leq c \leq C} \sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} \sum_{A<a\leq 2A} \sum_{E<e\leq 2E} \frac{( CXQY)^{\varepsilon}}{ AE^2L^2} \notag\\
& \times \sum_{\substack{1\leq g_1,g_2,g_3,g_4\ll \max\{Q^{\vartheta},X\}\\ g_1g_4 \leq Q^{\vartheta}, \ g_1g_3 \leq Q^{\vartheta}, \ g_2\ll X \\ (g_3,g_4)=(ec,g_1 g_2 g_3 g_4)=1 \\ a|g_1g_2g_3g_4 }} \frac{1}{ (g_1g_2g_3g_4)^{1-\varepsilon} } \sum_{\substack{d\leq Q^{\vartheta} \\ (d,g_3g_4cae )=1}} \frac{1}{d^{1-\varepsilon}} \notag\\
& \times\left( \frac{ L Q}{c}\right)\big( Q^{\vartheta}L + AEL^2 +AEL XCQ^{\vartheta-1} \big) \ + \ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y}.\label{eqn: Urbound2}
\end{align}
Our final task for this section is to evaluate the right-hand side of \eqref{eqn: Urbound2}. Observe that
\begin{equation}\label{eqn: dsumbound}
\sum_{\substack{d\leq Q^{\vartheta} \\ (d,g_3g_4cae )=1}} \frac{1}{d^{1-\varepsilon}} \cdot d \ll Q^{\varepsilon}.
\end{equation}
To evaluate the $g_1,g_2,g_3,g_4$-sum in \eqref{eqn: Urbound2}, we group together terms with the same product $g_1 g_2 g_3 g_4$ and use the divisor bound to write
\begin{align*}
\sum_{\substack{1\leq g_1,g_2,g_3,g_4\ll \max\{Q^{\vartheta},X\}\\ g_1g_4 \leq Q^{\vartheta}, \ g_1g_3 \leq Q^{\vartheta}, \ g_2\ll X \\ (g_3,g_4)=(ec,g_1 g_2 g_3 g_4)=1 \\ a|g_1g_2g_3g_4 }} \frac{1}{ (g_1g_2g_3g_4)^{1-\varepsilon} }
& \ll \sum_{\substack{\nu\ll XQ^{2\vartheta} \\ a|\nu }} \frac{1}{\nu^{1-\varepsilon}} \\
& \ll \frac{(XQa)^{\varepsilon}}{a} \ll \frac{(XQA)^{\varepsilon}}{A}.
\end{align*}
From this, \eqref{eqn: dsumbound}, and \eqref{eqn: Urbound2}, we deduce that
\begin{align*}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k) &\ll Q\sum_{1\leq c \leq C}\frac{1}{c} \sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} \frac{(CXQY)^{\varepsilon}}{ A E L}\\
&\hspace{.25in}\times\big( Q^{\vartheta}L + AEL^2 +AEL XCQ^{\vartheta-1} \big) + \ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y}.
\end{align*}
The condition $AL\ll XCQ^{\vartheta-1}$ implies that $AEL^2 \ll AEL XCQ^{\vartheta-1} $ because $A\gg 1$. Moreover, we have $\sum_{c\leq C} (1/c)\ll C^{\varepsilon}$. Hence
\begin{align}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k)
\ll (CXQY)^{\varepsilon} Q \sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} \frac{1}{ A E L } \big( LQ^{\vartheta} +AEL XCQ^{\vartheta-1} \big) \notag\\
+ \ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y}. \label{eqn: Urbound3}
\end{align}
Recall that, as stated below \eqref{eqn: Urshortbound}, each of the summation variables $A,E,L$ in \eqref{eqn: Urbound3} runs through the set $\{2^{\nu}:\nu\in \mathbb{Z}, \nu\geq -1\}$. We may thus evaluate the $A,E,L$-sum in \eqref{eqn: Urbound3} by writing
\begin{equation*}
\sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} \frac{Q^{\vartheta}}{ A E } \leq \sum_{\substack{A,E,L \\ AEL \ll Y }} 4Q^{\vartheta}
\ll Y^{\varepsilon} Q^{\vartheta}
\end{equation*}
and
\begin{equation*}
\sum_{\substack{A,E,L \\ AL \ll XCQ^{\vartheta-1} \\ AEL \ll Y }} XCQ^{\vartheta-1} \leq XCQ^{\vartheta-1}\sum_{\substack{A,E,L \\ AEL \ll Y }} 1
\ll Y^{\varepsilon} XCQ^{\vartheta-1}.
\end{equation*}
We conclude that
\begin{equation}\label{eqn: Urboundfinal}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{U}^r(h,k)
\ll_{\varepsilon} (XCQY)^{\varepsilon} Q ( Q^{\vartheta} + XCQ^{\vartheta-1}) \ + \ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y}.
\end{equation}
\section{Finishing the proof of Theorem~\ref{thm: main}}\label{sec: proofofthm}
We put together our estimates and deduce from \eqref{eqn: IandIstar}, \eqref{eqn: Ssplit}, \eqref{eqn: Dis0swap}, \eqref{eqn: Lsplit}, \eqref{eqn: Uready}, and \eqref{eqn: U2is1swap} that
\begin{equation}\label{eqn: SI0I1E}
\mathcal{S}(h,k) = \mathcal{I}_0(h,k) + \mathcal{I}_1(h,k) + \mathcal{E}(h,k),
\end{equation}
where
\begin{equation}\label{eqn: mathcalEdef}
\begin{split}
\mathcal{E}(h,k) = \mathcal{L}^r(h,k) + \mathcal{U}^r(h,k) + O \bigg( \bigg( Q + \frac{Q^2}{C}\bigg) \frac{(XCQhk)^{\varepsilon}(h,k)}{\sqrt{hk}} \bigg) \\
+O\Big((XCQhk)^{\varepsilon} \big( XC+ X^{-\frac{1}{2}} Q^{\frac{5}{2}} + Q^{\frac{3}{2}} +X^2hk Q^{-96} \big) \Big).
\end{split}
\end{equation}
For any $\vartheta>0$, we have
\begin{equation*}
\sum_{h,k\leq Q^{\vartheta}} \frac{(hk)^{\varepsilon} (h,k)}{hk} = \sum_{h,k\leq Q^{\vartheta}} \frac{(hk)^{\varepsilon} }{hk} \sum_{\substack{d|h \\ d|k}} \phi(d) = \sum_{d \leq Q^{\vartheta}} \frac{\phi(d)}{d^{2-\varepsilon}} \Bigg( \sum_{j\leq Q^{\vartheta}/d} \frac{1}{j^{1-\varepsilon}} \Bigg)^2 \ll Q^{\varepsilon},
\end{equation*}
\begin{equation*}
\sum_{h,k\leq Q^{\vartheta}} \frac{(hk)^{\varepsilon} }{\sqrt{hk}} \ll Q^{\vartheta+\varepsilon},
\end{equation*}
and
\begin{equation*}
\sum_{h,k\leq Q^{\vartheta}}\frac{(hk)^{\varepsilon} hk}{\sqrt{hk}} \ll Q^{3\vartheta+\varepsilon}.
\end{equation*}
From these bounds, \eqref{eqn: Lrbound}, \eqref{eqn: Urboundfinal}, and \eqref{eqn: mathcalEdef}, we deduce that if $\vartheta>0$ and $\{\lambda_h\}_{h=1}^{\infty}$ is any sequence of complex numbers such that $\lambda_h \ll_{\varepsilon} h^{\varepsilon}$ for all positive integers $h$, then
\begin{equation}\label{eqn: mathcalEbound}
\begin{split}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{E}(h,k) \ll (XCQ)^{\varepsilon} \bigg(Q^{1+\vartheta} + \frac{Q^2}{C} \bigg) + (XCQY)^{\varepsilon} ( Q^{1+\vartheta} + XCQ^{\vartheta}) \\
+ (XCQY)^{\varepsilon} \frac{X^2 CQ^{2\vartheta}}{Y} + (XCQ)^{\varepsilon} \big( XCQ^{\vartheta} + X^{-\frac{1}{2}} Q^{\frac{5}{2}+\vartheta} + Q^{\frac{3}{2}+\vartheta} + X^2Q^{-96+3\vartheta}\big).
\end{split}
\end{equation}
Recall our assumption that $X=Q^{\eta}$ with $1<\eta<2$. We optimize the upper bound \eqref{eqn: mathcalEbound} by choosing
$$
C = Q^{1-\frac{\vartheta}{2}-\frac{\eta}{2}},
$$
which implies $Q^2/C = XCQ^{\vartheta} = Q^{1+\frac{\vartheta}{2}+\frac{\eta}{2}}$. We impose the condition
\begin{equation*}
\vartheta<2-\eta
\end{equation*}
so that $C\gg Q^{\varepsilon}$. Note that $\vartheta<2-\eta$ implies $\vartheta<\eta$ since $\eta>1$. We also choose $Y$ to be a large power of $Q$, say $Y=Q^{99}$. With these choices for $C$ and $Y$ and the condition $\vartheta<2-\eta$, we deduce from \eqref{eqn: mathcalEbound} that
\begin{equation}\label{eqn: mathcalEbound2}
\sum_{h,k\leq Q^{\vartheta}} \frac{ \lambda_h \overline{\lambda_k}}{\sqrt{hk}}\mathcal{E}(h,k) \ll Q^{1+\frac{\vartheta}{2}+\frac{\eta}{2}+\varepsilon} + Q^{\frac{5}{2}-\frac{\eta}{2}+\vartheta+\varepsilon} .
\end{equation}
We have thus proved that the conclusion of Theorem~\ref{thm: main} holds under the additional assumption \eqref{eqn: orbitals}. To complete the proof of Theorem~\ref{thm: main}, it is left to show that \eqref{eqn: mathcalEbound2} holds for any multisets $A$ and $B$ of complex numbers with moduli $\leq C_1/\log Q$, where $C_1$ is an arbitrary fixed positive constant. We do this by showing for each $\ell=0,1$ that $\mathcal{I}_\ell(h,k)$ is holomorphic in each of the variables $\alpha\in A$ and $\beta\in B$ in the region where $|\alpha|,|\beta|\leq C_1/\log Q$ for all $\alpha\in A$ and $\beta\in B$ (or, more precisely, that the only singularities of $\mathcal{I}_\ell(h,k)$ in this region are removable singularities). The holomorphy of $\mathcal{I}_0(h,k)$ is immediate from \eqref{eqn: I_l(h,k)def2} with $\ell=0$: if $\ell=0$ then the integrand on the right-hand side of \eqref{eqn: I_l(h,k)def2} is holomorphic in each of the variables $\alpha\in A$ and $\beta\in B$ so long as $\alpha,\beta\ll \varepsilon$ for each $\alpha\in A$ and $\beta\in B$. To prove the holomorphy of $\mathcal{I}_1(h,k)$, define $I_{E,F}(n)$ for finite multisets $E,F$ of complex numbers by the Dirichlet series expression
\begin{equation*}
\frac{\prod_{\xi\in E}\zeta(\xi+s)}{\prod_{\rho\in F} \zeta(\rho+s)} = \sum_{n=1}^{\infty} \frac{I_{E,F}(n)}{n^s}.
\end{equation*}
This definition implies that if $\alpha\in A$ and $\re(s)$ is sufficiently large, then
\begin{equation*}
\sum_{n=1}^{\infty}\frac{I_{A \cup \{-\beta \}, \{\alpha \} }(n)}{n^s} = \zeta(-\beta +s) \prod_{\hat{\alpha}\neq \alpha} \zeta(\hat{\alpha} +s).
\end{equation*}
From this and the uniqueness of Dirichlet coefficients, we deduce that if $\alpha\in A$, then
\begin{equation}\label{eqn: IABidentity1}
I_{A \cup \{-\beta \}, \{\alpha \} }(n) = {\tau}_{A\smallsetminus \{\alpha \} \cup \{-\beta \} }(n)
\end{equation}
for every positive integer $n$. Similarly, if $\beta\in B$, then
\begin{equation}\label{eqn: IABidentity2}
I_{B \cup \{-\alpha \}, \{\beta \} } (n) = {\tau}_{B\smallsetminus \{\beta \} \cup \{-\alpha\} }(n)
\end{equation}
for every positive integer $n$. Now we claim that if $A$ and $B$ have no repeated elements and the elements of $A\cup B$ are distinct from each other and are $\ll 1/\log Q$, then
\begin{equation}\label{eqn: vandermonde1swap}
\begin{split}
\mathcal{I}_{1}(h,k) & = \sum_{\substack{q=1 \\ (q,hk)=1} }^{\infty} W\left( \frac{q}{Q}\right)\sideset{}{^\flat}\sum_{\chi \bmod q} \frac{1}{(2\pi i )^4} \int_{(\epsilon)} \int_{(\epsilon)} \oint_{|z|= \epsilon/4} \oint_{|y|=\epsilon/4} X^{s_1+s_2} \widetilde{V}(s_1) \widetilde{V}(s_2) \\
&\hspace{.25in} \times \mathscr{X} (\tfrac{1}{2}-z +s_1 )\mathscr{X} (\tfrac{1}{2}-y+s_2 ) q^{z-s_1+y-s_2} \\
&\hspace{.25in} \times \frac{ \prod_{\substack{\alpha\in A \\ \beta\in B}} \zeta(1+\alpha+\beta+s_1+s_2) \prod_{\alpha\in A} \zeta(1+\alpha+z) \prod_{\beta\in B} \zeta(1+\beta+y) }{ \prod_{\alpha\in A} \zeta(1+\alpha+s_1-y+s_2) \prod_{\beta\in B} \zeta(1-z+s_1+\beta+s_2) } \\
&\hspace{.25in} \times \zeta(1+y+z-s_1-s_2) \zeta(1-y-z+s_1+s_2) \prod_{p|q} P_0 \\
&\hspace{.05in} \times \prod_{p|hk}\Bigg\{ P_0 \!\!\!\!\!\! \sum_{ \substack{0\leq m,n<\infty\\ m+\text{ord}_p(h) = n+\text{ord}_p(k)}} \!\!\!\!\!\! \frac{I_{A_{s_1} \cup \{y-s_2\}, \{-z+s_1\} } (p^m) I_{B_{s_2} \cup \{z-s_1\}, \{-y+s_2\} } (p^n) }{p^{m/2}p^{n/2} } \Bigg\}\\
&\hspace{.05in} \times \prod_{p\nmid qhk} \Bigg\{P_0 \ \sum_{m=0}^{\infty} \frac{I_{A_{s_1} \cup \{y-s_2\}, \{-z+s_1\} } (p^m) I_{B_{s_2} \cup \{z-s_1\}, \{-y+s_2\} } (p^m) }{p^{m} } \Bigg\}\\
&\hspace{.25in} \times \,dy\,dz\,ds_2\,ds_1,
\end{split}
\end{equation}
where $P_0$ is defined by
\begin{align*}
P_0 = P_0(z,y,s_1,s_2;A,B)
& := \left( 1-\frac{1}{p}\right)^{-2} \left( 1 - \frac{1}{p^{ 1+y+z-s_1-s_2 }}\right) \left( 1 - \frac{1}{p^{ 1-y-z+s_1+s_2 }}\right) \\
& \times \prod_{ \substack{\alpha\in A \\ \beta\in B}} \left( 1 - \frac{1}{p^{1+\alpha+\beta+s_1+s_2 }}\right) \prod_{\alpha\in A} \left( 1 - \frac{1}{p^{1+\alpha+z }}\right) \prod_{\beta\in B}\left( 1 - \frac{1}{p^{ 1+\beta+y }}\right) \\
& \times \prod_{\alpha\in A} \left( 1 - \frac{1}{p^{1+\alpha+s_1-y+s_2 }}\right)^{-1} \prod_{\beta\in B} \left( 1 - \frac{1}{p^{ 1-z+s_1+\beta+s_2 }}\right)^{-1} .
\end{align*}
To see this, we use the residue theorem to evaluate the $z$- and $y$-integrals. The Euler product on the right-hand side of \eqref{eqn: vandermonde1swap} converges absolutely by an argument similar to the proof of Lemma~\ref{lem: 1swapeulerbound}. Thus the poles of the integrand that are enclosed by the circles $|z|=\epsilon/4$ and $|y|=\epsilon/4$ are precisely the poles of the factors
$$
\prod_{\alpha\in A}\zeta(1+\alpha+z)\prod_{\beta\in B} \zeta(1+\beta+y).
$$
After evaluating the $z$- and $y$-integrals using the residue theorem, we may simplify each residue by using \eqref{eqn: IABidentity1} and \eqref{eqn: IABidentity2} to see that the right-hand side of \eqref{eqn: vandermonde1swap} is equal to the right-hand side of \eqref{eqn: I_l(h,k)def2} with $\ell=1$. This proves our claim that \eqref{eqn: vandermonde1swap} holds if $A$ and $B$ have no repeated elements and the elements of $A\cup B$ are distinct from each other. Now the right-hand side of \eqref{eqn: vandermonde1swap} is holomorphic in each of the variables $\alpha\in A$ and $\beta\in B$ in any region with $\alpha,\beta\ll 1/\log Q$ for each $\alpha\in A$ and $\beta\in B$ because the Euler product in its integrand converges absolutely. Hence, by analytic continuation, it follows that $\mathcal{I}_1(h,k)$ is holomorphic in each of the variables $\alpha\in A$ and $\beta\in B$ in the region. As a side note, we remark that this argument can be generalized to show the holomorphy of $\mathcal{I}_{\ell}(h,k)$ for each $\ell $ with $0\leq \ell\leq \min\{|A|,|B|\}$.
We have now shown that $\mathcal{I}_0(h,k)$ and $\mathcal{I}_1(h,k)$ are each holomorphic in each of the variables $\alpha\in A$ and $\beta\in B$ in any given region such that $\alpha,\beta\ll 1/\log Q$ for each $\alpha\in A$ and $\beta\in B$. Now $\mathcal{S}(h,k)$ is holomorphic in the same region since its definition \eqref{eqn: S(h,k)def} has only finitely many nonzero terms by the assumption that $W$ and $V$ are compactly supported. It follows from these and \eqref{eqn: SI0I1E} that $\mathcal{E}(h,k)$ is also holomorphic in the same region. Thus, since \eqref{eqn: mathcalEbound} holds for $A,B$ satisfying the condition \eqref{eqn: orbitals}, the maximum modulus principle implies that \eqref{eqn: mathcalEbound} also holds for finite multisets $A,B$ satisfying $|\alpha|,|\beta|\leq C_0/\log Q$ for all $\alpha\in A$ and $\beta\in B$, where $C_0$ is the arbitrary positive constant in \eqref{eqn: orbitals}. This completes the proof of Theorem~\ref{thm: main}.
\appendix
\section{Proof of Lemma \ref{hybridlargesieve}}\label{hybridlargesievedetails}
In this section, we give the details of the proof of Lemma \ref{hybridlargesieve}, which is an analogue of Proposition~1 of \cite{CIS} and likewise a consequence of the hybrid large sieve in the form of Theorem~9.12 of \cite{IK}.
\begin{proof}[Proof of Lemma \ref{hybridlargesieve}]
To apply Theorem~9.12 of \cite{IK}, we need to express each $\chi$ mod~$qj$ in terms of a product of two characters, one with modulus $\tilde{q}$ and the other with modulus $\tilde{\jmath}$, where $\tilde{q}$ and $\tilde{\jmath}$ are factors of $qj$ such that $(\tilde{q}, \tilde{\jmath})=1 $. To this end, recall that each Dirichlet character $\chi$ mod~$qj$ is induced by a unique primitive Dirichlet character modulo some divisor of $qj$. We may write this divisor uniquely as $\tilde{q}\tilde{\jmath}$, where $(\tilde{q},j)=1$ and $\tilde{\jmath}$ is composed only of primes that divide $j$. Note that if $\chi$ is non-principal, then $\tilde{q}\tilde{\jmath}>1$. Since $\tilde{q}\tilde{\jmath}$ is a divisor of $qj$, it holds that $qj=D\tilde{q}\tilde{\jmath}$ for some positive integer $D$, and dividing both sides by $(j,\tilde{\jmath})$ implies
$$
q\frac{j}{(j,\tilde{\jmath})} = D\tilde{q}\frac{\tilde{\jmath}}{(j,\tilde{\jmath})}.
$$
It follows that $j/(j,\tilde{\jmath})$ divides $D$ because $j/(j,\tilde{\jmath})$ is relatively prime to both $\tilde{q}$ and $\tilde{\jmath}/(j,\tilde{\jmath})$. Thus we may write $D=dj/(j,\tilde{\jmath})$ for some positive integer $d$. Hence $q= d\tilde{q}\tilde{\jmath}/(j,\tilde{\jmath})$. We have thus shown that for each non-principal $\chi$ mod $qj$, there is a unique quadruple $(\tilde{\jmath},d,\tilde{q},\tilde{\chi})$ such that $\tilde{\jmath}$ is a positive integer composed only of the primes dividing $j$, $\tilde{q}$ is a positive integer with $(\tilde{q},j)=1$ and $\tilde{q}\tilde{\jmath}>1$, $d$ is a positive integer such that $q= d\tilde{q}\tilde{\jmath}/(j,\tilde{\jmath})$, and $\tilde{\chi}$ is a primitive character modulo $\tilde{q}\tilde{\jmath}$ such that $\chi=\tilde{\chi}\chi_0$, where $\chi_0$ is the principal character modulo $qj$. Therefore we have
\begin{align*}
\sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}} &\Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2\\
&\leq \sum_{q\leq R} \sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \sum_{\substack{ 1\leq d,\tilde{q} <\infty \\ (\tilde{q},j)=1 \\ \tilde{q}\tilde{j}>1 \\ q=d\tilde{q} {\tilde{\jmath}}/{(j,\tilde{\jmath})} }} \,\sideset{}{^*}\sum_{ \tilde{\chi} \bmod{\tilde{q}\tilde{\jmath}} } \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \tilde{\chi}(n)\chi_0(n) }{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2
\end{align*}
because the summand is nonnegative, where the * notation indicates that the sum is over primitive characters. We substitute $q=d\tilde{q} \tilde{\jmath}/(j,\tilde{\jmath})$ to write
\begin{align*}
\sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}} &\Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2\\
&\leq \sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \sum_{d \leq R \frac{(j,\tilde{\jmath})}{\tilde{\jmath}}} \sum_{ \substack{\tilde{q} \leq R \frac{(j,\tilde{\jmath})}{d\tilde{\jmath}} \\ (\tilde{q},j)=1 \\ \tilde{q}\tilde{\jmath}>1 }} \,\sideset{}{^*}\sum_{ \tilde{\chi} \bmod{\tilde{q}\tilde{\jmath}} } \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \tilde{\chi}(n)\chi_0(n) }{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2,
\end{align*}
where $\chi_0$ denotes the principal character modulo $qj=d\tilde{q}j\tilde{\jmath} /(j,\tilde{\jmath}) $. Now we may replace the function $\chi_0$ on the right-hand side with the characteristic function of the condition $(n,dj)=1$. Indeed, if $(n,dj)>1$, then $n$ and $d\tilde{q}j\tilde{\jmath}/(j,\tilde{\jmath})$ are not relatively prime, and so $\chi_0(n)=0$. If $(n,dj)=1$ and $(n,\tilde{q})>1$, then $\tilde{\chi}(n)\chi_0(n)=\tilde{\chi}(n)$ because both quantities are zero. If $(n,dj)=1$ and $(n,\tilde{q})=1$, then $n$ and $d\tilde{q}j\tilde{\jmath}/(j,\tilde{\jmath})$ are relatively prime, and so $\chi_0(n)=1$. Hence
\begin{align*}
\sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}}& \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2\\
&\leq \sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \sum_{d \leq R \frac{(j,\tilde{\jmath})}{\tilde{\jmath}}} \sum_{ \substack{\tilde{q} \leq R \frac{(j,\tilde{\jmath})}{d\tilde{\jmath}} \\ (\tilde{q},j)=1 \\ \tilde{q}\tilde{\jmath}>1 }} \,\sideset{}{^*}\sum_{ \tilde{\chi} \bmod{\tilde{q}\tilde{\jmath}} } \Bigg( \int_{-T}^T \Bigg|\sum_{\substack{n\leq N\\ (n,dj)=1 }} \frac{a_n \tilde{\chi}(n) }{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2.
\end{align*}
To bound the $\tilde{q},\tilde{\chi}$-sum, we apply the Cauchy-Schwarz inequality and then Theorem~9.12 of \cite{IK}. (There, take $k=\tilde{\jmath}$, $Q=R(j,\tilde{\jmath})/(d\tilde{\jmath})$, $T=T$, $N=N$, $a_n=a_n/n^{\sigma}$ if $(n,dj)=1$, and $a_n=0$ if $(n,dj)>1$. Note that we may apply the theorem because if $\tilde{\chi}$ is a primitive Dirichlet character modulo $\tilde{q}\tilde{\jmath}$, then $\tilde{\chi}$ equals the product of a primitive Dirichlet character modulo $\tilde{q}$ and a primitive Dirichlet character modulo $\tilde{\jmath}$ since $(\tilde{q},\tilde{\jmath})=1$.) This gives
\begin{align*}
\sum_{ \substack{\tilde{q} \leq R {(j,\tilde{\jmath})}/{(d\tilde{\jmath})} \\ (\tilde{q},j)=1 \\ \tilde{q}\tilde{\jmath}>1 }} \,\sideset{}{^*}\sum_{ \tilde{\chi} \bmod{\tilde{q}\tilde{\jmath}} } &\Bigg( \int_{-T}^T \Bigg|\sum_{\substack{n\leq N\\ (n,dj)=1 }} \frac{a_n \tilde{\chi}(n) }{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2 \\
& \leq 2T \sum_{ \substack{\tilde{q} \leq R {(j,\tilde{\jmath})}/{(d\tilde{\jmath})} \\ (\tilde{q},j)=1 \\ \tilde{q}\tilde{\jmath}>1 }} \,\sideset{}{^*}\sum_{ \tilde{\chi} \bmod{\tilde{q}\tilde{\jmath}} } \int_{-T}^T \Bigg|\sum_{\substack{n\leq N\\ (n,dj)=1 }} \frac{a_n \tilde{\chi}(n) }{n^{\sigma+it}} \Bigg|^2 \,dt \\
& \ll T(\log (jRTN))^3 \bigg(N+ \frac{(j,\tilde{\jmath})^2R^2T}{d^2\tilde{\jmath}} \bigg) \sum_{\substack{n\leq N\\ (n,dj)=1 }} \frac{|a_n|^2}{n^{2\sigma}},
\end{align*}
where the implied constant is absolute. Therefore
\begin{align*}
& \sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}} \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2\\
& \ll \sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \sum_{d \leq R {(j,\tilde{\jmath})}/{\tilde{\jmath}}}T(\log (jRTN))^3 \bigg(N+ \frac{(j,\tilde{\jmath})^2R^2T}{d^2\tilde{\jmath}} \bigg) \sum_{\substack{n\leq N\\ (n,dj)=1 }} \frac{|a_n|^2}{n^{2\sigma}}.
\end{align*}
We may ignore the condition $(n,dj)=1$ and then evaluate the $d$-sum to deduce that
\begin{align*}
\sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}}& \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2 \\
&\ll T(\log (jRTN))^3\sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \bigg(\frac{(j,\tilde{\jmath})RN}{ \tilde{\jmath}}+ \frac{(j,\tilde{\jmath})^2R^2T}{\tilde{\jmath}} \bigg) \sum_{ n\leq N } \frac{|a_n|^2}{n^{2\sigma}}.
\end{align*}
Now let $j=\prod_{p|j}p^{j_p}$ be the prime factorization of $j$. Multiplicativity implies
\begin{align*}
\sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \frac{(j,\tilde{\jmath})}{\tilde{\jmath}}
& =\prod_{p|j} \sum_{\nu=0}^{\infty} \frac{p^{\min\{j_p,\nu\}}}{p^{\nu}}
= \prod_{p|j} \bigg(j_p + \frac{1}{1-\frac{1}{p}} \bigg) \ll j^{\varepsilon}
\end{align*}
and
\begin{align*}
\sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \frac{(j,\tilde{\jmath})^2}{\tilde{\jmath} }
& \leq j \sum_{\substack{1\leq \tilde{\jmath}<\infty \\ p|\tilde{\jmath}\Rightarrow p|j}} \frac{(j,\tilde{\jmath})}{\tilde{\jmath}} \ll j^{1+\varepsilon}.
\end{align*}
Hence
\begin{align*}
\sum_{q\leq R} \sum_{\substack{\chi \bmod{qj} \\ \chi\neq \chi_0}} \Bigg( \int_{-T}^T \Bigg|\sum_{n\leq N} \frac{a_n \chi(n)}{n^{\sigma+it}} \Bigg| \,dt \Bigg)^2
\ll (jRNT)^{\varepsilon} (RNT+jR^2T^2 ) \sum_{ n\leq N } \frac{|a_n|^2}{n^{2\sigma}}.
\end{align*}
\end{proof}
\printbibliography
\end{document} |
\begin{document}
\title{Bit-Blasting ACL2 Theorems}
\begin{abstract}
Interactive theorem proving requires a lot of human guidance. Proving a
property involves (1) figuring out why it holds, then (2) coaxing the theorem
prover into believing it. Both steps can take a long time. We explain how
to use \emph{GL}, a framework for proving finite ACL2 theorems with BDD- or
SAT-based reasoning. This approach makes it unnecessary to deeply understand
why a property is true, and automates the process of admitting it as a
theorem. We use GL at Centaur Technology to verify execution units for x86
integer, MMX, SSE, and floating-point arithmetic.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
In hardware verification you often want to show that some circuit implements its
specification. Many of these problems are in the scope of fully automatic
decision procedures like SAT solvers. When these tools can be used, there are
good reasons to prefer them over \emph{The Method}~\cite{00-kaufmann-car} of
traditional, interactive theorem proving. For instance, these tools can:
\begin{itemize}
\item Reduce the level of human understanding needed in the initial
process of developing the proof;
\item Provide clear counterexamples, whereas failed ACL2 proofs can
often be difficult to debug; and
\item Ease the maintenance of the proof, since after the design changes
they can often find updated proofs without help.
\end{itemize}
\emph{GL}~\cite{10-swords-dissertation} is a framework for proving
\emph{finite} ACL2 theorems---those which, at least in principle, could be
established by exhaustive testing---by bit-blasting with a Binary Decision
Diagram (BDD) package or a SAT solver. These approaches have much higher
capacity than exhaustive testing. We are using GL heavily at Centaur
Technology~\cite{11-slobodova-framework,10-hardin-centaur,09-hunt-fadd}. So
far, we have used it to verify RTL implementations of floating-point addition,
multiplication, and conversion operations, as well as hundreds of bitwise and
arithmetic operations on scalar and packed integers.
This paper is an introduction to GL and a practical guide for using it to prove
ACL2 theorems. For a comprehensive treatment of the implementation of GL, see
Swords' dissertation~\cite{10-swords-dissertation}. Additional details about
particular commands can be found in the online documentation with \texttt{:doc
gl}.
GL is the successor of Boyer and Hunt's~\cite{09-boyer-g} \emph{G} system
(Section \ref{sec:related}), and its name stands for \emph{G in the Logic}.
The G system was written as a raw Lisp extension of the ACL2 kernel, so using
it meant trusting this additional code. In contrast, GL is implemented as ACL2
books and its proof procedure is formally verified by ACL2, so the only code we
have to trust besides ACL2 is the ACL2(h) extension that provides hash-consing
and memoization~\cite{06-boyer-acl2h}. Like the G system, GL can prove
theorems about ordinary ACL2 definitions; you are not restricted to some small
subset of the language.
How does GL work? You can probably imagine writing a bit-based encoding of
ACL2 objects. For instance, you might represent an integer with some structure
that contains a 2's-complement list of bits. GL uses an encoding like this,
except that Boolean expressions take the place of the bits. We call these
structures \emph{symbolic objects} (Section \ref{sec:symbolic-objects}).
GL provides a way to effectively compute with symbolic objects; e.g., it can
``add'' two integers whose bits are expressions, producing a new symbolic
object that represents their sum. GL can perform similar computations for most
ACL2 primitives. Building on this capability, it can \emph{symbolically
execute} terms (Section \ref{sec:symbolic-execution}). The result of a
symbolic execution is a new symbolic object that captures all the possible
values the result could take.
Symbolic execution can be used as a proof procedure (Section
\ref{sec:proving-theorems}). To prove a theorem, we first symbolically execute
its goal formula, then show the resulting symbolic object cannot represent
\texttt{nil}. GL provides a \texttt{def-gl-thm} command that makes it easy to
prove theorems with this approach (Section \ref{sec:def-gl-thm}). It handles
all the details of working with symbolic objects, and only needs to be told how
to represent the variables in the formula.
Like any automatic procedure, GL has a certain capacity. But when these limits
are reached, you may be able to increase its capacity by:
\begin{itemize}
\item Optimizing its symbolic execution strategy to use more efficient
definitions (Section \ref{sec:optimization}),
\item Decomposing difficult problems into easier subgoals using an automatic tool
(Section \ref{sec:def-gl-param-thm}), or
\item Using a SAT backend (Section \ref{sec:aig-mode}) that outperforms BDDs
on some problems.
\end{itemize}
There are also some good tools and techniques for debugging failed proofs
(Section \ref{sec:debugging}).
\subsection{Example: Counting Bits}
\label{sec:counting-bits}
Let's use GL to prove a theorem. The following C code, from Anderson's
\emph{Bit Twid\-dl\-ing Hacks}~\cite{11-anderson-bit-hacks} page, is a fast way
to count how many bits are set in a 32-bit integer.
\[
\begin{array}{l}
\texttt{v = v - ((v >> 1) \& 0x55555555);} \\
\texttt{v = (v \& 0x33333333) + ((v >> 2) \& 0x33333333);} \\
\texttt{c = ((v + (v >> 4) \& 0xF0F0F0F) * 0x1010101) >> 24;} \\
\end{array}
\]
We can model this in ACL2 as follows. It turns out that using
arbitrary-precision addition and subtraction does not affect the result, but we
must take care to use a 32-bit multiply to match the C code.
\[
\begin{array}{l}
\texttt{(defun 32* (x y)} \\
\texttt{~~(logand (* x y) (1- (expt 2 32))))} \\
\texttt{} \\
\texttt{(defun fast-logcount-32 (v)} \\
\texttt{~~(let* ((v (- v (logand (ash v -1) \#x55555555)))} \\
\texttt{~~~~~~~~~(v (+ (logand v \#x33333333) (logand (ash v -2) \#x33333333))))} \\
\texttt{~~~~(ash (32* (logand (+ v (ash v -4)) \#xF0F0F0F) \#x1010101) -24)))} \\
\end{array}
\]
We can then use GL to prove \texttt{fast-logcount-32} computes the same result
as ACL2's built-in \texttt{logcount} function for all unsigned 32-bit inputs.
\[
\begin{array}{l}
\texttt{(def-gl-thm fast-logcount-32-correct} \\
\texttt{~~:hyp~~~(unsigned-byte-p 32 x)} \\
\texttt{~~:concl (equal (fast-logcount-32 x)} \\
\texttt{~~~~~~~~~~~~~~~~(logcount x))} \\
\texttt{~~:g-bindings `((x ,(g-int 0 1 33))))} \\
\end{array}
\]
The \texttt{:g-bindings} form is the only help GL needs from the user.
It tells GL how to construct a symbolic object that can represent every value
for \texttt{x} that satisfies the hypothesis (we explain what it means in
later sections). No arithmetic books or lemmas are required---we actually don't
even know why this algorithm works. The proof completes in 0.09 seconds and
results in the following ACL2 theorem.
\[
\begin{array}{l}
\texttt{(defthm fast-logcount-32-correct} \\
\texttt{~~(implies (unsigned-byte-p 32 x)} \\
\texttt{~~~~~~~~~~~(equal (fast-logcount-32 x)} \\
\texttt{~~~~~~~~~~~~~~~~~~(logcount x)))} \\
\texttt{~~:hints ((gl-hint ...)))} \\
\end{array}
\]
Why not just use exhaustive testing? We wrote a fixnum-optimized
exhaustive-testing function that can cover the $2^{32}$ cases in 143 seconds.
This is slower than GL but still seems reasonable. On the other hand,
exhaustive testing is clearly incapable of scaling to the 64-bit and 128-bit
versions of this algorithm, whereas GL completes the proofs in 0.18 and 0.58
seconds, respectively.
Like exhaustive testing, GL can generate counterexamples to non-theorems. At
first, we didn't realize we needed to use a 32-bit multiply in
\texttt{fast-logcount-32}, and we just used an arbitrary-precision multiply
instead. The function still worked for test cases like \texttt{0}, \texttt{1},
\texttt{\#b111}, and \texttt{\#b10111}, but when we tried to prove its
correctness, GL showed us three counterexamples, \texttt{\#x80000000},
\texttt{\#xFFFFFFFF}, and \texttt{\#x9448C263}. By default, GL generates a
first counterexample by setting bits to 0 wherever possible, a second by
setting bits to 1, and a third with random bit settings.
\subsection{Example: UTF-8 Decoding}
\label{sec:utf-8}
Davis~\cite{06-davis-input} used exhaustive testing to prove lemmas toward the
correctness of UTF-8 processing functions. The most difficult proof carried
out this way was a well-formedness and inversion property for four-byte
UTF-8 sequences, which involved checking $2^{32}$ cases. Davis' proof takes 67
seconds on our computer. It involves four testing functions and five lemmas
about them; all of this is straightforward but mundane. The testing functions
are guard-verified and optimized with \texttt{mbe} and type declarations for
better performance.
We used GL to prove the same property. The proof (included in the supporting
materials) completes in 0.17 seconds and requires no testing functions or
supporting lemmas.
\subsection{Getting GL}
GL is included in ACL2 4.3, and the development version is available from the
ACL2 Books repository, \url{http://acl2-books.googlecode.com/}. Note that
using GL requires ACL2(h), which is best supported on 64-bit Clozure Common
Lisp. BDD operations can be memory intensive, so we recommend using a computer
with at least 8 GB of memory. Instructions for building GL can be found in
\texttt{centaur/README}, and it can be loaded with
\[
\texttt{(include-book "centaur/gl/gl" :dir :system)}.
\]
\section{GL Basics}
\label{sec:gl-basics}
At its heart, GL works by manipulating Boolean expressions. There are many
ways to represent Boolean expressions. GL currently supports a hons-based BDD
package~\cite{06-boyer-acl2h} and also has support for using a hons-based
And-Inverter Graph (AIG) representation with an external SAT solver.
For any particular proof, the user can choose to work in \emph{BDD mode} (the
default) or \emph{AIG mode}. Each representation has strengths and weaknesses,
and the choice of representation can significantly impact performance. We give
some advice about choosing proof modes in Section \ref{sec:aig-mode}.
\newcommand{\ensuremath{\mathit{true}}\xspace}{\ensuremath{\mathit{true}}\xspace}
\newcommand{\ensuremath{\mathit{false}}\xspace}{\ensuremath{\mathit{false}}\xspace}
The GL user does not need to know how BDDs and AIGs are represented; in this
paper we just adopt a conventional mathematical syntax to describe Boolean
expressions, e.g., $\ensuremath{\mathit{true}}\xspace$, $\ensuremath{\mathit{false}}\xspace$, $A \wedge B$, $\neg C$, etc.
\subsection{Symbolic Objects}
\label{sec:symbolic-objects}
\newcommand{\texttt{ }}{\texttt{ }}
GL groups Boolean expressions into \emph{symbolic objects}. Much like
a Boolean expression can be evaluated to obtain a Boolean value, a symbolic
object can be evaluated to produce an ACL2 object. There are several kinds of
symbolic objects, but numbers are a good start. GL represents symbolic, signed
integers as
\[
\texttt{(:g-number~$\mathit{lsb\textrm{-}bits}$)},
\]
where \emph{lsb-bits} is a list of Boolean expressions that represent the two's
complement bits of the number. The bits are in lsb-first order, and the last,
most significant bit is the sign bit. For instance, if $p$ is the following
\texttt{:g-number},
\[
p = \texttt{(:g-number (}\ensuremath{\mathit{true}}\xspace \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{ } A \wedge B \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{))},
\]
then $p$ represents a 4-bit, signed integer whose value is either 1 or 5,
depending on the value of $A \wedge B$.
GL uses another kind of symbolic object to represent ACL2 Booleans. In particular,
\[
\texttt{(:g-boolean~.~$\mathit{val}$)}
\]
represents \texttt{t} or \texttt{nil} depending on the Boolean expression
\emph{val}. For example,
\[
\texttt{(:g-boolean~.~$\neg(A \wedge B)$)}
\]
is a symbolic object whose value is \texttt{t} when $p$ has value 1, and
\texttt{nil} when $p$ has value 5.
GL has a few other kinds of symbolic objects that are also tagged with
keywords, such as \texttt{:g-var} and \texttt{:g-apply}. But an ACL2 object
that does not have any of these special keywords within it is \emph{also}
considered to be a symbolic object, and just represents itself. Furthermore, a
cons of two symbolic objects represents the cons of the two objects they
represent. For instance,
\[
\texttt{(1~.~(:g-boolean~.~$A \wedge B$))}
\]
represents either \texttt{(1~.~t)} or \texttt{(1~.~nil)}. Together, these
conventions allow GL to avoid lots of tagging as symbolic objects are
manipulated.
\newcommand{\ensuremath{\mathit{test}}}{\ensuremath{\mathit{test}}}
\newcommand{\ensuremath{\mathit{then}}}{\ensuremath{\mathit{then}}}
\newcommand{\ensuremath{\mathit{else}}}{\ensuremath{\mathit{else}}}
One last kind of symbolic object we will mention represents an if-then-else
among other symbolic objects. Its syntax is
\[
\texttt{(:g-ite~$\ensuremath{\mathit{test}}$~$\ensuremath{\mathit{then}}$~.~$\ensuremath{\mathit{else}}$)},
\]
where $\ensuremath{\mathit{test}}$, $\ensuremath{\mathit{then}}$, and $\ensuremath{\mathit{else}}$ are themselves symbolic objects. The
value of a \texttt{:g-ite} is either the value of $\ensuremath{\mathit{then}}$ or of $\ensuremath{\mathit{else}}$,
depending on the value of $\ensuremath{\mathit{test}}$. For example,
\[
\begin{array}{l}
\texttt{(:g-ite~(:g-boolean~.~$A$)} \\
\texttt{~~~~~~~~(:g-number~($B$~$A$~\ensuremath{\mathit{false}}\xspace))} \\
\texttt{~~~~~~~~.~\#\textbackslash{}C)}
\end{array}
\]
represents either 2, 3, or the character \texttt{C}.
GL doesn't have a special symbolic object format for ACL2 objects other than
numbers and Booleans. But it is still possible to create symbolic objects that
take any finite range of values among ACL2 objects, by using a nesting of
\texttt{:g-ite}s where the tests are \texttt{:g-boolean}s.
\subsection{Computing with Symbolic Objects}
\label{sec:symbolic-execution}
Once we have a representation for symbolic objects, we can perform symbolic
executions on those objects. For instance, recall the symbolic number $p$
which can have value 1 or 5,
\[
p = \texttt{(:g-number (}\ensuremath{\mathit{true}}\xspace \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{ } A \wedge B \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{))}.
\]
We might symbolically add 1 to $p$ to obtain a new symbolic number, say $q$,
\[
q = \texttt{(:g-number (}\ensuremath{\mathit{false}}\xspace \texttt{ } \ensuremath{\mathit{true}}\xspace \texttt{ } A \wedge B \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{))},
\]
which represents either 2 or 6. Suppose $r$ is another symbolic number,
\[
r = \texttt{(:g-number (}A \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{ } \ensuremath{\mathit{true}}\xspace \texttt{ } \ensuremath{\mathit{false}}\xspace \texttt{))},
\]
which represents either 4 or 5. We might add $q$ and $r$ to obtain $s$,
\[
s = \texttt{(:g-number (}A \texttt{ } \ensuremath{\mathit{true}}\xspace \texttt{ } \neg(A \wedge B) \texttt{ } A \wedge B \texttt{ }
\ensuremath{\mathit{false}}\xspace \texttt{))},
\]
whose value can be 6, 7, or 11. Why can't $s$ be 10 if $q$ can be 6 and $r$
can be 4? This combination isn't possible because $q$ and $r$ involve the same
expression, $A$. The only way for $r$ to be 4 is for $A$ to be false, but then
$q$ must be 2.
The underlying algorithm GL uses for symbolic additions is just a ripple-carry
addition on the Boolean expressions making up the bits of the two numbers.
Performing a symbolic addition, then, means constructing new
BDDs or AIGs, depending on which mode is being used.
GL has built-in support for symbolically executing most ACL2 primitives.
Generally, this is done by cases on the types of the symbolic objects being
passed in as arguments. For instance, if we want to symbolically execute
\texttt{consp} on $s$, then we are asking whether a \texttt{:g-number} may ever
represent a cons, so the answer is simply \texttt{nil}. Similarly, if we ever
try to add a \texttt{:g-boolean} to a \texttt{:g-number}, by the ACL2 axioms
the \texttt{:g-boolean} is simply treated as 0.
Beyond these primitives, GL provides what is essentially a McCarthy-style
interpreter~\cite{60-mccarthy-recursive} for symbolically executing terms. By
default, it expands function definitions until it reaches primitives, with some
special handling for \texttt{if}. For better performance, its
interpretation scheme can be customized with more efficient definitions and
other optimizations, as described in Section \ref{sec:optimization}.
\subsection{Proving Theorems by Symbolic Execution}
\label{sec:proving-theorems}
\newcommand{\ensuremath{x_{\mathit{best}}}\xspace}{\ensuremath{x_{\mathit{best}}}\xspace}
\newcommand{\ensuremath{x_{\mathit{init}}}\xspace}{\ensuremath{x_{\mathit{init}}}\xspace}
To see how symbolic execution can be used to prove theorems, let's return to
the bit-counting example, where our goal was to prove
\[
\begin{array}{l}
\texttt{(implies (unsigned-byte-p 32 x)} \\
\texttt{~~~~~~~~~(equal (fast-logcount-32 x)} \\
\texttt{~~~~~~~~~~~~~~~~(logcount x)))}. \\
\end{array}
\]
The basic idea is to first symbolically execute the above formula, and then
check whether it can ever evaluate to \texttt{nil}. But to do this symbolic
execution, we need some symbolic object to represent \texttt{x}.
We want our symbolic execution to cover all the cases necessary for proving the
theorem, namely all \texttt{x} for which the hypothesis
\texttt{(unsigned-byte-p 32 x)} holds. In other words, the symbolic object we
choose needs to be able to represent any integer from 0 to $2^{32}-1$.
Many symbolic objects cover this range. As notation, let $b_0,b_1,\dots$
represent independent Boolean variables in our Boolean expression
representation. Then, one suitable object is:
\[
\texttt{(:g-number ($b_0$~$b_1$~$\dots$~$b_{31}$~$b_{32}$))}.
\]
Why does this have 33 variables? The final bit, $b_{32}$, represents the sign,
so this object covers the integers from $-2^{32}$ to $2^{32}-1$. We could
instead use a 34-bit integer, or a 35-bit integer, or some esoteric creation
involving \texttt{:g-ite} forms. But perhaps the best object to use would be:
\[
\ensuremath{x_{\mathit{best}}}\xspace = \texttt{(:g-number ($b_0$~$b_1$~$\dots$~$b_{31}$~$\ensuremath{\mathit{false}}\xspace$))},
\]
since it covers exactly the desired range using the simplest possible Boolean
expressions.
Suppose we choose \ensuremath{x_{\mathit{best}}}\xspace to stand for \texttt{x}. We can now
symbolically execute the goal formula on that object.
What does this involve? First, \texttt{(unsigned-byte-p 32 x)} produces the
symbolic result \texttt{t}, since it is always true of the possible values of
\ensuremath{x_{\mathit{best}}}\xspace. It would have been equally valid for this to produce
\texttt{(:g-boolean~.~$\ensuremath{\mathit{true}}\xspace$)}, but GL prefers to produce constants
when possible.
Next, the \texttt{(fast-logcount-32 x)} and \texttt{(logcount x)} forms each
yield \texttt{:g-number} objects whose bits are Boolean
expressions in the variables $b_0, \dots, b_{31}$. For example, the least
significant bit will be an expression representing the XOR of all these
variables.
Finally, we symbolically execute \texttt{equal} on these two results. This
compares the Boolean expressions for their bits to determine if they are
equivalent, and produces a symbolic object representing the answer.
So far we have basically ignored the differences between using BDDs and AIGs as
our Boolean expression representation. But here, the two approaches produce
very different answers:
\begin{itemize}
\item Since BDDs are canonical, the expressions for the bits of the two numbers
are syntactically equal, and the result from \texttt{equal} is simply \texttt{t}.
\item With AIGs, the expressions for the bits are semantically equivalent but
not syntactically equal. The result is therefore
\texttt{(:g-boolean~.~$\phi$)}, where $\phi$ is a large Boolean expression in
the variables $b_0, \dots, b_{31}$. The fact that $\phi$ always evaluates to
\ensuremath{\mathit{true}}\xspace is not obvious just from its syntax.
\end{itemize}
At this point we have completed the symbolic execution of our goal formula,
obtaining either \texttt{t} in BDD mode, or this \texttt{:g-boolean} object in
AIG mode. Recall that to prove theorems using symbolic execution, the idea is
to symbolically execute the goal formula and then check whether its symbolic
result can represent \texttt{nil}. If we are using BDDs, it is obvious that
\texttt{t} cannot represent \texttt{nil}. With AIGs, we simply ask a SAT
solver whether $\phi$ can evaluate to \ensuremath{\mathit{false}}\xspace, and find that it cannot. This
completes the proof.
GL automates this proof strategy, taking care of many of the details relating
to creating symbolic objects, ensuring that they cover all the possible cases,
and ensuring that \texttt{nil} cannot be represented by the symbolic result.
When GL is asked to prove a non-theorem, it can generate counterexamples by
finding assignments to the Boolean variables that cause the result to become
\texttt{nil}.
\section{Using DEF-GL-THM}
\label{sec:def-gl-thm}
The \texttt{def-gl-thm} command is the main interface for using GL to prove
theorems. Here is the command we used in the bit-counting example.
\[
\begin{array}{l}
\texttt{(def-gl-thm fast-logcount-32-correct} \\
\texttt{~~:hyp~~~(unsigned-byte-p 32 x)} \\
\texttt{~~:concl (equal (fast-logcount-32 x)} \\
\texttt{~~~~~~~~~~~~~~~~(logcount x))} \\
\texttt{~~:g-bindings `((x ,(g-int 0 1 33))))} \\
\end{array}
\]
Unlike an ordinary \texttt{defthm} command, \texttt{def-gl-thm} takes separate
hypothesis and conclusion terms (its \texttt{:hyp} and \texttt{:concl}
arguments). This separation allows GL to use the hypothesis to limit the scope
of the symbolic execution it will perform. The user must also provide GL with
\texttt{:g-bindings} that describe the symbolic objects to use for each free
variable in the theorem (Section \ref{sec:writing-g-bindings}).
What are these bindings? In the \texttt{fast-logcount-32-corr\-ect} theorem, we
used a convenient function, \texttt{g-int}, to construct the
\texttt{:g-bindings}. Expanding this away, here are the actual bindings:
\[
\texttt{((x (:g-number (0 1 2 $\dots$ 32))))}.
\]
The \texttt{:g-bindings} argument uses a slight modification of the symbolic
object format where the Boolean expressions are replaced by distinct
natural numbers, each representing a Boolean variable. In this
case, our binding for \texttt{x} stands for the following symbolic object:
\[
\ensuremath{x_{\mathit{init}}}\xspace = \texttt{(:g-number ($b_0$~$b_1$~$\dots$~$b_{31}$~$b_{32}$))}.
\]
Note that \ensuremath{x_{\mathit{init}}}\xspace is not the same object as \ensuremath{x_{\mathit{best}}}\xspace from Section
\ref{sec:proving-theorems}---its sign bit is $b_{32}$ instead of \ensuremath{\mathit{false}}\xspace, so
\ensuremath{x_{\mathit{init}}}\xspace can represent any 33-bit signed integer whereas \ensuremath{x_{\mathit{best}}}\xspace only represents
32-bit unsigned values. In fact, the \texttt{:g-bindings} syntax does not even
allow us to describe objects like \ensuremath{x_{\mathit{best}}}\xspace, which has the constant \ensuremath{\mathit{false}}\xspace
instead of a variable as one of its bits.
There is a good reason for this restriction. One of the steps in our proof
strategy is to prove \emph{coverage}: we need to show the symbolic objects we
are starting out with have a sufficient range of values to cover all cases for
which the hypothesis holds (Section \ref{sec:proving-coverage}). The
restricted syntax permitted by \texttt{:g-bindings} ensures that the range of
values represented by each symbolic object is easy to determine. Because of
this, coverage proofs are usually automatic.
Despite these restrictions, GL will still end up using \ensuremath{x_{\mathit{best}}}\xspace to carry out the
symbolic execution. GL optimizes the original symbolic objects inferred from
the \texttt{:g-bindings} by using the hypothesis to reduce the space of objects
that are represented. In BDD mode this optimization uses \emph{BDD
parametrization}~\cite{99-aagaard-param}, which restricts the symbolic
objects so they cover exactly the inputs recognized by the hypothesis. In AIG
mode we use a lighter-weight transformation that replaces variables with
constants when the hypothesis sufficiently restricts them. In this example,
either optimization transforms \ensuremath{x_{\mathit{init}}}\xspace into \ensuremath{x_{\mathit{best}}}\xspace.
\subsection{Writing G-Bindings Forms}
\label{sec:writing-g-bindings}
In a typical \texttt{def-gl-thm} command, the \texttt{:g-bindings} should have
an entry for every free variable in the theorem. Here is an example that shows
some typical bindings.
\[
\begin{array}{l}
\texttt{:g-bindings~'((flag~~~(:g-boolean~.~0))} \\
\texttt{~~~~~~~~~~~~~~(a-bus~~(:g-number~(1~3~5~7~9)))} \\
\texttt{~~~~~~~~~~~~~~(b-bus~~(:g-number~(2~4~6~8~10)))} \\
\texttt{~~~~~~~~~~~~~~(mode~~~(:g-ite~(:g-boolean~.~11)~exact~.~fast))} \\
\texttt{~~~~~~~~~~~~~~(opcode~\#b0010100))} \\
\end{array}
\]
These bindings allow \texttt{flag} to take an arbitrary Boolean value,
\texttt{a-bus} and \texttt{b-bus} any five-bit signed integer values,
\texttt{mode} either the symbol \texttt{exact} or \texttt{fast}, and
\texttt{opcode} only the value 20.\footnote{Note that since \texttt{\#b0010100}
is not within a \texttt{:g-boolean} or \texttt{:g-number} form, it is
\emph{not} the index of a Boolean variable. Instead, like the symbols
\texttt{exact} and \texttt{fast}, it is just an ordinary ACL2 constant that
stands for itself, i.e., 20.}
Within \texttt{:g-boolean} and \texttt{:g-number} forms, natural number indices
take the places of Boolean expressions. The indices used throughout all of the
bindings must be distinct, and represent free, independent Boolean variables.
In BDD mode these indices have additional meaning: they specify the BDD
variable ordering, with smaller indices coming first in the order. This
ordering can greatly affect performance. In AIG mode the choice of indices has
no particular bearing on efficiency.
How do you choose a good BDD ordering? It is often good to interleave the bits
of data buses that are going to be combined in some way. It is also typically
a good idea to put any important control signals such as opcodes and mode
settings before the data buses.
Often the same \texttt{:g-bindings} can be used throughout several theorems,
either verbatim or with only small changes. In practice, we almost always
generate the \texttt{:g-bindings} forms by calling functions or macros. One
convenient function is
\[
\texttt{(g-int start by n)},
\]
which generates a \texttt{:g-number} form with \texttt{n} bits, using
indices that start at \texttt{start} and increment by \texttt{by}. This is
particularly useful for interleaving the bits of numbers, as we did for the
\texttt{a-bus} and \texttt{b-bus} bindings above:
\[
\begin{array}{l}
\texttt{(g-int 1 2 5)} \rightarrow \texttt{(:g-number (1 3 5 7 9))} \\
\texttt{(g-int 2 2 5)} \rightarrow \texttt{(:g-number (2 4 6 8 10))}.
\end{array}
\]
\subsection{Proving Coverage}
\label{sec:proving-coverage}
There are really two parts to any GL theorem. First, we need to symbolically
execute the goal formula and ensure it cannot evaluate to \texttt{nil}. But
in addition to this, we must ensure that the objects we use to represent the
variables of the theorem cover all the cases that satisfy the hypothesis. This
part of the proof is called the \emph{coverage obligation}.
For \texttt{fast-logcount-32-correct}, the coverage obligation is to
show that our binding for \texttt{x} is able to represent every integer
from 0 to $2^{32}-1$. This is true of \ensuremath{x_{\mathit{init}}}\xspace, and the coverage proof goes
through automatically.
But suppose we forget that \texttt{:g-number}s use a signed representation, and
attempt to prove \texttt{fast-log\-count-32-correct} using the following
(incorrect) g-bindings.
\[
\texttt{:g-bindings `((x ,(g-int 0 1 32)))}
\]
This looks like a 32-bit integer, but because of the sign bit it does not cover
the intended unsigned range. If we submit the \texttt{def-gl-thm} command
with these bindings, the symbolic execution part of the proof is still successful.
But this execution has only really shown the goal holds for 31-bit unsigned
integers, so \texttt{def-gl-thm} prints the message
\[
\texttt{ERROR: Coverage proof appears to have failed.}
\]
and leaves us with a failed subgoal,
\[
\begin{array}{l}
\texttt{(implies (and (integerp x)} \\
\texttt{~~~~~~~~~~~~~~(<= 0 x)} \\
\texttt{~~~~~~~~~~~~~~(< x 4294967296))} \\
\texttt{~~~~~~~~~(< x 2147483648))}. \\
\end{array}
\]
This goal is clearly not provable: we are trying to show \texttt{x} must be
less than $2^{31}$ (from our \texttt{:g-bindings}) whenever
it is less than $2^{32}$ (from the hypothesis).
Usually when the \texttt{:g-bindings} are correct, the coverage proof will be
automatic, so if you see that a coverage proof has failed, the first thing to
do is check whether your bindings are really sufficient.
On the other hand, proving coverage is undecidable in principle, so sometimes
GL will fail to prove coverage even though the bindings are appropriate. For
these cases, there are some keyword arguments to \texttt{def-gl-thm} that may
help coverage proofs succeed.
First, as a practical matter, GL does the symbolic execution part of the proof
\emph{before} trying to prove coverage. This can get in the way of debugging
coverage proofs when the symbolic execution takes a long time. You can use
\texttt{:test-side-goals t} to have GL skip the symbolic execution and go
straight to the coverage proof. Of course, no \texttt{defthm} is
produced when this option is used.
By default, our coverage proof strategy uses a restricted set of rules and
ignores the current theory. It heuristically expands functions in the
hypothesis and throws away terms that seem irrelevant. When this
strategy fails, it is usually for one of two reasons.
1. The heuristics expand too many terms and overwhelm ACL2. GL tries to avoid
this by throwing away irrelevant terms, but sometimes this approach is
insufficient. It may be helpful to disable the expansion of functions that are not
important for proving coverage. The \texttt{:do-not-expand} argument allows
you to list functions that should not be expanded.
2. The heuristics throw away a necessary hypothesis, leading to unprovable
goals. GL's coverage proof strategy tries to show that the binding for each
variable is sufficient, one variable at a time. During this process it throws
away hypotheses that do not mention the variable, but in some cases this can be
inappropriate. For instance, suppose the following is a coverage goal for
\texttt{b}:
\[
\begin{array}{l}
\texttt{(implies (and (natp a)} \\
\texttt{~~~~~~~~~~~~~~(natp b)} \\
\texttt{~~~~~~~~~~~~~~(< a (expt 2 15))} \\
\texttt{~~~~~~~~~~~~~~(< b a))} \\
\texttt{~~~~~~~~~(< b (expt 2 15))}.
\end{array}
\]
Here, throwing away the terms that don't mention \texttt{b} will cause the proof
to fail. A good way to avoid this problem is to separate type and size
hypotheses from more complicated assumptions that are not important for proving
coverage, along these lines:
\[
\begin{array}{l}
\texttt{(def-gl-thm~my-theorem} \\
\texttt{~~:hyp~(and~(type-assms-1~x)} \\
\texttt{~~~~~~~~~~~~(type-assms-2~y)} \\
\texttt{~~~~~~~~~~~~(type-assms-3~z)} \\
\texttt{~~~~~~~~~~~~(complicated-non-type-assms~x~y~z))} \\
\texttt{~~:concl~...} \\
\texttt{~~:g-bindings~...} \\
\texttt{~~:do-not-expand~'(complicated-non-type-assms))}.
\end{array}
\]
For more control, you can also use the \texttt{:cov-theory-add} argument to
enable additional rules during the coverage proof, e.g.,
\texttt{:cov-theory-add '(type-rule1 type-rule2)}.
\section{Optimizing Symbolic Execution}
\label{sec:optimization}
The scope of theorems GL can handle is directly impacted by its symbolic
execution performance. It is actually quite easy to customize the way
certain terms are interpreted, and this can sometimes provide
important speedups.
GL's symbolic interpreter operates much like a basic Lisp interpreter. To
symbolically interpret a function call, GL first eagerly interprets its
arguments to obtain symbolic objects for the actuals. Then GL symbolically
executes the function in one of three ways:
\begin{itemize}
\item As a special case, if the actuals evaluate to concrete objects, then GL
may be able to stop symbolically executing and just call the actual ACL2
function on these arguments (Section \ref{sec:concrete-execution}).
\item For primitive ACL2 functions like \texttt{+}, \texttt{consp},
\texttt{equal}, and for some defined functions like \texttt{logand} and
\texttt{ash} where performance is important, GL uses hand-written
functions called \emph{symbolic counterparts} that can operate on symbolic
objects. The advanced GL user can write new symbolic counterparts
(Section \ref{sec:custom-symbolic-counterparts}) to speed up symbolic
execution.
\item Otherwise, GL looks up the definition of the function, and recursively
interprets its body in a new environment binding the formals to the symbolic
actuals. The way a function is written can impact its symbolic execution
performance (Section \ref{sec:redundant-recursion}). It is easy to instruct
GL to use more efficient definitions for particular functions (Section
\ref{sec:preferred-definitions}).
\end{itemize}
GL symbolically executes functions strictly according to the ACL2 logic and
does not consider guards. An important consequence is that when \texttt{mbe}
is used, GL's interpreter follows the \texttt{:logic} definition instead of the
\texttt{:exec} definition, since it might be unsound to use the \texttt{:exec}
version of a definition without establishing the guard is met. Also, while GL
can symbolically simulate functions that take user-defined stobjs or even the
ACL2 \texttt{state}, it does not operate on ``real'' stobjs; instead, it uses
the logical definitions of the relevant stobj operations, which do not provide
the performance benefits of destructive operations.
Non-executable functions cannot be symbolically executed.
\subsection{Avoiding Redundant Recursion}
\label{sec:redundant-recursion}
Here are two ways to write a list-filtering function.
\[
\begin{array}{l}
\texttt{(defun~filter1~(x)} \\
\texttt{~~(cond~((atom~x)} \\
\texttt{~~~~~~~~~nil)} \\
\texttt{~~~~~~~~((element-okp~(car~x))~~~~~~~~~~~~~~~;;~keep~it} \\
\texttt{~~~~~~~~~(cons~(car~x)~(filter1~(cdr~x))))} \\
\texttt{~~~~~~~~(t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;;~skip~it} \\
\texttt{~~~~~~~~~(filter1~(cdr~x)))))} \\
\end{array}
\]
This definition can be inefficient for symbolic execution. Suppose we are
symbolically executing \texttt{filter1}, and the \texttt{element-okp} check has
produced a symbolic object that can take both \texttt{nil} and non-\texttt{nil}
values. Then, we proceed by symbolically executing both the keep- and
skip-branches, and construct a \texttt{:g-ite} form for the result. Since we
have to evaluate the recursive call twice, this execution becomes exponential
in the length of \texttt{x}.
We can avoid this blow-up by consolidating the recursive calls, as follows.
\[
\begin{array}{l}
\texttt{(defun~filter2~(x)} \\
\texttt{~~(if~(atom~x)} \\
\texttt{~~~~~~nil} \\
\texttt{~~~~(let~((rest~(filter2~(cdr~x))))} \\
\texttt{~~~~~~(if~(element-okp~(car~x))} \\
\texttt{~~~~~~~~~~(cons~(car~x)~rest)} \\
\texttt{~~~~~~~~rest))))} \\
\end{array}
\]
This is not a novel observation; Reeber~\cite{07-reeber-dissertation} suggests
the same sort of optimization for unrolling recursive functions in SULFA.
Of course, \texttt{filter1} is probably slightly better for concrete execution
since it has a tail call in at least some cases. If we do not want to change
the definition of \texttt{filter1}, we can simply tell GL to use the
\texttt{filter2} definition instead, as described in the next section. We
currently do not try to automatically apply this kind of optimization, though
we may explore this in future work.
\subsection{Preferred Definitions}
\label{sec:preferred-definitions}
To instruct GL to symbolically execute \texttt{filter2} in place of \texttt{filter1},
we can do the following:
\[
\begin{array}{l}
\texttt{(defthm~filter1-for-gl} \\
\texttt{~~(equal~(filter1~x)~(filter2~x))} \\
\texttt{~~:rule-classes~nil)} \\
\texttt{} \\
\texttt{(gl::set-preferred-def~filter1~filter1-for-gl)} \\
\end{array}
\]
The \texttt{gl::set-preferred-def} form extends a table that GL consults when
expanding a function's definition. Each entry in the table pairs a function
name with the name of a theorem. The theorem must state that a call of the
function is unconditionally equal to some other term. When GL encounters a
call of a function in this table, it replaces the call with the right-hand
side of the theorem, which is justified by the theorem. So after the above
event, GL will replace calls of \texttt{filter1} with \texttt{filter2}.
As another example of a preferred definition, GL automatically optimizes the
definition of \texttt{evenp}, which ACL2 defines as follows:
\[
\texttt{(evenp x)} = \texttt{(integerp (* x (/ 2)))}.
\]
This definition is basically unworkable since GL provides little support for
rational numbers. However, GL has an efficient, built-in implementation of
\texttt{logbitp}. So to permit the efficient execution of \texttt{evenp}, GL
proves the following identity and uses it as \texttt{evenp}'s preferred
definition.
\[
\begin{array}{l}
\texttt{(defthm~evenp-is-logbitp} \\
\texttt{~~(equal~(evenp~x)} \\
\texttt{~~~~~~~~~(or~(not~(acl2-numberp~x))} \\
\texttt{~~~~~~~~~~~~~(and~(integerp~x)} \\
\texttt{~~~~~~~~~~~~~~~~~~(equal~(logbitp~0~x)~nil)))))} \\
\end{array}
\]
\subsection{Executability on Concrete Terms}
\label{sec:concrete-execution}
Suppose GL is symbolically executing a function call. If the arguments to the
function are all concrete objects (i.e., symbolic objects that represent a
single value), then in some cases the interpreter can stop symbolically
executing and just run the ACL2 function on these arguments. In some
cases, this can provide a critical performance boost.
To actually call these functions, GL essentially uses a case statement along
the following lines.
\[
\begin{array}{l}
\texttt{(case~fn} \\
\texttt{~~(cons~~~~~(cons~(first~args)~(second~args)))} \\
\texttt{~~(reverse~~(reverse~(first~args)))} \\
\texttt{~~(member~~~(member~(first~args)~(second~args)))} \\
\texttt{~~...)} \\
\end{array}
\]
Such a case statement is naturally limited to calling a fixed set of functions.
To allow GL to concretely execute additional functions, you can use
\texttt{def-gl-clause-processor}, a special macro that defines a new version of
the GL symbolic interpreter and clause processor. GL automatically uses the
most recently defined interpreter and clause processor. For instance, here is
the syntax for extending GL so that it can execute \texttt{md5sum} and
\texttt{sets::mergesort}:
\[
\texttt{(def-gl-clause-processor my-cp '(md5sum sets::mergesort))}.
\]
\subsection{Full-Custom Symbolic Counterparts}
\label{sec:custom-symbolic-counterparts}
The advanced GL user can write custom symbolic counterparts to get better
performance. This is somewhat involved. Generally, such a function operates
by cases on what kinds of symbolic objects it has been given. Most of these
cases are easy; for instance, the symbolic counterpart for \texttt{consp} just
returns \texttt{nil} when given a \texttt{:g-boolean} or \texttt{:g-number}.
But in other cases the operation can require combining the Boolean
expressions making up the arguments in some way, e.g., the symbolic counterpart
for \texttt{binary-*} implements a simple binary multiplier.
Once the counterpart has been defined, it must be proven sound with respect to
the semantics of ACL2 and the symbolic object format. This is an ordinary ACL2
proof effort that requires some understanding of GL's implementation.
The most sophisticated symbolic counterpart we have written is an AIG to BDD
conversion algorithm~\cite{10-swords-bddify}. This function serves as a
symbolic counterpart for AIG evaluation, and at Centaur it is the basis for the
``implementation side'' of our hardware correctness theorems. This algorithm
and its correctness proof are publicly available; see
\texttt{centaur/aig/g-aig-eval}.
\section{Case-Splitting}
\label{sec:def-gl-param-thm}
BDD performance can sometimes be improved by breaking a problem into subcases.
The standard example is floating-point
addition~\cite{98-chen-adders,99-aagaard-param}, which benefits from separating
the problem into cases based on the difference between the two inputs'
exponents. For each exponent difference, the two mantissas are aligned
differently before being added together, so a different BDD order is necessary
to interleave their bits at the right offset. Without case splitting, a single
BDD ordering has to be used for the whole problem; no matter what ordering we
choose, the mantissas will be poorly interleaved for some exponent differences,
causing severe performance problems. Separating the cases allows the
appropriate order to be used for each difference.
GL provides a \texttt{def-gl-param-thm} command that supports this technique.
This command splits the goal formula into several subgoals and attempts to
prove each of them using the \texttt{def-gl-thm} approach, so for each subgoal
there is a symbolic execution step and coverage proof. To show the subgoals
suffice to prove the goal formula, it also does another
\texttt{def-gl-thm}-style proof that establishes that any inputs satisfying the
hypothesis are covered by some case.
Here is how we might split the proof
for \texttt{fast-logcount-32} into five subgoals.
One goal handles the case
where the most significant bit is 1. The other four goals assume the most
significant bit is 0, and separately handle the cases where the lower two bits
are 0, 1, 2, or 3. Each case has a different symbolic binding for \texttt{x},
giving the BDD variable order. Of course, splitting into cases and varying the
BDD ordering is unnecessary for this theorem, but it illustrates how the
\texttt{def-gl-param-thm} command works.
\[
\begin{array}{l}
\texttt{(def-gl-param-thm~fast-logcount-32-correct-alt} \\
\texttt{~:hyp~(unsigned-byte-p~32~x)} \\
\texttt{~:concl~(equal~(fast-logcount-32~x)} \\
\texttt{~~~~~~~~~~~~~~~(logcount~x))} \\
\texttt{~:param-bindings} \\
\texttt{~`((((msb~1)~(low~nil))~((x~,(g-int~32~-1~33))))} \\
\texttt{~~~(((msb~0)~(low~0))~~~((x~,(g-int~~0~~1~33))))} \\
\texttt{~~~(((msb~0)~(low~1))~~~((x~,(g-int~~5~~1~33))))} \\
\texttt{~~~(((msb~0)~(low~2))~~~((x~,(g-int~~0~~2~33))))} \\
\texttt{~~~(((msb~0)~(low~3))~~~((x~,(g-int~~3~~1~33)))))} \\
\texttt{~:param-hyp~(and~(equal~msb~(ash~x~-31))} \\
\texttt{~~~~~~~~~~~~~~~~~(or~(equal~msb~1)} \\
\texttt{~~~~~~~~~~~~~~~~~~~~~(equal~(logand~x~3)~low)))} \\
\texttt{~:cov-bindings~`((x~,(g-int~0~1~33))))} \\
\end{array}
\]
We specify the five subgoals to consider using two new variables, \texttt{msb}
and \texttt{low}. Here, \texttt{msb} will determine the most significant bit
of \texttt{x}; \texttt{low} will determine the two least significant bits of
\texttt{x}, but only when \texttt{msb} is 0.
The \texttt{:param-bindings} argument describes the five subgoals by assigning
different values to \texttt{msb} and \texttt{low}. It also gives the
\texttt{g-bindings} to use in each case. We use different bindings for
\texttt{x} for each subgoal to show how it is done.
The \texttt{:param-hyp} argument describes the relationship between
\texttt{msb}, \texttt{low}, and \texttt{x} that will be assumed in each
subgoal. In the symbolic execution performed for each subgoal, the
\texttt{:param-hyp} is used to reduce the space of objects represented by the
symbolic binding for \texttt{x}. For example, in the subgoal where
$\texttt{msb} = 1$, this process will assign \ensuremath{\mathit{true}}\xspace to $\texttt{x}[31]$. The
\texttt{:param-hyp} will also be assumed to hold for the coverage proof for
each case.
How do we know the case-split is complete? One final proof is needed to show
that whenever the hypothesis holds for some \texttt{x}, then at least one of
the settings of \texttt{msb} and \texttt{low} satisfies the \texttt{:param-hyp}
for this \texttt{x}. That is:
\[
\begin{array}{l}
\texttt{(implies~(unsigned-byte-p~32~x)} \\
\texttt{~~~~~~~~~(or~(let~((msb~1)~(low~nil))} \\
\texttt{~~~~~~~~~~~~~~~(and~(equal~msb~(ash~x~-31))} \\
\texttt{~~~~~~~~~~~~~~~~~~~~(or~(equal~msb~1)} \\
\texttt{~~~~~~~~~~~~~~~~~~~~~~~~(equal~(logand~x~3)~low))))} \\
\texttt{~~~~~~~~~~~~~(let~((msb~0)~(low~0))~...)} \\
\texttt{~~~~~~~~~~~~~(let~((msb~0)~(low~1))~...)} \\
\texttt{~~~~~~~~~~~~~(let~((msb~0)~(low~2))~...)} \\
\texttt{~~~~~~~~~~~~~(let~((msb~0)~(low~3))~...)))} \\
\end{array}
\]
This proof is also done in the \texttt{def-gl-thm} style, so we need we need
one last set of symbolic bindings, which is provided by the
\texttt{:cov-bindings} argument.
\section{AIG Mode}
\label{sec:aig-mode}
GL can optionally use And-Inverter Graphs (AIGs) to represent
Boolean expressions instead of BDDs. You can choose the
mode on a per-proof basis by running \texttt{(gl-bdd-mode)} or
\texttt{(gl-aig-mode)}, which generate \texttt{defattach} events.
Unlike BDDs, AIGs are non-canonical, and this affects performance in
fundamental ways. AIGs are generally much cheaper to construct than BDDs, but
to determine whether AIGs are equivalent we have to invoke a SAT solver,
whereas with BDDs we just need to use a pointer-equality check.
Using an external SAT solver raises questions of trust. For most verification
work in industry it is probably sufficient to just trust the solver. But Matt
Kaufmann has developed and reflectively verified an ACL2 function that
efficiently checks a resolution proof that is produced by the SAT solver. GL
can use this proof-checking capability to avoid trusting the SAT solver. This
approach is not novel: Weber and Amjad~\cite{09-weber-sat} have developed an
LCF-style integration of SAT in several HOL theorem provers, and Darbari, et
al~\cite{10-darbari-sat} have a reflectively verified SAT certificate checker
in Coq.
Recording and checking resolution proofs imposes significant overhead, but is
still practical in many cases. We measured this overhead on a collection of
AIG-mode GL theorems about Centaur's MMX/SSE module. These theorems take 10
minutes without proof recording. With proof-recording enabled, our SAT solver
uses a less-efficient CNF generation algorithm and SAT solving grows to 25
minutes; an additional 6 minutes are needed to check the recorded proofs.
The SAT solver we have been using, an integration of MiniSAT with an AIG package,
is not yet released, so AIG mode is not usable ``out of the box.''
As future work, we would like to make
it easier to plug in other SAT solvers. Versions of MiniSAT, PicoSAT, and ZChaff can
also produce resolution proofs, so this is mainly an interfacing issue.
A convenient feature of AIGs is that you do not have to come up with a good
variable ordering. This is especially beneficial if it avoids the need to
case-split. On the other hand, BDDs provide especially nice counterexamples,
whereas SAT produces just one, essentially random counterexample.
Performance-wise, AIGs are better for some problems and BDDs for others. Many
operations combine bits from data buses in a regular, orderly way; in these
cases, there is often a good BDD ordering and BDDs may be faster than SAT.
But when the operations are less regular, when no good BDD ordering is
apparent, or when case-splitting seems necessary to get good BDD performance,
SAT may do better. For many of our proofs, SAT works well enough that we
haven't tried to find a good BDD ordering.
\section{Debugging Failures}
\label{sec:debugging}
A GL proof attempt can fail in several ways. In the ``best'' case, the
conjecture is disproved and GL can produce counterexamples to help diagnose the
problem. However, sometimes symbolic execution simply runs forever (Section
\ref{sec:performance-problems}). In other cases, a symbolic execution may
produce an indeterminate result (Section \ref{sec:indeterminate-results}),
giving an example of inputs for which the symbolic execution failed. Finally,
GL can run out of memory or spend too much time in garbage collection (Section
\ref{sec:memory-problems}). We have developed some tools and techniques for
debugging these problems.
\subsection{Performance Problems}
\label{sec:performance-problems}
Any bit-blasting tool has capacity limitations. However, you may also run into
cases where GL is performing poorly due to preventable issues. When GL seems
to be running forever, it can be helpful to trace the symbolic interpreter to
see which functions are causing the problem. To trace the symbolic
interpreter, run
\[
\texttt{(gl::trace-gl-interp~:show-values t)}.
\]
Here, at each call of the symbolic interpreter, the term being interpreted and
the variable bindings are shown, but since symbolic objects may be too large to
print, any bindings that are not concrete are hidden. You can also get a trace
with no variable bindings using \texttt{:show-values nil}. It may also be
helpful to simply interrupt the computation and look at the Lisp backtrace,
after executing
\[\texttt{(set-debugger-enable t)}.\]
In many cases, performance problems are due to BDDs growing too large. This is
likely the case if the interpreter appears to get stuck (not printing any more
trace output) and the backtrace contains a lot of functions with
names beginning in \texttt{q-}, which is the convention for BDD operators. In
some cases, these performance problems may be solved by choosing a more
efficient BDD order. But note that certain operations like
multiplication are exponentially hard. If you run into these
limits, you may need to refactor or decompose your problem into simpler
sub-problems (Section \ref{sec:def-gl-param-thm}).
There is one kind of BDD performance problem with a special solution. Suppose
GL is asked to prove \texttt{(equal spec impl)} when this does not actually
hold. Sometimes the symbolic objects for \texttt{spec} and \texttt{impl} can
be created, but the BDD representing their equality is too large to fit in
memory. The goal may then be restated with \texttt{always-equal}
instead of \texttt{equal} as the final comparison. Logically,
\texttt{always-equal} is just \texttt{equal}. But \texttt{always-equal}
has a custom symbolic counterpart that returns \texttt{t} when its arguments
are equivalent, or else produces a symbolic object that captures just one
counterexample and is indeterminate in all other cases.
Another possible problem is that the symbolic interpreter never gets
stuck, but keeps opening up more and more functions. These
problems might be due to redundant recursion (see Section
\ref{sec:redundant-recursion}), which may be avoided by providing a
more efficient preferred definition (Section \ref{sec:preferred-definitions})
for the function. The symbolic interpreter might also be
inefficiently interpreting function calls on concrete arguments, in
which case a \texttt{def-gl-clause-processor} call may be used to allow GL
to execute the functions directly (Section \ref{sec:concrete-execution}).
\subsection{Indeterminate Results}
\label{sec:indeterminate-results}
Occasionally, GL will abort a proof and print a message saying it found
indeterminate results. In this case, the examples printed are likely
\emph{not} to be true counterexamples, and examining them may not be
particularly useful.
One likely reason for such a failure is that some of GL's built-in symbolic counterparts
have limitations. For example, most arithmetic primitives will
not perform symbolic computations on non-integer numbers. When ``bad'' inputs are
provided, instead of producing a new \texttt{:g-number} object, these functions
will produce a \texttt{:g-apply} object, which is a type of symbolic
object that represents a function call. A \texttt{:g-apply} object cannot be
syntactically analyzed in the way other symbolic objects can, so most
symbolic counterparts, given a \texttt{:g-apply} object, will simply create
another one wrapping its arguments.
To diagnose indeterminate results, it is helpful to know when the first
\texttt{:g-apply} object was created. If you run
\[
\texttt{(gl::break-on-g-apply)},
\]
then when a \texttt{:g-apply} object is constructed, the function and symbolic
arguments will be printed and an interrupt will occur, allowing you to inspect
the backtrace. For example, the following form produces an indeterminate result.
\[
\begin{array}{l}
\texttt{(def-gl-thm~integer-half} \\
\texttt{~~:hyp~(and~(unsigned-byte-p~4~x)} \\
\texttt{~~~~~~~~~~~~(not~(logbitp~0~x)))} \\
\texttt{~~:concl~(equal~(*~1/2~x)} \\
\texttt{~~~~~~~~~~~~~~~~(ash~x~-1))} \\
\texttt{~~:g-bindings~`((x~,(g-int~0~1~5))))} \\
\end{array}
\]
After running \texttt{(gl::break-on-g-apply)}, running the above form enters
a break after printing
\[
\texttt{(g-apply BINARY-* (1/2 (:G-NUMBER (NIL \# \# \# NIL)))}
\]
to signify that a \texttt{:g-apply} form was created after trying to multiply
some symbolic integer by $\frac{1}{2}$.
Another likely reason is that there is a typo in your theorem. When a variable
is omitted from the \texttt{:g-bindings} form, a warning is printed and the
missing variable is assigned a \texttt{:g-var} object. A \texttt{:g-var} can
represent any ACL2 object, without restriction. Symbolic counterparts
typically produce \texttt{:g-apply} objects when called on \texttt{:g-var}
arguments, and this can easily lead to indeterminate results.
\subsection{Memory Problems}
\label{sec:memory-problems}
Memory management can play a significant role in symbolic execution
performance. In some cases GL may use too much memory, leading to swapping and
slow performance. In other cases, garbage collection may run too frequently or
may not reclaim much space. We have several recommendations for managing
memory in large-scale GL proofs. Some of these suggestions are specific to
Clozure Common Lisp.
1. Load the \texttt{centaur/misc/memory-mgmt-raw} book and use the
\texttt{set-max-mem} command to indicate how large you would like the
Lisp heap to be. For instance, \[ \texttt{(set-max-mem (* 8 (expt 2
30)))} \] says to allocate 8 GB of memory. To avoid swapping, you should
use somewhat less than your available physical memory. This book disables
ephemeral garbage collection and configures the garbage collector to run only
when the threshold set above is exceeded, which can boost performance.
2. Optimize hash-consing performance. GL's representations of BDDs and AIGs
use \texttt{hons} for structure-sharing. The \texttt{hons-summary} command can
be used at any time to see how many honses are currently in use, and
hash-consing performance can be improved by pre-allocating space for these
honses with \texttt{hons-resize}. See the \texttt{:doc} topics for these
commands for more information.
3. Be aware of (and control) hash-consing and memoization overhead. Symbolic
execution can use a lot of hash conses and can populate the memoization tables
for various functions. The memory used for these purposes is \emph{not}
automatically freed during garbage collection, so it may sometimes be necessary
to manually reclaim it. A useful function is \texttt{(maybe-wash-memory~$n$)},
which frees this memory and triggers a garbage collection only when the amount
of free memory is below some threshold $n$. A good choice for $n$ might be
20\% of the \texttt{set-max-mem} threshold. It can be useful to call
\texttt{maybe-wash-memory} between proofs, or between the cases of
parametrized theorems; see \texttt{:doc def-gl-param-thm} for its
\texttt{:run-be\-fore-ca\-ses} argument.
\section{Related Work}
\label{sec:related}
GL is most closely related to Boyer and Hunt's~\cite{09-boyer-g} \emph{G}
system, which was used for earlier proofs about Centaur's floating-point unit.
G used a symbolic object format similar to GL's, but only supported BDDs. It
also included a compiler that could produce ``generalized'' versions of
functions, similar to symbolic counterparts. GL actually has such a compiler,
but the interpreter is more convenient since no compilation step is necessary,
and the performance difference is insignificant. In experimental comparisons,
GL performed as well or better than G, perhaps due to the change from G's
sign/magnitude number encoding to GL's two's-complement encoding.
The G system was written ``outside the logic,'' in Common Lisp. It could not
be reasoned about by ACL2, but an experimental connection was developed which
allowed ACL2 to trust G to prove theorems. In contrast, GL is written entirely
in ACL2, and its proof procedure is a reflectively-verified clause processor,
which provides a significantly better story of trust. Additionally, GL can be
safely configured and extended by users via preferred definitions and custom
symbolic counterparts.
Reeber~\cite{06-reeber-sulfa} identified a decidable subset of ACL2 formulas
called SULFA and developed a SAT-based procedure for proving theorems in this
subset. Notably, this subset included lists of bits and recursive functions of
bounded depth. The decision procedure for SULFA is not mechanically verified,
but Reeber's dissertation~\cite{07-reeber-dissertation} includes an argument
for its correctness. GL addresses a different subset of ACL2 (e.g., SULFA
includes uninterpreted functions, whereas GL includes numbers and arithmetic
primitives), but the goals of both systems are similar.
ACL2 has a built-in BDD algorithm (described in \texttt{:doc bdd}) that, like
SULFA, basically deals with Booleans and lists of Booleans, but not numbers,
addition, etc. This algorithm is tightly integrated with the prover; it can
treat provably Boolean terms as variables and can use unconditional rewrite
rules to simplify terms it encounters. The algorithm is written in program
mode (outside the ACL2 logic) and has not been mechanically verified. GL seems
to be significantly faster, at least on a simple series of
addition-commutativity theorems.
Fox~\cite{11-fox-blasting} has implemented a bit-blasting procedure in HOL4
that can use SAT to solve problems phrased in terms of a particular bit-vector
representation. This tool is based on an LCF-style integrations of
proof-producing SAT solvers, so it has a strong soundness story. We would
expect there to be some overhead for any LCF-style
solution~\cite{09-weber-sat}, and GL seems to be considerably faster on the
examples in Fox's paper; see the supporting materials for details.
Manolios and Srinivasan \cite{06-manolios-pipeline} describe a connection
between ACL2 and UCLID to verify that a pipelined processor implements its
instruction set. In this work, ACL2 is used to simplify the correctness
theorem for a bit-accurate model of the processor down to a more abstract,
term-based goal. This goal is then given to UCLID, a decision procedure for a
restricted logic of counter arithmetic, lambdas, and uninterpreted functions.
UCLID then proves the goal much more efficiently than, e.g., ACL2's rewriter.
This work seems complementary to GL, which deals with bit-level reasoning,
i.e., the parts of the problem that this strategy addresses using ACL2.
Srinivasan \cite{07-srinivasan-dissertation} additionally described ACL2-SMT, a
connection with the Yices SMT solver. The system attempts to unroll and
simplify ACL2 formulas until they can be translated into the input language of
the SMT solver (essentially linear integer arithmetic, array operations, and
uninterpreted integer and Boolean functions). It then calls Yices to discharge
the goal, and Yices is trusted. GL addresses a different subset of ACL2, e.g.,
GL supports list operations and more arithmetic operations like
\texttt{logand}, but ACL2-SMT has uninterpreted functions and can deal with,
e.g., unbounded arithmetic.
Armand, et. al~\cite{11-armand-sat} describe work to connect SAT and SMT
solvers with Coq. Unlike the ACL2-SMT work, the connection is carried out in a
verified way, with Coq being used to check proof witnesses generated by the
solvers. This connection can be used to prove Coq goals that directly fit into
the supported logic of the SMT solver. GL is somewhat different in that it
allows most any ACL2 term to be handled when its variables range over a finite
space.
\section{Conclusions}
GL provides a convenient and efficient way to solve many finite ACL2 theorems
that arise in hardware verification. It allows properties to be stated in a
straightforward manner, scales to large problems, and provides clear
counter-examples for debugging. At Centaur Technology, it
plays an important role in the verification of arithmetic units, and we make
frequent improvements to support new uses.
Beyond this paper, we encourage all GL users to see the online documentation,
which can be found under \texttt{:doc gl} after loading the GL library. If you
prefer, you can also generate an HTML version of the documentation; see
\texttt{centaur/README} for details. Finally, the documentation for ACL2(h)
may be useful, and can be found at \texttt{:doc hons-and-memoization}.
While we have described the basic idea of symbolic execution and how GL uses it
to prove theorems, Swords' dissertation~\cite{10-swords-dissertation} contains
a much more detailed description of GL's implementation. It covers tricky
topics like the handling of \texttt{if} statements and the details of BDD
parametrization. It also covers the logical foundations of GL, such as
correctness claims for symbolic counterparts, the introduction of symbolic
interpreters, and the definition and verification of the GL clause processor.
\subsection{Acknowledgments}
Bob Boyer and Warren Hunt developed the G system, which pioneered many of the
ideas in GL. Anna Slo\-bo\-do\-v\'{a} has carried out several sophisticated
proofs with GL and beta-tested many GL features. Matt Kaufmann and Niklas Een
have contributed to our verified SAT integration. Gary Byers has answered many
of our questions and given us advice about Clozure Common Lisp. We thank
Warren Hunt, Matt Kaufmann, David Rager, Anna Slo\-bo\-do\-v\'{a}, and the
anonymous reviewers for their corrections and feedback on this paper.
{}
\end{document} |
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\newtheorem{lemmaA}{Lemma~A}{\bf}{\it}
title{
Primal-dual path following method for
nonlinear semi-infinite programs with semi-definite constraints
thanks{
The work was supported by JSPS KAKENHI Grant Number [15K15943].
}
}
\author{Takayuki Okuno\and
Masao Fukushima
}
\institute{T. Okuno \at
RIKEN, The Center for Advanced Intelligence Project (AIP),
Nihonbashi 1-chome Mitsui Building, 15th floor,1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
\\ \email{[email protected]}
\and
M. Fukushima \at
Nanzan University, Faculty of Science and Engineering,
18 Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan
\email{[email protected]}
}
\date{Received: date / Accepted: date}
\title{
Primal-dual path following method for
nonlinear semi-infinite programs with semi-definite constraints
hanks{
The work was supported by JSPS KAKENHI Grant Number [15K15943].
}
\begin{abstract}
In this paper, we propose {two} algorithms for nonlinear semi-infinite semi-definite programs with infinitely many convex inequality constraints, called SISDP for short.
A straightforward approach to the SISDP is to use classical methods for semi-infinite programs such as discretization and exchange methods and solve a sequence of (nonlinear) semi-definite programs (SDPs).
However, it is often too demanding to find exact solutions of SDPs.
Our first approach does not rely on solving SDPs but on approximately following {a path leading to a solution}, which is formed on the intersection of the semi-infinite {region} and the interior of the semi-definite {region}.
We show weak* convergence of this method to a Karush-Kuhn-Tucker point
of the SISDP
under some mild assumptions and further provide with sufficient conditions for strong convergence.
Moreover,
as the second method, to achieve fast local convergence, we
integrate a two-step sequential quadratic programming method
{equipped} with Monteiro-Zhang scaling technique into the first method.
We particularly prove two-step superlinear convergence of the second
method using Alizadeh-Hareberly-Overton-like, Nesterov-Todd, and Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro scaling directions.
Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method through comparison with a discretization method that solves SDPs
obtained by finite relaxation of the SISDP.
\keywords{semi-infinite program
\and nonlinear semi-definite program
\and path-following method
\and superlinear convergence
\and global convergence
}
\subclass{90C22\and 90C26\and 90C34}
\end{abstract}
\section{Introduction}
In this paper, we consider the following nonlinear semi-infinite
semi-definite program with an infinite number of convex inequality constraints and one linear matrix inequality constraint, SISDP for short:
{\begin{align}
\begin{array}{ll}
\displaystyle{\mathop{\rm Minimize}} &f(x)
\\
{\rm subject~to} &g(x,tau)\elle 0\ \mbox{ for all } {tau}\in T, \\
&{F}(x)\in {S^m_{+}},\\
\end{array}\ellabel{lsisdp}
\end{align}
where
$f:\mathcal{R}^nto \mathcal{R}$ is a continuously differentiable function and $T$ is a compact metric space.
In addition, $g:\mathcal{R}^ntimes Tto \mathcal{R}$ is a continuous function, and $g(\cdot,tau)$ is supposed to be convex and continuously differentiable.
Moreover, $S^m$ and $S^m_{++} (S^m_{+})$ denote the sets of
$mtimes m$ symmetric matrices
and symmetric positive (semi-)definite matrices, respectively, and
$F(\cdot):\mathcal{R}^nto S^{m}$ is an affine function, i.e.,
$$
F(x):=F_0+\sum_{i=1}^nx_iF_i
$$
with $F_i\in S^m$ for $i=0,1,\elldots,n$ and $x=(x_1,x_2,\elldots,x_n)^{top}$.
We assume that the SISDP\,\eqref{lsisdp} has a nonempty solution set.
We may let the SISDP\,\eqref{lsisdp} include linear equality constraints, to which the algorithms and theories
given in the subsequent sections can be extended straightforwardly. But, for simplicity of expression, we omit them.
When $T$ comprises a finite number of elements, the SISDP reduces to a nonlinear semi-definite program (nonlinear SDP or NSDP).
Particularly when all the functions are affine with respect to $x$, it further reduces to the linear SDP (LSDP).
As is known broadly, studies on the LSDP have been crucially promoted in the aspects of theory, algorithms, and applications\,\cite{wolkowicz2012handbook}.
Compared with the LSDP, studies on the NSDP are still scarce, although
important applications are found in various areas\,\cite{freund2007nonlinear,konno2003cutting,leibfritz2009successive}.
Shapiro\,\cite{shapiro1997first} expanded an elaborate theory on the first and second order optimality conditions of the NSDP.
See \cite{BonSp} for a comprehensive description of the optimality conditions and duality theory of the NSDP.
Yamashita et al.\,\cite{yabe} proposed a primal-dual interior point-type method using the Monteiro-Zhang (MZ) directions family and showed its global convergence property. They further made local convergence analysis in \cite{yamashita2012local}.
The SQP method {for nonliear programs} was also {extended} to the NSDP by Freund et al.\,\cite{freund2007nonlinear}.
See the survey article\,\cite{yamashita2015survey} for {more} algorithms
designed to solve the NSDP.
In the absence of the semi-definite constraint, \eqref{lsisdp} becomes a nonlinear semi-infinite program (SIP) with an infinite number of convex constraints.
For solving nonlinear SIPs, many researchers proposed various kinds of algorithms, for example
discretization based methods\,\cite{reemtsen1991discretization,still2001discretization}, local reduction based methods\,\cite{gramlich1995local,pereira2011interior,pereira2009reduction,Tanaka}, Newton-type methods\,\cite{li2004smoothing,qi2003semismooth}, smoothing projection methods\,\cite{xu2014solving}, convexification based methods\,\cite{floudas2007adaptive,shiu2012relaxed,stein2012adaptive,wang2015feasible}, and so on.
For an overview of the SIP, see \cite{sip-recent,sip2,Reem} and the references therein.
Most closely related to the SISDP\,\eqref{lsisdp} are SIPs involving (possibly infinitely many) conic constraints.
Li et al.\,\cite{li2004solution} considered a linear SIP with semi-definite constraints and proposed a discretization based method.
Subsequently, Li et al.\,\cite{li2006relaxed} tackled the same problem and developed a relaxed cutting plane method.
Hayashi and Wu \cite{hayashi4} focused on a linear SIP involving second-order cone (SOC) constraints and proposed an exchange-type method. It is worth mentioning that the SISDP\,\eqref{lsisdp} can be viewed as a generalization of those problems.
More recently, Okuno et al.\,\cite{okuno2012regularized} considered a convex SIP with an infinite number of conic constraints, and proposed an exchange-type method combined with Tikhonov's regularization technique. Okuno and Fukushima\,\cite{okuno2014local} restricted themselves to a nonlinear SIP with infinitely many SOC constraints, and constructed a quadratically convergent sequential quadratic programming (SQP)-type method based on the local reduction method. One of common features of the algorithms mentioned above is to solve a sequence of certain conic constrained problems.
We can find some important applications of the SISDP.
For example,
semi-infinite eigenvalue optimization problems\,\cite{li2004solution},
finite impulse response (FIR) filter design problems\,\cite{spwu1996}, and
robust envelop-constrained filter design with orthonormal bases\,\cite{li2007robust} can be formulated as the SISDP whose functions are all affine with respect to $x$.
{Moreover,} robust beam forming problems\,\cite{yu2008novel} can be formulated as the SISDP with infinitely many nonlinear {inequality constraints}.
However, to the best of our knowledge, there is no existing work that deals with the SISDP\,\eqref{lsisdp} itself.
{In this paper, we propose two algorithms tailored to the SISDP.
In the first method, we generate a sequence approaching a Karush-Kuhn-Tucker (KKT) point of the SISDP
by approximately following a central path formed by barrier KKT (BKKT) points of the SISDP.
The BKKT points, whose definition will be provided in Section\,\ref{sec:3}, can be computed efficiently using the interior-point SQP-type method proposed in the authors' recent work\,\cite{okuno2018sc}.
Although it is possible to design a convergent algorithm that solves NSDPs iteratively like the existing algorithms mentioned in the previous paragraph, it is often too demanding to get an accurate solution of an NSDP at each iteration.
In contrast, the proposed path-following algorithm will only require solving quadratic programs if it is combined with the interior point SQP-type method.
In the second method, to accelerate the local convergence speed, we {further}
integrate a two-step {SQP} method into the first method.
{Specifically, we derive the scaled barrier KKT system of the SISDP by means of the local reduction method\,\cite{gramlich1995local,pereira2011interior,pereira2009reduction,Tanaka}
and the Monteiro-Zhang scaling technique\,\cite[Chapter~10]{wolkowicz2012handbook}.
We then perform a two-step SQP method to generate iteration points, while decreasing a barrier parameter to zero superlinearly.
In each step of the two-step SQP,
to produce a search direction,
we solve
a mixed linear complementarity system
approximating the aforementioned scaled barrier KKT system, which can be solved via a certain quadratic program.
We then adjust a step-size along the obtained search direction so that the next iteration point remains to lie in the interior of the semi-definite region.
We will show that, under some regularity conditions at a KKT point of the SISDP, a step-size of the unity is eventually adopted and two-step superlinear convergence is achieved.
The proposed methods may be viewed as an extension of the primal-dual interior point method\,\cite{yabe} for the NSDPs.
Nonetheless, the theoretical and algorithmic extensions are not straightforward because of the {presence} of infinitely many inequality constraints.
Furthermore, the results {obtained in the paper} have novelty not only in the field of the SIP but also the NSDP.
The paper is organized as follows:
In Section~\ref{sec:3}, we propose a primal-dual path-following method for the SISDP.
{We prove that {any} $\mbox{weak}^\ast$-accumulation point of the generated sequence is a KKT point of the SISDP under some mild assumptions. We also give a sufficient condition for strong convergence of the sequence.}
In Section~\ref{sec:4}, we further combine the local-reduction based SQP method with the prototype method and prove that it converges to a KKT point of the SISDP two-step superlinearly.
In Section~\ref{sec:5}, we conduct some numerical experiments to exhibit the efficiency of the proposed method.
Finally, we conclude this paper with some remarks.
\subsection*{Notations}
Throughout this paper, we use the following notations:
The identity matrix is denoted by $I$.
For any $P\in \mathcal{R}^{mtimes m}$, ${\rm Tr}(P)$ denotes the trace of $P$.
For any symmetric matrices $X,Y\in S^m$, we denote the Jordan product of $X$ and $Y$ by $X\circ Y:=(XY+YX)/2$ and the inner product of $X$ and $Y$ by
$X\bullet Y={\rm Tr}(XY)$.
Also, we denote the Frobenius norm $\|X\|_F:=\sqrt{X\bullet X}$
and
\begin{align*}
{\rm svec}(X):=&
(X_{11},\sqrt{2}X_{21},\elldots,\sqrt{2}X_{m1},X_{22},\\
&\hspace{3em}\sqrt{2}X_{32},\elldots,\sqrt{2}X_{m2},X_{33},\elldots,X_{mm})^{top}\in \mathcal{R}^{\frac{m(m+1)}{2}}
\end{align*}
for $X\in S^m$.
We write $\mathcal{F}iV:=\elleft(F_1\bullet V,F_2\bullet V,\elldots,F_n\bullet V\right)^{top}\in \mathcal{R}^n$
for $V,F_1,F_2,\elldots,F_n\in S^m$.
For any $X\in S^m$, we define the linear operator $\mathcal{L}_X:S^mto S^m$ by $\mathcal{L}_X(Z):=X\circ Z$.
We also denote $({\zeta})_+:=\max({\zeta},0)$ for any $\zeta\in \mathcal{R}$.
For sequences $\{y^k\}$ and $\{z^k\}$, if $\|y^k\|\elle M\|z^k\|$
for any $k$ with some $M>0$,
we write $\|y^k\|=O(\|z^k\|)$. If $M_1\|z^k\|\elle \|y^k\|\elle M_2\|z^k\|$
for any $k$ with some $M_1,M_2>0$, we represent $\|y^k\|=\Theta(\|z^k\|)$.
Moreover, if there exists a sequence $\{\alpha_k\}$ with $\ellim_{kto\infty}\alpha_k=0$ and $\|y^k\|\elle \alpha_k\|z^k\|$ for any $k$,
we write $\|y^k\|=o(\|z^k\|)$.
Finally, we let $\perp$ denote the perpendicularity.
\subsection*{Terminologies from functional analysis}
Let us review some terminologies from functional analysis briefly.
For more details, refer to the basic material \cite[Section~2]{BonSp} or suitable textbooks of functional analysis.
Let $\mathcal{C}(T)$ be the set of real-valued continuous functions defined on $T$ endowed with the supremum norm $\|h\|:=\max_{tau\in T}|h(tau)|$.
Let $\mathcal{M}(T)$ be the dual space of $\mathcal{C}(T)$, which can be identified with the space of (finite signed) regular Borel measures with the Borel sigma algebra $\mathcal{B}$ on $T$ equipped with the total variation norm, i.e.,
$\|y\|:=\sup_{A\in \mathcal{B}}y(A)-\inf_{A\in \mathcal{B}}y(A)$ for $y\in \mathcal{M}(T)$.
Denote by $\mathcal{M}_+(T)$ the set of all the nonnegative Borel measures of $\mathcal{M}(T)$.
Especially if $y\in \mathcal{M}_+(T)$, $\|y\|=y(T)$ since $\inf_{A\in \mathcal{B}}y(A)=y(\emptyset)=0$
and $\sup_{A\in \mathcal{B}}y(A)=y(T)$.
We say that $y\in \mathcal{M}(T)$ is a finite discrete measure if there exist a finite number of indices $tau_1,tau_2,\elldots,tau_q\in T$ and scalars $\alpha_1,\alpha_2,\elldots,\alpha_q\in \mathcal{R}$ such that $y(A)=\sum_{i=1}^q\alpha_i\delta_A(tau_i)$
for any Borel set $A\in \mathcal{B}$, where $\delta_S:Tto \mathcal{R}$ is the indicator function satisfying
$\delta_S(tau)=1$ if $tau\in S$ and $\delta_S(tau)=0$ otherwise.
Let
$\ellangle\cdot,\cdot\rangle: \mathcal{M}(T)times \mathcal{C}(T)to \mathcal{R}$
be the bilinear form defined by
$
\ellangle y,h\rangle:=\int_{T}h(tau)y^{(i)}dt
$
for $y\in \mathcal{M}(T)$ and $h\in \mathcal{C}(T)$.
We then endow $\mathcal{M}(T)$ with the $\mbox{weak}^\ast$-topology,
which is the minimum topology such that any seminorm
$p_{\mathcal{A}}$ on $\mathcal{M}(T)$ is continuous for any finite subset $\mathcal{A}\subseteq \mathcal{C}(T)$, where
$p_{\mathcal{A}}:\mathcal{M}(T)to \mathcal{R}$ is
defined by
$
p_{\mathcal{A}}(y):=\max_{h\in \mathcal{A}}|\ellangle y,h\rangle|.
$
Let us here specify the concept of accumulation points and limit points in the sense of the $\mbox{weak}^\ast$-topology.
Let $\{y^k\}$ be a sequence in $\mathcal{M}(T)$ and $y^{\ast}\in \mathcal{M}(T)$.
\begin{enumerate}
\item We call $y^{\ast}$ the $\mbox{weak}^\ast$ limit point of $\{y^k\}$
if for any neighborhood $\mathcal{N}(y^{\ast})$ of $y^{\ast}$ with respect to the $\mbox{weak}^\ast$-topology
there exists an integer $K\ge 0$ such that $y^k\in\mathcal{N}(y^{\ast})$ for any $k\ge K$.
We then say $\{y^k\}$ $\mbox{weakly}^\ast$ converges to $y^{\ast}$ and often write it as {${\rm w}^\ast\mbox{-}\ellim_{kto \infty}y^k=y^{\ast}$}.
\item
We call $y^{\ast}$ a $\mbox{weak}^\ast$ accumulation point of $\{y^k\}$
if for any integer $K\ge 0$ and neighborhood $\mathcal{N}(y^{\ast})$ of $y^{\ast}$ with respect to the $\mbox{weak}^\ast$-topology
there exists an integer $k\ge K$ such that $y^k\in\mathcal{N}(y^{\ast})$.
\end{enumerate}
\section{Primal-dual path-following method}\ellabel{sec:3}
\subsection{KKT conditions for the SISDP}
In this section, we present the Karush-Kuhn-Tucker (KKT) conditions for the SISDP together with Slater's constraint qualification, abbreviated as SCQ.
Here, SCQ for the SISDP is defined precisely as below:
\begin{definition}
We say that the Slater constraint qualification (SCQ) holds for the SISDP if there exists some $\bar{x}\in \mathcal{R}^n$ such that
$
F(\bar{x})\in S^m_{++}$ and $g(\bar{x},tau)<0\ (tau\in T).
$
\end{definition}
\begin{theorem}\ellabel{kkt}
Let $x^{\ast}\in \mathcal{R}^n$ be a local optimal solution of the SISDP\,\eqref{lsisdp}.
Then, under the SCQ,
there exists some finite Borel-measure $y\in \mathcal{M}(T)$ such that
\begin{align}
&\nabla f(x^{\ast})+\int_T\nabla_xg(x^{\ast},tau)y^{(i)}dt-
\mathcal{F}iV
=0,\ellabel{e1}\\
&{F}(x^{\ast})\circ V=O,\ F(x^{\ast})\in S^m_+,\ V\in S^m_+,\ellabel{e2}\\
&\int_Tg(x^{\ast},tau)y^{(i)}dt=0,\ g(x^{\ast},tau)\elle 0\ (tau\in T),\ y\in\mathcal{M}_+(T), \ellabel{e4}
\end{align}
where
$V\in S^m$ is a Lagrange multiplier matrix associated with the constraint $F(x)\in S^m_{+}$.
In particular, there exists some discrete measure $y\in \mathcal{M}_+(T)$ satisfying the above conditions and $|{\rm supp}(y)|\elle n$, where
${\rm supp}(y):=\{tau\in T\mid y(\{tau\})\neq 0\}$.
Conversely, when $f$ is convex, if the above conditions {\eqref{e1}--\eqref{e4}} hold, then $x^{\ast}$ is an optimum of the SISDP\,\eqref{lsisdp}.
\end{theorem}
\begin{proof}
Note that $F(x^{\ast})\bullet V=0,\ F(x^{\ast})\in S^m_+$ and $V\in S^m_+$ hold
if and only if $F(x^{\ast})\circ V=O,\ F(x^{\ast})\in S^m_+$, and $V\in S^m_+$.
Then, the claim is proved in a manner similar to \cite[Theorem~2.4]{okuno2012regularized}.
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
\noindent {The system \eqref{e1}--\eqref{e4} is called the Karush-Kuhn-Tucker (KKT) conditions for the SISDP\,\eqref{lsisdp}.}
{We call
$(x,y,V)$ satisfying the KKT conditions \eqref{e1}--\eqref{e4} a KKT point of the SISDP\,\eqref{lsisdp} in particular.}
\subsection{Description of the algorithm}\ellabel{sec:rmu}
In this section, we propose an algorithm for solving the SISDP\,\eqref{lsisdp}, whose
fundamental framework is analogous to the primal-dual interior point method developed for solving the nonlinear SDP in \cite{yabe}.
It aims to find a KKT point of the SISDP\,\eqref{lsisdp}, i.e., a point satisfying the {optimality} conditions \eqref{e1}--\eqref{e4} for the SISDP\,\eqref{lsisdp}.
{Let us define the} function $R_{\mu}:\mathcal{R}^ntimes \mathcal{M}(T)times S^m_{+}to \mathcal{R}$ with a parameter $\mu\ge 0$ by
\begin{equation}
R_{\mu}(x,y,V):=
\sqrt{
theta(x)^2+
\|\varphi_1(x,{y},{V})\|^2+\varphi_2(x,y)^2+
\|\varphi_3(x,V,\mu)\|^2},\notag
\end{equation}
where
\begin{align}
theta(x)&:=\max_{tau\in T}\,\elleft(g(x,tau)\right)_+,
\notag \\
\varphi_1(x,{y},{V})&:=\nabla f(x)+{\displaystyle \int_T\nabla_xg(x,tau)y^{(i)}dt} -\mathcal{F}iV,
\notag \\
\varphi_2(x,y)&:=\int_Tg(x,tau)y^{(i)}dt,\notag\\
\varphi_3(x,V,\mu)&:={{\rm svec}}\elleft(F(x)\circ V-\mu I\right).\notag
\end{align}
Notice that a point satisfying $R_0(x,y,V)=0$ with $F(x)\in S^m_+$
and $V\in S^m_+$ is nothing but a KKT point of the SISDP\,\eqref{lsisdp}.
In terms of the function $R_{\mu}$, we define a barrier KKT(BKKT) point by perturbing
the semi-definite complementarity condition in the KKT conditions\,\eqref{e1}--\eqref{e4}.
\begin{definition}\ellabel{def_bkkt}
Let $\mu>0$.
We call $\elleft(x,y,V\right)\in \mathcal{R}^ntimes \mathcal{M}(T){times S^m}$ a barrier Karush-Kuhn-Tucker (BKKT) point of the SISDP\,\eqref{lsisdp} if
$R_{\mu}(x,y,V)=0$,\ $y\in \mathcal{M}_+(T)$,\ $F(x)\in S^m_{++}$, $V\in S^m_{++}$.
\end{definition}
Additionally, given a positive parameter $\varepsilon$, we define a neighborhood of the BKKT points with barrier parameter $\mu$:
\begin{equation*}
{\mathcal{N}}_{\mu}^{\sharp\,\mathrm{var}epsilon}:=\elleft\{w:=(x,y,V)\in \mathcal{R}^ntimes \mathcal{M}_+(T)times S^m_{++}\mid R_{\mu}(w)\elle \sharp\,\mathrm{var}epsilon,\ F(x)\in S^m_{++}\right\}.
\end{equation*}
The algorithm generates a sequence of
approximate BKKT points $\{w^k\}$ for the SISDP\,\eqref{lsisdp}
{such that
$w^k\in \mathcal{N}_{\mu_k}^{\varepsilon_k}$} for each $k$
while driving the values of both parameters $\mu_k$ and $\varepsilon_k$ to $0$ as $k$ tends to $\infty$.
\\
{\bf Algorithm~1} (Primal-dual path following method)
\begin{description}
\item[Step 0 (Initial setting):]
Choose an initial iteration point $w^0:=(x^0,y^0,V_0)\in \mathcal{R}^ntimes \mathcal{M}_+(T)times S^m$ such that
$F(x^0)\in S^m_{++}$ and $V_0\in S^m_{++}$.
Choose the initial parameters $\mu_0>0$, $\varepsilon_0>0$
and $\beta\in (0,1)$.
Let $k:=0$.
\item[Step~1 (Stopping rule):]
Stop if
\begin{equation}
R_{0}(w^k)=0,\ {F(x^k)\in S^m_+,\ V_k\in S^m_+,\ y^k\in \mathcal{M}_+(T).}
\end{equation}
Otherwise, go to Step~2.
\item[Step~{2} (Computing an approximate BKKT point):]
Find {an} approximate BKKT point
$w^{k+1}$ such that
\begin{equation}
w^{k+1}\in \mathcal{N}_{\mu_{k}}^{\varepsilon_{k}}.
\ellabel{bkkt_eq}
\end{equation}
\item[Step~3 (Update):]
Set $\mu_{k+1}:=\beta \mu_k$ and $\varepsilon_{k+1}:=\beta \varepsilon_k$.
Let $k:=k+1$. Return to Step~1.
\end{description}
In the recent work\,\cite{okuno2018sc}, the authors propose the interior-point SQP method for computing a BKKT point and show its global convergence property.
If we use the interior-point SQP method as a subroutine to find an approximate BKKT point satisfying condition\,\eqref{bkkt_eq}, Step~2 of Algorithm~1 is well-defined, i.e., such an approximate BKKT point
can be found {in} finitely many steps.
\subsection{Convergence analysis}
In this section,
we suppose the well-definedness of Step~3 in Algorithm~1 and establish its $\mbox{weak}^\ast$ convergence to KKT points
of SISDP\,\eqref{lsisdp}. Furthermore, we will characterize $\mbox{weak}^\ast$ accumulation points
of the generated sequence
more precisely for some special cases.
For the sake of analysis, we assume that Algorithm~1 produces an infinite sequence and further make the following assumptions:
\\
{\bf Assumption~A}
\begin{enumerate}
\item\ellabel{A2} {The} feasible set of SISDP\,\eqref{lsisdp} is {nonempty and} compact.
\item Slater's constraint qualification holds for SISDP\,\eqref{lsisdp}.
\end{enumerate}
Let $S^{\ast}\subseteq \mathcal{R}^n$ be the optimal solution set of SISDP\,\eqref{lsisdp} and $\bar{v}\in \mathcal{R}$ be
a constant larger than the optimal value of the SISDP.
If $f$ is convex, Assumption~A-\ref{A2} can be replaced with the milder assumption
that $S^{\ast}$ is compact by adding a convex constraint $f(x)\elle \bar{v}$ to the SISDP without changing the shape of $S^{\ast}$.
Under the above assumptions, we first show that the generated sequences $\{x^k\}$ and $\elleft\{(y^k,V_k)\right\}$ are bounded.
\begin{proposition}\ellabel{bound1}
Suppose that Assumption A-\ref{A2} holds. Then, any sequence $\{x^k\}$
produced by Algorithm~1 is bounded.
\end{proposition}
\begin{proof}
Denote the feasible set of SISDP\,\eqref{lsisdp} by $\mathcal{F}$ and
define a proper closed convex function $\varphi:\mathcal{R}^nto \mathcal{R}$ by
$$
\varphi(x):={\max}\mathcal{B}ig(-\ellambda_{\min}(F(x)),\ \max_{tau\in T}g(x,tau)\mathcal{B}ig).
$$
Since the level set $\{x\in \mathcal{R}^n \mid \varphi(x)\elle 0\}(=\mathcal{F})$ is compact,
any level set $\{x\in \mathcal{R}^n \mid \varphi(x)\elle \eta\}$ with $\eta>0$ is also compact. From \eqref{bkkt_eq} and $\sharp\,\mathrm{var}epsilon_k\elle \sharp\,\mathrm{var}epsilon_0$
for all $k$ sufficiently large,
it is not difficult to show that
$\{x^k\}\subseteq \{x\in \mathcal{R}^n \mid \varphi(x)\elle \sharp\,\mathrm{var}epsilon_0\}$, where $\sharp\,\mathrm{var}epsilon_0$ is an algorithmic parameter given in Step~0, and thus $\{x^k\}$ is bounded.
\\
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
\begin{proposition}\ellabel{prop2}
Suppose that Assumption~A holds.
Then, the generated Lagrange multiplier sequences $\{V_k\}\subseteq S^m_{++}$ {and $\{y^k\}\subseteq M_+(T)$} are bounded.
\end{proposition}
\begin{proof}
For simplicity of expression,
denote $tilde{w}^k:=(V_k,y^k)\in S^mtimes \mathcal{M}_+(T)$ and
\begin{equation*}
W_k:=\frac{V_k}{\|tilde{w}^k\|},\ p^k:=\frac{y^k}{\|tilde{w}^k\|}
\end{equation*}
{where $\|\cdot\|$ is a suitable norm such that
$\|tilde{w}^k\|^2=\|V_k\|^2+\|y^k\|^2$
on $S^mtimes \mathcal{M}(T)$.}
For contradiction, suppose that there exists a subsequence
$\{tilde{w}^k\}_{k\in K}\subseteq \{tilde{w}^k\}$
such that $\|tilde{w}^k\|to \infty\ (k\in Kto \infty)$.
Note that $\{(W_k,p^k)\}$ is bounded.
Notice also that
the corresponding sequence
$\{x^k\}_{k\in K}$ is bounded from Proposition\,\ref{bound1}.
Recall that any bounded sequence in $\mathcal{M}(T)$ has at least one $\mbox{weak}^\ast$ accumulation point and one can extract a subsequence $\mbox{weakly}^\ast$ converging to that point. Thanks to this property, without loss of generality we can assume that there exists a point $\elleft(x^{\ast},W_{\ast},p^{\ast}\right)\in \mathcal{R}^ntimes S^m_+times \mathcal{M}_+(T)$ such that
{\begin{align*}
\ellim_{k\in Kto \infty}\elleft(x^k,W_k\right)
=\elleft(x^{\ast},W_{\ast}\right),\ {\rm w}^\ast\mbox{-}\ellim_{k\in Kto \infty} p^k=p^{\ast}.\notag
\end{align*}
Note, in particular, that $\|(W_{\ast},p^{\ast})\|=1$, since ${\rm w}^\ast\mbox{-}\ellim_{k\in Kto\infty}p^k=p^{\ast}$ entails
the relation that $$
\ellim_{k\in Kto \infty}\|p^k\|
=\ellim_{k\in Kto \infty}\int_Tdp^k(tau)=\int_Tdp^{\ast}(tau)=\|p^{\ast}\|
$$ and therefore
\begin{align*}
\|(W_{\ast},p^{\ast})\|^2&=\|W_{\ast}\|^2+\|p^{\ast}\|^2\\
&=\ellim_{k\in Kto\infty}\elleft(\|W_{k}\|^2+\|p^{k}\|^2\right)\\
&=1.
\end{align*}
From \eqref{bkkt_eq}, for {each} $k\ge 1$, we have
\begin{align}
&\elleft\|\frac{\nabla f(x^k)}{\|tilde{w}^k\|}-\mathcal{F}iWk
+\int_T\nabla_x g(x^k,tau)dp^k(tau)\right\|\elle \frac{\varepsilon_{k-1}}{\|tilde{w}^k\|},\notag \\
&\elleft|
\int_Tg(x^k,tau)dp^k(tau)
\right|\elle {\frac{\varepsilon_{k-1}}{\|tilde{w}^k\|}},\ p^k\in\mathcal{M}_+(T),\notag \\
&\elleft\|F(x^k)\circ W_k{-\frac{\mu_{k-1}}{\|tilde{w}^k\|}I}\right\|\elle \frac{\varepsilon_{k-1}}{\|tilde{w}^k\|},\ F(x^k)\in S^m_{++},\ W_k\in S^m_{++}.\notag
\end{align}
By letting $k\in Kto \infty$, we obtain
\begin{align}
&\mathcal{F}iWast
-\int_T\nabla_x g(x^{\ast},tau)dp^{\ast}(tau)=0,\ellabel{cp1} \\
&\int_Tg(x^{\ast},tau)dp^{\ast}(tau)=0,\ p^{\ast}\in\mathcal{M}_+(T),\ellabel{cp4} \\
&F(x^{\ast})\circ W_{\ast}=O,\ F(x^{\ast})\in S^m_+,\ W_{\ast}\in S^m_+.\ellabel{cp5}
\end{align}
Now, choose a Slater point $tilde{x}\in \mathcal{R}^n$ arbitrarily and let $tilde{d}:=tilde{x}-x^{\ast}$.
Notice here that
\begin{equation}
F(tilde{x})\bullet W_{\ast}\ge 0,\ \int_Tg(tilde{x},tau)dp^{\ast}(tau)\elle 0, \ellabel{eq:0807}
\end{equation}
since
\begin{equation}
F(tilde{x})\in S^m_{++},\ W_{\ast}\in S^m_{+},\ \max_{tau\in T}g(tilde{x},tau)<0,\ p^{\ast}\in \mathcal{M}_+(T). \ellabel{eq:0807-2}
\end{equation}
Then, it holds that
\begin{eqnarray}
\ellefteqn{F(tilde{x})\bullet W_{\ast}-\int_Tg(tilde{x},tau)dp^{\ast}(tau)}\notag \\
&= &F(x^{\ast}+tilde{d})\bullet W_{\ast}
-\int_Tg(x^{\ast}+tilde{d},tau)dp^{\ast}(tau)\notag \\
&\elle&F(x^{\ast}+tilde{d})\bullet W_{\ast}-\int_T\elleft(g(x^{\ast},tau)
+\nabla_xg(x^{\ast},tau)^{top}tilde{d}\right)dp^{\ast}(tau)\notag \\
&=&tilde{d}^{top}
\mathcal{F}iWast
-\int_T \elleft(\nabla _xg(x^{\ast},tau)^{top}tilde{d}\right)dp^{\ast}(tau)\notag \\
&=&tilde{d}^{top}\elleft(
\mathcal{F}iWast
-\int_T\nabla_xg(x^{\ast},tau)dp^{\ast}(tau)\right)\notag \\
&=&0, \ellabel{eqa:0807-2}
\end{eqnarray}
where the first inequality holds because $g(x^{\ast},tau)+\nabla_xg(x^{\ast},tau)^{top}tilde{d}\elle g(x^{\ast}+tilde{d},tau)\ (tau\in T)$ by the convexity of $g(\cdot,tau)$.
Moreover, the third equality is obtained from \eqref{cp4} and the fact that $F(x^{\ast})\bullet W_{\ast}=0$ {by} \eqref{cp5}.
The last equality is due to \eqref{cp1}.
Combining \eqref{eq:0807} and \eqref{eqa:0807-2} implies that
$F(tilde{x})\bullet W_{\ast}=0$ and
$\int_Tg(tilde{x})dp^{\ast}(tau)=0$, from which we can conclude
$W_{\ast}=O$ and $p^{\ast}=0$ by using \eqref{eq:0807-2} again.
However, this contradicts $\|(W_{\ast},p^{\ast})\|=1$.
The proof is complete.}\\
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
Now, we are ready to establish the global convergence property of Algorithm~1.
\begin{theorem}\ellabel{thm:0612}
Suppose that Assumption~A holds. Then, the sequence $\{(x^k,y^k,V_k)\}$
produced by Algorithm~1 is bounded.
Let $(x^{\ast},y^{\ast},V_{\ast})\in \mathcal{R}^ntimes \mathcal{M}_+(T)times S^m$ be a $\mbox{weak}^\ast$-accumulation point of $\{(x^k,y^k,V_k)\}$.
Then, $(x^{\ast},y^{\ast},V_{\ast})$ is a KKT point of SISDP\,\eqref{lsisdp}.
In particular, if $f$ is convex, $x^{\ast}$ is an optimum.
\end{theorem}
\begin{proof}
The boundedness of $\{(x^k,y^k,V_k)\}$ follows from Propositions~{\ref{bound1} and \ref{prop2}}.
It remains to show the second half of the theorem.
We can assume
$\ellim_{kto \infty}(x^k,V_k)=(x^{\ast},V_{\ast})$ and ${\rm w}^\ast\mbox{-}\ellim_{kto\infty}y^k=y^{\ast}$ without loss of generality.
Then, by letting $kto \infty$ in \eqref{bkkt_eq},
we see that the KKT conditions\,\eqref{e1}--\eqref{e4} hold with $V=V_{\ast}$ and $y=y^{\ast}$.
By the second half of Theorem~\ref{kkt}, $x^{\ast}$ is an optimum of SISDP\,\eqref{lsisdp} when $f$ is convex.
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
Subsequently, let us consider the situation where
the number of elements of ${\rm supp}(y^k)$ is bounded from above through execution of the algorithm.
In this case, we can find a more precise form of the $\mbox{weak}^\ast$ accumulation points of $\{y^k\}$.
To see this, we begin with assuming $|{\rm supp}(y^k)|\elle M$ for any $k\ge 0$ with some $M>0$, and consider a sequence $\{t^k\}\subseteq T^M:=\overbrace{Ttimes \cdots times T}^{M\mbox{ times}}$
with $t^k:=(tau^k_1,tau^k_2,\elldots,tau^k_M)$ such that $t^k$ has all elements of ${\rm supp}(y^k)$ as a sub-vector and
$y^k(tau^k_i)=0$ if $tau^k_i\notin {\rm supp}(y^k)$ for $i=1,2,\elldots,M$.
Denote $\zeta^k:=(y^k(tau_1^k),y^k(tau_2^k),\elldots,y^k(tau_M^k))^{top}\in \mathcal{R}^M_+$ for $k=1,2,\elldots$.
In a manner similar to Proposition\,\ref{prop2}, we can show that $\{t^k\}$ and the accompanying sequence $\{(x^k,V_k)\}$ are bounded and have accumulation points with regard to the norm topology.
Without loss of generality, we suppose that there exist $(x^{\ast},V_{\ast})\in \mathcal{R}^ntimes S^m_+$, $t^{\ast}=(tau^{\ast}_1,tau^{\ast}_2,\elldots,tau^{\ast}_M)\in T^M$, and $\zeta^{\ast}=(\zeta^{\ast}_1,\zeta^{\ast}_2,\elldots,\zeta^{\ast}_M)\in \mathcal{R}^M_+$ such that $\ellim_{kto \infty}(x^k,V_k,\zeta^k,t^k)=(x^{\ast},V_{\ast},\zeta^{\ast},t^{\ast})$. Then we can establish the following theorem concerning the explicit form of the $\mbox{weak}^\ast$-accumulation point of $\{(x^k,y^k,V_k)\}$. In the remainder of the section, we use the notations and symbols introduced in this paragraph.
\begin{theorem}\ellabel{thm:0912}
Denote the {distinct} elements of $\{tau_1^{\ast},tau_2^{\ast},\elldots,tau_M^{\ast}\}$ by $s_1,s_2,\elldots,s_p\in T$, where $p\elle M$,
and define a finite discrete measure $y^{\ast}:\mathcal{B}to \mathcal{R}_+$ by
\begin{equation}
y^{\ast}(A):=\sum_{j=1}^p\xi_j^{\ast}\delta_A(s_j)\ \ \ (A\in \mathcal{B}),\ellabel{eq:yast}
\end{equation}
where
$
\xi_j^{\ast}:=\sum_{i:tau_i^{\ast}=s_j}\zeta^{\ast}_i
$ for $j=1,2,\elldots,p$.
Then, ${\rm w}^\ast\mbox{-}\ellim_{kto \infty}y^k=y^{\ast}$ holds and $(x^{\ast},y^{\ast},V_{\ast})$ is a KKT point of SISDP\,\eqref{lsisdp}.
\end{theorem}
\begin{proof}
Since the proof is straightforward, we omit it.
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
Let us end the section with the most concise but practical version for Theorem~\ref{thm:0912}.
Let $y^{\ast}$ be the measure defined by \eqref{eq:yast} and consider the case where
$|{\rm supp}(y^{\ast})|=M$.
Then, we readily obtain the following corollary from Theorem~\ref{thm:0912}:
\begin{corollary}
Suppose that ${\rm supp}(y^{\ast})=\{tau_1^{\ast},tau_2^{\ast},\elldots,tau_M^{\ast}\}$ and $tau^{\ast}_i\neq tau^{\ast}_j$ for any $i\neq j$.
Then,
$\{y^k\}$ converges to $y^{\ast}$ strongly on $\mathcal{M}(T)$
and $(x^{\ast},y^{\ast},V_{\ast})$ is a KKT point for SISDP\,\eqref{lsisdp}.
\end{corollary}
\section{Two-step superlinearly convergent algorithm}\ellabel{sec:4}
In the section, for the sake of rapid local convergence,
we propose to integrate the local reduction method\,\cite{sip2,pereira2009reduction,okuno2014local,Tanaka}, which is a classical semi-infinite optimization method, with Algorithm~1.
Throughout this section, we assume that the compact metric space $T$ is a bounded closed set in $\mathcal{R}^q$
formed by finitely many sufficiently smooth inequality constraints.
Also, we often identify $X\in S^m$ with $\mathop{\rm svec}(X)\in \mathcal{R}^{\frac{m(m+1)}{2}}$.
\subsection{The overall structure of the proposed algorithm}
The proposed method is designed to converge to a KKT point of SISDP\,\eqref{lsisdp} at least two-step superlinearly while satisfying the interior point constraints.
More precisely, it generates a sequence $\elleft\{w^k\right\}:=\elleft\{(x^k,y^k,V_k)\right\}\subseteq \mathcal{R}^ntimes \mathcal{M}_+(T)times S^m_{++}$
together with two kinds of search directions
\begin{equation*}
\elleft\{\Delta_{\frac{1}{2}} w^{k}\right\}:=\elleft\{(\Delta_{\frac{1}{2}} x^{k},\Delta_{\frac{1}{2}} y^{k},\Delta_{\frac{1}{2}} V_{k})\right\}\mbox{ and }
\elleft\{\Delta_1w^{k}\right\}:=\elleft\{\elleft(\Delta_{1} x^{k},\Delta_{1} y^{k},\Delta_{1} V_{k}\right)\right\}
\end{equation*}
such that
\begin{align}
&\elleft\|w^{k}+s^k_1h\Delta_{\frac{1}{2}} w^{k}+s^k_1\Delta_1 w^{k}-w^{\ast}\right\|=o\elleft(\elleft\|w^k-w^{\ast}\right\|\right),\notag\\
&F(x^k+s^k_1h \Delta_{\frac{1}{2}}x^k)\in S^m_{++},\ F(x^k+s^k_1h \Delta_{\frac{1}{2}}x^k+s^k_1\Delta_1x^k)\in S^m_{++},\ellabel{al:1115-1}\\
&V_k+s^k_1h \Delta_{\frac{1}{2}}V_k\in S^m_{++},\ V_k+s^k_1h \Delta_{\frac{1}{2}}V_k+s^k_1\Delta_1V_k\in S^m_{++}\ellabel{al:1115-2}
\end{align}
for any $k$ sufficiently large, where $w^{\ast}$ is a KKT point satisfying a certain regularity condition and $s_{j}^k\ (j=\fr,1)$ are step-sizes determined by
\footnote{
For $X\in S^{m}_{++},Y\in S^m$, the eigenvalues of $X^{-1}Y$ are real numbers, and hence $\ellambda_{\rm min}(X^{-1}Y)\in \mathcal{R}$.}
\begin{equation}
s_j^{k}=\min(t^{k}_j,u^{k}_j),\ellabel{eq:s0}
\end{equation}
where
\begin{align*}
t^{k}_j&:=
\begin{cases}
- \displaystyle{\frac{\delta}{\ellambda_{\rm min}(F(x^{k+j-\fr})^{-1}\sum_{i=1}^n\Delta_jx^k_iF_i)}}\elle \delta
\hspace{1em}&\mbox{if }\ellambda_{\rm min}\elleft(F(x^{k+j-\fr})^{-1}\sum_{i=1}^n\Delta_jx^k_iF_i\right)\elle -1\\
1 &\mbox{otherwise},
\end{cases}\\
u^{k}_j&:=
\begin{cases}
- \displaystyle{\frac{\delta}{\ellambda_{\rm min}(V^{-1}_{k+j-\fr}\Delta V_{k+j-\fr})}}\elle \delta\hspace{1em}&\mbox{if }\ellambda_{\rm min}\elleft(V_{k+j-\fr}^{-1}\Delta_jV_{k+j-\fr}\right)\elle -1\\
1 &\mbox{otherwise},
\end{cases}
\end{align*}
for $j={\frac{1}{2}},1$, where $\delta\in (0,1)$ is a {prescribed} algorithmic constant.
Here, $(x^{k+\fr},V_{k+\fr})$ is defined as
\begin{equation*}
(x^{k+\fr},V_{k+\fr}):=\elleft(x^k+s^k_1h\Delta_{\fr}x^k,V_k+s^k_1h\Delta_{\fr}V_k\right).
\end{equation*}
By the above choice of the step-sizes,
$s_{\fr}^k,s_{1}^k\in (0,1]$ holds and
the interior point constraints \eqref{al:1115-1} and \eqref{al:1115-2} are valid since
\begin{align}
\bar{s}&:={\sup\elleft\{s\mid \ellambda_{\rm min}(X+s\Delta X)\ge 0,s\ge 0\right\}}\notag\\
&=\begin{cases}
-\displaystyle{\frac{1}{\ellambda_{\rm min}(X^{-1}\Delta X)}}\ &\mbox{if }\ellambda_{\rm min}(X^{-1}\Delta X)<0\\
{\infty}\ &\mbox{otherwise}
\end{cases}\ellabel{al:1116-lam}
\end{align}
for given $X\in S^m_{++}$ and $\Delta X\in S^m$.
{We remark that if $X+\Delta X\in S^m_{++}$, then $\bar{s}>1
$ and hence
the step-size rule along with \eqref{al:1116-lam} yields
\begin{equation}
\ellambda_{\min}(X^{-1}\Delta X)>-1.\ellabel{eq:1206-1}
\end{equation}}
So as to attain fast convergence speed as above,
we {try to} follow the central path closely by
updating the barrier parameter $\mu_k$ so that $\mu_{k+1}=o(\mu_k)$ and
solving certain nonlinear systems to have
the search directions $\Delta_{1}w^k$ and $\Delta_{\frac{1}{2}}w^k$.
When those directions turn out to be unsuccessful, a point near the central path is computed by the interior-point SQP method developed in the recent paper\,\cite{okuno2018sc}.
Before describing the details, we first show the overall structure of the proposed algorithm:
\\
{\bf Algorithm~2} (Superlinearly convergent primal-dual path following method)
\begin{description}
\item[Step 0 (Initial setting):]
Choose parameters
\begin{equation*}
{0<\alpha< 1},\ 0<\beta<1,\ \gamma_1,\gamma_2>0,\ \delta\in (0,1),\ \mu_0>0,\ 0<c\elle \frac{1}{\alpha+2}.
\end{equation*}
Set $\sharp\,\mathrm{var}epsilon_0:=\gamma_1\mu_0^{1+\alpha}$.
Choose the initial iteration point ${w^0:=}(x^0,y^0,V_0)\in \mathcal{R}^ntimes \mathcal{M}_+(T)times S^m$ such that $F(x^0)\in S^m_{++}$ and $V_0\in S^m_{++}$.
Let $k:=0$.
\item[Step~1 (Stopping rule):]
Stop if
\begin{equation*}
R_{0}({w^k})=0,\ F(x^k)\in S^m_+,\ V_k\in S^m_+,\ y^k\in \mathcal{M}_+(T).
\end{equation*}
Otherwise, go to Step~2.
\item[Step~{2} (Computing an approximate BKKT point):]
Find an approximate BKKT point ${w^{k+1}}\in \mathcal{N}_{\mu_k}^{\varepsilon_k}$
by the following procedure:
\begin{description}
\item[Step~2-1:]
Choose a scaling matrix $P_k$ and obtain $\Delta_{\frac{1}{2}}w^k$ by solving the mixed linear complementarity system\,\eqref{al:0506-1}, \eqref{al:0506-3}, and \eqref{al:0506-2}, which amounts to solving the QP\,\eqref{al:qp}
(see Section\,\ref{sec:QP}) with
$\mu=\mu_k$, $P=P_k$ and
$\bar{w}=w^k$.
Compute $s^k_1h$ by \eqref{eq:s0} with $j=\frac{1}{2}$ and set $w^{k+\fr}:=w^k+s^k_1h\Delta_{\fr}w^k$.
\item[Step~2-2:]
Choose a scaling matrix $P_{k+\frac{1}{2}}$.
If the linear equations \eqref{al:1201-1}--\eqref{al:1201-4}
(see Section\,\ref{sec:QP})
with $\mu=\mu_k$, $P=P_{k+\frac{1}{2}}$ and
$\bar{w}=w^{k+\fr}$ are solvable, then set a solution as $\Delta_{1}w^k$
and compute $s^k_1$ by \eqref{eq:s0} with $j=1$.
Otherwise, go to Step~2-4.
\item[Step~2-3:] If $w^k_+:=w^{k+\fr}+s^k_1\Delta_{1}w^k\in \mathcal{N}_{\mu_k}^{\varepsilon_k}$, set $w^{k+1}:=w^{k}_+$ and go to Step~3. Otherwise, go to Step~2-4.
\item[Step~2-4:]Find $w^{k+1}\in \mathcal{N}_{\mu_{k}}^{\varepsilon_{k}}$ using the interior-point SQP method.
\end{description}
\item[Step~3 (Update):]
Update the parameters as
\begin{equation}
\mu_{k+1}:=\min\elleft(\beta\mu_k, \gamma_2\mu_k^{1+c\alpha}\right),
\sharp\,\mathrm{var}epsilon_{k+1}:=\gamma_1\mu_{k+1}^{1+\alpha}.\ellabel{eq:update_mu}
\end{equation}
Set $k:=k+1$ and return to Step~1.
\end{description}
We will discuss the structure of the mixed linear complementarity system\,\eqref{al:0506-1}--\eqref{al:0506-3}
and the equations
\eqref{al:1201-1}--\eqref{al:1201-4}
in Steps~2-1 and 2-2
later in Section\,\ref{sec:QP}.
As is confirmed easily, Algorithm~2 is a variant of Algorithm~1.
Hence, by Theorem~\ref{thm:0612}, we ensure its global convergence to a KKT point.
In the subsequent convergence analysis, we will focus on the local convergence {rate} of Algorithm~2.
\subsection{Local reduction technique}\ellabel{sec:local}
We explain the local reduction method to the SISDP\,\eqref{lsisdp} briefly.
For more details, we refer {the} readers to \cite{sip2,pereira2009reduction,okuno2014local,Tanaka}.
Suppose that we are standing at a point $\bar{x}\in \mathcal{R}^n$.
The local reduction method represents
the semi-infinite region $D:=\{x\in\mathcal{R}^n\mid g(x,tau)\elle 0\ (tau\in T)\}$ with finitely many inequality constraints locally around $\bar{x}$.
Specifically,
in some open neighborhood of $\bar{x}$, say $U(\bar{x})$,
it expresses the region $D\cap U(\bar{x})$ as
$$
D\cap U(\bar{x})=\{x\in U(\bar{x})\mid g(x,tau^i_{\mathcal{B}ar{x}}(x))\elle 0\ (i=1,2,\elldots,p(\bar{x}))\}
$$
using smooth implicit functions $tau^i_{\mathcal{B}ar{x}}:U(\bar{x})to T\ (i=1,2,\elldots,p(\bar{x}))$ {with some nonnegative integer $p(\bar{x})$}.
Then, {SISDP\,\eqref{lsisdp} is locally equivalent to} the problem with finitely many inequality constraints in $U(\bar{x})$, namely,
\begin{align}
\begin{array}{ll}
\displaystyle{\mathop{\rm Minimize}_{x\in U(\bar{x})}} &f(x)
\\
{\rm subject~to} &\hat{g}_i(x):=g(x,tau^i_{\mathcal{B}ar{x}}(x))\elle 0\ (i=1,2,\elldots,p(\bar{x})), \\
&{F}(x)\in S^m_{+},
\end{array}\ellabel{reduced_LSISDP}
\end{align}
to which
standard nonlinear optimization algorithms such as the SQP-type method are conceptually applicable.
In what follows, we clarify the condition under which the functions $tau^i_{\mathcal{B}ar{x}}(\cdot)\ (i=1,2,\elldots,p(\bar{x}))$ and the {open neighborhood} $U(\bar{x})$ exist.
Let us denote by $S(x)$ the set of all local maximizers of $\max_{tau\in T} g(x,tau)$ and let
\begin{equation}
S_{\delta}(x):=\{tau\in S(x)\mid g(x,tau)>\max_{tau\in T} g(x,tau)-\delta\}
\ellabel{eq:Sdelta}
\end{equation}
for a given constant $\delta>0$.
Moreover, define the nondegeneracy of $\bar{x}$ as follows:
\begin{definition}\ellabel{def:nond}
We say that $\bar{x}$ is nondegenerate for $\max_{tau\in T}g(\bar{x},tau)$ and $\delta>0$ if
$|S_{\delta}(\bar{x})|<\infty$ and the linear independence constraint qualification, the second-order sufficient conditions, and the strict complementarity condition regarding $\max_{tau\in T}g(\bar{x},tau)$ hold at any $tau\in S_{\delta}(\bar{x})$.
\end{definition}
If $\bar{x}$ is nondegenerate, there exist an open neighborhood $U(\bar{x})\subseteq \mathcal{R}^n$, a nonnegative integer
$p(\bar{x}):=|S_{\delta}(\bar{x})|$, and twice continuously differentiable implicit functions $tau^i_{\mathcal{B}ar{x}}(\cdot):U(\bar{x})to T$
such that $S_{\delta}(x)=\{tau^i_{\mathcal{B}ar{x}}(x)\}_{i=1}^{p(\bar{x})}$ and $\{tau^i_{\mathcal{B}ar{x}}(x)\}_{i=1}^{p(\bar{x})}$ are strict local maximizers
in $\max_{tau\in T}g(x,tau)$ for any $x\in U(\bar{x})$.
With those implicit functions, it holds that $\max_{tau\in T}g(x,tau)= \max_{1\elle i\elle p(\bar{x})}\hat{g}_i(x)$ in $U(\bar{x})$ and thus SISDP\,\eqref{lsisdp} and nonlinear SDP\,\eqref{reduced_LSISDP} are equivalent locally.
The functions $\hat{g}_i(\cdot)\ (i=1,2,\elldots,p(\mathcal{B}ar{x}))$
are convex in $U(\mathcal{B}ar{x})$ when the functions $g(\cdot,tau)\ (tau\in T)$ are convex.
Indeed, for each $i=1,2,\elldots,p(\mathcal{B}ar{x})$,
there exists some neighborhood $T_i\subseteq T$ of $tau^i_{\mathcal{B}ar{x}}(\mathcal{B}ar{x})$
such that $\max_{tau\in T_i}g(x,tau)=\hat{g}_i(x)$ holds for any $x\in U(\mathcal{B}ar{x})$.
By noting that $\max_{tau\in T_i}g(\cdot,tau)\ (i=1,2,\elldots,p(\bar{x}))$ are convex,
we then ensure the convexity of
$\hat{g}_i(\cdot)\ (i=1,2,\elldots,p(\bar{x}))$
in $U(\mathcal{B}ar{x})$.
We can compute the values of {$\nablatau^i_{\mathcal{B}ar{x}}(\bar{x})$ for $i=1,2,\elldots,p(\bar{x})$ by solving a certain linear system derived from the implicit function theorem, from which we further obtain the values of $\nabla\hat{g}_i(\bar{x})$ and $\nabla^2\hat{g}_i(\bar{x})$ for each $i$.}
Thanks to {this} result,
we acquire the concrete forms of the quadratic programs (QPs) that arise in the SQP iterations
for \eqref{reduced_LSISDP}, although it is difficult in general to have explicit forms of the functions $tau^i_{\mathcal{B}ar{x}}(\cdot)\ (i=1,2,\elldots,p(\bar{x}))$.
\subsection{Computing the directions $\Delta_{\frac{1}{2}} w$ and $\Delta_1 w$}\ellabel{sec:QP}
\subsubsection{First direction $\dDelta_{\fr} w$}\ellabel{sec:gene}
Let $\bar{w}=(\mathcal{B}ar{x},\bar{y},\bar{V})\in \mathcal{R}^ntimes \mathcal{M}_+(T)times S^m_{++}$ be
the current point such that $F(\mathcal{B}ar{x})\in S^m_{++}$ and $\mathcal{B}ar{x}$ is nondegenerate in the sense of Definition\,\ref{def:nond}.
We show that a first search direction $\dDelta_{\fr} w=(\dDelta_{\fr} x,\dDelta_{\fr} y,\dDelta_{\fr} V)\in \mathcal{R}^ntimes \mathcal{M}(T)times S^m$
can be computed through the local reduction method
in a manner similar to the interior-point SQP method proposed in the recent work\,\cite{okuno2018sc}.
To start with, we apply the Monteiro-Zhang scaling to $F(x)$ and $V$, in which
we select a nonsingular matrix $P\in \mathcal{R}^{mtimes m}$ and scale
the matrices $F(x)$ and $V$ as
\begin{align}
&F_P(x):=PF(x)P^{top}=F_0+\sum_{i=1}^nx_iF_P^i,\ {V}_P:=P^{-top}V P^{-1},\ellabel{scal}
\end{align}
where
$F_P^i:=PF_iP^{top}$
for $i=0,1,2,\elldots,n$.
Let us consider the reduced NSDP\,\eqref{reduced_LSISDP} with $F(x)\in S^m_{+}$ replaced by $F_P(x)\in S^m_+$, called the scaled NSDP\,\eqref{reduced_LSISDP}.
Since $F(x)\circ V=\mu I,\ F(x)\in S^m_{++},V\in S^m_{++}$ if and only if $F_P(x)\circ V_P=\mu I,\ F_P(x)\in S^m_{++},V_P\in S^m_{++}$ for any $\mu\ge 0$, the KKT (BKKT) conditions of the reduced NSDP\,\eqref{reduced_LSISDP}
are equivalent to those of the scaled NSDP\,\eqref{reduced_LSISDP}.
Therefore, to produce a search direction, it is natural to solve the following mixed linear complementarity system approximating
the BKKT system of the scaled NSDP:
\begin{align}
&\nabla f(\mathcal{B}ar{x})+{\nabla_{xx}^2L}(\mathcal{B}ar{x},\bar{y})\dDelta_{\fr} x+\nabla\hat{g}(\mathcal{B}ar{x})(\bar{y}+\dDelta_{\fr} y)-
\elleft(F_P^i\bullet \elleft(\overline{V}_P+\dDelta_{\fr} V_P\right)\right)_{i=1}^n=0,\ellabel{al:0506-1}\\
&F_P(\mathcal{B}ar{x})\circ(\overline{V}_P+\dDelta_{\fr}{V}_P)+\mathcal{L}_{\overline{V}_P}\sum_{i=1}^n\dDelta_{\fr} x_iF_P^i=\mu I,\ellabel{al:0814-1}\\
&0\elle y+\dDelta_{\fr} y\perp \hat{g}(\mathcal{B}ar{x})+\nabla \hat{g}(\mathcal{B}ar{x})^{top}\dDelta_{\fr} x\elle 0.\ellabel{al:0506-3}
\end{align}
In our method, we make a slight modification to the above system. Specifically, we replace the second equation\,\eqref{al:0814-1} with the following equation:
\begin{equation}
F_P(\mathcal{B}ar{x})\circ(\overline{V}_P+\dDelta_{\fr}{V}_P)+\frac{1}{2}{\elleft(\mathcal{L}_{\overline{V}_P}+\mathcal{L}_{F_P(\mathcal{B}ar{x})}\mathcal{L}_{\overline{V}_P}\mathcal{L}_{F_P(\mathcal{B}ar{x})}^{-1}\right)\sum_{i=1}^n\dDelta_{\fr} x_iF_P^i}=\mu I.\ellabel{al:0506-2}
\end{equation}
The second term of the left hand side
approximates $\mathcal{L}_{\overline{V}_P}\sum_{i=1}^n\Delta x_iF^i_P$ around a BKKT point.
Actually, at any BKKT point,
those two expressions are identical to each other since $\mathcal{L}_{F_P(\mathcal{B}ar{x})}$ and $\mathcal{L}_{\overline{V}_P}$ commute there.
Particularly when choosing a scaling matrix $P$ so that $F_P(\mathcal{B}ar{x})$ and $\overline{V}_P$ commute, \eqref{al:0814-1} and \eqref{al:0506-2} become identical to each other.
{The reason for using the system \eqref{al:0506-1}, \eqref{al:0506-3}, and \eqref{al:0506-2}
is that it can be solved via a KKT system of the following quadratic program (QP):
\begin{align}
\begin{array}{ll}
\displaystyle{\mathop{\rm Minimize}_{\Delta x}} &\nabla f(\mathcal{B}ar{x})^{top}\Delta x+
\frac{1}{2}\Delta x^{top}B_P(\mathcal{B}ar{x},\bar{y},\overline{V})\Delta x-\mu \xi_P(\mathcal{B}ar{x})^{top}\Delta x
\\
{\rm subject~to} &\hat{g}(\mathcal{B}ar{x})+\nabla \hat{g}(\mathcal{B}ar{x})^{top}\Delta x\elle 0,
\end{array}\ellabel{al:qp}
\end{align}
where
$\xi_P(\cdot):=\nabla \ellog\det F_P(\cdot)=(F^i_P\bullet F_P(\cdot)^{-1})_{i=1}^n$, $\hat{g}(\cdot):=(\hat{g}_1(\cdot),\hat{g}_2(\cdot),\elldots,\hat{g}_{p(\mathcal{B}ar{x})}(\cdot))^{top}$, and
\begin{equation}
B_P(x,y,V):=\nabla_{xx}^2{L}(x,y)+{H}_{P}(x,V)\ellabel{eq:0913}
\end{equation}
with ${L}(x,y)$ being the Lagrangian $f(x)+\sum_{i=1}^{p(\mathcal{B}ar{x})}\hat{g}_i(x)y(tau^i_{\mathcal{B}ar{x}}(x))-F(x)\bullet V$
and ${H}_P(x,V)$ being the symmetric matrix whose elements are defined by
\begin{equation}
\elleft({H}_P(x,V)\right)_{i,j}:=\frac{1}{2}{F_P^i\bullet\elleft(\mathcal{L}_{F_P(x)}^{-1}\mathcal{L}_{V_P}
+\mathcal{L}_{V_P}\mathcal{L}_{F_P(x)}^{-1}
\right)F_P^j}\ellabel{eq:HP}
\end{equation}
for $i,j=1,2,\elldots,{n}$.
Note that the linear operator $\mathcal{L}_{F_P(x)}$ is invertible when $F(x)\in S^m_{++}$.
Denote a KKT pair of the QP\,\eqref{al:qp} by $\elleft(\dDelta_{\fr} x,y+\dDelta_{\fr} y\right)$ and define
\begin{align*}
\dDelta_{\fr} V&:=\mu{F}(\mathcal{B}ar{x})^{-1}-\overline{V}-\sum_{i=1}^n\dDelta_{\fr} x_iP^{top}\frac{1}{2}{\elleft(\mathcal{L}_{F_P(\mathcal{B}ar{x})}^{-1}\mathcal{L}_{\bar{V}_P}+\mathcal{L}_{\bar{V}_P}\mathcal{L}_{F_P(\mathcal{B}ar{x})}^{-1}\right)F_P^i}P,\notag\\
\dDelta_{\fr} V_P&:=P^{-top}\dDelta_{\fr} VP^{-1}.\notag
\end{align*}
Then, we can see that the triple $\elleft(\dDelta_{\fr} x,\dDelta_{\fr} y, \dDelta_{\fr} V\right)$ solves the system \eqref{al:0506-1}, \eqref{al:0506-3}, and \eqref{al:0506-2}.}
The QP\,\eqref{al:qp} is necessarily feasible
if the original problem \eqref{lsisdp} is feasible, since the functions
$\hat{g}_i(\cdot)\ (i=1,2,\elldots,p(\mathcal{B}ar{x}))$ are convex as mentioned
above. Furthermore, we have the following property concerning the strong convexity of the objective function of the QP\,\eqref{al:qp}
\begin{proposition}\ellabel{rem:0607}
Suppose that ${F}_P(x)\in S^m_{++}$, ${V}_P\in S^m_{++}$, and $F_1,F_2,\elldots, F_n$ are linearly independent in $S^m$.
Also, suppose that either of the following is true:
\begin{enumerate}
\item[(i)] $\|F_P(x)\circ {V}_P-\mu I\|\elle theta \mu$ with $0\elle theta<1$;
\item[(ii)] $F_P(x)$ and $V_P$ commute.
\end{enumerate}
Then,
${H}_P(x,V)$ is positive definite.
Especially, if $f$ is convex and ${\bar{y}}\in \mathcal{R}^{p(\mathcal{B}ar{x})}_{+}$, the objective function of the QP\,\eqref{al:qp} is strongly convex. Therefore, it has a unique optimum.
\end{proposition}
\begin{proof}
Note that ${F}_P(x)\circ {V}_P\in S^m_{++}$ holds if either of the assumptions (i) and (ii) holds.
Then, the operators $\mathcal{L}_{{F}_P(x)}\mathcal{L}_{V_P}$ and $\mathcal{L}_{{V}_P}\mathcal{L}_{{F}_P(x)}$ are positive definite. Actually, for any $D\in S^m\setminus \{O\}$,
$D\bullet \mathcal{L}_{{F}_P(x)}\mathcal{L}_{{V}_P}D=
D\bullet\mathcal{L}_{{V}_P}\mathcal{L}_{{F}_P(x)}D
={\rm Tr}(D({F}_P(x)\circ {V}_P)D)>0$.
Then, letting $\Delta F:=\sum_{i=1}^n\Delta x_i{F}_P^i$ and noting
the linear independence of $F_1,F_2,\elldots,$ and $F_n$ in $S^m$, we obtain
\begin{align*}
2\Delta x^{top}H_P(x,V)\Delta x&=\Delta F\bullet \elleft(\mathcal{L}_{F_P(x)}^{-1}\mathcal{L}_{{V}_P}+\mathcal{L}_{V_P}
\mathcal{L}_{F_P(x)}^{-1}\right)\Delta F\\
&=\mathcal{L}_{F_P(x)}^{-1}(\Delta F)\bullet \elleft(\mathcal{L}_{V_P}\mathcal{L}_{F_P(x)}+\mathcal{L}_{F_P(x)}\mathcal{L}_{V_P}\right)\mathcal{L}_{F_P(x)}^{-1}(\Delta F)>0
\end{align*}
for any $\Delta x\neq 0$. We omit the proof for the latter claim.
\end{proof}
Below, we list some particular choices for the scaling matrix $P$ and the corresponding directions ${\dDelta_{\fr} V}$:
\begin{enumerate}
\item[(i)] $P=I$: {In this case,} $F_P(\mathcal{B}ar{x})=F(\mathcal{B}ar{x})$ and
\begin{equation*}
\dDelta_{\fr} V=\mu{F}(\mathcal{B}ar{x})^{-1}-\overline{V}-\frac{1}{2}{\elleft(\mathcal{L}_{{F}(\mathcal{B}ar{x})}^{-1}\mathcal{L}_{\overline{V}}
+\mathcal{L}_{\overline{V}}\mathcal{L}_{{F}(\mathcal{B}ar{x})}^{-1}\right)\sum_{i=1}^n\dDelta_{\fr} x_iF_i}.
\end{equation*}
\item[(ii)] $P=F(\mathcal{B}ar{x})^{-\frac{1}{2}}$:
In this case, ${F}_P(\mathcal{B}ar{x})=I$ and
\begin{equation*}
\dDelta_{\fr} V=\mu{F}(\mathcal{B}ar{x})^{-1}-\overline{V}-\frac{1}{2}{F(\mathcal{B}ar{x})^{-1}\elleft(\sum_{i=1}^n\dDelta_{\fr} x_iF_i\right)\overline{V}+\overline{V}\elleft(\sum_{i=1}^n\dDelta_{\fr} x_iF_i\right)F(\mathcal{B}ar{x})^{-1}}.
\end{equation*}
\item[(iii)] $P=W^{-\frac{1}{2}},\ W:=F(\mathcal{B}ar{x})^{\frac{1}{2}}(F(\mathcal{B}ar{x})^{\frac{1}{2}}\overline{V} F(\mathcal{B}ar{x})^{\frac{1}{2}})^{-\frac{1}{2}}F(\mathcal{B}ar{x})^{\frac{1}{2}}$:
In this case, ${F}_P(\mathcal{B}ar{x})=\overline{V}_P$ and
\begin{equation*}
\dDelta_{\fr} {V}=\mu{F}(\mathcal{B}ar{x})^{-1}-\overline{V}-W^{-1}\elleft(\sum_{i=1}^n\dDelta_{\fr} x_i{F}_i\right)W^{-1}.
\end{equation*}
\end{enumerate}
The direction $\dDelta_{\fr} V$ obtained as above can be related to the family of Monteiro-Zhang (MZ) directions\, \cite{wolkowicz2012handbook}.
Actually, as for (i), if $F_P(\bar{x})$ and $\overline{V}_P$ commute, the generated direction
can be cast as the Alizadeh-Hareberly-Overton (AHO) direction.
On the other hand, the generated directions
in (ii) and (iii) are nothing but the
Helmberg-Rendle-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro (HRVW/KSH/M) and Nesterov-Todd (NT) directions, respectively, by themselves.
\subsubsection{Second direction $\Delta_{1} w$}
We next show how to compute the second direction $\Delta_{1} w$ at $\bar{w}+s\dDelta_{\fr} w$.
In a manner similar to $\dDelta_{\fr} w$,
we may compute
the second direction $\Delta_{1} w$ by solving the QP\,\eqref{al:qp} with $\bar{w}$
and $P$ replaced by $\bar{w}+s\dDelta_{\fr} w$ and
another scaling matrix $\hat{P}\in \mathcal{R}^{mtimes m}$, respectively.
However, by exploiting information associated to $\Delta_{\fr} x$, we can replace the QP with certain linear equations as follows:
Let $J_a(\mathcal{B}ar{x}):=\elleft\{i\in \{1,2,\elldots,p(\mathcal{B}ar{x})\}\mid \hat{g}_i(\mathcal{B}ar{x})+\nabla \hat{g}_i(\mathcal{B}ar{x})^{top}\dDelta_{\fr} x=0\right\}$.
If the current point $\mathcal{B}ar{x}$ is sufficiently close to a KKT point, we can expect that the inequality constraints $\hat{g}_i(x)\elle 0\ (i\in J_a(\mathcal{B}ar{x}))$ are also active at the KKT point.
Motivated by this observation,
we propose to solve the following linear equations
for $\Delta_{1} w_{\hat{P}}:=(\Delta_{1} x,\Delta_{1} y,\Delta_{1} V_{\hat{P}})$:
\begin{align}
&\nabla f(\hat{x})+{\nabla_{xx}^2L}(\hat{x},\hat{y})\Delta_{1} x+\nabla\hat{g}(\hat{x})(\hat{y}+\Delta_{1} y)-
\elleft(F_{\hat{P}}^i\bullet \elleft(\hat{V}_{\hat{P}}+\Delta_{1} V_{\hat{P}}\right)\right)_{i=1}^n=0,\ellabel{al:1201-1}\\
&F_{\hat{P}}(\hat{x})\circ(\hat{V}_{\hat{P}}+\Delta_{1}{V}_{\hat{P}})+\frac{1}{2}{\elleft(\mathcal{L}_{\hat{V}_{\hat{P}}}+\mathcal{L}_{F_{\hat{P}}(\hat{x})}\mathcal{L}_{\hat{V}_{\hat{P}}}\mathcal{L}_{F_{\hat{P}}(\hat{x})}^{-1}\right)\sum_{i=1}^n\Delta_{1} x_iF_{\hat{P}}^i}=\mu I,
\ellabel{al:1201-2}\\
&\hat{g}_i(\hat{x})+\nabla \hat{g}_i(\hat{x})^{top}\Delta_{1} x=0\ (i\in J_a(\mathcal{B}ar{x})),\ellabel{al:1201-3}\\
&\hat{y}_i+\Delta_{1} y_i=0\ (i\notin J_a(\mathcal{B}ar{x})),\ellabel{al:1201-4}
\end{align}
where $(\hat{x},\hat{y},\hat{V}_{\hat{P}}):=\bar{w}_{\hat{P}}+s\dDelta_{\fr} w_{\hat{P}}$.
{We then set $\Delta_{1} w:=\elleft(\Delta_{1} x,\Delta_{1} y,\Delta_{1} V\right)$ with $\Delta_{1} V:=\hat{P}^{top}\Delta_{1} V_{\hat{P}}\hat{P}$.}
If the above linear equations are not solvable or not well-defined
because
$\{tau^i_{\mathcal{B}ar{x}}(\cdot)\}_{i\in J_a(\mathcal{B}ar{x})}\subseteq \{tau^i_{\hat{x}}(\cdot)\}_{i=1}^{p(\hat{x})}$ does not hold, i.e.,
the {family of} functions ${\hat{g}}_i(\cdot)\ (i\in J_a(\mathcal{B}ar{x}))$ defined at $\mathcal{B}ar{x}$ is not valid at $\hat{x}$, then
we skip the above procedure and proceed to the next step.
\subsection{Local convergence analysis}\ellabel{sec:local}
\input{super_conv.tex}
\section{Numerical experiments}\ellabel{sec:5}
In this section, we conduct some numerical experiments to demonstrate the efficiency of the primal-dual path following method (Algorithm~2) by solving two kinds of SISDPs with a one-dimensional index set of the form $T=[T_{\rm min},T_{\rm max}]$: The first one is a linear SISDP where all functions are affine with respect to $x$; the second one is an SISDP with a nonlinear objective function.
Throughout the section, we identify a symmetric matrix variable $X\in S^m$ with a vector variable $x:=(x_{11},x_{12},\elldots,x_{1m},x_{12},x_{22},\elldots,x_{mm})^{top}\in\mathcal{R}^{\frac{m(m+1)}{2}}$ through
$$
X=\begin{pmatrix}
x_{11}&x_{12}&\elldots&x_{1m}\\
x_{12}&x_{22}&\elldots&x_{2m}\\
\vdots&\vdots&\ddots&\vdots\\
x_{1m}&x_{2m}&\elldots&x_{mm}
\end{pmatrix}.
$$
The program was coded in MATLAB~R2012a and run on a machine with Intel(R) Xeon(R) CPU E5-1620 [email protected] and 10.24GB RAM.
We compute the scaling matrices for the NT direction according to \cite[Section~4.1]{todd1998nesterov}.
As for SISDPs with a nonlinear objective function,
the matrix $B_P$ in the quadratic program\,\eqref{al:qp} is not necessarily positive-definite.
So as to assure its positive definiteness, we modified $B_P$ by lifting its negative eigenvalues to $1$.
Let $\bar{x}$ be a current point and $\{tau_{\bar{x}}^i(\cdot)\}_{i=1}^{p(\bar{x})}$
be the set of implicit functions defined in \eqref{reduced_LSISDP}.
As for the set $S_{\delta}(\bar{x})$ defined by \eqref{eq:Sdelta},
we set $\delta := 10^{-1}$ and put $N+1$ grids
$\{s_1,s_2,\elldots,s_{N+1}\}$ on $T$
uniformly with $N:=100$.
To specify the set $S_{\delta}(\bar{x})$, we apply Newton's method combined with the projection onto $T$
for the problem $\max_{tau\in T}g(\bar{x},tau)$
starting from each of the local maximizers $\bar{s}$ of $\max\{g(\bar{x},s)\mid s=s_1,s_2,\elldots,s_{N+1}\}$ such that
$g(\bar{x},\bar{s}) > \max_{1\elle i\elle N+1}g(\bar{x},s_i) - \delta$.
Let $\bar{y}\in \mathcal{R}^{p(\bar{x})}_+$ be a current estimate of Lagrange multiplier vector associated with the inequality constraints $g(x,tau_{\bar{x}}^i(x))\elle 0\ (i=1,2,\elldots,p(\bar{x}))$.
As $\bar{x}$ moves to $\bar{x}+\Delta \bar{x}$, we trace the value of the implicit function $tau_{\bar{x}}^i$ for each $i=1,2,\elldots,p(\bar{x})$, namely, we identify $tau_{\bar{x}}^i(\bar{x}+\Delta \bar{x})$ with an element in $S_{\delta}(\bar{x}+\Delta \bar{x})$
to examine the correspondence
between $\bar{y}_i\ (i=1,2,\elldots,p(\bar{x}))$ and the inequality constraints ${g}(x,tau_{\bar{x}+\Delta \bar{x}}^j(x))\elle 0\ (1\elle j\elle p(\bar{x}+\Delta \bar{x}))$.
For this purpose, for each element $tau^j_{\bar{x}_+}(\bar{x}_+)\in S_{\delta}(\bar{x}_+)$ with $\bar{x}_+:=\bar{x}+\Delta \bar{x}$, we search $S_{\delta}(\bar{x})=\{tau_{\bar{x}}^1(\bar{x}),tau_{\bar{x}}^2(\bar{x}),\elldots,tau_{\bar{x}}^{p(\bar{x})}(\bar{x})\}$
for an index $tilde{i}\in \{1,2,\elldots,p(\bar{x})\}$ such that
$\|tau^j_{\bar{x}_+}(\bar{x}_+)-tau_{\bar{x}}^{tilde{i}}(\bar{x})-\nabla tau_{\bar{x}}^{tilde{i}}(\bar{x})^{top}\Delta \bar{x}\|
(\approx\|tau^j_{\bar{x}_+}(\bar{x}_+)-tau_{\bar{x}}^{tilde{i}}(\bar{x}_+)
\|)
\elle 10^{-1}$.
If it is found, we regard ${tau}_{\bar{x}_+}^j(\bar{x}_+)$ as $tau_{\bar{x}}^{tilde{i}}(\bar{x}_+)$.
Otherwise, we treat ${tau}_{\bar{x}_+}^j(\cdot)$ as the implicit function that newly appears at $\bar{x}_+$, and set zero to be the Lagrange multiplier for the inequality constraint $g(x,tau^j_{\bar{x}_+}(x))\elle 0$.
Next, we explain how each step of the algorithm is implemented.
In Step~0, we set
$$\gamma_1 =\sqrt{\frac{m(m+1)}{2}},\ \gamma_2=5,\ c = \frac{1}{2.99},\ \alpha = 0.99,\ \beta = 0.8.$$
As for starting points, we set $y^0 = (1,1)^{top}, V_0=m I$, and $\mu_0=1$, while $x^0$ is chosen so that $X^0 = m^{-1}I$ for linear SISDPs, and $x^0 =0$ is chosen for SISDPs with a nonlinear objective function.
In Step~1, we terminate the algorithm if
$\mu_{k+1}<10^{-10}$ or the value of
the function $R_0$ is less than $10^{-8}$, where $R_0$ is
the function $R_{\mu}$ with $\mu =0$ defined in Section\,\ref{sec:rmu}.
In Step~2.4, we implement the interior-point SQP-type method proposed in \cite{okuno2018sc} by using
the implementation details described therein.
In Step~3, for the sake of numerical stability, we set $\sharp\,\mathrm{var}epsilon_{k+1}:=\max(10^{-7},\gamma_1\mu_{k+1}^{1+\alpha})$.
For $X\in S^m_{++}$ and $Y\in S^m$, we compute $\mathcal{L}_X^{-1}Y$ by solving the linear equation $\mathcal{L}_XZ=Y$
{for} $Z\in S^m$ with the Matlab built-in solver texttt{lyap2}. We moreover use texttt{quadprog} {to solve} quadratic programs in Step~2-1.
For the sake of comparison, we also implement a discretization method that solves finitely relaxed SISDPs sequentially until an approximate feasible solution is obtained. More precisely,
for solving the SISDP\,\eqref{lsisdp}, we use the following discretization algorithm:
\begin{description}
\item[Step~0:] Choose an initial index set $T_0\subseteq T$ with
$|T_0|<+\infty$. Choose $theta>0$. Set $r:=0$.
\item[Step~1:] Get a KKT point $x^r$ of the finitely relaxed SISDP with $T$ replaced by $T_r$.
\item[Step~2:] Find $\bar{tau} \in T$
such that $g(x^r,\bar{tau}) > theta$ and set $T_{r + 1}:=T_r\cup \{\bar{tau}\}$.
If such a point does not exist in $T$, terminate the algorithm.
\item[Step~3:] Increment $r$ by one and return to Step~1.
\end{description}
In Step~0, we choose $T_0=\{T_{\rm min},T_{\rm max}\}$. In Step~2,
{to find such a $\bar{tau}\in T$ we solve $\max_{tau\in T}g(x^r,tau)$ by applying Newton's method with a starting point $s\in {\rm argmax}\{
g(x^k,s)\mid s = s_1,s_2,\elldots,s_{N+1}\}$, where $\{s_1,s_2,\elldots,s_{N+1}\}$ is the set of grids defined earlier in this section.
\footnote{
There is no theoretical guarantee for global optimality of $tau$ thus found.
In practice, however, we may expect to have a global optimum by setting $N$ large enough.}
We set $theta:=10^{-6}$.
\subsection{Linear SISDPs}
In this section, we consider the linear SISDP\,\eqref{lsisdp}, called {LSISDP for short}.
{Specifically, we} solve the following problem taken from \cite[Section~4.2]{li2004solution}:
\begin{align}
\begin{array}{rcl}
\displaystyle{\mathop{\rm Maximize}_{X\in S^m}}& &A_0\bullet X\\
\mbox{subject to}& & A(tau)\bullet X\ge 0\ (tau\in T)\\
& & I\bullet X = 1\\
& & X\in S^m_+,
\end{array}\ellabel{eig_semi}
\end{align}
where $A_0\in S^m$ and $A:Tto S^m$ is a symmetric matrix valued function
whose elements are $q$-th order polynomials in $tau$, i.e., $(A(tau))_{i,j}=\sum_{l=0}^qa_{i,j,l}tau^l$ for $1\elle i,j\elle m$.
In this experiment,
we deal with the cases where $q = 9$, $m = 10, 20$, and $T=[0,1]$, i.e., $T_{\rm min}=0$ and $T_{\rm max}=1$.
We generate 10 test problems for each of $m=10,20$ as follows:
We choose all entries of $A_0$ and the coefficients $a_{i,j,l}$ in $A(tau)$ from the interval $[-1,1]$ randomly.
Among those generated data sets, we use only data such that the semi-infinite constraint
includes at least one active constraint
at an optimum of \eqref{eig_semi}.
{Specifically, for each generated data, we} compute an optimum, say $tilde{X}$, of the SDP obtained by removing the semi-infinite constraints.
If $\min_{1\elle i\elle 21}A\elleft(T_{\rm min} + \frac{(i-1)(T_{\rm max}-T_{\rm min})}{20}\right)\bullet tilde{X}\elle -10^{-3}$, which implies that $tilde{X}$ does not satisfy the semi-infinite constraints, we adopt it as a valid data set.
We examine the performance of Algorithm~2
by comparing it
with the discretization method that uses SDPT3\,\cite{sdpt3}
with the default setting to solve linear SDPs sequentially.
The obtained results are shown in Tables~\ref{ta1} and \ref{ta2}, in which
``ave.time(s)'' and ``$\Phi_0^{\ast}$'' stand for
the average running time in seconds and the average value of $\Phi_0$ at the solution output by the algorithm
``Disc." stands for the discretization method.
Moreover, ``AHO-like'', ``NT'', and ``H.K.M'' {stand for} Algorithm~2 combined with the scaling matrices $P=I,F(x^k)^{-\frac{1}{2}}$, and
$W^{-\frac{1}{2}}$, respectively.
From the tables, we observe that
computational time for ``AHO-like'' is largest among all.
Actually, it spends around 3 seconds for $m = 10$ and 40 seconds for $m = 20$, while the others spend less than 1 second in all cases. This is mainly due to
high computational costs for calculating the matrix $H_P$ defined by \eqref{eq:HP}, in which $\mathcal{L}_{F(x)}^{-1}$ must be dealt with. However, in the cases of ``NT'' and ``H.K.M'', $H_P$ can be handed more efficiently.
Second, we observe that ``Disc.'' solves problems faster than Algorithm~2.
This is because an
SDP is solved very quickly with SDPT3 at each iteration of ``Disc.", and the number of SDPs solved is very small. In fact, only three or four SDPs are solved on average per run. However, we can see that our methods gain KKT points with higher accuracy than the discretization method.
More specifically, the values of $\Phi_0^{\ast}$ for
Algorithm~2 lie between $1.0times 10^{-9}$ and $2.0times 10^{-9}$, while
those for the discretization method are around $10^{-6}$.
We also observed that
Algorithm~2 skips Step~2.4 in most iterations, namely, $w^{k+1}$ is determined by the directions $\Delta_{\fr}w^{k+1}$ and $\Delta_{1}w^{k+1}$. Actually, Step~2.4 was skipped in more than 90\% of iterations.
Skipping Step~2.4 is desirable since the interior point SQP method performed in Step~2.4 is likely to solve multiple QPs and
result in more computational cost than Steps~2.1 and 2.2.
Also, in most cases, the full step was accepted eventually and the value of $\Phi_{\mu_{k-1}}$ converged to 0 superlinearly.
\begin{table}[h]
\centering
\small
\begin{minipage}{0.43\hsize}
\begin{tabular}{|c|c|c|}\hline
& ave.time(s) & $\Phi_0^{\ast}$ \\ \hline\hline
AHO-like & 2.63 & $1.39\cdot 10^{-9}$ \\ \hline
NT & 0.44& $1.39\cdot 10^{-9}$ \\ \hline
H.K.M. & 0.45 & $1.39\cdot 10^{-9}$ \\ \hline\hline
Disc. & 0.54 & $2.06\cdot 10^{-6}$ \\ \hline
\end{tabular}
\caption{Results for linear SISDPs with $m=10$}
\ellabel{ta1}
\end{minipage}
\begin{minipage}{0.43\hsize}
\begin{tabular}{|c|c|c|}\hline
& ave.time(s)& $\Phi_0^{\ast}$ \\ \hline\hline
AHO-like & 46.3 & $1.97\cdot10^{-9}$ \\ \hline
NT & 0.90 & $1.97\cdot10^{-9}$ \\ \hline
H.K.M. & 0.90 & $1.97\cdot10^{-9}$ \\ \hline\hline
Disc. & 0.40 & $1.34\cdot 10^{-6}$ \\ \hline
\end{tabular}
\caption{Results for linear SISDPs with $m=20$}
\ellabel{ta2}
\end{minipage}
\end{table}
\subsection{Nonlinear SISDPs}
Next, we solve the following SISDP whose objective function is nonlinear:
\begin{align}
\begin{array}{rcl}
\displaystyle{\mathop{\rm Minimize}_{x\in \mathcal{R}^{\frac{m(m+1)}{2}}}}& &\frac{1}{2}x^{top}Mx+c^{top}x+\omega{\|x\|^4}\\
\mbox{subject to}& &
\sum_{i=1}^{n}tau^{i-1}x_i\elle \sum_{i=1}^ntau^{2i} + \sin(9\pi tau)+2 \ \ (tau\in T)\\
& & X + \kappa I\in S^m_+
\end{array}\ellabel{eig_semi2}
\end{align}
with $\omega>0$, $\kappa >0$, and $n:=m(m+1)/2$.
The objective function is
not convex in general but coercive in the sense that
$f(x)to \infty$ as $\|x\|to \infty$, and thus the considered problem is guaranteed to have at least one global optimum.
We deal with the cases of $m = 10,20$.
{For each of $m=10,20$,} all the elements of $M\in S^m$ and $c\in \mathcal{R}^n$ are randomly generated from the interval $[-1,1]$.
We set $T=[0,1]$ and $\kappa = \omega=0.01$. In Step~2 of the discretization method, we use
the primal-dual interior point method
\cite{yabe} to solve finitely relaxed SISDPs.
We show the results in Tables~\ref{ta3} and \ref{ta4},
where each column and row has the same meaning as in Tables~\ref{ta1} and \ref{ta2}.
From the tables, ``AHO-like'' spends the largest CPU-time like in linear SISDPs.
We observe that Algorithm~2 (AHO-like, NT, H.K.M.) successfully obtains KKT points with higher accuracy than the discretization method. Actually, the values of $\Phi_0^{\ast}$
obtained by Algorithm~2 lie between $10^{-9}$ and $2times 10^{-9}$, while those for the discretization method are around $10^{-6}$.
Compared with the case of linear SISDPs, we observed that the rate of skipping Step~2-4 was less.
Actually, Step~2-4 was used at about 15\% of iterations when $m=10$
and about 24\% when $m=20$,
while it was used only in a few early iterations for linear SISDPs.
This might be caused by the nonlinearity of the objective function.
\begin{table}[h]
\centering
\small
\begin{minipage}{0.43\hsize}
\begin{tabular}{|c|c|c|}\hline
& ave.time(s) & $\Phi_0^{\ast}$ \\ \hline\hline
AHO-like & 3.16 & $1.39\cdot 10^{-9}$ \\ \hline
NT & 0.86 & $1.39\cdot 10^{-9}$ \\ \hline
H.K.M. & 0.85 & $1.39\cdot 10^{-9}$ \\ \hline\hline
Disc. & 1.27 & $9.62\cdot 10^{-7}$ \\ \hline
\end{tabular}
\caption{Results for the nonlinear SISDP with $m=10$}
\ellabel{ta3}
\end{minipage}
\begin{minipage}{0.43\hsize}
\begin{tabular}{|c|c|c|}\hline
& ave.time(s)& $\Phi_0^{\ast}$ \\ \hline\hline
AHO-like & 50.3 & $1.97\cdot10^{-9}$ \\ \hline
NT & 4.06 & $2.32\cdot10^{-9}$ \\ \hline
H.K.M. & 4.00 & $2.32\cdot10^{-9}$ \\ \hline\hline
Disc. & 8.08 & $8.06\cdot 10^{-7}$ \\ \hline
\end{tabular}
\caption{Results for the nonlinear SISDP with $m=20$}
\ellabel{ta4}
\end{minipage}
\end{table}
\section{Conclusion}
In this paper, we proposed
two algorithms for solving the SISDP\,\eqref{lsisdp}: The first one (Algorithm~1) is a primal-dual path following method
designed to find a KKT point of the SISDP by following a path {formed by} BKKT points.
We showed that a sequence generated by the algorithm $\mbox{weakly}^{\ast}$ converges to a KKT point under some mild assumptions. To accelerate local convergence speed,
the second algorithm (Algorithm~2) integrates
a two-step SQP method into Algorithm~1.
Algorithm~2 solves a sequence of quadratic programs and Newton equations obtained by the local reduction method and Monteiro-Zhang scaling technique, while decreasing the value of the barrier parameter.
We established two-step superlinear convergence of Algorithm~2
for the particular case where the AHO-like directions is used.
As for the cases of the NT and H.K.M directions, we can show a two-step superlinear convergence in a manner analogous to
\cite[Theorems~3,4]{yamashita2012local}.
Finally, we conducted some numerical experiments to investigate the efficiency of Algorithm~2 by comparing it with the discretization method which solves (nonlinear) SDPs obtained by finite relaxation of the SISDP\,\eqref{lsisdp}.
In the experiments, we confirmed that the sequences generated by Algorithm~2 actually converged to a KKT point two-step superlinearly.
We also observed that it exhibited the numerical efficiency comparable to the discretization method.
In particular, it worked better in finding highly accurate solutions than the discretization method.
\section*{Appendix}
\defthesection{A}
In the appendix, we prove Proposions\,\ref{prop:0511}, \ref{lem:1203-2}, and Lemma\,\ref{prop:0604-1}.
We begin with
giving some lemmas that help to show Proposion\,\ref{prop:0511}.
\begin{lemma}\ellabel{lem:0424}
Let $X\in S^m_+$, $Y\in S^m$ and $\mu\ge 0$.
Then,
\begin{enumerate}
\item
$
\|XY-YX\|_F\elle 2\|X\circ Y-\mu I\|_F
$ and
\item
$
\|\mathcal{L}_X\mathcal{L}_Y-\mathcal{L}_Y\mathcal{L}_X\|_2\elle \|X\circ Y-\mu I\|_F.
$
\end{enumerate}
\end{lemma}
\begin{proof}
Using some orthogonal matrix $\mathcal{O}\in \mathcal{R}^{mtimes m}$, we make an eigenvalue decomposition of $X$: $\mathcal{O}^{top}X\mathcal{O}=D$ with $D\in \mathcal{R}^{mtimes m}$ being a diagonal matrix. Denote the $i$-th diagonal entry of $D$
by $d_i\ge 0$ for $i=1,2,\elldots,m$.
Let $tilde{Y}:=\mathcal{O}^{top}Y\mathcal{O}$ with the $(i,j)$-th entry $tilde{y}_{ij}$ for $1\elle i,j\elle m$.
\begin{enumerate}
\item We have the desired result from
\begin{align}
\|XY-YX\|_F^2&=\|\mathcal{O}^{top}X\mathcal{O}\mathcal{O}^{top}Y\mathcal{O}-\mathcal{O}^{top}Y\mathcal{O}\mathcal{O}^{top}X\mathcal{O}\|_F^2\notag\\
&=\|Dtilde{Y}-tilde{Y}D\|_F^2\notag\\
&=\sum_{1\elle i,j\elle m}(d_{i}-d_{j})^2tilde{y}_{ij}^2\notag\\
&\elle \sum_{1\elle i\neq j\elle m}(d_{i}+d_{j})^2tilde{y}_{ij}^2\notag\\
&\elle\sum_{1\elle i\neq j\elle m}(d_{i}+d_{j})^2tilde{y}_{ij}^2+\sum_{i=1}^m(2d_{i}tilde{y}_{ii}-2\mu)^2\notag\\
&=\|Dtilde{Y}+tilde{Y}D-2\mu I\|_F^2\notag\\
&=\|XY+YX-2\mu I\|_F^2\notag \\
&=4\|X\circ Y-\mu I\|_F^2,\notag
\end{align}
where the first inequality follows from $d_{i}\ge 0$ for $i=1,2,\elldots,m$.
\item
{By direct calculation, we have}
\begin{align}
\|\mathcal{L}_X\mathcal{L}_Y-\mathcal{L}_Y\mathcal{L}_X\|_2
&=\max_{\|Z\|_F=1}\|\mathcal{L}_X\mathcal{L}_YZ-\mathcal{L}_Y\mathcal{L}_XZ\|_F\notag \\
&=\max_{\|Z\|_F=1}\frac{\|(XY-YX)Z-Z(XY-YX)\|_F}{4}\notag \\
&\elle \frac{\|XY-YX\|_F}{2}\notag\\
&\elle \|X\circ Y-\mu I\|_F,\notag
\end{align}
where the {second} inequality follows from item~1.
\end{enumerate}
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
\begin{lemma}\ellabel{lem:0424-2}
{Let $(X_{\ast},Y_{\ast})\in S^m_+times S^m_+$ satisfy the strict complementarity condition that $X_{\ast}{\circ}Y_{\ast}=O$ and $X_{\ast}+Y_{\ast}\in S^m_{++}$.
Let $\{\mu_r\}\subseteq \mathcal{R}_{++}$ and $\{(X_r,Y_r)\}\subseteq S^m_{++}times S^m_{++}$ be sequences such that $\ellim_{rto\infty}\mu_r=0$ and $\ellim_{rto\infty}(X_r,Y_r)=(X_{\ast},Y_{\ast})$.}
Let spectral decompositions of $X_{\ast}$ and $Y_{\ast}$ be
$$
\mathcal{O}_{\ast}^{top}X_{\ast}\mathcal{O}_{\ast}=\begin{pmatrix}
D_{X_{\ast}}&O\\
O&O
\end{pmatrix},\
\mathcal{O}_{\ast}^{top}Y_{\ast}\mathcal{O}_{\ast}
=
\begin{pmatrix}
O&O\\
O&D_{Y_{\ast}}
\end{pmatrix}
$$
using some orthogonal matrix $\mathcal{O}_{\ast}\in \mathcal{R}^{mtimes m}$
and positive diagonal matrices $D_{X_{\ast}}\in S^p_{++}$ and $D_{Y_{\ast}}\in S^q_{++}$ with $p+q=m$.
Furthermore, suppose $p,q>0$ and
choose a sequence of orthogonal matrices $\{\mathcal{O}_{r}\}\subseteq \mathcal{R}^{mtimes m}$ such that
\begin{equation*}
\mathcal{O}_r^{top}X_r\mathcal{O}_r=\begin{pmatrix}
D_{X_{r}}&O\\
O&E_{X_{r}}
\end{pmatrix},\ \ellim_{rto\infty}\mathcal{O}_r=\mathcal{O}_{\ast}
\end{equation*}
with $D_{X_r}\in \mathcal{R}^{ptimes p}$ and $E_{X_r}\in \mathcal{R}^{qtimes q}$ being positive diagonal matrices for $r\ge 1$.
(Notice that $\ellim_{rto \infty}E_{X_r}$=O.)
If $\|X_r\circ Y_r-\mu_r I\|=o(\mu_r)$, then
\begin{equation}
\ellim_{rto\infty}\frac{1}{\mu_r}E_{X_r}=D_{Y_{\ast}}^{-1}.\ellabel{eq:0424-1}
\end{equation}
\end{lemma}
\begin{proof}
Let $tilde{Y}_r:=\mathcal{O}_r^{top}Y_r\mathcal{O}_r$ and
$tilde{y}^r_{ii}$ and
$e^r_i$ be the $i$-th diagonal entry of $tilde{Y}_r$ and $E_{X_r}$, respectively for any $i=p+1,p+2,\elldots,m$.
Since $\|X_r\circ Y_r-\mu_rI\|_F=o(\mu_r)$ and
\begin{align*}
\|X_r\circ Y_r-\mu_{r}I\|_F&=\elleft\|\begin{pmatrix}
D_{X_{r}}&O\\
O&E_{X_{r}}
\end{pmatrix}\circtilde{Y}_r-\mu_r I\right\|_F\notag \\
&\ge \sqrt{
\sum_{i=p+1}^{m}(e^r_{i}tilde{y}_{ii}^r-\mu_r)^2},
\end{align*}
we have
\begin{equation}
0=\ellim_{rto \infty}\frac{\sqrt{\sum_{i=p+1}^{m}\elleft(e^r_itilde{y}^r_{ii}-\mu_r\right)^2}}{\mu_r}
=\ellim_{rto \infty}\sqrt{\sum_{i=p+1}^{m}\elleft(\frac{e^r_i}{\mu_r}tilde{y}^r_{ii}-1\right)^2},
\notag
\end{equation}
which yields $\ellim_{rto\infty}\frac{e^r_i}{\mu_r}tilde{y}^r_{ii}=1$ for any $i=p+1,\elldots,m$.
Notice that,
for $i\ge p+1$,
$\{tilde{y}^r_{ii}\}$ converges to
the $i$-th positive diagonal entry of $D_{Y_{\ast}}$.
In view of these facts, we obtain \eqref{eq:0424-1}.
\hspace{\fill}$\mathcal{B}ox$
\end{proof}
\subsubsection*{Proof of Proposition\,\ref{prop:0511}}
For the case where $X_{\ast}\in S^m_{++}$, it is easy to prove the desired result.
So, we consider the case of $X_{\ast}\in S^m_+\setminus S^m_{++}$.
Let $\ellambda_r>0$ be the smallest eigenvalue of $X_r$.
Notice that $\ellambda_rto 0\ (rto \infty)$ and, by Lemma~\ref{lem:0424-2}, $\ellim_{rto\infty}\frac{\ellambda_r}{\mu_r}$ exists and is positive.
Thus, we also have
\begin{equation}
\ellim_{rto\infty}\frac{\mu_r}{\ellambda_r}>0.\ellabel{eq:0604}
\end{equation}
Note that, for any $X\in S^m$ having $m$ eigenvalues $\alpha_1\elle \alpha_2\elle\cdots \elle \alpha_m$,
the corresponding symmetric linear operator $\mathcal{L}_X$ has $m(m+1)/2$ eigenvalues
$
\alpha_1,\alpha_2,\elldots,\alpha_m,\{(\alpha_i+\alpha_j)/2\}_{i\neq j}.
$
This fact yields that the maximum eigenvalue of the operator $\mathcal{L}_{X_r}^{-1}$ is $\ellambda_r^{-1}$.
Therefore, we have $\|\mathcal{L}_{X_r}^{-1}\|_2=\ellambda_r^{-1}$ for any $r\ge 0$.
It then follows that
\begin{align}
\|
\mathcal{L}_{X_r}\mathcal{L}_{Y_r}\mathcal{L}_{X_r}^{-1}-\mathcal{L}_{Y_r}
\|_2
&\elle
\|\mathcal{L}_{Y_r}\mathcal{L}_{X_r}-\mathcal{L}_{X_r}\mathcal{L}_{Y_r}
\|_2\|\mathcal{L}_{X_r}^{-1}\|_2\notag \\
&\elle
\mu_r\|\mathcal{L}_{X_r}^{-1}\|_2
\frac{\|
X_r\circ Y_r-\mu_rI
\|_F}{\mu_r} \notag\\
&=\frac{\mu_r}{\ellambda_r}\frac{\|
X_r\circ Y_r-\mu_rI
\|_F}{\mu_r},
\end{align}
where the second inequality follows from Lemma\,\ref{lem:0424}.
This relation together with \eqref{eq:0604} and $\|X_r\circ Y_r-\mu_rI\|_F=O(\mu_r^{1+{\zeta}})$
implies
$
\|
\mathcal{L}_{X_r}\mathcal{L}_{Y_r}\mathcal{L}_{X_r}^{-1}-\mathcal{L}_{Y_r}
\|_2=O(\mu_r^{\zeta}).$
\\
\hspace{\fill}$\mathcal{B}ox$
\subsubsection*{Proof of Proposition\,\ref{lem:1203-2}}
Define $\Phi_r(s):=\elleft(X_r+s\Delta X_r\right)\circ \elleft(Y_r+s\Delta Y_r\right)$ for $s\in [0,1]$ and each $r$.
By using the fact that $\|X\|_F\ge |\ellambda_{\min}(X)|$ for any $X\in S^m$, the conditions\,\eqref{al:1202-1}--\eqref{al:1202-3} yield that there exists some $theta>0$ such that
\begin{align}
&\ellambda_{\min}\elleft(\Delta X_r\circ \Delta Y_r\right)\ge -theta\mu_r^2,\ellabel{al:1202-4}\\
&\ellambda_{\min}\elleft(X_r\circ Y_r\right)\ge \mu_r-theta \mu_r^{1+\zeta},\ellabel{al:1202-5} \\
&\ellambda_{\min}\elleft(Z_r-\hat{\mu}_rI\right)\ge -theta \hat{\mu}_r^{1+\hat{\zeta}}.\ellabel{al:1202-6}
\end{align}
Then, it holds that
\begin{align}
\ellambda_{\min}(\Phi_r(s))&=\ellambda_{\min}\elleft(
X_r\circ Y_r+s X_r\circ \Delta Y_r+s Y_r\circ \Delta X_r+s^2\Delta X_r\circ \Delta Y_r
\right)\notag\\
&=\ellambda_{\min}\elleft(
(1-s)X_r\circ Y_r+s(Z_r-\hat{\mu}_r I)+s\hat{\mu}_r I+s^2\Delta X_r\circ \Delta Y_r\right)\notag\\
&\ge (1-s)\ellambda_{\min}\elleft(X_r\circ Y_r\right)+s\ellambda_{\min}(Z_r-\hat{\mu}_r I)\notag\\
&\hspace{5em}+s\ellambda_{\min}(\hat{\mu}_r I)+s^2\ellambda_{\min}\elleft(\Delta X_r\circ \Delta Y_r\right)\notag\\
&\ge (1-s)\elleft(\mu_r-theta \mu_r^{1+\zeta}\right)
-stheta\hat{\mu}_r^{1+\hat{\zeta}}+s\hat{\mu}_r-s^2 theta\mu_r^2\notag\\
&=:\varphi_r(s) \notag
\end{align}
for any $r$ sufficiently large and $s\in [0,1]$, where the first inequality follows from the fact that $\ellambda_{\min}(A+B)\ge \ellambda_{\min}(A)+\ellambda_{\min}(B)$ for $A, B\in S^m$ and the second inequality is due to \eqref{al:1202-4}--\eqref{al:1202-6} and $s\in [0,1]$.
Notice that $\varphi_r(s)$ is concave and quadratic.
Then, for any $r$ sufficiently large, we have
$\varphi_r(s)>0\ (s\in [0,1])$ since $0<\zeta,\hat{\zeta}<1$, $\ellim_{rto \infty}(\mu_r,\hat{\mu}_r)=(0,0)$, and \eqref{al:1202-7} imply that
$\varphi_r(0)=\mu_r-theta \mu_r^{1+\zeta}>0$ and
$\varphi_r(1)=\hat{\mu}_r-theta\hat{\mu}_r^{1+\hat{\zeta}}-theta\mu_r^2>0$ for sufficiently large $r$.
This means that $\ellambda_{\min}(\Phi_r(s))\ge \varphi_r(s)>0\ (s\in [0,1])$ and therefore
\begin{equation}
\Phi_r(s)\in S^m_{++}\ (s\in [0,1]), \ellabel{eq:Phi}
\end{equation}
from which we can derive $X_r+\Delta X_r\in S^m_{++}$ and $Y_r+\Delta Y_r\in S^m_{++}$.
Actually, for contradiction, suppose that either one of these two conditions is not true.
We can assume $X_r+\Delta X_r\notin S^m_{++}$ without loss of generality.
Recall that $X_r\in S^m_{++}$.
Then, there exists some $\bar{s}\in (0,1]$ such that $X_r+\bar{s}\Delta X_r\in S^m_{+}\setminus S^m_{++}$.
Therefore, we can find some nonzero vector $d\in \mathcal{R}^n$ such that $(X_r+\bar{s}\Delta X_r)d=0$.
From this fact, we readily have
\begin{align}
d^{top}\Phi_r(\bar{s})d&=\frac{
d^{top}(X_r+\bar{s}\Delta X_r)(Y_r+\bar{s}\Delta Y_r)d+
d^{top}(Y_r+\bar{s}\Delta Y_r)(X_r+\bar{s}\Delta X_r)d
}{2}=0,\notag
\end{align}
which contradicts \eqref{eq:Phi}. Hence, we conclude that $X_r+\Delta X_r\in S^m_{++}$ and $Y_r+\Delta Y_r\in S^m_{++}$ for all $r$ sufficiently large. The proof is complete. \hspace{\fill}$\mathcal{B}ox$
\subsubsection*{Proof of Lemma\,\ref{prop:0604-1}}
To begin with, by $w^k\in \mathcal{N}_{\mu_{k-1}}^{\varepsilon_{k-1}}$ and $\varepsilon_{k-1}=\gamma_1\mu_{k-1}^{1+\alpha}$, it follows that
\begin{align}
&\elleft\|\nabla f(x^k)+\sum_{i=1}^{p(x^{\ast})}\nabla \hat{g}_i(x^k)y_i^k-(F_i\bullet V_k)_{i=1}^n\right\|=o(\mu_{k-1}),\ \|F(x^k)\circ V_k\|_F=\Theta(\mu_{k-1}),\ellabel{al:1012-1}\\
&\elleft|\sum_{i=1}^{p(x^{\ast})}y_i^k\hat{g}_i(x^k)\right|=o(\mu_{k-1}),\ \max_{1\elle i\elle p(x^{\ast})}(\hat{g}_i(x^k))_+=o(\mu_{k-1})\ellabel{al:1012}
\end{align}
together with $y_i^k\ge 0\ (i=1,2,\elldots,p(x^{\ast}))$.
Then, \eqref{al:1012} implies $|\hat{g}_i(x^k)|=o(\mu_{k-1})\ (i=1,2,\elldots,p(x^{\ast}))$, which together
with \eqref{al:1012-1} and \eqref{al:1012} yields $\|\Phi_0(tilde{w}^k)\|=\Theta(\mu_{k-1})$.
We then have $\mu_{k-1}=\Theta(\|\Phi_{0}(tilde{w}^k)\|)$.
We next prove $\mu_{k-1}=\Theta(\|w^k-w^{\ast}\|)$.
Notice that by Assumption~B-\ref{sc},
for sufficiently large $k$, $y^k_i>0\ (i\in I_a(x^{\ast}))$ and $y^k_i=0\ (i\in \{1,2,\elldots,p(x^{\ast})\}\setminus I_a(x^{\ast}))$, which together with $y_i^{\ast}=0\ (i\in \{1,2,\elldots,p(x^{\ast})\}\setminus I_a(x^{\ast}))$ implies $\|tilde{w}^k-tilde{w}^{\ast}\|=\|{w}^k-{w}^{\ast}\|$.
Thus, to show the desired result, we have only to prove $\|\Phi_{0}({w}^k)\|=\Theta(\|tilde{w}^k-tilde{w}^{\ast}\|)$.
In other words, it suffices to show that
the sequence of positive numbers $\{\zeta_k\}$ is bounded above and away from zero, where $\zeta_k:={\|\Phi_{0}(tilde{w}^k)\|}/{\|tilde{w}^k-tilde{w}^{\ast}\|}$. Note that
\begin{equation*}
\zeta_k=\frac{\|\Phi_{0}(tilde{w}^k)-\Phi_{0}(tilde{w}^{\ast})\|}{{\|tilde{w}^k-tilde{w}^{\ast}\|}}=
\elleft\|\mathcal{J}\Phi_{0}(tilde{w}^{\ast})\frac{tilde{w}^k-tilde{w}^{\ast}}{\|tilde{w}^k-tilde{w}^{\ast}\|}+\frac{O(\|tilde{w}^k-tilde{w}^{\ast}\|^2)}{\|tilde{w}^k-tilde{w}^{\ast}\|}\right\|.
\end{equation*}
Obviously, $\zeta_k$ is bounded from above.
To show $\zeta_k$ is bounded away from zero, suppose to the contrary.
Then, without loss of generality, we can assume that
$\ellim_{kto \infty }\zeta_k=0$,
and hence there exists some $d^{\ast}$ with $\|d^{\ast}\|=1$ such that $\ellim_{kto\infty}\frac{tilde{w}^k-tilde{w}^{\ast}}{\|tilde{w}^k-tilde{w}^{\ast}\|}=d^{\ast}$ and $\mathcal{J}\Phi_0(tilde{w}^{\ast})d^{\ast}=0$.
However, this contradicts
the nonsingularity of $\mathcal{J}\Phi_{0}(tilde{w}^{\ast})$ from Assumption~C-\ref{nonsing}.
We have the desired conclusion.
\hspace{\fill}$\mathcal{B}ox$
\end{document} |
\begin{document}
\title{Interpretable Predictions of Tree-based Ensembles via\\ Actionable Feature Tweaking}
\author{Gabriele Tolomei}
\affiliation{
\institution{Yahoo Research}
\city{London, UK}
}
\email{[email protected]}
\author{Fabrizio Silvestri}
\authornote{The author has contributed to this work while he was employed at Yahoo Research.}
\affiliation{
\institution{Facebook}
\city{London, UK}
}
\email{[email protected]}
\author{Andrew Haines}
\affiliation{
\institution{Yahoo Research}
\city{London, UK}
}
\email{[email protected]}
\author{Mounia Lalmas}
\affiliation{
\institution{Yahoo Research}
\city{London, UK}
}
\email{[email protected]}
\begin{abstract}
Machine-learned models are often described as ``black boxes''.
In many real-world applications however, models may have to sacrifice predictive power in favour of human-interpretability.
When this is the case, feature engineering becomes a crucial task, which requires significant and time-consuming human effort. Whilst some features are inherently static, representing properties that cannot be influenced (\emph{e.g.}, the age of an individual), others capture characteristics that could be \emph{adjusted} (\emph{e.g.}, the daily amount of carbohydrates taken).
Nonetheless, once a model is learned from the data, each prediction it makes on new instances is irreversible - assuming every instance to be a static point located in the chosen feature space.
There are many circumstances however where it is important to understand \emph{(i)} why a model outputs a certain prediction on a given instance, \emph{(ii)} which adjustable features of that instance should be modified, and finally \emph{(iii)} how to alter such a prediction when the mutated instance is input back to the model.
In this paper, we present a technique that exploits the internals of a tree-based ensemble classifier to offer \emph{recommendations} for transforming true negative instances into positively predicted ones.
We demonstrate the validity of our approach using an online advertising application.
First, we design a Random Forest classifier that effectively separates between two types of ads: \emph{low} (negative)
and \emph{high} (positive) quality ads (instances).
Then, we introduce an algorithm that provides recommendations that aim to transform a low quality ad (negative instance) into a high quality one (positive instance).
Finally, we evaluate our approach on a subset of the active inventory of a large ad network, \emph{Yahoo Gemini}.
\end{abstract}
\begin{CCSXML}
<ccs2012>
<concept>
<concept_id>10003752.10010070.10010071</concept_id>
<concept_desc>Theory of computation~Machine learning theory</concept_desc>
<concept_significance>500</concept_significance>
</concept>
<concept>
<concept_id>10010147.10010257.10010321.10010333</concept_id>
<concept_desc>Computing methodologies~Ensemble methods</concept_desc>
<concept_significance>500</concept_significance>
</concept>
<concept>
<concept_id>10002951.10003260.10003272</concept_id>
<concept_desc>Information systems~Online advertising</concept_desc>
<concept_significance>500</concept_significance>
</concept>
</ccs2012>
\end{CCSXML}
\ccsdesc[500]{Theory of computation~Machine learning theory}
\ccsdesc[500]{Computing methodologies~Ensemble methods}
\ccsdesc[500]{Information systems~Online advertising}
\keywords{Model interpretability; Actionable feature tweaking; Recommending feature changes; Altering model predictions; Random forest}
\maketitle
\section{Introduction}
\label{sec:intro}
An increasing number of organisations and governments rely on \emph{Machine Learning} (ML) techniques to extract knowledge from the large volumes of data they collect every day to optimise their operational effectiveness.
ML solutions are usually considered as ``black boxes''; they take some inputs and produce desired outputs, especially when their ultimate goal is \emph{prediction} rather than \emph{inference}. As long as ML models work properly, ``everybody'' is happy and little attention is devoted to understand why such surprisingly good results are obtained.
Still, it is beneficial to have available techniques supporting humans in interpreting and ``debugging'' these models, particularly when they fail~\cite{szegedy2013intriguing} or lead to some oddities.\footnote{\url{http://www.telegraph.co.uk/technology/2016/03/24/microsofts-teen-girl-ai-turns-into-a-hitler-loving-sex-robot-wit/}}
Excluding recent trends in ML such as Deep Learning~\cite{lecun2015nature}, typically the initial effort when designing an ML solution consists in modelling the objects of a given domain of interest, \emph{i.e.}, \emph{feature engineering}.
This step aims to describe each object in the domain using an appropriate set of properties (\emph{features}), which define a so-called \emph{feature space}.
For a given dataset, each object can be considered as a static point located in the feature space since each feature value is deemed to be fixed; once a model is learned from the data, each prediction it makes on new objects is irreversible.
Let us assume that we disagree with a prediction that the model returns for a given object or that we would like to enforce switching such a prediction. The research question we ask in this work is \emph{how can we understand what can be changed in the feature vector in order to modify the prediction accordingly?}
To better understand this challenge with an example, consider an ML application in the healthcare domain, where patients (objects) are mapped to a vector of clinical indicators (features), such as age, blood pressure, daily carbohydrates taken, \emph{etc.}\ Assume next that an ML model has been designed to accurately predict from these features whether a patient is at risk of a heart attack or not.
If for a given patient our model predicts that there is a high risk of a heart attack it would be of great advantage for medical physicians to also have a tool that suggests the most appropriate clinical treatment by offering targeted adjustments to specific indicators (\emph{e.g.}, reducing the daily amount of carbohydrates). In other words, to recommend the clinical treatment to switch a patient from being of high risk (negative instance) to low risk (positive instance).
In this work, we propose an algorithm for \emph{tweaking} input features so as to change the output predicted by an existing machine-learned model.
Our method is designed to operate on top of any tree-based ensemble binary classifier, although it can be extended to multi-class classification. Our proposed algorithm exploits the internals of the model to generate recommendations for transforming true negative instances into positively predicted ones (or vice versa).
We describe the theoretical framework along with experiments designed to validate the proposed algorithm.
Our approach is then evaluated in the commercial and more implementable setting of online advertisement recommendations to illustrate the generic nature of our framework and the many and varied domains it can be applied to.
After presenting an effective Random Forest classifier that is able to separate between \emph{low} and \emph{high} quality advertisements~\cite{Lalmas:2015:PPP:2783258.2788581}, we show how our algorithm can be used to automatically generate ``interpretable'' and ``actionable'' suggestions on how to convert a low quality ad (negative instance) into a high quality one (positive instance).
Such insights can be provided to advertisers who may turn them into actual changes to their ad campaigns with the aim of improving their return on investment.
Finally, we assess the quality of recommendations that our algorithm generates out of a dataset of advertisements served by the \emph{Yahoo Gemini} ad network.
\section{Problem Statement}
\label{sec:problem}
We start by considering the typical binary classification problem and focus on an ensemble of tree-based classifiers as an effective solution to the above problem.
Additionally, we define how the internals of an existing ensemble of trees can be used to derive a feedback loop for recommending how true negative instances can be turned into positively predicted ones (or vice versa).
The approach we propose can be easily extended to the more general multi-class classification problem. We plan to present this result in future extended work.
\subsection{Notation}
\label{subsec:notation}
Let $\mathcal{X}\subseteq \mathbb{R}^n$ be an $n$-di\-men\-sio\-nal vector space of real-valued features.
Any $\mathbf{x}\in \mathcal{X}$ is an $n$-di\-men\-sio\-nal feature vector, \emph{i.e.}, $\mathbf{x} = (x_1, x_2, \ldots, x_n)^T$, representing an object in the vector space $\mathcal{X}$.
Suppose that each $\mathbf{x}$ is associated with a binary \emph{class label} -- either {\tt neg} ({\em negative}) or {\tt pos} ({\em positive}) -- and let $\mathcal{Y} = \{-1,+1\}$ be the set encoding all such possible class labels.
We assume there exists an \emph{unknown target} function $f: \mathcal{X} \longmapsto \mathcal{Y}$ that maps any feature vector to its corresponding class label.
In addition, we let $\hat{f}\approx f$ which is learned from a labelled dataset of $m$ instances $\mathcal{D} = \{(\mathbf{x_1}, y_1), (\mathbf{x_2}, y_2), \ldots, (\mathbf{x_m}, y_m)\}$.
More specifically, $\hat{f}$ is the estimate that best approximates $f$ on $\mathcal{D}$, according to a specific \emph{loss function} $\ell$.
Such a function measures the ``cost'' of prediction errors we would make if we replaced the true target $f$ with the estimate $\hat{f}$.
The flexibility \emph{vs.}\ interpretability of $\hat{f}$ depends on the \emph{hypothesis space} which $\hat{f}$ has been picked from by the learning algorithm.
In this work, we focus on $\hat{f}$ represented as an \emph{ensemble} of $K$ tree-based classifiers, $\hat{f} = \phi(\hat{h}_1,\ldots, \hat{h}_K)$. Each $\hat{h}_k:\mathcal{X} \longmapsto \mathcal{Y}$ is a base estimate, and $\phi$ is the function responsible for combining the output of all the individual base classifiers into a single prediction.
A possible implementation of $\phi$ could use a \emph{majority voting} strategy.
In this setting, a given instance $\mathbf{x}$ would obtain a predicted class label $\hat{f}(\mathbf{x})$ based on the result of the majority of the base classifiers; this is the \emph{mode} of the base predictions.
Although other strategies may be used, this does not impact our proposed approach.
\subsection{Enforcing Positive Prediction}
\label{subsec:enforce-prediction}
In any ensemble of tree-based classifiers, each base estimate $\hat{h}_k$ is encoded by a decision tree $T_k$, and the ensemble is represented as a forest $\mathcal{T} = \{T_1, \ldots, T_K\}$.
Our aim is to identify how to transform a true negative instance into a positively predicted one.
Let $\mathbf{x}\in \mathcal{X}$ be a true negative instance such that $f(\mathbf{x}) = \hat{f}(\mathbf{x}) = -1$.
The task can now be defined as transforming the original input feature vector $\mathbf{x}$ into a new feature vector $\mathbf{x'}$ ($\mathbf{x} \leadsto \mathbf{x'}$) such that $\hat{f}(\mathbf{x'}) = +1$.
Moreover, we accomplish an optimised form of the problem by choosing $\mathbf{x'}$ as the best transformation among all the possible transformations $\mathbf{x^*}$, according to a \emph{cost function} $\delta: \mathcal{X}\times \mathcal{X} \longmapsto \mathbb{R}$.
This is defined as follows:
\[
\mathbf{x'} = \argmin_{\mathbf{x^*}} \Big\{\delta(\mathbf{x}, \mathbf{x^*})~|~\hat{f}(\mathbf{x}) = -1 \wedge \hat{f}(\mathbf{x^*}) = +1 \Big\}
\]
The cost function measures the ``effort'' of transforming $\mathbf{x}$ into $\mathbf{x'}$.
A possible choice of such a function is the number of features affected by the transformation or the Euclidean distance between the original and the transformed vector.
\subsection{Positive and Negative Paths}
\label{subsec:posneg-paths}
Any root-to-leaf path of a single decision tree can be interpreted as a cascade of \emph{if}-\emph{then}-\emph{else} statements, where every internal (non-leaf) node is a boolean test on a specific feature value against a threshold. We restrict the tree decisions to be binary representations as any multiway decision can be represented in a binary form and there is little performance benefit in n-ary splits.
An instance's feature value is then evaluated at each node to determine which branch to traverse. This is repeated until the leaves are reached whereby the {\tt pos}/{\tt neg} classification labels are defined and assigned.
Given a forest of $K$ decision trees $\mathcal{T} = \{T_1, \ldots, T_K\}$, we denote by $p_{k,j}$ the $j$-th path of the $k$-th tree $T_k$.
We refer to $p^{+}_{k,j}$ (or $p^{-}_{k,j}$) as the $j$-th path of $T_k$ that leads to a leaf node labelled as {\tt pos} (or {\tt neg}) -- a \emph{positive} (or \emph{negative}) path.
For simplicity, we assume that each path of a decision tree contains at most $n$ non-leaf nodes, which correspond to $n$ boolean conditions, one for each distinct feature.\footnote{\small{In general, there can be multiple boolean conditions associated with a single feature.}} We thus represent a root-to-leaf path as follows:
\begin{equation}
\label{eq:path}
p_{k,j} = \{(x_1 \lesseqgtr \theta_1), (x_2 \lesseqgtr \theta_2), \ldots, (x_n \lesseqgtr \theta_n)\}
\end{equation}
Let $P^{+}_k = \bigcup_{j\in T_k}p^{+}_{k,j}$ describe the set of all positive paths, and $P^{-}_k = \bigcup_{j\in T_k}p^{-}_{k,j}$ the set of all the negative paths in $T_k$. Also, let $P_k = P^{+}_k \cup P^{-}_k$ be the set of \emph{all} the paths in $T_k$.
We thus enumerate the possible paths in a single decision tree.
Even under the assumption that each $p_{k,j}\in P_k$ is at most a length-$n$ path then $T_k$ is a depth-$n$ binary tree, whose number of leaves is therefore bounded to~$2^n$.
As the total number of leaves coincides with the total number of possible paths, we obtain $|P_k| \leq 2^n$.
In general, we cannot ensure a bound on each $|P_k|$, as there might exist some paths $p_{k,j}\in P_k$ whose length is greater than $n$.
In practice though, we can specify the maximum number of paths at training time by bounding the depth of the generated trees to the number $n$ of features.
Even with such a relaxed condition, the total number of possible paths encoded by the forest $\mathcal{T}$ is equal to $\sum_{k=1}^K |P_k| \leq K2^n$, therefore still exponential in $n$.
We see later how this does not disrupt computational efficacy in practice, as our algorithm operates on a subset of those paths.
\subsection{Tweaking Input Features}
\label{subsec:tweaking}
Given our input feature vector $\mathbf{x}$, we know from our hypothesis that $f\left(\mathbf{x}\right) = \hat{f}(\mathbf{x}) = -1$.
If the overall prediction is obtained using a majority voting strategy, it follows:
\[
\hat{f}(\mathbf{x}) = -1 \iff \left( \sum_{k=1}^K \hat{h}_k(\mathbf{x})\right) \leq 0
\]
Furthermore, there must be at least $\left\lceil\frac{K}{2}\right\rceil$ decision trees (base classifiers) of the forest $\mathcal{T}$ whose output is $-1$. That is, there exists $K^- \subseteq \{1,\ldots,K\}$ with $|K^-| \geq \left\lceil\frac{K}{2}\right\rceil$, such that:
\[
\hat{h}_{k^-}(\mathbf{x}) = -1,~\forall k^-\in K^-
\]
As we are operating in a binary classification setting, there must also exist $K^+ = \{1,\ldots,K\}\setminus K^-$, which denote the set of classifier indices that output a positive label when input with $\mathbf{x}$, \emph{i.e.}, $\hat{h}_{k^+}(\mathbf{x}) = +1,~\forall k^+\in K^+$.
Our goal is to tweak the original input feature vector $\mathbf{x}$ so as to adjust the prediction made by the ensemble from negative ($-1$) to positive ($+1$).
We can skip all the trees indexed by $K^+$, as these are already encoding the (positive) prediction we ultimately want.
We therefore focus on each tree $T_k$ where $k \in K^-$, and consider the set $P^{+}_k$ of all its positive paths.
With each $p^{+}_{k,j}\in P^{+}_k$ we associate an instance $\mathbf{x}^+_{j} \in \mathcal{X}$ that \emph{satisfies} that path -- \emph{i.e.}, an instance whose adjusted feature values meet the boolean conditions encoded in $p^{+}_{k,j}$ to finish on a
{\tt pos}-labelled leaf, and therefore $\hat{h}_k(\mathbf{x}^+_{j}) = +1$.
Among all the possibly infinite instances satisfying $p^{+}_{k,j}$, we restrict to $\mathbf{x}_{j(\epsilon)}^+$ to be feature value changes with a ``tolerance'' of at most $\epsilon$. We call it the $\epsilon$-satisfactory instance of $p^{+}_{k,j}$.
We consider $p^{+}_{k,j}$ containing at most $n$ boolean conditions, as specified by Equation~\ref{eq:path}.
Therefore, for any (small) fixed $\epsilon > 0$, we build a feature vector $\mathbf{x}_{j(\epsilon)}^+$ as follows:
\begin{equation}
\label{eq:tweak}
\mathbf{x}_{j(\epsilon)}^+[i] = \left\{
\begin{array}{l l}
\theta_i - \epsilon & \quad \text{if the $i$-th condition is $(x_i \leq \theta_i)$}\\
\theta_i + \epsilon & \quad \text{if the $i$-th condition is $(x_i > \theta_i)$}
\end{array} \right.
\end{equation}
Having a single \emph{global} tolerance $\epsilon$ for all the features works as long as we standardise every feature (\emph{e.g.}, using z-score or min-max). If we use standard z-score for each feature, the actual magnitude of the change cannot just depend on $\epsilon$ but instead needs to be considered a multiple of a unit of standard deviation from the feature mean. Let $\theta_i = \frac{t_i - \mu_i}{\sigma_i}$ be the z-score of the threshold on the $i$-th feature value, where $t_i$ is the non-standardised value, $\mu_i$ and $\sigma_i$ are the mean and standard deviation of the $i$-th feature
respectively.\footnote{\small{In practice, $\mu_i$ and $\sigma_i$ are often unknown, so the sample mean and the sample standard deviation are used instead.}}
Now, suppose that $x_i = \theta_i \pm \epsilon$ is the tweaked value of the $i$-th feature according to the ongoing transformation of the input vector $\mathbf{x}$.
Therefore, $x_i = \frac{t_i - \mu_i}{\sigma_i}\pm \epsilon$.
Returning to the original (\emph{i.e.}, non-standardised) feature scale, we obtain $x_i = t_i - \mu_i \pm \epsilon \sigma_i$.
Depending on the sign, the tweaked feature is either moving closer to or farther away from the original feature mean $\mu_i$, pivoting around $t_i$.
For each $p^{+}_{k,j} \in P^{+}_k$ we transform our input feature vector $\mathbf{x}$ into the $\epsilon$-satisfactory instance $\mathbf{x}_{j(\epsilon)}^+$ that validates $p^{+}_{k,j}$.
This leads us to a set of transformations $\Gamma_k = \bigcup_{j\in P^{+}_k}\mathbf{x}_{j(\epsilon)}^+$, associated with the $k$-th tree $T_k$.
Each resulting transformation in $\Gamma_k$ may have an impact on other trees of the forest.
There might exist $l \in K^+$ whose corresponding tree already provides the correct prediction when this is input with $\mathbf{x}$, $\hat{h}_{l}(\mathbf{x}) = +1$.
It may also happen that by changing $\mathbf{x}$ into $\mathbf{x}'$ the prediction of the $l$-th tree is incorrect and now $\hat{h}_{l}(\mathbf{x}') = -1$.
In other words, by changing $\mathbf{x}$ into another instance $\mathbf{x}' \in \Gamma_k$ we are only guaranteed that the prediction of the $k$-th base classifier is correctly fixed, \emph{i.e.}, from $\hat{h}_k(\mathbf{x}) = -1$ to $\hat{h}_k(\mathbf{x}') = +1$.
The overall prediction for $\mathbf{x}'$ may or may not be fixed, where $\hat{f}(\mathbf{x}')$ may still output $-1$, exactly as $\hat{f}(\mathbf{x})$ did.
If the change from $\mathbf{x}$ to $\mathbf{x}'$ also leads to $\hat{f}(\mathbf{x}') = +1$, then $\mathbf{x}'$ will be a \emph{candidate} transformation for $\mathbf{x}$.
More formally, let $\Gamma = \bigcup_{k=1}^K \Gamma_k$ be the set of \emph{all} the $\epsilon$-satisfactory transformations of the original $\mathbf{x}$ from the positive paths of all the trees in the forest. Our feature
tweaking problem can then be generally defined as follows:
\[
\mathbf{x'} = \argmin_{\mathbf{x}_{j(\epsilon)}^+\in \Gamma~|~\hat{f}(\mathbf{x}_{j(\epsilon)}^+) = +1} \Big\{\delta(\mathbf{x}, \mathbf{x}_{j(\epsilon)}^+) \Big\}
\]
In~\cite{cui2015kdd}, it has already been proven that a problem similar to the one we define above is NP-hard as it reduces to DNF-MAXSAT. Our version, in fact, introduces an additional constraint ($\epsilon$) on the possible way features can be tweaked and thus it is itself NP-hard.
This problem definition is still valid for the base case when $K=1$.
There, the additional condition requiring $\hat{f}(\mathbf{x}_{j(\epsilon)}^+) = +1$ is not necessary because it is implicitly true by definition.
In that scenario, the ensemble is composed of a single base classifier -- \emph{i.e.}, the forest contains a single decision tree and tweaking its prediction also results in changing the overall prediction.
Note that when there is only one decision tree, our problem can be solved \emph{optimally}: We can enumerate all the positive paths, choose the one with the minimum cost, and check if the threshold of tolerance $\epsilon$ is satisfied on each feature.
Because base trees are interconnected through the features they share, simply enumerating positive paths does not work for an ensemble of trees since the output of a base tree may affect outputs of its sibling trees.
The presence of the condition $\hat{f}(\mathbf{x}_{j(\epsilon)}^+) = +1$ circumvents this issue by querying the model itself on the correctness of the $\epsilon$-transformation of an instance.
\subsection{The Feature Tweaking Algorithm}
\label{subsec:algorithm}
Our approach takes as input 4 key components: \emph{(i)} The trained ensemble model $\hat{f}$; \emph{(ii)} A feature vector $\mathbf{x}$ that represents a true negative instance; \emph{(iii)} A cost function $\delta$ measuring the ``effort'' required to transform the true negative instance into a positive one; and
\emph{(iv)} A positive threshold $\epsilon$ that bounds the tweaking of each single feature to pass every boolean test on a positive path of each tree.
The result being the transformation $\mathbf{x'}$ of the original $\mathbf{x}$ that exhibits the minimum cost according to $\delta$.
The detailed description is presented in Algorithm~\ref{algo:tweakfeat}.
\begin{algorithm}[t]
\small
\SetAlgoLined
\SetKwFunction{getPositivePaths}{getPositivePaths}
\SetKwFunction{buildPositiveInstance}{buildPositiveInst}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetCommentSty{emph}
\Input{\\
\noindent $\triangleright$ An estimate function $\hat{f}$ resulting from an ensemble of decision trees $\mathcal{T} = \{T_1,\ldots,T_K\}$, each one associated with a base estimate $\hat{h}_k,~k=1,\ldots,K$\\
\noindent $\triangleright$ A feature vector $\mathbf{x}$ representing a \emph{true negative} instance, such that $f(\mathbf{x}) = \hat{f}(\mathbf{x}) = -1$\\
\noindent $\triangleright$ A cost function $\delta$\\
\noindent $\triangleright$ A (small) threshold $\epsilon > 0$\\
}
\Output{\\
\noindent $\triangleright$ The optimal transformation $\mathbf{x}'$ with respect to $\delta$, such that $\hat{f}(\mathbf{x}') = +1$
}
\BlankLine
\Begin{
$\mathbf{x}' \longleftarrow \mathbf{x}$\;
$\delta_{\textit{min}} \longleftarrow +\infty$\;
\For {$k=1,\ldots, K$} {
\If {$\hat{f}(\mathbf{x}) == \hat{h}_k(\mathbf{x})~{\bf and}~\hat{h}_k(\mathbf{x}) == -1$ }{
\tcc{retrieve the set of \emph{positive} paths of the $k$-th decision tree}
$P^{+}_k \longleftarrow$ \getPositivePaths{$T_k$}\;
\ForEach {$p^{+}_{k,j} \in P^{+}_k$} {
\tcc{generate the $\epsilon$-satisfactory instance associated with the $j$-th positive path of the $k$-th decision tree}
$\mathbf{x}_{j(\epsilon)}^+ \longleftarrow $ \buildPositiveInstance{$\mathbf{x}, p^{+}_{k,j}, \epsilon$}\;
\If {$\hat{f}(\mathbf{x}_{j(\epsilon)}^+) == + 1$ }{
\If {$\delta(\mathbf{x}, \mathbf{x}_{j(\epsilon)}^+) < \delta_{\textit{min}}$ }{
$\mathbf{x}' \longleftarrow \mathbf{x}_{j(\epsilon)}^+$\;
$\delta_{\textit{min}} \longleftarrow \delta(\mathbf{x}, \mathbf{x}_{j(\epsilon)}^+)$\;
}
}
}
}
}
\mathbb{R}eturn{$\mathbf{x}'$}\;
}
\caption{The {\sc Feature Tweaking Algorithm}}
\label{algo:tweakfeat}
\end{algorithm}
In the worst case, our algorithm examines all $K$ trees of the forest, although it investigates only those trees whose base predictions are negatives ({\tt neg}-labelled).
Then, all the positive paths of each tree are considered, and for each of those a potential candidate $\epsilon$-transformation is built according to the scheme proposed in Equation~\ref{eq:tweak}.
As such, the number of steps depends on the number of positive paths on each tree, which in turn is related to the number of leaves.
We stated in Section~\ref{subsec:posneg-paths} that we cannot provide \emph{apriori} any limit to the depth of a decision tree, and therefore to its number of leaves.
However, in practice these can be bounded whilst training the model.
We therefore set the maximum length of each root-to-leaf path, \emph{i.e.}, the depth of each tree, to be at most equal to the number of input features $n$.
Considering our goal, this is not a limitation as each transformation should at most affect each feature exactly once.
The total number of positive paths to be examined will then be limited by $K2^n$, and so the worst case complexity is $O(2^n)$.
It follows that our method might be unsuitable for dealing with high-dimensional datasets, such as text, images, or videos.
In reality such a setting would not really make sense when transforming input instances in the original feature space (\emph{e.g.}, change a few words on a document or a few pixels of an image).
Although the search space can be exponential in the number of features, in practice the number of positive paths of each tree is significantly smaller.
This makes our method feasible on average input sizes (\emph{i.e.}, below 100 features).
For example, in our experiments we found that the maximum depth of each tree leading to the best classification results is significantly smaller than the total number $n=45$ of features (see Table~\ref{tab:cv-performance}).
In addition, many positive paths may share several boolean conditions, especially when extracted from the same decision tree.
This allows us to avoid tweaking the same input feature multiple times according to the same condition by using some caching mechanism.
Finally, our algorithm can be easily parallelised since each tree can be explored independently from the others for any given instance --\emph{i.e.}, we can adjust the $k$-th tree while keeping the remaining $K\setminus k$ trees simultaneously fixed for all $k\in \{1,\ldots, K\}$.
\section{Use Case: Improving Ad Quality}
\label{sec:adquality-prediction}
We demonstrate the utility of our method when applied to a real-world use case in online advertising.
We investigate how our algorithm can be used to improve the \emph{quality} of advertisements served by the \emph{Yahoo Gemini} ad network.
\subsection{Why Ad Quality?}
\label{subsec:adquality-why}
A main source of monetisation for online web services comes in the form of advertisements (\emph{ads} for short) impressed in dedicated real estate units of rendered web pages.
Online publishers operating these web services typically reserve predefined slots within their streams, utilising a third party ad network to deliver ad inventory to impress within them.
Ad networks free publishers from running their own ad servers as they decide for them which ads should be placed at which slots, when and to whom.
Advertisers rely on ad networks to optimise their \emph{return on investment} -- for example through targeting the right audience according to the advertiser budget and marketing strategy.
A trustworthy relationship between advertisers, publishers and ad networks is instrumental to the success of online advertising.
On the advertiser side, ad networks provide advertisers with tools for monitoring key performance indicators (\emph{e.g.}, number of ad \emph{impressions}, \emph{click-through rate} or CTR, \emph{bounce rate}) as well as effective mechanisms to overcome fraudulent activities such as \emph{click spam}~\cite{Daswani2007AC, StoneGross2011UFA}.
On the publisher side, ad networks provide mechanisms that allow users to hide ads they dislike and indicate the reasons for doing so.
This information can be used by ad networks to ensure that the ads they serve do not negatively affect user engagement on a publisher website, as well as reporting to advertisers on the \emph{quality} of their ads.
As with any self served content delivery platform, an ad network has a varying distribution of quality with many ads being of low quality. Not serving them may not be an option when supply is exhausted.
Therefore, another approach to positively shift the inventory quality distribution is to leverage the interpretability of the internal machinery of existing ad quality prediction models -- \emph{i.e.}, binary classifiers -- so as to offer actionable \emph{recommendations}. The intention of these programmatically-computed recommendations is to provide advertisers with guidance on how they can improve their ad quality at scale. Such a system yields value for all beneficiaries in this advertising ecosystem ultimately culminating with a better user experience.
Returning to our work, by applying our feature tweaking algorithm introduced in Section~\ref{subsec:algorithm}, we show in the rest of this paper how we can transform a low quality ad into a set of new ``proposed'' high quality ads by shifting their position in an ad quality feature space.
The algorithm employs the internals of a learned binary classifier to \emph{tweak} the feature-based representation of a low quality ad so that the new ``proposed'' ads are promoted to high quality ones when
re-input to the classifier.
Each transformation is associated with a \emph{cost}, allowing us to generate actionable suggestions from the ``proposed'' instances with the least cost, so as to improve low quality ads.
We validate our approach on a dataset of mobile native ads served by the \emph{Yahoo Gemini}\footnote{\small{\url{https://gemini.yahoo.com}}} ad network.
\subsection{A Definition of Ad Quality}
\label{subsec:adquality-definition}
Many factors can affect the quality of an ad: its \emph{relevance}, \emph{i.e.}, whether the ad matches the user interest~\cite{Raghavan:2009:RMB:1571941.1572116}; the \emph{pre-click} experience, \emph{i.e.}, whether the ad annoys a user~\cite{zhou2016predicting};
and finally the \emph{post-click} experience, \emph{i.e.}, whether the \emph{ad landing page}\footnote{\small{We refer to \emph{ad landing page} as the web page of the advertiser that a user is redirected to after clicking on an \emph{ad}.}}
meets the user click intent that brought them to the landing page~\cite{Lalmas:2015:PPP:2783258.2788581}.
We focus on the latter, the post-click experience, following from ~\cite{DBLP:conf/www/BarbieriSL16,Lalmas:2015:PPP:2783258.2788581}.
Inspired by these studies, we define and measure the quality of ads using the time spent on their landing pages as a proxy, referred to as \emph{dwell time}.
We know from~\cite{Lalmas:2015:PPP:2783258.2788581} that ad landing pages exhibiting long dwell times promote a positive long-term post-click experience. Based on this definition of ad quality, we design a binary classifier that effectively
separates between \emph{low} and \emph{high} quality ads, \emph{i.e.}, ad landing pages whose dwell time is below or above a threshold $\tau$, respectively.
We compute $\tau$ as the median of all the sample means of dwell times observed for a large set of ad landing pages.
Intuitively, an ad landing page is of high quality if its average dwell time is greater than $\tau$ --
\emph{i.e.}, if the average time users spent on the page is greater than the average time users spent on at least 50\% of any other ad landing page.
Although more sophisticated approaches can be designed~\cite{DBLP:conf/www/BarbieriSL16}, this is not the main goal of this research, and we leave it for future work.
\subsection{Predicting Ad Quality}
\label{subsec: adquality-prediction}
To apply our algorithm to our use case, we first need to learn a binary classifier that predicts whether an ad is of high quality or not, given a feature-based representation of the ad creative and the landing page.
\subsubsection{Ad Feature Engineering}
\label{subsubsec:features}
A sample of the set of features used in this work is listed in Table~\ref{tab:features}.
This is based on the same set used in~\cite{DBLP:conf/www/BarbieriSL16}, with the additional ``{\sf Language}'' category (marked with ``$\dagger$'' in the Table).
Due to space constraints we do not show all the features, and invite the reader to refer to~\cite{DBLP:conf/www/BarbieriSL16} for a complete description of them.
Each feature in the table is associated with a \emph{category} and a \emph{source}.
The former indicates the type of features whilst the latter specifies whether the feature is computed from the ad \emph{landing page} ({\sf LP}), the ad \emph{creative} ({\sf CR}), or a combination of the two ({\sf CR-LP}).
Although our focus is on the post-click experience (\emph{i.e.}, ad landing page), we aim to obtain the best performing model for predicting the quality of the ads, and hence include both pre-click (\emph{i.e.}, ad creative) and historical features; the latter not participating in the tweaking process as -- by definition -- they cannot be altered.
\begin{table*}[!htb]
\caption{\label{tab:features} {\small Set of features used to characterise ad \emph{creative} ({\sf CR}), ad \emph{landing page} ({\sf LP}), or both ({\sf CR-LP}).}}
\small
\begin{center}
\begin{tabu} to \textwidth {| X[2,c,m] | X[1,c,m] | X[5,c,m] |}
\hline
\tr{\bf Category}\br & \tr{\bf Source}\br & \tr{\bf Description}\br
\\ \hline
\multirow{1}{*} {\sf Language}{$^\dagger$} & {\sf CR} & {This set of features capture the extent to which the text of the ad creative may include adult, violent, or spam content (\emph{e.g.}, ADULT\_SCORE, HATE\_SCORE, and SPAM\_SCORE)}
\\ \hline
\multirow{1}{*} {\sf DOM} & {\sf LP} & {This set of features are derived from the elements extracted from the HTML DOM of the ad landing page, such as the main textual content (LANDING\_MAIN\_TEXT\_LENGTH), the total number of internal and external hyperlinks (LINKS\_TOTAL\_COUNT), the ratio of main text length to the total number of hyperlinks on the page (LINKS\_MAIN\_LENGTH\_TOTAL\_RATIO), \emph{etc.}}
\\ \hline
\multirow{1}{*} {\sf Readability} & {\sf CR-LP} & {These features range from a simple count of tokens (words) in the text of the ad creative and landing page to well-known scores for measuring the summarisability/readability of a text (\emph{e.g.}, READABILITY\_SUMMARY\_SCORE), \emph{etc.}}
\\ \hline
\multirow{1}{*} {\sf Mobile Optimising} & {\sf LP} & {This set of features describe the degree of mobile optimisation of the ad landing page by measuring the ability of it to be tuned to different screen sizes (VIEW\_PORT), testing for the presence of a click-to-call button (CLICK\_TO\_CALL), \emph{etc.}}
\\ \hline
\multirow{1}{*} {\sf Media} & {\sf LP} & {These features refer to any media content displayed within the ad landing page, such as the number of images (NUM\_IMAGES), \emph{etc.}}
\\ \hline
\multirow{1}{*} {\sf Input} & {\sf LP} & {This set of features represent all the possible input types available on the ad landing page, such as the number of checkboxes, drop-down menus, and radio buttons (NUM\_INPUT\_CHECKBOX, NUM\_INPUT\_DROPDOWN, NUM\_INPUT\_RADIO), \emph{etc.}}
\\ \hline
\multirow{1}{*} {\sf Content \& Similarity} & {\sf CR-LP} & {These features extract the set of Wikipedia \emph{entities} from the ad creative and landing page (NUM\_CONCEPT\_ANNOTATION), and measure the \emph{Jaccard} similarity between those two sets (SIMILARITY\_WIKI\_IDS), \emph{etc.}}
\\ \hline
\multirow{1}{*} {\sf History} & {\sf LP} & {These features measure historical indicators, such as the median \emph{dwell time} as computed from the last 28 days of observed ad clicks (HISTORICAL\_DWELLTIME), and the \emph{bounce rate} -- \emph{i.e.}, the proportion of ad clicks whose dwell time is below 5 seconds (HISTORICAL\_BOUNCE\_RATE), \emph{etc.} }
\\ \hline
\end{tabu}
\end{center}
\end{table*}
\subsubsection{Learning Binary Classifiers}
\label{subsubsec:classifiers}
As our feature tweaking algorithm is designed to work on tree-based ensemble classifiers, we train the following learning models to find our best estimate $\hat{f}$: Decision Trees ({\sf DT})~\cite{Quinlan1986DT}, which can be thought of as a special case of an ensemble with a single tree; Gradient Boosted Decision Trees ({\sf GBDT})~\cite{friedman02stochastic}, and Random Forests ({\sf RF})~\cite{Breiman2001RF}.
The original dataset $\mathcal{D} = \{(\mathbf{x_1}, y_1), (\mathbf{x_2}, y_2), \ldots, (\mathbf{x_m}, y_m)\}$ is split into two datasets $\mathcal{D}_{\text{train}}$ and $\mathcal{D}_{\text{test}}$ using stratified random sampling.
$\mathcal{D}_{\text{train}}$ is used for training the models and accounts for 80\% of the total number of instances in $\mathcal{D}$, whilst $\mathcal{D}_{\text{test}}$ contains the remaining held-out portion used for evaluating the models.
$\mathcal{D}_{\text{train}}$ is also used to perform model selection, which is achieved by tuning the hyperparameters specific for each.
With every combination of model and corresponding hyperparameters, we run a 10-fold cross validation to find the best settings for each model -- \emph{i.e.}, the one with the best cross validation performance.
We measure this performance using the \emph{Area Under the Curve of the Receiver Operating Characteristic} (ROC AUC).
Each model is in turn re-trained on the whole $\mathcal{D}_{\text{train}}$ using the best hyperparameter setting.
Finally, the overall best model is deemed to be the one that best performs on the test set $\mathcal{D}_{\text{test}}$.
\subsubsection{Labelled Dataset of Ads}
\label{subsubsec:dataset}
We collect a random sample of 1,500 ads served by the \emph{Yahoo Gemini} ad network on a mobile app during one month.
To ensure reliable estimates of dwell time, we only consider ads clicked at least 500 times.
We saw that the distribution was skewed with around 80\% of the instances having an average dwell time within approximately 100 seconds, whereas the remaining 20\% sat in the long tail of very long dwell times.
The median $\tau$ of those averages, calculated as described
in Section~\ref{subsec:adquality-definition}, is equal to $\approx 62.5$ seconds.
We therefore reach an evenly balanced ground truth, where 50\% of the instances have an average dwell time at most equal to $\tau$ and the remaining 50\% above $\tau$.
To build our labelled dataset $\mathcal{D}$, for each ad we extract the features listed in Section~\ref{subsubsec:features}.
As these are a mix of categorical (\emph{i.e.}, discrete) and continuous features, we apply \emph{one-hot encoding} to transform each $k$-valued categorical feature into a $k$-dimensional binary vector.
The $i$-th component of such a vector evaluates to 1 if and only if the value of the original feature is $i$, and 0 otherwise.
We also standardise continuous features by transforming their original values into their corresponding \emph{z-scores}~\cite{Kreyszig1979}.
Finally, we obtain a set of 45 features.
\subsubsection{Offline Evaluation}
\label{subsubsec:offlineval}
From the balanced labelled dataset above we derive two random partitions $\mathcal{D}_{\text{train}}$ and $\mathcal{D}_{\text{test}}$, which contain 80\% and 20\% of the total samples respectively.
We again validate our performance by running a 10-fold cross validation on $\mathcal{D}_{\text{train}}$ selecting between different models and their varying hyperparameter settings.
For {\sf DT}, we test two different node-splitting criteria $s$: \emph{Gini index} and \emph{entropy}. We also set the maximum depth of the tree $d$ to the total number of features.
For {\sf GBDT}, we use four values of the number $K$ of base trees in the ensemble, with $K =\{10, 100, 500, 1$,$000\}$ in combination with the learning rates $\alpha$, 0.001, 0.01, 0.05, 0.1, 1.
Finally, for {\sf RF}\ we test the same number of base trees as for {\sf GBDT}\ whilst again bounding the maximum depth of each base tree to the total number of features.
For each learning model we report in Table~\ref{tab:cv-performance} the hyperparameter settings leading to the best cross validation ROC AUC. The overall best performing model is {\sf RF}\ with an ensemble of $1$,$000$ base trees and maximum depth 16.
\begin{table}[t]
\centering
\small
\begin{tabular}[c]{|c||c|c|c|c|c|}
\hline
Model & $s$ & $\alpha$ & $d$ & $K$ & ROC AUC
\\ \hline
{\sf RF} & $-$ & $-$ & $16$ & $1$,$000$ & {\bf 0.93}
\\ \hline
{\sf GBDT} & $-$ & $0.1$ & $-$ & $100$ & 0.92
\\ \hline
{\sf DT} & \emph{entropy} & $-$ & $3$ & $-$ & 0.84
\\ \hline
\end{tabular}
\caption{\label{tab:cv-performance} {\small Best cross validation hyperparameter settings for each learning model (``$-$'' if the hyperparameter is not considered).}}
\end{table}
To avoid mixing model selection with model evaluation, we re-train each model on $\mathcal{D}_{\text{train}}$ using its best hyperparameter setting, and assess its validity on the held-out and unseen test set $\mathcal{D}_{\text{test}}$.
We measure two standard quality metrics, $\text{F}_{\text{1}}$ and Matthews Correlation Coefficient (MCC)~\cite{matthews11comparison}.
Table~\ref{tab:test-performance} shows the results.
{\sf RF}\ is the best performing model also with respect to the ability of generalising its predictive power to previously unseen examples.
\begin{table}[t]
\centering
\small
\begin{tabular}[c]{c|c|c|c|}
\cline{2-4}
& {\sf RF} & {\sf GBDT} & {\sf DT}
\\ \hline
\multicolumn{1}{|c|}{$\text{F}_{\text{1}}$} & {\bf 0.84} & $0.81$ & $0.75$
\\ \hline
\multicolumn{1}{|c|}{MCC} & {\bf 0.66} & $0.63$ & $0.49$
\\ \hline
\end{tabular}
\caption{\label{tab:test-performance} {\small Evaluation of best performing models on the test set $\mathcal{D}_{\text{test}}$.}}
\end{table}
Compared with the results reported in~\cite{Lalmas:2015:PPP:2783258.2788581}, we notice a remarkable increase of ROC AUC (+10.7\%) and a small improvement of $\text{F}_{\text{1}}$ (+1.2\%).
In their work, Logistic Regression ({\sf LogReg})~\cite{yu11dual} was the best model (ROC AUC = 0.84 and $\text{F}_{\text{1}}$ = 0.83). Our increased performance comes from a combination of a more rigorous procedure for determining the threshold $\tau$,\footnote{\small{In their work, the threshold was set as the \emph{flat} median of the observed dwell times of all ads.}} a more effective learning model, {\sf RF}, and a larger set of features.
We also calculate the ``importance'' of each feature from the learned {\sf RF}\ model. Figure~\ref{fig:featimp} lists the top-20 most important ones.
As in~\cite{DBLP:conf/www/BarbieriSL16,Lalmas:2015:PPP:2783258.2788581}, historical features have significant predictive power.
We keep historical features because we want the learned model for which we run our feature tweaking algorithm to be the most effective possible.
However, our feature tweaking algorithm will ignore historical features when generating recommendations, as it only considers \emph{adjustable} ad features, \emph{i.e.}, features that advertisers can actively alter to improve the quality of their ads.
\begin{figure}
\caption{{\small Top-20 most important features of our {\sf RF}
\label{fig:featimp}
\end{figure}
From now on, we use {\sf RF}\ as our learned model to generate actionable suggestions on which features to tweak to turn a low quality ad into a high quality one.
\section{Experiments: Ad Feature Recommendations}
\label{sec:ad-recommendations}
\begin{figure*}
\caption{{\small Distribution of per-ad $\epsilon$-transformations.}
\label{fig:transf_per_ad_distr_001}
\label{fig:transf_per_ad_distr_005}
\label{fig:transf_per_ad_distr_01}
\label{fig:transf_per_ad_distr_05}
\label{fig:transf_distr}
\end{figure*}
We validate the recommendations generated with our approach, by applying our feature tweaking algorithm to our learned {\sf RF}\ model.
Any $\mathbf{x}'$ that results from a valid (\emph{i.e.}, positive) $\epsilon$-trans\-for\-ma\-tion of the original negative instance $\mathbf{x}$ encapsulates a set of directives on how to positively change the ad features.
We compute the vector $\mathbf{r}$ resulting from the component-wise difference between $\mathbf{x}'$ and $\mathbf{x}$, which is $\mathbf{r}[i] = \mathbf{x}'[i] - \mathbf{x}[i]$.
Then for each feature $i$, such that $\mathbf{r}[i] \neq 0$ (\emph{i.e.}, $\mathbf{x}'[i] \neq \mathbf{x}[i]$), this vector provides the \emph{magnitude} and the \emph{direction} of the changes that should be made on feature~$i$.
The magnitude denotes the absolute value of the change (\emph{i.e.}, $|\mathbf{x}'[i] - \mathbf{x}[i]|$), whilst the direction indicates whether this is an \emph{increase} or a \emph{decrease} of the original value of feature $i$ (\emph{i.e.}, $\text{sgn}(\mathbf{x}'[i] - \mathbf{x}[i])$).
Finally, to derive the final list of recommendations, we sort $\mathbf{r}$ according to the feature ranking, as shown in Figure~\ref{fig:featimp}.
\subsection{The impact of hyperparameters $\delta$ and $\epsilon$}
\label{subsec:rec-hyperparams}
Our approach depends on a \emph{tweaking cost} ($\delta$) associated with transforming a negative instance (low quality ad) into a positive instance (high quality ad), and a \emph{tweaking tolerance} ($\epsilon$) used to change each individual ad feature.
We first explore how $\epsilon$ impacts on the ad \emph{coverage}, which is the percentage of ads for which our approach is able to provide recommendations.
We experiment with five values of $\epsilon$: 0.01, 0.05, 0.1, 0.5, and 1. These values can be thought of as multiples of a unit of standard deviation from each individual feature mean, as discussed in Section~\ref{subsec:tweaking}.
Table~\ref{tab:coverage} shows the highest coverage is when $\epsilon = 0.5$.
\begin{table}[t]
\centering
\small
\begin{tabular}[c]{|c|c|c|c|c|c|}
\hline
$\epsilon$ & 0.01 & 0.05 & 0.10 & 0.50 & 1.00
\\ \hline
\emph{ad coverage} (\%) & 58.5 & 64.2 & 72.3 & {\bf 77.4} & 63.2
\\ \hline
\end{tabular}
\caption{\label{tab:coverage} {\small The impact of tolerance threshold $\epsilon$ on ad coverage.}}
\end{table}
\begin{figure*}
\caption{{\small The impact of tolerance threshold $\epsilon$ on costs $\delta$.}
\label{fig:avg_costs_per_epsilon}
\label{fig:median_avg_costs_per_epsilon}
\label{fig:costs_per_epsilon}
\end{figure*}
Although some low quality ads cannot be transformed, those that can are often associated with multiple transformations.
Figure~\ref{fig:transf_distr} shows the distribution of $\epsilon$-transformations across the set of ads, generated using different values of~$\epsilon$ (except $\epsilon=1$ which is similar to $\epsilon=0.05$).
All the distributions are skewed offering a high number of transformations proposed for few ads.
Interestingly, the number of transformations is more evenly distributed across the ads when $\epsilon$ increases.
This is in agreement with the finding above, where larger values of $\epsilon$ result in a higher coverage before decreasing again between $\epsilon=0.5$ and 1.
To choose the most appropriate transformation for an ad, we experiment with several tweaking cost functions $\delta$, each taking as input the original ($\mathbf{x}$) and the transformed ($\mathbf{x}'$) feature vectors:
\begin{itemize}\itemsep0pt
\item {\sf tweaked\_feature\_rate}: proportion of features affected by the transformation of $\mathbf{x}$ into $\mathbf{x}'$ (range = [0, 1]);
\item {\sf euclidean\_distance}: Euclidean distance between $\mathbf{x}$ and $\mathbf{x}'$ (range = $\mathbb{R}$);
\item {\sf cosine\_distance}: 1 minus the cosine of the angle between $\mathbf{x}$ and $\mathbf{x}'$ (range = [0, 2]);
\item {\sf jaccard\_distance}: one's complement of the Jaccard similarity between $\mathbf{x}$ and $\mathbf{x}'$ (range = [0, 1]);
\item {\sf pearson\_correlation\_distance}: 1 minus the Pearson's correlation coefficient between $\mathbf{x}$ and $\mathbf{x}'$ (range = [0, 2]).
\end{itemize}
Up to a certain value, the tolerance $\epsilon$ is positively correlated with the ad coverage.
We explore how it impacts the five tweaking cost functions.
Figure~\ref{fig:costs_per_epsilon}(a) plots the micro-average costs and Figure~\ref{fig:costs_per_epsilon}(b) shows the median of all the individual per-ad average costs.
In general, the greater the tolerance the higher the cost (except for {\sf tweaked\_feature\_rate} and {\sf jaccard\_distance} when $\epsilon=1$); thus a trade-off between $\epsilon$ (\emph{i.e.}, ad coverage) and the cost of ad transformations $\delta$ is desirable.
\subsection{Evaluating Recommendations}
\label{subsec:rec-eval}
\begin{figure*}
\caption{{\small Top-5 most frequent features appearing in the top-1, top-2 and top-3 $\epsilon$-transformations ($\epsilon=0.05$).}
\label{fig:top-1}
\label{fig:top-2}
\label{fig:top-3}
\label{fig:top-k}
\end{figure*}
We first present descriptive statistics on the recommendations obtained with our approach on a set of 100 low quality ad landing pages, the true negative instances in $\mathcal{D}_{\text{test}}$.
Each recommendation either suggests to \emph{increase} or \emph{decrease} the value of a given feature.
Overall, the recommendations are almost evenly distributed over the two cases above.
In Figure~\ref{fig:top-k}, we list the top-5 most frequent features recommended to be tweaked according to the top-1, top-2 and top-3 proposed $\epsilon$-transformations by measuring the relative frequency of each feature appearing among each of the $\epsilon$-transformations. The most frequent feature is {\small LINKS\-\_TEXT\-\_LENGTH\-\_TOTAL\-\_RATIO} in all settings, whi\-ch measures the ratio of text length to the total number of hyperlinks in the ad landing page.
Interestingly, \emph{all} the recommendations concerning this feature suggest to \emph{decrease} its value.
This indicates that low quality ad landing pages generally exhibit an unbalanced ratio of text to hyperlinks suggesting that saturating a page in links rather than content has negative effects on dwell time.
We measure the Pearson's correlation coefficient ($\rho$) between feature rankings appearing in the top-1, top-2 and top-3 $\epsilon$-trans\-for\-ma\-tions.
All three rankings are strongly related with each other, with top-1 reaching $\rho$ = 0.93 and 0.81 when compared to top-2 and top-3, respectively. Similarly, top-2 is highly correlated to top-3 ($\rho$ = 0.79).\footnote{\small{All values are statistically significant at $\alpha$ = 0.01}.}
We also compute the correlation coefficient between top-1 $\epsilon$-transformations for all values of $\epsilon$, and $\delta=\text{{\sf cosine\_distance}}$.
The top-1 rankings derived from $\epsilon$ = 0.05 and 0.1 are the highest correlated ($\rho$ = 0.92).
However, there is no statistical significant correlation between top-1 rankings when $\epsilon$ = 0.1 and 0.5, indicating that higher values of tolerance may impact more on the features requiring change.
We now perform a \emph{qualitative} and \emph{quantitative} assessment of such recommendations.
For each low quality landing page, we focus on its top-$k$ $\epsilon$-transformations, \emph{i.e.}, the $k$ less costly transformations according to the cost function $\delta$.
In turn, each transformation contains a list of recommendations, sorted by the feature rank they refer to.
We set the hyperparameters to $\epsilon=0.05$ and $\delta=$ {\sf cosine\_distance}, as this combination provides the best trade-off between ad coverage and average cost. We consider the top-3 $\epsilon$-transformations suggested for each ad landing page.
Around $91.0\%$ of landing pages can be associated with all three $\epsilon$-transformations, whereas our algorithm provides the remaining $7.5\%$ and $1.5\%$ with two and one $\epsilon$-trans\-for\-ma\-tion, respectively.
We also asked an internal team of creative strategists (CS)\footnote{\small{Creative strategists work with advertisers' web masters on strategic choices to help them developing effective advertising messages.}} to validate the recommendations generated by our approach.
Each CS was assigned a set of ad landing pages with the corresponding $\epsilon$-trans\-for\-ma\-tions, and additional metadata useful for assessing the recommendations within each transformation.
The same set of ad landing pages -- and therefore the same list of recommendations -- was assessed by two CSs, who were asked to rate each recommendation as \emph{helpful}, \emph{non-helpful}, or \emph{non-actionable}.
A recommendation is deemed \emph{helpful} when it is likely to help the advertiser to improve the user experience of the ad, and \emph{non-helpful} otherwise.
A \emph{non-actionable} recommendation is one that cannot be practically implemented.
Whenever a disagreement occurred, a third CS was called to resolve the conflict.
Overall, $57.3\%$ of all the generated recommendations are rated helpful with an inter-agreement rate of $60.4\%$ and only $0.4\%$ result in a non-actionable suggestion.
We also look at the $42.3\%$ non-helpful recommendations, and saw that about $25\%$ can be considered ``neutral''; that is, they would not hurt the user experience if discarded as well as not adding any positive value if implemented.
Non-helpful tweaks might occur due to two reasons. First, the learned model we leverage for generating feature recommendations -- no matter how accurate it is -- is not perfect; therefore, a true negative instance that is transformed into a positive prediction does not necessarily mean it is \emph{actually} positive.
Second, tuning the hyperparameters ($\delta$ and $\epsilon$) of our algorithm affects the set of candidate transformations.
As such, limiting non-helpful tweaks can be achieved by improving the accuracy of the learned model and choosing values of the hyperparameters to minimize errors.
Furthermore, when we further look into the non-actionable recommendations we see that these are related to the features {\small ADULT\_SCORE} and {\small NUM\_INPUT \_DROPDOWN}.
Our algorithm suggests to decrease the value of those features; however, the ad landing pages do not contain adult words nor drop-downs.
Most likely, the ad copies and landing pages used to generate recommendations have changed before the CSs performed their assessment.
Finally, we measure the ``helpfulness'' of each feature recommendation as follows:
\[
\text{{\sf helpfulness}}(i) = \frac{|\text{{\sf helpful}}(i)|}{|\text{{\sf helpful}}(i)|+|\lnot \text{{\sf helpful}}(i)|}
\]
This computes the relative frequency of recommendations for feature $i$ as being described as helpful by the CS team.
In Figure~\ref{fig:helpfulness}, we report the ranked list of features involved in the top-10 most helpfulness recommendations.
A similar ranking is obtained if we weight the helpfulness score on the basis of the \emph{overall} relative recommendation frequency.
The majority of the most helpful recommendations were features extracted from the DOM structure and content of the ad landing page, indicating that high quality landing pages should exhibit a good balance between textual content and hyperlinks.
Those features were the most predictive in our {\sf RF}\ ad quality model (Figure~\ref{fig:featimp}).
\begin{figure}
\caption{{\small Top-10 most helpful feature recommendations according to the {\sf helpfulness}
\label{fig:helpfulness}
\end{figure}
\section{Related Work}
\label{sec:related}
The research challenge addressed in this work is largely unexplored. Although machine learning has received a lot of attention in recent years, the focus has been mainly on the accuracy, efficiency, scalability, and robustness of the proposed various techniques.
Works on extracting actionable knowledge from machine-learned models have been mostly
conducted within the business and marketing domains.
Early works have focused on the development of interestingness metrics as proxy measures of knowledge actionability~\cite{hilderman2000pkdd, cao2007ijbidm}.
Another line of research on actionable knowledge discovery concerns post-processing techniques. Liu \emph{et al.}\ propose methods for pruning and summarizing learned rules, as well as matching rules by similarity~\cite{liu1996aaai, liu1999kdd}.
Cao \emph{et al.}\ present domain-driven data
mining; a paradigm shift from a research-centered discipline to a practical tool for actionable knowledge~\cite{cao2006pakdd, cao2010tkde}.
The authors discuss several frameworks for handling different problems and applications.
Many works discuss post-processing techniques specifically tailored to decision trees~\cite{yang2003icdm, karim2013jsea, yang2007tkde, du2011lori}. Yang \emph{et al.}\ study the problem of proposing actions to maximise the expected profit for a group of input instances based on a single decision tree, and introduce a greedy algorithm to approximately solve such a problem~\cite{yang2003icdm}.
This is significantly different
from our work; in fact, our work is more related to the one presented by Cui \emph{et al.}~\cite{cui2015kdd}.
Here, the authors propose a method to support actionability for additive tree models (ATMs), which is to find the set of actions that can change the prediction of an input instance to a desired status with the minimum cost. The authors formulate the problem as an instance of integer linear programming (ILP) and solve it using existing techniques.
Similarly to Cui \emph{et al.}, we also consider transforming the prediction for a given instance output by an ensemble of trees, and we introduce an algorithm that finds the \emph{exact} solution to the problem.
Our work differs from theirs in several aspects:
\emph{(i)} We tackle the theoretical intractability (NP-hardness) of the problem by designing an algorithm that creates a feedback loop with the original model to build a set of candidate transformations without the need, in practice, to explore the entire exponential search space;
\emph{(ii)} We introduce another hyperparameter ($\epsilon$) to govern the amount of change that each feature can sustain;
\emph{(iii)} We experiment with five concrete functions describing the cost of each transformation ($\delta$);
\emph{(iv)} We leverage on the importance of each feature derived from the model to rank the final list of recommendations;
\emph{(v)} We focus on the actual recommendations generated, and how they impact in practice on a real use case, if properly implemented.
More recent work on related topics are those of Ribeiro \emph{et al.}~\cite{Ribeiro:2016:WIT:2939672.2939778, ribeiro2016model}. In particular, \cite{Ribeiro:2016:WIT:2939672.2939778} presents LIME, a method that aims to explain the predictions of any classifier by learning an interpretable model that is specifically built around the predictions of interest.
They frame this task as a submodular optimization problem, which the authors solved using a well-known greedy algorithm achieving performance guarantees.
They test their algorithm on different models for text (\emph{e.g.}, random forests) and image classification (\emph{e.g.}, neural networks), and validate the utility of generated explanations both via simulated and human-assessed experiments.
\section{Conclusions}
\label{sec:conclusions}
Machine-learned models are often designed to favour accuracy of prediction at the expense of human-interpretability.
However, in some circumstances it becomes important to understand why the model returns a certain prediction on a given instance and how such an instance could be transformed in such a way that the model changes its original prediction. We investigate this problem within the context of general ensembles of tree-based classifiers, which has been proven to be NP-hard. We then introduce an algorithm that is able to transform a true negative instance into a set of new ``proposed'' positive instances by shifting their position in the feature space.
The algorithm leverages the internals of the learned ensemble to \emph{tweak} the feature-based representation of a true negative instance so that the new ``proposed'' ones are promoted to a positive classification when re-input to the classifier.
Despite computationally intractable in the worst case, we demonstrate the applicability of our approach on a real-world use case in online advertising. The feasibility of our approach has been achieved by \emph{(i)} setting an upper bound to the maximum number of changes affecting each instance (\emph{i.e.}, at most equivalent to the number of features), which can be controlled at training time, and by \emph{(ii)} creating a feedback loop with the original model to build a set of candidate transformations without the need, in practice, to explore the entire exponential search space.
After designing an effective Random Forest classifier able to separate between \emph{low} and \emph{high} quality ads -- our application scenario -- we automatically provide ``actionable'' suggestions on how to optimally convert a low quality ad (negative instance) into a high quality one (positive instance) using our approach. To illustrate the outcomes of our algorithm, we assess the quality of the recommendations that our method generates from a dataset of ads served by a large ad network, \emph{Yahoo Gemini}.
An evaluation conducted by an internal team of creative strategists shows that 57.3\% of the provided recommendations are indeed helpful, and likely to improve the ad quality, if implemented.
In future work, we plan to extend the approach presented in this work to multi-class setting as well as to other learning models, and to encapsulate it into a reinforcement learning framework.
\begin{acks}
The authors would like to thank Huw Evans, Mahlon Chute, and all the Yahoo's internal team of creative strategists for their invaluable contributions in evaluating the quality of the method presented in this work.
\end{acks}
\end{document} |
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\ShortArticleName{Non-Hermitian Quantum Mechanics with
Minimal Length Uncertainty}
\ArticleName{Non-Hermitian Quantum Mechanics\\ with
Minimal Length Uncertainty\fracootnote{This paper is a
contribution to the Proceedings of the 5-th Microconference
``Analytic and Algebraic Me\-thods~V''. The full collection is
available at
\href{http://www.emis.de/journals/SIGMA/Prague2009.html}{http://www.emis.de/journals/SIGMA/Prague2009.html}}}
\Author{T.K. JANA~$^\deltaag$ and P. ROY~$^\deltadag$}
\AuthorNameForHeading{T.K. Jana and P. Roy}
\Address{$^\deltaag$~Department of Mathematics, R.S. Mahavidyalaya, Ghatal 721212, India}
\EmailD{\href{mailto:[email protected]}{[email protected]}}
\Address{$^\deltadag$~Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India}
\EmailD{\href{mailto:[email protected]}{[email protected]}}
\ArticleDates{Received June 30, 2009, in f\/inal form August 10, 2009; Published online August 12, 2009}
\Abstract{We study non-Hermitian quantum mechanics in the presence of a minimal length. In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty. The spectrum in both the cases are found to be real. It is also shown that the models are $\eta$ pseudo-Hermitian and the metric operator is found explicitly in both the cases.}
\Keywords{non-Hermitian; minimal length}
\Classification{81Q05; 81S05}
\renewcommand{\arabic{footnote}}{\alpharabic{footnote}}
\sigmaetcounter{footnote}{0}
\sigmaection{Introduction}
In recent years there have been growing interest on quantum systems with a minimal length \cite{kempf1,kempf2,kempf3,kempf4}. There are quite a few reasons for this. For example, the concept of minimal length has found applications in quantum gravity \cite{garay}, perturbative string theory \cite{gross2}, black holes \cite{magg} etc. Exact as well as perturbative solutions of various non relativistic quantum mechanical systems, e.g., harmonic oscillator \cite{kempf2,kempf3,chang,dadic,gemba}, Coulomb problem \cite{brau,tk,yao,chang1}, Pauli equation \cite{nou} etc., have been obtained in the presence of minimal length. Exact solutions of relativistic models like the Dirac oscillator have also been obtained \cite{tk1,nou1}. A novel approach based on momentum space supersymmetry was also used to obtain exact solutions of a number of problems \cite{tk1,tk2,spector}.
On the other hand, since the work of Bender et al.~\cite{bender} non-Hermitian quantum systems have been studied extensively over the past few years\fracootnote{See \url{http://gemma.ujf.cas.cz/~znojil/conf/}.}.
Many of these models, especially the $\cal{PT}$ symmetric and the $\eta$ pseudo-hermitian ones admit real spectrum in spite of being non-Hermitian. Recently some possible applications of non-Hermitian quantum mechanics have also been suggested \cite{markis,moi}. However all these studies have been made in the context of point particles. Here our aim is to examine non-Hermitian quantum mechanics in the presence of a~minimal length. In particular we shall obtain exact solutions of a displaced harmonic oscillator with a complex coupling and the Swanson model~\cite{swanson}. It will be shown that in both the cases the spectrum is entirely real (subject to the parameters in the later case satisfying some constraints depending on the minimal length) and both the models are in fact $\eta$ pseudo-Hermitian. Explicit representation of the metric will also be obtained in both the cases. The organization of the paper is as follows. In Section~\ref{QM} we present a few results concerning quantum mechanics with minimal length uncertainty. In~Section~\ref{dho} we present exact solutions of the displaced harmonic oscillator problem. Section \ref{swanson} contains exact solutions of the Swanson model. In Section~\ref{pseudo} we discuss $\eta$ pseudo-Hermiticity of the models and finally Section~\ref{con} is devoted to a discussion.
\sigmaection{Quantum mechanics with minimal length uncertainty}\leftarrowbel{QM}
In one dimensional quantum mechanics with a minimal length the canonical commutation relation between $\hat{x}$ and $\hat{p}$ is modified and reads \cite{kempf2}
\betaegin{gather}
[{\hat x},{\hat p}] = i\hbar\betaig(1+\betaeta p^2\betaig),\leftarrowbel{cano}
\end{gather}
where $\beta$ is a small parameter. A representation of $\hat{x}$ and $\hat{p}$ which realizes (\ref{cano}) is given by \cite{kempf2}
\betaegin{gather}
{\hat x} = i\hbar\left[\betaig(1+\betaeta p^2\betaig)\frac{\piartial}{\piartial p}+\gammaamma p\right],\qquad {\hat p} = p. \leftarrowbel{rep1}
\end{gather}
From (\ref{cano}) and (\ref{rep1}) it can be shown that
\betaegin{gather}
\Deltaelta{\hat x}\Deltaelta {\hat p} \gammaeq \frac{\hbar}{2}\betaig[1+\betaeta (\Deltaelta {\hat p})^2\betaig],\leftarrowbel{uncer}
\end{gather}
where in obtaining (\ref{uncer}) we have taken $\leftarrowngle p\rightarrowngle =0$. Thus the standard Heisenberg uncertainty relation (corresponding to $\beta\rightarrow 0$) is modified and it follows that there is UV/IR mixing. Furthermore from (\ref{uncer}) it follows that there also exist a minimal length given by
\betaegin{gather*}
(\Deltaelta {\hat x})_{\muin} = \hbar \sigmaqrt{\betaeta}.
\end{gather*}
In the space where position $({\hat x})$ and momentum $({\hat p})$ are given by (\ref{rep1}) the associated scalar product is defined by
\betaegin{gather}
\leftarrowngle \pihi(p)|\pisi(p)\rightarrowngle = \int \frac{\pihi^*(p)\pisi(p)}{(1+\betaeta p^2)^{1-\frac{\gammaamma}{\betaeta}}}\, dp.\leftarrowbel{scalar1}
\end{gather}
\sigmaection{Non-Hermitian displaced harmonic oscillator}\leftarrowbel{dho}
The Schr\"odinger equation for the displaced oscillator is given by
\betaegin{gather}
H\pisi(p) = E\pisi(p),\qquad H = \frac{1}{2\muu}{\hat p}^2 + \frac{1}{2}\muu \omegaega^2{\hat x}^2 + i\leftarrowm{\hat x},\leftarrowbel{sch1}
\end{gather}
where $\leftarrowm$ is a real constant. Now using (\ref{scalar1}) it can be shown that
\betaegin{gather*}
H \nueq H^\deltaagger,
\end{gather*}
so that $H$ is non-Hermitian.
Then we use (\ref{rep1}) to write the Schr\"odinger equation (\ref{sch1}) in momentum space as
\betaegin{gather}
\left[-f(p) \frac{d^2}{dp^2} + g(p)\frac{d}{dp}+h(p)\right]\pisi(p) = \epsilon \pisi(p),\leftarrowbel{sch2}
\end{gather}
where $f(p)$, $g(p)$, $h(p)$ and $\epsilon$ are given by
\betaegin{gather}
f(p) = \betaig(1+\betaeta p^2\betaig)^2,\qquad
g(p) = -2\betaig(1+\betaeta p^2\betaig)\left[(\gammaamma+\betaeta)p+\frac{\leftarrowm}{\muu\hbar \omegaega^2}\right],\nuonumber\\
h(p) = \left[\frac{1}{{\hbar}^2\muu^2\omegaega^2}-\gammaamma(\betaeta+\gammaamma)\right]p^2-\frac{2\leftarrowm\gamma}{\hbar\muu\omega^2}p,\qquad
\epsilon = \frac{2E}{{\hbar}^2\muu\omegaega^2}+\gamma.\leftarrowbel{fgh}
\end{gather}
It is now necessary to solve equation~(\ref{sch2}). To this end we perform a simultaneous change of wave function as well as the independent variable:
\betaegin{gather}
\pisi(p) = \rho(p)\pihi(p), \qquad q = \int \frac{1}{\sigmaqrt{f(p)}}\, dp, \leftarrowbel{t}
\end{gather}
where
\betaegin{gather}
\rho(p) = e^{\int \chi(p)\,dp},\qquad \chi(p) = \frac{f^\pirimeime+2g}{4f}. \leftarrowbel{rho}
\end{gather}
Using the transformation (\ref{t}) we obtain from (\ref{sch2})
\betaegin{gather}
\left[-\frac{d^2}{dq^2} + V(q)\right]\pihi(q) = \epsilon\pihi(q), \leftarrowbel{sch3}
\end{gather}
where $V(q)$ is given by
\betaegin{gather*}
V(q) = \left[\frac{4g^2+3{f^\pirimeime}^2+8gf^\pirimeime}{16f}-\frac{f^{\pirimeime\pirimeime}}{4}-\frac{g^\pirimeime}{2} + h(p)\right]_q.
\end{gather*}
It is easy to see that (\ref{sch3}) is a standard Schr\"odinger equation in the variable $q$ and $V(q)$ is the corresponding potential. In the present case we obtain on using (\ref{fgh})
\betaegin{gather}
q = \frac{1}{\sigmaqrt{\betaeta}}\, {\rm tan}^{-1}\betaig(\sigmaqrt{\betaeta}p\betaig),\qquad -\frac{\pii}{2\sigmaqrt{\betaeta}}<q<\frac{\pii}{2\sigmaqrt{\betaeta}},
\nuonumber\\
V(q) = \frac{{\rm sec}^2(\sigmaqrt{\betaeta}q)}{{\hbar}^2\muu^2\omegaega^2\betaeta} +\frac{\leftarrowm^2}{{\hbar}^2\muu^2\omegaega^4}-\frac{1}{{\hbar}^2\muu^2\omegaega^2\beta}+\gammaamma. \leftarrowbel{pot1}
\end{gather}
The potential $V(q)$ given above is a standard solvable potential. The energy eigenvalues and the wave functions are given by \cite{khare}
\betaegin{gather*}
\epsilon_n = \betaig(A+n\sigmaqrt{\beta}\,\betaig)^2+\frac{\leftarrowm^2}{{\hbar}^2\muu^2\omegaega^4}-\frac{1}{{\hbar}^2\muu^2\omegaega^2\beta}+\gammaamma,\qquad n=0,1,2,\deltaots,\\
\pihi_n(q) = N_n \betaig[\cos \betaig(q\sigmaqrt{\beta}\,\betaig)\betaig]^{\frac{A}{\sigmaqrt{\beta}}} P_n^{\betaig(\frac{A}{\sigmaqrt{\beta}}-\frac{1}{2},\frac{A}{\sigmaqrt{\beta}}-\frac{1}{2}\betaig)}\betaig(\sigmain \betaig(q\sigmaqrt{\beta}\,\betaig)\betaig),\\
A = \frac{\sigmaqrt{\beta}+\sigmaqrt{\beta+\frac{4}{\hbar^2\muu^2\omegaega^2\beta}}}{2},
\end{gather*}
where $N_n$ are normalization constants and $P_n^{(r,s)}(z)$ denotes Jacobi polynomials.
So from (\ref{fgh}) and (\ref{t}) we finally obtain ($n=0,1,2,\deltaots$)
\betaegin{gather}
E_n = {\hbar}\omegaega\left[\frac{\betaeta\hbar\omegaega\muu}{2}\left(n^2+n+\frac{1}{2}\right)
+\left(n+\frac{1}{2}\right)\sigmaqrt{1+\frac{\betaeta^2\hbar^2\omegaega^2\muu^2}{4}}\right]+\frac{\leftarrowm^2}{2\muu\omegaega^2},\nuonumber
\\
\pisi_n(p) = N_n e^{-\frac{\leftarrowm \,{\rm tan}^{-1}(\sigmaqrt{\beta}p)}{\hbar \muu\omegaega^2\sigmaqrt{\beta}}}\left(1+\betaeta p^2\right)^{-\betaig(\frac{\gammaamma}{2\beta}+\frac{A}{\sigmaqrt{\beta}}\betaig)} P_n^{\betaig(\frac{A}{\sigmaqrt{\beta}}-\frac{1}{2},
\frac{A}{\sigmaqrt{\beta}}-\frac{1}{2}\betaig)}\left(\frac{\sigmaqrt{\beta}p}{1+\beta p^2}\right).\leftarrowbel{wf}
\end{gather}
Thus we find that the spectrum is completely real and for $\leftarrowm=0$ it reduces to the known re\-sults~\cite{kempf2,kempf3,dadic,gemba}.
\sigmaection{Swanson model}\leftarrowbel{swanson}
We now consider another type of model, namely, the Swanson model with the Hamiltonian given by \cite{swanson}
\betaegin{gather}
H = \omegaega a^\deltaagger a + \leftarrowm a^2 + \delta{a^\deltaagger}^2 + \frac{\omega}{2},\leftarrowbel{swan1}
\end{gather}
where $\leftarrowm\nueq \delta$ are real numbers and $a$, $a^\deltaagger$ are annihilation and creation operators of the standard harmonic oscillator. Although the above Hamiltonian involves no complex coupling it is non-Hermitian and has real eigenvalues provided $(\omega^2-4\leftarrowm\delta)>0$ \cite{swanson}.
We shall now obtain exact solutions of the Swanson model in the presence of a minimal length. In this case the operators $a$, $a^\deltaagger$ are defined exactly as in the standard case except that~${\hat x}$ and~${\hat p}$ are given by (\ref{rep1}):
\betaegin{gather*}
a = \frac{1}{\sigmaqrt{2m\hbar \omegaega}}\left({\hat p}-i\omegaega{\hat x}\right),\qquad a^\deltaagger = \frac{1}{\sigmaqrt{2m\hbar \omegaega}}\left({\hat p}+i\omegaega{\hat x}\right).
\end{gather*}
Now using (\ref{scalar1}) it can be shown that $H\nueq H^\deltaagger$ so that the Hamiltonian (\ref{swan1}) is non-Hermitian.
In order to obtain the spectrum we now write the eigenvalue equation $H\pisi(p)=E\pisi(p)$ in momentum space as
\betaegin{gather}
H\pisi(p) = \left[-f(p) \frac{d^2}{dp^2} + g(p)\frac{d}{dp}+h(p)\right]\pisi(p) = \epsilon \pisi(p),\leftarrowbel{swan2}
\end{gather}
where $f(p)$, $g(p)$, $h(p)$ and $\epsilon$ are now given by
\betaegin{gather*}
f(p) = \betaig(1+\betaeta p^2\betaig)^2,\\
g(p) = {-2\left[\frac{2(\deltaelta-\leftarrowmbda)}{\hbar m\omegaega(\omegaega-\leftarrowmbda-\deltaelta)}+2(\beta+\gammaamma)\right]\betaig(1+\betaeta p^2\betaig)p},\\
h(p) = {\left[ \frac{\omegaega+\leftarrowmbda+\deltaelta}{\omegaega-\leftarrowmbda-\deltaelta} \frac{1}{m^2\hbar^2\omegaega^2}
-\frac{2\gammaamma(\deltaelta-\leftarrowmbda)}{(\omegaega-\leftarrowmbda-\deltaelta)\hbar m\omegaega}-\gamma^2\right]p^2}\\
\pihantom{h(p) =}{} -\left[\frac{\delta-\leftarrowm+\omega}{\hbar m\omega(\omega-\leftarrowm-\delta)}+\gamma\right]\betaig(1+\beta p^2\betaig),\\
\epsilon ={\frac{1}{{\hbar}m(\omega-\leftarrowm-\delta)}\left(\frac{2E}{\omega}-1\right)}.
\end{gather*}
Now performing the transformation (\ref{t}) we obtain from (\ref{swan2})
\betaegin{gather*}
\left[-\frac{d^2}{dq^2}+V(q)\right]\pihi(q) = \epsilon\pihi(q),
\end{gather*}
where the potential is given by
\betaegin{gather}
V(q) = \nu \,{\rm sec}^2(\sigmaqrt{\beta}q) + \frac{4\leftarrowm\delta-\omega^2}{\hbar^2m^2\omega^2\beta(\omega-\delta-\leftarrowm)^2},\leftarrowbel{v2}
\\
\nuu = {\frac{\omega^2-4\leftarrowm\delta-\hbar m\omega^2\beta(\omega-\delta-\leftarrowm)}{\hbar^2m^2\omega^2\beta(\omega-\delta-\leftarrowm)^2}}.\nuonumber
\end{gather}
Now proceeding as before the energy eigenvalues and the eigenfunctions are found to be
\betaegin{gather}
E_n=\fracrac{\hbar m \omegaega \betaeta(\omegaega-\leftarrowmbda-\deltaelta)}{2}\left(n^2+n+\fracrac{1}{2}\right) +\left(n+\fracrac{1}{2}\right)\sigmaqrt{\left[\omegaega-\fracrac{\hbar m \omegaega \betaeta(\omegaega\!-\!\leftarrowmbda\!-\!\deltaelta)}{2}\right]^2\!-4\leftarrowmbda \deltaelta},\!\!\leftarrowbel{ener2}
\\
\pisi_n(p) = N_n\betaig(1+\betaeta p^2\betaig)^\kappa~P_n^{(s,s)}\left(\frac{\sigmaqrt{\beta}p}{1+\beta p^2}\right),\qquad n=0,1,2,\deltaots,\nuonumber
\end{gather}
where
\betaegin{gather*}
s = \fracrac{\sigmaqrt{1+\fracrac{4\nuu}{\betaeta}}}{2},\qquad
\kappa = \fracrac{\leftarrowmbda-\deltaelta}{2\hbar m \omegaega}(\omegaega-\leftarrowmbda-\deltaelta)-\fracrac{\gammaamma}{2\betaeta}-\fracrac{1+\sigmaqrt{1+\fracrac{4\nuu}{\betaeta}}}{2}.
\end{gather*}
From (\ref{ener2}) it follows that the energy is real provided
\betaegin{gather}
\left[\omegaega-\fracrac{\hbar m \omegaega \betaeta(\omegaega-\leftarrowmbda-\deltaelta)}{2}\right]^2-4\leftarrowm\delta>0\leftarrowbel{constraint1}
\end{gather}
and for $\beta=0$ we recover the standard Swanson model constraint mentioned earlier. From (\ref{constraint1}) it also follows that for given $\omegaega$, $\leftarrowm$, $\delta$ (such that $\omega-2\sigmaqrt{\leftarrowm\delta}>0$) there is a critical value $\beta_c$ such that for $\beta<\beta_c$ the energy is real. This value is given by
\betaegin{gather}
\beta_c = \frac{2(\omega-2\sigmaqrt{\leftarrowm\delta})}{m{\hbar}\omega(\omega-\leftarrowm-\delta)}.\leftarrowbel{constraint}
\end{gather}
Thus in this case apart from the standard Swanson model constraint, there is an additional constraint (\ref{constraint}) involving the minimal length parameter.
\sigmaection[$\eta$ pseudo-Hermiticity]{$\betaoldsymbol{\eta}$ pseudo-Hermiticity}\leftarrowbel{pseudo}
We recall that a Hamiltonian $H$ is called $\eta$ pseudo-Hermitian if it satisfies the condition \cite{mostafa}
\betaegin{gather*}
\eta H\eta^{-1} = H^\deltaagger,
\end{gather*}
where $\eta$ is a Hermitian operator. It may be noted that for $\eta$ pseudo-Hermitian systems the usual scalar product (\ref{scalar1}) can not be used since it may lead to a norm with f\/luctuating sign. The scalar product for such systems is def\/ined as
\betaegin{gather}
\leftarrowngle \pihi(p)|\pisi(p)\rightarrowngle_{\eta} = \leftarrowngle \pihi(p)|\eta\pisi(p)\rightarrowngle . \leftarrowbel{scalar2}
\end{gather}
Thus in the present case scalar product reads
\betaegin{gather*}
\leftarrowngle\pihi(p)|\pisi(p)\rightarrowngle_{\eta}=\int \frac{\eta\pihi^*(p)\pisi(p)}{(1+\betaeta p^2)^{1-\frac{\gammaamma}{\betaeta}}}\,dp.
\end{gather*}
Also $\eta$ pseudo-Hermitian systems are characterized by the fact that their spectrum is either completely real or the eigenvalues occur in complex conjugate pairs \cite{mostafa}. Since in both the models considered here the eigenvalues are real it is natural to look for $\eta$ pseudo-Hermiticity of the Hamiltonians (\ref{sch1}) and (\ref{swan1}).
Next we take the metric as
\betaegin{gather}
\eta = \betaig(1+\beta p^2\betaig)^{-\frac{\gamma}{\beta}}\exp\left[-\int(\chi+\chi^*)\,dp\right].\leftarrowbel{metric}
\end{gather}
Then using (\ref{fgh}) and (\ref{rho}) the metric for
the displaced oscillator is found to be
\betaegin{gather}
\eta_{ho} = \exp\left[\frac{2\leftarrowm}{\hbar\muu\sigmaqrt{\betaeta}\omegaega^2}{\,{\rm tan}^{-1}\betaig(\sigmaqrt{\betaeta}p\betaig)}\right].\leftarrowbel{eta1}
\end{gather}
Now it can be shown that (\ref{eta1}) satisf\/ies
\betaegin{gather}
\eta_{ho} H\eta_{ho}^{-1} = H^\deltaagger,\leftarrowbel{prop1}
\end{gather}
so that $H$ is $\eta$ pseudo-Hermitian. It can also be verif\/ied that the wave functions (\ref{wf}) are orthonormal with respect to the scalar product (\ref{scalar2}):
\betaegin{gather}
\leftarrowngle\pisi_m(p)|\pisi_n(p)\rightarrowngle_{\eta} = \deltaelta_{mn}.\leftarrowbel{prop2}
\end{gather}
Similarly using (\ref{metric}) the metric for the Swanson model can be found to be
\betaegin{gather*}
\eta_s = \left(1+\beta p^2\right)^{\frac{(\delta-\leftarrowm)}{\hbar m\omega\beta(\omega-\leftarrowm-\delta)}}.
\end{gather*}
It can be verif\/ied that the Swanson Hamiltonian (\ref{swan1}) satisf\/ies the relations (\ref{prop1}) and (\ref{prop2}). Thus the Swanson model is also $\eta$ pseudo-Hermitian.
\sigmaection{Discussion}\leftarrowbel{con}
In this paper we have obtained exact solutions of a couple of non-Hermitian models in a space admitting a minimal length. In the case of the displaced oscillator the spectrum is real irrespective of the coupling strength and for the Swanson model the spectrum is real subject to certain constraints on the parameters.
In this context we note that non-Hermiticity can also be introduced in a model by considering non-Hermitian coordinates, i.e.~${\hat x}^\deltaagger\nueq {\hat x}$. This may be achieved by replacing ${\hat x}\rightarrow {\hat X}={\hat x} + i\epsilon$ so that ${\hat X}\nueq {\hat X}^\deltaagger$. However this case reduces to the model considered in Section~\ref{dho} once the parameters $\epsilon$ and $\leftarrowm$ are suitably related. A second possibility is to consider replacing $\gamma$ by $i\gamma$ in (\ref{rep1}). With such a replacement the harmonic oscillator Hamiltonian becomes non-Hermitian although the spectrum will still remain real. We would now like to mention about the symmetry of the problems considered here. Since the transformation (\ref{t}) of the variable $p$ to the variable $q$ is invertible, it is expected that the symmetry of the original problem is the same as that of the corresponding Schr\"odinger one~\cite{kamran}. Since the underlying symmetry of the potentials~(\ref{pot1}) and~(\ref{v2}) is a nonlinear algebraic one~\cite{quesne} we expect that the original problems to have the same symmetry. We feel it would be interesting to investigate the symmetry structure of these types of models.
Finally in view of the fact that the representation of the position operator in higher dimension is non trivial we feel it would be interesting to examine non-Hermitian interactions in higher dimensions and also to examine solvability of Schr\"odinger equation with other types of non-Hermitian interactions.
\sigmaubsection*{Acknowledgments}
The authors would like to thank the referees for suggesting improvements.
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\end{document} |
\begin{document}
\begin{abstract}
The $q$-semicircular distribution is a probability law that interpolates between the Gaussian law
and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings
where $q$ follows the number of crossings, whereas for the free cumulants one has to restrict the
enumeration to connected matchings.
The purpose of this article is to describe combinatorial properties of the classical cumulants.
We show that like the free cumulants, they are obtained by an enumeration of connected matchings,
the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph.
The case $q=0$ of these cumulants was studied by Lassalle using symmetric functions and
hypergeometric series. We show that the underlying combinatorics is explained through the
theory of heaps, which is Viennot's geometric interpretation of the Cartier-Foata monoid.
This method also gives results for the classical cumulants of the free Poisson law.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Let us consider the sequence $\{m_n(q)\}_{n\geq0}$ defined by the generating function
\[
\sum_{n\geq 0} m_n(q) z^n =
\cfrac{1}{1 -
\cfrac{ [1]_q z^2}{1 -
\cfrac{ [2]_q z^2}{1 - \ddots
}}}
\]
where $[i]_q=\frac{1-q^i}{1-q}$. For example, $m_0(q)=m_2(q)=1$, $m_4(q)=2+q$, and the odd values are 0.
The generating function being a Stieltjes continued fraction, $m_n(q)$ is the $n$th moment of a symmetric
probability measure on $\mathbb{R}$ (at least when $0\leq q\leq 1$). An explicit formula for the density
$w(x)$ such that $m_n(q)=\int x^n w(x) {\rm d}x$ is given by Szegő~\cite{szego}:
\[
w(x) =
\begin{cases} \frac 1\pi \sqrt{1-q} \sin\theta \prod\limits_{n=1}^\infty (1-q^n) |1-q^ne^{2i\theta}|^2 &
\text{ if } -2\leq x\sqrt{1-q} \leq 2, \\ 0 & \text{otherwise,}
\end{cases}
\]
where $\theta\in[0,\pi]$ is such that $2 \cos \theta = x \sqrt{1-q}$.
At $q=0$, it is the semicircular distribution with density $(2\pi)^{-1}\sqrt{4-x^2}$ supported on $[-2,2]$,
whereas at the limit $q\to 1$ it becomes the Gaussian distribution with density $(2\pi)^{-1/2}e^{-x^2/2}$.
This law is therefore known either as the $q$-Gaussian or the $q$-semicircular law.
It can be conveniently characterized by its orthogonal polynomials, defined by the relation
$xH_n(x|q) = H_{n+1}(x|q) + [n]_q H_{n-1}(x|q)$ together with $H_1(x|q)=x$ and $H_0(x|q)=1$,
and called the continuous $q$-Hermite polynomials (but we do not insist on this point of view
since the notion of cumulant is not particularly relevant for orthogonal polynomials).
The semicircular law is the analogue in free probability of the Gaussian law \cite{hiai,nica}.
More generally, the $q$-semicircular measure plays an important role in noncommutative probability
theories \cite{anshelevitch,blitvic,bozejko1,bozejko2,leeuwen1,leeuwen2}. This was initiated by Bożejko
and Speicher \cite{bozejko1,bozejko2} who used creation and annihilation operators in a twisted
Fock space to build generalized Brownian motions.
The goal of this article is to examine the combinatorial meaning of the classical
cumulants $k_n(q)$ of the $q$-semicircular law (we recall the definition in the next section).
The first values lead to the observation that
\begin{equation*}
\tilde k_{2n}(q) = \frac{ k_{2n}(q) }{ (q-1)^{n-1} }
\end{equation*}
is a polynomial in $q$ with nonnegative coefficients. For example:
\begin{equation*}
\tilde k_2(q)=\tilde k_4(q)=1, \qquad \tilde k_6(q)=q+5, \qquad
\tilde k_8(q)= q^3+7q^2+28q+56.
\end{equation*}
We actually show in Theorem~\ref{cumultutte} that this $ \tilde k_{2n}(q)$ can be given a meaning
as a generating function of connected matchings, i.e. the same objects that give a
combinatorial meaning to the free cumulants of the $q$-semicircular law. However, the weight function
that we use here on connected matching is not as simple as in the case of free cumulants, it is
given by the value at $(1,q)$ of the Tutte polynomial of a graph attached to each connected matching,
called the crossing graph.
There are various points where the evaluation of a Tutte polynomials has combinatorial meaning,
in particular $(1,0)$, $(1,1)$ and $(1,2)$. In the first and third case ($q=0$ and $q=2$), they
can be used to give an alternative proof of Theorem~\ref{cumultutte}. These will be provided
respectively in Section~\ref{sec:heaps} and Section~\ref{sec:q=2}. The integers $\tilde k_{2n}(0)$
were recently considered by Lassalle \cite{lassalle} who defines them as a sequence simply related with
Catalan numbers, and further studied in \cite{vignat}.
Being the (classical) cumulants of the semicircular law, it might seem unnatural to
consider this quantity since this law belongs to the world of free probability, but on the other side
the free cumulants of the Gaussian have numerous properties (see \cite{belinschi}). The interesting
feature is that this particular case $q=0$ can be proved via the theory of heaps \cite{cartier,viennot}.
As for the case $q=2$, even though the $q$-semicircular is only defined when $|q|<1$ its moments and
cumulants and the link between still exist because \eqref{relmk} can be seen as an identity between formal
power series in $z$. The particular proof for $q=2$ is an application of the exponential formula.
\section{Preliminaries}
Let us first precise some terms used in the introduction. Besides the moments $\{m_n(q)\}_{n\geq0}$, the
$q$-semicircular law can be characterized by its {\it cumulants} $\{k_n(q)\}_{n\geq1}$ formally defined by
\begin{equation} \label{relmk}
\sum_{n\geq 1} k_n(q) \frac{z^n}{n!} = \log \Bigg( \sum_{n\geq 0} m_n(q) \frac{z^n}{n!} \Bigg),
\end{equation}
or by its {\it free cumulants} $\{c_n(q)\}_{n\geq1}$ \cite{nica} formally defined by
\[
C(zM(z)) = M(z)\quad \text{ where } M(z)=\sum_{n\geq0} m_n(q)z^n,\quad C(z) = 1+\sum_{n\geq1} c_n(q) z^n.
\]
These relations can be reformulated using set partitions.
For any finite set $V$, let $\mathcal{P}(V)$ denote the lattice of set partitions of $V$, and let
$\mathcal{P}(n)=\mathcal{P}(\{1,\dots,n\})$. We will denote by $\hat 1$ the
maximal element and by $\mu$ the Möbius function of these lattices, without mentioning $V$ explicitly.
Although we will not use it, let us mention that $\mu(\pi,\hat 1) = (-1)^{\# \pi -1} (\#\pi -1)!$ where $\#\pi$
is the number of blocks in $\pi$. See \cite[Chapter~3]{stanley} for details.
When we have some sequence $(u_n)_{n\geq 1}$, for any $\pi\in\mathcal{P}(V)$ we will use the notation:
\[
u_\pi = \prod_{ b \in \pi } u_{\# b}.
\]
Then the relations between moments and cumulants read:
\begin{equation} \label{inversion}
m_n(q) = \sum_{ \pi \in \mathcal{P}(n) } k_\pi(q), \qquad
k_n(q) = \sum_{ \pi \in \mathcal{P}(n) } m_\pi(q) \mu(\pi,\hat 1).
\end{equation}
These are equivalent via the Möbius inversion formula and both can be obtained from \eqref{relmk} using
Faà di Bruno's formula.
When $V\subset\mathbb{N}$, let $\mathcal{NC}(V)\subset\mathcal{P}(V)$ denote the subset of {\it noncrossing partitions},
which form a sublattice with Möbius function $\mu^{NC}$. Then we have \cite{hiai,nica}:
\begin{equation} \label{inversionfree}
m_n(q) = \sum_{ \pi \in \mathcal{NC}(n) } c_\pi(q), \qquad
c_n(q) = \sum_{ \pi \in \mathcal{NC}(n) } m_\pi(q) \mu^{NC}(\pi,\hat 1).
\end{equation}
Equations \eqref{inversion} and \eqref{inversionfree} can be used to compute the first non-zero values:
\begin{equation*}\begin{array}{lll}
k_2(q)=1,\qquad & k_4(q)=q-1, \qquad & k_6(q) =q^3+3q^2-9q+5, \\[2mm]
c_2(q)=1,\qquad & c_4(q)=q, \qquad & c_6(q) =q^3+3q^2.
\end{array}\end{equation*}
Let $\mathcal{M}(V)\subset\mathcal{P}(V)$ denote the set of {\it matchings}, i.e. set partitions whose all blocks
have size 2. As is customary, a block of $\sigma\in\mathcal{M}(V)$ will be called an {\it arch}. When
$V\subset\mathbb{N}$, a {\it crossing} \cite{ismail} of $\sigma\in\mathcal{M}(V)$ is a pair of arches $\{i,j\}$ and $\{k,\ell\}$
such that $i<k<j<\ell$. Let $\cro(\sigma)$ denote the number of crossings of $\sigma\in\mathcal{M}(V)$. Let
$\mathcal{N}(V) = \mathcal{M}(V) \cap \mathcal{NC}(V)$ denote the set of {\it noncrossing matchings},
i.e. those such that $\cro(\sigma)=0$.
Let also $\mathcal{M}(2n) = \mathcal{M}(\{1,\dots,2n\})$ and $\mathcal{N}(2n) = \mathcal{N}(\{1,\dots,2n\})$.
Let $\mathcal{P}^c(n) \subset \mathcal{P}(n)$ denote the set of {\it connected} set partitions, i.e. $\pi$ such
that no proper interval of $\{1,\dots,n\}$ is a union of blocks of $\pi$, and let
$\mathcal{M}^c(2n) = \mathcal{M}(2n) \cap \mathcal{P}^c(2n)$ denote the set of connected matchings.
It is known \cite{ismail} that for any $n\geq0$, the moment $m_{2n}(q)$ count matchings on $2n$
points according to the number of crossings:
\begin{equation} \label{mucro}
m_{2n}(q) = \sum_{\sigma\in\mathcal{M}(2n)} q^{\cro(\sigma)}.
\end{equation}
It was showed by Lehner~\cite{lehner} that \eqref{inversionfree} and \eqref{mucro} gives a combinatorial meaning for the
free cumulants:
\[
c_{2n}(q) = \sum_{\sigma\in\mathcal{M}^c(2n)} q^{\cro(\sigma)}.
\]
See \cite{belinschi} for various properties of connected matchings in the context of free
probability. Let us also mention that both quantities $m_{2n}(q)$ and $c_{2n}(q)$
are considered in an article by Touchard \cite{touchard}.
\section{\texorpdfstring{A combinatorial formula for $k_n(q)$}{A combinatorial formula for kn(q)} }
We will use the Möbius inversion formula in Equation~\eqref{inversion},
but we first need to consider the combinatorial meaning of the products $m_{\pi}(q)$.
\begin{lem} \label{lemmpi}
For any $\sigma\in\mathcal{M}(2n)$ and $\pi\in\mathcal{P}(2n)$ such that $\sigma \leq \pi$,
let $\cro(\sigma,\pi)$ be the number of crossings $(\{i,j\},\{k,\ell\})$ of $\sigma$ such that $\{i,j,k,\ell\}\subset b$
for some $b\in\pi$. Then we have:
\begin{equation} \label{mpi}
m_\pi(q) = \sum_{ \substack{ \sigma \in\mathcal{M}(2n) \\ \sigma \leq \pi} } q^{\cro(\sigma,\pi)}.
\end{equation}
\end{lem}
\begin{proof}
Denoting $\sigma|_b = \{ x\in\sigma \; : \; x\subset b \}$,
the map $\sigma \mapsto (\sigma|_b)_{b\in\pi}$ is a natural bijection between the set
$\{\sigma \in\mathcal{M}(2n) \; : \; \sigma\leq\pi \}$ and the product
$\Pi_{b\in\pi} \mathcal{M}(b) $, in such a way that $\cro(\sigma,\pi) = \sum_{b\in\pi} \cro (\sigma|_b)$.
This allows to factorize the right-hand side in \eqref{mpi} and obtain $m_{\pi}(q)$.
\end{proof}
From Equation~\eqref{inversion} and the previous lemma, we have:
\begin{equation} \label{kW} \begin{split}
k_{2n}(q) &= \sum_{ \pi \in \mathcal{P}(2n) } m_\pi(q) \mu(\pi,\hat 1)
= \sum_{ \pi \in \mathcal{P}(2n) } \sum_{\substack{ \sigma \in \mathcal{M}(2n) \\ \sigma \leq \pi}}
q^{\cro(\sigma,\pi)} \mu(\pi,\hat 1) \\
&= \sum_{ \sigma \in \mathcal{M}(2n) } \sum_{ \substack{ \pi\in\mathcal{P}(2n) \\ \pi \geq \sigma}}
q^{\cro(\sigma,\pi)} \mu(\pi,\hat 1) = \sum_{ \sigma \in \mathcal{M}(2n) } W(\sigma),
\end{split}\end{equation}
where for each $\sigma\in\mathcal{M}(2n)$ we have introduced:
\begin{equation} \label{W1}
W(\sigma) = \sum_{\substack{ \pi\in\mathcal{P}(2n) \\ \pi \geq \sigma}} q^{\cro(\sigma,\pi)} \mu(\pi,\hat 1).
\end{equation}
A key point is to note that $W(\sigma)$ only depends on how the arches of $\sigma$ cross
with respect to each other, which can be encoded in a graph. This leads to the following:
\begin{defn}
Let $\sigma\in\mathcal{M}(2n)$. The {\it crossing graph} $G(\sigma)=(V,E)$ is as follows.
The vertex set $V$ contains the arches of $\sigma$ (i.e. $V=\sigma$), and the edge set $E$
contains the crossings of $\sigma$ (i.e. there is an edge between the vertices $\{i,j\}$ and $\{k,\ell\}$
if and only if $i<k<j<\ell$).
\end{defn}
See Figure~\ref{crogra} for an example. Note that the graph $G(\sigma)$ is connected if and only if
$\sigma$ is a connected matching in the sense of the previous section.
\begin{figure}
\caption{A matching $\sigma$ and its crossing graph $G(\sigma)$. \label{crogra}
\label{crogra}
\end{figure}
\begin{lem} \label{Wgraph}
Let $\sigma\in\mathcal{M}(2n)$ and $G(\sigma)=(V,E)$ be its crossing graph.
If $\pi\in\mathcal{P}(V)$, let $i(E,\pi)$ be the number of elements in the edge set $E$
such that both endpoints are in the same block of $\pi$. Then we have:
\begin{equation} \label{W2}
W(\sigma) = \sum_{\pi \in \mathcal{P}(V)} q^{i(E,\pi)} \mu(\pi,\hat 1).
\end{equation}
\end{lem}
\begin{proof}
There is a natural bijection between the interval $[\sigma,\hat 1]$ in $\mathcal{P}(2n)$ and the set $\mathcal{P}(V)$,
in such a way that $\cro(\sigma,\pi) = i(E,\pi)$. Hence Equation~\eqref{W2} is just a rewriting of \eqref{W1} in terms
of the graph $G(\sigma)$.
\end{proof}
Now we can use Proposition~\ref{proptutte} from the next section. It allows to recognize $(q-1)^{-n+1}W(\sigma)$
as an evaluation of the Tutte polynomial $T_{G(\sigma)}$, except that it is 0 when the graph is not connected.
Gathering Equations~\eqref{kW}, \eqref{W2}, and Proposition~\ref{proptutte} from the next section,
we have proved:
\begin{thm} \label{cumultutte}
For any $n\geq 1$,
\[
\tilde k_{2n}(q) = \sum_{\sigma \in \mathcal{M}^c(2n)} T_{G(\sigma)} (1,q).
\]
In particular $\tilde k_{2n}(q)$ is a polynomial in $q$ with nonnegative coefficients.
\end{thm}
\section{The Tutte polynomial of a connected graph}
For any graph $G=(V,E)$, let $T_G(x,y)$ denote its Tutte polynomial, we give here a short definition
and refer to \cite[Chapter~9]{aigner} for details. This graph invariant can be computed recursively
via edge deletion and edge contraction. Let $e\in E$, let $G\backslash e = (V,E\backslash e)$ and
$G/e = ( V/e , E\backslash e)$ where $V/e$ is the quotient set where both endpoints of the edge $e$
are identified. Then the recursion is:
\begin{equation} \label{recurtutte}
T_G(x,y) =
\begin{cases}
xT_{G/e}(x,y) & \text{if $e$ is a bridge,} \\
yT_{G\backslash e}(x,y) & \text{if $e$ is a loop,} \\
T_{G/e}(x,y)+T_{G\backslash e}(x,y) & \text{otherwise.}
\end{cases}
\end{equation}
The initial case is that $T_G(x,y)=1$ if the graph $G$ has no edge.
Here, a {\it bridge} is an edge $e$ such that $G\backslash e$ has one more connected component than $G$,
and a {\it loop} is an edge whose both endpoints are identical.
\begin{prop} \label{proptutte}
Let $G=(V,E)$ be a graph (possibly with multiple edges and loops). Let $n=\#V$.
With $i(E,\pi)$ defined as in Lemma~\ref{Wgraph}, we have:
\begin{equation} \label{tutte}
\frac{1}{(q-1)^{n-1}} \sum_{\pi\in\mathcal{P}(V)} q^{i(E,\pi)} \mu(\pi,\hat 1) =
\begin{cases}
T_G(1,q) & \hbox{ if $G$ is connected,} \\
0 & \hbox{otherwise.}
\end{cases}
\end{equation}
\end{prop}
\begin{proof}
Denote by $U_G$ the left-hand side in \eqref{tutte} and let $e$ be an edge of $G$.
Suppose $e\in E$ is a loop, it is then clear that $i(E\backslash e,\pi)=i(E,\pi)-1$, so $U_G = qU_{G\backslash e}$.
Then suppose $e$ is not a loop, and let $x$ and $y$ be its endpoints. We have:
\[
U_G - U_{G\backslash e} =
\frac{1}{(q-1)^{n-1}} \sum_{\pi\in\mathcal{P}(V)} \Big(q^{i(E,\pi)} - q^{i(E\backslash e,\pi)} \Big) \mu(\pi,\hat 1).
\]
In this sum, all terms where $x$ and $y$ are in different blocks
of $\pi$ vanish. So we can keep only $\pi$ such that $x$ and $y$ are in the same block, and these can be
identified with elements of $\mathcal{P}(V/e)$ and satisfy $i(E\backslash e,\pi)=i(E,\pi)-1$. We obtain:
\[
U_G - U_{G\backslash e} = \frac{1}{(q-1)^{n-2}} \sum_{\pi\in\mathcal{P}(V/e)} q^{i(E\backslash e,\pi)} \mu(\pi,\hat 1) = U_{G/e}.
\]
This is a recurrence relation which determines $U_G$, and it remains to describe the initial case.
So, suppose the graph $G$ has $n$ vertices and no edge, i.e. $G=(V,\emptyset)$. We have
$i(\emptyset,\pi)=0$. By the definition of the Möbius function, we have:
\[
\sum_{\pi\in\mathcal{P}(V)} \mu(\pi,\hat 1) = \delta_{n1},
\]
hence $U_G=\delta_{n1}$ as well in this case.
We have thus a recurrence relation for $U_G$, and it remains to show that the right-hand side of \eqref{tutte}
satisfies the same relation. This is true because when $x=1$, and when we consider a variant of the Tutte
polynomial which is 0 for a non-connected graph, then the first case of \eqref{recurtutte} becomes a particular
case of the third case.
\end{proof}
\begin{rem}
The proposition of this section can also be derived from results of Burman and Shapiro \cite{burman}, at least in
the case where $G$ is connected. More precisely, in the light of \cite[Theorem~9]{burman} we can recognize
the sum in the left-hand side of \eqref{tutte} as the {\it external activity polynomial} $C_G(w)$, where all edge
variables are specialized to $q-1$. It is known to be related with $T_G(1,q)$, see for example
\cite[Section 2.5]{sokal}.
\end{rem}
\section{\texorpdfstring{The case $q=0$, Lassalle's sequence and heaps}{The case q=0, Lassalle's sequence and heaps}}
\label{sec:heaps}
In the case $q=0$, the substitution $z \to iz$ recasts Equation~\eqref{relmk} as
\begin{equation} \label{relmk2}
- \log\bigg( \sum_{n\geq 0} (-1)^n C_n \frac{z^{2n}}{(2n)!} \bigg) = \sum_{n\geq 1} \tilde k_{2n}(0) \frac{z^{2n}}{(2n)!},
\end{equation}
where $C_n = \frac{1}{n+1}\tbinom {2n}n $ is the $n$th Catalan number, known to be the cardinal of
$\mathcal{N}(2n)$, see \cite{stanley}. The integer sequence $\{\tilde k_{2n}(0)\}_{n\geq1}=(1,1,5,56,\dots)$
was previously defined by Lassalle \cite{lassalle} via an equation equivalent to \eqref{relmk2},
and Theorem 1 from \cite{lassalle} states that the integers $\tilde k_{2n}(0)$ are positive and increasing
(stronger results are also true, see \cite{lassalle,vignat}).
The goal of this section is to give a meaning to \eqref{relmk2} in the context of the theory of heaps \cite{viennot}
\cite[Appendix 3]{cartier}. This will give an alternative proof of Theorem~\ref{cumultutte} for the case $q=0$, based on
a classical result on the evaluation $T_G(1,0)$ of a Tutte polynomial in terms of some orientations of the graph $G$.
\begin{defn}
A graph $G=(V,E)$ is {\it rooted} when it has a distinguished vertex $ r \in V$, called the {\it root}.
An orientation of $G$ is {\it root-connected}, if for any vertex $v\in V$ there exists a directed path
from the root to $v$.
\end{defn}
\begin{prop}[Greene \& Zaslavsky \cite{greene}] \label{tutte10}
If $G$ is a rooted and connected graph, $T_G(1,0)$ is the number of its root-connected acyclic orientations.
\end{prop}
The notion of heap was introduced by Viennot \cite{viennot} as a geometric interpretation of elements in
the Cartier-Foata monoid \cite{cartier}, and has various applications in enumeration. We refer to
\cite[Appendix 3]{cartier} for a modern presentation of this subject (and comprehensive bibliography).
Let $M$ be the monoid built on the generators $(x_{ij})_{1\leq i < j}$ subject to the relations
$x_{ij}x_{k\ell} = x_{k\ell} x_{ij} $ if $i<j<k<\ell$ or $i<k<\ell<j$. We call it the Cartier-Foata monoid (but
in other contexts it could be called a partially commutative free monoid or a trace monoid as well).
Following \cite{viennot}, we call an element of $M$ a {\it heap}.
Any heap can be represented as a ``pile'' of segments, as in the left part of Figure~\ref{heapposet}
(this is remindful of \cite{bousquet}).
This pile is described inductively: the generator $x_{ij}$ correspond to a single segment whose extremities
have abscissas $i$ and $j$, and multiplication $m_1m_2$ is obtained by placing the pile of segments
corresponding to $m_2$ above the one corresponding to $m_1$. In terms of segments, the relation
$x_{ij}x_{k\ell} = x_{k\ell} x_{ij} $ if $i<j<k<\ell$ has a geometric interpretation: segments are allowed to move
vertically as long as they do not intersect (this is the case of $x_{34}$ and $x_{67}$ in Figure~\ref{heapposet}).
Similarly, the other relation $x_{ij}x_{k\ell} = x_{k\ell} x_{ij} $ if $i<k<\ell<j$ can be treated by thinking of
each segment as the projection of an arch as in the central part of Figure~\ref{heapposet}. In this three-dimensional
representation, all the commutation relations are translated in terms of arches that are allowed to move
along the dotted lines as long as they do not intersect.
A heap can also be represented as a poset. Consider two segments $s_1$ and $s_2$ in a pile of segments,
then the relation is defined by saying that $s_1<s_2$ if $s_1$ is always below $s_2$, after any movement of the
arches (along the dotted lines and as long as they do not intersect, as above).
This way, a heap can be identified with a poset where each element is labeled by a generator of $M$, and
two elements whose labels do not commute are comparable.
See the right part of Figure~\ref{heapposet} for an example and \cite[Appendice 3]{cartier} for details.
\begin{figure}
\caption{The heap $m=x_{46}
\label{heapposet}
\end{figure}
\begin{defn}
For any heap $m\in M$, let $|m|$ denote its length as a product of generators.
Moreover, $m\in M$ is called a {\it trivial heap} if it is a product of pairwise commuting generators.
Let $M^\circ\subset M $ denote the set of trivial heaps.
\end{defn}
Let $\mathbb{Z}[[M]]$ denote the ring of formal power series in $M$, i.e. all formal sums
$\sum_{m\in M} \alpha_m m$ with multiplication induced by the one of $M$.
A fundamental result of Cartier and Foata \cite{cartier} is the identity in $\mathbb{Z}[[M]]$ as follows:
\begin{equation} \label{cartierfoata}
\bigg( \sum_{m \in M^\circ } (-1)^{|m|} m \bigg)^{-1} = \sum_{m\in M} m.
\end{equation}
Note that $M^\circ$ contains the neutral element of $M$ so that the sum in the left-hand side is invertible,
being a formal power series with constant term equal to 1.
\begin{defn} \label{defpyr}
An element $m\in M$ is called a {\it pyramid} if the associated poset has a unique maximal element.
Let $P\subset M$ denote the subset of pyramids.
\end{defn}
A fundamental result of the theory of heaps links the generating function of pyramids with the one of all
heaps \cite{cartier,viennot}. It essentially relies on the exponential formula for labeled combinatorial
objects, and reads:
\begin{equation} \label{logpyr1}
\log \bigg( \sum_{m\in M} m \bigg) =_{\text{comm}} \sum_{p \in P} \frac{1}{|p|} p,
\end{equation}
where the sign $=_{\text{comm}}$ means that the equality holds in any commutative
quotient of $\mathbb{Z}[[M]]$. Combining \eqref{cartierfoata} and \eqref{logpyr1}, we obtain:
\begin{equation} \label{logpyr2}
- \log \bigg( \sum_{m\in M^\circ} (-1)^{|m|} m \bigg) =_{\text{comm}} \sum_{p \in P} \frac{1}{|p|} p.
\end{equation}
Now, let us examine how to apply this general equality to the present case.
The following lemma is a direct consequence of the definitions, and permits
to identify trivial heaps with noncrossing matchings.
\begin{lem} \label{Phi}
The map
\begin{equation} \label{defphi}
\Phi : x_{i_1j_1} \cdots x_{i_nj_n} \mapsto \{\{i_1,j_1\},\dots,\{i_n,j_n\}\}
\end{equation}
defines a bijection between the set of trivial heaps $M^\circ$ and the disjoint union of $\mathcal{N}(V)$
where $V$ runs through the finite subsets (of even cardinal) of $\mathbb{N}_{>0}$.
\end{lem}
For a general heap $m\in M$, we can still define $\Phi(m)$ via \eqref{defphi} but it may not be a matching,
for example $\Phi(x_{1,2}x_{2,3}) = \{\{1,2\},\{2,3\}\}$. Let us first consider the case of $m\in M$ such that
$\Phi(m)$ is really a matching.
\begin{lem} \label{ac_or}
Let $\sigma\in\mathcal{M}(V)$ for some $V\subset \mathbb{N}_{>0}$. Then the heaps $m\in M$ such that
$\Phi(m)=\sigma$ are in bijection with acyclic orientations of $G(\sigma)$.
Thus, such a heap $m\in M$ can be identified with a pair $(\sigma,r)$ where $r$ is an acyclic orientation
of the graph $G(\sigma)$.
\end{lem}
\begin{proof}
An acyclic orientation $r$ on $G(\sigma)$ defines a partial order on $\sigma$ by saying that two arches $x$ and $y$
satisfy $x<y$ if there is a directed path from $y$ to $x$. In this partial order, two crossing arches are always comparable
since they are adjacent in $G(\sigma)$. We recover the description of heaps in terms of posets, as described above,
so each pair $(\sigma,r)$ corresponds to a heap $m\in M$ with $\Phi(m)=\sigma$.
\end{proof}
To treat the case of $m\in M$ such that $\Phi(m)$ is not a matching, such as $x_{12}x_{23}$,
we are led to introduce a set of commuting variables $(a_i)_{ i \geq 1}$ such that $a_i^2=0$, and consider the specialization
$x_{ij}\mapsto a_ia_j$ which defines a morphism of algebras $\omega : \mathbb{Z}[[M]] \to \mathbb{Z}[[a_1,a_2,\dots]] $.
This way, for any $m\in M$ we have either $\omega(m)=0$, or $\Phi(m) \in \mathcal{M}(V)$
for some $V\subset \mathbb{N}_{>0}$.
Let $m\in M$ such that $\omega(m)\neq0$. As seen in Lemma~\ref{ac_or}, it can be identified with the
pair $(\sigma,r)$ where $\sigma=\Phi(m)$, and $r$ is an acyclic orientation of $G(\sigma)$.
Then the condition defining pyramids is easily translated in terms of $(\sigma,r)$,
indeed we have $m\in P$ if and only if the acyclic orientation $r$ has a unique source
(where a {\it source} is a vertex having no ingoing arrows).
Under the specialization $\omega$, the generating function of trivial heaps is:
\begin{equation} \label{omega1}
\omega\bigg( \sum_{m \in M^\circ } (-1)^{|m|} m \bigg) = \sum_{n\geq 0} (-1)^n C_n e_{2n},
\end{equation}
where $e_{2n}$ is the $2n$th elementary symmetric functions in the $a_i$'s.
Indeed, let $V\subset \mathbb{N}_{>0}$ with $\# V = 2n$, then the coefficient of $\prod_{i\in V} a_i $
in the left-hand side of \eqref{omega1} is $(-1)^n \# \mathcal{N}(V)= (-1)^n C_n$, as can be seen
using Lemma~\ref{Phi}. In particular, it only depends on $n$ so that this generating function can be
expressed in terms of the $e_{2n}$. Moreover, since the variables $a_i$ have vanishing squares their
elementary symmetric functions satisfy
\[
e_{2n} = \frac{1}{(2n)!} e_1^{2n},
\]
so that the right-hand side of \eqref{omega1} is actually the exponential generating of the Catalan numbers
(evaluated at $e_1$). It remains to understand the meaning of taking the logarithm of the left-hand side of
\eqref{omega1} using pyramids and Equation~\eqref{logpyr2}.
Note that the relation $=_{\text{comm}}$ becomes a true equality after the specialization $x_{ij}\mapsto a_ia_j$.
So taking the image of \eqref{logpyr2} under $\omega$ and using \eqref{omega1}, this gives
\[
- \log\bigg( \sum_{n \geq 0} (-1)^n C_n e_{2n} \bigg)
= \sum_{p\in P} \frac{1}{|p|} \omega(p).
\]
The argument used to obtain \eqref{omega1} shows as well that the right-hand side of the previous equation
is
$\sum_{} \frac{x_n}n e_{2n}$
where $x_n=\#\{ p\in P \; : \; \omega(p)=a_1 \cdots a_{2n} \}$. So we have
\[
- \log\bigg( \sum_{n \geq 0} (-1)^n C_n e_{2n} \bigg)
= \sum_{n \geq 0} \frac{x_n}n e_{2n},
\]
and comparing this with \eqref{relmk2}, we obtain $ \tilde k_{2n} (0) = \frac {x_n}{n}$.
Clearly, a graph with an acyclic orientation always has a source, and it has a unique source
only when it is root-connected (for an appropriate root, viz. the source). So a pyramid
$p$ such that $\omega(p)\neq0$ can be identified with a pair $(\sigma,r)$ where $r$ is a
root-connected acyclic orientation of $G(\sigma)$. Then using Proposition~\ref{tutte10}, it follows that
\[
x_n = n \sum_{\sigma \in \mathcal{M}^c(2n) } T_{G(\sigma)}(1,0).
\]
Here, the factor $n$ in the right-hand side accounts for the $n$ possible choices of the source
in each graph $G(\sigma)$. Eventually, we obtain
\begin{equation} \label{cumultutte0}
\tilde k _{2n}(0) = \sum_{\sigma \in \mathcal{M}^c(2n) } T_{G(\sigma)} (1,0),
\end{equation}
i.e. we have proved the particular case $q=0$ of Theorem~\ref{cumultutte}.
Let us state again the result in an equivalent form. We can consider that if $\sigma\in\mathcal{M}(2n)$,
the graph $G(\sigma)$ has a canonical root which the arch containing 1. Then, Equation \eqref{cumultutte0}
gives a combinatorial model for the integers $\tilde k_{2n}(0)$:
\begin{thm}
The integer $\tilde k_{2n}(0)$ counts pairs $(\sigma,r)$ where $\sigma\in\mathcal{M}^c(2n)$, and $r$
is an acyclic orientation of $G(\sigma)$ whose unique source
is the arch of $\sigma$ containing 1.
\end{thm}
From this, it is possible to give a combinatorial proof that the integers $\tilde k_{2n}(0)$ are increasing,
as suggested by Lassalle \cite{lassalle} who gave an algebraic proof. Indeed, we can check that pairs
$(\sigma,r)$ where $\{1,3\}$ is an arch of $\sigma$ are in bijection with the same objects but of size one
less, hence $\tilde k_{2n}(0) \leq \tilde k_{2n+2}(0)$.
Before ending this section, note that the left-hand side of \eqref{relmk2} is $-\log( \frac 1z J_1(2z))$
where $J_1$ is the Bessel function of order 1. There are quite a few other cases where the combinatorics
of Bessel functions is related with the theory of heaps, see the articles of Fédou \cite{fedou1,fedou2},
Bousquet-Mélou and Viennot \cite{bousquet}.
\section{\texorpdfstring{The case $q=2$, the exponential formula}{The case q=2, the exponential formula}}
\label{sec:q=2}
The specialization at $(1,2)$ of a Tutte polynomial has combinatorial significance in terms of
connected spanning subgraphs (see \cite[Chapter 9]{aigner}), so it is natural to consider
the case $q=2$ of Theorem~\ref{cumultutte}. This case is particular because the factor $(q-1)^{n-1}$ disappears, so
that $\tilde k_{2n}(2) = k_{2n}(2)$. We can then interpret the logarithm in the sense of combinatorial species, by
showing that $\tilde k_{2n}(2)$ counts some {\it primitive} objects and $m_{2n}(2)$ counts {\it assemblies} of those,
just like permutations that are formed by assembling cycles (this is the exponential formula for labeled combinatorial
objects, see \cite[Chapter 3]{aigner}). What we obtain is another more direct proof of Theorem~\ref{cumultutte}, based
on an interpretation of $T_G(1,2)$ as follows.
\begin{prop}[Gioan \cite{gioan}] \label{propgioan}
If $G$ is a rooted and connected graph, $T_G(1,2)$ is the number of its root-connected orientations.
\end{prop}
This differs from the more traditional interpretation of $T_G(1,2)$ in terms of connected
spanning subgraphs mentioned above, but it is what naturally appears in this context.
\begin{defn}
Let $\mathcal{M}^+(2n)$ be the set of pairs $(\sigma,r)$ where $\sigma\in\mathcal{M}(2n)$ and $r$ is an
orientation of the graph $G(\sigma)$. Such a pair is called an {\it augmented matching}, and is depicted
with the convention that the arch $\{i,j\}$ lies above the arch $\{k,\ell\}$ if there is an oriented edge
$\{i,j\} \rightarrow \{k,\ell\}$, and behind it if there is an oriented edge $\{k,\ell \} \rightarrow \{i,j\}$
.
\end{defn}
See Figure~\ref{aug} for example. Clearly, $\#\mathcal{M}^+(2n) = m_{2n}(2)$. Indeed, each graph
$G(\sigma)=(V,E)$ has $2^{\# E}$ orientations, and $\# E= \cro(\sigma)$, so this follows from \eqref{mucro}.
\begin{figure}
\caption{An augmented matching $(\sigma,r)$ and the corresponding orientation of $G(\sigma)$. \label{aug}
\label{aug}
\end{figure}
Notice that if there is no directed cycle in the oriented graph $(G(\sigma),r)$, the augmented
matching $(\sigma,r)$ can be identified with a heap $m\in M$ as defined in the previous section.
The one in Figure~\ref{aug} would be $x_{3,5}x_{4,11}x_{10,12}x_{1,6}x_{7,9}x_{2,8}$.
Actually, the application of the exponential formula in the present section is quite reminiscent of
the link between heaps and pyramids as seen in the previous section.
\begin{defn}
Recall that each graph $G(\sigma)$ is rooted with the convention that the root is the arch containing 1.
Let $\mathcal{I}(2n) \subset \mathcal{M}^+(2n)$ be the set of augmented matchings $(\sigma,r)$ such that
$\sigma$ is connected and $r$ is a root-connected orientation of $G(\sigma)$. The elements of $\mathcal{I}(2n)$
are called {\it primitive} augmented matchings. For any $V\subset \mathbb{N}_{>0}$ with $\#V=2n$, we also
define the set $\mathcal{I}(V)$, with the same combinatorial description as $\mathcal{I}(2n)$ except that
matchings are based on the set $V$ instead of $\{1,\dots,2n\}$.
\end{defn}
Using Proposition~\ref{propgioan}, we have
\[
\# \mathcal{I}(2n) = \sum_{\sigma\in\mathcal{M}^c(2n)} T_{G(\sigma)}(1,2),
\]
so that the particular case $q=2$ of Theorem~\ref{cumultutte} is the equality $\# \mathcal{I}(2n) = k_{2n}(2)$.
To prove this from \eqref{relmk} and using the exponential formula, we have to see how an augmented
matching can be decomposed into an assembly of primitive ones, as stated in Proposition~\ref{propdecomp}
below. This decomposition thus proves the case $q=2$ of Theorem~\ref{cumultutte}.
Note also that the bijection given below is equivalent to the first identity in \eqref{inversion}.
\begin{prop} \label{propdecomp}
There is a bijection
\[
\mathcal{M}^+(2n) \longrightarrow \biguplus_{\pi\in\mathcal{P}(n)} \; \prod_{ V\in \pi } \mathcal{I}(V).
\]
\end{prop}
\begin{proof}
Let $(\sigma,r) \in \mathcal{M}^+(2n)$, the bijection is defined as follows. Consider the vertices of $G(\sigma)$ which are
accessible from the root. This set of vertices defines a matching on a subset $V_1\subset \{1,\dots,2n\}$. For example,
in the case in Figure~\ref{aug}, the root is $\{1,6\}$ and the only other accessible vertex is $\{2,8\}$, so $V_1=\{1,2,6,8\}$.
Together with the restriction of the orientation $r$ on this subset of vertices, this defines an augmented matching
$(\sigma_1,r_1)\in\mathcal{M}^+(V_1)$ which by construction is primitive. By repeating this operation on the set
$\{1,\dots,2n\}\backslash V_1$, we find $V_2\subset \{1,\dots,2n\}\backslash V_1$ and $(\sigma_2,r_2)\in\mathcal{I}(V_2)$,
and so on. See Figure~\ref{decomp} for the result, in the case of the augmented matching in Figure~\ref{aug}.
The inverse bijection is easily described. If $(\sigma_i,r_i)\in\mathcal{I}(V_i)$ for any $1\leq i\leq k$ where $\pi=\{V_1,\dots,V_k\}$,
let $\sigma=\sigma_1 \cup \dots \cup \sigma_k$, and the orientation $r$ of $G(\sigma)$ is as follows. Let $e$ be an edge of
$G(\sigma)$ and $x_1$, $x_2$ be its endpoints, with $x_1\in\sigma_{j_1}$ and $x_2\in\sigma_{j_2}$. If $j_1=j_2$, the edge
$e$ is oriented in accordance with the orientation $r_{j_1}=r_{j_2}$. Otherwise, say $j_1<j_2$, then the edge $e$ is oriented
in the direction $x_1 \leftarrow x_2$.
\end{proof}
\begin{figure}
\caption{Decomposition of an augmented matching into primitive ones. \label{decomp}
\label{decomp}
\end{figure}
\section{Cumulants of the free Poisson law}
The free Poisson law appears in free probability and random matrices theory, and can be characterized by the fact that
all free cumulants are equal to some $\lambda>0$, see \cite{nica}. It follows from \eqref{inversionfree} that its
moments $m_n(\lambda)$ count noncrossing partitions, and consequently the coefficients are given by the Narayana
numbers (see \cite{stanley}):
\[
m_n(\lambda)= \sum_{\pi\in\mathcal{NC}(n)} \lambda^{\# \pi} = \sum_{k=1}^n \frac {\lambda^k}{n} \binom{n}{k}\binom{n}{k-1}.
\]
The corresponding cumulants are as before defined by
\begin{equation} \label{defklam}
\sum_{n\geq1} k_n(\lambda) \frac{z^n}{n!} =\log \Bigg( \sum_{n\geq0} m_n(\lambda) \frac{z^n}{n!} \Bigg).
\end{equation}
For any set partition $\pi\in\mathcal{P}(V)$ for some $V\subset \mathbb{N}$, we can define a crossing graph $G(\pi)$,
whose vertices are the blocks of $\pi$, and there is an edge between $b,c\in\pi$ if $\{b,c\}$ is not a noncrossing
partition. Note that $\pi$ is connected if and only if the graph $G(\pi)$ is connected.
The two different proofs for the semicircular cumulants show as well the following:
\begin{thm} \label{cumulpoisson}
For any $n\geq1$, we have:
\[
k_n(\lambda) = - \sum_{\pi\in\mathcal{P}^c(n)} (-\lambda)^{\# \pi } T_{G(\pi)}(1,0).
\]
\end{thm}
Let us sketch the proofs. If $\pi\in\mathcal{P}(n)$, similar to Lemma~\ref{lemmpi} we have:
\[
m_{\pi}(\lambda) = \sum_{\substack{ \rho \in \mathcal{P}(n) \\ \rho \unlhd \pi } } \lambda^{\# \rho}
\]
where the relation $\rho \unlhd \pi $ means that $\rho \leq \pi $ and $\rho|_b$ is a noncrossing
partition for each $b\in\pi$. Indeed the map $\rho \mapsto (\rho|_b)_{b\in\pi}$ is a bijection
between $\{ \rho\in\mathcal{P}(n) : \rho \unlhd \pi \}$ and $\prod_{b\in\pi} \mathcal{NC}(b)$.
The same computation as in \eqref{kW} and \eqref{W1} gives
\begin{equation} \label{invpoisson}
k_n(\lambda) = \sum_{\pi\in\mathcal{P}(n)} m_{\pi}(\lambda) \mu(\pi,\hat 1)
= \sum_{ \substack{ \rho,\pi \in \mathcal{P}(n) \\ \rho \unlhd \pi } } \lambda^{ \# \rho } \mu(\pi,\hat 1)
= \sum_{\rho \in \mathcal{P}(n) } \lambda^{\# \rho} W(\rho),
\end{equation}
where
\[
W(\rho) = \sum_{ \substack{ \pi \in \mathcal{P}(n) \\ \rho \unlhd \pi } } \mu(\pi,\hat 1).
\]
Denoting $G(\rho)=(V,E)$ the crossing graph of $\rho$, the previous equality is rewritten
$W(\rho) = \sum \mu(\pi,\hat 1)$ where the sum is over $\pi\in\mathcal{P}(V)$ such that
for any $b\in\pi$, $e\in E$, the block $b$ does not contain both endpoints of the edge $e$.
Then the case $q=0$ of Proposition~\ref{proptutte} shows that
\[
W(\rho) = \begin{cases}
(-1)^{\# \rho +1 } T_{G(\rho)}(1,0) & \text{if } \rho\in\mathcal{P}^c(n), \\
0 & \text{otherwise.}
\end{cases}
\]
Together with \eqref{invpoisson}, this completes the first proof of Theorem~\ref{cumulpoisson}.
As for the second proof, we follow the outline of Section~\ref{sec:heaps}, but with another definition
for $M$, $M^\circ$, $P$ and $\omega$. Let $M$ be the monoid with generators
$(x_V)$ where $V$ runs through finite subsets of $\mathbb{N}_{>0}$, and with relations $x_Vx_W=x_Wx_V$ if
$\{V,W\}$ is a noncrossing partition. We also denote $M^\circ \subset M$ the corresponding set of trivial heaps,
i.e. products of pairwise commuting generators. The subset $P\subset M$ is characterized by Definition~\ref{defpyr}.
Now, we consider the morphism $\omega$ defined by
\[
\omega(x_V) = \lambda \prod_{i\in V} a_i.
\]
We have:
\begin{align*}
\omega \bigg( \sum_{m\in M^\circ} (-1)^{|m|} m \bigg)
&= \sum_{V} \sum_{\pi\in\mathcal{NC}(V)} (-1)^{\#\pi} \prod_{b\in \pi} \omega(x_b) \\
&= \sum_{V} \sum_{\pi\in\mathcal{NC}(V)} (-\lambda)^{\#\pi} \prod_{i\in V} a_i \\
&= \sum_{n\geq 0} m_n(-\lambda) e_{n}
= \sum_{n\geq 0} m_n(-\lambda) \frac{e_{1}^n}{n!}.
\end{align*}
We still understand that $V\subset \mathbb{N}_{>0}$ is finite,
$(a_i)_{i\geq 1}$ are commuting variables with vanishing squares, and $e_n$ is the $n$th
elementary symmetric function in the $a_i$'s.
Equation \eqref{logpyr2} is still valid as such with the new definition of $M^\circ$ and $P$,
and taking the image by $\omega$ gives:
\[
-\log \bigg( \sum_{n\geq 0} m_n(-\lambda) \frac{e_{1}^n}{n!} \bigg) = \sum_{p\in P} \frac{1}{|p|} \omega(p).
\]
Comparing with \eqref{defklam} and taking the coefficient of $e_1^n$, we get:
\[
-k_n(-\lambda) = \sum_{\substack{ p\in P \\ \omega(p) = a_1 \cdots a_n }} \frac{\lambda^{|p|}}{|p|} .
\]
Let $p\in P$ be such that $\omega(p) = a_1 \cdots a_n$. Following the idea in Lemma~\ref{ac_or},
we can write $p=x_{V_1}\cdots x_{V_k}$ where $V_1,\dots,V_k$ are the blocks of a set partition
$\pi\in\mathcal{P}(n)$, and $p$ is characterized by $\pi$ together with an acyclic orientation
of the graph $G(\pi)$ having a unique source. Following the idea at the end of Section~\ref{sec:heaps},
we thus complete the second proof of Theorem~\ref{cumulpoisson}.
\section{Final remarks}
It would be interesting to explain why the same combinatorial objects appear both for
$c_{2n}(q)$ and $k_{2n}(q)$. This suggests that there exists some quantity that interpolates between
the classical and free cumulants of the $q$-semicircular law, however, building a noncommutative
probability theory that encompasses the classical and free ones appear to be elusive (see \cite{leeuwen2}
for a precise statement). It means that building such an interpolation would rely not only on the
$q$-semicircular law and its moments, but on its realization as a noncommutative random variable.
This might be feasible using $q$-Fock spaces \cite{bozejko1,bozejko2} but is beyond the scope
of this article.
\section*{Acknowledgment}
This work was initiated during the trimester ``Bialgebras in Free Probability'' at the Erwin Schrödinger
Institute in Vienna. In particular I thank Franz Lehner, Michael Anshelevich and Natasha Blitvić for
their conversation.
\setlength{\parindent}{0pt}
\end{document} |
\begin{document}
\title{A generalisation of de la Vall\'{e}e-Poussin procedure to multivariate approximations}
\author{
Nadezda Sukhorukova, \\
Swinburne University of Technology, John St, Hawthorn VIC 3122,\\
Australia and Federation University Australia,\\
Postal address: PO Box 663. Ballarat VIC 3353\\
{[email protected]}
\and Julien Ugon, Federation University Australia,\\
Postal address: PO Box 663. Ballarat VIC 3353\\
{[email protected]}}
\maketitle
\abstract{The theory of Chebyshev approximation has been extensively studied. In most cases, the optimality conditions are based on the notion of alternance or alternating sequence (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical de la Vall\'{e}e-Poussin procedure. In this paper we demonstrate that under certain assumptions the classical de la Vall\'{e}e-Poussin procedure, developed for univariate polynomial approximation, can be extended to the case of multivariate approximation. The corresponding basis functions are not restricted to be monomials.
}
{\bf Keywords:} {Multivariate polynomial, Chebyshev approximation, de la Vall\'{e}e-Poussin procedure}
{\bf Subclass:} {41A10 \and 41A50 \and 41N10}
\section{Introduction}\label{sec:introduction}
The theory of Chebyshev approximation for univariate functions was developed in the late nineteenth (Chebyshev) and twentieth century (just to name a few \cite{nurnberger,rice67,Schumaker68}). Many papers are dedicated to polynomial and polynomial spline approximations, however, other types of functions (for example, trigonometric polynomials) have also been used. In most cases, the optimality conditions are based on the notion of alternance (that is, maximal deviation points with alternating deviation signs).
There have been several attempts to extend this theory to the case of multivariate functions. One of them is \cite{rice63}. The main obstacle in extending these results to the case of multivariate functions is that it is not very easy to extend the notion of monotonicity to the case of several variables.
The main contribution of this paper is the extention of the classical de la Vall\'{e}e-Poussin procedure (originally developed for univariate polynomial approximation \cite{valleepoussin:1911}) to the case of multivariate approximation under certain assumptions. The corresponding basis functions are not restricted to be monomials (that is, non-polynomial approximation).
The paper is organised as follows. In section~\ref{sec:convexObjective} we demonstrate that the corresponding optimisation problems are convex. Then, in section~\ref{sec:VPprocedure} we extend the classical de la Vall\'{e}e-Poussin procedure to the case of multivariate approximation. Finally, section~\ref{sec:conclusion} highlights our future research directions.
\section{Convexity of the objective function}\label{sec:convexObjective}
Let us now formulate the objective function. Suppose that a continuous function $f(\mathbf{x})$ is to be approximated by a function
\begin{equation}\label{eq:model_function}
L(\mathbf{A},\mathbf{x})=a_0+\sum_{i=1}^{n}a_ig_i(\mathbf{x}),
\end{equation}
where $L(\mathbf{A},\mathbf{x})$ is a modelling function, $g_i(\mathbf{x}),~i=1,\dots,n$ are the basis functions and the multipliers $\mathbf{A} = (a_0,a_1,\dots,a_n)$ are the corresponding coefficients. In the case of polynomial approximation, basis functions are monomials. In this paper, however, we do not restrict ourselves to polynomials. At a point \(\mathbf{x}\) the deviation between the function \(f\) (also referred as approximation function) and the approximation is:
\begin{equation}
d(\mathbf{A},\mathbf{x}) = |f(\mathbf{x}) - L(\mathbf{A},\mathbf{x})|.
\end{equation}
\label{eq:deviation}
Then we can define the uniform approximation error over the set \(Q\) by
\begin{equation}
\label{eq:uniformdeviation}
\Psi(\mathbf{A})=\sup_{\mathbf{x}\in Q} \max\{f(\mathbf{x})-a_0-\sum_{i=1}^{n}a_ig_i(\mathbf{x}),a_0+\sum_{i=1}^{n}a_ig_i(\mathbf{x})-f(\mathbf{x})\}.
\end{equation}
The approximation problem is
\begin{equation}\label{eq:obj_fun_con}
\mathrm{minimise~}\Psi(\mathbf{A}) \mathrm{~subject~to~} \mathbf{A}\in
\mathbb{R}^{n+1}.
\end{equation}
Since the function \(L(\mathbf{A},\mathbf{x})\) is linear in \(\mathbf{A}\), the approximation error function \(\Psi(\mathbf{A})\), as the supremum of affine functions, is convex. Furthermore, its subdifferential at a point~\(\mathbf{A}\) is trivially obtained using the gradients of the active affine functions in the supremum (see \cite{Zalinescu2002} for details):
\begin{equation}
\label{eq:subdifferentialObjective}
\partial \Psi(\mathbf{A}) = \mathrm{co}\left\{ \begin{pmatrix}
1\\
g_1(\mathbf{x})\\
g_2(\mathbf{x})\\
\vdots \\
g_n(\mathbf{x})
\end{pmatrix}: \mathbf{x} \in E_+(\mathbf{A}),-\begin{pmatrix}
1\\
g_1(\mathbf{x})\\
g_2(\mathbf{x})\\
\vdots \\
g_n(\mathbf{x})
\end{pmatrix}: \mathbf{x}\in E_-(\mathbf{A})\right\},
\end{equation}
where \(E_+(\mathbf{A})\) and \(E_-(\mathbf{A})\) are respectively the points of maximal positive and negative deviation (extreme points):
\begin{align*}
E^+(\mathbf{A}) &= \mathbf{B}ig\{\mathbf{x}\in Q: f(\mathbf{x})-L(\mathbf{A},\mathbf{x}) = \max_{\mathbf{y}\in Q} d(A,\mathbf{y})\mathbf{B}ig\},\\
E_- (\mathbf{A})&= \mathbf{B}ig\{\mathbf{x}\in Q: -f(\mathbf{x})+ L(\mathbf{A},\mathbf{x}) = \max_{\mathbf{y}\in Q} d(\mathbf{A},\mathbf{y})\mathbf{B}ig\}.
\end{align*}
Note that in the case of multivariate polynomial approximation, $g_i(\mathbf{x})$, $i=1,\dots,n$ are monomials.
Define by \(G^+\) and \(G^-\) the sets
\begin{align*}
G^+(\mathbf{A}) &= \mathbf{B}ig\{(1,g_1(\mathbf{x}),\dots,g_n(\mathbf{x}))^T: \mathbf{x}\in E^+(\mathbf{A})\mathbf{B}ig\}\\
G^-(\mathbf{A}) &= \mathbf{B}ig\{(1,g_1(\mathbf{x}),\dots,g_n(\mathbf{x}))^T: \mathbf{x}\in E^-(\mathbf{A})\mathbf{B}ig\}
\end{align*}
Assume that \(\mathrm{card}(E_+) + \mathrm{card}(E_-) = n+2\).
The following theorem holds. We present the proof for completeness.
\begin{theorem}\label{thm:main}(\cite{matrix})
$\mathbf{A}^*$ is an optimal solution to problem~(\ref{eq:obj_fun_con}) if and only if the convex hulls of the sets \(G^+(\mathbf{A}^*)\) and \(G^-(\mathbf{A}^*)\) intersect.
\end{theorem}
\begin{proof}
The vector \(\mathbf{A}^*\) is an optimal solution to the convex problem \eqref{eq:obj_fun_con} if and only if
\[
\mathbf{0}_{n+1} \in \partial \Psi(\mathbf{A}^*),
\]
where $\Psi$ is defined in \eqref{eq:uniformdeviation}.
Note that due to Carath\'eodory's theorem, $\mathbf{0}_{n+1}$ can be constructed as a convex combination of a finite number of points (one more than the dimension of the corresponding space). Since the dimension of the corresponding space is $n+1$, it can be done using at most $n+2$ points.
Assume that in this collection of $n+2$ points $k$ points ($h_i,~i=1,\dots,k$) are from~$G^+(\mathbf{A}^*)$ and $n+2-k$ ($h_i,~i=k+1,\dots,n+2$) points are from $G^-(\mathbf{A}^*)$. Note that $0<k<n+2$, since the first coordinate is either~1 or $-1$ and therefore $\mathbf{0}_{n+1}$ can only be formed by using both sets ($G^+(\mathbf{A}^*)$ and $-G^-(\mathbf{A}^*)$). Then
$$\mathbf{0}_{n+1}=\sum_{i=1}^{n+2}\alpha_ih_i,~0\leq\alpha\leq 1.$$
Let $0<\gamma=\sum_{i=1}^{k}\alpha_i$, then
$$\mathbf{0}_{n+1}=\sum_{i=1}^{n+2}\alpha_ih_i=\gamma\sum_{i=1}^{k}\frac{\alpha_i}{\gamma}h_i+(1-\gamma)\sum_{i=k+1}^{n+2}\frac{\alpha_i}{1-\gamma}h_i=\gamma h^+ +(1-\gamma)h^-,$$
where $h^+\in G^+(\mathbf{A}^*)$ and $h^-\in -G^-(\mathbf{A}^*)$. Therefore, it is enough to demonstrate that $\mathbf{0}_{n+1}$ is a convex combination of two vectors, one from $G^+(\mathbf{A}^*)$ and one from $-G^-(\mathbf{A}^*)$.
By the formulation of the subdifferential of \(\Psi\) given by \eqref{eq:subdifferentialObjective}, there exists a nonnegative number \(\gamma \leq 1\) and two vectors
\[
g^+ \in \mathrm{co}\left\{ \begin{pmatrix}
1\\
g_1(\mathbf{x})\\
g_2(\mathbf{x})\\
\vdots \\
g_n(\mathbf{x})
\end{pmatrix}: \mathbf{x} \in E^+(\mathbf{A}^*)\right\}, \]
and
\[
g^- \in \mathrm{co}\left\{ \begin{pmatrix}
1\\
g_1(\mathbf{x})\\
g_2(\mathbf{x})\\
\vdots \\
g_n(\mathbf{x})
\end{pmatrix}: \mathbf{x} \in E^-(\mathbf{A}^*)\right\}
\]
such that \(\mathbf{0} = \gamma g^+ - (1-\gamma) g^-\). Noticing that the first coordinates \(g^+_1 = g^-_1 = 1\), we see that \(\gamma = \frac{1}{2}\). This means that \(g^+ - g^- = 0\). This happens if and only if
\begin{equation}\label{eq:opt_main2}
\mathrm{co}\left\{
\left(
\begin{matrix}
1\\
g_1(\mathbf{x})\\
g_2(\mathbf{x})\\
\vdots
\\
g_n(\mathbf{x})\\
\end{matrix}
\right): \mathbf{x} \in E^+(\mathbf{A}^*)
\right
\}\cap
\mathrm{co}\left\{
\left(
\begin{matrix}
1\\
g_1(\mathbf{x})\\
g_2(\mathbf{x})\\
\vdots
\\
g_n(\mathbf{x})\\
\end{matrix}
\right): \mathbf{x} \in E^-(\mathbf{A}^*)
\right \}\ne\emptyset.
\end{equation}
As noted before, the first coordinates of all these vectors are the same, and therefore the theorem is true, since if $\gamma$ exceeds one, the solution where all the components are divided by $\gamma$ can be taken as the corresponding coefficients in the convex combination.
\end{proof}
\section{de la Vall\'{e}e-Poussin procedure for nonsingular basis}\label{sec:VPprocedure}
\underline{\partial}section{Definitions and existing results}
We start with necessary definitions from convex analysis.
\begin{definition}
The relative interior of a set $S$ (denoted by $\textrm{relint} (S)$) is defined as its interior within the affine hull of $S$.
That is,
$$
\textrm{relint}(S)= \{\textbf{x} \in S : \exists \varepsilon>0, B_\varepsilon(x)\cap \textrm{aff}(S)\underline{\partial}seteq S\},$$
where $B_\varepsilon(x)$ is a ball of radius $\varepsilon$ centred in $x$ and $\textrm{aff}(S)$ is the affine hull of $S$.
\end{definition}
A useful property of relative interiors of convex hulls of finite number of points is formulated in the following lemma.
\begin{lemma}
Any relative interior point of a convex combination of a finite number of points can be presented as a convex combination of all these points with strictly positive convex combination coefficients and vice versa.
\end{lemma}
In univariate case polynomial approximation, basis is an arbitrary collection of $n+2$ points, where $n$ is the number of monomials. What do we call basis in multivariate case? Based on necessary and sufficient optimality conditions (Theorem~\ref{thm:main}) the convex hulls built over positive and negative maximal deviation points should intersect. Is it always possible to partition $n+2$ points in to two subsets in such a way that the corresponding convex hulls are intersecting. The answer to this question is ``yes'', if $n\geq d$. The following theorem holds.
\begin{theorem}(Radon \cite{Radon1921})
Any set of $d+2$ points in $\mathbb{R}^d$ can be partitioned into two disjoint sets whose convex hulls intersect.
\end{theorem}
\begin{definition}
A point in the intersection of these convex hulls is called a Radon point of the set.
\end{definition}
In the rest of the paper we assume that $n\geq d$.
It will be demonstrated that it is not possible to extend de la Vall\'{e}e-Poussin procedure to multivariate approximations without imposing additional assumptions (non-singular basis). It may be possible that some (or all) of these assumptions can be removed if we restrict ourselves to a particular class of basis functions (for example, monomials). This research direction is out of scope of this paper.
\begin{definition}
Consider a set \(\mathcal{S}\) of \(n+2\) points partitioned into two sets, the sets \(\mathcal{Y}\) of points with positive deviation and \(\mathcal{Z}\) of points with negative deviation. These points are said to form a \emph{basis} if the convex hulls of \(\mathcal{Y}\) and \(\mathcal{Z}\) intersect. Furthermore, if the relative interiors of the convex hulls intersect and any $(n+1)$ point subset of this basis form an affine independent system then the basis is said to be \emph{non-singular}.
\end{definition}
\underline{\partial}section{de la Vall\'{e}e-Poussin procedure for multivariate approximations}
\underline{\partial}subsection{Classical univariate procedure}
The classical univariate de la Vall\'{e}e-Poussin procedure contains three steps.
\begin{enumerate}
\item For any basis ($n+2$ points) there exists a unique polynomial, such that the absolute deviation at the basis points is the same and the deviation sign is alternating. This polynomial is also called Chebyshev interpolation polynomial.
\item If there is a point (outside of the current basis), such that the absolute deviation at this point is higher than at the basis points then this point can be included in the basis by removing one of the current basis points and the deviation signs are deviating.
\item The absolute deviation of the new Chebyshev interpolating polynomial is at least as high as the absolute deviation for the original basis.
\end{enumerate}
In the rest of this section we extend the procedure for a non-singular basis.
\underline{\partial}subsection{Step one extension}
We start with constructing Chebyshev interpolation polynomials. The following theorem holds.
\begin{theorem}
Assume that a system of points $\mathbf{y}_i,~i=1,\dots,N_+$ and $\mathbf{z}_i,~i=1,\dots,N_-$ forms a non-singular basis. Then there exists a unique polynomial deviating from $f$ at the points $\mathbf{y}_i,~i=1,\dots,N_+$ and $\mathbf{z}_i,~i=1,\dots,N_-$ by the same value and the deviation signs are opposite for $\mathbf{y}_i$ and $\mathbf{z}_i$.
\end{theorem}
\begin{proof}
Consider the following linear system:
\begin{equation}\label{eq:main_system}
\left(
\begin{tabular}{ccc}
1&{$g(\mathbf{y}_1)$}&1\\
1&{$g(\mathbf{y}_2)$}&1\\
\vdots & \vdots & \vdots\\
1&{$g(\mathbf{y}_{N_+})$}&1\\
1&{$g(\mathbf{z}_1)$}&-1\\
1&{$g(\mathbf{z}_2)$}&-1\\
\vdots & \vdots & \vdots\\
1&{$g(\mathbf{z}_{N_-})$}&-1\\
\end{tabular}
\right)\left(
\begin{tabular}{c}
$\mathbf{A}$\\
$\sigma$\\
\end{tabular}
\right)
=\left(\
\begin{tabular}{c}
$f(\mathbf{y}_1)$\\
$f(\mathbf{y}_2)$\\
\vdots\\
$f(\mathbf{y}_{N_+})$\\
$f(\mathbf{z}_1)$\\
$f(\mathbf{z}_2)$\\
\vdots\\
$f(\mathbf{z}_{N_-})$\\
\end{tabular}
\right),
\end{equation}
where $\mathbf{A}$ represents the parameters of the polynomial, while $\sigma$ is the deviation. If $\sigma=0$, there exists a polynomial passing through the chosen points (interpolation).
Denote the system matrix in~(\ref{eq:main_system}) by $M$. Since the basis is non-singular, that is, the relative interiors of sets ${\cal{Y}}$ and ${\cal{Z}}$ are intersecting, there exist two sets of strictly positive coefficients $$\alpha_1,\dots,\alpha_{N_+}:~ \sum_{i=1}^{N_+}\alpha_i=1$$
and
$$\beta_1,\dots,\alpha_{N_-}:~ \sum_{i=1}^{N_-}\beta_i=1,$$
such that
\begin{equation}\label{eq:intersecting}
\sum_{i=1}^{N_+}\alpha_ig(\mathbf{y}_i)=\sum_{i=1}^{N_-}\beta_i g(\mathbf{z}_i).
\end{equation}
Multiply the first row of $M$ by the convex coefficient $\alpha_1$ from~(\ref{eq:intersecting}). For each remaining row of $M$ one can apply the following update:
\begin{itemize}
\item multiply by the corresponding convex coefficient and add all the rows that correspond to the vertices with the same deviation sign as the first row;
\item multiply by the corresponding convex coefficient and subtract all the rows that correspond to the vertices with the deviation sign opposite to the sign of the first row.
\end{itemize}
Then
\begin{equation}\label{eq:det_tilde_M}
\alpha_l\det(\tilde{M})=2(-1)^{l+2+i}\det(M^+_l),\;~l=1,\dots, N_{+},
\end{equation}
where $M^+_i$ is obtained from~$\tilde{M}$ by removing the last column and the $i-$th row and $M^-_j$ is obtained from~$\tilde{M}$ by removing the last column and the $(N_{+}+j)$-th row. Also note that
\begin{equation}
\det(M^+_i)=2(-1)^{l+2+N_{+}+j+1}\det(M^-_j),~l=1,\dots, N_{+}.
\end{equation}
If now we evaluate the the determinant of~$M$ directly, then
\begin{equation}\label{eq:directly_det_M}
\det M=\sum_{i=1}^{N_+}(-1)^{l+2+i}\Delta_i+\sum_{j=N_{+}+1}^{N_{+}+N_{-}}(-1)^{l+2+j+1}\Delta_j.
\end{equation}
Based of~(\ref{eq:det_tilde_M}), each component in the right hand side of~(\ref{eq:directly_det_M}) has the same sign.
Therefore, the linear system~(\ref{eq:main_system}) has a unique solution for any right hand side of the system.
\end{proof}
Note that the division into ``positive'' and ``negative'' basis points does not mean that the deviation sign is positive for ``positive'' basis points and negative for ``negative'' basis points. The actual deviation sign also depends on the sign of $\sigma$ from~(\ref{eq:main_system}).
Extending the notion of Chebyshev interpolating polynomial to the case of multivariate approximation and not restricting ourselves to polynomials, define the following.
\begin{definition}
A modelling function $L(\mathbf{A},\mathbf{x})$ from~(\ref{eq:model_function}) that deviates at the basis points by the same absolute value from its approximation function and the deviation signs are opposite for any two points if they are selected from different basis subsets (positive or negative) is called Chebyshev interpolation modelling function.
\end{definition}
The additional requirement for a basis to be non-singular may be removed by
\begin{itemize}
\item restricting to some particular types of basis functions (for example, polynomials);
\item allowing the system~(\ref{eq:main_system}) to have more than one solution.
\end{itemize}
These will be included in our future research directions.
\underline{\partial}subsection{Step two extension}
Our next step is to demonstrate
\begin{theorem}
Consider two intersecting sets \(\mathcal{Y}\) and~\(\mathcal{Z}\) such that the points in \(\mathcal{Y}\) all have the same deviation and opposite deviation to all the points in \(\mathcal{Z}\) (\(g(\tilde{y}) = -g(\tilde{z}), \forall \tilde{y}\in \mathcal{Y}, \tilde{z}\in\mathcal{Z}\)). Assume now that $g(\mathbf{y}) = g(\tilde{y}), \forall \tilde{y} \in \mathcal{Y}$, and that the set
$$\mathcal{K}=\textrm{relint}(\{{\cal{Y}}\cup g(\mathbf{y})\})\cap\textrm{relint}({\cal{Z}}) \neq \emptyset.$$ There exists a point in the combined collection of vertices of~${\cal{Y}}$ and ${\cal{Z}}$, that can be removed while~$\mathbf{y}$ is included in~${\cal{Y}}$, such that the updated sets~${\cal{\tilde{Y}}}$ and ${\cal{\tilde{Z}}}$ intersect.
\end{theorem}
\begin{proof}
Since ${\textrm{relint}}({\cal{Y}})\cap\text{relint}({\cal{Z}})\ne \emptyset$, there exist strictly positive coefficients
$$\alpha_i,~ i=1,\dots,N_+$$ and $$\beta_i,~j=1,\dots,N_-,$$ such that $\sum_{i=1}^{N_+}\alpha_i=1$ and $\sum_{j=1}^{N_-}\beta_j=1$.
Since
$\mathcal{K}\ne\emptyset$ there exist strictly positive coefficients $$\alpha,~\tilde{\alpha}_i,~i=1,\dots,N_+$$ such that $\alpha+\sum_{i=1}^{N_+}\tilde{\alpha}_i=1$ and $\tilde{\beta}_i$, $j=1,\dots,N_-$, such that $$\sum_{j=1}^{N_-}\tilde{\beta}_j=1.$$
Find
\begin{equation}\label{eq:gamma}
\gamma=\min\left\{\min_{i=1,\dots,N_+}{\tilde{\alpha}_i\over \alpha_i},\min_{j=1,\dots,N_-}{\tilde{\beta}_j\over \beta_j}\right\}.
\end{equation}
First, assume that $\gamma={\tilde{\alpha}_1\over\alpha_1}$. Note that $\alpha_1\ne 0$,
then~(\ref{eq:intersecting}) can be written as
$$\mathbf{y}_1={1\over\alpha_1}\left(\sum_{j=1}^{N_-}\beta_jg(\mathbf{z}_j)-\sum_{i=2}^{N_+}\alpha_i g(\mathbf{y}_i)\right).$$
Then, the convex hull with the new point $\mathbf{y}$ is
\begin{equation}
\alpha g(\mathbf{y})+{\tilde{\alpha}_1\over\alpha_1}\left(\sum_{j=1}^{N_-}\beta_jg(\mathbf{z}_j)-\sum_{i=2}^{N_+}\alpha_ig(\mathbf{y}_i)\right)+\sum_{i=2}^{N_+}\tilde{\alpha_i g(\mathbf{y}_i)}=\sum_{j=1}^{N_-}\tilde{\beta}_jg(\mathbf{z}_j)
\end{equation}
and finally
\begin{equation}
\alpha g(\mathbf{y})+\sum_{i=2}^{N_+}(\tilde{\alpha}_i-{\tilde{\alpha}_i\over \alpha_i})g(\mathbf{y}_i)=\sum_{j=1}^{N_-}(\tilde{\beta}_j-{\tilde{\alpha}_1\over \alpha_1})g(\mathbf{z}_j).
\end{equation}
Since $\alpha_i>0$, $i=1,\dots,N_+$ and the definition of $\gamma$, one can obtain that for any $i=1,\dots,N_+$
\begin{equation}
\tilde{\alpha}_i-{\tilde{\alpha}_1\over \alpha_1}\geq \tilde{\alpha}_i-{\tilde{\alpha}_i\over \alpha_i}=0.
\end{equation}
Similarly, for any $j=1,\dots,N_-$, $$\tilde{\beta}_j-{\tilde{\alpha}_1\over\alpha_1}\beta_j\geq 0.$$
Note that
\begin{equation}
\sum_{j=1}^{N_-}(\tilde{\beta}_j-{\tilde{\alpha}_1\over \alpha_1}\beta_j)=1-{\tilde{\alpha}_1\over \alpha_1}
\end{equation}
and
\begin{equation}
\alpha+\sum_{i=2}^{N_+}\tilde{\alpha}_i-{\tilde{\alpha}_1\over\alpha_1}\sum_{i=2}^{N_+}\alpha_i=\alpha+(1-\alpha\tilde{\alpha}_1)-{\tilde{\alpha}_1\over\alpha_1}=1-{\tilde{\alpha}_1\over\alpha_1}=1-\gamma.
\end{equation}
Since $\alpha$ is strictly positive, $\gamma<1$. Therefore, the new point can be included instead of $\mathbf{y}_1$ and the convex hulls of the updated sets are intersecting (and so their relevant interiors).
Second, assume that $\gamma={\tilde{\beta}_1\over\beta_1}$. Note that $\beta_1\ne 0$, otherwise $\mathbf{y}$ can be included instead of $\mathbf{z}_1$.
Similarly to part~1, obtain
\begin{equation}
\alpha g(\mathbf{y})+\sum_{i=1}^{N_+}(\tilde{\alpha}_i-{\tilde{\beta}_1\over\beta_1}\alpha_i)g(\mathbf{y}_i)=\sum_{j=2}^{N_-}(\tilde{\beta}_j-{\tilde{\beta}_1\over\beta_1}\beta_j)g(\mathbf{z}_j).
\end{equation}
Since
$$\alpha+1-\alpha-{\tilde{\beta}_1\over\beta_1}=1-\tilde{\beta}_1-{\tilde{\beta}_1\over\beta_1}(1-\beta_1)=1-{\tilde{\beta}_1\over\beta_1}>0,$$
the convex hulls of the updated sets are intersecting.
\end{proof}
Note that for the extension of this step we only need the assumption that the relative interiors are intersecting, moreover, if this is the case, the new basis preserves this property.
\underline{\partial}subsection{Step three extension}
The final step is to show that the proposed exchange rule leads to a modelling function whose deviation at the new basis is strictly higher than the deviation at the points of the original basis.
\begin{theorem}
Assume that a point with a higher absolute deviation is included in the basis instead of one of the points of the original basis (which is also non-singular). The absolute deviation of the Chebyshev interpolation modelling function that corresponds to the new basis is higher than the one of the Chebyshev interpolation modelling function on the original basis.
\end{theorem}
\begin{proof}
Denote by \[\mathcal{Y} = \{\mathbf{y}_i,~i=1,\dots,N_+\}\] and \[\mathcal{Z} = \{\mathbf{z}_j,~j=1,\dots,N_-\}\] respectively. Assume that \(\tilde{\mathcal{Y}} = \mathcal{Y}\cup \{y\}\setminus \{y_1\}\) and \(\tilde{Z} = \mathcal{Z}\) (when the a point from the set \(\mathcal{Z}\) is removed instead, the proof is similar.)
Since the convex hulls of positive and negative deviation points are intersecting, there exist nonnegative convex coefficients
\begin{itemize}
\item $\alpha_1,\dots,\alpha_{N_+}: \sum_{i=1}^{N_+}\alpha_i=1$ and $\beta_1,\dots,\beta_{N_-}: \sum_{j=1}^{N_-}\beta_j=1$ (original basis);
\item $\alpha,~\tilde{\alpha}_2,\dots,\tilde{\alpha}_{N_+}: \alpha+\sum_{i=2}^{N_+}\tilde{\alpha}_i=1$ and $\beta_1,\dots,\beta_{N_-}: \sum_{j=1}^{N_-}\beta_j=1$ (new basis),
\end{itemize}
such that on the original basis
\begin{equation}\label{eq:convex_hulls_new_basis}
\sum_{i=1}^{N_+}\alpha_i\mathbf{y}_i-\sum_{j=1}^{N_-}\beta_j\mathbf{z}_j=\mathbf{0}
\end{equation}
and on the new basis
\begin{equation}\label{eq:convex_hulls_original_basis}
\alpha\mathbf{y}+\sum_{i=2}^{N_+}\tilde{\alpha}_i\mathbf{y}_i-\sum_{j=1}^{N_-}\tilde{\beta}_j\mathbf{z}_j=\mathbf{0}
\end{equation}
Systems~(\ref{eq:convex_hulls_new_basis}) is equivalent to
\begin{equation}
\left[\alpha,\tilde{\alpha}_2,\dots,\tilde{\alpha}_{N_+},\tilde{\beta}_1,\dots,\tilde{\beta}_{N_-}\right]\left[\begin{matrix}
\mathbf{y}\\
\mathbf{y}_2\\
\vdots\\
\mathbf{y}_{N_+}\\
\mathbf{z}_1\\
\vdots\\
\mathbf{z}_{N_-}
\end{matrix}
\right]=\mathbf{0}.
\end{equation}
Then
\begin{equation}
\left[\alpha,\tilde{\alpha}_2,\dots,\tilde{\alpha}_{N_+},\tilde{\beta}_1,\dots,\tilde{\beta}_{N_-}\right]\left[\begin{matrix}
1&\mathbf{y}\\
1&\mathbf{y}_2\\
\vdots&\vdots\\
1&\mathbf{y}_{N_+}\\
1&\mathbf{z}_1\\
\vdots&\vdots\\
1&\mathbf{z}_{N_-}
\end{matrix}
\right]\mathbf{A}=\mathbf{0}
\end{equation}
for any $\mathbf{A}\in\mathbb{R}^{n+1}$.
Let $\mathbf{A}_{o}$ and $\mathbf{A}_{new}$ be parameter coefficients of the Chebyshev interpolation modelling functions that correspond to the original and new basis respectively. Then
\begin{equation}\label{eq:orig_cheb_inter_pol}
\alpha P_n(\mathbf{A}_{o},\mathbf{y})+\sum_{i=2}^{N_+}\tilde{\alpha}_iP_n(\mathbf{A}_{o},\mathbf{y}_i)-\sum_{j=1}^{N_-}\tilde{\beta}_jP_n(\mathbf{A}_{o},\mathbf{z}_j)=0
\end{equation}
and
\begin{equation}\label{eq:new_cheb_inter_pol}
\alpha P_n(\mathbf{A}_{new},\mathbf{y})+\sum_{i=2}^{N_+}\tilde{\alpha}_iP_n(\mathbf{A}_{new},\mathbf{y}_i)-\sum_{j=1}^{N_-}\tilde{\beta}_jP_n(\mathbf{A}_{new},\mathbf{z}_j)=0.
\end{equation}
Assume that
\begin{equation}
f(\mathbf{y}_1)-P_n(\mathbf{A}_{new},\mathbf{y}_1)=\sigma_{new}>0.
\end{equation}
Then
\begin{equation}
\sigma_{new}+P_n(\mathbf{A}_{new},\mathbf{y})=f(\mathbf{y}),
\end{equation}
\begin{equation}
\sigma_{new}+P_n(\mathbf{A}_{new},\mathbf{y}_i)=f(\mathbf{y}_i),~i=2,\dots,N_+,
\end{equation}
and
\begin{equation}
-\sigma_{new}+P_n(\mathbf{A}_{new},\mathbf{z}_j)=f(\mathbf{z}_j),~j=2,\dots,N_-.
\end{equation}
Due to~(\ref{eq:orig_cheb_inter_pol})-(\ref{eq:new_cheb_inter_pol})
\begin{align*}
2\sigma_{new}=&\\
=&\alpha(f(\mathbf{y})-P_n(\mathbf{A}_{o},\mathbf{y})+\sum_{i=2}^{N_+}\tilde{\alpha_i}(f(\mathbf{y}_i)-P_n(\mathbf{A}_{o},\mathbf{y}_i))-\sum_{j=1}^{N_-}\tilde{\beta_j}(f(\mathbf{z}_j)-P_n(\mathbf{A}_{o},\mathbf{z}_i))\\
>&2\sigma_{o}.\\
\end{align*}
Therefore, $\sigma_{new}>\sigma_{o}.$
\end{proof}
Therefore, the notion of basis and de la Vall\'{e}e-Poussin procedure is extended to multidimensional functions. Also, it has been extended to any basis functions (not only traditional polynomials). If the newly obtained basis is non-singular, one can make another de la Vall\'{e}e-Poussin procedure step.
\section{Further research directions}\label{sec:conclusion}
We will extend the results to the case when the basis is singular. In order to do this, we need to remove two assumptions.
\begin{enumerate}
\item Any $(n+1)$ point subset of the basis ($n+2$ points) form an affine independent system.
\item Relative interiors of the convex hulls of positive and negative maximal deviation points (restricted to basis) are intersecting.
\end{enumerate}
The first assumption may not be removed for an arbitrary type of basis function. However, it may be possible to remove this assumption for some special types of functions (for example, polynomials). The removal of the second assumption may lead to dimension reduction. These will be included in our future research directions.
\end{document} |
\begin{enumerate}gin{document}
\begin{enumerate}gin{center}
\mathbf LeftarrowRGE{\bf Describing groups using first-order language}\\[10pt]
\langlerge{Yuki Maehara}\\
\langlerge{Supervisor: Andr\'e Nies}
\mathbf{e}nd{center}
ace{20pt}
\section{Introduction}
How can large groups be described efficiently?
Of course one can always use natural language, or give presentations to be more rigorous, but how about using formal language?
In this paper, we will investigate two notions concerning such descriptions; \mathbf{e}mph{quasi-finite axiomatizability}, concerning infinite groups, and \mathbf{e}mph{polylogarithmic compressibility}, concerning classes of finite groups.
An infinite group is said to be \mathbf{e}mph{quasi-finitely axiomatizable} if it can be described by a single first-order sentence, together with the information that the group is finitely generated (which is not first-order expressible).
In first-order language, the only parts of a sentence that can contain an infinite amount of information are the quantifiers; $\mathbf{e}xistsists$ and $\forall$.
Since the variables correspond to the elements in the language of groups, we can only ``talk'' about elements, but not, for example, all subgroups of a given group.
We give several examples of groups that can be described in this restricted language, with proofs intended to be understandable even to undergraduate students.
We say a class of finite groups is \mathbf{e}mph{polylogarithmically compressible} if each group in the class can be described by a first-order sentence, whose length is order polylogarithmic (i.e.~polynomial in $\mathtt{log}$) to the size of the group.
We need a restriction on the length of the sentence because each finite group can be described by a first-order sentence.
The most standard (and inefficient) way to do so is to describe the whole Cayley table, in which case the length of the sentence has order square of the size of the group.
The examples given in this paper include the class of finite simple groups (excluding a certain family), and the class of finite abelian groups.
\section{Preliminaries} \langlebel{preliminaries}
\subseteqsection{Homomorphisms} \langlebel{homomorphisms}
A surjective homomorphism is called an {\it epimorphism}.
A homomorphism from a structure into itself is called an {\it endomorphism}.
If $G$ is an abelian group, then endomorphisms of $G$ form a ring under usual addition and composition (see \cite[\S 5]{Kar.Mer:79}).
In particular, if $x$ (not necessarily in $G$) is such that conjugation by $x$ is an automorphism of $G$, and if $P(X)=\sum_{i \in I}\mathbf{a}lpha_i X^i$ is a polynomial over $\mathbb{Z}$ where $I$ is a finite subset of $\mathbb{Z}$, then for any element $g \in G$, we write
\[
g^{P(x)}=\sum_{i \in I} \mathbf{a}lpha_i g^{x^i}
\]
if $G$ is written additively, and similarly
\[
g^{P(x)}=\preceqod_{i \in I} (g^{x^i})^{\mathbf{a}lpha_i}
\]
if $G$ is written multiplicatively.
If $G$ is a group and $x \in G$, then the map defined by $g \rightarrowto x^{-1}gx$ is an automorphism of $G$ called {\it conjugation} by $x$.
We denote $x^{-1}gx$ by $g^x$, or $\mathtt{Conj}(g,x)$.
\subseteqsection{Presentations} \langlebel{presentations}
\mathbf{e}mph{Presentations} are a way of describing groups.
A group $G$ has a presentation
\[
P = \langlengle~X~|~R~\ranglengle
\]
where $X$ is the set of generators and $R$ is the set of relators written in terms of the generators in $X$, iff
\[
G \cong F(X)/N
\]
where $F(X)$ is the free group generated by $X$ and $N$ is the least normal subgroup containing $R$.
$G$ is said to be \mathbf{e}mph{finitely presented} (\mathbf{e}mph{f.p.}) if $X,R$ can be chosen to be finite.
\subseteqsection{Metabelian groups} \langlebel{metabelian}
For any groups $G,A,C$, one says $G = A \rtimes C$ ($G$ is a {\it semidirect product} of $A$ and $C$) if
\begin{enumerate}gin{center}
$AC=G$, $A \triangleleftngleleft G$, and $A\cap C= \{1\}$.
\mathbf{e}nd{center}
In particular, $G$ is said to be {\it metabelian} if $G = A \rtimes C$ for some abelian groups $A, C$. This is equivalent to saying that $G'$ is abelian.
\begin{enumerate}gin{lemma}[Nies \cite{Nies:03}] \langlebel{metabelian commutators}
Let $G = A \rtimes C$ where $A$ and $C$ are abelian.
Then, the commutator subgroup of $G$ is $G' = [A, C]$.
Moreover, if $C$ is generated by a single element $d$, then $G' = \{[u,d]~|~u \in A \}$.
In particular, $G'$ coincides with the set of commutators.
\mathbf{e}nd{lemma}
\begin{enumerate}gin{proof}
Let $u,v \in A$ and $x,y \in C$.
Then, $[ux,vy] = [u^x, y][x, v^y]$ using the commutator rule ${[ab,c] = [a,c]^b[b,c]}$ (\cite[3.2.(3)]{Kar.Mer:79}) and since $A$ and $C$ are abelian.
Since both $[u^x, y]$ and $[x, v^y]$ are in $[A,C]$, $G' = [A, C]$.
Now suppose $C = \langlengle d \ranglengle$.
We use additive notation in $A$.
Then, clearly $S = {\{[u,d] \ | \ u \in A \}}$ is a subset of $G' = [A, \langlengle d \ranglengle]$.
Since $[u,d][v,d]=[u+v,d]$ and $[u,d]^{-1}=[-u,d]$, $S$ forms a subgroup of $G$.
Also, since $[u,x^{-1}]=[-u^{x^{-1}},x]$ and ${[u, d^{n+1}]=[u^{d^n}+ \ldots +u^d+u,d]}$ for all $x \in C$, $S$~contains all commutators.
Hence $G'=S$.
\mathbf{e}nd{proof}
\subseteqsection{Finitely generated groups} \langlebel{f.g.}
One says a group $G$ is {\it finitely generated} ({\it f.g.}) if $G$ is generated by some finite subset.
\begin{enumerate}gin{lemma} \langlebel{f.g. factor}
Let $G$ be a f.g.\ group and let $N$ be a normal subgroup of $G$.
Then the factor group $G/N$ is also f.g.
\mathbf{e}nd{lemma}
\begin{enumerate}gin{proof}
Let $S = \{g_1, \ldots, g_n\}$ be a finite generating set of $G$.
Then, $N$ is generated by $SN = \{g_i N \ | \ 1 \le i \le n \}$, which is clearly finite.
\mathbf{e}nd{proof}
\begin{enumerate}gin{lemma}[Kargapolov and Merzljakov \cite{Kar.Mer:79}] \langlebel{f.g. abelian}
Each f.g.\ abelian group is a finite direct sum of cyclic groups.
\mathbf{e}nd{lemma}
\begin{enumerate}gin{lemma} \langlebel{prod conj}
If a group $G$ is generated by a subset $X \cup Y$ of $G$, then each element $g \in G$ can be written as a product
\begin{enumerate}gin{center}
$g = w \cdot \preceqod \mathtt{Conj}(x_i, y_i)$ \ where $x_i \in X$, \ $w, y_i \in \langlengle Y \ranglengle$.
\mathbf{e}nd{center}
In particular, if $A$ is a normal subgroup of $G$ such that $X \subseteqseteq A$ and $\langlengle Y \ranglengle \cap A$ is trivial, then each element $u \in A$ can be written as
\begin{enumerate}gin{center}
$u = \preceqod \mathtt{Conj}(x_i, y_i)$ \ where $x_i \in X$, \ $y_i \in \langlengle Y \ranglengle$.
\mathbf{e}nd{center}
\mathbf{e}nd{lemma}
\begin{enumerate}gin{proof}
Let $g \in G$.
Then since $X \cup Y$ generates $G$, it can be written as a product
\[
g = \preceqod_{j=1}^n v_j u_j
\]
for some positive integer $n$, where $u_j \in \langlengle X \ranglengle$, $v_j \in \langlengle Y \ranglengle$ for each $j$.
The lemma can be proven by induction on $n$.
\mathbf{e}nd{proof}
\subseteqsection{Rings} \langlebel{rings}
In this paper, we mean by a \textquotedblleft ring\textquotedblright \ an associative ring with the multiplicative identity $1 \noindenteq 0$.
An ideal of a ring is said to be {\it principal} if it is generated by a single element.
One says a ring $\mathcal{R}$ is {\it principal} if every ideal of $\mathcal{R}$ is principal.
Let $\mathcal{R}$ be a ring.
A non-zero element $u \in \mathcal{R}$ is called a {\it zero-divisor} if there exists a non-zero element $v \in \mathcal{R}$ such that $uv=0$ or $vu=0$.
$\mathcal{R}$ is said to be {\it entire} if it is commutative and does not contain any zero-divisors.
Let $\mathcal{R}$ be a commutative ring and let $S$ be a subset of $\mathcal{R}$ containing $1$, closed under multiplication.
Define a relation $\sim$ on the set ${\{(a,s) \ | \ a \in \mathcal{R}, \ s \in S\}}$ by
\begin{enumerate}gin{center}
$(a,s) \sim (a',s')$ iff there exists $s_1 \in S$ such that $s_1(s'a-sa')=0$,
\mathbf{e}nd{center}
then clearly $\sim$ is an equivalence relation.
Let $S^{-1}\mathcal{R}$ be the set of equivalence classes and let $a/s$ denote the equivalence class containing $(a,s)$.
Then $S^{-1}\mathcal{R}$ forms a ring under the following operations;
\begin{enumerate}gin{itemize}
\item addition is defined by $a/s + a'/s' = (s'a+sa')/ss'$
\item multiplication is defined by $(a/s) \cdot (a'/s') = aa'/ss'$
\mathbf{e}nd{itemize}
This ring is called the {\it ring of fractions} of $\mathcal{R}$ by $S$.
For the proofs that these operations are well-defined and $S^{-1}\mathcal{R}$ forms a ring, see \cite[\S 3]{Lang:84}.
\subseteqsection{Modules} \langlebel{modules}
Let $\mathcal{R}$ be a ring.
A {\it module} over $\mathcal{R}$, or an {\it $\mathcal{R}$-module} $M$, written additively, is an abelian group with multiplication by elements in $\mathcal{R}$ defined in such a way that for any $a,b \in \mathcal{R}$ and for any $x,y \in M$,
\begin{enumerate}gin{itemize}
\item $(a+b)x=ax+bx$ \ and
\item $a(x+y)=ax+ay$.
\mathbf{e}nd{itemize}
Modules are generalization of abelian groups in the sense that every abelian group is a $\mathbb{Z}$-module.
A {\it generating set} $S$ of an $\mathcal{R}$-module $M$ is a subset of $M$ such that every element of $M$ can be written as a sum of terms in the form $a_i s_i$ where $a_i \in \mathcal{R}$, $s_i \in S$.
A module is said to be {\it finitely generated} if it possesses a finite generating set.
One says an $\mathcal{R}$-module $M$ is {\it torsion-free} if for any $a \in \mathcal{R}$, $x \in M$, $ax=0$ implies $a=0$ or $x=0$.
An $\mathcal{R}$-module $M$ is {\it free} if it is isomorphic to $\begin{itemize}goplus_{i \in I} R_i$ for some finite index $I$, where each $R_i$ is isomorphic to $\mathcal{R}$ seen as a module over itself in the natural way.
\section{Quasi-finitely axiomatizable groups} \langlebel{QFA}
\begin{enumerate}gin{definition} \langlebel{QFA def}
An infinite f.g.\ group $H$ is \mathbf{e}mph{quasi-finitely axiomatizable (QFA)} if there exists a first-order sentence $\psi$ such that $H \models \psi$ and if $G$ is a f.g.\ group and $G \models \psi$, then $G \cong H$.
\mathbf{e}nd{definition}
The idea of quasi-finite axiomatizability was introduced by Nies in \cite{Nies:03}.
It was originally used to determine the expressiveness of first-order logic in group theory.
Later, it turned out to be interesting even from an algebraic point of view.
For example, Oger and Sabbagh \cite{Oger.Sabbagh:06} showed that if $G$ is a f.g.~nilpotent group, then $G$ is QFA iff each element $z \in Z(G)$ satisfies $z^n \in G'$ for some positive integer $n$.
In each of the proofs below, we give the sentence $\psi$, suppose a f.g.~group $G$ satisfies $\psi$ and show that $G$ must be isomorphic to $H$.
Since we ``do not know'' whether $G \cong H$ holds until the end of the proof, we want to talk about $G$ and $H$ separately.
So we refer to the group $H$ as the {\it standard case}, as opposed to the {\it general case} $G$.
\subseteqsection{Finitely presented groups} \langlebel{BS}
Our first example is Baumslag-Solitar groups, which are finitely presented (see below for presentations).
They are relatively easy to describe in first-order because the whole presentation can be a part of the sentence.
Although most of the QFA groups we give in this paper are described as semidirect products, this is the only case where we can define the action in first-order.
A Baumslag-Solitar group is a group with a presentation of the form
\[
\langlengle \ a,d \ | \ d^{-1}a^n d= a^m \ranglengle
\]
for some integers $m,n$.
We show that each Baumslag-Solitar group is QFA for the cases where $m \ge 2$ and $n=1$.
For each $m \ge 2$, define
\[
H_m = \langlengle~a,d \ | \ d^{-1}ad = a^m \ranglengle.
\]
Then $H_m$ is the semidirect product of $A = \mathbb{Z}[1/m] = \{z m^{-i} \ |\ z \in \mathbb{Z}, i \in {\mathbb{N}} \}$ by $C = \langlengle d \ranglengle$, where the action of $d$ on $A$ is given by $d^{-1}ud = um$ for $u \in A$.
\begin{enumerate}gin{theorem}[Nies \cite{Nies:07}] \langlebel{BS thm}
$H_m$ is QFA for each integer $m \ge 2$.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{proof}
We prove the theorem by actually giving a sentence $\mathbf{e}xistsists d \ \varphi_m(d)$ describing the group.
We list sufficiently many first-order properties (P0)-(P9) of $H_m$ and its element $d$ so that whenever a f.g.\ group $G$ has an element $d$ satisfying the conjunction $\varphi_m(d) \mathbf{e}nd{quotation}uiv {{\rm[(P0)} \wedge \ldots \wedge {\rm(P9)]}}$, we must have $G \cong H_m$. That is, \ $G = A \rtimes C$ where each of $A,C$ is isomorphic to that of the standard case, and $C$ is generated by $d$.
The first two are properties of $d$.
\begin{enumerate}gin{itemize}
\item[(P0)] $d \noindenteq 1$
\item[(P1)] $\forall g \, [g^i \noindenteq d]$, for each $i$, $1 < i \le m$.
\mathbf{e}nd{itemize}
The next five formulas define the subgroups $G', A, C$ and describe some of their properties.
Fix a prime $q$ that does not divide $m$.
\begin{enumerate}gin{itemize}
\item[(P2)] The commutators form a subgroup (so that $G'$ is definable)
\item[(P3)] $A=\{g \ | \ g^{m-1} \in G'\}$ and $C=C(d)$ are abelian, and $G = A \rtimes C$
\item[(P4)] $|C:C^2|=2$
\item[(P5)] $|A:A^q|=q$
\item[(P6)] The map $u \rightarrowto u^q$ is 1-1 in $A$.
\mathbf{e}nd{itemize}
We know (P2) holds in $H_m$ from Lemma \ref{metabelian commutators}.
(P4) can be expressed as \textquotedblleft there is an element which is not the square of any element, and for any three elements $x_1, x_2, x_3$, the formula $\mathbf{e}xistsists y \ [x_i = x_j y^2]$ is satisfied for some $1 \le i < j \le 3$\textquotedblright, and similar for (P5).
We show that (P3) actually defines $A$ in the standard case.
If $g \in \mathbb{Z}[1/m]$, then $[g,d]=g^{-1}d^{-1}gd=g^{-1}g^{m}=g^{m-1}$, so $g^{m-1} \in H'_m$.
Conversely, since $\mathbb{Z}[1/m]$ is closed under taking roots (because $H_m/\mathbb{Z}[1/m]$ is torsion-free), $g \noindentotin \mathbb{Z}[1/m]$ implies $g^{m-1} \noindentotin \mathbb{Z}[1/m]$.
Since $H_m/\mathbb{Z}[1/m]$ is abelian, $H'_m \le \mathbb{Z}[1/m]$ and so $g^{m-1} \noindentotin H'_m$.
The last three describe how $C$ acts on $A$.
\begin{enumerate}gin{itemize}
\item[(P7)] $\forall u \in A \ [ d^{-1}u d= u^m]$
\item[(P8)] $u^x \in A-\{1,u\}$ for $u \in A- \{1\}$, $x \in C -\{1\}$
\item[(P9)] $u^x \noindenteq u^{-1}$ for $u \in A-\{1\}$, $x \in C$
\mathbf{e}nd{itemize}
(P8) says that $C-\{1\}$ acts on $A-\{1\}$ without fixed points, and (P9) says that the orbit of $u \in A$ under $C$ does not contain $u^{-1}$, unless $u=1$.
Now let $G$ be a f.g.\ group and suppose $d \in G$ satisfies (P0)-(P9).
First, we show that the order of $d$ is infinite.
If $d^r = 1$ for some $r > 0$, then for each $u \in A$ we have $u = d^{-r}ud^r = u^{mr}$.
So $A$ is a periodic group of some exponent $k \le mr-1$.
If $q$ divides $k$, then there exists an element $v \in A$ of order $q$.
This makes the map $u \rightarrowto u^q$ not 1-1, contrary to (P6).
If $q$ does not divide $k$, then the map is an automorphism of $A$ and so $A^q=A$, contrary to (P5).
Let $\mathcal{R} = \mathbb{Z}[1/m]$ viewed as a ring.
Then $A$ can be seen as an $\mathcal{R}$-module by defining $u(zm^{-i}) = \mathtt{Conj}(u^z,d^{-i}) \ ( = u^{zm^{-i}}$, so well-defined) for $z \in \mathbb{Z}$, $i \in {\mathbb{N}}$.
Now we show that $A$ is f.g.\ and torsion-free as an $\mathcal{R}$-module.
Since $C \cong G/A$ and $G$ is f.g., $C$ is f.g.\ abelian by Lemma \ref{f.g. factor} and (P3).
So $C$ is a direct sum of cyclic groups by Lemma \ref{f.g. abelian}, and has only one infinite cyclic factor by (P4).
Since $d$ has infinite order, we can choose a generator $c \in C$ of this factor that satisfies $c^s = d$ for some $s \ge 1$.
Then, $C = \langlengle c \ranglengle \times F$ where $F = T(C)$ is the torsion subgroup of C.
Since $G = AC$, $G$~has a finite generating set of the form $B \cup \{ c \} \cup F$, where $B \subseteqseteq A$. We may assume $B$ is closed under taking inverse, and under conjugation by elements of the set $F \cup \{ c^i \ | \ 1 \le i < s\}$.
If $u \in A$, then $u$ can be written as a product of the terms $\mathtt{Conj}(b, xc^z)$ where $x \in F$, $z \in \mathbb{Z}$, $b \in B$, by Lemma \ref{prod conj}.
Hence by the closure properties of $B$, $u$ is a product of the terms $\mathtt{Conj}(b', d^w)$ where $b' \in B$ and $w \in \mathbb{Z}$.
This shows that $A$ is f.g.\ as an $\mathcal{R}$-module.
Suppose $u(zm^{-i}) = \mathtt{Conj}(u^z, d^{-i}) = 1$ for some $u \noindenteq 1$, $i \ge 0$, $z \noindenteq 0$.
Then $u^z = 1$ by (P8), so conjugation by $d$ is an automorphism of the finite subgroup $\langlengle u \ranglengle$ by (P7).
Hence some power of $d$ has a fixed point, contrary to (P8).
Therefore $A$ is torsion-free as an $\mathcal{R}$-module.
Since $\mathcal{R}$ is a principal entire ring, $A$ is a free $\mathcal{R}$-module by \cite[Thm.\ XV.2.2]{Lang:84}, so that $A$ as a group is isomorphic to $\begin{itemize}goplus_{1 \le i \le k} R_i$ for some positive integer $k$, where each $R_i$ is isomorphic to the additive group of $\mathcal{R}$.
But then $|A:A^q| = q^k$, so $k = 1$ by (P5).
Now we show that $F$ is trivial.
For, suppose $x \in F$, then the action of $x$ is an automorphism of $\mathbb{Z} [1/m]$ of finite order.
Note $\mathtt{Aut}(\mathbb{Z}[1/m])$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}_2$ where the first factor is generated by the map $u \rightarrowto um$ and the second by the map $u \rightarrowto -u$.
So the action of $x$ is either the identity, or inversion.
But we know that $x$ cannot be inversion by (P9), so $F = \{1\}$.
Recall that $s$ is the positive integer satisfying $c^s=d$.
Then $s \le m$ because the automorphism $u \rightarrowto um$ is not an $i$th power in $\mathtt{Aut}(\mathbb{Z} [1/m])$ for any $i > m$.
So $i = 1$ by (P1), meaning $\langlengle d \ranglengle = C$.
Hence $G \cong H_m$.
\mathbf{e}nd{proof}
(P1) was needed in the last part because $d$ might be a proper power of $c$.
For example, if $m=4$, then $\mathbb{Z}[1/4]=\mathbb{Z}[1/2]$ and the map $u \rightarrowto 4u$ is clearly the square of the map $u \rightarrowto 2u$, which is an automorphism of $\mathbb{Z}[1/4]$.
\subseteqsection{Non-finitely presented groups} \langlebel{wreath}
As mentioned in the last subsection, the groups in this example are also described as semidirect products, but in this case we cannot define the action explicitly.
Instead, we use a relationship between (definable) subgroups to restrict our possibilities.
The restricted wreath product $\mathbb{Z}_p \wr \mathbb{Z}$ is the semidirect product $H_p = A \rtimes C$ where $A = \begin{itemize}goplus_{z \in \mathbb{Z}} \mathbb{Z}_p^{(z)}$,
$\mathbb{Z}_p^{(z)}$ is a copy of $\mathbb{Z}_p$, $C = \langlengle d \ranglengle$ with $d$ of infinite order,
and $d$ acts on $A$ by shifting, i.e.\ $(\mathbb{Z}_p^{(z)})^d = \mathbb{Z}_p^{(z+1)}$.
It has a presentation
\begin{enumerate}gin{equation} \langlebel{presentation 1}
\langlengle~a,d \ | \ a^p, [v_r, v_s] (r,s \in \mathbb{Z}, r<s) \ranglengle
\mathbf{e}nd{equation}
where $a$ corresponds to a generator of $\mathbb{Z}_p^{(0)}$ and $v_r = a^{d^r}$.
\begin{enumerate}gin{theorem}[Nies \cite{Nies:03}] \langlebel{wreath thm}
$\mathbb{Z}_p \wr \mathbb{Z}$ is QFA for each prime $p$.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{proof}
The general idea of the proof is the same as that of the previous example.
We express the group as a semidirect product of its subgroups $A,C$, and then show that each of them is isomorphic to that in the standard case.
We also need to make sure that $C$ acts on $A$ correctly in this case, since the action of $d$ cannot be expressed in first-order.
Let $H_p = \mathbb{Z}_p \wr \mathbb{Z}$.
Then the sentence describing $H_p$ is $\mathbf{e}xistsists a \mathbf{e}xistsists d \ \varphi_p(a,d)$ where $\varphi_p(a,d) \mathbf{e}nd{quotation}uiv$ [(P0) $\wedge \ \ldots \ \wedge$ (P6)].
We use additive notation in $A$.
The first formula says that neither of $a,d$ is the identity.
Note both 0 and 1 refer to the identity here, since $a$ is in $A$ where we use additive notation, while $d$ is in $C$ where we use multiplicative notation.
\begin{enumerate}gin{itemize}
\item[(P0)] $a \noindenteq 0$, $p \cdot a = 0$, $d \noindenteq 1$.
\mathbf{e}nd{itemize}
The next five define the subgroups $G', A, C$ and describe some of their properties.
Note that $\langlengle a \ranglengle$ is first-order definable since it is finite by (P0) above.
\begin{enumerate}gin{itemize}
\item[(P1)] The commutators form a subgroup (so that $G'$ is definable)
\item[(P2)] $A = G' + \langlengle a \ranglengle = G' \oplus \langlengle a \ranglengle$ and $C = C(d)$ are abelian, and $G = A \rtimes C$
\item[(P3)] $|C:C^2|=2$
\item[(P4)] $\forall u \in A \ [p \cdot u=0]$
\item[(P5)] No element in $C-\{1\}$ has order $< p$.
\mathbf{e}nd{itemize}
The last thing we need to say is that $C-\{1\}$ acts on $A-\{0\}$ without fixed points.
\begin{enumerate}gin{itemize}
\item[(P6)] $u^x \in A-\{0, u\}$ for $u \in A-\{0\}$, $x \in C-\{1\}$.
\mathbf{e}nd{itemize}
First, we show that ${A = H'_p \oplus \langlengle a \ranglengle}$ holds in the standard case.
We know that it has the form ${H'_p = \{[u,d] \ | \ u \in A \}}$ from Lemma~\ref{metabelian commutators}, and so it is a subgroup of $A$.
Consider the group $\widetilde{H}_p$ with a presentation
\begin{enumerate}gin{equation} \langlebel{presentation 2}
\langlengle~\widetilde{a}, \widetilde{d} \ | \ {\widetilde{a}}^p, [\widetilde{a},\widetilde{d}]~\ranglengle.
\mathbf{e}nd{equation}
Since for each relator in (\ref{presentation 1}), a corresponding relator is in (\ref{presentation 2}), there exists an epimorphism $\Psi: H_p \rightarrow \widetilde{H}_p$ mapping $a, d$ to $\widetilde{a}, \widetilde{d}$ respectively.
As $\mathtt{Ker}(\Psi)$ is properly contained in $A$ and $\widetilde{H}_p$ is abelian, $H'_p$ is properly contained in $A$.
Now for each $z \noindenteq 0$, $\mathbb{Z}_p^{(z)}$ is generated by $a^{d^z}=a+[a,d^z]$, so $H'_p+\langlengle a \ranglengle=A$.
If $a^r \in H'_p$ for some $0<r<p$, then there exists $u \in A$ such that $a^r=[u,d]$ or equivalently $u+a^r=u^d$ i.e.\ $a^r$ shifts $u$, which is impossible.
Hence $H'_p \cap \langlengle a \ranglengle$ is trivial and so $A = H'_p \oplus \langlengle a \ranglengle$.
Now let $G$ be a f.g.\ group and suppose $a,d \in G$ satisfy (P0)-(P6).
We first prove that $C$ is infinite cyclic.
Since $C$ is f.g.\ abelian of torsion-free rank 1 by (P3), it suffices to show that $C$ is torsion-free by Lemma \ref{f.g. abelian}.
For, suppose $t \in C-\{1\}$ has finite order $r$.
Then every orbit in $A-\{0\}$ under the action of $t$ has size $r$, because if some orbit has size $s<r$, then $t^s \in C-\{1\}$ has a fixed point.
Let $A$ be viewed as a vector space over $\mathbb{Z}_p$ and let $U$ be the $t$-invariant subspace of $A$ generated by $a$.
Then $|U| = p^n$ for some $1 \le n \le r$, because $U = \left\{\sum_{0 \le i < r} m_i \cdot a^{t^i} \ | \ m_i \in \mathbb{Z}_p \right\}$.
But $|U-\{0\}| \ge p$ by (P5) and so $n > 1$.
Now consider the size of $G' \cap U$, which is also $t$-invariant because $G'$ is normal in~$G$.
Since $a$ is not in $G'$, $|U:G' \cap U| > 1$.
We also know that $|U:G' \cap U| \le |A:G'|=p$ \mbox{from \cite[Exercise 2.4.4]{Kar.Mer:79}}.
Hence the only possible size is $p^{n-1}$ since it must divide $|U|=p^n$.
As every orbit has size $r$ and the orbits partition each $t$-invariant subspace excluding the identity, $r$ divides $p^n-1$ and $p^{n-1}-1$.
But $(p^n-1)-p(p^{n-1}-1)=p-1$, so $r$ also divides $p-1$.
In particular, $r \le p-1$, contrary to (P5).
Choose a generator $c$ of $C$ and let $\mathcal{R}$ be the ring of fractions of $\mathbb{Z}_p[c]$ by the multiplicative subset $\{c^n \ | \ n \ge 0\}$.
Then $\mathcal{R}$ is a principal entire ring because the polynomial ring $\mathbb{Z}_p[c]$ is principal entire (see \cite[Section II.3 and Exercise 4]{Lang:84}).
Now, $A$ can be seen as an $\mathcal{R}$-module by defining $u \cdot P = \sum_{i=r}^s\mathbf{a}lpha_i u^{c^i}$ for $u \in A$, $P = \sum_{i=r}^s\mathbf{a}lpha_ic^i \in \mathcal{R}$.
We show that $A$ is f.g.\ and torsion-free as an $\mathcal{R}$-module.
Let $B=\{b_1,\ldots,b_m\}$ be a finite generating set of $G$.
Then, since each $b_i \in B$ can be written in the form $b_i = u_i c^{z_i}$ where $u_i \in A$, $z_i \in \mathbb{Z}$, the set $S \cup \{c\}$ also generates $G$ where $S = \{u_1,\ldots,u_m\}$.
Hence every element $u$ in $A$ can be written as a sum of the terms $\mathtt{Conj}(u_j, c^{z_j})$ where $u_j \in S$, $z_j \in \mathbb{Z}$ by Lemma \ref{prod conj}, meaning $A$ is f.g.\ as an $\mathcal{R}$-module.
Suppose $u \cdot P=0$ for some $u \in A - \{0\}$, $P = \sum_{i=r}^s\mathbf{a}lpha_ic^i \in \mathcal{R} - \{0\}$.
Then $P$ must consist of more than one term, for if $\mathbf{a}lpha u^{c^z} = 0$ for some $\mathbf{a}lpha \noindenteq 0$, then $u^{c^z} = 0$ by~(P4), contrary to (P6).
We can assume that the leading coefficient of $P$ is $-1$, so that $u^{c^s} = \sum_{i=r}^{s-1} \mathbf{a}lpha_i u^{c^i}$.
But then, for each $w \ge s$, $u^{c^w}$ is in the finite subspace of~$A$ generated by ${\{ u^{c^i} \ | \ r \le i \le s-1 \}}$.
\footnote{e.g.
\[
\begin{enumerate}gin{split}
u^{c^{s+1}}&={\sum_{i=r}^{s-1} \mathbf{a}lpha_i u^{c^{i+1}}}\\
&={\mathbf{a}lpha_{s-1} u^{c^s} + \sum_{i=r}^{s-2} \mathbf{a}lpha_i u^{c^{i+1}}}\\
&={\sum_{i=r}^{s-1} [\mathbf{a}lpha_{s-1} \mathbf{a}lpha_i + \mathbf{a}lpha_{i-1}]u^{c^i}}
\mathbf{e}nd{split}
\]
for the same $\mathbf{a}lpha_i$ as above except $\mathbf{a}lpha_{r-1}=0$.}
Hence the action of some power of $c$ has a fixed point, contrary to (P6).
This shows that $A$ is torsion-free as an $\mathcal{R}$-module.
Recall $\mathcal{R}$ is a principal entire ring. Since $A$ f.g.\ and torsion-free as an $\mathcal{R}$-module, it is a free module by \cite[Thm.\ XV.2.2]{Lang:84} so that $A$ as a group is isomorphic to $\begin{itemize}goplus_{1 \le i \le k} R_i$ for some positive integer $k$, where each $R_i$ is isomorphic to the additive group of $\mathcal{R}$.
Observe that $R_i \rtimes C \cong H_p$ for each $i$, where the action of $c$ on $R_i$ is defined by $P \rightarrowto P \cdot c$, and so $|R_i:[R_i,C]| = p$.
If $k >1$, then $|A:G'| = \left|\begin{itemize}goplus_i R_i:\left[\begin{itemize}goplus_i R_i,C\right]\right| > p$, contrary to (P2).
The last thing we need to show is that the action of $c$ on $A$ is correct.
To avoid confusion, here we denote by $d_H$, $A_H$ one of the generators and the normal subgroup of $H_p$ respectively.
Since $c$ has infinite order and each power of $c$ (except the identity) acts without fixed points, the action of $c$ on $A$ is equivalent to the action of $d_H^m$ on $A_H$ for some $m \ge 1$.
But if $m > 1$, then $A \noindentot \subseteqsetseteq G' \oplus \langlengle a \ranglengle$, contrary to~(P2).
\mathbf{e}nd{proof}
\subseteqsection{Semidirect products of f.g.~groups} \langlebel{Oger}
In \cite{Oger:06}, Oger gave examples of QFA groups, which are semidirect products of $\mathbb{Z}[u]$ and infinite cyclic $\langlengle u \ranglengle$ where $u$ is a complex number satisfying certain conditions.
Since both $\mathbb{Z}[u], \langlengle u \ranglengle$ are f.g.~abelian, we can talk about the rank of $\mathbb{Z}[u]$ as a free abelian group, making the proof fairly different from the previous examples.
Let $\mathcal{R}$ be a commutative ring.
An element $\mathbf{a}lpha$ of $\mathcal{R}$ is said to be {\it integral} over $\mathcal{R}$ if there exists a monic (i.e.\ the leading coefficient is 1) polynomial $P$ over $\mathcal{R}$ such that $P(\mathbf{a}lpha)=0$.
Let $\mathcal{S}$ be a commutative ring containing $\mathcal{R}$ as a subring.
Then, the elements of $\mathcal{S}$ integral over $\mathcal{R}$ form a subring of $\mathcal{S}$.
This ring is called the {\it integral closure} of $\mathcal{R}$ in $\mathcal{S}$ (see \cite[IX, \S1]{Lang:84}).
\begin{enumerate}gin{theorem}[Oger \cite{Oger:06}] \langlebel{Oger thm}
Let $u$ be a complex number such that
\begin{enumerate}gin{itemize}
\item $\mathbb{Z}[u]$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[u]$
\item the multiplicative group $(\mathbb{Z}[u]^*, \times)$ is infinite and generated by $u$ and $-1$.
\mathbf{e}nd{itemize}
Then there exists a first-order sentence $\psi$ which characterizes, among f.g.\ groups, those which are isomorphic to semidirect products $A \rtimes \langlengle u \ranglengle$, where $A$ is a non-zero ideal of $\mathbb{Z}[u]$, and the action of $u$ on $A$ is defined by $x \rightarrowto xu$.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{corollary}[\cite{Oger:06}] \langlebel{Oger col}
If $u$ satisfies the conditions above and $\mathbb{Z}[u]$ is principal, then $\mathbb{Z}[u] \rtimes \langlengle u \ranglengle$ is QFA.
\mathbf{e}nd{corollary}
\begin{enumerate}gin{proof}
Let $u$ be a complex number that satisfies all of these conditions.
Let $A$ be a non-zero ideal of $\mathbb{Z}[u]$.
Then, since $\mathbb{Z}[u]$ is principal, there exists $a \in \mathbb{Z}[u]$ such that $A =a \cdot\mathbb{Z}[u]$.
If we define a map ${\Phi : \mathbb{Z}[u] \rightarrow A}$ by $\Phi(x) = ax$, then $\Phi$ is clearly a group isomorphism ${(\mathbb{Z}[u], +) \rightarrow (A, +)}$.
Since $\Phi$ also preserves the action of~$u$ (as $\Phi(xu)=axu=\Phi(x) \cdot u$), $\Phi$ can be extended to an isomorphism $\mathbb{Z}[u] \rtimes \langlengle u \ranglengle \rightarrow A \rtimes \langlengle u \ranglengle$.
\mathbf{e}nd{proof}
One example of such $u$ was given in \cite{Oger:06}, namely $u=2+\sqsubsetrt{3}$.
Clearly ${\mathbb{Z}[u]=\mathbb{Z}[\sqsubsetrt{3}]}$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqsubsetrt{3}]$.
One can show that each invertible element $x\in \mathbb{Z}[\sqsubsetrt{3}]$ has the form $x=\pm (2+\sqsubsetrt{3})^n$ for some integer $n$ by considering the sequence $\{x_k\}$ defined by $x_0=x$, $x_{k+1}=x_k \cdot (2+\sqsubsetrt{3})^{-1}$.
Since the (norm) function $N:\mathbb{Z}[\sqsubsetrt{3}] \rightarrow {\mathbb{N}}$ defined by $N(a+b\sqsubsetrt{3})=|a^2-3b^2|$ satisfies the conditions
\begin{enumerate}gin{itemize}
\item if $y_1, y_2 \noindenteq 0$ then $N(y_1) \le N(y_1 \cdot y_2)$
\item if $y_2 \noindenteq 0$ then there exist $q,r \in \mathbb{Z}[\sqsubsetrt{3}]$ such that $y_1=q \cdot y_2+r$ and ${f(r) < f(y_2)}$
\mathbf{e}nd{itemize}
for any $y_1, y_2 \in \mathbb{Z}[\sqsubsetrt{3}]$, the ring $\mathbb{Z}[\sqsubsetrt{3}]$ is an Euclidean domain and so is principal (see \cite[5.5]{Ribenboim:01}).
\begin{enumerate}gin{proof}[Proof of Theorem \ref{Oger thm}]
The first-order sentence describing the semidirect products is $\mathbf{e}xistsists y \mathbf{e}xistsists z \ \varphi(y,z)$ where $\varphi(y,z) \mathbf{e}nd{quotation}uiv$ [(P0) $\wedge \ \ldots \ \wedge$ (P6)].
Let $P$ be the minimal polynomial of $u$ over $\mathbb{Z}$, and let $n = \mathtt{deg}(P)$.
We use additive notation in $A$.
First, we state that $y,z$ are non-identity elements.
Note both $0,1$ refer to the identity element.
\begin{enumerate}gin{itemize}
\item[(P0)] $y \noindenteq 0$, $z \noindenteq 1$
\mathbf{e}nd{itemize}
Next, we define $A,C$ and describe some of their properties.
\begin{enumerate}gin{itemize}
\item[(P1)] $A = C(y)$ and $C = C(z)$ are abelian, and $G = A \rtimes C$
\item[(P2)] $|A:2A|=2^n$
\item[(P3)] $|C:C^2|=2$
\item[(P4)] $x^k \noindenteq 1$ for $x \in C-\{1\}$, $1 \le k \le n+1$
\mathbf{e}nd{itemize}
The rest is the following.
\begin{enumerate}gin{itemize}
\item[(P5)] $\mathtt{Conj}(w,x) \noindenteq w$ for $w \in A-\{0\}$, $x \in C-\{1\}$
\item[(P6)] $P(f)=0$ for the automorphism $f$ of $A$ defined by $w \rightarrowto w^z$
\mathbf{e}nd{itemize}
(P6) is equivalent to saying $P(z)=0$, but we need to express it this way because the group operation (which is multiplication when considering $C$) is the only operation we are allowed to use.
Let $G$ be a f.g.\ model of $\psi$.
First, we show that $z$ has infinite order.
For, suppose $z^t = 1$ for some positive integer $t>1$.
Then $f$ is a root of the polynomial $X^t-1$ and so $P$ divides $X^t-1$ since $P$ is also a minimal polynomial of $f$.
But then $u^t-1=0$, contrary to the fact that $u,-1$ generate the infinite multiplicative group $\mathbb{Z}[u]^*$.
Hence $z$ has infinite order, in particular, $|C:\langlengle z \ranglengle|$ is finite as $C$ is f.g.\ abelian of torsion-free rank 1 by (P1), (P3).
Let $w_1, \ldots, w_r \in A$, $x_1, \ldots, x_r \in C$ such that $\{ w_1 x_1, \ldots, w_r x_r \}$ generates $G$, and let $z_1, \ldots, z_s \in C$ such that $C = z_1 \langlengle z \ranglengle \cup \ldots \cup z_s \langlengle z \ranglengle$ i.e.\ $\{z_1, \ldots, z_s\}$ contains at least one representative from each coset of $\langlengle z \ranglengle$ in $C$.
Then by Lemma \ref{prod conj},
\[
\begin{enumerate}gin{split}
A&=\langlengle \{ \mathtt{Conj}(w_i,x) \ | \ 1 \le i \le r, \ x \in C \} \ranglengle\\
&=\langlengle \{ \mathtt{Conj}(\mathtt{Conj}(w_i,z_j),z^k) \ | \ 1 \le i \le r, \ 1 \le j \le s, \ k \in \mathbb{Z} \} \ranglengle
\mathbf{e}nd{split}
\]
because each $x \in C$ can be written in the form $x=z_j \cdot z^k$ for some \mbox{$1 \le j \le s$, $k \in \mathbb{Z}$.}
Since $\langlengle \{ \mathtt{Conj}(w,z^k) \ | \ k \in \mathbb{Z} \} \ranglengle$ is f.g.\ for each $w \in C$ by (P6), this means that $A$ is f.g.
Now we show that $A$ is torsion-free.
For, suppose $w \in A-\{0\}$ is a torsion element.
Then ${\{f^k(w) \ | \ k \in \mathbb{Z} \}}$ is contained in the torsion subgroup of $A$, which is finite since $A$ is f.g.\ abelian (see \cite[Exercise 8.1.5]{Kar.Mer:79}).
Hence there exist $k_1,k_2 \in \mathbb{Z}$ with $k_1 < k_2$ such that $f^{k_1}(w) = f^{k_2}(w)$.
But this means that $z^{k_2-k_1} \in C-\{1\}$ fixes $f^{k_1}(w)=\mathtt{Conj}(w,z^{k_1}) \in A-\{0\}$, contrary to (P5).
Since $A$ is f.g.\ torsion-free, it is free abelian of rank $n$ by (P2).
Also, the subgroup $A_{(y)} = \langlengle \{ f^k(y) \ | \ 0 \le k \le n-1 \} \ranglengle$ of $A$ has rank $n$ by (P6) and the minimality of~$P$.
Hence the action of $z$ on $A_{(y)}$, which has finite index in $A$, is equivalent to the action of $u$ on a non-zero ideal of $\mathbb{Z}[u]$, meaning that the action of $z$ on $A$ is also equivalent.
Now we show that $C$ is torsion-free.
Otherwise, there exists ${x \in C-\{1\}}$ of prime order $p \ge n+2$ by (P4).
But then $1 = y^{x^p-1} = (y^{x^{p-1} + \ldots + x + 1})^{x-1}$, or equivalently, $Y^x = Y$ where ${Y = y^{x^{p-1} + \ldots + x + 1}}$, and so $Y=1$ by (P2).
Since $A$ is torsion-free (in particular, $y$ has infinite order) and ${X^{p-1}+ \ldots + X+1}$ is irreducible \mbox{(see \cite[Exercise IV.5.6]{Lang:05})}, ${y^{x^{p-1} + \ldots + x + 1}=1}$ means that the set $\{y^{x^k} \ | \ 0 \le k \le p-2\}$ generates a free abelian group of rank $p-1 \ge n+1$, which is a subgroup of~$A$.
But $A$ is free abelian of rank $n$, contradiction.
We know $C$ is f.g.\ torsion-free abelian of rank 1, or equivalently, infinite cyclic.
Choose a generator $c$ of $C$.
Then there exists $k \in \mathbb{Z}$ such that $c^k = z$.
Define an automorphism $g$ of $A$ by $w \rightarrowto w^c$ and let $Q$ be the minimal polynomial of $g$ over $\mathbb{Z}$.
We show that $\mathtt{deg}(Q)=n$.
Because $g$ is an automorphism of a free abelian group of rank~$n$, $\mathtt{deg}(Q)\le n$.
Also, since ${\langlengle\{g^{kr}(y) \ | \ 0 \le r \le n-1\}\ranglengle} = {\langlengle\{ f^r(y) \ | \ 0 \le r \le n-1 \}\ranglengle}$ has rank $n$, $\mathtt{deg}(Q) \ge n$.
Choose a root $v \in \mathbb{C}$ of $Q$ and an ideal $I$ of the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[v]$ so that the action of $c$ on $A$ is equivalent to the action of $v$ on $I$.
Because $g^k = f$, we can assume $v^k = u$ and so $\mathbb{Q}[u] \subseteqseteq \mathbb{Q}[v]$.
But since both fields have dimension~$n$ over $\mathbb{Q}$, $\mathbb{Q}[u] = \mathbb{Q}[v]$.
Now $v$ belongs to $\mathbb{Z}[v]=\mathbb{Z}[u]=\langlengle u,-1 \ranglengle \cup \{0\}$, so $k = \pm 1$ and $C= \langlengle c \ranglengle= \langlengle z \ranglengle$.
\mathbf{e}nd{proof}
\subseteqsection{Nilpotent groups} \langlebel{UT}
Our last example is a nilpotent group.
We give the definition of (class 2) nilpotency later, and for now we only mention that it has a non-trivial center.
This fact stops us from describing it as a semidirect product, because we do not have the main weapon ``no action has a fixed point'' any more.
Let $U$ be the discrete Heisenberg group $UT_3(\mathbb{Z})$, the group of upper unitriangular matrices (i.e.~the entries on the main diagonal are all $1$ and the entries below the diagonal are all $0$) over $\mathbb{Z}$.
Then $U$ is a nilpotent group of \mbox{class 2}.
That is, the $U'$ is contained in the center $Z$.
In fact, by \cite[Exercise 16.1.3]{Kar.Mer:79}, $U$ is isomorphic to the free class 2 nilpotent group $F$ with two generators.
Let $t_{mn}(k)$ denote the 3-by-3 matrix with $1$ in its diagonal entries, $k$ in the $m$-th row $n$-th column entry and $0$ everywhere else.
Then the generators of $F$ correspond to $a = t_{23}(1)$ and $b = t_{12}(1)$.
The following is a well-known fact about nilpotent groups.
\begin{enumerate}gin{lemma} \langlebel{f.g. nilpotent}
If $G$ is a nilpotent group such that $G/G'$ is f.g., then every subgroup of $G$ is f.g.
In particular, every subgroup of a f.g.\ nilpotent group is f.g.
\mathbf{e}nd{lemma}
\begin{enumerate}gin{proof}
See Robinson \cite[3.1.6, 5.2.17]{Robinson:82} for the proof of the first part.
The second part follows because every factor group of a f.g.~group is f.g.~(Lemma \ref{f.g. factor}).
\mathbf{e}nd{proof}
QFAness of $U$ can be shown using Oger and Sabbagh's criterion (\cite[Thm.10]{Oger.Sabbagh:06}), but here we give a sentence describing $U$ to make it easier to see how $U$ can be characterized in first-order.
The following facts will be used in the proof of \mbox{Theorem \ref{UT thm}}.
\begin{enumerate}gin{lemma} \langlebel{nilpotent commutators}
Let $G$ be a nilpotent group of class 2 and let $x,y \in G$.
Then, $[x^{m_1}y^{n_1},x^{m_2}y^{n_2}]=[x,y]^{m_1n_2-m_2n_1}$ for any $m_1,m_2,n_1,n_2 \in \mathbb{Z}$.
\mathbf{e}nd{lemma}
\begin{enumerate}gin{proof}
First, we show that $[x^m,y^n]=[x,y]^{mn}$ for any $m,n \in \mathbb{Z}$.
Note
\[
\begin{enumerate}gin{split}
y^{-1}x&=xx^{-1} \cdot y^{-1}x \cdot yy^{-1}\\
&=x \cdot [x,y] \cdot y^{-1}
\mathbf{e}nd{split}
\]
(i.e.\ we get $[x,y]$ every time we swap $y^{-1}$ and~$x$).
Since $G$ is class 2 nilpotent, $[x,y] \in G' \subseteqseteq Z(G)$ and so
\[
\begin{enumerate}gin{split}
[x^m,y^n]&=x^{-m}y^{-n}x^my^n\\
&=x^{-m}y^{-(n-1)}xyx^{m-1}y^n[x,y]\\
&\hspace{50pt}\vdots\\
&=x^{-m}x^my^{-n}y^n[x,y]^{mn}\\
&=[x,y]^{mn}
\mathbf{e}nd{split}
\]
holds for positive $m,n$.
Also, since
\[
\begin{enumerate}gin{split}
[x^{-1},y]&=xy^{-1}x^{-1}y \cdot xx^{-1}\\
&=[y,x]^{x^{-1}}\\
&=[y,x]=[x,y]^{-1}
\mathbf{e}nd{split}
\]
and similarly $[x,y^{-1}]=[x,y]^{-1}$ holds in $G$, $[x^m,y^n]=[x,y]^{mn}$ holds for any $m,n \in \mathbb{Z}$.
Now, since $G' \subseteqseteq Z(G)$, the commutator rule ${[ab,c] = [a,c]^b[b,c]}$ (\cite[3.2.(3)]{Kar.Mer:79}) can be reduced to ${[ab,c] = [a,c][b,c]}$, and by taking inverse we also get $[c,ab]=[c,b][c,a]$.
So we have
\[
\begin{enumerate}gin{split}
[x^{m_1}y^{n_1},x^{m_2}y^{n_2}]&=[x^{m_1},x^{m_2}y^{n_2}][y^{n_1},x^{m_2}y^{n_2}]\\
&=[x^{m_1},y^{n_2}][x^{m_1},x^{m_2}][y^{n_1},y^{n_2}][y^{n_1},x^{m_2}]\\
&=[x,y]^{m_1n_2} \cdot 1 \cdot 1 \cdot [y,x]^{m_2n_1}\\
&=[x,y]^{m_1n_2-m_2n_1}
\mathbf{e}nd{split}
\]
as required.
\mathbf{e}nd{proof}
\begin{enumerate}gin{lemma} \langlebel{UT center}
The center $Z$ of $UT_3(\mathbb{Z})$ is the infinite cyclic group generated by $c = [a,b] = t_{13}(1)$, which coincides with the set of commutators.
\mathbf{e}nd{lemma}
\begin{enumerate}gin{proof}
Since \[ \left[ \left(
\begin{enumerate}gin{array}{@{}ccc@{}}
1 & \mathbf{a}lpha_1 & \begin{enumerate}ta_1 \\
0 & 1 & \gamma_1 \\
0 & 0 & 1
\mathbf{e}nd{array}
\right), \left(
\begin{enumerate}gin{array}{@{}ccc@{}}
1 & \mathbf{a}lpha_2 & \begin{enumerate}ta_2 \\
0 & 1 & \gamma_2 \\
0 & 0 & 1
\mathbf{e}nd{array}
\right) \right] = t_{13}(\mathbf{a}lpha_1\gamma_2-\mathbf{a}lpha_2\gamma_1), \]
the center is precisely
$Z = \{t_{13}(z) \ | \ z \in \mathbb{Z}\} = \langlengle c \ranglengle$.
Now by \cite[Exercise 16.1.3]{Kar.Mer:79}, each element $u \in U$ can be written as $u = a^mb^nc^l$ for some $m,n,l \in \mathbb{Z}$ and so
\[
\begin{enumerate}gin{split}
[u,v]&=[a^{m_1}b^{n_1}c^{l_1},a^{m_2}b^{n_2}c^{l_2}]\\
&=[a^{m_1}b^{n_1},a^{m_2}b^{n_2}]\\
&=[a,b]^{m_1n_2-m_2n_1}\\
&=c^{m_1n_2-m_2n_1} \in \langlengle c \ranglengle
\mathbf{e}nd{split}
\]
for any $u=a^{m_1}b^{n_1}c^{l_1},v=a^{m_2}b^{n_2}c^{l_2} \in U$ by Lemma \ref{nilpotent commutators}.
Hence $U' = \langlengle c \ranglengle$.
\mathbf{e}nd{proof}
As a part of the sentence describing $U$, we use the modified version of a formula first introduced by Mal'cev \cite{Malcev:71}.
The formula $\mu(x,y;a,b)$ with parameters $a,b$ defines the ``square'' operation $M_{a,b}$ on the center $Z$ in the sense that $(Z, \circ, M_{a,b}) \cong (\mathbb{Z}, +, Q)$ where $Q=\{ (t,t^2) | \ t \in \mathbb{Z}\}$.
The formula is
\[
\begin{enumerate}gin{split}
\mu(x,y;a,b)\mathbf{e}nd{quotation}uiv\mathbf{e}xistsists u \mathbf{e}xistsists v\{&[u,a]=[v,b]=1 \ \wedge\\
&x=[a,v]=[u,b] \ \wedge\\
&y=[u,v]\}.
\mathbf{e}nd{split}
\]
This defines the ``square'' because $[a^m,b^n]=[a,b]^{mn}$ holds in $U$ by Lemma \ref{nilpotent commutators}.
\begin{enumerate}gin{theorem}[Nies \cite{Nies:03}] \langlebel{UT thm}
$UT_3(\mathbb{Z})$ is QFA.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{proof}
The sentence $\psi_U$ consists of four formulas;
\begin{enumerate}gin{itemize}
\item[(P1)] the center $Z$ coincides with the set of commutators
\item[(P2)] $\mathbf{e}xistsists r \mathbf{e}xistsists s \ \gamma(r,s)$ where $\gamma(r,s)$ is as described below
\item[(P3)] $|Z:Z^2|=2$
\item[(P4)] $|B:B^2|=4$ where $B=G/Z$.
\mathbf{e}nd{itemize}
Roughly speaking, $\gamma(r,s)$ says $Z$ is linearly orderable, using Lagrange's theorem: an integer is non-negative iff it is the sum of four squares of integers.
Formally, $\gamma(r,s)$ is a formula expressing
\begin{enumerate}gin{itemize}
\item $\mu(x,y;r,s)$ defines a unary operation $M_{r,s}$ on $Z$
\item let $P_{r,s}=\{u\ |\ \mathbf{e}xistsists v_1 \ldots \mathbf{e}xistsists v_4 \ u=M_{r,s}(v_1) \circ \ldots \circ M_{r,s}(v_4)\}$. Then $x \le y \leftrightarrow y-x \in P_{r,s}$ defines a linear order which turns $Z$ into an ordered abelian group with $[r,s]$ being the least positive element.
\mathbf{e}nd{itemize}
Let $G$ be a f.g.\ model of $\psi_U$. Since $Z$ is linearly orderable by (P2), it is torsion-free.
Now we show that $B$ is also torsion-free.
If $u \in G-Z$, there exists $v \in G$ such that $[u,v] \noindenteq 1$.
Then for each positive integer $n$, $[u^n, v]=[u,v]^n\noindenteq 1$ by Lemma \ref{nilpotent commutators}.
Hence $u^n \noindentotin Z$.
Since $G$ is f.g.\ and (class 2) nilpotent by (P1), $Z$ is f.g.\ by Lemma \ref{f.g. nilpotent}.
Also, since $Z=G'$ by (P1), $B=G/Z=G/G'$ is abelian.
So we know that $Z, B$ are both f.g.\ torsion-free abelian and have rank 1,2 respectively by (P3) and (P4)
i.e.~$Z \cong \mathbb{Z}$, $B \cong \mathbb{Z} \oplus \mathbb{Z}$.
Now we show that $G$ is generated by two elements.
Let $c,d \in G$ such that the cosets $Zc,Zd$ generate $B$, and let $g,h \in G$ such that the commutator $[g,h]$ generates~$Z$.
Then, there exist $x,y,z,w \in \mathbb{Z}$ and $u,v \in Z$ such that $g=uc^xd^y$ and $h=vc^zd^w$.
Hence $[g,h]=[c^xd^y,c^zd^w]=[c,d]^{xw-yz}$ by Lemma \ref{nilpotent commutators}.
But also $[g,h]^r=[c,d]$ for some $r \in \mathbb{Z}$ because $Z$ is generated by $[g,h]$.
Since $Z$ is torsion-free, it follows that $xw-yz=r=\pm1$.
Thus $[c,d]$ also generates $Z$ and so the two elements $c,d$ generate $G$.
Because $U$ is the free class 2 nilpotent group of rank 2, there exists an epimorphism $h:U \rightarrow G$ mapping $a,b$ to $c,d$ respectively.
If $h$ is not $1-1$, then $\mathtt{Ker}(h)$ is non-trivial and so it must intersect $Z(U)$ non-trivially as $U$ is nilpotent, \mbox{by \cite[Thm.\ 16.2.3]{Lang:84}}.
But this is impossible, because $h([a,b])=[c,d]$ and so $h$ induces an isomorphism $Z(U) \rightarrow Z(G)$.
Hence $h$ is $1-1$, or equivalently, $h$ is itself an isomorphism.
\mathbf{e}nd{proof}
\section{Polylogarithmic compressibility} \langlebel{PLC}
As an analogue of quasi-finite axiomatizability, we define \mathbf{e}mph{polylogarithmic compressibility} below as a property of a class of finite groups, that the groups can be described by ``short'' first-order sentences in the sense described below.
It makes sense to define it as a property of a class of groups rather than a single group, because the length of the sentence is always constant (and so cannot be compared to the size of the group) if we have only one group.
We define the length $|\psi|$ of a first-order formula $\psi$ to be the number of symbols used in $\psi$.
We assume we have infinitely many variables and so each variable is counted as one symbol.
It usually reduces the length of each (sufficiently long) formula by the factor of $O(\mathtt{log} \ n)$ where $n$ is the number of variables used in the sentence.
This is because, if we have only finitely many variables, then (when $n$ is sufficiently large) the variables in the sentence require extra indices, which have length $O(\mathtt{log} \ n)$.
It can be avoided in some cases by repeating the same variables. e.g.\ the sentence
\[
\forall x_1 \forall x_2 [x_1,x_2]=1 \rightarrow \mathbf{e}xistsists x_3 \mathbf{e}xistsists x_4 [x_3,x_4]=1
\]
is equivalent to
\[
\forall x \forall y [x,y]=1 \rightarrow \mathbf{e}xistsists x \mathbf{e}xistsists y [x,y]=1.
\]
\begin{enumerate}gin{definition} \langlebel{PLC def}
A class $\mathcal{C}$ of finite groups is \mathbf{e}mph{polylogarithmically compressible (PLC)} if for any $H \in \mathcal{C}$, there exists a first-order sentence $\psi_H$ such that $H \models \psi_H$, $|\psi_H| = O(\mathtt{log}^k|H|)$ for some fixed $k$, and if $G \models \psi_H$ then $G \cong H$.
In particular, we say $\mathcal{C}$ is \mathbf{e}mph{logarithmically-compressible (LC)} if $k = 1$.
\mathbf{e}nd{definition}
Since we allow the polynomial change in the length, PLCness is independent of the particular way we define first-order language.
For example, it does not matter whether we use parentheses or Polish notation (which allows us to write parenthesis-free formulas without ambiguity).
Here we give an example of an LC class to illustrate the definition, namely the cyclic groups of order $2^n$.
The sentence describing $\mathbb{Z}_{2^n}$, written additively, consists of three formulass; $\psi \mathbf{e}nd{quotation}uiv \forall x[\psi_1 \wedge \psi_2 \wedge \psi_3]$ where
\[
\begin{enumerate}gin{split}
\psi_1(x) &\mathbf{e}nd{quotation}uiv \forall y [2y \noindenteq x] \ \vee \ \mathbf{e}xistsists z \mathbf{e}xistsists w [(2z=x) \wedge (2w=x) \wedge \forall t [(2t=x) \rightarrow (t=z \vee t=w)]]\\
\psi_2(x) &\mathbf{e}nd{quotation}uiv \noindenteg \mathbf{e}xistsists x_2 \ldots \mathbf{e}xistsists x_{n+1} \left[2x=x_2 \wedge \begin{itemize}gwedge_{2 \leq i < n+1} 2x_i=x_{i+1}\wedge x_{n+1} \noindenteq 0\right]\\
\psi_3(x) &\mathbf{e}nd{quotation}uiv \mathbf{e}xistsists x_1 \ldots \mathbf{e}xistsists x_n \left[\begin{itemize}gwedge_{1 \leq i < n} 2x_i=x_{i+1}\wedge x_n \noindenteq 0 \right].
\mathbf{e}nd{split}
\]
Note that each part has length $O(n)$.
The first formula $\psi_1$ says that for each element $x$ of the group, either no element $y$ satisfies $2y=x$, or there are exactly 2 such $y$.
This is true in $\mathbb{Z}_{2^n}$ because, if $x$ is odd then no $y$ satisfies $2y=x$, and if $2m=x$ for some $m$ in $\mathbb{Z}$ then precisely $y_1=m$ and $y_2=m+2^{n-1}$ satisfy the equation in $\mathbb{Z}_{2^n}$.
The next formula says $2^n x=0$ for any element $x$ (i.e.\ every element has order $2^i$ where $i \le n$), and the last formula says there exists an element $x_1$ such that $2^{n-1} x_1 \noindenteq 0$.
Clearly both of them hold in $\mathbb{Z}_{2^n}$.
Now let $G$ be a group written additively such that $G \models \psi$.
Then since $0 \in G$ and $0+0=0$, there exists exactly one element of order $2^1$ from $\psi_1$.
Similarly, it can be shown that $G$ has at most $2^{i-1}$ elements of order $2^i$ for each $i$.
Since every element of $G$ has order $2^i$ for some $i \le n$ from $\psi_2$, the maximum number of elements $G$ can have is $1+\sum_{1 \le i \le n} 2^{i-1}=2^n$.
But there exists an element of order $2^n$ from~$\psi_2,\psi_3$ and so the cyclic subgroup generated by this element must coincide with the whole group $G$.
In other words, $G \cong \mathbb{Z}_{2^n}$.
In this section, we give more examples of PLC and LC classes.
The proofs follow the scheme described below except for the last example:
\begin{enumerate}gin{itemize}
\item[(i)] We give a presentation for the group $H$ so that if $G \models \psi_H$, then $G$ contains a subgroup $\widetilde{G}$ isomorphic to some factor of $H$.
\item[(ii)] We express that the generators of $\widetilde{G}$ generate the whole group $G$.
\item[(iii)] We express that $\widetilde{G} \cong H$.
\mathbf{e}nd{itemize}
The following lemmas are used repeatedly.
\begin{enumerate}gin{lemma} \langlebel{presentation}
Given a finite presentation for a group $H$ with generators $a_1,\ldots,a_m$, there exists a first-order formula $\zeta(x_1,\ldots,x_m)$ such that $H \models \zeta(a_1,\ldots,a_m)$, and if $G \models \zeta(b_1,\ldots,b_m)$ for some group $G$ and its elements $b_1,\ldots,b_m$, then the subgroup $\langlengle b_1,\ldots,b_m \ranglengle$ of $G$ is isomorphic to $H/N$ for some normal subgroup $N$ of~$H$.
\mathbf{e}nd{lemma}
Of course, this lemma is used in the part (i) of the scheme.
The length of the formula $\zeta$ depends on the length of the relators.
\begin{enumerate}gin{proof}
Let $P=\langlengle~a_1,\ldots,a_m~|~t_1,\ldots,t_n~\ranglengle$ be a presentation for $H$.
Note that each relator $t_i$ is first-order definable with parameters $a_1,\ldots,a_m$ since it is a product of the gerenators and their inverses \mbox{(i.e.~$t_i=t_i(a_1,\ldots,a_m)$)}.
Then the formula is
\[
\zeta(x_1,\ldots,x_m) \mathbf{e}nd{quotation}uiv \begin{itemize}gwedge_{1 \le i \le n} t_i(x_1,\ldots,x_m)=1.
\]
If $G \models \zeta(b_1,\ldots,b_m)$ for some group $G$ and its elements $b_1,\ldots,b_m$, then the subgroup $\widetilde{G}=\langlengle b_1,\ldots,b_m \ranglengle$ of $G$ has a presentation
\[\langlengle~x_1,\ldots,x_m~|~t_1,\ldots,t_n, u_1,\ldots~\ranglengle\]
where each $x_j$ corresponds to $b_j$, and $t_i=t_i(x_1,\ldots,x_m)$, $u_k=u_k(x_1,\ldots,x_m)$ for each $i,k$. Hence $\widetilde{G} \cong H/N$ where $N$ is the normal subgroup of $H$ generated \mbox{by $\{u_k(a_1,\ldots,a_m)~|~1 \le k\}$.}
In particular, if $N$ is trivial then $\widetilde{G} \cong H$.
\mathbf{e}nd{proof}
\begin{enumerate}gin{lemma} \langlebel{repeated squaring}
For each positive integer $n$, there exists a first-order formula $\theta_n(x,y)$ of length $O(\mathtt{log}\ n)$ such that $G \models \theta_n(x,y)$ iff $x^n=y$ in the group $G$.
\mathbf{e}nd{lemma}
The method used here is called \mathbf{e}mph{repeated squaring}.
The formulas $\psi_2, \psi_3$ in the example above are also using this technique.
\begin{enumerate}gin{proof}
Let $n=\mathbf{a}lpha_1\ldots\mathbf{a}lpha_k$ written in binary where $k=\lfloor \mathtt{log_2}\ n \rfloor$.
Then the formula $\theta_n$ is
\[
\theta_n(x,y) \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists y_1 \ldots \mathbf{e}xistsists y_k \left[y_1=x \ \wedge \ y_k=y \ \wedge \ \begin{itemize}gwedge_{1 \le i < k} y_{i+1}=y_i \cdot y_i \cdot x^{\mathbf{a}lpha_{i+1}}\right]
\]
where $x^{\mathbf{a}lpha_i}=x$ if ${\mathbf{a}lpha_i}=1$ and $x^{\mathbf{a}lpha_i}=1_G$ if ${\mathbf{a}lpha_i}=0$. Clearly $\theta_n$ has length $O(\mathtt{log}\ n)$.
Now we show that the formula is correct, by induction on $k$.
If $k=1$, then the only possibility is $n=1$ and correctness is obvious because the formula is reduced to ${\theta_1(x,y) \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists y_1 [x=y_1=y]}$.
Suppose $\theta_n(x,y)$ is correct for all $n < 2^{k}$ for some~$k$.
Let $N \in {\mathbb{N}}$ such that $2^{k} \le N < 2^{k+1}$ and let $N=\begin{enumerate}ta_1 \ldots \begin{enumerate}ta_k$ written in binary.
Then,
\[
\begin{enumerate}gin{split}
\theta_N(x,y)&\mathbf{e}nd{quotation}uiv \mathbf{e}xistsists y_1 \ldots \mathbf{e}xistsists y_{k} \left[\begin{itemize}gwedge_{1 \le i < k} y_{i+1}=y_i \cdot y_i \cdot x^{\mathbf{a}lpha_{i+1}} \ \wedge \ y_1=x \ \wedge \ y_{k}=y\right]\\
&\mathbf{e}nd{quotation}uiv \mathbf{e}xistsists y_k \left[\theta_{\widetilde{N}}(x,y_{k-1}) \ \wedge \ y_k=y_{k-1} \cdot y_{k-1} \cdot x^{\begin{enumerate}ta_k} \ \wedge \ y_k=y \right]
\mathbf{e}nd{split}
\]
where $\widetilde{N}=\begin{enumerate}ta_1 \ldots \begin{enumerate}ta_{k-1}$.
If $\theta_N(x,y)$ holds in $G$ with witnesses $y_1,\ldots,y_k$, then we have ${y_{k-1}=x^{\widetilde{N}}}$ by the inductive hypothesis because $\widetilde{N} < 2^k$.
Since $N = 2\widetilde{N}+{\begin{enumerate}ta_k}$, it follows that $y_k=y_{k-1} \cdot y_{k-1} \cdot x^{\begin{enumerate}ta_k}=x^{2\widetilde{N}+{\begin{enumerate}ta_k}}=x^N$, as required.
\mathbf{e}nd{proof}
\begin{enumerate}gin{lemma} \langlebel{finite product}
Given a generating set $S$ of a finite group $G$, every element of $G$ can be written as a product of at most $|G|$ generators in $S$.
\mathbf{e}nd{lemma}
This lemma, combined with the next one, is used in the part (ii) of the scheme.
The basic idea of the proof is the pigeonhole principle.
\begin{enumerate}gin{proof}
Let $S=\{s_1,\ldots,s_n\}$ be a generating set of $G$.
Then, each element $g \in G$ can be written as a product
\[
g=\preceqod_{1 \le i \le m}t_i
\]
for some $m$ where $t_i \in S$ for each $i$.
If $m > |G|$, then there exist $j,k \in {\mathbb{N}}$ with $j < k \le m$ and
\[
\preceqod_{1 \le i \le j}t_i = \preceqod_{1 \le i \le k}t_i
\]
and so $g$ can also be written as
\[
g=\left(\preceqod_{1 \le i \le j}t_i\right) \cdot \left(\preceqod_{k < i \le m}t_i\right)
\]
which is a product of $m-(k-j)$ generators.
We can repeat the same procedure until $g$ is written as a product of no more than $|G|$ generators.
\mathbf{e}nd{proof}
\begin{enumerate}gin{lemma} \langlebel{generation}
Let $G$ be a finite group.
Then for each positive integer $n$, there exists a first-order formula $\pi_n(g;x_1,\ldots,x_n)$ with parameters $x_1,\ldots,x_n$ of length $O(n+\mathtt{log}|G|)$ such that ${G \models \pi_n(g;x_1,\ldots,x_n)}$ iff ${g \in \langlengle x_1,\ldots,x_n \ranglengle}$.
In other words, $\pi_n$ defines the subgroup ${\langlengle x_1,\ldots,x_n \ranglengle}$ of $G$.
\mathbf{e}nd{lemma}
As mentioned above, this lemma is usually used in the part (ii) of the scheme.
A modified version of the formula is also used in the proof of Theorem \ref{abelian thm} to define a subset of the group consisting of some powers of a certain element.
\begin{enumerate}gin{proof}
We use a device that originated in computational complexity to show that the set of true quantified boolean formulas is PSPACE complete \cite[Thm 8.9]{Sipser:97}.
We define the formulas $\delta_i(g;x_1,\ldots,x_n)$ with parameters $x_1,\ldots,x_n$ for each $i \in {\mathbb{N}}$ inductively.
For $i=0$,
\[
\delta_0(g;x_1,\ldots,x_n) \mathbf{e}nd{quotation}uiv \begin{itemize}gvee_{1 \le j \le n}g=x_j \ \vee \ g=1
\]
and for $i>0$,
\[
\begin{enumerate}gin{split}
\delta_i(g;x_1,\ldots,x_n) \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists u_i \mathbf{e}xistsists v_i [&g = u_i v_i \ \wedge\\
&\forall w_i [(w_i = u_i \vee w_i = v_i) \rightarrow \delta_{i-1}(w_i;x_1,\ldots,x_n)]].
\mathbf{e}nd{split}
\]
Note $\delta_i$ has length $O(n+i)$, and $G \models \delta_i(g;x_1,\ldots,x_n)$ iff $g$ can be written as a product of at most $2^i$ $x$'s.
Now let $\widetilde{G}=\langlengle x_1,\ldots,x_n \ranglengle$.
Then by Lemma \ref{finite product}, each $g \in \widetilde{G}$ can be written as a product of at most $|\widetilde{G}|$ generators of $\widetilde{G}$ (i.e.\ $x_1,\ldots,x_n$).
Hence by defining $\pi_n(g;x_1,\ldots,x_n) \mathbf{e}nd{quotation}uiv \delta_k(g;x_1,\ldots,x_n)$ where $k = \lceil \mathtt{log_2}|G| \rceil$ (and so $2^k \ge |G| \ge |\widetilde{G}|$), we get the required formula.
\mathbf{e}nd{proof}
\subseteqsection{Simple groups} \langlebel{simple}
It is known that finite simple groups can be classified into 18 infinite families, with exceptions of 26 so-called sporadic groups.
In \cite{Babai:97}, L.~Babai {\it et al.}\ showed that all finite simple groups have `short' presentations, with possible exception of the three families: the projective special unitary groups $PSU_3(q) = {^2A_2(q)}$ where $q$ is a prime-power, the Suzuki groups $Sz(q)={^2B_2(q)}$ where $q=2^{2e+1}$ for some positive integer $e > 1$, and the Ree groups $R(q)={^2G_2(q)}$ where $q=3^{2e+1}$ for some positive integer $e > 1$.
They defined the length $l(P)$ of a presentation $P$ to be the number of characters required to write all the relations (or equivalently relators) in $P$, where the exponents are written in binary, and proved that each of these groups have a presentation of length $O(\mathtt{log}^2|G|)$ where $|G|$ is the size of the group.
Note that $l(P)$ is the maximum number of generators in $P$ the relations can `talk' about.
Since each generator must appear in the relations at least once (otherwise it will have infinite order), this means the number of generators in $P$ is also $O(\log^2|G|)$.
Among the families they missed, two of them were shown to have short presentations by other people.
One is $PSU_3(q)$, shown by Hulpke and Seress \cite{Hulpke:01}.
Given a finite field $F_{q^2}$ for some prime-power $q$, an order 2 automorphism $\mathbf{a}lpha$ of $F_{q^2}$ can be defined by $x \rightarrowto x^q$ and it can be extended in the natural way to the (multiplicative) groups of matrices over $F_{q^2}$.
The {\it special unitary group} $SU_3(q)$ is
\[
SU_3(q) = \{A \in SL_3(q^2) \ | \ A \omega \overline{A}^T = \omega\}
\]
where $\overline{A}=A^{\mathbf{a}lpha}$ and $\omega =
\left(
\begin{enumerate}gin{array}{@{}ccc@{}}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\mathbf{e}nd{array}
\right)$, and the {\it projective special unitary group} $PSU_3(q)$ is the factor of $SU_3(q)$ by its center.
The other family shown to have short presentations is the Suzuki groups.
In fact, the presentation was given in the original paper by Suzuki \cite{Suzuki:62}, and that was observed by J.~Thompson (personal communication to W.~Kantor) according to Hulpke and Seress \cite{Hulpke:01}.
We use these results to prove the following theorem.
\begin{enumerate}gin{theorem}
The class of finite simple groups, excluding the family of the Ree groups $R(q)={^2G_2(q)}$, is PLC.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{proof}
Let $H$ be a finite simple group not belonging to the family $R(q)$ and let $P=\langlengle~a_1,\ldots,a_m~|~t_1,\ldots,t_n~\ranglengle$ be a presentation for $H$ with $l(P)=O(\mathtt{log}^2|H|)$.
Then the sentence describing $H$ is $\psi \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists a_1 \ldots \mathbf{e}xistsists a_m [\psi_1 \wedge \psi_2 \wedge \psi_3]$ where formulas $\psi_1,\psi_2,\psi_3$ correspond to (i),(ii),(iii) in the scheme respectively.
First, we show that $\psi_1$, the `presentation' for the group (which corresponds to the formula $\zeta$ in Lemma \ref{presentation}), has length $O(\mathtt{log}^2|H|)$.
It suffices to show that we can write each relator in appropriate length.
If $t=a_{\varphi(1)}^{z_1} \ldots a_{\varphi(k)}^{z_k}$ is a relator, where each $a_{\varphi(i)}$ is a generator in $P$, then the number of characters required to write $t$ is $l(t)=k+\sum_{1 \le i \le k} \lfloor \mathtt{log}_2 z_i \rfloor$.
Since the formula
\[
\tau(a_1,\ldots,a_m) \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists b_1 \ldots \mathbf{e}xistsists b_k \left[\begin{itemize}gwedge_{1 \le i \le k} \theta_{z_i}(a_{\varphi(i)},b_i) \wedge \preceqod_{1 \le i \le k} b_i=1\right]
\]
expresses $t=1$, and has length $\simeq 5k + 10 \sum_{1 \le i \le k} \lfloor \mathtt{log_2}~z_i \rfloor$ by Lemma \ref{repeated squaring}, we obtain the required result.
I.e.\ the formula
\[
\psi_1(a_1,\ldots,a_m) \mathbf{e}nd{quotation}uiv \begin{itemize}gwedge_{1 \le j \le n} \tau_j(a_1,\ldots,a_{m})
\]
where each $\tau_j$ corresponds to the relator $t_j$, has length approximately
\[
\sum_{1 \le j \le n} \left[ 5k + 10 \sum_{1 \le i \le k} \lfloor \mathtt{log_2}~z_i \rfloor \right]
\]
where $k$ is dependent on $j$.
The second formula $\psi_2$ expresses that $a_1,\ldots,a_n$ generate $H$, using Lemma \ref{generation};
\[
\psi_2(a_1,\ldots,a_m) \mathbf{e}nd{quotation}uiv \forall h [\pi_k(h;a_1,\ldots,a_m)]
\]
where $k = \lceil \mathtt{log}|H| \rceil$.
(Recall that $H \models \pi_k(h;a_1,\ldots,a_m)$ iff $h \in \langlengle a_1,\ldots,a_m \ranglengle$.)
Clearly $\psi_2$ has length $O(\mathtt{log}|H|)$.
Now let $G$ be a group and let $x_1,\ldots,x_m \in G$ such that $G \models \psi_1 \wedge \psi_2(x_1,\ldots,x_m)$.
Then we know that $G$~is generated by the elements $x_1,\ldots,x_m$ and is isomorphic to some factor group of $H$.
But since $H$ is simple, we must have $G \cong H$ unless $G$~is trivial.
Hence the last formula $\psi_3$ is
\[
\psi_3(a_1,\ldots,a_m) \mathbf{e}nd{quotation}uiv [a_1 \noindenteq 1]
\]
assuming $a_1$ is not the identity element in $H$.
We can make this assumption safely because if $a_1=1$, then we can get a shorter presentation for $H$ by excluding $a_1$ from $P$.
Clearly $\psi_3$ has constant length and so the length of the whole sentence $\psi$ is $O(\mathtt{log}^2|H|)$.
\mathbf{e}nd{proof}
\subseteqsection{Symmetric groups} \langlebel{symmetric}
In the previous subsection, it was easy to obtain the formula $\psi_2$ because the groups considered were simple.
Something similar happens for the case of the symmetric groups, because for each $n \ge 5$, the alternating group $A_n$ is the only non-trivial normal subgroup of $S_n$ (see \cite[3.2.3]{Robinson:82}).
In \cite{Bray:11}, Bray {\it et al.}\ found presentations of length $O(\mathtt{log}(n))$ for the symmetric groups $S_n$, one of whose generators corresponds to the $n$-cycle $(1,\ldots,n)$.
They defined the length of a presentation in a slightly different way from Babai {\it et al.}~\cite{Babai:97}, but it does not affect the order of a presentation (they included the number of generators).
Note that their presentation for $S_n$ has length $O(\mathtt{log \ log}|S_n|)$ because $|S_n| = n!$.
\begin{enumerate}gin{theorem} \langlebel{symmetric thm}
The class of symmetric groups $S_n$ is LC.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{proof}
We can assume $n \ge 5$ and so $A_n$ is the only non-trivial normal subgroup of $S_n$.
The sentence is $\psi \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists \mathbf{e}ta \mathbf{e}xistsists \sigma_2 \ldots \mathbf{e}xistsists \sigma_k [\psi_1 \wedge \psi_2 \wedge \psi_3]$ where $k$ is the number of generators in the short presentation for $S_n$, $\mathbf{e}ta$ corresponds to the $n$-cycle $(1,\ldots,n)$ and $\sigma_2, \ldots, \sigma_k$ correspond to the rest of the generators.
The constructions of $\psi_1, \psi_2$ are exactly the same as that in the previous theorem, and so $\psi_1,\psi_2$ have length $O(\mathtt{log} \ n)$, $O(\mathtt{log}|S_n|)$ respectively.
Now let $G$ be a group and let $x_1, \ldots, x_k \in G$ such that $G \models \psi_1 \wedge \psi_2 (x_1, \ldots, x_k)$.
Then, since $\{1\}$, $A_n$, $S_n$ are the only normal subgroups of $S_n$, the group $G$ is isomorphic to $S_n$, $\mathbb{Z}_2$ or $\{1\}$.
So the formula
\[
\psi_3(\mathbf{e}ta, \sigma_2, \ldots, \sigma_k) \mathbf{e}nd{quotation}uiv [\mathbf{e}ta \noindenteq 1 \ \wedge \ \mathbf{e}ta^2 \noindenteq 1]
\]
guarantees that if $G \models \psi$ then $G \cong S_n$.
Since $\psi_3$ has constant length, the length of the whole sentence $\psi$ is $O(\mathtt{log}|S_n|)$.
\mathbf{e}nd{proof}
\subseteqsection{Abelian groups} \langlebel{abelian}
It is known that each finite abelian group is isomorphic to a direct product of (finite) cyclic groups, and $\mathbb{Z}_m \oplus \mathbb{Z}_n \cong \mathbb{Z}_{mn}$ iff $m,n$ are coprime.
Hence each finite abelian group can be written as a unique direct product of cyclic groups of prime-power order, up to permutation of the factors.
\begin{enumerate}gin{theorem} \langlebel{abelian thm}
The class of finite abelian groups is LC.
\mathbf{e}nd{theorem}
\begin{enumerate}gin{proof}
Let $H=\begin{itemize}goplus_{1 \le i \le n} \mathbb{Z}_{q_i}$ where each $q_i$ has the form $q_i = p_i^{z_i}$ for some prime $p_i$ and some positive integer $z_i$.
Then $H$ has a presentation
\[
\langlengle~a_1,\ldots,a_n~|~a_1^{q_1},\ldots,a_n^{q_n}, [a_j,a_k]~(1 \le j < k \le n)~\ranglengle
\]
where each $a_i$ corresponds to a generator of $\mathbb{Z}_{q_i}$.
We follow the scheme again (i.e.\ the sentence $\psi$, written additively, has the form ${\psi \mathbf{e}nd{quotation}uiv\mathbf{e}xistsists a_1 \ldots \mathbf{e}xistsists a_n [\psi_1 \wedge \psi_2 \wedge \psi_3]}$), but $\psi_1$ is slightly different here.
Since saying ``every element commutes with each other'' requires a shorter formula than saying ``every commutator commutes with each other'', $\psi_1$ is
\[
\psi_1(a_1,\ldots,a_n) \mathbf{e}nd{quotation}uiv \forall g \forall h [g,h]=0 \ \wedge \ \begin{itemize}gwedge_{1 \le i \le n} \theta_{q_i}(a_i,0).
\]
(Recall that $H \models \theta_n(x,y)$ iff $x^n=y$ holds in $H$.)
Clearly it has length $O(\mathtt{log}|H|)$.
The second formula $\psi_2$ is exactly the same as that of the previous example;
\[
\psi_2(a_1,\ldots,a_n) \mathbf{e}nd{quotation}uiv \forall g [\pi_k(g;a_1,\ldots,a_n)]
\]
where $k = \lceil \mathtt{log}|H| \rceil$.
It has length $O(\mathtt{log}|H|)$.
Now let $G$ be a group written additively and let $x_1,\ldots,x_n \in G$ such that ${G \models \psi_1 \wedge \psi_2(x_1,\ldots,x_n)}$, then we know $G$ is abelian and generated by $x_1,\ldots,x_n$.
For $G$ to be isomorphic to $H$, it suffices that $p_i^{z_i-1} \cdot x_i \noindenteq 0$ for each $i$, and $x_1,\ldots,x_n$ form an independent set in the sense that if $\sum_{1 \le i \le n} \mathbf{a}lpha_i x_i = 0$ for some integers $\mathbf{a}lpha_i$, then $\mathbf{a}lpha_i x_i = 0$ for each~$i$.
Recall that each $x_i$ satisfies $q_i x_i = p_i^{z_i} x_i = 0$ where $p_i$ is prime.
We define a relation $\sim$ on $\{1,\ldots,n\}$ by $i \sim j$ iff $p_i=p_j$.
Now clearly $\sim$ is an equivalence relation.
We denote by $[i], \Omegaega$ the equivalence class containing $i$ and the set of all the equivalence classes respectively.
Then it is easy to see that if $S_{[i]} = \{x_j~|~j \in [i]\}$ is independent for each $[i] \in \Omegaega$, then the whole group $G$ is independent.
So the last formula $\psi_3$ is
\[
\psi_3(a_1,\ldots,a_n) \mathbf{e}nd{quotation}uiv \begin{itemize}gwedge_{1 \le i \le n} \noindenteg \left[ \theta_{p_i^{z_i-1}}(a_i,0) \right] \ \wedge \ \begin{itemize}gwedge_{[i] \in \Omegaega} \xi_{[i]}(a_1,\ldots,a_n)
\]
where each $\xi_{[i]}$ expresses that $S_{[i]}+p_i G = \{x_j + p_i G~|~j \in [i]\}$ is independent.
This formula is correct because $S_i$ is independent iff $G/p_i G \cong \begin{itemize}goplus_{j \in [i]} \mathbb{Z}_{p_i}^{(j)}$ where each $\mathbb{Z}_{p_i}^{(j)}$ is a copy of $\mathbb{Z}_{p_i}$ iff $S_{[i]}+p_i G$ is independent.
(Strictly speaking, we need the assumption that $S_{[i]}+p_i G$ does not contain the identity element $p_i G$, which is the first part of $\xi_{[i]}$ below.)
As a part of $\xi_{[i]}$, we use a modified version of the formula~$\pi_n$;
\[
\pi'_i(g;x) = \delta_k(g;x)
\]
where $k = \lceil \mathtt{log_2} \ p_i \rceil$ and $\delta_k$ is as defined in Lemma \ref{generation}.
So $G \models \pi'_i(g;x)$ iff $g=z \cdot x$ for some non-negative integer $z \le r$ where $r$ is the smallest power of $2$ not smaller than $p_i$.
In particular, $G \models \pi'_i(z \cdot x;x)$ for all $0 \le z < p_i$.
Note that $\pi'_i$ has length $O(\mathtt{log} \ p_i)$.
Now we are ready to write~$\xi_{[i]}$;
\[
\begin{enumerate}gin{split}
\xi_{[i]}(a_1,\ldots,a_n) \mathbf{e}nd{quotation}uiv \ &\begin{itemize}gwedge_{j \in [i]} \noindentexists b \left[ \theta_{p_i}(b,a_j) \right] \ \wedge\\
&\forall b_1 \ldots \forall b_{\langlembda(i)} \left[ \left( \begin{itemize}gwedge_{j \in [i]} \pi'_i(a_j,b_{\varphi(j)}) \wedge \mathbf{e}xistsists c \left[ \theta_{p_i} \left(c,\sum_{j \in [i]} b_{\varphi(j)}\right) \right] \right) \right.\rightarrow\\
&\hspace{125pt}\left. \mathbf{e}xistsists c_1, \ldots, \mathbf{e}xistsists c_{\langlembda(i)} \begin{itemize}gwedge_{j \in [i]} \theta_{p_i} (c_{\varphi(j)},b_{\varphi(j)}) \right]
\mathbf{e}nd{split}
\]
where $\langlembda(i)$ is the size of $[i]$ and $\varphi$ is a bijection from $[i]$ to $\{1,\ldots,\langlembda(i)\}$.
The second part says that if a linear combination $\sum_{j \in [i]} z_j a_j$ is in $p_i G$ for some non-negative integers ${z_j \le r = 2^{\lceil \mathtt{log_2}~p_i \rceil}}$, then $z_j a_j \in p_i G$ for each $j$.
Now consider the length of $\psi_3$.
For each $i$, $\psi_3$ contains:
\begin{enumerate}gin{itemize}
\item one $\theta_{p_i^{z_i-1}}$, which has length $\simeq 10(z_i-1)\lfloor \mathtt{log_2}~p_i \rfloor$
\item three $\theta_{p_i}$, each of which has length $\simeq 10\lfloor \mathtt{log_2}~p_i \rfloor$
\item one $\pi'_i$, which has length $\simeq 26 \lceil \mathtt{log_2}~p_i \rceil$.
\mathbf{e}nd{itemize}
Hence the length of the formula $\psi_3$ has order of
\[
\sum_{1 \le i \le n} z_i \ \mathtt{log} \ p_i = \mathtt{log} \left( \preceqod_{1 \le i \le n} p_i^{z_i} \right) = \mathtt{log}|H|
\]
and so the length of the whole sentence $\psi$ is $O(\mathtt{log}|H|)$.
\mathbf{e}nd{proof}
\subseteqsection{Upper unitriangular matrix groups} \langlebel{unitriangular}
In Subsection \ref{UT}, we analyzed the structure and some properties of the group $UT_3(\mathbb{Z})$.
Using some of those results, and also some part of the previous theorem, we consider similar finite groups, namely ${UT_3(n)=UT_3(\mathbb{Z}_n)}$ where $n$ is any positive integer.
Each $UT_3(n)$ is isomorphic to the free 2-generated class 2 nilpotent group with exponent $n$.
Their freeness can be shown in a similar way to the case of $UT_3(\mathbb{Z})$ (see \cite[Exercise 16.1.3]{Kar.Mer:79}).
\begin{enumerate}gin{proposition} \langlebel{UT finite thm}
The class of the unitriangular groups $UT_3(n)$ is LC.
\mathbf{e}nd{proposition}
\begin{enumerate}gin{proof}
Let ${a=t_{23}(1)}$, ${b=t_{12}(1)}$.
(Recall that $t_{mn}(k)$ denotes the 3-by-3 matrix with $1$ in its diagonal entries, $k$ in the $m$-th row $n$-th column entry and $0$ everywhere else.)
We begin by analyzing the structure of the group $H=UT_3(n)$.
Since \[ \left[ \left(
\begin{enumerate}gin{array}{@{}ccc@{}}
1 & \mathbf{a}lpha_1 & \begin{enumerate}ta_1 \\
0 & 1 & \gamma_1 \\
0 & 0 & 1
\mathbf{e}nd{array}
\right), \left(
\begin{enumerate}gin{array}{@{}ccc@{}}
1 & \mathbf{a}lpha_2 & \begin{enumerate}ta_2 \\
0 & 1 & \gamma_2 \\
0 & 0 & 1
\mathbf{e}nd{array}
\right) \right] = t_{13}(\mathbf{a}lpha_1\gamma_2-\mathbf{a}lpha_2\gamma_1), \]
holds in $H$, the center $Z$ of $H$ is
\[
Z=\{t_{13}(z)~|~z \in \mathbb{Z}_n\}=\langlengle c \ranglengle
\]
where $c=[a,b]=t_{13}(1)$.
Now $H/Z$ is isomorphic to $\mathbb{Z}_n \oplus \mathbb{Z}_n$ (generated by $aZ,bZ$) which is abelian, so $H$ is class 2 nilpotent.
Hence each element $h \in H$ can be written as a product of the form $h=xyz$ where $x \in \langlengle a \ranglengle$, $y \in \langlengle b \ranglengle$, $z \in Z$, and $H'$~coincides with the set of commutators by Lemma \ref{nilpotent commutators}.
Let $\varphi$ be the formula
\[
\begin{enumerate}gin{split}
\varphi(h,x,y,z;a,b) \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists u \mathbf{e}xistsists v \{&\pi'_n(u;a)~\wedge~\pi'_n(v;b)~\wedge~\forall w~[z,w]=1~\wedge\\
&h=uvz~\wedge~x=[u,b]~\wedge~y=[a,v]\}
\mathbf{e}nd{split}
\]
with parameters $a,b$ where $\pi'_n(r;s)=\delta_k(r;s)$, $k=\lceil \mathtt{log_2}~n \rceil$
(i.e.~$H \models \pi'_n(r;s)$ iff $s=r^z$ for some $z \in \mathbb{Z}_n$).
Then it defines a bijection $\Phi : H \rightarrow Z \times Z \times Z$ such that $\Phi(h)=(x,y,z)$ for
${h = \left(
\begin{enumerate}gin{array}{@{}ccc@{}}
1 & y & z \\
0 & 1 & x \\
0 & 0 & 1
\mathbf{e}nd{array}
\right) \in H}$.
Note that $\varphi$ has length $O(\mathtt{log}~n)$.
Now we are ready to write the sentence $\psi$ describing $H$.
Let $n = \preceqod_{1 \le i \le m} p_i^{z_i}$ be the prime decomposition of $n$.
Then the sentence is $\psi \mathbf{e}nd{quotation}uiv \mathbf{e}xistsists a \mathbf{e}xistsists b [\psi_1 \wedge \ldots \wedge \psi_6]$ where $\psi_1$ says that $a,b$ have order dividing $n$
\[
\psi_1(a,b) \mathbf{e}nd{quotation}uiv \theta_n(a,1)~\wedge~\theta_n(b,1)
\]
$\psi_2$ says that $c$ has order $n$ (using the previous result)
\[
\begin{enumerate}gin{split}
\psi_2(a,b) \mathbf{e}nd{quotation}uiv~&\theta_n([a,b],1)~\wedge \\
&\mathbf{e}xistsists c_1 \ldots \mathbf{e}xistsists c_m \left[ \begin{itemize}gwedge_{1 \le i \le m} \left\{ \pi'_n(c_i;[a,b])~\wedge~\theta_{p_i^{z_i}}(c_i,1)~\wedge~\noindenteg \theta_{p_i^{z_i-1}}(c_i,1) \right\} \right]
\mathbf{e}nd{split}
\]
$\psi_3$ says that $H'=Z=\langlengle c \ranglengle$ coincides with the set of commutators
\[
\begin{enumerate}gin{split}
\psi_3(a,b) \mathbf{e}nd{quotation}uiv~&\forall r \forall s \forall t \forall u \mathbf{e}xistsists v \mathbf{e}xistsists w~[r,s][t,u]=[v,w]~\wedge~\forall r \forall s \forall h [[r,s],h]=1~\wedge \\
&\forall z \{ \forall h~[z,h]=1 \rightarrow (\mathbf{e}xistsists r \mathbf{e}xistsists s~[r,s]=z~\wedge~\pi'_n([a,b],z)\}
\mathbf{e}nd{split}
\]
$\psi_4,\psi_5$ say that $\Phi(h)$ is a function $H \rightarrow Z \times Z \times Z$
\[
\begin{enumerate}gin{split}
\psi_4(a,b) \mathbf{e}nd{quotation}uiv~&\forall h \mathbf{e}xistsists x \mathbf{e}xistsists y \mathbf{e}xistsists z~\varphi(h,x,y,z;a,b)\\
\psi_5(a,b) \mathbf{e}nd{quotation}uiv~&\forall h \forall x_1 \forall x_2 \forall y_1 \forall y_2 \forall z_1 \forall z_2\\
&[\{\varphi(h,x_1,y_1,z_1;a,b)~\wedge~\varphi(h,x_2,y_2,z_2;a,b)\} \rightarrow\\ &\hspace{100pt}\{x_1=x_2~\wedge~y_1=y_2~\wedge~z_1=z_2\}]
\mathbf{e}nd{split}
\]
and $\psi_6$ says that $\Phi$ is surjective
\[
\psi_6(a,b) \mathbf{e}nd{quotation}uiv~\forall x \forall y \forall z \{\forall g~[x,g]=[y,g]=[z,g]=1 \rightarrow \mathbf{e}xistsists h~\varphi(h,x,y,z;a,b)\}.
\]
It can be easily seen that $\psi$ has length $O(\mathtt{log}~n)$.
Now let $G$ be a group satisfying $\psi$ with witnesses $a,b \in G$.
Then from $\psi_2,\psi_3$, the center $Z$ of $G$ is cyclic of order $n$ generated by $c=[a,b]$.
Since $\varphi$ defines an surjective function $\Phi : G \rightarrow Z \times Z \times Z$ from $\psi_4,\psi_5,\psi_6$, $G$ has size at least $n^3$.
But since $a,b$ have order at most $n$ from $\psi_1$ and each element $g \in G$ can be written as a product of the form $g=uvz$ where $u \in \langlengle a \ranglengle$, $v \in \langlengle b \ranglengle$, $z \in Z$ from $\psi_4$, $G$ cannot have more than $n^3$ elements.
Hence $G$ has precisely $n^3$ elements, and has the form $G=\{a^{\mathbf{a}lpha} b^{\begin{enumerate}ta} c^{\gamma}~|~\mathbf{a}lpha, \begin{enumerate}ta, \gamma \in \mathbb{Z}_n \}$.
Since $G$ is class 2 nilpotent from $\psi_3$ and $c=[a,b]$, one can deduce the equation
\[
a^{\mathbf{a}lpha_1} b^{\begin{enumerate}ta_1} c^{\gamma_1} \cdot a^{\mathbf{a}lpha_2} b^{\begin{enumerate}ta_2} c^{\gamma_2} = a^{\mathbf{a}lpha_1+\mathbf{a}lpha_2} b^{\begin{enumerate}ta_1+\begin{enumerate}ta_2} c^{\gamma_1+\gamma_2-\mathbf{a}lpha_2 \begin{enumerate}ta_1}
\]
and it determines the group uniquely up to isomorphism.
\mathbf{e}nd{proof}
The short presentation conjecture \cite{Babai:97} asks whether there exists a constant $C$ such that every finite group $G$ has a presentation of length $O({\mathtt {log}}^c~|G|)$. In analogy, we ask:
\begin{enumerate}gin{question} Is the class of finite groups polylogarithmically compressible (PLC)? Is it in fact logarithmically compressible? \mathbf{e}nd{question}
\def$'${$'$}
\begin{enumerate}gin{thebibliography}{10}
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\noindentewblock Short presentations for three-dimensional unitary groups.
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M.~I. Kargapolov and Ju.~I. Merzljakov.
\noindentewblock {\mathbf{e}m Fundamentals of the theory of groups}, volume~62 of {\mathbf{e}m
Graduate Texts in Mathematics}.
\noindentewblock Springer-Verlag, New York, 1979.
\noindentewblock Translated from the second Russian edition by Robert G. Burns.
\begin{itemize}bitem{Lang:84}
Serge Lang.
\noindentewblock {\mathbf{e}m Algebra}.
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second edition, 1984.\begin{itemize}bitem{Lang:05}Serge~Lang. \noindentewblock {\mathbf{e}m Undergraduate algebra}. \noindentewblock Springer Science+Business Media, New York, third edition, 2005.
\begin{itemize}bitem{Malcev:71}
Anatoli{\u\i}~Ivanovi{\v{c}} Mal{$'$}cev.
\noindentewblock {\mathbf{e}m The metamathematics of algebraic systems. {C}ollected papers:
1936--1967}.
\noindentewblock North-Holland Publishing Co., Amsterdam, 1971.
\noindentewblock Translated, edited, and provided with supplementary notes by Benjamin
Franklin Wells, III, Studies in Logic and the Foundations of Mathematics,
Vol. 66.
\begin{itemize}bitem{Nies:03}
Andr{\'e} Nies.
\noindentewblock Separating classes of groups by first-order sentences.
\noindentewblock {\mathbf{e}m Internat. J. Algebra Comput.}, 13(3):287--302, 2003.
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Andre Nies.
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F.~Oger.
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models.
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Paulo Ribenboim.
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\mathbf{e}nd{thebibliography}
\mathbf{e}nd{document} |
\begin{document}
\begin{abstract}
The Directed Power Graph of a group is a graph whose vertex set is the elements of the group, with an edge from $x$ to $y$ if $y$ is a power of $x$. The \textit{Power Graph} of a group can be obtained from the directed power graph by disorienting its edges. This article discusses properties of cliques, cycles, paths, and coloring in power graphs of finite groups. A construction of the longest directed path in power graphs of cyclic groups is given, along with some results on distance in power graphs. We discuss the cyclic subgroup graph of a group and show that it shares a remarkable number of properties with the power graph, including independence number, completeness, number of holes etc., with a few exceptions like planarity and Hamiltonian.
\end{abstract}
\title{Power Graphs of Finite Group}
\section{Introduction}
The power graph of finite groups has been studied in \cite{Power finite}, \cite{power semi}, \cite{power combi}, \cite{Power finiteII}.
In this work, we denote the group of integers mod $n$ under addition by $\mathbb{Z}_n$, or $\mathbb{Z} / n\mathbb{Z}$. These notations are used interchangeably in this article. It can be shown that any cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n$.
A $\textit{graph}$ is a pair $\Gamma=(V(\Gamma),E(\Gamma))$ where $V(\Gamma)$ is a non-empty set, called the \textit{vertex set} whose elements are called \textit{vertices}, and $E(\Gamma)$ is a (possibly empty) set consisting of sets of pairs of elements of $V(\Gamma)$ called the $\textit{edge set}$, whose elements are called $\textit{edges}.$ If $e\in E(\Gamma)$ is an edge and $v\in e$, then $e$ is said to be $\textit{incident }$to $v$. The \textit{degree} of a vertex $v$ is the number of edges incident to $v$. If $\{v_1,v_2\}\in E(\Gamma)$, then $v_1$ and $v_2$ are said to be adjacent. Where there is no ambiguity, $V(\Gamma)$ is sometimes denoted as just $V$, and similarly $E(\Gamma)$ as E. A $\textit{directed graph}$ or digraph is a pair $\Gamma=(V(\Gamma),E(\Gamma))$ where $V(\Gamma)$ is a non-empty vertex set, and $E$ is a (possible empty) set consisting of ordered pairs of elements in $V(\Gamma)$ called the edge set.
Let $G$ be a group. The \textit{power graph} of $G$, denoted by $\mathfrak{g}(G)$ is defined as the graph with vertex set consisting of the elements of $G$, and edge set $E=\{\{x,y\}\mid x\neq y $ and $\langle x\rangle \leq \langle y\rangle$ or $\langle y\rangle \leq \langle x\rangle \}$. The \textit{directed power graph} of $G$, denoted $\vec{\mathfrak{g}}(G)$ is defined as the graph with vertex set consisting of the elements of $G$, and edge set $E=\{(x,y)\mid x\neq y $ and $\langle y\rangle \leq \langle x\rangle$\}.
\section{Some Properties of Power Graphs of Groups}
A graph $\Gamma$ is called $\textit{connected}$ if for any two vertices $u$ and $v$ in $\Gamma$, there is a path joining $u$ and $v$. In a connected graph $\Gamma$, the distance between $u$ and $v$, denoted $d_{\Gamma}(u,v)$, can be defined as the length of the shortest path joining $u$ and $v$. The $\textit{eccentricity}$ of a vertex $u$ is the maximum distance between $u$ and any other vertex in $\Gamma$. The $\textit{radius}$ of $\Gamma$ is the minimum eccentricity of a vertex in $\Gamma$, and the $\textit{diameter}$ of $\Gamma$ is the maximum eccentricity of a vertex in $\Gamma$. A central vertex in $\Gamma$ is a vertex with eccentricity equal to the radius of $\Gamma$. The $\textit{center}$ of $\Gamma$ is the set of central vertices in $\Gamma$.
\begin{proposition}
It is well known that the Power graphs of finite groups are connected.
\end{proposition}
\begin{proof}
Let $G$ be a group with power graph $\mathfrak{g}(G)$ and identity $e$. Let $g$ be an arbitrary element of $G$. Since $G$ is a group, $\langle e\rangle \leq \langle g\rangle$, so in the power graph of a group there is an edge between the identity and every other group element.
\end{proof}
Note that the distance between the identity and any other vertex is $1$, in the power graph of a group, giving the following corollary.
\begin{Corollary}
The radius of the power graph of a group is $1$, and the center of the power graph of a group of order $n$ is the set of vertices with degree $n-1$.
\end{Corollary}
The $\textit{center}$ of a group $G$ is the set of elements of $G$ which commute with all other elements of $G$. The following proposition gives a relation between the center of a power graph of a group and the center of the group.
\begin{proposition}
Let $G$ be a group of order $n$ with power graph $\mathfrak{g}(G)$. The vertices in the center of the power graph $\mathfrak{g}(G)$ are in the center of the group $G$.
\end{proposition}
\begin{proof}
Let $g\in G$ and suppose $deg_{\mathfrak{g}(G)}(g)=n-1$. Then for any other $h\in G$, either $\langle h\rangle \leq \langle g\rangle$, or $\langle g\rangle \leq \langle h\rangle$, either way $g$ and $h$ commute. Hence, $g$ is in the center of $G$.
\end{proof}
Since power graphs of groups are connected and the identity is adjacent to every other element, there is a path of length at most $2$ between any two vertices of a power graph of a group, so the diameter of the power graph of a group is $2$, unless the power graph is complete, in which case it is $1$.
\begin{proposition}
Let $G$ be a non trivial Abelian group of order $n$. The center of $\mathfrak{g}(G)$ has cardinality
\begin{enumerate}
\item $\phi(n)+1$, if $G$ is cyclic and $n\neq p^k$ for any prime $p$ and positive integer $k$.
\item $n$, if $G$ is cyclic and $n=p^k$ for some prime $p$ and positive integer $k$.
\item $1$, if $G$ is non-cyclic.
\end{enumerate}
\end{proposition}
\begin{proof}$\phantom e$
\begin{enumerate}
\item[(1):] First let $G$ be a cyclic group whose order is $n$, and suppose $n$ is not a power of a prime. Note, all the $\phi(n)$ many generators of $G$ and the identity are in the center of $\mathfrak{g}(G)$, so the center has size at least $\phi(n)+1$. Since $n$ is not a power of a prime, $n$ has at least two distinct prime factors. Consider any $b\in G$, where $b$ is neither a generator of $G$, nor the identity of $G$. So, if order of $b$ is $m$, then $m<n$. Hence, by converse of Lagrange's Theorem over finite Abelian group, there exists $c\in G$, where order of $c$ is $r$ and $r$ does not divide $m$, $m$ does not divide $r$. So, there is no edge between $b$ and $c$. Since $b$ is an arbitrary non-generator of $G$, which is not the identity of $G$ either, the cardinality result follows.
\item[(2):] Now suppose $G$ is cyclic and it's order is $n=p^k$ for some prime $p$ and positive integer $k$. Then, the graph being complete, by \cite{power semi}, the center has cardinality $n$.
\item[(3):] Let, $G$ be not cyclic. Then, the result follows by the proposition \ref{prop 8} in the next section.
\end{enumerate}
\end{proof}
\section{Some results relating Composition series in power graphs}
The Jordan-H\"{o}lder theorem asserts that any two composition series of a given group have the same length, and isomorphic factor groups up to permutation. The length of a composition series of a group $G$ is called its \textit{composition length} and denoted as $\ell$(G). The composition length $\ell(G)$ is also the maximum length a normal series of $G$ can have. Then a group $G$ can be partitioned in the following way, let $X_i$ be the set of all elements of $G$ which generate a cyclic subgroup with composition length $i$. Then $G=\bigcup_i X_i$, and $X_i\cap X_j=\varnothing$ for any $i\neq j$.
\begin{Lemma}
The composition length of an Abelian group is the sum of the exponents of the order of the group, when written as a product of prime numbers.
\end{Lemma}
\begin{proof}
An Abelian group is simple only if its order is prime. Then if $\lvert G\rvert=p_1^{k_1}p_2^{k_2}\cdots p_n^{k_n}$, then each $G_{i}/G_{i+1}$ must have prime order, that is $\lvert G_{i}\rvert=p_i\lvert G_{i+1}\rvert$. Then each $G_{i-1}$ has an order whose exponents sum to one less than those in the order of $G_i$, so there must be $k_1+k_2+\cdots+k_n$ inclusions in a maximal length normal series of an Abelian group.
\end{proof}
\begin{proposition}
If $X_i$ is non-empty, then $X_{i-1}$ is non-empty.
\end{proposition}
\begin{proof}
Let $x\in X_i$ and $\langle x\rangle\supset \langle x_{i-1}\rangle\supset\cdots\langle x_2\rangle \supset \{e\}$ be a composition series. Then there is a normal series of length $i-1$ from $\langle x_{i-1}\rangle$. Suppose there were a normal series of length $i$ or greater from $\langle x_{i-1}\rangle$, then $\langle x\rangle \supset \langle x_{i-1}\rangle \supset\cdots\supset \{e\}$ is a normal series from $\langle x\rangle$ of length greater than $i$, contradicting the assumption that $x\in X_i$.
\end{proof}
\begin{Corollary}
Removing subgroups from the beginning of a composition series of length greater than zero from a cyclic subgroup leaves a composition series.
\end{Corollary}
\begin{proposition}
If some vertex in $X_i$ is adjacent to all other vertices in $X_i$, then $X_i$ is a clique in $\mathfrak{g}(G)$.
\end{proposition}
\begin{proof}
If $x_1,x_2\in X_i$ and they are adjacent in $\mathfrak{g}(G)$, then either $\langle x_1\rangle \leq \langle x_2\rangle$ or $\langle x_2\rangle \leq \langle x_1\rangle$. Without loss of generality suppose $\langle x_2\rangle$ is a proper subgroup of $\langle x_1\rangle$. Then there exists a composition series $\langle x_2\rangle\supset \langle x_3\rangle\supset\cdots\supset\{e\}$ of length $i$ and $\langle x_1\rangle\supset \langle x_2\rangle\supset\cdots\supset\{e\}$ is a normal series of length $i+1$, contradicting the assumption that $x_1\in X_i$, so it must be the case that $\langle x_2\rangle=\langle x_1\rangle$. Then, if any element of $X_i$ is adjacent to all other elements of $X_i$ in $\mathfrak{g}(G)$, then all elements of $X_i$ generate the same subgroup and form a clique in $\mathfrak{g}(G).$
\end{proof}
\begin{proposition}
If $X_i$ is a clique in $\mathfrak{g}(G)$, then $X_{i+1}$ is a clique in $\mathfrak{g}(G)$.
\end{proposition}
\begin{proof}
Suppose $X_i$ is a clique in $\mathfrak{g}(G)$ and $X_{i+1}$ is not a clique in $\mathfrak{g}(G)$, and let $x\in X_i$. Then there exist two distinct elements in $X_{i+1}$, say $y$ and $z$, with orders $p\lvert x\rvert$ and $q\lvert x\rvert$ respectively where $p$ and $q$ are prime and $p\neq q$. Write $\lvert x\rvert=p_1^{k_1}p_2^{k_2}\cdots p_n^{k_n}$, with $\sum_{t=1}^n k_t=i$. Then $\langle y\rangle$ has a subgroup of order $pp_1^{k_1-1}p_2^{k_2}\cdots p_n^{k_n}$, and $\langle z\rangle$ has a subgroup of order $qp_1^{k_1-1}p_2^{k_2}\cdots p_n^{k_n}$. Both of these subgroups are in $X_i$, but since $p\neq q$ at least one of the subgroups is not equal to $\langle x\rangle$, contradicting the assumption that $X_i$ is a clique in $\mathfrak{g}(G)$.
\end{proof}
\begin{proposition}
If $X_i$ is a clique in $\mathfrak{g}(G)$ and $x\in X_i$ and $\langle x\rangle \neq p^k$ for some prime $p$ and positive integer $k$, then $X_{i+1}$ is empty.
\end{proposition}
\begin{proof}
Suppose $x\in X_i$, $\langle x\rangle \neq p^k$, so $\lvert x\rvert$ has at least two distinct prime factors, say $p$ and $q$. Suppose there exists a $y\in X_{i+1}$, then $pq \mid \lvert y\rvert$, but then $\lvert y\rvert=pqm$ for some $m$ (possible equal to $p$ or $q$), and in that case $\langle y\rangle$ has subgroups of orders $pm$ and $qm$, both in $X_i$, contradicting the assumption that $X_i$ is a clique in $\mathfrak{g}(G)$.
\end{proof}
\begin{proposition}\label{prop 8} If $G$ is a finite Abelian group and not cyclic, then $max_iX_i$ is not a clique in $\mathfrak{g}(G)$.
\end{proposition}
\begin{proof}
Since $G$ is a finite non-cyclic Abelian group, $G\cong \mathbb{Z}_{p_1}^{k_1}\times\mathbb{Z}_{p_2}^{k_2}\times\cdots\times\mathbb{Z}_{p_n}^{k_n}$, where not all of the $p_{i}'s$ are distinct. Then, $G\cong \mathbb{Z}_m\times\mathbb{Z}_{p_i}^{k_i}\times\cdots\times\mathbb{Z}_{p_t}^{k_t}$, where $p_i^{k_i}$ $\mid$ $m,\cdots$ $p_t^{k_t}$ $\mid$ $m$, and $m=q_1^{l_1}q_2^{l_2}\cdots q_r^{l_r}$ where $q_1,q_2,\cdots,q_r$ are relatively prime. Notice that $g=(1,0,\cdots,0)$ and $g'=(1,1,0,\cdots,0)$ both have the maximum possible order in $G$, that is $m$. Also, notice that $\langle g\rangle \neq \langle g'\rangle$, so $g$ and $g'$ are both in $max_iX_i$ but not adjacent to each other, so $max_iX_i$ is not a clique in $\mathfrak{g}(G)$.
\end{proof}
Since $max_iX_i$ is not a clique in $\mathfrak{g}(G)$, then no $X_i$ is a clique in $\mathfrak{g}(G)$. Since $G=\bigcup_iX_i$, no non-identity vertex in $\mathfrak{g}(G)$ has degree $n-1$.
\section{Perfect Graphs}
A \textit{clique} in a graph $\Gamma$ is a subset $C$ of the vertices of $\Gamma$, such that any two vertices in $C$ are adjacent. The size of the largest clique in $\Gamma$ is called the \textit{clique number} of $\Gamma$, and is denoted $\omega(\Gamma)$.
A \textit{vertex coloring} of a graph is an assignment of labels (called colors) to the vertices of a graph $\Gamma$ such that no two adjacent vertices share the same color. The smallest number of colors which can be used in a coloring of $\Gamma$ is called the \textit{chromatic number} of $\Gamma$, and is denoted $\chi(\Gamma)$.
A graph $\Gamma$ is called \textit{perfect} if for every induced subgraph $\Gamma_i$ of $\Gamma$, $\omega(\Gamma_i)=\chi(\Gamma_i).$ It was conjectured by Berge in 1961, and was proved by Chudnovsky et. al. in \cite{PerfectGraphs} that graphs are perfect if and only if they contain no holes or anti-holes of odd length greater than $3$. A \textit{hole} in a graph is a cycle such that no two vertices in the cycle are joined by an edge which does not itself belong to the cycle. An anti-hole is the edge complement of a hole.
\begin{Theorem}[Strong Perfect Graph Theorem]
A graph is perfect if and only if it contains no odd holes or odd anti-holes of length greater than 3.
\end{Theorem}
Let $G$ be a group with power graph $\mathfrak{g}(G)$. Then $\mathfrak{g}(G)$ can contain no holes of odd length. To prove this a few short lemmas are used.
\begin{Lemma}[Path]
Let $G$ be a group with directed power graph $\vec{\mathfrak{g}}(G)$. If a path exists between two vertices then they are adjacent.
\begin{proof}
Denote the start of the path as vertex $a$ and label the vertices along the path as $a_1,a_2,\cdots a_m$. Then $a_1=a^n,a_2={a_1}^{n_1},a_3={a_2}^{n_2}\cdots a_m={a_{m-1}}^{n_{m-1}}$. Then $a_m=a^{nn_1n_2\cdots n_{m-1}}$, so there is an edge from $a$ to $a_m$.
\end{proof}
\end{Lemma}
\begin{Lemma}[Strong Path]
Let $G$ be a group with directed power graph $\vec{\mathfrak{g}}(G)$, and let $\vec{\mathfrak{g}}(G)$ contain a directed path of length $n$, then the vertices making up the directed path form a clique of size $n$ in $\mathfrak{g}(G)$.
\end{Lemma}
\begin{proof}
The proof follows from the previous (Path) Lemma.
\end{proof}
The strong path lemma shows that whenever there is a path of length $n$ in $\vec{\mathfrak{g}}(G)$ there is a clique of size $n$ consisting of the same vertices in $\mathfrak{g}(G)$. Here it will be shown that the converse is true as well, that is, whenever there is a clique of size $n$ in $\mathfrak{g}(G)$, then those vertices are traversable by a path in $\vec{\mathfrak{g}}(G)$. Path-clique equivalence in power graphs of finite groups is a consequence of R\'{e}dei's theorem~\cite{Redei}.
As per our need, later in our work, we reproduced the proof of the following well known theorem here one more time.
\begin{Theorem}[R\'{e}dei's Theorem]
Every orientation of a complete graph contains a directed Hamiltonian path.
\end{Theorem}
\begin{Theorem}[Path-clique Equivalence]\label{theo 3}
Let $G$ be a group with directed power graph $\vec{\mathfrak{g}}(G)$ and undirected power graph $\mathfrak{g}(G)$. Then whenever there is a path in $\vec{\mathfrak{g}}(G)$, its constituent vertices form a clique in $\mathfrak{g}(G)$, and whenever there is a clique in $\mathfrak{g}(G)$, its vertices are traversable by a path in $\vec{\mathfrak{g}}(G)$.
\end{Theorem}
\begin{proof}
The proof of the first direction is given above, here we show that a clique in $\mathfrak{g}(G)$ is traversable by a path in $\vec{\mathfrak{g}}(G)$. Let $\mathfrak{g}(G)$ contain a clique of size $\alpha$. The proof is by induction on $\alpha$. If $\alpha=1$, then the clique is traversable by the path consisting of only the single vertex in the clique.
Suppose the result holds for $0<\alpha \leq k$, and let there exist in $\mathfrak{g}(G)$ a clique of size $k+1$. By the induction hypothesis, a clique of size $k$ is traversable by a directed path, so at most one vertex is excluded from the longest path through the clique. Proceed along that directed path through these vertices and label them in the order they are encountered, $v_1, v_2, v_3,\cdots,v_k$. If for any $v_i$ in the clique both $(v_i,v_{k+1})$ and $(v_{k+1},v_i)$ are edges in $\vec{\mathfrak{g}}(G)$, then the sequence can be modified from $(\cdots v_i,v_{i+1}\cdots)$ to $(\cdots v_i, v_{k+1}, v_{i+1}\cdots)$ adding $v_{k+1}$ to the path. Suppose no two-sided edges exist in the clique. If $(v_{k+1}, v_1)$ is an edge in $\vec{\mathfrak{g}}(G)$, then $v_{k+1}$ can be added to the beginning of the existing path. If $(v_k,v_{k+1}) $ is an edge in $\vec{\mathfrak{g}}(G)$, then $v_{k+1}$ can be added to the end of the existing path. Suppose neither of these are edges in $\vec{\mathfrak{g}}(G)$. If $(v_{k+1}, v_2)$ is an edge is $\vec{\mathfrak{g}}(G)$, then $v_{k+1}$ can be inserted in the path between $v_1$ and $v_2$. Then, if $(v_{k+1}, v_2)$ is not an edge in $\vec{\mathfrak{g}}(G)$, $(v_2,v_{k+1})$ must be an edge in $\vec{\mathfrak{g}}(G)$. Proceeding in this way we get the desired directed path in $\vec{\mathfrak{g}}(G)$.
\end{proof}
\begin{figure}
\caption{An illustration of Theorem \ref{theo 3}
\caption{An edge in both directions between $v_{k+1}
\caption{If there is an edge from $v_k$ to $v_{k+1}
\caption{If there is an edge from $v_{k+1}
\caption{If none of the other cases hold, then there must exist a pair of vertices in between which $v_{k+1}
\end{figure}
\setcounter{figure}{2}
It is well known that
Power graphs of groups are perfect, which has been proved by using
Theorem~\cite{PerfectGraphs}.
\section{The Cyclic Subgroup Graph}
Let $G$ be a group and define the relation $\sim$ on $G$ by $x\sim y$ if $\langle x\rangle=\langle y\rangle.$ Define the graph $\vec{C}(G)$ by $V(\vec{C}(G))= G / \mathord{\sim}$ and $(A,B)\in E(\vec{C}(G))$ if there exists elements $b\in B$ and $a\in A$ such that $\langle b\rangle \leq \langle a\rangle$ and $\langle b \rangle \neq \langle a\rangle.$ Also define a weight function $w:V(\vec{C}(G))\to \mathbb{N}$ by $w(A)=\lvert A\rvert$. Then $\vec{C}(G)$ is a directed acyclic graph with a similar structure to the directed power graph $\vec{\mathfrak{g}}(G).$ \begin{proposition}The weight of the path with the largest weight in $\vec{C}(G)$ is the length of the longest path in $\vec{\mathfrak{g}}(G).$
\end{proposition}
\begin{proof}
The vertices in $\vec{C}(G)$ with only out-edges represent generators of the maximal cyclic subgroups of $G$. As in the proof of Theorem 3 above any longest path in $G$ must start with these vertices. If this maximal cyclic subgroup has order $n$ then a vertex adjacent to it represents generators of a cyclic subgroup of order $\frac{n}{d}$ where $d$ is a divisor of $n$, and if the subgroup is maximal $d$ will be a prime divisor of $n$. Then the sum of the weights of the vertices in a path from a maximal cyclic subgroup of $G$ of order $n$ through all of its maximal subgroups of maximum order to the trivial subgroup will be given by $\Psi(n)$, the length of the longest path in $\vec{\mathfrak{g}}(G)$.
\end{proof}
When $G$ is cyclic there is no ambiguity in naming vertices in $\vec{C}(G)$ by their corresponding isomorphic group $\mathbb{Z}_n$, for example
\begin{figure}
\caption{$\vec{C}
\end{figure}
The non-oriented graph $C(G)$ also contains a lot of the information in the power graph in a smaller form. For example $d_{\mathfrak{g}(G)}(u,v)=d_{C(G)}([u]_{\sim},[v]_{\sim})$ as long as $[u]_{\sim}\neq [v]_{\sim}$ and the independence number $\alpha(\mathfrak{g}(G))=\alpha(C(G))$.
\begin{proposition}
The cyclic subgroup graph is isomorphic to an induced subgraph of the power graph.
\end{proposition}
\begin{proof}
Define $f:V(C(G))\to(V\mathfrak{g}(G))$ by $f([x]_{\sim})=x^{'}$, for a fixed choice of the corresponding equivalence class representative $x^{'}$. The result follows by the definition of cyclic subgroup graph.
\end{proof}
\begin{proposition}
$d_{\mathfrak{g}(G)}(u,v)=d_{C(G)}([u]_{\sim},[v]_{\sim})$ as long as $[u]_{\sim}\neq [v]_{\sim}$.
\end{proposition}
\begin{proof}
let $G$ be a group and let $u,v\in G$ with $d_{\mathfrak{g}(G)}(u,v)=k$.
Let $P=\{u, u_1,u_2,\cdots,u_k, v\}$ denote the shortest path from $u$ to $v$ in $\mathfrak{g}(G)$. It must be the case that for each $u_i,u_j\in P$, $[u_i]_{\sim}\neq[u_j]_{\sim}$ otherwise deleting the $u_i$ or $u_j$ we get a shorter path in $\mathfrak{g}(G)$ which is a contradiction. Then $\{[u]_{\sim},[u_1]_{\sim},\cdots,[u_k]_{\sim}\}$ is a path from $[u]_{\sim}$ to $[v]_{\sim}$ in $C(G)$. Suppose there is a shorter path from $[u]_{\sim}$ to $[v]_{\sim}$ in $C(G)$, and denote that path $P_1=\{[u]_{\sim},[y_1]_{\sim},\cdots[y_j]_{\sim},[v]_{\sim}\}$ where $j<k$. Then $\{u,y_1,\cdots, y_j,v\}$ is a path in $\mathfrak{g}(G)$ which is shorter than $P$, a contradiction. Then $\{[u]_{\sim},[u_1]_{\sim},\cdots,[u_k]_{\sim}\}$ is the shortest path from $[u]_{\sim}$ to $[v]_{\sim}$ in $C(G)$ meaning $d_{C(G)}([u]_{\sim},[v]_{\sim})=k=d_{\mathfrak{g}(G)}(u,v)$.
\end{proof}
\begin{proposition}
Let $G$ be a group. Elements $\{g_1,g_2,\cdots,g_k\}$ form an independent set in $\mathfrak{g}(G)$ if and only if $\{[g_1]_{\sim},[g_2]_{\sim},\cdots,[g_k]_{\sim}\}$ is an independent set in $C(G)$.
\end{proposition}
\begin{proof}
Suppose $I=\{g_1,g_2,\cdots,g_k\}$ forms an independent set in $\mathfrak{g}(G)$. Then for each $g_i,g_j\in I$, $\langle g_i\rangle \nleq \langle g_j\rangle$ and $\langle g_j\rangle \nleq \langle g_i\rangle$. So, $\{[g_i]_{\sim},[g_j]_{\sim}\}\notin E(C(G))$, giving an independent set of size $k$ in $C(G)$.
Now suppose $\{[g_1]_{\sim},[g_2]_{\sim},\cdots,[g_k]_{\sim}\}$ is an independent set in $C(G)$. Then $\{g_1,g_2,\cdots,g_k\}$ is an independent set in $\mathfrak{g}(G)$.
\end{proof}
\begin{Corollary}
$\alpha(\mathfrak{g}(G))=\alpha(C(G))$.
\end{Corollary}
\begin{proposition}
$\mathfrak{g}(G)$ is complete if and only if $C(G)$ is complete.
\end{proposition}
\begin{proof}
Suppose $\mathfrak{g}(G)$ is complete and let $[u]_{\sim}$ and $[v]_{\sim}$ be arbitrary vertices in $C(G)$ Since $u\in [u]_{\sim}$ and $v\in [v]_{\sim}$ and $\{u,v\}\in E(\mathfrak{g}(G))$, $\langle u\rangle \leq \langle v\rangle$ or $\langle v\rangle \leq \langle u\rangle$, so $\{[u]_{\sim},[v]_{\sim}\}\in E(C(G))$. Since $[u]_{\sim}$ and $[v]_{\sim}$ were arbitrary, $C(G)$ must be complete.
Now suppose $\mathfrak{g}(G)$ is not complete, then there exist vertices $u$ and $v$ such that $\{u,v\}\notin E(\mathfrak{g}(G))$. Then $\{u,v\}$ is an independent set in $\mathfrak{g}(G)$ and consequently $\{[u]_{\sim},[v]_{\sim}\}$ is an independent set in $C(G)$. Then there exist elements $[u]_{\sim}$ and $[v]_{\sim}$ in $C(G)$ such that $\{[u]_{\sim},[v]_{\sim}\}\notin E(C(G))$ so $C(G)$ is not complete.
\end{proof}
\begin{proposition}
Let $u_1,u_2,\cdots u_{2k}$ be vertices in $\mathfrak{g}(G)$, then $u_1, \cdots u_{2k}$ is a hole in $\mathfrak{g}(G)$ if and only if $[u_1]_{\sim},[u_2]_{\sim},\cdots,[u_{2k}]_{\sim}$ is a hole in $C(G)$.
\end{proposition}
\begin{proof}
Let $P=\{u_1,u_2,\cdots,u_{2k}\}$ be a hole in $\mathfrak{g}(G)$, then each $u_i\in P$ has exactly two neighbors in $P$, call them $u_{i+1}$ and $u_{i-1}$. Then either $\langle u_i\rangle \leq \langle u_{i+1}\rangle$ and $\langle u_i\rangle \leq \langle u_{i-1}\rangle$ or $\langle u_{i+1}\rangle \leq \langle u_{i}\rangle$ and $\langle u_{i-1}\rangle \leq \langle u_{i}\rangle$. Either way, the following are true, $\langle u_i\rangle \neq \langle u_{i+1}\rangle$ and $\langle u_i\rangle \neq \langle u_{i-1}\rangle$, so $[u_i]_{\sim}\neq [u_{i+1}]_{\sim}$ and $[u_i]_{\sim}\neq [u_{i-1}]_{\sim}$, so $[u_{i+1}]_{\sim}$ and $[u_{i-1}]_{\sim}$ are neighbors of $[u_i]_{\sim}$ in $C(G)$. There is no chord in $C(G)$ inside that hole in $\mathfrak{g}(G)$, for if so, then there will be a chord inside the hole in $\mathfrak{g}(G)$ giving a contradiction. So $[u_1]_{\sim},[u_2]_{\sim},\cdots,[u_{2k}]$ forms a hole in $C(G)$.The other direction also follows via a similar argument.
\end{proof}
\begin{Corollary}
$C(G)$ are perfect graphs.
\end{Corollary}
\begin{proposition}
$\mathfrak{g}(G)$ is claw free if and only if $C(G)$ is so.
\end{proposition}
\begin{proposition}
$\mathfrak{g}(Z_n)$ is chordal if and only if $C(Z_n)$ is chordal.
\begin{proof}
The result follows since a hole exists in $C(Z_n)$ if and only if a hole exists in $\mathfrak{g}(Z_n).$
\end{proof}
\end{proposition}
\begin{proposition}
A vertex $x\in \mathfrak{g}(Z_n)$ is simplicial if and only if $[x]_{\sim}\in C(Z_n)$ is simplicial.
\begin{proof}
The result follows as induced subgraph of a complete graph is complete.
\end{proof}
\end{proposition}
\begin{proposition}
If $C(G)$ is Hamiltonian, then $\mathfrak{g}(G)$ is also Hamiltonian.
\end{proposition}
\begin{proof}
Suppose $C(G)$ is Hamiltonian, and let $\{[u_1]_{\sim},[u_2]_{\sim},\cdots,[u_2]_{\sim}$, $[u_1]_{\sim}$ be a Hamiltonian cycle in $C(G)$. $[u_i]_{\sim}$ is a clique in $\mathfrak{g}(G)$ since for $g_1,g_2\in [u_i]_{\sim}$, $\langle g_1\rangle =\langle g_2\rangle$, so $\langle g_1\rangle \leq \langle g_2\rangle$, so $\{g_1,g_2\}\in E(\mathfrak{g}(G))$, so denote $[u_i]_{\sim}$ by $\{u_{1i},u_{2i},\cdots,u_{k_ii}\}$, then $\{u_{11},u_{21},u_{31},\cdots,u_{k_11},u_{12},u_{22},$\\
$\cdots, u_{k_22},\cdots,u_{1n},u_{2n},\cdots u_{k_nn},u_{11}\}$ is a Hamiltonian cycle in $\mathfrak{g}(G).$
\end{proof}
Singh and Devi showed in \cite{CyclicSubgroupGraph} that the cyclic subgroup graph of cyclic groups of non-prime order is Hamiltonian in. Power graphs of groups of prime order are complete, and therefore Hamiltonian for all orders except for $2$, so we note the following corollary by using R\'{e}dei's theorem,~\cite{Redei}.
\begin{Corollary}
Let $G\cong$ $\mathbb{Z}_n$ be a cyclic group with $n\neq 2$, then $\mathfrak{g}(G)$ is Hamiltonian.
\end{Corollary}
Since the cyclic subgroup subgraph is isomorphic to an induced subgraph of the power graph, if $\mathfrak{g}(G)$ is planner, then so is $C(G)$. For example, consider $G= \mathbb{Z}_9$. Here $\mathfrak{g}(G)$ being a complete graph with $9$ vertices it is not planner. $C(G)$ being a triangle is planner.
\section{Chordless Cycles}
It has now been shown that power graphs contain no holes of odd-length. Here it will be shown that for arbitrary even integer $n$, there exists a finite group whose power graph contains a hole of length $n$.
\begin{proposition}
Let $n$ be an even integer, then for even $n>4$, the power graph of a cyclic group will contain a hole of length $n$, if the order of the group has $\frac{n}{2}$ distinct prime factors. The power graph of the group will contain a hole of length $4$, if the order of the group has at least two prime factors of multiplicity two or more.
\end{proposition}
\begin{proof}
First consider the case that $n=4$. Then in the group $\mathbb{Z}_{p^2q^2}$ there is a subgraph consisting of the vertices $p,pq^2,q,$ and $p^2q$. This subgraph will be a hole of length $4$.
Now consider the case that $n\geq 6$, and take primes $p_1,p_2,\cdots p_{\frac{n}{2}}$. Then the group contains a hole of length $n$ namely subgraph consisting of vertices $$p_1-p_1p_2- p_2-p_2p_3-p_3\cdots p_{\frac{n}{2}} \cdots p_{\frac{n}{2}}p_{1}-p_1$$
Then this subgraph is a hole of length $n$ in $\mathfrak{g}(\mathbb{Z}_{p_1p_2...p_{\frac{n}{2}}})$.
\end{proof}
Note: Following the proof of the theorem, it is possible to create many holes of even length permuting the positions of the primes and allowing various exponent of them.
\begin{proposition}
If the power graph of a finite cyclic group $G$ contains a hole of length $n$, then $\lvert G \rvert$ has at least $\frac{n}{2} $ distinct prime factors.
\end{proposition}
\begin{proof}
Suppose $\mathfrak{g}(\mathbb{Z}_m)$ contains a hole of length $n$. This cycle in the corresponding directed power graph consists of $\frac{n}{2}$ vertices with only out-edges and $\frac{n}{2}$ vertices with only in-edges. Each vertex with out-edges is non-adjacent to each other vertex with out-edges, so certainly if $x, y$ are group elements represented by vertices in the hole with out-edges, then $\lvert x\rvert $ does not divide $\lvert y \rvert$, and $\lvert y \rvert$ does not divide $\lvert x\rvert $. Then the order of each of these $\frac{n}{2}$ vertices with out-edges has a prime factor which is not shared by the other $\frac{n}{2}$ vertices with out-edges. By Lagrange's theorem, the order of each element must divide the order of the group, so the order of the group must contain at least $\frac{n}{2}$ prime factors.
\end{proof}
It has been shown that for an arbitrary even integer $n$, a finite group can be found whose power graph contains a hole of length $n$. The proof relied on the fact that there were $\frac{n}{2}$ primes dividing the order of the group so that the multiples of the primes were a subset of their multiples in $\mathbb{Z}$. Then $\mathfrak{g}(\mathbb{Z})$ contains holes of any even length. In fact since there are infinitely many prime numbers, there will be an infinite number of holes of any even length in $\mathfrak{g}(\mathbb{Z})$.
A necessary and sufficient condition for the existence of a hole of length $n$ in finite cyclic groups has been given above. This is not a necessary condition for the existence of holes of length $n$ in general Abelian groups. Consider the Abelian group $\mathbb{Z}_{12}\times \mathbb{Z}_{12}$. This group has order $144=2^4\cdot 3^2$. consider the subgraph of $\mathbb{Z}_{12}\times \mathbb{Z}_{12}$ consisting of the elements $(1,0),(2,0),(1,6),(3,6),(1,2),(2,4),(1,8),(3,0)$. These elements (in order around the cycle) form a hole of length $8$.
\begin{proposition}
Let $G$ be an Abelian group whose order is $p^k$ for some prime $p$ and positive integer $k$. Then $\mathfrak{g}(G)$ can contain no hole.
\end{proposition}
\begin{proof}
The result follows as the power graph is complete in this case.
\end{proof}
\begin{Theorem}
An element whose order is a power of a prime cannot be a vertex with out-edges in a hole of even length.
\end{Theorem}
\begin{proof}
Suppose, that is not the case. Then, $\vec{\mathfrak{g}}(G)$ contains a hole with an element $x$ of order $p^n$ as a vertex with out-edges in the hole. Let $y$ and $z$ be the elements adjacent to $x$. Then $y$ and $z$ are of orders $p^l$ and $p^m$ respectively, where $l,m<n$. $l\neq m$. Without loss of generality, suppose $l<m$, then $\langle y \rangle \subset \langle z \rangle$ as each are proper subgroups of $\langle x \rangle$ which is a primary cyclic group. So there is a chord from $y$ to $z$ giving a contradiction.
\end{proof}
\begin{proposition}
Power graphs of groups can contain no anti-holes of length greater than $4$.
\end{proposition}
\begin{proof}
Let $G$ be a group and suppose that $\mathfrak{g}(G)$ contains an anti-hole of length $n$ greater than $4$. Claim: All vertices in the anti hole have in-degree zero or out-degree zero in the corresponding directed power graph. Proof of the claim:
First note that, if we consider $n=4$, then the claim follows clearly. Arbitrarily choose a vertex $d$ in the anti hole. There must be a vertex $s_1$ in the antihole adjacent to $d$. Without loss of generality, let the source of the corresponding edge be $s_1$ and the destination vertex $d$. Now choose an edge between a vertex $s_2$, which is non-adjacent to vertex $s_1$, and adjacent to $d$. Such an edge must exist as we assume $n>4$, since no two vertices can share the same pair of non-adjacent vertices. The direction of this edge must be from $s_2$ to $d$ since if it were from $d$ to $s_2$ then a directed path would exist between $s_1$ and $s_2$, which by the path lemma would make them adjacent. Repeat the process for a vertex non-adjacent to one of either $s_1$ or $s_2$ and adjacent to $d$ again. The stopping point of this process is when all $n-3$ vertices adjacent to vertex $d$ have been selected. Following this procedure we see that $d$ has out-degree zero. Hence the claim follows.
Now power graph of a group being perfect, here $n>4$ means $n\geq 6$, as it can't have any anti-hole of odd length. So, for any arbitrary vertex $d$ in the antihole, degree of $d=n-3\geq 3$. So, it is possible to choose two vertices $v_1$ and $v_2$ in the neighborhood of $d$ that are adjacent to each other. Without loss of generality, if we assume $d$ has in degree zero, then as there is at least one directed edge between $v_1$ and $v_2$, we get a contradiction to the above claim. Hence the result follows.
\end{proof}
\section{Completeness}
Here an alternative proof of the well known result regarding completeness of power graphs of cyclic groups of prime-power order in terms of the strong path lemma is presented.
\begin{proposition}
The power graph of a cyclic group of order $p^n$ where $p$ is prime and $n$ is a non-negative integer is complete.
\end{proposition}
\begin{proof}
The proof is by induction on $n$. By the Strong path lemma it suffices to show that there exists a directed path through all vertices in the power graph. When $n=0$ the graph consists of one vertex and the result follows.
Suppose a directed path exists through all vertices in the power graphs of cyclic groups of order $p^k$ for $0\leq k < n$. Let $G\cong \mathbb{Z}_{p^k}$. Let $x,y$ denote two arbitrary generators of $G$, then in $\vec{\mathfrak{g}}(G)$ there is an edge from $x$ to $y$ and there is an edge from $y$ to $x$, as both elements are members of $G$ and therefore generated by each other. Then, there is a directed path between all generators of $G$. This path can be extended towards a generator of a subgroup of order $p^{k-1}$, which contains a directed path through all of it's vertices by the induction hypothesis. As every subgroup of $G$ divides the order of $G$ by Lagrange's theorem, every subgroup of $G$ is properly contained inside the subgroup of order $p^{k-1}$, so there exists a directed path through the entire vertex set of $\mathfrak{g}(G)$.
By the strong path lemma, there is a clique of size $p^n$ in the cyclic group of order $p^n$, so the graph is complete.
\end{proof}
\section{Chromatic Number of Power Graphs of Cyclic Groups}
\begin{proposition}
Let $G$ be the cyclic group of order $n$ and let $\mathfrak{g}(G)$ be its power graph. Let $H$ denote the set of non-generators of $G$. Let $\chi(\Gamma)$ denote the chromatic number of a graph $\Gamma$. Then $\chi(\mathfrak{g}(G))=\phi(n)+\chi(\mathfrak{H})$ where $\mathfrak{H}$ is the subgraph of $G$ consisting of vertices representing non-generators and the edges between them and $\phi$ is Euler's totient function.
\begin{proof}
Since elements in $G\setminus H$ generate $G$, for any $g\in G\setminus H$ and $x\in G$, $x\in\langle g \rangle$, so vertex $g$ is adjacent to all elements in $\mathfrak{g}(G)$. Then no color used in a coloring of the portion of $\mathfrak{g}(G)$ consisting of elements of $G\setminus H$ can be used in a coloring of $\mathfrak{H}$. Additionally that portion of the graph is internally complete as well so the subgraph of $\mathfrak{g}(G)$ consisting of elements of $G\setminus H$ is isomorphic to $K_{\phi(n)}$ and has chromatic number $\phi(n)$. Since every color used in the coloring of $\mathfrak{H}$ is distinct from every color in the subgraph of $\mathfrak{g}(G)$ consisting of elements of $G\setminus H$, and $\mathfrak{H}$ has chromatic number $\chi(\mathfrak{H})$, $\chi(\mathfrak{g}(G))$ is at most $\phi(n)+\chi(\mathfrak{H})$. Also since the colors used in the colorings of the two subgraphs of $\mathfrak{g}(G)$ are distinct, $\chi(\mathfrak{g}(G))$ cannot be less than $\phi(n)+\chi(\mathfrak{H})$, so $\chi(\mathfrak{g}(G))=\phi(n)+\chi(\mathfrak{H})$.
\end{proof}
\end{proposition}
\begin{Corollary}
Let $G\cong \mathbb{Z}_n$ then $\phi(n)<\chi(\mathfrak{g}(G))\leq n$
\end{Corollary}
The result above gives a lower bound for the chromatic number of a power graph of a cyclic group by identifying a clique of a known size in every cyclic group. By the strong path lemma, the existence of a directed path of length $k$ implies that the vertices along the path also make up a clique of length $k$. If $m$ is the length of the longest directed path in a group $G\cong \mathbb{Z}_n$, then the chromatic number of the cyclic group of order $n$ is at least as large as the length of that path.
\begin{proposition}
Let $G$ be a group and let $m$ be the length of a directed path in $\vec{\mathfrak{g}}(G)$. Then $m\leq \chi(\mathfrak{g}(G)) \leq n$.
\end{proposition}
In a cyclic group $G$ of order $n$, a path $m$ of length longer than $\phi(n)$ can be constructed as follows. Follow the path of length $\phi(n)$ through the generators of of $G$. After visiting the last generator, follow the path to a generator of a proper subgroup of $G$, which must exist as a path exists from a generator of $G$ to every element of $G$. As any two generators of any group have edges from each vertex to the other, each of the generators of this subgroup can be added to the path. This process can than be continued for a subgroup of this subgroup and so on until the the identity element is added to the path. In fact, the longest path through any cyclic subgroup will be of this form, a descending chain of generators of proper subgroups.
\begin{Theorem}
Let $\mathbb{Z} / n\mathbb{Z}$ be the cyclic group of order $n$, and let $S_k$ be the set of generators of $\mathbb{Z} / k\mathbb{Z}$. The longest path through $\mathbb{Z} / n\mathbb{Z}$ will be of the form \\$s_{n1},s_{n2},...s_{n\phi(n)}, d\cdot s_{\frac{n}{d}1}, d\cdot s_{\frac{n}{d}2}, d\cdot s_{\frac{n}{d}\phi(\frac{n}{d})}, ...(dd_1...d_j)\cdot s_{\frac{n}{dd_1d_2...d_j}}\phi(\frac{n}{dd_1d_2...d_j})$ where $d, d_1, d_2...d_j$ are prime divisors of $n$ with $d\leq d_1 \leq d_2 \leq ... \leq d_j$. and $s_{ix}\in S_i$.
\end{Theorem}
\begin{proof}
Suppose a path longer than the one given exists. Such a path necessarily begins with the set of elements of order $n$, that is the elements of $S_n$, since if it did not these elements could simply be added to the beginning of the path to make a new longer path. After every element of order $n$ is added to the path an element $g_1$ from a proper subgroup of $\mathbb{Z} / n\mathbb{Z}$ can be added to the path, but this element will have order $\frac{n}{d}$ where $d$ is a divisor of $n$. In fact this element will generate a subgroup $\langle g_1 \rangle$ of order $\frac{n}{d}$ which is isomorphic to $\mathbb{Z} / \frac{n}{d}\mathbb{Z}$ with the mapping given by $z\in \mathbb{Z} / \frac{n}{d}\mathbb{Z} \mapsto d\cdot g\in \langle g_1 \rangle$. Then for each of the $\phi(\frac{n}{d})$ generators $z\in z\in \mathbb{Z} / \frac{n}{d}\mathbb{Z}$, the corresponding element $d\cdot z \in \langle g_1 \rangle$must be added to the path, since if it were not added to the path a new path could be constructed with these elements following (or preceding) $g_1$ which is longer. The same process can be repeated from $\langle g_1 \rangle$, add an element from a subgroup of $\langle g_1 \rangle$ and all of the other generators of the same subgroup. In this way generators of a descending chain of subgroups are added to the path, terminating with the generator of the trivial group, the identity element.
Since the longest path must be in the form of generators of a chain of subgroups, it remains to be shown that by always choosing the largest possible proper subgroup whose generators to add to the path, the path size is maximized. That is, by traversing the subgroups of $\mathbb{Z} / n\mathbb{Z}$ in the order \\$\mathbb{Z} / n\mathbb{Z}\to d\mathbb{Z} / n\mathbb{Z}\to d\cdot d_1\mathbb{Z} / n\mathbb{Z}\to...\to (d\cdot d_1\cdot ... \cdot d_j)\mathbb{Z} / n\mathbb{Z}$ where $d\leq d_1 \leq ... \leq d_j$, the number of elements added to the path is as large as possible. To see this, some properties of Euler's totient function are examined. First observe that increasing any single prime factor in a number will increase the totient of that number, that is if $n=p_{11}p_{12}...p_{1k}$ where $p_1 \leq p_2 \leq... \leq p_k$ and $m=p_{21}p_{22}...p_{2k}$ where $p_{1x}=p_{2x}$ for all $x$ except one, and at that one index $p_{2x}>p_{1x}$, then $\phi(m)\geq \phi(n)$. It is known that the totient function is multiplicative over relatively prime arguments, that is $\phi(ab)=\phi(a)\phi(b)$ if $\gcd(a,b)=1$, so we can write $\phi(n)$ and $\phi(m)$ as $\Pi_{x\in X} \phi(p_{1x}^{k_x})$ and $\Pi_{y\in Y} \phi(p_{2y}^{k_y})$ respectively where $X$ is the set of prime factors of $n$ and $Y$ is the set of prime factors of $m$. Then both $\phi(n)$ and $\phi(m)$ can be divided by the prime factors and multiplicities for which they agree, leaving only the prime factors which differ between $n$ and $m$. Then observe that $\phi(p^k) < \phi(q^k)$ if $p < q$ are distinct primes. Also $\phi(p^{k})\leq \phi(p^{k-1})q$ if $p< q$ since $p^k-p^{k-1} \leq (p^{k-1}-p^{k-2})(q-1)$. Then it is clear that if $n$ is the order of a cyclic group the subgroups must be traversed in the order above to make the number of elements in the path as large as possible.
\end{proof}
Here the largest path through $\vec{\mathfrak{g}}(\mathbb{Z} / n\mathbb{Z})$ has been constructed, which will have a length equal to the size of the largest clique in $\mathfrak{g}(\mathbb{Z} / n\mathbb{Z})$, by the path-clique equivalence theorem. Since power graphs are perfect, the size of the largest clique in a power graph is also equal to its chromatic number. Then the following result is true for power graphs of cyclic groups
\begin{Corollary}
Let $G\cong \mathbb{Z} / n\mathbb{Z}$ be a group with power graph $\mathfrak{g}(G)$. Also let $d_1\leq d_2\leq... \leq d_k$ be (not necessarily distinct) prime divisors of $n$ then $\chi(\mathfrak{g}(G))=\phi(n)+\phi(\frac{n}{d_1})+\phi(\frac{n}{d_1d_2})+...+\phi(\frac{n}{d_1d_2...d_k})$
\end{Corollary}
Here the proof of Theorem \ref{theo 3} did not depend on the fact that $G$ was a cyclic group in any way, except that the path constructed through the elements of $G$ could always be started with an element of order $n$, where $n$ is the order of $G$. In a general group, there are no elements with order equal to the order of the group, so there are many possibilities for where to begin the longest path. Let $g_1, g_2,...g_n$ be elements of a group $G$, and define $\Psi(n)=\phi(n)+\phi(\frac{n}{d_1})+\phi(\frac{n}{d_1d_2})+...+\phi(\frac{n}{d_1d_2...d_k})$, where $d_1, d_2, ...d_k$ are prime divisors of $n$, then $\chi(G)=\max_{i}\{\Psi(g_i)\}$
\section{Chordallity of Power Graph of cyclic groups}
\begin{proposition}
Consider $n\neq p^{m}$ for some prime $p$ and a positive integer $m$. If a vertex $k \in \mathfrak g(\mathbb Z_{n})$ is simplicial then $gcd(k,n)\neq 1$.
\begin{proof}
Let $k \in \mathfrak g(\mathbb Z_{n})$ be simplicial with $gcd(k,n)= 1$. Then $k$ generates $\mathbb Z_{n}$. So, $k$ being adjacent to every $x\in \mathfrak g(\mathbb Z_{n})$, $\mathfrak g(\mathbb Z_{n})$ turns to be complete, which is a contradiction as $n\neq p^m$.
\end{proof}
\end{proposition}
Converse of the above result is not true in general. For example: Consider $Z_{12}, gcd(6,12)\neq 1$. Though, $6$ is not a simplicial vertex in $\mathfrak g(\mathbb Z_{12})$ as because $2,3$ are adjacent to $6$, but they are not adjacent to each other.
\begin{proposition}\label{prop 27}
$\mathfrak g(\mathbb Z_{n})$ is chordal if and only if $n=p^{m}$ for some prime $p$ and positive integer $m$ or $n=p^{m}q$, for two distinct primes $p,q$ and positive integer $m$.
\end{proposition}
\begin{proof}
If $n=p^{m}$, then $\mathfrak g(\mathbb Z_{n})$ being complete is chordal. If $n=p^{m}q$, then subgroup diagram of the group $\mathbb Z_{n}$ is chordal. And hence, $C(Z_n)$ is also so, as the subgroup diagram is a subgraph of $C(Z_n)$. Thus, $\mathfrak g(\mathbb Z_{n})$ is chordal.
Conversely, if $n\neq p^{m}, p^{m}q$, then $n$ has at least two distinct prime factors $p,q$ with powers $m,n$ where both $m,n$ are positive integers bigger than or equal to $2$. In that case, $p-pq^{2}-q-p^{2}q-p$ is a chordless cycle in $\mathfrak g(\mathbb Z_{n})$ giving the graph as non-chordal.
\end{proof}
\begin{proposition}
A vertex in $C(Z_{n})$ other than $[n]_\sim$ and $[1]_\sim$ is simplicial iff it has only one parent and one child.
\begin{proof}
Let $x$ be a vertex in $C(Z_{n})$ other than $[n]_\sim$ and $[1]_\sim$. If it has more than one parent, then any two parents are not adjacent. Similarly, if it has more than one child then any two children are not adjacent to each other.\\
Conversely let $x$ be a simplicial vertex. Then, it can neither have more one parent, nor more than one child.
\end{proof}
\end{proposition}
Note that, even if $\mathfrak g(\mathbb Z_{n})$ is not chordal for $n= p^{m}q^{r}$ where $m,r \geq 2$ and $p,q$ are distinct primes, they have simplicial vertices namely $p^{m}, q^{r}$, as because they are simplicial in the corresponding $C(Z_n)$.
Thus, we can now state the following result:
\begin{Theorem}
If $n = {\prod_{i=1}^{k} p_{i}^{\alpha_{i}}}$, then whenever $k\geq 3, \mathfrak g(\mathbb Z_{n})$ does not have any simplicial vertex.
\end{Theorem}
\begin{proof}
$n$ and $1$ are not simplicial. Other wise, $\mathfrak g(\mathbb Z_{n})$ will be complete, giving more than one prime factor of $n$ which contradicts the hypothesis of the theorem. On the other hand, if we consider any vertex in $\mathfrak g(\mathbb Z_{n})$ namely $ m={\prod_{i=1}^{k} p_{i}^{\beta_{i}}}$,
then the equivalence classes of $p_{j}p_{i},p_{j}p_{k}$ are two parents of that of $p_{j}$ in $C(\mathbb Z_{n})$. So, the vertex is not simplicial in $C(\mathbb Z_{n})$ and hence is not simplicial in $\mathfrak g(\mathbb Z_{n})$.\\
\end{proof}
\section{Power graph of $U_n$ and $Q_n$}
For any positive integer $n$, let $\mathbb U_{n}$ be the multiplicative group of integers modulo $n$ and let $ \mathbb Q_n$ be it's subgroup of quadratic residue modulo $n$. Then, we have the following results.
\begin{proposition}
$\mathfrak g(\mathbb Q_{n})$ is not planner whenever,
\begin{enumerate}
\item [i.] $n=p$ or $2p$, where $p$ is a prime bigger than $37$.
\item[ii.] $n=p^{m}$ or $2p^{m}$, where $p\geq 7, m\geq 2$ or else, $p=3$ or $5$ and $m\geq 2$.
\end{enumerate}
\end{proposition}
\begin{proof}$\phantom e$
\begin{enumerate}
\item [i.] For $n=p$ or $2p$, the cardinality of $\mathbb Q_{n}$ is $\mu =\frac {\phi(n)}{2}= \frac{p-1}{2}$ by Lemma 7.3 of \cite{book}. In that case $\mathbb Q_{n}$ is cyclic as $\mathbb U_{n}$ is so. Hence by lemma 4.7 in \cite{power semi}, $\mathfrak g(\mathbb Q_{n})$ is not planner if $\phi{(\frac{p-1}{2})}>7$, that is if $\frac{p-1}{2}>18$, that is if $p>37$.
\item[ii.] Now let $n=p^{m}$, where $p$ is an odd prime, then by lemma 4.7 in \cite{power semi}, $\mathfrak g(\mathbb Q_{n})$ is not planner if $\frac {\phi(n)}{2}= \frac{p^{m}}{2}= \frac {p^{m-1}(p-1)}{2}= p^{m-2}(p-1){\phi(\frac{p-1}{2})}>7$. Hence, the result follows.
\end{enumerate}
\end{proof}
\begin{proposition}
$\mathfrak g(\mathbb Q_{n})$ is planner if $n$ divides $240$.
\end{proposition}
\begin{proof} If $n$ divides $240$, then $\mathfrak g(\mathbb U_{n})$ is planner by theorem 4.10 in \cite{power semi}. Hence by proposition 4.5 in \cite {power semi}, $\mathfrak g(\mathbb Q_{n})$ is also planner.
\end{proof}
\begin{proposition}
Let $n= p^{m}$ or $n=2p^{m}$, where $p$ is a Fermat prime. Then, $\mathfrak g(\mathbb U_{n})$ is chordal iff $m\leq 2$ or $p=F_{0}=3$.
\end{proposition}
\begin{proof}
Let $n= p^2$ or $n=2p^2$ where $p$ is a Fermat prime. Then, by Corollary 6.14 in \cite{book}, $\mathbb U_{n}$ is isomorphic to $\mathbb Z_{p}\times \mathbb Z_{p-1}= \mathbb Z_{p}\times \mathbb Z_{2^{m}}=\mathbb Z_{2^{m}p}$, for some positive integer $m$ as $p$ is a Fermat Prime. Hence the graph is chordal by proposition \ref{prop 27}. In a similar way, for $n=p$ or $n=2p$ where $p$ is a Fermat prime, $\mathbb U_{n}$ is isomorphic to $\mathbb Z_{p-1}=\mathbb Z_{2^{m}}$ for some positive integer $m$ and hence is chordal by proposition \ref{prop 27}. Now let $p=F_{0}=3$. Then, if $n=p^{m}$ or $n=2p^{m}, U_{n}$ is isomorphic to $\mathbb Z_{p^{m-1}}\times\mathbb Z_{2}=\mathbb Z_{2p^{m-1}}$, as $p$ is odd. Hence the graph is chordal.\\
Conversely, let $n\neq p,2p,p^{2},2p^{2}$ where $p\neq 3$ and let $n\neq p^{m}, 2p^{m}$ where $p=F_{0}$ and $m$ is any positive integer. So, here if $p>F_{0}, m\geq 3$, then as in \cite{book}, $\mathbb U_{n}$ is isomorphic to $Z_{p^{m-1}}\times\mathbb Z_{2^{i}}, i\geq 2$. Thus the power graph is not chordal by proposition~\ref{prop 27}.
\end{proof}
\begin{Corollary}
For a positive integer $n$, if $p^{m}$ or $2p^{m}$ divides $n$, where $p$ is a Fermat prime bigger than $3$ and $m\geq 3$, then $\mathfrak g(\mathbb U_{n})$ is not chordal.
\end{Corollary}
\begin{proposition}\label{prop 32}
Let, $n$ be an odd integer which is not square free and no Fermat prime be a factor of $n$. Then, $\mathfrak g(\mathbb U_{n})$ is not Chordal.
\end{proposition}
\begin{proof}
As $n$ is not square free, there is an odd prime $p$, which is not a Fermat prime and $p^{m}$ divides $n$ for some $m\geq 2$. In that case, as in Corollary 6.14 of \cite{book}, $\mathbb Z_{p^{m-1}(p-1)}$ appears inside the direct product decomposition of $\mathbb U_{n}$, where $p-1$ is not a power of $2$ since $p$ is not a Fermat prime. So, $p-1$ being it has an odd prime factor $q$ where $p\neq q$, thus $p^{m-1}(p-1)$ contains at least $3$ distinct odd primes and hence the power graph is not chordal.
\end{proof}
\begin{proposition}\label{prop 33}
Let $n$ be an even integer. Then, if
\begin{enumerate}
\item [i.] $2^{f}\parallel n$ (that means $f$ is the largest integer so that $2^{f}$ divides $n$), where $f\geq 4$ and if there are at least two distinct primes $p$ and $q$ and positive integers $\alpha, \beta$, where neither of $p,q$ are Fermat's prime, then $\mathfrak g(\mathbb U_{n})$ is not chordal.
\item[ii.] If $2^{f}\parallel n, f\geq 1$ and $p^{\alpha}, q^{\beta} \parallel n$, where $\alpha,\beta \geq 2$ and neither of $p,q$ are Fermat's prime, then
$\mathfrak g(\mathbb U_{n})$ is not chordal.
\end{enumerate}
\end{proposition}
\begin{proof}
The proof follows by Corollary 6.14 in \cite{book} and by our proposition \ref{prop 27}.
\end{proof}
We have analogous result regarding chordality for $\mathfrak g(\mathbb Q_{n})$.
\begin{proposition}
Let $n= p^{m}$ or $n=2p^{m}$, where $p$ is a Fermat prime. Then, $\mathfrak g(\mathbb Q_{n})$ is chordal iff $m\leq 2$ or $p=F_{0}=3$ or $p=F_{1}=5$.
\end{proposition}
\begin{proof}
The proof follows by using Lemma 7.1 and Corollary 6.14 in \cite{book} as in that case, $\mathbb Q_{n} =\mathbb Z_{\frac{\phi(p^{m})}{2}}= Z_{p^{m-1}(\frac{p-1}{2})}$. Hence, by similar method as in the proof of proposition \ref{prop 32}, the result follows.
\end{proof}
\begin{Corollary}
For a positive integer $n$, if $p^{m}$ or $2p^{m}$ divides $n$, where $p$ is a Fermat prime bigger than $3, 5$ and $m\geq 3$, then $\mathfrak g(\mathbb Q_{n})$ is not chordal.
\end{Corollary}
\begin{proposition}
Let, $n$ be an odd integer which is not square free and no Fermat prime be a factor of $n$. Then, $\mathfrak g(\mathbb Q_{n})$ is not Chordal.
\end{proposition}
\begin{proof}
The proof follows as in proposition \ref{prop 32} and Lemma 7.1 in \cite{book}.
\end{proof}
\begin{proposition}
Let $n$ be an even integer. Then, if \begin{enumerate}
\item[i.] $2^{f}\parallel n$ where $f\geq 5$ and if there are at least two distinct primes $p$ and $q$ and positive integers $\alpha, \beta$, where neither of $p,q$ are Fermat's prime, then $\mathfrak g(\mathbb Q_{n})$ is not chordal.
\item[ii.] If $2^{f}\parallel n, f\geq 2$ and $p^{\alpha}, q^{\beta} \parallel n$, where $\alpha,\beta \geq 2$ and neither of $p,q$ are Fermat's prime, then
$\mathfrak g(\mathbb Q_{n})$ is not chordal.
\end{enumerate}
\end{proposition}
\begin{proof}
The proof follows as in proposition \ref{prop 33}.
\end{proof}
Finally, we conclude with a question: Precisely, for what positive integers $n,\mathfrak g(\mathbb U_{n})$ and $\mathfrak g(\mathbb Q_{n})$ are chordal?
\textit{Acknowledgment:\/} Authors acknowledge Dr. M. K. Sen for his helpful suggestions and advice towards this work.
\end{document} |
\begin{document}
\begin{abstract}
In this paper, we show all $k$-linear abelian 1-Calabi-Yau categories over an algebraically closed field $k$ are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of $k[[t]]$. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all $k$-linear abelian 1-Calabi-Yau categories up to equivalence.
\end{abstract}
\title{Abelian 1-Calabi-Yau Categories}
\section{Introduction}
In this paper, we classify \emph{abelian 1-Calabi-Yau categories}
over an algebraically closed field~$k$. Recall that an abelian
1-Calabi-Yau category is a $k$-linear $\operatorname {Hom}/\operatorname {Ext}$-finite abelian category
together with
natural isomorphisms $\operatorname {Hom}(X,Y) \cong \operatorname {Ext}(Y,X)^\ast$ for $X,Y\in {\cal A}$.
Our main result (reformulated in the body of the text as Theorem
\ref{theorem:Main}) is the following.
\begin{theorem}
\label{mainth}
Let $\mathcal{A}$ be an indecomposable abelian 1-Calabi-Yau
category. Then $\mathcal{A}$ is \emph{derived} equivalent to one of the
following two categories.
\begin{enumerate}
\item Finite dimensional
representations of $k[[t]]$.
\item The category of coherent sheaves on an elliptic
curve.
\end{enumerate}
\end{theorem}
There is a general interest in the classification of categories which
are homologically small in some sense (see e.g.\ \cite{Happel01}, \cite{Keller06}
\cite{ReVdB02}, \cite{vanRoosmalen06}). The above
theorem represents an enhancement of our knowledge in this area.
Besides this general motivation we mention the following particular
application. Recently Polishchuk and Schwartz \cite{Polishchuk03} constructed a
category ${\cal C}$ of holomorphic vector bundles on a
\emph{non-commutative 2-torus}. Polishchuk subsequently showed that
${\cal C}$ is derived equivalent to the category of coherent sheaves on an
elliptic curve \cite{Polishchuk03}. Part of Polishchuk's proof amounts to
establishing the highly non-trivial fact that ${\cal C}$ is $1$-Calabi-Yau
\cite[Cor 2.12]{Polishchuk03}. Once one knows this, one could now finish the
proof by simply invoking Theorem \ref{mainth} (with ${\cal A}$ being a
suitable abelian hull of~${\cal C}$).
We briefly outline some steps in the proof of Theorem \ref{mainth}. Some of
our tools come from representation theory of algebras and non-commutative
algebraic geometry. Other
tools were already employed by Polishchuk, but are now used in a more
abstract setting.
Fix a connected abelian 1-Calabi-Yau category ${\cal A}$. First, we prove
the existence of \emph{endo-simple} objects in ${\cal A}$, i.e.\ objects
$X\in {\cal A}$ such that $\operatorname {End} X \cong k$. Associated to such objects
there are \emph{twist functors} \cite{Seidel01} $T_A,
T_A^\ast$. These functors are mutually
inverse auto-equivalences of $D^b({\cal A})$ which on objects take the values
$T_X Y =
\operatorname {cone} (X \otimes \operatorname {RHom}(X,Y) \longrightarrow Y)$ and $T_X^\ast Y =
\operatorname {cone} (Y[-1] \longrightarrow X[-1] \otimes \operatorname {RHom}(Y,X)^\ast)$.
Using twist functors we establish various useful facts. Most notably, we
prove that the subcategory of endo-simple objects in ${\cal A}$ has no cycles of
non-zero maps (Proposition \ref{proposition:EndoSimplesDirected}) and
hence is ordered. We also show that all Auslander-Reiten components
of ${\cal A}$ are homogeneous tubes based on endo-simple objects (Proposition
\ref{proposition:Building}).
We may assume that ${\cal A}$ has at least two non-isomorphic endo-simple
objects as the remaining case is easily disposed with. Using
connectedness and the results mentioned in the previous paragraph we
may in fact select non-isomorphic endo-simple objects $E$ and $B$ such
that $\operatorname {Hom}(E,B)\neq 0$. After doing so we consider the sequence of
objects ${\cal E}=(T_B^{n} E)_{n \in {\blb Z}}$ in $\Db {\cal A}$. We construct a
certain associated $t$-structure on $D^b({\cal A})$ with heart ${\cal H}$ such
that ${\cal E}$ is an ample sequence in the sense of \cite{Polishchuk05} in
${\cal H}$. Hence ${\cal E}$ defines a finitely presented graded coherent
algebra $A$ such that ${\cal H}$ is equivalent to the category
$\operatorname{qgr}(A)$ of finitely presented graded $A$-modules modulo
finite dimensional ones.
We then show that $A$ is a domain of Gelfand-Kirillov dimension two
and we invoke the celebrated Artin and Stafford classification theorem
\cite{Artin95} which shows that $\operatorname{qgr}(A)$ is of the form
$\operatorname{coh}(X)$ for a projective curve $X$. Since ${\cal H}$ is
1-Calabi-Yau this implies that $X$ must be an elliptic curve,
finishing the proof.
It is not hard to describe the abelian 1-Calabi-Yau
categories that occur within the derived equivalence classes in
Theorem \ref{mainth} (see e.g. \cite{Burban06}). We discuss this using the
language of this paper in \S\ref{subsection:Abelian}.
\subsection*{Acknowledgment}
The author thanks Michel Van den Bergh for useful discussions as well as
for contributing some ideas.
\section{Preliminaries}
Throughout this paper, fix an algebraically closed field $k$ of arbitrary characteristic. All algebras and categories are assumed to be $k$-linear.
We will also assume all abelian categories are \emph{connected} in the sense that between two indecomposable objects there is an unoriented path of non-zero maps between indecomposables.
If ${\cal A}$ is abelian, we write $\Db {\cal A}$ for the bounded derived category of ${\cal A}$. The category $\Db {\cal A}$ has the structure of a triangulated category. Whenever we use the word "triangle" we mean "distinguished triangle".
An abelian or triangulated category ${\cal A}$ is \emph{Ext-finite} if for all objects $X,Y \in \operatorname{Ob}({\cal A})$ one has that $\dim_{k}\operatorname {Ext}^{i}(X,Y)<\infty$ for all $i \in {\blb N}$. We say that ${\cal A}$ is \emph{hereditary} if $\operatorname {Ext}^{i}(X,Y)=0$ for all $i \geq 2$.
\subsection{Serre duality}
If ${\cal C}$ is triangulated category, we will say that ${\cal C}$ satisfies \emph{Serre duality} if there exists an auto-equivalence $F:{\cal C} \to {\cal C}$, called the \emph{Serre functor}, such that, for all $X,Y\in\operatorname{Ob} {\cal C}$, there is an isomorphism
$$\operatorname {Hom}_{{\cal C}}(X,Y) \cong \operatorname {Hom}_{{\cal C}}(Y,FX)^{*}$$
which is natural in $X$ and $Y$ and where $(-)^{*}$ denotes the vector space dual.
We will say an abelian category ${\cal A}$ has Serre duality if the category $\Db {\cal A}$ has a Serre functor.
It has been proven in \cite{ReVdB02} that an abelian category ${\cal A}$ without non-zero projectives has a Serre functor if and only if the category $\Db{\cal A}$ has Auslander-Reiten triangles. In this case the action of the Serre functor on objects coincides with $\tau[1]$, where $\tau$ is the Auslander-Reiten translation.
\subsection{Calabi-Yau categories}
Let ${\cal A}$ be an Ext-finite abelian category with Serre duality. We will say ${\cal A}$ is \emph{Calabi-Yau of dimension $n$} or shorter that ${\cal A}$ is \emph{$n$-Calabi-Yau} if $F \cong [n]$ for a certain $n \in {\blb N}$, thus if the $n^\text{th}$ shift is a Serre functor. We write $\operatorname {CYdim} {\cal A} = n$.
The following well-known property relates the Calabi-Yau dimension and the homological dimension.
\begin{proposition}\label{proposition:Dimension}
Let ${\cal A}$ be an abelian Calabi-Yau category. Then $\operatorname {CYdim} {\cal A} = \operatorname {gl\,dim} {\cal A}$.
\end{proposition}
\begin{proof}
Let $n = \operatorname {CYdim} {\cal A}$, then for every $X \in \operatorname{Ob} {\cal A}$ we have $\operatorname {Hom}(X,X) \cong \operatorname {Ext}^n(X,X)^{*}$. Since the former is non-zero, we see $\operatorname {CYdim} {\cal A} \leq \operatorname {gl\,dim} {\cal A}$.
Let $i \in {\blb N}$ and $X,Y \in \operatorname{Ob} {\cal A}$ be chosen such that $\operatorname {Ext}^{i}(X,Y) \not = 0$. Using the Calabi-Yau property, we find $\operatorname {Ext}^{i}(X,Y) \cong \operatorname {Ext}^{n-i}(Y,X)$, hence $n \geq i$. We find $\operatorname {CYdim} {\cal A} \geq \operatorname {gl\,dim} {\cal A}$.
\end{proof}
In particular, if ${\cal A}$ is a 1-Calabi-Yau category, then ${\cal A}$ is hereditary. Since $F \cong [1]$ and $F$ coincides with $\tau[1]$ on indecomposables of ${\cal A}$, it follows that $\tau$ is naturally isomorphic to the identity functor on ${\cal A}$ and hence $\operatorname {Hom}_{\cal A}(X,Y) \cong \operatorname {Ext}_{\cal A}(Y,X)^\ast$, for all objects $X,Y \in {\cal A}$.
\subsection{Twist functors}
Let ${\cal A}$ be an abelian 1-Calabi-Yau category. For an object $A \in \Db {\cal A}$, we may consider the \emph{twist functors}, $T_A$ and $T_A^\ast$, in $\Db {\cal A}$ whose values on objects are up to isomorphism characterized by the following triangles
$$T_A X[-1] \longrightarrow A \otimes \operatorname {RHom}(A,X) \stackrel{\epsilon}{\longrightarrow} X \longrightarrow T_A X$$
and
$$T_A^\ast X \longrightarrow X \stackrel{\epsilon^\ast}{\longrightarrow} A \otimes \operatorname {RHom}(X,A)^\ast \longrightarrow T_A^\ast X[1]$$
where $\epsilon : A \otimes \operatorname {RHom}(A,X) \longrightarrow X$ and $\epsilon^\ast : X \longrightarrow A \otimes \operatorname {RHom}(X,A)^\ast$ are the canonical morphisms.
Let $S$ be an \emph{endo-simple} object, i.e.\ $\operatorname {End}(S) \cong k$. Since ${\cal A}$ is 1-Calabi-Yau, we know from \cite[Proposition 2.10]{Seidel01} that $T_S$ and $T_S^\ast$ are inverses. In particular, they are autoequivalences.
\subsection{Ample sequences}
For the benefit of the reader, we will recall some definitions and results from \cite{Polishchuk05} which will be used in the rest of this paper. Throughout, let ${\cal A}$ be a Hom-finite abelian category.
We begin with the definition of ample sequences.
\begin{enumerate}
\item A sequence ${\cal E} = (E_i)_{i\in {\blb Z}}$ is called \emph{projective} if for every epimorphism $X \to Y$ in ${\cal A}$ there is an $n\in {\blb Z}$ such that $\operatorname {Hom}(E_i,X) \to \operatorname {Hom}(E_i,Y)$ is surjective for $i < n$.
\item A projective sequence ${\cal E} = (E_i)_{i\in {\blb Z}}$ is called \emph{coherent} if for every $X \in \operatorname{Ob} {\cal A}$ and $n \in {\blb Z}$, there are integers $i_1, \ldots, i_s \leq n$ such that the canonical map
$$\bigoplus_{j=1}^s \operatorname {Hom}(E_i, E_{i_j}) \otimes \operatorname {Hom}(E_{i_j},X) \longrightarrow \operatorname {Hom}(E_i,X)$$
is surjective for $i<<0$.
\item A coherent sequence ${\cal E} = (E_i)_{i\in {\blb Z}}$ is \emph{ample} if for all $X \in {\cal A}$ the map $\operatorname {Hom}(E_i,X) \not=0$ for $i<<0$.
\end{enumerate}
Let $A_{ij} = \operatorname {Hom}(E_i,E_j)$ for $i \leq j$. We may define an algebra $A = A({\cal E}) = \oplus_{i \leq j} A_{ij}$ in a natural way. If $A_{ii} \cong k$, then $A$ is a \emph{coherent ${\blb Z}$-algebra} in the sense of \cite{Polishchuk05} (see \cite[Proposition 2.3]{Polishchuk05}).
We will refer to the right $A$-modules having a resolution by finitely generated projectives as \emph{coherent modules}. These modules form an abelian category, $\mathop{\text{\upshape{coh}}} A$, and the finite dimensional modules form a Serre subcategory denoted by $\mathop{\text{\upshape{coh}}}^{b} A$. We define the quotient
$$\mathop{\text{\upshape{coh}}}proj A \cong \mathop{\text{\upshape{coh}}} A / {\mathop{\text{\upshape{coh}}}}^{b} A.$$
We may use this to give a description of Ext-finite abelian categories with an ample sequence.
\begin{theorem}\label{theorem:Polishchuk}
\cite[Theorem 2.4]{Polishchuk05} Let ${\cal E}=(E_i)$ be an ample sequence, $A=A({\cal E})$ the corresponding ${\blb Z}$-algebra, then there is a equivalence of categories ${\cal A} \cong \mathop{\text{\upshape{coh}}}proj A$.
\end{theorem}
We will be interested in the special case where there is an automorphism $t : \Db {\cal A} \longrightarrow \Db {\cal A}$ such that $E_i \cong t^i E$. We let $R = R({\cal E}) = \oplus_{i \in {\blb N}} R_i$ where $R_i = \operatorname {Hom}(E,t^i E)$ and make it into a ${\blb Z}$-graded algebra in an obvious way.
If $R$ is noetherian then the coherent $R$-modules correspond to the finitely generated ones and $\mathop{\text{\upshape{coh}}}proj R$ corresponds to $\operatorname{qgr} R$, the finitely generated modules modulo the finite dimensional ones.
We will use following corollary of Theorem \ref{theorem:Polishchuk}.
\begin{corollary}\label{corollary:Polishchuk}
Let ${\cal A}$ be a Hom-finite abelian category, $t$ be an autoequivalence of ${\cal A}$ and $E$ an object of ${\cal A}$. If ${\cal E}=(t^i E)$ is an ample sequence and the corresponding graded algebra $R = R({\cal E})$ is noetherian, then ${\cal A} \cong \operatorname{qgr} R$.
\end{corollary}
\subsection{$t$-structures}
In order to find derived equivalent categories, we will use the theory of $t$-structures \cite{Beilinson82}.
\begin{definition}\label{definition:t}
A \emph{$t$-structure} on a triangulated category ${\cal C}$ is a pair $(D^{\geq 0}, D^{\leq 0})$ of non-zero full subcategories of ${\cal C}$ satisfying the following conditions, where we denote $D^{\leq n} = D^{\leq 0} [-n]$ and $D^{\geq n} = D^{\geq 0} [-n]$
\begin{enumerate}
\item $D^{\leq 0} \subseteq D^{\leq 1}$ and $D^{\geq 1} \subseteq D^{\geq 0}$
\item $\operatorname {Hom}(D^{\leq 0}, D^{\geq 1}) = 0$
\item\label{split} $\operatorname {For}all Y \in {\cal C}$, there exists a triangle $X \to Y \to Z \to X[1]$ with $X \in D^{\leq 0}$ and $Z \in D^{\geq 1}$.
\end{enumerate}
\end{definition}
Furthermore, we will say the $t$-structure is \emph{bounded} if $\bigcap_n D^{\leq n} = \bigcap_n D^{\geq n} = \{0\}$.
We will say a $t$-structure is \emph{split} if all triangles in (\ref{split}) are split, or equivalently, if $\ind {\cal C} = \ind D^{\geq 1} \cup \ind D^{\leq 0}$. We have following result.
\begin{theorem}\cite{Berg}\label{theorem:Berg}
Let ${\cal A}$ be an abelian category and let $(D^{\geq 0}, D^{\leq 0})$ be a bounded $t$-structure on $\Db {\cal A}$. Then the heart ${\cal H}$ is hereditary if and only if $(D^{\geq 0}, D^{\leq 0})$ is a split $t$-structure. In this case, ${\cal A}$ and ${\cal H}$ are derived equivalent.
\end{theorem}
\input{Elliptic}
\section{Endo-simple objects}
Let ${\cal A}$ be a connected $k$-linear abelian 1-Calabi-Yau category. It will turn out that the endo-simple objects are the building blocks of ${\cal A}$. Therefore, in this section, we will give some properties of endo-simple objects. Recall that $X$ is an endo-simple object if $\operatorname {End} X \cong k$. It follows from the Calabi-Yau property that every endo-simple object is 1-spherical in the sense of \cite{Seidel01}.
\begin{proposition}\label{proposition:EndoSimple}
Let ${\cal C}$ be a Hom-finite abelian category. For every object $X \in \operatorname{Ob}{\cal C}$ there exists an endo-simple object occurring both as subobject and quotient object of $X$. In particular, ${\cal C}$ has an endo-simple object.
\end{proposition}
\begin{proof}
Assume $X$ is not endo-simple and let $f : X \to X$ be a non-invertible endomorphism. We show that $\dim \operatorname {End} I < \dim \operatorname {End} X$ where $I = \operatorname {im} f$.
Indeed, since we have an epimorphism $X \to I$ and monomorphism $I \to X$, we get a composition of monomorphisms $\operatorname {Hom}(I,I) \to \operatorname {Hom}(X,I) \to \operatorname {Hom}(X,X)$. Since the image of this composition has to be in $\operatorname {rad}(X,X)$, we know $\dim \operatorname {Hom}(I,I) < \dim \operatorname {Hom}(X,X)$. Iteration finishes the proof.
\end{proof}
\begin{proposition}\label{proposition:CanonicalMap}
Let $S$ be an endo-simple object and $X \in \ind {\cal A}$. Each of the canonical maps $S \otimes \operatorname {Hom}(S,X) \longrightarrow X$ and $X \longrightarrow S \otimes \operatorname {Hom}(X,S)^\ast$ is either a monomorphism or an epimorphism. If $\operatorname {Hom}(X,S) \not= 0$, then the first map is a monomorphism. If $\operatorname {Hom}(S,X) \not= 0$, then the latter is an epimorphism.
\end{proposition}
\begin{proof}
Consider in the derived category $\Db {\cal A}$ the twist functor $T_S$ characterized by
$$T_S X[-1] \longrightarrow S \otimes \operatorname {RHom}(S,X) \stackrel{\epsilon}{\longrightarrow} X \longrightarrow T_S X.$$
It is shown in \cite{Seidel01} that this is an equivalence. Applying the homological functor $H^0$ gives the long exact sequence
$$0 \to H^{-1}(T_S X) \to S \otimes \operatorname {Hom}(S,X) \stackrel{H^0 \epsilon}{\longrightarrow} X \to H^0(T_S X) \to S \otimes \operatorname {Ext}(S,X) \to 0.$$
Since $X$ is indecomposable and $T_S$ is an equivalence, either $H^{-1}(T_S X)$ or $H^0(T_S X)$ is zero, hence $H^0 \epsilon$ is a monomorphism or an epimorphism, respectively.
If we assume furthermore $\operatorname {Hom}(X,S) \not= 0$, and hence by the Calabi-Yau property $\operatorname {Ext}(S,X) \not= 0$, we find $H^0(T_S X) \not= 0$. Hence $H^{-1}(T_S X) = 0$ and the canonical map $S \otimes \operatorname {Hom}(S,X) \to X$ is a monomorphism.
The other case is dual.
\end{proof}
\begin{proposition}\label{proposition:EndoSimplesDirected}
The subcategory of endo-simples is a directed category.
\end{proposition}
\begin{proof}
Let $S_0 \to S_1 \to \operatorname {cd}ots \to S_n \to S_0$ be a cycle of non-zero non-isomorphisms between endo-simple objects. We will assume $n$ is minimal with the property that such a cycle exists.
By Proposition \ref{proposition:CanonicalMap} we know the canonical map $\epsilon:S_0 \otimes \operatorname {Hom}(S_0,S_1) \to S_1$ is either a monomorphism or an epimorphism. If $\epsilon$ is a monomorphism, then we know the composition
$$S_n \otimes \operatorname {Hom}(S_0,S_1) \longrightarrow S_0 \otimes \operatorname {Hom}(S_0,S_1) \stackrel{\epsilon}{\longrightarrow} S_1$$
is non-zero. This induces a non-zero morphism $f: S_n \longrightarrow S_1$. Since $f$ factors through $S_0 \otimes \operatorname {Hom}(S_0,S_1)$, we know $f$ is not invertible.
Likewise, if $\epsilon$ is an epimorphism, we find a non-zero non-invertible morphism $S_0 \to S_2$. In both cases we have found a shorter cycle, contradicting with the minimality of $n$.
\end{proof}
We now wish to show that every object has a composition series with endo-simple quotients. Even more so, every indecomposable object has a composition series in which only one isomorphism class of endo-simple objects occur. We start with a lemma.
\begin{lemma}\label{lemma:Cone}
Let $X \in \ind {\cal A}$ such that the endo-simple object $S$ occurs both as subobject and quotient object of $X$. If $C = \operatorname {coker}(S \otimes\operatorname {Hom}(S,X) \longrightarrow X)$ is not zero, then S occurs as both subobject and quotient object of every direct summand of $C$.
\end{lemma}
\begin{proof}
Assume $C \not\cong 0$. Consider the exact sequence
$$0 \to S \otimes \operatorname {Hom}(S,X) \longrightarrow X \longrightarrow H^0(T_S X) \longrightarrow S \otimes \operatorname {Ext}(S,X) \to 0.$$
from the proof of Proposition \ref{proposition:CanonicalMap}. We may splice this as
$$0 \longrightarrow S \otimes \operatorname {Hom}(S,X) \longrightarrow X \longrightarrow C \longrightarrow 0$$
and
$$0 \longrightarrow C \longrightarrow H^0(T_S X) \longrightarrow S \otimes \operatorname {Ext}(S,X) \longrightarrow 0.$$
Since $T_S$ is an automorphism and $X$ is indecomposable, we know $H^0(T_S X)$ is indecomposable. It now follows directly from \cite[Lemma 2*]{Ringel05} that $\operatorname {Hom}(S,C_1) \cong \operatorname {Ext}(C_1,S)^* \not=0$ and $\operatorname {Hom}(C_1,S) \cong \operatorname {Ext}(S,C_1)^* \not=0$ for every direct summand $C_1$ of $C$. Proposition \ref{proposition:CanonicalMap} now yields that $S$ is both a subobject and a quotient object of every direct summand of $C$.
\end{proof}
\begin{proposition}\label{proposition:Building}
Every indecomposable object is obtained by repeatedly extending a given endo-simple with itself.
\end{proposition}
\begin{proof}
Let $S$ be an endo-simple object and denote by ${\cal A}_S$ the full subcategory of ${\cal A}$ spanned by the objects $Z$ which can be obtained from $S$ by taking a finite amount of extensions with itself. The number of such extensions needed, will be denoted by $l_S(Z)$, and we will refer to it as the length of $Z$.
Since ${\cal A}_S$ is a hereditary category with a unique simple $S$ such that $\dim \operatorname {Ext}(S,S)=1$, it follows easily that ${\cal A}_S$ is equivalent to the finite dimensional representations of $k[[t]]$.
We will prove that if $X$ is an indecomposable object of ${\cal A}$ such that $S$ occurs both as quotient and subobject, then $X \in {\cal A}_S$. Note that by Proposition \ref{proposition:EndoSimple} we may assume such an $S$ exists.
For every subobject $A$ of $X$ in ${\cal A}_S$ and quotient object $B$ of $X$ in ${\cal A}_S$, we have
\begin{equation*}
\dim \operatorname {End}_{\cal A} X \ge \min (l_S(A),l_S(B)),
\end{equation*}
thus we may deduce either the length of such subobjects or the length of such quotient objects is bounded. Assume that the length of $A$ is bounded, the other case is dual.
We will now construct in ${\cal A}_S$ an ascending sequence of subobjects of $X$. Let $A_0 = S \otimes \operatorname {Hom}(S,X)$ and denote $C_0 = \operatorname {coker}(S \otimes\operatorname {Hom}(S,X) \longrightarrow X)$. We will assume $C_0 \not\cong 0$.
We choose a decomposition $C_0 \cong X_1 \oplus C'_0$ where $X_1$ is indecomposable, hence by Lemma \ref{lemma:Cone} we know $S$ occurs both as subobject and as quotient object of $X_1$ and of every direct summand of $C'_0$. Consider the following diagram with exact rows and columns
$$\xymatrix{
&& 0\ar[d] & 0\ar[d] & \\
0 \ar[r] & A_0 \ar[r]\ar@{=}[d] & A_1 \ar[r]\ar[d] & S \otimes \operatorname {Hom}(S,X_1) \ar[r]\ar[d] & 0 \\
0 \ar[r] & A_0 \ar[r] & X \ar[r]\ar[d] & X_1 \oplus C'_0 \ar[r]\ar[d] & 0 \\
&& C_1 \oplus C'_0\ar@{=}[r]\ar[d] & C_1 \oplus C'_0 \ar[d] & \\
&& 0 & 0 &
}$$
It follows from Lemma \ref{lemma:Cone} that $S$ occurs both as subobject and quotient of every indecomposable of $C_1 \oplus C'_0$ where $C_1 = \operatorname {coker} (S \otimes\operatorname {Hom}(S,X_1) \longrightarrow X_1)$. Hence, using $C_0 \not\cong 0$, we have found a subobject $A_1 \in {\cal A}_S$ of $X$ such that $l_S(A_0) < l_S(A_1)$. Iteration and using that the length is bounded, we see that $X \in {\cal A}_S$.
\end{proof}
\begin{remark}
It follows from previous proposition that all Auslander-Reiten components of ${\cal A}$ are \emph{homogeneous tubes}, i.e. they are of the form ${\blb Z} A_\infty / \langle \tau \rangle$, cfr. Figure \ref{figure:Tube}, were the bottom element is endo-simple.
\end{remark}
Finally, we will formulate a useful corollary.
\begin{corollary}\label{corollary:TubesDirected}
Every cycle $X_0 \to X_1 \to \operatorname {cd}ots \to X_n \to X_0$ of non-zero non-isomorphisms between indecomposable objects belongs to a single homogeneous tube.
\end{corollary}
\begin{proof}
Directly from Propositions \ref{proposition:EndoSimplesDirected} and \ref{proposition:Building}.
\end{proof}
\begin{remark}
It follows that the set of homogeneous tubes of the category ${\cal A}$ are directed, thus there can be no cycle containing two objects from different homogeneous tubes.
\end{remark}
\section{Classification}
Let ${\cal A}$ be a connected $k$-linear abelian Ext-finite 1-Calabi-Yau category. In this section, we wish to classify all such categories up to derived equivalence. If every two endo-simples of ${\cal A}$ are isomorphic, then ${\cal A}$ is equivalent to the finite dimensional nilpotent representations of the one loop quiver.
So, assume there are at least two non-isomorphic endo-simples, $E$ and $B$. By connectedness and Proposition \ref{proposition:Building}, we assume $\operatorname {Hom}(E,B) \not= 0$. First, we will find a $t$-structure in $\Db {\cal A}$ such that the heart ${\cal H}$ admits an ample sequence ${\cal E}$. Then we will use Theorem \ref{theorem:Polishchuk} to show ${\cal A} \cong \operatorname{qgr} R({\cal E})$. A discussion of $R({\cal E})$ will then complete the classification of abelian 1-Calabi-Yau categories up to derived equivalence.
From here on, we will always denote $\operatorname {Hom}(E,B)$ by $V$ and its dimension by $d$.
\subsection{The sequence ${\cal E}$ and a $t$-structure in $\Db {\cal A}$}\label{subsection:t}
With $E$ and $B$ as above, associate the autoequivalence $t=T_B : \Db {\cal A} \longrightarrow \Db {\cal A}$ and the sequence ${\cal E} = (E_i)$ where $E_i = t^i E$.
The following will define a $t$-structure in ${\cal C}$ with a hereditary heart ${\cal H}$.
\begin{eqnarray*}
\ind D^{\leq 0} &=& \{X \in \ind {\cal C} \mid \mbox{there is a path from $E_i$ to $X$, for an $i \in {\blb Z}$}\} \\
\ind D^{\geq 1} &=& \ind {\cal C} \setminus \ind D^{\leq 0}
\end{eqnarray*}
If follows directly from this definition that $t$ restricts to an autoequivalence on ${\cal H}$, which we will also denote by $t$. Note that this implies $E_i \in \operatorname{Ob} {\cal H}$, for all $i \in {\blb Z}$. Also, since $\operatorname {Hom}(B[-1],E_i) \not= 0$, there is no path from $E_i$ to $B[-1]$ and hence we have $B \in \operatorname{Ob} {\cal H}$.
It follows from Theorem \ref{theorem:Berg} that ${\cal H}$ is hereditary and $\Db {\cal H} \cong \Db {\cal A}$. Since ${\cal H}$ is a 1-Calabi-Yau category, the results we have proved about ${\cal A}$ apply to ${\cal H}$ as well.
Note that, since $t^i B \cong B$, we find there is a natural isomorphism $\operatorname {Hom}(E,B) \cong \operatorname {Hom}(E_i,B)$ and as such, we get triangles of the form $B[-1] \otimes V^\ast \longrightarrow E_{i-1} \longrightarrow E_i \longrightarrow B \otimes V^\ast$. Such a triangle in $\Db {\cal A}$ gives rise to an exact sequence
$$0 \longrightarrow E_{i-1} \longrightarrow E_i \longrightarrow B \otimes V^\ast \longrightarrow 0$$
in ${\cal H}$, which is the universal extension of $E_{i-1}$ with $B$ and all these exact sequences lie in the same $t$-orbit.
Using Proposition \ref{proposition:EndoSimplesDirected}, we may prove following easy lemma.
\begin{lemma}\label{lemma:E} Let ${\cal E} = (E_i)_{i\in I}$ and $B$ as above, then
\begin{enumerate}
\item $\operatorname {Hom}(E_i,E_j) = \operatorname {Ext}(E_j,E_i) = 0$ for $i>j$,
\item $\operatorname {Hom}(B,E_i) = \operatorname {Ext}(B,E_i) = 0$ for all $i \in I$.
\end{enumerate}
\end{lemma}
If ${\cal H}$ is of the form $\mathop{\text{\upshape{coh}}} X$ for an elliptic curve $X$ (which we will show below to be the case) one may verify that $E$ corresponds to a stable vector bundle of rank $\dim V$ and $B$ to the structure sheaf $k(P)$ of a point $P$. The $E_i$ are equal to $E(-iP)$.
\subsection{${\cal E}$ is an ample sequence in ${\cal H}$}
We now wish to show the sequence ${\cal E} = (E_i)_{i \in {\blb Z}}$ is ample. The following lemma will be useful.
\begin{lemma}\label{lemma:EiX}
If $\operatorname {Hom}(E_i,X) \not=0$, then $\operatorname {Hom}(E_j,X) \not=0$ for all $j \leq i$.
\end{lemma}
\begin{proof}
It suffices to show that $\operatorname {Hom}(E_{i-1},X) \not=0$. Since $\operatorname {Hom}(E_i,X) \not=0$ and $t$ is an auto-equivalence, we know $\operatorname {Hom}(E_{i-1},t^{-1}X) \not=0$. Applying the functor $\operatorname {Hom}(E_{i-1},-)$ to the exact sequence
$$0 \longrightarrow t^{-1} X \longrightarrow X \longrightarrow B \otimes \operatorname {Hom}(X,B)^\ast \longrightarrow 0$$
yields $\operatorname {Hom}(E_{i-1},X) \not= 0$.
\end{proof}
\begin{proposition}\label{proposition:EE}
In ${\cal H}$ the sequence ${\cal E} = (E_i)$ is ample.
\end{proposition}
\begin{proof}
First, we will show ${\cal E}$ is projective. Therefore, let $X \to Y$ be an epimorphism and let $K$ be the kernel. By the construction of ${\cal H}$ in \S\ref{subsection:t}, we know there are paths from the sequence ${\cal E}$ to every direct summand of $K$. Hence, by Corollary \ref{corollary:TubesDirected}, we know $\operatorname {Hom}(K, E_i)=0$ for $i<<0$ and, by the Calabi-Yau property, $\operatorname {Ext}(E_i,K)=0$. Thus $\operatorname {Hom}(E_i,X) \to \operatorname {Hom}(E_i,Y)$ is surjective for $i < n$.
Next, we will show ${\cal E}$ is coherent. Thus we consider an object $X \in {\cal H}$ and we may assume there is a $j \in {\blb Z}$ such that $\operatorname {Hom}(E_{j+2},X) \not= 0$, and hence by Lemma \ref{lemma:EiX}, that $\operatorname {Hom}(E_i,X) \not= 0$ for all $i < j+2$. Fix an $i < j$, we will prove that $f: E_{i-1}\longrightarrow X$ factors through $E_i \oplus E_{j}$. Iteration then implies $f$ factors through a number of copies of $E_{j-1} \oplus E_j$, and hence ${\cal E}$ is coherent.
To prove previous claim, it will be convenient to work in the derived category. The following two triangles in $\Db {\cal H}$ will be used
\begin{equation}\label{equation:Triangle1}
\xymatrix@1{B \otimes V^\ast [-1] \ar[r]^-{\theta} & E_{i-1} \ar[r] & E_i \ar[r] & B \otimes V^\ast}
\end{equation}
and
\begin{equation}\label{equation:Triangle2}
\xymatrix@1{B \otimes V^\ast [-1] \ar[r]^-{\varphi} & E_{j} \ar[r] & E_{j+1} \ar[r] & B \otimes V^\ast}
\end{equation}
where $V = \operatorname {Hom}(E_i,B) \cong \operatorname {Hom}(E_{j+1},B)$. We may assume $f: E_{i-1}\longrightarrow X$ does not factor though $E_i$, hence from triangle (\ref{equation:Triangle1}) it follows that the composition $f \circ \theta \not= 0$.
Note that, since $\operatorname {Hom}(E_{j+1},X) \not= 0$, we may use Corollary \ref{corollary:TubesDirected} to see $\operatorname {Hom}(X,E_{j+1}) = 0$, and hence also $\operatorname {Ext}(E_{j+1},X)=0$.
Applying the functor $\operatorname {Hom}(-,X)$ on triangle (\ref{equation:Triangle2}) and using $\operatorname {Ext}(E_{j+1},X)=0$, shows that every map $B \otimes V^\ast [-1] \longrightarrow X$ factors though $\varphi$. Hence there is a morphism $g : E_{j} \longrightarrow X$ such that the following diagram commutes.
$$\xymatrix{
B \otimes V^\ast [-1] \ar[r]^-{\theta} \ar[d]_{\varphi}& E_{i-1} \ar[d]^f\\
E_{j} \ar[r]^-{g} & X
}$$
Furthermore, applying $\operatorname {Hom}(-,E_{j})$ to triangle (\ref{equation:Triangle1}) yields that $\varphi$ factors through $\theta$, hence there is a map $h : E_{i-1} \longrightarrow E_{j}$ such that $g \circ h \circ \theta = f \circ \theta$, or $(g \circ h - f) \circ \theta = 0$.
Summarizing, $f = g \circ h + f'$, where $f' : E_{i-1} \longrightarrow X$ lies in $\operatorname {ker}(\theta,X)$ and as such factors through $E_i$. The map $f$ factors though $E_i \oplus E_{j}$ and we may conclude the sequence ${\cal E}$ is coherent.
Finally, we show the sequence ${\cal E}$ is ample. Let $X$ be an indecomposable object. Due to the construction of ${\cal H}$, we know that there is an oriented path from $E_n$ to $X$, for a certain $n \in {\blb Z}$. Thus it suffices to prove that if $\operatorname {Hom}(E_n,X) \not= 0$, then there is a finite set $I \subset {\blb Z}$ such that
$$\bigoplus_{i\in I} E_i \otimes \operatorname {Hom}(E_i,X) \longrightarrow X$$
is an epimorphism.
Let $i_1, \ldots, i_m \in {\blb Z}$ be as in the definition of coherence. Consider the map
\begin{equation}\label{equation:Ample}
\theta : \bigoplus_{j=1}^{m} E_{i_j} \otimes \operatorname {Hom}(E_{i_j},X) \longrightarrow X
\end{equation}
and let $C = \operatorname {coker} \theta$. To ease notation, we will refer to the domain of $\theta$ by $\operatorname {dom} \theta$.
There is an exact sequence $0 \longrightarrow \operatorname {im} \theta \longrightarrow X \longrightarrow C \longrightarrow 0$. Using the Calabi-Yau property, we see $\operatorname {Hom}(\operatorname {im} \theta, C) \not= 0$, and since $\operatorname {im} \theta$ is a quotient object of $\operatorname {dom} \theta$, this yields $\operatorname {Hom}(\operatorname {dom} \theta, C) \not= 0$. Hence we may assume there is an $i_j$ such that $\operatorname {Hom}(E_{i_j}, C) \not= 0$.
Since ${\cal E}$ is projective, there is an $l<<0$ such that the induced map in $\operatorname {Hom}(E_l,C)$ lifts to a map in $\operatorname {Hom}(E_l,X)$. Again using coherence, this map should factor through $\operatorname {dom} \theta$. We may conclude $C=0$, and hence $\theta$ is an epimorphism.
\end{proof}
\subsection{Description of $R = R({\cal E})$}
Having shown in Proposition \ref{proposition:EE} that ${\cal E}$ is an ample sequence, we may invoke Proposition \ref{theorem:Polishchuk} to see the that ${\cal H} \cong \mathop{\text{\upshape{coh}}}proj A({\cal E})$.
We will now proceed to discuss the graded algebra $R = R({\cal E})$. In particular, we wish to show $R$ is a finitely generated domain of Gelfand-Kirillov dimension 2 which admits a Veronese subalgebra generated in degree one. It would then follow from \cite{Artin95} that $R$ is noetherian and that $\operatorname{qgr} R$ is equivalent to $\mathop{\text{\upshape{coh}}} X$ where $X$ is a curve, while it would follow from Corollary \ref{corollary:Polishchuk} that ${\cal H} \cong \operatorname{qgr} R$.
We start by showing $\operatorname {GKdim} R = 2$.
\begin{lemma}
Let ${\cal E}= (E_i)_{i \in I}$ and $B$ be as before. If $j > i$, then
$$\dim \operatorname {Hom}(E_i,E_j) = (j-i)d^2$$
where $d=\dim\operatorname {Hom}(E_0,B)$.
\end{lemma}
\begin{proof}
We apply $\operatorname {Hom}(E_i,-)$ to the short exact sequence
$$0 \longrightarrow E_{j-1} \longrightarrow E_{j} \longrightarrow B \otimes \operatorname {Hom}(E_0,B)^* \longrightarrow 0.$$
We will proceed by induction on $j > i$. Note that $\dim \operatorname {Hom}(E_i,B) = \dim \operatorname {Hom}(E_0,B)^* = d$ and Lemma \ref{lemma:E} implies that $\operatorname {Ext}(E_i,E_j)=0$.
If $j=i+1$, then it follows from $\dim \operatorname {Hom}(E_i,E_i) = \dim\operatorname {Ext}(E_i,E_i)=1$ that $\dim \operatorname {Hom}(E_i,E_j) = d^2$.
For higher $j$, we find by induction $\dim \operatorname {Hom}(E_i,E_j) = (j-i)d^2$.
\end{proof}
\begin{lemma}\label{lemma:Domain}
Assume $E$ and $B$ are non-isomorphic endo-simple objects of $\Db {\cal A}$ chosen such that $d=\dim \operatorname {Hom}_{\Db {\cal A}}(E,B)$ is minimal and $d \not= 0$. Then $R$ is a domain.
\end{lemma}
\begin{proof}
It suffices to show every non-zero non-isomorphism $f : E_0 \longrightarrow E_i$ is a monomorphism. We will prove this by induction on $i$. The case $i=0$ is trivial. So let $i \geq 1$.
Since $\operatorname {im} f$ is a quotient object of $E_0$ and $\dim \operatorname {Hom}(E,B)=d$, we see that $\dim \operatorname {Hom}(\operatorname {im} f, B) \leq d$, and due to the minimality of $d$, we know that either $\dim \operatorname {Hom}(\operatorname {im} f, B)=0$, or $\dim \operatorname {Hom}(\operatorname {im} f, B)=d$ and $\operatorname {im} f$ is an endo-simple object.
If $\dim \operatorname {Hom}(\operatorname {im} f, B)=0$, the inclusion $\operatorname {im} f \hookrightarrow E_i$ has to factor through a map $j:\operatorname {im} f \longrightarrow E_{i-1}$.
$$\xymatrix{
&&E_0 \ar@{->>}[d]\\
&&{\operatorname {im} f} \ar@{^{(}->}[d]\ar[ld]_{\exists j}\\
0\ar[r]&E_{i-1}\ar[r]&E_i\ar[r]&B\otimes\operatorname {Hom}(E_i,B)^*\ar[r]&0}$$
Composition gives a non-zero map $E_0 \longrightarrow E_{i-1}$ which is a monomorphism by the induction hypothesis. We conclude that $f$ is a monomorphism.
We are left with $\dim \operatorname {Hom}(\operatorname {im} f,B)=d$, and hence $\dim \operatorname {Hom}(K,B)=0$ where $K = \operatorname {ker} f$. With ${\cal E}$ being ample, we may assume there is a $k \in {\blb Z}$, such that $E_k$ maps non-zero to every direct summand of $K$. Using the exact sequence $0 \to E_k \to E_{k+1} \to B \otimes \operatorname {Hom}(E_{k+1},B)^* \to 0$, we find that for every $l \in {\blb Z}$, $E_l$ maps non-zero to every direct summand of $K$. Hence $\operatorname {Hom}(K,E_i) = 0$ and thus $K=0$. We conclude that $f$ is a monomorphism.
\end{proof}
In general, however, $R$ will not be generated in degree 1. We show that the Veronese subalgebra $R^{(3)} = \oplus_k R_{3k}$ of $R$ is generated in degree 1.
\begin{lemma}\label{lemma:Degree1}
The sequence ${\cal E}^{(3)}=(E_{3k})_{k\in {\blb Z}}$ is an ample sequence. Furthermore $R^{(3)} = R({\cal E}^{(3)})$ is generated in degree 1.
\end{lemma}
\begin{proof}
The sequence ${\cal E}^{(3)}$ is projective and ample since ${\cal E}$ is. Coherence of ${\cal E}^{(3)}$ may be shown as in the proof of Proposition \ref{proposition:EE}.
Next, we prove $R^{(3)}$ is generated in degree one. Therefore, it suffices to show that for every $k>1$ every map $E_0 \to E_{3k}$ factors through the canonical map $\theta : E_0 \to E_3 \otimes \operatorname {Hom}(E_0,E_3)^\ast$. We write $V = \operatorname {Hom}(E_0,E_3)$ and we consider the triangle
$$\xymatrix@1{C \ar[r] & {E_0} \ar[r]^-{\theta} & E_3 \otimes V^\ast \ar[r]& C[1]}$$
where $C = T_{E_3} E_0$ is an endo-simple object since $T_{E_3}$ is an automorphism. Applying the functor $\operatorname {Hom}(-,E_{3k})$ to this triangle gives the exact sequence
$$0 \longrightarrow \operatorname {Hom}(C[1],E_{3k}) \longrightarrow \operatorname {Hom}(E_3 \otimes V^\ast,E_{3k}) \longrightarrow \operatorname {Hom}(E_0,E_{3k}) \longrightarrow \operatorname {Hom}(C,E_{3k}) \longrightarrow 0.$$
We now consider the dimensions of these vector spaces. Since
$$\dim \operatorname {Hom}(E_0,E_{3k}) = (3k)d^2 < \dim \operatorname {Hom}(E_3 \otimes V^\ast,E_{3k}) = 9(k-1)d^4$$
we may see $\operatorname {Hom}(C[1],E_{3k}) \not= 0$ and $\dim \operatorname {Hom}(C,E_{3k}) \not= \dim \operatorname {Hom}(C[1],E_{3k})$, hence $E_{3k} \not\cong C[1]$.
Using Proposition \ref{proposition:EndoSimplesDirected}, we obtain $\operatorname {Hom}(C,E_{3k})=0$, hence every map $E_0 \longrightarrow E_{3k}$ lifts through $\theta$ and the algebra $R^{(3)}$ is generated in degree one.
\end{proof}
\subsection{Classification up to derived equivalence}
We are now ready to prove the main result of this article.
\begin{theorem}\label{theorem:Main}
Let ${\cal A}$ be a connected $k$-linear abelian Ext-finite 1-Calabi-Yau category, then ${\cal A}$ is derived equivalent to either
\begin{enumerate}
\item the category of finite dimensional representations of $k[[t]]$, or
\item the category of coherent sheaves on an elliptic curve $X$.
\end{enumerate}
\end{theorem}
\begin{proof}
By Proposition \ref{proposition:EndoSimple} we know there are endo-simple objects. First, assume all endo-simple objects are isomorphic. Using Proposition \ref{proposition:Building} we easily see that ${\cal A}$ is equivalent to $\operatorname{Mod}fd k[[t]]$.
Next, assume there are at least two non-isomorphic endo-simple objects. Since ${\cal A}$ is connected and using Proposition \ref{proposition:Building}, we may choose two endo-simples, $E$ and $B$, such that $\operatorname {Hom}(E,B) \not= 0$, yet with a minimal dimension. Let ${\cal H}$ be the abelian category constructed in \S\ref{subsection:t}.
By Lemmas \ref{lemma:Domain} and \ref{lemma:Degree1}, we know $R^{(3)} = R({\cal E}^{(3)})$ is a domain of GK-dimension 2 which is finitely generated by elements of degree one, hence by \cite{Artin95} we find that $R^{(3)}$ is noetherian and $\operatorname{qgr} R^{(3)}$ is equivalent to the coherent sheaves on a curve $X$.
Since $R$ is noetherian, it follows from \ref{theorem:Polishchuk} that ${\cal H}$ is equivalent to $\operatorname{qgr} R^{(3)}$.
The structure sheaf ${\cal O}_X$ of $X$ is an endo-simple object. Since the genus of $X$ is $\dim H^1({\cal O}_X) = \dim \operatorname {Ext}({\cal O}_X, {\cal O}_X) = \dim \operatorname {Hom}({\cal O}_X,{\cal O}_X) = 1$, we know $X$ is an elliptic curve.
\end{proof}
\subsection{Classification of abelian categories}\label{subsection:Abelian}
We will now combine Theorem \ref{theorem:Main} with \cite[Proposition 5.1]{Burban06} to obtain a description of all abelian 1-Calabi-Yau categories. First, we recall some results from \cite{Happel96}.
Let ${\cal A}$ be any hereditary abelian category. A \emph{torsion theory} on ${\cal A}$, $({\cal F},{\cal T})$, is a pair of full additive subcategories of ${\cal A}$, such that $\operatorname {Hom}({\cal T},{\cal F})=0$ and having the additional property that for every $X \in \operatorname{Ob} {\cal A}$ there is a short exact sequence
$$0 \longrightarrow T \longrightarrow X \longrightarrow F \longrightarrow 0$$
with $F \in {\cal F}$ and $T \in {\cal T}$.
We will say the torsion theory $({\cal F},{\cal T})$ is \emph{split} if $\operatorname {Ext}({\cal F},{\cal T})=0$. In case of a split torsion theory we obtain, by \emph{tilting}, a hereditary category ${\cal H}$ derived equivalent to ${\cal A}$ with an induced split torsion theory $({\cal T},{\cal F}[1])$. Furthermore, the category ${\cal H}$ will only be hereditary if and only if $({\cal F},{\cal T})$ is a split torsion theory.
We now discuss all possible torsion theories when ${\cal A}$ is equivalent to $\mathop{\text{\upshape{coh}}} X$. Note that, since ${\cal H}$ will be 1-Calabi-Yau and hence hereditary, all torsion theories on ${\cal A}$ will be split.
Let $({\cal F},{\cal T})$ be a torsion theory on ${\cal A}$, and let ${\cal E}$ be an indecomposable of ${\cal T}$. Then every indecomposable ${\cal F}$ with slope strictly larger than $\mu({\cal E})$ has to be in ${\cal T}$ since $\operatorname {Hom}({\cal E},{\cal F}) \not= 0$. Furthermore, if ${\cal E}$ is in ${\cal T}$ and there is a path from ${\cal E}$ to an indecomposable ${\cal E}'$, then ${\cal E}' \in \ind {\cal T}$.
We may now give a characterization of all possible torsion theories.
\begin{theorem}\label{theorem:Abelian}\cite{Burban06}
Let $X$ be an elliptic curve. Every category ${\cal H}$ derived equivalent to ${\cal A} = \mathop{\text{\upshape{coh}}} X$ may be obtained by tilting with respect to a torsion theory. Moreover, all torsion theories on $\mathop{\text{\upshape{coh}}} X$ are split and may be described as follows. Let $\theta \in {\blb R} \cup \{\infty\}$. Denote by ${\cal A}_{> \theta}$ and ${\cal A}_{\geq \theta}$ the subcategory of ${\cal A}$ generated by all indecomposables ${\cal E}$ with $\mu({\cal E}) > \theta$ and $\mu({\cal E}) \geq \theta$, respectively. All full subcategories ${\cal T}$ of ${\cal A}$ with ${\cal A}_{\geq \theta} \subseteq {\cal T} \subseteq {\cal A}_{> \theta} \subseteq {\cal A}$ give rise to a torsion theory $({\cal F},{\cal T})$, with $\ind {\cal F} = \ind {\cal A} \setminus \ind {\cal T}$.
\end{theorem}
\begin{proof}
That these are all possible torsion theories, follows from the above discussion. That all categories ${\cal H}$ may be obtained in this way, is shown in \cite[Proposition 5.1]{Burban06}. Alternatively, it is straightforward to check these torsion theories generate all bounded $t$-structures on $\Db {\cal A}$ up to shifts.
\end{proof}
\begin{example}
We give some examples of torsion theories. In here ${\cal H}$ always stands for the category tilted with respect to the described torsion theory.
\begin{enumerate}
\item If $\theta \in {\blb Q} \cup \{\infty\}$ and ${\cal T} = {\cal A}_{> \theta}$, then the tilted category ${\cal H}$ is equivalent to $\mathop{\text{\upshape{coh}}} X$. If ${\cal T} = {\cal A}_{\geq \theta}$, then ${\cal H}$ is dual to ${\cal A}$.
\item If $\theta \in {\blb R} \setminus {\blb Q}$ and ${\cal T} = {\cal A}_{> \theta} = {\cal A}_{\geq \theta}$ then ${\cal H}$ is equivalent to the category of holomorphic bundles on a noncommutative two-torus (\cite{Polishchuk04}).
\end{enumerate}
\end{example}
Theorem \ref{theorem:Abelian} classifies all categories derived equivalent to $\mathop{\text{\upshape{coh}}} X$. We further need to classify all categories derived equivalent to ${\cal B} = \operatorname{Mod}fd k[[t]]$.
Let ${\cal H}$ be such a category derived equivalent to ${\cal B}$. Then ${\cal H}$ induces a $t$-structure $(D^{\geq 0}, D^{\leq 0})$ on $\Db {\cal B}$. Since this $t$-structure is split, we may assume the heart ${\cal H} = D^{\leq 0} \cap D^{\geq 0}$ contains the endo-simple object $E$ of ${\cal B}[0]$ and, since ${\cal B}$ has only one endo-simple object, this is the unique endo-simple object of ${\cal H}$, up to isomorphism.
Moreover, for every $X \in {\cal B}$ we have $\operatorname {Hom}(X,B) \not=0$ and $\operatorname {Hom}(B,X) \not=0$, thus we have ${\cal B} \subseteq D^{\leq 0} \cap D^{\geq 0} = {\cal H}$.
Since ${\cal B}$ has only one endo-simple object, $E$ is the unique endo-simple object of ${\cal H}$, up to isomorphism. From this we infer ${\cal B} = {\cal H}$ as subcategories of $\Db {\cal B}$.
We conclude that every category derived equivalent to $\operatorname{Mod}fd k[[t]]$ is in fact equivalent to $\operatorname{Mod}fd k[[t]]$.
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\end{document} |
\begin{document}
\runninghead{Nyaga \textit{et al.}}
\title{ANOVA model for network meta-analysis of diagnostic test accuracy data}
\author{Victoria Nyaga\affilnum{1,2}, Marc Aerts\affilnum{2} and Marc Arbyn\affilnum{1}}
\affiliation{\affilnum{1}Scientific Institute of Public Health, Unit of Cancer Epidemiology/Belgian Cancer Center, Belgium\\
\affilnum{2}Hasselt University, Center for Statistics, Belgium}
\corrauth{Victoria Nyaga
Scientific Institute of Public Health, Unit of Cancer Epidemiology/Belgian Cancer Center,
Juliette Wytsmanstreet 14,
1050, Brussels,
Belgium.}
\email{[email protected]}
\begin{abstract}
Network meta-analysis (NMA) allow combining efficacy information from multiple comparisons from trials assessing different therapeutic interventions for a given disease and to estimate unobserved comparisons from a network of observed comparisons. Applying NMA on diagnostic accuracy studies is a statistical challenge given the inherent correlation of sensitivity and specificity.
A conceptually simple and novel hierarchical arm-based (AB) model which expresses the logit transformed sensitivity and specificity as sum of fixed effects for test, correlated study-effects and a random error associated with various tests evaluated in given study is proposed. We apply the model to previously published meta-analyses assessing the accuracy of diverse cytological and molecular tests used to triage women with minor cervical lesions to detect cervical precancer and the results compared with those from the contrast-based (CB) model which expresses the linear predictor as a contrast to a comparator test.
The proposed AB model is more appealing than the CB model in that it yields the marginal means which are easily interpreted and makes use of all available data and easily accommodates more general variance-covariance matrix structures.
\end{abstract}
\keywords{meta-analysis, network meta-analysis, diagnostic test accuracy, hierarchical model, ANOVA}
\maketitle
\section{Introduction}
Network meta-analyses (NMA) have classically been used to extend conventional pairwise meta-analyses by combining and summarizing direct and indirect evidence on multiple `therapeutic' interventions for a given condition when the set of evaluated interventions/treatments differs among studies. By borrowing strength from the indirect evidence, there is a potential gain in precision of the estimates \cite{Higgins}. Furthermore, the estimates may be less biased and more robust. Such an approach uses the data efficiently and is line with the principle of intention-to-treat (ITT) \cite{Fisher} in randomized clinical trials which requires that all valid available data should be used even when a part of the data is missing.
In a diagnostic test accuracy study, an index test and possibly one or more comparator tests are administered to each tested subject. A standard or reference test or procedure is also applied to all the patients to classify them as having the target condition or not. The patients results are then categorized by the index and reference test as true positive, false positive, true negative and false negative. The diagnostic accuracy of the index test is represented as a bivariate outcome and is typically expressed as sensitivity and specificity at a defined test cutoff. Differences due to chance, design, conduct, patients/participants, interventions, tests and reference test imply there will be heterogeneity often in opposite direction for the two typical accuracy outcomes: sensitivity and specificity. While traditional meta-analyses allow for comparison between two tests, there are often multiple tests for the diagnosis of a particular disease outcome. To present the overall picture, inference about all the tests for the same condition and patient characteristics is therefore required. The simultaneous analysis of the variability in the accuracy of multiple tests within and between studies may be approached through a network meta-analysis.
In combining univariate summaries from studies where the set of tests differs among studies two types of linear mixed models have been proposed. The majority of network meta-analyses express treatment effects in each study as a contrast relative to a baseline treatment in the respective study \cite{Higgins, Lumley}. This is the so called contrast-based (CB) model. Inspired by the CB models developed for interventional studies, Menten and Lesaffre (2015) \cite{Menten} introduced a CB model for diagnostic test accuracy data to estimate the average log odds ratio for sensitivity and specificity of the index test relative to a baseline or comparator test.
The second type of models is the classical two-way ANOVA model with random effects for study and fixed effect for tests \cite{Senn, Whitehead, Piepho12}, the so called arm-based (AB) model. The AB model is based on the assumption that the missing arms or tests are missing at random. While the two types of models yield similar results for the contrasts with restricted maximum likelihood (REML) procedures, the CB model is generally not invariant to changes in the baseline test in a subset of studies and yields an odds ratio (OR) making it difficult to recover information on the absolute diagnostic accuracy (the marginal means), relative sensitivity or specificity of a test compared to another or differences in accuracy between tests, measures that are easily interpretable and often used in clinical epidemiology. It is common knowledge that the OR is only a good approximation of relative sensitivity/specificity when the outcome is rare but this is often not the case in diagnostic studies. Moreover, the AB model is simpler when the baseline/comparator treatment varies from one study to another or when the number of tests varies substantially among studies. By accommodating more complex variance-covariance structures AB models have been shown to be superior to CB models \cite{Zhang}.
We apply the two-way ANOVA model in a diagnostic data setting by extending the AB model in two ways: 1. using two independent binomial distributions to describe the distribution of true positives and true negatives among the diseased and the healthy individuals, 2. inducing a correlation between sensitivity and specificity by introducing correlated and shared study effects. The resulting generalized linear mixed model is analogous to randomized trials with complete block designs or repeated measures in analysis of variance models where studies are equivalent to blocks. The main assumption is that, results missing for some tests and studies are missing at random. This approach is efficient because the correlation structure allows the model to borrow information from the `imputed' missing data to obtain adjusted sensitivity and specificity estimates for all the tests.
\section{Motivating dataset}
To illustrate the use of the proposed model in network meta-analysis of diagnostic test accuracy data, we analyse data on a diversity of cytological or molecular tests to triage women with equivocal or mildly abnormal cervical cells \cite{Arbyn12, Arbyn13a, Arbyn13b, Roelens, Verdoodt}. A Pap smear is a screening test used to detect cervical precancer. When abnormalities in the Pap smear are not high grade, a triage test is needed to identify the women who need referral for further diagnostic work-up. There are several triage options, such as repetition of the Pap smear or HPV DNA or RNA assays. HPV is the virus causing cervical cancer \cite{Bosch}. Several other markers can be used for triage as well, such as p16 or the combinations of p16/Ki67 which are protein markers indicative for a transforming HPV infection~\cite{Roelens, Arbyn09} .
The data are derived from a comprehensive series of meta-analyses on the accuracy of triage with HPV assays, cervical cytology or molecular markers applied on cervical specimens in women with minor cervical abnormalities \cite{Arbyn12, Arbyn13a, Arbyn13b, Roelens, Verdoodt}. Two patient groups with minor cytological abnormalities were distinguished: women with ASC-US (atypical squamous cells of unspecified significance) and LSIL (low-grade squamous intraepithelial lesions). Studies were included in the analysis if they performed besides one or more triage test a verification with a reference standard based on colposcopy and biopsies.
In total, the accuracy of 11 tests for detecting cervical precancer were evaluated. Labelled 1 to 11 the tests were: hrHPV DNA testing with HC2 (HC2), Conventional Cytology (CC), Liquid-Based Cytology (LBC), generic PCRs targeting hrHPV DNA (PCR) and commercially available PCR-based hrHPV DNA assays such as: Abbott RT PCR hrHPV, Linear Array, and Cobas-4800; assays detecting mRNA transcripts of five (HPV Proofer) or fourteen (APTIMA) HPV types HPV types; and protein markers identified by cytoimmunochestry such as: p16 and p16/Ki67, which are over-expressed as a consequence of HPV infection. Two levels of precancer (disease) were considered: intraepithelial neoplasia lesion of grade two or worse (CIN2+) or of grade three or worse (CIN3+). 125 studies with at least one test and maximum of six tests were included allowing assessment of the accuracy of the eleven triage tests. In
\begin{figure}
\caption{Network plot of all included tests\protect\endnotemark[1] by triage\protect\endnotemark[2] group (women with ASC-US or LSIL cytology) and the outcome\protect\endnotemark[3] (CIN2+ or CIN3+).\label{Fig:1}
\label{Fig:1}
\end{figure}
The size of the nodes in figure~\ref{Fig:1} is proportional to the number of studies evaluating a test and thickness of the lines between the nodes is proportional to the number of direct comparisons between tests. The size of the node and the amount of information in a node consequently influence the standard errors of the marginal means and the relative measures. From the network plot, test 1 (HC2) and test 11 (APTIMA) were the most commonly assessed tests. The network in figure~\ref{Fig:1} is connected.
\section{Methodology}
Suppose there are \textit{K} tests and \textit{I} studies. Studies assessing two tests ($ k = 2 $) are called `two-arm' studies while those with $k > 2 $ are ‘multi-arm’ studies. For a certain study \textit{i}, let $(Y_{i1k}, ~Y_{i2k})$ denote the true positives and true negatives, $(N_{i1k}, ~N_{i2k})$ the diseased and healthy individuals and $(\pi_{i1k}, ~\pi_{i2k})$ the `unobserved' sensitivity and specificity respectively with test \textit{k} in study \textit{i}. Given study-specific sensitivity and specificity, two independent binomial distributions describe the distribution of true positives and true negatives among the diseased and the healthy individuals as follows;
\begin{equation}\label{Eq:0}
Y_{ijk} ~|~ \pi_{ijk}, ~x_i ~\sim~ bin(\pi_{ijk}, ~N_{ijk}), ~i ~=~ 1, ~\ldots I, ~j ~=~ 1, ~2, ~k~ = 1,~ \ldots ~K,
\end{equation}
where $x_i$ generically denotes one or more covariates, possibly affecting $\pi_{ijk}$. In the next section, we present the recently introduced contrast-based model~\cite{Menten} followed by our proposed arm-based model to estimate the mean as well as comparative measures of sensitivity and specificity.
\subsection{Contrast-based model}
By taking diagnostic test $T_K$ as the baseline, Menten and Lessafre (2015) \cite{Menten} proposed the following model,
\begin{align}\label{Eq:1}
logit(\pi_{ijk}) ~=~ \theta_{ijk} \nonumber\\
\theta_{ij1} ~=~ \mu_{ij} ~+~ (K ~-~ 1)\times \frac{\delta_{ij1}}{K} ~-~ \frac{\delta_{ij2}}{K} ~-~ \frac{\delta_{ij3}}{K} ~-~ \ldots ~\frac{\delta_{ij(k-1)}}{K} \nonumber\\
\theta_{ij2} ~=~ \mu_{ij} ~-~ \frac{\delta_{ij1}}{K} ~+~ (K ~-~ 1)\times\frac{\delta_{ij2}}{K} ~-~ \frac{\delta_{ij3}}{K} ~-~ \ldots ~\frac{\delta_{ij(k-1)}}{K} \nonumber\\
\theta_{ijK} ~=~ \mu_{ij} ~-~ \frac{\delta_{ij1}}{K} ~-~ \frac{\delta_{ij2}}{K} ~-~ \frac{\delta_{ij3}}{K} ~-~ \ldots ~\frac{\delta_{ij(k-1)}}{K} \nonumber\\
with \nonumber\\
(\delta_{i11}, ~\delta_{i12}, ~\delta_{i21}, ~\delta_{i22}, ~\ldots, ~\delta_{i1(K-1)}, ~\delta_{i2(K-1)}) ~\sim~ N(\boldsymbol{\nu}_\delta, ~\boldsymbol{\Sigma}) \nonumber\\
and \nonumber \\
\nu_\delta ~=~ (\nu_{\delta11}, ~\nu_{\delta21}, ~\nu_{\delta12}, ~\nu_{\delta22} ~\ldots, ~\nu_{\delta1(K-1)}, ~\nu_{\delta2(K-1)})
\end{align}
The $\boldsymbol{\nu}_\delta$ represents the average log odds ratio for sensitivity and specificity of the \textit{K~-~1} tests compared to the baseline test $T_K$. There are known difficulties in estimating the variance-covariance matrix $\boldsymbol{\Sigma}$ since each sampled matrix should be positive-definite \cite{Daniels}. The authors therefore recommend a diagonal or block diagonal variance-covariance matrix $\boldsymbol{\Sigma}$. While this reduces model complexity and difficulty in estimation, such a covariance matrix accounts for the correlation between contrasts but ignores correlation between sensitivity and specificity. Moreover, the model identification becomes difficult as the number of tests included increases.
The authors estimate the absolute accuracy of the tests from the estimated $logit^{-1}( \mu_{jk})$ as follows
\begin{align}\label{Eq:2}
\mu_{j1} ~=~ logit^{-1}(E(\mu_j)) ~+
&~ \frac{K-1}{K}\times \nu_{\delta{j1}} ~-~ \frac{1}{K}\times\nu_{\delta{j2}} ~-~ \frac{1}{K}\times\nu_{\delta{j3}} ~-~ \ldots ~-~ \frac{1}{K}\times\nu_{\delta{j(K-1)}} \nonumber\\
\mu_{j2} ~=~ logit^{-1}(E(\mu_j)) ~-
&~ \frac{1}{K}\times\nu_{\delta{j1}} ~+~ \frac{K-1}{K}\times\nu_{\delta{j2}} ~-~ \frac{1}{K}\times\nu_{\delta{j3}} ~-~ \ldots ~-~ \frac{1}{K}\times\nu_{\delta{j(K-1)}} \nonumber\\
\mu_{j3} ~=~ logit^{-1}(E(\mu_j)) ~-
&~ \frac{1}{K}\times\nu_{\delta{j1}} ~-~ \frac{1}{K}\times\nu_{\delta{j2}} ~+~ \frac{K-1}{K}\times\nu_{\delta{j3}} ~-~ \ldots ~-~ \frac{1}{K}\times\nu_{\delta{j(K-1)}} \nonumber\\
\mu_{jK} ~=~ logit^{-1}(E(\mu_j)) ~-
&~ \frac{1}{K}\times\nu_{\delta{j1}} ~-~ \frac{1}{K}\times\nu_{\delta{j2}} ~-~ \frac{1}{K}\times\nu_{\delta{j3}} ~-~ \ldots ~-~ \frac{1}{K}\times\nu_{\delta{j(K-1)}}
\end{align}
where $logit^{-1}(E(\mu_j))$ is the average probability of testing positive/negative. Equation~\ref{Eq:2} estimates the accuracy of tests for a hypothetical study with random-effects equal to zero but not the meta-analytic estimates as will be explained in the next section.
\subsection{Arm-based model}
Consider a design where there is at least one test per study. The study serves as a block where all diagnostic accuracy tests are hypothetically evaluated of which some are missing. This modelling approach has potential gain in precision by borrowing strength from studies with single tests as well as multi-arm studies. The proposed single-factor design with repeated measures model is written as follows
\begin{align}\label{Eq:3}
logit(\pi_{ijk}) = \mu_{jk} + \eta_{ij} + \delta_{ijk} \nonumber\\
\begin{pmatrix}
\eta_{i1} \nonumber\\
\eta_{i2}
\end{pmatrix} \sim N \bigg (\begin{pmatrix}
0 \nonumber\\
0
\end{pmatrix}, \boldsymbol{\Sigma} \bigg ) \nonumber\\
\boldsymbol{\Sigma} = \begin{bmatrix}
\sigma^2_1 ~~~~ \rho\sigma_1\sigma_2 \nonumber\\
\rho\sigma_1\sigma_2 ~~~~ \sigma^2_2
\end{bmatrix} \nonumber\\
(\delta_{ij1}, \delta_{ij1}, \ldots \delta_{ijK}) \sim N(\textbf{0}, diag(\tau^2_j))
\end{align}
where $\mu_{1k}$ and $\mu_{2k}$ are the mean sensitivity and specificity in a hypothetical study with random-effects equal to zero respectively. $\eta_{ij}$ is the study effect for healthy individuals \textit{(j = 1)} or diseased individuals \textit{(j = 2)} and represents the deviation of a particular study \textit{i} from the mean sensitivity (j=1) or specificity (j=2), inducing between-study correlation. The study effects are assumed to be a random sample from a population of such effects. The between-study variability of sensitivity and specificity and the correlation thereof is captured by the parameters $\sigma_1^2$, $\sigma_2^2$, and $\rho$ respectively. $\delta_{ijk}$ is the error associated with the sensitivity (\textit{j=1}) or specificity (\textit{j=2}) of test \textit{k} in the $i^{th}$ study. Conditional on study \textit{i}, the repeated measurements are independent with variance constant across studies such that $\boldsymbol{\tau}_j^2$ = $(\tau_{j1}^2, \ldots, \tau_{jK}^2)$ is a \textit{K} dimensional vector of homogeneous variances.
In case $\tau_{jk}^2 = \tau_j^2$ (variances homogenous across tests), the shared random element $\eta_{ij}$ within study \textit{i} induce a non-negative correlation between any two test results \textit{k} and $k\prime$ from healthy individuals \textit{(j = 1)} or from diseased individuals \textit{(j = 2)} equal to $\rho_j = \frac{\sigma_j^2}{\sigma_j^2 + \tau_j^2}$ (implying that a covariance matrix with compound symmetry). While it might seem logical to expect and allow for similar correlation between any two sensitivities or specificities in a given study, the variances $\tau_{jk}^2$ of different sensitivities or specificities of the same study may be different. In such instances, the unstructured covariance matrix is more appropriate as it allows varying variances between the tests (in which case $\boldsymbol{\tau}_j^2$ is a \textit{K} dimension vector of the unequal variances). The correlation between the $k^{th}$ and $k^{'th}$ test result is then equal to $\rho_{jkk}' = \frac{\sigma_j^2}{\sqrt{\sigma_j^2 + \tau_{jk}^2~\times (\sigma_j^2 + \tau_{jk'}^2 )}}$. $\rho_j$ or $\rho_{jkk}'$ is called the intra-study correlation coefficient which also measures the proportion of the variability in $logit(\pi_{ijk})$ that is accounted for by the between study variability. It takes the value 0 when $\sigma_j^2 = 0$ (if study effects convey no information) and values close to 1 when $\sigma_j^2$ is large relative to $\tau_j^2$ and the studies are essentially all identical. When all components of $\boldsymbol{\tau}_j^2$ equal to zero, the model reduces to fitting separate bivariate random-effect meta-analysis (BRMA)~\cite{reitsma, chu} model for each test.
In essence, the model separates the variation in the studies into two components: the within-study variation $diag(\tau_j^2)$ referring to the variation in the repeated sampling of the study results if they were replicated, and the between-study variation $\boldsymbol{\Sigma}$ referring to variation in the studies true underlying effects.
The study-level covariate information is included in the linear predictor in Equation~\ref{Eq:3} as follows
\begin{equation} \label{Eq:17}
logit(\pi_{ijk}) = \mu_{jk} + \sum_{p = 1}^{P} \theta_{pjk} X_{pi} + \eta_{ij} + \delta_{ijk}
\end{equation}
where $\theta_{pjk}$ is the $p^{th}$ coefficients corresponding to the $X_{pi}$ covariate in a hypothetical study with random-effects equal to zero respectively.
The population-averaged or the marginal sensitivity/specificity in the intercept-only model for test \textit{k} is estimated as
\begin{align}\label{Eq:4}
E(\pi_{ijk}) ~=~ & E(logit^{-1}(\mu_{jk} ~+~ \eta_{ij} ~+~ \delta_{ijk})) \nonumber\\
=& \int_{-\infty}^{\infty}logit^{-1}(\mu_{jk} ~+~ \eta_{ij} ~+~ \delta_{ijk})~f(\eta_{ij})~f(\delta_{ijk})~d\eta_{ij}~d\delta_{ijk}.
\end{align}
The relative sensitivity and specificity and other relative measures of test \textit{k} (relative to test $k\prime, k \ne k\prime$) are then estimated from the marginal sensitivity or specificity of test \textit{k} and \textit{k’}.
In most practical situations, the mean structure is of primary interest and not the covariance structure. Nonetheless, appropriate covariance modelling is critical in the interpretation of the random variation in the data as well as obtaining valid model-based inference for the mean structure. Compound symmetry assumes homogeneity of variance and covariance and such restriction could invalidate inference for the mean structure when the assumed covariance structure is misspecified \cite{Altham}. When the primary objective of the analysis is on estimating the marginal means of sensitivity and specificity, the choice between compound symmetry and unstructured covariance structure is not critical because the inference procedure for the marginal means are the same. Moreover, over-parameterisation of the covariance structure might lead to inefficient estimation and potentially poor assessment of standard errors of the marginal means \cite{Verbeke}.
\subsection{Ranking of the tests}
While ranking of tests using rank probabilities and rankograms is an attractive feature of univariate NMA, it is still a challenge to rank competing diagnostic tests especially when a test does not outperform the others on both sensitivity and specificity.
Consider the diagnostic odds ratio (DOR) \cite{Glas} which is expressed in terms of sensitivity and specificity as
\begin{equation} \label{Eq:5}
DOR_k = \frac{sensitivity_k \times specificity_k}
{(1 - sensitivity_k) \times (1 - specificity_k)}.
\end{equation}
and ranges from 0 to $\infty$ with: $DOR_k > 1$ or higher indicating better discriminatory test performance, $DOR_k = 1$ indicating a test that does not discriminate between the healthy and diseased, and $DOR_k < 1$ indicating an improper test. The DOR is a single indicator combining information about sensitivity and specificity and is invariant of disease prevalence. However, the measure cannot distinguish between tests with high sensitivity but low specificity or vice-versa.
Alternatively, the superiority of a diagnostic test could be quantified using a superiority index introduced by Deutsch et al. \cite{Deutsch} expressed as
\begin{equation} \label{Eq:6}
S_k = \frac{2a_k + c_k}
{2b_k + c_k},
\end{equation}
where $a_k$ is the number of tests to which test \textit{k} is superior (higher sensitivity and specificity), $b_k$ is the number of tests to which test \textit{k} is inferior (lower sensitivity and specificity), and $c_k$ the number of tests with equal performance as test \textit{k} (equal sensitivity and specificity). \textit{S} ranges from 0 to $\infty$ with; $S$ tending to $\infty $ and $S $ tending to $0$ as the number of tests to which test \textit{k} is superior and inferior increases respectively, and $S$ tending to $1$ the more the tests are equal. Since the number of tests not comparable to test \textit{k} do not enter into the calculation of \textit{S} the index for different tests may be based on different sets of tests.
\subsection{Missing data and exchangeability }
In the models above, not all the studies provide estimates of all effects of interest because some of the components of the vector $\textbf{Y}_{ij} = (Y_{ij1}, \ldots, Y_{ijK})$ are missing. The $\textbf{Y}_{ij}$ vector can be partitioned into the observed $\textbf{Y}_{ij}^o$ and the missing $\textbf{Y}_{ij}^m$. For each component of $\textbf{Y}_{ij}$ let $\textbf{R}_{ij}$ denote a vector of missingness indicator with
\begin{equation} \label{Eq:7}
R_{ijk} = \bigg \{ \begin{matrix}
1 ~ if ~Y_{ijk} ~is ~observed,\nonumber\\
0 ~ otherwise.
\end{matrix}
\end{equation}
The joint distribution of (\textbf{Y}, \textbf{R}) given the parameters ($\beta, \phi$) is given by
\begin{equation} \label{Eq:8}
p(y_{ij}, R_{ij}| \boldsymbol{\beta}_{j}, \boldsymbol{\phi}_{j})
\end{equation}
where
$\boldsymbol{\phi}_{j}$ contains the missingness paramaters and $\boldsymbol{\beta}_{j}$ contains $(\boldsymbol{\pi}_{ij}, ~\boldsymbol{\Sigma}, ~\rho, ~\sigma_j, ~ diag(\boldsymbol{\tau}_j))$. In a selection framework \cite{Rubin, Little} the joint distribution in Equation~\ref{Eq:8} is factorised as
\begin{equation} \label{Eq:9}
p(y_{ij}~|~ \boldsymbol{\beta}_{j}), \boldsymbol{\phi}_{j}~p(R_{ij}~|~ Y_{ij}, ~\boldsymbol{\phi}_{j}) =
p(y_{ij}^o, ~y_{ij}^m ~|~ \boldsymbol{\beta}_{j}, ~\boldsymbol{\phi}_{j})p(R_{ij}~|~ Y_{ij}^o, ~Y_{ij}^m,~ \boldsymbol{\phi}_{j})
\end{equation}
where $p(R_{ij}~|~y_{ij}^o,~y_{ij}^m, ~\boldsymbol{\phi_{j}})$ describes the mechanism for data missingness. Assuming that the probability of missingness is conditionally independent of the unobserved data given the observed (so called missing at random (MAR)), the second part of Equation~\ref{Eq:9} simplifies to
\begin{equation} \label{Eq:10}
p(R_{ij}~|~y_{ij}^o,~y_{ij}^m,~\boldsymbol{\phi}_{j})~=~ p(R_{ij}~|~y_{ij}^o, ~ \boldsymbol{\phi}_{j}).
\end{equation}
When the parameters $\beta_{ij}$ and $\phi_{ij}$ are distinct and functionally independent, the missing data mechanism is ignorable and the Expression~\ref{Eq:10} can be dropped from the joint distribution in Equation~\ref{Eq:8}.
Intergrating over the unknown missing values in the first part of Equation~\ref{Eq:9} yields a marginal density with the observed information which is to be evaluated
\begin{equation} \label{Eq:11}
\int_{}^{}p(y_{ij}^o, ~y_{ij}^m~|~ \boldsymbol{\beta}_{j})~dy_{ij}^m ~=~ p(y_{ij}^o~|~\boldsymbol{\beta}_{j}).
\end{equation}
Since the main objective is to be able to make valid and efficient inference about the parameters of interest and not to estimate or predict the missing data, the ignorability condition validates inference based on the observed data likelihood only.
Conditional on $\pi_{ijk}$ the studies are assumed to be exchangeable. The observed information $Y_{ijk}$ on a given test/arm \textit{k} generically represents a point estimate of $\pi_{ijk}$ and contributes to the estimation of the fixed effects $\mu_{jk}$.
At the second level of the hierarchy (Equation~\ref{Eq:2} and ~\ref{Eq:4}), exchangeable normal prior distributions with mean zero split the variability into between- and within-study variability. The observed data in each study contributes to the estimation of $\eta_{ij}$ while all the studies all-together contribute to the estimation of $\delta_{ijk}$ where $\delta_{ijk}$ and $\eta_{ij}$ are considered independent samples from a population controlled by the hyper-parameters $\boldsymbol{\Sigma}$ and $\tau_j^2$ which are estimated from the observed data. The hyper-parameters also have exchangeable vague or non-informative prior distributions.
The exchangeability assumption is applied in both the CB and the AB models but in a different manner. The CB model assumes exchangeability of tests contrasts (odds ratios) across the studies while the AB assumes exchangeability of tests effects (means) across the studies.
\subsection{Prior distributions}
We decompose the covariance matrix $\boldsymbol{\Sigma}$ into a variance and correlation matrix such that
\begin{equation} \label{Eq:12}
\boldsymbol{\Sigma} = diag(\sigma_1, \sigma_2) \times ~\boldsymbol{\Omega} \times~diag(\sigma_1, \sigma_2),
\end{equation}
where
\begin{equation} \label{Eq:13}
\boldsymbol{\Omega} = \begin{bmatrix}
1 ~~ \rho \nonumber\\
\rho ~~ 1
\end{bmatrix}.
\end{equation}
The model is completed by specifying vague priors on the mean, variance and correlation parameters as follows
\begin{align}\label{Eq:14}
tanh^{-1}(\rho), ~\mu_{jk} \sim N(0, ~25) \nonumber\\
\tau_j, ~\sigma_j \sim U(0, ~5).
\end{align}
Since it is not clear when certain choices of prior distributions are vague and non-informative, it is necessary to vary the prior distribution and assess their influence on the parameter estimates. The following prior distributions were also used as part of a sensitivity analysis
\begin{align}\label{Eq:15}
\rho ~\sim U(-1, ~1) \nonumber\\
\tau_j, ~\sigma_j \sim cauchy(0, ~2.5).
\end{align}
An alternative prior distribution for the correlation matrix $\boldsymbol{\Omega}$ is the LKJ prior distribution with shape parameter $\nu~=~1$ or $\nu~=~2$ \cite{Lewandowski}.
\begin{equation}\label{Eq:16}
\boldsymbol{\Omega} ~\sim ~LKJcorr(\nu) ~\propto det(\boldsymbol{\Omega})^{\nu-1} ~ for ~ \nu \ge 1,
\end{equation}
where $\nu$ controls the expected correlation with larger values favouring less correlation and vice-versa.
Other possible prior distributions for $\boldsymbol{\Sigma}$ are: the Inverse-Wishart distribution having the advantage of computational convenience but being difficult to interpret or the more relaxed scaled inverse Wishart which is a conjugate to the multivariate normal making Gibbs sampling simpler \cite{Gelman07}.
\subsection{Implementation}
The models are fitted in the Bayesian framework using Stan \cite{Stan}, a probabilistic programming language which has implemented Hamilton Monte Carlo (MHC) and No-U-Turn sampler (NUTS) \cite{Hoffman} within R 3.2.3 \cite{R} using the rstan 2.8.2 package \cite{rstan}. The Stan code for the model is provided alongside the supplementary material. We run three chains in parallel until there is convergence. Trace plots are used to visually check whether the distributions of the three simulated chains mix properly and are stationary. For each parameter, convergence is assessed by examining the potential scale reduction factor $\hat{R}$, the effective number of independent simulation draws ($n_{eff}$) and the MCMC error. It is common practice to run simulations until $\hat{R}$ is no greater than 1.1 for all the parameters. Since Markov chain simulations tend to be autocorrelated, $n_{eff}$ is usually smaller compared to the total number of draws. To reduce autocorrelation and consequently increase $n_{eff}$, it is necessary to do thinning by keeping every $n^{th}$ (e.g. every $10^{th}$, $20^{th}$, $30^{th}$ \ldots) draw and discarding the rest of the samples. Besides, thinning saves memory especially when the total number of iterations is large.
\section{Results}
Figure~\ref{Fig:2} presents the study-specific sensitivity and specificity of all the eleven used to detect CIN2+ in ASC-US triage from all available studies and from studies that evaluated at least two tests one of them being test 1. We successively present the sensitivity and specificity of the eleven tests in triage of ASC-US and LSIL for outcomes CIN2+ and CIN3+, in figures~\ref{Fig:3}, \ref{Fig:4} and \ref{Fig:5} respectively.
Representing the pooled sensitivity and specificity, the black diamonds are estimated by the AB model from all the available studies, the red diamond by the same model but from studies with at least two tests with one of them being test 1 while the blue diamonds are estimated by the CB model from studies with at least two tests with one of them being test 1. The vertical lines represent the 95\% credible intervals. In each instance, the studies included in estimating the diagnostic accuracy estimates are in grey points underlying the diamonds.
From the study-specific grey points there was substantial variation in both sensitivity and specificity between the studies and some studies had outlying values. It is also apparent that the number of tests evaluated differed among studies (see also supplementary material : Additional-tables.docx).
\subsection{All available data (black diamonds)}
\subsubsection{Triage of women with ASC-US to detect CIN2+}
According to figure~\ref{Fig:2}, test 6 (Linear Array) was the most sensitive (0.91[0.87, 0.94]) but among the least specific (0.44 [0.36, 0.51]) tests while test 10 (HPV Proofer) the least sensitive (0.68 [0.59, 0.76]) and the most specific (0.79 [0.73, 0.84]) test.
Both the diagnostic odds ratio and superiority index in the supplementary material (Results1.xlsx) indicate that test 9 (p16/Ki67) had the best discriminatory power with a sensitivity of 0.84 [0.76, 0.91] and specificity of 0.74 [0.66, 0.81].
\begin{figure}
\caption{Plot of study-specific sensitivity (top) and specificity (bottom) in grey points and their corresponding pooled (diamonds) estimates with their 95\% credible intervals (vertical lines) of tests\protect\endnotemark[1] detecting CIN2+\protect\endnotemark[3] in ASC-US\protect\endnotemark[2] triage.
The black diamonds and vertical lines are estimated by the AB model from all the available studies, the red by the same model but from studies with at least two tests with one of them being test 1 while the blue are estimated by the CB model from studies with at least two tests with one of them being test 1.\label{Fig:2}
\label{Fig:2}
\end{figure}
Compared to test 1 (HC2), tests 3 (LBC), 8 (p16) and 10 (HPV Proofer) were less sensitive but more specific, while tests 9 (p16/Ki67) was equally as sensitive but more specific. All other tests had similar sensitivity and specificity as test 1 (HC2) (see table~\ref{Tab:1}).
\begin{table}[h]
\centering
\caption{Posterior relative sensitivity and specificity and the corresponding 95\% credible intervals of other tests relative to test 1 (HC2) in detecting CIN2+\protect\endnotemark[3] in ASC-US\protect\endnotemark[2] triage as estimated by the AB model}
\label{Tab:1}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& & \multicolumn{3}{l|}{Relative sensitivity} & \multicolumn{3}{l|}{Relative specificity} \\ \hline
Label & Index test & Mean & Lower & Upper & Mean & Lower & Upper \\ \hline
2 & Conventional Cytology (CC) & 0.83 & 0.73 & 0.91 & 1.08 & 0.93 & 1.24 \\ \hline
3 & Liquid-Based Cytology (LBC) & 0.81 & 0.70 & 0.90 & 1.30 & 1.12 & 1.47 \\ \hline
4 & Non-Commercial PCR assays & 0.96 & 0.90 & 1.02 & 1.02 & 0.87 & 1.18 \\ \hline
5 & Abbott RT PCR hrHPV & 1.00 & 0.90 & 1.06 & 0.97 & 0.75 & 1.18 \\ \hline
6 & Linear Array & 1.00 & 0.96 & 1.05 & 0.82 & 0.67 & 0.96 \\ \hline
7 & Cobas-4800 & 1.01 & 0.94 & 1.06 & 1.01 & 0.80 & 1.21 \\ \hline
8 & P16 & 0.89 & 0.81 & 0.95 & 1.32 & 1.19 & 1.44 \\ \hline
9 & P16/Ki67 & 0.93 & 0.84 & 1.00 & 1.39 & 1.22 & 1.54 \\ \hline
10 & HPV Proofer(mRNA) & 0.75 & 0.65 & 0.84 & 1.48 & 1.36 & 1.59 \\ \hline
11 & APTIMA(mRNA) & 0.97 & 0.91 & 1.02 & 1.14 & 1.00 & 1.26 \\ \hline
\end{tabular}
\end{table}
\subsubsection{Triage of women with ASC-US to detect CIN3+}
It can be seen in figure~\ref{Fig:3} that test 5 (Abbott RT PCR hrHPV) was the most sensitive (0.97 [0.89, 1.00]) but among least specific (0.48 [0.35, 0.60]) tests. The diagnostic odds ratio and the superiority index (see supplementary material :Results1.xlsx) indicate that test 9 (p16/Ki67) had the best discriminatory power with sensitivity and specificity of 0.96 [0.85, 1.00] and 0.66 [0.53, 0.78] respectively.
\begin{figure}
\caption{Plot of study-specific sensitivity (top) and specificity (bottom) in grey points and their corresponding pooled (diamonds) estimates with their 95\% credible intervals (vertical lines) of tests\protect\endnotemark[1] detecting CIN3+\protect\endnotemark[3] in ASC-US triage\protect\endnotemark[2]. The black diamonds and vertical lines are estimated by the AB model from all the available studies, the red by the same model but from studies with at least two tests with one of them being test 1 while the blue are estimated by the CB model from studies with at least two tests with one of them being test 1.\label{Fig:3}
\label{Fig:3}
\end{figure}
Relative to test 1 (HC2), test 3 (LBC), 4 (Non-commercial PCR assays), 8 (p16) and 10 (HPV Proofer) were less sensitive but more specific while tests 2 (CC), 5 (Abbott RT PCR hrHPV), 7 (Cobas-4800) were as sensitive and specific (see table~\ref{Tab:2}).
\begin{table}[h]
\centering
\caption{Posterior relative sensitivity and specificity and the corresponding 95\% credible interval of other tests relative to test 1 (HC2) in detecting CIN3+\protect\endnotemark[3] in ASC-US\protect\endnotemark[2] triage as estimated by the AB model}
\label{Tab:2}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& & \multicolumn{3}{l|}{Relative sensitivity} & \multicolumn{3}{l|}{Relative specificity} \\ \hline
Label & Index test & Mean & Lower & Upper & Mean & Lower & Upper \\ \hline
2 & Conventional Cytology (CC) & 0.79 & 0.29 & 1.06 & 1.17 & 0.62 & 1.65 \\ \hline
3 & Liquid-Based Cytology (LBC) & 0.84 & 0.71 & 0.93 & 1.45 & 1.25 & 1.63 \\ \hline
4 & Non-Commercial PCR assays & 0.83 & 0.70 & 0.95 & 1.13 & 0.83 & 1.43 \\ \hline
5 & Abbott RT PCR hrHPV & 1.03 & 0.95 & 1.08 & 0.97 & 0.70 & 1.24 \\ \hline
6 & Linear Array & 1.03 & 0.99 & 1.06 & 0.81 & 0.65 & 0.99 \\ \hline
7 & Cobas-4800 & 1.03 & 0.97 & 1.07 & 0.99 & 0.74 & 1.24 \\ \hline
8 & P16 & 0.87 & 0.79 & 0.94 & 1.34 & 1.14 & 1.51 \\ \hline
9 & P16/Ki67 & 1.03 & 0.91 & 1.08 & 1.34 & 1.08 & 1.60 \\ \hline
10 & HPV Proofer(mRNA) & 0.87 & 0.77 & 0.95 & 1.59 & 1.43 & 1.74 \\ \hline
11 & APTIMA(mRNA) & 0.99 & 0.94 & 1.03 & 1.15 & 1.02 & 1.28 \\ \hline
\end{tabular}
\end{table}
\subsubsection{Triage of women with LSIL to detect CIN2+}
Figure~\ref{Fig:4} and the absolute diagnostic estimates presented in the supplementary material (Results1.xlsx) show that test 1 (HC2) was the most sensitive (0.94 [0.93, 0.95]) test but among the least specific (0.29 [0.27, 0.31]) tests while test 10 (HPV proofer) was the least sensitive (0.64 [0.54, 0.73]) and the most specific (0.73 [0.67, 0.78]) test detecting CIN2+ in LSIL cytology. Both the diagnostic odds ratio and superiority index presented in the supplementary material (Results1.xlsx) indicate once more that test 9 (p16/Ki67) had the best discriminatory power with an estimated sensitivity and specificity of 0.86 [0.79, 0.91] and 0.63 [0.57, 0.69].
\begin{figure}
\caption{Plot of study-specific sensitivity (top) and specificity (bottom) in grey points and their corresponding pooled (diamonds) estimates with their 95\% credible intervals (vertical lines) of tests\protect\endnotemark[1] detecting CIN2+\protect\endnotemark[3] in LSIL\protect\endnotemark[2] triage. The black diamonds and vertical lines are estimated by the AB model from all the available studies, the red by the same model but from studies with at least two tests with one of them being test 1 while the blue are estimated by the CB model from studies with at least two tests with one of them being test 1.\label{Fig:4}
\label{Fig:4}
\end{figure}
\begin{table}[h]
\centering
\caption{Posterior relative sensitivity and specificity and the corresponding 95\% credible interval of other tests relative to test 1 (HC2) in detecting CIN2+\protect\endnotemark[2] in LSIL\protect\endnotemark[2] triage as estimated by the AB model}
\label{Tab:3}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& & \multicolumn{3}{l|}{Relative sensitivity} & \multicolumn{3}{l|}{Relative specificity} \\ \hline
Label & Index test & Mean & Lower & Upper & Mean & Lower & Upper \\ \hline
2 & Conventional Cytology (CC) & 0.86 & 0.72 & 0.96 & 1.50 & 1.15 & 1.90 \\ \hline
3 & Liquid-Based Cytology (LBC) & 0.82 & 0.69 & 0.93 & 1.88 & 1.52 & 2.22 \\ \hline
4 & Non-Commercial PCR assays & 0.87 & 0.77 & 0.95 & 1.26 & 0.95 & 1.59 \\ \hline
5 & Abbott RT PCR hrHPV & 0.98 & 0.90 & 1.03 & 1.22 & 0.93 & 1.55 \\ \hline
6 & Linear Array & 0.98 & 0.93 & 1.02 & 0.98 & 0.77 & 1.21 \\ \hline
7 & Cobas-4800 & 0.96 & 0.89 & 1.02 & 1.13 & 0.85 & 1.45 \\ \hline
8 & P16 & 0.83 & 0.76 & 0.89 & 2.07 & 1.84 & 2.30 \\ \hline
9 & P16/Ki67 & 0.91 & 0.84 & 0.96 & 2.18 & 1.91 & 2.42 \\ \hline
10 & HPV Proofer(mRNA) & 0.68 & 0.58 & 0.78 & 2.49 & 2.24 & 2.74 \\ \hline
11 & APTIMA(mRNA) & 0.95 & 0.89 & 0.99 & 1.43 & 1.23 & 1.64 \\ \hline
\end{tabular}
\end{table}
\subsubsection{Triage of women with LSIL to detect CIN3+}
The forest plot presented in figure~\ref{Fig:5} (see also supplementary material: Results1.xlsx) shows that tests 5 (Abbott RT PCR hrHPV) and 6 (Linear Array) were the most sensitive but among the least specific tests in detecting CIN3+ in women with LSIL. The diagnostic odds ratio indicate that test 5 (Abbott RT PCR hrHPV) had the best discriminatory power (sensitivity = 0.99 [0.96, 1.00], specificity = 0.28 [0.20, 0.37]) while test 9 (p16/Ki67) best discriminatory test (sensitivity = 0.94 [0.88, 0.98], specificity = 0.45 [0.34, 0.56]) according to the superiority index.
\begin{figure}
\caption{Plot of study-specific sensitivity and specificity in grey points and their corresponding marginal (black points) sensitivity and specificity with their 95\% credible intervals (vertical lines) of diagnostic tests\protect\endnotemark[1] detecting CIN3+\protect\endnotemark[3] in LSIL\protect\endnotemark[2] triage. The black squares are estimated by the AB model from all the available studies (underlying in grey), the black diamond by the same model but from studies with at least two tests with one of them being test 1 while the black triangles are estimated by the CB model from studies with at least two tests with one of them being test 1.\label{Fig:5}
\label{Fig:5}
\end{figure}
According to table~\ref{Tab:3} and \ref{Tab:4} test 5 (Abbott RT PCR hrHPV), 6 (Linear Array) and 7 (Cobas-4800) were as sensitive and as specific while most of the rest of the tests were less sensitive but more specific as test 1 (HC2) in detecting CIN2+ and CIN3+ in triage of women with LSIL cytology.
\begin{table}[h]
\centering
\caption{Posterior relative sensitivity and specificity and the corresponding 95\% credible interval of other tests relative to test 1 (HC2) in detecting CIN3+\protect\endnotemark[3] in LSIL\protect\endnotemark[2] triage as estimated by the AB model}
\label{Tab:4}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& \multicolumn{3}{l|}{Relative sensitivity} & \multicolumn{3}{l|}{Relative specificity} \\ \hline
Label & Index test & Mean & Lower & Upper & Mean & Lower & Upper \\ \hline
2 & Conventional Cytology (CC) & 0.71 & 0.31 & 0.99 & 1.91 & 1.13 & 2.76 \\ \hline
3 & Liquid-Based Cytology (LBC) & 0.85 & 0.71 & 0.94 & 2.07 & 1.62 & 2.48 \\ \hline
4 & Non-Commercial PCR assays & 0.80 & 0.67 & 0.91 & 1.71 & 1.22 & 2.30 \\ \hline
5 & Abbott RT PCR hrHPV & 1.03 & 0.99 & 1.05 & 1.11 & 0.79 & 1.50 \\ \hline
6 & Linear Array & 1.03 & 1.00 & 1.05 & 0.90 & 0.63 & 1.19 \\ \hline
7 & Cobas-4800 & 0.99 & 0.94 & 1.03 & 1.14 & 0.80 & 1.54 \\ \hline
8 & P16 & 0.86 & 0.77 & 0.93 & 2.16 & 1.79 & 2.48 \\ \hline
9 & P16/Ki67 & 0.98 & 0.91 & 1.02 & 1.81 & 1.32 & 2.28 \\ \hline
10 & HPV Proofer(mRNA) & 0.79 & 0.67 & 0.87 & 2.73 & 2.38 & 3.08 \\ \hline
11 & APTIMA(mRNA) & 1.00 & 0.97 & 1.03 & 1.39 & 1.16 & 1.65 \\ \hline
\end{tabular}
\end{table}
\subsubsection{Variance Components}
The total variability in sensitivity (in the logit scale) from a compound symmetry working variance-covariance structure ranged from 0.24 [0.04, 0.64] (see supplementary material: Additional-tables.xlsx) in tests used to detect CIN3+ in ASC-US triage to 0.66 [0.41, 1.04] in tests used to detect CIN2+ in LSIL triage. The percentage of total variability in logit sensitivity attributable to between study variability ranged from 21.86\% [0.01\%, 81.19\%] in tests used to detect CIN3+ in LSIL triage to 74.09\% [22.51\%, 99.57\%] in tests used to detect CIN2+ in ASC-US triage.
Similarly for logit specificity, the total variability ranged from 0.39 [0.24, 0.61] in tests used to detect CIN3+ in LSIL to 0.54 [0.40, 0.73] in tests detecting CIN2+ in ASC-US triage. Of the total variability in logit specificitiy, as low as 59.09\% [29.32\%, 79.23\%] in tests used to detect CIN2+ in ASC-US triage and as high as 75.79\% [59.18\%, 88.14\%] in tests used to detect CIN2+ in LSIL triage was due to between study heterogeneity. In other words, there was a stronger correlation between any two logit specificities in a given study than between any two logit sensitivities.
There was in general negative but insignificant correlation between sensitivity and specificity except among tests used to detect CIN2+ in LSIL triage group ($\rho$~ = -0.80 [-1.00, -0.41]). The insignificant correlation parameters suggest absence of overall study effect in the respective data.
\subsubsection{Sensitivity Analysis}
The sensitivity analysis did not highlight any particular change on the mean structure for different priors of the variance-covariance parameters. Based on the MCMC error sampling the variance-covariance $\boldsymbol{\Sigma}$ was better sampled and less auto-correlated with LKJ and Cauchy distributions.
\subsection{AB versus CB model (black and red vs. blue)}
The data were re-analysed to compare the estimates from the AB and CB was performed. Studies included in the re-analysis evaluated at least two tests with one of them being test 1 (HC2) set as the common comparator (based on the high number of studies that evaluated test 1 (HC2) besides any another test). Test 1 (HC2) was set as the comparator because most of the studies evaluated it besides another diagnostic test. The network plot of the studies included in the re-analyses is shown in figure~\ref{Fig:6}.
\begin{figure}
\caption{Network plot with studies that evaluated at least two tests\protect\endnotemark[1] with test 1 (HC2) as the common comparator by triage\protect\endnotemark[2] group and outcome\protect\endnotemark[3].\label{Fig:6}
\label{Fig:6}
\end{figure}
A graphical summary of the results from the second and third analysis are presented in figures~\ref{Fig:2}, ~\ref{Fig:3}, ~\ref{Fig:4} and ~\ref{Fig:5} and represented by the red and blue diamonds respectively. Overall, there are discrepancies between the locations of the black, red and blue diamonds.
Firstly, while the black and blue diamonds represent the marginal means, the blue diamonds represent the accuracy estimate for a hypothetical study with random-effects equal to zero. This explains why the black and red diamonds are closer while the blue diamonds are more deviating.
Secondly, the location of the black diamonds is estimated from all available data, including studies evaluating single tests while the location of the red and blue diamonds are determined by studies evaluating at least two studies with one of them being test 1 (HC2). As a consequence of the reduced number of studies, the credible intervals presented in vertical lines are wider especially for the CB approach. As a cascade effect, the ranking of the tests based on the DOR and the superiority also changes (see supplementary material: Results1.xlsx vs. Results2.xlsx vs. Results3.xlsx).
\section{Discussion}
In this paper, we propose a conceptually simple model to estimate sensitivity and specificity of multiple tests within a network meta-analysis framework analogous to a single-factor analysis of variance method with repeated measures.
The model is based on the assumption that all the tests were hypothetically used but missing at random. When the mechanism of missing data is not a crucial aspect of inference, models ignoring the missing value mechanism and only using the observed data as the proposed model does provide valid answers under a missing at random (MAR) process. In contrast to the CB model, the proposed AB model uses all available data in line with principle of intention-to-treat (ITT) \cite{Fisher}. The missing `unobservable' sensitivities and specificities are parameters are estimated along with the other parameters in the model based on the exchangeability assumption. The cost however is that the model assumptions cannot be formally checked from the data under analysis.
When the data were never intended to be collected in the first place, the MAR assumption has been shown to hold as is the case in diagnostic studies where older tests become less used and new tests progressively more available with time\cite{Schafer}.
In the analysis, we included studies with at least one test. This is still acceptable because such studies still provide partial information allowing estimation of the mean and the variance-covariance parameters and only the study effects estimates might have larger standard errors~\cite{Gelman07}.
The proposed AB model allows for easy estimation of the marginal means and credible intervals for the intra-class correlation. Bayesian methods are known to be computationally intensive but with efficient sampling algorithms such as Hamilton Monte Carlo sampling implemented in Stan~\cite{Stan} convergence to a stationary distribution is accelerated even with poor initial values. Furthermore parallel chain processing greatly reduces computational time.
With the logit transformation, it is assumed that the transformed data is approximately normal with constant variance. For binary data as well as proportions, the mean and variance depend on the underlying probability. Therefore, any factor affecting the probability will change the mean and the variance. This implies that a linear model where the predictors affect the mean but assume a constant variance will not be adequate. Nonetheless, when the model for the mean is correct but the true distribution is not normal, the maximum likelihood (ML) estimates of the model parameters will be consistent but the standard errors will be incorrect \cite{Agresti}. An alternative to the logit transformation would be a variance stabilizing angular transformation; however the variance stabilizing property of the transform depends on each \textit{n} being large \cite{Crowder}.
The natural and optimal modelling approach would be to use the beta distribution. This was the motivation behind our work on copula based bivariate beta distribution in meta-analysis of diagnostic data ~\cite{Nyagaa, Nyagab}. Our further research will focus on how different mean and correlation structures are accommodated and modelled using the beta-binomial distribution in network meta-analysis of diagnostic data.
There were discrepancies in identifying the best test between the DOR and the superiority index. While the range of values estimated by the two measures range from 0 to infinity, the DOR yield larger values than the superiority index. From the full dataset, the superiority index consistently identified test 9 (p16/Ki67) as the best test. From the reduced data, the DOR identified tests with very low sensitivity but high specificity or vice-versa as the best and in disagreement with the superiority index. This illustrates that DOR cannot distinguish between tests with high sensitivity but low specificity or vice-versa. In contrast, the superiority index gives more weight to tests performing relatively well on both diagnostic accuracy measures and less weight on tests performing poorly on both diagnostic measures or tests performing better on one measure but poorly on the other\cite{Deutsch}. Nonetheless, both measures do not allow to prioritise one parameter which may be clinically appropriate.
Incoherence or inconsistency within NMA is a major concern where for the same contrast, the direct and indirect evidence differ substantially. Lu and Ades (2006) \cite{Lu}, Dias \textit{et al}. (2010) \cite{Dias} and Krahn \textit{et al}.(2014)\cite{Krahn}, explain how to visualize, detect and handle inconsistencies. Since the AB model implicitly assumes consistency, the methods used to detect and quantify inconsistency in CB need not be used in the AB models.
For the AB models, Hong \textit{et al}. (2015) \cite{Hong} measure inconsistency by data-driven magnitude of bias, the discrepancy between observed and imputed treatment(test) effects while Piepho (2014) \cite{Piepho14} classifies grouping of studies according to the set of tests included and introduce an interaction term: designs by test, to represent inconsistency. We caution against the grouping of studies into designs and including the interaction between designs and test into the model as proposed by Piepho (2014) \cite{Piepho14} because the design variable is an observational factor which will only complicate the `cause-and-effect' inference on test and design.
From our viewpoint, inconsistency/incoherence is a form of heterogeneity between the studies which is often due to missing information in an outlying or influential study. In our model, the influence of the study on the mean is adequately captured by study-effects and the fact that the model hypothetically allows any two tests to be compared directly within each study makes inconsistency less an issue. That said, it is important to assess and identify influence of certain observations on the marginal mean. Detection of influential observations within the Bayesian framework is a computationally involved exercise and still an active research area.
This article does not consider individual-level data for which the model adaptation is automatic. Future research includes a study on impact of various aspects of data missingness on the robustness of the models.
\section{Conclusion}
The proposed AB model contributes to the knowledge on methods used in systematic reviews of diagnostic data in presence of more than two competing tests. The AB model is more appealing than the CB model for meta-analyses of diagnostic studies because it yields marginal means which are easily interpreted and uses all available data. Furthermore, the model is superior since more general variance-covariance matrix structures can be easily accommodated.
\endnotetext[1]{Tests labels: 1-HC2, 2-CC, 3-LBC, 4-Generic PCR, 5-Abbott RT PCR hrHPV, 6-Linear Array, 7-Cobas-4800, 8-p16, 9-p16/ki67, 10-HPV Proofer, 11-APTIMA.}
\endnotetext[2]{ASC-US: atypical squamous cells of unspecified significance, LSIL :low-grade squamous intraepithelial lesions, CIN: cervival intraepithelial neoplasia}
\endnotetext[3]{CIN2+: cervival intraepithelial neoplasia of grade two or worse, CIN3+: cervival intraepithelial neoplasia of grade three or worse,}
\theendnotes
\begin{funding}
Nyaga V received financial support from the Scientific Institute of Public Health (Brussels) through the OPSADAC project. Arbyn M was supported by the COHEAHR project funded by the 7th Framework Programme of the European Commission (grant No 603019). Aerts M was supported by the IAP research network nr P7/06 of the Belgian Government (Belgian Science Policy).
\end{funding}
\begin{sm}
Contact the corresponding author for the supplementary materials.
\begin{enumerate}
\item Model-code.txt A text file with code of the fitted models in Stan language.
\item Model-fitting.txt A text file with code to fit the models and reproduce the results.
\item mydata.csv A file with the data used in this analysis.
\item Additional-tables.docx A file with additional tables.
\item Results1.xlsx A file with posterior diagnostic accuracy estimates as estimated by the AB model from all the available data.
\item Results2.xlsx A file with posterior diagnostic accuracy estimates as estimated by the AB model from studies that evaluated at least two studies with one of them being test 1.
\item Results3.xlsx A file with posterior diagnostic accuracy estimates as estimated by the CB model from studies that evaluated at least two studies with one of them being test 1.
\end{enumerate}
\end{sm}
\end{document} |
\begin{document}
\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\noindent {\Large From Many-Valued Consequence to Many-Valued Connectives
\footnote{Acknowledgements: We are very grateful to Benjamin Spector for providing inspiration and support to this project. We thank two anonymous referees for detailed and helpful comments. We also thank Denis Bonnay, Keny Chatain, Christian Ferm\"uller, Jo\~ao Marcos, Hitoshi Omori, Francesco Paoli, David Ripley, Lorenzo Rossi, Hans Rott, Philippe Schlenker, Jan Sprenger, Shane Steinert-Threlkeld, Heinrich Wansing for helpful conversations, as well as audiences in Regensburg (workshop ``New Perspectives on Conditionals and Reasoning''), Bochum (Logic in Bochum IV) and Dagstuhl (Dagstuhl Seminar 19032 ``Conditional logics and conditional reasoning''). The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.313610, and from the ANR program ``Trivalence and Natural Language Meaning'' (ANR-14-CE30-0010). We also thank the Ministerio de Econom\'ia, Industria y Competitividad, Gobierno de Espana, as part of the project ``Logic and substructurality" (Grant. FFI2017-84805-P), as well as grant FrontCog, ANR-17-EURE-0017 for research conducted in the Department of Cognitive Studies at ENS.
We dedicate this paper to the memory of Carolina Blasio.
}}\\
\renewcommand*{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}\\
Emmanuel Chemla$^\textrm{a}$ \& Paul Egr\'e$^\textrm{b}$ \\
{\scriptsize a. Laboratoire de Sciences Cognitives et Psycholinguistique, D\'epartement d'\'etudes cognitives, ENS, PSL University, EHESS, CNRS, 75005 Paris, France}\\
{\scriptsize b. Institut Jean Nicod, D\'epartement d'\'etudes cognitives \& D\'epartement de philosophie, ENS, PSL University, EHESS, CNRS, 75005 Paris, France}
\begin{abstract}
\noindent Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary $N$-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting.
The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.
\end{abstract}
\noindent{\small {\bf Keywords:} logical consequence; mixed consequence; compositionality; {truth-functionality}; many-valued logic; algebraic logic; conditionals; connectives; sequent calculus; deduction theorem; truth value}
\section{Introduction: matching conditionals and consequence relations}
In 2-valued logic, logical consequence is defined as the preservation of the value True (=$1$) from premises to conclusions in an argument. A fundamental feature of 2-valued logic is the existence of a binary sentential connective {representing or internalizing} the consequence relation in the object-language, namely the material conditional $\supset$ (or material implication, taking the value 1 exactly when the value of the antecedent is less or equal to the value of the consequent). As established by the \emph{deduction theorem}, $A\vdash B$ iff {$A\supset B$ is valid}, and more generally, when $\Gamma$ is a finite set of premises $A_{1}$ to $A_{n}$: $\Gamma\vdash B$ iff {$(A_{1} \supset (A_{2} \supset \cdots (A_{n} \supset B))\cdots)$ is valid}.
{This work extends the question of the representability of consequence relations by means of adequate conditional operators to logics with more truth-values, following a question originally raised by Benjamin Spector and posed in our joint article \cite{chemla2017charac}.
}
In many-valued logics, more notions of consequence become available as the set of truth values expands, and similarly the space of binary operators quickly increases. Perhaps as a result of the greater freedom in the choice of those parameters -- though sometimes due to {independent} desiderata -- various popular systems of many-valued logics rest on the choice of a conditional operator that fails one or the other direction of the deduction theorem, and thereby fails to internalize logical consequence adequately in the object-language (see \citealt{avron1991natural, cobreros2015vagueness, wintein2016kleene}).
{For instance, a popular logic such as Strong Kleene's $\mathsf{K3}$, in which $v(A\rightarrow B)=v(\neg A \vee B)= max(1-v(A), v(B))$, and where consequence is defined as the preservation of the value 1, loses conditional introduction ($A\vdash A$, but $\not\vdash A\rightarrow A$).
Similarly, the dual logic $\mathsf{LP}$ which uses the same conditional operator but in which consequence is defined as the preservation of non-zero values ($\{1, \half\}$) fails the converse direction akin to the modus ponens ($\vdash A \rightarrow ((A \rightarrow B) \rightarrow B)$, but $ A, (A\rightarrow B)\not\vdash B$). {Both of those counterexamples are forestalled in \L ukasiewicz's 3-valued system $\mathsf{\L3}$, who defines logical consequence as the preservation of the value $1$, and in which $v(A\rightarrow B)$ equals $1$ whenever $v(A)\leq v(B)$, and equals $1-(v(A)-v(B))$ otherwise (\citealt{luk1920}). Even there, however, conditional introduction fails, for whereas $A\wedge \neg A\vdash B$, the schema $A\wedge \neg A\to B$ is not valid (as can be seen by assigning $A$ the value $\half$ and $B$ the value $0$).}\footnote{{See \cite{pogorzelski1964deduction} and \cite{avron1991natural} for more on the deduction theorem in relation to \L ukasiewicz's three-valued conditional. Pogorzelski shows that \L ukasiewicz's conditional satisfies a more complex form of the deduction theorem relative to consequence defined as the preservation of the value 1. Avron shows that it can satisfy the deduction theorem in standard form if the definition of logical consequence is modified in a way that rules out the above counterexample, by giving up structural contraction. We note that relative to the mixed consequence relation $st$ (introduced below), \L ukasiewicz's conditional would satisfy the deduction theorem (but not what we call premise Gentzen-regularity, unlike with Avron's consequence)}.}
}
This situation of mismatch between consequence and conditional raises two natural questions: (i)~Given a consequence relation in $N$-valued logic, which conditional operators satisfy the deduction theorem, if any? The question, we will see, does not always have an obvious answer for $N \geq 3$. (ii)~Moving away from conditional operators, which connectives in general are naturally associated with a consequence relation, and what do the corresponding operators say, taken together, about this consequence relation? The first issue provides the motivation and a guiding direction for this paper, but we provide important general results pertaining to question~(ii). {By so doing, we follow the general perspective that \cite{bonnay2012consequence} aptly call ``extracting or mining'' connectives from consequence relations.
As a first step toward that goal, we are interested in a wider space of consequence relations than those standardly defined in terms of the preservation of a constant set of designated values from premises to conclusion. One motivation for this comes from the consideration of so-called \emph{mixed} consequence relations (\citealt{cobreros2012tolerant, chemla2017charac, frankowski2004, malinowski1990q}). By way of illustration, consider the semantics of the Strong Kleene conditional given above: re.~(ii), we just observed that it internalizes logical consequence neither as the preservation of the value $1$ (so-called $ss$-consequence), nor as the preservation of non-zero values (so-called $tt$-consequence). However, it does adequately internalize so-called \emph{strict-tolerant} consequence ($st$), defined in terms of the impossibility for the premises to take the value $1$ and for the conclusion to take the value $0$ (see \citealt{cobreros2012tolerant, cobreros2015vagueness}).
In what follows, we thus propose to investigate the relationship between consequence relations and conditional operators over the space of \emph{intersections of mixed consequence relations}. As established in previous work (see \citealt{chemla2018suszko}), intersective mixed relations fundamentally correspond to monotone consequence relations (see also \citealt{blasio2017inferentially,french2017valuations}). This space includes non-Tarskian relations which may fail either to be reflexive ($ts$) or transitive ($st$). On the other hand, it does include some Tarskian relations that are not obtainable in terms of the preservation of a fixed set of designated values (see \citealt{chemla2017charac}). This includes in particular the \emph{order-theoretic} consequence relations (definable as the intersection of all pure relations), an example of which is $ss\cap tt$ in 3-valued logic (requiring the preservation of both 1 and of non-zero values from premises to conclusion). While previous work exists concerning the relationship between Tarksian consequence operations and the existence of conditional operators internalizing them (see \citealt{avron1991natural}), we are not aware of a similar systematic investigation over the wider space of relations here considered.\footnote{A recent exception is \cite{wintein2016kleene} looking at 3-valued and 4-valued mixed consequence relations, but not at intersective mixed relations.}
Before we start, let us make a few more remarks concerning our research agenda. Firstly, like \cite{avron1991natural}, in this paper we will consider not just conditional operators, but more logical connectives, in particular negation, conjunction and disjunction, and we will be interested in the internalization of consequence relations by operators that not only satisfy the deduction theorem, but that satisfy further conditions that seem just as natural.
Secondly, one of the ambitions of this paper is to serve as a repository of results that can help logicians to find out a map of consequence relations and conditionals for the special case of 3-valued and 4-valued logics in particular. There is a sense in which every logician knows the map of 2-valued logic: they are completely familiar with all 16 binary operators, and they know of several arguments to select the horseshoe as the best candidate for being a conditional in that space. For 3-valued logic, where the number of binary truth-functional operators approximates 20,000, no similar map is available, and even for the best-known logics, it can be unclear which set of connectives is to be paired with a given consequence relation, or conversely. We propose to fill this gap, namely to chart the land of 3-valued and 4-valued logic in a systematic manner.
Thirdly, to serve that goal we will present several results based on computer-aided methods. This means that for several of the results we will state regarding 3-valued and 4-valued logic, we have found useful to do an exhaustive search of the space of consequence relations and the associated binary operators.
The search was used both as a heuristic to discover generalizations as well as a way to demonstrate results, and we think such computer-aided exploration can be of value to answer further questions.\footnote{Computer-aided investigations of this kind still seem quite rare, which is striking considering that some pioneers such as \cite{Foxley1962:computer3valued} had bravely started deploying them for very related tasks, when much more ingenuity was needed to compensate for the lower power of computers.}
The paper is structured as follows.
In Section~\ref{sec:consequence}, we lay out the ground for the rest of the paper: we define consequence relations and introduce \emph{intersective mixed consequence truth-relations} as our framework.
In Section~\ref{sec:operators}, we put forward the notion of a regular connective, and focus on what we call \emph{Gentzen-regular connectives}, that is connectives obeying a biconditional version of Gentzen's classic sequent calculus rules. We show that classical connectives are Gentzen-regular, but that Gentzen-regular connectives are more general: they correspond to {a subset of} all truth-functional connectives in a given 4-valued logic. The problem we pose in this paper, generally put, comes down to determining which class of Gentzen-regular connectives is admitted by a given intersective mixed consequence truth-relation.
In Sections~\ref{sec:three} and~\ref{sec:four}, we list extensively which consequence relations in 3-valued and 4-valued logics admit conjunctions, disjunctions, negations and conditionals, providing computer programs to reproduce and extend this inventory.
In Section~\ref{sec:pureandorderresults}, we move on to $N$-valued logics and show that the problem admits a simple and stable solution for two important classes of consequence relations, \emph{mixed} consequence relations (and even more so for the subclass of \emph{pure} consequence relations) and \emph{order-theoretic} consequence relations. Finally, we lay out algebraic characterization results for all finite-valued logics in Section~\ref{sec:N}.
\section{Consequence relations}\label{sec:consequence}
In this section, we introduce the framework we will be using to represent and interpret consequence relations and connectives.
\subsection{Languages and semantics}
We work with sentential languages, and restrict attention to truth-functional interpretations of formulae. Throughout the paper, we use Roman capitals $A, B, ...$ to denote formulae of the language, and Greek capitals $\Gamma$, $\Delta$,..., to denote sets of formulae. We use small Greek letters, $\gamma, \delta$, to denote subsets of truth values.
\begin{definition}[Sentential Language] A \emph{sentential language} $\mathcal{L}$ consists of a denumerable set of atoms $p_{1}, p_{2},...$, together with a set $\mathcal{C}$ of sentential connectives, where formulae are generated in the usual way.
\end{definition}
\begin{definition}[Semantic interpretation]
Given a sentential language $\mathcal{L}$, and a set of truth values $\mathcal{V}$ containing at least the special values $1$ and $0$, a \emph{semantic interpretation} (or valuation) is a morphism $v$ such that
\begin{itemize}
\item $v$ maps atoms on truth values
\item $v$ maps $n$-ary sentential connectives (a.k.a. operators) to truth-functions from $\mathcal{V}^{n}$ to $\mathcal{V}$
\item for any $n$-ary connective $C$, $v(C(A_{1},...,A_{n}))=v(C)(v(A_{1}),...,v(A_{n}))$.
\end{itemize}
\end{definition}
\begin{definition}[Semantics]\label{def:semantics}
A \emph{semantics} is a set of valuations such that
\begin{itemize}
\item for any connective $C$ and any two valuations $v_1$, $v_2$: $v_1(C)=v_2(C)$,
\item for any list of pairs of (distinct) atoms and truth values, one can find a valuation $v$ which assigns to each of the atoms in the list the relevant truth value.
\end{itemize}
\end{definition}
In this paper, unless otherwise noted, we will also suppose that the semantics is `sufficiently' expressive. In the following Definition~\ref{def:constantsandmaximalexpressiveness}, we present three levels of expressiveness: \emph{maximal expressiveness}, \emph{constant expressiveness}, and \emph{atomic expressiveness}. The first one is the most stringent, the second one may be obtained by ensuring the presence of relevant $0$-ary connectives in the language. The last one, atomic expressiveness, is the least demanding. We call it atomic expressiveness because it holds for instance if the language has atomic formulae whose semantic values are meant to cover the whole space of truth-values and with maximal variation across the different atomic formulae, as in the columns of a truth-table (this situation corresponds to a `valuational' semantics in the sense of \citealp{chemla2018suszko}).
Maximal expressiveness and constant expressiveness play a useful role in the rest of the paper, and by default we assume that they hold. We will make it explicit when results hold for the more inclusive class of atomic expressive semantics.
\begin{definition}[Maximal expressiveness, Constant expressiveness, Atomic expressiveness]\label{def:constantsandmaximalexpressiveness}
$\phantom{x}$
\begin{itemize}
\item The semantics is \emph{maximally expressive} if for every function $P$ from the set of valuations to the set of truth values (that is, for every `proposition'), the language contains a formula $F_P$ such that for all valuation $v$: $v(F_P)=P(v)$.
\item The semantics is \emph{constant expressive} if for every truth value $\alpha$, the language contains a formula $F_\alpha$ such that for all valuation $v$, $v(F_\alpha)=\alpha$.
\item The semantics is \emph{atomic expressive} if for every set of truth values $\gamma$, there is a set of formulae $\Gamma$ and a valuation $v$ such that $v(\Gamma)=\gamma$.
\end{itemize}
\end{definition}
\subsection{Consequence relations: intersective mixed}
We define a consequence relation to be any relation between subsets of formulae of the language (so potentially a relation between several premises and several conclusions, following \citealt{gentzen1935investigations, scott1974completeness, shoesmith1978multiple}). At the semantic level, we restrict attention to consequence relations that are \emph{truth-relational}, that is, consequence relations interpretable as relations between sets of truth values:
\begin{definition}[Consequence Relation]
Given $\mathcal{L}$ a sentential language,
we call a \emph{consequence relation} a subset $\vdash$ of $\mathcal{P}(\mathcal{L})\times\mathcal{P}(\mathcal{L})$.\end{definition}
\noindent We note that this way of defining consequence relations, through sets rather than lists, imposes structural contraction (namely $\Gamma, A, A\vdash \Delta$ iff $\Gamma, A\vdash \Delta$, and similarly $\Gamma\vdash A, A,\Delta$ iff $\Gamma \vdash A, \Delta$).
\begin{definition}[Truth-relations, truth-relational consequence relations] A \emph{truth-relation} is a subset of $\mathcal{P}(\mathcal{V})\times\mathcal{P}(\mathcal{V})$. We say that a consequence relation $\vdash$ is \emph{truth-relational} if there exists a truth-relation $\mathrel|\joinrel\equiv$ such that $\Gamma \vdash \Delta$ iff for every semantic interpretation $v$ in the semantics, $v(\Gamma) \mathrel|\joinrel\equiv v(\Delta)$.
\end{definition}
We will restrict attention to
specific types of truth-relations, namely \emph{intersective mixed consequence relations}. The notion of a \emph{mixed consequence relation} constitutes a generalization of the classic semantic notion of consequence relation in the sense of Tarski (see \citealt{cobreros2012tolerant, chemla2017charac}). The classic notion is defined as the preservation of designated values from premises to conclusion in an argument. For a mixed consequence relation, the set of designated values is allowed to vary between premises and conclusions:
\begin{definition}[Designated values]
A set of designated values is a set of truth values including $1$ and not $0$.
\end{definition}
\begin{definition}[Mixed consequence, pure consequence and intersective mixed consequence relations]\
\begin{itemize}
\item A \emph{mixed consequence truth-relation} is a truth-relation noted $\mathrel|\joinrel\equiv_{\mathcal{D}_p,\mathcal{D}_c}$, where
$\mathcal{D}_p$ is a \emph{premise-set of designated values},
and $\mathcal{D}_c$ is a \emph{conclusion-set of designated values}, such that
for all sets of truth values
$\gamma, \delta: \gamma \mathrel|\joinrel\equiv_{\mathcal{D}_p,\mathcal{D}_c} \delta$
iff
$\gamma\subseteq\mathcal{D}_p$ implies $\delta\cap\mathcal{D}_c\not=\emptyset$.
\item If the sets of designated values are the same for premise and conclusion, that is $\mathcal{D}_p=\mathcal{D}_c$, the relation is called a \emph{pure consequence relation}.
\item An \emph{intersective mixed consequence truth-relation} is an intersection of mixed consequence relations: $\mathrel|\joinrel\equiv_{\mathcal{D}_p^1,\mathcal{D}_c^1}\cap...\cap\mathrel|\joinrel\equiv_{\mathcal{D}_p^K,\mathcal{D}_c^K}$.
\end{itemize}
\end{definition}
\begin{definition}[Representation, Minimal Representation]
A \emph{representation} of an intersective mixed consequence truth-relation is a set of mixed consequence relations {whose intersection} is the relation in question.
A \emph{minimal representation} of an intersective mixed consequence truth-relation is a representation based on the least possible number of mixed truth-relations whose intersection gives that relation.
\end{definition}
\begin{example}[Mixed relations]
Let $\mathcal{V}=\{0, \half,1\}$. Let $\mathcal{D}_{s}=\{1\}$, and $\mathcal{D}_{t}=\{1,\half\}$. Then $\mathrel|\joinrel\equiv_{\mathcal{D}_{s}, \mathcal{D}_{s}}$, also called $ss$, and $\mathrel|\joinrel\equiv_{\mathcal{D}_{t}, \mathcal{D}_{t}}$, also called $tt$, are mixed consequence relations corresponding to standard consequence relations (also called \emph{pure} relations, because the premise and conclusion sets of designated values are identical, see \citealt{chemla2017charac}). The relation $\mathrel|\joinrel\equiv_{\mathcal{D}_{s}, \mathcal{D}_{t}}$ is a mixed consequence relation, also known as {p-consequence} (\citealt{frankowski2004}), or $st$ (\citealt{cobreros2012tolerant}), and likewise for $\mathrel|\joinrel\equiv_{\mathcal{D}_{t}, \mathcal{D}_{s}}$, {also known as q-consequence} (\citealt{malinowski1990q}), or $ts$ (\citealt{cobreros2012tolerant}).
\end{example}
\begin{example} [Intersective Mixed relations]\label{ex:sstt}
Consider the intersective mixed relation given by $\mathrel|\joinrel\equiv_{\mathcal{D}_{s}, \mathcal{D}_{s}}\cap \mathrel|\joinrel\equiv_{\mathcal{D}_{t}, \mathcal{D}_{t}}$. This relation, also known as $ss\cap tt$, cannot be expressed as a mixed relation (for a proof, see \citealt{chemla2017charac}). The latter representation is a minimal representation for it.
A nonminimal representation for it is for example: $\mathrel|\joinrel\equiv_{\mathcal{D}_s, \mathcal{D}_s}\cap \mathrel|\joinrel\equiv_{\mathcal{D}_t, \mathcal{D}_t} \cap \mathrel|\joinrel\equiv_{\mathcal{D}_s, \mathcal{D}_t}$, i.e. $ss\cap tt \cap st$. (It is a representation for it because $ss\subseteq st$, so $st$ plays no role in the representation, except for making it nonminimal).
\end{example}
Why focus on intersective mixed relations? For three main reasons. First, even aside from our purpose in this paper, mixed consequence relations have had a wide range of applications in recent years (see in particular \citealt{cobreros2015vagueness} for a review). Secondly, as shown in \citealt{chemla2018suszko}, intersective mixed consequence relations correspond exactly to the class of \emph{monotonic} consequence relations, that is, of relations $\vdash$ such that $\Gamma'\vdash \Delta'$ whenever $\Gamma \vdash \Delta$ and $\Gamma\subseteq \Gamma', \Delta\subseteq \Delta'$. This class, importantly, includes consequence relations that are not necessarily reflexive (like $ts$, in which $A\nvdash A$) or transitive (like $st$, in which $A\vdash B$, and $B\vdash C$ need not imply $A\vdash C$). That is, it includes non-Tarskian consequence relations. But it thereby gives a more general perspective on Tarskian relations, by setting reflexivity and transitivity as distinct parameters (on the same perspective, see \citealt{blasio2017inferentially, french2017valuations}). Our third and specific motivation, finally, which we repeat from the introduction, is that for some operators standardly used as conditionals in 3-valued logic, like the Strong Kleene conditional, no pure consequence relation (i.e. $ss$ or $tt$) can be such that this conditional satisfies the full deduction theorem relative to it. But a mixed relation such as $st$ does. Mixed consequence relations, which are relatively nonstandard, thus form a natural class to anyone concerned with the proof-theoretic behavior of operators that are standardly admitted.
\subsection{Order-theoretic consequence relations}
Before moving on, we also show that this approach to consequence relations subsumes another natural way to define a consequence relation, namely through an ordering on truth values:
\begin{definition}[Order theoretic relations]\label{def:ordertheoretic}
Suppose the set of truth values is equipped with an order $\leq$ {(a reflexive, transitive, antisymmetric relation{, be it partial or total})}, with $1$ ranked higher than any other value, and $0$ ranked lower. One may then define the \emph{order-theoretic consequence relation (associated to this order)}, requesting that premises have globally lower values than conclusions, through the formula:
\centerline{$\gamma\mathrel|\joinrel\equiv_{\leq}\delta$ $\quad$ iff $\quad$ ($\exists x\in\gamma,\exists y\in\delta: x\leq y$) or ($0\in\gamma$ or $1\in\delta$).
\footnote{One may entertain other ways to extend an order on truth values (possibly with more properties, such as the systematic presence of infimums and/or supremums) onto a truth-relation between subsets of truth values. Below are some examples, close to descriptions in \cite{chemla2017charac}, but which we will not attend to specifically here:
$\gamma\mathrel|\joinrel\equiv\delta$ iff $\inf(\gamma)\leq\sup(\delta)$,
or
$\gamma\mathrel|\joinrel\equiv\delta$ iff $\exists d\in\delta: \inf(\gamma)\leq d$.
}}
\end{definition}
\noindent {We call truth-values other than 1 and 0 \emph{indeterminates}}. For illustration purposes, we distinguish two specific cases:
\begin{definition}[Total and degenerate order-theoretic relations]\label{def:specificordertheoretic}
$\phantom{x}$
\begin{itemize}
\item An order-theoretic relation is called \emph{total} if it is based on a total order of truth values: for all pairs of {indeterminates}, $\#_i\leq\#_j$ or $\#_j\leq\#_i$.
\item An order-theoretic relation is called \emph{degenerate} if it is based on the order such that no two {indeterminates} are {comparable}.
\end{itemize}
\end{definition}
We can here extend a result in the single-conclusion setting from \cite{chemla2017charac} to the current multi-conclusion framework:
\begin{theorem}\label{th:ordertheoric=intersectionpure}
An order-theoretic truth-relation derived from the order $\leq$ is equivalent to the intersection of all pure consequence relations based on designated values (containing $1$ but not $0$) that are upsets for this order.\footnote{Given an ordering $\leq$, an upset is a set that is closed under $\leq$, namely such that $y$ belongs to the set whenever $x$ belongs and $x\leq y$.}
\end{theorem}
\begin{myproof}
Choose $\gamma, \delta$. We must prove that the following statements are equivalent:
(i)~$\exists x\in\gamma, \exists y\in\delta: x\leq y$ or $0\in\gamma$ or $1\in\delta$ (that is, $\gamma\mathrel|\joinrel\equiv_\leq\delta$),
and
(ii)~$\forall \mathcal{D} \textrm{ an upset of designated values}: \gamma\subseteq\mathcal{D}\Rightarrow\delta\cap\mathcal{D}\not=\emptyset$.
\begin{description}
\item[(i) entails (ii).]
If $0\in\gamma$ or $1\in\delta$, the result follows because $\mathcal{D}s$ are sets of designated values (contain $1$ and not $0$). Otherwise, choose $(x_0, y_0)$ according to (i), and let $\mathcal{D}$ be an upset such that $\gamma\subseteq\mathcal{D}$. Then $x_0\in\gamma\subseteq\mathcal{D}$, so $y_0\in\mathcal{D}$, and therefore $y_0\in\delta\cap\mathcal{D}$, which is not empty.
\item[(ii) entails (i).]
Let $\mathcal{D}=\{y | \exists x \in \gamma: x\leq y \}$. Then $\mathcal{D}$ is an upset (by transitivity of $\leq$) and $\gamma\subseteq\mathcal{D}$ (by reflexivity of $\leq$). Hence, if $\mathcal{D}$ is a set of designated values, then applying (ii) yields $y_0$ in $\delta\cap\mathcal{D}$, and it follows that there is $x_0\in\gamma$ such that $x_0\leq y_0$. If $\mathcal{D}$ is not a set of designated values, that would be either because $0\in\mathcal{D}$ or $1\not\in\mathcal{D}$. In the former case, that means that $0\in\gamma$. In the latter, it follows that $\gamma=\emptyset$, and so that $\gamma\subseteq\{1\}$, which is an upset of designated values and so, by (ii), that $\{1\}\cap\delta\not=\emptyset$, i.e.~$1\in\delta$.
\qedhere
\end{description}
\end{myproof}
This Theorem can be applied to specific examples:
\begin{corollary}\label{ex:ordertheoretictotal}
A total order-theoretic relation, with $0\leq\#_1\leq...\leq\#_N\leq 1$, is represented by:
$$\mathrel|\joinrel\equiv_{\{1\},\{1\}}\cap\mathrel|\joinrel\equiv_{\{1,\#_1\},\{1\,\#_1\}}\cap\mathrel|\joinrel\equiv_{\{1,\#_1,\#_2\},\{1\,\#_1,\#_2\}}\cap...$$
\end{corollary}
\begin{corollary}\label{ex:ordertheoreticdegenerate}
The degenerate order-theoretic relation is represented by:
\begin{gather*}
\mathrel|\joinrel\equiv_{\{1\},\{1\}} \\
\cap \\
\mathrel|\joinrel\equiv_{\{1,\#_1\},\{1,\#_1\}} \cap \mathrel|\joinrel\equiv_{\{1,\#_2\},\{1,\#_2\}} \cap ... \cap \mathrel|\joinrel\equiv_{\{1,\#_N\},\{1,\#_N\}} \\
\cap \\
\mathrel|\joinrel\equiv_{\{1,\#_1,\#_2\},\{1,\#_1,\#_2\}} \cap \mathrel|\joinrel\equiv_{\{1,\#_1,\#_3\},\{1,\#_1,\#_3\}} ... \cap \mathrel|\joinrel\equiv_{\{1,\#_{N-1},\#_N\},\{1,\#_{N-1},\#_N\}}\\
\cap\\ ...\\
\cap \\
\mathrel|\joinrel\equiv_{\{1,\#_1,\#_2,..., \#_N\},\{1,\#_1,\#_2,..., \#_N\}}
\end{gather*}
\end{corollary}
The total and degenerate order-theoretic relations (and all other order-theoretic relations) in fact collapse in 3-valued logic, because there is no need to worry about the ordering among indeterminates when there is only one. The single order-theoretic relation in 3-valued logic is in fact one that has already been presented in Example~\ref{ex:sstt}, under the label $ss \cap tt$.
With more than 3 truth-values, the notions come apart: an example of a degenerate (and non-total) order-theoretic relation is given by Belnap's 4-valued logic (\citealt{belnap1977useful}; see \citealt{shramko2011truth} for details on the order-theoretic aspect of the consequence relation, and \citealt{omori2015generalizing} for details on 4-valued connectives).
Order-theoretic relations form an important subclass of the intersective mixed consequence relations, and we will be able to provide extensive results concerning the types of connectives that they allow (see Theorem~\ref{th:fullordertheoretic}).
\subsection{Summary and assumptions}
We have here defined and delimited a space of consequence relations that we will explore systematically: unless otherwise noted, we will be interested in languages equipped with an intersective mixed consequence truth-relation and maximal or constant expressiveness.
In the next section, we turn to an examination of a particular type of constraints that can be put on logical connectives.
\section{Regular connectives\label{sec:operators}}
We are ultimately interested in specific connectives, such as conditionals, and their relation to consequence relations. Toward that goal, however, we shall take a broader view of other logical connectives, and ask what defines a connective from the point of view of a consequence relation. We want to find out how a given connective ought to interact with a consequence relation: what does it mean to have a formula headed by this connective in premise position? in conclusion position? These questions are of central importance in proof theory, and they were posed for the first time by \cite{gentzen1935investigations} when Gentzen introduced the sequent calculus for classical logic and for intuitionistic logic. We first review the way in which Gentzen's sequent rules arise for conjunction, disjunction, negation, and conditional, and then go on to introduce the class of what we call regular connectives for a consequence relation. We will provide various results about these connectives, revealing their behavior in classical logic, as well as their non-classical behavior in general.
{We finally show that every Gentzen connective can be exemplified by a truth-function in a particular $4$-valued logic.}
\subsection{Gentzen's operational rules, first examples}\label{sec:gentzen}
Gentzen's sequent calculus for classical logic can be viewed as a systematic framework to represent the way in which the meaning of a logical connective ought to be understood in relation to a given consequence relation. Gentzen distinguished \emph{structural rules}, concerned with the general behavior of the consequence relation (is it contractive, commutative, monotone, etc), from \emph{operational rules}, concerned with the specific behavior of logical connectives. For each connective, Gentzen's operational rules specify how the connective is to be treated as a premise in an argument, or as a conclusion in an argument.
Consider conjunction and disjunction first. In Gentzen's approach, the concatenation of premises corresponds to their conjunction, and the concatenation of conclusions correspond to their disjunction. This yields the following rules when a conjunctive formula appears as a premise, and a disjunctive formula as a conclusion:
\begin{itemize}
\item $\forall\Gamma,\Delta: \Gamma, P \wedge Q \vdash \Delta$ iff $\Gamma, P, Q \vdash \Delta$
\item $\forall\Gamma,\Delta: \Gamma \vdash P \vee Q, \Delta$ iff $\Gamma, \vdash P, Q, \Delta$
\end{itemize}
\noindent When a conjunctive formula appears as a conclusion, or a disjunctive formula as a premise, we get the following dual rules:
\begin{itemize}
\item $\forall\Gamma,\Delta: \Gamma \vdash P \wedge Q, \Delta$ iff $\Gamma \vdash P, \Delta$ and $\Gamma \vdash Q, \Delta$
\item $\forall\Gamma,\Delta: \Gamma, P \vee Q \vdash \Delta$ iff $\Gamma, P \vdash \Delta$ and $\Gamma, Q \vdash \Delta$
\end{itemize}
For negation, we get the following sequent rules for when negation appears in premise position, or in conclusion position:
\begin{itemize}
\item $\forall\Gamma,\Delta: \Gamma, \neg P \vdash \Delta$ iff $\Gamma \vdash P, \Delta$
\item $\forall\Gamma,\Delta: \Gamma \vdash \neg P, \Delta$ iff $\Gamma, P \vdash \Delta$
\end{itemize}
For the conditional {(or implication)}, Gentzen proposed the following rules:\footnote{Gentzen originally stated only the right-to-left direction of those rules, but it is natural to use invertible rules.}
\begin{itemize}
\item $\forall\Gamma,\Delta: \Gamma \vdash P \rightarrow Q, \Delta$
iff $\Gamma, P \vdash Q, \Delta$
\item $\forall\Gamma,\Delta: \Gamma , P \rightarrow Q\vdash \Delta$
iff ($\Gamma \vdash P, \Delta$ and $\Gamma, Q \vdash \Delta$)
\end{itemize}
The first of these rules basically corresponds to the full deduction theorem: it tells us that the conditional internalizes the consequence relation in the object-language. The rule for the conditional in premise position is best understood in relation to the standard definition of logical consequence in classical logic: a conditional is not designated (false) provided its antecedent is designated and its consequent is not.
\subsection{Regularity: definition and first application}
An important feature of the Gentzen rules is their analytic character, namely the fact that the meaning of a connective, whether in premise position or in conclusion position, is explained fully in terms of the (possibly empty) conjunction of sequent rules involving the component subformulae of the formula build from that connective. In \cite{chemla2018suszko}, we put forward the notion of a \emph{regular connective} to describe logical connectives obeying such constraints:
\begin{definition}[Gentzen-regular connectives]\label{def:regconn}
Given a consequence relation $\vdash$, an $n$-ary connective $C$ is \emph{regular} for it if there exist
$\mathcal{B}^p\subseteq\mathcal{P}(\{1,..., n\})\times\mathcal{P}(\{1,..., n\})$ and
$\mathcal{B}^c\subseteq\mathcal{P}(\{1,..., n\})\times\mathcal{P}(\{1,..., n\})$
such that
$\forall\Gamma, \Delta, \forall F_1, ..., F_n:$
\[\begin{array}{c@{\textrm{ iff }}c}
\Gamma, C(F_1, ..., F_n) \vdash \Delta
& \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\Gamma, \{F_i: i\in B_p\}\vdash \{F_i: i\in B_c\}, \Delta}\\
\Gamma \vdash C(F_1, ..., F_n) , \Delta
& \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^c}
{\Gamma, \{F_i: i\in B_p\}\vdash \{F_i: i\in B_c\}, \Delta}\\
\end{array}\]
\end{definition}
To immediately provide examples, we state that every truth-function in classical logic defines a regular connective. We simply provide the regularity rules associated with binary truth-functions, we will prove the general result later in Section~\ref{sec:bivalentregconnectives}.
\begin{example}[Regular rules associated to all binary truth-functional connectives in classical logic]\label{regrulesclassicalconnectives} All 16 binary truth-functions of classical logic can be associated with a regularity rule, as indicated in the following table.\footnote{This is not to imply that a regular connective may satisfy only one regularity rule. For instance, in reflexive logics, as in classical logic, adding a conjunct of the form $\Gamma, A \vdash A, \Delta$ to a regularity rule produces a new rule, but it is essentially the same rule and certainly it is satisfied by the same connectives.}
\centering\noindent {\small\framebox{\begin{tabular}{r|l|ll}
$O(P,Q)$ & Premise-regularity rules & Conclusion-regularity rules\\
& $\Gamma, O(P,Q) \vdash \Delta$ iff: & $\Gamma \vdash O(P,Q), \Delta$ iff:\\
\hline
\hline
$\top$
& $\Gamma \vdash \Delta$
& \emph{always true} (an empty conjunction)\\
$\bot$
& \emph{always true} (an empty conjunction)
& $\Gamma \vdash \Delta$\\
\hline
$P$
& $\Gamma, P \vdash \Delta$
& $\Gamma \vdash P, \Delta$\\
$\neg P$
& $\Gamma \vdash P, \Delta$
& $\Gamma, P \vdash \Delta$\\
\hline
$Q$
& $\Gamma, Q \vdash \Delta$
& $\Gamma \vdash Q, \Delta$\\
$\neg Q$
& $\Gamma \vdash Q, \Delta$
& $\Gamma, Q \vdash \Delta$\\
\hline
$P\vee Q$
& $\Gamma, P \vdash \Delta$ and $\Gamma, Q \vdash \Delta$
& $\Gamma \vdash P, Q, \Delta$\\
$\neg (P\vee Q)$
& $\Gamma \vdash P, Q, \Delta$
& $\Gamma, P \vdash \Delta$ and $\Gamma, Q \vdash \Delta$\\
\hline
$P\wedge Q$
& $\Gamma, P, Q \vdash \Delta$
& $\Gamma \vdash P, \Delta$ and $\Gamma \vdash Q, \Delta$\\
$\neg (P\wedge Q)$
& $\Gamma \vdash P, \Delta$ and $\Gamma \vdash Q, \Delta$
& $\Gamma, P, Q \vdash \Delta$\\
\hline
$P\rightarrow Q$
& $\Gamma, Q \vdash \Delta$ and $\Gamma \vdash P, \Delta$
& $\Gamma, P \vdash Q, \Delta$\\
$P\wedge \neg Q$
& $\Gamma, P \vdash Q, \Delta$
& $\Gamma, Q \vdash \Delta$ and $\Gamma \vdash P, \Delta$\\
\hline
$P\leftarrow Q$
& $\Gamma, P \vdash \Delta$ and $\Gamma \vdash Q, \Delta$
& $\Gamma, Q \vdash P, \Delta$\\
$\neg P\wedge Q$
& $\Gamma, Q \vdash P, \Delta$
& $\Gamma, P \vdash \Delta$ and $\Gamma \vdash Q, \Delta$\\
\hline
$P\leftrightarrow Q$
& $\Gamma, P, Q \vdash \Delta$ and $\Gamma \vdash P, Q, \Delta$
& $\Gamma, P \vdash Q, \Delta$ and $\Gamma, Q \vdash P, \Delta$\\
$P~\underline{\vee}~Q$
& $\Gamma, P \vdash Q, \Delta$ and $\Gamma, Q \vdash P, \Delta$
& $\Gamma, P, Q \vdash \Delta$ and $\Gamma \vdash P, Q, \Delta$
\\
\end{tabular}}}
\end{example}
\subsection{Interactions between G-connectives: some classical rules}
We state here a few properties concerning the interactions between Gentzen-regular connectives. These highlight, again, the similarity of behavior with classical, truth-functional connectives. We start with a general closure condition.
\begin{theorem}
A combination of Gentzen-regular connectives is Gentzen-regular.
\end{theorem}
\begin{myproof}
This can be proved by induction, a conjunction of conjunctions being a conjunction itself.
\end{myproof}
\noindent
Furthermore, every $n$-ary projection $pr_i^n$ is Gentzen regular, which together with the previous result establishes that Gentzen regular connectives form what is called a \emph{clone} (see \citealp{KERKHOFF2014107} for a full presentation).
\begin{theorem}
A projection $pr_i^n$, which associates $pr_i^n(X_1,..., X_n)$ to $X_i$, satisfies the regularity rules:
\begin{itemize}
\item $\Gamma, pr_i^n(X_1,..., X_n) \vdash \Delta \textrm{ iff } \Gamma, X_i \vdash \Delta$
\item $\Gamma \vdash pr_i^n(X_1,..., X_n), \Delta \textrm{ iff } \Gamma \vdash X_i, \Delta$
\end{itemize}
\end{theorem}
\begin{myproof}
Immediate.
\end{myproof}
Let us define now a couple of Gentzen-regular connectives of particular salience:
\begin{definition}[Gentzen-connectives]
We call a G-conjunction / G-disjunction / G-negation / G-conditional a truth-functional connective which obeys the appropriate Gentzen rules stated in Section~\ref{sec:gentzen}.
\end{definition}
\noindent The interactions between these operators are classical. For instance, the usual De Morgan's laws and the contraposition rule hold:
\begin{theorem}\label{th:demorgan}[De Morgan's laws]
$\phantom{x}$
\begin{itemize}
\item If $\neg$ is a G-negation and $\wedge$ is a G-conjunction, then $\neg(\neg\_\_\wedge\neg\_\_)$ is a G-disjunction.
\item If $\neg$ is a G-negation and $\vee$ is a G-disjunction, then $\neg(\neg\_\_\vee\neg\_\_)$ is a G-conjunction.
\end{itemize}
\end{theorem}
\begin{theorem}\label{th:contraposition}(Contraposition)
If $\to$ is a G-conditional and $\neg$ a G-negation, then the contraposition rules apply:
\begin{itemize}
\item $\Gamma, A\to B \vdash \Delta$ iff $\Gamma, (\neg B \to \neg A) \vdash \Delta$
\item $\Gamma \vdash A\to B, \Delta$ iff $\Gamma \vdash (\neg B \to \neg A), \Delta$
\end{itemize}
\end{theorem}
\begin{myproof}[Theorem~\ref{th:demorgan}]
For instance, we can show the premise-regularity rule for the G-disjunction defined from a G-negation and a G-conjunction:
$\Gamma, \neg(\neg A \wedge\neg B)\vdash\Delta$
iff
$\Gamma \vdash (\neg A \wedge\neg B), \Delta$
(negation premise-regularity rule)
iff
$\Gamma \vdash \Delta$ and $\Gamma \vdash \neg B, \Delta$
(conjunction conclusion-regularity rule)
iff
$\Gamma, A \vdash \Delta$ and $\Gamma, B \vdash \Delta$
(negation conclusion-regularity rule twice), which is the disjunction premise-regularity rule.
\end{myproof}
\begin{myproof}[Theorem~\ref{th:contraposition}]
\begin{itemize}
\item $\Gamma, (\neg B \to \neg A) \vdash \Delta$
iff $\Gamma, \neg A \vdash \Delta$ and $\Gamma \vdash \neg B, \Delta$ (premise Gentzen-regularity rule for $\to$)
iff $\Gamma \vdash A, \Delta$ and $\Gamma, B \vdash \Delta$ (Gentzen-regularity rules for $\neg$)
iff $\Gamma, (A \to B) \vdash \Delta$ (premise Gentzen-regularity rule for $\to$).
\item $\Gamma \vdash (\neg B \to \neg A) , \Delta$
iff $\Gamma, \neg B \vdash \neg A, \Delta$ (conclusion Gentzen-regularity rule for $\to$)
iff $\Gamma, A \vdash B, \Delta$ (Gentzen-regularity rules for $\neg$)
iff $\Gamma \vdash (A \to B), \Delta$ (conclusion Gentzen-regularity rule for $\to$).
\qedhere\end{itemize}
\end{myproof}
Given that G-conditionals are our core interest, we here introduce a couple of specific results. We show in particular that the existence of a G-conditional may be guaranteed by the existence of other G-connectives, and that it actually guarantees the existence of the other G-connectives assuming that there is a formula $\bot$ with constant value $0$ (which is one of our default assumptions for the language and its semantics, see Definition~\ref{def:constantsandmaximalexpressiveness}). Theorem~\ref{th:conditionalissufficient} will later provide a more general result of complete expressiveness based on G-conditionals however, within section~\ref{sec:bivalentregconnectives} which discusses bivalent connectives more systematically.
\begin{theorem}[Interactions between G-conditionals and other G-connectives]\label{th:combinationsforconditionals}
$\phantom{x}$
\begin{itemize}
\item If $\neg$ is a G-negation and $\vee$ is a G-disjunction, then $(\neg\_\_)\vee\_\_$ is a G-conditional,
\item If $\neg$ is a G-negation and $\wedge$ is a G-conjunction, then $\neg(\_\_\wedge\neg\_\_)$ is a G-conditional,
\item If $\to$ is a G-conditional, $\_\_\to\bot$ is a G-negation,
\item If $\to$ is a G-conditional, $(\_\_\to\bot)\to \_\_$ is a G-disjunction,
\item If $\to$ is a G-conditional, $(\_\_\to (\_\_\to\bot))\to\bot$ is a G-conjunction.
\end{itemize}
\end{theorem}
\begin{myproof}
The proof is essentially the same as before and obtained by combining regularity rules with one another, here using in particular the regularity rule of $\bot$ as given in Example~\ref{regrulesclassicalconnectives}.
\end{myproof}
\subsection{Non-classical behavior}
Crucially, G-regularity does not entail a completely classic behavior, unless the consequence relation itself plays its part: for instance, a number of classical rules will hold with G-regular connectives if and only if the consequence relation is reflexive.
\begin{theorem}[Reflexivity and G-connectives]
Consider a language with a monotonic
consequence relation, a constant expressive semantics, and G-connectives $\neg$, $\vee$, $\wedge$ and $\to$. The following statements are all equivalent:
\begin{enumerate}
\item $\forall A:\ A \vdash A$
\emph{(Reflexivity)}
\item $\forall A, B:\ (A \to B) \vdash (A \to B)$
\emph{(Reflexivity restricted to conditional formulae)}
\item $\forall A:\ \neg A \vdash \neg A$
\emph{(Reflexivity restricted to negative formulae)}
\item $\forall A, B:\ (A \wedge B) \vdash (A \wedge B)$
\emph{(Reflexivity restricted to conjunctive formulae)}
\item $\forall A, B:\ (A \vee B) \vdash (A \vee B)$
\emph{(Reflexivity restricted to disjunctive formulae)}
\item $\forall A, B:\ A, (A \to B) \vdash B$
\emph{(Modus Ponens)}
\item $\forall A:\ \vdash A, \neg A$
\emph{(Law of Excluded Middle, version 1)}
\item $\forall A:\ \vdash (A \vee \neg A)$
\emph{(Law of Excluded Middle, version 2)}
\item $\forall A:\ A, \neg A \vdash $
\emph{(Principle of Explosion, version 1)}
\item $\forall A:\ (A \wedge \neg A)\vdash$
\emph{(Principle of Explosion, version 2)}
\item $\forall A, B:\ A \vdash A, B$
\emph{(Intermediate step)}
\item $\forall A, B:\ A, B \vdash A$
\emph{(Intermediate step)}
\item $\forall A, B:\ A, B \vdash A, B$
\emph{(Intermediate step)}
\end{enumerate}
\end{theorem}
\begin{myproof}
We start by showing the equivalence between 1, 11, 12 and 13: By monotonicity, 1 entails 11, 12 and 13; conversely, if we choose $B=A$ in either 11, 12 or 13, we see that either of them entails 1. From there, all the rest follows, because every statement from 2 to 10 all can be reduced to a conjunction of statements among 1, 11, 12, 13 by applying the regularity rule of the relevant connective whenever it appears.
Let us take three examples. First, Modus Ponens (6):
$\forall A, B:\ A, (A \to B) \vdash B$
is true iff
$\forall A, B:\ A, B \vdash A, B$, by applying the premise regularity rule of the G-conditional.
Second, the Law of Excluded Middle, version 2 (8):
$\forall A:\ \vdash (A \vee \neg A)$. There is a G-disjunction in the conclusions, so we apply the conclusion regularity for a G-disjunction and obtain that this is true iff
$\forall A:\ \vdash A, \neg A$. This is (7). Since it contains a G-negation in conclusion, we can apply the relevant rule and obtain that this is equivalent to $\forall A:\ A \vdash A$.
Finally, we can show why reflexivity restricted to conditional formulae (2) is equivalent to plain reflexivity:
$\forall A, B:\ (A \to B) \vdash (A \to B)$
iff
$\forall A, B:\ (A \to B), A \vdash B$ (conclusion regularity rule for the G-conditional)
iff
$\forall A, B:\ B, A \vdash B \textrm{ and } A \vdash A, B$ (premise regularity rule for the G-conditional).
iff
$\forall A, B:\ B, A \vdash B \textrm{ and } \forall A, B: A \vdash A, B$ (distributivity of conjunction and universal quantifiers).
And this last statement is indeed a conjunction of the statements 12 and 13 (which are, remember, equivalent).
\end{myproof}
Not all intersective mixed consequence relations are reflexive, so this result reveals that Gentzen-regular rules are not sufficient to impose a classical behavior for the connectives if the consequence relation itself is not structurally classical in the first place. Similarly, we know that not all intersective mixed consequence relations are transitive. Hence, we could investigate what laws of classical logic are equivalent to transitivity, or any other such property for that matter. Instead we now turn to the issue of whether G-connectives always exist for a consequence relation.
\subsection{Regularity: key results}
This section presents more consequences of regularity. These properties are fundamental, and they will help to prove important results later on. Some of them are rather technical and can be skipped on a first reading.
To begin with, in order to investigate the connection between regular connectives and truth-relational consequence relations, it is useful to assume constant expressiveness. Under that assumption, regular connectives can be translated at the level of truth values without any loss.
\begin{theorem}\label{th:reducetotruthvalues}
In a language with maximal or constant expressiveness (see Definition~\ref{def:constantsandmaximalexpressiveness}), a truth-functional connective $C$ with truth-function $\underline{C}$ satisfies a regularity rule iff it satisfies it at the level of truth values:
\begin{tabular}{lll}
&
$\forall\Gamma, \Delta, \forall{F_1, ..., F_n}:$ &
$\begin{array}[t]{c@{\quad\textrm{ iff }\quad}c}
\Gamma, C(F_1, ..., F_n) \vdash \Delta
& \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\Gamma, \{F_i: i\in B_p\}\vdash \{F_i: i\in B_c\}, \Delta}\\
\Gamma \vdash C(F_1, ..., F_n) , \Delta
& \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^c}
{\Gamma, \{F_i: i\in B_p\}\vdash \{F_i: i\in B_c\}, \Delta}\\
\end{array}$
\\
iff \\
&
$\forall\gamma, \delta, \forall x_1, ..., x_n:$ &
$\begin{array}[t]{c@{\quad\textrm{ iff }\quad}c}
\gamma, \underline{C}(x_1, ..., x_n) \mathrel|\joinrel\equiv \delta
& \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\gamma, \{x_i: i\in B_p\}\mathrel|\joinrel\equiv \{x_i: i\in B_c\}, \delta}\\
\gamma \mathrel|\joinrel\equiv \underline{C}(x_1, ..., x_n) , \delta
& \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^c}
{\gamma, \{x_i: i\in B_p\}\mathrel|\joinrel\equiv \{x_i: i\in B_c\}, \delta}\\
\end{array}$
\end{tabular}
\end{theorem}
\begin{myproof}
The downward to upward direction is clear: if the rule holds for every possible list of truth values, it holds for every semantic interpretation taken as a whole. Conversely, because the language is constant expressive, the fact that the rule holds for \emph{all} formulae, therefore for all constant formulae, guarantees that it holds in the second form too (when the formulae are constant, the universal statement that reduces $\vdash$ to $\mathrel|\joinrel\equiv$ is trivial).
\end{myproof}
Secondly, regularity and the truth-relation together put very explicit constraints on the truth-function of a regular connective, of which we can extract a useful constructive presentation:
\begin{theorem}\label{th:truthconstraintforregconn}
Consider a logic equipped with a truth-relation with a minimal representation
$\mathrel|\joinrel\equiv{\mathcal{D}_p^1,\mathcal{D}_c^1}\cap...\cap\mathrel|\joinrel\equiv{\mathcal{D}_p^K,\mathcal{D}_c^K}$, and an $n$-ary connective $C$ with truth-function $\underline{C}$. $C$ follows a regularity rule (defined as above) iff
$\forall k, \forall x_1, ..., x_n$:
\[\begin{array}{c@{\quad\textrm{ iff }\quad}c}
\underline{C}(x_1, ..., x_n) \not\in\mathcal{D}_p^k
&
\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\{x_i: i\in B_p\}\subseteq\mathcal{D}_p^k \Rightarrow \{x_i: i\in B_c\}\cap\mathcal{D}_c^k\not=\emptyset}
\\
\underline{C}(x_1, ..., x_n) \in\mathcal{D}_c^k
&
\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^c}
{\{x_i: i\in B_p\}\subseteq\mathcal{D}_p^k \Rightarrow \{x_i: i\in B_c\}\cap\mathcal{D}_c^k\not=\emptyset}
\\
\end{array}\]
\end{theorem}
\begin{myproof}
Consider a connective satisfying these constraints, and choose $\gamma, \delta, x_1, ..., x_n$.
$\gamma, \underline{C}(x_1,...,x_n)\mathrel|\joinrel\equiv\delta$
\begin{minipage}[t]{.8\textwidth}
iff
$\forall k:\gamma, \underline{C}(x_1,...,x_n)\mathrel|\joinrel\equiv_k\delta$
iff
$\forall k: [\underline{C}(x_1,...,x_n)\not\in\mathcal{D}_p^k
\textrm{ or }
\gamma \mathrel|\joinrel\equiv_k \delta]$
iff
$\forall k: [
(\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\{x_i: i\in B_p\}\subseteq\mathcal{D}_p^k \Rightarrow \{x_i: i\in B_c\}\cap\mathcal{D}_c^k\not=\emptyset})
\textrm{ or }
\gamma \mathrel|\joinrel\equiv_k \delta]$
iff
$\forall k:
\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\{x_i: i\in B_p\}, \gamma \mathrel|\joinrel\equiv_k \{x_i: i\in B_c\}, \delta}$
iff
$\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\{x_i: i\in B_p\}, \gamma \mathrel|\joinrel\equiv \{x_i: i\in B_c\}, \delta}$.
\end{minipage}
\noindent This proves that the premise regularity rule holds, and an analogous derivation would prove that the conclusion regularity rule also holds.
Conversely, consider a connective satisfying the relevant regularity rule. Then pick $k_0, x_1, ..., x_n$. First note that
$\forall k\not=k_0:
\mathcal{D}_p^{k_0}\mathrel|\joinrel\equiv_{\mathcal{D}_p^k,\mathcal{D}_c^k} \overline{\mathcal{D}_c^{k_0}}$, for otherwise
$\mathcal{D}_p^{k_0}\subseteq\mathcal{D}_p^k$ and
$\mathcal{D}_c^k\subseteq\mathcal{D}_c^{k_0}$, in which case $\mathrel|\joinrel\equiv_{k_0}$ should have been dropped from the minimal representation of $\mathrel|\joinrel\equiv$.
Then consider the statement $(\mathcal{S}): \mathcal{D}_p^{k_0}, C(x_1, ..., x_n)\mathrel|\joinrel\equiv \overline{\mathcal{D}_c^{k_0}}$.
\begin{itemize}
\item On the one hand, given that $\forall k\not=k_0: \mathcal{D}_p^{k_0}\mathrel|\joinrel\equiv_k \overline{\mathcal{D}_c^{k_0}}$ is already true,
$(\mathcal{S})$ is true iff
$\mathcal{D}_p^{k_0}, C(x_1, ..., x_n)\mathrel|\joinrel\equiv_{k_0} \overline{\mathcal{D}_c^{k_0}}$, which boils down to whether $C(x_1, ..., x_n)\not\in\mathcal{D}_p^{k_0}$.
\item On the other hand, by regularity, $(\mathcal{S})$ is true
\begin{minipage}[t]{.6\textwidth}
iff
$\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\mathcal{D}_p^{k_0}, \{x_i: i\in B_p\}\mathrel|\joinrel\equiv \{x_i: i\in B_c\}, \overline{\mathcal{D}_c^{k_0}}}$
iff
$\forall k: \bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\mathcal{D}_p^{k_0}, \{x_i: i\in B_p\}\mathrel|\joinrel\equiv_k \{x_i: i\in B_c\}, \overline{\mathcal{D}_c^{k_0}}}$
iff
$\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^p}
{\mathcal{D}_p^{k_0}, \{x_i: i\in B_p\}\mathrel|\joinrel\equiv_{k_0} \{x_i: i\in B_c\}, \overline{\mathcal{D}_c^{k_0}}}$
iff
$\{x_i: i\in B_p\}\subseteq\mathcal{D}_p^{k_0} \Rightarrow \{x_i: i\in B_c\}\cap\mathcal{D}_c^{k_0}\not=\emptyset$.
\end{minipage}
\end{itemize}
\noindent
Putting together the final step of the two equivalence derivations above provides the first half of the result, corresponding to the premise regularity rule (first of the two conditions). The second half is obtained similarly by manipulating the statement
$(\mathcal{S'}): \mathcal{D}_p^{k_0} \mathrel|\joinrel\equiv C(x_1, ..., x_n), \overline{\mathcal{D}_c^{k_0}}$.
\end{myproof}
Corollary~\ref{th:formcond} illustrates how this Theorem~\ref{th:truthconstraintforregconn} constrains the form of G-conjunctions, G-disjunctions and G-conditionals. A useful paraphrase of the regularity rules for the conditional emerges: when the value of the first argument of a conditional is designated, then the value of the second argument must be designated too (abstracting away from the difference between premise and conclusion designated values, or from the various mixed consequence relations potentially involved). This paraphrase surely is reminiscent of the intuition behind the definition of a mixed consequence relation in the first place.
\begin{corollary}\label{th:formcond}
Consider a truth-relation $\mathrel|\joinrel\equiv$ minimally represented as $\mathrel|\joinrel\equiv_{\mathcal{D}_p^1, \mathcal{D}_c^1}\cap...\cap\mathrel|\joinrel\equiv_{\mathcal{D}_p^K, \mathcal{D}_c^K}$.
\begin{itemize}
\item A connective $\wedge$ is a G-conjunction for $\mathrel|\joinrel\equiv$ iff $\forall a, b$, for all sets of designated values $\mathcal{D}$:
\begin{itemize}
\item $(a \wedge b) \in \mathcal{D} \quad$ iff $\quad$ ($a\in\mathcal{D} \textrm{ and } b\in\mathcal{D}$).
\end{itemize}
\item A connective $\vee$ is a G-disjunction for $\mathrel|\joinrel\equiv$ iff $\forall a, b$, for all sets of designated values $\mathcal{D}$:
\begin{itemize}
\item $(a \vee b) \in \mathcal{D} \quad$ iff $\quad$ ($a\in\mathcal{D} \textrm{ or } b\in\mathcal{D}$).
\end{itemize}
\item A connective $\to$ is a G-conditional for $\mathrel|\joinrel\equiv$ iff $\forall a, b:\forall i:$
\begin{itemize}
\item $(a \to b) \in \mathcal{D}_p^i \quad$ iff $\quad$ ($a\in\mathcal{D}_c^i \Rightarrow b\in\mathcal{D}_p^i$)
\item $(a \to b) \in \mathcal{D}_c^i \quad$ iff $\quad$ ($a\in\mathcal{D}_p^i \Rightarrow b\in\mathcal{D}_c^i$)
\end{itemize}
\end{itemize}
\end{corollary}
In section~\ref{sec:N}, we will capitalize on these constraints on G-connectives to characterize what truth-relations can admit a truth-function for them. For now, we can mention a more general consequence of this Theorem: the regular connectives of an \emph{intersective} mixed consequence truth-relation are inherited from the regular connectives of the mixed consequence truth-relations they are made of. Consider an intersective mixed consequence truth-relation, say $ss\cap tt$. A connective $C$ can be a regular connective for $ss \cap tt$, say a G-conjunction, if and only if $\underline{C}$ is a common G-conjunction of $ss$ and of $tt$ in the first place. More generally:
\begin{corollary}\label{cor:intersectioncoincideR}
Consider an intersective mixed consequence truth-relation with a minimal representation written as $\mathrel|\joinrel\equiv_{\mathcal{D}_p^1,\mathcal{D}_c^1}\cap...\cap\mathrel|\joinrel\equiv_{\mathcal{D}_p^K,\mathcal{D}_c^K}$. And consider some regularity rule $\mathcal{R}$. Then
\begin{tabular}{ll}
& $f$ is the truth-function of a regular connective satisfying $\mathcal{R}$ for $\mathrel|\joinrel\equiv$ \\
iff & $f$ is the truth-function of a regular connective satisfying $\mathcal{R}$ for each $\mathrel|\joinrel\equiv_{\mathcal{D}_p^k,\mathcal{D}_c^k}$.
\end{tabular}
\end{corollary}
In a similar note, we can consider the reverse question: given some operator, say a conditional operator, what consequence relation may be associated to it? Concretely, the following result entails in a compact logic {(see Definition \ref{def:regularcompactlogic})} that $ss$ and $ts$ have no conditional in common (because they have the same `conclusion-tautologies' ($\{1\}$)); and that $ss$ and $st$ neither (because they have the same `premise-contradictions' ($\{0,\half\}$)).
\begin{theorem}\label{th:fromcondtoLC}
Two truth-relational consequence relations $\vdash_1$ and $\vdash_2$ in a compact logic are identical as soon as they share (i)~a truth-functional conditional and (ii)~the same conclusion-tautologies ($\{x: \vdash_i x\}$) or the same premise-contradictions ($\{x: x \vdash_i\}$).
\end{theorem}
\begin{myproof}
Assume that $\to$ is a conditional for $\vdash_1$ and $\vdash_2$, then the two relations also share a G-conjunction, a G-disjunction and a G-negation. For a given finite argument $P_1, ..., P_n\vdash Q_1, ..., Q_n$, all premises can be moved one by one to the conclusions, using the regularity rule for the G-negation, hence the argument holds iff
$\vdash \neg P_1, ..., \neg P_n, Q_1, ..., Q_n$.
We can continue and collapse all conclusions with the regularity rule for disjunction, so the argument holds iff
$\vdash \neg P_1 \vee ... \vee \neg P_n \vee Q_1 \vee ... \vee Q_n$.
Hence, given G-disjunctions and G-negations, whether {a finite} argument holds only depends on what {arguments} of the form {$\vdash Q$} {hold}. If two consequence relations share a G-disjunction and a G-negation, and have the same conclusion tautologies, they are identical {on finite arguments, and if the logic is compact they are thus identical}.
Similarly, if two consequence relations share a G-conjunction and G-negation, and have the same premise-contradictions, they are identical because every argument $P_1, ..., P_n\vdash Q_1, ..., Q_n$ boils down to
$P_1 \wedge ... \wedge P_n \wedge \neg Q_1 \wedge ... \wedge \neg Q_n \vdash $.
\end{myproof}
To conclude this section, we explore the (limited) multiplicity of regular connectives of any given type, for a given truth-relation.
Whether a truth-relation admits a connective satisfying a particular regularity rule is in general a difficult question. Before diving into it in the following sections, we address here a different problem: \emph{given a consequence relation and a regularity rule, how many truth-functions can produce a connective satisfying this rule, if any?}
The solution is mostly guided by the truth-relation, not by the specifics of the regularity rule.
Theorem \ref{th:systematicmulitplicity} states conditions under which a truth-relation allows for a multiplicity of regular connectives. The converse result (Theorem~\ref{th:limitedmulitplicity}) shows that this is the total extent to which there could be multiplicity.
\begin{theorem}\label{th:systematicmulitplicity}
Suppose two truth values $y_1$ and $y_2$ play the same role throughout the truth-relation (that is,
$\forall\gamma, \delta: \gamma, y_1\mathrel|\joinrel\equiv \delta$ iff $\gamma, y_2\mathrel|\joinrel\equiv \delta$
and
$\forall\gamma, \delta: \gamma \mathrel|\joinrel\equiv y_1, \delta$ iff $\gamma \mathrel|\joinrel\equiv y_2, \delta$). If $f$ is a truth-function for a connective that follows some regularity rule and such that $f(\vec{x})=y_1$, then the truth-function $f_{\vec{x}\to{y_2}}$ which is just like $f$ except that $f_{\vec{x}\to{y_2}}(\vec{x})=y_2$ follows the same regularity rule.
\end{theorem}
\begin{myproof}
Consider a connective $C$ to which the truth-function $f$ is assigned and which satisfies some regularity rule, for instance a premise-regularity rule:
$\Gamma, C(A_1,..., A_n)\vdash\Delta$
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^p} \Gamma, \{F_i : i \in \mathcal{B}_p\} \vdash\{F_i : i \in \mathcal{B}_c\}, \Delta$.
Now we can show that a connective $C'$ which has $f_{\vec{x}\to{y_2}}$ has a truth-function will follow the same regularity rule.
$\Gamma, C'(A_1,..., A_n)\vdash\Delta$
iff
$\forall v: v(\Gamma), v(C')(v(A_1),..., v(A_n))\mathrel|\joinrel\equiv v(\Delta)$ (by definition of a truth-relation)
iff
$\forall v: v(\Gamma), f_{\vec{x}\to{y_2}}(v(A_1),..., v(A_n))\mathrel|\joinrel\equiv v(\Delta)$ (applying truth-functionality)
iff
$\forall v: v(\Gamma), f(v(A_1),..., v(A_n))\mathrel|\joinrel\equiv v(\Delta)$ (because the difference between $f$ and $f_{\vec{x}\to{y_2}}$ is irrelevant to such arguments)
iff
$\forall v: v(\Gamma), v(C)(v(A_1),..., v(A_n))\mathrel|\joinrel\equiv v(\Delta)$ (by truth-functionality of $C$)
iff
$\Gamma, C(A_1,..., A_n)\vdash\Delta$ (by definition of the truth-relation)
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^p} \Gamma, \{F_i : i \in \mathcal{B}_p\} \vdash\{F_i : i \in \mathcal{B}_c\}, \Delta$ (applying the regularity rule to $C$).
\end{myproof}
\begin{theorem}[Limited multiplicity]\label{th:limitedmulitplicity}
Consider $f_1$ and $f_2$, two truth-functions associated with a given regularity rule. Suppose that they differ on the value they assign to some input $\vec{x}$: $f_1(\vec{x})\not=f_2(\vec{x})$. Then these two values $f_1(\vec{x})$ and $f_2(\vec{x})$ play the same role throughout the truth-relation.
\end{theorem}
\begin{myproof}
The following two equivalences are obtained by plugging in each time the relevant regularity rule:
$\forall \gamma, \delta: \gamma, f_1(\vec{x}) \mathrel|\joinrel\equiv \delta$ iff $\gamma, f_2(\vec{x}) \mathrel|\joinrel\equiv \delta$
and
$\forall \gamma, \delta: \gamma \mathrel|\joinrel\equiv f_1(\vec{x}), \delta$ iff $\gamma \mathrel|\joinrel\equiv f_2(\vec{x}), \delta$.
\end{myproof}
\subsection{Regularity: classical logic, bivalent connectives}\label{sec:bivalentregconnectives}
With the previous technical results in place, we can prove a result announced earlier:
\begin{theorem}\label{th:classicalallregular}
Every truth-functional connective in classical logic is regular.
\end{theorem}
\begin{myproof}
Suppose $C$ is an $n$-ary truth-functional connective in classical logic, with $n>0$ ($n=0$ is unproblematic). $\underline{C}$ can be written in conjunctive normal form, in which each conjunct can be written as a disjunction of the form ``$x_{n_1}=\alpha_1 \textrm{ or } x_{n_2}=\alpha_2 \textrm{ or } ...$'', with $\alpha_i\in\{0,1\}$. Such disjunctions translate in rules of the form
``$0\in\{x_i: \alpha_i=0 \}\textrm{ or }1\in\{x_i: \alpha_i=1 \}$'',
that is ``$\{x_i: \alpha_i=0 \}\subseteq\{1\} \Rightarrow \{x_i: \alpha_i=1 \}\cap\{1\}\not=\emptyset$'',
that is statements of the same form as those used in a regularity rule when $\mathcal{D}_p=\mathcal{D}_c=\{1\}$. Put simply, the conjunctive normal form of $\underline{C}$ provides the conclusion regularity rule for $C$, following the format of Theorem~\ref{th:truthconstraintforregconn}. The premise regularity rule is provided by the conjunctive normal form for the negation of $C$.
\end{myproof}
Furthermore, a somewhat converse result also holds:
\begin{theorem}\label{th:bivconnarereg}
Every regular connective in $N$-valued logics with a weakly bivalent truth-function (one which takes bivalent values for bivalent inputs) shares a regularity rule with a connective from classical logic.
\end{theorem}
\begin{myproof}
Suppose that $C$ is a regular, bivalent connective. It is clear that the rule of Theorem~\ref{th:truthconstraintforregconn} holds not only for $\underline{C}$, but also for $\widetilde{\underline{C}}$, the restriction of $\underline{C}$ to inputs in $\{0,1\}$. This last truth-function thus satisfies the same regularity rule as the original one, which is thus one of a classical truth-function: the output values are in $\{0,1\}$, by hypothesis, and the question of whether a value belongs to some set of designated values or not is the same question as whether the value is $1$ or not ($\{1\}$ is the set of designated values for classical logic).
\end{myproof}
Finally, we show a result of expressive completeness: a G-conditional is sufficient to deliver all regular rules from classical connectives.
\begin{theorem}\label{th:conditionalissufficient}
Assume that a logic is constant expressive and has a G-conditional. Then for any classical connective $C$, the logic has a regular connective $C^+$ that shares a regularity rule with this connective $C$.
\end{theorem}
\begin{myproof}
The logic has a G-conditional and is constant expressive, so it also has a G-conjunction and a G-disjunction (Theorem~\ref{th:combinationsforconditionals}).
Let us write the conclusion regularity rule of a classical $n$-ary connective $C$ as usual with a set $\mathcal{B}^c$ of pairs $(B_p,B_c)$ of sets of indices. Consider the connective $C^+$ with a truth-function defined as:
$\underline{C^+}(x_1, ..., x_n)=\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} [ \bigwedge\{x_i:i\in B_p\} \to \bigvee\{x_i:i\in B_c\}]$ (here using the relevant truth-functions for conjunctions, disjunctions and conditionals, including for $\bigwedge$ and $\bigvee$, for which order does not matter and can be held fixed). This connective surely satisfies the conclusion regularity rule; for all $i$ an index of one mixed truth-relation in the intersective mixed truth-relation:
$\underline{C^+}(x_1, ..., x_n)\in \mathcal{D}_c^i$
iff
$(\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} [ \bigwedge\{x_i:i\in B_p\} \to \bigvee\{x_i:i\in B_c\}] )\in \mathcal{D}_c^i$
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \bigwedge\{x_i:i\in B_p\} \to \bigvee\{x_i:i\in B_c\}] \in \mathcal{D}_c^i)$
(G-conjunction, see~\ref{th:formcond})
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \bigwedge\{x_i:i\in B_p\}\in\mathcal{D}_p^i \Rightarrow \bigvee\{x_i:i\in B_c\}\in\mathcal{D_c^i}] $
(G-conditional, see~\ref{th:formcond})
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \{x_i:i\in B_p\}\subset\mathcal{D}_p^i \Rightarrow \{x_i:i\in B_c\}\cap\mathcal{D}_c^i\not=\emptyset] $
(G-conjunction and G-disjunction, see~\ref{th:formcond}).
This provides the result by Theorem~\ref{th:truthconstraintforregconn}.
As for the premise regularity rule, it will follow because for classical connectives it is constrained by the conclusion regularity rule in the first place. We start with a derivation similar to the one above:
$\underline{C^+}(x_1, ..., x_n)\in \mathcal{D}_p^i$
iff
$(\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} [ \bigwedge\{x_i:i\in B_p\} \to \bigvee\{x_i:i\in B_c\}] )\in \mathcal{D}_p^i$
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \bigwedge\{x_i:i\in B_p\} \to \bigvee\{x_i:i\in B_c\}] \in \mathcal{D}_p^i)$
(G-conjunction, see~\ref{th:formcond})
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \bigwedge\{x_i:i\in B_p\}\in\mathcal{D}_c^i \Rightarrow \bigvee\{x_i:i\in B_c\}\in\mathcal{D}_p^i] $
(G-conditional, see~\ref{th:formcond}, the main difference between this derivation and the previous one is that here we apply a different bullet point for this step)
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \forall i\in B_p: x_i\in\mathcal{D}_c^i] \Rightarrow [\exists i\in B_c: x_i\in\mathcal{D}_p^i])$
(G-conjunction and G-disjunction, see~\ref{th:formcond}).
Although it uses the pairs $(B_p,B_c)$ from the $\mathcal{B}^c$ in the conclusion regularity rule, this last statement can be translated in the premise regularity rule: for classical connectives, the premise and conclusion regularity rules are dual of one another: whenever $C(x_1,..., x_n)\not\in\{1\}$ (that is, the premise rule holds), then it cannot be that $C(x_1,..., x_n)\in\{1\}$ (that is, conclusion rule does not hold). Thus, the premise and conclusion regularity rules are such that:
($\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \forall i\in B_p: \alpha_i] \Rightarrow [\exists i\in B_c: \alpha_i])$ is true
iff
$\bigwedge_{(B_p,B_c)\in\mathcal{B}^c} ([ \forall i\in B_p: \alpha_i] \Rightarrow [\exists i\in B_c: \alpha_i])$ is false),
whenever $\alpha_1, ..., \alpha_n$ is a list of Booleans.
Chaining this at the end of the previous derivation helps translate from the formula with the $\mathcal{B}^c$ of the conclusion regularity rule to a formula with the $\mathcal{B}^p$ of the premise regularity rule, which provides the result by Theorem~\ref{th:truthconstraintforregconn}.
\end{myproof}
\subsection{Regularity: reductions to 4-valued logics and the specific case of $\mathrel|\joinrel\equiv_{\{1,\#_p\},\{1,\#_c\}}$
}
\label{sec:reg=4value}
{Some regularity rules are not exemplified by connectives in classical logic, and one may thus like to investigate regular connectives in logic with more truth values. Here we establish that every regularity rule can be exemplified by a connective in 4-valued logic.}
To see this, let us first remind that one motivation for the notion is that regularity allows one to represent a broad class of $N$-valued logics (Definition~\ref{def:regularcompactlogic}) within $4$-valued logic, as stated in Theorem~\ref{thm:reg}.
\begin{definition}[Logic, Compact, Regular]\label{def:regularcompactlogic}
We define a \emph{logic} to be
{a language equipped with a consequence relation.}
A logic is \emph{compact} if $\Gamma \vdash \Delta$ implies the existence of finite subsets $\Gamma'$ and $\Delta'$ of $\Gamma$ and $\Delta$ respectively, such that $\Gamma'\vdash \Delta'$. A logic is \emph{regular} if it has only regular connectives. \end{definition}
\noindent
Beyond Gentzen's approach, regularity matters because of the following result from \cite{chemla2018suszko}:
\begin{theorem}\label{thm:reg} Every monotonic, compact, and regular logic is semantically representable by means of an at-most 4-valued truth-relational and truth-functional semantics.
\end{theorem}
The fact that at most 4 values are needed to semantically characterize a monotonic consequence relation was also established by \cite{blasio2017inferentially} and by \cite{french2017valuations}, but with no heed paid to truth-functionality. The addition from Theorem \ref{thm:reg} was that when the consequence relation admits only regular connectives, regularity ensures that the reduction to four values is moreover truth-functional, that is compositional on truth values. This is not the case in general, however, and the reduction of a semantics for a monotonic logic to only four values (or three, or two) may otherwise lead to violations of truth-functionality (as originally stressed by \citealt{suszko1977fregean} for the semantic representation of Tarskian logics by means of just two values). This is one aspect in which regularity is an important constraint: it guarantees that a logical connective behaves truth-functionally over a \emph{minimal} semantic representation of its associated consequence relation.
In fact, the previous result can be made more precise, establishing that all regular connectives have a representative in the 4-valued logic equipped with the truth-relation $\mathrel|\joinrel\equiv_{\{1,\#_p\},\{1,\#_c\}}$:
\begin{theorem}
Consider the truth-relation $\mathrel|\joinrel\equiv_{\{1,\#_p\},\{1,\#_c\}}$ in 4-valued logic with truth-values noted $\{1,\#_p,\#_c,0\}$. Every pair of premise/conclusion regularity rules for an $n$-ary connective is represented by a unique truth-functional $n$-ary connective.
\end{theorem}
\begin{myproof}
For every regularity rule, Theorem~\ref{th:truthconstraintforregconn} shows the constraints that need to be satisfied by a truth-function for the connective to obey the regularity rule. These constraints can always be satisfied for the current truth-relation, because belonging or not to $\mathcal{D}_p$ (here $\{1,\#_p\}$) is independent from belonging or not to $\mathcal{D}_c$ (here $\{1,\#_c\}$). (See also \citealp[Theorem~4.15]{chemla2018suszko}). Furthermore, Theorem~\ref{th:limitedmulitplicity} establishes that no two truth-functions can satisfy the same regularity rules in $\mathrel|\joinrel\equiv_{\{1,\#_p\},\{1,\#_c\}}$, because no two truth-values play the same role.
\end{myproof}
\subsection{Summary}
We are now equipped with a definition of the family of consequence relations of interest to us, namely intersective mixed truth-relations, and with a definition of the family of connectives of relevance to us, namely Gentzen-regular connectives. We started with a presentation mostly related to classical logic, and then argued that Gentzen-regularity rules are best understood in some specific $4$-valued logic. In the coming sections, we will focus on four types of connectives: disjunction, conjunction, negation, and conditional. An exhaustive review of the situation will be provided for 3-valued logics (Section~\ref{sec:three}), and for 4-valued logics (Section~\ref{sec:four}). General results for all pure consequence relations, and all order-theoretic relations are presented in Section~\ref{sec:pureandorderresults}. Necessary and sufficient algebraic conditions for the presence of conditionals are eventually obtained for all $N$-valued logics in the final Section~\ref{sec:N}.
\section{Exhaustive search in three-valued logics}\label{sec:three}
\newcommand{\truthtablethreevalues}[9]{
\begin{tabular}[t]{c|ccc}
& 1 & \half & 0\\
\hline
1 & #1 & #2 & #3\\
\half & #4 & #5 & #6\\
0 & #7 & #8 & #9\\
\end{tabular}
}
In three-valued logics, there are exactly five intersective mixed relations,
namely: $ss$, $tt$, $st$, $ts$ and $ss \cap tt$. In this section, we consider which G-connectives among negation, conjunction, disjunction, and the conditional, can be defined for them. We distinguish two cases:
first, we look at maximally or constant expressive languages (with $\top, \#, \bot$ for $1, \half,0$), then we generalize the results to less expressive languages, namely atomic expressive languages (see Definition~\ref{def:constantsandmaximalexpressiveness}).
\subsection{Maximally (or constant) expressive languages}
With the exception of $ss\cap tt$, all intersective mixed relations in 3-valued logic admit Gentzen-regular connectives. More specifically:
\begin{theorem}
G-disjunctions and G-conjunctions can be found for all 3-valued intersective mixed relations. G-negations and G-conditionals can be found for all intersective mixed relations except $ss \cap tt$, which admits neither.
\end{theorem}
\begin{myproof}
These results can be proven by hand, but they follow from the companion computer program available at \url{https://arxiv.org/src/1809.01066v1/anc}.
The results are obtained as follows: the computer program lists all possible truth-functions and, for each such truth-function $c$ and each truth-relation in 3-valued logic, it checks whether, for all pairs of sets of truth values $\gamma$, $\delta$, the truth-function satisfies the regularity rules associated with conjunction, disjunction, and the conditional. The result is thus first obtained at the level of truth values, but it can be shown to be similar for whole propositions, and therefore formulae
(see Theorem~\ref{th:reducetotruthvalues}).
The exhaustive list of G-connectives for each consequence relation is as follows:
\begin{description}
\item[Conjunctions] All consequence relations share the following (Strong Kleene) conjunction:
\[
\truthtablethreevalues{1}{\half}{0}{\half}{\half}{0}{0}{0}{0}
\]
Compatible with Theorems~\ref{th:systematicmulitplicity} and \ref{th:limitedmulitplicity}, we observe that
this is the unique conjunction for $st$, $ts$ and $ss \cap tt$, while, in addition to these,
$ss$ allows for all variants of these tables where a $0$ is replaced by $\half$ or the reverse (hence a total of $2^8$ conjunctions for $ss$);
$tt$ allows for all variants of these tables where a $1$ is replaced by $\half$ or the reverse (hence a total of $2^4$ conjunctions for $tt$).
\item[Disjunctions] All consequence relations share the following (Strong Kleene) disjunction:
\[
\truthtablethreevalues{1}{1}{1}{1}{\half}{\half}{1}{\half}{0}
\]
Compatible with Theorems~\ref{th:systematicmulitplicity} and \ref{th:limitedmulitplicity}, we observe that
this is the unique disjunction for $st$, $ts$ and $ss \cap tt$, whereas, in addition to these,
$ss$ allows for all variants of these tables where a $0$ is replaced (nonuniformly) by $\half$ or the reverse (hence a total of $2^4$ disjunctions for $ss$);
$tt$ allows for all variants of these tables where a $1$ is replaced (nonuniformly) by $\half$ or the reverse (hence a total of $2^8$ disjunctions for $tt$).
\item[Conditionals] The relation $ss \cap tt$ does not have a conditional.
We can provide a direct proof of this rather central and surprising fact. Suppose that $\to$ were a regular conditional for $ss \cap tt$. Then, we can prove that no value would work for $\half\to0$:
(i)~$\half\not\mathrel|\joinrel\equiv_{ss\cap tt} 0$,
so $\not\mathrel|\joinrel\equiv_{ss\cap tt} (\half\to0)$,
so $\half\to0$ cannot be $1$.
(ii)~$\half, \half\not\mathrel|\joinrel\equiv_{ss\cap tt} 0$,
so $\half\not\mathrel|\joinrel\equiv_{ss\cap tt} (\half\to0)$,
so $\half\to0$ cannot be $\half$.
(iii)~$\not\mathrel|\joinrel\equiv_{ss\cap tt} 0$,
so $\not\mathrel|\joinrel\equiv_{ss\cap tt}\half$,
so it's not the case that $0 \mathrel|\joinrel\equiv_{ss\cap tt}$ and $\mathrel|\joinrel\equiv_{ss\cap tt}\half$,
so $(\half\to0) \not\mathrel|\joinrel\equiv_{ss\cap tt}$,
so $\half\to0$ cannot be $0$.
Moving away from $ss \cap tt$ then, the following tables are conditionals for the other four consequence relations. Compatible with Theorems~\ref{th:systematicmulitplicity} and \ref{th:limitedmulitplicity}, these tables exhaust the possibilities, except for $ss$ and for $tt$, for which the same replacements as above produce other conditionals ($2^2$ for $ss$, and $2^7$ for $tt$).\footnote{See \cite{jeffrey1963indeterminate} for a related (more restricted) result concerning the conditionals internalizing $tt$-validity.}
\noindent\begin{tabular}{c@{$\quad$}c@{$\quad$}c@{$\quad$}c}
$ss$: \truthtablethreevalues{1}{0}{0}{1}{1}{1}{1}{1}{1} &
$tt$: \truthtablethreevalues{1}{1}{0}{1}{1}{0}{1}{1}{1} &
$st$: \truthtablethreevalues{1}{\half}{0}{1}{\half}{\half}{1}{1}{1} &
$ts$: \truthtablethreevalues{1}{\half}{0}{1}{\half}{\half}{1}{1}{1}
\end{tabular}
\item[Negations] The relation $ss \cap tt$ does not have a negation.
This follows from the absence of a conditional and the presence of a disjunction (Theorem \ref{th:combinationsforconditionals}). Alternatively, this can be obtained with essentially the same proof as the one above for conditionals: no value can be assigned to $\neg(\half)$.
For other consequence relations, the admissible negations are given by the last column of the tables for the conditionals, with variability obtained through the relevant replacements for $ss$ and $tt$, again because of Theorems~\ref{th:systematicmulitplicity} and \ref{th:limitedmulitplicity}.
\qedhere
\end{description}
\end{myproof}
We note, however, that the absence of a G-conditional for $ss\cap tt$ fundamentally depends on the multi-conclusion setting and on the premise regularity rule. In other settings a conditional may be well-behaved, although the options may be more restricted than one can think of given the range of 3-valued conditionals that have been studied:
\begin{theorem}
The relation $ss \cap tt$ has a unique conditional satisfying a single-conclusion version of the conclusion-Gentzen regularity rule (i.e.~the deduction theorem): $\forall \Gamma: \Gamma \vdash A \to B$ iff $\Gamma, A\vdash B$. It is defined as follows:\footnote{This conditional corresponds to a three-valued version of the so-called G\"odel implication, see \cite{hajek1998meta}.}
\[
\truthtablethreevalues{1}{\half}{0}{1}{1}{0}{1}{1}{1}
\]
However, the relation $ss \cap tt$ has no conditional satisfying a single-conclusion version of the premise-Gentzen regularity rule, which may be expressed as follows (note that this requires us to accept empty conclusion sets): $\forall \Gamma,A,B: \Gamma, A\to B\vdash $ iff ($\Gamma, B \vdash$ and $\Gamma \vdash A$). The previous conditional does, however, satisfy the right-to-left direction of that rule.
\end{theorem}
\begin{myproof} This result follows from the companion program, but we prove pieces of it by hand to illustrate the process:
\begin{itemize}
\item Show that $\Gamma, A \vdash B $ iff $\Gamma \vdash A\to B$.
For all semantic interpretation, the right-hand side holds
iff $v(A\to B)=1$ or there is $x\in v(\Gamma)$ lower than $v(A\to B)$.
That is, iff $v(A\to B)=1$ or ($v(A\to B)\not=1$ and there is $x\in v(\Gamma)$ lower than $v(A\to B)$).
That is, iff $v(A)\leq v(B)$ (see truth-table) or there is $x\in v(\Gamma)$ lower than $v(B)$ (in all cases where $v(A\to B)\not=1$, $v(A\to B)=v(B)$, see truth-table).
That is, iff $v(A)\mathrel|\joinrel\equiv v(B)$ or $v(\Gamma)\mathrel|\joinrel\equiv v(B)$.
That is, iff there is $x$ in $v(\Gamma)$ or $v(A)$ lower than $v(B)$, that is, iff the left-hand side holds.
\item Uniqueness:
For any such a conditional, now in a constant expressive setting, let us denote by $\overline{x}$ the constant proposition with value $x$, for any $x$ a truth value. So, $\top$ is $\overline{1}$, $\bot$ is $\overline{0}$, $\#$ is $\overline{\half}$.
Then, $x\leq y$ iff $x\mathrel|\joinrel\equiv y$ iff $\overline{x}\models\overline{y}$ iff $\models\overline{x}\to\overline{y}$ iff $\mathrel|\joinrel\equiv x\to y$ iff $x\to y=1$. That accounts for the diagonal $1$ in the truth-table, the $1$s below that diagonal, and the absence of $1$ above the diagonal.
Furthermore, $\#, \top \vdash \#$, hence it $\# \vdash \top \to \#$, and it is thus not possible that $1\to \half=0$.
Finally, $\#, \overline{x} \not\vdash \bot$, for $x=1$ or $x=\half$, hence $\# \not\vdash \overline{x}\to \bot$, and therefore
$x\to 0$ cannot be $\half$.
\item Show that if $\Gamma, B\vdash $ and $\Gamma \vdash A$, then $\Gamma, A\to B \vdash $.
From the hypotheses, it follows that for every semantic interpretation $v$, then
($0$ belongs to $v(\Gamma)$ or $v(B)=0$) and (there is $x$ in $v(\Gamma)$ such that $x\leq v(A)$ or $v(A)=1$).
If $0$ belongs to $v(\Gamma)$, the result follows easily. So, we may suppose that $0\not\in v(\Gamma)$. It follows that $v(B)=0$ (simplifying the first parenthesis in the condition above), and that $v(A)>0$ (because $\Gamma\vdash A$). Hence, we are in one of the two top cells in the last column of the table, that is $v(A \to B)=0$, and the result follows.
\qedhere
\end{itemize}
\end{myproof}
\subsection{Less expressive languages}
We have answered our question for languages in which all constants are expressible.
What happens when the language is less expressive?
When a consequence relation admits a Gentzen-regular connective in a maximally expressive setting, the truth-function can be used to define a regular connective in a less expressive setting too. In a sense, respecting the regularity rule in a maximally expressive setting is like respecting it \emph{intensionally}, so the rule will continue to be satisfied whether or not all relevant propositions are actually expressible.
But what happens if a given connective does not exist in a maximally expressive setting, can we find a connective that would be appropriate in a \emph{less} expressive one?
To answer this, we shall consider $ss \cap tt$, which is the only intersective mixed relation lacking regular connectives in a fully expressive setting, namely G-negations and G-conditionals. More general results will soon be obtained (Theorem~\ref{th:fullordertheoretic}), but we can here directly show that even in a less expressive setting $ss \cap tt$ admits neither a G-negation nor a G-conditional.
\begin{theorem}\label{th:no negation for sstt}
Even with an atomic expressive semantics, $ss \cap tt$ can never admit a G-negation.
\end{theorem}
\begin{myproof}
Suppose $\neg$ is a G-negation for a given language, with $ss \cap tt$ as the associated consequence relation. Assume that we can find a formula $A$ and a semantic interpretation $v$ such that $v(A)=\half$. $A\vdash A$, because $ss \cap tt$ is reflexive. By Gentzen-regularity, $A ,\neg A\vdash$ and $\vdash \neg A, A$. From the former, it follows from $tt$-validity that $\neg(\half)=0$, and from the latter it follows from $ss$-validity that $\neg(\half)=1$.
\end{myproof}
\begin{theorem}\label{th:no conditional for sstt}
Even with an atomic expressive semantics, $ss \cap tt$ can never admit a G-conditional.
\end{theorem}
\begin{myproof}
Suppose $\to$ is a G-conditional for a given language, with $ss \cap tt$ as the associated consequence relation, choose $p$ and $q$ two distinct atomic propositions.
\begin{itemize}
\item $p \vdash q, p$ by reflexivity and monotonicity of $ss \cap tt$. Hence, by Gentzen-regularity: $\vdash p\to q, p$.
Hence, for all $v$ st $v(p)\not=1$, $v(p\to q)=1$. From this, it follows that $\half\to0=1$.
\item $p, q \vdash q$ and $p \vdash p, q$, again by reflexivity of $ss \cap tt$; hence, if $\to$ is Gentzen-regular, $p, p\to q \vdash q$.
From a valuation $v$ in which $v(q)=0$ and $v(p)=\half$, we infer that necessarily, $\half\to0=0$.
\end{itemize}
Assuming truth-functionality, the two constraints above are incompatible.
\end{myproof}
\subsection{Summary}
The upshot is that, in a multi-premise multi-conclusion setting, all mixed consequence relations admit a G-conjunction, a G-disjunction, a G-negation and a G-conditional. The only `strict intersective' mixed consequence relation, $ss\cap tt$, only admits a G-conjunction and a G-disjunction, but neither a G-conditional nor a G-negation.
\section{Exhaustive search in four-valued logics}\label{sec:four}
Regular connectives have a close connection to 4-valued logics: they correspond to all connectives going with a particular truth-relation (see Section~\ref{sec:reg=4value}). However, there are many intersective mixed consequence relations in 4-valued logic, more than in the 3-valued case. In this section, we provide a computer-aided exhaustive search of all of these consequence relations, and of the G-connectives they admit, for languages that are constant expressive.
\newcommand{\truthtablefourvaluescond}[6]{
\begin{tabular}{c|cccc}
& 1 & $\#_1$ & $\#_2$ & 0\\
\hline
1 & 1 & $\#_1$ & $\#_2$ & 0\\
$\#_1$ & 1 & #1 & #2 & #3\\
$\#_2$ & 1 & #4 & #5 & #6\\
0 & 1 & 1 & 1 & 1\\
\end{tabular}
}
\begin{theorem}\label{th:4-valued-all}
There are 167 distinct 4-valued intersective mixed consequence relations.
Among these:
\begin{itemize}
\item 18 admit a G-conditional (and, therefore, a G-negation, a G-disjunction and a G-conjunction). These are the 16 mixed consequence relations as well as the following two for which we provide the truth-tables of their unique conditionals:
\[
\begin{tabular}{cc}
$\mathrel|\joinrel\equiv_{\{1,\#_1\},\{1,\#_1\}} \cap \mathrel|\joinrel\equiv_{\{1,\#_2\},\{1,\#_2\}}$
&
$\mathrel|\joinrel\equiv_{\{1,\#_1\},\{1,\#_2\}} \cap \mathrel|\joinrel\equiv_{\{1,\#_2\},\{1,\#_1\}}$
\\[1ex]
\truthtablefourvaluescond{1}{$\#_2$}{$\#_2$}{$\#_1$}{1}{$\#_1$}
&
\truthtablefourvaluescond{$\#_1$}{1}{$\#_1$}{1}{$\#_2$}{$\#_2$}
\end{tabular}
\]
\item 28 admit G-conjunctions and G-disjunctions, but no G-negation (nor G-conditional).
\item
27 admit G-conjunctions, but no G-disjunction (and therefore no G-negation, nor G-conditional);
and
27 others admit G-disjunctions, but no G-conjunction (and therefore no G-negation, nor G-conditional).
\item
The remaining 67 admit no G-disjunction, G-conjunction, G-negation nor G-conditional.
\end{itemize}
\end{theorem}
\begin{myproof}
The proof is provided by the companion computer program available at \url{https://arxiv.org/src/1809.01066v1/anc}.
This program explores arguments at the level of truth values, and results are thus obtained for constant expressive semantics (see Theorem~\ref{th:reducetotruthvalues}).
The program does not explore the whole set of truth-tables as in 3-valued logic above, but only a subset in which we can be sure to find regular conditionals, negations, conjunctions and disjunctions, as long as one exists. It does so by restricting attention to connectives of the following form, in which the $\star$ indicates a place for which all choices of truth values were scanned:
\[
\begin{tabular}{cccc}
Negations & Conditionals & Conjunctions & Disjunctions \\
\begin{tabular}{c|cccc}
& Neg \\
\hline
1 & 0 \\
$\#_1$ & $\star$\\
$\#_2$ & $\star$\\
0 & 1\\
\end{tabular}
&
\truthtablefourvaluescond{$\star$}{$\star$}{$\star$}{$\star$}{$\star$}{$\star$}
&
\begin{tabular}{c|cccc}
& 1 & $\#_1$ & $\#_2$ & 0\\
\hline
1 & 1 & $\#_1$ & $\#_2$ & 0\\
$\#_1$ & $\#_1$ & $\star$ & $\star$ & 0\\
$\#_2$ & $\#_2$ & $\star$ & $\star$ & 0\\
0 & 0 & 0 & 0 & 0\\
\end{tabular}
&
\begin{tabular}{c|cccc}
& 1 & $\#_1$ & $\#_2$ & 0\\
\hline
1 & 1 & 1 & 1 & 1\\
$\#_1$ & 1 & $\star$ & $\star$ & $\#_1$\\
$\#_2$ & 1 & $\star$ & $\star$ & $\#_2$\\
0 & 1 & $\#_1$ & $\#_2$ & 0\\
\end{tabular}
\end{tabular}
\]
It can be shown that if there is a relevant Gentzen-regular connective, one can obtain another Gentzen-regular connective of the same kind by making the necessary replacements to obtain the same values as those fixed in the above tables: the reason is simply that the values above make the Gentzen-regularity rules work.
The only thing left to prove then is the uniqueness of the conditionals provided above, but this follows from Theorem~\ref{th:limitedmulitplicity} (one can check that no two truth values play the same role for the relevant truth-relations).
\end{myproof}
In the previous theorem, consequence relations for which the role of the indeterminates would be switched (e.g., $\mathrel|\joinrel\equiv_{\{1,\#_1\},\{1,\#_1\}}$ and $\mathrel|\joinrel\equiv_{\{1,\#_2\},\{1,\#_2\}}$) are considered distinct, even though the difference only hinges on the name given to the indeterminates. By collapsing consequence relations in which those indeterminates play the same role, the numbers decrease slightly:
\begin{theorem}
There are 97 distinct 4-valued intersective mixed relations, if we assimilate those which are identical except for the name given to the indeterminates. Among these:
\begin{itemize}
\item 12 admit a G-conditional (and, therefore, G-negation, G-disjunction and G-conjunction). These include 10 mixed consequence relations as well as the two intersective mixed relations identified above.
\item 15 admit G-conjunctions and G-disjunctions, but no G-negations (nor G-conditionals).
\item
16 admit G-conjunctions, but no G-disjunction (and therefore no G-negations, nor G-conditionals);
and 27 others admit G-disjunctions, but no G-conjunction (and therefore no G-negations, nor G-conditionals).
\item
The remaining 38 admit no G-disjunction, G-conjunction, G-negation nor G-conditional.
\end{itemize}
\end{theorem}
\begin{myproof}
The proof is provided by the companion computer program available at \url{https://arxiv.org/src/1809.01066v1/anc}. The program here is similar to the one used for the proof of the previous Theorem~\ref{th:4-valued-all}, but additionally it compares all consequence relations and prunes those that are similar to another up to permutation of the two indeterminate truth-values.
\end{myproof}
$N$-valued logic therefore exhibits a wide variety of situations. To motivate looking higher up into $N$-valued logics, we note that it is necessary to go at least to $N=5$ to discover a consequence relation with a G-negation but not a G-conditional.
\begin{fact}
The following truth-relation in 5-valued logic admits a G-negation but no G-conditional:
$$
\mathrel|\joinrel\equiv_{\{1,\#_1,\#_2\},\{1,\#_1\}}
\cap
\mathrel|\joinrel\equiv_{\{1,\#_1,\#_3\},\{1,\#_2\}}
\cap
\mathrel|\joinrel\equiv_{\{1,\#_2,\#_3\},\{1,\#_3\}}
$$
\end{fact}
\begin{myproof}
This truth-relation has been found and demonstrated to have a G-negation and no G-conditional by a computer program. But we can use later results to verify this result.
\begin{description}
\item[There is a G-negation]
We observe that the operator below (provided by the same computer program) is such that
$\forall x:$ ($x\in\mathcal{D}_p^i$ iff $\neg(x)\in\mathcal{D}_c^i$) and ($x\in\mathcal{D}_c^i$ iff $\neg(x)\in\mathcal{D}_p^i$), and (the proof of) Theorem~\ref{th:disjunctioncondition} shows that this makes it a G-negation.
\begin{center}
\begin{tabular}{r|ccccc}
$X$ & 1 & $\#_3$ & $\#_2$ & $\#_1$ & 0\\ \hline
$\neg X$ & 0 & $\#_1$ & $\#_2$ & $\#_3$ & 1
\end{tabular}
\end{center}
\item[There is no G-conditional]
The necessary condition DC1 from Definition~\ref{def:disjconjcompatibility} to have a G-conjunction and a G-disjunction is not satisfied: $\{1,\#_1,\#_2\}$ is included in no other set of designated values, yet all of its truth values belong to other (distinct) sets of designated values in the representation.
\qedhere
\end{description}
\end{myproof}
\section{General results for mixed, pure, and order-theoretic relations}\label{sec:pureandorderresults}
In the previous sections, we have presented an exhaustive investigation of truth-relations of all kinds in 3-valued logics and 4-valued logics. Here, we propose a more in-depth look at specific kinds of consequence relations in $N$-valued logics for all $N$s: \emph{mixed} consequence relations (and among them \emph{pure} consequence relations) and \emph{order-theoretic} consequence relations.
\subsection{Mixed and pure consequence relations in $N$-valued logics: classical connectives}\label{sec:pureresults}
\begin{theorem}\label{th:mixedhasclassical}
Every mixed consequence relation admits a G-conditional and all regular connectives from classical logic.
\end{theorem}
\begin{myproof}
Starting from a $\mathrel|\joinrel\equiv_{\mathcal{D}_p,\mathcal{D}_c}$, we will show that it admits all classical regular connectives, by showing that it admits a G-conditional (see Theorem~\ref{th:conditionalissufficient}).
One could then use a shortcut: $ss$, $st$, $ts$ and $\mathrel|\joinrel\equiv_{\{1,\#_p\},\{1,\#_c\}}$ all admit a G-conditional, and since they exhaust the possibilities of how the two sets of designated values can be in a superset/subset relation, we could reason from them, that is, $\mathrel|\joinrel\equiv_{\mathcal{D}_p,\mathcal{D}_c}$ will behave like one of these, depending on which (if any) of $\mathcal{D}_p$ and $\mathcal{D}_c$ is included in the other.
Another approach is to exhibit a G-conditional, or a pattern to construct one. In the table below, consider that
$\#_p$ stands for any truth-value that belongs to $\mathcal{D}_p$ but not $\mathcal{D}_c$,
$\#_c$ stands for any truth-value that belongs to $\mathcal{D}_c$ but not $\mathcal{D}_p$,
$1$ stands for any truth-value that belongs to both, and
$0$ for any truth-value that belongs to neither of the sets of designated values.
The schema is underspecified, e.g., $\#_p$ means any value of the relevant type, and which specific value it is may differ depending on the column, line or output, but the choice does not matter. Crucially, one can always define a proper truth-function from this schema: it never outputs a value that is not $1$ or $0$ (which exist in all logics), or of the same type as one of the input values that produce this output. One can finally verify easily that a conditional constructed from this schema will always respect the G-conditional regularity rules (this is most easily seen from Theorem~\ref{th:truthconstraintforregconn}, or simply because it is a G-conditional for the most complete $\mathrel|\joinrel\equiv_{\{1,\#_p\},\{1,\#_c\}}$).
\[
\begin{tabular}[b]{c|cccc}
& 1 & $\#_p$ & $\#_c$ & 0\\ \hline
1 & 1 & $\#_p$ & $\#_c$ & 0\\
$\#_p$ & 1 & $\#_p$ & 1 & $\#_p$\\
$\#_c$ & 1 & 1 & $\#_c$ & $\#_c$\\
0 & 1 & 1 & 1 & 1
\end{tabular}\qedhere\]
\end{myproof}
The next two theorems separate out mixed consequence relations from pure consequence relations, and reveal further details:
\begin{theorem}\label{th:pureisclassical}
Every pure consequence relation admits exactly the same regular connectives as classical logic.
\end{theorem}
\begin{myproof}
Consider a regular connective for a pure consequence relation. It can be turned into an equivalently regular connective that would be bivalent, by changing all non classical value into $1$ when they belong to the set of designated values, and $0$ otherwise, as these changes comply with the constraints from Theorem~\ref{th:systematicmulitplicity}. Hence, any such connective satisfies a regularity rule also satisfied by a classical connective, by Theorem~\ref{th:bivconnarereg}.
\end{myproof}
\begin{theorem}\label{th:mixedisnotclassical}
Every non-pure mixed consequence relation admits regular connectives with no counterpart in classical logic (as well as all regular connectives from classical logic).
\end{theorem}
\begin{myproof}
Consider a non-pure mixed consequence relation based on $\mathrel|\joinrel\equiv_{\mathcal{D}_p,\mathcal{D}_c}$. If it is non-pure, we can find an element that is in one of the set of designated values and not in the other, call it $\alpha$. The $0$-ary connective with constant value $\alpha$ is a regular connective, that does not correspond to a regular connective in classical logic: let us show this by showing the regularity rules corresponding to the two possibilities for $\overline{\alpha}$, and note that they are not possible regularity rules for $0$-ary connectives in classical logic (see Example~\ref{regrulesclassicalconnectives})
\begin{itemize}
\item $\alpha\in\mathcal{D}_p\setminus\mathcal{D}_c$.
($\Gamma, \overline{\alpha} \vdash \Delta$ iff $\Gamma \vdash \Delta$)
\phantom{\emph{true}}
($\Gamma \vdash \overline{\alpha}, \Delta$ iff $\Gamma \vdash \Delta$)
\item $\alpha\in\mathcal{D}_c\setminus\mathcal{D}_p$.
($\Gamma, \overline{\alpha} \vdash \Delta$ iff \emph{true})
$\phantom{\Gamma\vdash\Delta}$
($\Gamma \vdash \overline{\alpha}, \Delta$ iff \emph{true})
\qedhere
\end{itemize}
\end{myproof}
One may thus wonder whether the converse of Theorem~\ref{th:pureisclassical} would hold, yielding a powerful characterization of all pure consequence relations from the set of regular connectives they admit, as in Conjecture~\ref{conjecture:pureconsclass}. But this conjecture is false and the counter-example has already been mentioned, as one of the intersective mixed consequence relations that admit a conditional in 4-valued logics:
\begin{conjecture}[false]\label{conjecture:pureconsclass}
Every relation which admits exactly the same regular connectives as classical logic is a pure consequence relation.
\end{conjecture}
\begin{counterexample}
The 4-valued logic based on the truth-relation $\mathrel|\joinrel\equiv_{\{1,\#_1\},\{1,\#_2\}}\cap\mathrel|\joinrel\equiv_{\{1,\#_2\},\{1,\#_1\}}$ admits exactly the same regular connectives as classical logic.
\end{counterexample}
\begin{myproof}
First, it admits all the connectives from classical logic, because it admits a G-conditional (see Theorem~\ref{th:conditionalissufficient}). Conversely, assume that $C$ is a regular connective for this logic, we will show that its restriction to classical inputs is classical (what we called `weakly bivalent' earlier) and therefore, by Theorem~\ref{th:bivconnarereg}, follows the same regularity rules. Consider conclusion regularity rules of the usual form, and apply Theorem~\ref{th:truthconstraintforregconn} to $C$ and each of the members in the intersection forming the truth-relation:
\[\begin{array}{c@{\quad\textrm{ iff }\quad}c}
\underline{C}(x_1, ..., x_n) \in\{1,\#_2\}
&
\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^c}
{\{x_i: i\in B_p\}\subseteq\{1,\#_1\} \Rightarrow \{x_i: i\in B_c\}\cap\{1,\#_2\}\not=\emptyset}
\\
\underline{C}(x_1, ..., x_n) \in\{1,\#_1\}
&
\bigwedge\limits_{(B_p,B_c)\in \mathcal{B}^c}
{\{x_i: i\in B_p\}\subseteq\{1,\#_2\} \Rightarrow \{x_i: i\in B_c\}\cap\{1,\#_1\}\not=\emptyset}
\\
\end{array}\]
\noindent
Consider classical inputs $x_1, ..., x_n$ for now. We can drop the indeterminate values from the right-hand sides. The two right-hand sides from above become equivalent, and we thus obtain that ($\underline{C}(x_1, ..., x_n)\in\{1,\#_1\}$ iff $\underline{C}(x_1, ..., x_n)\in\{1,\#_2\}$), that is, $\underline{C}(x_1, ..., x_n)\in\{1,0\}$.
\end{myproof}
\subsection{Order-theoretic relations in $N$-valued logics}
\label{sec:ordertheoreticresults}
In 3-valued logics, the order-theoretic relation $ss\cap tt$ stands out as one that does not admit a G-regular negation or conditional. What about order-theoretic relations in general? We start by capitalizing on the previous section about pure consequence relations.
\begin{theorem}
Every regular connective for an order-theoretic relation shares a regularity rule with a connective from classical logic.
\end{theorem}
\begin{myproof}
Theorem~\ref{th:pureisclassical} shows that pure consequence relations only satisfy classical regularity rules,
Theorem~\ref{th:ordertheoric=intersectionpure} shows that order-theoretic relations are intersections of pure consequence relations,
and Corollary~\ref{cor:intersectioncoincideR} shows that for an intersection of truth-relations to admit a connective satisfying some regularity rule, all its members should have a connective satisfying this regularity rule.
\end{myproof}
However, for the restricted set of G-connectives we have been interested in, we can offer a complete generalization for every $N\geq 3$, regardless of whether the semantics is maximally expressive (as soon as it is atomic expressive).
\begin{theorem}\label{th:fullordertheoretic}
Consider $N$-valued logics with $N\geq 3$, and assume at least atomic expressiveness.
An order-theoretic relation has no G-negation and no G-conditional.
It has a G-conjunction and a G-disjunction if and only if it is a total order-theoretic relation.
\end{theorem}
\begin{myproof}
All elements of the proof work in the same way. For each connective $C$, we start by using the fact that the order-theoretic relation is reflexive, and state for atomic propositions that $C(p_1,p_2,...)\vdash C(p_1,p_2,...)$. Then we apply first the premise-regularity rule for $C$, and then the conclusion-regularity rule for $C$. The two together constrain the values that $C$ can take.
\begin{itemize}
\item Assume that there is a G-negation $\neg$. By reflexivity, for every atomic formula $p$: $\neg p \vdash \neg p$. The following two consequences are incompatible.
\begin{itemize}
\item Premise-regularity rule: $\vdash p, \neg p$. So, $\forall v: \mathrel|\joinrel\equiv v(p), v(\neg p)$. Choosing $v$ such that $v(p)$ is an indeterminate $\#_i$, it follows that $\neg \#_i=1$.
\item Conclusion-regularity rule: $p, \neg p\vdash $. So, $\forall v: v(p), v(\neg p) \mathrel|\joinrel\equiv $. Choosing the same $v$ as above, it follows that $\neg \#_i=0$.
\end{itemize}
\item Assume that there is a G-conditional $\to$. (We would not need this part of the proof in a constant expressive setting, for the absence of a G-negation would guarantee the absence of a G-conditional). By reflexivity, for all atomic formulae $p, q$: $(p\to q) \vdash (p\to q)$. The following two consequences are incompatible.
\begin{itemize}
\item Premise-regularity rule: $\vdash p, (p\to q)$ and $q \vdash (p\to q)$.
So, in particular, $\forall v: \mathrel|\joinrel\equiv v(p), v(p\to q)$. Choosing $v$ such that $v(p)$ is an indeterminate $\#_i$ and $v(q)=0$, it follows that $v(\#_i\to 0)=1$.
\item Conclusion-regularity rule: $p, (p\to q) \vdash q$.
So, $\forall v: v(p), v(p\to q)\mathrel|\joinrel\equiv v(q)$. Choosing the same $v$ as above, it follows that $v(\#_i\to 0)=0$.
\end{itemize}
\item Assume that there is a G-conjunction $\wedge$. By reflexivity, for all atomic formulae $p, q$: $(p\wedge q) \vdash (p\wedge q)$. The following two consequences are incompatible.
\begin{itemize}
\item Premise-regularity rule: $p, q \vdash (p\wedge q)$.
For any two indeterminates $\#_i$ and $\#_j$, choose $v_{i,j}$ such that $v_{i,j}(p)=\#_i$ and $v_{i,j}(q)=\#_j$. It follows that
$v_{i,j}(p), v_{i,j}(p) \mathrel|\joinrel\equiv v_{i,j}(p\wedge q)$.
That is $\#_i \leq v_{i,j}(p\wedge q)$ or $\#_j \leq v_{i,j}(p\wedge q)$.
\item Conclusion-regularity rule: $(p\wedge q) \vdash p$ and $(p\wedge q) \vdash q$.
For any two indeterminates $\#_i$ and $\#_j$, choose $v_{i,j}$ such that $v_{i,j}(p)=\#_i$ and $v_{i,j}(q)=\#_j$. It follows that
$v_{i,j}(p\wedge q) \mathrel|\joinrel\equiv v_{i,j}(p)$ and $v_{i,j}(p\wedge q) \mathrel|\joinrel\equiv v_{i,j}(q)$.
That is $v_{i,j}(p\wedge q) \leq \#_i$ and $v_{i,j}(p\wedge q) \leq \#_j$.
\end{itemize}
Summarizing, there is an $x$ such that ($\#_i\leq x$ or $\#_j\leq x$) and ($x\leq\#_i$ and $x\leq\#_j$).
It follows that ($\#_i\leq\#_j$ or $\#_j\leq\#_i$), that is, any two indeterminates are ordered.
\item The proof for G-disjunctions is analogous.
\qedhere
\end{itemize}
\end{myproof}
Order-theoretic relations thus show a very stable behavior across the number of truth values and the expressive resources of the language. In short, they do not admit G-conditionals or G-negations, and they rarely admit G-conjunctions or G-disjunctions.
\section{Algebraic characterization of consequence relations admitting G-conditionals}\label{sec:N}
In the previous sections, we have looked at particular (kinds of) consequence relations (3- or 4-valued logics, pure or order-theoretic consequence relations), and asked whether they admitted certain G-connectives. Here, we show that one may start from a G-connective, and seek a general algebraic characterization of the conditions under which a given intersective mixed relation admits this particular G-connective, for all $N$. We proceed with G-disjunctions and G-conjunctions, then examine G-negations to finally get at G-conditionals. Let us stress from the outset that the algebraic characterizations we obtain are simple for disjunctions and conjunctions, but not so for conditionals, even though we thought that conditionals were the best candidate for a connective to closely resemble its consequence relation.
\subsection{Disjunctions}
\begin{definition}[disjunction-compatible]\label{def:disjunctioncompatible}
A list of sets of designated values $\mathcal{D}_1, ..., \mathcal{D}_n$ is called \emph{disjunction-compatible} if
for all truth values $x$ and $y$, there is a truth value $z$ that belongs to all the sets of designated values to which $x$ belongs, all the sets of designated values to which $y$ belongs, and to no other set of designated values.
\end{definition}
\begin{theorem}\label{th:disjunctioncondition}
The following statements are equivalent:
\begin{itemize}
\item The truth-relational consequence relation $\vdash$ admits a G-disjunction in a constant expressive setting.
\item All the minimal representations of $\mathrel|\joinrel\equiv$ are based on a list of sets of designated values which is disjunction-compatible.
\item One of the representations of $\mathrel|\joinrel\equiv$ is based on a list of sets of designated values which is disjunction-compatible.
\end{itemize}
\end{theorem}
\begin{myproof}
Corollary~\ref{th:formcond} says that a connective $C$ will be a G-disjunction iff its truth-function $f$ is such that for all sets of designated values taking part in a minimal representation, and for all truth-values:
$f(x,y)\in\mathcal{D}$ iff $x\in\mathcal{D}$ or $y\in\mathcal{D}$.
Disjunction compatibility is equivalent to the possibility to define such a function.
\end{myproof}
\subsection{Conjunctions}
\begin{definition}[conjunction-compatible]
A list of sets of designated values $\mathcal{D}_1, ..., \mathcal{D}_n$ is called \emph{conjunction-compatible} if
for all truth values $x$ and $y$, there is a truth value $z$ that belongs to all the sets of designated values to which both $x$ and $y$ belong, and to no other.
\end{definition}
\begin{theorem}\label{th:conjunctioncondition}
The following statements are equivalent:
\begin{itemize}
\item The truth-relational consequence relation $\vdash$ admits a G-conjunction in a constant expressive setting.
\item All the minimal representations of $\mathrel|\joinrel\equiv$ are based on a list of sets of designated values which is conjunction-compatible.
\item One of the representations of $\mathrel|\joinrel\equiv$ is based on a list of sets of designated values which is conjunction-compatible.
\end{itemize}
\end{theorem}
\begin{myproof}
Corollary~\ref{th:formcond} says that a connective $C$ will be a G-conjunction iff its truth-function $f$ is such that for all sets of designated values taking part in a minimal representation, and for all truth-values:
$f(x,y)\in\mathcal{D}$ iff $x\in\mathcal{D}$ and $y\in\mathcal{D}$.
Disjunction compatibility is equivalent to the possibility to define such a function.
\end{myproof}
\subsection{Conjunction and disjunction}
Interestingly, the conditions on conjunctions and disjunctions only depend on the list of sets of designated values, independently of whether they are repeated, or whether they are premise sets or conclusion sets. We can describe the situation differently, seeing that what is needed is, roughly, that the sets are pairwise distinct, as well as maximally intersecting with one another.
\begin{definition}[disjunction-conjunction compatible]\label{def:disjconjcompatibility}
A list of sets of designated values $\mathcal{D}_1, ..., \mathcal{D}_n$ is called \emph{disjunction-conjunction-compatible} if
\begin{description}
\item[DC1:] For each set $\mathcal{D}_i$, there is a truth value that belongs only to $\mathcal{D}_i$, or $\mathcal{D}_i$ is included in another, distinct set of designated values.
\item[DC2:]
For all non-empty sublists of sets of designated values: $\mathcal{D}'_1, ..., \mathcal{D}'_{n'}$, there is an element $x$ that belongs to all of these sets ($\forall i', x\in\mathcal{D}'_{i'}$), as well as to any set $\mathcal{D}_i$ in which one of the $\mathcal{D}'_{i'}$ would be fully included, but to no other of the original sets of designated values $\mathcal{D}_i$s. In other words, $x\in\mathcal{D}_i$ iff $\exists i': \mathcal{D}'_{i'}\subseteq\mathcal{D}_i$.
\end{description}
\end{definition}
\begin{theorem}\label{char:disj/conj}
The following statements are equivalent:
\begin{itemize}
\item The truth-relational consequence relation $\mathrel|\joinrel\equiv$ admits a G-disjunction and a G-conjunction in a constant expressive setting.
\item All minimal representations of $\mathrel|\joinrel\equiv$ are based on a list of sets of designated values which is disjunction-conjunction compatible.
\item One of the representations of $\mathrel|\joinrel\equiv$ is based on a list of sets of designated values which is disjunction-conjunction-compatible.
\end{itemize}
\end{theorem}
\begin{myproof}
Firstly, suppose that the relation admits a G-disjunction and a G-conjunction, then any minimal representation is based on disjunction-conjunction-compatible sets of designated values:
\begin{itemize}
\item The first condition DC1 follows from there being a conjunction (no need for a disjunction). Consider $x_1$, the conjunction of all truth values in $\mathcal{D}_1$ (no matter the order in which the conjunction is taken, which may vary given that the conjunction is a binary operator here):
$x_1=(x_1^1 \wedge (x_1^2 \wedge ( ... \wedge x_1^{n_1})...)$, if $\mathcal{D}_1$ has $n_1$ elements $x_1^1, ..., x_1^{n_1}$.
Clearly, $x_1$ belongs to $\mathcal{D}_1$ (as a conjunction of elements that do).
If $x_1$ also belongs to $\mathcal{D}_2$, then all the $x_1^1, ..., x_1^{n_1}$ also belong to $\mathcal{D}_2$ (for the same reason).
\item As for the second condition DC2, pick a list of sets of designated values, consider the disjunction of the elements made of the conjunctions of all elements in each of these sets. That element satisfies the constraint DC2.
\end{itemize}
Secondly, suppose that the third statement above is true: there is a representation {of $\mathrel|\joinrel\equiv$} based on a disjunction-conjunction compatible list of sets of designated values.
The condition DC2 by itself guarantees disjunction-compatibility / conjunction-compatibility: for any two truth values $x$ and $y$,
pick the list of sets of designated values to which one or the other or both belong (if any), and you can construct the necessary $z$ for disjunction-compatibility,
pick the list of sets of designated values to which both belong (if any), and you can construct the necessary $z$ for conjunction-compatibility.
\end{myproof}
\subsection{Negations}
For G-negation, we simply state a set of necessary conditions, which will prove useful later on:
\begin{definition}[negation-necessity]\label{def:negationnecessary}
A representation $\mathrel|\joinrel\equiv_{\mathcal{D}_p^1,\mathcal{D}_c^1} \cap ... \cap \mathrel|\joinrel\equiv_{\mathcal{D}_p^K,\mathcal{D}_c^K}$ satisfies \emph{negation-necessity} if
\begin{description}
\item[N1] There is no inclusion between two $\mathcal{D}_p$s or two $\mathcal{D}_c$s.
\item[N2] If there is a superset/subset relation between, a $\mathcal{D}_p^i$ and a $\mathcal{D}_c^j$, then there is the same superset/subset relation between $\mathcal{D}_p^j$ and $\mathcal{D}_c^i$.
\end{description}
\end{definition}
\begin{theorem}\label{char:negation:necessary}
If an intersective mixed relation admits a G-negation, then all of its minimal representations satisfy negation-necessity.
\end{theorem}
\begin{myproof}
Assume there is a G-negation $\neg$. We will make use of Theorem~\ref{th:truthconstraintforregconn} applied to negation: ($x\in\mathcal{D}_p^i$ iff $\neg x\not\in\mathcal{D}_c^i$) and ($x\in\mathcal{D}_c^i$ iff $\neg x\not\in\mathcal{D}_p^i$).
\begin{description}
\item[For N1:] Suppose $\mathcal{D}_p^1\subseteq\mathcal{D}_p^2$. Then pick $x\in\mathcal{D}_c^2$. Then $\neg x\not\in\mathcal{D}_p^2$. Hence, $\neg x\not\in\mathcal{D}_p^1$. Hence, $x\in\mathcal{D}_c^1$. So, $\mathcal{D}_p^1\subseteq\mathcal{D}_p^2$, and $\mathcal{D}_c^2\subseteq\mathcal{D}_c^1$, we can drop $\mathrel|\joinrel\equiv_{\mathcal{D}_p^1,\mathcal{D}_c^1}$ from the representation, contradicting minimal representation.
\item[For N2:] Suppose, for instance, that $\mathcal{D}_p^1 \subseteq \mathcal{D}_c^2$ (similar proof for $\mathcal{D}_c^1 \subseteq \mathcal{D}_p^2$.). Pick $x\in\mathcal{D}_p^2$.
Then $\neg x\not\in\mathcal{D}_c^2$.
Hence, $\neg x\not\in\mathcal{D}_p^1$.
Hence, $x\in\mathcal{D}_c^1$ and, abstracting away from the initial choice of $x$, $\mathcal{D}_p^2\subseteq\mathcal{D}_c^1$.
\qedhere
\end{description}
\end{myproof}
\subsection{Conditionals}
Seeking necessary and sufficient conditions for the existence of a G-conditional, we could start from the form a G-conditional ought to have, as described in Theorem~\ref{th:formcond}. Instead, we capitalize on the fact that allowing for a G-conditional is equivalent to allowing for any of the following set of G-connectives (see Theorem~\ref{th:combinationsforconditionals}):
(a G-negation and a G-disjunction),
(a G-negation and a G-conjunction),
(a G-negation, a G-disjunction and a G-conjunction).
As a consequence, we can combine the previous exploration of conditions of existence for a G-disjunction and a G-conjunction (Corollary~\ref{char:disj/conj}) and of conditions for a G-negation (Theorem~\ref{char:negation:necessary}), and obtain necessary and sufficient conditions for the existence of a G-conditional:
\begin{theorem}\label{theorem:fullalgcond}
An intersective mixed relation admits a G-conditional iff all of its minimal representations satisfy disjunction-conjunction-compatibility (Definition~\ref{def:disjconjcompatibility}) and negation-necessity (Definition~\ref{def:negationnecessary}). We repeat these conditions here:
\begin{description}
\item[DC1:] For each set $\mathcal{D}_i$, there is a truth value that belongs only to $\mathcal{D}_i$, or $\mathcal{D}_i$ is included in another, distinct set of designated values.
\item[DC2:]
For all non-empty sublists of sets of designated values: $\mathcal{D}'_1, ..., \mathcal{D}'_{n'}$, there is an element $x$ that belongs to all of these sets ($\forall i', x\in\mathcal{D}'_{i'}$), as well as to any set $\mathcal{D}_i$ in which one of the $\mathcal{D}'_{i'}$ would be fully included, but to no other of the original sets of designated values $\mathcal{D}_i$s. In other words, $x\in\mathcal{D}_i$ iff $\exists i': \mathcal{D}'_{i'}\subseteq\mathcal{D}_i$.
\item[N1] There is no inclusion between two $\mathcal{D}_p$s or two $\mathcal{D}_c$s.
\item[N2] If there is a superset/subset relation between, a $\mathcal{D}_p^i$ and a $\mathcal{D}_c^j$, then there is the same superset/subset relation between $\mathcal{D}_p^j$ and $\mathcal{D}_c^i$.
\end{description}
\end{theorem}
\begin{myproof}
These conditions are necessary and sufficient for the existence of a G-disjunction and a G-conjunction by Theorem~\ref{char:disj/conj}, and necessary for the existence of a G-negation by Theorem~\ref{char:negation:necessary}. To prove that they are plainly necessary and sufficient for the existence of a G-conditional, we thus need to prove that they are sufficient for the existence of a G-negation.
Assuming that the conditions are met, we will show that we can meet the requirements imposed by Theorem~\ref{char:negation:necessary} for a G-negation, namely that for all $x$, there is $y$ such that
($y \in \mathcal{D}_c^i$ iff $x \not\in \mathcal{D}_p^i$)
and
($y \in \mathcal{D}_p^i$ iff $x \not\in \mathcal{D}_c^i$).
Consider a minimal representation of the intersective mixed relation and let $x$ be a truth value. We would like to find $y$ satisfying the conditions above.
The desiderata then is that $y$ belongs to a given list of sets of designated values $\mathcal{D}_1, ..., \mathcal{D}_n$. Consider $y=\alpha_1 \vee ... \vee \alpha_n$ (the order for the disjunction does not matter), with $\alpha_k$ being the conjunction of all elements in $\mathcal{D}_k$. Surely, $y$ is in all the relevant sets: each $\alpha_k$ is in $\mathcal{D}_k$ (as a conjunction of elements that are), and their disjunction $y$ is therefore in all $\mathcal{D}_k$s (because at least one element in the disjunction is).
Now, conversely, we should prove that $y$ does not belong to an unwanted set of designated values. Suppose $y$ belongs to some set of designated values, let us say a premise set $\mathcal{D}_p^*$ (for a conclusion-set, the proof would proceed in the same manner). Then, there is an $\alpha_k$ such that $\alpha_k\in\mathcal{D}_p^*$ (because of the disjunctive nature of $y$). Hence, $\mathcal{D}_k\subseteq\mathcal{D}_p^*$ (because of the conjunctive nature of $\alpha_k$). One option is that $\mathcal{D}_k$ is $\mathcal{D}_p^*$, which is fine for our needs for $y$. If not, by N1 from Definition \ref{def:negationnecessary}, it follows that $\mathcal{D}_k$ is a conclusion-set in the representation, call it $\mathcal{D}_c^1$. So we have $\mathcal{D}_c^1\subseteq\mathcal{D}_p^*$, and by N2 from Definition \ref{def:negationnecessary}, we also have $\mathcal{D}_c^*\subseteq\mathcal{D}_p^1$.
Now, the reason why $\mathcal{D}_c^1$, aka $\mathcal{D}_k$ was in the list of desiderata was that $x\not\in\mathcal{D}_p^1$, from which it now follows that $x\not\in\mathcal{D}_c^*$, and it is therefore fine that $y$ belongs to $\mathcal{D}_p^*$. That is, we have proved that any $\mathcal{D}^*$ to which $y$ belongs is actually one in the list of the $\mathcal{D}_1,..., \mathcal{D}_n$.
\end{myproof}
Theorem~\ref{theorem:fullalgcond} offers a complete, if not simple, algebraic characterization of the truth-relations allowing for G-conditionals. Let us say again that, with constant expressiveness, these are also conditions under which a truth-relation admits regular connectives corresponding to all classical truth-functions.
We have seen that all mixed consequence relations do admit all classical connectives; some (properly) intersective mixed relations do as well, but not all, and here we can tease apart those that do from those that don't.
If Gentzen-regularity can be motivated as a criterion of logicality for a set of connectives, by extension this result may be used in favor of the logicality of mixed consequence relations, as well as all others which may admit a conditional. In previous work (\citealt{chemla2017charac}), we had
{treated order-theoretic relations as equally worthy of being called logical as pure consequence relations}, but based on criteria that did not take into consideration the interaction of logical consequence relations with sentential connectives, namely on the fact that {order-theoretic relations are} reflexive and transitive. In light of our results, the lack of an internalizing conditional for order-theoretic relations may be used to single those out: they satisfy classical structural properties such as reflexivity and transitivity, but the non-classical truth values may be used to allow or prevent the existence of certain connectives, chief among which are conditionals.
\section{Conclusions and open issues}
We have presented general results concerning the definability of Gentzen-regular connectives for intersective mixed consequence relations. In the 3-valued and 4-valued case, moreover, we can enumerate which intersective relations admit Gentzen-regular conjunctions, disjunctions, negations, and conditionals, and exhibit the truth-tables for those operators. For classical logic and some $4$-valued logics, and in $N$-valued logics for mixed and pure consequence relations as well as for order-theoretic consequence relations, we have offered general results that inform us either about these logics, or about the Gentzen-regular connectives.
The value of those results is not just combinatorial. They are a step toward a better understanding of the nature of logic and the way in which consequence relations and logical constants interact. To buttress this point, we hereby mention further directions in which the present framework, its results and its methods, may be fruitfully extended:
\begin{itemize}
\item What happens with other Gentzen-regular connectives? We have explored in depth the situation for four types of binary connectives {but there are in principle much more of them (a subset of the $4^{4 \times 4}$ of all truth-functional connectives in the 4-valued logic described in Section \ref{sec:reg=4value}).} If all of these connectives turn out to be equally important, then the current exploration should be pursued. Alternatively, one may find reasons to focus on a subclass of these connectives: the set we looked at is familiar, albeit arbitrary. One may investigate the whole set of connectives in classical logic for instance, or some other subset of connectives defined by putting algebraic constraints on what may count as a regularity rule (in fact, classical connectives are so constrained because premise and conclusion regularity rules are essentially the conjunctive normal forms of two formulae that are negations from one another, see the proof of Theorem~\ref{th:classicalallregular}). Our hope is that if some subclass of Gentzen regular connectives is found to be important, it should be possible to reproduce our analysis and study this subclass systematically, possibly in a computer-aided fashion as we have proposed.
\item Similarly, we have focussed attention on determining what class of logics admit a particular G-connective, e.g., a conjunction or a conditional. One can ask a broader question: what class of logics admit a particular \emph{set} of G-connectives? For this reason, in the (false) Conjecture~\ref{conjecture:pureconsclass} we asked whether pure consequence relations could be characterized as the relations which admit classical regular connectives and no other regular connectives.
From the $4^{4^n}$ possible G-connectives of arity $n$, not all combinations are possible (e.g., when there is a G-conditional, there is a G-negation).
One can raise a functional completeness problem: which subset is possible (we know they would have to be clones, see \citealp{KERKHOFF2014107})? More relevant to our current enterprise, what rank is needed to admit exactly a given subset: to obtain all regular connectives, a 4-valued logic is sufficient, what about various proper subsets of all the Gentzen regular connectives? We have seen that the first logic with a G-negation but no G-conditional is found with $N=5$; this pushes the limit of previous results, which tried to reduce the minimal number of truth-values to just $4$ (\citealt{chemla2018suszko}).
\item Conversely, what happens when Gentzen-regularity is relaxed? The methods here employed can serve to refine the conditions put on logical connectives. While Gentzen-regularity is a natural criterion on the interplay between a connective and a consequence relation, for some applications we may want to investigate which consequence relations admit conditional operators satisfying weaker constraints (for example: only the full deduction theorem), or specific constraints (not necessarily implied by Gentzen-regularity).
\item What happens in settings with limited expressiveness? In this paper, we have assumed logics that are expressively complete. We have shown that when expressive completeness is relaxed the admissible Gentzen-regular connectives stay the same in 3-valued logics and, in fact, for all $N$-valued order theoretic relations. However, it remains an open question to discover how different the situation will be if we consider \emph{all} intersective mixed relations.
\item What happens if we move to settings that are single-conclusion, or single-premise? We have seen that, with three truth values already, things change in a single-conclusion setting, in particular $ss\cap tt$ admits a conditional satisfying at least part of the full Gentzen-regularity condition. Since multi-conclusion inference is sometimes viewed with suspicion (\citealt{steinberger2011conclusions}), it is natural to explore the issues we have covered in that setting. {Note, however, that this may require adjusting the regularity rules, which often rely on the possibility to add up conclusions.}
\item
Finally, while we have started from consequence relations and looked for associated G-connectives, one may consider the reverse problem: given a Gentzen-regular sentential operator, which consequence relations admit it (see Theorem~\ref{th:fromcondtoLC} for a statement of unicity but only under some conditions)?
The question is of importance to determine how tightly a conditional operator is associated to a specific consequence relation.
\end{itemize}
\end{document} |
\begin{document}
\title{Optimized dynamical control of state transfer through noisy spin
chains}
\author{Analia Zwick, Gonzalo A. Álvarez, Guy Bensky and Gershon Kurizki}
\address{Weizmann Institute of Science, Rehovot 76100, Israel}
\begin{abstract}
We propose a method of optimally controlling the tradeoff of speed
and fidelity of state transfer through a noisy quantum channel (spin-chain).
This process is treated as qubit state-transfer through a fermionic
bath. We show that dynamical modulation of the boundary-qubits levels
can ensure state transfer with the best tradeoff of speed and fidelity.
This is achievable by dynamically optimizing the transmission spectrum
of the channel. The resulting optimal control is robust against both
static and fluctuating noise in the channel's spin-spin couplings.
It may also facilitate transfer in the presence of diagonal disorder
(on site energy noise) in the channel.
\end{abstract}
\maketitle
One dimensional (1D) chains of spin-$\frac{1}{2}$ systems with nearest-neighbor
couplings, nicknamed spin chains, constitute a paradigmatic quantum
many-body system of the Ising type \cite{ising_beitrag_1925}. As
such, spin chains are well suited for studying the transition from
quantum to classical transport and from mobility to localization of
excitations as a function of disorder and temperature \cite{kramer_localization:_1993}.
In the context of quantum information (QI), spin chains are envisioned
to form reliable quantum channels for QI transmission between nodes
(or blocks) \cite{bose_quantum_2003,Bose_review_2007}. Contenders
for the realization of high-fidelity QI transmission are spin chains
comprised of superconducting qubits \cite{lyakhov_quantum_2005,majer_coupling_2007},
cold atoms \cite{duan_controlling_2003,hartmann_effective_2007,fukuhara_quantum_2013,simon_quantum_2011},
nuclear spins in liquid- or solid-state NMR \cite{madi_time-resolved_1997,doronin_multiple-quantum_2000,zhang_simulation_2005,zhang_iterative_2007,cappellaro_dynamics_2007,rufeil-fiori_effective_2009,alvarez_perfect_2010,ajoy_algorithmic_2012},
quantum dots \cite{petrosyan_coherent_2006}, ion traps \cite{lanyon_universal_2011,blatt_quantum_2012}
and nitrogen-vacancy (NV) centers in diamond \cite{cappellaro_coherence_2009,neumann_quantum_2010,yao_scalable_2012,ping_practicality_2013}.
The distribution of coupling strengths between the spins that form
the quantum channel, determines the state transfer-fidelities \cite{bose_quantum_2003,zwick_quantum_2011,christandl_perfect_2005,karbach_spin_2005,kay_perfect_2006,kay_review_2010}.
Perfect state-transfer (PST) channels can be obtained by precisely
engineering each of those couplings \cite{christandl_perfect_2004,Albanese_mirror_2004,christandl_perfect_2005,karbach_spin_2005,kay_perfect_2006,paternostro_perfect_2008,kay_review_2010,zwick_robustness_2011}.
Such engineering is however highly challenging at present, being an
unfeasible task for long channels that possess a large number of control
parameters and are increasingly sensitive to imperfections as the
number of spins grows \cite{Alvarez_NMR_2010,zwick_robustness_2011,zwick_spin_2012,Zwick_Chapt_2013}.
A much simpler control may involve \textit{only} the boundary (source
and target) qubits that are connected via the channel. Recently, it
has been shown that if the boundary qubits are weakly-coupled to a
uniform (homogeneous) channel (\textit{i.e.}, one with identical couplings),
quantum states can be transmitted with arbitrarily high fidelity at
the expense of increasing the transfer time \cite{wojcik_unmodulated_2005,wojcik_multiuser_2007,Venuti_Qubit_2007,Venuti_Long-distance_2007,Giampolo_entanglement_2009,Giampaolo_Long-distance_2010,yao_robust_2011,zwick_spin_2012}.
Yet such slowdown of the transfer may be detrimental because of omnipresent
decoherence.
To overcome this problem, we here propose a hitherto unexplored approach
for optimizing the tradeoff between fidelity and speed of state-transfer
in quantum channels. This approach employs temporal modulation of
the couplings between the boundary qubits and the rest of the channel.
This kind of control has been considered before for a different purpose,
namely to implement an effective optimal encoding of the state to
be transferred \cite{Haselgrove_Optimal_2005}. Instead, we treat
this modulation as dynamical control of the boundary system which
is coupled to a fermionic bath that is treated as a source of noise.
The goal of our modulation is to realize an optimal spectral filter
\cite{clausen_bath-optimized_2010,clausen_task-optimized_2012,escher_optimized_2011,bensky_optimizing_2012,petrosyan_reversible_2009,gordon_universal_2007,gordon_optimal_2008,kofman_universal_2001,kofman_unified_2004}
that blocks transfer via those channel eigenmodes that are responsible
for noise-induced leakage of the QI \cite{wu_master_2009}. We show
that under optimal modulation, the fidelity and the speed of transfer
can be improved \textit{by several orders of magnitude}, and the fastest
possible transfer is achievable (for a given fidelity).
Our approach allows to reduce the complexity of a large system to
that of a simple and small open system where it is possible to apply
well developed tools of quantum control to optimize state transfer
with few universal control requirements on the source and target qubits.
In this picture, the complexity of the channel is simply embodied
by correlation functions in such a way that we obtain a universal,
simple, analytical expression for the optimal modulation. While in
this article we optimize the tradeoff between speed and fidelity so
as to avoid decoherence as much as possible, this description \cite{clausen_bath-optimized_2010,clausen_task-optimized_2012,escher_optimized_2011,bensky_optimizing_2012,petrosyan_reversible_2009,gordon_universal_2007,gordon_optimal_2008,kofman_universal_2001,kofman_unified_2004,wu_master_2009}
allows one to actively suppress decoherence and dissipation in a simple
manner, since it may be viewed as a generalization of dynamical decoupling
protocols \cite{Viola_Dynamical_1998,viola_dynamical_1999,Viola_RobustDD_2003,Lidar_QDynDec_2005}.
In what follows, we explicitly deal with a spin-chain quantum channel,
but point out that our control may be applicable to a broad variety
of other quantum channels.
\section{\label{sec:Quantum channel and state transfer fidelity}Quantum channel
and state transfer fidelity}
\subsection{Hamiltonian and boundary control}
We consider a chain of $N\!+\!2$ spin-$\frac{1}{2}$ particles with
XX interactions between nearest neighbors, which is a candidate for
a variety of state-transfer protocols \cite{bose_quantum_2003,Bose_review_2007,lyakhov_quantum_2005,majer_coupling_2007,duan_controlling_2003,hartmann_effective_2007,fukuhara_quantum_2013,simon_quantum_2011,madi_time-resolved_1997,zhang_simulation_2005,zhang_iterative_2007,cappellaro_dynamics_2007,rufeil-fiori_effective_2009,doronin_multiple-quantum_2000,ajoy_algorithmic_2012,alvarez_perfect_2010,petrosyan_coherent_2006,lanyon_universal_2011,blatt_quantum_2012,cappellaro_coherence_2009,neumann_quantum_2010,yao_scalable_2012,ping_practicality_2013,zwick_quantum_2011,christandl_perfect_2005,karbach_spin_2005,kay_perfect_2006,kay_review_2010,zwick_robustness_2011,christandl_perfect_2004,Albanese_mirror_2004,paternostro_perfect_2008}.
The Hamiltonian is given by
\begin{equation}
H=H_{0}+H_{bc}(t),\label{eq:hamiltonian}
\end{equation}
\begin{equation}
H_{0}=\sum_{i=1}^{N-1}\frac{J_{i}}{2}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}\right),\: H_{bc}(t)=\alpha(t)\sum_{i\in\{0,N\}}\frac{J_{i}}{2}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}\right),
\end{equation}
where $H_{0}$ and $H_{bc}$ stand for the chain and boundary-coupling
Hamiltonians, respectively, $\sigma_{i}^{x(y)}$ are the appropriate
Pauli matrices and $J_{i}$ are the corresponding exchange-interaction
couplings.
\subsection{Mapping to a few-body open-quantum system}
The magnetization-conserving Hamiltonian $H$ can be mapped onto a
non-interacting fermionic Hamiltonian \cite{lieb_two_1961} that has
the particle-conserving form
\begin{equation}
H_{0}=\sum_{i=1}^{N-1}\frac{J_{i}}{2}\left(c_{i}^{\dagger}c_{i+1}+c_{i}c_{i+1}^{\dagger}\right),\: H_{bc}(t)=\alpha(t)\sum_{i\in\{0,N\}}\frac{J_{i}}{2}\left(c_{i}^{\dagger}c_{i+1}+c_{i}c_{i+1}^{\dagger}\right),
\end{equation}
where $c_{j}=\frac{1}{2}e^{i\frac{\pi}{4}\sum_{0}^{j-1}\sigma_{i}^{+}\sigma_{i}^{-}}\sigma_{j}^{-}$
create a fermion at site $j$ and $\sigma^{\pm}=\sigma^{x}\pm i\sigma^{y}$.
The Hamiltonian $H_{0}$ can be diagonalized as $H_{0}=\sum_{k=1}^{N}\omega_{k}b_{k}^{\dagger}b_{k}$,
where $b_{k}^{\dagger}=\sum_{j=1}^{N}\langle j|\omega_{k}\rangle c_{j}^{\dagger}$
populates a single-particle fermionic eigenstate $\vert\omega_{k}\rangle$
of energy $\omega_{k}$, and $\vert j\rangle=\vert0..01_{j}0..0\rangle$
denote the single-excitation subspace. Under the assumption of mirror
symmetry of the couplings with respect to the source and target qubits
$J_{i}=J_{N-i}$, the energies $\omega_{k}$ are not degenerate, $\omega_{k}<\omega_{k+1}$,
and the eigenvectors have a definite parity that alternates as $\omega_{k}$
increases \cite{karbach_spin_2005}. This property implies that $\langle j|\omega_{k}\rangle=(-1)^{k-1}\langle N-j+1|\omega_{k}\rangle$
and allows us to rewrite the boundary-coupling Hamiltonian as
\begin{equation}
H_{bc}(t)=\alpha(t)J_{0}c_{0}^{\dagger}\underset{k=1}{\overset{N}{\sum}}\langle1|\omega_{k}\rangle b_{k}+\alpha(t)J_{N}c_{N+1}^{\dagger}\underset{k=1}{\overset{N}{\sum}}(-1)^{k-1}\langle N|\omega_{k}\rangle b_{k}+\mathrm{h.c.}
\end{equation}
For an odd $N$, there exists a single non-degenerate, zero-energy
fermionic mode in the quantum channel, labelled by $k=z=\frac{N+1}{2}$
\cite{wojcik_multiuser_2007,yao_robust_2011,ping_practicality_2013}.
As a consequence, the two boundary qubits ($0$ and $N+1$) are resonantly
coupled to this mode. Therefore, we consider these three resonant
fermionic modes as the ``system'' $S$ and reinterpret the other
fermionic modes as a ``bath'' $B$. In this picture, the system-bath
$SB$ interaction is off-resonant. Then, we rewrite the total Hamiltonian
as
\begin{equation}
H=H_{S}(t)+H_{B}+H{}_{SB}(t),\label{eq:H}
\end{equation}
where
\begin{equation}
H_{B}=\sum_{k\ne z,k=1}^{N}\omega_{k}b_{k}^{\dagger}b_{k},\: H_{S}(t)=s_{+}(t)\tilde{J}_{z}b_{z}+\mathrm{h.c.},
\end{equation}
\begin{equation}
H_{SB}(t)=s_{+}(t)\sum_{k\in k_{odd}}\tilde{J}_{k}b_{k}+s_{-}(t)\sum_{k\in k_{even}}\tilde{J}_{k}b_{k}+h.c.,\label{eq:Hsb}
\end{equation}
with $s_{\pm}(t)=\alpha(t)(c_{0}^{\dagger}\pm c_{N+1}^{\dagger})$,
$\tilde{J}_{k}=J_{1}\langle1\vert\omega_{k}\rangle$, $k_{odd}=\{1,3,..,N\},$
provided $k_{odd}\ne z$, and $k_{even}=\{2,4,..,N-1\}$.
The form (\ref{eq:H}) is amenable to the application of optimal dynamical
control of the multipartite system \cite{clausen_bath-optimized_2010,clausen_task-optimized_2012,gordon_scalability_2011,gordon_dynamical_2009,kurizki_universal_2013,Schulte-Herbruggen_Optimal_2009}:
such control would be a generalization of the single-qubit dynamical
control by modulation of the qubit levels \cite{escher_optimized_2011,bensky_optimizing_2012,petrosyan_reversible_2009,gordon_universal_2007,gordon_optimal_2008,kofman_universal_2001,kofman_unified_2004}.
To this end, we rewrite Eq. (\ref{eq:Hsb}) in the interaction picture
as a sum of tensor products between system $S_{j}$ and bath $B_{j}$
operators (see \ref{sec:Appendix-A:-Interaction})
\begin{equation}
H_{SB}^{I}(t)=\sum_{j=1}^{4}S_{j}(t)\otimes B_{j}^{^{\dagger}}(t).\label{eq:HSB_int-pict}
\end{equation}
From this form one can derive the system density matrix of the system,
$\rho_{S}(t)$, in the interaction picture, under the assumption of
weak system-bath interaction, to second order in $H_{SB}$, as \cite{clausen_bath-optimized_2010,escher_optimized_2011}
\begin{equation}
\rho_{S}(t)=\rho_{S}(0)-t\sum_{i,i'=1}^{6}R_{i,i'}(t)[\hat{\nu}_{i},\hat{\nu}_{i'}\rho_{S}(0)]+h.c.,\label{eq:rho_s}
\end{equation}
where
\begin{equation}
R_{i,i'}(t)=\frac{1}{t}\sum_{j,j'=1}^{4}\int_{0}^{t}dt'\int_{0}^{t'}dt"\Phi_{j,j'}(t'-t")\Omega_{j,i}(t')\Omega_{j',i'}^{*}(t"),
\end{equation}
with $\Phi_{j,j^{'}}(\tau)=\mathrm{Tr}_{B}\left\{ B_{j}(\tau)B_{j^{'}}(0)\rho_{B}(0)\right\} $
denoting the correlation functions of bath operators and $\Omega_{j,i}(t)$
being a rotation-matrix in a chosen basis of operators $\hat{\nu}_{i}$
used to represent the evolving system operators, $S_{j}(t)=\underset{i=1}{\overset{6}{\sum}}\Omega_{j,i}(t)\hat{\nu}_{i}$
(\ref{sec:Appendix-A:-Interaction}). The solution (\ref{eq:rho_s})
will be used to calculate and optimize the state-transfer fidelity
in what follows.
\begin{figure}
\caption{\label{fig:chain-FilterFunction}
\label{fig:chain-FilterFunction}
\end{figure}
\subsection{Fidelity derivation}
We are interested in transferring a qubit state $\vert\psi_{0}\rangle$
initially stored on the $0$ qubit to the $N+1$ qubit . Here $\vert\psi_{0}\rangle$
is an arbitrary normalized superposition of the spin-down $\vert0_{0}\rangle$
and spin-up $\vert1_{0}\rangle$ (single-spin) states. To assess the
state transfer over time $T$, we calculate the averaged fidelity
$F(T)=\frac{f_{0,N+1}^{2}(T)}{6}+\frac{f_{0,N+1}(T)}{3}+\frac{1}{2}$
\cite{bose_quantum_2003}, which is the state-transfer fidelity averaged
over all possible input states $\vert\psi_{0}\rangle$. In the interaction
picture, $f_{0,N+1}(T)=\left|_{S}\left\langle \psi\right|\rho_{S}(T)\left|\psi\right\rangle _{S}\right|$
where $\vert\psi\rangle_{S}=\vert1_{0}\rangle\otimes\vert0_{z}0_{N+1}\rangle{}_{S}$
and $\vert\psi\rangle_{S}\otimes\vert\psi\rangle_{B}$ is the initial
state of $S+B$.
In the ideal regime of an isolated 3-level system, perfect state transfer
occurs when the accumulated phase due to the modulation control
\begin{equation}
\phi(T)=\tilde{J_{z}}\int_{0}^{T}\alpha(t)dt\label{eq:phi_phase}
\end{equation}
satisfies $\phi(T)=\frac{\pi}{\sqrt{2}}$. Obviously, this condition
does not strictly hold when the system-bath interaction is accounted
for, yet it is still adequate within the second-order approximation
in $H_{SB}$ used in Eq. (\ref{eq:rho_s}). In this approximation,
$f_{0,N+1}(T)$ takes the form
\begin{equation}
f_{0,N+1}(T)=1-\zeta(T),\label{eq:f_0,N+1}
\end{equation}
where
\begin{equation}
\zeta(T)=\Re\int_{0}^{T}\!\! dt\!\int_{0}^{t}\! dt'\underset{\pm}{\sum}\Omega_{\pm}(t)\Omega_{\pm}(t')\Phi_{\pm}(t-t')).\label{eq:eta_t}
\end{equation}
Here, $\Phi_{\pm}(t)=\sum_{k\in k{}_{odd(even)}}|\tilde{J}_{k}|^{2}e^{-i\omega_{k}t}$
are the bath-correlation functions, while $\Omega_{+}(t)=\alpha(t)cos(\sqrt{2}\phi(t))$
and $\Omega_{-}(t)=\alpha(t)$ are the corresponding dynamical-control
functions (\ref{sec:Appendix-A:-Interaction}-\ref{sec:Appendix-B:-Interaction}).
In the calculations we considered $\vert\psi\rangle_{B}=\vert0\rangle_{B}$.
However, in the weak-coupling regime the transfer fidelity remains
the same for a completely unpolarized state \cite{danieli_quantum_2005,yao_robust_2011}
or any other initial state \cite{ping_practicality_2013} of the bath.
In the energy domain, Eq. $\!$(\ref{eq:eta_t}) has the convolutionless
form \cite{escher_optimized_2011,bensky_optimizing_2012,petrosyan_reversible_2009,gordon_universal_2007,gordon_optimal_2008,kofman_universal_2001,kofman_unified_2004}
\begin{equation}
\zeta(T)=\int_{-\infty}^{\infty}d\omega\underset{\pm}{\sum}F_{T,\pm}(\omega)G_{\pm}(\omega),\label{eq:eta_w}
\end{equation}
where the Fourier-transforms
\begin{equation}
G_{\pm}(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dt\Phi_{\pm}(t)e^{i\omega t},\: F_{T,\pm}(\omega)=\frac{1}{2\pi}|\int_{0}^{T}dt\Omega_{\pm}(t)e^{i\omega t}|^{2}\label{eq:G(w) F(w)}
\end{equation}
are the bath-correlation spectra, $G_{\pm}(\omega)$, associated
with odd(even) parity modes and the spectral filter functions, $F_{T,\pm}(\omega)$,
which can be designed by the modulation control.
\begin{figure}
\caption{\label{fig:infidelity-Gw-semicircle}
\label{fig:infidelity-Gw-semicircle}
\end{figure}
\section{\label{sec:Optimization-method}Optimization method}
To ensure the best possible state-transfer fidelity, we use modulation
as a tool to minimize the infidelity $\zeta(T)$ in (\ref{eq:eta_t}-\ref{eq:eta_w})
by rendering the overlap between the interacting bath- and filter-spectrum
functions as small as possible \cite{clausen_bath-optimized_2010,clausen_task-optimized_2012}.
\subsection{\label{sub:Optimizing-the-modulation}Optimizing the modulation control
for non-Markovian baths}
The minimization of $\zeta(T)$ in (\ref{eq:eta_t}) can be done for
a specific bath-correlation function of a given channel which represents
a non-Markovian bath. The Euler-Lagrange (E-L) equation for minimizing
$\zeta(T)$ with the energy constraint
\begin{equation}
E(T)=\tilde{J}_{z}^{2}\int_{0}^{T}|\alpha(t)|^{2}dt\label{eq:Energy}
\end{equation}
turns out to be
\begin{equation}
\frac{d}{dt}(\frac{\partial\zeta}{\partial\dot{\phi}}-\lambda\frac{\partial E}{\partial\dot{\phi}})-(\frac{\partial\zeta}{\partial\phi}-\lambda\frac{\partial E}{\partial\phi})=0,\label{eq:E-L}
\end{equation}
where $\lambda$ is the Lagrange multiplier and $\dot{\phi}=\tilde{J_{z}}\alpha$.
The optimal modulation can be obtained by solving the integro-differential
equation
\begin{equation}
\begin{array}{cc}
\ddot{\phi}(t) & =\frac{\sqrt{E}Q(t,\phi(t),\dot{\phi}(t))}{\tilde{J}_{z}\sqrt{\int_{0}^{T}dt\left|\int_{0}^{t}dt'Q(t',\phi(t'),\dot{\phi}(t'))\right|^{2}}},\end{array}\label{eq:phi''(E)}
\end{equation}
where
\begin{equation}
\begin{array}{l}
Q(t,\phi(t),\dot{\phi}(t))=\int_{0}^{T}dt'\Theta(t-t')\frac{\dot{\phi}(t')}{2\tilde{J}_{z}^{4}}\!\left(\frac{d\Phi_{+}(t-t')}{dt}cos(\sqrt{2}\phi(t))cos(\sqrt{2}\phi(t'))+\frac{d\Phi_{-}(t-t')}{dt}\right)\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\frac{\dot{\phi}(t)}{2\tilde{J}_{z}^{4}}\left(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi(t))+\Phi_{-}(0)\right).
\end{array}\label{eq:Q}
\end{equation}
The solution of Eq. (\ref{eq:phi''(E)}) should satisfy the boundary
conditions $\phi(0)=0$ and $\phi(T)=\frac{\pi}{\sqrt{2}}$ to ensure
the required state transfer.
In general the bath-correlations have recurrences and time fluctuations
due to mesoscopic revivals in finite-length channels. Therefore, it
is not trivial to solve Eqs. (\ref{eq:phi''(E)}-\ref{eq:Q}) analytically
rather than solving them numerically for each specific channel. We
however are interested in obtaining universal analytical solutions
for state-transfer in the presence of non-Markovian noise sources.
To this end, we here discuss suitable criteria for optimizing the
state transfer in such cases.
We require the channel to be symmetric with respect to the source
and target qubits and the number of eigenvalues to be odd. These requirements
allow for a central eigenvalue that is \textit{invariant under noise}
on the couplings. This holds provided a \textit{gap exists} between
the central eigenvalue and the adjacent ones, \textit{i.e.} they are
not strongly blurred (mixed) by the noise, so as not to make them
overlap. At the same time, we assume that the discreteness of the
bath spectrum of the quantum channel is smoothed out by the noise,
since it tends to affect more strongly the higher frequencies \cite{zwick_robustness_2011,zwick_spin_2012,Zwick_Chapt_2013}.
Then, if we consider the central eigenvalue as part of the system,
a common characteristic of $G_{\pm}(\omega)$ is to have a central
gap (as exemplified in Fig. \ref{fig:chain-FilterFunction}b).
Therefore, in order to minimize the overlap between $G_{\pm}(\omega)$
and $F_{T,\pm}(\omega)$ for general gapped baths, and thereby the
transfer infidelity in (\ref{eq:eta_w}), we will design a narrow
bandpass filter centered on the gap.
We present a universal approach that allows us to obtain analytical
solutions for a narrow bandpass filter around $\omega_{z}$. Since
$G_{-}(\omega)$ has a narrower gap than $G_{+}(\omega)$, we optimize
the filter $F_{T,-}(\omega)$ under the variational E-L method. We
seek a narrow bandpass filter, whose form on time-domain via Fourier-transform
decays as slowly as possible, so as to filter out the higher frequencies.
This amounts to maximizing
\begin{equation}
F_{T,-}(\tau)=\int_{-\infty}^{\infty}F_{T,-}(\omega)e^{-i\omega\tau}d\omega=\int_{0}^{T}\alpha(t)\alpha(t+\tau)dt,\label{eq:Fiter_tau}
\end{equation}
subject to the variational E-L equation (\ref{eq:E-L}), upon replacing
$\zeta$ by $F_{T,-}$. Since there is no explicit dependence on $\phi$,
the second term therein is null, $\frac{\partial}{\partial\phi}(F_{T,-}-\lambda_{E}E)=0$,
yielding
\begin{equation}
\alpha(t+\tau)+\alpha(t-\tau)=\lambda_{E}\alpha(t)+\lambda_{\phi},\label{eq:alpha_tau}
\end{equation}
where $\lambda_{E}$ is the Lagrange multiplier and $\lambda_{\phi}$
is an integration constant chosen to satisfy the boundary conditions
obeyed by the accumulated phase (\ref{eq:phi_phase}).
Analytical solutions of (\ref{eq:alpha_tau}) are obtainable for small
$\tau$, corresponding to the differential equation
\begin{equation}
\overset{..}{\alpha}(t)=-\tilde{\lambda}_{E}\alpha(t)+\tilde{\lambda}_{\phi},\label{eq:alpha_dif_Eq}
\end{equation}
with $\tilde{\lambda}_{E}=\frac{-(\lambda_{E}-2)}{\tau^{2}}$ and
$\tilde{\lambda}_{\phi}=\frac{\lambda_{\phi}}{\tau^{2}}$ . It has
a general solution
\begin{equation}
\alpha(t)=Asin(\omega_{v}t)+Bcos(\omega_{v}t)+C.\label{eq:alpha_gral_sol}
\end{equation}
The unknown parameters are then optimized under chosen constraints,
e.g. on the boundary coupling, the transfer time, the energy, etc.
The frequencies $\omega_{v}$ that give a low and flat filter $F_{T,-}(\omega)$
outside a small range around $\omega=\omega_{z}=0$ are $\omega_{v}=\frac{\pi n}{T}$,
$n\epsilon\mathbb{Z}$, since the components of $\alpha(t)$ that
oscillate with $\omega_{v}$ then interfere destructively. Only if
$n=0,1,2$ will the filter have a \textit{single} central peak around
$\omega=0$, and the contribution of larger frequencies will be suppressed,
while the filter-overlap with the central energy level will be maximized;
for larger $n$, the central peak splits and additional peaks appear
at larger frequencies.
Therefore, the analytical expressions for the optimal solutions satisfying
$\phi(0)=0$ and $\phi(T)=\frac{\pi}{\sqrt{2}}$ are found to be
\begin{equation}
\alpha_{p}(t)=\alpha_{M}sin^{p}\left(\frac{\pi t}{T}\right),\label{eq:Opt Mod alphap}
\end{equation}
where $p=0,1,2$\textcolor{black}{,
\begin{equation}
\alpha_{M}=c_{p}\frac{\pi}{\sqrt{2}\tilde{J}_{z}T}\label{eq:alpha_Mp,Tp}
\end{equation}
}
\noindent and $c_{p}=\frac{\sqrt{\pi}\Gamma(\frac{1+p}{2})}{\Gamma(\frac{1+p}{2})}(c_{0}=1,\, c_{1}=\frac{\pi}{2},\, c_{2}=2)$.
Here $p=0$ means static control, while $p=1,2$ stand for dynamical
control. Note that $T$ and $\alpha_{M}=max\{\alpha_{p}(t)\}$ cannot
be independently chosen. If the transfer time is fixed, then the maximum
amplitude depends on $p$, $\alpha_{M}=\alpha_{M_{p}}$, according
to Eq. (\ref{eq:alpha_Mp,Tp}). Similarly, if the maximum amplitude
is kept constant, then the transfer time will depend on $p$, $T=T_{p}$,
by Eq. (\ref{eq:alpha_Mp,Tp}).
The different solutions in Eq. (\ref{eq:Opt Mod alphap}) are sinc-like
bandpass filter functions around $0$ that become narrower as $T$
increases. For $p=0$, which satisfies the minimal-energy condition
$E_{min}(T_{0})=\frac{\pi^{2}}{2T_{0}}$, the corresponding filter
is the narrowest around $0$, but it has many wiggles on the filter
tails (Fig. \ref{fig:chain-FilterFunction}b) which overlap with bath-energies
that hamper the transfer. In contrast, the $p=1,2$ bandpass filters
are wider (for the same $T$) and require more energy, $E_{1}=\frac{\pi^{2}}{8}E_{{\scriptstyle min}}$
and $E_{2}=\frac{3}{2}E_{min}$ respectively, but these filters are
flatter and lower throughout the bath-energy domain.
Hence, the bandpass filter width (\textit{i.e.} full width at half
maximum) and the overlap of its tail-wiggles with bath-energies as
a function of $T$, determine which modulations $\alpha_{p}(t)$ are
optimal, as shown in the inset of Fig. \ref{fig:chain-FilterFunction}b
($F_{T,+}(\omega)$ filters out a similar spectral range). The shorter
$T$, the lower is $p$ that yields the highest fidelity, because
the central peak of the filter that produces the dominant overlap
with the bath spectrum is then the narrowest. However, as $T$ increases,
larger $p$ will give rise to higher fidelity, because now the tails
of the filter make the dominant contribution to the overlap. As shown
in Fig. \ref{fig:infidelity-Gw-semicircle}, the filter for $p=1,2$
can improve the transfer fidelity \textit{by orders of magnitude}
in a noisy gapped bath bounded by the Wigner-semicircle, which is
representative of fully randomized channels \cite{wigner_distribution_1958}
(\ref{sec:Appendix-C:-Considerations}).
\subsection{Optimizing the modulation control for a Markovian Bath}
We next consider the worst-case scenario of a Markovian bath, where
the bath-correlation functions $\Phi_{\pm}(\tau)$ vanish for $\tau>0$.
This is the case when the gap is closed by a noise causing the bath
energy levels to fluctuate faster than the system dynamics. We note
that, finding optimal solutions for the noise spectrum of a Markovian
bath is important for the case where the gap is reduced or even lost
in static cases.
The infidelity function (\ref{eq:eta_t}) that must be minimized when
the correlation time $\tau_{c}=0$, \textit{i.e.} $\Phi_{\pm}(\tau)=\delta(\tau)$,
is
\begin{equation}
\zeta(T)=\Re\int_{0}^{T}dt\frac{\dot{\phi}^{2}(t)}{\tilde{J}_{z}^{2}}\bigl(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi(t))+\Phi_{-}(0)\bigr).\label{eq:Mark}
\end{equation}
The E-L equation under energy constraint (\ref{eq:E-L}), is now
\begin{equation}
\ddot{\phi}(t)\!\left(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi(t))\!+\!\Phi_{-}(0)\!-\!2\lambda\tilde{J}_{z}^{2}\right)\!-\!\sqrt{2}\dot{\phi}^{2}(t)\,\Phi_{+}(0)cos(\sqrt{2}\phi(t))sin(\sqrt{2}\phi(t))\!=\!0.\label{eq:phi Mark bath-1-1}
\end{equation}
This equation has a non-trivial analytical solution and the modulation
that minimizes $\zeta(T)$ is given by the following transcendental
equation
\begin{equation}
{\normalcolor \begin{array}{c}
T\intop_{0}^{\phi(t)}\sqrt{cos(2\sqrt{2}\phi)\Phi_{+}(0)+\Phi_{+}(0)+2\Phi_{-}(0)-2\lambda\tilde{J}_{z}^{2}}d\phi\\
-t\intop_{0}^{\phi(T)}\sqrt{2(\Phi_{+}(0)cos^{2}(\sqrt{2}\phi)+\Phi_{-}(0)-\lambda\tilde{J}_{z}^{2})}d\phi=0.
\end{array}}\label{eq:analitycal solution Mark Bath}
\end{equation}
The infidelity for this optimal modulation almost coincides with the
one obtained for static control ($\alpha_{p=0}(t)=\alpha_{M}$ from
Eq. (\ref{eq:Opt Mod alphap})), \textit{i.e.}
\begin{equation}
1-F(T)\approx\frac{\pi^{2}N}{6\sqrt{2}JT}(1-\frac{\pi^{2}N}{16\sqrt{2}JT}),\: T=\frac{\pi\sqrt{N}}{2\alpha_{M}J},
\end{equation}
and they only differ by about 0.1\%. This optimal modulation can
be phenomenologically approximated by
\begin{equation}
\alpha(t)\approx a\alpha_{M}+b\, sin^{q}(\frac{t\pi}{T})),\: q\sim3.5,\:\frac{b}{a}\sim\frac{1}{3},\: a\sim0.84,
\end{equation}
assuming no constraints (\textit{$\lambda=0$}). An example of the
performance of this solution is discussed below and shown in Fig.
\ref{fig:Mark-Noise}.
\begin{figure}
\caption{\label{fig:Fmax-alp}
\label{fig:Fmax-alp}
\end{figure}
\section{\label{sec:Optimal-control-of}Optimal control of transfer in a homogeneous
spin-chain channel }
Consider a \textit{uniform} (homogeneous) spin-chain channel, \textit{i.e.}
$J_{i}\equiv J$ in Eq. (\ref{eq:hamiltonian}), whose energy eigenvalues
are $\omega_{k}=2Jcos(\frac{k\pi}{N+1})$ \cite{wojcik_unmodulated_2005}.
In Fig. \ref{fig:Fmax-alp}, we show the performance of the general
optimal solutions (\ref{eq:Opt Mod alphap}) for this specific channel
as a function of $\alpha_{M}$ and $T$.
The approach based on Eq. (\ref{eq:eta_t}) strictly holds in the
weak-coupling regime $(\alpha_{M}\ll1)$ \cite{clausen_bath-optimized_2010,clausen_task-optimized_2012,escher_optimized_2011,bensky_optimizing_2012,petrosyan_reversible_2009,kofman_universal_2001,kofman_unified_2004}.
In this regime (marked with arrows in Fig. \ref{fig:Fmax-alp}b),
we found that the transfer time is $T_{p}\!\approx\! c_{p}\frac{\pi\sqrt{N}}{2\alpha_{M}J}$,
and the infidelity decreases by reducing $\alpha_{M}$ according to
a power law, aside from the oscillations due to the discrete nature
of the bath-spectrum (see \ref{sec:Appendix-C:-Considerations}).
The filter tails are sinc-like functions, so that when a zero of the
filter matches a bath-energy eigenvalue, the infidelity exhibits a
dip. Aside from oscillations, the best tradeoff between speed and
fidelity within this regime is given by the optimal modulation with
$p=2$ (for the system described in Fig. \ref{fig:Fmax-alp}a).
However, this approach can also be extended \textit{to strong couplings}
$\alpha_{M}$, since it \textit{becomes compatible with the weak-coupling
regime} under the optimal filtering process that increases the state
fidelity in the interaction picture \cite{gordon_universal_2007,gordon_dynamical_2009,kurizki_universal_2013}.
The bandpass filter width increases as $T$ decreases; consequently,
in the strong coupling regime $(\alpha_{M}\sim1)$ the filter may
now overlap the bath energies closest to $\omega_{z}$, but still
block the higher bath energies, which are the most detrimental for
the state transfer \cite{zwick_robustness_2011,zwick_spin_2012,Zwick_Chapt_2013}.
Then, the participation of the closest bath energies yields a transfer
time $T_{p}\!\approx\! c_{p}\frac{N}{2J}$. There is a clear minimal
infidelity value at the point that we denote as $\alpha_{M_{p}}^{opt}$
which depends on $p$ (Fig. \ref{fig:Fmax-alp}b); thus extending
the previous static-control ($p=0$) results, where an optimal $\alpha_{M_{0}}^{opt}$
was found \cite{zwick_quantum_2011,banchi_long_2011,banchi_optimal_2010,zwick_spin_2012}.
The infidelity dip corresponds to a better filtering-out (suppression)
of the higher energies, retaining only those that correspond to an
almost equidistant spectrum of $\omega_{k}$ around $\omega_{z}$,
which allow for coherent transfer \cite{zwick_spin_2012}.
Figure \ref{fig:Fmax-alp}b shows that by fixing $\alpha_{M}$, the
dynamical control ($p=1,2$) of the boundary-couplings reduces the
transfer infidelity \textit{by orders of magnitude} only at the expense
of slowing down the transfer time $T_{p}$ at most by a factor of
2, $\frac{T_{p}}{T_{0}}\approx\frac{c_{p}}{c_{0}}\leq2$. If the constraint
on $\alpha_{M}$ can be relaxed, \textit{i.e.} more energy can be
used, the advantages of dynamical control can be even more appreciated
for both infidelity decrease and transfer-time reduction by orders
of magnitude, as shown in Fig. \ref{fig:Fmax-alp}a. Hence, our main
result is that the speed-fidelity tradeoff can be drastically improved
under optimal dynamical control.
\section{Robustness against different noises}
We now explicitly consider the effects of optimal control on noise
affecting the coupling strengths, also called off-diagonal noise,
causing: $J_{i}\rightarrow J_{i}+J_{i}\Delta_{i}(t),\; i=1,...,N$
with $\Delta_{i}$ being a uniformly distributed random variable in
the interval $\left[-\varepsilon_{J},\varepsilon_{J}\right]$. Here
$\varepsilon_{J}>0$ characterizes the noise or disorder strength.
When $\Delta_{i}$ is time-independent, it is called \textit{static
noise}, as was considered in other state-transfer protocols \cite{de_chiara_perfect_2005,ronke_effect_2011,zwick_robustness_2011,zwick_spin_2012}.
When $\Delta_{i}(t)$ is time-dependent, we call it \textit{fluctuating
noise} \cite{Burgarth_fluctuating}\textit{.} These kinds of noises
will affect the bath energy levels, while the central energy $\omega_{z}$
remains invariant \cite{zwick_robustness_2011,Zwick_Chapt_2013}.
In the following we analyse the performance of the control solutions
obtained in Sec. \ref{sec:Optimization-method} for these types of
noise and later on, in Sec. \ref{sub:Other-sources-of} we discuss
briefly the effects of other sources of noise.
\begin{figure}
\caption{\label{fig:Noise}
\label{fig:Noise}
\end{figure}
\subsection{\textit{Static noise} }
Static control on the boundary-couplings can suppress static noise
\cite{zwick_robustness_2011,zwick_spin_2012} but here we show that
dynamical boundary-control makes the channel even more robust, because
it filters out the bath-energies that damage the transfer. To illustrate
this point, we compare the effect of modulations $\alpha_{p}(t)$
with $\alpha_{M}=\alpha_{M_{p}}^{opt}$ for $p=0$ and 2 in the strong-coupling
regime (Fig. \ref{fig:Noise}a). There is an evident advantage of
dynamical control with $p=2$ compared to static control ($p=0$),
at the expense of increasing the transfer time by only a factor of
2, $\frac{T_{2}}{T_{0}}\approx2$. In the weak-coupling regime, if
we choose $\alpha_{M}$ such that the transfer fidelity is similar
for $p=0$ and $p=2$, then both cases are similarly robust under
static disorder, but the modulated case $p=2$ is an order of magnitude
faster. Remarkably, because of disorder-induced localization \cite{Porter1965,Imry2002,akulin_spectral_1993,pellegrin_mie_2001},
regardless of how small is $\alpha_{M}$, the averaged fidelity under
static noise cannot be improved beyond the bound
\begin{equation}
1-\bar{F}\propto N\varepsilon_{J}^{2},\:(\varepsilon_{J}\ll1).
\end{equation}
\subsection{\textit{Markovian noise}}
The worst scenario for quantum state transfer is the absence of an
energy gap around $\omega_{z}$. This case corresponds to Markovian
noise characterized by $\left\langle \Delta_{i}(t)\Delta_{i}(t+\tau)\right\rangle =\delta(\tau)$,
where the brackets denote the noise ensemble average, or equivalent
to a bath correlation-function that vanishes at $\tau>0$. In this
case there is an analytical solution for the optimal modulation given
by Eq. (\ref{eq:analitycal solution Mark Bath}), although the infidelity
achieved by it almost coincides with the one obtained by the static
($p=0$) optimal control (Fig. \ref{fig:Mark-Noise}). Counterintuitively,
arbitrarily high fidelities can be achieved for such noise by decreasing
$max\left|\alpha(t)\right|$ and thereby slowing down the transfer.
This comes about because in a Markovian bath, the very fast coupling
fluctuations suppress the disorder-localization effects that hamper
the transfer fidelity as we show below for a typical case.
\begin{figure}
\caption{\label{fig:Mark-Noise}
\label{fig:Mark-Noise}
\end{figure}
\subsection{\textit{Non-Markovian noise}}
We now consider a non-Markovian noise of the form $J_{i}+J_{i}\Delta_{i}(t)$,
where $\Delta_{i}(t)=\Delta_{i}\left(\left[t/\tau_{c}\right]\right)$,
where the integer part $\left[t/\tau_{c}\right]=n$ defines a noise
$\Delta_{i}\left(n\right)$ that randomly varies between the interval
$\left[-\varepsilon_{J},\varepsilon_{J}\right]$ at time-intervals
of $\tau_{c}$ during the transfer. We observe a convergence of the
transfer fidelity to its value without noise as the noise correlation
time $\tau_{c}$ decreases (Fig. \ref{fig:Noise}b). Consequently
the fidelity can be substantially improved by reducing $\alpha_{M}$.
The effective noise strength scales down as $\tau_{c}^{1/2}$ (Fig.
\ref{fig:Noise}b, inset). By contrast to the Markovian limit $\tau_{c}\rightarrow0$,
dynamical control can strongly reduce the infidelity in the non-Markovian
regime that lies between the static and Markovian limits and whose
bath-spectrum is gapped.
\subsection{\textit{\label{sub:Other-sources-of}Other sources of noise}}
\textit{Timing errors}: In addition to resilience to noise affecting
the spin-spin couplings, there is another important characteristic
of the transfer robustness, namely, the length of the time window
in which high fidelity is obtained. The fidelity $F(t)$ under optimal
dynamical control ($p=1,2)$, yields a wider time-window around $T$
where the fidelity remains high compared with its static ($p=0)$
counterpart. This allows more time for determining the transferred
state or using it for further processing. Consequently, the \textit{robustness
}against timing imperfections \cite{kay_perfect_2006,zwick_robustness_2011}
\textit{is increased} under optimal dynamical control.
\noindent \textit{On-site energy noise}: This kind of noise, alias
diagonal-noise, can be either static or fluctuating. The static one
can give rise to the emergence of quasi-degenerate central states.
Then, the dynamical control approach introduced in this work is still
capable of isolating the ``system'' defined here (Sec. \ref{sec:Quantum channel and state transfer fidelity})
from the remaining ``bath'' levels. It may happen that the spin
network is not symmetric with respect to the source and target spins,
and then the effective couplings of the source and target qubits with
the central level will not be symmetric. This asymmetry can be effectively
eliminated by boundary control. On the other hand, a fluctuating diagonal-noise
that may produce a fluctuation of the central energy level is here
fought by optimizing the tradeoff between speed and fidelity as detailed
above. Additional dynamical control of only the source and target
spins can be applied to avoid these decoherence effects, by the mapping
to an effective 3-level system, as a variant of dynamical decoupling
\cite{Viola_Dynamical_1998,viola_dynamical_1999,Viola_RobustDD_2003,Lidar_QDynDec_2005}.
\section{Conclusions}
We have proposed a general, optimal dynamical control of the tradeoff
between the speed and fidelity of qubit-state transfer through the
central-energy global mode of a quantum channel in the presence of
either static or fluctuating noise. Dynamical boundary-control has
been used to design an optimal spectral filter realizable by universal,
simple, modulation shapes. The resulting transfer infidelity and/or
transfer time can be reduced by orders of magnitude, while their robustness
against noise on the spin-spin couplings is maintained or even improved.
Transfer-speed maximization is particularly important in our strive
to reduce the random phase accumulated during the transfer when energy
fluctuations (diagonal noise) affect the spins \cite{Ajoy_perfect_2013}.
We have shown that, counterintuitively, static noise is more detrimental
than fluctuating noise on the spin-spin couplings. This general approach
is applicable to quantum channels that can be mapped to Hamiltonians
quadratic in bosonic or fermionic operators \cite{cappellaro_dynamics_2007,rufeil-fiori_effective_2009,doronin_multiple-quantum_2000,yao_robust_2011,yao_quantum_2013}.
We note that our control is complementary to the recently suggested
control aimed at balancing possible asymmetric detunings of the boundary
qubits from the channel resonance \cite{Ajoy_perfect_2013,yao_quantum_2013}.
\ack We acknowledge the support of ISF-FIRST (Bikura) and the EC
Marie Curie (Intra-European) Fellowship (G.A.A.).
\appendix
\section{The Hamiltonian in the interaction picture\label{sec:Appendix-A:-Interaction}}
The system-bath Hamiltonian (Eq. (\ref{eq:Hsb}) of the main text)
splits into a sum of symmetric and antisymmetric system operators
that are coupled to odd- and even-bath modes: $H_{SB}(t)=\underset{j=1}{\overset{4}{\sum}}\tilde{S}_{j}\otimes\tilde{B}_{j}^{\dagger},$
where $\tilde{S}_{1(3)}=\alpha(t)(c_{0}+(-)c_{N+1})$, $\tilde{S}_{2(4)}=\tilde{S}_{1(3)}^{\dagger}$,
$\tilde{B}_{1(3)}=\underset{k\in k_{odd(even)}}{\sum}\tilde{J}_{k}b_{k}$
and $\tilde{B}_{2(4)}=\tilde{B}_{1(3)}^{\dagger}$. In the interaction
picture $H_{SB}(t)$ becomes
\begin{equation}
H_{SB}^{I}(t)=\sum_{j=1}^{4}S_{j}(t)\otimes B_{j}^{\dagger}(t),\label{eq:HI_SB}
\end{equation}
where
\begin{equation}
\begin{array}{c}
S_{j}(t)=U_{S}^{\dagger}(t)\tilde{S}_{j}(t)U_{S}(t),\, U_{S}(t)=\mathcal{T}e^{-i\intop_{0}^{t}dt^{'}H_{S}(t^{'})},\\
B_{j}(t)=U_{B}^{\dagger}(t)\tilde{B}_{j}U_{B}(t),\, U_{B}(t)=e^{-iH_{B}t};
\end{array}
\end{equation}
and the evolution operators are
\begin{equation}
\begin{array}{l}
\begin{array}{cl}
U_{S}(t)= & \vert0\rangle_{SS}\langle0\vert+\left(\frac{\cos(\sqrt{2}\phi(t))+1}{2}\right)\left(\vert0\rangle\langle0\vert+\vert N+1\rangle\langle N+1\vert\right)\\
& +\left(\frac{\cos(\sqrt{2}\phi(t))-1}{2}\right)\left(\vert0\rangle\langle N+1\vert+\vert N+1\rangle\langle0\vert\right)\\
& +\cos(\sqrt{2}\phi(t))\vert z\rangle\langle z\vert-i\,\frac{\sin(\sqrt{2}\phi(t))}{2}\left(\vert0\rangle\langle z\vert+\vert N+1\rangle\langle z\vert+h.c.\right),\\
U_{B}(t)= & \overset{N}{\underset{k=1,k\neq z}{\sum}}e^{-i\omega_{k}t}\vert k\rangle\langle k\vert+\vert0\rangle_{BB}\langle0\vert,
\end{array}\end{array}
\end{equation}
where the states $\vert0\rangle_{S}=\vert0_{0}0_{z}0_{N+1}\rangle_{S}$
and $\vert0\rangle_{B}=\vert0_{1}...0_{N}\rangle_{B}$ refer to the
zero-excitation states in the system (S) and bath (B) respectively.
Therefore, the bath operators are $B_{1(3)}(t)=\underset{k\in k_{odd(even)}}{\sum}\vert\tilde{J}_{k}\vert^{2}e^{-i\omega_{k}t}\vert k\rangle{}_{B}\langle0\vert,\, B_{2(4)}(t)=B_{1(3)}^{\dagger}(t)$.
We define a basis of operators $\hat{\nu}_{i}$ to describe the rotating
system operators $S_{j}(t)$ via a rotation-matrix $\Omega_{j,i}(t)$.
They are given by
\begin{equation}
\begin{array}{cc}
\hat{\nu}_{1}=\vert0\rangle_{S}\left(\langle0\vert+\langle N+1\vert\right) & \hat{\nu}_{2}=\hat{\nu}_{1}^{\dagger},\\
\hat{\nu}_{3}=\vert0\rangle_{S}\langle z\vert & \hat{\nu}_{4}=\hat{\nu}_{3}^{\dagger},\\
\hat{\nu}_{5}=\vert0\rangle_{S}\left(\langle0\vert-\langle N+1\vert\right) & \hat{\nu}_{6}=\hat{\nu}_{5}^{\dagger},
\end{array}\label{eq:nu_i}
\end{equation}
such that $S_{j}(t)=\overset{6}{\underset{i=1}{\sum}}\Omega_{j,i}(t)\hat{\nu}_{i}.$
Given that $S_{1}(t)=\dot{\phi}(t)\left(cos(\sqrt{2}\phi(t))\hat{\nu}_{1}-i\,\sqrt{2}sin(\sqrt{2}\phi(t))\hat{\nu}_{3}\right)$,
$S_{3}(t)=\dot{\phi}(t)\hat{\nu}_{5},\, S_{2(4)}(t)=S_{1(3)}^{\dagger}(t)$
the rotation-matrix vectors are
\begin{equation}
\begin{array}{c}
\begin{array}{l}
\Omega_{1,i}(t)=\dot{\phi}(t)\left(cos(\sqrt{2}\phi(t)),0,-i\,\sqrt{2}sin(\sqrt{2}\phi(t)),0,0,0\right)\\
\Omega_{2,i}(t)=\dot{\phi}(t)\left(0,cos(\sqrt{2}\phi(t)),0,i\,\sqrt{2}sin(\sqrt{2}\phi(t)),0,0\right)\\
\Omega_{3,i}(t)=\dot{\phi}(t)(0,0,0,0,1,0)\\
\Omega_{4,i}(t)=\dot{\phi}(t)(0,0,0,0,0,1).
\end{array}\end{array}\label{eq:RotationMatrix_Omega}
\end{equation}
\section{The fidelity in the interaction picture\label{sec:Appendix-B:-Interaction}}
Here we derive Eqs. (\ref{eq:f_0,N+1}-\ref{eq:eta_t}) from Eq. (\ref{eq:rho_s})
of the main text. Considering $\vert\psi\rangle=\vert100...0\rangle_{SB}=\vert\psi\rangle_{S}\otimes\vert0\rangle_{B}$
with $\vert\psi\rangle_{S}=\vert1_{0}0_{z}0_{N+1}\rangle_{S}$ as
the initial state, the fidelity is reduced to
\begin{equation}
\begin{array}{cc}
f_{0,N+1}(T)= & \left|_{S}\left\langle \psi\right|\rho_{S}(T)\left|\psi\right\rangle _{S}\right|=1-\zeta(T)\end{array},
\end{equation}
where $\zeta(T)=T\underset{i,i'=1}{\overset{6}{\sum}}R_{i,i'}(T)\Gamma_{i,i^{'}}$,
with
\begin{equation}
\Gamma_{i,i^{'}}={}_{S}\left\langle \psi\right|[\hat{\nu}_{i},\hat{\nu}_{i'}\left|\psi\right\rangle _{S}{}_{S}\left\langle \psi\right|]\left|\psi\right\rangle _{S}=\delta_{i,2}\delta_{1,i'}+\delta_{i,2}\delta_{5,i'}+\delta_{i,6}\delta_{1,i'}+\delta_{i,6}\delta_{5,i'}
\end{equation}
and
\begin{equation}
R_{i,i'}(T)\!=\!\frac{1}{T}\int_{0}^{T}dt\!\int_{0}^{t}dt'(\Phi_{2,1}(t-t')\Omega_{2,i}(t)\Omega_{1,i'}(t')+\Phi_{4,3}(t-t')\Omega_{4,i}(t)\Omega_{3,i'}(t')).
\end{equation}
Here $\hat{\nu}_{i'}$ and $\Omega_{j,i}$ are as defined in Eqs.
(\ref{eq:nu_i}-\ref{eq:RotationMatrix_Omega}), while the correlation
functions are
\begin{equation}
\Phi_{j,j'}(t-t')=\sum_{k\in k_{odd}}\vert\tilde{J}_{k}\vert^{2}e^{-i\omega_{k}(t-t')}\delta_{j,2}\delta_{1,j'}+\sum_{k\in k_{even}}\vert\tilde{J}_{k}\vert^{2}e^{-i\omega_{k}(t-t')}\delta_{j,4}\delta_{3,j'}.
\end{equation}
This leads to the infidelity $\zeta(T)$ of Eq. (\ref{eq:eta_t}).
\section{Considerations for a specific non-Markovian bath: the uniform spin-channel\label{sec:Appendix-C:-Considerations}}
Consider a \textit{uniform} (homogeneous) spin-chain channel, \textit{i.e.}
$J_{i}\equiv J$ in Eq.(\ref{eq:hamiltonian}), whose energy eigenvalues
are $\omega_{k}=2Jcos(\frac{k\pi}{N+1})$. In the weak-coupling regime
where $\alpha_{M}\ll1$, the coupling strength in the interaction
$H_{bc}$, $\tilde{J}_{z}=\sqrt{\frac{2}{N+1}}J$ and $\tilde{J}_{k}=\tilde{J}_{z}sin(\frac{k\pi}{N+1})$,
are always much smaller than the nearest eigenvalue gap $\vert\omega_{z}-\omega_{z\pm1}\vert\sim\frac{2J}{N}$
\cite{wojcik_unmodulated_2005,wojcik_multiuser_2007,yao_robust_2011}.
The correlation function of the bath is
\begin{equation}
\Phi_{\pm}(\tau)=\underset{k{}_{odd(even)}}{\sum}\left|\sqrt{\frac{2}{N\!+\!1}}Jsin(\frac{k\pi}{N\!+\!1})\right|^{2}e^{-i2Jcos(\frac{k\pi}{N+1})\tau}
\end{equation}
and has recurrences and time fluctuations due to mesoscopic revivals,
while at short times $t$, it behaves as a Bessel function {\small ${\color{black}{\color{red}{\color{black}\Phi(t)=\frac{2(\alpha_{0}J)^{2}}{J\tau}\mathtt{\mathcal{J}_{1}(}2Jt)}}}$.
}The latter correlation function represents the limiting case of an
infinite channel and it gives a continuous bath-spectrum that becomes
a semicircle. In the case of a finite channel, $G(\omega)$ will be
discrete but modulated by the semicircle with a central gap. If disorder
is considered, the position of the spectrum lines fluctuates from
channel to channel but they are essentially modulated by the semicircle
with a central gap as was considered in the Fig. \ref{fig:chain-FilterFunction}b
of the main text, where
\begin{equation}
G_{\pm}(\omega)=\frac{1}{2}\sqrt{4J^{2}-\omega^{2}}(1-\Theta(\omega-\omega_{l})\Theta(\omega+\omega_{l})),\:\omega_{l}=\frac{3\omega_{z+1}}{4}.
\end{equation}
This is the Wigner-distribution for fully randomized channels \cite{wigner_distribution_1958}
with a central gap.
\section*{References}
\end{document} |
\begin{document}
\title[Distribution of Number Fields with Wreath Products as Galois Groups]
{The Distribution of Number Fields with Wreath Products as Galois Groups}
\author{J\"urgen Kl\"uners}
\email{[email protected]}
\address{Universit\"at Paderborn, Institut f\"ur Mathematik, D-33095 Paderborn, Germany.}
\subjclass{Primary 11R29; Secondary 11R16, 11R32}
\begin{abstract}
Let $G$ be a wreath product of the form $C_2 \wr H$, where $C_2$ is
the cyclic group of order 2. Under mild conditions for $H$ we
determine the asymptotic behavior of the counting functions for
number fields $K/k$ with Galois group $G$ and bounded discriminant.
Those counting functions grow linearly with the norm of the
discriminant and this result coincides with a conjecture of Malle.
Up to a constant factor these groups have the same asymptotic
behavior as the conjectured one for symmetric groups.
\end{abstract}
\maketitle
\section{Introduction}
Let $k$ be a number field and $K=k(\alpha)$ be a finite extension of
degree $n$ with minimal polynomial $f$ of $\alpha$. By abuse of
notation we define $\Gal(K/k):=\Gal(f)$. This means that we associate
a Galois group even to a non-normal extension. Therefore the Galois group of
$K/k$ is a transitive permutation group $G\leq S_n$.
Denote by $\Norm=\Norm_{k/\Q}$ the norm function. Let
$$Z(k,G;x):=\#\left\{K/k : \Gal(K/k)=G,\ \Norm(d_{K/k})\le
x\right\}$$ be the number of field extensions of $k$ (inside a fixed
algebraic closure $\bar\Q$) of relative degree~$n$ with Galois group
permutation isomorphic to $G$ and norm of the
discriminant $d_{K/k}$ bounded above by $x$. It is well known that the
number of extensions of $k$ with bounded norm of the discriminant is
finite, hence $Z(k,G;x)$ is finite for all $G$, $k$ and $x\in\R$. We
are interested in the asymptotic behavior of this function for
$x\rightarrow\infty$. Gunter Malle \cite{Ma4,Ma5} has given a precise
conjecture how this asymptotics should look like. Before we can state
it we need to introduce some group theoretic definitions.
\begin{definition}
Let $1\ne G\leq S_n$ be a transitive subgroup acting on $\Omega=\{1,\ldots,n\}$.
\begin{enumerate}
\item For $g\in G$ we define the index $\ind(g):= n- \mbox{ the number of orbits of $g$ on }\Omega.$
\item $\ind(G):=\min\{\ind(g): 1\ne g\in G\}.$
\item $a(G):=\ind(G)^{-1}$.
\item Let $C$ be a conjugacy class of $G$ and $g\in C$. Then $\ind(C):=\ind(g)$.
\end{enumerate}
\end{definition}
The last definition is independent of the choice of $g$ since all
elements in a conjugacy class have the same cycle shape. We define an
action of the absolute Galois group of $k$ on the
$\bar{\Q}$-characters of $G$. The orbits under this action are called
$k$--conjugacy classes. Note that we get the ordinary conjugacy
classes when $k$ contains all $N$-th roots of unity for $N=|G|$.
\begin{definition}
For a number field $k$ and a transitive subgroup $1\ne G\leq S_n$ we define:
$$b(k,G):=\#\{C : C\; k\mbox{-conjugacy class of minimal index }\ind(G)\}.$$
\end{definition}
Now we can state the conjecture of Malle \cite{Ma5}, where we write $f(x) \sim g(x)$ for
$\lim\limits_{x\rightarrow\infty} \frac{f(x)}{g(x)} =1$.
\begin{conjecture}\label{con}(Malle)
For all number fields $k$ and all transitive permutation groups
$1\ne G\leq S_n$ there exists a constant $c(k,G)>0$ such that
$$Z(k,G;x) \sim c(k,G)x^{a(G)} \log(x)^{b(k,G)-1},$$
where $a(G)$ and $b(k,G)$ are given as above.
\end{conjecture}
We remark that at the time when the conjecture was stated it was only
known for all abelian groups and the groups $S_3\leq S_3$ and $D_4\leq S_4$.
Let us state some easy properties of the constants $a(G)$ and $b(k,G)$
which are already given in \cite{Ma4,Ma5}. It is easy to see that
$a(G)\leq 1$ and equality occurs if and only if $G$ contains a
transposition. It is an easy exercise (see Lemma
\ref{lem:transposition}) that all transpositions are conjugated
in a transitive permutation group. Therefore we obtain $b(k,G)=1$,
if $a(G)=1$. Since the symmetric group always contains a
transposition, Malle's conjecture implies that the counting function
$Z(k,n;x)$ for degree $n$ extensions with bounded discriminant as
above behaves like $c(n) x$. The latter conjecture is proven for
$n\leq 5$, see \cite{DaHe,Bh1,Bh2}, but nothing is known for $n\geq
6$.
One result of this paper is that for every even $n$ there exists a
group $G$ such that $Z(k,G;x) \sim c(k,G) x.$ This group $G$ will be a
wreath product of type $C_2 \wr H$, where $H\leq S_{n/2}$, see
Corollaries \ref{Cor1} and \ref{Cor2}. There are mild conditions for $H$,
but those are fulfilled if $H$ is nilpotent or regular for instance.
The main results will be Theorems \ref{Satz:kranz} and \ref{mainwreath}.
Let $H$ be a permutation group which fulfills the mild conditions of
Theorem \ref{Satz:kranz}. Then the counting function of $G:=C_2\wr H$ behaves
like
$$Z(k,C_2\wr H;x) \sim c(k,G) x.$$
Furthermore, the corresponding Dirichlet series has a simple pole at 1
and has a meromorphic continuation to real part larger than $7/8$.
Note that in \cite{Kl5} we have given a counter example to
Conjecture \ref{con}. In these counter examples it might happen that the
$\log$-factor is bigger than expected when certain subfields of
cyclotomic extensions occur as intermediate fields. Nevertheless, the
main philosophy of this conjecture should be true.
\section{Zeta functions, Hecke $L$--series, and ray class groups}
\label{sec:hecke}
In this section we collect some properties about Hecke $L$--series.
For a number field $k$ we denote by $\PP(k)$ the set of prime ideals
of the ring of integers $\OO_k$ of $k$. We denote by
$$\zeta_k(s) :=\prod_{\idp\in\PP(k)}\left(1-\frac{1}{\Norm(\idp)^s}\right)^{-1},\;\;\Re(s)>1$$
the Dedekind zeta function of $k$ which converges absolutely and locally uniformly for
$\Re(s)>1$. This function has a simple pole at $s=1$ and we get the following estimates.
\begin{lemma}\label{residuum}
Let $k$ be a number field of degree $m$ with absolute discriminant
$d_k$. Then:
\begin{enumerate}
\item $|\zeta_k(s)|\leq \zeta_\Q(\Re(s))^m$ for all $s$ with $\Re(s)>1$.
\item For all $0<\epsilon\leq 1$:
$$\res_{s=1} \zeta_k(s)\leq 2^{1+m} (d_k\pi^{-m/2})^\epsilon \epsilon^{1-m} \leq 2^{1+m}d_k^\epsilon \epsilon^{1-m}.$$
\end{enumerate}
\end{lemma}
\begin{proof}
The first assertion is Corollary 3 in \cite[p. 326]{Nar}. The second
one is Corollary 3 in \cite[p. 332]{Nar}.
\end{proof}
For an ideal $\idc\subseteq \OO_k$ we consider a character $\chi$ of the ray class group
$\Cl_\idc$, i.e. a homomorphism from $\Cl_\idc$ to $\C^*$. This character is only defined for
ideals coprime to $\idc$. Let $S:=\{\idp\in \PP(k): \idp \mid \idc\}$ be the exceptional set. For
$\idp\in S$ we define $\chi(\idp)=0$. Therefore we multiplicatively extend this character to
all ideals. Now we are able to define the Hecke $L$--series:
$$L_k(\chi,s):=\prod_{\idp\in\PP(k)}\left(1-\frac{\chi(\idp)}{\Norm(\idp)^s} \right)^{-1}.$$
As the Dedekind zeta function this product converges absolutely and locally uniformly for
$\Re(s)>1$. For further properties we refer the reader to \cite[p. 343]{Nar}.
The Hecke $L$--series have a meromorphic continuation to the left. In the
following we need upper estimates for
$L_k(\chi,s)$ in strips of the form $a<\Re(s)\leq 1$. The following theorem follows
directly from
\cite[equation 5.20]{IwKo}. The proof is similar to the proof of Theorem 7.4. in
\cite[p. 350]{Nar}, where we need to apply the convexity principle
\cite[p. 265]{Lang}.
\begin{theorem}\label{bound_heckeL}
Let $k$ be a number field of degree $m$, $\idf$ be an ideal of $\OO_k$, $\chi$ be an character of the
ray class group $\Cl_\idf$, and
$D:=d_k\Norm(\idf)$. Define $\delta:=1$ if $\chi$ is the trivial character
and $\delta:=0$ otherwise. Then for all $\epsilon>0$ and all $s$ with
$0\leq\sigma:=\Re(s)\leq 1$ we get the following estimate:
$$|(s-1)^\delta L_k(s,\chi)| \leq c(\epsilon,m)(D|1+s|^m)^{(1-\sigma)/2+\epsilon}.$$
\end{theorem}
We can prove the following corollary.
\begin{corollary}\label{korbound}
With the same notations as in Theorem \ref{bound_heckeL} we get for all $\epsilon>0$:
$$|L_k(s,\chi)-\frac{R(\chi)}{s-1}| \leq c(\epsilon,m) (D|1+s|^m)^{(1-\sigma)/2+\epsilon},$$
where $R(\chi)$ denotes the residue of $L_k(s,\chi)$ at $s=1$.
We define $R(\chi)=0$, if $\chi$ is not the trivial character.
\end{corollary}
\begin{proof}
If $\chi$ is not trivial this is Theorem \ref{bound_heckeL}.
For the trivial character $\chi$ with exceptional set $S$ we get:
$$L_k(s,\chi)= \zeta_k(s)\prod_{\idp\in S} \left(1-\frac{1}{\Norm(\idp)^s}\right).$$
Using Lemma \ref{residuum} we get for our residue:
$$|R(\chi)|\leq \tilde c(\epsilon,m) d_k^\epsilon \mbox{ for all }\epsilon>0.$$
Using Theorem \ref{bound_heckeL} and by applying the triangular inequality we find
a new constant $c(\epsilon,m)$ with
$$(s-1)L_k(s,\chi)-R(\chi)\leq c(\epsilon,m)
(D|1+s|^m)^{(1-\sigma)/2+\epsilon}.$$
Since $L_k(s,\chi)-R(\chi)/(s-1)$ is analytic in $s=1$, we get the wanted estimate
for small $|s-1|$ using the maximum principle.
\end{proof}
For our main results we need upper bounds for the number of cyclic
extensions of a number field $k$ which are at most ramified in a given
finite set $S$ of prime ideals. We refer the reader to
\cite[p.123-126]{Lang} for properties of ray class groups which we use
in the proof of the next theorem. In the following we denote by
$\rk_\ell(\Cl_k)$ the {$\ell$--rank} of the class group of $k$. We
remark that we need the following result only for $\ell=2$.
\begin{theorem}\label{upper_Zl_bound}
Let $k$ be an algebraic number field of degree $m$ with $r_1$ real embeddings,
$\ell$ be a prime number, $S$ be a finite set of prime ideals of
$\OO_k$, and
$$S_1:=\{\idp \in S\mid \ell \notin\idp \}.$$
Define $$s:=\begin{cases} \rk_\ell(\Cl\nolimits_k)+|S_1| +2m& \ell>2\\
\rk_\ell(\Cl\nolimits_k)+|S_1|+2m +r_1& \ell=2
\end{cases}.
$$
Then there exist at most $\frac{\ell^s-1}{\ell-1}$
$C_\ell$--extensions of $k$ which are at most ramified in $S$.
\end{theorem}
\begin{proof}
The idea of the proof is to choose $\idm$ in such a way that all $C_\ell$--extensions
are subfields of the ray class field of $\idm$. The infinite places are only important
when $\ell=2$. Each real infinite place may increase the $2$--rank by at most 1.
In case $\ell=2$ we insert all real infinite places in $\idm_\infty$ and define
$$\idm_0:=\prod_{\idp\in S} \idp^{e_\idp},$$
where $e_\idp=1$ for $\idp\in S_1$. For $\idp\in S\setminus S_1$ we have wild ramification
and the following estimates are valid for arbitrary $e_{\idp}>1$.
In the following we compute upper bounds for the $\ell$--rank of
$(\OO_k/\idm_0)^*$. Using the chinese remainder theorem we get:
$$(\OO_k/\idm_0)^* \cong \prod_{\idp\in S} (\OO_k/\idp^{e_\idp})^*
\mbox{ for }\idm_0=\prod_{\idp\in S} \idp^{e_\idp}.$$ In case
$e_\idp=1$ we get that $(\OO_k/\idp)^*$ is the multiplicative group
of a finite field which is therefore cyclic. This explains the
$|S_1|$-part in our formula. In case $e_\idp>1$ we get
$(\OO_k/\idp^{e_\idp})^* \cong (\OO_k/\idp)^* \times
(1+\idp)/(1+\idp^{e_\idp})$. This case can only occur when $\idp$ is
wildly ramified and therefore lies over $\ell$. In this case the
order of the multiplicative group of the residue field is coprime to
$\ell$. The second factor is an $\ell$--group which can be generated
by at most $[k_\idp:\Q_\ell]+1$ elements (see e.g. \cite{HePaPo}).
Since
$$\sum_{\ell\in\idp} [k_\idp:\Q_\ell] = m$$
we get the worst case when all prime ideals above $\ell$ are
contained in $S$ and all corresponding completions have degree 1. In
that case we can estimate the contribution of those prime ideals by
$2m$. The contribution of the unramified extensions to the
$\ell$--rank is estimated by the $\ell$--rank of the class group.
\end{proof}
Unfortunately we do not know good estimates for the $\ell$--rank of
the class group. The best thing we can do in general is to bound
$\ell^{\rk_\ell(\Cl_k)} \leq |\Cl_k|$. The latter expression can be
bounded by the following (see \cite[p. 153]{Nar}).
\begin{theorem}\label{boundclass}
For all $\epsilon>0$ and all $m\in\N$ there exist constants $c(m)$ and $c(m,\epsilon)$ such that
for all number fields $k/\Q$ of degree $m$ we have:
$$|\Cl\nolimits_k| \leq c(m) d_k^{1/2}\log(d_k)^{m-1} \mbox{ and }$$
$$|\Cl\nolimits_k| \leq c(m,\epsilon) d_k^{1/2+\epsilon}.$$
\end{theorem}
\section{Quadratic extensions}
The asymptotics of quadratic extensions of a number field $k$ is well
studied and known. Let us define the following Dirichlet series
corresponding to $Z(k,C_2;x)$:
$$\Phi_{k,C_2}(s) := \sum_{[K:k]=2} \frac{1}{\Norm(d_{K/k})^s}=\sum_{N=1}^{\infty} \frac{a_N}{N^s}.$$
It is known that this Dirichlet series converges for $\Re(s)>1$. Here $a_N$ is the number of
quadratic extensions $K/k$ such that $\Norm(d_{K/k})=N$. This means that $a_N\geq 0$ for all
$N\in\N$. The following theorem is proved in \cite{CoDiOl2}:
\begin{theorem}[Cohen, Diaz y Diaz, Olivier]\label{phiZ2}
Let $k$ be a number field with $i(k)$ complex embeddings. Then we get for $\Re(s)>1$:
$$\Phi_{k,C_2}(s)= -1 +\frac{2^{-i(k)}}{\zeta_k(2s)}\sum_{\idc \mid 2\OO_k}
\Norm(2\OO_k/\idc)^{1-2s}\sum_{\chi} L_k(s,\chi),$$
where $\chi$ runs over the quadratic characters of the ray class group
$\Cl_{\idc^2}$ and $ L_k(s,\chi)$ is the Hecke $L$--series of $k$ corresponding to $\chi$.
\end{theorem}
Using a Tauberian theorem (see e.g. \cite[p. 121]{Nar2}) the following corollary is proved
in \cite{CoDiOl2}.
\begin{corollary}[Cohen, Diaz y Diaz, Olivier]\label{phiZ2res}
$$Z(k,C_2;x) \sim 2^{-i(k)}\frac{\res_{s=1}\zeta_k(s)}{\zeta_k(2)}x,$$
where $2^{-i(k)}\frac{\res_{s=1}\zeta_k(s)}{\zeta_k(2)}$ equals the residue in $s=1$
of $\Phi_{k,C_2}$.
\end{corollary}
Our Dirichlet series has a simple pole at $s=1$ and has a meromorphic
continuation to the left. The proof of the following theorem comes
from the properties of Hecke $L$--series. The number of characters,
i.e. the number of summands can be bounded by the size of the ray
class group which can be bounded up to a constant term depending on
$[K:k]$ by the size of the class group of $k$. The latter one we bound by
$O_{\epsilon,m}(d_k^{1/2+\epsilon})$, where $m=[k:\Q]$. Altogether we get:
\begin{theorem}\label{bound_phi}
$\Phi_{k,C_2}(s)$ has a meromorphic continuation for
$\Re(s)>1/2$. In this area it has only one pole at $s=1$ with residue
$R(k)=\frac{2^{-i(k)}\res_{s=1}\zeta_k(s)}{\zeta_k(2)}$. Furthermore, the
function $g_k(s):=\Phi_{k,C_2}(s)-\frac{R(k)}{s-1}$ is analytic for
$\Re(s)>1/2$ and we get for all $\epsilon>0$ and $\Re(s)>1/2$:
$$|g_k(s)| \leq c(\epsilon,m) (d_k|1+s|^{m})^{(1-\sigma)/2+\epsilon}d_k^{1/2}.$$
\end{theorem}
\section{Wreath products}
\label{sec:wr}
Let $H_1\leq S_e$ and $H_2\leq S_d$ be two transitive groups and
assume $n=ed$. Then the wreath product $H_1\wr H_2 \cong H_1^d
\rtimes H_2 \leq S_n$ is a semidirect product, where $H_2\leq S_d$
permutes the $d$ copies of $H_1^d$. For a formal definition we refer
the reader to \cite[p. 46]{DiMo}. The wreath product has a nice field
theoretic interpretation in Galois theory. Assume that we have a field
tower $L/K/k$ such that $\Gal(L/K)=H_1$ and $\Gal(K/k)=H_2$. Then we
get that $\Gal(L/k)\leq H_1\wr H_2$.
We want to study the asymptotic behavior of our counting function
$Z(k,G;x)$ for wreath products $G=H_1\wr H_2$ when we assume that we
have some information for the corresponding counting functions for
$H_1$ and $H_2$. First results in this direction already appear in
\cite{Ma4}. The $a(G)$-part of the following lemma is \cite[Lemma
5.1]{Ma4}.
\begin{lemma}
Let $k$ be a number field and $H_1\leq S_e,H_2\leq S_d$ be
transitive groups. Let $G:=H_1\wr H_2$. Then
$$a(G) = a(H_1) \mbox{ and }b(k,G) = b(k,H_1).$$
\end{lemma}
\begin{proof}
Let $g=(h_1,h_2)\in H_1\wr H_2$ where
$h_1=(h_{1,1},\ldots,h_{1,d})\in H_1^d$ and $h_2$ is the image of
$g$ under the projection to the complement $H_2$. If $h_2\ne 1$ then
$g$ interchanges at least two blocks. Therefore the number of orbits
is at most $(d-2)e+e=(d-1)e$. On the other hand, if $h_2=1,
h_{1,2}=\cdots=h_{1,d}=1$ then $g$ has at least $(d-1)e+1$ orbits.
Thus we may assume that $h_2=1$ and elements with minimal index have
the property that $d-1$ of the $h_{1,i}$ equal 1. By conjugating
with a suitable element of type $(1,\tilde h_2)\in G$ we can assume
that $h_{1,2}=\cdots=h_{1,d}=1$. Now let $h\in H_1$ be an element of
minimal index $e-\ell$. Then $\ind(((h,1,\ldots,1),1))=
n-(d-1)e-\ell=e-\ell$. This shows $a(H_1)=a(G)$. It is clear that
$h$ and $\tilde h\in H_1$ are conjugated in $h_1$ if and only if
$((h,1,\ldots,1),1)$ and $((\tilde h,1,\ldots,1),1)$ are conjugated
in $G=H_1\wr H_2$. $h$ and $\tilde{h}$ are in the same
$k$--conjugacy class if a suitable power $\tilde{h}^a$ is conjugated
to $h$. This statement remains true in the wreath product
representation. Therefore we get the second statement.
\end{proof}
\section{Wreath products of the form $C_2 \wr H$}
\label{sec:C2wr}
In this section we prove Conjecture \ref{con} for groups $G=C_2 \wr H$,
where we need to assume weak properties of the asymptotic function for $H\leq S_d$. The proofs
are inspired by the methods described in \cite{CoDiOl2}, where the corresponding results
were shown for $G=D_4 \cong C_2 \wr C_2$.
Let $L/k$ be an extension with Galois group $G=C_2\wr H$. Then there exists a subfield
$K\leq L$ such that $\Gal(L/K)=C_2$ and $\Gal(K/k)=H$. In a first step of our proof we will
count all "field towers" of this type, i.e. we count all extensions $L/k$ such that there
exists an intermediate field $K$ with $\Gal(L/K)=C_2$ and $\Gal(K/k)=H$. We remark that
$\Gal(L/k)\leq C_2 \wr H$ using a theorem of Krasner and Kaloujnine \cite{KraKal}. In a second
step of the proof we show that the asymptotics of proper subgroups which occur in such field
towers is strictly less.
In \cite[Proposition 8.3]{KlMa2} we already proved the following upper bound for wreath
products of this type. We remark that we weakened the assumption by replacing the
exponent $a(H)+\delta$ by $1+\delta$. The same proof gives the new result.
\begin{proposition} \label{upper_wreath}
Let $k$ be a number field, $H\le S_d$ be a transitive permutation group such
that $Z(k,H;x)\le c(k,H,\delta)\,x^{1+\delta}$ for all $\delta>0$. Then
for any $\epsilon>0$ there exists a constant $c(k,C_2\wr H,\epsilonilon)$ such that
$$Z(k,C_2\wr H;x)\leq c(k,C_2\wr H,\epsilonilon)\, x^{a(C_2\wr H)+\epsilonilon}\,.$$
\end{proposition}
We remark that $a(C_2\wr H)= a(C_2) =1$. Furthermore we remark that the
proof counts all fields towers $L/K/k$ as above. Therefore the same upper bound
applies.
In the following let us assume that for all $\epsilon>0$ we have
$$Z(k,H;x) \leq c(k,H,\epsilon) x^{1+\epsilon}.$$
We remark that using the results in \cite{KlMa2} this assumption is true for
all $p$-groups. Using results proved in \cite{ElVe} this assumption is also
true for all regular $H$, i.e. when $K/k$ is normal.
For the first step we define the corresponding counting function
$$\tilde{Z}(k,C_2\wr H;x):=
\#\{L/k\mid \exists K: \Gal(L/K)=C_2,
\Gal(K/k)=H, \Norm(d_{L/k}) \leq x\}.$$
Using our assumption on $H$ and Proposition \ref{upper_wreath} we get for all $\epsilon>0$
that
$$\tilde{Z}(k,C_2\wr H;x) \leq c(k,H,\epsilon) x^{1+\epsilon}.$$
Let us associate the corresponding Dirichlet series to $\tilde Z(k,C_2\wr H)$. Define
$$\K_H:=\{K/k \mid \Gal(K/k)= H\}$$ and
\begin{equation}\label{eq:phi}
\Phi(s) := \sum_{K\in \K_H} \frac{\Phi_{K,C_2}(s)}{\Norm(d_{K/k})^{2s}}
= \sum_{N=1}^{\infty} \frac{a_N}{N^s},
\end{equation}
where $\Phi_{K,C_2}(s)$ is the Dirichlet series associated to $Z(K,C_2;x)$.
Since we know that $\tilde{Z}(k,C_2\wr H;x) \leq c(k,H,\epsilon) x^{1+\epsilon}$ we get
that the Dirichlet series $\Phi(s)$ converges for $\Re(s)>1$.
\begin{theorem}\label{Satz:kranz}
Assume that there exists at least one extension of $k$ with Galois group $H$ and
that the following estimate holds:
$$Z(k,H;x)=O_{k,H,\epsilon}(x^{1+\epsilon}).$$
Then the function $\Phi(s)$ defined in equation \eqref{eq:phi} has a meromorphic
continuation to $\Re(s)>7/8$. In this area it has exactly one pole at $s=1$.
\end{theorem}
\begin{proof}
Using Theorem \ref{bound_phi} the result is trivial if there are only finitely many extensions
of $k$ with Galois group $H$. We remark that $d_K$ and $\Norm(d_{K/k})$ only differ by a constant depending
on $k$ and $H$ since $d_K=d_k^{[K:k]}\Norm(d_{K/k})$. Using our assumption we get that the Dirichlet series
\begin{equation} \label{eq:d_K}
\sum_{K\in \K_H} \frac{1}{\Norm(d_{K/k})^s}
\end{equation}
converges absolutely and locally uniformly for $\Re(s)>1$. We consider the function
$$g(s):=\sum_{K\in \K_H}
\frac{\Phi_{K,C_2}(s)-R(K)/(s-1)}{\Norm(d_{K/k})^{2s}},$$
where $R(K)$ is the residue of $\Phi_{K,C_2}$ at $s=1$.
Using Theorem \ref{bound_phi} we get that $g_K(s):=\Phi_{K,C_2}(s)-R(K)/(s-1)$
is an analytic function for $\Re(s)>7/8$. For all
$\epsilon>0$ we derive the following estimate
$|g_K(s)|=O_{\epsilon,[k:\Q]}(|d_K(s+1)^{[K:\Q]}|^{1/2+1/16+\epsilon})$.
Since
$$2\frac{7}{8}-\frac{9}{16}=\frac{19}{16}>1\mbox{ and \eqref{eq:d_K}}$$
we get that the Dirichlet series
$$\sum_{K\in \K_H} \frac{g_K(s)}{\Norm(d_{K/k})^{2s}}$$
converges absolutely and locally uniformly for
$\Re(s)>7/8$. Therefore
$g(s)$ is an analytic function for $\Re(s)>7/8$.
Using Lemma \ref{residuum} we have $R(K) = O_{\epsilon,[k:\Q]}(d_K^\epsilon)$ for
all $\epsilon>0$. Since $d_K=d_k^{[K:k]}\Norm(d_{K/k})$ we get that
$$\frac{1}{s-1}\sum_{K\in \K_H} \frac{R(K)}{\Norm(d_{K/k})^{2s}}$$
converges absolutely and locally uniformly for all regions which are contained in
$\{s\in\C \mid \Re(s)>7/8 \mbox{ and }s\ne 1\}$. The absolute convergence of all considered
series gives the wished result for
$$\Phi(s)=g(s)+\sum_{K\in \K_H} \frac{R(K)/(s-1)}{\Norm(d_{K/k})^{2s}}.$$
\end{proof}
As an application of a suitable Tauberian theorem we immediately get:
\begin{corollary}
Using the same assumptions as in Theorem \ref{Satz:kranz} we get:
$$\tilde{Z}(k,C_2 \wr H;x) \sim \res_{s=1}(\Phi(s)) x.$$
\end{corollary}
In the following we would like to show that
$$\tilde{Z}(k,C_2 \wr H;x) \sim Z(k,C_2 \wr H;x)$$ holds, i.e. extensions which do not have
the wreath product as Galois group do not contribute to the main term. We need some group
theory.
\begin{definition}\label{block}
Let $G\leq S_n$ be a transitive group operating on $\Omega=\{1,\ldots,n\}$.
Then $\Delta\subseteq \Omega$ is called a block of $G$, if
$\Delta^g \cap \Delta\in\{\Delta,\emptyset\}$ for all $g\in G$. If $G$ only
contains blocks of size 1 or $n$ we call $G$ primitive. Otherwise $G$ is called
imprimitive.
\end{definition}
\index{primitiv}\index{imprimitiv}\index{Block}
We remark that a field extension $L/k$ contains non-trivial subfields if and only if
$\Gal(L/k)$ is imprimitive. The blocks containing 1 are in 1-1 correspondence to the
subfields of $L/k$.
\begin{lemma}\label{lem:transposition}
Let $G\leq S_n$ be a transitive group containing a transposition. Then:
\begin{enumerate}
\item All transpositions are conjugated in $G$, i.e. $b(k,G)=1$.
\item $G= S_e \wr H$ for $1\ne e$, $e\mid n$ and $H\leq S_{n/e}$ transitive.
\end{enumerate}
\end{lemma}
\begin{proof}
The first part is \cite[Lemma 2.2]{Ma5}. If $G$ is primitive the
second statement is \cite[Theorem 3.3A]{DiMo}. Assume that
$\tau=(i,j)$ is a transposition of $G$ and $B$ is a minimal block of
size larger than 1 containing $i$. Then $\tau(i)=j\in B$ since all
the other elements in $B$ are fixed by $\tau$. Therefore $G|_B$
contains a transposition and operates primitively on $B$ ($B$ is a
minimal block). Therefore the operation of $G|_B$ on $B$ is
isomorphic to $S_{|B|}$. Let $\tilde{B}$ be a conjugated block of
$B$. By conjugating $\tau$ we can find a transposition in $\tilde
B$. Therefore we find $n/|B|$ different copies of $S_{|B|}$.
Therefore $G\cong S_{|B|} \wr H$, where $H$ is the image of the natural
homomorphism $\varphi: G \rightarrow S_{n/|B|}$ which permutes the
conjugated blocks.
\end{proof}
Now we apply this lemma to our situation of field towers. Having a subfield $K$ with
$L/K$ of degree $e=2$ means that $\Gal(f)$ contains a block system of blocks of size 2.
\begin{lemma}\label{lem:tower}
Let $L/K/k$ be extensions of number fields with $\Gal(K/k)=H$ and $[L:K]=2$. Let $p$ be
a prime which is unramified in $K/k$ and assume $p||\Norm(d_{L/K})$. Then $\Gal(L/k)=C_2\wr H$.
\end{lemma}
Note that $p$ unramified in $K/k$ and $p||\Norm(d_{L/K})$ is equivalent to $p||\Norm(d_{L/k})$.
\begin{proof}
$\Gal(L/k)$ contains a transposition since $p||\Norm(d_{L/K})$. Let $\tau=(i,j)$ be such
a transposition and $B$ a minimal block of $\Gal(f)$ corresponding to $K$ which contains $i$.
When we apply the proof of Lemma \ref{lem:transposition} to this situation we get the
wanted result.
\end{proof}
We remark that we can replace the prime $p$ in the above lemma by an unramified prime
ideal $\idp\subseteq \OO_k$. This does not improve the following estimates.
In the following we would like to count all field towers $L/K/k$
counted by $\tilde{Z}(k,C_2\wr H;x)$ such that $\Gal(L/k)$ is a proper subgroup of $C_2\wr H$.
Therefore we define
$$Y(k,C_2\wr H;x):=$$
$$\#\{L/K/k\mid \Gal(L/k)\ne C_2\wr H,\Gal(K/k)=H,[L:K]=2,\Norm(d_{L/k})\leq x\}.$$
We find upper bounds for this function when we count all field towers $L/K/k$ which do not satisfy
the assumptions of Lemma \ref{lem:tower}. Before we examine those field towers we need a
definition.
\begin{definition}
Let $a\in\N$ be a positive integer and $S\subseteq \PP$ be a set of primes. Then
$a^S$ is defined to be the largest divisor of $a$ coprime to $S$.
\end{definition}
For a field tower $k\subset K \subset L$ we get:
$$\Norm(d_{L/k})=\Norm(d_{K/k}^2)\Norm(d_{L/K})
\geq \Norm(d_{K/k}^2) \Norm(d_{L/K})^{S_K},$$
where $S_K:=\{p\in\PP\mid p | \Norm(d_{K/k})\}$.
We define
$$\hat Z^{S_K}(K,C_2;x):=\#\{L/K \mid \Gal(L/K)=C_2, \Norm(d_{L/K})^{S_K}\leq x, $$
$$p \mid (\Norm(d_{L/K}))^{S_k} \Rightarrow p^2 \mid (\Norm(d_{L/K}))^{S_k} \forall p\in\PP\}$$
and get
$$Y(k,C_2\wr H;x) \leq \sum_{K\in \K_H(x^{1/2})} \hat Z^{S_K}(K,C_2;x/\Norm(d_{K/k}^2)),$$
where $\K_H(x):=\{K\in \K_H \mid \Norm(d_{K/k}) \leq x\}$.
We need an estimate for $\hat Z^{S_K}(K,C_2;x)$. We denote by $a_N$ the number of fields $L$
such that $\Norm(d_{L/K})^{S_K}=N$. Since we ignore all primes in $S_K$ and all other prime
divisors occur with multiplicity at least 2, we get that $a_N=0$ if there exists a prime
$p$ with $p||N$. We choose $S\subseteq \PP(K)$ as the smallest set containing all prime ideals
which lie over a prime in $S_K$ or over a prime dividing $N$. We are interested in the number
of quadratic extensions of $K$ which are at most ramified in prime ideals contained in $S$.
We get $|S|\leq (\omega(N)+|S_K|)t$, where $\omega(N)$ is the number of different prime factors
and $t:=[K:\Q]$.
Using Theorems \ref{upper_Zl_bound} and \ref{boundclass} we get
$$a_N \leq 2^{\rk_2(\Cl_K)}2^{t (\omega(N)+|S_K|)}2^{3t} \leq c(t,\epsilon)d_K^{1/2+\epsilon}2^{t \omega(N)}.$$
Therefore we get:
$$\sum_{N\leq x} a_N \leq c(t,\epsilon)d_K^{1/2+\epsilon} \sum_{N\leq x^{1/2}} 2^{t\omega(N)}.$$
Using $\sum_{N\leq x}(2^t)^{\omega(N)} =O(x^{1+\epsilon})$ we get with a new constant $c(t,\epsilon)$:
$$\hat Z^{S_K}(K,C_2;x) \leq c(t,\epsilon)d_K^{1/2+\epsilon} x^{1/2+\epsilon}.$$
Inserting this in the above estimate for $Y(k,X_2\wr H;x)$ we get using $d_K=d_k^2\Norm(d_{K/k})$:
$$Y(k,C_2\wr H;x)
\leq \sum_{K\in \K_H(x^{1/2})} c(t,\epsilon)(d_k^2\Norm(d_K))^{1/2+\epsilon}
\left(\frac{x}{\Norm(d_{K/k}^2)}\right)^{1/2+\epsilon} $$
$$\leq c(t,\epsilon) d_k^{1+2\epsilon}x^{1/2+\epsilon}
\sum_{K\in \K_H(x^{1/2})} \frac{\Norm(d_{K/k})^{1/2+\epsilon}}{\Norm(d_{K/k})^{1+2\epsilon}}$$
Using $\Norm(d_{K/k})\leq x^{1/2}$ we get:
$$Y(k,C_2\wr H;x)
\leq c(t,\epsilon) d_k^{1+2\epsilon}x^{1/2+\epsilon} x^{1/4+\epsilon} \sum_{K\in \K_H(x^{1/2})}
\frac{1}{\Norm(d_{K/k})^{1+2\epsilon}}.$$
The last sum converges under the assumption for $H$ of Theorem \ref{Satz:kranz}.
This proves for all $\epsilon>0$ the following estimate:
$$Y(k,C_2\wr H;x) \leq c(k,H, t,\epsilon) x^{3/4+2\epsilon}.$$
Since $Z(k,C_2\wr H;x) + Y(k,C_2\wr H;x) =\tilde{Z}(k,C_2\wr H;x)$ and Theorem
\ref{Satz:kranz} we proved the following:
\begin{theorem}\label{mainwreath}
Assume the same as in Theorem \ref{Satz:kranz}. Then the Dirichlet
series corresponding to $Z(k,C_2\wr H)$ has a meromorphic
continuation to $\Re(s)>7/8$, where $s=1$ is the only pole in that
region. The residue $r$ of that pole coincides with the one of the
function $\Phi(s)$. We get:
$$Z(k,C_2\wr H;x) \sim \res_{s=1}(\Phi(s))x.$$
\end{theorem}
We are able to give an expression for this residue as a convergent sum.
\begin{corollary}
$$\res_{s=1}(\Phi(s)) = \sum_{K\in \K_H} \frac{\res_{s=1}\zeta_K(s)}{2^{i(K)}d_K^2\zeta_K(2)}.$$
\end{corollary}
These results support our main conjecture.
\begin{corollary}\label{Cor1}
Conjecture \ref{con} is true for all
$C_2\wr H$ and all number fields $k$ such that $H$ fulfills the assumptions of Theorem
\ref{Satz:kranz}.
\end{corollary}
We are already remarked that this assumption is true for all $p$--groups and all
permutation groups in regular representation. Therefore we get the following
corollary.
\begin{corollary}\label{Cor2}
For even $n$ there always exists a group $G\leq S_n$ with $a(G)=1$ and
$$Z(k,G;x) \sim c(k,G) x =c(k,G)x^{a(G)}.$$
\end{corollary}
\section*{Acknowledgments}
I would like to thank Gunter Malle for many discussions about this topic. This project was partially supported
by the Deutsche Forschungsgemeinschaft (DFG).
\end{document} |
\begin{document}
\title{How Sampling Impacts the Robustness of Stochastic Neural Networks}
\begin{abstract}
Stochastic neural networks (SNNs) are random functions
whose predictions are gained by averaging over multiple realizations.
Consequently, a gradient-based adversarial example is calculated based on one set of samples and its classification on another set.
In this paper, we derive a sufficient condition for such a stochastic prediction
to be robust against a given sample-based attack.
This allows us to identify the factors that lead to an increased robustness of SNNs and gives theoretical explanations for:
(i) the well known observation, that increasing the amount of samples drawn for the estimation of adversarial examples increases the attack's strength,
(ii) why increasing the number of samples during an attack can not fully reduce the effect of stochasticity,
(iii) why the sample size during inference does not influence the robustness, and
(iv) why a higher gradient variance and a shorter expected value of the gradient relates to a higher robustness.
Our theoretical findings give a unified view on the mechanisms underlying previously proposed approaches for increasing attack strengths or model robustness
and are verified by an extensive empirical analysis.
\end{abstract}
\section{Introduction}
Since the discovery of adversarial examples~\citep{biggio_adv_att, intruding_Szegedy},
a significant amount
of research was dedicated to hinder
attacks~\citep[e.g.][]{ madry2018towards,papernot_distillation, trades}, to enhance attack strategies~\citep[e.g.][]{ obfuscated_grad, survey_adv_vision, carlini_bypassing, obscurity_robustness} or to derive ways to certify model robustness~\citep[e.g.][]{cohen2019certified, lecuyer2019certified}.
Robustness guarantees often
specify an $\epsilon$-ball around input points in which perturbations do not lead to a label change \citep{hein_formal_gurantees, Croce2020Provable}.
The maximal possible radius of such an $\epsilon$-ball corresponds to the distance of the input point to the nearest decision boundary, which on the other hand is equal to the length of the smallest perturbation vector that leads to a misclassification (c.f. figure~\ref{fig:decision_boundaries}a)).
Such a robustness analysis assumes
that the decision boundaries are fixed and
that the attacker is able to estimate (at least approximately) this minimal perturbation vector, which is a reasonable assumption for deterministic networks but usually does not hold for stochastic neural networks (SNNs).
Stochastic neural networks, and stochastic classifiers more generally, are random functions and predictions are given by the expected value
of the random function for the given input. In practice, this expectation is usually
not tractable and hence it is approximated by averaging over multiple realizations of the random function. This approximation leads to the challenging setting where predictions, decision boundaries, and gradients become random variables themselves.
Hence, under an adversarial attack, the decision boundaries used for calculating the
adversarial example
and those used when predicting the label
of
the resulting
adversarial example differ.
This means that the attacker can not estimate the optimal perturbation direction
i.e., the direction to the closest
decision boundary
of
the network that will be sampled during inference c.f. figure~\ref{fig:decision_boundaries} b), c).
In this paper,
we study how robustness of SNNs arises from this misalignment of the attack direction and the optimal perturbation direction during inference that results from the stochasticity inherent to stochastic classifiers.
We make the following contributions:
First, we derive a
sufficient condition for a SNN prediction relying on one set of samples to be robust against
an attack that was calculated on a second set of samples.
Second, we discuss how
model properties and sample sizes impact this condition
which does not only allows us to answer the questions stated in the abstract but also to explain the success of recently proposed defense mechanism from a simple unifying geometric perspective.
Lastly, we conduct an empirical analysis that demonstrates that the novel theoretical insights perfectly match what we observe in practice.
\begin{figure}
\caption{
(Un-)successful attacks on a binary stochastic classifier with a linear decision boundary.
a) an adversarial example
$x_{\text{adv}
\label{fig:decision_boundaries}
\end{figure}
\section{Related work}
Several works proposed stochastic defense mechanism to increase adversarial robustness~\citep[e.g.][]{BART, xie2018mitigating}. \citet{obfuscated_grad} linked their success to gradient obfuscating during the attack and showed
that increasing the number of samples for approximating the gradient during attack leads to a sever decrease in adversarial accuracy.
However, SNNs were still found
to have an increased robustness even w.r.t.~stronger attacks~\citep{simpleSNN, He_2019_CVPR, weight_covariance-eustratiadis21a,learn2perturb}.
Their success
was attributed to different effects of stochasticity, e.g.~model smoothing~\citep{liu_selfensemble, Addepalli_2021_CVPR} or
diversification of the gradients~\citep{GradDiv, diverse_directions_bender20a}.
While a lot of work analyzed the robustness of deterministic neural networks \cite[e.g.][]{madry2018towards, croce_max_lin_regions, Croce2020Provable, cvpr_robust, yang2021ensemble_robust},
the robustness of
SNNs
is less well understood. One line of research
focused on the robustness of Bayesian neural networks (BNNs).
\citet{wicker2020probabilistic} certified robustness of BNNs using
interval bound propagation techniques which they later also employed
to derive guarantees for the robustness of BNNs
with modified adversarial training \citep{wicker_adv_train_BNN}.
Moreover, \citet{carbone2020robustness} investigate robustness
in the infinite-width infinite-sample limit.
Lastly, \citet{nips19_theory_robustness_randomization} derived theoretical robustness guarantees for randomized networks,
where the randomization
is based on additive noise from an exponential family distribution.
This leads to a generalization of the robustness guarantees derived previously by~\citet{lecuyer2019certified} and~\citet{cohen2019certified}. Those certified robustness guarantees specify the radius of an $\ell_2$-ball in which the prediction does not change. In contrast, our results specify the robustness that results from the difficulty to identify the ideal attack direction and imply that even for perturbations outside this confidence ball, a gradient-based attack has a chance of not being successful.
To the best of our knowledge, no
existing theoretical analysis of SNNs explicitly discusses either the effect of stochasticity during inference nor the
impact
of the sample size during attack and prediction on the robustness.
\section{Preliminaries}
\label{sec:preliminaries}
We first clarify the terminology before we state the main theoretical results of our paper.
\paragraph{Stochastic classifiers}
\label{subsec:stoch_classifiers}
We use the term \textit{stochastic classifiers} for all classifiers which have an inherent stochasticity
through the use of random variables in the model. Formally, we define them as follows:
\begin{definition}[Stochastic classifiers]
A stochastic classifier with $k$ classes corresponds to a function
$ f:\mathbb{R}^d\times \Omega^h \rightarrow \mathbb{R}^k$ that maps a pair $(x, \Theta)$ to the output $f(x, \Theta)=( f_1(x,\Theta), \dots f_k(x,\Theta))^T$, where $x \in \mathbb{R}^d$ is an input vector, $\Theta \in \Omega^h, \Theta \sim p(\Theta)$ is a random vector, and $f_c(x,\Theta)$ with $c \in \{1,\dots,k\}$ are the discriminant functions for each class.
The prediction of a stochastic classifier for
an input $x$
is given by $
\mathbb{E}_{\Theta}
[f(x,\Theta)]$
and the predicted class by
$
\arg \max_{c}
\mathbb{E}_{\Theta}
[f_c(x,\Theta)] .
$
\end{definition}
This generic definition of stochastic classifiers
covers linear models with random
weights, but also more complicated methods like
BNNs~\citep{neal}, infinite mixtures~\citep{daubener2020investigating}, Monte Carlo dropout networks \citep{mc_dropout_gal},
randomized smoothing as proposed by \citet{lecuyer2019certified}f^{\mathcal{I}}_{y-c}ootnote{Note, that,
in contrast to \citet{lecuyer2019certified} the variant of randomized smoothing proposed by
\citet{cohen2019certified}
does not
define
the
prediction of the SNN to be given by the
expectation over $\Theta$
but
by
$\arg \max _{c \in \mathcal{Y}} P(f(x+ \epsilon) = c)$, where $\epsilon \sim \mathcal{N}(0, \sigma^2 I)$.
This makes our results not directly applicable to their networks.
However,
they can probably be transferred to the decision boundaries and attacks corresponding to their form of generating predictions.
},
and any other class
of neural networks which use stochasticity at the input level~\citep[e.g.][]
{
BART}
or within the network~\citep[e.g.][]{He_2019_CVPR, learn2perturb, liu_selfensemble, simpleSNN, weight_covariance-eustratiadis21a}.
In cases where
$
\mathbb{E}_{\Theta} [f(x,\Theta)]$
is not tractable --- which is in practice usually the case ---
the prediction of the stochastic classifier is
approximated by its Monte Carlo (MC) estimate
$
f^{\mathcal{S}}(x) := f^{\mathcal{I}}_{y-c}rac{1}{S}\sum_{s=1}^S f(x, \theta_s)
$,
where the
samples in the sample set $\mathcal{S}= \{\theta_1,\dots, \theta_S\}$
are drawn from $p(\Theta)$.
\paragraph{Adversarial attacks on stochastic classifiers}
\label{subsec:adv_attacks}
Informally speaking, adversarial examples are inputs that
are modified such that
the network predicts wrong classes even though
the changes to the inputs are not
perceptible
for a human.
More precisely, let $x$ be an
input with corresponding true label $y \in \{1,\dots,k\}$ that is classified correctly by the multi-class classifier $f(\cdot)$, that is $\arg \max_c f_c(x)=y.$
We consider the most common attack form
which
aims at
misclassifying
$x$ by allowing for some predefined maximum magnitude of perturbation. That is, the attacker targets the optimization problem
\begin{align} \label{eq:max_allowable_attack}
&\mathit{maximize} \;\;\; \mathcal{L}(f(x+\delta), y) \enspace , \; \; \mathit{s.t.} \;\;\; \|\delta\|_p \leq \eta \enspace,
\end{align}
where $\mathcal{L}(\cdot, \cdot)$
is the loss function,
$\|\cdot\|_p$ with $p \geq 1$ is the $\ell_p$-norm,
and $\eta$ is the \textit{perturbation strength}, i.e.~the maximal allowed magnitude of the attack (see
figure~\ref{fig:decision_boundaries}a) for an illustration).
Common choices of loss functions include
the cross-entropy loss and the negative margin loss $\mathcal{L}_{\text{margin}}(f(x+\delta), y) = -(f_y(x+\delta )- \max_{c\neq y} f_c(x+\delta))$.
In practice, targeting the optimization problem in eq.~\eqref{eq:max_allowable_attack} usually involves estimating $\delta$ by performing some kind of gradient-based optimization on the loss function~\citep[e.g.][]{Goodfellow_fgsm, madry2018towards}.
If $f^{\mathcal{S}}(\cdot)$ is a stochastic classifier (as introduced in the previous paragraph) approximated by its MC estimate, the loss gradient with respect to the input is stochastic as well and given by
\begin{equation}
\label{eq:stochastic_gradient}
\nabla_x \mathcal{L}(f^{\mathcal{S}}(x), y) = \nabla_x \mathcal{L}\left( f^{\mathcal{I}}_{y-c}rac{1}{S}\sum_{s=1}^S f(x,\theta_s), y\right) \enspace .
\end{equation}
Note, that for linear $\mathcal{L}$, as for example the margin loss for a fixed class $c$, the loss gradient of the mean prediction is equivalent to the mean of the loss gradients for single sample predictions. However, in general (e.g.~for the cross entropy loss) this is not the case.
\section{Geometrical robustness analysis}
\label{sec:theory}
After clarifying our understanding of stochastic classifiers and on how gradient-based adversarial attacks are conducted on them, we are now able to present
a
simple
but general geometrical, adversarial robustness analysis for
stochastic classifiers. It is motivated by the following observation:
Each time a stochastic classifier is used for
a prediction, another set of realizations from the random vectors
are drawn, resulting in another set of discriminant functions and corresponding decision boundaries.
To put it into other words, the classifier gets a random variable itself.
As a consequence, calculating a gradient-based adversarial attack for an input $x$ is done with respect to a drawn set of realizations
$\mathcal{A}=\{\theta^a_1, \theta^a_2, \dots, \theta^a_{S^{\mathcal{A}}} \}$, and thus $\delta$ from eq.~\eqref{eq:max_allowable_attack} is specific for a
realization $f^{\mathcal{A}}(x)$
of the classifier,
which we specify by writing $\delta^{\mathcal{A}}$.
During inference, the resulting adversarial example $x_{\text{adv}} = x+ \delta^{\mathcal{A}}$
is then fed to another random classifier $f^{\mathcal{I}}$, which is based on a different set of realizations from the random vector
$\mathcal{I}=\{\theta^i_1,\theta^i_2, \dots, \theta^i_{S^{\mathcal{I}}}\}$.
From this perspective, the prediction
model
$f^{\mathcal{I}}$ is robust against the attack if the distance from $x$ to the decision boundary (given by $f^{\mathcal{I}}_y(x)- f^{\mathcal{I}}_{c \neq y}(x) =0$ ) in the \emph{direction of $\delta^{\mathcal{A}}$} is larger than the length of $\delta^{\mathcal{A}}$, as illustrated in figures~\ref{fig:decision_boundaries}b) and c).
\paragraph{Robustness conditions for stochastic attacks}
For a classifier with linear discriminant functions, we are able to turn the previously described observation
into a theorem in which we
derive a sufficient and necessary condition for the prediction
model
to be robust against a given attack.
\begin{theorem}[Sufficient and necessary robustness condition for linear classifiers]
\label{thm:linear}
Let $f:\mathbb{R}^d\times \Omega^h \rightarrow \mathbb{R}^k$ be a stochastic classifier with linear discriminant functions and $f^{\mathcal{A}}$ and $f^{\mathcal{I}}$ be two MC estimates of the classifier.
Let
$x \in \mathbb{R}^d $ be a data point with label $y \in \{1,\dots,k\}$ and $\arg \max_c f_c^{\mathcal{A}}(x)= \arg \max_c f_c^{\mathcal{I}}(x)=y$, and let $x_{\text{adv}}=x+ \delta^{\mathcal{A}}$ be an adversarial example computed for solving the minimization problem~\eqref{eq:max_allowable_attack}
for $f^{\mathcal{A}}$.
It holds that $\arg \max_{c} f^{\mathcal{I}}_c(x+ \delta^{\mathcal{A}}) =y $ if and only if
\begin{equation}\label{eq:r_linear}
\min_{c\neq y} \tilde{r}^{\mathcal{I}}_c > \|\delta^{\mathcal{A}}\|_2 \enspace, \text{ with }
\end{equation}
\begin{equation}
\tilde{r}^{\mathcal{I}}_c = \begin{cases}
\infty \enspace , &\text{if} \;\; \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})= f^{\mathcal{I}}_{y-c}rac{\langle -\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)), \delta^{\mathcal{A}} \rangle}{\|\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)) \|_2 \cdot \|\delta^{\mathcal{A}} \|_2} \leq 0 \\
f^{\mathcal{I}}_{y-c}rac{ f^{\mathcal{I}}_y \left(x \right) - f^{\mathcal{I}}_c\left(x \right) }{ \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) ) \|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})} \enspace ,
&\text{otherwise \enspace,
}
\nonumber
\end{cases}
\end{equation}
where $\alpha_c^{\mathcal{I}, \mathcal{A}}$ is the angle between $-\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) )$ and $\delta^{\mathcal{A}}$.
\end{theorem}
The proof which is based on Taylor expansion is given in supplement~A.
The conditions for $\cos(\alpha_c^{\mathcal{I}, \mathcal{A}})$ have a nice geometrical interpretation:
An angle of more than 90°
(which corresponds to a negative cosine value)
indicates that the gradient $-\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) )$ and
the perturbation
$\delta^{\mathcal{A}}$ point into ``opposite'' directions and thus even for infinitely long moves into the direction of $\delta^{\mathcal{A}}$, the predicted label for $x$ will not change to class $c \neq y$.
For positive cosine values,
$\tilde{r}^{\mathcal{I}}_c$
specifies the distance to the decision boundary in the attack direction. It looks similar to the minimal distance to the decision boundary in a deterministic
setting which is given by $f^{\mathcal{I}}_{y-c}rac{ f_y \left(x \right) - f_c\left(x \right) }{ \| \nabla_x (f_y(x) - f_c(x) ) \|_2 }$ and which
is recovered if $\cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) =1$.
A cosine value of one however can only occur if the gradient and perturbation direction are identical
i.e.,~if the margin loss is used for calculating the attack direction and
if attack and inference model are identical.
In practice, the latter is almost surely not the case due to
the finite sample approximation, and thus
$\cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) < 1$.
This illustrates the robustness advantage induces by stochasticity.
The derived conditions may locally hold for classifiers which can be reasonably well approximated by a first-order Taylor approximation. To derive further guarantees, we can relax the linearity assumption by assuming discriminant functions which are $L$-smooth as defined in the following.
\begin{definition}[$L$-smoothness~\citep{yang2021ensemble_robust}]
A differentiable function $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ is $L$-smooth, if for any $x_1, x_2\in \mathbb{R}^d$ and any output
$c\in \{1,\dots,k\}$:
\[
f^{\mathcal{I}}_{y-c}rac{\| \nabla_{x_1} f_c(x_1)- \nabla_{x_2} f_c(x_2)\|_2}{\| x_1 - x_2\|_2} \leq L \enspace.
\]
\end{definition}
For such smooth discriminant function we can derive a sufficient (but not necessaryf^{\mathcal{I}}_{y-c}ootnote{The necessity of this condition is not given since for non-linear decision boundaries it is possible that behind a region of a different class there exists another region of class $y$ that an attack ends in if $\delta^{\mathcal{A}}$ is long enough.}) robustness condition specified by the following theorem, which is proven in supplement~A.
\begin{theorem}[Sufficient condition for the robustness of a L-smooth stochastic classifier]
\label{thm:robustness_beta}
In the setting of Theorem~\ref{thm:linear}, let $f^{\mathcal{A}}$ and $f^{\mathcal{I}}$ be $L$-smooth (instead of linear) discriminant functions.
Then it holds that $\arg \max_c f_c^{\mathcal{I}}(x+ \delta^{\mathcal{A}}) = y$ if $\min_{c \neq y} r_c^{\mathcal{I}}> \| \delta^{\mathcal{A}}\|_2$ with
\begin{equation*}
r_c^{\mathcal{I}} = \begin{cases}
&\infty \enspace , \enspace \text{if} \enspace \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x))\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 \leq 0 \enspace , \\
&f^{\mathcal{I}}_{y-c}rac{f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)}{ \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x))\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 } \enspace , \enspace \text{otherwise
} \end{cases} \\
\end{equation*}
with $\cos(\alpha^{\mathcal{I}, \mathcal{A}}_c)$ as in theorem~\ref{thm:linear}.
\end{theorem}
\paragraph{Factors influencing the robustness of stochastic classifiers}
Since in practice neither the parameter set $\mathcal{A}$ nor $\mathcal{I}$ is fixed, the quantity
better
describing the practical robustness of a stochastic classifier with linear discriminant functions is
\begin{equation}\label{eq:probability}
\mathbb{P}(\min_{c \neq y} \tilde r^{\mathcal{I}}_c >\|\delta^{\mathcal{A}}\|_2) \enspace ,
\end{equation}
where $\tilde{r}^{\mathcal{I}}_c$ and $\delta^{\mathcal{A}}$ are random variables.
For $L$-smooth models,
replacing $\tilde r^{\mathcal{I}}_c$ by $ r^{\mathcal{I}}_c$ leads to a lower
bound on the robustness.
Deriving an analytic expression for this probability is a hard problem.
However, based on
theorems~\ref{thm:linear} and \ref{thm:robustness_beta}
it becomes clear
that a larger $\min_{c \neq y} \tilde{r}_c^{\mathcal{I}}$
relates to increasing the probability in eq.~\eqref{eq:probability} and thus to an increased robustness.
We note
that
$\tilde{r}_c^{\mathcal{I}}$ with $c \neq y$ grows with i) larger prediction margins $f^{\mathcal{I}}_y(x) -f^{\mathcal{I}}_c(x)$,
ii) smaller gradient norms $\|\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) )\|_2$, and
iii) larger angles $\alpha_c^{\mathcal{I}, \mathcal{A} }$.
Larger prediction margins and smaller gradient norms were also found to positively impact the robustness of deterministic networks~\citep[e.g.][]{Ross_Doshi-Velez_2018}. In contrast, the dependency on the angle is unique to the stochastic setting. We therefore focus on the analysis of this factor in the following.
\paragraph{
Analyzing the
expected angle}
The angle $\alpha_c^{\mathcal{I}, \mathcal{A}}$ depends on both terms $-\nabla_x (f_y^{\mathcal{I}}(x)- f_c^{\mathcal{I}}(x))$
(for which we use the shorthand $ -\nabla_x f^{\mathcal{I}}_{y-c}(x)$ in the following)
and $\delta^{\mathcal{A}}$.
For further analysis, we first rewrite the gradient as
\begin{equation*}
-\nabla_x f^{\mathcal{I}}_{y-c}(x) = \nabla_x \left( f^{\mathcal{I}}_{y-c}rac{1}{S^{\mathcal{I}}}\sum_{s=1}^{S^{\mathcal{I}}} -f_{y-c}(x, \theta_s^i) \right)
= f^{\mathcal{I}}_{y-c}rac{1}{S^{\mathcal{I}}}\sum_{s=1}^{S^{\mathcal{I}}} -\nabla_x f_{y-c}(x, \theta_s^i)
\enspace.
\end{equation*}
Let $\mu= \mathbb{E}_{\Theta}[-\nabla_x f_{y-c}(x, \theta_s^i)]$ and $\Sigma$ be the covariance of $-\nabla_x f_{y-c}(x, \theta_s^i$), then it follows from the central limit theorem that for sufficiently many samples $-\nabla_x f^{\mathcal{I}}_{y-c}(x)
\sim \mathcal{N} \left( \mu, f^{\mathcal{I}}_{y-c}rac{\Sigma}{S^{\mathcal{I}}} \right)$.
For simplicity let us assume that
the attack is based on the margin loss and only one iteration of gradient ascent. In this case,
$\delta^{\mathcal{A}}= f^{\mathcal{I}}_{y-c}rac{\eta}{\|\hat{\delta}\|_2} \cdot \hat{\delta}$, with
$\hat{\delta} \sim \mathcal{N} \left( \mu, f^{\mathcal{I}}_{y-c}rac{\Sigma}{S^{\mathcal{A}}} \right).$
Note, that we can neglect the scaling of the attack vector, since the angle only depends on $\hat{\delta}$.
Therefore, estimating the
distribution of
$\alpha_c^{\mathcal{I}, \mathcal{A}}$
corresponds to estimating the
distribution of the
angle
between two independent multivariate Gaussian random vectors with the same mean and (potentially) differently scaled versions of the same covariance.f^{\mathcal{I}}_{y-c}ootnote{In the case of more complicated attacks the mean will differ as well but we expect the findings of these section to generalize
also to this scenario.
}
It is known for vectors from normalized standard multivariate Gaussian distributions that the
mode of the distribution of the angle between two random vectors is equal to 90° and that
with increased
dimension the concentration around this mode gets tighter ~\citep{cai_dist_angle}.
Unfortunately, deriving a closed form expression for the distribution or expectation in the general case is a challenging task and beyond the scope of this paper. However, we conjecture that the expectation of the angle increases proportionally with the variance and anti-proportionally with the norm of the mean.
We illustrate
the intuition behind this hypothesis in
figure~\ref{fig:exp_angle}.
To empirically verify the correctness of this hypothesis
we conducted an
experiment where we estimated the expected angle between two identically distributed
1,000-dimensional Gaussian random vectors for different
choices of
means and diagonal covariances.
More precisely, we sampled the mean and variances uniformly from $[0, t]$, where $t$ increased from zero to ten with step size $0.2$, and
estimated
the expected angle based on 10,000 vector pairs drawn from the resulting distributions. Results are shown in figure~\ref{fig:exp_angle} on the right side.
As hypothesized,
the
smaller the length of $\mu$ and the higher the
average variances, the higher the expected angle.
\begin{figure}
\caption{Dependence of the angle w.r.t~mean and variance of the gradient. \textbf{Left, top}
\label{fig:exp_angle}
\end{figure}
\paragraph{Implications for the attack}
Based on the previous discussion, the only way the attacker can influence
the probability of a successful attack, and in this sense the
robustness of the model against the attack,
is by increasing the amount of samples and thereby reducing the variance of the attack vector.
A reduction of the variance
leads to a decrease of the expected angle as described in the previous section. This gives a more elaborated explanation of what was often loosely described as finding the ``correct'' gradient direction in previous work~\citep{obfuscated_grad}.
However, even if the attacker would be able to take infinitely many samples, and thus the variance
of the attack direction
would be reduced to zero,
the expected value of the angle will be larger than zero because of the still existing stochasticity in the inference process. This stochasticity can even lead to an expected angle close to 90° if $\mu$ is short, and/or the
covariance $\Sigma$
is high. That is, the advantage of obfuscating the optimal attack direction
by incorporating stochasticity into the classifier can be decreased but not fully counterbalanced by taking more samples during the attack.
This might explain the finding in~\citet{He_2019_CVPR}, that the accuracy under attack stagnates at a higher level than the deterministic counterpart when increasing the number of iterations of iterative gradient-based attack methods.
\paragraph{Implications for the model}
From the perspective of the defender,
the analysis of the angle shows that models with an increased gradient variance (which is often associated to
a high
prediction variance) and a small norm of the mean gradient
are connected to larger values of
$\alpha_c^{\mathcal{I}, \mathcal{A}}$ and thus to a higher probability of unsuccessfully attacks.
This explains why including the norm of the mean gradient, the gradient variance, or the angle between gradients as regularization terms in the training of SNNs, as for example proposed by~\citet{diverse_directions_bender20a} and~\citet{GradDiv}f^{\mathcal{I}}_{y-c}ootnote{In these works the regularization terms were motivated by maintaining input sensitivity and bounding the expected loss increase, respectively.},
lead to an increased empirical robustness.
It would be naturally to suspect that increasing the number of samples used during inference also decreases the robustness, since it decreases the expected angle. However, this is not the case, as it is counterbalanced by an decrease of the norm of the gradient estimate (i.e.~the second term in the denominator of $\tilde r_c$) with growing sample size.f^{\mathcal{I}}_{y-c}ootnote{
We present a preposition showing that the
interval incorporating the gradient norm decreases to the norm of $\mu$ with increasing amount of samples in supplement~A.}
This can be seen by
rewriting
the expected denominator with respect to the
two sample sets $\mathcal{I}$ and $\mathcal{A}$
as
\begin{align*}
& \mathbb{E}_{\mathcal{I},\mathcal{A}} \left[\| \nabla_x f^{\mathcal{I}}_{y-c} (x)\|_2 \cdot \cos(\alpha^{\mathcal{I}, \mathcal{A}}_c)\right]
= \mathbb{E}_{\mathcal{I},\mathcal{A}}\left[\| \nabla_x f^{\mathcal{I}}_{y-c} (x)\|_2 \cdot f^{\mathcal{I}}_{y-c}rac{\langle -\nabla_x f^{\mathcal{I}}_{y-c} (x), \delta^{\mathcal{A}} \rangle}{ \| \nabla_x f^{\mathcal{I}}_{y-c} (x)\|_2 \cdot \| \delta^{\mathcal{A}} \|_2 } \right]\\
&= \mathbb{E}_{\mathcal{I},\mathcal{A}} \left[ \langle -\nabla_x f^{\mathcal{I}}_{y-c} (x), f^{\mathcal{I}}_{y-c}rac{\delta^{\mathcal{A}}}{\| \delta^{\mathcal{A}} \|_2} \rangle \right] = \sum_{i=1}^p \mathbb{E}_{\mathcal{I}} \left[-\nabla_{x_i} f^{\mathcal{I}}_{y-c} (x) \right]\cdot \mathbb{E}_{\mathcal{A}} \left[f^{\mathcal{I}}_{y-c}rac{\delta_i^{\mathcal{A}}}{\| \delta^{\mathcal{A}} \|_2} \right]\enspace ,
\end{align*}
where the last equation
holds
due to the independence of $\delta^{\mathcal{A}}$ and $-\nabla_x f^{\mathcal{I}}_{y-c}(x)$.
The expectation of
$f^{\mathcal{I}}_{y-c}rac{\delta^{\mathcal{A}}}{\| \delta^{\mathcal{A}} \|_2}$
does not depend on the samples taken during inference and the expectation of the estimate of the derivative w.r.t.~the $i$-th input $ \mathbb{E}_{\mathcal{I}} \left[-\nabla_{x_i} f^{\mathcal{I}}_{y-c} (x) \right]$ is the same for different amounts of samples. Therefore, the expectation of the denominator does not change when changing the sample size during inference. This observation explains why the robustness of a stochastic classifier does not depend on the amount of samples taken during inference and gives an justification for picking an arbitrary sample size that allows for a good trade-off between efficiency and reduction of the variance of the MC estimate used for prediction.
\section{Experimental robustness analysis}
\label{sec:experiments}
In this section we empirically demonstrate that the findings of our theoretical analysis are transferable to SNNs and help to explain the mechanisms leveraging the experimentally observed robustness of previously proposed SNNs.
\paragraph{Experimental setup}
Our experiments are conducted on two different image datasets: FashionMNIST~\citep{fashionmnist} and CIFAR10~\citep{cifar10}.f^{\mathcal{I}}_{y-c}ootnote{Additional experiments for CIFAR100 are presented in the supplement~C.5.
}
For experiments on FashionMNIST we used
feedforward neural networks (FNN) with
two stochastic hidden layers, each with 128 neurons.
We trained the FNN as a Variational Matrix Gaussian (VMG, the BNN proposed by~\citet{louizos16}) via variational inference or as an infinite mixture (IM) with the maximum likelihood objective proposed by~\citet{daubener2020investigating},
with matrix variate normal distribution placed over the weights. We also trained FNNs of the same architecture, where we added Gaussian noise
with $\sigma^2 = \{ 0.05, 0.1 \}$
to the input,
by minimizing the
cross-entropy.
We refer to these models as \textit{stochastic input networks} (SINs) 0.05 and SIN 0.1, respectively.
Note, that these networks correspond to the basic networks proposed for randomized smoothing by \citet{lecuyer2019certified}.
For experiments on CIFAR10
we trained two wide residual networks (ResNet) with MC dropout layers~\citep{mc_dropout_gal}
applied after
the convolution blocks
and dropout probabilities
$p=0.3$ and $p=0.6$. If not specified otherwise we used 100 samples of $p(\Theta)$ for inference on all datasets and calculated
adversarial attacks
with the fast gradient (sign)
method (FGM)~\citep{Goodfellow_fgsm},
\begin{wrapfigure}{r}{0.5\textwidth}
\begin{center}
\includegraphics[width=0.5\textwidth]{images/r_plot_BNN.png}
\end{center}
\caption{Adversarial accuracy of the smoothed BNN model for attacks based on 10 samples vs percentage of images with
$\min_c r_c^{\mathcal{I}} > \| \delta^{\mathcal{A}} \|_2$ (smooth) and $\min_c \tilde{r}_c^{\mathcal{I}} > \| \delta^{\mathcal{A}} \|_2$ (linear)
for the first 100 images from the FashionMNIST dataset.}
\label{fig:r_accuracy_fmnist}
\end{wrapfigure}
the cross-entropy loss for IMs, BNNs, and ResNets, the margin loss $\mathcal{L}_{\text{margin}}$ specified in section~\ref{sec:preliminaries} for SINs, and the $\ell_2$-norm constraint
based on the CleverHans repository~\citep{papernot2018cleverhans}. All
experiments were run on a single NVIDIA GeForce RTX 2080 Ti.
We refer the reader to supplement~B for
more
details on the datasets, models, and the training procedure.
\paragraph{Accuracy of robustness conditions}
\label{subsec:adv_r_acc}
We first investigated the practical transferability of
the derived theorems.
For enforcing the smoothness condition used in theorem \ref{thm:robustness_beta},
we built on the
result from
~\citet{yang2021ensemble_robust}, who showed that the $L$-smoothness parameter of a classifier $g:x \rightarrow \mathbb{E}_{\epsilon}[f(x+\epsilon)]$ smoothed with random noise $\epsilon \sim \mathcal{N}(0, \sigma^2)$, is bounded by $L \leq 2/ \sigma^2$.
We therefore applied randomized smoothing during training of the models.
For the BNN, on which we focus in this section due to space restrictions (results for the other networks look qualitatively similar and can be found in supplement~C),
we replaced
each image in the batch by two noisy copies with Gaussian noise $\epsilon \sim \mathcal{N}(0, 0.1)$ which ensures $L\leq 20$.
During prediction we estimated the expectation under Gaussian noise with
10
samples and also used 10 samples for calculating the FGM attack.
We estimated the percentage of resulting attacks for which
$\min_c r_c^{\mathcal{I}} > \| \delta^{\mathcal{A}} \|_2$ and $\min_c \tilde{r}_c^{\mathcal{I}} > \| \delta^{\mathcal{A}} \|_2$
and compared this to the adversarial accuracy (i.e. the percentage of perturbed samples classified correctly) in figure~\ref{fig:r_accuracy_fmnist}. The percentage of samples fulfilling the condition $\min_c r_c^{\mathcal{I}} > \| \delta^{\mathcal{A}} \|_2$ approaches zero with growing perturbation strength, indicating that the lower bound provided by $r_c^{\mathcal{I}}$ is rather loose due to the high (upper bound of the) L-smoothness constant. On the other hand
the percentage of samples for which~\eqref{eq:r_linear} is fulfilled is a good approximation of the real accuracy for small attack length, which suggests that the discriminant functions are approximately linear in a small neighborhood of the input.
\begin{figure}
\caption{Accuracy under attack
for different perturbation strength and amount of samples used for calculating the attack. The dashed line shows the adversarial accuracy for the
models with the same architecture but less prediction variance.}
\label{fig:acc_under_attack}
\end{figure}
\paragraph{Stronger attacks by increased sample size}
In this section we investigate the effect of varying the amount of samples used for calculating the attack, e.g. for $S^{\mathcal{A}} \in \{ 1, 5, 10, 100, (1000)\}$.
Figure~\ref{fig:acc_under_attack} shows the resulting adversarial robustness of the
IM, SIN 0.1, and the ResNet with dropout probability 0.6.
The accuracy under attack decreases for all models
with increasing amount of samples used for calculating the attack as conjectured.
We found this to hold also
for stronger attacks, i.e.~attacks based on logits for IM and BNN (where we observe
highly
confident softmax predictions), attacks with $L_{\infty}$ constraint, or \textit{projected gradient descent} (PGD)~\citep{madry2018towards} attacks with 100 iterations as shown in supplement~C. Note, that iterative attacks already increase the sample size due to their iterative nature, and therefore only few samples per iteration may
sufficiently increase attack strength.
Simultaneously to the accuracy we
estimated
the corresponding values of $\cos(\alpha_j^{\mathcal{I}, \mathcal{A}})$, for
$j= \arg \min_c \tilde{r}_c^{\mathcal{I}}$,
for the first 1,000 test images and depicted them in figure~\ref{fig:angle_attack}.
It can be seen that the cosine values are increasing (which corresponds to decreasing angles) with growing sample size and that the larger the observed values, the lower the
accuracy under attack as shown in figure~\ref{fig:acc_under_attack}.
This observation
is in accordance to our theoretical analysis
which predicts that an increased amount of samples leads to higher cosine values and in turn to less adversarial robustness.
Note however, that
even when taking
many samples $\cos(\alpha_j^{\mathcal{I}, \mathcal{A}})<1$, which underlines the fact that the optimal attack direction can not be recovered due to the still existing stochasticity in the inference procedure.
Further results on the angle under different amounts of samples during attack can be found in supplement~C.2.
\begin{figure}
\caption{Cosine
values
for adversarial examples with perturbation strength $1.5$ for a) and b) and $0.3$ for c)
and
different amounts of samples. White crosses indicate mean values.}
\label{fig:angle_attack}
\end{figure}
\paragraph{Prediction variance as robustness indicator}
In this section we compare the properties of SNNs that have
the same network architecture and a similar training procedure but different prediction variances.
That is, we compare the BNN against the IM, SIN 0.05 against SIN 0.1, and
the two ResNets with different dropout probabilities against each other.
First, we estimated the standard deviation of the prediction and the average standard deviation of the gradient entries of the models by
calculating the average of the empirical estimates
over the
first 1,000 examples from the respective test sets
(see table~\ref{tab:pred_variance}).
As expected,
the IM, SIN 0.1, and ResNet with dropout probability 0.6 have a
larger standard deviation of the prediction and hence prediction variance than their respective counterpart. The higher standard deviation of the prediction
translates to an increased standard deviation of the gradient.
This explains why we observe smaller cosine values for these models compared to their counterparts (see right most boxplots in figure~\ref{fig:angle_attack} a) and c)) which in turn translate to an increased robustness (as indicated by the dotted accuracy curves in figure~\ref{fig:acc_under_attack}).
We also found
that the length of the
mean of the gradient that we estimated based on 1,000 samples is smaller for the models with higher variance.f^{\mathcal{I}}_{y-c}ootnote{For SINs, it is known that a higher variance used during training relates to a lower Lipschitz-constant~\cite[e.g.][]{yang2021ensemble_robust, nips_smoothness_bound} which leads to a stronger smoothing effect.} This
might be another reason for the observed smaller angles
as discussed in the previous section.
\begin{table}
\caption{
Empirical standard deviation
of the prediction (for the correct class)
based on 1,000 single predictions (each with $1$ inference sample) and
average length and variance of the corresponding
gradient.
We report averages over the first 1,000 images from the respective test set.
}
\label{tab:pred_variance}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{l|cc|cc|cc}
\toprule
& \multicolumn{4}{c|}{FashionMNIST} & \multicolumn{2}{c}{CIFAR10}\\
& BNN & IM & SIN 0.05 & SIN 0.1 & dr 0.3 & dr 0.6 \\
\hline
Avg. std of predictions & 0.0186 & 0.0473 & 85.4657 & 186.5599 & 0.0146 & 0.0196 \\
\hline
Avg. gradient length & 0.5218 & 0.5044 & 49.0120 & $\phantom{1}$48.8312 & 0.0713 & 0.0599 \\
Avg. Std of gradient & 0.0316& 0.0959 & $\phantom{1}$1.5741 & $\phantom{1}\phantom{1}$1.8323 & 0.0000 & 0.0000 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
\paragraph{Robustness in dependence of the amount of samples used during inference}
\label{subsec:ex_sample_inference}
In practice the amount of samples $S^{\mathcal{I}}$
drawn during inference is fixed to an arbitrary number.
In this section we investigate the impact of varying $S^{\mathcal{I}} \in \{1, 5, 10, 100 \}$ which are values frequently used.
First, we observed that only few samples
are necessary to get reliable predictions on clean data for all models as shown by the test accuracies in table~\ref{tab:accuracy_inference_samples}.
Second, we found that increasing the amount of samples did not affect the adversarial accuracy. This can be explained by
the observation that the decrease of $\alpha_c^{\mathcal{I}, \mathcal{A}}$ caused by increasing the sample set is counterbalanced by an simultaneous decrease of the average norm of the gradient estimate, as can be seen by inspecting the results in figure \ref{fig:num_samples_inference}. That is,
the product $\| \nabla_x f^{\mathcal{I}}_{y-c} \|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})$ stays approximately the same
regardless of the inference sample size as predicted by our theoretical analysis.
Interestingly, the effect of increasing the number of samples during inference has almost no effect on the prediction margin (more results are shown in
supplement~C.4).
\begin{table}[h]
\caption{Test set and adversarial accuracy with 100 samples during the attack and allowed perturbation strength of $1.5$ on FashionMNIST and $0.3$ on CIFAR10 for increasing number of samples used during prediction. We estimated the average accuracy 10 times and report the average and standard deviation.}
\label{tab:accuracy_inference_samples}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{r|cc|cc|cc}
\toprule
& \multicolumn{4}{c|}{FashionMNIST} & \multicolumn{2}{c}{CIFAR10} \\
$|\mathcal{I}|$& BNN & IM & SIN 0.05 & SIN 0.1 & dr 0.3 & dr 0.6 \\
\hline
\multicolumn{7}{c}{Test set accuracy}\\
\hline
1 & $83.03 \pm 0.12$ & $79.15 \pm 0.27$ & $86.98 \pm 0.17$ & $85.76 \pm 0.17$ & $92.08 \pm 0.13$ & $92.68 \pm 0.15$ \\
5 & $84.81 \pm 0.14$ & $84.29 \pm 0.21$ & $88.21 \pm 0.09$ &
$87.49 \pm 0.15$ & $92.57 \pm 0.05$ & $93.51 \pm 0.09$ \\
10 & $85.03 \pm 0.13$ & $85.02 \pm 0.18$ & $88.47 \pm 0.06$ & $87.92 \pm 0.10$ & $92.66 \pm 0.06$ & $93.60 \pm 0.06$ \\
100 & $85.21 \pm 0.06$ & $85.72 \pm 0.08$ & $88.63 \pm 0.07$ & $88.27 \pm 0.06$ & $92.74 \pm 0.03$ & $93.69 \pm 0.04$ \\
\hline
\multicolumn{7}{c}{Adversarial accuracy}\\
\hline
1 & $ 35.76 \pm 0.92 $ & $ 44.67 \pm 1.19 $& $ 36.15 \pm 0.79 $ & $ 40.63 \pm 0.79 $ & $42.64 \pm 0.39$ & $ 47.18 \pm 0.70 $ \\
5 & $ 36.19 \pm 0.47 $ &$ 47.08 \pm 0.73 $ & $ 35.92 \pm 0.51 $ & $ 40.57 \pm 0.44 $ & $42.73 \pm 0.31$ & $ 47.00 \pm 0.38 $ \\
10 & $ 36.30 \pm 0.50 $ & $ 47.23 \pm 0.62 $ & $ 36.06 \pm 0.41 $ & $ 39.93 \pm 0.66 $ & $42.69 \pm 0.37$ & $ 47.01 \pm 0.33 $ \\
100 & $ 36.50 \pm 0.39 $ & $ 47.96 \pm 0.24 $& $ 35.91 \pm 0.23 $ & $ 39.67 \pm 0.29 $ & $ 42.82 \pm 0.17 $ & $ 47.02 \pm 0.20 $ \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
\begin{figure}
\caption{
Values of single factors
of $\tilde{r}
\label{fig:num_samples_inference}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this work we have stressed the fact that
stochastic neural networks (SNNs) (and stochastic classifiers in general) often depend on samples and thus their predictions are random variables themselves. For gradient-based adversarial attacks this means that the attack is calculated based on one realization of the stochastic network (which depends on multiple samples
of the random variables used in the network) and applied to another which is used for inference.
We derived a sufficient condition for this inference network to be robust against the calculated attack. This allowed us to identify the factors that lead to an increased robustness of stochastic classifiers:
i) larger prediction margins
ii) a smaller norm of the gradient estimates
and
iii) higher angles between the attack direction and the direction to the closest decision boundary during inference.
The observed angles depend inverse proportionally on the norm of the expected gradient and proportionally on the variance of the gradient estimates.
This variance can be reduced by increasing the sample size.
These insights enable us to explain previously
reported empirical findings for SNNs from a geometrical perspective,
e.g.~that the robustness of SNNs is higher than the robustness of their deterministic counterparts
even for strong attacks that are based on several samples~\citep{He_2019_CVPR, weight_covariance-eustratiadis21a}, why regularization of the gradient variance~\citep{diverse_directions_bender20a}, norm of the mean gradient, and angle~\citep{GradDiv} improves the adversarial robustness, and last but not least why increasing the sample size during attack is important to exploit its potential~\citep{obfuscated_grad}.
Therefore, our work poses a general applicable and simple framework,
that helps understanding the mode of operation of existing stochastic defense mechanisms, even if they were motivated from a different point of view.
Moreover, we derived a justification for the common practice of choosing the sample size during inference in a way that balances its prediction certainty against the computational cost.
Finally, we
believe
our findings will be useful to evaluate and compare
the robustness of different models, since they point out that they might require different amounts of samples during attack to sufficiently reduce variance and hope that they will help to improve the robustness of stochastic classifiers in future.
\paragraph{Potential negative societal impact and limitations.} Since we do not propose a new attack strategy, but
contribute to a better
understanding of the robustness of SNNs and advise to cautiousness when determining sample sizes, we do not see negative impact.
\section*{Acknowledgments}
We would like to thank Denis Lukovnikov for his useful comments on our work and beautifying our graphics.
This work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2092 CASA - 390781972.
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\appendix
\section*{Supplement}
\section{Proofs}
\label{app:proofs}
In this section we proof the theorems of the main paper and recap the needed definitions to do so.
\begin{definition}[Maximum magnitude attack]
Given a multi-class classifier $f(\cdot)$, a loss function $\mathcal{L}(\cdot, \cdot)$, $\ell_p$-norm $\| \cdot \|_p$ and a given perturbation strength $\eta$ the optimization problem for a maximum magnitude attack can be written as
\begin{equation}\label{eq:app_max_allowable_attack}
\textit{maximize} \enspace \mathcal{L}(f(x+ \delta), y) \ , \ w.r.t. \enspace \delta \enspace s.t. \ \| \delta\|_p \leq \eta \enspace.
\end{equation}
\end{definition}
\begin{theorem}[Sufficient and necessary robustness condition for linear classifiers]
\label{thm:app_linear}
Let $f:\mathbb{R}^d\times \Omega^h \rightarrow \mathbb{R}^k$ be a stochastic classifier with linear discriminant functions and $f^{\mathcal{A}}$ and $f^{\mathcal{I}}$ be two MC estimates of the classifier.
Let
$x \in \mathbb{R}^d $ be a data point with label $y \in \{1,\dots,k\}$ and $\arg \max_c f_c^{\mathcal{A}}(x)= \arg \max_c f_c^{\mathcal{I}}(x)=y$, and let $x_{\text{adv}}=x+ \delta^{\mathcal{A}}$ be an adversarial example computed for solving the minimization problem~\eqref{eq:app_max_allowable_attack}
for $f^{\mathcal{A}}$.
It holds that $\arg \max_{c} f^{\mathcal{I}}_c(x+ \delta^{\mathcal{A}}) =y $ if and only if
\begin{equation}\label{eq:app_r_linear}
\min_{c \neq y} \tilde{r}^{\mathcal{I}}_c > \|\delta^{\mathcal{A}}\|_2 \enspace, \text{ with }
\end{equation}
\begin{equation}
\tilde{r}^{\mathcal{I}}_c = \begin{cases}
\infty \enspace , &\text{if} \;\; \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})= f^{\mathcal{I}}_{y-c}rac{\langle -\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)), \delta^{\mathcal{A}} \rangle}{\|\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)) \|_2 \cdot \|\delta^{\mathcal{A}} \|_2} \leq 0 \\
f^{\mathcal{I}}_{y-c}rac{ f^{\mathcal{I}}_y \left(x \right) - f^{\mathcal{I}}_c\left(x \right) }{ \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) ) \|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})} \enspace ,
&\text{otherwise \enspace,
}
\nonumber
\end{cases}
\end{equation}
where $\alpha_c^{\mathcal{I}, \mathcal{A}}$ is the angle between $-\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) )$ and $\delta^{\mathcal{A}}$.
\end{theorem}
\begin{proof}
An adversarial attack on $f^{\mathcal{I}}$ with the adversarial example $x+\delta^{\mathcal{A}}$ is not successful iff $f^{\mathcal{I}}_{y-c}orall c \in \{1,2,\dots,k\}, c \neq y:$
\begin{equation*} \label{eq:app_unsucc_attack}
f^{\mathcal{I
}}_y \left(x + \delta^{\mathcal{A}} \right) - f^{\mathcal{I}}_c\left(x + \delta^{\mathcal{A}} \right) >0 \enspace .
\end{equation*}
With Taylor expansion around $x$ we can rewrite
$ f^{\mathcal{I
}}_y \left(x + \delta^{\mathcal{A}} \right) - f^{\mathcal{I}}_c\left(x + \delta^{\mathcal{A}} \right) $
as
\begin{align}
& f^{\mathcal{I}}_y \left(x \right) +\langle \nabla_x f^{\mathcal{I}}_y(x), \delta^{\mathcal{A}} \rangle -
f^{\mathcal{I}}_c\left(x \right)
- \langle \nabla_x
f^{\mathcal{I}}_c(x), {\delta^{\mathcal{A}}} \rangle \nonumber \\
=& f^{\mathcal{I}}_y \left(x \right)
- f^{\mathcal{I}}_c\left(x \right)
+\langle \nabla_x f^{\mathcal{I}}_y(x)- \nabla_x
f^{\mathcal{I}}_c(x), \delta^{\mathcal{A}} \rangle \nonumber \\
=& f^{\mathcal{I}}_y \left(x \right)
- f^{\mathcal{I}}_c\left(x \right) - \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) ) \|_2 \cdot \| \delta^{\mathcal{A}} \|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) \label{eq:app_taylor-lin}
\end{align}
where $\alpha_c^{\mathcal{I}, \mathcal{A}} := \alpha_c^{\mathcal{I}, \mathcal{A}}gle (- \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) ), \delta^{\mathcal{A}} ) $.
We can distinguish two cases for each $c$.
\paragraph{Case 1: $\cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) \leq 0$.}
In this case last term
of eq.~\eqref{eq:app_taylor-lin} is negative or zero and thus
\begin{equation*}
f^{\mathcal{I
}}_y \left(x + \delta^{\mathcal{A}} \right) - f^{\mathcal{I}}_c\left(x + \delta^{\mathcal{A}} \right) \geq f^{\mathcal{I}}_y \left(x \right)
- f^{\mathcal{I}}_c\left(x \right) > 0 \enspace,
\end{equation*}
where the second inequality holds since
$ \arg \max_c f_c^{\mathcal{I}}(x)=y$.
\paragraph{Case 2: $\cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) > 0$.}
In this case the last term of
equation eq.~\eqref{eq:app_taylor-lin} is positive and thus
\begin{equation*}\label{eq:app_cos_alpha<1}
f^{\mathcal{I
}}_y \left(x + \delta^{\mathcal{A}} \right) - f^{\mathcal{I}}_c\left(x + \delta^{\mathcal{A}} \right) \leq f^{\mathcal{I}}_y \left(x \right) - f^{\mathcal{I}}_c\left(x \right) \enspace .
\end{equation*}
As we see from rearranging eq.~\eqref{eq:app_taylor-lin}, it holds $f^{\mathcal{I}}_y \left(x + \delta^{\mathcal{A}} \right) - f^{\mathcal{I}}_c\left(x + \delta^{\mathcal{A}} \right) >0$ if
\begin{equation}
\tilde{r}_c^{\mathcal{I}} := f^{\mathcal{I}}_{y-c}rac{ f^{\mathcal{I}}_y \left(x \right) - f^{\mathcal{I}}_c\left(x \right) }{ \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x) ) \|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})} > \| \delta^{\mathcal{A}} \|_2 \label{eq:app_eta_value} \enspace .
\end{equation}
For each class $c$ either case 1 holds and we define $\tilde{r}_c^{\mathcal{I}}:=\infty$, or condition \eqref{eq:app_eta_value} is fulfilled, which yields the condition stated in the theorem.
\end{proof}
In the main part of the paper we relaxed the linear classifier assumption by $L$-smoothness, which was defined as follows:
\begin{definition}[$L$-smoothness~\citep{yang2021ensemble_robust}]
A differentiable function $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ is $L$-smooth, if for any $x_1, x_2\in \mathbb{R}^d$ and any output dimension $c\in \{1,\dots,k\}$:
\[
f^{\mathcal{I}}_{y-c}rac{\| \nabla_{x_1} f_c(x_1)- \nabla_{x_2} f_c(x_2)\|_2}{\| x_1 - x_2\|_2} \leq L \enspace.
\]
\end{definition}
Next we restate one property of L-smooth functions which we will use in our proof of theorem 2.
\begin{proposition}[\cite{bubeck2015convex}]
\label{prop:app_smooth}
Let $f$ be an $L$-smooth function on $\mathbb{R}^n$. For any $x,y \in \mathbb{R}^n$ it holds:
\begin{equation*}\label{eq:app_decent_lemma}
|f(y) -f(x) - \langle \nabla_x f(x) , y-x \rangle | \leq f^{\mathcal{I}}_{y-c}rac{L}{2} \| y-x\|_2^2 \enspace.
\end{equation*}
\end{proposition}
\begin{proof}
From the fundamental theorem of calculus we know that for a differentiable function $f$ it holds that $f(y)- f(x) = \int_x^y \nabla_t f(t) dt$. By substituting $x_t = x + t(y-x)$ we see that $x_0 = x$ and $x_1 = y$ and thus we can write $f(y)- f(x) = \int_0^1 \nabla f(x + t(y-x)) ^T \cdot (y-x) dt$. This allows the following approximations
\begin{align*}
&|f(y) -f(x) - \langle \nabla_x f(x) , y-x \rangle | \\
= &| \int_0^1 \nabla f(x + t(y-x))^T \cdot (y-x) dt - (\nabla_x f(x))^T (y-x) | \\
\leq& \int_0^1 | (\nabla f(x + t(y-x)) - \nabla_x f(x))^T ) \cdot (y-x) | dt \\
\overset{\text{Cauchy-Schwarz}}{\leq }& \int_0^1 \| \nabla f(x + t(y-x)) - \nabla_x f(x) \|_2 \cdot \| y-x \|_2 dt\\
\overset{L \text{-smoothness}}{\leq }& L \cdot \| y-x \|_2^2 \cdot \int_0^1 t dt \\
=& f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| y-x \|_2^2 \enspace .
\end{align*}
\end{proof}
\begin{theorem}[Sufficient condition for the robustness of an L-smooth stochastic classifier]
Let $f:\mathbb{R}^d\times \Omega^h \rightarrow \mathbb{R}^k$ be a stochastic classifier with $L$-smooth discriminant functions and $f^{\mathcal{A}}$ and $f^{\mathcal{I}}$ be two MC estimates of the prediction. Let
$x\in \mathbb{R}^d$ be a data point with label $y \in \{1,\dots ,k \}$ and $\arg \max_c f_c^{\mathcal{A}}(x)=\arg \max_c f_c^{\mathcal{I}}(x)=y$, and let $x_{\text{adv}}=x+ \delta^{\mathcal{A}}$ be an adversarial example computed for solving the minimization problem~\eqref{eq:app_max_allowable_attack}
for $f^{\mathcal{A}}$.
It holds that $\arg \max_{c} f^{\mathcal{I}}_c(x+ \delta^{\mathcal{A}}) =y $ if
\begin{equation*}
\min_{c \neq y} r_c^{\mathcal{I}} > \|\delta^{\mathcal{A}}\|_2 \enspace,
\end{equation*}
with
\begin{equation*}\label{eq:app_ri_beta}
r_c^{\mathcal{I}} = \begin{cases}
&\infty \enspace \text{, if} \enspace \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x))\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 \leq 0 , \\
&f^{\mathcal{I}}_{y-c}rac{f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)}{ \| \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x))\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 } \enspace \text{, else} \end{cases} \\
\end{equation*}
and
\begin{equation*}
\cos(\alpha^{\mathcal{I}, \mathcal{A}}_c)= f^{\mathcal{I}}_{y-c}rac{\langle - \nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)), \delta^{\mathcal{A}} \rangle}{\|\nabla_x (f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)) \|_2 \cdot \|\delta^{\mathcal{A}} \|_2} \nonumber \enspace .
\end{equation*}
\end{theorem}
\begin{proof}
For better readability we write $f^{\mathcal{I}}_{y-c}(x):=f^{\mathcal{I}}_y(x) - f^{\mathcal{I}}_c(x)$.
Using the result from proposition~\ref{prop:app_smooth} and reordering the terms,
we get the following lower bound:
\begin{align}
& f_{y-c}^{\mathcal{I}}(x + \delta^{\mathcal{A}}) \nonumber \\
& \geq f^{\mathcal{I}}_{y-c}(x) + \langle \nabla_x ( f^{\mathcal{I}}_{y-c}(x) ), \delta^{\mathcal{A}} \rangle - f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2^2 \nonumber \\
&= f^{\mathcal{I}}_{y-c}(x) - \langle - \nabla_x ( f^{\mathcal{I}}_{y-c}(x) ), \delta^{\mathcal{A}} \rangle - f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2^2 \nonumber \\
&=f^{\mathcal{I}}_{y-c}(x) - \|-\nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \| \delta^{\mathcal{A}} \|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}})
- f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2^2 \nonumber \\
& = f^{\mathcal{I}}_{y-c}(x) - \left(\| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 \right) \cdot \| \delta^{\mathcal{A}} \|_2 \label{eq:app_lower} \enspace .
\end{align}
If eq.~\eqref{eq:app_lower} is bigger than zero, the attack cannot be successful.
Hence,
\begin{align}
& f^{\mathcal{I}}_{y-c}(x) - \bigg( \| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2}\cdot \| \delta^{\mathcal{A}} \|_2 \bigg) \cdot \| \delta^{\mathcal{A}} \|_2 \overset{!}{>} 0 \label{eq:app_pre}\\
&f^{\mathcal{I}}_{y-c}(x) > \bigg(\| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 \bigg) \cdot \| \delta^{\mathcal{A}} \|_2 \label{eq:app_calc_bound} \enspace .
\end{align}
\paragraph{Case 1: $ \| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 > 0$.}
Transforming eq.~\eqref{eq:app_calc_bound} leads to
\begin{align*}
f^{\mathcal{I}}_{y-c}rac{f^{\mathcal{I}}_{y-c}(x)}{ \| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 } >& \| \delta^{\mathcal{A}} \|_2 \enspace.
\end{align*}
\paragraph{Case 2 : $\| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 = 0$. }
In this case eq.~\eqref{eq:app_pre} is trivially fulfilled because $f^{\mathcal{I}}_{y-c}(x)>0$ per definition.
\paragraph{Case 3: $\| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 < 0$.}
For this case we get, that
\begin{align*}
-f^{\mathcal{I}}_{y-c}rac{f^{\mathcal{I}}_{y-c}(x)}{ | \| \nabla_x f^{\mathcal{I}}_{y-c}(x)\|_2 \cdot \cos(\alpha_c^{\mathcal{I}, \mathcal{A}}) + f^{\mathcal{I}}_{y-c}rac{L}{2} \cdot \| \delta^{\mathcal{A}} \|_2 | } <& \| \delta^{\mathcal{A}} \|_2 \enspace ,
\end{align*}
which is always guaranteed based on the initial assumption that the benign input was classified correctly, which concludes the proof.
\end{proof}
The following proposition relates to footnote~5 from the main paper. We show, that the interval, in which the expectation of the gradient norm lies, decreases to the true length of $\mu$ with reducing the covariance.
\begin{proposition}
Let X be an $n$-dimensional random vector following a multivariate normal distribution with mean vector $\mu$ and diagonal covariance matrix $\Sigma$. Then the expectation of $\| X \|_2$ can be upper and lower bounded by
\begin{equation*}
\| \mu \|_2 \leq \mathbb{E}[\| X \|_2 ] \leq \sqrt{ \| \mu \|_2^2 + tr\left(\Sigma\right)} \enspace .
\end{equation*}
\end{proposition}
\begin{proof}
We first look at the lower bound which by convexity of the norm and Jensen inequality can be derived via
\begin{equation*}
\mathbb{E}[\| X \|_2 ] \leq \| \mathbb{E}[ X ] \|_2 = \| \mu \|_2 \enspace .
\end{equation*}
Let $X'\sim \mathcal{N}(0,\textbf{1}_n)$, with $\textbf{1}_n$ an $n\times n$-dimensional unit matrix.
With Jensen inequality and concavity of the square-root function we derive the upper bound
\begin{align*}
&\mathbb{E}[ \| X \|_2 ]
=\mathbb{E}\left[ \| \mu + X' \cdot
\Sigma^{1/2}
\|_2 \right]\\
&\leq \left(
\mathbb{E}\left[\left\| \mu + X' \cdot
\Sigma^{1/2} \right\|_2^2 \right] \right)^{1/2} \\
&= \bigg ( \| \mu \|_2^2 + 2 \mu^T
\Sigma^{1/2} \mathbb{E}[ X'] + \mathbb{E}\left[X'^T \Sigma^{{1/2}^{T}}
\Sigma^{1/2} X'\right] \bigg )^{1/2} \\
&= \sqrt{ \| \mu \|_2^2 + tr\left(\Sigma\right) } \enspace .
\end{align*}
\end{proof}
\section{Additional information on datasets, models and training}\label{app:hyper}
In the following we describe additional details on the datasets, models and training procedures which were not stated in the main paper due to space restrictions. Additionally, please find the code for the results in the main paper attached in the supplementary material.
\subsection{Datasets}
We used three well know datasets: FashionMNIST~\citep{fashionmnist}, CIFAR10 and CIFAR100~\citep{cifar10-100}, which consist out of 60,000 training and 10,000 test images of dimension $28\times 28$ or $32 \times 32 \times 3$ in case of CIFAR where each image is uniquely associated to one out of 10 or 100 possible labels. We took all datasets from the torchvision package with the predefined training and test split.
\subsection{Models trained on FashionMNIST}
\label{app:hyper_fashion}
For training the BNN and IM we used the exact same hyperparameters. First, we assumed a standard normal prior decomposed as matrix variate normal distributions for the BNN and approximated the posterior distribution via maximizing the evidence lower bound (ELBO). For the IM we
added a Kullback-Leibler distance from the trained parameter distribution to a standard matrix variate normal distribution as a regularization term.
For both models we used a batch size of 100 and trained for 50 epochs with Adam~\citep{adam} and an initial learning rate of $0.001$. To leverage the difference between IM and BNN we used 5 samples to approximate the expectation in the ELBO/IM-objective.
We used the same learning rate, batch size, optimizer and amount of epochs for training the stochastic input networks. During a forward pass in training we created and used five noisy versions of each input, where the noise was drawn from a centered Gaussian distribution with variance $0.05$ or $0.1$ and the average prediction was fed into the cross-entropy loss.
\subsection{Models trained on CIFAR10}
\label{app:hyper_cifar10}
As stated in the main part, we used the wide ResNet~\citep{zagoruyko2017wide} of depth 28 and widening factor 10 provided by \url{https://github.com/meliketoy/wide-resnet.pytorch} with dropout probabilities 0.3 and 0.6 and also used the learning hyperparameters provided with the code which are: training for 200 epochs with batch size 100, stochastic gradient descent as optimizer with momentum 0.9, weight decay 5e-4 and a scheduled learning rate decreasing from an initial 0.1 for epoch 0-60 to 0.02 for 60-120 and lastly 0.004 for epochs 120-200.
\section{Additional experimental results}
\label{app:further_experiments}
In this section we present the results which were not shown in the main part due to space restrictions.
\begin{figure}
\caption{Adversarial accuracy of the a) smoothed IM on FashionMNIST and b),c) smoothed ResNet with dropout probability 0.3 and 0.6 on CIFAR10 vs percentage of images for which $\min_c r^{\mathcal{I}
\label{fig:app_r_value_BNN_drop03}
\end{figure}
\subsection{Complementary experiments on accuracy of robustness conditions}
\label{app:com_accuracy_of_robustness_condition}
For completeness we attached the results on the transferability of our derived sufficient conditions to: the IM on FashionMNIST and the two ResNet with dropout probability 0.3 and 0.6 on CIFAR10. For the BNN we used the same setting as described in the main paper and derive similar results (c.f. figure~\ref{fig:app_r_value_BNN_drop03}): while the percentage of samples fulfilling the condition $\min_c r_c^{\mathcal{I}} > \| \delta^{\mathcal{A}} \|_2$ approaches zero with growing perturbation strength the percentage of samples fulfilling the condition from theorem 1 in the main paper closely matches the real adversarial accuracy in a narrow environment. For models on CIFAR10 we had to adapt the noise added for the smooth classifier to $0.01$ and reduce the amount of samples during inference to 50 such that it fits on one GPU.
\subsection{Complementary experiments on stronger attacks}
\label{app:comp_experiments_attack_sample}
We first present the results not shown in the main paper. That is, we investigate the accuracy under FGM attack with an increasing amount of samples used during the attack (c.f. figure~\ref{fig:app_acc_under_attack_BNN_drop03}). Similar to the observations in the main paper, the accuracy under attack is reduced by an increased amount of samples. However,
we observe only a very small decrease in accuracy
when increasing the amount of samples from
100 and 1,000 for the BNN, from 1 to 5 or above for the SIN 0.05 and
when using 5 instead of
10 or 100 samples for the attack on the ResNet trained with dropout probability 0.3.
This observation is mirrored by the reduction of $\cos(\alpha_c^{\mathcal{I}, \mathcal{A}})$ displayed in figure~\ref{fig:app_angle_BNN_drop03} where we observe an increase of the cosine boxplots which matches the decrease of the adversarial accuracy when taking more samples during the attack.
\begin{figure}
\caption{Accuracy under FGM attack for a) the BNN and b) the SIN 0.05 on FashionMNIST and c) ResNet with dropout probability 0.3 on CIFAR10 for different perturbation strengths and amount of samples used for calculating the attack. During inference we used 100 samples.}
\label{fig:app_acc_under_attack_BNN_drop03}
\end{figure}
\begin{figure}
\caption{Cosine of the angle for the first 1,000 test set images from the
FashionMNIST and CIFAR10 test set for an a) BNN, b) SIN 0.05 and c) ResNet trained with dropout probability 0.3 when attacked with
different amounts of samples and attack strength 1.5 and 0.3 with FGM respectively. White crosses indicate the mean value.}
\label{fig:app_angle_BNN_drop03}
\end{figure}
\begin{figure}
\caption{Accuracy under FGSM attack under $\ell_\infty$-norm constraint for a) the BNN, b) the IM, c) SIN 0.05 and d) SIN 0.1 on the first 1,000 test set images from FashionMNIST for different perturbation strengths and amount of samples used for calculating the attack. Predictions during inference are based on 100 samples.}
\label{fig:app_fgsm_l_infty_fmnist}
\end{figure}
\begin{figure}
\caption{Accuracy under FGSM attack under $\ell_\infty$-norm constraint for the ResNet model with a dropout probability a) of 0.3 and b) of 0.6 on the first 1,000 test set images from CIFAR10 for different perturbation strengths and amount of samples used for calculating the attack. Predictions during inference are based on 100 samples.}
\label{fig:app_fgsm_l_infty_cifar}
\end{figure}
\subsubsection{Attacks with FGSM ($\ell_\infty$- norm)}
All adversarial examples in the main part of the paper were based on an $\ell_2$-norm constraint, which we chose for the nice geometric distance interpretation. However, the first proposed attack scheme~\citep{Goodfellow_fgsm} was based on $\ell_\infty$-norm, which we test in the following. In figure~\ref{fig:app_fgsm_l_infty_fmnist} and~\ref{fig:app_fgsm_l_infty_cifar} we see the respective results on the different data sets. For all models the robustness is decreased with multiple samples and models with higher prediction variance also have a higher accuracy under this attack. Note, that the values of the perturbation strength are not comparable to the values under $\ell_2$- norm constraint, since $\|x \|_\infty \leq \| x \|_2$.
\subsubsection{Attacks with PGD}
\label{app:pgd_results}
Projected gradient descent~\citep{madry2018towards} is a strong iterative attack, where multiple small steps of size $\nu$ of fast gradient method are applied. Specifically, we used the same $\ell_2$-norm length constraint on $\eta$ as in the experiments of the main part of the paper but chose step size $\nu = \eta/50$ and 100 iterations. Note that at each iteration a new network is sampled such that for an attack based on 1 samples, 100 different attack networks were seen, for an attack based on 5 samples 500 different networks and so on. In figure~\ref{fig:app_pgd_fmnist} and~\ref{fig:app_pgd_cifar} we see that the overall accuracy is decreased compared to the results for FGM,
but still, IM has a higher accuracy under attack than the BNN and so does the ResNet with a higher dropout probability. Further, the attacks get stronger with taking more samples, so the general observations made in the main paper also hold for strong attacks and are not due to a sub-optimal attack.
\begin{figure}
\caption{Accuracy under PGD attack with 100 iterations for a) the BNN and b) the IM on the first 1,000 test set images from FashionMNIST for different perturbation strengths and amount of samples used for calculating the attack. }
\label{fig:app_pgd_fmnist}
\end{figure}
\begin{figure}
\caption{Accuracy under PGD attack with 100 iterations for the ResNet with a dropout probability of a) 0.3 and b) of 0.6 on CIFAR10 for different perturbation strengths and amount of samples used for calculating the attack.}
\label{fig:app_pgd_cifar}
\end{figure}
\subsection{Discussing the impact of extreme prediction values}
\label{app:comp_extrem_pred_and_length}
Attacks based on softmax predictions can be unsuccessful for deterministic and stochastic neural networks alike when encountering overly confident predictions. That is, predictions where the softmax output is equal to 1, since these lead to zero gradients.
A practical solution to circumvent this problem is to calculate the gradient based on the logits~\citep{carlini2017evaluating}.
This approach is feasible for classifier whose predictions do not depend on the scaling of the output, that is, outputs which are equally expressive in both intervals $[0,1]$ and $[-\infty, \infty]$. In Bayesian neural networks, where $f_c(x, \Theta)$ is per definition a probability the shortcut over taking the gradient over logits leads to distorted gradients.
For completeness, we nevertheless look at the performance of an attack based on the logits for the two different models trained on FashionMNIST with softmax outputs: IM and BNN.
We conducted an adversarial FGM attack based on 100 samples, but instead of using the cross-entropy loss we used the Carlini-Wagner (CW) loss~\citep{carlini2017evaluating} on the averaged logits, given by:
\begin{equation*}
CW(x, \Theta^{\mathcal{A}}) = \max\left(\max_{i \neq t}( Z(x, \Theta^{\mathcal{A}})_i) - Z(x, \Theta^{\mathcal{A}})_t, 0\right) \enspace,
\end{equation*}
where $ Z(x, \Theta^{\mathcal{A}})_t = f^{\mathcal{I}}_{y-c}rac{1}{S^A} \sum_{s=1}^{S^A} Z(x, \theta_s)_t$ is an arbitrary averaged logit of an output node for input $x$.
In figure~\ref{fig:app_carlini_wagner_loss} it is shown, that the attack with the CW loss on the infinite mixture model improves upon the original attack scheme (FGM), whereas it did not improve the attack's success for the BNN. We additionally conducted an attack based on the logit margin loss $\mathcal{L}(x+\delta^\mathcal{A}, y) = - (\min_{c \neq y} f_{y-c}(x) )$ equivalent to the attack conducted on the SIN, but we found that it performs similar to the CW loss.
\begin{figure}
\caption{Accuracy under attack for the first 1,000 test set images from FashionMNIST with varying perturbation strengths and different attack objectives. Each attack was calculated based on 100 samples and $\ell_2$-constraint for models trained on FashionMNIST.}
\label{fig:app_carlini_wagner_loss}
\end{figure}
\subsection{Complementary experiments on robustness in dependence of the amount of samples
used during inference}
\label{app:comp_inference}
In the main part of the paper we argued why, surprisingly, the amount of samples during inference does not influence the robustness, even though we see in figure~\ref{fig:app_angle_inference} that less sample lead to the smallest values for $\cos(\alpha_c^{\mathcal{I}, \mathcal{A}})$.
As stated in the main part, the increased gradient norm when using only few samples seems to compensate the assumed benefits with regard to the angle for using few samples (c.f. figure~\ref{fig:app_norm_inference}). This can also be seen in table 2 and figure~6 from the main paper, where a (negative) effect on the robustness can only be observed for one inference sample, which also leads to the worst test set accuracy.
The benign prediction margins are also hardly effected by the increased number of samples during prediction (c.f.~figure~\ref{fig:app_margin_inference}).
\begin{figure}
\caption{Cosine of the angle for models trained on FashionMNIST ( a), b), d), e) ) and trained on CIFAR10 ( c) and f) ) for different amounts of samples used during inference. Used attack direction $\delta$ was calculated based on 100 sample of FGM under $\ell_2$-norm constraint with $\eta=1.5$ and $0.3$ respectively.}
\label{fig:app_angle_inference}
\end{figure}
\begin{figure}
\caption{Norm of the gradient for models trained on FashionMNIST ( a), b), d), e) ) and trained on CIFAR10 ( c) and f) ) for different amounts of samples used during inference.}
\label{fig:app_norm_inference}
\end{figure}
\begin{figure}
\caption{Prediction margin for the benign inputs for models trained on FashionMNIST ( a), b), d), e) ) and trained on CIFAR10 ( c) and f) ) for different amounts of samples used during inference. }
\label{fig:app_margin_inference}
\end{figure}
\subsection{Experiments for CIFAR100}
\label{App:subsec:cifar100}
For the experiments on CIFAR100 we used yet another method to create stochastic neural networks, namely by applying a Laplace approximation~\citep{mackay92} to an already trained network. This is archived by adapting a Gaussian distribution over the network's parameters such that the mean is given by the maximum a posterior estimate and the covariance are calculated to match the local loss curvature.
Specifically, we used the GitHub library from~\citet{laplace2021} on top of the adversarial trained wide ResNet70-16 with clean accuracy $69.15$ provided by~\citet{gowal2020uncovering}.
Because of the high amount of parameters in this network we used a last-layer diagonal Gaussian approximation for fitting our posterior distribution from which we sample the $\theta_i$'s for deriving an approximate expected prediction. As in the previous experiments we observe, that the adversarial accuracy decreases with more samples during attack, while the angle between the attack and negative gradient during inference decreases which leads to a cosine increase.
\begin{table}[h]
\label{tab:app_cifar100}
\caption{Adversarial accuracy decrease and $\cos(\alpha_c^{\mathcal{I},\mathcal{A}})$ increase on CIFAR100 with increased amount of samples during attack, where the attack was conducted with $\ell_{\infty}$ norm constraint and perturbation strength 8/225.}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{r|c|c}
\toprule
\# samples & Adversarial accuracy & Average $\cos(\alpha_c^{\mathcal{I},\mathcal{A}}) \pm $ std\\
\hline
1& 48.00& 0.2028 $\pm$ 0.112 \\
5& 45.00& 0.2550 $\pm$ 0.100 \\
10& 43.90& 0.2680 $\pm$ 0.095 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
\end{document} |
\begin{document}
\title{The Schur functor on tensor powers}
\author{Kay Jin Lim}
\author{Kai Meng Tan}
\address{Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076.}
\email[K. J. Lim]{[email protected]}
\email[K. M. Tan]{[email protected]}
\date{February 2011}
\thanks{Supported by MOE Academic Research Fund R-146-000-135-112.}
\thanks{2010 {\em Mathematics Subject Classification.} 20G43, 20C30}
\begin{abstract}
Let $M$ be a left module for the Schur algebra $S(n,r)$, and let $s \in \mathbb{Z}^+$. Then $M^{\otimes s}$ is a $(S(n,rs), F\sym{s})$-bimodule, where the symmetric group $\sym{s}$ on $s$ letters acts on the right by place permutations. We show that the Schur functor $f_{rs}$ sends $M^{\otimes s}$ to the $(F\sym{rs},F\sym{s})$-bimodule $F\sym{rs} \otimes_{F(\sym{r} \wr \sym{s})} ((f_rM)^{\otimes s} \otimes F\sym{s})$. As a corollary, we obtain the image under the Schur functor of the Lie power $L^s(M)$, exterior power $\boldsymbol{/\!\backslash}^s(M)$ of $M$ and symmetric power $S^s(M)$.
\end{abstract}
\maketitle
\section{Introduction} \label{S:intro}
The representations of general linear groups and symmetric groups are classical objects of study. Following the work by Schur in 1901, there is an important connection between the polynomial representations of general linear groups and the representations of symmetric groups via the Schur functor. In this short article, we examine the images of tensor powers, Lie powers, symmetric powers and exterior powers under the Schur functor.
Our motivation comes from our study of the Lie module $\operatorname{\mathrm{Lie}}(s)$ of the symmetric group $\sym{s}$ on $s$ letters. This may be defined as the left ideal of the group algebra $F\sym{s}$ generated by the Dynkin-Specht-Wever element
$$
\upsilon_s = (1-c_2)\dotsm (1-c_s),
$$
where $c_k$ is the descending $k$-cycle $(k, k-1, \dotsc, 1)$ (note that we compose the elements of $\sym{s}$ from right to left). This module is also the image under the Schur functor of the Lie power $L^s(V)$, which is the homogeneous part of degree $s$ of the free Lie algebra on an $n$-dimensional vector space $V$ with $n \geq s$.
In our study, we found that the knowledge of the image of $L^s(M)$ (where $M$ is a general $S(n,r)$-module; note that $V$ is naturally an $S(n,1)$-module) under the Schur functor will be most useful. For example, the formula we provide here is used by Bryant and Erdmann \cite{BE} to understand the summands of $\operatorname{\mathrm{Lie}}(s)$ lying in non-principal blocks. It is also used by Erdmann and the authors \cite{ELT} in their study of the complexity of $\operatorname{\mathrm{Lie}}(s)$.
As we shall see, the image of $L^s(M)$ under the Schur functor can be easily obtained as a corollary by understanding the image under the Schur functor of the tensor power $M^{\otimes s}$ as a $(F\sym{rs},F\sym{s})$-bimodule. The latter result can be regarded as a refinement of a special case of \cite[2.5, Lemma]{DE}, although our proof is independent of their result. Besides obtaining the image of $L^s(M)$ under the Schur functor from this refinement, we can also get those of the symmetric powers $S^s(M)$ and exterior powers $\boldsymbol{/\!\backslash}^s(M)$. Our proofs are fairly elementary.
The organisation of the paper is as follows: in the next section, we give a background on Schur algebras and a summary of the results we need. We then proceed in Section \ref{S:main} to state and prove our main results.
\section{Schur algebras}
We briefly discuss Schur algebras and the results we need in this section. The reader may refer to \cite{G} for more details.
Throughout, we fix an infinite field $F$ of arbitrary characteristic.
Let $n,r \in \mathbb{Z}^+$. The Schur algebra $S(n,r)$ has a distinguished set $\{ \xi_{\alpha} \mid \alpha \in \Lambda(n,r)\}$ of pairwise orthogonal idempotents which sum to 1, where $\Lambda(n,r)$ is the set of compositions of $r$ with $n$ parts \cite[(2.3d)]{G}. Thus each left $S(n,r)$-module $M$ has a vector space decomposition
$$
M = \bigoplus_{\alpha \in \Lambda(n,r)} \xi_\alpha M.
$$
We write $M^{\alpha}$ for $\xi_{\alpha}M$, and call it the $\alpha$-weight space of $M$.
Let $\mathrm{GL}_n(F)$ be the general linear group. There is a surjective algebra homomorphism $e_r : F\mathrm{GL}_n(F) \to S(n,r)$ \cite[(2.4b)(i)]{G}.
If $s$ is another positive integer, and $M_1$ and $M_2$ are left $S(n,r)$- and $S(n,s)$-modules respectively, then $M_1 \otimes_F M_2$ can be endowed with a natural left $S(n,r+s)$-module structure, which satisfies
\begin{equation*} \label{E:tensor}
e_{r+s}(g) (m_1 \otimes m_2) = (e_r(g)m_1) \otimes (e_s(g)m_2)
\end{equation*}
for all $g \in \mathrm{GL}_n(F)$, $m_1 \in M_1$ and $m_2 \in M_2$. The weight spaces of $M_1 \otimes_F M_2$ can be described \cite[(3.3c)]{G} in terms of the weight spaces of $M_1$ and $M_2$, as follows:
\begin{equation} \label{E:weight}
(M_1 \otimes_F M_2)^{\gamma} = \bigoplus_{\substack{\alpha \in \Lambda(n,r) \\ \beta \in \Lambda(n,s) \\ \alpha+\beta = \gamma}} M_1^{\alpha} \otimes_F M_2^{\beta}.
\end{equation}
(Here, and hereafter, if $\alpha = (\alpha_1,\dotsc, \alpha_n) \in \Lambda(n,r)$ and $\beta = (\beta_1,\dotsc, \beta_n) \in \Lambda(n,s)$, then $\alpha + \beta = (\alpha_1 + \beta_1,\dotsc, \alpha_n + \beta_n) \in \Lambda(n,r+s)$.)
The symmetric group $\sym{n}$ on $n$ letters acts on $\Lambda(n,r)$ by place permutation:
$\tau \cdot (\alpha_1,\dotsc, \alpha_n) = (\alpha_{\tau^{-1}(1)},\dotsc, \alpha_{\tau^{-1}(n)})$.
We also view $\sym{n}$ as the subgroup of $\mathrm{GL}_n(F)$ consisting of permutation matrices. Thus, $\sym{n}$ also acts naturally on left $S(n,r)$-modules via $e_r$. We have the following lemma:
\begin{lem}\label{L:iso}
Let $n,r \in \mathbb{Z}^+$, and let $M$ be a left $S(n,r)$-module.
Let $\sigma \in \sym{n}$ and $\alpha = (\alpha_1,\dotsc, \alpha_n) \in \Lambda(n,r)$.
\begin{enumerate}
\item[(i)] $e_r(\sigma)$ maps $M^{\alpha}$ bijectively onto $M^{\sigma \cdot \alpha}$.
\item[(ii)] If $\sigma(i) = i$ for all $i$ such that $\alpha_i \ne 0$, then $e_r(\sigma)$ acts as identity on $M^{\alpha}$. (Equivalently, if $\sigma_1(i) = \sigma_2(i)$ for all $i$ such that $\alpha_i \ne 0$, then $e_r(\sigma_1)m = e_r(\sigma_2)m$ for all $m \in M^{\alpha}$.)
\end{enumerate}
\end{lem}
\begin{proof}
Part (i) is (3.3a) of \cite{G} (and its proof). For part (ii), it follows from the definition of $e_r$ in \cite[\S2.4]{G} that $e_r(\sigma) m = \xi_{\sigma\mathbf{i},\mathbf{i}}m$ for $m$ lying in a weight space associated to $\mathbf{i}$, so that $e_r(\sigma) m = m$ when $m \in M^{\alpha}$, and $\sigma$ satisfies the condition in (ii).
\end{proof}
In the case where $n \geq r$, let $$\omega_r = (\underbrace{1,\dotsc, 1}_{r \text{ times}}, \underbrace{0,\dotsc, 0}_{n-r \text{ times}}) \in \Lambda(n,r),$$
The subalgebra $\xi_{\omega_r} S(n,r) \xi_{\omega_r}$ of $S(n,r)$ is isomorphic to $F\sym{r}$ \cite[(6.1d)]{G}. This induces the Schur functor $f_r : {}_{S(n,r)}\textbf{mod} \to {}_{F\sym{r}}\textbf{mod}$ which sends a left $S(n,r)$-module $M$ to its weight space $M^{\omega_r}$. The $\sym{r}$-action on $f_rM = M^{\omega_r}$ is that via $e_r$ and viewing $\sym{r}$ as a subgroup of $\mathrm{GL}_n(F)$ via the embedding $\sym{r} \subseteq \sym{n} \subseteq \mathrm{GL}_n(F)$, i.e. if $m \in f_rM$ and $\sigma \in \sym{r}$, then
\begin{equation*} \label{E:symaction}
\sigma \cdot m = e_r(\sigma) m.
\end{equation*}
\section{Main results} \label{S:main}
Let $M$ be a left $S(n,r)$-module, and let $s \in \mathbb{Z}^+$. The $s$-fold tensor product $M^{\otimes s}$ is then a left $S(n,rs)$-module, and it also admits another commuting right action of $\sym{s}$ by place permutations, i.e.\ $(m_1 \otimes \dotsb \otimes m_s) \cdot \sigma = m_{\sigma(1)} \otimes \dotsb \otimes m_{\sigma(s)}$ where $m_1,\dotsc, m_s \in M$, $\sigma \in \sym{s}$. As such, if $n \geq rs$, then $f_{rs} M^{\otimes s}$ is a $(F\sym{rs}, F\sym{s})$-bimodule.
On the other hand, $(f_r M)^{\otimes s}$ is a left $F(\sym{r} \wr \sym{s})$-module via
$$ (\sigma_1,\dotsc,\sigma_s) \tau \cdot (m_1 \otimes \dotsb \otimes m_s) = \sigma_1 m_{\tau^{-1}(1)} \otimes \dotsb \otimes \sigma_s m_{\tau^{-1}(s)}$$ and we can make it into a $(F(\sym{r} \wr \sym{s}), F\sym{s})$-bimodule by allowing $\sym{s}$ to act trivially on its right, while $F\sym{s}$ is naturally a $(F\sym{s},F\sym{s})$-bimodule and we can make it into a $(F(\sym{r} \wr \sym{s}), F\sym{s})$-bimodule by allowing $(\sym{r})^s$ to act trivially on its left. Thus, $(f_r M)^{\otimes s} \otimes_F F\sym{s}$ is a $(F(\sym{r} \wr \sym{s}), F\sym{s})$-bimodule via the diagonal action.
For each $1\leq i\leq s$ and $\sigma\in \sym{r}$, we write $\sigma[i]\in \sym{rs}$ for the permutation sending $(i-1)r+j$ to $(i-1)r+\sigma(j)$ for each $1\leq j\leq r$, and fixing everything else pointwise; also, let $\sym{r}[i] = \{ \sigma[i] \mid \sigma \in \sym{r} \}$. For $\tau\in \sym{s}$, we write $\tau^{[r]}\in\sym{rs}$ for the permutation sending $(i-1)r+j$ to $(\tau(i)-1)r+j$ for each $1\leq i\leq s$ and $1\leq j \leq r$; also, let $\sym{s}^{[r]} = \{ \tau^{[r]} \mid \tau \in \sym{s} \}$. We identify $\sym{r}\wr \sym{s}$ with the subgroup $(\prod_{i=1}^s \sym{r} [i])\sym{s}^{[r]}$ of $\sym{rs}$.
With the above understanding, we have the following result.
\begin{thm} \label{T:main}
Let $n, r,s \in \mathbb{Z}^+$ with $n \geq rs$, and let $M$ be an $S(n,r)$-module. Then
$$
f_{rs} M^{\otimes s} \cong \operatorname{Ind}_{\sym{r} \wr \sym{s}}^{\sym{rs}} ((f_r M)^{\otimes s} \otimes_F F\sym{s})$$ as $(F\sym{rs}, F\sym{s})$-bimodules.
\end{thm}
\begin{proof}
Firstly, by \eqref{E:weight},
\begin{equation} \label{E:weight2}
f_{rs} M^{\otimes s} = (M^{\otimes s})^{\omega_{rs}} = \bigoplus_{(\alpha^{[1]}, \dotsc, \alpha^{[s]}) \in \Lambda} M^{\alpha^{[1]}} \otimes_F \dotsb \otimes_F M^{\alpha^{[s]}},
\end{equation}
where $\Lambda = \{ (\alpha^{[1]}, \dotsc, \alpha^{[s]}) \mid \alpha^{[i]} \in \Lambda(n,r)\ \forall i,\ \sum_{i=1}^s \alpha^{[i]} = \omega_{rs} \}$.
Also,
\begin{align*}
\operatorname{Ind}_{\sym{r} \wr \sym{s}}^{\sym{rs}} ((f_r M)^{\otimes s} \otimes_F F\sym{s}) &= F\sym{rs} \otimes_{F(\sym{r} \wr \sym{s})} ((f_r M)^{\otimes s} \otimes_F F\sym{s}) \\
&= \bigoplus_{t \in T} t \otimes ((f_r M)^{\otimes s} \otimes 1),
\end{align*}
where $T$ is a fixed set of left coset representatives of $\prod_{i=1}^s \sym{r}[i]$ in $\sym{rs}$.
The symmetric group $\sym{rs}$ acts naturally and transitively on the set
$$\Omega = \left\{ (A_1,\dotsc, A_s) \mid |A_i| = r\ \forall i,\ \bigcup_{i=1}^s A_i = \{ 1,\dotsc, rs\} \right\},$$
with $\prod_{i=1}^s \sym{r}[i]$ being the stabiliser of $(\{1,\dotsc, r\},\dotsc, \{(s-1)r + 1,\dotsc, rs\})$. As such, the function $\theta : T \to \Omega$ defined by
$$t \mapsto (\{t(1),\dotsc, t(r)\},\dotsc, \{t((s-1)r+1), \dotsc, t(rs)\})$$ is a bijection.
On the other hand, each $r$-element subset $A$ of $\{1,\dotsc, rs\}$ corresponds naturally to a distinct element $\alpha_A = ((\alpha_A)_1,\dotsc, (\alpha_A)_n) \in \Lambda(n,r)$ defined by $(\alpha_A)_j = 1$ if $j \in A$, and $0$ otherwise. This induces a bijection $\chi: \Omega \to \Lambda$ defined by
$$
(A_1,\dotsc, A_s) \mapsto (\alpha_{A_1}, \dotsc, \alpha_{A_s}).$$
For each $t \in T$ and $i = 1,\dotsc, s$, let $\tau_{t,i}$ be any fixed element of $\sym{rs}$ satisfying $\tau_{t,i}(j) = t((i-1)r + j)$ for $j = 1, \dotsc, r$ (we shall see below that how $\tau_{t,i}$ acts on other points is immaterial for our purposes). Let $\alpha^{[i]} = \tau_{t,i} \cdot \omega_r$; then $\alpha^{[i]} = \alpha_{A_i}$, where
$$
A_i = \{ \tau_{t,i}(1), \dotsc, \tau_{t,i}(r) \} = \{ t((i-1)r + 1), \dotsc, t(ir) \}.
$$
Thus $(\alpha^{[1]},\dotsc, \alpha^{[s]}) = \chi(\theta (t)) \in \Lambda$.
Let
\begin{align*}
\phi_t : t \otimes ((f_r M)^{\otimes s} \otimes 1) &\to M^{\alpha^{[1]}} \otimes_F \dotsb \otimes_F M^{\alpha^{[s]}} \\
t \otimes ((x_1 \otimes \dotsb \otimes x_s) \otimes 1) &\mapsto e_r(\tau_{t,1}) x_1 \otimes \dotsb \otimes e_r(\tau_{t,s}) x_s \quad (x_1, \dotsc, x_s \in f_rM).
\end{align*}
Since each $\tau_{t,i}$ maps $f_r M = M^{\omega_r}$ bijectively onto $M^{\tau_{t,i} \cdot \omega_r} = M^{\alpha^{[i]}}$ by Lemma \ref{L:iso}(i), we see that $\phi_t$ is bijective. Now let
$$\phi = \bigoplus_{t \in T} \phi_t : \operatorname{Ind}_{\sym{r} \wr \sym{s}}^{\sym{rs}} ((f_r M)^{\otimes s} \otimes_F F\sym{s}) \to f_{rs} M^{\otimes s}.$$ This is well-defined and bijective by \eqref{E:weight2} (note that $\chi \circ \theta : T \to \Lambda$ is a bijection).
Let $g \in \sym{rs}$ and $h \in \sym{s}$. Then if $t \in T$ and $x_1,\dotsc, x_s \in f_rM$, we have
\begin{align*}
g (t \otimes ((x_1 \otimes \dotsb \otimes x_s) \otimes 1)) h &=
gth^{[r]} \otimes ((x_{h(1)} \otimes \dotsb \otimes x_{h(s)}) \otimes 1) \\
&= t' \otimes ((e_r(g_1)x_{h(1)} \otimes \dotsb \otimes e_r(g_s)x_{h(s)}) \otimes 1),
\end{align*}
where $gth^{[r]} = t' \prod_{i=1}^s g_i[i]$ with $t' \in T$ and $g_i \in \sym{r}$ for all $i$. Thus it is sent by $\phi$ to $e_r(\tau_{t',1})e_r(g_1) x_{h(1)} \otimes \dotsb \otimes e_r(\tau_{t',s})e_r(g_s) x_{h(s)}$. Note that
\begin{multline*}
\tau_{t',i}(j) = t'((i-1)r + j) = gth^{[r]} (\prod_{i=1}^s g_i[i])^{-1} ((i-1)r + j) \\
= gth^{[r]} ((i-1) + g_i^{-1}(j)) = gt((h(i)-1) + g_i^{-1}(j)) = g\tau_{t,h(i)}g_i^{-1}(j),
\end{multline*}
so that $e_r(\tau_{t',i})e_r(g_i) x_{h(i)} = e_r(g\tau_{t,h(i)}g_i^{-1})e_r(g_i) x_{h(i)}$ by Lemma \ref{L:iso}(ii). Hence,
\begin{align*}
\phi(g (t \otimes ((x_1 \otimes \dotsb \otimes x_s) \otimes 1)) h) &=
e_r(g) e_r(\tau_{t,h(1)}) x_{h(1)} \otimes \dotsb \otimes e_r(g) e_r(\tau_{t,h(s)}) x_{h(s)} \\
&= e_{rs}(g) (e_r(\tau_{t,1})x_1 \otimes \dotsb \otimes e_r(\tau_{t,s})x_s) h \\
&= g(\phi(t \otimes ((x_1 \otimes \dotsb \otimes x_s) \otimes 1))) h.
\end{align*}
Thus $\phi$ is an $(F\sym{rs},F\sym{s})$-bimodule isomorphism.
\end{proof}
The $s$-th Lie power $L^s(M)$, the $s$-th exterior power $\boldsymbol{/\!\backslash}^s(M)$ and the $s$-th symmetric power $S^s(M)$ of the left $S(n,r)$-module $M$ may be defined as follows:
\begin{align*}
L^s(M) &= (M^{\otimes s}) \upsilon_s; \\
\boldsymbol{/\!\backslash}^s(M) &= (M^{\otimes s}) (\sum_{\sigma \in \sym{s}} \operatorname{sgn}(\sigma) \sigma); \\
S^s(M) &= M^{\otimes s} \otimes_{F\sym{s}} F.
\end{align*}
Here, $\upsilon_s$ is the Dynkin-Specht-Wever element mentioned in Section \ref{S:intro}, and $\operatorname{sgn}$ is the signature representation of $\sym{s}$.
\begin{cor} \label{C:isom}
Let $n, r,s \in \mathbb{Z}^+$ with $n \geq rs$, and $M$ be an $S(n,r)$-module. Then
\begin{align*}
f_{rs} L^s(M) &\cong \operatorname{Ind}_{\sym{r} \wr \sym{s}}^{\sym{rs}} ((f_r M)^{\otimes s} \otimes_F \operatorname{\mathrm{Lie}}(s)); \\
f_{rs} \boldsymbol{/\!\backslash}^s(M) &\cong \operatorname{Ind}_{\sym{r} \wr \sym{s}}^{\sym{rs}} ((f_r M)^{\otimes s} \otimes_F \operatorname{sgn}); \\
f_{rs} S^s(M) &\cong \operatorname{Ind}_{\sym{r} \wr \sym{s}}^{\sym{rs}} (f_r M)^{\otimes s}
\end{align*}
as left $F\sym{rs}$-modules.
\end{cor}
\begin{proof}
Post-multiply $\upsilon_s$ and $\sum_{\sigma \in \sym{s}} \operatorname{sgn}(\sigma) \sigma$ to both sides of the isomorphism in Theorem \ref{T:main} to obtain the first two isomorphisms. The third isomorphism is obtained by taking tensor product with $F$ over $F\sym{s}$ on the right of both sides of the same isomorphism.
\end{proof}
\end{document} |
\begin{document}
\title{\bfseries Regularized Interior Point Methods for Constrained Optimization and Control}
\author{Alberto De~Marchi\thanks{\TheAffiliation. \email{[email protected]}, \orcid{0000-0002-3545-6898}.}}
\date{}
\maketitle
\begin{abstract}
Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications.
This paper discusses the interactions between these techniques and proposes an algorithm that synergistically combines them.
Building a sequence of closely related subproblems and approximately solving each of them, this approach inherently exploits warm-starting, early termination, and the possibility to adopt subsolvers tailored to specific problem structures.
Moreover, by relaxing the equality constraints with a proximal penalty, the regularized subproblems are feasible and satisfy a strong constraint qualification by construction, allowing the safe use of efficient solvers.
We show how regularization benefits the underlying linear algebra and a detailed convergence analysis indicates that limit points tend to minimize constraint violation and satisfy suitable optimality conditions.
Finally, we report on numerical results in terms of robustness, indicating that the combined approach compares favorably against both interior point and augmented Lagrangian codes.
\end{abstract}
\keywords{\TheKeywords}
\subclass{\TheAMSsubj}
\section{Introduction}
Mathematical optimization plays an important role in model-based and data-driven control systems, forming the basis for advanced techniques such as optimal control, nonlinear model predictive control (MPC) and parameters estimation.
Significant research effort on computationally efficient real-time optimization algorithms contributed to the success of MPC over the years and yet the demand for fast and reliable methods for a broad spectrum of applications is growing; see \cite{sopasakis2020open,saraf2022efficient} and references therein.
In order to tackle these challenges, it is desirable to have an algorithm that benefits from warm-starting information, can cope with infeasibility, is robust to problem scaling, and exploits the structure of the problem.
In order to reduce computations and increase robustness, a common approach is to relax the requirements on the solutions, in terms of optimality, constraint violation, or both \cite{diehl2009efficient,saraf2022efficient}.
In this work, we propose to address such features by combining proximal regularization and interior point techniques, for developing a stabilized, efficient and robust numerical method.
We advocate for this strategy by bringing together and combining a variety of ideas from the nonlinear programming literature.
Let us consider the constrained nonconvex problem
\begin{align}\label{eq:P}\tag{P}
\minimize_x\quad&f(x) \\
\stt\quad&c(x) = 0 ,\qquad
x \geq 0 ,\nonumber
\end{align}
where functions $\func{f}{\R^n}{\mathbb{R}}$ and $\func{c}{\R^n}{\R^m}$ are (at least) continuously differentiable.
Problems with general inequality constraints on $x$ and $c(x)$ can be reformulated as \eqref{eq:P} by introducing auxiliary variables.
Nonlinear programming (NLP) problems such as \eqref{eq:P} have been extensively studied and there exist several approaches for their numerical solution.
Interior point (IP) \cite{vanderbei1999interior,waechter2006implementation},
penalty and augmented Lagrangian (AL) \cite{conn1991globally,andreani2008augmented,birgin2014practical}
and sequential programming \cite{fiacco1968nonlinear}
schemes are predominant ideas and have been re-combined in many ways \cite{curtis2012penalty,birgin2016sequential,armand2019rapid}.
Starting from linear programming, IP methods had a significant impact on the field of mathematical optimization \cite{gondzio2012interior}.
By solving a sequence of barrier subproblems, they can efficiently handle inequality constraints and scale well with the problem size.
The state-of-the-art solver \ipopt{}, described by \cite{waechter2006implementation}, is an emblem of this remarkable success.
However, relying on Newton-type schemes for approximately solving the subproblems, IP algorithms may suffer degeneracy and lack of constraint qualifications if suitable counter-mechanisms are not implemented.
On the contrary, proximal techniques naturally cope with these scenarios thanks to their inherent regularizing action.
Widely investigated in the convex setting \cite{rockafellar1976monotone}, their favorable properties have been exploited to design (primal-dual) stabilized methods building on the proximal point algorithm \cite{friedlander2012primal,liaomcpherson2020fbstab,demarchi2022qpdo}.
The analysis of their close connection with the AL framework \cite{rockafellar1974augmented} led to the development of techniques applicable to more general problems \cite{ma2018stabilized,demarchi2021augmented,potschka2021sequential}.
The combination of IP and proximal strategies has been successfully leveraged in the context of convex quadratic programming \cite{altman1999regularized,cipolla2022proximal} and for linear \cite{dehghani2020regularized} and nonlinear \cite{siqueira2018regularized} least-squares problems.
With this work we address general NLPs and devise a method for their numerical solution, which can be seen as an extension of a regularized Lagrange--Newton method to handle bound constraints via a barrier function \cite{demarchi2021augmented}, or as a proximally stabilized IP algorithm, generalizing the ideas put forward by \cite{cipolla2022proximal}.
\footnote{Although beyond the scope of this paper, topics such as infeasibility detection \cite{armand2019rapid}, ill-posed subproblems \cite{andreani2008augmented}, and feasibility restoration \cite[\S 3.3]{waechter2006implementation} are of great relevance in this context.
Appropriate mechanisms should be incorporated in practical implementations, as they can significantly improve the performances.}
\paragraph*{Outline}
The paper is organized as follows.
In \cref{sec:stationarity_concepts} we provide and comment on some relevant optimality notions.
The methodology is discussed in \cref{sec:methods} detailing the proposed algorithm, whose convergence properties are investigated in \cref{sec:convergence}.
We report numerical results on benchmark problems in \cref{sec:numerics} and conclude the paper in \cref{sec:conclusion}.
\paragraph*{Notation}
With $\mathbb{N}$, $\mathbb{R}$, and $\mathbb{R}inf := \mathbb{R} \cup \{\infty\}$ we denote the natural, real and extended real numbers, respectively.
We denote the set of real vectors of dimension $n\in\mathbb{N}$ as $\mathbb{R}^n$; a real matrix with $m\in\mathbb{N}$ rows and $n\in\mathbb{N}$ columns as $A \in \mathbb{R}^{m\times n}$ and its transpose as $A^\top \in \mathbb{R}^{n\times m}$.
For a vector $a \in \mathbb{R}^n$, its $i$-th element is $a_i$ and its squared Euclidean norm is $\|a\|^2 = a^\top a$.
A vector or matrix with all zero elements is represented by $0$.
The gradient of a function $\func{f}{\R^n}{\mathbb{R}}$ at a point $\bar{x} \in \mathbb{R}^n$ is denoted by $\nabla f(\bar{x}) \in \R^n$; the Jacobian of a vector function $\func{c}{\R^n}{\R^m}$ by $\nabla c(\bar{x}) \in \mathbb{R}^{m\times n}$.
\section{Optimality and Stationarity}\label{sec:stationarity_concepts}
In this section we introduce some optimality concepts, following \cite[Ch. 3]{birgin2014practical} and \cite[\S 2]{demarchi2022interior}.
\begin{defn}[Feasibility]
Relative to \eqref{eq:P}, we say a point $x^\ast\in\R^n$ is \emph{feasible} if $x^\ast \geq 0$ and $c(x^\ast) = 0$; it is \emph{strictly feasible} if additionally $x^\ast > 0$.
\end{defn}
\begin{defn}[Approximate KKT stationarity]\label{defn:epsKKTstationary}
Relative to \eqref{eq:P}, a point $x^\ast\in\R^n$ is \emph{$\varepsilon$-KKT stationary} for some $\varepsilon \geq 0$ if there exist multipliers $y^\ast\in\R^m$ and $z^\ast\in\R^n$ such that
\begin{subequations}\label{eq:epsKKTstationary}
\begin{align}
\norm{ \nabla f(x^\ast) + \nabla c(x^\ast)^\top y^\ast + z^\ast } {}\leq{}& \varepsilon , \label{eq:epsKKTstationary:x}\\
\norm{ c(x^\ast) } {}\leq{}& \varepsilon , \\
x^\ast \geq - \varepsilon ,\quad
z^\ast \leq \varepsilon ,\quad
\min\{x^\ast,-z^\ast\} {}\leq{}& \varepsilon . \label{eq:epsKKTstationary:z}
\end{align}
\end{subequations}
When $\varepsilon = 0$, the point $x^\ast$ is said \emph{KKT stationary}.
\end{defn}
Notice that \eqref{eq:epsKKTstationary:z} provides a condition modeling the approximate satisfaction of the (elementwise) complementarity condition $\min\{x,-z\} = 0$ within some tolerance $\varepsilon \geq 0$.
The IP algorithm discussed in \cref{sec:methods} satisfies a stronger version of these conditions, since the iterates it generates meet the constraints $x \geq 0$ and $z \leq 0$ by construction.
Furthermore, we point out that the condition $\min\{x_i,-z_i\}\leq \varepsilon$ is analogous to $- x_i z_i \leq \varepsilon$, more typical for interior point methods, but does not depend on a specific barrier function, e.g., the logarithmic barrier in \cite[\S 2.1]{waechter2006implementation}.
We shall consider the limiting behavior of approximate KKT stationary points when the tolerance $\varepsilon$ vanishes.
In fact, having $x^k \to x^\ast$ with $x^k$ $\varepsilon_k$-KKT stationary for \eqref{eq:P} and $\varepsilon_k \searrow 0$ does not guarantee KKT stationarity of a limit point $x^\ast$ of $\{x^k\}$.
This issue raises the need for defining KKT stationarity in an asymptotic sense \cite[Def. 3.1]{birgin2014practical}.
\begin{defn}[Asymptotic KKT stationarity]\label{defn:asympKKTstationary}
Relative to \eqref{eq:P}, a feasible point $x^\ast\in\R^n$ is \emph{AKKT stationary} if there exist sequences $\{x^k\}, \{z^k\}\subset\R^n$, and $\{y^k\}\subset\R^m$ such that $x^k \to x^\ast$ and
\begin{subequations}\label{eq:asympKKTstationary}
\begin{align}
\nabla f(x^k) + \nabla c(x^k)^\top y^k + z^k {}\to{}& 0 , \label{eq:asympKKTstationary:x} \\
\min\{ x^k, -z^k \} {}\to{}& 0 . \label{eq:asympKKTstationary:z}
\end{align}
\end{subequations}
\end{defn}
Any local minimizer $x^\ast$ for \eqref{eq:P} is AKKT stationary, independently of constraint qualifications \cite[Thm 3.1]{birgin2014practical}.
\section{Approach and Algorithm}\label{sec:methods}
The methodology presented in this section builds upon the AL framework, interpreted as a proximal point scheme in the nonconvex regime, and IP methods.
The basic idea is to construct a sequence of proximally regularized subproblems and to approximately solve each of them as a single barrier subproblem, effectively merging the AL and IP outer loops.
Reduced computational cost can be achieved with an effective warm-starting of the IP iterations and with the tight entanglement of barrier and proximal penalty strategies, by monitoring and updating the parameters' values alongside with the inner tolerance.
A classical approach is to consider a sequence of bound-constrained Lagrangian (BCL) subproblems \cite{conn1991globally,birgin2014practical}
\begin{equation}\label{eq:Pbcl}
\minimize_{x\geq 0}\quad f(x) + \frac{1}{2 \rho_k} \| c(x) + \rho_k \hat{y}^k \|^2
\end{equation}
where $\rho_k > 0$ and $\hat{y}^k \in \R^m$ are some given penalty parameter and dual estimate, respectively.
The nonlinearly-constrained Lagrangian (NCL) scheme \cite{ma2018stabilized} considers equality-constrained subproblems by introducing an auxiliary variable $s \in \R^m$ and the constraint $c(x) = s$.
Analogously, a proximal point perspective yields the equivalent reformulation
\begin{align}\label{eq:Pproxy}
\minimize_{x ,\, \lambda}\quad&f(x) + \frac{\rho_k}{2} \| \lambda \|^2 \\
\stt\quad&c(x) + \rho_k (\hat{y}^k - \lambda) = 0 ,\qquad
x \geq 0 , \nonumber
\end{align}
recovering the dual regularization term obtained, e.g., by \cite{potschka2021sequential,demarchi2021augmented,demarchi2022qpdo}.
By construction, these regularized subproblems are always feasible and satisfy a strong constraint qualification, namely the LICQ, at all points.
The regularized subproblems \eqref{eq:Pbcl}--\eqref{eq:Pproxy} can be numerically solved via IP algorithms.
Let us consider a barrier parameter $\mu_k > 0$ and barrier functions $\func{b_i}{\mathbb{R}}{\mathbb{R}inf}$, $i=1,\ldots,n$, each with domain $\dom b_i = (0,\infty)$, and such that $b_i(t) \to \infty$ as $t\to 0^+$ and $b_i^\prime \leq 0$.
Exemplarily, the logarithmic function $x \mapsto - \ln(x)$ is one of such barrier functions.
Other choices can be considered as well, e.g., to handle bilateral constraints \cite{bertolazzi2007real}.
We collect these barrier functions to define $\func{b}{\R^n}{\mathbb{R}inf}$, $b \colon x \mapsto \sum_{i=1}^n b_i(x_i)$, whose domain is $\dom b = (0,\infty)^n$.
Thus, analogously to \cite{armand2019rapid}, a barrier counterpart for the BCL subproblem \eqref{eq:Pbcl} reads
\begin{equation}\label{eq:Pbarrier_BCL}
\minimize_x \quad f(x) + \frac{1}{2 \rho_k} \| c(x) + \rho_k \hat{y}^k \|^2 + \mu_k b(x) ,
\end{equation}
whereas for the constrained subproblem \eqref{eq:Pproxy} this leads to
\begin{align}\label{eq:PproxyBarrier}
\minimize_{x ,\, \lambda}\quad&f(x) + \frac{\rho_k}{2} \| \lambda \|^2 + \mu_k b(x) \\
\stt\quad&c(x) + \rho_k (\hat{y}^k - \lambda) = 0 , \nonumber
\end{align}
which is a regularized version of \cite[Eq. 3]{waechter2006implementation} and reminiscent of \cite[Eq. 2]{birgin2016sequential}.
It should be stressed that, in stark contrast with classical AL and IP schemes, we intend to find an (approximate) solution to the regularized subproblem \eqref{eq:Pproxy} by (approximately) solving only one barrier subproblem \eqref{eq:PproxyBarrier}.
Inspired by \cite{curtis2012penalty,cipolla2022proximal}, our rationale is to drive $\rho_k, \mu_k$ and the inner tolerance $\epsilon_k$ concurrently toward zero, effectively knitting together proximal and barrier strategies.
It should be noted that a primal (Tikhonov-like) regularization term is not explicitly included in \eqref{eq:Pbcl}--\eqref{eq:PproxyBarrier}.
In fact, the original objective $f$ could be replaced by a (proximal) model of the form $x \mapsto f(x) + \frac{\sigma_k}{2} \| x - \hat{x}^k \|^2$, with some given primal regularization parameter $\sigma_k \geq 0$ and reference point $\hat{x}^k \in \R^n$.
However, as this term can be interpreted as an inertia correction, we prefer the subsolver to account for its contribution; cf. \cite[\S 3.1]{waechter2006implementation}.
In this way, the subsolver can surgically tune the primal correction term as needed, possibly improving the convergence speed, and surpassing the issue that suitable values for $\sigma_k$ are unknown a priori.
\input{TeX/Alg/ProxIP.tex}
The overall procedure is detailed in \cref{alg:RIPM}.
At every outer iteration, indexed by $k$, \cref{step:xy} requires to compute an approximate stationary point, with the associated Lagrange multiplier, for the regularized barrier subproblem \eqref{eq:PproxyBarrier}.
As the dual estimate $\hat{y}^k$ is selected from some bounded set $Y\subset\R^m$ at \cref{step:yhat}, the AL scheme is \emph{safeguarded} and has stronger global convergence properties \cite[Ch. 4]{birgin2014practical}.
The assignment of $z^k$ at \cref{step:z} follows from comparing and matching the stationarity conditions for \eqref{eq:P} and \eqref{eq:PproxyBarrier}.
After checking termination, we monitor progress in constraint violation and complementarity, based on \eqref{eq:epsKKTstationary}, and update parameters $\rho_k$ and $\mu_k$ accordingly, as well as the inner tolerance $\varepsilon_k$.
At \cref{step:updatePenalty:if,step:updateBarrier:if} we consider relaxed conditions for \emph{satisfactory} feasibility and complementarity as it is preferable to have the sequences $\{\rho_k\}$, $\{\mu_k\}$, and $\{\epsilon_k\}$ bounded away from zero, in order to avoid unnecessary ill-conditioning and tight tolerances.
Sufficient conditions to guarantee boundedness of the penalty parameter $\{\rho_k\}$ away from zero are given, e.g., by \cite[\S 5]{andreani2008augmented}.
Remarkably, as established by \cref{lem:asympComplementarity} in \cref{sec:convergence}, there is no need for the barrier parameter $\mu_k$ to vanish in order to achieve $\varepsilon$-complementarity in the sense of \eqref{eq:epsKKTstationary:z}, for $\varepsilon > 0$.
We shall mention that considering equivalent yet different subproblem formulations may affect the practical performance of the subsolver.
It is enlightening to pinpoint the effect of the dual regularization in \eqref{eq:PproxyBarrier} and to appreciate its interactions with the linear algebra routines used to solve the linear systems arising in Newton-type methods.
Although \eqref{eq:PproxyBarrier} has more (possibly many more) variables than \eqref{eq:Pbarrier_BCL}, a simple reordering yields matrices with the same structure \cite{demarchi2021augmented,potschka2021sequential}.
Let us have a closer look.
Defining the Lagrangian function $\mathcal{L}_k(x,y) := f(x) + \mu_k b(x) + \innprod{y}{c(x)}$, the stationarity condition for \eqref{eq:Pbarrier_BCL} reads $0 = \nabla_x \mathcal{L}_k( x, y_k(x) )$, where $y_k(x) := \hat{y}^k + \rho_k^{-1} c(x)$, and the corresponding Newton system is
\begin{equation}
\begin{bmatrix}
H_k( x, y_k(x) ) + \frac{1}{\rho_k} \nabla c(x)^\top \nabla c(x)
\end{bmatrix} \delta x = - \nabla_x \mathcal{L}_k(x,y_k(x)) ,
\end{equation}
where $H_k(x,y) \in \mathbb{R}^{n \times n}$ denotes the Hessian matrix $\nabla_{xx}^2 \mathcal{L}_k(x,y)$ or a symmetric approximation thereof.
A linear transformation yields the equivalent linear system
\begin{equation}
\begin{bmatrix}
H_k( x, y_k(x) ) & \nabla c(x)^\top \\
\nabla c(x) & - \rho_k I
\end{bmatrix} \begin{bmatrix}
\delta x \\ \delta y
\end{bmatrix} = - \begin{bmatrix}
\nabla_x \mathcal{L}_k(x,y_k(x)) \\
0
\end{bmatrix} .
\end{equation}
Analogous Newton systems for \eqref{eq:PproxyBarrier} read
\begin{equation}
\begin{bmatrix}
H_k(x,y) & \cdot & \nabla c(x)^\top \\
\cdot &\rho_k I & - \rho_k I \\
\nabla c(x) & - \rho_k I & \cdot
\end{bmatrix} \begin{bmatrix}
\delta x \\ \delta \lambda \\ \delta y
\end{bmatrix} = - \begin{bmatrix}
\nabla_x \mathcal{L}_k(x,y) \\
\rho_k (\lambda - y) \\
c(x) + \rho_k (\hat{y}^k - \lambda)
\end{bmatrix}
\end{equation}
and formally solving for $\delta\lambda$ gives the condensed system
\begin{equation}
\label{eq:linsys}
\begin{bmatrix}
H_k(x,y) & \nabla c(x)^\top \\
\nabla c(x) & - \rho_k I
\end{bmatrix} \begin{bmatrix}
\delta x \\ \delta y
\end{bmatrix} = - \begin{bmatrix}
\nabla_x \mathcal{L}_k(x,y) \\
c(x) + \rho_k (\hat{y}^k - y)
\end{bmatrix} .
\end{equation}
The resemblances between these linear systems are apparent, as well as the differences.
The AL relaxation in \eqref{eq:Pbarrier_BCL} introduces a dual regularization for both the linear algebra and nonlinear solver, whose \emph{hidden} constraint $c(x) + \rho_k (\hat{y}^k - y) = 0$ holds pointwise due to the identity $y = y_k(x)$.
We remark that, entering the (2,2)-block, the dual regularization prevents issues due to linear dependence.
Furthermore, the primal regularization is left to the inertia correction strategy of the subsolver, affecting the (1,1)-block as in \cite[\S 3.1]{waechter2006implementation}.
If the approximation $H_k(x,y)$ is positive definite, e.g., by adopting suitable quasi-Newton techniques, the matrix in \eqref{eq:linsys} is symmetric quasi-definite and can be efficiently factorized with tailored linear algebra routines \cite{vanderbei1995symmetric}.
\section{Convergence Analysis}\label{sec:convergence}
In this section we analyze the asymptotic properties of the iterates generated by \cref{alg:RIPM} under the following blanket assumptions:
\begin{enumerate}[label=(\textsc{a}\arabic*)]
\item Functions $\func{f}{\R^n}{\mathbb{R}}$ and $\func{c}{\R^n}{\R^m}$ in \eqref{eq:P} are continuously differentiable.
\item\label{ass:wellposed} Subproblems \eqref{eq:PproxyBarrier} are well-posed for all parameters' values, namely for any $\mu_k \leq \mu_0$, $\rho_k \leq \rho_0$, and $\hat{y}^k \in Y$.
\end{enumerate}
First, we characterize the iterates in terms of stationarity.
\begin{lem}\label{lem:xyzSignKKT}
Consider a sequence $\{x^k,y^k,z^k\}$ generated by \cref{alg:RIPM}.
Then, for all $k\in\mathbb{N}$, it is $x^k > 0$, $z^k \leq 0$, and the following conditions hold:
\begin{subequations}\label{eq:xyzSignKKT}
\begin{align}
\| \nabla f(x^k) + \nabla c(x^k)^\top y^k + z^k \|
{}\leq{}&
\epsilon_k , \label{eq:xyzSignKKT:x}\\
\| c(x^k) + \rho_k (\hat{y}^k - y^k) \|
{}\leq{}&
2 \epsilon_k \label{eq:xyzSignKKT:y}.
\end{align}
\end{subequations}
\end{lem}
\begin{proof}
Positivity of $x^k$ follows from the barrier function $b$ having domain $\dom b = (0,\infty)^n$, whereas nonpositivity of $z^k$ is a consequence of $b_i^\prime \leq 0$ for all $i$ and $\mu_k > 0$.
Based on \cref{defn:epsKKTstationary} and \cref{step:z} of \cref{alg:RIPM}, the $\epsilon_k$-KKT stationarity of $(x^k,\lambda^k)$ for \eqref{eq:PproxyBarrier}, with multiplier $y^k$, yields \eqref{eq:xyzSignKKT:x} along with
\begin{subequations}\label{eq:epsKKTstationary:PproxyBarrier}
\begin{align}
\rho_k \norm{ \lambda^k - y^k } {}\leq{}& \epsilon_k , \label{eq:epsKKTstationary:PproxyBarrier:lambda}\\
\norm{ c(x^k) + \rho_k (\hat{y}^k - \lambda^k) } {}\leq{}& \epsilon_k . \label{eq:epsKKTstationary:PproxyBarrier:y}
\end{align}
\end{subequations}
By the triangle inequality, \eqref{eq:epsKKTstationary:PproxyBarrier:lambda}--\eqref{eq:epsKKTstationary:PproxyBarrier:y} imply \eqref{eq:xyzSignKKT:y}.
\qedhere
\end{proof}
Patterning \cite[Thm 4.2(ii)]{demarchi2022interior}, we establish asymptotic complementarity.
\begin{lem}\label{lem:asympComplementarity}
Consider a sequence $\{ x^k,y^k,z^k \}$ of iterates generated by \cref{alg:RIPM} with $\varepsilon = 0$.
Then, it holds $\lim\limits_{k\to\infty} \min\{x^k,-z^k\} = 0$.
\end{lem}
\begin{proof}
The algorithm can terminate in finite time only if the returned triplet $(x^\ast,y^\ast,z^\ast)$ satisfies $\min\{ x^\ast,-z^\ast \} = 0$.
Excluding this ideal situation, we may assume that it runs indefinitely and that consequently $\mu_k \searrow 0$, owing to \cref{step:updateBarrier:if,step:updateBarrier:failed} and recalling that $x^k > 0$ and $z^k \leq 0$ for all $k\in\mathbb{N}$ by \cref{lem:xyzSignKKT}.
Consider now an arbitrary index $i\in\{1,\ldots,n\}$ and the two possible cases.
If $x_i^k \to 0$, then the statement readily follows from $z_i^k \leq 0$.
If instead a subsequence $\{x_i^k\}_K$ remains bounded away from zero, then $\{ b_i^\prime(x_i^k) \}_K$ is bounded and therefore $z_i^k = \mu_k b_i^\prime(x_i^k) \to 0$ as $k\to_K \infty$, proving the statement since $x_i^k>0$.
The claim then follows from the arbitrarity of the index $i$ and the subsequence.
\qedhere
\end{proof}
Like all penalty-type methods in the nonconvex setting, \cref{alg:RIPM} may generate limit points that are infeasible for \eqref{eq:P}.
Patterning standard arguments, the following result gives sufficient conditions for the feasibility of limit points; cf. \cite[Ex. 4.12]{birgin2014practical}.
\begin{prop}
Consider a sequence $\{ x^k,y^k,z^k \}$ of iterates generated by \cref{alg:RIPM}.
Then, each limit point $x^\ast$ of $\{x^k\}$ is feasible for \eqref{eq:P} if one of the following conditions holds:
\begin{enumerate}[label=(\roman*)]
\item the sequence $\{\rho_k\}$ is bounded away from zero, or
\item there exists some $B \in \mathbb{R}$ such that for all $k\in\mathbb{N}$
\begin{equation*}
f(x^k) + \frac{1}{2\rho_k} \norm{c(x^k) + \rho_k \hat{y}^k}^2 \leq B .
\end{equation*}
\end{enumerate}
\end{prop}
These conditions are generally difficult to check a priori.
Nevertheless, in the situation where each iterate $x^k$ is actually a (possibly inexact) global minimizer of \eqref{eq:PproxyBarrier}, then limit points generated by \cref{alg:RIPM} have minimum constraint violation and tend to minimize the objective function subject to minimal infeasibility \cite[Thm 5.1, Thm 5.3]{birgin2014practical}.
In particular, limit points are indeed feasible if \eqref{eq:P} admits feasible points.
However, these properties cannot be expected by solving the subproblems only up to stationarity.
Nonetheless, even in the case where a limit point is not necessarily feasible, the next result shows that it is at least a stationary point for a feasibility problem associated to \eqref{eq:P}.
\begin{prop}
Consider a sequence $\{x^k,y^k,z^k\}$ generated by \cref{alg:RIPM} with $\varepsilon = 0$.
Then each limit point $x^\ast$ of $\{x^k\}$ is KKT stationary for the problem
\begin{equation}\label{eq:feasibilityP}
\minimize_{x\geq 0} \quad \frac{1}{2} \|c(x)\|^2 .
\end{equation}
\end{prop}
\begin{proof}
We may consider two cases, depending on the sequence $\{\rho_k\}$.
If $\{\rho_k\}$ remains bounded away from zero, then \cref{step:updatePenalty:if,step:updatePenalty:failed} of \cref{alg:RIPM} imply that $\|c(x^k)\|\to0$ for $k\to\infty$.
Continuity of $c$ and properties of norms yield $c(x^\ast) = 0$.
Furthermore, by construction, we have $x^k > 0$ for all $k\in\mathbb{N}$, hence $x^\ast \geq 0$.
Altogether, this shows that $x^\ast$ is feasible for \eqref{eq:P}, namely a global minimizer for the feasibility problem \eqref{eq:feasibilityP} and, therefore, a KKT stationary point thereof.
Assume now that $\rho_k \to 0$.
Define $\delta^k\in\R^n$ and $\eta^k \in \R^m$ as
\begin{align*}
\delta^k {}:={}& \nabla f(x^k) + \nabla c(x^k)^\top y^k + z^k \\
\eta^k {}:={}& c(x^k) + \rho_k (\hat{y}^k - y^k)
\end{align*}
for all $k\in\mathbb{N}$.
In view of \cref{lem:xyzSignKKT}, we have that $\norm{\delta^k} \leq \epsilon_k$ and $\norm{\eta^k}\leq2\epsilon_k$ hold for all $k\in\mathbb{N}$.
Multiplying $\delta^k$ by $\rho_k$, substituting $y^k$ and rearranging, we obtain
\begin{equation*}
\rho_k \delta^k
{}={}
\rho_k \nabla f(x^k) + \nabla c(x^k)^\top \left[ \rho_k \hat{y}^k + c(x^k) - \eta^k \right] + \rho_k z^k .
\end{equation*}
Now, let $x^\ast$ be a limit point of $\{x^k\}$ and $\{x^k\}_K$ a subsequence such that $x^k \to_K x^\ast$.
Then the sequence $\{\nabla f(x^k)\}_K$ is bounded, and so is $\{\hat{y}^k\}_K \subset Y$ by construction.
Recalling from \cref{lem:xyzSignKKT} that $x^k > 0$ and $z^k \leq 0$, and observing that $0 \leq \norm{\delta^k}, \norm{\eta^k} \leq 2 \epsilon_k \to 0$, we shall now take the limit of $\rho_k \delta^k$ for $k\to_K\infty$, resulting in
\begin{equation*}
0
{}={}
\nabla c(x^\ast)^\top c(x^\ast) + \tilde{z}^\ast
\end{equation*}
for some $\tilde{z}^\ast \leq 0$.
As a limit point of $\{ \rho_k z^k\}$, $\tilde{z}^\ast$ together with $x^\ast$ satisfy $\min\{x^\ast,-\tilde{z}^\ast\}=0$ by \cref{lem:asympComplementarity}.
Since we also have $x^\ast \geq 0$, it follows that $x^\ast$ is KKT stationary for \eqref{eq:feasibilityP} according to \cref{defn:epsKKTstationary}.
\qedhere
\end{proof}
Finally, we qualify the output of \cref{alg:RIPM} in the case of feasible limit points.
In particular, it is shown that any feasible limit point is AKKT stationary for \eqref{eq:P} in the sense of \cref{defn:asympKKTstationary}.
Under some additional boundedness conditions, feasible limit points are KKT stationary, according to \cref{defn:epsKKTstationary}.
\begin{thm}\label{thm:subseqKKTstationarity}
Let $\{x^k,y^k,z^k\}$ be a sequence of iterates generated by \cref{alg:RIPM} with $\varepsilon = 0$.
Let $x^\ast$ be a feasible limit point of $\{x^k\}$ and $\{x^k\}_K$ a subsequence such that $x^k \to_K x^\ast$.
Then,
\begin{enumerate}[label=(\roman{*})]
\item\label{thm:subseqKKTstationarity:asymptotic} $x^\ast$ is an AKKT stationary point for \eqref{eq:P}.
\item\label{thm:subseqKKTstationarity:bounded} If $\{y^k,z^k\}_K$ remain bounded, then $x^\ast$ is KKT stationary for \eqref{eq:P}.
\end{enumerate}
\end{thm}
\begin{proof}
\ref{thm:subseqKKTstationarity:asymptotic}
Together with the fact that $\epsilon_k \to 0$, \cref{lem:xyzSignKKT} ensures that the sequence $\{x^k\}_K$ satisfies condition \eqref{eq:asympKKTstationary:x}, whereas \cref{lem:asympComplementarity} implies \eqref{eq:asympKKTstationary:z}.
Feasibility of $x^\ast$ completes the proof.
\ref{thm:subseqKKTstationarity:bounded}
By boundedness, the subsequences $\{y^k\}_K$ and $\{z^k\}_K$ admit some limit points $y^\ast$ and $z^\ast$, respectively.
Thus, from the previous assertion and with continuity arguments on $f$ and $c$, it follows that $x^\ast$ is KKT stationary for \eqref{eq:P}, not only asymptotically.
\qedhere
\end{proof}
Provided that the iterates admit a feasible limit point, finite termination of \cref{alg:RIPM} with an $\varepsilon$-KKT stationary point can be established as a direct consequence of \cref{thm:subseqKKTstationarity}.
\section{Numerical Results}\label{sec:numerics}
In this section we test an instance of the proposed regularized interior point approach, denoted \regip{}, on the CUTEst benchmark problems \cite{gould2015cutest}.
\regip{} is compared in terms of robustness against the IP solver \ipopt{} \cite{waechter2006implementation} and the AL solver \percival{} \cite{santos2020percivaljl}, which is based on a BCL method \cite{conn1991globally} coupled with a trust-region matrix-free solver \cite{lin1999newton} for the subproblems.
We do not report runtimes nor iteration counts since a fair comparison would require close inspection of heuristics and fallbacks \cite[\S 3]{waechter2006implementation}.
We implemented \regip{} in Julia and set up the numerical experiments adopting the JSO software infrastructure by \cite{orban2019jso}.
The IP solver \ipopt{} acts as subsolver to execute \cref{step:xy}, warm-started at the current primal $(x^{k-1}, y^{k-1})$ and dual $(y^{k-1}$, $z^{k-1})$ estimates.
We use its parameter \ipoptOption{tol} to set the (inner) tolerance $\epsilon_k$, disabling other termination conditions, and let \ipopt{} control the barrier parameter as needed to approximately solve the regularized subproblem.
\footnote{Solving a sequence of barrier subproblems may hinder the computational efficiency of \regip{} compared to the approach behind \cref{alg:RIPM}, but does not degrade its reliability.
Ongoing research focuses on solving \eqref{eq:PproxyBarrier} and letting the IP subsolver update the barrier parameter after warm-starting at the current primal-dual estimate, in the spirit of \cite[Alg. 3]{cipolla2022proximal}.\label{foot:barriersubsolver}}
We let the safeguarding set be $Y := \{ v\in\R^m \,|\, \|v\|_\infty \leq 10^{20} \}$ and choose $\hat{y}^k$ by projecting the current estimate $y^{k-1}$ onto $Y$.
We set the initial penalty parameter to $\rho_0 = 10^{-6}$, the inner tolerance $\epsilon_0 = \sqrt[3]{\varepsilon}$, and parameters $\theta_\rho = 0.5$, $\kappa_\rho = 0.5$, and $\kappa_\epsilon = 0.5$.
\regip{} declares success, and returns a $\varepsilon$-KKT stationary point, as soon as $\epsilon_k \leq \varepsilon$ and $C^k \leq \varepsilon$.
\footnote{The condition $\epsilon_k \leq \varepsilon$ implies both $\varepsilon$-stationarity \eqref{eq:epsKKTstationary:x} and $\varepsilon$-complementarity \eqref{eq:epsKKTstationary:z} required for $\varepsilon$-KKT stationarity. This follows from the observation that, in \regip{}, the subsolver approximately solves \eqref{eq:Pproxy} at \cref{step:xy}, not \eqref{eq:PproxyBarrier}; see Footnote~\ref{foot:barriersubsolver}.}
Instead, if $\epsilon_k \leq \varepsilon$, $C^k > \varepsilon$ and $\rho_k \leq \rho_{\min} := 10^{-20}$, \regip{} stops declaring (local) infeasibility.
For \ipopt{}, we set the tolerance \ipoptOption{tol} to $\varepsilon$, remove the other desired thresholds, and disable termination based on acceptable iterates.
For \percival{}, we set absolute and relative tolerances \ipoptOption{atol}, \ipoptOption{rtol}, and \ipoptOption{ctol} to $\varepsilon$.
We select all CUTEst problems with at most 1000 variables and constraints, obtaining a test set with 609 problems.
All solvers are provided with the primal-dual initial point available in CUTEst, a time limit of $60$ seconds, the maximum number of iterations set to $10^9$, and a tolerance $\varepsilon \in \{10^{-3}, 10^{-5} \}$.
A solver is deemed to solve a problem if it returns with a successful status; it fails otherwise.
The source codes for the numerical experiments have been archived on Zenodo at
\begin{center}
\href{https://doi.org/10.5281/zenodo.7109904}{\textsc{doi}: 10.5281/zenodo.7109904}.
\end{center}
\Cref{tab:cutest} summarizes the results, stratified by solver, termination tolerance ($\varepsilon$) and problem size ($n$, $m$).
For each combination, we indicate the number of times \regip{} solves a problem that the other solver fails (``W'') or solves (``T+'') and the number of times \regip{} fails on a problem that the other one fails (``T-'') or solves (``L'').
The results show that \regip{} succeeds on more problems than the other solvers, consistently for both low and high accuracy, indicating that the underlying regularized IP approach can form the basis for reliable and scalable solvers.
\footnote{We agree with \cite[\S 5]{birgin2016sequential} on the fact that ``strong statements concerning the relative efficiency or robustness [$\ldots$] are not possible in nonlinear optimization.''}
\begin{table}
\begin{center}
\renewcommand{1.2}{1.2}
\caption{Comparison on CUTEst problems with $n$ variables and $m$ constraints}
\label{tab:cutest}
\begin{tabular}{cc|ccc|ccc}
\hline
\multicolumn{8}{c}{\regip{} against \ipopt{}} \\ \hline
\multicolumn{2}{c|}{Size $n, m$} & \multicolumn{3}{c|}{Tolerance $\varepsilon = 10^{-3}$} & \multicolumn{3}{|c}{Tolerance $\varepsilon = 10^{-5}$} \\
Min & Max & W & T & L & W & T & L \\ \hline
0 & 10 & 19 & 417+/25- & 3 & 15 & 417+/29- & 3 \\
11 & 100 & 6 & 60+/7- & 1 & 7 & 58+/8- & 1 \\
101 & 1000 & 6 & 57+/8- & 0 & 7 & 53+/10- & 1 \\ \hline
\multicolumn{8}{c}{\regip{} against \percival{}} \\ \hline
\multicolumn{2}{c|}{Size $n, m$} & \multicolumn{3}{c|}{Tolerance $\varepsilon = 10^{-3}$} & \multicolumn{3}{|c}{Tolerance $\varepsilon = 10^{-5}$} \\
Min & Max & W & T & L & W & T & L \\ \hline
0 & 10 & 17 & 419+/20- & 8 & 19 & 413+/24- & 8 \\
11 & 100 & 12 & 54+/8- & 0 & 18 & 47+/8- & 1 \\
101 & 1000 & 37 & 26+/8- & 0 & 37 & 23+/11- & 0 \\ \hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}\label{sec:conclusion}
This paper has presented a regularized interior point approach to solving constrained nonlinear optimization problems.
Operating as an outer regularization layer, a quadratic proximal penalty provides robustness whilst consuming minimal computation effort once embedded into existent interior point codes as a principled inertia correction strategy.
Furthermore, regularizing the equality constraints allows to safely adopt more efficient linear algebra routines, while waiving the need for an infeasibility detection mechanism within the subsolver.
Preliminary numerical results indicate that a close integration of proximal regularization within interior point schemes is key to provide efficient and robust solvers.
We encourage further research in this direction.
{\small
}
\end{document} |
\begin{document}
\title{On the unfolding of simple closed curves}
\author{John Pardon}
\address{Durham Academy Upper School\\3601 Ridge Road\\Durham, North Carolina 27705}
\curraddr{Princeton University\\Princeton, New Jersey 08544}
\email{[email protected]}
\subjclass[2000]{Primary 53C24; Secondary 53A04}
\date{January 2, 2007}
\begin{abstract}
I show that every rectifiable simple closed curve in the
plane can be continuously deformed into a convex
curve in a motion which preserves arc length and does
not decrease the Euclidean distance between any pair of
points on the curve. This result is obtained by approximating
the curve with polygons and invoking the result of
Connelly, Demaine, and Rote that such a motion exists
for polygons. I also formulate a generalization of their
program, thereby making steps toward a fully continuous
proof of the result. To facilitate this, I generalize
two of the primary tools used in their program:
the Farkas Lemma of linear programming to Banach
spaces and the Maxwell-Cremona Theorem of rigidity theory
to apply to stresses represented by measures on the plane.
\end{abstract}
\maketitle
\section{Introduction}\label{intro}
Imagine a loop of string lying flat on a table without
crossing itself. Now suppose the loop is slowly
deformed until it becomes convex, without stretching
or breaking it, in an {\it expansive} motion.
By expansive, I mean that
if you pick any pair of points on the string, then during
the deformation, the distance between them will be
nondecreasing. Then we can ask whether, given
an initial loop, there always exists an expansive motion
which deforms that loop until it becomes convex. If the
loop is a polygon, then the answer is yes, as proved by
Connelly, Demaine, and Rote \cite{cdr}. The first theorem
of this paper (Theorem \ref{main}) is that the answer is yes for any rectifiable
curve, no matter how complicated (in section \ref{path},
we give some examples of pathological curves to which the
theorem applies). This solves Problem 4 listed by Ghomi
\cite[p. 1]{open}.
My proof of the main theorem uses a limiting process,
relying on the result of \cite{cdr}. I next generalize
the program used in \cite{cdr}, which
relies on techniques of linear programming, specifically the Farkas
Lemma. This approach naturally lends itself to computation;
an example of research on the computation of nonexpansive unfoldings
of polygons is given by \cite{energy}. In my continuous
analogue of the program, I develop a version of the Farkas
Lemma for Banach spaces (Theorem \ref{farkas}) as well as a
continuous version of the Maxwell-Cremona Theorem (Theorem
\ref{maxwell}), a combinatorial version of which was used in
the program in \cite{cdr}. A different version of the Farkas Lemma
in Banach spaces and specifically in $\Lp$ spaces has been studied
in \cite{farkas}. I am not aware of any previous
generalization of the Maxwell-Cremona Theorem to the
case I consider here. Finally, I use the continuous
version of the program to give a different proof of the existence
of infinitesimal expansions for polygons. The hope is that
a continuous analogue of the discrete program could yield
a direct proof (one which does not rely on approximation by
polygons) of the main theorem for some class of curves more
general than polygons.
I would like to thank Robert Bryant for many useful
conversations about the work in this paper, regarding both
its content and presentation, and Robert Connelly for
suggesting some reogranization to clarify the results.
I also thank Andrew Ferrari for introducing me to many of
the techniques used here.
\subsection{Notation}
We will use the following function spaces:
\begin{description}
\item[$C(X,Y)$] the Banach space of continuous functions from
$X$ to $Y$ given the supremum norm.
\item[$C_c(X,Y)$] the subspace of $C(X,Y)$ consisting of
functions of compact support.
\item[$C_0(X,Y)$] the Banach space completion of $C_c(X,Y)$
with respect to the supremum norm. These are the functions
that ``vanish at infinity''.
\item[$C_0^\infty(X,Y)$] the subspace of $C_c(X,Y)$ consisting
of infinitely differentiable functions.
\item[$\Lp(X,Y)$] the Banach space of $\Lp$ functions from
$X$ to $Y$.
\end{description}
If $Y$ is left out, it is assumed to be $\R$, except in
section \ref{cpx}, where it is assumed to be $\C$.
All Hilbert and Banach spaces are implicitly assumed to
be over $\R$, except in section \ref{cpx}, where they will
be over $\C$. If $E$ is a Banach space, $E^*$ is its dual.
The duality bracket $\langle x,y\rangle$ will be used both
in the case that $x\in E^*$ and $y\in E$, and in the case
that $x,y\in H$, a Hilbert space. We will write $\Lin(X,Y)$
for the Banach space of bounded linear transformations from
$X$ to $Y$ given the operator norm.
\section{Proof for General Curves using \cite{cdr}}\label{cpx}
\subsection{Preliminaries}
Consider a simple closed curve in the plane. I wish to prove
the existence of a continuous deformation of the curve into
a convex curve, so that the intrinsic distance between every
pair of points on the curve stays constant, and the extrinsic
distance between every pair of points on the curve is
nondecreasing. Here, by intrinsic distance I mean the distance
along the curve, and by extrinsic distance I mean the Euclidean
distance in $\R^2$.
A curve is called {\it rectifiable} if a finite intrinsic
distance can be defined between every pair of points, that is,
the supremum of the lengths of all inscribed
polygons is finite:
\begin{equation}
L_x^y(\f):=\sup_{x=a_0<a_1<\cdots<a_k=y}\sum_{j=1}^k|\f(a_j)-\f(a_{j-1})|<\infty
\end{equation}
We will only consider rectifiable curves in
this paper. If a curve is rectifiable, then it has a {\it unit
speed parameterization}, that is $\f(s)=\int_0^s\f'(s')\,ds'$ and
$|\f'(s)|=1$ almost everywhere. Since a homothety will scale the
arc length of a curve, it suffices to consider simple closed
curves of length $2\pi$. Thus, given $\f_0$, we seek a
continuous family of simple closed curves $\f_t:\R/2\pi\map\R^2$
parameterized by $t\in[0,1]$ such that each curve is of unit
speed, $|\f_{t_1}(x)-\f_{t_1}(y)|\leq|\f_{t_2}(x)-\f_{t_2}(y)|$
whenever $t_1\leq t_2$, and $\f_1$ is convex.
\subsection{Main Result}
For this section, it will be natural to consider curves
in $\C$ (rather than $\R^2$). Thus Banach spaces will be
over $\C$. It will be convenient to have our curves
reside in the following space:
\begin{equation}
\mathcal D:=\left\{\f:\R/2\pi\map\C\Bigm|
\text{$\f(0)=0$, $\f$ absolutely continuous, $\f'\in\Linfty(\R/2\pi)$}\right\}
\end{equation}
There is, of course, the natural correspondence between
$\f\in\mathcal D$ and $\f'\in\{u\in\Linfty(\R/2\pi):\int u=0\}$.
Thus $\mathcal D$ is a Banach space with norm $\|\f'\|_\infty$. Now
topologize $\mathcal D$ using the weak-$*$ topology on
$\Linfty(\R/2\pi)$. Since $\Lone(\R/2\pi)$ is separable,
the Banach-Alaoglu Theorem implies that any norm bounded
sequence in $\mathcal D$ has a convergent subsequence.
The choice of topology on $\mathcal D$ is justified by
the following lemma.
\begin{lemma}\label{unif}
Suppose $\f_n\to\f$ in $\mathcal D$, then $\f_n\to\f$ uniformly.
\end{lemma}
\begin{proof}
By the Uniform Boundedness Principle, we know that
$\|\f_n\|$ is bounded. Thus there exists $M$
with $|\f_n'|\leq M$, hence $\{\f_n\}$ is an equicontinuous family.
It is clear that $\f_n\to\f$ pointwise since we have
$\int\chi_{[0,x]}\f_n'\to\int\chi_{[0,x]}\f'$. And
an equicontinuous sequence of functions converges
pointwise if and only if it converges uniformly.
\end{proof}
Define the continuous function $\mathcal E:\mathcal D\map\R$ by
$\mathcal E(\f)=\iint_{(\R/2\pi)^2}|\f(x)-\f(y)|$. Also define the following
order relation on $\mathcal D$: we say that $\f\trianglelefteq\g$
if and only if $|\f(x)-\f(y)|\leq|\g(x)-\g(y)|$
for all $x$ and $y$.
\begin{theorem}\label{main}
Given a unit speed simple closed curve $\f:\R/2\pi\map\C$,
there exists a continuous function $\h:[0,1]\map\mathcal D$ such
that:
\begin{itemize}
\item[(1)] $\h(0)=\f$.
\item[(2)] $\h(1)$ is convex.
\item[(3)] $\h(t)$ has unit speed for all $t$.
\item[(4)] If $t_1\leq t_2$, then $\h(t_1)\trianglelefteq\h(t_2)$.
\end{itemize}
\end{theorem}
\begin{proof}
For $n\geq 3$, consider the polygon $\mathcal P_n$
inscribed in $\f$ which has $n$ vertices spaced
out at multiples of $2\pi/n$ starting at zero.
Explicitly:
\begin{equation}
\mathcal P_n(x):=\left(1-\left\{\frac{nx}{2\pi}\right\}\right)
\f\left(\frac{2\pi}n\left\lfloor\frac{nx}{2\pi}\right\rfloor\right)
+\left\{\frac{nx}{2\pi}\right\}
\f\left(\frac{2\pi}n\left(\left\lfloor\frac{nx}{2\pi}\right\rfloor+1\right)\right)
\end{equation}
This polygon may or may not be simple. It will,
however, divide the plane into a finite number of
simply connected regions. Let $\mathcal P_n'$ be a constant
speed $s_n\leq 1$ parameterization of the boundary
of that region which has greatest area. Then let
$\h_n:[0,1]\map\mathcal D$ be continuous and satisfy:
\begin{itemize}
\item[(1$'$)] $\h_n(0)=\mathcal P_n'$.
\item[(2$'$)] $\h_n(1)$ is convex.
\item[(3$'$)] $\h_n(t)$ has speed $s_n\leq 1$ for all $t$.
\item[(4$'$)] If $t_1\leq t_2$, then $\h_n(t_1)\trianglelefteq\h_n(t_2)$.
\item[(5$'$)] $\h_n(t)(\pi)\in\R_{>0}$ for all $t$.
\item[(6$'$)] $\mathcal E(\h_n(t))$ is a linear function of $t$.
\end{itemize}
The existence of an $\h_n$ satisfying (1$'$)--(4$'$) is
implied by Theorem 1 of \cite[p. 207]{cdr}.
Condition (5$'$) can be achieved by properly rotating
each curve. The motion of \cite{cdr} is {\it strictly}
expansive, so $\mathcal E(\h_n(t))$ will be strictly
increasing, so a simple reparameterization in $t$
suffices to make it linear and satisfy (6$'$).
Let $\Q_{[0,1]}=\Q\cap[0,1]$.
This set is countable; suppose $\{r_i\}_{i=1}^\infty$
is a counting of it. Let $\h_n^{(0)}=\h_n$.
Inductively, let $\h_n^{(i)}$ be a subsequence of
$\h_n^{(i-1)}$ such that $\h_n^{(i)}(r_i)$ converges.
(Such a subsequence is guaranteed to exist since
$\|\h_n^{(i)}(r_i)\|=s_n$ is bounded). Now $\h_j^{(j)}$
converges pointwise to a function $\tilde{\h}:\Q_{[0,1]}\map\mathcal D$
which satisfies:
\begin{itemize}
\item[(1$''$)] $\tilde{\h}(0)(\R/2\pi)=\f(\R/2\pi)$.
\item[(2$''$)] $\tilde{\h}(1)$ is convex.
\item[(3$''$)] $\tilde{\h}(t)$ has speed $\leq 1$.
\item[(4$''$)] If $t_1\leq t_2$, then $\tilde{\h}(t_1)\trianglelefteq\tilde{\h}(t_2)$.
\item[(5$''$)] $\tilde{\h}(t)(\pi)\in\R_{>0}$ for all $t$.
\item[(6$''$)] $\mathcal E(\tilde{\h}(t))$ is a linear function of $t$.
\end{itemize}
We will now construct $\h:[0,1]\map\mathcal D$. For every
$t\in[0,1]$, we set $\h(t)$ to be some arbitrary subsequential
limit of $\tilde{\h}(q_j)$ where $q_j$ is some sequence of
rationals converging to $t$. Clearly $\h$ satisfies
(1$''$)--(6$''$) as well. Now (1$''$) and (3$''$) together
mean that $\h(0)(s)=\f(s+\Delta)$ for some
$\Delta$. We can take $\Delta=0$. Hence we have
(1), (2), and (4). To prove (3), note that:
\begin{equation}
\left|\frac{\f(x+h)-\f(x)}h\right|\leq\left|\frac{\h(t)(x+h)-\h(t)(x)}h\right|\leq 1
\end{equation}
As $h\to 0$, the left hand side approaches $1$ for
almost all $x$, hence $|\h(t)'(x)|=1$ almost
everywhere as desired.
Finally, we must show that $\h$ is in fact continuous.
This follows from (5$''$) and (6$''$) in the following way.
Suppose the contrary, that there is some $t$ where
$\h$ is not continuous. Then there exists a sequence
$q_j\to t$ with either $q_j<t$ for all $j$ or $q_j>t$
for all $j$, and a neighborhood $N$ of $\h(t)$ such that
$\h(q_j)\notin N$ for all $j$. Now a subsequence of
$\h(q_j)$ will converge in $\mathcal D$ to a limit $\g$.
Now we have:
\begin{itemize}
\item[(i)] $\mathcal E(\g)=\mathcal E(\h(t))$
\item[(ii)] $\g\trianglelefteq\h(t)$ or $\g\trianglerighteq\h(t)$
depending on whether $q_j<t$ or $q_j>t$
\item[(iii)] $\g(0)=0=\h(t)(0)$
\item[(iv)] $\g(\pi),\h(t)(\pi)\in\R_{>0}$
\end{itemize}
The conditions (i) and (ii) imply
that $|\g(x)-\g(y)|=|\h(t)(x)-\h(t)(y)|$ for
all $x$ and $y$. This means that the curves are
rigid motions of each other. Then (iii) and (iv)
imply that they are actually the same
curve since they have the same orientation. Thus a
subsequence of $\h(q_j)$ converges to $\h(t)$. This
is of course a contradiction since each $\h(q_j)$
is outside the neighborhood $N$ of $\h(t)$.
This contradiction proves that $\h$ is continuous.
\end{proof}
\subsection{Pathological Rectifiable Curves}\label{path}
Define $f_-$ and $f_+$:
\begin{equation}
f_\pm(x)=\begin{cases}x^2\sin x^{-1}\pm e^{-1/x}&x>0\cr 0&x=0\end{cases}
\end{equation}
If we plot $f_-$ and $f_+$ on $[0,\pi^{-1}]$ and add
line segments around the left side of the curve to close
it, we get an infinite number of interlocking ``teeth''.
This example is based on a polygon with a finite number
of such teeth unfolded by Erik Demaine. We also have:
\begin{equation}
g(t)=\begin{cases}t^2e^{i/t}&t>0\cr 0&t=0\cr-t^2e^{-i/t}&t<0\end{cases}
\end{equation}
Plotting $g$ on $[-\pi^{-1},\pi^{-1}]$ and adding line
segments to close the curve gives a simple closed curve
with an infinite spiral. By Theorem \ref{main},
both of these curves can be unfolded in an expansive
motion, something which is not at all intuitive considering
their geometry.
\section{A Generalization of the CDR Program}
The program in \cite{cdr} proves the existence of an infinitesimal
expansion for any polygon. That is, if a nonconvex polygon has
verticies $\p_i$, it shows the existence of velocities $\vv_i$ satisfying:
\begin{align}
(\p_i-\p_{i+1})\cdot(\vv_i-\vv_{i+1})&=0\\
(\p_i-\p_j)\cdot(\vv_i-\vv_j)&>0\text{ for $i$ and $j$ not adjacent}
\end{align}
From this, it is relatively straightforward to solve a
differential equation of the form $\frac d{dt}\{\p_i\}=\{\tilde{\vv}_i\}$
(where the $\{\tilde{\vv}_i\}$ depend continuously on the $\{\p_i\}$),
thus constructing an expansive motion of the polygon. Clearly,
if we have a curve $\f$, then the analogue is to find a variation
$\varphi$ satisfying:
\begin{align}
\f'(x)\cdot\varphi'(x)&=0\text{ for all $x$}\\
(\f(x)-\f(y))\cdot(\varphi(x)-\varphi(y))&\geq 0\text{ for all $x$ and $y$}
\end{align}
The generalized program developed here will be able to
prove the existence of infinitesimal expansions for
polygons, a hard theoretical result of \cite{cdr}.
It also proves the existence of ``almost'' expansive variations
for all rectifiable curves which in a neighborhood of any point
look like the rotated graph of a function from $\R$ to $\R$.
By this I mean that for every $x\in\R/2\pi$, there exists
$\vv\in\R^2$ such that $\f(y)\cdot\vv$ is one to one in a
neighborhood of $x$. The final result of this generalized
program is Theorem \ref{gen}.
The generalizations of the Farkas Lemma and the Maxwell-Cremona
Theorem, the tools used in the program, are stated and proved
in sections \ref{farkass} and \ref{maxwells} respectively.
\subsection{Notation}
Let $H:=\{u\in C(\R/2\pi,\R^2):u(0)=0$, $u$ absolutely
continuous, and $\int|u'|^2<\infty\}$. So that $H$ is a Hilbert
space, equip it with the norm $\sqrt{\int|u'|^2}$
and inner product $\int u'\cdot v'$. Topologize $H$ with
the weak topology. We will need the sets:
\begin{align}
Q_\f&:=\{u\in H:u'\cdot\f'\equiv 0\}\text{ (a closed subspace)}\\
T&:=\{t\in C((\R/2\pi)^2)^*:t\geq 0\}
\end{align}
Note that we will be looking for $\varphi\in Q_\f$, since it is
these variations which preserve arc length. Also note that in
this section, we do not assume that $\f$ is parameterized by
arc length.
\begin{lemma}\label{unif2}
If $\g_n\to\g$ in the weak topology on $H$, then
$\g_n\to\g$ uniformly.
\end{lemma}
\begin{proof}
This is completely analogous to Lemma \ref{unif}. We know that
$\g_n\to\g$ pointwise. Observing that $\|\g_n\|$ is bounded, we
have the inequality:
\begin{equation}
\int_a^b|\g_n'|=\int_{\R/2\pi}\g_n'\frac{\g_n'}{|\g_n'|}\chi_{[a,b]}\leq
\sqrt{\int_{\R/2\pi}|\g_n'|^2}\sqrt{\int_a^b\left|\frac{\g_n'}{|\g_n'|}\right|^2}
\leq M\sqrt{b-a}
\end{equation}
This shows that $\g_n$ are uniformly continuous, and hence
converge uniformly.
\end{proof}
Define $D\subset H$, the set of curves we will consider,
to be the set of $\f\in H$ satisfying:
\begin{itemize}
\item[(1)] $\f$ is a simple closed curve, that is $\f$ is injective.
\item[(2)] $\f'\ne 0$ almost everywhere (this in fact is not implied by (1)).
\item[(3)] For every $x$, there exists $\delta>0$ and $\vv$
such that $\f(y)\cdot\vv$ is one to one for $|y-x|<\delta$. (locally graph-like)
\end{itemize}
The symbol $\f$ will always denote a member of $D$.
The following bounded operator will be essential
to the program; it is called the {\it Rigidity
Operator}:
\begin{gather}
R_\f:H\map C((\R/2\pi)^2)\cr
(R_\f\varphi)(x,y)=(\f(x)-\f(y))\cdot(\varphi(x)-\varphi(y))
\end{gather}
\subsection{Outline of the Program}
Before we state the final result of the program in its full
generality (Theorem \ref{gen}), it is useful to state the
following corollary which gives the general idea of the result.
\begin{corollary}\label{cgen}
Let $\f\in D$ not be convex. Let $V\subset(\R/2\pi)^2$ be
closed and have the property that for all $(x,y)\in V$,
the line segment between $\f(x)$ and $\f(y)$ is not competely
contained in $\f(\R/2\pi)$ (for example, if $\f$ has no
straight sections, we can take $V=\{(x,y)\in(\R/2\pi)^2:|x-y|>\epsilon\}$).
Then there exists a $\varphi\in Q_\f$ such that:
\begin{equation}\label{expan}
(\f(x)-\f(y))\cdot(\varphi(x)-\varphi(y))>0\text{ for all $(x,y)\in V$}
\end{equation}
\end{corollary}
This result includes the result \cite{cdr} of the
existence of infinitesimal expansions for nonconvex
polygons.
\begin{corollary}[Theorem 3 of {\cite[p. 215]{cdr}}]
If $\{\p_i\}$ is a nonconvex simple polygon with no
straight verticies, then there exist $\{\vv_i\}$ satisfying:
\begin{align}
(\p_i-\p_{i+1})\cdot(\vv_i-\vv_{i+1})&=0\\
(\p_i-\p_j)\cdot(\vv_i-\vv_j)&>0\text{ for $i$ and $j$ not adjacent}
\end{align}
\end{corollary}
\begin{proof}
Apply Corollary \ref{cgen} to $\f=\text{the polygon}$ and:
\begin{equation}
\begin{split}
V=\{(x,y)\in(\R/2\pi)^2:\text{ }&\text{there are two full edges
separating $x$ and $y$}\cr&\text{in both directions}\}
\end{split}
\end{equation}
Then we have a $\varphi$. Set $\vv_i=\varphi(\f^{-1}(\p_i))$.
Then $(\p_i-\p_j)\cdot(\vv_i-\vv_j)>0$ for $i$ and $j$ not
adjacent is clear from (\ref{expan}). Now:
\begin{equation}
(\vv_{i+1}-\vv_i)\cdot(\p_{i+1}-\p_i)=
\int_{\f^{-1}(\p_i)}^{\f^{-1}(\p_{i+1})}\!\varphi'(x)\cdot(\p_{i+1}-\p_i)\,dx=0
\end{equation}
since $\varphi\in Q_\f$.
\end{proof}
Theorem \ref{gen}, the main result of the generalization of
the program of \cite{cdr} is essentially Corollary \ref{cgen}
made uniform over some suitable set of curves.
\begin{theorem}[Analogue of Theorem 3 of {\cite[p. 215]{cdr}}]\label{gen}
Suppose $D_1\subset D$ is (weakly) closed and contains no
convex curves, and that $V\subset D_1\cross (\R/2\pi)^2$
is closed. Additionally, suppose that for every $(\f,x,y)\in V$,
the line segment joining $\f(x)$ and $\f(y)$ is not completely
contained in $\f(\R/2\pi)$. Then there exists $\epsilon>0$ such
that for each $\f\in D_1$, there exists $\varphi\in Q_\f$ with:
\begin{itemize}
\item[(1)] $\|\varphi\|=1$.
\item[(2)] $R_\f\varphi(x,y)\geq\epsilon$ whenever $(\f,x,y)\in V$.
\end{itemize}
\end{theorem}
We can see that Corollary \ref{cgen} is obtained by taking $D_1$
to consist of a single curve. A corollary which does not lose
the uniformity is the following:
\begin{corollary}
Suppose $D_1\subset D$ is weakly closed, contains no convex
curves, and contains no curves with straight sections. Then
for every $\delta>0$, there exists $\epsilon>0$ such that for
every $\f\in D_1$, there exists $\varphi\in Q_\f$ satisfying:
\begin{itemize}
\item[(1)] $\|\varphi\|=1$.
\item[(2)] $(\varphi(x)-\varphi(y))\cdot(\f(x)-\f(y))\geq\epsilon$ if $|x-y|\geq\delta$.
\end{itemize}
\end{corollary}
\begin{proof}
Choose $V=D_1\cross\{(x,y)\in(\R/2\pi)^2:|x-y|\geq\delta\}$ and apply
Theorem \ref{gen}.
\end{proof}
The main difficulty in showing the existence of a $\varphi$
which is expansive for {\it all} pairs $x$ and $y$
is the fact $V$ being closed is critical to the proof. Clearly
$(\f,x,x)$ can never be in $V$ since then we would conclude
that $(\varphi(x)-\varphi(x))\cdot(\f(x)-\f(x))>0$. Hence, we
must always exclude a neighborhood of the ``diagonal'' of
$(\R/2\pi)^2$. This means that we will not have shown that
$(\varphi(x)-\varphi(y))\cdot(\f(x)-\f(y))>0$ for all
pairs $x$ and $y$.
The following theorem is the essence of why expansive variations
exist. It relies on the generalization of the Maxwell-Cremona
Theorem (Theorem \ref{maxwell}).
\begin{theorem}[Analogue of Theorem 4 of {\cite[p. 216]{cdr}}]\label{needmax}
If $\f\in D$ and $t\in T$ such that
$\langle t,R_\f\alpha\rangle=0$ for all $\alpha\in Q_\f$, then
either:
\begin{itemize}
\item[(1)] The curve $\f$ is convex.\newline OR
\item[(2)] For all $(x,y)\in\operatorname{supp}t$, the line
segment connecting $\f(x)$ and $\f(y)$
is completely contained in $\f(\R/2\pi)$.
\end{itemize}
\end{theorem}
In the spirit of the generalization of the Farkas Lemma (Theorem
\ref{farkas}), it is possible to prove that Theorem \ref{needmax} implies
Theorem \ref{gen}.
\begin{proposition}[Analogue of Lemma 3 of {\cite[p. 216]{cdr}}]\label{impl}
Theorem \ref{needmax} implies Theorem \ref{gen}.
\end{proposition}
\subsection{Proof of Theorem \ref{needmax}}
\begin{proof}
Suppose that we have some $\f\in D$ and $t\in T$
with $\langle t,R_\f\alpha\rangle=0$ for all $\alpha\in Q_\f$.
Let $\mathbf{\hat f}'$ denote $\f'/|\f'|$.
First, let us show that there exists
$\beta\in\Ltwo(\R/2\pi)$ such that:
\begin{equation}
\langle t,R\alpha\rangle=\int_{\R/2\pi}\beta(x)\mathbf{\hat f}'(x)\cdot\alpha'(x)\,dx
\end{equation}
Clearly there exists $\mu\in\Ltwo(\R/2\pi,\R^2)$
such that
$\langle t,R\alpha\rangle=\int_{\R/2\pi}\mu(x)\cdot\alpha'(x)\,dx$.
Now we can constrain $\mu$ as follows. For any
$\lambda\in\Ltwo(\R/2\pi)$ satisfying
$\int\lambda\mathbf{\hat f}'=0$, we know that:
\begin{equation}
\int_{\R/2\pi}\left(\mu(x)\cdot i\mathbf{\hat f}'(x)\right)\lambda(x)\,dx=0
\end{equation}
The set $H$ of such $\lambda$ is of codimension $2$ in
$\Ltwo(\R/2\pi)$. Now
$\mu(x)\cdot i\mathbf{\hat f}'(x)\in H^\perp$,
which is of dimension $2$. But we can exercise
two dimensions of freedom by
adding constants to $\mu(x)$. Thus we can assume
$\mu(x)\cdot i\mathbf{\hat f}'(x)\equiv 0$,
in other words $\mu\parallel\f'$, and hence is
of the form $\beta(x)\mathbf{\hat f}'(x)$.
We will consider the operators $A_1,A_2\in\Lin(C_0(\R^2,\R^2),\R^2)$
defined by:
\begin{align}
A_1\U&:=\iint_{(\R/2\pi)^2}t(x,y)(\f(x)-\f(y))
\int_{\f(y)}^{\f(x)}\U\cdot\dd s\\
A_2\U&:=\int_{\R/2\pi}\beta(x)\mathbf{\hat f}'(x)[\U(\f(x))\cdot\f'(x)]\,dx
\end{align}
Since $A_1$ and $A_2$ are linear combinations of projections, they
are symmetric, that is there exist $a_j,b_j,e_j\in\Lin(C_0(\R^2,\R),\R)=C_0(\R^2)^*$
such that $A_j=\left(\smallmatrix a_j&b_j\cr b_j&e_j\endsmallmatrix\right)$.
Then $A:=A_1-A_2=\left(\smallmatrix a&b\cr b&e\endsmallmatrix\right)$,
where $a,b,e\in\Lin(C_0(\R^2,\R),\R)=C_0(\R^2)^*$. We have:
\begin{equation}
\begin{split}
A_1\grad g&=\iint_{(\R/2\pi)^2}t(x,y)(\f(x)-\f(y))(g(\f(x))-g(\f(y)))\cr
&=\Bigl(\langle t,R(\e_1g(\f(\cdot)))\rangle,\langle t,R(\e_2g(\f(\cdot)))\rangle\Bigr)\cr
&=\int_{\R/2\pi}\beta(x)\mathbf{\hat f}'(x)[\grad g(\f(x))\cdot\f'(x)]\,dx=A_2\grad g
\end{split}
\end{equation}
Hence $A\grad g=0$ for all $g\in C_0^\infty(\R^2)$.
By the generalization of the Maxwell-Cremona Theorem,
Theorem \ref{maxwell}, there exists a $c\in C_c(\R^2)$
such that we have (in the distributional sense):
\begin{equation}
A\U=\iint_{\R^2}\left(\begin{matrix}
c_{yy}&-c_{xy}\cr-c_{xy}&
c_{xx}\end{matrix}\right)\U\,dx\,dy
\end{equation}
Now the matrices $\left(\begin{smallmatrix}
c_{yy}&-c_{xy}\cr
-c_{xy}&
c_{xx}\end{smallmatrix}\right)$ and
$\left(\begin{smallmatrix}c_{xx}&c_{xy}\cr
c_{xy}&c_{yy}\end{smallmatrix}\right)$ are
related by a similarity transform. The former is
a positive linear combination of projections at
every point in $\R^2-\f(\R/2\pi)$, hence the latter
is positive at every point not on the curve as well. Hence
$c$ is locally convex on the interior of the curve
and on the exterior of the curve.
Now let $M=\sup_{\p\in\R^2}c(\p)$ and define the nonempty
closed set $S=\{\p\in\R^2:c(\p)=M\}$.
Suppose $\p\in\partial S$ and $\p\notin\f(\R/2\pi)$.
Then there is a neighborhood of $\p$ which is disjoint
from $\f(\R/2\pi)$. In this neighborhood, $c$ will
be convex. Hence the whole neighborhood will belong
to $S$, a contradiction. Thus
$\partial S\subseteq\f(\R/2\pi)$. We thus have four
cases:
\begin{itemize}
\item[(1)] $S$ is the closure of the exterior of the curve.
\item[(2)] $S$ is the closure of the interior of the curve.
\item[(3)] $S$ is a closed subset of the curve.
\item[(4)] $S$ is the whole plane.
\end{itemize}
If (1) is true, then $c$ is zero on the curve. This
implies that $\f$ is a level curve of a function with positive
hessian and as such must be convex. If (4) is true,
then $c\equiv 0$. Then for every $(x,y)\in\operatorname{supp}t$,
we will necessarily have the line segment joining
$\f(x)$ and $\f(y)$ completely contained in $\f(\R/2\pi)$.
This is because if not, then there would be a point
in $\R^2-\f(\R/2\pi)$ where the matrix
$\left(\begin{smallmatrix}
c_{yy}&-c_{xy}\cr
-c_{xy}&
c_{xx}\end{smallmatrix}\right)$ would be
positive, giving $c$ upward convexity. The case (2) is
easily disposed of since $c=0$ outside the convex hull
of the curve and hence will be zero on
at least one point of the curve. Hence the maximum
value $c$ attains is zero, a contradiction. Thus it
suffices to show that case (3) cannot happen.
Assume (3) is true. We have two cases:
\begin{itemize}
\item[(1$'$)] There exists $x\in\R/2\pi$ such that for every
$\delta>0$, $\f([x,x+\delta])\nsubseteq S$ and $\f([x-\delta,x])\nsubseteq S$.
\item[(2$'$)] There does not exist such an $x\in\R/2\pi$.
\end{itemize}
I will deal with the easier case (1$'$) first. WLOG $x=0$.
Also, WLOG, $\f(x)\cdot\e_1$ is one to one for $|x|<\epsilon$.
Choose $\delta_1,\delta_2>0$ such that the curve in the square
$[-\delta_1,\delta_1]\cross[-\delta_2,\delta_2]\subset\R^2$ looks like the graph of
a function, that is, $\f^{-1}([-\delta_1,\delta_1]\cross[-\delta_2,\delta_2])\subseteq[-\epsilon,\epsilon]$.
Let $-\delta_1<x_-<0<x_+<\delta_1$ have $c(x_-,0)\ne M$ and
$c(x_+,0)\ne M$.
Now let:
\begin{equation}
M'=\frac 12\left(M+\max_{\p\in\partial[x_-,x_+]\cross[-\delta_2,\delta_2]}c(\p)\right)<M
\end{equation}
Let $y_+$ be the least $y>0$ such that $c(0,y)=M'$ and let
$y_-$ be the highest $y<0$ such that $c(0,y)=M'$. Consider
the level curves passing through $y_+$ and $y_-$. By the
convexity of $c$ they must curve away from $(0,0)$ where
the maximum occurs, but they must meet the curve on both
sides of $(0,0)$ at some $x_-'$ and $x_+'$. This is a
contradiction.
Now suppose (2$'$) is true. Let $[x,y]\subset\R/2\pi$
satisfy $c(\f([x,y]))=M$ and for every $\delta>0$,
$\f([x-\delta,x])\nsubseteq S$ and $\f([y,y+\delta])\nsubseteq S$.
Then $\f([x,y])$ is a level curve of $c$ restricted to
the interior of the curve. As the level curve of a
convex function it must be curved towards the interior
of the curve. But by the same reasoning, $\f([x,y])$
is a level curve of $c$ restricted to the outside of
the curve, and hence must be curved towards the outside
of the curve. Hence $\f([x,y])$ is a line segment.
As above, we can rotate $\f$ so it looks like the graph
of a function $\R\map\R$ near $\f(x)$ and near $\f(y)$. Using
the same procedure as above, we get a contradiction
by considering level curves of $M-\eta$ for a suitably small $\eta>0$.
\end{proof}
We have now justified every step in the proof of Theorem
\ref{gen} except for Proposition \ref{impl} and the generalized
Maxwell-Cremona Theorem. We will prove these next.
\section{A Generalization of the Farkas Lemma}\label{farkass}
The Farkas Lemma from linear programming is as follows:
\begin{lemma}[Farkas Lemma]\label{farkasl}
Let $A:\R^n\to\R^m$ be a linear transformation. Then exactly
one of the following two statements holds:
\begin{itemize}
\item[(1)] There exists a nonzero $y\in\R^m$ whose components
are all nonnegative and which satisfies $A^{\operatorname{T}}y=0$.
\item[(2)] There exists an $x\in\R^n$ such that every component
of $Ax$ is positive.
\end{itemize}
\end{lemma}
The generalization of the Farkas Lemma that we
will need will have the basic form:
\begin{theorem}\label{farkas}
Let $X$ be a compact Hausdorff space and $Y$ a (real) Hilbert space.
Let $A:Y\map C(X)$ be linear and bounded. Also let $A':C(X)^*\map Y$
denote its adjoint, that is
$\langle\lambda,Ay\rangle=\langle A'\lambda,y\rangle$. Then
exactly one of the following two statements holds:
\begin{itemize}
\item[(1)] There exists a nonzero positive $t\in C(X)^*$ such that $A't=0$.
\item[(2)] There exists a $y\in Y$ such that $Ay>0$.
\end{itemize}
\end{theorem}
We remark that if we take $Y$ to be finite dimensional and $X$ to consist
of a finite number of points, then we recover Lemma \ref{farkasl}.
\begin{proof}
It is trivial that (1) and (2) cannot simultaneously hold,
for if so, $0=\langle A't,y\rangle=\langle t,Ay\rangle>0$.
It remains to show that $\sim$(1)$\implies$(2). Let
$T:=\{t\in C(X)^*:t\geq 0\}$.
I claim that there exists $\epsilon>0$ such that
$\|A't\|\geq\epsilon\|t\|$ for all $t\in T$. If we
suppose the contrary, then there exists a sequence $t_n\in T$ with
$\|t_n\|=1$ such that $A't_n\to 0$. By the Banach-Alaoglu
Theorem, there exists a subnet $t_\alpha$ which converges to
$t\in T$ (in the weak-$*$ topology on $T$). We know that we will
have $t\in T$ and $\|t\|=1$. Also, for all $y\in Y$, we have:
\begin{equation}
0=\lim_\alpha\langle A't_\alpha,y\rangle=
\lim_\alpha\langle t_\alpha,Ay\rangle=\langle t,Ay\rangle=\langle A't,y\rangle
\end{equation}
Thus $A't=0$, contradicting $\sim$(1). Thus the claim is
true. I now can show (2).
Let $t_n\in T$ be a sequence such that $\|t_n\|=1$ and:
\begin{equation}
\|A't_n\|\to\inf_{\begin{smallmatrix}t\in T\cr\|t\|=1\end{smallmatrix}}
\|A't\|=:w\geq\epsilon
\end{equation}
Then a subnet $t_\alpha$ will converge in the weak-$*$
topology to a limit $t_{\infty}$. Now:
\begin{equation}
w\leq\|A't_{\infty}\|\leq\liminf_\alpha\|A't_\alpha\|=w
\end{equation}
Hence $\|A't_{\infty}\|=w$.
Let $y:=A't_{\infty}/\|A't_{\infty}\|$. I claim that
$(Ay)(x)\geq\epsilon$ for all $x\in X$. It suffices to
show that $\langle t,Ay\rangle\geq w$ for all $t\in T$
with $\|t\|=1$. But if $\langle t,Ay\rangle<w$ for some
$t\in T$ with $\|t\|=1$, then
$\langle A't,y\rangle<w$. Consider then:
\begin{equation}
\begin{split}
\left.\frac d{d\eta}\right|_{\eta=0}
&\|A'((1-\eta)t_{\infty}+\eta t)\|^2\cr
&=\left.\frac d{d\eta}
\left[(1-\eta)^2\|A't_{\infty}\|^2+2\eta(1-\eta)\langle A't,A't_{\infty}\rangle
+\eta^2 \|A't\|^2\right]\right|_{\eta=0}\cr
&=-2w^2+2\langle A't,wy\rangle<0
\end{split}
\end{equation}
This is a contradiction since $\|(1-\eta)t_{\infty}+\eta t\|=1$.
Hence the proof is complete.
\end{proof}
We can prove Proposition \ref{impl} using the same proof outline
from Theorem \ref{farkas}. We will, however, need the following
approximation lemma.
\begin{lemma}\label{approx}
Suppose $\f_n\to\f$ in $D$ and that $q\in Q_\f$ is of the form
$q'=\lambda i\f'$ where $\lambda$ is smooth. Then there exist
$q_n\in Q_{\f_n}$ such that $q_n\to q$ (weakly).
\end{lemma}
\begin{proof}
We will search for $q_n$ of the form $q_n'=(\lambda+\nu_n)i\f_n'$.
We will have $\|q_n\|$ bounded if $\|\nu_n\|_\infty$ is bounded.
Hence we will have $q_n\to q$ weakly if $\|\nu_n\|_\infty$ is
bounded and $\langle\ell,q-q_n\rangle\to 0$ for all smooth
$\ell\in H$. Now $|\langle\ell,q-q_n\rangle|$ is equal to:
\begin{equation}
\begin{split}
\left|\int_{\R/2\pi}\ell'\cdot(q'-q_n')\right|&=\left|\int_{\R/2\pi}\ell'\cdot
(\lambda i\f'-\lambda i\f_n'-\nu_n i\f_n')\right|\cr
&\leq\left|\int_{\R/2\pi}\ell'\lambda\cdot
i(\f'-\f_n')\right|+\left|\int_{\R/2\pi}\nu_n\ell'\cdot i\f_n'\right|\cr
&=\left|\int_{\R/2\pi}[\ell''\lambda+\ell'\lambda']\cdot
i[\f-\f_n]\right|+\left|\int_{\R/2\pi}\nu_n\ell'\cdot i\f_n'\right|\cr
&\leq 2\pi\|\ell''\lambda+\ell'\lambda'\|_\infty\|\f-\f_n\|_\infty
+\|\nu_n\|_\infty\|\ell'\|_\infty\sqrt{2\pi}\|\f_n\|
\end{split}
\end{equation}
By Lemma \ref{unif2}, $\|\f-\f_n\|_\infty\to 0$. Thus in order
for $q_n\to q$ weakly, all we need is $\|\nu_n\|_\infty\to 0$ and
$\int_{\R/2\pi}(\lambda+\nu_n)\f_n'=0$ (because clearly we
must have $\int_{\R/2\pi}q_n'=0$). Using integration
by parts, this last equality can be written:
\begin{equation}\label{needforc}
\int_{\R/2\pi}\f_n\nu_n'=\int_{\R/2\pi}[\f-\f_n]\lambda'
\end{equation}
We can pick $a_1$, $a_2$, and $a_3$ in $\R/2\pi$ such that:
\begin{equation}
\left|\begin{matrix}1&1&1\cr\f(a_1)\cdot\e_1&\f(a_2)\cdot\e_1&\f(a_3)\cdot\e_1
\cr\f(a_1)\cdot\e_2&\f(a_2)\cdot\e_2&\f(a_3)\cdot\e_2\end{matrix}\right|\geq 2\epsilon>0
\end{equation}
There exists an $N$ such that for every $n\geq N$, the determinant
with $\f$ replaced with $\f_n$ is greater than $\epsilon$. It
suffices to choose $\nu_n$ for $n\geq N$. Set
$C_n=\int_{\R/2\pi}[\f-\f_n]\lambda'$. We know that
$|C_n|\leq 2\pi\|\lambda'\|_\infty\|\f-\f_n\|_\infty$. We solve
the following system of equations for $b_{n,i}\in\R$:
\begin{align}
\hphantom{\f_n(a_1)}b_{n,1}+\hphantom{\f_n(a_2)}b_{n,2}+\hphantom{\f_n(a_3)}b_{n,3}&=0\\
\f_n(a_1)b_{n,1}+\f_n(a_2)b_{n,2}+\f_n(a_3)b_{n,3}&=C_n
\end{align}
For $n\geq N$, we can use Cramer's Rule to give the follwing
bound on the solution:
\begin{equation}
|b_{n,i}|\leq\epsilon^{-1}2[2\pi\|\lambda'\|_\infty\|\f-\f_n\|_\infty]2[\sqrt{2\pi}\|\f_n\|]
\end{equation}
Set $\nu_n(0)=0$ and:
\begin{equation}
\nu_n'(x)=b_{n,1}\delta(x-a_1)+b_{n,2}\delta(x-a_2)+b_{n,3}\delta(x-a_3)
\end{equation}
Then we will guarantee $\int_{\R/2\pi}\nu_n'=0$, equation
(\ref{needforc}), and $\|\nu_n\|_\infty\to 0$. Thus we
will have $q_n\to q$ (weakly).
\end{proof}
\begin{proof}[Proof of Proposition \ref{impl}]
We will write $V(\f)$ for
$\{(x,y)\in(\R/2\pi)^2:(\f,x,y)\in V\}$. Also, if
$Z\subset(\R/2\pi)^2$, we will write $T_Z$ for
$\{t\in C((\R/2\pi)^2):t\geq 0\text{ and }\supp t\subseteq Z\}$.
We assume Theorem \ref{needmax}. Let
$\pi_\f:H\map Q_\f$ be the orthogonal projection and let
$J_\f=\pi_\f\circ R_\f'$. Then Theorem \ref{needmax} implies
``If $\f\in D_1$, $t\in T_{V(\f)}$, and $J_\f t=0$, then
$t=0$''.
I claim that there exists $\epsilon>0$ such that
$\|J_\f t\|\geq\epsilon\|t\|$ for
all $\f\in D_1$ and $t\in T_{V(\f)}$. If we suppose
the contrary, then there exist two sequences, $\f_n\in D_1$
and $t_n\in T_{V(\f_n)}$ with $\|t_n\|=1$
such that $\|J_{\f_n}t_n\|\to 0$. Since
$D_1$ is weakly closed, it is compact by the
Banach-Alaoglu Theorem, hence there exists a convergent
subsequence of $\f_n$ which we assume WLOG is the
whole sequence, so that $\f_n\to\f$. Since
this means that $\f_n\to\f$ uniformly, we will have
$\|R'_{\f_n}-R'_\f\|\to 0$. Thus:
\begin{equation}
\|\pi_{\f_n}R'_{\f_n}t_n\|\to 0\implies\|\pi_{\f_n}R'_\f t_n\|\to 0
\end{equation}
Now there is also a weak-$*$ convergent subsequence of the
$t_n$ by the Banach-Alaoglu Theorem, which again WLOG
is the whole sequence. Thus $t_n\to t\in T_{V(\f)}$
since $V$ is closed; also $\|t\|=1$. Pick some $q\in Q_\f$
which can be written as $q'=\lambda i\f'$ where $\lambda$
is smooth (such $q$ are dense in $Q_\f$). Let $q_n\in Q_{\f_n}$
be the sequence guaranteed to exist by Lemma \ref{approx}. We note
that since $q_n$ is weakly convergent, it is bounded. Now:
\begin{equation}\label{feq1}
0=\lim_{n\to\infty}\langle\pi_{\f_n}R'_\f t_n,q_n\rangle
=\lim_{n\to\infty}\langle R'_\f t_n,q_n\rangle
=\lim_{n\to\infty}\langle t_n,R_\f q_n\rangle
\end{equation}
Now by Lemma \ref{unif2}, $R_\f q_n\to R_\f q$ strongly.
Thus the final limit in equation (\ref{feq1}) is equal to
$\langle t,R_\f q\rangle$. This means that $\langle R_\f't,q\rangle=0$
for a dense subset of $q\in Q_\f$. Thus $J_\f t=0$ where
$\f\in D_1$ and $t\in T_{V(\f)}-\{0\}$, contradicting
Theorem \ref{needmax}. Thus the claim is proved.
We can now show the existence of an appropriate $\varphi$
for every $\f\in D_1$ exactly as in the proof of Theorem
\ref{farkas}.
Fix some $\f\in D_1$. Let $t_n\in T_{V(\f)}$ be a
sequence such that $\|t_n\|=1$ and:
\begin{equation}
\|J_\f t_n\|\to\inf_{\begin{smallmatrix}t\in T_{V(\f)}\cr\|t\|=1\end{smallmatrix}}\|J_\f t\|
=:w\geq\epsilon
\end{equation}
A subsequence is weak-$*$ convergent (WLOG the whole
sequence) to a limit $t_{\infty}$. Using the same reasoning
as above, we conclude that $J_\f t_n\to J_\f t_{\infty}$ in
the weak topology, so:
\begin{equation}
w\leq\|J_\f t_{\infty}\|\leq\liminf\|J_\f t_n\|=w
\end{equation}
Thus $\|J_\f t_{\infty}\|=w$. Let $q:=J_\f t_{\infty}/\|J_\f t_{\infty}\|$.
Now I claim that $\langle J_\f t,q\rangle\geq w\|t\|$
for all $t\in T_{V(\f)}$. Suppose not, that we have $t\in T_{V(\f)}$
with $\|t\|=1$ and $\langle J_\f t,q\rangle<w$.
Then $\langle J_\f t,J_\f t_{\infty}\rangle<w^2$. But
consider then:
\begin{equation}
\begin{split}
\left.\frac d{d\eta}\right|_{\eta=0}
&\|J_\f((1-\eta)t_{\infty}+\eta t)\|^2\cr
&=\left.\frac d{d\eta}
\left[(1-\eta)^2\|J_\f t_{\infty}\|^2+2\eta(1-\eta)\langle J_\f t,J_\f t_{\infty}\rangle
+\eta^2 \|J_\f t\|^2\right]\right|_{\eta=0}\cr
&=-2w^2+2\langle J_\f t_{\infty}, J_\f t\rangle<0
\end{split}
\end{equation}
This is a contradiction since $\|(1-\eta)t_{\infty}+\eta t\|=1$.
Hence the claim is proved.
Let $\varphi=q$. Then:
\begin{equation}
\langle t,R_\f\varphi\rangle=\langle J_\f t,q\rangle\geq w\|t\|\geq\epsilon\|t\|
\text{ for all $t\in T_{V(\f)}$}
\end{equation}
This means that $R_\f\varphi(x,y)\geq\epsilon$ for all $(x,y)\in V(\f)$.
\end{proof}
\section{A Generalization of the Maxwell-Cremona Theorem}\label{maxwells}
Let $A\in\Lin(C_0(\R^2,\R^2),\R^2)$
have compact support. Then by the Riesz Representation Theorem,
$A$ can be thought of as a matrix of measures on $\R^2$:
\begin{equation}
A=\left(\begin{matrix}a&b\cr d&e\end{matrix}\right)
\end{equation}
We are concerned with the case when $A$ is symmetric, that
is $b=d$. For the moment, suppose
$a$, $b$, and $e$ are continuous functions. In this case,
at each point $A$ has orthogonal eigenvectors $\vv_1$ and $\vv_2$
with eigenvalues $\lambda_1$ and $\lambda_2$. We think of
$A$ as representing a ``stress'' on the plane, where at each
point, there is tension in the $\vv_i$ direction of magnitude
$\lambda_i$. It turns out that it is right to call such a
stress is an ``equilibrium stress'' if:
\begin{equation}\label{eqstress}
A\grad g=0\text{ for all $g\in C_0^\infty(\R^2)$}
\end{equation}
In the case that $a$, $b$, and $e$ are continuous, it is
straightforward to show that in fact:
\begin{equation}\label{maxc}
A=\left(\begin{matrix}a&b\cr b&e\end{matrix}\right)=
\left(\begin{matrix}
c_{yy}&-c_{xy}\cr-c_{xy}&
c_{xx}\end{matrix}\right)
\end{equation}
The function $c$ will be in $C_c(\R^2)$. This is the
Maxwell-Cremona ``lifting'' of the stress represented by
$A$.
However, the notion of being an equilibrium stress (\ref{eqstress})
makes sense for any compactly supported $A$, so one would
expect that (\ref{maxc}) should hold in some sense for all
equilibrium stresses $A$. If $\U$ is a smooth vector field
and we integrate
$\iint_{\R^2}\left(\begin{smallmatrix}
c_{yy}&-c_{xy}\cr
-c_{xy}&
c_{xx}\end{smallmatrix}\right)\U\,dx\,dy$ by parts, we get
$\iint_{\R^2}c[i\grad\curl\U]\,dx\,dy$, so if (\ref{maxc}) holds
in the distributional sense, we would like this last integral
to give $A\U$ for smooth $\U$. This is the intuition for
the following theorem.
\begin{theorem}\label{maxwell}
Let $A\in\Lin(C_0(\R^2,\R^2),\R^2)$ have compact support.
Suppose $A$ is symmetric, that is there exist
$a,b,c\in C_0(\R^2)^*$ such that:
\begin{equation}
A=\left(\begin{matrix}a&b\cr b&e\end{matrix}\right)
\end{equation}
Additionally, suppose that for every $g\in C_0^\infty(\R^2)$,
$A\grad g=0$. Then there exists $c\in C_c(\R^2)$ such that
for all $\U\in C_0^\infty(\R^2,\R^2)$:
\begin{equation}\label{mconc}
A\U=\iint_{\R^2}c[i\grad\curl\U]\,dx\,dy
\end{equation}
\end{theorem}
\begin{proof}
First, let us show that (the matrix of measures associated with)
$A$ has no pure point part. Let $\p$
and $\vv$ be arbitrary. Choose $g\in C_0^\infty(\R^2)$ so
that $\grad g(\p)=\vv$. Then $0=A(\grad g)(\epsilon(\cdot)+\p)$,
but as $\epsilon\to 0$, right hand side approaches the pure
point part of $A$ at $\p$ applied to $\vv$. Hence $A$ has
no pure point part.
Consider the measure $|A|\in C_0(\R^2)^*$, where the
$|\cdot|$ of a matrix is its operator norm. In other words,
for $f\geq 0$, we define:
\begin{equation}
|A|f:=\sup_{\begin{smallmatrix}\theta:\R^2\map\R\cr\psi:\R^2\map\R\end{smallmatrix}}\iint_{\R^2}
\left(\begin{matrix}\cos\theta&\sin\theta\end{matrix}\right)
\left(\begin{matrix}a&b\cr b&e\end{matrix}\right)
\left(\begin{matrix}\cos\psi\cr \sin\psi\end{matrix}\right)f
\end{equation}
We know $|A|$ comes from a measure, which we will also denote
$|A|$. Let $\mu(\theta)$
be the measure on the real line $\R$ at angle $\theta$ passing
through the origin, obtained by projecting the
measure $|A|$ orthogonally onto the line. In other words:
\begin{equation}
\int_\R f(x)\,d\mu(\theta)=\iint_{\R^2}f((x,y)\cdot(\cos\theta,\sin\theta))|A|
\end{equation}
Now let $\mu_{\text{pp}}(\theta)$ be the pure point part of
$\mu(\theta)$. I claim that $\mu_{\text{pp}}(\theta)\ne 0$ for
at most countably many $\theta$. We note that this is implied
by the following:
\begin{equation}\label{crit}
\sum_{i=1}^N\|\mu_{\text{pp}}(\theta_i)\|\leq\||A|\|\text{ whenever $\theta_i$ are distinct}
\end{equation}
But (\ref{crit}) is true because any part of $|A|$ which
contributes to both $\|\mu_{\text{pp}}(\theta_i)\|$ and
$\|\mu_{\text{pp}}(\theta_j)\|$ would have to be supported on
a countable set of points, and hence would have to be pure point,
which we know $A$, and hence $|A|$ does not have. Now let
$m(\theta,h)=\sup_{x\in\R}\int_x^{x+h}\mu(\theta)(y)\,dy$.
Now $m(\theta,h)\to 0$ as $h\to 0$ if $\mu(\theta)$ has no
pure point part, thus $m(\theta,h)\to 0$ for almost all $\theta$.
This fact being proved, we can proceed to the construction of $c$.
Let $\phi$ be a smooth real valued even function on
$\R^2$ with support contained in the unit disc which satisfies
$\phi\geq 0$ and $\iint_{\R^2}\phi=1$. Let
$\phi_\eta(\p)=\eta^{-2}\phi(\eta^{-1}\p)$. We can then define
the operator:
\begin{equation}
A_\eta=A*\phi_\eta=\left(\begin{matrix}
a^{(\eta)}&b^{(\eta)}\cr b^{(\eta)}&e^{(\eta)}\end{matrix}\right)
\end{equation}
Now we know that:
\begin{equation}
a^{(\eta)},b^{(\eta)},e^{(\eta)}\in C_0^\infty(\R^2)\text{ and that
$A_\eta\grad g=0$ for all $g\in C_0^\infty(\R^2)$}
\end{equation}
Thus the vector fields $(a^{(\eta)},b^{(\eta)})$ and
$(b^{(\eta)},e^{(\eta)})$ have zero divergence. That
means there exist $f^{(\eta)},g^{(\eta)}\in C_0^\infty(\R^2)$ such
that $a^{(\eta)}=f^{(\eta)}_y$,
$b^{(\eta)}=-f^{(\eta)}_x=-g^{(\eta)}_y$, and
$e^{(\eta)}=g^{(\eta)}_x$. The equality $f^{(\eta)}_x=g^{(\eta)}_y$ implies
that there exists $c^{(\eta)}\in C_0^\infty(\R^2)$ such
that $f^{(\eta)}=c^{(\eta)}_y$ and $g^{(\eta)}=c^{(\eta)}_x$.
In other words:
\begin{equation}
A_\eta=\left(\begin{matrix}
c^{(\eta)}_{yy}&-c^{(\eta)}_{xy}
\cr-c^{(\eta)}_{xy}&
c^{(\eta)}_{xx}\end{matrix}\right)
\end{equation}
{\bf Claim: For every $\epsilon>0$, there exist $\delta>0$
and $\eta_0>0$ such that:
\begin{equation}
\eta_0>\eta>0\text{ and }|\q-\p|<\delta\implies|c^{(\eta)}(\p)-c^{(\eta)}(\q)|<\epsilon
\end{equation}}
Let $\epsilon>0$ be given. Suppose $\p=(x_0,y)\in\R^2$ and
$\q\in\R^2$ and we wish to bound
$|c^{(\eta)}(\p)-c^{(\eta)}(\q)|$ given $|\q-\p|<\delta$.
To simplify notation, we will for the moment assume
that $\q=(x,y)$. Then:
\begin{equation}
\begin{split}
\left|c^{(\eta)}(\p)-c^{(\eta)}(\q)\right|&=\left|\int_{x_0}^xc^{(\eta)}_x(t,y)\,dt\right|
=\left|\int_{x_0}^x\int_{-\infty}^yc^{(\eta)}_{xy}(t,z)\,dz\,dt\right|\cr
&\leq\int_{x_0}^x\int_{-\infty}^\infty\left|b^{(\eta)}(t,z)\right|\,dz\,dt
\leq\int_{x_0-\eta}^{x+\eta}\int_{-\infty}^\infty\left|A(t,z)\right|\,dz\,dt\cr
&\leq m(0,\delta+2\eta)
\end{split}
\end{equation}
Similary, if $\theta_\p^\q$ is the angle of the segment from
$\p$ to $\q$, then we have:
\begin{equation}
\left|c^{(\eta)}(\p)-c^{(\eta)}(\q)\right|\leq m(\theta_\p^\q,2\eta+\delta)
\end{equation}
Now since $m(\theta,h)\to 0$ as $h\to 0$ for all but at most
countably many $\theta$, there exists $h>0$ such that the
measure of the set $\{\theta:m(\theta,h)<\epsilon/4\}$
is more than $\frac{5\pi}3$. Then if $2\eta+\delta<
\min(\epsilon/(4\pi\|t\|),h)$ and the slope the segment from
$\p$ to $\q$ is not in the exceptional set of $\theta$
(which has measure less than $\frac\pi 3$),
then $|c^{(\eta)}(\p)-c^{(\eta)}(\q)|\leq\epsilon/2$. But
for any $\p$ and $\q$ within $\delta$ of each other,
we can find a $\rr$ within $\delta$ of both $\p$ and
$\q$ so that neither of the segments $\p$ to $\rr$ and
$\rr$ to $\q$ are in the exceptional set of $\theta$.
Hence by the triangle inequality,
$|c^{(\eta)}(\p)-c^{(\eta)}(\q)|\leq\epsilon$ if we set
$\eta_0=\delta=\frac 14\min(\epsilon/(4\pi\|t\|),h)$.
Thus the claim is true.
Now by the Arzel\`a-Ascoli Theorem, there exists a
subsequence of $c^{(1/n)}$ which converges uniformly
to a continous function $c\in C_c(\R^2)$. Thus let
$\eta_i\to 0$ and satisfy $c^{(\eta_i)}\to c\in C_c(\R^2)$
uniformly as $i\to\infty$. As remarked before, if $\U$
is smooth compactly supported vector field, then it is
a straightforward integration by parts to show:
\begin{equation}
A(\U*\phi_{\eta_i})=A_{\eta_i}\U=\iint_{\R^2}c^{(\eta_i)}[i\grad\curl\U]\,dx\,dy
\end{equation}
Taking the limit as $i\to\infty$, we obtain (\ref{mconc})
as was to be shown.
\end{proof}
\section{Open Problems}
Now, I can state some conjectures on possible strengthening
of Theorem \ref{main}. For example, we can
conjecture that there exists an $\h$ which is not only
continuous, but in fact smooth. Also, if the initial curve is
smooth, we can require that the curve be smooth at every
time during the deformation.
\begin{conjecture}
Given a unit speed simple closed curve $\f:\R/2\pi\map\C$,
there exists a smooth function $\h:[0,1]\map\mathcal D$ satisfying
(1)--(4).
\end{conjecture}
\begin{conjecture}
Given a smooth unit speed simple closed curve $\f:\R/2\pi\map\C$,
there exists a continuous function $\h:[0,1]\map\mathcal D$ satisfying
(1)--(4) as well as:
\begin{itemize}
\item[(5)] $\h(t)(x)$ is a smooth function of $x$ for all $t\in[0,1]$.
\end{itemize}
\end{conjecture}
I also conjecture that it is possible to extend Corollary \ref{cgen}
to something resembling the following.
\begin{conjecture}
Suppose $\f:\R/2\pi\map\R^2$ is a rectifiable simple closed
curve which is not convex. Then there exists
$\varphi:\R/2\pi\map\R^2$ which is absolutely continuous and
satisfies $\f'\cdot\varphi'\equiv 0$, as well as
$(\f(x)-\f(y))\cdot(\varphi(x)-\varphi(y))>0$ whenever the
line segment connecting $\f(x)$ and $\f(y)$ is not
completely contained in $\f(\R/2\pi)$.
\end{conjecture}
Of course, this would be in preparation to prove:
\begin{mconjecture}
There exists a proof of Theorem \ref{main} which does not rely
on approximation by polygons.
\end{mconjecture}
\end{document} |
\begin{document}
\ellinespread{1.5}
\begin{frontmatter}
\tauitle{Asymptotics for infinite server queues with fast/slow Markov switching and fat tailed service times}
\runtitle{Infinite server queues with switching and fat tailed service times}
\begin{aug}
\author{\fnms{{\cal L}arge{Landy}}
\snm{{\cal L}arge{Rabehasaina}}
\ead[label=e2]{[email protected]}}
\runauthor{L.Rabehasaina}
\address{\hspace*{0cm}\\
Laboratory of Mathematics, University Bourgogne Franche-Comt\'e,\\
16 route de Gray, 25030 Besan\c con cedex, France.\\[0.2cm]
\printead{e2}}
\end{aug}
\begin{abstract}
We study a general $k$ dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index $\alpha\in (0,1)$. When the arrival rate is sped up by a factor $n^\gamma$, the transition probabilities of the underlying Markov chain are divided by $n^\gamma$ and the service times are divided by $n$, we identify two regimes ("fast arrivals", when $\gamma>\alpha$, and "equilibrium", when $\gamma=\alpha$) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third "slow arrivals" regime, $\gamma<\alpha$, we show the convergence of the two first joint moments of the rescaled process.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60G50}
\kwd{60K30}
\kwd{62P05}
\kwd{60K25}
\end{keyword}
\begin{keyword}
Infinite server queues, Incurred But Not Reported (IBNR) claims, Markov modulation, Rescaled process
\end{keyword}
\end{frontmatter}
\normalsize
\section{Introduction and notation}\ellambdabel{sec:model}
\subsection{Model and related work}
The classical infinite server queue consists of a system where tasks or customers arrive according to a general arrival process and begin receiving service immediately. Such a model was studied extensively, under various assumptions on the interarrival and service time distributions, in \cite[Chapter 3, Section 3]{T62}. Several variants or extensions have been considered, in particular where arrivals and service times are governed by an external background Markovian process \cite{OP86, D08, FA09, BKMT14, MDT16, BDTM17}, or where customers arrive in batches \cite{MT02}. An extension to a network of infinite-server queues where arrival and service rates are Markov modulated is studied in \cite{JMDW19}.
We consider yet another generalization of this model with Markov switching described as follows. Let $\{ N_t,\ t\ge 0\}$ be a Poisson process with intensity $\ellambdambda>0$, corresponding jump times $(T_i)_{i\in\mathbb{N}}$ satisfying $T_0=0$, such that $(T_i-T_{i-1})_{i\ge 1}$ is a sequence of independent and identically distributed (iid) random variables with same exponential distribution with parameter $\ellambdambda>0$, denoted by ${\cal E(\ellambdambda)}$. Let $(L_{ij})_{i\in \mathbb{N}, j=1,\elldots,k}$ be a sequence of independent random variables such that the sequence of vectors $\elleft((L_{i1},\elldots,L_{ik})\right)_{i\in \mathbb{N}}$ is iid (with entries $L_{i1}$,\elldots,$L_{ik}$ having different distributions for each $i$). Finally, for some $K$ and $k$ in $\mathbb{N}^*$ we consider the discrete set ${\cal S}=\{ 0,\elldots,K\}^k$ and a stationary finite Markov chain $(X_i)_{i\in\mathbb{N}}$ with state space ${\cal S}$. Then, for all $i$, $X_i$ is a vector of the form $X_i=(X_{i1},\elldots,X_{ik})$ with $X_{ij}\in \{ 0,\elldots,K\}$, $j=1,\elldots,k$. We then define the following $k$ dimensional process $\{Z(t)=(Z_1(t),\elldots,Z_k(t)),\ t\ge 0 \}$ with values in $\mathbb{N}^k$ as
\begin{equation}\ellambdabel{def_Z_t}
Z_j(t)=\sum_{i=1}^{N_t}X_{ij} \mathbbm{1}_{[t<L_{ij}+T_i]}=\sum_{i=1}^{\infty}X_{ij} \mathbbm{1}_{[T_i\elle t<L_{ij}+T_i]},\quad j=1,...,k.
\end{equation}
The process defined by \eqref{def_Z_t} has many applications, of which we list two most important ones:
\begin{itemize}
\item {\it incurred but not reported correlated claims: } in an actuarial context, $Z(t)=(Z_1(t),\elldots,Z_k(t))$ represents a set of branches where $Z_j(t)$ is the number of incurred but non reported (IBNR) claims in the $j$th branch of an insurance company. Here $X_{ij}$ is the number of such claims arriving in that branch at time $T_i$, and $L_{ij}$ is the related delay time before the claim $j$ is reported. From another point of view, $X_{ij}\in [0,+\infty)$ may also represent the amount (say, in euros) of the claim occurring at time $T_i$ in the $j$th branch, in which case $Z_j(t)$ is the total amount of undeclared claims which have occurred by time $t$. Another application is when $K=1$, in which case $X_{ij}=0$ means that the claim in branch $j$ occurring at time $T_i$ is reported and dealt with immediately by the policyholder, whereas $X_{ij}=1$ means that some effective lag in the report is observed. The Markovian nature of $(X_i)_{i\in\ensuremath{\mathbb{N}}}$ here is important from a practical point of view, as a claim amount at time $T_i$ may impact the one at time $T_{i+1}$, or because a policyholder may decide to grant a long report delay for the claim at time $T_{i+1}$ with high probability if the claim at time $T_i$ is reported immediately.
\item {\it infinite server queues with batch arrivals and Markov switching: } $Z(t)=(Z_1(t),\elldots,Z_k(t))$ represents a set of $k$ correlated queues with an infinite number of servers, such that customers arrive at each time $T_i$, with $X_{ij}$ customers arriving in queue $j\in\{1,...,k\}$, with corresponding (same) service times $L_{ij}$ (as an example, the basic case where $X_{ij}=1$ for all $i\in \mathbb{N}$ and $j=1,...,k$ corresponds to $k$ customers arriving simultaneously at each instant $T_i$). $Z_j(t)$ can also be seen as the number of customers of class $j$ in a (single) infinite-server queue, as illustrated in \cite[Figure 1]{RW16}. Other infinite-server queue, such as one where the customers within a batch arriving at time $T_i$ have different service times, may be inferred from the model \eqref{def_Z_t} by choosing an appropriate value of $k$ and Markov chain $(X_i)_{i\in\ensuremath{\mathbb{N}}}$, see \cite[Section 6]{RW16}. Here, the Markov switching is a major novelty in the present model because it allows for some dependence between the successive number of incoming customers. One simple example is when $K=1$, so that $X_{ij}=0$ means that an incoming customer at time $T_i$ in queue $j$ is rejected from the system, whereas $X_{ij}=1$ means that it is accepted: a classical situation would then be that if a customer is rejected at time $T_i$ then the next one could be accepted with high probability, at time $T_{i+1}$. In other words, this Markov switching can help model traffic regulation mechanisms.
\end{itemize}
The present paper follows \cite{RW18}, which studies the transient or limiting distribution of a discounted version of $Z(t)$ of the form
\begin{equation}\ellambdabel{discounted_Z(t)}
Z_j(t)=\sum_{i=1}^{N_t}X_{ij} e^{-a (L_{ij}+T_i)}\mathbbm{1}_{[t<L_{ij}+T_i]}=\sum_{i=1}^{\infty}X_{ij} e^{-a (L_{ij}+T_i)} \mathbbm{1}_{[T_i\elle t<L_{ij}+T_i]},\quad j=1,...,k,
\end{equation}
for $t\ge 0$. The main difference with \cite{RW18} is that the latter has more general assumptions on the interarrival and service distributions, whereas we focus here on Poisson arrivals. Even though the assumptions are more restrictive than in \cite{RW18}, the goal here is different in that we are trying to exhibit different behaviours for the limiting models when the arrival rate is increased and the service times are decreased by suitable factors, whereas \cite{RW18} is more focused on analytical stochastic properties such as the moments of $Z(t)$ or its limiting distribution as $t\tauo\infty$. The discounting factor $a \ge 0$ in \eqref{discounted_Z(t)} is important in situations e.g. where, in an actuarial context, $X_{ij} e^{-a (L_{ij}+T_i)}$ represents the value of the claim amount at the actual realization time $L_{ij}+T_i$.
Furthermore, the state space ${\cal S}=\{ 0,\elldots,K\}^k$, although seemingly artificially complex, allows in fact for some flexibility and enables us to retrieve some known models. In particular, consider a Markov-modulated infinite-server queue, i.e. a queueing process $\{{\cal Z}(t),\ t\ge 0\}$ of which interarrivals and service times are modulated by a background continuous time Markov chain $\{Y(t),\ t\ge 0\}$ with state space say $\{1,...,\kappappa\}$, i.e. such that customers arrive on the switching times of the Markov chain, with service times depending on the state of the background process (see \cite{MT02}, \cite[Model II]{MDT16}). Then \cite[Section 6]{RW18} explains how this process $\{{\cal Z}(t),\ t\ge 0\}$ can be embedded in a process $\{Z(t),\ t\ge 0\}$ defined by \eqref{def_Z_t} with an appropriate choice of $k$, $K$ in function of $\kappappa$, as well as of the Markov chain $(X_i)_{i\in\mathbb{N}}$ and the sequence $(L_{ij})_{i\in \mathbb{N}, j=1,\elldots,k}$ of service times. Thus, studying a general process $\{Z(t),\ t\ge 0\}$ in \eqref{def_Z_t} allows to study a broad class of infinite server queue models in a similar Markov modulated context.
We now proceed with some notation related to the model and used throughout the paper. Let $P=(p(x,x'))_{(x,x')\in {\cal S}^2}$ and $\pi=(\pi(x))_{x\in {\cal S}}$ (written as a row vector) be respectively the transition matrix and stationary distribution of the Markov chain. We next define for all $r\ge 0$ and $s=(s_1,\elldots,s_k)\in (-\infty,0]^k $,
\begin{align}
\tauilde{\pi}(s,r)&:= \displaystyle\mbox{diag}\elleft[ \mathbb{E} \elleft( \exp\elleft\{ \sum_{j=1}^k s_jx_j \mathbbm{1}_{[L_j>r]}\right\}\right),\ x=(x_1,\elldots,x_k)\in{\cal S}\right],\ellambdabel{def_pi_Q_tilda}\\
\Delta_i&:=\mbox{diag} \elleft[ x_i,\ x=(x_1,\elldots,x_k)\in{\cal S}\right],\quad i=1,\elldots,k,\ellambdabel{Di}
\end{align}
where $P'$ denotes the transpose of matrix $P$. $I$ is the identity matrix, ${\bf 0}$ is a column vector with zeroes, and ${\bf 1}$ is a column vector with $1$'s, of appropriate dimensions.
The Laplace Transform (LT) of the process $Z(t)$ jointly to the state of $X_{N_t}$ given the initial state of $X_0$ is denoted by
\begin{equation}\ellambdabel{def_mgf}
\psi(s,t):=\elleft[ \mathbb{E}\elleft( \elleft. e^{<s,Z(t)>}\mathbbm{1}_{[X_{N_t}=y]}\right| X_0=x\right)\right]_{(x,y)\in {\cal S}^2},\quad t\ge 0,\ s=(s_1,\elldots,s_k)\in (-\infty,0]^k
\end{equation}
where $<\cdot, \cdot>$ denotes the Euclidian inner product on $\mathbb{R}^k$. Note that $X_0$ has no direct physical interpretation here, as the claims sizes/customer batches are given by $X_i$, $i\ge 1$, and is rather introduced for technical purpose.
We finish this section with the following notation. For two sequences of random variables $(A_n)_{n\in\mathbb{N}}$ and $(B_n)_{n\in\mathbb{N}}$ and two random variables $A$ and $B$, the notation ${\cal D}\elleft(\elleft. A_n\right|B_n\right)\ellongrightarrow_{n\tauo\infty} {\cal D}\elleft(\elleft. A\right|B\right)$ indicates that, as $n\tauo \infty$, the conditional distribution of $A_n$ given $B_n$ converges weakly to the conditional distribution of $A$ given $B$.
\subsection{Rescaling}\ellambdabel{sec:rescale}
We arrive at the main topic of the paper, which is to be able to provide some information on the distribution of $Z(t)$ in \eqref{def_Z_t}. In the particular case of Poisson arrivals, and since $Z(t)$ in \eqref{def_Z_t} is a particular case of the process in \eqref{discounted_Z(t)} with discount factor $a=0$, the LT $\psi(s,t)$ defined in \eqref{def_mgf} is characterized by \cite[Proposition 4]{RW18}, which we rewrite here:
\begin{prop}\ellambdabel{prop_Poisson_psi}
When $\{ N_t,\ t\ge 0\}$ is a Poisson process with intensity $\ellambdambda>0$, then $\psi(s,t)$ is the unique solution to the first order linear (matrix) differential equation
\begin{equation}\ellambdabel{Poisson_ODE}
\partial_t \psi(s,t) =[\ellambdambda (P-I) + \ellambdambda P(\tauilde{\pi}(s,t)-I)]\psi(s,t)
\end{equation}
with the initial condition $\psi(s,0)= I$.
\end{prop}
Unfortunately, the first order ordinary differential equation \eqref{Poisson_ODE} does not have an explicit expression in general, so that studying the (transient or stationary) distribution of the couple $(Z(t),X_{N_t})$ is difficult. In that case, as in \cite{BDTM17, BKMT14, MDT16}, it is appealing to study the process when the intensity of the Poisson process is sped up and the switching rates of the Markov chain are modified. Similarly to those papers, the goal of this paper is thus to study the behaviour of the queue/IBNR process in "extreme conditions" for the arrival rates, transition rates and delays, while trying to maintain minimal assumptions on the service time distributions. For this we will suppose that the rescalings of the parameters, denoted by ${\bf (S1)}$, ${\bf (S2)}$ and ${\bf (S3)}$ hereafter, are performed as follows:
\begin{itemize}
\item the arrival rate is multiplied by $n^\gamma$ for some $\gamma>0$, denoted by $${\bf (S1)}\quad \ellambdambda_n= \ellambdambda n^\gamma ,$$
with associated Poisson process $\{ N_t^{(n)},\ t\ge 0\}$ and jump times $(T_i^n)_{i\in\mathbb{N}}$,
\item the transition probabilities $p(x,y)$ are divided by $n^\gamma$ when $x\neq y$, $x$, $y$ in ${\cal S}$, i.e. the new transition matrix is given by $${\bf (S2)} \quad P_n=P/n^\gamma + (1-1/n^\gamma) I,$$
with corresponding stationary Markov chain $(X_i^{(n)})_{i\in\mathbb{N}}$, having the same distribution $\pi$ as $(X_i)_{i\in\mathbb{N}}$.
\end{itemize}
Since the transition matrix $P_n$ verifies $P_n\ellongrightarrow_{n\tauo\infty} I$, such normalizing assumptions imply that, as $n\tauo\infty$, one is close to a model where the arriving customers or claims come in the $k$ queues in batches with same fixed size: those queues are nonetheless correlated because the customers arrive according to the same Poisson process. Also, observe that
$\ellambdambda_n (P_n-I)$ is the infinitesimal generator of the continuous time Markov chain $\elleft\{Y^{(n)}(t)=X^{(n)}_{N^{(n)}_t},\ t\ge 0\right\}$ of which embedded Markov chain is the underlying Markov chain i.e. $(Y^{(n)}(T_i^n))_{i\in\mathbb{N}}=(X_i^{(n)})_{i\in\mathbb{N}}$. Thus, since the rescaling is such that
$$\ellambdambda_n (P_n-I)=\ellambdambda (P-I)$$
for all $n$ (a property which will be extensively used in the paper), we remark that the rescalings {\bf (S1)} and {\bf (S2)} for the
arrival rate and the transition probabilities are such that the transition rates between the states of $\cal S$ of $\{Y^{(n)}(t),\ t\ge 0\}$ are independent from $n$, which allows for enough dynamics in the model that compensates the fact that $P_n$ tends to $I$, and yielding non trivial asymptotics in the convergence results in this paper as $n\tauo\infty$.
The assumptions for the service times/delays distribution are the following. We first suppose that the base model features fat tailed distributed service times with same index $\alpha \in (0,1)$, i.e. such that
$$\mathbb{P}(L_j >t)\sigmam 1/t^\alpha,\quad t\tauo \infty,$$
for all $j=1,...,k$. This kind of distribution (included in the wider class of heavy tailed distributions) mean that the service times are "large". In particular, those service times have {\it infinite expectation}. Furthermore, the rescaling for the service times is such that they are divided by $n$, denoted by
$$
{\bf (S3)}\quad L_j^{(n)}=L_j/n.
$$
Hence, the situation is the following: the arrivals are sped up by factor $n^\gamma$, but this is compensated by the fact that the delay times are diminished with factor $n$, so that one expects one of the three phenomena to occur at time $t$ for the limiting model: the arrivals occur faster than it takes time for customers to be served and the corresponding queue content $Z^{(n)}(t)$ grows large as $n\tauo\infty$, the arrivals occur slower and services are completed fast so that $Z^{(n)}(t)$ tends to $0$ as $n\tauo\infty$, or an equilibrium is reached. Those three cases will be studied in the forthcoming sections. Some limiting behaviour was studied in \cite{BKMT14, MDT16}, where the authors identified three regimes for different scalings in a Markov modulating context and obtained a Central Limit Theorem for a renormalized process,
when the service times have general distribution with finite expectation or are exponentially distributed. \cite{BDTM17} provides some precise asymptotics on the tail probability of the queue content for exponentially distributed service times. \cite{JMDW19} provides a diffusion approximation for a model with exponentially distributed service times. A novelty in this paper is that we restrict here the class of distributions to that of fat tailed distributions in order to exhibit (under different scalings) a different behaviour and different limiting distribution which is not gaussian. Also note that the class of fat tailed distributions is interesting in itself as, in actuarial practice, this corresponds to {\it latent claims}, i.e. very long delays which are incidentally in practice often not observed (as the case $\alpha\in(0,1)$ corresponds to the $L_j$'s having infinite expectation), see \cite[Section 6.6.1]{H17}. This motivates the convergence results in this paper, which feature the exponent $\alpha$ as the only information required on those delays. This in itself is a noticeable difference from the Central Limit Theorems obtained in \cite[Section 4]{BKMT14}, where the normalization and limiting distribution require the explicit cumulative distribution function of the service times. Not only that, but the scaling is rather done in those references \cite{BKMT14, MDT16} on the transition rates of the underlying continuous time Markov chain modulating the arrival and service rates, whereas here these are constant, as we saw that $\ellambdambda_n (P_n-I)=\ellambdambda (P-I)$ is independent of $n$, and the scaling is rather done on the service times instead. When the service times are heavy tailed, this particular model can also be seen as a generalization of the {\it infinite source model}, see \cite[Section 2.2]{MRRS02}. Since the class of fat tailed distributions is a sub-class of the set of heavy tailed distributions, the normalizations {\bf (S1)} and {\bf (S3)} can be directly compared to \cite[Section 3.1]{MRRS02}, which studies limiting distributions of such normalized processes, and where the authors introduce the notion of so-called Slow and Fast Growth conditions when the arrival rate of customers is respectively negligible or dominant, compared to the service times. The reader is also referred to \cite{GK03} for a a similar model where the interarrivals are heavy tailed.
All in all, what is going to be studied hereafter is, when $t$ is fixed in say $[0,1]$ w.l.o.g., the limiting distribution as $n\tauo\infty$ of the $\mathbb{N}^k\tauimes {\cal S}$ valued random vector
$$
\elleft(Z^{(n)}(t),X^{(n)}_{N^{(n)}_t}\right)
$$
under rescaling ${\bf (S1)}$, ${\bf (S2)}$ and ${\bf (S3)}$, or of a renormalized version of it in the "fast" or "slow" arriving customers case. Note that that the convergence is proved on the interval $[0,1]$, but all proofs can be adapted to show the convergence on any interval $[0,M]$ for $M>0$. The corresponding joint Laplace Transform is given by
\begin{equation}\ellambdabel{def_LT_n}
\psi^{(n)}(s,t)=\elleft[ \mathbb{E}\elleft( \elleft. e^{<s,Z^{(n)}(t)>}\mathbbm{1}_{\elleft[X^{(n)}_{N^{(n)}_t}=y\right]}\right| \ X_0^{(n)}=x\right)\right]_{(x,y)\in {\cal S}^2},\ s=(s_1,...,s_j)\in (-\infty, 0]^k,
\end{equation}
where we recall that $(X^{(n)}_i)_{i\in \mathbb{N}}$ is the underlying Markov chain with generating matrix $P_n$, stationary distribution $\pi$, and $\elleft\{N^{(n)}_t,\ t\ge 0\right\}$ is a Poisson process representing the arrivals, with scaled intensity $\ellambdambda_n$. We also introduce the first and second joint matrix moments defined by
\begin{equation}\ellambdabel{def_moments}
\begin{array}{rcl}
M_j^{(n)}(t)&:= &\elleft[ \mathbb{E}\elleft(\elleft. Z^{(n)}_j(t) \mathbbm{1}_{\elleft[X^{(n)}_{N^{(n)}_t}=y\right]}\right| \ X_0^{(n)}=x\right) \right]_{(x,y)\in {\cal S}^2},\quad j=1,...,k,\\
M_{jj'}^{(n)}(t)&:= &\elleft[ \mathbb{E}\elleft(\elleft. Z^{(n)}_j(t)\ Z^{(n)}_{j'}(t) \mathbbm{1}_{\elleft[X^{(n)}_{N^{(n)}_t}=y\right]}\right| \ X_0^{(n)}=x\right) \right]_{(x,y)\in {\cal S}^2},\quad j,j'=1,...,k .
\end{array}
\end{equation}
\section{Statement of results and organization of paper}
The core results of the paper concerning the different regimes are given in the following two Theorems \ref{theo_regimes} and \ref{theo_slow_arrival}:
\begin{theorem}\ellambdabel{theo_regimes}
Let $\{{\cal X}^\alpha (t)=({\cal X}^\alpha_1 (t),...,{\cal X}^\alpha_k (t)),\ t\in[0,1] \}$ be a $\{0,...,K\}^k$ valued continuous time inhomogeneous Markov chain with infinitesimal generating matrix $\frac{1}{1-\alpha} (1-t)^{\frac{\alpha}{1-\alpha}}\ellambdambda (P-I)$ with ${\cal X}^\alpha (0)\sigmam \pi$, and $\{\nu_j^\alpha(t),\ t\in [0,1] \}$, $j=1,...,k$, be $k$ independent Poisson processes with same intensity $\frac{\ellambdambda}{1-\alpha}$, independent from $\{{\cal X}^\alpha (t),\ t\in[0,1] \}$. Let $t\in[0,1]$ fixed.
\begin{itemize}
\item {\bf Fast arrivals: }If $ \gamma >\alpha$ then, as $n\tauo\infty$,
\begin{multline}\ellambdabel{convergence_fast}
{\cal D}\elleft(\elleft. \elleft(\frac{Z^{(n)}(t)}{n^{\gamma-\alpha}},X^{(n)}_{N^{(n)}_t} \right)\right|\ X^{(n)}_0\right)\\
\ellongrightarrow {\cal D}\elleft(
\elleft.
\elleft(\frac{\ellambdambda}{1-\alpha} \int_{1-t^{1-\alpha}}^{1} {\cal X}^\alpha(v)\ dv,\ {\cal X}^\alpha (1)\right)
\right|\ {\cal X}^\alpha\elleft(1-t^{1-\alpha}\right)
\right),
\end{multline}
\item {\bf Equilibrium: } If $\gamma =\alpha$ then, as $n\tauo\infty$,
\begin{multline}\ellambdabel{convergence_equilibrium}
{\cal D}\elleft(\elleft. \elleft(Z^{(n)}(t),X^{(n)}_{N^{(n)}_t} \right)\right|\ X^{(n)}_0\right)\\
\ellongrightarrow {\cal D}\elleft(
\elleft.
\elleft( \elleft(\int_{1-t^{1-\alpha}}^{1} {\cal X}^\alpha_j(v)\ \nu_j^\alpha(dv)\right)_{j=1,...,k},\ {\cal X}^\alpha (1)\right)
\right|\ {\cal X}^\alpha\elleft(1-t^{1-\alpha}\right)
\right).
\end{multline}
\end{itemize}
\end{theorem}
We note that the terms in the limits on the right hand side of \eqref{convergence_fast} and \eqref{convergence_equilibrium} feature simple objects (in regards to the complexity of the original model) where the only characteristic parameters needed are $\ellambdambda$, $P$ and $\alpha$; in particular, and apart from $\alpha$, characteristics of the service times $L_j$, $j=1,...,k$, such as their cumulative distribution functions, do not show up in the limiting distributions \eqref{convergence_fast} and \eqref{convergence_equilibrium}.
The convergences in distribution \eqref{convergence_fast} and \eqref{convergence_equilibrium} give some information on convergence of the (possibly renormalized) joint distribution of the couple $\elleft(Z^{(n)}(t),X^{(n)}_{N^{(n)}_t}\right)$, $t\in [0,1]$ given the initial state $X_0^{(n)}$.
Intuitively, for fixed $t\in [0,1]$, in the right hand sides of \eqref{convergence_fast} and \eqref{convergence_equilibrium} we may interpret the inhomogeneous continuous time Markov chain $\{{\cal X}^\alpha (v),\ v\in[1-t^{1-\alpha},1] \}$ as the limiting counterpart of the modulating process $\elleft\{ X^{(n)}_{N^{(n)}_v},\ v\in [0,t]\right\}$. On an even cruder level, we observe in the Fast arrivals case from \eqref{convergence_fast} that each entry of $Z^{(n)}(t)=(Z^{(n)}_1(t),...,Z^{(n)}_k(t))$ behaves roughly like $n^{\gamma-\alpha}t^{1-\alpha}$. The intuition behind this behaviour may be explained as follows. Within queue $j=1,...k$, there are approximately $\ellambdambda n^{\gamma} t$ arrivals in the interval $[0,t]$, each arriving customer with service time distributed as $L^{(n)}_j$, so that we may very grossly consider that a customer is still present at time $t$ with probability $\mathbb{P} (L^{(n)}_j>t)=\mathbb{P} (L_j/n>t) $. Hence the number of customers in queue $j$ is approximately
$$
Z^{(n)}_j(t)\approx \ellambdambda n^{\gamma} t \tauimes \mathbb{P} (L^{(j)}/n>t) = \ellambdambda n^{\gamma} t \tauimes\mathbb{P} (L^{(j)}>nt) \approx \ellambdambda n^{\gamma} t \tauimes \frac{1}{(nt)^\alpha}=\ellambdambda n^{\gamma-\alpha}t^{1-\alpha}
$$
which is the expected order of growth $n^{\gamma-\alpha}t^{1-\alpha}$. Of course, such approximations are very crude, however this enables us to justify the presence of the normalizing factor $n^{\gamma-\alpha}$ as well as the time dilated factor $t^{1-\alpha}$ in \eqref{convergence_fast}.
In the case when $ \gamma <\alpha$, proving the convergence in distribution of an adequate normalization of $Z^{(n)}(t)$ seems more difficult. The following result show that the two first moments of $Z^{(n)}(t)$ converge under respective normalization $n^{\alpha-\gamma}$ and $n^{(\alpha-\gamma)/2}$:
\begin{theorem}[\bf Slow arrivals]\ellambdabel{theo_slow_arrival}
If $ \gamma <\alpha$ then the following convergences of the two joint moments hold as $n\tauo \infty$
\begin{eqnarray}
n^{\alpha-\gamma} M_j^{(n)}(t)&\ellongrightarrow & \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t \frac{1}{v^\alpha} e^{-\ellambdambda v (P-I)} \Delta_j e^{\ellambdambda v (P-I)} dv, \ellambdabel{convergence_slow_M1}\\
n^{\alpha-\gamma} M_{jj}^{(n)}(t)&\ellongrightarrow & \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t \frac{1}{v^\alpha} e^{-\ellambdambda v (P-I)} \Delta_j^2 e^{\ellambdambda v (P-I)} dv, \ellambdabel{convergence_slow_M20}\\
n^{\alpha-\gamma} M_{jj'}^{(n)}(t)&\ellongrightarrow & 0 \quad j\neq j', \ellambdabel{convergence_slow_M21}
\end{eqnarray}
for all $j$, $j'\neq j$, in $1,...,k$, $t\in [0,1]$, where we recall that $\Delta_j$ is defined in \eqref{Di}.
\end{theorem}
One interesting by-product of Theorem \ref{theo_regimes} is that it in particular gives some insight on the (non conditional) limiting distribution of $Z^{(n)}(t)$, with a limit in a somewhat simpler form. More precisely, the following corollary follows from the proofs of \eqref{convergence_fast} and \eqref{convergence_equilibrium}:
\begin{cor}\ellambdabel{rem:marginal}
In the Fast arrivals case $\gamma >\alpha$, the following convergence holds
\begin{equation}\ellambdabel{remark_conv_distrib_simpler_fast}
\frac{Z^{(n)}(t)}{n^{\gamma-\alpha}}\stackrel{\cal D}{\ellongrightarrow} \ellambdambda \int_{0}^{t} \frac{{\cal Y}(v)}{v^\alpha}\ dv,\ n\tauo\infty, \quad t\in [0,1],
\end{equation}
where $\{{\cal Y}(t)=({\cal Y}_1(t),...,{\cal Y}_k(t)),\ t\in [0,1]\}$ is a (time homogeneous) stationary continuous time Markov chain on the state space ${\cal S}$, with infinitesimal generator matrix defined by
\begin{equation}\ellambdabel{generator_Y}
\ellambdambda (\Delta_\pi^{-1} P' \Delta_\pi-I),\quad \Delta_\pi:=\mbox{\normalfont diag}(\pi(x),\ x\in {\cal S}).
\end{equation}
In the Equilibrium case $\gamma =\alpha$, one has
\begin{equation}\ellambdabel{remark_conv_distrib_simpler_equilibirum}
Z^{(n)}(t)\stackrel{\cal D}{\ellongrightarrow}\elleft( \int_0^t {\cal Y}_j(v) \ \tauilde{\nu}_j^\alpha(dv)\right)_{j=1,...,k},\ n\tauo\infty, \quad t\in [0,1].
\end{equation}
Here $\{\tauilde{\nu}_j^\alpha(t),\ t\in [0,1] \}$, $j=1,...,k$, are the inhomogeneous Poisson processes defined by $\tauilde{\nu}_j^\alpha(t)=\nu_j^\alpha(t^{1-\alpha})$, $t\in [0,1]$, where $\{\nu_j^\alpha(t),\ t\in [0,1] \}$, $j=1,...,k$, are defined in Theorem \ref{theo_regimes}.
\end{cor}
As mentioned in Section \ref{sec:rescale}, \cite{MRRS02, GK03} introduced a notion of Fast and Slow growth similar to the Fast and Slow arrivals presented in Theorems \ref{theo_regimes} and \ref{theo_slow_arrival}, for a process of interest which is either a superposition of renewal processes with heavy tailed interarrivals or the cumulative input of an infinite source Poisson model with heavy tailed services. In those references, the process is shown to converge weakly or in finite dimensional distributions towards specific limit processes under appropriate scaling, see \cite[Theorem 1]{MRRS02} and \cite[Theorem 1]{GK03}. Here, the outline of the proof of Theorem \ref{theo_regimes} is the following:
\begin{itemize}
\item We will first expand the LT of the left hand side of \eqref{convergence_fast} and \eqref{convergence_equilibrium} as $n\tauo \infty$ and prove that the limit satisfies a particular ODE thanks to Proposition \ref{prop_Poisson_psi}, which will be referred as Steps 1 and 2 in the proofs in the forthcoming Sections \ref{sec:fast} and \ref{sec:equilibrium}.
\item Then, we will identify this limit as the LT of the right hand side of \eqref{convergence_fast} and \eqref{convergence_equilibrium} thanks to a proper use of the Feynman-Kac or Campbell formula. This step will be referred as Step 3 in the proofs.
\end{itemize}
This is to be compared with the approach in \cite{BKMT14, MDT16}, where the authors derive ODEs for the limiting moment generating function and identify a gaussian limiting distribution for the normalized process.
The paper is organized in the following way. Section \ref{main_proofs} is dedicated to the proofs of the main results, with Subsections \ref{sec:fast}, \ref{sec:equilibrium} and \ref{sec:slow} giving the proofs of the convergences in distribution of Theorem \ref{theo_regimes} in fast arrivals and equilibrium cases, and of Theorem \ref{theo_slow_arrival} in the slow arrivals case. The proof of Corollary \ref{rem:marginal} is included in Subsections \ref{sec:fast}, for the convergence \eqref{remark_conv_distrib_simpler_fast}, and \ref{sec:equilibrium}, for the convergence \eqref{remark_conv_distrib_simpler_equilibirum}. As a concluding remark, we will discuss in Section \ref{sec:remark_compute} some computational aspect for the limiting distributions mentioned in those different regimes in Theorem \ref{theo_regimes} in the particular case when $\alpha$ is a rational number lying in $(0,1)$.
\section{Proofs of Theorems \ref{theo_regimes}, \ref{theo_slow_arrival} and Corollary \ref{rem:marginal}}\ellambdabel{main_proofs}
\subsection{Preliminary results}\ellambdabel{sec:preliminary}
We will repeatedly use the following general lemma in the proofs:
\begin{lemm}\ellambdabel{lemma_convergence}
Let $\elleft(t\in [0,1]\mapsto A_n(t) \right)_{n\in\mathbb{N}}$ be a sequence of continuous functions with values in $\mathbb{R}^{{\cal S}\tauimes {\cal S}}$, and let us assume that there exists some continuous function $t\in [0,1]\mapsto A(t)\in \mathbb{R}^{{\cal S}\tauimes {\cal S}}$ such that $\int_0^1 || A_n(v)-A(v)|| dv \ellongrightarrow 0$ as $n\tauo\infty$ for any matrix norm $||.||$. Let $y\in \mathbb{R}^{{\cal S}\tauimes {\cal S}}$ and $t\in [0,1]\mapsto Y_n(t)\in \mathbb{R}^{{\cal S}\tauimes {\cal S}}$ be the solution to the following differential equation
\begin{equation}\ellambdabel{lemma_EDOn}
\elleft\{
\begin{array}{rcl}
\frac{d}{dt}Y_n(t) &=& A_n(t) Y_n(t),\quad t\in [0,1],\\
Y_n(0)&=& y\in \mathbb{R}^{{\cal S}\tauimes {\cal S}} ,
\end{array}
\right. \quad n\in \mathbb{N} .
\end{equation}
Then one has $Y_n(t)\ellongrightarrow Y(t)$ uniformly in $t\in[0,1]$, as $n\tauo \infty$, where $t\in [0,1]\mapsto Y(t)\in \mathbb{R}^{{\cal S}\tauimes {\cal S}}$ is the solution to the following differential equation
\begin{equation}\ellambdabel{lemma_EDO}
\elleft\{
\begin{array}{rcl}
\frac{d}{dt}Y(t) &=& A(t) Y(t),\quad t\in [0,1],\\
Y(0)&=& y .
\end{array}
\right.
\end{equation}
\end{lemm}
\begin{proof}
We first observe that, because of continuity of $t\in [0,1]\mapsto A_n(t)$ and $t\in [0,1]\mapsto A(t)$, \eqref{lemma_EDOn} and \eqref{lemma_EDO} read in integral form
\begin{equation}\ellambdabel{equ_diff_Y}
Y_n(t)= y + \int_0^t A_n(v) Y_n(v) dv,\quad Y(t)= y + \int_0^t A(v) Y(v) dv
\end{equation}
for all $t\in [0,1]$. Since the norm $||.||$ may be arbitrary, we pick a submultiplicative one on the set of ${\cal S}\tauimes {\cal S}$ matrices. \eqref{equ_diff_Y} implies the following inequality
$$
||Y_n(t)|| \elle ||y||+ \int_0^t ||A_n(v)||.|| Y_n(v)|| dv,\quad \forall t\in [0,1] .
$$
Gronwall's lemma thus implies that $ ||Y_n(t)|| \elle ||y|| \exp\elleft( \int_0^t ||A_n(v)|| dv \right)$ for all $t\in [0,1]$. Since by assumption $\int_0^1 || A_n(v)-A(v)|| dv \ellongrightarrow 0$ as $n\tauo\infty$, one has that $\elleft(\int_0^1 ||A_n(v)|| dv\right)_{n\in \ensuremath{\mathbb{N}}}$ is a bounded sequence. We deduce the following finiteness
\begin{multline*}
M_Y:= \sup_{n\in\ensuremath{\mathbb{N}}}\sup_{t\in [0,1]}||Y_n(t)|| \elle \sup_{n\in\ensuremath{\mathbb{N}}}\sup_{t\in [0,1]} ||y|| \exp\elleft( \int_0^t ||A_n(v)|| dv \right)\\
\elle ||y|| \exp\elleft( \int_0^1 \sup_{n\in\ensuremath{\mathbb{N}}} ||A_n(v)|| dv \right) <+\infty .
\end{multline*}
Let us then introduce $M_A:= \sup_{v\in [0,1]}||A(v)||$, which is a finite quantity. Then one obtains that
\begin{multline*}
||Y_n(t)-Y(t)||\elle \int_0^t ||A_n(v)-A(v)||. ||Y_n(v)|| dv + \int_0^t ||A(v)||.||Y_n(v)-Y(v)||dv\\
\elle M_Y \int_0^t ||A_n(v)-A(v)|| dv + M_A \int_0^t ||Y_n(v)-Y(v)||dv,\quad \forall t\in[0,1].
\end{multline*}
Gronwall's lemma thus implies that, for all $t\in [0,1]$,
\begin{multline*}
||Y_n(t)-Y(t)||\elle M_Y \elleft[ \int_0^t ||A_n(v)-A(v)|| dv\right].\ e^{M_A t}\\ \elle M_Y \elleft[ \int_0^1 ||A_n(v)-A(v)|| dv \right].\ e^{M_A}\ellongrightarrow 0 \mbox{ as } n\tauo \infty.
\end{multline*}
Since the right hand side of the above inequality is independent from $t\in [0,1]$, this proves the uniform convergence result.
\end{proof}
We finish this subsection by stating the differential equation satisfied by the Laplace Transform $\psi^{(n)}(s,t)$ of $\elleft(Z^{(n)}(t),X^{(n)}_{N^{(n)}_t}\right)$ defined in \eqref{def_LT_n}, which will be the central object studied in Subsections \ref{sec:fast} and \ref{sec:equilibrium}.
Thanks to equation\eqref{Poisson_ODE} with the new parameters $\ellambdambda_n$, $P_n$ instead of $\ellambdambda$ and $P$ (and remembering that $\ellambdambda_n (P_n-I)=\ellambdambda(P-I)$), this reads here
\begin{equation}\ellambdabel{Poisson_ODEn}
\elleft\{
\begin{array}{rcl}
\partial_t \psi^{(n)}(s,t) &=& [\ellambdambda (P-I) + \ellambdambda n^\gamma P_n(\tauilde{\pi}_n(s,t)-I)]\psi^{(n)}(s,t),\quad t\ge 0,\\
\psi^{(n)}(s,0)&=& I,
\end{array}
\right.
\end{equation}
for all $s=(s_1,...,s_k)\in (-\infty,0]^k$. And, from \eqref{def_pi_Q_tilda}, using the expansion $\prod_{j=1}^k (a_j+1)=1+ \sum_{I\subset \{1,...,k\}} \prod_{\ell \in I} a_\ell$ for all real numbers $a_1,...,a_k$, we have the following expansion which will be useful later on:
\begin{multline}\ellambdabel{def_pi_n}
\tauilde{\pi}_n(s,t)-I=\mbox{diag} \elleft( \prod_{j=1}^k \elleft( (e^{s_j x_j}-1)\mathbb{P} \elleft[L_j^{(n)}>t\right]+1\right)-1, \ x=(x_1,...,x_k)\in {\cal S}\right)\\
= \mbox{diag} \elleft( \sum_{I\subset \{1,...,k\}} \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell}-1)\mathbb{P}\elleft[L_\ell^{(n)}>t\right]\right], \ x=(x_1,...,x_k)\in {\cal S}\right).
\end{multline}
\subsection{Case $\gamma>\alpha$: Fast arriving customers}\ellambdabel{sec:fast}
We now proceed to show convergence \eqref{convergence_fast} in Theorem \ref{theo_regimes}. In the present case, it is sensible to guess that $Z^{(n)}(t)$ converges towards infinity as $n\tauo\infty$, hence it is natural to find a normalization such that a convergence towards a proper distribution occurs. We renormalize the queue content by dividing it by $n^{\gamma-\alpha}$, i.e. we are here interested in $\elleft(Z^{(n)}(t)/n^{\gamma-\alpha},X^{(n)}_{N^{(n)}_t} \right)$, of which Laplace transform is given by $\psi^{(n)}(s/n^{\gamma-\alpha},t)$, $s=(s_1,...,s_k)\in (-\infty,0]^k$. In order to avoid cumbersome notation, we introduce the quantity
$$
\beta:= \frac{1}{1-\alpha}\in (1,+\infty).
$$
We observe then that
\begin{equation}\ellambdabel{time_transfo}
t\in[0,1]\mapsto t^\beta \in [0,1]
\end{equation}
is a one to one mapping. Hence, studying the limiting distribution of $\elleft(Z^{(n)}(t)/n^{\gamma-\alpha},X^{(n)}_{N^{(n)}_t} \right)$ for all $t\in [0,1]$ amounts to study the limiting distribution of
\begin{equation}\ellambdabel{fast_renormalized_beta}
\elleft(Z^{(n)}(t^\beta)/n^{\gamma-\alpha},X^{(n)}_{N^{(n)}_{t^\beta }} \right)
\end{equation}
for all $t\in [0,1]$, then changing variable $t:=t^{1/\beta}$. The time transformation \eqref{time_transfo} may at this point look artificial, but this is a key step which will later on enable us to use the convergence result in Lemma \ref{lemma_convergence}.The LT of \eqref{fast_renormalized_beta} is given by
$$\chi^{(n)}(s,t):= \psi^{(n)}(s/n^{\gamma-\alpha},t^\beta),\quad t\in [0,1].$$
From \eqref{Poisson_ODEn}, $\chi^{(n)}(s,t)$ satisfies
\begin{equation}\ellambdabel{Poisson_ODEn_fast}
\elleft\{
\begin{array}{rcl}
\partial_t \chi^{(n)}(s,t) &=& \beta t^{\beta-1}[\ellambdambda (P-I) + \ellambdambda n^\gamma P_n(\tauilde{\pi}_n(s/n^{\gamma-\alpha},t^\beta)-I)] \chi^{(n)}(s,t),\quad t\in [0,1],\\
\chi^{(n)}(s,0)&=& I.
\end{array}
\right.
\end{equation}
The starting point for proving \eqref{convergence_fast} is the following: we will set to prove that
\begin{equation}\ellambdabel{def_An_fast}
A_n(s,t)=\beta t^{\beta-1}[\ellambdambda (P-I) + \ellambdambda n^\gamma P_n(\tauilde{\pi}_n(s/n^{\gamma-\alpha},t^\beta)-I)]
\end{equation}
converges to some limit $A(s,t)$ as $n\tauo\infty$, use Lemma \ref{lemma_convergence}, then identify the limit $\chi(s,t):=\ellim_{n\tauo\infty}\chi^{(n)}(s,t)$ as the Laplace Transform of a known distribution.\\
{\bf Step 1: Determining $A(s,t)$.} This step is dedicated to finding the limit function $t\in[0,1]\mapsto A(s,t)$ of \eqref{def_An_fast}. Recalling that $\ellim_{n\tauo \infty}P_n=I$, studying the limit of \eqref{def_An_fast} amounts to studying that of $\beta t^{\beta-1}\ellambdambda n^\gamma(\tauilde{\pi}_n(s/n^{\gamma-\alpha},t^\beta)-I)$ as $n\tauo\infty$. In view of \eqref{def_pi_n}, the $x$th diagonal element of this latter term is
\begin{equation}\ellambdabel{term1_fast}
\beta t^{\beta-1}\ellambdambda n^\gamma \sum_{I\subset \{1,...,k\}} \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)\mathbb{P}\elleft[L_\ell^{(n)}>t^\beta\right]\right]
\end{equation}
of which we proceed to find the limit as $n\tauo \infty$. In order to study its convergence, we are going to isolate the terms in the sum \eqref{term1_fast} for which $\mbox{Card}(I)=1$ and $\mbox{Card}(I)\ge 2$, and show that the former admit a non zero limit and the latter tend to $0$. We thus write \eqref{term1_fast} as
\begin{eqnarray}
&& \beta t^{\beta-1}\ellambdambda n^\gamma \sum_{I\subset \{1,...,k\}} \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)\mathbb{P}\elleft[L_\ell^{(n)}>t^\beta\right]\right]=J_n^1(s,t) + J_n^2(s,t),\quad \mbox{where}\nonumber\\
&&J_n^1(s,t)=J_n^1(s,t,x):= \beta t^{\beta-1}\ellambdambda n^\gamma \sum_{\ell =1}^k (e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)\mathbb{P}\elleft[L_\ell^{(n)}>t^\beta\right],\ellambdabel{def_J1n}\\
&&J_n^2(s,t)=J_n^2(s,t,x):= \beta t^{\beta-1}\ellambdambda n^\gamma \sum_{\mbox{\tauiny Card}(I)\ge 2} \ \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)\mathbb{P}\elleft[L_\ell^{(n)}>t^\beta\right]\right].\ellambdabel{def_J2n}
\end{eqnarray}
Both terms $J_n^1(s,t)$ and $J_n^2(s,t)$ are studied separately. Using that $e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1\sigmam s_\ell x_\ell/n^{\gamma-\alpha}$ as $n\tauo\infty$ and
\begin{equation}\ellambdabel{basic_fat_tail}
\mathbb{P}\elleft[L_\ell^{(n)}>t^\beta\right]=\mathbb{P}\elleft[L_\ell>nt^\beta\right]\sigmam \frac{1}{n^\alpha t^{\beta \alpha}}
\end{equation}
when $t>0$, and since $\beta\alpha=\alpha/(1-\alpha)=\beta-1$, we arrive at
$$
J_n^1(s,t)\sigmam \beta \ellambdambda \sum_{\ell =1}^k t^{\beta-1} n^\gamma \frac{s_\ell x_\ell}{n^{\gamma-\alpha}} \frac{1}{n^\alpha t^{\beta \alpha}}\sigmam \beta \ellambdambda \sum_{\ell =1}^k s_\ell x_\ell,\quad n\tauo \infty,
$$
when $t>0$, and is $0$ when $t=0$.
Next we show that $J_n^2(s,t)$ tends to $0$ by showing that each term on the right hand side of \eqref{def_J2n} tend to $0$. So, if $I\subset \{1,...,k\}$ is such that $I=\{\ell_1,\ell_2\}$, i.e. $\mbox{Card}(I) =2$, then
\begin{multline}\ellambdabel{J2n_tends_zero}
\elleft| \beta t^{\beta-1}\ellambdambda n^\gamma \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)\mathbb{P}\elleft[L_\ell^{(n)}>t^\beta\right]\right]\right|= \beta t^{\beta-1}\ellambdambda n^\gamma \elleft|e^{s_{\ell_1} x_{\ell_1}/n^{\gamma-\alpha}}-1\right|\mathbb{P}\elleft[L_{\ell_1}>nt^\beta\right]\\
. \elleft|e^{s_{\ell_2} x_{\ell_2}/n^{\gamma-\alpha}}-1\right|\mathbb{P}\elleft[L_{\ell_2}>nt^\beta\right]\elle \beta t^{\beta-1}\ellambdambda n^\gamma |s_{\ell_1} x_{\ell_1}|.|s_{\ell_2} x_{\ell_2}| \frac{1}{n^{2(\gamma-\alpha)}}\mathbb{P}\elleft[L_{\ell_1}>nt^\beta\right]\\
= \beta t^{\beta-1}\ellambdambda |s_{\ell_1} x_{\ell_1}|.|s_{\ell_2} x_{\ell_2}| \frac{1}{n^{\gamma-\alpha}} n^\alpha \mathbb{P}\elleft[L_{\ell_1}>nt^\beta\right],
\end{multline}
where we used the inequality $|e^u-1|\elle |u|$ for $u\elle 0$ and $\mathbb{P}\elleft[L_{\ell_2}>nt^\beta\right]\elle 1$. Thanks to \eqref{basic_fat_tail}, the right hand side of \eqref{J2n_tends_zero} thus tends to zero when $t\in (0,1]$. The case $\mbox{Card}(I) >2$ is dealt with similarly. Finally, all terms on the right hand side of \eqref{def_J2n} tend to $0$ as $n\tauo\infty$, i.e. $\ellim_{n\tauo\infty}J_n^2(s,t)=0$ for all $t\in (0,1]$. When $t=0$ then $J_n^2(s,t)=0$, so that the limit holds for all $t\in [0,1]$.\\
Hence we have that \eqref{term1_fast} tends to $\ellim_{n\tauo\infty}J_n^1(s,t)+\ellim_{n\tauo\infty}J_n^2(s,t)$, i.e. to $ \beta \ellambdambda\sum_{\ell =1}^k s_\ell x_\ell$ when $t\in (0,1]$, and to $0$ when $t=0$. The candidate for the continuous function $A(s,t)$ is then
\begin{equation}\ellambdabel{candidate_Ast}
t\in [0,1]\mapsto A(s,t):= \beta t^{\beta-1}\ellambdambda (P-I) + \beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell
\end{equation}
where we recall from \eqref{Di} that $\Delta_\ell=\mbox{diag} \elleft[ x_\ell ,\ x=(x_1,\elldots,x_k)\in{\cal S}\right]$. This is where the time transformation \eqref{time_transfo} described previously is important, as without it it would not have been possible to exhibit the limit \eqref{candidate_Ast} for $A_n(s,t)$. Note that the limit when $t=0$ for $A_n(s,t)$ in \eqref{def_An_fast} differs from $A(s,0)=\beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell$, as indeed a closer look from the study of the limits of $J_n^1(s,t)$ and $J_n^2(s,t)$ would yield that $\ellim_{n\tauo \infty}A_n(s,0)$ should rather be the $0$ matrix. This is due to the fact that the limit $t\in [0,1]\mapsto A(s,t)$ in Lemma \ref{lemma_convergence} has to be {\it continuous} so that the lemma holds.\\
{\bf Step 2: Determining $\chi(s,t)=\ellim_{n\tauo}\chi_n(s,t)$.} We now need to prove that $\int_0^1 || A_n(s,v)-A(s,v)|| dv \ellongrightarrow 0$ as $n\tauo\infty$ in order to apply Lemma \ref{lemma_convergence}. Thanks to \eqref{def_An_fast} and \eqref{candidate_Ast}, and by the definitions \eqref{def_J1n} and \eqref{def_J2n} of $J^1_n(s,t,x)$ and $J^1_n(s,t,x)$, we observe that $A_n(s,t)$ can be decomposed as
\begin{multline*}
A_n(s,t)=A(s,t) + P_n\ \mbox{diag}\elleft(J_n^1(s,t,x) - \beta \ellambdambda \sum_{\ell =1}^k s_\ell x_\ell,\ x\in {\cal S}\right) \\
+ P_n\ \mbox{diag}\elleft(J_n^2(s,t,x),\ x\in {\cal S}\right) + (P_n-I)\beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell, \quad t\in[0,1].
\end{multline*}
Hence, since $\ellim_{n\tauo \infty}P_n=I$, proving $\ellim_{n\tauo\infty}\int_0^1 || A_n(s,v)-A(s,v)|| dv = 0$ amounts to proving that
\begin{equation}\ellambdabel{to_prove_limits_J12}
\begin{array}{rcl}
\displaystyle\int_0^1 \elleft|J_n^1(s,v,x) - \beta \ellambdambda \sum_{\ell =1}^k s_\ell x_\ell\right| dv & \ellongrightarrow & 0, \ \mbox{and}\\
\displaystyle\int_0^1 |J_n^2(s,v,x)|dv=\int_0^1 J_n^2(s,v,x)dv & \ellongrightarrow & 0,
\end{array}
\end{equation}
as $n\tauo\infty$, for each fixed $x\in {\cal S}$. Let us first focus on $\int_0^1 \elleft|J_n^1(s,v,x) - \beta \ellambdambda \sum_{\ell =1}^k s_\ell x_\ell\right| dv$. We have
\begin{eqnarray}
\int_0^1 \elleft|J_n^1(s,v,x) - \beta \ellambdambda \sum_{\ell =1}^k s_\ell x_\ell\right| dv&\elle & \sum_{\ell=1}^k (I^1_n(\ell) + I^2_n(\ell)), \mbox{ where, for all } \ell =1,...,k, \ellambdabel{term3_fast}\\
I^1_n(\ell) &:=& \int_0^1 \ellambdambda \beta v^{\beta-1}\elleft| n^\gamma(e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)- n^\alpha s_\ell x_\ell\right| \mathbb{P}\elleft[L_\ell^{(n)}>v^\beta\right]dv \nonumber\\
I^2_n(\ell) &:=& |s_\ell x_\ell| \int_0^1 \ellambdambda \elleft|\beta v^{\beta-1} n^\alpha \mathbb{P}\elleft[L_\ell^{(n)}>v^\beta\right]-\beta \right|dv .\nonumber
\end{eqnarray}
Expanding the exponential function, one has that $|e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1-s_\ell x_\ell/n^{\gamma-\alpha}|\elle M_\ell/n^{2(\gamma-\alpha)}$ where $M_\ell>0$ only depends on $s_\ell$ and $x_\ell$. Thus, one deduces the following upper bounds for $I^1_n(\ell)$, $\ell =1,...,k$:
\begin{eqnarray}
I^1_n(\ell) &=& \int_0^1 \ellambdambda \beta v^{\beta-1}\elleft| n^\gamma(e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1)- n^\alpha s_\ell x_\ell\right| \mathbb{P}\elleft[L_\ell> n v^\beta\right]dv\nonumber\\
&\elle & n^\gamma \frac{M_\ell}{n^{2(\gamma-\alpha)}} \ellambdambda \int_0^1 \beta v^{\beta-1} \mathbb{P}\elleft[L_\ell> n v^\beta\right]dv = \frac{M_\ell}{n^{\gamma-\alpha}} \beta \ellambdambda \int_0^1 n^\alpha v^{\beta-1} \mathbb{P}\elleft[L_\ell> n v^\beta\right]dv \nonumber\\
&=& \frac{M_\ell}{n^{\gamma-\alpha}} \beta \ellambdambda \int_0^1 (nv^{\beta})^\alpha \ \mathbb{P}\elleft[L_\ell> n v^\beta\right]dv,\ellambdabel{term4_fast}
\end{eqnarray}
the last equality holding because $\beta-1=\beta \alpha$ implies that the integrand verifies $n^\alpha v^{\beta-1} =(nv^{\beta})^\alpha$. A consequence of the fact that $L_\ell$ is fat tailed with index $\alpha$ is that $\sup_{u\ge 0}u^\alpha \mathbb{P}(L_\ell>u)<+\infty$, from which one deduces immediately that
\begin{equation}\ellambdabel{bound_sup_L_l}
\sup_{j\in\mathbb{N},\ v \in [0,1]} (jv^{\beta})^\alpha \ \mathbb{P}\elleft[L_\ell> j v^\beta\right]<+\infty
\end{equation}
(note that those two latter suprema are in fact equal). One then gets from \eqref{term4_fast} that
\begin{equation}\ellambdabel{term5_fast}
I^1_n(\ell)\elle \frac{M_\ell}{n^{\gamma-\alpha}} \beta \ellambdambda \elleft[\sup_{j\in\mathbb{N},\ v \in [0,1]} (jv^{\beta})^\alpha \ \mathbb{P}\elleft[L_\ell> j v^\beta\right]\right]\ellongrightarrow 0,\quad n\tauo\infty .
\end{equation}
We now turn to $I^2_n(\ell)$, $\ell =1,...,k$. Using again $\beta-1=\beta \alpha$, one may write in the integrand of $I^2_n(\ell)$ that $v^{\beta-1} n^\alpha =(nv^{\beta})^\alpha$: hence
$$ I^2_n(\ell) = |s_\ell x_\ell| \int_0^1 \ellambdambda \elleft|\beta (nv^{\beta})^\alpha \mathbb{P}\elleft[L_\ell>nv^\beta\right]-\beta \right|dv .$$
Since $L_\ell$ is fat tailed with index $\alpha$, estimates similar to the ones leading to the upper bound \eqref{term5_fast} for $I_n^1(\ell)$ yield that
$$
\sup_{n\in\mathbb{N},\ v \in [0,1]}\elleft|\beta (nv^{\beta})^\alpha \mathbb{P}\elleft[L_\ell>nv^\beta\right]-\beta \right|<+\infty .
$$
Furthermore, again because $L_\ell$ is fat tailed, one has $\mathbb{P}\elleft[L_\ell>nv^\beta\right] \sigmam 1/(nv^{\beta})^\alpha$ as $n\tauo \infty$ when $v>0$. Hence $\elleft|\beta (nv^{\beta})^\alpha \mathbb{P}\elleft[L_\ell>nv^\beta\right]-\beta \right|\ellongrightarrow 0 $ as $n\tauo \infty$ when $v\in (0,1]$, and is equal to $\beta$ when $v=0$. The dominated convergence theorem thus implies that
\begin{equation}\ellambdabel{term6_fast}
I^2_n(\ell)\ellongrightarrow 0, \quad n\tauo\infty .
\end{equation}
Gathering \eqref{term3_fast}, \eqref{term5_fast} and \eqref{term6_fast}, we thus deduce finally that $\int_0^1 \elleft|J_n^1(s,v,x) - \beta \ellambdambda \sum_{\ell =1}^k s_\ell x_\ell\right| dv$ tends to $0$ as $n\tauo\infty$ for each $x\in{\cal S}$. \\
We now prove that $\int_0^1 J_n^2(s,v,x)dv\ellongrightarrow 0$ as $n\tauo \infty$.
In view of the definition \eqref{def_J2n}, it suffices to prove that
\begin{equation}\ellambdabel{to_prove_J2}
\int_0^1\beta v^{\beta-1}\ellambdambda n^\gamma \ \prod_{\ell \in I}\elleft[|e^{s_\ell x_\ell/n^{\gamma-\alpha}}-1|\ \mathbb{P}\elleft[L_\ell^{(n)}>v^\beta\right]\right]dv
\end{equation}
tends to $0$ as $n\tauo\infty$ for $I\subset \{1,...,k\}$ such that $\mbox{ Card}(I)\ge 2$. Let us prove the convergence for $\mbox{ Card}(I)= 2$, i.e. for $I=\{\ell_1,\ell_2\}$ for some $\ell_1\neq\ell_2$ in $1,...,k$, the case $\mbox{ Card}(I)> 2$ being dealt with similarly. By the basic inequality $|e^u-1|\elle |u|$ for $u\elle 0$ we deduce that $|e^{s_{\ell_i} x_{\ell_i}/n^{\gamma-\alpha}}-1|\elle |s_{\ell_i} x_{\ell_i}|/n^{\gamma-\alpha}$, $i=1,2$. Since $\mathbb{P}\elleft[L_{\ell_1}^{(n)}>v^\beta\right]\elle 1$ for all $v\in [0,1]$, we then deduce that \eqref{to_prove_J2} is upper bounded by
$$\frac{|s_{\ell_1} x_{\ell_1}|}{n^{\gamma-\alpha}} |s_{\ell_2} x_{\ell_2}|
\int_0^1 \beta v^{\beta-1}\ellambdambda n^\alpha \mathbb{P}\elleft[L_{\ell_2}^{(n)}>v^\beta\right] dv.
$$
As $v^{\beta-1} n^\alpha =(nv^{\beta})^\alpha$, and thanks to \eqref{bound_sup_L_l}, the latter quantity is in turn written then bounded as follows
\begin{multline*}
\frac{|s_{\ell_1} x_{\ell_1}|}{n^{\gamma-\alpha}} |s_{\ell_2} x_{\ell_2}|
\int_0^1 \beta \ellambdambda (nv^{\beta})^\alpha \mathbb{P}\elleft[L_{\ell_2}^{(n)}>v^\beta\right] dv
= \frac{|s_{\ell_1} x_{\ell_1}|}{n^{\gamma-\alpha}} |s_{\ell_2} x_{\ell_2}|
\int_0^1 \beta \ellambdambda (nv^{\beta})^\alpha \mathbb{P}\elleft[L_{\ell_2}>nv^\beta\right] dv\\
\elle \frac{|s_{\ell_1} x_{\ell_1}|}{n^{\gamma-\alpha}} |s_{\ell_2} x_{\ell_2}| \beta \ellambdambda \elleft[\sup_{j\in\mathbb{N},\ v \in [0,1]} (jv^{\beta})^\alpha \ \mathbb{P}\elleft[L_{\ell_2}> j v^\beta\right] \right] \ellongrightarrow 0,\quad n\tauo\infty,
\end{multline*}
proving that \eqref{to_prove_J2} tends to $0$ as $n\tauo\infty$ when $I=\{\ell_1,\ell_2\}$.\\
Hence we just proved \eqref{to_prove_limits_J12}, which implies $\int_0^1 || A_n(s,v)-A(s,v)|| dv \ellongrightarrow 0$. We may then use Lemma \ref{lemma_convergence} to deduce that $\chi^{(n)}(s,t)$ convgerges to $\chi(s,t)$ which satisfies
\begin{equation}\ellambdabel{Poisson_ODE_fast}
\elleft\{
\begin{array}{rcl}
\partial_t \chi(s,t) &=& A(s,t)\chi(s,t)=\elleft[\beta t^{\beta-1}\ellambdambda (P-I) + \beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell \right]\chi(s,t) ,\quad t\in [0,1],\\
\chi(s,0)&=& I.
\end{array}
\right.
\end{equation}
{\bf Step 3: Identifying the limit in distribution.} Let us note that \eqref{Poisson_ODE_fast} does not admit an explicit expression. However, since we purposely chose $s=(s_1,...,s_k)$ with $s_j\elle 0$, $j=1,...,k$, one has that $\sum_{j=1}^k s_j \Delta_j = \sum_{j=1}^k s_j\ \mbox{diag}(x_j,\ x\in {\cal S})$ is a diagonal matrix with non positive entries. Let $\Delta_\pi:=\mbox{\normalfont diag}(\pi(x),\ x\in {\cal S})$ and let us introduce the matrix $P^{(r)}$ defined by $P^{(r)}=\Delta_\pi^{-1}P' \Delta_\pi\iff P=\Delta_\pi^{-1}P^{(r)'} \Delta_\pi$. It is standard that $P^{(r)}$ is the transition matrix of the reversed version of the stationary Markov chain $\{X_i,\ i\in\mathbb{N} \}$ with distribution $\pi$, and that $\beta t^{\beta-1}\ellambdambda (P^{(r)}-I)$ is the infinitesimal generator matrix of an inhomogeneous Markov process
\begin{equation}\ellambdabel{def_U}
\{U(t)=(U_j(t))_{j=1,...,k}\in {\cal S},\ t\in [0,1]\}
\end{equation}
with values in ${\cal S}$, and initial distribution $U(0)\sigmam\pi$.
In fact, it turns out that the conditional distribution of $U(t)$ given $U(0)$ is given by $\elleft[\mathbb{P}(U(t)=y|\ U(0)=x)\right]_{(x,y)\in {\cal S}}=\exp(t^\beta \ellambdambda(P^{(r)}-I))$, which results in $U(t)\sigmam\pi$ for all $t\in[0,1]$, i.e. that $\{U(t),\ t\in [0,1]\}$ is stationary. Since $\sum_{j=1}^k s_j \Delta_j$ is diagonal, one checks easily that $A(s,t)=\Delta_\pi^{-1} \elleft[\beta t^{\beta-1}\ellambdambda ({P^{(r)}}'-I)+\sum_{j=1}^k s_j \Delta_j\right]\Delta_\pi$ and that $Y(t)=Y(s,t):=\Delta_\pi^{-1} \chi(s,t)'\Delta_\pi $ satisfies the differential equation
$$
\elleft\{
\begin{array}{rcl}
\partial_t Y(t) &=& Y(t)\elleft[\beta t^{\beta-1}\ellambdambda ({P^{(r)}}-I) + \beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell \right] ,\quad t\in [0,1],\\
Y(0)&=& I.
\end{array}
\right.
$$
The Feynman-Kac formula ensures that one has the representation
$$Y(t)=Y(s,t)=\elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[U(t)=y]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_0^t U_j(v) dv \right)\right| U(0)=x\right]\right]_{(x,y)\in {\cal S}^2},\quad \forall t\in [0,1],$$
see \cite[Chapter III, 19, p.272]{Rogers_Williams00} for the general theorem on this formula, or \cite[Section 5, Expression (5.2) and differential equation (5.3)]{BH96} for the particular case of a finite Markov chain, adapted here to an inhomogeneous Markov process. Also, the reversed process $\{U(1-t),\ t\in [0,1]\}$ admits ${\Delta_\pi}^{-1}\beta (1-t)^{\beta-1}\ellambdambda ({P^{(r)}}'-I)\Delta_\pi= \beta (1-t)^{\beta-1}\ellambdambda (P-I)$ as infinitesimal generator matrix, which is the generator of the process $\{{\cal X}^\alpha (t)=({\cal X}_1^\alpha (t),...,{\cal X}_k^\alpha (t))\in {\cal S},\ t\in[0,1] \}$ introduced in the statement of Theorem \ref{theo_regimes}, so that $\{{\cal X}^\alpha (t),\ t\in[0,1] \}\stackrel{\cal D}{=}\{U(1-t),\ t\in [0,1]\}$ pathwise. Hence, one obtains for all $x$ and $y$ in ${\cal S}$ that
\begin{eqnarray}
&&\mathbb{E}\elleft[\elleft. {\bf 1}_{[U(t)=y]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_0^t U_j(v) dv \right)\right| U(0)=x\right]\nonumber\\
&=& \mathbb{E}\elleft[\elleft. {\bf 1}_{[{\cal X}^\alpha (1-t)=y]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_{1-t}^1 {\cal X}^\alpha_j (v) dv \right)\right|{\cal X}^\alpha (1)=x\right]\nonumber\\
&=& \mathbb{E}\elleft[\elleft. {\bf 1}_{[{\cal X}^\alpha (1)=x]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_{1-t}^1 {\cal X}^\alpha_j (v) dv \right)\right| {\cal X}^\alpha (1-t)=y\right] \frac{\pi(y)}{\pi(x)},\ellambdabel{relation_U_chi}
\end{eqnarray}
the last line coming from the fact that $U(0)$, $U(t)$, $ {\cal X}^\alpha (1-t)$ and ${\cal X}^\alpha (1)$ all have same distribution $\pi$. Switching the role of $x$ and $y$ in the above results in the following relationship:
\begin{eqnarray*}
&&\elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[{\cal X}^\alpha (1)=y]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_{1-t}^1 {\cal X}^\alpha_j (v) dv \right)\right| {\cal X}^\alpha (1-t)=x\right] \right]_{(x,y)\in {\cal S}^2}\\
&=& \elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[U(t)=x]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_0^t U_j(v) dv \right)\right| U(0)=y\right] \frac{\pi(y)}{\pi(x)}\right]_{(x,y)\in {\cal S}^2}\\
&=& \Delta_\pi^{-1} Y(t)'\Delta_\pi=\chi(s,t).
\end{eqnarray*}
Since we just proved that $\chi^{(n)}(s,t):= \psi^{(n)}(s/n^{\gamma-\alpha},t^\beta)$ converges as $n\tauo \infty$ towards $\chi(s,t)$, expressed above, for all $s=(s_1,...,s_k)\in (-\infty, 0]^k$, and identifying Laplace transforms, we obtained in conclusion that
\begin{equation}\ellambdabel{conv_fast_final}
{\cal D}\elleft(\elleft. \elleft(Z^{(n)}(t^\beta)/n^{\gamma-\alpha},X^{(n)}_{N^{(n)}_{t^\beta }} \right)\right|\ X^{(n)}_0 \right)\ellongrightarrow {\cal D}\elleft(
\elleft.
\elleft( \beta\ellambdambda \int_{1-t}^{1} {\cal X}^\alpha(v)\ dv,\ {\cal X}^\alpha (1)\right)
\right|\ {\cal X}^\alpha(1-t)
\right)
\end{equation}
as $n\tauo\infty$ for all $t\in[0,1]$. Changing $t$ into $t^{1/\beta}$ yields \eqref{convergence_fast}.\\
{\bf Proof of the convergence \eqref{remark_conv_distrib_simpler_fast} in Corollary \ref{rem:marginal}. }With the previous definitions of processes $\{U(t),\ t\in [0,1]\}$ in \eqref{def_U} and $\{{\cal X}^\alpha (t),\ t\in[0,1] \}$, \eqref{relation_U_chi} implies the following matrix equality
\begin{multline}\ellambdabel{relation_U_chi_matrix}
\elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[U(t)=y]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_0^t U_j(v) dv \right)\right| U(0)=x\right] \right]_{(x,y)\in {\cal S}^2}\\
= \elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[{\cal X}^\alpha (1)=x]}\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_{1-t}^1 {\cal X}^\alpha_j (v) dv \right)\right| {\cal X}^\alpha (1-t)=y\right] \frac{\pi(y)}{\pi(x)}\right]_{(x,y)\in {\cal S}^2}
\end{multline}
Left-multiplying and right-multiplying \eqref{relation_U_chi_matrix} respectively by the row vector $(\pi(x))_{x\in {\cal S}}$ and the column vector ${\bf 1}$ results in in the following equality of LT
\begin{multline}\ellambdabel{LT_reverse_fast}
\mathbb{E}\elleft[ \exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_0^t U_j(v) dv \right) \right]\\
= \mathbb{E}\elleft[\exp\elleft(\sum_{j=1}^k s_j \beta \ellambdambda\int_{1-t}^1 {\cal X}^\alpha_j (v) dv \right)\right],\quad s=(s_1,...,s_k)\in (-\infty, 0]^k,
\end{multline}
which, combined with \eqref{conv_fast_final}, yields the convergence $\frac{Z^{(n)}\elleft(t^\beta\right)}{n^{\gamma-\alpha}}\stackrel{\cal D}{\ellongrightarrow} \beta\ellambdambda \int_{0}^{t} U(v)\ dv$ as $n\tauo \infty$. Changing $t$ into $t^{1/\beta}$ and performing the change of variable $ v:=v^{1/\beta}=v^{1-\alpha} $, we obtain
\begin{equation}\ellambdabel{conv_distribution_reversed_fast}
\frac{Z^{(n)}\elleft(t\right)}{n^{\gamma-\alpha}}\stackrel{\cal D}{\ellongrightarrow} \ellambdambda \int_{0}^{t} \frac{U(v^{1-\alpha})}{v^\alpha}\ dv,\ n\tauo\infty, \quad t\in [0,1].
\end{equation}
Since the ${\cal S}$ valued Markov process $\{U(t),\ t\in [0,1]\} $ admits $\beta t^{\beta-1}\ellambdambda (P^{(r)}-I)$ as the infinitesimal generator matrix, the time changed Markov process $\{{\cal Y}(t):=U(t^{1-\alpha})=U(t^{1/\beta}),\ t\in [0,1]\} $ admits \eqref{generator_Y} as generator, so that \eqref{remark_conv_distrib_simpler_fast} follows from \eqref{conv_distribution_reversed_fast}.
\subsection{Equilibrium case $\gamma=\alpha$}\ellambdabel{sec:equilibrium}
We now proceed to show convergence \eqref{convergence_equilibrium} in Theorem \ref{theo_regimes}. Intuitively, we are in the critical case where customers should arrive just fast enough such that the queue at time $t$ converges as $n\tauo\infty$. We are here interested in the behaviour of ${\cal D}\elleft(
\elleft. \elleft(Z^{(n)}(t),X^{(n)}_{N^{(n)}_t} \right) \right|\ X^{(n)}_0 \right)$ as $n\tauo\infty$ when $t\in[0,1]$ is fixed. As in Section \ref{sec:fast}, we first consider $t^\beta$ instead of $t$ and let
$$\chi^{(n)}(s,t):= \psi^{(n)}(s,t^\beta),\quad t\in [0,1],$$
the corresponding Laplace transform, where $s=(s_1,...,s_k)\in (-\infty,0]^k$. $t\in [0,1]\mapsto \chi^{(n)}(s,t)$ then satisfies, thanks to \eqref{Poisson_ODEn}, the following differential equation
\begin{equation}\ellambdabel{Poisson_ODEn_equilibrium}
\elleft\{
\begin{array}{rcl}
\partial_t \chi^{(n)}(s,t) &=& \beta t^{\beta-1}[\ellambdambda (P-I) + \ellambdambda n^\gamma P_n(\tauilde{\pi}_n(s,t^\beta)-I)] \chi^{(n)}(s,t),\quad t\in [0,1],\\
\chi^{(n)}(s,0)&=& I.
\end{array}
\right.
\end{equation}
The present case has the same roadmap as Subsection \ref{sec:fast}: we will study the behaviour as $n\tauo\infty$ of $\ellambdambda n^\gamma(\tauilde{\pi}_n(s,t^\beta))-I)$ in order to obtain a limit as $n\tauo\infty$ of
\begin{equation}\ellambdabel{def_An_equilibirum}
A_n(s,t)=\beta t^{\beta-1}[\ellambdambda (P-I) + \ellambdambda n^\gamma P_n(\tauilde{\pi}_n(s,t^\beta))-I)]
\end{equation}
then getting a limiting matrix differential equation with solution $\chi(s,t)=\ellim_{n\tauo\infty} \chi^{(n)}(s,t)$. Then we will identify $\chi(s,t)$ as the Laplace transform of a (conditional) distribution, yielding \eqref{convergence_equilibrium}.\\
{\bf Step 1: Determining $A(s,t)=\ellim_{n\tauo \infty }A_n(s,t)$.} We recall that the $(x,x)$th diagonal element of $\ellambdambda n^\gamma(\tauilde{\pi}_n(s,t^\beta))-I)$ is (from \eqref{def_pi_n})
$\sum_{I\subset \{1,...,k\}} \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell}-1)\mathbb{P}\elleft[L_\ell>nt\right]\right]$, which we decompose as in Section \ref{sec:fast} as $K_n^1(s,t)+K_n^2(s,t)$ with
\begin{eqnarray}
K_n^1(s,t)&=&K_n^1(s,t,x):= \beta t^{\beta-1}\ellambdambda n^\gamma \sum_{\ell =1}^k (e^{s_\ell x_\ell}-1)\mathbb{P}\elleft[L_\ell >n t^\beta\right],\ellambdabel{def_K1n}\\
K_n^2(s,t)&=&K_n^2(s,t,x):= \beta t^{\beta-1}\ellambdambda n^\gamma \sum_{\mbox{\tauiny Card}(I)\ge 2} \ \prod_{\ell \in I}\elleft[(e^{s_\ell x_\ell}-1)\mathbb{P}\elleft[L_\ell >nt^\beta\right]\right].\ellambdabel{def_K2n}
\end{eqnarray}
The important point here is that, throughout this subsection, we have $\gamma=\alpha$ in the expressions \eqref{def_An_equilibirum}, \eqref{def_K1n} and \eqref{def_K2n}, which will impact on the convergences and limiting results we are going to prove. Using that $\mathbb{P}\elleft[L_\ell >n t^\beta\right]\sigmam \frac{1}{n^\alpha}\frac{1}{t^{\alpha\beta}}$, $n\tauo\infty$, when $t>0$, and since $\alpha\beta=\beta-1$, and $\gamma=\alpha$, one here finds that
$$
K_n^1(s,t)=K_n^1(s,t,x)\ellongrightarrow \elleft\{
\begin{array}{cl}
\beta\ellambdambda \sum_{\ell =1}^k (e^{s_\ell x_\ell}-1), & t>0,\\
0, & t=0,
\end{array}\right.
\quad n\tauo \infty .
$$
As to $K_n^2(s,t)$, one proves easily that it tends to $0$ as $n\tauo\infty$ for all $t\in [0,1]$, as the sum in \eqref{def_K2n} is over $\mbox{Card}(I)\ge 2$, and using the fat tailed property of the service times. The candidate for the continuous function is thus
\begin{equation}\ellambdabel{candidate_Ast_equilibrium}
t\in [0,1]\mapsto A(s,t):= \beta t^{\beta-1}\ellambdambda (P-I) + \beta \ellambdambda \sum_{\ell =1}^k \mbox{diag}(e^{s_\ell x_\ell}-1,\ x=(x_1,...,x_k)\in {\cal S}).
\end{equation}
{\bf Step 2: Determining $\chi(s,t)=\ellim_{n\tauo}\chi_n(s,t)$.} We now wish to apply Lemma \ref{lemma_convergence} and prove that $\int_0^1 || A_n(s,v)-A(s,v)|| dv \ellongrightarrow 0$ where $A_n(s,t)$ and $A(s,t)$ are defined in \eqref{def_An_equilibirum} and \eqref{candidate_Ast_equilibrium}. The method is very similar as to proving \eqref{to_prove_limits_J12} in Step 2 of Section \ref{sec:fast}, as this is equivalent to proving for all $x\in {\cal S}$ that
\begin{eqnarray}
\int_0^1 \elleft| K_n^1(s,v,x)- \beta\ellambdambda \sum_{\ell =1}^k (e^{s_\ell x_\ell}-1)\right| dv &\ellongrightarrow & 0,\ellambdabel{limits_K1n}\\
\int_0^1 K_n^2(s,v,x)dv &\ellongrightarrow & 0 \ellambdabel{limits_K2n}
\end{eqnarray}
as $n\tauo\infty$. In view of the expression \eqref{def_K2n} of $K^2_n(s,t)$, \eqref{limits_K2n} is proved the same way as for proving that $\ellim_{n\tauo\infty}\int_0^1 J_n^2(s,v,x)dv=0$ in Step 2 of Section \ref{sec:fast}. More precisely, it suffices from \eqref{def_K2n} to prove that
\begin{equation}\ellambdabel{proof_limit_K2n}
\ellim_{n\tauo \infty} \int_0^1 \beta t^{\beta-1}\ellambdambda n^\alpha \prod_{\ell \in I} \mathbb{P}\elleft[L_\ell >nt^\beta\right] dt =0
\end{equation}
for all $I\subset \{1,...,k\}$, $\mbox{Card}(I)\ge 2$. We prove it for $I=\{\ell_1,\ell_2\}$, $\ell_1\neq \ell_2$, the proof for $\mbox{Card}(I)> 2$ being very similar. The trick is again to use that $v^{\beta-1} n^\alpha =(nv^{\beta})^\alpha$ as well as the previously established upper bound \eqref{bound_sup_L_l}, resulting in
\begin{multline*}
\int_0^1 \beta t^{\beta-1}\ellambdambda n^\alpha \mathbb{P}\elleft[L_{\ell_1} >nt^\beta\right] \mathbb{P}\elleft[L_{\ell_2} >nt^\beta\right]dt \\
\elle \beta\ellambdambda \elleft[\sup_{j\in\mathbb{N},\ v \in [0,1]} (jv^{\beta})^\alpha \ \mathbb{P}\elleft[L_{\ell_1}> j v^\beta\right] \right]\ \int_0^1 \mathbb{P}\elleft[L_{\ell_2} >nt^\beta\right]dt,
\end{multline*}
which converges to zero as $n\tauo\infty$ by the dominated convergence theorem, proving \eqref{proof_limit_K2n} when $\mbox{Card}(I)= 2$.
As to \eqref{limits_K1n}, this is proved, in view of the expression \eqref{def_K1n} of $K^1_n(s,t)$, by showing that $\int_0^1 \ellambdambda \elleft|\beta v^{\beta-1} n^\alpha \mathbb{P}\elleft[L_\ell>nv^\beta\right]-\beta \right|dv$ tends to $0$ as $n\tauo\infty$ for all $\ell=1,...,k$, as again we have that $\gamma=\alpha$; However,
this was already proved in Step 2 of Section \ref{sec:fast} when proving that $\ellim_{n\tauo\infty}I_n^2(\ell)=0$, $\ell=1,...,k$, see the arguments leading to the convergence \eqref{term6_fast}. All in all, one has the convergence $\int_0^1 || A_n(s,v)-A(s,v)|| dv \ellongrightarrow 0$, and Lemma \ref{lemma_convergence} is applicable so that $\chi^{(n)}(s,t)$ convgerges to $\chi(s,t)$ which satisfies
\begin{equation}\ellambdabel{Poisson_ODE_equilibrium}
\elleft\{
\begin{array}{rcl}
\partial_t \chi(s,t) &=& A(s,t)\chi(s,t)=\elleft[\beta t^{\beta-1}\ellambdambda (P-I) \right.\\
&+&\elleft. \beta \ellambdambda \sum_{\ell =1}^k \mbox{diag}(e^{s_\ell x_\ell}-1,\ x=(x_1,...,x_k)\in {\cal S}) \right]\chi(s,t) ,\quad t\in [0,1],\\
\chi(s,0)&=& I.
\end{array}
\right.
\end{equation}
{\bf Step 3: Identifying the limit in distribution.} With the same notation as in Step 3 of Section \ref{sec:fast} for process $\{{\cal X}^\alpha (t)=({\cal X}_1^\beta (t),...,{\cal X}_k^\beta (t))\in {\cal S},\ t\in[0,1] \}$, one finds this time that
\begin{equation}\ellambdabel{equilibrium_chi}
\chi(s,t)= \elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[{\cal X}^\alpha (1)=y]}\exp\elleft(\sum_{j=1}^k \beta \ellambdambda\int_{1-t}^1 \elleft(e^{s_j{\cal X}^\alpha_j (v)}-1\right) dv \right)\right| {\cal X}^\alpha (1-t)=x\right] \right]_{(x,y)\in {\cal S}^2}
\end{equation}
for all $s=(s_1,...,s_k)\in (-\infty,0]^k$. We recall the Campbell formula which states that for all measurable function $f:t\in [0,+\infty) \mapsto f(t)\in \mathbb{R}$ such that $\int_0^\infty (e^{f(v)}-1) \xi \ dv$ is finite for some $\xi>0$ then one has the identity
$$
\exp \elleft(\int_0^\infty \elleft(e^{f(v)}-1\right) \xi \ dv \right) =\mathbb{E} \elleft[ \exp\elleft( \int_0^\infty f(v)\ \nu(dv)\right)\right],
$$
where $\{ \nu(x),\ x\ge 0\}$ is a Poisson process with intensity $\xi$, see \cite[Section 3.2]{K93}. Conditioning on $\{{\cal X}^\alpha(v),\ v\in [0,1] \}$, this results in \eqref{equilibrium_chi} being written as
$$
\chi(s,t)= \elleft[ \mathbb{E}\elleft[\elleft. {\bf 1}_{[{\cal X}^\alpha (1)=y]}\exp\elleft(\sum_{j=1}^k s_j\int_{1-t}^{1} {\cal X}^\alpha_j(v)\ \nu_j^\alpha(dv) \right)\right| {\cal X}^\alpha (1-t)=x\right] \right]_{(x,y)\in {\cal S}^2}
$$
where $\{\nu_j^\alpha(t),\ t\ge 0 \}$, $j=1,...,k$, are independent Poisson processes with intensities $\beta\ellambdambda=\ellambdambda/(1-\alpha)$, and independent from $\{{\cal X}^\alpha(t),\ t\in[0,1] \}$. Identifying Laplace Transforms, we obtain in conclusion that
\begin{multline}\ellambdabel{conv_equilibrium_final}
{\cal D}\elleft(\elleft. \elleft(Z^{(n)}(t^\beta),X^{(n)}_{N^{(n)}_{t^\beta }} \right)\right|\ X^{(n)}_0 \right)\\
\ellongrightarrow {\cal D}\elleft(
\elleft.
\elleft( \elleft( \int_{1-t}^{1} {\cal X}^\alpha_j(v)\ d\nu_j^\alpha(v)\right)_{j=1,...,k},\ {\cal X}^\alpha (1)\right)
\right|\ {\cal X}^\alpha(1-t)
\right)
\end{multline}
as $n\tauo\infty$ for all $t\in [0,1]$. Changing $t$ into $t^{1/\beta}$ completes the proof of \eqref{convergence_equilibrium}.\\
{\bf Proof of the convergence \eqref{remark_conv_distrib_simpler_equilibirum} in Corollary \ref{rem:marginal}. }This follows the same pattern as the proof of \eqref{remark_conv_distrib_simpler_fast}, to which we refer here. More precisely, one verifies this time that, from \eqref{equilibrium_chi}, the analog of \eqref{LT_reverse_fast} in the Fast arrival case is here
\begin{multline}\ellambdabel{LT_reverse_equilibrium}
\mathbb{E}\elleft[ \exp\elleft(\sum_{j=1}^k s_j\int_0^t U_j(v)\ \nu_j^\alpha (dv) \right) \right]\\
= \mathbb{E}\elleft[\exp\elleft(\sum_{j=1}^k s_j\int_{1-t}^1 {\cal X}^\alpha_j(v)\ d\nu_j^\alpha(v) \right)\right],\quad s=(s_1,...,s_k)\in (-\infty, 0]^k,
\end{multline}
which, combined with \eqref{conv_equilibrium_final}, yields the convergence $Z^{(n)}\elleft(t^\beta\right)\stackrel{\cal D}{\ellongrightarrow} \elleft( \int_{0}^{t} U_j(v)\ \nu_j^\alpha (dv)\right)_{j=1,...k}$ as $n\tauo \infty$. Changing $t$ into $t^{1/\beta}$ and performing the change of variable $ v:=v^{1/\beta}=v^{1-\alpha} $, we obtain
\begin{equation}\ellambdabel{conv_distribution_reversed_equilibrium}
Z^{(n)}\elleft(t\right)\stackrel{\cal D}{\ellongrightarrow} \elleft( \int_{0}^{t} U_j(v^{1-\alpha})\ \tauilde{\nu}_j^\alpha(dv)\right)_{j=1,...,k},\ n\tauo\infty, \quad t\in [0,1],
\end{equation}
where $\{\tauilde{\nu}_j^\alpha(t),\ t\in [0,1] \}$, $j=1,...,k$, are the inhomogeneous independent Poisson processes given by $\tauilde{\nu}_j^\alpha(t)=\nu_j^\alpha(t^{1/\beta})=\nu_j^\alpha(t^{1-\alpha})$, i.e. Poisson processes with non constant intensity $\ellambdambda t^{-\alpha}$. Arguing, as in the Fast arrival case, the time changed Markov process $\{{\cal Y}(t):=U(t^{1-\alpha})=U(t^{1/\beta}),\ t\in [0,1]\} $ admits \eqref{generator_Y} as generator, hence \eqref{remark_conv_distrib_simpler_equilibirum} follows from \eqref{conv_distribution_reversed_equilibrium}.
\subsection{Proof of Theorem \ref{theo_slow_arrival}: Slow arriving customers.}\ellambdabel{sec:slow}
We now consider the case $\gamma<\alpha$. Section 4.2 of \cite{RW18} provide the first two joint moments of the $Z_j^{(n)}(t)$, $j=1,...,k$, with a particular discount factor $a \ge 0$ (recall the notation \eqref{discounted_Z(t)} in Section \ref{sec:model} for the discounted counterpart of the queueing process \eqref{def_Z_t}). Recalling that the rescaling implies that $\ellambdambda_n(P_n-I)=\ellambdambda(P-I)$, we get from \cite[Theorems 14 and 15 with $a=0$ discount factor]{RW18}, that those moments are given by
\begin{eqnarray}
M_j^{(n)}(t) &=& \ellambdambda_n e^{\ellambdambda t (P-I)}\int_0^t {\mathbb P} \elleft(L_j^{(n)}>v\right) e^{-\ellambdambda v (P-I)} \Delta_j P_n e^{\ellambdambda v (P-I)} dv, \ellambdabel{m1_sec:slow}\\
M_{jj}^{(n)}(t) &=& \ellambdambda_n e^{\ellambdambda t (P-I)}\int_0^t {\mathbb P}\elleft(L_j^{(n)}>v\right) e^{-\ellambdambda v (P-I)}
\Delta_j^2 P_n e^{\ellambdambda v (P-I)}\nonumber\\
& + & 2 {\mathbb P}\elleft(L_j^{(n)}>v\right) \Delta_j P_n M_{j}^{(n)}(v)dv,\ellambdabel{m2_jj_sec:slow}\\
M_{jj'}^{(n)}(t) &=& \ellambdambda_n e^{\ellambdambda t (P-I)}\int_0^t {\mathbb P}\elleft(L_j^{(n)}>v\right){\mathbb P}\elleft(L_{j'}^{(n)}>v\right) e^{-\ellambdambda v (P-I)}
\Delta_j\Delta_{j'} P_n e^{\ellambdambda v (P-I)} \nonumber\\
& + & {\mathbb P}\elleft(L_j^{(n)}>v\right) \Delta_j P_n M_{j'}^{(n)}(v)
+ {\mathbb P}\elleft(L_{j'}^{(n)}>v\right) \Delta_{j'} P_n M_{j}^{(n)}(v) dv, \ellambdabel{m2_jj'_sec:slow}
\end{eqnarray}
for all $t\ge 0$ and $j\neq j'$, $j$ and $j'$ in $\{1,..,k\}$. We first show \eqref{convergence_slow_M1}. Since $\ellambdambda_n=\ellambdambda n^\gamma$, multiplying \eqref{m1_sec:slow} by $n^{\alpha-\gamma}$ yields for $j=1,...,k$
\begin{equation}\ellambdabel{m1_sec:slow_proof}
n^{\alpha-\gamma} M_j^{(n)}(t)= \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t n^\alpha {\mathbb P} \elleft(L_j^{(n)}>v\right) e^{-\ellambdambda v (P-I)} \Delta_j P_n e^{\ellambdambda v (P-I)} dv .
\end{equation}
By definition of $L_j^{(n)}$ and the fat tail property of $L_j$:
$$n^\alpha {\mathbb P} \elleft(L_j^{(n)}>v\right)=n^\alpha {\mathbb P} \elleft(L_j>nv\right)\sigmam n^\alpha \frac{1}{(nv)^\alpha}=\frac{1}{v^\alpha},\quad v\in (0,t),\quad n\tauo \infty.$$
Now, since $\ellim_{n\tauo\infty}P_n=P$ and
\begin{multline}\ellambdabel{m1_sec_argument}
\sup_{n\in \mathbb{N}}n^\alpha {\mathbb P} \elleft(L_j^{(n)}>v\right)=\sup_{n\in \mathbb{N}}n^\alpha {\mathbb P} \elleft(L_j>nv\right)=\frac{\sup_{n\in \mathbb{N}}(nv)^\alpha {\mathbb P} \elleft(L_j>nv\right)}{v^\alpha}\\
\elle \frac{\sup_{u\ge 0}u^\alpha {\mathbb P} \elleft(L_j>u\right)}{v^\alpha},\quad v\in(0,1),
\end{multline}
the dominated convergence enables us to let $n\tauo\infty$ in \eqref{m1_sec:slow_proof} to get \eqref{convergence_slow_M1}. We now turn to \eqref{convergence_slow_M20}. Multiplying \eqref{m2_jj_sec:slow} by $n^{\alpha-\gamma}$ yields
\begin{multline}\ellambdabel{m2_jj_sec:slow_proof}
n^{\alpha-\gamma} M_{jj}^{(n)}(t) = \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t n^\alpha {\mathbb P}\elleft(L_j^{(n)}>v\right) e^{-\ellambdambda v (P-I)}
\Delta_j^2 P_n e^{\ellambdambda v (P-I)}dv\\
+ 2 \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t n^\alpha {\mathbb P}\elleft(L_j^{(n)}>v\right) \Delta_j P_n M_{j}^{(n)}(v)dv.
\end{multline}
Since \eqref{convergence_slow_M1} in particular implies that $\ellim_{n\tauo \infty}M_{j}^{(n)}(v)=0$ for all $v\in(0,1)$, and thanks to the upper bound \eqref{m1_sec_argument}, a dominated convergence argument entails that the second integral on the right hand side of \eqref{m2_jj_sec:slow_proof} tends to $0$ as $n\tauo\infty$. We also conclude by a dominated convergence argument that the first integral on the right hand side of \eqref{m2_jj_sec:slow_proof} tends to the right hand side of \eqref{convergence_slow_M20}, and we are done. As to \eqref{m2_jj'_sec:slow}, we have for $j\neq j'$
\begin{multline}\ellambdabel{m2_jj'_sec:slow_proof}
n^{\alpha-\gamma}M_{jj'}^{(n)}(t) = \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t n^\alpha {\mathbb P}\elleft(L_j^{(n)}>v\right){\mathbb P}\elleft(L_{j'}^{(n)}>v\right) e^{-\ellambdambda v (P-I)}
\Delta_j\Delta_{j'} P_n e^{\ellambdambda v (P-I)} dv\\
+ \ellambdambda e^{\ellambdambda t (P-I)}\int_0^t \elleft\{n^\alpha {\mathbb P}\elleft(L_j^{(n)}>v\right) \Delta_j P_n M_{j'}^{(n)}(v)
+ n^\alpha{\mathbb P}\elleft(L_{j'}^{(n)}>v\right) \Delta_{j'} P_n M_{j}^{(n)}(v)\right\} dv.
\end{multline}
Similarly to the second integral on the right hand side of \eqref{m2_jj_sec:slow_proof}, we show that the second integral on the right hand side of \eqref{m2_jj'_sec:slow_proof} converges to $0$ as $n\tauo\infty$. As to the first integral, the fact that ${\mathbb P}\elleft(L_{j'}^{(n)}>v\right)={\mathbb P}\elleft(L_{j'}>nv\right)\ellongrightarrow 0$ as $n\tauo\infty$, combined with the upper bound \eqref{m1_sec_argument}, yields by the dominated convergence theorem that it tends to $0$ as $n\tauo\infty$, achieving the proof of \eqref{m2_jj'_sec:slow} and of the theorem.
\section{A remark on the computation of the limiting joint Laplace transform when $\alpha\in \mathbb{Q}$}\ellambdabel{sec:remark_compute}
We identified in Theorem \ref{theo_regimes} the different limiting regimes when $\gamma$ is larger or equal to $\alpha$ by obtaining the corresponding limiting joint Laplace transform $\chi(s,t)$ in each case. Even though the distributional limits \eqref{convergence_fast} and \eqref{convergence_equilibrium} involve simple processes $\{{\cal X}^\alpha (t),\ t\in[0,1] \}$ and $\{\nu_j^\alpha(t),\ t\ge 0 \}$, $j=1,...,k$, it turns out that the Laplace transforms $\chi(s,t)$, which are solutions to the differential equations \eqref{Poisson_ODE_fast} and \eqref{Poisson_ODE_equilibrium}, are in general not explicit in the fast or equilibrium arriving cases. We suggest to show that things are much simpler when $\alpha\in(0,1)$ is rational, say of the form $$\alpha=1-p/q$$ for some $p$ and $q\in\mathbb{N}^*$, with $p<q$. The idea here is quite simple and standard, and consists in expanding a transformation of the solution $t\in[0,1]\mapsto \chi(s,t)\in \mathbb{R}^{{\cal S}\tauimes {\cal S}}$ into a power series with matrix coefficients, as explained in \cite[Section 1.1]{Balser00}. Let us focus on the fast arrival case in Section \ref{sec:fast}, although the method is of course applicable to the equilibrium case, and let us put $\check\chi(s,t):= \chi(s,t^p)$, $t\in[0,1]$. In that case, we deduce from \eqref{Poisson_ODE_fast} that $t\in[0,1]\mapsto \check\chi(s,t)$ verifies the matrix differential equation
$$\elleft\{
\begin{array}{rcl}
\partial_t \check\chi(s,t) &=& \elleft[(p+q) t^{q}\ellambdambda (P-I) + pt^{p-1}\beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell \right]\check\chi(s,t) ,\quad t\in [0,1],\\
&=& [Q_1 t^q + Q_2(s) t^{p-1}]\check\chi(s,t),\\
\chi(s,0)&=& I.
\end{array}
\right.
$$
where $Q_1:=(p+q)\ellambdambda (P-I)$ and $Q_2(s):=p\beta \ellambdambda \sum_{\ell =1}^k s_\ell\Delta_\ell$, $s=(s_1,...,s_k)$. It is quite simple to check that $\check\chi(s,t)$ can then be expanded as
\begin{equation}\ellambdabel{expansion_chi}
\check\chi(s,t)=\sum_{j=0}^\infty U_j(s) t^j,\quad t\in [0,1],
\end{equation}
where the sequence of matrices $ (U_j(s))_{j\in\mathbb{N}}$ is defined from \cite[Relation (1.4)]{Balser00} by $U_0(s)=I$ and
\begin{equation}\ellambdabel{rel_U_j}
U_j(s)=
\elleft\{
\begin{array}{c l}
0, & 1\elle j <p,\\
Q_2(s)U_{j-p}(s)/j, & p\elle j<q+1,\\
\elleft[ Q_2(s)U_{j-p}(s)+Q_1 U_{j-q-1}(s) \right]/j, & j\ge q+1 ,
\end{array}
\right.
\end{equation}
and that \eqref{expansion_chi} converges for all $t$, as proved in \cite[Lemma 1 p.2]{Balser00}. The final solution is then expressed in that case as
$$
\chi(s,t)=\check\chi(s,t^{1/p})=\sum_{j=0}^\infty U_j(s) t^{j/p},\quad t\in [0,1].
$$
The $U_j(s)$'s, $j\in\mathbb{N}$, being simply expressed with the simple linear recurrence \eqref{rel_U_j}, this expansion for $\chi(s,t)$ is then easy to handle as it can be e.g. approximated by truncation.
\end{document} |
\begin{document}
\title[Nondivergence form degenerate parabolic equations]{Nondivergence form degenerate linear parabolic equations on the upper half space}
\author[H. Dong]{Hongjie Dong}
\address[H. Dong]{Division of Applied Mathematics, Brown University, 182 George Street, Providence RI 02912, USA}
\email{hongjie\[email protected]}
\author[T. Phan]{Tuoc Phan}
\address[T. Phan]{Department of Mathematics, University of Tennessee, 227 Ayres Hall,
1403 Circle Drive, Knoxville, TN 37996-1320, USA}
\email{[email protected]}
\author[H. V. Tran]{Hung Vinh Tran}
\address[H. V. Tran]{Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall
480 Lincoln Drive
Madison, WI 53706, USA}
\email{[email protected]}
\thanks{
H. Dong is partially supported by the NSF under agreement DMS-2055244 and the Simons Fellows Award 007638.
H. Tran is supported in part by NSF CAREER grant DMS-1843320 and a Vilas Faculty Early-Career Investigator Award.
}
\subjclass[2020]{35K65, 35K67, 35K20, 35D30}
\keywords{Degenerate linear parabolic equations; degenerate viscous Hamilton-Jacobi equations; nondivergence form; boundary regularity estimates; existence and uniqueness; weighted Sobolev spaces}
\begin{abstract}
We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times \mathbb{R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in \mathbb{R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given.
The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded positive definite matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain.
The divergence form equations in this setting were studied in \cite{DPT21}.
Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces.
Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations.
\end{abstract}
\dedicatory{Dedicated to Professor Mikhail Safonov on the occasion of his $70^{\text{th}}$ birthday}
\maketitle
\section{Introduction and main results}
\subsection{Settings}
Let $T\in (-\infty,\infty]$ and $\Omega_T=(-\infty,T)\times \mathbb{R}^d_+$.
We study the following degenerate parabolic equation in nondivergence form
\begin{equation}\label{eq:main}
\begin{cases}
\sL u=\mu(x_d) f \quad &\text{ in } \Omega_T,\\
u=0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+,
\end{cases}
\end{equation}
where $u: \Omega_T \rightarrow \mathbb{R}$ is an unknown solution, $f: \Omega_T \rightarrow \mathbb{R}$ is a given measurable forcing term, and
\begin{equation} \label{L-def}
\sL u = a_0(z) u_t+\lambda c_0(z)u-\mu(x_d) a_{ij}(z)D_i D_j u.
\end{equation}
Here in \eqref{L-def}, $\lambda\ge 0$ is a constant, $z=(t,x) \in \Omega_T$ with $x = (x', x_d) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$, $D_i$ denotes the partial derivative with respect to $x_i$, and $a_0, c_0: \Omega_T \rightarrow \mathbb{R}$ and $\mu: \mathbb{R}_+ \rightarrow \mathbb{R}$ are measurable and satisfy
\begin{equation} \label{con:mu}
a_0(z), \ c_0(z), \ \frac{\mu(x_d)}{x_d^\alpha} \in[\nu,\nu^{-1}], \quad \forall \ x_d \in \mathbb{R}_+, \quad \forall \ z \in \Omega_T,
\end{equation}
for some given $\alpha\in (0,2)$ and $\nu \in (0,1)$. Moreover, $(a_{ij}): \Omega_T \rightarrow \mathbb{R}^{d\times d}$ are measurable and satisfy the uniform ellipticity and boundedness conditions
\begin{equation} \label{con:ellipticity}
\nu|\xi|^2 \leq a_{ij}(z) \xi_i \xi_j, \quad |a_{ij}(z)| \leq \nu^{-1}, \quad \forall \ z \in \Omega_T,
\end{equation}
for all $\xi = (\xi_1, \xi_2, \ldots, \xi_d) \in \mathbb{R}^d$.
We observe that due to \eqref{con:mu} and \eqref{con:ellipticity}, the diffusion coefficients in the PDE in \eqref{eq:main} are degenerate when $x_d \rightarrow 0^+$, and singular when $x_d \rightarrow \infty$. We also note that the PDE in \eqref{eq:main} can be written in the form
\[
[a_0(z) u_t + \lambda c_0(z) u]/\mu(x_d) - a_{ij}(z) D_iD_j u = f \quad \text{in} \quad \Omega_T,
\]
in which the singularity and degeneracy appear in the coefficients of the terms involving $u_t$ and $u$. In the special case when $a_0 = c_0 =1, \mu(x_d) = x_d^\alpha$, and $(a_{ij})$ is an identity matrix, the equation \eqref{eq:main} is reduced to
\begin{equation} \label{simplest-eqn}
\left\{
\begin{array}{cccl}
u_t + \lambda u - x_d^\alpha \Delta u & = & x_d^\alpha f & \quad \text{in} \quad \Omega_T, \\
u & =& 0 & \quad \text{on} \quad (-\infty, T) \times \partial \mathbb{R}^d_+,
\end{array} \right.
\end{equation}
in which the results obtained in this paper are still new.
The theme of this paper is to study the existence, uniqueness, and regularity estimates for solutions to \eqref{eq:main}. To demonstrate our results, let us state the following theorem which gives prototypical estimates of our results in a special weighted Lebesgue space $L_p(\Omega, x_d^\gamma\, dz)$ with the power weight $x_d^\gamma$ and norm
\[
\|f\|_{L_p(\Omega, x_d^\gamma dz)} = \left( \int_{\Omega_T} |f(t,x)|^p x_d^\gamma\, dx dt \right)^{1/p}.
\]
{For any measurable function $f$ and $s \in \mathbb{R}$, we define the multiplicative operator $(\mathbf{M}^s f)(\cdot)=x_d^s f(\cdot)$.}
\begin{theorem} \label{thm:demo}
Let $\alpha \in (0,2), \lambda >0$, $p\in (1,\infty)$, and $\gamma \in \big(p(\alpha-1)_+-1,2p-1\big)$. Then, for every $f \in L_p(\Omega, x_d^\gamma\, dz)$, there exists a unique strong solution $u$ to \eqref{simplest-eqn}, which satisfies
\begin{align} \label{show-est-1}
\|\mathbf{M}^{-\alpha}u_t\|_{L_p}+\lambda\|\mathbf{M}^{-\alpha}u\|_{L_p}
+\|D^2 u\|_{L_p}\le N\|f\|_{L_p};
\end{align}
and for $\gamma\in (\alpha p/2-1,2p-1)$,
\begin{equation}\label{show-est-2}
\lambda^{1/2}\|\mathbf{M}^{-\alpha/2}Du\|_{L_p}
\le N\|f\|_{L_p},
\end{equation}
where $\|\cdot\|_{L_p} = \|\cdot \|_{L_p(\Omega, x_d^\gamma dz)}$
and $N=N(d,\nu,\alpha, \gamma, p)>0$.
\end{theorem}
\noindent
See Corollary \ref{cor1} and Theorem \ref{thm:xd} for more general results.
We note that the ranges of $\gamma$ in \eqref{show-est-1}--\eqref{show-est-2} are optimal as pointed out in Remarks \ref{remark-1-range}--\ref{remark-2-range} below.
In fact, in this paper, a much more general result in weighted mixed-norm spaces is established in Theorem \ref{main-thrm}.
As an application, we obtain a regularity result for solutions to degenerate viscous Hamilton-Jacobi equations in Theorem \ref{example-thrm}.
To the best of our knowledge, our main results (Theorems \ref{thm:demo}, \ref{main-thrm}, \ref{thm:xd}, Corollary \ref{cor1}, and Theorem \ref{example-thrm}) appear for the first time in the literature.
\subsection{Relevant literature}
The literature on regularity theory for solutions to degenerate elliptic and parabolic equations is extremely rich, and we only describe results related to \eqref{eq:main}.
The divergence form of \eqref{eq:main} was studied by us in \cite{DPT21} with motivation from the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations of the form
\begin{equation}\label{eq:HJ-intro}
u_t+\lambda u-\mu(x_d) \Delta u=H(z,Du) \qquad \text{ in } \Omega_T.
\end{equation}
Here, $H:\Omega_T \times \mathbb{R}^d \to \mathbb{R}$ is a given Hamiltonian.
Under some appropriate conditions on $H$, we obtain a regularity {and solvability} result for
\eqref{eq:HJ-intro} in Theorem \ref{example-thrm}.
Another class of divergence form equations, which is closely related to that in \cite{DPT21}, was analyzed recently in \cite{JinXiong2}
when $\alpha<1$. When $\alpha=2$ and $ d=1$, a specific version of \eqref{eq:HJ-intro} gives the well-known Black-Scholes-Merton PDE that appears in mathematical finance. The analysis for \eqref{eq:main} when $\alpha\geq 2$ is completely open.
A similar equation to \eqref{eq:main}, \eqref{simplest-eqn}, and \eqref{eq:HJ-intro}
\[
u_t+\lambda u-\beta D_du - x_d \Delta u = f \qquad \text{ in } \Omega_T
\]
with an additional structural condition $\beta>0$, an important prototype equation in the study of porous media equations and parabolic Heston equation, was studied extensively in the literature (see \cite{DaHa, FePo, Koch, JinXiong1, JinXiong2} and the references therein).
We stress that we do not require this structural condition in the analysis of \eqref{eq:main} and \eqref{eq:HJ-intro}, and thus, our analysis is rather different from those in \cite{DaHa, FePo, Koch}.
We note that similar results on the wellposedness and regularity estimates in weighted Sobolev spaces for a different class of equations with singular-degenerate coefficients were established in a series of papers \cite{DP-20, DP-21, DP-JFA, DP-AMS}.
There, the weights of singular/degenerate coefficients of $u_t$ and $D^2u$ appear in a balanced way, which plays a crucial role in the analysis and functional space settings.
If this balance is lost, then Harnack's inequalities were proved in \cite{Chi-Se-1, Chi-Se-2} to be false in certain cases. However, with an explicit weight $x_d^\alpha$ as in our setting, it is not known if some version of Harnack's inequalities and H\"{o}lder estimates of the Krylov-Safonov type as in \cite{K-S} still hold for in \eqref{eq:main}.
Of course, \eqref{eq:main} does not have this balance structure, and our analysis is quite different from those in \cite{DP-20, DP-21, DP-JFA, DP-AMS}.
Finally, we emphasize again that the literature on equations with singular-degenerate coefficients is vast. Below, let us give some references on other closely related results.
The H\"{o}lder regularity
for solutions to elliptic equations with singular and degenerate coefficients, which are in the $A_2$-Muckenhoupt {class}, were proved in the classical papers \cite{Fabes, FKS}. See also the books \cite{Fichera, OR},
the papers \cite{KimLee-Yun, Sire-1, Sire-2}, and the references therein for other results on the wellposedness, H\"{o}lder, and Schauder regularity estimates for various classes of degenerate equations. Note also that the Sobolev regularity theory version of the
results in \cite{Fabes, FKS} was developed and proved in \cite{Men-Phan}. In addition, we would like to point out that equations with degenerate coefficients also appear naturally in geometric analysis \cite{Lin, WWYZ}, in which H\"{o}lder and Schauder estimates for solutions were proved.
\subsection{Main ideas and approaches}
The main ideas of this paper are along the lines with those in \cite{DPT21}.
However, at the technical level, the proofs of our main results are quite different from those in \cite{DPT21}. More precisely, instead of the $L_2$-estimates as in \cite{DPT21}, the starting point in this paper is the weighted $L_p$-result in Lemma \ref{l-p-sol-lem} which is based on the weighted $L_p$ for divergence form equations established in \cite{DPT21}, an idea introduced by Krylov \cite{Kr99}, together with a suitable scaling. Moreover, while the proofs in \cite{DPT21} use the Lebesgue measure as an underlying measure, in this paper we make use of more general underlying measure $\mu_1 (dz) = x_d^{\gamma_1}$ with an appropriate parameter $\gamma_1$. In particular, this allows us to obtain an optimal range of exponents for power weights in Corollary \ref{cor1}. See Remarks \ref{remark-1-range} - \ref{remark-2-range}. Several new H\"{o}lder
estimates for higher order derivatives of solutions
to a class of degenerate homogeneous equations are proved in Subsections \ref{subsec:boundary}--\eqref{subsec:int}. The results and techniques developed in these subsections
{might be} of independent interest.
\subsection*{Organization of the paper}
The paper is organized as follows.
In Section \ref{sec:2}, we introduce various function spaces, assumptions, and then state our main results.
The filtration of partitions, a quasi-metric, the weighted mixed-norm Fefferman-Stein theorem and Hardy-Littlewood theorem are recalled in Section \ref{Feffer}.
Then, in Section \ref{sec:3}, we consider \eqref{eq:main} in the case when the coefficients in \eqref{eq:main} only depend on the $x_d$ variable.
A special version of Theorem \ref{main-thrm}, Theorem \ref{thm:xd}, will be stated and proved in this section.
The proofs of Theorem \ref{main-thrm} and Corollary \ref{cor1} are given in Section \ref{sec:4}.
Finally, we study the degenerate viscous Hamilton-Jacobi equation \eqref{eq:HJ-intro} in Section \ref{sec:5}.
\section{Function spaces, parabolic cylinders, mean oscillations, and main results}\label{sec:2}
\subsection{Function spaces}
Fix $p,q \in [1, \infty)$, $-\infty\le S<T\le +\infty$, and a domain $\cD \subset \mathbb{R}^d_+$. Denote by $L_p((S,T)\times \cD)$ the usual Lebesgue space consisting of measurable functions $u$ on $(S,T)\times \cD$ such that
\[
\|u\|_{L_p( (S,T)\times \cD)}= \left( \int_{(S,T)\times \cD} |u(t,x)|^p\, dxdt \right)^{1/p} <\infty.
\]
For a given weight $\omega$ on $(S,T)\times \cD$, let $L_{p}((S,T)\times \cD,\omega)$ be the weighted Lebesgue space on $(S,T)\times \cD$ equipped with the norm
\begin{equation*}
\|u\|_{L_{p}((S,T)\times \cD, \omega)}=\left(\int_{(S,T)\times \cD} |u(t,x)|^p \omega (t,x)\, dx dt\right)^{1/p} < \infty.
\end{equation*}
For the weights $\omega_0=\omega_0(t)$, $\omega_1=\omega_1(x)$, and a measure $\sigma$ on $\cD$, set $\omega(t,x)=\omega_0(t)\omega_1(x)$ and define $L_{q,p}((S,T)\times \cD,\omega d\sigma)$ to be the weighted and mixed-norm Lebesgue space on $(S,T)\times \cD$ equipped with the norm
\[
\|u\|_{L_{q,p}((S,T)\times \cD, \omega d\sigma)}=\left(\int_S^T \left(\int_{\cD} |u(t,x)|^p \omega_1(x)\, \sigma(dx)\right)^{q/p} \omega_0(t)\,dt \right)^{1/q} < \infty.
\]
\subsubsection{Function spaces for nondivergence form equations}
Consider $\alpha>0$.
We define the solution spaces as follows.
Firstly, define
\[
W_{p}^{1,2}((S,T)\times \cD, \omega)
=\left\{u \,:\, \mathbf{M}^{-\alpha} u, \mathbf{M}^{-\alpha} u_t, D^2u \in L_p((S,T) \times \cD,\omega)\right\},
\]
where, for $u\in W_{p}^{1,2}((S,T)\times \cD, \omega)$,
\begin{multline*}
\|u\|_{W^{1,2}_p((S,T)\times \cD,\omega)}\\
=\| \mathbf{M}^{-\alpha} u\|_{L_p((S,T)\times \cD,\omega)}+\| \mathbf{M}^{-\alpha} u_t\|_{L_p((S,T)\times \cD,\omega)}+\|D^2u\|_{L_p((S,T)\times \cD,\omega)}.
\end{multline*}
Let $\sW^{1,2}_p((S,T)\times \cD,\omega)$ be the closure in $W^{1,2}_p((S,T)\times \cD,\omega)$ of all compactly supported functions in $C^\infty((S,T)\times \overline{\cD})$ vanishing near $\overline{\cD} \cap \{x_d=0\}$ if $\overline{\cD} \cap \{x_d=0\}$ is not empty.
The space $\sW^{1,2}_p((S,T)\times \cD,\omega)$ is equipped with the same norm $\|\cdot\|_{\sW^{1,2}_p((S,T)\times \cD,\omega)}=\|\cdot\|_{W^{1,2}_p((S,T)\times \cD,\omega)}$.
When there is no time dependence, we write these two spaces as $W^2_p(\cD,\omega)$ and $\sW^2_p(\cD,\omega)$, respectively.
Next, denote by
\[
\begin{split}
& W_{q,p}^{1,2}((S,T)\times \cD, \omega\, d\sigma)\\
& \qquad =\left\{u \,:\, \mathbf{M}^{-\alpha} u, \mathbf{M}^{-\alpha} u_t,D^2u \in L_{q,p}((S,T) \times \cD,\omega\, d\sigma)\right\},
\end{split}
\]
which is equipped with the norm
\begin{multline*}
\|u\|_{W^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)}
=\| \mathbf{M}^{-\alpha} u\|_{L_{q,p}((S,T)\times \cD,\omega\, d\sigma)}\\
+\| \mathbf{M}^{-\alpha} u_t\|_{L_{q,p}((S,T)\times \cD,\omega\, d\sigma)}+\|D^2u\|_{L_{q,p}((S,T)\times \cD,\omega\, d\sigma)}.
\end{multline*}
Let $\sW^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)$ be the closure in $W^{1,2}_{q,p}((S,T)\times \cD,\omega d\sigma)$ of all compactly supported functions in $C^\infty((S,T)\times \overline{\cD})$ vanishing near $\overline{\cD} \cap \{x_d=0\}$ if $\overline{\cD} \cap \{x_d=0\}$ is not empty.
The space $\sW^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)$ is equipped with the same norm $\|\cdot\|_{\sW^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)}=\|\cdot\|_{W^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)}$.
\subsubsection{Function spaces for divergence form equations}
We also need function spaces for divergence form equations in this paper, which are taken from \cite{DPT21}.
Set
$$
W^1_p((S,T)\times \cD, \omega)=\left\{u\,:\, \mathbf{M}^{-\alpha/2} u, Du\in L_p((S,T)\times \cD, \omega)\right\},
$$
which is equipped with the norm
$$
\|u\|_{W^1_p((S,T)\times \cD,\omega)}=\| \mathbf{M}^{-\alpha/2} u\|_{L_p((S,T)\times \cD,\omega)}+\|Du\|_{L_p((S,T)\times \cD,\omega)}.
$$
We denote by $\sW^1_p((S,T)\times \cD,\omega)$ the closure in $W^1_p((S,T)\times \cD,\omega)$ of all compactly supported functions in $C^\infty((S,T)\times \overline{\cD})$ vanishing near $\overline{\cD} \cap \{x_d=0\}$ if $\overline{\cD} \cap \{x_d=0\}$ is not empty.
The space $\sW^1_p((S,T)\times \cD,\omega)$ is equipped with the same norm $\|\cdot\|_{\sW^1_p((S,T)\times \cD,\omega)}=\|\cdot\|_{W^1_p((S,T)\times \cD,\omega)}$.
Set
\[
\begin{split}
& \mathbb{H}_{p}^{-1}( (S,T)\times \cD, \omega) \\
& =\big\{u\,:\, u = \mu(x_d) D_iF_i +f_1+f_2, \ \ \text{where}\ \mathbf{M}^{1-\alpha} f_1,\mathbf{M}^{-\alpha/2}f_2\in L_{p}( (S,T)\times \cD, \omega)\\
& \qquad\text{and }
F= (F_1,\ldots,F_d) \in L_{p}((S,T)\times \cD, \omega)^{d}\big\},
\end{split}
\]
equipped with the norm
\begin{align*}
&\|u\|_{\mathbb{H}_{p}^{-1}((S,T)\times \cD, \omega)} \\
&=\inf\big\{\|F\|_{L_{p}((S,T)\times \cD, \omega)}
+\|| \mathbf{M}^{1-\alpha} f_1|+|\mathbf{M}^{-\alpha/2}f_2|\|_{L_{p}((S,T)\times \cD, \omega)}\,:\\
&\qquad u= \mu(x_d) D_iF_i +f_1+f_2\big\}.
\end{align*}
Define
\[
\cH_{p}^1((S,T)\times \cD, \omega)
=\big\{u \,:\, u \in \sW^1_p((S,T) \times \cD,\omega)),
u_t\in \mathbb{H}_{p}^{-1}( (S,T)\times \cD, \omega)\big\},
\]
where, for $u\in \cH_{p}^1((S,T)\times \cD, \omega)$,
\begin{align*}
\|u\|_{\cH_{p}^1((S,T)\times \cD, \omega)} &= \|\mathbf{M}^{-\alpha/2} u\|_{L_{p}((S,T)\times \cD, \omega)} + \|Du\|_{L_{p}((S,T)\times \cD, \omega)} \\
& \qquad +\|u_t\|_{\mathbb{H}_{p}^{-1}((S,T)\times \cD, \omega)}.
\end{align*}
\subsection{Parabolic cylinders}
We use the same setup as that in \cite{DPT21}.
For $x_0 = (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$ and $\rho>0$, denote by $B_\rho(x_0)$ the usual ball with center $x_0$ radius $\rho$ in $\mathbb{R}^d$, $B_\rho'(x_0')$ the ball center $x_0'$ radius $\rho$ in $\mathbb{R}^{d-1}$, and
\[
B_\rho^+(x_0) = B_\rho(x_0) \cap \mathbb{R}^d_+.
\]
We note that \eqref{eq:main} is invariant under the scaling
\begin{equation} \label{scaling}
(t,x) \mapsto (s^{2-\alpha} t, sx), \qquad s > 0.
\end{equation}
For $x_d \sim x_{0d} \gg 1$, $a_{ij} = \delta_{ij}$, and $\lambda =f=0$, then \eqref{eq:main} behaves like a heat equation
\[
u_t -x_{0d}^{\alpha} \Delta u = 0,
\]
which can be reduced to the heat equation with unit heat constant under the scaling
\[
(t,x) \mapsto (s^{2-\alpha} t, s^{1-\alpha/2} x_{0d}^{-\alpha/2}x), \quad s>0.
\]
It is thus natural to use the following parabolic cylinders in $\Omega_T$ in this paper.
For $z_0 = (t_0, x_0) \in (-\infty, T) \times \mathbb{R}^d_+$ with $x_0= (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$ and $\rho>0$, set
\begin{equation} \label{def:Q}
\begin{split}
& Q_{\rho}(z_0) = (t_0 - \rho^{2-\alpha}, t_0) \times B_{r(\rho, x_{0d})} (x_0), \quad \\
&Q_{\rho}^+(z_0) = Q_\rho(z_0) \cap \{x_d>0\},
\end{split}
\end{equation}
where
\begin{equation} \label{def:r}
r(\rho,x_{0d}) = \max\{\rho, x_{0d}\}^{\alpha/2} \rho^{1-\alpha/2}.
\end{equation}
Of course, $Q_{\rho}(z_0) = Q_{\rho}^+(z_0) \subset (-\infty, T) \times \mathbb{R}^d_+$ for $\rho \in (0,x_{0d})$.
Finally, for $z' = (t, x') \in \mathbb{R} \times \mathbb{R}^{d-1}$, we write
\[
Q_{\rho}'(z') = (t-\rho^{2-\alpha}, t_0) \times B_{\rho}'(x').
\]
\subsection{Mean oscillations and main results} \label{main-result-sect} Throughout the paper, for a locally integrable function $f$, a locally finite measure $\omega$, and a domain $Q\subset \mathbb{R}^{d+1}$, we write
\begin{equation} \label{everage-def}
(f)_{Q} = \fint_{Q} f(s,y)\, dyds, \qquad (f)_{Q,\omega} = \frac{1}{\omega(Q)}\int_{Q} f(s,y) \,\omega(dyds).
\end{equation}
Also, for a number $\gamma_1 \in (-1, \infty)$ to be determined, we define
\[
\mu_1(dz) = x_d^{\gamma_1}\, dxdt.
\]
We impose the following assumption on the partial mean oscillations of the coefficients $(a_{ij})$, $a_0$, and $c_0$.
\begin{assumption}[$\rho_0,\gamma_1, \delta$] \label{assumption:osc} For every $\rho \in (0, \rho_0)$ and $z_0= (z_0', z_{0d}) \in \overline{\Omega}_T$, there exist $[a_{ij}]_{\rho, z'}, [a_{0 }]_{\rho, z'}, [c_{0 }]_{\rho, z'}: ((x_{d} -r(\rho, x_d))_+, x_d + r(\rho, x_d)) \rightarrow \mathbb{R}$ such that \eqref{con:mu}--\eqref{con:ellipticity} hold on $((x_{d} -r(\rho, x_d))_+, x_d + r(\rho, x_d))$ with $[a_{ij}]_{\rho, z'}$, $[a_{0 }]_{\rho, z'}$, $[c_{0 }]_{\rho, z'}$ in place of $(a_{ij})$, $a_0$, $c_0$, respectively, and
\begin{align*}
a_\rho^{\#}(z_0):= & \max_{1 \leq i, j \leq d}\fint_{Q_\rho^+(z_0)} | a_{ij}(z) -[a_{ij}]_{\rho,z'}(x_d)|\, \mu_1(dz) \\
& \qquad + \fint_{Q_\rho^+(z)} | a_{0}(z) -[a_{0}]_{\rho,z'}(x_d)|\, \mu_1(dz) \\
& \qquad + \fint_{Q_\rho^+(z)} | c_{0}(z) -[c_{0}]_{\rho,z'}(x_d)|\, \mu_1(dz) < \delta.
\end{align*}
\end{assumption}
\noindent
We note that the un-weighted partial mean oscillation was introduced in \cite{Kim-Krylov} to study a class of elliptic equations with uniformly elliptic and bounded coefficients (i.e., $\gamma_1=\alpha=0$). Note also that by dividing the equation \eqref{eq:main} by $a_{dd}$ and adjusting $\nu$, we can assume without loss of generality throughout the paper that
\begin{equation} \label{add-assumption}
a_{dd} \equiv 1.
\end{equation}
The theorem below is the first main result of our paper, in which the definition of the $A_p$ Muckenhoupt class of weights can be found in Definition \ref{Def-Muck-wei} below.
\begin{theorem} \label{main-thrm} Let $T \in (-\infty, \infty]$, $\nu \in (0,1)$, $p, q, K \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$ for $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$. Then, there exist $\delta = \delta(d, \nu, p, q, K, \alpha, \gamma_1)>0$ sufficiently small and $\lambda_0 = \lambda_0(d, \nu, p, q, K, \alpha, \gamma_1)>0$ sufficiently large such that the following assertion holds. Suppose that \eqref{con:mu}, \eqref{con:ellipticity}, and \eqref{add-assumption} are satisfied, $\omega_0 \in A_q(\mathbb{R})$, $\omega_1 \in A_p(\mathbb{R}^d_+, \mu_1)$ with
\[
[\omega_0]_{A_q(\mathbb{R})} \leq K \quad \text{and} \quad [\omega_1]_{A_p(\mathbb{R}^d_+, \mu_1)} \leq K, \quad \text{where} \,\, \mu_1(dz) = x_d^{\gamma_1} dxdt.
\]
Suppose also that Assumption \ref{assumption:osc} $(\rho_0, \gamma_1,\delta)$ holds for some $\rho_0>0$. Then, for any function $f \in L_{q, p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$ and $\lambda \geq \lambda_0 \rho_0^{-(2-\alpha)}$, there exists a strong solution $u{\in \sW^{1,2}_{q, p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)}$ to the equation \eqref{eq:main}, which satisfies
\begin{equation} \label{main-est-1}
\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \| f\|_{L_{q,p}},
\end{equation}
where $\omega(t, x) = \omega_0(t) \omega_1(x)$ for $(t,x) \in \Omega_T$, $L_{q,p} = L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega \, d\mu_1)$,
and $N = N(d, \nu, p, q, \alpha, \gamma_1)>0$. Moreover, if $\beta_0 \in {(\alpha-1}, \alpha/2]$, then it also holds that
\begin{equation} \label{main-est-2}
\begin{split}
& \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \| D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} \\
& \leq N \|f\|_{L_{q,p}}.
\end{split}
\end{equation}
\end{theorem}
The following is an important corollary of Theorem \ref{main-thrm} in which $\omega_1$ is a power weight of the $x_d$ variable and $\beta_0$ and $\gamma_1$ are specifically chosen.
\begin{corollary} \label{cor1} Let $T \in (-\infty, \infty]$, $\nu \in (0,1)$, $p, q \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma \in (p(\alpha-1)_+ -1, 2p-1)$. Then, there exist $\delta = \delta(d, \nu, p, q, \alpha, \gamma)>0$ sufficiently small and $\lambda_0 = \lambda_0(d, \nu, p, q, \alpha, \gamma)>0$ sufficiently large such that the following assertion holds. Suppose that \eqref{con:mu}, \eqref{con:ellipticity} hold and suppose also that Assumption \ref{assumption:osc} $(\rho_0, 1-(\alpha-1)_+, \delta)$ holds for some $\rho_0>0$. Then, for any $f \in L_{q, p}(\Omega_T, x_d^{\gamma} dz)$ and $\lambda \geq \lambda_0 \rho_0^{-(2-\alpha)}$, there exists a strong solution $u \in \sW^{1,2}_{q, p}(\Omega_T, x_d^\gamma\, dz)$ to the equation \eqref{eq:main}, which satisfies
\begin{equation} \label{cor-est-1}
\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \|f\|_{L_{q,p}},
\end{equation}
where $L_{q,p} = L_{q,p}(\Omega_T, x_d^\gamma dz)$ and $N = N(d, \nu, p, q, \alpha, \gamma)>0$. Additionally, if Assumption \ref{assumption:osc} $(\rho_0, 1-\alpha/2, \delta)$ also holds and $\gamma \in (\alpha p/2 -1, 2p-1)$, then we have
\begin{equation} \label{cor-est-2}
\begin{split}
&\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} \\
& \leq N \|f\|_{L_{q,p}}.
\end{split}
\end{equation}
\end{corollary}
\begin{remark}
By viewing solutions to elliptic equations as stationary solutions to parabolic equations, from Theorem \ref{main-thrm} and Corollary \ref{cor1}, we derive the corresponding results for elliptic equations. Furthermore, it follows from Corollary \ref{cor1} and the weighted Morrey embedding (see, for instance, \cite[Theorem 5.3]{RO}), we obtain the $C^{1,\alpha}$ regularity of solutions to the corresponding elliptic equations when $p>d+\gamma$.
\end{remark}
In the remarks below, we give examples showing that the ranges of $\gamma$ in \eqref{show-est-1}--\eqref{show-est-2} as well as \eqref{cor-est-1}--\eqref{cor-est-2} are optimal.
We note that the range of $\gamma$ for the estimate of $Du$ in \eqref{show-est-2}, \eqref{main-est-2}, and \eqref{cor-est-2} is smaller than that for $u, u_t, D^2u$ in \eqref{show-est-1}, \eqref{main-est-1}, and \eqref{cor-est-1}.
See Remark \ref{remark-3-range} below to see the necessity of such different ranges.
\begin{remark} \label{remark-1-range}
When $\alpha \in (0,1)$, the range $(p(\alpha-1)_+ -1, 2p-1)$ for the power $\gamma$ in \eqref{show-est-1} becomes $(-1,2p-1)$, which agrees with the range in \cite{KN} for equations with uniformly elliptic and bounded coefficients.
See also \cite{DP-JFA} and \cite{MNS}
in which a similar range of the power $\gamma$ is also used in for a class of equations of extensional type.
When $\alpha \in [1,2)$, the lower bound $p(\alpha-1)_+ -1$ for $\gamma$ in \eqref{show-est-1} is optimal.
To see this, consider an explicit example when $d=1$, $\lambda>0$, $T < \infty$, and
\[
u(t,x)=\left(x + c x^{3-\alpha}\right) \xi(x) e^{\lambda t} \qquad \text{ for } (t,x) \in \Omega_T.
\]
Here, $\xi \in C^\infty([0,\infty),[0,\infty))$ is a cutoff function such that $\xi=1$ on $[0,1]$, $\xi=0$ on $[3,\infty)$, $\|\xi'\|_{L^\infty(\mathbb{R})} \leq 1$, and
\[
c=\frac{2 \lambda}{(3-\alpha)(2-\alpha)}.
\]
Set
\[
f(t,x)=x^{-\alpha}(u_t +\lambda u) - u_{xx}.
\]
Then, $u$ solves
\[
u_t + \lambda u - x^\alpha u_{xx} = x^\alpha f \qquad \text{ in } \Omega_T.
\]
We see that $\mathbf{M}^{-\alpha}u_t, \mathbf{M}^{-\alpha}u \in L_p(\Omega_T, x^\gamma)$ for $\gamma>p(\alpha-1)-1$.
On the other hand,
\[
\begin{split}
& \int_{\Omega_T} |x^{-\alpha}u|^p x^{p(\alpha-1)-1}\,dz
=\int_{\Omega_T} |x^{-1}u|^p x^{-1}\,dz \\
& \geq \int_{0}^1 \int_{-\infty}^T x^{-1} e^{p\lambda t}\,dt dx = N\int_{0}^1 x^{-1} \,dx=\infty.
\end{split}
\]
Thus, $\mathbf{M}^{-\alpha}u_t, \mathbf{M}^{-\alpha}u \notin L_p(\Omega_T, x^{p(\alpha-1)-1})$.
We next note that $f(t,x)=0$ for $(t,x) \in (-\infty,T] \times [3,\infty)$, and
\[
f(t,x)=2 c\lambda x^{3-2\alpha} e^{\lambda t} \qquad \text{ for } (t,x) \in (-\infty,T] \times [0,1].
\]
From this and
\[
\int_{0}^1 \int_0^T |x^{3-2\alpha}|^p x^{p(\alpha-1)-1} e^{p \lambda t} dt dx = N \int_0^1 x^{p(2-\alpha)-1} dx < \infty,
\]
it follows that $f\in L_p(\Omega_T, x^{p(\alpha-1)-1})$.
\end{remark}
\begin{remark} \label{remark-3-range}
When $\alpha \in (0,2)$, the lower bound $\alpha p/2 -1$ for $\gamma$ in \eqref{show-est-2} is optimal.
Indeed, consider the same example as that in Remark \ref{remark-1-range} above.
It is clear that $\mathbf{M}^{-\alpha/2}u_x \in L_p(\Omega_T, x^\gamma)$ for $\gamma> \alpha p/2-1$.
On the other hand, $\mathbf{M}^{-\alpha/2}u_x \notin L_p(\Omega_T, x^{\alpha p/2-1})$ as
\[
\int_{\Omega_T} |x^{-\alpha/2}u_x|^p x^{\alpha p/2-1}\,dz
=\int_{\Omega_T} |u_x|^p x^{-1}\,dz \geq \int_{0}^1 \int_{-\infty}^T x^{-1} e^{p\lambda t}\,dt dx =\infty.
\]
Besides, $f(t,x)=0$ for $(t,x) \in (-\infty,T] \times [3,\infty)$, and
\[
f(t,x)=2 c\lambda x^{3-2\alpha} e^{\lambda t} \qquad \text{ for } (t,x) \in (-\infty,T] \times [0,1].
\]
Hence, $f \in L_p(\Omega_T, x^{\alpha p/2-1})$ as
\[
\int_{0}^1 \int_0^T |x^{3-2\alpha}|^p x^{\alpha p/2-1} e^{p \lambda t} dt dx = N \int_0^1 x^{p(3-3\alpha/2)-1} dx < \infty.
\]
\end{remark}
\begin{remark} \label{remark-2-range}
We also have that the upper bound $\gamma < 2p-1$ in \eqref{show-est-1}--\eqref{show-est-2} is optimal.
Indeed, for $\gamma=2p-1$, the trace of $W_{p}^{2}(\cD,x_d^{2p-1})$ is not well defined.
For simplicity, let $d=1$, $\cD=[0,1/2]$, and consider
\[
\phi(x) = \log (|\log x|).
\]
Then,
\[
\phi_{xx} = \frac{1}{x^2} \left( |\log x|^{-1} - |\log x|^{-2} \right).
\]
It is clear that $\phi \in W_{p}^{2}([0,1/2],x^{2p-1})$, and $\phi$ is not finite at $0$.
\end{remark}
\section{A filtration of partitions and a quasi-metric} \label{Feffer}
We recall the construction of a filtration of partitions $\{\mathbb{C}_n\}_{n \in \mathbb{Z}}$ (i.e., dyadic decompositions) of $\mathbb{R}\times \mathbb{R}^d_+$ in \cite{DPT21},
which satisfies the following three basic properties (see \cite{Krylov}):
\begin{enumerate}[(i)]
\item The elements of partitions are ``large'' for big negative $n$'s and ``small''
for big positive $n$'s: for any $f\in L_{1,\text{loc}}$,
$$
\inf_{C\in \mathbb{C}_n}|C|\to \infty\quad\text{as}\,\,n\to -\infty,\quad
\lim_{n\to \infty}(f)_{C_n(z)}=f(z)\quad\text{a.e.},
$$
where $C_n(z)\in \mathbb{C}_n$ is such that $z\in C_n(z)$.
\item The partitions are nested: for each $n\in \mathbb{Z}$, and $C \in \mathbb{C}_n$, there exists a unique $C' \in \mathbb{C}_{n-1}$ such that $C \subset C'$.
\item The following regularity property holds: For $n,C, C'$ as in (ii), we have
$$
|C'|\le N_0|C|,
$$
where $N_0>0$ is independent of $n$, $C$, and $C'$.
\end{enumerate}
For $s\in \mathbb{R}$, denote by $\lfloor s \rfloor$ the integer part of $s$.
For a fixed $\alpha\in (0,2)$ and $n\in \mathbb{Z}$, let $k_0=\lfloor -n/(2-\alpha) \rfloor$.
The partition $\mathbb{C}_n$ contains boundary cubes in the form
$$
((j-1)2^{-n},j2^{-n}]\times (i_12^{k_0},(i_1+1)2^{k_0}]
\times\cdots\times (i_{d-1}2^{k_0},(i_{d-1}+1)2^{k_0}]\times (0, 2^{k_0}],
$$
where $j,i_1,\ldots,i_{d-1}\in \mathbb{Z}$, and interior cubes in the form
$$
((j-1)2^{-n},j2^{-n}]\times (i_12^{k_2},(i_1+1)2^{k_2}]
\times\cdots \times (i_d2^{k_2}, (i_d+1)2^{k_2}],
$$
where $j,i_1,\ldots,i_{d}\in \mathbb{Z}$ and
\begin{equation}
\label{eq:part1}
i_d2^{k_2}\in [2^{k_1},2^{k_1+1})\, \text{for some integer}\, k_1\ge k_0,
\quad k_2=\lfloor (-n+k_1\alpha)/2 \rfloor-1.
\end{equation}
It is clear that $k_2$ increases with respect to $k_1$ and decreases with respect to $n$.
As $k_1\ge k_0>-n/(2-\alpha)-1$, we have
$(-n+k_1\alpha)/2-1\le k_1$,
which implies $k_2\le k_1$ and $(i_d+1)2^{k_2}\le 2^{k_1+1}$.
According to \eqref{eq:part1}, we also have
$$
(2^{k_2}/2^{k_1})^2\sim 2^{-n}/(2^{k_1})^{2-\alpha},
$$
which allows us to apply the interior estimates after a scaling.
The quasi-metric $\varrho: \Omega_\infty\times \Omega_\infty\to [0,\infty)$ is defined as
$$
\varrho((t,x),(s,y))=|t-s|^{1/(2-\alpha)}
+\min\big\{|x-y|,|x-y|^{2/(2-\alpha)}\min\{x_d,y_d\}^{-\alpha/(2-\alpha)}\big\}.
$$
There exists a constant $K_1=K_1(d,\alpha)>0$ such that
$$
\varrho((t,x),(s,y))\le K_1\big(\varrho((t,x),(\hat t,\hat x))+ \varrho((\hat t,\hat x),(s,y))\big)
$$
for any $(t,x),(s,y),(\hat t,\hat x)\in \Omega_\infty$, and $ \varrho((t,x),(s,y))=0$ if and only if $(t,x)=(s,y)$.
Besides, the cylinder $Q_\rho^+(z_0)$ defined in \eqref{def:Q} is comparable to
$$
\{(t,x)\in \Omega: t<t_0,\, \varrho((t,x),(t_0,x_0))<\rho \}.
$$
Of course, $(\Omega_T, \varrho)$ equipped with the Lebesgue measure is a space of homogeneous type and we have the above dyadic decomposition.
The dyadic maximal function and sharp function of a locally integrable function $f$ and a given weight $\omega$ in $\Omega_\infty$ are defined as
\begin{align*}
\cM_{\text{dy},\omega} f(z)&=\sup_{n<\infty}\frac{1}{\omega(C_n(z))}\int_{C_n(z)\in \mathbb{C}_n}|f(s,y)| \omega(s,y)\,dyds,\\
f_{\text{dy},\omega}^{\#}(z)&=\sup_{n<\infty}\frac{1}{\omega(C_n(z))}\int_{C_n(z)\in \mathbb{C}_n}|f(s,y)-(f)_{C_n(z),\omega}|\omega(s,y)\,dyds.
\end{align*}
Observe that the average notation in \eqref{everage-def} is used in the above definition. Similarly, the maximal function and sharp function over cylinders are given by
\begin{align*}
\cM_\omega f(z)&=\sup_{z\in Q^+_\rho(z_0), z_0\in \overline{\Omega_\infty}} \frac{1}{\omega(Q_\rho^+(z_0))} \int_{Q_\rho^+(z_0)}|f(s,y)|\omega(s,y)\,dyds,\\
f^{\#}_\omega(z)&=\sup_{z\in Q^+_\rho(z_0),z_0\in \overline{\Omega_\infty}}\frac{1}{\omega(Q_\rho^+(z_0))}\int_{Q_\rho^+(z_0)}|f(s,y)-(f)_{Q^+_\rho(z_0)}|\omega(s,y)\,dyds.
\end{align*}
We have, for any $z\in \Omega_\infty$,
$$
\cM_{\text{dy},\omega} f(z)\le N\cM_{\omega} f(z) \qquad \text{ and } \qquad f_{\text{dy},\omega}^{\#}(z)\le Nf^{\#}_\omega(z),
$$
where $N=N(d,\alpha)>0$.
We also recall the following definition of the $A_p$ Muckenhoupt class of weights.
\begin{definition}
\label{Def-Muck-wei}
For each $p \in (1, \infty)$ and for a nonnegative Borel measure $\sigma$ on $\mathbb{R}^d$, a locally integrable function $\omega : \mathbb{R}^d \rightarrow \mathbb{R}_+$ is said to be in the $A_p( \mathbb{R}^d, \sigma)$ Muckenhoupt class of weights if and only if $[\omega]_{A_p(\mathbb{R}^d, \sigma)} < \infty$, where
\begin{equation}
\label{Ap.def}
\begin{split}
& [\omega]_{A_p(\mathbb{R}^d, \sigma)} \\
& =
\sup_{\rho >0,x =(x', x_d)\in \mathbb{R}^d } \bigg[\fint_{B_{\rho} (x)} \omega(y)\, \sigma(dy) \bigg]\bigg[\fint_{B_{\rho}(x)} \omega(y)^{\frac{1}{1-p}}\, \sigma(dy) \bigg]^{p-1}.
\end{split}
\end{equation}
Similarly, the class of weights $A_p(\mathbb{R}^d_+, \sigma)$ can be defined in the same way in which the ball $B_{\rho} (x)$ in \eqref{Ap.def} is replaced with $B_\rho^+(x)$ for $x\in \overline{\mathbb{R}^d_+}$. For weights with respect to the time variable, the definition is similar with the balls replaced with intervals $(t_0 -\rho^{2-\alpha}, t_0 + \rho^{2-\alpha})$ and $\sigma(dy)$ replaced with $dt$. If $\sigma$ is a Lebesgue measure, we simply write $A_p(\mathbb{R}^d_+) = A_p(\mathbb{R}^d_+, dx)$ and $A_p(\mathbb{R}^d) = A_p(\mathbb{R}^d, dx)$. Note that if $\omega \in A_p(\mathbb{R})$, then $\tilde{\omega} \in A_p(\mathbb{R}^d)$ with $[\omega]_{A_p(\mathbb{R})} = [\tilde{\omega}]_{A_p(\mathbb{R}^d)}$, where $\tilde{\omega}(x) = \omega(x_d)$ for $x = (x', x_d) \in \mathbb{R}^d$. Sometimes, if the context is clear, we neglect the spatial domain and only write $\omega \in A_p$.
\end{definition}
The following version of the weighted mixed-norm Fefferman-Stein theorem and Hardy-Littlewood maximal function theorem can be found in \cite{Dong-Kim-18}.
\begin{theorem} \label{FS-thm} Let $p, q \in (1,\infty)$, $\gamma_1 \in (-1, \infty)$, $K\geq 1$, and $\mu_1(dz) = x_d^{\gamma_1}\, dxdt$. Suppose that $\omega_0\in A_q(\mathbb{R})$ and $\omega_1 \in A_p(\mathbb{R}^{d}_{+},\mu_1)$ satisfy
$$
[\omega_0]_{A_q}, \,\, [\omega_{1}]_{A_p(\mathbb{R}_+^d, \mu_1)}\le K.$$
Then, for any $f \in L_{q, p}(\Omega_T, \omega\, d\mu_1)$, we have
\begin{align*}
& \|f\|_{L_{q, p}(\Omega_T, \omega\, d\mu_1)}
{\leq N \| f^{\#}_{\text{dy},\mu_1}\|_{L_{q,p}(\Omega_T, \omega\, d \mu_1)}}
\leq N \| f^{\#}_{\mu_1}\|_{L_{q,p}(\Omega_T, \omega\, d \mu_1)}, \\
& \|\mathcal{M}_{\mu_1}(f)\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \leq N \|f\|_{L_{q, p}(\Omega_T, \omega\, d \mu_1)},
\end{align*}
where $N = N(d, q, p, \gamma_1, K)>0$ and $\omega(t,x) = \omega_0(t)\omega_1(x)$ for $(t,x) \in \Omega_T$.
\end{theorem}
\section{Equations with coefficients depending only on the \texorpdfstring{$x_d$}{} variable} \label{sec:3}
In this section, we consider \eqref{eq:main} when the coefficients in \eqref{eq:main} only depend on the $x_d$ variable.
Let us denote
\begin{equation} \label{L0-def}
\sL_0 u = \bar{a}_0(x_d) u_t+\lambda \bar{c}_0(x_d) u-\mu(x_d) \bar{a}_{ij}(x_d)D_iD_j u.
\end{equation}
where $\mu, \bar{a}_0, \bar{c}_0, \bar{a}_{ij}: \mathbb{R}_+ \rightarrow \mathbb{R}$ are given measurable functions and they satisfy \eqref{con:mu}-\eqref{con:ellipticity}. We consider
\begin{equation}\label{eq:xd}
\left\{
\begin{array}{cccl}
\sL_0 u & = & \mu(x_d) f \quad &\text{ in } \Omega_T,\\
u & = & 0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+.
\end{array} \right.
\end{equation}
The main result of this section is the following theorem, which is a special case of Corollary \ref{cor1}.
\begin{theorem}\label{thm:xd}
Assume that $\bar{a}_0, \bar{c}_0, (\bar{a}_{ij})$ satisfy \eqref{con:mu}--\eqref{con:ellipticity} and assume further that $f \in {L_p(\Omega_T,x_d^\gamma\, dz)}$ for some given $p>1$ and
\[
\gamma \in \big(p(\alpha-1)_+-1,2p-1\big).
\]
Then, \eqref{eq:xd} admits a strong unique solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$.
Moreover,
\begin{align} \notag
& \|\mathbf{M}^{-\alpha}u_t\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} + \|D^2 u\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} \\ \label{eq:xd-main}
& \quad \qquad +\lambda\|\mathbf{M}^{-\alpha}u\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} \le N\|f\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)};
\end{align}
and if $\gamma\in (\alpha p/2-1,2p-1)$, we also have
\begin{equation}
\label{eq3.09}
\lambda^{1/2}\|\mathbf{M}^{-\alpha/2}Du\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)}
\le N\|f\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)},
\end{equation}
where $N=N(d,\nu,\alpha, \gamma, p)>0$.
\end{theorem}
The proof of Theorem \ref{thm:xd} requires various preliminary results and estimates.
Our starting point is Lemma \ref{l-p-sol-lem} below which gives Theorem \ref{thm:xd} when $\gamma$ is large. See Subsection \ref{subsec:L2} below.
Then, in Subsections \ref{subsec:boundary} and \ref{subsec:int}, we derive pointwise estimates for solutions to the corresponding homogeneous equations.
Afterwards, we derive the oscillation estimates for solutions in Subsection \ref{subsec:osc-est}.
The proof of Theorem \ref{thm:xd} will be given in the last subsection, Subsection \ref{proof-xd}.
Before starting, let us point out several observations as well as recall several needed definitions. Note that by dividing the PDE in \eqref{eq:xd} by $\bar{a}_0$ and then absorbing $\bar{a}_{dd}$ into $\mu(x_d)$, without loss of generality, we may assume that
\begin{equation} \label{add-cond}
\bar{a}_{dd}=1 \qquad \text{ and } \qquad \bar{a}_0 =1.
\end{equation}
Observe that \eqref{eq:xd} can be rewritten into a divergence form equation
\begin{equation}
\label{eq:xd-div}
\bar{a}_0 u_t+\lambda \bar{c}_0(x_d)u-\mu(x_d) D_i(\tilde a_{ij}(x_d) D_{j} u)=\mu(x_d)f \quad \text{ in } \Omega_T,
\end{equation}
where
\begin{equation}
\label{eq:change}
\tilde a_{ij}=\left\{
\begin{array}{ll}
\bar{a}_{ij}+ \bar{a}_{ji} & \hbox{for $i\neq d$ and $j=d$;} \\
0 & \hbox{for $i=d$ and $j\neq d$;} \\
\bar{a}_{ij} & \hbox{otherwise.}
\end{array}
\right.
\end{equation}
We note that even though $(\tilde a_{ij})$ is not symmetric, it still satisfies the ellipticity condition \eqref{con:ellipticity} and also $\tilde a_{dd} =1$ when \eqref{add-cond} holds.
Due to the divergence form as in \eqref{eq:xd-div}, we need the definition of its weak solutions.
In fact, sometimes in this section, we consider the following class of equations in divergence form which are slightly more general than \eqref{eq:xd-div}
\begin{equation} \label{eq:dx-loc}
u_t + \lambda \bar{c}_0(x_d) u - \mu(x_d)D_i (\tilde{a}_{ij}(x_d)D_j u - F_i) = \mu(x_d) f \quad \text{in} \quad (S, T) \times \mathcal{D}
\end{equation}
with the boundary condition
\begin{equation*}
u = 0 \quad \text{on} \quad (S, T) \times (\overline{\mathcal{D}} \cap \{x_d =0\})
\end{equation*}
for some open set $\mathcal{D} \subset \mathbb{R}^d_+$ and $-\infty \leq S < T \leq \infty$.
\begin{definition}
For a given weight $\omega$ defined on $(S, T) \times \mathcal{D} $ and for given $F= (F_1, F_2, \ldots, F_2) \in L_{p, \text{loc}}((S, T) \times \mathcal{D} )^d$ and $f \in L_{p, \text{loc}}((S, T) \times \mathcal{D} )$, we say that a function $u \in \cH_p^1((S, T) \times \mathcal{D} , \omega)$ is a weak solution of \eqref{eq:dx-loc} if
\begin{equation} \label{def-local-weak-sol}
\begin{split}
& \int_{(S, T) \times \mathcal{D} }\mu(x_d)^{-1}(-u \partial_t \varphi + \lambda \overline{c}_0 u \varphi)dz + \int_{(S, T) \times \mathcal{D} } (\tilde{a}_{ij} D_ju - F_i) D_i \varphi dz \\
& = \int_{(S, T) \times \mathcal{D} } f(z) \varphi(z) dz, \quad \forall \ \varphi \in C_0^\infty((S, T) \times \mathcal{D} ).
\end{split}
\end{equation}
\end{definition}
\subsection{\texorpdfstring{$L_p$}{} strong solutions when the powers of weights are large} \label{subsec:L2}
The following lemma is the main result of this subsection, which gives Theorem \ref{thm:xd} when $\gamma \in (p-1, 2p-1)$.
\begin{lemma} \label{l-p-sol-lem} Let $\nu \in (0,1)$, $\lambda>0$, $\alpha \in (0,2)$, $p \in (1, \infty)$, and $\gamma \in (p-1, 2p-1)$. Assume that $\bar{a}_0, \bar{c}_0, (\bar{a}_{ij})$, and $\mu$ satisfy the ellipticity and boundedness conditions \eqref{con:mu}--\eqref{con:ellipticity}. Then, for any $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, there exists a unique strong solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$ to \eqref{eq:xd}. Moreover, for every solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$ of \eqref{eq:xd} with $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, it holds that
\begin{align} \notag
& \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T, x_d^{\gamma}dz)} + \sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}Du \|_{L_p(\Omega_T, x_d^{\gamma}dz)} \\ \label{est-0405-1}
& \qquad + \| D^2u\|_{L_p(\Omega_T, x_d^{\gamma}dz)} + \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^{\gamma}dz)} \leq N \|f\|_{L_p(\Omega_T, x_d^{\gamma}dz)},
\end{align}
where $N = N(d, \alpha, \nu, \gamma, p)>0$.
\end{lemma}
\begin{proof} The key idea is to apply \cite[Theorem 2.4]{DPT21} to the divergence form equation \eqref{eq:xd-div}, and then use an idea introduced by Krylov in \cite[Lemma 2.2]{Kr99} with a suitable scaling.
To this end, we assume that \eqref{add-cond} holds, and let us denote $\gamma' = \gamma -p \in (-1, p-1)$ and we observe that
\[
x_d^{1-\alpha}\mu(x_d) |f(z)| \sim x_d |f(z)| \in L_p(\Omega_T, x_d^{\gamma'}dz).
\]
As $\gamma' \in (-1, p-1)$, we have $x_d^{\gamma'} \in A_p$.
Moreover, the equation \eqref{eq:xd} can be written in divergence form as \eqref{eq:xd-div}.
Therefore, we apply \cite[Theorem 2.4]{DPT21} to \eqref{eq:xd-div} with $f_1=\mu(x_d) f$ and $f_2= 0$ to yield the existence of a unique weak solution $u \in \mathscr{H}^1_p(\Omega_T, x_d^{\gamma'}dz)$ of \eqref{eq:xd-div} satisfying
\begin{align} \notag
& \|Du\|_{L_{p}(\Omega_T,x_d^{\gamma'} dz)} + \sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}u\|_{L_{p}(\Omega_T,x_d^{\gamma'}dz)} \\ \label{Du-0226}
& \le N\|{x_d^{1-\alpha}f_1}\|_{L_{p}(\Omega_T,x_d^{\gamma'}dz)} = N \|f\|_{L_p(\Omega_T, x_d^\gamma dz)},
\end{align}
with $N = N(d, \nu, \gamma, p)>0$. We note here that because the coefficients $\bar{c}_0, \bar{a}_{ij}$ only depend on $x_d$, \cite[Theorem 2.4]{DPT21} holds for any $\lambda>0$ by a scaling argument. From \eqref{Du-0226}, the zero boundary condition, and the weighted Hardy inequality (see \cite[Lemma 3.1]{DP-AMS} for example), we infer that
\begin{align} \notag
\|u\|_{L_p(\Omega_T, x_d^{\gamma -2p}dz)} & = \|\mathbf{M}^{-1}u\|_{L_p(\Omega_T, x_d^{\gamma'}dz)} \leq N \|Du\|_{L_{p}(\Omega_T,x_d^{\gamma'})} \\ \label{u-op-wei-0226}
& \leq \|f\|_{L_p(\Omega_T, x_d^\gamma)}.
\end{align}
It remains to prove that \eqref{est-0405-1} holds as it also implies that $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma)$.
We apply the idea introduced by Krylov in \cite[Lemma 2.2]{Kr99} and combine it with a scaling argument to remove the degeneracy of the coefficients. See also \cite[Theorem 3.5]{DK15} and \cite[Lemma 4.6]{DP-JFA}.
To this end, let us fix a standard non-negative cut-off function $\zeta \in C_0^\infty((1,2))$. For each $r >0$, let $\zeta_r(s) =\zeta(rs)$ for $s \in \mathbb{R}_+$. Note that with a suitable assumption on the integrability of a given function $v: \Omega_T \rightarrow \mathbb{R}$ and for $\beta \in \mathbb{R}$, by using the substitution $r^{\alpha}t \mapsto s$ for the integration with respect to the time variable, and then using the Fubini theorem, we have
\begin{equation} \label{weight-kry}
\begin{split}
& \int_0^\infty \left(\int_{\Omega_{r^{-\alpha}T}}|\zeta_r(x_d) v_r(z)|^p\, dz\right) r^{-\beta-1}\, dr =N_1\int_{\Omega_T} |v(z)|^p x_d^{\beta+\alpha}\, dz, \\
& \int_0^\infty \left(\int_{\Omega_{r^{-\alpha} T}}|\zeta_r'(x_d) v_r(z)|^p\, dz\right) r^{-\beta-1}\, dr =N_2\int_{\Omega_T} |v_r(z)|^p x_d^{\beta + \alpha -p}\, dz, \\
& \int_0^\infty \left(\int_{\Omega_{r^{-\alpha} T}}|\zeta_r''(x_d) v_r(z)|^p\, dz\right) r^{-\beta-1}\, dr =N_3\int_{\Omega_T} |v_r(z)|^p x_d^{\beta + \alpha-2p}\, dz,
\end{split}
\end{equation}
where $v_r(z) = v(r^{\alpha}t, x)$ for $z = (t, x) \in \Omega_{r^{-\alpha}T}$,
\[
N_1 = \int_0^\infty |\zeta (s)|^p s^{-\beta-\alpha-1} ds, \quad
N_2 = \int_0^\infty |\zeta'(s)|^p s^{p-\beta-\alpha-1} ds,
\]
and
\[
N_3 = \int_0^\infty |\zeta''(s)|^p s^{2p-\beta-\alpha -1} ds.
\]
Next, for $r>0$, we denote $u_r(z) = u(r^{\alpha}t,x)$,
\[
\hat{a}_{ij} (x_d)= r^{\alpha}\mu(x_d) \bar{a}_{ij}(x_d), \quad \bar{\lambda} = \lambda r^{\alpha}, \quad \text{and} \quad f_r (z) = r^{\alpha} \mu(x_d) f(r^{\alpha}t, x).
\]
Note that $u_r$ solves the equation
\[
\partial_t u_r + \bar{\lambda} \bar{c}_0 u_r - \hat{a}_{ij}(x_d) D_i D_j u_r = f_r \quad \text{in} \quad \Omega_{r^{-\alpha}T}.
\]
Let $w(z) =\zeta_r(x_d) u_r(z)$, which satisfies
\begin{equation} \label{w-eqn-0405-1}
w_t + \bar{\lambda} \bar{c}_0(x_d) w - \hat{a}_{ij}(x_d) D_i D_j w = \hat{g} \quad \text{in} \quad \Omega_{r^{-\alpha}T}
\end{equation}
with the boundary condition $w(z', 0) =0$ for $z' \in (-\infty, r^{-\alpha}T)\times \mathbb{R}^{d-1}$, where \[
\begin{split}
\hat{g}(z) & = \zeta_r f_r(z) - \hat{a}_{dd} \zeta''_r u_r - \sum_{i \neq d} \big(\hat{a}_{id} + \hat{a}_{di}\big)\zeta '_rD_i u_r.
\end{split}
\]
We note that $\text{supp}(w) \subset (-\infty, r^{-\alpha}T) \times \mathbb{R}^{d-1} \times(1/r, 2/r)$, and on this set the coefficient matrix $(\hat{a}_{ij})$ is uniformly elliptic and bounded as $r^{\alpha}\mu (x_d) \sim 1$ due to \eqref{con:mu}.
We now prove \eqref{est-0405-1} with the extra assumption that $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}dz)$. Under this assumption and as $\zeta_r$ is compactly supported in $(0, \infty)$, we see that $w \in W^{1,2}_p(\Omega_{r^{\alpha}T})$, the usual parabolic Sobolev space. Then by applying the $W^{1, 2}_{p}$-estimate for the uniformly elliptic and bounded coefficient equation \eqref{w-eqn-0405-1} (see, for instance, \cite{D12}), we obtain
\[
\bar{\lambda} \|w\| + \bar{\lambda}^{1/2}\, \|Dw\| + \|D^2 w\| + \|w_t\| \leq N \|\hat{g}\|,
\]
where $\|\cdot \| = \| \cdot \|_{L_p(\Omega_{r^{-\alpha}T})}$ and $N = N(d, \nu, p)>0$. From this, the definition of $\hat{g}$, and a simple manipulation, we obtain
\[
\begin{split}
& \lambda r^{\alpha} \|\zeta_r u_r \| + \sqrt{\lambda} r^{\alpha/2} \|\zeta_r Du_r\| + \|\zeta_r D^2u_r\| +\|\zeta_r \partial_t u_r \| \\
& \leq N\Big[ \|\zeta_r f_r\| + \sqrt{\lambda} r^{\alpha/2} \|\zeta_r' u_r\| + \|\zeta''_r u_r\| + \|\zeta_r' Du_r\| \Big].
\end{split}
\]
Now, we raise this last estimate to the power $p$, multiply both sides by $r^{-(\gamma-\alpha)-1}$, integrate the result with respect to $r$ on $(0,\infty)$, and then apply \eqref{weight-kry} to obtain
\[
\begin{split}
& \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} +\sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}Du \|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \\
& \qquad + \| D^2u\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)}\\
& \leq N \Big[ \|f\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} +\sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}u\|_{L_p(\Omega_T, x_d^{\gamma -p}\, dz)} + \|u\|_{L_p(\Omega_T, x_d^{\gamma -2p}\, dz)} \\
& \qquad + \|Du\|_{L_p(\Omega_T, x_d^{\gamma -p}\, dz)}\Big].
\end{split}
\]
From the last estimate, \eqref{Du-0226}, \eqref{u-op-wei-0226}, and the fact that $\gamma' = \gamma-p$, we infer that
\[
\begin{split}
& \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} + \sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}Du \|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} \\
& \qquad + \| D^2u\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} + \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} \leq N \|f\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)}.
\end{split}
\]
This proves \eqref{est-0405-1} under the additional assumption that $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}\,dz)$.
It remains to remove the extra assumption that $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}dz)$. By mollifying the equation \eqref{eq:xd} in $t$ and $x'$ and applying \cite[Theorem 2.4]{DPT21} to the equations of $u^{(\varepsilon)}_t$ and $D_{x'}u^{(\varepsilon)}$, we obtain
$$
\mathbf{M}^{-\alpha}u^{(\varepsilon)}, \mathbf{M}^{-\alpha} u_t^{(\varepsilon)}, DD_{x'}u^{(\varepsilon)} \in L_p(\Omega_T, x_d^{\gamma'} dz).
$$
This and the PDE in \eqref{eq:xd} for $u^{(\varepsilon)}$ imply that
\[
D_{dd} u^{(\varepsilon)} \in L_p(\Omega_T, x_d^{\gamma'} dz).
\]
Therefore $u^{(\varepsilon)} \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}dz)$ is a strong solution of \eqref{eq:xd} with $f^{(\varepsilon)}$ in place of $f$. From this, we apply the a priori estimate \eqref{est-0405-1} that we just proved for $u^{(\varepsilon)}$ and pass to the limit as $\varepsilon \rightarrow 0^+$ to obtain the estimate \eqref{est-0405-1} for $u$. The proof of the lemma is completed.
\end{proof}
\subsection{Boundary H\"older estimates for homogeneous equations} \label{subsec:boundary}
Recall the operator $\sL_0$ defined in \eqref{L0-def}. In this subsection, we consider the homogeneous equation
\begin{equation}
\label{eq:hom}
\begin{cases}
\sL_0u=0 \quad &\text{ in } Q_1^+,\\
u=0 \quad &\text{ on } Q_1\cap \{x_d=0\}.
\end{cases}
\end{equation}
As above, {without loss of generality we assume \ref{add-cond} so that} \eqref{eq:hom} can be written in divergence form as
\begin{equation}
\label{eq:hom-div}
\left\{
\begin{array}{cccl}
u_t+\lambda \bar{c}_0(x_d)u-\mu(x_d) D_i(\tilde a_{ij}(x_d) D_{j} u) & = &0 & \quad \text{ in } Q_1^+,\\
u & = & 0 & \quad \text{on} \quad Q_1\cap \{x_d=0\}.
\end{array} \right.
\end{equation}
A function $u \in \cH_{p}^1(Q_1^+)$ with $p \in (1,\infty)$ is said to be a weak solution of \eqref{eq:hom} if it is a weak solution of \eqref{eq:hom-div} in the sense defined in \eqref{def-local-weak-sol} and $u=0$ on $Q_1\cap \{x_d=0\}$ in the sense of trace.
For each $\beta \in (0,1)$, the $\beta$-H\"older semi-norm in the spatial variable of a function $u$ on an open set $Q\subset \mathbb{R}^{d+1}$ is given by
\[
\llbracket u\rrbracket_{C^{0, \beta}(Q)} = \sup\left\{ \frac{|u(t,x) - u(t,y)|}{|x-y|^{\beta}}: x \not =y, \ (t,x), (t,y) \in Q \right\}.
\]
For $k, l \in \mathbb{N} \cup \{0\}$, we denote
\[
\|u\|_{C^{k, l}(Q)} = \sum_{i=0}^k \sum_{|j| \leq l}\|\partial_t^i D_{x}^j u\|_{L_\infty(Q)} .
\]
We also use the following H\"{o}lder norm of $u$ on $Q$
\[
\|u\|_{C^{k, \beta}(Q)} = \|u\|_{C^{k,0}(Q)} + \sum_{i=0}^{k} \llbracket \partial_t^i u\rrbracket_{C^{0, \beta}(Q)}.
\]
We begin with the following Caccioppoli type estimate.
\begin{lemma} \label{caccio}
Suppose that $u\in \cH^1_2(Q_1^+)$ is a weak solution of \eqref{eq:hom}.
Then, for any integers $k,j\ge 0$ and $l=0,1$,
\begin{equation}
\label{eq:hom-b1}
\int_{Q_{1/2}^+}|\partial_t^k D_{x'}^j D_{d}^l u|^2 \,dz
\le N \int_{Q_{1}^+} u^2 \,dz
\end{equation}
where $N = N(d,\nu,\alpha,k,j,l)>0$.
\end{lemma}
\begin{proof} Again, we can assume \eqref{add-cond} holds. The estimate \eqref{eq:hom-b1} follows from \cite[(4.12)]{DPT21} applied to \eqref{eq:hom-div}.
\end{proof}
\begin{lemma}
\label{lem:boundary}
Let $p_0 \in (1, \infty)$ and suppose that $u\in \cH^1_{p_0}(Q_1^+)$ is a weak solution of \eqref{eq:hom}.
Then,
\begin{equation}
\label{eq:hom-b2}
\begin{split}
& \|u\|_{C^{1,1}(Q_{1/2}^+)}+\|D_{x'}u\|_{C^{1,1}(Q_{1/2}^+)}+\|D_d u\|_{C^{1,\delta_0}(Q_{1/2}^+)} \\
& + \sqrt{\lambda}\|\mathbf{M}^{-\alpha/2}u\|_{C^{1,1-\alpha/2}(Q_{1/2}^+)} \le N \|Du\|_{L_{p_0}(Q_1^+)},
\end{split}
\end{equation}
where $N=N(d,\nu,\alpha, p_0)>0$ and $\delta_0=\min\{2-\alpha,1\}$.
\end{lemma}
\begin{proof} As explained, we can assume that \eqref{add-cond} holds. We apply \cite[Lemma 5.5]{DPT21} to \eqref{eq:hom-div} by noting that $U:=\tilde a_{dj}D_j u=D_du$ in view of \eqref{add-cond} and \eqref{eq:change}.
\end{proof}
\begin{lemma}
\label{prop:boundary}
Let $p_0 \in (1, \infty)$, $\beta_0\in {(-\infty}, \min\{1,\alpha\}]$, and $\alpha_0 > -1$ be fixed constants.
There exists a number $\beta_1=\beta_1(\alpha,\beta_0) \in (0,1]$ such that for every weak solution $u\in \cH^1_{p_0}(Q_1^+)$ to \eqref{eq:hom}, it holds that
\begin{align}
\label{eq:hom-b3}
\|\mathbf{M}^{-\beta_0} u\|_{C^{1,\beta_1}(Q_{1/2}^+)}&\le N\|\mathbf{M}^{-\beta_0} u\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}\,dz)},\\
\label{eq:hom-b4}
\|\mathbf{M}^{-\beta_0} u_t\|_{C^{1,\beta_1}(Q_{1/2}^+)}&\le N\|\mathbf{M}^{-\beta_0} u_t\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}\,dz)},\\
\label{eq:hom-b5}
\|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u\|_{C^{1,\beta_1}(Q_{1/2}^+)}&\le N\|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}\,dz)},
\end{align}
and
\begin{equation}
\label{eq:hom-b-Du}
\|\mathbf{M}^{\beta_0}Du\|_{C^{1,\beta_1}(Q_{1/2}^+)} \le N\|\mathbf{M}^{\beta_0}Du\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}dz)},
\end{equation}
where $N=N(d,\nu,\alpha,\alpha_0,\beta_0, p_0)>0$.
\end{lemma}
\begin{proof}
Again, we assume \eqref{add-cond}.
Note that once the lemma with $\alpha_0 \geq 0$ is proved, the case when $\alpha_0 \in (-1, 0)$ will follow immediately.
Hence, we only need to prove the lemma with the assumption that $\alpha_0 \geq 0$. We first assume $p_0 =2$.
Since $\beta_0\le \min\{1,\alpha\}$, by \eqref{eq:hom-b1} and the boundary Poincar\'e inequality, the right-hand sides of \eqref{eq:hom-b3}, \eqref{eq:hom-b4}, and \eqref{eq:hom-b5} are all finite.
We consider two cases.
\noindent
{\em Case 1: $\beta_0=0$.} When $\alpha_0=0$, \eqref{eq:hom-b3} and \eqref{eq:hom-b-Du} follow from \eqref{eq:hom-b2} and \eqref{eq:hom-b1}. For general $\alpha_0\ge 0$, by \eqref{eq:hom-b3} with $\beta_0=0$ and $\alpha_0=0$ and H\"older's inequality, we have
\begin{align*}
\|u\|_{L_\infty(Q_{1/2}^+)}& \le N \|u\|_{L_2(Q_{2/3}^+)} \leq N\|u\|^{2\alpha_0/(1+2\alpha_0)}_{L_2(Q_{2/3}^+,x_d^{-1/2}\,dz)}
\|u\|^{1/(1+2\alpha_0)}_{L_2(Q_{2/3}^+,x_d^{\alpha_0}\,dz)}\\
&\le N\|u\|^{2\alpha_0/(1+2\alpha_0)}_{L_\infty(Q_{2/3}^+)}
\|u\|^{1/(1+2\alpha_0)}_{L_2(Q_{2/3}^+,x_d^{\alpha_0}\,dz)} \\
& \leq \frac{1}{2} \|u\|_{L_\infty(Q_{2/3}^+)} + N\|u\|_{L_2(Q_{2/3}^+,x_d^{\alpha_0}\,dz)},
\end{align*}
where $N = N(d, \nu, \alpha, \alpha_0)>0$. From this and the standard iteration argument (see \cite[p. 75]{HanLin} for example), we obtain
\begin{equation} \label{est.alpha-zero}
\|u\|_{L_\infty(Q_{1/2}^+)} \leq N \|u\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}.
\end{equation}
The above, together with Lemma \ref{caccio}, yields
\begin{equation}
\label{eq:hom-b6}
\int_{Q_{1/2}^+}|\partial_t^k D_{x'}^j D_{d}^l u|^2 \,dz
\le N(d,\nu,\alpha, \alpha_0,k,j,l) \int_{Q_{3/4}^+} u^2 x_d^{\alpha_0}\,dz
\end{equation}
for any integers $k,j\ge 0$ and $l=0,1$.
Using this last estimate, \eqref{eq:hom-b2}, and by suitably adjusting the sizes of the cylinders, we obtain \eqref{eq:hom-b3} with $\beta_1 = 1$.
Similar to \eqref{est.alpha-zero}, we have
\[
\|Du\|_{L_\infty(Q_{1/2}^+)} \leq N \|Du\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}.
\]
From this, \eqref{eq:hom-b2}, and by shrinking the cylinders, we obtain
\[
\|Du\|_{C^{1,\delta_0}(Q_{1/2}^+)} \leq N \|Du\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}dz)}, \quad \text{where} \,\, \delta_0=\min\{2-\alpha,1\},
\]
which is \eqref{eq:hom-b-Du} when $\beta_0 =0$.
Since $u_t$ and $D_{x'} u$ satisfy the same equation with the same boundary condition, similarly we also obtain \eqref{eq:hom-b4} as well as
\begin{equation}
\label{eq:hom-b7}
\|DD_{x'}u\|_{C^{1,\delta_0}(Q_{1/2}^+)}\le N\|DD_{x'}u\|_{L_2(Q_{2/3}^+)},
\end{equation}
by Lemma \ref{lem:boundary}.
This together with \eqref{eq:hom-b6} implies \eqref{eq:hom-b5} with
$$\beta_1=\min\{\delta_0,\alpha \} = \min\{\alpha, 2-\alpha,1\}.$$
\noindent
{\em Case 2: $\beta_0\neq 0$.}
We first prove \eqref{eq:hom-b5}. By \eqref{eq:hom-b7} and by using the iteration argument as in \eqref{est.alpha-zero}, we have
\[
\|DD_{x'} u\|_{L_\infty(Q_{1/2}^+)} \leq N\|DD_{x'} u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\, dz)},
\]
where $N = N(d, \nu, \alpha, \alpha_0)>0$. Then, it follows from \eqref{eq:hom-b7} that
\begin{equation} \label{eq:hom-b7-bis}
\|DD_{x'}u\|_{C^{1,\delta_0}(Q_{1/2}^+)}\le N\|DD_{x'}u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\, dz)}.
\end{equation}
Therefore, if $\beta_0=\alpha$, \eqref{eq:hom-b5} with $\beta_1=\delta_0$ follows from \eqref{eq:hom-b7-bis}. If $\beta_0<\alpha$, it follows from \eqref{eq:hom-b7-bis} that
\[
\|DD_{x'}u\|_{C^{1,\delta_0}(Q_{1/2}^+)}\le N\|\mathbf{M}^{{\alpha-\beta_0}}DD_{x'}u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\, dz)}
\]
where $N = N(d, \nu, \alpha, \beta_0, \alpha_0)>0$.
Then we also have \eqref{eq:hom-b5} with
\[ \beta_1=\min\{\delta_0,\alpha-\beta_0\} = \min\{2-\alpha,1,\alpha-\beta_0\}. \]
Similarly, \eqref{eq:hom-b-Du} can be deduced from \eqref{eq:hom-b-Du} when $\beta_0 =0$ by taking $\beta_1 = \min\{\delta_0, \beta_0\}$. Hence, both \eqref{eq:hom-b5} and \eqref{eq:hom-b-Du} hold with
\[
\beta_1=\min\{\delta_0,\alpha-\beta_0,{\beta_0}\} = \min\{2-\alpha,1,\alpha-\beta_0, \beta_0\}.
\]
Next we show \eqref{eq:hom-b3}.
Since $\beta_0\le 1$, using the zero boundary condition, \eqref{eq:hom-b2}, and \eqref{eq:hom-b6}, we get
\begin{equation}
\label{eq:hom-b8}
\|\mathbf{M}^{-\beta_0} u\|_{L_\infty(Q_{1/2}^+)}
\le N \|D_d u\|_{L_\infty(Q_{1/2}^+)}\le N\|u\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}.
\end{equation}
Since $u_t$ and $D_{x'} u$ satisfy the same equation and the same boundary condition, we have
\begin{align}
\label{eq:hom-b9}
&\|\mathbf{M}^{-\beta_0} u_t\|_{L_\infty(Q_{1/2}^+)}
+\|\mathbf{M}^{-\beta_0}D_{x'} u\|_{L_\infty(Q_{1/2}^+)}\notag\\
&\le \ N \|D_d u_t\|_{L_\infty(Q_{1/2}^+)}+N \|D_dD_{x'} u\|_{L_\infty(Q_{1/2}^+)}\notag\\
&\le \ N \|D u\|_{L_2(Q_{2/3}^+,x_d^{\alpha_0})}\le N\|u\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)},
\end{align}
where we used \eqref{eq:hom-b2}.
To estimate the H\"older semi-norm of $\mathbf{M}^{-\beta_0} u$ in $x_d$, we write
$$
x_d^{-\beta_0} u(t,x)=x_d^{1-\beta_0}\int_0^1 (D_d u)(t,x',sx_d)\,ds
$$
and use \eqref{eq:hom-b2} and \eqref{eq:hom-b6}. Then we see that
\[
\llbracket \mathbf{M}^{-\beta_0} u \rrbracket_{C^{0, \beta_1}(Q_{1/2}^+)} +
\llbracket \mathbf{M}^{-\beta_0} \partial_t u \rrbracket_{C^{0, \beta_1}(Q_{1/2}^+)} \leq N \|\mathbf{M}^{-\beta_0} u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\,dz)}
\]
where $\beta_1 = \min\{\delta_0, 1-\beta_0\}$. Combining this with \eqref{eq:hom-b8} and \eqref{eq:hom-b9}, we reach \eqref{eq:hom-b3}.
Note that $u_t$ satisfies the same equation and the same boundary condition, we deduce \eqref{eq:hom-b4} from \eqref{eq:hom-b3}. The proof of the lemma when $p_0 =2$ is completed.
Next, we observe that when $p_0 >2$, the estimates \eqref{eq:hom-b3}--\eqref{eq:hom-b-Du} can be derived directly from the case $p_0 =2$ that we just proved using H\"{o}lder's inequality. On the other hand, when $p_0 \in (1, 2)$, it follows from Lemma \ref{lem:boundary} that $u \in \cH_2^1(Q_{3/4}^+)$. Then, by shrinking the cylinders, we apply the assertion when $p_0=2$ that we just proved and an iteration argument as in the proof of \eqref{est.alpha-zero} to obtain the estimates \eqref{eq:hom-b3}--\eqref{eq:hom-b-Du}.
\end{proof}
\begin{remark} The number $\beta_1$ defined in Lemma \ref{prop:boundary} can be found explicitly.
However, we do not need to use this in the paper.
\end{remark}
\subsection{Interior H\"older estimates for homogeneous equations} \label{subsec:int}
Fix a point $z_0 = (t_0, x_0) \in \Omega_T$, where $x_0 = (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$.
For $0<\rho < x_{0d}$ and $\beta \in (0,1)$, we define the weighted $\beta$-H\"{o}lder semi-norm of a function $u$ on $Q_\rho(z_0)$ by
\[
\begin{split}
\llbracket u\rrbracket_{C^{\beta/2, \beta}_{\alpha}(Q_\rho(z_0))} & = \sup \Big\{ \frac{|u(s,x) - u(t, y)|}{\big(x_{0d}^{-\alpha/2}|x-y| + |t-s|^{1/2}\big)^{\beta}}: (s,x) \not=(t,y) \\
& \qquad \qquad \qquad \text{and } (s,x), (t,y) \in Q_\rho(z_0) \Big\}.
\end{split}
\]
As usual, we denote the corresponding weighted norm by
\[
\|u\|_{C^{\beta/2, \beta}_{\alpha}(Q_\rho(z_0))} = \|u\|_{L_\infty(Q_\rho(z_0))} + \llbracket u\rrbracket_{C^{\beta/2, \beta}_{\alpha}(Q_\rho(z_0))}.
\]
The following result is the interior H\"older estimates of solutions to the homogeneous equation \eqref{eq:hom-div}.
\begin{lemma}\label{prop:int}
Let $z_0 = (t_0, x_0) \in \Omega_T$, where $x_0 = (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$, and $\rho \in (0, x_{0d}/4)$.
Let $u \in \sW_{p_0}^{1,2}(Q_{2\rho}(z_0))$ be a strong solution of
\begin{equation*}
\sL_0 u=0 \quad \text{ in } Q_{2\rho}(z_0)
\end{equation*}
with some $p_0 \in (1, \infty)$. Then for any $\beta \in \mathbb{R}$,
\begin{align*}
& \|\mathbf{M}^{\beta} u\|_{L_\infty(Q_{\rho}(z_0))} + \rho ^{(1-\alpha/2)/2} \llbracket \mathbf{M}^{\beta} u\rrbracket_{C^{1/4, 1/2}_\alpha(Q_{\rho}(z_0))} \\
&\leq N \left(\fint_{ Q_{2\rho}( z_0)} |x_{d}^{\beta}u|^{p_0} \mu_0(dz)\right)^{1/p_0},
\end{align*}
and
\begin{align*}
& \|\mathbf{M}^{\beta}Du\|_{L_\infty(Q_{\rho}(z_0))}
+ \rho^{(1-\alpha/2)/2} \llbracket {\mathbf{M}^{\beta} Du}\rrbracket_{C^{1/4, 1/2}_\alpha(Q_{\rho}(z_0))} \\
&\leq N \left(\fint_{ Q_{2\rho}( z_0)} |x_d^{\beta}Du|^{p_0} \mu_0(dz) \right)^{1/p_0},
\end{align*}
where $\mu_0(dz) = x_d^{\alpha_0}dtdx$ with some $\alpha_0 > -1$, and $N = N(\nu, d,\alpha, \alpha_0)>0$.
\end{lemma}
\begin{proof} As in the proof of Lemma \ref{prop:boundary}, we may assume that $p_0 =2$. Without loss of generality, we assume that $x_{0d}=1$. Note that when $\beta = 0$, the assertions follow directly from \cite[Proposition 4.6]{DPT21}. In general, the assertions follow from the case when $\beta =0$ and the fact that
\[
\left(\fint_{ Q_{2\rho}( z_0)} |\mathbf{M}^{\beta}f(z)|^{p_0} \mu_0(dz) \right)^{1/p_0} \approx \left(\fint_{ Q_{2\rho}( z_0)} |f(z)|^{p_0} dz \right)^{1/p_0}.
\]
The lemma is proved.
\end{proof}
\subsection{Mean oscillation estimates} \label{subsec:osc-est} In this subsection, we apply Lemmas \ref{prop:boundary} and \ref{prop:int} to derive the mean oscillation estimates of
$$
U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u) \qquad \text{and} \qquad Du
$$
respectively with the underlying measure
\begin{equation} \label{mu-1-def}
\mu_1(dz) = x_d^{\gamma_1}\,dx dt \qquad \text{and} \qquad \bar{\mu}_1(dz) = x_d^{\bar{\gamma}_1} dxdt,
\end{equation}
where $u$ is a strong solution of \eqref{eq:xd},
\[ \gamma_1 \in (p_0(\beta_0-\alpha +1)-1, p_0(\beta_0-\alpha+2)-1) \quad \text{and} \quad \bar{\gamma}_1 =\gamma_1 + p_0 (\alpha /2-\beta_0) \]
with some $p_0 \in (1, \infty)$ and $\beta_0 \in (\alpha-1, \min\{1, \alpha\}]$. The main result of the subsection is Lemma \ref{oscil-lemma-2} below.
Let us point out that both $\mu_1$ and $\bar{\mu}_1$ depend on the choice of $\beta_0$, and
\begin{equation} \label{mu1=bar-mu-1}
\mu_1= \bar{\mu}_1 \quad \text{when} \quad \beta_0 = \alpha/2.
\end{equation}
To get the weighted estimate of $U$ in $L_p(\Omega_T, x_d^\gamma\, dz)$ with the optimal range for $\gamma$ as in Theorem \ref{thm:xd}, we will use $\beta_0 = \min\{1, \alpha\}$. On the other hand, to derive the estimate for $Du$, we will use $\beta_0 = \alpha/2$ and \eqref{mu1=bar-mu-1}.
For the reader's convenience, let us also recall that for a cylinder $Q \subset \mathbb{R}^{d+1}$, a locally finite measure $\omega$, and an $\omega$-integrable function $g$ on $Q$, we denote the average of $g$ on $Q$ with respect to the measure $\omega$ by
\[
(g)_{Q, \omega} = \frac{1}{\omega(Q)}\int_{Q} g(z)\, \omega(dz)
\]
and the average of $g$ on $Q$ with respect to the Lebesgue measure by
\[
(g)_{Q} =\frac{1}{|Q|} \int_{Q} g(z)\, dz.
\]
We begin with the following lemma on the mean oscillation estimates of solutions to the homogeneous equations.
\begin{lemma} \label{oscil-lemma-1} Let $\nu \in (0,1)$, $\alpha \in (0,2)$, $p_0 \in (1, \infty)$, $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$, and $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha+2)-1)$. There exists $N = N(d,\nu, \alpha, \gamma_1, p_0)>0$ such that if $u \in \sW^{1,2}_{p_0}(Q_{14\rho}^+(z_0), x_d^{\gamma_1'}\, dz)$ is a strong solution of
\[
\left\{
\begin{array}{cccl}
\sL_0 u & =& 0 & \quad \text{in} \quad Q_{14\rho}^+(z_0) \\
u & = & 0 & \quad \text{on} \quad Q_{14\rho}(z_0) \cap \{x_d =0\}
\end{array}
\right.
\]
for some $\lambda>0, \rho>0$, $z_0 =(z_0', x_{d0}) \in \overline{\Omega}_T$, and for $\gamma_1' = \gamma_1 -p_0(\beta_0-\alpha)$, then
\begin{equation} \label{osc-h}
(|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0}
\end{equation}
and
\begin{equation} \label{Du-osc-h}
(|Du - (Du)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1}|)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1} \leq N \kappa^{\theta} (|Du|^{p_0})_{Q_{14 \rho}^+(z_0), \bar{\mu}_1}^{1/p_0}
\end{equation}
for every $\kappa \in (0,1)$, where $\mu_1, \bar{\mu}_1$ are defined in \eqref{mu-1-def},
\[
U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u),
\]
and $\theta = \min\{\beta_1(\alpha, \beta_0), (2-\alpha)/4, 2-\alpha, 1\} \in (0,1)$ in which $\beta_1$ is defined in lemma \ref{prop:boundary}.
\end{lemma}
\begin{proof}
By using the scaling \eqref{scaling}, we assume that $\rho =1$. We consider two cases: the boundary case and the interior one.
\noindent
{\em Boundary case.} Consider $x_{0d} <4$. Let $\bar{z} =(t_0, x_0', 0)$ and note that from the definition of cylinders in \eqref{def:Q}, we have
\[
Q_{1}^+(z_0) \subset Q_{5}^+(\bar{z}_0) \subset Q_{10}^+(\bar{z}_0) \subset Q_{14}^+(z_0).
\]
Then, we apply the mean value theorem and the estimates \eqref{eq:hom-b3}-\eqref{eq:hom-b5} in Lemma \ref{prop:boundary} with $\gamma_1$ in place of $\alpha_0$, $\beta_0$ in place of $\beta$ in the estimates of $u$, $u_t$, and $DD_{x'}u$.
We infer that
\[
\begin{split}
& (|U - (U)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\
& \leq N \kappa^{2-\alpha}
\|\partial_t U\|_{L^\infty(Q_1(z_0))} + N \kappa^{\beta_1} \llbracket U \rrbracket_{C^{0, \beta_1}(Q_{1}^+(z_0))} \\
& \leq N \kappa^{\theta}\big[ \|\partial_t U\|_{L^\infty(Q_{5}^+(\bar{z}))} + \llbracket U \rrbracket_{C^{0, \beta_1}(Q_{5}^+(\bar{z}))} \big] \\
& \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{10}^+(\bar{z}), \mu_1}^{1/p_0} \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14}^+(z_0), \mu_1}^{1/p_0},
\end{split}
\]
where we used the doubling property of $\mu_1$ in the last step.
This implies the estimate \eqref{osc-h} as $\kappa \in (0,1)$. To estimate the oscillation of $Du$ as asserted in \eqref{Du-osc-h}, we note that
$$\bar{\gamma}_1 = \gamma_1 - p_0(\beta_0 -\alpha/2) > p_0 (1-\alpha/2)-1 >-1. $$
Therefore, \eqref{Du-osc-h} can be proved in a similar way as that of \eqref{osc-h} using the estimate \eqref{eq:hom-b-Du} in Lemma \ref{prop:boundary} with $\beta =0$ and $\alpha_0 = \bar{\gamma}_1 >-1$.
\ \\ \noindent
{\em Interior case.} Consider $x_{0d} >4\rho=4$. By using Lemma \ref{prop:int} with $\beta = -\beta_0$ and the doubling property of $\mu_1$, we see that
\[
\begin{split}
& (|\mathbf{M}^{-\beta_0} u - (\mathbf{M}^{-\beta_0} u)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\
& \leq N \kappa^{1/2-\alpha/4}\llbracket \mathbf{M}^{-\beta_0} u \rrbracket_{C_\alpha^{1/4, 1/2}(Q_{1}^+(z_0))} \\
& \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{2}^+(z_0)}|\mathbf{M}^{-\beta_0} u|^{p_0}\mu_1(dz) \right)^{1/p_0} \\
& \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{14}^+(z_0)}|\mathbf{M}^{-\beta_0} u|^{p_0} \mu_1(dz) \right)^{1/p_0}.
\end{split}
\]
Similarly, by using the finite difference quotient, we can apply Lemma \ref{prop:int} to $u_t$ and obtain
\[
\begin{split}
& (|\mathbf{M}^{-\beta_0} u_t - (\mathbf{M}^{-\beta_0} u_t)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\
& \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{14}^+(z_0)}|\mathbf{M}^{-\beta_0} u_t|^{p_0} \mu_1(dz) \right)^{1/p_0}.
\end{split}
\]
In the same way, by applying Lemma \ref{prop:int} to $D_{x'}u$ with $\gamma=\alpha-\beta_0$ and $\alpha_0 = \gamma_1$, we infer that
\[
\begin{split}
& (|\mathbf{M}^{\alpha -\beta_0} DD_{x'} u - (\mathbf{M}^{\alpha -\beta_0} DD_{x'} u)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\
& \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{14}^+(z_0)}|\mathbf{M}^{\alpha -\beta_0} DD_{x'} u|^{p_0} \mu_1(dz) \right)^{1/p_0}.
\end{split}
\]
The oscillation estimate of $Du$ can be proved in a similar way. Therefore, we obtain \eqref{osc-h}. The proof of the lemma is completed.
\end{proof}
Now, we recall that for a given number $a \in \mathbb{R}$,
$a_+ = \max\{a, 0\}$.
We derive the oscillation estimates of solutions to the non-homogeneous equation \eqref{eq:xd}, which is the main result of the subsection.
\begin{lemma} \label{oscil-lemma-2} Let $\nu \in (0,1)$, $\alpha \in (0,2)$, $p_0 \in (1, \infty)$, $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$, and $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha+2)-1)$. There exists $N = N(d,\nu, \alpha, \gamma_1, p_0)>0$ such that the following assertions hold. Suppose that $u \in \sW^{1,2}_{p_0, \textup{loc}}(\Omega_T, x_d^{\gamma_1'}\, dz)$ is a strong solution of \eqref{eq:xd} with $f \in L_{p_0, \textup{loc}}(\Omega_T, x_d^{\gamma_1'}\, dz)$ and $\gamma_1' = \gamma_1 - p_0(\beta_0-\alpha)$. Then, for every $z_0 \in \overline{\Omega}_T$, $\rho \in (0, \infty)$, $\kappa \in (0,1)$, we have
\[
\begin{split}
& (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\
&\leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}
\end{split}
\]
and
\[
\begin{split}
& \lambda^{1/2}(|Du - (Du)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1}|)_{Q_{\kappa \rho}^+(z_0),\bar{\mu}_1} \\
& \leq N \kappa^{\theta} \lambda^{1/2} (|Du|^{p_0})_{Q_{14 \rho}^+(z_0), \bar{\mu}_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha/2} f|^{p_0})_{Q_{14\rho}^+(z_0), \bar{\mu}_1}^{1/p_0},
\end{split}
\]
where $\theta{\in (0,1)}$ is defined in Lemma \ref{oscil-lemma-1},
\[
U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u),
\]
and $\mu_1, \bar{\mu}_1$ are defined in \eqref{mu-1-def}.
\end{lemma}
\begin{proof} As $\gamma_1 \in (p_0(\beta_0-\alpha +1)-1, p_0(\beta_0-\alpha+2)-1)$, we see that
\[
\gamma_1' = \gamma_1 - p_0(\beta_0-\alpha) \in (p_0 -1, 2p_0 -1).
\]
Therefore, by Lemma \ref{l-p-sol-lem}, there is a strong solution $v \in \sW^{1,2}_{p_0}(\Omega_T, x_d^{\gamma_1'}\, dz)$ to
\begin{equation} \label{v-sol-1}
\left\{
\begin{array}{cccl}
\sL_0 v & =& f \mathbf{1}_{Q_{14\rho}^+(z_0)} & \quad \text{in} \quad \Omega_T,\\
v & = & 0 & \quad \text{on} \quad \{x_d =0\}
\end{array} \right.
\end{equation}
satisfying
\begin{equation} \label{v-sol-est-1}
\begin{split}
\|\mathbf{M}^{-\alpha} v_t\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)} & + \|D^2 v\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Dv\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)}\\
& + \lambda \|\mathbf{M}^{-\alpha} v\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)} \leq N \|f\|_{L_{p_0}(Q_{14\rho}^+(z_0), x_d^{\gamma_1'} dz)}.
\end{split}
\end{equation}
Let us denote
\[
V = (\mathbf{M}^{-\beta_0} v_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} v, \lambda \mathbf{M}^{-\beta_0}v).
\]
Then, it follows from \eqref{v-sol-est-1} and the definitions of $\mu_1$ and $\gamma_1'$ that
\begin{equation} \label{V-osc-est-1}
(|V|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \leq N (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}.
\end{equation}
Note also that due to \eqref{v-sol-est-1} and the definition of $\bar{\gamma}_1$,
\[
\begin{split}
\lambda^{1/2} \left( \int_{Q_{14\rho}^+(z_0)} | Dv|^{p_0} x_d^{\bar{\gamma}_1}dz \right)^{1/p_0} & = \lambda^{1/2} \left( \int_{Q_{14\rho}^+(z_0)} |\mathbf{M}^{-\alpha/2}Dv|^{p_0} x_d^{\gamma_1'} dz \right)^{1/p_0} \\
& \leq N \left( \int_{Q_{14\rho}^+(z_0)} |\mathbf{M}^{ \alpha/2} f|^{p_0} x_d^{\bar{\gamma}_1}dz \right)^{1/p_0}.
\end{split}
\]
Then,
\begin{equation} \label{os-Dv-18}
\lambda^{1/2} (|Dv|^{p_0})_{Q_{14\rho}^+(z_0), \bar{\mu}_1}^{1/p_0} \leq N (|\mathbf{M}^{\alpha/2}f|^{p_0})_{Q_{14\rho}^+(z_0), \bar{\mu}_1}^{1/p_0}.
\end{equation}
Now, let $w = u- v$. From \eqref{v-sol-1}, we see that $w \in \sW^{1,2}_{p_0}(Q_{14\rho}^+(z_0), x_d^{\gamma_1'}\, dz)$ is a strong solution of
\[
\left\{
\begin{array}{cccl}
\sL_0 w & =& 0 & \quad \text{in} \quad Q_{14\rho}^+(z_0),\\
w & = & 0 & \quad \text{on} \quad Q_{14\rho}(z_0) \cap \{x_d =0\}.
\end{array}
\right.
\]
Then, by applying Lemma \ref{oscil-lemma-1} to $w$, we see that
\begin{equation} \label{W-osc-est-1}
(|W - (W)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \leq N \kappa^{\theta} (|W|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0}
\end{equation}
and
\begin{equation} \label{os-Dw-18}
(|Dw - (Dw)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1}|)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1} \leq N \kappa^{\theta} (|Dw|^{p_0})_{Q_{14 \rho}^+(z_0), \bar{\mu}_1}^{1/p_0},
\end{equation}
where
\[
\begin{split}
& W= (\mathbf{M}^{-\beta_0} w_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} w, \lambda \mathbf{M}^{-\beta_0}w).
\end{split}
\]
Now, note that from \eqref{def:Q} and \eqref{def:r} we have
\begin{align} \notag
\frac{\mu_1(Q_{14 \rho}^+(z_0))}{\mu_1(Q_{\kappa \rho}^+(z_0))} & = N(d) \kappa^{\alpha-2} \Big(\frac{r(14\rho, x_{0d})}{r(\kappa \rho, x_{0d})}\Big)^{d + (\gamma_1)_+} \\ \label{Q-compared}
& \leq N (d)\kappa^{-(d+ (\gamma_1)_+ + 2-\alpha)}.
\end{align}
Then, it follows from the triangle inequality, H\"{o}lder's inequality, \eqref{W-osc-est-1}, and \eqref{Q-compared} that
\[
\begin{split}
& (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \\
& \leq (|W - (W)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} + (|V - (V)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \\
& \leq (|W - (W)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \\
& \qquad \quad+ N(d) \kappa^{-(d+ (\gamma_1)_+ +2-\alpha)/p_0} (|V|^{p_0})^{1/p_0}_{Q_{14\rho}^+(z_0), \mu_1}\\
& \leq N \kappa^{\theta} (|W|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N(d) \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|V|^{p_0})^{1/p_0}_{Q_{14\rho}^+(z_0), \mu_1}.
\end{split}
\]
As $W = U -V$ and $\kappa \in (0,1)$, we apply the triangle inequality again to see that
\[
\begin{split}
& (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\
&\leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0) , \mu_1}^{1/p_0} \\
& \qquad + N\big( \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} + \kappa^{\theta}\big)(|V|^{p_0})_{Q_{14\rho}^+(z_0) , \mu_1}^{1/p_0} \\
& \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|V|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}.
\end{split}
\]
From this and \eqref{V-osc-est-1}, it follows that
\[
\begin{split}
& (|U - (U)_{Q_{\kappa \rho}^+(z_0) , \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\
& \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0) , \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_++2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0) , \mu_1}^{1/p_0},
\end{split}
\]
where $N = N(d,\nu, \alpha, \gamma_1, p_0)>0$. This proves the assertion on the oscillation of $U$. The oscillation estimate of $Du$ can be proved similarly using \eqref{os-Dv-18} and \eqref{os-Dw-18}. The proof of the lemma is completed.
\end{proof}
We now conclude this subsection by pointing out the following important remark, which can be proved in the same way as Lemma \ref{oscil-lemma-2} with minor modifications.
\begin{remark} \label{all-oscilla-est} Under the assumptions as in Lemma \ref{oscil-lemma-2}, and if $\beta_0 \in {(\alpha-1}, \alpha/2]$, it holds that
\[
\begin{split}
& (|U' - (U')_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\
&\leq N \kappa^{\theta} (|U'|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}
\end{split}
\]
where
\[
U' = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda^{1/2}\mathbf{M}^{\alpha/2-\beta_0} Du, \lambda \mathbf{M}^{-\beta_0}u).
\]
\end{remark}
\subsection{Proof of Theorem \ref{thm:xd}} \label{proof-xd} We are now ready to give the proof of Theorem \ref{thm:xd}.
\begin{proof}[Proof of Theorem \ref{thm:xd}] We begin with the proof of the a priori estimates \eqref{eq:xd-main}--\eqref{eq3.09} assuming that $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$ is a strong solution to the equation \eqref{eq:xd} with
\begin{equation} \label{gamma-alla-range}
\gamma \in (p (\alpha-1)_{+} -1, 2p-1), \quad \text{where} \,\, (\alpha -1)_+ = \max\{\alpha-1, 0\}.
\end{equation}
In our initial step, we prove \eqref{eq:xd-main}--\eqref{eq3.09} with an extra assumption that $u$ is compactly supported. We first prove \eqref{eq:xd-main}. Let $\beta_0 = \min\{1, \alpha\}$, and we will apply Lemma \ref{oscil-lemma-2} with this $\beta_0$. Let $p_0 \in (1, p)$ and $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha+2)-1)$. We choose $p_0$ to be sufficiently close to $1$ and $\gamma_1$ to be sufficiently close to $p_0(\beta_0-\alpha+2)-1$ so that
\begin{equation} \label{nature-choice-2}
\gamma - [\gamma_1 +p(\alpha-\beta_0)] < (1+\gamma_1)(p/p_0 -1).
\end{equation}
We note that this is possible because $\alpha-\beta_0 = (\alpha-1)_+$ and
\[
\gamma - [ \gamma_1 + p(\alpha-\beta_0)] < p[2 - (\alpha-1)_+] -1 -\gamma_1,
\]
and also from our choices of $p_0$ and $\gamma_1$,
\[
\begin{split}
(1+\gamma_1)(p/p_0 -1) & \sim p (1+\gamma_1) -1 -\gamma_1 \sim p[2 - (\alpha-1)_+] -1 -\gamma_1.
\end{split}
\]
Now, let us denote
\begin{equation} \label{gamma-1-pri}
\gamma_1' : = \gamma_1 + p(\alpha-\beta_0) = \gamma_1 + p (\alpha-1)_+.
\end{equation}
Due to \eqref{gamma-alla-range} and the definition of $\gamma_1'$, it follows that
\begin{equation} \label{nature-choice-1}
\gamma - \gamma_1' = \gamma - p(\alpha-1)_+ - \gamma_1 > -1 - \gamma_1.
\end{equation}
From \eqref{nature-choice-1} and \eqref{nature-choice-2}, it holds that
\begin{equation} \label{gamma-0414}
\gamma' : = \gamma - \gamma_1' \in (-1-\gamma_1, (1+\gamma_1)(p/p_0 -1)).
\end{equation}
Now, since $u$ has compactly support in $\Omega_T$, we have $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma_1'} dz)$. Therefore, it follows from Lemma \ref{oscil-lemma-2} that
\[
U^{\#}_{\mu_1} \leq N \Big[ \kappa^{\theta} \cM_{\mu_1}(|U|^{p_0}) ^{1/p_0} + \kappa^{-(d+ (\gamma_1)_+ + 2-\alpha)/2} \cM_{\mu_1}(|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})^{1/p_0} \Big],
\]
where $\mu_1(dz) = x_d^{\gamma_1}dxdt$ and
\[
U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u).
\]
Next, due to \eqref{gamma-0414}, we see that $x_d^{\gamma'} \in A_{p/p_0}(\mu_1)$. It then follows from the weighted Fefferman-Stein theorem and Hardy-Littlewood theorem (i.e., Theorem \ref{FS-thm}) that
\begin{align} \notag
& \|U\|_{L_p(\Omega_T, x_d^{\gamma'}\, d\mu_1)} \leq N \|U^{\#}_{\mu_1}\|_{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)} \\ \notag
& \leq N \Big[\kappa^{\theta}\|\cM_{\mu_1}(|U|^{p_0})^{1/p_0}\|_{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1 )} \\ \notag
& \qquad + \kappa^{-(d + (\gamma_1)_++2-\alpha)/2} \|\cM_{\mu_1}(|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})^{1/p_0}\|_{L_p(\Omega_T, x_d^{\gamma'}\, d\mu_1)} \Big] \\ \label{est:0414-1}
& \leq N \Big[ \kappa^{\theta} \|U\|_{{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)}} + \kappa^{-(d+ (\gamma_1)_++2-\alpha)/2} \| \mathbf{M}^{\alpha-\beta_0} f\|_{{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)}} \Big].
\end{align}
From the definition of $U$, the choices of $\gamma'$ in \eqref{gamma-0414} and $\gamma_1'$ in \eqref{gamma-1-pri}, we have
\[
\begin{split}
\|U\|_{{L_p(\Omega_T, x_d^{\gamma'}d\mu_1)}} & = \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|DD_{x'}u\|_{L_p(\Omega_T, x_d^\gamma dz)} \\
& \quad + \lambda \|\mathbf{M}^{-\alpha} u \|_{L_p(\Omega_T, x_d^\gamma\, dz)}
<\infty.
\end{split}
\]
Then, by choosing $\kappa \in (0,1)$ sufficiently small so that $N \kappa^{\theta} < 1/2$, we obtain from \eqref{est:0414-1} that
\[
\begin{split}
& \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|DD_{x'}u\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \lambda \|\mathbf{M}^{-\alpha} u \|_{L_p(\Omega_T, x_d^\gamma\, dz)} \\
& \leq N \| f\|_{{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)}} = N \| f\|_{{L_p(\Omega_T, x_d^{\gamma}\,dz)}}.
\end{split}
\]
Also, from the PDE in \eqref{eq:xd}, we see that
\[
|D_{dd} u| \leq N[|DD_{x'}u| + (|u_t| + \lambda |u|)x_d^{-\alpha} + |f|],
\]
and therefore
\[
\begin{split}
& \|\mathbf{M}^{-\alpha}u_t\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)}+\lambda\|\mathbf{M}^{-\alpha}u\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)}
+\|D^2 u\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)}\\
&\le N\|f\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)},
\end{split}
\]
which is \eqref{eq:xd-main}.
Next, we prove the estimate \eqref{eq3.09} also with the extra assumption that $u$ has compact support. We observe that if $\gamma \in (p -1, 2p -1)$, \eqref{eq3.09} follows from \eqref{est-0405-1}. Therefore, it remains to consider the case that $\gamma \in (\alpha p/2-1, p -1]$ or equivalently
\begin{equation} \label{gamma-range-2}
\gamma - \alpha p/2 \in (-1, p(1-\alpha/2) -1].
\end{equation}
The main idea is to apply Lemma \ref{oscil-lemma-2} with this $\beta_0 = \alpha/2$. Let $p_0, \gamma_1$ be as before but with the new choice of $\beta_0$. As noted in \eqref{mu1=bar-mu-1}, we have
\[ \bar{\gamma}_1 = \gamma_1 - p_0(\beta_0 -\alpha/2) = \gamma_1 \qquad \text{and} \qquad \bar{\mu}_1 = \mu_1.
\]
Because of \eqref{gamma-range-2}, we can perform the same calculation as the one that yields \eqref{gamma-0414} to obtain
\[
\bar{\gamma}' := \gamma - (\bar{\gamma}_1 + p\alpha/2 ) \in (-1 - \bar{\gamma}_1, (1+\bar{\gamma}_1)(p/p_0 -1))
\]
and therefore $x_d^{\bar{\gamma}'} \in A_{p/p_0}(\bar{\mu}_1)$. By using Lemma \ref{oscil-lemma-2} , we have
\begin{equation} \label{Du-sharp}
\begin{split}
\lambda^{1/2} (Du)^{\#}_{\bar{\mu}_1} & \leq N \Big[ \kappa^{\theta} \lambda^{1/2}\cM_{\bar{\mu}_1}(|Du|^{p_0}) ^{1/p_0} \\
& \quad + \kappa^{-(d+ \bar{\gamma}_1 + 2-\alpha)/2} \cM_{\bar{\mu}_1}(|\mathbf{M}^{\alpha/2} f|^{p_0})^{1/p_0} \Big],
\end{split}
\end{equation}
where $\bar{\mu}_1(dz) = x_d^{\bar{\gamma}_1}dxdt$. We apply Theorem \ref{FS-thm} to \eqref{Du-sharp}, and then choose $\kappa>0$ sufficiently small as in the proof of \eqref{eq:xd-main} to obtain
\[
\lambda^{1/2}\|Du\|_{L_p(\Omega_T, x_d^{\bar{\gamma}'} d\bar{\mu}_1)} \leq N \|\mathbf{M}^{\alpha/2} f\|_{L_p(\Omega_T, x_d^{\bar{\gamma}'}\, d\bar{\mu}_1)}.
\]
This implies
\[
\lambda^{1/2} \|\mathbf{M}^{-\alpha/2}Du\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \leq N \| f\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)}
\]
as $\gamma - p\alpha/2= \bar{\gamma}' + \bar{\gamma}_1$. The estimate \eqref{eq3.09} is proved.
Now, we prove \eqref{eq:xd-main}--\eqref{eq3.09} without the assumption that $u$ is compactly supported. As $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma dz)$, there is a sequence $\{u_n\}$ in $C_0^\infty(\Omega_T)$ such that
\begin{equation} \label{u-approx-compact}
\lim_{n\rightarrow \infty} \|u_n -u\|_{\sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)} =0.
\end{equation}
Let $f_n = f + \sL_0 (u_n - u)/\mu(x_d)$ and observe that $u_n$ is a strong solution of
\[
\sL_0 u_n = \mu(x_d) f_n \quad \text{in} \quad \Omega_T \quad \text{and} \quad u_n =0 \quad \text{on} \quad \{x_d =0\}.
\]
Then, applying the estimates \eqref{eq:xd-main}--\eqref{eq3.09} to $u_n$, we obtain
\begin{equation} \label{un-supported}
\|u_n\|_{\sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)} \leq N\|f_n\|_{L_p(\Omega_T, x_d^\gamma\, dz)}.
\end{equation}
Note that
\[
\begin{split}
& \|f_n\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \leq \|f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + N\lambda \|\mathbf{M}^{-\alpha} (u-u_n)\|_{L_p(\Omega_T, x_d^\gamma\, dz)}\\
& \qquad + N \Big [ \|D^2(u-u_n)\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha}(u-u_n)_t\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \| \Big] \\
& \rightarrow \|f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \quad \text{as} \quad n \rightarrow \infty.
\end{split}
\]
Therefore, by taking $n\rightarrow \infty$ in \eqref{un-supported} and using \eqref{u-approx-compact}, we obtain the estimates \eqref{eq:xd-main}--\eqref{eq3.09} for $u$. Hence, the proof of \eqref{eq:xd-main}--\eqref{eq3.09} is completed.
It remains to prove the existence of a strong solution $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ to \eqref{eq:xd} assuming that $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, for $p \in (1, \infty)$ and $\gamma \in (p (\alpha-1)_{+} -1, 2p-1)$. We observe when $\gamma \in (p-1, 2p-1)$, the existence of solution is already proved in Lemma \ref{l-p-sol-lem}. Therefore, it remains to consider the case when
$$\gamma \in (p (\alpha-1)_{+} -1, p-1].$$
We consider two cases.
\noindent
{\em Case} 1.
Consider $\gamma \in (p(\alpha-1)_+ -1, p-1)$.
As $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, there is a sequence $\{f_k\}_k \subset C_0^\infty(\Omega_T)$ such that
\begin{equation} \label{fk-approx}
\lim_{k\rightarrow \infty}\|f_k - f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} =0.
\end{equation}
For each $k \in \mathbb{N}$, because $f_k$ has compact support, we see that
\[ x_d^{1-\alpha} \mu(x_d) f_k \sim x_d f_k \in L_p(\Omega_T, x_d^{\gamma}\, dz). \]
Then, as in the proof of Lemma \ref{l-p-sol-lem}, we apply \cite[Theorem 2.4]{DPT21} to find a weak solution $u_k \in \cH^1_p(\Omega_T, x_d^{\gamma}\, dz)$ to the divergence form equation \eqref{eq:xd-div} with $f_k$ in place of $f$. Moreover,
\begin{equation} \label{uk-Hp-est}
\|Du_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} < \infty.
\end{equation}
We claim that $u_k \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ for each $k \in \mathbb{N}$. Note that if the claim holds, we can apply the a priori estimate that we just proved for the equations of $u_k$ and of $u_k - u_l$ to get
\[
\begin{split}
& \|u_k\|_{\sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)} \leq N \|f_k\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \quad \text{and} \\
& \|u_k - u_l\|_{\sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)} \leq N \|f_k - f_l\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)}
\end{split}
\]
for any $k, l \in \mathbb{N}$, where $N = N(\nu, \gamma, \alpha, p)>0$ which is independent of $k, l$. The last estimate and \eqref{fk-approx} imply that the sequence $\{u_k\}_k$ is convergent in $\sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$. Let $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ be the limit of such sequence, we see that $u$ solves \eqref{eq:xd}.
Hence, in this case, it remains to prove the claim that $u_k \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ for every $k \in \mathbb{N}$. Also, let us fix $k \in \mathbb{N}$, and let us denote $\Omega_T' = (-\infty, T) \times \mathbb{R}^{d-1}$. Let $0 < r_0 <R_0$ such that
\begin{equation} \label{fk-support}
\text{supp}(f_k) \subset {\Omega_T'} \times (r_0, R_0).
\end{equation}
Without loss of generality, we assume that $
r_0 =2$.
From \eqref{uk-Hp-est}, it follows directly that
\[
\begin{split}
& \|Du_k\|_{L_p(\Omega_T' \times (1, \infty), x_d^{\gamma -p}\, dz)} + \|u_k\|_{L_p(\Omega_T' \times (1, \infty), x_d^{\gamma-2p}\, dz)} \\
& \qquad + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (1,\infty), x_d^{\gamma-p}\, dz)} <\infty.
\end{split}
\]
Then, we can follow the proof of Lemma \ref{l-p-sol-lem} to show that
\[
\|u_k\|_{\sW^{1,2}_{p}(\Omega_T' \times (1,\infty), x_d^{\gamma}\, dz)} <\infty.
\]
It now remains to prove that $u_k \in \sW^{1,2}_{p}({\Omega_T'\times (0,1)}, x_d^{\gamma}\, dz)$ and
\begin{equation} \label{near-est-uk}
\|u_k\|_{\sW^{1,2}_{p}(\Omega_T' \times (0, 1), x_d^{\gamma}\, dz)} < \infty.
\end{equation}
To this end, because of \eqref{fk-support}, we note that $u_k$ solves the homogeneous equation
\begin{equation} \label{uk-ne-zero}
\sL_0 u_k =0 \quad \text{in} \quad \Omega_T' \times (0, 2)
\end{equation}
with the boundary condition $u_k =0$ on $\{x_d =0\}$. Let us denote
\[
\begin{split}
& C_r = [-1, 0) \times \big\{ x = (x_1, \ldots, x_d) \times \mathbb{R}^{d}_+ : {\max_{1 \leq i \leq d}|x_i|}<r\big\}, \\
& C_r(t,x) = C_r + (t,x), \quad r >0.
\end{split}
\]
Consider $\alpha \in (0, 1)$. By using Lemmas \ref{caccio}, and \ref{prop:boundary} with a scaling argument and translation, we obtain
\begin{equation*}
\begin{split}
& \|\mathbf{M}^{-\alpha} u_k\|_{L_\infty(C_{1}(z_0))} + \|Du_k\|_{L_\infty(C_{1}(z_0))} + \|\mathbf{M}^{-\alpha} \partial_t u_k\|_{L_\infty(C_{1}(z_0))} \\
& \quad + \|DD_{x'} u_k\|_{L_\infty(C_{1}(z_0))} \leq N \Big[\|Du_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma dz)} \Big]
\end{split}
\end{equation*}
for every $z_0 = (t_0, x_0', 0) \in \Omega_T' \times\{0\}$. Note that $N$ depends on $k$, but is independent of $z_0$. This and the PDE in \eqref{uk-ne-zero} imply that
\[
\begin{split}
& \|\mathbf{M}^{-\alpha} u_k\|_{L_\infty(C_{1}(z_0))} + \|Du_k\|_{L_\infty(C_{1}(z_0))} + \|\mathbf{M}^{-\alpha} \partial_t u_k\|_{L_\infty(C_{1}(z_0))} \\
& \quad + \|D^2 u_k\|_{L_\infty(C_{1}(z_0))} \leq N \Big[\|Du_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} \Big].
\end{split}
\]
Then, as $\gamma > -1$, we see that
\[
\begin{split}
& \|\mathbf{M}^{-\alpha}u_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)}
+ \|Du_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)}
+ \|\mathbf{M}^{-\alpha}\partial_t u_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)} \\
& + \|\ D^2u_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)} \leq N \Big[\|Du_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} \Big].
\end{split}
\]
Then, with $z_0 = (t_0, x_0', 0)$ and with $\mathcal{I} = ({(\mathbb{Z}+T)} \cap (-\infty, T{]}) \times (2\mathbb{Z})^{d-1}$, we have
\[
\begin{split}
\|u_k\|^p_{\sW^{1,2}_p(\Omega_T' \times (0,1))} & = \sum_{(t_0', x_0') \in \mathcal{I} } \|u_k\|^p_{\sW^{1,2}_p(C_1(z_0))} \\
& \leq N \sum_{(t_0', x_0') \in \mathcal{I} }\Big[\|Du_k\|^p_{L_{p}(C_{2}(z_0))} + \|\mathbf{M}^{-\alpha/2} u_k\|^p_{L_{p}(C_{2}(z_0))} \Big] \\
& = N\Big[ \|Du_k\|^p_{L_{p}(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|^p_{L_{p}(\Omega_T, x_d^\gamma\, dz)}\Big] <\infty.
\end{split}
\]
Hence, \eqref{near-est-uk} holds.
Now, we consider $\alpha \in [1, 2)$.
As $\gamma + p (1-\alpha) >-1$, we see that
\[
\begin{split}
& \int_{C_1(z_0)} |x_d^{-\alpha} u_k(z)|^p x_d^\gamma dz = \int_{C_1(z_0)} |x_d^{-1}u_k(z)|^p x_d^{\gamma + p (1-\alpha)} dz \\
& \leq N \|Du_k\|^p_{L_\infty(C_1(z_0))} \\
& \leq N\Big[ \|Du_k\|^p_{L_{p}(C_2(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|^p_{L_{p}(C_2(z_0), x_d^\gamma\, dz)}\Big].
\end{split}
\]
Then, by taking the sum of this inequality for $(t_0, x_0') \in \mathcal{I}$, we also obtain
\[
\|\mathbf{M}^{-\alpha} u_k\|_{L_p(\Omega_T' \times (0,1), x_d^\gamma\, dz)} \leq N \Big[ \|Du_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha}u_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \Big].
\]
Similarly, we also have $\mathbf{M}^{-\alpha} (u_k)_t, Du_k\in L_p(\Omega_T' \times (0,1), x_d^\gamma\,dz)$.
By using the different quotient, we also get $DD_{x'} u_k \in L_p(\Omega_T' \times (0,1), x_d^\gamma\,dz)$.
From this, and the PDE of $u_k$, we have $D^2u_k \in L_p(\Omega_T' \times (0,1), x_d^\gamma\, dz)$.
Therefore, \eqref{near-est-uk} holds. The proof of the claim in this case is completed.
\noindent
{\em Case} 2.
We consider $\gamma =p-1$. Let $\{f_k\}_k$ be as in \eqref{fk-approx} and let $\bar{\gamma} \in (p(\alpha-1)_+ -1, p-1)$. As in {\em Case 1}, we can find a weak solution $u_k \in \cH^1_p(\Omega_T, x_d^{\bar{\gamma}}\, dz)$ to the divergence form equation \eqref{eq:xd-div} with $f_k$ in place of $f$, and
\begin{equation} \label{uk-Hp-est-b}
\|Du_k\|_{L_p(\Omega_T, x_d^{\bar{\gamma}}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T, x_d^{\bar{\gamma}}\, dz)} < \infty.
\end{equation}
We claim that for each $k \in \mathbb{N}$,
\begin{equation} \label{uk-Hp-est-b-1}
\|Du_k\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} < \infty.
\end{equation}
Once this claim is proved, we can follow the proof in {\em Case 1} to obtain the existence of a solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$. Therefore, we only need to prove \eqref{uk-Hp-est-b-1}.
Let us fix $k \in \mathbb{N}$ and let $0 < r_0 < R_0$ such that \eqref{fk-support} holds. As $\bar{\gamma} < \gamma$, we see that
\[
\begin{split}
& \|Du_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^\gamma\, dz)} \\
& \leq N\Big[ \|Du_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^{\bar\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^{\bar\gamma}\, dz)}\Big] <\infty
\end{split}
\]
due to \eqref{uk-Hp-est-b}. Hence, it remains to prove
\begin{equation} \label{uk-Hp-est-b-2}
\|Du_k\|_{L_p(\Omega_T' \times (2R_0, \infty), x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (2R_0, \infty), x_d^\gamma\, dz)} < \infty.
\end{equation}
To prove \eqref{uk-Hp-est-b-2}, we use the localization technique along the $x_d$ variable.
See \cite[Proof of Theorem 4.5, Case II]{DP-JFA}. We skip the details.
\end{proof}
\section{Equations with partially VMO coefficients} \label{sec:4}
We study \eqref{eq:main} in this section.
Precisely, we consider the equation
\begin{equation}\label{eq:main-1}
\begin{cases}
\sL u=\mu(x_d) f \quad &\text{ in } \Omega_T,\\
u=0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+,
\end{cases}
\end{equation}
where $\sL$ is defined in \eqref{L-def} in which the coefficients $a_0$, $c_0$, and $a_{ij}$ are measurable functions depending on $z = (z', x_d) \in \Omega_T$. We employ the perturbation method by freezing the coefficients. For $z_0 = (z'_0, x_{0d}) \in \overline{\Omega}_T$, let $[{a}_{ij}]_{Q_{\rho}'(z'_0)}, [a_{0}]_{Q_{\rho}'(z'_0)}$, and $[c_{0}]_{Q_{\rho}'(z'_0)}$ be functions defined in Assumption \ref{assumption:osc} $(\delta, \gamma_1, \rho_0)$, and we denote
\begin{equation} \label{a-sharp-def}
\begin{split}
a^{\#}_{\rho_0}(z_0) & =\sup_{\rho\in(0,\rho_0)}\left[ \max_{i,j=1, 2,\ldots, d}\fint_{Q_{\rho}^+(z_0)}|a_{ij}(z) -[{a}_{ij}]_{Q_{\rho}'(z'_0)}(x_d)| \mu_1(dz) \right. \\
& \qquad + \fint_{Q_{\rho}^+(z_0)}|a_{0}(z) -[{a}_{0}]_{Q_{\rho}'(z'_0)}(x_d)| \mu_1(dz) \\
& \qquad \left. + \fint_{Q_{\rho}^+(z_0)}|c_{0}(z) -[{c}_{0}]_{Q_{\rho}'(z'_0)}(x_d)| \mu_1(dz) \right].
\end{split}
\end{equation}
For the reader's convenience, recall that $\mu_1, \bar{\mu}_1$ are defined in \eqref{mu-1-def}. We also recall that for a given $u$, we denote
\[
U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u).
\]
We also denote
\[
U' = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda^{1/2} \mathbf{M}^{\alpha/2-\beta_0}Du, \lambda \mathbf{M}^{-\beta_0}u).
\]
We begin with the following oscillation estimates for solutions to \eqref{eq:main-1} that have small supports in the time-variable.
\begin{lemma} \label{osc-est-small} Let $\nu, \rho_0 \in (0,1)$, $p_0 \in (1, \infty)$, $\alpha \in (0,2)$, $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$, $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha +2)-1)$, and $\gamma_1' = \gamma_1-p_0(\beta_0-\alpha)\in (p_0-1,2p_0-1)$. Assume that $u \in \sW^{1,2}_{p}(Q_{6\rho}^+(z_0), x_d^{\gamma_1'}dz)$ is a strong solution of
\[
\left\{
\begin{array}{cccl}
\sL u & = & \mu(x_d) f & \quad \text{in} \quad Q_{6\rho}^+(z_0),\\
u & = & 0 & \quad \text{on} \quad Q_{6\rho}(z_0) \cap \{x_d =0\}
\end{array} \right.
\]
for $f \in L_{p_0}(Q_{6\rho}^+(z_0), x_d^{{\gamma_1'}}dz)$. Assume in addition that $\textup{supp}(u) \subset (t_1 -(\rho_0 \rho_1)^{2-\alpha}, t_1 +(\rho_0 \rho_1)^{2-\alpha})$ for some $t_1 \in \mathbb{R}$ and $\rho_0 >0$. Then,
\begin{align} \notag
&\big (|U - (U)_{Q_{\kappa\rho}^+(z_0), \mu_1}|\big)_{Q_{\kappa\rho}^+(z_0), \mu_1} \\ \notag
& \leq N \Big[\kappa^{\theta} + \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} \big( a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] (|U|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} \\ \label{U-osc-gen}
& \qquad + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0},
\end{align}
where $\theta>0$ is defined in Lemma \ref{oscil-lemma-1}, $p\in (p_0,\infty)$, and $N = N(p, p_0, \gamma_1, \alpha, \beta_0, d, \nu)>0$. In addition, if $\beta_0 \in {(\alpha-1}, \alpha/2]$, we also have
\begin{align} \notag
&\big (|U' - (U')_{Q_{\kappa\rho}^+(z_0), \mu_1}|\big)_{Q_{\kappa\rho}^+(z_0), \mu_1} \\ \notag
& \leq N \Big[\kappa^{\theta} + \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} \big( a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] (|U'|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} \\ \label{Uall-osc-gen}
& \qquad + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}.
\end{align}
\end{lemma}
\begin{proof} We split the proof into two cases.
\noindent
{\em Case 1.} We consider $\rho < \rho_0/14$. We denote
\[
\sL_{\rho, z_0} u =[a_0]_{Q_{6\rho}'(z_0')}(x_d) u_t + \lambda [c_0]_{Q_{6\rho}'(z_0')}(x_d)u - \mu(x_d) [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d) D_i D_j u
\]
and
\[
\begin{split}
\tilde{f}(z) & = f(z) + [a_{ij} - [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d)] D_i D_ju \\
& \qquad + \big[ \lambda ([c_0]_{Q_{6\rho}'(z_0')} - c_0) u + ([a_0]_{Q_{6\rho}'(z_0')} - a_0) u_t \big]/\mu(x_d).
\end{split}
\]
Then, $u \in \sW^{1,2}_{p}(Q_{6\rho}^+(z_0), x_d^{\gamma_1'}dz)$ is a strong solution of
\[
\left\{
\begin{array}{cccl}
\sL_{\rho, z_0} u & = &\mu(x_d) \tilde{f} & \quad \text{in} \quad Q_{6\rho}^+(z_0)\\
u & = & 0 & \quad \text{on} \quad Q_{6\rho}^+(z_0) \cap \{x_d =0\}.
\end{array} \right.
\]
We note that due to \eqref{add-assumption}, the term $a_{dd} - \bar{a}_{dd} =0$.
Therefore, by using H\"{o}lder's inequality and \eqref{con:ellipticity}, we obtain
\[
\begin{split}
& \left(\fint_{Q_{14\rho}^+(z_0)}|\mathbf{M}^{\alpha-\beta_0} \big(a_{ij} - [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d)\big) D_i D_ju|^{p_0} \mu_1(dz) \right)^{1/p_0} \\
& \leq \left(\fint_{Q_{14\rho}^+(z_0)}|a_{ij} - [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d)|^{pp_0/(p-p_0)} \mu_1(dz) \right)^{\frac{1}{p_0} -\frac{1}{p}} \\
& \qquad \times \left(\fint_{Q_{14\rho}^+(z_0)}|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u|^{p} \mu_1(dz)\right)^{1/p} \\
& \leq N a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} \left(\fint_{Q_{14\rho}^+(z_0)}|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u|^{p} \mu_1(dz)\right)^{1/p}.
\end{split}
\]
By a similar calculation using \eqref{con:mu}, we also obtain the estimate for the term $\big[ \lambda ([c_0]_{Q_{6\rho}'(z_0')}(x_d) - c_0) u + ([a_{0}]_{Q_{6\rho}'(z_0')}(x_d) - a_0) u_t \big]/\mu(x_d)$. Thus,
\[
\begin{split}
(|\mathbf{M}^{\alpha-\beta_0} \tilde{f}|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} & \leq (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \\
& \qquad + N a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} (|U|^{p})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p}.
\end{split}
\]
Then, applying Lemma \ref{oscil-lemma-2}, we obtain
\[
\begin{split}
& (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\
&\leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} \tilde{f}|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \\
& \leq N \big(\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} \big) (|U|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} \\
& \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}.
\end{split}
\]
Therefore, \eqref{U-osc-gen} holds. In a similar way but applying Remark \ref{all-oscilla-est}, we also obtain \eqref{Uall-osc-gen}.
\noindent
{\em Case 2.} Consider $\rho \geq \rho_0/14$. Denoting $\Gamma = (t_1 -(\rho_0 \rho_1)^{2-\alpha}, t_1 + (\rho_0 \rho_1)^{2-\alpha})$, we apply \eqref{Q-compared} and the triangle inequality to infer that
\[
\begin{split}
& \fint_{Q_{\kappa\rho}^+(z_0)} |U - (U)_{Q_{\kappa\rho}^+(z_0), \mu_1}|
\mu_1(dz)\leq 2 \fint_{Q_{\kappa\rho}^+(z_0)} |U(z)|\mu_1(dz) \\
& \leq N \kappa^{-(d+2 -\alpha + (\gamma_1)_+)} \left(\fint_{Q_{14\rho}^+(z_0)} |U(z)|^{p_0}\mu_1(dz)\right)^{\frac 1 {p_0}} \left(\fint_{Q_{14\rho}^+(z_0)} \mathbf{1}_{\Gamma}(z) \mu_1(dz)\right)^{1-\frac 1 {p_0}} \\
& \leq N \kappa^{-(d+2 -\alpha + (\gamma_1)_+)} \rho_1^{(2-\alpha)(1-1/p_0)}\left(\fint_{Q_{14\rho}^+(z_0)} |U(z)|^{p_0}\mu_0(dz)\right)^{1/p_0} \\
& \leq N \kappa^{-(d+2 -\alpha + (\gamma_1)_+)} \rho_1^{(2-\alpha)(1-1/p_0)}(|U|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} .
\end{split}
\]
Therefore, \eqref{U-osc-gen} follows. Similarly, \eqref{Uall-osc-gen} can be proved.
\end{proof}
Our next lemma gives the a priori estimates of solutions having small supports in $t$.
\begin{lemma}[Estimates of solutions having small supports] \label{small-support-sol} Let $T \in (-\infty, \infty]$, $\nu \in (0,1)$, $p, q, K \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$ for $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$. Then, there exist sufficiently small positive numbers $\delta$ and $\rho_1$, depending on $d, \nu, p, q, K, \alpha{,\beta_0}$, and $\gamma_1$, such that the following assertion holds. Suppose that $\omega_0 \in A_q(\mathbb{R})$, $\omega_1 \in A_p(\mathbb{R}^d_+, \mu_1)$ with
\[
[\omega_0]_{A_q(\mathbb{R})} \leq K \qquad \text{and} \qquad [\omega_1]_{A_p(\mathbb{R}^d_+, \mu_1)} \leq K.
\]
Suppose that \eqref{con:mu}, \eqref{con:ellipticity}, and \eqref{add-assumption} hold, and \textup{Assumption \ref{assumption:osc}}$(\delta, \gamma_1, \rho_0)$ holds with some $\rho_0>0$. If $u \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$ is a strong solution to \eqref{eq:main} with some $\lambda>0$ and a function $f\in L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$, and $u$ vanishes outside $(t_1 - (\rho_0\rho_1)^{2-\alpha}, t_1+(\rho_0\rho_1)^{2-\alpha})$ for some $t_1 \in \mathbb{R}$, then
\begin{equation} \label{est-1-small-supp}
\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \|f\|_{L_{q,p}},
\end{equation}
where $N = N(d,\nu, p, q, \alpha,{ \beta_0,}\gamma_1, K)>0$, $L_{q,p}=L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$, $\omega(t,x) =\omega_0(t)\omega_1(x)$ for $(t,x) \in \Omega_T$, and $\mu_1(dz) = x_d^{\gamma_1}\, dxdt$. Moreover, if $\beta_0 \in [0, \alpha/2]$, then it also holds that
\begin{equation} \label{est-2-small-supp}
\begin{split}
& \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \\
& \leq N \|f\|_{L_{q,p}}.
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
As $\omega_0 \in A_q((-\infty,T))$ and $\omega_1 \in A_p(\mathbb{R}^d_+, d\mu_1)$, by the reverse H\"older's inequality \cite[Theorem 3.2]{MS1981}, we find $p_1=p_1(d,p,q,\gamma_1,K)\in (1,\min(p,q))$ such that
\begin{equation} \label{eq0605_13}
\omega_0 \in A_{q/p_1}((-\infty,T)),\quad
\omega_1 \in A_{p/p_1}(\mathbb{R}^d_+, \mu_1).
\end{equation}
Because $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$, we can choose $p_0 \in (1, p_1)$ sufficiently closed to $1$ so that
\[
\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha +2) -1).
\]
By \eqref{U-osc-gen} of Lemma \ref{osc-est-small} and H\"{o}lder's inequality, we have
\[
\begin{split}
U^{\#}_{\mu_1} \leq & N \Big[\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] \cM_{\mu_1}(|U|^{p_1})^{1/p_1} \\
& \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \cM_{\mu_1}(|\mathbf{M}^{\alpha-\beta_0} f|^{p_1})^{1/p_1}
\quad \text{in} \quad \overline{\Omega_T}
\end{split}
\]
for any $\kappa\in (0,1)$, where $N = N(\nu, d, p_0, p_1, \alpha,\beta_0, \gamma_1) >0$ and $a_{\rho_0}^{\#}$ is defined in \eqref{a-sharp-def}. Therefore, it follows from Theorem \ref{FS-thm} and \eqref{eq0605_13} that
\[
\begin{split}
& \norm{U}_{L_{q,p}(\Omega_T, \omega\, d\mu_1)}\\
& \leq N \Big[\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(\delta^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] \times \\
& \qquad \qquad \times \|\cM_{\mu_1}(|U|^{p_1})^{1/p_1}\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \\
& \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \| \cM_{\mu_1} (|\mathbf{M}^{\alpha -\beta_0} f|^{p_1})^{\frac 1 {p_1}}\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \\
& \leq N \Big[\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(\delta^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] \|U\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \\
& \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \|\mathbf{M}^{\alpha-\beta_0} f\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)},
\end{split}
\]
where $N = N(d,\nu, p, q, \alpha,\beta_0, \gamma_1, K)>0$. Now, by choosing $\kappa$ sufficiently small and then $\delta$ and $\rho_1$ sufficiently small depending on $d,\nu, p, q,\alpha, \gamma_1$, and $K$ such that
\[
N\Big [\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(\delta^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] <1/2,
\]
we obtain
\[
\begin{split}
& \norm{U}_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \leq N(d, \nu, p, q, \alpha,\beta_0 \gamma_0, K) \|\mathbf{M}^{\alpha -\beta_0} f\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)}.
\end{split}
\]
From this and the PDE in \eqref{eq:main}, we obtain
\[
\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \| f\|_{L_{q,p}}.
\]
This proves \eqref{est-1-small-supp}. The proof of \eqref{est-2-small-supp} is similar by applying \eqref{Uall-osc-gen} instead of \eqref{U-osc-gen}.
\end{proof}
Below, we provide the proof of Theorem \ref{main-thrm}.
\begin{lemma}[A priori estimates of solutions] \label{apriori-est-lemma} Let $T \in (-\infty, \infty]$, $ \nu \in (0,1)$, $p, q, K \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$ for $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$. Then, there exist $\delta = \delta(d, \nu, p, q, K, \alpha, \beta_0,\gamma_1)>0$ sufficiently small and $\lambda_0 = \lambda_0(d, \nu, p, q, K, \alpha,\beta_0, \gamma_1)>0$ sufficiently large such that the following assertions hold. Let $\omega_0 \in A_q(\mathbb{R})$, $\omega_1 \in A_p(\mathbb{R}^d_+, \mu_1)$ satisfy
\[
[\omega_0]_{A_q(\mathbb{R})} \leq K \qquad \text{and} \qquad [\omega_1]_{A_p(\mathbb{R}^d_+, \mu_1)} \leq K.
\]
Suppose that \eqref{con:mu}, \eqref{con:ellipticity}, and \eqref{add-assumption} hold, and suppose that \textup{Assumption \ref{assumption:osc}}$ (\delta, \gamma_1, \rho_0)$ holds with some $\rho_0>0$. If $u \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega\,d\mu_1)$ is a strong solution to \eqref{eq:main} with some $\lambda{\ge \lambda_0\rho_0^{\alpha-2}}$ and $f \in L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega\, d\mu_1)$, then
\begin{equation} \label{main-est-1-b}
\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \|f\|_{L_{q,p}},
\end{equation}
where $\omega(t, x) = \omega_0(t) \omega_1(x)$ for $(t,x) \in \Omega_T$, $L_{q,p} = L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega \,d\mu_1)$, and $N = N(d, \nu, p, q, \alpha,{\beta_0,K,} \gamma_1)>0$. Moreover, if $\beta_0 \in {(\alpha-1}, \alpha/2]$, then it also holds that
\begin{equation} \label{main-est-2-b}
\begin{split}
& \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \\
& \leq N \|f\|_{L_{q,p}}.
\end{split}
\end{equation}
\end{lemma}
\begin{proof} Let $\delta, \rho_1$ be positive numbers in Lemma \ref{small-support-sol}, and let $\lambda_0>$ be a number sufficiently large to be determined, depending on $d, p, q, \alpha,{\beta_0,\nu,} \gamma_1, K$. As the proof of \eqref{main-est-1-b} and of \eqref{main-est-2-b} are similar, we only prove the a priori estimate \eqref{main-est-1-b}. We use a partition of unity argument in the time variable.
Let $\delta>0$ and $\rho_1>0$ be as in Lemma \ref{small-support-sol} and let
$$
\xi=\xi(t) \in C_0^\infty( -(\rho_0\rho_1)^{2-\alpha}, (\rho_0\rho_1)^{2-\alpha})
$$
be a non-negative cut-off function satisfying
\begin{equation} \label{xi-0702}
\int_{\mathbb{R}} \xi(s)^q\, ds =1 \qquad \text{and} \qquad \int_{\mathbb{R}}|\xi'(s)|^q\,ds \leq \frac{N}{(\rho_0\rho_1)^{q(2-\alpha)}}.
\end{equation}
For fixed $s \in (-\infty, \infty)$, let $u^{(s)}(z) = u(z) \xi(t-s)$ for $z = (t, x) \in \Omega_T$.
We see that $u^{(s)} \in \sW^{1,2}_p(\Omega_T,x_d^{p(\alpha-\beta_0)}\omega\, d\mu_1)$ is a strong solution of
\[
\sL u^{(s)}(z) =\mu(x_d) f^{(s)} (z) \quad \text{in} \quad \Omega_T
\]
with the boundary condition $u^{(s)} =0$ on $\{x_d =0\}$, where
\[
f^{(s)}(z) = \xi(t-s) f(z) + \xi'(t-s) u(z)/\mu(x_d).
\]
As $\text{spt}(u^{(s)}) \subset (s -(\rho_0\rho_1)^{2-\alpha}, s+ (\rho_0\rho_1)^{2-\alpha}) \times \mathbb{R}^{d}_{+}$, we apply Lemma \ref{small-support-sol} to get
\[
\begin{split}
\|\mathbf{M}^{-\alpha} \partial_tu^{(s)}\|_{L_{q,p}} + \|D^2u^{(s)}\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u^{(s)}\|_{L_{q,p}} \leq N \|f^{(s)}\|_{L_{q,p}}.
\end{split}
\]
Then, by integrating the $q$-th power of this estimate with respect to $s$, we get
\begin{align}\notag
& \int_{\mathbb{R}}\Big(|\mathbf{M}^{-\alpha} \partial_tu^{(s)}\|_{L_{q,p}}^q + \|D^2u^{(s)}\|_{L_{q,p}}^q + \lambda^q \|\mathbf{M}^{-\alpha} u^{(s)}\|_{L_{q,p}}^q\Big)\, ds\\ \label{par-int-0515}
& \leq N\int_{\mathbb{R}} \|f^{(s)}\|_{L_{q,p}}^q\, ds.
\end{align}
Now, by the Fubini theorem and \eqref{xi-0702}, it follows that
\[
\begin{split}
& \int_{\mathbb{R}}\|\mathbf{M}^{-\alpha} \partial_tu^{(s)}\|_{L_{q,p}}^q\, ds\\
& = \int_{\mathbb{R}} \left(\int_{-\infty}^T \|\mathbf{M}^{-\alpha}u_t(t,\cdot)\|_{L_p(\mathbb{R}^d_+, x_d^{p(\alpha-\beta_0)} \omega_1\, d\mu_1)}^q \omega_0(t) \xi^q(t-s)\, dt \right)\, ds \\
&= \int_{-\infty}^T \left( \int_{\mathbb{R}}\xi^q(t-s)\, ds \right) \|\mathbf{M}^{-\alpha}u_t(t,\cdot)\|_{L_p(\mathbb{R}^d_+,x_d^{p(\alpha-\beta_0)} \omega_1\, d\mu_1)}^q \omega_0(t)\, dt \\
& = \|\mathbf{M}^{-\alpha}u_t \|_{L_{q,p}(\mathbb{R}^d_+,x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)}^q,
\end{split}
\]
and similarly
\[
\begin{split}
& \int_{\mathbb{R}} \| D^2u^{(s)}\|_{L_{q,p}}^q\, ds = \|\mathbf{M}^{\alpha-\beta_0} D^2u\|_{L_{q,p}}^q, \\
& \int_{\mathbb{R}} \|\mathbf{M}^{-\alpha} u^{(s)}\|_{L_{q,p}}^q\, ds = \|\mathbf{M}^{-\beta_0} u\|_{L_{q,p}}^q .
\end{split}
\]
Moreover,
\[
\int_{\mathbb{R}} \|f^{(s)}\|_{L_{q,p}}^q\, ds \leq \|f\|_{L_{q,p}}^q + \frac{N}{(\rho_0\rho_1)^{q(2-\alpha)}} \|\mathbf{M}^{-\alpha}u\|_{L_{q,p}}^q,
\]
where \eqref{xi-0702} is used and $N = N(q)>0$. As $\rho_1$ depends on $d, \nu, p, q, K, \alpha{,\beta_0,\gamma_1}$, by combining the estimates we just derived, we infer from \eqref{par-int-0515} that
\[
\begin{split}
& \|\mathbf{M}^{-\alpha} \partial_tu \|_{L_{q,p}} + \| D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N\Big(\|f\|_{L_{q,p}} + \rho_0 ^{\alpha-2} \|\mathbf{M}^{-\alpha}u\|_{L_{q,p}} \Big)
\end{split}
\]
with $N=N(d, \nu, \alpha, p, q, \gamma_1) >0$.
Now we choose $\lambda_0 = 2N$.
Then, with $\lambda \geq \lambda_0 \rho_0^{\alpha-2}$, we have
\[
\begin{split}
& \|\mathbf{M}^{-\alpha} \partial_tu \|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \| f\|_{L_{q,p}} .
\end{split}
\]
This estimate yields \eqref{main-est-1-b}.
\end{proof}
Now, we have all ingredients to complete the proof of Theorem \ref{main-thrm}.
\begin{proof}[Proof of Theorem \ref{main-thrm}] The a priori estimates \eqref{main-est-1} and \eqref{main-est-2} follow from Lemma \ref{apriori-est-lemma}. Hence, it remains to prove the existence of solutions. We employ the the technique introduced in \cite[Section 8]{Dong-Kim-18}. See also \cite[Proof of Theorem 2.3]{DP-JFA}. The proof is split into two steps, and we only outline the key ideas in each step.
\noindent
{\em Step 1.} We consider the case $p =q$, $\omega_0 \equiv 1$, and $\omega_1 \equiv 1$. We employ the method of continuity. Consider the operator
\[
\sL_\tau = (1-\tau)\big(\partial_t + \lambda - \mu(x_d) \Delta\big) + \tau \sL, \qquad \tau \in [0, 1].
\]
It is a simple calculation to check that the assumptions in Theorem \ref{main-thrm} are satisfied uniformly with respect to $\tau \in [0,1]$. Then, using the solvability in Theorem \ref{thm:xd} and the a priori estimates obtained in Lemma \ref{apriori-est-lemma}, we get the existence of a solution $u \in \sW^{1,2}_p(\Omega_T, x_d^{p(\alpha-\beta_0)}\, d\mu_1)$ to \eqref{eq:main} when $\lambda \geq \lambda_0 \rho_0^{\alpha-2}$, where $\lambda_0$ is the constant in Lemma \ref{apriori-est-lemma}.
\noindent
{\em Step 2.} We combine {\em Step 1} and Lemma \ref{apriori-est-lemma} to prove the existence of {a strong} solution $u$ satisfying \eqref{main-est-1}.
Let $p_1 > \max\{p,q\}$ be sufficiently large and let $\varepsilon_1, \varepsilon_2 \in (0,1)$ be sufficiently small depending on $K, p, q$, and $\gamma_1$ such that
\begin{equation} \label{epsilon12-def}
1-\frac{p}{p_1} = \frac{1}{1+\varepsilon_1} \qquad \text{and} \qquad 1 - \frac{q}{p_1} = \frac{1}{1+\varepsilon_2},
\end{equation}
and both $\omega_1^{1+\varepsilon_1}$ and $\omega_0^{1+\varepsilon_2}$ are locally integrable and satisfy the doubling property. Specifically, there is $N_0>0$ such that
\begin{equation} \label{omega-0}
\int_{\Gamma_{2r}(t_0)} \omega_0^{1+\varepsilon_2}(s)\, ds \leq N_0 \int_{\Gamma_{r}(t_0)} \omega_0^{1+\varepsilon_2}(s)\, ds
\end{equation}
for any $r>0$ and $t_0 \in \mathbb{R}$, where $\Gamma_{r}(t_0) = (t_0 -r^{2-\alpha}, \min\{t_0 + r^{2-\alpha}, T\})$. Similarly
\begin{equation} \label{omega-1-0308}
\int_{B_{2r}^+(x_0)} \omega_1^{1+\varepsilon_1}(x)\, d\mu_1 \leq N_0\int_{B_{r}^+(x_0)} \omega_1^{1+\varepsilon_1}(x)\, d\mu_1
\end{equation}
for any $r >0$ and $x_0 \in \overline{\mathbb{R}^d_+}$.
Next, let $\{f_k\}$ be a sequence in $C_0^\infty(\Omega_T)$ such that
\begin{equation} \label{f-k-converge-0227}
\lim_{k\rightarrow \infty} \|f_k - f\|_{L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega\, d\mu_1)} =0.
\end{equation}
By {\em Step 1}, for each $k \in \mathbb{N}$, we can find a solution $u_k \in \sW^{1,2}_{p_1}(\Omega_T,x_d^{p_1(\alpha-\beta_0)}\, d\mu_1)$ of \eqref{eq:main} with $f_k$ in place of $f$, where $\lambda \geq \lambda_0 \rho_0^{\alpha-2}$ for $\lambda_0 = \lambda_0(d, \nu, p_1, p_1,\alpha,\beta_0, \gamma_1, K)>0$. Observe that if the sequence $\{u_k\}$ is in $\sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega \, d\mu_1)$, then by applying the a priori estimates in Lemma \ref{apriori-est-lemma}, \eqref{f-k-converge-0227}, and the linearity of the equation \eqref{eq:main}, we conclude that $\{u_k\}$ is Cauchy in $\sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$.
Let $u \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$ be the limit of the sequence $\{u_k\}$. Then, by letting $k \rightarrow \infty$ in the equation for $u_k$, we see that $u$ solves \eqref{eq:main}.
It remains to prove that for each fixed $k \in \mathbb{N}$, $u_k \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$. To this end, let us denote
\[
D_{R} = (-R^{2-\alpha}, \min\{R^{2-\alpha}, T\}) \times B_R^+.\]
Then, let $R_0>0$ be sufficiently large such that
\begin{equation} \label{fk-spt}
\operatorname{supp}(f_k) \subset D_{R_0}.
\end{equation}
We note that $R_0$ depends on $k$. It follows from \eqref{epsilon12-def}, \eqref{omega-0}, \eqref{omega-1-0308}, and H\"{o}lder's inequality that
\[
\begin{split}
& \|u_k\|_{\sW^{1,2}_{q,p}(D_{2R_0},x_d^{p(\alpha-\beta_0)} \omega d\mu_1)} \\
& \leq N(d, p, q, p_1, \alpha, \gamma_1, R_0) \|u_k\|_{\sW^{1,2}_{p_1}(D_{2R_0}, x_d^{p_1(\alpha-\beta_0)} d\mu_1)} <\infty.
\end{split}
\]
Hence, we only need to prove
\[
\|u_k\|_{\sW^{1,2}_{q,p}(\Omega_T\setminus D_{R_0},x_d^{p(\alpha-\beta_0)} \omega d\mu_1)} <\infty.
\]
This is done by the localization technique employing \eqref{epsilon12-def}, \eqref{omega-0}, \eqref{omega-1-0308}, \eqref{fk-spt}, and H\"{o}lder's inequality, using the fast decay property of solutions when the right-hand side is compactly supported.
We skip the details as the calculation is very similar to that of \cite[Section 8]{Dong-Kim-18}, and also of \cite[Step II - Proof of Theorem 2.3]{DP-JFA}. The proof of Theorem \ref{main-thrm} is completed.
\end{proof}
Next, we prove Corollary \ref{cor1}.
\begin{proof}[Proof of Corollary \ref{cor1}] It is sufficient to show that we can make the choices for $\gamma_1, \beta_0$, and $\omega_1$ to apply Theorem \ref{main-thrm} to obtain \eqref{cor-est-1} and \eqref{cor-est-2}. Indeed, the choices are similar to those in the proof of Theorem \ref{thm:xd}. To obtain \eqref{cor-est-1}, we take $\beta_0 = \min\{1, \alpha\}$, and with this choice of $\beta_0$, we have
\[
\alpha - \beta_0 = (\alpha-1)_+ \quad \text{and} \quad (\beta_0 -\alpha, \beta_0-\alpha +1] = (-(\alpha -1)_+, 1- (\alpha-1)_+].
\]
Then, let $\gamma_1 = 1- (\alpha-1)_+$ and $\gamma' = \gamma - [\gamma_1 + p(\alpha-1)_+]$.
From the choice of $\gamma_1$ and the condition on $\gamma$, we see that
\begin{equation} \label{cond-gamma-1}
-1-\gamma_1 < \gamma' < (1+\gamma_1) (p-1).
\end{equation}
Now, let
$ \omega_1(x) = x_d^{\gamma'}$ for $x \in \mathbb{R}^d_+$.
It follows from \eqref{cond-gamma-1} that $\omega_1 \in A_p(\mu_1)$.
As Assumption $(\rho_0, \gamma_1, \delta)$ holds,
we can apply \eqref{main-est-1} to obtain \eqref{cor-est-1}.
Next, we prove \eqref{cor-est-2}. In this case, we choose $\beta_0 = \alpha/2$, $\gamma_1 = 1-\alpha/2$, and
\begin{equation} \label{cond-gamma-2}
\gamma' =\gamma - [\gamma_1 + p\alpha/2].
\end{equation}
We use the fact that $\gamma \in (p\alpha/2-1, 2p-1)$ and \eqref{cond-gamma-2} to get \eqref{cond-gamma-1}.
As Assumption $(\rho_0, 1-\alpha/2, \delta)$ holds, by taking $\omega_1(x) = x_d^{\gamma'}$, we obtain \eqref{cor-est-2} from \eqref{main-est-2}.
The proof is complete.
\end{proof}
\section{Degenerate viscous Hamilton-Jacobi equations}\label{sec:5}
To demonstrate an application of the results in our paper, we consider the following degenerate viscous Hamilton-Jacobi equation
\begin{equation}\label{eq:nonlinear}
\begin{cases}
u_t+\lambda u-\mu(x_d) \Delta u=H(z,Du) \quad &\text{ in } \Omega_T,\\
u=0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+,
\end{cases}
\end{equation}
where $\mu$ satisfies \eqref{con:mu} and $H:\Omega_T \times \mathbb{R}^d \to \mathbb{R}$ is a given Hamiltonian.
We assume that there exist $\beta, \ell >0$, and $h:\Omega_T \to {\overline{\mathbb{R}_+}}$ such that, for all $(z,P) \in \Omega_T \times \mathbb{R}^d$,
\begin{equation} \label{G-cond}
|H(z,P)| \leq{ \nu^{-1} (\min\{x_d^\beta,1\} |P|^{\ell}+x_d^\alpha h(z))}.
\end{equation}
The following is the main result in this section.
\begin{theorem} \label{example-thrm}
Let $p \in (1, \infty)$, $\alpha \in (0,2)$, and $\gamma \in (p(\alpha-1)_+-1, 2p-1)$.
Assume that \eqref{G-cond} holds with $\ell =1$, $\beta \geq 1$, and $h \in L_p(\Omega_T, x_d^\gamma\, dz)$.
Then, there exists $\lambda_0 = \lambda_0(d, p, \alpha, \beta, \gamma)>0$ sufficiently large such that the following assertion holds.
For any $\lambda \geq \lambda_0$, there exists a unique solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma \,dz)$ to \eqref{eq:nonlinear} such that
\[
\|\mathbf{M}^{-\alpha} u_t\|_{L_p} + \|D^2 u\|_{L_p} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p} \leq N \|h\|_{L_p}
\]
where $\|\cdot\|_{L_p} = \|\cdot \|_{L_p(\Omega_T, x_d^\gamma\, dz)}$ and $N = N(d, p, \alpha, \beta, \gamma)>0$.
\end{theorem}
\begin{proof}
The proof follows immediately from Theorem \ref{thm:xd} and the interpolation inequality in Lemma \ref{interpolation-inq} (i) below.
\end{proof}
\begin{remark}
Overall, it is meaningful to study \eqref{eq:nonlinear} for general Hamiltonians $H$.
It is typically the case that if we consider \eqref{eq:nonlinear} in $(0,T)\times \mathbb{R}^d_+$ with a nice given initial data, then we can obtain Lipschitz a priori estimates on the solutions via the classical Bernstein method or the doubling variables method under some appropriate conditions on $H$.
See \cite{CIL, AT, LMT} and the references therein.
In particular, $\|Du\|_{L^\infty([0,T]\times \mathbb{R}^d_+)} \leq N$, and hence, the behavior of $H(z,P)$ for $|P|>2N+1$ is unrelated and can be modified according to our purpose.
As such, if we assume \eqref{G-cond}, then it is natural to require that $\ell=1$ because of the above.
We note however that assuming \eqref{G-cond} with $\ell=1$ and $\beta \geq 1$ in Theorem \ref{example-thrm} is rather restrictive.
It is not yet clear to us what happens when $0\leq \beta<1$, and we plan to revisit this point in the future work.
\end{remark}
To obtain a priori estimates for solutions to \eqref{eq:nonlinear}, we consider the nonlinear term $H$ as a perturbation.
We prove the following interpolation inequalities when the nonlinear term satisfies \eqref{G-cond} with $\ell=1$ and $\ell=2$, which might be of independent interests.
\begin{lemma} \label{interpolation-inq} Let $p \in (1, \infty), \beta\ge 0, \gamma>-1$, $1 \leq \ell \leq \frac{d}{d-p}$, and $\theta = \frac{1}{2}(1+\frac{d}{p}-\frac{d}{\ell p})$. Assume that $H$ satisfies \eqref{G-cond}. The following interpolation inequalities hold for every $u \in C_0^\infty(\Omega_T)$ and $\tilde f(z) = x_d^{-\alpha} {\min\{x_d^\beta,1\}|Du|^\ell}$,
\begin{itemize}
\item[(i)] If $\ell=1$ and $\beta \geq 1$,
\[
\begin{split}
\|\tilde f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} & \leq N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{1/2}\|D^2 u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{1/2} \\
& \qquad +N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)},
\end{split}
\]
where $N = N(d, p, \beta, \gamma) >0$.
\item[(ii)] If $\ell=2$, $p \geq \frac{d}{2}$, and $\beta \geq \max\{\frac{\gamma}{p}+\frac{d\alpha}{2p}, \frac{\gamma}{p}+2 +\frac{\alpha}{d} - \frac{d\alpha}{p}\}$, then
\[
\begin{split}
\|\tilde f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} & \leq N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{2(1-\theta)}\|D^2 u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{2 \theta} \\
& \qquad +N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^2,
\end{split}
\]
where $N = N(d, p, \beta, \gamma) >0$.
\end{itemize}
\end{lemma}
\begin{proof}
For $m\in \mathbb{Z}$, set $
\Omega_m=\{z\in \Omega_T\,:\, 2^{-m-1} < x_d \leq 2^{-m}\}$.
By the Gagliado-Nirenberg interpolation inequality, for $m\in \mathbb{Z}$,
\[
\|Du\|_{L_{p\ell}(\Omega_m)} \leq N \left (\|u\|_{L_p(\Omega_m)}^{1-\theta} \|D^2u\|_{L_p(\Omega_m)}^\theta + 2^{2m\theta} \|u\|_{L_p(\Omega_m)} \right).
\]
Hence, for $m\geq 0$,
\begin{align*}
&\|\mathbf{M}^{\beta-\alpha} |Du|^\ell\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^p
=\int_{\Omega_m} x_d^{p(\beta-\alpha)+\gamma}|Du|^{p \ell}\,dz\\
&\leq \, 2^{-m(p(\beta-\alpha)+\gamma)} \int_{\Omega_m} |Du|^{p\ell}\,dz\\
&\leq \, N2^{-m(p(\beta-\alpha)+\gamma)} \left(\int_{\Omega_m} |u|^{p}\,dz\right)^{\ell(1-\theta)} \left(\int_{\Omega_m} |D^2u|^{p}\,dz\right)^{\ell \theta} \\
&\qquad+ N2^{-m(p(\beta-\alpha)+\gamma+d-p\ell-d\ell)} \left(\int_{\Omega_m} |u|^{p}\,dz\right)^\ell\\
&\leq \, N2^{-m(p(\beta-\alpha)+\gamma+ p \ell \alpha(1-\theta)-\ell\gamma )} \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell (1-\theta)}\|D^2 u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell\theta} \\
&\qquad+ N2^{-m(p(\beta-\alpha)+\gamma+d-p \ell-d\ell+p \ell\alpha-\ell \gamma)}\|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell}.
\end{align*}
By performing similar computations, we get that, for $m< 0$,
\begin{align*}
&\|\mathbf{M}^{-\alpha} |Du|^\ell \|_{L_p(\Omega_m,x_d^\gamma)}^p\\
&\leq \, N2^{-m(-p\alpha+\gamma+ p \ell \alpha(1-\theta)-\ell \gamma )} \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell(1-\theta)}\|D^2 u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell \theta} \\
&\qquad + N2^{-m(-p\alpha+\gamma+d-p\ell -d\ell +p\ell\alpha-\ell \gamma)}\|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p\ell}.
\end{align*}
Then, if $\ell=1$ and $\beta \geq 1$, we have
\[
\begin{cases}
p(\beta-\alpha)+\gamma+ p \ell \alpha(1-\theta)-\ell \gamma =p(\beta - \frac{\alpha}{2}) \geq 0,\\
p(\beta-\alpha)+\gamma+d-p\ell-d\ell+p\ell\alpha-\ell\gamma =p(\beta-1) \geq 0,\\
-p\alpha+\gamma+ p \ell \alpha(1-\theta)-\ell \gamma = -\frac{p\alpha}{2} \leq 0,\\
-p\alpha+\gamma+d-p\ell -d\ell +p\ell \alpha-\ell \gamma=-p \leq 0.
\end{cases}
\]
We thus obtain (i).
Similarly, the above four inequalities hold true when $\ell=2$, $p \geq \frac{d}{2}$, and $\beta \geq \max\{\frac{\gamma}{p}+\frac{d\alpha}{2p}, \frac{\gamma}{p}+2 +\frac{\alpha}{d} - \frac{d\alpha}{p}\}$, which yield (ii).
\end{proof}
\def$'${$'$}
\end{document} |
\begin{document}
\author{Giorgio Ottaviani, Alicia Tocino}
\address{Dipartimento di Matematica e Informatica ``Ulisse Dini'', University of Florence, Italy}
\email{[email protected], [email protected]}
\title{\textbf{Best rank k approximation for binary forms}}
\begin{abstract}
In the tensor space $\mathrm{Sym}^d \mathbb{R}^2$ of binary forms we study the best rank $k$ approximation problem. The critical points of the best rank $1$ approximation problem are the eigenvectors and it is known that they span a hyperplane.
We prove that the critical points of the best rank $k$ approximation problem lie in the same hyperplane.
\end{abstract}
\maketitle
\mathbb{S}ection{Introduction}
The symmetric tensor space $\mathrm{Sym}^dV$, with $V=\mathbb{R}^2$ (resp. $V=\mathbb{C}^2$), contains real (resp. complex) binary forms,
which are homogeneous polynomials in two variables. The forms which can be written as $v^d$, with $v\in V$, correspond to polynomials which are the $d$-power of a linear form,
they have rank one.
We denote by $C_d\mathbb{S}ubset \mathrm{Sym}^dV$ the variety of forms of rank one.
The $k$-secant variety $\mathbb{S}igma_k(C_d)$ is the closure of the set of forms which can be written as $\mathbb{S}um_{i=1}^k\lambda_iv_i^d$ with $\lambda_i\in\mathbb{R}$ (resp. $\lambda_i\in\mathbb{C}$).
We say that a nonzero rank $1$ tensor is a critical rank one tensor for $f\in \mathrm{Sym}^d V$ if it is a critical point of the distance function from $f$ to the variety of rank $1$ tensors.
Critical rank one tensors are important to determine the best rank one approximation of $f$, in the setting of optimization
\cite{FriTam, Lim, Stu}. Critical rank one tensors may be written as $\lambda v^d$ with $\lambda\in\mathbb{C}$ and $v\cdot v=1$, the last scalar product is the Euclidean scalar product.
The corresponding vector $v\in V$ has been called tensor eigenvector, independently by Lim and Qi, \cite{Lim, Qi}. In this paper we concentrate on
critical rank one tensors $\lambda v^d$, which live in $\mathrm{Sym}^dV$ (not in $V$ like the eigenvectors), for a better comparison with critical rank $k$ tensors, see Definition \ref{def:kcritical} .
There are exactly $d$ critical rank one tensor (counting with multiplicities)
for any $f$ different from $c(x^2+y^2)^{d/2}$ (with $d$ even), while
there are infinitely many critical rank one tensors for $f=(x^2+y^2)^{d/2}$ (see Prop. \ref{prop:eigendisc}).
The critical rank one tensors for $f$ are contained in the hyperplane $H_f$
(called the singular space, see \cite{OP}), which is orthogonal to the vector $D(f)=yf_x-xf_y$. We review this statement at the beginning of \S \ref{sec:singularspace}.
The main result of this paper is the following extension of the previous statement to critical rank $k$ tensors, for any $k\ge 1$.
\begin{thm}\label{thm:main} Let $f\in \mathrm{Sym}^d\mathbb{C}^2$ .
i) All critical rank $k$ tensors for $f$ are contained in the hyperplane $H_f$, for any $k\ge 1$.
ii) Any critical rank $k$ tensor for $f$ may be written as a linear combination of the critical rank $1$ tensors for $f$.
\end{thm}
Theorem \ref{thm:main} follows after Theorem \ref{mainTheorem} and Proposition \ref{prop:main2}. Note that Theorem \ref{thm:main} may applied to the best rank $k$ approximation of $f$, which turns out to be contained in $H_f$
and may then be written as a linear combination of the critical rank $1$ tensors for $f$. This statement may be seen as a weak extension of the Eckart-Young Theorem to tensors. Indeed, in the case of matrices, the best rank $k$ approximation is exactly the sum of the first $k$ critical rank one tensors, by the Eckart-Young Theorem, see \cite{OP}. The polynomial $f$ itself may be written as linear combination of its critical rank $1$ tensors, see Corollary \ref{cor:corf}, this statement may be seen as a {\it spectral decomposition for $f$}. All these statements may be generalized to the larger class of tensors, not necessarily symmetric, in any dimension, see \cite{DOT}.
In \S\ref{sec:lastreal} we report about some numerical experiments regarding the number of real critical rank $2$ tensors in $\mathrm{Sym}^4\mathbb{R}^2$.
\mathbb{S}ection{Preliminaries}
Let $V=\mathbb{R}^2$ equipped with the Euclidean scalar product.
The associated quadratic form has the coordinate expression $x^2+y^2$,
with respect to the orthonormal basis $x, y$.
The scalar product can be extended to a scalar product on the tensor space $\mathrm{Sym}^dV$ of binary forms,
which is $SO(V)$-invariant.
For powers $l^d$, $m^d$ where $l, m\in V$, we set
$\langle l^d, m^d\rangle : = \langle l, m\rangle^d$
and by linearity this defines the scalar product on the whole $\mathrm{Sym}^dV$ (see Lemma \ref{lema:scalarproduct}).
Denote as usual $\left\|{f}\right\|=\mathbb{S}qrt{\langle f, f\rangle}$.
For binary forms which split in the product of linear forms we have the formula
\begin{equation}\label{eq:decomp}\langle l_1l_2\cdots l_d, m_1m_2\cdots m_d\rangle =
\frac{1}{d!}\mathbb{S}um_{\mathbb{S}igma}\langle l_1,m_{\mathbb{S}igma(1)}\rangle \langle l_2,m_{\mathbb{S}igma(2)}\rangle
\cdots \langle l_d,m_{\mathbb{S}igma(d)}\rangle \end{equation}
The powers $l^d$ are exactly the tensors of rank one in $\mathrm{Sym}^dV$,
they make a cone $C_d$ over the rational normal curve.
The sums $l_1^d+\ldots +l_k^d$ are the tensors of rank $\le k$, and equality holds when the number of summands is minimal. The closure of the set of tensors of rank $\le k$, both in the Euclidean or in the Zariski topology, is a cone $\mathbb{S}igma_kC_d$,
which is the $k$-secant variety of $C_d$.
The Euclidean distance function $d(f,g)=\left\|f-g\right\|$ is our objective function.
The optimization problem we are interested is, given a real $f$, to minimize $d(f,g)$ with the
constraint that $g\in \left(\mathbb{S}igma_kC_d\right)_\mathbb{R}$. This is equivalent to minimize the square function $d^2(f,g)$,
which has the advantage to be algebraic. The number of complex critical points of the square distance function $d^2$
is called the Euclidean distance degree (EDdegree \cite{DHOST}) of $\mathbb{S}igma_kC_d$ and has been computed for small values of
$k, d$ in the rightmost chart in Table 4.1 of \cite{OSS}. We do not know a closed formula for these values,
although \cite[Theorem 3.7]{OSS} computes them in the case of a general quadratic distance function, not
$SO(2)$-invariant.
\mathbb{S}ection{Critical points of the distance function}
Let us recall the notion of eigenvector for symmetric tensors (see \cite{Lim, Qi},\cite[Theorem 4.4]{OP}).
\begin{defn}\label{def:eigentensor}
Let $f\in \mathrm{Sym}^d V$. We say that a nonzero rank $1$ tensor is a critical rank one tensor for $f$ if it is a critical point of the distance function from $f$ to the variety of rank $1$ tensors. It is convenient to write a critical rank one tensor in the form $\lambda v^d$ with $\left\|{v}\right\|=1$, in this way
$v$ is defined up to $d$-th roots of unity and is called an eigenvector of $f$ with eigenvalue $\lambda$.
\end{defn}
\begin{remark}
Let $d=2$ and let $f$ be a symmetric matrix.
All the critical rank one tensors of $f$ have the form $\lambda v^2$ where $v$ is a classical eigenvector of norm $1$ for the symmetric matrix $f$, with eigenvalue $\lambda$.
\end{remark}
\begin{lem}\label{lem:eigentensor}
Given $f\in\mathrm{Sym}^d V$, the point $\lambda v^d$ of rank $1$, with $\left\|{v}\right\|=1$, is a critical rank one tensor for $f$ if and only if $\langle f,v^{d-1}w\rangle=
\lambda \langle v,w\rangle$ $\forall w\in V$, which can be written (identifying $V$ with $V^\vee$ according to the Euclidean scalar product) as
$$f\cdot v^{d-1}= \lambda v,$$ with $\lambda=\langle f, v^d\rangle $.
\end{lem}
\begin{proof} The property of critical point is equivalent to $f-\lambda v^d$ being orthogonal to
$v^{d-1}w$ $\forall w\in V$, which gives
$\langle f, v^{d-1}w\rangle =\langle \lambda v^d,v^{d-1}w\rangle$
$\forall w\in V$. The right-hand side is $\left\|{v}\right\|^{2d-2} \lambda \langle v,w\rangle=\lambda \langle v,w\rangle$, as we wanted.
Setting $w=v$ we get $\langle f, v^{d}\rangle =\lambda$.
\end{proof}
On the other hand, eigenvectors correspond to critical points of the function
$f(x,y)$ restricted on the circle $S^1=\{(x,y)|x^2+y^2=1\}$ (\cite{Lim, Qi}).
By Lagrange multiplier method, we can compute the eigenvectors of $f$ as the normalized solutions $(x,y)$ of:
\begin{equation}\label{eq1}
\mathrm{rank}
\begin{bmatrix}
f_{x} & f_{y} \\
x & y
\end{bmatrix} \leq 1
\end{equation}
This corresponds with the roots of discriminant polynomial $D(f)=yf_x-xf_y$.
$D$ is a well known differential operator which satisfies the Leibniz rule, i.e. $D(fg)=D(f)g+fD(g)$
$\forall f, g\in \mathrm{Sym}^d V$.
For any $l=ax+by\in V$ denote $l^\perp=D(l)=-bx+ay$. Note that $\langle l,l^\perp\rangle =0$.
We have the following:
\begin{prop}\label{prop:eigendisc} Consider $f(x,y)\in\mathrm{Sym}^dV$:
\begin{itemize}
\item If $v$ is eigenvector of $f$ then $D(v)=v^\perp$ is a linear factor of $D(f)$.
\item Assume that $D(f)$ splits as product of distinct linear factors and $v^\perp|D(f)$, then $\frac{v}{\left\|{v}\right\|}$ is an eigenvector of $f$.
\end{itemize}
\end{prop}
We postpone the proof after Prop. \ref{prop:critical}.
Now let us differentiate some cases in terms of $D(f)$ (see Theorem $2.7$ of \cite{ASS}):
\begin{itemize}
\item if $d$ is odd: $D(f)= 0$ if and only if $f= 0$, in particular $D:\mathrm{Sym}^dV\rightarrow \mathrm{Sym}^dV$ is an isomorphism.
\item if $d$ is even: $D(f)=0$ if and only if $f=c(x^2+y^2)^{d/2}$ for some $c\in\mathbb{R}$. We will show in Lemma \ref{lemma2} which are the eigenvectors in this case. The image of $D:\mathrm{Sym}^dV\rightarrow \mathrm{Sym}^dV$ is the space orthogonal to $f=(x^2+y^2)^{d/2}$.
\end{itemize}
\begin{lem} (\cite{LS}, Section $2$)\label{lema:scalarproduct}
Suppose $f=\mathbb{S}um_{i=0}^{d} \binom{d}{i} a_i x^iy^{d-i}$ and $g=\mathbb{S}um_{i=0}^{d} \binom{d}{i} b_i x^iy^{d-i}$. Then we get:
\begin{equation}\label{eq:scalar}
\langle f,g\rangle:=\mathbb{S}um_{i=0}^{d} \binom{d}{i} a_ib_i
\end{equation}
where $\langle\,,\,\rangle$ is the scalar product defined in the introduction.
\end{lem}
\begin{proof}
By linearity we may assume $f=(\alpha x+\beta y)^d$ and $g=(\alpha'x+\beta' y)^d$. The right-hand side of (\ref{eq:scalar}) gives
$$\langle f,g\rangle=\mathbb{S}um_{i=0}^{d} \binom{d}{i}(\alpha\alpha')^i(\beta \beta')^{d-i}=(\alpha\alpha'+\beta\beta')^d$$
which agrees with $\langle \alpha x+\beta y,\alpha' x+\beta' y\rangle^d$.
\end{proof}
\begin{lem}\label{remark:2}
Let $f=(x^2+y^2)^{d/2}\in \mathrm{Sym}^d V$ with $d$ even, and $v=\alpha x+\beta y\in V$, $v\neq 0$, then $\langle v^d,f\rangle=\left\|{v}\right\|^d$.
\end{lem}
\begin{proof}
By applying (\ref{eq:decomp}) with a grain of salt (e.g. decomposing $x^2+y^2$
into two conjugates linear factors) we get
$$\langle v^d,f\rangle=\langle (x^2+y^2),v^2\rangle^{d/2} =
(\alpha^2+\beta^2)^{d/2}=\left\|{v}\right\|^d.$$
\end{proof}
\begin{lem}\label{lemma2}
If $f=(x^2+y^2)^{d/2}\in \mathrm{Sym}^d V$ then, for every nonzero $v\in V$,
$\langle f, v^{d-1}w\rangle=\left\|{v}\right\|^{d-2}\langle v, w\rangle$.
In particular every vector $v$ of norm $1$ is eigenvector of $f$ with eigenvalue $1$.
\end{lem}
\begin{proof}
As in Lemma \ref{remark:2} we get
$$\langle f, v^{d-1}w\rangle = {\langle (x^2+y^2),v^2\rangle}^{d/2-1}
\langle (x^2+y^2),vw\rangle = \left\|{v}\right\|^{d-2}\langle v, w\rangle.$$
The second part follows by putting $w=v$ and equating with Lemma \ref{remark:2}. We get
$\langle f, v^{d-1}w\rangle=\langle v^d,f\rangle\langle v, w\rangle$ just in the case $|v|=1$.
\end{proof}
\begin{remark}
Lemma \ref{lemma2} extends the fact that every vector of norm $1$ is eigenvector of the identity matrix with eigenvalue $1$.
The geometric interpretation of this lemma
is that the $2$-dimensional cone
of rank $1$ degree $d$ binary forms cuts any sphere centered in $(x^2+y^2)^{d/2}$
in a curve. This curve
\end{remark}
\begin{lem}\label{lem}
The normal space at $l^d\in C_d$ coincides with $\left(l^\perp\right)^2\cdot \mathrm{Sym}^{d-2}V$
\end{lem}
\begin{proof}
The tangent space at $l^d$ is spanned by $l^{d-1}V$ and has dimension $2$. The elements in $\left(l^\perp\right)^2\cdot \mathrm{Sym}^{d-2}V$ are orthogonal to the tangent space,
moreover the dimension of this space is the expected one $d-1$.
\end{proof}
\begin{defn}\label{def:kcritical}
We say that $g\in\mathrm{Sym}^dV$ is a critical rank $k$ tensor for $f$ if it is a critical point of the distance function $d(f,\_)$ restricted on $\mathbb{S}igma_kC_d$.
\end{defn}
\begin{prop}\label{prop:critical} Let $2k\le d$. A polynomial $g=\mathbb{S}um_{i=1}^k \mu_i l_i^d \in \mathbb{S}igma_kC_d$ is a critical rank $k$ tensor for $f$ if and only if there exist $h\in\mathrm{Sym}^{d-2k}V$ such that
\begin{equation}\label{eq}
f=\mathbb{S}um_{i=1}^k \mu_i l_i^d +h\cdot \prod_{i=1}^k\left(l_i^\perp\right)^2
\end{equation}
\end{prop}
\begin{proof}
By Terracini Lemma, the tangent space of the point $g\in\mathbb{S}igma_kC_d$ is given by the sum of $k$ tangent spaces at $l_i^d=(a_ix+b_iy)^d$. By Lemma \ref{lem} the normal space of each of these tangent spaces are given by $\left(l_i^\perp\right)^2\cdot \mathrm{Sym}^{d-2}V$. Hence, the normal space to $g$ is given by intersection of the $k$ normal spaces, which is given by
polynomials $\prod_{i=1}^k\left(l_i^\perp\right)^2 \cdot h$ where $h\in\mathrm{Sym}^{d-2k}V$.
Now suppose that $g$ is a critical rank $k$ tensor for $f$. This means that $f-g$ is in the normal space. Hence, $f-g$ is of the form $\prod_{i=1}^k\left(l_i^\perp\right)^2 \cdot h$ for some $h\in\mathrm{Sym}^{d-2k}V$.
Conversely, if $(\ref{eq})$ holds, we need that $f-g$ belongs to the normal space at $g$ which is also true by the construction of the normal space.
\end{proof}
\begin{proof} [Proof of Prop. \ref{prop:eigendisc}]
If $v$ is eigenvector of $f$ then $\langle f,v^d\rangle v^d$ is critical rank $1$ tensor for $f$ (by Lemma \ref{lem:eigentensor}). By Prop. \ref{prop:critical}
$f=\langle f,v^d\rangle v^d+h \left(v^\perp\right)^2$ where $h\in\mathrm{Sym}^{d-2}V$. Applying the operator $D$ to $f$ we get by Leibniz rule, since $D(v)=v^{\perp}$ and $D(v^{\perp})=-v$:
$$D(f)=\langle f,v^d\rangle dv^{d-1}v^\perp+D(h)\left(v^\perp\right)^2-2vv^\perp h\Longrightarrow v^\perp|D(f)$$
Conversely, since we assume there are $d$ distinct eigenvectors, then we find all the linear factors of $D(f)$.
\end{proof}
This proposition is connected with Theorem $2.5$ of \cite{LS}.
\mathbb{S}ection{The singular space}\label{sec:singularspace}
In \cite{OP} it was considered the singular space $H_f$ as the hyperplane orthogonal to $D(f)=yf_x-xf_y$. It follows from Prop. \ref{prop:eigendisc}
that the critical rank $1$ tensor for $f$ belong to $H_f$ (since the eigenvectors of $f$ can be computed as the solutions of (\ref{eq1}) that coincides with $D(f)$ for binary forms), see \cite[Def. 5.3]{OP}.
It is worth to give a direct proof that the critical rank $1$ tensors for $f$ belong to $H_f$, the hyperplane orthogonal to $D(f)$,
based on Prop.
\ref{prop:critical}.
Let $\mu l^d$ be a critical rank $1$ tensors for $f$, then by Prop.
\ref{prop:critical} there exist $h\in\mathrm{Sym}^{d-2}V$ such that
$f= \mu l^d +h\left(l^\perp\right)^2$.
We have to prove $\langle D(f), l^d\rangle =0$ which follows immediately from (\ref{eq:decomp})
since $l^{\perp}$ divides $D(f)$ by Prop. \ref{prop:eigendisc}.
\begin{lem}\label{lem:lmperp}
Let $l, m\in V$, Then $\langle l^\perp, m\rangle +\langle m^\perp, l\rangle =0$.
\end{lem}
\begin{proof}
Straightforward.
\end{proof}
Our main result is
\begin{thm}\label{mainTheorem}
The critical points of the form $\mathbb{S}um_{i=1}^{k}\mu_i l_i^d$ of the distance function $d(f,-)$ restricted on $\mathbb{S}igma_kC_d$
belong to $H_f$.
\end{thm}
\begin{proof}
Given a decomposition
$f= \mathbb{S}um_{i=1}^k \mu_i l_i^d +h\cdot \prod_{i=1}^k\left(l_i^\perp\right)^2$, with $h\in\mathrm{Sym}^{d-2k}V$,
we compute
\begin{equation}\label{eq:3sum}
D(f)=d\mathbb{S}um_{i=1}^k \mu_il_i^{\perp}l_i^{d-1}-\mathbb{S}um_{i=1}^k2l_il_i^{\perp}\prod_{j\neq i}^k\left(l_j^\perp\right)^2h+D(h)\prod_{i=1}^k\left(l_i^\perp\right)^2\end{equation}
and we have to prove \begin{equation}\label{eq:dfli}\langle D(f),\mathbb{S}um_{j=1}^k l_j^d\rangle =0.\end{equation}
We compute separately the contribution of the three summands in (\ref{eq:3sum}) to the scalar product with $l_j^d$.
We have for the first summand
$$\langle \left(\mathbb{S}um_{i=1}^kl_i^{\perp}l_i^{d-1}\right), l_j^d\rangle = \mathbb{S}um_{i=1}^k\langle l_i^\perp, l_j\rangle\langle l_i\cdot l_j\rangle^{d-1}$$
Summing over $j$ we get zero by Lemma \ref{lem:lmperp}.
We have for the second summand
$$\langle\left(\mathbb{S}um_{i=1}^kl_i,l_i^{\perp}\prod_{p\neq i}^k\left(l_p^\perp\right)^2h\right),l_j^d \rangle =
\langle\left(l_jl_j^{\perp}\prod_{p\neq j}^k\left(l_p^\perp\right)^2h\right),l_j^d \rangle=0 $$
We have for the third summand
$$\langle\left(D(h)\prod_{i=1}^k\left(l_i^\perp\right)^2\right), l_j^d\rangle = 0$$
Summing up, this proves (\ref{eq:dfli}) and then the thesis.
\end{proof}
\begin{example}
If $f=x^3y+2y^4$ then there are $6$ critical points of the form $l_1^4+l_2^4$ and $x^3y$ which lies on the tangent line at $x^4$. It cannot be written as $l_1^4+l_2^4$ and indeed it has rank $4$.
\end{example}
\mathbb{S}ection{The scheme of eigenvectors for binary forms}
Suppose $f\in \mathrm{Sym}^d V$ a symmetric tensor and $\mathrm{dim} V=2$. We denote by $Z$ the scheme defined by the polynomial $D(f)$, embedded in $\mathbb{P}(\mathrm{Sym}^d V)$ by the $d$-Veronese embedding in $\mathbb{P} V$ (see \cite{AEKP} for the case of matrices).
\begin{prop}\label{prop:main2}
$\langle Z \rangle = H_f$.
\end{prop}
\begin{proof}
$(i)$ If $D(f)$ has $d$ distinct roots then it is known that $\langle Z \rangle\mathbb{S}ubseteq H_f$, since $H_f$ is the hyperplane orthogonal to $D(f)$ (Theorem \ref{mainTheorem} with $k=1$). Hence $\langle Z \rangle\mathbb{S}ubseteq H_f$.
$(ii)$ Now let us suppose that $D(f)$ has multiple roots but $f\neq (x^2+y^2)^{d/2}$. We show that $\langle Z\rangle\mathbb{S}ubseteq H_f$ by a limit argument. For every tensor $f$ such that $f\neq 0$ and $f\neq(x^2+y^2)^{d/2}$ there exists a sequence $(f_n)$ such that $f_n\rightarrow f$ and $D(f_n)$ has distinct roots for all $n$. Then, $H_{f_n}\rightarrow H_f$ because the differential operator is continuous. Moreover $H(f_n)$ is a hyperplane for all $n$. On the other hand, by definition we have that $\langle Z_{f_n}\rangle$ is the spanned of the roots of $D(f_n)$. When $f_n$ goes to the limit we get that $\langle Z_{f_n}\rangle\rightarrow \langle Z\rangle$. Hence, $\langle Z\rangle\mathbb{S}ubseteq H_f$.
$(iii)$ In the case that $f=(x^2+y^2)^{d/2}$ with $d$ even, then by Lemma \ref{lemma2} we know that every unitary vector is an eigenvector and $H_f$ is the ambient space. Hence, $\langle Z\rangle=H_f$.
We prove now that $\mathrm{dim} \langle Z \rangle=\mathrm{dim} H_f$ for $(i)$ and $(ii)$.
Since $\mathcal{I}_{Z,\mathbb{P}^1}=\mathcal{O}_{\mathbb{P}^1}(-d)$,
$$\mathrm{codim}\langle Z\rangle=h^0(\mathcal{I}_{Z,\mathbb{P}^1}(d))=h^0(\mathcal{O}_{\mathbb{P}^1}(-d+d))=h^0(\mathcal{O}_{\mathbb{P}^1})=1$$
which coincides with the codimension of $H_f$.
\def\niente{
First we suppose that $D(f)$ has $d$ distinct roots.
It is known that $\langle Z \rangle\mathbb{S}ubseteq H_f$, since $H_f$ is the hyperplane orthogonal to $D(f)$.
We prove that the equality holds by showing that $\mathrm{dim} \langle Z \rangle=\mathrm{dim} H_f$.
If we embedded the $d$ distinct eigenvectors $v_1,\ldots,v_d$ into the rational normal curve of degree $d$, $C_d$, it turns out $d$ independent elements $v_1^d,\ldots,v_d^d$. The embedding of each of the eigenvectors in $C_d$ is of the form:
$$(1:v_i)\mapsto (1:v_i:v_i^2:\ldots:v_i^d)\quad i=1,\ldots,d$$
We know that these points are independent by using \textit{Vandermonde determinant} since
\begin{equation*}
\mathrm{det}
\begin{bmatrix}
1 & v_1 & v_1^2 & \ldots & v_1^d\\
1 & v_2 & v_2^2 & \ldots & v_2^d\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & v_d & v_d^2 & \ldots & v_d^d
\end{bmatrix}
=\prod_{i<j}(v_j-v_i)\neq 0
\end{equation*}
Hence, the dimension of each of the spaces are equal.
Let us suppose that $D(f)$ has multiple roots but $f\neq (x^2+y^2)^{d/2}$. First we show that $\langle Z\rangle\mathbb{S}ubseteq H_f$ by a limit argument. For every tensor $f$ such that $f\neq 0$ and $f\neq(x^2+y^2)^{d/2}$ there exists a sequence $(f_n)$ such that $f_n\rightarrow f$ and $D(f_n)$ has distinct roots for all $n$. Then, $H_{f_n}\rightarrow H_f$ because the differential operator is continuous. Moreover $H(f_n)$ is a hyperplane for all $n$. On the other hand, by definition we have that $\langle Z_{f_n}\rangle$ is the spanned of the roots of $D(f_n)$. When $f_n$ goes to the limit we get that $\langle Z_{f_n}\rangle\rightarrow \langle Z\rangle$. Hence, $\langle Z\rangle\mathbb{S}ubseteq H_f$.
Now we prove the other inclusion. Suppose that we have $r$ distinct eigenvectors $v_1,\ldots,v_r$ with multiplicities $m_1,\ldots,m_r$ and $m_1+\ldots +m_r=d$.
If we embedded the $r$ distinct eigenvectors into the rational normal curve of degree $d$, $C_d$, it turns out $r$ independent elements $v_1^d,\ldots,v_r^d$. The embedding of each of the eigenvectors in $C_d$ is of the form:
$$(1:v_i)\mapsto (1:v_i:v_i^2:\ldots:v_i^{d-1}:v_i^d)\quad i=1,\ldots,r$$
We consider also its derivatives:
$$(1:v_i)\mapsto (0:1:2v_i:\ldots:(d-1)v_i^{d-2}:dv_i^{d-1})$$
$$(1:v_i)\mapsto (0:0:2:\ldots:(d-1)(d-2)v_i^{d-3}:d (d-1)v_i^{d-2})$$
$$\vdots$$
$$(1:v_i)\mapsto (0:0:0:\ldots:\binom{d-1}{m_i-1}v_i^{d-m_i}:\binom{d}{m_i-1}v_i^{d+1-m_i})$$
We know that these points are independent by using the \textit{Confluent Vandermonde determinant} since,
\begin{equation*}
\mathrm{det}
\begin{bmatrix}
1 & v_1 & v_1^2 & v_1^3 & \ldots & v_1^{d-1} & v_1^d\\
0 & 1 & 2v_1 & 3 v_1^2 & \ldots & (d-1)v_1^{d-2} & dv_1^{d-1}\\
0 & 0 & 2 & 6 v_1 & \ldots & (d-1)(d-2)v_1^{d-3} & d (d-1)v_1^{d-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & v_2 & v_2^2 & v_2^3 & \ldots & v_2^{d-1} & v_2^d\\
0 & 1 & 2v_2 & 3 v_2^2 & \ldots & (d-1)v_2^{d-2} & dv_2^{d-1}\\
0 & 0 & 2 & 6 v_2 & \ldots & (d-1)(d-2)v_2^{d-3} & d (d-1)v_2^{d-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & v_r & v_r^2 & v_r^3 & \ldots & v_r^{d-1} & v_r^d\\
0 & 1 & 2v_r & 3 v_r^2 & \ldots & (d-1)v_r^{d-2} & dv_r^{d-1}\\
0 & 0 & 2 & 6 v_r & \ldots & (d-1)(d-2)v_r^{d-3} & d (d-1)v_r^{d-2}\\
\vdots & \vdots & \vdots & \ddots & \vdots
\end{bmatrix}
=\prod_{1\leq i<j\leq r}(v_j-v_i)^{m_i m_j}\neq 0
\end{equation*}
Hence, the dimension of each of the spaces are equal.
Finally, if $f=(x^2+y^2)^{d/2}$ with $d$ even, then by Lemma \ref{lemma2} we know that every nonzero vector is an eigenvector and $H_f$ is the ambient space. Hence, $\langle Z\rangle=H_f$.}
\end{proof}
As a consequence we obtain the following corollary, which may be seen
as a {\it Spectral Decomposition} of any binary form $f$.
\begin{cor}\label{cor:corf}
Any binary form $f\in \mathrm{Sym}^d V$ with $\mathrm{dim} V=2$ can be written as a linear combination of the critical rank one tensors for $f$.
\end{cor}
The previous statement holds even in the special case $d$ even
and $f=(x^2+y^2)^{d/2}$, since from \cite[Theorem 9.5]{Rez} there exists $c_d\in{\mathbb R}$ such that the following decomposition holds $\forall\phi\in{\mathbb R}$
$$(x^2+y^2)^{d/2}=c_d\mathbb{S}um_{k=0}^{d/2}\left[\cos(\frac{2k\pi}{d+2}+\phi)x+\mathbb{S}in(\frac{2k\pi}{d+2}+\phi)y\right]^d$$
In this decomposition the summands on the right-hand side correspond to
$(d+2)/2$ consecutive vertices of a regular $(d+2)$-gon.
In the $d=2$ case, the Spectral Theorem asserts any binary quadratic form $f\in\mathrm{Sym}^2\mathbb{R}^2$
can be written as sum of its rank one critical tensors. This statement fails for $d\ge 3$, as it can be checked already on the examples $f=x^d+y^d$ for $d\ge 3$,
where only two among the $d$ rank one critical tensors are used, namely $x^d$ and $y^d$, and the coefficients of the remaining $d-2$ rank one critical tensors
in the Spectral Decomposition of $f$ are zero.
\mathbb{S}ection{Real critical rank $2$ tensors for binary quartics}
\label{sec:lastreal}
We recall the following result by M. Maccioni.
\begin{thm}\label{thm:maccioni}(Maccioni, \cite[Theorem 1]{Mac})
Let $f$ be a binary form.
$$\# \text{ real roots of f }\leq\# \text{ real critical rank 1 tensors for\ } f$$
The inequality is sharp, moreover it is the only constraint between the number of real roots and the number of real critical rank $1$ tensors, beyond parity mod $2$.
\end{thm}
As a consequence, as it was first proved in \cite{ASS}, hyperbolic binary forms (i.e. with only real roots) have all real critical rank $1$ tensors.
We attempted to extend Theorem \ref{thm:maccioni} to rank $2$ critical tensors.
Our description is not yet complete and we report about some numerical experiments in the space $\mathrm{Sym}^4\mathbb{R}^2$. From these experiments it seems that the constraints
about the number of real rank $2$ critical tensors are weaker than for rank $1$ critical tensors.
For quartic binary forms the computation of the critical rank $2$ tensors
is
easier since the dual variety of the secant variety $\mathbb{S}igma_2(C_4)$ is given by quartics which
are squares, which make a smooth variety.
The number of complex critical rank $2$ tensors for a general binary form of degree d
was guessed in \cite{OSS} to be $3/2d^2-9d/2+1$. For $d=4$
this number is $7$, which can be confirmed by a symbolic computation
on a rational random binary quartic.
In conclusion, for a general binary quartic there are $4$ complex critical rank $1$ tensors
and $7$
complex rank $2$ critical tensors.
The following table reports some computation done for the case of binary quartic forms, by testing several different quartics. The appearance of ``yes'' in the last column means that we have found an example
of a binary quartic with the prescribed number of distinct and simple
real roots, real rank $1$ critical tensors and real critical rank $2$ tensors.
Note that we have not found any quartic with the maximum number of seven real
rank $2$ critical tensors, we wonder if they exist.
\begin{center}
\begin{tabular}{c | c| c |c| c}
&\#\text{real roots}& \#\text{real critical rank 1 tensors} & \#\text{real critical rank 2
tensors} & \\
\hline
& $0$ & $2$ & $3$ & yes\\
& $2$ & $2$ & $3$ & yes\\
& $0$ & $2$ & $5$ & yes\\
& $2$ & $2$ & $5$ & yes\\
& $0$ & $4$ & $3$ & yes\\
& $2$ & $4$ & $3$ & yes\\
& $4$ & $4$ & $3$ & yes\\
& $0$ & $4$ & $5$ & yes\\
& $2$ & $4$ & $5$ & yes\\
& $4$ & $4$ & $5$ & ?\\
& * & * & 7 & ?
\end{tabular}
\end{center}
\mathbb{S}ection{Acknowledgement} Giorgio Ottaviani is member of INDAM-GNSAGA. This paper has been partially supported by the Strategic Project ``Azioni di Gruppi su variet\'a e tensori'' of the University of Florence.
\end{document} |
\begin{document}
\begin{center}
\large
{THE CHIRAL OSCILLATOR AND ITS APPLICATIONS IN QUANTUM THEORY}
\end{center}
\vskip 2cm
R. Banerjee\footnote{E-mail:[email protected]}\\
S. N. Bose National Centre for Basic Sciences\\
Block JD, Sector III, Calcutta 700091, India\\
\vskip .5cm
\noindent and\\
\vskip .5cm
\noindent Subir Ghosh\\
Dinabandhu Andrews College\\
Garia, West Bengal, India.\\
\vskip 2cm
\noindent Abstract\\
The fundamental importance of the chiral oscillator is elaborated. Its quantum
invariants are computed. As an application the Zeeman effect is analysed.
We also show that the chiral oscillator is the most basic
example of a duality invariant model, simulating the effect of the familiar
electric-magnetic duality.
It is well known that the Harmonic Oscillator (HO) pervades our understanding
of quantum mechanical as well as field theoretical
models in various contexts. An interesting thrust in
this direction was
recently made in \cite{rprl} where the quantum invariants of the HO
were computed. It was also opined that this approach could be used for
developing a technique \cite{rjmp} to study interacting and time dependent
(open) systems.
In this paper we argue that, in some instances, the Chiral Oscillator (CO)
instead of the usual HO captures the essential physics of the problem. This
is tied to the fact
that the CO simulates the left-right symmetry. Consequently the CO
has a decisive role in those cases where this symmetry is significant.
The CO is first systematically derived from the HO and the issue of
symmetries is clarified. Indeed, it is explicitly shown that the
decomposition of the HO leads to a pair of left-right symmetric CO's. The
soldering of these oscillators to reobtain the HO is an instructive
exercise. Following the methods of \cite{rprl,rjmp}, the quantum invariants
of the CO's are computed and their connection with the HO invariant is
illuminated. As an application, the Zeeman splitting \cite{pk} for the
Hydrogen atom
electron energy levels under the influence of a constant magnetic field is
studied. The interaction of the atom with a time-dependent magnetic field,
constituting an open system, can also be analysed from the general expressions.
In a completely different setting we show that the CO is
the most basic example of a
duality invariant theory
\cite{az}. By reexpressing the computations in a
suggestive electromagnetic notation, the mapping
of this duality with Maxwell's
electromagnetic duality is clearly established.
The Lagrangean for the one dimensional HO is given by
\begin{equation}
L={M\over 2}(\dot {x}^2-\omega^2x^2).
\label{eqlho}
\end{equation}
To obtain the CO, the basic step is to convert (\ref{eqlho}) in a first order
form by introducing an auxiliary variable $\Lambda$ in a symmetrised form,
\begin{equation}
L={M\over 2}(\Lambda\dot x-x\dot{\Lambda}-{\Lambda^2}-\omega^2x^2).
\label{eqlf}
\end{equation}
There are now two distinct classes for relabelling these variables
corresponding to proper and improper rotations generated by the matrices
with determinant $\pm1$,
\[ \left (
\begin{array}{c}
x\\
{{\Lambda}\over{\omega}}\\
\end{array}
\right )=\left (
\begin{array}{cc}
cos\theta & sin\theta\\
-sin\theta & cos\theta
\end{array}
\right )\left (
\begin{array}{c}
x_1 \\
x_2\\
\end{array}\right ),
~~ \left (
\begin{array}{c}
x\\
{{\Lambda}\over{\omega}}\\
\end{array}
\right )=\left (
\begin{array}{cc}
sin\phi & cos\phi\\
cos\phi & -sin\phi
\end{array}
\right )\left (
\begin{array}{c}
x_1 \\
x_2\\
\end{array}\right )
\]
leading to the structures,
\begin{equation}
L_{\pm}={M\over 2}(\pm\omega\epsilon_{\alpha\beta}x_\alpha \dot{x}_\beta
-\omega^2x_\alpha^2),
\label{eqlco}
\end{equation}
where $\alpha=1,2$ is an internal index with $\epsilon_{12}=1$.
The basic Poisson brackets
of the above model are read off from the symplectic structure,
\begin{equation}
\{x_\alpha ,x_\beta \}_{\pm}=\mp{1\over{\omega M}}\epsilon_{\alpha\beta}.
\label{eqbr}
\end{equation}
The corresponding Hamiltonians are,
\begin{equation}
H_{\pm}={{M\omega^2}\over 2}(x_1^2+x_2^2) =\tilde{H}.
\label{eqhco}
\end{equation}
The above Lagrangeans in (\ref{eqlco}) are interpreted as
two bi-dimensional CO's rotating
in either a clockwise or an anti-clockwise sense. A
simple way to verify this property is to look at the spectrum of the
angular momentum operator,
\begin{equation}
\omega J_{\pm}=\omega\epsilon_{\alpha\beta}
x_\alpha p_\beta=\pm{1\over 2}M\omega^2x_\alpha^2
=\pm\tilde{H},
\label{eqJ}
\end{equation}
where $\tilde{H}$ is defined above.
To complete the picture it is desirable to show the mechanism of
combining the left and right CO's to reproduce the usual HO. This is achieved
by the soldering technique \cite{adc,abc} introduced recently. Let us then begin with two {\it independent} chiral Lagrangeans
$L_+(x)$ and $L_-(y)$. Consider
the following gauge transforms,
$\delta x_\alpha=\delta y_\alpha =\eta_\alpha$ under which
$$
\delta L_{\pm}(z)=M\omega\epsilon_{\alpha\beta}\eta_\alpha
(\pm\dot{z}_\alpha
+\omega\epsilon_{\alpha\beta}z_\beta),~~~z=x,y.
$$
Introduce a new variable $B_\alpha$, which will effect the soldering,
transforming as,
$\delta B_\alpha =\epsilon_{\alpha\beta}\eta_\beta$. This new Lagrangean
\begin{equation}
L=L_{+}(x)+L_{-}(y)-M\omega B_\alpha(\dot{x}_\alpha
+\omega\epsilon_{\alpha\beta}x_\beta-\dot{y}_\alpha
+\omega\epsilon_{\alpha\beta}y_\beta),
\label{eqlinv}
\end{equation}
is invariant under the above transformations.
Eliminating $B_\alpha$ by the equations of motion, we obtain the final soldered
Lagrangean,
$$L(w)={M\over 4}(\dot{w}^2_{\alpha}-\omega^2 w^2_\alpha),$$
which is no longer a function of $x$ and $y$ independently, but only on
their gauge invariant combination, $w_\alpha=x_\alpha -y_\alpha$. The
soldered Lagrangean just corresponds to a bi-dimensional
simple harmonic oscillator.
Thus, by starting from two distinct Lagrangeans containing the opposite
aspects of chiral symmetry, it is feasible to combine them into a single
Lagrangean.
The connection between the CO and HO is now used to obtain the invariants
of the former by exploiting known results \cite{rprl} for the latter.
For the positive CO,
\begin{equation}
I^+={1\over 2}\tan^{-1}({x_1}^{-1}x_2)+ {1\over 2}\tan^{-1}(x_2{x_1}^{-1}),
\label{eqco+}
\end{equation}
is the invariant, while $I^-$ is given by interchanging $x_1$ and $x_2$. Note
that non-commutativity of the variables has already been taken into
account. Incorporating the "soldering" prescription
\cite {abc} whereby we were able
to construct a bi-dimensional oscillator
from the two CO's, another
quantum invariant can also be obtained,
\begin{equation}
I^+(x_1,x_2) \oplus I^-(y_1,y_2)=I(x_1-y_1,x_2-y_2),
\label{eqsol}
\end{equation}
where, the right hand side of the equation is a simple sum of two terms,
obtained by
substituting $x_1-y_1,~~M(\dot x_1-\dot y_1)/2$ and
$x_2-y_2,~~M(\dot x_2-\dot y_2)/2$
in place of $x$ and $p$ in the corresponding expression for HO in \cite{rprl}.
We stress that the above
invariant operators are independent as they pertain to completely
different systems and were not present in the literature so far. In the next
section, we will put the CO invariants into direct use in interacting and
open quantum systems by considering the Zeeman effect.
Let us consider the simplistic Bohr model of Hydrogen atom, where the
(non-relativistic) electron is moving in the presence of a repulsive
centrifugal barrier and the attractive Coulomb potential. The effective
central potential has a well like structure and we consider the
standard HO approximation about the potential minimum. The excitations are
the HO states above the minimum. Hence the electron, at a particular
stationary state, is approximated to an oscillator with a
frequency
$\omega$, obtained from the effective potential
seen by the atomic electron without
the magnetic field.This
yields $\omega=(Me^4)/l^3$, with $l=Mr^2\dot{\phi}$
being the angular momentum, when expressed in plane polar coordinates.
In the presence of a magnetic field ${\bf B}$, the motion of the electron
can be broken into components parallel and perpendicular to ${\bf B}$. The
Lorentz force acting on the electron affects the motion in the normal plane of
${\bf B}$ only, the motion being two rotational modes in the clockwise
and anti-clockwise sense, or more succintly two CO's of opposite chirality.
In this setup, ${\bf B}$ splits the
original level into three levels, one of
frequency $\omega$ remaining unchanged and the other two frequencies
changed to $\omega\pm(eB)/(2Mc)$ \cite{pk}. This clearly shows that there is a
redundancy in the number of degrees of freedom in treating the electron
as a HO, whereas the CO representation is more elegant and economical
whenever the degeneracy between the right and left movers is lifted such as
in the presence of magnetic field.
The Hamiltonian of a charged HO in an axially symmetric magnetic field is,
$${\bf A}={1\over 2}B(t){\bf k}\times{\bf r},~~{\bf B}(t)=\nabla\times{\bf A}
=B(t){\bf k}$$
$$H={1\over{2M}}({\bf p}-e{\bf A})^2 +{1\over 2}M\omega^2r^2
={1\over{2M}}({p_1}^2+{p_2}^2)+{1\over 2}M\omega^2({x_1}^2+{x_2}^2)$$
\begin{equation}
+{{eB(t)}\over{2Mc}}(x_2p_1-x_1p_2)
+{{e^2}\over{8Mc^2}}{B(t)}^2({x_1}^2+{x_2}^2).
\label{eqcho}
\end{equation}
For the semi-classical reasoning (regarding the Zeeman effect) to hold,
$\mid{\bf B}\mid$ must be small in the sense that the radius
of gyration $r=~(cMv)/(eB)=~(cl)/(eBr)$, which simplifies to
$r=~\sqrt{(cl)/(eB)}=~\sqrt{(nc\hbar)/(eB)}$
is much larger than the Bohr radius of the (Hydrogen) atom \cite{yk}
$r_{Bohr}=~\hbar^2/(Me^2).$
This condition is expressed as
\begin{equation}
{{\hbar^3B}\over{cM^2e^3}}<<1.
\label{eqsb}
\end{equation}
In our Hamiltonian (\ref{eqcho}), this condition will hold if
\begin{equation}
\mid{1\over 2}M\omega^2({x_1}^2+{x_2}^2)\mid>>
\mid{{eB(t)}\over{2Mc}}(x_2p_1-x_1p_2)\mid.
\label{eqsm}
\end{equation}
To verify this, substitute $\omega=~Me^4/l^3 $ and $({x_1}^2+{x_2}^2)=~r_
{Bohr}$ in the left hand side, and $(x_2p_1-x_1p_2)=~l$ in the right hand
side. This reproduces (\ref{eqsb}). The quadratic
$B$-term in (\ref{eqcho}) is still smaller.
The above structure of the Hamiltonian is very similar
to the model of a charged particle in a
specified electromagnetic field, considered in \cite{rprl,
rjmp}. The idea there is
to look for the invariants of the full interacting Hamiltonian,
and to construct eigenstates of the invariant operator. The
solutions of the time dependent Schrodinger equation are related uniquely
to these eigenstates via a time dependent phase,
$$
\mid\lambda,k,t>_{Sch}=e^{i\alpha_{\lambda k}(t)}\mid\lambda,k,t>_{I},~~~~
I(t)\mid\lambda,k,t>_{I}=\lambda \mid\lambda,k,t>_{I},$$
satisfying,
$$i\hbar{{d\alpha_{\lambda k}}\over{dt}}=<\lambda,k\mid_I(i\hbar{{\partial}
\over{\partial t}}-H)\mid\lambda,k>_I.$$
Next we define,
$$H_o={1\over{2M}}({p_1}^2+{p_2}^2)+{1\over 2}M\omega^2({x_1}^2+{x_2}^2)$$
and the rest of the $B$-dependent terms
appearing in (\ref{eqcho}) as small perturbations. In the framework
of \cite{rpla}, the invariant
operator is also expressible as a power series in the small
parameter $B\hbar^3/(cM^2e^3)$
and the zeroth order invariant $I_0$ is identical
to $H_o$. Hence the
eigenstates of $H_o$ and $I_o$ will be same and $\mid\lambda,k,t>_I=
exp(-i(n+{1\over 2})\omega t)\mid\lambda,k>_I$. As in the conventional
scenario, the total energy is also expressed as a series with the zeroth term
being $(n+{1\over 2})\hbar\omega$. Thus we will compute the $B$-dependent
corrections only by the scheme of \cite{rjmp}, which actually comprises
the task of calculating the phase $\alpha_{\lambda k}$. Here the CO's will
come into play.
As we have already established the connection between the results of
HO and CO models, we simply replace the HO variables by the CO ones in
the final result. From the symplectric structure, the following
identifications are consistent,
\begin{equation}
CO^+:~~\{{x_1}^+,~{x_2}^+\}=-{1\over{\omega M}}
,~~\to p_1\equiv -\omega M{x_2}^+,~p_2\equiv \omega M{x_1}
^+,
\label{px+}
\end{equation}
\begin{equation}
CO^-:~~\{{x_1}^-,~{x_2}^-\}={1\over{\omega M}}
,~~\to p_1\equiv \omega M{x_2}^-,~p_2\equiv -\omega M{x_1}
^-.
\label{px-}
\end{equation}
Introducing these in (\ref{eqcho}), we get,
\begin{equation}
H_{\pm}={M\over 2}({x_1}^2+{x_2}^2)(1+{{e^2B^2}\over
{4M^2c^2}}\mp{{eB}\over{Mc}}).
\label{cco}
\end{equation}
The above splitting in the energy is one of our main results. This underlines
the economy in the CO formulation since one CO is sufficient to obtain the
correct results. Obviously it is easier to work with less number of degrees
of freedom in cases of more complicated systems. Essentially
this change in the relative sign of the linear $B$ term
can also be interpreted as a consequence of the
opposite angular momenta of the CO's, as demonstrated before. This brings
us to the cherished expression of the phase for the two CO's,
\begin{equation}
\alpha_{jn}^{\pm}=\mp[n+(j+{1\over 2})]{e\over{Mc}}\int^t dt'[{1\over 2}
B(t')-\rho^{-2}(t')],
\label{eqpco}
\end{equation}
where the quantum numbers $j$ and $n$ are explained in \cite{rjmp} and $\rho
(t')$ satisfies the equation,
$$({{Mc}\over e})^2\ddot{\rho}+{{B^2(t)}\over 2}\rho -\rho^{-3}=0.$$
Considering the simplest case, that is normal Zeeman effect, where $B$ is a
constant, we find a time-independent solution of $\rho$,
$\rho^2=\pm{\sqrt 2}/B$.
When $\rho^2=-{\sqrt 2}/B$ is
substituted in (\ref{eqpco}), the standard
Zeeman level splitting is reproduced.
\begin{equation}
E_n^{\pm}=(n+{1\over 2})\hbar\omega\pm[n+(j+{1\over 2})]{{eB}\over{Mc}}.
\label{eqzee}
\end{equation}
On the other hand, $\rho^2={\sqrt 2}/B$ reveals no shift in the
energy eigenvalue. Clearly this is reminiscent of the fact that the
energy of the mode parallel to ${\bf B}$ remains unaffected.
For time dependent magnetic field, one has to obtain the appropriate solution
for $\rho$. Inserting this in (\ref{eqpco}) it is then possible to obtain
the solutions of the corresponding Schr$\ddot o$dinger equation.
We next show the possibility of interpreting the CO as a prototyype of a
duality invariant model characteristic of the electric-magnetic duality
\cite{az}. For convenience, we set $M=~\omega=~1$ in (\ref{eqlho}).
Introduce a change of variables,
$E=\dot x,~~~B=x$,
so that
\begin{equation}
\dot B -E=0
\label{eqeb}
\end{equation}
is identically satisfied. In these variables, the Lagrangian (\ref{eqlho}) and
the corresponding equation of motion are expressed as
\begin{equation}
L={1\over 2}(E^2-B^2),
~~~\dot E+B=0.
\label{eqem}
\end{equation}
It is simple to observe that the transformations, \footnote{Note that these
are the discrete cases $(\theta=~\pm{{\pi}\over 2})$ for a general $SO(2)$
rotation matrix parametrised by the angle $\theta$.}
$E\rightarrow \pm B;~B\rightarrow\pm E$,
swap the equation of motion
in (\ref{eqem}) with the identity (\ref{eqeb}) although the
Lagrangean (\ref{eqem}) is not invariant. The similarity with the
corresponding analysis in Maxwell theory is quite striking, with $x$ and
$\dot x$ simulating the roles of the magnetic and electric fields,
respectively. There is a duality among the equation of motion and the
"Bianchi" identity (\ref{eqeb}), which is not manifested in the Lagrangean.
Let us now consider the Lagrangean for the CO,
\begin{equation}
L_{\pm}={1\over 2}(\pm\epsilon_{\alpha\beta}x_\alpha\dot{x}_\beta
-x^{2}_\alpha)
={1\over 2}(\pm\epsilon_{\alpha\beta}B_\alpha E_\beta
-B^{2}_\alpha).
\label{eqdu}
\end{equation}
These chiral Lagrangeans are manifestly invariant under the duality
transformations,
\begin{equation}
x_\alpha\rightarrow~R^{+}_{\alpha\beta}(\theta)x_\beta.
\label{eqxr}
\end{equation}
Thus, the CO's represent a quantum
mechanical example of a duality invariant model. Indeed, the expressions for
$L_{\pm}$ given in the second line of (\ref{eqdu}) closely resemble the
analogous structure for the Maxwell theory deduced in \cite{bw}.
The generator of the infinitesimal symmetry transformmation is given by,
$Q=x_\alpha x_\alpha/2,$
so that the complete transformations (\ref{eqxr}) are generated by,
$$x_\alpha\rightarrow~x_{\alpha}'=e^{-i\theta Q}x_\alpha e^{i\theta Q}
=~R^{+}_{\alpha\beta}(\theta)x_\beta.$$
This follows by exploiting the basic bracket of the theory given in
(\ref{eqbr}).
To conclude, certain interesting properties of the CO were
illustrated. A systematic method of obtaining this oscillator from the
usual simple HO was given. It was also shown that the
distinct left and right components of the CO were combined by the soldering
formalism \cite{adc, abc} to yield a bi-dimensional HO. In this way the
symmetries of the model were highlighted. The importance in the CO lies in
the fact that in some cases it has a concrete and decisive role than the
usual simple HO in illuminating the basic physics of the problem. This was
particularly well seen in the derivation of the Zeeman splitting by
exploiting the perturbation theory technique based on quantum invariant
operators \cite{rprl, rjmp}. An explicit computation of the quantum
invariants for the CO was also performed. Apart from the study of the Zeeman
effect, such CO invariants can find applications in other quantum mechanical
examples, particularly where a left-right symmetry is significant. Another
remarkable feature of the present analysis has been the elucidation of the
fundamental nature of the duality symmetry currently in vogue either in
quantum field theory or the string theory \cite{az, bw}. It was shown that
the CO was a duality symmetric model, contrary to the usual HO. Expressed
in the "electromagnetic" notation, this difference was seen to be the origin
of the presence or absence of duality symmetry in electrodynamics.
It may be remarked that the explicit demonstration of duality symmetry in
a quantum mechanical world is nontrivial since conventional analysis
\cite{az} considers two types of duality invariance confined to either
$D=4k$ or $D=4k+2$ dimensions, thereby leaving no room for $D=1$ dimension.
Nevertheless, since most field theoretical problems can be understood on
the basis of the HO, it is reassuring to note that the origin of
electromagnetic duality invariance is also contained in a variant of the
HO- the chiral oscillator. Our study clearly reveals that the CO complements
the usual HO in either understanding or solving various problems in quantum
theory.
\end{document} |
\begin{document}
\thispagestyle{empty}
\printtitle
\printauthor
\mathbb Setcounter{page}{1}
\mathbb Section{Introduction}\label{intro}
The aim of these notes is to introduce to the geometers useful tools from the \textit{Theory of partial differential equations} which are used in order to obtain hypersurfaces with prescribed mean curvature. For instance, graphs with prescribed mean curvature can be obtained by solving the Dirichlet problem for a particular quasilinear elliptic partial differential equation of second order.
The Dirichlet problem for the prescribed mean curvature equation consists on find a function satisfying the prescribed mean curvature equation in a bounded domain of the $n-$dimensional Euclidean ambient space and that continuously takes on given boundary values.
Precisely, given a smooth bounded domain $\W\mathbb Subset\R^n$ ($n\geq 2$) and $\varepsilonphi\in\cl^0({\partial\W})$, we ask if for a prescribed smooth function {$H$} there exists some $u\in\cl^2(\W)\cap\cl^0(\overline{\W})$ satisfying
\begin{equation}\tanhg{$P$}\label{ProblemaP}
\left\{
\begin{split}
\text{div} \,er \left(\dfrac{\nabla u}{\mathbb Sqrt{1+\norm{\nabla u}^2}} \right)&=n H(x) \ \mbox{in}\ \W,\\
u&=\varepsilonphi \ \mbox{in}\ \partial\W.
\varepsilonnd{split}\right.
\varepsilonnd{equation}
If this is the case, then the graph of $u$
is an hypersurface in $\R^{n+1}$ of mean curvature $H(x)$ at each point $(x,u(x))$.
This problem traces its roots to the Plateau's problem that was first posed by Lagrange \cite{lagrange} in 1760. Lagrange wanted to find the surface with the least area among all the surfaces having the same boundary. This problem arose as an example of the Calculus of Variations that he was developing. In order to find a minimum of the area functional, Lagrange derived the Euler-Lagrange equation for the solutions of this problem. That is, if an area minimizing surface in the three dimensional Euclidean space is a graph of a smooth function $u$ over a bounded domain, then $u$ necessarily satisfies
\begin{equation}\label{eq_minima_n2}
\left(1+u_y^2 \right)u_{xx} - 2u_xu_yu_{xy}\left(1+u_x^2 \right)u_{yy}=0
\varepsilonnd{equation}
in that domain.
It was Meusnier \cite{Meusnier} in 1776 who gave a geometrical interpretation for equation \varepsilonqref{eq_minima_n2}. He realized that the connexion between this partial differential equation and Geometry is given by the concept of \textit{mean curvature} that was formally introduced on his work. Indeed, the mean curvature of the graph of the solutions of this equation must vanishes. All those surfaces are called minimal surfaces even though a surface having vanish mean curvature is not necessarily globally area minimizing.
Find examples of functions satisfying the minimal surface equation \varepsilonqref{eq_minima_n2} is not an easy task. As a matter of fact, Lagrange could only give the constants functions as examples. It was Meusnier \cite{Meusnier} in 1776 who gave another example of such a function that locally represents the helicoid. He also proved that the catenoid, discovered by Euler in 1741, can be seen locally as the graph of a function satisfying \varepsilonqref{eq_minima_n2}. In 1834, almost seventy years later of the discovering of the catenoid and the helicoid, H. F. Scherk \cite{scherk} gave new examples of minimal surfaces, the most famous one is the Sherk's surface.
At this point, the interest was about the geometry of the domains over which minimal graphs can actually exist rather than find explicit expressions of the solutions of \varepsilonqref{eq_minima_n2}. In 1910, Bernstein \cite{Bernstein} showed that the Dirichlet problem for equation \varepsilonqref{eq_minima_n2} has solution in disks of $\R^2$ for continuous boundary data. But Bernstein also realized that the disks could be replaced by convex domains (see \cite[p. 236]{Bernstein}). Proofs of this fact were given independently in 1930 by Douglas \cite{Douglas1931} and Radó \cite[p. 795]{Rado1930} who also ensured the uniqueness of the solution. In 1965 Finn \cite{Finn1965} made an important contribution when he proved that the Dirichlet problem for equation \varepsilonqref{eq_minima_n2} may not be solvable if the domain is non-convex. That is, this convexity conditions is sharp in order to obtain minimal graphs for arbitray continuous boundary values.
Using the results from Douglas and Radó, and following a suggestion from Osserman, Finn stated the following {sharp} theorem:
\begin{taggedtheorem}{A}[Douglas-Rad\'o-Finn {\cite[T. 4a p. 146]{Finn1965}}]\label{SharpFinn}
The Dirichlet problem for equation \varepsilonqref{eq_minima_n2} has solution in $\W$ for arbitrary continuous boundary data if, and only if, $\W$ is convex.
\varepsilonnd{taggedtheorem}
In 1966, Jenkins and Serrin \cite{Serrin1968} generalized Theorem \ref{SharpFinn} to higher dimensions:
\begin{taggedtheorem}{B}[Jenkins-Serrin {\cite[T. 1 p. 171]{Serrin1968}}]\label{SharpJenkinsSerrin}
Let $\W$ be a bounded domain in $\R^n$ whose boundary is of class $\cl^2$. Then the Dirichlet problem for the minimal surface equation ($H=0$) in $\W$ is uniquely solvable for arbitrary $\cl^2$ boundary values if, and only if, the mean curvature of $\partial\W$ is non-negative.
\varepsilonnd{taggedtheorem}
It can be observed that, although the convexity of $\W$ is not the appropriated generalization of the two-dimensional case (as it was thought), it is again a geometric property. We recall that a domain whose boundary has non-negative mean curvature is called a \textit{mean convex} domain. Throughout the text $\mathcal H_{\partial\W}$ will denote the mean curvature of $\partial\W$.
Mathematicians were also questioning about the existence of graphs whose mean curvature not necessarily vanishes. It was Serrin \cite{Serrin} who gave a complete answer:
\begin{taggedtheorem}{C}[Serrin {\cite[p. 416]{Serrin}}]\label{SharpSerrin}
Let $\W$ be a bounded domain in $n$-dimensional Euclidean space whose boundary is of class $\cl^2$. Then for every constant $H$ the Dirichlet problem \varepsilonqref{ProblemaP} has a unique solution for arbitrary $\cl^2$ boundary data if, and only if,
\begin{equation}\label{cond_serrin_constante}
(n-1) \mathcal H_{\partial\W}(y) \geq n H \ \ \forall\ y\in\partial\W.
\varepsilonnd{equation}
\varepsilonnd{taggedtheorem}
This work of Serrin was actually focused on the study of a more general class of Dirichlet problems within which is problem \varepsilonqref{ProblemaP}. In fact, Theorem \ref{SharpSerrin} is a direct conclusion of the following result:
\begin{taggedtheorem}{D}[Serrin {\cite[T. p. 484]{Serrin}}]\label{T_Serrin_Ricci}
Let $\W$ be a bounded domain in $n$-dimensional Euclidean space, whose boundary is of class $\cl^2$.
Let $H\in\cl^{1}(\overline{\W})$ and suppose that
\begin{equation}\label{cond_Ricc_Serrin}
\norm{\nabla H(x)}\leq \dfrac{n}{n-1}(H(x))^2\ \forall\ x\in\W.
\varepsilonnd{equation}
Then problem \varepsilonqref{ProblemaP} is uniquely solvable for arbitrarily given $\cl^2$ boundary values if, and only if,
\begin{equation}\label{SerrinCondition}
(n-1)\mathcal H_{\partial\W}(y)\geq n\mathopen d{H(y)} \ \forall \ y\in\partial\W.
\varepsilonnd{equation}
\varepsilonnd{taggedtheorem}
Since the non-divergence form of the minimal operator is a quasilinear elliptic operator of second order, the basic tools for the solvability of problem \varepsilonqref{ProblemaP} come from the theory of partial differential equations. However, it can be seen in Theorem \ref{SharpSerrin} how the existence of graphs with constant mean curvature imposes, as in the minimal case, a geometric restriction over the domain. In the more general case, the existence of a function $H$ satisfying the hypothesis of Theorem \ref{T_Serrin_Ricci} leads to {additional geometric implications on $\W$}.
This relation between the solvability of problem \varepsilonqref{ProblemaP} and the geometry of the domain is due to the fact that the minimal operator is non-uniformly elliptic.
In order to solve problem \varepsilonqref{ProblemaP} the Leray-Schauder fixed point theorem is used in these notes. However, the beautiful Continuity Methods is another tool that can be used for the same purpose. In any case, the application of the method strongly depends on the existence of a priori estimates for the solutions of some ``related problems''.
These notes are organized as follows. In section \ref{apendice_Anal_equac} is obtained the non-divergence form of the equation of the prescribed mean curvature and some analytical properties about the minimal operator are stated.
Section \ref{chapterExistence} treats about the existence of solutions of problem \varepsilonqref{ProblemaP}. The estimates needed for the existence program are stated in section \ref{chapter_estimates}. The sharpness of the Serrin condition is proved in section \ref{cap_NaoExis}. With the intention of having an understandable text, some of the results from the theory of partial differential equations needed were established in the placed they are used. {At the end of the text it can be found some recent results in more general ambient spaces.
Finally, we want to point out that these notes definitely do not represent the whole subject. For instance, some regularity issues are left to the reader as a motivation for further studies.}
\mathbb Section{Analysis of the mean curvature equation}\label{apendice_Anal_equac}
\mathbb Subsection{The non-divergence form of the mean curvature equation}
Let $S$ be an oriented hypersurface in $\R^{n+1}$. First, recall that if $N$ is a normal vector field along $S$, then the \textit{principal curvatures} of $S$ at a point $p\in S$ are the eigenvalues of the Weingarten map (or shape operator) $A_N$ at $p$ and the \textit{mean curvature} $H$ at $p$ is the average of the principal curvatures at $p$. Therefore,
\begin{equation}\label{curv_media_metrica}
nH=\tr\left(A_N\right)=\ds\mathbb Sum_{i,j=1}^{n-1} g^{ij}b_{ij},
\varepsilonnd{equation}
where $g$ is the induced metric on $S$ and $b_{ij}$ are the coefficients of the second fundamental form\footnote{Recall that the second fundamental form is a symmetric bilinear form and that the Weingarten map is the self-adjoint linear transformation associated. Recall also that if we replace $N$ by $-N$, the principal curvatures change the sign, but the corresponding principal directions remain the same.}.
In the case where $S$ is the graph over a domain $\W\mathbb Subset M$ of a function $u\in\cl^2(\W)$ the map
$$ \funcionesp{F}{\W}{\R^{n+1}}{x}{(x,u(x))}$$
is a global parametrization of $S$. The tangents vectors on $S$ are
$$X_i=\parcial{F}{x_i}=e_i+ \partial_i u e_{n+1}, \ 1\leq i \leq n,$$
where $\{e_1,\downarrowts, e_{n+1}\}$ is the canonical basis on $\R^{n+1}$. Hence, the coefficients of the induced metric $g$ on $S$ and of the inverse matrix $g^{-1}$ are given, respectively, by
\[ g_{ij}=\varepsilonscalar{X_i}{X_j}=\delta_{ij}+ \partial_i u\partial_j u\]
and
\begin{equation}\label{inv_metr_S}
g^{ij}=\delta_{ij}- \frac{\partial_i u\partial_j u}{W^2}.
\varepsilonnd{equation}
Besides, the unit normal field along $S$ is
\[N=\dfrac{-\nabla u (x) + e_{n+1}}{W}, \]
where (and from now on)
\[W=\mathbb Sqrt{1+\norm{\nabla u (x)}^2}.\]
Therefore, once
$$\parcialll{F}{x_i}{x_j}=\partial_ij u e_{n+1},$$
the expression for the coefficients of the second fundamental form is
\begin{equation}\label{bij_eq_curv_media_graf_MxR}
b_{ij}=II(X_i,X_j)=\varepsilonscalar{N}{\parcialll{F}{x_i}{x_j}}=\frac{1}{W}\partial_ij u.
\varepsilonnd{equation}
Using \varepsilonqref{inv_metr_S} and \varepsilonqref{bij_eq_curv_media_graf_MxR} in \varepsilonqref{curv_media_metrica} it follows that $u$ satisfies the equation
\begin{equation}\label{operador_minimo_1_anal_eq}
\ds\frac{1}{W}\mathbb Sum_{i,j=1}^n \left(\delta_{ij} - \frac{\partial_i u \partial_j u}{W^2} \right)\partial_ij u=nH.
\varepsilonnd{equation}
Finally, some algebraic computations show that
$$n H = \text{div} \,\left(\dfrac{\nabla u}{\mathbb Sqrt{1+\norm{\nabla u}^2}}\right). $$
\mathbb Subsection{Ellipticity of the minimal operator}\label{secao_eq_cur_media}
Note that equation \varepsilonqref{operador_minimo_1_anal_eq} is equivalent to
\begin{equation}\label{operador_minimo_1_coord}
\mathcal{M} u:=\ds\mathbb Sum_{i,j=1}^n \left(W^2\delta_{ij} - {\partial_i u \partial_j u} \right)\partial_ij u=nH(x){W^3}.
\varepsilonnd{equation}
We call $\mathcal{M}$ of minimal operator. From now on will also be considered the equation
\begin{equation}\label{operador_Q}
\mathfrak{Q} u:=\mathcal{M} u-nH(x)W^3=0.
\varepsilonnd{equation}
The operator $\mathcal{M}$ is a quasilinear operator of second order. In fact the coefficients of the second order derivatives are given by
\begin{equation}\label{coeff_aij}
a_{ij}(p)=\left((W(p))^2\delta_{ij}-{p_i p_j}\right),
\varepsilonnd{equation}
where $p$ stands for $\nabla u$ and $W(p)=\mathbb Sqrt{1+\norm{p}^2}$.
The aim of this section is to prove that the minimal operator $\mathcal{M}$ is strictly elliptic. That is, $A(p)=(a_{ij}(p))$ is positive definite for each fixed $p$, and the infimum of the smallest eigenvalue $\lambda(p)$ of $A(p)$ is positive\footnote{For a precise definition of ellipticity and strict ellipticity we refer that in \cite[p. 259]{GT}. Observe that in the case of the minimal operator $\mathcal{M}$ the matrix $A$ does not depend on $x$.}.
This is an important property which is required in many of the results from the theory of partial differential equations used in this text.
In order to prove that $A(p)$ is positive definite, it is just to find the eigenvalues of
the quadratic form
\begin{align*}
Q_p(q)=\varepsilonscalar{{A}(p) q}{q}&=\mathbb Sum_{i,j=1}^n\left((W(p))^2\delta_{ij}-p_i p_j\right)q_iq_j =\left(1+\norm{p}^2\right)\norm{q}^2-\varepsilonscalar{p}{q}^2.
\varepsilonnd{align*}
If $q=p$ it follows
\begin{align*}
\varepsilonscalar{{A}(p) p}{p}&=\left(1+\norm{p}^2\right)\norm{p}^2-\norm{p}^4=\norm{p}^2,
\varepsilonnd{align*}
and for $q\bot p$
\begin{align*}
\varepsilonscalar{{A}(p) q}{q}&=\left(1+\norm{p}^2\right)\norm{q}^2.
\varepsilonnd{align*}
Therefore, the smallest eigenvalues of ${A}(p)$ is $\lambda(p)=\lambda= 1$ with associated eigenspace $\mathbb Spann\{p\}$. Consequently, $\mathcal{M}$ and $\mathfrak{Q}$ are strictly elliptic operators.
Observe also that $\Lambda(p)=1+\norm{p}^2$ is the largest {(and there is no other)} eigenvalue of ${A}(p)$ whose associated eigenspace is ${p}^{\bot}$. Since $\Lambda(p)\rightarrow\infty$ as $\norm{p}\rightarrow\infty$, $\mathcal{M}$ and $\mathfrak{Q}$ are non-uniformly elliptic operators. {This is the reason why the geometry of the domain is important for the solvability of problem \varepsilonqref{ProblemaP} (see \cite[\S 12.4 p. 309, p. 345]{GT}).}
\mathbb Subsection{Maximum Principles}
{
The maximum principles are important tools used in the study of second order elliptic equations. The following theorem is restricted to the particular case of the operator $\mathfrak{Q}$ defined in \varepsilonqref{operador_Q}.
\begin{teo}[Comparison principle {\cite[Th. 10.1 p. 263]{GT}}]\label{PM_quasilineares}
Let $\W\in\R^n$ be a bounded domain and $u,\ v\in\cl^2(\W)\cap\cl^0(\overline{\W})$ satisfying
$$\left\{
\begin{split}
\mathfrak{Q} u&\geq \mathfrak{Q} v \mbox{ in } \W,\\
u&\leq v \mbox{ in } \partial\W.
\varepsilonnd{split}\right.$$
Then $u\leq v$ in $\W$.
\varepsilonnd{teo}
\begin{proof} Let $w=u-v$.
Recalling the expression for the coefficients $a_{ij}$ given in \varepsilonqref{coeff_aij} it follows
\begin{align*}
\mathfrak{Q} u - \mathfrak{Q} v
=&\ds\mathbb Sum_{ij}a_{ij}(\nabla u)\partial_ij w+\ds\mathbb Sum_{ij}a_{ij}(\nabla u)\partial_ij v-\ds\mathbb Sum_{ij}a_{ij}(\nabla v)\partial_ij v.
\varepsilonnd{align*}
For each $x\in\W$ let us define
$$f_x(p)=\ds\mathbb Sum_{ij}a_{ij}(p)D_{ij}v(x)-nH(x)(W(p))^3.$$
Applying the mean value theorem to the function
$$\varepsilonphi_x(t)=f((1-t)\nabla u(x)+t\nabla v(x)),$$
we obtain
\begin{align*}
\mathfrak{Q} u-\mathfrak{Q} v = & \ds\mathbb Sum_{ij}a_{ij}(\nabla u(x))\partial_ij w + \ds\mathbb Sum_{i}b_i(x)\partial_i w,
\varepsilonnd{align*}
where
\begin{align*}
{b}_i(x)=&\ds\mathbb Sum_{kl} \dfrac{\partial a_{kl}}{\partial p_i}(x,(1-t(x))\nabla u(x)+t(x)\nabla v(x))D_{ij}v(x)\\&-nH(x)\dfrac{\partial \left(W^3\right)}{\partial p_i}(x,(1-t(x))\nabla u(x)+t(x)\nabla v(x)).
\varepsilonnd{align*}
So, $\mathfrak{L} w:=\mathfrak{Q} u-\mathfrak{Q} v\geq 0$. Since also $w\leq 0$ in $\partial\W$, then $w\leq 0$ in $\W$ as a direct consequence of the following theorem for linear operators.
\noindent\colorbox{shadecolor}{
\begin{minipage}{.975\textwidth}
\begin{teorema}[Week maximum principle {\cite[Th. 3.1 p. 32]{GT}}]\label{WMPrinciple}
Let $\Omegaega\mathbb Subset\mathbb{R}^n$ be a bounded domain. Assume that $u\in\cl^2(\W)\cap\cl^0(\overline{\W})$ satisfies
$$\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i(x)\partial_i u \geq 0 \mbox{ in }\W,$$
where $\mathfrak{L}$ is elliptic and the coefficients $a_{ij}$ and $b_i$ are locally bounded.
Then
$$ \mathbb Sup_{\W} u =\mathbb Sup_{\partial\W}u.$$
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
\noindent Indeed, it can easily be verified that $\mathfrak{L}$ satisfies the hypothesis of this theorem.
\varepsilonnd{proof}
}
The following proposition is an important tool in section \ref{cap_NaoExis}.
This variant of the comparison principle traces its roots back to the work of Finn \cite[Lemma p. 139]{Finn1965} about the minimal surface equation in domains of the plane.
His lemma was extended by Jenkins-Serrin {\cite[Prop. III p. 182]{Serrin1968}} for the minimal hypersurface equation in $\R^n$, and subsequently by Serrin \cite[Th. 1 p. 459]{Serrin} for more general quasilinear elliptic operators (see also \cite[Th. 14.10 p. 347]{GT}). {We just refer here to the operator $\mathfrak{Q}$ defined in \varepsilonqref{operador_Q}.}
\begin{prop}[{\cite[L. 3.4.3 p. 109]{han2016nonlinear}}]\label{M_prop_gen_JS}
Let $\W\in M$ be a bounded domain and $\Gamma$ a relative open portion of $\partial\W$ of class $\cl^1$. For $H\in\cl^{0}(\overline{\W})$, let $u\in\cl^2(\W)\cap\cl^1(\W\cup \Gamma)\cap\cl^0(\overline{\W})$ and $v\in\cl^2(\W)\cap\cl^0(\overline{\W})$ satisfying
$$
\left\{ \begin{array}{cl}
\mathfrak{Q} u \geq \mathfrak{Q} v &\mbox{in } \W,\\
u \leq v &\mbox{in } \partial\W\mathbb Setminus\Gamma,\\
\parcial{v}{N}=-\infty &\mbox{in }\Gamma,
\varepsilonnd{array} \right.
$$
where $N$ is the inner unit normal to $\Gamma$. Then $u\leq v$ in $\Gamma$. Consequently, $u\leq v$ in $\W$.
\varepsilonnd{prop}
\begin{proof}
Suppose by contradiction that $m=\ds\max_{\Gamma} (u-v) > 0.$
Hence, $u \leq v + m $ in $\Gamma$. Then $u\leq v+m$ in $\partial\W$ since $u\leq v$ in $\partial\W\mathbb Setminus\Gamma$ by hypotheses.
Also,
$$\mathfrak{Q}(v+m)=\mathfrak{Q} v \leq \mathfrak{Q} u.$$
As a consequence of Theorem \ref{PM_quasilineares}, $u\leq v+m$ in $\W$.
Let now $y_0\in \Gamma$ be such that $m = u(y_0)-v(y_0)$.
Then, for $t>0$ near $0$ one has
\begin{align*}
u(y_0+tN_{y_0})-u(y_0)
\leq \left(v\left(y_0+tN_{y_0}\right)+m\right)-\left(v(y_0)+m\right)= v(y_0+tN_{y_0})-v(y_0).
\varepsilonnd{align*}
Dividing the expression by $t$ and passing to the limit as $t$ goes to zero it follows that
$$ \parcial{u}{N}\leq \parcial{v}{N} =-\infty,$$
which is a contradiction since $u\in\cl^1(\Gamma)$. Hence, $u\leq v$ in $\Gamma$. Using again Theorem \ref{PM_quasilineares} one has $u\leq v$ in $\W$.
\varepsilonnd{proof}
\mathbb Subsection{Transformation formulas for the mean curvature equation}\label{apend_form_transform}
The goal in this section is to derive some transformation formulas that are used in the next sections. Note first that the operator $\mathcal{M}$ defined in \varepsilonqref{operador_minimo_1_coord} can be written as
\begin{equation}\label{equ_curv_media_MxR_Delta_2}
\mathcal{M} u
=W^2\Deltalta u(x) - \varepsilonscalar{\Hess u \cdot \nabla u}{\nabla u},
\varepsilonnd{equation}
where $\Hess$ denotes the Hessian of a function (the matrix of second derivatives).
Let $\W\in \R^n$ be a bounded domain and $\varepsilonphi\in\cl^2(\overline{\W})$. Let $I$ be an interval and $\psi\in\cl^2(I)$.
Let us now define
$$w=\psi\circ \varepsilonrho + \varepsilonphi,$$
where $\varepsilonrho$ is a distance function\footnote{Recall that if $\varepsilonrho:\W\rightarrow \R$ is a distance function, then $\nabla \varepsilonrho \varepsilonquiv 1$ on $\W$ \cite[p. 41]{petersen1998}.}.
Then, $\partial_i w=\psi'\partial_i\varepsilonrho+\partial_i\varepsilonphi$ and $\nabla w=\psi'\nabla \varepsilonrho+\nabla \varepsilonphi$. Thus,
\begin{align}
\partial_{ij} w &=\psi'' \partial_i\varepsilonrho\partial_j\varepsilonrho + \psi' \partial_{ij} \varepsilonrho + \partial_{ij} \varepsilonphi\label{hess_coord_1},
\varepsilonnd{align}
and
\[\Hess w=\psi'' \nabla \varepsilonrho \otimes \nabla \varepsilonrho + \psi' \Hess \varepsilonrho +\Hess \varepsilonphi .
\]
Since $\varepsilonrho$ is a distance function it follows
\begin{equation}\label{Laplaciano_w}
\Deltalta w(x)=\tr \left(\Hess w\right) =\psi''\norm{\nabla \varepsilonrho}^2+\psi'\Deltalta \varepsilonrho + \Deltalta\varepsilonphi=\psi''+\psi'\Deltalta \varepsilonrho + \Deltalta\varepsilonphi.
\varepsilonnd{equation}
Besides\footnote{Recall that $\left(u\otimes v\right)w=\varepsilonscalar{v}{w}u$ for any vectors $u$, $v$ and $w$.},
\begin{align*}
&\varepsilonscalar{\Hess w \cdot \nabla w}{\nabla w} \\
=&\psi'' \varepsilonscalar{\nabla \varepsilonrho}{\nabla w}^2 + \psi' \varepsilonscalar{\Hess \varepsilonrho\cdot \nabla w}{\nabla w}+\varepsilonscalar{\Hess \varepsilonphi \cdot \nabla w}{\nabla w}\\
=&\psi'' \varepsilonscalar{\nabla \varepsilonrho}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}^2 + \psi' \varepsilonscalar{\Hess \varepsilonrho \cdot \left(\psi' \nabla\varepsilonrho + \nabla \varepsilonphi\right)}{ \psi' \nabla\varepsilonrho + \nabla \varepsilonphi}\\
&+\varepsilonscalar{ \Hess \varepsilonphi\cdot\left(\psi' \nabla\varepsilonrho + \nabla \varepsilonphi\right)}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}\\
=&\psi'' \varepsilonscalar{\nabla \varepsilonrho}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}^2\\
&+\psi'\left(\psi'^2 \varepsilonscalar{ \Hess \varepsilonrho \cdot \nabla\varepsilonrho}{\nabla\varepsilonrho} + 2\psi' \varepsilonscalar{\Hess \varepsilonrho\cdot\nabla\varepsilonrho}{\nabla \varepsilonphi} + \varepsilonscalar{\Hess \varepsilonrho\cdot\nabla \varepsilonphi}{\nabla \varepsilonphi}\right)\\
&+\varepsilonscalar{\Hess \varepsilonphi\cdot\left(\psi' \nabla\varepsilonrho + \nabla \varepsilonphi\right)}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}.
\varepsilonnd{align*}
{Recalling again that $\varepsilonrho$ is a distance function one has
\begin{align*}
\varepsilonscalar{\Hess \varepsilonrho\cdot \nabla \varepsilonrho}{e_j}=\ds\mathbb Sum_{i} \partial_ij \varepsilonrho \partial_i \varepsilonrho =\varepsilonscalar{\partial_j \nabla\varepsilonrho}{\nabla\varepsilonrho} = \frac{1}{2} \partial_j \left(\norm{\nabla \varepsilonrho }^2\right)=0\ \ \forall \ 1\leq j \leq n.
\varepsilonnd{align*}}
As a consequence,
\begin{equation}\label{D2w}
\begin{array}{r}
\varepsilonscalar{\Hess w \cdot \nabla w}{\nabla w} = \psi'' \varepsilonscalar{\nabla \varepsilonrho}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}^2 + \psi' \varepsilonscalar{\Hess \varepsilonrho \cdot \nabla \varepsilonphi}{\nabla \varepsilonphi}\\
+ \varepsilonscalar{\Hess \varepsilonphi \cdot \left(\psi' \nabla\varepsilonrho + \nabla \varepsilonphi\right)}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}.
\varepsilonnd{array}
\varepsilonnd{equation}
{From \varepsilonqref{equ_curv_media_MxR_Delta_2}, \varepsilonqref{Laplaciano_w} and \varepsilonqref{D2w} we derive}
\begin{align*}
\mathcal{M} w=&W^2(\psi''+\psi'\Deltalta \varepsilonrho + \Deltalta\varepsilonphi)-\left(\psi''\varepsilonscalar{\nabla \varepsilonrho}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}^2 +
\psi'\varepsilonscalar{\Hess \varepsilonrho\cdot \nabla \varepsilonphi}{\nabla \varepsilonphi}\right.\\
&\pushright{\left.+ \varepsilonscalar{\Hess \varepsilonphi\cdot \left(\psi' \nabla\varepsilonrho + \nabla \varepsilonphi\right)}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}\right),}
\varepsilonnd{align*}
where $W=W(\nabla w)=\mathbb Sqrt{1+\norm{\psi'\nabla \varepsilonrho +\nabla \varepsilonphi}^2}$. Therefore,
\begin{equation}\label{Eq_trans_varphi}
\begin{split}
\mathcal{M} w=&\psi'W^2\Deltalta\varepsilonrho -\psi'\varepsilonscalar{\Hess \varepsilonrho\cdot\nabla \varepsilonphi}{\nabla \varepsilonphi} \\
&\psi''W^2-\psi''\varepsilonscalar{\nabla \varepsilonrho}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi}^2\\
& \Deltalta \varepsilonphi W^2 - \varepsilonscalar{\Hess \varepsilonphi\cdot\left(\psi' \nabla\varepsilonrho + \nabla \varepsilonphi\right)}{\psi' \nabla\varepsilonrho + \nabla \varepsilonphi} .
\varepsilonnd{split}
\varepsilonnd{equation}
Furthermore, if $\varepsilonphi$ is constant, then $W=\mathbb Sqrt{1+\psi'^2}$ and
\begin{equation}\label{calc_Q_w}
\mathcal{M} w = \psi'(1+\psi'^2) \Deltalta \varepsilonrho + \psi''.
\varepsilonnd{equation}
\mathbb Section{The existence program}\label{chapterExistence}
In this section we carry out with the existence program for the Dirichlet problem \varepsilonqref{ProblemaP}. It will be seen how the elliptic theory assures that the solvability of problem \varepsilonqref{ProblemaP} strongly depends on $\cl^{1}$ a priori estimates for the family of related problems
\begin{equation}\tanhg{$P_{\tanhu}$}\label{ProblemaPsigma}
\left\{\begin{array}{l}
\mathcal{M} u = \tanhu nH(x)W^3\ \mbox{ in }\ \W, \\
\phantom{\mathcal{M}} u = \tanhu \varepsilonphi \ \mbox{ in }\ \partial\Omegaega,
\varepsilonnd{array}\right.
\varepsilonnd{equation}
not depending on $\tanhu$ or $u$\footnote{Note that problem \varepsilonqref{ProblemaP} is equivalent to problem $(P_{\tanhu=1})$.}. Actually, the following theorem (which is also valid for more general elliptic operators) holds.
\begin{teo}[{\cite[T. 11.4 p. 281]{GT}}]\label{T_Exist_quaselineares}
Let $\Omegaega\mathbb Subset \R^n$ be a bounded domain with $\partial\W$ of class $\cl^{2,\alpha}$, $\varepsilonphi\in \cl^{2,\alpha}(\overline{\Omegaega})$ and {$H\in\cl^{\alpha}(\overline{\W})$} for some $\alpha\in(0,1)$. Assume there exists $M>0$ independent of $u$ and $\tanhu$ such that any solution $u$ of the related problems \varepsilonqref{ProblemaPsigma} satisfies $\norm{u}_{\cl^{1}(\overline{\W})}<M$.
Then the Dirichlet problem \varepsilonqref{ProblemaP} has a unique solution in $\cl^{2,\alpha}(\overline{\W})$.
\varepsilonnd{teo}
\begin{proof}
Let $\beta\in[0,1]$ to be made precise later.
For each $v\in \cl^{1,\beta}(\overline{\Omegaega})$ we consider the linear operator
\begin{equation}\label{definitionLL}
\mathfrak{L}^v u: =\ds\mathbb Sum_{ij} a_{ij}^v(x)\partial_ij u ,
\varepsilonnd{equation}
where
$$ a_{ij}^v(x)=({W_v(x)})^2\delta_{ij}-{\partial_i v(x) \partial_j v(x)},\ W_v(x)=W(\nabla v(x))=\mathbb Sqrt{1+\nabla v(x)}.$$
We now want to study the following linear problem
\begin{equation}\tanhg{$LP_v$}\label{ProblemaPv}
\left\{\begin{array}{rl}
\mathfrak{L}^v u & = nH(x)({W_v(x)})^3\ \mbox{ in }\ \W, \\
u & = \varepsilonphi \ \mbox{ in }\ \partial\Omegaega.
\varepsilonnd{array}\right.
\varepsilonnd{equation}
The following theorem for linear operators guaranties the existence of a unique solution $u^v\in\cl^{2,\alpha\beta}(\overline{\W})\mathbb Subset\cl^{1,\beta}(\overline{\W})$ of problem \varepsilonqref{ProblemaPv}.
\noindent\colorbox{shadecolor}{
\begin{minipage}{.97\textwidth}
\begin{teorema}[Schauder {\cite[T. 6.14 p. 107]{GT}}]
Let $\Omegaega\mathbb Subset\mathbb{R}^n$ be a $\cl^{2,\alpha}$ bounded domain and $\varepsilonphi\in \cl^{2,\alpha}(\overline{\Omegaega})$ for some $\alpha\in(0,1)$. Let
$\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i(x)\partial_i u +c(x)u$
be a strictly elliptic operator in $\W$ with coefficients in $\cl^{\alpha}(\overline{\W})$ and $c\leq 0$. Let also $f$ be a function in $\cl^{\alpha}(\overline{\W})$. Then the problem
\begin{align*}
\left\{\begin{array}{l}
\mathfrak{L} u=f \textrm{ in } \Omegaega,\\
\phantom{\mathfrak{L}} u=\varepsilonphi\textrm{ in } \partial\Omegaega,
\varepsilonnd{array}\right.
\varepsilonnd{align*}
has a unique solution in $\cl^{2,\alpha}(\overline{\Omegaega})$.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
In fact, notice that $\partial\W$ is of class $\cl^{2,\alpha\beta}$ and $\varepsilonphi\in\cl^{2,\alpha\beta}(\overline{\W})$.
Also, $nHW_v^3$ and $a_{ij}^v$ belong to $\cl^{\alpha\beta}(\overline{\W})$ since $v\in\cl^{1,\beta}(\overline{\W})$, $H\in\cl^{\alpha}(\overline{\W})$ and the coefficients $a_{ij}$ are regular.
Besides, $\mathfrak{L}^v$ is strictly elliptic because it was proved in section \ref{secao_eq_cur_media} that the infimum of the smallest eigenvalue of the matrix $(a_{ij}(\nabla v(x)))=(a_{ij}^v(x))$ is 1.
Therefore, the operator
$$\funciones{T}{\cl^{1,\beta}(\overline{\W})}{\cl^{1,\beta}(\overline{\W})}{v}{u^v} $$
is well defined.
In addition, the solvability of problem \varepsilonqref{ProblemaP} is equivalent to the existence of a fixed point of $T$. Indeed, since $Tv$ is the only solution of \varepsilonqref{ProblemaPv}, that is,
\begin{equation*}
\left\{
\begin{split}
\mathbb Sum_{i,j=1}^n \left(\left({W_v(x)}\right)^{2}\delta_{ij} - {\partial_i v \partial_j v} \right)\partial_ij (Tv) &=nH(x) \left({W_v(x)}\right)^{3}\ \mbox{in}\ \W,\\
Tv&=\varepsilonphi \ \mbox{in}\ \partial\W,
\varepsilonnd{split}\right.
\varepsilonnd{equation*}
then the existence of a function $u\in\cl^{1,\beta}(\overline{\W})$ satisfying $Tu=u$ is exactly problem \varepsilonqref{ProblemaP}. {So, it gets evident the need for a theorem guaranteeing the existence of a fixed point of $T$. The following theorem is enough for our purpose.}
\noindent\colorbox{shadecolor}{
\begin{minipage}{.975\textwidth}
\begin{teorema}[Leray-Schauder {\cite[T. 11.3 p. 280]{GT}}]
Let $\mathcal{B}$ be a Banach space and $T: \mathcal{B} \rightarrow \mathcal{B}$ a completely continuous operator\footnotemark.
If there exists $M>0$ such that
\begin{equation*}
\norm{x}_\mathcal{B} < M
\varepsilonnd{equation*}
for each $\tanhu\in [0,1]$ and $x\in \mathcal{B}$ satisfying $x=\tanhu T x$, then $T$ has a fixed point.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
\footnotetext{We recall that a continuous mapping between two Banach spaces is called \textit{compact} or \textit{completely continuous} if the images of bounded sets are precompact.}
In order to use the Leray-Schauder fixed point theorem we observe first that the family of solutions of the related problems \varepsilonqref{ProblemaPsigma} is not empty once $u = 0$ obviously satisfies $(P_0)$. Also $\mathcal{B}=\cl^{1,\beta}(\overline{\W})$ is a Banach space.
In addition, since \varepsilonqref{ProblemaPv} holds for each $v\in\cl^{1,\beta}(\overline{\W})$, then for $\tanhu \in[0,1]$ it follows
\begin{equation}
\label{ProblemaPvtau}
\left\{
\begin{split}
\mathbb Sum_{i,j=1}^n \left(\left({W_v(x)}\right)^2\delta_{ij} - {\partial_i v \partial_j v} \right)\partial_ij (\tanhu Tv) &=\tanhu nH(x) \left({W_v(x)}\right)^{3}\ \mbox{in}\ \W,\\
\tanhu Tv&=\tanhu \varepsilonphi \ \mbox{in}\ \partial\W.
\varepsilonnd{split}\right.
\varepsilonnd{equation}
So, equation $\tanhu T u=u$ is equivalent to problem \varepsilonqref{ProblemaPsigma}, and the family of solutions of these problems is bounded in $\cl^1(\overline{\W})$ from the hypotheses.
The following global Hölder estimate for quasilinear operators of second order guarantees that we actually have an a priori bound in $\cl^{1,\beta}(\overline{\W})$ for some $\beta\in(0,1)$.
\noindent\colorbox{shadecolor}{
\begin{minipage}{.985\textwidth}
\begin{teorema}[Ladyzhenskaya-Ural'tseva {\cite[T. 13.7 p. 331]{GT}}]
Let $\Omegaega\mathbb Subset \R^n$ be a bounded domain with $\partial\W$ of class $\cl^2$ and $\varepsilonphi\in\cl^2(\overline{\W})$. Assume that
$u\in \cl^2( \overline{\Omegaega})$ satisfies
\begin{equation*}\left\{\begin{array}{l}\mathfrak{Q} u =\ds\mathbb Sum_{ij}a_{ij}(x,u,\nabla u)\partial_ij u + b(x,u,\nabla u)=0 \mbox{ in } \W,\\
\phantom{\mathfrak{Q} u =\ds\mathbb Sum_{ij}a_{ij}(x,u,\nabla u)\partial_ij u + b(x,u,\nabla u)} u = \varepsilonphi \mbox{ in } \partial\W,\varepsilonnd{array}\right.
\varepsilonnd{equation*}
where $\mathfrak{Q}$ is an elliptic operator such that $a_{ij}\in \cl^1(\overline{\Omegaega} \times \R\times \R^n)$ and $b\in \cl^0(\overline{\Omegaega} \times \R\times \R^n)$.
Then
\begin{equation*}
[u_i]_{\alpha, \Omegaega} < C \ \ \forall i=1,\downarrowts,n,
\varepsilonnd{equation*}
where $C=C\left(n,\norm{u}_{\cl^1(\overline{\W})},\norm{\varepsilonphi}_{\cl^2(\overline{\W})},\W,\lambda\right)$ and
$\alpha=\alpha\left(n,\norm{u}_{\cl^1(\overline{\W})},\W,\lambda\right)>0$.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
\noindent Indeed, the operator $\mathfrak{Q} $ defined in \varepsilonqref{operador_Q} obviously satisfies the hypothesis of the previous theorem. Observe that this estimate does not depend on $u$ or $\tanhu$ since we already have an a priori estimate in $\cl^1(\overline{\W})$ for the family of solutions of \varepsilonqref{ProblemaPsigma}. {This is the constant $\beta$ that should be fixed at the beginning.}
{
It still remains to prove that $T$ is continuous and compact. We recall first the following theorem for linear elliptic operators.
\noindent\colorbox{shadecolor}{
\begin{minipage}{.985\textwidth}
\begin{teorema}[Global Schauder estimate {\cite[Th. 6.6 p. 98]{GT}}]
Let $\Omegaega\mathbb Subset\mathbb{R}^n$ be a bounded domain with $\partial\W$ of class $\cl^{2,\alpha}$
and $\varepsilonphi\in \cl^{2,\alpha}(\overline{\Omegaega})$ for some $\alpha\in(0,1)$. Assume that $u\in \cl^{2,\alpha}(\overline{\Omegaega})$ satisfies
\begin{align*}\label{probl}
\left\{\begin{array}{l}
\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i(x)\partial_i u +c(x)u=f(x) \textrm{ in } \Omegaega,\\
\phantom{\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i(x)\partial_i u +c(x)} u=\varepsilonphi\textrm{ in } \partial\Omegaega,
\varepsilonnd{array}\right.
\varepsilonnd{align*}
where $f$ and the coefficients of the strictly elliptic operator $\mathfrak{L}$ belong to $\cl^{\alpha}(\overline{\W})$.
Then
\begin{equation}\label{tezsches}
\norm{u}_{\cl^{2,\alpha}(\overline{\Omegaega})}\leq C\left(\|u\|_{\cl^0(\overline{\Omegaega})}+\|f\|_{\cl^{\alpha}(\overline{\Omegaega})}+\|\varepsilonphi\|_{\cl^{2,\alpha}(\overline{\Omegaega})}\right)
\varepsilonnd{equation}
for $C=C(\Omegaega, n, \alpha, \lambda, K )>0$ {where $K \geq \ds\max\left\{\norm{a_{ij}}_{\cl^{\alpha}(\overline{\Omegaega})},\norm{b_{i}}_{\cl^{\alpha}(\overline{\Omegaega})},\norm{c}_{\cl^{\alpha}(\overline{\Omegaega})}\right\}$}.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
In view of the previous theorem
$$\norm{T v}_{\cl^{2,\alpha\beta}(\overline{\Omegaega})}\leq C\left(\|T v\|_{\cl^0(\overline{\Omegaega})}+\norm{n H W_v^{3}}_{\cl^{\alpha\beta}(\overline{\Omegaega})}+\|\varepsilonphi\|_{\cl^{2,\alpha\beta}(\overline{\Omegaega})}\right)
$$
for every $v\in \cl^{1,\beta}(\overline{\W}) $. Besides,
$$\|T v\|_{\cl^0(\overline{\Omegaega})}\leq \mathbb Sup_{\partial\W}\mathopen d{\varepsilonphi} + \tilde{C} \norm{nH W_v^{3}}_{\cl^{0}(\overline{\W})}$$
as a direct consequence of the following theorem for linear elliptic operators.
\noindent\colorbox{shadecolor}{
\begin{minipage}{0.985\textwidth}
\begin{teorema}[Height estimate {\cite[Th. 3.7 p. 36]{GT}}]
Let $\W\mathbb Subset\R^n$ be a bounded domain. Assume that $u\in \cl^{2}(\W)\cap\cl^0(\overline{\Omegaega})$ satisfies
$$\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i(x)\partial_i u +c(x)u=f(x)$$
where $f$ and the coefficients $b_i$ of the strictly elliptic operator $\mathfrak{L}$ are bounded in $\overline{\W}$ and $c\leq 0$.
Then
\begin{equation}\label{eq_est_apriori_linear}
\ds\mathbb Sup_{\W}\mathopen d{u}\leq \mathbb Sup_{\partial\W}\mathopen d{u}+C\mathbb Sup_{\W}\mathopen d{f}.
\varepsilonnd{equation}
where $C=C\left(\lambda,\diam(\W),\ds\max_{i}\mathbb Sup_{\W}\mathopen d{b_i}\right)$.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
Observe that the constants $C$ and $\tilde{C}$ are independent of $v$. Hence, $T$ maps bounded sets in $\cl^{1,\beta}(\overline{\W})$ into bounded sets in $\cl^{2,\alpha\beta}(\overline{\W})\hookrightarrow \cl^{1,\beta}(\overline{\W})\hookrightarrow \cl^{\alpha}(\overline{\W})$.
Let now $\left\{ T v_m\right\}$ be a sequence in $T\left(\cl^{1,\beta}(\overline{\W})\right)$. We affirm that there exists some subsequence that converges in $\cl^{1,\beta}(\overline{\W})$.
Indeed, since for every $x$ and $y$ in $\overline{\W}$ we have
$$ \mathopen d{T v_m(x)-T v_m(y)} \leq \norm{T v_m}_{\cl^{\alpha}(\overline{\W})} \norm{x-y}^{\alpha}, $$
so $\left\{T v_m\right\}$ is equicontinuous. This sequence is also uniformly bounded, then there exists a subsequence $\left\{Tv_{\xi(m)}\right\}$ that converges uniformly to a continuous function $w$ by the Arzela-Ascoli theorem.
Besides, for every $1\leq i\leq n$,
$$ \mathopen d{\partial_i T v_m(x)-\partial_i T v_m(y)}\leq \norm{Tv_m}_{\cl^{1,\beta}(\overline{\W})} \norm{x-y}^{\beta},$$
then $\left\{\partial_i Tv_m\right\}$ is also equicontinuous. By the Arzela-Ascoli theorem there exists a subsequence $\left\{\partial_i Tv_{\phi_i(m)}\right\}$ that converges uniformly to a continuous function $w_i$ since $\left\{\partial_i Tv_m\right\}$ is uniformly bounded. This implies that $\partial_i w$ exists, is continuous and $w_i=\partial_i w$.
Furthermore, for every $1\leq i, j\leq n$,
$$ \mathopen d{\partial_ij T v_m(x)-\partial_ij T v_m(y)}\leq \norm{Tv_m}_{\cl^{2,\alpha\beta}(\overline{\W})} \norm{x-y}^{\alpha\beta}.$$ Because of the $\cl^{2,\alpha\beta}$ estimate $\left\{\partial_ij T v_m\right\}$ is equicontinuous. Since it is also a bounded sequence, then there exists a subsequence $\left\{\partial_ij Tv_{\psi_{ij}(m)}\right\}$ that converges uniformly to $\partial_ij w$ (which necessarily exists and is continuous).
This proves that $w\in\cl^{2}(\overline{\W})$ and that the subsequence $Tv_{\psi(\phi(\xi(m)))}$, with $\phi=\phi_1\circ\downarrowts\circ\phi_n$ and $\psi=\psi_{nn}\circ\psi_{n,n-1}\circ\downarrowts\circ\psi_{11}$, converges to $w$ in $\cl^{2}(\overline{\W})$ and also in $\cl^{1,\beta}(\overline{\W})$.
In order to prove the continuity of $T$ let $\{v_m\}$ be a sequence converging to $v$ in $\cl^{1,\beta}(\overline{\W})$. Since $\{Tv_m\}$ is precompact in $\cl^2(\overline{\W})$, every subsequence has a convergent subsequence. Let $\left\{Tv_{\xi(m)}\right\}$ such a convergent subsequence with limit $w\in\cl^2(\overline{\W})$. Then
$$\left\{
\begin{split}
\mathbb Sum_{i,j=1}^n \left(\left(W_{v_{\xi(m)}}\right)^2\delta_{ij} - {\partial_i v_{\xi(m)} \, \partial_j v_{\xi(m)}}\right)\partial_ij \left(Tv_{\xi(m)}\right) &=nH(x) \left(W_{v_{\xi(m)}}\right) ^{3}\ \mbox{in}\ \W,\\
Tv_{\xi(m)}&=\varepsilonphi \ \mbox{in}\ \partial\W.
\varepsilonnd{split}\right.
$$
Passing to the limits and recalling the definition of $T$ one has $w=Tv$.
Since every subsequence of $\{Tv_m\}$ has at least one subsequence which converges to $Tv$, then $\{Tv_m\}$ also converges to $Tv$.
}
So, it had been proved that $T$ has a fixed point $u\in\cl^{2,\alpha\beta}(\overline{\W})\hookrightarrow\cl^{1,\beta}(\overline{\W})$. To see that $u\in\cl^{2,\alpha}(\overline{\W})$ recall that $u$ is a solution of the linear problem $LP_u$ (see problem \varepsilonqref{ProblemaPv})
and use the following global regularity theorem for linear operators.
\noindent\colorbox{shadecolor}{
\begin{minipage}{.97\textwidth}
\begin{teorema}[Global regularity {\cite[Th. 6.18 p. 111]{GT}}]
Let $\Omegaega\mathbb Subset\mathbb{R}^n$ be a bounded domain with $\partial\W$ of class $\cl^{2,\alpha}$ and $\varepsilonphi\in \cl^{2,\alpha}(\overline{\Omegaega})$ for some $\alpha\in(0,1)$. Suppose that $u\in\cl^{2}(\W)$ satisfies
\begin{align*}
\left\{\begin{array}{l}
\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i\partial_i u +c(x)u=f \textrm{ in } \Omegaega,\\
\phantom{\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i\partial_i u +c(x)} u=\varepsilonphi\textrm{ in } \partial\Omegaega,
\varepsilonnd{array}\right.
\varepsilonnd{align*}
where $f$ and the coefficients of the strictly elliptic operator $\mathfrak{L}$ belong to $\cl^{\alpha}(\overline{\W})$. Then $u\in\cl^{2,\alpha}(\overline{\W})$.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
Finally, notice that the solution is unique as a consequence of the comparison principle, Theorem \ref{PM_quasilineares}.
\varepsilonnd{proof}
\begin{obs}
Applying the same argument to every fixed $\tanhu\in[0,1]$ we ensure the existence of a solution for each problem \varepsilonqref{ProblemaPsigma}.
\varepsilonnd{obs}
\mathbb Section{Fulfillment of the existence program's requirements}\label{chapter_estimates}
The goal in this section is to obtain the a priori estimates for the family of solutions of the related problems \varepsilonqref{ProblemaPsigma} required by the existence program.
In order to derive the a priori global gradient estimate the techniques introduced by Caffarelli-Nirenberg-Spruck \cite[p. 51]{CNSpruck} are used.
\begin{teo}[A priori global gradient estimate {\cite[Th. 3.2.4 p. 94]{han2016nonlinear}}]\label{teo_Est_global_gradiente}
Let $\W\in M$ be a bounded domain with $\partial\W$ of class $\cl^2$. For $H\in\cl^{1}\left(\W\right)$, let $u\in\cl^3(\W)\cap\cl^1(\overline{\W})$ be a solution of \varepsilonqref{operador_minimo_1_coord}.
Then
\begin{equation}\label{est_global_final}
\mathbb Sup_{\W}\norm{\nabla u(x)}\leq\left(\mathbb Sqrt{3}+\mathbb Sup\limits_{\partial\W}\norm{\nabla u}\right)\varepsilonxp\left(2\mathbb Sup\limits_{\W}\mathopen dulo{u}\left(1+8n\left(\norm{H}_1\right)\right)\right).
\varepsilonnd{equation}
\varepsilonnd{teo}
\begin{proof}
Let $w(x)=\norm{\nabla u(x)}e^{Au(x)}$ where $A\geq 1$. Suppose $w$ attains a maximum at $x_0\in\overline{\W}$. If $x_0\in\partial\W$, then
$$w(x)\leq w(x_0) =\norm{\nabla u(x_0)}e^{Au(x_0)}.$$
So,
\begin{equation}\label{est_global_1}
\mathbb Sup_{\W}\norm{\nabla u(x)}\leq\mathbb Sup_{\partial\W}\norm{\nabla u}e^{2A\mathbb Sup\limits_{\W}\mathopen dulo{u}}.
\varepsilonnd{equation}
Suppose now that $x_0\in\W$ and that $\nabla u(x_0)\neq 0$. {It can be assumed that}
$e_1=\frac{\nabla u(x_0)}{\norm{\nabla u(x_0)}}$. Then,
\begin{equation}\label{der_u_x0}
\partial_k u(x_0)=\varepsilonscalar{e_k}{\nabla u(x_0)}
=\norm{\nabla u(x_0)}\delta_{k1}.
\varepsilonnd{equation}
Differentiating \varepsilonqref{operador_minimo_1_coord} with respect to $x_1$ we derive
\begin{equation}\label{der_MCE}
\begin{split}
\partial_1 \left(W^2\right)\Deltalta u+W^2 \ds\mathbb Sum_{i=1}^n\partial_1ii u -2\ds\mathbb Sum_{i,j=1}^n\partial_i u \partial_1j u \partial_{ij} u -\mathbb Sum_{i,j=1}^n \partial_i u \partial_j u \partial_1ij u
\\
=n\partial_1 H W^3 + nH\partial_1\left(W^3\right).
\varepsilonnd{split}
\varepsilonnd{equation}
The goal now is to estimate the third derivatives in \varepsilonqref{der_MCE}. Observe first that the function $\tilde{w}(x)=\ln w(x)=Au(x)+\ln \norm{\nabla u(x)}$ also attains a maximum at $x_0$. Therefore, for each $0\leq k \leq n$,
\begin{equation}\label{derk}
\partial_k \tilde{w}(x_0)= A\partial_k u(x_0) + \dfrac{\partial_k \left(\norm{\nabla u}^2\right)(x_0)}{2\norm{\nabla u(x_0)}^2}=0,
\varepsilonnd{equation}
and
$$\partial_kk \tilde{w}(x_0)=A\partial_kk u(x_0) +\dfrac{1}{2} \partial_k\left({\norm{\nabla u}^{-2}}\right)(x_0)\partial_k \left(\norm{\nabla u}^2\right)(x_0)+\dfrac{\partial_kk \left(\norm{\nabla u}^2\right)(x_0)}{2\norm{\nabla u(x_0)}^2}\leq 0.$$
Since
$$\partial_k\left({\norm{\nabla u}^{-2}}\right)=\partial_k\left({\norm{\nabla u}^{2}}\right)^{-1}=-\left({\norm{\nabla u}^{2}}\right)^{-2} \partial_k\left({\norm{\nabla u}^{2}}\right),$$
then
\begin{equation}\label{derkk}
A\partial_kk u(x_0) - \dfrac{\left(\partial_k \left(\norm{\nabla u}^2\right)(x_0)\right)^2}{2\norm{\nabla u(x_0)}^{4}}+\dfrac{\partial_kk \left(\norm{\nabla u}^2\right)(x_0)}{2\norm{\nabla u(x_0)}^2}\leq 0.
\varepsilonnd{equation}
Once
\begin{equation}\label{Dknorma}
\partial_k \left(\norm{\nabla u}^2\right)=\partial_k \left(\mathbb Sum_{i=1}^n(\partial_i u(x))^2\right)= 2 \ds\mathbb Sum_{i=1}^n \partial_i u \partial_ki u,
\varepsilonnd{equation}
it follows from \varepsilonqref{der_u_x0}
\begin{equation}\label{Dknorm}
\partial_k \left(\norm{\nabla u}^2\right)(x_0)=2 \norm{\nabla u(x_0)}\partial_1k u(x_0).
\varepsilonnd{equation}\vspace*{.1cm}
\noindent Substituting \varepsilonqref{der_u_x0} and \varepsilonqref{Dknorm} in \varepsilonqref{derk} one has
{
$$ A\norm{\nabla u (x_0)}\delta_{k1} + \dfrac{2 \norm{\nabla u(x_0)}\partial_1k u(x_0)}{2\norm{\nabla u(x_0)}^2} =0,$$
thus,}
\begin{equation}\label{derk_2}
\partial_1k u(x_0)=-A\norm{\nabla u (x_0)}^2\delta_{k1}.
\varepsilonnd{equation}
\noindent Substituting also \varepsilonqref{derk_2} in \varepsilonqref{Dknorm} we obtain
\begin{equation}\label{Dknorm_2}
\partial_k \left(\norm{\nabla u}^2\right)(x_0)=-2A\norm{\nabla u (x_0)}^3\delta_{k1}.
\varepsilonnd{equation}
Besides, from \varepsilonqref{Dknorma} we get
\begin{align*}
\partial_kk \left(\norm{\nabla u}^2\right)(x) =&2\ds\mathbb Sum_{i=1}^n \left( \partial_kki u \partial_i u + (\partial_ki u)^2\right),
\varepsilonnd{align*}
and from \varepsilonqref{der_u_x0} we conclude
\begin{equation}\label{dkknorma_2}
\begin{split}
\partial_kk \left(\norm{\nabla u}^2\right)(x_0)=2\norm{\nabla u(x_0)}\partial_kkum u +2 \ds\mathbb Sum_{i=1}^n (\partial_ki u(x_0))^2.
\varepsilonnd{split}
\varepsilonnd{equation}
Using expressions \varepsilonqref{Dknorm_2} and \varepsilonqref{dkknorma_2} in \varepsilonqref{derkk} we verify that
\[
\begin{split}
A\partial_kk u(x_0)-2A^2\norm{\nabla u (x_0)}^2\delta_{k1}+\ds \dfrac{\partial_kkum u(x_0)}{\norm{\nabla u(x_0)}}
+\dfrac{\ds\mathbb Sum_{i=1}^n\left(\partial_ki u(x_0)\right)^2}{\norm{\nabla u (x_0)}^2} &\leq 0.
\varepsilonnd{split}
\]
From \varepsilonqref{derk_2} we have for $k=1$
\[\begin{split}
-A^2\norm{\nabla u (x_0)}^2-2A^2\norm{\nabla u (x_0)}^2+\ds \dfrac{\partial_1umum u(x_0)}{\norm{\nabla u(x_0)}}
+\dfrac{\ds\mathbb Sum_{i=1}^n\left(-A\norm{\nabla u(x_0)}^2\right)^2\delta_{i1}}{\norm{\nabla u (x_0)}^2}& \leq 0,
\varepsilonnd{split}\]
then,
\begin{equation}\label{est_Dumumum}
\partial_1umum u(x_0)\leq 2A^2\norm{\nabla u (x_0)}^3.
\varepsilonnd{equation}
\noindent If $k>1$, then
\begin{align*}
A\partial_kk u(x_0)+\ds \dfrac{\partial_kkum u(x_0)}{\norm{\nabla u(x_0)}}\leq-\dfrac{\ds\mathbb Sum_{i=1}^n\left(\partial_ki u(x_0)\right)^2}{\norm{\nabla u (x_0)}^2} \leq 0,
\varepsilonnd{align*}
so,
\begin{equation}\label{est_Dkkum}
\partial_kkum u(x_0)\leq -A\partial_kk u(x_0)\norm{\nabla u (x_0)}, \ 1 < k\leq n .
\varepsilonnd{equation}
On the other hand, since \varepsilonqref{Dknorm_2} holds we deduce
\begin{equation}\label{dW2}
\partial_1\left(W^2\right)(x_0)=\partial_1\left(\norm{\nabla u}^2\right)(x_0)=-2A\norm{\nabla u (x_0)}^3,
\varepsilonnd{equation}
and
\begin{equation}\label{dW3}
\partial_1\left(W^3\right)(x_0)=\frac{3}{2}W_0\partial_1 \left(W^2\right)(x_0)=-3AW_0\norm{\nabla u (x_0)}^3.
\varepsilonnd{equation}
Using \varepsilonqref{der_u_x0}, \varepsilonqref{derk_2}, \varepsilonqref{est_Dumumum}, \varepsilonqref{est_Dkkum}, \varepsilonqref{dW2} and \varepsilonqref{dW3} in \varepsilonqref{der_MCE} it follows
\begin{align*}
&n\partial_1 H(x_0) W_0^3 -3nA H_0 W_0\norm{\nabla u(x_0)}^3\\[1em]
=&-2A\norm{\nabla u(x_0)}^3\Deltalta u(x_0)+W_0^2\ds\mathbb Sum_{i=1}^n \partial_1ii u(x_0) \\
&+2A\norm{\nabla u(x_0)}^3\partial_1um u(x_0)-\norm{\nabla u(x_0)}^2 \partial_1umum u(x_0) \\[1em]
=&-2A\norm{\nabla u(x_0)}^3\ds\mathbb Sum_{i>1}\partial_ii u(x_0)+W_0^2\ds\mathbb Sum_{i>1} \partial_1ii u(x_0) +\partial_1umum u(x_0) \\[1em]
\leq &-2A\norm{\nabla u(x_0)}^3\ds\mathbb Sum_{i>1}\partial_ii u(x_0)-A\norm{\nabla u(x_0)}W_0^2\ds\mathbb Sum_{i>1}\partial_ii u(x_0) +2A^2\norm{\nabla u(x_0)}^3 \\[1em]
=&-A\norm{\nabla u(x_0)}\left(1+3\norm{\nabla u(x_0)}^2\right)\ds\mathbb Sum_{i>1}\partial_ii u(x_0) +2A^2\norm{\nabla u(x_0)}^3,
\varepsilonnd{align*}
where we have used the notation $H_0=H(x_0)$ and $W_0=\mathbb Sqrt{1+\norm{\nabla u(x_0)}^2}$.
In addition, since \varepsilonqref{der_u_x0} and \varepsilonqref{derk_2} holds, the mean curvature equation \varepsilonqref{operador_minimo_1_coord} at $x_0$ takes the form
\begin{align*}
nH_0 W_0^3=W_0^2 \Deltalta u(x_0)-\norm{\nabla u(x_0)}^2\partial_1um u(x_0)
=W_0^2\ds\mathbb Sum_{i>1}\partial_ii u(x_0) -A \norm{\nabla u(x_0)}^2,
\varepsilonnd{align*}
so,
\begin{equation}\label{equ_curv_media_MxR_Delta_x0}
\ds\mathbb Sum_{i>1}\partial_ii u(x_0) = nH_0W_0+\dfrac{A\norm{\nabla u(x_0)}^2}{W_0^2}.
\varepsilonnd{equation}
Therefore,
\begin{align*}
0\leq&-A\norm{\nabla u(x_0)}\left(1+3\norm{\nabla u(x_0)}^2\right)\left(nH_0W_0+\dfrac{A\norm{\nabla u(x_0)}^2}{W_0^2} \right)\\
&+2A^2\norm{\nabla u(x_0)}^3 + 3nA H_0 W_0\norm{\nabla u(x_0)}^3 - n\partial_1 HW_0^3 \\[1em]
=&-A n H_0 W_0\norm{\nabla u(x_0)}-n\partial_1 HW_0^3+\dfrac{A^2\norm{\nabla u(x_0)}^3}{W_0^2}\left(1-\norm{\nabla u(x_0)}^2\right).
\varepsilonnd{align*}
Then
\begin{align*}
\dfrac{A^2\norm{\nabla u(x_0)}^3}{W_0^2}\left(\norm{\nabla u(x_0)}^2-1\right)
\leq A n h_0 W_0\norm{\nabla u(x_0)} + n h_1 W_0^3,
\varepsilonnd{align*}
where $h_0=\mathbb Sup\limits_{\W}\mathopen dulo{H}$ and $h_1=\mathbb Sup\limits_{\W} \norm{\nabla H}$. Dividing by $A^2 W_0^3$ and noticing that $W_0^2> W_0 >\norm{\nabla u(x_0)}$ it follows
$$ \dfrac{\norm{\nabla u(x_0)}^3}{W_0^5}\left(\norm{\nabla u(x_0)}^2-1\right)< \dfrac{n}{A} \norm{H}_1. $$
Obviously we can suppose that $\norm{\nabla u(x_0)}>1$. Since
$$W_0^3=\left(1+\norm{\nabla u(x_0)}^2\right)^{3/2}<\left(2\norm{\nabla u(x_0)}^2\right)^{3/2}
<4\norm{\nabla u(x_0)}^3, $$
we see that
$$\dfrac{1}{4}\dfrac{\norm{\nabla u(x_0)}^2-1}{W_0^2}<\dfrac{\norm{\nabla u(x_0)}^3}{W_0^3}\dfrac{\norm{\nabla u(x_0)}^2-1}{W_0^2}< \dfrac{n}{A} \norm{H}_1, $$
that is,
$$\dfrac{\norm{\nabla u(x_0)}^2-1}{\norm{\nabla u(x_0)}^2+1}< \dfrac{4n}{A} \norm{H}_1. $$
Choosing $A=1+8n\norm{H}_1$ it follows
$$\dfrac{\norm{\nabla u(x_0)}^2-1}{\norm{\nabla u(x_0)}^2+1}< \dfrac{1}{2}, $$
so,
$$ \norm{\nabla u(x_0)}<\mathbb Sqrt{3}.$$
As a consequence,
$$w(x)\leq w(x_0) =\norm{\nabla u(x_0)}e^{Au(x_0)}\leq\mathbb Sqrt{3}e^{Au(x_0)},$$
thus
\begin{equation}\label{est_global_grad_interior}
\mathbb Sup_{\W}\norm{\nabla u(x)}\leq\mathbb Sqrt{3}e^{2A\mathbb Sup\limits_{\W}\mathopen dulo{u}}.
\varepsilonnd{equation}
Putting together \varepsilonqref{est_global_1} and \varepsilonqref{est_global_grad_interior} we obtain
\begin{equation*}
\mathbb Sup_{\W}\norm{\nabla u(x)}\leq\mathbb Sqrt{3}e^{2A\mathbb Scriptstyle\mathbb Sup\limits_{\W}\mathopen dulo{u}} + \mathbb Sup\limits_{\partial\W}\norm{\nabla u}e^{2A\mathbb Scriptstyle\mathbb Sup\limits_{\W}\mathopen dulo{u}},
\varepsilonnd{equation*}
which yields the desired estimate.
\varepsilonnd{proof}
The following theorem due to Serrin \cite[\S 9 p. 434]{Serrin} ensures the boundary gradient estimate. For the proof is used the classical idea of finding an upper and a lower barriers for $u$ on $\partial\W$ to get a control for $\nabla u$ along $\partial\W$.
\begin{teo}[A priori boundary gradient estimate {\cite[Th. 3.2.2 p. 89]{han2016nonlinear}}]\label{teo_Est_gradiente_fronteira}
Let $\W\in \R^n$ be a bounded domain with $\partial\W$ of class $\cl^2$ and $\varepsilonphi\in\cl^2(\overline{\W})$.
Let
$H\in\cl^{1}\left(\overline{\W}\right)$ satisfying
\begin{equation}\label{cond_Serrin}
(n-1)\mathcal H_{\partial\W}(y)\geq n \mathopen dulo{H(y)} \ \forall \ y\in\partial\W.
\varepsilonnd{equation}
If $u\in\cl^2(\W)\cap\cl^1(\overline{\W})$ is a solution of \varepsilonqref{ProblemaP}, then
\begin{equation}\label{EstGradFront}
\mathbb Sup\limits_{\partial\W}\norm{\nabla u}\leq \norm{\varepsilonphi}_1 + \ds e^{C\left(1+ \norm{H}_1+ \norm{\varepsilonphi}_2\right)\left(1+\norm{\varepsilonphi}_1\right)^3\left(\norm{u}_0+\norm{\varepsilonphi}_0\right)}
\varepsilonnd{equation}
for some $C=C(n,\W)$.
\varepsilonnd{teo}
\begin{proof}
We set $d(x)=\dist(x,\partial\W)$ for $x\in\W$. Let $\tanhu>0$ be such that $d$ is of class $\cl^2$ over the set of points in $\W$ for which $d(x)\leq\tanhu$ (see \cite[L. 14.16 p. 355]{GT}, \cite[L. 3.1.8 p. 84]{han2016nonlinear} and \cite[L. 1 p. 420]{Serrin}).
Let $\psi\in\cl^2([0,\tanhu])$ be a non-negative function satisfying
\begin{multicols}{3}
\begin{enumerate}
\item[P1.] $\psi'(t)\geq 1$,
\item[P2.] $\psi''(t) \leq 0$,
\item[P3.] $t\psi'(t)\leq 1$.
\varepsilonnd{enumerate}
\varepsilonnd{multicols}
For $a<\tanhu$ to be fixed latter on we consider the set
$$\W_{a}=\left\{x\in M; d(x)<a \right\} .$$
We now define $w^{\pm}=\pm \psi\circ d + \varepsilonphi$. Firstly, we estimate $\pm\mathcal{M} w^{\pm}$ in $\W_a$.
Using the transformation formula \varepsilonqref{Eq_trans_varphi} one has
\begin{equation}\label{eq_barreira_superior}
\begin{split}
\pm \mathcal{M} w^{\pm}=&\psi'W_{\pm}^2\Deltalta d -\psi'\varepsilonscalar{\Hess d \cdot \nabla \varepsilonphi}{ \nabla \varepsilonphi} \\
& + \psi''W_{\pm}^2-\psi''\varepsilonscalar{\nabla d }{\pm \psi' \nabla d + \nabla \varepsilonphi}^2\\
&\pm W_{\pm}^2\Deltalta \varepsilonphi \mp \varepsilonscalar{\Hess\varepsilonphi\cdot (\pm\psi' \nabla d + \nabla \varepsilonphi)}{\pm\psi' \nabla d + \nabla \varepsilonphi},
\varepsilonnd{split}
\varepsilonnd{equation}
where
$$
W_{\pm}=\mathbb Sqrt{1+\norm{\nabla w^{\pm}}^2}=\mathbb Sqrt{1+\norm{\pm \psi'\nabla d + \nabla \varepsilonphi}^2}.
$$
Once $d$ is of class $\cl^2$ in $\W_a$ and $\psi'\geq 1$ we have
\begin{equation}\label{BBB}
\psi'\mathopen dulo{\varepsilonscalar{\Hess d \cdot \nabla \varepsilonphi}{ \nabla \varepsilonphi}} \leq \psi'^2 \norm{d}_2\norm{\varepsilonphi}_1^2.
\varepsilonnd{equation}
Since $\psi''<0$ and $\varepsilonscalar{\nabla d}{\pm\psi' \nabla d + \nabla \varepsilonphi}^2\leq \norm{\pm\psi' \nabla d + \nabla \varepsilonphi}^2$, then
\begin{equation}\label{AAA}
\psi''W_{\pm}^2-\psi''\varepsilonscalar{\nabla d }{\pm\psi' \nabla d + \nabla \varepsilonphi}^2\leq \psi''.
\varepsilonnd{equation}
Also $\varepsilonphi$ is of class $\cl^2$ in $\W_a$ by hypothesis, so
\begin{align*}
&\mathopen dulo{\pm \Deltalta \varepsilonphi W_{\pm}^2\mp \varepsilonscalar{\Hess \varepsilonphi \cdot (\pm\psi' \nabla d + \nabla \varepsilonphi)}{\pm\psi' \nabla d + \nabla \varepsilonphi} } \\
& \leq n\norm{\varepsilonphi}_2 W_{\pm}^2+ \norm{\varepsilonphi}_2\norm{\pm\psi' \nabla d + \nabla \varepsilonphi}^2\\
& \leq 2 n \norm{\varepsilonphi}_2 W_{\pm}^2.
\varepsilonnd{align*}
Notice now that
$$
\norm{\pm\psi' \nabla d + \nabla \varepsilonphi}^2=\left(\psi'^2+2 \psi'\varepsilonscalar{\pm\nabla d}{\nabla \varepsilonphi}+\norm{\nabla \varepsilonphi}^2\right)
\leq \left(1+\norm{\varepsilonphi}_1\right)^2\psi'^2,
$$
hence
\begin{equation}\label{est_W2}
W_{\pm}^2 \leq 1 + \left(1+\norm{\varepsilonphi}_1\right)^2\psi'^2 \leq 2\left(1+\norm{\varepsilonphi}_1\right)^2\psi'^2.
\varepsilonnd{equation}
Therefore,
\begin{equation}\label{CCC}
\begin{split}
\mathopen dulo{\pm \Deltalta \varepsilonphi W_{\pm}^2 \mp \varepsilonscalar{\Hess \varepsilonphi \cdot (\pm\psi' \nabla d + \nabla \varepsilonphi)}{\pm\psi' \nabla d + \nabla \varepsilonphi} } \phantom{.}\\
\leq 4n\norm{\varepsilonphi}_2\left(1+\norm{\varepsilonphi}_1\right)^2\psi'^2.
\varepsilonnd{split}
\varepsilonnd{equation}
Substituting \varepsilonqref{AAA}, \varepsilonqref{BBB}, \varepsilonqref{CCC} in \varepsilonqref{eq_barreira_superior} it follows
\begin{equation}\label{est_Mwpm}
\pm\mathcal{M} w^{\pm}\leq \psi' W_{\pm}^2 \Deltalta d + \psi''+ c \psi'^2,
\varepsilonnd{equation}
where
\begin{equation}\label{constantec0}
c=\norm{d}_2\norm{\varepsilonphi}_1^2+4n\norm{\varepsilonphi}_2\left(1+\norm{\varepsilonphi}_1\right)^2.
\varepsilonnd{equation}
From \varepsilonqref{est_Mwpm} we obtain
\begin{equation}\label{eq_barreira_superior_Q_2}
\begin{array}{r}
\pm \mathfrak{Q}_{ } w^{\pm} \leq \psi'W_{\pm}^2\Deltalta d + \psi''+ c \psi'^2 + n \mathopen dulo{H(x)}W_{\pm}^{3}.
\varepsilonnd{array}
\varepsilonnd{equation}
Let now $y=y(x)$ be the point of $\partial\W$ nearest to $x$. Then $x$ belongs to the segment $\{y+tN_y; 0\leq t \leq a\}$, where $N$ is the inner unit normal to $\partial\W$. Let us denote by $\Gamma_t$ the hypersurface parallel to $\partial\W$ at a distance $t$ contained in $\W_a$. We define $\mathcal H(t):=\mathcal H_{\Gamma_t}(y + t N_y)$ where $\mathcal H_{\Gamma_t}$ is the mean curvature of $\Gamma_t$ with respect to the normal that coincides with $\nabla d(x)=\frac{x-y}{\norm{x-y}}$ at $x$.
Recall that $\Deltalta d(x)=-(n-1)\mathcal H_{\Gamma_{d(x)}}(x)$ (see \cite[\S 14.6 p 354]{GT}, \cite[\S 3.1 p. 80]{han2016nonlinear} and \cite[\S 3 p. 420]{Serrin}). In addition,
$$\mathcal H_{\Gamma_t}(y+tN_y) \geq \mathcal H_{\partial\W}(y) $$
since $\mathcal H'(t)\geq (\mathcal H(t))^2 \geq 0$ in $[0,a)$ (see \cite[p. 485]{Serrin}).
Using also the Serrin condition \varepsilonqref{cond_Serrin} we get
\begin{equation}\label{est_usar_hiperb}
\Deltalta d(x) \leq \Deltalta d(y) \leq -n \mathopen dulo{H(y)} \ \forall \ x\in\W_a
\varepsilonnd{equation}
Substituting \varepsilonqref{est_usar_hiperb} in \varepsilonqref{eq_barreira_superior_Q_2} we obtain
{\mathbb Small
\begin{equation}\label{Qw_medio}
\begin{split}
\pm \mathfrak{Q}_{ } w^{\pm} \leq & n \psi'W_{\pm}^2( \mathopen dulo{H(x)} -\mathopen dulo{H(y)}) +n \mathopen dulo{H(x)}W_{\pm}^2 \left(W_{\pm}-\psi'\right) + \psi''+ c \psi'^2 .
\varepsilonnd{split}
\varepsilonnd{equation}}
Besides,
$$
\mathopen dulo{H(x)}-\mathopen dulo{H(y)}\leq h_1(1+\norm\varepsilonphi_1)d(x),
$$
where $h_1=\mathbb Sup\limits_{\W}\norm{\nabla H }.$
Recalling also of \varepsilonqref{est_W2} one gets
\[n \psi'W_{\pm}^2( \mathopen dulo{H(x)} -\mathopen dulo{H(y)} )\leq 2nh_1\left(1+\norm{\varepsilonphi}_1\right)^3 d(x)(\psi'(d(x)))^3.\]
Using the assumption P3 it follows
\begin{equation}\label{Termo2}
\begin{array}{c}
n \psi'W_{\pm}^2( \mathopen dulo{H(x)} -\mathopen dulo{H(y)} )\leq 2 n h_1 \left(1+\norm{\varepsilonphi}_1\right)^3 \psi'^2.
\varepsilonnd{array}
\varepsilonnd{equation}
On the other hand,
\begin{equation}\label{Wmenospsi_0}
W_{\pm}-\psi'\leq 1+\norm{\pm \psi'\nabla d +\nabla \varepsilonphi} -\psi' \leq 1+\norm{\varepsilonphi}_1.
\varepsilonnd{equation}
From \varepsilonqref{est_W2} and \varepsilonqref{Wmenospsi_0} we obtain
\begin{equation}\label{Wmenospsi}
n \mathopen dulo{H(x)} \left(W_{\pm}-\psi'\right)W_{\pm}^2\leq 2 n h_0\left(1+\norm{\varepsilonphi}_1\right)^3\psi'^2,
\varepsilonnd{equation}
where $h_0=\mathbb Sup\limits_{\W}\mathopen dulo{H}.$
Using \varepsilonqref{Termo2} and \varepsilonqref{Wmenospsi} in \varepsilonqref{Qw_medio} we get
$$ \pm \mathfrak{Q}_{ } w^{\pm} \leq \left(c+2n\norm{H}_{1}\left(1+\norm{\varepsilonphi}_1\right)^3\right)\psi'^2+\psi''.$$
Recalling the expression for $c$ given in \varepsilonqref{constantec0} and making some algebraic computations we infer that
\begin{align*}
c+2n\norm{H}_{1}\left(1+\norm{\varepsilonphi}_1\right)^3 < C \left(1+\norm{\varepsilonphi}_2 + \norm{H}_{1}\right)\left(1+\norm{\varepsilonphi}_1\right)^3,
\varepsilonnd{align*}
where
\begin{equation}\label{C_kappa}
C= 4n\left(1+\norm{d}_2+1/\tanhu\right).
\varepsilonnd{equation}
Choosing
\begin{equation}\label{nu}
\nu= C \left(1+ \norm{H}_1+ \norm{\varepsilonphi}_2\right)\left(1+\norm{\varepsilonphi}_1\right)^3
\varepsilonnd{equation}
we define $\psi$ by
$$
\psi(t)=\dfrac{1}{\nu}\log(1+kt).
$$
So,
\begin{equation}\label{dpsi}
\psi'(t)=\dfrac{k}{\nu(1+kt)}
\varepsilonnd{equation}
and
\begin{equation}\label{ddpsi}
\psi''(t)=-\dfrac{k^2}{\nu(1+kt)^2},
\varepsilonnd{equation}
hence
$$
\pm \mathfrak{Q} w^{\pm} < \nu\psi'^2+\psi''=0, \ \mbox{ in } \ \W_a.
$$
Besides
$$ t\psi'(t)=\dfrac{kt}{\nu(1+kt)}\leq \dfrac{1}{\nu}<1,$$
which is property P3. From \varepsilonqref{ddpsi} we see that property P2 is also satisfied. Another consequence of \varepsilonqref{ddpsi} is that $\psi'(t)>\psi'(a)$ for all $t\in[0,a]$, thus property P1 is ensured provided that
\begin{equation}\label{paraP1}
\psi'(a)=\dfrac{k}{\nu(1+ka)}=1.
\varepsilonnd{equation}
Furthermore, choosing
\begin{equation}\label{esc_psia_curv_media}
\psi(a) = \dfrac{1}{\nu}\log(1+ka) = \norm{u}_0+\norm{\varepsilonphi}_0,
\varepsilonnd{equation}
we have
$$\pm w^{\pm}(x)=\psi(a)\pm \varepsilonphi(x)=\norm{u}_0+\norm{\varepsilonphi}_0 \pm \varepsilonphi(x) \geq \pm u(x) \ \forall \ x\in \partial\W_a\mathbb Setminus\partial\W.$$
By combining \varepsilonqref{paraP1} and \varepsilonqref{esc_psia_curv_media} we see that
\begin{equation}\label{cte_k}
k=\nu\ds e^{\nu(\norm{u}_0+\norm{\varepsilonphi}_0)}
\varepsilonnd{equation}
and, therefore,
$$
a
=\dfrac{e^{\nu(\norm{u}_0+\norm{\varepsilonphi}_0)}-1}{\nu\ds e^{\nu(\norm{u}_0+\norm{\varepsilonphi}_0)}}.
$$
Note also that $a<\frac{1}{\nu}<\tanhu$ as required.
Finally, if $x\in\partial\W$, then $w^{\pm}(x)=\pm \psi(0)+\varepsilonphi(x)=u(x)$. By the maximum principle we can conclude that $w^-\leq u \leq w^+ $ in $\W_a$, thus
$$ -\psi\circ d \leq u - \varepsilonphi \leq \psi\circ d \mbox{ in } \W_a.$$
Observe that
$$ -\psi\circ d = u - \varepsilonphi = \psi\circ d =0 \mbox{ in } \partial\W,$$
thus, for $y\in\partial\W$ and $0 \leq t \leq a$, we have
$$-\psi(t) + \psi(0) \leq (u-\varepsilonphi) (y+t N_y) - (u-\varepsilonphi)(y) \leq \psi(t)-\psi(0).$$
Dividing by $t>0$ and passing to the limit as $t$ goes to zero we infer that
$$ -\psi'(0) \leq \varepsilonscalar{\nabla u(y)-\nabla\varepsilonphi(y)}{N_y}\leq \psi'(0),$$
so,
$$ \mathopen d{\varepsilonscalar{\nabla u(y)-\nabla \varepsilonphi(y)}{N_y}}\leq \psi'(0).$$
Therefore,
\begin{equation}\label{est_grad_fronteira_1}
\mathopen dulo{\varepsilonscalar{\nabla u(y)}{N_y}}\leq \mathopen dulo{\varepsilonscalar{\nabla \varepsilonphi(y)}{N_y}} + \psi'(0).
\varepsilonnd{equation}
Since $u\varepsilonquiv\varepsilonphi$ in $\partial\W$
$$ \nabla u(y)=(\nabla \varepsilonphi(y))^{T}+\varepsilonscalar{\nabla u(y)}{N_y} N_y.$$
Using \varepsilonqref{est_grad_fronteira_1} we derive
\begin{align*}
\norm{\nabla u(y)}^2=&\norm{(\nabla \varepsilonphi(y))^{T}}^2+{\varepsilonscalar{\nabla u(y)}{N}}^2\\
\leq&\norm{(\nabla \varepsilonphi(y))^{T}}^2+ \left(\mathopen d{\varepsilonscalar{\nabla \varepsilonphi(y)}{N}} + \psi'(0)\right)^2\\
=&\norm{(\nabla \varepsilonphi(y))^{T}}^2+ \varepsilonscalar{\nabla \varepsilonphi(y)}{N}^2 + 2\mathopen d{\varepsilonscalar{\nabla \varepsilonphi(y)}{N}} \psi'(0)+ (\psi'(0))^2\\
\leq&\norm{\nabla \varepsilonphi(y)}^2+ 2\norm{\nabla \varepsilonphi(y)} \psi'(0)+(\psi'(0))^2\\
=&\left(\norm{\nabla \varepsilonphi(y)} + \psi'(0)\right)^2.
\varepsilonnd{align*}
which yields the desired estimate.
\varepsilonnd{proof}
{The combination of Theorems \ref{teo_Est_global_gradiente} and \ref{teo_Est_gradiente_fronteira} with Theorem \ref{T_Exist_quaselineares} implies that the existence program for the Dirichlet problem \varepsilonqref{ProblemaP} is reduced to finding an a priori height estimate. In fact, the following theorem holds.}
\begin{teo}[{\cite[Th. 16.9 p. 407]{GT}}]\label{T_Exist_quaselineares_C0}
Let $\Omegaega\mathbb Subset \R^n$ be a bounded domain with $\partial\W$ of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$, $\varepsilonphi\in \cl^{2,\alpha}(\overline{\Omegaega})$ and {$H\in\cl^{1,\alpha}(\overline{\W})$}. Assume that the family of solutions of the related problems \varepsilonqref{ProblemaPsigma} is uniformly bounded. If
\begin{equation}\label{StrongSerrinCondition_exist_c0}
(n-1)\mathcal H_{\partial\W}(y)\geq n \mathopen dulo{H\left(y\right)} \ \forall \ y\in\partial\W,
\varepsilonnd{equation}
then the Dirichlet problem \varepsilonqref{ProblemaP} has a unique solution in $\cl^{2,\alpha}(\overline{\W})$.
\varepsilonnd{teo}
\begin{proof} Let $M$ an a priori bound for the family of solutions of the related problems \varepsilonqref{ProblemaPsigma}. For $\tanhu\in[0,1]$ fixed, let $u$ a solution of \varepsilonqref{ProblemaPsigma}.
Note now that Theorem \ref{teo_Est_global_gradiente} can be applied provided $u\in\cl^3({\W})$. Taking into account that $H\in\cl^{1,\alpha}(\overline{\W})$, that is a consequence of applying twice the following interior regularity theorem for linear operators.
\noindent\colorbox{shadecolor}{
\begin{minipage}{.98\textwidth}
\begin{teorema}[Interior regularity {\cite[Th. 6.17 p. 109]{GT}}]
Let $\Omegaega\mathbb Subset\mathbb{R}^n$ a domain. Suppose that $u\in\cl^{2}(\W)$ satisfies
$$\mathfrak{L} u=\ds\mathbb Sum_{ij}a_{ij}(x)\partial_ij u+\ds\mathbb Sum_{i}b_i(x)\partial_i u +c(x)u=f(x)$$
where $f$ and the coefficients of the elliptic operator $\mathfrak{L}$ belong to $\cl^{k,\alpha}({\W})$. Then $u\in\cl^{k+2,\alpha}({\W})$.
\varepsilonnd{teorema}
\varepsilonnd{minipage}}
\noindent{In fact, $u$ can be seen as a solution of the linear equation $\mathfrak{L}^u u= \tanhu nH(x)$ where $\mathfrak{L}^u$ is defined in \varepsilonqref{definitionLL}.} Therefore,
\begin{align*}
\mathbb Sup_{\W}\norm{\nabla u(x)}\leq&\left(\mathbb Sqrt{3}+\mathbb Sup\limits_{\partial\W}\norm{\nabla u}\right)\varepsilonxp\left(2\mathbb Sup\limits_{\W}\mathopen dulo{u}\left(1+8n\left(\tanhu\norm{H}_1\right)\right)\right)\\
\leq&\left(\mathbb Sqrt{3}+\mathbb Sup\limits_{\partial\W}\norm{\nabla u}\right)\varepsilonxp\left(2M\left(1+8n\left(\norm{H}_1\right)\right)\right).
\varepsilonnd{align*}
But, on account of assumptions \varepsilonqref{StrongSerrinCondition_exist_c0} one has for any $y\in\partial\W$
$$(n-1)\mathcal H_{\partial\W}(y)\geq n \mathopen dulo{H(y)} \geq \tanhu n \mathopen dulo{H(y)}.$$
Then $u$ satisfies the estimate \varepsilonqref{EstGradFront} stated on Theorem \ref{teo_Est_gradiente_fronteira}. That is, there exists some constant $C=C(n,\W)$ such that
\begin{align*}
\mathbb Sup\limits_{\partial\W}\norm{\nabla u}\leq & \norm{\tanhu\varepsilonphi}_1 + \ds e^{C\left(1+ \norm{H}_1+ \norm{\tanhu\varepsilonphi}_2\right)\left(1+\norm{\tanhu\varepsilonphi}_1\right)^3\left(\norm{u}_0+\norm{\tanhu\varepsilonphi}_0\right)}\\
\leq & \norm{ \varepsilonphi}_1 + \ds e^{C\left(1+ \norm{H}_1+ \norm{ \varepsilonphi}_2\right)\left(1+\norm{ \varepsilonphi}_1\right)^3\left(M+\norm{ \varepsilonphi}_0\right)}.
\varepsilonnd{align*}
Thus, the family of solutions of the related problems \varepsilonqref{ProblemaPsigma} is bounded in $\cl^1(\overline{\W})$ independently of $\tanhu$. Theorem \ref{T_Exist_quaselineares} ensures the existence of a unique solution $u\in\cl^{2,\alpha}(\overline{\W})$ for our problem \varepsilonqref{ProblemaP}.
\varepsilonnd{proof}
The following theorem guarantees an a priori height estimate if the function $H$ satisfies a further hypothesis in addition to the Serrin condition.
\begin{teo}[A priori height estimate {\cite[p. 484]{Serrin}}]\label{teo_Est_altura}
Let $\W\in \R^n$ be a bounded domain with $\partial\W$ of class $\cl^2$. Let $H\in\cl^{1}(\overline{\W})$ satisfying
\begin{equation}\label{cond_H_Ricci_sup}
\norm{\nabla H(x)} \leq \dfrac{n}{n-1}\left(H(x)\right)^2 \ \forall \ x\in\W
\varepsilonnd{equation}
and
\begin{equation}\label{cond_Serrin_hightest_teo}
(n-1)\mathcal H_{\partial\W}(y)\geq n \mathopen dulo{H(y)} \ \forall \ y\in\partial\W.
\varepsilonnd{equation}
If $u\in\cl^2(\W)\cap\cl^0(\overline{\W})$ is a solution of the mean curvature equation \varepsilonqref{operador_minimo_1_coord} in $\W$, then
\begin{equation}\label{Est_Altura}
\mathbb Sup\limits_{\W}\mathopen dulo{u}\leq \mathbb Sup\limits_{\partial\W} \mathopen dulo{u} +\dfrac{e^{\mu\delta}-1}{\mu},
\varepsilonnd{equation}
where $\mu>n\mathbb Sup\limits_{\overline{\W}}\mathopen dulo{H}$ and $\delta=\diam(\W)$.
\varepsilonnd{teo}
\begin{proof}
Let $d(x)=\dist(x,\partial\W)$ for $x\in\W$. Let $\W_0$ be the biggest open subset of $\W$ having the unique nearest point property,
{then $d\in\cl^2(\W_0)$ (see \cite[p. 409, Lemmas 14.16 and 14.17 p. 355]{GT}, \cite[p. 481, \S 3 p. 420]{Serrin}).}
We now define $w=\phi\circ d + \mathbb Sup\limits_{\partial\W}\mathopen dulo{u}$ over $\W$, where
$$\phi(t)=\dfrac{\ds e^{\mu\delta}}{\mu}\left(1-e^{-\mu t}\right).$$
If we prove that $\mathopen d{u}\leq w$ in $\overline{\W}$ we obtain the desired estimate. By the sake of contradiction we suppose first that the function $v=u-w$ attains a maximum $m>0$ at $x_0\in{\W}$ (note that $u\leq w$ in $\partial\W$).
Let $y_0\in\partial\W$ be such that $d(x_0)=\dist(x_0,y_0)=t_0$ and $\gamma$ the straight line segment joining $x_0$ to $y_0$. Restricting $u$ and $w$ to $\gamma$ we see that $v'(t_0)=0$.
Hence,
$u'(t_0)=w'(t_0)=\phi'(t_0)>0$ which implies that $\nabla u(x_0)\neq 0$. Therefore,
$\Gamma_0=\left\{x\in\W;u(x)=u(x_0)\right\}$ is of class $\cl^2$ near $x_0$. Then, there exists a small ball $B_{\varepsilonpsilon}(z_0)$ tangent to $\Gamma_0$ in $x_0$ such that
\begin{equation}\label{eq_bola_1}
u > u(x_0) \mbox{ in } \overline{B_{\varepsilonpsilon}(z_0)}\mathbb Setminus\{x_0\}.
\varepsilonnd{equation}
We note that
$$\dist(z_0,y_0)\leq \dist(z_0,x_0)+\dist(x_0,y_0)=\varepsilonpsilon + d(x_0).$$
Hence, for $\tilde{z}$ lying in the intersection of $\partial B_{\varepsilonpsilon}(z_0)$ with the straight line segment joining $z_0$ to $y_0$, we have
$$d(\tilde{z})\leq \dist(\tilde{z},y_0)=\dist(z_0,y_0)-\varepsilonpsilon \leq d(x_0)+\varepsilonpsilon -\varepsilonpsilon = d(x_0).$$
Thus, $w(\tilde{z})\leq w(x_0)$ since $\phi$ is increasing. Consequently,
$$ u(\tilde{z})-w(x_0) \leq u(\tilde{z})-w(\tilde{z}) \leq u(x_0)-w(x_0)$$
and $u(\tilde{z})\leq u(x_0)$. By \varepsilonqref{eq_bola_1} one has that $\tilde{z}=x_0$, so $z_0$ belongs to $\gamma$ and $\gamma$ is orthogonal to $\Gamma_0$.
This ensures that $x_0\in\W_0$ because if there exists $y_1\neq y_0$ satisfying $d(x_0)=\dist(x_0,y_1)$, then
the straight line joining $y_1$ and $x_0$ is also orthogonal to $\Gamma_0$, which is a contradiction.
However, let us show that this is also impossible. Using the transformation formula \varepsilonqref{calc_Q_w} one has
\begin{equation}\label{Mw_est_altura_0}
\mathcal{M} w = \phi'(1+\phi'^2) \Deltalta d + {\phi''} \ \mbox{ in } \W_0.
\varepsilonnd{equation}
We first estimate $\Deltalta d$ in $\W_0$. For $x\in\W_0$, let $y=y(x)$ in $\partial\W$ be the nearest point to $x$, so $x$ belongs to the segment $\{y + t N_y; t>0\}$, where $N$ is the inner normal to $\partial\W$. Note that $y$ is now fixed. {Let us denote by $\{\Gamma_t\}$ the hypersurface parallel to some portion of $\partial\W$ containing $y$ at distance $t$. So $x$ belongs to $\Gamma_{d(x)}$.
Let
$$h(t)=\frac{n}{n-1}H\left(y + t N_y\right).$$
Therefore
$$ h'(t)=\dfrac{n}{n-1}\varepsilonscalar{\nabla H(y + t N_y)}{N_y}. $$
Taking into account the additional hypothesis \varepsilonqref{cond_H_Ricci_sup} we see that
$$\mathopen dulo{h'(t)}\leq \dfrac{n}{n-1}\norm{\nabla H(y + t N_y)} \leq (h(t))^2 ,$$
hence
$$\mathopen dulo{h'(t)}- (h(t))^2 \leq 0.$$
Recalling again that $\mathcal H'(t)\geq (\mathcal H(t))^2 $ (see \cite[p. 485]{Serrin}) it follows
\begin{equation}\label{desig_sem_ricc}
\mathcal H'(t) \geq \left(\mathcal H(t)\right)^2 + \mathopen dulo{h'(t)} - (h(t))^2 .
\varepsilonnd{equation}
Then,
\begin{equation}\label{desig_sem_ricc_v}
(\mathcal H(t) - h(t))'\geq \left(\mathcal H(t)+h(t)\right)\left(\mathcal H(t)-h(t)\right)
\varepsilonnd{equation}
and
\begin{equation}\label{desig_sem_ricc_g}
(\mathcal H(t) + h(t))'\geq \left(\mathcal H(t)-h(t)\right)\left(\mathcal H(t)+h(t)\right).
\varepsilonnd{equation}
Let us define $v(t)=\mathcal H(t)-h(t)$ and $g(t)=\mathcal H(t)+h(t)$.
From \varepsilonqref{desig_sem_ricc_v} one has
$$v'(t) \geq g(t) v(t)$$
Multiplying this inequality by $\ds e^{\int_0^t g(s)ds}$, it results
$$\left(\dfrac{v(t)}{\ds e^{\int_0^t g(s)ds}}\right)'\geq 0,$$
so
$$\ds\dfrac{v(t)}{\ds e^{\int_0^t g(s)ds}}\geq v(0)=\mathcal H(0)-h(0).$$
From the Serrin condition \varepsilonqref{cond_Serrin_hightest_teo} it follows that
$$\mathopen dulo{h(0)} = \dfrac{n}{n-1} \mathopen dulo{H\left(y\right)}\leq \mathcal H_{\partial\W}(y) =\mathcal H(0) .$$
thus, $v(t)\geq 0$ and $\mathcal H(t)\geq h(t).$
Using \varepsilonqref{desig_sem_ricc_g} we obtain in a similar way that $\mathcal H(t)\geq -h(t).$
Therefore,
$$
\mathcal H(t)\geq\mathopen dulo{h(t)},
$$
that is,
$$ n \mathopen dulo{H(y+tN_y)}\leq (n-1)\mathcal H_{\Gamma_t}(y+tN_y).$$
Consequently,
\begin{equation}\label{forRemark}
n \mathopen dulo{H(x)}\leq (n-1)\mathcal H_{\Gamma_{d(x)}}(x).
\varepsilonnd{equation}
This proves that
$$\Deltalta d(x)\leq-n\mathopen dulo{H\left(x\right)} \ \forall \ x\in\W_0.$$
}
Using this estimate in \varepsilonqref{Mw_est_altura_0} we have in $\W_0$
$$\mathcal{M} w \leq -n\mathopen dulo{H(x)} {\phi'}{(1+\phi'^2)} + {\phi''}.$$
Also
$$
\phi''(t)=-\mu \ds e^{\mu(\delta-t)}=-\mu\phi'(t)<-n\mathopen dulo{H(x)}\phi'(t)
$$
and $\phi'\geq 1$, so
\begin{equation}\label{est_Mw_est_alt}
\mathcal{M} w \leq -n\mathopen dulo{H\left(x\right)} {\phi'(2+\phi'^2)}< -n\mathopen dulo{H\left(x\right)}{\left(1+\phi'^2\right)^{3/2}}.
\varepsilonnd{equation}
From \varepsilonqref{est_Mw_est_alt} we conclude that
\begin{align*}
\mathfrak{Q} (w+m) =\mathfrak{Q} w = & \mathcal{M} w - nH\left(x\right) {\left(1+\phi'^2\right)^{3/2}}\leq 0=\mathfrak{Q} u.
\varepsilonnd{align*}
Moreover, $u \leq w + m$ and $u(x_0)=w(x_0)+m$. By the maximum principle $u\varepsilonquiv w+m$ in $\W_0$ which is a contradiction since $u<w+m$ in $\partial\W$. This proves that $u\leq w$ in $\overline{\W}$.
Applying the same argument to the function $-u$ we also obtain that $-u\leq w$ in $\overline{\W}$.
\varepsilonnd{proof}
\begin{obs}
The proof shows that if there exists a function $H$ satisfying the hypothesis \varepsilonqref{cond_H_Ricci_sup} in addition to the Serrin condition \varepsilonqref{cond_Serrin_hightest_teo}, then any hypersurface that is parallel to some portion of $\partial\W$ ``inherit'' the Serrin condition (see \varepsilonqref{forRemark}). This geometric implication is the key to obtain the height estimate for solutions of the mean curvature equation in terms of its boundary values.
\varepsilonnd{obs}
\begin{obs}
The proof shows that condition \varepsilonqref{cond_H_Ricci_sup} only needs to be valid in $\W_0$.
\varepsilonnd{obs}
We are able to prove the following theorem from Serrin.
\begin{teo}[Serrin {\cite[p. 484]{Serrin}}]\label{T_exist_Ricci}
Let $\Omegaega \mathbb Subset \R^n$ be a bounded domain with $\partial\W$ of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$.
Let $H\in\cl^{1,\alpha}(\overline{\W})$ satisfying
\begin{equation}\label{cond_H_Ricci_exist}
\norm{\nabla H(x)} \leq \dfrac{n}{n-1}\left(H(x)\right)^2 \ \forall \ x\in\W.
\varepsilonnd{equation}
and
\begin{equation}\label{StrongSerrinCondition_exist}
(n-1)\mathcal H_{\partial\W}(y)\geq n \mathopen dulo{H\left(y\right)} \ \forall \ y\in\partial\W.
\varepsilonnd{equation}
Then for every $\varepsilonphi\in\cl^{2,\alpha}(\overline{\W})$ there exists a unique solution $u\in\cl^{2,\alpha}(\overline{\W})$ of the Dirichlet problem \varepsilonqref{ProblemaP}.
\varepsilonnd{teo}
\begin{proof}
By Theorem \ref{T_Exist_quaselineares_C0} it only remains to prove that the family of solutions of the related problems \varepsilonqref{ProblemaPsigma} is uniformly bounded. Let $u$ be a solution of problem \varepsilonqref{ProblemaPsigma} for arbitrary $\tanhu\in[0,1]$ and let $w=\phi\circ d + \mathbb Sup\limits_{\partial\W}\mathopen dulo{\varepsilonphi}$ as in the proof of Theorem \ref{teo_Est_altura}. Analogously as in the proof of that theorem, if the function $u-w$ attains a positive maximum $m$ at $x_0\in \overline{\W}$, then $x_0$ would be in $\W_0$ (the biggest open subset of $\W$ having the unique nearest point property).
But for $x\in\W_0$ we have
\begin{align*}
\mathfrak{Q}_{\tanhu}(w + m )= \mathfrak{Q}_{\tanhu}(w)
\leq \mathcal{M} w + \tanhu n\mathopen dulo{H(x)}(1+\phi'^2)^{3/2}\leq 0
\varepsilonnd{align*}
once \varepsilonqref{est_Mw_est_alt} holds and $\tanhu\in[0,1]$. Proceeding as in the proof of Theorem \ref{teo_Est_altura}, we get that $u\leq w$ also in $\W_0$. Analogously it follows that $-u \leq w$ in $\overline{\W}$. Hence, $u$ satisfies the estimate \varepsilonqref{Est_Altura}, that is,
$$\mathbb Sup\limits_{\W}\mathopen dulo{u}\leq \mathbb Sup\limits_{\partial\W} \mathopen dulo{\tanhu\varepsilonphi} +\dfrac{e^{\mu\delta}-1}{\mu}
\leq \mathbb Sup\limits_{\partial\W} \mathopen dulo{\varepsilonphi} +\dfrac{e^{\mu\delta}-1}{\mu},$$
where $\mu>n\mathbb Sup\limits_{\overline{\W}}\mathopen dulo{H}$ and $\delta=\diam(\W)$.
\varepsilonnd{proof}
\begin{obs}
Observe that Theorem \ref{T_exist_Ricci} is not exactly the existence part in Theorem \ref{T_Serrin_Ricci} stated in the introduction. In order to reduce the differentiability assumptions on the domain and the boundary data, the original problem can be approximated by new problems having the $\cl^{2,\alpha}$ differentiability requirements. For instance, $\partial\W$ can be approximated by $\cl^{2,\alpha}$ hypersurfaces. However, it can only be guaranteed that the mean curvature of any approximating surface $\Sigma$ satisfies $(n-1)\mathcal H_{\Sigma}(x)\geq n\mathopen d{H(x)}-\varepsilonepsilon$ for every $x\in\Sigma$, where $\varepsilonepsilon$ is a positive constante. Hence, a boundary gradient estimate sharper than that obtained in Theorem \ref{teo_Est_gradiente_fronteira} is needed. Furthermore, it is also required an interior gradient estimate which yields a compactness result also used in this argument. We refer the work of Serrin \cite[\S 14 p. 451]{Serrin} for further studies.
\varepsilonnd{obs}
\begin{obs}
Although the additional hypothesis \varepsilonqref{cond_H_Ricci_exist} is required in order to obtain an a priori height estimate, the fundamental role in Theorem \ref{T_exist_Ricci} is played by the Serrin condition \varepsilonqref{StrongSerrinCondition_exist}.
If fact, if the function $H$ satisfies the integral condition
$$ \norm{H}_{L^n(\W)}<n\left(\ds\int_{\R^n}\left(1+\norm{p}^2\right)^{-\frac{n+2}{2}}dp\right)^{\frac{1}{p}}$$
instead of \varepsilonqref{cond_H_Ricci_exist}, the height estimate is guaranteed and the same conclusion of Theorem \ref{T_exist_Ricci} holds as a consequence of Theorem \ref{T_Exist_quaselineares_C0} (see \cite[Ths. 3.2.1 p. 87 and 3.4.1 p. 105]{han2016nonlinear} and \cite[Th. 16.10 p. 408]{GT}).
\varepsilonnd{obs}
\mathbb Section{Sharpness of the Serrin condition}\label{cap_NaoExis}
The goal in this section is to prove that the Serrin condition,
\begin{equation}\label{SerrinCondition_naoexist}
(n-1)\mathcal H(y)\geq n\mathopen d{H(y)} \ \forall \ y\in\partial\W,
\varepsilonnd{equation}
is actually sharp for the solvability of the Dirichlet problem \varepsilonqref{ProblemaP}.
That is, if \varepsilonqref{SerrinCondition_naoexist} fails, then there exists boundary values for which problem \varepsilonqref{ProblemaP} has no possible solution.
The next lemma is an important peace. In this lemma is established a height a priori estimate for solutions of equation
\varepsilonqref{operador_minimo_1_coord} in $\W$ in those points of $\partial\W$ on which the Serrin condition \varepsilonqref{SerrinCondition_naoexist} fails.
\begin{lema}[{\cite[L. 3.4.4 p. 109]{han2016nonlinear}}]\label{M_nao_exist_MxR_estimativa_Hxz}
Let $\W\mathbb Subset M$ be a bounded domain whose boundary is of class $\cl^2$. Let $H\in\cl^0(\overline{\W})$ be a non-negative function and $u\in\cl^2(\W)\cap\cl^0(\overline{\W})$ satisfying \varepsilonqref{operador_minimo_1_coord}.
Assume that there exists $y_0\in\partial\W$ such that
\begin{equation}\label{cond_Serrin_negac_M_pos}
(n-1)\mathcal H_{\partial\W}(y_0)<nH(y_0).
\varepsilonnd{equation}
Then for each $\varepsilonepsilon>0$ there exists $a>0$ depending only on $\varepsilonepsilon$, $\mathcal H_{\partial\W}(y_0)$, the geometry of $\W$ and the modulus of continuity of $H$ in $y_0$, such that
\begin{equation}\label{est_nao_exist_Hxz}
u(y_0) < \ds\mathbb Sup_{\partial\W\mathbb Setminus B_a(y_0)}~u+\varepsilonepsilon.
\varepsilonnd{equation}
\varepsilonnd{lema}
\begin{proof}
The proof is done in two steps. Firstly, it will be find an estimate for $u(y_0)$
depending on $\mathbb Sup\limits_{\partial B_a(y_0)\cap \W}u$ for some $a$ that does not depend on $u$. Secondly, an upper bound for $\ds \mathbb Sup_{\partial B_a(y_0)\cap \W}u$ in terms of $\mathbb Sup\limits_{\partial\W\mathbb Setminus B_a(y_0)} u $ is stated.
\noindent\textbf{Step 1.}
First of all note that from \varepsilonqref{cond_Serrin_negac_M_pos} there exists $\nu>0$ such that
\begin{equation}\label{M_condcomigual}
(n-1)\mathcal H_{\partial\W}(y_0) < n H(y_0)-4\nu.
\varepsilonnd{equation}
Let $R_1>0$ be such that $\partial B_{R_1}(y_0)\cap\W$ is connected and
\begin{equation}\label{M_cond_H}
\mathopen dulo{H(x)-H(y_0)}<\dfrac{\nu}{n}, \ \forall \ x\in B_{R_1}(y_0)\cap\W.
\varepsilonnd{equation}
Let $S$ be a quadric hypersurface inside $\W$,
tangent to $\partial\W$ at $y_0$ and whose mean curvature calculated with respect to the normal field $N$ which coincides with the inner normal to $\partial\W$ at $y_0$
satisfies
\begin{equation}\label{M_curv_S_Gamma_ponto}
\mathcal H_{S}(y_0)<\mathcal H_{\partial\W}(y_0)+\dfrac{\nu}{(n-1)}.
\varepsilonnd{equation}
{
Let $d(x)=\dist(x,S)$ for $x\in\W$. It is known that $d$ is of class $\cl^2$ over the strip
$$\Sigma_{\tanhu}=\{x+tN_x; x \in S , \ t\in[0,\tanhu) \}$$
for some $\tanhu>0$ (see \cite[L. 3.1.8 p. 84]{han2016nonlinear} and \cite[L. 1 p. 420]{Serrin}).
Besides, for each $t\in[0,\tanhu)$ fixed,
$$S_{t}=\{x+tN_x; x \in S \}$$
is parallel to $S$.
Let $0<R_2<\min\{\tanhu,R_1\}$ be such that
\begin{equation}\label{est_Laplaciano_d}
\mathopen dulo{\Deltalta d(x)-\Deltalta d(y_0)}<\nu \ \ \forall \ x\in B_{R_2}(y_0)\cap \Sigma_{\tanhu}.
\varepsilonnd{equation}
}
Let us fix $a<R_2$ to be made precise later. For $0<\varepsilonpsilon<a$ let
$$\W_{\varepsilonpsilon}=\{x \in B_a(y_0)\cap\Sigma_{\tanhu}; d(x)>\varepsilonpsilon\}.$$
Let $\phi\in\cl^2(\varepsilonpsilon, a)$ satisfying
{\mathbb Small
\begin{multicols}{4}
\begin{enumerate}
\item[P1.] $\phi(a)=0$,
\item[P2.] $\phi'\leq0$,
\item[P3.] $\phi''\geq0$,
\item[P4.] $\phi'(\varepsilonpsilon)=-\infty$.
\varepsilonnd{enumerate}
\varepsilonnd{multicols}}
\noindent It is also required that
\begin{equation}\label{assumption_phi}
\nu (\phi'(t))^3 + \phi''(t)=0, \ t\in (\varepsilonpsilon,a).
\varepsilonnd{equation}
Let us define $v = \ds\mathbb Sup_{\partial B_a(y_0)\cap\W} u + \phi\circ d$. So, $v\geq u$ in $\partial \W_{\varepsilonpsilon}\mathbb Setminus S_{\varepsilonpsilon}$. If $u\leq v$ in $S_{\varepsilonpsilon}$, then an estimate for $u(y_0+\varepsilonpsilon N_{y_0})$ is obtained. Observe now that if $N_{\varepsilonpsilon}$ is the normal to $S_{\varepsilonpsilon}$ inwards $\W_{\varepsilonpsilon}$ and $x\in S_{\varepsilonpsilon}\cap B_a(y_0)$, then
\begin{align*}
\parcial{v}{N_{\varepsilonpsilon}}(x)&=\varepsilonscalar{\nabla v(x)}{N_{\varepsilonpsilon}(x)}=\varepsilonscalar{\phi'(d(x))\nabla d(x)}{\nabla d(x)}=\phi'(\varepsilonpsilon)=-\infty.
\varepsilonnd{align*}
So, Proposition \ref{M_prop_gen_JS} can be used if $\mathfrak{Q} u \geq \mathfrak{Q} v$ in $\W_\varepsilonpsilon$. This will be proved in the sequel.
For $x\in \W_{\varepsilonpsilon}$ the transformation formula \varepsilonqref{calc_Q_w} yields
$$\mathfrak{Q} v = \phi'(1+\phi'^2) \Deltalta d + \phi''-n H(x) (1+\phi'^2)^{3/2}.$$
The assumptions on $\phi$ immediately gives
$$(1+\phi'^2)^{3/2}
=(1+\phi'^2)^{1/2}(1+\phi'^2)
>(\phi'^2)^{1/2}(1+\phi'^2)=\mathopen dulo{\phi'}(1+\phi'^2)=-\phi'(1+\phi'^2).$$
Since $H\geq 0$, then
$$-nH(x)(1+\phi'^2)^{3/2} < nH(x){\phi'}{(1+\phi'^2)} .$$
Therefore,
\begin{equation}\label{M_exp_Qv_3}
\mathfrak{Q} v < {\phi'}{(1+\phi'^2)} \left(\Deltalta d(x) + n H(x) \right) + {\phi''}.
\varepsilonnd{equation}
Furthermore,
\begin{align*}
\Deltalta d(x) + n H(x) = & \Deltalta d(x) - \Deltalta d(y_0) -(n-1)\mathcal H_S(y_0) + n H(x)\\
> & -\nu -(n-1)\mathcal H_S(y_0) + n H(x)\tanhg{a}\\
> & -2\nu -(n-1)\mathcal H_{\partial\W}(y_0) + n H(x)\tanhg{b}\\
> & 2\nu - n H(y_0) + n H(x) \tanhg{c}\\
> & \nu \tanhg{d},
\varepsilonnd{align*}
where (a) follows directly from \varepsilonqref{est_Laplaciano_d}, (b) from \varepsilonqref{M_curv_S_Gamma_ponto}, (c) from \varepsilonqref{M_condcomigual} and (d) from \varepsilonqref{M_cond_H}. Using this estimate in \varepsilonqref{M_exp_Qv_3} it follows
\begin{align*}
\mathfrak{Q} v < \phi'(1+\phi'^2) \nu + \phi'' < \phi'^3 \nu + \phi''.
\varepsilonnd{align*}
Assumption \varepsilonqref{assumption_phi} yields $\mathfrak{Q} v <0$ in $\W_{\varepsilonpsilon}$. From Proposition \ref{M_prop_gen_JS} it is deduced that
$$ u \leq v = \ds\mathbb Sup_{\partial B_a(y_0)\cap\W} u + \phi(\varepsilonpsilon) \ \ \mbox{in}\ \ S_{\varepsilonpsilon}\cap B_a(y_0). $$
Let us now define $\phi$ explicitly by (see also \cite[\S 14.4]{GT})
\begin{equation}\label{M_exp_phi}
\phi(t)=\mathbb Sqrt{\dfrac{2}{\nu}}\left((a-\varepsilonpsilon)^{1/2}-(t-\varepsilonpsilon)^{1/2}\right).
\varepsilonnd{equation}
Observe that $\phi$ satisfies P1--P4 and that $\phi'^3 \nu + \phi''=0$ in $(\varepsilonpsilon,a)$.
{
Indeed,
$$ \phi'(t)=-\dfrac{1}{2}\mathbb Sqrt{\dfrac{2}{\nu}}(t-\varepsilonpsilon)^{-1/2}=-\dfrac{1}{2}\left(\dfrac{2}{\nu(t-\varepsilonpsilon)}\right)^{1/2}$$
and
$$ \phi''(t)=\dfrac{1}{4}\mathbb Sqrt{\dfrac{2}{\nu}}(t-\varepsilonpsilon)^{-3/2}=\dfrac{\nu}{8}\left(\dfrac{2}{\nu(t-\varepsilonpsilon)}\right)^{3/2}=-\nu \phi'(t)^3.$$
}
Therefore,
$$ u(y_0 + \varepsilonpsilon N_{y_0}) \leq \ds\mathbb Sup_{\partial B_a(y_0)\cap\W} u + \mathbb Sqrt{\dfrac{2}{\nu}}\left((a-\varepsilonpsilon)^{1/2}\right).$$
Since this estimate holds for each $0<\varepsilonpsilon<a$, we can pass to the limit as $\varepsilonpsilon$ goes to zero, so
\begin{equation}\label{est_u0_1}
u(y_0) \leq \ds\mathbb Sup_{\partial B_a(y_0)\cap\W} u + \mathbb Sqrt{\dfrac{2a}{\nu}}.
\varepsilonnd{equation}
\noindent\textbf{Step 2.}
Let $\delta=\diam(\W)$ and $\psi\in\cl^2(a,\delta)$ satisfying
{\mathbb Small
\begin{multicols}{4}
\begin{enumerate}
\item[P5.] $\psi(\delta)=0$,
\item[P6.] $\psi'\leq0$,
\item[P7.] $\psi''\geq0$,
\item[P8.] $\psi'(a)=-\infty$.
\varepsilonnd{enumerate}
\varepsilonnd{multicols}}
\noindent It is also needed that
\begin{equation}\label{assumptionpsi}
(n-1)\frac{(\psi'(t))^3}{t}+\psi''(t)\leq 0, \ t\in(a,\delta).
\varepsilonnd{equation}
Let $w=\ds\mathbb Sup_{\partial\W\mathbb Setminus B_a(y_0)} u + \psi\circ\rho$, where $\rho(x)=\dist(x,y_0)$. Remind that $\rho\in\cl^2(\R^n\mathbb Setminus\{y_0\})$, so $w\in\cl^2(\W\mathbb Setminus B_a(y_0))$.
The idea is to use Proposition \ref{M_prop_gen_JS} again.
Note that $w\geq u$ in $\partial\W\mathbb Setminus B_a(y_0)$.
Also, if $N_a$ is the normal to $\partial B_a(y_0)\cap\W$ inwards $\W\mathbb Setminus B_a(y_0)$ and $x\in\partial B_a(y_0)\cap\W$, then
\begin{align*}
\parcial{w}{N_a}(x)&=\varepsilonscalar{\nabla w(x)}{N_a(x)}=\varepsilonscalar{\psi'(\rho(x))\nabla \rho(x)}{\nabla \rho(x)}=\psi'(a)=-\infty.
\varepsilonnd{align*}
On the other hand, the transformation formula \varepsilonqref{calc_Q_w} gives
$$ \mathfrak{Q} w = {\psi'}{(1+\psi'^2)} \Deltalta \rho + {\psi''}-n H(x) {(1+\psi'^2)^{3/2}}.$$
Since $H\geq 0$ and $\Deltalta \rho(x) = \dfrac{n-1}{\rho(x)}$
it follows
\begin{align*}
\mathfrak{Q} w \leq \dfrac{n-1}{\rho} \psi'(1+\psi'^2) + \psi''<\dfrac{n-1}{\rho} \psi'^3 + \psi''.
\varepsilonnd{align*}
Assumption \varepsilonqref{assumptionpsi} yields $\mathfrak{Q} w <0$ in $\W\mathbb Setminus B_a(y_0)$.
From Proposition \ref{M_prop_gen_JS} we conclude that $u \leq w$ in $\partial B_a(y_0)\cap\W$, so\footnote{Observe that no other assumption that the connectedness of $\partial B_a(y_0) \cap \W$ was required in step 2.}
\begin{equation}\label{M_desigualdade_bordo}
\ds\mathbb Sup_{\partial B_a(y_0)\cap \W} u \leq \ds\mathbb Sup_{\partial\W\mathbb Setminus B_a(y_0)} u + \psi(a).
\varepsilonnd{equation}
Using \varepsilonqref{M_desigualdade_bordo} in \varepsilonqref{est_u0_1} from step 1 one gets
$$u(y_0) \leq \ds\mathbb Sup_{\partial\W\mathbb Setminus B_a(y_0)} u + \psi(a) +\mathbb Sqrt{\dfrac{2a}{\nu}}.$$
Let us define $\psi$ by (see also \cite[\S 14.4]{GT})
\begin{equation}\label{M_expres_psi}
\psi(t)=\left(\dfrac{2}{n-1}\right)^{1/2}\ds\int_t^{\delta} \left(\log \frac{r}{a}\right)^{-1/2}dr.
\varepsilonnd{equation}
Such a function satisfies P5--P8, and also $\dfrac{n-1}{t} \psi'(t)^3 + \psi''(t)<0$ for each $t\in(a,\delta)$.
{
In fact,
$$\psi'(t)=-\left(\dfrac{2}{n-1}\right)^{1/2}\left(\log \frac{t}{a}\right)^{-1/2}$$
and
\begin{align*}
\psi''(t)=&-\left(\dfrac{2}{n-1}\right)^{1/2}\left(-\dfrac{1}{2}\left(\log \frac{t}{a}\right)^{-3/2}\dfrac{a}{t}\dfrac{1}{a}\right)\\
=&\dfrac{1}{2t}\left(\dfrac{2}{n-1}\right)^{1/2}\left(\log \frac{t}{a}\right)^{-3/2}\\
=&\dfrac{n-1}{4t}\left(\dfrac{2}{n-1}\right)^{3/2}\left(\log \frac{t}{a}\right)^{-3/2}\\
=&-\dfrac{n-1}{4t}\psi'(t)^3\\
<&-\dfrac{n-1}{t}\psi'(t)^3.
\varepsilonnd{align*}
}
Additionally, it is easy to see that $\ds\lim_{a\rightarrow 0}\psi(a)=0$.
Hence, for each $\varepsilonepsilon>0$, $a$ can be chosen small enough to satisfy
\[
\psi(a)+\mathbb Sqrt{\dfrac{2a}{\nu}}<\varepsilonepsilon.\qedhere
\]
\varepsilonnd{proof}
Now we are able to prove the following non-existence result.
\begin{teo}[{\cite[Th. 3.4.5 p. 112]{han2016nonlinear}}]\label{M_nao_exist_MxR_Hxz}
Let $\W\mathbb Subset M$ be a bounded domain whose boundary is of class $\cl^2$. Let $H\in\cl^0(\overline{\W})$ be a function either non-positive or non-negative. Assume that there exists $y_0\in\partial\W$ such that
$$(n-1) \mathcal H_{\partial\W}(y_0) < n \mathopen dulo{H(y_0)}.$$
Then, for any $\varepsilonepsilon>0$, there exists $\varepsilonphi\in\cl^{\infty}(\overline{\W})$ with $\mathopen dulo{\varepsilonphi}<\varepsilonepsilon$ on $\partial\W$, such that there exists no $u \in \cl^2 (\Omegaega)\cap \cl^0(\overline{\Omegaega})$ satisfying problem \varepsilonqref{ProblemaP}.
\varepsilonnd{teo}
\begin{proof}
Obviously it can be supposed that $H\geq 0$.
For any $\varepsilonepsilon>0$ take $a$ as in the previous lemma. Let $\varepsilonphi\in\cl^{\infty}(\overline{\W})$ such that $\varepsilonphi=0$ in $\partial\W\mathbb Setminus B_a(y_0)$, $0\leq \varepsilonphi \leq \varepsilonepsilon$ on $\partial\W\cap B_a(y_0)$ and $\varepsilonphi(y_0)=\varepsilonepsilon$. Hence, no solution of equation \varepsilonqref{operador_minimo_1_coord} in $\W$ could have $\varepsilonphi$ as boundary values because such a function does not satisfy \varepsilonqref{est_nao_exist_Hxz}.
\varepsilonnd{proof}
\mathbb Section{Prescribed mean curvature equations in Riemannian manifolds}
Dirichlet problems for equations whose solutions describe hypersurfaces of prescribed mean curvature have been also studied outside of the Euclidean space. However, Serrin type solvability criteria have been obtained only in {few} cases.
For instance, P.-A Nitsche \cite{Nitsche2002} was concerned with graph-like prescribed mean curvature hypersurfaces in hyperbolic space $\HH^{n+1}$. In the half-space setting, he studied radial graphs over the totally geodesic hypersurface $S = \{x \in \R^{n+1}_+; (x_0)^2 + \downarrowts + (x_n)^2 = 1\}$. He established an existence result if $\W$ is a bounded domain of $S$ of class $\cl^{2,\alpha}$ and $H\in\cl^1(\overline{\W})$ is a function satisfying $\mathbb Sup\limits_{\overline{\W}}\mathopen dulo{H}\leq 1$ and $\mathopen dulo{H(y)}<\mathcal H_{C}(y)$ everywhere on $\partial\W$, where $\mathcal H_{C}$ denotes the hyperbolic mean curvature of the cylinder $C$ over $\partial\W$. Furthermore, he showed the existence of smooth boundary data such that no solution exists in case of $\mathopen dulo{H(y)}>\mathcal H_{C}(y)$ for some $y\in\partial\W$ under the assumption that $H$ has a sign. We observe that these results do not provide Serrin type solvability criterion.
Also in the half space model of the hyperbolic space, E. M. Guio-R. Sa Earp \cite{elias,eliasarticle} considered a bounded domain $\W$ contained in a vertical totally geodesic hyperplane $P$ of $\HH^{n+1}$ and studied the Dirichlet problem for the mean curvature equation for horizontal graphs over $\W$, that is, hypersurfaces which intersect at most only once the horizontal horocycles orthogonal to $\W$. They considered the hyperbolic cylinder $C$ generated by horocycles cutting ortogonally $P$ along the boundary of $\W$ and the Serrin condition, $\mathcal H_C(y) \geq \mathopen dulo{H(y)}$ $\forall\ y\in\partial\W$. They obtained a Serrin type solvability criterion for prescribed mean curvature $H=H(x)$ and also proved a sharp solvability criterion for constant $H$.
There are also some results of this type in the Riemannian product $M\times\R$, where $M$ is a complete Riemannian manifold of dimension $n\geq 2$. Analogously to the Euclidean setting, the solutions of the equation
\begin{equation}\label{operador_minimo_1_manifolds}
\text{div} \,er_M\left(\dfrac{\nabla_M u}{\mathbb Sqrt{1+\norm{\nabla_M u}_M^2}}\right) = nH
\varepsilonnd{equation}
are vertical graphs in $M\times\R$ with mean curvature $H$ at each point of the graph.
However, even though the study of the Dirichlet problem for equation \varepsilonqref{operador_minimo_1_manifolds} inherits the techniques from the Euclidean setting, it is more difficult.
For instance, in a coordinate system $(x_1,\downarrowts,x_n)$ in $M$, the non-divergence form of equation \varepsilonqref{operador_minimo_1_manifolds} is equivalent to
\begin{equation}\label{operador_minimo_1_manifolds_coord}
\mathcal{M} u:=\mathbb Sum_{i,j=1}^n \left(W^2\mathbb Sigma^{ij} - {u^iu^j} \right)\partial_{ij} u=nH{W^3},
\varepsilonnd{equation}
where $(\mathbb Sigma^{ij})$ is the inverse of the metric $(\mathbb Sigma_{ij})$ of $M$, $u^i=\ds\mathbb Sum_{j=1}^n\mathbb Sigma^{ij} \partial_j u$ are the coordinates of $\nabla u$ and $\partial_{ij} u(x)=\Hess u(x){\left(e_i,e_j\right)}$.
In this context Aiolfi-Ripoll-Soret {\cite[Th. 1 p. 72]{Aiolfi}} proved that there always exists a vertical minimal graph ($H=0$) in $M\times\R$ over a mean convex, smooth and bounded domain $\W$ in $M$ for arbitrary continuous boundary data. This result generalizes the existence part in Theorem \ref{SharpJenkinsSerrin} stated in the introduction.
In the case where $M$ is a Hadamard manifold whose sectional curvature is bounded above by $-1$, then the mean convexity condition is sharp due to a work of M. Telichevesky {\cite[Th. 6 p. 246]{miriam}}.
The combination of these two results gives a sharp solvability criterion for the minimal hypersurface equation in bounded domains of these types of Hadamard manifolds.
{The author of these notes have generalized the aforementioned non-existence result in the $M\times\R$ context on her PhD thesis \cite{minhatese} supervised by professor R. Sa Earp}. More precisely, it was proved that if $H$ is a continuous function and non-decreasing in the variable $z$, then the \textit{strong Serrin condition}
\begin{equation}\label{SerrinConditionGeneral}
(n-1)\mathcal H_{\partial\W}(y)\geq n\ds\mathbb Sup_{z\in\R}\mathopen dulo{H(y,z)} \ \forall \ y\in\partial\W
\varepsilonnd{equation}
is necessary for the solvability of the Dirichlet problem
\begin{equation}\tanhg{$P_{M\times\R}$}\label{ProblemaP_manifolds}
\left\{
\begin{split}
\mathcal{M} u &=n H(x,u)W^3 \ \mbox{in}\ \W,\\
u&=\varepsilonphi \ \mbox{in}\ \partial\W.
\varepsilonnd{split}\right.
\varepsilonnd{equation}
in every Hadamard manifold {\cite[Th. 2.5 p. 26]{minhatese} (see also \cite[Cor. 2 p. 3]{AlvarezNonExistence})} and in every compact and simply connected manifold which is strictly $1/4-$pinched\footnote{A Riemannian manifold is said to be strictly $1/4-$pinched if the sectional curvature $K$ of $M$ satisfies $\frac{1}{4} K_0 < K \leq K_0$ for a positive constant $K_0$.} {\cite[Th. 2.6 p. 27]{minhatese} (see also \cite[Cor. 3 p. 4]{AlvarezNonExistence})}.
Some direct consequences derived from these non-existence results are the following. Firstly, the combination of the non-existence result for Hadamard manifolds \cite[Th. 2.5 p. 26]{minhatese}
with the existence theorem from Aiolfi-Ripoll-Soret {\cite[Th. 1 p. 72]{Aiolfi}} for the minimal case shows that the sharp solvability criterion of Jenkins-Serrin (see Theorem \ref{SharpJenkinsSerrin} stated in the introduction) actually holds in every Cartan-Hadamard manifolds:
\begin{teo}[Sharp Jenkins-Serrin-type solvability criterion]\label{cond_nece_suf_minimo_Hadamard}
Let $M$ be a Cartan-Hadamard manifold and $\W\mathbb Subset M$ a bounded domain whose boundary is of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$. Then the Dirichlet problem for the minimal equation in $\W$ has a unique solution for arbitrary continuous boundary data if, and only if, $\W$ is {mean convex}.
\varepsilonnd{teo}
Secondly, combining the non-existence result for positively curved manifolds \cite[Th. 2.6 p. 27]{minhatese}
with an existence result of Spruck {\cite[Th. 1.4 p. 787]{spruck}} we infer the following:
\begin{teo}[Sharp Serrin-type solvability criterion]\label{cond_nece_suf_minimo_curv_positivo_este_este}
Let $M$ be a compact and simply connected manifold which is strictly $1/4-$pinched. Let $\W\mathbb Subset M$ be a domain with $\diam(\W)<\frac{\pi}{2\mathbb Sqrt{K_0}}$ and whose boundary is of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$. Then for every constant $H$ the Dirichlet problem \varepsilonqref{ProblemaP_manifolds} has a unique solution for arbitrary continuous boundary data if, and only if, $(n-1)\mathcal H_{\partial\W}\geq n\mathopen dulo{H}$.
\varepsilonnd{teo}
Notice that it was not derived directly
a sharp Serrin type result (see Theorem \ref{SharpSerrin} stated in the introduction) for arbitrary constant $H$ in every Cartan-Hadamard manifold. This is due to the fact that in Hadamard manifolds we have some geometric restrictions. For example, not in every mean convex domain of a Hadamard manifolds the mean curvature of the parallel hypersurfaces is increasing along the geodesics normals to $\partial\W$.
By way of illustrating better this fact let $M=\HH^n$. It follows from the existence result of Spruck \cite[Th. 1.4 p. 787]{spruck} that the Serrin condition is a sufficient condition if $H\geq \frac{n-1}{n}$. In the opposite case $0<H<\frac{n-1}{n}$, Spruck noted the existence of an entire graph of constant mean curvature $\frac{n-1}{n}$ in $\HH^n\times\R$ (see \cite{BerardRicardo} for explicit formulas) whose vertical translations and reflexions are barrier for the solutions of the problem. Having this height estimate it was possible to establish an a priori boundary gradient estimate if the strict inequality $(n-1)\mathcal H_{\partial\W} > n {H}$ holds {since in this case there exists a tubular neighborhood of $\partial\W$ on which $(n-1)\mathcal H_{\Gamma_t} > n {H}$ for every hypersurface parallel to $\partial\W$ contained on it.} This restriction over the Serrin condition in the last case did not allows us to establish a Serrin type solvability criterion for every constant $H$ directly from the combination of the existence result of Spruck {\cite[Th. 5.4 p. 797]{spruck}} with our non-existence result for Hadamard manifods {\cite[Th. 2.5 p. 26]{minhatese}} when the ambient is the hyperbolic space.
We have also established an existence result {\cite[Th. 4.4 p. 51]{minhatese} (see also \cite[Th. 5 p. 4]{AlvarezExistence})} for prescribed $H\in\cl^{1,\alpha}(\overline{\W}\times\R)$ which extends the existence result of Spruck and that yields the following Serrin type solvability criterion when combined with the non-existence theorem for Hadamard manifold {\cite[Th. 2.5 p. 26]{minhatese}} mentioned before:
\begin{teo}[Serrin type solvability criterion in $\HH^n\times\R$]\label{cond_nece_suf_HnxR}
Let $\W\mathbb Subset \HH^n$ be a bounded domain with $\partial\W$ of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$. Let $H\in\cl^{1,\alpha}(\overline{\W}\times\R)$ be a function satisfying $\partial_z H\geq 0$ and $0 \leq {H} \leq \frac{n-1}{n}$ in ${\W\times\R}$. Then the Dirichlet problem \varepsilonqref{ProblemaP_manifolds} has a unique solution $u\in\cl^{2,\alpha}(\overline{\W})$ for every $\varepsilonphi\in\cl^{2,\alpha}(\overline\W)$ if, and only if, the strong Serrin condition \varepsilonqref{SerrinConditionGeneral} holds.
\varepsilonnd{teo}
Combining Theorem \ref{cond_nece_suf_HnxR} with the non-existence result for Hadamard manifolds {\cite[Th. 2.5 p. 26]{minhatese}} and the existence result of Spruck {\cite[Th. 1.4 p. 787]{spruck}} we deduce that the sharp solvability criterion of Serrin {(Theorem \ref{SharpSerrin})} also holds in the $\cl^{2,\alpha}$ class if we replace $\R^n$ by $\HH^n$:
\begin{teo}[Sharp Serrin type solvability criterion in $\HH^n\times\R$]\label{cond_nece_suf_HnxR_constate}
Let $\W\mathbb Subset \HH^n$ be a bounded domain whose boundary is of class {$\cl^{2,\alpha}$}. Then for every constant $H$ the Dirichlet problem \varepsilonqref{ProblemaP_manifolds} has a unique solution for arbitrary continuous boundary data if, and only if, $(n-1) \mathcal H_{\partial\W}(y) \geq n \mathopen dulo{H}$.
\varepsilonnd{teo}
We have also proved \cite[Th. 4.1 p. 40]{minhatese} (see also \cite[Th. 4 p. 4]{AlvarezExistence}) a generalization of the Spruck's existence result {\cite[Th. 1.4 p. 787]{spruck}} for constant mean curvature.
Putting together this result with the non-existence theorem for Hadamard manifolds {\cite[Th. 2.5 p 26]{minhatese}} we derive the following generalization in the $\cl^{2,\alpha}$ class of Theorem \ref{T_Serrin_Ricci} of Serrin stated in the introduction:
\begin{teo}[Serrin type solvability criterion 2]\label{cond_nece_suf_hadamard}
Let $M$ be a Cartan-Hadamard manifold and $\W\mathbb Subset M$ a bounded domain whose boundary is of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$. Suppose that $H\in\cl^{1,\alpha}(\overline{\W}\times\R)$ is either non-negative or non-positive in $\overline{\W}\times\R$, $\partial_z H\geq 0$ and
$$\Ricc_x\geq n\ds\mathbb Sup_{z\in\R}\norm{\nabla_x H(x,z)}-\dfrac{n^2}{n-1}\ds\inf_{z\in\R}\left(H(x,z)\right)^2, \ \forall \ x\in\W.$$
Then the Dirichlet problem \varepsilonqref{ProblemaP_manifolds} has a unique solution $u\in\cl^{2,\alpha}(\overline{\W})$ for every $\varepsilonphi\in\cl^{2,\alpha}(\overline{\W})$ if, and only if, the strong Serrin condition \varepsilonqref{SerrinConditionGeneral} holds.
\varepsilonnd{teo}
Finally, using the non-existence result for the types of positively curved manifold mentioned above we also derive:
\begin{teo}[Serrin type solvability criterion 3]
\label{cond_nece_suf_curv_posit}
Let $M$ be a compact and simply connected manifold which is strictly $1/4-$pinched. Let $\W\mathbb Subset M$ be a domain with $\diam(\W)<\frac{\pi}{2\mathbb Sqrt{K_0}}$ and whose boundary is of class $\cl^{2,\alpha}$ for some $\alpha\in(0,1)$. Suppose that $H\in\cl^{1,\alpha}(\overline{\W}\times\R)$ is either non-negative or non-positive in $\overline{\W}\times\R$, $\partial_z H\geq 0$ and
$$\Ricc_x\geq n\ds\mathbb Sup_{z\in\R}\norm{\nabla_x H(x,z)}-\dfrac{n^2}{n-1}\ds\inf_{z\in\R}\left(H(x,z)\right)^2, \ \forall \ x\in\W.$$
Then the Dirichlet problem \varepsilonqref{ProblemaP_manifolds} has a unique solution $u\in\cl^{2,\alpha}(\overline{\W})$ for every $\varepsilonphi\in\cl^{2,\alpha}(\overline{\W})$ if, and only if, the strong Serrin condition \varepsilonqref{SerrinConditionGeneral} holds.
\varepsilonnd{teo}
Other works have considered a Serrin type condition that provides some existence theorems in more general context (see \cite{Alias2008}, \cite{Dajczer2008}, \cite{Dajczer2005} as examples). However, to the best of our knowledge, no other Serrin-type solvability criterion has been proved in settings different from the Euclidean one.
\phantomsection
\addcontentsline{toc}{section}{Bibliography}
\varepsilonnd{document} |
\begin{document}
\title{
Permutations of the integers induce only the
trivial automorphism of the Turing degrees
}
\author{
Bj{\o}rn Kjos-Hanssen\\ Department of Mathematics\\ University of Hawai\textquoteleft i at M\=anoa\\ {\tt [email protected]}\footnote{
This work was partially supported by
a grant from the Simons Foundation (\#315188 to Bj\o rn Kjos-Hanssen).
The author acknowledges the support of the Institut f\"ur Informatik at the University of Heidelberg, Germany
during the workshop on \emph{Computability and Randomness}, June 15 -- July 9, 2015.
}
}
\maketitle{}
\begin{abstract}
Let $\pi$ be an automorphism of the Turing degrees induces by
a homeomorphism $\varphi$ of the Cantor space $2^\omega$ such that $\varphi$ preserves all Bernoulli measures.
It is proved that $\pi$ must be trivial.
In particular, a permutation of $\omega$ can only induce the trivial automorphism of the Turing degrees.
\end{abstract}
\tableofcontents
\section{Introduction}
Let $\mathscr D_{\mathrm{T}}$ denote the set of Turing degrees and let $\le$ denote its ordering.
This article gives a partial answer to the following famous question.
\begin{que}\langlebel{rigid}
Does there exist a nontrivial automorphism of $\mathscr D_{\mathrm{T}}$?
\end{que}
\begin{df}
A bijection $\pi:\mathscr D_{\mathrm{T}}\to\mathscr D_{\mathrm{T}}$ is an \emph{automorphism} of $\mathscr D_{\mathrm{T}}$ if
for all $\mathbf x, \mathbf y\in\mathscr D_{\mathrm{T}}$, $\mathbf x\le\mathbf y$ iff $\pi(\mathbf x)\le\pi(\mathbf y)$.
If moreover there exists an $\mathbf x$ with $\pi(\mathbf x)\ne\mathbf x$ then $\pi$ is \emph{nontrivial}.
\end{df}
Question \ref{rigid} has a long history.
Already in 1977, Jockusch and Solovay \cite{MR0432434} showed that each jump-preserving automorphism of the Turing degrees is the identity above $\mathbf 0^{(4)}$.
Nerode and Shore 1980 \cite{Nerode.Shore:80} showed that
each automorphism (not necessarily jump-preserving) is equal to the identity on some cone.
Slaman and Woodin \cite{Slaman.Woodin:08}
showed that each automorphism is equal to the identity on the cone above $\mathbf 0''$.
Haught and Slaman \cite{Haught.Slaman:97} used permutations of the integers to obtain
automorphisms of the polynomial-time Turing degrees in an ideal (below a fixed set).
\begin{thm}[Haught and Slaman \cite{Haught.Slaman:97}]
There is a permutation of $2^{<\omega}$, or equivalently of $\omega$, that induces a nontrivial automorphism of
\[
(\mathsf{PTIME}^A,\le_{\mathrm{pT}}).
\]
for some $A$.
\end{thm}
Our result can be seen as a contrast to the following work of Kent.
\begin{df}
$A\subset \omega$ is \emph{cohesive} if
for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite.
\end{df}
\begin{thm}[Kent {\cite[Theorem 12.3.IX]{Rogers}, \cite{KentTAMS,KentPhD}}]\langlebel{Kent}
There exists a permutation $f$ such that
\begin{enumerate}[(i)]
\item for all recursively enumerable $B$, $f(B)$ and $f^{-1}(B)$ are recursively enumerable
(and hence for all recursive $A$, $f(A)$ and $f^{-1}(A)$ are recursive);
\item $f$ is not recursive.
\end{enumerate}
\end{thm}
\begin{proof}
Kent's permutation is just any permutation of a cohesive set
(and the identity off the cohesive set).
\end{proof}
\section{Universal algebra setup}
\begin{df}\langlebel{pullback}
The \emph{pullback} of $f:\omega\rightarrow\omega$ is $f^*:\omega^\omega\rightarrow \omega^\omega$ given by
\[
f^*(A)(n) = A(f(n)).
\]
We often write $F=f^*$.
Given a set $S\subseteq\omega$ let $\mathscr D_S = S^\omega/\equiv_{\mathrm{T}}$.
Thus the elements of $\mathscr D_S$ are of the form
\[
[g]_S = \{\,h\in S^\omega \mid h\equiv_{\mathrm{T}} g\,\},\qquad g\in S^\omega.
\]
Given $F:S^\omega\to S^\omega$, let $F_S:\mathscr D_S\rightarrow\mathscr D_S$ be defined by
\[
F_S([A]_S) = [F(A)]_S.
\]
If $F=f^*_S$ then we say that $F_S$ and $F$ are both \emph{induced} by $f$.
\end{df}
\begin{lem}\langlebel{vivaldi}
For each $f:\omega\rightarrow\omega$ and each $S\subseteq\omega$,
the pullback $f^*$ maps $S^\omega$ into $S^\omega$.
\end{lem}
\begin{proof}
\[
A\in S^\omega,\, n\in\omega \quad\Longrightarrow\quad f^*(A)(n) = A(f(n)) \in S.\qedhere
\]
\end{proof}
In light of Lemma \ref{vivaldi}, we can define:
\begin{df}
$f^*_S:\mathscr D_S\to\mathscr D_S$ is the map given by
\[
f^*_S([g]_S) = [f^*(g)]_S.
\]
\end{df}
For $S\subseteq\omega$ (with particular attention to $S\in\{2,\omega\}$), let
\[
\mathscr D_S = S^\omega / \equiv_{\mathrm{T}}.
\]
Our main result concerns $\mathscr D_2$; the corresponding result for $\mathscr D_\omega$ is much easier:
\begin{thm}
Let $f:\omega\to\omega$ be a bijection and let $f^*$ be its pullback.
If $f^*_S$ is an automorphism of $\mathscr D_S$ for some infinite computable set $S$, then $f$ is computable.
\end{thm}
\begin{proof}
Let $\eta:\omega\rightarrow S$ be a computable bijection between $\omega$ and $S$.
Then for all $x\in\omega$,
\[
f^*(\eta\circ f^{-1})(x)=(\eta\circ f^{-1})(f(x)) = \eta(f^{-1}(f(x))) = \eta(x).
\]
Since $\eta\in S^\omega$ is computable and $f^*_S$ is an automorphism,
$\eta\circ f^{-1}\in S^\omega$ must be computable.
Hence $f$ is computable.
\end{proof}
\section{Permutations preserve randomness}
\begin{thm}\langlebel{bioquant}
If $B$ is $f$-$\mu_p$-random, $F=f^*$ and $A=F(B)$ or $A=F^{-1}(B)$, then $A$ is $f$-$\mu_p$-random.
\end{thm}
\begin{proof}
First note that
$f^{-1}$-$\mu_p$-randomness is the same as $f$-$\mu_p$-randomness since $f\equiv_{\mathrm{T}} f^{-1}$.
Thus the result for $A=F^{-1}(B)$ follows from the result for $A=F(B)$.
So suppose $A=F(B)$ and $A$ is not $f$-$\mu_p$-random.
So $A\in\cap_n U_n$ where $\{U_n\}_n$ is an $f$-$\mu_p$-ML test. Then
\[
B\in \{X \mid F(X)\in\cap_n U_n\} = \cap_n V_n
\]
where
\[
V_n = \{X \mid F(X)\in U_n\} = F^{-1}(U_n)
\]
We claim that $V_n$ is $\Sigma^0_1(f)$ (uniformly in $n$) and $\mu_p(V_n)=\mu_p(U_n)$.
Write $U_n=\cup_k [\sigma_k]$ where the strings $\sigma_k$ are all incomparable.
Then
\[
V_n = \cup_k F^{-1}([\sigma_k])
\]
and
\[
\mu_p [\sigma_k] = \mu_p F^{-1}([\sigma_k])
\]
and the $F^{-1}([\sigma_k])$, $k\in\omega$ are still disjoint and clopen.
(If we think of $\sigma\in 2^{<\omega}$ as a partial function from $\omega$ to $2$ then
\[
F^{-1}([\sigma]) = \{X \mid F(X) \in [\sigma]\}
\]
\[
= \{X\mid X(f(n)) = \sigma(n), n<|\sigma|\}
= [\{\langlengle f(n), \sigma(n)\ranglengle \mid n<|\sigma|\}].)
\]
Thus $\{V_n\}_n$ is another $f$-$\mu_p$-ML test, and so
$B$ is not $f$-$\mu_p$-random, which completes the proof.
\end{proof}
\begin{thm}\langlebel{Claim1}
$\mu_p(\{A: A\ge_{\mathrm{T}} p\})=1$, in fact if $A$ is $\mu_p$-ML-random then $A$ computes $p$.
\end{thm}
\begin{proof}
Kjos-Hanssen \cite{K:2010} showed that each Hippocratic $\mu_p$-random set computes $p$.
In particular, each $\mu_p$-random set computes $p$.
\end{proof}
\section{Cones have small measure}
\begin{df}[Bernoulli measures]
For each $n\in\omega$,
$$\mu_p(\{X\in 2^\omega: X(n)=1\})=p$$
and $X(0),X(1),X(2),\ldots$ are mutually independent random variables.
\end{df}
\begin{df}
An \emph{ultrametric} space is a metric space with metric $d$ satisfying the strong triangle inequality
\[
d(x, y)\le\max\{d(x, z), d(z, y)\}.
\]
\end{df}
\begin{df}
A \emph{Polish space} is a separable completely metrizable topological space.
\end{df}
\begin{df}
In a metric space, $B(x,\varepsilon)=\{y: d(x,y) < \varepsilon\}$.
\end{df}
\begin{thm}[\texorpdfstring{\cite[Proposition 2.10]{BenMiller}}{}]\langlebel{LDT}
Suppose that $X$ is a Polish ultrametric space,
$\mu$ is a probability measure on $X$, and
$\mathcal A\subseteq X$ is Borel. Then
\[
\lim_{\varepsilon\to 0}\frac{\mu(\mathcal A\cap B(x,\varepsilon))}{\mu(B(x,\varepsilon))}=1
\]
for $\mu$-almost every $x\in \mathcal A$.
\end{thm}
\begin{df}
For any measure $\mu$ define the conditional measure by
\[
\mu(\mathcal A\mid\mathcal B) = \frac{\mu(\mathcal A\cap\mathcal B)}{\mu(\mathcal B)}.
\]
A measurable set $\mathcal A$ has density $d$ at $X$ if
\[
\lim_n \mu_p(\mathcal A\mid [X\upharpoonright n]) = d.
\]
\end{df}
Let $\Xi(\mathcal A) = \{X: \mathcal A\text{ has density }1\text{ at }X\}$.
\begin{thm}[Lebesgue Density Theorem for $\mu_p$]\langlebel{cold-brewed}
For Cantor space with Bernoulli($p$) product measure $\mu_p$, the Lebesgue Density Theorem holds:
\[
\lim_{n\to\infty}\frac{\mu_p(\mathcal A\cap [x\upharpoonright n])}{\mu_p([x\upharpoonright n])} = 1
\]
for $\mu$-almost every $x\in \mathcal A$.
If $\mathcal A$ is measurable then so is $\Xi(\mathcal A)$.
Furthermore, the measure of the symmetric difference of $\mathcal A$ and $\Xi(\mathcal A)$ is zero, so
$\mu(\Xi(\mathcal A))=\mu(\mathcal A)$.
\end{thm}
\begin{proof}
Consider the ultrametric $d(x,y)=2^{-\min\{n:x(n)\ne y(n)\}}$.
It induces the standard topology on $2^\omega$. Apply Theorem \ref{LDT}.
\end{proof}
Sacks \cite{Sacks:63} and de Leeuw, Moore, Shannon, and Shapiro \cite{deLeeuw} showed that
each cone in the Turing degrees has measure zero. Here we use Theorem \ref{cold-brewed} to extend this to $\mu_p$.
\begin{thm}\langlebel{ce}
If $\mu_p(\{X: W_e^X=A\})>0$ then $A$ is c.e. in $p$.
\end{thm}
\begin{proof}
Suppose $\mu_p(\{X: W_e^X=A\})>0$.
Then $S := \{X \mid W_e^X = A\}$ has positive measure, so $\Xi(S)$ has positive measure,
and hence by Theorem \ref{LDT} there is an $X$ such that $S$ has density 1 at $X$.
Thus, there is an $n$ such that
$\mu_p(S\mid [X\upharpoonright n]) > \frac12$.
Let $\sigma = X\upharpoonright n$.
We can now enumerate $A$ using $p$ by taking a ``vote'' among the sets extending $\sigma$.
More precisely, $n\in A$ iff
\[
\mu_p(\{Y: \sigma\prec Y\wedge n\in W_e^Y\}) > \frac12,
\]
and the set of $n$ for which this holds is clearly c.e. in $p$.
\end{proof}
\begin{thm}\langlebel{Claim3}
Each cone strictly above $p$ has $\mu_p$-measure zero:
\[
\mu_p(\{A: A\ge_{\mathrm{T}} q\})=1\qquad\Longrightarrow\qquad q\le_{\mathrm{T}} p.
\]
\end{thm}
\begin{proof}
If $A$ can compute $q$ then $A$ can enumerate both $q$ and the complement of $q$.
Hence by Theorem \ref{ce}, $q$ is both c.e. in $p$ and co-c.e. in $p$; hence $q\le_{\mathrm{T}} p$.
\end{proof}
\section{Main result}
We are now ready to prove our main result Theorem \ref{main} that
no nontrivial automorphism of the Turing degrees is induced by a permutation of $\omega$.
\begin{thm}\langlebel{main}
If $\pi$ is an automorphism of $\mathscr D_2$ which is induced by a permutation of $\omega$ then
$\pi(\mathbf p)=\mathbf p$ for each $\mathbf p\in\mathscr D_{\mathrm{T}}$.
\end{thm}
\begin{proof}
Fix a permutation $f:\omega\to\omega$ and let $F=f^*\upharpoonright 2^\omega$.
Let $B$ be $f$-$\mu_p$-random.
We claim that $B$ computes $F(p)$.
By Theorem \ref{Claim1}, for any $f$-$\mu_p$ random $A$, we have
$p\le_{\mathrm{T}} A$, hence $F(p)\le_{\mathrm{T}} F(A)$. So it suffices to represent $B$ as $F(A)$.
Now $B = F(F^{-1}(B))$. Let $A = F^{-1}(B)$. By Theorem \ref{bioquant}, $A$ is $f$-$\mu_p$-random.
Thus every $f$-$\mu_p$-random computes $F(p)$.
Thus we have completed the proof of our claim that $\mu_p$-almost every real computes $F(p)$.
By
Theorem \ref{Claim3} it follows that $F(p)\le_{\mathrm{T}} p$.
By considering the inverse $f^{-1}$ we also obtain $F^{-1}(p)\le_{\mathrm{T}} p$ and hence $p\le_{\mathrm{T}} F(p)$.
So $F(p)\equiv_{\mathrm{T}} p$ and $F$ induces the identity automorphism.
\end{proof}
\section{Computing the permutation}
\begin{thm}\langlebel{GuestHouse4426}
Let $f:\omega\to\omega$ be a permutation.
Let $F=f^*$ be its pullback (Definition \ref{pullback}) to $2^\omega$.
If for positive Lebesgue measure many $G$, $F(G)\le_{T} G$,
then $f$ is recursive.
\end{thm}
\begin{proof}
By the Lebesgue Density Theorem we can get a $\mathbb{P}hi$ and a $\sigma$ such that,
if $\mu_\sigma$ denotes conditional probability on $\sigma$
and $E = \{A: F(A)=\mathbb{P}hi^A\}$, then
\[
\mu_\sigma(E)\ge 95\%.
\]
For simplicity let us write $p_n(A) = A+n=A\cup\{n\}$ and $m_n(A) = A-n=A\setminus\{n\}$.
Then $p_n^{-1}E = \{A: p_n(A)\in E\}$. Note that
\[
E \subseteq p_n^{-1}(E)\cup m_n^{-1}(E)
\]
and
\[
E^c \subseteq p_n^{-1}(E^c)\cup m_n^{-1}(E^c)
\]
Then
\[
\mu_{\sigma}(E) \le \mu_{\sigma}(p_n^{-1}(E)\cup m_n^{-1}(E))
\le \mu_{\sigma}(p_n^{-1}(E)) + \mu_{\sigma}(m_n^{-1}(E))
\]
We now have
\[
\mu_\sigma\{A:F(A+n)=\mathbb{P}hi^{A+n}\}\ge 90\%
\]
and
\[
\mu_\sigma\{A:F(A-n)=\mathbb{P}hi^{A-n}\}\ge 90\%;
\]
Indeed,
the events $m_n^{-1}(A)$, $p_n^{-1}(A)$ are each independent of the event $n\in A$,
so for $n>|\sigma|$,
\begin{eqnarray*}
95\% \le \mu_{\sigma}(E) &=& \mu_{\sigma}(p_n^{-1}(E)\mid n\in A)\mu_{\sigma}(n\in A)
+ \mu_{\sigma}(p_n^{-1}(E)\mid n\notin A)\mu_{\sigma}(n\notin A)\\
&=& \frac12\left(
\mu_{\sigma}(p_n^{-1}(E)\mid n\in A)
+ \mu_{\sigma}(m_n^{-1}(E)\mid n\notin A)
\right)\\
&=& \frac12\left(\mu_{\sigma}(p_n^{-1}(E))+\mu_{\sigma}(m_n^{-1}(E))\right)
\end{eqnarray*}
which gives
\[
1.9 \le \mu_{\sigma}(p_n^{-1}(E)) + \mu_{\sigma}(m_n^{-1}(E))
\le 1 + \min\{\mu_{\sigma}(p_n^{-1}(E)),\mu_{\sigma}(m_n^{-1}(E))\}.
\]
Also $F(A-n)$ and $F(A+n)$ differ in exactly one bit, namely $f^{-1}(n)$, for all $A$:
\begin{eqnarray*}
F(A-n)(b)\ne F(A+n)(b)&\Longleftrightarrow& (A-n)(f(b))\ne (A+n)(f(b))\\
&\Longleftrightarrow& n=f(b)\Longleftrightarrow b=f^{-1}(n),
\end{eqnarray*}
that is
\[
\{A: (\forall b)(F(A+n)(b)\ne F(A-n)(b)\leftrightarrow b=f^{-1}(n))\} = 2^\omega.
\]
Let $D_{n,b} = \{A: \mathbb{P}hi^{A+n}(b)\downarrow\ne\mathbb{P}hi^{A-n}(b)\downarrow\}$.
For $n>|\sigma|$,
\[
\mu_\sigma \left(D_{n,f^{-1}(n)}\setminus\bigcup_{b\ne f^{-1}(n)} D_{n,b}\right)
= \mu_\sigma\{A: (\forall b)(A\in D_{n,b}\leftrightarrow b=f^{-1}(n))\}\ge 80\%
\]
since
\[
\mu_\sigma\{A: \neg(\forall b)(A\in D_{n,b}\quad\leftrightarrow\quad b=f^{-1}(n))\}
\]
\[
\le
\mu_\sigma(\neg p_n^{-1}(E)) + \mu_\sigma(\neg m_n^{-1}(E)) \le 10\%+10\%=20\%.
\]
Therefore, given any $n$, we can compute $f^{-1}(n)$:
enumerate computations until we have found some bit $b$ such that
\[
\mu_\sigma D_{n,b}\ge 80\%.
\]
Then $b=f^{-1}(n)$.
Thus $f^{-1}$ is computable and hence so is $f$.
\end{proof}
\begin{thm}\langlebel{abstract}
If $\pi$ is an automorphism of $\mathscr D_{\mathrm{T}}$ which is induced by
a permutation $f$ of $\omega$ then $f$ is recursive.
\end{thm}
\begin{proof}
By Theorem \ref{main}, $f^*(G)\equiv_{\mathrm{T}} G$ for each $G\in 2^\omega$.
By Theorem \ref{GuestHouse4426}, $f$ is recursive.
\end{proof}
\section{Measure-preserving homeomorphisms of the Cantor set}
\begin{pro}
A permutation of $\omega$ induces a homeomorphism of $2^\omega$ that is $\mu_p$-preserving for each $p$.
\end{pro}
\begin{pro}
There exist homeomorphisms of $2^\omega$ that are $\mu_p$-preserving for each $p$, but are not induced by a permutation.
\end{pro}
\begin{proof}
Map
\[
[1]\mapsto [111]\cup [001]\cup [101]\cup [110]
\]
(more generally, any collection of cylinders of strings of length 3 including 2 strings of Hamming weight 2 and 1 of Hamming weight 1).
Another way to express this is that the homeomorphism preserves the fraction of 1s in a certain sense.
More precisely,
\begin{eqnarray*}
100 \mapsto 001,\\
101 \mapsto 101,\\
110 \mapsto 110,\\
111 \mapsto 111.
\end{eqnarray*}
\end{proof}
\begin{thm}
Suppose $\varphi$ is a homeomorphism of $2^\omega$ which is $\mu_p$-preserving for all $p$
(it suffices to require this for infinitely many $p$, or for a single transcendental $p$).
Suppose $\varphi$ induces an automorphism $\pi$ of the Turing degrees. Then $\pi = \mathrm{id}$.
\end{thm}
We omit the proof which follows along the same lines as before.
\end{document} |
\begin{document}
\author[M. El Bachraoui and J. S\'{a}ndor]{Mohamed El Bachraoui and J\'{o}zsef S\'{a}ndor}
\address{Dept. Math. Sci,
United Arab Emirates University, PO Box 15551, Al-Ain, UAE}
\email{[email protected]}
\address{Babes-Bolyai University, Department of Mathematics and Computer Science, 400084 Cluj-Napoca, Romania}
\email{[email protected]}
\keywords{$q$-trigonometric functions; $q$-digamma function; transcendence.}
\subjclass{33B15, 11J81, 33E05, 11J86}
\begin{abstract}
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting
$q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions.
Our key argument is a theta product formula of Jacobi
and Gosper's $q$-trigonometric identities.
\end{abstract}
\date{\thetaxtit{\today}}
\maketitle
\section{Introduction}\label{sec-introduction}
Throughout we let $\tau$ be a complex number in the upper half plane and let $q=e^{\pi i\tau}$.
Note that the assumption $\mathrm{Im}(\tau)>0$ implies
that $|q|<1$.
The $q$-shifted factorials of a complex number $a$ are defined by
\[
(a;q)_0= 1,\quad (a;q)_n = \prod_{i=0}^{n-1}(1-a q^i),\quad
(a;q)_{\infty} = \lim_{n\to\infty}(a;q)_n.
\]
For convenience we write
\[
(a_1,\ldots,a_k;q)_n = (a_1;q)_n\cdots (a_k;q)_n,\quad
(a_1,\ldots,a_k;q)_{\infty} = (a_1;q)_{\infty} \cdots (a_k;q)_{\infty}.
\]
The $q$-gamma function is given by
\[
\Gamma_q(z) = \dfrac{(q;q)_\infty}{(q^{z};q)_\infty} (1-q)^{1-z} \quad (|q|<1)
\]
and it is well-known that $\Gamma_q (z)$ is a $q$-analogue for the gamma function $\Gamma (z)$, see
\cite{Andrews-Askey-Roy, Askey, Gasper-Rahman, Jackson-1, Jackson-2} for details on the function $\Gamma_q(z)$.
The digamma function $\psi(z)$ and the $q$-digamma function $\psi_q (z)$ are given by
\[
\psi(z) = \big(\log\Gamma(z)\big)' = \frac{\Gamma'(z)}{\Gamma(z)} \quad\thetaxt{and\quad}
\psi_q (z) = \big(\log\Gamma_q(z)\big)' = \frac{\Gamma_q '(z)}{\Gamma_q (z)}.
\]
By Krattenthaler and Srivastava~\cite{Krattenthaler-Srivastava} one has
$\lim_{q\to 1} \psi_q(z) = \psi(z)$, showing that the function $\psi_q(z)$ is the $q$-analogue for
the function $\psi (z)$.
Jacobi first theta function is defined as follows:
\[
\theta_1(z \mid \tau) = 2\sum_{n=0}^{\infty}(-1)^n q^{(2n+1)^2/4}\sin(2n+1)z
= i q^{\fracac{1}{4}}e^{-iz} (q^2 e^{-2iz},e^{2iz},q^2; q^2)_{\infty}.
\]
Jacobi theta functions have been extensively studied by mathematicians during the last two centuries with hundreds of
properties and formulas as a result.
Standard references on theta functions include Lawden~\cite{Lawden} and Whittaker~and~Watson~\cite{Whittaker-Watson}.
Among the well-known properties of the function $\theta_1(z\mid\tau)$ which we need in this paper
we have
\begin{equation}\label{theta-cot}
\frac{\theta_1'(z|\tau)}{\theta_1(z|\tau)} = \cot z + 4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}} \sin (2nz).
\end{equation}
Gosper~\cite{Gosper} introduced $q$-analogues of $\sin z$ and $\cos z$ as follows
\begin{equation}\label{sine-cosine-q-gamma}
\begin{split}
\sin_q \pi z
&=
q^{\frac{1}{4}} \Gamma_{q^2}^2\left(\frac{1}{2}\right) \fracac{q^{z(z-1)}}{\Gamma_{q^2}(z) \Gamma_{q^2}(1-z)} \\
\cos_q \pi z
&=
\Gamma_{q^2}^2\left(\frac{1}{2}\right) \frac{q^{z^2}}{\Gamma_{q^2}\left(\frac{1}{2}-z \right) \Gamma_{q^2}\left(\frac{1}{2}+z\right)}.
\end{split}
\end{equation}
and proved that
\begin{equation}\label{sine-cosine-theta}
\begin{split}
\sin_q (z) = \fracac{\theta_1(z\mid \tau')}{\theta_1\left( \fracac{\pi}{2}\bigm| \tau' \right)} \qquad \thetaxt{and \quad}
\cos_q (z) = \fracac{\theta_1\left( z+\fracac{\pi}{2} \bigm| \tau' \right)}
{\theta_1 \left( \fracac{\pi}{2} \bigm| \tau' \right)} \quad \quad (\tau' = \fracac{-1}{\tau}).
\end{split}
\end{equation}
It can be shown that
$\lim_{q\to 1}\sin_q z = \sin z$ and $\lim_{q\to 1}\cos_q z = \cos z$.
Moreover, from (\ref{sine-cosine-q-gamma}), one can easily verify by differentiating logarithms that
$\sin_q' (z)$ is the $q$-analogue of $\sin' (z) = \cos z$ and that
$\cos_q' (z)$ is the $q$-analogue of $\cos' (z) = -\sin z$.
We mention that there are known other examples of $q$-analogues for the functions $\sin z$
and $\cos z$, see for instance the book by Gasper~and~Rahman~\cite{Gasper-Rahman}.
A function which is very important for our current purpose is
\begin{equation}\label{Cotan-q}
\Ct_q(z) = \frac{\sin_q' z}{\sin_q z}
\end{equation}
for which we clearly have $\lim_{q\to 1} \Ct_q(z) = \cot z$. In addition, by taking in
(\ref{sine-cosine-q-gamma}) logarithms and differentiating with respect to $z$ we get
\begin{equation}\label{reflection}
\psi_{q^2}(z)-\psi_{q^2}(1-z) = (2z-1)\log q - \pi\Ct_q (\pi z),
\end{equation}
which is the $q$-analogue of the well-known reflection formula
\[
\psi(z)-\psi(1-z) = -\pi \cot(\pi z).
\]
Jacobi~\cite{Jacobi} proved that
\begin{equation}\label{MainProd}
\frac{(q^{2n};q^{2n})_{\infty}}{(q^2;q^2)_{\infty}^n}
\prod_{k=-\frac{n-1}{2}}^{\frac{n-1}{2}}\theta_1 \left(z+\frac{k\pi}{n} \bigm| \tau \right) =
\theta_1(nz \mid n\tau),
\end{equation}
see also Enneper~\cite[p. 249]{Enneper}.
This formula turns out to be equivalent to the following $q$-trigonometric identity
of Gosper~\cite[p. 92]{Gosper}:
\begin{equation}\label{SineProd}
\prod_{k=0}^{n-1}\sin_{q^n}\pi \left(z+\frac{k}{n} \right) =
q^{\fracac{(n-1)(n+1)}{12}} \fracac{(q;q^2)_{\infty}^2}{(q^n;q^{2n})_{\infty}^{2n}} \sin_q n\pi z
\end{equation}
which he apparently was not aware of as
he stated the identity without proof or reference.
Unlike many of Jacobi's results, the formula (\ref{MainProd})
seems not to have received much attention by mathematicians. This is probably due to the lack of applications.
The authors recently in~\cite{Bachraoui-Sandor} offered a new proof for~(\ref{SineProd}) and as an application they established a
$q$-analogue for the Gauss multiplication formula for the gamma function as well as for
an identity of S\'{a}ndor~and~T\'{o}th~\cite{Sandor-Toth} for a short product on Euler gamma function.
Our purpose in this note is to apply~(\ref{SineProd}) in order to evaluate finite and infinite sums involving
the function $\Ct_q (z)$ along with
the functions $h_{q,M,a}(k)$ and $f_{q,M,a}(k)$ both defined on integers $k$ as follows:
\begin{equation}\label{h-f}
\begin{split}
h_{q,M,a}(k) &= \frac{1}{\pi}\Big(
(\log q)\frac{2k+a-2M}{2M}-\psi_q \big(\frac{2k+a}{2M} \big) - \psi_q \big(1-\frac{2k+a}{2M} \big) \Big) \\
f_{q,M,a}(k) &= \sum_{n=1}^{\infty}\frac{q^{\frac{2n}{M}}}{1-q^{\frac{2n}{M}}}\sin \frac{(2k+a)n\pi}{M}.
\end{split}
\end{equation}
More specifically, we shall prove the following main results which are new, up
to the authors' best knowledge.
\begin{theorem}\label{thm-main-1}
Let $M>1$ be an integer and let $a$ be an odd integer. Then
\noindent
\emph{(a)\ }
\[
\sum_{n=1}^{\infty} \frac{1}{n} \Ct_q\Big(\frac{(2n+a)\pi}{2M}\Big)
= -\frac{1}{M} \sum_{k=1}^M \Ct_q\Big(\frac{(2k+a)\pi}{2M}\Big) \psi\big(\frac{k}{M}\big).
\]
\noindent
\emph{(b)\ } The function $h_{q,M,a}(k)$ is periodic with period $M$ and we have
\[
\sum_{n=1}^{\infty} \frac{h_{q,M,a}(n)}{n} = -\frac{1}{M} \sum_{k=1}^M h_{q,M,a}(k) \psi\big(\frac{k}{M}\big).
\]
\noindent
\emph{(c)\ } The function $f_{q,M,a}(k)$ is periodic with period $M$ and we have
\[
\sum_{n=1}^{\infty} \frac{f_{q,M,a}(n)}{n} = -\frac{1}{M} \sum_{k=1}^M f_{q,M,a}(k) \psi\big(\frac{k}{M}\big).
\]
\end{theorem}
\begin{theorem}\label{thm-main-2}
Let $M$ be a positive integer and let $a$ be an odd integer. Then
\begin{align*}
\emph{(a)\quad } &
\sum_{k=1}^M \Big( \psi_q \big( \frac{2k+a}{2M} \big) - \psi_q \big( 1- \frac{2k+a}{2M} \big) \Big)
= \frac{a+1}{2} \log q. \\
\emph{(b)\quad } &
\sum_{k=1}^{M}\Big( \psi_{q}\big( \frac{4k+a}{4M} \big) - \psi_{q}\big( 1-\frac{4k+a}{4M} \big) \Big) \\
& \quad = \frac{(a+2)\log q}{4} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{4}\big) \\
& \quad = \begin{cases}
\frac{(a+2)\log q}{4} -\frac{\log q}{4}\frac{\Pi_{q^{1/(4M)}}^2}{\Pi_{q^{1/(2M)}}}
& \thetaxt{if\ } a\equiv 1,-3 \pmod{8} \\
\frac{(a+2)\log q}{4} +\frac{\log q}{4}\frac{\Pi_{q^{1/(4M)}}^2}{\Pi_{q^{1/(2M)}}}
& \thetaxt{if\ } a\equiv -1,3 \pmod{8},
\end{cases} \\
\emph{(c)\quad } &
\sum_{k=1}^{M}\Big( \psi_{q}\big( \frac{6k+a}{6M} \big) - \psi_{q}\big( 1-\frac{6k+a}{6M} \big) \Big) \\
& \quad = \frac{(a+3)\log q}{6} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{6}\big) \\
& \quad = \begin{cases}
\frac{(a+3)\log q}{6} +\frac{\log q}{3}\frac{\Pi_{q^{1/(6M)}}^{3/2}}{\Pi_{q^{1/(2M)}}^{1/2}}
& \thetaxt{if\ } a\equiv 1,-5 \pmod{12} \\
\frac{(a+3)\log q}{6} -\frac{\log q}{3}\frac{\Pi_{q^{1/(6M)}}^{3/2}}{\Pi_{q^{1/(2M)}}^{1/2}}
& \thetaxt{if\ } a\equiv -1,5 \pmod{12}. \\
\end{cases}
\end{align*}
\end{theorem}
\begin{remark}\label{rmk-main-1}
By letting $q\to 1$ in Theorem~\ref{thm-main-1} and Theorem~\ref{thm-main-2} one gets
related sums for the functions $\cot z$ and $\psi(z)$. For instance, from Theorem~\ref{thm-main-1}(a)
we obtain for $M>1$ and odd integer $a$
\begin{equation}\label{q-to-1}
\sum_{n=1}^{\infty} \frac{1}{n} \cot\Big(\frac{(2n+a)\pi}{2M}\Big)
= -\frac{1}{M} \sum_{k=1}^M \cot \Big(\frac{(2k+a)\pi}{2M}\Big) \psi\big(\frac{k}{M}\big)
\end{equation}
and from Theorem~\ref{thm-main-2}(c) we deduce
\[
\sum_{k=1}^M \Big( \psi \big( \frac{6k+a}{6M} \big) - \psi \big( 1- \frac{6k+a}{6M} \big) \Big)
= - M\pi \cot\big(\frac{a\pi}{6}\big)
\]
\[
= \qquad \begin{cases}
- \sqrt{3} M\pi & \thetaxt{if\ } a\equiv 1,-5 \pmod{12} \\
\sqrt{3} M\pi & \thetaxt{if\ } a\equiv -1,5 \pmod{12} \\
0 & \thetaxt{if\ } a\equiv -3,3 \pmod{12}.
\end{cases}
\]
\end{remark}
\begin{remark}\label{rmk-transcendence}
By the well-known fact that $\cot r\pi$ is an algebraic number for any rational number $r$ and the relation
(\ref{q-to-1}) we deduce by a result of Adhikari~\emph{et al.}~\cite{Adhikari-et-al} that the sum
$\sum_{n=1}^{\infty} \frac{1}{n} \cot\Big(\frac{(2n+a)\pi}{2M}\Big)$ is either zero or transcendental.
A similar statement can be made about the $q$-analogue of the sum given in
Theorem~\ref{thm-main-1}(b).
\end{remark}
\noindent
Blagouchine~\cite{Blagouchine} recently evaluated a variety of finite sums involving the digamma function
and the trigonometric functions. For instance, he proved that for any positive integer $M$
\[
\sum_{k=1}^{M-1} \big(\cot\frac{k\pi}{M}\big) \psi\big(\frac{k}{M}\big) = -\frac{\pi(M-1)(M-2)}{6}.
\]
We have the following related contribution.
\begin{theorem}\label{thm-main-3}
For any integer $M>1$ and any odd integer $a$ we have
\[
\sum_{k=1}^{M-1} \Big( \cot\frac{(2k+a)\pi}{2M} + \cot\frac{(2k-a)\pi}{2M} \Big) \psi\big(\frac{k}{M}\big)
= - \sum_{k=1}^{M-1}\big(\cot\frac{k\pi}{M}\big) \cot\frac{(2k+a)\pi}{2M}.
\]
\end{theorem}
\noindent
The rest of the paper is organized as follows. In Section~\ref{Sec:q-trig} we review Gosper's $q$-trigonometry and collect the facts
which are needed for our discussion. In Section~\ref{sec:proof-main-1} we give the proof of Theorem~\ref{thm-main-1},
Section~\ref{sec:proof-main-2} is devoted to the proof for Theorem~\ref{thm-main-2}, and Section~\ref{sec:proof-main-3}
is devoted to the proof of Theorem~\ref{thm-main-3}.
\section{Facts on Gosper's $q$-trigonometry}\label{Sec:q-trig}
\noindent
Just as for the function $\sin z$ and $\cos z$, it is easy to verify that
\begin{align}\label{sine-cos-basics}
\sin_q (\frac{\pi}{2}-z)=\cos_q z,\ \sin_q \pi = 0,\
\sin_q \frac{\pi}{2} = \cos_q 0= 1, \\
\sin_q(z+\pi)= -\sin_q z=\sin_q(-z), \ \thetaxt{and\ }
-\cos_q(z+\pi) = \cos_q z = \cos_q(-z), \nonumber
\end{align}
from which it follows that for any odd integer $a$,
\begin{equation}\label{sin-cos-aux}
\begin{split}
\sin_{q}\frac{a\pi}{4} &=
\begin{cases}
\sin_q\frac{\pi}{4} & \thetaxt{if\ } a\equiv 1, 3 \pmod{8} \\
- \sin_q \frac{\pi}{4}& \thetaxt{if\ } a\equiv -1, -3 \pmod{8},
\end{cases} \\
\sin_{q}\frac{a\pi}{6} &=
\begin{cases}
\sin_q\frac{\pi}{6} & \thetaxt{if\ } a\equiv 1, 5 \pmod{12} \\
-\sin_q\frac{\pi}{6} & \thetaxt{if\ } a\equiv -1, -5 \pmod{12} \\ \\
1 & \thetaxt{if\ } a\equiv 3 \pmod{12} \\
-1 & \thetaxt{if\ } a\equiv -3 \pmod{12}.
\end{cases}
\end{split}
\end{equation}
Also, by using (\ref{sine-cos-basics}) we have
\begin{align}\label{special-deriv}
\sin_q'\big(\frac{\pi}{2}-z \big)= -\cos_q' z,\ \cos_q'\big(z-\frac{\pi}{2}\big)= \sin_q' z,\\
- \sin_q'(\pi-z) = \sin_q' z,\ \thetaxt{and\ }
-\cos_q'(\pi-z) = \cos_q' z \nonumber
\end{align}
where the derivatives here and in what follows are with respect to $z$.
We can easily see from (\ref{special-deriv}) that for any odd integer $a$ we have
\begin{equation}\label{q-sine-derive-2}
\sin_q' \frac{a\pi}{2} = \cos_q' 0 = 0.
\end{equation}
The following $q$-constant appears frequently in Gosper's manuscript~\cite{Gosper}
\[
\Pi_q = q^{\frac{1}{4}} \frac{(q^2;q^2)_{\infty}^2}{(q;q^2)_{\infty}^2}.
\]
Gosper stated many identities involving $\sin_q z$ and $\cos_q z$ which easily follow just
from the definition and basic properties of other related functions. To mention an example,
he derived that
\begin{equation}\label{q-sine-derive-1}
\sin_q' 0 =- \cos_q'\frac{\pi}{2} = \frac{-2 \log q}{\pi}\Pi_q.
\end{equation}
On the other hand, Gosper~\cite{Gosper} using the computer facility \emph{MACSYMA} stated without proof a variety of identities involving $\sin_q z$ and $\cos_q z$ and he
asked the natural question whether his formulas hold true.
For instance, based on his conjectures, he stated
\[
\label{q-Double-2} \tag{$q$-Double$_2$}
\sin_q(2z) = \frac{\Pi_q}{\Pi_{q^2}} \sin_{q^2} z \cos_{q^2} z,
\]
\[
\label{q-Double-3} \tag{$q$-Double$_3$}
\cos_q(2z) = (\cos_{q^2} z)^2 - (\sin_{q^2} z)^2,
\]
\[ \label{q-Triple-2} \tag{$q$-Triple$_2$}
\sin_q(3z) = \fracac{\Pi_q}{\Pi_{q^3}} (\cos_{q^3} z)^2 \sin_{q^3}z - (\sin_{q^3}z)^3,
\]
and
\[ \label{q-Double-5} \tag{$q$-Double$_5$}
\cos_q(2z) = (\cos_{q}z)^4- (\sin_{q}z)^4.
\]
\noindent
A proof for (\ref{q-Double-2}) can be found in Mez\H{o}~\cite{Mezo-1} and proofs for (\ref{q-Double-2})
(\ref{q-Triple-2}), and (\ref{q-Double-5}) were obtained in~\cite{Bachraoui-1, Bachraoui-2, Bachraoui-3}.
Proofs for other identities of Gosper can be found in~\cite{Touk-Houchan-Bachraoui, He-Zhai, He-Zhang}.
Furthermore, Gosper deduced the following special values:
\begin{equation}\label{sin-cos-values}
\begin{split}
\sin_{q^2} \frac{\pi}{4} &= \cos_{q^2} \frac{\pi}{4} = \frac{\Pi_{q^2}^{\frac{1}{2}}}{\Pi_{q}^{\frac{1}{2}}} \\
\left(\sin_{q^3}\frac{\pi}{3} \right)^3 &= \left(\cos_{q^3}\frac{\pi}{6} \right)^3 = \frac{ \left(\frac{\Pi_q}{\Pi_{q^3}} \right)^{\frac{3}{2}}}{\left(\frac{\Pi_q}{\Pi_{q^3}} \right)^2 -1} \\
\left(\sin_{q^3}\frac{\pi}{6} \right)^3 &= \left(\cos_{q^3}\frac{\pi}{3} \right)^3 =
\frac{ 1}{\left(\frac{\Pi_q}{\Pi_{q^3}} \right)^2 -1}.
\end{split}
\end{equation}
\noindent
As to special values for derivatives we have the following list.
\begin{lemma} \label{lem-1-special}
Let $a$ be an odd integer. Then we have
\[
\begin{split}
\emph{(a)\ } & \sin_{q^2}'\frac{a\pi}{4} =
\begin{cases}
\frac{\log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^2}^{\frac{1}{2}}} & \thetaxt{if\ } a\equiv -1, 1 \pmod{8} \\
- \frac{\log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^2}^{\frac{1}{2}}} & \thetaxt{if\ } a\equiv -3, 3 \pmod{8}.
\end{cases} \\
\emph{(b)\ } &
\sin_{q^3}'\frac{a\pi}{3} =
\begin{cases}
\frac{\log q}{\pi}
\frac{\Pi_{q^3}^{\frac{1}{3}} (\Pi_q^2 - \Pi_{q^3}^2)^{\frac{2}{3}} (3\Pi_{q^3}^2-\Pi_q^2)}
{\Pi_{q}^2 - \Pi_{q^3}^2} & \thetaxt{if\ } a\equiv -1, 1 \pmod{6} \\
-\frac{2\log q}{\pi} \Pi_q
& \thetaxt{if\ } a\equiv 3 \pmod{6}
\end{cases} \\
\emph{(c)\ } &
\sin_{q^3}'\frac{a\pi}{6} =
\begin{cases}
-\frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}} \Pi_{q^3}^{\frac{1}{6}} }{ (\Pi_q^2- \Pi_{q^3}^2)^{\frac{1}{3}}}
& \thetaxt{if\ } a\equiv 1, -5 \pmod{12} \\
\frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}} \Pi_{q^3}^{\frac{1}{6}} }{ (\Pi_q^2- \Pi_{q^3}^2)^{\frac{1}{3}}}
& \thetaxt{if\ } a\equiv -1, 5 \pmod{12} \\
0 & \thetaxt{if\ } a\equiv -3, 3 \pmod{12}.
\end{cases}
\end{split}
\]
\end{lemma}
\begin{proof}
(a)\ From (\ref{special-deriv}) we have
\begin{equation}\label{q-sine-derive-quarter}
\sin_{q}' \frac{\pi}{4} = - \cos_{q}' \frac{\pi}{4}.
\end{equation}
On the other hand, from (\ref{q-Double-3})
we have
\[
2\cos_q' 2z = 2(\cos_{q^2} z) \cos_{q^2}' z - 2(\sin_{q^2} z) \sin_{q^2}'z,
\]
where if we let $z=\frac{\pi}{4}$ and use (\ref{q-sine-derive-quarter}) we deduce
\[
2\cos_q'\frac{\pi}{2} = -4 \sin_{q^2}\frac{\pi}{4} \sin_{q^2}'\frac{\pi}{4}.
\]
Now, combine the previous identity with (\ref{q-sine-derive-1}), (\ref{sin-cos-values}), and
(\ref{q-sine-derive-quarter}) to obtain the desired identity for $\sin_{q^2}'\frac{\pi}{4}$. Finally, note that
by (\ref{sine-cos-basics}) we have
\[
\sin_q'\frac{a\pi}{4} = \begin{cases}
\sin_q'\frac{\pi}{4} & \thetaxt{if\ } a\equiv \pm 1 \pmod{8}, \\
-\sin_q'\frac{\pi}{4} & \thetaxt{if\ } a\equiv \pm 3 \pmod{8}
\end{cases}
\]
to complete the proof of part (a).
\noindent
As to part (b),
from (\ref{special-deriv}), we easily find
\begin{equation}\label{sixth-third}
\cos_q'\frac{\pi}{6} = -\sin_q'\frac{\pi}{3} \ \thetaxt{and\ }
\cos_q'\frac{\pi}{3} = -\sin_q'\frac{\pi}{6}.
\end{equation}
Now differentiating (\ref{q-Double-5}) we have
\[
2\cos_q' 2z = 4 (\cos_q z)^3 \cos_q'z - 4 (\sin_q z)^3 \sin_q'z.
\]
Then taking $z=\frac{\pi}{6}$ in the previous identity, using (\ref{sixth-third}), and simplifying yield
\[
\big(2(\sin_q\frac{\pi}{6})^3 - 1 \big) \sin_q'\frac{\pi}{6} =
2 (\cos_q\frac{\pi}{6})^3 \cos_q'\frac{\pi}{6},
\]
in other words,
\begin{equation}\label{help1-lem-2-special}
\cos_q'\frac{\pi}{6} = \frac{2(\sin_q\frac{\pi}{6})^3 - 1}{2 (\cos_q\frac{\pi}{6})^3} \sin_q'\frac{\pi}{6}.
\end{equation}
On the other hand, differentiate (\ref{q-Triple-2}) to derive
\[
3 \sin_q 3z = \frac{\Pi_q}{\Pi_{q^3}}\big(\sin_{q^3}'z (\cos_{q^3} z)^2 +
2\sin_{q^3}z\cos_{q^3}z\cos_{q^3}'z \big) - 3 (\sin_{q^3} z)^2 \sin_{q^3}'z,
\]
which for $z=\frac{\pi}{3}$ and after simplification gives
\[
-\sin_q'0 = \Big(\frac{\Pi_q}{\Pi_{q^3}} (\cos_{q^3}\frac{\pi}{3})^2 - 3(\sin_{q^3}\frac{\pi}{3})^2 \Big)
\sin_{q^3}'\frac{\pi}{3} + 2 \frac{\Pi_q}{\Pi_{q^3}} \sin_{q^3}\frac{\pi}{3}\cos_{q^3}\frac{\pi}{3}\cos_{q^3}'\frac{\pi}{3}.
\]
It follows by virtue of (\ref{help1-lem-2-special}) and with the help of (\ref{sin-cos-values}) that
\[
\frac{6 \log q}{\pi}\Pi_q =\Big(\frac{\Pi_q}{\Pi_{q^3}} (\cos_{q^3}\frac{\pi}{3})^2 -3(\sin_{q^3}\frac{\pi}{3})^2
+ \frac{4 \frac{\Pi_q}{\Pi_{q^3}} \big(\sin_{q^3}\frac{\pi}{3}\big)^4 \cos_{q^3}\frac{\pi}{3}}
{2 \big(\sin_{q^3}\frac{\pi}{3}\big)^3 - 1} \Big) \sin_{q^3}'\frac{\pi}{3}.
\]
Now solving in the previous identity for $\sin_{q^3}'\frac{\pi}{3}$ and using (\ref{sin-cos-values}),
after a long but straightforward calculation, we derive the desired formula for $\sin_{q^3}'\frac{\pi}{3}$.
Finally, note from (\ref{sine-cos-basics}) that
\[
\sin_q'\frac{a\pi}{3} = \begin{cases}
\sin_q'\frac{\pi}{3} & \thetaxt{if\ } a\equiv \pm 1 \pmod{6}, \\
-\sin_q' 0 & \thetaxt{if\ } a\equiv \pm 3 \pmod{6}
\end{cases}
\]
to complete the proof of part (b).
The proof for part (c) is similar to the previous parts and it is therefore omitted.
\end{proof}
\noindent
By a combination of Lemma~\ref{lem-1-special} with (\ref{sin-cos-aux}) and
(\ref{sin-cos-values}), we arrive at the main result of this section.
\begin{corollary}\label{cor-special-C}
Let $a$ be an odd integer. Then we have
\[
\begin{split}
\emph{(a)\quad } & \Ct_{q^2} \big(\frac{a\pi}{4}\big) =
\begin{cases}
\frac{\log q}{\pi} \frac{\Pi_{q}^2}{\Pi_{q^2}} & \thetaxt{if\ } a\equiv 1, -3 \pmod{8} \\
-\frac{\log q}{\pi} \frac{\Pi_{q}^2}{\Pi_{q^2}} & \thetaxt{if\ } a\equiv -1, 3 \pmod{8}.
\end{cases} \\
\emph{(b)\quad } & \Ct_{q^3} \big(\frac{a\pi}{6}\big) =
\begin{cases}
-\frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^3}^{\frac{1}{2}}} & \thetaxt{if\ } a\equiv 1, -5 \pmod{12} \\
\frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^3}^{\frac{1}{2}}} & \thetaxt{if\ } a\equiv -1, 5 \pmod{12} \\
0 & \thetaxt{if\ } a\equiv -3, 3 \pmod{12}.
\end{cases}
\end{split}
\]
\end{corollary}
\section{Proof of Theorem~\ref{thm-main-1}}\label{sec:proof-main-1}
\noindent
We need the following result of Ram~Murty~and~Saradha~\cite{Murty-Saradha} which we record as a lemma.
\begin{lemma}\label{lem-MurtSara}
Let $f$ be any function defined on the integers and with period $M>1$.
\\
\noindent
The infinite series $\sum_{n=1}^{\infty} \frac{f(n)}{n}$ converges if and only if
$\sum_{k=1}^M f(k) = 0$. In case of convergence, we have
\[
\sum_{n=1}^{\infty} \frac{f(n)}{n} = -\frac{1}{M} \sum_{k=1}^M f(k) \psi\big(\frac{k}{M}\big).
\]
\end{lemma}
\noindent
\emph{Proof of Theorem~\ref{thm-main-1}(a)}\ Let
\[
f(k) = \Ct_q \Big( \frac{(2k+a)\pi}{2M} \Big)
\]
which is clearly well-defined on the integers and it is periodic with period $M$.
Then based on Lemma~\ref{lem-MurtSara}, all we need is prove that $\sum_{k=1}^{M} f(k) = 0$.
To do so, note that from (\ref{SineProd}) and the fact that $\sin_q(z+\pi) = -\sin_q z$ we find
\begin{equation}\label{SineProd-2}
\prod_{k=1}^{M}\sin_{q^M}\pi \left(z+\frac{k}{M} \right) =
- q^{\fracac{(M-1)(M+1)}{12}} \fracac{(q;q^2)_{\infty}^2}{(q^M;q^{2M})_{\infty}^{2M}} \sin_q M\pi z.
\end{equation}
Take logarithms and differentiate with respect to $z$ to derive
\[
\pi \sum_{k=1}^M \Ct_{q^M}\Big(z+\frac{k}{M}\Big) = M\pi\Ct_q(M\pi z)= M\pi \frac{\sin_q' M\pi z}{\sin_q M\pi z}.
\]
Now replace $q^M$ with $q$, let $z=\frac{a}{2M}$, and use (\ref{q-sine-derive-2}) to deduce that
that
\begin{equation}\label{sum-Ctq-zero}
\pi \sum_{k=1}^M \Ct_q \Big( \frac{(2k+a)\pi}{2M} \Big) = 0,
\end{equation}
or equivalently,
\[
\sum_{k=1}^{M} f(k) = 0,
\]
as desired.
\noindent
\emph{Proof of Theorem~\ref{thm-main-1}(b)}\ By the relation (\ref{sine-cosine-q-gamma}) we have
\[
\sin_q\pi\big(z+\frac{k}{M}\big) = q^{\frac{1}{4}}\Gamma_{q^2}^2\left(\frac{1}{2}\right)
\fracac{q^{(z+\frac{k}{M})(z+\frac{k}{M} -1)}}{\Gamma_{q^2}\big(z+\frac{k}{M} \big)
\Gamma_{q^2}\big(1-z-\frac{k}{M}\big)}
\]
which after taking logarithms and differentiating with respect to $z$ gives
\[
\pi \Ct_q\pi\big(z+\frac{k}{M}\big)
= (\log q)
\big( 2\big(z+\frac{k}{M}\big)-1 \big) - \psi_{q^2}\big(z+\frac{k}{M}\big) - \psi_{q^2}\big(1-z-\frac{k}{M}\big).
\]
Replacing in the previous relation $q^2$ by $q$ and letting $z=\frac{a}{2M}$, we get
\[
\Ct_{q^{1/2}}\big(\frac{(2k+a)\pi}{2M}\big)
=
\frac{1}{\pi}\Big( (\log q)\frac{2k+a-2M}{2M} - \psi_{q}\big(\frac{2k+a}{2M}\big) - \psi_{q}\big(1-\frac{2k+a}{2M}\big) \Big).
\]
As the left-hand side of the previous identity is evidently periodic with period $M$, the same holds for its right-hand side
which is nothing else but $h_{q,M,a}(k)$. We now claim that $\sum_{k=1}^M h_{q,M,a}(k) = 0$. Indeed,
apply~(\ref{sine-cosine-q-gamma}) to the factors in identity~(\ref{SineProd-2}), then take logarithms and finally differentiate with respect to $z$ to
obtain
\begin{align}\label{q-psi-key}
M(\log q) & \Big(\sum_{k=1}^M 2\big(z+ \frac{k}{M}\big) -1 \Big) -
\sum_{k=1}^M \Big(\psi_{q^{2M}}\big(z+\frac{k}{M}\big) - \psi_{q^{2M}}\big(1-z-\frac{k}{M}\big) \Big)
\nonumber \\
& = M\pi \Ct_q(M\pi z) .
\end{align}
Next replace $q^{2M}$ by $q$, let $z=\frac{a}{2M}$, and use
(\ref{q-sine-derive-2}) to deduce that
\begin{equation}\label{help1-cor-psiq-1}
\sum_{k=1}^M \Big( (\log q)\frac{2k+a-M}{2M} - \psi_{q}\big(\frac{2k+a}{2M}\big) -
\psi_{q}\big(1-\frac{2k+a}{2M}\big) \Big) = 0.
\end{equation}
That is,
\[
\sum_{k=1}^M h_{q,M,a}(k) = 0,
\]
and the claim is confirmed.
Finally, apply Lemma~\ref{lem-MurtSara} to the function $h_q(M,a,k)$ to complete the proof of part (b).
\noindent
\emph{Proof of Theorem~\ref{thm-main-1}(c)}\ Note that the function $f_{q,M,a}(k)$ is clearly periodic with period $M$.
By virtue of~(\ref{sine-cosine-theta}) and~(\ref{SineProd-2}) and after taking logarithm and differentiating we find
\[
\pi \sum_{k=1}^M \frac{\theta_1'\big(\pi z +\frac{k\pi}{M} | \frac{\tau'}{M}\big)}{\theta_1\big(\pi z +\frac{k\pi}{M} | \frac{\tau'}{M}\big)}
= M\pi \Ct_q(M\pi z),
\]
which upon substituting $z$ by $\frac{a}{2M}$ and $\tau'$ by $\tau$ yields
\[
\sum_{k=1}^M \frac{\theta_1'\big(\frac{(2k+a)\pi}{2M} | \frac{\tau}{M}\big)}{\theta_1\big(\frac{(2k+a)\pi}{2M} | \frac{\tau}{M}\big)} = 0.
\]
Now combine the foregoing identity with (\ref{theta-cot}) and the $q$-analogue of (\ref{sum-Ctq-zero}) to derive
\[
\sum_{k=1}^M f_{q,M,a}(k)
= \sum_{k=1}^M \sum_{n=1}^{\infty}\frac{q^{\frac{2n}{M}}}{1-q^{\frac{2n}{M}}}\sin \frac{(2k+a)n\pi}{M} =0.
\]
Finally apply Lemma~\ref{lem-MurtSara} to the function $f_{q,M,a}(k)$ to complete the proof.
\section{Proof of Theorem~\ref{thm-main-2}}\label{sec:proof-main-2}
\noindent
(a)\ If $M=1$, then the desired formula
\[
\psi_q\big(1+\frac{a}{2}\big) - \psi_q\big(-\frac{a}{2}\big) = \frac{(a+1)\log q}{2}
\]
follows by virtue of (\ref{reflection}). If $M>1$, then an immediate consequence of (\ref{SineProd-2}) is
\[
\sum_{k=1}^M \Big(\psi_{q}\big(\frac{2k+a}{2M}\big) - \psi_{q}\big(1-\frac{2k+a}{2M}\big)\Big) =
\frac{(a+1) \log q}{2},
\]
which is the desired relation.
\noindent
(b)\ If $M=1$, the statement follows by (\ref{reflection}). Now suppose that $M>1$.
Then by letting in (\ref{q-psi-key}) $z=\frac{a}{4M}$ and after simplification we find
\[
\frac{(a+2)M\log q}{2} -
\sum_{k=1}^M \Big( \psi_{q^{2M}}\big(\frac{4k+a}{4M}\big) - \psi_{q^{2M}}\big(\frac{4(M-k)-a}{4M}\big)\Big)
= M\pi \Ct_q\big(\frac{a\pi}{4}\big).
\]
Then uopn replacing $q^{2M}$ by $q$ and rearranging becomes
\[
\sum_{k=1}^M \Big(\psi_{q}\big(\frac{4k+a}{4M}\big) - \psi_{q}\big(\frac{4(M-k)-a}{4M}\big)\Big)
= \frac{(a+2)\log q}{4} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{4}\big),
\]
which is the first identity of this part. As to the second formula,
simply use Corollary~\ref{cor-special-C} to evaluate the right-hand-side of the previous identity
and rearrange in the appropriate way.
\noindent
(c)\ If $M=1$, the statement follows by (\ref{reflection}). Now suppose that $M>1$.
Let in (\ref{q-psi-key}) $z=\frac{a}{6M}$ and simplify to obtain
\[
\frac{(a+3)M\log q}{3} -
\sum_{k=1}^M \Big(\psi_{q^{2M}}\big(\frac{6k+a}{6M}\big) - \psi_{q^{2M}}\big(\frac{6(M-k)-a}{6M}\big)\Big)
= M\pi \Ct_q\big(\frac{a\pi}{6}\big).
\]
Then replace in the foregoing formula $q^{2M}$ by $q$ and rearrange to get
\[
\sum_{k=1}^M \Big(\psi_{q}\big(\frac{6k+a}{6M}\big) - \psi_{q}\big(\frac{6(M-k)-a}{6M}\big)\Big)
= \frac{(a+3)\log q}{6} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{6}\big).
\]
This establishes the first formula of this part.
Finally apply Corollary~\ref{cor-special-C} to complete the proof.
\section{Proof of Theorem~\ref{thm-main-3}}\label{sec:proof-main-3}
\noindent
In our proof we shall make an appeal to a result of
Weatherby~in~\cite{Weatherby} for which we need the following notation.
For any real number $\alpha$ and any positive integer $l$, let
\[
A_{\alpha,l} := \frac{(-1)^{l-1} \big(\pi\cot(\pi \alpha)\big)^{(l-1)}}{\pi^l(l-1)!}
\]
and let
\[
Z(l) = \begin{cases}
0 & \thetaxt{if $l$\ is odd,} \\
\frac{\zeta(l)}{\pi^l} & \thetaxt{otherwise.}
\end{cases}
\]
Notice that $Z(l)\in\mathbb{Q}$ for all positive integer $l$.
\begin{lemma}\label{lem-Weatherby}
Let $f$ be an algebraic valued function defined on the integers with period $M>1$ and let $l$ be a positive integer. Then
\[
\sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{f(n)}{n^l} =
\Big(\frac{\pi}{M}\Big)^l \Big( \sum_{k=1}^{M-1} f(k) A_{\frac{k}{M},l}+ 2f(M) Z(l) \Big).
\]
\end{lemma}
\noindent
\emph{Proof of Theorem~\ref{thm-main-3}.\ }
Let
\[
f(k) = \cot \frac{(2k+a)\pi}{2M}.
\]
Clearly $f(k)$ is well-defined on the integers since is $a$ is odd and it is periodic with period $M$.
It is a well-known fact that for any rational number $r$ we have that $\cot\pi r$ is an algebraic number.
Then by virtue of Lemma~\ref{lem-Weatherby} we get
\begin{equation}\label{key-sum-main-3}
\sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot\pi\big(\frac{2n+a}{2M}\big)}{n^l}
=\Big(\frac{\pi}{M}\Big)^l \Big( \sum_{k=1}^{M-1} \big(\cot\pi\frac{2k+a}{2M}\big) A_{\frac{k}{M},l}+ 2f(M) Z(l) \Big),
\end{equation}
which for $l=1$ reduces to
\begin{equation}\label{help2-cor-cot-1}
\begin{split}
\sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot \frac{(2n+a)\pi}{2M}}{n}
&=
\frac{\pi}{M}\sum_{k=1}^{M-1}A_{\frac{k}{M},1}\cot\frac{(2k+a)\pi}{2M} \\
&=
\frac{\pi}{M}\sum_{k=1}^{M-1} \frac{1}{\pi} \big(\cot\frac{k\pi}{M} \big) \cot\frac{(2k+a)\pi}{2M}.
\end{split}
\end{equation}
On the other hand, with the help of the $q$-analogue of Theorem~\ref{thm-main-1}(a) we deduce
\[
\sum_{n=1}^{\infty}\frac{\cot \frac{(-2n+a)\pi}{2M}}{-n}
= \sum_{n=1}^{\infty}\frac{\cot \frac{(2n-a)\pi}{2M}}{n}
= -\frac{1}{M} \sum_{k=1}^{M} \Big(\cot\frac{(2k-a)\pi}{2M}\Big)\psi\big(\frac{k}{M}\big),
\]
which implies that
\[
\begin{split}
\sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot \frac{(2n+a)\pi}{2M}}{n}
&=
\sum_{n=1}^{\infty}\frac{\cot \frac{(2n+a)\pi}{2M}}{n} + \sum_{n=1}^{\infty}\frac{\cot \frac{(2n-a)\pi}{2M}}{n} \\
&=
-\frac{1}{M} \sum_{k=1}^{M} \psi\big(\frac{k}{M}\big)\Big(\cot\frac{(2k+a)\pi}{2M} + \cot\frac{(2k-a)\pi}{2M} \Big).
\end{split}
\]
As
\[\cot\frac{(2M+a)\pi}{2M} + \cot\frac{(2M-a)\pi}{2M} = \frac{\sin 2\pi}
{\frac{1}{2}\big(\cos\frac{a\pi}{M} - \cos\frac{2M\pi}{M} \big)} = 0,
\]
the $M$-th term in the last summation vanishes and so we get
\begin{equation}\label{help1-cor-cot-1}
\sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot \frac{(2n+a)\pi}{2M}}{n} =
-\frac{1}{M} \sum_{k=1}^{M-1} \psi\big(\frac{k}{M}\big)\Big(\cot\frac{(2k+a)\pi}{2M} + \cot\frac{(2k-a)\pi}{2M} \Big).
\end{equation}
Now combine (\ref{help1-cor-cot-1}) and (\ref{help2-cor-cot-1}) to obtain the desired formula.
\end{document} |
\boldsymbol{b}egin{document}
\title{\Large $L^1$-optimality conditions for circular restricted three-body problems
}
\author{Zheng Chen\thanks{Laboratoire de Math\'ematiqeus d'Orsay, Univ. Paris-Sud, CNRS, Universit\'e Paris-Saclay, 91405 Orsay, France. E-mail:{[email protected]}
}
}
\maketitle
\boldsymbol{b}egin{abstract}
In this paper, the $L^1$-minimization for the translational motion of a spacecraft in a circular restricted three-body problem (CRTBP) is considered. Necessary conditions are derived by using the Pontryagin Maximum Principle, revealing the existence of bang-bang and singular controls. Singular extremals are detailed, recalling the existence of the Fuller phenomena according to the theories developed by Marchal in Ref.~\boldsymbol{b}oldsymbol{c}ite{Marchal:73} and Zelikin {\it et al.} in Refs.~\boldsymbol{b}oldsymbol{c}ite{Zelikin:94,Zelikin:03}. The sufficient optimality conditions for the $L^1$-minimization problem with fixed endpoints have been solved in Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15}. In this paper, through constructing a parameterised family of extremals, some second-order sufficient conditions are established not only for the case that the final point is fixed but also for the case that the final point lies on a smooth submanifold. In addition, the numerical implementation for the optimality conditions is presented. Finally, approximating the Earth-Moon-Spacecraft system as a CRTBP, an $L^1$-minimization trajectory for the translational motion of a spacecraft is computed by employing a combination of a shooting method with a continuation method of Caillau {\it et al.} in Refs.~\boldsymbol{b}oldsymbol{c}ite{Caillau:12,Caillau:12time}, and the local optimality of the computed trajectory is tested thanks to the second-order optimality conditions established in this paper.
\end{abstract}
\boldsymbol{b}oldsymbol{s}ection{ Introduction}
\label{intro}
As an increasing number of artificial satellites or spacecrafts have been and are being launched into deeper space since 1960s, the problem of controlling the translational motion of a spacecraft in the gravitational field of multiple celestial bodies such that some cost functionals are minimized or maximized arises in astronautics. The circular restricted three-body problem (CRTBP), which though as a degenerate model in celestial mechanics can capture the chaotic property of $n$-body problem, is extensively used in the literature in recent years to study optimal trajectories in deeper space. The controllability properties for the translational motion in CRTBPs are studied by Caillau {\it et al.} in Ref. \boldsymbol{b}oldsymbol{c}ite{Caillau:12time}, showing that there exist admissible controlled trajectories in an appropriate subregion of state space. The present paper is concerned with the $L^1$-minimization problem for the translational motion of a spacecraft in a CRTBP, which aims at minimizing the $L^1$-norm of control. Therefore, if the control is generated by propulsion systems which expel mass in a high speed to generate an opposite reaction force according to Newton's third law of motion, the $L^1$-minimization problem is referred to as the well-known fuel-optimal control problem in astronautics. The existence of the $L^1$-minimization solutions in CRTBPs can be obtained by a combination of Filippov theorem in Ref. \boldsymbol{b}oldsymbol{c}ite{Agrachev:04} and the technique in Ref. \boldsymbol{b}oldsymbol{c}ite{Gergaud:06} if we assume that admissible controlled trajectories remain in a fixed compact, see Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:12}.
While in the planar case where the translational motion is restricted in a 2-dimensional (2D) plane, the singular extremals and the corresponding chattering arcs are analyzed by Zelikin and Borisov in Ref.~\boldsymbol{b}oldsymbol{c}ite{Zelikin:03}, the synthesis of the solutions of singular extremals in 3-dimensional (3D) case, to the author's knowledge, is not covered up to the present time. Therefore, in this paper, in addition to an emphasis on the necessary conditions arising from the Pontryagin Maximum Principle (PMP), which reveals the existences of bang-bang and singular controls, the solutions of singular extremals are investigated to show that the $L^1$-minimization trajectories in 3D case can exhibit Fuller or chattering phenomena according to the theories developed by Marchal in Ref.~\boldsymbol{b}oldsymbol{c}ite{Marchal:73} as well as by Zelikin and Borisov in Ref.~\boldsymbol{b}oldsymbol{c}ite{Zelikin:94}.
Even though one does not consider singular and chattering controls, the bang-bang type of control as well as the chaotic property in CRTBPs makes the computation of the $L^1$-minimization solutions a big challenge. To address this challenge, various numerical methods, {\it e.g.,} direct methods~\boldsymbol{b}oldsymbol{c}ite{Mingotti:09,Ross:07}, indirect methods~\boldsymbol{b}oldsymbol{c}ite{Caillau:12,Caillau:12time}, and hybrid methods~\boldsymbol{b}oldsymbol{c}ite{Ozimek:10}, have been developed recently. In this paper, the indirect method, proposed by Caillau {\it et al.} in Refs. \boldsymbol{b}oldsymbol{c}ite{Caillau:12,Caillau:12time} to combine a shooting method with a continuation method, is employed to compute the extremal trajectories of the $L^1$-minimization problem. Based on this method, some kinds of fuel-optimal trajectories in a CRTBP are computed recently as well in Ref. \boldsymbol{b}oldsymbol{c}ite{Zhang:15}. Whereas, one can notice that the extremal trajectories computed by this indirect method cannot be guaranteed to be at least locally optimal unless sufficient optimality conditions are satisfied. Thus, it is indeed crucial to test sufficient conditions to check if a computed trajectory realizes a local optimality, which is what is missing in the research of optimal trajectories in CRTBPs.
The sufficient conditions for optimal control problems are widely studied in the literature in recent years, see Refs.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:02,Poggiolini:04,Schattler:12,Noble:02,Caillau:15,Agrachev:04,Kupka:87,Sarychev:82,Sussmann:85,Bonnard:07} and the references therein. Through defining an accessory finite dimensional problem in Refs.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:02,Poggiolini:04}, some sufficient conditions are developed for optimal control problems with a polyhedral control set. In Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15}, two no-fold conditions are established for the $L^1$-minimization problem, which generalises the results of Refs.~\boldsymbol{b}oldsymbol{c}ite{Schattler:12,Noble:02}. Assuming the endpoints are fixed, these two no-fold conditions are sufficient to guarantee a bang-bang extremal of the $L^1$-minimization problem to be a strong local optimizer (cf. Subsection \boldsymbol{b}oldsymbol{r}ef{Subse:sufficient1}). Whereas, in addition to the two no-fold conditions, a third condition has to be established once the dimension of the constraint submanifold of final states is not zero, see Refs.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:02,Brusch:70,Wood:74}. In this paper, a parameterized family of extremals around a given extremal is constructed such that the third condition is managed to be related with Jacobi field under some regularity assumptions (cf. Subsection \boldsymbol{b}oldsymbol{r}ef{Subse:sufficient2}). Then, it is shown that the propagation of Jacobi field is enough to test the sufficient optimality conditions (cf. Sect. \boldsymbol{b}oldsymbol{r}ef{SE:Procedure}).
The paper is organized as follows. In Sect. \boldsymbol{b}oldsymbol{r}ef{SE:Problem_Formulation}, the $L^1$-minimization problem is formulated in CRTBPs. Then, the necessary conditions are derived with an emphasis on singular solutions in Sect. \boldsymbol{b}oldsymbol{r}ef{SE:Necessary}. In Sect. \boldsymbol{b}oldsymbol{r}ef{SE:Sufficient}, a parameterized family of extremals is first constructed. Under some regularity assumptions, the sufficient conditions for the strong-local optimality of the nonsingular extremals with bang-bang controls are established. In Sect. \boldsymbol{b}oldsymbol{r}ef{SE:Procedure}, a numerical implementation for the optimality conditions is derived. In Sect. \boldsymbol{b}oldsymbol{r}ef{SE:Numerical}, consider the Earth-Moon-Spacecraft system as a CRTBP, a transfer trajectory of a spacecraft from a circular geosynchronous orbit of the Earth to a circular orbit around the Moon is calculated to provide a bang-bang extremal, whose local optimality is tested thanks to the second-order optimality conditions developed in this paper.
\boldsymbol{b}oldsymbol{s}ection{Definitions and notations}\label{SE:Problem_Formulation}
A CRTBP in celestial mechanics is generally defined as an isolated dynamical system consisting of three gravitationally interacting bodies, $P_1$, $P_2$, and $P_3$, whose masses are denoted by $m_1$, $m_2$, and $m_3$, respectively, such that 1) the third mass $m_3$ is so much smaller than the other two that its gravitational influence on the motions of the other two is negligible and 2) the two bodies, $P_1$ and $P_2$, move on their own circular orbits around their common centre of mass.
Without loss of generality, we assume $m_1 > m_2$ and consider a rotating frame $OXYZ$ such that its origin is located at the barycentre of the two bodies $P_1$ and $P_2$, see Fig.~\boldsymbol{b}oldsymbol{r}ef{Fig:rotating_frame}.
\boldsymbol{b}egin{figure}[!ht]
\includegraphics[trim=4.0cm 1.5cm 3.0cm 1.5cm, clip=true, width=4in]{crtbp_figure.eps}
\boldsymbol{b}oldsymbol{c}aption[]{Rotating frame $OXYZ$ of the CRTBP.}
\label{Fig:rotating_frame}
\end{figure}
The unit vector of $X$-axis is defined in such a way that it is collinear to the line between the two primaries $P_1$ and $P_2$ and points toward $P_2$, the unit vector of $Z$-axis is defined as the unit vector of the momentum vector of the motion of $P_1$ and $P_2$, and the $Y$-axis is defined to complete a right-hand coordinate system. It is advantageous to use non-dimensional parameters. Let $d_*$ be the distance between $P_1$ and $P_2$, and let $m_* = m_1 + m_2$, we denote by $d_*$ and $m_*$ the unit of length and mass, respectively. We also define the unit of time $t_*$ in such a way that the gravitational constant $G>0$ equals to one. Accordingly, one can obtain $$t_* = \boldsymbol{b}oldsymbol{s}qrt{\boldsymbol{b}oldsymbol{f}rac{d_*^{3}}{Gm_*}}$$ through the usage of Kepler's third low. Then, denote by the superscript ``~$T$~" the transpose of matrices, if $\mu = m_2/m_*$, the two constant vectors $\boldsymbol{b}oldsymbol{r}_1 = (-\mu,0,0)^T$ and $\boldsymbol{b}oldsymbol{r}_2 = (1-\mu,0,0)^T$ denote the position of $P_1$ and $P_2$ in the rotating frame $OXYZ$, respectively.
\boldsymbol{b}oldsymbol{s}ubsection{Dynamics}
{{In this paper, we denote the space of $n$-dimensional column vectors by $\mathbb{R}^n$ and the space of $n$-dimensional row vectors by $(\mathbb{R}^n)^*$.}} Let $t\in\mathbb{R}_+$ be the non-dimensional time and let $\boldsymbol{b}oldsymbol{r} \in \mathbb{R}^3$ and $\boldsymbol{b}oldsymbol{v} \in \mathbb{R}^3$ be the non-dimensional position vector and velocity vector of $P_3$, respectively, in the rotating frame $OXYZ$. Then, consider a spacecraft as the third mass point $P_3$ controlled by a finite-thrust propulsion system and let $m= m_3/m_*$, its state $\boldsymbol{b}oldsymbol{x}\in\mathbb{R}^n$ ($n=7$) consists of position vector $\boldsymbol{b}oldsymbol{r}$, velocity vector $\boldsymbol{b}oldsymbol{v}$, and mass $m$, i.e., $\boldsymbol{b}oldsymbol{x} = (\boldsymbol{b}oldsymbol{r},\boldsymbol{b}oldsymbol{v},m)$. Denote by the two constants $r_{m_1}>0$ and $r_{m_2}>0$ the radiuses of the two bodies $P_1$ and $P_2$, respectively, and denote by the constant $m_c>0$ the mass of the spacecraft without any fuel, we define the admissible subset for state $\boldsymbol{b}oldsymbol{x}$ as
\boldsymbol{b}egin{eqnarray}
\mathcal{X}=\boldsymbol{b}ig\{(\boldsymbol{b}oldsymbol{r},\boldsymbol{b}oldsymbol{v},m)\in\mathbb{R}^3 \times \mathbb{R}^3\times\mathbb{R}_+\ \arrowvert\ \|\boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{r}_1\| > r_{m_1},\ \|\boldsymbol{b}oldsymbol{r}-\boldsymbol{b}oldsymbol{r}_2\| >r_{m_2},\ m \boldsymbol{b}oldsymbol{g}eq m_c\boldsymbol{b}ig\},\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where ``~$\|\boldsymbol{b}oldsymbol{c}dot \|$~" denotes the Euclidean norm. Then, the differential equations for the controlled translational motion of the spacecraft in the CRTBP in the admissible set $\mathcal{X}$ for positive times can be written as
\boldsymbol{b}egin{eqnarray}
\Sigma:
\boldsymbol{b}egin{cases}
\dot{\boldsymbol{b}oldsymbol{r}}(t) = \boldsymbol{b}oldsymbol{v}(t),\\
\dot{\boldsymbol{b}oldsymbol{v}}(t) = \boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v}(t)) + \boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}(t)) + \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{\tau}(t)}{m(t)},\\
\dot{m}(t) = -\boldsymbol{b}eta{\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{\tau}(t)\boldsymbol{b}oldsymbol{p}arallel},\label{EQ:mass}
\end{cases}
\label{EQ:Sigma}
\end{eqnarray}
with
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v}) = \left[\boldsymbol{b}egin{array}{ccc}
0 & 2 & 0\\
-2 & 0 & 0\\
0 & 0 & 0
\end{array}\boldsymbol{b}oldsymbol{r}ight]\boldsymbol{b}oldsymbol{v}, \ \boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}) = \left[\boldsymbol{b}egin{array}{ccc}1& 0 & 0\\
0 & 1 & 0\\
0 & 0 & 0\end{array}\boldsymbol{b}oldsymbol{r}ight] \boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{f}rac{1- \mu}{\|\boldsymbol{b}oldsymbol{r}-\boldsymbol{b}oldsymbol{r}_1\|^3}(\boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{r}_1) - \boldsymbol{b}oldsymbol{f}rac{\mu}{\|\boldsymbol{b}oldsymbol{r} -\boldsymbol{b}oldsymbol{r}_2\|^3}(\boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{r}_2),\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where $\boldsymbol{b}eta \boldsymbol{b}oldsymbol{g}eq 0$ is a scalar constant determined by the specific impulse of the engine equipped on the spacecraft and $\boldsymbol{b}oldsymbol{\tau}\in\mathbb{R}^3$ is the thrust vector, taking values in
\boldsymbol{b}egin{eqnarray}
\mathcal{T} = \{\boldsymbol{b}oldsymbol{\tau} \in\mathbb{R}^3\ \arrowvert \ \|\boldsymbol{b}oldsymbol{\tau}\|\leq \tau_{max}\},\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where the constant $\tau_{max}>0$, in unit of ${m_*d_*}/{t_*^2}$, denotes the maximum magnitude of the thrust of the engine.
Denote by $\boldsymbol{b}oldsymbol{r}ho\in[0,1]$ the normalized mass flow rate of the engine, i.e., $\boldsymbol{b}oldsymbol{r}ho = \|\boldsymbol{b}oldsymbol{\tau}\|/\tau_{max}$, and $\boldsymbol{b}oldsymbol{\omega}\in\mathbb{S}^2$ the unit vector of the thrust direction, i.e., $\boldsymbol{b}oldsymbol{\tau} = \boldsymbol{b}oldsymbol{r}ho\tau_{max}\boldsymbol{b}oldsymbol{\omega}$, we then have that $\boldsymbol{b}oldsymbol{r}ho$ and $\boldsymbol{b}oldsymbol{\omega}$ are control variables in the dynamics $\Sigma$ in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Sigma}). Let $\boldsymbol{b}oldsymbol{u}=(\boldsymbol{b}oldsymbol{r}ho,\boldsymbol{b}oldsymbol{\omega})$ and $\mathcal{U}=[0,1]\times\mathbb{S}^2$, we say $\mathcal{U}$ is the admissible set for the control $\boldsymbol{b}oldsymbol{u}$.
Let us define the controlled vector field $\boldsymbol{b}oldsymbol{f}$ on $\mathcal{X}\times\mathcal{U}$ by
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{f}:\mathcal{X}\times\mathcal{U}\boldsymbol{b}oldsymbol{r}ightarrow \mathbb{R}^n,\ \boldsymbol{b}oldsymbol{f}(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{r}ho,\boldsymbol{b}oldsymbol{\omega}) = \boldsymbol{b}oldsymbol{f}_0(\boldsymbol{b}oldsymbol{x}) + \boldsymbol{b}oldsymbol{r}ho \boldsymbol{b}oldsymbol{f}_1(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{\omega}),\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{f}_0(\boldsymbol{b}oldsymbol{x}) = \left(\boldsymbol{b}egin{array}{c}
\boldsymbol{b}oldsymbol{v}\\
\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v}) + \boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r})\\
{0}\end{array}\boldsymbol{b}oldsymbol{r}ight),\ \
\boldsymbol{b}oldsymbol{f}_1(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{\omega}) = \left(\boldsymbol{b}egin{array}{c}
\boldsymbol{b}oldsymbol{0}\\
{\tau_{max}} \boldsymbol{b}oldsymbol{\omega}/{m}\\
- {\tau_{max}}\boldsymbol{b}eta \end{array}\boldsymbol{b}oldsymbol{r}ight).\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Then, the dynamics in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Sigma}) can be rewritten as the control-affine form
\boldsymbol{b}egin{eqnarray}
{\Sigma}:
\dot{\boldsymbol{b}oldsymbol{x}}(t) =\boldsymbol{b}oldsymbol{f}(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{r}ho(t),\boldsymbol{b}oldsymbol{\omega}(t)) = \boldsymbol{b}oldsymbol{f}_0(\boldsymbol{b}oldsymbol{x}(t)) + \boldsymbol{b}oldsymbol{r}ho(t) \boldsymbol{b}oldsymbol{f}_1(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{\omega}(t)).
\label{EQ:system}
\end{eqnarray}
\boldsymbol{b}oldsymbol{s}ubsection{$L^1$-minimization problem}
Given an $l\in\mathbb{N}$ such that $0< l \leq n$, we define the $l$-codimensional constraint submanifold on final state as
\boldsymbol{b}egin{eqnarray}
\mathcal{M}=\{\boldsymbol{b}oldsymbol{x}\in\mathcal{X}\ \arrowvert\ \boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{x})=0\},
\label{EQ:final_manifold}
\end{eqnarray}
where $\boldsymbol{b}oldsymbol{p}hi:\mathcal{X}\boldsymbol{b}oldsymbol{r}ightarrow \mathbb{R}^l$ denotes a twice continuously differentiable function of $\boldsymbol{b}oldsymbol{x}$ and its expression depends on specific mission requirements, see an explicit example in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:function_phi}). Then, given a fixed initial state $\boldsymbol{b}oldsymbol{x}_0\in\mathcal{X}$ and a fixed final time $t_f>0$, the $L^1$-minimization problem~\boldsymbol{b}oldsymbol{c}ite{Caillau:15} for the translational motion in the CRTBP consists of steering the system $\Sigma$ in $\mathcal{X}$ by a measurable control $(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot))\in\mathcal{U}$ on $[0,t_f]$ from the initial point $\boldsymbol{b}oldsymbol{x}_0\in\mathcal{X}$ to a final point $\boldsymbol{b}oldsymbol{x}_f\in\mathcal{M}$ such that the $L^1$-norm of control is minimized, i.e.,
\boldsymbol{b}egin{eqnarray}
\int_0^{t_f} \boldsymbol{b}oldsymbol{r}ho(t) dt \boldsymbol{b}oldsymbol{r}ightarrow \text{min}.
\label{EQ:cost_functional}
\end{eqnarray}
Note that the $L^1$-minimization problem is referred to as the fuel-minimum problem if $\boldsymbol{b}eta > 0$.
Controllability for the translational motion of the spacecraft in a CRTBP holds in an appropriate subregion of state space, see Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:12}. Let $t_m > 0$ be the minimum time to steer the system $\Sigma$ by measurable controls $(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot))\in\mathcal{U}$ from the point $\boldsymbol{b}oldsymbol{x}_0\in\mathcal{X}$ to a point $\boldsymbol{b}oldsymbol{x}_f\in\mathcal{M}$. Then, assuming $t_f > t_m$ and that the admissible controlled trajectories of $\Sigma$ remain in a fixed compact, the existence of the $L^1$-minimization solutions can be obtained by Filippov theorem \boldsymbol{b}oldsymbol{c}ite{Agrachev:04} since the convexity issues due to the $\boldsymbol{b}oldsymbol{r}ho$ term in the integrand of the cost in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:cost_functional}) can be dealt with as in Ref. \boldsymbol{b}oldsymbol{c}ite{Gergaud:06}. Therefore, the PMP is applicable to formulate the following necessary conditions.
\boldsymbol{b}oldsymbol{s}ection{Necessary conditions}\label{SE:Necessary}
\boldsymbol{b}oldsymbol{s}ubsection{Pontryagin Maximum Principle}
According to the PMP in Ref. \boldsymbol{b}oldsymbol{c}ite{Pontryagin}, if a trajectory ${\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{X}$ associated with a measurable control ${\boldsymbol{b}oldsymbol{u}}(\boldsymbol{b}oldsymbol{c}dot)=(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot))$ in $\mathcal{U}$ on $[0,t_f]$ is an optimal one of the $L^1$-minimization problem, there exists a nonpositive real number $p^0$ and an absolutely continuous mapping ${t\mapsto\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot)\in T^*_{\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot)}\mathcal{X}}$ on $[0,t_f]$, satisfying $(\boldsymbol{b}oldsymbol{p},p^0) \boldsymbol{b}oldsymbol{n}eq 0$ and called adjoint state, such that almost everywhere on $[0,t_f]$ there holds
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}egin{cases}
\dot{\boldsymbol{b}oldsymbol{x}}(t) = \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial H}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t),p^0,\boldsymbol{b}oldsymbol{u}(t)),\\
\dot{\boldsymbol{b}oldsymbol{p}}(t) = -\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial H}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t),p^0,\boldsymbol{b}oldsymbol{u}(t)),
\end{cases}
\label{EQ:cannonical}
\end{eqnarray}
and
\boldsymbol{b}egin{eqnarray}
H({\boldsymbol{b}oldsymbol{x}}(t),{\boldsymbol{b}oldsymbol{p}}(t),{p}^0,\boldsymbol{b}oldsymbol{u}(t)) =\boldsymbol{b}oldsymbol{u}nderset{\boldsymbol{b}oldsymbol{\eta}(t)\in\mathcal{U}}{\text{max}} H({\boldsymbol{b}oldsymbol{x}}(t),{\boldsymbol{b}oldsymbol{p}}(t),{p}^0,\boldsymbol{b}oldsymbol{\eta}(t)) ,
\label{EQ:maximum_condition}
\end{eqnarray}
where
\boldsymbol{b}egin{eqnarray}
H(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p},p^0,\boldsymbol{b}oldsymbol{u}) = \boldsymbol{b}oldsymbol{p}\left[\boldsymbol{b}oldsymbol{f}_0(\boldsymbol{b}oldsymbol{x}) + \boldsymbol{b}oldsymbol{r}ho \boldsymbol{b}oldsymbol{f}_1(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{\omega})\boldsymbol{b}oldsymbol{r}ight] + p^0 \boldsymbol{b}oldsymbol{r}ho,
\label{EQ:Hamiltonian}
\end{eqnarray}
is the Hamiltonian. Moreover, the transversality condition asserts
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{p}(t_f) = \boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u} d \boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{x}(t_f)),\label{EQ:Transversality_1}
\end{eqnarray}
where $\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}\in(\mathbb{R}^l)^*$ is a constant vector whose elements are Lagrangian multipliers.
The 4-tuple $t\mapsto(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t),p^0,\boldsymbol{b}oldsymbol{u}(t))$ on $[0,t_f]$ is called an extremal. Furthermore, an extremal is called a normal one if $p^0\boldsymbol{b}oldsymbol{n}eq 0$ and it is called an abnormal one if $p^0 = 0$. The abnormal extremals have been ruled out by Caillau {\it et al.} in Ref. \boldsymbol{b}oldsymbol{c}ite{Caillau:12}. Thus, in this paper only normal extremals are considered and $(\boldsymbol{b}oldsymbol{p},p^0)$ is normalized such that $p^0 = -1$. According to the maximum condition in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:maximum_condition}), for every extremal $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot),p^0,\boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$, the corresponding extremal control $\boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{c}dot)$ is a function of $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$, i.e., $\boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{c}dot) = \boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$. Thus, in the remainder of this paper, with some abuses of notations, we denote by $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))\in T^*\mathcal{X}$ and $\boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))\in\mathcal{U}$ on $[0,t_f]$ the normal extremal and the corresponding extremal control, respectively. And, we denote by ${H}(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$ the maximized Hamiltonian of the extremal $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$, which is written as
\boldsymbol{b}egin{eqnarray}
{H}(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}) := {H}_0(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}) + \boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}) {H}_1(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}),\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where ${H}_0 (\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}) = \boldsymbol{b}oldsymbol{p}\boldsymbol{b}oldsymbol{f}_0(\boldsymbol{b}oldsymbol{x})$ and ${H}_1(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}) = \boldsymbol{b}oldsymbol{p}\boldsymbol{b}oldsymbol{f}_1(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p})) - 1$.
Let us define by $\boldsymbol{b}oldsymbol{p}_r\in T_{\boldsymbol{b}oldsymbol{r}}\mathbb{R}^3$, $\boldsymbol{b}oldsymbol{p}_v\in T_{\boldsymbol{b}oldsymbol{v}}\mathbb{R}^3$, and $p_m\in T_{m}\mathbb{R}_+$ in such a way that $\boldsymbol{b}oldsymbol{p}=(\boldsymbol{b}oldsymbol{p}_r,\boldsymbol{b}oldsymbol{p}_v,p_m)$, the maximum condition in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:maximum_condition}) implies
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{\omega}= \boldsymbol{b}oldsymbol{p}_v / \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel ,\ \text{if}\ \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v\boldsymbol{b}oldsymbol{p}arallel\boldsymbol{b}oldsymbol{n}eq 0,
\label{EQ:Max_condition2}
\end{eqnarray}
and
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}egin{cases}\boldsymbol{b}oldsymbol{r}ho =1,\ \ \ \ \ \text{if}\ H_1 > 0,
\\
\boldsymbol{b}oldsymbol{r}ho = 0, \ \ \ \ \ \text{if}\ H_1 < 0.\\
\end{cases}
\label{EQ:Max_condition1}
\end{eqnarray}
Thus, the optimal direction of the thrust vector $\boldsymbol{b}oldsymbol{\tau}$ is collinear to $\boldsymbol{b}oldsymbol{p}_v$ that is well-known as the primer vector of Lawden~\boldsymbol{b}oldsymbol{c}ite{Lawden:63}.
If the switching function $H_1$ has only isolated zeros along an extremal $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$, this extremal is called a bang-bang one.
\boldsymbol{b}egin{definition}
Along a bang-bang extremal $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$, an arc on a finite interval $[t_1,t_2]\boldsymbol{b}oldsymbol{s}ubset [0,t_f]$ with $t_1 < t_2$ is called a maximum-thrust (or burn) arc if $\boldsymbol{b}oldsymbol{r}ho = 1$, otherwise it is called a zero-thrust (or coast) arc.
\end{definition}
\boldsymbol{b}oldsymbol{s}ubsection{Singular solutions and chattering arcs}
An extremal $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$ is said to be a singular one if $H_1(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot)) \equiv 0$ for a finite interval $[t_1,t_2]\boldsymbol{b}oldsymbol{s}ubseteq[0,t_f]$ with $t_1 < t_2$. Note that the maximum condition in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:maximum_condition}) is trivially satisfied for every $\boldsymbol{b}oldsymbol{r}ho \in[0,1]$ if $H_1 \equiv 0$. One can compute the optimal value of $\boldsymbol{b}oldsymbol{r}ho$ on singular arcs by repeatedly differentiating the identity $H_1 \equiv 0$ until $\boldsymbol{b}oldsymbol{r}ho$ explicitly appears. It is known from Ref.~\boldsymbol{b}oldsymbol{c}ite{Kelley:66} that $\boldsymbol{b}oldsymbol{r}ho$ explicitly appears in the differentiation ${d^q H_1}/{d t^q}$ if and only if $q$ is an even integer, and the order of the singular arc is then designated as $q/2$.
\boldsymbol{b}egin{proposition}
Given a singular extremal $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot))$ on $[t_1,t_2]\boldsymbol{b}oldsymbol{s}ubseteq[0,t_f]$ with $t_1 < t_2$, assume $\|\boldsymbol{b}oldsymbol{p}_v (\boldsymbol{b}oldsymbol{c}dot)\| \boldsymbol{b}oldsymbol{n}eq 0$ on $[t_1,t_2]$, we have that the order of the singular extremal is at least two.
\label{PR:singular_order}
\end{proposition}
\boldsymbol{b}egin{proof}
Since $H_1 \equiv 0$ along a singular arc, differentiating $H_1$ with respect to time and using Poisson bracket, one obtains
\boldsymbol{b}egin{eqnarray}
0 = H_{01}:=\boldsymbol{b}ig\{H_0,H_1\boldsymbol{b}ig\} = -\tau_{max}\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}_v^T [ \boldsymbol{b}oldsymbol{p}_r + d{ \boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v})} \boldsymbol{b}oldsymbol{p}_v]}{m \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel},
\label{EQ:H01}
\end{eqnarray}
where the notation ``~$\{\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{c}dot\}$~" denotes the Poisson bracket. Using Leibniz rule, Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:H01}) implies
\boldsymbol{b}egin{eqnarray}
H_{101} &:=& \boldsymbol{b}ig\{H_1,H_{01}\boldsymbol{b}ig\} = 0,\boldsymbol{b}oldsymbol{n}onumber\\
H_{1001} &:=& \boldsymbol{b}ig\{H_1,\boldsymbol{b}ig\{H_0,H_{01}\boldsymbol{b}ig\}\boldsymbol{b}ig\}\boldsymbol{b}oldsymbol{n}onumber\\
&=& \boldsymbol{b}ig\{-H_{01},H_{01}\boldsymbol{b}ig\} + \boldsymbol{b}ig\{H_0,H_{101}\boldsymbol{b}ig\} = 0.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Then, the equality, $0 = H_{001} + \boldsymbol{b}oldsymbol{r}ho H_{101}$,
implies $H_{001} = 0$, whose implicit equation is
\boldsymbol{b}egin{eqnarray}
H_{001} = \tau_{max} \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}_v^T d\boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}) \boldsymbol{b}oldsymbol{p}_v + [\boldsymbol{b}oldsymbol{p}_r + 2 d\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v})\boldsymbol{b}oldsymbol{p}_v]^T[\boldsymbol{b}oldsymbol{p}_r + d\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v})\boldsymbol{b}oldsymbol{p}_v ] }{m \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel}.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
A direct calculation on this equation yields
\boldsymbol{b}egin{eqnarray}
H_{0001} &:=& \boldsymbol{b}ig\{H_0,H_{001}\boldsymbol{b}ig\}\boldsymbol{b}oldsymbol{n}onumber\\
&=&\boldsymbol{b}oldsymbol{f}rac{\tau_{max}}{m\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel} \boldsymbol{b}oldsymbol{B}ig\{ {{ \boldsymbol{b}ig[ \boldsymbol{b}oldsymbol{p}_v^T d^2\boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}) \boldsymbol{b}oldsymbol{p}_v\boldsymbol{b}ig] }}\boldsymbol{b}oldsymbol{v} - \boldsymbol{b}oldsymbol{p}_v^T d\boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}) [2\boldsymbol{b}oldsymbol{p}_r + 3 d\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v}) \boldsymbol{b}oldsymbol{p}_v]\boldsymbol{b}oldsymbol{n}onumber\\
&-& [2 d\boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}) \boldsymbol{b}oldsymbol{p}_v + 3 d\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{r})\boldsymbol{b}oldsymbol{p}_r + 4(d\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v}))^2\boldsymbol{b}oldsymbol{p}_v ]^T[\boldsymbol{b}oldsymbol{p}_r + d\boldsymbol{b}oldsymbol{h}(\boldsymbol{b}oldsymbol{v})\boldsymbol{b}oldsymbol{p}_v ]\boldsymbol{b}oldsymbol{B}ig\}.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Eventually, one has $0 = \dot{H}_{0001} = H_{00001} + \boldsymbol{b}oldsymbol{r}ho H_{10001}$. Let
$\alpha_i$ ($i=1,2$) be defined by
$$\boldsymbol{b}oldsymbol{c}os(\alpha_i) = \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}_v^T (\boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{r}_i)}{\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{r}-\boldsymbol{b}oldsymbol{r}_i \boldsymbol{b}oldsymbol{p}arallel},$$
the explicit expression of $H_{10001}:=\{H_1,H_{0001}\}$, therefore, is
\boldsymbol{b}egin{eqnarray}
H_{10001} &=&\tau_{max} \boldsymbol{b}oldsymbol{f}rac{ \boldsymbol{b}ig[ \boldsymbol{b}oldsymbol{p}_v^T d^2\boldsymbol{b}oldsymbol{g}(\boldsymbol{b}oldsymbol{r}) \boldsymbol{b}oldsymbol{p}_v\boldsymbol{b}ig] \boldsymbol{b}oldsymbol{p}_v}{m^2 \boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel^2}\boldsymbol{b}oldsymbol{n}onumber\\
& =& 3\tau_{max} \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel}{m^2} \left[\mu \boldsymbol{b}oldsymbol{c}os \alpha_2\boldsymbol{b}oldsymbol{f}rac{3 - 5\boldsymbol{b}oldsymbol{c}os^2 \alpha_2}{\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{r}_2 \boldsymbol{b}oldsymbol{p}arallel^4} + (1-\mu)\boldsymbol{b}oldsymbol{c}os \alpha_1 \boldsymbol{b}oldsymbol{f}rac{3 - 5 \boldsymbol{b}oldsymbol{c}os^2 \alpha_1}{\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{r} - \boldsymbol{b}oldsymbol{r}_1\boldsymbol{b}oldsymbol{p}arallel^4}\boldsymbol{b}oldsymbol{r}ight].\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Note that the term $H_{10001}$ does not vanish identically on a singular extremal. Thus, the singular extremal is of order two according to Kelley's definition in Ref.~\boldsymbol{b}oldsymbol{c}ite{Kelley:66}, which proves the proposition.
\end{proof}
This proposition for the 3D case expands the work in Ref.~\boldsymbol{b}oldsymbol{c}ite{Zelikin:03} where the motion of the spacecraft is restricted into a 2D plane and the work in Ref.~\boldsymbol{b}oldsymbol{c}ite{Robbins:65} where model of two-body problem ($\mu = 0$) is considered.
Note that Kelley's second-order necessary condition \boldsymbol{b}oldsymbol{c}ite{Kelley:66} in terms of $\boldsymbol{b}oldsymbol{r}ho$ on singular arcs is $H_{10001} \leq 0$.
Let us define the singular submanifold $\mathcal{S}$ as
\boldsymbol{b}egin{eqnarray}
\mathcal{S} = \boldsymbol{b}ig\{(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p})\in T^*\mathcal{X}\ \arrowvert\ H_1=H_{01}=H_{001} = H_{0001} = 0,\ H_{10001} \leq 0\boldsymbol{b}ig\},\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
we then obtain the following result.
\boldsymbol{b}egin{corollary}[Fuller phenomenon, Zelikin and Borisov \boldsymbol{b}oldsymbol{c}ite{Zelikin:94}]
Let $\text{int}(\mathcal{S})$ be the interior of $\mathcal{S}$. Then, given every point $(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p})\in\text{int}(\mathcal{S})$, there exists a one parameter family of chattering solutions of Eqs.~(\boldsymbol{b}oldsymbol{r}ef{EQ:cannonical}--\boldsymbol{b}oldsymbol{r}ef{EQ:Hamiltonian}) passing through the point $(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p})$ and another one parameter family of chattering solutions of Eqs.~(\boldsymbol{b}oldsymbol{r}ef{EQ:cannonical}--\boldsymbol{b}oldsymbol{r}ef{EQ:Hamiltonian}) coming out from the point $(\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p})$.
\label{CO:fuller_phenomonon}
\end{corollary}
\boldsymbol{b}oldsymbol{n}oindent Though the efficient computation of chattering solutions is an open problem, see Ref.~\boldsymbol{b}oldsymbol{c}ite{Ghezzi:15}, Corollary \boldsymbol{b}oldsymbol{r}ef{CO:fuller_phenomonon} shows an insight into the control structure of the $L^1$-minimization trajectory, i.e., there exists a chattering arc when concatenating a singular arc with a nonsingular arc. The chattering arcs may not be found by direct numerical methods when concatenating singular arcs with nonsingular arcs~\boldsymbol{b}oldsymbol{c}ite{Park:13}.
\boldsymbol{b}oldsymbol{s}ection{Sufficient optimality conditions for bang-bang extremals}\label{SE:Sufficient}
Before studying the sufficient conditions for local optimality, we firstly give the definition of local optimality.
\boldsymbol{b}egin{definition}[Local Optimality \boldsymbol{b}oldsymbol{c}ite{Poggiolini:04,Agrachev:02}]
Given a fixed final time $t_f > 0$, an extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)\in \mathcal{X}$ associated with the extremal control $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{u}}(\boldsymbol{b}oldsymbol{c}dot)=(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{r}ho}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\omega}}(\boldsymbol{b}oldsymbol{c}dot))$ in $\mathcal{U}$ on $[0,t_f]$ is said to realize a weak-local optimality in $L^{\infty}$-topology (resp. a strong-local optimality in $C^0$-topology) if there exists an open neighborhood $\mathcal{W}_{\boldsymbol{b}oldsymbol{u}}\boldsymbol{b}oldsymbol{s}ubseteq \mathcal{U}$ of $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{u}}(\boldsymbol{b}oldsymbol{c}dot)$ in $L^{\infty}$-topology (resp. an open neighborhood $\mathcal{W}_{\boldsymbol{b}oldsymbol{x}}\boldsymbol{b}oldsymbol{s}ubseteq \mathcal{X}$ of $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ in $C^{0}$-topology) such that for every admissible controlled trajectory $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot)\boldsymbol{b}oldsymbol{n}ot\equiv\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ in $\mathcal{X}$ associated with the measurable control $\boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{c}dot)=(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot))$ in $\mathcal{W}_{\boldsymbol{b}oldsymbol{u}}$ on $[0,t_f]$ (resp. for every admissible controlled trajectory $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot)\boldsymbol{b}oldsymbol{n}ot\equiv\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ in $\mathcal{W}_{\boldsymbol{b}oldsymbol{x}}$ associated with the measurable control $\boldsymbol{b}oldsymbol{u}(\boldsymbol{b}oldsymbol{c}dot)=(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot)) $ in $\mathcal{U}$ on $[0,t_f]$) with the boundary conditions $\boldsymbol{b}oldsymbol{x}(0) = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(0)$ and $\boldsymbol{b}oldsymbol{x}(t_f)\in\mathcal{M}$, there holds
$$\int_{0}^{t_f}\boldsymbol{b}oldsymbol{r}ho(t) dt \boldsymbol{b}oldsymbol{g}eq \int_{0}^{t_f}\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{r}ho}(t) dt.$$
We say it realizes a strict weak-local (resp. strong-local) optimality if the strict inequality holds.
\label{DE:optimality}
\end{definition}
\boldsymbol{b}oldsymbol{n}oindent Note that if a trajectory $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{X}$ on $[0,t_f]$ realizes a strong-local optimality, it automatically realizes a weak-local optimality. This section is concerned with establishing the sufficient conditions for the strong-local optimality.
\boldsymbol{b}oldsymbol{s}ubsection{Parameterized family of extremals}
In this subsection, a family of extremals is constructed to be parameterized by $\boldsymbol{b}oldsymbol{p}(0)\in T_{\boldsymbol{b}oldsymbol{x}_0}^*\mathcal{X}$ such that the Poincar\'e-Cartan form $\boldsymbol{b}oldsymbol{p} d\boldsymbol{b}oldsymbol{x} - Hdt$ is exact on this family, which will be used to establish the sufficient optimality conditions later.
Let $\boldsymbol{b}oldsymbol{p}_0 = \boldsymbol{b}oldsymbol{p}(0)$, we define by
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{g}amma:[0,t_f]\times T^*_{\boldsymbol{b}oldsymbol{x}_0}\mathcal{X} \boldsymbol{b}oldsymbol{r}ightarrow T^*\mathcal{X},\ \boldsymbol{b}oldsymbol{g}amma(t,\boldsymbol{b}oldsymbol{p}_0) = (\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t)),\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
the solution trajectory of Eqs.~(\boldsymbol{b}oldsymbol{r}ef{EQ:cannonical}--\boldsymbol{b}oldsymbol{r}ef{EQ:Hamiltonian}) such that $(\boldsymbol{b}oldsymbol{x}_0,\boldsymbol{b}oldsymbol{p}_0) = \boldsymbol{b}oldsymbol{g}amma(0,\boldsymbol{b}oldsymbol{p}_0)$. For every $\boldsymbol{b}oldsymbol{p}_0\in T^*_{\boldsymbol{b}oldsymbol{x}_0}\mathcal{X}$, we say $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$ is an extremal. Note that at this moment we do not restrict any conditions on the final point of the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$ for every $\boldsymbol{b}oldsymbol{p}_0\in T^*\mathcal{X}$.
\boldsymbol{b}egin{definition}
We define $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0\in T^*_{\boldsymbol{b}oldsymbol{x}_0}\mathcal{X}$ in such a way that the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ at $t_f$ satisfies the final condition in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:final_manifold}) and transversality condition in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Transversality_1}).
\end{definition}
\boldsymbol{b}egin{definition}[Parameterized family of extremals]
Given the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, let $\mathcal{P}\boldsymbol{b}oldsymbol{s}ubset T_{\boldsymbol{b}oldsymbol{x}_0}^*\mathcal{X}$ be an open neighbourhood of $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0$, we say the subset
\boldsymbol{b}egin{eqnarray}
\mathcal{F} = \boldsymbol{b}ig\{(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t))\in T^*\mathcal{X}\ \arrowvert\ (\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t))=\boldsymbol{b}oldsymbol{g}amma(t,\boldsymbol{b}oldsymbol{p}_0),\ t\in[0,t_f],\ \boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}\boldsymbol{b}ig\},\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
is a $\boldsymbol{b}oldsymbol{p}_0$-parameterized family of extremals around the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$.
\end{definition}
\boldsymbol{b}oldsymbol{n}oindent Note that the open neighborhood $\mathcal{P}$ of $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0$ in this paper can be shrunk whenever necessary.
Let
\boldsymbol{b}egin{eqnarray}
\Pi: T^*\mathcal{X} \boldsymbol{b}oldsymbol{r}ightarrow \mathcal{X},\ \ (\boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{p}) \mapsto \boldsymbol{b}oldsymbol{x},\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
be the mapping that mapps a submanifold from the cotangent space $T^*\mathcal{X}$ onto the state space $\mathcal{X}$, we say the mapping $\Pi$ is a canonical projection.
An extremal ceases to be locally optimal if a focal point (or called a conjugate point if $l=n$ since in this case the endpoints are fixed) occurs~\boldsymbol{b}oldsymbol{c}ite{Bonnard:07}. According to Agrachev's approach in Ref.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04}, a focal point occurs on the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ at a time $t_c\in(0,t_f]$ if the projection of the family $\mathcal{F}$ loses its local diffeomorphism at $t_c$. We say the projection of the family $\mathcal{F}$ at $t_c\in(0,t_f]$ is a fold singularity if it loses its local diffeomrophism at $t_c$.
Thus, focal points are related to the fold singularities of the projection of the family $\mathcal{F}$.
\boldsymbol{b}oldsymbol{s}ubsection{Sufficient conditions for the case of $l=n$}\label{Subse:sufficient1}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, without loss of generality, let the positive integer $k\in\mathbb{N}$ be the number of switching times $t_i$ ($i =1,2,\boldsymbol{b}oldsymbol{c}dots,k$) such that $0 < t_1 < t_2 < \boldsymbol{b}oldsymbol{c}dots < t_k < t_f$.
\boldsymbol{b}egin{assumption}
Along the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, each switching point at the switching time ${t}_i\in (0,t_f)$ is assumed to be a regular one, i.e., $H_1(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}({t}_i),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}({t}_i)) = 0$ and $H_{01}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}({t}_i),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}({t}_i))\boldsymbol{b}oldsymbol{n}eq 0$ for $i=1,2,\boldsymbol{b}oldsymbol{c}dots,k$.
\label{AS:Regular_Switching}
\end{assumption}
\boldsymbol{b}oldsymbol{n}oindent As a result of this assumption, if the subset $\mathcal{P}$ is small enough, the number of switching times on each extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)\in\mathcal{F}$ on $[0,t_f]$ keeps as $k$ and the $i$-th switching time of the extremals $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)\in\mathcal{F}$ on $[0,t_f]$ is a smooth function of $\boldsymbol{b}oldsymbol{p}_0$. Thus, we define by
\boldsymbol{b}egin{eqnarray}
t_i: \mathcal{P}\boldsymbol{b}oldsymbol{r}ightarrow \mathbb{R}_+,\ \boldsymbol{b}oldsymbol{p}_0 \mapsto t_i(\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
the $i$-th switching time of the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)\in\mathcal{F}$ on $[0,t_f]$.
Let
\boldsymbol{b}egin{eqnarray}
\mathcal{F}_i &=& \boldsymbol{b}ig\{(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t))\in T^*\mathcal{X}\ \arrowvert\ \boldsymbol{b}oldsymbol{n}onumber\\
&& (\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t)) = \boldsymbol{b}oldsymbol{g}amma(t,\boldsymbol{b}oldsymbol{p}_0),\ t \in (t_{i-1}(\boldsymbol{b}oldsymbol{p}_0),t_i(\boldsymbol{b}oldsymbol{p}_0)],\ \boldsymbol{b}oldsymbol{p}_0 \in \mathcal{P}\boldsymbol{b}ig\}, \boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
for $\ i = 1,\ 2,\ \boldsymbol{b}oldsymbol{c}dots,\ k,\ k+1$ with $t_0 = 0$ and $t_{k+1} = t_f$. If the subset $\mathcal{P}$ is small enough, there holds
\boldsymbol{b}egin{eqnarray}
\mathcal{F} = \mathcal{F}_1 \boldsymbol{b}oldsymbol{c}up \mathcal{F}_2\boldsymbol{b}oldsymbol{c}up \boldsymbol{b}oldsymbol{c}dots \boldsymbol{b}oldsymbol{c}up \mathcal{F}_k\boldsymbol{b}oldsymbol{c}up \mathcal{F}_{k+1}.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Let $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0))=\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$ be the extremals in $\mathcal{F}$. In order to avoid heavy notations, denote by $\delta(\boldsymbol{b}oldsymbol{c}dot)$ the determinant of the matrix $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, i.e.,
$$\delta(\boldsymbol{b}oldsymbol{c}dot) = \det\left[\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)\boldsymbol{b}oldsymbol{r}ight],$$
on $[0,t_f]$. Note that the projection of the subset $\mathcal{F}_i$ at a time $t_c\in(t_i,t_{i+1})$ is a fold singularity if $\delta(t_c) = 0$, as is shown by the typical picture for the occurrence of a conjugate point in Fig.~\boldsymbol{b}oldsymbol{r}ef{Fig:smooth_fold}. If $\delta(\boldsymbol{b}oldsymbol{c}dot)\boldsymbol{b}oldsymbol{n}eq 0$ on $(t_i,t_{i+1})$, the projection of the subset $\mathcal{F}_i$ restricted to the domain $(t_i,t_{i+1})\times\mathcal{P}$ is a diffeomorphism, see Refs.~\boldsymbol{b}oldsymbol{c}ite{Schattler:12,Agrachev:04}.
Let us define the following condition.
\boldsymbol{b}egin{condition}
$\delta (\boldsymbol{b}oldsymbol{c}dot) \boldsymbol{b}oldsymbol{n}eq 0$ on the open subintervals $({t}_i,{t}_{i+1})$ for $i=0,1,\boldsymbol{b}oldsymbol{c}dots,k-1$ as well as on the semi-open subinterval $(t_{k},t_f]$.
\label{AS:Disconjugacy_bang}
\end{condition}
\boldsymbol{b}egin{figure}[!ht]
\includegraphics[trim=2.0cm 2.0cm 3.0cm 1.0cm, clip=true, width=3.0in, angle=0]{smooth_fold.eps}
\boldsymbol{b}oldsymbol{c}aption[]{A typical picture for a fold singularity of the projection of $\mathcal{F}$ onto the state space $\mathcal{X}$~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04}.}
\label{Fig:smooth_fold}
\end{figure}
\boldsymbol{b}oldsymbol{n}oindent Though this condition guarantees that both the restriction of $\Pi(\mathcal{F}_i)$ on $(t_{i-1},t_{i})\times\mathcal{P}$ for $i=1,2,\boldsymbol{b}oldsymbol{c}dots,k$ and the restriction of $\Pi(\mathcal{F}_{k+1})$ on $(t_k,t_f]\times\mathcal{P}$ are local diffeomorphisms, it is not sufficient to guarantee that the projection of the family $\mathcal{F}$ restricted to the whole domain $(0,t_f]\times\mathcal{P}$ is a diffeomorphism as well, as Fig.~\boldsymbol{b}oldsymbol{r}ef{Fig:trans} shows that the flows $\boldsymbol{b}oldsymbol{x}(t,\boldsymbol{b}oldsymbol{p}_0)$ may intersect with each other near a switching time $t_i(\boldsymbol{b}oldsymbol{p}_0)$.
\boldsymbol{b}egin{remark}
The behavior that the projection of $\mathcal{F}$ at a switching time $t_i$ is a fold singularity can be excluded by a transversal condition established by Noble and Sch\"attler in Ref.~\boldsymbol{b}oldsymbol{c}ite{Noble:02}. This transversal condition is reduced as $\delta(t_i-)\delta(t_i+)>0$ by Chen {\it et al.} in Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15}.
\end{remark}
\boldsymbol{b}egin{figure}[!ht]
\includegraphics[trim=2.0cm 1.5cm 1.0cm 1.0cm, clip=true, width=3.5in, angle=0]{broken_fold.eps}
\boldsymbol{b}oldsymbol{c}aption[]{The left plot denotes a diffeomorphism for the projection of $\mathcal{F}$ around the switching time $t_i(\boldsymbol{b}oldsymbol{p}_0)$, and the right plot denotes a fold singularity for the projection around the switching time $t_i(\boldsymbol{b}oldsymbol{p}_0)$ \boldsymbol{b}oldsymbol{c}ite{Noble:02,Schattler:12}.}
\label{Fig:trans}
\end{figure}
\boldsymbol{b}egin{condition}
$\delta (t_i-) \delta (t_{i}+) > 0$ for each switching time $t_i$ for $i=1,2,\boldsymbol{b}oldsymbol{c}dots,k$.
\label{AS:Transversality}
\end{condition}
\boldsymbol{b}oldsymbol{n}oindent If this condition is satisfied, the projection of the family $\mathcal{F}$ around each switching time $t_i(\boldsymbol{b}oldsymbol{p}_0)$ is a diffeomorphism at least for a sufficiently small subset $\mathcal{P}$, see Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15}.
\boldsymbol{b}egin{remark}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that every switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}) and Conditions \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are satisfied, if the subset $\mathcal{P}$ is small enough, every extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$ for $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$ does not contain conjugate points. Then, for every $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$, we are able to
construct a perturbed Lagrangian submanifold $\mathcal{L}_{\boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{s}ubset T^*\mathcal{X}$ (cf. Theorem 21.3 in Ref.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04} or Appendix A in Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15}) around the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$ such that
\boldsymbol{b}egin{description}
\item $1)$ the projection of the Lagrangian submanifold $\mathcal{L}_{\boldsymbol{b}oldsymbol{p}_0}$ onto its image is a diffeomorphism; and
\item $2)$ the domain $\Pi(\mathcal{L}_{\boldsymbol{b}oldsymbol{p}_0})$ is a tubular neighborhood of the extremal trajectory ${\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)=\Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0))$ on $[0,t_f]$.
\end{description}
\label{RE:lagrangian}
\end{remark}
\boldsymbol{b}oldsymbol{n}oindent As a result of this remark, one obtains the following remark.
\boldsymbol{b}egin{remark}
If the subset $\mathcal{P}$ is small enough, let
\boldsymbol{b}egin{eqnarray}
\mathcal{L} = \boldsymbol{b}oldsymbol{u}nderset{\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}}{\boldsymbol{b}igcap} \mathcal{L}_{\boldsymbol{b}oldsymbol{p}_0},
\label{EQ:neighborhood}
\end{eqnarray}
it follows that
\boldsymbol{b}egin{description}
\item $1)$ the projection of $\mathcal{L}$ onto its image is a diffeomorphism;
\item $2)$ the projection of $\mathcal{L}$ is a tubular neighborhood of the extremal trajectory $\Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0))$ on $[0,t_f]$; and
\item $3)$ there holds $\Pi(\mathcal{F})\boldsymbol{b}oldsymbol{s}ubset \Pi(\mathcal{L})$ at every time $t\in[0,t_f]$.
\end{description}
\label{RE:neighborhood}
\end{remark}
\boldsymbol{b}oldsymbol{n}oindent Then, directly applying the theory of field of extremals (cf. Theorem 17.1 in Ref.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04}), one obtains the following result.
\boldsymbol{b}egin{theorem}[Agrachev and Sachkov~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04}]
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that every switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}), let $(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0))\in\mathcal{U}$ be the optimal control function associated with the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)\in\mathcal{F}$ on $[0,t_f]$. Then, if {\it Conditions \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}} are satisfied and if the subset $\mathcal{P}$ is small enough, every extremal trajectory ${\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)=\Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0))$ on $[0,t_f]$ for $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$ realizes a strict minimum cost among every admissible controlled trajectory $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)\in\Pi(\mathcal{L})$ associated with the measurable control $(\boldsymbol{b}oldsymbol{r}ho_*(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}_*(\boldsymbol{b}oldsymbol{c}dot))\in\mathcal{U}$ on $[0,t_f]$ with the same endpoints $\boldsymbol{b}oldsymbol{x}(0,\boldsymbol{b}oldsymbol{p}_0)=\boldsymbol{b}oldsymbol{x}_*(0)$ and $\boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0) = \boldsymbol{b}oldsymbol{x}_*(t_f)$, i.e.,
\boldsymbol{b}egin{eqnarray}
\int_0^{t_f}{\boldsymbol{b}oldsymbol{r}ho}(t,\boldsymbol{b}oldsymbol{p}_0) dt \leq \int^{t_f}_0 \boldsymbol{b}oldsymbol{r}ho_*(t)dt,\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where the equality holds if and only if $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)\equiv \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$.
\label{CO:cor1}
\end{theorem}
\boldsymbol{b}egin{proof}
According to Theorem $17.1$ in Ref.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04}, under the hypotheses of this theorem, every extremal trajectory $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$ for $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$ realizes a strict minimum cost among every admissible controlled trajectory $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)\in\Pi(\mathcal{L}_{\boldsymbol{b}oldsymbol{p}_0})$ on $[0,t_f]$ with the same endpoints. Notice from Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:neighborhood}) that $\Pi(\mathcal{L})\boldsymbol{b}oldsymbol{s}ubseteq\Pi(\mathcal{L}_{\boldsymbol{b}oldsymbol{p}_0})$ at each time $t\in[0,t_f]$ for every $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$, one proves this theorem.
\end{proof}
\boldsymbol{b}oldsymbol{n}oindent Note that the endpoints of the $L^1$-minimization problem are fixed if $l=n$.
\boldsymbol{b}egin{remark}
As a combination of Remark \boldsymbol{b}oldsymbol{r}ef{RE:neighborhood} and Theorem \boldsymbol{b}oldsymbol{r}ef{CO:cor1}, one obtains that {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are sufficient to guarantee the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$ is a strict strong-local optimum (cf. Definition \boldsymbol{b}oldsymbol{r}ef{DE:optimality}) if $l=n$.
\end{remark}
Under Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}, the projection of the family $\mathcal{F}$ near the switching time $t_i(\boldsymbol{b}oldsymbol{p}_0)$ is a fold singularity if the strict inequality
$\delta (t_i-)\delta (t_{i}+) < 0$
is satisfied~\boldsymbol{b}oldsymbol{c}ite{Caillau:15}.
\boldsymbol{b}egin{remark}
Given the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that each switching point is regular (cf. Assumption~\boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}), conjugate points can occur not only on each smooth bang arc at a time $t_c\in (t_{i-1},t_i)$ if $\delta (t_c) = 0$ but also at each switching time $t_i$ if $\delta (t_i-) \delta (t_{i}+) < 0$.
\end{remark}
\boldsymbol{b}oldsymbol{n}oindent The fact that conjugate points can occur at switching times generalizes the conjugate point theory developed by the classical variational methods for totally smooth extremals, see Refs.~\boldsymbol{b}oldsymbol{c}ite{Bryson:69,Breakwell:65,Mermau:76,Wood:74}.
\boldsymbol{b}oldsymbol{s}ubsection{Sufficient conditions for the case of $l<n$}\label{Subse:sufficient2}
In this subsection, we establish the sufficient optimality conditions for the case that the dimension of the final constraint submanifold $\mathcal{M}$ is not zero.
\boldsymbol{b}egin{remark}
If $l<n$, to ensure the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$ is a strict strong-local optimum, in addition to Conditions \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}, a further second-order condition (cf. Refs.~\boldsymbol{b}oldsymbol{c}ite{Wood:74,Brusch:70}) is required to guarantee that every admissible controlled trajectory $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)\in\Pi(\mathcal{L})$ on $[0,t_f]$, not only with the same endpoints $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(0)=\boldsymbol{b}oldsymbol{x}_*(0)$ and $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f) = \boldsymbol{b}oldsymbol{x}_*(t_f)$ but also with the boundary conditions $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(0)=\boldsymbol{b}oldsymbol{x}_*(0)$ and $\boldsymbol{b}oldsymbol{x}_*(t_f)\in\mathcal{M}\boldsymbol{b}ackslash\{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)\}$, has a bigger cost than the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$.
\end{remark}
\boldsymbol{b}oldsymbol{n}oindent Let $\mathcal{N}\boldsymbol{b}oldsymbol{s}ubset\mathcal{X}$ be the restriction of $\Pi(\mathcal{F})$ on $\{t_f\}\times\mathcal{P}$, i.e.,
$$\mathcal{N} = \boldsymbol{b}ig\{\boldsymbol{b}oldsymbol{x}\in\mathcal{X}\ \arrowvert\ \boldsymbol{b}oldsymbol{x} = \Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0)),\ \boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}\boldsymbol{b}ig\}.$$
Note that the mapping $\boldsymbol{b}oldsymbol{p}_0\mapsto \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0)$ on the sufficiently small subset $\mathcal{P}$ is a diffeomorphism if $\delta (t_f)\boldsymbol{b}oldsymbol{n}eq 0$, which indicates that the subset $\mathcal{N}$ is an open neighborhood of $\boldsymbol{b}ar{x}({t}_f)$ if Condition \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} is satsfied. Thus, in the case of $l<n$, the subset $\mathcal{M}\boldsymbol{b}oldsymbol{c}ap \mathcal{N}\boldsymbol{b}ackslash\{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f\}$ is not empty if $\delta(t_f)\boldsymbol{b}oldsymbol{n}eq 0$, see the sketch for a 2-dimensional state space in Fig~\boldsymbol{b}oldsymbol{r}ef{Fig:terminal_transversality}.
\boldsymbol{b}egin{figure}[!ht]
\includegraphics[trim=1cm 1cm 1cm 1cm, clip=true,width=0.8\textwidth]{terminal_transversality.eps}
\boldsymbol{b}oldsymbol{c}aption[]{The relationship between $\mathcal{N}$ and $\mathcal{M}$.}
\label{Fig:terminal_transversality}
\end{figure}
\let\emptyset\boldsymbol{b}oldsymbol{v}arnothing
For every sufficiently small subset $\mathcal{P}$, let us define by $\mathcal{Q}\boldsymbol{b}oldsymbol{s}ubseteq \mathcal{P}$ a subset of all $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$ satisfying $\Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0))\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$, i.e.,
\boldsymbol{b}egin{eqnarray}
\mathcal{Q} = \boldsymbol{b}ig\{\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}\ \arrowvert\ \Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0))\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}\boldsymbol{b}ig\}.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
\boldsymbol{b}oldsymbol{n}oindent Note that for every $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{Q}$ there holds $\boldsymbol{b}oldsymbol{x}_0 =\Pi(\boldsymbol{b}oldsymbol{g}amma(0,\boldsymbol{b}oldsymbol{p}_0))$ and $\Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0))\in\mathcal{M}$.
\boldsymbol{b}egin{remark}
For every $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{Q}$, the extremal trajectory $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0) = \Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0))$ on $[0,t_f]$ is an admissible controlled trajectory of the $L^1$-minimization problem.
\label{RE:admissible_control_trajectory}
\end{remark}
\boldsymbol{b}egin{definition}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ and a small $\boldsymbol{b}oldsymbol{v}arepsilon > 0$, let $l< n$. Then, we define by $\boldsymbol{b}oldsymbol{y}: [-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}oldsymbol{r}ightarrow \mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N},\ \eta\mapsto \boldsymbol{b}oldsymbol{y}(\eta)$ a twice continuously differentiable curve on $\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ such that $\boldsymbol{b}oldsymbol{y}(0)=\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)$.
\label{DE:smooth_curve}
\end{definition}
\boldsymbol{b}egin{lemma}
Given the extremal $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that each switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}) and {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are satisfied, let $l<n$. Then, if the subset $\mathcal{P}$ is small enough, for every smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$, there exists a smooth path $\eta\mapsto \boldsymbol{b}oldsymbol{p}_0(\eta)$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ in $\mathcal{Q}$ such that $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot) = \Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{c}dot)))$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$.
\label{LE:smooth_path}
\end{lemma}
\boldsymbol{b}egin{proof}
Note that the mapping $\boldsymbol{b}oldsymbol{p}_0\mapsto \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0)$ restricted to the subset $\mathcal{Q}$ is a diffeomorphism under the hypotheses of the lemma. Then, according to the {\it inverse function theorem}, the lemma is proved.
\end{proof}
\boldsymbol{b}egin{definition}
Define a path ${\boldsymbol{b}oldsymbol{\lambda}}:[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}oldsymbol{r}ightarrow T^*_{\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)}\mathcal{X},\ \eta\mapsto \boldsymbol{b}oldsymbol{\lambda}(\eta)$ in such a way that $(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot),{\boldsymbol{b}oldsymbol{\lambda}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{c}dot))$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$. Then, for every $\boldsymbol{b}oldsymbol{x}i\in[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$, we define by $J:[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}oldsymbol{r}ightarrow \mathbb{R},\ \boldsymbol{b}oldsymbol{x}i\mapsto J(\boldsymbol{b}oldsymbol{x}i)$ the integrand of the Poincar\'e-Cartan form $\boldsymbol{b}oldsymbol{p} d\boldsymbol{b}oldsymbol{x} - H dt$ along the extremal lift $(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\lambda}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,\boldsymbol{b}oldsymbol{x}i]$, i.e.,
\boldsymbol{b}egin{eqnarray}
J(\boldsymbol{b}oldsymbol{x}i) = \int_0^{\boldsymbol{b}oldsymbol{x}i}\boldsymbol{b}oldsymbol{\lambda}(\eta)\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\eta) - H(\boldsymbol{b}oldsymbol{y}(\eta),\boldsymbol{b}oldsymbol{\lambda}(\eta))\boldsymbol{b}oldsymbol{f}rac{d t_f}{d\eta}d\eta,\ \boldsymbol{b}oldsymbol{x}i\in[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon].
\label{EQ:J_xi}
\end{eqnarray}
\end{definition}
\boldsymbol{b}egin{proposition}
In the case of $l<n$, given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that each switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}) and {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are satisfied, assume $\boldsymbol{b}oldsymbol{v}arepsilon > 0$ is small enough. Then, the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$ is a strict strong-local optimality (cf. Definition \boldsymbol{b}oldsymbol{r}ef{DE:optimality}) if and only if there holds
\boldsymbol{b}egin{eqnarray}
J(\boldsymbol{b}oldsymbol{x}i) > J(0),\ \boldsymbol{b}oldsymbol{x}i\in[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}ackslash\{0\},
\label{EQ:lemma1_compare}
\end{eqnarray}
for every smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$.
\label{CO:J_xi_J_0}
\end{proposition}
\boldsymbol{b}egin{proof}
Let us first prove that, under the hypotheses of this proposition, Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:lemma1_compare}) is a sufficient condition for the strict strong-local optimality of the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$. Denote by $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)$ in $\Pi(\mathcal{L})$ on $[0,t_f]$ be an admissible controlled trajectory with the boundary conditions $\boldsymbol{b}oldsymbol{x}_*(0) = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(0)$ and $\boldsymbol{b}oldsymbol{x}_*(t_f)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}\boldsymbol{b}ackslash\{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)\}$. Let $(\boldsymbol{b}oldsymbol{r}ho_*(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\omega}_*(\boldsymbol{b}oldsymbol{c}dot))\in\mathcal{U}$ and $(\boldsymbol{b}oldsymbol{r}ho(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{\omega}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0))\in\mathcal{U}$ on $[0,t_f]$ be the measurable control and the optimal control associated with $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)$ and $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0)$ on $[0,t_f]$, respectively. According to Definition \boldsymbol{b}oldsymbol{r}ef{DE:smooth_curve} and Lemma \boldsymbol{b}oldsymbol{r}ef{LE:smooth_path}, for every final point $\boldsymbol{b}oldsymbol{x}_*(t_f)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}\boldsymbol{b}ackslash\{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)\}$, there must exist a $\boldsymbol{b}oldsymbol{x}i\in[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}ackslash\{0\}$ and a smooth path $\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{Q}$ associated with the smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ such that $\boldsymbol{b}oldsymbol{y}(0) = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f) = \Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)))$ and $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i) = \boldsymbol{b}oldsymbol{x}_*(t_f)=\Pi(\boldsymbol{b}oldsymbol{g}amma(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)))$. Since the trajectory $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$ has the same endpoints with the extremal trajectory $\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))=\Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)))$ on $[0,t_f]$, according to Theorem \boldsymbol{b}oldsymbol{r}ef{CO:cor1}, one obtains
\boldsymbol{b}egin{eqnarray}
\int_0^{t_f} \boldsymbol{b}oldsymbol{r}ho_*(t)dt \boldsymbol{b}oldsymbol{g}eq \int_0^{t_f}\boldsymbol{b}oldsymbol{r}ho(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))dt,
\label{EQ:rho_*_rho_xi}
\end{eqnarray}
where the equality holds if and only if $\boldsymbol{b}oldsymbol{x}_*(\boldsymbol{b}oldsymbol{c}dot)\equiv {\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))$ on $[0,t_f]$.
Note that the four paths $(\boldsymbol{b}oldsymbol{x}_0,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,\boldsymbol{b}oldsymbol{x}i]$, $\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, $(\boldsymbol{b}oldsymbol{x}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)),\boldsymbol{b}oldsymbol{p}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))$ on $[0,t_f]$, and $(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}oldsymbol{\lambda}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,\boldsymbol{b}oldsymbol{x}i]$ constitute a closed curve on the family $\mathcal{F}$. Since the integrand of the Poincar\'e-Cartan form $\boldsymbol{b}oldsymbol{p} d\boldsymbol{b}oldsymbol{x} - H dt$ is closed on $\mathcal{F}$, see Refs.~\boldsymbol{b}oldsymbol{c}ite{Agrachev:04,Schattler:12,Caillau:15}, one obtains
\boldsymbol{b}egin{eqnarray}
&&J(\boldsymbol{b}oldsymbol{x}i) + \int_0^{t_f}\boldsymbol{b}ig[\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t)\dot{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}}(t) - H(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t))\boldsymbol{b}ig]dt\boldsymbol{b}oldsymbol{n}onumber\\
&=& \int_0^{t_f}\boldsymbol{b}ig[\boldsymbol{b}oldsymbol{p}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))\dot{\boldsymbol{b}oldsymbol{x}}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)) - H(\boldsymbol{b}oldsymbol{x}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)),\boldsymbol{b}oldsymbol{p}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)))\boldsymbol{b}ig]dt\boldsymbol{b}oldsymbol{n}onumber\\
&+& \int_0^{\boldsymbol{b}oldsymbol{x}i} \boldsymbol{b}oldsymbol{B}ig[\boldsymbol{b}oldsymbol{p}_0(\eta)\boldsymbol{b}oldsymbol{f}rac{d {\boldsymbol{b}oldsymbol{x}}_0}{d\eta} - H(\boldsymbol{b}oldsymbol{x}_0,\boldsymbol{b}oldsymbol{p}_0(\eta))\boldsymbol{b}oldsymbol{f}rac{d t_0}{d\eta}\boldsymbol{b}oldsymbol{B}ig]d\eta,
\label{EQ:compare1111}
\end{eqnarray}
where $t_0 = 0$. Since $\boldsymbol{b}oldsymbol{x}_0$ is fixed, one obtains $$ \int_0^{\boldsymbol{b}oldsymbol{x}i} \boldsymbol{b}oldsymbol{B}ig[\boldsymbol{b}oldsymbol{p}_0(\eta)\boldsymbol{b}oldsymbol{f}rac{d {\boldsymbol{b}oldsymbol{x}}_0}{d\eta} - H(\boldsymbol{b}oldsymbol{x}_0,\boldsymbol{b}oldsymbol{p}_0(\eta))\boldsymbol{b}oldsymbol{f}rac{d t_0}{d\eta}\boldsymbol{b}oldsymbol{B}ig]d\eta = 0$$ for every $\boldsymbol{b}oldsymbol{x}i\in[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$.
Then, taking into account Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Hamiltonian}), a combination of Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:compare1111}) with Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:rho_*_rho_xi}) leads to
\boldsymbol{b}egin{eqnarray}
\int_0^{t_f} \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{r}ho}(t) dt &=& \int_0^{t_f}\boldsymbol{b}ig[\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t)\dot{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}}(t) - H(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t))\boldsymbol{b}ig]dt\boldsymbol{b}oldsymbol{n}onumber\\
&=& - J(\boldsymbol{b}oldsymbol{x}i) + \int_0^{t_f}\boldsymbol{b}ig[\boldsymbol{b}oldsymbol{p}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))\dot{\boldsymbol{b}oldsymbol{x}}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)) - H(\boldsymbol{b}oldsymbol{x}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)),\boldsymbol{b}oldsymbol{p}(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)))\boldsymbol{b}ig]dt\boldsymbol{b}oldsymbol{n}onumber\\
&=& - J(\boldsymbol{b}oldsymbol{x}i) + \int_0^{t_f}\boldsymbol{b}oldsymbol{r}ho(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))dt\boldsymbol{b}oldsymbol{n}onumber\\
&\leq & - J(\boldsymbol{b}oldsymbol{x}i) + \int_0^{t_f}\boldsymbol{b}oldsymbol{r}ho_*(t)dt.
\label{EQ:lemma1_compare_new}
\end{eqnarray}
Since $J(0) = 0$, Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:lemma1_compare}) implies the strict inequality
\boldsymbol{b}egin{eqnarray}
\int_0^{t_f} \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{r}ho}(t) dt < \int_0^{t_f}\boldsymbol{b}oldsymbol{r}ho_*(t)dt,
\label{EQ:compare111}
\end{eqnarray}
holds if $\boldsymbol{b}oldsymbol{x}i\boldsymbol{b}oldsymbol{n}eq 0$ or $\boldsymbol{b}oldsymbol{x}_*(t_f)\boldsymbol{b}oldsymbol{n}eq \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)$. For the case of $\boldsymbol{b}oldsymbol{x}_*(t_f)=\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)$, Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:compare111}) is satisfied as well according to Theorem \boldsymbol{b}oldsymbol{r}ef{CO:cor1}, which proves that Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:lemma1_compare}) is a sufficient condition.
Next, let us prove that Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:lemma1_compare}) is a necessary condition. Assume Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:lemma1_compare}) is not satisfied, i.e., there exists a smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ and a $\boldsymbol{b}oldsymbol{x}i\in[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}ackslash\{0\}$ such that $J(\boldsymbol{b}oldsymbol{x}i) \leq J(0) = 0$. Then, according to Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:lemma1_compare_new}), one obtains
$$\int_0^{t_f}\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{r}ho}(t)dt \boldsymbol{b}oldsymbol{g}eq \int_0^{t_f}\boldsymbol{b}oldsymbol{r}ho(t,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)) dt.$$
Note that the extremal trajectory $\Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i)))$ in $\Pi(\mathcal{F})\boldsymbol{b}oldsymbol{s}ubset\Pi(\mathcal{L})$ is an admissible controlled trajectory of the $L^1$-minimization problem (cf. Remark \boldsymbol{b}oldsymbol{r}ef{RE:admissible_control_trajectory}). Thus, the proposition is proved.
\end{proof}
\boldsymbol{b}egin{proposition}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot))=\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that each switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}) and {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are satisfied, let $l<n$. Then, if $\boldsymbol{b}oldsymbol{v}arepsilon>0$ is small enough, the inequality $J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}(0) \boldsymbol{b}oldsymbol{g}eq 0$ (resp. the strict inequality $J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}(0) > 0$) for every smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ is a necessary condition (resp. a sufficient condition) for the strict strong-local optimality of the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$.
\label{PR:proposition_J2}
\end{proposition}
\boldsymbol{b}egin{proof}
Since the final time $t_f$ is fixed, Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:J_xi}) is reduced as
\boldsymbol{b}egin{eqnarray}
J(\boldsymbol{b}oldsymbol{x}i) = \int_{0}^{\boldsymbol{b}oldsymbol{x}i}\boldsymbol{b}oldsymbol{\lambda}(\eta){\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}}(\eta)d\eta.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Taking derivative of $J(\boldsymbol{b}oldsymbol{x}i)$ with respect to $\boldsymbol{b}oldsymbol{x}i$ yields
\boldsymbol{b}egin{eqnarray}
J^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i) = \boldsymbol{b}oldsymbol{\lambda}(\boldsymbol{b}oldsymbol{x}i)\boldsymbol{b}oldsymbol{c}dot \boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i).
\label{EQ:J_prime}
\end{eqnarray}
Note that ${\boldsymbol{b}oldsymbol{\lambda}}(0) = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_f)$. Taking into account Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Transversality_1}), for every smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$, we have ${ J^{\boldsymbol{b}oldsymbol{p}rime}}(0) = {\boldsymbol{b}oldsymbol{\lambda}}(0) \boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0) = 0$ since ${\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}}(0)$ is a tangent vector of the submanifold $\mathcal{M}$ at $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)$. Then, according to Proposition \boldsymbol{b}oldsymbol{r}ef{CO:J_xi_J_0}, this proposition is proved.
\end{proof}
\boldsymbol{b}egin{definition}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, denote by $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}\in(\mathbb{R}^l)^*$ the vector of the Lagrangian multipliers of this extremal such that
$$\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_f) = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}{d\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))}.$$
\label{DE:Lagrangian_Multiplier}
\end{definition}
\boldsymbol{b}egin{proposition}
In the case of $l< n$, given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) =\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that each switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}), assume {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are satisfied. Then, the inequality ${ J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}}(0) \boldsymbol{b}oldsymbol{g}eq 0$ (resp. strict inequality ${ J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}}(0) > 0$) is satisfied for every smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ if and only if there holds
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{z}eta}^T\left\{\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1} - \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))\boldsymbol{b}oldsymbol{r}ight\}\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{z}eta} \boldsymbol{b}oldsymbol{g}eq 0\ \text{(resp.}\ >0\text{)},\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
for every tangent vector $\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{z}eta}\in T_{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)}\mathcal{M}\boldsymbol{b}ackslash\{0\}$.
\label{LE:lemma2}
\end{proposition}
\boldsymbol{b}egin{proof}
Differentiating $J^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)$ in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:J_prime}) with respect to $\boldsymbol{b}oldsymbol{x}i$ yields
\boldsymbol{b}egin{eqnarray}
{J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}}(\boldsymbol{b}oldsymbol{x}i) &=& {\boldsymbol{b}oldsymbol{\lambda}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}{\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)} + \boldsymbol{b}oldsymbol{\lambda}(\boldsymbol{b}oldsymbol{x}i){\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}.
\label{EQ:d2Jdxi2}
\end{eqnarray}
Then, differentiating $\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i))$ with respect to $\boldsymbol{b}oldsymbol{x}i$ yields
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{f}rac{d }{d\boldsymbol{b}oldsymbol{x}i}\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i)) &=&{{ d\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i))}} {\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}= 0,\label{EQ:dphidxi}\boldsymbol{b}oldsymbol{n}onumber\\
\boldsymbol{b}oldsymbol{f}rac{d^2 }{d\boldsymbol{b}oldsymbol{x}i^2}\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i)) &=& [d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i))\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)]\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)
+ {d \boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}= 0.
\label{EQ:dphi2dxi2}
\end{eqnarray}
Since $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_f)) = (\boldsymbol{b}oldsymbol{y}(0),\boldsymbol{b}oldsymbol{\lambda}(0))$, according to the definition of the vector $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}$ in Definition \boldsymbol{b}oldsymbol{r}ef{DE:Lagrangian_Multiplier}, one immediately has $\boldsymbol{b}oldsymbol{\lambda}(0) = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}{d\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(0))}$.
Thus, multiplying $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}$ on both sides of Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:dphi2dxi2}) and fixing $\boldsymbol{b}oldsymbol{x}i = 0$, we obtain
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}\boldsymbol{b}oldsymbol{f}rac{d^2 \boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(0))}{d\boldsymbol{b}oldsymbol{x}i^2} &=& \boldsymbol{b}oldsymbol{\lambda}(0){\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}(0)} + \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}
\left[d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(0))\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)\boldsymbol{b}oldsymbol{r}ight] \boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)
\boldsymbol{b}oldsymbol{n}onumber\\
&=& {\boldsymbol{b}oldsymbol{\lambda}(0){\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}(0)} +
\left[\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)\boldsymbol{b}oldsymbol{r}ight]^{T}\left[ \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(0))\boldsymbol{b}oldsymbol{r}ight]\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0) }
\boldsymbol{b}oldsymbol{n}onumber\\
&=& 0.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Substituting this equation into Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:d2Jdxi2}) yields
\boldsymbol{b}egin{eqnarray}
{ J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime}}(0) = \boldsymbol{b}oldsymbol{\lambda}^{\boldsymbol{b}oldsymbol{p}rime}(0){\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)} -\left[\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)\boldsymbol{b}oldsymbol{r}ight]^{T}\left[\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{y}(0))\boldsymbol{b}oldsymbol{r}ight]\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0).
\label{EQ:d2Jdxi20}
\end{eqnarray}
Note that we have
\boldsymbol{b}egin{eqnarray}
{\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)} &=& \boldsymbol{b}oldsymbol{f}rac{d\boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{d\boldsymbol{b}oldsymbol{x}i} = \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\left[{\boldsymbol{b}oldsymbol{p}_0^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}\boldsymbol{b}oldsymbol{r}ight]^T,\boldsymbol{b}oldsymbol{n}onumber\\
\left[{\boldsymbol{b}oldsymbol{\lambda}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)} \boldsymbol{b}oldsymbol{r}ight]^T&=&\boldsymbol{b}oldsymbol{f}rac{d\boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{d\boldsymbol{b}oldsymbol{x}i}= \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[{\boldsymbol{b}oldsymbol{p}_0^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}\boldsymbol{b}oldsymbol{r}ight]^T.
\label{EQ:dlambdadxi}
\end{eqnarray}
Since the matrix $ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}$ is nonsingular if {\it Condition} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} is satisfied, we have $$\left[{\boldsymbol{b}oldsymbol{p}_0^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)} \boldsymbol{b}oldsymbol{r}ight]^T= \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1}{\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}.$$
Substituting this equation into Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:dlambdadxi})
yields
$$\left[{\boldsymbol{b}oldsymbol{\lambda}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}\boldsymbol{b}oldsymbol{r}ight]^T = \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}oldsymbol{p}_0(\boldsymbol{b}oldsymbol{x}i))}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1}{\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(\boldsymbol{b}oldsymbol{x}i)}.$$
Again, substituting this equation into Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:d2Jdxi20}) and taking into account $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0=\boldsymbol{b}oldsymbol{p}_0(0)$ and $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f) = \boldsymbol{b}oldsymbol{y}(0)$, we eventually get that for every smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ there holds
\boldsymbol{b}egin{eqnarray}
J^{\boldsymbol{b}oldsymbol{p}rime\boldsymbol{b}oldsymbol{p}rime} (0) = \left[\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)\boldsymbol{b}oldsymbol{r}ight]^T\boldsymbol{b}oldsymbol{B}ig\{\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1} - \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)) \boldsymbol{b}oldsymbol{B}ig\}\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0).\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Note that the vector $\boldsymbol{b}oldsymbol{y}^{\boldsymbol{b}oldsymbol{p}rime}(0)$ can be an arbitrary vector in the tangent space $T_{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)}\mathcal{X}\boldsymbol{b}ackslash\{0\}$, one proves this proposition.
\end{proof}
\boldsymbol{b}egin{condition}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, let
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{z}eta}^T\left\{\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1} - \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))\boldsymbol{b}oldsymbol{r}ight\}\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{z}eta} > 0,\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
be satisfied for every vector $\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{z}eta}\in T_{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)}\mathcal{M}\boldsymbol{b}ackslash\{0\}$.
\label{AS:terminal_condition}
\end{condition}
\boldsymbol{b}oldsymbol{n}oindent Then, as a combination {\it Propositions} \boldsymbol{b}oldsymbol{r}ef{PR:proposition_J2} and \boldsymbol{b}oldsymbol{r}ef{LE:lemma2}, we eventually obtain the following result.
\boldsymbol{b}egin{theorem}
Given the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot)) = \boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ such that every switching point is regular (cf. Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}), let $l<n$. Then, if {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang}, \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}, and \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition} are satisfied, the extremal trajectory $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$ realizes a strict strong-local optimality (cf. Definition \boldsymbol{b}oldsymbol{r}ef{DE:optimality}).
\label{TH:optimality}
\end{theorem}
Consequently, in the case of $l<n$, {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang}, \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}, and \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition} are sufficient to guarantee a bang-bang extremal with regular switching points to be a strict strong-local optimum. In next section, the numerical implementation for these three conditions will be derived.
\boldsymbol{b}oldsymbol{s}ection{Numerical implementation for sufficient optimality conditions}\label{SE:Procedure}
Once the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot))=\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ is computed, according to Definition \boldsymbol{b}oldsymbol{r}ef{DE:Lagrangian_Multiplier}, the vector $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}$ of Lagrangian multipliers in Condition \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition} can be computed by
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}} = \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_f){d\boldsymbol{b}oldsymbol{p}hi^T(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))} \left[{d\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))} {d\boldsymbol{b}oldsymbol{p}hi^T(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))}\boldsymbol{b}oldsymbol{r}ight]^{-1}.\label{EQ:numerical_nu}
\end{eqnarray}
\boldsymbol{b}egin{definition}
We define by $\boldsymbol{b}oldsymbol{C}\in\mathbb{R}^{n\times (n-l)}$ a full-rank matrix such that its columns constitute a basis of the tangent space $T_{\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)}\mathcal{M}$.\label{DE:definition_C}
\end{definition}
\boldsymbol{b}oldsymbol{n}oindent Then, one immediately gets that Condition \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition} is satisfied if and only if there holds
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{C}^T\left\{\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1} - \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))\boldsymbol{b}oldsymbol{r}ight\}\boldsymbol{b}oldsymbol{C} \boldsymbol{b}oldsymbol{s}ucc 0.
\label{EQ:positive_definite}
\end{eqnarray}
Note that the matrix $\boldsymbol{b}oldsymbol{C}$ can be computed by a simple Gram--Schmidt process once one derives the explicit expression of the matrix ${d\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))}$. Thus, it suffices to compute the matrix $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$ and the matrix $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ at $t_f$ in order to test {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang}, \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}, and \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition}.
It follows from the classical results about solutions to ODEs that the extremal trajectory $(\boldsymbol{b}oldsymbol{x}(t,{\boldsymbol{b}oldsymbol{p}}_0),\boldsymbol{b}oldsymbol{p}(t,\boldsymbol{b}oldsymbol{p}_0))$ and its time derivative are continuously differentiable with respect to $\boldsymbol{b}oldsymbol{p}_0$ on $[0,t_f]$. Thus, taking derivative of Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:cannonical}) with respect to $\boldsymbol{b}oldsymbol{p}_0$ on each segment $(t_i,t_{i+1})$, we obtain
\boldsymbol{b}egin{eqnarray}
\left[\boldsymbol{b}egin{array}{c}
\boldsymbol{b}oldsymbol{f}rac{d}{dt}\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial\boldsymbol{b}oldsymbol{p}_0}(t,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) \\
\boldsymbol{b}oldsymbol{f}rac{d}{dt}\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial\boldsymbol{b}oldsymbol{p}_0}(t,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)
\end{array}
\boldsymbol{b}oldsymbol{r}ight]=
\left[\boldsymbol{b}egin{array}{cc}
H_{\boldsymbol{b}oldsymbol{p}\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t)) & H_{\boldsymbol{b}oldsymbol{p}\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t))\\
-H_{\boldsymbol{b}oldsymbol{x}\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t)) &- H_{\boldsymbol{b}oldsymbol{x}\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t))
\end{array}\boldsymbol{b}oldsymbol{r}ight]
\left[
\boldsymbol{b}egin{array}{c}
\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(t,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)\\
\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(t,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)
\end{array}
\boldsymbol{b}oldsymbol{r}ight].
\label{EQ:Homogeneous_matrix}
\end{eqnarray}
Since the initial point $\boldsymbol{b}oldsymbol{x}_0$ is fixed, one can obtain the initial conditions as
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(0,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) = \boldsymbol{b}oldsymbol{0}_n, \ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(0,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) = I_n,
\label{EQ:initial_condition}
\end{eqnarray}
where $\boldsymbol{b}oldsymbol{0}_n$ and $I_n$ denote the zero and identity matrix of $\mathbb{R}^{n\times n}$.
Note that the two matrices $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ and $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ are discontinuous at the each switching time $t_i$. Comparing with the development in Refs. \boldsymbol{b}oldsymbol{c}ite{Schattler:12,Noble:02,Caillau:15}, the updating formulas for the two matrices $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ and $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ at each switching time $t_i$ can be written as
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(t_i+,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) &=& \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(t_i-,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) - \boldsymbol{b}oldsymbol{D}elta \boldsymbol{b}oldsymbol{r}ho_i \boldsymbol{b}oldsymbol{f}_1(\boldsymbol{b}oldsymbol{x}(t_i),\boldsymbol{b}oldsymbol{\omega}(t_i)){d t_i(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)},\label{EQ:update_formula_x}\\
\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(t_i+,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) &=& \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(t_i-,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0) + \boldsymbol{b}oldsymbol{D}elta \boldsymbol{b}oldsymbol{r}ho_i \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{f}_1}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{x}(t_i),\boldsymbol{b}oldsymbol{\omega}(t_i)\boldsymbol{b}oldsymbol{p}^T(t_i){d t_i(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)},\label{EQ:update_formula_p}
\end{eqnarray}
where $\boldsymbol{b}oldsymbol{D}elta \boldsymbol{b}oldsymbol{r}ho_i = \boldsymbol{b}oldsymbol{r}ho(t_i+) - \boldsymbol{b}oldsymbol{r}ho(t_i -)$. Up to now, except for ${d t_i(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}$, all necessary quantities can be computed. Note that for every $\boldsymbol{b}oldsymbol{p}_0\in\mathcal{P}$ there holds
\boldsymbol{b}egin{eqnarray}
H_1(\boldsymbol{b}oldsymbol{x}(t_i(\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{p}(t_i(\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{p}_0)) = 0.
\label{EQ:H_1(t_i)}
\end{eqnarray}
Taking into account $\dot{H}_1(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t)) = H_{01}(\boldsymbol{b}oldsymbol{x}(t),\boldsymbol{b}oldsymbol{p}(t))$, see Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:H01}), and differentiating Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:H_1(t_i)}) with respect to $\boldsymbol{b}oldsymbol{p}_0$ yields
\boldsymbol{b}egin{eqnarray}
0 &=& {H}_{01}(\boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{p}(t_i,\boldsymbol{b}oldsymbol{p}_0))dt_i({\boldsymbol{b}oldsymbol{p}_0}) + \boldsymbol{b}oldsymbol{p}(t_i,\boldsymbol{b}oldsymbol{p}_0)\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{f}_1}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{\omega}(t_i,\boldsymbol{b}oldsymbol{p}_0))\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}oldsymbol{p}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{n}onumber\\
& +& \boldsymbol{b}oldsymbol{f}_1^T(\boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{\omega}(t_i,\boldsymbol{b}oldsymbol{p}_0))\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_i,\boldsymbol{b}oldsymbol{p}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}.\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
According to Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching}, there holds $H_{01}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_i),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_i)) \boldsymbol{b}oldsymbol{n}eq 0$ for $i=1,2,\boldsymbol{b}oldsymbol{c}dots,k$. Thus, we obtain
\boldsymbol{b}egin{eqnarray}
{d t_i}{( \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)} &=& -\boldsymbol{b}oldsymbol{B}ig[\boldsymbol{b}oldsymbol{p}(t_i,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{f}_1}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}oldsymbol{p}_0),\boldsymbol{b}oldsymbol{\omega}(t_i,\boldsymbol{b}oldsymbol{p}_0))\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \boldsymbol{b}oldsymbol{n}onumber\\
&+& \boldsymbol{b}oldsymbol{f}_1^T(\boldsymbol{b}oldsymbol{x}(t_i,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0),\boldsymbol{b}oldsymbol{\omega}(t_i,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0))\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_i,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{B}ig]/{H}_{01}(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_i),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_i)).\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
Therefore, in order to compute the two matrices $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ and $\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)$ on $[0,t_f]$, it is sufficient to choose the initial condition in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:initial_condition}), then to numerically integrate Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Homogeneous_matrix}) and to use the updating formulas in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:update_formula_x}) and Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:update_formula_p}) once a switching point is encountered.
According to the approach of Chen {\it et al.} in Ref.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15},
given every bang-bang extremal $\Pi(\boldsymbol{b}oldsymbol{g}amma(\boldsymbol{b}oldsymbol{c}dot,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0))$ on $[0,t_f]$, $\delta(\boldsymbol{b}oldsymbol{c}dot)$ is a constant on every zero-thrust arc. Hence, to test focal points (or conjugate points for $l=n$), it suffices to test the zero of $\delta(\boldsymbol{b}oldsymbol{c}dot)$ on each maximum-thrust arc and to test the non-positivity of $\delta(t_i-)\delta(t_i+) $ at each switching time $t_i$.
\boldsymbol{b}oldsymbol{s}ection{Orbital Transfer Computation}\label{SE:Numerical}
In this numerical section, we consider the three-body problem of the Earth, the Moon, and an artificial spacecraft. Since the orbits of the Earth and the Moon around their common centre of mass are nearly circular, i.e., the eccentricity is around $5.49\times 10^{-2}$, and the mass of an artificial spacecraft is negligible compared with that of the Earth and the Moon, the Earth-Moon-Spacecraft (EMS) system can be approximately considered as a CRTBP, see Ref.~\boldsymbol{b}oldsymbol{c}ite{Szebehely:67}. Then, we have the below physical parameters corresponding to the EMS, $\mu = 1.2153\times 10^{-2}$, $d_* = 384,400.00$ km, $t_* = 3.7521\times 10^{5}$ seconds, and $m_* = 6.045\times 10^{24}$ kg. The initial mass of the spacecraft is specified as $500$ kg, the maximum thrust of the engine equipped on the spacecraft is taken as $1.0$ N, i.e., $$\tau_{max} =1.0 \boldsymbol{b}oldsymbol{f}rac{ t_*^{2}}{m_*d_*},$$ such that the initial maximum acceleration is $2.0\times 10^{-3}$ m$^2$/s. The spacecraft initially moves on a circular Earth geosynchronous orbit lying on the $XY$-plane such that the radius of the initial orbit is $r_g = 42,165.00$ km. When the spacecraft moves to the point on $X$-axis between the Earth and the Moon, i.e., $\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{r}(0) \boldsymbol{b}oldsymbol{p}arallel = r_g/d_* - \mu$, we start to control the spacecraft to fly to a circular orbit around the Moon with radius $r_m = 13,069.60$ km such that the $L^1$-norm of control is minimized at the fixed final time $t_f = 38.46$ days. Accordingly, the initial state $\boldsymbol{b}oldsymbol{x}_0 = (\boldsymbol{b}oldsymbol{r}_0,\boldsymbol{b}oldsymbol{v}_0,m_0)$ is given as
$$\boldsymbol{b}oldsymbol{r}_0 = (r_g/d_*-\mu,0,0)^T,\ \boldsymbol{b}oldsymbol{v}_0=(0,v_g,0)^T,\ \text{and}\ m_0 = 500/m_*,$$
where $v_g$ is the non-dimensional velocity of the spacecraft on the initial orbit, and the explicit expression of the function $\boldsymbol{b}oldsymbol{p}hi$ in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:final_manifold}) can be written as
\boldsymbol{b}egin{eqnarray}
\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{x}_f) = \left[
\boldsymbol{b}egin{array}{c}
\boldsymbol{b}oldsymbol{f}rac{1}{2}\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{r}(t_f) - [1-\mu,0,0]^T\boldsymbol{b}oldsymbol{p}arallel^2 - \boldsymbol{b}oldsymbol{f}rac{1}{2}(r_m/d_*)^2 \\
\boldsymbol{b}oldsymbol{f}rac{1}{2}\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{v}(t_f) \boldsymbol{b}oldsymbol{p}arallel^2 - \boldsymbol{b}oldsymbol{f}rac{1}{2}v_m^2 \\
\boldsymbol{b}oldsymbol{v}^T(t_f)\boldsymbol{b}oldsymbol{c}dot(\boldsymbol{b}oldsymbol{r}(t_f) - [1-\mu,0,0]^T) \\
\boldsymbol{b}oldsymbol{r}^T(t_f)\boldsymbol{b}oldsymbol{c}dot \boldsymbol{b}oldsymbol{1}_{Z} \\
\boldsymbol{b}oldsymbol{v}^T(t_f)\boldsymbol{b}oldsymbol{c}dot \boldsymbol{b}oldsymbol{1}_Z
\end{array}
\boldsymbol{b}oldsymbol{r}ight],
\label{EQ:function_phi}
\end{eqnarray}
where $\boldsymbol{b}oldsymbol{1}_z=[0,0,1]^T$ denotes the unit vector of the $Z$-axis of the rotating frame $OXYZ$ and $v_m$ is the non-dimensional velocity of the spacecraft on the circular orbit around the Moon with radius $r_m$.
We consider the constant mass model in which $\boldsymbol{b}eta = 0$ since this constant mass model can capture the main features of the original problem, see Refs.~\boldsymbol{b}oldsymbol{c}ite{Caillau:15,Caillau:12,Caillau:12time}. In this case, the mass $m$ is a constant parameter instead of a state in the system $\Sigma$, it follows that $\boldsymbol{b}oldsymbol{x} = (\boldsymbol{b}oldsymbol{r},\boldsymbol{b}oldsymbol{v})$ and $\boldsymbol{b}oldsymbol{p} = (\boldsymbol{b}oldsymbol{p}_r,\boldsymbol{b}oldsymbol{p}_v)$. Firstly, we compute the extremal $(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(\boldsymbol{b}oldsymbol{c}dot),\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(\boldsymbol{b}oldsymbol{c}dot))$ on $[0,t_f]$. It suffices to solve a shooting function corresponding to a two-point boundary value problem~\boldsymbol{b}oldsymbol{c}ite{Pan:13}. A simple shooting method is not stable to solve this problem because one usually does not know a priori the structure of the optimal control, and the numerical computations of the shooting function and its differential may be intricate since the shooting function is not continuous differentiable. We use a regularization procedure \boldsymbol{b}oldsymbol{c}ite{Caillau:12} by smoothing the control corner to get an energy-optimal trajectory firstly, then use a homotopy method to solve the real trajectory with a bang-bang control. Note that both the initial point $\boldsymbol{b}oldsymbol{x}_0$ and the final constraint submanifold $\mathcal{M}$ lie on the $XY$-plane, it follows that the whole trajectory lies on the $XY$-plane as well. Fig. \boldsymbol{b}oldsymbol{r}ef{Fig:Transferring_Orbit3_1} illustrates the non-dimensional profile of the position vector $ \boldsymbol{b}oldsymbol{r} $ along the computed extremal trajectory.
\boldsymbol{b}egin{figure}[!ht]
\boldsymbol{b}oldsymbol{c}entering\includegraphics[width=\textwidth, angle=0]{trajectory_Tmax3_bang.eps}
\boldsymbol{b}oldsymbol{c}aption[]{The non-dimensional profile of the position vector $ \boldsymbol{b}oldsymbol{r} $ of the $L^1$-minization trajectory in the rotating frame $OXYZ$ of the EMS system. The thick curves are the maximum-thrust arcs and the thin curves are the zero-thrust arcs. The bigger dashed circle and the smaller one are the initial and final circular orbits around the Earth and the Moon, respectively.}
\label{Fig:Transferring_Orbit3_1}
\end{figure}
The profiles of $\boldsymbol{b}oldsymbol{r}ho$, $\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel$, and $H_1$ with respect to non-dimensional time are shown in Fig. \boldsymbol{b}oldsymbol{r}ef{Fig:Transferring_Orbit4_1}, from which we can see that the number of maximum-thrust arcs is 15 with 29 switching points and that the ragularity condition in Assumption \boldsymbol{b}oldsymbol{r}ef{AS:Regular_Switching} at every switching point is satisfied.
\boldsymbol{b}egin{figure}[!ht]
\boldsymbol{b}oldsymbol{c}entering\includegraphics[ width=\textwidth, angle=0]{control_H1_pv.eps}
\boldsymbol{b}oldsymbol{c}aption[]{The profiles of $\boldsymbol{b}oldsymbol{r}ho$, $\boldsymbol{b}oldsymbol{p}arallel \boldsymbol{b}oldsymbol{p}_v \boldsymbol{b}oldsymbol{p}arallel $, and $H_1$ with respect to non-dimensional time along the $L^1$-minimization trajectory.}
\label{Fig:Transferring_Orbit4_1}
\end{figure}
Since the extremal trajectory is computed based on necessary conditions, one has to check sufficient optimality conditions to make sure that it is at least locally optimal. According to what has been developed in Section \boldsymbol{b}oldsymbol{r}ef{SE:Sufficient}, it suffices to check the satisfaction of {\it Conditions} \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang}, \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}, and \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition}.
Using Eqs.~(\boldsymbol{b}oldsymbol{r}ef{EQ:Homogeneous_matrix}--\boldsymbol{b}oldsymbol{r}ef{EQ:update_formula_p}), one can compute $\delta(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$. In order to have a clear view, the profile of $\delta(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$ is rescaled
\boldsymbol{b}egin{figure}[!ht]
\boldsymbol{b}oldsymbol{c}entering\includegraphics[width=\textwidth, angle=0]{det_Tmax3_bang.eps}
\boldsymbol{b}oldsymbol{c}aption[]{The profile of $sgn(\delta(t))|\delta(t)|^{1/12}$ with respect to non-dimensional time along the $L^1$-minimization extremal in EMS.}
\label{Fig:det}
\end{figure}
by $\text{sgn}(\delta(\boldsymbol{b}oldsymbol{c}dot))*|\delta(\boldsymbol{b}oldsymbol{c}dot)|^{1/12}$, which can capture the sign property of $\delta(\boldsymbol{b}oldsymbol{c}dot)$ on $[0,t_f]$, as is illustrated in Fig. \boldsymbol{b}oldsymbol{r}ef{Fig:det}. We can see that there exist no sign changes at each switching point and no zeros on each smooth bang arc. Thus, Conditions \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality} are satisfied along the computed extremal. To check Condition \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition}, differentiating $\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{c}dot)$ in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:function_phi}) yields
\boldsymbol{b}egin{eqnarray}
{d \boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)) }=
\left[\boldsymbol{b}egin{array}{ccccc}
\boldsymbol{b}oldsymbol{r}(t_f) - [1-\mu,0,0]^T & \boldsymbol{b}oldsymbol{0}_{3\times 1} & \boldsymbol{b}oldsymbol{v}(t_f)& \boldsymbol{b}oldsymbol{1}_Z& \boldsymbol{b}oldsymbol{0}_{3\times 1}\\
\boldsymbol{b}oldsymbol{0}_{3\times 1} &\boldsymbol{b}oldsymbol{v}(t_f) &\boldsymbol{b}oldsymbol{r}(t_f) - [1-\mu,0,0]^T & \boldsymbol{b}oldsymbol{0}_{3\times 1}&\boldsymbol{b}oldsymbol{1}_Z
\end{array}
\boldsymbol{b}oldsymbol{r}ight]^T,
\label{EQ:dphi_numerical}
\end{eqnarray}
and
\boldsymbol{b}egin{eqnarray}
d^2{\boldsymbol{b}oldsymbol{p}hi_1(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))} &=& \left(\boldsymbol{b}egin{array}{cc} I_3 &\boldsymbol{b}oldsymbol{0}_3\\
\boldsymbol{b}oldsymbol{0}_3 & \boldsymbol{b}oldsymbol{0}_3
\end{array}\boldsymbol{b}oldsymbol{r}ight),\ \ d^2{\boldsymbol{b}oldsymbol{p}hi_2(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))}= \left(\boldsymbol{b}egin{array}{cc} \boldsymbol{b}oldsymbol{0}_3 &\boldsymbol{b}oldsymbol{0}_3\\
\boldsymbol{b}oldsymbol{0}_3 & I_3
\end{array}\boldsymbol{b}oldsymbol{r}ight),\boldsymbol{b}oldsymbol{n}onumber\\
d^2{\boldsymbol{b}oldsymbol{p}hi_3(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))} &=& \left(\boldsymbol{b}egin{array}{cc} \boldsymbol{b}oldsymbol{0}_3 & I_3\\
I_3 & \boldsymbol{b}oldsymbol{0}_3
\end{array}\boldsymbol{b}oldsymbol{r}ight),\ \ d^2{\boldsymbol{b}oldsymbol{p}hi_4(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))} = d^2{\boldsymbol{b}oldsymbol{p}hi_5(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))}= \boldsymbol{b}oldsymbol{0}_6,\boldsymbol{b}oldsymbol{n}onumber
\end{eqnarray}
where $\boldsymbol{b}oldsymbol{p}hi_i(\boldsymbol{b}oldsymbol{c}dot):\mathcal{X}\boldsymbol{b}oldsymbol{r}ightarrow \mathbb{R},\ \boldsymbol{b}oldsymbol{x}\mapsto \boldsymbol{b}oldsymbol{p}hi_i(\boldsymbol{b}oldsymbol{x})$ for $i=1,2,\boldsymbol{b}oldsymbol{c}dots,l$ are the elements of the vector-valued function $\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}oldsymbol{x})$.
Then, substituting the values of $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)$ and $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}(t_f)$ into Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:numerical_nu}), the vector $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}}$ can be computed. Up to now, except the matrix $\boldsymbol{b}oldsymbol{C}$, all the quantities in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:positive_definite}) are obtained. Actually, one can use a Gram-Schmidt process to compute the matrix $\boldsymbol{b}oldsymbol{C}$ associated with the matrix in Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:dphi_numerical}). Then, substituting numerical values into Eq.~(\boldsymbol{b}oldsymbol{r}ef{EQ:positive_definite}), we obtain
$$\boldsymbol{b}oldsymbol{C}^T\left\{\boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}^T(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0} \left[ \boldsymbol{b}oldsymbol{f}rac{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{x}(t_f,\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{p}}_0)}{\boldsymbol{b}oldsymbol{p}artial \boldsymbol{b}oldsymbol{p}_0}\boldsymbol{b}oldsymbol{r}ight]^{-1} - \boldsymbol{b}ar{\boldsymbol{b}oldsymbol{\boldsymbol{b}oldsymbol{n}u}} d^2\boldsymbol{b}oldsymbol{p}hi(\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f))\boldsymbol{b}oldsymbol{r}ight\}\boldsymbol{b}oldsymbol{C} \approx 0.5292 \boldsymbol{b}oldsymbol{s}ucc 0.$$
Thus, Condition \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition} is satisfied. Note that the dimension of the submanifold $\mathcal{M}$ is one, it follows that the smooth curve $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]$ for every $\boldsymbol{b}oldsymbol{v}arepsilon > 0$ is a one-dimensional curve restricted on the final circular orbit around the Moon.
Fig. \boldsymbol{b}oldsymbol{r}ef{Fig:J_xi}
\boldsymbol{b}egin{figure}[!ht]
\boldsymbol{b}oldsymbol{c}entering\includegraphics[width=\textwidth, angle=0]{J_xi.eps}
\boldsymbol{b}oldsymbol{c}aption[]{Let $X(\boldsymbol{b}oldsymbol{x}i)$ and $Y(\boldsymbol{b}oldsymbol{x}i)$ be the projection of the position vector $\boldsymbol{b}oldsymbol{r}(\boldsymbol{b}oldsymbol{x}i)$ on $X$- and $Y$-axis of the rotating frame $OXYZ$, respectively, and let $V_x(\boldsymbol{b}oldsymbol{x}i)$ and $V_y(\boldsymbol{b}oldsymbol{x}i)$ be the projection of the velocity vector $\boldsymbol{b}oldsymbol{v}(\boldsymbol{b}oldsymbol{x}i)$ on $X$- and $Y$-axis of the rotating frame $OXYZ$, respectively. The figure plots the profiles $J(\boldsymbol{b}oldsymbol{x}i)$ with respect to $X(\boldsymbol{b}oldsymbol{x}i)$, $Y(\boldsymbol{b}oldsymbol{x}i)$, $V_x(\boldsymbol{b}oldsymbol{x}i)$, and $V_y(\boldsymbol{b}oldsymbol{x}i)$. The dots on each plot denote $(J(0),\boldsymbol{b}oldsymbol{y}(0))$.}
\label{Fig:J_xi}
\end{figure}
illustrates the profile of $J(\boldsymbol{b}oldsymbol{c}dot)$ with respect to $\boldsymbol{b}oldsymbol{y}(\boldsymbol{b}oldsymbol{c}dot)\in\mathcal{M}\boldsymbol{b}oldsymbol{c}ap\mathcal{N}$ in a small neighbourhood of $\boldsymbol{b}ar{\boldsymbol{b}oldsymbol{x}}(t_f)$. we can clearly see that $J(\boldsymbol{b}oldsymbol{c}dot)>J(0)$ on $[-\boldsymbol{b}oldsymbol{v}arepsilon,\boldsymbol{b}oldsymbol{v}arepsilon]\boldsymbol{b}ackslash\{0\}$. Up to now, all the conditions in Theorem \boldsymbol{b}oldsymbol{r}ef{TH:optimality} are satisfied. So, the computed $L^1$-minimization trajectory realizes a strict strong-local optimality in $C^{0}$-topology.
\boldsymbol{b}oldsymbol{s}ection{Conclusions}
In this paper, the PMP is first employed to formulate the Hamiltonian system of the $L^1$-minimization problem for the translational motion of a spacecraft in the CRTBP, showing that the optimal control functions can exhibit bang-bang and singular behaviors. Moreover, the singular extremals are of at least order two, revealing the existence of Fuller or chattering phenomena. To establish the sufficient optimality conditions, a parameterized family of extremals is constructed. As a result of analyzing the projection behavior of this family, we obtain that conjugate points may occur not only on maximum-thrust arcs between switching times but also at switching times. Directly applying the theory of field of extremals, we obtain that the disconjugacy conditions (cf. Conditions \boldsymbol{b}oldsymbol{r}ef{AS:Disconjugacy_bang} and \boldsymbol{b}oldsymbol{r}ef{AS:Transversality}) are sufficient to guarantee an extremal to be locally optimal if the endpoints are fixed. For the case that the dimension of the final constraint submanifold is not zero, we establish a further second-order condition (cf. Condition \boldsymbol{b}oldsymbol{r}ef{AS:terminal_condition}), which is a necessary and sufficient one for the strict strong-local optimality of a bang-bang extremal if disconjugacy conditions are satisfied. In addition, the numerical implementation for these three sufficient optimality conditions is derived. Finally, an example of transferring a spacecraft from a circular orbit around the Earth to an orbit around the Moon is computed and the second-order sufficient optimality conditions developed in this paper are tested to show that the computed extremal realizes a strict strong-local optimum. The sufficient optimality conditions for open-time problems will be considered in future work.
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\begin{document}
\title{Axiomatizing Analog Algorithms}
\author{Olivier Bournez\inst{1}\thanks{This author's research was
partially supported by a French National Research
Agency's grant (ANR-15-CE40-0016-02).}
\and Nachum
Dershowitz\inst{2}\thanks{This author's research benefited from
a fellowship at the Institut d'\'Etudes Avanc\'ees de Paris
(France), with the financial support of the French National Research Agency's ``Investissements
d'avenir'' program (ANR-11-LABX-0027-01 Labex RFIEA+).}
\and Pierre N\'eron\inst{3}}
\institute{Laboratoire d'Informatique de l'X (LIX), \'Ecole
Polytechnique, France \and
School of Computer Science, Tel Aviv University, Ramat Aviv, Israel
\and French Network and Information Security Agency (ANSSI), France \\
\email{[email protected]}~~\email{[email protected]}~~\email{[email protected]}}
\authorrunning{O. Bournez and N. Dershowitz and P. N\'eron}
\maketitle
\begin{abstract}
We propose a formalization of generic algorithms that includes analog
algorithms. This is achieved by reformulating and extending the framework of abstract
state machines to include continuous-time models of computation.
{We prove that every hybrid algorithm satisfying some reasonable postulates may be expressed precisely by
a program in a simple and expressive language.}
\end{abstract}
\section{Introduction}
In \cite{Gur00}, Gurevich showed
that any algorithm that satisfies three intuitive ``Sequential
Postulates'' can be step-by-step emulated by an {abstract state machine (ASM)}. These postulates
formalize the following intuitions: (I) one is dealing with
discrete deterministic state-transition systems; (II) the information in states
suffices to determine future transitions and may be captured by
logical structures that respect isomorphisms; and (III) transitions
are governed by the values of a finite and input-independent set of
ground
terms.
All notions of algorithms for ``classical'' \emph{discrete-time} models of
computation in computer science are covered by this formalization.
This includes Turing machines, random-access memory (RAM) machines,
and their sundry extensions.
The geometric constructions in \cite{Reisig}, for example, are
loop-free examples of discrete-step continuous-space (real-number) algorithms.
The ASM formalization also covers general discrete-time models evolving over continuous
space like the Blum-Shub-Smale machine model \cite{BSS}.
However, capturing \emph{continuous-time} models of computation is
still a challenge, that is to say, capturing models of computation that operate
in continuous (real) time and with real values.
Examples of continuous-time models of computations include models of analog
machines like the General Purpose Analog Computer (GPAC) of Claude Shannon
\cite{Sha41}, proposed as a mathematical model of the Differential
Analyzers, built for the first time in 1931 \cite{Bush31}, and
used to solve various problems ranging from
ballistics to aircraft design -- before the era of the digital computer
\cite{Nyc96}.
Others include
Pascal's 1642 \textit{Pascaline},
Hermann's 1814 \textit{Planimeter},
as well as
Bill Phillips' 1949 water-run \textit{Financephalograph}.
Continuous-time computational models also include neural networks
and systems built using electronic
analog devices.
Such systems begin in some initial state and evolve over time;
results are read off from the evolving
state and/or from a terminal state. More generally, determining which
systems can actually be considered to be computational models is an
intriguing question and relates to philosophical discussions about
what constitutes a programmable machine.
Continuous-time
computation theory is far less understood than its discrete-time
counterpart \cite{Survey}.
Another line of development of continuous-time models was
motivated by hybrid systems, particularly by questions related to
the hardness of their verification and control. In hybrid
systems, the dynamics change in response to changing
conditions, so there are discrete transitions as well as continuous
ones.
Here, models are not
seen as necessarily modeling analog machines, but,
rather, as abstractions of systems about
which one would like to establish properties or derive
verification algorithms \cite{Survey}.
Some work on ASM models dealing with
continuous-time systems has been accomplished
for specific cases \cite{Cohen,Cohen2}.
Rust \cite{rust2000hybrid} specifies forms of continuous-time evolution
based on ASMs using infinitesimals.
However, we find that a comprehensive framework
capturing general analog systems is still wanting.
Our goal is to capture all such analog and hybrid models within one uniform notion of
computation and of algorithm.
To this end, we formalize a generic notion of
continuous-time algorithm. The proposed
framework is an extension of \cite{Gur00}, as discrete-time
algorithms are a simple special case of analog algorithms.
(The initial attempt \cite{BournezDF12} was not fully satisfactory, as no
completeness theorem nor general-form result was obtained. Here, we
indeed achieve both.)
We
provide postulates defining continuous-time algorithms, in the spirit of
those of \cite{Gur00},
and we
prove some completeness results.
We define a simple notion of an analog ASM
program and prove that all models satisfying the postulates
have
corresponding analog programs (Lemma
\ref{mlemma} and Theorem \ref{thone}).
Furthermore, we provide conditions guaranteeing that said program is unique up
to equivalence (Theorem \ref{thtwo} and Corollary
\ref{coromain}).
All of this seamlessly extends the
results of \cite{Gur00} to analog and hybrid systems.
The proposed framework covers all classes of
continuous-time systems that can be modeled by ordinary differential
equations or have hybrid dynamics, including the models in \cite{Survey} and
the examples in \cite{BournezDF12}.
It is a first step towards a general understanding of
computability theory for continuous-time models, taken in the hope that it will also lead to a
formalization of a ``Church-Turing thesis'' for analog systems in the spirit of what has been achieved for discrete-time models~\cite{BD,dershowitz2008natural,3P}.
Systems with continuous input signals
and other means of specifying continuous behavior are left for future
work.
Some of our ideas were inspired by the way the semantics
of hybrid systems are given in the approach of
Platzer~\cite{Platzer08}.
Among attempts at studying the semantics of analog systems within a general
framework is
\cite{TZ}. Recent results on comparing analog models include
\cite{fu2015models}. Soundness and (relative) completeness results for
a programming language with infinitesimals have also been obtained in
\cite{suenaga2011programming}. Applications to verification have been
explored \cite{hasuo2012exercises}.
\section{General Algorithms}
We want to generalize the notion of algorithms introduced by Gurevich in \cite{Gur00} in
order to capture not only the sequential case but also continuous
behavior. (For lack of place, we assume some familiarity with
\cite{Gur00}.)
However, when evolving continuously, an algorithm can no longer be viewed as a discrete sequence of
states, and we need a notion of evolution that can capture both kinds of behavior. This
is based on a notion of a \emph{timeline} that corresponds
to algorithm execution.
\begin{definition}[Time]
\emph{Time} $\ensuremath{\mathcal {T}}T$ corresponds
to a totally ordered monoid: there is an associative binary
operation $+$, with some neutral element $0$, and a total relation $\le$ preserved
by $+$: $t \le t'$ implies $t+t'' \le t'+t''$ for all $t''\in\ensuremath{\mathcal {T}}T$.
\end{definition}
An element of $\ensuremath{\mathcal {T}}T$ will be called a \emph{moment}. Examples of time \ensuremath{\mathcal {T}}T\@
are $\ensuremath{\mathbb{R}}^{\geq 0}$ and $\ensuremath{\mathbb{N}}$. As expected, $t < t'$ will mean $t
\le t'$ but not $t = t'$.
\begin{definition}[Timeline]
A timeline is a subset of $\ensuremath{\mathcal {T}}T$ containing $0$. We let $\ensuremath{\mathbb{I}}$ denote the set of all timelines.
\end{definition}
For a moment $i \in I$ of timeline $I$, we write $\notContinuous{i}$ if there exists
$t \in I$ with $i<t$, and there is no $t' \in I$ with $i < t' < t$.
We write $\Continuous{i}$ otherwise: that means that for all $t$, $i<t$,
there is some in-between $t' \in I$ with $i < t' < t$.
\nd{A moment $i$ with $\notContinuous{i}$ is meant to indicate a
discrete transition. In this case, we write $i^+$ for the smallest
$t$ greater than $i$.}
A timeline $I$ is non-Zeno if for any moment $i \in I$, there is a
finite number of moments $j \le i$ with $\notContinuous{j}$. $\ensuremath{\mathbb{I}}$ is
non-Zeno if all its timelines are.
For timelines $\ensuremath{\mathbb{I}}=\ensuremath{\mathbb{R}}^{\geq 0}$, for instance, we have $\Continuous{i}$ for all $i \in
\ensuremath{\mathbb{I}}$.
For $\ensuremath{\mathbb{I}}=\ensuremath{\mathbb{N}}$, we have $\notContinuous{i}$ for all $i \in \ensuremath{\mathbb{I}}$, and
$i^+=i+1$.
We intend (for hybrid systems, in particular) to also
consider timelines mixing both properties, that is, with
$\Continuous{i}$ for some $i$ and $\notContinuous{i}$ for
other $i$. Formally building such timelines is easy (for
example $\bigcup_{n \in \ensuremath{\mathbb{N}}} [n,n+0.5]$). All these examples are non-Zeno.
\begin{definition}[Truncation]
Given a timeline $I\in\ensuremath{\mathbb{I}}$ and a moment $i$ of $I$, the
\emph{truncated} timeline $I[i]$ is the timeline defined by
$I[i] = \{t \mid i+t \in I\}$.
\end{definition}
With timelines in hand, we can define hybrid dynamical systems.
\begin{definition}[Dynamical System]\label{LTS} \label{def:ts}
A \emph{dynamical system} $\langle \ensuremath{\mathcal {S}}, \ensuremath{\mathbb{I}}n, \iota, \varphi
\rangle$
consists of the following:
\itm{a} a nonempty set (or class)
$\ensuremath{\mathcal {S}}$ of \emph{states};
\itm{b} a nonempty subset (or subclass) $\ensuremath{\mathbb{I}}n \subseteq \ensuremath{\mathcal {S}}$, called \emph{initial} states;
\itm{c} a \emph{timeline} map $\iota : \ensuremath{\mathcal {S}} \to \ensuremath{\mathbb{I}}$, with $\ensuremath{\mathbb{I}}$ non-Zeno;
\itm{d} a \emph{trajectory} map $\varphi: (X : \ensuremath{\mathcal {S}}) \times \iota(X) \to \ensuremath{\mathcal {S}}$.
We require that,
for any state $X$ and moments $i, i+i' \in \iota(X)$, one has
$$\varphi(X,0) = X\,, \hspace*{8mm}
\iota(\varphi(X,i)) = \iota(X)[i]\,, \hspace*{8mm}
\varphi(X,i+i') = \varphi(\varphi(X,i),i')\,.
$$
\end{definition}
Together, the \emph{timeline} and \emph{trajectory} maps associate to each state its future
evolution. For a state $X$, $\iota(X)$ defines the timeline corresponding to the system behavior
starting from $X$, and $\varphi(X)$ defines its concrete evolution by associating to each moment in
$\iota(X)$ its corresponding state. The third condition ensures that evolution during $i+i'$ is similar to
first evolving during $i$ and then during $i'$; the preceding condition ensures a similar
property for timelines (and ensures consistency of the last condition).
\begin{postulatep} An algorithm is a dynamical system.
\end{postulatep}
A vocabulary $\ensuremath{\mathcal {V}}$ is a finite collection of fixed-arity (possibly
nullary) function
symbols,
some functions of which may be tagged \emph{relational}. A term whose
outermost function symbol is relational is termed \emph{Boolean}. We
assume that $\ensuremath{\mathcal {V}}$ contains the scalar (nullary) function \textit{true}.
A (first-order) \emph{structure} $X$ of vocabulary $\ensuremath{\mathcal {V}}$ is a nonempty set $S$, the
\emph{base set (domain)} of $X$, together with interpretations of the function
symbols in $\ensuremath{\mathcal {V}}$ over $S$: A $j$-ary function symbol $f$ is interpreted as a function,
denoted $\val{f}{X}$, from $S^j$ to $S$. Elements of $S$ are also called elements of
$X$, or \emph{values}. Similarly, the interpretation of a term $f(t_1,\dots ,t_n)$ in $X$
is recursively defined by $\val{f(t_1,\dots ,t_n)}{X} = \val{f}{X}(\val{t_1}{X},\dots ,\val{t_n}{X})$.
Let $X$ and $Y$ be structures of the same vocabulary $\ensuremath{\mathcal {V}}$. An
\emph{isomorphism} from $X$ onto $Y$ is a one-to-one function $\zeta$
from the base set of $X$ onto the base set of $Y$ such that $f(\zeta
x_1,\dots,\zeta x_j)= \zeta x_0$ in $Y$ whenever $f(x_1,\dots,x_j)=x_0$
in $X$.
\begin{definition}[Abstract Transition System]\label{ATS}
An \emph{abstract transition system} is a dynamical system
whose states $\ensuremath{\mathcal {S}}$ are (first-order) structures over some finite vocabulary $\ensuremath{\mathcal {V}}$,
such that the following hold:
\begin{enumerate}
\item States are closed under isomorphism, so if $X\in \ensuremath{\mathcal {S}}$ is a state of the system, then any structure $Y$ isomorphic to $X$ is also a state in $\ensuremath{\mathcal {S}}$, and $Y$ is an initial state if $X$ is.
\item Transformations preserve the base set: that is, for every state $X \in \ensuremath{\mathcal {S}}$, for any $i \in \iota(X)$, $\varphi(X,i)$ has the same base set as $X$.
\item Transformations respect isomorphisms: if $X\cong_\zeta Y$
is an isomorphism of states $X,Y\in \ensuremath{\mathcal {S}}$, then
$\iota(X)=\iota(Y)$
and for all $i \in \iota(X)$, $X_i \cong_\zeta Y_i$,
where $X_i = \varphi(X,i)$, and $Y_i=\varphi(Y,i)$.
\end{enumerate}
\end{definition}
\begin{postulatep} \label{postulateats} An
algorithm is an abstract
transition system.
\end{postulatep}
When $\iota(X)$ is $\ensuremath{\mathbb{N}}$ (or order-isomorphic to $\ensuremath{\mathbb{N}}$) for all $X$, this corresponds precisely to the concepts
introduced by \cite{Gur00}, considering that $\varphi(X,n) =
\tau^{[n]}(X)$.
It is convenient to think of a structure $X$ as a memory of some kind: If
$f$ is a $j$-ary function symbol in vocabulary $\ensuremath{\mathcal {V}}$, and
$\overline{a}$ is a $j$-tuple of elements of the base set of $X$, then
the pair $\LOCATION{f}{\overline{a}}$ is called a
\emph{location}. We denote by $\vall{X}{f(\overline{a})}$ its interpretation
in $X$, i.e.\@ $\val{f}{X}(\overline{a})$.
If $\LOCATION{f}{\overline{a}}$ is a
location of $X$ and $b$ is an element of $X$ then
$\update{f}{\overline{a}}{b}$ is an \emph{update} of $X$.
When $Y$ and $X$ are structures over the same domain and vocabulary,
$Y\setminus X$ denotes the set of updates $\Delta^+=
\{ \update{f}{\overline{a}}{\vall{Y}{f(\overline{a})}} \mid
\vall{Y}{f(\overline{a})} \neq \vall{X}{f(\overline{a})}
\}.$
We want instantaneous evolution to be describable by updates:
\begin{definition} \label{defmachin}
An \emph{infinitesimal generator} is \itm{a} a function $\Delta$ that maps states $X$ to a set $\Delta(X)$ of updates,
and \itm{b} preserves
isomorphisms: if $X\cong_\zeta Y$ is an isomorphism of states
$X,Y\in \ensuremath{\mathcal {S}}$, then for all updates
$\update{f}{\overline{a}}{b} \in \Delta(X)$, we have an isomorphic update
$\update{f}{\overline{\zeta a}}{\zeta b} \in \Delta(Y)$.
\end{definition}
\nd{We write $\notContinuous{X}$ and say that $X$ is a \emph{jump} when $\notContinuous{0}$ in
timeline
$\iota(X)$; otherwise, we write $\Continuous{X}$ and say that it is a \emph{flow}.}
For states $X$ with
$\notContinuous{X}$, the following is natural:
\begin{definition} \label{defupdate}
The \emph{update generator} is the infinitesimal generator defined on
jump states $X$ as
$\Delta(X)=\Delta^+(X)$, where $\Delta^+(X)$ stands for
$\varphi(X,0^+) \setminus X$.
\end{definition}
To deal with flow states, we will also define some corresponding infinitesimal
generator $\Deltapsi$. Before doing so, let's see how to go from semantics to
generators.
An \emph{initial evolution} over $S$ is a function
whose
domain of definition is a timeline and whose range is $S$.
An initial evolution is said to be \emph{initially constant} if it has
a constant prefix: that is to say, there is some $0<t$ such that
the function is constant over $[0\mathbin{..}t]$.
\begin{definition}[Semantics]
A \emph{semantics} $\psi$ over a class $\mathcal{C}$ of sets $S$
is a partial function mapping
initial evolutions over some $S \in \mathcal{C}$
to an element of $S$.
\end{definition}
\begin{remark} \label{rq:ex}
When $\ensuremath{\mathcal {T}}T=\ensuremath{\mathbb{R}}^{\geq 0}$, an example of
semantics over the class of sets $S$ containing $\ensuremath{\mathbb{R}}$ is the derivative $\psi_{\textrm{der}}$, mapping a function
$f$ to its
derivative at $0$ when that exists. When $\ensuremath{\mathcal {T}}T=\ensuremath{\mathbb{N}}$, an example of
semantics over the class of all sets would be the function $\psi_\ensuremath{\mathbb{N}}$ mapping
$f$ to $f(1)$. More generally, when $0 \in \ensuremath{\mathcal {T}}T$ is such that $\notContinuous{0}$, an example of
semantics over the class of all sets is the function $\psi_\ensuremath{\mathbb{N}}$ mapping
$f$ to $f(0^+)$.
\end{remark}
Consider a semantics $\psi$ over a class of sets $S$.
Let $X$ be a state whose domain is in the class and a location $\LOCATION{f}{\overline{a}}$
of $X$.
Denote by $Evolution({X,\LOCATION{f}{\overline{a}}})$ the corresponding initial
evolution: that is to say, the
function given formally by $Evolution({X,\LOCATION{f}{\overline{a}})}: t \mapsto
\val{f(\overline{a})}{\varphi(X,t)}$ for $0 \le t \le I_1, t \in
\iota(X)$, for some $I_1 \in \iota(X)$, with $I_1=0^+$ for a jump.
We use $\psi[X,f,\overline{a}]$ to denote the image of this evolution under $\psi$ (when it exists).
\begin{definition}[Infinitesimal generator associated with
$\psi$] The infinitesimal generator associated with $\psi$,
maps each state $X$, such that $\psi[X,f,\overline{a}]$ is defined for all locations, to
the set: $\Deltapsi(X) =\{ \update{f}{\overline{a}}{\psi[X,f,\overline{a}]} \mid \LOCATION{f}{\overline{a}} \mbox{ is a location of
$X$} ,~
Evolution({X,\LOCATION{f}{\overline{a}}}) \text{ is not initially constant} \}.$
\end{definition}
The update generator $\Delta^+$ (see Definition
\ref{defupdate}) is the infinitesimal generator associated with the
semantics $\psi_\ensuremath{\mathbb{N}}$ (of Remark \ref{rq:ex}) over flow states.
From now on, we assume that some semantics $\psi$ is fixed to deal
with flow states. It could be
$\psi_{\textrm{der}}$, but it could also be another one (for example:
talking about integrals or built using infinitesimals as in \cite{rust2000hybrid}). We denote by
$\Deltapsi$ the associated infinitesimal generator.
We are actually discussing algorithms relative to some
$\psi$, and to be more precise, we should be refering to $\psi$-algorithms.
The point is that not every infinitesimal generator is appropriate
and that appropriateness is actually relative to a time domain and
to the class of allowed dynamics over this time domain.
\nd{To see this, keep in mind that -- when $\Deltapsi$ corresponds to derivative
-- to be able to talk about
derivatives, one implicitly restricts oneself to dynamics that are
differentiable, hence non-arbitrary. In other words, one is restricting
to a particular class of possible dynamics, and not all dynamics
are allowed. Restricting to other classes of dynamics (for example, analytic
ones) may lead to different notions of algorithm.}
From the update generator $\Delta^+$ and $\Deltapsi$, we build a
generator also tagging states by the fact that they correspond to a
jump or a flow:
\begin{definition}[Generator of a State] \label{def:behaviord}
We define the \emph{tagged generator} of a state $X$, denoted $\gen{X}$, as a
function that maps state $X$ to $\{\mathcal{F}\} \times \Deltapsi(X)$ when
$\Continuous{X}$ and $\Deltapsi(X)$ is defined and to $\{\mathcal{J}\} \times \Delta^+(X)$ when
$\notContinuous{X}$.
\end{definition}
Let $T$ be a set of ground terms.
We say that states $X$ and $Y$ \emph{coincide} over $T$,
if $\val{s}{X}=\val{s}{Y}$ for all $s\in T$.
This will be abbreviated $X =_{T} Y $.
The fact that $X$ and $Y$ {coincide} over $T$ implies that $X$ and $Y$
necessarily share some common elements in their
respective base sets, at least all the $\val{s}{X}$ for $s \in T$.
An algorithm should have a finite imperative description. Intuitively, the evolution
of an algorithm from a given state is only determined by inspecting part of this
state by means of the terms appearing in the algorithm
description. The following
corresponds to the \emph{Bounded Exploration} postulate in
\cite{Gur00}.
\begin{postulatep} \label{postulatetrois}
For any algorithm, there exists a finite set $T$ of ground terms over
vocabulary $\ensuremath{\mathcal {V}}$ such that for all states $X$ and $Y$ that coincide
for $T$, $\gen{X}$ and $\gen{Y}$ both exist and $\gen{X} = \gen{Y}$.
\end{postulatep}
A ground term of $T$ is a \emph{critical term} and a \emph{critical
element} is the value (interpretation) of a critical term.
\begin{definition}[Analog Algorithm] \label{defalgo}
An \emph{algorithm} is an object satisfying Postulates I through III.
\end{definition}
\section{Characterization Theorem}
We now go on to define the rules of our programs (adding to those of ASM
programs in \cite{Gur00}).
\begin{definition}
\begin{itemize}
\item \textbf{Update Rule:} An \emph{update rule} of vocabulary $\ensuremath{\mathcal {V}}$ has the
form $f(t_1,\dots,t_j):=t_0$ where $f$ is a $j$-ary function symbol
in $\ensuremath{\mathcal {V}}$ and $t_1,\dots,t_j$ are ground terms over $\ensuremath{\mathcal {V}}$.
\item \textbf{Parallel Update Rule: }
If $R_1,\dots,R_k$ are update rules of vocabulary $\ensuremath{\mathcal {V}}$, then
\begin{center}
\begin{minipage}{8cm}
\begin{ttcode}
par \\
\s $R_1$ \\
\s $R_2$ \\
\s $\dots$ \\
\s $R_k$ \\
endpar \\
\end{ttcode}
\end{minipage}
\end{center}
is a \emph{parallel update rule} of vocabulary $\ensuremath{\mathcal {V}}$.
\end{itemize}
\end{definition}
$\gen{R_i,X}$ denotes the interpretation of a rule $R$ in state $X$
and is defined as expected: If $R$ is an update rule $f (t_1,\dots,t_j) := t_0$ then
$\gen{R,X} = \J{\update{f}{{(\val{t_i}{X},\dots,\val{t_j}{X})}}{\val{t_0}{X}}}$
and when $R$ is $\text{\tt par }R_1,\dots,R_k\ \text{\tt endpar}$ then $\gen{R,X} = \J{(d_1 \cup \dots \cup d_k)}$ where $\gen{R_i,X} = \J{d_i }$ for all $i$.
Next, we introduce rules to deal with $\mathit{Flows}$.
\begin{definition}
\begin{itemize}
\item \textbf{Basic Continuous Rule:}
A \emph{basic continuous rule} of vocabulary $\ensuremath{\mathcal {V}}$ has the
form $\textsc{Dynamic}(f (t_1,\dots,t_j) , t_0) $ where $f$ is a symbol of arity $j$ and $t_0,t_1,\dots,t_j$ are ground terms of vocabulary $\ensuremath{\mathcal {V}}$.
\item \textbf{Flow Rule:}
If $R_1,\dots,R_k$ are basic continuous rules of vocabulary $\ensuremath{\mathcal {V}}$, then
\begin{center}
\begin{minipage}{8cm}
\begin{ttcode}
flow \\
\s $R_1$ \\
\s $R_2$ \\
\s $\dots$ \\
\s $R_k$ \\
endflow \\
\end{ttcode}
\end{minipage}
\end{center}
is a \emph{flow rule} of vocabulary $\ensuremath{\mathcal {V}}$.
\end{itemize}
\end{definition}
Their semantics are then defined as follows.
If $R$ is a basic continuous rule $\textsc{Dynamic}(f (t_1,\dots,t_j) ,t_0)$ then
$\gen{R,X} = \C{\{\update{f}{{(a_1,\dots,a_j)}}{a_0}\}}$
where each $a_i= \val{t_i}{X}$.
If $R$ is a flow rule with constituents $R_1,\dots,R_k$, then
$\gen{R,X} = \C{(d_1 \cup \dots \cup d_k)}$
where $\gen{R_i,X} = \C{d_i }$.
Finally, we allow conditionals:
\begin{definition}
\begin{itemize}
\item \textbf{Selection Rule:}
If $\varphi$ is a ground boolean term over vocabulary $\ensuremath{\mathcal {V}}$ and
$R_1$ and $R_2$ are rules of vocabulary $\ensuremath{\mathcal {V}}$ then:
\begin{center}
\begin{minipage}{8cm}
\begin{ttcode}
if $\varphi$ then \\
\s $R_1$ \\
else\\
\s $R_2$ \\
endif \\
\end{ttcode}
\end{minipage}
\end{center}
is a rule of vocabulary $\ensuremath{\mathcal {V}}$.
\end{itemize}
\end{definition}
Given such a rule $R$ and a state $X$, if $\varphi$ evaluates to
\textit{true} (the interpretation of scalar function \textit{true}) in $X$ then
$\gen{R,X} = \gen{R_1,X}$ else $\gen{R,X} = \gen{R_2,X}$.
An ASM program of vocabulary $\ensuremath{\mathcal {V}}$ is a rule of vocabulary $\ensuremath{\mathcal {V}}$.
The first key result is the following, which can be seen as a
completeness result.
\begin{theorem}[Completeness] \label{mlemma} \nd{For every algorithm of vocabulary $\ensuremath{\mathcal {V}}$, there
is an ASM program $\Pi$ over $\ensuremath{\mathcal {V}}$ with the identical behavior:
$\gen{\Pi,X} = \gen{X}$ for all states $X$.}
\end{theorem}
\section{Proof of Theorem \ref{mlemma}}
Before turning to the proof of our main theorem, we reformulate and
extend several of the constructions in \cite{Gur00}.
\begin{lemma}[{\cite[Lemma 6.2]{Gur00}}] \label{lem1}
Consider an algorithm $A$, consider a state $X$ of $A$ and assume
$\notContinuous{X}$. By definition, $\gen{X}=\J{\Delta^+(X)}$.
Consider $\update{f}{a_1,\dots,a_j}{a_0}$,
an update of $\Delta^+(X)$. Then all elements $a_0,a_1,\dots,a_j$
are critical elements of $X$, that is, they correspond to values
(interpretations) of critical terms.
\end{lemma}
\begin{proof}
The proof proceeds by contradiction. Assume that some $a_k$ does not correspond to the
value of any critical term. One can easily consider a structure $Y$ isomorphic to $X$ which is obtained from $X$ by
replacing $a_k$ with a fresh element $b$.
By Postulate
\ref{postulateats}, $Y$ is a state and $\val{t}{Y}=\val{t}{X}$ for every
critical term $t$.
By Postulate \ref{postulatetrois}, we know that
$\notContinuous{Y}$, and we must have:
$\gen{Y}=\J{\Delta^+(Y)}=\J{\Delta^+(X)}$.
By Postulate \ref{postulateats},
$a_k$ does not occur in (the base set of) $\varphi(Y,0^+)$ either. Hence, it cannot occur in
$\Delta^+(Y)= \varphi(Y,0^+) \setminus Y$. This gives the desired
contradiction.
\end{proof}
\begin{lemma}[Generalization of {\cite[Lemma 6.2]{Gur00}}] \label{lem2}
Consider an algorithm $A$ and assume $\Continuous{X}$. Then
by definition
$\gen{X}=\C{\Deltapsi(X)}$.
Consider $\update{f}{a_1,\dots,a_j}{a_0}$,
an element of $\Deltapsi(X)$. Then all elements $a_0,a_1,\dots,a_j$
are critical elements of $X$, that is, they correspond to values of critical terms.
\end{lemma}
\begin{proof}
The proof proceeds by contradiction. Assume that some $a_k$ does not correspond to the
value of any critical term. One can easily consider a structure $Y$ isomorphic to $X$ which is obtained from $X$ by
replacing $a_k$ with a fresh element $b$.
By Postulate
\ref{postulateats}, $Y$ is a state. Observe that $\val{t}{Y}=\val{t}{X}$ for every
critical term $t$.
By Postulate \ref{postulatetrois}, we know that
$\Continuous{Y}$, and we must have:
$$\gen{Y}=\C{\Deltapsi(Y)}=\C{\Deltapsi(X)}.$$
By Postulate \ref{postulateats},
$a_k$ does not occur in (the base set of) $Y$. Hence it cannot occur in
$\Deltapsi(Y)$, since by Definition \ref{defmachin} elements in $\Deltapsi(Y)$ are
elements of the base set of $Y$. This gives the desired
contradiction.
\end{proof}
The following follows directly from Lemmas \ref{lem1} and \ref{lem2}.
\begin{corollary}[Corollary 6.6 of \cite{Gur00}]
For every state $X$, there exists a
rule $R^X$ such that $\gen{X}=\gen{R^X,X}$.
\end{corollary}
We now generalize some of the other lemmas from \cite{Gur00} to apply to our
more general setting.
\begin{lemma}[Generalization of {\cite[Lemma 6.7]{Gur00}}]
\label{sixsept}
If states $X$ and $Y$ coincide over the set $T$ of critical terms,
then:
$$\gen{R^X,Y} = \gen{Y}.$$
\end{lemma}
\begin{proof}
We have $\gen{R^X,Y}=\gen{R^X,X}=\gen{X}=\gen{Y}$.
The first equality holds because $R^X$ involves only critical terms
and because critical terms have the same values in $X$ and $Y$. The
second equality holds by the definition of $R^X$ (that is to say, this
is the previous corollary). The third equality
holds because of the choice of $T$ and because $X$ and $Y$ coincide
over $T$.
\end{proof}
\begin{lemma}[Generalization of {\cite[ Lemma 6.8]{Gur00}}] \label{prevle}
Suppose that $X,Y$ are states and that $\gen{R^X,Z}=\gen{Z}$ for some
state $Z$ isomorphic to $Y$ then:
$$\gen{R^X,Y}=\gen{Y}.$$
\end{lemma}
\begin{proof}
Let $\zeta$ be an isomorphism from $Y$ onto an appropriate $Z$. Extend $\zeta$ to tuples,
locations, updates and set of updates. It is easy to check that $\zeta(\gen{R^X,Y}) = \gen{R^X,Z}$.
By the choice of $Z$, $\gen{R^X,Z} = \gen{A,Z}$.
By Definition \ref{defmachin}, generators preserve isomorphisms,
thus $\gen{A, Z} = \zeta(\gen{A, Y })$ and then $\zeta(\gen{R^X , Y}) = \zeta(\gen{A, Y})$.
It remains to apply $\zeta^{-1}$ to both sides of the last equality.
\end{proof}
At each state $X$, the equality relation between critical elements
induces an equivalence relation
$$E_X(t_1,t_2) \mbox{ iff } \val{t_1}{X}=\val{t_2}{X}$$
over critical terms.
States $X$ and $Y$ are $T$-similar if $E_X=E_Y$.
\begin{lemma}[Generalization of {\cite[Lemma 6.9]{Gur00}}]\label{lemma:lem62gur}
Let $X$ be a state. Then, for every
state $Y$ that is $T$-similar to $X$, we have:
$$\gen{R^X,Y}=\gen{Y}.$$
\end{lemma}
\begin{proof}
Replace every element of $Y$ that
belongs to $X$ with a fresh element. This gives a structure $Z_1$ that
is isomorphic to $Y$ and disjoint from $X$. By Postulate
\ref{postulateats}, $Z_1$ is a state. Since $Z_1$ is isomorphic to $Y$,
it is $T$-similar to $Y$ and therefore $T$-similar to $X$.
Let $Z_2$ be the structure isomorphic to $Z_1$ that is obtained
from $Z_1$ by replacing $\val{t}{Y}$ with $\val{t}{X}$ for all critical
term $t$ (the definition of $Z_2$ is coherent because $X$ and $Z_1$ are
$T$-similar). By Lemma \ref{sixsept}, we have $\gen{R^X,Z_2}=\gen{Z_2}$.
Since $Z_2$ is isomorphic to $Z_1$ isomorphic to $Y$, then $Z_2$ is isomorphic to $Y$
and by Lemma \ref{prevle}, we conclude $\gen{R^X,Y}=\gen{Y}$.
\end{proof}
By previous Lemma \ref{lemma:lem62gur}, for every state $X$, there exists a boolean term $\varphi^X$ that
evaluates to $\ensuremath{\mathcal {T}}rue$ in a structure $Y$ if and only if $Y$ is
$T$-similar to $X$. Indeed, the desired term asserts that the equality
relation on the critical terms is exactly the equivalence relation
$E_X$.
Since there are only finitely many critical terms, there are
only finitely many possible equivalence relations $E_X$. Hence there
exists a finite set $\{X_1,\dots,X_m,Y_1,\dots,Y_n\}$ of states such that every
state is $T$-similar to one of the state $X_i$ or $Y_i$, and such that
$\notContinuous{X_i}$ and $\Continuous{Y_i}$ for all $i$ (recall that the
property of being $\Continuous$ is preserved by $T$-similarity
from the previous lemma). States $\{X_1,\dots,X_m,Y_1,\dots,Y_n\}$ can
be chosen mutually exclusive, that is to say in different equivalence
relations. Boolean terms $(\varphi^{X_i})_i$ and $(\varphi^{Y_i})_i$ then realize a partition of the
set of states.
We can then go to the proof of Theorem \ref{mlemma}:
Let $X_1,\dots,X_m, Y_1, \dots, Y_n$ be as above.
The desired $\Pi$ is
\begin{center}
\begin{minipage}{5cm}
\begin{ttfamily}
\noindent
if $\varphi^{X_1}$ then\\ \s $R^{X_1}$\\ else \\
\s if $\varphi^{X_2}$ then\\ \s\s $R^{X_2}$ \\ \s else \\
\s\s\s\s $\dots$ \\
\s\s\s\s if $\varphi^{X_m}$ then \\
\s\s\s\s \s $R^{X_m}$ \\
\s\s\s\s else \\
\s\s\s\s \s\s
if $\varphi^{Y_1}$ then\\
\s\s\s\s \s\s\s $R^{Y_1}$ \\
\s\s\s\s \s\s else \\
\s\s\s\s \s\s\s if $\varphi^{Y_2}$ then \\
\s\s\s\s \s\s\s\s $R^{Y_2} $\\
\s\s\s\s \s\s\s else \\
\s\s\s\s \s\s\s\s\s\s\s $\dots$ \\
\s\s\s\s \s\s\s\s \s\s\s if $\varphi^{Y_{n-1}}$ then\\
\s\s\s\s \s\s\s\s \s\s\s\s $R^{Y_{n-1}}$
\s\s\s\s \s\s\s\s \s\s\s else
\s\s\s\s \s\s\s\s \s\s\s\s $R^{Y_{n}}$
\s\s\s\s \s\s\s\s \s\s\s endif \\
\s\s\s\s \s\s\s endif \\
\s\s\s\s \s\s endif\\
\s\s\s\s endif\\
\s endif\\
endif
\end{ttfamily}
\end{minipage}
\end{center}
where the $R^{X_i}$ are (possibly parallel) update rules, and the $R^{Y_i}$ are flow
rules.
\section{Extended Statements}
We are now very close to formulating our other theorems. First we define an
abstract state machine relative to semantics $\psi$.
\begin{definition} A $\psi$-abstract state machine $B$ comprises the following:
\itm{a} an ASM program $\Pi$;
\itm{b} a set $\ensuremath{\mathcal {S}}$ of (first-order) structures over some finite
vocabulary $\ensuremath{\mathcal {V}}$ closed under isomorphisms, and a subset $\ensuremath{\mathbb{I}}n \subseteq \ensuremath{\mathcal {S}}$ closed under isomorphisms;
\itm{c} a map $\iota$ and a map $\varphi$ such that $\langle \ensuremath{\mathcal {S}}, \ensuremath{\mathbb{I}}n, \iota, \varphi
\rangle$ is an algorithm, where $\Deltapsi$ is fixed to be the infinitesimal
generator associated with $\psi$, and
for all
states $X$ in $\ensuremath{\mathcal {S}}$,
$\gen{\Pi,X} = \gen{X}$.
\end{definition}
By definition, a $\psi$-abstract state machine $B$ satisfies all the
postulates and hence is an algorithm.
\begin{definition}
An ASM program $\Pi$ is \emph{$\psi$-solvable} for a set $\ensuremath{\mathcal {S}}$ of (first-order) structures over some finite
vocabulary $\ensuremath{\mathcal {V}}$ closed under isomorphisms and a subset $\ensuremath{\mathbb{I}}n
\subseteq \ensuremath{\mathcal {S}}$ closed under isomorphisms if there exists a unique
$\iota$ and $\varphi$ such that $(\Pi,\ensuremath{\mathcal {S}},\ensuremath{\mathbb{I}}n,\iota,\varphi)$ is a $\psi$-abstract
state machine.
\end{definition}
\begin{definition}
A semantics $\psi$ is \emph{unambiguous} if for all sets $\ensuremath{\mathcal {S}}$ of (first-order) structures over some finite
vocabulary $\ensuremath{\mathcal {V}}$ closed under isomorphisms, and for all subsets $\ensuremath{\mathbb{I}}n
\subseteq \ensuremath{\mathcal {S}}$ closed under isomorphisms, whenever there exists some $\iota$
and $\varphi$ such that $(\Pi,\ensuremath{\mathcal {S}},\ensuremath{\mathbb{I}}n,\iota,\varphi)$ is a $\psi$-abstract
state machine, then $\iota$ and $\varphi$ are unique.
\end{definition}
Our main results follow.
\begin{theorem} \label{thone}
For every $\psi$-definable
algorithm $A$, there exists an equivalent $\psi$-abstract state machine $B$.
\end{theorem}
\begin{proof}
By construction, $A$ is a hybrid dynamical
system such that $\gen{A,X}=\gen{\Pi,X}$ for some $\Pi$ given by
previous discussions. Set the states of $B$ to be the
states of $A$ and the initial states of
$B$ to the initial states of $A$.
\end{proof}
\begin{theorem} \label{thtwo}
Assume that $\psi$ is unambiguous.
For every $\psi$-definable
algorithm $A$, there exists a unique equivalent $\psi$-abstract
state machine $B$ with same states and initial states.
\end{theorem}
\begin{proof}[of Theorem \ref{thtwo}]
This is exactly the same proof as for Theorem \ref{thone}. Unicity comes from the definition of
unambiguity.
\end{proof}
\begin{corollary} \label{coromain}
Assume that $\psi$ is unambiguous.
For every $\psi$-definable
algorithm $A$, there exists an equivalent $\psi$-solvable ASM
program.
\end{corollary}
To any
algorithm $A$ that is $\psi$-definable there corresponds an
equivalent $\psi$-abstract state machine $B$, and hence a
$\psi$-solvable program $\Pi$. Conversely, a $\psi$-abstract state
machine $B$ corresponds to a $\psi$-definable algorithm.
However, not every program $\Pi$ is $\psi$-solvable.
When $\psi$-corresponds to $\psi_{\textrm{der}}$, unambiguity comes from
(unicity in)
the Cauchy-Lipschitz theorem. The fact that
not every program $\Pi$ is $\psi$-solvable is due to the fact that
not all differential equations have a solution.
\section{Examples}
The examples in this section are for semantics $\psi_{\textrm{der}}$.
Our settings cover, first of all, analog algorithms that are pure flow, in particular all systems that can be modeled as ordinary differential equations.
A very simple, classical example is the pendulum: the motion of an idealized simple pendulum is governed by the second-order
differential equation $
\theta''+\frac{g}{L}\theta = 0\,,
$
where
$\theta$ is angular displacement, $g$ is gravitational acceleration,
and $L$ is the length of the pendulum rod.
This can indeed be modeled as the program
\begin{center}
\begin{minipage}{8cm}
\begin{ttcode}
flow \\
\s $\textsc{Dynamic}(\theta,\theta_1)$ \\
\s $\textsc{Dynamic}(\theta_1,-\frac{g}{L}\cdot \theta)$ \\
endflow
\end{ttcode}
\end{minipage}
\end{center}
\noindent
using the fact that any ordinary differential equation can be put in the form of a vectorial first-order equation, here $\theta_1$ corresponding to the derivative of $\theta$.
As a consequence, our formalism covers very generic classes of continuous-time models of computation, including the GPAC, which corresponds to ordinary differential equations with polynomial right-hand sides \cite{GC03,GBC09}.
Recall that the GPAC was proposed as a mathematical model of
differential analyzers (DAs), one of the most famous analog computer
machines in history.
Figure~\ref{gpac} (left) depicts a (non-minimal) GPAC that generates sine and cosine.
In this
picture, $\int$ signifies some integrator, and $-1$ denotes
some constant block.
This simple GPAC
can be modeled by the program
\begin{center}
\begin{minipage}{8cm}
\begin{ttcode}
flow \\
\s $\textsc{Dynamic}(x,z)$ \\
\s $\textsc{Dynamic}(y,x)$ \\
\s $\textsc{Dynamic}(z,-x)$ \\
endflow
\end{ttcode}
\end{minipage}
\end{center}
\begin{figure}
\caption{A GPAC for sine and cosine (left). Corresponding evolution (right).}
\label{gpac}
\end{figure}
Our proposed model can also adequately describe hybrid
systems, made of alternating sequences of continuous evolution and discrete transitions.
This includes, for example, a simple model of a bouncing ball,
the physics of which are given by the flow equations
$x'' = - g m $,
where $g$ is the gravitational constant and $v=x'$ is the velocity,
except that upon impact, each time $x=0$, the velocity changes according to
$
v' = -k \cdot v',$
where $k$ is the coefficient of impact. Every time the ball
bounces, its speed is reduced by a factor $k$.
This system can be described by a program like
\noindent
\qquad\begin{minipage}{15cm}
\begin{ttcode}
if $x=0$ then \\
\s $v := -k \cdot v$ \\
else \\
\s flow \\
\s\s $ \textsc{Dynamic}(x,v)$ \\
\s\s $\textsc{Dynamic}(v,-g.m)$ \\
\s endflow \\
endif
\end{ttcode}
\end{minipage}
Our setting is an extension of classical discrete-time algorithms; hence, all classical discrete-time algorithms can also be modeled.
As for examples with semantics other than $ \psi_{\textrm{der}}$: Observe
that one can consider timelines like $\mathbb{Q}$ instead of $\ensuremath{\mathbb{R}}$. (For such
a timeline, we have $\Continuous{i}$ for all $i \in \mathbb{Q}$.) One
can define a semantics on such a timeline where for every state $X$
we have $\Continuous{X}$ by first extending the evolution function
to $\mathbb{R}$ (for example by
restricting to continuous
dynamics) and then using the derivative.
Constructions of \cite{rust2000hybrid} are also covered by our
settings: In some sense, the example at the beginning of the paragraph is the
spirit of the constructions from \cite{rust2000hybrid}, where the
timeline is the set of hyperreals obtained by multiplying some fixed
infinitesimal by some hyperinteger (using hyperreals and
infinitesimals).
Notice that there is no need to consider derivatives or similar
notions: we could also consider analytic dynamics, and consider a
semantics related to the family of Taylor coefficients. Weaker
notions of solution, like variational approaches, can also be considered.
\end{document} |
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